The exact analytical solution of the problem on the average number of spikes of the narrowband Gaussian stochastic process

Abstract

In this article, the problem of the number of spikes (level crossings) of the stationary narrowband Gaussian process has been considered. The process was specified by an exponentially-cosine autocorrelation function. The problem had been solved earlier by Rice in terms of the joint probabilities’ density of the process and its derivative with respect to time, but in our article we obtained the solution using the functional of probabilities’ density (the functional was obtained by Amiantov), as well as an expansion of the canonical stochastic process. In this article, the optimal canonical expansion of a narrowband stochastic process based on the work of Filimonov and Denisov was also considered to solve the problem. The application of all these resources allowed obtaining an exact analytical solution of the problem on spikes of stationary narrowband Gaussian process. The obtained formulae could be used to solve, for example, some problems about the residual resource of some radiotechnical products, about the breaking sea waves and others.