A New Structure of Linear Recursive Estimator

A new structure of linear recursive estimator which minimizes the mean square error is derived for a system with a multiplicative noise included in the measurement model. The signal is modeled in the same way as in the Kalman filter. The conventional form of a recursive estimator (the new estimate is the linear combination of the new data and the previous estimate) is not appropriate for the above system. In contrast, according to our new form of estimator, the new estimate is the linear combination of the previous estimate and the new innovation which is recursively obtained by a linear combination of the new data, the previous data, and the previous innovation. The new recursive form of the innovation process gives the new form of a linear MMSE estimator. Not only is the on-line estimation recursive, but also the off-line computation of the coefficients (which are the counterparts of the Kalman gain) is recursive. A constraint on the system matrices should be satisfied and its limitation is justified. A physical interpretation of this constraint is also given.