Abstract
The frequency estimation of the sinusoidal signal in noise is a problem of prime importance. A usual estimation method uses the Discrete Fourier transform to obtain the coarse estimation which is improved by a fine estimation stage. In this paper, we propose an efficient iterative estimation algorithm using the phase shift Fourier interpolation. We derive the performance of the new estimator. We show that the estimator is asymptotically unbiased and its mean squared error is slightly above the asymptotical Cramer-Rao bound over the whole frequency estimation range. To reduce the calculation, we study the estimation errors of the coarse estimation caused by finite data bits and short signal sequence. An approximate model is proposed by which we can obtain the estimation errors of different data bits and sequence length. Then we choose the proper sequence length and data bits to achieve the coarse search. This allows for a dramatic reduction in the calculation of coarse estimation, while still attaining the same performance comparable to that of a full length and data bits. In the numerical results we show that, in the proposed framework, when the signal to noise ratio is bigger than 11 dB, the computation speed is 13.45 times of the original with 512 sampling points.
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig1_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig2_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig3_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig4_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig5_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig6_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig7_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig8_HTML.gif)
![](http://media.springernature.com/m312/springer-static/image/art%3A10.1007%2Fs00034-016-0453-x/MediaObjects/34_2016_453_Fig9_HTML.gif)
Similar content being viewed by others
References
T.J. Abatzoglou, A fast maximum likelihood algorithm for frequency estimation of a sinusoid based on Newton’s method. IEEE Trans. Acoust. Speech Signal Process. 33(1), 77–89 (1985)
E. Aboutanios, B. Mulgrew, Iterative frequency estimation by interpolation on Fourier coefficients. IEEE Trans. Signal Process. 53(4), 1237–1241 (2005)
E. Aboutanios, S. Ye, Efficient iterative estimation of the parameters of a damped complex exponential in noise. IEEE Signal Process. Lett. 21(8), 975–979 (2014)
J. Berent, P.L. Dragotti, T. Blu, Sampling piecewise sinusoidal signals with finite rate of innovation methods. IEEE Trans. Signal Process. 58(2), 613–625 (2010)
T. Brown, M. Wang, An iterative algorithm for single frequency estimation. IEEE Trans. Signal Process. 50(11), 2671–2682 (2002)
K.W. Chan, H.C. So, A Novel Iterative Approach for Complex Single-Tone Frequency Estimation. Proceedings of 2006 International Conference on Acoustics, Speech, and Signal Processing, vol. 3 (Toulouse, France 2006), pp. 77–80
V. Clarkson, P.J. Kootsookos, B.G. Quinn, A analysis of the variance threshold of Kay’s weighted linear predictor frequency estimator. IEEE Trans. Signal Process. 42(9), 2370–2379 (1994)
M.L. Fowler, J.A. Johnson, Extending the threshold and frequency range for phase-based frequency estimation. IEEE Trans. Signal Process. 47(10), 2857–2863 (1999)
K. Gedalyahu, Y.C. Eldar, Time-delay estimation from low-rate samples: a union of subspaces approach. IEEE Trans. Signal Process. 58(6), 3017–3031 (2010)
P. Handel, On the performance of the weighted linear predictor frequency estimator. IEEE Trans. Signal Process. 43(12), 3070–3071 (1995)
S.M. Kay, A fast and accurate single frequency estimator. IEEE Trans. Acoust. Speech Signal Process. 37(12), 1987–1990 (1989)
D. Kim, M.J. Narasimha, D.C. Cox, An improved single frequency estimator. IEEE Signal Process. Lett. 3(7), 212–214 (1996)
S.H. Leung, Y. Xiong, W.H. Lau, Modified Kay’s method with improved frequency estimation. Electron. Lett. 36(10), 918–920 (2000)
B.C. Lovell, R.C. Williamson, The statistical performance of some instantaneous frequency estimators. IEEE Trans. Signal Process. 40(7), 1708–1723 (1992)
M.D. Macleod, Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones. IEEE Trans. Signal Process. 46(1), 141–148 (1998)
A. Masmoudi, F. Bellili, S. Affes, A non-data-aided maximum likelihood time delay estimator using importance sampling. IEEE Trans. Signal Process. 59(10), 4505–4514 (2011)
A. Masmoudi, F. Bellili, S. Affes, A maximum likelihood time delay estimator in a multipath environment using importance sampling. IEEE Trans. Signal Process. 61(1), 182–193 (2013)
L.C. Palmer, Coarse frequency estimation using the discrete Fourier transform. IEEE Trans. Inf. Theory. 20(1), 104–109 (1974)
B.G. Quinn, Estimating frequency by interpolation using Fourier coefficients. IEEE Trans. Signal Process. 42(5), 1264–1268 (1994)
D.C. Rife, R.R. Boorstyn, Single-tone parameter estimation from discrete-time observation. IEEE Trans. Inform. Theory. 20(5), 591–598 (1974)
D.C. Rife, R.R. Boorstyn, Multiple tone parameter estimation from discrete time observations. Bell Syst. Tech. J. 55(9), 1389–1410 (1976)
D.C. Rife, G.A. Vincent, Use of the discrete Fourier transform in the measurement of frequencies and levels of tones. Bell. Syst. Tech J. 49(2), 197–228 (1970)
P. Stoica, H. Li, J. Li, Amplitude estimation of sinusoidal signals: survey, new results, and an application. IEEE Trans. Signal Process. 48(2), 338–352 (2000)
S.A. Tretter, Estimating the frequency of a noisy sinusoid by linear regression. IEEE Trans. Inf. Theory 31(6), 832–835 (1985)
Y.V. Zakharov, T.C. Tozer, Frequency estimator with dichotomous search of periodogram peak. Electron. Lett. 35(19), 1608–1609 (1999)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhao, S., Li, Xg. & Zhang, Ll. Efficient Iterative Frequency Estimator of Sinusoidal Signal in Noise. Circuits Syst Signal Process 36, 3265–3288 (2017). https://doi.org/10.1007/s00034-016-0453-x
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00034-016-0453-x