Page 115 - Schaum's Outline of Theory and Problems of Signals and Systems

1 04 LINEAR TIME-INVARIANT SYSTEMS [CHAP. 2

2.44. Consider the discrete-time system in Prob. 2.43 for an initially at rest condition.
(a) Find in impulse response h[n] of the system.
(b) Find the step response s[n] of the system.
(c) Find the impulse response h[n] from the result of part (b).

(a) Setting K = 1 and y[- 11 = a = 0 in Eq. (2.166), we obtain

(b) Setting K = 1, b = 1, and y[- 1] = y-, = 0 in Eq. (2.161), we obtain

(c) From Eqs. (2.41) and (2.168) the impulse response h[n] is given by

When n = 0.

When n r 1,

Thus, h[n] = anu[n]

which is the same as Eq. (2.167).

2.45. Find the impulse response h[n] for each of the causal LTI discrete-time systems
satisfying the following difference equations and indicate whether each system is a FIR
or an IIR system.

(a) y[n] = x[n] - 2x[n - 21 + x[n - 31
(b) y[n] + 2y[n - 11 = x[n] + x[n - 11
(c) y[n] - ty[n - 21 = 2x[n] -x[n - 21

(a) By definition (2.56)

Since h[n] has only four terms, the system is a FIR system.