Digital Signal Processing Reference
In-Depth Information
Hence,
z
[f[n]g[n]] Z
z
[f[n]]
z
[g[n]].
(11.9)
The
z
-transform of a product of two functions is
not
equal to the product of the
z
-transforms of the functions.
In this section, the unilateral and bilateral
z
-transforms are defined. These
transforms are a series in the variable
z
; however, we will see that the series for
many useful signals can be expressed in closed form. The complex inversion integral
for the inverse
z
-transform is also given, but we generally use tables for finding in-
verse transforms. The linearity properties of the
z
-transform are derived in this sec-
tion. In the remainder of the chapter, we develop
z
-transform analysis from the
definitions given here.
11.2
EXAMPLES
In this section, we introduce
z
-transform system analysis with a simple application.
First, two examples of derivations of
z
-transforms are presented. Next, the
z
-transform
is used to find the step response of a first-order digital filter. The
z
-transform is then
developed in more detail in the sections that follow.
Before presenting the first example, we consider the convergent power series from
Appendix C:
q
1
1 - a
;
+
Á
=
a
n
= 1 + a + a
2
ƒ
a
ƒ
6 1.
(11.10)
a
n= 0
Any function in Appendix C can be verified by dividing the numerator by the de-
nominator; for (11.10), this division yields
+
Á
1 + a + a
2
+ a
3
1 - a
1
1
-
a
a
a
-
a
2
(11.11)
a
2
a
2
- a
3
a
3
Á .
The series of (11.10) is useful in expressing certain
z
-transforms in closed form
(not as a series), as we illustrate subsequently. We prefer to express
z
-transforms
in closed form because of the resulting simplifications in manipulating these
transforms.
