Rectangular pulse and its Fourier transform

: source.nb
: source.cdf
: source code in HTML
: source code in PDF

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 $t0$ .

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.