Rectangular function

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The rectangular function (also known as the rectangle function, rect function, unit pulse, or the normalized boxcar function) is defined as:

$\mathrm{rect}(t) = \sqcap(t) = \begin{cases} 0 & \mbox{if } |t| > \frac{1}{2} \\[3pt] \frac{1}{2} & \mbox{if } |t| = \frac{1}{2} \\[3pt] 1 & \mbox{if } |t| < \frac{1}{2}. \end{cases}$

Alternate definitions of the function define $\mathrm{rect}(\pm \begin{matrix} \frac{1}{2} \end{matrix})$ to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, $u (t)$ :

$\mathrm{rect}\left(\frac{t}{\tau}\right) = u \left( t + \frac{\tau}{2} \right) - u \left( t - \frac{\tau}{2} \right),\,$

or, alternatively:

$\mathrm{rect}(t) = u \left( t + \frac{1}{2} \right) \cdot u \left( \frac{1}{2} - t \right).\,$

The unitary Fourier transforms of the rectangular function are:

$\int_{-\infty}^\infty \mathrm{rect}(t)\cdot e^{-i 2\pi f t} \, dt =\frac{\sin(\pi f)}{\pi f} = \mathrm{sinc}(f),\,$

and:

$\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \mathrm{rect}(t)\cdot e^{-i \omega t} \, dt =\frac{1}{\sqrt{2\pi}}\cdot \mathrm{sinc}\left(\frac{\omega}{2\pi}\right),\,$