Rectangular function

Rectangular function

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

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

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

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

or, alternatively:

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

The unitary Fourier transforms of the rectangular function are:

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

and:

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

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

Note that as long as the definition of the pulse function is only motivated by the time-domain experience of it, there is no reason to believe that the oscillatory interpretation (i.e. the Fourier transform function) should be intuitive, or directly understood by humans. However, some aspects of the theoretical result may be understood intuitively, such as the infinite bandwidth requirement incurred by the indefinitely-sharp edges in the time-domain definition.

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.

See also

• Fourier transform
• Square wave
• Triangular function