Signals and Systems - Electrical Engineering

Problems 295

4.19. Using properties to find the Fourier series
Use the Fourier series of a square train of pulses (done in this chapter) to compute the Fourier series of the
triangular signalx(t)with a period,

x 1 (t)=r(t)− 2 r(t− 1 )+r(t− 2 )

(a) Find the derivative ofx(t)ory(t)=dx(t)/dtand carefully plot it. Plot alsoz(t)=y(t)+ 1. Use the

Fourier series of the square train of pulses to compute the Fourier series coefficients ofy(t)andz(t).

(a) Obtain the trigonometric Fourier series ofy(t)andz(t)and explain why they are represented by sines

and whyz(t)has a nonzero mean.

(c) Obtain the Fourier series coefficients ofx(t)from those ofy(t).

(d) Obtain the sinusoidal form ofx(t)and explain why the cosine representation is more appropriate for

this signal than a sine representation.

4.20. Applying Parseval’s result—MATLAB
We wish to approximate the triangular signalx(t)in Problem 4.19 by its Fourier series with a finite number
of terms, let’s say 2 N. This approximation should have95%of the average power of the triangular signal.
Use MATLAB to find the value ofN.

4.21. Fourier series of multiplication of periodic signals
Consider the Fourier series of two periodic signals,

x(t)=

∑∞

k=−∞

Xkejokt

y(t)=

∑∞

k=−∞

Ykej^1 kt

(a) Let 1 = 0. Isz(t)=x(t)y(t)periodic? If so, what is its period and its Fourier series coefficients?

(b) If 1 = 2  0. Isw(t)=x(t)y(t)periodic? If so, what is its period and its Fourier series coefficients?

4.22. Integration of periodic signals
Consider now the integral of the Fourier series of the pulse signalp(t)=x(t)− 1 of periodT 0 = 1 , where
x(t)is given in Figure 4.20.

FIGURE 4.20
Problem 4.23: train of
rectangular pulses.

· · · · · ·

2

x(t)

−1.25 −0.75 −0.25 0.25 0.75 1.25

t

T 0 = 1