Page 268 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

Thus, by the time convolution theorem (5.58) and Eq. (5.154) we obtain

since X(o)S(o) = X(O)S(w) by Eq. (1.25). Thus,

5.34. Using the integration property (5.57) and Eq. (1.31), find the Fourier transform of
u(t).
From Eq. (1.31) we have

(t) = / S(r) dr
- m
S(t) -
Now from Eq. (5.140) we have
1
Setting x(r) = S(7) in Eq. (5.571, we have
x(t) = s(t) -X(o) = 1 and X(0) = 1
and

5.35. Prove the frequency convolution theorem (5.591, that is,

1
xdf )x2W " %XI(4 * X2(4

By definitions (5.31) and (5.32) we have

.F[xl(t)x2(t)] = jm ~~(t)x,(t)e-~"'dt
- m
1
X,(h)eJ" dA ~,(t)e-~"'dt

m
1 /
= - X,(A) x,(t)e-j("-*)'dt
2~ -w - m
1 m 1
= -/ X,(A)X,(o - A) dl = -X,(o) * X,(o)
27~ -, 27r
Hence,