A new approach for analyzing the limiting behavior of the normalized LMS algorithm under weak assumptions

This paper presents a new and simple approach to analyzing the limiting behavior of the normalized LMS algorithm under weak assumptions. No restrictions are made on the dependence between successive regressors, the dependence among the regressor elements, the length of the adaptive filter, or the distribution types of the filter input and the noise. The analysis holds for all values of the algorithm step-size in the range between 0 and 2. The analysis is carried out using a new performance measure, based on the time evolution of the component of the weight deviation vector in the direction of the regressor. This component is termed as the effective weight deviation since it is the only component that contributes to the excess estimation error at the output of the adaptive filter. The paper derives upper bounds for the long-term averages of the mean-square effective weight deviation, mean absolute excess estimation error, and of the mean-square excess estimation error. The analytical results of the paper are supported by simulations.