The z-transform - Continuous and Discrete Time Signals and Systems - page 568

Example 13.1 derives the bilateral z-transform of the exponential sequence
x [ k ] = α k u [ k ]:

1
1 − α z − 1 ,

←→

α k u [ k ] with ROC z > α.

Since no restriction is imposed on the magnitude of α , the bilateral
z-transform of the exponential sequence exists for all values of α within the
specified ROC. Recall that the DTFT of an exponential sequence exists only
for α< 1. For α ≥ 1, the exponential sequence is not summable and its DTFT
does not exist. This is an important distinction between the DTFT and the bilat-
eral z-transform. While the DTFT exists for a limited number of absolutely
summable sequences, no such restrictions exist for the z-transform. By associ-
ating an ROC with the bilateral z-transform, we can evaluate the z-transform
for a much larger set of sequences.

Example 13.2
Calculate the bilateral z-transform of the left-hand-sided exponential sequence
x [ k ] =−α k u [ − k − 1].

Solution
For the DT sequence x [ k ] =−α k u [ − k − 1], Eq. (13.4) reduces to

∞

k =−∞ ( α z

− 1

−α k u [ − k − 1] z

− k − 1 ) k .

X ( z ) = =−

k =−∞

To make the limits of summation positive, we substitute m =− k in the above

equation to obtain

−

− 1 z
1 − α

α

∞

− 1 z < 1

α

− 1 z ) m

X ( z ) =− ( α =

− 1 z

undefined elsewhere ,

m = 1

which simplifies to

1
1 − α z − 1

z < α

X ( z ) =

undefined elsewhere .

The DT sequence x [ k ] =−α k u [ − k − 1] and the ROC associated with its
z-transform are illustrated in Fig. 13.2.

In Examples 13.1 and 13.2, we have proved the following z-transform pairs:

1
1 − α z − 1 ,

←→

α k u [ k ] with ROC z > α,