Page 291 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

from which we get

Thus.

(b) From Eq. (5.137)

sin at Iwl <a
~(t) =pu(w) =
= - -X(w)
7~t Iwl >a
Using Parseval's identity (5.64), we have

Then, by Eq. (5.180)

from which we get
Ww = 0.9a rad/s
Note that the absolute bandwidth of x(t) is a (radians/second).

5.58. Let x(t) be a real-valued band-limited signal specified by [Fig. 5-34(b)]

Let x,(t be defined by

(a) Sketch x$t) for T, < r/o, and for T, > r/oM.
(b) Find and sketch the Fourier spectrum X$o) of xJr) for T, < r/oM and for
T, > n/w,.
(a) Using Eq. (I.26), we have

The sampled signal x,(r) is sketched in Fig. 5-34(c) for Tq < r/w,, and in Fig. 5-34(i) for
T, > T/w~.
The signal x,(t) is called the ideal sampled signal, T, is referred to as the sampling
interr.al (or period), and f, = 1/T, is referred to as the sampling rate (or frequency ).