Databases Reference
In-Depth Information
Dividing the numerator by the denominator we get the following:
az
−
1
a
2
z
−
2
+
+
···
1
a
z
−
z
z
−
a
a
a
a
2
z
−
1
a
2
z
−
1
−
Thus, the quotient is
∞
az
−
1
a
2
z
−
2
a
n
z
−
n
1
+
+
+··· =
n
=
0
We can easily see that the sequence for which
F
(
z
)
is the Z-transform is
a
n
u
f
n
=
[
n
]
12.9.4 Z-Transform Properties
Analogous to the continuous linear systems, we can define the transfer function of a discrete
linear system as a function of
z
that relates the Z-transform of the input to the Z-transform of
the output. Let
f
n
}
n
=−∞
g
n
}
n
=−∞
{
be the input to a discrete linear time-invariant system, and
{
be the output. If
F
(
z
)
is the Z-transform of the input sequence, and
G
(
z
)
is the Z-transform
of the output sequence, then these are related to each other by
G
(
z
)
=
H
(
z
)
F
(
z
)
(116)
and
H
is the transfer function of the discrete linear time-invariant system.
If the input sequence
(
z
)
f
n
}
n
=−∞
{
had a Z-transform of one, then
G
(
z
)
would be equal to
(
)
H
z
. It is an easy matter to find the requisite sequence:
1
n
∞
0
0 otherwise
=
f
n
z
−
n
F
(
z
)
=
=
1
⇒
f
n
=
(117)
n
=−∞
This particular sequence is called the
discrete delta function
. The response of the system to
the discrete delta function is called the impulse response of the system. Obviously, the transfer
function
H
(
z
)
is the Z-transform of the impulse response.
12.9.5 Discrete Convolution
In the continuous time case, the output of the linear time-invariant systemwas a convolution of
the input with the impulse response. Does the analogy hold in the discrete case? We can check
this out easily by explicitly writing out the Z-transforms in Equation (
116
). For simplicity
