Semiparametric CRB and Slepian-Bangs formulas for Complex Elliptically Symmetric Distributions

The main aim of this paper is to extend the semiparametric inference methodology, recently investigated for Real Elliptically Symmetric (RES) distributions, to Complex Elliptically Symmetric (CES) distributions. The generalization to the complex field is of fundamental importance in all practical applications that exploit the complex representation of the acquired data. Moreover, the CES distributions has been widely recognized as a valuable and general model to statistically describe the non-Gaussian behaviour of datasets originated from a wide variety of physical measurement processes. The paper is divided in two parts. In the first part, a closed form expression of the constrained Semiparametric Cram\'{e}r-Rao Bound (CSCRB) for the joint estimation of complex mean vector and complex scatter matrix of a set of CES-distributed random vectors is obtained by exploiting the so-called \textit{Wirtinger} or $\mathbb{C}\mathbb{R}$-\textit{calculus}. The second part deals with the derivation of the semiparametric version of the Slepian-Bangs formula in the context of the CES model. Specifically, the proposed Semiparametric Slepian-Bangs (SSB) formula provides us with a useful and ready-to-use expression of the Semiparametric Fisher Information Matrix (SFIM) for the estimation of a parameter vector parametrizing the complex mean and the complex scatter matrix of a CES-distributed vector in the presence of unknown, nuisance, density generator. Furthermore, we show how to exploit the derived SSB formula to obtain the semiparametric counterpart of the Stochastic CRB for Direction of Arrival (DOA) estimation under a random signal model assumption. Simulation results are also provided to clarify the theoretical findings and to demonstrate their usefulness in common array processing applications.Comment: Submitted to IEEE Transactions on Signal Processing. arXiv admin note: substantial text overlap with arXiv:1807.08505, arXiv:1807.0893

Matched direction detectors and estimators for array processing with subspace steering vector uncertainties

In this paper, we consider the problem of estimating and detecting a signal whose associated spatial signature is known to lie in a given linear subspace but whose coordinates in this subspace are otherwise unknown, in the presence of subspace interference and broad-band noise. This situation arises when, on one hand, there exist uncertainties about the steering vector but, on the other hand, some knowledge about the steering vector errors is available. First, we derive the maximum-likelihood estimator (MLE) for the problem and compute the corresponding Cramer-Rao bound. Next, the maximum-likelihood estimates are used to derive a generalized likelihood ratio test (GLRT). The GLRT is compared and contrasted with the standard matched subspace detectors. The performances of the estimators and detectors are illustrated by means of numerical simulations

Subspace Methods for Joint Sparse Recovery

We propose robust and efficient algorithms for the joint sparse recovery problem in compressed sensing, which simultaneously recover the supports of jointly sparse signals from their multiple measurement vectors obtained through a common sensing matrix. In a favorable situation, the unknown matrix, which consists of the jointly sparse signals, has linearly independent nonzero rows. In this case, the MUSIC (MUltiple SIgnal Classification) algorithm, originally proposed by Schmidt for the direction of arrival problem in sensor array processing and later proposed and analyzed for joint sparse recovery by Feng and Bresler, provides a guarantee with the minimum number of measurements. We focus instead on the unfavorable but practically significant case of rank-defect or ill-conditioning. This situation arises with limited number of measurement vectors, or with highly correlated signal components. In this case MUSIC fails, and in practice none of the existing methods can consistently approach the fundamental limit. We propose subspace-augmented MUSIC (SA-MUSIC), which improves on MUSIC so that the support is reliably recovered under such unfavorable conditions. Combined with subspace-based greedy algorithms also proposed and analyzed in this paper, SA-MUSIC provides a computationally efficient algorithm with a performance guarantee. The performance guarantees are given in terms of a version of restricted isometry property. In particular, we also present a non-asymptotic perturbation analysis of the signal subspace estimation that has been missing in the previous study of MUSIC.Comment: submitted to IEEE transactions on Information Theory, revised versio

DOA estimation of two targets using beamformer based methods with application to automotive radar

Projecte final de carrera fet en col.laboració amb Technische Universität DarmstadtEnglish: Direction-of-arrival (DOA) estimation of two targets plays an important role in automotive radar. Two cases are distinguished: when the targets are closely spaced and the conventional beamformer is not able to resolve them, and when the targets are widely spaced and the beamformer is able to resolve them. In the first case, accurate estimates can be obtained using high-resolution techniques. In the second case, estimates are typically biased. Automotive radar applications demand real-time processing and therefore the computational cost has to be addressed. For the resolved scenario, we propose a procedure to reduce the bias of the beamformer estimates, thus avoiding the use of iterative algorithms. The final estimates are obtained after applying a correction term, which is calculated off-line and stored in a look-up table. For the non-resolved scenario, we consider a practicable implementation of the maximum likelihood estimator. A simplified version of the cost function is used to reduce the complexity. The peak location from the beamformer can also be used to delimit the search range. The results of the mentioned methods are compared with other iterative algorithms, in terms of performance and computational cost. Applying the correction factors, the bias of the beamformer estimates are successfully reduced, making them accurate enough for the automotive radar application. The simplified implementation of the ML cost function reduces significantly the computational cost, allowing its use in real-time applications. Moreover, the performance obtained is also within the acceptable range for the automotive radar application, even for narrow angular separations. A block diagram containing the proposed methods is finally given, which is proposed as a suitable DOA estimation system for the automotive radar application.Castellano: La estimación del ángulo de llegada (DOA estimation) para dos objetivos juega un papel importante dentro de las aplicaciones radar para la automoción. Para este caso, distinguimos entre dos escenarios: cuando los objetivos se encuentran muy cerca el uno del otro y el beamformer convencional no es capaz de resolverlos, y cuando los objetivos se encuentran bastante separados y éste sí es capaz de resolverlos. En el primer escenario, podemos conseguir estimaciones más precisas mediante el uso de técnicas de alta resolución. En el segundo escenario, las estimaciones obtenidas son típicamente sesgadas. Las aplicaciones radar para la automoción requieren del procesado de datos en tiempo real y, por lo tanto, la carga computacional debe ser reducida. Para el escenario resuelto, proponemos un procedimiento que permite reducir el sesgo de las estimaciones del beamformer, evitando así el uso de algoritmos iterativos. Las estimaciones finales se obtienen tras aplicar un término de corrección, que es calculado previamente off-line y almacenado en una tabla (look-up table). Para el caso no resuelto, consideramos una implementación factible del estimador de máxima verosimilutud (MLE). Por tal de reducir la complejidad de los cálculos, usamos una versión simplificada de la función de coste del estimador. Además, el máximo obtenido del beamformer es usado también para delimitar el rango de búsqueda, reduciendo aún más la carga computacional. Los resultados obtenidos tras aplicar los métodos mencionados son contrastados con otros algoritmos iterativos, en términos de rendimiento y carga computacional. Aplicando los factores de corrección, el sesgo de las estimaciones obtenidas mediante el beamformer se ve reducido considerablemente, permitiendo tasas de error aptas para su uso en aplicaciones radar para la automoción. La versión simplificada del MLE reduce significantemente la carga computacional, haciendo posible también su uso para aplicaciones en tiempo real. Además, el comportamiento obtenido se encuentra de igual manera dentro del rango aceptado en aplicaciones radar para la automoción, incluso para ángulos de llegada muy próximos entre si. Finalmente, proporcionamos un diagrama de bloques que combina las técnicas descritas, el cual es propuesto como un sistema apropiado para la estimación del ángulo de llegada en aplicaciones radar para la automoción.Català: L'estimació de l'angle d'arribada (DOA estimation) per dos objectius juga un paper important dintre de les aplicacions radar per a l'automoció. En aquest cas, podem distingir entre dos escenaris diferents: quan els objectius es troben molt a prop l'un de l'altre i el beamformer convencional no és capaç de resoldre'ls, i quan els objectius es troben bastant separats i aquest si és capaç de resoldre'ls. En el primer escenari, podem aconseguir estimacions més precises mitjançant l'ús de tècniques d'alta resolució. En el segon escenari, les estimacions obtingudes son típicament esbiaixades. Les aplicacions radar per a l'automoció requereixen de processament de dades en temps real i, per tant, la carga computacional ha de ser reduïda. Per a l'escenari resolt, proposem un procediment que fa possible reduir el biaix de les estimacions del beamformer, evitant així l'ús d'algoritmes iteratius. Les estimacions finals s'obtenen després d'aplicar un terme de correcció, que és calculat prèviament off-line i emmagatzemat en una taula (look-up table). Per al cas no resolt, considerem una implementació factible de l'estimador de màxima versemblança (MLE). Per tal de reduir la complexitat dels càlculs, utilitzem una versió simplificada de la funció de cost de l'estimador. A més, utilitzem el màxim obtingut del beamformer per delimitar el rang de cerca, reduint encara més la carga computacional. Els resultats obtinguts després d'aplicar els mètodes mencionats són contrastats am altres algoritmes iteratius, en termes de rendiment i carga computacional. Aplicant els factors de correcció, el biaix de les estimacions del beamformer es veu reduït considerablement, produint tasses d'error aptes per el seu ús en aplicacions radar per a l'automoció. La versió simplificada del MLE redueix significativament la carga computacional, fent possible també el seu ús per aplicacions en temps real. A més, el comportament obtingut es troba d'igual manera dins del marge acceptable en aplicacions radar per a l'automoció, fins i tot per angles d'arribada molt pròxims entre si. Finalment, proporcionem un diagrama de blocs que combina les tècniques descrites, el qual es proposat com a sistema apropiat per a l'estimació de l'angle d'arribada en aplicacions radar per a l'automoció