Digital Signal Processing Reference
In-Depth Information
The initial gain factor of (3.47) can be shown to be
A
2
/
B
2
. The quadratic
factors can be factored into first-order factors, as indicated in (3.48):
A
(
s
+
z
)
⋅
(
s
+
z
)
⋅
(
s
+
z
)
⋅
(
s
+
z
)
2
1
a
1
b
1
c
1
d
H
(
s
)
=
⋅
(3.48)
B
(
s
+
p
)
⋅
(
s
+
p
)
⋅
(
s
+
p
)
⋅
(
s
+
p
)
2
1
a
1
b
1
c
1
d
We then reconstruct two quadratics in the numerator and denominator by
matching the complex conjugate pairs:
2
2
A
(
s
+
a
⋅
s
+
a
)(
s
+
a
⋅
s
+
a
)
2
1
2
4
5
H
(
s
)
=
⋅
(3.49)
2
2
B
(
s
+
b
⋅
s
+
b
)(
s
+
b
⋅
s
+
b
)
2
1
2
4
5
This result is valid for the rational approximation functions (inverse
Chebyshev and elliptic), but for the all-pole approximations (Butterworth and
Chebyshev), we must develop a slightly different version. Equations (3.50) and
(3.51) show the factoring and substitution of (3.39). After the quadratics of (3.51)
are factored and the matching complex conjugate terms are combined, the final
form is (3.52) .
A
A
2
2
H
(
s
)
=
=
(3.50)
2
*
1
S
+
B
⋅
S
+
B
(
S
+
p
)(
S
+
p
)
1
2
1
2
2
S
=
(
BW
⋅
s
)
(
s
+
ω
)
2
2
2
A
(
s
+
ω
)
2
o
H
(
s
)
=
⋅
(3.51)
*
1
2
2
2
*
1
2
p
⋅
p
[
s
+
(
BW
/
p
)
⋅
s
+
ω
]
⋅
[
s
+
(
BW
/
p
)
⋅
s
+
ω
]
1
1
o
o
2
2
2
A
(
s
+
ω
)
2
o
H
(
s
)
=
⋅
(3.52)
2
2
B
(
s
+
b
⋅
s
+
b
)(
s
+
b
⋅
s
+
b
)
2
1
2
4
5
Complete numerical examples of the bandstop unnormalization process are
given next.
