Study on the concept of the generalized convolution

• Autorzy: Korohoda P. , Dąbrowski A.

Warianty tytułu

PL

Studium pojęcia splotu uogólnionego

• Konferencja: Signal Processing ; 07.09.2007 ; Poznań, Poland
• Języki publikacji: EN

Abstrakty

EN

In the paper study of the generalized convolution, called also the A-convolution, is described, indicating that this approach may be very promising from the point of view of its applications. The basic idea consists in the observation that filtering of the finite length discrete signal using a filter defined in the transform domain of any linear transform can be performed in the primary domain. After defining and developing the concept, examples are provided as hints for selection among 20 discrete transforms: Haar, Hadamard, Hartley, Karhunen-Loeve, and all 16 Discrete Trigonometric Transforms. The main motivation for the study has been to provide tools for the discrete-domain signal processing in the way similar to that developed and commonly used for the Fourier analysis, but allowing for application of any other discrete transform, appropriately selected for a given task.

PL

Artykuł zawiera studium pojęcia splotu uogólnionego, nazwanego również splotem A. Wykazano, że tak rozumiane pojęcie może być przydatne w wielu aplikacjach. Podstawowa idea polega na obserwacji, że filtracja dyskretnego skończonego sygnału za pomocą filtru określonego w dziedzinie dowolnej transformacji liniowej może być zrealizowana w dziedzinie pierwotnej. Po określeniu zaproponowanych pojęć przedstawiono przykłady, które służą do ilustracji problemu wyboru między 20 transformacjami: Haara, Hadamarda, Hartleya, Karhunena-Loeve i 16 transformacjami trygonometrycznymi. Głównym celem tej pracy było opracowanie narzędzi do przetwarzania sygnałów dyskretnych w sposób podobny do tego, jaki jest stosowany w przypadku analizy Fouriera, ale pozwalający na użycie innych, odpowiednio dobranych transformacji dyskretnych.

Słowa kluczowe

EN

generalized convolution; Haar; Hadamard; Hartley; Kar-hunen-Loeve; DTT transform; DCT transform; DST transform

PL

splot uogólniony; transformacje Haara; Karhunena-Loeve; DTT; DCT; DST

• Wydawca: Wydawnictwo SIGMA-NOT
• Czasopismo: Elektronika : konstrukcje, technologie, zastosowania
• Rocznik: 2007
• Tom: Vol. 48, nr 4
• Strony: 16--20
• Opis fizyczny: Bibliogr. 20 poz., tab.

Twórcy

autor

Korohoda P.

autor

Dąbrowski A.

• University of Science and Technology (AGH), Institute of Electronics, Cracow, Poland

Bibliografia

• [1] Aizenberg N. N., Trofimiuk O. T.: Shift, convolution and correlation finction of the discrete signals in an arbitrary basis. (In Russian) Rep. Acad. Science USSR (Doklady Akademii. Nauk SSSR) vol. 250, no 1., pp. 47-51, 1980.
• [2] Aizenberg I., Astola J. T.: Discrete generalized Fresnel functions and transform in an arbitrary discrete basis. IEEE Trans. on signal processingin Proc., vol. 54, no 11, pp. 4261-4270, Nov. 2006.
• [3] Aizenberg N. N.: Spectrum of the convolution of the discrete signals in an arbitrary basis. (In Russian) Rep. Acad. Science USSR (Doklady Akademii. Nauk SSSR) vol. 241, no 3, pp. 551-554, 1978.
• [4] Bovik A. (ed.): Handbook of Image and Video Processing. Academic Press, 2000.
• [5] Bracewell R. N.: The Fourier Transform and Its Applications. McGraw-Hill, 2000.
• [6] Clarke R. J.: Transform Coding of Images. Academic Press, 1985.
• [7] Korohoda P., Dąbrowski A.: Generalized primary domain interpretation of product filtering of digital signals in the transform domain. Proc. of the IEEE Workshop on Signal Processing, (Poznań, Poland), pp. 75-80, Oct. 2003.
• [8] Korohoda P., Dąbrowski A.: Generalized convolution concept based on DCT. 12th European Signal Processing Conference - EUSIPCO'04, (Vienna, Austria), pp. 973-976, Sept. 2004.
• [9] Korohoda P.: Generalized convolution with windowing in the primary domain. Proc. of the 11th International Workshop on Systems, Signals and Image Processing, (Poznań, Poland), pp. 383-386, Sept. 2004.
• [10] Korohoda P., Dąbrowski A.: Generalized convolution concept for filtering of random signals. Proc. of the IEEE Workshop on Signal Processing, (Poznań, Poland), pp. 55-60, Sept. 2004.
• [11] Korohoda P.: Equivalent of impulse response notion for selected discrete transformations. (In Polish) Automatyka, Semiannual of the AGH University of Science and Technology, (Cracow, Poland), vol. 8, no 3, pp. 351-359, 2004.
• [12] Korohoda P.: Discussion of possible adjustment of the adaptive filtering algorithms description to selected discrete transformation. (In Polish), Automatyka, Semiannual of the AGH University of Science and Technology, (Cracow, Poland), vol. 9, no 3, pp. 401-407, 2005.
• [13] Korohoda P., Dąbrowski A.: Generalized convolution concept extension to multidimensional signals. Proc. of the IEEE Workshop on Multidimensional Signal Processing, (Wuppertal, Germany), pp. 187-192, July 2005.
• [14] Korohoda P., Dąbrowski A.: Approximation of the generalized convolution - case study example. Proc. of the IEEE Workshop on Signal Processing, (Poznań, Poland), pp. 41-44, Sept. 2005.
• [15] Madisetti V. K., Williams D. B. (ed.): The Digital Signal Processing Handbook. CRC Press & IEEE Press, 1998.
• [16] Margrave G. F.: Theory of nonstationary linear filtering in the Fourier domain with application to time-variant filtering. Geophysics, vol. 63, no 1, pp. 244-259, 1998.
• [17] Martucci S. A: Symmetric convolution and the discrete sine and cosine transforms. IEEE Transactions on Signal Processing, vol. 42, no 5, pp. 1038-1051, May 1994.
• [18] Mitra S. K.: Digital Signal Processing: A Computer Based Approach. McGraw-Hill, 1998.
• [19] Proakis J. G., Manolakis D. G.: Digital Signal Processing: Principles, Algorithms and Applications. 3nd ed. Prentice Hall, 1996.
• [20] Vetterli M., Kovacevic J.: Wavelets and Subband Coding. Prentice Hall, 1995.