Separability, positivity, and minimum uncertainty in time-frequency energy distributions

Journal of Mathematical Physics

Volumn 39, Issue 8, 1998, Pages 4016-4040

  Flandrin, Patrick ^a  

^a UNIVERSITÉ DE LYON   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0032399295     PISSN: 00222488     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.532483     Document Type: Article

References (34)

1. E. P. Wigner, "On the quantum correction for thermodynamic equilibrium," Phys. Rev. 40, 749-759 (1932).
2. J. Ville, "Théorie et applications de la notion de signal analytique," Câbles Transm. 2ème A, 61-74 (1948).
3. L. Cohen, Time-Frequency Analysis (Prentice-Hall, Englewood Cliffs, 1995).
4. P. Flandrin, Temps-Fréquence (Hermès, Paris, 1993).
5. D. Gabor, "Theory of communication," J. Inst. Electr. Eng. 93, 429-457 (1946).
6. R. L. Hudson, "When is the Wigner quasi-probability density non-negative?," Rep. Math. Phys. 6, 249-252 (1974).
7. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "On the time-frequency discrimination of energy distributions -Can they look sharper than Heisenberg?," IEEE-ICASSP'84, San Diego, CA, 1984, pp. 41B7.1-41B7.4.
8. K. Urbanik, "Joint probability distributions of observables in quantum mechanics," Stud. Math. XXI, 117-133 (1961).
9. A. J. E. M. Janssen, "Positivity of weighted Wigner distributions," SIAM J. Math. Anal. 12, 752-758 (1981).
10. A. J. E. M. Janssen, "Positivity properties of phase-plane distribution functions," J. Math. Phys. 25, 2240-2252 (1984).
11. A. J. E. M. Janssen, "Bilinear phase-plane distribution functions and positivity," J. Math. Phys. 26, 1986-1994 (1985).
12. A. J. E. M. Janssen, "Positivity of time-frequency distribution functions," Signal Process. 14, 243-252 (1988).
13. J. Aczél, Lectures on Functional Equations and Their Applications (Academic, New York, 1966).
14. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, "The Wigner distribution - A tool for time-frequency signal analysis. Part I: Continuous-time signals," Philips J. Res. 35, 217-250 (1980).
15. A. J. E. M. Janssen, "On the locus and spread of pseudo-density functions in the time-frequency plane," Philips J. Res. 37, 79-110 (1982).
16. R. A. Altes, "Methods of wide-band signal design for radar and sonar systems," Ph.D. dissertation, University of Rochester, Rochester, NY, 1970.
17. N. J. De Bruijn, "Uncertainty principles in Fourier analysis," in Inequalities, edited by O. Shisha (Academic, New York, 1967), pp. 55-71.
18. E. H. Lieb, "Integral bounds for radar ambiguity functions and Wigner distributions," J. Math. Phys. 31, 594-599 (1990).
19. A. J. E. M. Janssen, "Optimality property of the Gaussian window spectrogram and an application to lattice expansions," IEEE Trans. Signal Process. SP-39, 202-204 (1991).
20. J. Bertrand and P. Bertrand, "A class of affine Wigner distributions with extended covariance properties," J. Math. Phys. 33, 2515-2527 (1992).
21. J. Ph. Ovarlez, "La transformation de Mellin - Un outil pour l'analyse des signaux a large bande," Thèse de doctorat, Univ. Paris 6, 1992.
22. L. Cohen, "The scale representation," IEEE Trans. Signal Process. 41, 3275-3292 (1993).
23. J. R. Klauder, "Path integrals for affine variables," in Functional Integration: Theory and Applications, edited by J.-P. Antoine and E. Tirapegui (Plenum, New York, 1980), pp. 101-119.
24. K. B. Stolarsky, "Generalizations of the logarithmic mean," Math. Mag. 48, 87-92 (1975).
25. P. Flandrin and P. Gonçalvès, "Geometry of affine time-frequency distributions," Appl. Comp. Harm. Anal. 3, 10-39 (1996).
26. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1972).
27. R. A. Altes, "Sonar for generalized target description and its similarity to animal echolocation systems," J. Acoust. Soc. Am. 59, 97-105 (1976).
28. E. Lukacs, Characteristic Functions (Griffin, London, 1970).
29. I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic, New York, 1980).
30. F. Oberhettinger, Fourier Transforms of Distributions and Their Inverses - A Collection of Tables (Academic, New York, 1973).
31. B. Escudié and A. Grossmann, "Une représentation bilinéaire en temps et échelle," 13ème Coll. GRETSI, Juan-les-Pins, France, 1991, pp. 33-36.
32. A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, "The hyperbolic class of quadratic time-frequency representations. Part I: Constant-Q warping, the hyperbolic peradigm, properties and members," IEEE Trans. Signal Process. SP-41, 3425-3444 (1993).
33. R. G. Baraniuk and D. L. Jones, "Unitary equivalence: A new twist on signal processing," IEEE Trans. Signal Process. SP-43, 2269-2282 (1995).
34. R. A. Altes, "Wideband, proportional-bandwidth Wigner-Ville analysis," IEEE Trans. Acoust., Speech, Signal Process. ASSP-38, 1005-1012 (1990).