Digital Signal Processing Reference
In-Depth Information
In the case of the quadratic terms that are used to describe our coefficients,
the transformation process is shown in (6.23) to (6.25). Again,
G
represents the
gain adjustment necessary for each of the quadratic factors for the filter.
2
2
z
−
1
2
z
−
1
⎛
⎞
⎛
⎞
A
⋅
⎜
⎝
⋅
⎟
⎠
+
A
⋅
⎜
⎝
⋅
⎟
⎠
+
A
0
1
2
T
z
+
1
T
z
+
1
H
(
z
)
=
2
2
z
−
1
2
z
−
1
⎛
⎞
⎛
⎞
B
⋅
⎜
⎝
⋅
⎟
⎠
+
B
⋅
⎜
⎝
⋅
⎟
⎠
+
B
0
1
2
T
z
+
1
T
z
+
1
(6.23)
⎛
⎞
⎛
⎞
N
N
1
−
1
2
−
2
1
+
⎜
⎝
⎟
⎠
⋅
z
+
⎜
⎝
⎟
⎠
⋅
z
−
1
−
2
N
N
N
a
+
a
⋅
z
+
a
⋅
z
0
0
0
0
1
2
(6.24)
H
(
z
)
=
⋅
=
G
⋅
−
1
−
2
D
⎛
D
⎞
⎛
D
⎞
b
+
b
⋅
z
+
b
⋅
z
0
−
1
−
2
0
1
2
⎜
⎝
1
⎟
⎠
⎜
⎝
2
⎟
⎠
1
+
⋅
z
+
⋅
z
D
D
0
0
where
2
N
=
A
+
2
f
A
+
4
f
A
0
2
s
1
s
0
2
N
=
2
⋅
(
A
−
4
f
A
)
1
2
0
s
2
N
=
A
−
2
f
A
+
4
f
A
2
2
s
1
s
0
(6.25)
2
D
=
B
+
2
f
B
+
4
f
B
0
2
s
1
s
0
2
D
=
2
⋅
(
B
−
4
f
B
)
1
2
0
s
2
D
=
B
−
2
f
B
+
4
f
B
2
2
s
1
s
0
Example 6.6 Butterworth Bilinear Transform Filter Design
Problem:
Determine the digital filter to meet the specifications given in
Example 6.5 using the bilinear transformation.
Solution:
By entering the attenuations and the prewarped
analog
frequencies
in WFilter for analog filter design, we determine the following analog transfer
function:
7
7
8877
⋅
10
H
(
s
)
=
2
4
7
s
+
1
.
2560
⋅
10
s
+
7
8877
⋅
10
