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.

Filters primarily concerned with gain response

Polynomial	Introduced by	Filter	Introduced by	Properties	Examples/definition
Butterworth polynomials, $B_{n}(x)$	Stephen Butterworth (1930)	Butterworth filter (Maximally-flat filter)	Stephen Butterworth (1930)	The Butterworth filter is maximally flat. That is, it is the flattest response possible in the passband with a monotonically increasing insertion loss with frequency. However, the transition band is relatively wide in the Butterworth. Other filter types can achieve a much sharper cutoff in exchange for variations (ripple) in loss in the passband.	$F_{n}(\omega )=\omega ^{n}\ .$ ${\begin{aligned}&{\frac {1}{H_{3}(s)}}=B_{3}(s)=s^{3}+2s^{2}+2s+1\\&{\frac {1}{H_{4}(s)}}=s^{4}+2.613s^{3}+3.414s^{2}+2.613s+1\ .\end{aligned}}$
Chebyshev polynomials, $T_{n}(x)$	Pafnuty Chebyshev (1854)	Chebyshev filter (Chebyshev type I)	Wilhelm Cauer (1958, published posthumously)	The Chebyshev filter has the steepest possible roll-off for a given passband ripple and monotonically increasing loss in the stopband. The sharpness of the transition of the Chebyshev can only be improved by relaxing the latter requirement – that is, allowing stopband ripple. Increasing the allowed passband ripple results in increased steepness of the transition, but even an insignificantly small ripple produces a great improvement over the Butterworth. The ripple is equiripple; that is, the peaks in insertion loss are all equal.	${\begin{aligned}&F_{n}(\omega )=T_{n}(\omega )\\&F_{3}(\omega )=T_{3}(\omega )=4\omega ^{3}-3\omega \\&F_{4}(\omega )=T_{4}(\omega )=8\omega ^{4}-8\omega ^{2}+1\ .\end{aligned}}$
		Inverse Chebyshev filter (Chebyshev type II)	-	The inverse Chebyshev filter is maximally flat at low frequency and monotonic in the passband. It has superior roll-off to the Butterworth, but not as good as the regular Chebychev (also known as the type I Chebyshev). The stopband has equiripple set by $/varepsilon$ and consequently does not have such good stopband rejection since the insertion loss will start to fall again at some frequency above the transition band.	$\varepsilon F_{n}(\omega )={\dfrac {1}{\varepsilon 'T_{n}(\omega )}}\ .$
Elliptic rational functions, $R_{n}(\xi ,x)$	Yegor Ivanovich Zolotarev (1877)	Elliptic filter (Cauer filter, or sometimes Zolotarev filter)	Wilhelm Cauer (1931)	The elliptic filter has the steepest possible roll-off of any filter in exchange for equiripple in both the passband and stopband. Because the transition is much sharper than even a Chebyshev filter, the same performance can be achieved with a lower order filter. The ripple in the passband and stopband can be set independently if desired, but maximum roll-off is achieved when they are equal.	$F_{n}(\omega )=R_{n}(\xi ,\omega )$
Legendre polynomials, $P_{n}(x)$	Adrien-Marie Legendre (written 1782, published 1785)	Legendre filter (Legendre-Papoulis filter, Optimum "L" filter, or Papoulis filter)	Athanasios Papoulis (1958)	The Legendre filter is a compromise between Butterworth and Chebyshev responses. First and second-order Legendre filters are identical to Butterworth filters; they only differ from third-order onwards. Like the Butterworth, the Legendre filter has a monotonically increasing loss with frequency. However, it is not maximally flat, meaning its gradient does not approach zero at low frequency, which rules it out for some hi-fi audio applications. The Legendre filter has a faster roll-off than the Butterworth, but still not as good as the Chebyshev. In fact, the Legendre has the fastest possible roll-off at the cutoff frequency while maintaining a monotonic response. It is in this sense that it is described as "optimum".	${\begin{aligned}&F_{n}(\omega )=P_{n}(\omega )\\&F_{3}(\omega )=P_{3}(\omega )={\frac {1}{2}}(5\omega ^{3}-3\omega )\\&F_{4}(\omega )=P_{4}(\omega )={\frac {1}{8}}(35\omega ^{4}-30\omega ^{2}+3)\ .\end{aligned}}$
Associated Legendre polynomials, $P_{n}^{(m)}(x)$		Associated Legendre filter	Yu Hsiu Ku and Meier Drubin (1962)	Associated Legendre filters are another compromise between Butterworth and Chebyshev filters. With $m=0$ they are identical to Legendre filters. First-modified and second-modified associated Legendre filters (that is $m=1$ and $m=2$ respectively) are useful in filter design because these have the an improved group delay response in the passband. These filters have some ripple. It is not equiripple, but is generally less than the equivalent Chebyshev.	$F_{n}^{(m)}(\omega )=P_{n}^{(m)}(\omega )$ ${\begin{aligned}&F_{3}^{(2)}(\omega )=P_{3}^{(2)}(\omega )=15\omega ^{3}-3\omega )\\&F_{4}^{(2)}(\omega )=P_{4}^{(2)}(\omega )={\frac {15}{2}}(7\omega ^{2}-1)(1-\omega ^{2})\ .\end{aligned}}$
Zolotarev polynomials, $Z_{n}(x|\kappa )$	Yegor Ivanovich Zolotarev (1868)	Achieser-Zolotarev filter (Zolotarev filter)	Ralph Levy (1970)	The Zolotarev filter has similar properties to the Chebyshev filter. There is equiripple response in the passband which can be increased by the designer in exchange for increased roll-off. The big difference to the Chebyshev is that the lowest frequency ripple is a peak in attenuation larger than the rest. The Zolotarev has a better impedance match than the Chebyshev in some applications, and when implemented in waveguide has more practical component dimensions.	$F_{n}(\omega )=Z_{n}(\omega |\kappa )$
Gegenbauer polynomials (Ultraspherical polynomials), $\varGamma _{n}^{(\alpha )}(x)$	Leopold Gegenbauer (1877)	Ultraspherical filter	David E. Johnson and Johnny R. Johnson (1966)	The Gegenbauer polynomials are a generalisation of several other classes of polynomial (see table below). With a suitable choice of $\alpha$, they can be used to construct filters that are a compromise between two other classes, thus obtaining some of the good features of both.	$F_{n}^{(\alpha )}(\omega )={\dfrac {n!\varGamma _{n}^{(\alpha )}(\omega )}{\Pi _{k=1}^{n}(k+\alpha )}}$
Hermite polynomials, $H_{n}(x)$	Pafnuty Chebyshev (1859) Charles Hermite (1864)		Yu Hsiu Ku and Meier Drubin (1962)	Filters built have Hermite polynomials have ripple in the passband with the ripple nearest the cutoff frequency being the largest (compare with Zolotarev polynomials where the largest ripple is furthest from cutoff). The steepness of the roll-off is comparable to $m=2$ Associated Legendre (only marginally worse) but the group delay is comparable to the Chebyshev. From this perspective, there is little benefit in using it over Legendres.	${\begin{aligned}&r_{n}F_{n}(\omega )=H_{n}(\omega )\\&r_{3}F_{3}(\omega )=H_{3}(\omega )=8(\omega ^{3}-{\frac {3}{2}}\omega )\\&r_{4}F_{4}(\omega )=H_{4}(\omega )=16(\omega ^{4}-3\omega ^{2}+{\frac {3}{4}})\ .\end{aligned}}$

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.

Filters primarily concerned with delay response

Polynomial	Introduced by	Filter	Introduced by	Properties	Examples/definition
Reverse Bessel polynomials, $\theta _{n}(x)$	Leonard Carlitz (1957), named after Friedrich Bessel and Bessel polynomials	Bessel filter (Bessel–Thomson filter)	W. E. Thomson (1949)	Bessel filters have maximally flat time delay in the passband in the same way that Butterworth filters have maximally flat gain. Like the Butterworth filters, the Bessel filter has monotonically increasing loss with frequency. The roll-off and stopband rejection are both worse in the Bessel filter than the Butterworth filter which itself has the worst of these characteristics amongst the common filter types. The Bessel filter has the best phase linearity and flattest group delay.	$H_{n}(s)={\frac {\theta (0)}{\theta _{n}(s)}}.$
Polynomials not named		Gaussian filter	M. Dishal (1959)	The Gaussian filter has a transcendental transfer function and hence cannot be expressed exactly as a polynomial or realised in a practical filter. However, it can be very closely approximated with a polynomial with various approximation theory techniques. The Gaussian filter has less time-delay than the Bessel filter, but a slightly worse shaping factor. The flatness of the group delay and phase delay is slightly worse than the Bessel, but still good. The Gaussian filter has applications in radar and image processing due to its pulse shaping properties.
Polynomials not named		Hourglass filter	Byron J. Bennett (1988)	Hourglass is more a design technique than a specific class of filter. The technique explicitly places the transfer and reflection functions zeros. With this method, a filter having both equiripple magnitude response (Chebyshev filter) and equiripple time-delay in the passband can be achieved. Bennett called these filters hourglass because the most basic invocations of hourglass filters (but by no means oall of them) have coefficients that are symmetrical about the centre term and increase in magnitude towards each end.

Classification

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

Jacobi polynomials $\varGamma _{n}^{(\alpha ,\beta )}(\omega )$	$\alpha =\beta$	Gegenbauer polynomials $\alpha >-1$	Butterworth polynomials $\alpha =\infty$
			Chebyshev polynomials $\alpha =-0.5$
			Legendre polynomials $\alpha =0$
			Associated Legendre polynomials $\alpha =m\in \mathbb {N}$
		$\alpha \leq -1$	Filters using polynomials in this range are not guaranteed to have responses with all the zeroes in the passband and monotonically increasing loss in the stopband.
	$\alpha \neq \beta$		Polynomials with this condition cannot be even functions, and are thus not so useful for analogue filter design.
Laguerre polynomials	Hermite polynomials For the relationship, see Hermite polynomials § Laguerre polynomials.

Notes

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.

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Unprocessed sources

• More on Legendre filters [1][2]
• Legendre and Laguerre filters are used in transversal filters [3]. Laguerre polynomials
• Mentions parabolic and Halpern approximations (mostly academic interest, related to Jacobi polynomials) as well as Legendre [4]. Also mentions Laguerre [5]
  • 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

• 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.

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.