Signal, Image and Video Processing

, Volume 13, Issue 3, pp 617–625 | Cite as

Adaptive importance sampling for estimating multi-component chirp signal parameters in colored noise

  • Peng YangEmail author
  • Hong Ding
  • Hui Xiong
  • Linhua Zheng
Original Paper


The problem of estimating the center frequency and the chirp rate of multi-component chirp signals from data corrupted with colored noise often occurs in many signal processing applications. In the Bayesian framework, an algorithm based on adaptive importance sampling (AIS) and modified discrete chirp Fourier transform (MDCFT) was introduced in the present paper in order to calculate the center frequency and the chirp rate of chirp signals from observations corrupted by additive autoregressive (AR) random noise. First, importance functions (IFs) were constructed to approximate the marginal posterior distributions of interested parameters with observed data, and AIS was then employed to evaluate the posterior distributions. MDCFT was adopted to detect the spectral peaks of chirp signals, and the obtained results were applied to construct IFs with multivariate student-t distribution. It was found that the quality of the samples generated from multivariate student-t distribution was gradually improved. Finally, computer simulations validated the results obtained from the proposed algorithm. In comparison with the Markov chain-based method, the proposed method had better SNR adaptability and excellent convergence rate and required fewer samples. Furthermore, the proposed algorithm can be considered as a further expansion of previous works related to AIS method for estimating multi-component chirp signal parameters in white noise.


Colored noise Multi-component chirp signals Bayesian inference Parameter estimation Modified discrete chirp Fourier transform Adaptive importance sampling 


  1. 1.
    Li, H.S., Djuric, P.M.: MMSE estimation of nonlinear parameters of multiple linear & quadratic chirps. IEEE Trans. Signal Process. 46(3), 796–800 (1998)CrossRefGoogle Scholar
  2. 2.
    Djuric, P.M., Kay, S.M.: Parameter estimation of chirp signals. IEEE Trans. Acoust. Speech Signal Process. 38(12), 2118–2126 (1999)CrossRefGoogle Scholar
  3. 3.
    Sucic, V., Lerga, J., Boashash, B.: Multicomponent noisy signal adaptive instantaneous frequency estimation using components time support information. IET Signal Process. 8(3), 277–284 (2013)CrossRefGoogle Scholar
  4. 4.
    Lerga, J., Sucic, V., Boashash, B.: An improved method for nonstationary signals components extraction based on the ICI rule. In: International Workshop on Systems, Signal Processing and Their Applications, Tipaza, Algeria, pp. 307–310 (2011)Google Scholar
  5. 5.
    Lerga, J., Sucic, V.: An instantaneous frequency estimation method based on the improved sliding pair-wise ICI rule. In: International Conference on Information Science, Signal Processing and Their Applications, Kuala Lumpur, Malaysia, pp. 161–164 (2010)Google Scholar
  6. 6.
    Lerga, J., Sucic, V.: Nonlinear IF estimation based on the pseudo WVD adapted using the improved sliding pairwise ICI rule. IEEE Signal Process. Lett. 16(11), 953–956 (2009)CrossRefGoogle Scholar
  7. 7.
    Li, Y., Zhang, F., Li, Y., Tao, R.: Application of linear canonical transform correlation for detection of linear frequency modulated signals. IET Signal Process. 10(4), 351–358 (2016)CrossRefGoogle Scholar
  8. 8.
    Jensen, T.L., Nielsen, J.K., Jensen, J.R., Christensen, M.G., Jensen, S.H.: A fast algorithm for maximum-likelihood estimation of harmonic chirp parameters. IEEE Trans. Signal Process. 65(19), 5137–5152 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Serbes, A.: On the estimation of LFM signal parameters: analytical formulation. IEEE Trans. Aerosp. Electron. Syst. 54(2), 848–860 (2018)CrossRefGoogle Scholar
  10. 10.
    Guo, J., Zou, H., Yang, X., Liu, G.: Parameter estimation of multicomponent chirp signals via sparse representation. IEEE Trans. Aerosp. Electron. Syst. 47(3), 2261–2268 (2011)CrossRefGoogle Scholar
  11. 11.
    Millioz, F., Davies, M.: Sparse detection in the chirplet transform: application to FMCW radar signals. IEEE Trans. Signal Process. 60(6), 2800–2813 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Bugallo, M.F., Hong, M., Djuric, P.M.: Marginalized population Monte Carlo. In: IEEE International Conference on Acoustics, Speech and Signal Processing, Taipei, Taiwan, pp. 2925–2928 (2009)Google Scholar
  13. 13.
    Saha, S., Kay, S.M.: Maximum likelihood parameter estimation of superimposed chirps using Monte Carlo importance sampling. IEEE Trans. Signal Process. 50(2), 224–230 (2002)CrossRefGoogle Scholar
  14. 14.
    Cho, C.M.: Detection and estimation of multiple cisoids in colored noise by Bayesian predictive densitied. In: IEEE International Conference on Acoustics, Speech and Signal Processing, Adelaide, SA, Australia, pp. 501–504 (1994)Google Scholar
  15. 15.
    Cho, C.M., Djuric, P.M.: Bayesian detection and estimation of cisoids in colored noise. IEEE Trans. Signal Process. 43(12), 2943–2952 (1995)CrossRefGoogle Scholar
  16. 16.
    Mallat, S.G., Zhang, Z.: Matching pursuits with time-frequency dictionaries. IEEE Trans. Signal Process. 41(12), 3397–3415 (1993)CrossRefzbMATHGoogle Scholar
  17. 17.
    Andrieu, C., Doucet, A.: Joint Bayesian model selection and estimation of noisy sinusoids via reversible jump MCMC. IEEE Trans. Signal Process. 47(10), 2667–2676 (1999)CrossRefGoogle Scholar
  18. 18.
    Davy, M., Doncarli, C., Tourneret, J.Y.: Classification of chirp signals using hierarchical Bayesian learning and MCMC methods. IEEE Trans. Signal Process. 50(2), 377–388 (2002)CrossRefGoogle Scholar
  19. 19.
    Lin, C.C., Djuric, P.M.: Estimation of chirp signals by MCMC. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, Istanbul, Turkey, Turkey, pp. 265–268 (2000)Google Scholar
  20. 20.
    Lin, Y., Peng, Y., Wang, X.: Maximum likelihood parameter estimation of multiple chirp signals by a new Markov Chain Monte Carlo approach. In: Proceedings of the 2004 IEEE Radar Conference, Philadelphia, pp. 559–562 (2004)Google Scholar
  21. 21.
    Cappe, O., Douc, R., Guillin, A., Marin, J.M., Robert, C.P.: Adaptive importance sampling in general mixture classes. Stat. Comput. 18(2), 447–459 (2008)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Darren, W., Martin, K., Karim, B., Olivier, C., Jean, F.C., Gersende, F., Simon, P., Christian, P.R.: Estimation of cosmological parameters using adaptive importance sampling. Phys. Rev. D 80(2), 101–117 (2009)Google Scholar
  23. 23.
    Hong, M., Bugallo, M.F., Djuric, P.M.: Joint model selection and parameter estimation by population Monte Carlo simulation. IEEE J. Sel. Top. Signal Process. 4(3), 526–539 (2010)CrossRefGoogle Scholar
  24. 24.
    Xia, X.G.: Discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Trans. Signal Process. 48(11), 3122–3133 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Guo, X., Sun, H.B., Wang, S.L.: Comments on discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Trans. Signal Process. 50(12), 3115–3116 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Xia, X.G.: Response to comments on discrete chirp-Fourier transform and its application to chirp rate estimation. IEEE Trans. Signal Process. 50(12), 3116 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Tsung, I.: Bayesian analysis of mixture modelling using the multivariate t distribution. Stat. Comput. 14(2), 447–459 (2004)MathSciNetGoogle Scholar
  28. 28.
    Yang, P., Liu, Z., Jiang, W.L.: Parameter estimation of multi-component Chirp signals based on discrete Chirp Fourier transform and population Monte Carlo. Signal Image Video Process. 9(5), 1137–1149 (2015)CrossRefGoogle Scholar
  29. 29.
    Robert, C.P.: The Bayesian Choice. Springer, New York (2007)Google Scholar
  30. 30.
    Larocque, J.R., Reilly, J.P.: Reversible jump MCMC for joint detection and estimation of sources in colored noise. IEEE Trans. Signal Process. 50(2), 231–240 (2002)CrossRefGoogle Scholar
  31. 31.
    Petre, S., Andreas, J., Jian, L.: Cisoid parameter estimation in the colored noise case: asymptotic Cramer-Rao bound, maximum likelihood, and nonlinear least-squares. IEEE Trans. Signal Process. 45(8), 2048–2059 (1997)CrossRefGoogle Scholar

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© Springer-Verlag London Ltd., part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Electronic ScienceSchool of National University of Defense TechnologyChangshaChina

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