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-L i o u v i l l ed e f i n i t i o no ff r a c t i o n a ld e r i v a t i v e sa n di t sa p p l i c a t i o nt ot h em e a ns q u a r e de r r o ri na d d i t i o nt ot h ef i r s to r d e r 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.

Implementation of LMS algorithm for system identification

“…Weiner filter is the fundamental concept in the theory of adaptive filter. It is an optimal filter and was derived by two researchers independently, namely Wiener in 1942, and Kolmogorov in 1939 [3][4]. Actually, the filter coefficients as well as input signal both may be complex quantities.…”

“…Weiner gives the optimal solution in the sense that no other method can achieve lower value of the MSE cost function than the Weiner solution. Still it is not practically possible in real time scenario due to various reasons that can be identified from (4). Equation (4) indicates that computation of optimal solution requires correlation matrix inversion which is quite computationally intensive.…”

“…Still it is not practically possible in real time scenario due to various reasons that can be identified from (4). Equation (4) indicates that computation of optimal solution requires correlation matrix inversion which is quite computationally intensive. The computation of optimal solution also requires correlation matrix R and cross correlation vector p those are not directly available.…”

“…The computation of optimal solution also requires correlation matrix R and cross correlation vector p those are not directly available. The above mentioned points suggest an iterative approach to calculate Wiener solution instead of direct solution using (4). A number of algorithms exist for finding the optimal solution iteratively.…”

A New Variable Length LMS Algorithm for Partial Update Adaptive Filtering Driven by Cyclostaionary Signal

“…Such algorithms can reduce computational complexity and on the other hand perform close to the full-update methods in term of convergence rate and Mean Error Square (MSE). There are several variants of the LMS algorithms with partial update methods [3][4][5][6][7][8][9][10][11][12][13][14]. In addition, partial-update adaptive filters may suffer from stability or convergence problems when the input signal is cyclostationary or periodic [15].…”