Chapter 12. Filtering Processes with Random Structure in Discrete Time
Serhii Ya. Zhuk and Igor О. Tovkach
Department of Radio Engineering Devices and Systems, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine
Part of the book: Stochastic Processes: Fundamentals and Emerging Applications
The urgency of the problem of filtering processes with a random jump-like change in probabilistic characteristics in information-measuring systems is shown. Stochastic dynamical systems with a random structure with Markov switches represent a wide class of mathematical models of such processes. Algorithms of filtering processes with a random structure with Markov switches, based on the mathematical apparatus of mixed Markov processes in continuous time, as well as of the Bayesian method of adaptive estimation in discrete time, have been analyzed. The expediency of using the mathematical apparatus of mixed Markov processes in discrete time for solving this problem is displayed. The analysis of a priori probabilistic characteristics of an extended mixed process, including a continuous-valued process with a random structure in discrete time and a discrete-valued Markov switching variable, is carried out. An optimal recurrent filtering algorithm for processes with a random structure in the presence of white Gaussian noise, which describes the evolution of the posterior probability density of the mixed process, has been synthesized. On its basis, quasi-optimal adaptive filtering algorithms are achieved by the method of Gaussian approximation. Structures of optimal and quasi-optimal filters, which belong to the class of devices with feedbacks between channels, are considered. The two-functional Bayesian decisive rule for determing the estimates of mixed Markov process is found, and its analysis for given loss funtion is performed. The analysis of the proposed quasi-optimal algorithms was carried out on the examples of solving the problems of adaptive trajectory filtering of a target moving with a maneuver and in the presence of anomalous measurements. Optimal and quasi-optimal recurrent algorithms for filtering processes with a random structure in the presence of Markov interferences are synthesized. The structure of the quasi-optimal filter is considered. The analysis of the proposed quasi-optimal algorithm is carried out on the example of solving the problem of joint filtering and recognition of the type of stationarity sections of speech (voice) signals in the presence of correlated interference.
Keywords: dynamical system, random structure, Markov switching’s, discrete time,
mixed Markov process, optimal algorithm, quasi-optimal algorithm, Markov interference,
multichannel device, poly-Gaussian approximation
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