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

SPARSE ECHO CANCELLATION USING VARIANTS OF LEAST MEAN FOURTH AND LEAST MEAN SQUARE ALGORITHMS

Echo cancellation is the most essential and indispensable component of telephone networks. The impulse responses of most of the networks are sparse in nature; that is, the impulse response has a small percentage of its components with a significant magnitude (large energy), while the rest are zero or small. In these sparse environments, conventional adaptive algorithms like least mean square (LMS) and normalized LMS (NLMS) show substandard and inferior performances. In this paper, the performances of the normalized least mean square (NLMS) algorithm, the normalized least mean fourth (NLMF) and the proportionate normalized least mean fourth (PNLMF) are compared for sparse echo cancellation. The sparseness of both the echo response and the input signal is exploited in this algorithm to achieve improved results at a low computational cost. The PNLMF algorithm showed better results and faster convergence in sparse and non sparse systems, but its results in sparse environments are more impressive. The NLMF algorithm shows good results in sparse environments but not in non-sparse environments. The PNLMS algorithm can be considered superior to the NLMF and NLMS algorithms with respect to the error profile. A modified algorithm, the sparse controlled modified proportionate normalized LMF (SCMPNLMF) algorithm, is proposed, and its performances are compared with the other algorithms