Hilbert Transform of Multicomponent Narrow-Band Periodically Non-Stationary Random Signal

The periodical non-stationary random signal (PNRS), which is a superposition of amplitude and phase stochastically modulated harmonics with multiple frequencies, is considered. It is assumed that the modulation spectrum is concentrated in a band width of which is less than the basic frequency of the signal. In vibrodiagnostics determining of the modulation parameters for the vibration signals from damaged mechanisms makes it possible to identify defects and extract their characteristic features on the early stages of their initiation. The covariance and spectral properties of multicomponent narrow-band PNRS and its Hilbert transform are considered. Formulae for Fourier series coefficients of auto- and cross-covariance functions, spectral densities and their Hilbert transforms are obtained, relationships between them are found. The covariance and spectral structure of the analytical signal are analyzed. It is shown that, in contrast to a monocomponent narrow-band signal, the analytical signal is a periodically non-stationary random process. The number of harmonics of its covariance function is twice less than that of the covariance function of the signal. It is established that the amplitudes of the harmonics of the covariance function of a multicomponent narrow-band signal are determined only by correlations between the quadratures of harmonics with different orders.