Page 298 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

5.69. Find the inverse Fourier transform of
1
X(w) =
2 - w2 + j3w
Hint: Note that

2 - w2 + j3w = 2 + ( jw12 + j3w = (1 + jw)(2 + jw)
and apply the technique of partial-fraction expansion.
Am. x(t) = (e-' - e-2')u(t)

5.70. Verify the frequency differentiation property (5.561, that is,

Hint: Use definition (5.31) and proceed in a manner similar to Prob. 5.28.

5.71. Find the Fourier transform of each of the following signals:
(a) x(t) = cos wotu(t)
(b) x(t) = sin wotu(t)
(c) x(t)=e-"'~~~w~tu(t),
a>O
(dl x(t) = e-"'sin w,tu(t), a > 0
Hint: Use multiplication property (5.59).

X X iw
Am. (a) X(w) = -S(w - wo) + -S(w + w,) +
2 2 ( jw)' + wi

00
(dl X(w) =
(a + jo12 +
5.72. Let x(t) be a signal with Fourier transform X(w) given by

Consider the signal

Find the value of

Hint: Use Parseval's identity (5.64) for the Fourier transform.