The z-transform - Continuous and Discrete Time Signals and Systems - page 573

unit circle. Equation (13.8) is, therefore, not valid for the unit step function.
This may also be verified from Table 11.2, where the DTFT of the unit step
function is different from the value obtained by substituting z = e j Ω in its z-
transform. The DTFT of the unit step function in Table 11.2 contains the Dirac
delta functions, which makes the amplitude of the DTFT infinite at certain
frequencies. No Dirac delta functions exist in the z-transform of the unit step
function. Likewise, the ROCs for the z-transforms of the sine and cosine waves
do not contain the unit circle, and Eq. (13.8) is also not valid in these cases.

13.2.2 Region of convergence

As a side note to our discussion, we observe that the z-transform is guaranteed
to exist at all points within the ROC. For example, consider the causal sinusoidal
sequence x [ k ] = cos(0 . 2 π k ) u [ k ], whose z-transform is given in Table 13.1 as
follows:

− 1

1 − cos( Ω 0 ) z

X ( z ) =

− 2 ,

ROC: z > 1 ,

1 − cos( Ω 0 ) z

− 1 + z

with Ω 0 = 0 . 2 π . We are interested in calculating the values of its z-transform

at two points z 1 = 0 . 8 + j0 . 6. Since z 1 lies within the ROC,

z > 1, the value of the z-transform at z 1 is given by

= 2 + j0 . 6 and z 2

− 1

1 − cos(0 . 2 π ) z

X ( z ) = = 1 . 39 − j0 . 05 .

1 − cos(0 . 2 π ) z − 1 + z − 2

z = 2 + j0 . 6

However, the point z 2 = 0 . 8 + j0 . 6 lies outside the ROC, z > 1. Therefore,

the z-transform of the causal sinusoidal sequence cannot be computed for z 2 .In
the following, we list the important properties of the ROC for the z-transform.

(1) The ROC consists of a 2D plane of concentric circles of the form z > z 0
or z < z 0 . All entries in Table 13.1 have ROCs that are concentric circles.
(2) The ROC does not include any poles of the z-transform.

The poles of a z-transform are defined as the roots of its denominator poly-
nomial. Since the value of the z-transform is infinite at the location of a pole,
the ROC cannot include any pole. Property (2) can be verified for all entries in
Table 13.1. Consider, for example, the unit step function, which has a single
pole at z = 1. The ROC of the z-transform of the unit step function is given by
z > 1 and does not include its pole ( z = 1).

(3) The ROC of a right-hand-sided sequence ( x [ k ] = 0 for k < k 0 ) is defined
by the region outside a circle. In other words, the ROC of a right-hand-sided
sequence has the form z > z 0 .

Entries (3)-(12) in Table 13.1 are right-hand-sided sequences. Consequently, it
is observed that the ROC for all these sequences is of the form z > z 0 .