Digital Signal Processing Reference
In-Depth Information
0
05
⋅
a
G
=
10
(2.32)
pass
We are now ready to define a generalized transfer function for the Chebyshev
approximation function as shown below:
0
05
⋅
a
∏
(
10
)
⋅
(
B
)
pass
2
m
m
H
(
S
)
=
,
C
,
n
∏
(2.33a)
2
(
S
+
B
⋅
S
+
B
)
1
m
2
m
m
m
=
0
1
…
,
(
n
/
2
−
1
(
n
even)
∏
sinh(
D
)
⋅
(
B
)
2
m
m
H
(
S
)
=
,
C
,
n
∏
2
(2.33b)
(
S
+
sinh(
D
))
⋅
(
S
+
B
S
+
B
)
1
m
2
m
m
m
=
0
1
…
,
[(
n
−
1
)/
2
−
1
(
n
odd)
It is again time to consider some numerical examples before using WFilter to
determine the filter coefficients.
Example 2.4 Chebyshev Third-Order Normalized Transfer Function
Problem:
Determine the order, pole locations, and coefficients of the transfer
function for a Chebyshev filter to satisfy the following specifications:
a
pass
= −1 dB,
a
stop
= −22 dB, ω
pass
= 1 rad/sec, and ω
stop
= 2 rad/sec
Solution:
First, we determine the fundamental constants needed from (2.20),
(2.22), and (2.23):
ε = 0.508847
n
= 2.96 (3rd order)
D
= 0.475992
cosh(
D
) = 1.115439
sinh(
D
) = 0.494171
Next, we find the locations of the first-order pole and the complex pole in the
second quadrant from (2.24)-(2.27). A plot of the poles is shown in Figure 2.9:
(1st order) σ
R
= −0.494171 ω
R
= 0.0
φ
0
= 1π/6 σ
0
= −0.247085 ω
0
= +0.965999
Finally, we generate the transfer function from (2.28)-(2.33). The results
from WFilter are shown in Figure 2.10.
