Tracking analysis of the normalized LMS algorithm without the independence and small step size assumptions

The paper provides a rigorous tracking analysis of the normalized LMS algorithm under weak assumptions. The analysis is done in the context of identifying a randomly time-varying plant. No restrictions are made on the dependence between successive regressors, the dependence among the regressor elements, the length of the adaptive filter, the distributions of the filter input, noise, and plant parameter increments. 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 recently presented performance measure, which is 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 provides boundedness results concerning the effective weight deviation and the excess estimation error. An expression of the optimum step size is derived. The derived results hold over a wide range of the non-stationarity degree of the plant parameters. The analytical results of the paper are supported by simulations.