Page 374 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 61 FOURIER ANALYSIS OF DISCRETE-TIME SIGNALS AND SYSTEMS 361

Find the Fourier transform of the sequence x[n] shown in Fig. 6-41
Am. X(R) = j2(sin R + 2 sin 2R + 3 sin 3R)

Fig. 6-41

Find the inverse Fourier transform of each of the following Fourier transforms:
(a) X(R) = cos(2R)
(6) X(R) = jR

Am. (a) x[nl= f8[n - 21 + 3[n + 21

Consider the sequence y[n] given by
n even
n odd
Express y(R) in terms of X(R).

Ans. Y(R) = $X(R) + $x(R - 7)

Let

(a) Find y[n 1 = x[n] * x[n].
(b) Find the Fourier transform Y(0) of y[n].

In15 5
Am. (a) y[n] =
In1 > 5

Verify Parseval's theorem [Eq. (6.66)] for the discrete-time Fourier transform, that is,
m 1

Hint: Proceed in a manner similar to that for solving Prob. 5.38.