Introduction to Digital Signal and System Analysis - Tutorsindia

Introduction to Digital Signal and System Analysis
Z Domain Analysis
5 Z Domain Analysis
5.1 z-transform and inverse z-transform
In the previous chapters, time domain and frequency domain analysis have been introduced. In each of those domains
different insights of digital signals are revealed. It is useful to introduce another domain: the z domain. A digital time
signal can be transferred into z domain by the z-transform. The z-transform is defined as
X ( z)
=
∑ ∞
n=
0
x[
n]
z
−n
(5.1)
where z is a complex variable. The transform defined by Eq. (5.1) is a unilateral transform as defined on one side of the
axis 0 ≤ n < ∞ . In the transform, each sample x [n]
is multiplied by the complex variable , i.e.
n
z −
x[0]
z
0
x[1]
z
−1
x[2]
z
−2
x[3]
z
−3
...
...
↑
There is advantage in this unilateral transform definition as it can avoid mathematical inconvenience. One can shift the
signal of interest to obtain a required origin in its analysis, thus usually causing no trouble in applications.
The inverse z-transform can be found by
x[
n]
1
2π
j
n−1
= X ( z)
z dz
(5.2)
It involves contour integration, and further discussion is beyond the scope of this basic content. However, an alternative
approach is available using partial fractions together with z-transform formulas of basic functions. Table 5.1 lists the basic
properties of the z transform and Table 5.2 lists some basic z- transform pairs.
Example 5.1 Find the z transform for a signal and reconstruct a signal from its z-transform.
2 3
a) x[n]=[ 1 0.8 0.8 0.8 ... ] is shown in Figure 5.1(a), find the z-transform.
↑
Using the definition Eq.(5.1),
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