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

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.

Multiband SHEPWM Control Technology Based on Walsh Functions

The drawback of modulation ratio limitation in selective harmonic elimination pulse width modulation (SHEPWM) technology based on Walsh function transformation. In order to solve this problem, a multiband SHEPWM control technology method for a multilevel inverter based on Walsh functions is proposed. By analyzing multiband SHEPWM for multilevel inverter voltage waveforms under a Fourier series transform, the unified nonlinear multiband SHEPWM equations for a multilevel inverter with a modulation index varying from 0 to 1 can be generalized. The equations can be solved by Walsh function transform. In this way, the difficulties faced in expanding the modulation index, online calculation, and real-time control are resolved simultaneously. A seven-level inverter is taken as the example, for which the piecewise linear equations of Walsh functions under different bands and the trajectories of switching angles are given. Meanwhile, the conditions for multiple sets of solutions at such a point where the modulation index is switched over are also taken into account. The feasibility of the proposed method is verified by simulation based on MATLAB and SIMULINK. Finally, the feasibility of the practical application is proved by the experiment based on Digital Signal Processor (DSP).

Vector-valued nonuniform multiresolution analysis related to Walsh function

In this paper, we introduce vector-valued nonuniform multiresolution analysis on positive half-line related to Walsh function. We obtain the necessary and sufficient condition for the existence of associated wavelets.