Digital Signal Processing Reference
In-Depth Information
Example 2.5 Chebyshev Fourth-Order Normalized Transfer Function
Problem:
Determine the order, pole locations, and transfer function
coefficients for a Chebyshev filter to satisfy the following specifications:
a
pass
= −1 dB,
a
stop
= −33 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
= 3.92 (4th order)
D
= 0.356994
cosh(
D
) = 1.064402
sinh(
D
) = 0.364625
Next, we find the locations of the two complex poles in the second quadrant
from (2.24)-(2.27). A plot of the poles is shown in Figure 2.11.
θ
0
= 1π/8
σ
0
= −0.139536
ω
0
= +0.983379
θ
1
= 3π/8
σ
1
= −0.336870
ω
1
= +0.407329
Finally, we generate the transfer function from (2.28)-(2.33). Note that in this
even-order case, the gain constant of 0.891251 is included. The results from
WFilter for this Chebyshev specification are shown in Figure 2.12.
0
89125
⋅
0
98650
⋅
0
27940
H
C
(
S
)
=
,
4
2
2
(
S
+
0
27907
⋅
S
+
0
98650
)
⋅
(
S
+
0
67374
⋅
S
+
0
27940
)
Figure 2.11
Pole locations for fourth-order Chebyshev normalized filter.
