Estimation de fréquences panorama des méthodes paramétriques
Frequency Estimation : Survey of Parametric Methods
OPEN ACCESS
This paper is a survey of parametric modeling applied to frequency estimation of sinusoidal signals corrupted by an additive noise. A summary of major estimation methods using second or higher order statistics is presented . Discussed techniques include classical AR modeling (based on Yule-Walker equations) and "High Resolution" methods (Truncated SVD, Root-MUSIC and ESPRIT). The maximum likelihood estimator and Bayesian methods are also under interest. But the selection of the model order in these methods is often a critical one. Thus major model order estimation methods are reviewed. Finally, the efficiency of each method is examined through a simulation example.
Résumé
Le but de cet article est de dresser un panorama des méthodes paramétriques appliquées à l'estimation de fréquences de signaux sinusoïdaux bruités (bruit additif) . Nous rappelons les principales méthodes d'estimation basées sur les moments du signal : méthodes classiques déduites des équations de Yule-Walker et méthodes «Haute Résolution » (SVD tronquée, Root-MUSIC et ESPRIT). Nous présentons aussi l'estimateur du maximum de vraisemblance et l'estimation Bayésienne. L'efficacité de ces méthodes étant souvent liée au choix de l'ordre du modèle, nous rappelons les principaux estimateurs de l'ordre. Enfin, une comparaison des méthodes met en évidence les performances de chacune.
Sinusoids, frequency estimation, parametric modeling, AR model, «High Resolution » methods
Mots clés
Sinusoïdes, estimation de fréquences, méthodes paramétriques, modèle AR, méthodes « Haute Résolution »
[Abatzoglou, 19851 Abatzoglou T .J . : "A fast ML algorithm for frequency estimation of a sinusoid based Newton's method", IEEE Trans. on ASSP, Vol . 33, No 1, Feb. 1985, pp . 77-89.
[Anderson, 1995] Anderson, J .M .M ., Giannakis G .B . & Swami A . : "Harmonic retrieval using higher order statistics : A deterministic formulation", IEEE Trans . on SP, Vol . 43, No 8, Aug . 1995, pp . 1880-1889 .
[Barabell, 1983] Barabell A .J . : " Improving the resolution performance of eigenstructure-based direction-finding algorithms", Proceedings ICASS P 1983, pp . 336-339 .
[Besson, 1992] Besson, O . : "Analyse spectrale paramétrique et modèles multiplicatifs : Application aux signaux de radar Doppler", Thèse de l'INPT, N o 527, 1992 .
[Bienvenu, 1979] Bienvenu G . & Kopp . L . : "Principle of adaptive localization in passive listening", Actes du GRETSI 1979, pp . 106/1-106/5 .
[Bresler, 1986] Bresler Y. & Macovski A . : "Exact maximum likelihood parameter estimation of superimposed exponential signals in noise", IEEE Trans . on ASSP, Vol . 34, No 5, Oct . 1986, pp . 1081-1089 .
[Burg, 1975] Burg J .P. : "Maximum entropy spectral analysis", Ph .D . dissertation , Standford University, 1975 .
[Dadzow, 1982 ] Cadzow J .A . : " Spectral estimation : An overdetermined rational model equation Approach", Proceedings of the IEEE, Vol . 70, NO 9, Sep. 1982, pp . 907-939.
[Dadzow, 1991] Cadzow J .A. & Wilkes D .M . : " Enhanced sinusoidal and exponential data modeling", SVD and Signal Processing II, Ed . Vaccaro, Elsevier Science Publishers B .V., 1991, pp . 335-352 .
[Castanie, 1987] Castanié F. & Gasmi F. : "Extraction of sinusoids from noise Prony modeling of high-order correlation functions", Proceedings DSP 1987 .
[Chan, 1982] Chan Y.T., Langford R.P. : "Spectral estimation via the HOYW equations", IEEE Trans . on ASSP, Vol . 30, No 5, Oct . 1982, pp . 689-698 .
[Cho, 1995] Cho C-M. & Djuric P.M . : "Bayesian dectection and estimation of cisoids in colored noise", IEEE Trans . on SP, Vol . 43, No 12, Dec . 1995 , pp2943-2952 .
[Clergeot, 1989] Clergeot H., Tressens S. & Ouamri A . : "Performance of high resolution frequencies estimation methods compared to the Cramér-Ra o bounds", IEEE Trans . on ASSP, Vol . 37, No 11, Nov. 1989, pp . 1703-1720 .
[Djuric, 1993] Djuric P.M . : " Simultaneous detection and frequency estimation of sinusoidal signals", Proceedings ICASSP 1993, Vol . IV, pp . 53-56.
[Djuric, 1995] Djuric P.M . & Li H .T. : "Bayesian spectrum estimation of harmonic signals", Signal Processing Letters, Vol. 2, No 11, Nov. 1995, pp . 213-215.
[Djuric, 1996] Djuric P.M. : "A model selection rule for sinusoids in white gaussian noise", IEEE Trans . on ASSP, Vol . 44, No 7, Jul. 1996, pp . 1744-1751 .
[Dublanchet, 1995] Dublanchet E, Duvaut P ., Idier J . & Demoment G . : "Expulse Complexe, estimation bayésienne de sinusoïdes par déconvolution de la transformée de Fourier discrète du signal", Actes du GRETSI 1995, pp . 37-40 .
[Ducasse, 1997] Ducasse A . : "Estimation de sous-harmoniques à l'aide de méthodes paramétriques", Thèse de l'INPT, No 1269, Jan . 1997 .
[Duvaut, 1991] Duvaut P. : "Traitement du signal : concepts et applications" , éditions Hermès, 1991.
[Fuchs, 1988] Fuchs J-J . : "Estimating the number of sinusoids in additive white noise", IEEE Trans . on ASSP, Vol . 36, No 12, Dec . 1988, pp . 1846-1853 .
[Gasmi, 1988] Gasmi F. & Castanié F. : "Successive correlation based Prony modeling very low SNR behavior analysis", Proceedings ICASSP 1988 , pp. 2420-2423 .
[Gingras, 1988] Gingras D .F. : "Asymptotic properties of HOYW estimates of the AR parameters of an time series", IEEE Trans . on ASSP, Vol. 33, No 4 , Oct . 1985, pp. 1095-1101 .
[Golub, 1989] Golub G .H . & Van Loan C.F. : "Matrix Computations", ed . The Johns Hopkins University Press, 1989 .
[Haykin, 1989] Haykin S . : "Modern filters", MacMillan Publishing Compagny , 1989 .
[Hsieh, 1990] Hsieh S .F., Liu K .J .R. & Yao K. : "Applications of truncated QR methods to sinusoidal frequency estimation", Proceedings ICASSP 1990 , pp . 2571-2574.
[Hua, 1990] Hua Y. & Sakar T.K . : "Matrix Pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise", IEEE Trans . on ASSP, Vol . 38, No 5, May 1990, pp . 814-824 .
[James, 1994] James B ., 2 B .D .O . & Williamson R.C. : "Conditional mean and maximum likelihood frequency estimation", IEEE Trans . on ASSP, Vol . 42 , No 6, Jun . 1994, pp . 1366-1375 .
[Johnston, 1994]] Johnston D .E. & Djuric P.M . : "An efficient bayes solution to AR signal modelling for short sequences", Proceedings ICASSP 1994, Vol . IV, pp . 345-348 .
[Kay, 1980] Kay S .M . : "Noise compensation for autoregressive spectral estimates", IEEE Trans . on ASSP, Vol . 28, No 3, Jun . 1980, pp . 292-303 .
[Kay, 1981] Kay S .M. & Marple S .L. : "Spectrum analysis : A modern perspective", Proceedings of the IEEE, Vol . 69, Nov . 1981, pp . 1380-1419 .
[Kay, 1988] Kay S .M. : "Modem Spectral Estimation : Theory and Application" . Prentice Hall, chap. 7.
[Kumaresan, 1983] Kumaresan R . & Tufts D.W. : "Estimating the angles of arriva l of multiple plane waves", Trans . Aerosp. Electron. Syst ., Vol. 19, Jan . 1983 , pp . 134-139 .
[Lacoume, 1988] Lacoume J .L. & Martin M . : "Statistical properties of AR estimators", Proceedings EUSIPCO 1988, pp . 523-526 .
[Lang, 1980] S .W. Lang & J .H . McClellan : "Frequency estimation with maximum entropy spectral estimators", IEEE Trans . on ASSP, Vol . 28, No 6, Dec . 1980 , pp. 716-724 .
[Levinson, 1947] Levinson N. : "The Wiener RMS error criterion in filter design and prediction", J . Math. Phys ., Vol . 25, pp . 261-278 .
[Makhoul, 1975] Makhoul J . : "Linear prediction : a tutorial review", Proceedings of the IEEE, Vol. 63, Apr. 1975, pp . 561-580 .
[Makhoul, 1977] Makhoul J. : "Stable and efficient Lattice methods for linear prediction", IEEE Trans . on ASSP, Vol . 25, No 5, Oct. 1977, pp. 423-428 .
[McGinn, 1983] McGinn D . & Johnson D .H . : "Reduction of all-pole parameter estimator bias by successive autocorrelation", Proceedings ICASSP 1983 , pp . 1088-1091 .
[McWhorter, 1995] McWhorter L.T. & Scharf L .L . : "Nonlinear ML estimation of autoregressive time series", IEEE Trans . on ASSP, Vol . 43, No 12, Dec . 1995, pp . 2909-2919.
[Mendel, 1991] Mendel J.M . : "Tutorial on high-order statistics in signal processing and system theory : Theorical results and some applications" , Proceedings of the IEEE, Vol. 79, No 3, Mar. 1991, pp . 277-305 .
[Morf, 1977] Morf M ., Dickinson B ., Kailath T. & Vieira A . : "Efficient solution of covariance equations for linear prediction", IEEE Trans. on ASSP, Vol . 25 , No 5, Oct . 1977, pp . 429-433.
[Nagesha, 1996] Nagesha V. & Kay S . : "Spectral analysis based on the canonical autoregressive decomposition", IEEE Trans . on ASSP, Vol . 44, No 7, Jul . 1996, pp. 1719-1733 .
[Neider, 1965] Neider J .A. & Mead R . "A Simplex method for function minimization", Computer Journal, Vol . 7, pp . 308-313 .
[Palmer, 1974] Palmer L.C . : "Coarse frequency estimation using the discrete Fourier transform", IEEE Trans. on IT, Vol. 20, Jan . 1974, pp . 104-109 .
[Park, 1989] Park S . & Gerhardt L .A . : "A robust spectral estimation by modeling an estimated autocovariance with an ARMA model", IEEE Trans . on ASSP, Vol. 37, No 2, Feb . 1989, pp . 181-191 .
[Parker, 1990] Parker P.J. & 2 B .D .O . : "Frequency tracking of nonsinusoidal periodic signals in noise", Signal Processing 20, 1990, pp . 127-152 .
[Pisarenko, 1973] Pisarenko V.F. : "The retrieval of harmonics from covariance function", Geophys. J. Roy. Astron . Soc ., Vol . 33, 1973, pp . 347-366.
[Porat, 1994] Porat B . : "Digital processing of random signals, Theory & Methods", Prentice Hall, 1994 .
[Proakis, 1992] Proakis J .G ., Rader C .M., Ling F. and Nikias C .L . : "Advanced digital signal processing", Macmillan Publishing Company, 1992 .
[Rao, 1988] Rao B .D . : "Perturbation analysis of an SVD-based linear prediction method for estimating the frequencies of multiple sinusoids", IEEE Trans . on ASSP, Vol . 36, No 7, Jul . 1988, pp . 1026-1035 .
[Roy, 1986] Roy R ., Paulraj A. and Kailath T. : "ESPRIT - A subspace rotation approach to estimation of parameters of cisoids in noise", IEEE Trans . on ASSP, Vol . 34, No 4, Oct . 1986, pp . 1340-1342 .
[Roy, 1987] Roy R. & Kailath T. : "Total least squares ESPRIT", Proceedings 21th Asilomar Conf. Circuits Syst. Computing, Nov. 1986 .
[Roy, 1988] Roy R. Paulraj A . & Kailath T : "Direction of arrival estimation by subspace rotation methods - ESPRIT", Proceedings ICASSP 1986, pp . 2495- 2498.
[Sakai, 1979] Sakai H . & Arase M. : "Recursive parameter estimation of an autoregressive process disturbed by white noise", Int . J . Contr., Vol . 30, 1979 , pp. 949-966 .
[Sakai, 1984] Sakai H . : "Statistical analysis of Pisarenko's method for sinusoidal frequency estimation", IEEE Trans . on AS SP, Vol . 32, No 1, Feb . 1984, pp . 95 -101 .
[Schmidt, 1981] Schmidt R .O ., "A signal subspace approach to multiple emitter location and spectral estimation", Ph . D . dissertation, Standford University, 1981 .
[Shaw, 1995] Shaw A.K . : "ML estimation of multiple frequencies with constraints to guarantee unit circle roots", IEEE Trans . on SP, Vol . 43, No 3, Mar. 1995 , pp . 796-799.
[Soderstrom, 1991] Söderström T. & Stoïca P. : "On accuracy of HOYW methods for cisoids", Proceedings ICASSP 1991, pp . 3573-3576 .
[Starer, 1992] Starer D . & Nehorai A. : "Newton algorithm for conditional and unconditional ML estimation of the parameters of exponential signals in noise", IEEE Trans . on SP, Vol. 40, No 6, Jun . 1992, pp . 1528-1533 .
[Stoica, 1989a] Stoïca P., Söderström T. & Ti F. : "Asymptotic properties of HOYW estimates of sinusoidal frequencies", IEEE Trans . on ASSP, Vol 37, No 11, Nov. 1989, pp . 1721-1734 .
[Stoica, 1989b] Stoïca P. & Nehorai A. : "MUSIC, Maximum Likelihood, and Cramér-Rao bound", IEEE Trans. on ASSP, Vol . 37, No 5, May 1989, pp . 720 -741 .
[Stoica, 1989c] Stoïca P., Friedlander B . & Söderström T. : "Maximum likelihood estimation of the parameters of multiple sinusoids from noisy measurements" , IEEE Trans . on ASSP, Vol 37, No 3, Mar. 1989, pp. 378-392 .
[Stoica, 1991] Stoïca P. & Söderström T. : "Statistical analysis of MUSIC and subspace Rotation estimates of sinusoïdal frequencies", IEEE Trans . on SP, Vol. 39, No 8, Aug . 1991, pp. 1836-1847 .
[Stoica, 1992] Stoïca P., Söderström T. & Van Huffel S . : "On SVD-based and TLS-based high-order Yule-Walker methods of frequency estimation", Signal Processing, Vol . 29, 1992, pp . 309-317 .
[Sudhakar, 1985] Sudhakar R ., Agarwal R.C. & Duttaroy S .C . : "Frequency estimation based on iterated autocorrelation function", IEEE Trans . on ASSP, Vol . 33, No 1, Fev . 1985, pp. 70-76 .
[Swami, 1991] Swami A . & Mendel J.M. : "Cumulant-based approach to the harmonic retrieval and related problems", IEEE Trans . on SP, Vol . 39, No 5 , May 1991, pp . 1099-1108 .
[Swingler, 1980] Swingler D.N . : "Frequency errors in MEM processing", IEEE Trans . on ASSP, Vol . 28, No 2 Apr. 1980, pp . 257-259.
[Tufts, 1982] Tufts D .W. & Kumaresan R . : "Estimation of frequencies of multiple sinusoids : Making linear prediction perform like maximum likelihood" , Proceedings of the IEEE, Vol. 70, No 9, Sept . 1982 .
[Umesh, 1996] Umesh S . & Tufts D .W. : "Estimation of parameters of exponentially damped sinusoids using fast ML estimation with application to NMR spectroscopy data", IEEE Trans . on SP, Vol . 44, No 9, Sep . 1996, pp . 2245 -2259 .
[Van Huffel, 1991] Van Huffel S. & Vandewalle J . : "The total least squares problem", ed . SIAM, 1991 .
[Wax, 1985] Wax M . & Kailath T. : "Detection of signals by information theoretic criteria", IEEE Trans . on ASSP, Vol . 33, No 2, Apr. 1985, pp. 387-392 .
[Wilkes, 1993] Wilkes D .M . & Cadzow J .A . : " The effects of phase on highresolution frequency estimators", IEEE Trans . on SP, Vol. 41, No 3, Mar. 1993, pp . 1319-1330.