Algebraic foundations of split hypercomplex nonlinear adaptive filtering

TL;DR

This paper introduces a hypercomplex learning algorithm for nonlinear adaptive filtering that rigorously incorporates hypercomplex algebra, promising improved performance especially for quaternionic signals.

Contribution

It presents a novel split hypercomplex adaptive filtering algorithm that fully respects hypercomplex algebra and calculus, with rigorous convergence analysis.

Findings

Improved performance predicted for quaternionic processes.

Algorithm rigorously derived from hypercomplex algebra laws.

Convergence of the proposed method is proven.

Abstract

A split hypercomplex learning algorithm for the training of nonlinear finite impulse response adaptive filters for the processing of hypercomplex signals of any dimension is proposed. The derivation strictly takes into account the laws of hypercomplex algebra and hypercomplex calculus, some of which have been neglected in existing learning approaches (e.g. for quaternions). Already in the case of quaternions we can predict improvements in performance of hypercomplex processes. The convergence of the proposed algorithms is rigorously analyzed. Keywords: Quaternionic adaptive filtering, Hypercomplex adaptive filtering, Nonlinear adaptive filtering, Hypercomplex Multilayer Perceptron, Clifford geometric algebra