Performance Analysis of l_0 Norm Constraint Least Mean Square Algorithm

As one of the recently proposed algorithms for sparse system identification, $l_0$ norm constraint Least Mean Square ($l_0$-LMS) algorithm modifies the cost function of the traditional method with a penalty of tap-weight sparsity. The performance of $l_0$-LMS is quite attractive compared with its various precursors. However, there has been no detailed study of its performance. This paper presents all-around and throughout theoretical performance analysis of $l_0$-LMS for white Gaussian input data based on some reasonable assumptions. Expressions for steady-state mean square deviation (MSD) are derived and discussed with respect to algorithm parameters and system sparsity. The parameter selection rule is established for achieving the best performance. Approximated with Taylor series, the instantaneous behavior is also derived. In addition, the relationship between $l_0$-LMS and some previous arts and the sufficient conditions for $l_0$-LMS to accelerate convergence are set up. Finally, all of the theoretical results are compared with simulations and are shown to agree well in a large range of parameter setting.Comment: 31 pages, 8 figure

Sparseness-controlled adaptive algorithms for supervised and unsupervised system identification

In single-channel hands-free telephony, the acoustic coupling between the loudspeaker and the microphone can be strong and this generates echoes that can degrade user experience. Therefore, effective acoustic echo cancellation (AEC) is necessary to maintain a stable system and hence improve the perceived voice quality of a call. Traditionally, adaptive filters have been deployed in acoustic echo cancellers to estimate the acoustic impulse responses (AIRs) using adaptive algorithms. The performances of a range of well-known algorithms are studied in the context of both AEC and network echo cancellation (NEC). It presents insights into their tracking performances under both time-invariant and time-varying system conditions. In the context of AEC, the level of sparseness in AIRs can vary greatly in a mobile environment. When the response is strongly sparse, convergence of conventional approaches is poor. Drawing on techniques originally developed for NEC, a class of time-domain and a frequency-domain AEC algorithms are proposed that can not only work well in both sparse and dispersive circumstances, but also adapt dynamically to the level of sparseness using a new sparseness-controlled approach. As it will be shown later that the early part of the acoustic echo path is sparse while the late reverberant part of the acoustic path is dispersive, a novel approach to an adaptive filter structure that consists of two time-domain partition blocks is proposed such that different adaptive algorithms can be used for each part. By properly controlling the mixing parameter for the partitioned blocks separately, where the block lengths are controlled adaptively, the proposed partitioned block algorithm works well in both sparse and dispersive time-varying circumstances. A new insight into an analysis on the tracking performance of improved proportionate NLMS (IPNLMS) is presented by deriving the expression for the mean-square error. By employing the framework for both sparse and dispersive time-varying echo paths, this work validates the analytic results in practical simulations for AEC. The time-domain second-order statistic based blind SIMO identification algorithms, which exploit the cross relation method, are investigated and then a technique with proportionate step-size control for both sparse and dispersive system identification is also developed