Tracking random walk systems with vector space adaptive filters

Tracking properties are considered for adaptive filters that are adjusted within a vector space of filtering operations. Such vector space adaptive filters have both the desirable convergence properties of adaptive finite impulse response filters, and additionally some of the modeling flexibility of adaptive infinite impulse response filters. For a given vector space of systems, the adaptive filter structure is determined by a choice of basis for the vector space. Basis dependent expressions 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 variations. It is shown that under these conditions, the minimum achievable mean square error using least mean square adaptation is equal to the optimal value provided by the Kalman filtering algorithm, but that recursive least squares adaptation does not in general attain this minimum error