the local polynomial approximation
elimination of boundary effects of filtering
V.I. Marchuk A.I. Sherstobitov, D.S. Gavrilenko
Currently, there is a wide class of linear and nonlinear methods of processing one-and multidimensional signals to solve the problems of filtration with the mean-square objective function (Wiener filter). For the synthesis of the Wiener filter requires a priori information about the functional dependence of emitted signal and the statistical characteristics of the current additive noise. Using the practice of classical optimal linear estimates is extremely limited. Nonlinear filtering estimate of digital signals found its extensive development and application in solving a wide range of problems in the past 10–12 years, but remain highly specialized tools. Not unimportant is the ratio of the initial implementation of the processed digital signal and the length of the impulse response of the synthesized filter. In the event that they are comparable, then there is a significant increase in the error signal estimation at the beginning and end of implementation. The difference in error estimates of the signal at various sites can reach several times. Within the imposed initial conditions, the use of most existing methods of treatment is extremely limited.
In this paper the method of processing one-dimensional signals in solving the problem of weakening of the additive noise and the selection of useful low frequency signal. The essence of the proposed method consists of dividing the initial implementation of the signal in a sliding window of fixed length for the overlapping intervals and approximation on each system of orthogonal polynomials. The resulting set of estimates in each section of the original process, averaged and the resulting score is formed for each sample processed signal realization. Using a system of orthogonal polynomials for solving the problem of approximation in each interval of the partition of the initial implementation of the signal, yields a generalized approach of signal processing at all its sites and to obtain a mathematical model of the processing method for an arbitrary degree polynomial approximating function. The expressions relating the values of the initial implementation and evaluation of the values of the desired signal for an arbitrary degree of the approximating polynomial and the value of a sliding window of the partition of the initial implementation of the signal, using a system of orthogonal polynomials. Using the properties of orthogonal polynomials has ample opportunities for modification of the proposed method of treatment by adapting the degree of the approximating function at each grazing interval. In this paper, an analytical expression for the system function digital filter that implements the proposed method of treatment, depending on the parameters and.