Rectangular function

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

Rectangular function

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

Alternate definitions of the function define ${\displaystyle \mathrm {rect} (\pm 1/2)}$ to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, ${\displaystyle u(t)}$:

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

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:

${\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\,dt=1}$

The unitary Fourier transforms of the rectangular function are,

${\displaystyle {\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i\omega t}\,dt={\frac {1}{\sqrt {2\pi }}}\cdot \mathrm {sinc} \left({\frac {\omega }{2\pi }}\right)}$,

and, in terms of the normalized sinc function,

${\displaystyle \int _{-\infty }^{\infty }\mathrm {rect} (t)\cdot e^{-i2\pi ft}\,dt=\mathrm {sinc} (f)}$

We can define the triangular function as the convolution of two rectangular functions:

${\displaystyle \mathrm {tri} (t)=\mathrm {rect} (t)*\mathrm {rect} (t)}$

Viewing the rectangular function as a probability distribution function, its characteristic function is,

${\displaystyle \varphi (k)={\frac {\sin(k/2)}{k/2}}\,}$

and its moment generating function is,

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

where ${\displaystyle \mathrm {sinh} (t)}$ is the hyperbolic sine function.

See also

W T F

• Fourier transform
• Square wave
• Triangular function