In this section, a design procedure is developed that uses
a Chebyshev error criterion in both the passband and the stopband.
This is the fourth possible combination of Chebyshev and Taylor's
series approximations in the passband and stopband. The resulting
filter is called an elliptic-function filter, because elliptic
functions are normally used to calculate the pole and zero
locations. It is also sometimes called a Cauer filter or a rational
Chebyshev filter, and it has equal ripple approximation error in
both pass and stopbands Entry 6, Entry 5, Entry 4, Entry 7.
The error criteria of the elliptic-function filter are
particularly well suited to the way specifications for filters
are often given. For that reason, use of the elliptic-function
filter design usually gives the lowest order filter of the four
classical filter design methods for a given set of
specifications. Unfortunately, the design of this filter is the
most complicated of the four. However, because of the efficiency
of this class of filters, it is worthwhile gaining some
understanding of the mathematics behind the design procedure.
This section sketches an outline of the theory of elliptic-
function filter design. The details and properties of the elliptic
functions themselves should simply be accepted, and attention put on
understanding the overall picture. A more complete development is
available in Entry 6, Entry 3. Straightforward design of elliptic-function
filters can be accomplished by skipping this section and going directly to
Program 8 in the appendix or by using Matlab. However, it is important to
understand the basics of the underlying theory to use the packaged design
programs intelligently.
Because both the passband and stopband approximations are
over the entire bands, a transition band between the two must be
defined. Using a normalized passband edge, the bands are defined by
0
<
ω
<
1
passband
0
<
ω
<
1
passband
(1)
1
<
ω
<
ω
s
transition
band
1
<
ω
<
ω
s
transition
band
(2)
ω
s
<
ω
<
∞
stopband
ω
s
<
ω
<
∞
stopband
(3)
This is illustrated in Figure .
The characteristics of the elliptic function
filter are best described in terms of the four parameters that
specify the frequency response:
- The maximum variation or ripple in the passband δ1δ1,
- The width of the transition band (ωs-1)(ωs-1),
- The maximum response or ripple in the stopband δ2δ2, and
- The order of the filter NN.
The result of the design is that for any three of the parameters
given, the fourth is minimum. This is a very flexible and
powerful description of a filter frequency response.
The form of the frequency-response function is a
generalization of that for the Chebyshev filter
F
F
(
j
ω
)
=
|
F
(
j
ω
)
|
2
=
1
1
+
ϵ
2
G
2
(
ω
)
F
F
(
j
ω
)
=
|
F
(
j
ω
)
|
2
=
1
1
+
ϵ
2
G
2
(
ω
)
(4)
where
F
F
(
s
)
=
F
(
s
)
F
(
-
s
)
F
F
(
s
)
=
F
(
s
)
F
(
-
s
)
(5)
with F(s)F(s) being the prototype analog filter transfer
function similar to that for the Chebyshev filter. G(ω)G(ω) is a
rational function that approximates zero in the passband and infinity in
the stopband. The definition of this function is a generalization of the
definition of the Chebyshev polynomial.
In order to develop analytical expressions for equal-ripple
rational functions, an interesting class of transcendental
functions, called the Jacobian elliptic functions, is outlined.
These functions can be viewed as a generalization of the normal
trigonometric and hyperbolic functions. The elliptic integral of the
first kind Entry 1 is defined as
u
(
φ
,
k
)
=
∫
0
φ
d
y
1
-
k
2
sin
2
(
y
)
u
(
φ
,
k
)
=
∫
0
φ
d
y
1
-
k
2
sin
2
(
y
)
(6)
The trigonometric sine of the inverse of this
function is defined as the Jacobian elliptic sine of uu with
modulus kk, and is denoted
s
n
(
u
,
k
)
=
sin
(
φ
(
u
,
k
)
)
s
n
(
u
,
k
)
=
sin
(
φ
(
u
,
k
)
)
(7)
A special evaluation of (Equation 6) is known as the complete elliptic
integral K=u(π/2,k)K=u(π/2,k). It can be shown Entry 1 that sn(u)sn(u) and
most of the other elliptic functions are periodic with periods 4K4K if uu
is real. Because of this, KK is also called the “quarter period". A plot
of sn(u,k)sn(u,k) for several values of the modulus kk is shown in
Figure 2.
For k=0, sn(u,0)=sin(u)sn(u,0)=sin(u). As kk approaches 1, the sn(u,k)sn(u,k)
looks like a "fat" sine function. For k=1k=1, sn(u,1)=tanh(u)sn(u,1)=tanh(u)
and is not periodic (period becomes infinite).
The quarter period or complete elliptic integral KK is a
function of the modulus kk and is illustrated in Figure 3.
For a modulus of zero, the quarter period is K=π/2K=π/2 and it
does not increase much until k nears unity. It then increases
rapidly and goes to infinity as kk goes to unity.
Another parameter that is used is the complementary modulus
k'k' defined by
k
2
+
k
'
2
=
1
k
2
+
k
'
2
=
1
(8)
where both kk and k'k' are assumed real and between 0 and 1. The
complete elliptic integral of the complementary modulus is denoted
K'K'.
In addition to the elliptic sine, other elliptic functions
that are rather obvious generalizations are
c
n
(
u
,
k
)
=
c
o
s
(
φ
(
u
,
k
)
)
c
n
(
u
,
k
)
=
c
o
s
(
φ
(
u
,
k
)
)
(9)
s
c
(
u
,
k
)
=
t
a
n
(
φ
(
u
,
k
)
)
s
c
(
u
,
k
)
=
t
a
n
(
φ
(
u
,
k
)
)
(10)
c
s
(
u
,
k
)
=
c
t
n
(
φ
(
u
,
k
)
)
c
s
(
u
,
k
)
=
c
t
n
(
φ
(
u
,
k
)
)
(11)
n
c
(
u
,
k
)
=
s
e
c
(
φ
(
u
,
k
)
)
n
c
(
u
,
k
)
=
s
e
c
(
φ
(
u
,
k
)
)
(12)
n
s
(
u
,
k
)
=
c
s
c
(
φ
(
u
,
k
)
)
n
s
(
u
,
k
)
=
c
s
c
(
φ
(
u
,
k
)
)
(13)
There are six other elliptic functions that have no trigonometric
counterparts Entry 1. One that is needed is
d
n
(
u
,
k
)
=
1
-
k
2
s
n
2
(
u
,
k
)
d
n
(
u
,
k
)
=
1
-
k
2
s
n
2
(
u
,
k
)
(14)
Many interesting properties of the elliptic functions exist
Entry 1. They obey a large set of identities such as
s
n
2
(
u
,
k
)
+
c
n
2
(
u
,
k
)
=
1
s
n
2
(
u
,
k
)
+
c
n
2
(
u
,
k
)
=
1
(15)
They have derivatives that are elliptic functions. For example,
d
s
n
d
u
=
c
n
d
n
d
s
n
d
u
=
c
n
d
n
(16)
The elliptic functions are the solutions of a set of nonlinear
differential equations of the form
x
'
'
+
a
x
±
b
x
3
=
0
x
'
'
+
a
x
±
b
x
3
=
0
(17)
Some of the most important properties for the elliptic functions
are as functions of a complex variable. For a purely imaginary
argument
s
n
(
j
v
,
k
)
=
j
s
c
(
v
,
k
'
)
s
n
(
j
v
,
k
)
=
j
s
c
(
v
,
k
'
)
(18)
c
n
(
j
v
,
k
)
=
n
c
(
v
,
k
'
)
c
n
(
j
v
,
k
)
=
n
c
(
v
,
k
'
)
(19)
This indicates that the elliptic functions, in contrast to the
circular and hyperbolic trigonometric functions, are periodic in
both the real and the imaginary part of the argument with periods
related to KK and K'K', respectively. They are the only class
of functions that are “doubly periodic".
One particular value that the snsn function takes on that is
important in creating a rational function is
s
n
(
K
+
j
K
'
,
k
)
=
1
/
k
s
n
(
K
+
j
K
'
,
k
)
=
1
/
k
(20)
The rational function G(ω)G(ω) needed in (Equation 4) is sometimes called a
Chebyshev rational function because of its equal-ripple properties.
It can be defined in terms of two elliptic functions with moduli kk
and k1k1 by
G
(
ω
)
=
s
n
(
n
s
n
-
1
(
ω
,
k
)
,
k
1
)
G
(
ω
)
=
s
n
(
n
s
n
-
1
(
ω
,
k
)
,
k
1
)
(21)
In terms of the intermediate complex variable φφ, G(ω)G(ω)
and ωω become
G
(
ω
)
=
s
n
(
n
φ
,
k
1
)
G
(
ω
)
=
s
n
(
n
φ
,
k
1
)
(22)
ω
=
s
n
(
φ
,
k
)
ω
=
s
n
(
φ
,
k
)
(23)
It can be shown Entry 3 that G(ω)G(ω) is a real-valued rational
function if the parameters kk, k1k1, and nn take on special
values. Note the similarity of the definition of G(ω)G(ω) to
the definition of the Chebyshev polynomial CN(ω)CN(ω). In this case,
however, n is not necessarily an integer
and is not the order of the filter. Requiring that G(ω)G(ω) be a
rational function requires an alignment of the imaginary periods
Entry 3 of the two elliptic functions in (Equation 22,Equation 23). It also requires
alignment of an integer multiple of the real periods. The integer
multiplier is denoted by NN and is the order of the resulting
filter Entry 3. These two requirements are stated by the following
very important relations:
n
K
'
=
K
1
'
alignment
of
imaginary
periods
n
K
'
=
K
1
'
alignment
of
imaginary
periods
(24)
n
K
=
N
K
1
alignment
of
a
multiple
of
the
real
periods
n
K
=
N
K
1
alignment
of
a
multiple
of
the
real
periods
(25)
which, on removing the parameter nn, become
K
1
K
N
=
K
1
'
K
'
K
1
K
N
=
K
1
'
K
'
(26)
or
N
=
K
K
1
'
K
'
K
1
N
=
K
K
1
'
K
'
K
1
(27)
These relationships are central to the design of elliptic- function
filters. NN is an odd integer which is the order of the filter. For N=5N=5, the resulting rational function is shown in Figure 4.
This function is the basis of the approximation necessary
for the optimal filter frequency response. It approximates zero over the
frequency range -1<ω<1-1<ω<1 by an equal-ripple oscillation between
±1±1. It also approximates infinity over the range 1/k<|ω|<∞1/k<|ω|<∞ by a reciprocal oscillation that keeps |F(ω)|>1/k1|F(ω)|>1/k1. The
zero approximation is normalized in both the frequency range and the
F(ω)F(ω) values to unity. The infinity approximation has its frequency
range set by the choice of the modulus kk, and the minimum value of
|F(ω)||F(ω)| is set by the choice of the second modulus k1k1.
If kk and k1k1 are determined from the filter specifications,
they in turn determine the complementary moduli k'k' and k1'k1',
which altogether determine the four values of the complete elliptic
integral KK needed to determine the order NN in (Equation 27). In
general, this sequence of events will not result in an integer. In
practice, however, the next larger integer is used, and either kk
or k1k1 (or perhaps both) is altered to satisfy (Equation 27).
In addition to the two-band equal-ripple characteristics,
G(ω)G(ω) has another interesting and valuable property. The pole and
zero locations have a reciprocal relationship that can be expressed
by
G
(
ω
)
G
(
ω
s
/
ω
)
=
1
/
k
1
G
(
ω
)
G
(
ω
s
/
ω
)
=
1
/
k
1
(28)
where
ω
s
=
1
/
k
ω
s
=
1
/
k
(29)
This states that if the zeros of G(ω)G(ω) are located at
ωziωzi, the poles are located at
ω
p
i
=
1
/
(
k
ω
z
i
)
ω
p
i
=
1
/
(
k
ω
z
i
)
(30)
If the zeros are known, the poles are known, and vice versa. A
similar relation exists between the points of zero derivatives in
the 0 to 1 region and those in the 1/k1/k to infinity region.
The zeros of G(ω)G(ω) are found from (Equation 22) by requiring
G
(
ω
)
=
s
n
[
n
φ
,
k
1
]
=
0
G
(
ω
)
=
s
n
[
n
φ
,
k
1
]
=
0
(31)
which implies
n
φ
=
2
K
1
i
nφ=2
K
1
i
for
i
=
0,1,...
i=0,1,...
From Equation 21, this gives
ω
zi
=
sn
[
2
K
1
i
/
n,k
],
ω
zi
=sn[2
K
1
i/n,k],
i
=
0,1,...
i=0,1,...
This can be reformulated using (Equation 25) so that nn and K1K1 are not
needed. For NN odd, the zero locations are
ω
zi
=
sn
[
2
K
1
i
/
N,k
],
ω
zi
=sn[2
K
1
i/N,k],
i
=
0,1,...
i=0,1,...
The pole locations are found from these zero locations using
(Equation 30). The locations of the zero-derivative points are given by
ω
d
i
=
s
n
[
K
(
2
i
+
1
)
/
N
,
k
]
ω
d
i
=
s
n
[
K
(
2
i
+
1
)
/
N
,
k
]
(32)
in the 0 to 1 region, and the corresponding points in
the 1/k to infinity region are found from (Equation 30).
The above relations assume N to be an odd integer. A
modification for N even is necessary. For proper alignment of the
real periods, the original definition of G(ω)G(ω) is
changed to
G
(
ω
)
=
s
n
[
φ
+
K
1
,
k
1
]
G
(
ω
)
=
s
n
[
φ
+
K
1
,
k
1
]
(33)
which gives for the zero locations with N even
ω
z
i
=
s
n
[
(
2
i
+
1
)
K
1
/
n
,
k
]
ω
z
i
=
s
n
[
(
2
i
+
1
)
K
1
/
n
,
k
]
(34)
The even and odd N cases can be combined to
give
ω
z
i
=
±
s
n
(
i
K
/
N
,
k
)
ω
z
i
=
±
s
n
(
i
K
/
N
,
k
)
(35)
for
i
=
0
,
2
,
4
,
.
.
.
,
N
-
1
for
N
odd
i
=
0
,
2
,
4
,
.
.
.
,
N
-
1
for
N
odd
(36)
i
=
1
,
3
,
5
,
.
.
.
,
N
-
1
for
N
even
i
=
1
,
3
,
5
,
.
.
.
,
N
-
1
for
N
even
(37)
with the poles determined from (Equation 30).
Note that it is possible to determine G(ω)G(ω) from k and N
without explicitly using k1k1 or nn. Values for k1k1 and nn are
implied by the requirements of (Equation 29) or (Equation 28).
Zero Locations
The locations of the zeros of the filter transfer function
F(ω)F(ω) are easily found since they are the same as the poles of
G(ω)G(ω), given in (Equation 35).
ω
z
i
=
±
1
k
s
n
(
i
K
/
N
,
k
)
ω
z
i
=
±
1
k
s
n
(
i
K
/
N
,
k
)
(38)
for
i
=
0
,
2
,
4
,
.
.
.
,
N
-
1
N
odd
i
=
0
,
2
,
4
,
.
.
.
,
N
-
1
N
odd
(39)
i
=
1
,
3
,
5
,
.
.
.
,
N
-
1
N
even
i
=
1
,
3
,
5
,
.
.
.
,
N
-
1
N
even
(40)
These zeros are purely imaginary and lie on the ωω axis.
Pole Locations
The pole locations are somewhat more complicated to find. An
approach similar to that used for the Chebyshev filter is
used here. FF(s)FF(s) becomes infinite when
1
+
ϵ
2
G
2
=
0
1
+
ϵ
2
G
2
=
0
(41)
or
G
=
±
j
(
1
/
ϵ
)
G
=
±
j
(
1
/
ϵ
)
(42)
Using (Equation 22) and the periodicity of sn (u,k) , this implies
G
=
s
n
(
n
φ
+
2
K
1
i
,
k
1
)
=
±
j
1
/
ϵ
G
=
s
n
(
n
φ
+
2
K
1
i
,
k
1
)
=
±
j
1
/
ϵ
(43)
or
φ
=
(
-
2
K
1
i
+
s
n
-
1
(
j
1
/
e
,
k
1
)
)
/
n
φ
=
(
-
2
K
1
i
+
s
n
-
1
(
j
1
/
e
,
k
1
)
)
/
n
(44)
Define ν0ν0 to be the second term in (Equation 44) by
j
ν
0
=
(
s
n
-
1
(
j
1
/
e
,
k
1
)
)
/
n
j
ν
0
=
(
s
n
-
1
(
j
1
/
e
,
k
1
)
)
/
n
(45)
which is similar to the equation for the Chebyshev case. Using properties
of snsn of an imaginary variable and (Equation 26), ν0ν0 becomes
ν
0
=
(
K
/
N
K
1
)
s
c
-
1
(
1
/
ϵ
,
k</
