The Mathematical Model of Cyclic Signals in Dynamic Systems as a Cyclically Correlated Random Process

Abstract

This work is devoted to the procedure for constructing of a cyclically correlated random process of a continuous argument as a mathematical model of cyclic signals in dynamic systems, which makes it possible to consistently describe cyclic stochastic signals, both with regular and irregular rhythms, not separating them, but complementing them within the framework of a single integrated model. The class of cyclically correlated random processes includes the subclass of cyclostationary (periodically) correlated random processes, which enable the use of a set of powerful methods of analysis and the forecasting of cyclic signals with a stable rhythm. Mathematical structures that model the cyclic, phase and rhythmic structures of a cyclically correlated random process are presented. The sufficient and necessary conditions that the structural function and the rhythm function of the cyclically correlated random process must satisfy have been established. The advantages of the cyclically correlated random process in comparison with other mathematical models of cyclic signals with a variable rhythm are given. The obtained results contribute to the emergence of a more complete and rigorous theory of this class of random processes and increase the validity of the methods of their analysis and computer simulation.

Isomorphic Multidimensional Structures of the Cyclic Random Process in Problems of Modeling Cyclic Signals with Regular and Irregular Rhythms

“…Therefore, in order to ensure the integrity of the content of the article, here we present only the main elements of this stage of construction. As shown in the work of [61], in general, a cyclic signal is random process ξ(ω, t), ω ∈ Ω, t ∈ R ( ξ : R → L 2 (Ω, P) ) which is given as a set of pairs (argument t, value ξ(ω, t)) ξ = {(t, ξ(ω, t)) : t ∈ R} with the same probability space (Ω, F, P). A necessary prerequisite for building a one-dimensional and multidimensional cyclic structure of a CRP is the ordered (ordered by m) countable partition D c R = {W c m , m ∈ Z} of domain R, then, for the elements of D c R , the following can be determined [61]:…”

“…In order to build a mathematical construction based on a strict definition of the CRP, we formalize intuitive (informal) fundamental concepts such as the multidimensional cycle and phase structures of a cyclic process within the framework of the theory of random processes. The first stage of the procedure for the CRP's construction coincides with the first stage of constructing a cyclically correlated random process, which is described in detail in the work of [61]. Therefore, in order to ensure the integrity of the content of the article, here we present only the main elements of this stage of construction.…”

“…∈ Z is a subset of R whose elements correspond to the moments at the beginning of the cycles of a cyclic signal. In the work of [61], the elements W c m of partition D c R are interpreted as carriers of relational systems ⟨W c m , ≤⟩ with the linear order ≤ and are ordered by the m countable family RS c R = {⟨W c m , ≤⟩, m ∈ Z} of the subrelational systems of a relational system ⟨R, ≤⟩, between which there is an isomorphism with respect to the linear order ≤ (see Figure 1).…”

“…In work [61] it is shown that by bijection R ⇐⇒ ξ from partition D c R = {W c m , m ∈ Z} of domain R can be built countable family RS c ξ = {⟨ξ c m , ≤ 2 ⟩, m ∈ Z} of the isomorphic with respect to binary relation of linear order ≤ 2 subrelational systems ⟨ξ c m , ≤ 2 ⟩ of relational system ⟨ξ, ≤ 2 ⟩. Linear order ≤ 2 here is generated in ξ = {(t, ξ(ω, t)): t ∈ R} by linear order ≤ in R ( ⟨R, ≤⟩ ⇐⇒ ⟨ξ, ≤ 2 ⟩ ).…”

“…However, there are no research studies devoted to the method of formation such processes and procedures for constructing multidimensional cyclic and phase structures. In contrast to the cyclic random process in the strict sense, the construction procedure and fundamental properties of a cyclically correlated random process in a broad sense were presented in the work of [61]. The purpose of this work is to construct isomorphic time-invariant structures of a CRP in an explicit, meaningful, interpretable and mathematical form, to describe its multidimensional cyclic and phase structures and to form the basis for a procedure for the construction and definition of CRPs.…”