Fourier Series and Fourier Transforms.

Given a rectangular pulse as shown in Figure 13–4, with amplitude A, width T, and period T0, we can compute and plot the coefficients in the corresponding Fourier series. If we allow T0 to increase to infinity, the waveform is a single pulse and the Fourier series approaches a scaled version of the Fourier transform. To see this graphically, use MATLAB to create the following series of plots. Let A ¼

5, T ¼ 1, and T0 ¼ 2. Compute the Fourier series coefficients for n ¼ 0, 1, 2, . . . 10 T0. Create a stem plot of (an  T0) on the vertical axis versus the (n/T0) on the horizontal axis.

Increment T0 by one and repeat the stem plot. Create plots up until T0 ¼ 20 and comment on the behavior of the results.

Now compute the Fourier transform of f(t)¼A[u(t þT/2) u

(tT/2)]. Evaluate F(v) for v ¼ 0 to 20p rad/s. On the same axes as your final stem plot, plot 2F(v) versus v/2p.

Comment on the results.