A mathematical model of an extrapolator based on a stochastic equation that connects processes with a rational and quasi-rational spectrum
Abstract
The paper establishes a stochastic differential-difference equation that relates a random process with a rational spectral density to a process with a quasi-rational spectral density. The solution to this equation allows us to understand the mechanism of formation and structure of the spectral characteristics and optimal extrapolation of a process with a quasi-rational spectral density based on the known formulas for processes with a rational spectrum. It is shown that a process with a quasi-rational spectrum is an L-Markov process. It has been established that the forecast of an L-Markov process depends on the values of the process at several points, the so-called L-boundary. A method is described to determining the numerical values of optimal extrapolation operator by approximating the studied L-Markov process with a trigonometric polynomial with an optimal number of harmonics. The developed models and algorithms can be used in various radio engineering applications for processes with a quasi-rational spectrum.
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