Elliptic filter

An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not).^{[citation needed]} Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations.

As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter.

The gain of a lowpass elliptic filter as a function of angular frequency ω is given by:

${\displaystyle G_{n}(\omega )={1 \over {\sqrt {1+\epsilon ^{2}R_{n}^{2}(\xi ,\omega /\omega _{0})}}}}$

where R_n is the nth-order elliptic rational function (sometimes known as a Chebyshev rational function) and

${\displaystyle \omega _{0}}$ is the cutoff frequency

${\displaystyle \epsilon }$ is the ripple factor

${\displaystyle \xi }$ is the selectivity factor

The value of the ripple factor specifies the passband ripple, while the combination of the ripple factor and the selectivity factor specify the stopband ripple.

Properties[edit]

• In the passband, the elliptic rational function varies between zero and unity. The gain of the passband therefore will vary between 1 and ${\displaystyle 1/{\sqrt {1+\epsilon ^{2}}}}$.
• In the stopband, the elliptic rational function varies between infinity and the discrimination factor ${\displaystyle L_{n}}$ which is defined as:

${\displaystyle L_{n}=R_{n}(\xi ,\xi )\,}$

The gain of the stopband therefore will vary between 0 and ${\displaystyle 1/{\sqrt {1+\epsilon ^{2}L_{n}^{2}}}}$.

• In the limit of ${\displaystyle \xi \rightarrow \infty }$ the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor ε
• Since the Butterworth filter is a limiting form of the Chebyshev filter, it follows that in the limit of ${\displaystyle \xi \rightarrow \infty }$, ${\displaystyle \omega _{0}\rightarrow 0}$ and ${\displaystyle \epsilon \rightarrow 0}$ such that ${\displaystyle \epsilon \,R_{n}(\xi ,1/\omega _{0})=1}$ the filter becomes a Butterworth filter
• In the limit of ${\displaystyle \xi \rightarrow \infty }$, ${\displaystyle \epsilon \rightarrow 0}$ and ${\displaystyle \omega _{0}\rightarrow 0}$ such that ${\displaystyle \xi \omega _{0}=1}$ and ${\displaystyle \epsilon L_{n}=\alpha }$, the filter becomes a Chebyshev type II filter with gain

${\displaystyle G(\omega )={\frac {1}{\sqrt {1+{\frac {1}{\alpha ^{2}T_{n}^{2}(1/\omega )}}}}}}$

Poles and zeroes[edit]

The zeroes of the gain of an elliptic filter will coincide with the poles of the elliptic rational function, which are derived in the article on elliptic rational functions.

The poles of the gain of an elliptic filter may be derived in a manner very similar to the derivation of the poles of the gain of a type I Chebyshev filter. For simplicity, assume that the cutoff frequency is equal to unity. The poles ${\displaystyle (\omega _{pm})}$ of the gain of the elliptic filter will be the zeroes of the denominator of the gain. Using the complex frequency ${\displaystyle s=\sigma +j\omega }$ this means that:

${\displaystyle 1+\epsilon ^{2}R_{n}^{2}(-js,\xi )=0\,}$

Defining ${\displaystyle -js=\mathrm {cd} (w,1/\xi )}$ where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:

${\displaystyle 1+\epsilon ^{2}\mathrm {cd} ^{2}\left({\frac {nwK_{n}}{K}},{\frac {1}{L_{n}}}\right)=0\,}$

where ${\displaystyle K=K(1/\xi )}$ and ${\displaystyle K_{n}=K(1/L_{n})}$. Solving for w

${\displaystyle w={\frac {K}{nK_{n}}}\mathrm {cd} ^{-1}\left({\frac {\pm j}{\epsilon }},{\frac {1}{L_{n}}}\right)+{\frac {mK}{n}}}$

where the multiple values of the inverse cd() function are made explicit using the integer index m.

The poles of the elliptic gain function are then:

${\displaystyle s_{pm}=i\,\mathrm {cd} (w,1/\xi )\,}$

As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form (Lutovac & et al. 2001, § 12.8)

${\displaystyle s_{pm}={\frac {a+jb}{c}}}$

${\displaystyle a=-\zeta _{n}{\sqrt {1-\zeta _{n}^{2}}}{\sqrt {1-x_{m}^{2}}}{\sqrt {1-x_{m}^{2}/\xi ^{2}}}}$

${\displaystyle b=x_{m}{\sqrt {1-\zeta _{n}^{2}(1-1/\xi ^{2})}}}$

${\displaystyle c=1-\zeta _{n}^{2}+x_{i}^{2}\zeta _{n}^{2}/\xi ^{2}}$

where ${\displaystyle \zeta _{n}}$ is a function of ${\displaystyle n,\,\epsilon }$ and ${\displaystyle \xi }$ and ${\displaystyle x_{m}}$ are the zeroes of the elliptic rational function. ${\displaystyle \zeta _{n}}$ is expressible for all n in terms of Jacobi elliptic functions, or algebraically for some orders, especially orders 1,2, and 3. For orders 1 and 2 we have

${\displaystyle \zeta _{1}={\frac {1}{\sqrt {1+\epsilon ^{2}}}}}$

${\displaystyle \zeta _{2}={\frac {2}{(1+t){\sqrt {1+\epsilon ^{2}}}+{\sqrt {(1-t)^{2}+\epsilon ^{2}(1+t)^{2}}}}}}$

where

${\displaystyle t={\sqrt {1-1/\xi ^{2}}}}$

The algebraic expression for ${\displaystyle \zeta _{3}}$ is rather involved (See Lutovac & et al. (2001, § 12.8.1)).

The nesting property of the elliptic rational functions can be used to build up higher order expressions for ${\displaystyle \zeta _{n}}$:

${\displaystyle \zeta _{m\cdot n}(\xi ,\epsilon )=\zeta _{m}\left(\xi ,{\sqrt {{\frac {1}{\zeta _{n}^{2}(L_{m},\epsilon )}}-1}}\right)}$

where ${\displaystyle L_{m}=R_{m}(\xi ,\xi )}$.

Minimum Q-factor elliptic filters[edit]

See Lutovac & et al. (2001, § 12.11, 13.14).

Elliptic filters are generally specified by requiring a particular value for the passband ripple, stopband ripple and the sharpness of the cutoff. This will generally specify a minimum value of the filter order which must be used. Another design consideration is the sensitivity of the gain function to the values of the electronic components used to build the filter. This sensitivity is inversely proportional to the quality factor (Q-factor) of the poles of the transfer function of the filter. The Q-factor of a pole is defined as:

${\displaystyle Q=-{\frac {|s_{pm}|}{2\mathrm {Re} (s_{pm})}}=-{\frac {1}{2\cos(\arg(s_{pm}))}}}$

and is a measure of the influence of the pole on the gain function. For an elliptic filter, it happens that, for a given order, there exists a relationship between the ripple factor and selectivity factor which simultaneously minimizes the Q-factor of all poles in the transfer function:

${\displaystyle \epsilon _{Qmin}={\frac {1}{\sqrt {L_{n}(\xi )}}}}$

This results in a filter which is maximally insensitive to component variations, but the ability to independently specify the passband and stopband ripples will be lost. For such filters, as the order increases, the ripple in both bands will decrease and the rate of cutoff will increase. If one decides to use a minimum-Q elliptic filter in order to achieve a particular minimum ripple in the filter bands along with a particular rate of cutoff, the order needed will generally be greater than the order one would otherwise need without the minimum-Q restriction. An image of the absolute value of the gain will look very much like the image in the previous section, except that the poles are arranged in a circle rather than an ellipse. They will not be evenly spaced and there will be zeroes on the ω axis, unlike the Butterworth filter, whose poles are arranged in an evenly spaced circle with no zeroes.

Construction from Chebyshev transmission zeros[edit]

Elliptic filter stop bands are essentially Chebyshev filters with transmission zeros where the transmission zeros are arranged in a manner that yields an equi-ripple stop band. Given this, it is possible to convert a Chebyshev filter characteristic equation, ${\displaystyle K(s)}$ containing the Chebyshev reflection zeros in the numerator and no transmission zeros in the denominator, to an Elliptic filter ${\displaystyle K(s)}$ containing the Elliptic reflection zeros in the numerator and Elliptic transmission zeros in the denominator, by iteratively creating transmission zeros from the scaled inverse of the Chebyshev reflection zeros, and then reestablishing an equi-ripple Chebyshev pass band from the transmission zeros, and repeating until the iterations produce no further changes of significance to ${\displaystyle K(s)}$^[1]. The scaling factor used, ${\displaystyle \Omega _{c}}$, is the stop band to pass band cutoff frequency ratios and is also known as the inverse of the "selectivity factor"^[2]. Since Elliptic designs are generally specified from the stop band attenuation requirements, ${\displaystyle \Omega _{c}}$, may be derived from the equations that establish the minimum order, n^[2], for given stop and pass band attenuation and frequency requirements.

The characteristic polynomials, ${\displaystyle K(s)}$, may then be translated to the transfer function polynomials, ${\displaystyle G(s)}$ with the classic translation, ${\displaystyle G(s)={\sqrt {1/(1+\varepsilon ^{2}K(s)K(-s))}}{\bigg |}_{\text{LHP poles}}}$ where ${\displaystyle \varepsilon ^{2}=10^{Ap/10.}-1.}$ and ${\displaystyle A_{p}}$ is the pass band ripple^[1][3].

${\displaystyle {\begin{aligned}{\text{Given:}}&\\A_{p}&={\text{ pass band ripple in dB}}\\A_{s}&={\text{ stop band ripple in dB}}\\\omega _{p}&={\text{ maximum pass band frequency}}\\\omega _{s}&={\text{ minimun stop band frequency}}\\D&={\frac {10^{A_{s}/10}-1}{10^{A_{p}/10}-1}}\\\\{\text{Then:}}&\\k&={\text{ the selectivity factor }}={\frac {\omega _{p}}{\omega _{s}}}\\u&={\frac {1-{\sqrt[{4}]{1-k^{2}}}}{2(1+{\sqrt[{4}]{1-k^{2}}})}}\\q&=u+2u^{5}+15u^{9}+150u^{13}\\D&={\frac {10^{A_{s}/10}-1}{10^{A_{p}/10}-1}}\\n&={\text{ minimum order }}=ceil{\bigg (}{\frac {\log {16D}}{\log(1/q)}}{\bigg )}\\\end{aligned}}}$

the ${\displaystyle \omega _{s}/\omega _{p}}$ ratio, ${\displaystyle \Omega _{c}}$ may be derived by working the problem above backwards from n to find ${\displaystyle \Omega _{c}}$.

${\displaystyle {\begin{aligned}q&=(16D)^{(-1/n)}\\0&=-q+u+2u^{5}+15u^{9}+150u^{13}\\u&={\text{ real root of above equation}}\\k&={\text{ the selectivity factor }}={\sqrt {1-{\bigg (}{\frac {1-2u}{1+2u}}{\bigg )}^{4}}}\\\Omega _{c}&={\frac {\omega _{s}}{\omega _{p}}}={\frac {1}{k}}={\frac {1}{\sqrt {1-{\bigg (}{\frac {1-2u}{1+2u}}{\bigg )}^{4}}}}\\\end{aligned}}}$

Simple example[edit]

Design an Elliptic filter with a pass band ripple of 1 dB from 0 to 1 rad/sec and a stop band ripple of 40 dB from at least 1.25 rad/sec to ${\displaystyle \infty }$.

Applying the calculations above for the value for n prior to applying the ceil() function, n is found to be 4.83721900 rounded up to the next integer, 5, by applying the ceil() function, which means a 5 pole Elliptic filter is required to meet the specified design requirements. Applying the calculations above for ${\displaystyle \Omega _{c}}$ needed to design a stop band of exactly 40dB of attenuation, ${\displaystyle \Omega _{c}}$ is found to be 1.2186824.

The polynomial scaled inversion function may be performed by translating each root, s, to ${\displaystyle \Omega _{c}/s}$, which may be easily accomplished by inverting the polynomial and scaling it by ${\displaystyle \Omega _{c}}$, as shown.

${\displaystyle {\begin{aligned}&as^{n}+bs^{n-1}\dots cs^{2}+ds^{1}+es^{0}\Longrightarrow ({\frac {e}{\Omega _{c}^{n}}})s^{n}+({\frac {d}{\Omega _{c}^{n-1}}})s^{n-1}\dots ({\frac {c}{\Omega _{c}^{2}}})s^{2}+({\frac {b}{\Omega _{c}^{1}}})s^{1}+({\frac {a}{\Omega _{c}^{0}}})s^{0}\\\end{aligned}}}$

The Elliptic design steps are then as follows^[1]:

1. Design a Chebyshev filters with 1 dB pass band ripple.
2. Invert all the reflections zeros about ${\displaystyle \Omega _{c}}$to create transmission zeros
3. Create an equi-ripple pass band from the transmission zeros using the process outlined is Chebyshev transmission zeros
4. Repeat steps 2 and 3 until both the pass band and stop band no longer change by any appreciable amount. Typically, 15 to 25 iterations produce coefficient differences in the order of than 1.e-15.

To illustrate the steps, the below K(s) equations begin with a standard Chebyshev K(s), then iterate through the process. Visible differences are seen in the first three iterations. By time 18 iterations have been reached, the differences in K(s) become negligible. Iterations may be discontinued when the change in K(s) coefficients becomes sufficiently small so as to meet design accuracy requirements. The below K(s) iterations have all been normalized such that ${\displaystyle |K(j)|=1}$, however, this step may be postponed until the last iteration, if desired.

${\displaystyle {\begin{aligned}{\text{iteration 0: }}K(s)&={\frac {16s^{5}+20s^{3}+5s}{1}}\\{\text{iteration 1: }}K(s)&={\frac {9.2965947s^{5}+12.999133s^{3}+4.0025668s}{0.14167325s^{4}+0.84164496s^{2}+1}}\\{\text{iteration 2: }}K(s)&={\frac {8.6496472s^{5}+12.270597s^{3}+3.8746611s}{0.19518773s^{4}+0.94147634s^{2}+1}}\\&\vdots \\{\text{iteration 17: }}K(s)&={\frac {8.550086786383502s^{5}+12.157269873073034s^{3}+3.854163602012615s}{0.2043607336740334s^{4}+0.9573802183509494s^{2}+1}}\\{\text{iteration 18: }}K(s)&={\frac {8.550086786383422s^{5}+12.157269873072942s^{3}+3.854163602012599s}{0.2043607336740334s^{4}+0.9573802183509494s^{2}+1}}\\\end{aligned}}}$

To find the ${\displaystyle G(s)}$ transfer function, do the following^[1].

${\displaystyle {\begin{aligned}\varepsilon ^{2}&=10^{1dB/10.}-1.=.25892541\\G(s)&={\sqrt {G(s)G(-s)}}{\bigg |}_{\text{LHP poles}}={\sqrt {\frac {1}{1+\varepsilon ^{2}K(s)K(-s)}}}{\bigg |}_{\text{LHP poles}}={\sqrt {\frac {K(s)_{den}K(-s)_{den}}{K(s)_{den}K(-s)_{den}+\varepsilon ^{2}K(s)_{num}K(-s)_{num}}}}{\bigg |}_{\text{LHP poles}}\\&={\frac {0.20436073s^{4}+0.95738022s^{2}+1}{{\sqrt {-18.928479s^{10}-53.78661s^{8}-54.942632s^{6}-22.939175s^{4}-1.931467+1}}{\bigg |}_{\text{LHP poles}}}}\\\end{aligned}}}$

To obtain ${\displaystyle G(s)}$ from the left half plane, factor the numerator and denominator to obtain the roots using a root finding algorithm. Discard all roots from the right half plane of the denominator, half the repeated roots in the numerator, and rebuild ${\displaystyle G(s)}$ with the remaining roots^[1][3]. Generally, normalize ${\displaystyle |G(s)|}$ to 1 at ${\displaystyle s=0}$.

${\displaystyle {\begin{aligned}&G(s)={\frac {0.20436073s^{4}+0.95738022s^{2}+1}{4.3506872s^{5}+4.0174213s^{4}+8.0362343s^{3}+4.9129149s^{2}+3.4288915s+1}}\\\end{aligned}}}$

To confirm that the example ${\displaystyle G(s)}$ is correct, the plot of ${\displaystyle G(s)}$ along ${\displaystyle j\omega }$ is shown below with a pass band ripple of 1 dB, a cut off frequency of 1 rad/sec, a stop band attenuation of 40 dB beginning at 1.21868 rad/sec

Comparison with other linear filters[edit]

Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients:

As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth.

References[edit]

• Daniels, Richard W. (1974). Approximation Methods for Electronic Filter Design. New York: McGraw-Hill. ISBN 0-07-015308-6.
• Lutovac, Miroslav D.; Tosic, Dejan V.; Evans, Brian L. (2001). Filter Design for Signal Processing using MATLAB and Mathematica. New Jersey, USA: Prentice Hall. ISBN 0-201-36130-2.