HILBERT TRANSFORM FOR ANALYZING STOCHASTIC MODULATION OF BI-PERIODICALLY NON-STATIONARY RANDOM SIGNAL

Abstract

Stochastic signals of natural and technological origin often have a clearly expressed rhythmic structure, which manifests itself in first- and second-order statistical characteristics. In the simplest case, such a structure is described by periodically non-stationary random processes (PNRP) with a single base frequency. In practice, there are also frequent cases when several rhythms are simultaneously present in the same signal. That is, the stochastic repeatability of one period interacts with the repeatability of another. A probabilistic model of stochastic variability with double rhythmicity is a bi-periodically non-stationary random process (BPNRP). It was previously shown that for the analysis of signals with stochastic modulation, in cases where the harmonics of the carrier are characterized by a single fundamental frequency and its multiples, the use of Hilbert transform allows the hidden structure of periodically non-stationary modulating oscillations to be extracted. However, despite the existence of a well-developed PNRP theory and initial results on its demodulation using Hilbert transform for single-carrier signals, the properties of Hilbert transform for BPNRP have not yet been sufficiently studied. The paper discusses in detail the properties of the signal model in the form of BPNRP. It is shown that the correlation structure of BPNRP is determined by stationary correlated random processes that modulate the amplitude and phase of carrier harmonics, whose frequencies are combinations of two fundamental frequencies. An analysis of stochastic modulation using the Hilbert transform is carried out. The characteristic features of the correlation and spectral structure of the Hilbert transform of a signal have been established, which must be taken into account when processing real data.