Rectangular function

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:

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

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

<math>mathrmleft(fracright) = u left( t + frac right) - u left( t - frac right),</math>


or, alternatively:

<math>mathrm(t) = u left( t + frac right) - u left( t - frac right).</math>


In more general form:

<math>operatornameleft(frac right) = u(t - X + Y/2) - u(t - X - Y/2)</math>


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

The unitary Fourier transforms of the rectangular function are:

<math>int_^infty mathrm(t)cdot e^ , dt
=frac = mathrm(f),,</math>

and:

<math>fracint_^infty mathrm(t)cdot e^ , dt
=fraccdot mathrmleft(fracright),,</math>

where <math>mathrm</math> 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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