Eric Chaumette

On the High-SNR Conditional Maximum-Likelihood Estimator Full Statistical Characterization

In the field of asymptotic performance characterization of the conditional maximum-likelihood (CML) estimator, asymptotic generally refers to either the number of samples or the signal-to-noise ratio (SNR) value. The first case has been already fully characterized, although the second case has been only partially investigated. Therefore, this correspondence aims to provide a sound proof of a result, i.e., asymptotic (in SNR) Gaussianity and efficiency of the CML estimator in the multiple parameters case, generally regarded as trivial but not so far demonstrated

A New Barankin Bound Approximation for the Prediction of the Threshold Region Performance of Maximum Likelihood Estimators

It is well known that the ML estimator exhibits a threshold effect, i.e., a rapid deterioration of estimation accuracy below a certain signal-to-noise ratio (SNR) or number of snapshots. This effect is caused by outliers and is not captured by standard tools such as the Cramer-Rao bound (CRB). The search of the SNR threshold value (where the CRB becomes unreliable for prediction of maximum likelihood estimator variance) can be achieved with the help of the Barankin bound (BB), as proposed by many authors. The major drawback of the BB, in comparison with the CRB, is the absence of a general analytical formula, which compels one to resort to a discrete form, usually the Mcaulay-Seidman bound (MSB), requesting the search of an optimum over a set of test points. In this paper, we propose a new practical BB discrete form that provides, for a given set of test points, an improved SNR threshold prediction in comparison with existing approximations (MSB, Abel bound, Mcaulay-Hofstetter bound) at the expense of the computational complexity increased by a factor les (P+1)3 , where P is the number of unknown parameters. We have derived its expression for the general Gaussian observation model to be used in place of existing approximations.

New Results on Deterministic Cramér–Rao Bounds for Real and Complex Parameters

The Cramér-Rao bounds (CRB) is a lower bound of great interest for system analysis and design in the asymptotic region [high signal-to-noise ratio (SNR) and/or large number of snapshots], as it is simple to calculate and it is usually possible to obtain closed form expressions. The first part of the paper is a generalization to complex parameters of the Barankin rationale for deriving MSE lower bounds, that is the minimization of a norm under a set of linear constraints. With the norm minimization approach the study of Fisher information matrix (FIM) singularity, constrained CRB and regularity conditions become straightforward corollaries of the derivation. The second part provides new results useful for system analysis and design: a general reparameterization inequality, the equivalence between reparameterization and equality constraints, and an explicit relationship between parameters unidentifiability and FIM singularity.

On the influence of a detection step on lower bounds for deterministic parameter estimation

A wide variety of actual processing requires a detection step, whose main effect is to restrict the set of observations available for parameter estimation. Therefore, as a contribution to the theoretical formulation of the joint detection and estimation problem, we address the derivation of lower bounds for deterministic parameters conditioned by a binary hypothesis testing problem. The main result is the introduction of a general scheme-detailed in the particular case of CRB-enabling the derivation of conditional deterministic MSE lower bounds. To prove that it is meaningful, we also show, with the help of a fundamental application, that the problem of lower bound tightness at low SNR may arise from an incorrect lower bound formulation that does not take into account the true nature of the problem under investigation: a joint detection-estimation problem.

Statistical Performance Prediction of Generalized Monopulse Estimation

Monopulse is an established array processing technique for fast and accurate angle estimation. This technique has been generalized to space-time array processing of any dimension. The statistical performance of this generalized monopulse parameter estimation has been characterized for several target fluctuation models but not for all cases. This gap is filled by this paper. We derive the mean and variance of the complex and real and imaginary part of the averaged monopulse ratio for all Swerling target models, deterministic targets (Swerling 0 case), χ2 distributed targets with 4 degrees of freedom (Swerling 3 or 4 case) including a given detection threshold, for an arbitrary number of difference beams, and for an arbitrary number of noncoherent averaging of time snapshots. For completeness we give also the already known results for Rayleigh targets (Swerling 1 or 2 case) in the same notation. From these means and variances, the performance of all kinds of parameter estimates with the generalized monopulse formula can be calculated. Applications of these statistical descriptions are presented for planar arrays and adaptive beamforming and for space-time adaptive processing (STAP) for broadband interference suppression. From these examples some interesting conclusions can be drawn.

Monopulse-radar tracking of swerling III-IV targets using multiple observations

This paper proposes a novel statistical prediction of monopulse errors (Levanon, 1988) for a radar Swerling III-IV target embedded in noise or noise jamming where multiple observations are available. First, the study of the maximum likelihood estimator (MLE) of the complex monopulse ratio for a Swerling III-IV target embedded in spatially white noise allows us to extend the use of the MLE practical approximate form introduced by Mosca (1969) for Swerling 0-I-II cases. Afterward, we derive analytical formulas for both the mean and variance of the MLE in approximate form conditioned by the usual detection step performed on the sum channel of a monopulse antenna. Last, we provide a comparison of target direction of arrival (DOA) estimation performance based on monopulse ratio estimation as a function of the Swerling model in the context of a multifunction radar.

A Direct Method to Generate Approximations of the Barankin Bound

The search for an easily computable but tight approximation of the Barankin bound (BB) is important for the prediction of the signal-to-noise ratio (SNR) value where the Cramer-Rao bound (CRB) becomes unreliable for prediction of maximum likelihood estimators (MLE) variance. In this paper we propose a method for the derivation of a general class of BB approximations which has the advantage of a clear interpretation. This method suggests a new practical BB approximation, whose computational complexity does not exceed that of the CRB but which seems tighter than existing approximations

Cramér-Rao Bound Conditioned by the Energy Detector

A wide variety of processing incorporates a binary detection test that restricts the set of observations available for parameter estimation and requires to take this statistical conditioning into account to compute the Cramer-Rao bound (CRB). Therefore, we propose a derivation of the CRB for the deterministic signal model conditioned by the energy detector widely used in signal processing applications. This derivation has lead us to introduce novel identities on some conditional expectations of complex circular Gaussian random vectors that may be useful for other derivations.

A parameterized maximum likelihood method for multipaths channels estimation

In paths localization, a resolution that goes beyond the classical Rayleigh beamwidth is of great interest. To improve the resolution, model based techniques have been introduced (high resolution methods), but they are very sensitive to noise correlation and they assume underlying data model. We develop a parameterized maximum likelihood (PML) technique, based on a knowledge of the transmitted signal. We develop the exact PML approach and present its implementation by a Gauss Newton procedure. Simulations on data sets are examined. The performances are compared to the Cramer Rao bound. Its superiority over the traditional matched filter (MF) and the conditional maximum likelihood (CML) is shown. The paper concludes with the improvements introduced by a knowledge of the transmitted signals.

Application of ICA to airport surveillance

As air traffic gets more and more dense, it becomes very difficult to locate and recognize planes in the neighborhood of civil airports. The technique proposed resorts to a particular device, the monopulse radar, and to a tool called independent component analysis (ICA), in order to separate messages falling in the same radar beam. The algorithms utilized to compute the ICA use second and fourth order cumulants of the observed signals.<<ETX>>