Demodulation of non-stationary random signal using Hilbert transform

A narrow-band high frequency amplitude modulation as a model of vibration signal is considered. Use of Hilbert transform for the demodulation of periodically non-stationary random signal (PNRS) is discussed. Relations for spectral and covariance components of model signal, its Hilbert transform and cross-covariance components are obtained. Quadratures for modulation signal are extracted and analyzed. It is shown, that the Fourier coefficients of the auto-covariance functions of a signal and its Hilbert transform are the same and its cross-covariance functions differ only by a sign. The square of the modulus of the analytical signal is not a “squared envelope” in the known sense. A “squared envelope” in this case is a random process, whose mathematical expectation is equal to twice the variance of the raw signal. This results in an identity of cyclic spectrums of variances for analytic and raw signals. Thus, the Hilbert transform cannot be used directly as a demodulation procedure, and the “squared envelope” can be analyzed only as the implementation of a random process using PNVP methods. It is shown that band-pass filtering and the Hilbert transform can be used for extraction of modulating signal quadratures.