Rectangular function

The rectangular function is defined as

$\mathbf {rect} (t)=\left\{{\begin{matrix},&\left|x\right|>{\frac {1}{2}}\\{\frac {1}{2}},&\left|x\right|={\frac {1}{2}}\\1,&\left|x\right|<{\frac {1}{2}}\end{matrix}}\right.$

or in terms of the Heaviside step function

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

The Fourier transform is

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

where $sinc$ is the sinc function.

See also

• Fourier transform