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Time-Frequency Analysis: What We Know and What We Don’t

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Landscapes of Time-Frequency Analysis

Part of the book series: Applied and Numerical Harmonic Analysis ((ANHA))

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Abstract

We discuss the basic ideas and motivations of time-frequency analysis from the point of view of what is known, what is not, what is mathematically possible, and what is not. We address the issue of whether a fully consistent theory of time-varying spectra is possible and the arguments given against the possible formulation of a proper theory. Historically, discussions of joint densities in time and frequency have been intimately tied with the question of the existence of manifestly positive joint densities, the marginals, and the uncertainty principle. We discuss the relations between these ideas.

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References

  1. R. G. Baraniuk and D. L. Jones, IEEE Transactions on Signal Processing, 43, 2269–2282, 1995.

    Article  Google Scholar 

  2. J. S. Ben-Benjamin and L. Cohen, “Pulse propagation and windowed wave functions,” Journal of Modern Optics, 61, 36–42, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. S. Ben-Benjamin, M. B. Kim, W. P. Schleich, W. B. Case, L. Cohen, “Working in phase space with Wigner and Weyl,” Fortschr. Phys. 65 , 1–11, 2017.

    Article  MathSciNet  MATH  Google Scholar 

  4. J. S. Ben-Benjamin, L. Cohen , N. C. Dias, P. Loughlin, and J. N. Prata, “What is the Wigner function closest to a given square integrable function?”, Siam J. Math. Anal., 50, 5161–5197, 2018.

    Article  MathSciNet  MATH  Google Scholar 

  5. J. S. Ben-Benjamin, L. Cohen, M. O. Scully, “From von Neumann to Wigner and beyond,” Eur. Phys. J., 227, 2171–2182, 2019.

    Google Scholar 

  6. G.F. Boudreaux-Bartels, “Mixed Time-Frequency Signal Transformations.” in: The Transforms and Applications Handbook: Second Edition, Ed. Alexander D. Poularikas, Boca Raton: CRC Press LLC, 2000.

    Google Scholar 

  7. H. Choi and W. Williams, “Improved time-frequency representation of multicomponent signals using exponential kernels” IEEE Trans. ASSP, 37, 862–871, 1989.

    Article  Google Scholar 

  8. T. Claasen and W. Mecklenbrauker, “The Wigner distribution - A tool for time-frequency analysis, Part I-III,” Phil. J. Res., vol. 35, nos. 3–6, pp. 217–250, 276–300, 372–389, 1980.

    Google Scholar 

  9. L. Cohen, “Generalized phase–space distribution functions,” Jour. Math. Phys., 7, 781–786, 1966.

    Article  MathSciNet  Google Scholar 

  10. L. Cohen and Y. Zaparovanny, “Positive Quantum Joint Distributions,” J. Math. Phys., 21, 794–796, 1980.

    Article  MathSciNet  Google Scholar 

  11. L. Cohen, “Time-Frequency Distributions – A Review,” Proc. of the IEEE, 77, 941–981, 1989.

    Article  Google Scholar 

  12. L. Cohen, Time-Frequency Analysis, Prentice-Hall, Englewood Cliffs, 1995.

    Google Scholar 

  13. L. Cohen, “The Uncertainty principle for Windowed Wave Functions”, Optics Communication, 179, 221–229, 2000.

    Article  Google Scholar 

  14. L. Cohen, “The History of Noise”, IEEE Signal Processing Magazine, 22, 20–45, 2005.

    Article  Google Scholar 

  15. L. Cohen and P. Loughlin, “Bohmian and quantum phase space distribution expansions and approximations”, Physics Letters A 378, 931–936, 2014.

    Article  MathSciNet  MATH  Google Scholar 

  16. K. Davidson and P. Loughlin, “Compensating for window effects in the calculation of spectrographic instantaneous bandwidth,” IEEE Trans. Biomed. Engr., 47, 556–558, 2000.

    Article  Google Scholar 

  17. K. Davidson, Instantaneous moments of signals, Ph.D. dissertation, University of Pittsburgh, 2001.

    Google Scholar 

  18. N.C. Dias and J.N. Prata, “Bohmian trajectories and quantum phase space distributions”, Phys. Lett. A 302, 261–272, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  19. M. Emresoy and P. Loughlin, “Weighted least squares Cohen-Posch time-frequency distributions with specified conditional and joint moment constraints,” IEEE Trans. Sig. Process., 47, 893–900, 1999.

    Article  Google Scholar 

  20. M. Emresoy and A. El-Jaroudi, “Evolutionary spectrum estimation by positivity constrained deconvolution,” IEEE Trans. Sig. Process., 47, 889–893, 1999

    Article  Google Scholar 

  21. B.-G. Englert, and Wodkiewicz, K., “Intrinsic and operational observables in quantum mechanics” Phys. Rev. A 51, R2661, 1995.

    Google Scholar 

  22. J. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Sig. Proc., vol. 44, no. 8, pp. 2086–2091, 1996.

    Article  Google Scholar 

  23. L. Galleani and L. Cohen, “The Wigner distribution for classical systems,” Physics Letters A, 302, 149–155, 2002.

    Article  MathSciNet  MATH  Google Scholar 

  24. L. Galleani and L. Cohen, “Time-Frequency Wigner Distribution Approach To Differential Equations,” in K. Barner and G. Arce (eds.), Nonlinear Signal and Image Processing: Theory, Methods, and Applications, CRC Press, 2003.)

    Google Scholar 

  25. L. Galleani and L. Cohen, “Direct Time-Frequency Characterization of Linear Systems Governed by Differential Equations,” Signal Processing Letters, 11 , 721–724, 2004.

    Article  Google Scholar 

  26. L. Galleani and L. Cohen, “Nonstationary stochastic differential equations”, in: Advances of nonlinear signal and image processing, S. Marshall and G. Sicuranza (Eds.), Hindowi Publishing, pp. 1–13, 2006.

    Google Scholar 

  27. D. Groutage, D.Bennink, P. Loughlin, and L. Cohen, “Positive Time-Frequency Distributions and Acoustic Echoes,” in A. Papandreaux (ed.), Applications in Time-Frequency Signal Processing, Ch. 4, pp. 163–178, CRC Press, 2003.

    Google Scholar 

  28. F. Hlawatsch and G.F. Boudreaux-Bartels, “Linear and quadratic time-frequency signal representations”, IEEE Signal Processing Magazine, 9, 21–67, 1992.

    Article  Google Scholar 

  29. J. E. M. Janssen, “On the locus and spread of pseudo–density functions in the time–frequency plane,” Philips Journal of Research, 37, 79–110, 1982.

    MathSciNet  MATH  Google Scholar 

  30. J. Jeong and W. Williams, “Kernel design for reduced interference distributions,” IEEE Trans. Sig. Process., 40, 402–412, 1992.

    Article  Google Scholar 

  31. R. Marks, Handbook of Fourier Analysis & Its Applications, Oxford University Press, New York, 2009.

    Book  MATH  Google Scholar 

  32. Lee H. W., “Theory and application of the quantum phase-space distribution-functions”, Physics Reports, 259, 147–211, 1995.

    Article  MathSciNet  Google Scholar 

  33. P. Loughlin, J. Pitton and L. Atlas, “An information-theoretic approach to positive time-frequency distributions,” Proc. IEEE Intl. Conf. Acous., Speech and Sig. Process.’92, vol. V, 125–128, 1992.

    Google Scholar 

  34. P. Loughlin, J. Pitton and L. Atlas, “Bilinear time-frequency representations: new insights and properties,” IEEE Trans. Sig. Process., 41, 750–767, 1993.

    Article  MATH  Google Scholar 

  35. P. Loughlin, J. Pitton and L. Atlas, “Construction of positive time-frequency distributions,” IEEE Trans. Sig. Proc., 42, 2697–2705, 1994.

    Article  Google Scholar 

  36. P. Loughlin, “Cohen-Posch (positive) time-frequency distributions: development and applications,” Applied Sig. Process., vol. 4, pp. 122–130, 1997.

    Google Scholar 

  37. P. Loughlin and K. Davidson, “Positive local variances of time-frequency distributions and local uncertainty,” Proc. IEEE-SP Intl. Symp. Time-Frequency and Time-Scale Analysis, pp. 541–544, 1998.

    Google Scholar 

  38. P. Loughlin and G. Bernard, “Cohen-Posch (positive) time-frequency distributions and their application to machine vibration analysis,” Mech. Syst. Sig. Proc., 11, 561–576, 1997.

    Article  Google Scholar 

  39. P. Loughlin and L. Cohen, “Positive Time-Frequency Distributions,” in A. Papandreou (Ed.), Applications in Time-Frequency Signal Processing, Ch. 3, pp. 121–162, CRC Press, 2003.

    Google Scholar 

  40. P. Loughlin and L. Cohen, “A Wigner approximation method for wave propagation,” J. Acoust. Soc. Amer., 118, 1268–1271, 2005.

    Article  Google Scholar 

  41. P. Loughlin and L. Cohen, “Phase-space approach to wave propagation with dispersion and dam**,” Proc. SPIE, 5559, 221–231, 2004.

    Article  Google Scholar 

  42. P. Loughlin and L. Cohen, “Approximate wave function from approximate non-representable Wigner distributions,” J. Modern Optics, 55, 3379–3387, 2008

    Article  MATH  Google Scholar 

  43. H. Margenau, Measurements and Quantum States: Part I and II, Philosophy of Science, Vol. 30 pp. 1–16; 138–157, 1963.

    Google Scholar 

  44. Moyal J. E., “Quantum mechanics as a statistical theory,” Proc. Camb. Phil. Soc., 45, 99–124, 1949

    Article  MathSciNet  MATH  Google Scholar 

  45. Mugur-Schächter, M., “A study of Wigner’s theorem on joint probabilities,” Found. of Phys., 9, 389–404, 1979.

    Article  MathSciNet  Google Scholar 

  46. R. Nickel, T. Sang and W. Williams, “A new signal adaptive approach to positive time-frequency distributions with suppressed interference terms,” IEEE Proc. ICASSP’98, vol. 3 pp. 1777–1780, 1998.

    Google Scholar 

  47. J. Pitton, L. Atlas and P. Loughlin, “Deconvolution for positive time-frequency distributions,” Proc. 27th Asilomar Conf. on Sigs., Syst. and Comps., pp. 1450–1454, 1993.

    Google Scholar 

  48. J. Pitton, P. Loughlin and L. Atlas, “Positive time-frequency distributions via maximum entropy deconvolution of the evolutionary spectrum,” Proc. IEEE Intl. Conf. Acous., Speech and Sig. Proc.’93, vol. IV, pp. 436–439, 1993.

    Google Scholar 

  49. J. Pitton, L. Atlas and P. Loughlin, “Applications of positive time-frequency distributions to speech processing,” IEEE Trans. Speech and Audio Proc., 2, 554–566, 1994.

    Article  Google Scholar 

  50. J. Pitton, “Linear and quadratic methods for positive time-frequency distributions,” IEEE Proc. ICASSP’97, vol. V, pp. 3649–3652, 1997.

    Google Scholar 

  51. J. Pitton,“Positive time-frequency distributions via quadratic programming,” Multidimensional Systems and Signal Processing, 9, 439–445, 1998.

    Article  MATH  Google Scholar 

  52. T. Sang, W. Williams and J. O’Neil, “An algorithm for positive time-frequency distributions,” IEEE-SP Proc. Intl. Symp. Time-Freq./Time-Scale Analysis, 165–168, 1996.

    Google Scholar 

  53. S. Shah, A. El-Jaroudi, P. Loughlin and L. Chaparro, “Signal synthesis and positive time-frequency distributions,” J. Franklin Institute, 337, 317–328, 2000.

    Article  MATH  Google Scholar 

  54. H. Tohyama, S. Kikkawa, and K. Ohara, “On an optimum positive time-frequency distribution by the least squares method” Trans. Inst. Electr., Inform. Commun. Eng., J75-A1, 661–663, 1992.

    Google Scholar 

  55. J. Ville, “Theorie et applications de la notion de signal analytique,”Cables et Transmissions, 2A, 61–74, 1948.

    Google Scholar 

  56. M. C. Wang and G. E. Uhlenbeck, “On the Theory of the Brownian Motion II,” Rev. of Mod. Phys., 17, 323–342, 1945; 61–74, 1948.

    Google Scholar 

  57. E. P. Wigner, “On the quantum correction for thermodynamic equilibrium,” Physical Review, 40, 749–759, 1932.

    Article  MATH  Google Scholar 

  58. E. P. Wigner, “Quantum–mechanical distribution functions revisited,” Perspectives in Quantum Theory, W. Yourgrau and A. van der Merwe, eds., pp. 25–36, MIT Press, 1971.

    Google Scholar 

  59. K. Wodkiewicz, “On the operational uncertainty relation”, Phys. Lett., A124, 207–210, 1987.

    Article  Google Scholar 

  60. Y. Zhao, L. Atlas and R. Marks II, “The use of cone-shaped kernels for generalized time-frequency representations of nonstationary signals,” IEEE Trans. ASSP, 38, 1084–1091, 1990.

    Article  Google Scholar 

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Authors and Affiliations

  1. Hunter College of The City University of New York, New York, NY, USA

    Leon Cohen

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  1. Leon Cohen
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Correspondence to Leon Cohen .

Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Turin, Torino, Italy

    Paolo Boggiatto

  2. Department of Mathematics, Ghent University, Ghent, Belgium

    Tommaso Bruno

  3. Department of Mathematics, University of Turin, Torino, Italy

    Elena Cordero

  4. Institute of Mathematics, University of Vienna, Wien, Austria

    Hans G. Feichtinger

  5. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Fabio Nicola

  6. Department of Mathematics, University of Turin, Torino, Italy

    Alessandro Oliaro

  7. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Anita Tabacco

  8. Department of Mathematical Sciences, Polytechnic University of Turin, Torino, Italy

    Maria Vallarino

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Cohen, L. (2020). Time-Frequency Analysis: What We Know and What We Don’t. In: Boggiatto, P., et al. Landscapes of Time-Frequency Analysis. Applied and Numerical Harmonic Analysis. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-56005-8_5

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