The z-transform - Continuous and Discrete Time Signals and Systems - page 574

(4) The ROC of a left-hand-sided sequence ( x [ k ] = 0 for k > k 0 ) is defined
by the inside region of a circle. Mathematically, this implies that the ROC
of a left-sided sequence has the form z < z 0 .

In Example 13.2, we computed the ROC for the left-hand-sided exponential
sequence x [ k ] =−α k u [ − k − 1] as z <α , which satisfies Property (4).

(5) The ROC of a double-sided (or bilateral) sequence, which extends to infinite
values of k in both directions, is confined to a ring with a finite area and
has the form z 1 < z < z 2 .

x [ k ] = β k u [ k ] − α k u [ − k −

An example of a double-sided sequence is

1]. By applying the linearity property, which is formally derived in

Section 13.4.1, it is observed that the z-transform is given by

1
1 − α z − 1

1
1 − β z − 1 ,

←→

β k u [ k ] − α k u [ − k − 1]

+ ROC: β< z <α,

which satisfies Property (5).

(6) The ROC of a finite-length sequence ( x [ k ] = 0 for k < k 1 , k > k 2 ) is the
entire z-plane except for the possible exclusion of the points z = 0 and

z =∞ .

As an example of Property (6), we consider entries (1) and (2) of Table 13.1.
Also, sequence x 5 [ k ] defined in Example 13.3 is a finite-length sequence. In
such cases, we note that the ROC consists of the entire z-plane except for the
possible exclusion of z = 0 and z =∞ .

13.3 Inverse z-tran sform

Evaluating the inverse of z-transform is an important step in the analysis of
LTID systems. There are four commonly used methods to evaluate the inverse
z-transform:

(i) table look-up method;
(ii) inversion formula method;
(iii) partial fraction expansion method;
(iv) power series method.

Evaluating the inverse z-transform using the inversion formula (method (ii))
involves contour integration, which is fairly complex and beyond the scope of
the text. In this section, we cover the remaining three methods in more detail.

13.3.1 Table look-up method

In this method, the z-transform function X ( z ) is matched with one of the entries
in Table 13.1. As the transform pairs are unique, the inverse transform is readily
obtained from the time-domain entry. For example, if the inverse z-transform