Page 284 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 51 FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS

-

Fig. 5-31 -n/2 rad phase shifter.

where

Now from Eq. (5.153)
2
sgn(t) - jo

and by the duality property (5.54) we have

1
- C, - jsgn(w) (5.172)
Tt
since sgn(w) is an odd function of w. Thus, the impulse response h(t) is given by
1
h(t) = F-'[~(w)] = F-'[-jsgn(w)] = - (5.173)
nt

(b) By Eq. (2.6)

The signal y(t) defined by Eq. (5.174) is called the Hilbert transform of x(t) and is
usually denoted by f (l).
(c) From Eq. (5.144)
cos wet H T[S(W - w,,) + a( w + wO)]
Then

Y(w) = X(w)H(w) =T[s(~ - w,) + S(w + w,)][-jsgn(w)]
= -j~ sgn(w,)S(w - w,,) - jT sgn( -w,,)S(w + w,)
= -jd(w - w,) + j.rrS(w + w,)
since sgn(oo) = 1 and sgn( -w,) - 1. Thus, from Eq. (5.145) we get
y(t) =sin w,t
Note that cos(w,t - 7/21 = sin wot.

5.49. Consider a causal continuous-time LTI system with frequency response
H(o) =A(w) + jB(o)
Show that the impulse response h(t) of the system can be obtained in terms of A(w)
or B(w) alone.