The rectangular function (also known as the rectangle function , rect function or the normalized boxcar function
Rectangular function Rectangular function
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{\displaystyle \mathrm {rect} (x)=\sqcap (x)={\begin{cases}0&{\mbox{if }}|x|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|x|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|x|<{\frac {1}{2}}\end{cases}}}
or in terms of the Heaviside step function , u(t) :
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{\displaystyle \mathrm {rect} (x)=u\left(x+{\frac {1}{2}}\right)-u\left(x-{\frac {1}{2}}\right)}
or, alternatively:
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{\displaystyle \mathrm {rect} (x)=u\left(x+{\frac {1}{2}}\right)\cdot u\left({\frac {1}{2}}-x\right)}
The rectangular function is normalized:
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{\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (x)\,dx=1}
The unitary Fourier transforms of the rectangular function are:
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{\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt {2\pi }}}\cdot \mathrm {sinc} (\omega /2)}
unnormalized sinc function .
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π
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{\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i2\pi ft}\,dt=\mathrm {sinc} (f)}
normalized sinc function probability distribution function, its characteristic function is therefore written
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{\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}}\,}
and its moment generating function is:
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{\displaystyle M(k)={\frac {\mathrm {sinh} (k/2)}{k/2}}\,}
where "sinh" is the hyperbolic sine function.
See also 
![{\displaystyle \mathrm {rect} (x)=\sqcap (x)={\begin{cases}0&{\mbox{if }}|x|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|x|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|x|<{\frac {1}{2}}\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18968fc51424e9c28296c1ff41c1164994f55f8c)
