Angular resolution limit for deterministic correlated sources

This paper is devoted to the analysis of the angular resolution limit (ARL), an important performance measure in the directions-of-arrival estimation theory. The main fruit of our endeavor takes the form of an explicit, analytical expression of this resolution limit, w.r.t. the angular parameters of interest between two closely spaced point sources in the far-field region. As by-products, closed-form expressions of the Cram\'er-Rao bound have been derived. Finally, with the aid of numerical tools, we confirm the validity of our derivation and provide a detailed discussion on several enlightening properties of the ARL revealed by our expression, with an emphasis on the impact of the signal correlation

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 target's spacing parameter estimation in the SIRP clutter context. Then we derive and compare various Cram\'er-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 Smith's 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 ARL's insightful properties.Comment: 34 pages, 12 figure

CRLB under K-distributed observation with parameterized mean

International audienceA semi closed-form expression of the Fisher information matrix in the context of K-distributed observations with parameterized mean is given and related to the classical, i.e. Gaussian case. This connection is done via a simple multiplicative factor, which only depends on the intrinsic parameters of the texture and the size of the observation vector. Finally, numerical simulation is provided to corroborate the theoretical analysi

Fast sequential source localization using the projected companion matrix approach

International audienceThe sequential forms of the spectral MUSIC algorithm, such as the Sequential MUSIC (S-MUSIC) and the Recursively Applied and Projected MUSIC (RAP-MUSIC) algorithms, use the previously estimated DOA (Direction Of Arrival) to form an intermediate array gain matrix and project both the array manifold and the signal subspace estimate into its orthogonal complement. By doing this, these methods avoid the delicate search of multiple maxima and yield a more accurate DOA estimation in difficult scenarios. However, these high-resolution algorithms adapted to a general array geometry suffer from a high computational cost. On the other hand, for linear equispaced sensor array, the root- MUSIC algorithm is a fast and accurate high-resolution scheme which also avoids the delicate search of multiple maxima but a sequential scheme based on the root-MUSIC algorithm does not exist. This paper fills this need. Thus, we present a new sequential high-resolution estimation method, called the Projected Companion Matrix MUSIC (PCM-MUSIC) method, in the context of source localisation in the case of linear equispaced sensor array. Remark that the proposed algorithm can be used without modification in the context of spectral analysis

Bayesian Lower Bounds for Dense or Sparse (Outlier) Noise in the RMT Framework

Robust estimation is an important and timely research subject. In this paper, we investigate performance lower bounds on the mean-square-error (MSE) of any estimator for the Bayesian linear model, corrupted by a noise distributed according to an i.i.d. Student's t-distribution. This class of prior parametrized by its degree of freedom is relevant to modelize either dense or sparse (accounting for outliers) noise. Using the hierarchical Normal-Gamma representation of the Student's t-distribution, the Van Trees' Bayesian Cram\'er-Rao bound (BCRB) on the amplitude parameters is derived. Furthermore, the random matrix theory (RMT) framework is assumed, i.e., the number of measurements and the number of unknown parameters grow jointly to infinity with an asymptotic finite ratio. Using some powerful results from the RMT, closed-form expressions of the BCRB are derived and studied. Finally, we propose a framework to fairly compare two models corrupted by noises with different degrees of freedom for a fixed common target signal-to-noise ratio (SNR). In particular, we focus our effort on the comparison of the BCRBs associated with two models corrupted by a sparse noise promoting outliers and a dense (Gaussian) noise, respectively

Joint ML calibration and DOA estimation with separated arrays

This paper investigates parametric direction-of-arrival (DOA) estimation in a particular context: i) each sensor is characterized by an unknown complex gain and ii) the array consists of a collection of subarrays which are substantially separated from each other leading to a structured noise covariance matrix. We propose two iterative algorithms based on the maximum likelihood (ML) estimation method adapted to the context of joint array calibration and DOA estimation. Numerical simulations reveal that the two proposed schemes, the iterative ML (IML) and the modified iterative ML (MIML) algorithms for joint array calibration and DOA estimation, outperform the state of the art methods and the MIML algorithm reaches the Cram\'er-Rao bound for a low number of iterations

Relaxed concentrated MLE for robust calibration of radio interferometers

In this paper, we investigate the calibration of radio interferometers in which Jones matrices are considered to model the interaction between the incident electromagnetic field and the antennas of each station. Specifically, perturbation effects are introduced along the signal path, leading to the conversion of the plane wave into an electric voltage by the receptor. In order to design a robust estimator, the noise is assumed to follow a spherically invariant random process (SIRP). The derived algorithm is based on an iterative relaxed concentrated maximum likelihood estimator (MLE), for which closed-form expressions are obtained for most of the unknown parameters

GLRT-Based Framework for the Multidimensional Statistical Resolution Limit

International audienceRecently, a criterion for Multidimensional Statistical Resolution Limit (MSRL) evaluation, which is defined as the minimal separation to resolve two closely spaced signals depending on several parameters, was empirically proposed in [1] but without a statistical analysis. In this paper, we fill this lack by demonstrating that this MSRL criterion is asymptotically equivalent (upon to a translator factor) to a UMP (Uniformly Most Powerful) test among all invariant statistical tests. This result is an extension of a previous work on mono-dimensional SRL (i.e., when the signals only depend on one parameter). As an illustrative example, the 3-D harmonic retrieval case for wireless channel sounding is treated to show the good agreement of the proposed result

Statistical Resolution Limit for the Multidimensional Harmonic Retrieval Model: Hypothesis Test and Cramer-Rao Bound Approaches

Special issue on Advances in Angle-of-Arrival and Multidimensional Signal Processing for Localization and CommunicationsInternational audienceThe 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 paper we generalize the concept of the SRL to the Multidimensional SRL (MSRL) applied to the multidimensional harmonic retrieval model. In this paper, we derive the SRL for the so-called multidimensional harmonic retrieval model by 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

NONMATRIX CLOSED-FORM EXPRESSIONS OF THE CRAMER-RAO BOUNDS FOR NEAR-FIELD LOCALIZATION PARAMETERS

International audienceNear-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 Cram´er-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