Optimal Constraint Vectors for Set-Membership Proportionate Affine Projection Algorithms

Sparsity is an inherent feature of certain practical systems and appears in problems such as channel equalization and echo cancellation. Designed for exploiting the intrinsic structure of sparse environments, while also taking advantage of the data reuse and selection strategies, the set-membership proportionate affine projection algorithm (SM-PAPA) relies on the choice of a constraint vector (CV) that affects the behavior of the adaptive system. Although the selection of this CV has been based on some heuristics, a recent work proposes an optimal CV for the set-membership affine projection algorithm, a particular instance of the SM-PAPA. This paper adopts a convex optimization framework and generalizes the optimal CV concept for the SM-PAPA, allowing its use in sparse systems. Moreover, by using the gradient projection method for solving the related constrained convex problem, this paper demonstrates that the optimal CV can indeed be applied in real-time applications.