Rectangular function: Difference between revisions

Revision as of 18:40, 5 August 2006

Latest revision as of 21:11, 28 May 2025

(208 intermediate revisions by more than 100 users not shown)

Line 1:

{{Short description|Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way}}

[[Image:Rectangular function.svg|thumb|right|Rectangular function]]

{{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 a = 1]]

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 '''normalized [[boxcar function]]''') is defined as,<ref name="wolfram">{{MathWorld |title=Rectangle Function |id=RectangleFunction}}</ref>

:<math display="block">\mathrmoperatorname{rect}\left(\frac{t}{a}\right) = \sqcapPi\left(\frac{t}{a}\right) = \begin{cases}

\left\{\begin{array}{rl}

0 & \mbox{if } |t| > \frac{1}{2} \\[3pt]

\frac{1}{2} 0, & \mboxtext{if } |t| => \frac{1a}{2} \\[3pt]

\frac{1 }{2}, & \mboxtext{if } |t| <= \frac{1a}{2} \\

1, & \text{if } |t| < \frac{a}{2}.

\end{cases} </math>

\end{array}\right.</math>

Alternative definitions of the function define <math display="inline">\operatorname{rect}\left(\pm\frac{1}{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.

Alternate definitions of the function define <math>\mathrm{rect}(\pm 1/2)</math> to be 0, 1, or undefined. We can also express the rectangular function in terms of the [[Heaviside step function]], <math>u(t)</math>:

Its periodic version is called a ''[[rectangular wave]]''.

:<math>\mathrm{rect}(t) = u \left( t + \frac{1}{2} \right) - u \left( t - \frac{1}{2} \right) </math>

==History==

or, alternatively:

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==

:<math>\mathrm{rect}(t) = u \left( t + \frac{1}{2} \right) \cdot u \left( \frac{1}{2} - t \right) </math>

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>

The rectangular function is normalized:

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>

:<math>\int_{-\infty}^\infty \mathrm{rect}(t)\,dt=1</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 [[Continuous_Fourier_transformFourier transform#Table_of_important_Fourier_transformsTables 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>\fracoperatorname{1}{\sqrt{2\pi}rect} (x/a)</math>, its Fourier transform is<math display="block">\int_{-\infty}^\infty \mathrmoperatorname{rect}\left(\frac{t}{a}\right)\cdot e^{-i 2\omegapi f t} \, dt

=a \frac{1\sin(\pi af)}{\sqrt{2\pi af}} = a\cdot \mathrmoperatorname{sinc}\left(\frac{\omega}{2_\pi}\right{(a f)}.</math>,

and,==Relation in terms ofto the normalized [[sinctriangular function]],==

We can define the [[triangular function]] as the [[convolution]] of two rectangular functions:

<math display=block>\operatorname{tri(t/T)} = \operatorname{rect(2t/T)} * \operatorname{rect(2t/T)}.\,</math>

==Use in probability==

:<math>\int_{-\infty}^\infty \mathrm{rect}(t)\cdot e^{-i 2\pi f t} \, dt

{{Main |Uniform distribution (continuous)}}

= \mathrm{sinc}(f)</math>

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>

We can define the [[triangular function]] as the convolution of two rectangular functions:

and its [[moment-generating function]] is

:<math>\mathrm{tri}(t) = \mathrm{rect}(t) * \mathrm{rect}(t) </math>

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

Viewing the rectangular function as a [[probability distribution]] function, its [[characteristic function (probability theory)|characteristic function]] is,

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

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

==Rational approximation==

and its [[moment generating function]] is,

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

:<math display="block">M\Pi(kt) = \fraclim_{n\mathrmrightarrow \infty, n\in \mathbb(Z)} \frac{sinh1}{(k/22t)}^{k/22n}\,+1}.</math>

===Demonstration of validity===

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

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:

==See also==

<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]]

*[[TriangularStep function]]

*[[Top-hat filter]]

*[[Boxcar function]]

==References==

{{Reflist}}

{{DEFAULTSORT:Rectangular Function}}

[[Category:Elementary special functions]]

[[Category:Special functions]]