Hilbert transform of a periodically non-stationary random signal: Low-frequency modulation

The Hilbert transform of mono- and multicomponent periodically non-stationary random signals (PNRSs), the carrier harmonics of which are narrow-band low-frequency modulated, are considered here. The relations for the auto- and cross-covariance and cross-spectral functions of a PNRS and its Hilbert transform are analyzed and the conditions for stationarity of the analytical signal are obtained. The covariance and spectral properties of the quadratures for monocomponent PNRS are analyzed, it is shown that correlation of the quadratures of a polycomponent PNRS causes the periodic non-stationarity of the analytic signal, and the specific features of this are mentioned. Extraction and analysis of the quadratures are carried out, and examples are given of the use of the Hilbert transform to analyze simulated and real-life polycomponent periodically non-stationary time series.