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## Rectangular pulse and its Fourier transform

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 .

Mathematically, a rectangular pulse delayed by seconds is defined as

And its Fourier transform or spectrum is defined as

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

As the pulse becomes more flat (i.e. the width 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 -axis) move closer to the origin. In the limit, as T becomes very large, the magnitude spectrum approaches a Dirac delta function located at the origin.

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

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

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