Geometric-Algebra Adaptive Filters

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.