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). 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:

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

1. is the cutoff frequency

2. is the ripple factor

3. 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

• .

• which is defined as:

• .

• the elliptic rational function becomes a Chebyshev polynomial, and therefore the filter becomes a Chebyshev type I filter, with ripple factor ε

• the filter becomes a Butterworth filter

• , the filter becomes a Chebyshev type II filter with gain

Poles and zeroes

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 this means that:

where cd() is the Jacobi elliptic cosine function and using the definition of the elliptic rational functions yields:

. Solving for w

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:

As is the case for the Chebyshev polynomials, this may be expressed in explicitly complex form

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

where

The algebraic expression is rather involved (See ).

The nesting property of the elliptic rational functions:

.

Design considerations

See .

Elliptic filters are generally specified by requiring a particular value for the passband ripple, αp, stopband ripple, αs, and the sharpness of the cutoff. These specify a minimum order for the rational elliptic function describing the filter behavior. Specifically, consider a filter passing frequencies below and stopping frequencies above To constrain the passband gain into and stopband gain into , the order must satisfyThat equation arises from an approximation to the complete elliptic integral function K; an exact formulation isAnother 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 filter transfer function. The Q-factor of a pole is defined as:

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:

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.

Comparison with other linear filters

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.

Construction from Chebyshev transmission zeros

Elliptic filter stop bands are essentially Chebyshev filters with transmission zeros where the transmission zeros., is the stop band to pass band cutoff frequency ratios and is also known as the inverse of the "selectivity factor"., may be derived from the equations that establish the minimum order, n, above.

may be derived by working the minimum order, n, problem above backwards from n.

is the pass band ripple.

Simple example

Design an Elliptic filter with a pass band ripple of 1dB from 0 to 1 rad/sec and a stop band ripple of 40.

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() needed to design a stop band of exactly 40 is found to be 1.2186824.

The polynomial scaled inversion function may be performed by translating each root, s, as shown.

The Elliptic design steps are then as follows:

1. Design a Chebyshev filters with 1dB pass band ripple.

2. to create transmission zeros

3. Create an equi-ripple pass band from the transmission zeros using the process outlined in 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.

, however, this step may be postponed until the last iteration, if desired.

transfer function, do the following.

from the left half plane, factor the numerator and denominator to obtain the roots using a root finding algorithm with the remaining roots..

is shown below with a pass band ripple of 1dB, a cut off frequency of 1 rad/sec, a stop band attenuation of 40dB beginning at 1.21868 rad/sec

Even order modifications

Even order Elliptic filters implemented with passive elements, typically inductors, capacitors, and transmission lines, with terminations of equal value on each side cannot be implemented with the traditional Elliptic transfer function without the use of coupled coils, which may not be desirable or feasible. This is due to the physical inability to accommodate the even order Chebyshev reflection zeros and transmission zeros that result in the scattering matrix while maintaining the equi-ripple response of the pass band and stop band.

denominator.

..

Where:

is the original Elliptic function zero or pole

is the mapped zero or pole for the modified even order transfer function.

is the lowest frequency reflection zero in the pass band.

.

..

Where:

is the original Elliptic function zero or pole

is the mapped zero or pole for the modified even order transfer function.

is the highest frequency transmission zero in the pass band.

must be negative to conform to the left half plane requirement.

All applications require both pass and stop translations. Passive network diplexers, for example, only require even order stop band translations, and perform more efficiently with untranslated even order pass bands.

is completed, an equi-ripple transfer function is created with scattering matrix, which may be implements with passive equally terminated networks.

while maintaining an equi-ripple pass band and stop band frequency response. . Likewise, an order, n, computation may result in a smaller value than the actual required order.

Hourglass implementation

An Hourglass filter is a special case of filter where the reflection zeros, are the reciprocal of the transmission zeros about a 3.01dB normalized cut-off attenuation frequency of 1 rad/sec, resulting all poles of the filter residing on the unit circle. The Elliptic Hourglass implementation has an advantage over an Inverse Chebyshev filter in that the pass band is flatter, and has an advantage over traditional Elliptic filters in that the group delay has a less sharp peak at the cut-off frequency.

Syntheses process

The most straightforward way to synthesize an Hourglass filter is to design an Elliptic filter with a specified design stop band attenuation, As, and a calculated pass band attenuation that meets the lossless two-port network requirement that scattering parameters., algebraic manipulation yields the following pass band attenuation calculated requirement.

The Ap, defined above will produce reciprocal reflection and transmission zeros about a yet unknown 3.01dB cut-off frequency. to Design an Elliptic filter with a pass band frequency of 1 rad/sec the 3.01dB attenuation frequency needs to be determined and that frequency needs to be used to inversely scale the Elliptic design polynomials. The result will be polynomials with an attenuation of 3.01dB at a normalized frequency of 1 rad/sec. Newton's method or solving the equations directly with a root finding algorithm may be used to determine the 3.01dB attenuation frequency.

Frequency scaling with Newton's method

is the Hourglass transfer function to find the 3.01 is the 3

1. .

2. . For example, 3.010dB converts to 0.5, 1dB converts to 0.79432823 and so on.

3. . Always take the absolute value.

4. . DO NOT take the absolute value of the derivative.

When steps 1) through 4) are complete, the expression involving Newton's method may be written as:

will then be virtually the exact desired value at 1 rad/sec. If performed properly, only a handful of iterations are needed to set the attenuation through a wide range of desired attenuation values for both small and very large order filters.

Frequency scaling with root finding

to be real at all frequencies, and then finding the frequency that result on the square of the desired attention.

1. .

2. . For example, 3.010dB converts to 0.5, 1dB converts to 0.79432823 and so on.

3. Find the roots of P(S) using a root finding algorithm.

4. .

Scaling the transfer function

has been determined, the Hourglass transfer function polynomial may be scaled as follows:

Even order modifications

Even order Hourglass filters have the same limitations regarding equally terminated passive networks as other Elliptic filters. The same even order modifications that resolve the problem with Elliptic filters also resolve the problem with Hourglass filters.

References