The conventional least-mean-square (LMS) algorithm has a poor performance when the input autocorrelation’s eigenvalue spread is quite large. For instance, the cost function is inadequately described when impulsive noise is present, making it impossible for the LMS approach to correctly identify the system. The cost function of the conventional LMS method is enhanced by the addition of a $l_{0}$ or $l_{1}$ -norm penalty component. This extra term exploits system sparsity and improves the performance of the LMS algorithm. In this work, we propose a sparse variable step-size LMS approach in a code-division multiple access (CDMA) system. The proposed algorithm makes use of an arctan constraint in the cost function of the algorithm. This constraint enforces a zero attraction of the filter coefficients based on the relative value of each filter coefficient among all entries. Convergence analysis of the proposed algorithm in the mean sense is derived. The CDMA is employed by spreading the transmitted symbols using an m-sequence during the transmission process where the channel state is considered constant over a symbol period. At the receiver, decorrelation stage is considered prior to the channel estimation. This step improves the input signal-to-noise ratio (SNR) at the filter’s input, thereby accelerating the adaptive filter’s convergence. However, this gain, in the rate of convergence, comes at the expense of the system’s spectral efficiency. The proposed method shows high performance, in terms of steady-state mean-square deviation (MSD), compared to those of the zero attracting LMS (ZA-LMS) and the reweighted ZA-LMS (RZA-LMS) algorithms in a system identification setting with an impulsive noise environment. In addition, the proposed algorithm has shown significant performance in estimating the channel for CDMA communication systems in terms of bit-error rate (BER).
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