Rectangular function

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

:

Alternate definitions of the function define to be 0, 1, or undefined. We can also express the rectangular function in terms of the Heaviside step function, $u(t)$:

:$mathrm\{rect\}left(frac\{t\}\{\; au\}\; ight)\; =\; u\; left(\; t\; +\; frac\{\; au\}\{2\}\; ight)\; -\; u\; left(\; t\; -\; frac\{\; au\}\{2\}\; ight),,$

or, alternatively:

:$mathrm\{rect\}(t)\; =\; u\; left(\; t\; +\; frac\{1\}\{2\}\; ight)\; cdot\; u\; left(\; frac\{1\}\{2\}\; -\; t\; ight).,$

The unitary Fourier transforms of the rectangular function are:

:$int\_\{-infty\}^infty\; mathrm\{rect\}(t)cdot\; e^\{-i\; 2pi\; f\; t\}\; ,\; dt=frac\{sin(pi\; f)\}\{pi\; f\}\; =\; mathrm\{sinc\}(f),,$

and:

:$frac\{1\}\{sqrt\{2piint\_\{-infty\}^infty\; mathrm\{rect\}(t)cdot\; e^\{-i\; omega\; t\}\; ,\; dt=frac\{1\}\{sqrt\{2picdot\; mathrm\{sinc\}left(frac\{omega\}\{2pi\}\; ight),,$

where $mathrm\{sinc\}$ is the normalized form.

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

:$mathrm\{tri\}(t)\; =\; mathrm\{rect\}(t)\; *\; mathrm\{rect\}(t).,$

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

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

and its moment generating function is:

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

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

ee also

*Fourier transform
*Square wave
*Triangular function