ON THE CONVERGENCE BEHAVIOR OF THE LMS AND THE NORMALIZED LMS ALGORITHMS

This paper has three parts. First, we indicate that the normalized least mean square (NLMS) algorithm is a potentially faster converging algorithm compared to the LMS algorithm, when the design of the adaptive filter is based on the usually quite limited knowledge of its input signal statistics. Second, we propose a very simple model for the input signal vectors that greatly simplifies analysis of the convergence behavior of the LMS and NLMS algorithms. Using this model, answers can be obtained to questions for which no answers are currently available using other (perhaps more realistic) models. The answers thus obtained can only acclaim a qualitative value, but we give examples to illustrate that even quantitatively, they can be good approximations. Finally, we want to emphasize that the convergence of the NLMS algorithm can be speeded up significantly by employing a time-varying step size. We are able to specify a priori the optimal step-size sequence for the case of a white input signal with arbitrary distribution.