Document Type : Research Paper
Authors
1 Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics, University of Vienna, Vienna, Austria.
2 Faculty of Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.
Abstract
The article introduces cyclic dilation groups and finite affine groups for prime integers, and as an application of this theory it presents a unified group theoretical approach for the cyclic wavelet transform (CWT) of prime dimensional periodic signals.
Keywords
[1] S. T. Ali, J. P. Antoine and J. P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, New York, 2000.
[2] J. P. Antoine, R. Murenzi and P. Vendergheynst, S. T. Ali, Two Dimensional Wavelets and Their Relatives, Cambridge University Press, Cambridge. 2003.
[3] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter bands, Appl. Comput. Harmon. Anal., 5 (1998) 389-427.
[4] J. J. Benedetto and O. Treiber. Wavelet frames: Multiresolution analysis and extension principles. Debnath, Lokenath, Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhauser. Applied and Numerical Harmonic Analysis, 3-36 (2001)., 2001.
[5] G. Caire, R. L. Grossman and H. Vincent Poor, Wavelet transforms associated with nite cyclic Groups, IEEE Transaction On Information Theory, Vol. 39, No. 4, July 1993.
[6] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
[7] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992.
[8] H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, Gabor analysis and algorithms, 123-170, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
[9] H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Gabor analysis and algorithms, 233-266, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
[10] H. G. Feichtinger and T. Strohmer, Advances in Gabor Analysis, Series: Applied and Numerical Harmonic Analysis, Birkhuser 2002.
[11] H. Feichtinger and F. Luef. Gabor analysis and time-frequency methods, Encyclopedia of Applied and Computational Mathematics, 2012.
[12] H. G. Feichtinger, T. Strohmer, and O. Christensen. A group-theoretical approach to Gabor analysis, Opt. Eng., 34 (1697) 1704, 1995.
[13] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC press, 1995.
[14] D. Gabor, Theory of communication, JIEEE, 93(26), Part III, 1964, 429-457.
[15] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations I. general results, J. Math. Phys., 26 (10) (1985) 2473-2479
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[16] K. Grochenig, Foundation of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 2001.
[17] K. Grochenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and Algorithms, 211-231, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 1998.
[18] G. H. Hardy and E.M.Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press (1979).
[19] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, 1963.
[20] R. Reiter and J. D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.
[21] S. Mallat, A Wavelet Tour of Signal Processing, 2nd Edition, Academic Press, 1999.
[22] S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315 (1989), 69-87.
[23] G. L. Mullen and D. Panario, Handbook of Finite Fields, Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.
[24] G. Pfander, Gabor Frames in Finite Dimensions. In G. E. Pfander, P. G. Casazza, and G. Kutyniok, editors, Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
[25] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, Vol. 10 Academic Press, San Diego, 1999.
[26] K. Flornes, A. Grossmann, M. Holschneider and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1 (1994) 137-146.
[27] H. Riesel, Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhuser, (1994).
[28] T. Strohmer., Numerical algorithms for discrete Gabor expansions, Gabor Analysis and Algorithms: Theory and Applications, pages 267{294. Birkhauser Boston, Boston, 1998.
[29] T. Strohmer. A Uni ed Approach to Numerical Algorithms for Discrete Gabor Expansions, In Proc. SampTA - Sampling Theory and Applications, Aveiro/Portugal, pages 297{302, 1997.
[30] S. Sarkar, H. Vincent Poor, Cyclic Wavelet Transforms for Arbitrary Finite Data Lengths, Signal Processing, 80 (2000) 2541-2552.
[31] G. Strang and T. Nguyen, Wavelets and Filter Banks,Wellesley-Cambridge Press, Wellesley, MA, 1996.
[32] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, 1995.
[33] M.V. Wickerhauser, Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, Department of Mathematics, 1992.
[34] M.W. Wong, Discrete Fourier Analysis, Pseudo-di erential Operators Theory and applications Vol. 5, Birkhauser 2010.
[35] Z. Zhang. Periodic wavelet frames. Adv. Comput. Math., 22 (2) (2005) 165-180.
[2] J. P. Antoine, R. Murenzi and P. Vendergheynst, S. T. Ali, Two Dimensional Wavelets and Their Relatives, Cambridge University Press, Cambridge. 2003.
[3] J. J. Benedetto and S. Li, The theory of multiresolution analysis frames and applications to filter bands, Appl. Comput. Harmon. Anal., 5 (1998) 389-427.
[4] J. J. Benedetto and O. Treiber. Wavelet frames: Multiresolution analysis and extension principles. Debnath, Lokenath, Wavelet transforms and time-frequency signal analysis. Boston, MA: Birkhauser. Applied and Numerical Harmonic Analysis, 3-36 (2001)., 2001.
[5] G. Caire, R. L. Grossman and H. Vincent Poor, Wavelet transforms associated with nite cyclic Groups, IEEE Transaction On Information Theory, Vol. 39, No. 4, July 1993.
[6] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
[7] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992.
[8] H. G. Feichtinger and G. Zimmermann, A Banach space of test functions for Gabor analysis, Gabor analysis and algorithms, 123-170, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
[9] H. G. Feichtinger and W. Kozek, Quantization of TF lattice-invariant operators on elementary LCA groups, Gabor analysis and algorithms, 233-266, Applied and Numerical Harmonic Analysis, Birkhauser, Boston, MA 1998.
[10] H. G. Feichtinger and T. Strohmer, Advances in Gabor Analysis, Series: Applied and Numerical Harmonic Analysis, Birkhuser 2002.
[11] H. Feichtinger and F. Luef. Gabor analysis and time-frequency methods, Encyclopedia of Applied and Computational Mathematics, 2012.
[12] H. G. Feichtinger, T. Strohmer, and O. Christensen. A group-theoretical approach to Gabor analysis, Opt. Eng., 34 (1697) 1704, 1995.
[13] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC press, 1995.
[14] D. Gabor, Theory of communication, JIEEE, 93(26), Part III, 1964, 429-457.
[15] A. Grossmann, J. Morlet and T. Paul, Transforms associated to square integrable group representations I. general results, J. Math. Phys., 26 (10) (1985) 2473-2479
.
[16] K. Grochenig, Foundation of Time-Frequency Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 2001.
[17] K. Grochenig, Aspects of Gabor analysis on locally compact abelian groups, Gabor analysis and Algorithms, 211-231, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, MA 1998.
[18] G. H. Hardy and E.M.Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press (1979).
[19] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol. 1, 1963.
[20] R. Reiter and J. D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.
[21] S. Mallat, A Wavelet Tour of Signal Processing, 2nd Edition, Academic Press, 1999.
[22] S. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315 (1989), 69-87.
[23] G. L. Mullen and D. Panario, Handbook of Finite Fields, Series: Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.
[24] G. Pfander, Gabor Frames in Finite Dimensions. In G. E. Pfander, P. G. Casazza, and G. Kutyniok, editors, Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
[25] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, Vol. 10 Academic Press, San Diego, 1999.
[26] K. Flornes, A. Grossmann, M. Holschneider and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1 (1994) 137-146.
[27] H. Riesel, Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhuser, (1994).
[28] T. Strohmer., Numerical algorithms for discrete Gabor expansions, Gabor Analysis and Algorithms: Theory and Applications, pages 267{294. Birkhauser Boston, Boston, 1998.
[29] T. Strohmer. A Uni ed Approach to Numerical Algorithms for Discrete Gabor Expansions, In Proc. SampTA - Sampling Theory and Applications, Aveiro/Portugal, pages 297{302, 1997.
[30] S. Sarkar, H. Vincent Poor, Cyclic Wavelet Transforms for Arbitrary Finite Data Lengths, Signal Processing, 80 (2000) 2541-2552.
[31] G. Strang and T. Nguyen, Wavelets and Filter Banks,Wellesley-Cambridge Press, Wellesley, MA, 1996.
[32] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, 1995.
[33] M.V. Wickerhauser, Lectures on Wavelet Packet Algorithms, Technical Report, Washington University, Department of Mathematics, 1992.
[34] M.W. Wong, Discrete Fourier Analysis, Pseudo-di erential Operators Theory and applications Vol. 5, Birkhauser 2010.
[35] Z. Zhang. Periodic wavelet frames. Adv. Comput. Math., 22 (2) (2005) 165-180.
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