Parallel combinations of adaptive filters have been effectively used to improve the performance of adaptive algorithms and address well-known trade-offs, such as convergence rate vs. steady-state error. Nevertheless, typical combinations suffer from a convergence stagnation issue due to the fact that the component filters run independently. Solutions to this issue usually involve conditional transfers of coefficients between filters, which although effective, are hard to generalize to combinations with more filters or when there is no clearly faster adaptive filter. In this work, a more natural solution is proposed by cyclically feeding back the combined coefficient vector to all component filters. Besides coping with convergence stagnation, this new topology improves tracking and supervisor stability, and bridges an important conceptual gap between combinations of adaptive filters and variable step size schemes. We analyze the steady-state, tracking, and transient performance of this topology for LMS component filters and supervisors with generic activation functions. Numerical examples are used to illustrate how coefficients feedback can improve the performance of parallel combinations at a small computational overhead.

This paper exploits Geometric (Clifford) Algebra (GA) theory in order to devise and introduce a new adaptive filtering strategy. From a least-squares cost function, the gradient is calculated following results from Geometric Calculus (GC), the extension of GA to handle differential and integral calculus. The novel GA least-mean-squares (GA-LMS) adaptive filter, which inherits properties from standard adaptive filters and from GA, is developed to recursively estimate a rotor (multivector), a hypercomplex quantity able to describe rotations in any dimension. The adaptive filter (AF) performance is assessed via a 3D pointclouds registration problem, which contains a rotation estimation step. Calculating the AF computational complexity suggests that it can contribute to reduce the cost of a full-blown 3D registration algorithm, especially when the number of points to be processed grows. Moreover, the employed GA/GC framework allows for easily applying the resulting filter to estimating rotors in higher dimensions.

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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 wellsuited 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.

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