Page 257 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

(b) In Prob. 2.8(c), x,(r) =x2(t) =x(t), as shown in Fig. 2-12, which is the same as Fig. 5-8
(Prob. 5.5). From Eq. (5.105) we have

Thus, by Eq. (5.133) the complex Fourier coefficients ck of f(t) are

Note that in Prob. 2.8(c), f(t) =x,(r) @x,(t), shown in Fig. 2-13(b), is proportional to
x(t), shown in Fig. 5-13(a). Thus, replacing A by A~T,/~ the results from Prob. 5.9,
in
we get

which are the same results obtained by using Eq. (5.133).

FOURIER TRANSFORM

5.16. (a) Verify the time-shifting property (5.50), that is,

(
)
x(t - to) H e-jw'~lX o
By definition (5.31)

By the change of variable T = r - t,,, we obtain

Hence,

5.17. Verify the frequency-shifting property (5.511, that is,
x(t) eiwuJ HX(O - wO)

By definition (5.31)