A new time-frequency transform for non-stationary signals with any nonlinear instantaneous phase



A new generic adaptive time-frequency transform based on the Wigner distribution is proposed for amplitude estimation of transient signals with any nonlinear non-polynomial variation of the instantaneous frequency. It is for use in situations where independent synchronous measurements of the instantaneous frequency are available. It shows that the new transform tracks the instantaneous frequency and estimates signal amplitude without errors along the curve of the instantaneous frequency. The paper also applies the new transform to signals with the non-linear sinusoidal and exponential variations in the instantaneous phase and determines formulae for transform in these cases. The paper compares the new transform with the Wigner distribution in several cases and demonstrates that the new transform is more effective at amplitude estimation of signals with nonlinear variation of the instantaneous frequency. New analytic formulae are obtained for the new transform and the Wigner distribution in both the sinusoidal and the exponential cases. An analytic formula is obtained which relates the new transform to the Wigner distribution in the sinusoidal case. This formula is inverted to obtain a previously unknown formula for the Wigner distribution of any signal. It is shown that the new transform could be used for adaptive processing of transient signals with instantaneous frequency variations. The paper studies the performance of the new transform with no adaptation, partial adaptation and complete adaptation.


Wigner distribution Time-frequency analysis Non-linear variation of the instantaneous phase Adaptation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Akan A. (2005). Signal-adaptive evolutionary spectral analysis using instantaneous frequency estimation. FREQUENZ Journal of RF-Engineering and Telecommunications, 59(7–8): 201–205Google Scholar
  2. Akan A., Chaparro L.F. (2001). Evolutionary chirp representation of non-stationary signals via Gabor transform. Signal Processing, 81(11): 2429–2436MATHCrossRefGoogle Scholar
  3. Angrisani L., D’Arco M. (2002). A measurement method based on a modified version of the chirplet transform for instantaneous frequency estimation. IEEE Transactions on Instrumentation and Measurement, 51(4): 704–711CrossRefGoogle Scholar
  4. Baraniuk R., Jones D. (1996). Wigner-based formulation of the chirplet transform. IEEE Transactions on Signal Processing, 44(12): 3129–3135CrossRefGoogle Scholar
  5. Blödt, M., Chabert, M., Regnier, J., Faucher, J., & Dagues, B. (2005). Detection of mechanical load faults in induction motors at variable speed using stator current time–frequency analysis. In IEEE International Symposium on Diagnostics for Electric Machines, Power Electronics and Drives (SDEMPED ’05), Vienna, Austria, pp. 229–234.Google Scholar
  6. Blough J. (2003). Development ands analysis of time variant discrete Fourier transform order tracking. Mechanical Systems and Signal Processing, 17(6): 1185–1199CrossRefGoogle Scholar
  7. Boashash B., O’Shea P. (1995). Polynomial Wigner–Ville distributions and their relationship to time-varying higher order spectra. IEEE Transactions on Signal Processing, 42(1): 216–220CrossRefGoogle Scholar
  8. Boashash, B., & Ristic, B. (1993). Polynomial WVD’s and time-varying polyspectra. In B. Boashash et al. (Eds.), Higher order statistical processing. Longman Cheshire.Google Scholar
  9. Bossley K.M., Mckendrick R.J. (1999). Hybrid computed order tracking. Mechanical Systems and Signal Processing, 13(4):627–641CrossRefGoogle Scholar
  10. Bultan A. (1999). A four-parameter atomic decomposition of chirplets. IEEE Transactions on Signal Processing, 47(3): 731–745MATHCrossRefMathSciNetGoogle Scholar
  11. Fan P., Xian X. (2000). A modified discrete chirp-Fourier transform scheme. World Computer Congress, 1, 57–60Google Scholar
  12. Fyfe K.R., Munck E.D. (1997). Analysis of computed order tracking. Mechanical Systems and Signal Processing, 11(2): 187–205CrossRefGoogle Scholar
  13. Gelman L. (2007). Adaptive time–frequency transform for non-stationary signals with nonlinear polynomial frequency variation. Mechanical Systems and Signal Processing, 21(6): 2684–2687CrossRefGoogle Scholar
  14. Gelman, L., & Adamenko, D. (2000). Usage of the Wigner-Ville distribution for vibro-acoustcal diagnostics of cracks. Proceedings of the Meeting of the Acoustical Society of America, Journal of the Acoustical Society of America, 108(5), Pt. 2, pp. 452–453.Google Scholar
  15. Gelman L., Gould J. (2007). Time–frequency chirp-Wigner transform for signals with any nonlinear polynomial time varying instantaneous frequency. Mechanical Systems and Signal Processing, 21(8): 2980–3002CrossRefGoogle Scholar
  16. Gelman L., Ottley M. (2006). New processing technique for transient signals with nonlinear variation of the instantaneous frequency in time. Mechanical Systems and Signal Processing, 20(5): 1254–1262CrossRefGoogle Scholar
  17. Katkovnik V. (1996). Local polynomial approximation of the instantaneous frequency: Asymptotic accuracy. Signal Processing, 52(3): 343–356MATHCrossRefGoogle Scholar
  18. Katkovnik V. (1997). Erratum to “Local polynomial approximation of the instantaneous frequency: Asymptotic accuracy”. Signal Processing, 59(2): 251–252CrossRefGoogle Scholar
  19. Leuridan, J., Vold, H., Kopp, G., & Moshrefi, N. (1995). High resolution order tracking using Kalman tracking filters-theory and applications. In Proceedings of the SAE Noise and Vibration Conference. SAE Paper no. 951332.Google Scholar
  20. Mann, S., & Haykin, S. (1991). The chirplet transform-a generalization of Gabor’s Logon transform. In: International Conference “Vision Interface-91”.Google Scholar
  21. Mann S., Haykin S. (1992a). Chirplets’ and ‘warblets’: Novel time–frequency methods. Electronics Letters, 28(2): 114–116CrossRefGoogle Scholar
  22. Mann S., Haykin S. (1992b). Time–frequency perspectives: The chirplet transform. Proceedings IEEE International Conference Acoustics, Speech. Signal Processing, 3, 417–420Google Scholar
  23. Mann S., Haykin S. (1995). The chirplet transform: Physical considerations. IEEE Transactions on Signal Processing, 43(11): 2745–2761CrossRefGoogle Scholar
  24. Mecklenbrauker, W., & Hlawatsch. (1997). The Wigner distribution-theory and application in signal processing. Elsevier Science B.V.Google Scholar
  25. Mercado E., Myers C., Gluck M. (2000). Modeling auditory cortical processing as an adaptive chirplet tranfrom. Neurocomputing, 32–33, 913–919CrossRefGoogle Scholar
  26. Mihovilovic D., Bracewell R. (1991). Adaptive chirplet representation of signals on time–frequency plane. Electronics Letters, 27(13): 1159–1161CrossRefGoogle Scholar
  27. Mihovilovic D., Bracewell R. (1992). Whistler analysis in the time–frequency plane using chirplets. Journal Geophysics Research, 97(A11): 17199–17204CrossRefGoogle Scholar
  28. Mitra, S. (2005). Digital signal proessing. Tata McGraw-Hill.Google Scholar
  29. Poletti, M. (1992). Linearly swept frequency measurements and the Wigner–Ville distribution, chapter 19. In B. Boashash (Ed.), Time–Frequency signal analysis (pp. 424–444).Google Scholar
  30. Proakis, J., & Manolakis, D. (1996). Digital signal processing. Prentice Hall.Google Scholar
  31. Rabiner L., Gold B. (1975). Theory and application of digital signal processing. London, Prentice Hall InternationalGoogle Scholar
  32. Shao H., Jin W., Qian S. (2003). Order tracking by discrete Gabor expansion. IEEE Transactions on Instrumentation and measurement, 52(3): 754–761CrossRefGoogle Scholar
  33. Stankovic L. (1997). Local polynomial Wigner distribution. Signal Processing, 59, 123–128MATHCrossRefGoogle Scholar
  34. Stankovic L. (1998). On the realization of the polynomial Wigner–Ville distribution for multicomponent signals. IEEE Signal Processing Letters, 5(7): 157–159CrossRefMathSciNetGoogle Scholar
  35. Van Trees H.L. (1968). Detection, estimation, and modulation theory. New York, WileyMATHGoogle Scholar
  36. Vold, H., & Leuridan, J. (1993). High resolution order tracking at extreme slow rates, using Kalman tracking filters. In Proceedings of the SAE Noise and Vibration Conference. SAE Paper no. 931288.Google Scholar
  37. Vold, H., Mains, M., & Blough, J. R. (1997). Theoretical foundations for high performance order tracking with the Vold–Kalman filter. In Proceedings of Society of Automotive Engineers, Noise and Vibration Conference. SAE Paper no. 972007.Google Scholar
  38. Wahl T.J., Bolton J.S. (1993). The application of the Wigner distribution to the identification of structure-borne noise components. Journal of Sound and Vibration, 163(1): 101–122MATHCrossRefGoogle Scholar
  39. Wang G., Bao Z. (1999). Inverse synthetic aperture radar imaging of maneuvering targets based on chirplet decomposition. Optical Engineering, 39(9): 1534–1541Google Scholar
  40. Wang G., Xia X.-G., Root B.T., Chen V.C., Zhang Y., Amin M. (2003). Maneuvering target detection in over-the-horizon radar using adaptive clutter refection and adaptive chirplet transform. IEE Proceedings—Radar Sonar Navigation, 150(4): 292–298CrossRefGoogle Scholar
  41. Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge University Press.Google Scholar
  42. Wexler J., Raz S. (1990). Discrete Gabor expansions. Signal Processing, 21(3): 207–221CrossRefGoogle Scholar
  43. Xian X. (2000). Discret e chirp-Fourier transform and its application to chirp rate estimation. IEEE Transactions on Signal Processing, 48(11): 3122–3133CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Cranfield UniversityCranfieldUK

Personalised recommendations