system of the integro-differential equations
method of an establishment
There are many practical problems of optimal estimation of a signal corrupted by noise are frequently reduced to synthesis of an adaptive filter because of limited a priori knowledge and probable nonstationary statistical characteristics of the signal and noise. There are many adaptive filtering algorithms existing up to now are modification of least mean squares algorithm which are in steady state made a Widrow Hoff estimator of the signal. This estimator is based on using cumulants up to the second order and it is optimum only for Gaussian signal and noise.
Nowadays an actual problem is transformation performance function of adaptive filter so that it takes into account cumulants higher second order to increase accuracy of adaptive filtering estimator. Therefore it is interest of a quantitative estimation of efficiency of using cumulants the higher orders in problems of an adaptive filtering. For this purpose in paper the problem of an estimating of the central moments of non-Gaussian stochastic process which is modeled as output of nonlinear system is driven by white Gaussian noise is considered. Namely, calculation of a dispersion of the stochastic process which is cubic transformation of a normal Markov process and a dispersion of the process which is nonlinear transformation of white Gaussian noise are considered. If probability distribution function of stochastic process is approximate by excess distribution the error of the dispersion estimator will be much less in comparison with estimator for Gaussian approximation of distribution. Thus, if to take into account cumulants higher order then second it will allow increasing accuracy of estimating of the central moments of stochastic process.
The given fact gives the basis for development of effective algorithms of the adaptive filtering based on using cumulants of high orders.