Abstract
The Fourier transform is a classical method in mathematical modeling of systems. Assuming finite non-Abelian groups as the underlying mathematical structure might bring advantages in modeling certain systems often met in computer science and information technologies. Frequent computing of the inverse Fourier transform is usually required in dealing with such systems. These computations require for each function value to compute many times traces of certain matrices. These matrices are products of matrix-valued entries of unitary irreducible representations and matrix-valued Fourier coefficients. In the case of large non-Abelian groups the complexity of these computations can be a limiting factor in applications. In this paper, we present a method for speeding-up computing the traces by using decision diagrams to operate on matrix-valued group representations and related Fourier coefficients.
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References
Astola, J.T., Stanković, R.S.: Fundamentals of Switching Theory and Logic Design. Springer, Heidelberg (2006)
Berghammer, R., Leoniuk, B., Milanese, U.: Implementation of relational algebra using binary decision diagrams. In: de Swart, H. (ed.) RelMiCS 2001. LNCS, vol. 2561, pp. 241–257. Springer, Heidelberg (2002)
Butler, J.T.: Bent function discovery by reconfigurable computer. In: Proc. 9th Int. Workshop on Boolean Problems, Freiberg, Germany, September 16-17, pp. 1–12 (2010)
Miller, D.M., Stanković, R.S.: An Heterogeneous decision diagram. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds.) EUROCAST 2009. LNCS, vol. 5717, pp. 540–547. Springer, Heidelberg (2009)
Miller, D., Thornton, M.: “QMDD: A decision diagram structure for reversible and quantum circuits. In: Proc. Int. Symp. on Multiple-Valued Logic (CD), p. 6 (2006)
Roziner, T.D., Karpovsky, M.G., Trachtenberg, L.A.: Fast Fourier transform over finite groups by multiprocessor systems. IEEE Trans. Acoust., Speech, Signal Processing ASSP-38(2), 226–240 (1990)
Sasao, T., Fujita, M. (eds.): Representations of Discrete Functions. Kluwer Academic Publishers, Dordrecht (1996)
Stanković, S., Astola, J.: XML Framework for various types of decision diagrams for discrete functions. IEICE - Transactions on Information and Systems E90-D(11), 1731–1740 (2007)
Stanković, R.S., Moraga, C., Astola, J.T.: Fourier Analysis on Finite Non-Abelian Groups with Applications in Signal Processing and System Design. Wiley/IEEE Press (2005)
Stanković, S., Astola, J.T., Miller, D.M., Stanković, R.S.: Heterogeneous decision diagrams for applications in Harmonic Analysis on finite non-Abelian groups. In: Proc. 40th Int. Symp. on Multiple-Valued Logic, Barcelona, Spain, May 26-18, pp. 307–312. IEEE-CS-Press, Los Alamitos (2010)
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Stanković, R.S., Astola, J.T., Moraga, C., Stanković, S. (2012). Remarks on Efficient Computation of the Inverse Fourier Transforms on Finite Non-Abelian Groups. In: Moreno-Díaz, R., Pichler, F., Quesada-Arencibia, A. (eds) Computer Aided Systems Theory – EUROCAST 2011. EUROCAST 2011. Lecture Notes in Computer Science, vol 6928. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27579-1_37
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DOI: https://doi.org/10.1007/978-3-642-27579-1_37
Publisher Name: Springer, Berlin, Heidelberg
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