The periodically non-stationary random signals (PNRSs), whose carrier harmonics are mo-dulated by jointly stationary high-frequency random processes are analyzed. A representation of the signal in the form of a superposition of high-frequency components is obtained and it is shown that these components are jointly periodically non-stationary random processes. The random process is periodically non-stationary of the second order only in the case when some of the cross-covariance functions of its modulation processes are not equal to zero. The correlations of the PNRP spectral harmonics and the correlations of the modulating processes in series representation are equivalent. Evaluating the specific features of the auto- and cross-covariances for modulating processes as well as contribution of each pair to the covariance component values allows us to detect defects at early stages.

1. Dragan Y.; Javorskyj I. Rhythmics of sea waving and underwater acoustic signals; Naukova Dumka, 1982 (in Russian).

2. Gardner, W.A. Cyclostationarity in Communications and Signal Processing; IEEE Press, 1994.

3. Antoni, J. Cyclostationarity by examples, Mech. Syst. Signal Process., 2009, 23, 987-1036. https://doi.org/10.1016/j.ymssp.2008.10.010

4. Javorskyj, I.; Yuzefovych, R.; Matsko, I.; Kurapov, P. Hilbert transform of a periodically non-stationary random signal: Low-frequency modulation, Digit. Signal Process., 2021, 116, 103113. https://doi.org/10.1016/j.dsp.2021.103113

5. McFadden, P.D.; Smith, J.D. Vibration monitoring of rolling element bearings by the high frequency resonance technique - A review, Tribol. Int., 1984, 17, 3-10. https://doi.org/10.1016/0301-679X(84)90076-8

6. Ho, D.; Randall, R.B. Optimization of bearing diagnostic techniques using simulated and actual bearing fault signals, Mech. Syst. Signal Process., 2000, 14, 763-788. https://doi.org/10.1006/mssp.2000.1304

7. Antoni, J. Cyclic spectral analysis of rolling-element bearing signals: Facts and Fictions, J. Sound Vib., 2007, 304, 497-529. https://doi.org/10.1016/j.jsv.2007.02.029

8. Randall, R.B.; Antoni, J. Rolling element bearing diagnostics - A tutorial, Mech. Syst. Signal Process., 2011, 25, 485-520. https://doi.org/10.1016/j.ymssp.2010.07.017

9. Napolitano, A. Cyclostationary processes and time series: Theory, applications, and generali-zations; Elsevier, 2020.

10. Mykhailyshyn, V.; Javorskyj, I.; Vasylyna, Y.; Drabych, O.; Isaev, I. Probabilistic models and statistical methods for the analysis of vibration signals in the problems of diagnostics of machines and structures, Materials Science, 1997, 33, 655-672. https://doi.org/10.1007/BF02537594

11. McCormick, A.C.; Nandi, A.K. Cyclostationarity in rotating machine vibrations, Mech. Syst. Signal Process., 1998, 12, 225-242. https://doi.org/10.1006/mssp.1997.0148

12. Capdessus, C.; Sidahmed, M.; Lacoume, J.L. Cyclostationary processes: Application in gear fault early diagnostics, Mech. Syst. Signal Process., 2000, 14, 371-385. https://doi.org/10.1006/mssp.1999.1260

13. Antoniadis, I.; Glossiotis, G. Cyclostationary analysis of rolling-element bearing vibration signals, J. Sound Vib., 2001, 248, 829-845. https://doi.org/10.1006/jsvi.2001.3815

14. Antoni, J.; Bonnardot, F.; Raad, A.; El Badaoui, M. Cyclostationary modeling of rotating machine vibration signals, Mech. Syst. Signal Process., 2004, 18, 1285-1314. https://doi.org/10.1016/S0888-3270(03)00088-8

15. Antoni, J.; Randall, R.B. A stochastic model for simulation and diagnostics of rolling element bearings with localized faults, ASME J. Vib. Acoust., 2003, 125, 282-289 https://doi.org/10.1115/1.1569940

16. Javorskyj, I.; Kravets, I.; Matsko, I.; Yuzefovych, R. Periodically correlated random processes: Application in early diagnostics of mechanical systems, Mech. Syst. Signal Process., 2017, 83, 406-438. https://doi.org/10.1016/j.ymssp.2016.06.022

17. Borghesani, P.; Pennacchi, P.; Ricci, R.; Chatterton, S. Testing second order cyclostationarity in the squared envelope spectrum of non-white vibration signals, Mech. Syst. Signal Process., 2013, 40, 38-55. https://doi.org/10.1016/j.ymssp.2013.05.012

18. Wang, H. Early detection of gear tooth cracking using the resonance demodulation technique, Mech. Syst. Signal Process., 2001, 15, 887-903. https://doi.org/10.1006/mssp.2001.1416

19. Kay, S.M. Modern spectral estimation: Theory and application. Prentice Hall, 1988.

20. Randall, R.B.; Sawalhi, N.: Coats, M. A comparison of methods for separation of deterministic and random signals, Int. J. Cond. Monitoring, 2011, 1, 11-19. https://doi.org/10.1784/204764211798089048

21. Borghesani, P.; Pennacchi, P.; Randall, R.B.; Sawalhi, N.; Ricci, R. Application of cepstrum pre-whitening for the diagnosis of bearing faults under variable speed conditions, Mech. Syst. Signal Process., 2013, 36, 370-384. https://doi.org/10.1016/j.ymssp.2012.11.001

22. Hurd, H.L.; Miamee, A. Periodically Correlated Random Sequences: Spectral Theory and Prac-tice; Wiley, 2007. https://doi.org/10.1002/9780470182833

23. Antoni, J.; Borghesani, P. A statistical methodology for the design of condition indicators, Mech. Syst. Signal Process., 2019, 114, 290-327. https://doi.org/10.1016/j.ymssp.2018.05.012

24. Wiener, N. Generalized harmonic analysis, Acta Math., 1930, 55, 117-258. https://doi.org/10.1007/BF02546511

25. Javorskyj, I.; Isayev, I.; Majewski, J.; Yuzefovych, R. Component covariance analysis for periodically correlated random processes, Signal Process., 2010, 90(4), 1083-1102. https://doi.org/10.1016/j.sigpro.2009.07.031