Norm-constrained adaptive algorithms for sparse system identification based on projections onto intersections of hyperplanes

• A new approach for developing norm-constrained adaptive algorithms is presented.
• The new approach uses projections onto intersections of hyperplanes.
• Adaptive algorithms based on the l0 and the l1 norms are developed.
• Enhanced algorithms with reduced number of parameters are also developed.
• Simulation results ratify the effectiveness of the proposed algorithms.

This paper introduces a novel approach to derive norm-constrained adaptive algorithms for sparse system identification. In contrast to other similar approaches found in the literature, the proposed approach is focused primarily on keeping the a posteriori error equal to zero (which is a characteristic of the normalized least-mean-square algorithm) while seeking to satisfy a norm constraint. To this end, the proposed algorithms look directly for a vector belonging to the intersection of a zero-error hyperplane and a hyperplane resulting from a relaxed norm constraint. This somewhat simpler strategy leads to effective sparsity-promoting adaptive algorithms that exhibit low computational complexity and use parameters that are easy to adjust. In this context, a general framework that allows obtaining adaptive algorithms using different norm functions is devised. From this framework, two norm-constrained algorithms based on the ℓ1 and ℓ0 norms are obtained. Moreover, enhanced versions of these algorithms are developed aiming to make them independent of user-defined norm-bound parameters. Numerical simulation results corroborate the effectiveness of the proposed framework as well as the very good performance of the obtained algorithms.