Page 252 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

(b) In a similar fashion, differentiating Eq. (5.81, we obtain
OC
x'(r) = z kwo(bkcos kwot -aksin kw,r) (5.122)
k-1
Equation (5.122) shows that the Fourier cosine coefficients of xf(t) equal nw,,b, and that the
sine coefficients equal -nwOak. Hence, from Eq. (5.112), replacing A by 2A/T0, we have

Equating Eqs. (5.122) and (5.1231, we have
b, = 0 a, =O k=2m#O

From Eqs. (5.61 and (5.10) and Fig. 5-13(a) we have

Substituting these values into Eq. (5.8), we get

5.10. Consider the triangular wave x(t) shown in Fig. 5-14(a). Using the differentiation
technique, find the triangular Fourier series of x(t ).
From Fig. 5-14(a) the derivative x'(t) of the triangular wave x(t) is, as shown in
Fig. 5-l4( b),

Using Eq. (5.1171, Eq. (5.125) becomes

Equating Eqs. (5.126) and (5.122), we have

From Fig. 5-14(a) and Eq. (5.9~) we have

Thus, substituting these values into Eq. (5.81, we get

A AWL 2 T
x(t) = - + - z -sin kw,t W,, = -
k
2 Tk=, To