6 Elliptic approximation
Elliptic filters have equiripple characteristics in both the pass-band and the stop-band. The elliptic filters are optimal in terms of a minimum width of transition band; they provide the fastest transition from the band-pass to the band-stop. Elliptic filters are also well known as Cauer filters or Zolotarev filters.
The typical magnitude response of elliptic filters is provided on the Fig. 6.1.
Fig. 6.1. Elliptic filters. Typical magnitude response.
The squared magnitude response of the elliptic approximation is defined as
where:
-
- Is an elliptic rational function of the order 
. -
- Parameter, that characterizes the loss of the filter in the pass-band -
- Radian frequency -
- Scaling frequency
Generally, the rational function is a ratio of polynomials. The behavior of the elliptic rational function and the squared elliptic rational function for the order of 5, plotted at the same frequency scale, is given on the Fig. 6.2. It is easy to see (compare Fig. 6.1. and 6.2) that the plot of the loss of elliptic filters is similar to the plot of the squared elliptic rational function. The squared elliptic rational function varies in the pass-band (
) in the range from 0 to +1(Fig. 6.2). The number of zeros in the pass-band is equal to the order 
of the rational function. In the stop-band (
), the squared rational function varies from 
to infinity.
Fig 6.2. The elliptic rational function (Chebyshev rational function) and squared elliptic rational function.
It is very convenient to define the scale frequency for the elliptic approximation as
Then normalized frequency 
can be introduced. In this case, the normalized edge frequencies can be expressed as follows
The normalized frequencies 
and 
must satisfy inequality 
. The selectivity factor of the elliptic filter is defined as
From (6.2) and (6.3) follows that normalized edge frequencies 
and 
can be expressed via selectivity factor
The normalized transfer function for the elliptic approximation can be designed when the ratio of the pass-band edge and stop-band edge frequencies is known. To de-normalize the transfer function, the scale frequency 
must be used.
The normalized rational function 
may be presented in a simple form
Parameter
is called a discrimination factor. Parameters 
(selectivity factor) and 
(discrimination factor) are very important for the elliptic filters design. Critical frequencies 
in (6.5) must satisfy inequality 
, so all zeros of the rational function occur in the pass-band. It is easy to see that the poles of the rational function (6.5) are inverse of zeros.
Elliptic rational functions (6.5) must have the following properties:
-
Function
is odd when 
is odd, and even when is even. Consequently, the elliptic rational function is either even or odd. -
The function
in the pass-band 
varies in the range 
. 
-
All critical frequencies
(zeros of rational function) are located in the pass-band, 
. -
All poles of the rational function are inverse of zeros:
. All poles are located in the stop-band 
. -
In the stop-band
function 
varies in the range 
. 
-
The derivative
has 
zeros in the pass-band (see Fig 6.2) that occur when 
, and 
zeros in the stop-band when 
.
The rational function that has the above listed properties may be described by a differential equation (see [2], [3])
Signal attenuation for elliptic approximation is given as
Using property #2, the attenuation at the pass-band can be expressed as follows
Therefore, parameter 
can be expressed as
The attenuation at the stop-band edge can be found using property #5
Solving (6.11) yields
From (6.6) and (6.12) follows that
Now, using this expression, the differential equation (6.7) can be written in the form
The integral form of equation (6.14) is as follows
Replacing variable 
with 
, in such a way that 
equation (6.15) can be rearranged into the form
These integrals are known as elliptic integrals of the first kind in the form of Jacobi. So, the elliptic rational function (Chebyshev rational function) can be presented in integral form by (6.16).
Elliptic functions and integrals
How to express elliptic rational function via Jacobi elliptic functions
Elliptic rational functions of an odd order
Elliptic rational functions of an even order
Minimum order determination
Zero and pole locations
How to obtain the transfer function of an elliptic filter
References
is the modulus. By the variables' replacement 
and 
, this integral can be written in the trigonometric form
is defined as inverse function to (6.17)
and 
are defined as follows
integral
is called the complementary modulus.
- integers.
and the imaginary period 
.
and 
are also double periodical
has the real period 
and imaginary period 
.The function 
has the real period 
and imaginary period 
.
by the lines 
and 
, where 
are real and imaginary periods of the elliptic function. Note that main periods 
, is called a fundamental period parallelogram of the elliptic function with main periods 
.
and 
and 
(Fig 6.3). If the values of the elliptic function 
are known within the period parallelogram, the elliptic function is known over the entire 
due to the property of double periodicity.
and imaginary period 
. The period parallelogram includes all internal points plus two adjacent sides 
and 
without vertices 
and 
.
can be expressed as follows
and 
that follow from the definition (6.17) are listed below:
, 
, 
stand for 
, 
, 
respectively.
- Property (6.33)
- Property (6.34)
- Properties (6.35) and (6.28)
- Properties (6.35) and (6.30)
- Properties (6.37), (6.30), (6.31), (6.32)
. Therefore, the main periods are not independent. But the ratio 
is arbitrary. This ratio can be used to characterize elliptic functions. In practice, this ratio is determined via parameter
varies in the range 
, and parameter 
must vary in the range 
as well. Therefore, 
. The series (6.41) converges rapidly, and can be used to compute 
precisely.
is known, the modulus 
and the complete integral of the first kind 
can be determined by the series
is a solution to the differential equation (6.14)
,
is expressed as a Jacobi elliptic sine with modulus 
, and the normalized frequency 
is expressed as an elliptic sine with modulus 
.
is an odd number, the arbitrary constant 
. This is following from property (6.33)(
when 
, 
). For the odd 
, the rational function is expressed as
has the property of double periodicity (see 6.28) with the real period 
and imaginary period 
. The elliptic function 
(considered as a function of 
) is also a doubly periodic function with the real period 
and imaginary period 
.
and 
are not independent of each other. It can be shown (see [2], [3]) that the real period of 
times the real period of 
can be obtained from (6.48) 
, the elliptic rational function can be expressed as
is interpreted as a function of 
, where 
is a solution to equation (6.46).
in (6.46) is a function of 
as well, it will be useful to plot period parallelograms for these functions at the same 
scale.
presented on the Fig. 6.4.
.
- Period parallelogram for the normalized frequency 
- Period parallelogram for the elliptic rational function 
and 
evaluated for the real argument 
are given in the table (6.1).
of real argument is a periodic function, the behavior of which is very similar to the behavior of trigonometric sine 
. The similarity is very close when parameter 
is small (from the trigonometric definition of the elliptic sine follows that 
); when modulus 
is close to 1, the shape of 
becomes more rectangular.
and 
plotted for 
.
, the properties of elliptic function 
are:
occur at critical frequencies: 
when 
when 
maps segment [0,K] of the 
onto the pass-band on the real positive 
axis.
maps conformally the 
onto the complex 
. Due to the double periodicity of the elliptic sine, the period parallelogram maps onto the entire 
as well. The practical interest for the filters design is a real frequency.
gets mapped onto the real positive 
axis, and the path around rectangle 0EDC- gets mapped onto the real negative 
axis. To verify this property, we will consequently map individual path segments onto the 
axis.
maps this path onto the segment on the real positive axis 
bounded by the points 
and 
. Consequently, the path 0A is mapping to the pass-band 
on the real positive frequency axis.
, bounded by the points 
, belonging to the path 0A, in which the elliptic function takes the values 0, -1, and +1.
can be found as 
are the zeros of an elliptic rational function on the real positive axis.
in the pass-band varies in the range 
. To find real positive frequencies when 
the following equation must be solved first
. Now, corresponding normalized frequencies can be found as
, the following equation must be solved first
presented in the table 6.2
. After using the addition formulas (6.37), the expression for the frequency yields
and 
with (6.31) and (6.32) respectively, yields
of real argument 
maps the real argument 
onto the segment on the real axis. Since 
, and 
, function 
varies in the range
and 
of real argument 
.
and considering (6.65), we can conclude that (6.64) maps points on the real axis 
onto the line segment on the axis 
, bounded by the points 
and 
maps onto the transition band on the real 
axis. The elliptic rational function on this path is
is odd, 
. Now we can evaluate the elliptic rational function at the bounder points A and B
. After using the addition formulas (6.37) and formulas (6.30), (6.31), (6.32) for imaginary arguments, the expression for the frequency yields
, this expression becomes
frequencies are:
maps to the stop-band on the real positive axis 
:
varies in the same range 
for both paths 0A and BC+, we can write down the expressions for the normalized frequency and elliptic rational function:
, the quantities of the elliptic rational function in the stop-band vary in the ranges 
. At the bound points 
frequencies are:
maps the path 0E onto the real negative axis 
, bounded by the points 
and 
. This is the pass-band segment on the negative frequency axis 
.
at the bound points are
, can be transformed to the form
maps onto the segment 
on the real 
axis. The corresponding elliptic rational function on this path can be represented as
, frequencies are:
maps to the stop-band segment on the real axis 
:
is an odd number, the following conclusions can be made:
maps onto the real positive 
axis. Path 0EDC- in the 
maps onto the real negative 
axis. The mapping function is
, the constant 
is the same that for the case of an odd order. The constant 
, which is following from the property 
. Therefore, the elliptic rational function for the even order 
can be expressed as follows
is a solution to 
. Function (6.87) is an even function of the variable 
. It follows from the periodical properties of elliptic sine:
.
. Since function 
is an odd function of argument 
, the critical frequencies on the negative real axis can be expressed by (6.90) with a negative sign.
with real variable 
yields
can be obtained as 
, where brackets [] stand for the nearest integer exceeding 
.
and 
. These parameters are expressed as
, are required to calculate the minimum filter order.
can be expressed as a power series (see 6.41)
is very small, so 
. Therefore, from (6.94) and (6.93) follows
, expression (6.92) can be rearranged to the following form
is rounded up to the next integer, the degree equation is not satisfied precisely for the given parameters 
and 
. To satisfy it exactly, one of these parameters should be recalculated.
, equation (6.92) can be used. First, parameter 
can be computed as
can be calculated using the power series (6.43). 
can be updated to 
. The resulting parameter 
is larger then the initial one, therefore the updated stop-band frequency is smaller then the original one, making the transitional band narrower.
can be recalculated using the same procedure. In this case, the stop-band attenuation must be updated using equation
is to be zero, the others are nonzero and finite.
Elliptic cosine function can be defined as follows
evaluated for 
takes the following set of values
evaluated for 
takes the set of values
. This function evaluated for 
takes the set of values
evaluated for 
takes the set of values
of (6.100) in the complex s-plane can be expressed as follows
in (6.100) is squared, the set of zeros (6.111) must be doubled. Consequently, (6.100) has 
zeros. Each of the functions 
in (6.100) has 
of identical sets of zeros (6.111), or 
complex conjugate imaginary pairs in the complex frequency plane (s-plane).
is expressed by (6.52)
to this equation must also be a purely imaginary number that follows from the property (6.30). The solution 
is a real solution. Consequently, expression (6.116) represents two real poles of the squared magnitude response in the complex s-plane; one of them being positive and the other negative.
is considered as a function of argument 
, the following identity is true for any integer 
. 
is expressed by
, this expression can be rearranged as follows
. In order to do this, two expressions can be considered
in the first expression in (6.131) and 
in the second expression, these expressions can be written in the form
has even values and in the second expression index 
is an even for any index n and therefore the expression (6.137) can be written as
is an even for any index n and therefore this expression can be arranged to (6.138).
is an even number, the transfer function of the elliptic filter has 
finite purely complex conjugate pairs of zeros and 
, which follows directly from (6.126). Therefore, the gain at DC is as follows
, the transfer function of the elliptic filter must have the factor 
. The transfer function in this case can be presented as follows
purely complex conjugate pairs of zeroes and 