Method of synthesis of phase-correcting networks

A method is presented for determining the transfer functions of phase-corrective networks of both minimum-phase and all-pass types. The method is based on manipulation of a class of polynomials containing one or more variable parameters, which is derived from Bessel polynomials and represents basically an extension of the application of Bessel polynomials and related Bessel functions to solving the problem of approximation to the ideal constant-group-delay characteristic. Some simple phase-correcting networks, useful in those applications where the phase distortion occurs mainly at high frequencies, are first discussed, and the formulas and graphs for the evaluation of their group-delay responses are given. The extension of the method to allow for determination of the transfer function of the correcting network matching a prescribed phase characteristic is then presented. Both maximally fiat and equal-ripple types of approximation are considered, and explicit formulas for the evaluation of the coefficients of transfer functions of order n<6 are derived. The conditions on which the order of equalisation can be increased without affecting the complexity of the correcting network are also stated explicitly.