This thesis discusses mathematical optimization techniques for waveform design in cognitive radars. These techniques have been designed with an increasing level of sophistication, starting from a bistatic model (i.e. two transmitters and a single receiver) and ending with a cognitive network (i.e. multiple transmitting and multiple receiving radars). The environment under investigation always features strong signal-dependent clutter and noise. All algorithms are based on an iterative waveform-filter optimization. The waveform optimization is based on convex optimization techniques and the exploitation of initial radar waveforms characterized by desired auto and cross-correlation properties. Finally, robust optimization techniques are introduced to account for the assumptions made by cognitive radars on certain second order statistics such as the covariance matrix of the clutter. More specifically, initial optimization techniques were proposed for the case of bistatic radars. By maximizing the signal to interference and noise ratio (SINR) under certain constraints on the transmitted signals, it was possible to iteratively optimize both the orthogonal transmission waveforms and the receiver filter. Subsequently, the above work was extended to a convex optimization framework for a waveform design technique for bistatic radars where both radars transmit and receive to detect targets. The method exploited prior knowledge of the environment to maximize the accumulated target return signal power while keeping the disturbance power to unity at both radar receivers. The thesis further proposes convex optimization based waveform designs for multiple input multiple output (MIMO) based cognitive radars. All radars within the system are able to both transmit and receive signals for detecting targets. The proposed model investigated two complementary optimization techniques. The first one aims at optimizing the signal to interference and noise ratio (SINR) of a specific radar while keeping the SINR of the remaining radars at desired levels. The second approach optimizes the SINR of all radars using a max-min optimization criterion. To account for possible mismatches between actual parameters and estimated ones, this thesis includes robust optimization techniques. Initially, the multistatic, signal-dependent model was tested against existing worst-case and probabilistic methods. These methods appeared to be over conservative and generic for the considered signal-dependent clutter scenario. Therefore a new approach was derived where uncertainty was assumed directly on the radar cross-section and Doppler parameters of the clutters. Approximations based on Taylor series were invoked to make the optimization problem convex and {subsequently} determine robust waveforms with specific SINR outage constraints. Finally, this thesis introduces robust optimization techniques for through-the-wall radars. These are also cognitive but rely on different optimization techniques than the ones previously discussed. By noticing the similarities between the minimum variance distortionless response (MVDR) problem and the matched-illumination one, this thesis introduces robust optimization techniques that consider uncertainty on environment-related parameters. Various performance analyses demonstrate the effectiveness of all the above algorithms in providing a significant increase in SINR in an environment affected by very strong clutter and noise.

Dans le domaine du radar aéroporté, les enjeux industriels actuels sont nombreux et portent,entre autres, sur l'établissement de profils distance de cibles aériennes, terrestres et maritimes pour leur identification. Cela implique en particulier la mise en oeuvre de chaînes d'émission/réception pour des modes de fonctionnement haute résolution. Dans ce contexte, les problématiques à traiter comprennent alors la conception et l'analyse de performances de détecteurs pour des modèles de cibles étendues, la conception de formes d'ondes multi-résolutions et le développement des traitements associés, l'optimisation de formes d'onde robustes au fouillis, etc. Le travail de cette thèse, qui s'intègre dans ce cadre, se décompose en deux parties. Dans un premier temps, nous traitons la détection d'une cible dite "étendue", c'est-à-dire caractérisée par plusieurs réflecteurs élémentaires prépondérants répartis sur plusieurs cases distance non nécessairement consécutives. Ce modèle est notamment approprié lorsque la résolution en distance est suffisamment fine, et s'intègre dans les problématiques d'identification de cible. Dans ce cadre, nous étudions un test de détection fondé sur le rapport de vraisemblances généralisé (GLRT) intégrant la localisation inconnue des réflecteurs, et lorsque la perturbation est du bruit blanc gaussien. En utilisant des résultats issus des statistiques d'ordre, nous déduisons des approximations de la probabilité de fausse alarme et de la probabilité de détection. Des comparaisons numériques avec des détecteurs existants sont fournies. Dans un second temps, nous étudions une forme d'onde correspondant à un train d'impulsions contenant deux codes de phase, l'un intra impulsion et l'autre inter impulsion. Pour un modèle de cible ponctuelle et un fouillis gaussien, nous proposons de sélectionner ces codes en tenant compte de différents critères tels que la maximisation de la probabilité de détection ou encore la minimisation des lobes secondaires du signal reçu après traitement. Pour un type donné de fouillis modélisé par un processus autorégressif (AR), nous abordons le problème d'optimisation multi-objectifs en utilisant les fronts de Pareto. La modélisation AR permettant de considérer plusieurs types de fouillis à partir d'un nombre réduit de paramètres, nous étudions alors la robustesse des codes de phase optimaux à des variations de fouillis. / In the field of airborne radar, one of the current industrial stakes, among others, is the identification of a target, whether airborne, terrestrial or maritime, through the establishment of its range profile. This implies to set up a transmit/receive processing for high resolution modes. In this context, the issues to be addressed include the design and the performance analysis of detectors for extended target models, the design of multi-resolution waveforms and the associated processing, the optimization of waveforms that are robust to clutter, etc. Within this frame, the work of this thesis is twofold. The first part deals with the detection of a so-called "extended" target, i.e. which is characterized by a few main scatterers spread over several range gates not necessarily consecutive. This model is appropriate when the range resolution is thin enough and it is suited for target identification issues. In this context, we study a detection test based on the generalized likelihood test (GLRT) which includes the unknown locations of the scatterers, and when the disturbance is white Gaussian noise. By using ordered statistics, we deduce approximations of the probability of false alarm and the probability of detection. Numerical comparison with existing detectors are also provided. Secondly, we study a waveform based on a pulse train which contains two phase codes: the first one is intrapulse whereas the second one is interpulse. Assuming a point target and Gaussian clutter, we propose to select these codes taking into account several criteria such as the maximization of the probability of detection or the minimization of the sidelobes of the received signal after processing. For a given type of clutter modeled by an autoregressive (AR) process, we address this multi-objective optimization problem using the Pareto fronts. Since the AR modeling makes it possible to consider several types of clutter from a reduced number of parameters, we study the robustness of optimal phase codes to clutter variations.

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