Performance Analysis of l_0 Norm Constraint Least Mean Square Algorithm

TL;DR

This paper provides a comprehensive theoretical analysis of the $l_0$-LMS algorithm for sparse system identification, including steady-state performance, parameter optimization, and convergence conditions, validated by simulations.

Contribution

It offers the first detailed theoretical performance analysis of $l_0$-LMS, including expressions for steady-state MSD and convergence conditions, enhancing understanding of its behavior.

Findings

Derived steady-state MSD expressions

Established parameter selection rules

Confirmed theoretical results with simulations

Abstract

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.