=Paper=
{{Paper
|id=Vol-3309/short2
|storemode=property
|title=Property Analysis of Conditional Linear Random Process as a Mathematical Model of Cyclostationary Signal
|pdfUrl=https://ceur-ws.org/Vol-3309/short2.pdf
|volume=Vol-3309
|authors=Mykhailo Fryz,Bogdana Mlynko
|dblpUrl=https://dblp.org/rec/conf/ittap/FryzM22
}}
==Property Analysis of Conditional Linear Random Process as a Mathematical Model of Cyclostationary Signal==
https://ceur-ws.org/Vol-3309/short2.pdf
Property Analysis of Conditional Linear Random Process as a
Mathematical Model of Cyclostationary Signal
Mykhailo Fryz a and Bogdana Mlynko a
a
Ternopil Ivan Puluj National Technical University, Ruska st. 56, Ternopil, 46001, Ukraine
Abstract
The properties of probability characteristics of the conditional linear random process have been
analyzed in the context of its application for the mathematical modelling of cyclostationary
information signals. The class of conditional linear periodically correlated random processes
has been characterized using the properties of their mathematical expectation and covariance
function. The characteristic function method has been used to describe the class of conditional
linear cyclostationary random processes represented in the form of the stochastic integral of
cyclostationary kernel driven by the random process with independent periodic increments.
Keywords 1
Mathematical model, information signal, conditional linear cyclostationary random process,
stochastic integral, kernel, independent increments, characteristic function, moment functions.
1. Introduction
Many information signals in technical, medical, or economical systems, the mathematical model of
which can be represented in the form of a linear or conditional linear random process, also have the
property of cyclostationarity, which can be caused by various factors, for example, daily, weekly, or
seasonal cyclicity of electricity loads, gas or water consumption, the cyclicity of heart beats in the
analysis of electro-cardio signals, the periodicity of photo stimulation in tasks of analysis of visually
evoked biopotentials of the brain, etc.
The most general model of such kind of information signals is a cyclostationary random process
whose finite-dimensional distribution functions (or characteristic functions) are periodic with respect
to their time arguments [1, 2]. As the author [3] noted, the idea of stochastic periodicity belongs to E.E.
Slutsky and is presented in his work "The Summation of Random Causes as the Source of Cyclic
Processes". Therefore, in some works, cyclostationary processes are also called periodic by Slutsky [3].
A very important subclass of cyclostationary random processes is a periodically correlated random
processes [3, 4], which have periodic moment functions of the first and second order.
A conditional linear random process (CLRP) is an important instrument for the problems of
mathematical modelling of information signals that can be represented as a sum of large number of
stochastically dependent random impulses occurring at Poisson times [5 - 7]. Thus, such kind of
processes can be physically reasonable model of radar noise, many electrophysiological information
signals, resource consumptions, vibration noises etc.
The constructive properties of linear and conditional linear random processes allow taking into
account the causes of the rhythmic or cyclic properties of the studied information signals in the
corresponding mathematical models [3, 6].
The main goal of the paper is to justify the conditions for CLRP to be periodically correlated random
process, which is extension of the results of [6], and to describe the class of conditional linear
cyclostationary random processes using the characteristic function expressions obtained in [7].
ITTAP’2022: 2nd International Workshop on Information Technologies: Theoretical and Applied Problems, November 22–24, 2022,
Ternopil, Ukraine
EMAIL: mykh.fryz@gmail.com (M. Fryz); mlynko@ukr.net (B. Mlynko)
ORCID: 0000-0002-8720-6479 (M. Fryz); 0000-0003-0780-5365 (B. Mlynko)
© 2022 Copyright for this paper by its authors.
Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR Workshop Proceedings (CEUR-WS.org)
2. Conditional Linear Periodically Correlated Random Processes
A real-valued conditional linear random process ξ(ω, t ), ω∈ Ω, t ∈ (−∞, ∞) (where {Ω, F, P} is
some probability space) is defined in the following form [6, 7]:
∞
ξ(ω, t )=
−∞
∫ ϕ(ω, τ, t )d η(ω, τ), ω∈ Ω, t ∈ , (1)
where ϕ(ω, τ, t ), τ, t ∈ is a real-valued stochastic kernel of CLRP; η(ω, τ), τ∈ (−∞, ∞) is a
stochastically continuous Hilbert process with independent increments, satisfying the following
conditions: Eη(ω, τ)= a (τ) < ∞ and Var [ η(ω, τ) ]= b(τ) < ∞ , ∀τ ; random functions ϕ(ω, τ, t ) and
η(ω, τ) are stochastically independent.
The stochastic integral (1) is assumed to be exist in the mean-square convergence sense [6].
Mathematical expectation Eξ(ω, t ) and covariance function Rξ (t1 , t2 ) of conditional linear random
process (1) are represented as:
∞
Eξ(ω, t ) = ∫ φ(τ, t )da(τ),
−∞
(2)
∞ ∞ ∞
(t1 , t2 )
Rξ = ∫ ∫ R (τ , τ ; t , t )da(τ )da(τ ) + ∫ E ( ϕ(ω, τ, t )ϕ(ω, τ, t ) ) db(τ).
−∞ −∞
ϕ 1 2 1 2 1 2
−∞
1 2 (3)
where φ(τ, t ) = Eϕ(ω, τ, t ) is the mathematical expectation of the kernel of CLRP;
Rϕ (τ1 , τ2 ; t1 , t2 )= E ( ϕ0 (ω, τ1 , t1 )ϕ0 (ω, τ2 , t2 ) ) is the covariance function of the kernel of conditional
linear random process ( ϕ0 (ω, τ, t ) =ϕ(ω, τ, t ) − φ(τ, t ) is the centered kernel).
Let there exist the least real number (period) T > 0 and number α ∈ (−∞, ∞) such that the process
with independent increments η(ω, τ) satisfies the following conditions:
da (τ= ) da (τ + αT ) and db(τ= ) db(τ + αT ) ,
and mathematical expectation and covariance function of the kernel have the following properties:
φ(τ, t ) =φ(τ + αT , t + T ), (4)
Rϕ (τ1 , τ2 ; t1 , t2=
) Rϕ (τ1 + αT , τ2 + αT ; t1 + T , t2 + T ). (5)
then CLRP (1) is periodically correlated random process.
Indeed, mathematical expectation of CLRP in this case has the following property:
∞ ∞ ∞
Eξ(ω, t )= ∫ φ(τ, t )da(τ)=
−∞ −∞
∫ φ(τ + αT , t + T )da(τ + αT )= ∫ φ(s, t + T )da(s)= Eξ(ω, t + T ).
−∞
From (3), (4), and (5) it follows that the covariance function is expressed as:
∞ ∞
(t1 , t2 )
Rξ= ∫ ∫ R (τ + αT , τ + αT ; t + T , t + T )da(τ + αT )da(τ + αT ) +
−∞ −∞
ϕ 1 2 1 2 1 2
∞
+ ∫ Rϕ (τ + αT , τ + αT ; t1 + T , t2 + T )db(τ + αT ) +
−∞
∞
+ ∫ φ(τ + αT , t1 + T )φ(τ + αT , t2 + T )db(τ + αT ).
−∞
Let us denote: τ1 + αT = s1 , τ2 + αT = s2 , τ + αT = s , then we obtain:
∞ ∞
Rξ (t1 , t2 )
= ∫ ∫ R (s , s ; t + T , t + T )da(s )da(s ) +
−∞ −∞
ϕ 1 2 1 2 1 2
∞ ∞
+ ∫ Rϕ ( s, s; t1 + T , t2 + T )db( s ) + ∫ φ( s, t1 + T )φ( s, t2 + T )db( s ) =
−∞ −∞
= Rξ (t1 + T , t2 + T ).
The mathematical expectation and the covariance function of the considered process are periodic
with respect to their arguments. Thus, it is a conditional linear periodically correlated random process.
3. Conditional Linear Cyclostationary Random Processes
Let Fϕ ⊂ F be a σ -subalgebra generated by the random kernel ϕ(ω, τ, t ) satisfying the following
conditions [7]:
∞ ∞
2
∫ ϕ(ω, τ, t ) da(τ) < ∞,
−∞
∫ ϕ(ω, τ, t ) db(τ) < ∞ , ∀t with probability 1.
−∞
The m-dimensional characteristic function of CLRP is represented as [7]:
m
tm ) E exp i ∑ uk ξ(ω, t=
F
f ξ (u1 , u2 ,..., um ; t1 , t2 ,...,= k ) Ef ξ ϕ (ω, u1 , u2 ,..., um ; t1 , t2 ,..., tm ),
k =1
m
; t1 , t2 ,..., tm ) E exp i ∑ uk ξ(ω, tk ) Fϕ is conditional (with respect to Fϕ )
F
where f ξ ϕ (ω, u1 , u2 ,..., um=
k =1
characteristic function ( Fϕ -characteristic function) of CLRP (1), which is expressed as follows [7]:
m ∞
t1 , t2 ,..., tm ) exp i ∑ uk ∫ ϕ(ω, τ, tk )da (τ) +
F
f ξ ϕ (ω, u1 , u2 ,..., um ;=
k =1 −∞
∞ ∞
m m
d d K ( x; τ) (6)
+ ∫ ∫ exp ix ∑ uk ϕ(ω, τ, tk ) − 1 − ix ∑ uk ϕ(ω, τ, tk ) x τ 2 ,
=−∞ −∞ k 1= k 1 x
uk , tk ∈ (−∞, ∞), k = 1, m ,
where K ( x; τ) is the Poisson jump spectrum in Kolmogorov’s form of infinitely divisible random
process with independent increments η(ω, τ) .
Let us now consider the general case of conditional linear cyclostationary random process. Namely,
using the method of characteristic functions, we prove the sufficient conditions (which are important
for applications) that the kernel ϕ(ω, τ, t ) and the process η(ω, τ) have to satisfy in order for the
conditional linear random process to be cyclostationary.
Let there exist the least real number (period) T > 0 and number α ∈ (−∞, ∞) such that:
• random functions (fields) ϕ(ω, τ, t ) and ϕ(ω, τ + αT , t + T ) are stochastically equivalent in the
wide sense, that is, their finite-dimensional distributions are equal:
n m n m
P {ω : ϕ(ω, τi , t j )= < xij } P {ω : ϕ(ω, τi + αT , t j + T ) < xij } , xij ∈ ; (7)
=i 1 =j 1 =i 1 =j 1
• η(ω, τ) is a random process with independent increments satisfying the following properties:
da (τ=) da (τ + αT ),
(8)
d x d τ=K ( x; τ) d x d τ K ( x; τ + αT ).
Then the characteristic function of CLRP (1) is T-periodic by its time arguments, that is
f ξ (u1 , u2 ,..., um ;=
t1 , t2 ,..., tm ) f ξ (u1 , u2 ,..., um ; t1 + T , t2 + T ,..., tm + T ). (9)
Thus, the process (1), satisfying (7) and (8) is conditional linear cyclostationary random process.
Note that, the process with independent increments η ( ω, τ ) in the above statement is the random
process with independent αT − periodical increments [3]. That is, the characteristic function
f ∆η ( u; s, τ ) =Me
iu ∆ s η( ω, τ )
of the increments ∆ s η ( ω, τ ) =η ( ω, τ ) − η ( ω, s ) , s < τ of such kind of process
η ( ω, τ ) satisfies the following condition:
f ∆η (=
u; s, τ ) f ∆η ( u; s + αT , τ + αT ) .
Parameter α ∈ (−∞, ∞) is a ratio of the period of increments of the random process η ( ω, τ ) and
period of the random process ξ(ω, t ) (see also the same property in [3]). We should note, that in many
problems of mathematical modelling of information signals in technical, medical, or economical
applications α =1 , that is, period of increments of the process η ( ω, τ ) is equal to the period of the
kernel ϕ(ω, τ, t ) in the sense of (7) and (8).
We can prove the above statement analyzing the properties of m-dimensional characteristic function
of CLRP taking into account the conditions (7) and (8).
Like in [7] we further write Law(ξ(ω = )) Law(η(ω)) if random variables ξ(ω) and η(ω) have the
same distribution functions (distribution laws). So, we can write the following:
m ∞
m ∞
Law ∑ uk ∫ ϕ(ω,= τ, tk )da (τ) Law ∑ uk ∫ ϕ(ω, τ + αT , tk + t )da = ( τ + αT )
= k 1= −∞ k 1 −∞
m ∞
= Law ∑ uk ∫ ϕ(ω, s, tk + t )da ( s ) ,
k =1 −∞
∞ ∞ ix ∑ uk ϕ( ω, τ,tk )
m
m d d K ( x; τ)
Law ∫ ∫ e k =1 − 1 − ix ∑ uk ϕ(ω, τ, tk ) x τ 2 =
−∞ −∞ k =1 x
∞ ∞ ix ∑ m
uk ϕ ( ω, τ+αT ,tk +T ) m d d K ( x; τ + αT )
Law ∫ ∫ e k =1 − 1 − ix ∑ uk ϕ(ω, τ + αT , =tk + T ) x τ
−∞ −∞ k =1 x 2
∞ ∞ ix ∑ uk ϕ( ω, s ,tk +T )
m
m d d K ( x; s )
= Law ∫ ∫ e k =1 − 1 − ix ∑ uk ϕ(ω, s, tk + T ) x τ 2 ,
−∞ −∞ k =1 x
where s = τ + αT .
From the above expressions we can conclude that the probability distribution of random m-
dimensional Fϕ -characteristic function of conditional linear random process has the following
property:
F F
Law( f ξ ϕ (ω, u1 , u2 ,..., um ; t1 , t2 ,..., t=
m )) Law( f ξ ϕ (ω, u1 , u2 ,..., um ; t1 + T , t2 + T ,..., tm + T )) .
F
Taking into account the f ξ (u1 , u2 ,..., um ; t1 , t2 ,...,
= tm ) Ef ξ ϕ (ω, u1 , u2 ,..., um ; t1 , t2 ,..., tm ) , we can
conclude that characteristic function of the CLRP satisfies (9).
4. Discussion
The obtained results can be used for mathematical model identification of information signals which
are physically generated as a sum of a large amount of stochastically dependent impulses occurring at
random Poisson times, that is when the model of conditional linear random process is applicable.
Comparing with the results of [7] (where conditions have been proven of CLRP to be stationary) the
sufficient conditions of cyclostationarity of such kind of models have been proven in the paper.
In papers [5-7] the connection of CLRP with models of stochastic linear dynamic systems is noted.
Their identification is carried out on the basis of observation and analysis of signals at the input and
output of the systems. For the CLRP model, obviously, the generating process with independent
increments is not available for observation. Identification is carried out on the basis of a theoretical
analysis of the characteristics of CLRP, as well as their statistical estimation based on the results of
registration of discrete time information signals.
However, the CLRP model with continuous time (including the cyclostationary case) makes it
possible to take into account some important a priori information about the research object and the
corresponding stochastic information signal. In particular, identification of the class of the model
(stationary process or cyclostationary, etc.), formulation of hypotheses regarding the distribution of the
generating process, its homogeneity, justification of the properties of the model kernel, etc. can be
carried out on the basis of the analysis of the physical (biophysical, economic, etc.) mechanism of
generation of the simulated information signal. Thus, the identified model with continuous time
becomes adequate for the studied information signal, because it is based on its physical nature. It is
clear that theoretical conclusions or hypotheses made on the basis of the model constructed in this way
can be further confirmed on the basis of experimental data.
The structure of identification of information signal mathematical models based on conditional
linear cyclostationary random process has been represented on the Figure 1.
Figure 1: Structure of identification of information signal mathematical models based on conditional
linear cyclostationary random process
We can see that the identification of the appropriate model with discrete time, as well as its
informative characteristics detection, the justification of the methods of their statistical estimation is
carried out on the basis of the primary model with continuous time.
In the context of model characteristics estimation, we should emphasize the great role of periodic
autoregressive and moving average methods [8, 9] for the cyclostationary linear model identification
and informative features selection. The corresponding approach for the case of CLRP should consists
of the following. The discrete-time conditional linear cyclostationary random process can be interpreted
as the response of a digital filter with random cyclostationary parameters to the input cyclostationary
white noise. If this filter is built so that it has only a recursive structure, then the random signal at its
output will be a cyclostationary autoregressive process with random coefficients [10]. So, application
of the random coefficient periodic autoregressive methods for conditional linear cyclostationary random
process identification is a prospective task.
5. Conclusion
The class of conditional linear periodically correlated random processes has been characterized.
Each element of the class is the conditional linear random process driven by the process with
independent periodic increments, the random kernel of the process has periodic mathematical
expectation, the covariance function of the kernel is periodic by its time arguments with the same period.
The class of conditional linear cyclostationary random processes has been characterized using the
characteristic functions method. Each element of the class is the conditional linear random process
driven by the process with independent periodic increments, the random kernel of the process is the
cyclostationary bivariate random field. Also, the considered processes are the mixtures of infinitely
divisible distributions.
The general approach for applied identification of information signal mathematical models based on
conditional linear cyclostationary random process has been analyzed based on the above theoretical
results.
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