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Wiener filter - Wikipedia
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class="vector-toc-numb">2</span> <span>Wiener filter solutions</span> </div> </a> <button aria-controls="toc-Wiener_filter_solutions-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>Toggle Wiener filter solutions subsection</span> </button> <ul id="toc-Wiener_filter_solutions-sublist" class="vector-toc-list"> <li id="toc-Noncausal_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Noncausal_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Noncausal solution</span> </div> </a> <ul id="toc-Noncausal_solution-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Causal_solution" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Causal_solution"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Causal solution</span> </div> </a> <ul id="toc-Causal_solution-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Finite_impulse_response_Wiener_filter_for_discrete_series" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Finite_impulse_response_Wiener_filter_for_discrete_series"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Finite impulse response Wiener filter for discrete series</span> </div> </a> <button aria-controls="toc-Finite_impulse_response_Wiener_filter_for_discrete_series-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>Toggle Finite impulse response Wiener filter for discrete series subsection</span> </button> <ul id="toc-Finite_impulse_response_Wiener_filter_for_discrete_series-sublist" class="vector-toc-list"> <li id="toc-Relationship_to_the_least_squares_filter" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Relationship_to_the_least_squares_filter"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Relationship to the least squares filter</span> </div> </a> <ul id="toc-Relationship_to_the_least_squares_filter-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Complex_signals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Complex_signals"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Complex signals</span> </div> </a> <ul id="toc-Complex_signals-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Applications</span> </div> </a> <ul id="toc-Applications-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>Further reading</span> </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-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="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <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="Toggle the table of contents" > <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">Toggle the table of contents</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">Wiener filter</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="Go to an article in another language. Available in 14 languages" > <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-14" 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">14 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Filtre_de_Wiener" title="Filtre de Wiener – Catalan" lang="ca" hreflang="ca" data-title="Filtre de Wiener" data-language-autonym="Català" data-language-local-name="Catalan" 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/Wiener%C5%AFv_filtr" title="Wienerův filtr – Czech" lang="cs" hreflang="cs" data-title="Wienerův filtr" data-language-autonym="Čeština" data-language-local-name="Czech" 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/Wiener-Filter" title="Wiener-Filter – German" lang="de" hreflang="de" data-title="Wiener-Filter" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Wieneri_filter" title="Wieneri filter – Estonian" lang="et" hreflang="et" data-title="Wieneri filter" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Filtro_de_Wiener" title="Filtro de Wiener – Spanish" lang="es" hreflang="es" data-title="Filtro de Wiener" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%81%DB%8C%D9%84%D8%AA%D8%B1_%D9%88%DB%8C%D9%86%D8%B1" title="فیلتر وینر – Persian" lang="fa" hreflang="fa" data-title="فیلتر وینر" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Filtre_de_Wiener" title="Filtre de Wiener – French" lang="fr" hreflang="fr" data-title="Filtre de Wiener" data-language-autonym="Français" data-language-local-name="French" 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_di_Wiener" title="Filtro di Wiener – Italian" lang="it" hreflang="it" data-title="Filtro di Wiener" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Wiener-filter" title="Wiener-filter – Dutch" lang="nl" hreflang="nl" data-title="Wiener-filter" data-language-autonym="Nederlands" data-language-local-name="Dutch" 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_Wienera" title="Filtr Wienera – Polish" lang="pl" hreflang="pl" data-title="Filtr Wienera" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%92%D0%B8%D0%BD%D0%B5%D1%80%D0%BE%D0%B2%D1%81%D0%BA%D0%BE%D0%B5_%D0%BE%D1%86%D0%B5%D0%BD%D0%B8%D0%B2%D0%B0%D0%BD%D0%B8%D0%B5" title="Винеровское оценивание – Russian" lang="ru" hreflang="ru" data-title="Винеровское оценивание" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sk mw-list-item"><a href="https://sk.wikipedia.org/wiki/Wienerov_filter" title="Wienerov filter – Slovak" lang="sk" hreflang="sk" data-title="Wienerov filter" data-language-autonym="Slovenčina" data-language-local-name="Slovak" class="interlanguage-link-target"><span>Slovenčina</span></a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%88%DB%8C%D9%86%D8%B1_%D9%85%D8%B5%D9%81%D8%A7%DB%81" title="وینر مصفاہ – Urdu" lang="ur" hreflang="ur" data-title="وینر مصفاہ" 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print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Not to be confused with <a href="/wiki/Wien_filter" title="Wien filter">Wien filter</a> or <a href="/wiki/Wien_bridge" title="Wien bridge">Wien bridge</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Lead_too_short plainlinks metadata ambox ambox-content ambox-lead_too_short" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Wiki_letter_w.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/40px-Wiki_letter_w.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/60px-Wiki_letter_w.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/6/6c/Wiki_letter_w.svg/80px-Wiki_letter_w.svg.png 2x" data-file-width="44" data-file-height="44" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article's <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Length" title="Wikipedia:Manual of Style/Lead section">lead section</a> <b>may be too short to adequately <a href="/wiki/Wikipedia:Summary_style" title="Wikipedia:Summary style">summarize</a> the key points</b>.<span class="hide-when-compact"> Please consider expanding the lead to <a href="/wiki/Wikipedia:Manual_of_Style/Lead_section#Provide_an_accessible_overview" title="Wikipedia:Manual of Style/Lead section">provide an accessible overview</a> of all important aspects of the article.</span> <span class="date-container"><i>(<span class="date">October 2023</span>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, the <b>Wiener filter</b> is a <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filter</a> used to produce an estimate of a desired or target random process by linear time-invariant (<a href="/wiki/Linear_filter" title="Linear filter">LTI</a>) filtering of an observed noisy process, assuming known <a href="/wiki/Stationary_process" title="Stationary process">stationary</a> signal and noise spectra, and additive noise. The Wiener filter minimizes the mean square error between the estimated random process and the desired process. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Description">Description</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=1" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The goal of the wiener filter is to compute a <a href="/wiki/Estimation_theory" title="Estimation theory">statistical estimate</a> of an unknown signal using a related signal as an input and filtering it to produce the estimate. For example, the known signal might consist of an unknown signal of interest that has been corrupted by additive <a href="/wiki/Noise" title="Noise">noise</a>. The Wiener filter can be used to filter out the noise from the corrupted signal to provide an estimate of the underlying signal of interest. The Wiener filter is based on a <a href="/wiki/Statistical" class="mw-redirect" title="Statistical">statistical</a> approach, and a more statistical account of the theory is given in the <a href="/wiki/Minimum_mean_square_error" title="Minimum mean square error">minimum mean square error (MMSE) estimator</a> article. </p><p>Typical deterministic filters are designed for a desired <a href="/wiki/Frequency_response" title="Frequency response">frequency response</a>. However, the design of the Wiener filter takes a different approach. One is assumed to have knowledge of the spectral properties of the original signal and the noise, and one seeks the <a href="/wiki/LTI_system_theory" class="mw-redirect" title="LTI system theory">linear time-invariant</a> filter whose output would come as close to the original signal as possible. Wiener filters are characterized by the following:<sup id="cite_ref-Brown1996_1-0" class="reference"><a href="#cite_note-Brown1996-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Assumption: signal and (additive) noise are stationary linear <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a> with known spectral characteristics or known <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> and <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a></li> <li>Requirement: the filter must be physically realizable/<a href="/wiki/Causal_system" title="Causal system">causal</a> (this requirement can be dropped, resulting in a non-causal solution)</li> <li>Performance criterion: <a href="/wiki/Minimum_mean-square_error" class="mw-redirect" title="Minimum mean-square error">minimum mean-square error</a> (MMSE)</li></ol> <p>This filter is frequently used in the process of <a href="/wiki/Deconvolution" title="Deconvolution">deconvolution</a>; for this application, see <a href="/wiki/Wiener_deconvolution" title="Wiener deconvolution">Wiener deconvolution</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Wiener_filter_solutions">Wiener filter solutions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=2" title="Edit section: Wiener filter solutions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <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(t+\alpha )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s(t+\alpha )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a20d31cab4235d1716c4147403cef84faf899ee0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.067ex; height:2.843ex;" alt="{\displaystyle s(t+\alpha )}"></span> be an unknown signal which must be estimated from a measurement signal <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 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></span><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)}"></span>, where <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }"></span> is a tunable parameter. <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 \alpha >0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>></mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha >0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edd4f784b6e8bb68fa774213ceacbab2d97825dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha >0}"></span> is known as prediction, <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 \alpha =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30cc00f65bbc630448311dd2dc82e7ce5e90985a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha =0}"></span> is known as filtering, and <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 \alpha <0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α<!-- α --></mi> <mo><</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha <0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e9d48dc3d4d98b4c949bf36f18559a74bc3d87b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.749ex; height:2.176ex;" alt="{\displaystyle \alpha <0}"></span> is known as smoothing (see Wiener filtering chapter of <sup id="cite_ref-Brown1996_1-1" class="reference"><a href="#cite_note-Brown1996-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> for more details). </p><p>The Wiener filter problem has solutions for three possible cases: one where a noncausal filter is acceptable (requiring an infinite amount of both past and future data), the case where a <a href="/wiki/Causal_system" title="Causal system">causal</a> filter is desired (using an infinite amount of past data), and the <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR) case where only input data is used (i.e. the result or output is not fed back into the filter as in the IIR case). The first case is simple to solve but is not suited for real-time applications. Wiener's main accomplishment was solving the case where the causality requirement is in effect; <a href="/wiki/Norman_Levinson" title="Norman Levinson">Norman Levinson</a> gave the FIR solution in an appendix of Wiener's book. </p> <div class="mw-heading mw-heading3"><h3 id="Noncausal_solution">Noncausal solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=3" title="Edit section: Noncausal solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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(s)={\frac {S_{x,s}(s)}{S_{x}(s)}}e^{\alpha 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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mi>s</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)={\frac {S_{x,s}(s)}{S_{x}(s)}}e^{\alpha s},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06357e486cbd48b2a95526916bc7bf8ac5c5d07d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:19.172ex; height:6.509ex;" alt="{\displaystyle G(s)={\frac {S_{x,s}(s)}{S_{x}(s)}}e^{\alpha s},}"></span></dd></dl> <p>where <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}"></span> are <a href="/wiki/Spectral_density" title="Spectral density">spectral densities</a>. Provided that <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(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></span><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)}"></span> is optimal, then the <a href="/wiki/Minimum_mean-square_error" class="mw-redirect" title="Minimum mean-square error">minimum mean-square error</a> equation reduces to </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 E(e^{2})=R_{s}(0)-\int _{-\infty }^{\infty }g(\tau )R_{x,s}(\tau +\alpha )\,d\tau ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mn>0</mn> <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>τ<!-- τ --></mi> <mo stretchy="false">)</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ<!-- τ --></mi> <mo>+</mo> <mi>α<!-- α --></mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>d</mi> <mi>τ<!-- τ --></mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(e^{2})=R_{s}(0)-\int _{-\infty }^{\infty }g(\tau )R_{x,s}(\tau +\alpha )\,d\tau ,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c7fddeb7cd78a3819b02abbf91ea0d7790905fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:41.317ex; height:6.009ex;" alt="{\displaystyle E(e^{2})=R_{s}(0)-\int _{-\infty }^{\infty }g(\tau )R_{x,s}(\tau +\alpha )\,d\tau ,}"></span></dd></dl> <p>and the solution <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(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></span><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)}"></span> is the inverse two-sided <a href="/wiki/Laplace_transform" title="Laplace transform">Laplace transform</a> of <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(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></span><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)}"></span>. </p> <div class="mw-heading mw-heading3"><h3 id="Causal_solution">Causal solution</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=4" title="Edit section: Causal solution"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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(s)={\frac {H(s)}{S_{x}^{+}(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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G(s)={\frac {H(s)}{S_{x}^{+}(s)}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab1b64064bce759d564f5f8bf4cdb679dbfa5b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:15.24ex; height:6.676ex;" alt="{\displaystyle G(s)={\frac {H(s)}{S_{x}^{+}(s)}},}"></span></dd></dl> <p>where </p> <ul><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 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></span><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)}"></span> consists of the causal part of <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_{x,s}(s)}{S_{x}^{-}(s)}}e^{\alpha s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {S_{x,s}(s)}{S_{x}^{-}(s)}}e^{\alpha s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aa7d70444e1777ed4114f2155b239196996f9ccb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.7ex; height:6.676ex;" alt="{\displaystyle {\frac {S_{x,s}(s)}{S_{x}^{-}(s)}}e^{\alpha s}}"></span> (that is, that part of this fraction having a positive time solution under the inverse Laplace transform)</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 S_{x}^{+}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{+}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f736c54316906d5b4331d883ba5d60566cef9a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{+}(s)}"></span> is the causal component of <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_{x}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d3551eaa2b2df844bfc5b827dc9761d4800243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:2.843ex;" alt="{\displaystyle S_{x}(s)}"></span> (i.e., the inverse Laplace transform of <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_{x}^{+}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{+}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f736c54316906d5b4331d883ba5d60566cef9a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{+}(s)}"></span> is non-zero only for <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 t\geq 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\geq 0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/248525429e9cd266f53ab8c52d17bc206c546060" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.101ex; height:2.343ex;" alt="{\displaystyle t\geq 0}"></span>)</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 S_{x}^{-}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{-}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9ed407db041671eb87223ff59a7f6f5bae513d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{-}(s)}"></span> is the anti-causal component of <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_{x}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d3551eaa2b2df844bfc5b827dc9761d4800243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:2.843ex;" alt="{\displaystyle S_{x}(s)}"></span> (i.e., the inverse Laplace transform of <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_{x}^{-}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{-}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9ed407db041671eb87223ff59a7f6f5bae513d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{-}(s)}"></span> is non-zero only for <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 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></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c8875f14d87cb6daa44307512a91eceb5f34d87" 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}"></span>)</li></ul> <p>This general formula is complicated and deserves a more detailed explanation. To write down the solution <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(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></span><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)}"></span> in a specific case, one should follow these steps:<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p> <ol><li>Start with the spectrum <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_{x}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/24d3551eaa2b2df844bfc5b827dc9761d4800243" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.497ex; height:2.843ex;" alt="{\displaystyle S_{x}(s)}"></span> in rational form and factor it into causal and anti-causal components: <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_{x}(s)=S_{x}^{+}(s)S_{x}^{-}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}(s)=S_{x}^{+}(s)S_{x}^{-}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd69fa1779b752f0c9e1606e60ed7028b9b3af75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.46ex; height:3.009ex;" alt="{\displaystyle S_{x}(s)=S_{x}^{+}(s)S_{x}^{-}(s)}"></span> where <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^{+}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{+}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8fb8e791330b1f1aabb7d9ab514afa5b879190a9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.032ex; height:2.509ex;" alt="{\displaystyle S^{+}}"></span> contains all the zeros and poles in the left half plane (LHP) and <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^{-}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S^{-}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/269e3e6c1fe2d2dfa225d8535e76cc8846a5c8fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.032ex; height:2.509ex;" alt="{\displaystyle S^{-}}"></span> contains the zeroes and poles in the right half plane (RHP). This is called the <a href="/wiki/Wiener%E2%80%93Hopf_method" title="Wiener–Hopf method">Wiener–Hopf factorization</a>.</li> <li>Divide <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_{x,s}(s)e^{\alpha s}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>,</mo> <mi>s</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α<!-- α --></mi> <mi>s</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x,s}(s)e^{\alpha s}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50b36c1219a0eb61d723042919e4db5cadae8a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.864ex; height:3.009ex;" alt="{\displaystyle S_{x,s}(s)e^{\alpha s}}"></span> by <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_{x}^{-}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{-}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a9ed407db041671eb87223ff59a7f6f5bae513d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{-}(s)}"></span> and write out the result as a <a href="/wiki/Partial_fraction_decomposition" title="Partial fraction decomposition">partial fraction expansion</a>.</li> <li>Select only those terms in this expansion having poles in the LHP. Call these terms <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)}"> <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></span><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)}"></span>.</li> <li>Divide <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)}"> <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></span><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)}"></span> by <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_{x}^{+}(s)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>s</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{x}^{+}(s)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f736c54316906d5b4331d883ba5d60566cef9a6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.932ex; height:3.009ex;" alt="{\displaystyle S_{x}^{+}(s)}"></span>. The result is the desired filter transfer function <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(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></span><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)}"></span>.</li></ol> <div class="mw-heading mw-heading2"><h2 id="Finite_impulse_response_Wiener_filter_for_discrete_series">Finite impulse response Wiener filter for discrete series</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=5" title="Edit section: Finite impulse response Wiener filter for discrete series"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wiener_block.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Wiener_block.svg/350px-Wiener_block.svg.png" decoding="async" width="350" height="105" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/00/Wiener_block.svg/525px-Wiener_block.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/00/Wiener_block.svg/700px-Wiener_block.svg.png 2x" data-file-width="600" data-file-height="180" /></a><figcaption>Block diagram view of the FIR Wiener filter for discrete series. An input signal <i>w</i>[<i>n</i>] is convolved with the Wiener filter <i>g</i>[<i>n</i>] and the result is compared to a reference signal <i>s</i>[<i>n</i>] to obtain the filtering error <i>e</i>[<i>n</i>].</figcaption></figure> <p>The causal <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> (FIR) Wiener filter, instead of using some given data matrix X and output vector Y, finds optimal tap weights by using the statistics of the input and output signals. It populates the input matrix X with estimates of the auto-correlation of the input signal (T) and populates the output vector Y with estimates of the cross-correlation between the output and input signals (V). </p><p>In order to derive the coefficients of the Wiener filter, consider the signal <i>w</i>[<i>n</i>] being fed to a Wiener filter of order (number of past taps) <i>N</i> and with coefficients <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 \{a_{0},\cdots ,a_{N}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{a_{0},\cdots ,a_{N}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/784e249e9bc26655a202c10365ddbb250f328367" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.708ex; height:2.843ex;" alt="{\displaystyle \{a_{0},\cdots ,a_{N}\}}"></span>. The output of the filter is denoted <i>x</i>[<i>n</i>] which is given by the expression </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 x[n]=\sum _{i=0}^{N}a_{i}w[n-i].}"> <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> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x[n]=\sum _{i=0}^{N}a_{i}w[n-i].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b52e807887e91ea452fcc1d5c0b8037593336d05" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:21.53ex; height:7.343ex;" alt="{\displaystyle x[n]=\sum _{i=0}^{N}a_{i}w[n-i].}"></span></dd></dl> <p>The residual error is denoted <i>e</i>[<i>n</i>] and is defined as <i>e</i>[<i>n</i>] = <i>x</i>[<i>n</i>] − <i>s</i>[<i>n</i>] (see the corresponding block diagram). The Wiener filter is designed so as to minimize the mean square error (<a href="/wiki/Minimum_mean_square_error" title="Minimum mean square error">MMSE</a> criteria) which can be stated concisely as follows: </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 a_{i}=\arg \min E\left[e^{2}[n]\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>arg</mi> <mo>⁡<!-- --></mo> <mo movablelimits="true" form="prefix">min</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}=\arg \min E\left[e^{2}[n]\right],}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9c1479360ebceaf703b055388dcd8a3f1d251ca6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.976ex; height:3.343ex;" alt="{\displaystyle a_{i}=\arg \min E\left[e^{2}[n]\right],}"></span></dd></dl> <p>where <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 E[\cdot ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">[</mo> <mo>⋅<!-- ⋅ --></mo> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E[\cdot ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57d488a22bc9f41e976d3afb6036190bcbb36b2e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.716ex; height:2.843ex;" alt="{\displaystyle E[\cdot ]}"></span> denotes the expectation operator. In the general case, the coefficients <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 a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> may be complex and may be derived for the case where <i>w</i>[<i>n</i>] and <i>s</i>[<i>n</i>] are complex as well. With a complex signal, the matrix to be solved is a <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a>, rather than <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a>. For simplicity, the following considers only the case where all these quantities are real. The mean square error (MSE) may be rewritten as: </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 {\begin{aligned}E\left[e^{2}[n]\right]&=E\left[(x[n]-s[n])^{2}\right]\\&=E\left[x^{2}[n]\right]+E\left[s^{2}[n]\right]-2E[x[n]s[n]]\\&=E\left[\left(\sum _{i=0}^{N}a_{i}w[n-i]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{i=0}^{N}a_{i}w[n-i]s[n]\right]\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>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>−<!-- − --></mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>+</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">[</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>+</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}E\left[e^{2}[n]\right]&=E\left[(x[n]-s[n])^{2}\right]\\&=E\left[x^{2}[n]\right]+E\left[s^{2}[n]\right]-2E[x[n]s[n]]\\&=E\left[\left(\sum _{i=0}^{N}a_{i}w[n-i]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{i=0}^{N}a_{i}w[n-i]s[n]\right]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fade3732a89c3c06d14e7532d4e5ac8e8ba5e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.171ex; margin-top: -0.245ex; width:74.741ex; height:15.509ex;" alt="{\displaystyle {\begin{aligned}E\left[e^{2}[n]\right]&=E\left[(x[n]-s[n])^{2}\right]\\&=E\left[x^{2}[n]\right]+E\left[s^{2}[n]\right]-2E[x[n]s[n]]\\&=E\left[\left(\sum _{i=0}^{N}a_{i}w[n-i]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{i=0}^{N}a_{i}w[n-i]s[n]\right]\end{aligned}}}"></span></dd></dl> <p>To find the vector <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 [a_{0},\,\ldots ,\,a_{N}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…<!-- … --></mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [a_{0},\,\ldots ,\,a_{N}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23b15c3b4639ed827b254c60c133606cec814d33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.451ex; height:2.843ex;" alt="{\displaystyle [a_{0},\,\ldots ,\,a_{N}]}"></span> which minimizes the expression above, calculate its derivative with respect to each <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 a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span> </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 {\begin{aligned}{\frac {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]&={\frac {\partial }{\partial a_{i}}}\left\{E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{j=0}^{N}a_{j}w[n-j]s[n]\right]\right\}\\&=2E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)w[n-i]\right]-2E[w[n-i]s[n]]\\&=2\left(\sum _{j=0}^{N}E[w[n-j]w[n-i]]a_{j}\right)-2E[w[n-i]s[n]]\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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂<!-- ∂ --></mi> <mrow> <mi mathvariant="normal">∂<!-- ∂ --></mi> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mrow> <mo>{</mo> <mrow> <mi>E</mi> <mrow> <mo>[</mo> <msup> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>]</mo> </mrow> <mo>+</mo> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>s</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">]</mo> </mrow> <mo>)</mo> </mrow> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <mi>E</mi> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">]</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <mi>E</mi> <mo stretchy="false">[</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]&={\frac {\partial }{\partial a_{i}}}\left\{E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{j=0}^{N}a_{j}w[n-j]s[n]\right]\right\}\\&=2E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)w[n-i]\right]-2E[w[n-i]s[n]]\\&=2\left(\sum _{j=0}^{N}E[w[n-j]w[n-i]]a_{j}\right)-2E[w[n-i]s[n]]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32d0c093c419d914a8d0b81eb7b53aec3d2e77bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.505ex; width:88.159ex; height:24.176ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]&={\frac {\partial }{\partial a_{i}}}\left\{E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)^{2}\right]+E\left[s^{2}[n]\right]-2E\left[\sum _{j=0}^{N}a_{j}w[n-j]s[n]\right]\right\}\\&=2E\left[\left(\sum _{j=0}^{N}a_{j}w[n-j]\right)w[n-i]\right]-2E[w[n-i]s[n]]\\&=2\left(\sum _{j=0}^{N}E[w[n-j]w[n-i]]a_{j}\right)-2E[w[n-i]s[n]]\end{aligned}}}"></span></dd></dl> <p>Assuming that <i>w</i>[<i>n</i>] and <i>s</i>[<i>n</i>] are each stationary and jointly stationary, the sequences <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 R_{w}[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{w}[m]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fd433a0ac1fe24b6b95d7bad92cea453be5c7f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.507ex; height:2.843ex;" alt="{\displaystyle R_{w}[m]}"></span> and <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 R_{ws}[m]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{ws}[m]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e7790e74796ce594e25ef54c65a4af6ea895803" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.278ex; height:2.843ex;" alt="{\displaystyle R_{ws}[m]}"></span> known respectively as the autocorrelation of <i>w</i>[<i>n</i>] and the cross-correlation between <i>w</i>[<i>n</i>] and <i>s</i>[<i>n</i>] can be defined as follows: </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 {\begin{aligned}R_{w}[m]&=E\{w[n]w[n+m]\}\\R_{ws}[m]&=E\{w[n]s[n+m]\}\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>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{w}[m]&=E\{w[n]w[n+m]\}\\R_{ws}[m]&=E\{w[n]s[n+m]\}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/013fa6d6fbd4e75c4c4ffefdc98dde2ba69d0aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.814ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}R_{w}[m]&=E\{w[n]w[n+m]\}\\R_{ws}[m]&=E\{w[n]s[n+m]\}\end{aligned}}}"></span></dd></dl> <p>The derivative of the MSE may therefore be rewritten as: </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 {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]=2\left(\sum _{j=0}^{N}R_{w}[j-i]a_{j}\right)-2R_{ws}[i]\qquad i=0,\cdots ,N.}"> <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>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> </mfrac> </mrow> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> <mo>=</mo> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>j</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−<!-- − --></mo> <mn>2</mn> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]=2\left(\sum _{j=0}^{N}R_{w}[j-i]a_{j}\right)-2R_{ws}[i]\qquad i=0,\cdots ,N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b838dec6ea00072ce0d8b4ec7e69c4c24768811" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:64.031ex; height:7.676ex;" alt="{\displaystyle {\frac {\partial }{\partial a_{i}}}E\left[e^{2}[n]\right]=2\left(\sum _{j=0}^{N}R_{w}[j-i]a_{j}\right)-2R_{ws}[i]\qquad i=0,\cdots ,N.}"></span></dd></dl> <p>Note that for real <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 w[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a4e3e5afc2a8c6da9020b8c6b21450959101a18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.353ex; height:2.843ex;" alt="{\displaystyle w[n]}"></span>, the autocorrelation is symmetric:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{w}[j-i]=R_{w}[i-j]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>j</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>i</mi> <mo>−<!-- − --></mo> <mi>j</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{w}[j-i]=R_{w}[i-j]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a7c44b249eb0d5faa8fd08516814ecedbebc451" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.234ex; height:2.843ex;" alt="{\displaystyle R_{w}[j-i]=R_{w}[i-j]}"></span>Letting the derivative be equal to zero results in: </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 \sum _{j=0}^{N}R_{w}[j-i]a_{j}=R_{ws}[i]\qquad i=0,\cdots ,N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>j</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{N}R_{w}[j-i]a_{j}=R_{ws}[i]\qquad i=0,\cdots ,N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a9984c26d8ebeb301f22cbeeaa797ee44ef7632" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.685ex; height:7.676ex;" alt="{\displaystyle \sum _{j=0}^{N}R_{w}[j-i]a_{j}=R_{ws}[i]\qquad i=0,\cdots ,N.}"></span></dd></dl> <p>which can be rewritten (using the above symmetric property) in matrix form </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 \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}[1]&\cdots &R_{w}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}[N-1]\\\vdots &\vdots &\ddots &\vdots \\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}\\a_{1}\\\vdots \\a_{N}\end{bmatrix}} _{\mathbf {a} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd> <mo>⋱<!-- ⋱ --></mo> </mtd> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}[1]&\cdots &R_{w}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}[N-1]\\\vdots &\vdots &\ddots &\vdots \\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}\\a_{1}\\\vdots \\a_{N}\end{bmatrix}} _{\mathbf {a} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/22ca4ea294dd47b8d634cb78e5be93073547626f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -10.171ex; margin-right: -0.028ex; width:62.448ex; height:18.009ex;" alt="{\displaystyle \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}[1]&\cdots &R_{w}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}[N-1]\\\vdots &\vdots &\ddots &\vdots \\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}\\a_{1}\\\vdots \\a_{N}\end{bmatrix}} _{\mathbf {a} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}"></span></dd></dl> <p>These equations are known as the <a href="/wiki/Wiener%E2%80%93Hopf_equations" class="mw-redirect" title="Wiener–Hopf equations">Wiener–Hopf equations</a>. The matrix <b>T</b> appearing in the equation is a symmetric <a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz matrix</a>. Under suitable conditions on <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 R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, these matrices are known to be positive definite and therefore non-singular yielding a unique solution to the determination of the Wiener filter coefficient vector, <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 \mathbf {a} =\mathbf {T} ^{-1}\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} =\mathbf {T} ^{-1}\mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a449e7502e4f85491ef71c65d1c8cdcd6fe707ce" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.001ex; height:2.676ex;" alt="{\displaystyle \mathbf {a} =\mathbf {T} ^{-1}\mathbf {v} }"></span>. Furthermore, there exists an efficient algorithm to solve such Wiener–Hopf equations known as the <a href="/wiki/Levinson_recursion" title="Levinson recursion">Levinson-Durbin</a> algorithm so an explicit inversion of <b>T</b> is not required. </p><p>In some articles, the cross correlation function is defined in the opposite way:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{sw}[m]=E\{w[n]s[n+m]\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>E</mi> <mo fence="false" stretchy="false">{</mo> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>+</mo> <mi>m</mi> <mo stretchy="false">]</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{sw}[m]=E\{w[n]s[n+m]\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/674ae192cf58157ffa143e5a78c60150f9e0ec96" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.489ex; height:2.843ex;" alt="{\displaystyle R_{sw}[m]=E\{w[n]s[n+m]\}}"></span>Then, the <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 \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"></span> matrix will contain <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 R_{sw}[0]\ldots R_{sw}[N]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> <mo>…<!-- … --></mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{sw}[0]\ldots R_{sw}[N]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74f1a47c80f31bc0f228f14d5efc080ecfb440e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.199ex; height:2.843ex;" alt="{\displaystyle R_{sw}[0]\ldots R_{sw}[N]}"></span>; this is just a difference in notation. </p><p>Whichever notation is used, note that for real <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 w[n],s[n]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>w</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>,</mo> <mi>s</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w[n],s[n]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47e41c9959e941c5694db0af5c0634c4537c574c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.165ex; height:2.843ex;" alt="{\displaystyle w[n],s[n]}"></span>:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{sw}[k]=R_{ws}[-k]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{sw}[k]=R_{ws}[-k]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/185aafdce950ee0f36925021ae3c93f5c5023d5b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.805ex; height:2.843ex;" alt="{\displaystyle R_{sw}[k]=R_{ws}[-k]}"></span> </p> <div class="mw-heading mw-heading3"><h3 id="Relationship_to_the_least_squares_filter">Relationship to the least squares filter</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=6" title="Edit section: Relationship to the least squares filter"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The realization of the causal Wiener filter looks a lot like the solution to the <a href="/wiki/Least_squares" title="Least squares">least squares</a> estimate, except in the signal processing domain. The least squares solution, for input matrix <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 \mathbf {X} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {X} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f75966a2f9d5672136fa9401ee1e75008f95ffd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.019ex; height:2.176ex;" alt="{\displaystyle \mathbf {X} }"></span> and output vector <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 \mathbf {y} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {y} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb25a040b592282dc2a254c3117e792c3c81161f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.411ex; height:2.009ex;" alt="{\displaystyle \mathbf {y} }"></span> is </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 {\boldsymbol {\hat {\beta }}}=(\mathbf {X} ^{\mathbf {T} }\mathbf {X} )^{-1}\mathbf {X} ^{\mathbf {T} }{\boldsymbol {y}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi mathvariant="bold-italic">β<!-- β --></mi> <mo mathvariant="bold" stretchy="false">^<!-- ^ --></mo> </mover> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">X</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">y</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\hat {\beta }}}=(\mathbf {X} ^{\mathbf {T} }\mathbf {X} )^{-1}\mathbf {X} ^{\mathbf {T} }{\boldsymbol {y}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da645660956477eb039cf1cde40125c6e745bce6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.069ex; height:3.343ex;" alt="{\displaystyle {\boldsymbol {\hat {\beta }}}=(\mathbf {X} ^{\mathbf {T} }\mathbf {X} )^{-1}\mathbf {X} ^{\mathbf {T} }{\boldsymbol {y}}.}"></span></dd></dl> <p>The FIR Wiener filter is related to the <a href="/wiki/Least_mean_squares_filter" title="Least mean squares filter">least mean squares filter</a>, but minimizing the error criterion of the latter does not rely on cross-correlations or auto-correlations. Its solution converges to the Wiener filter solution. </p> <div class="mw-heading mw-heading3"><h3 id="Complex_signals">Complex signals</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=7" title="Edit section: Complex signals"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>For complex signals, the derivation of the complex Wiener filter is performed by minimizing <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 E\left[|e[n]|^{2}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>e</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[|e[n]|^{2}\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff9d46fcdf6b205655f7182df44cc92ec004ce5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.477ex; height:4.843ex;" alt="{\displaystyle E\left[|e[n]|^{2}\right]}"></span> =<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 E\left[e[n]e^{*}[n]\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mrow> <mo>[</mo> <mrow> <mi>e</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\left[e[n]e^{*}[n]\right]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/660e20966e374f35618b457d09451af5d0b1d7b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.054ex; height:2.843ex;" alt="{\displaystyle E\left[e[n]e^{*}[n]\right]}"></span>. This involves computing partial derivatives with respect to both the real and imaginary parts of <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 a_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bc77764b2e74e64a63341054fa90f3e07db275f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.029ex; height:2.009ex;" alt="{\displaystyle a_{i}}"></span>, and requiring them both to be zero. </p><p>The resulting Wiener-Hopf equations are: </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 \sum _{j=0}^{N}R_{w}[j-i]a_{j}^{*}=R_{ws}[i]\qquad i=0,\cdots ,N.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munderover> <mo>∑<!-- ∑ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>j</mi> <mo>−<!-- − --></mo> <mi>i</mi> <mo stretchy="false">]</mo> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo>=</mo> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>i</mi> <mo stretchy="false">]</mo> <mspace width="2em" /> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mo>⋯<!-- ⋯ --></mo> <mo>,</mo> <mi>N</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{j=0}^{N}R_{w}[j-i]a_{j}^{*}=R_{ws}[i]\qquad i=0,\cdots ,N.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a382076f2533f0688f1875d8f3a133be3d531f1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:41.83ex; height:7.676ex;" alt="{\displaystyle \sum _{j=0}^{N}R_{w}[j-i]a_{j}^{*}=R_{ws}[i]\qquad i=0,\cdots ,N.}"></span></dd></dl> <p>which can be rewritten in matrix form: </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 \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}^{*}[1]&\cdots &R_{w}^{*}[N-1]&R_{w}^{*}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}^{*}[N-2]&R_{w}^{*}[N-1]\\\vdots &\vdots &\ddots &\vdots &\vdots \\R_{w}[N-1]&R_{w}[N-2]&\cdots &R_{w}[0]&R_{w}^{*}[1]\\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[1]&R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}^{*}\\a_{1}^{*}\\\vdots \\a_{N-1}^{*}\\a_{N}^{*}\end{bmatrix}} _{\mathbf {a^{*}} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N-1]\\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd> <mo>⋱<!-- ⋱ --></mo> </mtd> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>2</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <mo>⋯<!-- ⋯ --></mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> </mrow> </munder> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mtd> </mtr> <mtr> <mtd> <msubsup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <msup> <mi mathvariant="bold">a</mi> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">∗<!-- ∗ --></mo> </mrow> </msup> </mrow> </mrow> </munder> <mo>=</mo> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋮<!-- ⋮ --></mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo>−<!-- − --></mo> <mn>1</mn> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>N</mi> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> <mo>⏟<!-- ⏟ --></mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </munder> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}^{*}[1]&\cdots &R_{w}^{*}[N-1]&R_{w}^{*}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}^{*}[N-2]&R_{w}^{*}[N-1]\\\vdots &\vdots &\ddots &\vdots &\vdots \\R_{w}[N-1]&R_{w}[N-2]&\cdots &R_{w}[0]&R_{w}^{*}[1]\\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[1]&R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}^{*}\\a_{1}^{*}\\\vdots \\a_{N-1}^{*}\\a_{N}^{*}\end{bmatrix}} _{\mathbf {a^{*}} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N-1]\\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5189a3cd5eb8558934169b1ed63016fa0f5b14a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.005ex; margin-right: -0.028ex; width:85.41ex; height:21.676ex;" alt="{\displaystyle \underbrace {\begin{bmatrix}R_{w}[0]&R_{w}^{*}[1]&\cdots &R_{w}^{*}[N-1]&R_{w}^{*}[N]\\R_{w}[1]&R_{w}[0]&\cdots &R_{w}^{*}[N-2]&R_{w}^{*}[N-1]\\\vdots &\vdots &\ddots &\vdots &\vdots \\R_{w}[N-1]&R_{w}[N-2]&\cdots &R_{w}[0]&R_{w}^{*}[1]\\R_{w}[N]&R_{w}[N-1]&\cdots &R_{w}[1]&R_{w}[0]\end{bmatrix}} _{\mathbf {T} }\underbrace {\begin{bmatrix}a_{0}^{*}\\a_{1}^{*}\\\vdots \\a_{N-1}^{*}\\a_{N}^{*}\end{bmatrix}} _{\mathbf {a^{*}} }=\underbrace {\begin{bmatrix}R_{ws}[0]\\R_{ws}[1]\\\vdots \\R_{ws}[N-1]\\R_{ws}[N]\end{bmatrix}} _{\mathbf {v} }}"></span></dd></dl> <p>Note here that:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}R_{w}[-k]&=R_{w}^{*}[k]\\R_{sw}[k]&=R_{ws}^{*}[-k]\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>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>s</mi> <mi>w</mi> </mrow> </msub> <mo stretchy="false">[</mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>w</mi> <mi>s</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msubsup> <mo stretchy="false">[</mo> <mo>−<!-- − --></mo> <mi>k</mi> <mo stretchy="false">]</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}R_{w}[-k]&=R_{w}^{*}[k]\\R_{sw}[k]&=R_{ws}^{*}[-k]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b747272d7edd8e56a4fa0ae17bc829245cefd5c2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.593ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}R_{w}[-k]&=R_{w}^{*}[k]\\R_{sw}[k]&=R_{ws}^{*}[-k]\end{aligned}}}"></span> </p><p>The Wiener coefficient vector is then computed as:<span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {a} ={(\mathbf {T} ^{-1}\mathbf {v} )}^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">a</mi> </mrow> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗<!-- ∗ --></mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {a} ={(\mathbf {T} ^{-1}\mathbf {v} )}^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abdafbd24f558f53fcabded30c867413de284053" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.864ex; height:3.343ex;" alt="{\displaystyle \mathbf {a} ={(\mathbf {T} ^{-1}\mathbf {v} )}^{*}}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=8" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Wiener filter has a variety of applications in signal processing, image processing,<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> control systems, and digital communications. These applications generally fall into one of four main categories: </p> <ul><li><a href="/wiki/System_identification" title="System identification">System identification</a></li> <li><a href="/wiki/Deconvolution" title="Deconvolution">Deconvolution</a></li> <li><a href="/wiki/Noise_reduction" title="Noise reduction">Noise reduction</a></li> <li><a href="/wiki/Detection_theory" title="Detection theory">Signal detection</a></li></ul> <style data-mw-deduplicate="TemplateStyles:r1273380762/mw-parser-output/.tmulti">.mw-parser-output .tmulti .multiimageinner{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}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner span:not(.skin-invert-image):not(.skin-invert):not(.bg-transparent) img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:204px;max-width:204px"><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Astronaut-noise.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Astronaut-noise.png/200px-Astronaut-noise.png" decoding="async" width="200" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Astronaut-noise.png/300px-Astronaut-noise.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Astronaut-noise.png/400px-Astronaut-noise.png 2x" data-file-width="537" data-file-height="356" /></a></span></div><div class="thumbcaption">Noisy image of an astronaut</div></div></div><div class="trow"><div class="tsingle" style="width:202px;max-width:202px"><div class="thumbimage"><span typeof="mw:File"><a href="/wiki/File:Astronaut-denoised.png" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Astronaut-denoised.png/200px-Astronaut-denoised.png" decoding="async" width="200" height="132" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Astronaut-denoised.png/300px-Astronaut-denoised.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Astronaut-denoised.png/400px-Astronaut-denoised.png 2x" data-file-width="539" data-file-height="357" /></a></span></div><div class="thumbcaption">The image after a Wiener filter is applied (full-view recommended)</div></div></div></div></div> <p>For example, the Wiener filter can be used in image processing to remove noise from a picture. For example, using the Mathematica function: <code>WienerFilter[image,2]</code> on the first image on the right, produces the filtered image below it. </p><p>It is commonly used to denoise audio signals, especially speech, as a preprocessor before <a href="/wiki/Speech_recognition" title="Speech recognition">speech recognition</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=9" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The filter was proposed by <a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Norbert Wiener</a> during the 1940s and published in 1949.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Wiener1949_5-0" class="reference"><a href="#cite_note-Wiener1949-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> The discrete-time equivalent of Wiener's work was derived independently by <a href="/wiki/Andrey_Kolmogorov" title="Andrey Kolmogorov">Andrey Kolmogorov</a> and published in 1941.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> Hence the theory is often called the <i>Wiener–Kolmogorov</i> filtering theory (<i>cf.</i> <a href="/wiki/Kriging" title="Kriging">Kriging</a>). The Wiener filter was the first statistically designed filter to be proposed and subsequently gave rise to many others including the <a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a>. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=10" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1184024115">.mw-parser-output .div-col{margin-top:0.3em;column-width:30em}.mw-parser-output .div-col-small{font-size:90%}.mw-parser-output .div-col-rules{column-rule:1px solid #aaa}.mw-parser-output .div-col dl,.mw-parser-output .div-col ol,.mw-parser-output .div-col ul{margin-top:0}.mw-parser-output .div-col li,.mw-parser-output .div-col dd{page-break-inside:avoid;break-inside:avoid-column}</style><div class="div-col" style="column-width: 25em;"> <ul><li><a href="/wiki/Wiener_deconvolution" title="Wiener deconvolution">Wiener deconvolution</a></li> <li><a href="/wiki/Least_mean_squares_filter" title="Least mean squares filter">Least mean squares filter</a></li> <li><a href="/wiki/Similarities_between_Wiener_and_LMS" title="Similarities between Wiener and LMS">Similarities between Wiener and LMS</a></li> <li><a href="/wiki/Linear_prediction" title="Linear prediction">Linear prediction</a></li> <li><a href="/wiki/Minimum_mean_square_error" title="Minimum mean square error">MMSE estimator</a></li> <li><a href="/wiki/Kalman_filter" title="Kalman filter">Kalman filter</a></li> <li><a href="/wiki/Generalized_Wiener_filter" title="Generalized Wiener filter">Generalized Wiener filter</a></li> <li><a href="/wiki/Matched_filter" title="Matched filter">Matched filter</a></li> <li><a href="/wiki/Information_field_theory" title="Information field theory">Information field theory</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=11" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-Brown1996-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Brown1996_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Brown1996_1-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFBrownHwang1996" class="citation book cs1">Brown, Robert Grover; Hwang, Patrick Y.C. (1996). <i>Introduction to Random Signals and Applied Kalman Filtering</i> (3 ed.). New York: John Wiley & Sons. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-12839-7" title="Special:BookSources/978-0-471-12839-7"><bdi>978-0-471-12839-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Random+Signals+and+Applied+Kalman+Filtering&rft.place=New+York&rft.edition=3&rft.pub=John+Wiley+%26+Sons&rft.date=1996&rft.isbn=978-0-471-12839-7&rft.aulast=Brown&rft.aufirst=Robert+Grover&rft.au=Hwang%2C+Patrick+Y.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWiener+filter" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWelch" class="citation web cs1">Welch, Lloyd R. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060920081221/http://csi.usc.edu/PDF/wienerhopf.pdf">"Wiener–Hopf Theory"</a> <span class="cs1-format">(PDF)</span>. Archived from <a rel="nofollow" class="external text" href="http://csi.usc.edu/PDF/wienerhopf.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-09-20<span class="reference-accessdate">. Retrieved <span class="nowrap">2006-11-25</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Wiener%E2%80%93Hopf+Theory&rft.aulast=Welch&rft.aufirst=Lloyd+R&rft_id=http%3A%2F%2Fcsi.usc.edu%2FPDF%2Fwienerhopf.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWiener+filter" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBoulfelfelRangayyanHahnKloiber1994" class="citation journal cs1">Boulfelfel, D.; Rangayyan, R. M.; Hahn, L. J.; Kloiber, R. (1994). "Three-dimensional restoration of single photon emission computed tomography images". <i>IEEE Transactions on Nuclear Science</i>. <b>41</b> (5): <span class="nowrap">1746–</span>1754. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1994ITNS...41.1746B">1994ITNS...41.1746B</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F23.317385">10.1109/23.317385</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:33708058">33708058</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Nuclear+Science&rft.atitle=Three-dimensional+restoration+of+single+photon+emission+computed+tomography+images&rft.volume=41&rft.issue=5&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1746-%3C%2Fspan%3E1754&rft.date=1994&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A33708058%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2F23.317385&rft_id=info%3Abibcode%2F1994ITNS...41.1746B&rft.aulast=Boulfelfel&rft.aufirst=D.&rft.au=Rangayyan%2C+R.+M.&rft.au=Hahn%2C+L.+J.&rft.au=Kloiber%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWiener+filter" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text">Wiener N: The interpolation, extrapolation and smoothing of stationary time series', Report of the Services 19, Research Project DIC-6037 MIT, February 1942</span> </li> <li id="cite_note-Wiener1949-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-Wiener1949_5-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWiener1949" class="citation book cs1"><a href="/wiki/Norbert_Wiener" title="Norbert Wiener">Wiener, Norbert</a> (1949). <a rel="nofollow" class="external text" href="https://direct.mit.edu/books/oa-monograph/4361/Extrapolation-Interpolation-and-Smoothing-of"><i>Extrapolation, Interpolation, and Smoothing of Stationary Time Series: With Engineering Applications</i></a>. <a href="/wiki/MIT_Press" title="MIT Press">MIT Press</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262257190" title="Special:BookSources/9780262257190"><bdi>9780262257190</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Extrapolation%2C+Interpolation%2C+and+Smoothing+of+Stationary+Time+Series%3A+With+Engineering+Applications&rft.pub=MIT+Press&rft.date=1949&rft.isbn=9780262257190&rft.aulast=Wiener&rft.aufirst=Norbert&rft_id=https%3A%2F%2Fdirect.mit.edu%2Fbooks%2Foa-monograph%2F4361%2FExtrapolation-Interpolation-and-Smoothing-of&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWiener+filter" class="Z3988"></span></span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text">Kolmogorov A.N: 'Stationary sequences in Hilbert space', (In Russian) Bull. Moscow Univ. 1941 vol.2 no.6 1-40. English translation in Kailath T. (ed.) <i>Linear least squares estimation</i> Dowden, Hutchinson & Ross 1977 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-87933-098-8" title="Special:BookSources/0-87933-098-8">0-87933-098-8</a></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=12" title="Edit section: Further reading"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Thomas_Kailath" title="Thomas Kailath">Thomas Kailath</a>, <a href="/wiki/Ali_H._Sayed" title="Ali H. Sayed">Ali H. Sayed</a>, and <a href="/wiki/Babak_Hassibi" title="Babak Hassibi">Babak Hassibi</a>, Linear Estimation, Prentice-Hall, NJ, 2000, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-022464-4" title="Special:BookSources/978-0-13-022464-4">978-0-13-022464-4</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Wiener_filter&action=edit&section=13" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Mathematica <a rel="nofollow" class="external text" href="http://reference.wolfram.com/mathematica/ref/WienerFilter.html">WienerFilter</a> function</li></ul> <!-- NewPP limit report Parsed by mw‐web.codfw.main‐84749c7844‐5vcjv Cached time: 20250210050533 Cache expiry: 2592000 Reduced expiry: false Complications: [vary‐revision‐sha1, show‐toc] CPU time usage: 0.354 seconds Real time usage: 0.550 seconds Preprocessor visited node count: 1447/1000000 Post‐expand include size: 16630/2097152 bytes Template argument size: 813/2097152 bytes Highest expansion depth: 16/100 Expensive parser function count: 5/500 Unstrip recursion depth: 1/20 Unstrip post‐expand size: 27890/5000000 bytes Lua time usage: 0.200/10.000 seconds Lua memory usage: 4915882/52428800 bytes Number of Wikibase entities loaded: 0/400 --> <!-- Transclusion expansion time report (%,ms,calls,template) 100.00% 356.985 1 -total 46.51% 166.018 1 Template:Reflist 31.40% 112.084 2 Template:Cite_book 22.95% 81.938 1 Template:Short_description 14.38% 51.335 2 Template:Pagetype 13.46% 48.043 1 Template:Lead_too_short 11.92% 42.570 1 Template:Ambox 6.98% 24.907 2 Template:ISBN 6.35% 22.669 1 Template:Distinguish 6.01% 21.458 1 Template:Multiple_image --> <!-- Saved in parser cache with key enwiki:pcache:1216721:|#|:idhash:canonical and timestamp 20250210050533 and revision id 1269313226. 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