Recursibility, Batch and Recursive Forms of Optimal Linear Estimation and Filtering

The Kalman filter is the workhorse of target tracking. As is well known, it is the recursive linear minimum mean-square error (LMMSE) filter for a linear system under its stated assumptions. It is little known, however, that for many linear systems the LMMSE filter does not have a recursive form—in other words, it is not recursible. In this paper, we introduce the concept of recursibility, that is, whether an estimator of a parameter or a filter for a linear or nonlinear system has a recursive form. We present necessary and sufficient conditions for the recursibility of LMMSE estimation and filtering. More important, for arbitrary autoand cross-correlations without the Kalman filter assumptions we present recursive LMMSE estimators and filters that are not necessarily equivalent to the batch LMMSE estimators and filters, but are optimal within the recursive class. Both the cases with and without knowledge about the prior moments of the estimatee are considered.