The proportionate normalized least mean square (PNLMS) algorithm and its variants are by far the most popular adaptive filters that are used to identify sparse systems. The convergence speed of the PNLMS algorithm, though very high initially, however, slows down at a later stage, even becoming worse than sparsity agnostic adaptive filters like the NLMS. In this paper, we address this problem by introducing a carefully constructed l1norm (of the coefficients) penalty in the PNLMS cost function which favors sparsity. This results in certain zero attracting terms in the PNLMS weight update equation which help in the shrinkage of the coefficients, especially the inactive taps, thereby arresting the slowing down of convergence and also producing lesser steady state excess mean square error (EMSE). A rigorous convergence analysis of the proposed algorithm is presented that expresses the steady state mean square deviation of both the active and the inactive taps in terms of a zero attracting coefficient of the algorithm. The analysis reveals that further reduction of the EMSE is possible by deploying a variable step size (VSS) simultaneously with a variable zero attracting coefficient in the weight update process. Simulation results confirm superior performance of the proposed VSS zero attracting PNLMS algorithm over existing algorithms, especially in terms of having both higher convergence speed and lesser steady state EMSE simultaneously.

The proportionate normalized least mean square (PNLMS) algorithm and its variants are by far the most popular adaptive filters that are used to identify sparse systems. The convergence speed of the PNLMS algorithm, though very high initially, however, slows down at a later stage, even becoming worse than sparsity agnostic adaptive filters like the NLMS. In this paper, we address this problem by introducing a carefully constructed l1norm (of the coefficients) penalty in the PNLMS cost function which favors sparsity. This results in certain “zero attractor” terms in the PNLMS weight update equation which help in the shrinkage of the coefficients, especially the inactive taps, thereby arresting the slowing down of convergence and also producing lesser steady state excess mean square error (EMSE). We also demonstrate both analytically and also intuitively, that the EMSE can not, however, be reduced significantly by the zero attractors due to some fundamental shortcoming of the PNLMS algorithm, and propose methods to counter it by deploying a variable step size and also a variable proportionality constant for the zero attractors. Simulation results confirm excellent performance of the proposed algorithm vis-a-vis existing methods.

The generalized Orthogonal Matching Pursuit (gOMP) is a recently proposed compressive sensing greedy recovery algorithm which generalizes the OMP algorithm by selecting N( ≥ 1) atoms in each iteration. In this letter, we demonstrate that the gOMP can successfully reconstruct a K-sparse signal from a compressed measurement y=Φx by a maximum of K iterations if the sensing matrix Φ satisfies the Restricted Isometry Property (RIP) of order NK, with the RIP constant δNKsatisfying δNKNK. We also show that by increasing the RIP order just by one (i.e., NK+1 from NK), it is possible to refine the bound further to δNK+1K+1in OMP.

In this paper, we provide an overview of the major developments in the area of sparse adaptive filters, starting from the celebrated works on PNLMS algorithm and its several variants to more recent approaches that use compressed sensing framework, more specifically, LASSO and basis pursuit or matching pursuit, to develop sparse adaptive algorithms with improved mean square error and tracking properties. Subsequently, we also present a new approach to identify sparse systems with time varying sparseness, for which a novel scheme of cooperative learning involving a PNLMS and a NLMS based adaptive filters is developed.