A Sparse-Aware Variable Step-Size NLMS Equalizer with Dynamic Reweighted Regularization for Time-Varying Multipath Channels

Sparse adaptive equalization is widely used for time-varying multipath channels due to its ability to exploit channel sparsity and reduce computational complexity. However, conventional normalized least mean square (NLMS)-based algorithms usually rely on fixed step-size strategies or manually selected regularization parameters, which limits their capability to simultaneously achieve fast convergence, low steady-state error, and robust tracking performance under varying channel conditions. This paper proposes a sparsity-aware variable step-size NLMS (SA-VSS-NLMS) equalizer with dynamic sparsity-adaptive regularization for time-varying sparse channels. The proposed framework employs online Hoyer sparsity estimation as a unified feedback signal to jointly adjust the adaptive step size and regularization strength. By incorporating sparsity information into both adaptation processes, the proposed method enables rapid convergence during channel variations while maintaining accurate steady-state tracking in sparse environments. In addition, an anomaly-aware tracking mechanism is introduced to improve recovery performance under abrupt channel changes. The convergence behavior and stability properties of the proposed algorithm are analyzed, and extensive simulations are conducted under stationary, time-varying, and fractional-delay sparse channel conditions. The results demonstrate that the proposed SA-VSS-NLMS achieves faster convergence, lower steady-state error, improved tracking robustness, and enhanced bit error rate (BER) performance compared with conventional NLMS-based and sparse adaptive filtering methods. Moreover, the proposed algorithm maintains linear computational complexity with respect to the filter length, making it suitable for practical adaptive communication systems.