Page 209 - Schaum's Outline of Theory and Problems of Signals and Systems

198 THE Z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

4.29. The output y[n] of a discrete-time LTI system is found to be 2(f )"u[n] when the input
x[n] is u[n].

(a) Find the impulse response h[n] of the system.
(b) Find the output y[n] when the input x[n] is (;)"u[n].

Hence, the system function H(z) is

Using partial-fraction expansion, we have

where

Thus,

Taking the inverse z-transform of H(z), we obtain

h[n] = 66[n] - 4(;)'u[n]

Then,

Again by partial-fraction expansion we have

2(z - 1) 2(z - 1)
where C, = ,I =-6 c2 =
2-7 r-L/2
Thus,

Taking the inverse z-transform of Y(z), we obtain

y[n] = [-6(;)' + 8($)'lu[n]