Page 248 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

and we obtain
m
A A "
(-I)
ej(2m+1)q,t
x(t)=-+- C -
2 ~,,-,Zrn+l
(b) From Eqs. (5.108), (5.10), and (5.12) we have

Substituting these values into Eq. (5.81, we obtain

cos wOt - -cos3w0t + -cos5wot - . ) (5.110)
1
1
-
2 n- 3 5
Note that x(t) is even; thus, x(t) contains only a dc term and cosine terms. Note also that
x(t) in Fig. 5-9 can be obtained by shifting x(t) in Fig. 5-8 to the left by T0/4.

5.7. Consider the periodic square wave x(t) shown in Fig. 5-10.

(a) Determine the complex exponential Fourier series of x(t).
(b) Determine the trigonometric Fourier series of x(t).
Note that x(t) can be expressed as
~(t) =x1(t) -A

where x,(t) is shown in Fig. 5-11. Now comparing Fig. 5-11 and Fig. 5-8 in Prob. 5.5, we
see that x,(t) is the same square wave of x(t) in Fig. 5-8 except that A becomes 2A.
j

-A

Fig. 5-10