Mohammed Nabil El Korso

Conditional and Unconditional Cramér–Rao Bounds for Near-Field Source Localization

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 In this context, the far-field assumption (i.e., planar wavefront) is, of course, no longer valid and one has to consider a more complicated model parameterized by the bearing (as in the far-field case) and by the distance, named range, between the source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 Cramer-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.

Statistical resolution limit for the multidimensional harmonic retrieval model: hypothesis test and Cramér-Rao Bound approaches

The statistical resolution limit (SRL), which is defined as the minimal separation between parameters to allow a correct resolvability, is an important statistical tool to quantify the ultimate performance for parametric estimation problems. In this article, we generalize the concept of the SRL to the multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this article, we derive the SRL for the so-called multidimensional harmonic retrieval model using a generalization of the previously introduced SRL concepts that we call multidimensional SRL (MSRL). We first derive the MSRL using an hypothesis test approach. This statistical test is shown to be asymptotically an uniformly most powerful test which is the strongest optimality statement that one could expect to obtain. Second, we link the proposed asymptotic MSRL based on the hypothesis test approach to a new extension of the SRL based on the Cramér-Rao Bound approach. Thus, a closed-form expression of the asymptotic MSRL is given and analyzed in the framework of the multidimensional harmonic retrieval model. Particularly, it is proved that the optimal MSRL is obtained for equi-powered sources and/or an equi-distributed number of sensors on each multi-way array.

Statistical Resolution Limit for Source Localization With Clutter Interference in a MIMO Radar Context

During the last decade, multiple-input multiple-ouput (MIMO) radar has received an increasing interest. One can find several estimation schemes in the literature related to the direction of arrivals and/or direction of departures, but their ultimate performance in terms of the statistical resolution limit (SRL) have not been fully investigated. In this correspondence, we fill this lack. Particulary, we derive the SRL to resolve two closely spaced targets in clutter interference using a MIMO radar with widely separated antennas. Toward this end, we use a hypothesis test formulation based on the generalized likelihood ratio test (GLRT). Furthermore, we investigate the link between the SRL and the minimum signal-to-noise ratio (SNR) required to resolve two closely spaced targets for a given probability of false alarm and for a given probability of detection. Finally, theoretical and numerical analysis of the SRL are given for several scenarios (with/without clutter interference, known/unknown parameters of interest and known/unknown noise variance).

Subspace Methods and Exploitation of Special Array Structures

Abstract In this chapter we provide an overview of subspace-based parameter estimation schemes for uniform arrays, non-uniform arrays, and other specific array structures. The popularity of many of these special array structures is due to the availability of search-free low computational complexity direction of arrival (or spatial frequency) estimation algorithms to exploit the particularly structure of the array. More precisely, we mainly focus on the study and the comparison between several subspace-based algorithms. The latter can be classified into spectral search-based and search-free techniques. The spectral searching schemes include MUSIC, weighted subspace fitting algorithms, and rank reduction schemes divised for sensor arrays composed of multiple fully-calibrated subarrays with unknown subarray displacements. On the other hand, the search-free schemes can be partitioned into two subclasses: (i) Polynomial-rooting techniques, in which, we describe and compare MODE, the root-MUSIC algorithm and its variants for the uniform linear array (ULA) configuration, and the interpolated root-MUSIC, the manifold separation, and the Fourier domain root-MUSIC schemes in the non-uniform array context. (ii) Matrix-shifting techniques, in which, we present and compare the ESPRIT algorithm adapted for array geometries that exhibit a shift invariance structure and its variants, as the GESPRIT and Unitary ESPRIT algorithms. By numerical simulations, we show that in the one-dimensional case, the threshold of the root-MUSIC algorithm occurs at a higher SNR than for ESPRIT-based algorithms, in which the Unitary ESPRIT scheme performs best among all ESPRIT-based schemes. For the case of multidimensional parameter estimation, we introduce R -D matrix-based and tensor-based algorithms. We demonstrate that multidimensional signals can be represented by tensors which provide a natural formulation of the R -dimensional signals and their properties (such as the R -D shift invariances needed for matrix shifting techniques). Based on this representation, an improved HOSVD-based signal subspace estimate is proposed. We show that this subspace estimate performs a more efficient denoising of the data which leads to a tensor gain in terms of an enhanced estimation accuracy. This subspace estimate can be combined with arbitrary existing multidimensional subspace-based parameter estimation schemes. Then we discuss the tensor-based schemes R -D Standard Tensor-ESPRIT and R -D Unitary Tensor-ESPRIT. They outperform the matrix based R -D ESPRIT-type algorithms due to the enhanced subspace estimate obtained from the HOSVD. We also show that strict-sense non-circular sources can be exploited to virtually double the number of available sensors by an augmentation of the measurement matrix. Based on this idea, the R -D NC Standard ESPRIT and the R -D NC Unitary ESPRIT algorithm are derived. As a result, the number of resolvable wavefronts is doubled and the achievable estimation accuracy is improved. Finally, the family of NC Tensor-ESPRIT-type algorithms is introduced to combine both benefits, the strict-sense non-circular source symbols and the multidimensional structure of the signals. This is a non-trivial task, since the augmentation of the measurement matrix performed for R -D NC Unitary ESPRIT destroys the structure needed for the Tensor-ESPRIT-type algorithms. This challenge can be solved by defining a mode-wise augmentation of the measurement tensor.

Statistical Resolution Limit for multiple parameters of interest and for multiple signals

The concept of Statistical Resolution Limit (SRL), which is defined as the minimal separation to resolve two closely spaced signals, is an important tool to quantify performance in parametric estimation problems. This paper generalizes the SRL based on the Cramér-Rao bound to multiple parameters of interest per signal and for multiple signals. We first provide a fresh look at the SRL in the sense of Smiths criterion by using a proper change of variable formula. Second, based on the Minkowski distances, we extend this criterion to the important case of multiple parameters of interest per signal and to multiple signals. The results presented herein can be applied to any estimation problem and are not limited to source localization problems.

MIMO radar target localization and performance evaluation under SIRP clutter

Multiple-input multiple-output (MIMO) radar has become a thriving subject of research during the past decades. In the MIMO radar context, it is sometimes more accurate to model the radar clutter as a non-Gaussian process, more specifically, by using the spherically invariant random process (SIRP) model. In this paper, we focus on the estimation and performance analysis of the angular spacing between two targets for the MIMO radar under the SIRP clutter. First, we propose an iterative maximum likelihood as well as an iterative maximum a posteriori estimator, for the targets spacing parameter estimation in the SIRP clutter context. Then we derive and compare various Cramer-Rao-like bounds (CRLBs) for performance assessment. Finally, we address the problem of target resolvability by using the concept of angular resolution limit (ARL), and derive an analytical, closed-form expression of the ARL based on Smiths criterion, between two closely spaced targets in a MIMO radar context under SIRP clutter. For this aim we also obtain the non-matrix, closed-form expressions for each of the CRLBs. Finally, we provide numerical simulations to assess the performance of the proposed algorithms, the validity of the derived ARL expression, and to reveal the ARLs insightful properties. HighlightsTwo MIMO radar target parameter estimators are designed under SIRP clutter.Various Cramer-Rao-like bounds are derived and compared in the same context.A closed-form expression for the target resolution limit in this context is proposed.

Statistical analysis of achievable resolution limit in the near field source localization context

In this fast communication, we derive the statistical resolution limit (SRL), characterizing the minimal parameter separation, to resolve two closely spaced known near-field sources impinging on a linear array. Toward this goal, we conduct on the first-order Taylor expansion of the observation model a Generalized Likelihood Ratio Test (GLRT) based on a Constrained Maximum Likelihood Estimator (CMLE) of the SRL. More precisely, the minimum separation between two near-field sources, that is detectable for a given probability of false alarm and a given probability of detection, is derived herein. Finally, numerical simulations are done to quantify the impact of the array geometry of the signal sources power distribution and of the array aperture on the statistical resolution limit.

Angular resolution limit for vector-sensor arrays: Detection and information theory approaches

The Angular Resolution Limit (ARL) on resolving two closely spaced polarized sources using vector-sensor arrays is considered in this paper. The proposed method is based on the information theory. In particular, the Steins lemma provides, asymptotically, a link between the probability of false alarm and the relative entropy between two hypothesis of a given statistical binary test. We show that the relative entropy can be approximated by a quadratic function in the ARL. This property allows us to derive and analyze a closed-form expression of the ARL. To illustrate the interest of our approach, the ARL, in the sense of the detection theory, is also derived. Finally, we show that the ARL is only sensitive to the norm of the polarization state vector and not to the particular values of the polarization parameters.

Nonmatrix closed-form expressions of the Cramér-Rao Bounds for near-field localization parameters

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 situation, the far-field assumption (planar wavefront) is no longer valid and we have 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 sensor. We can find a plethora of estimation schemes in the literature but the ultimate performance has not been fully investigated. In this paper, we derive and analyze the Cramér-Rao Bound (CRB) for a single time-varying source. In this case, we obtain nonmatrix closed-form expressions. Our approach has two advantages: (i) the computational cost for a large number of snapshots of a matrix-based CRB can be high while our approach is cheap and (ii) some useful informations can be deduced from the behavior of the bound. In particular, we show that closer is the source from the array and/or higher is the carrier frequency, better is the estimation of the range.

Maximum likelihood and maximum a posteriori direction-of-arrival estimation in the presence of sirp noise

The maximum likelihood (ML) and maximum a posteriori (MAP) estimation techniques are widely used to address the direction-of-arrival (DOA) estimation problems, an important topic in sensor array processing. Conventionally the ML estimators in the DOA estimation context assume the sensor noise to follow a Gaussian distribution. In real-life application, however, this assumption is sometimes not valid, and it is often more accurate to model the noise as a non-Gaussian process. In this paper we derive an iterative ML as well as an iterative MAP estimation algorithm for the DOA estimation problem under the spherically invariant random process noise assumption, one of the most popular non-Gaussian models, especially in the radar context. Numerical simulation results are provided to assess our proposed algorithms and to show their advantage in terms of performance over the conventional ML algorithm.