Generalized Functions and Hartley Transforms in Number Systems with a Permanent Base

To solve the problems of digital signal processing in the spectral domain, a new method for analytical synthesis of discrete real parametric basis functions is proposed that uses the Hartmut procedure generalization to the case of number systems with an arbitrary constant-value basis. The mathematical description of functions obtained is given; their basic properties and methods for construction of different orthogonal-basis systems and transformations on their basis are investigated. In terms of these functions, all the basic theorems of spectral analysis used in the theory and practice of digital processing are formulated and proved. The obtained results are original and form the basis of theory of the signal representation and conversion in the new orthogonal basis.

References

[1] Ifeachor E.C, Jervis B.W. Digital Signal Processing: A Practical Approach. 2nd Ed. 2002. 933 p. (Russ. Ed.: Ayficher E., Dzhervis B. Tsifrovaya obrabotka signalov: prakticheskiy podkhod. 2-e izd. Moscow, St. Petersburg., Kiev, Izd. dom "Vil’yams" Publ., 2004. 992 p.)

[2] Zalmanzon L.A. Preobrazovaniya Fur’e, Uolsha, Khaara i ikh primenenie v upravlenii, svyazi i drugikh oblastyakh [Fourier, Walsh and Haar transforms, and their application in control circuit, communication and other fields]. Moscow, Nauka Publ., 1989. 496 p.

[3] Kravchenko V.F. Tsifrovaya obrabotka signalov i izobrazheniy v radiofizicheskikh prilozheniyakh [Digital signal and image in radiophysical applications]. Moscow, Fizmatlit Publ., 2007. 554 p.

[4] Trakhtman A.M., Trakhtman V.A. Osnovy teorii diskretnykh signalov na konechnykh intervalakh [The heart of the theory of discrete signals on finite intervals]. Moscow, Sovetskoe Radio Publ., 1975. 208 p.

[5] Trakhtman A.M. Vvedenie v obobshchennuyu spektral’nuyu teoriyu signalov [Introduction to the generalized spectrum theory of signals]. Moscow, Sovetskoe Radio Publ., 1972. 352 p.

[6] Syuzev V.V. The energy spectra of signals in the Vilenkin-Christenson functions basis invariant to the cyclic shift. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Estestv. Nauki [Herald of the Bauman Moscow State Tech. Univ., Nat. Sci.], 2007, no. 2, pp. 64-77 (in Russ.).

[7] Syuzev V.V. Association functional of spectra in the complex exponential and Vilenkin-Christenson functions bases. Inzhenernyy zhurnal: nauka i innovatsii [Eng. J.: Science and Innovation], 2013, iss. 11. pp. 1-7 (in. Russ.). Sc. Educ: El. Sc. Techn. Publ. Available at: http://engjournal.ru/catalog/it/hidden/1051.html (accessed 22.02.2014).

[8] Syuzev V.V. Presentation methods and signal conversion in the generalized Christenson functions basis. Nauka i obrazovanie: elektronnoe nauchno-tekhnicheskoe izdanie [SCIENCE & EDUCATION: El. Sc. Techn. Periodical], 2012, no. 3, pp. 1-28 (in Russ.). Available at: http://technomag.edu.ru/doc/372760.html (accessed 22.02.2014).

[9] Vlasenko V.A., Lappa Yu.M., Yaroslavskiy L.P. Metody sinteza bystrykh algoritmov svertki i spektral’nogo analiza signalov [Synthesis technique of fast algorithms of fold and spectral analysis of signals]. Moscow, Nauka Publ., 1990. 180 p.

[10] Bracewell R. Hartley Transform: theory and applications. New York: Oxford University Press, 1986. (Russ. Ed.: Braysuell R. Preobrazovanie Khartli Moscow, Mir Publ., 1990. 175 p.)

[11] Syuzev V.V. Theoretical foundations of spectral analysis in the Hartley basis. Nauka i obrazovanie: elektronnoe nauchno-tekhnicheskoe izdanie [SCIENCE & EDUCATION: El. Sc. Techn. Periodical], 2011, no. 10, pp. 1-47. Available at: http://technomag.edu.ru/doc/230816.html (accessed 22.02.2014) (in Russ.).

[12] Dwight H.B. Tables of Integrals and Other Mathematical Data. 3th Ed. The Macmillan Co., N.Y., 1947. 198 p. (Russ. Ed.: Dvayt G.B. Tablitsy integralov i drugie matematicheskie formuly. Moscow, Nauka Publ., 1969. 227 p.)

[13] Syuzev V.V. Scalar method for the synthesis of the fast Walsh-Hadamard transform. Vestn. Mosk. Gos. Tekh. Univ. im. N.E. Baumana, Priborostr., Spetsvyp. "Informatsionnye tekhnologii i komp’yuternye sistemy" [Herald of the Bauman Moscow State Tech. Univ., Instrum. Eng., Spec. Issue "Information technologies and computer systems"], 2011, pp. 128-137 (in Russ.).