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In this paper, we study the performance of two algorithms for tensor reconstruction in the presence of random uncertainties: the first one is the non-blind state-of-the-art, and the second one, new, is blind and is designed to deal with such uncertainties. Using coupled tensor LL1 models, we show that the proposed algorithm performs better than the stateof-the-art, which naturally raises the question of its efficiency. In that perspective, the standard approach is to resort to the usual constrained Cramér-Rao bound (CCRB), which appears to be only partially informative when addressing the asymptotic achievable performance of the considered model. Indeed, the usual CCRB is not able to account for randomness in the set of constraints. To fill this gap, we also introduce a new randomlyconstrained Cramér-Rao bound and we illustrate its relevance to analyze the relative efficiency of the proposed algorithm. As a by-product, we provide closed-form expressions for the Fisher information matrices based on the LL1 model.

The use of carrier phase data is the main driver for high-precision Global Navigation Satellite Systems (GNSS) positioning solutions, such as Real-Time Kinematic (RTK). However, carrier phase observations are ambiguous by an unknown number of cycles, and their use in RTK relies on the process of mapping real-valued ambiguities to integer ones, so-called Integer Ambiguity Resolution (IAR). The main goal of IAR is to enhance the position solution by virtue of its correlation with the estimated integer ambiguities. With the deployment of new GNSS constellations and frequencies, a large number of observations is available. While this is generally positive, positioning in medium and long baselines is challenging due to the atmospheric residuals. In this context, the process of solving the complete set of ambiguities, so-called Full Ambiguity Resolution (FAR), is limiting and may lead to a decreased availability of precise positioning. Alternatively, Partial Ambiguity Resolution (PAR) relaxes the condition of estimating the complete vector of ambiguities and, instead, finds a subset of them to maximize the availability. This article reviews the state-of-the-art PAR schemes, addresses the analytical performance of a PAR estimator following a generalization of the Cramér–Rao Bound (CRB) for the RTK problem, and introduces Precision-Driven PAR (PD-PAR). The latter constitutes a new PAR scheme which employs the formal precision of the (potentially fixed) positioning solution as selection criteria for the subset of ambiguities to fix. Numerical simulations are used to showcase the performance of conventional FAR and FAR approaches, and the proposed PD-PAR against the generalized CRB associated with PAR problems. Real-data experimental analysis for a medium baseline complements the synthetic scenario. The results demonstrate that (i) the generalization for the RTK CRB constitutes a valid lower bound to assess the asymptotic behavior of PAR estimators, and (ii) the proposed PD-PAR technique outperforms existing FAR and PAR solutions as a non-recursive estimator for medium and long baselines.

This contribution analyzes the fundamental performance limits of traditional two-step Global Navigation Satellite System (GNSS) receiver architectures, which are directly linked to the achievable time-delay estimation performance. In turn, this is related to the GNSS baseband signal resolution, i.e., bandwidth, modulation, autocorrelation function, and the receiver sampling rate. To provide a comprehensive analysis of standard point positioning techniques, we consider the different GPS and Galileo signals available, as well as the signal combinations arising in the so-called GNSS meta-signal paradigm. The goal is to determine: (i) the ultimate achievable performance of GNSS code-based positioning systems, and (ii) whether we can obtain a GNSS code-only precise positioning solution and under which conditions. In this article, we provide clear answers to such fundamental questions, leveraging on the analysis of the Cram&eacute, r&ndash, Rao bound (CRB) and the corresponding Maximum Likelihood Estimator (MLE). To determine such performance limits, we assume no external ionospheric, tropospheric, orbital, clock, or multipath-induced errors. The time-delay CRB and the corresponding MLE are obtained for the GPS L1 C/A, L1C, and L5 signals, the Galileo E1 OS, E6B, E5b-I, and E5 signals, and the Galileo E5b-E6 and E5a-E6 meta-signals. The results show that AltBOC-type signals (Galileo E5 and meta-signals) can be used for code-based precise positioning, being a promising real-time alternative to carrier phase-based techniques.

International audience; In statistical signal processing, hybrid parameter estimation refers to the case where the parameters vector to estimate contains both non-random and random parameters. Numerous works have shown the versatility of deterministic constrained Cramér-Rao bound for estimation performance analysis and design of a system of measurement. However in many systems both random and non-random parameters may occur simultaneously. In this communication, we propose a constrained hybrid lower bound which take into account of equality constraint on deterministic parameters. The usefulness of the proposed bound is illustrated with an application to radar Doppler estimation.

The general theme of my research deals with the characterization of the joint detection-estimation problem arising frequently both in passive and active systems of measurement (radar, telecoms, sonar): the estimation of the deterministic (non random) parameters of an intermittent source of signal of interest embedded in a permanent noisy environment. This problem can be modelled as a binary hypothesis testing: H0 (noise signal only) and H1 (noise signal and signal of interest). My primary interest for deterministic parameters originates in my involvement in numerous studies dedicated to active radar estimation performance where the deterministic parametric model is the privileged model. Under the framework of deterministic parametric modelling, the problem under consideration can be addressed progressively, in terms of theoretical and computational complexity, in two steps: * the assessment of unconditional estimation performance, that is without a prior detection test, by resorting to lower bounds on estimation performance. In this first step, there is a single observation model H1. * the assessment of conditional estimation performance, that is including a prior detection test and leading to the characterization of the joint detection-estimation problem. This far more complex problem is investigated by studying the influence of a detection step on lower bounds on estimation performance when applied to two models of observations (the monopulse antenna and the Gaussian conditional observation model) for which some analytical expressions exist., Le thème général de ma recherche est la caractérisation des problèmes conjoints détection-estimation les plus fréquemment rencontrés en écoute active ou passive (radar, télécoms, sonar, ...) : l'estimation des paramètres déterministes (non aléatoires) d'un signal d'intérêt intermittent en présence d'un environnement permanent. Ce problème peut être modélisé par un test d'hypothèses binaires : H0 (environnement seul) et H1 (environnement et signal d'intérêt). L'intérêt premier pour l'estimation des paramètres déterministes provient de mon domaine d'application professionnel : le radar actif où l'approche à paramètres déterministe est privilégiée. Dans le cadre de paramètres déterministes, cette recherche peut s'aborder graduellement (en terme de difficulté théorique et calculatoire) sous deux axes : * l'étude des performances en estimation non conditionnelle (sans test de détection préalable) par le biais des bornes de performance en estimation. Dans ce cas il n'y a qu'un seul modèle d'observation H1. * l'étude des performances en estimation conditionnelle (avec test de détection préalable), c'est à dire la caractérisation des problèmes conjoints détection-estimation. Cette étude est réalisée par le biais des bornes de performance conditionnelle appliquées à deux modèles d'observation particuliers (l'antenne monopulse et le modèle d'observation gaussien déterministe) pour lesquels certains calculs analytiques sont accessibles.

Publication in the conference proceedings of EUSIPCO, Barcelona, Spain, 2011

This paper presents a simple approach for deriving computable lower bounds on the MSE of deterministic parameter estimators with a clear interpretation of the bounds. We also address the issue of lower bounds tightness in comparison with the MSE of ML estimators and their ability to predict the SNR threshold region. Last, as many practical estimation problems must be regarded as joint detection-estimation problems, we remind that the estimation performance must be conditional on detection performance.

Publication in the conference proceedings of EUSIPCO, Lausanne, Switzerland, 2008

In non-linear estimation problems three distinct regions of operation can be observed. In the asymptotic region, the Mean Square Error (MSE) of Maximum Likelihood Estimators (MLE) is small and, in many cases,close to the Cramer-Rao bound (CRB). In the a priory performance region where the number of independent snapshots and/or the SNR are very low, the MSE is close to that obtained from the prior knowledge about the problem. Between these two extremes, there is an additional transition region where MSE of estimators deteriorates with respect to CRB. The present paper provides exemples of improvement of MSE prediction by CRB, not only in the transition region but also in the a priori region, resulting from introduction of a detection step, which proves that this renement in MSE lower bounds derivation is worth investigating.

Publication in the conference proceedings of EUSIPCO, Viena, Austria, 2004

A wide variety of actual processing (Radar, Sonar, Telecoms ...) requires a detection step, who main effect is to restrict the set of observations available for unknown parameter estimation. Therefore, we address the derivation of Mean Square Error (MSE) lower bounds for determinii parameters conditioned by a binary hypothesis testing problem using a didactic approach of wi scope. To prove mat it is meaningful, we also show, with the help of a fundamental applicatior 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: joint detection-estimation problem., Un grand nombre d'applications concrètes (Radar, Sonar, Télécoms ...) requièrent une étape de détection dont l'effet principal est de restreindre l'ensemble des observations disponibles pour l'estimation des paramètres inconnus. Par conséquent, nous établissons l'expression des bornes inférieures de l'Erreur Quadratique Moyenne (EQM) conditionnées par un test d'hypothèse binaire au moyen d'une approche didactique générale. Pour valider l'intérêt de cette démarche, nous montrons également à l'aide d'une application fondamentale, que le problème de la précision de prédiction de l'EQM vraie par une borne inférieure à faible RSB, peut provenir d'une formulation incorrecte de la borne inférieure ne prenant pas en compte la vraie problématique, à savoir les problèmes conjoints de détection-estimation.

L'avènement des Système de Positionnement par Satellites, également désigné sous le sigle GNSS, a révolutionné le monde d'aujourd'hui avec des applications diverses, de la localisation à la télédétection. Le principe du GNSS est basé sur la trilatération, qui dépend de l'estimation du retard de propagation et de l'effet Doppler. En effet, la synchronisation retard-Doppler est un sujet de recherche important pour de nombreux domaines, avec des applications pratiques telles que le radar, le sonar, l'ultrason, la télécommunication et la navigation. Or, pour la conception et l'évaluation des techniques d'estimation, il est important de connaître la meilleure performance accessible au sens de l'erreur quadratique moyenne (EQM), ce qui est fourni par le calcul des bornes inférieures (BI) sur l'EQM. Par rapport aux autres BI, les bornes de Cramér-Rao (BCR) sont plus simples à calculer et fournissent une estimation précise de l'EQM de l'estimateur au sens du Maximum de Vraisemblance (EMV) dans la région d'opération asymptotique, sous certaines conditions. Aussi, les architectures des récepteurs GNSS reposent sur des approches d'acquisition et de suivi, effectuée de manière scalaire, qui peuvent être considérées comme des instances particulières de l'EMV. Cependant, malgré une littérature fournie sur les BCR relatives à l'estimation retard-Doppler, la plupart de ces expressions de la BCR sont trop restrictives et ne concernent que le modèle de signal à bande étroite, sans tenir compte de l'impact de l'effet du Doppler sur le signal en bande de base. En effet, une expression analytique de BCR, suffisamment générale et facile à utiliser, pour n'importe quel signal à bande limitée, n'était pas disponible au début de cette thèse. L'objectif principal de la présente thèse porte sur la caractérisation des performances asymptotiques de l'estimation du retard et du Doppler par un EMV cohérent. La première contribution est l'obtention d'une nouvelle expression analytique de la BCR pour l'estimation du retard, en considérant un signal générique à bande limitée et un retard de propagation constant, cette expression offrant de nouvelles possibilités pour la conception d'un signal optimal. Cette expression de la BCR est ensuite utilisée pour caractériser l'estimation conjointe du retard et de la phase du signal. Cette approche est pertinente pour la localisation précise, exploitant la phase de la porteuse, telles que PPP et RTK. Par ailleurs, ces travaux de recherche ont permis de mettre en évidence l'absence d'une analyse complète des performances des signaux GNSS dans la littérature scientifique, d'un point de vue de l'estimation optimale. L'analyse présentée ici vise à combler cette lacune et fournit également les limites de performance du positionnement standard. Pour franchir cette limite, il devient alors nécessaire de recourir aux techniques de positionnement basées sur la phase de la porteuse. Enfin, nous étendons la formulation analytique de la BCR générique pour inclure l'estimation conjointe du retard et du Doppler, d'abord pour les signaux à bande étroite, puis pour leurs homologues à large bande, incluant également l'amplitude et la phase. The advent of Global Navigation Satellite Systems (GNSS) has revolutionized today's world, with applications in positioning, timing and remote sensing. The principle of GNSS is based on multilateration, which first involves the estimation of both signal propagation time-delay and Doppler effect for each visible satellite. In fact, delay-Doppler synchronization is an active research topic of significant practical importance in many fields, with applications in radar, sonar, ultrasonics, communications and navigation. Now, when designing and assessing estimation techniques, it is of fundamental importance to know the ultimate achievable performance in the mean square error (MSE) sense, information which is brought by the computation of performance lower bounds (LBs) on the MSE. Compared to other LBs, Cramér-Rao Bounds (CRB) are simple to calculate and give an accurate estimation of the MSE of the maximum likelihood estimator (MLE) in the asymptotic region of operation under certain conditions. Indeed, GNSS receiver architectures rely on a scalar acquisition and tracking approach, which can be seen as particular instances of a MLE solution. However, even if the delay-Doppler CRB literature is abundant, most of these CRB expressions are unnecessarily restrictive and only address the standard narrowband signal model, without considering the impact of the Doppler effect on the baseband signal. Most importantly, a sufficiently generalized and easy-to-use compact CRB for generic band-limited signals is not available, being an important missing point of practical interest. The primary objective of the present thesis is the characterization of the asymptotic performance of GNSS time-delay and Doppler estimation, i.e., synchronization, within a coherent integration time. The first contribution is the derivation of a new compact closed-form CRB expression for time-delay estimation, considering a generic band-limited transmitted signal and constant transmitter-to-receiver propagation delay, completed with insights on an optimal signal design as well as a performance loss metric. This CRB derivation is then extended to include the joint time-delay and phase estimation, which is of interest for standard precise positioning approaches exploiting the carrier phase, such as PPP and RTK. A complimentary problem is then addressed: the lack of a comprehensive performance analysis of GNSS signals in literature, from an optimal estimation point of view. The work further provides the performance thresholds that require a need for carrier phase-based positioning techniques. Finally, we extend the derivation of the generic compact closed-form CRB expression to include the joint time-delay and Doppler stretch estimation, first for narrowband signals and then for their wideband counterpart, followed by the introduction of a general compact closed-form CRB expression for the amplitude and phase. Such results set the theoretical basis for the full GNSS receiver chain characterization.

Les avancées récentes dans le domaine de navigation par satellites (GNSS) ont conduit à une prolifération des applications de géolocalisation dans les milieux urbains. Pour de tels environnements, les applications GNSS souffrent d’une grande dégradation liée à la réception des signaux satellitaires en lignes indirectes (NLOS) et en multitrajets (MP). Ce travail de thèse propose une méthodologie originale pour l’utilisation constructive des signaux dégradés MP/NLOS, en appliquant des techniques avancées de traitement du signal ou à l’aide d’une assistance d’un simulateur 3D de propagation des signaux GNSS. D’abord, nous avons établi le niveau maximal réalisable sur la précision de positionnement par un système GNSS "Stand-Alone" en présence de conditions MP/NLOS, en étudiant les bornes inférieures sur l’estimation en présence des signaux MP/NLOS. Pour mieux améliorer ce niveau de précision, nous avons proposé de compenser les erreurs NLOS en utilisant un simulateur 3D des signaux GNSS afin de prédire les biais MP/NLOS et de les intégrer comme des observations dans l’estimation de la position, soit par correction des mesures dégradées ou par sélection d’une position parmi une grille de positions candidates. L’application des approches proposées dans un environnement urbain profond montre une bonne amélioration des performances de positionnement dans ces conditions. Recent trends in Global Navigation Satellite System (GNSS) applications in urban environments have led to a proliferation of studies in this field that seek to mitigate the adverse effect of non-line-of-sight (NLOS). For such harsh urban settings, this dissertation proposes an original methodology for constructive use of degraded MP/NLOS signals, instead of their elimination, by applying advanced signal processing techniques or by using additional information from a 3D GNSS simulator. First, we studied different signal processing frameworks, namely robust estimation and regularized estimation, to tackle this GNSS problem without using an external information. Then, we have established the maximum achievable level (lower bounds) of GNSS Stand-Alone positioning accuracy in presence of MP/NLOS conditions. To better enhance this accuracy level, we have proposed to compensate for the MP/NLOS errors using a 3D GNSS signal propagation simulator to predict the biases and integrate them as observations in the estimation method. This could be either by correcting degraded measurements or by scoring an array of candidate positions. Besides, new metrics on the maximum acceptable errors on MP/NLOS errors predictions, using GNSS simulations, have been established. Experiment results using real GNSS data in a deep urban environment show that using these additional information provides good positioning performance enhancement, despite the intensive computational load of 3D GNSS simulation.