Almost-Sure Convergence of the Continuous-Time LMS Algorithm

In this paper we consider the stability properties of the conventional continuous-time least mean square (LMS) algorithm. We investigate the algorithm for the case of stationary ergodic inputs, and present a necessary and sufficient condition for exponential almost-sure convergence. This condition is shown to be less restrictive than the well-known persistency of excitation condition. Also, we point out and clarify an apparently common error regarding the connection between persistency of excitation and positive definite autocorrelation in stationary ergodic vector waveforms.