Authors A.M. Makarov, A.S. Ermakov
Month, Year 11, 2015 @en
Index UDC 621.37; 575.3; 519-21.519-27
Abstract To date, one of the unsolved problems of the theory of signal detection is the problem of synthesis of optimal signal detection in noise with completely unknown autocorrelation function. Especially true solution of this problem is the detection of complex wideband signals. In this article we propose an approach based on the synthesis of the matched filter in the signal space of the Mellin integral transform of the original additive mixture of signal and noise. As shown by the authors in a previous article this option appears on the basis of the fact that the correlation function of the random processes after integral Mellin transform is invariant to the form of the original autocorrelation function. The article presents the basic mathematical equations to prove the invariance property and is the main theorem, the results of which form the basis of this work. The proof shows that the value of the real part of the basic core functions of the Mellin transform based on the fundamental equality in signal processing theory – Parseval"s equality. Its value is equal to Ѕ. Next, using the classic mathematical apparatus of optimum synthesized in integral space of Mellin transform we matched filter algorithm and its structure. Block diagrams of the optimum matched filter is also provided. As an example, consider the spectral representation of harmonic vibrations in the basis of the integral Mellin transform. It is shown that the spectral density of the signal is permanent basis in the frequency spectrum Mellin. Changing the frequency of the harmonic signal source causes a change in amplitude of the response at the output of the Mellin transform. Received a spectral representation of a segment of harmonic oscillation having a finite duration. Thus, there is an opportunity for further study of the properties of the optimal filter for signals with a relative phase telegraphy. Phase shift keyed signals are widely used in most modern data transmission systems. These signals have the highest noise immunity among others manipulated signals. In general it can be concluded on the establishment of a new scientific direction in the theory of the development of a new generation of information transfer systems, possessing the property of invariance to the form of the correlation function of noise.

Download PDF

Keywords Mellin transform; aprioristic indeterminacy of the correlation function of the noise; optimal matched filters; power spectrum density; Parseval equality; theorem of Wiener-Hinchey.
References 1. Akimov P.S., Evstratov F.F., Zakharov S.I. i dr. Obnaruzhenie radiosignalov [Detection of radio signals], Ed. by A.A. Kolosova. Moscow: Radio i svyaz', 1989, 288 p.
2. Van Tris G. Teoriya obnaruzheniya, otsenok i modulyatsii [Theory of detection, assessment and modulation]. Vol. 1. Moscow: Sovetskoe radio, 1972, 744 p.
3. Gonorovskiy I.S. Radiotekhnicheskie tsepi i signaly [Radio circuits and signals]. Part 1. Moscow: Sovetskoe Radio, 1966, 438 p.
4. Gonorovskiy I.S. Radiotekhnicheskie tsepi i signaly [Radio circuits and signals]. Part 2. Moscow: Sovetskoe Radio, 1967, 327 p.
5. Kotel'nikov V.A. Teoriya potentsial'noy pomekhoustoychivosti [Theory of potential noise immunity]. Moscow: Sovetskoe radio, 1956, 152 p.
6. Makarov A.M. Vzaimosvyaz' avtokorrelyatsionnoy funktsii statsionarnykh sluchaynykh protsessov v bazise preobrazovaniya Fur'e so spektral'noy plotnost'yu moshchnosti v bazise preobrazovaniya Mellina (analog teoremy Vinera-Khinchina) [Interrelation of autocorrelated function of stationary casual processes in basis of the Furye transformation from the spectral density of power in basis of the Mellin transformation (analog of viner-hinchin theorem)], Izvestiya YuFU. Tekhnicheskie nauki [Izvestiya SFedU. Engineering Sciences], 2014, No. 11 (160), pp. 52-57.
7. Makarov A.M. Primenenie integral'nogo preobrazovaniya Mellina v issledovanii svoystv gamma funktsiy [The application of the integral Mellin transform in the study of the properties of gamma functions], Materialy Mezhdunarodnoy molodezhnoy NK «Matematicheskie funktsii i ee prilozheniya (MFP-12) v ramkakh federal'noy tselevoy programmy «Nauchnye i nauchno-
pedagogicheskie kadry informatsionnoy Rossii, Pyatigorsk, SKFU 2012-2013» [proceedings of the International youth NC "Mathematical function and its applications (MFP-12) in the framework of the Federal target program "scientific and Scientific-pedagogical personnel of information of Russia, Pyatigorsk, NCFU in 2012-2013"]. Vol. 1, pp. 100-106.
8. Makarov A.M. Spektral'noe predstavlenie garmonicheskikh signalov v bazise integral'nogo preobrazovaniya Mellina [Spectral representation of harmonic signals in the basis of the integral of the Mellin transform], Upravlenie i informatsionnye tekhnologii: Mezhvuzovskiy nauchnyy sbornik [Management and information technologies: interuniversity scientific collection]. Pyatigorsk: Reklamno-informatsionnoe agentstvo na KMV, 2010, 216 p.
9. Tikhonov V.I. Statisticheskaya radiotekhnika [Statistical radio engineering]. 2nd. ed. Moscow: Radio i svyaz', 1982, 624 p.
10. Frenks L. Teoriya signalov [Signal theory]: Translation from English, Ed. by D.E. Vakmana. Moscow: Covetskoe radio, 1974, 344 p.
11. Shapiro D.A. Uravneniya v chastnykh proizvodnykh. Spetsial'nye funktsii. Asimptotiki. Konspekt lektsiy po matematicheskim metodam fiziki [Equation in partial derivatives. Special functions. The asymptotics. Lecture notes on mathematical methods of physics]. Novosibirsk: Novosibirskiy gosudarstvennyy universitet, 2004, 122 p.
12. Oberhettinger F. Tables of Mellin Transforms. Springer-Verlag Berlin Heidelberg New York, 1974, 278 p.
13. Antonio de Sena, Davide Rocchesso. The Mellin Pizzicator. Proc. of 9 Int. Conference on Digital Audio Effects(DAFx-06). Montreal, Canada. September 18-20. 2006.
14. Norbert Soullonol, Gerol Boumonn. Of the Mellin transforms of dirac’s delta dunction, the Housdorff dimension function, and the theorem by Mellin, Fractional Calculus Applied Analysis, 2004, Vol. 7, No. 4.
15. Bertrand J., Bertrand P., Ovarlez j. The Mellin Transform. The Transformsand Applications Handbook: Second Edition. Ed. Alexander D. Poularikas. Boca Raton: CRCPress LLC, 2000.
16. Philippe Flajolet, Xavier Gourdon, Philippe Dullas. Mellin transforms and asymtotics: Harmonic sums, Theoretical Computer Science, 1995, No. 144, pp. 3-38.
17. Ovarlez J., Bertrand P., Bertrand P. Computation of offine time – frequency distributions using the Fast Mellin transform. Proc IEEE – ICASSP. 1992.
18. Sheng Y., Arsenault H. Experiments on pattern recognition using invariant Fourier – Mellin descriptors, J. Opt. Soc. Am., 1986, No. 3 (6), pp. 885-887.
19. Glanni Pagnini, Yang Quon Chen. Mellin convolution for signal filtering and ITC application to the Gaussianization of Lewy noise. Proceedings of the ASME 2011, International Design Engineering Technical Conferences & Computers and Information in Engineering Conference IDETC/CIE 2011, August 26-31, 2011, Washington DC, USA.
20. Shuvanov R.I. Raspoznavanie obrazov na tsifrovykh izobrazheniyakh na osnove teorii invariantov [Pattern recognition on digital images based on invariant theory], Vestnik MGTU im. N.E. Baumana. Ser. Estestvennye nauki (modelirovanie v informatike) [Herald of the Bauman Moscow State Technical University. Series Natural Sciences (modeling in computer science)], 2012, pp. 158-165.

Comments are closed.