The Wavelet Transform, Time-Frequency Localization and Signal Analysis

IEEE Transactions on Information Theory

Volumn 36, Issue 5, 1990, Pages 961-1005

  Daubechies, Ingrid ^a  

^a UNIVERSITY OF MICHIGAN   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

MATHEMATICAL TRANSFORMATIONS--FOURIER TRANSFORMS;

LOCALIZATION THEOREMS; SIGNAL ANALYSIS; TIME-FREQUENCY LOCALIZATION; WAVELET TRANSFORM; WINDOWED FOURIER TRANSFORM;

SPECTRUM ANALYSIS;

EID: 0025482241     PISSN: 00189448     EISSN: 15579654     Source Type: Journal    
DOI: 10.1109/18.57199     Document Type: Article

References (71)

1. D. Gabor, “Theory of communication,” J. Inst. Elect. Eng. (London), vol. 93, no. III, pp. 429–457, 1946.
2. M. J. Bastiaans, “Gabor’s signal expansion and degrees of freedom of a signal,” Proc. IEEE, vol. 68, pp. 538–539, 1980. ——, “A sampling theorem for the complex spectrogram and Gabor’s expansion of a signal in Gaussian elementary signals,” Optical Eng., vol. 20, pp. 594–598, 1981.
3. A. J. E. M. Janssen, “Gabor representation of generalized functions,” J. Math. Appl., vol. 80, pp. 377–394, 1981. ——, “Gabor representation and Wigner distribution of signals,” in Proc. IEEE, 1984, pp. 41.B.2.1–41.B.2.4.
4. M. J. Davis and E. J. Heller, J. Chem. Phys., vol. 71, pp. 3383–3395, 1979.
5. J. R. Klauder and E. Sudarshan, Fundamentals of Quantum Optics, New York: Benjamin, 1968.
6. R. J. Glauber, “The quantum theory of coherence,” Phys. Rev., vol. 130, pp. 2529–2539, 1963; “Coherent and incoherent states of the radiation field,” Phys. Ret., vol. 131, pp. 2766–2788, 1963.
7. J. R. Klauder and B.-S. Skagerstam, “Coherent states,” Singapore: World Scientific, 1985.
8. E. Lieb, “Thomas-Fermi theory and related theories of atoms and molecules,” Rei. Mod. Phys., vol. 53, pp. 603–641, 1981.
9. G. A. Hagedorn, “Semiclassical quantum mechanics I: The ̄→ limit for coherent states,” Comm. Math. Phys., vol. 71, pp. 77–93, 1980.
10. C. W. Helstrom, “An expansion of a signal in Gaussian elementary signals,” IEEE Trans. Inform. Theory, vol. 12, pp. 81–82, 1966.
11. N. G. de Bruijn, “Uncertainty principles in Fourier analysis,” in Inequalities, O. Shisha, Ed. New York: Academic Press, pp. 57–71, 1967.
12. T. A. C. M. Claassen and W. F. G. Mecklenbrauker, “The Wigner distribution—A tool for time-frequency signal analysis,” Philips J. Res., vol. 35, pp. 217–250, 276–300, 372–389, 1980.
13. C. P. Janse and A. J. M. Kaiser, “Time-frequency distributions of loudspeakers: The application of the Wigner distribution,” J. Audio Eng. Soc, vol. 31, pp. 198–223, 1983.
14. R. J. Duffin and A. C. Schaeffer, “A class of nonharmonic Fourier series,” Trans. Am. Math. Soc, vol. 72, pp. 341–366, 1952.
15. R. M. Young, An Introduction to Nonharmonic Fourier Series. New York: Academic Press, 1980.
16. I. Daubechies, A. Grossmann, and Y. Meyer, “Painless non-orthogonal expansions,” J. Math. Phys., vol. 27, pp. 1271–1283, 1986.
17. A. Grossmann and J. Morlet, “Decomposition of Hardy functions ito square integrable wavelets of constant shape,” SIAM J. Math. Anal., vol. 15, pp. 723–736, 1984. P. Goupillaud, A. Grossmann, and J. Morlet, “Cycle-octave and related transforms in seismic signal analysis,” Geoexploration, vol. 23, p. 85, 1984.
18. A. Grossmann, J. Morlet, and T. Paul, “Transforms associated to square integrable group representations,” J. Math. Phys., vol. 26, pp. 2473–2479, 1985; Ann. Inst. Henri Poincaré, vol. 45, 293–309, 1986.
19. E. W. Aslaksen and J. R. Klauder, “Unitary representations of the affine group,” J. Math. Phys., vol. 9, pp. 206–211, 1968; “Continuous representation theory using the affine group,” J. Math. Phys., vol. 10, pp. 2267–2275, 1969.
20. T. Paul, “Functions analytic on the half-plane as quantum mechanical states,” J. Math. Phys., vol. 25, pp. 3252–3263, 1984.
21. ——, “Affine coherent states and the radial Schrodinger equation. I. Radial harmonic oscillator and hydrogen atom,” to be published.
22. C. Fefferman and R. de la Llave, “Relativistic stability of matter,” Rev. Math. Iberoamericana, vol. 2, pp. 119–213, 1986.
23. I. Daubechies, “Time-frequency localization operators—A geometric phase space approach,” IEEE Trans. Inform. Theory, vol. 34, pp. 605–612, 1988.
24. I. Daubechies and T. Paul, “Time-frequency localization operators —A geometric phase space approach II. The use of dilations,” Inverse Problems, vol. 4, pp. 661–680, 1988.
25. J. Morlet, G. Arens, I. Fourgeau, and D. Giard, “Wave propagation and sampling theory,” Geophys., vol. 47, pp. 203–236, 1982; “Sampling theory and wave propagation,” NATO ASI Series, vol. 1, Issues in Acoustic Signal/Image Processing and Recognition, C. H. Chen, Ed. Berlin: Springer, pp. 233–261.
26. R. Kronland-Martinet, J. Morlet, and A. Grossmann, “Analysis of sound patterns through wavelet transforms,” preprint CTP-87/p. 1981, Centre de Physique Theorique, CNRS, Marseille, France, 1987.
27. A. Grossmann, “Wavelet transforms and edge detection,” to be published in Stochastic Processes in Physics and Engineering, Ph. Blanchard, L. Streit, and M. Hasewinkel, Eds.
28. A. Grossmann, M. Holschneider, R. Kronland-Martinet, and J. Morlet, “Detection of abrupt changes in sound signals with the help of wavelet transforms,” in Inverse Problems: An Interdisciplinary Study, Advances in Electronics and Electron Physics, Supp. 19. New York: Academic, 1987.
29. S. Mallat, “A theory for multiresolution signal decomposition,” Ph.D. thesis, Univ. of Pennsylvania, Depts. of Elect. Eng. and Comput. Sci., 1988.
30. A. Grossmann and J. Morlet, “Decomposition of functions into wavelets of constant shape and related transforms,” in Mathematics and Physics, Lectures on Recent Results. Singapore: World Scientific, 1985.
31. Y. Meyer, “Principe d’incertitude, bases hilbertiennes et algebres d’operateurs,” Séminaire Bourbaki, nr. 662, 1985–1986.
32. P. G. Lemarie and Y. Meyer, “Ondelettes et bases hilbertinennes,” Rev. Math. Iberoamericana, vol. 2, pp. 1–18, 1986.
33. G. Battle, “A block spin construction of ondelettes. Part I: Lemarie functions,” Comm. Math. Phys., vol. 110, pp. 601–615, 1987.
34. P. G. Lemarie, “Une nouvelle base d’ondelettes de L2(∝n),” to be published in J. de Math. Pures et Appl.
35. a) Ph. Tchamitchian, “Calcul symbolique sur les operateurs de Caldéron-Zygmund et bases inconditionnelles de L2(∝n),” C. R. Acad. Sc. Paris, vol. 303, série 1, pp. 215–218, 1986; b) Ph. Tchamitchian, “Biorthogonalité et théorie des operateurs,” to be published in Rev. Math. Iberoamericana.
36. S. Mallat, “Multiresolution approximation and wavelets,” Trans. Am. Math. Soc, vol. 135, pp. 69–88, 1989.
37. Y. Meyer, “Ondelettes, function splines, et analyses graduées,” Univ. of Torino, 1986.
38. S. Mallat, “A Theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., vol. 31, pp. 674–693, 1989.
39. I. Daubechies, “Orthonormal basis of compactly supported wavelets,” Comm. Pure Applied Math., vol. 41, pp. 909–996, 1988.
40. S. G. Mallat, “Multifrequency channel decompositions of images and wavelet models,” IEEE Trans. Acoust. Speech Signal Processing, vol. 37, no. 12, pp. 2091–2110, Dec. 1987.
41. G. Battle and P. Federbush, “Ondelettes and phase cell cluster expansions: A vindication,” Comm. Math. Phys., 1987.
42. I. Daubechies, “Discrete sets of coherent states and their use in signal analysis,” in Proc. 1986 Int. Conf. Diff. Equations Math. Phys., Birmingham, AL, Springer Lecture Notes in Mathematics nr. 1285, pp. 73–82.
43. I. Daubechies and T. Paul, “Wavelets—Some applications,” Proc. Int. Conf. Math. Phys., M. Mebkhout and R. Seneor, Eds. Singapore: World Scientific, 1987.
44. I. Daubechies, “How to frame Bargmann and Hardy—Frame bounds for Hilbert spaces of entire functions,” unpublished memorandom, AT&T Bell Laboratories, 1987.
45. I. Daubechies and A. Grossmann, “Frames of entire functions in the Bargmann space,” Comm. Pure Appl. Math., vol. 41, pp. 151–164, 1988.
46. R. R. Coifman and R. Rochberg, “Representation theorems for holomorphic and harmonic functions in Lp,” Asterisque, vol. 77, pp. 11–66, 1980.
47. S. Janson, J. Peetre, and R. Rochberg, “Hankel forms and the Fock space,” Uppsala Univ., Math, report, 1986.
48. H. G. Feichtinger and K. Grochenig, “A unified approach to atomic decompositions via integrable group representations,” in Proc. Function Spaces Applications, Lund, 1986, to appear in Springer Led. Notes in Math. no. 1302, pp. 52–73; “Banach spaces related to integrable group representations and their atomic decomposition I, II, and III,” to be published, J. Funct. Anal., vol. 83, 1989.
49. M. Rieffel, “Von Neumann algebras associated with pairs of lattices in Lie groups,” Math. Ann., vol. 257, pp. 403–418, 1981.
50. G. Kaiser, “A sampling theorem for signals in the joint time-frequency domain,” to be published.
51. R. Balian, “Un principe d’incertitude fort en théorie du signal on en mécanique quantique,” C. R. Acad. Sc. Paris, vol. 292, série 2, 1981.
52. F. Low, “Complete sets of wave-packets,” in A Passion for Physics — Essays in Honor of Geoffrey Chew,” pp. 17–22. Singapore: World Scientific, 1985.
53. H. Elbrond Jensen, T. Hoholdt, and J. Justesen, “Double séries representation of bounded signals,” IEEE Trans. Inform. Theory, vol. 34, pp. 613–624, 1988.
54. J. Zak, “Finite translations in solid state physics,” Phys. Rev. Lett., vol. 19, pp. 1385–1397, 1967; “Dynamics of electrons in solids in external fields,” Phys. Rev., vol. 168, pp. 686–695; “The kq-representation in the dynamics of electrons in solids,” Solid State Phys., vol. 27, pp. 1–62, 1972.
55. W. Schempp, “Radar ambiguity functions, the Heinsenberg group, and holomorphic theta séries,” Proc. of the Am. Math. Soc, vol. 92, 1984.
56. M. Reed and B. Simon, Methods and Modem Mathematical Physics. IV Analysis of Operators. New York: Academic Press, 1978.
57. M. Boon, J. Zak, and J. Zucker, “Rational von Neumann lattices,” J. Math. Phys., vol. 24, pp. 316–323, 1983.
58. J. Zak, “Lattice operators in crystals for Bravais and reciprocal vectors,” Phys. Rev., vol. B 12, pp. 3023–3026, 1975.
59. A. J. E. M. Janssen, “Bargmann transform, Zak transform, and coherent states,” J. Math. Phys., vol. 23, pp. 720–731, 1982.
60. A. J. E. M. Janssen, “The Zak transform: A signal transform for sampled time-continuous signals,” Philips J. Res., vol. 42, 1987.
61. V. Bargmann, P. Butera, L. Girardello, and J. R. Klauder, “On the completeness of coherent states,” Rep. Math. Phys., vol. 2, pp. 221–228, 1971.
62. A. M. Perelomov, “Note on the completeness of systems of coherent states,” Teor. i. Matem. Fis., vol. 6, pp. 213–224, 1971.
63. H. Bacry, A. Grossmann, and J. Zak, “Proof of the completeness of lattice states in the kq-representation,” Phys. Rev., vol. B 12, pp. 1118–1120, 1975.
64. M. Reed and B. Simon, Methods of Modem Mathematical Physics, II. Fourier Analysis and Self-Adjointness. New York: Academic Press, 1975.
65. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty,” I, Bell. Syst. Tech. J., vol. 40, pp. 43–64, 1961; H. J. Landau and H. O. Pollak, II, Bell. Syst. Techn. J., vol. 40, pp. 65–84, 1961; H. J. Landau and H. O. Pollak, III, Bell. Syst. Techn. J., vol. 41, pp. 1295–1336, 1962.
66. D. Slepian, “On bandwidth,” Proc. IEEE, vol. 64, pp. 292–300, 1976.
67. A. Annéodo, G. Grasseau, and M. Holschneider, “Wavelet transform analysis of invariant measures for some dynamical systems,” Phys. Rec. Lett., vol. 61, pp. 2281–2287, 1988.
68. J. Ville, “Theorie et applications de la notion de signal analytique,” Revue Cables et Transmissions, vol. 1, pp. 61–74, 1948.
69. G. Battle, “Heisenberg proof of the Balian-low theorem,” Lett. Math. Phys., vol. 15, pp. 175–177, 1988.
70. I. Daubechies and A. J. E. M. Janssen, “Two theorems on lattice expansions,” submitted for publication.
71. H. Landau, “On the density of phase space functions,” submitted for publication.