Cette présentation a pour but d’introduire le radar météorologique et de présenter son fonctionnement global.

In this talk, Colin Cros will focus on the problem of cooperative positioning in the context of GNSS (Global Navigation Satellite Systems). The presentation is divided into two parts. The first examines the solvability of the problem from a theoretical point of view, where the specificity comes from the type of measurements made: pseudo-distances. The approach adopted is based on a study of the measurement graph and the theory of rigidity. The second part deals with practical aspects, presenting how to integrate a cooperative measurement into a Kalman-type navigation filter. The difficulty arises from the lack of knowledge of the correlations between the agents' errors, which means that so-called conservative filters have to be used. This presentation is based on my doctoral thesis, which is available at: https://theses.fr/2024GRALT032

Fuzz testing is a method used in software testing that involves inputting random or unexpected data into a system to identify vulnerabilities. Unlike deterministic methods, which test performance under controlled and predictable conditions, fuzz testing introduces variability to uncover hidden issues. This variability simulates real-world scenarios, uncovering weaknesses that might otherwise remain unnoticed. For instance, fuzz testing can effectively reveal how GNSS receivers respond to rapid signal fluctuations and other anomalous behaviors, situations often overlooked by standard tests. Unlike traditional methods that rely on predefined inputs, Collins Aerospace works on a new fuzz testing framework for GNSS, which employs advanced techniques such as automated input generation and real-time response monitoring. This approach not only facilitates a comprehensive assessment of receiver resilience but also allows for the dynamic adaptation of test scenarios in real-time, ensuring that a wide range of operational conditions is explored. The navigation equipment minimum testing procedures must be defined and need scenarios definitions as well as test steps and pass/fail criteria to provide minimum guidance to manufacturers for future equipment certification. The limitations of current testing methods further highlight the necessity of adopting fuzz testing. These methods predominantly rely on deterministic approaches, which do not effectively simulate the unpredictable nature of real-world signal degradation or complex interference scenarios posed by advanced spoofing techniques. As technology advances, the techniques utilized by malevolent actors likewise evolve, emphasizing the necessity for adaptive testing methodologies capable of responding to these changes. By introducing randomness and variability, fuzz testing plays a critical role in bolstering the reliability and operational integrity of GNSS systems by rigorously assessing their ability to withstand both known and unknown threats. The anticipated results from this fuzz testing framework are expected to identify vulnerabilities and enhance the resilience of GNSS receivers, suggesting that fuzz testing can play a transformative role in GNSS validation.

If P1, . . . , Pn and Q1, . . . , Qn are probability measures on Rd and P1 ∗ · · · ∗ Pn and Q1 ∗ · · · ∗ Qn are their respective convolutions, the Rényi divergence Dλ of order λ ∈ (0, 1] satisfies Dλ(P1 ∗ · · · ∗ Pn||Q1 ∗ · · · ∗ Qn) ≤ ni=1 Dλ(Pi ||Qi ). When Pi belongs to the natural exponential family generated by Qi , with the same natural parameter θ for any i = 1, . . . , n, the equality sign holds. The present note tackles the inverse problem, namely “does the equality Dλ(P1 ∗ · · · ∗ Pn||Q1 ∗ · · · ∗ Qn) = ni=1 Dλ(Pi ||Qi ) imply that Pi belongs to the natural exponential family generated by Qi for every i = 1, . . . , n?” The answer is not always positive and depends on the set of solutions of a generalization of the celebrated Cauchy functional equation. We discuss in particular the case P1 = · · · = Pn = P and Q1 = · · · = Qn = Q, with n = 2 and n = ∞, the latter meaning that the equality holds for all n. Our analysis is mainly devoted to P and Q concentrated on non-negative integers, and P and Q with densities with respect to the Lebesgue measure. The results cover the Kullback– Leibler divergence (KL), this being the Rényi divergence for λ = 1. We also show that the only f -divergences such that Df (P∗2||Q∗2) = 2Df (P||Q), for P and Q in the same exponential family, are mixtures of KL divergence and its dual.

This article addresses the problem of computing a Cramér-Rao bound when the likelihood of Euclidean observations is parameterized by both unknown Lie group (LG) parameters and covariance matrix. To achieve this goal, we leverage the LG structure of the space of positive definite matrices. In this way, we can assemble a global LG parameter that lies on the product of the two groups, on which LG's intrinsic tools can be applied. From this, we derive an inequality on the intrinsic error, which can be seen as the equivalent of the Slepian-Bangs formula on LGs. Subsequently, we obtain a closed-form expression of this formula for Euclidean observations. The proposed bound is computed and implemented on two real-world problems involving observations lying in $\mathbb{R}^{p}$, dependent on an unknown LG parameter and an unknown noise covariance matrix: the Wahba's estimation problem on $SE(3)$, and the inference of the pose in $SE(3)$ of a camera from pixel detections.

High resolution spectrometers are composed of different optical elements and detectors that must be modeled as accurately as possible. Specifically, accurate estimates of Instrument Spectral Response Functions (ISRFs) are critical in order not to compromise the retrieval of trace gas concentrations from spectral measurements. Currently, parametric models are used to estimate these response functions. However, these models cannot always take into account the diversity of ISRF shapes that are encountered in practical applications. This paper studies a new ISRF estimation method based on a sparse representation of the ISRF in a dictionary. The proposed method is shown to be very competitive when compared to parametric models, yielding up to one order of magnitude smaller normalized ISRF estimation errors. The method is applied to different high-resolution spectrometers, demonstrating its reproducibility for multiple remote sensing missions.

Nonlinear regression plays a crucial role in various engineering applications. For the sake of mathematical tractability and ease of implementation, most of the existing inference procedures are derived under the assumption of independent and identically distributed (i.i.d.) Gaussian-distributed data. However, real-world situations often deviate from this assumption, with the true data generating process being a correlated, heavy-tailed and non-Gaussian one. The paper aims at providing the Misspecified Cramér–Rao Bound (MCRB) on the Mean Squared Error (MSE) of any unbiased (in a proper sense) estimator of the parameters of a nonlinear regression model derived under the i.i.d. Gaussian assumption in the place of the actual correlated, non-Gaussian data generating process. As a special case, the MCRB for an uncorrelated, i.i.d. Complex Elliptically Symmetric (CES) data generating process under Gaussian assumption is also provided. Consistency and asymptotic normality of the related Mismatched Maximum Likelihood Estimator (MMLE) will be discussed along with its connection with the Nonlinear Least Square Estimator (NLLSE) inherent to the nonlinear regression model. Finally, the derived theoretical findings will be applied in the well-known problem of time-delay and Doppler estimation for GNSS.

In this Ph.D. thesis, we explore machine learning-based solutions for channel decoding in Machine-to-Machine type communications, where achieving ultra-reliable lowlatency communications (URLLC) is essential. Their primary issue arises from the exponential growth in the decoder’s complexity as the packet size increases. This curse of dimensionality manifests itself in three different aspects: i) the number of correctable noise patterns, ii) the codeword space to be explored, and iii) the number of trainable parameters in the models. To address the first limitation, we explore solutions based on a Support Vector Machine (SVM) framework and suggest a bitwise SVM approach that significantly reduces the complexity of existing SVM-based solutions. To tackle the second limitation, we investigate syndromebased neural decoders and introduce a novel message-oriented decoder, which improves on existing schemes both in the decoder architecture and in the choice of the parity check matrix. Regarding the neural network size, we develop a recurrent version of a transformer-based decoder, which reduces the number of parameters while maintaining efficiency, compared to previous neural-based solutions. Lastly, we extend the proposed decoder to support higherorder modulations through Bit-Interleaved and generic Coded Modulations (BICM and CM, respectively), aiding its application in more realistic communication environments.

In this Ph.D. thesis, we explore machine learning-based solutions for channel decoding in Machine-to-Machine type communications, where achieving ultra-reliable lowlatency communications (URLLC) is essential. Their primary issue arises from the exponential growth in the decoder’s complexity as the packet size increases. This curse of dimensionality manifests itself in three different aspects: i) the number of correctable noise patterns, ii) the codeword space to be explored, and iii) the number of trainable parameters in the models. To address the first limitation, we explore solutions based on a Support Vector Machine (SVM) framework and suggest a bitwise SVM approach that significantly reduces the complexity of existing SVM-based solutions. To tackle the second limitation, we investigate syndromebased neural decoders and introduce a novel message-oriented decoder, which improves on existing schemes both in the decoder architecture and in the choice of the parity check matrix. Regarding the neural network size, we develop a recurrent version of a transformer-based decoder, which reduces the number of parameters while maintaining efficiency, compared to previous neural-based solutions. Lastly, we extend the proposed decoder to support higherorder modulations through Bit-Interleaved and generic Coded Modulations (BICM and CM, respectively), aiding its application in more realistic communication environments.

Cette thèse s’intéresse à la construction d’une échelle de temps autonome et robuste aux erreurs d’horloge pour des essaims de satellites. Prévue pour une utilisation dans un essaim de nanosatellites, cette nouvelle échelle de temps appelée ATST (Autonomous Time Scale using the Student’s T-distribution) peut traiter les anomalies dues aux imperfections des horloges et aux liens inter-satellites manquants dans un environnement hostile. Plus précisément, les types d’anomalies traités incluent les sauts de phase, les sauts de fréquence, un bruit de mesure élevé dans certains liens et d’éventuelles données manquantes. En calculant la moyenne pondérée des résidus issus de l’équation de l’échelle de temps de base (BTSE), la contribution des satellites avec des mesures anormales est réduite pour la génération de l’échelle de temps. Les poids attribués à chaque horloge sont basés sur l’hypothèse que les résidus de l’ensemble suivent une loi de Student, ce qui permet d’utiliser des méthodes d’estimation robustes à la présence d’éléments aberrants. La performance de l’algorithme ATST est équivalente à celle de l’algorithme AT1 oracle, qui est une version de l’échelle de temps AT1 avec la capacité de détecter parfaitement toutes les anomalies dans des données simulées. Bien que l’algorithme n’ait pas de méthode de détection explicite, l’algorithme ATST affiche toujours un niveau de robustesse comparable à celui d’un détecteur parfait. Cependant, l’algorithme ATST est concu pour un essaim avec de nombreuses horloges de types homogènes et est limité par une complexité numérique élevée. De plus, les anomalies sont toutes traitées de la même manière sans distinction entre les différents types d’anomalies. Malgré ces limitations identifiées, ce nouvel algorithme ATST représente une contribution prometteuse dans le domaine des échelles de temps grâce a la robustesse atteinte. Une méthode de traitement des horloges ajoutées ou retirées de l’ensemble des horloges disponibles est également proposée dans cette thèse en conjonction avec la méthode ATST. La méthode obtenue préserve la continuité de phase et de fréquence de l’échelle de temps en attribuant un poids nul aux horloges concernées lorsque le nombre total d’horloges est modifié. Un estimateur des moindres carrés (Least Squares, LS) est présenté pour montrer comment les mesures des liens inter-satellites peuvent être traitées en amont pour réduire le bruit de mesure et en même temps remplacer les mesures manquantes. L’estimateur LS peut être utilisé avec une méthode de détection qui élimine les mesures anormales. Il peut alors remplacer les mesures supprimées par les estimations correspondantes. Cette thèse étudie également les performances de l’estimateur du maximum de vraisemblance (MLE) pour les paramètres des lois de probabilités à queues lourdes, plus précisément pour la loi de Student et pour un mélange de lois gaussiennes. Les améliorations obtenues en supposant que ces lois sont effectivement à queues lourdes par rapport à l’hypothèse de la loi gaussienne sont démontrées avec les bornes de Cramér-Rao mal-spécifiées (MCRB). Les expressions obtenues des MCRB pour une loi de Student et un mélange de lois gaussiennes confirment que les lois à queues lourdes sont meilleures pour l’estimation de la moyenne en présence de valeurs aberrantes. Elles permettent également de montrer que l’estimation des paramètres des lois à queues lourdes nécessite au moins 25 horloges pour obtenir une erreur d’estimation proche de la MCRB correspondante, c’est-à-dire que l’estimateur atteigne son efficacité asymptotique. Des propositions de pistes de recherche futures incluent le traitement des limitations de l’algorithme ATST concernant les types et le nombre d’horloges. Une nouvelle définition des pondérations des résidus issue d’une méthode d’apprentissage statistique utilisant des données d’apprentissage est envisageable grâce `a l’utilisation des résidus de l’échelle de temps de base BTSE. Une autre piste de recherche est le traitement des anomalies transitoires qui pose actuellement problème pour l’algorithme ATST. Un traitement de ce type d’erreurs pourrait être envisagé avec un algorithme d’apprentissage statistique ou avec un estimateur robuste de la fréquence des horloges sur une fenêtre de données passées. Mots clés: Estimation robuste, échelles de temps, détection des anomalies, bornes de Cramér-Rao mal-spécifiées.