A.Yu. Grishentsev – Dr.Sc.(Eng.), Associate Professor, ITMO University (Saint Petersburg)

E-mail: grishentcev@yandex.ru

In the work produced by the study of the autocorrelation and fractal properties of matrices and multidimensional arrays, linear unitary Fourier transform. In telecommunication and radar, the use of such matrices, relevant for the decision of problems of synthesis and recognition of wideband signals hidden in the noise of the radio. Therefore, the search for and study of matrices with a special form of the autocorrelation function is the concern not only of theoretical studies but the practical task of development.

The objects of study are matrices, and multidimensional arrays linear unitary Fourier transform and the broadband signals received on the basis of matrices of linear unitary Fourier transform. The subject of the study are autocorrelation and fractal properties of the matrix linear unitary Fourier transform and some properties of the autocorrelation of broadband signals received on the basis of matrices of linear unitary Fourier transform. The structure of the work. The work contains four sections: «Introduction»; «Correlation and fractal properties of the matrix Fourier transform»; «Synthesis of wideband signals and coding based on the matrix Fourier transform»; «Finally, the conclusions». In the first section «Introduction» the goals and objectives of the study, the basic mathematical apparatus of the confining area of research and defines the formal theoretical aspects of the study. In the second section «Correlation and fractal properties of the matrix Fourier transform» formulated and proved the theorem about the autocorrelation properties of the matrix linear unitary Fourier transforms are investigated and systematized fractal patterns that occur when increasing the space dimension and perform the conversions on the basis of matrices of linear unitary Fourier transform. In the third section, «Synthesis of wideband signals and coding based on the matrix of Fourier transformation» produced a scoping study of the signals obtained on the basis of matrices of linear unitary Fourier transform. In the fourth section, «Conclusion of conclusions» conclusions and opinions on the work.

The results of the research. Studies have shown that the autocorrelation function matrix of a linear unitary Fourier transform is a single impulse. Formulated and proved the theorem about the autocorrelation properties of the matrix linear unitary Fourier transform. The autocorrelation function matrix of a linear unitary Fourier transform has the form of a two-dimensional discrete single pulse. Identified and systematized fractal patterns that occur when increasing the space dimension and perform the conversions on the basis of matrices of linear unitary Fourier transform. A brief review of the comparative analysis of the use of matrices of linear unitary Fourier transform for synthesis of wideband correlation of signals. In conclusion, the work concludes. In the work produced by the study of the autocorrelation properties of matrices linear unitary Fourier transform. It is shown that the autocorrelation function matrix of a linear unitary Fourier transform is a single impulse. Formulated and proved the theorem about the autocorrelation properties of the matrix linear unitary Fourier transform. Identified and systematized fractal patterns that occur when increasing the space dimension and perform the conversions on the basis of matrices of linear unitary Fourier transform. A brief overview of the use of linear unitary matrices Fourier transform for synthesis of wideband correlation of signals. The obtained results can find application in solving problems of radio communication and radar, if needed, to transmit and receive radio signals hidden in the noise of the ether. The work is performed on the basis of the Federal state Autonomous educational institution of higher professional education «SaintPetersburg national research University of information technologies, mechanics and optics» (ITMO University) in the framework of research methods for the synthesis and recognition of signals hidden in the noise of the radio (radiotherapy).

1. Arslan X., Chen Chzh. N., Benedetto M. Sverxshirokopolosnaya besprovodnaya svyaz’. M.: Texnosfera. 2012. 640 s.
2. Goldsmit A. Besprovodny’e kommunikaczii. M.: Texnosfera. 2011. 904 s.
3. By’vshev M.E., Izvekov A.Yu., Kabakov I.V. Optimal’ny’j priem signalov na fone pomex i shumov / Pod red. Yu.I. Savvateeva. M.: Radiotexnika. 2011. 424 s.
4. Ipatov V. Shirokopolosny’e sistemy’ i kodovoe razdelenie signalov. Princzipy’ i prilozheniya. M.: Texnosfera. 2007. 488 s.
5. Bendat Dzh., Pirsol A. Primeneniya korrelyaczionnogo i spektral’nogo analiza: Per. s angl. M.: Mir. 1983. 312 s.
6. Grishenczev A.Yu. Sposob sinteza i primenenie shumopodobny’x shirokopolosny’x signalov v zadachax organizaczii zashhishhenny’x kanalov svyazi // Radiotexnika. 2017. № 9. S. 91−101.
7. Novikov S.P., Tajmanov I.A. Sovremenny’e geometricheskie struktury’ i polya. M.: MCzNMO. 2005. 584 s.
8. Pontryagin L.S., Novikov S.P., Smejl S.O. i dr. Topologicheskaya biblioteka. Tom III. Spektral’ny’e posledovatel’nosti v topologii // Moskva-Izhevsk: Institut komp’yuterny’x issledovanij. 2005. 640 s.
9. Dubrovin B.A., Novikov S.P., Fomenko A.T. Sovremennaya geometriya: Metody’ i prilozheniya. T. 1. Geometriya poverxnostej, grupp preobrazovanij i polej // Izd. 6-e. M.: URSS: Knizhny’j dom «LIBROKOM». 2013. 336 s.
10. Grishenczev A.Yu., Korobejnikov A.G. Ponizhenie razmernosti prostranstva pri korrelyaczii i svertke czifrovy’x signalov // Izvestiya VUZov. Priborostroenie. 2016. T. 59. № 3. S. 211−218.
11. Sheluxin O.I., Tenyakshev A.M., Osin A.V. Fraktal’ny’e proczessy’ v telekommunikacziyax / Pod red. O.I. Sheluxina. M.: Radiotexnika. 2003. 480 s.
12. Grishenczev A.Yu., Korobejnikov A.G. Primenenie nekotory’x vejvletov dlya generaczii shirokopolosny’x signalov // Izvestiya VUZov. Priborostroenie. 2017. T. 60. № 8. S. 712−720.
13. Raevskij A.S., Raevskij S.B. Kompleksny’e volny’. M.: Radiotexnika. 2010. 224 s.
14. Yarly’kov M.S. Meandrovy’e i shumopodobny’e signaly’ (VOS-signaly’) i ix raznovidnosti v sputnikovy’x radionavigaczionny’x sistemax.M.: Radiotexnika. 2017. 416 s.
15. Grishenczev A.Yu., Korobejnikov A.G. Algoritm poiska, nekotory’e svojstva i primenenie matricz s kompleksny’mi znacheniyami e’lementov dlya steganografii i sinteza shirokopolosny’x signalov // Zhurnal radioe’lektroniki. 2016. № 5. 9 s. URL = http://jre.cplire.ru/jre/may16/11/text.pdf.
16. Grishenczev A.Yu., Korobejnikov A.G., Velichko E.N., Nepomnyashhaya E’.K., Rozov S.V. Sintez binarny’x matricz dlya formirovaniya signalov shirokopolosnoj svyazi // Radiotexnika. 2015. № 9. S. 51−58