The z-transform - Continuous and Discrete Time Signals and Systems - page 571

In the above transform pair, the ROC α z

− 1 < 1 is equivalent to z >α and
consists of the region outside the circle of radius z =α in the complex z-
plane. Example 13.1 derives the bilateral z-transform for the function x 3 [ k ] =
α k u [ k ]. Since the function is causal, the bilateral and unilateral z-transforms
are identical.
(iv) By definition,

∞ ∞

k α k u [ k ] z

− k − 1 ) k .

X ( z ) = = k ( α z

k = 0 k = 0

Using the following result:
∞

r
(1 − r ) 2 ,

kr k

= provided r < 1 ,

k = 0

the above summation reduces to

− 1
(1 − α z − 1 ) 2 ,

α z

− 1 < 1 .

X ( z ) = ROC: α z

The z-transform pair for a time-rising, complex exponential is given by

− 1

α z α z

( z − α ) 2 ,

←→

k α k u [ k ]

or ROC: z > α.

(1 − α z − 1 ) 2

≤ ≤

(v) Since the input sequence x 5 [ k ] is zero outside the range 0 k 5,

Eq. (13.4) reduces to

∞

− k = x [0] + x [1] z − 1 + x [2] z − 2 + x [3] z − 3 + x [4] z − 4 + x [5] z − 5 .

X ( z ) = x [ k ] z

k = 0

Substituting the values of x 5 [ k ] for the range 0 ≤ k

≤ 5, we obtain

− 1 + 2 z − 2 + 2 z − 5

X ( z ) = 1 + z ROC: entire z-plane, except z = 0 .

For finite-duration sequences, the ROC is always the entire z-plane except for
the possible exclusion of z

= 0 and z =∞ .

13.2.1 Relationship between the DTFT and the z-transform

Comparing Eq. (13.2) with Eq. (13.4), the DTFT can be expressed in terms of
the bilateral z-transform as follows:

k =−∞ x [ k ] z

∞

− k

X ( Ω ) = = X ( z )

z = e j Ω .

(13.8)

Since, for causal functions, the bilateral and unilateral z-transforms are the
same, Eq. (13.8) is also valid for the unilateral z-transform for causal functions.
Equation (13.8) shows that the DTFT is a special case of the z-transform
with z = e j Ω . The equality z = e j Ω corresponds to the circle of unit radius
( z = 1) in the complex z-plane. Equation (13.8) therefore implies that the