Databases Reference
In-Depth Information
The theorem is easy to prove. Suppose we have a sequence
{
f
n
}
with Z-transform
F
(
z
)
.
Let us look at the Z-transform of the sequence
{
f
n
−
n
0
}
:
∞
f
n
−
n
0
z
−
n
Z
[{
f
n
−
n
0
}] =
(120)
n
=−∞
∞
f
m
z
−
m
−
n
0
=
(121)
m
=−∞
∞
z
−
n
0
f
m
z
−
m
=
(122)
m
=−∞
z
−
n
0
F
=
(
)
(123)
z
Assuming
G
(
z
)
is the Z-transform of
{
g
n
}
and
F
(
z
)
is the Z-transform of
{
f
n
}
, we can take
the Z-transform of both sides of the difference Equation (
119
):
a
1
z
−
1
F
a
2
z
−
2
F
b
1
z
−
1
G
b
2
z
−
2
G
G
(
z
)
=
a
0
F
(
z
)
+
(
z
)
+
(
z
)
+
(
z
)
+
(
z
)
(124)
From this we obtain the relationship between
G
(
z
)
and
F
(
z
)
as
a
1
z
−
1
a
2
z
−
2
a
0
+
+
G
(
z
)
=
b
2
z
−
2
F
(
z
)
(125)
1
−
b
1
z
−
1
−
By definition the transfer function
H
(
z
)
is therefore
G
(
z
)
H
(
z
)
=
(126)
F
(
z
)
a
1
z
−
1
a
2
z
−
2
a
0
+
+
=
(127)
b
1
z
−
1
b
2
z
−
2
1
−
−
12.10 Summary
In this chapter we have reviewed some of the mathematical tools we will be using throughout
the remainder of this topic. We started with a review of vector space concepts, followed by a
look at a number of ways we can represent a signal, including the Fourier series, the Fourier
transform, the discrete Fourier series, the discrete Fourier transform, and the Z-transform. We
also looked at the operation of sampling and the conditions necessary for the recovery of the
continuous representation of the signal from its samples.
Further Reading
1.
There are a large number of topics that provide a much more detailed look at the concepts
described in this chapter. A nice one is
Signal Processing and Linear Systems
,byB.P.
Lathi [
275
].
