User:Spinningspark/Work in progress/List of polynomials used in filter design

Polynomials, or rational functions (the ratio of two polynomials), are commonly used in the design of electronic filters. Design usually begins with a low-pass prototype filter. That is, a low-pass filter whose cutoff frequency is set to $$\omega = 1$$ where $$\omega$$ is the angular frequency. The ideal behaviour of such a filter is to pass all frequencies below $$\omega = 1$$ (the passband) with zero insertion loss, stop all frequencies above $$\omega = 1$$ (the stopband) with infinite insertion loss, and have a sharp transition between the two (a transition band of zero width, or, equivalently, infinite roll-off). Such a filter is known as a brick-wall filter, but this ideal behaviour is not possible to achieve, even in theory, for causality reasons. It can only be approximated to, and this list is of filters that use polynomial function approximations.

Defining the gain of the filter, $$G(\omega)$$, as the magnitude of the transfer function, $$H(s)$$, such that,


 * $$ G(\omega) = |H(j \omega)|$$

then the gain of many filters takes the form,


 * $$G(\omega)^2 = \dfrac {1}{1+[\varepsilon F(\omega)]^2}$$


 * where
 * $$F(\omega)$$ is some polynomial or rational function. The choice of $$F(\omega)$$ depends on the features that are most desired in the filter.  It is not possible to simultaneously optimise all desirable features.
 * $$\varepsilon$$ is a parameter related to the peaks of passband insertion loss ripple in filters that have this feature. The relationship between this loss, $$L$$, in decibels and $$\varepsilon$$ is given by,


 * $$ L = 10 \log (\varepsilon^2 +1) \ .$$

Throughout this article, the transfer functions and gain functions given are for prototype filters with the cutoff frequency scaled to $$\omega_c = 1$$.

List of polynomials
In the following table, examples are generally only given for third and fourth order ($$n=3$$ and $$n=4$$ respectively) polynomials. For more extensive tables, generating functions, and recurrence relationships, see the individual articles.

Meaningless heading
In some applications, the time-delay response of the filter is more important than its gain response. These are applications where it is important to preserve the wave shape of the signal. For these applications, the ideal response is to have the same delay at all passband frequencies. This is crucial in analogue video and television applications and has some importance in radar and data transmission.

Classification
Many filter types are special cases of more general polynomials. Some of this heirarchy is captured in the table below.

Unincorporated notes

 * Paarmann p.16


 * Halpern extended Papoulis for monotonic band falloff using Jacobi polynomials (1969)
 * Scanlan introduced filters with poles lying on an ellipse equally spaced in frequency (1965) The eccentricity of the ellipse trades magnitude response with time-domain response.
 * Attikiouzel and Phuc (1978) - ultraspherical and modified ultraspherical polynomials with a single parameter determining transitional forms.
 * Rabrenovic and Lutovac (1992) - extension to Cauer filters using quasi-elliptical functions and elliptical filters without the need to invoke elliptical functions.


 * Paarmann p.16 - time-delay filters


 * Macnee (1963) introduced a filter with a Chebychev approximation to constant time-delay.
 * Bunker (1970) Chebychev polynomials, ripple in time-delay or phase. (How is this different from Macnee's filter?)
 * Ariga and Masamitsu (1970)
 * Halpern (1976) used hyperbolic function approximation to improve low-order Bessel filters.

Unprocessed sources

 * More on Legendre filters
 * Legendre and Laguerre filters are used in transversal filters . Laguerre polynomials
 * Mentions parabolic and Halpern approximations (mostly academic interest, related to Jacobi polynomials) as well as Legendre . Also mentions Laguerre
 * Also diminishing ripple (good for component accuracy variations), Lerner, Gauss (Wanhammar, pp.57-58). Not much call for Gauss and Bessel as modern numerical programmes can simultaneously optimise multiple parameters. Hilbert filter is an interesting one, approximates to +/- 90°.

Sources to request from library
The last one is not on IEEEXplore. As far as I can discover, they only have this conference's papers back to 1989. Paper copies are shown in several institutions in Worldcat.
 * Macnee, A., "Chebyshev approximation of a constant group delay", IEEE Transactions on Circuit Theory, vol. 10, iss. 2, pp. 284-285, June 1963.
 * Bennett, B.J., "The hourglass and constant time delay", Conference Proceedings of the 28th Midwest Symposium on Circuits and Systems, pp. 364-368, August 1985.