  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.
