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Rectangular function

From Wikipedia, the free encyclopedia

Function whose graph is 0, then 1, then 0 again, in an almost-everywhere continuous way

"Box function" redirects here. For the Conway box function, seeMinkowski's question-mark function § Conway box function.

Rectangular function with ${\textstyle T=1}$

Rectangular function with ${\textstyle T=1}$

Therectangular function (also known as therectangle function,rect function,Pi function,Heaviside Pi function,^[1]gate function,unit pulse, or thenormalizedboxcar function) is defined as^[2]

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

Alternative definitions of the function define ${\textstyle \operatorname {rect} \left(t=\pm {\frac {T}{2}}\right)}$ to be 0,^[3] 1,^[4]^[5] or undefined. The area under the curve does not change for the different definitions of the functions at ${\textstyle t=\pm {\frac {T}{2}}}$ .

The rectangular function can be used as the basis for arectangular wave.

History

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Therect function has been introduced 1953 byWoodward^[6] in "Probability and Information Theory, with Applications to Radar"^[7] as an idealcutout operator, together with thesinc function^[8]^[9] as an idealinterpolation operator, and their counter operations which aresampling (comb operator) andreplicating (rep operator), respectively.

Relation to the boxcar function

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The rectangular function is a special case of the more generalboxcar function:

$\operatorname {rect} \left({\frac {t-X}{Y}}\right)=H(t-(X-Y/2))-H(t-(X+Y/2))=H(t-X+Y/2)-H(t-X-Y/2)$

where $H(x)$ is theHeaviside step function; the function is centered at $X {\displaystyle X}$ and has duration $Y {\displaystyle Y}$ , from $X-Y/2$ to $X+Y/2.$

Fourier transform of the rectangular function

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Plot of normalized $\operatorname {sinc} (x)$ function (i.e. $\operatorname {sinc} (\pi x)$ ) with its spectral frequency components.

{\displaystyle \operatorname {sinc} (x)} — Plot of normalized $\operatorname {sinc} (x)$ function (i.e. $\operatorname {sinc} (\pi x)$ ) with its spectral frequency components.

Theunitary Fourier transforms of the rectangular function are^[2] $\int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} (\pi f)=\operatorname {sinc} _{\pi }(f),$ using ordinary frequencyf, where $\operatorname {sinc} _{\pi }$ is the normalized form^[10] of thesinc function and ${\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt {2\pi }}}\cdot {\frac {\sin \left(\omega /2\right)}{\omega /2}}={\frac {1}{\sqrt {2\pi }}}\cdot \operatorname {sinc} \left(\omega /2\right),$ using angular frequency $\omega$ , where $\operatorname {sinc}$ is the unnormalized form of thesinc function.

For $\operatorname {rect} (x/a)$ , its Fourier transform is $\int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=a{\frac {\sin(\pi af)}{\pi af}}=a\ \operatorname {sinc} _{\pi }{(af)}.$

Self convolution of the Rectangular function

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The unit Rectangular function (in which ${\textstyle T=1}$ ) along with thepiecewise defined splines that result from successive convolutions of the Rectangular function with itself.

Theself convolution of the dis-continuous rectangular function results in thetriangular function, apiecewise defined spline that is continuous, but not continuouslydifferentiable. Successive convolutions of the rectangular function result in piecewise defined pulses with lower maximums which are wider and smoother, with "smoother" meaninghigher-order derivatives are coninuous.^[11]

Aconvolution of the discontinuous rectangular function with itself results in the triangular function, which is a continuous function:

${\begin{aligned}\operatorname {rect(2t/T)} *\operatorname {rect(2t/T)} =\operatorname {tri(t/T)} ={\begin{cases}1+t,&-T<t<0\\1-t,&\,\,\,\,\,0<t<T\\0&\,\,\,\,\,{\text{otherwise}}\\\end{cases}}\end{aligned}}$

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

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

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

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

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

${\begin{aligned}4^{th}\,{\text{order spline}}={\begin{cases}{\frac {625}{384}}+{\frac {125}{48}}t+{\frac {25}{16}}t^{2}+{\frac {5}{12}}t^{3}+{\frac {1}{24}}t^{4},&-{\frac {5}{2}}T<t<-{\frac {3}{2}}T\\{\frac {55}{96}}-{\frac {5}{24}}t-{\frac {5}{4}}t^{2}-{\frac {5}{6}}t^{3}-{\frac {1}{6}}t^{4},&-{\frac {3}{2}}T<t<-{\frac {1}{2}}T\\{\frac {115}{192}}-{\frac {5}{8}}t^{2}+{\frac {1}{4}}t^{4},&-{\frac {1}{2}}T<t<{\frac {1}{2}}T\\{\frac {55}{96}}+{\frac {5}{24}}t-{\frac {5}{4}}t^{2}+{\frac {5}{6}}t^{3}-{\frac {1}{6}}t^{4},&\,\,\,\,\,{\frac {1}{2}}T<t<{\frac {3}{2}}T\\{\frac {625}{384}}-{\frac {125}{48}}t+{\frac {25}{16}}t^{2}-{\frac {5}{12}}t^{3}+{\frac {1}{24}}t^{4},&\,\,\,\,\,{\frac {3}{2}}T<t<{\frac {5}{2}}T\\0&\,\,\,\,\,{\text{otherwise}}\\\end{cases}}\end{aligned}}$

Since theFourier Transform of the Rectangular function is theSinc function, theConvolution 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 Sinc², the Fourier transform of parabolic spline resulting from two successive convolutions of the Rectangular function with itself is Sinc³, etc.)

Use in probability

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Main article:Uniform distribution (continuous)

Viewing the rectangular function as aprobability density function, it is a special case of thecontinuous uniform distribution with $a=-1/2,b=1/2.$ Thecharacteristic function is

$\varphi (k)={\frac {\sin(k/2)}{k/2}},$

and itsmoment-generating function is

$M(k)={\frac {\sinh(k/2)}{k/2}},$

where $\sinh(t)$ is thehyperbolic sine function.

Rational approximation

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The pulse function may also be expressed as a limit of arational function:

$\Pi (t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}.$

Demonstration of validity

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First, we consider the case where ${\textstyle |t|<{\frac {1}{2}}.}$ Notice that the term ${\textstyle (2t)^{2n}}$ is always positive for integer $n . {\displaystyle n.}$ However, $2t<1$ and hence ${\textstyle (2t)^{2n}}$ approaches zero for large $n . {\displaystyle n.}$

It follows that: $\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{0+1}}=1,|t|<{\tfrac {1}{2}}.$

Second, we consider the case where ${\textstyle |t|>{\frac {1}{2}}.}$ Notice that the term ${\textstyle (2t)^{2n}}$ is always positive for integer $n . {\displaystyle n.}$ However, $2t>1$ and hence ${\textstyle (2t)^{2n}}$ grows very large for large $n . {\displaystyle n.}$

It follows that: $\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\frac {1}{+\infty +1}}=0,|t|>{\tfrac {1}{2}}.$

Third, we consider the case where ${\textstyle |t|={\frac {1}{2}}.}$ We may simply substitute in our equation:

$\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{1^{2n}+1}}={\frac {1}{1+1}}={\tfrac {1}{2}}.$

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

$\operatorname {rect} (t)=\Pi (t)=\lim _{n\rightarrow \infty ,n\in \mathbb {(} Z)}{\frac {1}{(2t)^{2n}+1}}={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\1&{\mbox{if }}|t|<{\frac {1}{2}}.\\\end{cases}}$

Dirac delta function

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The rectangle function can be used to represent theDirac delta function $\delta (x)$ .^[12] Specifically, $\delta (x)=\lim _{a\to 0}{\frac {1}{a}}\operatorname {rect} \left({\frac {x}{a}}\right).$ For a function $g(x)$ , its average over the width $a {\displaystyle a}$ around 0 in the function domain is calculated as,

$g_{avg}(0)={\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right).$ To obtain $g(0)$ , the following limit is applied,

$g(0)=\lim _{a\to 0}{\frac {1}{a}}\int \limits _{-\infty }^{\infty }dx\ g(x)\operatorname {rect} \left({\frac {x}{a}}\right)$ and this can be written in terms of the Dirac delta function as, $g(0)=\int \limits _{-\infty }^{\infty }dx\ g(x)\delta (x).$ The Fourier transform of the Dirac delta function $\delta (t)$ is

$\delta (f)=\int _{-\infty }^{\infty }\delta (t)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}{\frac {1}{a}}\int _{-\infty }^{\infty }\operatorname {rect} \left({\frac {t}{a}}\right)\cdot e^{-i2\pi ft}\,dt=\lim _{a\to 0}\operatorname {sinc} {(af)}.$ where thesinc function here is the normalized sinc function. Because the first zero of the sinc function is at $f=1/a$ and $a {\displaystyle a}$ goes to infinity, the Fourier transform of $\delta (t)$ is

$\delta (f)=1,$ 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.