The Weyl Correspondence and Time-Frequency Analysis

IEEE Transactions on Signal Processing

Volumn 42, Issue 2, 1994, Pages 318-331

  Parks, T W ^b  

^a SCHLUMBERGER   (United States)

^b Department of Obstetrics Gynecology   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

ACOUSTIC SURFACE WAVE FILTERS; FOURIER TRANSFORMS; FREQUENCY DOMAIN ANALYSIS; FREQUENCY RESPONSE; FUNCTION EVALUATION; LEAST SQUARES APPROXIMATIONS; MATHEMATICAL OPERATORS; SIGNAL DETECTION; TIME SERIES ANALYSIS; WHITE NOISE;

SHORT TIME FOURIER TRANSFORM (STFT); TIME-FREQUENCY DOMAIN; WEYL CORRESPONDENCE;

ACOUSTIC SIGNAL PROCESSING;

EID: 0028378193     PISSN: 1053587X     EISSN: 19410476     Source Type: Journal    
DOI: 10.1109/78.275605     Document Type: Article

References (79)

1. M. G. DeBruijn, “Uncertainty principles in Fourier analysis,” in Inequalities (O. Shisa, Ed.). New York: Academic, 1967, pp. 57–71.
2. D. Slepian and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-1,” AT&T Bell Labs. Tech. J., vol. 40., pp. 43–63, Jan. 1961.
3. H. J. Landau and H. O. Pollak, “Prolate spheroidal wave functions, Fourier analysis and uncertainty-2,” AT&T Bell Labs. Tech. J., vol. 40, pp. 65–84, Jan. 1961.
4. “Prolate spheroidal wave functions, Fourier analysis and un-certainty-3: The dimension of the space of essentially time- and band-limited signals,” AT&T Bell Labs. Tech. J., vol. 41, pp. 1295–1336, Apr. 1962.
5. I. Daubechies, “Time-frequency localization operators: A geometric phase space approach,” IEEE Trans. Inform. Theory, vol. 34, pp. 605–612, July 1988.
6. T. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distri-bution-A tool for time-frequency analysis-Part 1: Continuous time signals,” Philips J. Res., vol. 35, pp. 217–250, 1980.
7. P. Flandrin, “Maximum signal energy concentration in a time-frequency domain,” in Proc. IEEE ICASSP ‘88, 1988, pp. 2176–2179.
8. F. Hlawatsch and W. Kozek, “Time-frequency analysis of linear signal spaces,” in Proc. ICASSP ‘91, May 1991, pp. 2045–2048.
9. H. Weyl, The Theory of Groups and Quantum Mechanics. London: Methuen, 1931, 1952.
10. R. Howe, “On the role of the Heisenberg group in harmonic analysis,” Bull. Amer. Math. Soc. (N.S.), vol. 3, pp. 821–843, Sept. 1980.
11. __, “Quantum mechanics and partial differential equations,” J. Funct. Anal., vol. 38, pp. 188–254, 1980.
12. “A symbolic calculus for nilpotent groups,” in Operator Algebras and Group Representations: Vol. 1 (G. Arsene, S. Stratila, A. Verona, and D. Voiculescu, Eds.). New York: Pitman, 1984, pp. 254–277.
13. A. A. Kirillov, Elements of the Theory of Representations. New York: Springer-Verlag, 1976.
14. H. I. Choi and W. J. Williams, “Improved time-frequency representation. of multicomponent signals using exponential kernels,” IEEE Trans. Acoust. Speech Signal Processing, vol. 37, p. 1, 1989.
15. A. H. Nuttall, “Wigner distribution function: Relation to short-term spectral estimation, smoothing, and performance in noise,” NUSC Tech. Rep. 8225, Naval Underwater Syst. Cent., Feb. 1988.
16. The Wigner distribution function with minimum spread,” NUSC Tech. Rep. 8317, Naval Underwater Syst. Cent., June 1988.
17. D. J. Thomson, “Spectrum estimation and harmonic analysis,” Proc. IEEE, vol. 70, pp. 1055–1096, 1982.
18. W. Kozek and F. Hlawatsch, “Time-frequency filter banks with perfect reconstruction,” in Proc. ICASSP ‘91, May 1991, pp. 2049–2052.
19. “Time-frequency representation of linear time-varying systems using the Weyl symbol,” in Proc. IEE Sixth Int. Conf.Digital Signal Processing Commun., Sept. 1991.
20. W. Kozek, “Time-frequency signal processing based on the Wigner-Wey1 framework,” EURASIP Signal Processing, vol. 29, pp. 77–92, Oct. 1992.
21. A. J. E. M. Janssen, “Wigner weight functions and Weyl symbols of non-negative definite linear operators,” Philips J. Res., vol. 44, pp. 2049–2052, 1989.
22. J. Ramanathan and P. Topiwala, “Time frequency localization via the Weyl correpsondence,” SIAM J. Math. Anal., vol. 24, no. 5, pp. 1378–1393, Sept. 1993.
23. L. Cohen, “Generalized phase space distributions,” J. Math. Phys., vol. 7, pp. 781–786, May 1966.
24. __, “Quantization problem and variational principle in the phase-space space formulation of quantum mechanics,” J. Math. Phys., vol. 17, pp. 1863–1866, Oct. 1976.
25. Time-frequency distributions-A review,” Proc. IEEE, vol. 77, pp. 941–981, July 1989.
26. M. O. Scully and L. Cohen, “Quasi-probability distributions for arbitrary operators,” in Phys. Phase Space: Proc. First Int. Conf. Phase Space, 1987, pp. 253–260.
27. L. Cohen, “A general approach for obtaining joint representations in signal analysis and an application to scale,” in Proc. SPIE Adv. Signal Processing, Algorithms Implementations 2, 1991, pp. 109–133, vol. 1566.
28. __, “Scale and inverse frequency representations,” in Role Wavelets Signal Processing Represent., pp. 118–129, Mar. 1992.
29. __, “A general approach for obtaining joint representations in signal analysis,” to be published.
30. __, “The scale representation,” to be published.
31. J. Kohn and L. Nirenberg, “An algebra of pseudodifferential operators,” Comm. Pure Appl. Math., vol. 18, pp. 169–305, 19 6 5.
32. L. Hormander, “The Weyl calculus of pseudodifferential operators,” Comm. Pure Appl. Math., vol. 32, pp. 359–443, 1979.
33. C. L. Fefferman, “The uncertainty principle,” Bull. Amer. Math. Soc. (N.S.), vol. 9, pp. 129–206, Sept. 1983.
34. R. F. V. Anderson, “The Weyl functional calculus,” J. Funct. Anal., vol. 4, pp. 240–267, 1969.
35. R. F. V. Anderson, “On the Weyl functional calculus,” J. Funct. Anal., vol. 6, pp. 110–115, 1970.
36. The multiplicative Weyl functional calculus,” J. Funct. Anal., vol. 9, pp. 423–440, 1972.
37. M. E. Taylor, “Functions of several self-adjoint operators,” Proc. Amer. Math. Soc., vol. 19, pp. 91–98, 1968.
38. E. Nelson, “Operants: A functional calculus for non-commuting operators,” in Functional Analysis and Related Fields, (F. E. Browder, Ed.). New York: 1970, 172–187.
39. T. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution-A tool for time-frequency analysis-Part 3: Relations with other time-frequency signal transformations,” Philips J. Res., vol. 35, pp. 372–389, 1980.
40. S. S. Haykin, Detection and Estimation: Applications to Radar. New York: Halsted, 1976.
41. F. Hlawatsch and G. F. Boudreaux-Bartles, “Linear and quadratic time-frequency frequency representations,” IEEE Signal Processing Mag., vol. 9, pp. 21–67, Apr. 1992.
42. F. Hlawatsch, W. Kozek, and W. Krattenthaler, “Time-frequency sub-spaces and their application to time-varying filtering,” in Proc. ICASSP ‘90, Apr. 1990, pp. 1607–1610.
43. R. G. Shenoy and T. W. Parks, “Time-frequency concentrated basis functions,” in Proc. ICASSP ‘90, Apr. 1990, 2459–2462.
44. G. B. Folland, Harmonic Analysis in Phase Space. Princeton, N J : Princeton University Press, 1989.
45. J. C. T. Pool, “Mathematical aspects of the Weyl correspondence,” J. Math. Phys, vol. 7, pp. 66–76, 1966.
46. N. G. D. Bruijn, “A theory of generalized functions, with applications to Wigner distribution and Weyl correspondence,” Nieuw Arch. Wisk. (4), vol. 3, no. 21, pp. 205–280, 1973.
47. A. Grossman, G. Loupias, and E. M. Stein, “An algebra of pseudodifferential ferential operators and quantum mechanics in phase space,” Ann. Inst. Fourier (Grenoble), vol. 18, no. 2, 343–368, 1968.
48. L. Zadeh, “Time-varying networks-1,” in Proc. IRE, pp. 1488–1503, 1961.
49. __. “Frequency analysis of variable networks,” Proc. IRE, vol. 38, 1950, pp. 291–299.
50. I. E. Segal, “Transforms for operators and symplectic automorphisms over a locally compact Abelian group,” Math. Scand., vol. 13, pp. 31–43, 1963. [51] O. Rioul and P. Flandrin, “Time-scale distributions: A general class extending the wavelet transform,” IEEE Trans. Signal Processing, vol. 46, pp. 1746–1757, May 1992.
51. O. Rioul and P. Flandrin “Time-scale distributions: A general class extending the wavelet transform”, IEEE Trans. Signal Processing, vol. 46, pp. 1746–1757, May 1992
52. R. G. Baraniuk and D. L. Jones, “New dimensions in wavelet analysis,” in Proc. IEEE ICASSP ‘92, Mar. 1992.
53. S. Mann and S. Haykin, “Time-frequency perspectives: The chirplet transform,” in Proc. IEEE ICASSP ‘92, Mar. 1992.
54. A. A. Melkman, “n-widths and optimal interpolation of time and band. limited functions,” in Optimal Estimation in Approximation Theory. New York: Plenum, 1976, pp. 55–68.
55. A. A. Mellunan and C. A. Michelli, “Optimal estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal., vol. 16, pp. 87–105, Feb. 1979.
56. J. Peetre, “The Weyl transform and Laguerre polynomials,” Le Mathe-matiche matiche (Catania), vol. 27, pp. 301–323, 1972.
57. A Grossman, J. Morlet, and T. Paul, “Transforms associated to square integrable group representations 1: General results,” J. Math. Phys., vol. 26, pp. 2473–2479, Oct. 1985.
58. L. Cohen, “Generalized phase-space distribution functions,” J. Math. Phys. vol. 7, pp. 781–786, 1966.
59. T. C. M. Claasen and W. F. G. Mecklenbrauker, “The Wigner distribution—A tool for time-frequency analysis—Part 2: Discrete time signals,” Philips J. Res., vol. 35, pp. 276–300, 1980.
60. W. Schemmp, Harmonic Analysis on the Heisenberg Nilpotent Lie Group, with Applications to Signal Processing. White Plains, NY: Longman, 1986.
61. L. Auslander and R. Tolimieri, “Radar ambiguity functions and group theory,” SIAM J. Math. Anal., vol. 16, pp. 577–601, May 1985.
62. Characterizing the radar ambiguity functions,” IEEE Trans. Inform. Theory, vol. IT-30, pp. 832–836, Nov. 1984.
63. L. H. Loomis, An Introduction To Abstract Harmonic Analysis. Prince-ton, NJ: Van Nostrand, 1953.
64. S. A. Gaal, Linear Analysis And Representation Theory. New York: Springer-Verlag, 1973.
65. E. Hewitt and K. A. Ross, Abstract Harmonic Analysis 1. New York: Springer-Verlag, 1963.
66. W. Rudin, Fourier Analysis on Groups. New York: Wiley, 1962, 1990.
67. A. Grossman, J. Monet, and T. Paul, “Transforms associated to square integrable group representations 2: Examples,” Ann. Inst. H. Poincare. Phys. Theor., vol. 45, no. 3, pp. 293–309, 1986.
68. I. Daubechies, “The wavelet transform, time-frequency localization and signal analysis,” IEEE Trans. Inform. Theory, vol. 36, pp. 961–1005, Sept. 1990.
69. H. Moscovici, “Coherent state representations of nilpotent Lie groups,” Comm. Math. Phys., vol. 54, pp. 63–68, 1977.
70. H. Moscovici and A. Verona, “Coherent states and square integrable representations,” Ann. Inst. H. Poincare. Phys. Theor.: Section A, vol. 29, no. 2, pp. 139–156, 1978.
71. R. G. Shenoy, “Group representations and optimal recovery in signal modeling,” Ph.D. dissertation, Cornell Univ., Ithaca, NY, 1991.
72. I. Daubechies and T. Paul, “Time-frequency localisation operators-A geometric phase space approach: 2: The use of dilations,” Inverse Problems, vol. 4, pp. 661–680, 1988.
73. R. G. Shenoy and T. W. Parks, “Wideband ambiguity functions and affine Wigner distributions,” Schlumberger-Doll Research, Ridgefield
74. J. R. Higgins, “Five short stories about the cardinal series,” Bull. Amer. Math. Soc. (N.S.), vol. 12, no. 1, pp. 45-89, 1985.
75. M. M. Dodson and A. M. Silva, “Fourier analysis and the sampling theorem,” Proc. Royal Irish Acad. Sect. A, vol. 85, no. 1, pp. 81-108, 1985.
76. M. M. Dodson, A. M. Silva, and V. Soucek, “A Note on Whittakers cardinal series in harmonic analysis,” Proc. Edinburgh Math. Soc. (2), vol. 29, pp. 349-357, 1986.
77. A. Weil,“Sur certains groupes oprateurs unitaires,” Acta Math., vol. 111, pp. 143-211, 1964.
78. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. Providence RI: Amar. Math Soc., 1969
79. K. Maurin, General Eigenfunction Expansions and Unitary Representations of Topological Groups. Polish Scientific, 1968.