巴特沃斯滤波器 - 维基百科，自由的百科全书

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class="vector-icon mw-ui-icon-ellipsis mw-ui-icon-wikimedia-ellipsis"></span> <span class="vector-dropdown-label-text">个人工具</span> </label> <div class="vector-dropdown-content"> <div id="p-personal" class="vector-menu mw-portlet mw-portlet-personal user-links-collapsible-item" title="用户菜单" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="pt-sitesupport" class="user-links-collapsible-item mw-list-item"><a href="https://donate.wikimedia.org/?wmf_source=donate&wmf_medium=sidebar&wmf_campaign=zh.wikipedia.org&uselang=zh"><span>资助维基百科</span></a></li><li id="pt-createaccount" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E5%88%9B%E5%BB%BA%E8%B4%A6%E6%88%B7&returnto=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8" title="我们推荐您创建账号并登录，但这不是强制性的"><span class="vector-icon mw-ui-icon-userAdd mw-ui-icon-wikimedia-userAdd"></span> <span>创建账号</span></a></li><li id="pt-login" class="user-links-collapsible-item mw-list-item"><a href="/w/index.php?title=Special:%E7%94%A8%E6%88%B7%E7%99%BB%E5%BD%95&returnto=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8" title="建议你登录，尽管并非必须。[o]" accesskey="o"><span class="vector-icon mw-ui-icon-logIn mw-ui-icon-wikimedia-logIn"></span> <span>登录</span></a></li> </ul> </div> </div> <div id="p-user-menu-anon-editor" class="vector-menu mw-portlet mw-portlet-user-menu-anon-editor" > <div class="vector-menu-heading"> 未登录编辑者的页面 <a href="/wiki/Help:%E6%96%B0%E6%89%8B%E5%85%A5%E9%97%A8" aria-label="了解有关编辑的更多信息"><span>了解详情</span></a> </div> <div class="vector-menu-content"> <ul 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"><span>贡献</span></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"><span>讨论</span></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-巴特沃斯滤波器的特性" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#巴特沃斯滤波器的特性"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>巴特沃斯滤波器的特性</span> </div> </a> <ul id="toc-巴特沃斯滤波器的特性-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-传递函数" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#传递函数"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>传递函数</span> </div> </a> <button aria-controls="toc-传递函数-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> <span class="vector-icon mw-ui-icon-wikimedia-expand"></span> <span>开关传递函数子章节</span> </button> <ul id="toc-传递函数-sublist" class="vector-toc-list"> <li id="toc-根据衰减度求滤波器的阶数" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#根据衰减度求滤波器的阶数"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>根据衰减度求滤波器的阶数</span> </div> </a> <ul id="toc-根据衰减度求滤波器的阶数-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-幅度最平坦的滤波器" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#幅度最平坦的滤波器"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>幅度最平坦的滤波器</span> </div> </a> <ul id="toc-幅度最平坦的滤波器-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-高频衰减" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#高频衰减"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>高频衰减</span> </div> </a> <ul id="toc-高频衰减-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-实例" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#实例"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>实例</span> </div> </a> <ul id="toc-实例-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-归一化的巴特沃斯多项式" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#归一化的巴特沃斯多项式"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>归一化的巴特沃斯多项式</span> </div> </a> <ul id="toc-归一化的巴特沃斯多项式-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-与其他类型滤波器的比较" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#与其他类型滤波器的比较"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>与其他类型滤波器的比较</span> </div> </a> <ul id="toc-与其他类型滤波器的比较-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参见" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参见"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>参见</span> </div> </a> <ul id="toc-参见-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-参考文献" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#参考文献"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>参考文献</span> </div> </a> <ul id="toc-参考文献-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" title="目录" > <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" ><span class="vector-icon mw-ui-icon-listBullet mw-ui-icon-wikimedia-listBullet"></span> <span class="vector-dropdown-label-text">开关目录</span> </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"><span class="mw-page-title-main">巴特沃斯滤波器</span></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。20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-20" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">20种语言</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D8%B1%D8%B4%D8%AD_%D8%A8%D8%A7%D8%AA%D8%B1%D9%88%D9%88%D8%B1%D8%AB" title="مرشح باتروورث – 阿拉伯语" lang="ar" hreflang="ar" data-title="مرشح باتروورث" data-language-autonym="العربية" data-language-local-name="阿拉伯语" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Filtre_de_Butterworth" title="Filtre de Butterworth – 加泰罗尼亚语" lang="ca" hreflang="ca" data-title="Filtre de Butterworth" data-language-autonym="Català" data-language-local-name="加泰罗尼亚语" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Butterworth%C5%AFv_filtr" title="Butterworthův filtr – 捷克语" lang="cs" hreflang="cs" data-title="Butterworthův filtr" data-language-autonym="Čeština" data-language-local-name="捷克语" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Butterworth-Filter" title="Butterworth-Filter – 德语" lang="de" hreflang="de" data-title="Butterworth-Filter" data-language-autonym="Deutsch" data-language-local-name="德语" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-en mw-list-item"><a href="https://en.wikipedia.org/wiki/Butterworth_filter" title="Butterworth filter – 英语" lang="en" hreflang="en" data-title="Butterworth filter" data-language-autonym="English" data-language-local-name="英语" class="interlanguage-link-target"><span>English</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Filtrilo_de_Butterworth" title="Filtrilo de Butterworth – 世界语" lang="eo" hreflang="eo" data-title="Filtrilo de Butterworth" data-language-autonym="Esperanto" data-language-local-name="世界语" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Filtro_de_Butterworth" title="Filtro de Butterworth – 西班牙语" lang="es" hreflang="es" data-title="Filtro de Butterworth" data-language-autonym="Español" data-language-local-name="西班牙语" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Butterworthi_filter" title="Butterworthi filter – 爱沙尼亚语" lang="et" hreflang="et" data-title="Butterworthi filter" data-language-autonym="Eesti" data-language-local-name="爱沙尼亚语" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-eu mw-list-item"><a href="https://eu.wikipedia.org/wiki/Butterworth_iragazki" title="Butterworth iragazki – 巴斯克语" lang="eu" hreflang="eu" data-title="Butterworth iragazki" data-language-autonym="Euskara" data-language-local-name="巴斯克语" class="interlanguage-link-target"><span>Euskara</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Filtre_de_Butterworth" title="Filtre de Butterworth – 法语" lang="fr" hreflang="fr" data-title="Filtre de Butterworth" data-language-autonym="Français" data-language-local-name="法语" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Filtro_Butterworth" title="Filtro Butterworth – 意大利语" lang="it" hreflang="it" data-title="Filtro Butterworth" data-language-autonym="Italiano" data-language-local-name="意大利语" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%83%90%E3%82%BF%E3%83%BC%E3%83%AF%E3%83%BC%E3%82%B9%E3%83%95%E3%82%A3%E3%83%AB%E3%82%BF" title="バターワースフィルタ – 日语" lang="ja" hreflang="ja" data-title="バターワースフィルタ" data-language-autonym="日本語" data-language-local-name="日语" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Butterworth-filterprototype" title="Butterworth-filterprototype – 荷兰语" lang="nl" hreflang="nl" data-title="Butterworth-filterprototype" data-language-autonym="Nederlands" data-language-local-name="荷兰语" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Filtr_Butterwortha" title="Filtr Butterwortha – 波兰语" lang="pl" hreflang="pl" data-title="Filtr Butterwortha" data-language-autonym="Polski" data-language-local-name="波兰语" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Filtro_Butterworth" title="Filtro Butterworth – 葡萄牙语" lang="pt" hreflang="pt" data-title="Filtro Butterworth" data-language-autonym="Português" data-language-local-name="葡萄牙语" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A4%D0%B8%D0%BB%D1%8C%D1%82%D1%80_%D0%91%D0%B0%D1%82%D1%82%D0%B5%D1%80%D0%B2%D0%BE%D1%80%D1%82%D0%B0" title="Фильтр Баттерворта – 俄语" lang="ru" hreflang="ru" data-title="Фильтр Баттерворта" data-language-autonym="Русский" data-language-local-name="俄语" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Batervortov_filter" title="Batervortov filter – 塞尔维亚语" lang="sr" hreflang="sr" data-title="Batervortov filter" data-language-autonym="Српски / srpski" data-language-local-name="塞尔维亚语" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Butterworthfilter" title="Butterworthfilter – 瑞典语" lang="sv" hreflang="sv" data-title="Butterworthfilter" data-language-autonym="Svenska" data-language-local-name="瑞典语" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Butterworth_filtre" title="Butterworth filtre – 土耳其语" lang="tr" hreflang="tr" data-title="Butterworth filtre" data-language-autonym="Türkçe" data-language-local-name="土耳其语" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A4%D1%96%D0%BB%D1%8C%D1%82%D1%80_%D0%91%D0%B0%D1%82%D1%82%D0%B5%D1%80%D0%B2%D0%BE%D1%80%D1%82%D0%B0" title="Фільтр Баттерворта – 乌克兰语" lang="uk" hreflang="uk" data-title="Фільтр Баттерворта" data-language-autonym="Українська" data-language-local-name="乌克兰语" class="interlanguage-link-target"><span>Українська</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q1018162#sitelinks-wikipedia" title="编辑跨语言链接" class="wbc-editpage">编辑链接</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="命名空间"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" > <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li id="ca-nstab-main" class="selected vector-tab-noicon mw-list-item"><a href="/wiki/%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8" title="浏览条目正文[c]" accesskey="c"><span>条目</span></a></li><li id="ca-talk" class="vector-tab-noicon mw-list-item"><a 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class="noprint">维基百科，自由的百科全书</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="zh" dir="ltr"><div id="noteTA-5341a9fa" class="noteTA"><div class="noteTA-group"><div data-noteta-group-source="module" data-noteta-group="Communication"></div><div data-noteta-group-source="module" data-noteta-group="Electronics"></div></div></div> <p><b>巴特沃斯滤波器</b>（英語：<span lang="en">Butterworth filter</span>）是一种<span class="ilh-all" data-orig-title="通频带" data-lang-code="en" data-lang-name="英语" data-foreign-title="passband"><span class="ilh-page"><a href="/w/index.php?title=%E9%80%9A%E9%A2%91%E5%B8%A6&action=edit&redlink=1" class="new" title="通频带（页面不存在）">通频带</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/passband" class="extiw" title="en:passband"><span lang="en" dir="auto">passband</span></a></span><span class="ilh-paren">）</span></span></span>之<a href="/wiki/%E9%A2%91%E7%8E%87%E5%93%8D%E5%BA%94" title="频率响应">频率响应</a>曲线平坦無漣波的<span class="ilh-all" data-orig-title="滤波器 (信号处理)" data-lang-code="en" data-lang-name="英语" data-foreign-title="Filter (signal processing)"><span class="ilh-page"><a href="/w/index.php?title=%E6%BB%A4%E6%B3%A2%E5%99%A8_(%E4%BF%A1%E5%8F%B7%E5%A4%84%E7%90%86)&action=edit&redlink=1" class="new" title="滤波器 (信号处理)（页面不存在）">信号处理滤波器</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/Filter_(signal_processing)" class="extiw" title="en:Filter (signal processing)"><span lang="en" dir="auto">Filter (signal processing)</span></a></span><span class="ilh-paren">）</span></span></span>。它也被称作<b>最大平坦滤波器</b>。这种滤波器最先由<a href="/wiki/%E8%8B%B1%E5%9B%BD" title="英国">英国</a><a href="/wiki/%E5%B7%A5%E7%A8%8B%E5%B8%88" title="工程师">工程师</a>、<a href="/wiki/%E7%89%A9%E7%90%86%E5%AD%A6%E5%AE%B6" title="物理学家">物理学家</a><a href="/wiki/%E5%8F%B2%E8%92%82%E8%8A%AC%C2%B7%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF" title="史蒂芬·巴特沃斯">史蒂芬·巴特沃斯</a>在1930年发表的论文《滤波器放大器理论研究》中提出的。<sup id="cite_ref-1930paper_1-0" class="reference"><a href="#cite_note-1930paper-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="巴特沃斯滤波器的特性"><span id=".E5.B7.B4.E7.89.B9.E6.B2.83.E6.96.AF.E6.BB.A4.E6.B3.A2.E5.99.A8.E7.9A.84.E7.89.B9.E6.80.A7"></span>巴特沃斯滤波器的特性</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=1" title="编辑章节：巴特沃斯滤波器的特性"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Butterworth_filter_bode_plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Butterworth_filter_bode_plot.png/350px-Butterworth_filter_bode_plot.png" decoding="async" width="350" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/66/Butterworth_filter_bode_plot.png/525px-Butterworth_filter_bode_plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/66/Butterworth_filter_bode_plot.png/700px-Butterworth_filter_bode_plot.png 2x" data-file-width="1250" data-file-height="875" /></a><figcaption>一阶巴特沃斯低通滤波器的<a href="/wiki/%E6%B3%A2%E7%89%B9%E5%9B%BE" class="mw-redirect" title="波特图">波特图</a></figcaption></figure> <p>巴特沃斯滤波器的特点是<span class="ilh-all" data-orig-title="通频带" data-lang-code="en" data-lang-name="英语" data-foreign-title="passband"><span class="ilh-page"><a href="/w/index.php?title=%E9%80%9A%E9%A2%91%E5%B8%A6&action=edit&redlink=1" class="new" title="通频带（页面不存在）">通频带</a></span><span class="noprint ilh-comment"><span class="ilh-paren">（</span><span class="ilh-lang">英语</span><span class="ilh-colon">：</span><span class="ilh-link"><a href="https://en.wikipedia.org/wiki/passband" class="extiw" title="en:passband"><span lang="en" dir="auto">passband</span></a></span><span class="ilh-paren">）</span></span></span>内的<a href="/wiki/%E9%A2%91%E7%8E%87%E5%93%8D%E5%BA%94" title="频率响应">频率响应</a>曲线最大限度平坦，没有<a href="/wiki/%E6%BC%A3%E6%B3%A2" title="漣波">漣波</a>，而在阻频带则逐渐下降为零。<sup id="cite_ref-bianchi_2-0" class="reference"><a href="#cite_note-bianchi-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> 在对数<a href="/wiki/%E6%B3%A2%E7%89%B9%E5%9B%BE" class="mw-redirect" title="波特图">波特图</a>上，从某一边界角频率开始，振幅随着角频率的增加而线性减少至负无穷。 </p><p><style data-mw-deduplicate="TemplateStyles:r83946278">.mw-parser-output .template-facttext{background-color:var(--background-color-neutral,#eaecf0);color:inherit;margin:-.3em 0;padding:.3em 0}</style><mark class="template-facttext" title="需要提供文献来源">一阶巴特沃斯滤波器的衰减率为每<a href="/wiki/%E7%B4%94%E5%85%AB%E5%BA%A6" class="mw-redirect" title="純八度">倍频</a>6 dB，每十倍频20 dB（所有一阶低通滤波器具有相同的归一化频率响应）。</mark><sup class="noprint Template-Fact"><a href="/wiki/Wikipedia:%E5%88%97%E6%98%8E%E6%9D%A5%E6%BA%90" title="Wikipedia:列明来源"><span style="white-space: nowrap;" title="来源请求开始于2022年1月23日。">[來源請求]</span></a></sup>二阶巴特沃斯滤波器的衰减率为每倍频12 dB、三阶巴特沃斯滤波器的衰减率为每倍频18 dB、如此类推。巴特沃斯滤波器的振幅是 ω 的一个单调函数，并且也是唯一的无论阶数，振幅对角频率曲线都保持同样的形状的滤波器。只不过滤波器阶数越高，在阻频带振幅衰减速度越快。其他滤波器高阶的振幅对角频率图和低级数的振幅对角频率有不同的形状。 </p> <div class="mw-heading mw-heading2"><h2 id="传递函数"><span id=".E4.BC.A0.E9.80.92.E5.87.BD.E6.95.B0"></span>传递函数</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=2" title="编辑章节：传递函数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Butterworth_Filter_Orders.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Butterworth_Filter_Orders.svg/350px-Butterworth_Filter_Orders.svg.png" decoding="async" width="350" height="247" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Butterworth_Filter_Orders.svg/525px-Butterworth_Filter_Orders.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cd/Butterworth_Filter_Orders.svg/700px-Butterworth_Filter_Orders.svg.png 2x" data-file-width="850" data-file-height="600" /></a><figcaption>一阶至五阶巴特沃斯低通滤波器增益图，<a href="/wiki/%E6%88%AA%E6%AD%A2%E9%A0%BB%E7%8E%87" title="截止頻率">截止频率</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee04380d93e624497ceace7e45108a63b1cd2cd3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.761ex; height:2.509ex;" alt="{\displaystyle \omega _{0}=1}"></span>。注意到斜率是 20<i>n</i> dB/decade，其中 <i>n</i> 为滤波器阶数。</figcaption></figure> <p><i>n</i> 阶巴特沃斯低通滤波器的增益 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2675b545a63f2b8124f1088e17139dfc0552bf68" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.082ex; height:2.843ex;" alt="{\displaystyle G(\omega )}"></span> 为： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G^{2}(\omega )=\left|H(j\omega )\right|^{2}={\frac {{G_{0}}^{2}}{1+\left({\frac {\omega }{\omega _{c}}}\right)^{2n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msup> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G^{2}(\omega )=\left|H(j\omega )\right|^{2}={\frac {{G_{0}}^{2}}{1+\left({\frac {\omega }{\omega _{c}}}\right)^{2n}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e295e81aaa062776b0cd6023bb91ca06106d9523" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.005ex; width:33.213ex; height:9.009ex;" alt="{\displaystyle G^{2}(\omega )=\left|H(j\omega )\right|^{2}={\frac {{G_{0}}^{2}}{1+\left({\frac {\omega }{\omega _{c}}}\right)^{2n}}}}"></span></dd></dl> <p>其中, </p> <ul><li>n = 滤波器的阶数</li> <li>ω<sub>c</sub> =截止频率 =功率下降为 -3分贝时的频率</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccc7daa92085410e0eaef658bb5b07173816962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{0}}"></span> 是直流增益（零频率增益）</li></ul> <p>可以看出随着 <i>n</i> 趋近于无穷，增益变为一个矩形函数，频率低于 ω<sub>c</sub> 的会以 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ccc7daa92085410e0eaef658bb5b07173816962" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.881ex; height:2.509ex;" alt="{\displaystyle G_{0}}"></span> 的增益通过，而频率高于 ω<sub>c</sub> 的就会被抑制。对于较小的 <i>n</i> 值，截止就会变得不十分尖锐。 </p><p>我们希望能够（通过<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>）确定传递函数 <i>H(s)</i>，其中 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=\sigma +j\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>j</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\sigma +j\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b9a99cfe16a2003294604462f41acc7863fe66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.763ex; height:2.509ex;" alt="{\displaystyle s=\sigma +j\omega }"></span>。根据 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left|H(s)\right|^{2}=H(s){\overline {H(s)}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|H(s)\right|^{2}=H(s){\overline {H(s)}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f830df75745673a6b8ee6cff26eb7bab931e5a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.451ex; height:3.676ex;" alt="{\displaystyle \left|H(s)\right|^{2}=H(s){\overline {H(s)}}}"></span>，及拉普拉斯变换在虚轴 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=j\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>j</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=j\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114e5dcfb07325a06ed4329e3690d8225eadf422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.509ex;" alt="{\displaystyle s=j\omega }"></span> 上的性质 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(-j\omega )={\overline {H(j\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(-j\omega )={\overline {H(j\omega )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf46981fe7d173571bd2f570f582f217b1117ea5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.575ex; height:3.676ex;" alt="{\displaystyle H(-j\omega )={\overline {H(j\omega )}}}"></span>，若选取 H(s) 满足： </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)H(-s)={\frac {{G_{0}}^{2}}{1+\left({\frac {-s^{2}}{\omega _{c}^{2}}}\right)^{n}}},}"> <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> <mi>H</mi> <mo stretchy="false">(</mo> <mo>−</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)H(-s)={\frac {{G_{0}}^{2}}{1+\left({\frac {-s^{2}}{\omega _{c}^{2}}}\right)^{n}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99cfaf330787b6a62cff9ed6d71f3269c2184d71" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:28.03ex; height:8.676ex;" alt="{\displaystyle H(s)H(-s)={\frac {{G_{0}}^{2}}{1+\left({\frac {-s^{2}}{\omega _{c}^{2}}}\right)^{n}}},}"></span></dd></dl> <p>则对于虚数输入 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s=j\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>j</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=j\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/114e5dcfb07325a06ed4329e3690d8225eadf422" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.593ex; height:2.509ex;" alt="{\displaystyle s=j\omega }"></span>，我们就有了巴特沃斯滤波器的频率响应。 </p><p>上述表达式的 <i>n</i> 个<a href="/wiki/%E6%9E%81%E7%82%B9" class="mw-disambig" title="极点">极点</a>等距离地分布在<a href="/wiki/%E5%8D%8A%E5%BE%84" title="半径">半径</a>为 ω<sub>c</sub> 的<a href="/wiki/%E5%9C%86" title="圆">圆</a>上，并关于虚轴对称。为了具有稳定性，传递函数 H(s) 要选择只包含 <i>s</i> 负实半平面的极点。第 <i>k</i> 个极点为 </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\frac {s_{k}^{2}}{\omega _{c}^{2}}}=(-1)^{\frac {1}{n}}=e^{\frac {j(2k-1)\pi }{n}}\qquad k=1,2,3,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msubsup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> </mfrac> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>n</mi> </mfrac> </mrow> </msup> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>π</mi> </mrow> <mi>n</mi> </mfrac> </mrow> </msup> <mspace width="2em" /> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {s_{k}^{2}}{\omega _{c}^{2}}}=(-1)^{\frac {1}{n}}=e^{\frac {j(2k-1)\pi }{n}}\qquad k=1,2,3,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3f84317881fe09989c3af35fa09d60a20962c1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:46.646ex; height:6.676ex;" alt="{\displaystyle -{\frac {s_{k}^{2}}{\omega _{c}^{2}}}=(-1)^{\frac {1}{n}}=e^{\frac {j(2k-1)\pi }{n}}\qquad k=1,2,3,\ldots ,n}"></span></dd></dl> <p>因此， </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s_{k}=\omega _{c}e^{\frac {j(2k+n-1)\pi }{2n}}\qquad k=1,2,3,\ldots ,n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>c</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>j</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>+</mo> <mi>n</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>π</mi> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> </msup> <mspace width="2em" /> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s_{k}=\omega _{c}e^{\frac {j(2k+n-1)\pi }{2n}}\qquad k=1,2,3,\ldots ,n.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d99536a865c8971df200bf729c74eae9c320c22" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:38.809ex; height:4.343ex;" alt="{\displaystyle s_{k}=\omega _{c}e^{\frac {j(2k+n-1)\pi }{2n}}\qquad k=1,2,3,\ldots ,n.}"></span></dd></dl> <p><i>n</i>阶巴特沃斯低通滤波器的振幅和频率关系可用如下的公式表示： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+(\omega /\omega _{\mathrm {c} })^{2n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+(\omega /\omega _{\mathrm {c} })^{2n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf7f6435869f41fbd13edbd9dcd9d90ab3a4ae3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:37.183ex; height:6.509ex;" alt="{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+(\omega /\omega _{\mathrm {c} })^{2n}}}}}"></span> </p><p>其中： </p> <ul><li><i>G</i> 表示滤波器的放大率，</li> <li><i>H</i> 表示 <a href="/wiki/%E4%BC%A0%E9%80%92%E5%87%BD%E6%95%B0" title="传递函数">傳遞函數</a>，</li> <li><i>j</i> 是 <a href="/wiki/%E8%99%9A%E6%95%B0%E5%8D%95%E4%BD%8D" class="mw-redirect" title="虚数单位">虚数单位</a>，</li> <li><i>n</i> 表示滤波器的级数，</li> <li>ω 是信号的 <a href="/wiki/%E8%A7%92%E9%A2%91%E7%8E%87" title="角频率">角频率</a>，以弧度/秒为单位，</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {c} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {c} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa3b6d88c873503cc5417029f4426950dcfb32fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.408ex; height:2.009ex;" alt="{\displaystyle \omega _{\mathrm {c} }}"></span> 是振幅下降3分贝时的截止频率。</li></ul> <p>令截止频率<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{\mathrm {c} }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">c</mi> </mrow> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega _{\mathrm {c} }=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da590e1c2a473e346f1d19be17f91a3e2f45b584" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.669ex; height:2.509ex;" alt="{\displaystyle \omega _{\mathrm {c} }=1}"></span>), 将上列公式规定一化成为： </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+\omega ^{2n}}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <msub> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msup> </msqrt> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+\omega ^{2n}}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1d5d35f8c6aa802d2073848f72dd5fa66c6c54e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:31.803ex; height:6.509ex;" alt="{\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={1 \over {\sqrt {1+\omega ^{2n}}}}}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="根据衰减度求滤波器的阶数"><span id=".E6.A0.B9.E6.8D.AE.E8.A1.B0.E5.87.8F.E5.BA.A6.E6.B1.82.E6.BB.A4.E6.B3.A2.E5.99.A8.E7.9A.84.E9.98.B6.E6.95.B0"></span>根据衰减度求滤波器的阶数</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=3" title="编辑章节：根据衰减度求滤波器的阶数"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>令 1/A=<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7c1f4c577ef32e928711f4c525b038229e01c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.3ex; height:2.843ex;" alt="{\displaystyle G_{n}(\omega )}"></span> </p><p><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n={\frac {log_{10}(A^{2}-1)}{2log_{10}(\omega )}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n={\frac {log_{10}(A^{2}-1)}{2log_{10}(\omega )}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec27e6865278433f575e64ee9a3e1d79a4cb1d43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.745ex; height:6.676ex;" alt="{\displaystyle n={\frac {log_{10}(A^{2}-1)}{2log_{10}(\omega )}}}"></span> </p><p>例：在 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\omega )=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\omega )=2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc073d62e235eca600ca5c7519c1eb7bec3cadf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.516ex; height:2.843ex;" alt="{\displaystyle (\omega )=2}"></span> 时 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{n}(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>G</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G_{n}(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6b7c1f4c577ef32e928711f4c525b038229e01c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.3ex; height:2.843ex;" alt="{\displaystyle G_{n}(\omega )}"></span>=0.005 </p><p>A= 200, n=7.6, 取大一号整数，即需要 8 阶巴特沃斯滤波器。 </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Butterworth_II_Order_LPF_Bode_Plot.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Butterworth_II_Order_LPF_Bode_Plot.png/250px-Butterworth_II_Order_LPF_Bode_Plot.png" decoding="async" width="250" height="131" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Butterworth_II_Order_LPF_Bode_Plot.png/375px-Butterworth_II_Order_LPF_Bode_Plot.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f6/Butterworth_II_Order_LPF_Bode_Plot.png/500px-Butterworth_II_Order_LPF_Bode_Plot.png 2x" data-file-width="1152" data-file-height="602" /></a><figcaption><p align="justify">二阶巴特沃斯低通滤波器的<a href="/wiki/%E6%B3%A2%E7%89%B9%E5%9B%BE" class="mw-redirect" title="波特图">波特图</a></p></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="幅度最平坦的滤波器"><span id=".E5.B9.85.E5.BA.A6.E6.9C.80.E5.B9.B3.E5.9D.A6.E7.9A.84.E6.BB.A4.E6.B3.A2.E5.99.A8"></span>幅度最平坦的滤波器</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=4" title="编辑章节：幅度最平坦的滤波器"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><i><b>g</b></i>的头<b>(2n-1)</b>次导数在<i>ω = 0</i>时为零，说明放大率对 ω 是常数。因此巴特沃斯滤波器又被称为最平坦的滤波器。 </p> <div class="mw-heading mw-heading3"><h3 id="高频衰减"><span id=".E9.AB.98.E9.A2.91.E8.A1.B0.E5.87.8F"></span>高频衰减</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=5" title="编辑章节：高频衰减"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {{\left|H(j\omega )\right|^{2}}_{dB}}={40n}{log_{10}{\omega }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>j</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mi>B</mi> </mrow> </msub> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>40</mn> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>l</mi> <mi>o</mi> <msub> <mi>g</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>10</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mi>ω</mi> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {{\left|H(j\omega )\right|^{2}}_{dB}}={40n}{log_{10}{\omega }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0fd70c9fe7a621cba392c6492c8afc3f6af66d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.034ex; height:3.509ex;" alt="{\displaystyle {{\left|H(j\omega )\right|^{2}}_{dB}}={40n}{log_{10}{\omega }}}"></span></center> <p>因此，<b>n</b>阶巴特沃斯低通滤波器的高频衰减为每十倍频20n 分贝。 </p> <div class="mw-heading mw-heading2"><h2 id="实例"><span id=".E5.AE.9E.E4.BE.8B"></span>实例</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=6" title="编辑章节：实例"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:Butterworth_Cauer_1_form.PNG" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/3/35/Butterworth_Cauer_1_form.PNG" decoding="async" width="359" height="125" class="mw-file-element" data-file-width="359" data-file-height="125" /></a><figcaption></figcaption></figure> <p><b>k</b>阶巴特沃斯滤波器的<a href="/w/index.php?title=%E8%80%83%E5%B0%94%E7%AC%AC%E4%B8%80%E5%9E%8B&action=edit&redlink=1" class="new" title="考尔第一型（页面不存在）">考尔第一型</a>电子线路图如下: 其中： </p> <ul><li>电容<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>π</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50d89f2453b8a712f3162f798c07856760f0730d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.495ex; height:6.343ex;" alt="{\displaystyle C_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}"></span>; k = 奇数</li> <li>电感<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>s</mi> <mi>i</mi> <mi>n</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>2</mn> <mi>n</mi> </mrow> </mfrac> </mrow> <mi>π</mi> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ffa00d054b88288f1eb485483ac4dc8867dde0a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:23.416ex; height:6.343ex;" alt="{\displaystyle L_{k}=2sin\left[{\frac {(2k-1)}{2n}}\pi \right]}"></span>; k = 偶数</li></ul> <div class="mw-heading mw-heading2"><h2 id="归一化的巴特沃斯多项式"><span id=".E5.BD.92.E4.B8.80.E5.8C.96.E7.9A.84.E5.B7.B4.E7.89.B9.E6.B2.83.E6.96.AF.E5.A4.9A.E9.A1.B9.E5.BC.8F"></span>归一化的巴特沃斯多项式</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=7" title="编辑章节：归一化的巴特沃斯多项式"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <center> <table> <tbody><tr> <td> <table class="wikitable" style="text-align: center;"> <tbody><tr> <th>n</th> <th>多项式因子 <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B_{n}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B_{n}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5403d5abe1bb8af88861556e48400ade8cc260fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.882ex; height:2.843ex;" alt="{\displaystyle B_{n}(s)}"></span> </th></tr> <tr> <th>1 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7a6432c4f00809d751c3b7bbc04c095ec41f7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.903ex; height:2.843ex;" alt="{\displaystyle (s+1)}"></span> </td></tr> <tr> <th>2 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s^{2}+1.414s+1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.414</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s^{2}+1.414s+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3767a5b0f618d3a4b974e0f9d2e3f9993aa1e7d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:15.375ex; height:2.843ex;" alt="{\displaystyle s^{2}+1.414s+1}"></span> </td></tr> <tr> <th>3 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+1)(s^{2}+s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+1)(s^{2}+s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/111f06609bdddcf635e1c78d43e2d8bc95bf7acd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.79ex; height:3.176ex;" alt="{\displaystyle (s+1)(s^{2}+s+1)}"></span> </td></tr> <tr> <th>4 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{2}+0.7654s+1)(s^{2}+1.8478s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.7654</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.8478</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s^{2}+0.7654s+1)(s^{2}+1.8478s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16e670485a46d587f0314a2a4851237b78b1be58" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.694ex; height:3.176ex;" alt="{\displaystyle (s^{2}+0.7654s+1)(s^{2}+1.8478s+1)}"></span> </td></tr> <tr> <th>5 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+1)(s^{2}+0.6180s+1)(s^{2}+1.6180s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.6180</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.6180</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+1)(s^{2}+0.6180s+1)(s^{2}+1.6180s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a7c486822d79a05d5aecb9fbeffb1979e9b3a1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.596ex; height:3.176ex;" alt="{\displaystyle (s+1)(s^{2}+0.6180s+1)(s^{2}+1.6180s+1)}"></span> </td></tr> <tr> <th>6 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{2}+0.5176s+1)(s^{2}+1.414s+1)(s^{2}+1.9318s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.5176</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.414</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.9318</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s^{2}+0.5176s+1)(s^{2}+1.414s+1)(s^{2}+1.9318s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58c9f954a3adf555dc4276b8975b544d7d376525" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:53.878ex; height:3.176ex;" alt="{\displaystyle (s^{2}+0.5176s+1)(s^{2}+1.414s+1)(s^{2}+1.9318s+1)}"></span> </td></tr> <tr> <th>7 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s+1)(s^{2}+0.4450s+1)(s^{2}+1.247s+1)(s^{2}+1.8022s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.4450</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.247</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.8022</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s+1)(s^{2}+0.4450s+1)(s^{2}+1.247s+1)(s^{2}+1.8022s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eb9d32cb264909a591af2b392b67114e5e9d07e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:60.781ex; height:3.176ex;" alt="{\displaystyle (s+1)(s^{2}+0.4450s+1)(s^{2}+1.247s+1)(s^{2}+1.8022s+1)}"></span> </td></tr> <tr> <th>8 </th> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (s^{2}+0.3986s+1)(s^{2}+1.111s+1)(s^{2}+1.6630s+1)(s^{2}+1.9622s+1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>0.3986</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.111</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.6630</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>1.9622</mn> <mi>s</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (s^{2}+0.3986s+1)(s^{2}+1.111s+1)(s^{2}+1.6630s+1)(s^{2}+1.9622s+1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9e424b92b162464157d56df99ad68894a6649d4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:72.225ex; height:3.176ex;" alt="{\displaystyle (s^{2}+0.3986s+1)(s^{2}+1.111s+1)(s^{2}+1.6630s+1)(s^{2}+1.9622s+1)}"></span> </td></tr></tbody></table> </td></tr></tbody></table></center> <div class="mw-heading mw-heading2"><h2 id="与其他类型滤波器的比较"><span id=".E4.B8.8E.E5.85.B6.E4.BB.96.E7.B1.BB.E5.9E.8B.E6.BB.A4.E6.B3.A2.E5.99.A8.E7.9A.84.E6.AF.94.E8.BE.83"></span>与其他类型滤波器的比较</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=8" title="编辑章节：与其他类型滤波器的比较"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>下图是巴特沃斯滤波器（左上）和同阶I型切比雪夫滤波器（右上）、II型切比雪夫滤波器（左下）、椭圆函数滤波器（右下）的频率响应图。 </p> <figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Electronic_linear_filters.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Electronic_linear_filters.svg/500px-Electronic_linear_filters.svg.png" decoding="async" width="500" height="400" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Electronic_linear_filters.svg/750px-Electronic_linear_filters.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Electronic_linear_filters.svg/1000px-Electronic_linear_filters.svg.png 2x" data-file-width="1200" data-file-height="960" /></a><figcaption></figcaption></figure> <p>由图可见，巴特沃斯滤波器的衰减速度比其他类型滤波器缓慢，但十分平坦，没有幅度变化。 </p> <div class="mw-heading mw-heading2"><h2 id="参见"><span id=".E5.8F.82.E8.A7.81"></span>参见</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=9" title="编辑章节：参见"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/%E8%B4%9D%E5%A1%9E%E8%80%B3%E6%BB%A4%E6%B3%A2%E5%99%A8" class="mw-redirect" title="贝塞耳滤波器">贝塞耳滤波器</a></li> <li><a href="/wiki/%E5%88%87%E6%AF%94%E9%9B%AA%E5%A4%AB%E6%BB%A4%E6%B3%A2%E5%99%A8" title="切比雪夫滤波器">切比雪夫滤波器</a></li> <li><a href="/wiki/%E6%A2%B3%E7%8A%B6%E6%BB%A4%E6%B3%A2%E5%99%A8" title="梳状滤波器">梳状滤波器</a></li> <li><a href="/wiki/%E6%A4%AD%E5%9C%86%E5%87%BD%E6%95%B0%E6%BB%A4%E6%B3%A2%E5%99%A8" title="椭圆函数滤波器">椭圆函数滤波器</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="参考文献"><span id=".E5.8F.82.E8.80.83.E6.96.87.E7.8C.AE"></span>参考文献</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&action=edit&section=10" title="编辑章节：参考文献"><span>编辑</span></a><span class="mw-editsection-bracket">]</span></span></div> <ol class="references"> <li id="cite_note-1930paper-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1930paper_1-0">^</a></b></span> <span class="reference-text">In <i>Wireless Engineer</i> (also called <i>Experimental Wireless and the Wireless Engineer</i>), vol. 7, 1930, pp. 536–541 - <a rel="nofollow" class="external text" href="http://www.gonascent.com/papers/butter.pdf">"On the Theory of Filter Amplifiers"-S. Butterworth</a> （<a rel="nofollow" class="external text" href="//web.archive.org/web/20140521031249/http://www.gonascent.com/papers/butter.pdf">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>）</span> </li> <li id="cite_note-bianchi-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-bianchi_2-0">^</a></b></span> <span class="reference-text"> <cite class="citation book">Giovanni Bianchi and Roberto Sorrentino. <a rel="nofollow" class="external text" href="http://books.google.com/books?id=5S3LCIxnYCcC&pg=PT32&dq=Butterworth-approximation+maximally-flat#v=onepage&q=Butterworth-approximation%20maximally-flat&f=false">Electronic filter simulation & design</a>. McGraw-Hill Professional. 2007: 17–20. <a href="/wiki/Special:%E7%BD%91%E7%BB%9C%E4%B9%A6%E6%BA%90/978-0-07-149467-0" title="Special:网络书源/978-0-07-149467-0"><span title="国际标准书号">ISBN</span> 978-0-07-149467-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rfr_id=info%3Asid%2Fzh.wikipedia.org%3A%E5%B7%B4%E7%89%B9%E6%B2%83%E6%96%AF%E6%BB%A4%E6%B3%A2%E5%99%A8&rft.au=Giovanni+Bianchi+and+Roberto+Sorrentino&rft.btitle=Electronic+filter+simulation+%26+design&rft.date=2007&rft.genre=book&rft.isbn=978-0-07-149467-0&rft.pages=17-20&rft.pub=McGraw-Hill+Professional&rft_id=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3D5S3LCIxnYCcC%26pg%3DPT32%26dq%3DButterworth-approximation%2Bmaximally-flat%23v%3Donepage%26q%3DButterworth-approximation%2520maximally-flat%26f%3Dfalse&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook" class="Z3988"><span style="display:none;"> </span></span></span> </li> </ol> <ul><li>Matthaei, George L.; Young, Leo and Jones, E. M. T., <i>Microwave Filters, Impedance-Matching Networks, and Coupling Structures</i>, McGraw-Hill, 1964 <div class="noprint"><a href="/wiki/%E7%BE%8E%E5%9B%BD%E5%9B%BD%E4%BC%9A%E5%9B%BE%E4%B9%A6%E9%A6%86%E6%8E%A7%E5%88%B6%E5%8F%B7" class="mw-redirect" title="美国国会图书馆控制号">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/64007937">64-7937</a></div>.</li></ul>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">检索自“<a dir="ltr" href="https://zh.wikipedia.org/w/index.php?title=巴特沃斯滤波器&oldid=79409290">https://zh.wikipedia.org/w/index.php?title=巴特沃斯滤波器&oldid=79409290</a>”</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Special:%E9%A1%B5%E9%9D%A2%E5%88%86%E7%B1%BB" 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</div> </div> </nav> <div class="vector-sticky-header-context-bar-primary" aria-hidden="true" ><span class="mw-page-title-main">巴特沃斯滤波器</span></div> </div> </div> <div class="vector-sticky-header-end" aria-hidden="true"> <div class="vector-sticky-header-icons"> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-talk-sticky-header" tabindex="-1" data-event-name="talk-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbles mw-ui-icon-wikimedia-speechBubbles"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--icon-only" id="ca-subject-sticky-header" tabindex="-1" data-event-name="subject-sticky-header"><span class="vector-icon mw-ui-icon-article mw-ui-icon-wikimedia-article"></span> <span></span> </a> <a href="#" class="cdx-button cdx-button--fake-button cdx-button--fake-button--enabled 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