A new recursive filter for systems with multiplicative noise

An optimal linear recursive minimum mean-square-error estimator was previously developed by the authors (see IEEE Trans. Autom. Control, vol.34, no.5, p.568-74, May 1989) for a zero-mean signal corrupted by multiplicative noise in its measurement model. This recursive filter cannot be obtained by the recursive structure of a conventional Kalman filter where the new estimate is a linear combination of the previous estimate and the new data. Instead, the recursive structure was achieved by combining the previous estimate with recursive innovation, a linear combination of the most recent two data samples and the previous estimate. In this work the signal is extended to be nonzero-mean. In the conventional Kalman filter, the superposition principle can be applied to both the signal and the measurement models for this nonzero-mean extension. However, when multiplicative noise exists, the measurement model becomes nonlinear. Therefore, a new recursive structure for the innovation process needs to be developed to achieve a recursive filter