If we can multiply several z-transforms to get a higher-order system,we can also
factor z-transform polynomials to break down a large system into smaller sections.
Since a cascading system is equivalent to multiplying each individual system transfer
function,the factors of a higher-order polynomial,H(z),would represent component
systems that make up H(z) in a cascade connection. The concept of parallel and
cascade implementation will be further discussed in the realization of IIR filters in
Chapter 6.
Example 4.10: The following LTI system has the transfer function:
Hz 1 À 2z
À1
z
À3
:
This transfer function can be factored as
Hz 1 À z
À1
À
Á
1 À z
À1
À z
À2
À
Á
H
1
zH
2
z:
Thus the overall system H(z) can be realized as the cascade of the first-order
system H
1
z 1 À z
À1
and the second-order system H
2
z 1 À z
À1
À z
À2
.
4.3.2 Digital Filters
The general I/O difference equation of an FIR filter is given in (3.1.16). Taking the
z-transform of both sides,we have
Yz b
0
Xz b
1
z
À1
Xz Á Á Á b
LÀ1
z
ÀLÀ1
Xz
b
0
b
1
z
À1
Á Á Á b
LÀ1
z
ÀLÀ1
h
i
Xz:
4:3:7
Therefore the transfer function of the FIR filter is expressed as
Hz
Yz
Xz
b
0
b
1
z
À1
Á Á Á b
LÀ1
z
ÀLÀ1
LÀ1
l0
b
l
z
À1
:
4:3:8
The signal-flow diagram of the FIR filter is shown in Figure 3.6. FIR filters can be
implemented using the I/O difference equation given in (3.1.16),the transfer function
defined in (4.3.8),or the signal-flow diagram illustrated in Figure 3.6.
Similarly,taking the z-transform of both sides of the IIR filter defined in (3.2.18)
yields
Yz b
0
Xz b
1
z
À1
Xz Á Á Á b
LÀ1
z
ÀL1
Xz À a
1
z
À1
Yz À Á Á Á À a
M
z
ÀM
Yz
LÀ1
l0
b
l
z
Àl
2
3
Xz À
M
m1
a
m
z
Àm
2
3
Yz:
4:3:9
SYSTEMS CONCEPTS
143