A Square Root Normalized LMS Algorithm for Adaptive Identification with Non-Stationary Inputs

The conventional normalized least mean square (NLMS) algorithm is the most widely used for adaptive identification within a non-stationary input context. The convergence of the NLMS algorithm is independent of environmental changes. However, its steady state performance is impaired during input sequences with low dynamics. In this paper, we propose a new NLMS algorithm which is, in the steady state, insensitive to the time variations of the input dynamics. The square soot (SR)-NLMS algorithm is based on a normalization of the LMS adaptive filter input by the Euclidean norm of the tap-input. The tap-input power of the SR-NLMS adaptive filter is then equal to one even during sequences with low dynamics. Therefore, the amplification of the observation noise power by the tap-input power is cancelled in the misadjustment time evolution. The harmful effect of the low dynamics input sequences, on the steady state performance of the LMS adaptive filter are then reduced. In addition, the square root normalized input is more stationary than the base input. Therefore, the robustness of LMS adaptive filter with respect to the input non stationarity is enhanced. A performance analysis of the first- and the second-order statistic behavior of the proposed SR-NLMS adaptive filter is carried out. In particular, an analytical expression of the step size ensuring stability and mean convergence is derived. In addition, the results of an experimental study demonstrating the good performance of the SR-NLMS algorithm are given. A comparison of these results with those obtained from a standard NLMS algorithm, is performed. It is shown that, within a non-stationary input context, the SR-NLMS algorithm exhibits better performance than the NLMS algorithm.