rectangular function

The rectangular function (also known as the rectangle function or the normalized boxcar function) is defined as

\mathbf{rect}(t) = \left \{ \begin{matrix}

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{matrix} \right. or in terms of the Heaviside step function

\mathbf{rect}(t) = \mathbf{H}(t + 1/2) - \mathbf{H}(t - 1/2)

The rectangular function is normalized:

\int_{-\infty}^\infty \textrm{rect}(x)\,dx=1

The Fourier transform of the rectangular function is

\frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \textrm{rect}(x)e^{-ikx}\,dx

=\frac{\textrm{sinc}(k/2)}{\sqrt{2\pi}} where "sinc" is the sinc function. Viewing the rectangular function as a probability distribution function, its characteristic function is therefore written

\varphi(k)=\textrm{sinc}(k/2)\,

and its moment generating function is:

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

where "sinh" is the hyperbolic sine function.

Rectangular Function

See also