Total convergence analysis of recursive least-squares adaptive filters

Abstract

Recursive least-squares adaptive filters have been utilized in many fields of applications for a long time, but their convergence properties have not been known completely yet. In this dissertation, we study on the total convergence properties of the recursive least-squares adaptive filters. First we make a global analysis from the initialization stage to the steady-state stage. Next we give a detailed analysis on initial and asymptotic behaviors. Finally, we devise a fast parameter tracking and noise-immunized recursive least-squares adaptive filter and study its convergence properties as well. For the global analysis, we find an upper bound of convergence rate by deriving the exact difference equation of a stochastic Lyapunov function. The total convergence properties of the recursive least-squares adaptive filters can be shown from the upper bound of convergence rate obtained in this dissertation because it bounds above overall stage. For the initial behaviors, we find relation of initial values to initial overshoot phenomena which can be seen during the initialization period in computer simulations of the filter. For the asymptotic behaviors, we analyze and compare asymptotic behaviors of the a posteriori estimation error with those of the a priori estimation error. For fast tracking ability of the filter, we devise a fast parameter tracking recursive least-squares adaptive filter and derive its upper bound of convergence rate. We validate the results from our analysis through computer simulations.