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Parametrizations of All Wavelet Filters: Input-Output and State-Space

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Abstract

We here use notions from the theory linear shift-invariant dynamical systems to provide an explicit characterization, both practical and computable, of all rational wavelet filters. For a given N, (N ≥ 2) the number of inputs, the construction is based on a factorization to an elementary wavelet filter along with of m elementary unitary matrices. We shall call this m the index of the filter. It turns out that the resulting wavelet filter is of McMillan degree .

Moreover, beyond the parameters N and m, one confine the spectrum of the filters to lie in an open disk of radius ρ (stable filters mean ρ ∊ [0,1] and for FIR take ρ = 0). Then all filters can be described by a convex set of parameters ([0, π) × [0, 2π)2(N–1) × [0, ρ))m.

Rational wavelet filters bounded at infinity, admit state space realization. The above input-output parametrization is exploited for a step-by-step construction (where in each, the index m is increased by one) of state space model of wavelet filters.

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References

  1. D. Alpay and I. Gohberg, “Unitary rational matrix functions” In I. Gohberg, editor, Topics in interpolation theory of rational matrix-valued functions, Operator Theory: Advances and Applications, Vol. 33, pp. 175–222. Birkhäuser Verlag, Basel, 1988.

  2. D. Alpay, P. Jorgensen and I. Lewkowicz, “Extending Wavelet filters. Infinite Dimensions, the Non-Rational Case and Indefinite-Inner Product Spaces”, Excursions in Harmonic Analysis Book Series, Volume 2, Chapter 5, pp. 71–113, Springer-Birkhäuser, 2012.

  3. D. Alpay, P. Jorgensen, I. Lewkowicz and I. Marziano. “Representation Formulas for Hardy space functions through the Cuntz relations and new interpolation problems”, to appear in the volume: Multiscale Signal Analysis and Modeling (editors Xiaoping Shen and Ahmed Zayed), in the series Lecture Notes in Electrical Engineering, Springer. Also available at http://arxiv.org/abs/1104.0621

  4. R. Ashino, T. Mandai and A. Morimoto, “Applications of Wavelet Transforms to System Identification”, Operator Theory: Advances & Application, Vol. 155, pp. 203–218, Birkhäuser, 2004.

  5. M. Atteia and J. Gaches, Approximation hilbertienne: Splines, Ondelettes, Fractales, Presses Universitaires de Grenoble, 1999.

  6. L.W. Baggett, V. Furst, K.D. Merrill and J.A. Packer, “Generalized Filters, the Low-Pass Condition, and Connections to Multiresolution Analyses”, J. Funct. Analysis, Vol. 257, pp. 2760–2779, 2009.

    Article  MathSciNet  Google Scholar 

  7. J.D. Blanchard and I.A. Krishtal, “Matricial filters and crystallographic composite dilation wavelets”, Math. Comp., Vol. 81, pp. 905–922, 2012.

    Article  MathSciNet  Google Scholar 

  8. A. Benveniste, “Multiscale Signal Processing: from QMF to wavelets”, IRISA-INRIA, Publication interne No. 550, September 1990.

  9. A. Boggess and F.J. Narcowich, A First Course in Wavelets with Fourier Analysis, second edition, Wiley, 2009.

  10. O. Bratteli and P.E.T. Jorgensen, “Isometries, Shifts, Cuntz Algebras and Multiresolution Wavelet Analysis of Scale N”, Integral Equations & Operator Theory, Vol. 28, pp. 382–443, 1997.

    Article  MathSciNet  Google Scholar 

  11. O. Bratteli and P.E.T. Jorgensen, “Wavelet filters and infinite-dimensional unitary groups”, Proceedings of the International Conference on Wavelet Analysis and Applications (Guangzhou, China 1999), AMS/IP Stud. Adv. Math., Amer. Math. Soc. Vol. 25, pp. 35–65, 2002.

  12. O. Bratteli and P.E.T. Jorgensen, Wavelets through the Looking Glass, Birkhäuser, 2002.

  13. L. Chai, J. Zhang, C. Zhang and E. Mosca, “Frame-Theory-Based Analysis and Design of Oversampled Filter Banks: Direct Computational Method”, IEEE Trans. Sig. Proc., Vol. 55, pp. 507–519, 2007.

    Article  MathSciNet  Google Scholar 

  14. L. Chai, J. Zhang, C. Zhang and E. Mosca, “Efficient Computation of Frame Bounds Using LMI-Based Optimization”, IEEE Trans. Sig. Proc., Vol. 56, pp. 3029–3033, 2008.

    Article  MathSciNet  Google Scholar 

  15. L. Chai, J. Zhang, C. Zhang and E. Mosca, “Optimal Noise Reduction in Oversampled PR Filter Banks” IEEE Trans. Sig. Proc., Vol. 57, pp. 3844–3857, 2009.

    Article  MathSciNet  Google Scholar 

  16. L. Chai, J. Zhang, C. Zhang and E. Mosca, “Bound Ratio Minimization of Filter Banks Frames”, IEEE Trans. Sig. Proc., Vol. 58, pp. 209–220, 2010.

    Article  MathSciNet  Google Scholar 

  17. D. Courtney, P.S. Muhly, and S.W. Schmidt, “Composition operators and endomorphisms”, Comp. Anal. & Oper. The., Vol. 6, pp. 163–188, 2012.

    Article  Google Scholar 

  18. I. Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, Vol. 61, SIAM, 1992.

  19. D.E. Dutkay and P.E.T. Jorgensen, “Wavelets on fractals”, Rev. Mat. Iberoam., Vol. 22, pp. 131–180, 2006.

    Article  MathSciNet  Google Scholar 

  20. D.E. Dutkay and P.E.T. Jorgensen, “Methods from multiscale theory and wavelets applied to nonlinear dynamics”, Wavelets, multiscale systems and hypercomplex analysis, Oper. Theory Adv. Appl., Vol. 167, pp. 87–126, Birkhäuser, 2006.

  21. D.E. Dutkay and P.E.T. Jorgensen, “Fourier series on fractals: a parallel with wavelet theory”, Radon transforms, geometry, and wavelets, Contemp. Math. Vol. 464, pp. 75–101, Amer. Math. Soc., 2008.

  22. Y. Genin, P. Van Dooren and T. Kailath, J.M. Delosme and Martin Morf, “On Σ–Lossless Transfer Functions and related Questions”, Lin. Alg. & Appl., Vol. 50, pp. 251–275, 1983.

    Article  MathSciNet  Google Scholar 

  23. D. Gleich and M. Datcu, “Wavelet-based SAR image despeckling and information extraction, using particle filter”, IEEE Trans. Image Process., Vol. 18, pp. 2167–2184, 2009.

    Article  MathSciNet  Google Scholar 

  24. Guido, Rodrigo Capobianco, “A note on a practical relationship between filter coefficients and scaling and wavelet functions of discrete wavelet transforms”, Appl. Math. Lett., Vol. 24, pp. 1257–1259, 2011.

    Article  MathSciNet  Google Scholar 

  25. Han Bin, “Symmetric orthogonal filters and wavelets with linear-phase moments”, J. Comput. Appl. Math., Vol. 236, pp. 482–503, 2011.

    Article  MathSciNet  Google Scholar 

  26. E. Hernández and G. Weiss, A First Course on Wavelets, Studies in Advanced Mathematics, CRC Press, 1996.

  27. H.G. Hoang, H.D. Tuan and T.Q. Nguyen, “FrequencySelective KYP Lemma, IIR Filter and Filter Bank Design”, IEEE Trans. Sign. Proc., Vol. 57, pp. 956–965, 2009.

    Article  Google Scholar 

  28. R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985.

  29. P.E.T. Jorgensen, “Dense analytic subspaces in fractal L 2–spaces”, J. Anal. Math., Vol. 75, pp. 185–228, 1998.

    Article  MathSciNet  Google Scholar 

  30. P.E.T. Jorgensen, “Matrix Factorizations, Algorithms, Wavelets”, Notices of the American Mathematical Society, Vol. 50, pp. 880–894, 2003.

    MathSciNet  MATH  Google Scholar 

  31. P.E.T. Jorgensen, Analysis and probability: wavelets, signals, fractals, Graduate Texts in Mathematics, Vol. 234, Springer, 2006.

  32. P.E.T. Jorgensen, “Use of operator algebras in the analysis of measures from wavelets and iterated function systems”, Operator theory, operator algebras, and applications, Contemp. Math., Vol. 414, pp. 13–26, Amer. Math. Soc., 2006.

  33. P.E.T. Jorgensen, “Unitary matrix functions, wavelet algorithms, and structural properties of wavelets”, Gabor and wavelet frames, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., Vol. 10, pp. 107–166, World Sci. Publ., Hackensack, NJ, 2007.

  34. P.E.T. Jorgensen, K.D. Merrill and J.D. Packer (editors) Representations, wavelets, and frames, A celebration of the mathematical work of Lawrence W. Baggett, Applied and Numerical Harmonic Analysis, Papers from the conference held in Boulder, CO, USA, May 18–20, 2006, Birkhäuser, 2008.

  35. P.E.T. Jorgensen and M-S Song, “Analysis of fractals, image compression, entropy encoding, Karhunen-Lo`eve transforms”, Acta Appl. Math., Vol. 108, pp. 489–508, 2009.

    Article  MathSciNet  Google Scholar 

  36. T. Kailath, Linear Systems, Prentice-Hall, 1980.

  37. N.S. Larsen, and I. Raeburn, “From filters to wavelets via direct limits”, In: Operator theory, operator algebras, and applications, Contemp. Math., Vol. 414, pp. 35–40, Am. Math. Soc., 2006.

  38. Lin, Jianyu and Smith, Mark J. T., “New perspectives and improvements on the symmetric extension filter bank for subband/wavelet image compression”, IEEE Trans. Image Process., Vol. 17, pp. 177–189, 2008.

    Article  MathSciNet  Google Scholar 

  39. S. Mallat, A Wavelet Tour of Signal Processing, 3rd Edition, Academic Press, 2009.

  40. C.A. Micchelli and Q. Sun, “Interpolating filters with prescribed zeros and their refinable functions”, Commun. Pure Appl. Anal., Vol. 6, pp. 789–808, 2007.

    Article  MathSciNet  Google Scholar 

  41. H. Olkkonen, and J.T. Olkkonen, “Shifted B-spline interpolation filter”, Curr. Dev. Theory Appl. Wavelets, Vol. 4, pp. 289–305, 2010.

    MathSciNet  MATH  Google Scholar 

  42. S. Oraintara, T.D. Tran, P.N. Heller and T.Q. Nguyen, “Lattice structure for regular para-unitary linear-phase filterbanks and M-band orthogonal symmetric wavelets”, IEEE Trans. Sig. Proc., Vol. 49, pp. 2659–2672, 2001.

    Article  Google Scholar 

  43. Raeburn, Iain, “From filters to wavelets via direct limits. II. Wavelet-packet bases”, Acta Appl. Math., Vol. 108, pp. 509–514, 2009.

    Article  MathSciNet  Google Scholar 

  44. E.D. Sontag, Mathematical Control Theory - Deterministic Finite Dimensional Systems, Springer, 1990.

  45. P. Steffen, P.N. Heller, R.A. Gopinath and C.S. Burrus, “Theory of Regular M-Band Wavelet Bases”, IEEE Trans. Sig. Proc., Vol. 41, pp. 3497–3511, 1993.

    Article  Google Scholar 

  46. G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, 1996

  47. N.C. Suganya and G. A. Vijayalakshmi Pai, , “Evolution Based Hopfield Neural Network with Wavelet Based Filter for Complex-Constrained Portfolio Optimization”, Dyn. Contin. Discrete Impuls. Syst. Ser. B Appl. Algorithms, Vol. 17, pp. 175–205, 2010.

    MathSciNet  MATH  Google Scholar 

  48. J. Tuqan and P.P. Vaidyanathan, “A State Space Approach to the Design of Globally Optimal FIR Energy Compaction Filters”, IEEE Trans. Sig. Proc., Vol. 48, pp. 2822–2838, 2000.

    Article  Google Scholar 

  49. P.P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice-Hall, Signal Processing Series, 1993.

  50. H. Vikalo, B. Hassibi, A. Erdogan and T. Kailath, “On Robust Signal Reconstruction in Noisy Filter Banks”, Eurasip Signal Processing, Vol. 85, pp. 1–14, 2005.

    Article  Google Scholar 

  51. B. Wahlberg, “Orthogonal Rational Functions: A Transformation Analysis”, Siam Review, Vol. 45, pp. 689–705, 2003.

    Article  MathSciNet  Google Scholar 

  52. N.F.D Ward and J.R. Partington, “Rational Wavelet Decompositions of Transfer Functions in Hardy-Sobolev Classes”, Math. Cont. Sig. & Sys, Vol. 8, pp. 257–278, 1995.

    Article  MathSciNet  Google Scholar 

  53. P. Wojtaszczyk, A Mathematical Introduction to Wavelets, London Mathematical Society, Student Texts 37, Cambridge University Press, 1997.

  54. Z. Xu and A. Makur, “Frame-Theoretic Analysis of Robust Filter Bank Frames to Quantization and Erasures”, IEEE Trans. Sig. Proc., Vol. 58, pp. 3531–3544, 2010.

    Article  MathSciNet  Google Scholar 

  55. Yang, Guoan and Zheng, “Nanning, An Optimization Algorithm for biorthogonal Wavelet Filter Banks Design”, Int. J. Wavelets Multiresolut. Inf. Process., Vol. 6, pp. 51–63, 2008.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Ben-Gurion University of the Negev, Be’er Sheva, 84105, Israel

    Daniel Alpay

  2. Department of Mathematics, The University of Iowa, Iowa City, IA, 52242-1419, USA

    Palle Jorgensen

  3. Department of Electrical Engineering, Ben-Gurion University of the Negev, Be’er Sheva, 84105, Israel

    Izchak Lewkowicz

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Alpay, D., Jorgensen, P. & Lewkowicz, I. Parametrizations of All Wavelet Filters: Input-Output and State-Space. STSIP 12, 159–188 (2013). https://doi.org/10.1007/BF03549566

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