Rectangular function

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Rectangular function
Rectangular function

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 1/2) 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 rectangular function is normalized:

\int_{-\infty}^\infty \mathrm{rect}(t)\,dt=1

The unitary Fourier transforms of the rectangular function are,

\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),

and, in terms of the normalized sinc function,


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

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

tri(t) = rect(t) * rect(t)

Viewing the rectangular function as a probability distribution function, its characteristic function is,

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

and its moment generating function is,

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

where sinh(t) is the hyperbolic sine function.

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