Digital Signal Processing Reference
In-Depth Information
So in this lowpass case, the first step in the unnormalization procedure
doesn't require any work at all. We simply determine the order of the filter as we
have in the past. The second step of the unnormalization procedure, determination
of the normalized approximation function, has already been developed in the
previous chapter. It appears that we are ready to determine the third and final step
of the procedure, which is to unnormalize the normalized approximation function.
In the lowpass case, this simply requires a scaling of the frequency characteristic
from 1 rad/sec to a more usable frequency. A simple substitution for the
normalized variable
S
is all that is necessary, as shown in (3.7). (A subscript of
L
is used to indicate that this substitution is for lowpass filters only.) The frequency
constant ω
o
will be ω
stop
for the inverse Chebyshev approximation, as discussed in
Chapter 2, and ω
pass
for all other approximations.
s
S
=
(3.7)
L
ω
o
3.1.1 Handling a First-Order Factor
We will be developing code to implement the unnormalization process, so it is
important to carefully describe the substitution process. For the first-order factor,
the process begins with (3.8), where the
B
1
coefficient is typically 1:
A
⋅
S
+
A
A
⋅
(
s
ω
)
+
A
1
2
1
o
2
H
(
s
)
=
=
(3.8)
B
⋅
S
+
B
B
⋅
(
s
ω
)
+
B
1
2
1
o
2
S
=
s
ω
o
In this equation, uppercase
A
and
B
represent the coefficients of the
normalized approximation function. After simplification, (3.9) results in a new set
of coefficients. In this equation, lowercase
a
and
b
represent the unnormalized
coefficients that will be used in our final approximation function:
A
⋅
s
+
A
⋅
ω
a
⋅
s
+
a
1
2
o
1
2
H
(
s
)
=
=
(3.9)
B
⋅
s
+
B
⋅
ω
b
⋅
s
+
b
1
2
o
1
2
We can generalize these results for the first-order factor below:
• The gain constant is unchanged.
• The
s
-term coefficients become
a
1
=
A
1
,
b
1
=
B
1
• The constant term coefficients become
a
2
=
A
2
ω
o
,
b
2
=
B
2
ω
o
