Comb filter - Wikipedia

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class="vector-toc-numb">2.2.3</span> <span>Pole–zero interpretation</span> </div> </a> <ul id="toc-Pole–zero_interpretation_2-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Continuous_time_implementation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Continuous_time_implementation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Continuous time implementation</span> </div> </a> <button aria-controls="toc-Continuous_time_implementation-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 Continuous time implementation subsection</span> </button> <ul id="toc-Continuous_time_implementation-sublist" class="vector-toc-list"> <li id="toc-Feedforward_form_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feedforward_form_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.1</span> <span>Feedforward form</span> </div> </a> <ul id="toc-Feedforward_form_2-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Feedback_form_2" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Feedback_form_2"> <div class="vector-toc-text"> <span class="vector-toc-numb">3.2</span> <span>Feedback form</span> </div> </a> <ul id="toc-Feedback_form_2-sublist" class="vector-toc-list"> </ul> </li> </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">4</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">5</span> <span>References</span> </div> </a> <ul id="toc-References-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">6</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" 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title="Kammfilter – German" lang="de" hreflang="de" data-title="Kammfilter" 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/Kammfilter" title="Kammfilter – Estonian" lang="et" hreflang="et" data-title="Kammfilter" 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_comb" title="Filtro comb – Spanish" lang="es" hreflang="es" data-title="Filtro comb" 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_%D8%B4%D8%A7%D9%86%D9%87%E2%80%8C%D8%A7%DB%8C" title="فیلتر شانه‌ای – Persian" lang="fa" hreflang="fa" data-title="فیلتر شانه‌ای" 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_en_peigne" title="Filtre en peigne – French" lang="fr" hreflang="fr" data-title="Filtre en peigne" 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_comb" title="Filtro comb – Italian" lang="it" hreflang="it" data-title="Filtro comb" 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/Kamfilter" title="Kamfilter – Dutch" lang="nl" hreflang="nl" data-title="Kamfilter" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B3%E3%83%A0%E3%83%95%E3%82%A3%E3%83%AB%E3%82%BF" title="コムフィルタ – Japanese" lang="ja" hreflang="ja" data-title="コムフィルタ" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Filtr_grzebieniowy" title="Filtr grzebieniowy – Polish" lang="pl" hreflang="pl" data-title="Filtr grzebieniowy" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Filtro_Comb" title="Filtro Comb – Portuguese" lang="pt" hreflang="pt" data-title="Filtro Comb" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%93%D1%80%D0%B5%D0%B1%D0%B5%D0%BD%D1%87%D0%B0%D1%82%D1%8B%D0%B9_%D1%84%D0%B8%D0%BB%D1%8C%D1%82%D1%80" title="Гребенчатый фильтр – Russian" lang="ru" hreflang="ru" data-title="Гребенчатый фильтр" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Kamfilter" title="Kamfilter – Swedish" lang="sv" hreflang="sv" data-title="Kamfilter" data-language-autonym="Svenska" data-language-local-name="Swedish" 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class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Signal processing filter</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_feedforward.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/400px-Comb_filter_feedforward.svg.png" decoding="async" width="400" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/600px-Comb_filter_feedforward.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/800px-Comb_filter_feedforward.svg.png 2x" data-file-width="512" data-file-height="200" /></a><figcaption>Feedforward comb filter structure</figcaption></figure> <p>In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, a <b>comb filter</b> is a <a href="/wiki/Filter_(signal_processing)" title="Filter (signal processing)">filter</a> implemented by adding a delayed version of a <a href="/wiki/Signal_processing" title="Signal processing">signal</a> to itself, causing constructive and destructive <a href="/wiki/Interference_(wave_propagation)" class="mw-redirect" title="Interference (wave propagation)">interference</a>. The <a href="/wiki/Frequency_response" title="Frequency response">frequency response</a> of a comb filter consists of a series of regularly spaced notches in between regularly spaced <i>peaks</i> (sometimes called <i>teeth</i>) giving the appearance of a <a href="/wiki/Comb" title="Comb">comb</a>. </p><p>Comb filters exist in two forms, <i>feedforward</i> and <i><a href="/wiki/Feedback" title="Feedback">feedback</a></i>; which refer to the direction in which signals are delayed before they are added to the input. </p><p>Comb filters may be implemented in <a href="/wiki/Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a> or <a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> forms which are very similar. </p> <meta property="mw:PageProp/toc" /> <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=Comb_filter&action=edit&section=1" title="Edit section: Applications"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg/220px-Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/48/Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg/330px-Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/48/Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg/440px-Sony_VPL-HS1_-_EP-GW_1-682-352-12_-_Motorola_MC141627FT-1.jpg 2x" data-file-width="3053" data-file-height="3053" /></a><figcaption>Advanced PAL Comb Filter-II (APCF-II, Motorola MC141627FT)</figcaption></figure> <p>Comb filters are employed in a variety of signal processing applications, including: </p> <ul><li><a href="/wiki/Cascaded_integrator%E2%80%93comb_filter" title="Cascaded integrator–comb filter">Cascaded integrator–comb</a> (CIC) filters, commonly used for <a href="/wiki/Anti-aliasing_filter" title="Anti-aliasing filter">anti-aliasing</a> during <a href="/wiki/Interpolation" title="Interpolation">interpolation</a> and <a href="/wiki/Decimation_(signal_processing)" class="mw-redirect" title="Decimation (signal processing)">decimation</a> operations that change the <a href="/wiki/Sample_rate" class="mw-redirect" title="Sample rate">sample rate</a> of a discrete-time system.</li> <li>2D and 3D comb filters implemented in hardware (and occasionally software) in <a href="/wiki/PAL" title="PAL">PAL</a> and <a href="/wiki/NTSC" title="NTSC">NTSC</a> analog television decoders, reduce artifacts such as <a href="/wiki/Dot_crawl" title="Dot crawl">dot crawl</a>.</li> <li><a href="/wiki/Audio_signal_processing" title="Audio signal processing">Audio signal processing</a>, including <a href="/wiki/Delay_(audio_effect)" title="Delay (audio effect)">delay</a>, <a href="/wiki/Flanging" title="Flanging">flanging</a>, <a href="/wiki/Physical_modelling_synthesis" title="Physical modelling synthesis">physical modelling synthesis</a> and <a href="/wiki/Digital_waveguide_synthesis" title="Digital waveguide synthesis">digital waveguide synthesis</a>. If the delay is set to a few milliseconds, a comb filter can model the effect of <a href="/wiki/Acoustics" title="Acoustics">acoustic</a> <a href="/wiki/Standing_waves" class="mw-redirect" title="Standing waves">standing waves</a> in a cylindrical cavity or <a href="/wiki/Karplus-Strong_string_synthesis" class="mw-redirect" title="Karplus-Strong string synthesis">in a vibrating string</a>.</li> <li>In astronomy the <a href="/wiki/Astro-comb" title="Astro-comb">astro-comb</a> promises to increase the precision of existing <a href="/wiki/Spectrograph" class="mw-redirect" title="Spectrograph">spectrographs</a> by nearly a hundredfold.</li></ul> <p>In <a href="/wiki/Acoustics" title="Acoustics">acoustics</a>, comb filtering can arise as an unwanted artifact. For instance, two <a href="/wiki/Loudspeakers" class="mw-redirect" title="Loudspeakers">loudspeakers</a> playing the same signal at different distances from the listener, create a comb filtering effect on the audio.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> In any enclosed space, listeners hear a mixture of direct sound and reflected sound. The reflected sound takes a longer, delayed path compared to the direct sound, and a comb filter is created where the two mix at the listener.<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> Similarly, comb filtering may result from mono mixing of multiple mics, hence the 3:1 <a href="/wiki/Rule_of_thumb" title="Rule of thumb">rule of thumb</a> that neighboring mics should be separated at least three times the distance from its source to the mic.<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> </p> <div class="mw-heading mw-heading2"><h2 id="Discrete_time_implementation">Discrete time implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=2" title="Edit section: Discrete time implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Feedforward_form">Feedforward form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=3" title="Edit section: Feedforward form"><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:Comb_filter_feedforward.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/400px-Comb_filter_feedforward.svg.png" decoding="async" width="400" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/600px-Comb_filter_feedforward.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/13/Comb_filter_feedforward.svg/800px-Comb_filter_feedforward.svg.png 2x" data-file-width="512" data-file-height="200" /></a><figcaption>Feedforward comb filter structure in discrete time</figcaption></figure> <p>The general structure of a feedforward comb filter is described by the <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>: </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 y[n]=x[n]+\alpha x[n-K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=x[n]+\alpha x[n-K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/407d8d32def12f79789bde99eccb4d859fd98281" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.213ex; height:2.843ex;" alt="{\displaystyle y[n]=x[n]+\alpha x[n-K]}" /></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 K}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>K</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle K}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b76fce82a62ed5461908f0dc8f037de4e3686b0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.066ex; height:2.176ex;" alt="{\displaystyle K}" /></span> is the delay length (measured in samples), and <span class="texhtml"><i>α</i></span> is a scaling factor applied to the delayed signal. The <a href="/wiki/Z_transform" class="mw-redirect" title="Z transform"><span class="texhtml"><i>z</i></span> transform</a> of both sides of the equation yields: </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 Y(z)=\left(1+\alpha z^{-K}\right)X(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mi>α</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>K</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Y(z)=\left(1+\alpha z^{-K}\right)X(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfa5d7709737a8ddbf3391d7b9b044f81a89662" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.716ex; height:3.343ex;" alt="{\displaystyle Y(z)=\left(1+\alpha z^{-K}\right)X(z)}" /></span></dd></dl> <p>The <a href="/wiki/Z_transform#Transfer_function" class="mw-redirect" title="Z transform">transfer function</a> is defined 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 H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>K</mi> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mo>+</mo> <mi>α</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eff258dac98ab60794691f7d70e7599c717910d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:37.47ex; height:6.509ex;" alt="{\displaystyle H(z)={\frac {Y(z)}{X(z)}}=1+\alpha z^{-K}={\frac {z^{K}+\alpha }{z^{K}}}}" /></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Frequency_response">Frequency response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=4" title="Edit section: Frequency response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_response_ff_pos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Comb_filter_response_ff_pos.svg/220px-Comb_filter_response_ff_pos.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/99/Comb_filter_response_ff_pos.svg/330px-Comb_filter_response_ff_pos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/99/Comb_filter_response_ff_pos.svg/440px-Comb_filter_response_ff_pos.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Feedforward magnitude response for various <i>positive</i> values of <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>K</i> = 1</span> in discrete time</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_response_ff_neg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Comb_filter_response_ff_neg.svg/220px-Comb_filter_response_ff_neg.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Comb_filter_response_ff_neg.svg/330px-Comb_filter_response_ff_neg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f3/Comb_filter_response_ff_neg.svg/440px-Comb_filter_response_ff_neg.svg.png 2x" data-file-width="512" data-file-height="512" /></a><figcaption>Feedforward magnitude response for various <i>negative</i> values of <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>K</i> = 1</span> in discrete time</figcaption></figure> <p>The frequency response of a discrete-time system expressed in the <span class="texhtml"><i>z</i></span>-domain is obtained by substitution <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 z=e^{j\omega },}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=e^{j\omega },}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa76f71f544ba22f3c3b9b34318264aa7ecdffb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.849ex; height:3.009ex;" alt="{\displaystyle z=e^{j\omega },}" /></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 j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}" /></span> is the <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> 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 \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" /></span> is <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>. Therefore, for the feedforward comb filter: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>j</mi> <mi>ω</mi> <mi>K</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/193cdd759f4a92677d980906fde29a2753c1f1e7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:21.94ex; height:3.343ex;" alt="{\displaystyle H\left(e^{j\omega }\right)=1+\alpha e^{-j\omega K}}" /></span></dd></dl> <p>Using <a href="/wiki/Euler%27s_formula" title="Euler's formula">Euler's formula</a>, the frequency response is also given by </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\left(e^{j\omega }\right)={\bigl [}1+\alpha \cos(\omega K){\bigr ]}-j\alpha \sin(\omega K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>−</mo> <mi>j</mi> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(e^{j\omega }\right)={\bigl [}1+\alpha \cos(\omega K){\bigr ]}-j\alpha \sin(\omega K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a4500af1719c579363396c75f6f633b01eea232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:40.793ex; height:3.343ex;" alt="{\displaystyle H\left(e^{j\omega }\right)={\bigl [}1+\alpha \cos(\omega K){\bigr ]}-j\alpha \sin(\omega K)}" /></span></dd></dl> <p>Often of interest is the <i>magnitude</i> response, which ignores phase. This is defined 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 \left|H\left(e^{j\omega }\right)\right|={\sqrt {\Re \left\{H\left(e^{j\omega }\right)\right\}^{2}+\Im \left\{H\left(e^{j\omega }\right)\right\}^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi mathvariant="normal">ℜ</mi> <msup> <mrow> <mo>{</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi mathvariant="normal">ℑ</mi> <msup> <mrow> <mo>{</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\sqrt {\Re \left\{H\left(e^{j\omega }\right)\right\}^{2}+\Im \left\{H\left(e^{j\omega }\right)\right\}^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0be41ed9b90a0e2d8253142ef3c9c9399c6a51ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:43.086ex; height:4.843ex;" alt="{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\sqrt {\Re \left\{H\left(e^{j\omega }\right)\right\}^{2}+\Im \left\{H\left(e^{j\omega }\right)\right\}^{2}}}}" /></span></dd></dl> <p>In the case of the feedforward comb filter, this 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 {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega 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> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mo stretchy="false">(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> </msqrt> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega K)}}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1d3d832ebcc48e7fbace11b4b1ee3cf7e9813ae" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.338ex; width:47.284ex; height:9.843ex;" alt="{\displaystyle {\begin{aligned}\left|H(e^{j\omega })\right|&={\sqrt {(1+\alpha \cos(\omega K))^{2}+(\alpha \sin(\omega K))^{2}}}\\&={\sqrt {(1+\alpha ^{2})+2\alpha \cos(\omega K)}}\end{aligned}}}" /></span></dd></dl> <p>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 (1+\alpha ^{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1+\alpha ^{2})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8db092a02a8d91ab21f8e0a86ec7519df1c490" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.354ex; height:3.176ex;" alt="{\displaystyle (1+\alpha ^{2})}" /></span> term is constant, whereas 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 2\alpha \cos(\omega K)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\alpha \cos(\omega K)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ffee3bc128afee932a0caa76a1b59c324bf055e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.469ex; height:2.843ex;" alt="{\displaystyle 2\alpha \cos(\omega K)}" /></span> term varies <a href="/wiki/Periodic_function" title="Periodic function">periodically</a>. Hence the magnitude response of the comb filter is periodic. </p><p>The graphs show the periodic magnitude response for various values 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 \alpha .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>.</mo> </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/794850adc0db51d11a6d8cfa857538183424909c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.134ex; height:1.676ex;" alt="{\displaystyle \alpha .}" /></span> Some important properties: </p> <ul><li>The response periodically drops to a <a href="/wiki/Local_minimum" class="mw-redirect" title="Local minimum">local minimum</a> (sometimes known as a <i>notch</i>), and periodically rises to a <a href="/wiki/Local_maximum" class="mw-redirect" title="Local maximum">local maximum</a> (sometimes known as a <i>peak</i> or a <i>tooth</i>).</li> <li>For positive values 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 \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> </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/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="{\displaystyle \alpha ,}" /></span> the first minimum occurs at half the delay period and repeats at even multiples of the delay frequency thereafter:</li></ul> <dl><dd><dl><dd><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}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\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>f</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>2</mn> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>5</mn> <mrow> <mn>2</mn> <mi>K</mi> </mrow> </mfrac> </mrow> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>π</mi> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>π</mi> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>5</mn> <mi>π</mi> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>⋯</mo> <mspace width="thinmathspace"></mspace> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31a0ba61e6582590143c70e574a8e80b1d63076b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.524ex; margin-bottom: -0.314ex; width:22.668ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}f&={\frac {1}{2K}},{\frac {3}{2K}},{\frac {5}{2K}}\cdots \\\omega &={\frac {\pi }{K}},{\frac {3\pi }{K}},{\frac {5\pi }{K}}\cdots \,\end{aligned}}}" /></span></dd></dl></dd></dl> <ul><li>The levels of the maxima and minima are always equidistant from 1.</li> <li>When <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 =\pm 1,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =\pm 1,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d6592e980097036d911f0981f694127deda38984" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.204ex; height:2.509ex;" alt="{\displaystyle \alpha =\pm 1,}" /></span> the minima have zero amplitude. In this case, the minima are sometimes known as <i>nulls</i>.</li> <li>The maxima for positive values 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>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> coincide with the minima for negative values 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>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>, and vice versa.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Impulse_response">Impulse response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=5" title="Edit section: Impulse response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The feedforward comb filter is one of the simplest <a href="/wiki/Finite_impulse_response" title="Finite impulse response">finite impulse response</a> filters.<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> Its response is simply the initial impulse with a second impulse after the delay. </p> <div class="mw-heading mw-heading4"><h4 id="Pole–zero_interpretation"><span id="Pole.E2.80.93zero_interpretation"></span>Pole–zero interpretation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=6" title="Edit section: Pole–zero interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Looking again at the <span class="texhtml"><i>z</i></span>-domain transfer function of the feedforward comb filter: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)={\frac {z^{K}+\alpha }{z^{K}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mo>+</mo> <mi>α</mi> </mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {z^{K}+\alpha }{z^{K}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df4138f256655ad857b8a7e2457bd0ddf01578c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:16.007ex; height:6.009ex;" alt="{\displaystyle H(z)={\frac {z^{K}+\alpha }{z^{K}}}}" /></span></dd></dl> <p>the numerator is equal to zero whenever <span class="texhtml"><i>z<sup>K</sup></i> = −<i>α</i></span>. This has <span class="texhtml"><i>K</i></span> solutions, equally spaced around a circle in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>; these are the <a href="/wiki/Zero_(complex_analysis)" class="mw-redirect" title="Zero (complex analysis)">zeros</a> of the transfer function. The denominator is zero at <span class="texhtml"><i>z<sup>K</sup></i> = 0</span>, giving <span class="texhtml"><i>K</i></span> <a href="/wiki/Pole_(complex_analysis)" class="mw-redirect" title="Pole (complex analysis)">poles</a> at <span class="texhtml"><i>z</i> = 0</span>. This leads to a <a href="/wiki/Pole%E2%80%93zero_plot" title="Pole–zero plot">pole–zero plot</a> like the ones shown. </p> <table> <tbody><tr> <td><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_pz_ff_pos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Comb_filter_pz_ff_pos.svg/200px-Comb_filter_pz_ff_pos.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Comb_filter_pz_ff_pos.svg/300px-Comb_filter_pz_ff_pos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1f/Comb_filter_pz_ff_pos.svg/400px-Comb_filter_pz_ff_pos.svg.png 2x" data-file-width="370" data-file-height="370" /></a><figcaption>Pole–zero plot of feedforward comb filter with <span class="texhtml"><i>K</i> = 8</span> and <span class="texhtml"><i>α</i> = 0.5</span> in discrete time</figcaption></figure> </td> <td><figure class="mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_pz_ff_neg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Comb_filter_pz_ff_neg.svg/200px-Comb_filter_pz_ff_neg.svg.png" decoding="async" width="200" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/Comb_filter_pz_ff_neg.svg/300px-Comb_filter_pz_ff_neg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/Comb_filter_pz_ff_neg.svg/400px-Comb_filter_pz_ff_neg.svg.png 2x" data-file-width="370" data-file-height="370" /></a><figcaption>Pole–zero plot of feedforward comb filter with <span class="texhtml"><i>K</i> = 8</span> and <span class="texhtml"><i>α</i> = −0.5</span> in discrete time</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Feedback_form">Feedback form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=7" title="Edit section: Feedback form"><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:Comb_filter_feedback.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Comb_filter_feedback.svg/400px-Comb_filter_feedback.svg.png" decoding="async" width="400" height="156" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Comb_filter_feedback.svg/600px-Comb_filter_feedback.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Comb_filter_feedback.svg/800px-Comb_filter_feedback.svg.png 2x" data-file-width="512" data-file-height="200" /></a><figcaption>Feedback comb filter structure in discrete time</figcaption></figure> <p>Similarly, the general structure of a feedback comb filter is described by the <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equation</a>: </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 y[n]=x[n]+\alpha y[n-K]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo stretchy="false">]</mo> <mo>+</mo> <mi>α</mi> <mi>y</mi> <mo stretchy="false">[</mo> <mi>n</mi> <mo>−</mo> <mi>K</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y[n]=x[n]+\alpha y[n-K]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aab660dee16e819787a26d5de6162532f6da0da3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.039ex; height:2.843ex;" alt="{\displaystyle y[n]=x[n]+\alpha y[n-K]}" /></span></dd></dl> <p>This equation can be rearranged so that all terms in <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span> are on the left-hand side, and then taking the <span class="texhtml"><i>z</i></span> transform: </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 \left(1-\alpha z^{-K}\right)Y(z)=X(z)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>K</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(1-\alpha z^{-K}\right)Y(z)=X(z)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c91c76b4f72669cceaa7a5755d855ff9072d78eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:24.716ex; height:3.343ex;" alt="{\displaystyle \left(1-\alpha z^{-K}\right)Y(z)=X(z)}" /></span></dd></dl> <p>The transfer function is therefore: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>Y</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>K</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fd393568974a4f56acb9fca943bc96e3a70e9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:38.306ex; height:6.509ex;" alt="{\displaystyle H(z)={\frac {Y(z)}{X(z)}}={\frac {1}{1-\alpha z^{-K}}}={\frac {z^{K}}{z^{K}-\alpha }}}" /></span></dd></dl> <div class="mw-heading mw-heading4"><h4 id="Frequency_response_2">Frequency response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=8" title="Edit section: Frequency response"><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:Comb_filter_response_fb_pos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Comb_filter_response_fb_pos.svg/400px-Comb_filter_response_fb_pos.svg.png" decoding="async" width="400" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Comb_filter_response_fb_pos.svg/600px-Comb_filter_response_fb_pos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c1/Comb_filter_response_fb_pos.svg/800px-Comb_filter_response_fb_pos.svg.png 2x" data-file-width="748" data-file-height="367" /></a><figcaption>Feedback magnitude response for various <i>positive</i> values of <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>K</i> = 2</span> in discrete time</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_response_fb_neg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Comb_filter_response_fb_neg.svg/400px-Comb_filter_response_fb_neg.svg.png" decoding="async" width="400" height="196" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Comb_filter_response_fb_neg.svg/600px-Comb_filter_response_fb_neg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Comb_filter_response_fb_neg.svg/800px-Comb_filter_response_fb_neg.svg.png 2x" data-file-width="748" data-file-height="367" /></a><figcaption>Feedback magnitude response for various <i>negative</i> values of <span class="texhtml"><i>α</i></span> and <span class="texhtml"><i>K</i> = 2</span> in discrete time</figcaption></figure> <p>By substituting <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 z=e^{j\omega }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=e^{j\omega }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e38263aebb505f6e3125c67a9c0776db679dfb02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.202ex; height:2.676ex;" alt="{\displaystyle z=e^{j\omega }}" /></span> into the feedback comb filter's <span class="texhtml"><i>z</i></span>-domain 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 H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>j</mi> <mi>ω</mi> <mi>K</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff31bbe743d6ae7d09231148be90989d27b03830" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:23.81ex; height:5.676ex;" alt="{\displaystyle H\left(e^{j\omega }\right)={\frac {1}{1-\alpha e^{-j\omega K}}}\,,}" /></span></dd></dl> <p>the magnitude response becomes: </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 \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>|</mo> <mrow> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mi>ω</mi> </mrow> </msup> <mo>)</mo> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>2</mn> <mi>α</mi> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>K</mi> <mo stretchy="false">)</mo> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/369b1eae51ef51289cc27d400504163fd59c70e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:38.846ex; height:6.509ex;" alt="{\displaystyle \left|H\left(e^{j\omega }\right)\right|={\frac {1}{\sqrt {\left(1+\alpha ^{2}\right)-2\alpha \cos(\omega K)}}}\,.}" /></span></dd></dl> <p>Again, the response is periodic, as the graphs demonstrate. The feedback comb filter has some properties in common with the feedforward form: </p> <ul><li>The response periodically drops to a local minimum and rises to a local maximum.</li> <li>The maxima for positive values 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>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> coincide with the minima for negative values 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 \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> </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/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="{\displaystyle \alpha ,}" /></span> and vice versa.</li> <li>For positive values 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 \alpha ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>,</mo> </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/4b2cc8f6d373595f06dcd33f127dadf2b9d5727f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.134ex; height:2.009ex;" alt="{\displaystyle \alpha ,}" /></span> the first maximum occurs at 0 and repeats at even multiples of the delay frequency thereafter:</li></ul> <dl><dd><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}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \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>f</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mi>K</mi> </mfrac> </mrow> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd> <mi>ω</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>4</mn> <mi>π</mi> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>6</mn> <mi>π</mi> </mrow> <mi>K</mi> </mfrac> </mrow> <mo>⋯</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/425ca342b7798c0a5f2b507afbab8818da6c8ec1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.537ex; margin-bottom: -0.301ex; width:22.662ex; height:10.843ex;" alt="{\displaystyle {\begin{aligned}f&=0,{\frac {1}{K}},{\frac {2}{K}},{\frac {3}{K}}\cdots \\\omega &=0,{\frac {2\pi }{K}},{\frac {4\pi }{K}},{\frac {6\pi }{K}}\cdots \end{aligned}}}" /></span></dd></dl></dd></dl> <p>However, there are also some important differences because the magnitude response has a term in the <a href="/wiki/Denominator" class="mw-redirect" title="Denominator">denominator</a>: </p> <ul><li>The levels of the maxima and minima are no longer equidistant from 1. The maxima have an amplitude of <span class="texhtml"><style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style><span class="sfrac">⁠<span class="tion"><span class="num">1</span><span class="sr-only">/</span><span class="den">1 − <i>α</i></span></span>⁠</span></span>.</li> <li>The filter is only <a href="/wiki/BIBO_stability" title="BIBO stability">stable</a> if <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>α</i></span>|</span> is strictly less than 1. As can be seen from the graphs, as <span class="texhtml">|<span class="nowrap" style="padding-left:0.1em; padding-right:0.1em;"><i>α</i></span>|</span> increases, the amplitude of the maxima rises increasingly rapidly.</li></ul> <div class="mw-heading mw-heading4"><h4 id="Impulse_response_2">Impulse response</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=9" title="Edit section: Impulse response"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The feedback comb filter is a simple type of <a href="/wiki/Infinite_impulse_response" title="Infinite impulse response">infinite impulse response</a> filter.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> If stable, the response simply consists of a repeating series of impulses decreasing in amplitude over time. </p> <div class="mw-heading mw-heading4"><h4 id="Pole–zero_interpretation_2"><span id="Pole.E2.80.93zero_interpretation_2"></span>Pole–zero interpretation</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=10" title="Edit section: Pole–zero interpretation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Looking again at the <span class="texhtml"><i>z</i></span>-domain transfer function of the feedback comb filter: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(z)={\frac {z^{K}}{z^{K}-\alpha }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mrow> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </msup> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(z)={\frac {z^{K}}{z^{K}-\alpha }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04eff753ef5f20efb9db031d3b3986eb38d6de27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:16.007ex; height:6.176ex;" alt="{\displaystyle H(z)={\frac {z^{K}}{z^{K}-\alpha }}}" /></span></dd></dl> <p>This time, the numerator is zero at <span class="texhtml"><i>z<sup>K</sup></i> = 0</span>, giving <span class="texhtml"><i>K</i></span> zeros at <span class="texhtml"><i>z</i> = 0</span>. The denominator is equal to zero whenever <span class="texhtml"><i>z<sup>K</sup></i> = <i>α</i></span>. This has <span class="texhtml"><i>K</i></span> solutions, equally spaced around a circle in the <a href="/wiki/Complex_plane" title="Complex plane">complex plane</a>; these are the poles of the transfer function. This leads to a pole–zero plot like the ones shown below. </p> <table> <tbody><tr> <td><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_pz_fb_pos.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Comb_filter_pz_fb_pos.svg/220px-Comb_filter_pz_fb_pos.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Comb_filter_pz_fb_pos.svg/330px-Comb_filter_pz_fb_pos.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e3/Comb_filter_pz_fb_pos.svg/440px-Comb_filter_pz_fb_pos.svg.png 2x" data-file-width="370" data-file-height="370" /></a><figcaption>Pole–zero plot of feedback comb filter with <span class="texhtml"><i>K</i> = 8</span> and <span class="texhtml"><i>α</i> = 0.5</span> in discrete time</figcaption></figure> </td> <td><figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Comb_filter_pz_fb_neg.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Comb_filter_pz_fb_neg.svg/220px-Comb_filter_pz_fb_neg.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Comb_filter_pz_fb_neg.svg/330px-Comb_filter_pz_fb_neg.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/9/9f/Comb_filter_pz_fb_neg.svg/440px-Comb_filter_pz_fb_neg.svg.png 2x" data-file-width="370" data-file-height="370" /></a><figcaption>Pole–zero plot of feedback comb filter with <span class="texhtml"><i>K</i> = 8</span> and <span class="texhtml"><i>α</i> = −0.5</span> in discrete time</figcaption></figure> </td></tr></tbody></table> <div class="mw-heading mw-heading2"><h2 id="Continuous_time_implementation">Continuous time implementation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=11" title="Edit section: Continuous time implementation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Comb filters may also be implemented in <a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> which can be expressed in the <a href="/wiki/Laplace_domain" class="mw-redirect" title="Laplace domain">Laplace domain</a> as a function of the <a href="/wiki/Complex_number" title="Complex number">complex</a> frequency domain 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 s=\sigma +j\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>=</mo> <mi>σ</mi> <mo>+</mo> <mi>j</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s=\sigma +j\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/52b9a99cfe16a2003294604462f41acc7863fe66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.763ex; height:2.509ex;" alt="{\displaystyle s=\sigma +j\omega }" /></span> analogous to the z domain. <a href="/wiki/Analog_circuits" class="mw-redirect" title="Analog circuits">Analog circuits</a> use some form of <a href="/wiki/Analog_delay_line" title="Analog delay line">analog delay line</a> for the delay element. Continuous-time implementations share all the properties of the respective discrete-time implementations. </p> <div class="mw-heading mw-heading3"><h3 id="Feedforward_form_2">Feedforward form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=12" title="Edit section: Feedforward form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The feedforward form may be described by the equation: </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 y(t)=x(t)+\alpha x(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=x(t)+\alpha x(t-\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5bf3b9bbc46490c172faf2b4aa3229082a91a8af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.23ex; height:2.843ex;" alt="{\displaystyle y(t)=x(t)+\alpha x(t-\tau )}" /></span></dd></dl> <p>where <span class="texhtml"><i>τ</i></span> is the delay (measured in seconds). This has the following transfer function: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)=1+\alpha e^{-s\tau }}"> <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> <mo>=</mo> <mn>1</mn> <mo>+</mo> <mi>α</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>τ</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)=1+\alpha e^{-s\tau }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4c63079104fb45331cf0e8793157e2c4c90e4dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.768ex; height:3.009ex;" alt="{\displaystyle H(s)=1+\alpha e^{-s\tau }}" /></span></dd></dl> <p>The feedforward form consists of an infinite number of zeros spaced along the jω axis (<a href="/wiki/Laplace_transform#Fourier_transform" title="Laplace transform">which corresponds to the Fourier domain</a>). </p> <div class="mw-heading mw-heading3"><h3 id="Feedback_form_2">Feedback form</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Comb_filter&action=edit&section=13" title="Edit section: Feedback form"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The feedback form has the equation: </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 y(t)=x(t)+\alpha y(t-\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>α</mi> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)=x(t)+\alpha y(t-\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc9b316e35ef8a76d639ed2b9e59ccfa588a0389" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.056ex; height:2.843ex;" alt="{\displaystyle y(t)=x(t)+\alpha y(t-\tau )}" /></span></dd></dl> <p>and the following transfer function: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(s)={\frac {1}{1-\alpha e^{-s\tau }}}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>s</mi> <mi>τ</mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(s)={\frac {1}{1-\alpha e^{-s\tau }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fef50233bd61d68e49769e98b080059572cfb9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:18.604ex; height:5.509ex;" alt="{\displaystyle H(s)={\frac {1}{1-\alpha e^{-s\tau }}}}" /></span></dd></dl> <p>The feedback form consists of an infinite number of poles spaced along the jω axis. </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=Comb_filter&action=edit&section=14" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Dirac_comb" title="Dirac comb">Dirac comb</a></li> <li><a href="/wiki/Fabry%E2%80%93P%C3%A9rot_interferometer" title="Fabry–Pérot interferometer">Fabry–Pérot interferometer</a></li></ul> <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=Comb_filter&action=edit&section=15" 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-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></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="CITEREFRoger_Russell" class="citation web cs1">Roger Russell. <a rel="nofollow" class="external text" href="http://www.roger-russell.com/columns/combfilter2.htm">"Hearing, Columns and Comb Filtering"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">2010-04-22</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Hearing%2C+Columns+and+Comb+Filtering&rft.au=Roger+Russell&rft_id=http%3A%2F%2Fwww.roger-russell.com%2Fcolumns%2Fcombfilter2.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComb+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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100507124237/http://www.asc-hifi.com/acoustic_basics.htm">"Acoustic Basics"</a>. Acoustic Sciences Corporation. Archived from <a rel="nofollow" class="external text" href="http://www.asc-hifi.com/acoustic_basics.htm#2">the original</a> on 2010-05-07.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Acoustic+Basics&rft.pub=Acoustic+Sciences+Corporation&rft_id=http%3A%2F%2Fwww.asc-hifi.com%2Facoustic_basics.htm%232&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComb+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="CITEREFBurroughs1974" class="citation book cs1">Burroughs, Lou (1974). <i>Microphones: Design and Application</i>. Sagamore Publishing Company.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Microphones%3A+Design+and+Application&rft.pub=Sagamore+Publishing+Company&rft.date=1974&rft.aulast=Burroughs&rft.aufirst=Lou&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComb+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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith" class="citation web cs1">Smith, J. O. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110606210608/https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html">"Feedforward Comb Filters"</a>. Archived from <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/waveguide/Feedforward_Comb_Filters.html">the original</a> on 2011-06-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Feedforward+Comb+Filters&rft.aulast=Smith&rft.aufirst=J.+O.&rft_id=https%3A%2F%2Fccrma.stanford.edu%2F~jos%2Fwaveguide%2FFeedforward_Comb_Filters.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComb+filter" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSmith" class="citation web cs1">Smith, J.O. <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110606203008/https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html">"Feedback Comb Filters"</a>. Archived from <a rel="nofollow" class="external text" href="https://ccrma.stanford.edu/~jos/waveguide/Feedback_Comb_Filters.html">the original</a> on 2011-06-06.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Feedback+Comb+Filters&rft.aulast=Smith&rft.aufirst=J.O.&rft_id=https%3A%2F%2Fccrma.stanford.edu%2F~jos%2Fwaveguide%2FFeedback_Comb_Filters.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AComb+filter" class="Z3988"></span></span> </li> </ol></div></div> <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=Comb_filter&action=edit&section=16" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Commons-logo.svg" class="mw-file-description"><img alt="" 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Comb filter - Wikipedia