1. Introduction
It is well known that the intensive development of the digital signal processing industry has penetrated into all spheres of human existence, automating the previously routine, resource-consuming and long-term procedures of information processing, ensuring a high level of accuracy, validity and reliability of such procedures in telecommunications, energy, transport, medicine, the economy and other areas. This state of affairs also occurs in the problems of processing (estimation of characteristics, detection, classification, clustering and forecasting) the signals (processes and phenomena) of space-time structure, in particular, cardiac signals (electrocardiosignals, magnetocardiosignals, phonocardiosignals, photoplethysmocardiosignals, etc.), vibration signals of mechanisms and machines in rotation modes, astrophysical radiation signals of space objects (i.e., stars, quasars), modulated and demodulated telecommunication signals, gas and electricity consumption processes and cyclical economic processes.
The modern technologies (methods, algorithms, hardware and software) of digital processing of signals with a cyclic space-time structure can be conditionally divided into three large groups, namely, model-based, data-based and hybrid digital technologies. Model-based technologies are clearly oriented to previously developed mathematical models of the investigated signals; data-based technologies mostly do not require the prior development of a mathematical model of a cyclic signal, since they use machine learning methods based on a large amount of input data. Hybrid technologies, in a certain way, integrate processing procedures within the framework of model-based and data-based approaches. Despite the fact that data-based methods and algorithms for the processing of cyclic signals do not explicitly require the construction of a mathematical model of cyclic signals, they require the preliminary development of a mathematical metamodel of the process of automated generation of an optimal or quasi-optimal data-based algorithm for processing cyclic signals. This metamodel should clearly reflect the architecture of the information and computing environment of the data-based technology and the procedure for evaluating the optimal (quasi-optimal) parameters of the signal processing algorithm. In order to ensure the efficiency (accuracy, reliability and low time computational complexity) of processing cyclic signals, it is desirable that the architecture of the information and computing environment of hybrid technology also has formal means of taking into account the regularities of the space-time structure of the investigated signals, which also requires the development of appropriate mathematical-model cyclic signals.
In this article, we will focus directly on the model-based approach to cyclic signal processing technologies. Therefore, according to the work [
1], we will present the main components and stages of the development of information systems for processing cyclic signals using a model-based approach as shown in
Figure 1.
As can be seen from this figure, the initial and main stage of designing information systems for processing and simulating cyclic signals is the creation of their mathematical models that adequately reflect the important, from the point of view of research tasks, aspects of their space-time structure. In the framework of the model-based approach, the mathematical model largely determines the potential and effectiveness of the information technologies being created, and to some extent, determines the structure of the software and hardware components of the designed information system. The quality of the mathematical model of cyclic signals (see
Figure 2) significantly determines the accuracy and reliability of the methods of their processing in the information system and the level of informativeness of the diagnostic and authentication methods, as well as the reliability of the decisions made.
Today, there are a large number of mathematical models of cyclic signals, which are used as a logical–formal core of model-based technologies for digital processing of these signals. Mathematical models of cyclic signals can be conditionally divided into two large and interrelated types: constructive mathematical models that clearly reflect the regularities and mechanisms of the construction (i.e, formation, generation) of cyclic signals, and structural mathematical models that clearly reflect the regularities and organization of the structure (space and/or time and/or frequency structures) of the studied cyclic signals. Constructive mathematical models of cyclic signals are mainly presented as differential and integral equations for continuous parameter signals (linear and nonlinear, deterministic and stochastic differential equations and linear periodic random processes and fields) [
2,
3,
4] or as difference equations, autoregression and moving-average models for cyclic signals of a discrete parameter [
5,
6]. In the article, we will focus only on structural mathematical models of cyclic signals, since often, in the practical tasks of developing digital technologies, the mechanisms of the generation of these signals may not be taken into account. However, the structure of the cyclic signal itself and its parameters are mainly important objects of the structural–parametric identification procedures of these signals, since they are the carriers of information necessary to solve the corresponding problems of signal processing.
Table 1,
Table 2,
Table 3 and
Table 4 present the main (not all of them) structural mathematical models of cyclic signals and some of their properties. The theory of mathematical modeling of cyclic signals received the greatest development through the use of the idea of periodicity, namely, periodic deterministic and stochastic functions.
Table 1 presents the main (not all of them) structural mathematical models based on a periodic pattern.
The structural mathematical models of cyclic signals that generalize deterministic and stochastic periodic functions in the direction that mainly concerns the expansion of the spectral properties of periodic deterministic functions and periodic random processes are presented in
Table 2.
Table 3 presents the structural mathematical models of cyclic signals, which, in a certain way, take into account the variability (irregularity) of their rhythm.
Table 4 presents the cyclic functional relations that, in an explicit form, reflect the cyclic structure of signals and have formal means of taking into account the irregularity of their rhythm.
The abstract cyclic functional relation (first introduced and explored in papers [
1]) presented in
Table 4 is a generalized mathematical model of signals with a cyclic space-time structure. Such a generalized nature of this mathematical object made it possible to explicitly express the intuitive concept of a cyclic signal (cyclic process, cyclic phenomenon and cyclic movement) by formal means, which is common to deterministic, stochastic, fuzzy and interval model approaches. Despite the importance of this mathematical object for the creation of a general theory of structural mathematical models and model-based methods for processing cyclic signals, the construction procedure, possible alternative definitions and fundamental properties of the abstract cyclic functional relation in works [
1] has remained almost unexplored. In works [
45,
46], procedures for constructing partial cases of an abstract cyclic functional relation were developed, namely, the procedure for constructing a cyclically correlated random process and a cyclic (cyclically distributed) random process, which are mathematical models of cyclic stochastic signals. In this article, we will extend the procedures for constructing a cyclically correlated random process and a cyclic (cyclically distributed) random process developed in works [
45,
46] to a more general case, namely, to the case of an abstract cyclic functional relation (cyclic relative attributes functional relation). Also, the article will present and investigate a number of the fundamental properties of abstract cyclic functional relations and alternative definitions of this class of functions. Such a broad generalization of the results of the works [
45,
46] in the case of an arbitrary type of cyclic functional relation will provide a more systematic and broad view of the modeling of cyclic signals within the framework of various paradigms of mathematical modeling and will become the basis of a general theory of mathematical modeling and processing of signals with a cyclic space-time structure. In addition, the work is aimed at building a developed taxonomy of subclasses of abstract cyclic functional relations that forms a broad system of classes of potential mathematical models of cyclic signals within the framework of deterministic, stochastic, fuzzy and interval approaches to their modeling.
The work is organized as follows.
Section 2 is devoted to the conceptual (informal) foundations of the theory of mathematical modeling and processing with a cyclic space-time structure.
Section 3 is devoted to the procedure of construction of an abstract cyclic functional relation.
Section 4 presents the cyclic, phase and rhythmic structures of an abstract cyclic functional relation; the theorem on the necessary and sufficient conditions for the rhythm function of an abstract cyclic functional relation is given.
Section 5 presents some of the examples of cyclic functional relations as potential mathematical models of cyclic signals within deterministic, stochastic, fuzzy and interval modeling paradigms by means of concretization of the range of values and attributes of cyclicity of an abstract cyclic functional relation.
Section 6 is devoted to the method of generating (induction) taxonomies of classes of cyclic functional relations.
Section 7 presents a discussion of the main advantages of an abstract cyclic functional relation in comparison with known mathematical models of cyclic signals. Chapter 8 concludes the article, where the main conclusions regarding the results obtained in the article are given.
3. Procedure of Construction of an Abstract Cyclic Functional Relation
Having correctly matched the above basic informal concepts of the conceptual model with such formal objects as a set, partition of a set, linearly ordered set, functional relation (function), bijective mapping, relational system and isomorphism of relational systems, we will develop a procedure of the construction of an abstract cyclic functional relation and we will give it an appropriate definition.
Based on works [
45,
46], let us demonstrate the procedure for building a cyclic relative to a set of attributes’
(in certain cases, it is allowed that
) functional relation (other names are abstract cyclic functional relation or abstract cyclic function)
, which adequately and consistently reflects the cyclic structure of a wide class of signals of different nature and space-time structures. The domain of definition
of the cyclic functional relation
is an ordered discrete set
or set of real numbers
. In the case of the discreteness of the domain of definition
, the following type of linear ordering takes place for its elements:
if
or if
and
in other cases,
(
,
). The range of values of a cyclic functional relation is some linear space
(
or
) over the field of real or complex numbers, the elements of which can be numbers, fuzzy numbers, vectors, matrices, tensors, intervals, functions, random variables, random vectors, random matrices, random functions and random operators, etc. Since the functional relation
is a set of pairs (argument
t, value
)
, then, in the further presentation of the material, we will also mark it
. Since the range of values
of function
is an arbitrary linear space, then such a function, by analogy with an abstract function in the sense of Bochner, will be called an abstract function (abstract functional relation) or an abstractly valued function (abstractly valued functional relation).
In order to expand the mathematical means of the formalized description of cyclic signals and generalize the set of possible properties relative to which repetition occurs in the structure of the studied cyclic signal, the concept of set of attributes of cyclicity is introduced. Functions maps the -th Cartesian power into some set , which is the set of possible values of the cyclicity attribute of the signal. Elements of the set can be numbers, vectors, functions, etc., and, therefore, functions can be numerical functions, functionals or operators.
In other notations functions
, we will present it as follows:
. In order to exclude non-cyclic functions, we will consider only such functional relations
from
, for which exists such a number
, that there are such inequalities:
Let us have countable partition
of definition domain
, then, for the elements of partition
, the following relations are performed [
1]:
where
in the case
, and set
is a subset of
which correspond to the moments of the beginning of cycles of a cyclic signal. In the works [
45,
46], the elements
of partition
are interpreted as carriers of the relational systems
with a binary relation of the linear order
, and introduced ordered by
countable family
of subrelational systems of the relational system
, between which there is an isomorphism with respect to the linear order
. For the case when
,
Figure 1 in [
46] conditionally shows this type of isomorphism.
According to the works [
45,
46], it is easy to show that by bijective mapping
from a countable family
, a countable family
of the isomorphic with respect to the binary relation of the linear order
subrelational systems
of the relational system
can be built. The linear order
here is generated in
by the linear order
in
(
).
The countable family represents one-dimensional isomorphic structures of . To display multidimensional (-dimensional) isomorphic structures of , let us consider the Cartesian degree of the -th order () of , and consider the bijective mapping , which can always be constructed, because any -dimensional vector corresponds to one and only one -dimensional vector and vice versa, and for the two different -dimensional vectors and , the corresponding two -dimensional vectors and are also different, and vice versa. The bijective mapping induces (generates) a linear order in the Cartesian degree itself, which, in this case, can be considered as a carrier of the relational system with a binary relation of the linear order . The ordinal type of coincides with the ordinal type of the set . Namely, for any two -dimensional vectors and , it is always possible to specify their order: if or if . In the case when , we will have such an order: if or if . In general, in the case when (), we will have such an order: if or if . In other words, the bijective mapping is an isomorphism between the relational system and the relational system with respect to the binary relations of the linear order and (). That is, we will talk about as about a linear ordered Cartesian power by the type of ordering of Cartesian power .
According to the work [
46], let us form an ordered by
countable partition
of
based on the ordered countable partition
of domain
. Due to the linear ordering
of the set
, elements
of the partition
are also linearly ordered sets. Let us consider the elements
of partition
as carriers of the relational systems
with a binary relation of the linear order
. Thus, the partition
generates an ordered, by
, countable family
of the subrelational systems of the relational system
, between which there is an isomorphism with respect to the linear order
. For the case when
,
Figure 2 in [
46] conditionally shows this type of isomorphism.
Due to the bijective mapping , partition of generates an ordered countable partition of Cartesian power of the -th order, where every is the truncation of the to the set . Namely, each set matches the , which is its image according to bijective mapping That is, every is the set of those ordered -dimensional vectors of the , the argument of which belongs to , and the arguments of which belongs to .
Since the Cartesian product is the carrier of , then with its partition it is always possible to connect the countable family of the subrelational systems of . From the isomorphism between the subrelational systems with respect to the binary relation of the linear order due to the isomorphism follows the isomorphism between the subrelational systems with respect to the binary relation of the linear order . Namely, for any , the arbitrary subrelational systems and from are isomorphic with respect to the binary relation of the linear order , and for any Cartesian product, is a Cartesian product, linearly ordered by the type of ordering of its domain .
So, taking into account that mentioned above, it can be argued that there is (1) the isomorphism with respect to the binary relations of the linear order and between the relational systems and ; (2) the isomorphism with respect to the binary relation of the linear order between elements of the countable family of the subrelational systems of the relational system ; (3) the isomorphism with respect to the binary relation of the linear order between the elements of the countable family of the subrelational systems of the relational system ; and (4) the isomorphism with respect to the binary relations of the linear order and between arbitrary pair and , taken from the countable partition of the Cartesian power and from the countable partition of the Cartesian power .
Let us introduce a relational system:
where
,
are sets of carriers and
are sets of the relations of the relational system (3).
The partition of the Cartesian power of generates the family of subrelational systems.
of the relational system (3), where are carriers of the subrelational system . In the case when in Formula (3), assume that .
Let us amplify the isomorphism between the relational systems of the family by adding the requirements of the equality of values of functions for the bijective connected vectors and from two different arbitrary Cartesian products, and . Namely, the isomorphism with respect to the binary relations of the linear order , for the arbitrary two relational systems and , must be supplemented by an isomorphism between them with respect to functional relations .
This kind of isomorphism between the relational systems and will be called an isomorphism with respect to the linear order and to the set of attributes . Let us give a strict definition of this type of isomorphism between the relational systems and for any .
Definition 1. The set of bijective mappings between the appropriate Cartesian products and , which are carriers of the relational systems and , will be called the set of isomorphisms with respect to the relations of the linear order and with respect to the set of attributes between relational systems and , if the following statements are true.
There are isomorphisms between the relational systems and with respect to the relation of the linear order for any .
There are isomorphisms between the relational systems
and
with respect to the set of attributes
, namely, for all the bijective connected vectors
and
for any
, there are equal values of the functions
, namely,
Definition 2. The Cartesian products and , which are carriers of the isomorphic relational systems and , will be called the isomorphic Cartesian products with respect to the relation of linear order and with respect to the attribute or will be called more compact—the isomorphic Cartesian products.
The family of isomorphic subrelational systems, the carriers of which are the elements of the ordered countable partitions from sequences constructed above, makes it possible to give the definition of a cyclic, by attributes , functional relation.
Definition 3. Ordered by the domain of the definition functional relation with a range of values, is a cyclic, by attributes , functional relation (or an abstract cyclic functional relation or abstract cyclic function), if, for each of its ordered -th Cartesian power exists the ordered countable partition from set , the elements of which are carriers of isomorphic relational systems with respect to the relations of linear order and with respect to the set of attributes .
4. The Cycle, Phase and a Rhythm Structures of Abstract Cyclic Functional Relation
Since the multidimensional cycle and phase structures of an abstract cyclic functional relation are essentially similar to the multidimensional cycle and phase structures of the cyclic random process, which were studied in [
46], in this article, we will present only the main results for the abstract cyclic functional relation in a concise form.
Definition 4. The minimal ordered partition of the Cartesian product of the abstract cyclic functional relation into the isomorphic Cartesian products with respect to the relation of linear order and with respect to the attribute , will be called the partition into -dimensional cycles of the abstract cyclic functional relation , and the Cartesian product will be called the m-th -dimensional cycle of abstract cyclic functional relation .
Thus, the cyclic structure of the abstract cyclic functional relation is given by the set , elements of which are partitions into -dimensional cycles of the abstract cyclic functional relation .
Based on the results obtained above, let us present an abstract cyclic functional relation
and its Cartesian product
through their respective cycles, namely:
Since we require that the range of values of a cyclic functional relation is some linear space
over the field of real or complex numbers, then taking into account the property of linearity an abstract cyclic functional relation
and its Cartesian product
can be given in another form, namely:
where
in the areas
coincides with
, but in the areas
, the functional relation
is identically equal to zero in the linear space
.
Similarly to the representations of the abstract cyclic functional relation
and its Cartesian product
according to formulas (5) and (6) can be given representations of the attribute of cyclicity
(in another designation,
) of the functional relation
:
where
is a
-dimensional attribute of
m-th
-dimensional cycle
of an abstract cyclic functional relation
.
Similarly to the approach developed in [
46], we will construct the phase structure of the abstract cyclic functional relation
. Let us have the domain
of the
-dimensional
-th cycle
of an abstract cyclic functional relation
. Due to isomorphism between relational systems
and
(
), for any
in the domain
of arbitrary
-dimensional
-th cycle
, exists only one element
, which is bijectively connected with the element
. Since for an abstract cyclic functional relation
, exists a countable set
of
-dimensional cycles, then for any
-dimensional vector
, exists a countable set
of
-dimensional vectors
, which are bijectively connected to it. Set
of all bijectively connected vectors with a vector
is defined as follows:
If vector runs the all set , then we obtain the ordered in the indexes partition of the domain of the Cartesian product of the abstract cyclic functional relation .
By bijective mapping of elements
from the partition
into subsets
of the Cartesian product
(
), let us create an uncountable partition
of the Cartesian product
of the abstract cyclic functional relation
. According to [
46],
is a countable ordered by
set, defined as follows:
According to [
46],
is a countable set of the
-dimensional vectors of the Cartesian product
, among which there are no two vectors belonging to the same
-dimensional cycle; that is, among the elements of
, there are no two vectors
where
and
where
for which
.
Let us give the definition of the -dimensional phase of the cyclic functional relation .
Definition 5. Ordered by the indexes , the
partition of the Cartesian product is called the partition into sets of -dimensional phases of same type and the set is called the set of -dimensional phases of same type of an abstract cyclic functional relation , if are countable sets formed according to (10) and for different elements and from there is the following such equality of the cyclicity attribute : Definition 6. The m-th element of the set is called the -dimensional phase in the -dimensional m-th cycle of an abstract cyclic functional relation .
Definition 7. The set , determined according to the expressionis called the -set of single-phase values of an abstract cyclic functional relation .
It should be noted that according to [
46], there is not one, but a whole set
of possible partitions into
-dimensional cycles
of an abstract cyclic functional relation
. However, for the abstract cyclic functional relation
exists only one partition
into sets of
-dimensional phases of same type.
Let us represent the Cartesian product
of the abstract cyclic functional relation
through the elements of their phase structures, namely, through the elements of the partition
:
Let us represent of attribute of cyclicity
of the functional relation
:
where
is a
-dimensional attribute of the set
of
-dimensional phases of the same type of an abstract cyclic functional relation
.
The
-dimensional
m-th cycle
of an abstract cyclic functional relation
can be presented as follows:
or
Let us represent the Cartesian degree
through the set
:
Let us represent the set of
-dimensional phases of same type of an abstract cyclic functional relation
:
Based on the results of works [
1,
45,
46], let us formulate and prove the following theorem.
Theorem 1. For the cyclic, by attributes , functional relation , there exists a numerical function , for which the following properties occur:
- (1)
- (2)
for any and , for which , and for function a strict inequality holds:and for each attribute from the set of attributes there is the following equality:
On the contrary, I,f for the functional relation , there exists a numerical function with all mentioned above properties (16), (17), and if equalities (18) are hold for any , then it is a cyclic, by attributes , functional relation (an abstract cyclic functional relation).
Proof of Theorem 1. The first part of the procedure for proving this theorem coincides with the first part of proving a similar theorem for a cyclically correlated random process given in [
45]. In order to ensure the integrity of the description of the procedure for proving Theorem 1, we will use the results of [
1,
45]. □
According to the definition of the cyclic, by attributes
, functional relation
, any of its two cycles,
and
, are isomorphic with respect to the binary relation of the linear order
, and this isomorphism is due to the isomorphism with respect to the binary relation of the linear order
of the domains
and
, which are ordinary sets of real numbers. Between the isomorphic numerical sets
and
, some additive numeric functions
and
can always be constructed, namely [
45], there is the bijection between
and
, and the same type of linear ordering of sets
and
takes place, that is
,
, that
and there is a strong order relation:
and, vice versa,
, that
and there is a strong order relation:
Taking into account the isomorphism between all possible pairs of cycles of the cyclic, by attributes
, functional relation
, we introduce a countable-dimensional matrix of increasing numerical functions which specifies bijective mapping between the domains of its corresponding cycles, i.e., the following matrix:
moreover, on the diagonal of the functional matrix (21) when
we will have numerical functional relations of identity which are automorphisms with respect to the binary relation of the linear order
of the domains
, and at the permutation of the places of indices
and
of the function
, we obtain the inverse numerical increasing function
.
Entering the notations
and taking them into account in the indices of the elements of the matrix (21), i.e.,
, from the matrix (21), we obtain the following countable-dimensional vector of increasing numerical functions from two arguments
and
:
Note that the increase of these numerical functions takes place in relation to both the argument
and the argument
. Each element of the countable-dimensional vector (22) establishes an isomorphism between the domains of the definition of the arbitrary
-th cycle and
-th cycle, which is remote from
-th cycle on
cycles. In addition, for all elements of the countable-dimensional vector (22), there are the following inequalities [
1,
45]:
The first property follows from the followingfacts: , and . Moreover, , whereas , and therefore, and .
The second property follows from the fact that , since and .
The third property can be proved similarly to the first: , and , and , whereas and therefore, and .
Since, for an abstract cyclic functional relation
, there is a set
of its possible partitions into one-dimensional cycles, therefore, there exists a set of countable-dimensional vectors (22) corresponding to this partitions. However, since, for the cyclic functional relation
, there exists only one its partition
into sets of one-dimensional phases of same type, therefore, for all possible countable-dimensional vectors (22) corresponding to partitions from set
, there is the one and only one numerical function
, which is equal to the ordered union (sum) of the elements of the countable-dimensional vector (22) at a fixed
:
Due to the order of the union (24), the numerical function , similarly to the elements of the countable-dimensional vector of functions (22), is an isomorphism with respect to the binary relation of the linear order , and therefore, for it a strict inequality (17) holds, i.e., for any fixed function , there is an increasing numerical function. The requirement of the limited function necessarily follows from the fact that the duration of one-dimensional cycles is limited, that is, formally reflected in the inequalities when considering the partition . The properties (16) follow from the properties (23) of the components of the vector (22), since the numerical function , in fact, is “stitched” from these components.
For the
-dimensional vector
(
),
there exists a bijective connected to them, the
-dimensional vector
(
), and there are such equalities of attributes
, namely,
If the vector
runs through the entire set
and the vector
runs through the entire set
, then equalities (25) will turn into equalities (18), because
It is easy to see that, if for some an abstract functional relation , there exists a numerical function , which satisfies the conditions (16), (17), and equations (18) are hold, then such a functional relation is a cyclic, by attributes functional relation, because, in this case, there always exists the sequence , the elements of which are the partitions into the -dimensional cycles of the cyclic, by attributes functional relation , which are carriers of isomorphic relational systems with respect to the relations of linear order and with respect to the set of attributes .
This concludes the proof of the Theorem 1.
Similar to the results of work [
45], the function
,which is the smallest in modulus
among all such functions
which satisfy (16)–(18), is called a
rhythm function of an abstract cyclic functional relation .
The rhythm function
is a mathematical representation of the law of changing the time intervals between the single-phase values of the signals with a cyclic structure. In more detail, the properties of the rhythm function
of a cyclic functional relation are studied in works [
1,
45,
46].
The value of the Theorem 1 is that it provides the sufficient and necessary conditions that the function must satisfy, so that the functional relation was the abstract cyclic functional relation. The possibility of another way of defining an abstract cyclic functional relation directly follows from the Theorem 1.
Definition 8. The functional relation is called the cyclic, by attributes , functional relation (or abstract cyclic functional relation or abstract cyclic function), if, for it, there exists such a function , which satisfies the conditions of the rhythm function, namely,
- (1)
- (2)
For any and , for which , and for function a strict inequality holds: - (3)
The function is the smallest in modulus among all such functions which satisfy (27), (28), namely,and for each attribute from set there is the following equality:
A partial case of an abstract cyclic functional relation, when its rhythm function (), is a periodi, c by attributes functional relation (or abstract periodic functional relation or abstract periodic function or abstract cyclic functional relation with regular rhythm). A partial case of abstract cyclic functional relation, when its rhythm function , is a cyclic, by attributes functional relation with an irregular rhythm (or abstract cyclic functional relation with irregular rhythm or abstract cyclic function with irregular rhythm).
Most of the practical problems of processing cyclic signals on the basis of their mathematical model in the form of an abstract cyclic function is reduced to the identification (estimating) of a set of attributes of cyclicity and rhythm function . In this context, it is possible to conditionally distinguish two types of methods for processing cyclic signals, namely, methods of morphoanalysis of cyclic signals, the purpose of which is to identify (estimat) attributes of cyclicity and methods of analyzing the rhythm of cyclic signals, the purpose of which is to identify (estimate) the rhythm function .
5. The Simplest Examples of Deterministic, Stochastic, Fuzzy and Interval Cyclic Functional Relations
Let us consider some of the simplest examples of cyclic functional relations as potential mathematical models of signals with a cyclic space-time structure. We will consider only such cases when set of attributes
, that is, all cyclicity attributes depend on the values of the cyclic function:
First, consider deterministic cyclic functions. Cyclic deterministic functions are a broad subclass of cyclic functions. These functions can be used as mathematical models of cyclic processes and signals if they have a regular repeating structure, or as models in the first approximation, when uncertainty in the structure of the signals can be neglected.
The simplest representative of cyclic deterministic functions is a cyclic numerical function, which is a generalization of a periodic numerical function. If in Definition 8 of the cyclic functional relation
, the range of its values
is the set of real numbers (
), and a set of attributes
is a set that contains only one function
(
), namely,
, then, the definition of a cyclic real-valued function follows from the general definition of an abstract cyclic functional relation. According to work [
1], we will give the following definition.
Definition 9. The real-valued function is called the cyclic real-valued function, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equality takes place: An example of a cyclic real-valued function with a constant rhythm is an arbitrary periodic real-valued function. An example of a cyclic real-valued function with a variable rhythm, that is, when
, is an angularly modulated harmonic function, provided that the modulation is carried out without changing its phase ordering type in all cycles of the modulated function. A typical example of such cyclic function is a function
with rhythm function
(see
Figure 3).
If, in Definition 8 of the cyclic functional relation
, the range of its values
is the set of complex numbers (
), and a set of attributes
is a set that contains only one function
(
), namely,
, then, the definition of a cyclic complex-valued function follows from the general definition of an abstract cyclic functional relation. According to work [
1], we will give the following definition.
Definition 10. The complex-valued function () is called the cyclic complex-valued function, if, for, there it exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equalities take place: Typical example of such cyclic function is a function
with rhythm function
(see
Figure 4).
If, in Definition 8 of the cyclic functional relation
, the range of its values
is the set of complex numbers (
), and a set of attributes
is a set that contains only one function
(
), namely,
, then, the definition of a cyclic, with respect to the modulus, complex-valued function follows from the general definition of an abstract cyclic functional relation [
1].
Definition 11. The complex-valued function () is called the cyclic, with respect to the modulus, complex-valued function, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for the module the following such equality takes place: If, in Definition 8 of the cyclic functional relation , the range of its values is a vector -dimensional space (), and a set of attributes is a set that contains only one function (), namely, , then, the definition of a cyclic vector function follows from the general definition of an abstract cyclic functional relation.
Definition 12. The vector function is called the cyclic vector function (cyclic vector-valued function), if for it exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equalities hold for all components of the vector function : A typical example of a cyclic vector function is a set of frequency-modulated harmonic functions with the same modulation law, provided there is no phase jump of the modulated function. A typical example of such a cyclic function is function
with rhythm function
, where
,
(see
Figure 5).
If, in Definition 8, of the cyclic functional relation , the range of its values is a vector -dimensional space (), and a set of attributes is a set that contains only one function (), namely, the attribute is the norm in this vector space, then, the definition of a cyclic, with respect to the norm vector, function follows from the general definition of an abstract cyclic functional relation.
Definition 13. The vector function is called the cyclic, with respect to the norm vector, function (cyclic, with respect to the norm, vector-valued function), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and equality holds for the norm : Depending on which norm of the vector is being considered, it is possible to give the definition of cyclic relative to different norms of the vector function . Namely, the cyclicity of the vector function can be introduced, for example, with respect to the quadratic norm , or the norms and . It is obvious that any cyclic vector function is also cyclic with respect to its norm, but not vice versa.
If, in Definition 8 of the cyclic functional relation , the range of its values is a vector Euclidean -dimensional space (), and a set of attributes is a set that contains only one function (), namely, the attribute is the scalar product in this Euclidean vector space, then, the definition of a cyclic, with respect to the scalar product vector function follows from the general definition of an abstract cyclic functional relation.
Definition 14. The vector function is called the cyclic, with respect to the scalar product , vector function, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and equality holds for the scalar product : From cyclicity with respect to the scalar product
vector function
followed by its cyclicity relative to the quadratic norm
, because
. Therefore, in this case, there is cyclicity of the vector function with respect to two attributes, namely, with respect to the attribute
and attribute
. It is obvious that any cyclic vector function is also cyclic with respect to the scalar product, but not vice versa. In particular, the cyclic vector-valued function, the graph of which is given in
Figure 3 is cyclic with respect to the scalar product
vector function (see
Figure 6) and is cyclic with respect to the quadratic norm
vector-valued function (see
Figure 7) with rhythm function
.
If, in Definition 8 of the cyclic functional relation , the range of its values is a linear matrix -dimensional space () (matrix elements are complex numbers), and a set of attributes is a set that contains only one function (), namely, , then, the definition of a cyclic matrix function (cyclic matrix-valued function) follows from the general definition of an abstract cyclic functional relation.
Definition 15. The matrix function is called the cyclic matrix function (or cyclic matrix-valued function), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equalities hold for all elements of the matrix function : If, in Definition 8 of the cyclic functional relation , the range of its values is a linear matrix -dimensional space () (matrix elements are complex numbers), and a set of attributes is a set that contains only one function (), namely, attribute is the norm in this matrix space, then, the definition of a cyclic, with respect to the norm, matrix function follows from the general definition of an abstract cyclic functional relation.
Definition 16. The matrix function is called the cyclic, with respect to the norm, matrix function, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and equality holds for the norm : Depending on which norm of the matrix is considered, it is possible to give a definition of the cyclic relative to different norms of the matrix function . Namely, the cyclicity of the matrix function can be introduced, for example, relative to the quadratic norm , or norms and . It is obvious that any cyclic matrix function is also cyclic with respect to its corresponding norm, but not vice versa.
If, in Definition 8 of the cyclic functional relation , the range of its values is a linear functional space , the elements of which are numerical functions of real variables (for example, three spatial coordinates) () (numerical values of the functions are complex numbers), i.e , and a set of attributes is a set that contains only one function (), namely, , then, the definition of a cyclic field by argument follows from the general definition of an abstract cyclic functional relation.
Definition 17. The function of real variables is called the cyclic field (cyclic field by argument ), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and equality takes place: If, in Definition 8 of the cyclic functional relation , the range of its values is a linear space of tensors (tensor elements are real numbers) of the same dimension () (the number of indices is the same and the same number of states that the indices can acquire), and a set of attributes is a set that contains only one function (), namely, , then, the definition of a cyclic tensor function (cyclic tensor-valued function) follows from the general definition of an abstract cyclic functional relation.
Definition 18. The tensor function is called the cyclic tensor function (or cyclic tensor-valued function), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equalities hold for all elements of the tensor function : To represent cyclic random objects, we will denote the cyclic functional relation
as
. If, in Definition 8 of the cyclic functional relation
, the range of its values
is a space of random variables (for example, the Hilbert space
), which are given on the same probability space
, and a set of attributes
is a family of consistent
-dimensional distribution functions (here
), namely,
the definition of a cyclic random process follows from the general definition of an abstract cyclic functional relation.
According to works [
1,
46,
47], we will give the definition of a cyclic random process.
Definition 19. The random process is called the cyclic random process (cyclically distributed random process), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for k-dimensional distribution function from the family of consistent distribution functions of a cyclic random process there are there following equalities: If we consider the cyclicity of a random process within the framework of the correlation theory, namely, if, in Definition 8 of an abstract cyclic functional relation
, the range of its values
is a space of random variables (for example, the Hilbert space
), which are given on the same probability space
, and a set of attributes
is a two-element set
, where
the definition of a cyclically correlated random process follows from the general definition of an abstract cyclic functional relation. Operator
is the mathematical expectation operator.
According to the works [
1,
45], we will give the definition of a cyclically correlated random process.
Definition 20. The random process is called the cyclically correlated random process, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for it mathematical expectation and autocovariance function there are there following equalities: If we consider the cyclicity of a random process within the framework of its higher moment functions, namely, if, in Definition 8 of an abstract cyclic functional relation
the range of its values
is a space of random variables (for example, the Hilbert space
), which are given on the same probability space
, and a set of attributes
is a set that contains only one function
(
), namely,
the definition of a cyclic with respect to the mixed initial moment function
of order
follows from the general definition of an abstract cyclic functional relation.
Definition 21. The random process is called the cyclic with respect to the mixed initial moment function random process, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and, for it, the mixed initial moment function of order , the following such equality takes place: If, in Definition 8 of an abstract cyclic functional relation
, the range of its values
is a space of random variables (for example, the Hilbert space
), which are given on the same probability space
, and a set of attributes
is a set that contains only one function
(
), namely,
the definition of a cyclic, with respect to the mixed central moment, function
of order
follows from the general definition of an abstract cyclic functional relation.
Definition 22. The random process is called the cyclic, with respect to the mixed central moment, function random process, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and, for it, the mixed central moment function of order , the following such equality takes place: Based on the above results, in a similar way, it is possible to define different subclasses of cyclical according to different probabilistic characteristics random processes, namely, according to the domain of definition (discrete and continuous), according to the type of their distribution functions (Gaussian, Poisson, uniformly, exponential distributed, etc.), by belonging to the corresponding class of random processes, the defining feature of which does not contradict the idea of cyclicity (i.e., cyclic white noise, processes with independent cyclic increments, linear cyclic random processes, Markov cyclic random processes), by the appearance of the rhythm function (i.e., the rhythm is stable, the rhythm is variable), by probabilistic characteristics in which a cyclic structure is postulated. One of the simplest examples of a cyclic random processes is the random process with a cyclic mathematical expectation. An example of a random process with a cyclic mathematical expectation is the process , where and is a white noise with a normal distribution with zero mathematical expectation and unit variance. The mathematical expectation of the random process is equal: .
If, in Definition 8 of an abstract cyclic functional relation
, the range of its values
is a space of random vectors of dimension
(for example,
), which are given on the same probability space
, and a set of attributes
is a family of consistent
-dimensional distribution functions (here
), namely,
the definition of a vector of cyclic rhythmically connected random processes follows from the general definition of an abstract cyclic functional relation.
According to works [
1,
48], we will give the definition of a vector of cyclic rhythmically connected random processes.
Definition 23. Vector of random processes is called the vector of cyclic rhythmically connected random processes (and the processes themselves is called the cyclic rhythmically connected random processes), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for compatible -dimensional distribution function from the family of consistent distribution functions of a vector there are the following equalities: If we consider the cyclicity of a random vector within the framework of the correlation theory, namely, if, in Definition 8 of an abstract cyclic functional relation
, the range of its values
is a space of random vectors of dimension
(for example,
), which are given on the same probability space
, and a set of attributes
is a two-element set
, where
the definition of a vector of cyclically correlated rhythmically connected random processes follows from the general definition of an abstract cyclic functional relation.
Definition 24. Vector of random processes is called the vector of cyclically correlated rhythmically connected random processes (and the processes themselves is called the cyclically correlated rhythmically connected random processes), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for it mathematical expectations and autocovariance functions there are the following equalities: Similarly, it is possible to define the vector of cyclic, with respect to the mixed initial moment, functions rhythmically connected random processes and the vector of cyclic, with respect to the mixed central moment, functions rhythmically connected random processes.
If, in Definition 8 of an abstract cyclic functional relation
, the range of its values
is the set of fuzzy numbers (
), and a set of attributes
is a set that contains only one function
(
), where
is a set of membership functions (indicator functions), the definition of a cyclic fuzzy function follows from the general definition of an abstract cyclic functional relation [
1].
Definition 25. The fuzzy function is called the cyclic fuzzy function, if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and for its membership function the following such equality takes place: The cyclic fuzzy function makes it possible to take into account the non-exact repeatability of cyclic signals within the framework of the theory of fuzzy sets in the sense of L. Zade and can be fruitfully applied in the problems of modeling cyclic processes, provided that the oscillatory process is significantly irregular and its number of registered cycles is small, when deterministic and stochastic approaches are inefficient because they lead to low accuracy and reliability of cyclic signal analysis methods under such conditions.
If, in Definition 8 of the cyclic functional relation
(we will denote the cyclic functional relation
as
), the range of its values
is the set of open intervals (
), that is, the set
is the set of all possible open intervals, and a set of attributes
is a set that contains only one function
(
), namely,
, then, the definition of a cyclic interval function (cyclic interval-valued function) follows from the general definition of an abstract cyclic functional relation. According to work [
1], we will give the following definition.
Definition 26. The interval-valued function is called the cyclic interval function (cyclic interval-valued function), if, for it, there exists such a function , which satisfies the conditions (27)–(29) of the rhythm function and the following such equality takes place: Since a cyclic interval function
can be specified in terms of two cyclic real-valued functions
and
(
) with equal rhythm functions
, namely,
, then equality (59) is equivalent to the following two such equalities:
With cyclic interval function
it is possible to associate some real-valued function
, for which the following such inequality takes place:
Function
is not a cyclic function with respect to its values, but it is cyclic with respect to the intervals
from the set
, which can be useful in the problems of modeling (within the deterministic approach) cyclic signals, the single-phase values of which are generally different (not equal) in different cycles, but it is possible to specify the maximum magnitude (upper limit)
of the deviation of the single-phase signal values from each other, namely,
Although property (62) is quite similar to the property of an almost periodic function, in general, these properties are different.