Kolmogorov–Wiener Filter Weight Function for Stationary Traffic Forecasting: Polynomial and Trigonometric Solutions

Context. We investigate the Kolmogorov-Wiener filter weight function for the prediction of a continuous stationary random process with a power-law structure function. Objective. The aim of the work is to develop an algorithm of obtaining an approximate solution for the weight function without recourse to numerical calculation of integrals. Method. The weight function under consideration obeys the Wiener-Hopf integral equation. A search for an exact analytical solution for the corresponding integral equation meets difficulties, so an approximate solution for the weight function is sought in the framework of the Galerkin method on the basis of a truncated Walsh function series expansion. Results. An algorithm of the weight function obtaining is developed. All the integrals are calculated analytically rather than numerically. Moreover, it is shown that the accuracy of the Walsh function approximations is significantly better than the accuracy of polynomial approximations obtained in the authors’ previous papers. The Walsh function solutions are applicable in wider range of parameters than the polynomial ones. Conclusions. An algorithm of obtaining the Kolmogorov-Wiener filter weight function for the prediction of a stationary continuous random process with a power-law structure function is developed. A truncated Walsh function expansion is the basis of the developed algorithm. In opposite to the polynomial solutions investigated in the previous papers, the developed algorithm has the following advantages. First of all, all the integrals are calculated analytically, and any numerical calculation of the integrals is not needed. Secondly, the problem of the product of very small and very large numbers is absent in the framework of the developed algorithm. In our opinion, this is the reason why the accuracy of the Walsh function solutions is better than that of the polynomial solutions for many approximations and why the Walsh function solutions are applicable in a wider range of parameters than the polynomial ones. The results of the paper may be applied, for example, to practical traffic prediction in telecommunication systems with data packet transfer.

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POLYNOMIAL SOLUTIONS FOR THE KOLMOGOROV-WIENER FILTER WEIGHT FUNCTION FOR FRACTAL PROCESSES

Radio Electronics Computer Science Control ◽

10.15588/1607-3274-2019-2-5 ◽

2019 ◽

Vol 0 (2) ◽

pp. 44-52 ◽

Cited By ~ 1

Author(s):

V. N. Gorev ◽

A. Yu. Gusev ◽

V. I. Korniienko

Keyword(s):

Weight Function ◽

Wiener Filter ◽

Polynomial Solutions

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INVESTIGATION OF THE KOLMOGOROV-WIENER FILTER FOR CONTINUOUS FRACTAL PROCESSES ON THE BASIS OF THE CHEBYSHEV POLYNOMIALS OF THE FIRST KIND

Informatyka Automatyka Pomiary w Gospodarce i Ochronie Środowiska ◽

10.35784/iapgos.912 ◽

2020 ◽

Vol 10 (1) ◽

pp. 58-61

Author(s):

Vyacheslav Gorev ◽

Alexander Gusev ◽

Valerii Korniienko

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Weight Function ◽

Structure Function ◽

Power Law ◽

Chebyshev Polynomials ◽

Wiener Filter ◽

Expansion Method ◽

Solution Convergence ◽

Polynomial Expansion Method

This paper is devoted to the investigation of the Kolmogorov-Wiener filter weight function for continuous fractal processes with a power-law structure function. The corresponding weight function is sought as an approximate solution to the Wiener-Hopf integral equation. The truncated polynomial expansion method is used. The solution is obtained on the basis of the Chebyshev polynomials of the first kind. The results are compared with the results of the authors’ previous investigations devoted to the same problem where other polynomial sets were used. It is shown that different polynomial sets present almost the same behaviour of the solution convergence.

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

Radio Electronics Computer Science Control ◽

10.15588/1607-3274-2021-1-3 ◽

2021 ◽

Vol 1 (1) ◽

pp. 29-35

Author(s):

V. N. Gorev ◽

A. Yu. Gusev ◽

V. I. Korniienko

Keyword(s):

Integral Equation ◽

Approximate Solution ◽

Weight Function ◽

Fredholm Integral Equation ◽

Gaussian Noise ◽

Wiener Filter ◽

Expansion Method ◽

Approximate Solutions ◽

Fractional Gaussian Noise ◽

Polynomial Expansion Method

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.

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Density-based discrimination of protein and RNA in the ribosome

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100122721 ◽

1992 ◽

Vol 50 (1) ◽

pp. 464-465

Author(s):

Joachim Frank

Keyword(s):

Transfer Function ◽

Spatial Frequency ◽

Three Dimensional ◽

Wiener Filter ◽

Dark Field Image ◽

Dark Field ◽

Frequency Range ◽

Bright Field ◽

Contrast Transfer Function ◽

Pass Filter

Cryo-electron microscopy combined with single-particle reconstruction techniques has allowed us to form a three-dimensional image of the Escherichia coli ribosome.In the interior, we observe strong density variations which may be attributed to the difference in scattering density between ribosomal RNA (rRNA) and protein. This identification can only be tentative, and lacks quantitation at this stage, because of the nature of image formation by bright field phase contrast. Apart from limiting the resolution, the contrast transfer function acts as a high-pass filter which produces edge enhancement effects that can explain at least part of the observed variations. As a step toward a more quantitative analysis, it is necessary to correct the transfer function in the low-spatial-frequency range. Unfortunately, it is in that range where Fourier components unrelated to elastic bright-field imaging are found, and a Wiener-filter type restoration would lead to incorrect results. Depending upon the thickness of the ice layer, a varying contribution to the Fourier components in the low-spatial-frequency range originates from an “inelastic dark field” image. The only prospect to obtain quantitatively interpretable images (i.e., which would allow discrimination between rRNA and protein by application of a density threshold set to the average RNA scattering density may therefore lie in the use of energy-filtering microscopes.

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