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A deterministic analysis of linearly constrained adaptive filtering algorithms

This paper presents a mathematically rigorous analysis of linearly constrained adaptive filtering algorithms based on the adaptive projected subgradient method. We provide the novel concept of constraint-embedding functions that enables to analyze certain classes of linearly constrained adaptive algorithms in a unified manner. Trajectories of the linearly constrained adaptive filters always lie in the affine constraint set, a translation of a closed subspace. Based on this fact, we translate all the points on the constraint set to its underlying subspace - which we regard as a Hilbert space - thereby making the analysis feasible. Derivations of the linearly constrained adaptive filtering algorithms are finally presented in connection with the analysis.