Skip to main content
SpringerLink
Log in

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

  • Published:
Multidimensional Systems and Signal Processing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akan A. (2005). Signal-adaptive evolutionary spectral analysis using instantaneous frequency estimation. FREQUENZ Journal of RF-Engineering and Telecommunications, 59(7–8): 201–205

    Google Scholar 

  • Akan A., Chaparro L.F. (2001). Evolutionary chirp representation of non-stationary signals via Gabor transform. Signal Processing, 81(11): 2429–2436

    Article  MATH  Google Scholar 

  • 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): 704–711

    Article  Google Scholar 

  • Baraniuk R., Jones D. (1996). Wigner-based formulation of the chirplet transform. IEEE Transactions on Signal Processing, 44(12): 3129–3135

    Article  Google Scholar 

  • 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.

  • Blough J. (2003). Development ands analysis of time variant discrete Fourier transform order tracking. Mechanical Systems and Signal Processing, 17(6): 1185–1199

    Article  Google Scholar 

  • 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): 216–220

    Article  Google Scholar 

  • Boashash, B., & Ristic, B. (1993). Polynomial WVD’s and time-varying polyspectra. In B. Boashash et al. (Eds.), Higher order statistical processing. Longman Cheshire.

  • Bossley K.M., Mckendrick R.J. (1999). Hybrid computed order tracking. Mechanical Systems and Signal Processing, 13(4):627–641

    Article  Google Scholar 

  • Bultan A. (1999). A four-parameter atomic decomposition of chirplets. IEEE Transactions on Signal Processing, 47(3): 731–745

    Article  MATH  MathSciNet  Google Scholar 

  • Fan P., Xian X. (2000). A modified discrete chirp-Fourier transform scheme. World Computer Congress, 1, 57–60

    Google Scholar 

  • Fyfe K.R., Munck E.D. (1997). Analysis of computed order tracking. Mechanical Systems and Signal Processing, 11(2): 187–205

    Article  Google Scholar 

  • Gelman L. (2007). Adaptive time–frequency transform for non-stationary signals with nonlinear polynomial frequency variation. Mechanical Systems and Signal Processing, 21(6): 2684–2687

    Article  Google Scholar 

  • 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.

  • 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): 2980–3002

    Article  Google Scholar 

  • 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): 1254–1262

    Article  Google Scholar 

  • Katkovnik V. (1996). Local polynomial approximation of the instantaneous frequency: Asymptotic accuracy. Signal Processing, 52(3): 343–356

    Article  MATH  Google Scholar 

  • Katkovnik V. (1997). Erratum to “Local polynomial approximation of the instantaneous frequency: Asymptotic accuracy”. Signal Processing, 59(2): 251–252

    Article  Google Scholar 

  • 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.

  • Mann, S., & Haykin, S. (1991). The chirplet transform-a generalization of Gabor’s Logon transform. In: International Conference “Vision Interface-91”.

  • Mann S., Haykin S. (1992a). Chirplets’ and ‘warblets’: Novel time–frequency methods. Electronics Letters, 28(2): 114–116

    Article  Google Scholar 

  • Mann S., Haykin S. (1992b). Time–frequency perspectives: The chirplet transform. Proceedings IEEE International Conference Acoustics, Speech. Signal Processing, 3, 417–420

    Google Scholar 

  • Mann S., Haykin S. (1995). The chirplet transform: Physical considerations. IEEE Transactions on Signal Processing, 43(11): 2745–2761

    Article  Google Scholar 

  • Mecklenbrauker, W., & Hlawatsch. (1997). The Wigner distribution-theory and application in signal processing. Elsevier Science B.V.

  • Mercado E., Myers C., Gluck M. (2000). Modeling auditory cortical processing as an adaptive chirplet tranfrom. Neurocomputing, 32–33, 913–919

    Article  Google Scholar 

  • Mihovilovic D., Bracewell R. (1991). Adaptive chirplet representation of signals on time–frequency plane. Electronics Letters, 27(13): 1159–1161

    Article  Google Scholar 

  • Mihovilovic D., Bracewell R. (1992). Whistler analysis in the time–frequency plane using chirplets. Journal Geophysics Research, 97(A11): 17199–17204

    Article  Google Scholar 

  • Mitra, S. (2005). Digital signal proessing. Tata McGraw-Hill.

  • 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).

  • Proakis, J., & Manolakis, D. (1996). Digital signal processing. Prentice Hall.

  • Rabiner L., Gold B. (1975). Theory and application of digital signal processing. London, Prentice Hall International

    Google Scholar 

  • Shao H., Jin W., Qian S. (2003). Order tracking by discrete Gabor expansion. IEEE Transactions on Instrumentation and measurement, 52(3): 754–761

    Article  Google Scholar 

  • Stankovic L. (1997). Local polynomial Wigner distribution. Signal Processing, 59, 123–128

    Article  MATH  Google Scholar 

  • Stankovic L. (1998). On the realization of the polynomial Wigner–Ville distribution for multicomponent signals. IEEE Signal Processing Letters, 5(7): 157–159

    Article  MathSciNet  Google Scholar 

  • Van Trees H.L. (1968). Detection, estimation, and modulation theory. New York, Wiley

    MATH  Google Scholar 

  • 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.

  • 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.

  • 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): 101–122

    Article  MATH  Google Scholar 

  • Wang G., Bao Z. (1999). Inverse synthetic aperture radar imaging of maneuvering targets based on chirplet decomposition. Optical Engineering, 39(9): 1534–1541

    Google Scholar 

  • 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): 292–298

    Article  Google Scholar 

  • Watson, G. N. (1944). A treatise on the theory of Bessel functions. Cambridge University Press.

  • Wexler J., Raz S. (1990). Discrete Gabor expansions. Signal Processing, 21(3): 207–221

    Article  Google Scholar 

  • Xian X. (2000). Discret e chirp-Fourier transform and its application to chirp rate estimation. IEEE Transactions on Signal Processing, 48(11): 3122–3133

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Cranfield University, College Road, Cranfield, MK43 0AL, UK

    Len Gelman & Jeremy Gould

Authors
  1. Len Gelman
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jeremy Gould
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Len Gelman.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Gelman, L., Gould, J. A new time-frequency transform for non-stationary signals with any nonlinear instantaneous phase. Multidim Syst Sign Process 19, 173–198 (2008). https://doi.org/10.1007/s11045-007-0042-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11045-007-0042-z

Keywords

Search

Navigation