The error variance of LMS with time-varying weights

A calculation of weight error variance of the LMS (least mean square) algorithm is made in the presence of time-varying true weights. With time-varying weights the LMS error system has in general several time scales operating at once. This causes difficulties in the variance calculation which seem hitherto to have passed unnoticed. To handle this problem a sort of perturbation expansion is developed based on weak convergence methods or `stochastic averaging´. The main concern in carrying out the error variance calculation is to study the effect on LMS performance of adaptation speed as it relates to true speed of parameter change. Three cases are covered with reference to considering adaptation speed with respect to true speed: first, where the adaptation speed is too slow; second, where it is matched: and third, where it is faster