Digital Signal Processing Reference
In-Depth Information
Let us illustrate the decomposition of an IIR filter as the product of second-
order functions; consider the transfer function
0
.
5
+
0
.
2
z
−
1
+
0
.
3
z
−
2
+
0
.
1
z
−
4
H(z)
=
0
.
3667
z
−
5
(6.63)
The MATLAB program used to obtain the factorized form to realize the cas-
cade structure for an IIR filter is
1
.
965
−
3
.
202
z
−
1
+
4
.
435
z
−
2
−
3
.
14
z
−
3
+
1
.
591
z
−
4
−
num=[0.5 0.2 0.3 0.0 0.1]
den=[1.965 -3.202 4.435 -3.14 1.591 -0.3667];
[z,p,k]=tf2zp(num,den);
sos=zp2sos(z,p,k)
sos =
0
0.2545
0
1.0000
-0.4166
0
1.0000
0.8204
0.6247
1.0000
-0.6766
0.5094
1.0000
-0.4204
0.3201
1.0000
-0.5363
0.8794
Using the entries in this
sos
matrix, we write the factorized form of
H(z)
as
follows:
0
.
2545
z
−
1
1
−
0
.
4166
z
−
1
1
+
0
.
8204
z
−
1
+
0
.
5094
z
−
2
+
0
.
6247
z
−
2
1
−
0
.
6766
z
−
1
1
−
0
.
4204
z
−
1
0
.
8794
z
−
2
+
0
.
3201
z
−
2
×
(6.64)
1
−
0
.
5363
z
−
1
+
Note that the numerator in this expression seems to be a fifth-order polynomial
in inverse powers of
z
whereas the numerator of the transfer function (6.63)
is a fourth-order polynomial. But the factorization of a polynomial is carried
out when it is expressed in positive powers of
z
since the polynomials are of
the form
(z
z
i
)
,where
z
i
are the zeros. So when the preceding factorized
form is converted to the ratio of polynomials in positive powers of
z
,wegeta
fourth-order numerator polynomial and the fifth-order denominator:
0
.
2545
z
−
z
2
z
2
+
0
.
8204
z
+
0
.
6247
−
0
.
4204
z
+
0
.
3201
−
0
.
4166
z
2
−
0
.
6766
z
+
0
.
5094
z
2
−
0
.
5363
z
+
0
.
8794
0
.
5
z
4
+
0
.
2
z
3
+
0
.
3
z
2
+
0
.
1
=
1
.
965
z
5
−
3
.
202
z
4
+
4
.
435
z
3
−
3
.
14
z
2
+
1
.
591
z
−
0
.
3667
This agrees with the result of expressing
H(z)
as the ratio of a fourth-order
numerator polynomial and a fifth-order denominator polynomial in positive pow-
ers of
z
. So care is to be taken to express the transfer function in positive powers
of
z
and then check the results after constructing the factorized form, because the
function
zp2sos
works only if the zeros are inside the unit circle of the
z
plane.
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