Page 288 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

From Eq. (5.155)

Then

The normalized energy of x(t) is

Using Parseval's identity (5.641, the normalized energy of y(t) is

1 w, do 1 w,
1
1
=-/ -- -- tan-' - = -E = -
T O 4+02 27 2 2" 8
from which we obtain
'" c IT
-= tan - = 1 and o, = 2 rad/s
2 4

5.55. The equivalent bandwidth of a filter with frequency response H(o) is defined by

where IH(w)lm, denotes the maximum value of the magnitude spectrum. Consider the
low-pass RC filter shown in Fig. 5-6(a).
(a) Find its 3-dB bandwidth W, ,,.
(b) Find its equivalent bandwidth We,.

(a) From Eq. (5.91) the frequency response H(w) of the RC filter is given by
1 1
-
H(o) = -
l+joRC l+j(o/o,)
where o, = 1 /RC. Now

The amplitude spectrum lH(w)l is plotted in Fig. 5-6(b). When w = o, = 1/RC,
IH(o,)l = I/&. Thus, the 3-dB bandwidth of the RC filter is given by