Circuits, Systems, and Signal Processing

, Volume 37, Issue 8, pp 3441–3456 | Cite as

A Closed-Form ARMA-Based ML-Estimator of a Single-Tone Frequency

  • Igor Omelchuk
  • Iurii ChyrkaEmail author


The new synthesis methodology for single-tone frequency estimation is proposed on the basis of the maximum likelihood (ML) method and autoregressive moving average (ARMA) models. The novel frequency estimator for the scenario with a strongly correlated interference and an additive white Gaussian noise is synthesized in a closed form. The assumption about a constant value of that interference allows to simplify the mixture and transform it to ARMA (3,3) model. The synthesized estimator is invariant to the interference power when it is actually constant (has a unitary correlation coefficient). The particular and much more usual scenario without the interference and just with the noise is considered as well. For that case, the ML-estimator synthesized using the proposed methodology is equal to the existing modified least squares estimator. It is shown that the proposed ML-estimator has an advantage over the known one when the interference is present in the mixture.


Single-tone harmonic Autoregressive moving average model Frequency estimation Maximum likelihood Strongly correlated interference 


  1. 1.
    E. Aboutanios, B. Mulgrew, Iterative frequency estimation by interpolation on Fourier coefficients. IEEE Trans. Signal Process. 4, 1237–1242 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    E. Anarim, Y. Istefanopulos, Statistical analysis of Pisarenko type tone frequency estimator. Signal Process. 3, 291–298 (1991)CrossRefGoogle Scholar
  3. 3.
    J. Angeby, P. Stoika, T. Soderstrom, Asymptotic statistical analysis of autoregressive frequency estimates. Signal Process. 39, 277–292 (1994)CrossRefzbMATHGoogle Scholar
  4. 4.
    A. De Sabata, L. Toma, S. Mischie, Two-step Pisarenko harmonic decomposition for single tone frequency estimation. in International Conference on Electric Power Systems, High Voltages, Electric Machines, pp. 243–246 (2007)Google Scholar
  5. 5.
    R. Elasmi-Ksibi, S. Cherif, R. Lopez-Valcarce, H. Besbes, Closed-form real single-tone frequency estimator based on a normalized IIR notch filter. Signal Process. 90, 1905–1915 (2010)CrossRefzbMATHGoogle Scholar
  6. 6.
    R. Elasmi-Ksibi, R. Lopez-Valcarce, H. Besbes, S. Cherif, A family of real single-tone frequency estimators using higher-order sample covariance lags. in European Signal Processing Conference, (2008)Google Scholar
  7. 7.
    A. Eriksson, P. Stoica, On statistical analysis of Pisarenko tone frequency estimator. Signal Process. 3, 349–353 (1993)CrossRefzbMATHGoogle Scholar
  8. 8.
    H. Fuu, P.Y. Kam, MAP/ML estimation of the frequency and phase of a single sinusoid in noise. IEEE Trans. Signal Process. 55, 834–845 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    S. Jaggi, A.B. Martinez, A modified autoregressive spectral estimator for a real sinusoid in white noise. in Proceedings of Energy and Information Technologies, pp. 467–470 (1989)Google Scholar
  10. 10.
    R.J. Kenefic, A.H. Nuttall, Maximum likelihood estimation of the parameters of tone using real discrete data. IEEE J. Ocean. Eng. 1, 279–280 (1987)CrossRefGoogle Scholar
  11. 11.
    Y.F. Pisarenko, The retrieval of harmonics from a covariance function. Geophys. J. R. Astrophys. Soc. 33, 347–366 (1973)CrossRefzbMATHGoogle Scholar
  12. 12.
    B.G. Quinn, J.M. Fernandes, A fast efficient technique for the estimation of frequency. Biometrika 3, 489–497 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    R. Roy, T. Kailath, ESPRIT estimation of signal parameters via rotational invariance techniques. IEEE Trans. Acoust. Speech Signal Process. 7, 984–995 (1989)CrossRefzbMATHGoogle Scholar
  14. 14.
    H.C. So, K.W. Chan, Reformulation of Pisarenko harmonic decomposition method for single-tone frequency estimation. IEEE Trans. Signal Process. 4, 618–620 (2004)MathSciNetzbMATHGoogle Scholar
  15. 15.
    H.C. So, K.W. Chan, Y.T. Chan, K.C. Ho, Linear prediction approach for efficient frequency estimation of multiple real sinusoids: algorithms and analyses. IEEE Trans. Signal Process. 7, 2290–2305 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    H.C. So, F.K.W. Chan, W. Sun, Efficient frequency estimation of a single real tone based on principal singular value decomposition. Digit. Signal Process. 22, 1005–1009 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    P. Stoika, R.L. Moses, Spectral Analysis of Signals (Prentice Hall, Upper Saddle River, 2005)Google Scholar
  18. 18.
    D.W. Tufts, P.D. Fiore, Simple, effective estimation of frequency based on Prony’s method. IEEE Trans. Acoust. Speech 5, 2801–2804 (1996)Google Scholar
  19. 19.
    S. Ye, D.L. Kocherry, E. Aboutanios, A novel algorithm for the estimation of the pa-rameters of a real sinusoid in noise. in European Signal Processing Conference, pp. 2311–2315 (2015)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.National Aviation UniversityKyivUkraine

Personalised recommendations