卷积 - 维基百科，自由的百科全书

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class="vector-menu-content-list"> <li id="pt-anoncontribs" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%B4%A1%E7%8C%AE" title="来自此IP地址的编辑列表[y]" accesskey="y">贡献</a></li><li id="pt-anontalk" class="mw-list-item"><a href="/wiki/Special:%E6%88%91%E7%9A%84%E8%AE%A8%E8%AE%BA%E9%A1%B5" title="对于来自此IP地址编辑的讨论[n]" accesskey="n">讨论</a></li> </ul> </div> </div> </div> </div> </nav> </div> </header> </div> <div class="mw-page-container"> <div class="mw-page-container-inner"> <div class="vector-sitenotice-container"> <div id="siteNotice"></div> </div> <div class="vector-column-start"> <div class="vector-main-menu-container"> <div id="mw-navigation"> <nav id="mw-panel" class="vector-main-menu-landmark" aria-label="站点"> <div id="vector-main-menu-pinned-container" class="vector-pinned-container"> </div> </nav> </div> </div> <div class="vector-sticky-pinned-container"> <nav id="mw-panel-toc" aria-label="目录" data-event-name="ui.sidebar-toc" class="mw-table-of-contents-container vector-toc-landmark"> <div id="vector-toc-pinned-container" class="vector-pinned-container"> <div id="vector-toc" class="vector-toc vector-pinnable-element"> <div class="vector-pinnable-header vector-toc-pinnable-header vector-pinnable-header-pinned" data-feature-name="toc-pinned" data-pinnable-element-id="vector-toc" > <h2 class="vector-pinnable-header-label">目录</h2> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-toc.pin">移至侧栏</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">隐藏</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">序言</div> </a> </li> <li id="toc-定义" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#定义"> <div class="vector-toc-text"> 1 定义 </div> </a> <ul id="toc-定义-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-历史" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#历史"> <div class="vector-toc-text"> 2 历史 </div> </a> <ul id="toc-历史-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-简介" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#简介"> <div class="vector-toc-text"> 3 简介 </div> </a> <ul id="toc-简介-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-图解" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#图解"> <div class="vector-toc-text"> 4 图解 </div> </a> <ul id="toc-图解-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-周期卷积" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#周期卷积"> <div class="vector-toc-text"> 5 周期卷积 </div> </a> <ul id="toc-周期卷积-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-离散卷积" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#离散卷积"> <div class="vector-toc-text"> 6 离散卷积 </div> </a> <button aria-controls="toc-离散卷积-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 开关离散卷积子章节 </button> <ul id="toc-离散卷积-sublist" class="vector-toc-list"> <li id="toc-多维离散卷积" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#多维离散卷积"> <div class="vector-toc-text"> 6.1 多维离散卷积 </div> </a> <ul id="toc-多维离散卷积-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-离散周期卷积" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#离散周期卷积"> <div class="vector-toc-text"> 6.2 离散周期卷积 </div> </a> <ul id="toc-离散周期卷积-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-性质" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#性质"> <div class="vector-toc-text"> 7 性质 </div> </a> <button aria-controls="toc-性质-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 开关性质子章节 </button> <ul id="toc-性质-sublist" class="vector-toc-list"> <li id="toc-代数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#代数"> <div class="vector-toc-text"> 7.1 代数 </div> </a> <ul id="toc-代数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-积分" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#积分"> <div class="vector-toc-text"> 7.2 积分 </div> </a> <ul id="toc-积分-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-微分" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#微分"> <div class="vector-toc-text"> 7.3 微分 </div> </a> <ul id="toc-微分-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-卷积定理" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#卷积定理"> <div class="vector-toc-text"> 8 卷积定理 </div> </a> <button aria-controls="toc-卷积定理-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 开关卷积定理子章节 </button> <ul id="toc-卷积定理-sublist" class="vector-toc-list"> <li id="toc-周期卷积_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#周期卷积_2"> <div class="vector-toc-text"> 8.1 周期卷积 </div> </a> <ul id="toc-周期卷积_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-离散卷积_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#离散卷积_2"> <div class="vector-toc-text"> 8.2 离散卷积 </div> </a> <ul id="toc-离散卷积_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-离散周期卷积_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#离散周期卷积_2"> <div class="vector-toc-text"> 8.3 离散周期卷积 </div> </a> <ul id="toc-离散周期卷积_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-推广" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#推广"> <div class="vector-toc-text"> 9 推广 </div> </a> <ul id="toc-推广-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-离散卷積的計算方法" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#离散卷積的計算方法"> <div class="vector-toc-text"> 10 离散卷積的計算方法 </div> </a> <button aria-controls="toc-离散卷積的計算方法-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> 开关离散卷積的計算方法子章节 </button> <ul id="toc-离散卷積的計算方法-sublist" class="vector-toc-list"> <li id="toc-方法1：直接計算" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#方法1：直接計算"> <div class="vector-toc-text"> 10.1 方法1：直接計算 </div> </a> <ul id="toc-方法1：直接計算-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-方法2：快速傅立葉轉換" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#方法2：快速傅立葉轉換"> <div class="vector-toc-text"> 10.2 方法2：快速傅立葉轉換 </div> </a> <ul id="toc-方法2：快速傅立葉轉換-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-方法3：分段卷積" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#方法3：分段卷積"> <div class="vector-toc-text"> 10.3 方法3：分段卷積 </div> </a> <ul id="toc-方法3：分段卷積-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-應用時機" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#應用時機"> <div class="vector-toc-text"> 10.4 應用時機 </div> </a> <ul id="toc-應用時機-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-例子" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#例子"> <div class="vector-toc-text"> 10.5 例子 </div> </a> <ul id="toc-例子-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-应用" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#应用"> <div class="vector-toc-text"> 11 应用 </div> </a> <ul id="toc-应用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> 12 参见 </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-引用" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#引用"> <div class="vector-toc-text"> 13 引用 </div> </a> <ul id="toc-引用-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-延伸阅读" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#延伸阅读"> <div class="vector-toc-text"> 14 延伸阅读 </div> </a> <ul id="toc-延伸阅读-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-外部链接" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#外部链接"> <div class="vector-toc-text"> 15 外部链接 </div> </a> <ul id="toc-外部链接-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="目录" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" > <input type="checkbox" id="vector-page-titlebar-toc-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-vector-page-titlebar-toc" class="vector-dropdown-checkbox " aria-label="开关目录" > <label id="vector-page-titlebar-toc-label" for="vector-page-titlebar-toc-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only " aria-hidden="true" > 开关目录 </label> <div class="vector-dropdown-content"> <div id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading">卷积</h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="前往另一种语言写成的文章。40种语言可用" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-40" aria-hidden="true" > 40种语言 </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-af mw-list-item"><a href="https://af.wikipedia.org/wiki/Konvolusie" title="Konvolusie – 南非荷兰语" lang="af" hreflang="af" data-title="Konvolusie" data-language-autonym="Afrikaans" data-language-local-name="南非荷兰语" class="interlanguage-link-target">Afrikaans</a></li><li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D8%A7%D9%84%D8%B7%D9%8A_(%D8%B1%D9%8A%D8%A7%D8%B6%D9%8A%D8%A7%D8%AA)" title="الطي (رياضيات) – 阿拉伯语" lang="ar" hreflang="ar" data-title="الطي (رياضيات)" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target">العربية</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B2%D0%BE%D0%BB%D1%8E%D1%86%D0%B8%D1%8F" title="Конволюция – 保加利亚语" lang="bg" hreflang="bg" data-title="Конволюция" data-language-autonym="Български" data-language-local-name="保加利亚语" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Convoluci%C3%B3" title="Convolució – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Convolució" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Konvoluce" title="Konvoluce – 捷克语" lang="cs" hreflang="cs" data-title="Konvoluce" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Foldning" title="Foldning – 丹麦语" lang="da" hreflang="da" data-title="Foldning" data-language-autonym="Dansk" data-language-local-name="丹麦语" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Faltung_(Mathematik)" title="Faltung (Mathematik) – 德语" lang="de" hreflang="de" data-title="Faltung (Mathematik)" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A3%CF%85%CE%BD%CE%AD%CE%BB%CE%B9%CE%BE%CE%B7" title="Συνέλιξη – 希腊语" lang="el" hreflang="el" data-title="Συνέλιξη" data-language-autonym="Ελληνικά" data-language-local-name="希腊语" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Convolution" title="Convolution – 英语" lang="en" hreflang="en" data-title="Convolution" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target">English</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Kunfalda%C4%B5o" title="Kunfaldaĵo – 世界语" lang="eo" hreflang="eo" data-title="Kunfaldaĵo" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target">Esperanto</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Convoluci%C3%B3n" title="Convolución – 西班牙语" lang="es" hreflang="es" data-title="Convolución" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Konvolutsioon" title="Konvolutsioon – 爱沙尼亚语" lang="et" hreflang="et" data-title="Konvolutsioon" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%87%D9%85%E2%80%8C%DA%AF%D8%B4%D8%AA" title="هم‌گشت – 波斯语" lang="fa" hreflang="fa" data-title="هم‌گشت" data-language-autonym="فارسی" data-language-local-name="波斯语" class="interlanguage-link-target">فارسی</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Konvoluutio" title="Konvoluutio – 芬兰语" lang="fi" hreflang="fi" data-title="Konvoluutio" data-language-autonym="Suomi" data-language-local-name="芬兰语" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Produit_de_convolution" title="Produit de convolution – 法语" lang="fr" hreflang="fr" data-title="Produit de convolution" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target">Français</a></li><li class="interlanguage-link interwiki-he mw-list-item"><a href="https://he.wikipedia.org/wiki/%D7%A7%D7%95%D7%A0%D7%91%D7%95%D7%9C%D7%95%D7%A6%D7%99%D7%94" title="קונבולוציה – 希伯来语" lang="he" hreflang="he" data-title="קונבולוציה" data-language-autonym="עברית" data-language-local-name="希伯来语" class="interlanguage-link-target">עברית</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B8%E0%A4%82%E0%A4%B5%E0%A4%B2%E0%A4%A8" title="संवलन – 印地语" lang="hi" hreflang="hi" data-title="संवलन" data-language-autonym="हिन्दी" data-language-local-name="印地语" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Konvol%C3%BAci%C3%B3" title="Konvolúció – 匈牙利语" lang="hu" hreflang="hu" data-title="Konvolúció" data-language-autonym="Magyar" data-language-local-name="匈牙利语" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Convoluzione" title="Convoluzione – 意大利语" lang="it" hreflang="it" data-title="Convoluzione" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E7%95%B3%E3%81%BF%E8%BE%BC%E3%81%BF" title="畳み込み – 日语" lang="ja" hreflang="ja" data-title="畳み込み" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%ED%95%A9%EC%84%B1%EA%B3%B1" title="합성곱 – 韩语" lang="ko" hreflang="ko" data-title="합성곱" data-language-autonym="한국어" data-language-local-name="韩语" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Konvoliucija" title="Konvoliucija – 立陶宛语" lang="lt" hreflang="lt" data-title="Konvoliucija" data-language-autonym="Lietuvių" data-language-local-name="立陶宛语" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Convolutie" title="Convolutie – 荷兰语" lang="nl" hreflang="nl" data-title="Convolutie" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target">Nederlands</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Folding_i_signalhandsaming" title="Folding i signalhandsaming – 挪威尼诺斯克语" lang="nn" hreflang="nn" data-title="Folding i signalhandsaming" data-language-autonym="Norsk nynorsk" data-language-local-name="挪威尼诺斯克语" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Folding_(matematikk)" title="Folding (matematikk) – 书面挪威语" lang="nb" hreflang="nb" data-title="Folding (matematikk)" data-language-autonym="Norsk bokmål" data-language-local-name="书面挪威语" class="interlanguage-link-target">Norsk bokmål</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Splot_(analiza_matematyczna)" title="Splot (analiza matematyczna) – 波兰语" lang="pl" hreflang="pl" data-title="Splot (analiza matematyczna)" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Convolu%C3%A7%C3%A3o" title="Convolução – 葡萄牙语" lang="pt" hreflang="pt" data-title="Convolução" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Convolu%C8%9Bie" title="Convoluție – 罗马尼亚语" lang="ro" hreflang="ro" data-title="Convoluție" data-language-autonym="Română" data-language-local-name="罗马尼亚语" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%B2%D1%91%D1%80%D1%82%D0%BA%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%B5%D1%81%D0%BA%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D0%B8%D0%B7)" title="Свёртка (математический анализ) – 俄语" lang="ru" hreflang="ru" data-title="Свёртка (математический анализ)" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target">Русский</a></li><li class="interlanguage-link interwiki-sd mw-list-item"><a href="https://sd.wikipedia.org/wiki/%D9%BE%D9%8A%DA%86%D9%8F" title="پيچُ – 信德语" lang="sd" hreflang="sd" data-title="پيچُ" data-language-autonym="سنڌي" data-language-local-name="信德语" class="interlanguage-link-target">سنڌي</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Konvolucioni" title="Konvolucioni – 阿尔巴尼亚语" lang="sq" hreflang="sq" data-title="Konvolucioni" data-language-autonym="Shqip" data-language-local-name="阿尔巴尼亚语" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%9A%D0%BE%D0%BD%D0%B2%D0%BE%D0%BB%D1%83%D1%86%D0%B8%D1%98%D0%B0" title="Конволуција – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Конволуција" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-su mw-list-item"><a href="https://su.wikipedia.org/wiki/Konvolusi" title="Konvolusi – 巽他语" lang="su" hreflang="su" data-title="Konvolusi" data-language-autonym="Sunda" data-language-local-name="巽他语" class="interlanguage-link-target">Sunda</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Faltning" title="Faltning – 瑞典语" lang="sv" hreflang="sv" data-title="Faltning" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-tl mw-list-item"><a href="https://tl.wikipedia.org/wiki/Konbolusyon" title="Konbolusyon – 他加禄语" lang="tl" hreflang="tl" data-title="Konbolusyon" data-language-autonym="Tagalog" data-language-local-name="他加禄语" class="interlanguage-link-target">Tagalog</a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Konvol%C3%BCsyon" title="Konvolüsyon – 土耳其语" lang="tr" hreflang="tr" data-title="Konvolüsyon" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%97%D0%B3%D0%BE%D1%80%D1%82%D0%BA%D0%B0_(%D0%BC%D0%B0%D1%82%D0%B5%D0%BC%D0%B0%D1%82%D0%B8%D1%87%D0%BD%D0%B8%D0%B9_%D0%B0%D0%BD%D0%B0%D0%BB%D1%96%D0%B7)" title="Згортка (математичний аналіз) – 乌克兰语" lang="uk" hreflang="uk" data-title="Згортка (математичний аналіз)" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/T%C3%ADch_ch%E1%BA%ADp" title="Tích chập – 越南语" lang="vi" hreflang="vi" data-title="Tích chập" data-language-autonym="Tiếng Việt" data-language-local-name="越南语" class="interlanguage-link-target">Tiếng Việt</a></li><li class="interlanguage-link interwiki-wuu mw-list-item"><a href="https://wuu.wikipedia.org/wiki/%E5%8D%B7%E7%A7%AF" title="卷积 – 吴语" lang="wuu" 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href="/wiki/File:Comparison_convolution_correlation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Comparison_convolution_correlation.svg/400px-Comparison_convolution_correlation.svg.png" decoding="async" width="400" height="300" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/21/Comparison_convolution_correlation.svg/600px-Comparison_convolution_correlation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/21/Comparison_convolution_correlation.svg/800px-Comparison_convolution_correlation.svg.png 2x" data-file-width="512" data-file-height="384" /></a><figcaption>卷积、<a href="/wiki/%E4%BA%92%E7%9B%B8%E5%85%B3" title="互相关">互相关</a>和<a href="/wiki/%E8%87%AA%E7%9B%B8%E5%85%B3%E5%87%BD%E6%95%B0" title="自相关函数">自相关</a>的图示比较。运算涉及函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">，并假定<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">的高度是1.0，在5个不同点上的值，用在每个点下面的阴影面积来指示。<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">的对称性是卷积<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g*f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∗</mo> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g*f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d2fbdc97f570309b80e28109854afcacd85cce7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle g*f}">和互相关<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\star g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>⋆</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\star g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/371d3161cd7e094182a184d7601b49880228385c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\star g}">在这个例子中相同的原因。</figcaption></figure> 在<a href="/wiki/%E6%B3%9B%E5%87%BD%E5%88%86%E6%9E%90" title="泛函分析">泛函分析</a>中，捲積（convolution），或译为疊積、褶積或旋積，是透過两个<a href="/wiki/%E5%87%BD%E6%95%B0" title="函数">函数</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">生成第三个函数的一种数学<a href="/wiki/%E7%AE%97%E5%AD%90" title="算子">算子</a>，表徵函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">与经过翻转和平移的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">的乘積函數所圍成的曲邊梯形的面積。如果将参加卷积的一个函数看作<a href="/wiki/%E5%8C%BA%E9%97%B4" class="mw-redirect" title="区间">区间</a>的<a href="/wiki/%E6%8C%87%E7%A4%BA%E5%87%BD%E6%95%B0" title="指示函数">指示函数</a>，卷积还可以被看作是“<a href="/wiki/%E6%BB%91%E5%8B%95%E5%B9%B3%E5%9D%87" class="mw-redirect" title="滑動平均">滑動平均</a>”的推廣。 <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="定义">定义</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=1" title="编辑章节：定义">编辑</a>]</div> 卷积是<a href="/wiki/%E6%95%B0%E5%AD%A6%E5%88%86%E6%9E%90" title="数学分析">数学分析</a>中一种重要的运算。设：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f700860ee7af27797d11ddfad3d185eb7af0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}">是<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {R} }">上的两个<a href="/wiki/%E5%8F%AF%E7%A7%AF%E5%87%BD%E6%95%B0" title="可积函数">可积函数</a>，定义二者的卷积<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3f22d92993aba0b3be3ff9238f5e5373c7ca7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle (f*g)(t)}">为如下特定形式的<a href="/wiki/%E7%A7%AF%E5%88%86" title="积分">积分</a><a href="/wiki/%E7%A7%AF%E5%88%86%E5%8F%98%E6%8D%A2" title="积分变换">变换</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,\mathrm {d} \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,\mathrm {d} \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a9342b735529a84cbd078ea905f94c79df9a4103" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.958ex; height:6.009ex;" alt="{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,\mathrm {d} \tau }"></dd></dl> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa3f22d92993aba0b3be3ff9238f5e5373c7ca7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.048ex; height:2.843ex;" alt="{\displaystyle (f*g)(t)}">仍为可积函数，并且有着： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau =(g*f)(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau =(g*f)(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/996dac5cd8ac0a45e329465bb173d0b7661a5d2c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:44.028ex; height:6.009ex;" alt="{\displaystyle (f*g)(t)\triangleq \int _{-\infty }^{\infty }f(t-\tau )g(\tau )\,d\tau =(g*f)(t)}"></dd></dl> 函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">，如果只<a href="/wiki/%E6%94%AF%E6%92%91%E9%9B%86" title="支撑集">支撑</a>在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,\infty ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,\infty ]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52088d5605716e18068a460dec118214954a68e9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.814ex; height:2.843ex;" alt="{\displaystyle [0,\infty ]}">之上，则积分界限可以截断为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mspace width="1em" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d86de421b390b1f783148a1f282bf7600304c2cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:33.048ex; height:6.176ex;" alt="{\displaystyle (f*g)(t)=\int _{0}^{t}f(\tau )g(t-\tau )\,d\tau \quad }">对于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ f,g:[0,\infty )\to \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mi>f</mi> <mo>,</mo> <mi>g</mi> <mo>:</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi mathvariant="normal">∞</mi> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ f,g:[0,\infty )\to \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2d942d6ec12f97e43318383c3cef93ff6f80db7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.31ex; height:2.843ex;" alt="{\displaystyle \ f,g:[0,\infty )\to \mathbb {R} }"></dd></dl> 对于两个得出<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>值的<a href="/w/index.php?title=%E5%A4%9A%E5%85%83%E5%AE%9E%E5%8F%98%E5%87%BD%E6%95%B0&action=edit&redlink=1" class="new" title="多元实变函数（页面不存在）">多元实变函数</a>（英语：<a href="https://en.wikipedia.org/wiki/Function_of_several_real_variables" class="extiw" title="en:Function of several real variables">Function of several real variables</a>），可以定义二者的卷积为如下形式的<a href="/wiki/%E5%A4%9A%E9%87%8D%E7%A7%AF%E5%88%86" title="多重积分">多重积分</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(f*g)(t_{1},t_{2},\cdots ,t_{n})&\triangleq \int \int \cdots \int _{\mathbb {R} ^{n}}f(\tau _{1},\tau _{2},\cdots ,\tau _{n})g(t_{1}-\tau _{1},t_{2}-\tau _{2},\cdots ,t_{n}-\tau _{n},)\,d\tau _{1}d\tau _{2}\cdots d\tau _{n}\\&\triangleq \int _{\mathbb {R} ^{n}}f(\tau )g(t-\tau )\,d^{n}\tau \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mo>∫</mo> <mo>∫</mo> <mo>⋯</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>,</mo> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>d</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <mi>d</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>≜</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>τ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(f*g)(t_{1},t_{2},\cdots ,t_{n})&\triangleq \int \int \cdots \int _{\mathbb {R} ^{n}}f(\tau _{1},\tau _{2},\cdots ,\tau _{n})g(t_{1}-\tau _{1},t_{2}-\tau _{2},\cdots ,t_{n}-\tau _{n},)\,d\tau _{1}d\tau _{2}\cdots d\tau _{n}\\&\triangleq \int _{\mathbb {R} ^{n}}f(\tau )g(t-\tau )\,d^{n}\tau \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15f1bbe98a1e8ac3d6887ca614b036e2f9a22489" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:96.948ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}(f*g)(t_{1},t_{2},\cdots ,t_{n})&\triangleq \int \int \cdots \int _{\mathbb {R} ^{n}}f(\tau _{1},\tau _{2},\cdots ,\tau _{n})g(t_{1}-\tau _{1},t_{2}-\tau _{2},\cdots ,t_{n}-\tau _{n},)\,d\tau _{1}d\tau _{2}\cdots d\tau _{n}\\&\triangleq \int _{\mathbb {R} ^{n}}f(\tau )g(t-\tau )\,d^{n}\tau \end{aligned}}}"></dd></dl> 卷积有一个通用的工程上的符号约定<a href="#cite_note-1">[1]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)*g(t)\triangleq \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)*g(t)\triangleq \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8752156813649a7c66de74a2b3d691491c438c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:32.721ex; height:10.009ex;" alt="{\displaystyle f(t)*g(t)\triangleq \underbrace {\int _{-\infty }^{\infty }f(\tau )g(t-\tau )\,d\tau } _{(f*g)(t)}}"></dd></dl> 它必须被谨慎解释以避免混淆。例如：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)*g(t-t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)*g(t-t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da85a8202af7a2978ff30530efb5477cc2852334" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.621ex; height:2.843ex;" alt="{\displaystyle f(t)*g(t-t_{0})}">等价于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t-t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t-t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/234cb6bf010ce9f18c6fdbfef1a75d87661c52ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.782ex; height:2.843ex;" alt="{\displaystyle (f*g)(t-t_{0})}">，而<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t-t_{0})*g(t-t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t-t_{0})*g(t-t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c225f7585abf969b8bd4319b078d10f44437abde" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.356ex; height:2.843ex;" alt="{\displaystyle f(t-t_{0})*g(t-t_{0})}">却实际上等价于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(t-2t_{0})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mn>2</mn> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(t-2t_{0})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/792889872f217f9e95757a97da74f00cf27f2e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.944ex; height:2.843ex;" alt="{\displaystyle (f*g)(t-2t_{0})}"><a href="#cite_note-2">[2]</a>。 <div class="mw-heading mw-heading2"><h2 id="历史">历史</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=2" title="编辑章节：历史">编辑</a>]</div> 卷积运算的最早使用出现在<a href="/wiki/%E8%AE%A9%C2%B7%E5%8B%92%E6%9C%97%C2%B7%E8%BE%BE%E6%9C%97%E8%B4%9D%E5%B0%94" title="让·勒朗·达朗贝尔">达朗贝尔</a>于1754年出版的《宇宙体系的几个要点研究》中对<a href="/wiki/%E6%B3%B0%E5%8B%92%E5%AE%9A%E7%90%86" class="mw-redirect" title="泰勒定理">泰勒定理</a>的推导之中<a href="#cite_note-3">[3]</a>。还有<a href="/w/index.php?title=%E8%A5%BF%E5%B0%94%E7%BB%B4%E6%96%AF%E7%89%B9%C2%B7%E5%BC%97%E6%9C%97%E7%B4%A2%E7%93%A6%C2%B7%E6%8B%89%E5%85%8B%E9%B2%81%E7%93%A6&action=edit&redlink=1" class="new" title="西尔维斯特·弗朗索瓦·拉克鲁瓦（页面不存在）">西尔维斯特·拉克鲁瓦</a>（英语：<a href="https://en.wikipedia.org/wiki/Sylvestre_Fran%C3%A7ois_Lacroix" class="extiw" title="en:Sylvestre François Lacroix">Sylvestre François Lacroix</a>），将<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \int f(u)\cdot g(x-u)\,du}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mo>∫</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>u</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \int f(u)\cdot g(x-u)\,du}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d6d36b7751cb0766e53e6f139179baa44567c68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.259ex; height:3.176ex;" alt="{\textstyle \int f(u)\cdot g(x-u)\,du}">类型的表达式，用在他的1797年–1800年出版的著作《微分与级数论文》中<a href="#cite_note-4">[4]</a>。此后不久，卷积运算出现在<a href="/wiki/%E7%9A%AE%E5%9F%83%E5%B0%94-%E8%A5%BF%E8%92%99%C2%B7%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF" title="皮埃尔-西蒙·拉普拉斯">皮埃尔-西蒙·拉普拉斯</a>、<a href="/wiki/%E7%BA%A6%E7%91%9F%E5%A4%AB%C2%B7%E5%82%85%E9%87%8C%E5%8F%B6" title="约瑟夫·傅里叶">约瑟夫·傅里叶</a>和<a href="/wiki/%E8%A5%BF%E6%A2%85%E7%BF%81%C2%B7%E5%BE%B7%E5%B0%BC%C2%B7%E6%B3%8A%E6%9D%BE" title="西梅翁·德尼·泊松">西梅翁·泊松</a>等人的著作中。这个运算以前有时叫做“Faltung”（德语中的折叠）、合成乘积、叠加积分或卡森积分<a href="#cite_note-5">[5]</a>。 “卷积”这个术语早在1903年就出现了，然而其定义在早期使用中是相当生僻的<a href="#cite_note-6">[6]</a><a href="#cite_note-7">[7]</a>，直到1950年代或1960年代之前都未曾广泛使用。 <div class="mw-heading mw-heading2"><h2 id="简介">简介</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=3" title="编辑章节：简介">编辑</a>]</div> 如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">都是在<a href="/wiki/Lp%E7%A9%BA%E9%97%B4" title="Lp空间">Lp 空间</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d919ffb3de31d39cecd5f28a2992e96b37dcb9d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.343ex; height:3.176ex;" alt="{\displaystyle L^{1}(\mathbb {R} ^{n})}">内的<a href="/wiki/%E5%8B%92%E8%B2%9D%E6%A0%BC%E7%A9%8D%E5%88%86" title="勒貝格積分">勒贝格可积函数</a>，则二者的卷积存在，并且在这种情况下<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de088e4a3777d3b5d2787fdec81acd91e78a719e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}">也是可积的<a href="#cite_note-8">[8]</a>。这是<a href="/wiki/%E5%AF%8C%E6%AF%94%E5%B0%BC%E5%AE%9A%E7%90%86#非负函数的Tonelli定理" title="富比尼定理">托內利定理</a>的结论。对于在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}">中的函数在离散卷积下，或更一般的对于在任何群的上的卷积，这也是成立的。同样的，如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d18f6a375b29725466fd705ddd852c78b6969a9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.462ex; height:3.176ex;" alt="{\displaystyle f\in L^{1}(\mathbb {R} ^{n})}">而<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\in L^{p}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g\in L^{p}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de000d6e94bc4b702e579b94e7251b4ed2128800" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.305ex; height:2.843ex;" alt="{\displaystyle g\in L^{p}(\mathbb {R} ^{n})}">，这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq p\leq \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <mi>p</mi> <mo>≤</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq p\leq \infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7d63bb8c8def80f4fb709fbb2aae6a11c9cd41d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.853ex; height:2.509ex;" alt="{\displaystyle 1\leq p\leq \infty }">，则<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g\in L^{p}(\mathbb {R} ^{n})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>∈</mo> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g\in L^{p}(\mathbb {R} ^{n})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a8b04e5267aaa143352c2f0cae4cdf5a94055fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.778ex; height:2.843ex;" alt="{\displaystyle f*g\in L^{p}(\mathbb {R} ^{n})}">，并且其<a href="/wiki/Lp%E8%8C%83%E6%95%B0" title="Lp范数">Lp </a><a href="/wiki/%E8%8C%83%E6%95%B0" title="范数">范数</a>间有着<a href="/wiki/%E4%B8%8D%E7%AD%89%E5%BC%8F" title="不等式">不等式</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> <mo>∗</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> <mo>≤</mo> <mo fence="false" stretchy="false">‖</mo> <mi>f</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">‖</mo> <mi>g</mi> <msub> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac167e998466ca09086bc78af3450441a3a79b8a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.23ex; height:3.009ex;" alt="{\displaystyle \|{f}*g\|_{p}\leq \|f\|_{1}\|g\|_{p}}"></dd></dl> 在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c29a2f2fb3f642618036ed7a79712202e7ada924" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:5.52ex; height:2.509ex;" alt="{\displaystyle p=1}">的特殊情况下，这显示出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}">是在卷积下的<a href="/wiki/%E5%B7%B4%E6%8B%BF%E8%B5%AB%E4%BB%A3%E6%95%B0" title="巴拿赫代数">巴拿赫代数</a>（并且如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}"><a href="/wiki/%E5%B9%BE%E4%B9%8E%E8%99%95%E8%99%95" title="幾乎處處">几乎处处</a>非负则两边间等式成立）。 卷积与<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">傅里叶变换</a>有着密切的关系。例如两函数的傅里叶变换的乘积等于它们卷积后的傅里叶变换，利用此一性質，能簡化傅里叶分析中的许多问题。 由卷积得到的函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de088e4a3777d3b5d2787fdec81acd91e78a719e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}">，一般要比<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">都光滑。特别当<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">为具有紧支集的<a href="/wiki/%E5%85%89%E6%BB%91%E5%87%BD%E6%95%B0" title="光滑函数">光滑函数</a>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">为局部可积时，它们的卷积<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de088e4a3777d3b5d2787fdec81acd91e78a719e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}">也是光滑函数。利用这一性质，对于任意的可积函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">，都可以简单地构造出一列逼近于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">的光滑函数列<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{s}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/34f933ad7a8dc310b3fa8e9f7b0b2558cba136db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.143ex; height:2.509ex;" alt="{\displaystyle f_{s}}">，这种方法称为函数的光滑化或<a href="/wiki/%E6%AD%A3%E5%88%99%E5%8C%96_(%E6%95%B0%E5%AD%A6)" title="正则化 (数学)">正则化</a>。 函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f700860ee7af27797d11ddfad3d185eb7af0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}">的<a href="/wiki/%E4%BA%92%E7%9B%B8%E5%85%B3" title="互相关">互相关</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\star g)(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋆</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\star g)(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35266c2d9dbb26a94586f73e383ac149d4c08af7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.41ex; height:2.843ex;" alt="{\displaystyle (f\star g)(\tau )}">，等价于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d950c236d26a2bd8e7b3216c5d488d8f187219a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.098ex; height:2.843ex;" alt="{\displaystyle f(-\tau )}">的<a href="/wiki/%E5%85%B1%E8%BD%AD%E5%A4%8D%E6%95%B0" title="共轭复数">共轭复数</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f(-\tau )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f(-\tau )}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa777133e1e9df2383724c7d0799c0de1019a83c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.213ex; height:3.676ex;" alt="{\displaystyle {\overline {f(-\tau )}}}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ca3806d8f1456510d15896379772656cd465da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.127ex; height:2.843ex;" alt="{\displaystyle g(\tau )}">的卷积： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f\star g)(\tau )\triangleq \int _{-\infty }^{\infty }{\overline {f(t-\tau )}}g(t)\,dt={\overline {f(-\tau )}}*g(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>⋆</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f\star g)(\tau )\triangleq \int _{-\infty }^{\infty }{\overline {f(t-\tau )}}g(t)\,dt={\overline {f(-\tau )}}*g(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e6365fc831a45d617192b3c1595aa3ea6cdf7dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.268ex; height:6.009ex;" alt="{\displaystyle (f\star g)(\tau )\triangleq \int _{-\infty }^{\infty }{\overline {f(t-\tau )}}g(t)\,dt={\overline {f(-\tau )}}*g(\tau )}"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">叫做移位（displacement）或滞后（lag）。 对于<a href="/wiki/%E7%8B%84%E6%8B%89%E5%85%8B%CE%B4%E5%87%BD%E6%95%B0" title="狄拉克δ函数">單位脈衝</a>函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c050147a97868286252447ef73515c8108edd398" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.698ex; height:2.843ex;" alt="{\displaystyle \delta (t)}">和某个函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}">，二者得到的捲積就是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}">本身，此<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}">被稱為<a href="/wiki/%E5%86%B2%E6%BF%80%E5%93%8D%E5%BA%94" title="冲激响应">衝激響應</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\delta *h)(t)=\int _{-\infty }^{\infty }\delta (\tau )h(t-\tau )\,d\tau =h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>δ</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\delta *h)(t)=\int _{-\infty }^{\infty }\delta (\tau )h(t-\tau )\,d\tau =h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ab321a0bbd1a4487255d88feaced4ee25a7fed7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:38.954ex; height:6.009ex;" alt="{\displaystyle (\delta *h)(t)=\int _{-\infty }^{\infty }\delta (\tau )h(t-\tau )\,d\tau =h(t)}"></dd></dl> 在连续时间<a href="/wiki/%E7%BA%BF%E6%80%A7%E6%97%B6%E4%B8%8D%E5%8F%98%E7%B3%BB%E7%BB%9F%E7%90%86%E8%AE%BA" title="线性时不变系统理论">线性非时变系统</a>中，输出信号<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}">被描述为输入<a href="/wiki/%E4%BF%A1%E5%8F%B7_(%E4%BF%A1%E6%81%AF%E8%AE%BA)" title="信号 (信息论)">信号</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}">与<a href="/wiki/%E5%86%B2%E6%BF%80%E5%93%8D%E5%BA%94" title="冲激响应">冲激响应</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}">的卷积<a href="#cite_note-9">[9]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)=(x*h)(t)\ \triangleq \ \int \limits _{-\infty }^{\infty }x(t-\tau )\cdot h(\tau )\,\mathrm {d} \tau =\int \limits _{-\infty }^{\infty }x(\tau )\cdot h(t-\tau )\,\mathrm {d} \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>≜</mo> <mtext> </mtext> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> <mo>=</mo> <munderover> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">d</mi> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=(x*h)(t)\ \triangleq \ \int \limits _{-\infty }^{\infty }x(t-\tau )\cdot h(\tau )\,\mathrm {d} \tau =\int \limits _{-\infty }^{\infty }x(\tau )\cdot h(t-\tau )\,\mathrm {d} \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/724a2ccb3ccd5206c06b2ba6a75d631aaf44d52c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:64.344ex; height:8.843ex;" alt="{\displaystyle y(t)=(x*h)(t)\ \triangleq \ \int \limits _{-\infty }^{\infty }x(t-\tau )\cdot h(\tau )\,\mathrm {d} \tau =\int \limits _{-\infty }^{\infty }x(\tau )\cdot h(t-\tau )\,\mathrm {d} \tau }"></dd></dl> 两个<a href="/wiki/%E7%8B%AC%E7%AB%8B_(%E6%A6%82%E7%8E%87%E8%AE%BA)" title="独立 (概率论)">独立</a>的<a href="/wiki/%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F" title="随机变量">随机变量</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>U</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle U}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">，每个都有一个<a href="/wiki/%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E5%87%BD%E6%95%B0" class="mw-redirect" title="概率密度函数">概率密度函数</a>，二者之和的概率密度，是它们单独的密度函数的卷积： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{U+V}(x)=\int _{-\infty }^{\infty }f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}\right)(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> <mo>+</mo> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>y</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>U</mi> </mrow> </msub> <mo>∗</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>V</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{U+V}(x)=\int _{-\infty }^{\infty }f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}\right)(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebb120a9824eba38d4162458ca90bf2c7f075c98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.266ex; height:6.009ex;" alt="{\displaystyle f_{U+V}(x)=\int _{-\infty }^{\infty }f_{U}(y)f_{V}(x-y)\,dy=\left(f_{U}*f_{V}\right)(x)}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="图解">图解</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=4" title="编辑章节：图解">编辑</a>]</div> <table class="wikitable"> <tbody><tr style="vertical-align:top"> <td> <ol><li>已知右侧第一行图中两个函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf044fe2fbfc4bd8d6d7230f4108430263f9fd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.927ex; height:2.843ex;" alt="{\displaystyle f(t)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b84f700860ee7af27797d11ddfad3d185eb7af0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.765ex; height:2.843ex;" alt="{\displaystyle g(t)}">。</li> <li>首先將兩個函數都用<a href="/wiki/%E7%BA%A6%E6%9D%9F%E5%8F%98%E9%87%8F" class="mw-redirect" title="约束变量">约束变量</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">來表示，并将<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ca3806d8f1456510d15896379772656cd465da" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.127ex; height:2.843ex;" alt="{\displaystyle g(\tau )}">翻转至纵轴另一侧，从而得到右侧第二行图中<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcba00f11285b589b0ff57beeaf118defab2cfe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7270aba4a907afb2c1e074efac5f90ccc6ede41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}">。</li> <li>向函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7270aba4a907afb2c1e074efac5f90ccc6ede41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}">增加一个时间偏移量<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">，得到函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-(\tau -t))=g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(-(\tau -t))=g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c359146c5ddafe39db1b991ac0436f96b022d062" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.33ex; height:2.843ex;" alt="{\displaystyle g(-(\tau -t))=g(t-\tau )}">。<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">不是<a href="/wiki/%E5%B8%B8%E6%95%B0" title="常数">常数</a>而是<a href="/wiki/%E8%87%AA%E7%94%B1%E5%8F%98%E9%87%8F" class="mw-redirect" title="自由变量">自由变量</a>，当<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">取不同值时，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}">能沿着<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">轴“滑动”。如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">是正值，则<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}">等于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7270aba4a907afb2c1e074efac5f90ccc6ede41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}">沿着<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">轴向右（朝向<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }">）滑动数量<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">。如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">是负值，则<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}">等价于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7270aba4a907afb2c1e074efac5f90ccc6ede41e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.935ex; height:2.843ex;" alt="{\displaystyle g(-\tau )}">向左（朝向<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">）滑动数量<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |t|}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |t|}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa9b1439497e4de838a6b1bcf724ef7a8fe48147" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.133ex; height:2.843ex;" alt="{\displaystyle |t|}">。</li> <li>讓<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">從<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca2608c4b5fd3bffc73585f8c67e379b4e99b6f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle -\infty }">变化至<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle +\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>+</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle +\infty }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bddbb0e4420a7e744cf71bd71216e11b0bf88831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.132ex; height:2.176ex;" alt="{\displaystyle +\infty }">，当兩個函數交會時，計算交會範圍中兩個函數乘積的積分值。換句話說，在时间<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">，计算函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bcba00f11285b589b0ff57beeaf118defab2cfe8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.29ex; height:2.843ex;" alt="{\displaystyle f(\tau )}">经过<a href="/wiki/%E6%9D%83%E9%87%8D" title="权重">权重函数</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}">施以权重后其下的面积。右侧第三、第四和第五行图中，分别是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43469ec032d858feae5aa87029e22eaaf0109e9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t=0}">、<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=2.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>2.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=2.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0952a2f5eed09daab8545cb001f2a2e1a357de29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.91ex; height:2.176ex;" alt="{\displaystyle t=2.5}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t=5.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <mn>5.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=5.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a902bad8de47ff2779b2898c3d4293fb304c15c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.91ex; height:2.176ex;" alt="{\displaystyle t=5.5}">时的情况，从<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t>1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>></mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t>1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/730f3de856e6f8850f89a9b990cfc7f7ba7c28bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.101ex; height:2.176ex;" alt="{\displaystyle t>1}">时开始有交会，例如在第四行图中，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4422051052da869dc5b1f0e1cfb06a045ee0c36a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau =0}">则<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )=g(2.5)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>2.5</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )=g(2.5)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c4d64523d9decfa1c0840f18163c1cdfd2f64d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.803ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )=g(2.5)}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =1.5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mn>1.5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =1.5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729234f1df0d6b8c95f6ceabcaced14a42e4de45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.272ex; height:2.176ex;" alt="{\displaystyle \tau =1.5}">则<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )=g(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )=g(1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7e5bdc32b37a2fb5becd7a53f412343dd004fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.993ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )=g(1)}">，对于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau \notin [0,1.5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>∉</mo> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mn>1.5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau \notin [0,1.5]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838205ce42d577093b941bc0a0fe5bcfef0b8f9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.504ex; height:2.843ex;" alt="{\displaystyle \tau \notin [0,1.5]}">有着<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(\tau )g(t-\tau )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(\tau )g(t-\tau )=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7d3069e4a526b82c4d8fdf1157bfccb344ad3f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.358ex; height:2.843ex;" alt="{\displaystyle f(\tau )g(t-\tau )=0}">。</li></ol> <dl><dd>最後得到的<a href="/wiki/%E6%B3%A2%E5%BD%A2" title="波形">波形</a>（未包含在此圖中）就是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">的捲積。</dd></dl> </td> <td><a href="/wiki/File:Convolution3.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Convolution3.svg/468px-Convolution3.svg.png" decoding="async" width="468" height="530" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Convolution3.svg/702px-Convolution3.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Convolution3.svg/936px-Convolution3.svg.png 2x" data-file-width="713" data-file-height="807" /></a> </td></tr> <tr style="vertical-align:top"> <td>两个<a href="/wiki/%E7%9F%A9%E5%BD%A2%E5%87%BD%E6%95%B0" title="矩形函数">矩形</a><a href="/wiki/%E8%84%88%E8%A1%9D%E6%B3%A2" class="mw-redirect" title="脈衝波">脈衝波</a>的捲積。其中函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">首先对<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4422051052da869dc5b1f0e1cfb06a045ee0c36a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.463ex; height:2.176ex;" alt="{\displaystyle \tau =0}">反射，接著平移<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">，成為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t-\tau )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3cb1c1c6de1ad02f14ce3217246b956b66f1c5a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.807ex; height:2.843ex;" alt="{\displaystyle g(t-\tau )}">。那么重叠部份的面积就相当于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">处的卷积，其中橫坐標代表待变量<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">以及新函數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\ast g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f\ast g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f646d7f404b2b4c55d2005540bfe7cf36ee0eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f\ast g}">的自變量<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">。 </td> <td><a href="/wiki/File:Convolution_of_box_signal_with_itself2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6a/Convolution_of_box_signal_with_itself2.gif" decoding="async" width="468" height="147" class="mw-file-element" data-file-width="468" data-file-height="147" /></a> </td></tr> <tr style="vertical-align:top"> <td><a href="/wiki/%E7%9F%A9%E5%BD%A2%E5%87%BD%E6%95%B0" title="矩形函数">矩形</a><a href="/wiki/%E8%84%88%E8%A1%9D%E6%B3%A2" class="mw-redirect" title="脈衝波">脈衝波</a>和<a href="/wiki/%E6%8C%87%E6%95%B0%E8%A1%B0%E5%87%8F" title="指数衰减">指數衰減</a><a href="/wiki/%E8%84%88%E8%A1%9D%E6%B3%A2" class="mw-redirect" title="脈衝波">脈衝波</a>的捲積（後者可能出現於<a href="/wiki/RC%E9%9B%BB%E8%B7%AF" title="RC電路">RC電路</a>中），同樣地重疊部份面積就相當於<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}">處的捲積。注意到因為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">是對稱的，所以在這兩張圖中，反射並不會改變它的形狀。 </td> <td><a href="/wiki/File:Convolution_of_spiky_function_with_box2.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/b/b9/Convolution_of_spiky_function_with_box2.gif" decoding="async" width="468" height="135" class="mw-file-element" data-file-width="468" data-file-height="135" /></a> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="周期卷积">周期卷积</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=5" title="编辑章节：周期卷积">编辑</a>]</div> <style data-mw-deduplicate="TemplateStyles:r84833064">.mw-parser-output .hatnote{font-size:small}.mw-parser-output div.hatnote{padding-left:2em;margin-bottom:0.8em;margin-top:0.8em}.mw-parser-output .hatnote-notice-img::after{content:"\202f \202f \202f \202f "}.mw-parser-output .hatnote-notice-img-small::after{content:"\202f \202f "}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}body.skin-minerva .mw-parser-output .hatnote-notice-img,body.skin-minerva .mw-parser-output .hatnote-notice-img-small{display:none}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%9C%86%E5%91%A8%E5%8D%B7%E7%A7%AF" class="mw-redirect" title="圆周卷积">圆周卷积</a></div> <style data-mw-deduplicate="TemplateStyles:r70089473/mw-parser-output/.tmulti">.mw-parser-output .tmulti .thumbinner{display:flex;flex-direction:column}.mw-parser-output .tmulti .trow{display:flex;flex-direction:row;clear:left;flex-wrap:wrap;width:100%;box-sizing:border-box}.mw-parser-output .tmulti .tsingle{margin:1px;float:left}.mw-parser-output .tmulti .theader{clear:both;font-weight:bold;text-align:center;align-self:center;background-color:transparent;width:100%}.mw-parser-output .tmulti .thumbcaption{background-color:transparent}.mw-parser-output .tmulti .text-align-left{text-align:left}.mw-parser-output .tmulti .text-align-right{text-align:right}.mw-parser-output .tmulti .text-align-center{text-align:center}@media all and (max-width:720px){.mw-parser-output .tmulti .thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}</style><div class="thumb tmulti tright"><div class="thumbinner" style="width:472px;max-width:472px"><div class="trow"><div class="tsingle" style="width:470px;max-width:470px"><div class="thumbimage" style="height:193px;overflow:hidden"><a href="/wiki/File:Splotc2.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Splotc2.gif/468px-Splotc2.gif" decoding="async" width="468" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/6/68/Splotc2.gif 1.5x" data-file-width="512" data-file-height="212" /></a></div></div></div><div class="trow"><div class="tsingle" style="width:470px;max-width:470px"><div class="thumbimage" style="height:193px;overflow:hidden"><a href="/wiki/File:Splotc1.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/02/Splotc1.gif/468px-Splotc1.gif" decoding="async" width="468" height="194" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/0/02/Splotc1.gif 1.5x" data-file-width="512" data-file-height="212" /></a></div></div></div><div class="trow" style="display:flex"><div class="thumbcaption"><a href="/wiki/%E7%9F%A9%E5%BD%A2%E5%87%BD%E6%95%B0" title="矩形函数">矩形</a><a href="/wiki/%E8%84%88%E8%A1%9D%E6%B3%A2" class="mw-redirect" title="脈衝波">脉冲波</a>和<a href="/wiki/%E6%8C%87%E6%95%B0%E8%A1%B0%E5%87%8F" title="指数衰减">指数衰减</a><a href="/wiki/%E8%84%88%E8%A1%9D%E6%B3%A2" class="mw-redirect" title="脈衝波">脉冲波</a>的周期卷积</div></div></div></div> 两个<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">周期的函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf745360cd1499bb246e4c4a0537bbe08dd82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.324ex; height:3.009ex;" alt="{\displaystyle h_{_{T}}(t)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14897f04bb0ca2d651dd456402e3c5795e26bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.314ex; height:3.009ex;" alt="{\displaystyle x_{_{T}}(t)}">的“周期卷积”定义为<a href="#cite_note-Jeruchim-10">[10]</a><a href="#cite_note-Udayashankara-11">[11]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41326470a2d36ed2810203d0db93abc1828ed17c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.792ex; height:6.509ex;" alt="{\displaystyle \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02d3006c4190b1939b04d9b9bb21006fb4e6fa4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.894ex; height:2.343ex;" alt="{\displaystyle t_{0}}">是任意参数。 任何<a href="/w/index.php?title=%E7%BB%9D%E5%AF%B9%E5%8F%AF%E7%A7%AF%E5%88%86%E5%87%BD%E6%95%B0&action=edit&redlink=1" class="new" title="绝对可积分函数（页面不存在）">可积分函数</a>（英语：<a href="https://en.wikipedia.org/wiki/Absolutely_integrable_function" class="extiw" title="en:Absolutely integrable function">Absolutely integrable function</a>）<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}">，都可以通过求函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c484de351ba40ccb9a5ad522c29c1aac5686c0df" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.739ex; height:2.843ex;" alt="{\displaystyle s(t)}">的所有<a href="/wiki/%E5%80%8D%E6%95%B8" title="倍數">整数倍</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">的<a href="/wiki/%E5%B9%B3%E7%A7%BB" title="平移">平移</a>的<a href="/wiki/%E6%B1%82%E5%92%8C%E7%AC%A6%E5%8F%B7" title="求和符号">总和</a>，从而制作出具有<a href="/wiki/%E5%91%A8%E6%9C%9F" class="mw-redirect" title="周期">周期</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">的<a href="/wiki/%E5%91%A8%E6%9C%9F%E5%87%BD%E6%95%B0" title="周期函数">周期函数</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{_{P}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{_{P}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c539843035279ddcb5a59f2ee95b0d09797d743" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.138ex; height:3.009ex;" alt="{\displaystyle s_{_{P}}(t)}">，这叫做<a href="/w/index.php?title=%E5%91%A8%E6%9C%9F%E6%B1%82%E5%92%8C&action=edit&redlink=1" class="new" title="周期求和（页面不存在）">周期求和</a>（英语：<a href="https://en.wikipedia.org/wiki/Periodic_summation" class="extiw" title="en:Periodic summation">Periodic summation</a>）： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{_{P}}(t)\triangleq \sum _{m=-\infty }^{\infty }s(t+mP)=\sum _{m=-\infty }^{\infty }s(t-mP),\quad m\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>m</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>m</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{_{P}}(t)\triangleq \sum _{m=-\infty }^{\infty }s(t+mP)=\sum _{m=-\infty }^{\infty }s(t-mP),\quad m\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f1c0e96964754d5f4768b459bb80b47f1a158d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.914ex; height:6.843ex;" alt="{\displaystyle s_{_{P}}(t)\triangleq \sum _{m=-\infty }^{\infty }s(t+mP)=\sum _{m=-\infty }^{\infty }s(t-mP),\quad m\in \mathbb {Z} }"></dd></dl> 对于无周期函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">，其周期<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">的周期求和分别是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf745360cd1499bb246e4c4a0537bbe08dd82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.324ex; height:3.009ex;" alt="{\displaystyle h_{_{T}}(t)}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14897f04bb0ca2d651dd456402e3c5795e26bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.314ex; height:3.009ex;" alt="{\displaystyle x_{_{T}}(t)}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">的周期卷积，可以定义为<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14897f04bb0ca2d651dd456402e3c5795e26bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.314ex; height:3.009ex;" alt="{\displaystyle x_{_{T}}(t)}">的常规卷积，或<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf745360cd1499bb246e4c4a0537bbe08dd82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.324ex; height:3.009ex;" alt="{\displaystyle h_{_{T}}(t)}">的常规卷积，二者都等价于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0dbf745360cd1499bb246e4c4a0537bbe08dd82e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.324ex; height:3.009ex;" alt="{\displaystyle h_{_{T}}(t)}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a14897f04bb0ca2d651dd456402e3c5795e26bc1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.314ex; height:3.009ex;" alt="{\displaystyle x_{_{T}}(t)}">的周期积分： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h*x_{_{T}})(t)\ \triangleq \ \ \int _{-\infty }^{\infty }h(\tau )x_{_{T}}(t-\tau )\,d\tau \ =\ \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mo>≜</mo> <mtext> </mtext> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> </mrow> </msubsup> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h*x_{_{T}})(t)\ \triangleq \ \ \int _{-\infty }^{\infty }h(\tau )x_{_{T}}(t-\tau )\,d\tau \ =\ \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed3f9f6cb530132f7eb6e616ca84c3ef7b9df0ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:66.669ex; height:6.509ex;" alt="{\displaystyle (h*x_{_{T}})(t)\ \triangleq \ \ \int _{-\infty }^{\infty }h(\tau )x_{_{T}}(t-\tau )\,d\tau \ =\ \int _{t_{0}}^{t_{0}+T}h_{_{T}}(\tau )x_{_{T}}(t-\tau )\,d\tau }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h*x_{_{T}})(t)=(x*h_{_{T}})(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>∗</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h*x_{_{T}})(t)=(x*h_{_{T}})(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b531b6fddc8d74b69d41f72c7442cab1f1073fd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.413ex; height:3.009ex;" alt="{\displaystyle (h*x_{_{T}})(t)=(x*h_{_{T}})(t)}"></dd></dl> <a href="/wiki/%E5%9C%86%E5%91%A8%E5%8D%B7%E7%A7%AF" class="mw-redirect" title="圆周卷积">圆周卷积</a>是周期卷积的特殊情况<a href="#cite_note-Udayashankara-11">[11]</a><a href="#cite_note-Priemer-12">[12]</a>，其中函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">二者的非零部份，都限定在区间<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,T]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ccef2d3dc751e081375d51c111709d8a1d7ac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle [0,T]}">之内，此时的周期求和称为“周期延拓”。<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h*x_{_{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h*x_{_{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b33366f8a6a4af042b06e2901487cfb39f7a040" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.199ex; height:2.843ex;" alt="{\displaystyle h*x_{_{T}}}">中函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/56fa15d468a0a0215f4fad40c34256bd48ba91db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.666ex; height:2.343ex;" alt="{\displaystyle x_{_{T}}}">可以通过取<a href="/wiki/%E6%80%A7%E8%B3%AA%E7%AC%A6%E8%99%9F" title="性質符號">非负</a><a href="/wiki/%E4%BD%99%E6%95%B0" title="余数">余数</a>的<a href="/wiki/%E6%A8%A1%E9%99%A4" title="模除">模除</a>运算表达为“圆周函数”： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{_{T}}(t)=x(t_{\mathrm {mod} \ T}),\quad t\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>t</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{_{T}}(t)=x(t_{\mathrm {mod} \ T}),\quad t\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d0f00f91b1a232ae69bdc58b0b56ee007f5e8419" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:26.181ex; height:3.009ex;" alt="{\displaystyle x_{_{T}}(t)=x(t_{\mathrm {mod} \ T}),\quad t\in \mathbb {R} }"></dd></dl> 而积分的界限可以缩简至函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">的长度范围<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,T]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>T</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,T]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35ccef2d3dc751e081375d51c111709d8a1d7ac6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle [0,T]}">： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h*x_{_{T}})(t)=\int _{0}^{T}h(\tau )x((t-\tau )_{\mathrm {mod} \ T})\ d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">m</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">d</mi> </mrow> <mtext> </mtext> <mi>T</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h*x_{_{T}})(t)=\int _{0}^{T}h(\tau )x((t-\tau )_{\mathrm {mod} \ T})\ d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/18ab3031f01325fa5355ba6220fd10be1b2d2da7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:40.25ex; height:6.176ex;" alt="{\displaystyle (h*x_{_{T}})(t)=\int _{0}^{T}h(\tau )x((t-\tau )_{\mathrm {mod} \ T})\ d\tau }"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="离散卷积">离散卷积</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=6" title="编辑章节：离散卷积">编辑</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Konvolutsioon.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/37/Konvolutsioon.png/200px-Konvolutsioon.png" decoding="async" width="200" height="136" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/3/37/Konvolutsioon.png 1.5x" data-file-width="300" data-file-height="204" /></a><figcaption>离散卷积示意图</figcaption></figure> 对于定义在整數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }">上且得出<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>值的函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">，离散卷积定义为<a href="#cite_note-13">[13]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)[n]\ \ \triangleq \ \sum _{m=-\infty }^{\infty }{f[m]g[n-m]}=\sum _{m=-\infty }^{\infty }f[n-m]\,g[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mtext> </mtext> <mo>≜</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mspace width="thinmathspace" /> <mi>g</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)[n]\ \ \triangleq \ \sum _{m=-\infty }^{\infty }{f[m]g[n-m]}=\sum _{m=-\infty }^{\infty }f[n-m]\,g[m]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bd9a55083d6a0d0aa8bda67d179ebc0c7970166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:56.456ex; height:6.843ex;" alt="{\displaystyle (f*g)[n]\ \ \triangleq \ \sum _{m=-\infty }^{\infty }{f[m]g[n-m]}=\sum _{m=-\infty }^{\infty }f[n-m]\,g[m]}"></dd></dl> 這裡一樣把函數定義域以外的值當成零，所以可以擴展函數到所有整數上（如果本來不是的話）。两个有限序列的卷积的定义，是将这些序列扩展成在整数集合上有限支撑的函数。在这些序列是两个<a href="/wiki/%E5%A4%9A%E9%A1%B9%E5%BC%8F" class="mw-redirect" title="多项式">多项式</a>的系数之时，这两个多项式的普通乘积的系数，就是这两个序列的卷积。这叫做序列系数的<a href="/wiki/%E6%9F%AF%E8%A5%BF%E4%B9%98%E7%A7%AF" title="柯西乘积">柯西乘积</a>。 當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">的<a href="/wiki/%E6%94%AF%E6%92%91%E9%9B%86" title="支撑集">支撐集</a>為有限長度的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{-M,-M+1,\dots ,M-1,M\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo>−</mo> <mi>M</mi> <mo>,</mo> <mo>−</mo> <mi>M</mi> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>M</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{-M,-M+1,\dots ,M-1,M\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83f986c5ada0f03f25b8ef914cb30da0f84dabaa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.962ex; height:2.843ex;" alt="{\displaystyle \{-M,-M+1,\dots ,M-1,M\}}">之时，上式會變成有限<a href="/wiki/%E6%B1%82%E5%92%8C%E7%AC%A6%E5%8F%B7" title="求和符号">求和</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi>M</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0a31f12e6771e6a8b1868f15214e43c8714c1d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.597ex; height:7.509ex;" alt="{\displaystyle (f*g)[n]=\sum _{m=-M}^{M}f[n-m]g[m]}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="多维离散卷积">多维离散卷积</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=7" title="编辑章节：多维离散卷积">编辑</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/w/index.php?title=%E5%A4%9A%E7%BB%B4%E7%A6%BB%E6%95%A3%E5%8D%B7%E7%A7%AF&action=edit&redlink=1" class="new" title="多维离散卷积（页面不存在）">多维离散卷积</a>（英语：<a href="https://en.wikipedia.org/wiki/Multidimensional_discrete_convolution" class="extiw" title="en:Multidimensional discrete convolution">Multidimensional discrete convolution</a>）</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:2D_Convolution_Animation.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/1/19/2D_Convolution_Animation.gif" decoding="async" width="390" height="345" class="mw-file-element" data-file-width="390" data-file-height="345" /></a><figcaption>用离散二维卷积对<a href="/wiki/%E6%95%B0%E5%AD%97%E5%9B%BE%E5%83%8F" title="数字图像">图像</a>进行<a href="/w/index.php?title=%E6%A0%B8_(%E5%9B%BE%E5%83%8F%E5%A4%84%E7%90%86)&action=edit&redlink=1" class="new" title="核 (图像处理)（页面不存在）">锐化</a>（英语：<a href="https://en.wikipedia.org/wiki/Kernel_(image_processing)" class="extiw" title="en:Kernel (image processing)">Kernel (image processing)</a>）<a href="/wiki/%E5%9B%BE%E5%83%8F%E5%A4%84%E7%90%86" title="图像处理">处理</a>的动画</figcaption></figure> 类似于一维情况，使用<a href="/wiki/%E6%98%9F%E5%8F%B7" class="mw-redirect" title="星号">星号</a>表示卷积，而维度体现在星号的数量上，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">维卷积就写为<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">个星号。下面是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">维信号的卷积的表示法： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(n_{1},n_{2},...,n_{_{M}})=h(n_{1},n_{2},...,n_{_{M}})*{\overset {M}{\cdots }}*x(n_{1},n_{2},...,n_{_{M}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mo>⋯</mo> <mi>M</mi> </mover> </mrow> <mo>∗</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(n_{1},n_{2},...,n_{_{M}})=h(n_{1},n_{2},...,n_{_{M}})*{\overset {M}{\cdots }}*x(n_{1},n_{2},...,n_{_{M}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8882d8be851e18fbfbd33e5e61d90591fc4fcf6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:62.347ex; height:3.843ex;" alt="{\displaystyle y(n_{1},n_{2},...,n_{_{M}})=h(n_{1},n_{2},...,n_{_{M}})*{\overset {M}{\cdots }}*x(n_{1},n_{2},...,n_{_{M}})}"></dd></dl> 对于离散值的信号，这个卷积可以直接如下这样计算： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{_{M}}=-\infty }^{\infty }h(k_{1},k_{2},...,k_{_{M}})x(n_{1}-k_{1},n_{2}-k_{2},...,n_{_{M}}-k_{_{M}})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mo>.</mo> <mo>.</mo> <mo>.</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">(</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mi>x</mi> <mo stretchy="false">(</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{_{M}}=-\infty }^{\infty }h(k_{1},k_{2},...,k_{_{M}})x(n_{1}-k_{1},n_{2}-k_{2},...,n_{_{M}}-k_{_{M}})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84ae275b6e8d1788a42b8e45cfa43153e4e78741" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.838ex; width:72.797ex; height:7.676ex;" alt="{\displaystyle \sum _{k_{1}=-\infty }^{\infty }\sum _{k_{2}=-\infty }^{\infty }...\sum _{k_{_{M}}=-\infty }^{\infty }h(k_{1},k_{2},...,k_{_{M}})x(n_{1}-k_{1},n_{2}-k_{2},...,n_{_{M}}-k_{_{M}})}"></dd></dl> 结果的离散多维卷积所支撑的输出区域，基于两个输入信号所支撑的大小和区域来决定。 <dl><dd><figure class="mw-halign-none" typeof="mw:File/Thumb"><a href="/wiki/File:Picture1_wiki.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/12/Picture1_wiki.png/600px-Picture1_wiki.png" decoding="async" width="600" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/1/12/Picture1_wiki.png 1.5x" data-file-width="748" data-file-height="244" /></a><figcaption>在两个二维信号之间的卷积的可视化</figcaption></figure></dd></dl> <div class="mw-heading mw-heading3"><h3 id="离散周期卷积">离散周期卷积</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=8" title="编辑章节：离散周期卷积">编辑</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Aperiodisch_periodische_Faltung.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Aperiodisch_periodische_Faltung.svg/390px-Aperiodisch_periodische_Faltung.svg.png" decoding="async" width="390" height="327" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Aperiodisch_periodische_Faltung.svg/585px-Aperiodisch_periodische_Faltung.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Aperiodisch_periodische_Faltung.svg/780px-Aperiodisch_periodische_Faltung.svg.png 2x" data-file-width="585" data-file-height="490" /></a><figcaption>对比离散无周期卷积（左列）与离散圆周卷积（右列）</figcaption></figure> 对于离散序列和一个参数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">，无周期函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">的“周期卷积”是为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h*x_{_{N}})[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\underbrace {x_{_{N}}[n-m]} _{\sum _{k=-\infty }^{\infty }x[n-m-kN]}\ =\ \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{h}[m-kN]\right)x_{_{N}}[n-m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mo>≜</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> </munder> <mtext> </mtext> <mo>=</mo> <mtext> </mtext> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mi>h</mi> </mrow> <mo stretchy="false">[</mo> <mi>m</mi> <mo>−</mo> <mi>k</mi> <mi>N</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h*x_{_{N}})[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\underbrace {x_{_{N}}[n-m]} _{\sum _{k=-\infty }^{\infty }x[n-m-kN]}\ =\ \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{h}[m-kN]\right)x_{_{N}}[n-m]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14cb0408cb58e39c596cab8ee9f1f63cb6e2fa3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.171ex; width:81.215ex; height:9.509ex;" alt="{\displaystyle (h*x_{_{N}})[n]\ \triangleq \ \sum _{m=-\infty }^{\infty }h[m]\underbrace {x_{_{N}}[n-m]} _{\sum _{k=-\infty }^{\infty }x[n-m-kN]}\ =\ \sum _{m=0}^{N-1}\left(\sum _{k=-\infty }^{\infty }{h}[m-kN]\right)x_{_{N}}[n-m]}"></dd></dl> 这个函数有周期<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">，它有最多<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">个唯一性的值。<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}">的非零范围都是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,N-1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [0,N-1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5d4099dcd12fea62ad089a8d26d2a14c54fc88f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.557ex; height:2.843ex;" alt="{\displaystyle [0,N-1]}">的特殊情况叫做<a href="/wiki/%E5%9C%86%E5%91%A8%E5%8D%B7%E7%A7%AF" class="mw-redirect" title="圆周卷积">圆周卷积</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (h*x_{_{N}})[n]=\sum _{m=0}^{N-1}h[m]x_{_{N}}[n-m]=\sum _{m=0}^{N-1}h[m]x[(n-m)_{\bmod {N}}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>h</mi> <mo>∗</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <msub> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo lspace="thickmathspace" rspace="thickmathspace">mod</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (h*x_{_{N}})[n]=\sum _{m=0}^{N-1}h[m]x_{_{N}}[n-m]=\sum _{m=0}^{N-1}h[m]x[(n-m)_{\bmod {N}}]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac37f1cfe50298bfdefdcd870a5627ef30db1cee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:61.654ex; height:7.343ex;" alt="{\displaystyle (h*x_{_{N}})[n]=\sum _{m=0}^{N-1}h[m]x_{_{N}}[n-m]=\sum _{m=0}^{N-1}h[m]x[(n-m)_{\bmod {N}}]}"></dd></dl> 离散圆周卷积可简约为<a href="/wiki/%E7%9F%A9%E9%98%B5%E4%B9%98%E6%B3%95" class="mw-redirect" title="矩阵乘法">矩阵乘法</a>，这里的<a href="/wiki/%E7%A7%AF%E5%88%86%E5%8F%98%E6%8D%A2" title="积分变换">积分变换</a>的核函数是<a href="/wiki/%E5%BE%AA%E7%8E%AF%E7%9F%A9%E9%98%B5" title="循环矩阵">循环矩阵</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}y_{0}\\y_{1}\\\vdots \\y_{_{N-1}}\end{bmatrix}}={\begin{bmatrix}h_{0}&h_{_{N-1}}&\cdots &h_{1}\\h_{1}&h_{0}&\cdots &h_{2}\\\vdots &\vdots &\ddots &\vdots \\h_{_{N-1}}&h_{_{N-2}}&\cdots &h_{0}\end{bmatrix}}{\begin{bmatrix}x_{0}\\x_{1}\\\vdots \\x_{_{N-1}}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}y_{0}\\y_{1}\\\vdots \\y_{_{N-1}}\end{bmatrix}}={\begin{bmatrix}h_{0}&h_{_{N-1}}&\cdots &h_{1}\\h_{1}&h_{0}&\cdots &h_{2}\\\vdots &\vdots &\ddots &\vdots \\h_{_{N-1}}&h_{_{N-2}}&\cdots &h_{0}\end{bmatrix}}{\begin{bmatrix}x_{0}\\x_{1}\\\vdots \\x_{_{N-1}}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79c33531e3128fd998256cdb7740cbbc592ca1a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:45.287ex; height:14.843ex;" alt="{\displaystyle {\begin{bmatrix}y_{0}\\y_{1}\\\vdots \\y_{_{N-1}}\end{bmatrix}}={\begin{bmatrix}h_{0}&h_{_{N-1}}&\cdots &h_{1}\\h_{1}&h_{0}&\cdots &h_{2}\\\vdots &\vdots &\ddots &\vdots \\h_{_{N-1}}&h_{_{N-2}}&\cdots &h_{0}\end{bmatrix}}{\begin{bmatrix}x_{0}\\x_{1}\\\vdots \\x_{_{N-1}}\end{bmatrix}}}"></dd></dl> 圆周卷积最经常出现的<a href="/wiki/%E5%BF%AB%E9%80%9F%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="快速傅里叶变换">快速傅里叶变换</a>的实现算法比如<a href="/wiki/%E9%9B%B7%E5%BE%B7%E6%BC%94%E7%AE%97%E6%B3%95" title="雷德演算法">雷德演算法</a>之中。 <div class="mw-heading mw-heading2"><h2 id="性质">性质</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=9" title="编辑章节：性质">编辑</a>]</div> <div class="mw-heading mw-heading3"><h3 id="代数">代数</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=10" title="编辑章节：代数">编辑</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF%E4%BB%A3%E6%95%B0&action=edit&redlink=1" class="new" title="卷积代数（页面不存在）">卷积代数</a>（英语：<a href="https://en.wikipedia.org/wiki/Convolution_algebra" class="extiw" title="en:Convolution algebra">Convolution algebra</a>）</div> 各种卷积算子都满足下列性质： <dl><dt><a href="/wiki/%E4%BA%A4%E6%8D%A2%E5%BE%8B" class="mw-redirect" title="交换律">交换律</a></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g=g*f\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>g</mi> <mo>∗</mo> <mi>f</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g=g*f\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e0702c60de9cfe34bad273e611a5f2503f521961" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.664ex; height:2.509ex;" alt="{\displaystyle f*g=g*f\,}"></dd></dl> <dl><dt><a href="/wiki/%E7%BB%93%E5%90%88%E5%BE%8B" title="结合律">结合律</a></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*(g*h)=(f*g)*h\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>h</mi> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*(g*h)=(f*g)*h\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f91da320ed0154cf328e6777c90f3c55b87d177" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.35ex; height:2.843ex;" alt="{\displaystyle f*(g*h)=(f*g)*h\,}"></dd></dl> <dl><dt><a href="/wiki/%E5%88%86%E9%85%8D%E5%BE%8B" title="分配律">分配律</a></dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*(g+h)=(f*g)+(f*h)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*(g+h)=(f*g)+(f*h)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2796399c899a369d328478676837f13a05beb7d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.924ex; height:2.843ex;" alt="{\displaystyle f*(g+h)=(f*g)+(f*h)\,}"></dd></dl> <dl><dt>数乘结合律</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a(f*g)=(af)*g=f*(ag)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi>a</mi> <mi>g</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(f*g)=(af)*g=f*(ag)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/15bcb3b0cc4a54bcc9028ebdd4071274628af33b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.469ex; height:2.843ex;" alt="{\displaystyle a(f*g)=(af)*g=f*(ag)\,}"></dd></dl> 其中<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">为任意<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>（或<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>）。 <dl><dt>复数共轭</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>f</mi> <mo accent="false">¯</mo> </mover> </mrow> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>g</mi> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/470c71208829c0057c4e5039c3d57c05ca753822" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.722ex; height:3.343ex;" alt="{\displaystyle {\overline {f*g}}={\overline {f}}*{\overline {g}}}"></dd></dl> <dl><dt>微分有关</dt> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)'=f'*g=f*g'}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <msup> <mo stretchy="false">)</mo> <mo>′</mo> </msup> <mo>=</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <msup> <mi>g</mi> <mo>′</mo> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)'=f'*g=f*g'}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb43d0eaccf5b46033e79f16a1a88d99eba34ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.872ex; height:3.009ex;" alt="{\displaystyle (f*g)'=f'*g=f*g'}"></dd></dl> <dl><dt>积分有关</dt> <dd>如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>f</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98473e8bdf1dcc8ae0fe58d7ca12d35d24467601" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:18.834ex; height:3.676ex;" alt="{\textstyle F(t)=\int _{-\infty }^{t}f(\tau )d\tau }">，并且<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23987b1ff80b3b0a21bcc674a65eb86126b9c1db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:19.145ex; height:3.676ex;" alt="{\textstyle G(t)=\int _{-\infty }^{t}g(\tau )\,d\tau }">，则有：</dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>F</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>G</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bf3e59d8b80a869159b7668c92cd3de8bd05c0b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:42.126ex; height:6.343ex;" alt="{\displaystyle (F*g)(t)=(f*G)(t)=\int _{-\infty }^{t}(f*g)(\tau )\,d\tau }"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="积分">积分</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=11" title="编辑章节：积分">编辑</a>]</div> 如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">是可积分函数，则它们在整个空间上的卷积的积分，简单的就是它们积分的乘积<a href="#cite_note-14">[14]</a>： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{\mathbb {R} ^{n}}(f*g)(t)\,d^{n}t=\left(\int _{\mathbb {R} ^{n}}f(t)\,d^{n}t\right)\left(\int _{\mathbb {R} ^{n}}g(t)\,d^{n}t\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{\mathbb {R} ^{n}}(f*g)(t)\,d^{n}t=\left(\int _{\mathbb {R} ^{n}}f(t)\,d^{n}t\right)\left(\int _{\mathbb {R} ^{n}}g(t)\,d^{n}t\right)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1e073a296110ce71299bf95518e57e45d6a7f3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.861ex; height:6.176ex;" alt="{\displaystyle \int _{\mathbb {R} ^{n}}(f*g)(t)\,d^{n}t=\left(\int _{\mathbb {R} ^{n}}f(t)\,d^{n}t\right)\left(\int _{\mathbb {R} ^{n}}g(t)\,d^{n}t\right)}"></dd></dl> 这是<a href="/wiki/%E5%AF%8C%E6%AF%94%E5%B0%BC%E5%AE%9A%E7%90%86" title="富比尼定理">富比尼定理</a>的结果。如果<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">只被假定为非负可测度函数，根据<a href="/wiki/%E5%AF%8C%E6%AF%94%E5%B0%BC%E5%AE%9A%E7%90%86#非负函数的Tonelli定理" title="富比尼定理">托内利定理</a>，这也是成立的。 <div class="mw-heading mw-heading3"><h3 id="微分">微分</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=12" title="编辑章节：微分">编辑</a>]</div> 在一元函数情况下，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">的卷积的<a href="/wiki/%E5%AF%BC%E6%95%B0" title="导数">导数</a>有着： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}(f*g)={\frac {df}{dt}}*g=f*{\frac {dg}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>f</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>g</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}(f*g)={\frac {df}{dt}}*g=f*{\frac {dg}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5daf1200a04c8cbd3164dee2b061b1f4cffa312" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:28.77ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}(f*g)={\frac {df}{dt}}*g=f*{\frac {dg}{dt}}}"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55e262410b19a3eeab08b01224e89815c110c8ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:2.892ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}}">是<a href="/wiki/%E5%BE%AE%E5%88%86%E7%AE%97%E5%AD%90" title="微分算子">微分算子</a>。更一般的说，在<a href="/wiki/%E5%A4%9A%E5%85%83%E5%87%BD%E6%95%B0" class="mw-redirect" title="多元函数">多元函数</a>的情况下，对<a href="/wiki/%E5%81%8F%E5%AF%BC%E6%95%B0" title="偏导数">偏导数</a>也有类似的公式： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial }{\partial t_{i}}}(f*g)={\frac {\partial f}{\partial t_{i}}}*g=f*{\frac {\partial g}{\partial t_{i}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>g</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial t_{i}}}(f*g)={\frac {\partial f}{\partial t_{i}}}*g=f*{\frac {\partial g}{\partial t_{i}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85ae3732580569e3e82fc24ce769e49437b9cd00" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:30.76ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial }{\partial t_{i}}}(f*g)={\frac {\partial f}{\partial t_{i}}}*g=f*{\frac {\partial g}{\partial t_{i}}}}"></dd></dl> 这就有了一个特殊结论，卷积可以看作“光滑”运算：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">的卷积可微分的次数，是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">的总数。 这些恒等式成立的严格条件，为<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">是绝对可积分的，并且至少二者之一有绝对可积分（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L^{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L^{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74c288d1089f1ec85b01b4de25c441fc792bd2d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.637ex; height:2.676ex;" alt="{\displaystyle L^{1}}">）弱导数，这是<a href="/w/index.php?title=Young%E5%8D%B7%E7%A7%AF%E4%B8%8D%E7%AD%89%E5%BC%8F&action=edit&redlink=1" class="new" title="Young卷积不等式（页面不存在）">Young卷积不等式</a>（英语：<a href="https://en.wikipedia.org/wiki/Young%27s_convolution_inequality" class="extiw" title="en:Young's convolution inequality">Young's convolution inequality</a>）的结论。 在离散情况下，<a href="/wiki/%E5%B7%AE%E5%88%86" title="差分">差分</a><a href="/wiki/%E7%AE%97%E5%AD%90" title="算子">算子</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta [f](n)=f(n+1)-f(n)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mi>n</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta [f](n)=f(n+1)-f(n)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/10ef9708b32ccda9a3be9ff9c1f8865c4107cf59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.619ex; height:2.843ex;" alt="{\displaystyle \Delta [f](n)=f(n+1)-f(n)}">满足类似的关系： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta (f*g)=(\Delta f)*g=f*(\Delta g)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>g</mi> <mo>=</mo> <mi>f</mi> <mo>∗</mo> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>g</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta (f*g)=(\Delta f)*g=f*(\Delta g)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a3054211a7bca858f948f7c9dfa3dc7f42d6557" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.2ex; height:2.843ex;" alt="{\displaystyle \Delta (f*g)=(\Delta f)*g=f*(\Delta g)}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="卷积定理">卷积定理</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=13" title="编辑章节：卷积定理">编辑</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r84833064"><div role="note" class="hatnote navigation-not-searchable">主条目：<a href="/wiki/%E5%8D%B7%E7%A7%AF%E5%AE%9A%E7%90%86" title="卷积定理">卷积定理</a></div> <a href="/wiki/%E5%8D%B7%E7%A7%AF%E5%AE%9A%E7%90%86" title="卷积定理">卷积定理</a>指出<a href="#cite_note-15">[15]</a>，在适当的条件下，两个函数（或<a href="/wiki/%E4%BF%A1%E5%8F%B7" class="mw-disambig" title="信号">信号</a>）的卷积的<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">傅里叶变换</a>，是它们的傅里叶变换的<a href="/wiki/%E9%80%90%E7%82%B9%E4%B9%98%E7%A7%AF" title="逐点乘积">逐点乘积</a>。更一般的说，在一个域（比如<a href="/wiki/%E6%99%82%E5%9F%9F" title="時域">时域</a>）中的卷积等于在其他域（比如<a href="/wiki/%E9%A2%91%E5%9F%9F" class="mw-redirect" title="频域">频域</a>）<a href="/wiki/%E9%80%90%E7%82%B9" title="逐点">逐点</a>乘法。 设两个函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02c07825dae28705df03d15daeb8844d49c4dbd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.478ex; height:2.843ex;" alt="{\displaystyle h(x)}">，分别具有<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">傅里叶变换</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}">： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\int _{-\infty }^{\infty }g(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\int _{-\infty }^{\infty }h(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>s</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>s</mi> <mi>x</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\int _{-\infty }^{\infty }g(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\int _{-\infty }^{\infty }h(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d46a4244a87c775849a7b9a41ebcaf2576b0d89" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:48.249ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\int _{-\infty }^{\infty }g(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\int _{-\infty }^{\infty }h(x)e^{-i2\pi sx}\,dx,\quad s\in \mathbb {R} \end{aligned}}}"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}"><a href="/wiki/%E7%AE%97%E5%AD%90" title="算子">算子</a>指示<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">傅里叶变换</a>。 卷积定理声称： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g*h\}(s)=G(s)H(s),\quad s\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∗</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g*h\}(s)=G(s)H(s),\quad s\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d51ac270a6b46716ddadd6df1260e48ffa2405e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.555ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{g*h\}(s)=G(s)H(s),\quad s\in \mathbb {R} }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g\cdot h\}(s)=G(s)*H(s),\quad s\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>⋅</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>∗</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g\cdot h\}(s)=G(s)*H(s),\quad s\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7c544d4e59f781cf6fd4903a144c59ea7169de8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:35.234ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{g\cdot h\}(s)=G(s)*H(s),\quad s\in \mathbb {R} }"></dd></dl> 应用逆傅里叶变换<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/798153399d91ed4f7c88fa012bd0fabe708c4de2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.336ex; height:2.676ex;" alt="{\displaystyle {\mathcal {F}}^{-1}}">产生推论： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g*h)(s)={\mathcal {F}}^{-1}\{G\cdot H\},\quad s\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>G</mi> <mo>⋅</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g*h)(s)={\mathcal {F}}^{-1}\{G\cdot H\},\quad s\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf440c01e9f89b2eb44ca3c2fd55a730d084cba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.653ex; height:3.176ex;" alt="{\displaystyle (g*h)(s)={\mathcal {F}}^{-1}\{G\cdot H\},\quad s\in \mathbb {R} }"></dd> <dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g\cdot h)(s)={\mathcal {F}}^{-1}\{G*H\},\quad s\in \mathbb {R} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>⋅</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mi>G</mi> <mo>∗</mo> <mi>H</mi> <mo fence="false" stretchy="false">}</mo> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\cdot h)(s)={\mathcal {F}}^{-1}\{G*H\},\quad s\in \mathbb {R} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be42ca7c0e1256d362473717850e2ee740395c8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:33.653ex; height:3.176ex;" alt="{\displaystyle (g\cdot h)(s)={\mathcal {F}}^{-1}\{G*H\},\quad s\in \mathbb {R} }"></dd></dl> 这里的算符<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \,\cdot \,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mspace width="thinmathspace" /> <mo>⋅</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \,\cdot \,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3081768f879578964ed0ead9a51abaa41c41d542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.421ex; height:1.176ex;" alt="{\displaystyle \,\cdot \,}">指示<a href="/wiki/%E9%80%90%E7%82%B9" title="逐点">逐点</a>乘法。 这一定理对<a href="/wiki/%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2" title="拉普拉斯变换">拉普拉斯变换</a>、<a href="/wiki/%E5%8F%8C%E8%BE%B9%E6%8B%89%E6%99%AE%E6%8B%89%E6%96%AF%E5%8F%98%E6%8D%A2" title="双边拉普拉斯变换">双边拉普拉斯变换</a>、<a href="/wiki/Z%E5%8F%98%E6%8D%A2" class="mw-redirect" title="Z变换">Z变换</a>、<a href="/wiki/%E6%A2%85%E6%9E%97%E5%8F%98%E6%8D%A2" title="梅林变换">梅林变换</a>和<a href="/w/index.php?title=Hartley%E5%8F%98%E6%8D%A2&action=edit&redlink=1" class="new" title="Hartley变换（页面不存在）">Hartley变换</a>（英语：<a href="https://en.wikipedia.org/wiki/Hartley_transform" class="extiw" title="en:Hartley transform">Hartley transform</a>）等各种傅里叶变换的变体同样成立。在<a href="/wiki/%E8%AA%BF%E5%92%8C%E5%88%86%E6%9E%90" title="調和分析">调和分析</a>中还可以推广到在局部紧致的<a href="/wiki/%E9%98%BF%E8%B4%9D%E5%B0%94%E7%BE%A4" title="阿贝尔群">阿贝尔群</a>上定义的傅里叶变换。 <div class="mw-heading mw-heading3"><h3 id="周期卷积_2">周期卷积</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=14" title="编辑章节：周期卷积">编辑</a>]</div> 对于周期为<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">的函数<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{_{P}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{_{P}}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67f61b19c957f7cd78069b1627daf01db33775e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.647ex; height:3.009ex;" alt="{\displaystyle g_{_{P}}(x)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{P}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{P}}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8172030295d1c18d63dc30c3452ccd9b9d4489ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.876ex; height:3.009ex;" alt="{\displaystyle h_{_{P}}(x)}">，可以被表达为二者的<a href="/w/index.php?title=%E5%91%A8%E6%9C%9F%E6%B1%82%E5%92%8C&action=edit&redlink=1" class="new" title="周期求和（页面不存在）">周期求和</a>（英语：<a href="https://en.wikipedia.org/wiki/Periodic_summation" class="extiw" title="en:Periodic summation">Periodic summation</a>）： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }g(x-mP),\quad m\in \mathbb {Z} \\h_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }h(x-mP),\quad m\in \mathbb {Z} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>m</mi> <mi>P</mi> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }g(x-mP),\quad m\in \mathbb {Z} \\h_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }h(x-mP),\quad m\in \mathbb {Z} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81b6f246e1f3f696a28abf6c295cfc054562b0f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:37.229ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}g_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }g(x-mP),\quad m\in \mathbb {Z} \\h_{_{P}}(x)\ &\triangleq \sum _{m=-\infty }^{\infty }h(x-mP),\quad m\in \mathbb {Z} \end{aligned}}}"></dd></dl> 它们的<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0" title="傅里叶级数">傅里叶级数</a><a href="/wiki/%E7%B3%BB%E6%95%B0" title="系数">系数</a>为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G[k]&\triangleq {\mathcal {F}}\{g_{_{P}}\}[k]={\frac {1}{P}}\int _{P}g_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \\H[k]&\triangleq {\mathcal {F}}\{h_{_{P}}\}[k]={\frac {1}{P}}\int _{P}h_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>P</mi> </mfrac> </mrow> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>P</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>d</mi> <mi>x</mi> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G[k]&\triangleq {\mathcal {F}}\{g_{_{P}}\}[k]={\frac {1}{P}}\int _{P}g_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \\H[k]&\triangleq {\mathcal {F}}\{h_{_{P}}\}[k]={\frac {1}{P}}\int _{P}h_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ff5ff5748d540ed77192958f5ec50cbe83ff9b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:53.672ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}G[k]&\triangleq {\mathcal {F}}\{g_{_{P}}\}[k]={\frac {1}{P}}\int _{P}g_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \\H[k]&\triangleq {\mathcal {F}}\{h_{_{P}}\}[k]={\frac {1}{P}}\int _{P}h_{_{P}}(x)e^{-i2\pi kx/P}\,dx,\quad k\in \mathbb {Z} \end{aligned}}}"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">算子指示<a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E7%BA%A7%E6%95%B0" title="傅里叶级数">傅里叶级数</a><a href="/wiki/%E7%A7%AF%E5%88%86" title="积分">积分</a>。 <a href="/wiki/%E9%80%90%E7%82%B9%E4%B9%98%E7%A7%AF" title="逐点乘积">逐点乘积</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{_{P}}(x)\cdot h_{_{P}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{_{P}}(x)\cdot h_{_{P}}(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e69957704e4722c546473ebc5729095f7a3467d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.202ex; height:3.009ex;" alt="{\displaystyle g_{_{P}}(x)\cdot h_{_{P}}(x)}">的周期也是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">，它的傅里叶级数系数为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g_{_{P}}\cdot h_{_{P}}\}[k]=(G*H)[k]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo>⋅</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mi>G</mi> <mo>∗</mo> <mi>H</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g_{_{P}}\cdot h_{_{P}}\}[k]=(G*H)[k]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec23600fe822e18dc005e7025b45caeb30a17f48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:27.178ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\{g_{_{P}}\cdot h_{_{P}}\}[k]=(G*H)[k]}"></dd></dl> 周期卷积<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{_{P}}*h)(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{_{P}}*h)(x)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01ed21bc66cdd18171cff8cd5c9c06753c5d11c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.989ex; height:3.009ex;" alt="{\displaystyle (g_{_{P}}*h)(x)}">的周期也是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">，周期卷积的卷积定理为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g_{_{P}}*h\}[k]=\ P\cdot G[k]\ H[k]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mtext> </mtext> <mi>P</mi> <mo>⋅</mo> <mi>G</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mtext> </mtext> <mi>H</mi> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g_{_{P}}*h\}[k]=\ P\cdot G[k]\ H[k]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d08db60e83b8282a9c6351dc2b556632ad6c336" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:29.382ex; height:3.009ex;" alt="{\displaystyle {\mathcal {F}}\{g_{_{P}}*h\}[k]=\ P\cdot G[k]\ H[k]}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="离散卷积_2">离散卷积</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=15" title="编辑章节：离散卷积">编辑</a>]</div> 对于作为两个连续函数<a href="/wiki/%E9%87%87%E6%A0%B7" class="mw-redirect" title="采样">采样</a>的<a href="/wiki/%E5%BA%8F%E5%88%97" title="序列">序列</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}">，它们具有<a href="/wiki/%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="离散时间傅里叶变换">离散时间傅里叶变换</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}">： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\sum _{n=-\infty }^{\infty }g[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\sum _{n=-\infty }^{\infty }h[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>s</mi> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>≜</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>⋅</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>s</mi> <mi>n</mi> </mrow> </msup> <mspace width="thickmathspace" /> <mo>,</mo> <mspace width="1em" /> <mi>s</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\sum _{n=-\infty }^{\infty }g[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\sum _{n=-\infty }^{\infty }h[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0173c3574eda0d0a0ab15d8aef57be616c6efd03" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:47.976ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}G(s)&\triangleq {\mathcal {F}}\{g\}(s)=\sum _{n=-\infty }^{\infty }g[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \\H(s)&\triangleq {\mathcal {F}}\{h\}(s)=\sum _{n=-\infty }^{\infty }h[n]\cdot e^{-i2\pi sn}\;,\quad s\in \mathbb {R} \end{aligned}}}"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">算子指示<a href="/wiki/%E7%A6%BB%E6%95%A3%E6%97%B6%E9%97%B4%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="离散时间傅里叶变换">离散时间傅里叶变换</a>（DTFT）。 离散卷积的卷积定理为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g*h\}(s)=\ G(s)H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo>∗</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mtext> </mtext> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g*h\}(s)=\ G(s)H(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f92345020b517d0b7c9da022e2cc133cde2e2462" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.17ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}\{g*h\}(s)=\ G(s)H(s)}"></dd></dl> <div class="mw-heading mw-heading3"><h3 id="离散周期卷积_2">离散周期卷积</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=16" title="编辑章节：离散周期卷积">编辑</a>]</div> 对于周期为<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">的<a href="/wiki/%E5%BA%8F%E5%88%97" title="序列">序列</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g_{_{N}}[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g_{_{N}}[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30368575be97a3a4a9213a5bace7faff1a4a6371" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.379ex; height:3.009ex;" alt="{\displaystyle g_{_{N}}[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h_{_{N}}[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h_{_{N}}[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50abe51c291125077df860525b4c6c1c8d767595" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.609ex; height:3.009ex;" alt="{\displaystyle h_{_{N}}[n]}">： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}g_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }g[n-mN],\quad m,n\in \mathbb {Z} \\h_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }h[n-mN],\quad m,n\in \mathbb {Z} \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mi>N</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mtext> </mtext> </mtd> <mtd> <mi></mi> <mo>≜</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mi>N</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mspace width="1em" /> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}g_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }g[n-mN],\quad m,n\in \mathbb {Z} \\h_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }h[n-mN],\quad m,n\in \mathbb {Z} \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a06f5589438df8b4574b428b5869c5178b9d2e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:39.258ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}g_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }g[n-mN],\quad m,n\in \mathbb {Z} \\h_{_{N}}[n]\ &\triangleq \sum _{m=-\infty }^{\infty }h[n-mN],\quad m,n\in \mathbb {Z} \end{aligned}}}"></dd></dl> 相较于离散时间傅里叶变换<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}">的周期是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/92d98b82a3778f043108d4e20960a9193df57cbf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.176ex;" alt="{\displaystyle 1}">，它们是按间隔<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5c2544725c51dfe75eea07ee1f487feb8664c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.389ex; height:2.843ex;" alt="{\displaystyle 1/N}">采样<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2e6025c8f4c9d44fb1dc2da68407e4eb56f9db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.726ex; height:2.843ex;" alt="{\displaystyle G(s)}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f1b91390324fc9c33ec00fe57e3924ad7118cc1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.963ex; height:2.843ex;" alt="{\displaystyle H(s)}">，并在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">个采样上进行了逆<a href="/wiki/%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="离散傅里叶变换">离散傅里叶变换</a>（DFT-1或IDFT）的结果。 离散周期卷积<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{_{N}}*h)[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{_{N}}*h)[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3f3e5554f1c3fe2b5f2edeff4e779ccedb445a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.722ex; height:3.009ex;" alt="{\displaystyle (g_{_{N}}*h)[n]}">的周期也是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">。离散周期卷积定理为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}\{g_{_{N}}*h\}[k]=\ \underbrace {{\mathcal {F}}\{g_{_{N}}\}[k]} _{G(k/N)}\cdot \underbrace {{\mathcal {F}}\{h_{_{N}}\}[k]} _{H(k/N)},\quad k,n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mtext> </mtext> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mo>⋅</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mrow> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>N</mi> <mo stretchy="false">)</mo> </mrow> </munder> <mo>,</mo> <mspace width="1em" /> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}\{g_{_{N}}*h\}[k]=\ \underbrace {{\mathcal {F}}\{g_{_{N}}\}[k]} _{G(k/N)}\cdot \underbrace {{\mathcal {F}}\{h_{_{N}}\}[k]} _{H(k/N)},\quad k,n\in \mathbb {Z} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88f84f790337809a2e0ba2a745b25083a4f3129c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.838ex; width:48.849ex; height:6.843ex;" alt="{\displaystyle {\mathcal {F}}\{g_{_{N}}*h\}[k]=\ \underbrace {{\mathcal {F}}\{g_{_{N}}\}[k]} _{G(k/N)}\cdot \underbrace {{\mathcal {F}}\{h_{_{N}}\}[k]} _{H(k/N)},\quad k,n\in \mathbb {Z} }"></dd></dl> 这里的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/205d4b91000d9dcf1a5bbabdfa6a8395fa60b676" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}">算子指示长度<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">的<a href="/wiki/%E7%A6%BB%E6%95%A3%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="离散傅里叶变换">离散傅里叶变换</a>（DFT）。 它有着推论： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g_{_{N}}\}\cdot {\mathcal {F}}\{h_{_{N}}\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <msub> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g_{_{N}}\}\cdot {\mathcal {F}}\{h_{_{N}}\}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ab487693fbf66da441eb710125e5ceff2194398e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:36.854ex; height:3.343ex;" alt="{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g_{_{N}}\}\cdot {\mathcal {F}}\{h_{_{N}}\}\}}"></dd></dl> 对于其非零时段小于等于<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">的<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b26be3e694314bc90c3215047e4a2010c6ee184a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}">，离散圆周卷积的卷积定理为： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g\}\cdot {\mathcal {F}}\{h\}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mrow> </msub> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mtext> </mtext> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>g</mi> <mo fence="false" stretchy="false">}</mo> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo fence="false" stretchy="false">{</mo> <mi>h</mi> <mo fence="false" stretchy="false">}</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g\}\cdot {\mathcal {F}}\{h\}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0defba14a548128651067acefb5121ba77e020c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:33.699ex; height:3.343ex;" alt="{\displaystyle (g_{_{N}}*h)[n]=\ {\mathcal {F}}^{-1}\{{\mathcal {F}}\{g\}\cdot {\mathcal {F}}\{h\}\}}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="推广">推广</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=17" title="编辑章节：推广">编辑</a>]</div> 卷积的概念还可以推广到<a href="/wiki/%E6%95%B0%E5%88%97" title="数列">数列</a>、<a href="/wiki/%E6%B5%8B%E5%BA%A6" title="测度">测度</a>以及<a href="/wiki/%E5%B9%BF%E4%B9%89%E5%87%BD%E6%95%B0" title="广义函数">广义函数</a>上去。<a href="/wiki/%E5%87%BD%E6%95%B0" title="函数">函数</a><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f,g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>,</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f,g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25b6ab1762925585cd7605809caa8b1b5284177b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.429ex; height:2.509ex;" alt="{\displaystyle f,g}">是定義在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">上的<a href="/wiki/%E5%8F%AF%E6%B5%8B%E5%87%BD%E6%95%B0" title="可测函数">可測函數</a>（measurable function），<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}">与<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3556280e66fe2c0d0140df20935a6f057381d77" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}">存在卷积并记作<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f*g}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>∗</mo> <mi>g</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f*g}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de088e4a3777d3b5d2787fdec81acd91e78a719e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.589ex; height:2.509ex;" alt="{\displaystyle f*g}">。如果函數不是定義在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">上，可以把函數定義域以外的值都規定成零，這樣就變成一個定義在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {R} ^{n}}">上的函數。 若G是有某m <a href="/wiki/%E6%B5%8B%E5%BA%A6" title="测度">测度</a>的<a href="/wiki/%E7%BE%A4" title="群">群</a>（例如<a href="/wiki/%E8%B1%AA%E6%96%AF%E5%A4%9A%E5%A4%AB%E7%A9%BA%E9%97%B4" title="豪斯多夫空间">豪斯多夫空间</a>上<a href="/wiki/%E5%93%88%E5%B0%94%E6%B5%8B%E5%BA%A6" title="哈尔测度">哈尔测度</a>下<a href="/wiki/%E5%B1%80%E9%83%A8%E7%B4%A7%E8%87%B4" class="mw-redirect" title="局部紧致">局部紧致</a>的<a href="/wiki/%E6%8B%93%E6%89%91%E7%BE%A4" title="拓扑群">拓扑群</a>），对于G上m-<a href="/wiki/%E5%8B%92%E8%B4%9D%E6%A0%BC%E5%8F%AF%E7%A7%AF" class="mw-redirect" title="勒贝格可积">勒贝格可积</a>的<a href="/wiki/%E5%AE%9E%E6%95%B0" title="实数">实数</a>或<a href="/wiki/%E5%A4%8D%E6%95%B0_(%E6%95%B0%E5%AD%A6)" title="复数 (数学)">复数</a>函数f和g，可定义它们的卷积： <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (f*g)(x)=\int _{G}f(y)g(xy^{-1})\,dm(y)\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>f</mi> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> <mi>f</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mi>g</mi> <mo stretchy="false">(</mo> <mi>x</mi> <msup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>m</mi> <mo stretchy="false">(</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (f*g)(x)=\int _{G}f(y)g(xy^{-1})\,dm(y)\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8cec355c5ad869ef4f885be24d690d057d56c589" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:34.826ex; height:5.676ex;" alt="{\displaystyle (f*g)(x)=\int _{G}f(y)g(xy^{-1})\,dm(y)\,}"></dd></dl> 对于这些群上定义的卷积同样可以给出诸如卷积定理等性质，但是这需要对这些群的<a href="/wiki/%E7%BE%A4%E8%A1%A8%E7%A4%BA" class="mw-redirect" title="群表示">表示理论</a>以及调和分析的<a href="/wiki/%E5%BD%BC%E5%BE%97-%E5%A4%96%E5%B0%94%E5%AE%9A%E7%90%86" class="mw-redirect" title="彼得-外尔定理">彼得-外尔定理</a>。 <div class="mw-heading mw-heading2"><h2 id="离散卷積的計算方法">离散卷積的計算方法</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=18" title="编辑章节：离散卷積的計算方法">编辑</a>]</div> 計算卷積<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]*g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]*g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/53f78eb841159a85c00fe9a4331825a77341e67c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.966ex; height:2.843ex;" alt="{\displaystyle f[n]*g[n]}">有三種主要的方法，分別為 <ol><li>直接計算（Direct Method）</li> <li><a href="/wiki/%E5%BF%AB%E9%80%9F%E5%82%85%E7%AB%8B%E8%91%89%E8%BD%89%E6%8F%9B" class="mw-redirect" title="快速傅立葉轉換">快速傅立葉轉換</a>（FFT）</li> <li>分段卷積（sectioned convolution）</li></ol> 方法1是直接利用定義來計算卷積，而方法2和3都是用到了FFT來快速計算卷積。也有不需要用到FFT的作法，如使用<a href="/wiki/%E6%95%B8%E8%AB%96%E8%BD%89%E6%8F%9B" title="數論轉換">數論轉換</a>。 <div class="mw-heading mw-heading3"><h3 id="方法1：直接計算">方法1：直接計算</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=19" title="编辑章节：方法1：直接計算">编辑</a>]</div> <ul><li>作法：利用卷積的定義</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=f[n]*g[n]=\sum _{m=0}^{M-1}f[n-m]g[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=f[n]*g[n]=\sum _{m=0}^{M-1}f[n-m]g[m]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1081139aae46ebd7421e11ec20351dc12688cedd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.519ex; height:7.343ex;" alt="{\displaystyle y[n]=f[n]*g[n]=\sum _{m=0}^{M-1}f[n-m]g[m]}"></dd></dl></dd></dl> <ul><li>若<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">皆為實數信號，則需要<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle MN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle MN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fa6819fea85a0b616a35d340ad6461ababdbeee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.506ex; height:2.176ex;" alt="{\displaystyle MN}">個乘法。</li></ul> <ul><li>若<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">皆為更一般性的複數信號，不使用複數乘法的快速演算法，會需要<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 4MN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>4</mn> <mi>M</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 4MN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b944e9abade872ed66faaa18325338aa04007d97" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.668ex; height:2.176ex;" alt="{\displaystyle 4MN}">個乘法;但若使用複數乘法的快速演算法，則可簡化至<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3MN}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>M</mi> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3MN}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d8312dbac32788daf932d2fd6ccc8dd35707f237" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.668ex; height:2.176ex;" alt="{\displaystyle 3MN}">個乘法。</li></ul> <dl><dd>因此，使用定義直接計算卷積的複雜度為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(MN)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>M</mi> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(MN)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee63b96e5439a02e97fd879b623275fbda6be74a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.088ex; height:2.843ex;" alt="{\displaystyle O(MN)}">。</dd></dl> <div class="mw-heading mw-heading3"><h3 id="方法2：快速傅立葉轉換">方法2：快速傅立葉轉換</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=20" title="编辑章节：方法2：快速傅立葉轉換">编辑</a>]</div> <ul><li>概念：由於兩個離散信號在時域（time domain）做卷積相當於這兩個信號的離散傅立葉轉換在頻域（frequency domain）做相乘：</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=f[n]*g[n]\leftrightarrow Y[f]=F[f]G[f]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">↔</mo> <mi>Y</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>F</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mi>G</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=f[n]*g[n]\leftrightarrow Y[f]=F[f]G[f]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/27f83ca717dff1b21da43e56127c50874843b8b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.679ex; height:2.843ex;" alt="{\displaystyle y[n]=f[n]*g[n]\leftrightarrow Y[f]=F[f]G[f]}"></dd></dl></dd></dl> <dl><dd>，可以看出在頻域的計算較簡單。</dd></dl> <ul><li>作法：因此這個方法即是先將信號從時域轉成頻域：</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F[f]=DFT_{P}(f[n]),G[f]=DFT_{P}(g[n])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>G</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F[f]=DFT_{P}(f[n]),G[f]=DFT_{P}(g[n])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af332366ed1070784c006329adcd40833e64c44d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:40.311ex; height:2.843ex;" alt="{\displaystyle F[f]=DFT_{P}(f[n]),G[f]=DFT_{P}(g[n])}"></dd></dl></dd></dl> <dl><dd>，於是</dd></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y[f]=DFT_{P}(f[n])DFT_{P}(g[n])}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">[</mo> <mi>f</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y[f]=DFT_{P}(f[n])DFT_{P}(g[n])}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93ec9e3130dcd720d4d6b1930bd492f29fee0faf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.812ex; height:2.843ex;" alt="{\displaystyle Y[f]=DFT_{P}(f[n])DFT_{P}(g[n])}"></dd></dl></dd></dl> <dl><dd>，最後再將頻域信號轉回時域，就完成了卷積的計算：</dd></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=IDFT_{P}{DFT_{P}(f[n])DFT_{P}(g[n])}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=IDFT_{P}{DFT_{P}(f[n])DFT_{P}(g[n])}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adc12d1e8c9712ca85bf040d7e8573842c545588" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:38.971ex; height:2.843ex;" alt="{\displaystyle y[n]=IDFT_{P}{DFT_{P}(f[n])DFT_{P}(g[n])}}"></dd></dl></dd></dl> <dl><dd>總共做了2次DFT和1次IDFT。</dd></dl> <ul><li>特別注意DFT和IDFT的點數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">要滿足<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq M+N-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq M+N-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07fd28bdc507c483207bbd025c5c746d72a70f40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:16.193ex; height:2.343ex;" alt="{\displaystyle P\geq M+N-1}">。</li></ul> <ul><li>由於DFT有快速演算法FFT，所以運算量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(P\log _{2}P)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>P</mi> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mi>P</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(P\log _{2}P)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74fc93c033d1c137f1f3444761adb5c57a78446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.874ex; height:2.843ex;" alt="{\displaystyle O(P\log _{2}P)}"></li></ul> <ul><li>假設<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">點DFT的乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">為一般性的複數信號，並使用複數乘法的快速演算法，則共需要<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3a+3P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3a+3P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8101617e2091a1781d290397f0f22d80cd102954" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.141ex; height:2.343ex;" alt="{\displaystyle 3a+3P}">個乘法。</li></ul> <div class="mw-heading mw-heading3"><h3 id="方法3：分段卷積">方法3：分段卷積</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=21" title="编辑章节：方法3：分段卷積">编辑</a>]</div> <ul><li>概念：將<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">切成好幾段（section），每一段分別和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">做卷積後，再將結果相加。</li></ul> <ul><li>作法：先將<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">切成每段長度為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">的區段（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L>M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>></mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L>M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44be94a8e4243927e4c8d84983ad8c94f785e151" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.124ex; height:2.176ex;" alt="{\displaystyle L>M}">），假設共切成S段：</li></ul> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n](n=0,1,...,N-1)\to f_{1}[n],f_{2}[n],f_{3}[n],...,f_{S}[n](S=\left\lceil {\frac {N}{L}}\right\rceil )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">→</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>=</mo> <mrow> <mo>⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>L</mi> </mfrac> </mrow> <mo>⌉</mo> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n](n=0,1,...,N-1)\to f_{1}[n],f_{2}[n],f_{3}[n],...,f_{S}[n](S=\left\lceil {\frac {N}{L}}\right\rceil )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b2758eba65ad177619e0d9198cc913abf649b9a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:67.499ex; height:6.176ex;" alt="{\displaystyle f[n](n=0,1,...,N-1)\to f_{1}[n],f_{2}[n],f_{3}[n],...,f_{S}[n](S=\left\lceil {\frac {N}{L}}\right\rceil )}"></dd></dl></dd></dl> <dl><dd><dl><dd>Section 1: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{1}[n]=f[n],n=0,1,...,L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{1}[n]=f[n],n=0,1,...,L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22401ad12747eb80e3ec79d0c5b189b3c186d912" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.589ex; height:2.843ex;" alt="{\displaystyle f_{1}[n]=f[n],n=0,1,...,L-1}"></dd></dl></dd></dl> <dl><dd><dl><dd>Section 2: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{2}[n]=f[n+L],n=0,1,...,L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mi>L</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{2}[n]=f[n+L],n=0,1,...,L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ccb4f0b33490a5fc67f57cb6d353087f35c3db5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.012ex; height:2.843ex;" alt="{\displaystyle f_{2}[n]=f[n+L],n=0,1,...,L-1}"></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8039d9feb6596ae092e5305108722975060c083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }"></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd>Section r: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{r}[n]=f[n+(r-1)L],n=0,1,...,L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>L</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{r}[n]=f[n+(r-1)L],n=0,1,...,L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0d987ef3bab08edc0d97642435c8f97a1383304" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.792ex; height:2.843ex;" alt="{\displaystyle f_{r}[n]=f[n+(r-1)L],n=0,1,...,L-1}"></dd></dl></dd></dl> <dl><dd><dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \vdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>⋮</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \vdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8039d9feb6596ae092e5305108722975060c083" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.647ex; height:3.676ex;" alt="{\displaystyle \vdots }"></dd></dl></dd></dl></dd></dl> <dl><dd><dl><dd>Section S: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f_{S}[n]=f[n+(S-1)L],n=0,1,...,L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>S</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>L</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>n</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{S}[n]=f[n+(S-1)L],n=0,1,...,L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b30b353213c930583b0b65c5598ea334fdfee777" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.561ex; height:2.843ex;" alt="{\displaystyle f_{S}[n]=f[n+(S-1)L],n=0,1,...,L-1}"></dd></dl></dd></dl> <dl><dd>，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">為各個section的和</dd></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]=\sum _{r=1}^{S}f_{r}[n+(r-1)L]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>L</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]=\sum _{r=1}^{S}f_{r}[n+(r-1)L]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ad8dbc7f1039ba3824c87225b58895e27a85453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:26.893ex; height:7.343ex;" alt="{\displaystyle f[n]=\sum _{r=1}^{S}f_{r}[n+(r-1)L]}">。</dd></dl></dd></dl> <dl><dd>因此，</dd></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=f[n]*g[n]=\sum _{r=1}^{S}\sum _{m=0}^{M-1}f_{r}[n+(r-1)L-m]g[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>L</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=f[n]*g[n]=\sum _{r=1}^{S}\sum _{m=0}^{M-1}f_{r}[n+(r-1)L-m]g[m]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e15254f6bff4b4db4f6899f0bb3e450f75e3625d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:53.38ex; height:7.343ex;" alt="{\displaystyle y[n]=f[n]*g[n]=\sum _{r=1}^{S}\sum _{m=0}^{M-1}f_{r}[n+(r-1)L-m]g[m]}">，</dd></dl></dd></dl> <dl><dd>每一小段作卷積則是採用方法2，先將時域信號轉到頻域相乘，再轉回時域：</dd></dl> <dl><dd><dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y[n]=IDFT(\sum _{r=1}^{S}\sum _{m=0}^{M-1}DFT_{P}(f_{r}[n+(r-1)L-m])DFT_{P}(g[m])),P\geq M+L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>I</mi> <mi>D</mi> <mi>F</mi> <mi>T</mi> <mo stretchy="false">(</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>S</mi> </mrow> </munderover> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mo stretchy="false">(</mo> <mi>r</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>L</mi> <mo>−</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>P</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>g</mi> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>,</mo> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=IDFT(\sum _{r=1}^{S}\sum _{m=0}^{M-1}DFT_{P}(f_{r}[n+(r-1)L-m])DFT_{P}(g[m])),P\geq M+L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b3c1ccb338a07511f8c82bbe1478beb62fafdea6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:81.94ex; height:7.343ex;" alt="{\displaystyle y[n]=IDFT(\sum _{r=1}^{S}\sum _{m=0}^{M-1}DFT_{P}(f_{r}[n+(r-1)L-m])DFT_{P}(g[m])),P\geq M+L-1}">。</dd></dl></dd></dl> <ul><li>總共只需要做<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">點FFT <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2S+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2S+1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3df223d40c7b1f05b258b61f6151fe74c5f4214" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.665ex; height:2.343ex;" alt="{\displaystyle 2S+1}">次，因為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">只需要做一次FFT。</li></ul> <ul><li>假設<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">點DFT的乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">為一般性的複數信號，並使用複數乘法的快速演算法，則共需要<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df54ba339c817baead4f20ca77644146ee448262" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.951ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP}">個乘法。</li></ul> <ul><li>運算量：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>L</mi> </mfrac> </mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4b5d72adc35b450cfcc40c0d13720fa6d6bd070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.74ex; height:5.176ex;" alt="{\displaystyle {\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}"></li></ul> <ul><li>運算複雜度：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(N)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <mi>N</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(N)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/78484c5c26cfc97bb3b915418caa09454421e80b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.646ex; height:2.843ex;" alt="{\displaystyle O(N)}">，和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">呈線性，較方法2小。</li> <li>分為 Overlap-Add 和 Overlap-Save 兩種方法。</li></ul> 分段卷積: Overlap-Add 欲做<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]*h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]*h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c97358d336f2e21de7913bc46fac6477eef3bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.24ex; height:2.843ex;" alt="{\displaystyle x[n]*h[n]}">的分段卷積分，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}"> 長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, Step 1: 將<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}">每 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 分成一段 Step 2: 再每段 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 點後面添加 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 個零，變成長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e33f15990cd14d3bf8a47611f4e5e0bafe4173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle L+M-1}"> Step 3: 把 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}"> 添加 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe61101dd489eb8e1a974ab6c409190a76541bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.586ex; height:2.343ex;" alt="{\displaystyle L-1}">個零，變成長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e33f15990cd14d3bf8a47611f4e5e0bafe4173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle L+M-1}">的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af9a0c1d0c9753adba0e2e13fc2c45e8d5698f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:3.009ex;" alt="{\displaystyle h'[n]}"> Step 4: 把每個 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 的小段和 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af9a0c1d0c9753adba0e2e13fc2c45e8d5698f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:3.009ex;" alt="{\displaystyle h'[n]}"> 做快速卷積，也就是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71884e63653a1501f77c376b182141bd9d8e86ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.286ex; height:3.009ex;" alt="{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}">，每小段會得到長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e33f15990cd14d3bf8a47611f4e5e0bafe4173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle L+M-1}"> 的時域訊號 Step 5: 放置第 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> 個小段的起點在位置 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af7e585ea7b2669445ce0e7068a7a168616de7c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.226ex; height:2.176ex;" alt="{\displaystyle L\times i}"> 上, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0,1,...,\lceil {\frac {N}{L}}\rceil -1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>.</mo> <mo>.</mo> <mo>.</mo> <mo>,</mo> <mo fence="false" stretchy="false">⌈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>L</mi> </mfrac> </mrow> <mo fence="false" stretchy="false">⌉</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=0,1,...,\lceil {\frac {N}{L}}\rceil -1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0e431850a09ed6a4c8846077aaf114c9920344" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.397ex; height:5.176ex;" alt="{\displaystyle i=0,1,...,\lceil {\frac {N}{L}}\rceil -1}"> Step 6: 會發現在每一段的後面 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點有重疊，將所有點都相加起來，顧名思義 Overlap-Add，最後得到結果 舉例來說: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]=[1,2,3,4,5,-1,-2,-3,-4,-5,1,2,3,4,5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mo>−</mo> <mn>2</mn> <mo>,</mo> <mo>−</mo> <mn>3</mn> <mo>,</mo> <mo>−</mo> <mn>4</mn> <mo>,</mo> <mo>−</mo> <mn>5</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]=[1,2,3,4,5,-1,-2,-3,-4,-5,1,2,3,4,5]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98580fd364a838db6ecc3c27068105c31f81d9a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:49.363ex; height:2.843ex;" alt="{\displaystyle x[n]=[1,2,3,4,5,-1,-2,-3,-4,-5,1,2,3,4,5]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=15}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92efb5a8a42b0cf920aeb6e437f8ab40420bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=15}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]=[1,2,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]=[1,2,3]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fdc55efaef23389bacd0b58cf59865377f46f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.975ex; height:2.843ex;" alt="{\displaystyle h[n]=[1,2,3]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f650bb93b4dc73fb000a07a4c9d7f3e3094e965d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M=3}"> 令 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0a13e71c8b4a72c47192e0197ae0faaa78240a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L=5}"> <ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Data_overlap_add.png" class="mw-file-description" title="x[n]和h[n]"><img alt="x[n]和h[n]" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Data_overlap_add.png/300px-Data_overlap_add.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fc/Data_overlap_add.png/450px-Data_overlap_add.png 1.5x, //upload.wikimedia.org/wikipedia/commons/f/fc/Data_overlap_add.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">x[n]和h[n]</div> </li> </ul> 令 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0a13e71c8b4a72c47192e0197ae0faaa78240a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L=5}"> 切成三段，分別為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{0}[n],x_{1}[n],x_{2}[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{0}[n],x_{1}[n],x_{2}[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d11383cb60875adb6faaedbf729e5956982bff0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.285ex; height:2.843ex;" alt="{\displaystyle x_{0}[n],x_{1}[n],x_{2}[n]}">, 每段填 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 個零，並將 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}"> 填零至長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e33f15990cd14d3bf8a47611f4e5e0bafe4173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle L+M-1}"><ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Seperate_x_overlap_add.png" class="mw-file-description" title="分段x[n]"><img alt="分段x[n]" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Seperate_x_overlap_add.png/300px-Seperate_x_overlap_add.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/30/Seperate_x_overlap_add.png/450px-Seperate_x_overlap_add.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/30/Seperate_x_overlap_add.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">分段x[n]</div> </li> </ul>將每一段做<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71884e63653a1501f77c376b182141bd9d8e86ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.286ex; height:3.009ex;" alt="{\displaystyle IDFT_{L+M-1}\{{DFT_{L+M-1}(x[n])DFT_{L+M-1}(h'[n])}\}}"><ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Seperate_result_overlap_add.png" class="mw-file-description" title="分段運算結果"><img alt="分段運算結果" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Seperate_result_overlap_add.png/300px-Seperate_result_overlap_add.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Seperate_result_overlap_add.png/450px-Seperate_result_overlap_add.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/2f/Seperate_result_overlap_add.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">分段運算結果</div> </li> </ul>若將每小段擺在一起，可以注意到第一段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\thicksim 6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo class="MJX-variant">∼</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\thicksim 6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff1f76f3b7fb25ff5032477f749b824f079377ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 0\thicksim 6}"> ，第二段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\thicksim 11}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo class="MJX-variant">∼</mo> <mn>11</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\thicksim 11}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73806cbb3abe04465e34f3360cdc864050bc169e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.586ex; height:2.176ex;" alt="{\displaystyle 5\thicksim 11}">，第三段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10\thicksim 16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo class="MJX-variant">∼</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10\thicksim 16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04af9c6fcca3e9f2c6ed48dbea15abec8deda045" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.748ex; height:2.176ex;" alt="{\displaystyle 10\thicksim 16}">，三段的範圍是有重疊的<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Summation_overlap_add.png" class="mw-file-description" title="合併分段運算結果"><img alt="合併分段運算結果" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Summation_overlap_add.png/300px-Summation_overlap_add.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Summation_overlap_add.png/450px-Summation_overlap_add.png 1.5x, //upload.wikimedia.org/wikipedia/commons/3/3c/Summation_overlap_add.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">合併分段運算結果</div> </li> </ul>最後將三小段加在一起，並將結果和未分段的卷積做比較，上圖是分段的結果，下圖是沒有分段並利用快速卷積所算出的結果，驗證兩者運算結果相同。<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Final_result_overlap_add.png" class="mw-file-description" title="結果比較圖"><img alt="結果比較圖" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Final_result_overlap_add.png/300px-Final_result_overlap_add.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c7/Final_result_overlap_add.png/450px-Final_result_overlap_add.png 1.5x, //upload.wikimedia.org/wikipedia/commons/c/c7/Final_result_overlap_add.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">結果比較圖</div> </li> </ul>分段卷積: Overlap-Save 欲做<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]*h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>∗</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]*h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c97358d336f2e21de7913bc46fac6477eef3bc6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.24ex; height:2.843ex;" alt="{\displaystyle x[n]*h[n]}">的分段卷積分，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}"> 長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">, Step 1: 將 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 前面填 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 個零 Step 2: 第一段 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a682d568ee6a5fe51d76423186057f625ada5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.063ex; height:2.176ex;" alt="{\displaystyle i=0}">, 從新的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 中 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\times i-(M-1)\times i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>×</mo> <mi>i</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\times i-(M-1)\times i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8bbe351cd9908206caef3805019887754d2a0ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.963ex; height:2.843ex;" alt="{\displaystyle L\times i-(M-1)\times i}"> 取到 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L\times (i+1)-(M-1)\times i-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>×</mo> <mo stretchy="false">(</mo> <mi>i</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>−</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>×</mo> <mi>i</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L\times (i+1)-(M-1)\times i-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b1d31a1f325d0a1909572da59315cebde6b8ace" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.778ex; height:2.843ex;" alt="{\displaystyle L\times (i+1)-(M-1)\times i-1}"> 總共 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 點當做一段，因此每小段會重複取到前一小段的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點，取到新的一段全為零為止 Step 3: 把 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/89981bbbb05ffd469eeadb828c18359965985e46" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.027ex; height:2.843ex;" alt="{\displaystyle h[n]}"> 添加 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>−</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L-M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6b200b4cc6237428dd16e147476bc0f465f53b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.865ex; height:2.343ex;" alt="{\displaystyle L-M}"> 個零，變成長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af9a0c1d0c9753adba0e2e13fc2c45e8d5698f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:3.009ex;" alt="{\displaystyle h'[n]}"> Step 4: 把每個 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 的小段和 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h'[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h'[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7af9a0c1d0c9753adba0e2e13fc2c45e8d5698f7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.712ex; height:3.009ex;" alt="{\displaystyle h'[n]}"> 做快速卷積，也就是<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c639c720d3a6a4922220ac66a9c31d6e8eb7987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.968ex; height:3.009ex;" alt="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}">，每小段會得到長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 的時域訊號 Step 5: 對於每個 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> 小段，只會保留末端的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-(M-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L-(M-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe931bb8cbd6173a0b7762d45dd6f5a0698a179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.678ex; height:2.843ex;" alt="{\displaystyle L-(M-1)}"> 點，因此得名 Overlap-Save Step 6: 將所有保留的點合再一起，得到最後結果 舉例來說: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>10</mn> <mo>,</mo> <mn>11</mn> <mo>,</mo> <mn>12</mn> <mo>,</mo> <mn>13</mn> <mo>,</mo> <mn>14</mn> <mo>,</mo> <mn>15</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f54af19067aacc4ba29d8c5a4e661d92c3f2ab24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:47.297ex; height:2.843ex;" alt="{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=15}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>15</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=15}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d92efb5a8a42b0cf920aeb6e437f8ab40420bd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.487ex; height:2.176ex;" alt="{\displaystyle N=15}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]=[1,2,3]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]=[1,2,3]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/67fdc55efaef23389bacd0b58cf59865377f46f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.975ex; height:2.843ex;" alt="{\displaystyle h[n]=[1,2,3]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f650bb93b4dc73fb000a07a4c9d7f3e3094e965d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M=3}"> 令 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ceb99e4503e826f7f0aa12251552f6193d5fdc7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L=7}"> <ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Data_overlap_save.png" class="mw-file-description" title="x[n]和h[n]"><img alt="x[n]和h[n]" src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Data_overlap_save.png/300px-Data_overlap_save.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9e/Data_overlap_save.png/450px-Data_overlap_save.png 1.5x, //upload.wikimedia.org/wikipedia/commons/9/9e/Data_overlap_save.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">x[n]和h[n]</div> </li> </ul>將 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/864cbbefbdcb55af4d9390911de1bf70167c4a3d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.018ex; height:2.843ex;" alt="{\displaystyle x[n]}"> 前面填 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 個零以後，按照 Step 2 的方式分段，可以看到每一段都重複上一段的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Seperate_x_overlap_save.png" class="mw-file-description" title="分段x[n]"><img alt="分段x[n]" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Seperate_x_overlap_save.png/300px-Seperate_x_overlap_save.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Seperate_x_overlap_save.png/450px-Seperate_x_overlap_save.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b2/Seperate_x_overlap_save.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">分段x[n]</div> </li> </ul>再將每一段做 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c639c720d3a6a4922220ac66a9c31d6e8eb7987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.968ex; height:3.009ex;" alt="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}">以後可以得到<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Seperate_result_overlap_save.png" class="mw-file-description" title="分段運算結果"><img alt="分段運算結果" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Seperate_result_overlap_save.png/300px-Seperate_result_overlap_save.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Seperate_result_overlap_save.png/450px-Seperate_result_overlap_save.png 1.5x, //upload.wikimedia.org/wikipedia/commons/0/0b/Seperate_result_overlap_save.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">分段運算結果</div> </li> </ul>保留每一段末端的 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L-(M-1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L-(M-1)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfe931bb8cbd6173a0b7762d45dd6f5a0698a179" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.678ex; height:2.843ex;" alt="{\displaystyle L-(M-1)}"> 點，擺在一起以後，可以注意到第一段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\thicksim 4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo class="MJX-variant">∼</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0\thicksim 4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5be3a806df1e8c89dfe1ad470b7de303c4f1bc65" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 0\thicksim 4}"> ，第二段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 5\thicksim 9}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>5</mn> <mo class="MJX-variant">∼</mo> <mn>9</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 5\thicksim 9}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d24e6c6441877394c4dc2cc57d53305610921e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.423ex; height:2.176ex;" alt="{\displaystyle 5\thicksim 9}">，第三段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10\thicksim 14}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>10</mn> <mo class="MJX-variant">∼</mo> <mn>14</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 10\thicksim 14}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978b95a17201951e2fb90fe7c3e71d9332e75abe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.748ex; height:2.176ex;" alt="{\displaystyle 10\thicksim 14}">，第四段的範圍是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 15\thicksim 16}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>15</mn> <mo class="MJX-variant">∼</mo> <mn>16</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 15\thicksim 16}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f914497bd0ad34c39e327c2144cae036ef85b88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.748ex; height:2.176ex;" alt="{\displaystyle 15\thicksim 16}">，四段的範圍是沒有重疊的<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Summation_overlap_save.png" class="mw-file-description" title="合併分段運算結果"><img alt="合併分段運算結果" src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Summation_overlap_save.png/300px-Summation_overlap_save.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5f/Summation_overlap_save.png/450px-Summation_overlap_save.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/5f/Summation_overlap_save.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">合併分段運算結果</div> </li> </ul>將結果和未分段的卷積做比較，下圖是分段的結果，上圖是沒有分段並利用快速卷積所算出的結果，驗證兩者運算結果相同。<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Final_compare_overlap_save.png" class="mw-file-description" title="結果比較圖"><img alt="結果比較圖" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Final_compare_overlap_save.png/300px-Final_compare_overlap_save.png" decoding="async" width="300" height="221" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/ba/Final_compare_overlap_save.png/450px-Final_compare_overlap_save.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/ba/Final_compare_overlap_save.png 2x" data-file-width="561" data-file-height="413" /></a></div> <div class="gallerytext">結果比較圖</div> </li> </ul>至於為什麼要把前面 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 丟掉? 以下以一例子來闡述: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>6</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> <mo>,</mo> <mn>9</mn> <mo>,</mo> <mn>10</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/197c03fad73159099a753aa21015d7a410c619ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.503ex; height:2.843ex;" alt="{\displaystyle x[n]=[1,2,3,4,5,6,7,8,9,10]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=10}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>10</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=10}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/822d0ac489846a3ba69441969952c7c7526e81e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.006ex; height:2.176ex;" alt="{\displaystyle L=10}">， <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[n]=[1,2,3,4,5]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mo stretchy="false">[</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[n]=[1,2,3,4,5]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01382de68d6838f4a938ab33a5ad465fa1ae6d0f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.368ex; height:2.843ex;" alt="{\displaystyle h[n]=[1,2,3,4,5]}">, 長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c0200217dd924af1a3fad45f994af52cb7306ea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.703ex; height:2.176ex;" alt="{\displaystyle M=5}">, 第一條藍線代表 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}"> 軸，而兩條藍線之間代表長度 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">，是在做快速摺積時的週期<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Original_ov_extra.png" class="mw-file-description" title="x[n]和h[n]"><img alt="x[n]和h[n]" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Original_ov_extra.png/300px-Original_ov_extra.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/Original_ov_extra.png/450px-Original_ov_extra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/2/25/Original_ov_extra.png 2x" data-file-width="561" data-file-height="420" /></a></div> <div class="gallerytext">x[n]和h[n]</div> </li> </ul>當在做快速摺積時<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> <mi>D</mi> <mi>F</mi> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>L</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>h</mi> <mo>′</mo> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">)</mo> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c639c720d3a6a4922220ac66a9c31d6e8eb7987" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.968ex; height:3.009ex;" alt="{\displaystyle IDFT_{L}\{{DFT_{L}(x[n])DFT_{L}(h'[n])}\}}">，是把訊號視為週期 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">，在時域上為循環摺積分, 而在一開始前 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點所得到的值，是 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[0],h[6],h[7],h[8],h[9]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>6</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>7</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>8</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>9</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[0],h[6],h[7],h[8],h[9]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c2e1f7ce57e74223a3049d5d803bd8af91b8019" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.111ex; height:2.843ex;" alt="{\displaystyle h[0],h[6],h[7],h[8],h[9]}"> 和 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x[0],x[6],x[7],x[8],x[9]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mn>6</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mn>7</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mn>8</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mn>9</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[0],x[6],x[7],x[8],x[9]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f004d095726b6a642d26dd6dc4bef9660ec8c17" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.065ex; height:2.843ex;" alt="{\displaystyle x[0],x[6],x[7],x[8],x[9]}"> 內積的值， 然而 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[6],h[7],h[8],h[9]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mn>6</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>7</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>8</mn> <mo stretchy="false">]</mo> <mo>,</mo> <mi>h</mi> <mo stretchy="false">[</mo> <mn>9</mn> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[6],h[7],h[8],h[9]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ccee59494bedbceae16aa4bce150fbcef8b2cf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.282ex; height:2.843ex;" alt="{\displaystyle h[6],h[7],h[8],h[9]}"> 這 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 個值應該要為零，以往在做快速摺積時長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85e33f15990cd14d3bf8a47611f4e5e0bafe4173" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.868ex; height:2.343ex;" alt="{\displaystyle L+M-1}"> 時不會遇到這些問題， 而今天因為在做快速摺積時長度為 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> 才會把這 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點算進來，因此我們要丟棄這 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點內積的結果<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Cir_conv_ov_extra.png" class="mw-file-description" title="循環摺積"><img alt="循環摺積" src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Cir_conv_ov_extra.png/300px-Cir_conv_ov_extra.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ab/Cir_conv_ov_extra.png/450px-Cir_conv_ov_extra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/a/ab/Cir_conv_ov_extra.png 2x" data-file-width="561" data-file-height="420" /></a></div> <div class="gallerytext">循環摺積</div> </li> </ul>為了要丟棄這 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點內積的結果，位移 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h[-n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">[</mo> <mo>−</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h[-n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2160e2740fe612954bc0c6d78203d595d29076e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.836ex; height:2.843ex;" alt="{\displaystyle h[-n]}"> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}"> 點，並把位移以後內積合的值才算有效。<ul class="gallery mw-gallery-nolines"> <li class="gallerybox" style="width: 305px"> <div class="thumb" style="width: 300px;"><a href="/wiki/File:Cir_conv_shift_ov_extra.png" class="mw-file-description" title="位移以後內積"><img alt="位移以後內積" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Cir_conv_shift_ov_extra.png/300px-Cir_conv_shift_ov_extra.png" decoding="async" width="300" height="225" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Cir_conv_shift_ov_extra.png/450px-Cir_conv_shift_ov_extra.png 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b8/Cir_conv_shift_ov_extra.png 2x" data-file-width="561" data-file-height="420" /></a></div> <div class="gallerytext">位移以後內積</div> </li> </ul> <div class="mw-heading mw-heading3"><h3 id="應用時機">應用時機</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=22" title="编辑章节：應用時機">编辑</a>]</div> 以上三種方法皆可用來計算卷積，其差別在於所需總體乘法量不同。基於運算量以及效率的考量，在計算卷積時，通常會選擇所需總體乘法量較少的方法。 以下根據<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b176b3dbcced447341ad5ab70001ef0e3231062e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.967ex; height:2.843ex;" alt="{\displaystyle f[n]}">和<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>g</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g[n]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c5e1d771a2385e9aeb71838a40425bb07c89525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle g[n]}">的長度（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N,M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>,</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N,M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3b84399043f9b62ad56f415c81ae8f0a3b6d9b1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.54ex; height:2.509ex;" alt="{\displaystyle N,M}">）分成5類，並列出適合使用的方法： <ol><li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">為一非常小的整數 - 直接計算</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\ll N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>≪</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\ll N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c456a544c3c1b8f408180fba0c01467fefcce5fe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.12ex; height:2.176ex;" alt="{\displaystyle M\ll N}"> - 分段卷积</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\approx N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>≈</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\approx N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8018282f8dfe8fc79adc3f6a7bc1cd37fc5e0ba9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.604ex; height:2.176ex;" alt="{\displaystyle M\approx N}"> - 快速傅里叶变换</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\gg N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> <mo>≫</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M\gg N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f7b8060d0395cb2ce841998907ac013b1535a4fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.12ex; height:2.176ex;" alt="{\displaystyle M\gg N}"> - 分段卷积</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}">為一非常小的整數 - 直接計算</li></ol> 基本上，以上只是粗略的分類。在實際應用時，最好還是算出三種方法所需的總乘法量，再選擇其中最有效率的方法來計算卷積。 <div class="mw-heading mw-heading3"><h3 id="例子">例子</h3>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=23" title="编辑章节：例子">编辑</a>]</div> Q1：當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=2000,M=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2000</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2000,M=17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb0a6a4b701aa466fc4896c8e3b5f3f7dbc5a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.711ex; height:2.509ex;" alt="{\displaystyle N=2000,M=17}">，適合用哪種方法計算卷積? Ans： <dl><dd>方法1：所需乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3MN=102000}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>M</mi> <mi>N</mi> <mo>=</mo> <mn>102000</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3MN=102000}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/176f76ca3518d89f694c6c0830be780083947e96" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:15.741ex; height:2.176ex;" alt="{\displaystyle 3MN=102000}"></dd></dl> <dl><dd>方法2：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq M+N-1=2016}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>2016</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq M+N-1=2016}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/747186637828661c1069ff4d99e79898992dddb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.941ex; height:2.343ex;" alt="{\displaystyle P\geq M+N-1=2016}">，而2016點的DFT最少乘法數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=12728}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo>=</mo> <mn>12728</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a=12728}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db42666547a18589485d2431de9dac2cc363a4bb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.141ex; height:2.176ex;" alt="{\displaystyle a=12728}">，所以總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3(a+P)=44232}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>44232</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3(a+P)=44232}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed765f9deb33d69a263ab630ad2d2961a902b732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.698ex; height:2.843ex;" alt="{\displaystyle 3(a+P)=44232}"></dd></dl> <dl><dd>方法3： <dl><dd>若切成8塊（<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S=8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b6fde353a7ad189d669d892caf328dd817b4476" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.76ex; height:2.176ex;" alt="{\displaystyle S=8}">），則<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=250,P\geq M+L-1=266}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>250</mn> <mo>,</mo> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>L</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>266</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=250,P\geq M+L-1=266}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61dc28bcf875e9833584fc94e53d2394dc05526c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:31.501ex; height:2.509ex;" alt="{\displaystyle L=250,P\geq M+L-1=266}">。選<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=288}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>288</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=288}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98dd9e9428e5a1fa8da1e7613f3667fd479ab8ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.331ex; height:2.176ex;" alt="{\displaystyle P=288}">，則總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=26632}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>26632</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=26632}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00be17dc0e4330f599346fa3f0bcac092bc3d309" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.862ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=26632}">，比方法1和2少了很多。</dd> <dd>但是若要找到最少的乘法量，必須依照以下步驟 <dl><dd>(1)先找出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">:解<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> : <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial {{\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}}{\partial L}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>N</mi> <mi>L</mi> </mfrac> </mrow> <mn>3</mn> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msub> <mi>log</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mrow> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial {{\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}}{\partial L}}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5212be896bcbc7febbe5e992d9650c2b0ea09c6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:44.551ex; height:6.676ex;" alt="{\displaystyle {\frac {\partial {{\frac {N}{L}}3(L+M-1)[\log _{2}(L+M-1)+1]}}{\partial L}}=0}"></dd> <dd>(2)由<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq L+M-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq L+M-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/73b07631ac40ea2ec55be227826919cb664de70e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.712ex; height:2.343ex;" alt="{\displaystyle P\geq L+M-1}">算出點數在<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}">附近的DFT所需最少的乘法量，選擇DFT的點數</dd> <dd>(3)最後由<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=P+1-M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=P+1-M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c90e2502147388084dc4cf0371d0f749e941142" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.712ex; height:2.343ex;" alt="{\displaystyle L=P+1-M}">算出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{opt}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>o</mi> <mi>p</mi> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{opt}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/251f8d17fb14ec764eb37937a75c9ea07333ed12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:4.033ex; height:2.843ex;" alt="{\displaystyle L_{opt}}"></dd></dl></dd> <dd>因此， <dl><dd>(1)由運算量對<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">的偏微分為0而求出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=85}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>85</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=85}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/915cf63e7c6ca7d395868ebfedf0b4f4c0b79b18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.006ex; height:2.176ex;" alt="{\displaystyle L=85}"></dd> <dd>(2)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq L+M-1=101}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>101</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq L+M-1=101}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6af236cd660084af132397f1f7d5ecbc547c4975" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:22.298ex; height:2.343ex;" alt="{\displaystyle P\geq L+M-1=101}">，所以選擇101點DFT附近點數乘法量最少的點數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=96}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>96</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=96}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8176b66e459f4e6c5b30aa8e0a7d6d3084e3edf1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.169ex; height:2.176ex;" alt="{\displaystyle P=96}">或<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=120}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>120</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=120}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a8443c256b1bb6fb70e7244854af9cde6fc5b45c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.331ex; height:2.176ex;" alt="{\displaystyle P=120}">。</dd> <dd>(3-1)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=96\to a=280,L=P+1-M=80\to S=25}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>96</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>280</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>80</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>25</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=96\to a=280,L=P+1-M=80\to S=25}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/96bfbfe34cf2f827d1e04b2fd258866fd025cec7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:51.305ex; height:2.509ex;" alt="{\displaystyle P=96\to a=280,L=P+1-M=80\to S=25}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=21480}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>21480</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=21480}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3021467cf4f4a5e193f6dab0d40177178dccc45a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.862ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=21480}">。</dd> <dd>(3-2)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=120\to a=380,L=P+1-M=104\to S=20}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>120</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>380</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>104</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>20</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=120\to a=380,L=P+1-M=104\to S=20}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c2a4c174e7734c4d60a041cff1c0bc2b581a60" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:53.63ex; height:2.509ex;" alt="{\displaystyle P=120\to a=380,L=P+1-M=104\to S=20}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=22780}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>22780</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=22780}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50f6435fe44b7ca39cea737bd3a3287c311bcd9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.862ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=22780}">。</dd></dl></dd> <dd>由此可知，切成20塊會有較好的效率，而所需總乘法量為21480。</dd></dl></dd></dl> <ul><li>因此，當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=2000,M=17}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>2000</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>17</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=2000,M=17}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb0a6a4b701aa466fc4896c8e3b5f3f7dbc5a80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.711ex; height:2.509ex;" alt="{\displaystyle N=2000,M=17}">，所需總乘法量：分段卷積<快速傅立葉轉換<直接計算。故，此時選擇使用分段卷積來計算卷積最適合。</li></ul> Q2：當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1024,M=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1024</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1024,M=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5534a8add9bf105b36e9435dd1143026807e89fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.549ex; height:2.509ex;" alt="{\displaystyle N=1024,M=3}">，適合用哪種方法計算卷積? Ans： <dl><dd>方法1：所需乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3MN=9216}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>M</mi> <mi>N</mi> <mo>=</mo> <mn>9216</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3MN=9216}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf603bb99b5d764d4aec0c0e634833257533d9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.417ex; height:2.176ex;" alt="{\displaystyle 3MN=9216}"></dd></dl> <dl><dd>方法2：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq M+N-1=1026}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1026</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq M+N-1=1026}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dbbfd2baea30d8e84f98f4086312da79dd075683" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.941ex; height:2.343ex;" alt="{\displaystyle P\geq M+N-1=1026}">，選擇1026點DFT附近點數乘法量最少的點數，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to P=1152,a=7088}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mi>P</mi> <mo>=</mo> <mn>1152</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>7088</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to P=1152,a=7088}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dde34876d36ec115c115fe52e369b8419e52a111" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.475ex; height:2.509ex;" alt="{\displaystyle \to P=1152,a=7088}">。 <dl><dd><dl><dd>因此，所需乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3(a+P)=24342}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>24342</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3(a+P)=24342}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a994d5254d9022c33244969331dc7c38c0ddc94c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.698ex; height:2.843ex;" alt="{\displaystyle 3(a+P)=24342}"></dd></dl></dd></dl></dd></dl> <dl><dd>方法3： <dl><dd><dl><dd>(1)由運算量對<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">的偏微分為0而求出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=5}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>5</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=5}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f0a13e71c8b4a72c47192e0197ae0faaa78240a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.844ex; height:2.176ex;" alt="{\displaystyle L=5}"></dd> <dd>(2)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq L+M-1=7}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>7</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq L+M-1=7}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1cb0f183d7913a4a80707929cc4b43bd985dab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:19.973ex; height:2.343ex;" alt="{\displaystyle P\geq L+M-1=7}">，所以選擇7點DFT附近點數乘法量最少的點數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=8}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>8</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=8}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2cb6b1c3b569d1947518cddea3fd4065d4efe6e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=8}">或<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f00880e5728dfc4be0109de6351a907d8000c04" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=6}">或<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=4}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>4</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=4}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a72557b268b516cc20a9a15b5158d36b1104d85" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle P=4}">。</dd> <dd>(3-1)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=8\to a=4,L=P+1-M=6\to S=171}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>8</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>6</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>171</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=8\to a=4,L=P+1-M=6\to S=171}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e2bd0611281705e987b5fd5f6c4829c38182285" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.817ex; height:2.509ex;" alt="{\displaystyle P=8\to a=4,L=P+1-M=6\to S=171}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=5476}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>5476</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=5476}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b77a1cbd93d23dd58652c35e66e1eea04599dc35" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.699ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=5476}">。</dd> <dd>(3-2)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=6\to a=4,L=P+1-M=4\to S=256}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>6</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>4</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>256</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=6\to a=4,L=P+1-M=4\to S=256}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/00487b92bb8b73fd6585e111fc33c54666255fa3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.817ex; height:2.509ex;" alt="{\displaystyle P=6\to a=4,L=P+1-M=4\to S=256}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=6660}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>6660</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=6660}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b03c72421a46510026463dc22b43e1eca843bd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.699ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=6660}">。</dd> <dd>(3-3)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=4\to a=0,L=P+1-M=2\to S=512}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>4</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>2</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>512</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=4\to a=0,L=P+1-M=2\to S=512}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3d34e4a20ff71c07ca2eb6a4267ad87348b14c70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:47.817ex; height:2.509ex;" alt="{\displaystyle P=4\to a=0,L=P+1-M=2\to S=512}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=6144}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>6144</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=6144}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb38ec554194d8a38c2a957ae717c73ed03c2880" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.699ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=6144}">。</dd></dl></dd> <dd>由此可知，切成171塊會有較好的效率，而所需總乘法量為5476。</dd></dl></dd></dl> <ul><li>因此，當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1024,M=3}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1024</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>3</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1024,M=3}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5534a8add9bf105b36e9435dd1143026807e89fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.549ex; height:2.509ex;" alt="{\displaystyle N=1024,M=3}">，所需總乘法量：分段卷積<直接計算<快速傅立葉轉換。故，此時選擇使用分段卷積來計算卷積最適合。</li> <li>雖然當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>M</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle M}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f82cade9898ced02fdd08712e5f0c0151758a0dd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}">是個很小的正整數時，大致上適合使用直接計算。但實際上還是將3個方法所需的乘法量都算出來，才能知道用哪種方法可以達到最高的效率。</li></ul> Q3：當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1024,M=600}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1024</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>600</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1024,M=600}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9995308e131b620743af80eba59c16a8b54f36d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.874ex; height:2.509ex;" alt="{\displaystyle N=1024,M=600}">，適合用哪種方法計算卷積? Ans： <dl><dd>方法1：所需乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3MN=1843200}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>M</mi> <mi>N</mi> <mo>=</mo> <mn>1843200</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3MN=1843200}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/55d14707009359954f78aaf073f6511619f9ad5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.904ex; height:2.176ex;" alt="{\displaystyle 3MN=1843200}"></dd></dl> <dl><dd>方法2：<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq M+N-1=1623}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>M</mi> <mo>+</mo> <mi>N</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1623</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq M+N-1=1623}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/03cd4af9b7fe6282e41e63d9f788b5cf58260668" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.941ex; height:2.343ex;" alt="{\displaystyle P\geq M+N-1=1623}">，選擇1026點DFT附近點數乘法量最少的點數，<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \to P=2016,a=12728}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">→</mo> <mi>P</mi> <mo>=</mo> <mn>2016</mn> <mo>,</mo> <mi>a</mi> <mo>=</mo> <mn>12728</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \to P=2016,a=12728}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79759b86348e7932321d78bfb6158cebb2823148" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:23.637ex; height:2.509ex;" alt="{\displaystyle \to P=2016,a=12728}">。 <dl><dd><dl><dd>因此，所需乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3(a+P)=44232}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mo stretchy="false">(</mo> <mi>a</mi> <mo>+</mo> <mi>P</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>44232</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3(a+P)=44232}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ed765f9deb33d69a263ab630ad2d2961a902b732" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.698ex; height:2.843ex;" alt="{\displaystyle 3(a+P)=44232}"></dd></dl></dd></dl></dd></dl> <dl><dd>方法3： <dl><dd><dl><dd>(1)由運算量對<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}">的偏微分為0而求出<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L=1024}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> <mo>=</mo> <mn>1024</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L=1024}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1dea891d99d45646054fa5a0b58b3e5dce6f25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.331ex; height:2.176ex;" alt="{\displaystyle L=1024}"></dd> <dd>(2)<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\geq L+M-1=1623}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>≥</mo> <mi>L</mi> <mo>+</mo> <mi>M</mi> <mo>−</mo> <mn>1</mn> <mo>=</mo> <mn>1623</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P\geq L+M-1=1623}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a7b1e052b8de53ec0deb988b462c5b199854baf2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:23.46ex; height:2.343ex;" alt="{\displaystyle P\geq L+M-1=1623}">，所以選擇1623點DFT附近點數乘法量最少的點數<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=2016}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>2016</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=2016}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bad33a15e07caab3ed1f6b04d28dbeeadc8a635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.494ex; height:2.176ex;" alt="{\displaystyle P=2016}">。</dd> <dd>(3)當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=2016\to a=12728,L=P+1-M=1417\to S=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mn>2016</mn> <mo stretchy="false">→</mo> <mi>a</mi> <mo>=</mo> <mn>12728</mn> <mo>,</mo> <mi>L</mi> <mo>=</mo> <mi>P</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi>M</mi> <mo>=</mo> <mn>1417</mn> <mo stretchy="false">→</mo> <mi>S</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=2016\to a=12728,L=P+1-M=1417\to S=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77d1e873d9a4334c86d1af7183efaf91daf1f3f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:57.117ex; height:2.509ex;" alt="{\displaystyle P=2016\to a=12728,L=P+1-M=1417\to S=1}">，總乘法量為<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (2S+1)a+3SP=44232}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>2</mn> <mi>S</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>a</mi> <mo>+</mo> <mn>3</mn> <mi>S</mi> <mi>P</mi> <mo>=</mo> <mn>44232</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (2S+1)a+3SP=44232}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4351a2a23dc1ab7d18ddb46126658ccf9ad52e08" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.862ex; height:2.843ex;" alt="{\displaystyle (2S+1)a+3SP=44232}">。</dd></dl></dd> <dd>由此可知，此時切成一段，就跟方法2一樣，所需總乘法量為44232。</dd></dl></dd></dl> <ul><li>因此，當<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N=1024,M=600}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> <mo>=</mo> <mn>1024</mn> <mo>,</mo> <mi>M</mi> <mo>=</mo> <mn>600</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N=1024,M=600}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9995308e131b620743af80eba59c16a8b54f36d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:19.874ex; height:2.509ex;" alt="{\displaystyle N=1024,M=600}">，所需總乘法量：快速傅立葉轉換 = 分段卷積<直接計算。故，此時選擇使用分段卷積來計算卷積最適合。</li></ul> <div class="mw-heading mw-heading2"><h2 id="应用">应用</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=24" title="编辑章节：应用">编辑</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Halftone,_Gaussian_Blur.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Halftone%2C_Gaussian_Blur.jpg/220px-Halftone%2C_Gaussian_Blur.jpg" decoding="async" width="220" height="337" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d7/Halftone%2C_Gaussian_Blur.jpg/330px-Halftone%2C_Gaussian_Blur.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/d/d7/Halftone%2C_Gaussian_Blur.jpg 2x" data-file-width="438" data-file-height="670" /></a><figcaption><a href="/wiki/%E9%AB%98%E6%96%AF%E6%A8%A1%E7%B3%8A" title="高斯模糊">高斯模糊</a>可被用来从<a href="/wiki/%E5%8D%8A%E8%89%B2%E8%AA%BF" title="半色調">半色调</a>印刷品复原出光滑灰度数字图像。</figcaption></figure> 卷积在科学、工程和数学上都有很多应用： <ul><li><a href="/wiki/%E4%BB%A3%E6%95%B0" title="代数">代数</a>中，整数乘法和多项式乘法都是卷积。</li> <li><a href="/wiki/%E5%8D%B7%E7%A7%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="卷积神经网络">卷积神经网络</a>应用了多重级联的卷积<a href="/w/index.php?title=%E6%A0%B8%E5%BF%83_(%E5%9B%BE%E5%83%8F%E5%A4%84%E7%90%86)&action=edit&redlink=1" class="new" title="核心 (图像处理)（页面不存在）">核心</a>（英语：<a href="https://en.wikipedia.org/wiki/Kernel_(image_processing)" class="extiw" title="en:Kernel (image processing)">Kernel (image processing)</a>），它被用于<a href="/wiki/%E6%9C%BA%E5%99%A8%E8%A7%86%E8%A7%89" title="机器视觉">机器视觉</a>和<a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a><a href="#cite_note-16">[16]</a><a href="#cite_note-17">[17]</a>，尽管在多数情况下实际上用的是<a href="/wiki/%E4%BA%92%E7%9B%B8%E5%85%B3" title="互相关">互相关</a>而非卷积<a href="#cite_note-18">[18]</a>。</li> <li>在非<a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a>的<a href="/wiki/%E5%9B%BE%E5%83%8F%E5%A4%84%E7%90%86" title="图像处理">图像处理</a>中，用作图像模糊、锐化、<a href="/wiki/%E8%BE%B9%E7%BC%98%E6%A3%80%E6%B5%8B" title="边缘检测">边缘检测</a>。</li> <li><a href="/wiki/%E7%BB%9F%E8%AE%A1%E5%AD%A6" title="统计学">统计学</a>中，加权的滑动平均是一种卷积。</li> <li><a href="/wiki/%E6%A6%82%E7%8E%87%E8%AE%BA" title="概率论">概率论</a>中，两个统计独立变量X与Y的和的<a href="/wiki/%E6%A6%82%E7%8E%87%E5%AF%86%E5%BA%A6%E5%87%BD%E6%95%B0" class="mw-redirect" title="概率密度函数">概率密度函数</a>是X与Y的概率密度函数的卷积。</li> <li><a href="/wiki/%E5%A3%B0%E5%AD%A6" title="声学">声学</a>中，<a href="/wiki/%E5%9B%9E%E8%81%B2" title="回聲">回声</a>可以用源声与一个反映各种反射效应的函数的卷积表示。</li> <li><a href="/wiki/%E7%94%B5%E5%AD%90%E5%B7%A5%E7%A8%8B" title="电子工程">电子工程</a>与<a href="/wiki/%E4%BF%A1%E5%8F%B7%E5%A4%84%E7%90%86" title="信号处理">信号处理</a>中，任一个线性系统的输出都可以通过将输入信号与系统函数（系统的<a href="/wiki/%E5%86%B2%E6%BF%80%E5%93%8D%E5%BA%94" title="冲激响应">冲激响应</a>）做卷积获得。</li> <li><a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6" title="物理学">物理学</a>中，任何一个线性系统（符合<a href="/wiki/%E5%8F%A0%E5%8A%A0%E5%8E%9F%E7%90%86" title="叠加原理">叠加原理</a>）都存在卷积。</li></ul> <div class="mw-heading mw-heading2"><h2 id="参见">参见</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=25" title="编辑章节：参见">编辑</a>]</div> <ul><li><a href="/wiki/%E5%8F%8D%E8%A4%B6%E7%A7%AF" title="反褶积">反卷积</a></li> <li><a href="/wiki/%E8%87%AA%E7%9B%B8%E5%85%B3%E5%87%BD%E6%95%B0" title="自相关函数">自相关函数</a></li> <li><a href="/wiki/%E5%82%85%E9%87%8C%E5%8F%B6%E5%8F%98%E6%8D%A2" title="傅里叶变换">傅里叶变换</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="引用">引用</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=26" title="编辑章节：引用">编辑</a>]</div> <div class="reflist columns references-column-count references-column-count-2" style="-moz-column-count: 2; -webkit-column-count: 2; column-count: 2; list-style-type: decimal;"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite class="citation book">Smith, Stephen W. <a rel="nofollow" class="external text" href="http://www.dspguide.com/ch13/2.htm">13.Convolution</a>. 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California Technical Publishing. 1997 [22 April 2016]. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-9660176-3-3" title="Special:网络书源/0-9660176-3-3">ISBN 0-9660176-3-3</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20230626124155/http://www.dspguide.com/ch13/2.htm">存档</a>于2023-06-26）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=13.Convolution&rft.aufirst=Stephen+W&rft.aulast=Smith&rft.btitle=The+Scientist+and+Engineer%27s+Guide+to+Digital+Signal+Processing&rft.date=1997&rft.edition=1&rft.genre=bookitem&rft.isbn=0-9660176-3-3&rft.pub=California+Technical+Publishing&rft_id=http%3A%2F%2Fwww.dspguide.com%2Fch13%2F2.htm&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite class="citation book"><a href="/w/index.php?title=J._David_Irwin&action=edit&redlink=1" class="new" title="J. 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Sullivan (编), The Early Years of Radio Astronomy: Reflections Fifty Years After Jansky's Discovery, Cambridge University Press: 172, 2005, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-521-61602-7" title="Special:网络书源/978-0-521-61602-7">ISBN 978-0-521-61602-7</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=Early+work+on+imaging+theory+in+radio+astronomy&rft.au=R.+N.+Bracewell&rft.btitle=The+Early+Years+of+Radio+Astronomy%3A+Reflections+Fifty+Years+After+Jansky%27s+Discovery&rft.date=2005&rft.genre=bookitem&rft.isbn=978-0-521-61602-7&rft.pages=172&rft.pub=Cambridge+University+Press&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dv2SqL0zCrwcC%26pg%3DPA172&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFJohn_Hilton_Grace_and_Alfred_Young1903" class="citation">John Hilton Grace and Alfred Young, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NIe4AAAAIAAJ&pg=PA40">The algebra of invariants</a>, Cambridge University Press: 40, 1903</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=John+Hilton+Grace+and+Alfred+Young&rft.btitle=The+algebra+of+invariants&rft.date=1903&rft.genre=book&rft.pages=40&rft.pub=Cambridge+University+Press&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNIe4AAAAIAAJ%26pg%3DPA40&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFLeonard_Eugene_Dickson1914" class="citation">Leonard Eugene Dickson, <a rel="nofollow" class="external text" href="https://books.google.com/books?id=LRGoAAAAIAAJ&pg=PA85">Algebraic invariants</a>, J. Wiley: 85, 1914</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Leonard+Eugene+Dickson&rft.btitle=Algebraic+invariants&rft.date=1914&rft.genre=book&rft.pages=85&rft.pub=J.+Wiley&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DLRGoAAAAIAAJ%26pg%3DPA85&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  </li> <li id="cite_note-8"><a href="#cite_ref-8">^</a> (<a href="#CITEREFSteinWeiss1971">Stein & Weiss 1971</a>，Theorem 1.3) </li> <li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFCrutchfield2010" class="citation">Crutchfield, Steve, <a rel="nofollow" class="external text" href="http://www.jhu.edu/signals/convolve/index.html">The Joy of Convolution</a>, Johns Hopkins University, October 12, 2010 [November 21, 2010], （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20130911022447/http://www.jhu.edu/signals/convolve/index.html">存档</a>于2013-09-11）</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=The+Joy+of+Convolution&rft.aufirst=Steve&rft.aulast=Crutchfield&rft.date=2010-10-12&rft.genre=article&rft.jtitle=Johns+Hopkins+University&rft_id=http%3A%2F%2Fwww.jhu.edu%2Fsignals%2Fconvolve%2Findex.html&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988">  </li> <li id="cite_note-Jeruchim-10"><a href="#cite_ref-Jeruchim_10-0">^</a> <cite class="citation book">Jeruchim, Michel C.; Balaban, Philip; Shanmugan, K. 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September 2020, 16 (9): 5769–5779 [2023-10-24]. <a rel="nofollow" class="external text" href="//www.worldcat.org/issn/1941-0050">ISSN 1941-0050</a>. <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:213010088">S2CID 213010088</a>. <a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTII.2019.2956078">doi:10.1109/TII.2019.2956078</a>. （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20230731120013/https://ieeexplore.ieee.org/document/8913613/">存档</a>于2023-07-31）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=Powder-Bed+Fusion+Process+Monitoring+by+Machine+Vision+With+Hybrid+Convolutional+Neural+Networks&rft.au=Fuh%2C+Jerry+Ying+Hsi&rft.au=Soon%2C+Hong+Geok&rft.au=Ye%2C+Dongsen&rft.au=Zhu%2C+Kunpeng&rft.aufirst=Yingjie&rft.aulast=Zhang&rft.date=2020-09&rft.genre=article&rft.issn=1941-0050&rft.issue=9&rft.jtitle=IEEE+Transactions+on+Industrial+Informatics&rft.pages=5769-5779&rft.volume=16&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A213010088&rft_id=https%3A%2F%2Fieeexplore.ieee.org%2Fdocument%2F8913613&rft_id=info%3Adoi%2F10.1109%2FTII.2019.2956078&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988">  </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite class="citation journal">Chervyakov, N.I.; Lyakhov, P.A.; Deryabin, M.A.; Nagornov, N.N.; Valueva, M.V.; Valuev, G.V. <a rel="nofollow" class="external text" href="https://linkinghub.elsevier.com/retrieve/pii/S092523122030583X">Residue Number System-Based Solution for Reducing the Hardware Cost of a Convolutional Neural Network</a>. 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Neural Information Processing Systems (NIPS 1987). （原始内容<a rel="nofollow" class="external text" href="https://web.archive.org/web/20210414091306/https://papers.nips.cc/paper/1987/file/98f13708210194c475687be6106a3b84-Paper.pdf">存档</a> (PDF)于2021-04-14）.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=An+Artificial+Neural+Network+for+Spatio-Temporal+Bipolar+Patterns%3A+Application+to+Phoneme+Classification&rft.au=Atlas%2C+Homma%2C+and+Marks&rft.genre=article&rft.jtitle=Neural+Information+Processing+Systems+%28NIPS+1987%29&rft.volume=1&rft_id=https%3A%2F%2Fpapers.nips.cc%2Fpaper%2F1987%2Ffile%2F98f13708210194c475687be6106a3b84-Paper.pdf&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988">  </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="延伸阅读">延伸阅读</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=27" title="编辑章节：延伸阅读">编辑</a>]</div> <ul><li><cite id="CITEREFBracewell1986" class="citation">Bracewell, R., The Fourier Transform and Its Applications 2nd, McGraw–Hill, 1986, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-07-116043-4" title="Special:网络书源/0-07-116043-4">ISBN 0-07-116043-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=R.&rft.aulast=Bracewell&rft.btitle=The+Fourier+Transform+and+Its+Applications&rft.date=1986&rft.edition=2nd&rft.genre=book&rft.isbn=0-07-116043-4&rft.pub=McGraw%E2%80%93Hill&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><cite id="CITEREFDamelinMiller2011" class="citation">Damelin, S.; Miller, W., The Mathematics of Signal Processing, Cambridge University Press, 2011, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1107601048" title="Special:网络书源/978-1107601048">ISBN 978-1107601048</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Miller%2C+W.&rft.aufirst=S.&rft.aulast=Damelin&rft.btitle=The+Mathematics+of+Signal+Processing&rft.date=2011&rft.genre=book&rft.isbn=978-1107601048&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><cite id="CITEREFDiggle1985" class="citation">Diggle, P. 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Wissenschaft. 256, Springer, 1983, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/3-540-12104-8" title="Special:网络书源/3-540-12104-8">ISBN 3-540-12104-8</a>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0717035">MR 0717035</a>, <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-96750-4">doi:10.1007/978-3-642-96750-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=L.&rft.aulast=H%C3%B6rmander&rft.btitle=The+analysis+of+linear+partial+differential+operators+I&rft.date=1983&rft.genre=book&rft.isbn=3-540-12104-8&rft.pub=Springer&rft.series=Grundl.+Math.+Wissenschaft.&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0717035&rft_id=info%3Adoi%2F10.1007%2F978-3-642-96750-4&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><cite id="CITEREFKassel1995" class="citation">Kassel, Christian, <a rel="nofollow" class="external text" href="https://archive.org/details/quantumgroups0000kass">Quantum groups</a><img alt="需要免费注册" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" />, Graduate Texts in Mathematics 155, Berlin, New York: <a href="/wiki/Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, 1995, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-387-94370-1" title="Special:网络书源/978-0-387-94370-1">ISBN 978-0-387-94370-1</a>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=1321145">MR 1321145</a>, <a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-1-4612-0783-2">doi:10.1007/978-1-4612-0783-2</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=Christian&rft.aulast=Kassel&rft.btitle=Quantum+groups&rft.date=1995&rft.genre=book&rft.isbn=978-0-387-94370-1&rft.place=Berlin%2C+New+York&rft.pub=Springer-Verlag&rft.series=Graduate+Texts+in+Mathematics&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D1321145&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fquantumgroups0000kass&rft_id=info%3Adoi%2F10.1007%2F978-1-4612-0783-2&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  含有內容需登入查看的頁面 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%85%A7%E5%AE%B9%E9%9C%80%E7%99%BB%E5%85%A5%E6%9F%A5%E7%9C%8B%E7%9A%84%E9%A0%81%E9%9D%A2" title="Category:含有內容需登入查看的頁面">link</a>).</li> <li><cite id="CITEREFKnuth1997" class="citation"><a href="/wiki/Donald_Knuth" class="mw-redirect" title="Donald Knuth">Knuth, Donald</a>, Seminumerical Algorithms 3rd., Reading, Massachusetts: Addison–Wesley, 1997, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-201-89684-2" title="Special:网络书源/0-201-89684-2">ISBN 0-201-89684-2</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=Donald&rft.aulast=Knuth&rft.btitle=Seminumerical+Algorithms&rft.date=1997&rft.edition=3rd.&rft.genre=book&rft.isbn=0-201-89684-2&rft.place=Reading%2C+Massachusetts&rft.pub=Addison%E2%80%93Wesley&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><a href="/w/index.php?title=Template:Narici_Beckenstein_Topological_Vector_Spaces&action=edit&redlink=1" class="new" title="Template:Narici Beckenstein Topological Vector Spaces（页面不存在）">Template:Narici Beckenstein Topological Vector Spaces</a></li> <li><cite id="CITEREFReedSimon1975" class="citation">Reed, Michael; <a href="/w/index.php?title=Barry_Simon&action=edit&redlink=1" class="new" title="Barry Simon（页面不存在）">Simon, Barry</a>, Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, New York-London: Academic Press Harcourt Brace Jovanovich, Publishers: xv+361, 1975, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-12-585002-6" title="Special:网络书源/0-12-585002-6">ISBN 0-12-585002-6</a>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0493420">MR 0493420</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Simon%2C+Barry&rft.aufirst=Michael&rft.aulast=Reed&rft.btitle=Methods+of+modern+mathematical+physics.+II.+Fourier+analysis%2C+self-adjointness&rft.date=1975&rft.genre=book&rft.isbn=0-12-585002-6&rft.pages=xv%2B361&rft.place=New+York-London&rft.pub=Academic+Press+Harcourt+Brace+Jovanovich%2C+Publishers&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0493420&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li> <li><cite id="CITEREFRudin1962" class="citation"><a href="/wiki/Walter_Rudin" class="mw-redirect" title="Walter Rudin">Rudin, Walter</a>, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics 12, New York–London: Interscience Publishers, 1962, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-471-52364-X" title="Special:网络书源/0-471-52364-X">ISBN 0-471-52364-X</a>, <a rel="nofollow" class="external text" href="//www.ams.org/mathscinet-getitem?mr=0152834">MR 0152834</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=Walter&rft.aulast=Rudin&rft.btitle=Fourier+analysis+on+groups&rft.date=1962&rft.genre=book&rft.isbn=0-471-52364-X&rft.place=New+York%E2%80%93London&rft.pub=Interscience+Publishers&rft.series=Interscience+Tracts+in+Pure+and+Applied+Mathematics&rft_id=%2F%2Fwww.ams.org%2Fmathscinet-getitem%3Fmr%3D0152834&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><a href="/w/index.php?title=Template:Schaefer_Wolff_Topological_Vector_Spaces&action=edit&redlink=1" class="new" title="Template:Schaefer Wolff Topological Vector Spaces（页面不存在）">Template:Schaefer Wolff Topological Vector Spaces</a></li> <li><cite id="CITEREFSteinWeiss1971" class="citation"><a href="/wiki/Elias_Stein" class="mw-redirect" title="Elias Stein">Stein, Elias</a>; Weiss, Guido, <a rel="nofollow" class="external text" href="https://archive.org/details/introductiontofo0000stei">Introduction to Fourier Analysis on Euclidean Spaces</a><img alt="需要免费注册" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" />, Princeton University Press, 1971, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-691-08078-X" title="Special:网络书源/0-691-08078-X">ISBN 0-691-08078-X</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Weiss%2C+Guido&rft.aufirst=Elias&rft.aulast=Stein&rft.btitle=Introduction+to+Fourier+Analysis+on+Euclidean+Spaces&rft.date=1971&rft.genre=book&rft.isbn=0-691-08078-X&rft.pub=Princeton+University+Press&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fintroductiontofo0000stei&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  含有內容需登入查看的頁面 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%85%A7%E5%AE%B9%E9%9C%80%E7%99%BB%E5%85%A5%E6%9F%A5%E7%9C%8B%E7%9A%84%E9%A0%81%E9%9D%A2" title="Category:含有內容需登入查看的頁面">link</a>).</li> <li><cite id="CITEREFSobolev2001" class="citation">Sobolev, V.I., <a rel="nofollow" class="external text" href="http://www.encyclopediaofmath.org/index.php?title=C/c026430">Convolution of functions</a>, Hazewinkel, Michiel (编), <a href="/wiki/%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6" title="数学百科全书">数学百科全书</a>, <a href="/wiki/Springer_Science%2BBusiness_Media" class="mw-redirect" title="Springer Science+Business Media">Springer</a>, 2001, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-1-55608-010-4" title="Special:网络书源/978-1-55608-010-4">ISBN 978-1-55608-010-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=Convolution+of+functions&rft.aufirst=V.I.&rft.aulast=Sobolev&rft.btitle=%E6%95%B0%E5%AD%A6%E7%99%BE%E7%A7%91%E5%85%A8%E4%B9%A6&rft.date=2001&rft.genre=bookitem&rft.isbn=978-1-55608-010-4&rft.pub=Springer&rft_id=http%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DC%2Fc026430&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><cite id="CITEREFStrichartz1994" class="citation">Strichartz, R., A Guide to Distribution Theory and Fourier Transforms, CRC Press, 1994, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-8493-8273-4" title="Special:网络书源/0-8493-8273-4">ISBN 0-8493-8273-4</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=R.&rft.aulast=Strichartz&rft.btitle=A+Guide+to+Distribution+Theory+and+Fourier+Transforms&rft.date=1994&rft.genre=book&rft.isbn=0-8493-8273-4&rft.pub=CRC+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><cite id="CITEREFTitchmarsh1948" class="citation"><a href="/w/index.php?title=Edward_Charles_Titchmarsh&action=edit&redlink=1" class="new" title="Edward Charles Titchmarsh（页面不存在）">Titchmarsh, E</a>, Introduction to the theory of Fourier integrals 2nd, New York, N.Y.: Chelsea Pub. Co., 19481986, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-8284-0324-5" title="Special:网络书源/978-0-8284-0324-5">ISBN 978-0-8284-0324-5</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.aufirst=E&rft.aulast=Titchmarsh&rft.btitle=Introduction+to+the+theory+of+Fourier+integrals&rft.date=1948&rft.edition=2nd&rft.genre=book&rft.isbn=978-0-8284-0324-5&rft.place=New+York%2C+N.Y.&rft.pub=Chelsea+Pub.+Co.&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><a href="/w/index.php?title=Template:Tr%C3%A8ves_Fran%C3%A7ois_Topological_vector_spaces,_distributions_and_kernels&action=edit&redlink=1" class="new" title="Template:Trèves François Topological vector spaces, distributions and kernels（页面不存在）">Template:Trèves François Topological vector spaces, distributions and kernels</a></li> <li><cite id="CITEREFUludag1998" class="citation"><a href="/w/index.php?title=A._Muhammed_Uludag&action=edit&redlink=1" class="new" title="A. Muhammed Uludag（页面不存在）">Uludag, A. M.</a>, On possible deterioration of smoothness under the operation of convolution, J. Math. Anal. Appl., 1998, 227 (2): 335–358, <a rel="nofollow" class="external text" href="https://doi.org/10.1006%2Fjmaa.1998.6091">doi:10.1006/jmaa.1998.6091</a> <img alt="可免费查阅" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/14px-Lock-green.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/18px-Lock-green.svg.png 2x" data-file-width="512" data-file-height="813" /></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.atitle=On+possible+deterioration+of+smoothness+under+the+operation+of+convolution&rft.aufirst=A.+M.&rft.aulast=Uludag&rft.date=1998&rft.genre=article&rft.issue=2&rft.jtitle=J.+Math.+Anal.+Appl.&rft.pages=335-358&rft.volume=227&rft_id=info%3Adoi%2F10.1006%2Fjmaa.1998.6091&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal" class="Z3988"> </li> <li><cite id="CITEREFvon_zur_GathenGerhard2003" class="citation">von zur Gathen, J.; Gerhard, J ., Modern Computer Algebra, Cambridge University Press, 2003, <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-521-82646-2" title="Special:网络书源/0-521-82646-2">ISBN 0-521-82646-2</a></cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Gerhard%2C+J+.&rft.aufirst=J.&rft.aulast=von+zur+Gathen&rft.btitle=Modern+Computer+Algebra&rft.date=2003&rft.genre=book&rft.isbn=0-521-82646-2&rft.pub=Cambridge+University+Press&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> .</li> <li><cite id="Oppenheim" class="citation book"><a href="/w/index.php?title=Alan_V._Oppenheim&action=edit&redlink=1" class="new" title="Alan V. Oppenheim（页面不存在）">Oppenheim, Alan V.</a>; <a href="/w/index.php?title=Ronald_W._Schafer&action=edit&redlink=1" class="new" title="Ronald W. Schafer（页面不存在）">Schafer, Ronald W.</a>; Buck, John R. <a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan">Discrete-time signal processing</a><img alt="需要免费注册" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/9px-Lock-blue-alt-2.svg.png" decoding="async" width="9" height="14" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/14px-Lock-blue-alt-2.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Lock-blue-alt-2.svg/18px-Lock-blue-alt-2.svg.png 2x" data-file-width="512" data-file-height="813" /> 2nd. Upper Saddle River,N.J.: Prentice Hall. 1999: <a rel="nofollow" class="external text" href="https://archive.org/details/discretetimesign00alan/page/548">548</a>, 571. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-13-754920-2" title="Special:网络书源/0-13-754920-2">ISBN 0-13-754920-2</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Buck%2C+John+R.&rft.au=Schafer%2C+Ronald+W.&rft.aufirst=Alan+V.&rft.aulast=Oppenheim&rft.btitle=Discrete-time+signal+processing&rft.date=1999&rft.edition=2nd&rft.genre=book&rft.isbn=0-13-754920-2&rft.pages=548%2C+571&rft.place=Upper+Saddle+River%2CN.J.&rft.pub=Prentice+Hall&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fdiscretetimesign00alan&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988">  含有內容需登入查看的頁面 (<a href="/wiki/Category:%E5%90%AB%E6%9C%89%E5%85%A7%E5%AE%B9%E9%9C%80%E7%99%BB%E5%85%A5%E6%9F%A5%E7%9C%8B%E7%9A%84%E9%A0%81%E9%9D%A2" title="Category:含有內容需登入查看的頁面">link</a>)</li> <li><cite id="McGillem" class="citation book">McGillem, Clare D.; Cooper, George R. <a rel="nofollow" class="external text" href="https://archive.org/details/continuousdiscre0000mcgi">Continuous and Discrete Signal and System Analysis</a> 2. Holt, Rinehart and Winston. 1984. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/0-03-061703-0" title="Special:网络书源/0-03-061703-0">ISBN 0-03-061703-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%8D%B7%E7%A7%AF&rft.au=Cooper%2C+George+R.&rft.aufirst=Clare+D.&rft.aulast=McGillem&rft.btitle=Continuous+and+Discrete+Signal+and+System+Analysis&rft.date=1984&rft.edition=2&rft.genre=book&rft.isbn=0-03-061703-0&rft.pub=Holt%2C+Rinehart+and+Winston&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fcontinuousdiscre0000mcgi&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"> </li></ul> <div class="mw-heading mw-heading2"><h2 id="外部链接">外部链接</h2>[<a href="/w/index.php?title=%E5%8D%B7%E7%A7%AF&action=edit&section=28" title="编辑章节：外部链接">编辑</a>]</div> <style data-mw-deduplicate="TemplateStyles:r82655521">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:#f9f9f9;display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><div class="side-box side-box-right plainlinks sistersitebox" style="font-size:small;"><style data-mw-deduplicate="TemplateStyles:r82655520">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></div> <div class="side-box-text plainlist"><a href="/wiki/%E7%BB%B4%E5%9F%BA%E5%85%B1%E4%BA%AB%E8%B5%84%E6%BA%90" title="维基共享资源">维基共享资源</a>上的相关多媒体资源：<a href="https://commons.wikimedia.org/wiki/Category:Convolution" class="extiw" title="commons:Category:Convolution">卷积</a></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="http://jeff560.tripod.com/c.html">Earliest Uses: The entry on Convolution has some historical information.</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20110501074959/http://jeff560.tripod.com/c.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20060221234856/http://rkb.home.cern.ch/rkb/AN16pp/node38.html#SECTION000380000000000000000">Convolution</a>, on <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060512020859/http://rkb.home.cern.ch/rkb/titleA.html">The Data Analysis BriefBook</a></li> <li><a rel="nofollow" class="external free" href="http://www.jhu.edu/~signals/convolve/index.html">http://www.jhu.edu/~signals/convolve/index.html</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20120831101935/http://www.jhu.edu/~signals/convolve/index.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Visual convolution Java Applet</li> <li><a rel="nofollow" class="external free" href="http://www.jhu.edu/~signals/discreteconv2/index.html">http://www.jhu.edu/~signals/discreteconv2/index.html</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20120831102747/http://www.jhu.edu/~signals/discreteconv2/index.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Visual convolution Java Applet for discrete-time functions</li> <li><a rel="nofollow" class="external free" href="https://get-the-solution.net/projects/discret-convolution">https://get-the-solution.net/projects/discret-convolution</a> discret-convolution online calculator</li> <li><a rel="nofollow" class="external free" href="https://lpsa.swarthmore.edu/Convolution/CI.html">https://lpsa.swarthmore.edu/Convolution/CI.html</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20230715160527/https://lpsa.swarthmore.edu/Convolution/CI.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Convolution demo and visualization in javascript</li> <li><a rel="nofollow" class="external free" href="https://phiresky.github.io/convolution-demo/">https://phiresky.github.io/convolution-demo/</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231029022044/https://phiresky.github.io/convolution-demo/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） Another convolution demo in javascript</li> <li><a rel="nofollow" class="external text" href="https://archive.org/details/Lectures_on_Image_Processing">Lectures on Image Processing: A collection of 18 lectures in pdf format from Vanderbilt University. Lecture 7 is on 2-D convolution.</a>, by Alan Peters</li> <li>* <a rel="nofollow" class="external free" href="https://archive.org/details/Lectures_on_Image_Processing">https://archive.org/details/Lectures_on_Image_Processing</a></li> <li><a rel="nofollow" class="external text" href="http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/kernelmaskoperation/">Convolution Kernel Mask Operation Interactive tutorial</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231210161008/http://micro.magnet.fsu.edu/primer/java/digitalimaging/processing/kernelmaskoperation/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/Convolution.html">Convolution</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20230315013334/http://mathworld.wolfram.com/Convolution.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） at <a href="/wiki/MathWorld" title="MathWorld">MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://www.nongnu.org/freeverb3/">Freeverb3 Impulse Response Processor</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231029093122/http://www.nongnu.org/freeverb3/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）: Opensource zero latency impulse response processor with VST plugins</li> <li>Stanford University CS 178 <a rel="nofollow" class="external text" href="http://graphics.stanford.edu/courses/cs178/applets/convolution.html">interactive Flash demo </a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231029022042/http://graphics.stanford.edu/courses/cs178/applets/convolution.html">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） showing how spatial convolution works.</li> <li><a rel="nofollow" class="external text" href="https://www.youtube.com/watch?v=IW4Reburjpc">A video lecture on the subject of convolution</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231029022044/https://www.youtube.com/watch?v=IW4Reburjpc">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） given by <a href="/w/index.php?title=Salman_Khan_(educator)&action=edit&redlink=1" class="new" title="Salman Khan (educator)（页面不存在）">Salman Khan</a></li> <li><a rel="nofollow" class="external text" href="http://www.dspguide.com/ch24/6.htm">Example of FFT convolution for pattern-recognition (image processing)</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20230319094048/http://www.dspguide.com/ch24/6.htm">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>）</li> <li><a rel="nofollow" class="external text" href="https://betterexplained.com/articles/intuitive-convolution/">Intuitive Guide to Convolution</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20231207202207/https://betterexplained.com/articles/intuitive-convolution/">页面存档备份</a>，存于<a href="/wiki/%E4%BA%92%E8%81%94%E7%BD%91%E6%A1%A3%E6%A1%88%E9%A6%86" title="互联网档案馆">互联网档案馆</a>） A blogpost about an intuitive interpretation of convolution.</li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r84265675">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl 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class="nv-edit"><a href="/wiki/Special:%E7%BC%96%E8%BE%91%E9%A1%B5%E9%9D%A2/Template:Differentiable_computing" title="Special:编辑页面/Template:Differentiable computing"><abbr title="编辑该模板">编</abbr></a></li></ul></div><div id="可微分计算" style="font-size:110%;margin:0 5em">可微分计算</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/%E5%8F%AF%E5%BE%AE%E5%87%BD%E6%95%B0" title="可微函数">概论</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%8F%AF%E5%BE%AE%E5%88%86%E7%BC%96%E7%A8%8B" title="可微分编程">可微分编程</a></li> <li><a href="/wiki/%E8%87%AA%E5%8B%95%E5%BE%AE%E5%88%86" title="自動微分">自動微分</a></li> <li><a href="/wiki/%E5%BC%B5%E9%87%8F%E5%BE%AE%E7%A9%8D%E5%88%86" class="mw-redirect" title="張量微積分">张量微积分</a>（英语：<a href="https://en.wikipedia.org/wiki/Tensor_calculus" class="extiw" title="en:Tensor calculus">Tensor calculus</a>）</li> <li><a href="/wiki/%E4%BF%A1%E6%81%AF%E5%87%A0%E4%BD%95" title="信息几何">信息几何</a></li> <li><a href="/wiki/%E7%BB%9F%E8%AE%A1%E6%B5%81%E5%BD%A2" title="统计流形">统计流形</a></li> <li><a href="/w/index.php?title=%E7%A5%9E%E7%BB%8F%E5%BD%A2%E6%80%81%E5%B7%A5%E7%A8%8B&action=edit&redlink=1" class="new" title="神经形态工程（页面不存在）">神经形态工程</a>（英语：<a href="https://en.wikipedia.org/wiki/Neuromorphic_engineering" class="extiw" title="en:Neuromorphic engineering">Neuromorphic engineering</a>）</li> <li><a href="/wiki/%E6%A8%A1%E5%BC%8F%E8%AF%86%E5%88%AB" title="模式识别">模式识别</a></li> <li><a href="/w/index.php?title=%E8%BF%90%E7%AE%97%E5%AD%A6%E4%B9%A0%E7%90%86%E8%AE%BA&action=edit&redlink=1" class="new" title="运算学习理论（页面不存在）">运算学习理论</a>（英语：<a href="https://en.wikipedia.org/wiki/Computational_learning_theory" class="extiw" title="en:Computational learning theory">Computational learning theory</a>）</li> <li><a href="/wiki/%E5%BD%92%E7%BA%B3%E5%81%8F%E7%BD%AE" title="归纳偏置">归纳偏置</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">概念</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%A2%AF%E5%BA%A6%E4%B8%8B%E9%99%8D%E6%B3%95" title="梯度下降法">梯度下降</a> <ul><li><a href="/w/index.php?title=%E9%9A%8F%E6%9C%BA%E6%A2%AF%E5%BA%A6%E4%B8%8B%E9%99%8D&action=edit&redlink=1" class="new" title="随机梯度下降（页面不存在）">SGD</a>（英语：<a href="https://en.wikipedia.org/wiki/Stochastic_gradient_descent" class="extiw" title="en:Stochastic gradient descent">Stochastic gradient descent</a>）</li></ul></li> <li><a href="/wiki/%E8%81%9A%E7%B1%BB%E5%88%86%E6%9E%90" title="聚类分析">聚类</a></li> <li><a href="/wiki/%E8%BF%B4%E6%AD%B8%E5%88%86%E6%9E%90" title="迴歸分析">回归</a> <ul><li><a href="/wiki/%E9%81%8E%E9%81%A9" title="過適">过拟合</a></li></ul></li> <li><a href="/wiki/%E5%B9%BB%E8%A7%89_(%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD)" title="幻觉 (人工智能)">幻觉</a></li> <li><a href="/w/index.php?title=%E5%AF%B9%E6%8A%97%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0&action=edit&redlink=1" class="new" title="对抗机器学习（页面不存在）">对抗</a>（英语：<a href="https://en.wikipedia.org/wiki/Adversarial_machine_learning" class="extiw" title="en:Adversarial machine learning">Adversarial machine learning</a>）</li> <li><a href="/wiki/%E6%B3%A8%E6%84%8F%E5%8A%9B%E6%9C%BA%E5%88%B6" title="注意力机制">注意力</a></li> <li><a class="mw-selflink selflink">卷积</a></li> <li><a href="/wiki/%E5%88%86%E9%A1%9E%E5%95%8F%E9%A1%8C%E4%B9%8B%E6%90%8D%E5%A4%B1%E5%87%BD%E6%95%B8" title="分類問題之損失函數">損失函數</a></li> <li><a href="/wiki/%E5%8F%8D%E5%90%91%E4%BC%A0%E6%92%AD%E7%AE%97%E6%B3%95" title="反向传播算法">反向传播</a></li> <li><a href="/wiki/%E6%BF%80%E6%B4%BB%E5%87%BD%E6%95%B0" title="激活函数">激活函数</a> <ul><li><a href="/wiki/Softmax%E5%87%BD%E6%95%B0" title="Softmax函数">softmax</a></li> <li><a href="/wiki/S%E5%9E%8B%E5%87%BD%E6%95%B0" title="S型函数">sigmoid</a></li> <li><a href="/wiki/%E7%BA%BF%E6%80%A7%E6%95%B4%E6%B5%81%E5%87%BD%E6%95%B0" title="线性整流函数">ReLU</a></li></ul></li> <li><a href="/wiki/%E6%AD%A3%E5%88%99%E5%8C%96_(%E6%95%B0%E5%AD%A6)" title="正则化 (数学)">正则化</a></li> <li><a href="/wiki/%E8%AE%AD%E7%BB%83%E9%9B%86%E3%80%81%E9%AA%8C%E8%AF%81%E9%9B%86%E5%92%8C%E6%B5%8B%E8%AF%95%E9%9B%86" title="训练集、验证集和测试集">数据集</a></li> <li><a href="/w/index.php?title=%E6%89%A9%E6%95%A3%E8%BF%87%E7%A8%8B&action=edit&redlink=1" class="new" title="扩散过程（页面不存在）">扩散</a>（英语：<a href="https://en.wikipedia.org/wiki/Diffusion_process" class="extiw" title="en:Diffusion process">Diffusion process</a>）</li> <li><a href="/wiki/%E8%87%AA%E6%88%91%E8%BF%B4%E6%AD%B8%E6%A8%A1%E5%9E%8B" title="自我迴歸模型">自回归</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">应用</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0" title="机器学习">机器学习</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="人工神经网络">人工神经网络</a> <ul><li><a href="/wiki/%E6%B7%B1%E5%BA%A6%E5%AD%A6%E4%B9%A0" title="深度学习">深度学习</a></li></ul></li> <li><a href="/wiki/%E8%BF%90%E7%AE%97%E7%A7%91%E5%AD%A6" title="运算科学">科学计算</a></li> <li><a href="/wiki/%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD" title="人工智能">人工智能</a></li> <li><a href="/wiki/%E8%AA%9E%E8%A8%80%E6%A8%A1%E5%9E%8B" title="語言模型">語言模型</a> <ul><li><a href="/wiki/%E5%A4%A7%E5%9E%8B%E8%AF%AD%E8%A8%80%E6%A8%A1%E5%9E%8B" title="大型语言模型">大型语言模型</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">硬件</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%BC%A0%E9%87%8F%E5%A4%84%E7%90%86%E5%8D%95%E5%85%83" title="张量处理单元">TPU</a></li> <li><a href="/wiki/%E8%A7%86%E8%A7%89%E5%A4%84%E7%90%86%E5%8D%95%E5%85%83" title="视觉处理单元">VPU</a></li> <li><a href="/w/index.php?title=Graphcore&action=edit&redlink=1" class="new" title="Graphcore（页面不存在）">IPU</a>（英语：<a href="https://en.wikipedia.org/wiki/Graphcore" class="extiw" title="en:Graphcore">Graphcore</a>）</li> <li><a href="/wiki/%E6%86%B6%E9%98%BB%E5%99%A8" title="憶阻器">憶阻器</a></li> <li><a href="/w/index.php?title=SpiNNaker&action=edit&redlink=1" class="new" title="SpiNNaker（页面不存在）">SpiNNaker</a>（英语：<a href="https://en.wikipedia.org/wiki/SpiNNaker" class="extiw" title="en:SpiNNaker">SpiNNaker</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">软件库</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Theano" title="Theano">Theano</a></li> <li><a href="/wiki/TensorFlow" title="TensorFlow">TensorFlow</a> <ul><li><a href="/wiki/Keras" title="Keras">Keras</a></li></ul></li> <li><a href="/wiki/PyTorch" title="PyTorch">PyTorch</a></li> <li><a href="/wiki/Google_JAX" title="Google JAX">JAX</a></li> <li><a href="/w/index.php?title=Flux_(%E6%9C%BA%E5%99%A8%E5%AD%A6%E4%B9%A0%E6%A1%86%E6%9E%B6)&action=edit&redlink=1" class="new" title="Flux (机器学习框架)（页面不存在）">Flux.jl</a>（英语：<a href="https://en.wikipedia.org/wiki/Flux_(machine-learning_framework)" class="extiw" title="en:Flux (machine-learning framework)">Flux (machine-learning framework)</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">实现</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">视觉·语音</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/AlexNet" title="AlexNet">AlexNet</a></li> <li><a href="/wiki/WaveNet" title="WaveNet">WaveNet</a></li> <li><a href="/wiki/%E4%BA%BA%E9%AB%94%E5%9C%96%E5%83%8F%E5%90%88%E6%88%90" title="人體圖像合成">人像合成</a></li> <li><a href="/wiki/%E6%89%8B%E5%86%99%E8%AF%86%E5%88%AB" title="手写识别">手寫识别</a></li> <li><a href="/wiki/%E5%85%89%E5%AD%A6%E5%AD%97%E7%AC%A6%E8%AF%86%E5%88%AB" title="光学字符识别">OCR</a></li> <li><a href="/wiki/%E8%AF%AD%E9%9F%B3%E5%90%88%E6%88%90" title="语音合成">语音合成</a></li> <li><a href="/wiki/%E8%AF%AD%E9%9F%B3%E8%AF%86%E5%88%AB" title="语音识别">语音识别</a></li> <li><a href="/wiki/%E8%87%89%E9%83%A8%E8%BE%A8%E8%AD%98%E7%B3%BB%E7%B5%B1" title="臉部辨識系統">人脸识别</a></li> <li><a href="/wiki/AlphaFold" title="AlphaFold">AlphaFold</a></li> <li><a href="/wiki/DALL-E" title="DALL-E">DALL-E</a></li> <li><a href="/wiki/Midjourney" title="Midjourney">Midjourney</a></li> <li><a href="/wiki/Stable_Diffusion" title="Stable Diffusion">Stable Diffusion</a></li> <li><a href="/wiki/Sora_(%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD%E6%A8%A1%E5%9E%8B)" title="Sora (人工智能模型)">Sora</a></li> <li><a href="/w/index.php?title=Whisper_(%E8%AF%AD%E9%9F%B3%E8%AF%86%E5%88%AB%E7%B3%BB%E7%BB%9F)&action=edit&redlink=1" class="new" title="Whisper (语音识别系统)（页面不存在）">Whisper</a>（英语：<a href="https://en.wikipedia.org/wiki/Whisper_(speech_recognition_system)" class="extiw" title="en:Whisper (speech recognition system)">Whisper (speech recognition system)</a>）</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">自然语言</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Word2vec" title="Word2vec">Word2vec</a></li> <li><a href="/wiki/Seq2Seq%E6%A8%A1%E5%9E%8B" title="Seq2Seq模型">Seq2seq</a></li> <li><a href="/wiki/BERT" title="BERT">BERT</a></li> <li><a href="/wiki/%E5%B0%8D%E8%A9%B1%E7%A8%8B%E5%BC%8F%E8%AA%9E%E8%A8%80%E6%A8%A1%E5%9E%8B" title="對話程式語言模型">LaMDA</a> <ul><li><a href="/wiki/Bard" class="mw-disambig" title="Bard">Bard</a></li></ul></li> <li><a href="/wiki/%E7%A5%9E%E7%BB%8F%E6%9C%BA%E5%99%A8%E7%BF%BB%E8%AF%91" title="神经机器翻译">NMT</a></li> <li><a href="/w/index.php?title=%E8%BE%A9%E6%89%8B%E9%A1%B9%E7%9B%AE&action=edit&redlink=1" class="new" title="辩手项目（页面不存在）">辩手项目</a>（英语：<a href="https://en.wikipedia.org/wiki/Project_Debater" class="extiw" title="en:Project Debater">Project Debater</a>）</li> <li><a href="/wiki/%E6%B2%83%E6%A3%AE_(%E4%BA%BA%E5%B7%A5%E6%99%BA%E8%83%BD%E7%A8%8B%E5%BA%8F)" title="沃森 (人工智能程序)">沃森</a></li> <li><a href="/wiki/%E5%9F%BA%E4%BA%8E%E8%BD%AC%E6%8D%A2%E5%99%A8%E7%9A%84%E7%94%9F%E6%88%90%E5%BC%8F%E9%A2%84%E8%AE%AD%E7%BB%83%E6%A8%A1%E5%9E%8B" title="基于转换器的生成式预训练模型">GPT</a> <ul><li><a href="/wiki/GPT-1" title="GPT-1">GPT-1</a></li> <li><a href="/wiki/GPT-2" title="GPT-2">GPT-2</a></li> <li><a href="/wiki/GPT-3" title="GPT-3">GPT-3</a></li> <li><a href="/wiki/GPT-4" title="GPT-4">GPT-4</a></li></ul></li> <li><a href="/w/index.php?title=GPT-J&action=edit&redlink=1" class="new" title="GPT-J（页面不存在）">GPT-J</a>（英语：<a href="https://en.wikipedia.org/wiki/GPT-J" class="extiw" title="en:GPT-J">GPT-J</a>）</li> <li><a href="/wiki/ChatGPT" title="ChatGPT">ChatGPT</a></li> <li><a href="/wiki/%E6%96%87%E5%BF%83%E4%B8%80%E8%A8%80" title="文心一言">文心一言</a></li> <li><a href="/w/index.php?title=Chinchilla_AI&action=edit&redlink=1" class="new" title="Chinchilla AI（页面不存在）">Chinchilla AI</a>（英语：<a href="https://en.wikipedia.org/wiki/Chinchilla_AI" class="extiw" title="en:Chinchilla AI">Chinchilla AI</a>）</li> <li><a href="/w/index.php?title=PaLM&action=edit&redlink=1" class="new" title="PaLM（页面不存在）">PaLM</a>（英语：<a href="https://en.wikipedia.org/wiki/PaLM" class="extiw" title="en:PaLM">PaLM</a>）</li> <li><a href="/w/index.php?title=BLOOM_(%E8%AF%AD%E8%A8%80%E6%A8%A1%E5%9E%8B)&action=edit&redlink=1" class="new" title="BLOOM (语言模型)（页面不存在）">BLOOM</a>（英语：<a href="https://en.wikipedia.org/wiki/BLOOM_(language_model)" class="extiw" title="en:BLOOM (language model)">BLOOM (language model)</a>）</li> <li><a href="/wiki/LLaMA" title="LLaMA">LLaMA</a></li> <li><a href="/wiki/TAIDE" title="TAIDE">TAIDE</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">决策</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/AlphaGo" title="AlphaGo">AlphaGo</a></li> <li><a href="/wiki/Q%E5%AD%A6%E4%B9%A0" title="Q学习">Q学习</a></li> <li><a href="/wiki/SARSA%E7%AE%97%E6%B3%95" title="SARSA算法">SARSA</a></li> <li><a href="/w/index.php?title=OpenAI_Five&action=edit&redlink=1" class="new" title="OpenAI Five（页面不存在）">OpenAI Five</a>（英语：<a href="https://en.wikipedia.org/wiki/OpenAI_Five" class="extiw" title="en:OpenAI Five">OpenAI Five</a>）</li> <li><a href="/wiki/%E8%87%AA%E5%8B%95%E9%A7%95%E9%A7%9B%E6%B1%BD%E8%BB%8A" title="自動駕駛汽車">自动驾驶</a></li> <li><a href="/wiki/MuZero" title="MuZero">MuZero</a></li> <li><a href="/w/index.php?title=%E8%A1%8C%E5%8A%A8%E9%80%89%E6%8B%A9&action=edit&redlink=1" class="new" title="行动选择（页面不存在）">行动选择</a>（英语：<a href="https://en.wikipedia.org/wiki/Action_selection" class="extiw" title="en:Action selection">Action selection</a>） <ul><li><a href="/wiki/Auto-GPT" title="Auto-GPT">Auto-GPT</a></li></ul></li> <li><a href="/w/index.php?title=%E6%9C%BA%E5%99%A8%E4%BA%BA%E6%8E%A7%E5%88%B6&action=edit&redlink=1" class="new" title="机器人控制（页面不存在）">机器人控制</a>（英语：<a href="https://en.wikipedia.org/wiki/Robot_control" class="extiw" title="en:Robot control">Robot control</a>）</li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">人物</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E7%BA%A6%E4%B9%A6%E4%BA%9A%C2%B7%E6%9C%AC%E5%B8%8C%E5%A5%A5" title="约书亚·本希奥">约书亚·本希奥</a></li> <li><a href="/wiki/%E6%9D%B0%E5%BC%97%E9%87%8C%C2%B7%E8%BE%9B%E9%A1%BF" title="杰弗里·辛顿">杰弗里·辛顿</a></li> <li><a href="/wiki/%E6%9D%A8%E7%AB%8B%E6%98%86" title="杨立昆">杨立昆</a></li> <li><a href="/wiki/%E8%89%BE%E5%8A%9B%E5%85%8B%E6%96%AF%C2%B7%E6%A0%BC%E9%9B%B7%E5%A4%AB%E6%96%AF_(%E8%A8%88%E7%AE%97%E6%A9%9F%E7%A7%91%E5%AD%B8%E5%AE%B6)" title="艾力克斯·格雷夫斯 (計算機科學家)">艾力克斯·格雷夫斯</a></li> <li><a href="/wiki/%E4%BC%8A%E6%81%A9%C2%B7%E5%8F%A4%E5%BE%B7%E8%B4%B9%E6%B4%9B" title="伊恩·古德费洛">伊恩·古德费洛</a></li> <li><a href="/wiki/%E5%90%B4%E6%81%A9%E8%BE%BE" title="吴恩达">吴恩达</a></li> <li><a href="/wiki/%E6%9D%B0%E7%B1%B3%E6%96%AF%C2%B7%E5%93%88%E8%90%A8%E6%AF%94%E6%96%AF" title="杰米斯·哈萨比斯">杰米斯·哈萨比斯</a></li> <li><a href="/wiki/%E5%A4%A7%E8%A1%9B%C2%B7%E5%B8%AD%E7%88%BE%E7%93%A6_(%E8%A8%88%E7%AE%97%E6%A9%9F%E7%A7%91%E5%AD%B8%E5%AE%B6)" title="大衛·席爾瓦 (計算機科學家)">大衛·席爾瓦</a></li> <li><a href="/wiki/%E6%9D%8E%E9%A3%9B%E9%A3%9B" title="李飛飛">李飛飛</a></li> <li><a href="/wiki/%E4%BA%8E%E5%B0%94%E6%A0%B9%C2%B7%E6%96%BD%E5%AF%86%E5%BE%B7%E8%83%A1%E4%BC%AF" title="于尔根·施密德胡伯">于尔根·施密德胡伯</a></li> <li><a href="/wiki/%E4%BC%8A%E7%88%BE%E4%BA%9E%C2%B7%E8%98%87%E8%8C%A8%E5%85%8B%E7%B6%AD" title="伊爾亞·蘇茨克維">伊爾亞·蘇茨克維</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">组织</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/Anthropic" title="Anthropic">Anthropic</a></li> <li><a href="/wiki/DeepMind" class="mw-redirect" title="DeepMind">DeepMind</a></li> <li><a href="/w/index.php?title=EleutherAI&action=edit&redlink=1" class="new" title="EleutherAI（页面不存在）">EleutherAI</a>（英语：<a href="https://en.wikipedia.org/wiki/EleutherAI" class="extiw" title="en:EleutherAI">EleutherAI</a>）</li> <li><a href="/wiki/%E8%B0%B7%E6%AD%8C%E5%A4%A7%E8%84%91" title="谷歌大脑">Google Brain</a></li> <li><a href="/w/index.php?title=Meta_AI&action=edit&redlink=1" class="new" title="Meta AI（页面不存在）">Meta AI</a>（英语：<a href="https://en.wikipedia.org/wiki/Meta_AI" class="extiw" title="en:Meta AI">Meta AI</a>）</li> <li><a href="/w/index.php?title=Mila_(%E7%A0%94%E7%A9%B6%E6%89%80)&action=edit&redlink=1" class="new" title="Mila (研究所)（页面不存在）">Mila</a>（英语：<a href="https://en.wikipedia.org/wiki/Mila_(research_institute)" class="extiw" title="en:Mila (research institute)">Mila (research institute)</a>）</li> <li><a href="/wiki/MIT%E8%A8%88%E7%AE%97%E6%A9%9F%E7%A7%91%E5%AD%B8%E8%88%87%E4%BA%BA%E5%B7%A5%E6%99%BA%E6%85%A7%E5%AF%A6%E9%A9%97%E5%AE%A4" title="MIT計算機科學與人工智慧實驗室">MIT CSAIL</a></li> <li><a href="/wiki/OpenAI" title="OpenAI">OpenAI</a></li> <li><a href="/wiki/Hugging_Face" title="Hugging Face">Hugging Face</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">架构</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0px"><div style="padding:0em 0.25em"> <ul><li><a href="/wiki/%E5%A4%9A%E5%B1%82%E6%84%9F%E7%9F%A5%E5%99%A8" title="多层感知器">多层感知器</a>（MLP）</li> <li><a href="/wiki/%E5%BE%AA%E7%8E%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="循环神经网络">循环神经网络</a>（RNN）</li> <li><a href="/wiki/%E9%95%B7%E7%9F%AD%E6%9C%9F%E8%A8%98%E6%86%B6" title="長短期記憶">長短期記憶</a>（LSTM）</li> <li><a href="/w/index.php?title=%E9%97%A8%E6%8E%A7%E5%BE%AA%E7%8E%AF%E5%8D%95%E5%85%83&action=edit&redlink=1" class="new" title="门控循环单元（页面不存在）">门控循环单元</a>（英语：<a href="https://en.wikipedia.org/wiki/Gated_recurrent_unit" class="extiw" title="en:Gated recurrent unit">Gated recurrent unit</a>）（GRU）</li> <li><a href="/wiki/%E5%8D%B7%E7%A7%AF%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="卷积神经网络">卷积神经网络</a>（CNN）</li> <li><a href="/wiki/%E6%AE%8B%E5%B7%AE%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C" title="残差神经网络">残差神经网络</a>（ResNet）</li> <li><a href="/wiki/Transformer%E6%A8%A1%E5%9E%8B" title="Transformer模型">变换器</a></li> <li><a href="/wiki/%E8%87%AA%E7%BC%96%E7%A0%81%E5%99%A8" title="自编码器">自编码器</a></li> <li><a href="/wiki/%E5%8F%98%E5%88%86%E8%87%AA%E7%BC%96%E7%A0%81%E5%99%A8" title="变分自编码器">变分自编码器</a>（VAE）</li> <li><a href="/wiki/%E7%94%9F%E6%88%90%E5%AF%B9%E6%8A%97%E7%BD%91%E7%BB%9C" title="生成对抗网络">生成对抗网络</a>（GAN）</li> <li><a href="/w/index.php?title=%E5%9B%BE%E7%A5%9E%E7%BB%8F%E7%BD%91%E7%BB%9C&action=edit&redlink=1" class="new" title="图神经网络（页面不存在）">图神经网络</a>（英语：<a href="https://en.wikipedia.org/wiki/Graph_neural_network" class="extiw" title="en:Graph neural network">Graph neural network</a>）（GNN）</li> <li><a href="/w/index.php?title=%E5%9B%9E%E5%93%8D%E7%8A%B6%E6%80%81%E7%BD%91%E7%BB%9C&action=edit&redlink=1" class="new" title="回响状态网络（页面不存在）">回响状态网络</a>（英语：<a href="https://en.wikipedia.org/wiki/Echo_state_network" class="extiw" title="en:Echo state network">Echo state network</a>）（ESN）</li> <li><a href="/wiki/%E7%A5%9E%E7%BB%8F%E5%9B%BE%E7%81%B5%E6%9C%BA" title="神经图灵机">神经图灵机</a>（NTM）</li> <li><a href="/w/index.php?title=%E5%8F%AF%E5%BE%AE%E5%88%86%E7%A5%9E%E7%BB%8F%E8%AE%A1%E7%AE%97%E6%9C%BA&action=edit&redlink=1" class="new" title="可微分神经计算机（页面不存在）">可微分神经计算机</a>（英语：<a href="https://en.wikipedia.org/wiki/Differentiable_neural_computer" class="extiw" title="en:Differentiable neural computer">Differentiable neural computer</a>）（DNC）</li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><img alt="主题" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/16px-Symbol_portal_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/23px-Symbol_portal_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Symbol_portal_class.svg/31px-Symbol_portal_class.svg.png 2x" data-file-width="180" data-file-height="185" /> 主题 <ul><li><a href="/wiki/Portal:%E9%9B%BB%E8%85%A6%E7%A8%8B%E5%BC%8F%E8%A8%AD%E8%A8%88" 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</div><script>(RLQ=window.RLQ||[]).push(function(){mw.config.set({"wgHostname":"mw-web.codfw.main-f69cdc8f6-ckrcl","wgBackendResponseTime":1701,"wgPageParseReport":{"limitreport":{"cputime":"1.124","walltime":"1.503","ppvisitednodes":{"value":6272,"limit":1000000},"postexpandincludesize":{"value":233451,"limit":2097152},"templateargumentsize":{"value":2479,"limit":2097152},"expansiondepth":{"value":11,"limit":100},"expensivefunctioncount":{"value":39,"limit":500},"unstrip-depth":{"value":0,"limit":20},"unstrip-size":{"value":63791,"limit":5000000},"entityaccesscount":{"value":1,"limit":400},"timingprofile":["100.00% 698.875 1 -total"," 25.85% 180.655 2 Template:Navbox"," 23.80% 166.345 1 Template:Reflist"," 22.83% 159.532 1 Template:Differentiable_computing"," 15.23% 106.424 22 Template:Citation"," 9.91% 69.263 7 Template:Cite_book"," 8.85% 61.856 1 Template:Commons_category"," 8.71% 60.894 25 Template:Le"," 8.64% 60.356 1 Template:Sister_project"," 8.63% 60.288 1 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[\"CITEREFSmith1997\"] = 1,\n [\"CITEREFSteinWeiss1971\"] = 1,\n [\"CITEREFStrichartz1994\"] = 1,\n [\"CITEREFTitchmarsh1948\"] = 1,\n [\"CITEREFUdayashankara2010\"] = 1,\n [\"CITEREFUludag1998\"] = 1,\n [\"CITEREFWeisstein\"] = 2,\n [\"CITEREFZhangSoonYeFuh2020\"] = 1,\n [\"CITEREFvon_zur_GathenGerhard2003\"] = 1,\n [\"McGillem\"] = 1,\n [\"Oppenheim\"] = 1,\n}\ntemplate_list = table#1 {\n [\"Citation\"] = 21,\n [\"Cite book\"] = 7,\n [\"Cite journal\"] = 3,\n [\"Cite web\"] = 2,\n [\"Commons category\"] = 1,\n [\"DFT_{L\"] = 3,\n [\"DFT_{L+M-1\"] = 2,\n [\"Differentiable computing\"] = 1,\n [\"En-link\"] = 11,\n [\"Harv\"] = 1,\n [\"Harvnb\"] = 1,\n [\"Main\"] = 3,\n [\"Main article\"] = 1,\n [\"Multiple image\"] = 1,\n [\"Narici Beckenstein Topological Vector Spaces\"] = 1,\n [\"NoteTA\"] = 1,\n [\"Reflist\"] = 1,\n [\"Schaefer Wolff Topological Vector Spaces\"] = 1,\n [\"Springer\"] = 1,\n [\"Trèves François Topological vector spaces, distributions and kernels\"] = 1,\n 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