Digital Signal Processing Reference
In-Depth Information
2
R
=
1
a
C
2
(4.55)
]
2
[
C
o
=
C
⋅
(
a
/
b
)
−
1
(4.56)
2
2
K
=
2
+
(
a
−
b
−
b
a
)
(
2
b
)
2
2
1
2
2
(4.57)
R
R
=
1
+
(
a
−
b
−
b
a
)
(
2
b
)
B
A
2
2
1
2
2
(4.58)
After these calculations are made, the overall gain of the active filter can be
determined by evaluating the transfer function at ω = 0 or ω = ∞. If the gain of the
circuit is to be determined at ω = 0, then the value of
K
for that stage is used
except for the highpass notch stage. In that case, an additional multiplication
factor of
R
o
/ (
R
o
+ 2
R
) should be included as seen from (4.49). If the gain is to
be determined at an infinite frequency, then the lowpass notch stage will have a
gain that must be increased by a value of
C
/ (
C
+ 2
C
o
), as indicated by (4.54).
These values will be the same and can be included in the circuit using a voltage
divider.
Example 4.4 Chebyshev Bandstop Active Filter Design
Problem:
Determine the resistor and capacitor values to implement a
Chebyshev bandstop active filter to meet the following specifications:
a
pass
= −1 dB,
a
stop
= −40 dB,
f
pass1
= 666.67 Hz,
f
pass2
= 1,500 Hz,
f
stop1
= 909.09 Hz, and
f
stop2
= 1,100 Hz
Solution:
A third-order lowpass equivalent function is required, which
indicates that a sixth-order unnormalized bandstop approximation function will be
necessary, as shown below:
2
6
3
(
s
+
39
.
48
⋅
10
)
H
a
(
s
)
=
2
6
2
6
2
6
(
s
+
10
,
600
⋅
s
+
39
.
48
⋅
10
)(
s
+
811
.
0
⋅
s
+
17
.
87
⋅
10
)(
s
+
1
792
⋅
s
+
87
.
21
⋅
10
)
By picking
C
= 0.01 µF and
R
A
= 10 kΩ, the remaining values can be
calculated by matching terms. Note that there are three denominator quadratics
with three different values of constant terms. One of these values is larger than the
numerator constant term, one is equal to the numerator constant term, and one is
smaller than the numerator constant term. This indicates that one of the stages will
