The z-transform of a sequence is defined as
Xz=∑n=-∞∞xnz-n
Xz
n
x
n
z
n
(1)
Sometimes this equation is referred to as the
bilateral z-transform. At times the z-transform is defined as
Xz=∑n=0∞xnz-n
X
z
n
0
x
n
z
n
(2)
which is known as the
unilateral z-transform.
There is a close relationship between the z-transform and the
Fourier transform of a discrete time signal,
which is defined as
Xⅇⅈω=∑n=-∞∞xnⅇ-ⅈωn
X
ω
n
x
n
ω
n
(3)
Notice that that when the
z-n
z
n
is replaced with
ⅇ-ⅈωn
ω
n
the z-transform reduces to the Fourier Transform. When the
Fourier Transform exists,
z=ⅇⅈω
z
ω
, which is to have the magnitude of
zz equal to unity.
In order to get further insight into the relationship between
the Fourier Transform and the Z-Transform it is useful to look
at the complex plane or z-plane. Take a look at
the complex plane:
The Z-plane is a complex plane with an imaginary and real axis
referring to the complex-valued variable
zz. The position on the complex
plane is given by
rⅇⅈω
r
ω
, and the angle from the positive, real axis around the plane is
denoted by ωω.
XzXz is defined
everywhere on this plane. Xⅇⅈω
Xω
on the other
hand is defined only where
|z|=1
z1
,
which is referred to as the unit circle. So for example,
ω=1ω1
at
z=1z1
and
ω=πω
at
z=-1z-1.
This is useful because, by representing the Fourier transform
as the z-transform on the unit circle, the periodicity of
Fourier transform is easily seen.
The region of convergence, known as the ROC, is
important to understand because it defines the region where
the z-transform exists. The ROC for a given
xn
x
n
, is defined as the range of
z
z
for which the z-transform converges. Since the z-transform is
a power series, it converges when
xnz-n
x
n
z
n
is absolutely summable. Stated differently,
∑n=-∞∞|xnz-n|<∞
n
x
n
z
n
(4)
must be satisfied for convergence. This is best illustrated
by looking at the different ROC's of the z-transforms of
αnun
α
n
u
n
and
αnun-1
α
n
u
n
1
.
For
xn=αnun
x
n
α
n
u
n
(5)
Xz=∑n=-∞∞xnz-n=∑n=-∞∞αnunz-n=∑n=0∞αnz-n=∑n=0∞αz-1n
Xz
n
x
n
z
n
n
α
n
u
n
z
n
n
0
α
n
z
n
n
0
α
z
1
n
(6)
This sequence is an example of a right-sided exponential
sequence because it is nonzero for
n≥0
n
0
.
It only converges when
|αz-1|<1
α
z
1
.
When it converges,
Xz=11-αz-1=zz-α
Xz
1
1
α
z
z
z
α
(7)
If
|αz-1|≥1
α
z
1
,
then the series,
∑n=0∞αz-1n
n
0
α
z
n
does not converge. Thus the ROC is the range of values where
|αz-1|<1
α
z
1
(8)
or, equivalently,
|z|>|α|
z
α
(9)
For
xn=-αnu-n-1
x
n
α
n
u
n
1
(10)
Xz=∑n=-∞∞xnz-n=∑n=-∞∞-αnu-n-1z-n=-∑n=-∞-1αnz-n=-∑n=-∞-1α-1z-n=-∑n=1∞α-1zn=1-∑n=0∞α-1zn
Xz
n
x
n
z
n
n
α
n
u
-n
1
z
n
n
-1
α
n
z
n
n
-1
α
-1
z
n
n
1
α
-1
z
n
1
n
0
α
-1
z
n
(11)
The ROC in this case is the range of values where
|α-1z|<1
α
-1
z
1
(12)
or, equivalently,
|z|<|α|
z
α
(13)
If the ROC is satisfied, then
Xz=1-11-α-1z=zz-α
Xz
1
1
1
α
-1
z
z
z
α
(14)
"My introduction to signal processing course at Rice University."