Spectral Analysis and Higher Order Statistics

2-D High Resolution Spectral Estimation Based on Multiple Regions of Support

Authors:

Stéphanie Rouquette, Equipe signal et Image, ENSERB, B.P. 99, F-33402 Talence Cedex, France (France)
Olivier Alata, IRCOM-SIC, UMR 6615, SP2MI, Téléport 2, B.P. 179, F-86960 Futuroscope Cedex, France (France)
Mohamed Najim, Equipe Signal et Image, ENSERB, B.P. 99, F-33 402 Talence Cedex, France (France)
Charles W. Therrien, Dept. of Electrical and Computer Engr., Naval Postgraduate School, Monterey, CA, USA (USA)

Page (NA) Paper number 1619

Abstract:

This paper deals with frequency estimation in the 2-D case when one has only few data points. We propose a method to estimate the frequencies of a sum of exponentials. This method is based on an original set of 2-D linear prediction models with new regions of support derived from the standard quarter plane support region. These models define various spectra which are finally combined by computing their harmonic mean. This method benefits from the subspace decomposition of the covariance matrix to perform well. It is demonstrated that the new regions of support improve the spectrum geometry and the estimation accuracy compared to the classical quarter plane (QP) support regions.

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Harmonic Retrieval in Colored Non-Gaussian Noise

Authors:

Yan Zhang, Duke University (U.K.)
Shu-Xun Wang,

Page (NA) Paper number 1084

Abstract:

This paper addresses the harmonic retrieval problem in colored linear non-Gaussian noise of unknown covariance and unknown distribution. The assumptions made in the reported studies, that the non-Gaussian noise is asymmetrically distributed and no quadratic phase coupling occurs ,are released. Using the elaborately defined fourth-order cumulants of the complex noisy observations which are obtained through Hilbert transform ,we can either estimate the noise correlation nonpapametrically via cumulant projections or obtain the AR polynomial of the non-Gaussian noise parametrically through ARMA modeling. Then it is shown that the prewhitening or prefiltering techniques can be employed to retrieve harmonics respectively. Simulation results are presented to demonstrate the performance of the proposed algorithms.

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Accurate ARMA Models with Durbin's Second Method

Authors:

Piet M.T Broersen,

Page (NA) Paper number 1134

Abstract:

Long intermediate AR models are used in Durbin's algorithms for ARMA estimation. The order of that long AR model is infinite in the asymptotical theory, but very high AR orders are known to give inaccurate ARMA models in practice. A theoretical derivation is given for two different finite AR orders, as a function of the sample size. The first is the AR order optimal for prediction with a purely autoregressive model. The second theoretical AR order is higher and applies if the previously estimated AR parameters are used for estimating the MA parameters in Durbin's second, iterative, ARMA method. A Sliding Window (SW) algorithm is presented that selects good long AR orders for data of unknown processes. With a proper choice of the AR order, the accuracy of Durbin's second method approaches the Cramér-Rao bound for the integrated spectrum and the quality remains excellent if less observations are available.

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Markovian High Resolution Spectral Analysis

Authors:

Philippe Ciuciu,
Jer^ome Idier,
Jean-François Giovannelli,

Page (NA) Paper number 1174

Abstract:

When short data records are available, spectral analysis is basically an undetermined linear inverse problem. One usually considers the theoretical setting of regularization to solve such ill-posed problems. In this paper, we first show that "nonparametric" and "high resolution" are not incompatible in the field of spectral analysis. To this end, we introduce non quadratic convex penalization functions, like in low level image processing. The spectral amplitudes estimate is then defined as the unique minimizer of a compound convex criterion. An original scheme of regularization to simultaneously retrieve narrow-band and wide-band spectral features is finally proposed.

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Spectral Analysis Of Discrete Signals Generated By Multiplicative And Additive Iterative Procedures

Authors:

Tatiana Alieva, Technische Universiteit Eindhoven, Netherlands (The Netherlands)
André M Barbé, Katholieke Universiteit Leuven, Belgium (Belgium)
Martin J Bastiaans, Technische Universiteit Eindhoven, Netherlands (The Netherlands)

Page (NA) Paper number 1185

Abstract:

The discrete Fourier transform of signals constructed through multiplicative and additive iterative procedures is determined and its specific features are considered. It is shown that - in spite of the rather different structure of multiplicative and additive signals - the Fourier transforms of both types of signals exhibit the property of self-affinity. The power spectra of additive signals produced by different generating vectors have similar forms and can be divided into similar branches. The number of branches depends on the generation level and the symmetry of the power spectrum of the generating vector.

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Frequency Estimation, Phase Unwrapping and the Nearest Lattice Point Problem

Authors:

Ian V Clarkson,

Page (NA) Paper number 1603

Abstract:

In this paper, we examine the relationship between frequency estimation and phase unwrapping and a problem in algorithmic number theory known as the nearest lattice point problem. After briefly reviewing the theory of these three topics, we introduce an interpretation of the maximum likelihood frequency estimation problem as a nearest lattice point problem. We develop an algorithm based on this approach and present numerical results to compare its performance with other estimation techniques. We find that the algorithm has good powers of estimation.

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A New Class of Affine Higher Order Time-Frequency Representations

Authors:

Robin L Murray,
Antonia Papandreou-Suppappola,
G. Faye Boudreaux-Bartels,

Page (NA) Paper number 2200

Abstract:

We propose a new class of affine higher order time-frequency representations (HO-TFRs) unifying HO-TFRs which satisfy the desirable properties of scale covariance and time-shift covariance. This new class extends to higher order (N > 2) the affine class of quadratic (N = 2) time-frequency representations. In this paper, we provide five alternative formulations of the class in terms of multi-dimensional smoothing kernels. We discuss important class members, including the new higher order scalogram that is related to the wavelet transform. We also list additional desirable properties and derive the associated kernel constraints. Finally, we consider a subclass of affine HO-TFRs that intersects with a Cohen's class of time and frequency shift covariant HO-TFRs. A formulation for HO-TFRs satisfying three covariances in this higher order affine-Cohen intersection is derived.

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Identification Of Noncausal Nonminimum Phase AR Models Using Higher-Order Statistics

Authors:

Hakan Tora,
D. M. Wilkes,

Page (NA) Paper number 2244

Abstract:

In this paper, we address the problem of estimating the parameters of a noncausal autoregressive (AR) signal from estimates of the higher-order cumulants of noisy observations. The proposed family of techniques uses both 3rd-order and 4th-order cumulants of the observed output data. Consequently, at low SNR, they provide superior performance to methods based on autocorrelations. The measurement noise is assumed to be Gaussian and may be colored. The AR model parameters here are directly related to the solution of a generalized eigenproblem. The performance is illustrated by means of simulation examples.

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