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class="vector-toc-list"> <li id="toc-Power-law_spectra" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power-law_spectra"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.1</span> <span>Power-law spectra</span> </div> </a> <ul id="toc-Power-law_spectra-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Distribution_of_point_values" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Distribution_of_point_values"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.2</span> <span>Distribution of point values</span> </div> </a> <ul id="toc-Distribution_of_point_values-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Autocorrelation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Autocorrelation"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3</span> <span>Autocorrelation</span> </div> </a> <ul id="toc-Autocorrelation-sublist" class="vector-toc-list"> <li id="toc-1D_signal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#1D_signal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.1</span> <span>1D signal</span> </div> </a> <ul id="toc-1D_signal-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-2D_signal" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#2D_signal"> <div class="vector-toc-text"> <span class="vector-toc-numb">4.3.2</span> <span>2D signal</span> </div> </a> <ul id="toc-2D_signal-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Occurrence" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Occurrence"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Occurrence</span> </div> </a> <button aria-controls="toc-Occurrence-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 Occurrence subsection</span> </button> <ul id="toc-Occurrence-sublist" class="vector-toc-list"> <li id="toc-Precision_timekeeping" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Precision_timekeeping"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Precision timekeeping</span> </div> </a> <ul id="toc-Precision_timekeeping-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Humans" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Humans"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Humans</span> </div> </a> <ul id="toc-Humans-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Electronic_devices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electronic_devices"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.3</span> <span>Electronic devices</span> </div> </a> <ul id="toc-Electronic_devices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_gravitational_wave_astronomy" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#In_gravitational_wave_astronomy"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.4</span> <span>In gravitational wave astronomy</span> </div> </a> <ul id="toc-In_gravitational_wave_astronomy-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Climate_dynamics" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Climate_dynamics"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.5</span> <span>Climate dynamics</span> </div> </a> <ul id="toc-Climate_dynamics-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diffusion_processes" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diffusion_processes"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.6</span> <span>Diffusion processes</span> </div> </a> <ul id="toc-Diffusion_processes-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Origin" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Origin"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>Origin</span> </div> </a> <ul id="toc-Origin-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Audio_testing" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Audio_testing"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Audio testing</span> </div> </a> <ul id="toc-Audio_testing-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">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Footnotes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Footnotes"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>Footnotes</span> </div> </a> <ul id="toc-Footnotes-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">10</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">11</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 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Available in 23 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-23" 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">23 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BE%D0%B2_%D1%88%D1%83%D0%BC" title="Розов шум – Bulgarian" lang="bg" hreflang="bg" data-title="Розов шум" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target"><span>Български</span></a></li><li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Rosa_Rauschn" title="Rosa Rauschn – Bavarian" lang="bar" hreflang="bar" data-title="Rosa Rauschn" data-language-autonym="Boarisch" data-language-local-name="Bavarian" class="interlanguage-link-target"><span>Boarisch</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Soroll_rosa" title="Soroll rosa – Catalan" lang="ca" hreflang="ca" data-title="Soroll rosa" 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/R%C5%AF%C5%BEov%C3%BD_%C5%A1um" title="Růžový šum – Czech" lang="cs" hreflang="cs" data-title="Růžový šum" 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/1/f-Rauschen" title="1/f-Rauschen – German" lang="de" hreflang="de" data-title="1/f-Rauschen" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Ruido_rosa" title="Ruido rosa – Spanish" lang="es" hreflang="es" data-title="Ruido rosa" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a href="https://eo.wikipedia.org/wiki/Rozkolora_bruo" title="Rozkolora bruo – Esperanto" lang="eo" hreflang="eo" data-title="Rozkolora bruo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Bruit_rose" title="Bruit rose – French" lang="fr" hreflang="fr" data-title="Bruit rose" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B6%84%ED%99%8D%EC%83%89_%EC%9E%A1%EC%9D%8C" title="분홍색 잡음 – Korean" lang="ko" hreflang="ko" data-title="분홍색 잡음" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Rumore_rosa" title="Rumore rosa – Italian" lang="it" hreflang="it" data-title="Rumore rosa" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/R%C3%B3zsasz%C3%ADn_zaj" title="Rózsaszín zaj – Hungarian" lang="hu" hreflang="hu" data-title="Rózsaszín zaj" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Hingar_merah_jambu" title="Hingar merah jambu – Malay" lang="ms" hreflang="ms" data-title="Hingar merah jambu" data-language-autonym="Bahasa Melayu" data-language-local-name="Malay" class="interlanguage-link-target"><span>Bahasa Melayu</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Roze_ruis" title="Roze ruis – Dutch" lang="nl" hreflang="nl" data-title="Roze ruis" 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%83%94%E3%83%B3%E3%82%AF%E3%83%8E%E3%82%A4%E3%82%BA" title="ピンクノイズ – Japanese" lang="ja" hreflang="ja" data-title="ピンクノイズ" 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-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Rosa_brus" title="Rosa brus – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Rosa brus" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Szum_r%C3%B3%C5%BCowy" title="Szum różowy – Polish" lang="pl" hreflang="pl" data-title="Szum różowy" 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/Ru%C3%ADdo_rosa" title="Ruído rosa – Portuguese" lang="pt" hreflang="pt" data-title="Ruído rosa" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru badge-Q70894304 mw-list-item" title=""><a href="https://ru.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B7%D0%BE%D0%B2%D1%8B%D0%B9_%D1%88%D1%83%D0%BC" title="Розовый шум – Russian" lang="ru" hreflang="ru" data-title="Розовый шум" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/1/f_noise" title="1/f noise – Simple English" lang="en-simple" hreflang="en-simple" data-title="1/f noise" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Vaaleanpunainen_kohina" title="Vaaleanpunainen kohina – Finnish" lang="fi" hreflang="fi" data-title="Vaaleanpunainen kohina" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Sk%C3%A4rt_brus" title="Skärt brus – Swedish" lang="sv" hreflang="sv" data-title="Skärt brus" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A0%D0%BE%D0%B6%D0%B5%D0%B2%D0%B8%D0%B9_%D1%88%D1%83%D0%BC" title="Рожевий шум – Ukrainian" lang="uk" hreflang="uk" data-title="Рожевий шум" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E7%B2%89%E7%BA%A2%E5%99%AA%E5%A3%B0" title="粉红噪声 – Chinese" lang="zh" hreflang="zh" data-title="粉红噪声" data-language-autonym="中文" data-language-local-name="Chinese" class="interlanguage-link-target"><span>中文</span></a></li> </ul> <div class="after-portlet after-portlet-lang"><span class="wb-langlinks-edit wb-langlinks-link"><a href="https://www.wikidata.org/wiki/Special:EntityPage/Q974572#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div class="vector-page-toolbar"> <div class="vector-page-toolbar-container"> <div id="left-navigation"> <nav aria-label="Namespaces"> <div id="p-associated-pages" class="vector-menu vector-menu-tabs mw-portlet mw-portlet-associated-pages" 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<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 with equal energy per octave</div> <style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol 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.mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar nomobile nowraplinks" style="width:auto;"><tbody><tr><th class="sidebar-title" style="background:lavender"><a href="/wiki/Colors_of_noise" title="Colors of noise">Colors of noise</a></th></tr><tr><td class="sidebar-content plainlist"> <ul><li><a href="/wiki/White_noise" title="White noise">White</a></li> <li><a class="mw-selflink selflink">Pink</a></li> <li><a href="/wiki/Brownian_noise" title="Brownian noise">Red (Brownian)</a></li> <li><a href="/wiki/Purple_noise" class="mw-redirect" title="Purple noise">Purple</a></li> <li><a href="/wiki/Grey_noise" title="Grey noise">Grey</a></li></ul></td> </tr><tr><td class="sidebar-navbar"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1239400231">.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}</style><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Colors_of_noise" title="Template:Colors of noise"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Colors_of_noise" title="Template talk:Colors of noise"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Colors_of_noise" title="Special:EditPage/Template:Colors of noise"><abbr title="Edit this template">e</abbr></a></li></ul></div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">For the album by Laura Mvula, see <a href="/wiki/Pink_Noise_(album)" title="Pink Noise (album)">Pink Noise (album)</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">This article is about 1/f noise. For quantum 1/f noise, see <a href="/wiki/Quantum_1/f_noise" title="Quantum 1/f noise">quantum 1/f noise</a>.</div><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"Fractal noise" redirects here. For the novel, see <a href="/wiki/Fractal_Noise" title="Fractal Noise">Fractal Noise</a>.</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:2D_pink_noise.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/25/2D_pink_noise.png/220px-2D_pink_noise.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/2/25/2D_pink_noise.png 1.5x" data-file-width="304" data-file-height="304" /></a><figcaption>A two-dimensional pink noise <a href="/wiki/Grayscale" title="Grayscale">grayscale</a> image, generated with a computer program; some fields observed in nature are characterized by a similar power spectrum<sup id="cite_ref-Field-1987_1-0" class="reference"><a href="#cite_note-Field-1987-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup></figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Pink_noise_cube.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Pink_noise_cube.gif/220px-Pink_noise_cube.gif" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Pink_noise_cube.gif/330px-Pink_noise_cube.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/1c/Pink_noise_cube.gif/440px-Pink_noise_cube.gif 2x" data-file-width="493" data-file-height="493" /></a><figcaption>A 3D pink noise image, generated with a computer program, viewed as an animation in which each frame is a 2D slice</figcaption></figure> <p><b>Pink noise</b>, <b><style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.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="frac"><span class="num">1</span>⁄<span class="den">f</span></span> noise</b>, <b>fractional noise</b> or <b>fractal noise</b> is a <a href="/wiki/Signal_(information_theory)" class="mw-redirect" title="Signal (information theory)">signal</a> or process with a <a href="/wiki/Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">frequency spectrum</a> such that the <a href="/wiki/Power_spectral_density" class="mw-redirect" title="Power spectral density">power spectral density</a> (power per frequency interval) is <a href="/wiki/Inversely_proportional" class="mw-redirect" title="Inversely proportional">inversely proportional</a> to the <a href="/wiki/Frequency" title="Frequency">frequency</a> of the signal. In pink noise, each <a href="/wiki/Octave_(electronics)" title="Octave (electronics)">octave</a> interval (halving or doubling in frequency) carries an equal amount of noise energy. </p><p>Pink noise sounds like a <a href="/wiki/Waterfall" title="Waterfall">waterfall</a>.<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> It is often used to tune <a href="/wiki/Loudspeaker" title="Loudspeaker">loudspeaker</a> systems in <a href="/wiki/Professional_audio" title="Professional audio">professional audio</a>.<sup id="cite_ref-Sound_3-0" class="reference"><a href="#cite_note-Sound-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> Pink noise is one of the most commonly observed signals in biological systems.<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> </p><p>The name arises from the pink appearance of visible light with this power spectrum.<sup id="cite_ref-Downey-2012_5-0" class="reference"><a href="#cite_note-Downey-2012-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This is in contrast with <a href="/wiki/White_noise" title="White noise">white noise</a> which has equal intensity per frequency interval. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=1" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1096940132">.mw-parser-output .listen .side-box-text{line-height:1.1em}.mw-parser-output .listen-plain{border:none;background:transparent}.mw-parser-output .listen-embedded{width:100%;margin:0;border-width:1px 0 0 0;background:transparent}.mw-parser-output .listen-header{padding:2px}.mw-parser-output .listen-embedded .listen-header{padding:2px 0}.mw-parser-output .listen-file-header{padding:4px 0}.mw-parser-output .listen .description{padding-top:2px}.mw-parser-output .listen .mw-tmh-player{max-width:100%}@media(max-width:719px){.mw-parser-output .listen{clear:both}}@media(min-width:720px){.mw-parser-output .listen:not(.listen-noimage){width:320px}.mw-parser-output .listen-left{overflow:visible;float:left}.mw-parser-output .listen-center{float:none;margin-left:auto;margin-right:auto}}</style><div class="side-box side-box-right listen noprint"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409" /> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-center" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/60px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/120px-Gnome-mime-sound-openclipart.svg.png 1.5x" data-file-width="160" data-file-height="160" /></span><figcaption></figcaption></figure></div> <div class="side-box-text plainlist"><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Pink_noise.ogg" title="File:Pink noise.ogg">Pink noise</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="10" data-mwtitle="Pink_noise.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/6/6c/Pink_noise.ogg/Pink_noise.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/6/6c/Pink_noise.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">10 seconds of pink noise, <a href="/wiki/Audio_normalization" title="Audio normalization">normalized</a> to −1 <a href="/wiki/DBFS" title="DBFS">dBFS</a> <a href="/wiki/Peak_amplitude" class="mw-redirect" title="Peak amplitude">peak amplitude</a></div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p>Within the scientific literature, the term "1/f noise" is sometimes used loosely to refer to any noise with a power spectral density of the form </p><p><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 S(f)\propto {\frac {1}{f^{\alpha }}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S(f)\propto {\frac {1}{f^{\alpha }}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c3327da06d2abda94a7a8db08e0ff7afa4421ea" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:11.773ex; height:5.676ex;" alt="{\displaystyle S(f)\propto {\frac {1}{f^{\alpha }}},}" /></span> </p><p>where <span class="texhtml mvar" style="font-style:italic;">f</span> is frequency, and <span class="texhtml">0 < <i>α</i> < 2</span>, with exponent <span class="texhtml mvar" style="font-style:italic;">α</span> usually close to 1. One-dimensional signals with <span class="texhtml"><i>α</i> = 1</span> are usually called pink noise.<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> </p><p>The following function describes a length <span class="texhtml mvar" style="font-style:italic;">N</span> one-dimensional pink noise signal (i.e. a <a href="/wiki/Gaussian_white_noise" class="mw-redirect" title="Gaussian white noise">Gaussian white noise</a> signal with zero mean and standard deviation <span class="texhtml mvar" style="font-style:italic;">σ</span>, which has been suitably filtered), as a sum of sine waves with different frequencies, whose amplitudes fall off inversely with the square root of frequency <span class="texhtml mvar" style="font-style:italic;">u</span> (so that power, which is the square of amplitude, falls off inversely with frequency), and phases are random:<sup id="cite_ref-Das-thesis_7-0" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><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 h(x)=\sigma {\sqrt {\frac {N}{2}}}\sum _{u}{\frac {\chi _{u}}{\sqrt {u}}}\sin \left({\frac {2\pi ux}{N}}+\phi _{u}\right),\quad \chi _{u}\sim \chi (2),\quad \phi _{u}\sim U(0,2\pi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>N</mi> <mn>2</mn> </mfrac> </msqrt> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <msqrt> <mi>u</mi> </msqrt> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>u</mi> <mi>x</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>∼</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> </mrow> </msub> <mo>∼</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x)=\sigma {\sqrt {\frac {N}{2}}}\sum _{u}{\frac {\chi _{u}}{\sqrt {u}}}\sin \left({\frac {2\pi ux}{N}}+\phi _{u}\right),\quad \chi _{u}\sim \chi (2),\quad \phi _{u}\sim U(0,2\pi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3ed02c83515f0aac61c4e5333fd3705a5f780ad" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:71.865ex; height:6.843ex;" alt="{\displaystyle h(x)=\sigma {\sqrt {\frac {N}{2}}}\sum _{u}{\frac {\chi _{u}}{\sqrt {u}}}\sin \left({\frac {2\pi ux}{N}}+\phi _{u}\right),\quad \chi _{u}\sim \chi (2),\quad \phi _{u}\sim U(0,2\pi ).}" /></span> </p><p><span class="texhtml mvar" style="font-style:italic;">χ<sub>u</sub></span> are <a href="/wiki/Iid" class="mw-redirect" title="Iid">independently and identically (<abbr title="">iid</abbr>)</a> <a href="/wiki/Chi_distribution" title="Chi distribution"> chi-distributed variables</a>, and <span class="texhtml mvar" style="font-style:italic;">ϕ<sub>u</sub></span> are uniform random. </p><p>In a two-dimensional pink noise signal, the amplitude at any orientation falls off inversely with frequency. A pink noise square of length <span class="texhtml mvar" style="font-style:italic;">N</span> can be written as:<sup id="cite_ref-Das-thesis_7-1" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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 h(x,y)={\frac {\sigma N}{\sqrt {2}}}\sum _{u,v}{\frac {\chi _{uv}}{\sqrt {u^{2}+v^{2}}}}\sin \left({\frac {2\pi }{N}}(ux+vy)+\phi _{uv}\right),\quad \chi _{uv}\sim \chi (2),\quad \phi _{uv}\sim U(0,2\pi ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>σ</mi> <mi>N</mi> </mrow> <msqrt> <mn>2</mn> </msqrt> </mfrac> </mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mo>,</mo> <mi>v</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <msqrt> <msup> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>u</mi> <mi>x</mi> <mo>+</mo> <mi>v</mi> <mi>y</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>χ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>∼</mo> <mi>χ</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="1em"></mspace> <msub> <mi>ϕ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>u</mi> <mi>v</mi> </mrow> </msub> <mo>∼</mo> <mi>U</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>2</mn> <mi>π</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(x,y)={\frac {\sigma N}{\sqrt {2}}}\sum _{u,v}{\frac {\chi _{uv}}{\sqrt {u^{2}+v^{2}}}}\sin \left({\frac {2\pi }{N}}(ux+vy)+\phi _{uv}\right),\quad \chi _{uv}\sim \chi (2),\quad \phi _{uv}\sim U(0,2\pi ).}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4228fb814e4e0939d57123ab1569ef63a8cd0819" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:87.519ex; height:6.843ex;" alt="{\displaystyle h(x,y)={\frac {\sigma N}{\sqrt {2}}}\sum _{u,v}{\frac {\chi _{uv}}{\sqrt {u^{2}+v^{2}}}}\sin \left({\frac {2\pi }{N}}(ux+vy)+\phi _{uv}\right),\quad \chi _{uv}\sim \chi (2),\quad \phi _{uv}\sim U(0,2\pi ).}" /></span> </p><p>General <span class="texhtml">1/<i>f<span style="padding-left:0.2em;"><sup>α</sup></span></i></span>-like noises occur widely in nature and are a source of considerable interest in many fields. Noises with <span class="texhtml mvar" style="font-style:italic;">α</span> near 1 generally come from <a href="/wiki/Condensed_matter_physics" title="Condensed matter physics">condensed-matter</a> systems in <a href="/wiki/Quasi-equilibrium" class="mw-redirect" title="Quasi-equilibrium">quasi-equilibrium</a>, as discussed below.<sup id="cite_ref-Kogan-1996_8-0" class="reference"><a href="#cite_note-Kogan-1996-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> Noises with a broad range of <span class="texhtml mvar" style="font-style:italic;">α</span> generally correspond to a wide range of <a href="/wiki/Non-equilibrium_thermodynamics" title="Non-equilibrium thermodynamics">non-equilibrium</a> driven <a href="/wiki/Dynamical_system" title="Dynamical system">dynamical systems</a>. </p><p>Pink noise sources include <i><a href="/wiki/Flicker_noise" title="Flicker noise">flicker noise</a></i> in electronic devices. In their study of <a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">fractional Brownian motion</a>,<sup id="cite_ref-Mandelbrot1968_9-0" class="reference"><a href="#cite_note-Mandelbrot1968-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Benoit_Mandelbrot" title="Benoit Mandelbrot">Mandelbrot</a> and Van Ness proposed the name <i>fractional noise</i> (sometimes since called <i>fractal noise</i>) to describe <span class="texhtml">1/<i>f<span style="padding-left:0.2em;"><sup>α</sup></span></i></span> noises for which the exponent <span class="texhtml mvar" style="font-style:italic;">α</span> is not an even integer,<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> or that are <a href="/wiki/Fractional_derivative" class="mw-redirect" title="Fractional derivative">fractional derivatives</a> of <a href="/wiki/Brownian_noise" title="Brownian noise">Brownian</a> (<span class="texhtml">1/<i>f</i><span style="padding-left:0.2em;"><sup>2</sup></span></span>) noise. </p> <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=Pink_noise&action=edit&section=2" title="Edit section: Description"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Pink_noise_spectrum.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Pink_noise_spectrum.svg/220px-Pink_noise_spectrum.svg.png" decoding="async" width="220" height="220" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Pink_noise_spectrum.svg/330px-Pink_noise_spectrum.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Pink_noise_spectrum.svg/440px-Pink_noise_spectrum.svg.png 2x" data-file-width="1024" data-file-height="1024" /></a><figcaption>Spectrum of a pink noise approximation on a log-log plot; power density falls off at 10 dB/decade of frequency</figcaption></figure> <figure class="mw-default-size mw-halign-left" typeof="mw:File/Thumb"><a href="/wiki/File:Noise.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Noise.jpg/220px-Noise.jpg" decoding="async" width="220" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/76/Noise.jpg/330px-Noise.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/76/Noise.jpg/440px-Noise.jpg 2x" data-file-width="1059" data-file-height="735" /></a><figcaption>Relative intensity of pink noise (left) and <a href="/wiki/White_noise" title="White noise">white noise</a> (right) on an <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">FFT</a> <a href="/wiki/Spectrogram" title="Spectrogram">spectrogram</a> with the vertical axis being linear frequency</figcaption></figure> <p>In pink noise, there is equal energy per <a href="/wiki/Octave" title="Octave">octave</a> of frequency. The energy of pink noise at each frequency level, however, falls off at roughly 3 <a href="/wiki/Decibels" class="mw-redirect" title="Decibels">dB</a> per octave. This is in contrast to <a href="/wiki/White_noise" title="White noise">white noise</a> which has equal energy at all frequency levels.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>The <a href="/wiki/Human_auditory_system" class="mw-redirect" title="Human auditory system">human auditory system</a>, which processes frequencies in a roughly logarithmic fashion approximated by the <a href="/wiki/Bark_scale" title="Bark scale">Bark scale</a>, does not perceive different frequencies with equal sensitivity; signals around 1–4 kHz sound <a href="/wiki/Loudness" title="Loudness">loudest</a> for a given intensity. However, humans still differentiate between white noise and pink noise with ease. </p><p><a href="/wiki/Graphic_equalizer" class="mw-redirect" title="Graphic equalizer">Graphic equalizers</a> also divide signals into bands logarithmically and report power by octaves; audio engineers put pink noise through a system to test whether it has a flat frequency response in the spectrum of interest. Systems that do not have a flat response can be equalized by creating an inverse filter using a graphic equalizer. Because pink noise tends to occur in natural physical systems, it is often useful in audio production. Pink noise can be processed, filtered, and/or effects can be added to produce desired sounds. Pink-noise generators are commercially available. </p><p>One parameter of noise, the peak versus average energy contents, or <a href="/wiki/Crest_factor" title="Crest factor">crest factor</a>, is important for testing purposes, such as for <a href="/wiki/Audio_power_amplifier" title="Audio power amplifier">audio power amplifier</a> and <a href="/wiki/Loudspeaker" title="Loudspeaker">loudspeaker</a> capabilities because the signal power is a direct function of the crest factor. Various crest factors of pink noise can be used in simulations of various levels of <a href="/wiki/Dynamic_range_compression" title="Dynamic range compression">dynamic range compression</a> in music signals. On some digital pink-noise generators the crest factor can be specified. </p> <div class="mw-heading mw-heading2"><h2 id="Generation">Generation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=3" title="Edit section: Generation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:White_to_pink_filter.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/White_to_pink_filter.png/500px-White_to_pink_filter.png" decoding="async" width="500" height="259" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ed/White_to_pink_filter.png/750px-White_to_pink_filter.png 1.5x, //upload.wikimedia.org/wikipedia/commons/e/ed/White_to_pink_filter.png 2x" data-file-width="922" data-file-height="478" /></a><figcaption>The spatial filter which is convolved with a one-dimensional white noise signal to create a pink noise signal<sup id="cite_ref-Das-thesis_7-2" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></figcaption></figure> <p>Pink noise can be computer-generated by first generating a white noise signal, Fourier-transforming it, then dividing the amplitudes of the different frequency components by the square root of the frequency (in one dimension), or by the frequency (in two dimensions) etc. <sup id="cite_ref-Das-thesis_7-3" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> This is equivalent to spatially filtering (convolving) the white noise signal with a white-to-pink-filter. For a length <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5e3890c981ae85503089652feb48b191b57aae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" /></span> signal in one dimension, the filter has the following form:<sup id="cite_ref-Das-thesis_7-4" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p><p><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 a(x)={\frac {1}{N}}\left[1+{\frac {1}{\sqrt {N/2}}}\cos \pi (x-1)+2\sum _{k=1}^{N/2-1}{\frac {1}{\sqrt {k}}}\cos {{\frac {2\pi k}{N}}(x-1)}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>N</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </msqrt> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>π</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>+</mo> <mn>2</mn> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> <mo>−</mo> <mn>1</mn> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mi>k</mi> </msqrt> </mfrac> </mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a(x)={\frac {1}{N}}\left[1+{\frac {1}{\sqrt {N/2}}}\cos \pi (x-1)+2\sum _{k=1}^{N/2-1}{\frac {1}{\sqrt {k}}}\cos {{\frac {2\pi k}{N}}(x-1)}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b72f36747c5fe30c08f543d3fd900b8800fdcf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:67.942ex; height:7.843ex;" alt="{\displaystyle a(x)={\frac {1}{N}}\left[1+{\frac {1}{\sqrt {N/2}}}\cos \pi (x-1)+2\sum _{k=1}^{N/2-1}{\frac {1}{\sqrt {k}}}\cos {{\frac {2\pi k}{N}}(x-1)}\right].}" /></span> </p><p>Matlab programs are available to generate pink and other power-law coloured noise in <a rel="nofollow" class="external text" href="http://www.maxlittle.net/software/">one</a> or <a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise">any number</a> of dimensions. </p> <div class="mw-heading mw-heading2"><h2 id="Properties">Properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=4" title="Edit section: Properties"><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:Pink_noise_1d_autocorrelation.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Pink_noise_1d_autocorrelation.png/440px-Pink_noise_1d_autocorrelation.png" decoding="async" width="440" height="649" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Pink_noise_1d_autocorrelation.png/660px-Pink_noise_1d_autocorrelation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a6/Pink_noise_1d_autocorrelation.png/880px-Pink_noise_1d_autocorrelation.png 2x" data-file-width="2402" data-file-height="3542" /></a><figcaption>The autocorrelation (Pearson's correlation coefficient) of one-dimensional (top) and two-dimensional (bottom) pink noise signals, across distance d (in units of the longest wavelength comprising the signal); grey curves are the autocorrelations of a sample of pink noise signals (comprising discrete frequencies), and black is their average, red is the theoretically calculated autocorrelation when the signal comprises these same discrete frequencies, and blue assumes a continuum of frequencies<sup id="cite_ref-Das-thesis_7-5" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></figcaption></figure> <div class="mw-heading mw-heading3"><h3 id="Power-law_spectra">Power-law spectra</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=5" title="Edit section: Power-law spectra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The power spectrum of pink noise is <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 {1}{f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>f</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8077aad244e433526a9268ce973d0cd68e8a6e07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:2.115ex; height:5.676ex;" alt="{\displaystyle {\frac {1}{f}}}" /></span> only for one-dimensional signals. For two-dimensional signals (e.g., images) the average power spectrum at any orientation falls as <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 {1}{f^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{f^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fc896b224bd42ddb7c4e026df88eb3f3d5c7d8d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:3.211ex; height:5.843ex;" alt="{\displaystyle {\frac {1}{f^{2}}}}" /></span>, and 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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> dimensions, it falls as <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 {1}{f^{d}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{f^{d}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36fcdca2aa00231006ceaedb9377a94ea8da8f0e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:3.249ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{f^{d}}}}" /></span>. In every case, each octave carries an equal amount of noise power. </p><p>The average amplitude <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_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e75d02a8103bdaddecca46e413d56145afa71a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.233ex; height:2.009ex;" alt="{\displaystyle a_{\theta }}" /></span> and power <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 p_{\theta }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\theta }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1aa4fdd3e5ed698ea7889602a0cbe756e670f4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.262ex; height:2.009ex;" alt="{\displaystyle p_{\theta }}" /></span> of a pink noise signal at any orientation <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 \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.09ex; height:2.176ex;" alt="{\displaystyle \theta }" /></span>, and the total power across all orientations, fall off as some power of the frequency. The following table lists these power-law frequency-dependencies for pink noise signal in different dimensions, and also for general power-law colored noise with power <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> (e.g.: <a href="/wiki/Brown_noise" class="mw-redirect" title="Brown noise">Brown noise</a> has <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 =2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha =2}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/938489e6428bb7959330df8c06c79a994811c4a9" 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 =2}" /></span>): <sup id="cite_ref-Das-thesis_7-6" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <table class="wikitable"> <caption>Power-law spectra of pink noise </caption> <tbody><tr> <th>dimensions</th> <th>avg. amp. <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_{\theta }(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a_{\theta }(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa1ea0d2f064205c3aeeee2acbf3a6c00c2aad3f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.321ex; height:2.843ex;" alt="{\displaystyle a_{\theta }(f)}" /></span></th> <th>avg. power <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 p_{\theta }(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>p</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>θ</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p_{\theta }(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a62b930f78a40c9b0c6f1d14c33c9764bcc350eb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:5.35ex; height:2.843ex;" alt="{\displaystyle p_{\theta }(f)}" /></span></th> <th>tot. power <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 p(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>p</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle p(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f4222bc1612c8fbd041de3ddcd8e7dc16eaeec" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:4.347ex; height:2.843ex;" alt="{\displaystyle p(f)}" /></span> </th></tr> <tr> <td>1</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/{\sqrt {f}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>f</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/{\sqrt {f}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b18cbe9f58d2142f733d2380243631a01046f6a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.927ex; height:3.509ex;" alt="{\displaystyle 1/{\sqrt {f}}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span> </td></tr> <tr> <td>2</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13dc85513fc9696a8044cc820d3919af33af5799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.7ex; height:3.176ex;" alt="{\displaystyle 1/f^{2}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span> </td></tr> <tr> <td>3</td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{3/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{3/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f323fedd0aa829d6aaf2dcc378d5d0bd09a17b82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.344ex; height:3.343ex;" alt="{\displaystyle 1/f^{3/2}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{3}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fff56fb14c997834ee111839e3a89aa0a7f56b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.7ex; height:3.176ex;" alt="{\displaystyle 1/f^{3}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{d/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{d/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74d36597cad501f5c57cbe7b9fa6dfbc809ee837" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.381ex; height:3.343ex;" alt="{\displaystyle 1/f^{d/2}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/340c82156e1ea4dd59be6f42f4f78fdaa179518d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.737ex; height:3.176ex;" alt="{\displaystyle 1/f^{d}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span> </td></tr> <tr> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span>, power <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></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{\alpha d/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{\alpha d/2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c1f2db44d75ee404cbd4e3687b6f38a124ef1754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.433ex; height:3.343ex;" alt="{\displaystyle 1/f^{\alpha d/2}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{\alpha d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{\alpha d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/066a91c8c0ab8e27389798c889f75de0e8f11bcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.789ex; height:3.176ex;" alt="{\displaystyle 1/f^{\alpha d}}" /></span></td> <td><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/f^{1+(\alpha -1)d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mo>+</mo> <mo stretchy="false">(</mo> <mi>α</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>d</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{1+(\alpha -1)d}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b792ade3d353ed05a9e521a5530dc392d2a4dd49" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.27ex; height:3.343ex;" alt="{\displaystyle 1/f^{1+(\alpha -1)d}}" /></span> </td></tr></tbody></table> <div class="mw-heading mw-heading3"><h3 id="Distribution_of_point_values">Distribution of point values</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=6" title="Edit section: Distribution of point values"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Consider pink noise of any dimension that is produced by generating a Gaussian white noise signal with mean <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> and sd <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 \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" /></span>, then multiplying its spectrum with a filter (equivalent to spatially filtering it with a filter <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 {a}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">a</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {a}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df0c47bb21acb94ccb65db3da51e075abc68a898" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.471ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {a}}}" /></span>). Then the point values of the pink noise signal will also be normally distributed, with mean <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 \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }" /></span> and sd <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 \lVert {\boldsymbol {a}}\rVert \sigma }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">‖</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">a</mi> </mrow> <mo fence="false" stretchy="false">‖</mo> <mi>σ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lVert {\boldsymbol {a}}\rVert \sigma }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93fa2accc1e62d3659e51b1cb8356d5ecdd91553" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.126ex; height:2.843ex;" alt="{\displaystyle \lVert {\boldsymbol {a}}\rVert \sigma }" /></span>.<sup id="cite_ref-Das-thesis_7-7" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Autocorrelation">Autocorrelation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=7" title="Edit section: Autocorrelation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unlike white noise, which has no correlations across the signal, a pink noise signal is correlated with itself, as follows. </p> <div class="mw-heading mw-heading4"><h4 id="1D_signal">1D signal</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=8" title="Edit section: 1D signal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pearson's correlation coefficient of a one-dimensional pink noise signal (comprising discrete frequencies <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/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" /></span>) with itself across a distance <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 d}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e85ff03cbe0c7341af6b982e47e9f90d235c66ab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" /></span> in the configuration (space or time) domain is:<sup id="cite_ref-Das-thesis_7-8" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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(d)={\frac {\sum _{k}{\frac {\cos {\frac {2\pi kd}{N}}}{k}}}{\sum _{k}{\frac {1}{k}}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> </mrow> <mi>k</mi> </mfrac> </mrow> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(d)={\frac {\sum _{k}{\frac {\cos {\frac {2\pi kd}{N}}}{k}}}{\sum _{k}{\frac {1}{k}}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e50eddd03e8c16dddfe8d1fa31e32dccbc43f906" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:19.669ex; height:9.843ex;" alt="{\displaystyle r(d)={\frac {\sum _{k}{\frac {\cos {\frac {2\pi kd}{N}}}{k}}}{\sum _{k}{\frac {1}{k}}}}.}" /></span> If instead of discrete frequencies, the pink noise comprises a superposition of continuous frequencies from <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_{\textrm {min}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\textrm {min}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5941d605fb9c94c13d6353614e23e8ec8c90d953" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.184ex; height:2.509ex;" alt="{\displaystyle k_{\textrm {min}}}" /></span> to <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_{\textrm {max}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k_{\textrm {max}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e687f3ab8110816978d557de2286caead922075b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.502ex; height:2.509ex;" alt="{\displaystyle k_{\textrm {max}}}" /></span>, the autocorrelation coefficient is:<sup id="cite_ref-Das-thesis_7-9" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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(d)={\frac {{\textrm {Ci}}({\frac {2\pi k_{\textrm {max}}d}{N}})-{\textrm {Ci}}({\frac {2\pi k_{\textrm {min}}d}{N}})}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ci</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </mrow> </msub> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ci</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </mrow> </msub> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>log</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>max</mtext> </mrow> </mrow> </msub> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>min</mtext> </mrow> </mrow> </msub> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(d)={\frac {{\textrm {Ci}}({\frac {2\pi k_{\textrm {max}}d}{N}})-{\textrm {Ci}}({\frac {2\pi k_{\textrm {min}}d}{N}})}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f0a599bed13a81429a4e2c782a895e30085b7e4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:33.433ex; height:9.009ex;" alt="{\displaystyle r(d)={\frac {{\textrm {Ci}}({\frac {2\pi k_{\textrm {max}}d}{N}})-{\textrm {Ci}}({\frac {2\pi k_{\textrm {min}}d}{N}})}{\log {\frac {k_{\textrm {max}}}{k_{\textrm {min}}}}}},}" /></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 {\textrm {Ci}}(x)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext>Ci</mtext> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\textrm {Ci}}(x)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/828665af23a2f447b9a8be0531edf8d59e15f740" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.464ex; height:2.843ex;" alt="{\displaystyle {\textrm {Ci}}(x)}" /></span> is the <a href="/wiki/Trigonometric_integral#Cosine_integral" title="Trigonometric integral">cosine integral function</a>. </p> <div class="mw-heading mw-heading4"><h4 id="2D_signal">2D signal</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=9" title="Edit section: 2D signal"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Pearson's autocorrelation coefficient of a two-dimensional pink noise signal comprising discrete frequencies is theoretically approximated as:<sup id="cite_ref-Das-thesis_7-10" class="reference"><a href="#cite_note-Das-thesis-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> <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(d)={\frac {\sum _{k}{\frac {J_{0}({\frac {2\pi kd}{N}})}{k}}}{\sum _{k}{\frac {1}{k}}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>r</mi> <mo stretchy="false">(</mo> <mi>d</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>k</mi> <mi>d</mi> </mrow> <mi>N</mi> </mfrac> </mrow> <mo stretchy="false">)</mo> </mrow> <mi>k</mi> </mfrac> </mrow> </mrow> <mrow> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>k</mi> </mfrac> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle r(d)={\frac {\sum _{k}{\frac {J_{0}({\frac {2\pi kd}{N}})}{k}}}{\sum _{k}{\frac {1}{k}}}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3d8e20abc18308c4572014d988c437e102e5e88" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:20.106ex; height:9.843ex;" alt="{\displaystyle r(d)={\frac {\sum _{k}{\frac {J_{0}({\frac {2\pi kd}{N}})}{k}}}{\sum _{k}{\frac {1}{k}}}},}" /></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_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>J</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle J_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/750fead02065cb001fe30d98d309652b642f8e8f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.344ex; height:2.509ex;" alt="{\displaystyle J_{0}}" /></span> is the <a href="/wiki/Bessel_function#Bessel_functions_of_the_first_kind" title="Bessel function">Bessel function of the first kind</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Occurrence">Occurrence</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=10" title="Edit section: Occurrence"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pink noise has been discovered in the <a href="/wiki/Statistical_fluctuations" title="Statistical fluctuations">statistical fluctuations</a> of an extraordinarily diverse number of physical and biological systems (Press, 1978;<sup id="cite_ref-Press-1978_12-0" class="reference"><a href="#cite_note-Press-1978-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> see articles in Handel & Chung, 1993,<sup id="cite_ref-Handel-1993_13-0" class="reference"><a href="#cite_note-Handel-1993-13"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> and references therein). Examples of its occurrence include fluctuations in <a href="/wiki/Tide" title="Tide">tide</a> and river heights, <a href="/wiki/Quasar" title="Quasar">quasar</a> light emissions, heart beat, firings of single <a href="/wiki/Neuron" title="Neuron">neurons</a>, <a href="/wiki/Resistivity" class="mw-redirect" title="Resistivity">resistivity</a> in <a href="/wiki/Solid-state_electronics" title="Solid-state electronics">solid-state electronics</a> and single-molecule conductance signals<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> resulting in <a href="/wiki/Flicker_noise" title="Flicker noise">flicker noise</a>. Pink noise describes the <a href="/wiki/Scene_statistics" title="Scene statistics">statistical structure of many natural images</a>.<sup id="cite_ref-Field-1987_1-1" class="reference"><a href="#cite_note-Field-1987-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>General 1/<i>f</i><sup> α</sup> noises occur in many physical, biological and economic systems, and some researchers describe them as being ubiquitous.<sup id="cite_ref-Bak-1987_15-0" class="reference"><a href="#cite_note-Bak-1987-15"><span class="cite-bracket">[</span>15<span class="cite-bracket">]</span></a></sup> In physical systems, they are present in some <a href="/wiki/Meteorological" class="mw-redirect" title="Meteorological">meteorological</a> data series, the <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> output of some astronomical bodies. In biological systems, they are present in, for example, <a href="/wiki/Cardiac_cycle" title="Cardiac cycle">heart beat</a> rhythms, neural activity, and the statistics of <a href="/wiki/DNA_sequence" class="mw-redirect" title="DNA sequence">DNA sequences</a>, as a generalized pattern.<sup id="cite_ref-16" class="reference"><a href="#cite_note-16"><span class="cite-bracket">[</span>16<span class="cite-bracket">]</span></a></sup> </p><p>An accessible introduction to the significance of pink noise is one given by <a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> (1978) in his <i>Scientific American</i> column "Mathematical Games".<sup id="cite_ref-Gardner-1978_17-0" class="reference"><a href="#cite_note-Gardner-1978-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> In this column, Gardner asked for the sense in which music imitates nature. Sounds in nature are not musical in that they tend to be either too repetitive (bird song, insect noises) or too chaotic (ocean surf, wind in trees, and so forth). The answer to this question was given in a statistical sense by Voss and Clarke (1975, 1978), who showed that pitch and loudness fluctuations in speech and music are pink noises.<sup id="cite_ref-Voss-1975_18-0" class="reference"><a href="#cite_note-Voss-1975-18"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Voss-1978_19-0" class="reference"><a href="#cite_note-Voss-1978-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> So music is like tides not in terms of how tides sound, but in how tide heights vary. </p> <div class="mw-heading mw-heading3"><h3 id="Precision_timekeeping">Precision timekeeping</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=11" title="Edit section: Precision timekeeping"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Allan_variance" title="Allan variance">Allan variance</a></div> <p>The ubiquitous 1/f noise poses a "noise floor" to precision timekeeping.<sup id="cite_ref-Press-1978_12-1" class="reference"><a href="#cite_note-Press-1978-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> The derivation is based on.<sup id="cite_ref-20" class="reference"><a href="#cite_note-20"><span class="cite-bracket">[</span>20<span class="cite-bracket">]</span></a></sup> </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:AllanDeviation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/AllanDeviation.svg/300px-AllanDeviation.svg.png" decoding="async" width="300" height="264" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/dd/AllanDeviation.svg/450px-AllanDeviation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/dd/AllanDeviation.svg/600px-AllanDeviation.svg.png 2x" data-file-width="535" data-file-height="471" /></a><figcaption>A clock is most easily tested by comparing it with a <i>far more accurate</i> reference clock. During an interval of time <i>τ</i>, as measured by the reference clock, the clock under test advances by <i>τy</i>, where <i>y</i> is the average (relative) clock frequency over that interval.</figcaption></figure> <p>Suppose that we have a timekeeping device (it could be anything from <a href="/wiki/Crystal_oscillator" title="Crystal oscillator">quartz oscillators</a>, <a href="/wiki/Atomic_clock" title="Atomic clock">atomic clocks</a>, and <a href="/wiki/Hourglass" title="Hourglass">hourglasses</a><sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>21<span class="cite-bracket">]</span></a></sup>). Let its readout be a real number <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> that changes with the actual time <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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" /></span>. For concreteness, let us consider a quartz oscillator. In a quartz oscillator, <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> is the number of oscillations, 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 {\dot {x}}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/867f9767dea7667cf8deac86ec6bbe3162c8d4b1" 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 {\dot {x}}(t)}" /></span> is the rate of oscillation. The rate of oscillation has a constant component <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 {\dot {x}}_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c80b788267e394f5d4d07097c62c1c73def4aec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.384ex; height:2.509ex;" alt="{\displaystyle {\dot {x}}_{0}}" /></span>and a fluctuating component <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 {\dot {x}}_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b49610541a460cf921da1f0540726267845eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.466ex; height:2.843ex;" alt="{\displaystyle {\dot {x}}_{f}}" /></span>, so <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="{\textstyle {\dot {x}}(t)={\dot {x}}_{0}+{\dot {x}}_{f}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\dot {x}}(t)={\dot {x}}_{0}+{\dot {x}}_{f}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2aa15f3670d96d71669f63017b2a0d1167c1c981" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.416ex; height:3.009ex;" alt="{\textstyle {\dot {x}}(t)={\dot {x}}_{0}+{\dot {x}}_{f}(t)}" /></span>. By selecting the right units 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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" /></span>, we can have <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 {\dot {x}}_{0}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}_{0}=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d3e4c5734ba49070b7e5579c97a25521c89beed3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.645ex; height:2.509ex;" alt="{\displaystyle {\dot {x}}_{0}=1}" /></span>, meaning that on average, one second of clock-time passes for every second of real-time. </p><p>The stability of the clock is measured by how many "ticks" it makes over a fixed interval. The more stable the number of ticks, the better the stability of the clock. So, define the average clock frequency over the interval <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\tau ,(k+1)\tau ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <mi>k</mi> <mi>τ</mi> <mo>,</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [k\tau ,(k+1)\tau ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32f570a329f13fa63895a8b6111abe00f5ff9bda" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.966ex; height:2.843ex;" alt="{\displaystyle [k\tau ,(k+1)\tau ]}" /></span> 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 y_{k}={\frac {1}{\tau }}\int _{k\tau }^{(k+1)\tau }{\dot {x}}(t)dt={\frac {x((k+1)\tau )-x(k\tau )}{\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>τ</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>τ</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>τ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>τ</mi> <mo stretchy="false">)</mo> </mrow> <mi>τ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{k}={\frac {1}{\tau }}\int _{k\tau }^{(k+1)\tau }{\dot {x}}(t)dt={\frac {x((k+1)\tau )-x(k\tau )}{\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8f41e3845101311c9575cddefb8444725642e988" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:45.647ex; height:6.343ex;" alt="{\displaystyle y_{k}={\frac {1}{\tau }}\int _{k\tau }^{(k+1)\tau }{\dot {x}}(t)dt={\frac {x((k+1)\tau )-x(k\tau )}{\tau }}}" /></span>Note 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 y_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b2ab0248723a410cc2c67ce06ad5c043dcbb933" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" /></span> is unitless: it is the numerical ratio between ticks of the physical clock and ticks of an ideal clock<sup id="cite_ref-22" class="reference"><a href="#cite_note-22"><span class="cite-bracket">[</span>note 1<span class="cite-bracket">]</span></a></sup>. </p><p>The <a href="/wiki/Allan_variance" title="Allan variance">Allan variance</a> of the clock frequency is half the mean square of change in average clock frequency:<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 \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>K</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9f33d97b9b85ac64ca8f0b6e1db8768c10478b6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:47.257ex; height:7.343ex;" alt="{\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}}" /></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 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 an integer large enough for the averaging to converge to a definite value. For example, a 2013 atomic clock<sup id="cite_ref-23" class="reference"><a href="#cite_note-23"><span class="cite-bracket">[</span>22<span class="cite-bracket">]</span></a></sup> achieved <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 \sigma (25000{\text{ seconds}})=1.6\times 10^{-18}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>σ</mi> <mo stretchy="false">(</mo> <mn>25000</mn> <mrow class="MJX-TeXAtom-ORD"> <mtext> seconds</mtext> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1.6</mn> <mo>×</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>18</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma (25000{\text{ seconds}})=1.6\times 10^{-18}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a204a627734cf2d24f7a0fcf2473164607afcac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.567ex; height:3.176ex;" alt="{\displaystyle \sigma (25000{\text{ seconds}})=1.6\times 10^{-18}}" /></span>, meaning that if the clock is used to repeatedly measure intervals of 7 hours, the standard deviation of the actually measured time would be around 40 <a href="/wiki/Femtosecond" title="Femtosecond">femtoseconds</a>. </p><p>Now we have<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 y_{k}-y_{k-1}=\int _{\mathbb {R} }g(k\tau -t){\dot {x}}_{f}(t)dt=(g\ast {\dot {x}}_{f})(k\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> </mrow> </msub> <mi>g</mi> <mo stretchy="false">(</mo> <mi>k</mi> <mi>τ</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y_{k}-y_{k-1}=\int _{\mathbb {R} }g(k\tau -t){\dot {x}}_{f}(t)dt=(g\ast {\dot {x}}_{f})(k\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d2e44019ab67303abe416f7878008356cdcfd520" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:46.69ex; height:5.676ex;" alt="{\displaystyle y_{k}-y_{k-1}=\int _{\mathbb {R} }g(k\tau -t){\dot {x}}_{f}(t)dt=(g\ast {\dot {x}}_{f})(k\tau )}" /></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 g(t)={\frac {-1_{[0,\tau ]}(t)+1_{[-\tau ,0]}(t)}{\tau }}}"> <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> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo>−</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mi>τ</mi> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">[</mo> <mo>−</mo> <mi>τ</mi> <mo>,</mo> <mn>0</mn> <mo stretchy="false">]</mo> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mi>τ</mi> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)={\frac {-1_{[0,\tau ]}(t)+1_{[-\tau ,0]}(t)}{\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4898839ed0a0650e20feca88d05dc0296d1588d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:27.802ex; height:5.843ex;" alt="{\displaystyle g(t)={\frac {-1_{[0,\tau ]}(t)+1_{[-\tau ,0]}(t)}{\tau }}}" /></span> is one packet of a <a href="/wiki/Square_wave_(waveform)" title="Square wave (waveform)">square wave</a> with height <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/\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8c91446b93e596a86f559418f3e66b0c9fac82" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.527ex; height:2.843ex;" alt="{\displaystyle 1/\tau }" /></span> and wavelength <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\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/229ebe5c2aeb0d94fb9a40c1f98f4794ea6574ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.364ex; height:2.176ex;" alt="{\displaystyle 2\tau }" /></span>. 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 h(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle h(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/66abbb8ae1d9f30bb529739b109e1e5bbe83c626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.988ex; height:2.843ex;" alt="{\displaystyle h(t)}" /></span> be a packet of a square wave with height 1 and wavelength 2, then <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)=h(t/\tau )/\tau }"> <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> <mo>=</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle g(t)=h(t/\tau )/\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/468d5f31972fed09ced0d5273b3af6610856797b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.58ex; height:2.843ex;" alt="{\displaystyle g(t)=h(t/\tau )/\tau }" /></span>, and its Fourier transform satisfies <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 {\mathcal {F}}[g](\omega )={\mathcal {F}}[h](\tau \omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mo stretchy="false">[</mo> <mi>h</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}[g](\omega )={\mathcal {F}}[h](\tau \omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43398d6af2132e5181c44857f2f4771938a29f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.706ex; height:2.843ex;" alt="{\displaystyle {\mathcal {F}}[g](\omega )={\mathcal {F}}[h](\tau \omega )}" /></span>. </p><p>The Allan variance is then <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 \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{2}}{\overline {(g\ast {\dot {x}}_{f})(k\tau )^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mover> <mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>τ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo accent="false">¯</mo> </mover> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{2}}{\overline {(g\ast {\dot {x}}_{f})(k\tau )^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a00ae94269e38cf66e691501f10cb16c19124a29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:40.943ex; height:5.176ex;" alt="{\displaystyle \sigma ^{2}(\tau )={\frac {1}{2}}{\overline {(y_{k}-y_{k-1})^{2}}}={\frac {1}{2}}{\overline {(g\ast {\dot {x}}_{f})(k\tau )^{2}}}}" /></span>, and the discrete averaging can be approximated by a continuous averaging: <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 {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}\approx {\frac {1}{K\tau }}\int _{0}^{K\tau }{\frac {1}{2}}(g\ast {\dot {x}}_{f})(t)^{2}dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>K</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>≈</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>K</mi> <mi>τ</mi> </mrow> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>K</mi> <mi>τ</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}\approx {\frac {1}{K\tau }}\int _{0}^{K\tau }{\frac {1}{2}}(g\ast {\dot {x}}_{f})(t)^{2}dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4683f5b57dd02584ee3bff31e6615368756772e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:49.617ex; height:7.343ex;" alt="{\displaystyle {\frac {1}{K}}\sum _{k=1}^{K}{\frac {1}{2}}(y_{k}-y_{k-1})^{2}\approx {\frac {1}{K\tau }}\int _{0}^{K\tau }{\frac {1}{2}}(g\ast {\dot {x}}_{f})(t)^{2}dt}" /></span>, which is the total power of the 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 (g\ast {\dot {x}}_{f})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>g</mi> <mo>∗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (g\ast {\dot {x}}_{f})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/49dffddbd5986355408bfffe5db327e57651290d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.586ex; height:3.009ex;" alt="{\displaystyle (g\ast {\dot {x}}_{f})}" /></span>, or the integral of its <a href="/wiki/Spectral_density" title="Spectral density">power spectrum</a>: </p> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Illustration_for_Allan_variance_of_1-f_noise.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Illustration_for_Allan_variance_of_1-f_noise.png/315px-Illustration_for_Allan_variance_of_1-f_noise.png" decoding="async" width="315" height="109" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/Illustration_for_Allan_variance_of_1-f_noise.png/473px-Illustration_for_Allan_variance_of_1-f_noise.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/57/Illustration_for_Allan_variance_of_1-f_noise.png/630px-Illustration_for_Allan_variance_of_1-f_noise.png 2x" data-file-width="846" data-file-height="294" /></a><figcaption><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 \sigma ^{2}(1)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(1)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0f70654ec48abf37813e41a1c3325d4d583bed1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.356ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(1)}" /></span> is approximately the area under the green curve; 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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span> increases, <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[g](\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[g](\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d90c39e25b5b51b1d49db0c0dc60a53fdc4698c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.164ex; height:2.843ex;" alt="{\displaystyle S[g](\omega )}" /></span> shrinks on the x-axis, and the green curve shrinks on the x-axis but expands on the y-axis; 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 S[{\dot {x}}_{f}](\omega )\propto \omega ^{-\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[{\dot {x}}_{f}](\omega )\propto \omega ^{-\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72cd3e6e323dabcb336cc668e3c32499d4677ebf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.621ex; height:3.176ex;" alt="{\displaystyle S[{\dot {x}}_{f}](\omega )\propto \omega ^{-\alpha }}" /></span>, the combined effect of both is 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 \sigma ^{2}(\tau )\propto \tau ^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )\propto \tau ^{\alpha -1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7066db16393b57eb14e204eb9062303841baa2a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.137ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(\tau )\propto \tau ^{\alpha -1}}" /></span></figcaption></figure> <p><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 \sigma ^{2}(\tau )\approx \int _{0}^{\infty }S[g\ast {\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[g](\omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[h](\tau \omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>S</mi> <mo stretchy="false">[</mo> <mi>g</mi> <mo>∗</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>S</mi> <mo stretchy="false">[</mo> <mi>g</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>ω</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>S</mi> <mo stretchy="false">[</mo> <mi>h</mi> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>τ</mi> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>⋅</mo> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )\approx \int _{0}^{\infty }S[g\ast {\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[g](\omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[h](\tau \omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3fe9b2f262bc5cd191be214ca8575ee0f6f6532" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:84.82ex; height:5.843ex;" alt="{\displaystyle \sigma ^{2}(\tau )\approx \int _{0}^{\infty }S[g\ast {\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[g](\omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega =\int _{0}^{\infty }S[h](\tau \omega )\cdot S[{\dot {x}}_{f}](\omega )d\omega }" /></span>In words, the Allan variance is approximately the power of the fluctuation after <a href="/wiki/Band-pass_filter" title="Band-pass filter">bandpass filtering</a> at <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 \sim 1/\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>∼</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega \sim 1/\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d8904ad84e2f3e9245d05a933e8aa995af89b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.071ex; height:2.843ex;" alt="{\displaystyle \omega \sim 1/\tau }" /></span> with bandwidth <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 \Delta \omega \sim 1/\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>ω</mi> <mo>∼</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta \omega \sim 1/\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/817a90961634ebe4aea49bcd11ef80b683dbecd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.007ex; height:2.843ex;" alt="{\displaystyle \Delta \omega \sim 1/\tau }" /></span>. </p><p><br /> 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 1/f^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0022e4992b34cbe00c305e1604ff084622c6d2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.93ex; height:2.843ex;" alt="{\displaystyle 1/f^{\alpha }}" /></span> fluctuation, we have <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[{\dot {x}}_{f}](\omega )=C/\omega ^{\alpha }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>S</mi> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> <mo stretchy="false">]</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>C</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S[{\dot {x}}_{f}](\omega )=C/\omega ^{\alpha }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fb71b113d453356f20e42e91da05726c652e493" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.271ex; height:3.009ex;" alt="{\displaystyle S[{\dot {x}}_{f}](\omega )=C/\omega ^{\alpha }}" /></span> for some constant <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>, so <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 \sigma ^{2}(\tau )\approx \tau ^{\alpha -1}\sigma ^{2}(1)\propto \tau ^{\alpha -1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>≈</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )\approx \tau ^{\alpha -1}\sigma ^{2}(1)\propto \tau ^{\alpha -1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d881c8c19009e50eab993f6f86cdfee19d062ac3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.234ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(\tau )\approx \tau ^{\alpha -1}\sigma ^{2}(1)\propto \tau ^{\alpha -1}}" /></span>. In particular, when the fluctuating component <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 {\dot {x}}_{f}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\dot {x}}_{f}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85b49610541a460cf921da1f0540726267845eaf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.466ex; height:2.843ex;" alt="{\displaystyle {\dot {x}}_{f}}" /></span> is a 1/f noise, then <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 \sigma ^{2}(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c884f1d6aeec078d52c2e03cf4c9a2cf74c6acca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.396ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(\tau )}" /></span> is independent of the averaging time <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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span>, meaning that the clock frequency does not become more stable by simply averaging for longer. This contrasts with a white noise fluctuation, in which case <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 \sigma ^{2}(\tau )\propto \tau ^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <mo>∝</mo> <msup> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sigma ^{2}(\tau )\propto \tau ^{-1}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2014bf9920cd119e17783f91605083f0d39c3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.085ex; height:3.176ex;" alt="{\displaystyle \sigma ^{2}(\tau )\propto \tau ^{-1}}" /></span>, meaning that doubling the averaging time would improve the stability of frequency 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 {\sqrt {2}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>2</mn> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\sqrt {2}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4afc1e27d418021bf10898eb44a7f5f315735ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.098ex; height:3.009ex;" alt="{\displaystyle {\sqrt {2}}}" /></span>.<sup id="cite_ref-Press-1978_12-2" class="reference"><a href="#cite_note-Press-1978-12"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The cause of the noise floor is often traced to particular electronic components (such as transistors, resistors, and capacitors) within the oscillator feedback.<sup id="cite_ref-24" class="reference"><a href="#cite_note-24"><span class="cite-bracket">[</span>23<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Humans">Humans</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=12" title="Edit section: Humans"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In <a href="/wiki/Brains" class="mw-redirect" title="Brains">brains</a>, pink noise has been widely observed across many temporal and physical scales from <a href="/wiki/Ion_channel" title="Ion channel">ion channel</a> gating to <a href="/wiki/EEG" class="mw-redirect" title="EEG">EEG</a> and <a href="/wiki/Magnetoencephalography" title="Magnetoencephalography">MEG</a> and <a href="/wiki/Local_field_potential" title="Local field potential">LFP</a> recordings in humans.<sup id="cite_ref-25" class="reference"><a href="#cite_note-25"><span class="cite-bracket">[</span>24<span class="cite-bracket">]</span></a></sup> In clinical EEG, deviations from this 1/f pink noise can be used to identify <a href="/wiki/Epilepsy" title="Epilepsy">epilepsy</a>, even in the absence of a <a href="/wiki/Seizure" title="Seizure">seizure</a>, or during the interictal state.<sup id="cite_ref-Kerr-2012_26-0" class="reference"><a href="#cite_note-Kerr-2012-26"><span class="cite-bracket">[</span>25<span class="cite-bracket">]</span></a></sup> Classic models of EEG generators suggested that dendritic inputs in <a href="/wiki/Gray_matter" class="mw-redirect" title="Gray matter">gray matter</a> were principally responsible for generating the 1/f power spectrum observed in EEG/MEG signals. However, recent computational models using <a href="/wiki/Cable_theory" title="Cable theory">cable theory</a> have shown that <a href="/wiki/Action_potential" title="Action potential">action potential</a> transduction along <a href="/wiki/White_matter" title="White matter">white matter</a> tracts in the brain also generates a 1/f spectral density. Therefore, white matter signal transduction may also contribute to pink noise measured in scalp EEG recordings, <sup id="cite_ref-Douglas-2019_27-0" class="reference"><a href="#cite_note-Douglas-2019-27"><span class="cite-bracket">[</span>26<span class="cite-bracket">]</span></a></sup> particularly if the effects of ephaptic coupling are taken into consideration.<sup id="cite_ref-Douglas-2024_28-0" class="reference"><a href="#cite_note-Douglas-2024-28"><span class="cite-bracket">[</span>27<span class="cite-bracket">]</span></a></sup> </p><p>It has also been successfully applied to the modeling of <a href="/wiki/Mental_representation" title="Mental representation">mental states</a> in <a href="/wiki/Psychology" title="Psychology">psychology</a>,<sup id="cite_ref-cognitive_2003_29-0" class="reference"><a href="#cite_note-cognitive_2003-29"><span class="cite-bracket">[</span>28<span class="cite-bracket">]</span></a></sup> and used to explain stylistic variations in music from different cultures and historic periods.<sup id="cite_ref-30" class="reference"><a href="#cite_note-30"><span class="cite-bracket">[</span>29<span class="cite-bracket">]</span></a></sup> Richard F. Voss and J. Clarke claim that almost all musical melodies, when each successive note is plotted on a scale of <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitches</a>, will tend towards a pink noise spectrum.<sup id="cite_ref-Kuittinen_31-0" class="reference"><a href="#cite_note-Kuittinen-31"><span class="cite-bracket">[</span>30<span class="cite-bracket">]</span></a></sup> Similarly, a generally pink distribution pattern has been observed in <a href="/wiki/Shot_(filmmaking)" title="Shot (filmmaking)">film shot</a> length by researcher <a href="/wiki/James_E._Cutting" title="James E. Cutting">James E. Cutting</a> of <a href="/wiki/Cornell_University" title="Cornell University">Cornell University</a>, in the study of 150 popular movies released from 1935 to 2005.<sup id="cite_ref-32" class="reference"><a href="#cite_note-32"><span class="cite-bracket">[</span>31<span class="cite-bracket">]</span></a></sup> </p><p>Pink noise has also been found to be endemic in human response. Gilden et al. (1995) found extremely pure examples of this noise in the time series formed upon iterated production of temporal and spatial intervals.<sup id="cite_ref-Gilden-1995_33-0" class="reference"><a href="#cite_note-Gilden-1995-33"><span class="cite-bracket">[</span>32<span class="cite-bracket">]</span></a></sup> Later, Gilden (1997) and Gilden (2001) found that time series formed from <a href="/wiki/Reaction_time" class="mw-redirect" title="Reaction time">reaction time</a> measurement and from iterated two-alternative forced choice also produced pink noises.<sup id="cite_ref-Gilden-1997_34-0" class="reference"><a href="#cite_note-Gilden-1997-34"><span class="cite-bracket">[</span>33<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gilden-2001_35-0" class="reference"><a href="#cite_note-Gilden-2001-35"><span class="cite-bracket">[</span>34<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Electronic_devices">Electronic devices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=13" title="Edit section: Electronic devices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Flicker_noise" title="Flicker noise">Flicker noise</a></div> <p>The principal sources of pink noise in electronic devices are almost invariably the slow fluctuations of properties of the condensed-matter materials of the devices. In many cases the specific sources of the fluctuations are known. These include fluctuating configurations of defects in metals, fluctuating occupancies of traps in semiconductors, and fluctuating domain structures in magnetic materials.<sup id="cite_ref-Kogan-1996_8-1" class="reference"><a href="#cite_note-Kogan-1996-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Weissman-1988_36-0" class="reference"><a href="#cite_note-Weissman-1988-36"><span class="cite-bracket">[</span>35<span class="cite-bracket">]</span></a></sup> The explanation for the approximately pink spectral form turns out to be relatively trivial, usually coming from a distribution of kinetic activation energies of the fluctuating processes.<sup id="cite_ref-Dutta-1981_37-0" class="reference"><a href="#cite_note-Dutta-1981-37"><span class="cite-bracket">[</span>36<span class="cite-bracket">]</span></a></sup> Since the frequency range of the typical noise experiment (e.g., 1 Hz – 1 kHz) is low compared with typical microscopic "attempt frequencies" (e.g., 10<sup>14</sup> Hz), the exponential factors in the <a href="/wiki/Arrhenius_equation" title="Arrhenius equation">Arrhenius equation</a> for the rates are large. Relatively small spreads in the activation energies appearing in these exponents then result in large spreads of characteristic rates. In the simplest toy case, a flat distribution of activation energies gives exactly a pink spectrum, because <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 \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>f</mi> </mrow> </mfrac> </mrow> <mi>ln</mi> <mo>⁡</mo> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>f</mi> </mfrac> </mrow> <mo>.</mo> </mstyle> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6352cf348468b92ee9c10c6d9df63a169f1e6413" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:12.078ex; height:4.176ex;" alt="{\displaystyle \textstyle {\frac {d}{df}}\ln f={\frac {1}{f}}.}" /></span> </p><p>There is no known lower bound to background pink noise in electronics. Measurements made down to 10<sup>−6</sup> Hz (taking several weeks) have not shown a ceasing of pink-noise behaviour.<sup id="cite_ref-Kleinpenning-1988_38-0" class="reference"><a href="#cite_note-Kleinpenning-1988-38"><span class="cite-bracket">[</span>37<span class="cite-bracket">]</span></a></sup> (Kleinpenning, de Kuijper, 1988)<sup id="cite_ref-39" class="reference"><a href="#cite_note-39"><span class="cite-bracket">[</span>38<span class="cite-bracket">]</span></a></sup> measured the resistance in a noisy carbon-sheet resistor, and found 1/f noise behavior over the range 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 [10^{-5.5}\mathrm {Hz} ,10^{4}\mathrm {Hz} ]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>5.5</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> <mo>,</mo> <msup> <mn>10</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">H</mi> <mi mathvariant="normal">z</mi> </mrow> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [10^{-5.5}\mathrm {Hz} ,10^{4}\mathrm {Hz} ]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/82eab7b65c0f4455b0e07dcea832c101df410780" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.195ex; height:3.176ex;" alt="{\displaystyle [10^{-5.5}\mathrm {Hz} ,10^{4}\mathrm {Hz} ]}" /></span>, a range of 9.5 decades. </p><p>A pioneering researcher in this field was <a href="/wiki/Aldert_van_der_Ziel" title="Aldert van der Ziel">Aldert van der Ziel</a>.<sup id="cite_ref-40" class="reference"><a href="#cite_note-40"><span class="cite-bracket">[</span>39<span class="cite-bracket">]</span></a></sup> </p><p>Flicker noise is commonly used for the reliability characterization of electronic devices.<sup id="cite_ref-41" class="reference"><a href="#cite_note-41"><span class="cite-bracket">[</span>40<span class="cite-bracket">]</span></a></sup> It is also used for gas detection in chemoresistive sensors <sup id="cite_ref-42" class="reference"><a href="#cite_note-42"><span class="cite-bracket">[</span>41<span class="cite-bracket">]</span></a></sup> by dedicated measurement setups.<sup id="cite_ref-43" class="reference"><a href="#cite_note-43"><span class="cite-bracket">[</span>42<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="In_gravitational_wave_astronomy">In gravitational wave astronomy</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=14" title="Edit section: In gravitational wave astronomy"><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:Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png/440px-Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png" decoding="async" width="440" height="280" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/af/Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png/660px-Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/af/Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png/880px-Gravitational-wave_detector_sensitivities_and_astrophysical_gravitational-wave_sources.png 2x" data-file-width="1000" data-file-height="637" /></a><figcaption>Noise curves for a selection of <a href="/wiki/Gravitational-wave_detector" class="mw-redirect" title="Gravitational-wave detector">gravitational-wave detectors</a> as a function of frequency</figcaption></figure> <p>1/<i>f</i><sup> α</sup> noises with α near 1 are a factor in <a href="/wiki/Gravitational-wave_astronomy" title="Gravitational-wave astronomy">gravitational-wave astronomy</a>. The noise curve at very low frequencies affects <a href="/wiki/Pulsar_timing_array" title="Pulsar timing array">pulsar timing arrays</a>, the <a href="/wiki/European_Pulsar_Timing_Array" title="European Pulsar Timing Array">European Pulsar Timing Array</a> (EPTA) and the future <a href="/wiki/International_Pulsar_Timing_Array" title="International Pulsar Timing Array">International Pulsar Timing Array</a> (IPTA); at low frequencies are space-borne detectors, the formerly proposed <a href="/wiki/Laser_Interferometer_Space_Antenna" title="Laser Interferometer Space Antenna">Laser Interferometer Space Antenna</a> (LISA) and the currently proposed evolved Laser Interferometer Space Antenna (eLISA), and at high frequencies are ground-based detectors, the initial <a href="/wiki/LIGO" title="LIGO">Laser Interferometer Gravitational-Wave Observatory</a> (LIGO) and its advanced configuration (aLIGO). The characteristic strain of potential astrophysical sources are also shown. To be detectable the characteristic strain of a signal must be above the noise curve.<sup id="cite_ref-44" class="reference"><a href="#cite_note-44"><span class="cite-bracket">[</span>43<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Climate_dynamics">Climate dynamics</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=15" title="Edit section: Climate dynamics"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pink noise on timescales of decades has been found in climate proxy data, which may indicate amplification and coupling of processes in the <a href="/wiki/Climate_system" title="Climate system">climate system</a>.<sup id="cite_ref-45" class="reference"><a href="#cite_note-45"><span class="cite-bracket">[</span>44<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-46" class="reference"><a href="#cite_note-46"><span class="cite-bracket">[</span>45<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading3"><h3 id="Diffusion_processes">Diffusion processes</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=16" title="Edit section: Diffusion processes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Many time-dependent stochastic processes are known to exhibit 1/<i>f</i><sup> α</sup> noises with α between 0 and 2. In particular <a href="/wiki/Brownian_motion" title="Brownian motion">Brownian motion</a> has a <a href="/wiki/Spectral_density" title="Spectral density">power spectral density</a> that equals 4<i>D</i>/<i>f</i><sup> 2</sup>,<sup id="cite_ref-47" class="reference"><a href="#cite_note-47"><span class="cite-bracket">[</span>46<span class="cite-bracket">]</span></a></sup> where <i>D</i> is the <a href="/wiki/Mass_diffusivity" title="Mass diffusivity">diffusion coefficient</a>. This type of spectrum is sometimes referred to as <a href="/wiki/Brownian_noise" title="Brownian noise">Brownian noise</a>. The analysis of individual Brownian motion trajectories also show 1/<i>f</i><sup> 2</sup> spectrum, albeit with random amplitudes.<sup id="cite_ref-48" class="reference"><a href="#cite_note-48"><span class="cite-bracket">[</span>47<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Fractional_Brownian_motion" title="Fractional Brownian motion">Fractional Brownian motion</a> with <a href="/wiki/Hurst_exponent" title="Hurst exponent">Hurst exponent</a> <i>H</i> also show 1/<i>f</i><sup> α</sup> power spectral density with α=2<i>H</i>+1 for subdiffusive processes (<i>H</i><0.5) and α=2 for superdiffusive processes (0.5<<i>H</i><1).<sup id="cite_ref-49" class="reference"><a href="#cite_note-49"><span class="cite-bracket">[</span>48<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Origin">Origin</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=17" title="Edit section: Origin"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Supersymmetry#Supersymmetry_in_dynamical_systems" title="Supersymmetry">Supersymmetry § Supersymmetry in dynamical systems</a></div> <p>There are many theories about the origin of pink noise. Some theories attempt to be universal, while others apply to only a certain type of material, such as <a href="/wiki/Semiconductor" title="Semiconductor">semiconductors</a>. Universal theories of pink noise remain a matter of current research interest. </p><p>A hypothesis (referred to as the Tweedie hypothesis) has been proposed to explain the genesis of pink noise on the basis of a mathematical convergence theorem related to the <a href="/wiki/Central_limit_theorem" title="Central limit theorem">central limit theorem</a> of statistics.<sup id="cite_ref-Kendal2011b_50-0" class="reference"><a href="#cite_note-Kendal2011b-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> The <a href="/wiki/Tweedie_convergence_theorem" class="mw-redirect" title="Tweedie convergence theorem">Tweedie convergence theorem</a><sup id="cite_ref-51" class="reference"><a href="#cite_note-51"><span class="cite-bracket">[</span>50<span class="cite-bracket">]</span></a></sup> describes the convergence of certain statistical processes towards a family of statistical models known as the <a href="/wiki/Tweedie_distribution" title="Tweedie distribution">Tweedie distributions</a>. These distributions are characterized by a variance to mean <a href="/wiki/Power_law" title="Power law">power law</a>, that have been variously identified in the ecological literature as <a href="/wiki/Taylor%27s_law" title="Taylor's law">Taylor's law</a><sup id="cite_ref-Taylor1961_52-0" class="reference"><a href="#cite_note-Taylor1961-52"><span class="cite-bracket">[</span>51<span class="cite-bracket">]</span></a></sup> and in the physics literature as <i>fluctuation scaling</i>.<sup id="cite_ref-Eisler2008_53-0" class="reference"><a href="#cite_note-Eisler2008-53"><span class="cite-bracket">[</span>52<span class="cite-bracket">]</span></a></sup> When this variance to mean power law is demonstrated by the method of expanding enumerative bins this implies the presence of pink noise, and vice versa.<sup id="cite_ref-Kendal2011b_50-1" class="reference"><a href="#cite_note-Kendal2011b-50"><span class="cite-bracket">[</span>49<span class="cite-bracket">]</span></a></sup> Both of these effects can be shown to be the consequence of <a href="/wiki/Convergence_in_distribution" class="mw-redirect" title="Convergence in distribution">mathematical convergence</a> such as how certain kinds of data will converge towards the <a href="/wiki/Normal_distribution" title="Normal distribution">normal distribution</a> under the central limit theorem. This hypothesis also provides for an alternative paradigm to explain <a href="/wiki/Power_law" title="Power law">power law</a> manifestations that have been attributed to <a href="/wiki/Self-organized_criticality" title="Self-organized criticality">self-organized criticality</a>.<sup id="cite_ref-Kendal2015_54-0" class="reference"><a href="#cite_note-Kendal2015-54"><span class="cite-bracket">[</span>53<span class="cite-bracket">]</span></a></sup> </p><p>There are various mathematical models to create pink noise. The superposition of exponentially decaying pulses is able to generate a signal with 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/f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cd0e60c02ddf533da2abed825389fe5a94b7d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.603ex; height:2.843ex;" alt="{\displaystyle 1/f}" /></span>-spectrum at moderate frequencies, transitioning to a constant at low frequencies 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 1/f^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/f^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/13dc85513fc9696a8044cc820d3919af33af5799" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.7ex; height:3.176ex;" alt="{\displaystyle 1/f^{2}}" /></span> at high frequencies.<sup id="cite_ref-55" class="reference"><a href="#cite_note-55"><span class="cite-bracket">[</span>54<span class="cite-bracket">]</span></a></sup> In contrast, the sandpile model of self-organized criticality, which exhibits quasi-cycles of gradual stress accumulation between fast rare stress-releases, reproduces the flicker noise that corresponds to the intra-cycle dynamics.<sup id="cite_ref-56" class="reference"><a href="#cite_note-56"><span class="cite-bracket">[</span>55<span class="cite-bracket">]</span></a></sup> The statistical signature of self-organization is justified in <sup id="cite_ref-57" class="reference"><a href="#cite_note-57"><span class="cite-bracket">[</span>56<span class="cite-bracket">]</span></a></sup> It can be generated on computer, for example, by filtering white noise,<sup id="cite_ref-58" class="reference"><a href="#cite_note-58"><span class="cite-bracket">[</span>57<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-59" class="reference"><a href="#cite_note-59"><span class="cite-bracket">[</span>58<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-60" class="reference"><a href="#cite_note-60"><span class="cite-bracket">[</span>59<span class="cite-bracket">]</span></a></sup> <a href="/wiki/Inverse_Fourier_transform" class="mw-redirect" title="Inverse Fourier transform">inverse Fourier transform</a>,<sup id="cite_ref-61" class="reference"><a href="#cite_note-61"><span class="cite-bracket">[</span>60<span class="cite-bracket">]</span></a></sup> or by multirate variants on standard white noise generation.<sup id="cite_ref-Voss-1978_19-1" class="reference"><a href="#cite_note-Voss-1978-19"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-Gardner-1978_17-1" class="reference"><a href="#cite_note-Gardner-1978-17"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup> </p><p>In <a href="/wiki/Supersymmetric_theory_of_stochastic_dynamics" title="Supersymmetric theory of stochastic dynamics">supersymmetric theory of stochastics</a>,<sup id="cite_ref-62" class="reference"><a href="#cite_note-62"><span class="cite-bracket">[</span>61<span class="cite-bracket">]</span></a></sup> an approximation-free theory of <a href="/wiki/Stochastic_differential_equation" title="Stochastic differential equation">stochastic differential equations</a>, 1/<i>f</i> noise is one of the manifestations of the spontaneous breakdown of topological <a href="/wiki/Supersymmetry" title="Supersymmetry">supersymmetry</a>. This supersymmetry is an intrinsic property of all stochastic differential equations and its meaning is the preservation of the continuity of the <a href="/wiki/Phase_space" title="Phase space">phase space</a> by continuous time dynamics. Spontaneous breakdown of this supersymmetry is the stochastic generalization of the concept of <a href="/wiki/Chaos_theory" title="Chaos theory">deterministic chaos</a>,<sup id="cite_ref-63" class="reference"><a href="#cite_note-63"><span class="cite-bracket">[</span>62<span class="cite-bracket">]</span></a></sup> whereas the associated emergence of the long-term dynamical memory or order, i.e., 1/<i>f</i> and <a href="/wiki/Crackling_noise" title="Crackling noise">crackling</a> noises, the <a href="/wiki/Butterfly_effect" title="Butterfly effect">Butterfly effect</a> etc., is the consequence of the <a href="/wiki/Goldstone_boson" title="Goldstone boson">Goldstone theorem</a> in the application to the spontaneously broken topological supersymmetry. </p> <div class="mw-heading mw-heading2"><h2 id="Audio_testing">Audio testing</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=18" title="Edit section: Audio testing"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Pink noise is commonly used to test the loudspeakers in <a href="/wiki/Sound_reinforcement_system" title="Sound reinforcement system">sound reinforcement systems</a>, with the resulting sound measured with a test <a href="/wiki/Microphone" title="Microphone">microphone</a> in the listening space connected to a <a href="/wiki/Spectrum_analyzer" title="Spectrum analyzer">spectrum analyzer</a><sup id="cite_ref-Sound_3-1" class="reference"><a href="#cite_note-Sound-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> or a computer running a real-time <a href="/wiki/Fast_Fourier_transform" title="Fast Fourier transform">fast Fourier transform</a> (FFT) analyzer program such as <a href="/wiki/Smaart" title="Smaart">Smaart</a>. The sound system plays pink noise while the audio engineer makes adjustments on an <a href="/wiki/Equalization_(audio)" title="Equalization (audio)">audio equalizer</a> to obtain the desired results. Pink noise is predictable and repeatable, but it is annoying for a concert audience to hear. Since the late 1990s, FFT-based analysis enabled the engineer to make adjustments using pre-recorded music as the test signal, or even the music coming from the performers in real time.<sup id="cite_ref-64" class="reference"><a href="#cite_note-64"><span class="cite-bracket">[</span>63<span class="cite-bracket">]</span></a></sup> Pink noise is still used by audio system contractors<sup id="cite_ref-65" class="reference"><a href="#cite_note-65"><span class="cite-bracket">[</span>64<span class="cite-bracket">]</span></a></sup> and by computerized sound systems which incorporate an automatic equalization feature.<sup id="cite_ref-66" class="reference"><a href="#cite_note-66"><span class="cite-bracket">[</span>65<span class="cite-bracket">]</span></a></sup> </p><p>In manufacturing, pink noise is often used as a <a href="/wiki/Burn-in" title="Burn-in">burn-in</a> signal for <a href="/wiki/Audio_amplifier" class="mw-redirect" title="Audio amplifier">audio amplifiers</a> and other components, to determine whether the component will maintain performance integrity during sustained use.<sup id="cite_ref-67" class="reference"><a href="#cite_note-67"><span class="cite-bracket">[</span>66<span class="cite-bracket">]</span></a></sup> The process of end-users burning in their <a href="/wiki/Headphones" title="Headphones">headphones</a> with pink noise to attain higher fidelity has been called an <a href="/wiki/Audiophile" title="Audiophile">audiophile</a> "myth".<sup id="cite_ref-68" class="reference"><a href="#cite_note-68"><span class="cite-bracket">[</span>67<span class="cite-bracket">]</span></a></sup> </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=Pink_noise&action=edit&section=19" 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: 20em;"> <ul><li><a href="/wiki/Architectural_acoustics" title="Architectural acoustics">Architectural acoustics</a></li> <li><a href="/wiki/Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li> <li><a href="/wiki/Brownian_noise" title="Brownian noise">Brownian noise</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li> <li><a href="/wiki/Colors_of_noise" title="Colors of noise">Colors of noise</a></li> <li><a href="/wiki/Crest_factor" title="Crest factor">Crest factor</a></li> <li><a href="/wiki/Fractal" title="Fractal">Fractal</a></li> <li><a href="/wiki/Flicker_noise" title="Flicker noise">Flicker noise</a></li> <li><a href="/wiki/Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">Johnson–Nyquist noise</a></li> <li><a href="/wiki/Noise_(electronics)" title="Noise (electronics)">Noise (electronics)</a></li> <li><a href="/wiki/Quantum_1/f_noise" title="Quantum 1/f noise">Quantum 1/f noise</a></li> <li><a href="/wiki/Self-organised_criticality" class="mw-redirect" title="Self-organised criticality">Self-organised criticality</a></li> <li><a href="/wiki/Shot_noise" title="Shot noise">Shot noise</a></li> <li><a href="/wiki/Sound_masking" title="Sound masking">Sound masking</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Footnotes">Footnotes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Pink_noise&action=edit&section=20" title="Edit section: Footnotes"><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 reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-Field-1987-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-Field-1987_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Field-1987_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="CITEREFField1987" class="citation journal cs1">Field, D. J. (1987). <a rel="nofollow" class="external text" href="http://redwood.psych.cornell.edu/papers/field_87.pdf">"Relations between the statistics of natural images and the response properties of cortical cells"</a> <span class="cs1-format">(PDF)</span>. <i>J. Opt. Soc. Am. A</i>. <b>4</b> (12): <span class="nowrap">2379–</span>2394. <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/1987JOSAA...4.2379F">1987JOSAA...4.2379F</a>. <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.136.1345">10.1.1.136.1345</a></span>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1364%2FJOSAA.4.002379">10.1364/JOSAA.4.002379</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/3430225">3430225</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=J.+Opt.+Soc.+Am.+A&rft.atitle=Relations+between+the+statistics+of+natural+images+and+the+response+properties+of+cortical+cells&rft.volume=4&rft.issue=12&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2379-%3C%2Fspan%3E2394&rft.date=1987&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.136.1345%23id-name%3DCiteSeerX&rft_id=info%3Apmid%2F3430225&rft_id=info%3Adoi%2F10.1364%2FJOSAA.4.002379&rft_id=info%3Abibcode%2F1987JOSAA...4.2379F&rft.aulast=Field&rft.aufirst=D.+J.&rft_id=http%3A%2F%2Fredwood.psych.cornell.edu%2Fpapers%2Ffield_87.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" 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://www.soundonsound.com/glossary/pink-noise">"Glossary: Pink Noise"</a>. <i><a href="/wiki/Sound_on_Sound" title="Sound on Sound">Sound on Sound</a></i><span class="reference-accessdate">. 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O'Reilly Media. p. 79. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4493-1463-7" title="Special:BookSources/978-1-4493-1463-7"><bdi>978-1-4493-1463-7</bdi></a>. <q>Visible light with this power spectrum looks pink, hence the name.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Think+Complexity&rft.pages=79&rft.pub=O%27Reilly+Media&rft.date=2012&rft.isbn=978-1-4493-1463-7&rft.aulast=Downey&rft.aufirst=Allen&rft_id=http%3A%2F%2Fgreenteapress.com%2Fcomplexity%2Fhtml%2Fthinkcomplexity010.html%23toc57&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" 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"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBaxandall1968" class="citation magazine cs1">Baxandall, P. 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Retrieved <span class="nowrap">2019-08-08</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Wireless+World&rft.atitle=Noise+in+Transistor+Circuits%3A+1+-+Mainly+on+fundamental+noise+concepts&rft.pages=%3Cspan+class%3D%22nowrap%22%3E388-%3C%2Fspan%3E392&rft.date=1968-11&rft.aulast=Baxandall&rft.aufirst=P.+J.&rft_id=http%3A%2F%2Fwww.keith-snook.info%2Fwireless-world-magazine%2FWireless-World-1968%2FNoise%2520in%2520Transistor%2520Circuits%2520-%2520P%2520J%2520Baxandall.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" class="Z3988"></span></span> </li> <li id="cite_note-Das-thesis-7"><span class="mw-cite-backlink">^ <a href="#cite_ref-Das-thesis_7-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-2"><sup><i><b>c</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-3"><sup><i><b>d</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-4"><sup><i><b>e</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-5"><sup><i><b>f</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-6"><sup><i><b>g</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-7"><sup><i><b>h</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-8"><sup><i><b>i</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-9"><sup><i><b>j</b></i></sup></a> <a href="#cite_ref-Das-thesis_7-10"><sup><i><b>k</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDas2022" class="citation thesis cs1">Das, Abhranil (2022). <a rel="nofollow" class="external text" href="https://dx.doi.org/10.13140/RG.2.2.10585.80487"><i>Camouflage detection & signal discrimination: theory, methods & experiments (corrected)</i></a> (PhD). The University of Texas at Austin. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.13140%2FRG.2.2.10585.80487">10.13140/RG.2.2.10585.80487</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Adissertation&rft.title=Camouflage+detection+%26+signal+discrimination%3A+theory%2C+methods+%26+experiments+%28corrected%29&rft.inst=The+University+of+Texas+at+Austin&rft.date=2022&rft_id=info%3Adoi%2F10.13140%2FRG.2.2.10585.80487&rft.aulast=Das&rft.aufirst=Abhranil&rft_id=http%3A%2F%2Fdx.doi.org%2F10.13140%2FRG.2.2.10585.80487&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" class="Z3988"></span></span> </li> <li id="cite_note-Kogan-1996-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-Kogan-1996_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-Kogan-1996_8-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFKogan,_Shulim1996" class="citation book cs1">Kogan, Shulim (1996). <i>Electronic Noise and Fluctuations in Solids</i>. 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"1/<i>ƒ</i> noise". <i><a href="/wiki/Proceedings_of_the_IEEE" title="Proceedings of the IEEE">Proceedings of the IEEE</a></i>. <b>70</b> (3): <span class="nowrap">212–</span>218. <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%2FPROC.1982.12282">10.1109/PROC.1982.12282</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:921772">921772</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+IEEE&rft.atitle=1%2F%C6%92+noise&rft.volume=70&rft.issue=3&rft.pages=%3Cspan+class%3D%22nowrap%22%3E212-%3C%2Fspan%3E218&rft.date=1982&rft_id=info%3Adoi%2F10.1109%2FPROC.1982.12282&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A921772%23id-name%3DS2CID&rft.aulast=Keshner&rft.aufirst=M.+S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFChortiBrookes2007" class="citation book cs1">Chorti, A.; Brookes, M. (2007). "Resolving near-carrier spectral infinities due to 1/<i>f</i> phase noise in oscillators". <i>2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07</i>. Vol. 3. pp. III–1005–III–1008. <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%2FICASSP.2007.366852">10.1109/ICASSP.2007.366852</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-4244-0727-9" title="Special:BookSources/978-1-4244-0727-9"><bdi>978-1-4244-0727-9</bdi></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:14339595">14339595</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Resolving+near-carrier+spectral+infinities+due+to+1%2Ff+phase+noise+in+oscillators&rft.btitle=2007+IEEE+International+Conference+on+Acoustics%2C+Speech+and+Signal+Processing+-+ICASSP+%2707&rft.pages=III-1005-III-1008&rft.date=2007&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14339595%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1109%2FICASSP.2007.366852&rft.isbn=978-1-4244-0727-9&rft.aulast=Chorti&rft.aufirst=A.&rft.au=Brookes%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3APink+noise" class="Z3988"></span></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=Pink_noise&action=edit&section=22" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.mathworks.com/matlabcentral/fileexchange/121108-coloured-noise">Coloured Noise: Matlab toolbox to generate power-law coloured noise signals of any dimensions.</a></li> <li><a rel="nofollow" class="external text" href="http://www.maxlittle.net/software/">Powernoise: Matlab software for generating 1/<i>f</i> noise, or more generally, 1/<i>f</i><sup>α</sup> noise</a></li> <li><a rel="nofollow" class="external text" href="http://www.scholarpedia.org/article/1/f_noise">1/f noise at Scholarpedia</a></li> <li><a rel="nofollow" class="external text" href="http://www.acousticfields.com/white-noise-definition-vs-pink-noise/">White Noise Definition Vs Pink Noise</a></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid 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.navbox{display:none!important}}</style></div><div role="navigation" class="navbox" aria-labelledby="Noise_(physics_and_telecommunications)164" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374" /><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231" /><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Noise" title="Template:Noise"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Noise" title="Template talk:Noise"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Noise" title="Special:EditPage/Template:Noise"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Noise_(physics_and_telecommunications)164" style="font-size:114%;margin:0 4em"><a href="/wiki/Noise_(spectral_phenomenon)" title="Noise (spectral phenomenon)">Noise</a> (physics and telecommunications)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Acoustic_quieting" title="Acoustic quieting">Acoustic quieting</a></li> <li><a href="/wiki/Distortion" title="Distortion">Distortion</a></li> <li><a href="/wiki/Active_noise_control" title="Active noise control">Noise cancellation</a></li> <li><a href="/wiki/Noise_control" title="Noise control">Noise control</a></li> <li><a href="/wiki/Noise_measurement" title="Noise measurement">Noise measurement</a></li> <li><a href="/wiki/Noise_power" title="Noise power">Noise power</a></li> <li><a href="/wiki/Noise_reduction" title="Noise reduction">Noise reduction</a></li> <li><a href="/wiki/Noise_temperature" title="Noise temperature">Noise temperature</a></li> <li><a href="/wiki/Phase_distortion" title="Phase distortion">Phase distortion</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Noise in...</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Noise" title="Noise">Audio</a></li> <li><a href="/wiki/Architectural_acoustics" title="Architectural acoustics">Buildings</a></li> <li><a href="/wiki/Noise_(electronics)" title="Noise (electronics)">Electronics</a></li> <li><a href="/wiki/Noise_pollution" title="Noise pollution">Environment</a></li> <li><a href="/wiki/Noise_regulation" title="Noise regulation">Government regulation</a></li> <li><a href="/wiki/Health_effects_from_noise" title="Health effects from noise">Human health</a></li> <li><a href="/wiki/Image_noise" title="Image noise">Images</a></li> <li><a href="/wiki/Noise_(radio)" class="mw-redirect" title="Noise (radio)">Radio</a></li> <li><a href="/wiki/Soundproofing" title="Soundproofing">Rooms</a></li> <li><a href="/wiki/Noise_and_vibration_on_maritime_vessels" title="Noise and vibration on maritime vessels">Ships</a></li> <li><a href="/wiki/Sound_masking" title="Sound masking">Sound masking</a></li> <li><a href="/wiki/Noise_barrier" title="Noise barrier">Transportation</a></li> <li><a href="/wiki/Noise_(video)" title="Noise (video)">Video</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Class of noise</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Additive_white_Gaussian_noise" title="Additive white Gaussian noise">Additive white Gaussian noise</a> (AWGN)</li> <li><a href="/wiki/Atmospheric_noise" title="Atmospheric noise">Atmospheric noise</a></li> <li><a href="/wiki/Background_noise" title="Background noise">Background noise</a></li> <li><a href="/wiki/Brownian_noise" title="Brownian noise">Brownian noise</a></li> <li><a href="/wiki/Burst_noise" title="Burst noise">Burst noise</a></li> <li><a href="/wiki/Cosmic_noise" title="Cosmic noise">Cosmic noise</a></li> <li><a href="/wiki/Flicker_noise" title="Flicker noise">Flicker noise</a></li> <li><a href="/wiki/Gaussian_noise" title="Gaussian noise">Gaussian noise</a></li> <li><a href="/wiki/Grey_noise" title="Grey noise">Grey noise</a></li> <li><a href="/wiki/Infrasound" title="Infrasound">Infrasound</a></li> <li><a href="/wiki/Jitter" title="Jitter">Jitter</a></li> <li><a href="/wiki/Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">Johnson–Nyquist noise</a> (thermal noise)</li> <li><a class="mw-selflink selflink">Pink noise</a></li> <li><a href="/wiki/Quantization_error" class="mw-redirect" title="Quantization error">Quantization error</a> (or q. noise)</li> <li><a href="/wiki/Shot_noise" title="Shot noise">Shot noise</a></li> <li><a href="/wiki/White_noise" title="White noise">White noise</a></li> <li>Coherent noise <ul><li><a href="/wiki/Value_noise" title="Value noise">Value noise</a></li> <li><a href="/wiki/Gradient_noise" title="Gradient noise">Gradient noise</a></li> <li><a href="/wiki/Worley_noise" title="Worley noise">Worley noise</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Engineering <br />terms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Channel_noise_level" title="Channel noise level">Channel noise level</a></li> <li><a href="/wiki/Circuit_noise_level" title="Circuit noise level">Circuit noise level</a></li> <li><a href="/wiki/Effective_input_noise_temperature" title="Effective input noise temperature">Effective input noise temperature</a></li> <li><a href="/wiki/Equivalent_noise_resistance" title="Equivalent noise resistance">Equivalent noise resistance</a></li> <li><a href="/wiki/Equivalent_pulse_code_modulation_noise" title="Equivalent pulse code modulation noise">Equivalent pulse code modulation noise</a></li> <li><a href="/wiki/Impulse_noise_(audio)" class="mw-redirect" title="Impulse noise (audio)">Impulse noise (audio)</a></li> <li><a href="/wiki/Noise_figure" title="Noise figure">Noise figure</a></li> <li><a href="/wiki/Noise_floor" title="Noise floor">Noise floor</a></li> <li><a href="/wiki/Noise_shaping" title="Noise shaping">Noise shaping</a></li> <li><a href="/wiki/Noise_spectral_density" title="Noise spectral density">Noise spectral density</a></li> <li><a href="/wiki/Noise,_vibration,_and_harshness" title="Noise, vibration, and harshness">Noise, vibration, and harshness</a> (NVH)</li> <li><a href="/wiki/Phase_noise" title="Phase noise">Phase noise</a></li> <li><a href="/wiki/Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a></li> <li><a href="/wiki/Statistical_noise" class="mw-redirect" title="Statistical noise">Statistical noise</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ratios</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Carrier-to-noise_ratio" title="Carrier-to-noise ratio">Carrier-to-noise ratio</a> (<i>C</i>/<i>N</i>)</li> <li><a href="/wiki/Carrier-to-receiver_noise_density" class="mw-redirect" title="Carrier-to-receiver noise density">Carrier-to-receiver noise density</a> (<i>C</i>/<i>kT</i>)</li> <li><i><a href="/wiki/DBrnC" class="mw-redirect" title="DBrnC">dBrnC</a></i></li> <li><i><a href="/wiki/Eb/N0" title="Eb/N0">E<sub>b</sub>/N<sub>0</sub></a></i> (energy per bit to noise density)</li> <li><i><a href="/wiki/Eb/N0#Relation_to_Es.2FN0" title="Eb/N0">E<sub>s</sub>/N<sub>0</sub></a></i> (energy per symbol to noise density)</li> <li><a href="/wiki/Modulation_error_ratio" title="Modulation error ratio">Modulation error ratio</a> (<i>MER</i>)</li> <li><a href="/wiki/SINAD" title="SINAD">Signal, noise and distortion</a> (<i>SINAD</i>)</li> <li><a href="/wiki/Signal-to-interference_ratio" title="Signal-to-interference ratio">Signal-to-interference ratio</a> (<i>S</i>/<i>I</i>)</li> <li><a href="/wiki/Signal-to-noise_ratio" title="Signal-to-noise ratio">Signal-to-noise ratio</a> (<i>S</i>/<i>N</i>, <i>SNR</i>)</li> <li><a href="/wiki/Signal-to-noise_ratio_(imaging)" title="Signal-to-noise ratio (imaging)">Signal-to-noise ratio (imaging)</a></li> <li><a href="/wiki/Signal-to-interference-plus-noise_ratio" title="Signal-to-interference-plus-noise ratio">Signal-to-interference-plus-noise ratio</a> (<i>SINR</i>)</li> <li><a href="/wiki/Signal-to-quantization-noise_ratio" title="Signal-to-quantization-noise ratio">Signal-to-quantization-noise ratio</a> (<i>SQNR</i>)</li> <li><a href="/wiki/Contrast-to-noise_ratio" title="Contrast-to-noise ratio">Contrast-to-noise ratio</a> (<i>CNR</i>)</li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/List_of_noise_topics" title="List of noise topics">List of noise topics</a></li> <li><a href="/wiki/Acoustics" title="Acoustics">Acoustics</a></li> <li><a href="/wiki/Colors_of_noise" title="Colors of noise">Colors of noise</a></li> <li><a href="/wiki/Interference_(communication)" title="Interference (communication)">Interference (communication)</a></li> <li><a href="/wiki/Noise_generator" title="Noise generator">Noise generator</a></li> <li><a href="/wiki/Spectrum_analyzer" title="Spectrum analyzer">Spectrum analyzer</a></li> <li><a href="/wiki/Thermal_radiation" title="Thermal radiation">Thermal radiation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Denoise <br />methods</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Low-pass_filter" title="Low-pass filter">Low-pass filter</a></li> <li><a href="/wiki/Median_filter" title="Median filter">Median filter</a></li> <li><a href="/wiki/Total_variation_denoising" title="Total variation denoising">Total variation denoising</a></li> <li><a href="/wiki/Wavelet#Wavelet_denoising" title="Wavelet">Wavelet denoising</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">2D (Image)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Gaussian_blur" title="Gaussian blur">Gaussian blur</a></li> <li><a href="/wiki/Anisotropic_diffusion" title="Anisotropic diffusion">Anisotropic diffusion</a></li> <li><a href="/wiki/Bilateral_filter" title="Bilateral filter">Bilateral filter</a></li> <li><a href="/wiki/Non-local_means" title="Non-local means">Non-local means</a></li> <li><a href="/wiki/Block-matching_and_3D_filtering" title="Block-matching and 3D filtering">Block-matching and 3D filtering</a> (BM3D)</li> <li><a href="/wiki/Shrinkage_Fields_(image_restoration)" title="Shrinkage Fields (image restoration)">Shrinkage Fields</a></li> <li><a href="/wiki/Autoencoder#Denoising_autoencoder_(DAE)" title="Autoencoder">Denoising autoencoder</a> (DAE)</li> <li><a href="/wiki/Deep_Image_Prior" class="mw-redirect" title="Deep Image Prior">Deep Image Prior</a></li></ul> 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