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Rectangular function - Wikiwand

The rectangular function (also known as the rectangle function, rect function, Pi function, Heaviside Pi function,^[1] gate function, unit pulse, or the normalized boxcar function) is defined as^[2]

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

Alternative definitions of the function define ${\textstyle \operatorname {rect} \left(\pm {\frac {1}{2}}\right)}$ to be 0,^[3] 1,^[4]^[5] or undefined.

Its periodic version is called a rectangular wave.

The rect function has been introduced by Woodward^[6] in ^[7] as an ideal cutout operator, together with the sinc function^[8]^[9] as an ideal interpolation operator, and their counter operations which are sampling (comb operator) and replicating (rep operator), respectively.

The rectangular function is a special case of the more general boxcar 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 the Heaviside step function; the function is centered at $X$ and has duration $Y$ , from $X-Y/2$ to $X+Y/2.$

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

The unitary Fourier transforms of the rectangular function are^[2] $\int _{-\infty }^{\infty }\operatorname {rect} (t)\cdot e^{-i2\pi ft}\,dt={\frac {\sin(\pi f)}{\pi f}}=\operatorname {sinc} _{\pi }(f),$ using ordinary frequency f, where $\operatorname {sinc} _{\pi }$ is the normalized form^[10] of the sinc 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 the sinc function.

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

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

$\operatorname {tri(t/T)} =\operatorname {rect(2t/T)} *\operatorname {rect(2t/T)} .\,$

Viewing the rectangular function as a probability density function, it is a special case of the continuous uniform distribution with $a=-1/2,b=1/2.$ The characteristic function is

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

and its moment-generating function is

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

where $\sinh(t)$ is the hyperbolic sine function.

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

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

Demonstration of validity

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

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 $n.$ However, $2t>1$ and hence ${\textstyle (2t)^{2n}}$ grows very large for large $n.$

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,

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

The rectangle function can be used to represent the Dirac delta function $\delta (x)$ .^[11] 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 $a$ 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 the sinc function here is the normalized sinc function. Because the first zero of the sinc function is at $f=1/a$ and $a$ 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.

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