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Abstract Cyclic Functional Relation and Taxonomies of Cyclic Signals Mathematical Models: Construction, Definitions and Properties

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  • Serhii Lupenko

    (Faculty of Electrical Engineering, Automatic Control and Informatics of Opole University of Technology, 45-758 Opole, Poland
    EPAM School of Digital Technologies, American University Kyiv, Poshtova Sq 3, 04070 Kyiv, Ukraine
    Institute of Telecommunications and Global Information Space, 02000 Kyiv, Ukraine)

Abstract

This work is devoted to the procedure of the construction of an abstract cyclic functional relation, which summarizes and extends the known results for a cyclically correlated random process and a cyclic (cyclically distributed) random process to the case of arbitrary cyclic functional relations. Two alternative definitions of the abstract cyclic functional relation are given, and the fundamental properties of its cyclic and phase structures are presented. The theorem on the invariance of cyclicity attributes of an abstract cyclic functional relation to shifts of its argument, and which are determined by the rhythm function of this functional relation, is formulated and proved. This theorem gives the sufficient and necessary conditions that the rhythm function of an abstract cyclic functional relation must satisfy. By specifying the range of values and attributes of the cyclicity of an abstract cyclic functional relation, the definitions of important classes of cyclic functional relations are formulated. A deductive approach to building a wide system of taxonomies of classes of deterministic, stochastic, fuzzy and interval cyclic functional relations as potential mathematical models of cyclic signals is demonstrated. A comparative analysis of an abstract cyclic functional relation with the known mathematical models of cyclic signals was carried out. The results obtained in the article significantly expand and systematize the mathematical tools of the description of cyclic signals and are the basis for the development of effective model-based technologies for processing and computer simulation of signals with a cyclic space-time structure.

Suggested Citation

  • Serhii Lupenko, 2024. "Abstract Cyclic Functional Relation and Taxonomies of Cyclic Signals Mathematical Models: Construction, Definitions and Properties," Mathematics, MDPI, vol. 12(19), pages 1-37, October.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:19:p:3084-:d:1490775
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    References listed on IDEAS

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    1. Eric Ghysels, 1993. "A time series model with periodic stochastic regime switching," Discussion Paper / Institute for Empirical Macroeconomics 84, Federal Reserve Bank of Minneapolis.
    2. Hurd, H. & Kallianpur, G. & Farshidi, J., 2004. "Correlation and spectral theory for periodically correlated random fields indexed on Z2," Journal of Multivariate Analysis, Elsevier, vol. 90(2), pages 359-383, August.
    3. Bai, Chunyan, 2018. "Time delay effects of stochastic resonance induced by multiplicative periodic signal in the gene transcriptional regulatory model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 507(C), pages 304-311.
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