V.M. Artyushenko – Dr.Sc.(Eng.), Professor, Head of Department of Information Technology and Management Systems, Technological University (Korolyov, Moscow region)

E-mail: artuschenko@mail.ru

V.I. Volovach – Dr.Sc.(Eng.), Associate Professor, Head of Department of Information and Electronic Service,  Volga Region State University of Service (Togliatti) E-mail: volovach.vi@mail.ru

The issues associated with the use of the statistical linearization method for describing non-Gaussian random processes, signals and noise are considered and analyzed and communication of the called method and the method of cumulant.

It is shown that approximate methods can be used to describe non-Gaussian random processes, since it is not always possible or extremely difficult to determine multidimensional probability density distributions (PDF) for these processes. One of the common approximate methods of describing non-Gaussian processes, signals and noise is the method of statistical linearization. In this case, if a large number of terms of a sequence are used, an acceptable error of description can be obtained.

The method of statistical linearization is described and analyzed. The method involves finding the best probability approximation of an instantaneous nonlinear transformation by a linear dependence. Here, the initial and approximating functions should have similar mathematical expectation and correlation functions. It is shown that there are two ways to determine parameters of a statistically linearized dependence, which, in turn, are based on two criteria. The first one is a criterion of equality of mathematical expectations and variances of the true and the approximating functions. The second one is the criterion of the minimum mean square of the approximation error. The statistical approximation method is described by using the first criterion; the conditions of its applicability are determined. It is shown that when specifying multidimensional PDFs, including two-dimensional PDFs, this method allows us to analyze vector random processes. The latter is very important in the analysis of instantaneous of nonlinear blocks. An example is given for the case when PDF is one-dimensional. The method of statistical approximation is described by using the second criterion; its efficiency is illustrated. An example of the applicability of this method to the analysis of the forming filter is considered.

We presented two methods of analyzing nonlinear systems with the use of the cumulants method for the scalar case. The first method uses a differential equation for semiinvariants, which are derived from a differential equation for the logarithm of the characteristic function. It is shown that differential equations for semiinvariants generally form an infinite system of interrelated ordinary differential equations when PDF is non-Gaussian. Since they can be found only approximately, it corresponds to the hypothesis of insignificant influence of higher cumulants on the accuracy of lower ones. The second method simplifies finding semiinvariants. Here, the PDFs are approximated by a series depending on semiinvariants, for example, in the case under consideration by a series of Edgeworth. A typical example is given. The connection between the cumulant method and the method of statistical linearization is analyzed. As a result, it is concluded that the method of statistical linearization is a particular case of the method of semiinvariants.

A comparative assessment of different methods of statistical linearization in the Gaussian approximation is made. For this purpose, the PDF of a stationary steady state random process were determined by using the first criterion (for approximate and exact solutions) and the second criterion of approximation (for approximate solutions), and also the corresponding graphical dependences were constructed. The analysis of these graphs allows us to assess advantages of the solutions obtained for both methods.

1. Levin B.R. Teoreticheskie osnovy’ statisticheskoj radiotexniki. M.: Sov. radio. 1974. 552 s.
2. Tixonov V.I., Kul’man N.K. Nelinejnaya fil’tracziya i kvazikogerentny’j priem signalov. M.: Sov. radio. 1975. 704 s.
3. Tixonov V.I. Nelinejnoe preobrazovanie sluchajny’x proczessov. M.: Radio i svyaz’. 1986. 296 s.
4. Kassam S.A. Signal Detection in non-Gaussian Noise. New York: Springer Verlag. 1989. 242 p.
5. Valeev V.G. Optimal’naya oczenka parametrov signala pri nalichii negaussovskix pomex // Izv. AN SSSR. Texnicheskaya kibernetika. 1974. № 2. S. 135−146.
6. Akishin N.S., Rumyanczev V.L., Mixajlov A.V., Meliticzkaya V.V. Statisticheskie xarakteristiki ogibayushhej additivnoj smesi negaussovskogo periodicheski nestaczionarnogo radiosignala i negaussovskoj pomexi // Izvestiya VUZov. Radioe’lektronika. 1988. T. 31. № 4. S. 84−87.
7. Artyushenko V.M., Volovach V.I. Oczenka pogreshnosti izmereniya vektornogo informaczionnogo parametra signala na fone mul’tiplikativny’x pomex // Radiotexnika. 2016. № 2. S. 72−82.
8. Artyushenko V.M., Volovach V.I. Kvazioptimal’naya obrabotka signalov na fone uzkopolosny’x korrelirovanny’x negaussovskix pomex // Radiotexnika. 2016. № 7. S. 125−132.
9. Artyushenko V.M., Volovach V.I. Kvazioptimal’naya diskretnaya demodulyacziya signalov na fone korrelirovanny’x negaussovskix flyuktuaczionny’x mul’tiplikativny’x pomex // Radiotexnika. 2016. № 6. S. 106−112.
10. Artyushenko V.M., Volovach V.I. Oczenka tochnosti izmereniya informaczionny’x parametrov signala v usloviyax odnovremennogo vozdejstviya mul’tiplikativny’x i additivny’x negaussovskix pomex // Radiotexnika. 2017. № 10. S. 95−102.
11. Malaxov A.N. Kumulyativny’j analiz sluchajny’x negaussovskix proczessov i ix preobrazovanij. M.: Sov. radio. 1978. 376 s.
12. Kazakov I.E., Mal’chikov S.V. Analiz statisticheskix sistem v prostranstve sostoyanij. M.: Radio i svyaz’. 1986. 296 s.
13. Pugachev V.S., Siniczy’n I.N. Stoxasticheskie differenczial’ny’e sistemy’. M.: Nauka. 1985. 560 s.
14. Korzhik V.I., Fink L.B., Shhelkunov K.I. i dr. Raschet pomexoustojchivosti sistem peredachi diskretny’x soobshhenij: Spravochnik / Pod red. L.M. Finka. M.: Radio i svyaz’. 1981. 232 s.
15. Dadashev M.L. Priblizhenny’j analiz tochnosti nestaczionarny’x nelinejny’x sistem metodom semiinvariantov // Avtomatika i telemexanika. 1967. № 11. S. 62−81.