A.D. Vinokurov1, N.A. Kupriyanov2, V.V. Makarenkov3, G.N. Ulyanov4, A.A. Shatalov5, V.A. Shatalova6, A.B. Yastrebkov7

1, 4, 5 Mikhailovsky Military Artillery Academy (St. Petersburg, Russia)
 2 Krasnodar Higher Military Aviation School of Pilots n.a. Hero of the Soviet Union A.K. Serov (Krasnodar, Russia)
 3, 7 Mozhaisky Military Space Academy (St. Petersburg, Russia)
 6 St. Petersburg State University of Telecommunications
 n.a. Professor M.A. Bonch-Bruevich (St. Petersburg, Russia)
 1 gonta-gv@yandex.ru, 3 makar8722@mail.ru

Problem statement. Currently, the main efforts in the development of the general theory of extrapolation and its practical applications are aimed at predicting processes, moments of time and fields. At the same time, as a result of increased processing quality requirements, it is necessary to take into account the effect of various kinds of interference and noise. Therefore, most of the existing approaches to solving extrapolation problems rely to some extent on analytical methods. At the same time, the formulation of tasks mainly concerns processes and sequences, the changes of which are largely determined by random components. Due to the fact that the mathematical description of objects and the environment (or their probabilistic characteristics) is often unknown in advance, and it is not possible to determine them, an adaptive approach is typical for solving such problems. With an adaptive approach, some suboptimal solution is sought, obtained, for example, by using maximum likelihood or minimum average risk criteria. It is from this point of view that the problems of filtering, interpolation and extrapolation are often considered. The main focus is on the application of the basic principles of the theory of systems and adaptive Kalman filtering. It is assumed that the Gaussian approximation, a linear model of the system, as well as linear processing of input sequences are used. The optimal joint assessment of the parameters of the state model leads to a nonlinear filter structure, the implementation of which is associated with high requirements for performance and computer memory. Therefore, preference is usually given to suboptimal methods, on the basis of which simplified algorithms are formed that do not require large amounts of computer memory.

Goal – development of adaptive algorithms for the functioning of multidimensional linear and nonlinear extrapolators implemented in the time and private domain by digital signal processing (DSP) methods, as well as analysis of the quality characteristics of their functioning and application features.

Algorithms for the functioning of multidimensional extrapolators are proposed using examples of the development of adaptive algorithms using the method of linear prediction "forward" and "backward" and their ombinations. Algorithms for the operation of nonlinear "forward" extrapolators based on Kolmogorov-Gabor series in the time and frequency domains have been developed. Adaptive algorithms of nonlinear extrapolators based on the use of partitioning methods for calculating convolution are proposed.

It is shown that algorithms for the functioning of nonlinear extrapolators based on the use of partitioning methods for calculating convolution are more efficient than linear shift-invariant systems and at the same time require less memory. The features of creating adaptive filtering, interpolation and extrapolation algorithms using neurocomputers built on the basis of existing microchips of domestic and foreign production are considered.

Vinokurov A.D., Kupriyanov N.A., Makarenkov V.V., Ulyanov G.N., Shatalov A.A., Shatalova V.A., Yastrebkov A.B. Features of creation and application of multidimensional adaptive filtering algorithms in time and frequency domains. Part 3: Extrapolation. Science Intensive Technologies. 2024. V. 25. № 4. P. 12−27. DOI: https://doi.org/ 10.18127/j19998465-202404-02 (in Russian)

1. Shahtarin B.I., Kovrigin V.A. Metody spektral'nogo ocenivaniya sluchajnyh processov. M.: Goryachaya liniya – Telekom, 2011. 256 s. (in Russian).
2. Makarenkov V.V., Shatalov A.A., Shatalova V.A., Yastrebkov A.B. Adaptivnyj algoritm raspoznavaniya signalov, prinimaemyh ot medlenno i bystro fluktuiruyushchih celej na fone pomekh v mnogodiapazonnyh mnogopozicionnyh RLS s FAR. Vestnik vozdushno-kosmicheskoj oborony. 2021. № 4(32). S. 56–65 (in Russian).
3. Oppengejm A., Shafer R. Cifrovaya obrabotka signalov. M.: Tekhnosfera. 2006. 856 s. (in Russian).
4. Makarenkov V.V., Kupriyanov N.A., Liferenko V.D. i dr. Adaptivnyj algoritm formirovaniya diagrammy napravlennosti fazirovannoj antennoj reshetki s ispol'zovaniem diskretnogo razlozheniya Karunena–Loeva. Elektromagnitnye volny i elektronnye sistemy. 2023. T. 28, № 4. S. 48–56. DOI 10.18127/j5604128-202304-05 (in Russian).
5. Abas V.M.A., Arutyunyan R.V. Analiz i optimizaciya nelinejnyh sistem s pamyat'yu na osnove integro-funkcional'nyh ryadov Vol'terra i metodov Monte-Karlo. Izv. vuzov. Severo-Kavkazskij region. Ser. Tekhnicheskie nauki. 2021. № 3(211). S. 30–34. DOI 10.17213/1560-3644-2021-3-30-34 (in Russian).
6. Galiaskarova G.R., Chernov I.G. Postroenie analiticheskih reshenij interal'nyh uravnenij Vol'terra metodom stepennyh ryadov. Vysshaya shkola. 2015. № 21. S. 40–42 (in Russian).
7. Sandakov E.B. Gordeev Yu.N. Metody resheniya linejnyh differencial'nyh uravnenij i sistem s postoyannymi koefficientami. M.: NIYaU MIFI. 2013. 64 s. (in Russian).
8. Lajons R. Cifrovaya obrabotka signalov. Izd. 2-e: Per s angl. M.: OOO «Binom press». 2006. 656 s. (in Russian)
9. Dzhigan V.I. Adaptivnaya fil'traciya signalov: teoriya i algoritmy. M.: Tekhnosfera. 2013. 528 s. (in Russian)
10. Spravochnik po radiolokacii. Pod red. M.I. Skolnika: Per. s angl. pod obshch. red. V.S. Verby. V 2 kn. Kn. 1. M.: Tekhnosfera. 2014. 672 s. (in Russian)
11. Leksachenko V.A., Shatalov A.A. Sintez mnogomernogo vybelivayushchego fil'tra po metodu Grama-Shmidta. Radiotekhnika i elektronika. 1976. T. 21. № 1. S. 112 (in Russian).
12. Shatalov A.A. Adaptivnye algoritmy mnogomernoj nelinejnoj obrabotki signalov i harakteristiki effektivnosti ih raboty. Radiotekhnika i elektronika. 2000. T. 45. № 3. S. 320–326 (in Russian).
13. Makarenkov V.V., Pod"yachev V.V., Luc'ko I.S. Adaptivnyj algoritm podstrojki vesovyh koefficientov fazirovannoj antennoj reshetki po kriteriyu naimen'shih kvadratov s ispol'zovaniem lemmy ob obrashchenii kletochnyh matric. Elektromagnitnye volny i elektronnye sistemy. 2022. T. 27. № 6. S. 13–20. DOI 10.18127/j5604128-202206-02 (in Russian).
14. Lankaster P. Teoriya matric: Per. s angl. M.: GRFML. Nauka. 1978. 280 s. (in Russian)
15. Makarenkov V.V., Moroz A.V., Sahno I.V. i dr. Metodika formirovaniya diagrammy napravlennosti i rascheta otnosheniya signal-shum na vyhode sintezirovannoj antennoj reshetki ul'trazvukovogo lokacionnogo stenda v usloviyah pomekhovogo vozdejstviya. Vestnik metrologa. 2021. № 3. S. 28–33 (in Russian).
16. Nikolenko S.I., Tulup'ev A.L. Samoobuchayushchiesya sistemy. MCNMO. 2009. 288 s. (in Russian)
17. Shahnov V.A., Vlasov A.I., Kuznecov A.A., Polyakov Yu.A. Nejrokomp'yutery: Arhitektura i realizaciya. ChipNews. 2000. № 6–10 (in Russian).
18. Shvecov N.I., Bogushevich E. V. Issledovanie oblasti primeneniya PLIS v sistemah upravleniya i svyazi. Innovacionnye nauchnye issledovaniya. 2021. № 11–3(13). S. 20–25. DOI 10.5281/zenodo.5846709 (in Russian).
19. Bajgutlina I.A., Zamyatin P.A. Reshenie zadach prostranstvennogo analiza s ispol'zovaniem nejroprocessorov rossijskogo proizvodstva. Slavyanskij forum. 2022. № 1(35). S. 301–313 (in Russian).