The paper considers a new class of filters with transfer functions using the polynomials introduced by the British physicist Sir G.G. Stokes. The filters are transitional between Bessel and Butterworth ones. When one compares the step-responses of Bessel, Stokes, and Butterworth filters of the same order and the same −3dB bandwidth then the result is following. The Bessel filters have a certain delay and (practically) no overshoot in their step-transient response. The Stokes filters have larger delay than Bessel filters and a small overshoot. Finally, the Butterworth filters have largest delay and largest overshoot. The paper illustrates these properties and describes the data required for synthesis of the Stokes filters.

Stokes polynomial filters

A method for generating an all-pole stable transfer function to approximate a constant group delay is developed. The mean square error between the actual group delay and the specified group delay is minimized. Also, the computational complexity is considerably reduced.

A new approximation of constant group delay in the analog domain

The properties of Bessel (Thomson) filters are well known'<\" and extensively described\" 7.8 and generalized'\"'!\". The usual approach'r? to obtain the transfer function of such filters and explain the maximalllatness of their delay characteristic involves Bessel functions. In the following we derive this result using trigonometric functions only. The ideal normalized delay network transfer function is given by H(s)=e-' and the phase f3(w) associated with this function is linear f3(w)= -w Thus tan f3(w) = -tan co It is known IS that tan w can be expanded into a continued fraction w tan w= -----------

One Simple Method for the Derivation of Bessel (Thomson) Filter Transfer Functions

Parallel filter banks are used for realtime signal analysis, e.g. spectrum or speech analysis. The problem to be solved in filter bank realisation is to achieve as high selectivity of individual filters as possible and simultaneously meet the amplitude and phase characteristic specification of the bank. A design method for a bank of analogue filters is presented in the paper. An algorithm for the analysis of the amplitude, phase and group delay characteristics of the bank is given. The applicability of various types of transfer function (Bessel, Scanlan, Coung-Neirynck, Butterworth and Cheby?chev linear phase approximation) is investigated. To study the other possibilities of realising a filter bank, the so-called modified Scanlan filter has been formed by applying an automated optimisation procedure. This filter retains the performance of the Scanlan filter, but the Q factor of the critical pole pair is reduced. The examples of uniform and constant percentage bandwidth filter banks are presented to illustrate the comparative characteristics of filter banks designed on the basis of these six approximations with 12th-order filters.

Analysis of filter bank transfer functions

A function that often takes place in the study of special functions is the Gamma function, which is needed particularly on Bessel function description Bessel functions arise in many problems possessing cylindrical symmetry, in the study of the wave and heat equations and in the study of the Kepler problem of describing planetary motion, and provide another orthogonal set of basis functions The Legendre polynomials are one of a set of classical orthogonal polynomials, which satisfy a second-order linear ODE, so-called Legendre ODE occurring naturally in the solution of initial value problems having spherical symmetry

Special Differential Equations, Functions, and Polynomials

The transfer function obtained from the tuncated expansion of exp(p) in series of Bessel polynomials yield an approximation of the ideal filter. Thanks to a freely chosen parameter, it is possible to control the compromise between optimal amplitude and phase or between the rise time and the overshoot of the step response. Compared to the classical Thomson characteristic, which is linked to a single Bessel polynomial, the delay deviation in the passband is smaller.

J. Neirynck

Papers

Constant group delay approximation by series of Bessel polynomials