ON THE KOLMOGOROV-WIENER FILTER FOR RANDOM PROCESSES WITH A POWER-LAW STRUCTURE FUNCTION BASED ON THE WALSH FUNCTIONS

Abstract

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.