TRACKING RANDOM-WALK SYSTEMS WITH VECTOR-SPACE ADAPTIVE FILTERS

Tracking properties are considered for adaptive filters that are adjus ted within a vector space of filtering operations. Such vector space a daptive filters have both the desirable convergence properties of adap tive finite impulse response filters, and additionally some of the mod eling flexibility of adaptive infinite impulse response filters. For a given vector space of systems, the adaptive filter structure is deter mined by a choice of basis for the vector space. Basis dependent expre ssions are developed for the asymptotic mean square error under Least Mean Square, Recursive Least Square, and Kalman Filtering adaptation, when the optimal filter specification is subject to random walk variat ions. It is shown that under these conditions, the minimum achievable mean square error using Least Mean Square adaptation is equal to the o ptimal value provided by the Kalman Filtering algorithm, but that Recu rsive Least Squares adaptation does not in general attain this minimum error.