The limiting behavior of LMS

A realization-oriented analysis is given of the gradient noise misadjustment and lag misadjustment performance of the LMS (least-mean-square) algorithm. New formulas are given for both of these components of excess mean-square error. It is shown that the traditional formula for lag misadjustment needs to be modified by adding further terms due to gradient noise and noise variance. To perform the analysis, it is necessary to study the convergence (with probability one) of the noise-free, fixed-parameter LMS algorithm. Convergence is found under simple conditions that improve on those previously obtained