Page 77 - Schaum's Outline of Theory and Problems of Signals and Systems

66 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

special case when N = 0, from Eq. (2.53) we have

which is a nonrecursice equation since previous output values are not required to compute
the present output. Thus, in this case, auxiliary conditions are not needed to determine
Y ~ I .

C. Impulse Response:
Unlike the continuous-time case, the impulse response h[n] of a discrete-time LTI
system described by Eq. (2.53) or, equivalently, by Eq. (2.54) can be determined easily as

For the system described by Eq. (2.55) the impulse response h[n] is given by

Note that the impulse response for this system has finite terms; that is, it is nonzero for
only a finite time duration. Because of this property, the system specified by Eq. (2.55) is
known as a finite impulse response (FIR) system. On the other hand, a system whose
impulse response is nonzero for an infinite time duration is said to be an infinite impulse
response (IIR) system. Examples of finding impulse responses are given in Probs. 2.44 and
2.45. In Chap. 4, we will find the impulse response by using transform techniques.

Solved Problems

RESPONSES OF A CONTINUOUS-TIME LTI SYSTEM AND CONVOLUTION

2.1. Verify Eqs. (2.7) and (2.8), that is,

(a) x(t)*h(t)=h(t)*x(t)
(b) (x(t)* h,(t)J* h,(t) =x(t)*(h,(t)* h,(t)J
(a) By definition (2.6)

By changing the variable t - T = A, we have