Page 110 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

This can be true only if
h[n] = 0 n<O
Now if h[n] = 0 for n < 0, then Eq. (2.139) becomes
m

which indicates that the value of the output y[n] depends on only the past and the present
input values.

2.36. Consider a discrete-time LTI system whose input x[n] and output y[n] are related by

Is the system causal?
By definition (2.30) and Eq. (1.48) the impulse response h[n] of the system is given by

k= - x k= - x k= -m
By changing the variable k + 1 = m and by Eq. (1.50) we obtain
n+ 1
h[n] = 2-("+') x S[m] = 2-("+"u[n + 1] ( 2.140)
,,,= -m
From Eq. (2.140) we have h[- 11 = u[O] = 1 + 0. Thus, the system is not causal.

2.37. Verify the BIBO stability condition [Eq. (2.49)] for discrete-time LTI systems.
Assume that the input x[n] of a discrete-time LTl system is bounded, that is,
Ix[n]l l kl all n (2.141)
Then, using Eq. (2.351, we have

Since lxIn - k)l i k, from Eq. (2.141). Therefore, if the impulse response is absolutely
summable, that is,

we have
ly[n]l~k,K=k~
<m
and the system is BIB0 stable.

2.38. Consider a discrete-time LTI system with impulse response h[n] given by

(a) Is this system causal?
(b) Is this system BIBO stable?