Page 214 - Schaum's Outline of Theory and Problems of Signals and Systems

CHAP. 41 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS

4.36. Verify Eqs. (4.50) and (4.51), that is, for m 2 0,

(a) x[n - m] -z-"x,(z) + Z-~+'X[- 11 + Z-"" 'x[-2] + a - +x[-m]

(a) By definition (4.49) with m 2 0 and using the change in variable k = n - m, we have

(b) With m 10
m m
8,{x[n + m]) = C x[n + m]z-" = C x[k]~-(~-~)
n =O k=m

4.37. Using the unilateral z-transform, redo Prob. 2.42.
The system is described by
-
~[n] ay[n - 11 =x[n]
with y[ - 11 = y - , and x[n 1 = Kbnu[n]. Let
~["l *YI(z)
Then from Eq. (4.50)

~[n
- 11 -z-~Y,(z) +y[-I] =Z-'&(Z) +y-I
From Table 4-1 we have
Z
x[n] -X,(Z) =K- Izl> lbl
t-b
Taking the unilateral z-transform of Eq. (4.931, we obtain

&(z) - a{z-'~(z) + y-J = KZ
z-b