1. MMC
  2. All Volumes and Issues
  3. Volume 12, Number 3, 2025
  4. Hilbert Transform of biperiodically nonstationary random signals

Hilbert Transform of biperiodically nonstationary random signals

MMC.
2025;
Volume 12, Number 3
: pp. 1042–1052
Received: May 27, 2025
Revised: October 09, 2025
Accepted: October 11, 2025

Javorskyj I. M., Yuzefovych R. M., Pelypets R. I., Lychak O. V.  Hilbert Transform of biperiodically nonstationary random signals.  Mathematical Modeling and Computing. Vol. 12, No. 3, pp. 1042–1052 (2025)

Authors:
  1. I. M. Javorskyj
  2. R. M. Yuzefovych
  3. R. I. Pelypets
  4. O. V. Lychak
1
Karpenko Physico-mechanical Institute of NAS of Ukraine; Bydgoszcz University of Science and Technology
2
Karpenko Physico-mechanical Institute of NAS of Ukraine; Lviv Polytechnic National University
3
Lviv Polytechnic National University
4
Karpenko Physico-mechanical Institute of NAS of Ukraine

An analysis of the covariance and spectral structure of the Hilbert transform of biperiodically nonstationary random processes, which model signals with double rhythmicity, is presented here.  The obtained relations connect the cross-covariance and cross-spectral characteristics of the signal and its Hilbert transform with the characteristics of the signal itself.  We examine the properties of the analytic signal and present characteristic special cases determined by the spectral features of carrier-harmonic modulation.

biperiodically nonstationary random process
Hilbert transform
covariance and spectral components
analytic signal
  1. Javorskyj I. M.  Mathematical models and analysis of stochastic oscillations.  Karpenko Physico-Mechanical Institute of Ukraine, Lviv (2013).
  2. Napolitano A.  Cyclostationary processes and time Series: Theory, applications, and generalizations.  Academic Press, Elsevier, London (2020).
  3. Gardner W. A.  Cyclostationarity in communications and signal processing.  IEEE Press, New York (1994).
  4. Hurd H. L., Miamee A.  Periodically Correlated Random Sequences: Spectral Theory and Practice.  Wiley, New York (2007).
  5. Napolitano A.  Cyclostationarity: New trends and applications.  Signal Processing.  120, 385–408 (2016).
  6. Antoni J.  Cyclostationarity by examples.  Mechanical Systems and Signal Processing.  23 (4), 987–1036 (2009).
  7. Javorskyj I., Yuzefovych R., Kravets I., Matsko I.  Methods of periodically correlated random processes and their generalizations.  Cyclostationarity: Theory and Methods.  73–93 (2014).
  8. Javorskyj I., Yuzefovych R., Kravets I., Matsko I.  Periodically correlated processes: Applications in early diagnostics of mechanical systems.  Mechanical Systems and Signal Processing.  83, 406–438 (2017).
  9. Ho D., Randall R. B.  Optimization of bearing diagnostic techniques using simulated and actual bearing fault signals.  Mechanical Systems and Signal Processing.  14 (5), 763–788 (2000).
  10. Antoni J.  Cyclic spectral analysis of rolling-element bearing signals: Facts and fictions.  Journal of Sound and Vibration.  304 (3–5), 497–529 (2007).
  11. Randall R. B., Antoni J.  Rolling element bearing diagnostics – A tutorial.  Mechanical Systems and Signal Processing.  25 (2), 485–520 (2011).
  12. Javorskyj I., Yuzefovych R., Matsko I., Kurapov P.  Hilbert transform of a periodically non-sationary random signal: Low-frequency modulation.  Digital Signal Processing.  116, 103–113 (2021).
  13. Javorskyj I., Yuzefovych R., Lychak O., Semenov P., Shyepko R.  Detection of distributed and localized faults in rotating machines using periodically nonstationary covariance analysis of vibrations.  Measurement Science and Technology.  34 (6), 065–102 (2023).
  14. Javorskyj I., Yuzefovych R., Lychak O., Matsko I.  Hilbert transform for covariance analysis of periodically nonstationary random signals with high-frequency modulation.  ISA Transactions. 452–481 (2023).
  15. Javorskyj I., Yuzefovych R., Lychak O., Trokhym G., Varyvoda M.  Methods of periodically nonstationary random processes for vibrations monitoring of rolling bearing with damaged outer race.  Digital Signal Processing.  145, 104343 (2024).
  16. Randall R. B., Antoni J.  Choosing the right signal processing tools for mechanical systems.  Mechanical Systems and Signal Processing.  224, 112090 (2025).
  17. Borghesani P., Pennacchi P., Ricci R., Chatterton S.  Testing second order cyclostationarity in the squared envelope spectrum of non-white vibration signals.  Mechanical Systems and Signal Processing.  40 (1), 38–55 (2013).
  18. Wang D., Tse P. W., Tsui K. L.  An enhanced Kurtogram method for fault diagnosis of rolling element bearings.  Mechanical Systems and Signal Processing.  35 (1–2), 176–199 (2013).
  19. Abboud D., Elbadaoui M., Smith W. A., Randall R. B.  Advanced bearing diagnostics: A comparative study of two powerful approaches.  Mechanical Systems and Signal Processing.  114, 604–627 (2019).
  20. Borghesani P.  The envelope-based cyclic periodogram.  Mechanical Systems and Signal Processing.  58–59, 245–270 (2015).
  21. Abboud D., Antoni J., Sieg-Zieba S., Eltabach M.  Envelope analysis of rotating machine vibrations in variable speed conditions: A comprehensive treatment.  Mechanical Systems and Signal Processing.  84 (A), 200–226 (2017).
  22. Wang D., Zhao X., Kou L. L., Qin Y., Zhao Y., Tsui K. L.  A simple and fast guideline for generating enhanced/squared envelope spectra from spectral coherence for bearing fault diagnosis.  Mechanical Systems and Signal Processing.  122, 754–768 (2019).