Signals and Systems - Electrical Engineering

Problems 293

(a) Find the Fourier series coefficients forx(t)andy(t).

(b) Use MATLAB to plot the magnitude line spectra of the two signals from 0 to 40 πrad/sec. Plot them on

the same figure so you can determine which has a broader support. Indicate which signal is smoother

and explain how it relates to its line spectrum.

4.13. Derivatives and Fourier Series
Given the Fourier series representation for a periodic signal,

x(t)=

∑∞

k=−∞

Xkejk^0 t

we can compute derivatives of it, just like for any other signal.

(a) Consider the periodic train of pulses shown in Figure 4.19. Compute its derivative

y(t)=

dx(t)

dt

and carefully plot it. Find the Fourier series ofy(t).

(b) Use the Fourier series representation ofx(t)and find its derivative to obtain the Fourier series ofy(t).

How does it compare to the Fourier series obtained above?

4.14. Fourier series of sampling delta
The periodic signal

δTs(t)=

∑∞

m=−∞

δ(t−mTs)

will be very useful in the sampling of continuous-time signals.

(a) Find the Fourier series of this signal—that is,

δTs(t)=

∑∞

k=−∞

(^1) kejkst
find the Fourier coefficients (^1) k.
(b) Plot the magnitude line spectrum of this signal.
(c) PlotδTs(t)and its corresponding line spectrum (^1) kas functions of time and frequency. Are they both
periodic? How are their periods related? Explain.
FIGURE 4.19
Problem 4.13: train of
rectangular pulses.
· · · · · ·
2
x(t)
−1.25 −0.75 −0.25 0.25 0.75 1.25
t
T 0 = 1