The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 144

Table 5.2 z-Transform Properties
Property Time Domain z-Transform

Linearity

ax 1 ðnÞþbx 2 ðnÞ aZðx 1 ðnÞÞ þ bZðx 2 ðnÞÞ

Shift theorem

xðn mÞ z m XðzÞ

x 1 ðnÞx 2 ðnÞ¼ P k ¼0 x 1 ðn kÞx 2 ðkÞ

Linear convolution

X 1 ðzÞX 2 ðzÞ

The properties of the z-transform discussed in this section are listed in Table 5.2 .

5.3 INVERSE Z-TRANSFORM
The z-transform of the sequence xðnÞ and the inverse z-transform for the function XðzÞ are defined as,
respectively

XðzÞ¼ZðxðnÞÞ (5.7)

and

xðnÞ¼Z 1 ðXðzÞÞ (5.8)

where ZðÞ is the z-transform operator, and Z 1 ðÞ is the inverse z-transform operator.

The inverse of the z-transform may be obtained by at least three methods:

1. partial fraction expansion and lookup table;
2. power series expansion;
3. residue method.

The first method is widely utilized, and it is assumed that the reader is well familiar with the partial
fraction expansion method in learning Laplace transform. Therefore, we concentrate on the first
method in this topic. As for the power series expansion and residue methods, the interested reader is
referred to the textbook by Oppenheim and Schafer (1975). The key idea of the partial fraction
expansion is that if XðzÞ is a proper rational function of z , we can expand it to a sum of the first-order
factors or higher-order factors using the partial fraction expansion that can be inverted by inspecting
the z-transform table. The partial fraction expansion method is illustrated via the following examples.
(For simple z-transform functions, we can directly find the inverse z-transform using Table 5.1 . )

EXAMPLE 5.8
Find the inverse z-transform for each of the following functions:

4z
z 1

z
z 0:5

a. X ðzÞ¼2 þ

5z
ðz 1Þ 2

2z
ðz 0:5Þ 2

b. X ðzÞ¼