APPROXIMATE SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR CONTINUOUS FRACTIONAL GAUSSIAN NOISE

Context. We consider the Kolmogorov-Wiener filter for forecasting of telecommunication traffic in the framework of a continuous fractional Gaussian noise model. Objective. The aim of the work is to obtain the filter weight function as an approximate solution of the corresponding WienerHopf integral equation. Also the aim of the work is to show the convergence of the proposed method of solution of the corresponding equation. Method. The Wiener-Hopf integral equation for the filter weight function is a Fredholm integral equation of the first kind. We use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding equation. A set of Chebyshev polynomials of the first kind is used. Results. We obtained approximate solutions for the Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise. The solutions are obtained in the approximations of different number of polynomials; the results are obtained up to the nineteen-polynomial approximation. It is shown that the proposed method is convergent for the problem under consideration, i.e. the accuracy of the coincidence of the left-hand and right-hand sides of the integral equation increases with the number of polynomials. Such convergence takes place due to the fact that the correlation function of continuous fractional Gaussian noise, which is the kernel of the corresponding integral equation, is a positively-defined function. Conclusions. The Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise is obtained as an approximate solution of the corresponding Fredholm integral equation of the first kind. The proposed truncated polynomial expansion method is convergent for the problem under consideration. As is known, one of the simplest telecommunication traffic models is the model of continuous fractional Gaussian noise, so the results of the paper may be useful for telecommunication traffic forecast.


NOMENCLATURE
T is the time interval along which the input data are observed; k is the time interval for which the forecast should be made; ( ) h t is the Kolmogorov-Wiener filter weight function; H is the Hurst exponent; ( ) n S t are the Chebyshev polynomials of the first kind which are orthogonal on the time interval (0, ) t T ∈ ; ( )

INTRODUCTION
The problem of telecommunication traffic forecast is a topical problem of telecommunications. Traffic in telecommunication systems with data burst transfer is a selfsimilar process [1]. It should be stressed that self-similar processes take place in a huge variety of different systems (see, for example, [2]), and their forecast is investigated not only for telecommunication traffic, but also for other systems (see, for example, [3,4]).
One of the simplest models of self-similar telecommunication traffic is the model where the traffic is con-sidered to be a fractional Gaussian noise [1]. In [5] it is stressed that it is reasonable to consider traffic as a continuous random process because of a large amount of data. So, in a simple model the traffic can be treated as a continuous fractional Gaussian noise.
Fractional Gaussian noise is a stationary random process, so the Kolmogorov-Wiener filter may be used in order to investigate the traffic forecast. This paper is devoted to the obtaining of the weight function of the corresponding filter. In this paper we consider only the case where the Hurst exponent 0.5 Н > . The weight function under consideration obeys a Fredholm integral equation of the first kind [6]. We propose to use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding integral equation. The correlation function of fractional Gaussian noise, which is the kernel of the considered integral equation, is a positively-defined function [7], so the proposed method should be convergent.
The obtained results may be important for forecasting of telecommunication self-similar traffic.
The object of study is the Kolmogorov-Wiener filter for continuous fractional Gaussian noise.
The subject of study is the weight function of the corresponding filter.
The aim of the work is to obtain the corresponding weight function as an approximate solution of the Fredholm integral equation of the first kind. Also the aim of the work is to show the convergence of the truncated polynomial expansion method for the problem under consideration.

PROBLEM STATEMENT
As is known [6], the Kolmogorov-Wiener weight function obeys the Wiener-Hopf integral equation The correlation function of continuous fractional Gaussian noise for 0.5 Н > is as follows [7] ( ) ( ) where σ is the process variance. After substitution of (2) into (1) one can obtain ( ) ( ) where the fact that 0 k > , and obviously 0 t k + > , is used. The problem is to obtain the function ( ) h τ as an approximate solution of the integral equation (3).

REVIEW OF THE LITERATURE
As is known, traffic in telecommunication systems with data burst transfer is a self-similar process, and one of the simplest self-similar traffic models is the model of fractional Gaussian noise [1]. Fractional Gaussian noise is a stationary random process [7], so its forecast may be obtained on the basis of the Kolmogorov-Wiener filter. According to [5], the traffic is considered as a continuous random process, which is reasonable in case of a large amount of data.
In the case of a continuous process the Kolmogorov-Wiener weight function obeys a Fredholm integral equation of the first kind [6]. An exact analytical solution of such an equation meets difficulties, so we use the truncated polynomial expansion method in order to obtain an approximate solution of the corresponding equation.
In our previous papers [8][9][10] we investigated the corresponding method for the case of fractal processes with a power-law structure function. In [8] we used polynomials which are orthogonal on the time interval (0, ) t T ∈ without weight, and in [9,10] we used the Chebyshev polynomials of the first and second kind, respectively. It was shown that the behavior of the method convergence is identical for all the polynomial sets investigated in [8][9][10], and the method is not necessarily convergent for frac-tal processes with a power-law structure function. The reason is as follows. The method convergence is guaranteed [11] if the kernel of the corresponding integral equation is a positively-defined function. As can be seen from eq. (1), the kernel of the integral equation under consideration coincides with the correlation function of the random process under consideration. The correlation function of a fractal random process with a power-law structure function is not a positively-defined function, so for that case the method convergence is not guaranteed.
But, as is known [7], in the case where 0.5 Н > the correlation function of fractional Gaussian noise, in contrast to the correlation function of a process with a powerlaw structure function, is a positively-defined function. So the truncated polynomial expansion method should be convergent for the problem under consideration. In this paper a set of Chebyshev polynomials of the first kind is used.

MATERIALS AND METHODS
In the framework of the truncated polynomial expansion method [8][9][10] the function ( ) h τ , which is the solution of the integral equation (3), is sought as a truncated orthogonal polynomial series where l is the number of polynomials and [ ] l m g are coefficients multiplying the polynomials. The function ( ) h τ in the form (4) is the solution in the l -polynomial approximation.
The polynomials ( ) m S τ should be orthogonal on (0, ) T τ ∈ . In this paper this set is constructed on the basis of the Chebyshev polynomials of the first kind. As is known [12], the explicit expressions for the Chebyshev polynomials of the first kind are where [ ] y is the integer part of y and As is shown in [9], the following orthogonality relation is valid: where ( ) is the Kronecker delta. So, the polynomials with the weight ( ) w t , and they may be used in the expansion (4).
By substituting (4) into (3) one can obtain and by multiplying both sides of (11) by ( ) n S t and integrating over t one can obtain the following set of linear algebraic equations where mn G are the so-called integral brackets ( ) ( ) the functions ( ) n S t are taken from (10). Let us discuss the properties of the integral brackets (13). First of all, if we interchange the integration variables in (13), we obtain so the integral brackets obey the property mn nm Let us make the following change of the variables: On the basis of (13) and (17) As can be seen from (5), the following property is valid: which leads to the fact that

EXPERIMENTS
Of course, an interesting question is whether the proposed method is convergent. The kernel of the integral equation (3) is a positively-defined function, so the trun-cated polynomial expansion method should be convergent for the problem under consideration.
In order to check the convergence of the method we numerically compare the left-hand and the right-hand sides of the integral equation (3) for different numbers of polynomials. The integral on the left-hand side of eq. (3) is numerically calculated with the help of the Wolfram Mathematica as The numerical investigation is made for the set of parameters In what follows the comparison of the left-hand and the right-hand sides of the integral equation (3) is illustrated by graphs where the dotted line illustrates the graph for the left-hand side and the solid line illustrates the right-hand side. As can be seen from Fig. 1 and Fig. 2, the approximations of a small number of polynomials are not accurate.
However, as can be seen from Fig. 1 -Fig. 7, the accuracy of coincidence of the left-hand and the right-hand sides of the integral equation (3) increases with the number of polynomials. So one can conclude that the proposed method of truncated polynomial expansion is convergent for the problem under consideration.

RESULTS
The Kolmogorov-Wiener filter weight function for forecasting of fractional Gaussian noise is investigated as an approximate solution of the corresponding integral equation (3). The truncated polynomial expansion method based on the Chebyshev polynomials of the first kind is used.
The results are investigated up to the nineteenpolynomial approximation. The approximations of higherthan-nineteen polynomials are not investigated because the Wolfram Mathematica package is not able to calculate them adequately. It is shown that the method is convergent, i.e. the accuracy of coincidence of the left-hand and the right-hand sides of (3) increases with the number of polynomials. However, it should be stressed that the approximations of small numbers of polynomials are not accurate, and one should use the approximation of a rather large number of polynomials.

DISCUSSION
This paper is devoted to the investigation of the Kolmogorov-Wiener filter weight function for forecasting of continuous fractional Gaussian noise. As is known [1], one of the simplest models of telecommunication traffic in systems with data burst transfer is the model of fractional Gaussian noise. So the results of the paper may be useful for telecommunication traffic forecasting.
The integral equation for the corresponding weight function is the Wiener-Hopf integral equation (1) which can be expressed in the form (3). In fact, we deal with the Fredholm integral equation of the first kind. The weight function is obtained with the help of the truncated polynomial expansion method. In this paper the method is based on the Chebyshev polynomials of the first kind which are orthogonal on the time interval (0, ) t T ∈ . The numerical investigation is made for the parameters (23). The approximations up to the 19-polynomial one are investigated. The approximations of small numbers of polynomials are not accurate, but it is shown that the approximation accuracy increases with the number of polynomials, the approximations of rather large numbers of polynomials are rather accurate. So the method convergence is illustrated for the parameters (23). The kernel of the integral equation (3) is a positively-defined function, so, according to [11], in the framework of the problem under consideration the truncated polynomial expansion method should be convergent not only for the parameters (23), but also for other numerical values of the parameters.

CONCLUSIONS
The Kolmogorov-Wiener weight function for forecasting of fractional Gaussian noise is investigated as an approximate solution of the corresponding Wiener-Hopf integral equation. The truncated polynomial expansion method is used, the Chebyshev polynmials of the first kind orthogonal on (0, ) t T ∈ are chosen.
The scientific novelty of the paper is the fact that in contrast to the previously investigated model [8][9][10], the method convergence is shown for the model of fractional Gaussian noise.
The practical significance is that the obtained results may be applied to the telecommunication traffic forecast in systems with data burst transfer.
Prospects for further research are to investigate an exact analytical solution of the corresponding integral equation.