A $$p$$ p -norm variable step-size LMS algorithm for sparse system identification

Abstract

The performance of the least mean square (LMS) algorithm in sparse system identification has been improved by exploiting the sparsity property. This improvement is gained by adding an \(l_1\)-norm constraint to the cost function of the LMS algorithm. However, the LMS algorithm by itself performs poorly when the condition number is relatively high due to the step size of the LMS, and the \(l_1\)-norm LMS is an approximate method which, in turn, may not provide optimum performance. In this paper, we propose a sparse variable step-size LMS algorithm employing the \(p\)-norm constraint. This constraint imposes a zero attraction of the filter coefficients according to the relative value of each filter coefficient among all the entries which in turn leads to an improved performance when the system is sparse. The convergence analysis of the proposed algorithm is presented and stability condition is derived. Different experiments have been done to investigate the performance of the proposed algorithm. Simulation results show that the proposed algorithm outperforms different \(l_1\)-norm- and \(p\)-norm-based sparse algorithms in a system identification setting.