Gabor and Wavelet Expansions

  • Christopher Heil
  • David Walnut
Part of the NATO ASI Series book series (ASIC, volume 315)

Abstract

This paper is an examination of techniques for obtaining Fourier series-like expansions of finite-energy signals using so-called Gabor and wavelet expansions. These expansions decompose a given signal into time a frequency localized components. The theory of frames in Hilbert spaces is used as a criteria for determining when such expansions are good representations of the signals. Some results on the existence of Gabor and wavelet frames in the Hilbert space of all finite-energy signals are presented.

Keywords

Hilbert Space Orthonormal Basis Wavelet Coefficient Mother Wavelet Wavelet Frame 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [Bal]
    R. Balian, Un principe d’incertitude fort en théorie du signal on en mécanique quan- tique, C. r. Acad. Sei. Paris 292 (1981), 1357–1362 MathSciNetGoogle Scholar
  2. [Bat]
    G. Battle, Heisenberg proof of the Balian-Low theorem, Lett. Math. Phys. 15 (1988), 175–177. MathSciNetCrossRefGoogle Scholar
  3. [Ben]
    J. Benedetto, “Gabor representations and wavelets”, Commutative Harmonic Analysis, D. Colella, ed., Contemp. Math. 19, American Mathematical Society, Providence, 1989, pp. 9–27 Google Scholar
  4. [BHW]
    J. Benedictto, C. Heil, D. Walnut, Remarks on the proof of the Balian theorem, preprint.Google Scholar
  5. [BCR]
    G. Beylkin, R. Coifman, V. Rokhlin, Fast wavelet transforms and numerical algorithms, I, preprint.Google Scholar
  6. [Dl]
    I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appi. Math. 41 (1988), 909–996 MathSciNetMATHCrossRefGoogle Scholar
  7. [D2]
    I. Daubechies Time-frequency localization operators: a geometric phase space approach, IEEE Trans. Inform. Theory 34 (1988), 605–612 MathSciNetMATHCrossRefGoogle Scholar
  8. [D3]
    I. Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory, to appear.Google Scholar
  9. [DGM]
    I. Daubechies, A. Grossmann,Y. Meyer, Painless nonrthogonol expansins, J. Math. Phys. 27 (1986), 1271–1283 MathSciNetMATHCrossRefGoogle Scholar
  10. [DJ]
    I. Daubechies, A. J. E. M. Janssen, Two theorems on lattice expansions, IEEE Trans. Inform. Theory, to appear.Google Scholar
  11. [DS]
    R. J. Duffin, A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366 MathSciNetMATHCrossRefGoogle Scholar
  12. [Gab]
    D. Gabor, Theory of communications, J. Inst. Elee. Eng. (London) 93 (1946), 429–457 Google Scholar
  13. [GG]
    I. Gohberg, S. Goldberg, “Basic Operator Theory”, Birkhäuser, Boston, 1981.Google Scholar
  14. [Gro]
    A Grossmman, “Wavelet transforms and edge detection”, Stochastic Processing in Physics and Engineering, S. Albeverio et al., eds., D. Reidel, Dordrecht, the Netherlands, 1988, pp. 149–157 Google Scholar
  15. [GGM]
    P. Goupillaud, A. Grossmann, J. Morlet, Cycle-octave and related transforms in seismic signal analysis, Geoexploration 23 (1984/85), 85–102 CrossRefGoogle Scholar
  16. [GM]
    A. Grossmman, J. Morlet, Decomposition of Hardy functions into square integrable wavelets of constant shape, SIAM J. Math. Anal. 15 (1984), 723–736 MathSciNetCrossRefGoogle Scholar
  17. [H1]
    C. Heil, Generalized harmonic analysis in higher dimensions; Weyl-Heisenberg frames and the Zok transform, Ph.D. thesis, University of Maryland, College Park, MD, 1990.Google Scholar
  18. [H2]
    C. Heil Wavelets and frames, Proceedings of the Institute for Mathematics and Its Applications, to appear.Google Scholar
  19. [HW]
    C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Review 34 (1989) Google Scholar
  20. [Jl]
    A. J. E. M. Janssen, Bargmann transform, Zak transform, and coherent states, J. Math. Phys. 23 (1982), 720–731 MathSciNetMATHCrossRefGoogle Scholar
  21. [J2]
    ,A. J. E. M. Janssen The Zak transform: a signal transform for sampled time-continuous signals, Philips J. Res. 43 (1988), 23–69 MathSciNetMATHGoogle Scholar
  22. [KMG]
    R. Kronland-Martinet, J. Morlet, A. Grossmann, Analysis of sound patterns through wavelet transforms, Internat. J. Pattern Recog. Artif. Int. 1 (1987), 273–302 CrossRefGoogle Scholar
  23. [LM]
    P. LemariIÉ, Y. Meyer, Ondelettes et bases hilbertiennes, Rev. Mat. Iberoamericana 2(1986), 1–18 MathSciNetGoogle Scholar
  24. [L]
    F. Low, “Complete sets of wave packets”, A Passion for Physics—Essays in Honor of Geoffrey Chew, World Scientific, Singapore, 1985, pp. 17–22 Google Scholar
  25. [Mal]
    S. Mallat, A theory for multiresolution signal decomposition: the wavelet representation, IEEE Trans, on Pattern Anal, and Machine Intel. 11 (1989), 674–693MATHCrossRefGoogle Scholar
  26. [Mey]
    Y. Meyer, Principe d’incertitude, bases hilbertiennes et algebres d’opérateurs, Seminaire Bourbaki 662 (1985–86) Google Scholar
  27. [RR]
    T. Rado, P. Reichelderfer, “Continuous Transformations in Analysis”, Springer- Verlag, Berlin, New York, 1955 Google Scholar
  28. [S]
    W. Schempp, äRadar ambiguity functions, the Heisenberg group, and holomorphic theta series, Proc. Amer. Math. Soc. 92 (1984), 103–110 MathSciNetMATHCrossRefGoogle Scholar
  29. [W]
    D. WALNUT, Weyl-Heisenberg wavelet expansions: existence and stability in weighted spaces, Ph.D. thesis, University of Maryland, College Park, MD, 1989.Google Scholar
  30. [Y]
    R. YOUNG, “An Introduction to Nonharmonic Fourier Series”, Academic Press, New York, 1980.Google Scholar
  31. [Z]
    J. Zak, Finite translations in solid state physics, Phys. Rev. Lett. 19 (1967), 1385–1397 CrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1990

Authors and Affiliations

  • Christopher Heil
    • 1
  • David Walnut
    • 1
  1. 1.The MITRE CorporationMcLeanUSA

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