Formulation of New Hyperbolic Time-shift Covariant Time-frequency Symbols and Its Applications

  • Iem, Byeong-Gwan (Department of Control and Instrumentation Engineering Kangnung National University)
  • Published : 2003.03.01

Abstract

We propose new time-frequency (TF) tools for analyzing linear time-varying (LTV) systems and nonstationary random processes showing hyperbolic TF structure. Obtained through hyperbolic warping the narrowband Weyl symbol (WS) and spreading function (SF) in frequency, the new TF tools are useful for analyzing LTV systems and random processes characterized by hyperbolic time shifts. This new TF symbol, called the hyperbolic WS, satisfies the hyperbolic time-shift covariance and scale covariance properties, and is useful in wideband signal analysis. Using the new, hyperbolic time-shift covariant WS and 2-D TF kernels, we provide a formulation for the hyperbolic time-shift covariant TF symbols, which are 2-D smoothed versions of the hyperbolic WS. We also propose a new interpretation of linear signal transformations as weighted superposition of hyperbolic time shifted and scale changed versions of the signal. Application examples in signal analysis and detection demonstrate the advantages of our new results.

Keywords

References

  1. F. Hlawatsch and G. F. Boudreaux-BarteIs, 'Linear and quadratic time-frequency signal representations,' IEEE Signal Proc. Magazine, 9, 21-67, April 1992 https://doi.org/10.1109/79.127284
  2. L. Cohen, 'Time-frequency distributions,' Proceedings of the IEEE, 77, 941-981, July 1989 https://doi.org/10.1109/5.30749
  3. W. Kozek, 'On the generalized Weyl correspondence and its application to time-frequency analysis of linear time-varying systems,' IEEE-SP Int. Symp. on TFTS, Canada, 167-170, Oct. 1992
  4. W. Kozek, F. Hlawatsch, H. Kirchauer and U. Trautwein, 'Correlative TF analysis and classification of nonstationary processes,' IEEE-SP Ini. Symp. on TFTS, Phil., PA, 417-420, Oct. 1994
  5. R. G. Shenoy and T. W. Parks, 'The Weyl Correspodence and TF analysis,' IEEE Trans. on Sig. Proc., 318-331, Feb. 1994
  6. B. lem, 'Generalizations of the Weyl symbol and the spreading function via TF warpings: Theory and application,' Ph.D dissertation, Univ. of RI, Kingston, RI, 1998
  7. B. lem, A. Papandreou-SuppappoIa and G. F. Boudreaux-Bartels, 'Wideband Weyl symbols for dispersive time-varying processing of systems and random signals,'IEEE Trans. on Signal Proc., 1077-1090, May 2002
  8. I. Gohberg and S. Goldberg, Basic Operator Theory, (Birkhauser, Boston, MA, 1980)
  9. G. Matz and F. Hlawatsch, 'Time-frequency formulation and design of optimal detectors,' Inf. SymP. on TFTS, Paris, 213-216, June 1996
  10. A. M. Sayeed and D. L. Jones, 'Optimal quadratic detection and estimation using generalized joint signal representations' IEEE Trans. on Signal Proc., 3031-3043, Dec. 1996
  11. G. Matz, F. Hlawatsch, and W. Kozek, 'Generalized evolutionary spectral analysis and the Weyl spectrum of nonstationary random processes,' IEEE Trans. on Signal Proc., 1520-1534, June 1997
  12. R. G. Shenoy and T. W. Parks, 'Wideband ambiguity functions and affine Wigner distributions,' Proc. EURASIP, 41, 339-363, 1995
  13. P. Bertrand and J. Bertrand, 'TF representation of broad band signals,' Proc. IEEE ICASSP, NY, 2196-2199, April 1988
  14. A. Papandreou, F. Hlawatsch and G. F. Boudreaux-Bartels, 'The hyperbolic class of QTFRs, Part I,' IEEE Trans. on Signal Proc., 3425-3444, Dec. 1993