Generic Weighted Filtering of Stochastic Signals

In this paper, a new theory of optimal weighted nonlinear filtering is presented. Two filter models are considered. The first model is based on a representation of the filter in the polynomial-like form with $q$ terms where each term consists of weighted matrices and the matrix determined from the error minimization problem. The second model extends the first one to the case of the filter concatenation. The filter models are given in terms of pseudo-inverse matrices, i.e., the requirement of invertibility for covariance matrices is omitted. Thus, our filters always exist. We develop methods which allow us to exploit advantages associated with the proposed nonlinear filter models. The methods consist of the orthogonalization procedure and the reduction of the original problem to $q$ individual minimization problems for smaller matrices. This leads to a considerable reduction in the required computational work. The error associated with the first filter model decreases when the number $q$ of terms of filter increases. Its compression ratio can be adjusted by varying a particular value of ranks in each of its $q$ terms. This means that the proposed filer structure provides the two degrees of freedom. The second filter model provides another degree of freedom, a number $k$ of filters in the concatenation. Variations of the degrees of freedom improve the performance of the proposed filters. In particular, the error associated with the filter concatenation decreases as the filter number $k$ in the concatenation increases.