The demand for outdoor precise location is increasing with the development of new applications such as autonomous vehicles, exploration robots and wireless sensor networks. Global Navigation Satellite System (GNSS) is the go-to system for outdoor localization. This thesis focuses on developing new methods for GNSS single-point positioning (SPP) model, where no access to a reference station or precise GNSS parameters is needed. We investigated the limitations of the standard method, least- squares adjustment (LSA), and we derived the Cramer-Rao bounds for the SPP estimation problem. We also investigated different techniques to formulate the positioning problem with the goal to increase the accuracy. A new method is developed by reformulating the problem as difference-of-convex program (DC program) and utilizing convex-concave procedure (CCCP) to solve the positioning problem without linearizing the observation equations. In addition, we examined the potential of multiple-receiver systems in increasing the accuracy. We formulated the multiple- receiver SPP estimation problem, and we proposed to configure the multiple receivers in a fixed equilateral triangle to exploit the symmetry and the geometrical constraints of the configuration. We extended the use of LSA in multiple-receiver system. We also developed a modification of LSA algorithm, named least-squares adjustment extension (LSAE), that utilizes attitude information and the constraints of the multiple-receiver system. In addition, we developed a new algorithm to optimizes the SPP estimates over the equilateral triangles Riemannian manifold, which enforces the geometrical constraints of the multiple-receiver system. Furthermore, we derived the constrained and the unconstrained Cramer-Rao bounds (CRB and CCRB) for the multiple-receiver SPP problem. Moreover, we investigated the influence of both attitude information and the equilateral triangle baseline length on the algorithms’ performances and the derived CCRB. Finally, we carried out a numerical analysis by implementing the algorithms and the bounds in MATLAB, where we tested the algorithms on simulated GNSS scenarios. The proposed multiple-receiver methods provide more precise estimates for the SPP problem in comparison to the single receiver methods.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第17968号 / 情博第512号 / 新制||情||91(附属図書館) / 30798 / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 西村 直志, 准教授 山下 信雄 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Spectrally sparse signals arise in many applications of signal processing. A spectrally sparse signal is a mixture of a few undamped or damped complex sinusoids. An important problem from practice is to reconstruct such a signal from partial time domain samples. Previous convex methods have the drawback that the computation and storage costs do not scale well with respect to the signal length. This common drawback restricts their applicabilities to large and high-dimensional signals. The reconstruction of a spectrally sparse signal from partial samples can be formulated as a low-rank Hankel matrix completion problem. We develop two fast and provable non-convex solvers, FIHT and PGD. FIHT is based on Riemannian optimization while PGD is based on Burer-Monteiro factorization with projected gradient descent. Suppose the underlying spectrally sparse signal is of model order r and length n. We prove that O(r^2log^2(n)) and O(r^2log(n)) random samples are sufficient for FIHT and PGD respectively to achieve exact recovery with overwhelming probability. Every iteration, the computation and storage costs of both methods are linear with respect to signal length n. Therefore they are suitable for handling spectrally sparse signals of large size, which may be prohibited for previous convex methods. Extensive numerical experiments verify their recovery abilities as well as computation efficiency, and also show that the algorithms are robust to noise and mis-specification of the model order. Comparing the two solvers, FIHT is faster for easier problems while PGD has a better recovery ability.

La diagonalisation conjointe approximée d’un ensemble de matrices permet de résoudre le problème de séparation aveugle de sources et trouve de nombreuses applications, notamment pour l’électroencéphalographie, une technique de mesure de l’activité cérébrale.La diagonalisation conjointe se formule comme un problème d’optimisation avec trois composantes : le choix du critère à minimiser, la contrainte de non-dégénérescence de la solution et l’algorithme de résolution.Les approches existantes considèrent principalement deux critères, les moindres carrés et la log-vraissemblance.Elles sont spécifiques à une contrainte et se restreignent à un seul type d’algorithme de résolution.Dans ce travail de thèse, nous proposons de formuler le problème de diagonalisation conjointe selon un modèle géométrique, qui généralise les travaux précédents et permet de définir des critères inédits, notamment liés à la théorie de l’information.Nous proposons également d’exploiter l’optimisation riemannienne et nousdéfinissons un ensemble d’outils qui permet de faire varier les trois composantes indépendamment, créant ainsi de nouvelles méthodes et révélant l’influence des choix de modélisation.Des expériences numériques sur des données simulées et sur des enregistrements électroencéphalographiques montrent que notre approche par optimisation riemannienne donne des résultats compétitifs par rapport aux méthodes existantes.Elles indiquent aussi que les deux critères traditionnels ne sont pas les meilleurs dans toutes les situations. / The approximate joint diagonalisation of a set of matrices allows the solution of the blind source separation problem and finds several applications, for instance in electroencephalography, a technique for measuring brain activity.The approximate joint diagonalisation is formulated as an optimization problem with three components: the choice of the criterion to be minimized, the non-degeneracy constraint on the solution and the solving algorithm.Existing approaches mainly consider two criteria, the least-squares and the log-likelihood.They are specific to a constraint and are limited to only one type of solving algorithms.In this thesis, we propose to formulate the approximate joint diagonalisation problem in a geometrical fashion, which generalizes previous works and allows the definition of new criteria, particularly those linked to information theory.We also propose to exploit Riemannian optimisation and we define tools that allow to have the three components varying independently, creating in this way new methods and revealing the influence of the choice of the model.Numerical experiments on simulated data as well as on electroencephalographic recordings show that our approach by means of Riemannian optimisation gives results that are competitive as compared to existing methods.They also indicate that the two traditional criteria do not perform best in all situations.