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).


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Elliptic functions and integrals

The elliptic integral of the first kind in the form of Jacobi is defined as

where the parameter is the modulus. By the variables' replacement and , this integral can be written in the trigonometric form

The Jacobi elliptic function is defined as inverse function to (6.17)

or, in trigonometric form

The Jacobi elliptic functions and are defined as follows

For the specific case of integral

is defined as a complete elliptic integral of the first kind. The complete integral of the first kind is a function only of modulus. Parameter is called the complementary modulus.

The complementary complete integral of the first kind is defined as

The following properties of the complete integral of the first kind follow directly from the definitions

The Jacobi elliptic functions have the important property of double periodicity. They have a real period and imaginary period. For the elliptic sine, the following expression is valid

where - integers.

The Jacobi elliptic sine has 2 main periods: the real period and the imaginary period .

Elliptic functions and are also double periodical

The function has the real period and imaginary period .The function has the real period and imaginary period .

The double periodical behavior of the elliptic functions can be described by the period parallelograms, which are defined in the by the lines and , where are real and imaginary periods of the elliptic function. Note that main periods are different for different elliptic functions.

The specific parallelogram, which is defined by the vertices , is called a fundamental period parallelogram of the elliptic function with main periods .

This parallelogram includes all internal points and two adjacent sides and without vertices 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.

Fig 6.3. Period parallelogram for the elliptic function with the real period and imaginary period . The period parallelogram includes all internal points plus two adjacent sides and without vertices and .

The elliptic functions of imaginary argument can be expressed as follows

Some important properties of elliptic functions and that follow from the definition (6.17) are listed below:

The double periodicity property of the elliptic functions can be written in the form

Addition formulas for Jacobi elliptic functions:

Note that in (6.37) the functions , , stand for , , respectively.

Using the properties of the elliptic functions, some values of the elliptic sine inside of the fundamental periodic parallelogram can be obtained.

Fig. 6.3. Some values of the elliptic sine inside of the fundamental periodic parallelogram.

  • - 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)




Elliptic functions have the property of double periodicity. As it was mentioned before, the Jacobi elliptic sine has 2 main periods

Both main periods in these expressions depend on the same parameter . 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

It can be shown that parameter

can be expanded as a power series of

Reversing the series (6.40) results in

For practical applications, 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.

If parameter is known, the modulus and the complete integral of the first kind can be determined by the series


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How to express elliptic rational function via Jacobi elliptic functions

The integral form of elliptic rational function is a solution to the differential equation (6.14)

Both integrals can be expressed via Jacobi elliptic functions. If we denote

,

then according to the definition (6.19) of elliptic sine, the elliptic rational function can be expressed as follows

The elliptic rational function is expressed as a Jacobi elliptic sine with modulus , and the normalized frequency is expressed as an elliptic sine with modulus .

The elliptic sine in (6.45) does not explicitly depend on frequency. The elliptic sine in (6.46) maps the z-plane onto the real frequency axis , and the elliptic sine in (6.45) maps the z-plane onto the real axis .

Arbitrary constants in (6.45) can be determined for two general cases: when the order of the filter is odd, and when the order of the filter is even.


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Elliptic rational functions of an odd order

When the order is an odd number, the arbitrary constant . This is following from property (6.33)( when , ). For the odd , the rational function is expressed as

where z is a solution to (6.46). The elliptic sine in (6.47) does not depend on frequency explicitly.

The elliptic sine 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 .

The period parallelograms for elliptic functions and are not independent of each other. It can be shown (see [2], [3]) that the real period of is times the real period of , which can be expressed as follows

Iimaginary periods are equal

The arbitrary constant can be obtained from (6.48)

Furthermore from (6.48) and (6.49), follows that

This equation is known as a degree equation and it is very important for elliptic approximations.

For the odd , the elliptic rational function can be expressed as

In (6.52) is interpreted as a function of , where is a solution to equation (6.46).

Since normalized frequency in (6.46) is a function of as well, it will be useful to plot period parallelograms for these functions at the same scale.

Period parallelograms for the fixed order presented on the Fig. 6.4.

Fig. 6.4. Period parallelograms for .

a) - Period parallelogram for the normalized frequency

b) - Period parallelogram for the elliptic rational function

The quantities of and evaluated for the real argument are given in the table (6.1).

To evaluate these functions, the following properties were used:

Table 6.1

Property (6.33)

Property (6.33)

Property (6.34)

Property (6.36)

This data demonstrates the periodical properties of the elliptic sine functions with different real main periods.

The elliptic sine 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.

Fig 6.5 Elliptic functions and plotted for .

The plot on Fig. 6.5 explains the behavior of the elliptic rational function in the pass-band. Considering the case of , the properties of elliptic function are:

  • The zeros of occur at critical frequencies:

  • when

  • when

It is important to note that function maps segment [0,K] of the onto the pass-band on the real positive axis.

In general, the function 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.

The important property of the elliptic sine is that the path around rectangle 0ABC+ in the 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.

Fig. 6.6

  • Path 0A:

    The equation 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.

    On the other hand, the mapping function

    maps the path 0A onto the real axis , bounded by the points

    The elliptic rational function (6.53) varies in the range [-1,1]. Practical interest for filter design are those points in the , belonging to the path 0A, in which the elliptic function takes the values 0, -1, and +1.

    The zeros of (6.53) can be obtained by solving the equation

    The solutions of this equation may be simply expressed as follows

    Now, the frequencies corresponding to the zeros of can be found as

    Critical frequencies are the zeros of an elliptic rational function on the real positive axis.

    The rational function in the pass-band varies in the range . To find real positive frequencies when the following equation must be solved first

    The solution of this equation is as follows

    Brackets [] stand for the integer part of the ratio . Now, corresponding normalized frequencies can be found as

    To find real positive frequencies when , the following equation must be solved first

    Solutions of this equation are

    Now, corresponding to (6.62) normalized frequencies can be found as

    Some examples of normalized frequencies obtained for the odd orders presented in the table 6.2

    Table 6.2

    3

    5

    7

    9

    11

  • Path AB:

    On this path, the normalized frequency is expressed as . After using the addition formulas (6.37), the expression for the frequency yields

    Replacing functions of imaginary argument and with (6.31) and (6.32) respectively, yields

    Function of real argument maps the real argument onto the segment on the real axis. Since , and , function varies in the range

    Fig. 6.7 Functions and of real argument .

    Noting that 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

    Consequently, the path AB in the maps onto the transition band on the real axis. The elliptic rational function on this path is

    Using degree equation (6.51), this equation can be written in the form

    Again, using the addition formulas (6.37) and replacing functions of imaginary argument with (6.31) and (6.32) yields

    Since is odd, . Now we can evaluate the elliptic rational function at the bounder points A and B

  • Path BC+:

    On this path, the frequency is expressed as . After using the addition formulas (6.37) and formulas (6.30), (6.31), (6.32) for imaginary arguments, the expression for the frequency yields

    Noting that , this expression becomes

    At the bounder points frequencies are:

    The path BC+ in the maps to the stop-band on the real positive axis :

    Corresponding quantities of the elliptic rational function on this path becomes

    Note that degree equation (6.51) was used.

    Now, using the addition formulas (6.37) and replacing functions of imaginary argument with (6.30), (6.31) and (6.32) yields

    Noting that variable 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:

    After comparing these expressions, we can conclude that frequencies and elliptic rational function quantities for these paths are connected with each other as follows

    This important relationship between pass-band and stop-band parameters, allow us to make the following conclusions:

    • Zeros of (6.55) are poles of (6.76)

    • Since , the quantities of the elliptic rational function in the stop-band vary in the ranges

  • Path 0E:

    On this path, the normalized frequency is expressed as . At the bound points frequencies are:

    Function 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 .

    Corresponding quantities of the elliptic rational function on this path

    The values of at the bound points are

    The further analysis of this path is similar to that which was made for the path 0A. The relationships between frequencies and corresponding elliptic rational function quantities for paths 0A and 0E can be expressed as follows

  • Path ED:

    On this path frequency, which is expressed as , can be transformed to the form

    The path ED in the maps onto the segment on the real axis. The corresponding elliptic rational function on this path can be represented as

    Comparing these expressions with (6.65) and (6.70), we can conclude that following relationship is held

  • Path DC-:

    On this path, the frequency is expressed (see 6.72) as follows

    At the bound points when , frequencies are:

    Consequently, the path DC- in the maps to the stop-band segment on the real axis :

    The elliptic rational function on this path (see 6.76) becomes

    Expressions for normalized frequency and corresponding elliptic rational function quantities on the path DC- are the same as those for the path BC+. Since the elliptic sine is an odd function, the following relationships must be held

Therefore, for the case when the order is an odd number, the following conclusions can be made:

  • Path 0ABC+ in the maps onto the real positive axis. Path 0EDC- in the maps onto the real negative axis. The mapping function is

  • The corresponding quantities of the elliptic rational function which is given by

    satisfy all requirements.

    Normalized critical frequencies (zeros) of the elliptic rational function are determined according to (6.57) and (6.81)

  • Degree equation (6.51) must be held


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Elliptic rational functions of an even order

As it was already established, the elliptic rational function and the normalized frequency can be expressed in the form of elliptic functions of the complex argument z (see 6.45 and 6.46)

For the filter approximation with an even order , 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

where is a solution to . Function (6.87) is an even function of the variable . It follows from the periodical properties of elliptic sine:

Also the degree equation

must be held.

By analogy, with the case of the odd degree, it can be shown that (6.87) satisfies all requirements to the rational functions .

To find the critical frequencies, the equation

must be solved first. The solutions to this equation are as follows

Consequently, the critical frequencies of the elliptic rational function of the even order are determined as

Note, that critical frequencies (6.90) are located on the positive real axis . 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.


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Minimum order determination

The minimum filter order that meets the specification can be obtained from the degree equation

Replacing integer variable with real variable yields

Now the minimum order can be obtained as , where brackets [] stand for the nearest integer exceeding .

The minimum order of the filter is determined by 2 parameters and . These parameters are expressed as

All 4 parameters, , are required to calculate the minimum filter order.

Expression (6.91) can be rearranged to the following form

Parameter can be expressed as a power series (see 6.41)

The second expression in (6.93) can be rearranged to the form

For the practical filters, parameter is very small, so . Therefore, from (6.94) and (6.93) follows

When parameter , expression (6.92) can be rearranged to the following form

Expression (6.96) can be used to determine the minimum required order for the practical elliptic filters. For the case when the discrimination factor is not small enough, the expressions (6.91) or (6.92) can be used for the minimum order determination. Since the order of the filter 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.

To recalculate parameter , equation (6.92) can be used. First, parameter can be computed as

and then parameter can be calculated using the power series (6.43).

Now, stop-band frequency 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.

The parameter can be recalculated using the same procedure. In this case, the stop-band attenuation must be updated using equation

The updated stop-band attenuation is slightly larger than the original one.


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Zero and pole locations

The squared magnitude response of the normalized elliptic filter is given by

The normalized critical frequencies (zeros) of the elliptic rational function can be respectively expressed by (6.57) and (6.90) for the cases of odd and even orders. Those critical frequencies are as follows

As it can be seen from (6.101), all roots for even N are finite nonzero quantities. When the order is an odd number, one of the roots corresponding to is to be zero, the others are nonzero and finite.

It is a common practice to express the critical frequencies of the elliptic rational function via elliptic cosine function Elliptic cosine function can be defined as follows

Some properties of the elliptic cosine function are listed below:

Normalized zeros of the elliptic rational function with an odd order

Elliptic rational function evaluated for takes the following set of values

On the other hand, the elliptic function evaluated for takes the set of values

Noting that (6.104) and (6.105) represent the same set of quantities in a different order, we can conclude that all nonzero critical frequencies of the elliptic rational function of an odd order can be given as follows

Normalized zeros of the elliptic rational function with an even order

When the order of the filter is an even number, all zeros of the elliptic rational function are nonzero finite numbers. According to (6.101), the critical frequencies in this case are determined by elliptic sine . This function evaluated for takes the set of values

Alternatively, the elliptic function evaluated for takes the set of values

Since sets (6.107) and (6.108) consist of the same elements positioned in a different order, all critical frequencies of the elliptic rational function with an even order can be presented as follows

Consequently, all nonzero critical frequencies of the elliptic rational function can be presented as

Note that for the filter with an odd order, there is an additional critical frequency (see 6.102) equal to zero, which is not included in (6.110). We do not need to count it because there is no pole of elliptic rational function corresponding to it.

De-normalized zeros of the squared magnitude response of elliptic filters

The poles of the elliptic rational function occur at normalized frequencies that are inversed of the zeros. Because the elliptic rational function corresponding to (6.100) is expressed by (6.5), the zeros of the squared magnitude response arise from the poles of the elliptic rational function. Therefore, the de-normalized zeros of (6.100) in the complex s-plane can be expressed as follows

Zeros (6.111) represent purely imaginary complex conjugate pairs in the s-plane. To be mathematically precise, note that due to the fact that rational function 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).

De-normalized poles of the squared magnitude response of elliptic filters with an odd order

To obtain the poles of (6.100), the equation

must be solved. If the order N of the filter is an odd number, the rational function is expressed by (6.52)

Therefore, equation (6.112) can be written in the following form

Since on the right of (6.114) there is a purely imaginary number, the solution to this equation must also be a purely imaginary number that follows from the property (6.30). The solution to (6.114) can be presented in the following form

De-normalized frequency corresponding to (6.115) can be obtained as

Again, observing that the inverse elliptic sine of the purely imaginary argument is also purely imaginary, it can be stated that 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.

The remaining roots of (6.114) can be found using the periodical properties of the elliptic sine. If the function is considered as a function of argument , the following identity is true for any integer .

Therefore, the set of unrepeatable roots of (6.114) can be found as

The de-normalized frequencies corresponding to those solutions are as follows

Since N is a given odd number, the expression (6.119) represents an even number of roots. The first half of those roots can be expressed as follows

The second half of those roots can be expressed as follows

This expression can be rearranged to the following

Therefore, all poles can be given as

This expression can be presented as

Observing that

the de-normalized poles can be rearranged to the following form

De-normalized poles of the squared magnitude response of elliptic filters with an even order

When the order of the filter is an even number, the rational function is expressed by

To compute the poles of the squared magnitude response (6.100), the following equation must be solved

All unrepeatable solutions to this equation can be given as

De-normalized frequencies corresponding to these solutions can be expressed in the following form

Noting that , this expression can be rearranged as follows

It is useful to rearrange this expression to the form where the second addition in the elliptic sine argument is presented as a factor of . In order to do this, two expressions can be considered

The first expression represents the following set of quantities

The second expression in (6.130) represents the following set of quantities

It is easy to see that those sets of quantities consist of the same elements in a different order. The first quantity of the first set is equal to the last quantity of the second set, the second quantity of the first set is equal to the one before the last quantity in the second set and so on and so forth.

Expressions (6.130) may be rearranged to the equivalent form

Denoting in the first expression in (6.131) and in the second expression, these expressions can be written in the form

Again, observing that these expressions represent the same quantities in the opposite order, and noting that in the first expression of (6.132), index has even values and in the second expression index has odd values, the unified expression for the whole set of quantities can be given as follows

Therefore, the whole set of poles (6.129) of the squared magnitude response for the filter with an even order N can be written as

Formulas (6.125) and (6.134) express the de-normalized poles of the elliptic filters for the cases of an odd and even order. Those expressions are submitted in the form that is convenient for the numeric evaluations.

De-normalized poles of the squared magnitude response of elliptic filters expressed via elliptic cosine function

It is also possible to express the poles of elliptic filters via elliptic cosine. The expression for the de-normalized poles of the squared magnitude response can be given as follows

Expression (6.135) can be obtained by direct transformations of (6.125) and (6.134). To verify that (6.135) represents all unrepeatable poles of the squared magnitude response of the elliptic approximation, it is enough to show that (6.135) are solutions to both (6.114) and (6.127).

Using the definition of the elliptic cosine, the expression (6.135) can be rearranged to the form

Now the values of the elliptic rational function corresponding to (6.136) can be computed. In the case when the order N is an odd number

Since N is an odd number, the sum is an even for any index n and therefore the expression (6.137) can be written as

In the case when the order N of the filter is an even number

Since N is an even number, the sum is an even for any index n and therefore this expression can be arranged to (6.138).

Consequently, the cosine form of poles (6.135) satisfies equations (6.114) and (6.127).


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How to obtain the transfer function of an elliptic filter

When the filter order is an even number, the transfer function of the elliptic filter has finite purely complex conjugate pairs of zeros and complex conjugate pairs of poles. The elliptic rational function at DC , which follows directly from (6.126). Therefore, the gain at DC is as follows

Consequently, in order to achieve the zero minimum pass-band loss at , the transfer function of the elliptic filter must have the factor . The transfer function in this case can be presented as follows

When the filter order is an odd number, the elliptic filter has purely complex conjugate pairs of zeroes and poles. One of the poles is a real number, the rest of them are complex conjugate pairs. The transfer function in this case can be written as follows

In order to obtain the transfer function of an elliptic filter using (6.140) or (6.141), we have to have efficient computational methods for the evaluation of elliptic functions and inverse elliptic functions.


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References

[1] M. A. Lavrentyev, B. V. Shabat, Methods of the Theory of Functions of a Complex Variable, Nauka, Moscow, 1973 (Russian)

[2] R. W. Daniels, Approximation Methods for Electronic Filter Design, McGraw-Hill, New York, 1974.

[3] A. Antoniou, Digital Filters, 2nd ed., McGraw-Hill, New York, 1993.

[4] N.I.Akhiezer, Elements of elliptic functions theory, Nauka, Moscow, 1970 (Russian)

[5] A. Zverev, Handbook of Filter Synthesis, John Wiley and Sons, 1967

[6] Larry D. Paarmann, Design and analysis of analog filters, A Signal Processing perspective, Wichita State University, 2003.


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