Digital Signal Processing Reference
In-Depth Information
3.1.2 Handling a Second-Order Factor
In the case of the quadratic terms that are used to describe our coefficients, the
unnormalization process is shown in (3.10) and (3.11):
2
2
A
⋅
S
+
A
⋅
S
+
A
A
⋅
(
s
ω
)
+
A
⋅
(
s
ω
)
+
A
0
1
2
0
o
1
o
2
H
(
s
)
=
=
(3.10)
2
2
B
⋅
S
+
B
⋅
S
+
B
B
⋅
(
s
ω
)
+
B
⋅
(
s
ω
)
+
B
0
1
2
0
o
1
o
2
S
=
s
ω
o
2
2
2
A
⋅
s
+
A
⋅
ω
⋅
s
+
A
⋅
ω
a
⋅
s
+
a
⋅
s
+
a
0
1
o
2
o
0
1
2
(3.11)
H
(
s
)
=
=
2
2
2
B
⋅
s
+
B
⋅
ω
⋅
s
+
B
⋅
ω
b
⋅
s
+
b
⋅
s
+
b
0
1
o
2
o
0
1
2
The coefficient
A
0
will be 1 or 0. A value of 1 will be present only if an
inverse Chebyshev or elliptic approximation is being unnormalized, while a 0 will
be used for Chebyshev and Butterworth.
A
1
will normally be 0 for all
approximations, but is included for completeness of the derivation in the event we
want to use any of our work at a later time when complex conjugate zeros will
occur off the
j
ω axis.
B
0
will typically be 1 for all cases, but is retained for
generality. By observation, we can determine the following relationships that can
be used in our C code:
• The gain constant is unchanged.
• The
s
2
-term coefficients become
a
0
=
A
0
,
b
0
=
B
0
• The
s
-term coefficients become
a
1
=
A
1
ω
o
,
b
1
=
B
1
ω
o
• The constant term coefficients become
2
2
aA
=
ω
,
bB
=
ω
2
2
o
2
2
o
Complete numerical examples of the lowpass unnormalization process are
now in order.
Example 3.1 Unnormalized Inverse Chebyshev Lowpass Filter
Problem:
Determine the transfer function for an inverse Chebyshev lowpass
filter to satisfy the specifications:
a
pass
= −0.25 dB,
a
stop
= −38.0 dB,
ω
pass
= 600 rad/sec,
ω
stop
= 1,000 rad/sec
