Spectral density - Wikipedia

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id="toc-One-sided_vs_two-sided" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#One-sided_vs_two-sided"> <div class="vector-toc-text"> <span class="vector-toc-numb">1.1</span> <span>One-sided vs two-sided</span> </div> </a> <ul id="toc-One-sided_vs_two-sided-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Definition" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definition"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definition</span> </div> </a> <button aria-controls="toc-Definition-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 Definition subsection</span> </button> <ul id="toc-Definition-sublist" class="vector-toc-list"> <li id="toc-Energy_spectral_density" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Energy_spectral_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.1</span> <span>Energy spectral density</span> </div> </a> <ul id="toc-Energy_spectral_density-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Power_spectral_density" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Power_spectral_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2</span> <span>Power spectral density</span> </div> </a> <ul id="toc-Power_spectral_density-sublist" class="vector-toc-list"> <li id="toc-Properties_of_the_power_spectral_density" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Properties_of_the_power_spectral_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.2.1</span> <span>Properties of the power spectral density</span> </div> </a> <ul id="toc-Properties_of_the_power_spectral_density-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Cross_power_spectral_density" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cross_power_spectral_density"> <div class="vector-toc-text"> <span class="vector-toc-numb">2.3</span> <span>Cross power spectral density</span> </div> </a> <ul id="toc-Cross_power_spectral_density-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Estimation" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Estimation"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Estimation</span> </div> </a> <ul id="toc-Estimation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Related_concepts" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Related_concepts"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Related concepts</span> </div> </a> <ul id="toc-Related_concepts-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Applications</span> </div> </a> <button aria-controls="toc-Applications-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 Applications subsection</span> </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Electrical_engineering" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Electrical_engineering"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.1</span> <span>Electrical engineering</span> </div> </a> <ul id="toc-Electrical_engineering-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Cosmology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Cosmology"> <div class="vector-toc-text"> <span class="vector-toc-numb">5.2</span> <span>Cosmology</span> </div> </a> <ul id="toc-Cosmology-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>Notes</span> </div> </a> <ul id="toc-Notes-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">8</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">9</span> <span>External links</span> </div> </a> <ul id="toc-External_links-sublist" class="vector-toc-list"> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header class="mw-body-header vector-page-titlebar"> <nav aria-label="Contents" class="vector-toc-landmark"> <div id="vector-page-titlebar-toc" class="vector-dropdown vector-page-titlebar-toc vector-button-flush-left" title="Table of Contents" > <input type="checkbox" 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Available in 19 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-19" 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">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-bar mw-list-item"><a href="https://bar.wikipedia.org/wiki/Spektrale_Leistungsdichtn" title="Spektrale Leistungsdichtn – Bavarian" lang="bar" hreflang="bar" data-title="Spektrale Leistungsdichtn" 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/Densitat_espectral" title="Densitat espectral – Catalan" lang="ca" hreflang="ca" data-title="Densitat espectral" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Spektrale_Leistungsdichte" title="Spektrale Leistungsdichte – German" lang="de" hreflang="de" data-title="Spektrale Leistungsdichte" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Spektraaltihedus" title="Spektraaltihedus – Estonian" lang="et" hreflang="et" data-title="Spektraaltihedus" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Densidad_espectral" title="Densidad espectral – Spanish" lang="es" hreflang="es" data-title="Densidad espectral" 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/Spektra_povuma_distribuo" title="Spektra povuma distribuo – Esperanto" lang="eo" hreflang="eo" data-title="Spektra povuma distribuo" data-language-autonym="Esperanto" data-language-local-name="Esperanto" class="interlanguage-link-target"><span>Esperanto</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%DA%86%DA%AF%D8%A7%D9%84%DB%8C_%D8%B7%DB%8C%D9%81%DB%8C" title="چگالی طیفی – Persian" lang="fa" hreflang="fa" data-title="چگالی طیفی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Densit%C3%A9_spectrale_de_puissance" title="Densité spectrale de puissance – French" lang="fr" hreflang="fr" data-title="Densité spectrale de puissance" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Spettro_di_potenza" title="Spettro di potenza – Italian" lang="it" hreflang="it" data-title="Spettro di potenza" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E3%82%B9%E3%83%9A%E3%82%AF%E3%83%88%E3%83%AB%E5%AF%86%E5%BA%A6" title="スペクトル密度 – Japanese" lang="ja" hreflang="ja" data-title="スペクトル密度" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target"><span>日本語</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Widmowa_g%C4%99sto%C5%9B%C4%87_mocy" title="Widmowa gęstość mocy – Polish" lang="pl" hreflang="pl" data-title="Widmowa gęstość mocy" 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/Densidade_espectral" title="Densidade espectral – Portuguese" lang="pt" hreflang="pt" data-title="Densidade espectral" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D0%BA%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%BF%D0%BB%D0%BE%D1%82%D0%BD%D0%BE%D1%81%D1%82%D1%8C_%D0%BC%D0%BE%D1%89%D0%BD%D0%BE%D1%81%D1%82%D0%B8" title="Спектральная плотность мощности – Russian" lang="ru" hreflang="ru" data-title="Спектральная плотность мощности" 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/Power_spectrum" title="Power spectrum – Simple English" lang="en-simple" hreflang="en-simple" data-title="Power spectrum" 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-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Effektspektrum" title="Effektspektrum – Swedish" lang="sv" hreflang="sv" data-title="Effektspektrum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Spektral_yo%C4%9Funluk" title="Spektral yoğunluk – Turkish" lang="tr" hreflang="tr" data-title="Spektral yoğunluk" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%A1%D0%BF%D0%B5%D0%BA%D1%82%D1%80%D0%B0%D0%BB%D1%8C%D0%BD%D0%B0_%D0%B3%D1%83%D1%81%D1%82%D0%B8%D0%BD%D0%B0" title="Спектральна густина – Ukrainian" lang="uk" hreflang="uk" data-title="Спектральна густина" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a href="https://vi.wikipedia.org/wiki/M%E1%BA%ADt_%C4%91%E1%BB%99_ph%E1%BB%95_n%C4%83ng_l%C6%B0%E1%BB%A3ng" title="Mật độ phổ năng lượng – Vietnamese" lang="vi" hreflang="vi" data-title="Mật độ phổ năng lượng" data-language-autonym="Tiếng Việt" data-language-local-name="Vietnamese" class="interlanguage-link-target"><span>Tiếng Việt</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E8%B0%B1%E5%AF%86%E5%BA%A6" title="谱密度 – Chinese" lang="zh" hreflang="zh" data-title="谱密度" 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/Q1331626#sitelinks-wikipedia" title="Edit interlanguage links" class="wbc-editpage">Edit links</a></span></div> </div> </div> </div> </header> <div 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.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">This article is about signal processing and relation of spectra to time-series. For further applications in the physical sciences, see <a href="/wiki/Spectrum_(physical_sciences)" title="Spectrum (physical sciences)">Spectrum (physical sciences)</a>.</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951" /><div role="note" class="hatnote navigation-not-searchable">"Spectral power density" redirects here; not to be confused with <a href="/wiki/Spectral_power" class="mw-redirect" title="Spectral power">Spectral power</a>.</div> <p class="mw-empty-elt"> </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fluorescent_lighting_spectrum_peaks_labelled.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Fluorescent_lighting_spectrum_peaks_labelled.svg/220px-Fluorescent_lighting_spectrum_peaks_labelled.svg.png" decoding="async" width="220" height="133" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Fluorescent_lighting_spectrum_peaks_labelled.svg/330px-Fluorescent_lighting_spectrum_peaks_labelled.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Fluorescent_lighting_spectrum_peaks_labelled.svg/440px-Fluorescent_lighting_spectrum_peaks_labelled.svg.png 2x" data-file-width="1279" data-file-height="771" /></a><figcaption>The spectral density of a <a href="/wiki/Fluorescent_light" class="mw-redirect" title="Fluorescent light">fluorescent light</a> as a function of optical wavelength shows peaks at atomic transitions, indicated by the numbered arrows.</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Voice_waveform_and_spectrum.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Voice_waveform_and_spectrum.png/250px-Voice_waveform_and_spectrum.png" decoding="async" width="220" height="88" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Voice_waveform_and_spectrum.png/330px-Voice_waveform_and_spectrum.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f1/Voice_waveform_and_spectrum.png/500px-Voice_waveform_and_spectrum.png 2x" data-file-width="1046" data-file-height="418" /></a><figcaption>The voice waveform over time (left) has a broad audio power spectrum (right).</figcaption></figure><style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></span></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>may be too technical for most readers to understand</b>.<span class="hide-when-compact"> Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Spectral_density&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details.</span> <span class="date-container"><i>(<span class="date">June 2024</span>)</i></span><span class="hide-when-compact"><i> (<small><a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a></small>)</i></span></div></td></tr></tbody></table> <p>In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, the power spectrum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab836aa57df700d834cc0f736e9f81152fa470e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" /></span> of a <a href="/wiki/Continuous_time" class="mw-redirect" title="Continuous time">continuous time</a> <a href="/wiki/Signal" title="Signal">signal</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> describes the distribution of <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> into frequency components <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> composing that signal.<sup id="cite_ref-P_Stoica_1-0" class="reference"><a href="#cite_note-P_Stoica-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> According to <a href="/wiki/Fourier_analysis" title="Fourier analysis">Fourier analysis</a>, any physical signal can be decomposed into a number of discrete frequencies, or a spectrum of frequencies over a continuous range. The statistical average of any sort of signal (including <a href="/wiki/Noise_(electronics)" title="Noise (electronics)">noise</a>) as analyzed in terms of its frequency content, is called its <a href="/wiki/Spectrum" title="Spectrum">spectrum</a>. </p><p>When the energy of the signal is concentrated around a finite time interval, especially if its total energy is finite, one may compute the <b>energy spectral density</b>. More commonly used is the <b>power spectral density</b> (PSD, or simply <b>power spectrum</b>), which applies to signals existing over <i>all</i> time, or over a time period large enough (especially in relation to the duration of a measurement) that it could as well have been over an infinite time interval. The PSD then refers to the spectral energy distribution that would be found per unit time, since the total energy of such a signal over all time would generally be infinite. <a href="/wiki/Summation" title="Summation">Summation</a> or integration of the spectral components yields the total power (for a physical process) or variance (in a statistical process), identical to what would be obtained by integrating <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^{2}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58eb8c006ec38666f7f0bb9106d809a3903345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:3.176ex;" alt="{\displaystyle x^{2}(t)}" /></span> over the time domain, as dictated by <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>.<sup id="cite_ref-P_Stoica_1-1" class="reference"><a href="#cite_note-P_Stoica-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>The spectrum of a physical process <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> often contains essential information about the nature 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 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>. For instance, the <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</a> and <a href="/wiki/Timbre" title="Timbre">timbre</a> of a musical instrument are immediately determined from a spectral analysis. The <a href="/wiki/Color" title="Color">color</a> of a light source is determined by the spectrum of the electromagnetic wave's electric field <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf5be62d9f63f5df52e8bc156f950d41e131d99f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.425ex; height:2.843ex;" alt="{\displaystyle E(t)}" /></span> as it fluctuates at an extremely high frequency. Obtaining a spectrum from time series such as these involves the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a>, and generalizations based on Fourier analysis. In many cases the time domain is not specifically employed in practice, such as when a <a href="/wiki/Dispersive_prism" title="Dispersive prism">dispersive prism</a> is used to obtain a spectrum of light in a <a href="/wiki/Spectrograph" class="mw-redirect" title="Spectrograph">spectrograph</a>, or when a sound is perceived through its effect on the auditory receptors of the inner ear, each of which is sensitive to a particular frequency. </p><p>However this article concentrates on situations in which the time series is known (at least in a statistical sense) or directly measured (such as by a microphone sampled by a computer). The power spectrum is important in <a href="/wiki/Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">statistical signal processing</a> and in the statistical study of <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic processes</a>, as well as in many other branches of <a href="/wiki/Physics" title="Physics">physics</a> and <a href="/wiki/Engineering" title="Engineering">engineering</a>. Typically the process is a function of time, but one can similarly discuss data in the spatial domain being decomposed in terms of <a href="/wiki/Spatial_frequency" title="Spatial frequency">spatial frequency</a>.<sup id="cite_ref-P_Stoica_1-2" class="reference"><a href="#cite_note-P_Stoica-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Units">Units</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=1" title="Edit section: Units"><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/Fourier_transform#Units" title="Fourier transform">Fourier transform § Units</a></div> <p>In <a href="/wiki/Physics" title="Physics">physics</a>, the signal might be a wave, such as an <a href="/wiki/Electromagnetic_wave" class="mw-redirect" title="Electromagnetic wave">electromagnetic wave</a>, an <a href="/wiki/Sound_wave" class="mw-redirect" title="Sound wave">acoustic wave</a>, or the vibration of a mechanism. The <i>power spectral density</i> (PSD) of the signal describes the <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> present in the signal as a function of frequency, per unit frequency. Power spectral density is commonly expressed in <a href="/wiki/SI_units" class="mw-redirect" title="SI units">SI units</a> of <a href="/wiki/Watt" title="Watt">watts</a> per <a href="/wiki/Hertz" title="Hertz">hertz</a> (abbreviated as W/Hz).<sup id="cite_ref-FOOTNOTEMaral2004_2-0" class="reference"><a href="#cite_note-FOOTNOTEMaral2004-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>When a signal is defined in terms only of a <a href="/wiki/Voltage" title="Voltage">voltage</a>, for instance, there is no unique power associated with the stated amplitude. In this case "power" is simply reckoned in terms of the square of the signal, as this would always be <i>proportional</i> to the actual power delivered by that signal into a given <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a>. So one might use units of V<sup>2</sup> Hz<sup>−1</sup> for the PSD. <i>Energy spectral density</i> (ESD) would have units of V<sup>2</sup> s Hz<sup>−1</sup>, since <a href="/wiki/Energy_(physics)" class="mw-redirect" title="Energy (physics)">energy</a> has units of power multiplied by time (e.g., <a href="/wiki/Watt-hour" class="mw-redirect" title="Watt-hour">watt-hour</a>).<sup id="cite_ref-FOOTNOTENortonKarczub2003_3-0" class="reference"><a href="#cite_note-FOOTNOTENortonKarczub2003-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>In the general case, the units of PSD will be the ratio of units of variance per unit of frequency; so, for example, a series of displacement values (in meters) over time (in seconds) will have PSD in units of meters squared per hertz, m<sup>2</sup>/Hz. In the analysis of random <a href="/wiki/Vibration" title="Vibration">vibrations</a>, units of <i>g</i><sup>2</sup> Hz<sup>−1</sup> are frequently used for the PSD of <a href="/wiki/Acceleration" title="Acceleration">acceleration</a>, where <i>g</i> denotes the <a href="/wiki/G-force" title="G-force">g-force</a>.<sup id="cite_ref-FOOTNOTEBirolini200783_4-0" class="reference"><a href="#cite_note-FOOTNOTEBirolini200783-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Mathematically, it is not necessary to assign physical dimensions to the signal or to the independent variable. In the following discussion the meaning of <i>x</i>(<i>t</i>) will remain unspecified, but the independent variable will be assumed to be that of time. </p> <div class="mw-heading mw-heading3"><h3 id="One-sided_vs_two-sided">One-sided vs two-sided</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=2" title="Edit section: One-sided vs two-sided"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A PSD can be either a <i>one-sided</i> function of only positive frequencies or a <i>two-sided</i> function of both positive and <a href="/wiki/Negative_frequency" title="Negative frequency">negative frequencies</a> but with only half the amplitude. Noise PSDs are generally one-sided in engineering and two-sided in physics.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p> <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=Spectral_density&action=edit&section=3" title="Edit section: Definition"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading3"><h3 id="Energy_spectral_density">Energy spectral density<span class="anchor" id="Energy"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=4" title="Edit section: Energy spectral density"><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">"Energy spectral density" redirects here; not to be confused with <a href="/wiki/Energy_spectrum" class="mw-redirect" title="Energy spectrum">Energy spectrum</a>.</div> <p>In <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, the <a href="/wiki/Energy_(signal_processing)" title="Energy (signal processing)">energy</a> of a signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" /></span> is given by <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 E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>≜</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/95e55b9d546c492e4ef00dd98b4a7dd2596f3dcd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:19.317ex; height:6.009ex;" alt="{\displaystyle E\triangleq \int _{-\infty }^{\infty }\left|x(t)\right|^{2}\ dt.}" /></span> Assuming the total energy is finite (i.e. <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 a <a href="/wiki/Square-integrable_function" title="Square-integrable function">square-integrable function</a>) allows applying <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a> (or <a href="/wiki/Plancherel%27s_theorem" class="mw-redirect" title="Plancherel's theorem">Plancherel's theorem</a>).<sup id="cite_ref-FOOTNOTEOppenheimVerghese201660_6-0" class="reference"><a href="#cite_note-FOOTNOTEOppenheimVerghese201660-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> That is, <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 \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0735faea40ac3f62518064fd0ce9e7edeeaa4a7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:31.829ex; height:6.009ex;" alt="{\displaystyle \int _{-\infty }^{\infty }|x(t)|^{2}\,dt=\int _{-\infty }^{\infty }\left|{\hat {x}}(f)\right|^{2}\,df,}" /></span> where <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 {\hat {x}}(f)=\int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt,}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mtext> </mtext> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f)=\int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f6ea198f4fcbff160930dbb5b78a8620d22cc35" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.035ex; height:6.009ex;" alt="{\displaystyle {\hat {x}}(f)=\int _{-\infty }^{\infty }e^{-i2\pi ft}x(t)\ dt,}" /></span> is the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> at <a href="/wiki/Frequency" title="Frequency">frequency</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> (in <a href="/wiki/Hz" class="mw-redirect" title="Hz">Hz</a>).<sup id="cite_ref-FOOTNOTEStein2000108,_115_7-0" class="reference"><a href="#cite_note-FOOTNOTEStein2000108,_115-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> The theorem also holds true in the discrete-time cases. Since the integral on the left-hand side is the energy of the signal, the value 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 \left|{\hat {x}}(f)\right|^{2}df}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e9cdfe3d7660f5c2ddae825eeaa9f52e90fd6a6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.26ex; height:3.343ex;" alt="{\displaystyle \left|{\hat {x}}(f)\right|^{2}df}" /></span> can be interpreted as a <a href="/wiki/Density_function" class="mw-redirect" title="Density function">density function</a> multiplied by an infinitesimally small frequency interval, describing the energy contained in the signal at frequency <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> in the frequency 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 f+df}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>+</mo> <mi>d</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f+df}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c1b25aa5ed1a0704987e497cd22a5a27ba8c8a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.613ex; height:2.509ex;" alt="{\displaystyle f+df}" /></span>. </p><p>Therefore, the <b>energy spectral density</b> 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 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 defined as:<sup id="cite_ref-FOOTNOTEOppenheimVerghese201614_8-0" class="reference"><a href="#cite_note-FOOTNOTEOppenheimVerghese201614-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <style data-mw-deduplicate="TemplateStyles:r1266403038">.mw-parser-output table.numblk{border-collapse:collapse;border:none;margin-top:0;margin-right:0;margin-bottom:0}.mw-parser-output table.numblk>tbody>tr>td{vertical-align:middle;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2){width:99%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table{border-collapse:collapse;margin:0;border:none;width:100%}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:first-child,.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:last-child{padding:0 0.4ex}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td:nth-child(2){width:100%;padding:0}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{padding:0}.mw-parser-output table.numblk>tbody>tr>td:last-child{font-weight:bold}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child{font-weight:unset}.mw-parser-output table.numblk>tbody>tr>td:last-child::before{content:"("}.mw-parser-output table.numblk>tbody>tr>td:last-child::after{content:")"}.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::before,.mw-parser-output table.numblk.numblk-raw-n>tbody>tr>td:last-child::after{content:none}.mw-parser-output table.numblk>tbody>tr>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:first-child>td{border:thin solid}.mw-parser-output table.numblk>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:none;border-right:none;border-bottom:none}.mw-parser-output table.numblk.numblk-border>tbody>tr>td:nth-child(2)>table>tbody>tr:last-child>td{border-left:thin solid;border-right:thin solid;border-bottom:thin solid}.mw-parser-output table.numblk:target{color:var(--color-base,#202122);background-color:#cfe8fd}@media screen{html.skin-theme-clientpref-night .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output table.numblk:target{color:var(--color-base,#eaecf0);background-color:#301702}}</style><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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 {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>≜</mo> <msup> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/df07f4a3693d52e3c254dfbe470bdcedfbc9d56d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.637ex; height:3.343ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)\triangleq \left|{\hat {x}}(f)\right|^{2}}" /></span></td> <td></td> <td class="nowrap"><span id="math_Eq.1" class="reference nourlexpansion" style="font-weight:bold;">Eq.1</span></td></tr></tbody></table> </div> <p>The function <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}_{xx}(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>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f43c62387fe5b01596533b3e850be9843e4e4618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.773ex; height:3.176ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)}" /></span> and the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> form a Fourier transform pair, a result also known as the <a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a> (see also <a href="/wiki/Periodogram#Definition" title="Periodogram">Periodogram</a>). </p><p>As a physical example of how one might measure the energy spectral density of a signal, suppose <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 V(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/383b47023708ac9df0e198448491048ec7acb2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.436ex; height:2.843ex;" alt="{\displaystyle V(t)}" /></span> represents the <a href="/wiki/Electric_potential" title="Electric potential">potential</a> (in <a href="/wiki/Volt" title="Volt">volts</a>) of an electrical pulse propagating along a <a href="/wiki/Transmission_line" title="Transmission line">transmission line</a> of <a href="/wiki/Electrical_impedance" title="Electrical impedance">impedance</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" /></span>, and suppose the line is terminated with a <a href="/wiki/Impedance_matching" title="Impedance matching">matched</a> resistor (so that all of the pulse energy is delivered to the resistor and none is reflected back). By <a href="/wiki/Ohm%27s_law" title="Ohm's law">Ohm's law</a>, the power delivered to the resistor at 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> is equal 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 V(t)^{2}/Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275d111396d456efa3ad9b9dc2e55de79e605bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" /></span>, so the total energy is found by integrating <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 V(t)^{2}/Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275d111396d456efa3ad9b9dc2e55de79e605bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" /></span> with respect to time over the duration of the pulse. To find the value of the energy spectral density <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 {\bar {S}}_{xx}(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>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f43c62387fe5b01596533b3e850be9843e4e4618" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.773ex; height:3.176ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)}" /></span> at frequency <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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span>, one could insert between the transmission line and the resistor a <a href="/wiki/Bandpass_filter" class="mw-redirect" title="Bandpass filter">bandpass filter</a> which passes only a narrow range of 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 \Delta f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff38db24bc80b80a0a1ecfce75f3dca3fc79d54e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.214ex; height:2.509ex;" alt="{\displaystyle \Delta f}" /></span>, say) near the frequency of interest and then measure the total energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}" /></span> dissipated across the resistor. The value of the energy spectral density 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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span> is then estimated to be <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)/\Delta f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)/\Delta f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6da71905d2242140e6720b4c263f1fe8736ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.24ex; height:2.843ex;" alt="{\displaystyle E(f)/\Delta f}" /></span>. In this example, since the 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 V(t)^{2}/Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(t)^{2}/Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/275d111396d456efa3ad9b9dc2e55de79e605bef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.333ex; height:3.176ex;" alt="{\displaystyle V(t)^{2}/Z}" /></span> has units of V<sup>2</sup> Ω<sup>−1</sup>, the energy <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d733654dab86d76c9ed9ba7f4473250dfea33d3e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.864ex; height:2.843ex;" alt="{\displaystyle E(f)}" /></span> has units of V<sup>2</sup> s Ω<sup>−1</sup> = <a href="/wiki/Joule" title="Joule">J</a>, and hence the estimate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E(f)/\Delta f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">Δ</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E(f)/\Delta f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e6da71905d2242140e6720b4c263f1fe8736ecd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.24ex; height:2.843ex;" alt="{\displaystyle E(f)/\Delta f}" /></span> of the energy spectral density has units of J Hz<sup>−1</sup>, as required. In many situations, it is common to forget the step of dividing 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 Z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1cc6b75e09a8aa3f04d8584b11db534f88fb56bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" /></span> so that the energy spectral density instead has units of V<sup>2</sup> Hz<sup>−1</sup>. </p><p>This definition generalizes in a straightforward manner to a discrete signal with a <a href="/wiki/Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a> number of values <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" /></span> such as a signal sampled at discrete times <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_{n}=t_{0}+(n\,\Delta t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06f68e578977964c65c052bda78bcdf7e0e5af40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.257ex; height:2.843ex;" alt="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}" /></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>S</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <munder> <mrow class="MJX-TeXAtom-OP MJX-fixedlimits"> <munder> <msup> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>⏟</mo> </munder> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </munder> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{2}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf2277e38dfe79f12e226de9b06a8ce3e8442b25" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:40.354ex; height:12.843ex;" alt="{\displaystyle {\bar {S}}_{xx}(f)=\lim _{N\to \infty }(\Delta t)^{2}\underbrace {\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}} _{\left|{\hat {x}}_{d}(f)\right|^{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 {\hat {x}}_{d}(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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}_{d}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d7bb68246cf565870b81b6236217966b9ca1c81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.51ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}_{d}(f)}" /></span> is the <a href="/wiki/Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">discrete-time Fourier transform</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ee2d0d918f9085dbdaaf812726427a039846070" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.195ex; height:2.009ex;" alt="{\displaystyle x_{n}.}" /></span>  The sampling 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 \Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c28867ecd34e2caed12cf38feadf6a81a7ee542" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.775ex; height:2.176ex;" alt="{\displaystyle \Delta t}" /></span> is needed to keep the correct physical units and to ensure that we recover the continuous case in the limit <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 t\to 0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">→</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta t\to 0.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3fd6d16a945fdcba737dbdf70c793c79bfaf4716" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.199ex; height:2.176ex;" alt="{\displaystyle \Delta t\to 0.}" /></span>  But in the mathematical sciences the interval is often set to 1, which simplifies the results at the expense of generality. (also see <a href="/wiki/Normalized_frequency_(unit)" class="mw-redirect" title="Normalized frequency (unit)">normalized frequency</a>) </p> <div class="mw-heading mw-heading3"><h3 id="Power_spectral_density">Power spectral density</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=5" title="Edit section: Power spectral density"><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">Not to be confused with <a href="/wiki/Spectral_power_distribution" title="Spectral power distribution">spectral power distribution</a>.</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:PowerSpectrumExt.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/PowerSpectrumExt.svg/300px-PowerSpectrumExt.svg.png" decoding="async" width="300" height="245" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/16/PowerSpectrumExt.svg/450px-PowerSpectrumExt.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/16/PowerSpectrumExt.svg/600px-PowerSpectrumExt.svg.png 2x" data-file-width="880" data-file-height="720" /></a><figcaption>The power spectrum of the measured <a href="/wiki/Cosmic_microwave_background_radiation" class="mw-redirect" title="Cosmic microwave background radiation">cosmic microwave background radiation</a> temperature anisotropy in terms of the angular scale. The solid line is a theoretical model, for comparison.</figcaption></figure> <p>The above definition of energy spectral density is suitable for transients (pulse-like signals) whose energy is concentrated around one time window; then the Fourier transforms of the signals generally exist. For continuous signals over all time, one must rather define the <i>power spectral density</i> (PSD) which exists for <a href="/wiki/Stationary_process" title="Stationary process">stationary processes</a>; this describes how the <a href="/wiki/Power_(physics)" title="Power (physics)">power</a> of a signal or time series is distributed over frequency, as in the simple example given previously. Here, power can be the actual physical power, or more often, for convenience with abstract signals, is simply identified with the squared value of the signal. For example, statisticians study the <a href="/wiki/Variance" title="Variance">variance</a> of a function over 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 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> (or over another independent variable), and using an analogy with electrical signals (among other physical processes), it is customary to refer to it as the <i>power spectrum</i> even when there is no physical power involved. If one were to create a physical <a href="/wiki/Voltage" title="Voltage">voltage</a> source which followed <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> and applied it to the terminals of a one <a href="/wiki/Ohm" title="Ohm">ohm</a> <a href="/wiki/Resistor" title="Resistor">resistor</a>, then indeed the instantaneous power dissipated in that resistor would be given 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 x^{2}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x^{2}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f58eb8c006ec38666f7f0bb9106d809a3903345" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.033ex; height:3.176ex;" alt="{\displaystyle x^{2}(t)}" /></span> <a href="/wiki/Watt" title="Watt">watts</a>. </p><p>The average 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}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> of a signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" /></span> over all time is therefore given by the following time average, where the period <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is centered about some arbitrary 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=t_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t=t_{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be6d7492e2d48bf34fdd5dffa189b188c140820c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.832ex; height:2.343ex;" alt="{\displaystyle t=t_{0}}" /></span>: <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>−</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5c079f31f87a218b27f6f30b0e92facd0259e07c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:29.891ex; height:6.676ex;" alt="{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{t_{0}-T/2}^{t_{0}+T/2}\left|x(t)\right|^{2}\,dt}" /></span> </p><p>Whenever it is more convenient to deal with time limits in the signal itself rather than time limits in the bounds of the integral, the average power can also be written 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 P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>t</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b3845035b95df2ba2d4d249378966bc63f5f1f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.173ex; height:6.009ex;" alt="{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left|x_{T}(t)\right|^{2}\,dt,}" /></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 x_{T}(t)=x(t)w_{T}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{T}(t)=x(t)w_{T}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ecf040c677a347ef0ecbf38bdbe3956e1ca34014" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.147ex; height:2.843ex;" alt="{\displaystyle x_{T}(t)=x(t)w_{T}(t)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{T}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{T}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3f71e75940d49cc0bdd1bb79455c8af96870c166" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.702ex; height:2.843ex;" alt="{\displaystyle w_{T}(t)}" /></span> is unity within the arbitrary period and zero elsewhere. </p><p>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 P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" /></span> is non-zero, the integral must grow to infinity at least as fast 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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> does. That is the reason why we cannot use the energy of the signal, which is that diverging integral. </p><p>In analyzing the frequency content 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 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>, one might like to compute the ordinary Fourier transform <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 {\hat {x}}(f)}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594eba73af7f4adf231cffd67129dbed8957e7e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(f)}" /></span>; however, for many signals of interest the ordinary Fourier transform does not formally exist.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>nb 1<span class="cite-bracket">]</span></a></sup> However, under suitable conditions, certain generalizations of the Fourier transform (e.g. the <a href="/wiki/Fourier_transform#Fourier–Stieltjes_transform_on_measurable_spaces" title="Fourier transform">Fourier-Stieltjes transform</a>) still adhere to <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>. As such, <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 P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>f</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04a0b03b197c1eab9538bc2d531406baeefabc1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:29.05ex; height:6.009ex;" alt="{\displaystyle P=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|{\hat {x}}_{T}(f)|^{2}\,df,}" /></span> where the integrand defines the <b>power spectral density</b>:<sup id="cite_ref-FOOTNOTEOppenheimVerghese2016422–423_10-0" class="reference"><a href="#cite_note-FOOTNOTEOppenheimVerghese2016422–423-10"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEMillerChilders2012429–431_11-0" class="reference"><a href="#cite_note-FOOTNOTEMillerChilders2012429–431-11"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace"></mspace> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/98910d03e3d1cbbab5ac14c87e1ff49634c9aed9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:25.569ex; height:5.343ex;" alt="{\displaystyle S_{xx}(f)=\lim _{T\to \infty }{\frac {1}{T}}|{\hat {x}}_{T}(f)|^{2}\,}" /></span></td> <td></td> <td class="nowrap"><span id="math_Eq.2" class="reference nourlexpansion" style="font-weight:bold;">Eq.2</span></td></tr></tbody></table> </div> <p>The <a href="/wiki/Convolution_theorem" title="Convolution theorem">convolution theorem</a> then allows regarding <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 |{\hat {x}}_{T}(f)|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |{\hat {x}}_{T}(f)|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a59cd3d6ccab529c49019269f66388b69b3fbc0c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.155ex; height:3.343ex;" alt="{\displaystyle |{\hat {x}}_{T}(f)|^{2}}" /></span> as the <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> of the time <a href="/wiki/Convolution" title="Convolution">convolution</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{T}^{*}(-t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{T}^{*}(-t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7fb1d39ac427e1dc9610a2e92d8b22e555687b20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.176ex; height:3.009ex;" alt="{\displaystyle x_{T}^{*}(-t)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{T}(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{T}(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509c25f2150c4c401a7d9874bf9b59df21ad5ab7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.368ex; height:2.843ex;" alt="{\displaystyle x_{T}(t)}" /></span>, where * represents the complex conjugate. </p><p>In order to deduce Eq.2, we will find an expression 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 [{\hat {x}}_{T}(f)]^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\hat {x}}_{T}(f)]^{*}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41fa28f6e7948273506683203a9253ae63e5ef39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.155ex; height:2.843ex;" alt="{\displaystyle [{\hat {x}}_{T}(f)]^{*}}" /></span> that will be useful for the purpose. In fact, we will demonstrate 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 [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79a982b983e0f902913897adf500331009768225" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.453ex; height:3.176ex;" alt="{\displaystyle [{\hat {x}}_{T}(f)]^{*}={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}}" /></span>. Let's start by noting that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16d724056e92d0f6d135c4d5ceadc38737ef34e6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.159ex; margin-bottom: -0.179ex; width:36.538ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\end{aligned}}}" /></span> and 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 z=-t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z=-t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/58b99b744d1707a023f6bba8fd709b440a32d2cf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.834ex; height:2.176ex;" alt="{\displaystyle z=-t}" /></span>, so 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 z\rightarrow -\infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> <mo stretchy="false">→</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z\rightarrow -\infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71c52bb6e2294389a4f8d341711fc48b20df4e4e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:8.834ex; height:2.176ex;" alt="{\displaystyle z\rightarrow -\infty }" /></span> 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 t\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b543f76f961ec3f52d78fa3d72c3d87a521dd3a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.777ex; height:2.009ex;" alt="{\displaystyle t\rightarrow \infty }" /></span> and vice versa. So <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt&=\int _{\infty }^{-\infty }x_{T}^{*}(z)e^{i2\pi fz}\left(-dz\right)\\&=\int _{-\infty }^{\infty }x_{T}^{*}(z)e^{i2\pi fz}dz\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>z</mi> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mi>d</mi> <mi>z</mi> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>z</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>z</mi> </mrow> </msup> <mi>d</mi> <mi>z</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt&=\int _{\infty }^{-\infty }x_{T}^{*}(z)e^{i2\pi fz}\left(-dz\right)\\&=\int _{-\infty }^{\infty }x_{T}^{*}(z)e^{i2\pi fz}dz\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50653428f2a0ed8fb42e21cef63d4463ac6a67b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:47.589ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt&=\int _{\infty }^{-\infty }x_{T}^{*}(z)e^{i2\pi fz}\left(-dz\right)\\&=\int _{-\infty }^{\infty }x_{T}^{*}(z)e^{i2\pi fz}dz\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\end{aligned}}}" /></span> Where, in the last line, we have made use of the fact 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 z}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>z</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle z}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bf368e72c009decd9b6686ee84a375632e11de98" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> are dummy variables. So, 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 {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo stretchy="false">[</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f847b32744eede2a5711bdb49e4f5e3efce4ea4" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -15.005ex; width:38.24ex; height:31.176ex;" alt="{\displaystyle {\begin{aligned}{\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}&=\int _{-\infty }^{\infty }x_{T}^{*}(-t)e^{-i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)e^{i2\pi ft}dt\\&=\int _{-\infty }^{\infty }x_{T}^{*}(t)[e^{-i2\pi ft}]^{*}dt\\&=\left[\int _{-\infty }^{\infty }x_{T}(t)e^{-i2\pi ft}dt\right]^{*}\\&=\left[{\mathcal {F}}\left\{x_{T}(t)\right\}\right]^{*}\\&=\left[{\hat {x}}_{T}(f)\right]^{*}\end{aligned}}}" /></span> q.e.d. </p><p>Now, let's demonstrate eq.2 by using the demonstrated identity. In addition, we will make the subtitution <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 u(t)=x_{T}^{*}(-t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u(t)=x_{T}^{*}(-t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce18a6c1feadfb3a0bfba270a7abfcb8fbd053ee" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.253ex; height:3.009ex;" alt="{\displaystyle u(t)=x_{T}^{*}(-t)}" /></span>. In this way, 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 {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">[</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">F</mi> </mrow> </mrow> <mrow> <mo>{</mo> <mrow> <mi>u</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-BIN"> <mrow class="MJX-TeXAtom-ORD"> <mo mathvariant="bold">∗</mo> </mrow> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mi>u</mi> <mo stretchy="false">(</mo> <mi>τ</mi> <mo>−</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mtext> </mtext> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05f235cae01011dbc96a86e1c13fae97366e2aec" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -11.983ex; margin-bottom: -0.188ex; width:51.672ex; height:25.509ex;" alt="{\displaystyle {\begin{aligned}\left|{\hat {x}}_{T}(f)\right|^{2}&=[{\hat {x}}_{T}(f)]^{*}\cdot {\hat {x}}_{T}(f)\\&={\mathcal {F}}\left\{x_{T}^{*}(-t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\right\}\cdot {\mathcal {F}}\left\{x_{T}(t)\right\}\\&={\mathcal {F}}\left\{u(t)\mathbin {\mathbf {*} } x_{T}(t)\right\}\\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }u(\tau -t)x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau \\&=\int _{-\infty }^{\infty }\left[\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau ,\end{aligned}}}" /></span> where the convolution theorem has been used when passing from the 3rd to the 4th line. </p><p>Now, if we divide the time convolution above by the period <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> and take the limit 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 T\rightarrow \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\rightarrow \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e25dc989701049391fbd92c5a4daa9bb15db65a8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle T\rightarrow \infty }" /></span>, it becomes the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a> function of the non-windowed signal <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d54c275db3a1e620737b58e143b0818107fa5f5c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.979ex; height:2.843ex;" alt="{\displaystyle x(t)}" /></span>, which is denoted 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 R_{xx}(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{xx}(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/595461d5cce972e6ef30d5e8903fcef2313223ac" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.888ex; height:2.843ex;" alt="{\displaystyle R_{xx}(\tau )}" /></span>, provided that <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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 <a href="/wiki/Ergodic" class="mw-redirect" title="Ergodic">ergodic</a>, which is true in most, but not all, practical cases.<sup id="cite_ref-12" class="reference"><a href="#cite_note-12"><span class="cite-bracket">[</span>nb 2<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 \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mtext> </mtext> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/62dc3089e7eba0d870dee75bc70ed54243cb0dba" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:89.184ex; height:6.176ex;" alt="{\displaystyle \lim _{T\to \infty }{\frac {1}{T}}\left|{\hat {x}}_{T}(f)\right|^{2}=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }\ d\tau =\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }d\tau }" /></span> </p><p>Assuming the ergodicity 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 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>, the power spectral density can be found once more as the Fourier transform of the autocorrelation function (<a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a>).<sup id="cite_ref-FOOTNOTEMillerChilders2012433_13-0" class="reference"><a href="#cite_note-FOOTNOTEMillerChilders2012433-13"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p> <div class="equation-box" style="margin: 0 0 0 1.6em;padding: 6px; border-width:2px; border-style: solid; border-color: #0073CF; color: inherit;text-align: center; display: table"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1266403038" /><table role="presentation" class="numblk" style="margin-left: 0em;"><tbody><tr><td class="nowrap"><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_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>R</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da9c6a2f23b198e55d8971514264d1ec3e24f42a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.993ex; height:6.009ex;" alt="{\displaystyle S_{xx}(f)=\int _{-\infty }^{\infty }R_{xx}(\tau )e^{-i2\pi f\tau }\,d\tau ={\hat {R}}_{xx}(f)}" /></span></td> <td></td> <td class="nowrap"><span id="math_Eq.3" class="reference nourlexpansion" style="font-weight:bold;">Eq.3</span></td></tr></tbody></table> </div> <p>Many authors use this equality to actually define the power spectral density.<sup id="cite_ref-14" class="reference"><a href="#cite_note-14"><span class="cite-bracket">[</span>12<span class="cite-bracket">]</span></a></sup> </p><p>The power of the signal in a given frequency band <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 [f_{1},f_{2}]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">[</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">]</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle [f_{1},f_{2}]}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1ce110719ca89eeeac9a2786c1fce4e86800dd41" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.715ex; height:2.843ex;" alt="{\displaystyle [f_{1},f_{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 0<f_{1}<f_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo><</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo><</mo> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0<f_{1}<f_{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d35da78371f3b4fc8ad5906e8ead3a04a87e5cc3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.746ex; height:2.509ex;" alt="{\displaystyle 0<f_{1}<f_{2}}" /></span>, can be calculated by integrating over frequency. Since <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_{xx}(-f)=S_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(-f)=S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028da7805e56429af9ae8ace88b80fab9692f2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.158ex; height:2.843ex;" alt="{\displaystyle S_{xx}(-f)=S_{xx}(f)}" /></span>, an equal amount of power can be attributed to positive and negative frequency bands, which accounts for the factor of 2 in the following form (such trivial factors depend on the conventions used): <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 P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">bandlimited</mtext> </mrow> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msubsup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e74983c41139cf820a7e42d031f592c847858b3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:28.599ex; height:6.676ex;" alt="{\displaystyle P_{\textsf {bandlimited}}=2\int _{f_{1}}^{f_{2}}S_{xx}(f)\,df}" /></span> More generally, similar techniques may be used to estimate a time-varying spectral density. In this case the time 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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is finite rather than approaching infinity. This results in decreased spectral coverage and resolution since frequencies of less than <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/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee06bfe8f48b840ea1c11f78977a90f661f2375e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" /></span> are not sampled, and results at frequencies which are not an integer multiple 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 1/T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1/T}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee06bfe8f48b840ea1c11f78977a90f661f2375e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.961ex; height:2.843ex;" alt="{\displaystyle 1/T}" /></span> are not independent. Just using a single such time series, the estimated power spectrum will be very "noisy"; however this can be alleviated if it is possible to evaluate the expected value (in the above equation) using a large (or infinite) number of short-term spectra corresponding to <a href="/wiki/Statistical_ensemble" class="mw-redirect" title="Statistical ensemble">statistical ensembles</a> of realizations 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 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> evaluated over the specified time window. </p><p>Just as with the energy spectral density, the definition of the power spectral density can be generalized to <a href="/wiki/Discrete_time" class="mw-redirect" title="Discrete time">discrete time</a> variables <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5ea190699149306d242b70439e663559e3ffbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.548ex; height:2.009ex;" alt="{\displaystyle x_{n}}" /></span>. As before, we can consider a window 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 -N\leq n\leq N}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>N</mi> <mo>≤</mo> <mi>n</mi> <mo>≤</mo> <mi>N</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -N\leq n\leq N}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa7a34ea9aa050c8697291eec5cdc576d23206f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.527ex; height:2.343ex;" alt="{\displaystyle -N\leq n\leq N}" /></span> with the signal sampled at discrete times <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_{n}=t_{0}+(n\,\Delta t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo stretchy="false">(</mo> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/06f68e578977964c65c052bda78bcdf7e0e5af40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.257ex; height:2.843ex;" alt="{\displaystyle t_{n}=t_{0}+(n\,\Delta t)}" /></span> for a total measurement period <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=(2N+1)\,\Delta t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mi>N</mi> <mo>+</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T=(2N+1)\,\Delta t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980dd75b7530aaf351daa6cd351838d5cade30fb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.936ex; height:2.843ex;" alt="{\displaystyle T=(2N+1)\,\Delta t}" /></span>. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">Δ</mi> <mi>t</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mi>T</mi> </mfrac> </mrow> <msup> <mrow> <mo>|</mo> <mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi>N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>N</mi> </mrow> </munderover> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>n</mi> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>t</mi> </mrow> </msup> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852a7ee38f1589f8cfaee67e7b0278d4dd9877a0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:40.783ex; height:8.009ex;" alt="{\displaystyle S_{xx}(f)=\lim _{N\to \infty }{\frac {(\Delta t)^{2}}{T}}\left|\sum _{n=-N}^{N}x_{n}e^{-i2\pi fn\,\Delta t}\right|^{2}}" /></span> Note that a single estimate of the PSD can be obtained through a finite number of samplings. As before, the actual PSD is achieved 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 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> (and thus <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>) approaches infinity and the expected value is formally applied. In a real-world application, one would typically average a finite-measurement PSD over many trials to obtain a more accurate estimate of the theoretical PSD of the physical process underlying the individual measurements. This computed PSD is sometimes called a <a href="/wiki/Periodogram" title="Periodogram">periodogram</a>. This periodogram converges to the true PSD as the number of estimates as well as the averaging time 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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> approach infinity.<sup id="cite_ref-FOOTNOTEBrownHwang1997_15-0" class="reference"><a href="#cite_note-FOOTNOTEBrownHwang1997-15"><span class="cite-bracket">[</span>13<span class="cite-bracket">]</span></a></sup> </p><p>If two signals both possess power spectral densities, then the <a href="#Cross-spectral_density">cross-spectral density</a> can similarly be calculated; as the PSD is related to the autocorrelation, so is the cross-spectral density related to the <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a>. </p> <div class="mw-heading mw-heading4"><h4 id="Properties_of_the_power_spectral_density">Properties of the power spectral density</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=6" title="Edit section: Properties of the power spectral density"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Some properties of the PSD include:<sup id="cite_ref-FOOTNOTEMillerChilders2012431_16-0" class="reference"><a href="#cite_note-FOOTNOTEMillerChilders2012431-16"><span class="cite-bracket">[</span>14<span class="cite-bracket">]</span></a></sup> </p> <div><ul><li>The power spectrum is always real and non-negative, and the spectrum of a real valued process is also an <a href="/wiki/Even_function" class="mw-redirect" title="Even function">even function</a> of frequency: <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_{xx}(-f)=S_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mo>−</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(-f)=S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/028da7805e56429af9ae8ace88b80fab9692f2f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.158ex; height:2.843ex;" alt="{\displaystyle S_{xx}(-f)=S_{xx}(f)}" /></span>.</li><li>For a continuous <a href="/wiki/Stochastic_process" title="Stochastic process">stochastic process</a> x(t), the autocorrelation function <i>R</i><sub><i>xx</i></sub>(<i>t</i>) can be reconstructed from its power spectrum S<sub>xx</sub>(f) by using the <a href="/wiki/Inverse_Fourier_transform" class="mw-redirect" title="Inverse Fourier transform">inverse Fourier transform</a></li><li>Using <a href="/wiki/Parseval%27s_theorem" title="Parseval's theorem">Parseval's theorem</a>, one can compute the <a href="/wiki/Variance" title="Variance">variance</a> (average power) of a process by integrating the power spectrum over all 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 P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo>=</mo> <mi>Var</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mspace width="negativethinmathspace"></mspace> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4988857ad1b752fb0b96ce06872fe07d6adb704d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:28.852ex; height:6.009ex;" alt="{\displaystyle P=\operatorname {Var} (x)=\int _{-\infty }^{\infty }\!S_{xx}(f)\,df}" /></span></li><li>For a real process <i>x</i>(<i>t</i>) with power spectral density <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_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab836aa57df700d834cc0f736e9f81152fa470e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" /></span>, one can compute the <i>integrated spectrum</i> or <i>power spectral distribution</i> <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 F(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28b338d0157928ab4d2fbcaedb56bff674b9c8b4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.829ex; height:2.843ex;" alt="{\displaystyle F(f)}" /></span>, which specifies the average <i>bandlimited</i> power contained in frequencies from DC to f using:<sup id="cite_ref-FOOTNOTEDavenportRoot1987_17-0" class="reference"><a href="#cite_note-FOOTNOTEDavenportRoot1987-17"><span class="cite-bracket">[</span>15<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 F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>F</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>f</mi> </mrow> </msubsup> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mspace width="thinmathspace"></mspace> <mi>d</mi> <msup> <mi>f</mi> <mo>′</mo> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b8e4615e7990e9c09a3ab11a595a52c16061092" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:25.071ex; height:6.343ex;" alt="{\displaystyle F(f)=2\int _{0}^{f}S_{xx}(f')\,df'.}" /></span> Note that the previous expression for total power (signal variance) is a special case where <span class="texhtml"><i>f</i> → ∞</span>.</li></ul></div> <div class="mw-heading mw-heading3"><h3 id="Cross_power_spectral_density">Cross power spectral density <span class="anchor" id="Cross"></span><span class="anchor" id="Cross_spectral_density"></span><span class="anchor" id="Cross-spectral_density"></span></h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=7" title="Edit section: Cross power spectral density"><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/Coherence_(signal_processing)" title="Coherence (signal processing)">Coherence (signal processing)</a></div> <p>Given two signals <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> 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 y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" /></span>, each of which possess power spectral densities <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_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab836aa57df700d834cc0f736e9f81152fa470e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{yy}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{yy}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c29b5f848b786dea29cf5f205fbbd8bc3498754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.379ex; height:3.009ex;" alt="{\displaystyle S_{yy}(f)}" /></span>, it is possible to define a <b>cross power spectral density</b> (<b>CPSD</b>) or <b>cross spectral density</b> (<b>CSD</b>). To begin, let us consider the average power of such a combined signal. <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>P</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow> <mo>[</mo> <mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> <mi>d</mi> <mi>t</mi> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>d</mi> <mi>t</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0714e881bda14334fc847bdb3a20dc25c7d77b3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:65.027ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}P&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }\left[x_{T}(t)+y_{T}(t)\right]^{*}\left[x_{T}(t)+y_{T}(t)\right]dt\\&=\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }|x_{T}(t)|^{2}+x_{T}^{*}(t)y_{T}(t)+y_{T}^{*}(t)x_{T}(t)+|y_{T}(t)|^{2}dt\\\end{aligned}}}" /></span> </p><p>Using the same notation and methods as used for the power spectral density derivation, we exploit Parseval's theorem and obtain <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <mrow> <mo>[</mo> <mrow> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>]</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2e03b6cd665eb568fa3d4ecbaba1eb4a88e20b61" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:65.738ex; height:5.509ex;" alt="{\displaystyle {\begin{aligned}S_{xy}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {x}}_{T}^{*}(f){\hat {y}}_{T}(f)\right]&S_{yx}(f)&=\lim _{T\to \infty }{\frac {1}{T}}\left[{\hat {y}}_{T}^{*}(f){\hat {x}}_{T}(f)\right]\end{aligned}}}" /></span> where, again, the contributions of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{xx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cab836aa57df700d834cc0f736e9f81152fa470e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.626ex; height:2.843ex;" alt="{\displaystyle S_{xx}(f)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S_{yy}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{yy}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c29b5f848b786dea29cf5f205fbbd8bc3498754" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.379ex; height:3.009ex;" alt="{\displaystyle S_{yy}(f)}" /></span> are already understood. 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 S_{xy}^{*}(f)=S_{yx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc82db12ccd633a5695fa8533e068f6d1f5867b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.103ex; height:3.009ex;" alt="{\displaystyle S_{xy}^{*}(f)=S_{yx}(f)}" /></span>, so the full contribution to the cross power is, generally, from twice the real part of either individual <b>CPSD</b>. Just as before, from here we recast these products as the Fourier transform of a time convolution, which when divided by the period and taken to the limit <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\to \infty }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\to \infty }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d124f9781b2ebb80044d6fb64b3a5f7e1cafdd4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.574ex; height:2.176ex;" alt="{\displaystyle T\to \infty }" /></span> becomes the Fourier transform of a <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a> function.<sup id="cite_ref-18" class="reference"><a href="#cite_note-18"><span class="cite-bracket">[</span>16<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 {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <mrow> <mo>[</mo> <mrow> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mi>T</mi> </mfrac> </mrow> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msubsup> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mi>d</mi> <mi>t</mi> </mrow> <mo>]</mo> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> <mo>=</mo> <msubsup> <mo>∫</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </msubsup> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>τ</mi> </mrow> </msup> <mi>d</mi> <mi>τ</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d463bb998215805b38e3c5fca96a26874d586635" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:80.733ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}S_{xy}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }x_{T}^{*}(t-\tau )y_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{xy}(\tau )e^{-i2\pi f\tau }d\tau \\S_{yx}(f)&=\int _{-\infty }^{\infty }\left[\lim _{T\to \infty }{\frac {1}{T}}\int _{-\infty }^{\infty }y_{T}^{*}(t-\tau )x_{T}(t)dt\right]e^{-i2\pi f\tau }d\tau =\int _{-\infty }^{\infty }R_{yx}(\tau )e^{-i2\pi f\tau }d\tau ,\end{aligned}}}" /></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 R_{xy}(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{xy}(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b9f7ddb3b7b684e1d9d52b97ff377c88b88b0623" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.765ex; height:3.009ex;" alt="{\displaystyle R_{xy}(\tau )}" /></span> is the <a href="/wiki/Cross-correlation" title="Cross-correlation">cross-correlation</a> of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{yx}(\tau )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>τ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{yx}(\tau )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8db09cdbf6012ec80ae4693a60e9b6d846b2f616" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.765ex; height:3.009ex;" alt="{\displaystyle R_{yx}(\tau )}" /></span> is the cross-correlation 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 y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" /></span> with <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>. In light of this, the PSD is seen to be a special case of the CSD 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(t)=y(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> <mo>=</mo> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e64fbe15b5c71503c97394db87bfb1922219c9c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.881ex; height:2.843ex;" alt="{\displaystyle x(t)=y(t)}" /></span>. If <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> 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 y(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y(t)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/397de1edef5bf2ee15c020f325d7d781a3aa7f50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.804ex; height:2.843ex;" alt="{\displaystyle y(t)}" /></span> are real signals (e.g. voltage or current), their Fourier transforms <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 {\hat {x}}(f)}"> <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 stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {x}}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/594eba73af7f4adf231cffd67129dbed8957e7e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle {\hat {x}}(f)}" /></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {y}}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>y</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\hat {y}}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/322b92da277b2088d4ebb87ceea703a7a3a0a1aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.39ex; height:2.843ex;" alt="{\displaystyle {\hat {y}}(f)}" /></span> are usually restricted to positive frequencies by convention. Therefore, in typical signal processing, the full <b>CPSD</b> is just one of the <b>CPSD</b>s scaled by a factor of two. <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 \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>CPSD</mi> <mrow class="MJX-TeXAtom-ORD"> <mtext>Full</mtext> </mrow> </msub> <mo>=</mo> <mn>2</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>2</mn> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> <mi>x</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/21eb7ff8ddff5a41948858811510897ccd146965" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:30.99ex; height:3.009ex;" alt="{\displaystyle \operatorname {CPSD} _{\text{Full}}=2S_{xy}(f)=2S_{yx}(f)}" /></span> </p><p>For discrete signals <span class="texhtml"><i>x<sub>n</sub></i></span> and <span class="texhtml"><i>y<sub>n</sub></i></span>, the relationship between the cross-spectral density and the cross-covariance is <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_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>S</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mo>−</mo> <mi mathvariant="normal">∞</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>y</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mn>2</mn> <mi>π</mi> <mi>f</mi> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mrow> </msup> <mspace width="thinmathspace"></mspace> <mi mathvariant="normal">Δ</mi> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/afac87db2f8010187959d644c0fa0e593acb1b5f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:34.01ex; height:6.843ex;" alt="{\displaystyle S_{xy}(f)=\sum _{n=-\infty }^{\infty }R_{xy}(\tau _{n})e^{-i2\pi f\tau _{n}}\,\Delta \tau }" /></span> </p> <div class="mw-heading mw-heading2"><h2 id="Estimation">Estimation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=8" title="Edit section: Estimation"><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/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></div> <p>The goal of spectral density estimation is to <a href="/wiki/Estimation_theory" title="Estimation theory">estimate</a> the spectral density of a <a href="/wiki/Random_signal" class="mw-redirect" title="Random signal">random signal</a> from a sequence of time samples. Depending on what is known about the signal, estimation techniques can involve <a href="/wiki/Parametric_statistics" title="Parametric statistics">parametric</a> or <a href="/wiki/Non-parametric_statistics" class="mw-redirect" title="Non-parametric statistics">non-parametric</a> approaches, and may be based on time-domain or frequency-domain analysis. For example, a common parametric technique involves fitting the observations to an <a href="/wiki/Autoregressive_model" title="Autoregressive model">autoregressive model</a>. A common non-parametric technique is the <a href="/wiki/Periodogram" title="Periodogram">periodogram</a>. </p><p>The spectral density is usually estimated using <a href="/wiki/Fourier_transform" title="Fourier transform">Fourier transform</a> methods (such as the <a href="/wiki/Welch_method" class="mw-redirect" title="Welch method">Welch method</a>), but other techniques such as the <a href="/wiki/Maximum_entropy_spectral_estimation" title="Maximum entropy spectral estimation">maximum entropy</a> method can also be used. </p> <div class="mw-heading mw-heading2"><h2 id="Related_concepts">Related concepts</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=9" title="Edit section: Related concepts"><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">Not to be confused with <a href="/wiki/Spectral_density_(physical_science)" class="mw-redirect" title="Spectral density (physical science)">spectral density (physical science)</a>.</div> <ul><li>The <i><a href="/wiki/Spectral_centroid" title="Spectral centroid">spectral centroid</a></i> of a signal is the midpoint of its spectral density function, i.e. the frequency that divides the distribution into two equal parts.</li> <li><span class="anchor" id="Spectral_edge_frequency"></span> The <b>spectral edge frequency</b> (<b>SEF</b>), usually expressed as "SEF <i>x</i>", represents the <a href="/wiki/Frequency" title="Frequency">frequency</a> below which <i>x</i> percent of the total power of a given signal are located; typically, <i>x</i> is in the range 75 to 95. It is more particularly a popular measure used in <a href="/wiki/EEG" class="mw-redirect" title="EEG">EEG</a> monitoring, in which case SEF has variously been used to estimate the depth of <a href="/wiki/Anesthesia" title="Anesthesia">anesthesia</a> and stages of <a href="/wiki/Sleep" title="Sleep">sleep</a>.<sup id="cite_ref-FOOTNOTEIranmaneshRodriguez-Villegas2017_19-0" class="reference"><a href="#cite_note-FOOTNOTEIranmaneshRodriguez-Villegas2017-19"><span class="cite-bracket">[</span>17<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-FOOTNOTEImtiazRodriguez-Villegas2014_20-0" class="reference"><a href="#cite_note-FOOTNOTEImtiazRodriguez-Villegas2014-20"><span class="cite-bracket">[</span>18<span class="cite-bracket">]</span></a></sup></li> <li><span class="anchor" id="Envelope"></span> A <b>spectral envelope</b> is the <a href="/wiki/Envelope_curve" class="mw-redirect" title="Envelope curve">envelope curve</a> of the spectrum density. It describes one point in time (one window, to be precise). For example, in <a href="/wiki/Remote_sensing" title="Remote sensing">remote sensing</a> using a <a href="/wiki/Spectrometer" title="Spectrometer">spectrometer</a>, the spectral envelope of a feature is the boundary of its <a href="/wiki/Electromagnetic_spectrum" title="Electromagnetic spectrum">spectral</a> properties, as defined by the range of brightness levels in each of the <a href="/wiki/Spectral_bands" class="mw-redirect" title="Spectral bands">spectral bands</a> of interest.</li> <li>The spectral density is a function of frequency, not a function of time. However, the spectral density of a small window of a longer signal may be calculated, and plotted versus time associated with the window. Such a graph is called a <i><a href="/wiki/Spectrogram" title="Spectrogram">spectrogram</a></i>. This is the basis of a number of spectral analysis techniques such as the <a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> and <a href="/wiki/Wavelets" class="mw-redirect" title="Wavelets">wavelets</a>.</li> <li><span class="anchor" id="Phase_spectrum"></span> A "spectrum" generally means the power spectral density, as discussed above, which depicts the distribution of signal content over frequency. For <a href="/wiki/Transfer_function" title="Transfer function">transfer functions</a> (e.g., <a href="/wiki/Bode_plot" title="Bode plot">Bode plot</a>, <a href="/wiki/Chirp#Relation_to_an_impulse_signal" title="Chirp">chirp</a>) the complete frequency response may be graphed in two parts: power versus frequency and <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> versus frequency—the <b>phase spectral density</b>, <b>phase spectrum</b>, or <b>spectral phase</b>. Less commonly, the two parts may be the <a href="/wiki/Real_and_imaginary_parts" class="mw-redirect" title="Real and imaginary parts">real and imaginary parts</a> of the transfer function. This is not to be confused with the <i><a href="/wiki/Frequency_response" title="Frequency response">frequency response</a></i> of a transfer function, which also includes a phase (or equivalently, a real and imaginary part) as a function of frequency. The time-domain <a href="/wiki/Impulse_response" title="Impulse response">impulse response</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 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> cannot generally be uniquely recovered from the power spectral density alone without the phase part. Although these are also Fourier transform pairs, there is no symmetry (as there is for the <a href="/wiki/Autocorrelation" title="Autocorrelation">autocorrelation</a>) forcing the Fourier transform to be real-valued. See <a href="/wiki/Ultrashort_pulse#Spectral_phase" title="Ultrashort pulse">Ultrashort pulse#Spectral phase</a>, <a href="/wiki/Phase_noise" title="Phase noise">phase noise</a>, <a href="/wiki/Group_delay" class="mw-redirect" title="Group delay">group delay</a>.</li> <li><span class="anchor" id="Amplitude"></span> Sometimes one encounters an <b>amplitude spectral density</b> (<b>ASD</b>), which is the square root of the PSD; the ASD of a voltage signal has units of V Hz<sup>−1/2</sup>.<sup id="cite_ref-21" class="reference"><a href="#cite_note-21"><span class="cite-bracket">[</span>19<span class="cite-bracket">]</span></a></sup> This is useful when the <i>shape</i> of the spectrum is rather constant, since variations in the ASD will then be proportional to variations in the signal's voltage level itself. But it is mathematically preferred to use the PSD, since only in that case is the area under the curve meaningful in terms of actual power over all frequency or over a specified bandwidth.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=10" title="Edit section: Applications"><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">Further information: <a href="/wiki/Spectrum" title="Spectrum">Spectrum</a></div> <p>Any signal that can be represented as a variable that varies in time has a corresponding frequency spectrum. This includes familiar entities such as <a href="/wiki/Visible_light" class="mw-redirect" title="Visible light">visible light</a> (perceived as <a href="/wiki/Color" title="Color">color</a>), musical notes (perceived as <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</a>), <a href="/wiki/Radio_frequency" title="Radio frequency">radio/TV</a> (specified by their frequency, or sometimes <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>) and even the regular rotation of the earth. When these signals are viewed in the form of a frequency spectrum, certain aspects of the received signals or the underlying processes producing them are revealed. In some cases the frequency spectrum may include a distinct peak corresponding to a <a href="/wiki/Sine_wave" title="Sine wave">sine wave</a> component. And additionally there may be peaks corresponding to <a href="/wiki/Harmonics" class="mw-redirect" title="Harmonics">harmonics</a> of a fundamental peak, indicating a periodic signal which is <i>not</i> simply sinusoidal. Or a continuous spectrum may show narrow frequency intervals which are strongly enhanced corresponding to resonances, or frequency intervals containing almost zero power as would be produced by a <a href="/wiki/Notch_filter" class="mw-redirect" title="Notch filter">notch filter</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Electrical_engineering">Electrical engineering</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=11" title="Edit section: Electrical engineering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Spectrogram-fm-radio.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Spectrogram-fm-radio.png/250px-Spectrogram-fm-radio.png" decoding="async" width="220" height="138" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Spectrogram-fm-radio.png/330px-Spectrogram-fm-radio.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c4/Spectrogram-fm-radio.png/500px-Spectrogram-fm-radio.png 2x" data-file-width="794" data-file-height="497" /></a><figcaption>Spectrogram of an <a href="/wiki/FM_radio" class="mw-redirect" title="FM radio">FM radio</a> signal with frequency on the horizontal axis and time increasing upwards on the vertical axis.</figcaption></figure> <p>The concept and use of the power spectrum of a signal is fundamental in <a href="/wiki/Electrical_engineering" title="Electrical engineering">electrical engineering</a>, especially in <a href="/wiki/Communication_systems" class="mw-redirect" title="Communication systems">electronic communication systems</a>, including <a href="/wiki/Radio_communication" class="mw-redirect" title="Radio communication">radio communications</a>, <a href="/wiki/Radar" title="Radar">radars</a>, and related systems, plus passive <a href="/wiki/Remote_sensing" title="Remote sensing">remote sensing</a> technology. Electronic instruments called <a href="/wiki/Spectrum_analyzer" title="Spectrum analyzer">spectrum analyzers</a> are used to observe and measure the <i><b>power spectra</b></i> of signals. </p><p>The spectrum analyzer measures the magnitude of the <a href="/wiki/Short-time_Fourier_transform" title="Short-time Fourier transform">short-time Fourier transform</a> (STFT) of an input signal. If the signal being analyzed can be considered a stationary process, the STFT is a good smoothed estimate of its power spectral density. </p> <div class="mw-heading mw-heading3"><h3 id="Cosmology">Cosmology</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=12" title="Edit section: Cosmology"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Primordial_fluctuations" title="Primordial fluctuations">Primordial fluctuations</a>, density variations in the early universe, are quantified by a power spectrum which gives the power of the variations as a function of spatial scale. </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=Spectral_density&action=edit&section=13" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Bispectrum" title="Bispectrum">Bispectrum</a></li> <li><a href="/wiki/Brightness_temperature" title="Brightness temperature">Brightness temperature</a></li> <li><a href="/wiki/Colors_of_noise" title="Colors of noise">Colors of noise</a></li> <li><a href="/wiki/Least-squares_spectral_analysis" title="Least-squares spectral analysis">Least-squares spectral analysis</a></li> <li><a href="/wiki/Noise_spectral_density" title="Noise spectral density">Noise spectral density</a></li> <li><a href="/wiki/Spectral_density_estimation" title="Spectral density estimation">Spectral density estimation</a></li> <li><a href="/wiki/Spectral_efficiency" title="Spectral efficiency">Spectral efficiency</a></li> <li><a href="/wiki/Spectral_leakage" title="Spectral leakage">Spectral leakage</a></li> <li><a href="/wiki/Spectral_power_distribution" title="Spectral power distribution">Spectral power distribution</a></li> <li><a href="/wiki/Whittle_likelihood" title="Whittle likelihood">Whittle likelihood</a></li> <li><a href="/wiki/Window_function" title="Window function">Window function</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=14" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">Some authors, e.g., (<a href="#CITEREFRiskenFrank1996">Risken & Frank 1996</a>, p. 30) still use the non-normalized Fourier transform in a formal way to formulate a definition of the power spectral density <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 \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">⟨</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo stretchy="false">^</mo> </mover> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msup> <mi>ω</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <msup> <mi>ω</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f05d4307c767baeb358f564234dd4550028555ed" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.766ex; height:3.009ex;" alt="{\displaystyle \langle {\hat {x}}(\omega ){\hat {x}}^{\ast }(\omega ')\rangle =2\pi f(\omega )\delta (\omega -\omega '),}" /></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (\omega -\omega ')}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>δ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo>−</mo> <msup> <mi>ω</mi> <mo>′</mo> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta (\omega -\omega ')}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d9412fe50622d943fadaf2ab8bd1a7bd9d9c3b51" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.275ex; height:3.009ex;" alt="{\displaystyle \delta (\omega -\omega ')}" /></span> is the <a href="/wiki/Dirac_delta_function" title="Dirac delta function">Dirac delta function</a>. Such formal statements may sometimes be useful to guide the intuition, but should always be used with utmost care.</span> </li> <li id="cite_note-12"><span class="mw-cite-backlink"><b><a href="#cite_ref-12">^</a></b></span> <span class="reference-text"> The <a href="/wiki/Wiener%E2%80%93Khinchin_theorem" title="Wiener–Khinchin theorem">Wiener–Khinchin theorem</a> makes sense of this formula for any <a href="/wiki/Wide-sense_stationary_process" class="mw-redirect" title="Wide-sense stationary process">wide-sense stationary process</a> under weaker hypotheses: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c57d170e7cb2b17a74d1fec2427a2a26836b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.877ex; height:2.509ex;" alt="{\displaystyle R_{xx}}" /></span> does not need to be absolutely integrable, it only needs to exist. But the integral can no longer be interpreted as usual. The formula also makes sense if interpreted as involving <a href="/wiki/Distribution_(mathematics)" title="Distribution (mathematics)">distributions</a> (in the sense of <a href="/wiki/Laurent_Schwartz" title="Laurent Schwartz">Laurent Schwartz</a>, not in the sense of a statistical <a href="/wiki/Cumulative_distribution_function" title="Cumulative distribution function">Cumulative distribution function</a>) instead of functions. If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{xx}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mi>x</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{xx}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44c57d170e7cb2b17a74d1fec2427a2a26836b39" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.877ex; height:2.509ex;" alt="{\displaystyle R_{xx}}" /></span> is continuous, <a href="/wiki/Bochner%27s_theorem" title="Bochner's theorem">Bochner's theorem</a> can be used to prove that its Fourier transform exists as a positive <a href="/wiki/Measure_(mathematics)" title="Measure (mathematics)">measure</a>, whose distribution function is F (but not necessarily as a function and not necessarily possessing a probability density).</span> </li> </ol></div></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626" /><div class="reflist"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-P_Stoica-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-P_Stoica_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-P_Stoica_1-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-P_Stoica_1-2"><sup><i><b>c</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 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Archived from <a rel="nofollow" class="external text" href="https://www.rp-photonics.com/power_spectral_density.html">the original</a> on 2024-04-15<span class="reference-accessdate">. Retrieved <span class="nowrap">2024-06-26</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=rp-photonics.com&rft.atitle=Power+Spectral+Density&rft.aulast=Paschotta&rft.aufirst=R%C3%BCdiger&rft_id=https%3A%2F%2Fwww.rp-photonics.com%2Fpower_spectral_density.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></span> </li> <li id="cite_note-FOOTNOTEOppenheimVerghese201660-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOppenheimVerghese201660_6-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, p. 60.</span> </li> <li id="cite_note-FOOTNOTEStein2000108,_115-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEStein2000108,_115_7-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFStein2000">Stein 2000</a>, pp. 108, 115.</span> </li> <li id="cite_note-FOOTNOTEOppenheimVerghese201614-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOppenheimVerghese201614_8-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, p. 14.</span> </li> <li id="cite_note-FOOTNOTEOppenheimVerghese2016422–423-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEOppenheimVerghese2016422–423_10-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFOppenheimVerghese2016">Oppenheim & Verghese 2016</a>, pp. 422–423.</span> </li> <li id="cite_note-FOOTNOTEMillerChilders2012429–431-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMillerChilders2012429–431_11-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMillerChilders2012">Miller & Childers 2012</a>, pp. 429–431.</span> </li> <li id="cite_note-FOOTNOTEMillerChilders2012433-13"><span class="mw-cite-backlink"><b><a href="#cite_ref-FOOTNOTEMillerChilders2012433_13-0">^</a></b></span> <span class="reference-text"><a href="#CITEREFMillerChilders2012">Miller & Childers 2012</a>, p. 433.</span> </li> <li id="cite_note-14"><span class="mw-cite-backlink"><b><a href="#cite_ref-14">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDennis_Ward_Ricker2003" class="citation book cs1">Dennis Ward Ricker (2003). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NF2Tmty9nugC&q=%22power+spectral+density%22+%22energy+spectral+density%22&pg=PA23"><i>Echo Signal Processing</i></a>. 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Harvey (2000). <a rel="nofollow" class="external text" href="http://www.lumerink.com/courses/ece697/docs/Papers/The%20Fundamentals%20of%20FFT-Based%20Signal%20Analysis%20and%20Measurements.pdf">"The Fundamentals of FFT-Based Signal Analysis and Measurement"</a> <span class="cs1-format">(PDF)</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=The+Fundamentals+of+FFT-Based+Signal+Analysis+and+Measurement&rft.date=2000&rft.au=Michael+Cerna&rft.au=Audrey+F.+Harvey&rft_id=http%3A%2F%2Fwww.lumerink.com%2Fcourses%2Fece697%2Fdocs%2FPapers%2FThe%2520Fundamentals%2520of%2520FFT-Based%2520Signal%2520Analysis%2520and%2520Measurements.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Spectral_density&action=edit&section=15" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBirolini2007" class="citation book cs1">Birolini, Alessandro (2007). <i>Reliability Engineering</i>. Berlin ; New York: Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-49388-4" title="Special:BookSources/978-3-540-49388-4"><bdi>978-3-540-49388-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Reliability+Engineering&rft.place=Berlin+%3B+New+York&rft.pub=Springer+Science+%26+Business+Media&rft.date=2007&rft.isbn=978-3-540-49388-4&rft.aulast=Birolini&rft.aufirst=Alessandro&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrownHwang1997" class="citation book cs1">Brown, Robert Grover; Hwang, Patrick Y. C. (1997). <i>Introduction to Random Signals and Applied Kalman Filtering with Matlab Exercises and Solutions</i>. New York: Wiley-Liss. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-12839-7" title="Special:BookSources/978-0-471-12839-7"><bdi>978-0-471-12839-7</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Random+Signals+and+Applied+Kalman+Filtering+with+Matlab+Exercises+and+Solutions&rft.place=New+York&rft.pub=Wiley-Liss&rft.date=1997&rft.isbn=978-0-471-12839-7&rft.aulast=Brown&rft.aufirst=Robert+Grover&rft.au=Hwang%2C+Patrick+Y.+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFDavenportRoot1987" class="citation book cs1">Davenport, Wilbur B. (Jr); Root, William L. (1987). <i>An Introduction to the Theory of Random Signals and Noise</i>. New York: Wiley-IEEE Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-87942-235-6" title="Special:BookSources/978-0-87942-235-6"><bdi>978-0-87942-235-6</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+Introduction+to+the+Theory+of+Random+Signals+and+Noise&rft.place=New+York&rft.pub=Wiley-IEEE+Press&rft.date=1987&rft.isbn=978-0-87942-235-6&rft.aulast=Davenport&rft.aufirst=Wilbur+B.+%28Jr%29&rft.au=Root%2C+William+L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFImtiazRodriguez-Villegas2014" class="citation journal cs1">Imtiaz, Syed Anas; Rodriguez-Villegas, Esther (2014). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4204008">"A Low Computational Cost Algorithm for REM Sleep Detection Using Single Channel EEG"</a>. <i>Annals of Biomedical Engineering</i>. <b>42</b> (11): <span class="nowrap">2344–</span>59. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2Fs10439-014-1085-6">10.1007/s10439-014-1085-6</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4204008">4204008</a></span>. <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/25113231">25113231</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annals+of+Biomedical+Engineering&rft.atitle=A+Low+Computational+Cost+Algorithm+for+REM+Sleep+Detection+Using+Single+Channel+EEG&rft.volume=42&rft.issue=11&rft.pages=%3Cspan+class%3D%22nowrap%22%3E2344-%3C%2Fspan%3E59&rft.date=2014&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4204008%23id-name%3DPMC&rft_id=info%3Apmid%2F25113231&rft_id=info%3Adoi%2F10.1007%2Fs10439-014-1085-6&rft.aulast=Imtiaz&rft.aufirst=Syed+Anas&rft.au=Rodriguez-Villegas%2C+Esther&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4204008&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFIranmaneshRodriguez-Villegas2017" class="citation journal cs1">Iranmanesh, Saam; <a href="/wiki/Esther_Rodriguez-Villegas" title="Esther Rodriguez-Villegas">Rodriguez-Villegas, Esther</a> (2017). "An Ultralow-Power Sleep Spindle Detection System on Chip". <i>IEEE Transactions on Biomedical Circuits and Systems</i>. <b>11</b> (4): <span class="nowrap">858–</span>866. <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%2FTBCAS.2017.2690908">10.1109/TBCAS.2017.2690908</a>. <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://hdl.handle.net/10044%2F1%2F46059">10044/1/46059</a></span>. <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/28541914">28541914</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:206608057">206608057</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Biomedical+Circuits+and+Systems&rft.atitle=An+Ultralow-Power+Sleep+Spindle+Detection+System+on+Chip&rft.volume=11&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E858-%3C%2Fspan%3E866&rft.date=2017&rft_id=info%3Ahdl%2F10044%2F1%2F46059&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A206608057%23id-name%3DS2CID&rft_id=info%3Apmid%2F28541914&rft_id=info%3Adoi%2F10.1109%2FTBCAS.2017.2690908&rft.aulast=Iranmanesh&rft.aufirst=Saam&rft.au=Rodriguez-Villegas%2C+Esther&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMaral2004" class="citation book cs1">Maral, Gerard (2004). <i>VSAT Networks</i>. West Sussex, England ; Hoboken, NJ: Wiley. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-470-86684-9" title="Special:BookSources/978-0-470-86684-9"><bdi>978-0-470-86684-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=VSAT+Networks&rft.place=West+Sussex%2C+England+%3B+Hoboken%2C+NJ&rft.pub=Wiley&rft.date=2004&rft.isbn=978-0-470-86684-9&rft.aulast=Maral&rft.aufirst=Gerard&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFMillerChilders2012" class="citation book cs1">Miller, Scott; Childers, Donald (2012). <i>Probability and Random Processes</i>. Boston, MA: Academic Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-12-386981-4" title="Special:BookSources/978-0-12-386981-4"><bdi>978-0-12-386981-4</bdi></a>. <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/696092052">696092052</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Probability+and+Random+Processes&rft.place=Boston%2C+MA&rft.pub=Academic+Press&rft.date=2012&rft_id=info%3Aoclcnum%2F696092052&rft.isbn=978-0-12-386981-4&rft.aulast=Miller&rft.aufirst=Scott&rft.au=Childers%2C+Donald&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFNortonKarczub2003" class="citation book cs1">Norton, M. P.; Karczub, D. G. (2003). <i>Fundamentals of Noise and Vibration Analysis for Engineers</i>. Cambridge: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-49913-2" title="Special:BookSources/978-0-521-49913-2"><bdi>978-0-521-49913-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fundamentals+of+Noise+and+Vibration+Analysis+for+Engineers&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2003&rft.isbn=978-0-521-49913-2&rft.aulast=Norton&rft.aufirst=M.+P.&rft.au=Karczub%2C+D.+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFOppenheimVerghese2016" class="citation book cs1">Oppenheim, Alan V.; Verghese, George C. (2016). <i>Signals, Systems & Inference</i>. Boston: Pearson. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-13-394328-3" title="Special:BookSources/978-0-13-394328-3"><bdi>978-0-13-394328-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Signals%2C+Systems+%26+Inference&rft.place=Boston&rft.pub=Pearson&rft.date=2016&rft.isbn=978-0-13-394328-3&rft.aulast=Oppenheim&rft.aufirst=Alan+V.&rft.au=Verghese%2C+George+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRiskenFrank1996" class="citation book cs1">Risken, Hannes; Frank, Till (1996). <i>The Fokker-Planck Equation</i>. New York: Springer Science & Business Media. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-61530-9" title="Special:BookSources/978-3-540-61530-9"><bdi>978-3-540-61530-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Fokker-Planck+Equation&rft.place=New+York&rft.pub=Springer+Science+%26+Business+Media&rft.date=1996&rft.isbn=978-3-540-61530-9&rft.aulast=Risken&rft.aufirst=Hannes&rft.au=Frank%2C+Till&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStein2000" class="citation book cs1">Stein, Jonathan Y. (2000). <i>Digital Signal Processing</i>. New York Weinheim: Wiley-Interscience. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-29546-4" title="Special:BookSources/978-0-471-29546-4"><bdi>978-0-471-29546-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Digital+Signal+Processing&rft.place=New+York+Weinheim&rft.pub=Wiley-Interscience&rft.date=2000&rft.isbn=978-0-471-29546-4&rft.aulast=Stein&rft.aufirst=Jonathan+Y.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASpectral+density" 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=Spectral_density&action=edit&section=16" 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="http://vibrationdata.wordpress.com/category/power-spectral-density/">Power Spectral Density Matlab scripts</a></li></ul> <div class="navbox-styles"><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 dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist 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(dB)</div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/DBm" title="DBm">dBm</a> (or dBmW)</li> <li><a href="/wiki/Decibel_watt" title="Decibel watt">dBW</a></li> <li><a href="/wiki/Decibel#Voltage" title="Decibel">dBV</a></li> <li><a class="mw-selflink-fragment" href="#Explanation">dBm/Hz</a> <ul><li><a class="mw-selflink-fragment" href="#Power_spectral_density">PSD</a></li></ul></li> <li><a href="/wiki/A-weighting#Environmental_noise_measurement" title="A-weighting">dBA</a></li> <li><a href="/wiki/DBZ_(meteorology)" title="DBZ (meteorology)">dBZ</a> (radar)</li> <li><a href="/wiki/Decibel#Radar" title="Decibel">dBsm</a></li> <li><a href="/wiki/DBc" title="DBc">dBc</a></li> <li><a href="/wiki/Decibel#Antenna_measurements" title="Decibel">dBi</a></li> <li><a href="/wiki/DBFS" title="DBFS">dBFS</a></li> <li><a href="/wiki/DBrn" title="DBrn">dBrn</a></li> <li><a href="/wiki/Decibel#Other_measurements" title="Decibel">dB-Hz</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <dl><dt>See also</dt> <dd><a href="/wiki/Logarithmic_unit" class="mw-redirect" title="Logarithmic unit">logarithmic unit</a></dd> <dd><a href="/wiki/Link_budget" title="Link budget">link budget</a></dd> <dd><a href="/wiki/Signal_noise" class="mw-redirect" title="Signal noise">signal noise</a></dd> <dd><a href="/wiki/Telecommunications" title="Telecommunications">telecommunications</a></dd></dl> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://auth.wikimedia.org/loginwiki/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" 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Spectral density - Wikipedia