Square wave - Wikipedia

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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">5</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</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">7</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 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id="vector-page-titlebar-toc-unpinned-container" class="vector-unpinned-container"> </div> </div> </div> </nav> <h1 id="firstHeading" class="firstHeading mw-first-heading"><span class="mw-page-title-main">Square wave</span></h1> <div id="p-lang-btn" class="vector-dropdown mw-portlet mw-portlet-lang" > <input type="checkbox" id="p-lang-btn-checkbox" role="button" aria-haspopup="true" data-event-name="ui.dropdown-p-lang-btn" class="vector-dropdown-checkbox mw-interlanguage-selector" aria-label="Go to an article in another language. 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href="https://ca.wikipedia.org/wiki/Ona_quadrada" title="Ona quadrada – Catalan" lang="ca" hreflang="ca" data-title="Ona quadrada" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Obd%C3%A9ln%C3%ADkov%C3%BD_pr%C5%AFb%C4%9Bh" title="Obdélníkový průběh – Czech" lang="cs" hreflang="cs" data-title="Obdélníkový průběh" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target"><span>Čeština</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Rechteckschwingung" title="Rechteckschwingung – German" lang="de" hreflang="de" data-title="Rechteckschwingung" 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/Nelinurklaine" title="Nelinurklaine – Estonian" lang="et" hreflang="et" data-title="Nelinurklaine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A4%CE%B5%CF%84%CF%81%CE%B1%CE%B3%CF%89%CE%BD%CE%B9%CE%BA%CF%8C_%CE%BA%CF%8D%CE%BC%CE%B1" title="Τετραγωνικό κύμα – Greek" lang="el" hreflang="el" data-title="Τετραγωνικό κύμα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Onda_cuadrada" title="Onda cuadrada – Spanish" lang="es" hreflang="es" data-title="Onda cuadrada" 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/Ortangula_ondo" title="Ortangula ondo – Esperanto" lang="eo" hreflang="eo" data-title="Ortangula ondo" 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/%D9%85%D9%88%D8%AC_%D9%85%D8%B1%D8%A8%D8%B9%DB%8C" title="موج مربعی – Persian" lang="fa" hreflang="fa" data-title="موج مربعی" 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/Signal_carr%C3%A9" title="Signal carré – French" lang="fr" hreflang="fr" data-title="Signal carré" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%B0%A9%ED%98%95%ED%8C%8C" title="방형파 – Korean" lang="ko" hreflang="ko" data-title="방형파" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%B5%E0%A4%B0%E0%A5%8D%E0%A4%97_%E0%A4%A4%E0%A4%B0%E0%A4%82%E0%A4%97%E0%A4%B0%E0%A5%82%E0%A4%AA" title="वर्ग तरंगरूप – Hindi" lang="hi" hreflang="hi" data-title="वर्ग तरंगरूप" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target"><span>हिन्दी</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Onda_quadra" title="Onda quadra – Italian" lang="it" hreflang="it" data-title="Onda quadra" 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title="Меандр (радиотехника) – Russian" lang="ru" hreflang="ru" data-title="Меандр (радиотехника)" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Kvadratni_val" title="Kvadratni val – Slovenian" lang="sl" hreflang="sl" data-title="Kvadratni val" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Kanttiaalto" title="Kanttiaalto – Finnish" lang="fi" hreflang="fi" data-title="Kanttiaalto" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target"><span>Suomi</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Fyrkantsv%C3%A5g" 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</div> </nav> <nav class="vector-appearance-landmark" aria-label="Appearance"> <div id="vector-appearance-pinned-container" class="vector-pinned-container"> <div id="vector-appearance" class="vector-appearance vector-pinnable-element"> <div class="vector-pinnable-header vector-appearance-pinnable-header vector-pinnable-header-pinned" data-feature-name="appearance-pinned" data-pinnable-element-id="vector-appearance" data-pinned-container-id="vector-appearance-pinned-container" data-unpinned-container-id="vector-appearance-unpinned-container" > <div class="vector-pinnable-header-label">Appearance</div> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-pin-button" data-event-name="pinnable-header.vector-appearance.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-appearance.unpin">hide</button> </div> </div> </div> </nav> </div> </div> <div id="bodyContent" class="vector-body" aria-labelledby="firstHeading" data-mw-ve-target-container> <div class="vector-body-before-content"> <div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of non-sinusoidal waveform</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about the waveform. For the type of ocean surface wave, see <a href="/wiki/Cross_sea" title="Cross sea">Cross sea</a>.</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-More_citations_needed plainlinks metadata ambox ambox-content ambox-Refimprove" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><span typeof="mw:File"><a href="/wiki/File:Question_book-new.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/50px-Question_book-new.svg.png" decoding="async" width="50" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/75px-Question_book-new.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/99/Question_book-new.svg/100px-Question_book-new.svg.png 2x" data-file-width="512" data-file-height="399" /></a></span></div></td><td class="mbox-text"><div class="mbox-text-span">This article <b>needs additional citations for <a href="/wiki/Wikipedia:Verifiability" title="Wikipedia:Verifiability">verification</a></b>.<span class="hide-when-compact"> Please help <a href="/wiki/Special:EditPage/Square_wave" title="Special:EditPage/Square wave">improve this article</a> by <a href="/wiki/Help:Referencing_for_beginners" title="Help:Referencing for beginners">adding citations to reliable sources</a>. Unsourced material may be challenged and removed.<br /><small><span class="plainlinks"><i>Find sources:</i> <a rel="nofollow" class="external text" href="https://www.google.com/search?as_eq=wikipedia&q=%22Square+wave%22">"Square wave"</a> – <a rel="nofollow" class="external text" href="https://www.google.com/search?tbm=nws&q=%22Square+wave%22+-wikipedia&tbs=ar:1">news</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?&q=%22Square+wave%22&tbs=bkt:s&tbm=bks">newspapers</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.google.com/search?tbs=bks:1&q=%22Square+wave%22+-wikipedia">books</a> <b>·</b> <a rel="nofollow" class="external text" href="https://scholar.google.com/scholar?q=%22Square+wave%22">scholar</a> <b>·</b> <a rel="nofollow" class="external text" href="https://www.jstor.org/action/doBasicSearch?Query=%22Square+wave%22&acc=on&wc=on">JSTOR</a></span></small></span> <span class="date-container"><i>(<span class="date">December 2019</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> <style data-mw-deduplicate="TemplateStyles:r1257001546">.mw-parser-output .infobox-subbox{padding:0;border:none;margin:-3px;width:auto;min-width:100%;font-size:100%;clear:none;float:none;background-color:transparent}.mw-parser-output .infobox-3cols-child{margin:auto}.mw-parser-output .infobox .navbar{font-size:100%}@media screen{html.skin-theme-clientpref-night .mw-parser-output .infobox-full-data:not(.notheme)>div:not(.notheme)[style]{background:#1f1f23!important;color:#f8f9fa}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .infobox-full-data:not(.notheme) div:not(.notheme){background:#1f1f23!important;color:#f8f9fa}}@media(min-width:640px){body.skin--responsive .mw-parser-output .infobox-table{display:table!important}body.skin--responsive .mw-parser-output .infobox-table>caption{display:table-caption!important}body.skin--responsive .mw-parser-output .infobox-table>tbody{display:table-row-group}body.skin--responsive .mw-parser-output .infobox-table tr{display:table-row!important}body.skin--responsive .mw-parser-output .infobox-table th,body.skin--responsive .mw-parser-output .infobox-table td{padding-left:inherit;padding-right:inherit}}</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="background:#e0e0e0;padding:0.15em 0.5em 0.25em;font-weight:bold;">Square wave</th></tr><tr><td colspan="2" class="infobox-image" style="padding-bottom:0.4em;"><span typeof="mw:File"><a href="/wiki/File:Waveforms.svg" class="mw-file-description"><img alt="Sine, square, triangle, and sawtooth waveforms" src="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Waveforms.svg/400px-Waveforms.svg.png" decoding="async" width="400" height="342" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/77/Waveforms.svg/600px-Waveforms.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/77/Waveforms.svg/800px-Waveforms.svg.png 2x" data-file-width="620" data-file-height="530" /></a></span><div class="infobox-caption"><a href="/wiki/Sine_wave" title="Sine wave">Sine</a>, square, <a href="/wiki/Triangle_wave" title="Triangle wave">triangle</a>, and <a href="/wiki/Sawtooth_wave" title="Sawtooth wave">sawtooth</a> waveforms</div></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">General information</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">General definition</th><td class="infobox-data"><span class="mwe-math-element"><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)=4\left\lfloor t\right\rfloor -2\left\lfloor 2t\right\rfloor +1,2t\notin \mathbb {Z} }"> <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> <mn>4</mn> <mrow> <mo>⌊</mo> <mi>t</mi> <mo>⌋</mo> </mrow> <mo>−</mo> <mn>2</mn> <mrow> <mo>⌊</mo> <mrow> <mn>2</mn> <mi>t</mi> </mrow> <mo>⌋</mo> </mrow> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mi>t</mi> <mo>∉</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=4\left\lfloor t\right\rfloor -2\left\lfloor 2t\right\rfloor +1,2t\notin \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b99fadb88e3444b91e7fc6636a3eb4fe3e581b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.418ex; height:2.843ex;" alt="{\displaystyle x(t)=4\left\lfloor t\right\rfloor -2\left\lfloor 2t\right\rfloor +1,2t\notin \mathbb {Z} }"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;">Fields of application</th><td class="infobox-data">Electronics, synthesizers</td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Domain, codomain and image</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Domain_of_a_function" title="Domain of a function">Domain</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} \setminus \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">R</mi> </mrow> <mo class="MJX-variant">∖</mo> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo>}</mo> </mrow> <mo>,</mo> <mi>n</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {R} \setminus \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cc90b29d255a9b773f5ca3c9264e48d262305abc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:15.612ex; height:3.343ex;" alt="{\displaystyle \mathbb {R} \setminus \left\{{\tfrac {n}{2}}\right\},n\in \mathbb {Z} }"></span></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Codomain" title="Codomain">Codomain</a></th><td class="infobox-data"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{-1,1\right\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mn>1</mn> </mrow> <mo>}</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{-1,1\right\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5434d24c557b1f04fcfe4ee0ae625d4383fab54f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \left\{-1,1\right\}}"></span></td></tr><tr><th colspan="2" class="infobox-header" style="background:#e0e0e0;padding-bottom:0.2em;">Basic features</th></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Even_and_odd_functions" title="Even and odd functions">Parity</a></th><td class="infobox-data">Odd</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Periodic_function" title="Periodic function">Period</a></th><td class="infobox-data">1</td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Antiderivative" title="Antiderivative">Antiderivative</a></th><td class="infobox-data"><a href="/wiki/Triangle_wave" title="Triangle wave">Triangle wave</a></td></tr><tr><th scope="row" class="infobox-label" style="padding-top:0.25em;line-height:1.2em; padding-right:0.65em;"><a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a></th><td class="infobox-data"><span class="mwe-math-element"><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)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {1}{2k-1}}\sin \left(2\pi \left(2k-1\right)t\right)}"> <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> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {1}{2k-1}}\sin \left(2\pi \left(2k-1\right)t\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/174cbfb1d54af453605efa9d7a63e8c0f4a49087" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:37.933ex; height:6.843ex;" alt="{\displaystyle x(t)={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {1}{2k-1}}\sin \left(2\pi \left(2k-1\right)t\right)}"></span></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1096940132">.mw-parser-output .listen .side-box-text{line-height:1.1em}.mw-parser-output .listen-plain{border:none;background:transparent}.mw-parser-output .listen-embedded{width:100%;margin:0;border-width:1px 0 0 0;background:transparent}.mw-parser-output .listen-header{padding:2px}.mw-parser-output .listen-embedded .listen-header{padding:2px 0}.mw-parser-output .listen-file-header{padding:4px 0}.mw-parser-output .listen .description{padding-top:2px}.mw-parser-output .listen .mw-tmh-player{max-width:100%}@media(max-width:719px){.mw-parser-output .listen{clear:both}}@media(min-width:720px){.mw-parser-output .listen:not(.listen-noimage){width:320px}.mw-parser-output .listen-left{overflow:visible;float:left}.mw-parser-output .listen-center{float:none;margin-left:auto;margin-right:auto}}</style><div class="side-box side-box-right listen noprint"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-center" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/50px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/75px-Gnome-mime-sound-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/100px-Gnome-mime-sound-openclipart.svg.png 2x" data-file-width="160" data-file-height="160" /></span><figcaption></figcaption></figure></div> <div class="side-box-text plainlist"><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:220_Hz_anti-aliased_square_wave.ogg" title="File:220 Hz anti-aliased square wave.ogg">Square wave sound sample</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_0" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="5" data-mwtitle="220_Hz_anti-aliased_square_wave.ogg" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/0/08/220_Hz_anti-aliased_square_wave.ogg" type="audio/ogg; codecs="vorbis"" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/0/08/220_Hz_anti-aliased_square_wave.ogg/220_Hz_anti-aliased_square_wave.ogg.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">5 seconds of square wave at 220 Hz</div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p>A <b>square wave</b> is a <a href="/wiki/Non-sinusoidal_waveform" class="mw-redirect" title="Non-sinusoidal waveform">non-sinusoidal periodic waveform</a> in which the amplitude alternates at a steady <a href="/wiki/Frequency" title="Frequency">frequency</a> between fixed minimum and maximum values, with the same duration at minimum and maximum. In an ideal square wave, the transitions between minimum and maximum are instantaneous. </p><p>The square wave is a special case of a <a href="/wiki/Pulse_wave" title="Pulse wave">pulse wave</a> which allows arbitrary durations at minimum and maximum amplitudes. The ratio of the high period to the total period of a pulse wave is called the <a href="/wiki/Duty_cycle" title="Duty cycle">duty cycle</a>. A true square wave has a 50% duty cycle (equal high and low periods). </p><p>Square waves are often encountered in <a href="/wiki/Electronics" title="Electronics">electronics</a> and <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a>, particularly <a href="/wiki/Digital_electronics" title="Digital electronics">digital electronics</a> and <a href="/wiki/Digital_signal_processing" title="Digital signal processing">digital signal processing</a>. Its <a href="/wiki/Stochastic" title="Stochastic">stochastic</a> counterpart is a <a href="/wiki/Two-state_trajectory" title="Two-state trajectory">two-state trajectory</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Origin_and_uses">Origin and uses</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=1" title="Edit section: Origin and uses"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Square waves are universally encountered in digital switching circuits and are naturally generated by binary (two-level) logic devices. They are used as timing references or "<a href="/wiki/Clock_signal" title="Clock signal">clock signals</a>", because their fast transitions are suitable for triggering <a href="/wiki/Synchronous_logic" class="mw-redirect" title="Synchronous logic">synchronous logic</a> circuits at precisely determined intervals. However, as the frequency-domain graph shows, square waves contain a wide range of harmonics; these can generate <a href="/wiki/Electromagnetic_radiation" title="Electromagnetic radiation">electromagnetic radiation</a> or pulses of current that interfere with other nearby circuits, causing <a href="/wiki/Noise" title="Noise">noise</a> or errors. To avoid this problem in very sensitive circuits such as precision <a href="/wiki/Analog-to-digital_converter" title="Analog-to-digital converter">analog-to-digital converters</a>, <a href="/wiki/Sine_wave" title="Sine wave">sine waves</a> are used instead of square waves as timing references. </p><p>In musical terms, they are often described as sounding hollow, and are therefore used as the basis for <a href="/wiki/Wind_instrument" title="Wind instrument">wind instrument</a> sounds created using <a href="/wiki/Subtractive_synthesis" title="Subtractive synthesis">subtractive synthesis</a>. They also make up the "beeping" alerts used in many household, commercial, and industrial contexts. Additionally, the distortion effect used on <a href="/wiki/Electric_guitar" title="Electric guitar">electric guitars</a> clips the outermost regions of the waveform, causing it to increasingly resemble a square wave as more distortion is applied. </p><p>Simple two-level <a href="/wiki/Rademacher_function" class="mw-redirect" title="Rademacher function">Rademacher functions</a> are square waves. </p> <div class="mw-heading mw-heading2"><h2 id="Definitions">Definitions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=2" title="Edit section: Definitions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The square wave in mathematics has many definitions, which are equivalent except at the discontinuities: </p><p>It can be defined as simply the <a href="/wiki/Sign_function" title="Sign function">sign function</a> of a sinusoid: <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}x(t)&=\operatorname {sgn} \left(\sin {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\sin 2\pi ft)\\v(t)&=\operatorname {sgn} \left(\cos {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\cos 2\pi ft),\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>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> </mtr> <mtr> <mtd> <mi>v</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cos</mi> <mo>⁡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mi>π</mi> <mi>t</mi> </mrow> <mi>T</mi> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mi>sgn</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(t)&=\operatorname {sgn} \left(\sin {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\sin 2\pi ft)\\v(t)&=\operatorname {sgn} \left(\cos {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\cos 2\pi ft),\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d34b9649f66863ccc847229db6d2edbe61359c9c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:39.326ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}x(t)&=\operatorname {sgn} \left(\sin {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\sin 2\pi ft)\\v(t)&=\operatorname {sgn} \left(\cos {\frac {2\pi t}{T}}\right)=\operatorname {sgn}(\cos 2\pi ft),\end{aligned}}}"></span> which will be 1 when the sinusoid is positive, −1 when the sinusoid is negative, and 0 at the discontinuities. Here, <i>T</i> is the <a href="/wiki/Period_(physics)" class="mw-redirect" title="Period (physics)">period</a> of the square wave and <i>f</i> is its frequency, which are related by the equation <i>f</i> = 1/<i>T</i>. </p><p>A square wave can also be defined with respect to the <a href="/wiki/Heaviside_step_function" title="Heaviside step function">Heaviside step function</a> <i>u</i>(<i>t</i>) or the <a href="/wiki/Rectangular_function" title="Rectangular function">rectangular function</a> Π(<i>t</i>): <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}x(t)&=2\left[\sum _{n=-\infty }^{\infty }\Pi \left({\frac {2(t-nT)}{T}}-{\frac {1}{2}}\right)\right]-1\\&=2\sum _{n=-\infty }^{\infty }\left[u\left({\frac {t}{T}}-n\right)-u\left({\frac {t}{T}}-n-{\frac {1}{2}}\right)\right]-1.\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>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <mrow> <mo>[</mo> <mrow> <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> <mi mathvariant="normal">Π</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>2</mn> <mo stretchy="false">(</mo> <mi>t</mi> <mo>−</mo> <mi>n</mi> <mi>T</mi> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>2</mn> <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> <mrow> <mo>[</mo> <mrow> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>u</mi> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>t</mi> <mi>T</mi> </mfrac> </mrow> <mo>−</mo> <mi>n</mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mo>−</mo> <mn>1.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(t)&=2\left[\sum _{n=-\infty }^{\infty }\Pi \left({\frac {2(t-nT)}{T}}-{\frac {1}{2}}\right)\right]-1\\&=2\sum _{n=-\infty }^{\infty }\left[u\left({\frac {t}{T}}-n\right)-u\left({\frac {t}{T}}-n-{\frac {1}{2}}\right)\right]-1.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64023feb7c2635877ec4a5e073e8ab92e0223396" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.671ex; width:53.427ex; height:14.509ex;" alt="{\displaystyle {\begin{aligned}x(t)&=2\left[\sum _{n=-\infty }^{\infty }\Pi \left({\frac {2(t-nT)}{T}}-{\frac {1}{2}}\right)\right]-1\\&=2\sum _{n=-\infty }^{\infty }\left[u\left({\frac {t}{T}}-n\right)-u\left({\frac {t}{T}}-n-{\frac {1}{2}}\right)\right]-1.\end{aligned}}}"></span> </p><p>A square wave can also be generated using the <a href="/wiki/Floor_function" class="mw-redirect" title="Floor function">floor function</a> directly: <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 x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1}"> <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> <mn>2</mn> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mo fence="false" stretchy="false">⌊</mo> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">⌋</mo> <mo>−</mo> <mo fence="false" stretchy="false">⌊</mo> <mn>2</mn> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b6a85883aec4b42c2ba01c4f6b8fa79b3739d375" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.97ex; height:2.843ex;" alt="{\displaystyle x(t)=2\left(2\lfloor ft\rfloor -\lfloor 2ft\rfloor \right)+1}"></span> and indirectly: <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 x(t)=\left(-1\right)^{\lfloor 2ft\rfloor }.}"> <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> <msup> <mrow> <mo>(</mo> <mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo fence="false" stretchy="false">⌊</mo> <mn>2</mn> <mi>f</mi> <mi>t</mi> <mo fence="false" stretchy="false">⌋</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x(t)=\left(-1\right)^{\lfloor 2ft\rfloor }.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7308582f14ed050fb0c59014accc7db598267f4d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.516ex; height:3.509ex;" alt="{\displaystyle x(t)=\left(-1\right)^{\lfloor 2ft\rfloor }.}"></span> </p><p>Using the fourier series (below) one can show that the floor function may be written in trigonometric form<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> <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 {\frac {2}{\pi }}\arctan \left(\tan \left({\frac {\pi ft}{2}}\right)\right)+{\frac {2}{\pi }}\arctan \left(\cot \left({\frac {\pi ft}{2}}\right)\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>tan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>2</mn> <mi>π</mi> </mfrac> </mrow> <mi>arctan</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mi>cot</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>π</mi> <mi>f</mi> <mi>t</mi> </mrow> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {2}{\pi }}\arctan \left(\tan \left({\frac {\pi ft}{2}}\right)\right)+{\frac {2}{\pi }}\arctan \left(\cot \left({\frac {\pi ft}{2}}\right)\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/089c0dc6af4e3a8d33802e136e544c3be59afc48" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:49.6ex; height:6.176ex;" alt="{\displaystyle {\frac {2}{\pi }}\arctan \left(\tan \left({\frac {\pi ft}{2}}\right)\right)+{\frac {2}{\pi }}\arctan \left(\cot \left({\frac {\pi ft}{2}}\right)\right)}"></span> </p> <div class="mw-heading mw-heading2"><h2 id="Fourier_analysis">Fourier analysis <span class="anchor" id="Fourier_Analysis"></span></h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=3" title="Edit section: Fourier analysis"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:SquareWaveFourierArrows.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/6/6b/SquareWaveFourierArrows.gif" decoding="async" width="300" height="90" class="mw-file-element" data-file-width="300" data-file-height="90" /></a><figcaption>The six arrows represent the first six terms of the Fourier series of a square wave. The two circles at the bottom represent the exact square wave (blue) and its Fourier-series approximation (purple).</figcaption></figure> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Spectrum_square_oscillation.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Spectrum_square_oscillation.jpg/350px-Spectrum_square_oscillation.jpg" decoding="async" width="350" height="193" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b5/Spectrum_square_oscillation.jpg/525px-Spectrum_square_oscillation.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/b/b5/Spectrum_square_oscillation.jpg 2x" data-file-width="553" data-file-height="305" /></a><figcaption>(Odd) harmonics of a 1000 Hz square wave</figcaption></figure> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Fourier_Series-Square_wave_3_H.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fourier_Series-Square_wave_3_H.png/220px-Fourier_Series-Square_wave_3_H.png" decoding="async" width="220" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fourier_Series-Square_wave_3_H.png/330px-Fourier_Series-Square_wave_3_H.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/79/Fourier_Series-Square_wave_3_H.png/440px-Fourier_Series-Square_wave_3_H.png 2x" data-file-width="640" data-file-height="480" /></a><figcaption>Graph showing the first 3 terms of the Fourier series of a square wave</figcaption></figure> <p>Using <a href="/wiki/Fourier_series" title="Fourier series">Fourier expansion</a> with cycle frequency <span class="texhtml"><i>f</i></span> over time <span class="texhtml"><i>t</i></span>, an ideal square wave with an amplitude of 1 can be represented as an infinite sum of sinusoidal waves: <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}x(t)&={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin \left(2\pi (2k-1)ft\right)}{2k-1}}\\&={\frac {4}{\pi }}\left(\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+{\frac {1}{5}}\sin(5\omega t)+\ldots \right),&{\text{where }}\omega =2\pi f.\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>x</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>sin</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow> <mn>2</mn> <mi>π</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mi>f</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> </mrow> <mrow> <mn>2</mn> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>4</mn> <mi>π</mi> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>3</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>5</mn> </mfrac> </mrow> <mi>sin</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mn>5</mn> <mi>ω</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>+</mo> <mo>…</mo> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mtext>where </mtext> </mrow> <mi>ω</mi> <mo>=</mo> <mn>2</mn> <mi>π</mi> <mi>f</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}x(t)&={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin \left(2\pi (2k-1)ft\right)}{2k-1}}\\&={\frac {4}{\pi }}\left(\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+{\frac {1}{5}}\sin(5\omega t)+\ldots \right),&{\text{where }}\omega =2\pi f.\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b60c1df3ddf156c6e2efbda7bc5279874bdb19fc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:74.168ex; height:13.176ex;" alt="{\displaystyle {\begin{aligned}x(t)&={\frac {4}{\pi }}\sum _{k=1}^{\infty }{\frac {\sin \left(2\pi (2k-1)ft\right)}{2k-1}}\\&={\frac {4}{\pi }}\left(\sin(\omega t)+{\frac {1}{3}}\sin(3\omega t)+{\frac {1}{5}}\sin(5\omega t)+\ldots \right),&{\text{where }}\omega =2\pi f.\end{aligned}}}"></span> </p> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1235681985"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1096940132"><div class="side-box side-box-right listen noprint"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1126788409"> <div class="side-box-flex"> <div class="side-box-image"><figure class="mw-halign-center" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/50px-Gnome-mime-sound-openclipart.svg.png" decoding="async" width="50" height="50" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/75px-Gnome-mime-sound-openclipart.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/87/Gnome-mime-sound-openclipart.svg/100px-Gnome-mime-sound-openclipart.svg.png 2x" data-file-width="160" data-file-height="160" /></span><figcaption></figcaption></figure></div> <div class="side-box-text plainlist"><div class="haudio"> <div class="listen-file-header"><a href="/wiki/File:Additive_220Hz_Square_Wave.wav" title="File:Additive 220Hz Square Wave.wav">Additive square demo</a></div> <div><span typeof="mw:File"><span><audio id="mwe_player_1" controls="" preload="none" data-mw-tmh="" class="mw-file-element" width="232" style="width:232px;" data-durationhint="23" data-mwtitle="Additive_220Hz_Square_Wave.wav" data-mwprovider="wikimediacommons"><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4b/Additive_220Hz_Square_Wave.wav/Additive_220Hz_Square_Wave.wav.ogg" type="audio/ogg; codecs="vorbis"" data-transcodekey="ogg" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/transcoded/4/4b/Additive_220Hz_Square_Wave.wav/Additive_220Hz_Square_Wave.wav.mp3" type="audio/mpeg" data-transcodekey="mp3" data-width="0" data-height="0" /><source src="//upload.wikimedia.org/wikipedia/commons/4/4b/Additive_220Hz_Square_Wave.wav" type="audio/wav" data-width="0" data-height="0" /></audio></span></span></div> <div class="description">220 Hz square wave created by harmonics added every second over sine wave</div></div></div></div> <div class="side-box-abovebelow"><hr /><i class="selfreference">Problems playing this file? See <a href="/wiki/Help:Media" title="Help:Media">media help</a>.</i></div> </div> <p>The ideal square wave contains only components of odd-integer <a href="/wiki/Harmonic" title="Harmonic">harmonic</a> frequencies (of the form <span class="texhtml">2π(2<i>k</i> − 1)<i>f</i></span>). </p><p>A curiosity of the convergence of the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> representation of the square wave is the <a href="/wiki/Gibbs_phenomenon" title="Gibbs phenomenon">Gibbs phenomenon</a>. <a href="/wiki/Ringing_artifacts" title="Ringing artifacts">Ringing artifacts</a> in non-ideal square waves can be shown to be related to this phenomenon. The Gibbs phenomenon can be prevented by the use of <a href="/wiki/Sigma_approximation" title="Sigma approximation">σ-approximation</a>, which uses the <a href="/wiki/Lanczos_sigma_factor" class="mw-redirect" title="Lanczos sigma factor">Lanczos sigma factors</a> to help the sequence converge more smoothly. </p><p>An ideal mathematical square wave changes between the high and the low state instantaneously, and without under- or over-shooting. This is impossible to achieve in physical systems, as it would require infinite <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a>. </p> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Fourier_series_for_square_wave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/bc/Fourier_series_for_square_wave.gif/350px-Fourier_series_for_square_wave.gif" decoding="async" width="350" height="263" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/b/bc/Fourier_series_for_square_wave.gif 1.5x" data-file-width="480" data-file-height="360" /></a><figcaption>Animation of the additive synthesis of a square wave with an increasing number of harmonics</figcaption></figure> <p>Square waves in physical systems have only finite bandwidth and often exhibit <a href="/wiki/Ringing_(signal)" title="Ringing (signal)">ringing</a> effects similar to those of the Gibbs phenomenon or ripple effects similar to those of the σ-approximation. </p><p>For a reasonable approximation to the square-wave shape, at least the fundamental and third harmonic need to be present, with the fifth harmonic being desirable. These bandwidth requirements are important in digital electronics, where finite-bandwidth analog approximations to square-wave-like waveforms are used. (The ringing transients are an important electronic consideration here, as they may go beyond the electrical rating limits of a circuit or cause a badly positioned threshold to be crossed multiple times.) </p> <div class="mw-heading mw-heading2"><h2 id="Characteristics_of_imperfect_square_waves">Characteristics of imperfect square waves</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=4" title="Edit section: Characteristics of imperfect square waves"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>As already mentioned, an ideal square wave has instantaneous transitions between the high and low levels. In practice, this is never achieved because of physical limitations of the system that generates the waveform. The times taken for the signal to rise from the low level to the high level and back again are called the <i><a href="/wiki/Rise_time" title="Rise time">rise time</a></i> and the <i><a href="/wiki/Fall_time" title="Fall time">fall time</a></i> respectively. </p><p>If the system is <a href="/wiki/Overdamped" class="mw-redirect" title="Overdamped">overdamped</a>, then the waveform may never actually reach the theoretical high and low levels, and if the system is underdamped, it will oscillate about the high and low levels before settling down. In these cases, the rise and fall times are measured between specified intermediate levels, such as 5% and 95%, or 10% and 90%. The <a href="/wiki/Bandwidth_(signal_processing)" title="Bandwidth (signal processing)">bandwidth</a> of a system is related to the transition times of the waveform; there are formulas allowing one to be determined approximately from the other. </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=Square_wave&action=edit&section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_periodic_functions" title="List of periodic functions">List of periodic functions</a></li> <li><a href="/wiki/Rectangular_function" title="Rectangular function">Rectangular function</a></li> <li><a href="/wiki/Pulse_wave" title="Pulse wave">Pulse wave</a></li> <li><a href="/wiki/Sine_wave" title="Sine wave">Sine wave</a></li> <li><a href="/wiki/Triangle_wave" title="Triangle wave">Triangle wave</a></li> <li><a href="/wiki/Sawtooth_wave" title="Sawtooth wave">Sawtooth wave</a></li> <li><a href="/wiki/Waveform" title="Waveform">Waveform</a></li> <li><a href="/wiki/Sound" title="Sound">Sound</a></li> <li><a href="/wiki/Multivibrator" title="Multivibrator">Multivibrator</a></li> <li><a href="/wiki/Ronchi_ruling" title="Ronchi ruling">Ronchi ruling</a>, a square-wave stripe target used in imaging.</li> <li><a href="/wiki/Cross_sea" title="Cross sea">Cross sea</a></li> <li><a href="/wiki/Clarinet" title="Clarinet">Clarinet</a>, a musical instrument that produces odd overtones approximating a square wave.</li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style 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Retrieved <span class="nowrap">9 July</span> 2023</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=www.wolframalpha.com&rft.atitle=Partial+sum+formula&rft_id=https%3A%2F%2Fwww.wolframalpha.com%2Finput%3Fi%3D%255Csum_%257Bn%253D1%257D%255E%257B%255Cinfty%257D%255Cfrac%257B%255Csin%255Cleft%2528%255Cleft%25282n-1%255Cright%2529x%255Cright%2529%257D%257B2n-1%257D&rfr_id=info%3Asid%2Fen.wikipedia.org%3ASquare+wave" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Square_wave&action=edit&section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://www.geogebra.org/m/wUanseCs">Fourier decomposition of a square wave</a> Interactive demo of square wave synthesis using sine waves, from GeoGebra site.</li> <li><a rel="nofollow" class="external text" href="http://sara.ng/apps/square-wave">Square Wave Approximated by Sines</a> Interactive demo of square wave synthesis using sine waves.</li> <li><a rel="nofollow" class="external text" href="http://www.electric1.es/armonicos/armonicosOC.html">Flash applets</a> Square wave.</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 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