Abstract
The coherent and the component methods for the estimation of the linear covariance invariants of vectorial periodically correlated random processes (PCRP) are considered. The coherent estimators are calculated by averaging of the samples taken through the non-stationary period. The component estimators are built in the form of trigonometric polynomials, Fourier coefficients of which are calculated by weighted averaging of PCRP realization. The properties of the continuous and the discrete estimators are investigated, the asymptotical unbiasedness and mean square consistency are proved. The formulae for their biases and variances described dependency of these quantities on realization length, time sampling and PCRP covariance components are obtained. The conditions of the absence of the aliasing effects of the first and the second kinds are given. The comparison of the coherent and component estimators is carried out for the case of the amplitude modulated signals. It is shown that the advantage of the component method over coherent grows as a rate of PCRP correlations clumping increases. The example of the using of vectorial statistical processing for diagnosis of rolling bearing are given. The investigation results show that using of the covariance function invariants allow to improve the efficiency of the fault detection and to establish of the defect spatial properties.
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References
Dragan Y, Rozhkov V, Javorśkyj I (1987) The methods of probabilistic analysis of oceanological rhythms. Gidrometeoizdat, Leningrad. (in Russian)
Javorśkyj I (2013) Mathematical models and analysis of stochastic oscillations. Physico-Mechanical Institute of NAS of Ukraine, Lviv. (in Ukrainian)
Javorśkyj I, Matsko I, Yuzefovych R, Dzeryn O (2017a) Vectorial periodically correlated random processes and their covariance invariant analysis. In: Chaari F, Leskow J, Napolitano A, Zimroz R, Wylomanska A (eds) Cyclostationarity: theory and methods III. Springer, New York, pp 121–150
McCormick AC, Nandi AK (1998) Cyclostationarity in rotating machine vibrations. Mech Syst Signal Process 12(2):225–242
Capdessus C, Sidahmed M, Lacoume JL (2000) Cyclostationary processes: application in gear faults early diagnosis. Mech Syst Signal Process 14(3):371–385
Antoni J, Bonnardot F, Raad A, El Badaoui M (2004) Cyclostationary modeling of rotating machine vibration signals. Mech Syst Signal Process 18:1285–1314
Antoni J (2009) Cyclostationarity by examples. Mech Syst Signal Process 23:987–1036
Javorśkyj I, Kravets I, Matsko I, Yuzefovych R (2017b) Periodically correlated random processes: application in early diagnostics of mechanical systems. Mech Syst Signal Process 83:406–438
Gardner WA (1987) Statistical spectral analysis: a nonprobabilistic theory. Prentice Hall, Englewood Cliffs
Sadler BM, Dandawate AV (1998) Nonparametric estimation of the cyclic cross spectra. IEEE Trans Inf Theory 44(1):351–358
Javorśkyj I, Kravets I, Yuzefovych R et al (2014a) Vectorial diagnosis of rolling bearing with growing defect on the outer race. Vibratciji v technici I technologijah 2(76):101–110. (in Ukrainian)
Javorśkyj I, Isayev I, Zakrzewski Z, Brooks S (2007) Coherent covariance analysis of periodically correlated random processes. Signal Process 87(1):13–32
Javorśkyj I, Yuzefovych R, Kravets I, Matsko I (2014b) Methods of periodically correlated random processes and their generalizations. In: Chaari F, Leśkow J, Sanches-Ramirez A (eds) Cyclostationarity: theory and methods. Springer, New York, pp 73–93
Javorśkyj I, Isayev I, Yuzefovych R, Majewski J (2010) Component covariance analysis for periodically correlated random processes. Signal Process 90:1083–1102
Javorśkyj I (1984) Application of Buys-Bullot scheme in statistical analysis of rhythmic signals. Radioelectron Telecommun Syst 27(11):28–33
Javorśkyj I, Mykhajlyshyn V (1996) Probabilistic models and investigation of hidden periodicities. Appl Math Lett 9:21–23
Javorśkyj I, Dehay D, Kravets I (2014c) Component statistical analysis of second order hidden periodicities. Digit Signal Process 26:50–70
Javorśkyj I, Yuzefovych R, Matsko I, Zakrzewski Z, Majewski J (2017c) Coherent covariance analysis of periodically correlated random processes for unknown non-stationarity period. Digit Signal Process 65:27–51
Javorśkyj I, Yuzefovych R, Matsko I, Zakrzewski Z, Majewski J (2016) Discrete estimators for periodically correlated time series. Digit Signal Process 53:25–40
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Javors’kyj, I., Matsko, I., Yuzefovych, R., Trokhym, G., Semenov, P. (2020). The Coherent and Component Estimation of Covariance Invariants for Vectorial Periodically Correlated Random Processes and Its Application. In: Chaari, F., Leskow, J., Zimroz, R., Wyłomańska, A., Dudek, A. (eds) Cyclostationarity: Theory and Methods – IV. CSTA 2017. Applied Condition Monitoring, vol 16. Springer, Cham. https://doi.org/10.1007/978-3-030-22529-2_4
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DOI: https://doi.org/10.1007/978-3-030-22529-2_4
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