imported>Neils51: continous → continuous (4)

2025-11-16T12:09:47Z

continous → continuous (4)

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{{Short description|Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way}}
{{Redirect|Box function|the Conway box function|Minkowski's question-mark function#Conway box function}}
{{Use American English|date = March 2019}}
[[Image:Rectangular function.svg|300px|thumb|right|Rectangular function with <math display="inline">T = 1</math>]]

The '''rectangular function''' (also known as the '''rectangle function''', '''rect function''', '''Pi function''', '''Heaviside Pi function''',<ref>{{cite web |url=https://reference.wolfram.com/language/ref/HeavisidePi.html |title=HeavisidePi, Wolfram Language function |author=Wolfram Research |date=2008 |access-date=October 11, 2022}}</ref> '''gate function''', '''unit pulse''', or the '''normalized [[boxcar function]]''') is defined as<ref name="wolfram">{{MathWorld |title=Rectangle Function |id=RectangleFunction}}</ref>

<math display="block">\operatorname{rect}\left(\frac{t}{T}\right) = \Pi\left(\frac{t}{T}\right) =
\left\{\begin{array}{rl}
0, & \text{if } |t| > \frac{T}{2} \\
\frac{1}{2}, & \text{if } |t| = \frac{T}{2} \\
1, & \text{if } |t| < \frac{T}{2}.
\end{array}\right.</math>

Alternative definitions of the function define <math display="inline">\operatorname{rect}\left(t=\pm\frac{T}{2}\right)</math> to be 0,<ref>{{Cite book |last=Wang |first=Ruye |title=Introduction to Orthogonal Transforms: With Applications in Data Processing and Analysis |pages=135–136 |publisher=Cambridge University Press |year=2012 |url=https://books.google.com/books?id=4KEKGjaiJn0C&pg=PA135 |isbn=9780521516884 }}</ref> 1,<ref>{{Cite book |last=Tang |first=K. T. |title=Mathematical Methods for Engineers and Scientists: Fourier analysis, partial differential equations and variational models |page=85 |publisher=Springer |year=2007 |url=https://books.google.com/books?id=gG-ybR3uIGsC&pg=PA85 |isbn=9783540446958 }}</ref><ref>{{Cite book |last=Kumar |first=A. Anand |title=Signals and Systems |publisher=PHI Learning Pvt. Ltd. |pages=258–260 |url=https://books.google.com/books?id=FGGa6BXhy3kC&pg=PA258 |isbn=9788120343108 |year=2011 }}</ref> or undefined. The area under the curve does not change for the different definitions of the functions at <math display="inline">t=\pm\frac{T}{2}</math>.

The rectangular function can be used as the basis for a ''[[rectangular wave]]''.

==History==
The ''rect'' function has been introduced 1953 by [[Philip Woodward|Woodward]]<ref>{{Cite journal |last=Klauder |first=John R |title=The Theory and Design of Chirp Radars |pages=745–808 |journal=Bell System Technical Journal |year=1960 |volume=39 |issue=4 |doi=10.1002/j.1538-7305.1960.tb03942.x |url=https://ieeexplore.ieee.org/document/6773600 |url-access=subscription }}</ref> in "Probability and Information Theory, with Applications to Radar"<ref>{{Cite book |last=Woodward |first=Philipp M |title=Probability and Information Theory, with Applications to Radar |publisher=Pergamon Press |pages=29 |year=1953 }}</ref> as an ideal [[Window function#Rectangular window|cutout operator]], together with the [[Sinc function|''sinc'' function]]<ref>{{Cite book |last=Higgins |first=John Rowland |title=Sampling Theory in Fourier and Signal Analysis: Foundations |pages=4 |publisher=Oxford University Press Inc. |year=1996 |isbn=0198596995 }}</ref><ref>{{Cite book |last=Zayed |first=Ahmed I |title=Handbook of Function and Generalized Function Transformations |pages=507 |publisher=CRC Press |year=1996 |isbn=9780849380761 }}</ref> as an ideal [[Whittaker–Shannon interpolation formula|interpolation operator]], and their counter operations which are [[Sampling (signal processing)|sampling]] ([[Dirac comb#Dirac-comb identity|''comb'' operator]]) and [[Periodic summation|replicating]] ([[Dirac comb#Dirac-comb identity|''rep'' operator]]), respectively.

==Relation to the boxcar function==
The rectangular function is a special case of the more general [[boxcar function]]:

<math display=block>\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)</math>

where <math>H(x)</math> is the [[Heaviside step function]]; the function is centered at <math>X</math> and has duration <math>Y</math>, from <math>X-Y/2</math> to <math>X+Y/2.</math>
==Fourier transform of the rectangular function==
[[File:Sinc_function_(normalized).svg|thumb|400px|right|Plot of normalized <math>\operatorname{sinc}(x)</math> function (i.e. <math>\operatorname{sinc}(\pi x)</math>) with its spectral frequency components.]]

The [[Fourier transform#Tables of important Fourier transforms|unitary Fourier transforms]] of the rectangular function are<ref name="wolfram"/>
<math display="block">\int_{-\infty}^\infty \operatorname{rect}(t)\cdot e^{-i 2\pi f t} \, dt
=\frac{\sin(\pi f)}{\pi f} = \operatorname{sinc}(\pi f) =\operatorname{sinc}_\pi(f),</math>
using ordinary frequency {{mvar|f}}, where [[sinc function|<math>\operatorname{sinc}_\pi</math>]] is the normalized form<ref>Wolfram MathWorld, https://mathworld.wolfram.com/SincFunction.html</ref> of the [[sinc function]] and
<math display="block">\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),
</math>
using angular frequency <math>\omega</math>, where [[sinc function|<math>\operatorname{sinc}</math>]] is the unnormalized form of the [[sinc function]].

For <math>\operatorname{rect} (x/a)</math>, its Fourier transform is<math display="block">\int_{-\infty}^\infty \operatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\pi f t} \, dt
=a \frac{\sin(\pi af)}{\pi af} = a\ \operatorname{sinc}_\pi{(a f)}.</math>

==Self convolution of the Rectangular function==
[[File:Splines resulting from successive convolution of the unit Rect function.png|thumb|The unit Rectangular function (in which <math display="inline">T=1</math>) along with the [[Piecewise function|piecewise defined]] [[spline (mathematics)|splines]] that result from successive convolutions of the Rectangular function with itself.]]
The [[self convolution]] of the dis-[[Continuous function|continuous]] rectangular function results in the [[triangular function]], a [[Piecewise function|piecewise defined]] [[spline (mathematics)|spline]] that is continuous, but not continuously [[Differentiable function|differentiable]]. Successive convolutions of the rectangular function result in piecewise defined pulses with lower maximums which are wider and smoother, with "smoother" meaning [[higher-order derivative]]s are coninuous.<ref>{{cite web|url=https://cyclostationary.blog/2021/01/28/sptk-convolution-and-the-convolution-theorem/|title=SPTK: Convolution and the Convolution Theorem| first= Chad |last= Spooner| date= January 28, 2021}}</ref>

A [[convolution]] of the discontinuous rectangular function with itself results in the triangular function, which is a continuous function:

<math display=block> \begin{align} \operatorname{rect(2t/T)} * \operatorname{rect(2t/T)} = \operatorname{tri(t/T)} =
\begin{cases}
1 + t, & -T < t < 0 \\
1 - t, & \,\,\,\,\, 0 < t < T \\
0 & \,\,\,\,\,\text{otherwise} \\
\end{cases}
\end{align}
</math>

Self convolution of the rectangular function applied twice yields a continuous and differentiably continuous parabolic spline:

<math display=block> \begin{align} \operatorname{rect(2t/T)} * \operatorname{rect(2t/T)} * \operatorname{rect(2t/T)} = \operatorname{tri(t/T)} * \operatorname{rect(2t/T)} =
\begin{cases}
\frac{9}{8} + \frac{3}{2}t + \frac{1}{2}t^2, & -\frac{3}{2}T < t < -\frac{1}{2}T \\
\frac{3}{4} - t^2, & -\frac{1}{2}T < t < \frac{1}{2}T \\
\frac{9}{8} - \frac{3}{2}t + \frac{1}{2}t^2, & \,\,\,\,\, \frac{1}{2}T < t < \frac{3}{2}T \\
0 & \,\,\,\,\,\text{otherwise} \\
\end{cases}
\end{align}
</math>

A self convolution of the rectangular function applied three times yields a continuous, and a second order differentiably continuous cubic spline:

<math display=block> \begin{align} \operatorname{tri(t/T)} * \operatorname{tri(t/T)} =
\begin{cases}
\frac{4}{3} + {2}t + t^2 +\frac{1}{6}t^3 , & -2T < t < -T \\
\frac{2}{3} - t^2 - \frac{1}{2}t^3, & -T < t < 0 \\
\frac{2}{3} - t^2 + \frac{1}{2}t^3, & \,\,\,\,\,0 < t < T \\
\frac{4}{3} - {2}t + t^2 -\frac{1}{6}t^3 , & \,\,\,\,\,T < t < 2T \\
0 & \,\,\,\,\,\text{otherwise} \\
\end{cases}
\end{align}
</math>

A self convolution of the rectangular function applied four times yields a continuous, and a third order differentiably continuous 4th order spline:

<math display=block> \begin{align} 4^{th}\,\text{order spline} =
\begin{cases}
\frac{625}{384} + \frac{125}{48}t + \frac{25}{16}t^2 +\frac{5}{12}t^3 + \frac{1}{24}t^4, & -\frac{5}{2}T < t < -\frac{3}{2}T \\
\frac{55}{96} - \frac{5}{24}t - \frac{5}{4} t^2 - \frac{5}{6}t^3 -\frac{1}{6}t^4, & -\frac{3}{2}T < t < -\frac{1}{2}T \\

\frac{115}{192} - \frac{5}{8}t^2 + \frac{1}{4}t^4, & -\frac{1}{2}T < t < \frac{1}{2}T \\

\frac{55}{96} + \frac{5}{24}t - \frac{5}{4} t^2 + \frac{5}{6}t^3 -\frac{1}{6}t^4, & \,\,\,\,\,\frac{1}{2}T < t < \frac{3}{2}T \\
\frac{625}{384} - \frac{125}{48}t + \frac{25}{16}t^2 -\frac{5}{12}t^3 + \frac{1}{24}t^4, & \,\,\,\,\,\frac{3}{2}T < t < \frac{5}{2}T \\

0 & \,\,\,\,\,\text{otherwise} \\
\end{cases}
\end{align}
</math>

Since the [[Fourier Transform]] of the Rectangular function is the [[Sinc function]], the [[Convolution theorem]] mean that the Fourier transform of pulses resulting from successive convolution of the Rectangular function with itself is simply the Sinc function to the order of the number of times that the convolution function was applied + 1 (i.e., the Fourier transform of the Triangular function is Sinc<sup>2</sup>, the Fourier transform of parabolic spline resulting from two successive convolutions of the Rectangular function with itself is Sinc<sup>3</sup>, etc.)

==Use in probability==
{{Main |Uniform distribution (continuous)}}
Viewing the rectangular function as a [[probability density function]], it is a special case of the [[Uniform distribution (continuous)|continuous uniform distribution]] with <math>a = -1/2, b = 1/2.</math> The [[characteristic function (probability theory)|characteristic function]] is

<math display=block>\varphi(k) = \frac{\sin(k/2)}{k/2},</math>

and its [[moment-generating function]] is

<math display=block>M(k) = \frac{\sinh(k/2)}{k/2},</math>

where <math>\sinh(t)</math> is the [[hyperbolic sine]] function.

==Rational approximation==
The pulse function may also be expressed as a limit of a [[rational function]]:

<math display="block">\Pi(t) = \lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1}.</math>

===Demonstration of validity===
First, we consider the case where <math display=inline>|t|<\frac{1}{2}.</math> Notice that the term <math display=inline>(2t)^{2n}</math> is always positive for integer <math>n.</math> However, <math>2t<1</math> and hence <math display=inline>(2t)^{2n}</math> approaches zero for large <math>n.</math>

It follows that:
<math display="block">\lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1} = \frac{1}{0+1} = 1, |t|<\tfrac{1}{2}.</math>

Second, we consider the case where <math display="inline">|t|>\frac{1}{2}.</math> Notice that the term <math display="inline">(2t)^{2n}</math> is always positive for integer <math>n.</math> However, <math>2t>1</math> and hence <math display="inline">(2t)^{2n}</math> grows very large for large <math>n.</math>

It follows that:
<math display="block">\lim_{n\rightarrow \infty, n\in \mathbb(Z)} \frac{1}{(2t)^{2n}+1} = \frac{1}{+\infty+1} = 0, |t|>\tfrac{1}{2}.</math>

Third, we consider the case where <math display="inline">|t| = \frac{1}{2}.</math> We may simply substitute in our equation:

<math display="block">\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}.</math>

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

<math display="block">\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}</math>

== Dirac delta function ==
The rectangle function can be used to represent the [[Dirac delta function]] <math>\delta (x)</math>.<ref name=":0">{{Cite book |last1=Khare |first1=Kedar |title=Fourier Optics and Computational Imaging |last2=Butola |first2=Mansi |last3=Rajora |first3=Sunaina |publisher=Springer |year=2023 |isbn=978-3-031-18353-9 |edition=2nd |pages=15–16 |chapter=Chapter 2.4 Sampling by Averaging, Distributions and Delta Function |doi=10.1007/978-3-031-18353-9}}</ref> Specifically,<math display="block">\delta (x) = \lim_{a \to 0} \frac{1}{a}\operatorname{rect}\left(\frac{x}{a}\right).</math>For a function <math>g(x)</math>, its average over the width ''<math>a</math>'' around 0 in the function domain is calculated as,

<math display="block">g_{avg}(0) = \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \operatorname{rect}\left(\frac{x}{a}\right).</math>
To obtain <math>g(0)</math>, the following limit is applied,

<math display="block">g(0) = \lim_{a \to 0} \frac{1}{a} \int\limits_{- \infty}^{\infty} dx\ g(x) \operatorname{rect}\left(\frac{x}{a}\right)</math>
and this can be written in terms of the Dirac delta function as,
<math display="block">g(0) = \int\limits_{- \infty}^{\infty} dx\ g(x) \delta (x).</math>The Fourier transform of the Dirac delta function <math>\delta (t)</math> is

<math display="block">\delta (f)
= \int_{-\infty}^\infty \delta (t) \cdot e^{-i 2\pi f t} \, dt
= \lim_{a \to 0} \frac{1}{a} \int_{-\infty}^\infty \operatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\pi f t} \, dt
= \lim_{a \to 0} \operatorname{sinc}{(a f)}.</math>
where the [[sinc function]] here is the normalized sinc function. Because the first zero of the sinc function is at <math>f = 1 / a</math> and <math>a</math> goes to infinity, the Fourier transform of <math>\delta (t)</math> is

<math display="block">\delta (f) = 1,</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.

==See also==
*[[Fourier transform]]
*[[Square wave (waveform)|Square wave]]
*[[Step function]]
*[[Top-hat filter]]
*[[Boxcar function]]

==References==
{{Reflist}}

{{DEFAULTSORT:Rectangular Function}}
[[Category:Special functions]]

Rectangular function - Revision history

imported>Neils51: continous → continuous (4)