Cyclostationarity: Theory and Methods pp 73-93 | Cite as

# Methods of Periodically Correlated Random Processes and Their Generalizations

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## Abstract

The results obtained by authors in the area of theory and methods of statistical analysis of periodically correlated random processes and their generalizations are presented in this article. The main methods for estimation of their correlation and spectral characteristics: coherent, component, least square method and linear filtration method are analyzed. The ways of generalization of these methods to the case of unknown a priori period of non-stationarity are considered and the possible algorithms of its estimation are presented.

## Keywords

Stochastic oscillation Periodically correlated random processes Estimation methods Hidden periodicities## References

- Antoni J (2009) Cyclostationarity by examples. Mech Syst Signal Process 23:987–1036CrossRefGoogle Scholar
- Aschoff Y (ed) (1981) The biological rhythms. Plenum Press, New YorkGoogle Scholar
- Bennet WR (1958) Statistics of regenerative digital transmission. Bell System Techn Jorn 37: 1501–1542Google Scholar
- Bjelyshev A, Klevantsov Y, Rozhkov V (1983) Probabilistic analysis of sea currents. Gidrometeoizdat, Leningrad (in Russian)Google Scholar
- Drabych O, Mykhajlyshyn V, Javors’kyj I (2000) Determination of correlation period of the periodically correlated random processes using covariation transformations. Vidbir i Obrobka Informatsiji 14(90):47–52 (in Ukrainian)Google Scholar
- Dragan Y (1980) The structure and representation of the stochastic signal models. Naukova Dumka, Kiyev (in Russian)Google Scholar
- Dragan Y (1972) On biperiodically correlated random processes. Otbor i Peredacha Informatsiyi 34:12–14 (in Russian)Google Scholar
- Dragan Y, Javors’kyj I (1975) About representation of sea waves by periodically correlated random processes and their statistical processing methods. Methods of representation and apparatus analysis of random processes and fields. XIII Ail-Union Symposium, Leningrad, pp 29–33 (in Russian)Google Scholar
- Dragan Y, Javors’kyj I (1982) Rhythmic of Sea Waving and Underwater Acoustic Signals. Naukova Dumka, Kyiv (in Russian)Google Scholar
- Dragan Y, Rozhkov V, Javors’kyj (1987) The methods of probabilistic analysis of oceanological rhythmic. Gidrometeoizdat, Leningrad (in Russian)Google Scholar
- Franks LE (1969) Signal theory. Prentice-Hall, Englewood CliffszbMATHGoogle Scholar
- Gardner WA (1994) Cyclostationarity in communications and signal processing. In: Gardner W (ed) IEEE, New YorkGoogle Scholar
- Gardner WA, Franks LE (1975) Characterization of cyclostationary random signal processes. IEEE Trans Inf Theory IT-21:4–14Google Scholar
- Gardner WA (1985) Introduction to random processes with applications to signals and systems. Macmillan, New-YorkGoogle Scholar
- Gardner WA (1986) The spectral correlation theory of cyclostationary time-series. Signal Process 1:3–36Google Scholar
- Gladyshev E (1959) Periodically and almost periodically correlated processes with continuous time. Teoriya Veroyatnostei i yeyo Primenenie 3(2):84–189 (in Russian)Google Scholar
- Groisman P (1977) The estimate of autocorrelation matrixes of the precipitation time series considered as periodically correlated random processes. Trudy GGL 247:119–127 (in Russian)Google Scholar
- Gruza G (1982) Some general problems of time series statistical analysis in climatology. In: Proceedings of all-union scientific research institute of hydrometeorologi-cal information—world data centre 83, Obninsk, pp 3–9 (in Russian)Google Scholar
- Gudzenko L (1961) The generalization of the ergodic theorem for nonstationarv random processes. Izvestia Vysshikh Uchebnykh Zavedenij Ser Radiofizika 4(2):265–274 (in Russian)Google Scholar
- Gudzenko L (1959) On periodically nonstationary processes. Radiotekhnika i Elektronika 6(6):1020–1040 (in Russian)Google Scholar
- Gudzenko L (1959) The small fluctuation in essentially nonlinear auto-oscillation system. Dokl Akad Nauk USSR 125(1):62–65 (in Russian)MathSciNetGoogle Scholar
- Hurd HL (1969) An investigation of periodically correlated stochastic processes. Ph.D. dissertation. Duke University Department of Electrical EngineeringGoogle Scholar
- Hurd HL (1989) Nonparametric time series analysis for periodically correlated processes. IEEE Trans Inf Theory 35:350–359CrossRefzbMATHMathSciNetGoogle Scholar
- Isayev I, Javors’kyj I (1995) Component analysis of the time series with rhythmical structure. Izvestia Vysshykh Uchebnykh Zavedeniy Ser Radioelektronika 38(1):34–45 (in Russian)Google Scholar
- Javors’kyj I, Mikhailyshyn V (1996) Probabilistic models and investigation of hidden periodicities. Appl Math Lett 9(2):21–23Google Scholar
- Javors’kyj I (1984) The application of Buys-Ballot scheme for statistical analysis of rhythmical signals. Izvestiya Vysshykh Uchebnykh Zavedenij, ser. Radio-elektronica 27(11):31–37 (in Russian)Google Scholar
- Javors’kyj I (1985) The computation of characteristics of periodically correlated random processes by data random selection. Thesis of reports of I acoustical seminar on modelling, algorithms and reception of decision. Moscow, pp 44–40 (in Russian)Google Scholar
- Javors’kyj I (1987) Biperiodically correlated random processes as model of bi-rhythmic signals. In: All-Union Conference on Information Acoustics, Moscow, pp 6–10 (in Russian)Google Scholar
- Javors’kyj I. (1988) The least squares method for statistical analysis of signals with rhythmical structure. In: II all-union acoustical seminar on modelling, algorithms and reception of decisions, Moscow, pp 5–6 (in Russian)Google Scholar
- Javors’kyj I (1985) The component sample estimates of probabilistic characteristics of periodically correlated random processes. Otbor i Peredacha Informatsivi 72:17–27 (in Russian)Google Scholar
- Javors’kyj I (1985) On statistical analysis of periodically correlated random processes. Radiotekhnika i Electronika 6:1096–1104 (in Russian)Google Scholar
- Javors’kyj I (1986) Statistical analysis of biperiodically correlated random processes. Otbor i Peredacha Informatsiyi 73:12–21 (in Russian)Google Scholar
- Javors’kyj I (1987) The statistical analysts of vector periodical correlated random processes. Otbor i Peredacha Informatsiyi 76:3–12 (in Russian)Google Scholar
- Javors’kyj I (1987) The interpolation of the estimates of periodicallly correlated random processes. Avtomatika I:36–41 (in Russian)Google Scholar
- Javorskyj I, Kravets I, Isayev I (2006) Parametric modeling of periodically correlated random processes by their representation through stationary random processes. Radio Electron Commun Syst 49(11):23–29Google Scholar
- Javorskyj I, Isaev I, Zakrzewski Z, Brooks SP (2007) Coherent covariance analysis of periodically correlated random processes. Signal Process 8(1):13–32CrossRefGoogle Scholar
- Javorskyj I, Isayev I, Majewski J, Yuzefovych R (2010) Component covariance analysis for periodically correlated random processes. Signal Process 90:1083–1102CrossRefzbMATHGoogle Scholar
- Javorskyj I, Leskow J, Kravets I, Isayev I, Gajecka E (2012) Linear filtration methods for statistical analysis of periodically correlated random processes—Part I: coherent and component methods. Signal Process 92:15591566CrossRefGoogle Scholar
- Javorskyj I, Isayev I, Kravets I, (2007) Algorithms for separating the periodically correlated random processes into harmonic series representation. In: Proceedings of 15th European signal processing conference (EUSIPCO 2007), Poznan, pp 1857–1861Google Scholar
- Javorskyj I, Yuzefovych R, Krawets I, Zakrzewski Z (2011) Least squares method in the statistic analysis of periodically correlated random processes. Radoi Electron Commun Syst 54 (1):45 59Google Scholar
- Jaworskyj IM, Drabych PP, Kravets IB, Matsko II, (2011a) Metod of vibration diagnostics of initial stages of rotation systems damage. Mater Sci 47(2):264–271Google Scholar
- Jaworskyj I, Leskow J, Kravets I, Isayev I, Gajecka E (2011b) Linear filtration methods for statistical analysis of periodically correlated random processes—Part II: harmonic series representation. Signal Process 91(2506):2519Google Scholar
- Kiselyeva T, Chudnovsky A (1968) Statistical investigation of the diurnal cycle of the air temperature. Bull Nauchno-tekhnicheskoi Informatsiyi po Agronom Fizike 11:17–38 (in Russian)Google Scholar
- Kolyesnikova V, Monin A (1965) On spectrum of meteorological field oscillations. Izvestiya AN USSR, Ser. Fizika Atmosfery i Okeana 1(7):653–669 (in Russian)Google Scholar
- Koronkevich O (1957) The linear dynamic systems under action of the random forces. Naukovi Zapyskj Lvivskoho Universytetu 44(8):175–183 (in Ukrainian)Google Scholar
- Kostjukov Y, Uljanich I, Mezentsev V, Javorskyj I (1987) The analysis of nonequidistant time series of Gulf of Riga temperature and salinity. In: Ail-Union Scientific Research Institute of Hydro meteorological World Data Centre, vol 134, pp 87–97 (in Russian)Google Scholar
- Kozel S (1959) The transformation of the periodically nonstationary fluctuations by linear filter. Trudy Moskovskoho Fiziko-tekhnicheskoho Instituta 4:10–l6 (in Russian)Google Scholar
- Kravets IB (2012) Parametric models of cyclostationary signals. Radio Electron Commun Syst 55(6):257–267Google Scholar
- Malakhov A (1968) The fluctuation in autooscillation systems. Nauka, Moscow (in Russian)Google Scholar
- Mamontov N (1968a) The standard deviation and coefficients of skewness of air temperature in south-east of West-Siberia plain. Trudy NIIAK 54:29–34 (in Russian)Google Scholar
- Mamontov N (1968b) The investigation of distribution statistics of the relative air humidity. Trudy NIIAK 54:18–28 (in Russian)Google Scholar
- Mezentsev V, Javors’kyj I (1988) The properties of the optimum estimates of rhythmic signal probabilistic characteristics. Thesis of the reports of All-union workshop On signal processing. Uljanovsk Polytechnical Institute, Uljanovsk, pp 16–18 (in Russian)Google Scholar
- Mikhailyshyn V, Fligel S, Javors’kyj I (1990) Statistical analysis of wave packets of geomagnetic pulsations Pcl type by the method of periodically correlated random processes. Geomagnetizm i Aeronomia 30(5):757–764 (in Russian)Google Scholar
- Mikhailyshyn V, Fligel S, Javors’kyj (1990) The probabilistic model of the signal periodicity of geomagnetic pulsations. Pel. The investigation of the structure and wave properties near-earth plasma. Nauka, Moscow, pp 76–88 (in Russian)Google Scholar
- Mikhailyshyn V, Javors’kyj I (1990) LSM-analysis while identifying poly-rhythmic structure of stochastic signals. In: Proceedings of the first international conference on information technologies for image analysis and pattern recognition, Lviv, pp321–325Google Scholar
- Mishchenko Z (1966) The air temperature diurnal cycle and its agroclimatic significance. Gidrometeoizdat, Leningrad (in Russian)Google Scholar
- Mishchenko Z (1960) The air temperature diurnal cycle and plant termo-periodicity. Trudy GGO 91:15–28 (in Russian)Google Scholar
- Mykhailyshyn VYu, Yavors’kyi IM, Vasylyna YuT, Drabych OP (1997) Probabilistic models and statistical methods for the analysis of vibrational signals in the problems of diagnostics of machine. Mater Sci 33(5):655–672CrossRefGoogle Scholar
- Myslovich M, Pryimak N, Shcherbak L (1980) Periodically correlated random processes in problems of acoustic information processing. Znaniye, Kiyev (in Russian)Google Scholar
- Ogura H (1971) Spectral representation of periodic nonstationary random processes. IEEE Trans Inf Theory 17(2):143–149CrossRefzbMATHMathSciNetGoogle Scholar
- Papoulis A (1983) Random modulation: a review. IEEE Trans Acoust Speech Signal Process 31(1):96–105Google Scholar
- Poljak I (1978) Methods of analysis of random processes and fields in climatology. Gidrometeoizdat, Leningrad (in Russian)Google Scholar
- Romanenko A, Sergeyev G (1968) The problems of the applied analysis of random processes. Sov Radio, Moscow (in Russian)Google Scholar
- Rytov S (1976) An introduction to statistical radio-physics. P.1. Nauka, Moscow (in Russian)Google Scholar
- Serebrjennikov M, Pervozvansk A (1965) The hidden periodicities detecting. Nauka, Moscow (in Russian)Google Scholar
- Stratonovich R (1961) Selected problems of the fluctuation theory in radioengneering. Sov Radio, Moscow (in Russian)Google Scholar
- Tikhonov V (1956) When nonstationary random process can be substituted for stationary process. Zhurnal Teoreticheskoi i Ekspcrimentalnoi Fiziki 31(9):2057–2059 (in Russian)Google Scholar
- Voichishyn K, Dragan Y (1971) On simple stochastic model of the natural rhythmic processes. Otbor i Peredacha Informatsiyi 29:7–15 (in Russian)Google Scholar
- Voichyshyn K (1975) The problems of statistical analysis of nonstationary (rhythmic) phenomena conformable to some geophysical object. Cand. Phys. and Math. Sci. Dissertation, Institute of Earth Physics, Moscow (in Russian)Google Scholar
- Ya Dragan (1978) The harmonizability and the decomposition of the random processes with finite average power. Dokl Akademiyi Nauk USSR ser A 8:679–684 (in Russian)Google Scholar
- Ya Dragan (1969) On periodically correlated random processes and systems with periodic parameters. Otbor i Peredacha Informatsiyi 22:27–33 (in Russian)Google Scholar
- Ya Dragan (1970) On spectral properties of periodically correlated random processes. Otbor i Peredacha Informatsiyi 30:16–24 (in Russian)Google Scholar
- Ya Dragan (1972) About basing of rhythmic stochastic model. Otbor i Peredacha Informatsiyi 31:12–21 (in Russian)Google Scholar
- Ya Dragan (1972) The properties of the periodically correlated random process samples. Otbor i Peredacha Informatsiyi 33:9–12 (in Russian)Google Scholar
- Ya Dragan (1975) On representation of periodically correlated random processes by stationary components. Otbor i Peredacha Informatsiyi 45:7–20 (in Russian)Google Scholar
- Ya Dragan (1985) Periodic and periodically nonstationary random processes. Otbor i Peredacha Informatsiyi 72:3–17 (in Russian)Google Scholar
- Yaglom A (1981) Correlation theory of stationary random functions. Gidrometeoizdat, Leningrad (in Russian)Google Scholar
- Yavors’kyi I, Isaev I, Kravets I, Drabych P, Mats’ko I (2009) Methods for enhancement of the efficiency of statistical analysis of vibration signals from the bearing supports of turbines at thermal-electric power plants. Mater Sci 45(3):378–391CrossRefGoogle Scholar
- Yavorskyj IN, Kravets IB, Mats’ko IY (2011) Spectral analysis of stationary components of periodically correlated random processes. Radio Electron Commun Syst 54 8):451 463Google Scholar
- Yavorskyj IN, Yuzefovych R, Kravets IB, Matsko IY (2012) Properties of characteristics estimators of periodically correlated random processes in preliminary determination of the period of correlation. Radio Electron Commun Syst 55(8):335–348Google Scholar
- Zabolotnyj O, Mykhajlyshyn V, Javors’kyj I (2000) The least squares method of polyrhythmic statistical analysis. Dopovidi NAN Ukrainy 8:93–100 (in Ukrainian)Google Scholar
- Zhukovsky E (1969) The investigation of the statistical characteristics of relative air humidity. Sbornik Trudov po Agronom Fizike 20:3–28 (in Russian)Google Scholar

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