Geometric-Algebra Adaptive Filters

TL;DR

This paper introduces Geometric-Algebra Adaptive Filters (GAAFs), a new class of adaptive filters based on Geometric Algebra that generalizes existing LMS filters across various algebraic data types, with demonstrated effectiveness in system identification.

Contribution

The paper develops GAAFs using Geometric Algebra and Calculus, enabling a unified derivation of LMS variants for different data types, which was not previously possible.

Findings

GAAFs generalize LMS filters to subalgebras of Geometric Algebra.

Simulations show GAAFs perform well in noisy system identification.

The approach unifies adaptive filtering across real, complex, and quaternion data.

Abstract

This paper presents a new class of adaptive filters, namely Geometric-Algebra Adaptive Filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of Geometric Algebra (GA), a comprehensive mathematical language well-suited for the description of geometric transformations. Also, differently from standard adaptive-filtering theory, Geometric Calculus (the extension of GA to differential calculus) allows for applying the same derivation techniques regardless of the type (subalgebra) of the data, i.e., real, complex numbers, quaternions, etc. Relying on those characteristics (among others), a deterministic quadratic cost function is posed, from which the GAAFs are devised, providing a generalization of regular adaptive filters to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least-mean squares (LMS) adaptive filter variants: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing very good agreement for different levels of measurement noise.