Page 74 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 21 LINEAR TIME-INVARIANT SYSTEMS

I. Commutative:
x[n] * h[n] = h[n] * x[n] (2.36)
2. Associative:
{~[n] h,[n]}* h2[nl =+I *(h,[nl *h,[nIl (2.37)
*
3. Distributive:
x[n] *{h,[n]] +h,[n]] =+I * h,[nl +xb] *h,[nl (2.38)

E. Convolution Sum Operation:
Again, applying the commutative property (2.36) of the convolution sum to Eq. (2.351,
we obtain

which may at times be easier to evaluate than Eq. (2.35). Similar to the continuous-time
case, the convolution sum [Eq. (2.391 operation involves the following four steps:

1. The impulse response h[k] is time-reversed (that is, reflected about the origin) to
obtain h[-k] and then shifted by n to form h[n - k] = h[-(k - n)] which is a
function of k with parameter n.
2. Two sequences x[k] and h[n - k] are multiplied together for all values of k with n
fixed at some value.
3. The product x[k]h[n - k] is summed over all k to produce a single output sample
y[nI.
4. Steps 1 to 3 are repeated as n varies over -GO to GO to produce the entire output y[n].
Examples of the above convolution sum operation are given in Probs. 2.28 and 2.30.

F. Step Response:
The step response s[n] of a discrete-time LTI system with the impulse response h[n] is
readily obtained from Eq. (2.39) as

From Eq. (2.40) we have

h[n] = s[n] - s[n - l] (2.41)
Equations (2,401 and (2.41) are the discrete-time counterparts of Eqs. (2.12) and (2,131,
respectively.

2.7 PROPERTIES OF DISCRETE-TIME LTI SYSTEMS

A. Systems with or without Memory:
Since the output y[n] of a memoryless system depends on only the present input x[n],
then, if the system is also linear and time-invariant, this relationship can only be of the