m22 - The Z-Transform Transfer Function
1.1
2006/08/01 13:35:29.477 GMT-5
2006/09/17 13:22:45.260 GMT-5
C.
Sidney
Burrus
csb@rice.edu
C.
Sidney
Burrus
csb@rice.edu
Doug
Kochelek
kochelek@rice.edu
linear system
transfer function
z transform
The transfer function for a linear, time invariant discrete time system is the z transform of the impulse response. It is also the ratio of the z transform of the output over the z transform of the input.
The Z-Transform Transfer Function
Although the time-domain convolution is the most basic relationship of the
input to the output for linear systems, the z-transform is a close second in
importance. It gives different insight and a different set of tools for
analysis and design of linear time-invariant discrete-time systems.
If our system in linear and time-invariant, we have seen that its output is
given by convolution.
y
(
n
)
=
∑
m
=
−
∞
∞
h
(
n
−
m
)
x
(
m
)
Assuming that
h
(
n
)
is such that the summation converges properly, we can calculate the output to
an input that we already know has a special relation with discrete-time
transforms. Let
x
(
n
)
=
z
n
which gives
y
(
n
)
=
∑
m
=
−
∞
∞
h
(
n
−
m
)
z
m
With the change of variables of
k
=
n
−
m
,
we have
y
(
n
)
=
∑
k
=
−
∞
∞
h
(
k
)
z
n
−
k
=
[
∑
k
=
−
∞
∞
h
(
k
)
z
−
k
]
z
n
or
y
(
n
)
=
H
(
z
)
z
n
We have the remarkable result that for an input of
x
(
n
)
=
z
n
,
we get an output of exactly the same form but multiplied by a constant that
depends on
z
and this constant is the z-transform of the impulse response of the system. In
other words, if the system is thought of as a matrix or operator,
z
n
is analogous to an eigenvector of the system and
H
(
z
)
is analogous to the corresponding eigenvalue.
We also know from the properties of the z-transform that convolution in the
n
domain corresponds to multiplication in the
z
domain. This means that the z-transforms of
x
(
n
)
and
y
(
n
)
are related by the simple equation
Y
(
z
)
=
H
(
z
)
X
(
z
)
The z-transform decomposes
x
(
n
)
into its various components along
z
n
which passing through the system simply multiplies that value time
H
(
z
)
and the inverse z-transform recombines the components to give the output. This
explains why the z-transform is such a powerful operation in linear
discrete-time system theory. Its kernel is the eigenvector of these systems.
The z-transform of the impulse response of a system is called its transfer
function (it transfers the input to the output) and multiplying it times the
z-transform of the input gives the z-transform of the output for any system
and signal where there is a common region of convergence for the transforms.