Bandpass magnitude and delay approximations using digital transfer functions

Abstract

AbstractThis paper proposes a method to approximate transfer functions of magnitude and delay digital equalizers for transmission channels with bandpass characteristics. In this method, an extrapolation band is added to the equalization band on the unit circle on the z‐plane, and the resulting extended approximation band is mapped on the overall unit circle on the auxiliary complex plane.In the magnitude equalization, the all‐pole type transfer function is used for the approximation to have a loss from the extended approximated band. On the auxiliary complex plane, the desired characteristics are squared and inverted. Then the extrapolation characteristics are determined, and the inverted desired characteristics are expanded into Fourier cosine series. This series is transformed inversely into the z‐plane to determine the denominator of the transfer function. If the extrapolation characteristics are determined improperly, the resulting transfer functions may become unrealizable.This paper proposes a method of determining the extrapolation characteristics so that it always gives realizable transfer functions. In the delay equalization using the auxiliary complex plane, the approximation cannot be performed in a straightforward manner. In the proposed method, the delay characteristics are treated graphically and the inverted characteristics are expanded into Fourier cosine series. After inverse transformation into the z‐plane, the inverse of the series is expanded into the Fourier series and truncated in some length to give the transfer function.