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Citation: Sidorov, D.; Tynda, A.;
Muratov, V.; Yanitsky, E. Volterra
Black-Box Models Identification
Methods: Direct Collocation vs. Least
Squares. Mathematics 2024,12, 227.
https://doi.org/10.3390/
math12020227
Academic Editors: Jaan Janno and
Hongyu Liu
Received: 12 October 2023
Revised: 28 November 2023
Accepted: 22 December 2023
Published: 10 January 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
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Attribution (CC BY) license (https://
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4.0/).
mathematics
Article
Volterra Black-Box Models Identification Methods:
Direct Collocation vs. Least Squares
Denis Sidorov 1,2,* , Aleksandr Tynda 3, Vladislav Muratov 4and Eugeny Yanitsky 5
1
Applied Mathematics Department, Melentiev Energy Systems Institute, Siberian Branch of Russian Academy
of Sciences, Irkutsk 664003, Russia
2Industrial Mathematics Lab, Baikal School of BRICS, Irkutsk National Research Technical University,
Irkutsk 664074, Russia
3Higher and Applied Mathematics Department, Penza State University, Penza 440026, Russia;
tyndaan@mail.ru
4Institute of Mathematics and Information Technologies, Irkutsk State University, Irkutsk 664003, Russia;
muratov428@gmail.com
5Intermediate Radio Frequency Lab, Huawei Russian Research Institute, Moscow 121096, Russia;
yanitsky.eugeny@huawei.com
*Correspondence: dsidorov@isem.irk.ru
Abstract: The Volterra integral-functional series is the classic approach for nonlinear black box
dynamical system modeling. It is widely employed in many domains including radiophysics, aerody-
namics, electronic and electrical engineering and many others. Identifying the time-varying functional
parameters, also known as Volterra kernels, poses a difficulty due to the curse of dimensionality.
This refers to the exponential growth in the number of model parameters as the complexity of the
input-output response increases. The least squares method (LSM) is widely acknowledged as the
standard approach for tackling the issue of identifying parameters. Unfortunately, the LSM suffers
with many drawbacks such as the sensitivity to outliers causing biased estimation, multicollinearity,
overfitting and inefficiency with large datasets. This paper presents an alternative approach based
on direct estimation of the Volterra kernels using the collocation method. Two model examples are
studied. It is found that the collocation method presents a promising alternative for optimization,
surpassing the traditional least squares method when it comes to the Volterra kernels identification
including the case when input and output signals suffer from considerable measurement errors.
Keywords: Volterra series; collocation method; kernels identification; Chebyshev polynomials;
memory effects
MSC: 65R30; 45D05; 93B30
1. Introduction
At the current stage of development of wireless technologies like 5G/6G commu-
nication system networks based on antenna arrays with digital beam forming (Massive
Multiple Input Multiple Output system), it is impossible to manage without such digital
signal processing algorithms as digital correction of the nonlinear distortion DPD (Digital
Predistortion). Nonlinear distortions of the signal occurring inside the transceiver path
strongly distort the spectrum of this signal, as shown in Figure 1, where it is shown in red,
and the main signal is blue in color, respectively.
However, the international wireless standards like 3GPP, ETSI impose strict require-
ments on the spectral power of the radiated signal. The use of digital nonlinear distortion
correction algorithms allows for meeting the requirements of standards and at the same
time positively affecting the overall efficiency, that is, the energy consumption of the entire
signal receiving and transmitting system. There are different approaches to the implemen-
tation of such algorithms, both purely digital and analog and mixed. One of them, a purely
Mathematics 2024,12, 227. https://doi.org/10.3390/math12020227 https://www.mdpi.com/journal/mathematics
Mathematics 2024,12, 227 2 of 13
mathematical approach to the description of nonlinear distortions, we will describe below.
However, let us consider the general statement of the problem of digital correction (DPD)
with the following structure of the model of correction as shown in Figure 2.
Figure 1. Power spectrum density.
Figure 2. Digital correction scheme.
Here,
FDPD(
.
)
is a nonlinear operator reflecting the essence of nonlinear correction—
imagine it as some function dependent on parameters
W= [w1
,
. . .
,
wp]T
,
W∈Cp
.
FPA(
.
)
is a nonlinear operator identified with a nonlinear device which generates some complex
vector
Y= [y1
,
. . .
,
yn]T
,
Y∈Cn
and also defines some vector from a complex field of
numbers
Yd= [yd,1
,
. . .
,
yd,n]T
,
Yd∈Cn
on which the operator
FDPD(
.
)
depends. Under the
error E∈Cnwe will understand the difference between vectors Yand Yd
E=
Yd−
Y.
Then we can formulate the requirements for the definition of parameters
W
as follows:
ω=arg minW∥E∥2
, where
∥.∥
is Euclidean norm. Considering
Y=FPA(FDPD (Yd))
, the
above introduced expression can be rewritten as
ω=arg min
W
∥Yd−FPA (FDPD (Yd))∥2.
This equation will be task of DPD (Digital Predistortion). Here, we can highlight
several important sub-tasks, which in themselves are quite complex both theoretically
and computationally:
Mathematics 2024,12, 227 3 of 13
(a) Since we have formulated, in fact, the problem of approximation of a function, we need
to derive the analytical regression dependence
FDPD(
.
)
on the parameters
W
. How this
function is defined will depend on the quality of the correction of nonlinear distortions;
(b)
The procedure of searching for the parameters
W
is a classical optimization problem,
which is a linear or nonlinear regression with respect to the parameters
W
. Finding
efficient methods of convex or non-convex optimization is one of the major difficulties
in this problem;
(c)
Compression of a function FDPD (.), i.e., reducing its computational complexity.
One of the methods to solve the problem (a) for the DPD task is the Volterra functional
series. And it is also the conventional tool to characterize the complex nonlinear dynam-
ics in various fields including the radiophysics, mechanical engineering, electronic and
electrical engineering, energy sciences (here, readers may refer, e.g., to review [
1
] or [
2
]).
Volterra series are widely employed to represent the input-output relationship of nonlinear
dynamical systems with memory. Volterra power series are among the best-understood
nonlinear system representations in signal processing. Such an integral functional series
(also called Fréchet-Volterra series)
y(t) = F(x(t)) :=
t
R0
K1(s)x(t−s)ds +
t
R0
t
R0
K2(s1,s2)x(t−s1)x(t−s2)ds1ds2+. . .
+
t
R0
t
R0
· · ·
t
R0
Kn(s1,s2, . . . , sn)x(t−s1)x(t−s2). . . x(t−sn)ds1ds2. . . dsn+. . . t∈[0, T]
(1)
was proposed by Maurice Fréchet for a continuous nonlinear dynamical systems repre-
sentation [
3
,
4
]. Here, readers may also refer to overview [
5
] and monograph [
6
] for more
details on relevant Lyapunov–Liechtenstein operator and Lyapunov–Schmidt methods in
the theory of non-linear equations.
The role of a reproducing kernel Hilbert space in the development of a unifying view
of the Volterra theory and polynomial kernel regression is presented in [7].
In (1),
x(t)
is the input signal and
y(t)
is the output of a single-input-single-output
(SISO) nonlinear system and
Kn(s1
,
s2
,
. . .
,
sn)
are the multidimensional Volterra kernels (or
transfer functions) to be identified based on nonlinear system’s response
y(t)
as a reaction
on input
x(t)
(Figure 3). It is to be noted that for the basic case
n=
1, we have a conventional
Finite Impulse Response (FIR) linear model which is optimal in the least-squares sense.
x(t)
F(x(t)) y(t)
- -
Figure 3. Behavioral modeling of the black box system.
The Fréchet theorem [
3
] generalizes the famous Weierstrass approximation theorem
which characterizes the set of continuous functions on a compact interval via uniform
approximation by algebraic polynomials.
Power series
(1)
characterize the stationary dynamical systems. Stationarity here
means that a transfer function does not vary during the transient process as
t∈[
0,
T]
. More
general power series
(2)
models nonstationary dynamics when transfer functions depend
explicitly on time t
y(t) =
t
R0
K1(t,s)x(s)ds +
t
R0
t
R0
K2(t,s1,s2)x(s1)x(s2)ds1ds2+. . .
+
t
R0
t
R0
· · ·
t
R0
Kn(t,s1,s2, . . . , sn)x(s1)x(s2). . . x(sn)ds1ds2. . . dsn+. . . t∈[0, T].
(2)
Mathematics 2024,12, 227 4 of 13
The Volterra series is an essential tool for the mathematical modeling of the nonlinear
dynamical systems appearing in the digital pre-distortion (DPD) iterative process [
8
]. DPD
as we described before is an important part of the digital signal processing algorithms
used in transmitters and receivers. Particularly for short-distance applications where the
limitations of the transceiver are more significant. DPD is used to improve performance
by compensating for the imperfect response of transmitter components, e.g., in [
9
] the
frequency selective DPD was proposed. In [
9
] the Volterra series model structure consists
of a basic linear part and a partial-band pre-compensation part, moreover, a generalized
indirect learning architecture is employed to extract the coefficients. Several methods have
been studied for DPD, with Volterra series-based methods being popular due to their ease
of implementation and the straightforward interpretation of their nonlinear terms. The key
issue with Volterra series is the curse of dimension: as the order of the series increases, the
number of terms involved in the expansion grows exponentially, making it computationally
demanding. On the other hand, estimating the functional coefficients (Volterra kernels) of
the Volterra integral functional series can be challenging. It is often considered in its discrete
form and requires a significant amount of data and complex optimization algorithms to
find the best fit for the model coefficients. An alternative approach based on problem
reduction to multi-dimensional integral equations solution [
10
,
11
] requires a special probe
signal design.
In present paper, the alternative approach for the identification of Volterra kernels is
proposed using the direct collocation method. The results are compared with the conven-
tional least squares method (LSM) widely employed for the Volterra series identification
problem in the telecommunication domain.
The rest of the paper is structured as follows: The subsequent section provides the
problem statement. Section 3focuses on the collocation method. Section 4carries out
computational experiments with LSM, while Section 5discusses concluding remarks and
future work.
2. Identification Problem Statement
Let us consider the following segment of the truncated Volterra series (1) for n=2
y(t) =
t
Z
0
K1(s)x(t−s)ds +
t
Z
0
t
Z
0
K2(s1,s2)x(t−s1)x(t−s2)ds1ds2,t∈[0, T]. (3)
Our current problem in this section is to determine the kernels
K1(s)
and
K2(s1
,
s2)
by
a known input and output pair x(t),y(t).
In contrast to the linear case
n=
1, when it is sufficient to specify a single pair
x(t)
,
y(t)
to determine the kernel
K1(s)
, in the nonlinear case
n=
2, for the unique
identification of the two-dimensional kernel
K2(s1
,
s2)
, it is necessary to specify a two-
dimensional continuum of equalities. This means that problem
(4)
has an infinite set
of solutions.
Remark 1. It should be noted that if we consider this problem as an integral equation with two
unknown functions
K1(s)
and
K2(s1
,
s2)
, then this problem is essentially ill-posed. There are an
infinite number of solutions and this problem is insufficiently defined. In this regard, no classical
numerical methods designed for integral equations are applicable in this case. And as a result, there
are no any attempts to solve the problem in this form in the literature.
Remark 2. A fundamentally different situation takes place in the problem of determining an
unknown input signal
x(t)
with a known output signal
y(t)
after kernels identification. It is to
be noted that in this case we have the problem of nonlinear Volterra integral equations’ solution.
Here, readers may refer to Section 9 in book [
11
], papers [
12
–
14
] and references therein regarding
the Kantorovich principal solutions and the blow-up phenomenon.
Mathematics 2024,12, 227 5 of 13
Within the framework of this paper, from a practical point of view, we will be sat-
isfied with any pair of approximately found kernels
e
K1(s)
and
e
K2(s1
,
s2)
that provides a
sufficiently small residual norm
ε=max
t∈[0,T]y(t)−
t
Z
0e
K1(s)x(t−s)ds −
t
Z
0
t
Z
0e
K2(s1,s2)x(t−s1)x(t−s2)ds1ds2. (4)
Denoted by
Bi(t)
,
i=
0, 1, 2,
. . .
, the basis functions form a complete orthogonal
system of functions on the segment [0, T].
We look for an approximate solution of the problem
(3)
in the form of segments of
series of expansions according to the selected system of basis functions
e
K1,m(s) =
m−1
∑
i=0
AiBi(s),e
K2,m1,m2(s1,s2) =
m1−1
∑
i=0
m2−1
∑
j=0
Cij Bi(s1)Bj(s2). (5)
3. Collocation Method
Collocation-type methods are widely used in the discretization of various kinds of
integro-functional equations [
15
]. With sufficiently good accuracy and stability, they are
also computationally less expensive in comparison with projection methods of the Galerkin
type requiring additional integration [16].
In order to determine the unknown coefficients
Ai
and
Cij
, we introduce a uniform
grid of nodes
tk∈[0, T],k=0, 1, . . . , N, (6)
where N+1 is number of nodes.
Substitute (5) in (3) and then demand that the equalities be fulfilled at the points (6)
y(tk) =
tk
Z
0e
K1,m(s)x(tk−s)ds +
tk
Z
0
tk
Z
0e
K2,m1,m2(s1,s2)x(tk−s1)x(tk−s2)ds1ds2,k=0, N. (7)
Denote for a simplicity y(tk) = yk, and transform the last equalities as follows
yk=
m−1
∑
i=0
Ai
tk
Z
0
Bi(s)x(tk−s)ds +
m1−1
∑
i=0
m2−1
∑
j=0
Cij
tk
Z
0
tk
Z
0
Bi(s1)Bj(s2)x(tk−s1)x(tk−s2)ds1ds2. (8)
As a system of basis functions
Bi(t)
,
i=
0, 1,
. . .
, we choose Chebyshev polynomials
of the first kind
T0(t) = 1, T1(t) = t,Ti+1(t) = 2tTi(t)−Ti−1(t),i=1, 2, . . . . (9)
Sufficient conditions for the applicability of Chebyshev polynomial expansions of the
form
(5)
are the limitation of the first derivatives of the approximated kernels. For more
detailed information about convergence, we refer, for example, to the book [17].
Since these polynomials are orthogonal on the segment
[−
1, 1
]
, we apply a linear
mapping to the segment [0, T].
The controlled norm of the residual corresponding to the selected values of
m
,
m1
and
m2takes the form
εN=max
t∈[0,T]y(t)−
m−1
∑
i=0
Ai
t
Z
0
Bi(s)x(t−s)ds−
m1−1
∑
i=0
m2−1
∑
j=0
Cij
t
Z
0
t
Z
0
Bi(s1)Bj(s2)x(t−s1)x(t−s2)ds1ds2.
(10)
Mathematics 2024,12, 227 6 of 13
Let us denote
N=m+m1m2−
1. The number of equalities (number of nodes in the
grid) equals the number of unknown coefficients.
Thus, we have the following system of linear algebraic equations
yk=
m−1
∑
i=0
Aiβik +
m1−1
∑
i=0
m2−1
∑
j=0
Cij γijk,k=0, m+m1m2−1, (11)
with respect to the unknown coefficients
Ai
,
i=
0, 1,
. . .
,
m−
1 and
Cij
,
i=
0, 1,
. . .
,
m1−1, j=0, 1, . . . , m2−1. Here,
βik =
tk
Z
0
Bi(s)x(tk−s)ds,γijk =
tk
Z
0
tk
Z
0
Bi(s1)Bj(s2)x(tk−s1)x(tk−s2)ds1ds2. (12)
4. Least–Square Method
Let us denote
N>m+m1m2−
1. We have the situation where number of equalities is
larger than number of unknown coefficients
Ai
and
Cij
. Thus we have the overdetermined
system of linear equations with respect to the unknown coefficients
Ai
,
i=
0, 1,
. . .
,
m−
1
and Cij ,i=0, 1, . . . , m1−1, j=0, 1, . . . , m2−1:
yk=
m−1
∑
i=0
Aiβik +
m1−1
∑
i=0
m2−1
∑
j=0
Cij γijk, (13)
where
βik =
tk
Z
0
Ti(s)x(tk−s)ds,γijk =
tk
Z
0
tk
Z
0
Ti(s1)Tj(s2)x(tk−s1)x(tk−s2)ds1ds2. (14)
The system is inconsistent. The least–square method is used to find the approximate
solution of the system. The point of the method is to find such coefficients
Ai
and
Cij
such
that the following criteria is minimized:
N−1
∑
k=0 yk−
m−1
∑
i=0
Aiβik −
m1−1
∑
i=0
m2−1
∑
j=0
Cij γijk!2
−→ min . (15)
5. Numerical Experiments
Let us illustrate the operation of the proposed identification methods on two pairs of
model signals.
5.1. Model 1. Periodic Signal
Let us consider the case of periodic input signal, where
x(t) = sin(20t),y(t) = 1
81002 199 cos2(20t)−15 sin(40t)−200 cos(20t)e−2t+1+
10 sin(20t)e−2t+20 sin(20t)e−t+1
409 3 sin(20t)−20 cos(20t) + 850920
40501 e−3t.
(16)
The Figure 4shows the graphs of the input x(t)and output signal y(t).
Mathematics 2024,12, 227 7 of 13
Figure 4. Input and output functions.
5.1.1. Collocation Method Results for the Model 1
Table 1demonstrates the dependence of the residual
εN
on the values
m=m1=m2
for the uniform mesh tk=k
N,k=0, 1, . . . , N, covering the segment [0, 1].
Table 1. Dependence of the residual εNon the values m,m1,m2for Model 1.
mεN
31.41 ×10−2
41.14 ×10−6
54.72 ×10−9
61.77 ×10−12
71.83 ×10−14
81.53 ×10−18
10 2.84 ×10−26
All calculations were performed in the Maple system with parameter
Digits:=30
(the
number of digits that Maple uses when making calculations with software floating-point
numbers). It should be noted that when using other values of parameter
Digits
, the order
of the residual changes on average in direct proportion to this value. Also note that the
integration during the formation of the system
(11)
was carried out analytically and did
not introduce additional error in the calculation results. This is due to the fact that the
input signal
x(t)
in most cases allows for the analytical calculation of the values
(12)
. In the
case of using input signals of a more complex structure, special approximation methods
should be applied to the integrals
(12)
, taking into account the possible fast oscillation of
x(t). Figures 5–8demonstrate residual error for fixed parameter m.
Mathematics 2024,12, 227 8 of 13
Figure 5. Residual for m=3.
Figure 6. Residual for m=5.
Figure 7. Residual for m=7.
Mathematics 2024,12, 227 9 of 13
Figure 8. Residual for m=10.
5.1.2. Least–Square Method Results for the Model (16)
For simplicity, we assume that
m=m1=m2
. Table 2demonstrates the dependence of
the residual εNon the parameters.
Table 2. Dependence of the residual εNon the values mand k.
m= 3 m= 5 m= 7
k= (m+m2)×28.07 ×10−44.92 ×10−10 2.50 ×10−16
k= (m+m2)×58.07 ×10−43.90 ×10−10 1.50 ×10−16
k= (m+m2)×10 8.07 ×10−44.90 ×10−10 2.87 ×10−15
All calculations for least–square method were performed in MATLAB. Overdeter-
mined matrix is solved using the
lsqminnorm
function.
lsqminnorm
solves the linear
equation
AX =B
and minimizes
∥AX −B∥
. The function uses the complete orthogonal
decomposition to find a low-rank approximation of a matrix. It also should be noted that
all the integrations during calculation were carried out analytically and did not introduce
additional error in the results. Residual error for fixed
m=
5 and
k= (m+m2)×
5 is
presented in Figure 9. Residual error for fixed
m=
7 and
k= (m+m2)×
5 is presented in
Figure 10.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
410-10
Figure 9. Residual for m=5 and k= (m+m2)×5.
Mathematics 2024,12, 227 10 of 13
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6 10-16
Figure 10. Residual for m=7 and k= (m+m2)×5.
5.2. Model 2. Fading Input Signal
Let us consider the case of fading input signal, where
x(t) = e−3tsin(10t),
y(t) =
t
Z
0
coss
2x(t−s)ds+
t
Z
0
t
Z
0
sin(s1+2s2)x(t−s1)x(t−s2)ds1ds2.
(17)
The Figure 11 shows the graphs of the input signal x(t)and output signal y(t).
Figure 11. Input and output signals.
5.2.1. Collocation Method Results for Model 2
Table 3demonstrates the dependence of the residual
εN
on the values
m=m1=m2
for the uniform mesh tk=k
N,k=0, 1, . . . , N, covering the segment [0, 1].
Mathematics 2024,12, 227 11 of 13
Table 3. Dependence of the residual εNon the values m,m1,m2.
mεN
33.16 ×10−5
49.85 ×10−9
58.58 ×10−12
62.17 ×10−16
75.37 ×10−20
Let us also discuss the stability of suggested numerical technique. Let the input data
of the problem
(17)
be determined with some random error
εrand
varying within the
δ
value, namely
|εrand|⩽δ
. Table 4shows the dependence of the averaged residual
εN
on
the
δ
value at a fixed
m=
3 based on the results of 10 measurements. Figures 12 and 13
demonstrate the dependence of residual error with fixed parameter m.
Table 4. Stability results for collocation.
δ εN
10−20.01729
10−32.71 ×10−3
10−42.56 ×10−4
10−57.54 ×10−5
10−61.66 ×10−5
Figure 12. Residual for m=3.
Figure 13. Residual for m=7.
Mathematics 2024,12, 227 12 of 13
It can be seen from the results of the Table 4that residual continuously depends on
the limits of random measurement errors of the input and output signals. Thus, we can
conclude about the stability of the suggested method.
5.2.2. Least–Square Method Results for Model 2
Table 5demonstrates the dependence of the residual
εN
on the parameters. Figure 14
demonstrates the residual error for fixed mand k.
Table 5. Dependence of the residual εNon the values mand k.
m= 3 m= 5 m= 7
k= (m+m2)×2 2.38 ×10−67.77 ×10−14 2.93 ×10−16
k= (m+m2)×5 2.63 ×10−67.46 ×10−14 3.05 ×10−16
k= (m+m2)×10 3.48 ×10−67.41 ×10−14 3.80 ×10−16
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.5
1
1.5
2
2.5
3
3.5
410-16
Figure 14. Residual for m=7 and k= (m+m2)×7.
As for collocation method, let us check the stability of the least-square method on this
model. For testing stability, 10 rounds of experiments were performed and the average
residual
εN
was calculated. Also,
m=
3,
k= (m+m2)×
5 were fixed. Table 6shows the
stability results.
Table 6. Stability results for LSM.
δ εN
10−20.00628
10−35.11 ×10−4
10−46.02 ×10−5
10−55.36 ×10−6
10−62.64 ×10−6
6. Conclusions
Two numerical approaches to solving the problem of identification of the Volterra
model were proposed in the paper. As can be seen from the presented results, both methods
showed stable convergence. Convergence here can be interpreted only as the dependence
of the residual on the increase in the number of terms in the expansions of kernels by
Chebyshev polynomials (5). And this dependence is presented in numerical results.
It is to be noted, from the point of view of the arithmetic complexity of calculations,
the collocation method turns out to be less expensive. And this factor is more pronounced
the more parameters of the model are to be determined. This is due to the need to calculate
a significantly larger number of integrals proportional to the square of the number of
measurements being processed.
Mathematics 2024,12, 227 13 of 13
Further development of research suggests an increase in the number of terms
n
in the
model
(1)
to identify a more accurate functional relationship between the input and output
signals. It is also planned to develop special methods for approximating integrals
(12)
for
the case of using input signals of a more complex structure, including fast oscillating signals.
Author Contributions: Conceptualization, D.S., A.T. and E.Y.; methodology, D.S., A.T. and E.Y.;
software, V.M. and A.T.; validation, D.S. and A.T.; formal analysis, D.S.; investigation, D.S.; data
curation, D.S.; writing—original draft preparation, D.S. and A.T.; writing—review and editing, D.S.,
A.T., V.M. and E.Y.; visualization, V.M. and A.T.; supervision, D.S. All authors have read and agreed
to the published version of the manuscript.
Funding: The research was carried out within the state assignment of the Ministry of Science and
Higher Education of the Russian Federation (project code: FZZS-2024-0003).
Data Availability Statement: The data presented in this study are available on request from the
corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.
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