Alternate Implementation of NSAF and NLMS Learning Rules for Adaptive Filters

From the perspective of the original time-domain, the normalized subband adaptive filtering (NSAF) algorithm leads to the freeze of the filter update in $N-1$ out of every $N$ iterations so that slowing convergence of the filter. To fill this flaw, this brief proposes to perform the normalized least mean square (NLMS) algorithm in this frozen interval and then obtain the NSAF-NLMS algorithm with faster convergence. Moreover, we analyze the mean-square behavior of the NSAF-NLMS algorithm and give its stability condition, transient model, and steady-state model. Then, to handle sparse systems, the proportionate NSAF-NLMS algorithm is developed further to speed up the filter’s convergence. Simulation results have demonstrated the performance analysis and the superiority of the proposed algorithms over the existing counterparts.