Page 293 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

(h) From Eq. (5.147) (Prob. 5.25) we have

Let

Then, according to the frequency convolution theorem (5.59), we have

Using Eq. (1.261, we obtain

which shows that X,(w) consists of periodically repeated replicas of X(w) centered about
kw, for all k. The Fourier spectrum X,(w) is shown in Fig. 5-34 f ) for T, < r/w, (or
w, > 2wM), and in Fig. 5-34( j) for T, > r/wM (or w, < 2wM), where w, = 27~/T,. It is
seen that no overlap of the replicas X(o - ko,) occurs in X,(o) for w, r 2wM and that
overlap of the spectral replicas is produced for w,$ < 2wM. This effect is known as
aliasing.

5.59. Let x(t ) be a real-valued band-limited signal specified by

Show that x( t) can be expressed as
sin wM(t - kT,)
cC
41) C x(kTs)
=
&- -m - kT-)
where T, = rr/w,.
Let

From Eq. ( 5.183 we have
Xi
T,X,(w) = C X(o - ko,)
k= -m
Then, under the following two conditions,
7T
(1) X(o)=O,IwI>o, and (2) T,=-
WM
we see from Eq. (5.1185 that