The z-Transform - Digital Signal Processing: Fundamentals and Applications - page 137

CHAPTER

5

The z-Transform

CHAPTER OUTLINE

5.1 Definition ..................................................................................................................................... 137
5.2 Properties of the z-Transform ........................................................................................................ 140
5.3 Inverse z-Transform...................................................................................................................... 144
5.3.1 Partial Fraction Expansion Using MATLAB...................................................................150
5.4 Solution of Difference Equations Using the z-Transform .................................................................. 152
5.5 Summary ..................................................................................................................................... 156

OBJECTIVES
This chapter introduces the z-transform and its properties; illustrates how to determine the inverse
z-transform using partial fraction expansion; and applies the z-transform to solve linear difference
equations.

5.1 DEFINITION
The z-transform is a very important tool in describing and analyzing digital systems. It also supports
the techniques for digital filter design and frequency analysis of digital signals. We begin with the
definition of the z-transform.
The z-transform of a causal sequence xðnÞ , designated by XðzÞ or ZðxðnÞÞ , is defined as
XðzÞ¼ZðxðnÞÞ ¼ N

n¼ 0 x n z n

(5.1)

¼ xð 0 Þz 0 þ xð 1 Þz 1 þ xð 2 Þz 2

þ/

where z is the complex variable. Here, the summation taken from n ¼ 0to n ¼ N is according to the
fact that for most situations, the digital signal xðnÞ is the causal sequence, that is, xðnÞ¼ 0 for n < 0.
Thus, the definition in Equation (5.1) is referred to as a one-sided z-transform or a unilateral transform .
In Equation (5.1) , all the values of z that make the summation exist form a region of convergence in the
z-transform domain, while all other values of z outside the region of convergence will cause the
summation to diverge. The region of convergence is defined based on the particular sequence xðnÞ
being applied. Note that we deal with the unilateral z-transform in this topic, and hence when