Design and Analysis of the Fractional-Order Complex Least Mean Square (FoCLMS) Algorithm

• Published in: Circuits, systems, and signal processing Vol. 40; no. 10; pp. 5152 - 5181
• Main Authors: Ahmad, Jawwad, Zubair, Muhammad, Rizvi, Syed Sajjad Hussain, Shaikh, Muhammad Shafique
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2021; Springer Nature B.V
• Subjects: Algorithms; Circuits and Systems; Electrical Engineering; Electronics and Microelectronics; Engineering; Fractional calculus; Instrumentation; Learning curves; Signal,Image and Speech Processing; Steady state; Steady-state analysis; Channel identification problem; Adaptive algorithms; Least mean squares algorithm; Fractional complex derivative; FoCLMS algorithm; Transient analysis; Fractional calculus
• ISSN: 0278-081X; 1531-5878
• DOI: 10.1007/s00034-021-01715-8

In this work, a new class of stochastic gradient algorithm is developed based on fractional calculus. Unlike the existing algorithms, the concept of complex fractional gradient is introduced by employing Caputo’s fractional derivative which results in a fractional steepest descent algorithm and a fractional-order complex LMS (FoCLMS) algorithm. We demonstrate that with the Caputo’s fractional gradient definition, the Weiner solution remains invariant. Convergence analysis of the proposed FoCLMS algorithm is presented for both transient and steady state scenarios. Consequently, expressions for the learning curves and steady state EMSE are derived. Our theoretical developments are validated by simulation experiments. Extensive simulations are presented to investigate all possible scenarios: channel with negative weights and real input data, channel with positive weights and complex input data, and channel with complex weights and complex input data.