Digital Signal Processing Reference
In-Depth Information
As indicated by (3.19), the unnormalization process will result in a bandpass
approximation function that has twice the order of the lowpass function used to
generate it. This seems reasonable when we consider that a bandpass filter must
provide a transition from a stopband to a passband (like a highpass filter) and
another transition from a passband to a stopband (like a lowpass filter). The
resulting function must therefore be twice the order of the original lowpass
function on which it is based.
3.3.1 Handling a First-Order Factor
For a first-order factor in the lowpass approximation function, (3.25) shows how
the substitution of (3.19) is made:
2
2
A
⋅
S
+
A
A
⋅
[
(
s
+
ω
)
(
BW
⋅
s
)
]
+
A
1
2
1
o
2
H
(
s
)
=
=
(3.25)
2
2
B
⋅
S
+
B
B
⋅
[
(
s
+
ω
)
(
BW
⋅
s
)
]
+
B
1
2
2
2
S
=
(
s
+
ω
)
(
BW
⋅
s
)
1
o
2
o
And after some simplification we have the result in (3.26). The relationships
between the coefficients are shown. Note that if
A
1
= 0, as will normally be the
case, the numerator will only have an
s
-term present.
2
2
2
A
⋅
s
+
A
⋅
BW
⋅
s
+
A
⋅
ω
a
⋅
s
+
a
⋅
s
+
a
1
2
1
o
0
1
2
H
(
s
)
=
=
(3.26)
2
2
2
B
⋅
s
+
B
⋅
BW
⋅
s
+
B
⋅
ω
b
⋅
s
+
b
⋅
s
+
b
1
2
1
o
0
1
2
• The gain constant is unchanged.
• The
s
2
-term bandpass coefficients become
a
0
=
A
1
,
b
0
=
B
1
• The
s
-term bandpass coefficients become
a
1
=
A
2
BW
,
b
1
=
B
2
BW
• The constant term bandpass coefficients become
2
2
aA bB
=
ω
,
=
ω
2
1
o
2
1
o
3.3.2 Handling a Second-Order Factor
Unnormalizing a second-order factor is a bit more of a challenge. When the
substitution variable
S
P
of (3.19) is inserted into a second-order lowpass
approximation, a fourth-order factor results. What do we do with a fourth-order
factor? All of our development to this point is based on quadratic factors and with
good reason. They represent a complex conjugate pair and they will be used to
efficiently implement the filters in later chapters. We could factor the fourth-order,
but this would require a numerical algorithm that is time-consuming and not
always accurate. There is another directed procedure that can be used.
