CHAPTER - 3
3.
THE Z TRANSFORM
3.1.
INTRODUCTION
n this chapter, we review the z transform and its properties. We then use the z transform to define the
transfer function of a discrete-time system. At the end of the chapter we give two sets of problems;
theory problems on the z transform and a series of J-DSP computer exercises.
Discrete-time signals are described in the transform domain with the z transform and the discrete-
time Fourier transform (DTFT). These two transformations have similar roles as the Laplace transform
and the CFT for analog signals respectively.
The z transform is defined as
( )
( ).
n
n
X z
x n
z
∞
−
=−∞
=
∑
(3.1)
where
z
is a complex variable.
For a causal (right-handed) signal, i.e., if
x(n)=0
for
n<0
then the lower
index on the sum starts from 0. The z transform describes the signal in the complex z domain and for
finite-length causal signals the transform converges everywhere in
z
except for
z=0
. Note that a unique z
transform pair requires that the range of values of z for which the z transform exists, i.e. the region of
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convergence (ROC), is specified.
For causal signals the ROC extends outwards from the outermost pole
of the z domain function.
Example
3
.
1
: Consider the z transform of the unit impulse,
δ
(n)
( )
1
0
0
n
for n
elsewhere
δ
=
=
=
(3.2)
( ).
1
( )
1
n
n
n z
hence
n
ROC
z
δ
δ
∞
−
=−∞
=
↔
∀
∑
Example
3
.
2
: Consider a finite-length signal
( )
n
x
n
5
0
.
−
2
5
1
.
5
0
.
5
2
.
−
1
−
1
0
2
3
4
5
This signal is non-zero only between the discrete-time indexes –2 to 3.
Mathematically this signal can be
written in terms of unit impulses as follows
(
)
.5
(
2)
1.5
(
1)
2
(
)
(
1)
.5
(
2)
2.5
(
3)
x n
n
n
n
n
n
n
δ
δ
δ
δ
δ
δ
= −
+
+
+
+
−
−
+
−
−
−
The z transform of this finite-length non-causal signal is given by
2
1
2
3
( )
.5
1.5
2
.5
2.5
X
z
z
z
z
z
z
−
−
−
= −
+
+
−
+
−
The z transform converges for all values of z except z=0 and z=
∞
, i.e., ROC: z
≠
0, z
≠
∞
.