Page 282 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

(c) First, we rewrite H(w) in standard form as
lO(1 + jw)
H(o) =
(1 + jw/lO)(l + jo/100)
Then

Note tha ~t there are three corner frequencies, o = 1, w = 10, and w = 100. A ,t corner
frequency w = 1
H(~)I,, = 20 + 20loglO& - 2010g,,m6- - 2010g,~~ % 23 dB

At corner frequency w = 10
~(10)~~,=20+20lo~,,~-201o~,,&- 2010glomb 37dB
At corner frequency w = 100

The Bode amplitude plot is sketched in Fig. 5-30(a). Each term contributing to the
overall amplitude is also indicated. Next,
w 0
OH(w) = tan-' w - tan-' - -tan-' -
10 100
Then

and
eH(l) = tan-'(1) - tan-'(0.1) - tan-'(0.01) = 0.676 rad

eH(lO) = tan-'(10) - tan-'(1) - tan-'(0.1) = 0.586 rad
8,(100) = tan-'(100) - tan-'(10) - tan-'(1) = -0.696 rad
The plot of eH(w) is sketched in Fig. 5-30(b).

5.48. An ideal ( -7r/2) radian (or -90") phase shifter (Fig. 5-31) is defined by the frequency
response

(a) Find the impulse response h(t ) of this phase shifter.
(6) Find the output y(t) of this phase shifter due to an arbitrary input x(t).
(c) Find the output y(t) when x(t) = cos oot.

(a) Since e-j"l2 = -j and eJ"/2 = j, H(w) can be rewritten as
H(w) = -jsgn(w) (5.170)