A Statistical Analysis of the NLMS and NLMS Algorithms for Correlated Gaussian Signals

Abstract

The Nonnalized LMS adaptive algorithm is widely used in real life adaptive filtering and control applications mainly due to its simplicity and robustness to input signal power variations. This last characteristic is a consequence of its intrinsic nonnalization procedure. However, in practical applications it is usual the addition of a small positive constant to the nonnalization factor in order to avoid divisions by zero or overflow. In this case, the adaptive algorithm is named e-NLMS. This work presents a statistical analysis of the e-NLMS for Gaussian input signals. Deterministic recursive equations are obtained for the mean weight behavior and mean square error for a large number of adaptive coefficients. In addition, a closed expression is provided for the steady state misadjustment. Monte Carlo simulations show an excellent agreement between theoretical predictions and the algorithm's behavior in steady state. During transient, the new model is conservative and more accurate than the existing models. The developed equations can be lIsed with e =0 to predict the behavior of the NLMS algorithm. Simulations illustrate the better quality of the new NLMS model when compared to others already available in the literature.