Page 221 - Schaum's Outline of Theory and Problems of Signals and Systems

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

4.57. Consider a discrete-time LTI system whose system function H(z) is given by

(a) Find the step response s[n].
(b) Find the output y[n] to the input x[n] = nu[n].
Ans. (a) s[n] = [2 - (~)"]u[n]
(6) y[nl = 2[(4)" + n - llu[nl

4.58. Consider a causal discrete-time system whose output y[n] and input x[n] are related by
y[n] - :y[n - 11 + iy[n -21 =x[n]

(a) Find its system function H(z).
(b) Find its impulse response h[n].
z * 1
Am. (a) H(z)= , lzl> -
(z - $)(z - 3) 2
(b) h[n] = [3($)" - 2(f)"]u[n]

4.59. Using the unilateral z-transform, solve the following difference equations with the given initial
conditions.
(a) y[n] - 3y[n - 11 =x[n], with x[n] = 4u[n], y[- 11 = 1
(b) y[n] - 5y[n - 11 + 6y[n - 21 =x[n], with x[n] = u[n], y[- I] = 3, y[-2]= 2
Am. (a) y[n]=-2+9(3)",nr -1
(b) y[n] = 4 + 8(2In - Z(3ln, n 2 -2

4.60. Determine the initial and final values of x[n] for each of the following X(z):

z
(b) X(z)= , 121 > 1
2z2 - 3.2 + 1
Ans. (a) x[O]=2, x[w]= 0
(b) x[Ol = 0, x[wl= 1