THE SYNTHESIS OF EXPLICIT ANALYTICAL FORMULAE FOR THE PROBABILISTIC MODELS OF SIGNALS USING FRACTION-OF-TIME APPROACH

Abstract

The paper presents the development of the theoretical foundations of the fraction-of-time approach which allows one to carry out constructing the probabilistic model of a random process based on its singular realization where a priori information of its ergodic property is absent. The synthesis of the analytical expression for time-varying FOT density was conducted in the form of an explicit function depending on two variables, which was based on the detailed analysis of its implicit initial expression. The resultant form was shown to be the sum consisting of no more than countable number of summands containing one-dimensional Dirac delta functions. The explicit analytical representation led to a significant simplification of the subsequent integration in time, which was conducted for the evaluation of the stationary component of FOT describing the statistical property of the signal being analyzed. In addition, it is shown in the paper that the intermediate result of the conducted derivation, namely, the time-value distribution, plays an important role in constructing nonlinear probabilistic models of signals within the framework of the FOT approach. The Fourier series representation of the time-value distribution was obtained by the example of a monoharmonic signal, where the Fourier series coefficients are derived in the form of explicit functions of the threshold variable. The Fourier series representation with explicit Fourier coefficients was also obtained for the time-varying indicator function. The properties of the stationary components, which are related to zero cyclic frequency, of the time-value distribution and time varying indicator function were shown to be resembling the properties of probability density and cumulative distribution function describing one-dimensional random variable, while the components related to non-zero cyclic frequencies exhibit different properties. The models developed in the current research are aimed at developing new methods of estimating probabilistic characteristics of analyzed signals which in turn leads to the synthesis of new digital signal processing algorithms.