Elliptic filter

[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.

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 (Lutovac & et al. 2001, § 12.8) where

The nesting property of the elliptic rational functions can be used to build up higher order expressions for

[1] The equations account for standard low pass Elliptic filters, only.

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.

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.

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

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.

[2] Since Elliptic designs are generally specified from the stop band attenuation requirements,

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

needed to design a stop band of exactly 40 dB of attenuation,

The polynomial scaled inversion function may be performed by translating each root, s, to

Iterations may be discontinued when the change in K(s) coefficients becomes sufficiently small so as to meet design accuracy requirements.

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 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 S12 values that exceed the S12 value at

If it is not feasible to design the filter with one of the terminations increased or decreased to accommodate the pass band S12, then the Elliptic transfer function must be modified so as to move the lowest even order reflection zero to

numerator, and the highest frequency transmission zero may be found be factoring the

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

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

It is important to note that all applications require both pass and stop translations.

is completed, an equi-ripple transfer function is created with scattering matrix values for S12 of 1

iterations complete if it is desired to preserve the pass and stop band attenuations.

[6] 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.

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 3.01 dB cut-off frequency.

Newton's method or solving the equations directly with a root finding algorithm may be used to determine the 3.01 dB attenuation frequency.

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

does not contain any phase information, directly factoring the transfer function will not produce usable results.

The frequency response of a fourth-order elliptic low-pass filter with ε = 0.5 and ξ = 1.05. Also shown are the minimum gain in the passband and the maximum gain in the stopband, and the transition region between normalized frequency 1 and ξ
A closeup of the transition region of the above plot.
Log of the absolute value of the gain of an 8th order elliptic filter in complex frequency space (s = σ + jω) with ε = 0.5, ξ = 1.05 and ω 0 = 1. The white spots are poles and the black spots are zeroes. There are a total of 16 poles and 8 double zeroes. What appears to be a single pole and zero near the transition region is actually four poles and two double zeroes as shown in the expanded view below. In this image, black corresponds to a gain of 0.0001 or less and white corresponds to a gain of 10 or more.
An expanded view in the transition region of the above image, resolving the four poles and two double zeroes.
The normalized Q-factors of the poles of an 8-th order elliptic filter with ξ = 1.1 as a function of ripple factor ε . Each curve represents four poles, since complex conjugate pole pairs and positive-negative pole pairs have the same Q-factor. (The blue and cyan curves nearly coincide). The Q-factor of all poles are simultaneously minimized at ε Qmin = 1 / √ L n = 0.02323...
Five pole Elliptic simulation
Even order modified Elliptic illustration
Even order modified Elliptic illustration
Hourglass reciprocal S11 and S12 frequency response
7 pole Hourglass reciprocal S11 and S12 frequency response