Rectangular function

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

Rectangular function

Rectangular function

${\displaystyle \mathrm {rect} (t)=\sqcap (t)={\begin{cases}0&{\mbox{if }}|t|>{\frac {1}{2}}\\[3pt]{\frac {1}{2}}&{\mbox{if }}|t|={\frac {1}{2}}\\[3pt]1&{\mbox{if }}|t|<{\frac {1}{2}}\end{cases}}}$

or in terms of the Heaviside step function, u(t):

${\displaystyle \mathrm {rect} (t)=u\left(t+{\frac {1}{2}}\right)-u\left(t-{\frac {1}{2}}\right)}$

or, alternatively:

${\displaystyle \mathrm {rect} (t)=u\left(t+{\frac {1}{2}}\right)\cdot u\left({\frac {1}{2}}-t\right)}$

The rectangular function is normalized:

${\displaystyle \int _{-\infty }^{\infty }{\textrm {rect}}(x)\,dx=1}$

The Fourier transform of the rectangular function is

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }{\textrm {rect}}(x)e^{-i\omega x}\,dx={\frac {{\textrm {sinc}}(\omega /2)}{\sqrt {2\pi }}}}$

where "sinc" is the sinc function. Viewing the rectangular function as a probability distribution function, its characteristic function is therefore written

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

and its moment generating function is:

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

where "sinh" is the hyperbolic sine function.

See also

• Fourier transform
• square wave