Page 75 - Schaum's Outline of Theory and Problems of Signals and Systems

LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

form
Y[.] =Kx[n] (2.42)

where K is a (gain) constant. Thus, the corresponding impulse response is simply
h[n] = K6[n] (2.43)
Therefore, if h[n,] # 0 for n, # 0, the discrete-time LTI system has memory.

B. Causality:
Similar to the continuous-time case, the causality condition for a discrete-time LTI
system is

Applying the causality condition (2.44) to Eq. (2.391, the output of a causal discrete-time
LTI system is expressed as
72
Y[.] = C h[k]x[n - k] (2.45)
k =O
Alternatively, applying the causality condition (2.44) to Eq. (Z..V), we have

Equation (2.46) shows that the only values of the input x[n] used to evaluate the output
n.
y[n] are those for k I
As in the continuous-time case, we say that any sequence x[n] is called causal if

and is called anticausal if

Then, when the input x[n] is causal, the output y[n] of a causal discrete-time LTI system
is given by

C. Stability:
It can be shown (Prob. 2.37) that a discrete-time LTI system is BIB0 stable if its
impulse response is absolutely summable, that is,

2.8 EIGENFUNCTIONS OF DISCRETE-TIME LTI SYSTEMS

In Chap. 1 (Prob. 1.45) we saw that the eigenfunctions of discrete-time LTI systems
represented by T are the complex exponentials zn, with z a complex variable. That is,
T(zn) = Azn (2.50)