Exact expectation analysis of the LMS adaptive filter

{In almost all analyses of the least-mean-square (LMS) adaptive lter, it is assumed that the lter coeecients are statistically independent of the input data currently in lter memory, an assumption that is incorrect for shift-input data. In this paper, we present a method for deriving a set of linear update equations that can be used to predict the exact statistical behavior of a nite-impulse-response (FIR) LMS adaptive lter operating upon nite-time correlated input data. Using our method, we can derive exact bounds upon the LMS step size to guarantee mean and mean-square convergence. Our equation-deriving procedure is recursive and algorithmic, and we describe a program written in the MAPLE symbolic-manipulation software package that automates the derivation for arbitrarily-long adaptive lters operating on input data with stationary statistics. Using our analysis, we present a search algorithm that determines the exact step size mean-square stability bound for a given lter length and input correlation statistics. Extensive computer simulations indicate that the exact analysis is more accurate than previous analyses in predicting adaptation behavior. Our results also indicate that the exact step size bound is much more stringent than the bound predicted by the independence assumption analysis for correlated input data. 0 Permission of the IEEE to publish this abstract separately is granted.