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

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