Page 285 - Schaum's Outline of Theory and Problems of Signals and Systems

FOURIER ANALYSIS OF TIME SIGNALS AND SYSTEMS [CHAP. 5

Since the system is causal, by definition
h(t) = 0 r<O
Accordingly,

Let

where he(r) and h,(t) are the even and odd components of h(r), respectively. Then from
Eqs. (1.5) and (1.6) we can write
h(r) = 2he(r) = 2h,(r)
From Eqs. (5.616) and (5.61~) we have
h,(t) -A(w) and h,(t) -jB(w)
Thus, by Eq. (5.175)

Equations (5.176~) and (5.176b) indicate that h(t) can be obtained in terms of A(w) or B(w)
alone.

5.50. Consider a causal continuous-time LTI system with frequency response
H(o) = A(o) + jB(o)
If the impulse response h(t) of the system contains no impulses at the origin, then
show that A(w) and B(w) satisfy the following equations:

As in Prob. 5.49, let

h(r) = he(r) + ho(t)
Since h(t) is causal, that is, h(r) = 0 for t < 0, we have
he(t) = -hO(t) t <O
Also from Eq. (5.175) we have
h,(t) = h,(t) r>O
Thus, using Eq. (5.1521, we can write

h,(t) =h,(r) sgn(r)
ho(r) =he(r) sgn(t)
Now, from Eqs. (5.6161, (5.61~1, and (5.153) we have