Digital Signal Processing Reference
In-Depth Information
If we factor the lowpass approximation quadratic into two complex conjugate
factors before making the substitution of (3.19), the result after the substitution
and simplification is two quadratic equations. However, each of these quadratics
would have a complex coefficient that would mean they could not be implemented
directly. However, if these two quadratics are again factored, we will find two sets
of complex conjugate pairs within the set of four factors. These complex
conjugate pairs could then be combined to produce two quadratics that have all
real coefficients.
Perhaps an easy example is in order. Consider the transfer function shown in
(3.27) that has already been factored:
2
2
H
(
S
)
=
=
(3.27)
2
(
S
+
1
+
j
1
⋅
(
S
+
1
−
j
1
S
+
2
⋅
S
+
2
Now if we assume that ω
o
= 1 and
BW
= 1, we can substitute
S
= (
s
2
+ 1) /
s
and simplify to produce the following:
2
2
⋅
s
H
(
s
)
=
(3.28)
2
2
[
s
+
(
+
j
1
⋅
s
+
1
⋅
[
s
+
(
−
j
1
⋅
s
+
1
The roots of the first quadratic can be determined to be
−
(
+
j
1
±
(
486
+
j
2
058
)
s
=
=
(
−
0
257
+
j
0
529
),
(
−
0
743
−
j
1
53
)
(3.29a)
1
2
2
and the two roots of the second quadratic pair up with the first.
s
=
(
−
0
.
257
−
j
0
.
529
),
(
−
0
.
743
+
j
1
.
53
)
(3.29b)
3
4
The resulting transfer function can then be written as (3.30) by combining the
complex conjugate roots from each quadratic:
2
2
⋅
s
H
(
s
)
=
(3.30)
2
2
(
s
+
0
514
⋅
s
+
0
346
)
⋅
(
s
+
1
.
486
⋅
s
+
2
.
890
)
This algorithm for finding the two quadratics in the bandpass approximation
from the single quadratic in the lowpass function will be used as the standard
method in this section. Unfortunately, it is very difficult to define the final
bandpass coefficients in terms of only the initial lowpass coefficients because of
