Page 219 - Schaum's Outline of Theory and Problems of Signals and Systems

208 THE z-TRANSFORM AND DISCRETE-TIME LTI SYSTEMS [CHAP. 4

4.49. Using the relation
Z
anu[n] H - Izl> la1
z-a
find the z-transform of the following x[n]:
(a) x[n] = nun-'u[n]
(b) x[n] = n(n - l)a"-2u[n]
(c) x[nl = n(n - 1). . . (n - k + l)~"-~u[n]
Hint: Differentiate both sides of the given relation consecutively with respect to a.

4.50. Using the z-transform, verify Eqs. (2.130) and (2.131) in Prob. 2.27, that is,

Hint: Use Eq. (4.26) of the z-transform and transform pairs 1 and 4 from Table 4-1.

4.51. Using the z-transform, redo Prob. 2.47.
Hint: Use Eq. (4.26) and Table 4-1.

4.52. Find the inverse z-transform of
X( z) = ea/' 1.4 > 0

Hint: Use the power series expansion of the exponential function er.

Am.

4.53. Using the method of long division, find the inverse z-transform of the following X(z):