Digital Signal Processing Reference
In-Depth Information
1
ω
=
(2.72)
[
]
zm
rt
⋅
sn
f
(
m
),
rt
Although the elliptic approximation requires a number of mathematical
functions which aren't in everyday usage, we have most of the hard work done in
determining the transfer function we need. Our primary objective in this section is
to develop an orderly manner to calculate the pole and zero locations.
2.5.4 Elliptic Transfer Functions
Now we are able to define the first-order and quadratic factors that will make up
the elliptic approximation function. The first-order factor for the elliptic
approximation is indicated in (2.73), where σ
R
is as indicated in (2.70). Again,
there is no matching finite zero for the first-order pole factor; it is located at
infinity.
σ
+
R
H
(
S
)
=
(2.73)
o
σ
S
R
The form of the quadratic components of the transfer function will also be
identical to the inverse Chebyshev case, as indicated below:
2
B
⋅
(
S
+
A
⋅
S
+
A
)
2
m
1
m
2
m
H
(
S
)
=
(2.74)
m
2
A
⋅
(
S
+
B
⋅
S
+
B
)
2
m
1
m
2
m
where
B
= 2
⋅
σ
(2.75)
1
m
m
2
2
B
=
σ +
ω
(2.76)
2
m
m
m
A
=
−
2
⋅
σ
=
0
.
(2.77)
1
m
zm
2
2
2
A
=
σ
+
ω
=
ω
(2.78)
2
m
zm
zm
zm
We are now ready to define a generalized transfer function for the elliptic
approximation function that is almost identical to the inverse Chebyshev case. The
difference lies in the ripple in the passband as in the standard Chebyshev case.
