On Chirp and Some Related Signals Analysis: A Brief Review and Some New Results


Chirp signals have played an important role in the statistical signal processing literature. An extensive amount of work has been done in analyzing different one dimensional chirp, two dimensional chirp and some related signal processing models. The main aim of this article is to introduce the challenges associated with these problems to the statistical community with a hope that it will generate enough interests among the statisticians to contribute in this area of research. We provide a comprehensive review of different one dimensional, two dimensional, generalized chirp signals and some related models. We discuss some new chirp signal like models which can be used quite effectively for analyzing real life signals. Several open problems are discussed through out the paper for future research.

This is a preview of subscription content, access via your institution.


  1. Abatzoglou, T. (1986). Fast maximum likelihood joint estimation of frequency and frequency rate. IEEE Trans. Aero. Elec. Syst., 22, 708–715.

  2. Amar, A., Leshem, A. and van der Veen, A. J. (2010). A low complexity blind estimator of narrowband polynomial phase signals. IEEE Trans. Sig. Proc., 58, 4674–4683.

  3. Bai, Z.D., Rao, C.R., Chow, M. and Kundu, D. (2003). An efficient algorithm for estimating the parameters of superimposed exponential signals. J. Stat. Plan. Inf., 110, 23–34.

  4. Bai, Z.D., Rao, C.R., Wu, Y., Zen, M-M. and Zhao, L. (1999). The simultaneous estimation of the number of signals and frequencies of multiple sinusoids when some observations are missing: I. Asymptotic. Proc. Nat. Acad. Sci., 96, 11106–11110.

  5. Benson, O., Ghogho, M. and Swami, A. (1999). Parameter estimation for random amplitude chirp signals. IEEE Trans. Sig. Proc., 47, 3208–3219.

  6. Besson, O., Giannakis, G.B. and Gini, F. (1999). Improved estimation of hyperbolic frequency modulated chirp signals. IEEE Trans. Sig. Proc., 47, 1384–1388.

  7. Cao, F., Wang, S. and Wang, F. (2006). Cross-Spectral Method Based on 2-D Cross Polynomial Transform for 2-D Chirp Signal Parameter Estimation. ICSP2006 Proceedings, https://doi.org/10.1109/ICOSP.2006.344475

  8. Djukanović, S. and Djurović, I. (2012). Aliasing detection and resolving in the estimation of polynomial-phase signal parameters. Sig. Proc., 92, 235–239.

  9. Dhar, S.S., Kundu, D. and Das, U. (2019). On testing parameters of chirp signal model. IEEE Trans. Sig. Proc., 67, 4291–4301.

  10. Djurić, P.M. and Kay, S.M. (1990). Parameter estimation of chirp signals. IEEE Trans. Acous. Speech. Sig. Proc., 38, 2118–2126.

  11. Djurović, I., Simeunović, M. and Wang, P. (2017). Cubic phase function: A simple solution for polynomial phase signal analysis. Sig. Proc., 135, 48–66.

  12. Djurović, I. and Stanković L. J. (2014). Quasi maximum likelihood estimators of polynomial phase signals. IET Sig. Proc., 13, 347–359.

  13. Djurović, I., Wang, P. and Ioana, C. (2010). Parameter estimation of 2-D cubic phase signal function using genetic algorithm. Sig. Proc., 90, 2698–270.

  14. Doweck, Y., Amar, A. and Cohen, I. (2015). Joint model order selection and parameter estimation of chirps with harmonic components. IEEE Trans. Sig. Proc., 63, 1765–1778.

  15. Farquharson, M., O’Shea, P. and Ledwich, G. (2005). A computationally efficient technique for estimating the parameters phase signals from noisy observations. IEEE Trans. Sig. Proc., 53, 3337–3342.

  16. Fourier, D., Auger, F., Czarnecki, K. and Meignen, S. (2017). Chirp rate and instantaneous frequency estimation: application to recursive vertical synchrosqueezing. IEEE Sig. Proc. Lett., 1-1, 99.

  17. Francos, J.M. and Friedlander, B. (1998). Two-dimensional polynomial phase signals: parameter estimation and bounds. Multi. Sys. Sig. Proc., 9, 173–205.

  18. Francos, J.M. and Friedlander, B. (1999). Parameter estimation of 2-D random amplitude polynomial phase signals. IEEE Trans. Sig. Proc., 47, 1795–1810.

  19. Friedlander, B. and Francos, J.M. (1996). An estimation algorithm for 2-D polynomial phase signals. IEEE Trans. Image. Proc., 5, 1084–1087.

  20. Gabor, D. (1946). Theory of communication. Part 1: The analysis of information” J. Ins. Elec. Eng. - Part III: Rad. Com. Eng., 93, 429–441.

  21. Gini, F., Montanari, M. and Verrazzani, L. (2000). Estimation of chirp signals in compound Gaussian clutter: a cyclostationary approach. IEEE Trans. Acous. Speech. Sign. Proc., 48, 1029–1039.

  22. Grover, R., Kundu, D. and Mitra, A. (2018a). On approximate least squares estimators of parameters of one-dimensional chirp signal. Statistics, 52, 1060–1085.

  23. Grover, R, Kundu, D. and Mitra, A. (2018b). Asymptotic of approximate least squares estimators of parameters of two-dimensional chirp signal. J. Multi. Anal., 168, 211–220.

  24. Grover, R., Kundu, D. and Mitra, A. (2018c). A chirp like model to analyze periodic and nearly periodic signals. arXiv.

  25. Guo, J., Zou, H., Yang, X. and Liu, G. (2011). Parameter estimation of multicomponent chirp signals via sparse representation. IEEE Trans. Aero. Elec. Sys., 47, 2261–2268.

  26. Hedley, M. and Rosenfeld, D. (1992). A new two-dimensional phase unwrapping algorithm for MRI images. Mag. Reso. Med., 24, 177–181.

  27. Ikram, M.Z., Abed-Meraim, K. and Hua, Y. (1998). Estimating the parameters of chirp signals: an iterative approach. IEEE Trans. Sig. Proc., 46, 3436–3441.

  28. Ikram, M.Z. and Zhou, G.T. (2001). Estimation of multicomponent phase signals of mixed orders. Sig. Pro., 81, 2293–2308.

  29. Irizarry, R. A. (2000). Asymptotic distribution of estimates for a time-varying parameter in a harmonic model with multiple fundamentals. Stat. Sinica., 10, 1041–1067.

  30. Jennrich, R.I. (1969). Asymptotic properties of the nonlinear least squares estimators. Annal. Math. Stat., 40, 633–643.

  31. Jensen, T.L., Nielsen, J.K., Jensen, J.R., Christensen, M.G. and Jensen, S.H. (2017). A fast algorithm for maximum likelihood estimation of harmonic chirp parameters. IEEE Trans. Sig. Proc., 65, 5137–5152.

  32. Kennedy, Jr., W.J. and Gentle, J.E (1980). Statistical Computing, Marcel Dekker, Inc., New York.

  33. Kundu, D. (2020). Professor C.R. Rao’s contribution in Statistical Signal Processing and its Longterm Implications. Proceedings of Indian Academy of Sciences (Mathematical Sciences), 130, Article No. 43.

  34. Kundu, D. and Nandi, S. (2003). Determination of discrete spectrum in a random field. Stat. Neerlandica., 57, 258–283.

  35. Kundu, D. and Nandi, S. (2008). Parameter estimation of chirp signals in presence of stationary Christensen, noise. Statistica Sinica, 18, 187–201.

  36. Lahiri, A. (2011). Estimators of parameters of chirp signals and their properties, Ph.D. thesis, Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India.

  37. Lahiri, A. and Kundu, D. (2017). On parameter estimation of two-dimensional polynomial phase signal model. Stat. Sinica., 27, 1779–1792.

  38. Lahiri, A., Kundu, D. and Mitra, A. (2012). Efficient algorithm for estimating the parameters of chirp signal. J. Multi. Anal., 108, 15–27.

  39. Lahiri, A., Kundu, D. and Mitra, A. (2013). Efficient algorithm for estimating the parameters of two dimensional chirp signal. Sankhya, Ser. B, 75, 65–89.

  40. Lahiri, A., Kundu, D. and Mitra, A. (2014). On least absolute deviation estimator of one dimensional chirp model. Statistics, 48, 405–420.

  41. Lahiri, A., Kundu, D. and Mitra, A. (2015). Estimating the parameters of multiple chirp signals. J. Multi. Anal., 139, 189–205.

  42. Lin, C.C. and Djurić, P.M. (2000). Estimation of chirp signals by MCMC. ICASSP-1998, 1, 265–268.

  43. Liu, X. and Yu, H. (2013). Time-domain joint parameter estimation of chirp signal based on SVR. Math. Prob. Eng., Article ID 952743, 1–9.

  44. Lu, Y., Demirli, R., Cardoso, G. and Saniie, J. (2006). A successive parameter estimation algorithm for chirplet signal decomposition. IEEE Trans. Ultra. Ferroelec. Freq. Cont., 53.

  45. Mazumder, S. (2017). Single-step and multiple-step forecasting in one-dimensional chirp signal using MCMC-based Bayesian analysis. Commu. Stat. - Simul. Comp., 46, 2529–2547.

  46. Montgomery, H.L. (1990). Ten Lectures on the Interface Between Analytic Number Theory and Harmonic Analysis, American Mathematical Society, 196.

  47. Nandi, S. and Kundu, D. (2003). Estimating the fundamental frequency of a periodic function. Stat. Met. App., 12, 341–360.

  48. Nandi, S. and Kundu, D. (2004). Asymptotic properties of the least squares estimators of the parameters of the chirp signals. Ann. Inst. Stat. Mathe., 56, 529–544.

  49. Nandi, S. and Kundu, D. (2006). Analyzing non-stationary signals using a cluster type model. J. Stat. Plan. Inf., 136, 3871–3903.

  50. Nandi, S. and Kundu, D. (2020). Random amplitude chirp model. Signal Processing, 168, Art. 107328.

  51. Nandi, S. and Kundu, D. (2020). Statistical Signal Processing, 2nd edition, Springer, London.

  52. Nandi, S., Kundu, D. and Grover, R. (2019). Estimation of Parameters of Multiple Chirp Signal in presence of Heavy Tailed Errors. submitted for publication.

  53. O’Shea, P. (2010). On refining polynomial phase signal parameter estimates. IEEE Trans. Aero. Elec. Sys., 4, 978–987.

  54. Pelag, S. and Porat, B. (1991). Estimation and classification of polynomial phase signals. IEEE Trans. Info. Theo., 37, 422–430.

  55. Pincus, M. (1968). A closed form solution of certain programming problems. Oper. Res., 16, 690–694.

  56. Quinn, B. G. and Thomson, P. J. (1991). Estimating the frequency of a periodic function. Biometrika, 78, 65–74.

  57. Rihaczek, A.W. (1969). Principles of high resolution radar, McGraw-Hill, New York.

  58. Robertson, S.D., Gray, H.L. and Woodward, W.A. (2010). The generalized linear chirp process. J. Stat. Plan. Infe., 140, 3676–3687.

  59. Saha, S. and Kay, S.M. (2002). Maximum likelihood parameter estimation of superimposed chirps using Monte Carlo importance sampling. IEEE Trans. Sig. Proc., 50, 224–230.

  60. Seber, G. A. F. and Wild, C.J. (1989). Nonlinear Regression, Wiley, New York.

  61. Ticahvsky, P. and Handel, P. (1999). Multicomponent polynomial phase signal analysis using a tracking algorithm. IEEE Trans. Sig. Proc., 47, 1390–1395.

  62. Volcker, B. and Ottersten, B. (2001). Chirp parameter estimation from a sample covariance matrix. IEEE Trans. Sig. Proc., 49, 603–612.

  63. Wang, P. and Yang, J. (2006). Multicomponent chirp signals analysis using product cubic phase function. Dig. Sig. Proc., 16, 654–669.

  64. Wang, Y. and Zhou, Y.G.T. (1998). On the use of high-order ambiguity function for multi-component polynomial phase signals. Sig. Proc., 5, 283–296.

  65. Wu, C.F.J. (1981). Asymptotic theory of the nonlinear least squares estimation. Ann. Stat., 9, 501–513.

  66. Wang, J. Z., Su, S. Y. and Chen, Z. (2015). Parameter estimation of chirp signal under low SNR. Sci. China: Info. Sci., 58, 020307:1–020307:13.

  67. Xinghao, Z., Ran, T. and Siyong, Z. (2003). A novel sequential estimation algorithm for chirp signal parameters. IEEE Conf. Neu. Netw. Sig. Proc., Nanjing, China, December 14-17, 2003, pp 628–631.

  68. Yang, P., Liu, Z. and Jiang, W.-L. (2015). parameter estimation of multicomponent chirp signals based on discrete chirp Fourier transform and population Monte Carlo. Sig. Img. Vid. Proc., 9, 1137–1149.

  69. Yaron, D., Alon, A. and Israel, C. (2015). Joint model order selection and parameter estimation of chirps with harmonic components. IEEE Trans. Sig. Proc., 63, 1765–1778.

  70. Zhang, H., Liu, H., Shen, S., Zhang, Y. and Wang, X. (2013). Parameter estimation of chirp signals based on fractional Fourier transform. J. China Univ. Posts Telecomm. 20, 95–100.

    Article  Google Scholar 

  71. Zhang, H. and Liu, Q. (2006). Estimation of instantaneous frequency rate for multicomponent polynomial phase signals. ICSP2006 Proceedings, 498–502, https://doi.org/10.1109/ICOSP.2006.344448

  72. Zhang, K., Wang, S. and Cao, F. (2008). Product Cubic Phase Function Algorithm for Estimating the Instantaneous Frequency Rate of Multicomponent Two-dimensional Chirp Signals. 2008 Congress on Image and Signal Processing, https://doi.org/10.1109/CISP.2008.352

Download references


The authors would like to thank the unknown reviewers for their helpful comments which have helped to improve the manuscript significantly. Part of the work of the first author has been supported by a grant from SERB, Government of India.

Author information



Corresponding author

Correspondence to Debasis Kundu.

Additional information

This paper has been dedicated to Professor C.R.Rao on his birth centenary

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kundu, D., Nandi, S. On Chirp and Some Related Signals Analysis: A Brief Review and Some New Results. Sankhya A (2021). https://doi.org/10.1007/s13171-021-00242-7

Download citation


  • Chirp signals
  • Sinusoidal signals
  • Periodogram function
  • Maximum likelihood estimators
  • MCMC model
  • Least squares estimators
  • Consistency
  • Asymptotic normality.

AMS Subject Classifications

  • 62F10
  • 62F03
  • 62H12