Rectangular function

The rectangular function (or rectangle function) is defined as

${\displaystyle \mathbf {rect} (t)=\left\{{\begin{matrix}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{matrix}}\right.}$

or in terms of the Heaviside step function

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

The Fourier transform is

${\displaystyle \mathbf {rect} \left({\frac {\omega }{2W}}\right)={\frac {W}{\pi }}\operatorname {sinc} (Wt)}$

where "sinc" is the sinc function.

See also

• boxcar function
• Fourier transform