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Time-Frequency Processing of Time-Varying Signals with Nonlinear Group Delay

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Wavelets and Signal Processing

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

Abstract

Quadratic time-frequency representations (QTFRs) can be successful processing tools for different applications depending on the signal changes they can preserve. This chapter provides a tutorial on classes of QTFRs which are covariant to time shifts that match changes in the group delay function of the analysis signal. These changes may be constant or depend linearly or nonlinearly on frequency, and they may be the result of the signal propagating through systems with dispersive time-frequency characteristics. The unitary warping relationships of these group delay shift covariant QTFR classes to the constant time-shift covariant ones are established. Specific QTFR members are also presented together with the signal properties they satisfy. Various simulation examples are provided to demonstrate the importance of matching the time-frequency characteristics of the signal with the group delay shift of the QTFR.

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References

  1. L. Cohen, Time-Frequency Analysis. Englewood Cliffs, New Jersey: Prentice-Hall, 1995.

    Google Scholar 

  2. P. Flandrin, Time-Frequency/Time-Scale Analysis. San Diego, California: Academic Press, 1999. (Translated from French, Temps fréquence. Paris: Hermès, 1993).

    Google Scholar 

  3. F. Hlawatsch and G. F. Boudreaux-Bartels, Linear and quadratic time-frequency signal representations, IEEE Signal Processing Magazine, vol. 9, pp. 21–67, April 1992.

    Article  Google Scholar 

  4. B. Boashash, ed., Time-Frequency Signal Analysis—Methods and Applications. Melbourne, Australia: Longman-Cheshire, 1992.

    Google Scholar 

  5. S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications. Englewood Cliffs, New Jersey: Prentice-Hall, 1996.

    Google Scholar 

  6. A. Papandreou-Suppappola, ed., Applications in Time-Frequency Signal Processing. Boca Raton, Florida: CRC Press, 2002.

    Book  Google Scholar 

  7. W. F. G. Mecklenbräuker and E. Hlawatsch, eds., The Wigner Distribution—Theory and Applications in Signal Processing. Amsterdam, The Netherlands: Elsevier, 1997.

    Google Scholar 

  8. L. Cohen, Generalized phase-space distribution functions, Journal of Mathematics and Physics, vol. 7, pp. 781–786, 1966.

    Article  Google Scholar 

  9. T. A. C. M. Claasen and W. F. G. Mecklenbräuker, The Wigner distribution—A tool for time-frequency signal analysis, Part III: Relations with other time-frequency signal transformations, Philips Journal of Research, vol. 35, pp. 372–389, 1980.

    MathSciNet  MATH  Google Scholar 

  10. B. Boashash, Time-frequency signal analysis, in Advances in Spectrum Estimation (S. Haykin, ed.), Englewood Cliffs, New Jersey: Prentice Hall, 1990.

    Google Scholar 

  11. G. F. Boudreaux-Bartels, Mixed time-frequency signal transformations, in The Transforms and Applications Handbook (A. Poularikas, ed.), Boca Raton, Florida: CRC Press, 1996.

    Google Scholar 

  12. A. Papandreou-Suppappola, F. Hlawatsch, and G. F. Boudreaux-Bartels, Quadratic time-frequency representations with scale covariance and generalized time-shift covariance: A unified framework for the affine, hyperbolic, and power classes, Digital Signal Processing: A Review Journal, vol. 8, pp. 3–48, January 1998.

    Article  Google Scholar 

  13. A. Papandreou-Suppappola, Time-varying processing: tutorial on principles and practice, in Applications in Time-Frequency Signal Processing (A. Papandreou-Suppappola, ed.), Boca Raton, Florida: CRC Press, 2002.

    Google Scholar 

  14. E. P. Wigner, On the quantum correction for thermo-dynamic equilibrium, Physics Review, vol. 40, pp. 749–759, 1932.

    Article  Google Scholar 

  15. T. A. C. M. Claasen and W. E G. Mecklenbräuker, The Wigner distribution—A tool for time-frequency signal analysis, Part I: Continuous-time signals, Philips Journal of Research, vol. 35, pp. 217–250,1980.

    MathSciNet  MATH  Google Scholar 

  16. T. A. C. M. Claasen and W. E G. Mecklenbräuker, The Wigner distribution—A tool for time-frequency signal analysis, Part II: Discrete-time signals, Philips Journal of Research, vol. 35, pp. 276–300, 1980.

    MathSciNet  MATH  Google Scholar 

  17. E Hlawatsch and P. Flandrin, The interference structure of the Wigner distribution and related time-frequency signal representations, The Wigner Distribution—Theory and Applications in Signal Processing (W. Mecklenbräuker and F. Hlawatsch), Amsterdam: North Holland Elsevier Science Publishers, 1997.

    Google Scholar 

  18. L. R. Rabiner and R. W. Schafer, Digital Processing of Speech Signals. Englewood Cliffs, New Jersey: Prentice-Hall, 1978.

    Google Scholar 

  19. R. A. Altes, Detection, estimation, and classification with spectrograms, Journal of the Acoustical Society of America, vol. 67, pp. 1232–1246, April 1980.

    Article  MathSciNet  MATH  Google Scholar 

  20. J. Bertrand and P. Bertrand, A class of affine Wigner functions with extended covariance properties, Journal of Mathematics and Physics, vol. 33, pp. 2515–2527,1992.

    Article  MathSciNet  MATH  Google Scholar 

  21. O. Rioul and P. Flandrin, Time-scale energy distributions: A general class extending wavelet transforms, IEEE Transactions on Signal Processing, vol. 40, pp. 1746–1757, July 1992.

    Article  MATH  Google Scholar 

  22. P. Flandrin and P. Gonçalvès, From wavelets to time-scale energy distributions, in Recent Advances in Wavelet Analysis (L. L. Schumaker and G. Webb, eds.), pp. 309–334, New York: Academic Press, 1994.

    Google Scholar 

  23. P. Flandrin and P. Gonçalvès, Geometry of affine time-frequency distributions, Applied and Computational Harmonic Analysis, vol. 3, pp. 10–39, January 1996.

    Article  MathSciNet  MATH  Google Scholar 

  24. O. Rioul and M. Vetterli, Wavelets and signal processing, IEEE Signal Processing Magazine, vol. 8, pp. 14–38, October 1991.

    Article  Google Scholar 

  25. A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, The hyperbolic class of quadratic time-frequency representations, Part I: Constant-Q warping, the hyperbolic paradigm, properties, and members, IEEE Transactions on Signal Processing, vol. 41, pp. 3425–3444, December 1993.

    Article  MATH  Google Scholar 

  26. F. Hlawatsch, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, The hyperbolic class of quadratic time-frequency representations, Part II: Subclasses, intersection with the affine and power classes, regularity, and unitarity, IEEE Transactions on Signal Processing, vol. 45, pp. 303–315, February 1997.

    Article  Google Scholar 

  27. A. Papandreou-Suppappola, R. L. Murray, B. G. Iem, and G. E Boudreaux-Bartels, Group delay shift covariant quadratic time-frequency representations, IEEE Transactions on Signal Processing, vol. 49, pp. 2549–2564, November 2001.

    Article  Google Scholar 

  28. A. Papandreou-Suppappola, Time-frequency representations covariant to group delay shifts, in lime-Frequency Signal Analysis and Processing (B. Boashash, ed.), Englewood Cliffs, New Jersey: Prentice-Hall, 2003.

    Google Scholar 

  29. R. A. Altes, Wide-band, proportional-bandwidth Wigner-Ville analysis, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 38, pp. 1005–1012, June 1990.

    Article  Google Scholar 

  30. N. M. Marinovich, The Wigner Distribution and the Ambiguity Function: Generalizations, Enhancement, Compression and Some Applications. Ph.D. thesis, The City University of New York, 1986.

    Google Scholar 

  31. E Hlawatsch, A. Papandreou, and G. R Boudreaux-Bartels, The power classes of quadratic time-frequency representations: A generalization of the affine and hyperbolic classes, in Proceedings Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, California), pp. 1265–1270, November 1993.

    Google Scholar 

  32. F. Hlawatsch, A. Papandreou-Suppappola, and G. E Boudreaux-Bartels, The power classes—quadratic time-frequency representations with scale covariance and dispersive time-shift covariance, IEEE Transactions on Signal Processing, vol. 47, pp. 3067–3083, November 1999.

    Article  MATH  Google Scholar 

  33. A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, A unified framework for the scale covariant affine, hyperbolic, and power class time-frequency representations using generalized time-shifts, in Proceedings 1995 International Conference on Acoustics, Speech and Signal Processing, May 1995.

    Google Scholar 

  34. A. Papandreou-Suppappola, F. Hlawatsch, and G. R Boudreaux-Bartels, Power class time-frequency representations: Interference geometry, smoothing, and implementation, in Proceedings IEEE International Symposium on Time-Frequency/Time-Scale Analysis, (Paris, France), pp. 193–196, June 1996.

    Google Scholar 

  35. A. Papandreou-Suppappola, Generalized time-shift covariant quadratic time-frequency representations with arbitrary group delays, in Proceedings Twenty-Ninth Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, California), pp. 553–557, October/November 1995.

    Google Scholar 

  36. A. Papandreou-Suppappola and G. E Boudreaux-Bartels, The exponential class and generalized time-shift covariant quadratic time-frequency representations, in Proceedings IEEE International Symposium on Time-Frequency/Time-Scale Analysis, (Paris, France), pp. 429–432, June 1996.

    Google Scholar 

  37. A. Papandreou-Suppappola, B. G. Iem, R. L. Murray, and G. F. Boudreaux-Bartels, Properties and implementation of the exponential class of quadratic time-frequency represen-tations, in Proceedings Thirtieth Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, California), pp. 237–241, November 1996.

    Google Scholar 

  38. A. Papandreou-Suppappola, R. L. Murray, and G. F. Boudreaux-Bartels, Localized subclasses of quadratic time-frequency representations, in Proceedings IEEE International Conference on Acoustics, Speech, and Signal Processing, (Munich, Germany), pp. 2041–2044, April 1997.

    Google Scholar 

  39. R. G. Baraniuk and D. L. Jones, Warped wavelet bases: Unitary equivalence and signal processing, in Proceedings IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, (Minneapolis, Minnesota), pp. 320–323, April 1993.

    Google Scholar 

  40. R. G. Baraniuk, Warped perspectives in time-frequency analysis, in Proceedings IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, (Philadelphia, Pennsylvania), pp. 528–531, October 1994.

    Google Scholar 

  41. R. G. Baraniuk and D. L. Jones, Unitary equivalence: A new twist on signal processing, IEEE Transactions on Signal Processing, vol. 43, pp. 2269–2282, October 1995.

    Article  Google Scholar 

  42. R. G. Baraniuk, Covariant time-frequency representations through unitary equivalence, IEEE Signal Processing Letters, vol. 3, pp. 79–81, March 1996.

    Article  Google Scholar 

  43. F. Hlawatsch and H. Bölcskei, Displacement-covariant time-frequency distributions based on conjugate operators, IEEE Signal Processing Letters, vol. 3, pp. 44–46, February 1996.

    Article  Google Scholar 

  44. F. Hlawatsch and T. Twaroch, Covariant (a, ß), time-frequency, and (a, b) representations, in Proceedings IEEE International Symposium on Time-Frequency/Time-Scale Analysis, (Paris, France), pp. 437–440, June 1996.

    Google Scholar 

  45. A. M. Sayeed and D. L. Jones, A canonical covariance-based method for generalized joint signal representations, IEEE Signal Processing Letters, vol. 3, pp. 121–123, April 1996.

    Article  Google Scholar 

  46. A. M. Sayeed and D. L. Jones, A simple covariance-based characterization of joint signal representations of arbitrary variables, in Proceedings IEEE International Symposium on Time-Frequency/Time-Scale Analysis, (Paris, France), pp. 433–436, June 1996.

    Google Scholar 

  47. F. Hlawatsch, Duality and classification of bilinear time-frequency signal representations, IEEE Transactions on Signal Processing, vol. 39, pp. 1564–1574, July 1991.

    Article  Google Scholar 

  48. L. Cohen, Time-Frequency Distribution—A review, Proceedings IEEE, vol. 77, pp. 941–981, July 1989.

    Google Scholar 

  49. G. F. Boudreaux-Bartels, On the use of operators vs warpings vs axiomatic derivations of new time-frequency-scale (operator) representations, in Proceedings Twenty-Eighth Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, California), October/November, 1994.

    Google Scholar 

  50. H. I. Choi and W. J. Williams, Improved time-frequency representation of multicomponent signals using exponential kernels, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 37, pp. 862–871, June 1989.

    Article  Google Scholar 

  51. A. Papandreou and G. F. Boudreaux-Bartels, Generalization of the Choi—Williams distribution and the Butterworth distribution for time-frequency analysis, IEEE Transactions on Signal Processing, vol. 41, pp. 463–472, January 1993.

    Article  Google Scholar 

  52. P. Flandrin, Scale-invariant Wigner spectra and self-similarity, in Proceedings European Signal Processing Conference, EUSIPCO-90, (Barcelona, Spain), pp. 149–152, September.

    Google Scholar 

  53. C. Braccini and G. Gambardella, Form-invariant linear filtering: Theory and applications, IEEE Transactions on Acoustics, Speech, and Signal Processing, vol. 34, pp. 1612–1628, December 1986.

    Article  Google Scholar 

  54. ] J. Bertrand and P. Bertrand, Affine time-frequency distributions, in Time-Frequency Signal Analysis—Methods and Applications (B. Boashash, ed.), ch. 5, pp. 118–140, Melbourne, Australia: Longman-Cheshire, 1992.

    Google Scholar 

  55. F. Hlawatsch and R. L. Urbanke, Bilinear time-frequency representations of signals: The shift-scale invariant class, IEEE Transactions on Signal Processing, vol. 42, pp. 357–366, February 1994.

    Article  Google Scholar 

  56. J. P. Sessarego, J. Sageloli, P. Flandrin, and M. Zakharia, Time-frequency Wigner–Ville analysis of echoes scattered by a spherical shell, in Wavelets, Time-Frequency Methods and Phase Space (J. M. Combes, A. Grossman, and P. Tchamitchian, eds.), pp. 147–153, Berlin: Springer-Verlag, December 1989.

    Google Scholar 

  57. G. C. Gaunaurd and H. C. Strifors, Signal analysis by means of time-frequency (Wigner-type) distributions—Applications to sonar and radar echoes, Proceedings of the IEEE, vol. 84, pp. 1231–1248, September 1996.

    Article  Google Scholar 

  58. L. R. Dragonette, D. M. Drumheller, C. F. Gaumond, D. H. Hughes, and B. T. O’Connor, The application of two-dimensional signal transformations to the analysis and synthesis of structural excitations observed in acoustical scattering, Proceedings of the IEEE, vol. 84, pp. 1249–1263, September 1996.

    Article  Google Scholar 

  59. I. Gohberg and S. Goldberg, Basic Operator Theory. Boston, Massachussets: Birkhäuser, 1980.

    Google Scholar 

  60. A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, A unified framework for the Bertrand distribution and the Altes distribution: The new hyperbolic class of quadratic time-frequency distributions, in Proceedings IEEE Symposium on Time-Frequency and Time-Scale Analysis, (Victoria, Canada), pp. 27–30, October 1992.

    Google Scholar 

  61. A. Papandreou-Suppappola and G. F. Boudreaux-Bartels, Distortion that occurs when the signal group delay does not match the time-shift covariance of a time-frequency representation, in Proceedings 30th Annual Conference on Information Sciences and Systems, (Princeton, New Jersey), pp. 520–525, March 1996.

    Google Scholar 

  62. A. Papandreou-Suppappola and G. F. Boudreaux-Bartels, The effect of mismatching analysis signals and time-frequency representations, in Proceedings IEEE International Symposium on Time-Frequency/Time-Scale Analysis, (Paris, France), pp. 149–152, June 1996.

    Google Scholar 

  63. A. Papandreou, F. Hlawatsch, and G. F. Boudreaux-Bartels, Quadratic time-frequency distributions: The new hyperbolic class and its intersection with the affine class, in Proceedings Sixth Signal Processing Workshop on Statistical Signal and Array Processing, (Victoria, Canada), pp. 26–29, October 1992.

    Google Scholar 

  64. R. A. Altes and E. L. Titlebaum, Bat signals as optimally Doppler tolerant waveforms, Journal of the Acoustical Society of America, vol. 48, pp. 1014–1020, October 1970.

    Article  Google Scholar 

  65. P. Guillemain and R. White, Wavelet transforms for the analysis of dispersive systems, in Proceedings IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, (University of Warwick, Coventry, UK), pp. 32–39, August 1995.

    Google Scholar 

  66. D. E. Newland, Time-frequency and time-scale analysis by harmonic wavelets, in Signal Analysis and Prediction (A. Prochazka, ed.), ch. 1, Boston, Massachusetts: Birkhäuser, 1998.

    Google Scholar 

  67. D. E. Newland, Practical signal analysis: Do wavelets make any difference? in Proceedings ASME Design Engineering Technical Conferences, 16th Biennial Conference on Vibration and Noise, (Sacramento, California), 1997.

    Google Scholar 

  68. M. J. Freeman, M. E. Dunham, and S. Qian, Trans-ionospheric signal detection by time-scale representation, in Proceedings IEEE UK Symposium on Applications of Time-Frequency and Time-Scale Methods, (University of Warwick, Coventry, UK), pp. 152–158, August 1995.

    Google Scholar 

  69. A. Papandreou-Suppappola and L. T. Antonelli, Use of quadratic time-frequency representations to analyze cetacean mammal sounds, Tech. Rep. 11,284, Naval Undersea Warfare Center, Newport, Rhode Island, December 2001.

    Google Scholar 

  70. K. G. Canfield and D. L. Jones, Implementing time-frequency representations for non-Cohen classes, Proceedings Twenty-Seventh Asilomar Conference on Signals, Systems and Computers, (Pacific Grove, California), November 1993.

    Google Scholar 

  71. V. S. Praveenkumar, Implementation of hyperbolic class of time frequency distributions and removal of cross-terms, Master’s thesis, University of Rhode Island, Kingston, Rhode Island, May 1995.

    Google Scholar 

  72. R. L. Murray, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, New higher order spectra and time-frequency representations for dispersive signal analysis, in Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 4, (Seattle, Washington), pp. 2305–2308, May 1998.

    Google Scholar 

  73. R. L. Murray. A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, A new class of affine higher order time-frequency representations, in Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 3, (Phoenix, Arizona), pp. 1613–1616, March 1999.

    Google Scholar 

  74. B. Iem, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, Classes of smoothed Weyl symbols, IEEE Signal Processing Letters, vol. 7, pp. 186–188, July 2000.

    Article  Google Scholar 

  75. B. G. Iem, A. Papandreou-Suppappola, and G. E Boudreaux-Bartels, Wideband Weyl symbols for dispersive time-varying processing of systems and random signals, IEEE Trans. on Signal Processing, vol. 50, pp. 1077–1090, May 2002.

    Article  MathSciNet  Google Scholar 

  76. B. G. Iem, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, New concepts in narrowband and wideband Weyl correspondence time-frequency techniques, in Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 3, (Seattle, Washington), pp. 1573–1576, May 1998.

    Google Scholar 

  77. B. G. Iem, A. Papandreou-Suppappola, and G. F. Boudreaux-Bartels, A wideband time-frequency Weyl symbol and its generalization, in Proceedings IEEE-SP International Symposium on Time-Frequency and Time-Scale Analysis, (Pittsburgh, Pennsylvania), pp. 29–32, October 1998.

    Google Scholar 

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  1. Antonia Papandreou-Suppappola
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Editors and Affiliations

  1. Department of Mathematics, The University of Texas-Pan American, Edinburg, TX, 78539-2999, USA

    Lokenath Debnath

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Papandreou-Suppappola, A. (2003). Time-Frequency Processing of Time-Varying Signals with Nonlinear Group Delay. In: Debnath, L. (eds) Wavelets and Signal Processing. Applied and Numerical Harmonic Analysis. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0025-3_10

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  • DOI: https://doi.org/10.1007/978-1-4612-0025-3_10

  • Publisher Name: Birkhäuser, Boston, MA

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