Geometric-Algebra Adaptive Filters

Abstract

This paper reformulates adaptive filters (AFs) in the framework of geometric algebra (GA), developing a complete study of the resulting geometric-algebra adaptive filters (GAAFs). They are generated by formulating the underlying minimization problem (a deterministic cost function) from the perspective of 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 AFs to subalgebras of GA. From the obtained update rule, it is shown how to recover the following least mean squares (LMS) AF variants via algebraic isomorphisms: real-entries LMS, complex LMS, and quaternions LMS. Mean-square analysis and simulations in a system identification scenario are provided, showing very good agreement. Real-data experiments highlight the good tracking performance of the GAAFs in a joint linear prediction of different signals.