Optimal constraint vectors for set-membership affine projection algorithms

There is a growing interest in adaptive filtering solutions whose learning processes are data selective, bringing about computational reduction and energy savings while improving estimation accuracy. The set-membership affine projection algorithms are a representative family of algorithms including data-selection mechanisms. The update process of these algorithms depends on the choice of a constraint vector (CV) which, up to now, is based on some heuristics. In this paper we propose an optimal CV and discuss some of its inherent properties. The resulting problem falls into a convex optimization framework, allowing some unexpected features to surface; for instance, the widely used simple choice CV is asymptotically optimal for statistically white stationary inputs. Simulations indicate the optimal CV outperforms the simple choice CV regarding update rates and steady-state mean squared errors for statistically colored inputs.