With the DTFT, we have a complexvalued function of a realvalued variable
ω
ω
(and 2
π
periodic).
The Ztransform is a complexvalued function of a complex valued variable z.
With the Fourier transform, we had a complexvalued
function of a purely imaginary
variable,
Fiω
F
ω
. This was something we could envision with two
2dimensional plots (real and imaginary parts or magnitude and
phase). However, with Z, we have a complexvalued
function of a complex variable.
In order to examine the magnitude and phase or real and
imaginary parts of this function, we must examine
3dimensional surface plots of each component.
Consider the ztransform given by H(z)=zH(z)=z, as illustrated below.
The corresponding DTFT has magnitude and phase given below.
While these are legitimate ways of looking at a signal in the
Z domain, it is quite difficult to draw and/or analyze.
For this reason, a simpler method has been developed.
Although it will not be discussed in detail here, the method
of
Poles and Zeros
is much easier to understand and is the way both the Z
transform and its continuoustime counterpart the
Laplacetransform are
represented graphically.
What could the system H be doing? It is a perfect allpass, linearphase
system. But what does this mean?
Suppose h[n]=δ[nn0]h[n]=δ[nn0]. Then
H
(
z
)
=
∑
n
=

∞
∞
h
[
n
]
z

n
=
∑
n
=

∞
∞
δ
[
n

n
0
]
z

n
=
z

n
0
.
H
(
z
)
=
∑
n
=

∞
∞
h
[
n
]
z

n
=
∑
n
=

∞
∞
δ
[
n

n
0
]
z

n
=
z

n
0
.
(7)Thus, H(z)=zn0H(z)=zn0 is the zztransform of a system that simply delays the input by n0n0. H(z)H(z) is the zztransform of a unitdelay.
Now consider x[n]=αnu[n]x[n]=αnu[n]
X
(
z
)
=
∑
n
=

∞
∞
x
[
n
]
z

n
=
∑
n
=
0
∞
α
n
z

n
=
∑
n
=
0
∞
(
α
z
)
n
=
1
1

α
z
(
i
f

α
z

<
1
)
(
G
e
o
m
e
t
r
i
c
S
e
r
i
e
s
)
=
z
z

α
.
X
(
z
)
=
∑
n
=

∞
∞
x
[
n
]
z

n
=
∑
n
=
0
∞
α
n
z

n
=
∑
n
=
0
∞
(
α
z
)
n
=
1
1

α
z
(
i
f

α
z

<
1
)
(
G
e
o
m
e
t
r
i
c
S
e
r
i
e
s
)
=
z
z

α
.
(8)What if αz≥1αz≥1? Then ∑n=0∞(αz)n∑n=0∞(αz)n does not converge! Therefore, whenever we compute a zztranform, we must also specify the set of zz's for which the zztransform exists. This is called the regionofconvergenceregionofconvergence(ROC).
Matlab has two functions,
ztrans
and
iztrans
, that are both part of the
symbolic toolbox, and will find the Z and inverse
Z transforms respectively. This method is generally
preferred for more complicated functions. Simpler and more
contrived functions are usually found easily enough by using
tables.
"My introduction to signal processing course at Rice University."