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Rectangular pulse and its Fourier transform

Nasser M. Abbasi

December 27 2009 compiled on — Wednesday July 06, 2016 at 08:35 AM
This Demonstration illustrates the relationship between a rectangular pulse signal and its Fourier transform. There are three parameters that define a rectangular pulse: its height A, width T in seconds, and center t0   .

Mathematically, a rectangular pulse delayed by t0   seconds is defined as

                (      )   {      ||t−t0||   1
g(t− t0) = A rect t−-t0  =   A   if  T   ≤ 2
                   T         0   otherwise

And its Fourier transform or spectrum is defined as

G (f) = AT sinc(πf T)exp (− i2πft0)

This Demonstration illustrates how changing g(t)  affects its spectrum. Both the magnitude and phase of the spectrum are displayed.

As the pulse becomes more flat (i.e. the width T  of the pulse increases), the magnitude spectrum loops become thinner and taller. In other words, the zeros (the crossings of the magnitude spectrum with the x  -axis) move closer to the origin. In the limit, as T becomes very large, the magnitude spectrum approaches a Dirac delta function δ(t)  located at the origin.

As the height of the pulse become larger and its width becomes smaller, it approaches a Dirac delta function δ(t)  and the magnitude spectrum flattens out and becomes a constant of magnitude 1  in the limit.

As t0   changes, the pulse shifts in time, the magnitude spectrum does not change, but the phase spectrum does.

We notice a    ∘
180 degree phase shift at each frequency defined by k-
T  where k is an integer other than zero, and T is the pulse duration. These frequencies are the zeros of the magnitude spectrum.