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<title>Rectangular function</title>

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Rectangular function

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</style><div role="note" class="hatnote navigation-not-searchable">"Box function" redirects here. For the Conway box function, see <a href="Minkowski's_question-mark_function#Conway_box_function" title="Minkowski's question-mark function">Minkowski's question-mark function § Conway box function</a>.</div>

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function,<a href="#cite_note-1">[1]</a> gate function, unit pulse, or the normalized <a href="Boxcar_function" title="Boxcar function">boxcar function</a>) is defined as<a href="#cite_note-wolfram-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} \left({\frac {t}{a}}\right)=\Pi \left({\frac {t}{a}}\right)=\left\{{\begin{array}{rl}0,&{\text{if }}|t|>{\frac {a}{2}}\\{\frac {1}{2}},&{\text{if }}|t|={\frac {a}{2}}\\1,&{\text{if }}|t|<{\frac {a}{2}}.\end{array}}\right.}</annotation>

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Alternative definitions of the function define <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}">

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<annotation encoding="application/x-tex">{\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}</annotation>

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</math><img src="./5486f8643413b09e09b48b5572516d8f88c5cee0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.477ex; height:3.509ex;" alt="{\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}" loading="lazy"> to be 0,<a href="#cite_note-3">[3]</a> 1,<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> or undefined.

Its periodic version is called a <a href="Rectangular_wave" class="mw-redirect" title="Rectangular wave">rectangular wave</a>.

<div class="mw-heading mw-heading2"><h2 id="History">History</h2></div>

The rect function has been introduced 1953 by <a href="Philip_Woodward" title="Philip Woodward">Woodward</a><a href="#cite_note-6">[6]</a> in "Probability and Information Theory, with Applications to Radar"<a href="#cite_note-7">[7]</a> as an ideal <a href="Window_function#Rectangular_window" title="Window function">cutout operator</a>, together with the <a href="Sinc_function" title="Sinc function">sinc function</a><a href="#cite_note-8">[8]</a><a href="#cite_note-9">[9]</a> as an ideal <a href="Whittaker%E2%80%93Shannon_interpolation_formula" title="Whittaker–Shannon interpolation formula">interpolation operator</a>, and their counter operations which are <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> (<a href="Dirac_comb#Dirac-comb_identity" title="Dirac comb">comb operator</a>) and <a href="Periodic_summation" title="Periodic summation">replicating</a> (<a href="Dirac_comb#Dirac-comb_identity" title="Dirac comb">rep operator</a>), respectively.

<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_boxcar_function">Relation to the boxcar function</h2></div>

The rectangular function is a special case of the more general <a href="Boxcar_function" title="Boxcar function">boxcar function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} \left({\frac {t-X}{Y}}\right)=H(t-(X-Y/2))-H(t-(X+Y/2))=H(t-X+Y/2)-H(t-X-Y/2)}</annotation>

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where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H(x)}">

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<annotation encoding="application/x-tex">{\displaystyle H(x)}</annotation>

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</math><img src="./11f241aa7195bebab9d0a3c248ea97ef0c78b1ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.203ex; height:2.843ex;" alt="{\displaystyle H(x)}" loading="lazy"> is the <a href="Heaviside_step_function" title="Heaviside step function">Heaviside step function</a>; the function is centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

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<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

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</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and has duration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

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<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

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</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy">, from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X-Y/2}">

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<annotation encoding="application/x-tex">{\displaystyle X-Y/2}</annotation>

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</math><img src="./7d9b15e36bb54cdf7c3ca029e98fecdc81f32f34.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.919ex; height:2.843ex;" alt="{\displaystyle X-Y/2}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X+Y/2.}">

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<annotation encoding="application/x-tex">{\displaystyle X+Y/2.}</annotation>

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</math><img src="./deb90b2e6da313830976a4e5c46e16ac06157345.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.565ex; height:2.843ex;" alt="{\displaystyle X+Y/2.}" loading="lazy">

<div class="mw-heading mw-heading2"><h2 id="Fourier_transform_of_the_rectangular_function">Fourier transform of the rectangular function</h2></div>

The <a href="Fourier_transform#Tables_of_important_Fourier_transforms" title="Fourier transform">unitary Fourier transforms</a> of the rectangular function are<a href="#cite_note-wolfram-2">[2]</a>

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} (\pi f)=\operatorname {sinc} _{\pi }(f),}</annotation>

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using ordinary frequency f, where <a href="Sinc_function" title="Sinc function"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} _{\pi }}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} _{\pi }}</annotation>

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</math><img src="./02e16b853b0c49d919aa5088d75df5dc9386c2f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.062ex; height:2.509ex;" alt="{\displaystyle \operatorname {sinc} _{\pi }}" loading="lazy"></a> is the normalized form<a href="#cite_note-10">[10]</a> of the <a href="Sinc_function" title="Sinc function">sinc function</a> and

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<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt {2\pi }}}\cdot {\frac {\sin \left(\omega /2\right)}{\omega /2}}={\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {sinc} \left(\omega /2\right),}</annotation>

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using angular frequency <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy">, where <a href="Sinc_function" title="Sinc function"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sinc} }">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {sinc} }</annotation>

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</math><img src="./876707323d2d558232894c2c9e776a93a1206ade.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.888ex; height:2.176ex;" alt="{\displaystyle \operatorname {sinc} }" loading="lazy"></a> is the unnormalized form of the <a href="Sinc_function" title="Sinc function">sinc function</a>.

For <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {rect} (x/a)}">

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} (x/a)}</annotation>

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</math><img src="./10ddbd1e1355241a79286cbc4276cd89b0a4d644.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.412ex; height:2.843ex;" alt="{\displaystyle \operatorname {rect} (x/a)}" loading="lazy">, its Fourier transform is<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=a{\frac {\sin(\pi af)}{\pi af}}=a\ \operatorname {sinc} _{\pi }{(af)}.}">

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<annotation encoding="application/x-tex">{\displaystyle \int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=a{\frac {\sin(\pi af)}{\pi af}}=a\ \operatorname {sinc} _{\pi }{(af)}.}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Relation_to_the_triangular_function">Relation to the triangular function</h2></div>

We can define the <a href="Triangular_function" title="Triangular function">triangular function</a> as the <a href="Convolution" title="Convolution">convolution</a> of two rectangular functions:

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<annotation encoding="application/x-tex">{\displaystyle \operatorname {tri(t/T)} =\operatorname {rect(2t/T)} *\operatorname {rect(2t/T)} .\,}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Use_in_probability">Use in probability</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">Uniform distribution (continuous)</a></div>

Viewing the rectangular function as a <a href="Probability_density_function" title="Probability density function">probability density function</a>, it is a special case of the <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">continuous uniform distribution</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a=-1/2,b=1/2.}">

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<annotation encoding="application/x-tex">{\displaystyle a=-1/2,b=1/2.}</annotation>

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</math><img src="./09ef5d9c7878e397ae80c00cdef971a11f57bf6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.888ex; height:2.843ex;" alt="{\displaystyle a=-1/2,b=1/2.}" loading="lazy"> The <a href="Characteristic_function_(probability_theory)" title="Characteristic function (probability theory)">characteristic function</a> is

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<annotation encoding="application/x-tex">{\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}},}</annotation>

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and its <a href="Moment-generating_function" title="Moment-generating function">moment-generating function</a> is

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<annotation encoding="application/x-tex">{\displaystyle M(k)={\frac {\sinh(k/2)}{k/2}},}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle \sinh(t)}</annotation>

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</math><img src="./0622f942176288814f9373e3ffabc67c61928352.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.797ex; height:2.843ex;" alt="{\displaystyle \sinh(t)}" loading="lazy"> is the <a href="Hyperbolic_sine" class="mw-redirect" title="Hyperbolic sine">hyperbolic sine</a> function.

<div class="mw-heading mw-heading2"><h2 id="Rational_approximation">Rational approximation</h2></div>

The pulse function may also be expressed as a limit of a <a href="Rational_function" title="Rational function">rational function</a>:

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<annotation encoding="application/x-tex">{\displaystyle \Pi (t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}.}</annotation>

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<div class="mw-heading mw-heading3"><h3 id="Demonstration_of_validity">Demonstration of validity</h3></div>

First, we consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |t|<{\frac {1}{2}}.}">

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<annotation encoding="application/x-tex">{\textstyle |t|<{\frac {1}{2}}.}</annotation>

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</math><img src="./7414ad08dce24f9298bf80f4c3a9d3c9854f50a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.537ex; height:3.509ex;" alt="{\textstyle |t|<{\frac {1}{2}}.}" loading="lazy"> Notice that the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2t)^{2n}}">

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</math><img src="./8f8b7b0c7ce639de28918b244e48b2a6e9e3cc61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.852ex; height:3.009ex;" alt="{\textstyle (2t)^{2n}}" loading="lazy"> is always positive for integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">

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<annotation encoding="application/x-tex">{\displaystyle n.}</annotation>

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</math><img src="./e59df02a9f67a5da3c220f1244c99a46cc4eb1c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}" loading="lazy"> However, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2t<1}">

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<annotation encoding="application/x-tex">{\displaystyle 2t<1}</annotation>

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</math><img src="./d4ebe1dbeb3069e1a10f686e539620db5d31fb06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.263ex; height:2.176ex;" alt="{\displaystyle 2t<1}" loading="lazy"> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2t)^{2n}}">

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<annotation encoding="application/x-tex">{\textstyle (2t)^{2n}}</annotation>

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</math><img src="./8f8b7b0c7ce639de28918b244e48b2a6e9e3cc61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.852ex; height:3.009ex;" alt="{\textstyle (2t)^{2n}}" loading="lazy"> approaches zero for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">

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It follows that:

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<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{0+1}}=1,|t|<{\tfrac {1}{2}}.}</annotation>

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Second, we consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |t|>{\frac {1}{2}}.}">

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<annotation encoding="application/x-tex">{\textstyle |t|>{\frac {1}{2}}.}</annotation>

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</math><img src="./6e2148653296411e29b96f0ec726558ca481a04f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.537ex; height:3.509ex;" alt="{\textstyle |t|>{\frac {1}{2}}.}" loading="lazy"> Notice that the term <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2t)^{2n}}">

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<annotation encoding="application/x-tex">{\textstyle (2t)^{2n}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle n.}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 2t>1}</annotation>

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</math><img src="./9133728cc0b737abe432954f1029debbfe0173cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.263ex; height:2.176ex;" alt="{\displaystyle 2t>1}" loading="lazy"> and hence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle (2t)^{2n}}">

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<annotation encoding="application/x-tex">{\textstyle (2t)^{2n}}</annotation>

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</math><img src="./8f8b7b0c7ce639de28918b244e48b2a6e9e3cc61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.852ex; height:3.009ex;" alt="{\textstyle (2t)^{2n}}" loading="lazy"> grows very large for large <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}">

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<annotation encoding="application/x-tex">{\displaystyle n.}</annotation>

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It follows that:

<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{+\infty +1}}=0,|t|>{\tfrac {1}{2}}.}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{+\infty +1}}=0,|t|>{\tfrac {1}{2}}.}</annotation>

</semantics>

</math>

Third, we consider the case where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle |t|={\frac {1}{2}}.}">

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\textstyle |t|={\frac {1}{2}}.}</annotation>

</semantics>

</math><img src="./922d84cfd7ca700f08739b7039c153606169ca8f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:7.537ex; height:3.509ex;" alt="{\textstyle |t|={\frac {1}{2}}.}" loading="lazy"> We may simply substitute in our equation:

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{1^{2n}+1}}={\frac {1}{1+1}}={\tfrac {1}{2}}.}</annotation>

</semantics>

</math>

We see that it satisfies the definition of the pulse function. Therefore,

</mrow>

</mrow>

</munder>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

<mrow>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

<mfrac>

</mfrac>

</mrow>

</mtd>

<mtd>

</mstyle>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mstyle>

</mrow>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {rect} (t)=\Pi (t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\1&{\mbox{if }}|t|<{\frac {1}{2}}.\\\end{cases}}}</annotation>

</semantics>

</math>

<div class="mw-heading mw-heading2"><h2 id="Dirac_delta_function">Dirac delta function</h2></div>

The rectangle function can be used to represent the <a href="Dirac_delta_function" title="Dirac delta function">Dirac delta function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (x)}</annotation>

</semantics>

</math><img src="./4457507451c205a7e6adda92d919ee4c4a369cea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.188ex; height:2.843ex;" alt="{\displaystyle \delta (x)}" loading="lazy">.<a href="#cite_note-:0-11">[11]</a> Specifically,<math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (x)=\lim _{a\to 0}{\frac {1}{a}}\operatorname {rect} \left({\frac {x}{a}}\right).}">

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (x)=\lim _{a\to 0}{\frac {1}{a}}\operatorname {rect} \left({\frac {x}{a}}\right).}</annotation>

</semantics>

</math>For a function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(x)}</annotation>

</semantics>

</math><img src="./c6ca91363022bd5e4dcb17e5ef29f78b8ef00b59.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.255ex; height:2.843ex;" alt="{\displaystyle g(x)}" loading="lazy">, its average over the width <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g_{avg}(0)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right).}</annotation>

</semantics>

</math>

To obtain <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g(0)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(0)}</annotation>

</semantics>

</math><img src="./ef9ac33acee4c0045521f091546f50540d8eb364.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.088ex; height:2.843ex;" alt="{\displaystyle g(0)}" loading="lazy">, the following limit is applied,

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</munderover>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(0)=\lim _{a\to 0}{\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right)}</annotation>

</semantics>

</math>

and this can be written in terms of the Dirac delta function as,

</mrow>

</mrow>

</munderover>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g(0)=\int \limits _{-\infty }^{\infty }dx\ g(x)\delta (x).}</annotation>

</semantics>

</math>The Fourier transform of the Dirac delta function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (t)}</annotation>

</semantics>

</math><img src="./c050147a97868286252447ef73515c8108edd398.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.698ex; height:2.843ex;" alt="{\displaystyle \delta (t)}" loading="lazy"> is

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</munder>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

</msubsup>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (f)=\int _{-\infty }^{\infty }\delta (t)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}{\frac {1}{a}}\int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}\operatorname {sinc} {(af)}.}</annotation>

</semantics>

</math>

where the <a href="Sinc_function" title="Sinc function">sinc function</a> here is the normalized sinc function. Because the first zero of the sinc function is at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=1/a}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f=1/a}</annotation>

</semantics>

</math><img src="./b1a822d469fbeda8de4088367d572c76fd00a358.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.932ex; height:2.843ex;" alt="{\displaystyle f=1/a}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a}</annotation>

</semantics>

</math><img src="./ffd2487510aa438433a2579450ab2b3d557e5edc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}" loading="lazy"> goes to infinity, the Fourier transform of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta (t)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (t)}</annotation>

</semantics>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \delta (f)=1,}</annotation>

</semantics>

</math>

means that the frequency spectrum of the Dirac delta function is infinitely broad. As a pulse is shorten in time, it is larger in spectrum.

<ul><li><a href="Fourier_transform" title="Fourier transform">Fourier transform</a></li>

<li><a href="Square_wave_(waveform)" title="Square wave (waveform)">Square wave</a></li>

<li><a href="Step_function" title="Step function">Step function</a></li>

<li><a href="Top-hat_filter" title="Top-hat filter">Top-hat filter</a></li>

<li><a href="Boxcar_function" title="Boxcar function">Boxcar function</a></li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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<li id="cite_note-wolfram-2">^ <a href="#cite_ref-wolfram_2-0">a</a> <a href="#cite_ref-wolfram_2-1">b</a> <cite id="CITEREFWeisstein" class="citation web cs1"><a href="Eric_W._Weisstein" title="Eric W. Weisstein">Weisstein, Eric W.</a> <a rel="nofollow" class="external text" href="https://mathworld.wolfram.com/RectangleFunction.html">"Rectangle Function"</a>. <a href="MathWorld" title="MathWorld">MathWorld</a>.</cite>

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<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFWang2012" class="citation book cs1">Wang, Ruye (2012). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=4KEKGjaiJn0C&pg=PA135">Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis</a>. Cambridge University Press. pp. 135–136. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521516884</bdi>.</cite>

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<li id="cite_note-6"><a href="#cite_ref-6">^</a> <cite id="CITEREFKlauder1960" class="citation journal cs1">Klauder, John R (1960). <a rel="nofollow" class="external text" href="https://ieeexplore.ieee.org/document/6773600">"The Theory and Design of Chirp Radars"</a>. Bell System Technical Journal. 39 (4): 745–808. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1002%2Fj.1538-7305.1960.tb03942.x">10.1002/j.1538-7305.1960.tb03942.x</a>.</cite>

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<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFWoodward1953" class="citation book cs1">Woodward, Philipp M (1953). Probability and Information Theory, with Applications to Radar. Pergamon Press. p. 29.</cite>

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<li id="cite_note-8"><a href="#cite_ref-8">^</a> <cite id="CITEREFHiggins1996" class="citation book cs1">Higgins, John Rowland (1996). Sampling Theory in Fourier and Signal Analysis: Foundations. Oxford University Press Inc. p. 4. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0198596995</bdi>.</cite>

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<li id="cite_note-9"><a href="#cite_ref-9">^</a> <cite id="CITEREFZayed1996" class="citation book cs1">Zayed, Ahmed I (1996). Handbook of Function and Generalized Function Transformations. CRC Press. p. 507. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780849380761</bdi>.</cite>

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<li id="cite_note-10"><a href="#cite_ref-10">^</a> Wolfram MathWorld, <a rel="nofollow" class="external free" href="https://mathworld.wolfram.com/SincFunction.html">https://mathworld.wolfram.com/SincFunction.html</a>

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</ol></div></div></div><div><div class="zim-footer">

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