Page 204 - Schaum's Outline of Theory and Problems of Signals and Systems

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

4.22. Find the inverse z-transform of
2

X( 2) can be rewritten as

lzl > 2
Since the ROC is JzJ > 2, x[n] is a right-sided sequence, and from Table 4-1 we have

Using the time-shifting property (4.181, we have
Z 1
2n-Lu[n - l] ~z-1

Thus, we conclude that
x[n] = 3(2)"-'u[n - 11

4.23. Find the inverse z-transform of
+
2 + z-~ 32-4
X(2) = 121 > 0
z2 + 4z + 3
We see that X(z) can be written as
X(z) = (2.7-I +z-~ + ~z-~)x,(z)

where

Thus, if

then by the linearity property (4.17) and the time-shifting property (4.18), we get

where

1 z 1 z
-
Then X,(z) = - - - - lz1 > 0
2z+1 2z+3
Since the ROC of X,(z) is Izl > 0, x,[n] is a right-sided sequence, and from Table 4-1 we get
x1[n] = ;[(-I)" - (-3)"]u[n]
Thus, from Eq. (4.84) we get
x[n] = [(-I)"-' - (-3)"-']u[n - I] + f [(-I)"-' - (-3)"-']u[n -31

+ f [( -I)"-' - (-3)"-']u[n - 51