Digital Signal Processing Reference
In-Depth Information
X(z)
Σ
Σ
z
−
1
z
−
1
−
1
Σ
Σ
−
0.4
z
−
1
z
−
1
0.1
0.16
Σ
−
0.5
Y(z)
z
−
1
Σ
0.02
−
0.18
Figure 6.14
Cascade connection of an IIR filter.
One of the realizations used to implement this transfer function is shown in
Figure 6.14.
Instead of combining the factors
(z
0
.
1
)
and getting
(z
2
−
0
.
2
)
and
(z
+
−
0
.
1
z
−
0
.
02
)
, in the denominator of (6.22), we can combine
(z
−
0
.
2
)
and
(z
+
0
.
4
)
or
(z
0
.
4
)
to generate new second-order polynomials and
select many pole-zero pairs and order of second-order sections connected in
cascade, adding to the many possible realizations of (6.22) in the cascade form.
The cascade connection of second-order sections, each realized in direct form II,
has been a popular choice for a long time and was investigated in great detail,
until other structures became known for their better performance with respect to
finite wordlength effects and practical applications.
+
0
.
1
)
and
(z
+
Example 6.10: Parallel Form
The IIR transfer function can also be expanded as the sum of second-order
structures. It is decomposed into its partial fraction form, combining the terms
with complex conjugate poles together such that we have an expansion with real
coefficients only. We will choose the same example as (6.22) to illustrate this
form of realization.
One form of the partial fraction expansion of (6.22) is
1
.
1314
z
2
5
.
31165
z
z
1
.
111
z
z
5
.
21368
z
z
−
0
.
15947
z
H(z)
=
+
0
.
1
−
−
0
.
2
−
+
0
.
4
+
(6.26)
z
2
+
z
+
0
.
5
+
0
.
4
)
in different pairs to get the cor-
responding denominator polynomials, we get the following expressions for the
By combining
(z
+
0
.
1
)
,
(z
−
0
.
2
)
,
(z
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