A fractional taylor series-Based least mean square algorithm, and its application to power signal estimation

Abstract

The definition of fractional derivative by Caputo and Riemann inspired the researchers to develop new adaptive algorithms having better convergence properties in comparison with integer-gradient based adaptive algorithms. As reported in many studies, the existing fractional gradient-based adaptive algorithms lack justification for using fractional derivative in addition to integer-gradient, may become inconsistent in the event of negative weights, and may yield more or less same performance as compared to the conventional integer-derivative-based algorithms by appropriate selection of step-size parameter. Accordingly, this paper presents a novel fractional adaptive algorithm based on Fractional Taylor Series. Unlike (most of) the existing fractional-derivative-based algorithms, the proposed algorithm only involves fractional-derivative and ensures convergence of mean square error (MSE) provided the step-size is chosen appropriately. Simulation results are presented in order to depict a scenario where exploitation of fractional-derivative in the weight-update equation yields better convergence as compared to LMS algorithm in the context of power signal parameters estimation.