A Sparse Variable Step-Size LMS Algorithm for Impulsive Noise

In sparse system identification, exploiting the sparsity property improves the performance of the least mean square (LMS) algorithm. The improvement is achieved by adding an $l_{0}$ or $l_{1}$-norm penalty term to the cost function of the conventional LMS algorithm. However, when the eignenvalue spread of the autocorrelation of the input is relatively large, the performance of the LMS algorithm is poor. For instance, the LMS algorithm cannot perform accurate system identification under the impulsive noise environment because its cost function is not well defined for such scenarios. In this paper, we propose a sparse variable step-size LMS algorithm that employs arctan constraint in the cost function of the algorithm. This constraint imposes a zero attraction of the filter coefficients according to the relative value of each filter coefficient among all the entries. This, in turn, leads to an improved performance when the system is sparse. Different experiments have been conducted to investigate the performance of the proposed algorithm. Simulation results show that the proposed algorithm outperforms the zero attracting LMS (ZA-LMS) and the reweighted ZA-LMS (RZA-LMS) algorithms in a system identification setting under impulsive noise environment.