Page 250 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

(a) Determine the complex exponential Fourier series of ST$t).
(6) Determine the trigonometric Fourier series of ST$t).

(a) Let

Since S(t) is involved, we use Eq. (5.1026) to determine the Fourier coefficients and we
obtain

Hence, we get

30 2n
a0
+
aTJt) = i z (a, cos kwot + bk sin ko,t) oo = -
k-1 To
Since aT,$t) is even, b, = 0, and by Eq. (5.9a), a, are given by

2 2
a, = -/"'/* S(t) cos kw0tdt = -
To - T0/2 To
Thus, we get

5.9. Consider the triangular wave x(t) shown in Fig. 5-13(a). Using the differentiation
technique, find (a) the complex exponential Fourier series of dt), and (6) the
trigonometric Fourier series of x( t 1.

The derivative xl(t) of the triangular wave x(t) is a square wave as shown in Fig. 5-13(b).
(a) Let

Differentiating Eq. (5.118), we obtain
m
xl(t) = z jko o c k ejkW~1'
&= - m
Equation (5.119) shows that the complex Fourier coefficients of xYt) equal jkw,c,. Thus, we
can find ck (k # 0) if the Fourier coefficients of xl(t) are known. The term c, cannot be
determined by Eq. (5.119) and must be evaluated directly in terms of x(t) with Eq. (5.6).