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Abstract

A method and an apparatus are disclosed for processing a measurement process to estimate a signal process. The method synthesizes realizations of a signal process and a measurement process into a primary filter for estimating the signal process and, if required, an ancillary filter for providing the primary filter's estimation error statistics. Both the primary and the ancillary filters are made out of artificial recurrent neural networks (RNNs). Their implementation results in the filtering apparatus. The synthesis is performed through training RNNs. The weights/parameters and initial dynamic state of an RNN are determined by minimizing a training criterion by the variation of the same. The training criterion, which is constructed on the basis of a selected estimation error criterion, incorporates the aforementioned realizations. An alternative way to determine the initial dynamic state of an RNN is to simply set it equal to a canonical initial dynamic state. After adequate training, both the primary and the ancillary filters are recursive filters optimal for the given respective RNN architectures with the lagged feedbacks carrying the optimal conditional statistics. If appropriate RNN paradigms and estimation error criteria are selected, the primary and the ancillary filters of such paradigms are proven to approximate the respective optimal filters in performance (with respect to the selected estimation error criteria) to any desired degree of accuracy, provided that the RNNs that constitute the primary and ancillary filters are of sufficient sizes.

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