In the field of asymptotic performance characterization of 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. Index TermsMaximum Likelihood, statistical efficiency, high Signal to Noise Ratio, array processing.

Near-field source localization problem by a passive antenna array makes the assumption that the time-varying sources are located near the antenna. In this context, the far-field assumption (i.e. planar wavefront) is, of course, no longer valid and one has to consider a more complicated model parameterized by the bearing (as in the far-field case) and by the distance, named range, between the source and a reference coordinate system. One can find a plethora of estimation schemes in the literature, but their ultimate performance in terms of Mean Square Error (MSE) have not been fully investigated. To characterize these performance, the Cramér-Rao Bound (CRB) is a popular mathematical tool in signal processing. The main cause for this is that the MSE of several high-resolution direction of arrival algorithms are known to achieve the CRB under quite general/weak conditions. In this correspondence, we derive and analyze the so-called conditional and unconditional CRBs for a single time-varying near-field source. In each case, we obtain non matrix closed-form expressions. Our approach has two advantages: (i) due to the fact that one has to inverse the Fisher information matrix, the computational cost for a large number of snapshots (in the case of the conditional CRB) and/or for a large number of sensors (in the case of the unconditional CRB), of a matrix-based CRB can be high while our approach is low and (ii) some useful information can be deduced from the behavior of the bound. In particular, an explicit relationship between the conditional and the unconditional CRBs is provided and one shows that closer is the source from the array and/or higher is the signal carrier frequency, better is the range estimation.

Abstract-This letter deals with the Cramér-Rao bound for the estimation of a hybrid vector with both random and deterministic parameters. We point out the specificity of the case when the deterministic and the random vectors of parameters are statistically dependent. The relevance of this expression is illustrated by studying a practical phase estimation problem in a non-data-aided communication context.

Abstract-Minimal bounds on the mean square error (MSE) are generally used in order to predict the best achievable performance of an estimator for a given observation model. In this paper, we are interested in the Bayesian bound of the Weiss-Weinstein family. Among this family, we have Bayesian Cramér-Rao bound, the Bobrovsky-MayerWolf-Zakaï bound, the Bayesian Bhattacharyya bound, the Bobrovsky-Zakaï bound, the Reuven-Messer bound, and the Weiss-Weinstein bound. We present a unification of all these minimal bounds based on a rewriting of the minimum mean square error estimator (MMSEE) and on a constrained optimization problem. With this approach, we obtain a useful theoretical framework to derive new Bayesian bounds. For that purpose, we propose two bounds. First, we propose a generalization of the Bayesian Bhattacharyya bound extending the works of Bobrovsky, Mayer-Wolf, and Zakaï. Second, we propose a bound based on the Bayesian Bhattacharyya bound and on the Reuven-Messer bound, representing a generalization of these bounds. The proposed bound is the Bayesian extension of the deterministic Abel bound and is found to be tighter than the Bayesian Bhattacharyya bound, the Reuven-Messer bound, the Bobrovsky-Zakaï bound, and the Bayesian Cramér-Rao bound. We propose some closed-form expressions of these bounds for a general Gaussian observation model with parameterized mean. In order to illustrate our results, we present simulation results in the context of a spectral analysis problem.Index Terms-Bayesian bounds on the MSE, Weiss-Weinstein family.

Likelihood Criterions at high Signal to Noise Ratio. Then, thanks to this equivalence we prove the non-Gaussianity and the non-efficiency of the Unconditional Maximum Likelihood estimator. We also rediscover the closed-form expressions of the probability density function and of the variance of the estimates in the one source scenario and we derive a closed-form expression of this estimator variance in the two sources scenario. Index TermsAsymptotic performance, Unconditional Maximum Likelihood, finite number of data, high Signal to Noise Ratio, Cramér-Rao bound.

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We compute lower bounds on the mean-square error of multiple change-point estimation. In this context, the parameters are discrete and the Cramér-Rao bound is not applicable. Consequently, we focus on computing the Barankin bound (BB), the greatest lower bound on the covariance of any unbiased estimator, which is still valid for discrete parameters. In particular, we compute the multi-parameter version of the Hammersley-Chapman-Robbins, which is a Barankin-type lower bound. We first give the structure of the so-called Barankin information matrix (BIM) and derive a simplified form of the BB. We show that the particular case of two change points is fundamental to finding the inverse of this matrix. Several closed-form expressions of the elements of BIM are given for changes in the parameters of Gaussian and Poisson distributions. The computation of the BB requires finding the supremum of a finite set of positive definite matrices with respect to the Loewner partial ordering. Though, each matrix in this set of candidates is a lower bound on the covariance matrix of the estimator, the existence of a unique supremum w.r.t. to this set, i.e., the tightest bound, might not be guaranteed. To overcome this problem, we compute a suitable minimal-upper bound to this set given by the matrix associated with the Lowner-John Ellipsoid of the set of hyper-ellipsoids associated to the set of candidate lower-bound matrices. Finally, we present some numerical examples to compare the proposed approximated BB with the performance achieved by the maximum likelihood estimator.

Among the family of polarization sensitive crossed-dipole arrays, we can find the so-called Cocentred Orthogonal Loop and Dipole Uniform and Linear Array (COLD-ULA). In this paper, we derive the Statistical Resolution Limit (SRL) characterizing the minimal separation, in terms of the direction of arrivals, to resolve two closely spaced sources. Toward this end, nonmatrix closed form expressions of the deterministic Cramér-Rao Bound (CRB) are derived and thus, the SRL is deduced by a proper change of variable formula. Finally, concluding remarks and a comparison between the SRL of the COLD-ULA and the ULA are given. Particularly, it has been shown that, in the case of orthogonal sources, the SRL for the COLD-ULA is equal to the SRL for the ULA, meaning that it is not a function of polarization parameters. Furthermore, thanks to the derived SRL, it has been shown that generally the SRL of the COLD-ULA is smaller than the one for the ULA.