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232 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
b0 = 1
B =
1.0000 2.8284 4.0000
1.0000 0.7071 0.2500
1.0000 -0.7071 0.2500
1.0000 -2.8284 4.0000
A =
1 0 0
1 0 0
1 0 0
1 0 0
The cascade form structure is shown in Figure 6.13c.
□ EXAMPLE 6.5 For the filter in Example 6.4, what would be the structure if we desire a cascade
form containing linear-phase components with real coefficients?
□
Solution
We are interested in cascade sections that have symmetry and real coefficients.
From the properties of linear-phase FIR filters (see Chapter 7), if such a filter
has an arbitrary zero at z = r̸ θ, then there must be 3 other zeros at (1/r)̸ θ,
r̸ − θ, and (1/r)̸ − θ to have real filter coefficients. We can now make use of
this property. First we will determine the zero locations of the given 8th-order
polynomial. Then we will group 4 zeros that satisfy this property to obtain
one (4th-order) linear-phase section. There are two such sections, which we will
connect in cascade.
MATLAB script:
>> b=[1,0,0,0,16+1/16,0,0,0,1]; broots=roots(b)
broots =
-1.4142 + 1.4142i
-1.4142 - 1.4142i
1.4142 + 1.4142i
1.4142 - 1.4142i
-0.3536 + 0.3536i
-0.3536 - 0.3536i
0.3536 + 0.3536i
0.3536 - 0.3536i
>> B1=real(poly([broots(1),broots(2),broots(5),broots(6)]))
B1 =
1.0000 3.5355 6.2500 3.5355 1.0000
>> B2=real(poly([broots(3),broots(4),broots(7),broots(8)]))
B2 =
1.0000 -3.5355 6.2500 -3.5355 1.0000
The structure is shown in Figure 6.14.
□
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