Stadnyk N. Modeling and effective methods of cyclic signal processing based on isomorphic cyclic random processes

This thesis deals with the solution of the important problem of the development of modeling and methods of statistical cyclic signals processing within the theory of cyclic random processes in the direction of improving the concept of their isomorphism and establishing basic properties and relations between different classes of their equivalence as well as in the development of the mathematical model of digital cyclic signals with double stochasticity and development of methods for their statistical processing with low computational complexity in portable digital systems with limited computing resources. Information on isomorphism of cyclic functional relations, especially, on different types of isomorphism between cyclic random processes is considered in details. The important role of the isomorphism concept in the theory of modeling and processing of cyclic signals is noted, as this concept plays a fundamental role in defining deterministic, stochastic, interval and fuzzy models of cyclic signals and defines their processing methods. The procedure of constructing the mathematical model of cyclic digital signals in the form of cyclic random process of the discrete argument is developed. It makes possible to take into account the stochasticity of cyclic signals both in their morphological statistical analysis and in statistical analysis of their rhythm. The definition of discrete random function of rhythm and random domain of conditional cyclic random process of discrete argument, which are the basis of rhythm analysis of cyclic signals within stochastic approach, are given. Approaches to morphological statistical analysis of cyclic signals with double stochasticity and statistical analysis of their rhythm are formulated. А new approach to statistical processing of cyclic random processes of discrete argument based on the procedures of their reduction to isomorphic periodic random sequences, which simplifies analytical calculations and formulas complexity of appropriate algorithms in the problems of statistical processing of cyclic signals in information systems for medicine, engineering and economics is developed. It is especially important for their implementation in portable systems with sufficiently limited computing power. Applying this new approach for solving statistical problem of evaluation the probabilistic characteristics of cyclic discrete argument processes, the new computation efficient methods for statistical evaluation of the initial moment function of the first order (mathematical expectation) and the correlation function of discrete argument cyclic random processes are obtained. Analytical expressions for functions of computational complexity of known and new methods of statistical evaluation of initial moment function of the first order (mathematical expectation) and correlation function of discrete argument cyclic random processes are constructed. The system of computer programs for statistical evaluation of probabilistic characteristics of cyclic signals based on new computationally efficient methods developed in the third section of the dissertation is worked out. Structural and functional schemes and functional capabilities of multifunctional software package for cyclic signals modeling and analysis, equipped with the developed of computer programs system are considered. Creation of computer programs system for statistical evaluation of probabilistic characteristics of cyclic signals is the basis for increasing the speed of statistical cyclic signals processing within their mathematical models such as cyclic random process and conditional cyclic random process of discrete argument, which takes place in computerized diagnostic systems cardiovascular system and information systems for analysis and forecasting of economic cyclical processes.