Two Bicomplex and One Multicomplex Least Mean Square algorithms

TL;DR

This paper introduces new gradient operators in bicomplex and multicomplex settings to develop extended LMS algorithms, broadening the applicability of adaptive filtering techniques beyond traditional real and complex domains.

Contribution

It presents novel gradient operators and formulates new bicomplex and multicomplex LMS algorithms, extending classical LMS methods to higher-dimensional number systems.

Findings

Developed bicomplex LMS algorithms based on new gradient operators.

Extended LMS algorithms to multicomplex number systems.

Provides a theoretical foundation for adaptive filtering in bicomplex/multicomplex domains.

Abstract

We study and introduce new gradient operators in the complex and bicomplex settings, inspired from the well-known Least Mean Square (LMS) algorithm invented in 1960 by Widrow and Hoff for Adaptive Linear Neuron (ADALINE). These gradient operators will be used to formulate new learning rules for the Bicomplex Least Mean Square (BLMS) algorithms and we will also formulate these learning rules will for the case of multicomplex LMS algorithms (MLMS). This approach extends both the classical real and complex LMS algorithms.