Rectangular Function

# Rectangular function

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

$mathrm(t) = sqcap(t) = begin 0 & mbox |t| > frac \frac & mbox |t| = frac \1 & mbox |t| < frac. \end$

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

$mathrmleft(fracright) = u left( t + frac right) - u left( t - frac right),$

or, alternatively:

$mathrm(t) = u left( t + frac right) - u left( t - frac right).$

In more general form:

$operatornameleft(frac right) = u(t - X + Y/2) - u(t - X - Y/2)$

Where the function is centred at X and has duration Y.

The unitary Fourier transforms of the rectangular function are:

$int_^infty mathrm(t)cdot e^ , dt =frac = mathrm(f),,$

and:

$fracint_^infty mathrm(t)cdot e^ , dt =fraccdot mathrmleft(fracright),,$

where $mathrm$ 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...

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