An estimation of cross-spectral components for periodically non-stationary random processes

Authors: 
R. M. Yuzefovych

Karpenko Physico-mechanical institute NAS Ukraine

Correlations between harmonic components of random processes is a feature of signal non-stationarity. In the case of periodically non-stationary random processes (PNRP) correlate harmonics distant one from another on frequencies, what multiple to value T, T - period of non-stationarity [1, 2]. Since faults appearance in rotating elements of mechanical systems leads to periodical non-stationarity of vibration signals when searching of correlations between harmonic components can lay in the base for defects detection [1, 3-5]. For description of such correlations used spectral components, which are Fourier coefficients of varying spectral density and at the same time are the Fourier transformations of correlation components. For widening of vibration diagnostics capabilities reasonable to provide a cross-spectral analysis of vibration signals measured at different points of mechanical system. It gives possibility to investigate spatial properties of signals and to solve tasks of defects localization with higher effectiveness [6-7].
The properties of cross-spectral components estimators, based on Fourier transformation of smoothed estimators of cross-correlation components, are analyzed. The formulae for estimator bias and variance, which describe dependence of these values on realization length, point of correlogram cutoff, smoothing window form and signal spectral characteristics, are derived. The examples of cross-spectral components estimation for amplitude- and phase modulated signals are given. It is proved that variances of cross-spectral components estimators depend on all of spectral components, which are present in Fourier decomposition of varying spectral density of periodically non-stationary random processes. It is shown that statistical analysis of stationary approximation of PNRP can not be made within stationary model but only within the PNRP, which approximation it is.