Riemann–Liouville operator‐based fractional normalised least mean square algorithm with application to decision feedback equalisation of multipath channels

Abstract

In this study, the authors modify the mapping function by introducing a non-linear term based on Riemann–Liouville definition of fractional derivatives and its application to the mean squared error in addition to the first-order partial derivatives; thus create fractional variants of the least mean square (LMS) algorithm and its normalised version. The introduction of fractional term helps increase the convergence rate through the non-linear update term which depends on the fractional order and the in-process LMS weights; for steep gradient of the error measure, large changes are made to the weights which help the equaliser filter to better track the effects of multipath fading channels. They verify and validate the working of the proposed technique in the decision feedback equalisation of multipath fading channels for higher-order quadrature amplitude modulations; comparative results are shown in terms of performance metrics as symbol-error rate for various fractional orders and step sizes, combined equaliser and channel responses for different number of training symbols; simulations show that the proposed approach outperforms the conventional counterpart.