M.S. Yarlykov, S.M. Yarlykova

The problem of optimal nonlinear filtration of vector discrete-continuous Markovian random processes is solved on the base of the Markovian estimation theory by the method of reassignment of the filtered continuous process parameters and the method of feedbacks of the filtered discrete processes. When solving optimal nonlinear filtration problems the a posteriori probability density (or corresponding a posteriori combined distributions) of a combined state vector plays a fundamental role. When calculating the a posteriori combined distribution we can represent it in one of two forms. In the former case, the a posteriori combined distribution of a discrete-continuous state vector is the product of unconditional a posteriori probability of the state of a vector of discrete processes and conditional (by the states of a vector of discrete processes) a posteriori probability density of a vector of continuous processes. In the second case, the a posteriori combined distribution of a discrete-continuous state vector is sought in the form of a product of the unconditional a posteriori probability density of a vector of continuous processes and conditional (with respect to a vector of continuous processes) a posteriori probability of the state of a vector of discrete processes. Although both representations are equivalent from standpoint of the Bayes theorem, under development of quasi-optimal algorithms and the corresponding block diagrams for nonlinear processing of radio signals they differ in some significant features. The synthesis is executed when continuous part of discrete-continuous random process is a vector diffusion Markovian process and whose discrete part is a vector, each component of which is characterized by a simple several-position Markovian chain. It is assumed that desired signals are observed in the presence of additive white Gaussian noise. Quasi-optimal algorithms and the corresponding block diagrams for nonlinear processing of radio signals typical for radionavigation and communications systems are presented.