Page 116 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

(b) h[nl = -2h[n - 11 + 6[n] + 6[n - 11
Since the system is causal, h[ - 1] = 0. Then
h[O] = -2h[ - 1] + 6[0] + 6[ - 11 = S[O] = 1

h[l] = -2h[O] + 6[1] + S[O] = -2 + 1 = - 1
h[2] = -2h[l] + 6[2] + S[l] = -2( - 1) = 2
h[3] = -2h[2] + 6[3] + 6[2] = -2(2) = -2'

Hence, h[n] = 6[n] + ( - 1)"2"-'u[n - 11
Since h[nl has infinite terms, the system is an IIR system.
(c) h[nl = ih[n - 21 + 26[n] - 6[n - 21
Since the system is causal, h[- 21 = h[ - 11 = 0. Then

Hence, h[n] = 26[n]
Since h[nl has only one term, the system is a FIR system.

Supplementary Problems

2.46. Compute the convolution y(t ) = x( t * h(t ) of the following pair of signals:
-a <t <a , h(t) = I0 -a <t la
1
(a) X(I) =
otherwise otherwise
O<rsT O<r52T
otherwise ' otherwise

2a - It1 It( < 2a
Am. (a) YO)=
111 L 2a
r<O
O<t_<T
T<rs2T
2T<rs3T
3T<t