Walsh functions and Gray codes
DOI:
https://doi.org/10.18372/2410-7840.15.4774Keywords:
system of the Walsh functions, indicator matrix, Gray codesAbstract
The paper developed a method of synthesis of symmetric systems Walsh based on their indicator matrices (the method is defined as a direct challenge Walsh) and calculating the indicator matrix of these systems (the inverse problem Walsh). The order of the indicator matrix is a logarithmic function of the base 2 of the binary-rational order systems Walsh. Introduced matrix forms a complete set of simple Gray codes. The set contains the classical direct and inverse transform (called left-Gray codes) and a new class of right transformations Gray, supplemented by the operator to maintain the original codeword (identity matrix) and the matrix inverse permutation. Proposed composite Gray codes, which are the multiplicative combination of an arbitrary set of simple codes. The relationship of simple and symmetrical composite Gray codes with indicator matrices of appropriate systems of Walsh functionsReferences
Залманзон Л.А. Преобразования Фурье, Уолша, Хаара и их применение в управлении, связи и других областях. / Л.А. Залманзон. – М.: Наука, 1989. – 496 с.
Карповский М.Г. Спектральные методы анализа и синтеза дискретных устройств. / М.Г. Карповский, Э.С. Москалев. – Ленинград: Энергия, 1973. – 142 с.
Никитин Г.И. Применение функций Уолша в сотовых системах связи с кодовым разделением каналов. / И.Г. Никитин. – Санкт-Петербург: СПбГУАП, 2003. – 86 с.
Hadamard Н.J. Résolution d'une question relative aux déterminants. “Bull. Sci. Math.” 17, 1893, р. 240-246.
Walsh J.L. A closed set of normal orthogonal functions. – “Amer. J. Math.”, 1923, v. 45, p. 5-24.
Артемьев М.Ю. Алгоритм формирования симметричных систем функцій Уолша / М.Ю. Артемьев, Г.П. Гаев, Т.Э. Кренкель, А.П. Скотников // Радиотехника и электроника, 1978, № 7. – С. 1432-1440.
Paley R.E. A remarkable series of orthogonal functions. – “Proc. London Math. Soc.”, 1932, v. 34, p. 241-279.
Зеленков А.В. О формировании симметрических систем функций Виленкина – Крестенсона. / А.В. Зеленков // Радиотехника и электроника, 1982, № 5. – С. 921-929.
Трахтман А.М. Основы теории конечных сигналов на конечных интервалах. / А.М. Трахтман, В.А. Трахтман. – М.: Сов. радио, 1975. – 208 с.
Kaczmarz S., Steinhaus H. Theorie der ortogonalreihen. – Warszava-Lvov, 1935. – 508 p.
Yen C. Walsh functions and Grey code. IEEE Trans., 1971. EMC-13, № 3, p. 68-73.
Gray F. Pulse code communication. – Pat. USA, № 2632058, 1953.
Курош А.Г. Лекции по общей алгебре. / А.Г. Курош – М.: Наука, ГРФМЛ, 1973. – 400 с.
Белецкий А.Я. Преобразования Грея. / А.Я. Белецкий, А.А. Белецкий, Е.А. Белецкий. Монография в двух томах. – К.: Книжное изд-во НАУ, 2007. – Т. 1. Основы теории. – 412 с. – Т. 2. Прикладные аспекты. – 644 с.
Белецкий А.Я. Syntesis and analysis of system of Wolsh-Cooly basis functions. / А.Я. Белецкий. – Материалы МК: NIKON-2000: XIII International Conference. – Wroclaw, 2000.
Cooley J.W., Tukey J.W. An algorithm for the machine computation of complex Fourier series. – Math. Comp., 1965, v. 19, p. 297-301.
Блейхут Р. Теория и практика кодов, контролирующих ошибки. / Р. Блейхут. – М.: Мир, 1986. – 576 с.
Zalmanzon L.A. Fourier, Walsh, Haar and their application in management, communication and other areas. / L. Zalmanzon, Moscow: Nauka, 1989, 496 p.
Karpovskiy M.G. Spectral methods of analysis and synthesis of discrete devices. / MG Karpovskiy, ES Moskalev. - Leningrad: Energy, 1973, 142 p.
Nikitin GI Application of Walsh functions in cellular communication systems, code division tion channels. / IG Nikitin. - St. Petersburg: SPbSUAI, 2003, 86 p.
Hadamard Н.J. Résolution d'une question relative aux déterminants. “Bull. Sci. Math.” 17, 1893, р. 240-246.
Walsh J.L. A closed set of normal orthogonal functions. – “Amer. J. Math.”, 1923, v. 45, p. 5-24.
Artemyev, M. The algorithm for generating symmetric systems funktsіy Walsh / M. Artemyev, GP Guai, TE Ernst, AP Cattlemen / / Technology and Electronics, 1978, № 7., p. 1432-1440.
Paley R.E. A remarkable series of orthogonal functions. – “Proc. London Math. Soc.”, 1932, v. 34, p. 241-279.
Zelenkov A.V. On the formation of symmetric systems of functions Vilenkin - a cross-son. / A. Zelenkov / / Technology and Electronics, 1982, № 5., pp. 921-929.
Trahtman A.M. Fundamentals of the theory of finite signals on finite intervals. / A.M. Trahtman. - Moscow: Sov. radio, 1975, 208 p.
Kaczmarz S., Steinhaus H. Theorie der ortogonalreihen. – Warszava-Lvov, 1935, 508 p.
Yen C. Walsh functions and Grey code. IEEE Trans., 1971. EMC-13, № 3, p. 68-73.
Gray F. Pulse code communication. – Pat. USA, № 2632058, 1953.
Kurosh AG Lectures on general algebra. / AG Kourosh - Nauka, GRFML, 1973, 400 p.
Beletsky A.Ya. Conversions Gray. / A.Ja Beletsky, A.A. Beletsky, E.A. Beletsky. The monograph is in two volumes. - K.: Book publishing house NAU, 2007, T. 1. Fundamentals of the theory, 412 p., T. 2. Applied aspects., 644 p.
Downloads
Published
Issue
Section
License
Authors who publish with this journal agree to the following terms:- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
