Weak Convergence and Properties of Adaptive Asymptotic F ilters w ith 177 Constant Gains

-The basic adaptive filtering algorithm XG,, = Xi c Y, ( YiXG $,) is analyzed using the theory of weak convergence. Apart from some very special cases, the analysis is hard when done for each fixed z > 0. But the weak convergence techniques are set up to provide much information for small r. The relevant facts from the theory are given. Define x’(.) by x’(t) = X; on [nr, nc + E). Then weak (distributional) convergence of { x’( .)} and of { xf (. + tC)} is proved under very weak assumptions, where t, + cc as r --f 0. The normalized errors {( X,t 0)/ &} are analyzed, where 6’ is a “stable” point for the “mean” algorithm. The asymptotic properties of a projection algorithm are developed, where the XG are truncated at each iteration, if they fall outside of a given set.