Keywords: Un système radar-communication coopératif. Radar target = cible radar, communication transmitter = émetteur des communications, receiver = récepteur, mutual infromation = information . . . . . . . . ASK Amplitude Shift Keying. 45 CPFSK Continuous Phase Frequency-Shift Keying. x, xi, DBF Digital Beam-Forming. 45 DOA Direction-of-Arrival. 47 DPSS Discrete Prolate Spheroidal Sequences. 89 DSL Digital Subscriber Line. 70 37 WiFi Wireless Fidelity. 37, 67 WiMAX Worldwide Interoperability for Microwave Access. 67

Lastly, I am indebted to my family and friends, who have supported me unconditionally and provided a source of comfort and inspiration throughout this journey, and especially to the people that were there at the final

Résumé: En raison de la croissance continue des applications électromagnétiques, le spectre devient de plus en plus encombré. Une solution possible pour limiter l'encombrement consiste à créer des systèmes radar-communication joints qui utilisent la même bande passante pour réaliser les deux applications. Dans cette thèse, nous nous intéressons à un système exploitant une forme d'onde issue du monde des communications pour réaliser conjointement de l'imagerie radar et un transfert d'information. Pour cela, parmi la multitude de codes existants, nous avons choisi d'utiliser des codes modulés continument en phase CPM, et plus particulièrement une sous-famille appelée Continuous Phase Frequency Shift-Keying codes. Les propriétés de ceux-ci, notamment en ce qui concerne l'occupation spectrale, sont d'abord étudiées et comparées à d'autres codes représentatifs des communications.

Cependant, ces formes d'ondes présentent des qualités de compression dégradées par rapport au chirp habituellement utilisé en radar. En particulier les lobes secondaires résultant de la compression avec le filtre adapté sont plus élevés, nuisant à la qualité de l'image SAR résultante. Un filtre désadapté qui minimise l'énergie des lobes secondaires est proposé, ainsi qu'un algorithme rapide qui fournit les filtres pour tous les signaux émis en un temps de calcul raisonnable. Ce filtre désadapté est amélioré pour pouvoir traiter des valeurs inconnues de décalage Doppler ou de retard hors grille qui peuvent s'appliquer sur le signal reçu. De tels problèmes peuvent être généralisés à d'autres applications radar que le SAR. Une fois le choix de la méthode de compression d'impulsion établi, une évaluation des résultats est proposée. D'une part, des images SAR re-synthétisées sont générées, reconstruites à partir de données réelles basées sur le chirp, en utilisant des codes modulés continument en phase et les filtres désadaptés, et différents outils de comparaison sont utilisés pour s'assurer de leurs performances. D'autre part, des données réelles sont acquises dans un cadre ISAR, afin de valider notre système dans un contexte réaliste. Finalement, nous pouvons apporter une réponse positive à la question suivante : pouvons-nous créer un système conjoint SAR-communication qui transmet des informations et fournit des images radar de haute qualité? Title: Processing Optimization for Continuous Phase Modulation-based Joint Radar-Communication System : Application on Imaging Radar Keywords: Synthetic Aperture Radar (SAR), Communications, Mismatched Filter, Continuous Phase Modulation, Waveform Abstract: Due to the continuous growth of electromagnetic applications, the spectrum gets more and more congested. A possible solution to this problem is the creation of joint radar-communication systems, because they can alleviate the spectrum occupancy by using the same bandwidth to perform both applications. In this thesis, a joint Synthetic Aperture Radar (SAR)-Communication system based on a communication waveform is considered. To do so, among all the existing codes, we chose Continuous Phase Modulated codes (CPM), and more specifically a sub-family called Phase Frequency Shift-Keying codes (CPFSK). Their properties, in particular the spectral occupation, are first studied and compared to other well-known communication codes.

However, these waveforms present degraded compression qualities when compared to the usual chirp used for radars. More specifically, the sidelobes generated from the Matched Filter compression are higher, and thus deteriorate the resulting SAR image. The mismatched filter that minimizes the sidelobe level is proposed along with a fast algorithm that provides the filters for all the transmitted signals during an acceptable computational cost. This mismatched filter is further improved so that it can deal with unknown parameters. More precisely, if unknown Doppler shift or off-grid delay values are applied to the received signal, then an improved mismatched filter is provided. Such problems can be extended to other radar applications. Once the range compression method choice is established, an evaluation of the results is proposed. On the one hand, re-synthesized SAR images are generated, reconstructed from real chirp-based data, using CPM codes and mismatched filters, and different comparison tools to evaluate their performance. On the other hand, real data are acquired in an ISAR framework, in order to validate our system in a realistic context. Finally, we provide a positive answer to the question: can we create a joint SAR-communication system that transmits information and provides an image of good radar quality?

To my family... I would like to thank from the bottom of my heart my PhD team. Starting from my PhD director, Hélène ORIOT, I would like to thank you for your punctual presence and help during my PhD. I really appreciate the fact that you were always finding time to explain to me any problem that had come up, and plan monthly meetings with the team. I would like to thank my two supervisors, Abigael TAYLOR and Olivier RABASTE; for your immense help during our work together. I extremely appreciate your patience, your hard work, and that you were there for me whenever I needed you! I will remember the ambiance of our meetings all my life! I don't think I can put in words how grateful I am that I have met you and worked with you! I would also like to thank my colleagues, from the MATS unit for the nice moments we spent together, and the cohesion that the team has. I would like to specifically thank Anthony; thank you for putting up with me every single day during my manuscript writing, I really appreciate it. In addition, Dodo, thank you so much for the funny moments in the office and I really appreciate your willingness to help in every situation. I would like also to thank our neighbors, Huy and Pierre, for our various scientific or not discussions during our breaks. I am deeply grateful to the DEMR that welcomed me in such warmth. This includes our colleagues in the TSRE and RMES unit. I would like to thank Milan and Nicolas from RMES unit for their valuable help on acquiring and formatting the data and their time to explain me in detail the experiment. Another big thank you to Fayin from RMES unit for the implementation of ISAR algorithm, his willingness to help, and the time he spent on our discussions. Thank you a lot, without you three the last chapter could not have been possible. From the TSRE team, I would also like to thank Fred for his help and useful information on the servers. The frequency bounds of the out-of-band power for a 99% power of the spectrum. Ideally these bounds should not exceed the half-bandwidth value ( B 2 = 176MHz as the spectrum is centered) so that there is no spectral loss. These bounds are also illustrated in Figure 4 ISLR small provides the ISLR values of the impulse responses when the energy is computed over a shortened window that corresponds to the number of points considered in Table 6.3 for the ISLR computation. The purple-colored highlighted values correspond to the best value of each column. . . . . . . . . . . . . . . . . . . 117 6.3 Range PSLR and ISLR for the corner reflector of every re-synthesized SAR image. 127 xiii 6.4 Mean values of the correlation coefficients for different areas presented in Figure 6.9. 6.5 MSE and SSIM values for the corner reflector and the building images. . . . . . . . 6.6 MSE and SSIM values for the lake and the forest images. . . . . . . . . . . . . . . 6.7 Average value of energy over an area of the lake. . . . . . . . . . . . . . . . . . . . En particulier, les systèmes radar tentent de conserver leur place dans le spectre, et de préserver la bande passante allouée à leurs applications. La lutte contre la congestion du spectre électromagnétique est donc une problématique importante. À l'origine, les systèmes de radar et de communication répondent à des objectifs différents et ont été affectés à des bandes distinctes. Cependant, une solution intéressante au problème de congestion, proposée dans la littérature, consiste à combiner les tâches d'un radar et d'un système de communication. Ce nouveau système peut favoriser, soit le radar, soit les communications, soit les deux missions simultanément. Plus précisément, un système peut tolérer l'autre en le traitant comme une interférence, optimiser les paramètres d'une des applications en préservant les performances de l'autre ou encore optimiser les performances des deux applications simultanément. RADIOLOC.

.3 cm 0.03 cm 3 x 105Å 3 x 104Å 3 x 103Å 3 x 102Å 3 x 1 0 Å 3 Å 3 x 1 0 -1Å 3 x 1 0 -2Å 3 x 1 0 -3Å 3 x 1 0 -4Å 3 x 1 0 -5Å 3 x 1 0 -6Å

Organisation de la thèse

Ce travail de thèse est divisé en 3 parties et 8 chapitres. La partie I est composée des chapitres 2 et 3, et résume l'état de l'art de la thèse.

• Le chapitre 2 présente les principes fondamentaux du radar et du SAR. Plus précisément, la définition et la présentation de quelques applications de systèmes radar sont données. Le signal et le traitement radar classiquement utilisés sont ensuite présentés. Enfin, les spécificités du SAR et les critères d'évaluation d'image radar habituels sont fournis.

• Le chapitre 3 donne un aperçu général du monde des systèmes conjoints radar-communication. Tout d'abord, les systèmes de communication sont indroduits, puis différentes stratégies de systèmes de communication radar tirées de la littérature sont expliquées.

La partie II contient les chapitres 4 et 5, et prépare la création du système conjoint SAR -communication puisqu'elle précise le choix de la forme d'onde et du traitement qui lui sera appliqué.

• L'objectif du chapitre 4 est de justifier le choix de la modulation Continuous Phase Frequency-Shift Keying (CPFSK) comme meilleur candidat pour notre système. Il présente tout d'abord une analyse bibliographique approfondie de ces signaux, ainsi que les caractéristiques de quelques autres codes de communication largement utilisés dans les systèmes de communication ou de radar-communication. Ensuite, toutes ces formes d'onde sont comparées, menant à la sélection des CPFSK pour la suite de ce travail.

• Le chapitre 5 est consacré à l'optimisation des filtres désadaptés. Ces filtres, dont nous donnerons d'abord la définition classique, permettent de minimiser l'énergie des lobes secondaires. Nous étudierons ensuite deux extensions du problème pour améliorer les performances en cas de décalages Doppler intra-impulsion inconnus ou de décalages de retard hors grille. Enfin, la dernière partie du chapitre présente notre contribution pour réduire le temps de calcul de l'algorithme générant les filtres désadaptés.

La partie III contient les chapitres 6 et 7, et présente des images résultant de simulations et donnéees réelles après application des signaux CPFSK et de leurs filtres désadaptés associés. Les performances sont comparées à celles du chirp selon différents critères d'évaluation.

• Le chapitre 6 explique comment des images SAR re-synthétisées sont créées à partir de la réponse impulsionnelle des codes de communication. Une fois générées, elles sont comparées au cas du chirp en utilisant différents critères d'évaluation du radar et des images.

Etat de l'art

Nous abordons cette thèse d'un point de vue plus radariste et le chapitre 2 est ainsi consacré aux fondamentaux du radar et au contexte SAR. Les radars peuvent être embarqués sur des avions, des bateaux ou sur le sol et peuvent servir à des applications diverses [Skolnik, 1980]. Un système radar simplifié est introduit afin d'expliquer brièvement la chaîne d'émission et de réception. La

Résumé en Français xxv

fonctionnalité principale d'un radar est présentée: Elle permetde de mesurer la distance entre le radar et une cible. La forme d'onde la plus utilisée en radar est l'impulsion modulée linéairement en fréquence, également appelée chirp. Ses principales propriétés sont la linéarité de la fréquence instantanée par rapport au temps, une amplitude constante et la robustesse au Doppler. Une fois que les signaux rétrodiffusés atteignent l'antenne du radar, ils sont analysés afin de fournir la distance et la vitesse de la cible. La figure 3 illustre la compression distance d'un chirp et le résultat obtenu après traitement distance-Doppler. Ce traitement est constitué par la compression distance et la compression Doppler. Une technique de compression d'impulsion en distance largement utilisée dans les applications radar est le filtre adapté. Le filtre adapté est connu pour maximiser le rapport signal à bruit de la cible, permettant ainsi d'extraire le signal d'intérêt du bruit [Wang, 2008, Levanon andMozeson, 2004]. Ensuite, la compression Doppler est effectuée par transformée de Fourier sur le train d'impulsions émises, pour déterminer la fréquence Doppler de la cible et par conséquent sa vitesse.

De même, le traitement SAR est proche de celui du radar classique, ce qui consiste la deuxième partie du chapitre 2. Plus précisément, le SAR permet de distinguer les caractéristiques physiques d'une zone en manipulant des signaux radar [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. Dans notre cadre, une plateforme aéroportée suivant une trajectoire supposée linéaire uniforme est équipée d'une antenne qui effectue à la fois la transmission et la réception [START_REF] Chan | An introduction to synthetic aperture radar (SAR)[END_REF]. Le but est de créer une antenne virtuelle, en exploitant le mouvement de la plateforme, afin d'obtenir une meilleure résolution sur l'image de sortie. La qualité des images SAR peut être évaluée par des critères comme le Integrated Sidelobe Level Ratio (ISLR) et le Peak-to-Sidelobe Level Ratio (PSLR) et qui seront essentiels pour analyser nos résultats. Ils donnent notamment une idée du contraste de l'image.

Les gammes d'ondes utilisées pour toutes les applications radars font partie du spectre électromagnétique. Une certaine largeur de bande leur est attribuée. Or, les applications de communication se développent à un rythme exceptionnellement rapide en raison du nombre croissant d'utilisateurs et du besoin de performances plus rapides. Eux aussi doivent s'inscrire dans le spectre éléctromagnétique. Ainsi, le spectre électromagnétique devient de plus en plus encombré et constitue une ressource limitée et précieuse. La cohabitation des radars et des communications est inévitable avec comme stratégie: la coexistence, la coopération et le co-design. La coexistence désigne la configuration dans laquelle deux ou plusieurs applications fonctionnent de manière autonome sans partager d'informations, la coopération se produit lorsqu'il y a un fonctionnement mutuel entre les deux systèmes, et le co-design consiste à créer des formes d'onde qui prennent en compte et optimisent les performances des deux applications [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF]. Un exemple d'un tel système peut-être visualisé sur la figure 4. De tels systèmes ont été principalement étudiés pour les véhicules autonomes [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF], pour remplir à la fois un rôle de détection pour la navigation et transmettre des informations pour alerter ou conseiller les passagers. Parmi ces trois grandes catégories de systèmes conjoints, nous avons retenu la coopération, en utilisant des signaux du monde des communications afin de réaliser les deux missions simultanément. Ce choix découle du fait qu'on souhaite transmettre des informations avec un débit efficace, sans optimiser la performance. D'un point de vue de radar, l'optimisation du traitement est par contre prise en compte.

Choix des codes à modulation de phase continue pour le système conjoint

Pour créer un système SAR-communication coopératif, il faut d'abord définir la forme d'onde de communication à utiliser.

Dans le chapitre 4, nous présentons le choix des formes d'ondes pour notre système, qui sont les codes à modulation de phase continue (CPM). Une étude des différents paramètres de ce type de modulation est fournie, car le comportement du signal peut changer radicalement en termes de phase, de spectre, de symboles d'intégration, etc. En restreignant certains de leurs paramètres, nous décidons d'utiliser un cas particulier de CPM appelé CPFSK. Un exemple de trajectoire de phase ainsi que la densité spectrale de puissance de la forme d'onde CPFSK sont représentées en figure 5. La raison principale du choix de cette sous-catégorie réside dans la possibilité d'exprimer théoriquement la densité spectrale et donc de fournir une expression explicite de leur largeur de bande. Une fois le code souhaité bien établi, on justifiera le choix de cette forme d'onde en la comparant avec différentes formes d'ondes de communication, choisies parmi les plus citées dans la littérature : • de propriétés spectrales.

En termes de propriétés spectrales, on souhaite que l'énergie hors bande soit la plus petite possible, puisque l'on cherche avant tout à répondre à une problématique d'encombrement spectral. Par ailleurs, les signaux pourront tout de même présenter une certaine énergie hors bande. On étudie également l'impact d'un filtrage par un gabarit fréquentiel sur l'autocorrélation des signaux. Sur ces critères spectraux, les CPFSK se montrent avantageux par rapport aux candidats restants, ce qui peut être aussi observé sur la figure 6 qui illustre les sprectres des différents signaux. On peut remarquer que celui des CPFSK a des lobes secondaires spectraux plus bas que les autres.

Cette étude tend à conclure que, parmi les signaux étidiés, la forme d'onde la plus appropriée, selon les critères indiqués est la modulation CPFSK.

Minimisation des lobes secondaires par filtre désadapté

Une fois la forme d'onde choisie, il s'agit de définir le traitement associé. Généralement, le traitement radar consiste en une compression d'impulsion avec filtre adapté pour déterminer la distance de la cible. Cependant, l'application de ce filtre aux signaux de communication conduit généralement à des niveaux de lobes secondaires inacceptables, comme observé en figure 8. Sur une image SAR, ce niveau de lobes secondaires mènerait à un contraste trop faible pour que l'image soit correctement interprétée. Une technique de compression différente est donc utilisée, appelée filtre désadapté, qui minimise l'énergie des lobes secondaires, au lieu de maximiser le SNR. Le résultat de la compression est aussi affichée sur la figure 8. Ce filtre est généralement défini comme un problème de minimisation convexe [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF], qui peut être résolu par un solveur convexe tel que CVX dans Matlab.

Deux extensions de ce problème sont proposées dans ce travail et ont donné lieu à deux publications dans des conférences internationales. La première extension considère la minimisation du niveau des lobes secondaires intégré sur un intervalle Doppler continu, afin de prendre en considération des décalages Doppler intra-pulse inconnus [START_REF] Chatzitheodoridi | A mismatched filter for integrated sidelobe level minimization over a continuous Doppler shift interval[END_REF], et la seconde traite des décalages de retard hors grille (cf. figure 7) en minimisant l'énergie des lobes secondaires sur la cellule de résolution dans le cas spécifique des CPFSK [START_REF] Chatzitheodoridi | A sidelobe level minimization mismatched filter using continuous phase frequency-shift keying codes for the off-grid delay problem[END_REF]. Les solutions à ces deux problèmes peuvent être utilisées pour limiter la remontée de l'énergie des lobes secondaires si les signaux reçus ne coïncident pas avec la grille considérée. La figure 9 illustre les résultats obtenus en appliquant les filtres calculés avec nos deux problèmes d'optimisation sur un code CPFSK. La première sous-figure correspond à l'optimisation sur un intervalle Doppler et la deuxième sous-figure correspond à l'optimisation du filtre sur la cellule de resolution qui correspond à un échantillon de temps. Globalement, le filtre qu'on a proposé donne des niveaux de lobes secondaires plus faibles en moyenne et principalement en dehors du voisinage du point de la grille. Pour le point "0", le filtre optimal est bien entendu celui qui minimise le niveau des lobes secondaires pour cette valeur.

Résumé en

Imagerie SAR et ISAR avec signaux CPFSK

Les signaux CPFSK étudiés précédemment ont été, dans un premier temps, utilisés sur des données re-synthétisées pour créer des images SAR et, dans un deuxième temps, sur des données réelles pour produire des images dans un cadre Inverse Synthetic Aperture Radar (ISAR). [Chatzitheodoridi et al., 2022a] et dans un journal [START_REF] Chatzitheodoridi | A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters[END_REF].

Images re-synthétisées

Résumé en Français xxxiii

Images obtenues sur des données réelles

Pour aller encore plus loin dans l'évaluation des performances de ces codes dans le domaine du radar, le chapitre 7 présente des résultats obtenus sur des données réelles à partir de formes d'ondes transmises et enregistrées par un radar au sol vers un répondeur et un avion en vol. Le radar considéré est le radar HYCAM de l'ONERA, qui fonctionne principalement en bande S [START_REF] Brouard | Hycam: A new S band surface radar testbed[END_REF].

Dans un premier temps, les codes sont transmis vers un répondeur. Comme le répondeur est stationnaire, les signaux rétrodiffusés qui reviennent à HYCAM ne devraient pas subir de décalage Doppler. Cependant les échos reçus contiennent des réponses de fouillis, ce qui masquerait une cible à Doppler nul. Pour faciliter la séparation de sa réponse avec le fouilli de sol, le répondeur crée un décalage Doppler artificiel sur les données avant de rétrodiffuser les signaux vers l'antenne d'HYCAM.

Les cartes distance-Doppler sont présentées sur la figure 12, générées soit dans le cas du chirp comprimé par un filtre adapté, soit pour la modulation CPFSK par des filtres adaptés et désadaptés. • Revue internationale avec comité de lecture:

Chatzitheodoridi, M.-E., [START_REF] Chatzitheodoridi | A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters[END_REF]. A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters. IEEE Transactions on Geoscience and Remote Sensing, 61:1-14

• Conférence internationale avec comité de lecture:

Chatzitheodoridi, M.-E., Taylor, A., Oriot, H., and Rabaste, O. (2022a)

Conclusion et Perspectives

Conclusion

Le travail de cette thèse nous permet d'apporter une réponse à notre question initiale : Pouvons-nous créer un système coopératif de SAR-communication en utilisant des signaux de communication et offrant une performance radar acceptable ? Nous pouvons maintenant envisager une réponse positive ! Notre travail répresente une petite contribution au problème de la congestion du spectre électromagnétique. En effet, nous avons créé un système coopératif qui utilise des signaux de communication inhabituels pour les applications radar afin d'assurer la transmission d'informations, et nous nous concentrons donc sur l'optimisation du traitement radar. Après une comparaison approfondie, la modulation CPFSK a été choisie pour son enveloppe constante et ses propriétés spectrales.

La compression par filtrage adapté avec de tels signaux génère des lobes secondaires élevés qui peuvent dégrader le contraste de l'image radar. Pour cette raison, nous avons proposé de remplacer le filtre adapté par le filtre désadapté car il minimise l'énergie des lobes secondaires et peut être calculé à partir d'un problème d'optimisation convexe. Nous avons également défini des problèmes d'optimisation qui peuvent traiter les décalages Doppler intra-impulsionnels ou les problèmes de retard hors grille. Le choix d'utiliser un tel filtre dépend de l'application.

Cependant, en considérant un système de SAR-communication, une caractéristique intéressante pour les communications est la maximisation du débit de données. Dans notre cadre, cela peut être réalisé en transmettant une impulsion non identique à chaque acquisition, ce qui implique qu'un filtre doit être calculé pour chaque signal. Ceci entraîne un temps de calcul irréalisable avec les algorithmes convexes génériques disponibles tels que CVX, ce qui nous amène à proposer un algorithme rapide qui peut calculer tous les filtres nécessaires dans un temps raisonnable.

De plus, afin de fournir des résultats réalistes, nous avons généré des images SAR re-synthétisées, qui ont été comparées au cas classique de chirp par divers critères d'évaluation. Nous avons conclu que la qualité d'image obtenue avec des signaux CPFSK compressés avec un filtrage désadapté n'est pas très éloignée de celle d'une image générée avec un chirp, même si cette dernière est légèrement meilleure. Cependant, il ne faut pas oublier que l'un des objectifs est de transmettrede l'information: cette légère dégradation est le prix à payer pour accomplir les deux tâches.

Enfin, étant dans l'impossibilité de fournir de véritables acquisitions aéroportées avec CPFSK, nous avons utilisé un radar terrestre pour valider notre système de la manière la plus réaliste possible. Tout d'abord, des cartes distance-Doppler ont été créées en illuminant un répondeur, qui nous permet de visualiser le comportement des signaux de communication sur un point brillant. Dans un deuxième temps, nous avons créé des images ISAR en illuminant une plateforme aéroportée Résumé en Français xxxvii à partir du même radar terrestre. Ces résultats re-synthétisés et réels concluent le travail de ccette thèse. Ils permettent de démontrer la faisabilité de la stratégie proposées en conditions réelles.

Perspectives

Les potentielles perspectives de cette thèse sont les suivantes :

• Le choix du signal pourrait être différent si l'on modifie les critères de sélection considérés.

Par exemple, si la contrainte d'amplitude constante était plus flexible, les codes OFDM pourraient être utilisés pour le système comme dans [START_REF] Liu | Design of integrated radar and communication system based on MIMO-OFDM waveform[END_REF], Dokhanchi et al., 2018], mais dans un cadre SAR-communication, qui est très peu examiné [START_REF] Baudais | MIMO-OFDM signal optimization for SAR imaging radar[END_REF].

• Si les codes CPM sont toujours choisis, l'optimisation de la forme d'onde par rapport au choix du symbole pourrait être envisagée par exemple pour de meilleures propriétés d'autocorrélation, voire d'intercorrélation si plusieurs formes d'onde CPFSK différentes sont considérées afin de permettre la transmission d'information.

• Plusieurs aspects de communication peuvent être approfondis, comme le Bit Error Rate (BER), ou la synchronisation.

• Du point de vue de la compression, le filtre désadapté qui minimise l'énergie intégrée des lobes secondaires a été choisi en raison du cadre SAR, mais d'autres options pourraient être envisagées comme par example l'utilisation du PSLR pour minimiser le plus fort lobe secondaire. On pourrait même ajouter des contraintes qui portent sur le rapport signal à bruit afin de le maximiser également.

• Une autre perspective pourrait être de combiner les deux problèmes d'optimisation que nous avons proposé (avec les décalages Doppler intra-impulses et les décalages de rétard hors-grille) en un seul problème commun en optimisant les deux cas simultanément.

CHAPTER 1 Introduction

. General Framework and Objectives

From the beginning of the second millennium until now, electromagnetic applications, and especially communication missions, have grown at a very high rate.

The first motive comes from the number of devices utilized over the years. In 2015, it was stated that the electronics market had an installed base of 15.4 billion devices and it was predicted that this number would reach 75.4 billion devices by 2025 [Lucero et al., 2016]. Secondly, the need of transferring more information per unit of time has been preponderant over the years. For example, the third generation of mobile systems made its appearance in the early '00s and just 20 years later the 5G network deployment has already started [START_REF] Patil | 5G wireless technology[END_REF].

With all these new technologies being available in the market, more bandwidth is occupied, and thus the radio frequency spectrum dedicated to telecommunications is becoming more valuable. This impact can already be remarked on the market of different countries. UK communication operators payed 119.3 million pounds for 1800MHz of bandwidth in 2015, Germany charges more than 5 billion euros for 4 frequency bands for communications and the US sold 28GHz of spectral allocation for 5G applications for 702 million dollars [START_REF] Liu | Joint radar and communication design: Applications, state-of-the-art, and the road ahead[END_REF]. This raise in prices comes from the fact that the electromagnetic spectrum is a limited resource, and it is also used for applications other than communications. Consequently, these circumstances lead to the congestion of the electromagnetic spectrum.

In particular, radar systems try to keep their place in the spectrum, and maintain or increase if needed the bandwidth allocated for their applications. Thus, tackling the electromagnetic spectral congestion is an important task, and researchers put effort into improving the spectral efficiency in both radar and communication domains. Originally, radar and communication systems, that fulfill different objectives, have been assigned to operate in separate bands. However, an efficient solution to this problem that is proposed in the literature is to combine the missions of a radar and a communication system [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF]]. This new system can operate in favor of the radar, the communications or both missions. More specifically, one system may tolerate the other one by treating it as an interference, or optimize the parameters of one system and maintain the characteristics of the other one or even optimize the performance of both applications simultaneously.

The general term "radar-communication system" can be adopted for such systems, without restraining to a specific radar application. In this work, a Synthetic Aperture Radar (SAR) framework is considered. In a SAR framework, the radar is mounted on an airborne platform, it transmits the signals towards a part of the ground illuminated by the antenna mainlobe, and the received signals are processed in order to create an image that represents the ground reflectivity. The particularity of SAR processing is the creation of a virtual antenna, by exploiting the plane motion during the pulse train transmission and reception. The advantage over other radar systems is a better azimuth resolution onto the output image, as the larger the antenna length, the better the resolution. The antenna can only be enlarged virtually as mechanical constraints prevent from carrying an antenna exceeding certain dimensions. In a classical SAR-only framework, mainly chirp pulses are used, for their good characteristics [START_REF] Levanon | Radar signals[END_REF]. But despite all of its radar qualities, a chirp signal does not carry any information. Few literature exists on the creation of SAR-communication systems, which has as an objective the generation of a high-quality radar image and the simultaneous transmission of information.

Considering this context, the creation of a SAR-communication system that simultaneously generates a good contrast radar image and transmits information is the main challenge of this thesis.

. Contributions

Limiting the electromagnetic spectrum, requirement of the proposed system, contributes to answering the spectral congestion problem. In the literature, researchers provide three main categories to alleviate the occupancy of the spectrum : co-existence, cooperation, and co-design. We chose to create a cooperative SAR-communication system that utilizes signals from the communication world in order to perform both missions simultaneously in the same frequency bandwidth.

The creation of such a system consists firstly in choosing the suitable communication waveform in order to ensure the information transmission towards one or more receivers. To do so, different well-known communication waveforms are compared to each other in order to find a suitable candidate. More precisely, we compare, in this work the Continuous Phase Frequency-Shift Keying (CPFSK) modulation, the Binary Phase Shift Keying (BPSK) modulation, the Linear Frequency Modulation with Phase Shift Keying Symbols (LFM-PSK), and the Orthogonal Frequency Division Multiplexing (OFDM). The selection criteria, induced by radar imposed constraints, are a constant amplitude and spectral containment through the smallest possible out-of-band energy and the smallest deformation of the waveform after spectral filtering. In the study presented in this work, CPFSK signals tend to be the more suitable for our application. The arising problem while using these signals is the high sidelobe energy of the pulse compression output. Indeed, in radar applications, the matched filter is usually used for compression, as it maximizes the Signal-to-Noise Ratio (SNR), but with communication waveforms, it mainly leads to unacceptable sidelobe levels.

Contributions 3

A different compression technique is thus used in our framework, called mismatched filter. It can minimize the sidelobe energy rather than maximize the Signal-to-Noise Ratio (SNR). Two extensions of this problem are proposed in this work and they led to two contributions in international conferences. The first one tackles the minimization of the integrated sidelobe level over a continuous Doppler interval, in order to take into consideration unknown intra-pulse Doppler shifts [START_REF] Chatzitheodoridi | A mismatched filter for integrated sidelobe level minimization over a continuous Doppler shift interval[END_REF], and the second one handles off-grid delay shifts by minimizing the sidelobe level energy over the resolution cell in the specific case of CPFSK [START_REF] Chatzitheodoridi | A sidelobe level minimization mismatched filter using continuous phase frequency-shift keying codes for the off-grid delay problem[END_REF]. The solution to these two problems can be used to allow lower sidelobe energy if the received signals do not coincide with the transmitted ones.

Returning to our SAR-communication framework, the waveforms transmitted from the aircraft antenna should differ from one another, in order to maximize the data rate. This implies that the mismatched filter needs to be computed for every different transmitted signal. We realized that it is impossible to compute all the filters in a compliant delay, using the convex toolbox at our disposal. It led us to propose an algorithm that reduces dramatically the computational time, by exploiting the dual optimization problem. We showed that this proposed algorithm can compute the filters 250 times faster than generic Matlab algorithms when considering parameters from the SAR framework. This work was published in an international conference [START_REF] Chatzitheodoridi | Fast computation of the optimal integrated sidelobe level filter for large-scaled problems[END_REF].

Once it was feasible to compute all the needed mismatched filters, it became possible to compare simulated SAR impulse responses using CPFSK with the regular chirp case compressed with the matched filter. Overall, the chirp provided slightly better radar performance compared to the CPFSK modulation, but using the latter enables to transmit information, while keeping the overall SAR performance acceptable. The next step was to verify that the generation of a realistic SAR image where multiple scatterers are present was feasible. Faced with the difficulty of conducting airborne acquisitions by transmitting CPFSK codes, re-synthesized SAR images were created that took into consideration raw data acquired with chirp pulses. Using a reconstruction algorithm, the integrated ground reflectivity was estimated and utilized in the processing to create the resynthesized images using the impulse responses of the CPFSK compressed with MF or Mismatched Filter (MMF). We observed that the combination of CPFSK modulation and mismatched filtering provided a good image contrast, and at the same time transmitted information. These results constitute contributions in an international conference [Chatzitheodoridi et al., 2022a] (Best student paper award) and in a journal article [START_REF] Chatzitheodoridi | A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters[END_REF].

To go even further in our evaluation of the performance of such codes in the radar field, real CPFSK data were transmitted from a ground-based radar towards both a transceiver and a flying aircraft. We were able to create a range-Doppler map from the transceiver in order to compare it with the chirp through classical radar criteria (ISLR, PSLR). Last but not least, it was also possible to create ISAR images of the aircraft. The main observed differences between the chirp and the CPFSK were that the chirp had higher sidelobes close to the mainlobes and lower away from them, keeping thus a low noise-clutter floor, whereas the CPFSK had lower sidelobes around the mainlobes that remained uniformly distributed through all the space, increasing thus the noiseclutter floor. Still the contrast of the CPFSK image was acceptable and CPFSK modulation allows to transmit information.

. Outline

This thesis work is divided in 3 parts and 8 chapters. Chapter 1 corresponds to the introduction that gives an overview of the context, the contributions and the publications of this work. Part I is composed of chapters 2 and 3, and provides the literature review of the thesis context.

• Chapter 2 provides the radar and SAR fundamentals. More precisely, the definition and the applications of radar systems, the definition of a chirp pulse, radar and SAR processing, and usual radar evaluation criteria are provided.

• Chapter 3 gives an overview of the radar-communication literature. Firstly, an introduction on communication systems and electromagnetic spectrum are provided, and then different strategies of radar-communication systems are explained.

Part II contains chapters 4 and 5, and prepares for the creation of the SAR-communication system as it provides the choice of the waveform and the processing.

• The aim of chapter 4 is to justify the choice of the CPFSK modulation as a good candidate for our joint system. It first provides an extensive literature review on these signals, along with the characteristics of some other communication codes largely utilized in communication or radar-communication systems. Afterwards, all considered waveforms are compared, and the CPFSK modulation is selected for the rest of this work.

• Chapter 5 is dedicated to the mismatched filter optimization. Firstly, we provide the classical definition via a convex optimization problem, and subsequently we study two extensions to improve the performance when it comes to unknown intra-pulse Doppler shifts or off-grid delay shifts. Finally, the last part of the chapter provides our contribution for reducing the algorithm computational cost that allows to compute all the necessary filters in a reasonable amount of time.

Part III contains chapters 6 and 7, and provides the simulations and experimental results obtained with CPFSK signals and their associated mismatched filters.

• Chapter 6 explains how re-synthesized SAR images can be created from the impulse response of the communication codes. Once they are generated, they are compared to the chirp case using different radar and image evaluation criteria.

• Chapter 7 provides results obtained on real acquisitions from an ONERA measurement system on a ground transceiver and a flying aircraft.

Finally, chapter 8 concludes the work of this thesis and provides some perspectives for potential future work.

. Publications

The articles published during this work are summarized in the following :

• Journal Article:

Chatzitheodoridi, M.-E., [START_REF] Chatzitheodoridi | A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters[END_REF]. A cooperative SAR-communication system using continuous phase modulation codes and mismatched filters. IEEE Transactions on Geoscience and Remote Sensing, 61:1-14 This chapter introduces the fundamentals in radar signal processing, along with the Synthetic Aperture Radar framework. Different applications are presented, a synopsis of a radar system is provided, as well as an insight on the radar signal with its characteristics and associated processing. Afterwards, more details are given about airborne platform specifications from a processing point of view, and some criteria to evaluate the image quality.

. Radar Basics

A radar, abbreviation for RAdio Detection And Ranging, is a system that transmits electromagnetic waves and is usually used for detection and localization of targets by processing the back-scattered signal received by an antenna. The motivation of constructing radars is their ability to "see" objects that cannot be seen by human vision or other sensors under special circumstances. Indeed, during adverse weather conditions, in the middle of the night or at long distance, radars can still determine the presence of possible targets or threats. They were first used in the 30's as detection devices in order to warn from threatening aerial territorial intrusions. The first civilian radar application was the frequency-modulated aircraft radio altimeter, that had the ability to measure the height of an airborne platform from the ground, but it was not considered as a radar at that time as the radar term was only introduced during World War II [Skolnik, 1980]. Nowadays, radars can be ground-based or mounted on planes, satellites, ships, cars, drones, and they are used in multiple domains, for military and civilian purposes, as listed below [Skolnik, 1980, Melvin and[START_REF] Melvin | [END_REF]]:

• Ground-based radars. These radars are mainly used for detecting and localizing airborne targets (planes or satellites), but also detecting missiles in a battle field, or vehicles. They can be used for civilian applications like air traffic control for safe landing in adverse weather or law enforcement. In addition, millimeter-wave radars are a key technology nowadays to enable intelligence and autonomy in vehicles. Radars can also be used for military satellites, surveillance and navigation. More sophisticated systems have been developed over the years like phased-arrays used for air and missile defense.

• Shipboard. Radars can be mounted on ships for safety reasons in order to prevent them from colliding with other ships or up-front obstacles. They can be used either for military or civilian applications.

• Airborne and spaceborne radars. SAR systems are mainly used to create all-day and night radar images of the ground. Using aircraft radars, navigation through bad weather can also be simplified, as well as measuring the height of the aircraft (radar altimeter). Moreover, military airborne applications exist in order to detect potential aircraft targets (air to air applications).

In the following, we will first describe a classical radar system. Then, we will detail the important characteristics of the signals used, and more precisely the classical Linear Frequency Modulation. Finally, we will introduce how these signals are usually processed in order to perform detection, and estimate the distance and the velocity of the targets.

. Radar Systems

Radars are complex systems that are composed of different elements such as antennas, transmitters, receivers, etc. Figure 2.1 illustrates a simplified version of the transmission and reception of a radar system [Skolnik, 1980]. During transmission, the waveform generator modulates the desired signals, for example via amplitude modulation or linear or stepped frequency modulation. Afterwards, the signals are converted from analog-to-digital and shifted to higher frequencies, in order to be transmitted by the transmitter. At the transmitter, the energy of the radar pulse is generated, and thus the signals have a given power and a given central frequency to be transmitted by the antenna. Radars can be composed of one antenna or an antenna array that is in charge of the signal transmission and reception. Such an antenna is called transceiver, and the corresponding system is said to be monostatic. If the transmitter and the receiver antennas were different and located at different locations then the system would be called bistatic radar system. In the simple example introduced in figure 2.1, one antenna is considered, which performs both signal transmission and reception. The duplexer is the bridge between these two functionalities, as it prevents the receiver from burning out from the high power of the transmitter. In bistatic configurations, the duplexer is not necessary. A clock connects the transmitter and the receiver to retrieve and display information about potential targets provide a reference for the time variable. After the emission, the back-scattered signals reach back to the receiving antenna and follow the receiving line of the system. The energy of these signals is attenuated with respect to the distance and thus the receiver performs amplifications, along with filtering. Then, the mixer shifts the signal frequency from the radio frequency band to an intermediate frequency, which limits the required sampling frequency needed to correctly digitalize the signal. Afterwards, an analog-to-digital converter is used to create the received digital signals. Finally, the signals are further processed to retrieve and display information about potential targets.

After transmission, the signal pulses propagate at the speed of light. When they reach an obstacle, they are reflected in different directions and proportions, according to the shape of the object and its electromagnetic properties. The echoes that return to the receiving antenna can be used to determine the position and the velocity of the target [Wang, 2008], as we will see in the following. Figure 2.2 shows such a configuration, where a ground-based radar can detect an airborne target. This distance evaluation can be achieved by measuring the time needed by the signal to travel along the two-way path between the radar and the target. The time associated to the two-way path is called delay [START_REF] Levanon | Radar signals[END_REF] and it is denoted by τ. The distance R between the source and the target, also called target range, is linked to the two-way path delay by

R = c τ 2 , (2.1)
where c ⋍ 3 • 10 8 m/s is the speed of light. Equation (2.1) is an approximation, as the velocity c can change according to the environment, but in radar applications these changes can be considered negligible [START_REF] Levanon | Radar signals[END_REF].

The need to evaluate the velocity of a target, to detect at far range, and to potentially discriminate echoes from different targets leads to consider a repetitive transmission procedure for a pulsed radar system. This succession of pulses is usually called a pulse train. More precisely, once a pulse is transmitted, the antenna switches to reception mode for a given duration in order to collect the back-scattered signals and then another pulse is transmitted again. Each pulse has a given duration known as Pulse Duration, denoted by T p . It is transmitted periodically with a time interval known as Pulse Repetition Interval (PRI), and denoted as T r . These two notions are illustrated in figure 2.3. During the transmission of the pulse, the receiver is shut down so as not to be saturated by the transmitted signal. This creates a blind range during which the radar is not able to 'see' anything. Choosing a small T p creates a short blind range. However, it should not be too small because the transmitted energy will decrease accordingly and result in the disability to perform the target detection. The same remark can be made for PRI, or similarly Pulse Repetition Frequency (PRF), which is defined as the inverse of the PRI. So, if the PRF is too large, the echo of a pulse n arrives after or in between the transmission of the pulse n + 1, as shown in illustration 2.3. Such a configuration results in having ambiguous ranges, which are created by observing targets that appear much closer that they actually are. Thus, error in localization occurs. In order to prevent such ambiguities, the PRF should be chosen (when possible) in such way that the desired range values are less than the range ambiguity, expressed as follows

R ambig = c 2PRF . (2.2)

. Radar Signal

The signal transmitted by the antenna can be decomposed into the baseband signal and the carrier frequency. The baseband signal is centered around the zero frequency. It is of bandwidth B and pulse duration T p , i.e. the frequencies of the signal spectrum are mostly between ± B 2 . Afterwards it is shifted in frequency at carrier frequency f 0 , which is the carrier frequency used to perform the transmission in the desired electromagnetic band. Usually f 0 is much larger than B.

The transmitted signal can be expressed as

s(t) = s pulse (t) baseband signal e j2π f 0 t carrier signal , (2.3)
where t is the time variable initialized at the start of the first transmitted pulse. Once s is backscattered by a target and returned to the antenna, the received signal r corresponds to a noisy, attenuated and delayed version of the original signal s, that can be expressed as

r(t) = a s pulse (t -τ(t))
baseband effect e j2π f 0 (t-τ(t))

carrier effect

+n(t), (2.4)
where n is an additive white Gaussian noise, τ(t) is the target delay, and a is the target complex amplitude. This delay τ depends on time as it can evolve with the relative motion between the radar and the target, expressed with a first order approximation as follows:

τ(t) = τ 0 + 2v r c t, (2.5)
for a uniformly linear motion, with τ 0 the initial delay and v r the radial velocity. This delay change can be assimilated to a contraction/dilatation of the signal back-scattered by the target that affects both the baseband part of the signal and the carrier frequency. When the source, the target or both are moving, it thus induces a Doppler frequency shift f d . Indeed, once the delay formula (2.5) is injected in equation (2.4), the received signal is written as follows:

r(t) = a s pulse t - 2v r c t -τ 0 e j2π f 0 t- 2v r c t-τ 0 + n(t) = a s pulse 1 - 2v r c t -τ 0 e j2π f 0 t e -j2π f 0 τ 0 e -j2π 2 v r c f 0 t + n(t).
(2.6)

The Doppler shift f d corresponds to the last exponential term of equation (2.6) and it is expressed as

f d = 2v r c f 0 . (2.7)
Generally in radar frameworks, the impact of the delay on the baseband signal is neglected because the relative velocity v r is assumed much smaller than the speed of light. Thus, the factor 1 -2v r c causes a time contraction/dilatation in the time domain, but as the ratio v r c << 1, it is considered negligible on the duration of the pulse in many radar applications. However the effect on the carrier which results to the Doppler shift cannot be ignored due to the generally large value of the carrier frequency that leads to fast phase rotation.

Finally the received signal is expressed as

r(t) = a s pulse (t -τ 0 ) e j2π f 0 (t-τ 0 ) e j2π f d t + n(t).
(2.8)

. Linear Frequency Modulated Pulse

In radar applications, the most widely used waveform is the linear frequency modulated pulse, also named chirp. As indicated by its name, the chirp belongs to the frequency modulation category, which is characterized by a variation of the signal carrier frequency [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. Its main properties are the linearity of its instant frequency with respect to time, a constant amplitude and a uniformly filled bandwidth, as shown in figure 2.4. A chirp signal is defined as

Chapter 2. Radar Basics and Synthetic Aperture Radar Fundamentals

s(t) = Rect t T p exp j 2π f 0 t + πkt 2 , ( 2.9)
where k = B T p is the frequency rate with respect to time and

Rect(t) =      1 for - 1 2 ≤ t ≤ 1 2 0 otherwise .
Using equation (2.9), the chirp phase can be expressed as a quadratic function of time,

ϕ (t) = 2π f 0 t + πkt 2 .
By considering the derivative of the phase with respect to time, the chirp instant frequency can be deduced

f (t) = 1 2π ∂ ϕ (t) ∂t = kt + f 0 ,
which indeed demonstrates the linearity of the frequency modulation. We present in figure 2.4 the frequency, phase and spectrum of a chirp signal with parameters presented in table 2.1. It can be observed, in the third sub-figure of figure 2.4, that the chirp spectrum is well contained in its bandwidth, which is one of the reasons for which it is widely used in radar.

Parameter

Value Bandwidth B 500 MHz Pulse Duration T p 1 µs Sampling Frequency f s 4B

Table 2.1: Chirp parameters for the generation of Figure 2.4.

. Matched Filter and Range Resolution

The signal back-scattered to the antenna, expressed in equation (2.8), should be analyzed in order to provide valuable information about the target. A widely used pulse compression technique in radar applications is the matched filter, that is applied to the received signal, in order to determine the target position. The matched filter is known for maximizing the SNR at the target delay [Wang, 2008, Levanon andMozeson, 2004]. More precisely, it corresponds to the maximization of the ratio between the signal energy and the noise energy. It is expressed as

arg max τ R s(t -τ 0 )h(τ -t)dt 2 E R n(t)h(τ -t)dt 2
where h is the matched filter, and E[•] the expectation of the associated variable.

For a white noise, the solution to this optimization problem [START_REF] Levanon | Radar signals[END_REF]] is provided by h(t) = γs * (-t), (2.10)

or in the Fourier space by

H( f ) = γS * ( f ), (2.11)
where γ is a constant, usually equal to 1 as, in most radar applications, a 'constant gain' constraint is added. Finally, the output compressed signal, denoted by y, can be written as the convolution of the received signal with this optimal filter in order to consider all the possible delays. In the case where the target shows no Doppler shift, it is expressed as,

y(τ) = (r ⋆ h)(τ) = a +∞ -∞ s(t -τ 0 )h(τ -t)dt + +∞ -∞ n(t)h(τ -t)dt = a +∞ -∞ s(t -τ 0 )s * (t -τ)dt + +∞ -∞ n(t)s * (t -τ)dt.
(2.12)

By testing all the possible delays, this pulse compression technique can be used to determine the delay of the target since y shall reach its maximum value for that delay. It can be seen that the matched filter naturally leads to providing the auto-correlation of the transmitted signal.

Before any processing, when we observe the raw data, it is difficult to distinguish targets that were captured in the same pulse. Indeed, the resolution is defined as the minimum distance for which the radar has the ability to distinguish two different targets, and it is dependent on the Chapter 2. Radar Basics and Synthetic Aperture Radar Fundamentals pulse duration in the range axis without range matched filtering. It can be deduced for the raw data from the equation (2.5), for τ = T p [Taylor, 2016],

ρ r = cT p 2 .
(2.13)

For instance, for T p = 1µs, ρ r = 1500m, so it is impossible to distinguish targets closer than 1500m.

The matched filter allows to improve the resolution in comparison to equation 2.13. In radar, the matched filter is also referred to as range compression.

The output of the compression for a chirp is illustrated in subfigure 2.5a, where the improvement in range resolution can be easily remarked over the pulse duration interval. More precisely, for the chirp pulse and the matched filter, the time resolution in the range direction is approximately equal to the inverse of the bandwidth, which leads to the following expression for the range resolution after compression [START_REF] Carrara | Spotlight synthetic aperture radar : signal processing algorithms[END_REF]:

ρ r = cK r 2B , (2.14)
where K r is a bandwidth factor that takes into consideration mainlobe broadening from weighting windows or phase errors. However, in radar usually the theoretical resolution is provided when K r = 1 for simplification. Hence, as long as two targets are at least separated by ρ r meters, they can be distinguished after range compression. Figure 2.7 provides an example where two targets are present, but we can distinguish them only if they are separated by ρ r .

Sub-figure 2.5b is a zoom in the middle of sub-figure 2.5a, that shows an illustration of what are called the mainlobe and the sidelobes. The mainlobe is defined as the points that belong to the larger peak of the compression output and whose energy maximum indicates the position of a target, whereas the sidelodes are all the remaining lobes of the compression around the mainlobe, that correspond to the compression output for other delay values. Intuitively, the range resolution depends on the width of the mainlobe after compression. The thinner the mainlobe, the easier two targets can be discriminated. From subfigure 2.5a, it can be observed that the sidelobes of the chirp decrease as they get further from the mainlobe. From a detection point of view, having low sidelobes away from the target helps not to mask further away targets, which is advantageous for radar. However, it should be noted that high sidelodes close to the mainlobe could cause a high-energy target to hide a low-energy one, as shown in figure 2.6. The low-energy target is hidden in the sidelobes of the strong target, so that we cannot detect it.

. Ambiguity Function

So far, only the response of the matched filter for a target without velocity has been presented. However, as mentioned in subsection 2.1.2, the received signal is affected by the radial velocity between the radar and the target, which results in a Doppler shift. This means that the matched filter, expressed in equation (2.10), is no longer optimal, as it is expressed at zero-Doppler. For this reason, the ambiguity function can be introduced, which expresses the output of the matched filter for a signal with a Doppler shift f d and delay τ [START_REF] Levanon | Radar signals[END_REF]. Considering a noiseless signal and using the matched filter expression in (2.12), the ambiguity function is expressed as

χ(τ, f d ) = +∞ -∞ s(t)s * (t -τ)e 2 jπ f d t dt.
(2.15)

Ambiguity functions are widely used to characterize which signals are suitable for different radar applications [Mahafza, 2000]. Clearly, χ(τ, 0) corresponds to the definition of the matched filter. For a given τ, the sign of the Doppler frequency is meaningful for physical interpretation. With the proposed convention, when it is positive, the source and the target are getting closer, Chapter 2. Radar Basics and Synthetic Aperture Radar Fundamentals where two targets cannot be separated because their relative delay is lower than the range resolution. On the contrary, in the the second sub-figure, the two targets are separated by a distance higher than the resolution and thus they can be discriminated.

otherwise they are drawing away. χ(0, 0) provides the energy of the signal s, and corresponds to the maximum of the ambiguity function. Elsewhere, the energy is reduced, and a fundamental property of the ambiguity function shows that [START_REF] Levanon | Radar signals[END_REF]]

|χ(τ, f d )| ≤ |χ(0, 0)|, ∀τ, f d .
The behavior of the ambiguity function depends on the choice of the signal. For example, the ambiguity functions along with the zero-Doppler and the zero delay cuts of a chirp and a phase code like a Gold code, illustrated in figures 2.8 and 2.9 respectively, exhibit very different behaviors. For the chirp, due to the linearity of the ambiguity function, the target detection can be achieved for all the considered Doppler and delay, but it can induce a shift in the time or Doppler domain. This property might be referred to as the chirp Doppler robustness, or delay-Doppler coupling. However, for Gold codes, for large delay and Doppler values, it is not possible to detect a target with a single range matched filter at f d = 0.

. Coherent Pulse Train and Doppler Resolution

As mentioned earlier, the radar antenna periodically transmits repeated pulses of a given duration T p and with a PRI T r . Using such a pulse train enables to increase the Doppler resolution, but depending on the chosen parameters, it can lead to disadvantageous ambiguities. Thus a coherent train of N pulses is transmitted and defined as a repetition of the transmitted pulse expressed in equation:

(2.3), u N (t) = 1 √ N N-1 ∑ n=0 s(t -nT r ) (2.16)
Such a pulse train is illustrated in figure 2.10. Usually in radar application, the same chirp is transmitted at each acquisition. In this case, the received signal for a given target at distance τ c (2.17)

In order to extract the required information, such as position and velocity of targets, a range-Doppler processing is applied. The order of the processing is inter-changeable, but the norm is to start with the range compression. As mentioned, the matched filter is usually chosen so that the signal after range compression can be expressed as

s c (τ) = T p 0 r(t)s * (t -τ)dt = a N-1 ∑ n=0 T p 0 s(t -τ c -nT r )s * (t -τ)e j2π f d t dt + T p 0 n(t)s * (t -τ)dt
Afterwards, the train of pulses is cut in intervals [kT r , (k + 1)T r ]. Assuming that the phase rotation due to the Doppler shift is negligible for the pulse duration T p , an approximation on the Doppler term can be performed as e j2π f d t = e j2π f d kT r on each interval. By applying this approximation in the previous expression, we have

s c (τ) = a N-1 ∑ n=0 e j2π f d nT r T p 0 s(t -τ c -nT r )s * (t -τ)dt + n ′ (τ) (2.18)
where n ′ (τ) = Here B = 352MHz and T p = 1.5µs.

After performing the range compression for each pulse acquisition, Doppler processing is applied, which can be easily performed by a Fast Fourier Transform (FFT), when observing equation (2.18), since it will consist in compensating the phases e j2pi f d kT r .

The Doppler resolution is directly dependent on the integration time. It is expressed as

ρ a = 1 NT r . (2.19)
It is important to note here that such a coherent pulse train can be ambiguous if T r is not chosen wisely, as shown in figure 2.3. This effect is cause by the lack of knowledge on which repetition interval a detected target belongs to. Moreover, the PRF should also be chosen larger than the maximum Doppler frequency that can be applied on the signal in order to avoid ambiguities in Doppler direction. These two points lead to the Doppler dilemma that states: "A good choice of PRF to achieve a large unambiguous range will be a poor choice to achieve a large unambiguous velocity and vice versa." 1The output of a range-Doppler processing using chirp pulses is illustrated in figure 2.11 for one scatterer. Afterwards, the zero-Doppler and zero-delay cuts are provided in figure 2.12. In both sub-figures, the output coincides with a sinus-cardinal for the chirp case.

22

Chapter 2. Radar Basics and Synthetic Aperture Radar Fundamentals

. SAR Fundamentals

. Synthetic Aperture Radar Principles

In the early ′ 50s, the technique known as Synthetic Aperture Radar (SAR) was developed. It is an imaging process that can distinguish the physical characteristics of an area by manipulating radar signals [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. It can be compared to aerial photography as the output of this procedure is an image.

The created image displays the amount of reflected energy provided by the received signals in each resolution cell. The back-scattered power depends on the material and the structure of the reflecting objects. For example, the reflection on water and trees provides less energy than the reflection on metallic objects as shown in figure 2.13. It represents a high-resolution SAR image created by the aircraft SETHI [Ruault du [START_REF] Du | The onera airborne multi-frequency sar imaging systems[END_REF] and illustrates an area close to the Nîmes-Garons airport. In this case, the range resolution after the SAR processing is 0.27 m and the azimuth resolution is 0.34 m. With such resolutions even small details can be visualized.

SAR came to light for scientific purposes around the 1970s, as in the beginning it was only used for military projects [Wang, 2008]. Nowadays, it has several applications in multiple domains, for instance:

• Geology : Visualizing geological structures under bad weather conditions.

• Oceanography, Hydrology, Glaciology : Analyzing the climate changes and simulating water and ice behavior.

• Agriculture, Forestry : Analyzing different types of flora.

• Infrastructure Stability : Assuring the safety of buildings or bridges after earthquakes.

• Volcanology : Studying the state of volcanoes using SAR interferometry.

• Military purposes : Surveillance.

In the SAR framework, an airborne or spaceborne platform is equipped with one or more antennas that transmit electromagnetic waves towards a given area on the Earth surface. Figure 2.14 represents the geometry of SAR framework with an aircraft. As it can be observed in figure 2.14, the platform trajectory defines the azimuth axis. Explanations of the SAR terminology, illustrated in Figure 2.14, are provided below.

▷ Range direction is considered as the axis orthogonal to the platform motion, and azimuth direction is considered as the axis along which the plane or the satellite moves.

▷ Target is defined as a cartesian point on the ground that is going to be imaged. In practice, a larger area is considered due to the antenna beamwidth, and it is called beam footprint. Its shape also depends on the antenna characteristics.

▷ Slant range is the axis between the platform and the target position.

▷ Ground range is the projection of the slant range onto the ground.

▷ Near range is the smallest observed distance from the plane to the ground. ▷ Far range is the furthest observed distance from the plane to the ground.

In our framework, we consider one antenna that performs both transmission and reception [START_REF] Chan | An introduction to synthetic aperture radar (SAR)[END_REF] and a uniform linear motion trajectory of the platform, which carries the antenna. It transmits the signals towards the line of sight direction, that illuminate a part of the ground inside the antenna mainlobe. These signals are usually chirps, defined in the subsection 2.1.3. The received signals, also called raw data, are analyzed in order to create an image that represents the landscape. These raw data consist of the summation of all the contributors in the (2.8), where τ 0 = 2R c . The specificity of SAR processing, as the name denotes, consists in creating a virtual antenna, by exploiting the plane motion during the train pulse transmission and reception. The motivation is to obtain a better resolution in the azimuth direction onto the output image, as the larger the antenna length, the better the resolution. The antenna can only be enlarged virtually as mechanical constraints prevent from carrying an antenna exceeding certain dimensions. The processing of the raw data in such manner consists of • a range compression step;

• an azimuth compression step; closely related to the range-Doppler processing described earlier.

Raw Data in SAR Framework

The raw signal that is back-scattered on the ground and retrieved by the antenna can be written as

I raw u (t) = y x a(x, y) * s t - 2r u c dxdy, (2.20)
where a(x, y) is the ground reflectivity of the (x, y) point of the ground, and

r u = x 2 + (y -v u) 2 + H 2
the distance between a scatterer located at (x, y, 0), a platform located at (0, v u, H), v the sensor velocity, and u a given instant linked to the plane position.

Range Compression in SAR Framework

Range compression aims at measuring the distance between the platform and the various objects present in the scene, or equivalently to determine the delay τ. A matched filter is usually applied to chirp signals, as presented in subsection 2.1.4. The result can be approximately expressed as a sinc-function [Cumming and Wong, 2005, Taylor, 2016] for a scatterer located at distance r u (or equivalently at delay τ) from the platform:

y(t) ≈ T p e -j2π f 0 τ sinc(πB(t -τ)).
(2.21)

From equation (2.21), the range compressed signal C u (r u ) for a given footprint can be expressed as

C u (r u ) = L ru a(x, y) * sinc r u ρ r dy, (2.22)
where L r u is the set of points located in the footprint and at range r u from the sensor. Let us denote g i (r u ) the integrated ground reflectivity that is expressed as

g i (r u ) = L ru a(x, y)dy, (2.23)
and thus the ranged compressed data can be expressed as

C u (r u ) = g i (r u ) * sinc r u ρ r .
(2.24)

The time axis for range compression is called fast time, as opposed to the slow time considered for the Doppler processing performed over samples separated by the PRI. Compressing the received signals in such a way allows to provide an enhanced range resolution that depends on the chirp bandwidth, as defined on equation (2.14), for the slant range axis. The range resolution projected on the ground depends on the incidence angle θ . It is defined as

ρ ground r = ρ r sin(θ ) . (2.25)
Azimuth Compression in SAR Framework Azimuth compression consists in combining multiple transmitted signals in the pulse train during the integration time in order to improve the resolution in azimuth direction. The processing goal is to image fixed targets, unlike the Range-Doppler processing described in 2.1.6, which is used for moving target detection. More precisely, the azimuth resolution of just one pulse, without azimuth compression, for the chirp is expressed as [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]:

ρ a = Rλ L real , (2.26)
where R is the slant range distance, and L real the antenna length. It is thus inversely proportional to the antenna length.

SAR processing aims at improving this resolution by overcoming the physical antenna length limitation. To do so, the phase modifications of the received signals induced by the platform motion are compensated. This effect creates a synthetic antenna, which can be expressed as

L synthetic = vT int , (2.27)
where v is the velocity of the platform.

Azimuth compression can be compared to angular compression in phased arrays as the pulses are transmitted with a fixed time step and the phase of each pulse rotates due to the platform motion. The classical Range Doppler Algorithm (RDA) consists in applying a FFT on each range line in order to compensate the phase modulation term created from the plane motion [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. In such a way, the data are represented in the range-Doppler domain.

The resulting output after the coherent summation through the FFT is also a sinc function, that depends on the synthetic antenna length expressed in (2.27) [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. The width of the mainlobe of the sinc function provides the new azimuth resolution, expressed for the chirp pulse as

ρ a = Rλ 2L synthetic .
(2.28)

This definition can be directly linked to the Doppler frequency. Taking into consideration the variation of the Doppler frequency:

∆ f d = f max d -f min d = 4v f 0 c sin θ 2 (2.29)
where θ is the platform opening angle, which can be expressed as

sin θ 2 = L synthetic 2R .
(2.30)

Combining equation (2.29) and 2.30, the link between the azimuth resolution and the Doppler frequency can be observed:

∆ f d = 2vL synthetic λ R = v ρ a .
(2.31)

After defining the azimuth resolution in equation (2.28), the SAR image for the chirp pulse can be written as the convolution between the ground reflectivity a of each point of the scene and the sinus-cardinals on the range and azimuth direction. It can be expressed mathematically as

s(r, y) = sinc y ρ a * sinc r ρ r * a(x, y), (2.32) with r = √ x 2 + H 2 .
Performing compression along both range and azimuth axes improves the resolution in both directions and therefore, the output image is more detailed. When simulating one scatterer and applying the SAR processing, the output is the same as in figure 2.11. As mentioned the behavior of the train of chirp pulses after processing corresponds to a sinus cardinal in both directions, with decreasing sidelobes when getting furhter from the mainlobe. This means that the sidelobes are high close to the mainlobe, which could have a negative impact on the resulting SAR image. This is usually taken into account by using tapering windows at the cost of degrading the resolution. In the next section, some criteria are presented to evaluate this sidelobe impact.

. Classical Evaluation Criteria for SAR Images

An intuitive criterion to evaluate a SAR image is the eye vision. If the contrast of the image is good enough, so that we can distinguish the different objects of the scene, then we would consider that an image is of good quality. But in order to provide a more accurate evaluation result, some classical criteria to quantify the SAR image quality are provided. Firstly, the resolution of the image is of great importance. Moreover, two very common criteria are the Integrated Sidelobe Level Ratio and the Peak-to-Sidelobe Level Ratio, which provide information on the sidelobe behavior. Finally, the Noise Equivalent Sigma Zero is introduced, that characterizes the noise floor of a SAR image.

. Experimental Range Resolution

The resolution is an important criteria for radar performance, as it provides the minimum needed distance between two points in order to be able to distinguish them. For the range compression, the resolution has been defined in (2.14) and it is widely used for evaluation in radar usually by posing the bandwidth factor K r to 1.

In SAR, the range resolution needs to be computed accurately. It is geometrically defined as the -3dB width of the compression peak. More precisely, after providing the compression output in dB, the spread of the two points that are situated at -3dB from the maximum value of the mainlobe corresponds to the resolution. For the chirp, which has a sinus-cardinal compression behavior, the -3dB width corresponds to

ρ mes r = 0.886 c 2B = 0.886 ρ r . (2.33)
This definition is used in the following for the evaluation of our results in chapter 6 and chapter 7.

. Integrated Sidelobe Level Ratio (ISLR)

The ISLR is defined as the ratio of the energy of all the sidelobes to the energy of the main lobe. It provides an indication on the minimum level under which a weak target may be undetected in the presence of uniformly distributed scatterers of the same level over the whole signal range [START_REF] Xin | [END_REF]Sun, 2007, Martinez and[START_REF] Martinez | [END_REF].

The ISLR is computed as

ISLR(y) = 10 log 10 a -∞ |y(τ)| 2 dτ + ∞ b |y(τ)| 2 dτ b a |y(τ)| 2 dτ , (2.34)
where y is the compressed signal in range or azimuth direction and the mainlobe is contained in the interval [a, b] .

As the signals are sampled at the reception, the discrete form of the ISLR definition can be cast as where Ω SL and Ω ML respectively define the sidelobe and mainlobe sample discrete sets. In the particular case where the sampling frequency f s = 1/T s is chosen such that f s = B with B the signal bandwidth, then the mainlobe set Ω ML is reduced to one single sample representing the signal energy. In that case, when the signal energy is normalized, or if we want to consider only the sidelobe energy, the criterion is called Integrated Sidelobe Level (ISL) and it is defined as :

ISLR (y) = 10 log 10     ∑ k∈Ω SL |y k | 2 ∑ k∈Ω ML |y k | 2     , ( 2
ISL (y) = 10 log 10 ∑ k∈Ω SL |y k | 2 . (2.36)
This criterion is very important for the work presented in this thesis, as the main goal of chapter 5 will be to minimize the sidelobe level created after range compression, for which we do not consider the mainlobe energy. Choosing this criterion in the following will simplify the definition of the optimization problem that needs to be solved.

. Peak-to-Sidelobe Level Ratio (PSLR)

The Peak-to-Sidelobe Level Ratio is defined as the ratio between the energy of the first highest sidelobe and the energy of the mainlobe. It is useful for determining whether low amplitude targets will be detected or masked by sidelobes created by nearby stronger targets [START_REF] Xin | Parameter assessment for SAR image quality evaluation system[END_REF].

The PSLR is computed as

PSLR = 10 log 10 E s E m , (2.37)
where E s is the energy of the strongest sidelobe and E m is the peak energy of the main lobe.

For a chirp pulse after range compression, the ISLR and PSLR values are displayed in Table 2.2.

Signal

ISLR PSLR (dB) (dB) Chirp -10 -13

Table 2.2: ISLR and PSLR values for a compressed chirp pulse.

. Noise Floor Level

The noise floor level is a very important information for radar applications, and in SAR framework it is called Noise Equivalent Sigma Zero (NESZ). It can be computed as

NESZ = ∑ k∈p |y k | 2 l , (2.38)
where p is an area of the SAR image where no signal is backscattered, and l is the size of the p area projected on the ground. Once it is estimated, it gives an insight on the energy level of the system when there is no signal back-scattered from the ground. This level is mainly generated by the thermal noise of the system [START_REF] Li | Analysis of scattering properties of continuous slow-release slicks on the sea surface based on polarimetric synthetic aperture radar[END_REF]. If the images are well calibrated, then the NESZ can be estimated from a very low-energy area, such as water regions for example seas, lakes, etc. In chapter 6, we will provide such an example to perform SAR image evaluation.

. Conclusion

In this chapter, the fundamentals of radar were presented as well as a specific airborne framework: the SAR. We saw that the chirp pulse and the matched filter are the main bricks in radar processing. However the goal of this thesis is to be able to create radar images while transmitting information. The chirp cannot be used as it cannot transmit information. Other ways of performing the radar imaging must thus be investigated. Hence, the following chapter provides and analyzes radar-communication systems.

CHAPTER 3 Radar-Communication Strategies

Communication applications grow at an exceptionally rapid rate due to the increasing number of users and the need for faster performance. A communication system aims to transmit binary information from a transmitter to one or more receivers. The transmitted information includes any type of data such as text, videos, sounds, etc.

Besides communication ones, many other systems operate in the electromagnetic spectrum, with a specific bandwidth attributed to them. However, the spectrum is a limited resource, and thus the different systems need to use the same frequency band to perform their missions without interfering with one another. Considering this problem and the enormous growth of communication applications, their co-existence with other systems is inevitable. The combination of communication applications with radar is a prospering research area as they usually function in the same frequency bands. The main strategies are co-existence, cooperation, and co-design of radarcommunication systems. Such a framework has been primarily investigated in autonomous vehicles in the milli-meter band [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF], designed to fulfill both a sensing aspect for navigation and transmit information to alert or advise the passengers. Our work, however, consists in using an airborne platform to perform a joint SAR-communication mission.

In this chapter, brief explanations on the functionality of a communication system are firstly provided in section 3.1 in order to inform the reader about its main points. The organization of the electromagnetic spectrum and the congestion problem are then detailed in section 3.2. In the following section 3.3, an extensive literature review of different existing radar-communications strategies is provided.

. Communication System Basics

The concept of communication has existed since prehistory through the mean of speaking. From a technological point of view, the communication era began during the 19th century with telegraphy, where words were replaced with dots, and dashes [Fitz, 2007]. Nowadays, communications have evolved and have transitioned from analog systems to digital ones. A communication system aims to transmit information to one or more receivers. For digital communications, the message that needs to be communicated is represented in binary language through bits 0 and 1.

. Communication System Description

The main elements of a communication system are the transmitter, the channel, and the receiver. Figure 3.1 represents a simple diagram of a communication system. A very brief description of the system [Grami, 2015] is provided as the communication aspect is not studied in this manuscript. However, it can give the reader an insight into the communication system functionality. The information source produces the message that the system is going to transmit. The input transducer, which could be a microphone, converts the source output into binary information. Afterwards, the transmitter is composed of three elements:

• the source encoder. It transforms the signal into a sequence of bits.

Communication System Basics

33

• the channel encoder. Its mission is to increase the robustness to overcoming noise and interference.

• the modulator. It maps the binary sequence into waveforms in order to ensure the data rate while limiting the BER.

The communication channel is the physical medium between the transmitter and the receiver, through which the signal propagates, and is degraded by noise and/or interference. When the signal reaches the receiver, the primary aim is the recovery of the information that the signal carries. It first requires a synchronization in order to provide a time reference for the decoding. Then, the performed tasks are the same as during transmission but in a reversed way, which means that the receiver is composed of the demodulator, the channel decoder, and the source decoder. Finally, a transducer transforms the resulting binary output into a signal interpretable from the user. These specifications represent a short simplified description of the system, as in realistic digital systems, more tasks need to be performed, but it is out of the scope of this work.

. Information Embedding in Communication Codes

We will now focus on how the sequence of bits is embedded in the signal during the modulation step performed by the encoder. An example of binary modulation associates the 0 bit to a signal s 0 (t) and the 1 to a signal s 1 (t) (as illustrated on the right side of figure 3.2). In such a way, there is one bit transmitted at a time. The left side of figure 3.2 shows that the 0 and 1 bits are represented on the 1D real line in order to distinguish them. If binary phase modulation is chosen, then the {0, π} phase values are used so that the real part is equal to {-1, 1} respectively as shown in the right side of figure 3.2. In this case, the value -1 maps the 0 bit. The link between the bit choice and the signal expression for phase modulation is through exp ( jb), where b is the chosen bit phase. This {0, π} doublet defines the simplest constellation that can be defined as a representation of the signal in the complex plane. However, this simple scheme strongly limits the quantity of information that can be transmitted, since only 1 bit is encoded in a given phase value. This quantity of transmitted information is measured by the data rate defined as the number of transmitted bits per second. Over the years, communication systems have faced the need for a data rate increase.

To do so, k consecutive bits can be mapped to a particular symbol taken from a given alphabet, also called constellation. More precisely, each combination of bits of the same length is called a symbol, and all the possible bit combinations create the alphabet. The total number of elements in the alphabet defines the size of the constellation, denoted by M = 2 k , where k is the number of bits per symbol. For example, if a symbol is composed of 3 bits, then we can create 8 possible combinations.

There are many different ways of mapping the bits into the symbols in the 2D space. Figure 3.3 provides two possible ways, that are mainly used in modulation configurations. Sub-figure 3.3a illustrates symbols distributed on the unitary circle. It corresponds to a modulation scheme called PSK. In this case, the information is contained in the phase of the signal. The number of bits per symbol is equal to 3, and thus it is a 8-constellation. Sub-figure 3.3b illustrates a scheme called Quadrature Amplitude Modulation (QAM). The symbols are positioned onto the 2D plane on various amplitudes. In this case, the signal undergoes both phase and amplitude modifications. This example has 4 bits per symbol, so the constellation is of size 16. The latter scheme is more used for high constellations, to avoid significant decoding errors that the circular configuration might face. Indeed in the PSK, the more the symbols, the closer their position on the circle, which means that their distance is small, and thus it is more probable to have a false estimation of the symbol at reception.

The larger the constellation, the more bits can be mapped to a symbol. However, there is a compromise between the number of symbols and the probability of estimating wrong symbols. For this reason, an essential criterion for evaluating a communication system is the Bit Error Rate (BER). It is defined as the rate at which errors occur in a transmission system [START_REF] Proakis | Communication systems engineering[END_REF], and it is directly linked to the minimum Euclidean distance between two symbols of the chosen scheme, under the assumption that the symbols are equiprobable. This criterion is not further specified as it is out of the scope of this work.

The bandwidth of PSK modulation is typically close to 1 T c , where T c is the duration during which a symbol is transmitted. For a fixed T c , a larger constellation thus increases the data rate, while maintaining the bandwidth. However, in order to preserve the BER, the constellation size must remain reasonable, limiting the transmission capacity. To increase the data rate nonetheless, communication operators need additional allocated bandwidth. This quest for increasing bandwidth has reached to the point where the electromagnetic spectrum has started to be congested, especially in the past years as the spectrum is a limited resource, and problems of attribution and pricing have appeared. The following section is thus dedicated to spectral congestion and possible ways to face this issue.

. Electromagnetic Spectrum and its Congestion

Every electromagnetic wave can be characterized by its wavelength λ or its carrier frequency through the relationship (3.1)

c = λ f c . (3.1)
Knowing the wavelength of a signal, it is possible to categorize it in the frequency spectrum. More precisely, the electromagnetic spectrum is divided in two parts, the radio and the optical spectrum, as it can be observed in figure 3.4. Radar systems operate in the radio spectrum. This latter one is divided in bands, that designate an interval of frequencies allocated by the International Telecommunication Union (ITU), in order to avoid interference between different applications. The radio spectrum ranges from 3 kHz to 300 GHz, but radars mainly function from about 5 MHz to 77 GHz [Skolnik, 1980]. The ITU divides the different bands every 10 n meter wavelength with n = -3, 4 going from extremely high frequency (EHF) to very low frequency (VLF).

During World War II, these frequency bands were denoted with a letter code, as shown in figure 3.5, for military defense. Since then, the Institute for Electrical and Electronic Engineers (IEEE) has made the use of these notations official [IEEE, 2020], so radar engineers still use them today for convenience reasons.

The radio frequency electromagnetic spectrum is a very precious and limited resource as it is not only used for radar applications, but also for communications, radio and television broadcasting, navigation, geolocalization and other civilian or military applications [START_REF] Griffiths | Challenge problems in spectrum engineering and waveform diversity[END_REF]. Especially, wireless communications are booming extremely fast, user-wise and quality/optimization- RADIOLOC. wise, and thus mobile applications demand more and more bandwidth, without the rest of the electromagnetic applications reducing their needs.

.3 cm 0.03 cm 3 x 105Å 3 x 104Å 3 x 103Å 3 x 102Å 3 x 1 0 Å 3 Å 3 x 1 0 -1Å 3 x 1 0 -2Å 3 x 1 0 -3Å 3 x 1 0 -4Å 3 x 1 0 -5Å 3 x 1 0 -6Å
For example in [Lucero et al., 2016], it is stated that in 2015 the Internet of Things (IoT) market had an installed base of 15.4 billion devices and it was predicted that this number would expand three times and reach 75.4 billion devices by 2025. In addition, the data rates of information transfers between users have increased over a short period of time. More precisely, the third generation (3G) of mobile systems made its appearance in the early '00s and just 20 years later the 5G network deployment has already started [START_REF] Patil | 5G wireless technology[END_REF].

Furthermore, as the demand increases, the auction price of any available bandwidth also rises. As mentioned in [START_REF] Liu | Joint radar and communication design: Applications, state-of-the-art, and the road ahead[END_REF], the UK asked communication operators a payment of 119.3 million pounds for 1800MHz of bandwidth in 2015, Germany charged more than 5 billion euros for 4 frequency bands for communications and the US sold 28GHz of spectral allocation for 5G applications for 702 million dollars.

Hence, this continuous need of supplementary bandwidth congests the electromagnetic spectrum, and over time it becomes impossible to allocate the additional frequency bandwidth that the applications would need. Figure 3.6 illustrates the frequency allocations in the electromagnetic spectrum in the United States, which was revised in 2016. We can remark that each different frequency band of the spectrum is associated to one or more applications or domains. It is thus evident that it is a difficult task to distribute extra bandwidth as the electromagnetic spectrum is not infinite. Therefore, it is critical to alleviate it in order to avoid interference between signals and degraded performance [START_REF] Griffiths | Challenge problems in spectrum engineering and waveform diversity[END_REF], in particular in radar applications. This problem was firstly looked into for military purposes, but then commercial application stakeholders started to investigate how to deal with the congestion problem in various applications such as autonomous vehicles, remote sensing, medical devices [START_REF] Paul | Survey of RF communications and sensing convergence research[END_REF]. A powerful solution that was proposed by the signal processing community is the spectral combination of different applications, which means creating a system that would operate jointly for two applications by allocating them more bandwidth and by allowing them to function in the same frequency domain.

All these arguments reinforce the need of combining different electromagnetic applications. In this context, radar and communication systems seem to be an appropriate solution to tackle the spectral congestion problem [START_REF] Liu | Joint radar and communication design: Applications, state-of-the-art, and the road ahead[END_REF].

. Different Spectrum Sharing Strategies for Radar-Communication Systems

Extensive literature exists on radar-communication systems, and for some applications, the two systems already co-exist or function together in specific bands of the electromagnetic spectrum [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF], Feng et al., 2020, Liu et al., 2020]. Table 3.1 displays different radar and communication applications, showing that some of them co-exist in their frequency bands [START_REF] Liu | Joint radar and communication design: Applications, state-of-the-art, and the road ahead[END_REF][START_REF] Satcom | Comparing satellite internet bands[END_REF], Mercier, 2021[START_REF] Esa) | Satellite frequency bands[END_REF], Griffiths et al., 2015]. Among them, one can mention Long Term Evolution (LTE) and 5G, which co-exist together with classical civilian applications such as air traffic control and air surveillance in the L-band [START_REF] Wang | Spectrum sharing between communications and atc radar systems[END_REF] and the S-band [START_REF] Hessar | Spectrum sharing between a surveillance radar and secondary wi-fi networks[END_REF]. Weather radars, occupying the Cband, now cohabit with Wireless Local Area Network (WLAN) [START_REF] Tercero | Impact of aggregate interference on meteorological radar from secondary users[END_REF]. In higher frequency bands such as Ka-band or higher, automotive radars share their band with 5G networks [START_REF] Choi | Millimeter-wave vehicular communication to support massive automotive sensing[END_REF].

When referring to joint radar-communication systems, three main categories are presented in the literature: co-existence, cooperation, and co-design [START_REF] Feng | Joint radar and communication: A survey[END_REF], Ma et al., 2020]. They are extensively described in the following.

. Co-existence

Co-existence refers to the configuration where two or more radio-frequency applications function autonomously without sharing information. The other systems are considered as secondary ones, and could potentially generate interference, which need to be mitigated. In such a configuration, primary and secondary users exist, and they are usually defined in such a way that the secondary does not affect the primary one while using the same spectrum [Hayvaci andTavli, 2014, Labib et al., 2017]. The primary user can be either the radar or the communication system, and to derive an optimal radar (respectively communication) system, a measure that quantifies the interference coming from the communication (respectively radar) signal should be defined and optimized. Figure 3.7 illustrates a configuration where co-existence occurs.

The simplest way of providing a co-existence system without any interference would be space, frequency or time separation. More precisely, in [START_REF] Hayvaci | Spectrum sharing in radar and wireless communication systems: A review[END_REF] and [START_REF] Bhat | Bandwidth sharing and scheduling for multimodal radar with communications and tracking[END_REF], authors propose a system where the radar changes its bandwidth depending on the needed resolution, leaving the unused bandwidth for the communication system. In addition, a time sharing solution is proposed in [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF], which consists in alternating the two applications in the time domain. But this type of solution is unsuitable for systems with limited allocated resources.

Discarding these trivial solutions, a few examples are provided in the following by taking either communications or radar as the primary user. The different systems that are introduced in this domain study the interference and/or provide lower or upper evaluation bounds between systems, which do not yet perform interference minimization, but they highlight the need for it. Authors in [START_REF] Khawar | A mathematical analysis of cellular interference on the performance of S-band military radar systems[END_REF] study the interference impact of a commercial cellular application on a seaborne military radar. They introduce an upper bound on the probability of detection, dependent on the Signal to Interference and Noise Ratio (SINR) 1 , to provide target detection performance under the interference caused by the communication system. On the contrary, having a communication system as primary, authors in [START_REF] Han | Analysis for the BER of LTE system with the interference from radar[END_REF] show that LTE systems deploying OFDM codes can co-exist with radar applications by studying the BER 2 to quantify the radar-induced interference. However, they highlight that large distance, frequency isolation, or interference mitigation is needed for enhanced performance. Authors in [START_REF] Chiriyath | Inner bounds on performance of radar and communications co-existence[END_REF] and [START_REF] Bell | Results on spectrum sharing between a radar and a communications system[END_REF] study the performance of a simple radar-communication co-existence system for both cases: when the radar system is primary, or the communication one is. The first article introduces bounds for each system by studying among others the performance criteria mentioned above (SINR and BER). The second investigates the mutual interference of a mono-static radar and a single-carrier frequency communication system. They both point out that interference created by the secondary system can severely degrade the performance of the primary one.

To tackle this problem, a co-existence radar-communication system aims to mitigate the interference of the secondary system to improve the performance of the primary one. It can be expressed as in the following generic form:

min system parameters f interference , s.t. optional constraints, (3.3)
where f interference quantifies the impact of the secondary system interference on the primary one. According to which system has the principal role, the interference is expressed differently. More precisely:

• when the radar is the primary user, the interference is minimized by maximizing the SINR, which is also directly linked to the probability of detection.

1 The SINR is defined as

SINR = P S P I + P N , ( 3.2)
where P S is the signal power, P I is the interference power and P N is the noise power. It is very close to the SNR with the difference that is also considers the interference of the system. 2 The BER is defined as the number of wrongly estimated received bits over the total number of bits.

Chapter 3. Radar-Communication Strategies

• when the communication system is the primary one, the interference is minimized by minimizing the BER or maximizing/ensuring the data rate.

As an example in [Deng and Himed, 2013a], the problem to be optimized is expressed as

max w |w H s| 2 E[|w H (n + x c )| 2 ] = SINR(w), s.t. ||w|| 2 = 1, (3.4)
where s is the transmitted radar signal, w is the filter to be optimized for interference reduction,

x C is the communication system interference with x C ∼ C N (0, R x )3 , and n is the noise.

In this example, authors maximize the SINR in a Multiple Input Multiple Output (MIMO)4 radar.

In [START_REF] Cui | Interference alignment based precoder-decoder design for radar-communication co-existence[END_REF], the signal is supposed to be a combination of sub-carrier frequencies, and the weights applied to these sub-carriers is determined by optimizing the SINR with respect to precoding and decoding matrices. A constraint on the rank of the precoder and the decoder has to be respected to avoid the basic unitary solution. Thanks to the efficient interference suppression, both radar and communication capabilities are improved. Some papers propose to combine their approach with a cognitive radio as it could provide improved performance for the system [Deng andHimed, 2013b, Haykin, 2005]. Indeed, cognitive radio is an intelligent system that can learn from its surrounding environment, adapt to it and attribute bandwidth to the system that needs it during specific intervals [Haykin, 2005]. In a radar-communication system, this approach could be used to attribute time-frequency resources, based on the knowledge that the radar will only operate sporadically.

Through these examples representative of the co-existence strategy, we have seen that the two systems do not have any task in common, and do not share information. In the two following strategies, this information exchange will be reinforced.

. Cooperation

Cooperation occurs when there is a mutual operation between the two systems. They are still assigned independent missions but they can exchange information in order to provide better performance, such as interference mitigation [START_REF] Feng | Joint radar and communication: A survey[END_REF], Chiriyath et al., 2017a].

Examples of cooperative systems include using different signals for each application but optimizing their combined performance [START_REF] Chalise | Performance tradeoff in a unified passive radar and communications system[END_REF], or using communication signals to perform radar applications [START_REF] He | Performance gains from cooperative MIMO radar and MIMO communication systems[END_REF] or vice versa [START_REF] Sahin | A novel approach for embedding communication symbols into physical radar waveforms[END_REF]. These systems can either be physically separated or not.

As mentioned in [START_REF] Feng | Joint radar and communication: A survey[END_REF], the performance of each system now relies on both, adjusting their system towards enhanced performance. More precisely, if the systems are physically separated, then the performance of both radar and communication systems is optimized with respect to the SINR and the data rate respectively. Thus, finding a trade-off for their performance sions using mutual information 6 . Finally, a MIMO framework is also studied in [START_REF] He | Performance gains from cooperative MIMO radar and MIMO communication systems[END_REF], where the authors use a combination of MIMO radar and MIMO communication waveforms. For the radar system, target localization performance is estimated through the Cramer-Rao bound 7 , while on the communication side, the mutual information is exploited to provide improved communication efficiency.

. Radar-based waveform cooperative systems

A lot of attention has been drawn to the creation of a cooperative system that transmits a waveform that performs both radar and communication missions. The following examples use radar waveforms where the communication information is embedded in an efficient way in the phase, or the frequency domain, or the sidelobes etc. The aim is thus to embed information sequences in the waveform without degrading the radar performance. Maintaining radar performance usually refers to preserving the detection probability, keeping desired ambiguity function properties, and minimizing the sidelobe level [START_REF] Feng | Joint radar and communication: A survey[END_REF]. Figure 3.8 illustrates such a framework, where the radar antenna transmits the radar waveform in which the symbols are embedded.

Attempts to use chirp pulses to embed information were made. In [START_REF] Gaglione | Waveform design for communicating radar systems using fractional fourier transform[END_REF], 6 Mutual information is the a priori known information of the system that can enhance the performance of the other. In this specific example, the direct path of the radar signals can be suppressed in order to decode more efficiently the communication codes. 7 The Cramer-Rao bound is an inferior bound on the variance of unbiased estimators θ of a deterministic parameter θ . It is greater than the inverse of the Fisher information [Cramér, 1999]:

var( θ ) ≥ 1 nI(θ )
,

where

I(θ ) = -E ∂ 2 ln f (X;θ ) ∂ θ 2
is the Fisher information, and f (X; θ ) is the likelihood function of the random variable X. In this context, the Cramer-Rao bound is used to provide an indicator on the best possible performance.

the authors propose a system where a composition of quasi-orthogonal chirp sub-carriers 8 is used to perform both radar and communications, allowing in such a way to embed information into the radar signals. It is an approach somehow similar to OFDM (a waveform extensively used in communications, presented thoroughly in the next chapter), except that instead of pure tone carriers, chirps are used. This enables to preserve the good radar properties of the chirp, with fewer severe ambiguities than OFDM. However, communication properties are degraded (increased BER, decreased data rate), and another drawback of the method is -like OFDM-high Peak-to-Average Power Ratio (PAPR). In a similar context, authors in [START_REF] Sahin | A novel approach for embedding communication symbols into physical radar waveforms[END_REF] use a polyphasecoded frequency modulated radar signal 9 , in the phase of which they add communication symbols through continuous phase modulated sequences. This model provides good radar properties but the interference of the additional communication term is inevitable. For this reason, the paper provides a trade-off between the radar and the communication performance, with the advantage of keeping the good spectral and power characteristics of the signal.

In [START_REF] Ji | A dual-function radarcommunication system using FDA[END_REF] is provided a radar-communication framework where a frequency diverse array-MIMO radar transmits orthogonal signals and the communication symbols are embedded in the frequency difference between the carriers. The communication aspect is a secondary system that is transparent to the radar one, but the data rate of the system is low as one bit can be embedded per frequency increment through the definition of data embedding that they defined. The Cramer-Rao bound is also computed to ensure that the radar performance remains unchanged. In a similar framework that preserves the signal form, authors in [START_REF] Hassanien | Dualfunction radar-communications: Information embedding using sidelobe control and waveform diversity[END_REF] create a system where they use the antenna mainlobe to perform radar detection and its array beam sidelobes to ensure information transmission for the communication application. This method has a data rate in the range of a few kbits, and it is dependent on the number of transmitted array elements. A joint frequency-hopped 10 radar communication system in a MIMO radar framework was also proposed in [START_REF] Baxter | Joint radar and communications for frequency-hopped MIMO systems[END_REF], where they use one or several schemes for embedding information which gives a lot of liberty from a communication point of view and allows an increased data rate. From a radar point of view, the authors examine the ambiguity function which provides high sidelobe level, that needs to be mitigated using different compression techniques. This separation can be performed not only in the frequency domain but also by using an antenna selection algorithm, in order to improve the shared aperture and reduce high interference between the applications [START_REF] Wang | Sparse transmit array design for dual-function radar communications by antenna selection[END_REF].

All the previously cited articles are different examples on how information can be embedded in the radar waveform. In the following, the roles are inverted, and systems using communicationbased waveforms are utilized to create cooperative radar-communication systems.

8 Quasi-orthogonal chirp sub-carriers are created through a fractional Fourier transform (a class of timefrequency representations that rotates the time-frequency plane by an arbitrary angle). By exploiting the quasi-orthogonality of different chirps, they embed information by mapping them in the in-phase and quadrature phase.

9 Polyphase-coded frequency modulated radar signal is a modified form of continuous phase modulated codes that is appropriate for transmitting polyphase radar codes [START_REF] Blunt | Polyphase-coded fm (pcfm) radar waveforms, part i: implementation[END_REF]. 10 The main idea of frequency hopping consists in switching the frequency carrier according to the spectral availability to avoid interference [START_REF] Baxter | Joint radar and communications for frequency-hopped MIMO systems[END_REF].

. Communication-based waveform cooperative systems

Similarly, researchers propose radar communication systems that use communication waveforms to perform both missions simultaneously. Usually the radar performance is analyzed a posteriori, but there are various propositions in the literature. Figure 3.9 represents such a framework, where a communication code is used for both radar and communication missions. The aim of a system that uses communication codes is to minimize the artifacts on radar detection while maintaining the BER under a desired threshold.

In the following, some literature contributions are provided in order to give an insight on the different ways of creating a communication-based waveform cooperation radar-communication system. Firstly, examples from communication schemes are provided and secondly, signals that are used in WLAN framework are specified. In [START_REF] Tedesso | Code shift keying based joint radar and communications for EMCON applications[END_REF], authors use Gold and Kasami11 codes through binary phase shift keying and utilize them for the creation of a radar-communication system to avoid mutual interference. In this context, the communication performance needs to be verified, and thus they remark that the error rate of the symbols is similar to classical frequency-shift keying schemes. However, they do not optimize the radar performance but only mention that the range sidelobe levels are close to classical binary phase codes, which could be problematic for radar performance. The auto-correlation of these signals are illustrated in figure 3.10. The sidelobe level is similar for Kasami, Gold, and random binary codes. In that spirit, some works also provide an analysis of the radar ambiguity function using communication waveforms, and as we studied the ambiguity functions of some of them, figure 3.11 illustrates the ambiguity functions of Kasami and Gold codes. The sidelobe level is uniform along the time and frequency axes, except for the zero-Doppler and zero-delay cuts that provide slightly lower sidelobes compared to the rest of the values.

Ambiguity function of Kasami

In [START_REF] Arik | Utilizing sidelobe ASK based joint radarcommunication system under fading[END_REF], the authors take advantage of a uniform linear array, to transmit a combination of orthogonal waveforms and perform Amplitude Shift Keying (ASK). Different combinations are obtained depending on the bit sequence to be transmitted. To guarantee the two functions, the weight coefficient of each antenna is optimized to obtain a certain beampattern with limited energy in the radar antenna sidelobe, and with constraint of obtaining a specific signal in the communication direction. Their optimization problem designs Digital Beam-Forming (DBF) weight vectors for amplitude modulation, denoted µ k with k ∈ 1 . . . K , by minimizing the energy in the sidelobe region of the antenna, while preserving the energy in the mainlobe direction. It is expressed as

min µ k max θ ||G k (θ )| -|µ k a(θ )||, θ ∈ Θ ML , s.t. ||µ H k a(θ )|| ≤ ε, θ ∈ Θ SL , µ H k a(θ c ) = G k , (3.6)
where G k is the desired transmit beam-pattern, a(θ ) is the steering vector for direction θ , Θ ML is the angular region of the antenna mainlobe, and Θ SL the sidelobe sector, ε ∈ R + is the threshold for the maximum sidelobe level for the radar mission, and θ c is a direction of the antenna where a communication receiver is positioned.

Additionally, OFDM codes are also used to create joint systems, and in [START_REF] Mercier | Correlation-based radar receivers with pulse-shaped ofdm signals[END_REF]] authors use OFDM codes that are correlated with a symbol-based receiver that takes into consideration the waveform structure and provides good SINR performance.

Massive work has been performed in WLAN in the communication field and thus some radarcommunication systems are also introduced using the IEEE 802.11ad standard, especially for vehicular applications in the mmWave band [START_REF] Choi | Millimeter-wave vehicular communication to support massive automotive sensing[END_REF]. An example of such a framework is introduced in [START_REF] Kumari | Adaptive virtual waveform design for millimeter-wave joint communication-radar[END_REF], where the authors create an adaptive frame structure12 of IEEE 802.11ad standard that combines a radar and communication part, and provides a trade-off performance by optimizing the Cramer-Rao bound of the Doppler variable (for the radar aspect) and the distortion minimum mean square error metric13 (for the communications).

As observed, cooperative systems contribute a big part of joint radar-communication systems, and the previous examples provide a small sample of the literature contributions on the topic. The last category for designing joint systems, called co-design, is presented in the following subsection 3.3.3.

. Co-design

Co-design consists in designing dual-function waveforms that take into consideration and optimize the performance of both applications in order to create a fully joint mutual radio-frequency system [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF]. A common framework of co-design joint radar system is considering MIMO waveforms, for instance using OFDM signals [START_REF] Liu | Design of integrated radar and communication system based on MIMO-OFDM waveform[END_REF], but there are also a lot of examples that combine radar and communication signals such as linear frequency modulated waveforms with phase modulation [START_REF] Bekar | Realization of a joint MIMO radar and communication system using a PSK-LFM waveform[END_REF], especially in intelligent transportation systems. Figure 3.12 illustrates a co-design system, where the transmitted waveform is a signal optimized for both radar and communication missions.

. MIMO-based co-design systems

The attention of MIMO co-designed systems that combine radar and communication applications has grown a lot. MIMO framework is utilized in radar and communication applications for different purposes. For radar, using orthogonal signals through several antennas enables to illuminate a wide area, while improving the angle accuracy [Gomez, 2014]. For communications, it is helpful for large data rate needs and multi-users configurations [START_REF] Proakis | Communication systems engineering[END_REF].

One way to create a co-design radar-communication system consists in providing a set of waveforms optimized on a criterion taking into account the performance of both radar and communication application as in [START_REF] Mccormick | Simultaneous radar and communications emissions from a common aperture, part i: Theory[END_REF]. More precisely, they transmit waveforms that are optimized for a given far-field radar and a given different communication direction. In such a design, the waveforms are transmitted in a given direction θ , and can be mathematically

expressed as, s MIMO (θ ) = v(θ ) H S, (3.7)
where v(θ ) is the steering vector in the θ direction of size N a × 1, S is the matrix of size N a × N s that contains all the temporal signals to be transmitted, with N s the signal size and N a the number of antennas. The MIMO waveforms are optimized so that s MIMO (θ rad ) = s rad and s MIMO (θ comm ) = s comm , where s rad is the desired radar signal in direction θ rad , and s comm the desired communication signal in direction θ comm . Furthermore, they include a PAPR constraint so that the proposed method shows good power efficiency and quasi-constant amplitude. However, depending on the chosen parameters, the beampattern radar performance or the BER of the communication system may be degraded when deviating from the communication direction θ comm , even if the overall performance allows the creation of the joint system. This degradation appears mainly because the signals, in a given direction, are the result of the summation of different transmitted waveforms, and can cause high level away from the directions of interest. To avoid this effect, in [Liu et al., 2018a] authors provide a design on the beampattern by optimizing parameters of the system, while respecting constraints regarding the SINR and the transmitted power. Additionally, authors in [START_REF] Sit | Automotive MIMO OFDM radar: Subcarrier allocation techniques for multiple-user access and DOA estimation[END_REF] provide a MIMO automotive system using OFDM signals that performs Direction-of-Arrival (DOA) estimation by exploiting the antenna positions, while allowing multi-user communication in the radar network. Finally, some articles deal with a posteriori processing like in [START_REF] Bai | OFDM MIMO radar waveform design with high range resolution and low sidelobe level[END_REF], where the sidelobe level is optimized and the carrier orthogonality is ensured by the OFDM properties.

Such systems draw a lot of attention as both MIMO systems and OFDM codes can be used for both radar and communications. In the following, completely different systems that combine Linear Frequency Modulation (LFM) pulses with other communication codes are examined.

. Co-design systems through LFM and communication code combination

Combining the chirp with other codes allows to create a bridge between the two domains of interest, and to optimize this hybrid waveform to ensure the performance of both. Figure 3.12 shows such a waveform transmitted by the antenna that corresponds to both applications. In order to create this type of waveforms, the LFM phase is subdivided in a certain amount of chips, and in each one of them, a piece of a communication code is included by adding it in the phase. The phase of the joint radar-communication system waveform is expressed mathematically as

ϕ n = ϕ LFM n + ϕ Comm n .
(3.8)

The main motivation of this method is to keep as well as possible the advantageous properties of the chirp, and transmit information. There is a fine line between this category and the subsection of the radar-based waveforms 3.3.2.2, and the detail lies in the system dependency. For the cooperation case, the two systems are usually independent, whereas the co-design systems have a complete joint unit. However, there is always a trade-off between the performance of each system. Different codes can be used as for example Minimum Shift Keying (MSK) in [START_REF] Chen | A novel radar waveform compatible with communication[END_REF], Continous Phase Modulation (CPM) in [START_REF] Zhang | Waveform design for joint radar-communication with nonideal power amplifier and outband interference[END_REF], BPSK in [START_REF] Nowak | Co-designed radarcommunication using linear frequency modulation waveform[END_REF], MIMO BPSK in [START_REF] Bekar | Realization of a joint MIMO radar and communication system using a PSK-LFM waveform[END_REF]. More precisely, combining the chirp with MSK codes (proposed in [START_REF] Chen | A novel radar waveform compatible with communication[END_REF]), which are a binary sub-category of CPM codes (in [START_REF] Zhang | Waveform design for joint radar-communication with nonideal power amplifier and outband interference[END_REF]), provide good spectral properties, although the data rate is rather small. In [START_REF] Zhang | Waveform design for joint radar-communication with nonideal power amplifier and outband interference[END_REF], they further define spectral bounds to contain the spectrum even more in the desired frequency band, which can degrade slightly the communication performance. This results in a wider mainlobe and fluctuating sidelobes after the waveform modifications. Similarly, authors in [START_REF] Nowak | Co-designed radarcommunication using linear frequency modulation waveform[END_REF] use Maximum Length Binary Sequences (MLBS) modulated with phase reduced BPSK in the chirp phase, that reduce the maximum phase difference and thus prevent abrupt phase changes and undesirable sidelobes. Choosing these codes, high auto-correlation and low cross-correlation can be achieved, and the ambiguity function shows that the system provides improved radar performance compared to using regular BPSK codes, but the data rate is low.

All these systems provide a global analysis on how to create a joint radar-communication system for different applications and frameworks. In the following, we focus on a specific SAR framework, and propose an overview of what is found in the literature in terms of waveform selection and processing choice.

. SAR-communication systems in the literature

In the literature, there is not a lot of research in the SAR framework when using signals other than chirps. Some articles address the SAR framework using OFDM signals, but they do not necessarily consider communication applications [START_REF] Kim | A novel ofdm chirp waveform scheme for use of multiple transmitters in sar[END_REF], Riché et al., 2014, Garmatyuk and Brenneman, 2011], even if these signals were originally introduced for communications. For example, authors in [START_REF] Kim | A novel ofdm chirp waveform scheme for use of multiple transmitters in sar[END_REF] provide a MIMO framework by combining the chirp and the OFDM, while in [START_REF] Riché | Investigations on OFDM Signal for Range Ambiguity Suppression in SAR Configuration[END_REF] they propose to use OFDM for range ambiguity suppression. However, in [START_REF] Garmatyuk | Adaptive multicarrier OFDM SAR signal processing[END_REF], authors provide an adaptive OFDM system for SAR and mention the possibility of a joint system with their method, but they do not further investigate it.

A true example of a joint SAR-communication using OFDM codes is introduced in [START_REF] Zhang | OFDM synthetic aperture radar imaging with sufficient cyclic prefix[END_REF], where the OFDM codes are separated from pulse to pulse by the PRI as in radar applications, and not continuous as usual in communications, and the cyclic prefix is also added. They show that as long as the cyclic prefix is well-defined, both the radar and communications can reach good performance from a sidelobe level point of view and inter-symbol interference respectively. To avoid having such a monostatic framework where the signals need to respect the PRI (which results in a lower data rate), authors in [START_REF] Herschfelt | Joint radar-communications waveform multiple access and synthetic aperture radar receiver[END_REF] provide a multi-static context where multi-user communication and SAR tasks are performed simultaneously through Direct Sequence Spread Spectrum (DSSS)14 . Firstly, the messages are decoded and extracted by estimating the user signals and the channel. Then the SAR image can be generated, since the localization of the users that transmit the signals is known.

The combination of OFDM and SAR is thus more common than using different communication codes.

. Conclusion

We want to underline that the combination of radar and communication has been developed only for the past few years, so they do not cover all the variety of radar applications yet. Nevertheless, we have seen across this chapter that radar-communication system design can take various aspects, as well as different dependencies between the systems through the three strategies explained in this chapter. From a radar perspective, the most natural solution to accomplish such a system was to start from communication waveforms and to determine which might be suitable among them for SAR applications. This guarantees that the communication task should be feasible. As for the radar performance, they will be maintained by considering other compression strategies than the classical matched filter. This is the path that we have followed in this work. It will be developed in the next chapters.

. Continuous Phase Modulation Formulation and Properties

In communication applications, diverse coding schemes may be used to encode the desired information, some of which are well known such as Frequency Shift Keying, Phase Shift Keying and Quadrature Amplitude Modulations [Proakis, 1983]. In these strategies, communication symbols are selected from a discrete set of values and they are embedded in frequency, phase or/and amplitude, according to the chosen modulation. Direct encoding of these symbols without specific phase shaping usually leads to abrupt phase changes when transitioning from one symbol to the following one.

Such abrupt transitions tend to increase the out-of-band energy by creating high spectral sidelobes [Proakis, 1983]. On the contrary, CPM signals ensure phase transition continuity from one symbol to another, which prevents the spectrum from spreading too much outside of the band. These signals belong to signaling schemes with memory, because each embedded symbol depends on the previous phase-coded symbols to ensure phase continuity [Proakis, 1983]. CPM codes are used in military applications, but also in the Bluetooth wireless standard and aeronautical telemetry [START_REF] Sahin | A novel approach for embedding communication symbols into physical radar waveforms[END_REF], as well as wideband code division multiple access1 [START_REF] Pandey | Analysis on cpfsk modulation technique for wideband code division multiple access[END_REF]] and for global system for mobile communications (GSM) through 2G digital cellular networks and satellite communications [START_REF] Rice | Estimation techniques for gmsk using linear detectors in satellite communications[END_REF].

. CPM Code Definition

Let I = [I 1 , . . . , I N chip ] be the set of N chip symbols that are embedded in a CPM code phase, each symbol belonging to the constellation set M = {±1, ±3, ..., ±(M -1)}, where M is the constellation number. The set is also called M-ary alphabet, and the constellation designates all possible symbols that can be embedded in the signal. The number of bits k per embedded symbol is given by k = log 2 M. The transmitted signal can be expressed as follows:

s(t) = 2E s T c e jϕ (t;I) , (4.1)
with E s the signal energy, T c the chip duration, and ϕ (t; I) the phase, expressed as [START_REF] Anderson | Digital phase modulation[END_REF]:

ϕ (t; I) = 2π ∑ i≤n I i h i t 0 g(τ -iT c )dτ for nT c < t ≤ (n + 1)T c , = 2π ∑ i≤n I i h i u(t -iT c ), for nT c < t ≤ (n + 1)T c . (4.2)
In the above expression, h = [h 1 , . . . , h N chip ] is the modulation index with values usually in [0, 1], I i is the i th information symbol, and u is a phase smoothing response function, defined as the primitive (a) PSD of one CPM code.

-1.5 -1 -0.5 0 0.5 1 1.5

Frequency (Hz) of a function g that may span L consecutive chips, expressed as

u(t) =      t 0 g(τ)dτ for t ≤ LT c , 1 2 for t > LT c . (4.
3)

The parameters that mainly define the uniqueness of a CPM code are the following:

the number M of possible symbols in the constellation;

the length L of the support of the frequency smoothing response g;

the phase smoothing function u (or similarly the frequency shaping function g);

the modulation index h.

. Size of the constellation M

The size of the constellation is a crucial parameter of the signal as it defines the number of symbols of the alphabet. It directly impacts both the spectrum shape and the data rate. Figure 4.1 illustrates the Power Spectral Density (PSD) of CPM codes for different constellation sizes. Subfigure 4.1a displays this result using one CPM code. For this example with different constellation sizes, an average PSD over 100 different PSDs of CPM codes is provided, in order to visualize a less fluctuated behavior, and it is plotted in sub-figure 4.1b. In each configuration, the number of transmitted symbols and their duration is the same, which means that the number of bits increases with M, as each symbol is constituted by log 2 (M) bits, by keeping constant the pulse duration T p and the chip duration T c . Observing the PSD, the width of the mainlobe changes according to the constellation, and thus a larger data rate also implies a larger bandwidth.

. The frequency pulse width L

The frequency pulse width L defines the support of the frequency smoothing function g. It is also called memory length, because it specifies how many previous symbols are required to determine the phase of the code from the current symbol. When L = 1, the CPM response is called full-response, whereas when L > 1, it is called partial-response [START_REF] Proakis | Communication systems engineering[END_REF]. For example, CPM code families such as CPFSK and MSK are full-response CPMs [START_REF] Proakis | Communication systems engineering[END_REF]. However if L = 2, the support of the frequency smoothing function is equal to two chip durations, and thus the memory also depends on the previous symbol. When increasing L, the signals have a more contained spectrum in the band as it can be observed in figure 4.2, where the PSD is provided for L = {1, 3}, but this also generates more difficulties in the decoding process. The data rate remains invariant with respect to the frequency pulse width L.

. The phase smoothing function u

The phase smoothing function u determines the evolution of the phase change between two consecutive symbols, and by extension g is the instantaneous frequency in each chip [START_REF] Anderson | Digital phase modulation[END_REF]. It can be defined as any continuous function that is bounded in time between 0 and LT c , such that [START_REF] Anderson | Digital phase modulation[END_REF]

LT c 0 g(τ)dτ = u(LT c ) = 1 2 , for t ≥ LT c .
By taking the integral of the frequency response, the continuity of the phase can be ensured. Regarding the frequency shaping response g, it can be chosen among different windows. The two most used ones are the rectangular and the raised-cosine window defined as follows: • Rectangular window (LRECT) :

g(t) =    1 2LT c for 0 ≤ t ≤ LT c , 0 otherwise. (4.4)
• Raised-cosine window (LRC) : corresponding functions u, computed after integrating g. The latter one illustrates how the phase transitions will appear in the code, which can be also observed in figure 4.4, that provides the unwrapped phase of two different CPM codes, one with a rectangular shaping function and one with a raised cosine one, in order to observe how their integration impact the phase. In the first case the transitions are linear, whereas in the second one, they have a curvy behavior.

g(t) =    1 2LT c 1 -cos 2πt LT c for 0 ≤ t ≤ LT c , 0 otherwise.
Besides the phase behavior of the code, the shaping window also has an impact on the spectrum of the code. For this reason, the PSD of these codes should be studied as well. Figure 4.5 illustrates the PSD of two CPM codes with a rectangular and a raised cosine shaping window with M = 8, h = 1 2 , and L = 1. The difference mainly lies in the spectral containment and the mainlobe width of the spectrum. The code obtained using an integrated raised cosine has a broaden mainlobe and the sidelobes are slightly lower than the code with linear phase transitions.

. The modulation index h

Finally, the modulation index h is positive, usually rational in [0, 1]. When the modulation index h varies in a cyclic way, the signals are called multi-h modulated codes [START_REF] Saleem | Symbol block construction for frequency domain equalization of multi-h CPM[END_REF]. However, in most applications, h is considered constant over the symbols, as it will also be in the following of this work. The modulation index can also take irrational values, but the decoding complexity at the receiver end is increased in that case [START_REF] Zarei | An iterative detection algorithm for coded CPFSK signals with irrational modulation index[END_REF]. Taking thus a constant rational number for h, it can be expressed as h = m p , with m, p relatively prime positive integers., (4.6) m and p values are crucial for the number of possible phase states of the code. The smaller h gets, the narrower the bandwidth of the code, as it can be observed in figure 4.6. It illustrates the PSD of CPM codes with different modulation indices. However, it cannot be chosen too small either as the complexity will be high at the receiver due to the high number of As the main characteristics of the CPM codes are their constant amplitude and continuous phase, a representation that allows to observe both at the same time is provided in figure 4.7, where the real and the imaginary part of the signal are plotted with respect to time in a three dimensional plan. The question that arises is how the phase states are defined, because the phase transitions are dependent on the shaping functions. The parameter that defines the number of possible phase states is the modulation index and more precisely, its numerator m and denominator p. Let us denote S ϕ the set of possible phase states of a given code, and thus |S ϕ | its cardinality, which is 60 Chapter 4. On the choice of CPM Codes for the Joint System defined as:

|S ϕ | = pM L-1 , if m is even, 2pM L-1 , if m is odd. (4.7)
For the case of full-response signals, the set of phase states S ϕ is determined and defined as [START_REF] Proakis | Communication systems engineering[END_REF] For example, for h = 1 2 , and a memory length of L = 1, the number of phase states is |S ϕ | = 4, with states in the set {0, π 2 , π, 3π 2 }. This configuration is presented in figure 4.8. Therefore, considering a very small h implies that p is high and then the phase states are numerous and close to each other which results in an increased complexity at the receiver.

. Continuous Phase Frequency-Shift Keying (CPFSK): A special category of CPM codes

Continuous Phase Frequency-Shift Keying [Anderson et al., 2013, Raveendra andGabrani, 1994] is a sub-family of CPM, which presents the same attractive properties, such as constant amplitude and well contained out-of-band energy but in addition an analytic expression of the PSD can be provided. The expression in the time domain is firstly introduced, followed by the expression of the PSD. Afterward, we provide a bandwidth formulation that is used in this work, and the ambiguity function of these codes.

. Time domain expression

CPFSK formulations can be derived from equation (4.2), by constraining some of the parameters as follows:

the modulation index h is constant in ]0, 1[, such that h i = h, ∀i; the frequency shaping function g is rectangular;

the memory length L is full-response (L = 1).

By extension, for a rectangular frequency shaping window, the phase smoothing function u is expressed as

u(t) =      0, for t < 0, t 2T c , for 0 ≤ t < T c , 1 2 , for t ≥ T c .
An explicit formulation for the CPFSK phase can thus be given by [START_REF] Anderson | Digital phase modulation[END_REF]:

ϕ (t; I) = πh ∑ i<n I i + πh T c (t -nT c )I n , for nT c < t ≤ (n + 1)T c . ( 4

. Frequency domain

Power spectral density function The expression of the spectrum or the PSD of CPM codes cannot generally be provided analytically. However for special categories, and in particular for CPFSK codes, the PSD has an explicit analytic formulation. Knowing such an expression allows to compute the bandwidth of the codes analytically, which is primordial from a radar point of view, as it is related to the range resolution. This is also one of the main reasons why we chose these codes for the creation of the joint SAR-communication system.

The average PSD of the CPFSK codes is expressed as [START_REF] Anderson | Coded modulation systems[END_REF]]

G( f ) = 2 M M ∑ i=1 1 2 sinc 2 (g i ) + 2 M 2 M ∑ i, j=1 a i j sinc(g i ) sinc(g j ), (4.10)
with

g i = f T c - h 2 (2i -M -1), a i j = cos (π(g i + g j )) -C cos (π(g i + g j -2 f T c )) 1 -2C cos (2π f T c ) +C 2 , C = sin (Mπh) M sin (πh) .
The main parameters involved in the PSD of the CPFSK codes are the modulation index h, the constellation M and the chip duration T c . The analytic formula in (4.10) for CPFSK codes is used to determine the signal bandwidth, which can become wider by increasing the modulation index h and/or the constellation M. This behavior can be observed in figure 4.10, where the PSD is plotted both analytically (sub-figure 4.10a) and empirically (sub-figure 4.10b). The two sub-figures are different as the simulated ones have a more fluctuated behavior. They can converge to the analytic ones if several simulated PSD are averaged.

It can also be noted from sub-figure 4.10a that the out-of-band energy of these PSD is low for these particular codes. Another observation that can be made is that increasing the constellation size M results in a larger bandwidth and a flatter PSD mainlobe, which is an important characteristic for radar applications. However, it can be observed that the simulated PSD fluctuates a lot around the theoretical one, which leads to the conclusion that some compression techniques, such as the Wiener filter 2 [START_REF] Dogariu | An insightful overview of the wiener filter for system identification[END_REF] (a rather common SAR processing technique) should be used with caution.

Bandwidth Computation

As the fluctuations of the PSD for a given code do not allow to compute the bandwidth very accurately, the analytic expression is used here to determine the width of the frequency band.

The bandwidth is a key parameter for radar applications and ideally it needs to be known explicitly for the system design. It can be defined in different ways with respect to the application and model characteristics. According to [Amoroso, 1980], 6 different definitions can be used in signal processing and communications. The two main definitions used in the radar field are either the -3dB bandwidth, which is determined as the frequency for which the PSD value reaches the maximum value -3dB, or the null-to-null bandwidth, which is expressed as the frequencies between the first zero-values before and after the mainlobe of the PSD. These methods can be applied either on a theoretical or a simulated PSD. In our case, the CPFSK codes have a fluctuating PSD, as it can be observed in sub-figure 4.10b, which strongly depends on the encoded information. Fortunately, since the analytic expression is known, the Root Mean Square Bandwidth can be computed, and it consists the choice of how the bandwidth is computed.

The Root Mean Square bandwidth is defined as [Rihaczek, 1969, Zhou et al., 2014]

B rms = R f 2 |G( f )| 2 d f R |G( f )| 2 d f 1 2 , (4.11)
where G is the PSD. This expression is also called Gabor bandwidth [Amoroso, 1980] if the PSD is normalized such that the denominator is equal to 1.

In our framework, even with an analytic expression of the PSD, the Gabor bandwidth is not easy to compute. However, by observing the theoretical PSD plotted in sub-figure 4.10a, we can remark that it is relatively flat for high constellations. We thus make the assumption that we have a quasi-rectangular spectrum in the interval -B 2 , B 2 as shown in figure 4.11 and we can deduce the following result:

B = 2 √ 3B rms , (4.12)
2 The Wiener filter is able to estimate a random process by extracting it from an observed noisy process, which is somehow correlated to the signal of reference [START_REF] Dogariu | An insightful overview of the wiener filter for system identification[END_REF]. It is denoted h W and it is usually expressed statistically as

h W = R -1 p
where R = E[ss H ] is the covariance of the transmitted signal s and p = E[sy H ] is the cross-correlation vector between s and y, which is the received signal. This filter could be described as the inverse of the matched filter as its computation in the Fourier domain coincides to the division of the spectrum of y with respect to the spectrum of s. This results to potential problems when it comes to dividing with fluctuated spectra such as the ones of communication codes. where B is the signal bandwidth and B rms can be easily computed as we have the analytic expression for the PSD under the assumption of a rectangular spectrum. This means that we can transform equation (4.11) into a discrete formulation and compute it over an interval of frequencies from equation (4.10), as long as the PSD has an approximately rectangular shape.

G(f) f B 2 v -B 2
Proof.

B rms = R f 2 |G( f )| 2 d f R |G( f )| 2 d f 1 2 ≈       B 2 -B 2 f 2 v 2 d f B 2 -B 2 v 2 d f       1 2 =       B 2 -B 2 f 2 d f B 2 -B 2 d f       1 2 = B 2 √ 3
where v is the maximum value of G.

. Ambiguity Function of CPM signals

Another key for radar performance is the ambiguity function. Figure 4.12 represents the ambiguity function of a CPFSK code with respect to the delay and Doppler. The vertical left plot corresponds to the zero-Doppler cut, thus providing the auto-correlation of the code as no Doppler shift is applied, while the horizontal bottom plot is the zero-delay cut, resulting in a sinuscardinal function along the Doppler axis. The creation of the CPFSK ambiguity function is performed by using a single CPFSK signal for the Doppler processing. It also resembles to the ambiguity function of a Gold phase code illustrated in figure 2.9 in chapter 2. It presents a narrow mainlobe in range and Doppler, and a relatively uniform sidelobe level. Figure 4.13 provides a zoom on the mainlobe of the ambiguity function. It can be observed that the target will be detected only when applying the matched filter for a Doppler hypothesis close to the true target Doppler shift. In fact, the width of the Doppler mainlobe is of the order of 1 T p . This could be potentially a disadvantage for the radar performance, as targets may not be detected, if the matched filter is only applied at zero-Doppler, which is often the case (as opposed to the chirp range-Doppler coupling, which in this case turns out to be beneficial). In that case, a target may suffer from a strong loss after compression if its Doppler shift is sufficiently large (equal or larger than 2 T p ), while it will remain detectable for Doppler shift smaller than roughly 2 T p . However, the advantage of working in a radar/SAR framework is that the duration of a pulse is usually some micro-seconds and the typical intra-pulse Doppler is dependent on the inverse of the pulse duration. In the SAR framework, the maximum Doppler value is provided by

f max d = 2v plane c f c sin( π 2 ) = 8060Hz ≪ 1 T p = 208333Hz, (4.13)
where v plane = 124m/s the plane radial velocity, f c = 9.75GHz the carrier frequency, T p = 4.8µs, and the maximum angle that is applied for the maximum Doppler is θ = π 2 . This calculation shows that the Doppler shift remains very low, and the pulse compression should not be sensible to such a shift. It can thus still be possible for the SAR application to use only a single matched filter at zero Doppler even when using CPFSK codes. This behavior is quite different from the chirp case, as in the latter one such a problem is not created because of its linear behavior for which the target can mostly be observed.

. Different Signal Candidates for a SAR-Communication System

The pool of waveforms used in communication systems is wide. To select an appropriate candidate for our joint radar-communication system, we consider here some of the most popular communication codes in the literature that we will compare to the CPFSK codes: ⇝ Phase Shift Keying (PSK) codes; ⇝ Phase Shift-Keying codes embedded in the phase of a chirp (LFM-PSK); ⇝ Orthogonal Division Frequency Multiplexing (OFDM) codes.

All of these codes have the ability to transmit information. We assume that, in the scene illuminated by the radar, communication receivers are present, but we will not evaluate their demodulation and decoding capacities.

Before debating on the choice of the waveform for our radar application, we will first provide a short review of the signals mentioned above. The characteristics of the different codes used for this study are listed in table 4.1. These parameters are chosen in such a way that the bandwidth and the pulse duration are the same, or very close. We decided to preserve the value of these parameters as they have a fundamental role in radar processing and it provides a framework for signal comparison.

Here a binary constellation was considered for the PSK code, because in such a way we can assure that the bandwidth is 1 T c . Afterwards, the LFM-PSK code is defined such that its bandwidth is the same as for the PSK code. This is also the reason why their chip duration is different compared to CPFSK. For OFDM, the chip duration simply coincides with the pulse duration, while the number of carrier will be provided by the bandwidth. In addition the number of bits embedded in the phase changes, as the pulse duration is identical, and the chip duration is not the same among all the codes. All these parameters are linked to each other so it is difficult and sometimes impossible to keep them identical. To summarize, the parameters that change between the codes are the constellation, the number of embedded bits and the chip duration.

. Phase Shift Keying (PSK)

Phase modulation is widely used in the communication field, for applications such as Global Positioning System (GPS), WLANs/Wireless Fidelity (WiFi), Worldwide Interoperability for Microwave Access (WiMAX) [Grami, 2016]. It is created by embedding information symbols in the phase, and it is also called phase shift keying (PSK). It is a memoryless modulation method, as opposed to the continuous phase modulation codes, introduced in the previous section 4.1, which means that the current phase value depends only on the current information symbol and it is independent from the previous or the forward one. In a PSK scheme, the pulse is divided into different chips. Each chip will be assigned to a different phase drawn from the constellation according to the information to be transmitted. The most basic PSK example is the Binary Phase Shift Keying (BPSK) which transmits one bit (0 or 1) per symbol and for which the two different symbols are separated by a phase of π. When the constellation exceeds 2, then each symbol contains log 2 (M) bits. The following equation (4.14) expresses how the symbols are embedded in the phase [Grami, 2016, Proakis, 1983]: and ∀k, I k = 1, M , and T p = N chip T c . Figure 4.14 presents the phase and the spectrum of a BPSK code. For an ideal BPSK code, the phase changes are instantaneous, which creates a sinus-cardinal spectral behavior. Moreover, figure 4.15 illustrates the ambiguity function of a PSK code, along with the zero-delay and zero-Doppler cuts. The energy of the mainlobe is concentrated in the neighborhood of zero-delay and zero-Doppler, as it can be observed from the zoom on the mainlobe of the ambiguity function in figure 4.16. The level of the sidelobes are quite uniform with slightly lower sidelobes at the zero-Doppler cut. A potential target, when applying the classical MF at zero-Doppler, can be detected only if it presents a Doppler shift close to zero.

s PSK (t) = N chip -1 ∑ k=0 ϕ PSK k Rect t -(k + 1 2 )T c T c , ( 4
PSK modulation can be used to encode specific sequences of symbols, for instance Gold codes, that we introduce in the following.

Special Case: Gold Codes The Gold codes constitute a specific family of binary codes, firstly introduced by Robert Gold [Gold, 1966] in order to be able to provide low cross-correlation codes [START_REF] Arlery | Utilization of spreading codes as dedicated waveforms for active multi-static primary surveillance radar[END_REF]. They are widely used in the communication field, for instance for Code-Division Multiple Access (CDMA) [START_REF] Dinan | Spreading codes for direct sequence CDMA and wideband CDMA cellular networks[END_REF]], and in geolocalization as in GPS [START_REF] Kaplan | Understanding GPS/GNSS: principles and applications[END_REF]. They are derived from a selected pair of MLBS3 of the same order N and the same sequence length 2 N -1 [START_REF] Arlery | Utilization of spreading codes as dedicated waveforms for active multi-static primary surveillance radar[END_REF].

The Gold family is computed by taking the modulo 2 sum of the first sequence with all the possible shifted versions of the second one [START_REF] Arlery | Utilization of spreading codes as dedicated waveforms for active multi-static primary surveillance radar[END_REF]. Each resulting sequence is a binary Gold sequence and the Gold family is composed of 2 N + 1 sequences. However, even if the Gold codes are a derivation of the MLBS code, the reciprocal is not true.

. Phase Shift-Keying Codes Embedded in Chirp Phase (LFM-PSK)

To allow the entrance of the widely used chirp in the communication world, its phase can be slightly changed in order to embed communication symbols. In [START_REF] Bekar | Realization of a joint MIMO radar and communication system using a PSK-LFM waveform[END_REF], Nowak et al., 2016, Hassanien et al., 2016], the authors utilize the chirp phase as a mean to transmit phase shift keying symbols:

s LFM-PSK (t) = exp j ϕ chirp (t) + M-1 ∑ k=0 ϕ PSK k Rect t -(k + 1 2 )T c T c , (4.16)
where the phase of a chirp is expressed as ϕ chirp (t) = π B T t 2 , and ϕ PSK is the phase as in equation (4.14).

Specifically in [START_REF] Bekar | Realization of a joint MIMO radar and communication system using a PSK-LFM waveform[END_REF], the bandwidth of the embedded PSK codes is chosen, so that it is contained within the chirp spectrum. Here we consider the same bandwidth for both chirp and PSK. It is defined so that it can be comparable to the PSK codes, by having the same parameters.

Figure 4.17 illustrates the phase and the spectrum of such a code. The phase of this code is a combination between the quadratic phase of the chirp and the phase jumps of the PSK. The spectrum is neither a sinus-cardinal nor quasi-rectangular, but it is a combination of both chirp and PSK spectra. It is possible to make the spectrum converge towards the chirp one, as in [START_REF] Bekar | Realization of a joint MIMO radar and communication system using a PSK-LFM waveform[END_REF], or the PSK one, but here we constraint the parameters so that they are as close to the PSK ones as possible, which makes them comparable. Figure 4.18 illustrates the ambiguity function of a LFM-PSK, along the zero-Doppler and zero-delay cuts. We can remark that the knife edge behavior of the chirp ambiguity function is slightly observed on the sidelobes far away from the mainlobe, but the energy is not preserved when large Doppler shifts are applied on the signal. The energy of the mainlobe is mainly concentrated in the vicinity of zero-delay and zero-Doppler, but still keeps the known delay-Doppler coupling of the chirp between the delay and Doppler axes for a small amount of samples, and with a reduced energy away from (0,0). This effect can be observed better on figure 4.19, that provides a zoom of the LFM-PSK ambiguity function on the mainlobe.

. Orthogonal Division Frequency Multiplexing (OFDM) Codes

Orthogonal Frequency Division Multiplexing is a modulation technique that embeds information over multiple frequency sub-carrier channels. It is widely used in the communication field [Proakis, 1983], in domains such as Digital Audio Broadcasting (DAB), Digital Video Broadcasting (DVB), Digital Subscriber Line (DSL), WLAN, mobile Wi-MAX networks [Grami, 2016], and even in the joint radar-communication field as observed in chapter 3.

In OFDM, information symbols are carried by closely spaced orthogonal sub-carrier signals that are transmitted simultaneously in the time domain, allowing the transmission of several symbols at once. A symbol is defined according to a constellation M that can be PSK or QAM. Each one of them is carried by a specific sub-carrier frequency, that is defined as where k corresponds to the index of the sub-carrier.

f k = k T p , (4.
The OFDM signal can be expressed as [Proakis, 1983] s (4.18) where c k are the embedded symbols with k ∈ 1, N chip .

OFDM (t) = N chip ∑ k=1 c k e 2 jπ f k t , for 0 ≤ t ≤ T p ,
The OFDM advantages lie in the orthogonality of the sub-carriers, which prevents inter-symbol interference at the reception point [START_REF] Nee | OFDM for wireless multimedia communications[END_REF], the high data transmission in multipath channels and the implementation ease of coding and decoding from (4.18) with a simple FFT. However, to operate in a multi-path channel configuration, a cyclic prefix needs to be introduced, whose length depends on the channel length. It enables to "absorb" the different delays of the propagation channel, so that the remaining of the signal is clean from any interference coming from the previous symbol. Figure 4.20 illustrates this effect. Despite being a drawback in terms of data rate, it can become advantageous for clutter cancellation purpose in radar applications [Poullin, 2010].

Figure 4.21 provides the phase, the spectrum and the amplitude of an OFDM code. As expected the spectrum of such codes is close to rectangular as we attribute a symbol to each orthogonal subcarrier. The phase of an OFDM symbol is random and takes values in [0, 2π]. The amplitude for OFDM codes is not constant, as it has been stated, unlike the other signals presented so far. Finally, figure 4.22 illustrates the ambiguity function of an OFDM code without cyclic prefix, along with both zero-delay and zero-Doppler cuts. As for the previous ones, the ambiguity function of this code concentrates the mainlobe energy in the neighborhood of zerodelay and zero-Doppler. Thus, if a target is submitted to a large Doppler shift, then it cannot be detected with a single matched filter at zero-Doppler, and a bank of filters would be required. This can be remarked easier from figure 4.23, where the zoom of the OFDM ambiguity function is illustrated.

. Comparison of the CPFSK Codes with Other Communication Codes for SAR Image Generation

After studying the characteristics of the previous waveforms, we can now compare these signals with the CPFSK codes in order to justify our waveform choice, which is to our point of view more suitable for the creation of the SAR-communication system.

From a communication point of view, the only constraint that we impose to our system is the possibility of information transmission, which is natural for any signal introduced before, and potentially be able to maximize the data rate through our system configuration. From a SAR/radar point of view, two main conditions are desired [Tierney, 2019]: constant amplitude, spectral containment.

Constant envelop of a signal helps the amplifier to function in a saturated mode, preventing non-linear distortions, that would impact the transmitting signal. Considering a constant amplitude not only limits this effect, but it also provides an increased power efficiency as we make use of the full amplitude during the pulse duration [Tierney, 2019, McCormick et al., 2017].

Spectral containment aims at limiting the out-of-band spectral energy as much as possible to prevent polluting neighboring bands. Hence, the spectrum of the waveform that we are going to choose should ideally be close to a rectangular shape. For example, a spectrum close to the chirp one would be a good selection. Such a spectrum, additionally to having a low out-of-band energy, enables to avoid a too brutal frequency cut at the transmission end, that could cause amplitude modulation.

Among the previously introduced signals, all have constant amplitude, except for the OFDM. The non-constant envelop of OFDM creates a high peak-to-average power ratio (PAPR) [START_REF] Paredes | The problem of peak-toaverage power ratio in OFDM systems[END_REF], which can degrade the hardware performance [START_REF] Ma | Joint radarcommunication strategies for autonomous vehicles: Combining two key automotive technologies[END_REF]]. Yet some strategies were developed to limit the PAPR for example by using Golay sequences [START_REF] Davis | Peak-to-mean power control in OFDM, golay complementary sequences, and reed-muller codes[END_REF], but this is beyond the scope of our work. We will thus not consider OFDM for our system due to its amplitude modulation, but we believe that it has its place in the radarcommunication world.

Studying the spectra, it is important to mention that the behavior of the phase is crucial. To sum up, if the phase is continuous, then the spectrum is better contained in the bandwidth. Among the introduced waveforms, the best spectrum is given by OFDM. It is well-contained by definition, so the out-of-band energy is negligible. But as these codes are not considered for our system, we now compare the rest of the spectra of the unimodulary signals. Intuitively from the individual study of the waveforms in previous sections, two good candidates seem to be the LFM-PSK and the CPFSK codes, because the sidelobes of their spectrum are relatively low, which means that the out-of-band energy is also reduced.

The candidate spectra are compared with respect to the two following criteria:

Out-of-band Power (P ob ) value;

Effect on the autocorrelation of a spectral filtering at two different frequency cuts.

Waveform Frequency Bound f ob (MHz) BPSK 250.9 LFM-PSK 327.2 CPFSK 169.5 Chirp 159.5

Table 4.2: The frequency bounds of the out-of-band power for a 99% power of the spectrum. Ideally these bounds should not exceed the half-bandwidth value ( B 2 = 176MHz as the spectrum is centered) so that there is no spectral loss. These bounds are also illustrated in Figure 4

. Out-of-band Power

First, the out-of-band (P ob ) power is calculated. It is expressed as [START_REF] Wolf | Modulation Theory[END_REF]]

P ob ( f ob ) = |ν|> f ob |S(ν)| 2 dν ∞ -∞ |S(ν)| 2 dν , f ob ≥ 0,
where S(ν) is the signal spectrum and f ob is the positive frequency limit after which we consider that the spectral energy is out-of-band. Figure 4.24 shows an out-of-band power of P ob = 0.01, which means that 99% of the power is inside the spectrum at the frequency limit f ob . It illustrates the spectrum of each signal introduced above along with the chirp spectrum that is chosen as a reference for visual comparison. Once all the spectra are plotted together, we can now visually remark that the CPFSK codes have a well-contained spectral energy compared to the other codes. This can also be remarked in Table 4.2, where the out-of-band power bound values f ob are displayed for each code. Ideally, this power should not exceed the half-bandwidth value ( B 2 = 176MHz as the spectrum is centered) in order to avoid losing spectral energy. This is the case only for the reference (chirp) and the CPFSK codes, which thus provide the best performance for this criterion. For the remaining constant amplitude signals, the BPSK code has a sinus-cardinal-shaped spectrum, and the out-of-band energy is noticeable, when observing figure 4.24, and also exceeds the bandwidth value. Last to the ranking comes the LFM-PSK codes, for which the frequency bound f ob , displayed in Table 4.2, is almost twice as the chirp and the CPFSK bounds, so the out-of-band energy is unacceptable. The small windows on each sub-figure is a zoom at the peak of the mainlobe to visualize the loss after the compression.

. Effect of spectral filtering on the resulting autocorrelations

It could be argued that to avoid out-of-band power, one could simply filter them at transmission to comply with the official imposed spectral containment. But when cutting the spectrum of such codes at given frequencies, energy loss can occur after matched filtering with the input signal and degrade the radar performance. To quantify this, two band-pass filters are considered, one at 1.2B which removes much of the out-of-band energy, and a second one at 2B. Table 4.3 presents the energy loss of the mainlobe peak after range compression with the initial non-filtered signal for the two cuts. We can observe that for the 1.2B cut, the LFM-PSK code shows a loss of 1.14 dB on the autocorrelation peak. Around 0.5 dB less loss is shown by the BPSK and Gold codes. As expected, the best results are provided when using the CPFSK signals, as in the worst case scenario there is a loss of 0.1 dB in ideal conditions. It is also worth mentioning that when it comes to the 2B cut, there is no loss for the CPFSK codes. To visualize these remarks, figure 4.25 provides the auto-correlation of the filtered signals for both cut 1.2B and 2B with the unfiltered input signal. Zoomed images on the peak are also provided, so that the energy can also be observed. Moreover, figure 4.26 provides the amplitude of the filtered signals, in order to observe to which extent the codes are distorted. Especially for the LFM-PSK and the PSK, the amplitude distortion is quite visible. This is created by the loss of spectral energy after the cut. As the CPFSK lose little spectral energy, the resulting amplitude modulation is much smaller and will not create severe problems compared to the rest of the codes at the transmission end. Based on these observations on amplitude and spectral containment, we selected the CPFSK to pursue our study on a dual function system.

. A CPFSK System from a Communication Point of View

The radar-suitable characteristics of the waveform for our joint SAR communication system have been already partially provided here. The signal processing aspect to obtain a satisfying image will be addressed in detail in the following chapters. Even if an extensive study of the communication performance is out of the scope of this work, this subsection provides some insights on how the system could be employed from a communication point of view. Inserting a radar system into a communication chain implies considering a bistatic system where the airborne radar transmits and receives a set of fully known waveforms while the communication receiver will deal with unknown signals that must be decoded to retrieve the useful information. Two different possible solutions are suggested here for the decoding procedure.

Firstly, from the CPFSK encoding procedure, it appears that N chip information symbols I = [I 1 , I 2 , . . . , I N chip ] are embedded continuously in the CPFSK phase during the pulse duration. Each symbol is mapped to k bits according to the given constellation of size M = 2 k . This enables to transmit up to kN chip bits in the pulse duration, corresponding to an overall N CPFSK tot = 2 kN chip number of different CPFSK waveforms. Decoding these information symbols from the direct path signal at reception can be performed using the Viterbi algorithm [START_REF] Viterbi | Error bounds for convolutional codes and an asymptotically optimum decoding algorithm[END_REF]. The Viterbi's decoding iterative technique is not trivial using CPFSK codes as the phase values need to be unrolled in order to determine the different states. This first solution maximizes the data rate but will be efficient only if the Signal-to-Noise Ratio at the chip level is sufficient. Besides it requires a quite demanding decoding procedure.

The second possible solution to decode the signals consists in considering only a subset of size N sub ≪ N CPFSK tot of CPFSK waveforms. In that case, each waveform of the subset can be directly assigned to a communication symbol corresponding to a specific binary word of log 2 (N sub ) successive bits. They are chosen from a predefined set of signals that are portrayed as symbols. Hence, the communication system uses the knowledge of this bank of waveforms at reception, in order to perform correlations between the received signal and all the waveforms from the set, allowing in such way to find the symbol with maximum likelihood. This second method provides an easier decoding procedure, but the quantity of information that can be transmitted is limited to log 2 (N sub ) bits compared to kN chip bits for the first one.

Choosing one of these two solutions thus resorts to finding a trade-off between the data rate, and the simplicity and quality of the decoding procedure. For applications where the signal-tonoise ratio is sufficient and a high data rate is desired, the first solution is preferable, where the code changes every PRF. On the contrary for low signal-to-noise ratio, the second solution should be adopted, at the price of a lower data rate, where the code could change every PRF.

. Conclusion

In this chapter, the continuous phase modulated family was examined and the special category of CPFSK is chosen to represent the waveform for our SAR-Communication system, because of their spectral properties. However, when observing figure 4.25, the sidelobe level of all the codes are quite high, and can potentially mask targets. This is the problem we will tackle in the next chapter 5, where sidelobe suppression will be performed through the help of mismatched filtering.

CHAPTER 5 Mismatched Filter Optimization for Sidelobe Reduction

In Chapter 2, we introduced the notion of range compression via the MF that enables to maximize the SNR of the target, while providing a range resolution of c 2B . In many radar applications, the chirp is frequently used. It has good auto-correlation properties, as it provides relatively low sidelobes. As seen in Chapter 4, it is not the case for communication codes like CPFSK codes. High sidelobes after matched filtering can potentially degrade the radar performance when using such codes. More precisely, they may mask targets or threats in detection systems or deteriorate a radar image for imaging radars. Thus, the need of using a different range compression technique arises, that will not optimize the SNR but on the contrary will control the sidelobes [START_REF] Blunt | Overview of radar waveform diversity[END_REF].

Sidelobe reduction has been studied for years, and the method used to achieve it is dependent on the different considered waveforms and applications. An easily implemented and early way of reducing sidelobes consists in applying weighting windows, such as Hanning, Hann, Chebyshev and others [Harris, 1978], which are commonly used for chirps. However, such a method reduces the sidelobe level at the cost of a wider mainlobe, which implies a degraded range compression, which may be unacceptable in specific configurations. To preserve resolution, solutions include mismatched filters, that are usually defined as the solution of an optimization problem [START_REF] Baden | [END_REF]Cohen, 1990, Stoica et al., 2008]. Authors in [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]] also showed that the optimization problem can be convex, and that some additional convex constraints can be included, such as a constraint on the loss-in-processing gain that enables to control the loss in SNR with respect to the matched filter.

The cost function and the associated constraints are dependent on the application. The most common optimization criteria are the Peak-to-Sidelobe Level Ratio (PSLR) and the Integrated Sidelobe Level (ISL) [START_REF] Stoica | Transmit codes and receive filters for radar[END_REF], De Maio et al., 2011, Cilliers and Smit, 2007, Aubry et al., 2016, Baden and Cohen, 1990, Aittomäki and Koivunen, 2015]. As the aim of our work is to create a SAR-communications system, we want to create an image as clean as possible which means 82 Chapter 5. Mismatched Filter Optimization for Sidelobe Reduction without blurring effects due to sidelobes. The ISL criterion will be thus chosen to be minimized in our framework in order to limit the sidelobe energy on the resulting image. The definition of the optimal filter that minimizes the sidelobe level energy will be reminded in section 5.1.

However, at the reception side, the signals may differ from the signal considered for the optimization since

• the signal may undergo a Doppler shift, and/or

• the received signals are sampled in a way that may not match the sampling grid of the transmitted signal due to an unknown delay.

In such configurations, the resulting optimal filter computed is no longer optimal, leading to an increase of the sidelobe level, and a reduction of the processing gain. In the following, we tackle this problem for an unknown intra-pulse Doppler shift and an unknown delay shift that can occur on the received signals. In both cases, the cost function is redefined in a continuous way and calculated in order to provide an expression applicable to sampled signals. The two different configurations are presented in 5.2.1 and 5.2.2 respectively. Finally, being able to compute mismatched filters for large scale problems is crucial in our work as we want to transmit different information at each pulse, to maximize the data rate, and obtain an image with sufficient resolution. Hence, to make this feasible from a computational cost point of view, we develop a fast algorithm in section 5.3.

In the following sections and sub-sections, we use the parameters of the following table 5.1 for the simulation results, in order to define the signal and the filter characteristics. The values in table 5.1 are chosen in such a way to provide relatively short signals, so that the filter computation is performed in a reasonable time. For the last section 5.3, where we introduce the fast algorithm, we use different parameters that match better with the ones used for real SAR applications as studied in the next chapter 6.

Parameters Value

Bandwidth

. Mismatched Filter Optimization Problem

The mismatched filter can be classically defined as the solution of a minimization problem [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF].

Let us denote by s, the N-length sampled signal expressed as

s = s 1 s 2 . . . s N T , (5.1)
where . T is the transpose operator. At the reception, the signal undergoes range compression with a Mismatched Filter (MMF), different from the transmitted signal, in order to optimize a different criterion than the SNR. This MMF will be denoted q 0 . The length K of the mismatched filter q 0 ∈ C K can be equal to or greater than the length of the signal (K ≥ N). In the following we will see that performance-wise it is preferable to use filters longer than the signal length, but at the cost of a higher complexity for the filter computation. The compression output can be expressed in a matrix formulation as [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]

y = Λ Λ Λ * K (s) q 0 , (5.2) with q 0 = q 0 1 q 0 2 . . . q 0 K T ,
where Λ Λ Λ K is a convolution matrix of size (K + N -1) × K, defined by

Λ Λ Λ K (s) =                          s N 0 . . . .
                         . (5.3)
If the filter is chosen as q 0 = s, then the output of the compression coincides with the classical matched filter, that maximizes the SNR. But q 0 can be chosen so as to optimize various criteria, the most common ones being the PSLR [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF], Blunt et al., 2006, Stoica et al., 2008, Cilliers and Smit, 2007, Baden and Cohen, 1990], and the ISL [START_REF] Stoica | Transmit codes and receive filters for radar[END_REF], De Maio et al., 2011, Cilliers and Smit, 2007, Aubry et al., 2016, Baden and Cohen, 1990, Aittomäki and Koivunen, 2015, Rabaste and Savy, 2015]. Both criteria refer to sidelobe behavior, so the cost function of the minimization can be defined as (5.4) where F is a diagonal matrix of ones except for some zero values which correspond to the mainlobe position indices, and || • || 2 p defines the ℓ p norm of the vector. Defining F in such a way enables to take into account only the sidelobes.

f (q 0 ) = ||Fy|| 2 p ,
Considering the PSLR criterion, as defined in equation (2.37), results in setting p = ∞ in the cost function (5.4), as using the infinity norm (p = ∞) implies that only the largest sidelobe is taken into account. On the contrary, the ISL criterion is obtained by setting p = 2 in the cost function (5.4), in order to consider the energy summation of all sidelobes, as expressed in the ISL definition given in (2.36).

In our SAR-communication framework, the aim is to minimize the sidelobe energy, thus the cost function of the optimization problem is defined as a ℓ 2 -norm (p = 2) [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]:

min q ||Fy|| 2 2 , s.t. s H q 0 = s H s.
(5.5)

The constraint that is added to the problem allows to discard the trivial null solution, and it is necessary for the problem formulation. This minimization problem (5.5) can be analytically solved [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF], and the solution can be obtained by applying the Lagrange multipliers method 1 . It is expressed as:

q analytic (s, M) = (s H s) M -1 s s H M -1 s , ( 5.7)
1 The Lagrangian function for the optimization problem (5.5) is defined as

L (q, λ ) = q H Mq + λ (s H q -s H s),
where λ is the Lagragian multiplier.

Taking its gradient with respect to the filter q and finding its root provides the optimal filter expression with respect to the dual variable λ :

∇ q L (q opt , λ ) = 0 ⇐⇒ 2Mq opt + λ s = 0 ⇐⇒ q opt = - 1 2 λ M -1 s. (5.6)
Now the λ value can be found by maximizing the dual function L (q opt , λ ), which is a concave function.

This maximization provides the final expression of the filter, which is expressed as:

L (q opt , λ ) = - 1 4 λ 2 s H M -1 s -λ s H s ⇒ ∇ λ L (q opt , λ ) = - 1 2 λ s H M -1 s -s H s,
with optimal λ value when taking the root of the gradient, expressed as

λ opt = - 2s H s s H M -1 s .
Finally, by inserting λ opt in the filter expression in equation (5.6), we have the final formulation of q analytic given in equation (5.7). where

Classical Formulation of the

M = Λ Λ Λ T K (s)FΛ Λ Λ * K (s).
This analytic solution provides the optimal sidelobe level but at the price of a non-controlled loss-in-processing gain, which is potentially unacceptable.

In order to avoid a too large loss-in-processing gain, a second constraint is added in the optimization problem to control this gain after compression [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF].

The minimization problem can be thus expressed as [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]

min q ||Fy|| 2 2 , s.t. s H q 0 = s H s q H 0 q 0 ≤ 10 β 10 s H s, (5.8)
where β is a positive constant in dB that expresses the loss-in-processing gain. It represents the maximum tolerated loss for the resulting filter with respect to the matched filter [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]. This problem will be considered for the rest of our work.

The new formulation of the optimization problem in equation (5.8) does not have an analytic solution. However, both constraints are convex. Indeed, the first one is convex because it is linear and the second one is a positive semi-definite quadratic form. Moreover, the cost function is convex, as it is defined as an ℓ 2 -norm. Thus, globally the optimization problem is convex, meaning that any local optimum of the optimization problem is necessarily a global optimum [START_REF] Boyd | Convex optimization[END_REF]. The solution can be found by using a convex solver, such as the CVX: Matlab Software for Disciplined Convex Programming [START_REF] Grant | CVX: Matlab Software for Disciplined Convex Programming[END_REF]. It is used for instance in [START_REF] Chatzitheodoridi | A mismatched filter for integrated sidelobe level minimization over a continuous Doppler shift interval[END_REF], Chatzitheodoridi et al., 2021].

In order to visualize the difference between the filters, figure 5.1 illustrates the MF and the MMFs with and without the second constraint. It can be observed that the matched filter presents the highest sidelobes, whereas the compression with the analytic filter provides the lowest sidelobe level at the cost of a loss-in-processing gain of 3.1 dB. The loss-in-processing gain of the mismatched filter when the second constraint is considered is set to 1 dB. The effect of this loss can be observed on the values of the ISL. For the matched filter, the ISL value is -0.6 dB, followed by the MMF of equation (5.8) with a value of -4.4 dB and the analytic one with ISL equal to -5.4 dB. Thus the larger the loss during the processing the better the ISL result. In the following, this loss is set to 2 dB, in order to be able to control the loss-in-processing gain and at the same time not constraint it too much. The previous expressions are based on the ISL criterion, as it is the one suitable for our work. However if one considers the PSLR criterion, as defined in equation (2.37), it results in setting p = ∞ in the cost function of equation (5.4), as using the infinity norm (p = ∞) implies that only the largest sidelobe is taken into account. It thus allows the minimization of the largest sidelobe. An example of the optimal filter output computed in this configuration is illustrated in figure 5.2 for a CPFSK code. It shows a very uniform level of sidelobes, with a lower maximum sidelobe level than for the MF.

Note that up to now, the signal s is assumed to be perfectly known. In the following, we will consider an alternative of this optimization problem (5.8), so that it takes into consideration a potential signal mismatch. It will minimize the ISL over a continuous interval, which can be either a Doppler or a delay interval, in order to improve the compression performance of off-grid shifted signals.

Mismatched Filter for the Integrated Sidelobe Level Minimization with Signal

Mismatch 87

. Mismatched Filter for the Integrated Sidelobe Level Minimization with Signal Mismatch

The classical formulation of the mismatched filter expressed in equation (5.8) provides the optimal filter that minimizes the ISL for the corresponding sampled transmitted signal given as input. After the signals are back-scattered and return to the antenna, they may differ from the reference signal, because they could undergo for example an intra-pulse Doppler shift or a delay shift, not multiple of the sample period. This signal mismatch can result in higher sidelobes after compression, as the mismatched filter computed from (5.8) is no more optimal on the Dopplershifted or delay-shifted signal. The next two subsections handle firstly a Doppler mismatch and secondly a delay mismatch with a mismatched filter optimized with respect to the ISL criterion.

. Mismatched Filter Optimization of Signal Mismatch due to Doppler Shift

The aim of this subsection is to create a mismatched filter that minimizes the integrated sidelobe level for an unknown intra-pulse Doppler shift. First, the mathematical background is provided and afterwards associated simulation results are presented.

. Mathematical Definition

Let s be any transmitted complex signal vector of length N, as defined in equation (5.1). In the presence of a moving source or target, the received signal is subject to a Doppler shift denoted f d . Neglecting the Doppler effect on the baseband signal and thus modeling it only via the shift on the carrier frequency, the i th component of the received signal r f d can be defined as the multiplication between the i th component of s and the Doppler shift applied to the signal after reception, and expressed as:

r f d i = r f d (t i ) = s i e 2 jπ f d t i , ∀i ∈ 1, N ,
(5.9)

where t i = i f s , with the last sampling point corresponding to the pulse duration T p . If we apply the mismatched filter optimized with respect to s as described in the previous section 5.1, it appears that the sidelobe level dramatically increases with non-zero f d , as it can be seen in figure 5.3, where a more than 2 dB of ISL augmentation is remarked. It thus appears that the MMF is not robust to Doppler mismatch, and it would be of interest to compute a different mismatched filter that can cope with unknown Doppler shifts over a given Doppler interval.

The compression output for the Doppler-shifted signal r f d can be defined in the same way as in expression (5.2), as (5.10) where q f d is the mismatched filter of size K (K ≥ N) to be optimized, and where Λ Λ Λ * K (r f d ) is defined as in equation (5.3).

y( f d ) = Λ Λ Λ * K (r f d ) q f d ,
If q f d is optimized with respect to r f d , it will not be robust to other values of Doppler shift. A solution proposed in the literature is to consider a discrete set of k Doppler shifts { f d 1 , . . . , f d k } and optimize the PSLR or ISL over the whole set [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF], thus solving the following (5.11)

In [START_REF] Rabaste | Robust mismatched filter for off-grid target[END_REF], the authors provide a different solution adapted only to the ISL criterion for signals that undergo unknown parameters with values within a given resolution cell, applied to inter-pulse Doppler and to delay-shifted chirp signals. We propose here to extend this solution in a different framework, where we will consider the intra-pulse Doppler on a predefined interval. Assuring that the Doppler shift will be modeled as a random variable, the aim is to obtain the optimal filter with respect to the expected ISL over all the possible Doppler shifts. In this setting the minimization problem can be expressed as [START_REF] Chatzitheodoridi | A mismatched filter for integrated sidelobe level minimization over a continuous Doppler shift interval[END_REF]: (5.12) where the notations coincide with those adopted in equation (5.8), and E f d • denotes the expectation over the random variable f d . Recall that both constraints are convex as mentioned in section 5.1. The difference between problem (5.12) and the definition of the classical mismatched filter in (5.8) is the application of the expectation on the norm for the Doppler values.

min q E f d ||Fy( f d )|| 2 2 s.t. s H q = s H s, q H q ≤ 10 β 10 s H s,
Let us consider here that the Doppler frequency is uniformly distributed over the interval

Ξ = - η T p , η T p , with η ∈ R * + .
For example, note that η = 1 2 corresponds to the -3dB width of the main lobe of the zero-delay cut of the ambiguity function, while η = 1 corresponds to the full main lobe, that stretches up to the first null on each side. Note that the interval Ξ is defined to be symmetrical, but it could be defined as non-symmetrical by adapting the problem accordingly.

The cost function of expression (5.12) needs to be further developed in order to be analyzed and implemented. By injecting the compressed signal expression (5.10) into the optimization problem (5.12) and by noting that F H F = F, we obtain under the assumption of a uniform distribution of the Doppler shifts f d over Ξ that:

E f d ||Fy( f d )|| 2 2 = T p 2η Ξ ||Fy( f d )|| 2 2 d f d ∝ q H Ξ Λ Λ Λ T ( f d )FΛ Λ Λ * ( f d )d f d q.
(5.13)

Let M( f d ) = Λ Λ Λ T ( f d )FΛ Λ Λ * ( f d ) and M Ξ = Ξ M( f d )d f d , then E f d ||Fy( f d )|| 2 2 ∝ q H M Ξ q = ||M 1 2 Ξ q|| 2 2 .
(5.14)

By observing equation (5.14), the cost function remains convex since it is a norm. As mentioned before, the constraints are also convex, which leads to a convex optimization problem, for which any local optimum is necessarily a global one. Moreover, an explicit expression for the (k, l) element of M Ξ can be obtained as

(M Ξ ) k,l = 2η T p M k,l (0) sinc 2π N η(k -l) .
(5.15)

Equation (5.15) provides the expression of the (k, l) entry of matrix M Ξ . The calculations that lead to this result can be found in Appendix A.1. Remark that this matrix is similar to the matrix whose eigenvectors are the Discrete Prolate Spheroidal Sequences (DPSS) [Slepian, 1978], which are used in signal processing and communications applications. Indeed the (k, l) component of the DPSS generating matrix is expressed as:

P k,l (ξ ) = ξ -ξ e (2 jπ(k-l) f ) d f = 2ξ sinc 2π(k -l)ξ ,
where k, l ∈ 1, K .

Finally, the matrix M Ξ can be written in matrix form as the element-wise multiplication of two matrices, as

M Ξ = N T p M(0) ⊙ P( η N ), (5.16)
where ⊙ corresponds to the element-wise multiplication. M Ξ thus corresponds to the element-wise multiplication of the zero-Doppler cost function and the Slepian generating matrix. It depends on the considered Doppler interval through the variable η and on the signal characteristics, such as the pulse duration T p , the signal length N and its auto-correlation. By injecting (5.16) into (5.14), the final form of the cost function is obtained and shall be minimized. By solving the new optimization problem (5.12), an optimal solution q f d opt can be found. It is also important to mention that this formulation does not require any assumption on the transmitted signal, and thus it can be used with any desired waveform.

It is worth mentioning that an analytic solution can also be deduced if the second constraint on the loss-in-processing gain is removed. The solution is simply obtained by replacing the matrix M with M Ξ in expression (5.7) and becomes

q f d analytic (s, M Ξ ) = (s H s) M -1 Ξ s s H M -1 Ξ s .
(5.17)

This analytic solution provides a sidelobe level bound for the given pulse over the whole Doppler interval in the sense that it provides the optimal mean ISL, whatever the value of the loss-inprocessing gain.

. Simulation Results

Once the mathematical background is well established, we can now provide some simulation results in order to evaluate the performance. For the simulation results, the CPFSK codes are considered for the uniformity of this document, even if the signal choice does not impact the problem formulation2 . The optimal solution is obtained using the convex toolbox CVX: Matlab Software for Disciplined Convex Programming [START_REF] Grant | CVX: Matlab Software for Disciplined Convex Programming[END_REF]. This matlab toolbox solves generic convex problems using an interior point method for the optimization problem.

In the following, we chose an allowed maximum loss-in-processing gain equal to β = 2dB. For the simulations, let us now consider two different intervals Ξ for the computation of the mismatched filter.

'MMF K

0 ' : the filter of length K that minimizes the sidelobe level energy that minimizes the sidelobe level only for the zero-Doppler value.

'MMF K f d ' : our proposed filter of length K that minimizes the sidelobe level energy defined over the continuous Doppler interval.

For the different MMFs, the exponent K can be replaced with different lengths. In our case, it will be either of length equal to the signal length N (exponent • N ) or three times the signal size 3N (exponent • 3N ). The Doppler interval depends on the η variable and the pulse duration of the signal. The smaller the η variable, the smaller the Doppler interval where the filter is optimized. The two different configurations that we consider correspond to η = { 1 2 , 1}. When η converges to 0, we deduce that the solution of the problem coincides with the classical definition of the MMF defined in section 5.1, as the interval collapses to a single point.

Observing figure 5.4, we can first remark that the sidelobe level provided by the matched filter is the worst in most cases, as the matched filter is not defined to deal with the energy of the sidelobes. Interestingly, it remains flat over the whole considered interval. The mismatched

Mismatched Filters with Signal Mismatch

91

-5 -4 -3 -2 -1 0 1 2 3 4 5
Frequency (Hz) filters provide in general better performance in terms of ISL. For a given filter length, the MMF K 0 minimized for zero-Doppler provides the best ISL performance at f d = 0, because all the degrees of freedom are dedicated to optimize this single Doppler value. But as a non-zero Doppler shift is added on the signal, these filters yield gradually worsening results. At higher Doppler shift, their ISL can be even worse than the matched filter one. As for the MMF K f d filters, it can be observed that their sidelobes are larger than the ones optimized for 0 in the neighborhood of zero-Doppler, but as higher Doppler shifts are added, the sidelobe level is low compared to the classical MMF definition. For example for η = 1 2 and the 3N-length filter, the interval where the MMF 3N 0 is the best is [0, 2 • 10 5 ] (and the symmetrical negative values), whereas the interval where the MMF 3N f d is the best is [2 • 10 5 , 5 • 10 5 ] (and the symmetrical negative values).

The choice of η is important on the performance of the MMF K f d . The smaller the η, the fewer the Doppler values on which we want to minimize the ISL of the filter output, and the more the problem can be relaxed leading to better performance. This can be illustrated in figure 5.4. In this figure, if we consider a Doppler shift that is part of both optimized intervals, for instance f d = -2e5 Hz, then for the MMF 3N f d output for η = 1 2 (sub-figure 5.4a) the ISL of the compressed output is equal to -6 dB, whereas with η = 1 (sub-figure 5.4b) the ISL is equal to -4 dB. So if we know the maximum range of Doppler values of a given application, it is favorable not to exceed it in the optimization process.

Another observation is that the choice of the filter length plays a crucial role on the output result after compression. When comparing the output of the filter of length K = 3N with the one of K = N, the larger one provides enhanced performance no matter what criteria the filters preserve for the optimization. However the difference lies in the time needed to compute the filters, with the longer one that needs more time because of the higher number of samples in the vector that we need to compute. So a compromise between the performance and the time complexity of the problem should be considered. Note also that if the filter is of length K and the signal of length N, then the size of the output is of length K + N -1. Thus the longer the filter, the larger the size of the output. In other words, a filter of length 3N will provide a non-zero output over a length 4N -1, instead of a filter of length N that will have a support of length 2N -1.

Apart from inter-pulse Doppler shifts that can occur on the signal, delay off-grid shifts can also happen whenever the applied sampling grid does not coincide with the sampling of the transmitted signal. Thus, the next sub-section aims at solving this issue by defining a problem similar to equation (5.12) but that handles the delay off-grid effect. Unlike the problem of this subsection that is signal independent, the off-grid effect in delay cannot be separated from the signal form, so we will restrain it to the case of CPFSK codes.

. Mismatched Filter Optimization of Signal Mismatch due to Off-grid Delay

In this subsection, our goal is to define a mismatched filter that minimizes the average ISL, while taking into consideration an unknown off-grid delay shift on the received signals. The problem of optimizing ISL under an off-grid delay effect has already been considered in [START_REF] Rabaste | Robust mismatched filter for off-grid target[END_REF] for the particular case of chirp signals. In our case, the definition of the mismatched filter is taking into consideration CPFSK codes with unknown off-grid delay shifts over the whole resolution cell, so that it is not necessary to compute different mismatched filters for different off-grid values. We will start by explaining the off-grid problem, afterwards deriving the mathematical formulations of the delayed received signal and showing that the corresponding mismatched filter depends on the emitted signal. Once the solution is provided, validation through simulation results is given.

. Off-Grid Problem

The off-grid effect is a phenomenon that occurs a lot in radar applications due to signal sampling at the receiver side. When the transmitted signals return to the antenna, they are sampled with a certain sampling rate. The sampling grid at the reception will usually not coincide with the parameters of the targets or areas of interest [START_REF] Fannjiang | Compressive radar with offgrid targets: a perturbation approach[END_REF]. Figure 5.5 illustrates the on-grid points of a time axis and the intermediate intervals that correspond to the off-grid values. As a result, after radar processing, less energy will be observed from these targets as the received signals will not be sampled in the same way as the transmitted ones. In addition, the output signal may also have higher sidelobes. The worst case scenario occurs when the offgrid shift that undergoes the received signal is in the middle of two sampling points. An example is provided in figure 5.6, that illustrates this effect for a chirp pulse sampled with a sampling frequency equal to 1.5B. The received signal undergoes an off-grid delay shift that corresponds to half of the sampling cell, and we can thus observe that sampled curves after MF compression do not coincide at all.

. Mathematical Definition

We recall that the transmitted signal of length N is denoted s, defined as in equation (5.1). In the following, s explicitly refers to a sampled CPFSK code.

Let us consider a given off-grid delay τ ∈ -T s 2 , T s 2 . From the definition of the CPFSK signal in chapter 4, the i th component of the noiseless delayed CPFSK signal r τ is expressed as

r τ i = s(t i -τ) = s i exp (- jπh T c I n i τ), ∀i ∈ 1, N , (5.18)
where t i is the i th time sample which depends on the sampling period T s , the chip duration T c is defined such that T c = ζ T s with ζ ∈ [1, +∞[, and n i is the symbol index associated to the i th time sample (the signal can be oversampled, so that a chip containing a certain symbol is spread over several samples). It is important to remark that the additional phase term due to the off-grid problem delay in equation (5.18) depends on the parameters of the transmitted CPFSK signal, such as h, T c and I.

As before, the noiseless output signal of a given mismatched filter q τ can be expressed as (5.19) where q τ ∈ C K with K ≥ N, and where Λ Λ Λ * K (r τ ) is the convolution matrix of size (K + N -1) × K of the delayed signal r τ , as defined in equation (5.3). If there is no delay shift (τ = 0), then the problem becomes the same as the classical MMF definition in section 5.1, and if in addition the mismatched filter is chosen to be the transmitted signal (q = s), then the matched filter is deduced.

y(τ) = Λ Λ Λ * K (r τ ) q τ ,
Similarly to the Doppler problem, the off-grid delay problem is tackled by seeking the mismatched filter that minimizes the average sidelobe level produced by CPFSK signals over all possible delay shifts in the resolution cell. Considering the ISL criterion, the cost function of the minimization problem is expressed as the ℓ 2 -norm of the signal sidelobes, as defined in both equations (5.8) and (5.12).

Let us assume that the delay shift variable τ is uniformly distributed over the sampling cell:

Θ = - T s 2 , T s 2 .
(5.20)

More precisely, the middle point of the interval Θ corresponds to a delay τ = 0 and it is the ongrid sample, while any other delay value in the interval corresponds to an off-grid delay. Having two different sampled signals r τ with different delay shifts, their compression outputs are going to differ, which may result to high and different sidelobe levels. The minimization problem is thus expressed as [START_REF] Chatzitheodoridi | A sidelobe level minimization mismatched filter using continuous phase frequency-shift keying codes for the off-grid delay problem[END_REF] min

q E τ ||Fy(τ)|| 2 2 , s.t. s H q = s H s, q H q ≤ 10 β 10 s H s, (5.21)
where the notations coincide with the notations of equation (5.8).

As the delay shift is uniformly distributed over the interval Θ, the cost function expectation can be expressed as

E τ ||Fy(τ)|| 2 2 = 1 T s ||M 1 2 Θ q|| 2 2 , (5.22) with (M Θ ) k,l = K+N-1 ∑ a=1 Λ T k,a (0)F a,a Λ * a,l (0)T s sinc h T c (I n(N+l-a) -I n(N+k-a) ) T s 2
. More details on the calculations can be found in Appendix A.2. Thus, the cost function strongly depends on some signal characteristics such as the modulation index h, the chip duration T c and the embedded symbols I. From equation (5.22), we can observe that the cost function of the minimization can be expressed as a ℓ 2 norm problem, and thus it is a convex function. Hence, as the constraints of the problem (5.21) are also convex, this results in a convex optimization problem, for which any optimum is a global minimum.

. Simulation Results

Having explicitly expressed the MMF cost function expression, we can now move onto the performance of the filter. As mentioned in the previous sub-section 5.2.1, the CVX: Matlab Software for Disciplined Convex Programming is used in order to compute the mismatched filter in (5.21). The allowed chosen maximum loss-in-processing gain is set to β = 2 dB as before.

For the simulations, we consider that the on-grid point is in the middle of the interval as expressed in (5.20), and the time sampling is defined as t i = T s 2 + (i -1)T s , with i ∈ 1, N , so that the boundary points coincide with time samples. At these time samples, a delay shift τ is applied, where τ ∈ Θ. Moreover, the same filters as in sub-section 5.2.1 are used for comparison, and they are also listed here as a reminder by adapting the notations: 'MF' : the classical matched filter.

'MMF K

0 ' : the filter of length K that minimizes the sidelobe level energy only for one on-grid value, which corresponds to τ = 0.

'MMF K

τ ' : our proposed filter of length K that minimizes the sidelobe level energy defined over the continuous delay interval.

For the mismatched filters, the size of the filter will be again either set to K = N or K = 3N. also illustrated for each delay shift. The need of providing this statistical analysis of the mean and the standard deviation comes from the randomness of the information embedded in the CPFSK phase.

Observing the classical matched filter MF in figure 5.7, the sidelobe level is quite higher compared to the mismatched filters, as it is not optimized for such a criterion. As previously mentioned, for a given filter length K, the filter optimized for no delay shift MMF K 0 provides better sidelobe energy performance for no delay shift (τ = 0) and shifts up to |τ| T s ≈ 0.15. As long as we consider higher delay shifts, this filter is neither optimal nor provides good results but the sidelobe energy increases, especially for shifts close to the edge of the cell. When observing the performance of the mismatched filter MMF K τ optimized over the continuous delay shift interval, it produces larger sidelobe levels near the zero delay shift than the MMF K 0 filter, but lower ones when it comes to higher delay shifts, as shown in figure 5.7. Its behavior is almost uniform and it is coherent since our filter is optimized with respect to the average off-grid energy. In addition, the averaged ISL for the filter optimized on the whole considered delay interval is the best for the 3N-length filter, as observed in table 5.2, followed by theMMF 3N 0 , and the worst case given from the MF.

Once again, the performance of the two different filter lengths ought to be mentioned and analyzed. The results are very close to the ones expressed in the previous sub-section 5.2.1 for the Doppler off-grid problem. More precisely, after compression with a longer filter, the sidelobe energy is spread along this filter length and decreased at the price of a greater computational cost, needed for the computation of the extra samples. For instance, the filters MMF 3N 0 and MMF 3N τ (length K = 3N), provide a lower ISL than the filters MMF N 0 and MMF N τ (length K = N), over the sampling cell, as indicated in figure 5.7. Finally, the best average ISL performance is provided by the 3N-mismatched filter optimized over the full resolution cell (MMF 3N τ filter), when it comes to deal with off-grid delay shift problems.

Figure 5.8 provides the mean ISL over the sampling cell for CPFSK codes with respect to the sampling frequency factor α. The sampling frequency factor α determines how many samples are present in a symbol. For more accurate results due to the random nature of the message embedded in the phase, we provide a statistical analysis with mean and standard deviation values over 50 Monte Carlo iterations. When α increases, more points sample the signal symbols, and thus the mean ISL of MMF K τ almost coincides with the mean ISL of MMF K 0 , for the same filter length K. More precisely for α ≥ 2.5, the mean ISL difference between the MMF K 0 and MMF K τ is insignificant, around 0.01 dB. It is hence easy to conclude that the interest of this method lies on configurations where the signals are not over-sampled. On the contrary, if the signals are over-sampled, the offgrid effect becomes negligible and the filter optimized for the on-grid points provides good results. It is important to mention that the ISL is computed as the sum of the sidelobe energy on the sampled signals. So if more sampling points are considered due to over-sampling, the ISL may be 1.5 1.7 1.9 2.1 2.3 2.5 higher. This is why the ISL values can increase with respect to the α variable in figure 5.8.

As the delay term in the off-grid problem depends on the number of symbols, an analysis with respect to this parameter is presented in figure 5.9. More precisely, it displays the mean ISL for different CPFSK signals and different number of symbols over 50 Monte Carlo iterations. As we can remark, the performance of the filter seems to be independent of the number of symbols, which is an interesting characteristic to perform simultaneous radar and communication. We were however expecting to have higher ISL values for longer signals, as the correlation supports also increases, as seen in figure 5.8. However, we brought up to the conclusion that the additional degrees of freedom provided by the larger number of samples seem to allow a better sidelobe reduction, leading to an almost constant global ISL with respect to the number of symbols.

Overall, this section dealt with the robustness of the MMF when signals undergo inter-pulse Doppler and off-grid delay shifts. The proposed solutions are efficient and advantageous when the signals are sampled with a small sampling frequency, making the off-grid shift potentially large, for sufficiently large inter-pulse Doppler shifts. This will be the case for the radar-communication application since radar signals are generally not over-sampled and may undergo strong Doppler shifts in the presence of fast moving targets. On the contrary, if signals are highly over-sampled and undergo small Doppler shift, the mismatch is negligible and the classical mismatched filter, even if no longer optimal, is still suitable. In the SAR-communication system as we conceived it, the signals are heavily sampled at the reception to provide an accurate phase compensation for the SAR processing. Thus for the SAR-communication application considered in chapter 6, we will apply the classical mismatched filter.

Note that, although oversampling the signals enables to remove the off-grid problem, it also leads to much larger number of samples, especially when the pulse duration is large, that creates large scale optimization problems with thousands of variables. Besides it should also be noted that in order to increase the data rate of the communication side of the system, the easier solution consists in sending a different message at each transmitted pulse. Thus, for the radar processing side, it is necessary to compute a different MMF for every pulse by solving a large-scale optimization problem. By taking into consideration these aspects, and having in mind that the CVX toolbox is not recommended for large-scale problems [START_REF] Grant | CVX: Matlab Software for Disciplined Convex Programming[END_REF], we will introduce in the following section a faster algorithm for computing the optimal mismatched filter.

. Fast Computation of the Mismatched Filter for Large-Scale Problems

Since we want a joint SAR-communication system providing both a good image quality, while maximizing the communication data rate, the CPFSK codes should satisfy some characteristics. They ought to present a reasonable bandwidth (to ensure a good range resolution), be over-sampled (for a good phase correction), and be transmitted at high PRF during a long integration time (to give a good azimuth resolution). Furthermore, the number of mismatched filters that need to be computed is equal to the number of transmitted signals. For these reasons, when choosing mismatched filters as the desired pulse compression technique, we need to ensure that it is feasible to compute the filters in a reasonable amount of time. In the following, a fast algorithm is proposed to provide the MMFs. Firstly, the mathematical background is established and afterwards the performance of the method is performed, along with the algorithm in pseudo-language.

. Mathematical Formulations

This subsection is dedicated to the mathematical calculations of the dual formulation of the optimization problem.

. Primal Problem

Recall that the ℓ 2 -norm problem of equation (5.8) is

min q f (q) = q H Mq, s.t. s H q = s H s, q H q ≤ γs H s, (5.23) where M = Λ Λ Λ T K (s)FΛ Λ Λ * K (s)
is the matrix used for the analytic filter in (5.7), and γ = 10 β 10 .

. Dual Problem

We will consider here the Lagrangian function associated to the primal optimization problem in (5.23). The Lagrangian dual function is expressed as the cost function augmented by the weighted sum of the two constraints in (5.23) [Boyd et al., 2004, Floudas and[START_REF] Floudas | [END_REF]. We denote by λ the Lagrange multiplier for the inequality constraint and by ν the Lagrangian multiplier for the equality constraint. Both λ and ν are called dual variables or Lagrange multiplier vectors [START_REF] Boyd | Convex optimization[END_REF]. It can already be noted that there are only two dual variables, whereas the number of variables of the primal problem is at least equal to the filter length, as indicated in figure 5.10. The Lagrangian function is expressed as

L (q, λ , ν) = q H Mq + λ (q H q -γs H s) + ν(s H q -s H s) (5.24) = -λ γs H s -νs H s independent of q + q H (M + λ I)q + νs H q dependent on q
, where λ ∈ R + and ν ∈ R.

Minimizing the Lagrangian function with respect to q provides a lower bound on the optimal solution of the primal problem (5.23) [START_REF] Boyd | Convex optimization[END_REF]. The output of such a minimization is called dual function and it is denoted by g g(λ , ν) = inf q L (q, λ , ν).

(5.25)

It can be shown that the dual function is always concave [START_REF] Boyd | Convex optimization[END_REF]. The primal is a minimization problem, whereas the dual is a maximization one. An optimal value of the primal problem denoted p opt is always an upper bound of the solution of the dual problem, denoted d opt [START_REF] Boyd | Convex optimization[END_REF], with d opt ≤ p opt .

Figure 5.10 illustrates this effect, and the difference between the two solutions is called duality gap.

. Optimization with respect to q

Minimizing the Lagrangian function with respect to q can be performed by calculating the root of its gradient. Since the gradient with respect to q of the Lagrangian function is given by ∇ q L (q, λ , ν) = 2(M + λ I)q + νs,

(5.26) its null can be easily calculated, which provides the optimal value:

∇ q L (q opt , λ , ν) = 0 ⇐⇒ q opt = - 1 2 ν(M + λ I) -1 s.
(5.27)

Finally, by inserting the optimal filter q opt in the Lagrangian function in (5.24), the dual function can be written as

g(λ , ν) = L (q opt , λ , ν) = L (- 1 2 ν(M + λ I) -1 s, λ , ν) = -λ γs H s -νs H s - 1 4 ν 2 s H (M + λ I) -1 s.
Afterwards the optimal pair (λ opt , ν opt ) needs to be found in order to deduce the filter from the equation (5.27). Since, the function g is always concave [START_REF] Boyd | Convex optimization[END_REF], its optimum can be easily found. We denote by λ opt and ν opt , the values of λ and ν maximizing the dual function g(λ , ν), such that (λ opt , ν opt ) = arg max λ ,ν g(λ , ν).

(5.28)

. Optimization with respect to ν variable

The dual function is a second order degree polynomial with respect to the ν variable, so the optimal value can be analytically found by computing the partial derivative of the dual function with respect to ν,

∂ g ∂ ν = -s H s - 1 2 νs H (M + λ I) -1 s.
Then calculating its root enables to get

∂ g ∂ ν = 0 ⇐⇒ ν opt = - 2s H s s H (M + λ I) -1 s .
(5.29)

. Optimization with respect to λ variable

Once the optimal value of ν is calculated, it still remains to find λ opt . Let us express the mono-dimensional dual function with respect to λ variable as

g(λ ) = max ν g(λ , ν) = g(λ , ν opt ) = -λ γs H s + (s H s) 2 s H (M + λ I) -1 s , (5.30)
which is still a concave function.

It does not seem easy to calculate the optimal value λ opt analytically because of the inverse term of the denominator of the function g in equation (5.30). Still the concavity of g(λ ) and the fact that it depends only on one parameter makes it easy to maximize it numerically. We chose to use the Newton-Raphson method [START_REF] Ortega | Iterative solution of nonlinear equations in several variables[END_REF], which is an iterative procedure expressed as

λ k+1 = λ k - g ′ (λ k ) g ′′ (λ k ) .
(5.31)

For the sake of simplicity, let us denote

h(λ ) = s H (M + λ I) -1 s.
(5.32)

The first and second derivatives of the function g, expressed in (5.30), are computed with respect to the λ variable in order to specify and perform the iterative algorithm using the formula (5.31):

g ′ (λ ) = -αs H s -(s H s) 2 h ′ (λ ) h 2 (λ ) , g ′′ (λ ) = -(s H s) 2 • h ′′ (λ )h(λ ) -2[h(λ )] 2 h 3 (λ ) .
The derivatives of g are dependent on function h, expressed in (5.32), and its first and second derivatives. This means that for every iteration, the function h and its associated derivatives need to be computed, that leads to inverse matrix computations, at each iteration. In order to avoid inverting matrices at each step of the gradient algorithm, which is more costly from a computational point of view, an eigenvalue decomposition of M is performed. More precisely, M can be written as M = P H DP, with P a square matrix, D a diagonal one and the conjugate transpose is used because M is symmetric. Then, the function h can be simplified as:

h(λ ) = s H P H (D + λ I) -1 Ps = s H P H diag 1 d + λ Ps, (5.33)
where d is the vector that determines the diagonal elements of the matrix D, and now the inverse matrix term can be easily calculated [Mirsky, 2012].

Similarly, the first and the second derivatives of the h function from equation (5.33) can be expressed after performing the eigenvalue decomposition as

h ′ (λ ) = s H P H diag - 1 (d + λ ) 2 Ps, h ′′ (λ ) = s H P H diag 2 (d + λ ) 3 Ps.
These expressions present the advantage that the square matrix P and the diagonal matrix D can be computed only once and for all for the optimization problem. Then, the function h and its derivatives are finally computed only by performing multiplications and no matrix inversions.

. Slater's Condition

The last remaining step that needs to be verified to provide the optimal filter is to ensure that the optimal pair (λ opt , ν opt ) of the dual problem enables to retrieve the optimal solution of the primal convex problem defined in (5.23). This property is called strong duality, and it consists in proving that the duality gap is equal to zero. This means that the difference between the solution of the primal problem (5.23) and the solution of the dual problem (5.24) in figure 5.10 is zero.

Strong duality can be ensured when the primal problem is convex and Slater's constraint condition is verified [START_REF] Boyd | Convex optimization[END_REF]. The first condition has already been justified in the first section of the chapter 5.1. Slater's condition is reformulated in the following lemma for our problem.

Lemma 5.3.1 (Slater's condition for our problem). [START_REF] Boyd | Convex optimization[END_REF] There exists an x 0 ∈ relint (D), where relint (D) is the relative interior of D, and D the set of feasible solutions of the primal problem, that verifies the following equality and strict inequality of the problem:

s H x 0 = s H s, (i)
x H 0 x 0 < γs H s.

(ii)

This lemma 5.3.1 is verified for x 0 = s. Indeed, when x 0 = s, the equality (i) is true. Besides, the strict inequality (ii) clearly holds whenever γ > 1, which is always the case for our problem since γ = 1 corresponds to no loss-in-processing gain, and hence to the matched filter, and thus we will always set γ > 1 to get a different filter.

Parameters

. Simulation Results

To illustrate all this procedure, algorithm 1 sums up all the implementation steps to obtain one optimal mismatched filter, by providing the optimal pair (λ opt , ν opt ). It is important to highlight again that the solution of algorithm 1 is the same as the optimal solution of any other convex toolbox that we can use to solve the problem, for example the CVX, thanks to strong duality. The difference lies in the computational time which is decreased with the proposed method. Generally this algorithm can be applied to any sampled signal, as there is no particular assumption on the signal nature, as long as we consider the integrated sidelobe level as the criterion to be optimized. In the following simulations, we illustrate the effectiveness of this implementation. The parameters used coincide with the ones used for the generation of the SAR images in the next chapter 6, and are listed in table 5.3.

Observing table 5.3, apart from the large number of transmitted signals ( 6522), each signal contains also a lot of samples (3000 samples) and the length of the associated mismatched filter is three times the size of the signal (9000 samples). For the algorithm, the initialization of the λ variable is fixed to 0.5, and the algorithm stops either when the maximum number of iterations (set to 100 iterations) is reached or when a given precision on the convergence (set to 10 -10 ) is attained.

After providing the necessary parameters to run the algorithm 1, we can compute all the 6522 filters that we need and eventually create the range compressed data using mismatched filtering. Table 5.4 shows the computational time of the algorithm 1 and of the CVX toolbox for one mismatched filter, and the extrapolated computational time for all the 6522 filters. Even though the two different methods provide the same optimal filter solution, the computational time difference is enormous. The computational cost of one filter using our method is estimated to be equal to 3.48 minutes, after averaging over 30 different randomly drawn sequences, to be compared Algorithm 1: Mismatched Filter Optimization using Gradient Descent

Input: Loss-in-progressing gain α, initialization λ 1 , max iterations max i , precision ε, Output: Optimal pair (λ opt , ν opt ), optimal mismatched filter q opt ; s ← code generation;

M ← Λ Λ Λ T K (s)FΛ Λ Λ * K (s); [P, D] ← SVD(M); for i ∈ [1, max i ] do h i ← s H P H diag 1 diag(D) + λ i Ps; h ′ i ← s H P H diag - 1 (diag(D) + λ i ) 2 Ps; h ′′ i ← s H P H diag 2 (diag(D) + λ i ) 3 Ps; g ′ i ← -αs H s -(s H s) 2 h ′ i h 2 i ; g ′′ i ← -(s H s) 2 • h ′′ i h i -2h 2 i h 3 i ; λ i+1 ← λ i - g ′ i g ′′ i ; if |g ′ i | < ε then break; end end ν opt ← - 2s H s s H (M + λ opt I) -1 s ; q opt ← - 1 2 ν opt (M + λ opt I) -1 s.
with 14.5 hours spent on average by the CVX algorithm. Extrapolated to the 6522 different signals that are transmitted at each acquisition, this computational time would be equal to 11 years, as shown in Table 5.4. On the contrary, the proposed solution obtains all 6522 mismatched filters in less than 16 days. These results show that the proposed method computes the solution of the convex problem (5.23) almost 250 times faster than the CVX, as shown in the Ratio column of table 5.4.

. Conclusion

This chapter handled sidelobe suppression through mismatched filtering, that is expressed as a convex optimization problem. Two extensions of the problem were proposed to deal with unknown inter-pulse Doppler shifts or off-grid delay shifts over a given continuous interval of values, that can occur on the received signals. Finally, a fast algorithm is provided that enables to compute all the needed MMFs in a feasible time in order to create the SAR image. However, even if the filters can be computed in a reasonable time, it is still not possible to perform real-time acquisitions. Thus, one way to tackle such problem is to utilizing a bank of pre-defined signals to be transmitted, for which the filters can be computed offline.

In the next chapter 6, these computed mismatched filters will be utilized in order to create re-synthesized SAR images. Creating our joint SAR-communication system becomes feasible in terms of computational time, but the question that rises : is the quality of such an image close to the one created with chirps? CHAPTER 6 Generation and Analysis of Re-Synthesized SAR Images using CPFSK Codes

In the previous chapter 5, we have proposed to consider the MMF solution in order to improve the sidelobe level provided by CPFSK codes. Along with the compression technique choice, we proposed an algorithm to compute these MMF in a feasible time, which is a mandatory step if we must perform a real application with thousands of impulses. We will now study the performance of the combined use of CPFSK codes and their optimized MMF. To characterize the impact of using communication codes compressed with MMF on a resulting SAR image, we will first consider the simple case of one corner reflector. It will enable to observe the behavior of the CPFSK codes in the range-Doppler plane. Afterwards, because of the unavailability of an airborne system able to transmit CPFSK codes at ONERA, but with the will of considering realistic data, we decided to create re-synthesized SAR images. More precisely, we make use of raw data that are already acquired with chirps, and apply on them the impulse response of the communication codes. The output is a hybrid result between simulation and real data, which allows us to understand the behavior of the CPFSK codes in the presence of multiple scatterers.

In the following, the generation of these results are explained in detail. Different evaluation criteria are used to provide an analysis on the obtained images.

. Generation of Re-Synthesized SAR Images

In the absence of real SAR data acquisitions, we create re-synthesized SAR images, in order to yield as realistic outputs as possible in comparison to real-data SAR images. To achieve this goal, the integrated ground reflectivity of a real-data SAR image created with chirp signals is utilized to create re-synthesized images from CPFSK codes. The general framework is firstly provided in sub-section 6.1.1 by listing the different configurations that we generate for comparison and the real data properties from which these images were created from. Then the reconstruction algorithm explanations for the image generation are introduced in sub-section 6.1.2. Afterwards, the resulting images are displayed in sub-section 6.1.4 and 6.1.5, along with some visual remarks. Finally, an extensive analysis with different evaluation criteria is provided in section 6.2, so that the best performance configuration would be selected for the creation of the joint SAR-communication system.

. Real Data Properties and Communication Code Characteristics

The main idea of the creation of re-synthesized images is the use of real raw data from a high-resolution SAR image acquired with chirps. The airborne platform used for the acquisitions is SETHI [Ruault du [START_REF] Du | The onera airborne multi-frequency sar imaging systems[END_REF], an airborne radar system owned by ONERA, the French Aerospace Lab. They are obtained in X-band from a pulse train of chirp signals at a PRF of 5 kHz as presented in Table 6.1, which provides the principal parameters and characteristics of the real system, along with the parameters of the CPFSK code generation. The illuminated scene includes a corner reflector, and different types of clutter (urban areas, crop fields, forest, etc.). It can be illustrated in figure 6.1. The bandwidth used for the generation of this SAR image is 500 MHz and the pulse duration is equal to 8µs.

It is important to mention that the bandwidth of the generated re-synthesized SAR images has been chosen smaller than the one of the real data. There is no limitation on choosing the bandwidth of the generated images as long as it is strictly lower than the real-data bandwidth. In our framework, on the one hand the bandwidth choice depends on the communication signal characteristics, and on the other hand, lies in the need of providing a small computational cost for the mismatched filters, while maintaining a sufficient resolution. In the end, the generated signals were created with a bandwidth of 300 MHz instead of the 500 MHz band of the real data.

In the following, the chirp images used for comparison with the CPFSK outputs are also resynthesized with this smaller bandwidth, so as to obtain the same range resolution. As mentioned in chapter 4, the constellation choice for the waveform also decides the bandwidth shape. In order to provide a wider and more uniform spectrum with lower sidelobes, we chose a large constellation of size C = 16. The modulation index is chosen 2 5 , which is a fraction of prime numbers to each other, so the number of phase transitions is discrete. Finally, the pulse duration is set to 4.8µs , which is smaller compared to the one used for the signal generation of the real data image set to 8µs, so as to reduce the bandwidth of the re-synthesized chirp.

For comparison sake, we introduce 4 different configurations using communication codes and one with chirps along with the compression technique choice:

One identical CPFSK pulse is transmitted for all the acquisitions, and compressed with the matched filter (MF). This case is denoted 1_CPFSK_MF.

One identical CPFSK pulse is transmitted for all the acquisitions, and compressed with the mismatched filter (MMF). This case is denoted 1_CPFSK_MMF.

A different CPFSK code is transmitted at each pulse, and each one of them is compressed with the MF. This case is denoted N_CPFSK_MF.

A different CPFSK code is transmitted at each pulse, and each one of them is compressed with the corresponding MMF. This configuration requires the computation of 6522 MMF, and it is denoted N_CPFSK_MMF.

Chirp with the same bandwidth as the CPFSK signals, to fairly compare the CPFSK SAR images.

Considering each one of the above CPFSK configurations, it is possible to find the data rate. The data rate D is the amount of information transmitted during a second and can be defined in our case as

D = log 2 (C) T p N s T c N pulses PRF, (6.1)
where N pulses is the total number of transmitted pulses, and N s expresses the number of nonidentical transmitted CPFSK codes.

Transmitting the same CPFSK signal over the whole pulse train (configurations and ) corresponds to having N s = 1, resulting in a data rate of 766 bit/s, using the parameters in Table 6.1. On the contrary, when the CPFSK signals differ from pulse to pulse (configurations ans ), then N s = 6522 and the resulting data rate becomes 5 10 6 bit/s (or 625 10 3 bytes/s or 610 kbytes/s).

. SAR Image Reconstruction Algorithm

A possible solution to evaluate the behavior of the communication codes along with the mismatched filtering compression, without performing acquisitions, is to simulate in the most reliable way the electromagnetic response of a scene. The purpose of the reconstruction algorithm is to retrieve a realistic-looking ground reflectivity using existing raw data from chirp acquisitions, and using it in order to generate a re-synthesized SAR image using CPFSK codes, which avoids complicated electromagnetic simulations. Once the estimated ground reflectivity is provided, the classical SAR processing is performed by applying our desired communication codes and compression technique.

Let us first look at how to retrieve the ground reflectivity (or at least an approximation of it). The noiseless received raw data after the transmission are expressed as a convolution between the transmitted waveform and the ground reflectivity [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]:

r u (t) = g r u (t) * s u (t), (6.2)
where r u (t) is the received signal, g r u (t) is the ground reflectivity, s u (t) is the transmitted signal, t denotes the fast time, related to the intra-pulse sampling time, and u the index of the transmitted pulse, related to the slow time.

The provided real raw data acquired with chirps are deconvolved with the impulse response of the chirp, illustrated in the first box of figure 6.2, in order to provide the estimation of the integrated ground reflectivity. It can be computed in the Fourier domain as:

Ĝi (k) =    R u (k) S u (k) , 2π f 0 c - πB chirp c < k < 2π f 0 c + πB chirp c 0, otherwise, (6.3)
where Ĝi is the spectrum of the estimated integrated ground reflectively, f 0 is the carrier frequency, B chirp is the chirp bandwidth, and k is the spatial frequency defined as 2π λ . Since this operation is conducted in the spectral domain, the earlier-mentioned constraint on the bandwidth of the re-synthesized images arises to avoid boundary effects, as it should be smaller than the bandwidth of the original image.

The SAR range compressed data are denoted by c u (r), and can be written as (refer to equation (2.24) in chapter 2) 6.4) where r is the distance variable.

c u (r) = sinc r ρ r * g i (r), (
In the Fourier domain, it is written as

C u (k) = Rect(kρ r ) • G i (k), (6.5)
where C u is the spectrum of the range compressed data at slow time u.

Once the equation (6.5) is established with respect to the integrated ground reflectivity Ĝi , we need to apply the impulse response of the communication codes, range compressed with matched or mismatched filter. Computationally, it is performed by multiplying the compressed data in the Fourier domain by the Fourier transform H comm of the communication impulse response h comm , that includes the response of the codes and the compression with the matched or mismatched filters:

S ′ u (k) = H comm (k) • Rect (kρ r ) • Ĝi (k). (6.6)
If the chirp bandwidth is larger than the communication bandwidth, we can assume that

H comm • Rect (kρ r ) = H comm , so that S ′ u (k) = H comm (k) • Ĝi (k), (6.7)
or in the time domain,

s ′ u (r) = h comm (r) * ĝi (r). (6.8)
We can thus retrieve the compressed signal of a communication system. Finally, we perform the azimuth compression. Figure 6.2 presents the synoptic we used. The output of the algorithm is a hybrid result between simulation and real data. From one side, it provides a more realistic result compared to simulations, as it uses measured quantities, but it is not identical to the exact behavior of the CPFSK codes on a real scene. This procedure is repeated for the five different configurations presented in the previous subsection 6.1.1, where the number of signals and the compression technique change, in order to compare the performance of the different outputs.

. Drawbacks of the Reconstruction Algorithm

This methodology gives a good approximation of what would happen if a communication code has been used. Nevertheless, there are 3 main differences from a real experiment:

• the communication code is not deformed by the hardware system. This means that the simulated signal and the one that is actually transmitted from the antenna may differ, which might create artifacts on the output, that cannot be considered through the reconstruction algorithm;

• We assume that there is no impact of the Doppler on the return signal, and thus we do not know its impact;

• The noise level is not well represented.

The two first points can be observed in chapter 7, where real data are provided. Elaborating the last point, in the frequency domain, the noise can be written as

N(k) S u (k) • H comm (k) = N(k) S u (k) • S CPFSK (k) • F MMF (k).
(6.9)

In the real experiment, we would have had N(k) • F MMF (k), where N(k) is the Gaussian white noise expressed in the frequency domain, and F MMF (k) is the response of the mismatched filter in the frequency domain.

. Generation of the Impulse Responses

Before creating the re-synthesized SAR images with the estimated ground reflectivity, we first generate the impulse responses of the four configurations of the CPFSK codes along with the chirp case for a single point reflector. A 2D impulse response is the result of the code response for a stationary target that undergoes a range compression and a classical azimuth compression by Inverse Fast Fourier Transform (IFFT).

Figure 6.3 illustrates the impulse responses for the chirp and the four different CPFSK configurations on the full range and azimuth domain, as well as on a zoomed window around the mainlobe and its nearest sidelobes along the range and azimuth axes, in order to provide a better visualization of the behavior close to the mainlobe. The first sub-figure 6.3a illustrates the impulse response of a chirp compressed with the MF and it is used as reference for comparison. The second row of sub-figures (sub-figures 6.3b & 6.3c) presents the output of a single transmitted CPFSK code compressed with the MF and the MMF, respectively. Finally, the third row of sub-figures (sub-figures 6.3d & 6.3e) provides the impulse responses when multiple communication codes are sent and compressed with the above methods.

By observing these impulse responses, we can remark that the sidelobe level is spread along the 2D plane, when transmitting several codes. On the contrary, when sending one single CPFSK signal, the output after compression presents sidelobes concentrated only on the range and azimuth axes. This effect is due to the difference between the transmitted codes, that carry non-identical messages embedded in their phase. When range compression is performed, each code thus creates different sidelobes. Along the azimuth axis of the single point reflector signature, the azimuth compression tends to average all these range sidelobes, thus leading in practice to a lower sidelobe level than when using an individual code. This effect is highlighted in figure 6.4, where 8 different codes are selected and compressed with their mismatched filter. The sidelobe level difference can be visualized, and the resulting average is also illustrated. Even for as few as 8 signals, we can see that the recombination of the sidelobes strongly smoothed the compression response. Another effect arises when using different CPFSK signals. Indeed, there is also an energy spread along the 2D plane, which is created from the variety of phases of the different codes. Through this effect, even though higher sidelobes along the range axis are limited, low energy targets could be masked elsewhere in the scene.

Comparing the columns of figure 6.3, it appears that the sidelobe level energy of the impulse responses is higher for sub-figures compressed with MF (6.3b and 6.3d) than the ones compressed with MMF (6.3c and 6.3e). This is an expected result as the MMF minimizes the sidelobe energy. Moreover, the mismatched filters used here are three times longer than the signal length, because the performance of the sidelobe energy is better compared to shorter filters (refer to Chapter 5), but it also results to longer sidelobes, and to a further diffused sidelobe level over the range axis. Figure 6.5 provides a better insight on the impulse response evaluation, as it illustrates the range and azimuth cuts in its two sub-figures respectively for the chirp and all the different configurations of CPFSK codes. Starting by the azimuth cuts that correspond to sub-figure 6.5b, we can remark that they are all identical, which arises from the fact that the azimuth processing is the same for all the different configurations provided in figure 6.3. As for the range cut in figure 6.5a, we can remark that the worst case scenario in terms of sidelobe level energy is provided by the transmission of one single CPFSK code compressed with its MF, followed by this same waveform compressed with the MMF, whose sidelobes are worse than the chirp performance. When transmitting different codes at each acquisition, the sidelobe level in the range direction is close to the chirp output, which is interesting for the communication output.

Signals

PSLR ISLR ISLR small (dB) (dB) (dB) Chirp with MF -13.2 -9.5 -9.6 One CPFSK with MF -14.9 0.2 -6.8 Different CPFSK with MF -14.8 -12.9 -12.9 One CPFSK with MMF -15.8 -8.1 -13.4 Different CPFSK with MMF -15.2 -14.0 -14.0 ISLR small provides the ISLR values of the impulse responses when the energy is computed over a shortened window that corresponds to the number of points considered in Table 6.3 for the ISLR computation. The purple-colored highlighted values correspond to the best value of each column.

However the diffused energy created from the compression cannot be neglected. This behavior can be quantified in Table 6.2, which provides the classical evaluation radar criteria. From an implementation point of view, the PSLR is computed along the range and azimuth directions by finding the highest sidelobe and subtracting it with the mainlobe energy, as expected. On the contrary for the ISLR, two different values are computed on the impulse responses. The first one provides the energy of all the sidelobes created after compression in the range direction. Instead, the second one takes a given number of points, usually around 10 resolution cells, and computes the sidelobe energy over this subset, as usually done for the chirp-generated images as explained in chapter 2. However, since the CPFSK impulse response energy does not have the same range decreasing behavior, the latter definition is not well adapted.

Observing table 6.2, the best ISLR value corresponds to the case of N_CPFSK_MMF (subfigure 6.3e), for both ISLR columns. When transmitting different signals, the sidelobe energy is not affected a lot on the range axis from further sidelobes, whereas when transmitting a single CPFSK code the image is polluted along the range direction as it can be remarked from table 6.2. For the energy lines appear, especially for high-energy scatterers, along the range axis for both compression techniques. The one created using MF has even higher sidelobes that result in the complete destruction of the image contrast. It is no longer possible to distinguish the elements of the scene! When the MMF is used (sub-figure 6.7c), these high-energy lines still appear, but they have a smaller impact, situated mainly around a restricted number of high-energy points along the range axis, thanks to the sidelobe level minimization. Still, the quality of the SAR image is not acceptable as image details can be potentially masked, because of these lines.

The output obtained when transmitting different codes at each acquisition and compressing them with MF and MMF is provided in sub-figures 6.7d and 6.7e respectively. A sidelobe energy spread in the 2D plane can be observed, as in the associated 2D impulse responses, which provides a more diffused sidelobe level over the scene. This leads to a loss in contrast, quite visible in 6.7d. Using MF, the sidelobe level is higher and more visible, and thus deteriorates the image quality more than with the use of MMF. This spreading is visualized as a blurring effect especially around high-energy points. It is created by the random sidelobe summation of the codes after range and azimuth compression. On the positive side, thanks to this averaging effect, the high-energy lines observed when transmitting a single code are not present anymore. The contrast of these figures is visually better than the one obtained for a single CPFSK, as elements of the scene can be clearly distinguished from one another. However, intense blurring around some areas can cause problem in detection. To the naked eye, the result in sub-figure 6.7e is satisfying, and the performance looks close to the chirp performance. It can be noted that the building borders are even better visualized in the last case than with the chirp, thanks to faster sidelobe decrease.

In order to provide some more insights on the sidelobe energy, figure 6.8 provides the range and the azimuth cuts of the corner reflector of the SAR image. Illustrating these sub-figures also allows to compare them with the 2D impulse responses provided in the previous section. Firstly, for the range direction, because of the clutter level contribution, no output reaches a sidelobe energy as low as the impulse responses. The case of 1_CPFSK_MF shows the highest sidelobes, whereas 6.2. Quantitative Analysis of Re-Synthesized SAR Images through Several Evaluation Criteria 121

the chirp and the N_CPFSK_MMF gives the lowest ones. The difference between these cases lies in the fact that the chirp has higher sidelobes close to the mainlobe that decrease away from them, and the N_CPFSK_MMF has an on average stable sidelobe level. On the azimuth side, the cuts are close to one another, especially around the mainlobe.

6.2 . Quantitative Analysis of Re-Synthesized SAR Images through Several Evaluation Criteria

In this section, different evaluation criteria are introduced that are used either in the radar field for SAR image evaluation or generally in image processing field. Subsequently, all these criteria are considered in order to quantitatively study which strategy provides the closest performance to the chirp one.

. Evaluation Criteria for SAR Image Comparison

This subsection is devoted to the presentation of the different assessment criteria that are utilized for the image comparison. The Integrated Sidelobe Level Ratio (ISLR) and the Peakto-Sidelobe Level Ratio (PSLR) are two classical radar criteria that provide two different insights concerning the behaviors of the sidelobes. They are not described here as they were already defined in chapter 2, but they are used in the following. Pearson correlation coefficient is also taken into consideration, that enables to analyze to which extent two SAR images are similar. The structural similarity index is considered as an error estimator for image comparison, as well as the mean square error. Moreover, the average noise level over an area of the image that has low intensity is computed, and it provides the noise sensitivity of the system for different waveforms or processing. Finally, an area of the image is displayed, where airplane shadows are present, in order to provide an additional visualization of the noise floor of the different outputs.

. Pearson Correlation Coefficient

The Pearson correlation coefficients, also called coherence coefficients, of two signals or images provide a degree of similarity of the inputs [START_REF] Asamoah | Measuring the performance of image contrast enhancement technique[END_REF]. Let us consider two images A and B. The coherence coefficients between A and B are defined on a given patch by [START_REF] Asamoah | Measuring the performance of image contrast enhancement technique[END_REF], Born et al., 1999]

C i, j = ∑ i j∈D (A i j -E [A D ]) (B i j -E [B D ]) * ∑ i j∈D |A i j -E [A D ] | 2 |B i j -E [B D ] | 2
, where i, j are the current position indices, D = {id : i + d, jd : j + d} are the indices considered for the computation with d an integer according to the desired patch size, and E is the mean value.

Usually the Pearson coefficients are computed over the whole image and thus the length of D is equal to the dimensions of the matrices. We chose in this work to provide both configurations. A window of size 11 × 11 (d = 5) is used to compute a local degree of similarity of the images. The results provide the amplitude of the correlation coefficients.

When the output values are close to 1, the inputs have a strong coherence, which means that the possible variations of the pixel intensity between the two images are small, making the images

. Results and Validation on the Re-Synthesized SAR Images

The evaluation criteria introduced in the previous subsection 6.2.1 are now applied to the re-synthesized images. In figure 6.7, a large area of the ground was considered, in order to provide an overall visualization of the scene. For the comparison with the different criteria, four different areas of the SAR image are considered, which are displayed in figure 6.9. These areas are chosen for their different clutter nature: urban clutter for the building, natural clutter with low reflectivity for the lake natural clutter with higher reflectivity for the forest, and an artificial corner reflector.

The corner reflector (sub-figure 6.9a) is chosen in order to compare its evaluation results with the performance of the 2D impulse responses presented previously. The building (sub-figure 6.9b) is a high energy area, so it will show how the energy is distributed around the mainlobe with comparison to the chirp, as well as the behavior of a set of high-energy points. Afterwards, the lake (sub-figure 6.9c) provides a very dissimilar configuration, as it is an area of low energy. Finally, the forest image (sub-figure 6.9d) is an area of intermediate level, that of between the building and the lake.

The following paragraphs provide the results of the different evaluation criteria on the areas of the re-synthesized SAR images. Firstly, the PSLR and the ISLR are computed on the corner reflector, and they are compared to the ones of the impulse responses. Then, the Pearson coefficient is computed over each area, along with the SSIM computation. Afterwards, the average energy of a low energy area, which in our case is the lake, is provided to observe the average noise level value for the different configurations. Finally, an area containing the shadows of two planes is presented, as illustrated in figure 6.10.

. PSLR and ISLR of the corner reflector

The first criteria that are presented are the PSLR and the ISLR. Their values are computed only for the corner reflector illustrated in table 6.3, as it does not make sense to compute them on any other area.

Even then, since the re-synthesized SAR images have many scatterers, the sidelobes of the corner reflector may be hidden by the surrounding reflectivity. For this reason the measured ISLR is generally higher than the values of the impulse responses, shown in table 6.2, but it represents a more realistic framework, closer to real acquisitions. Figure 6.11 represents the auto-correlation of a chirp when a clutter contribution of up to -40 dB is applied and the case when no clutter is present. It shows that the sidelobes of the chirp can be hidden under the clutter since it has higher energy than the sidelobes of the signal. This effect is present in our results when observing figure 6.12 that illustrates the corner reflector of every SAR image. Especially for the two cases for which different codes are transmitted, their sidelobes are hidden under the clutter. This means that the computations of the ISLR may not be representative of the actual sidelobe energy of the pulses.

The ISLR of the corner reflector is computed over a restricted number of points, inferior to the actual number over which the corner reflector energy is spread, to avoid integrating sidelobes that are polluted by the neighboring scattereres, and to remain close to the actual corner reflector contribution. This corresponds to the ISLR small of table 6.2 for the theoretical values. For the PSLR, the values are higher compared to the simulated impulse responses which may be due to a slightly non rectangular spectrum of the input data.

Both tables 6.2 and 6.3 show that communication codes compressed with MMF provide better values for the PSLR and the ISLR compared to the MF. For the ISLR, this remark also validates the criterion of the optimization problem (5.23), that minimizes the integrated sidelobe energy. The best results correspond to the case of 1_CPFSK_MMF. Because of the previous section, we would have expected that the best result would have been the case N_CPFSK_MMF. Nevertheless, the values for different CPFSK codes with their mismatched filters are very close to the 1_CPFSK_MMF. This effect can also be explained from the clutter contribution that hides the sidelobes of the corner reflector, as well as the choice of the window on which the ISL is computed. Figure 6.12 illustrates the same corner reflector of sub-figure 6.9a, but for the four cases of CPFSK codes with matched or mismatched filtering. We can observe that when one signal is transmitted, the sidelobes away from the mainlobe are higher compared to when transmitting different codes. This means that if the ISL window would have been larger then the sidelobe energy would have increased.

Analysis of

. Pearson Correlation Coefficient

The next evaluation criterion is the Pearson correlation coefficient. The reference image, or master image in image processing, is the re-synthesized SAR image created with chirp pulses. Results are presented in figures 6.13 to 6.16 for all the four areas of figure 6.9 and in the same order. Each sub-figure is created by computing the Pearson coefficient on a small window of size 11 × 11 on each area. The Pearson coefficients averaged over each whole sub-image are displayed in table 6.4, in order to quantify the impact of the clutter type on the coherence values. The purple-highlighted values indicate the best results for this coefficient on each compared area.

From figures 6.13 -6.16, a general tendency appears: the highest correlation with the reference image is obtained with the MMF. This is confirmed by table 6.4, which however reveals that the coherence is varying according to the SNR of the area. The high-energy areas have a higher coherence probably because the noise has less impact on them. In the following, every sub-area of figure 6.9 are compared more thoroughly.

Corner Reflector (Figure 6.13) In this configuration, the best result is obtained in the case 1_CPFSK_MMF (right top sub-figure), as indicated in Table 6.4. This is probably due to the fact that in the case of multiple signals (right down sub-figure), the correlation coefficients away from the mainlobe are lower, which could be explained by sidelobe averaging after the azimuth compression, and by the spreading of the sidelobes over the 2D plane rather that on range/azimuth axis as for the chirp. Note that the sidelobe behavior of the CPFSK codes near the mainlobe is not the same as the chirp one, because the first sidelobe of the chirp is around 2 dB higher than for the CPFSK codes. 6.9.

Signals

Building (Figure 6.14) We can observe that the points that backscatter the most of the energy of the building maintain their high energy in every configuration. Even in the first configuration, where one code is compressed with the matched filter and the contrast is of poor quality, the building energy is mostly preserved. The rest of the scene around the building has low similarity compared to the chirp energy distribution when using the MF (first column sub-figures), whereas the performance of the chirp and the MMF compressed communication codes (second column subfigures) is highly closer. The best similarity is reached when multiple codes are transmitted (right down sub-figure), as can also be observed in table 6.4.

Lake (Figure 6.15) The lake is an area of low intensity as the water is a specular reflector [Solbø and Solheim, 2005], which means that the incoming signals get mirrored at the surface of the water resulting to no backscatter. In such an area, most of the energy comes from the neighboring pixel sidelobes and from thermal noise. This is the reason why in all of the correlation images, the similarity of the lake is low, as shown in Table 6.4 compared to other areas. Thus, the remarks made on the lake performance should not be the similarity of the scene but to which extent the energy of neighboring pixels affect the lake ones. When observing the MF compression, the lake energy is completely different compared to the chirp, which means that the neighboring pixel energy greatly affects the pixel energy of the lake. From table 6.4, the best result is provided by the configuration N_CPFSK_MMF. Remarking the color scale of the image, the Pearson value should have been even lower for the lake image, but because the ground around the lake is taken to consideration in the correlation computation, the coefficient increases a bit. Forest (Figure 6.16) The vegetation area is a medium-intensity area. Similar results as before can be observed, as the MF outputs provide a bad correlation coefficient when compared to the chirp image forest intensity. Between the MMF outputs, the configuration where we transmit a different pulse at each acquisition is still the one providing the best value, shown in table 6.4.

. Structural Similarity Index and Mean Square Error

In the following, two tables provide the SSIM values for all four areas of the image, along with the MSE computations for comparison.

Corner Reflector

The MSE and the SSIM of the corner reflector are provided in table 6.5.

As previously, the energy of the corner reflector is concentrated on its range and azimuth axis as the reference chirp configuration. As mentioned from authors in [START_REF] Asamoah | Measuring the performance of image contrast enhancement technique[END_REF], if the structural similarity is greater than 0.4, then the two images are close. Using matched filtering, the SSIM is below this value, and thus the use of communication codes with such compression is unfavorable, in terms of preservation of the image characteristics. When transmitting a single code and compressing it with its mismatched filter, the similarity of the two images is really close, as both MSE and SSIM present improved results. Nevertheless, transmitting multiple signals provides close outputs too.

Building The MSE and the SSIM of the building are provided in table 6.5. In this area, we would have expected to have a higher similarity compared to the obtained values, as the building backscatters high-energy. A possible reason for this loss is the presence of some areas of the ground around the building that cause dissimilarity. The best configuration is given by the last case N_CPFSK_MMF. Like for the coherence, the performance of 1_CPFSK_MMF is still close to the best case. However when using MF, the results are not acceptable as the resemblance between the images is not achieved, as shown in table 6.5 (SSIM values Lake By taking into consideration the MSE and SSIM values of the lake in table 6.6, the obtained images differ a lot from the chirp one. This is justified by the very low back-scattered energy of the lake. In the 1_CPFSK_MF case, the result is the worst with 0 value, which means that this image and the chirp one are entirely different. In the N_CPFSK_MF case, the output result does not provide a much better performance, neither for the ground around the lake, nor by the samples from the lake itself. When it comes to MMF, the performance is slightly better. Yet it is mostly explained by the strong scatterers present around the lake.

Forest Finally, the MSE and SSIM values of the forest are displayed in table 6.6. Even if the energy is higher than the lake one, the results between the configurations do not have high similarity. Still, the case N_CPFSK_MMF provides the best performance, followed by the case 1_CPFSK_MMF. Using MF, the SSIM values are again below 0.4, which means that these images are not close to the performance of the chirp, due to the random nature of the clutter.

. Average energy of the lake

The last quantitative comparison criterion presented in this work is the average energy of a low-intensity area, chosen to be the lake, for the four configurations. In order to consider only lake points, a selection of points is considered and illustrated with a red box in figure 6.17. The points inside the red box are averaged to provide an insight into how these pixels with low energy are close to the noise level of the scene, and how they are influenced from different codes and compression techniques.

Table 6.7 provides the average value of the pixels of this area. Putting aside the chirp, the N_CPFSK_MMF configuration presents the lowest energy, followed by the 1_CPFSK_MMF configuration. The difference between the best communication code case and the chirp is 5 dB. This seems to indicate that because of the sidelobe spreading, the noise level increases, that can be explained by the way the noise is influenced by the simulation. When using the MF, the sidelobe pollution is even stronger resulting to a maximum loss of 16 dB compared to the chirp case, which is unacceptable. Table 6.7: Average value of energy over an area of the lake.

. Visual criterion on the shadow of aircrafts

Last but not least, we present an area where the previous criterion can be visualized. It is an area of the image where two planes are present, but their existence can only be guessed by their shadow. The aim is to observe in which configurations these shadows are visible or masked. Moreover, the ground on which the planes are displayed is made of asphalt which does not reflect a lot of energy ,but presents a higher reflectivity that for example water. Figure 6.18 illustrates this area for the chirp pulse compressed with MF (sub-figure 6.18a) and the different configurations of CPFSK codes (sub-figures 6.18b-6.18e). We can remark that the shadows are not visible for the configuration where MF is applied (sub-figure 6.18b & 6.18d). This signifies that the level of noise and clutter is augmented to the extent that the image contrast is degraded for very low-energy targets. However, examining the cases with MMF, the shadows can be noticed, especially for the aircraft on the left. This remark validates the efficiency of this filter on communication codes to preserve a satisfying contrast.

. Conclusion

Overall, in this chapter, re-synthesized SAR images are created through real raw data using CPFSK and compressing them with the MF or the MMF. From these images, we can quantify the behavior of such signals, and compare them with the chirp. The N_CPFSK_MMF configuration provides very good SAR performance, for different level areas, and the results can compete with the chirp case. Only the PSLR and the ISLR values are outperformed by the 1_CPFSK_MMF case, but they remain acceptable for the different signals too. However, these values may not be reliable as the ISLR window can bias the results. The disadvantage using these signals with MMF is the computational cost for the computation of all the needed filters. Still, we can conclude that transmitting different CPFSK codes and compressing them with their associated mismatched filter allows to create a joint SAR-communication system, with a well-contrasted SAR image and a good data rate. In the following, we will present real-data acquisition in an ISAR framework, in order to provide the CPFSK behavior in a real configuration, and towards a transciever to provide the range-Doppler map. Will the output image performance coincide with the re-synthesized results of this chapter? For our application, HYCAM is used in two different configurations. Firstly, we retrieve the raw data from a direction where a transceiver is positioned, and from which we can create its range-Doppler map. It is located on the roof of a building in ONERA, around 250 m away from the HYCAM radar. These maps allow us to observe how the impulse responses between the simulated and real data differ. The second case consists in creating an ISAR image of an airborne platform. This configuration is very close to a SAR framework. However, instead of having the antenna on the moving platform, the radar is located on the ground (or more precisely on a roof in our configuration), and exploits the motion of a target to create the synthetic aperture. This is why it is called Inverse SAR (ISAR) [START_REF] Cumming | Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation[END_REF]. Even if it is an inverse framework compared to SAR, it can provide important information and especially verify if it is feasible to create images using CPFSK codes that are comparable to the chirp ones.

In both configurations, the radar utilizes the same waveforms. The parameters with which the communication signals are generated, as well as some information on the radar acquisitions, such as integration time and PRI, are provided in table 7.1. As HYCAM parameters such as bandwidth and pulse duration differ from the airborne configuration studied in chapter 6, the CPFSK and the associated filters were redefined accordingly. In particular, the bandwidth is equal to B = 152 MHz and the pulse duration to T p = 10µs. As the bandwidth is smaller and the frequency sampling is less than 2B for the complex signal compared to the generation of the re-synthesized images in chapter 6, the sampling length of the codes will also be smaller. To best match the bandwidth, the remaining parameters were set in table 7.1. The above parameters refer to the CPFSK signals, however chirps were also transmitted in order to be able to provide a comparison between the two waveforms. The acquisitions were thus alternating between one chirp and one CPFSK code, as illustrated in figure 7.2. This means that for a duration of 0.1 s, the number of total transmitted pulses are 1000, composed by 500 chirps and 500 CPFSK alternating. The number of non-identical CPFSK transmitted signals is constrained by HYCAM memory, and thus in our configuration a maximum number of 500 different codes is considered. The PRF between a chirp and a CPFSK is thus equal to PRF = 10 kHz, and the block of 1000 signals is repeated until the end of the integration time.

For one given signal of 2000 samples, the time needed to compute the associated filter (of length three times the signal length) is 3.2 minutes. As the number of transmitted non-identical codes is N s = 500, the computational time needed for the generation of all mismatched filters is 19.3 hours. The number of transmitted signals are N pulses = 10500 CPFSK pulses as the integration time is 2.1s and PRF CPFSK = 5 kHz, with a block of these 500 different pulses transmitted every 0.1s. The data rate of the system can be thus computed through the equation (6.1). It is reformulated with the given notations as follows:

D real = log 2 (C) T p N CPFSK s T c N CPFSK pulses PRF CPFSK = 0.3Mbits/s. ( 7

.1)

In the following, we first provide the results obtained for the acquisitions performed towards the transceiver.

. Generation of SAR Images using CPFSK Codes on a Transceiver

In this section, we examine the configuration where HYCAM transmits alternating chirps and CPFSK towards a transceiver using the parameters displayed in table 7.1. Firstly, we present the processing applied on the signals in order to be able to provide the range-Doppler map of the point-scatterer-like signature of the transceiver, and afterwards the results are illustrated and analyzed. The range and Doppler cuts are provided for comparison along with some classical radar evaluation criteria.

. Signal Processing

The codes transmitted by HYCAM reach the transceiver with a time delay τ. This delay is produced by three different factors: the distance between the radar and the transceiver (around 250m), the radar internal delay (around 150m), and the additive distance that the transceiver adds to the signal (around 10 km). As the transceiver is stationary, the back-scattered signals that arrive back to HYCAM should not undergo Doppler shift. However, it can be easier for radar to detect moving targets, as the target can be naturally separated from the clutter, and as the processing usually includes clutter rejection which discards the zero Doppler axis to reduce echoes from stationary objects. For this reason, in this setting, the transceiver creates an artificial Doppler shift on the data before back-scattering the signals back to the HYCAM antenna, thus helping to separate the transceiver contribution from the ground echoes. This effect will be visualized as a shift on the Doppler axis of the range-Doppler map.

The amplitude and the spectrum of the received signal for the chirp and for one CPFSK code are illustrated in figure 7.3. The amplitude in sub-figure 7.3a shows a high energy block for a duration of T p , during an interval [ε, T p + ε]. This amount of energy observed during a pulse duration corresponds to the coupling created between the transmitter and the receiver when a signal is transmitted. ε corresponds to the small amount of time between the start of the timing and the signal transmission. In order to avoid polluting the remaining data with this coupling, it is removed from the beginning of the received pulse as it is visualized in figure 7.3a with a grey vertical line. Once the cut is performed, the spectrum of the received signal is plotted in sub-figure 7.3b. We can identify the rectangular behavior of the chirp spectrum and the mainlobe of the spectrum for the CPFSK that starts to decrease slightly.

Afterwards, range compression is performed either with the matched or the mismatched filter. Three different configurations are provided:

⋆ Chirp acquisitions with matched filter; ⋆ CPFSK acquisitions with matched filter; ⋆ CPFSK acquisitions with mismatched filter.

The chirp case is provided without apodization in order to avoid widening the mainlobe and thus degrading the resolution. For the transceiver configuration, we do not have to compensate any range migration as the radar is not moving. However, the environmental clutter, such as building and trees, is at zero-Doppler and could potentially be high. The zero-Doppler value can be rejected from the range compressed data, to enhance the visibility of the target, thanks to its artificial Doppler. To do so, an projection on its orthogonal sub-space (zero-Doppler sub-space) can be performed for each range cell in order to clear out the clutter. The problem can be expressed for a given delay as [START_REF] Colone | A multistage processing algorithm for disturbance removal and target detection in passive bistatic radar[END_REF],

ŵ = arg min w ||y -Xw|| 2 , (7.2)
where X is a matrix that gathers the different Doppler hypotheses to be rejected, y is a vector of size N p × 1, N p is the number of pulses. In a general rejection framework, if N r Doppler hypotheses must be rejected, then X is of size N p × N r . In our case, we only consider one hypothesis, the zero-Doppler response, and thus the matrix X is of size N p × 1. For comprehension ease, let X = d 0 , as matrix X is transformed to a vector, and d 0 = [1 . . . 1] T . Hence, w is transformed to a scalar value.

The solution of problem (7.2) is expressed as

1 ŵ = d H 0 y d H 0 d 0 . (7.4)
Finally, the signal after clutter rejection is defined as the signal y subtracted with its projection 1 Let f (w) = ||yd 0 w|| 2 , then its gradient with respect to w is expressed as

∇ w f (w) = -2d H 0 y + 2d H 0 d 0 w. (7.3)
By taking the root of the gradient, we can find the optimal value of w as on the zero-Doppler plane provided by d 0 ŵ and it can be expressed as

ŵ = d H 0 d 0 -1 d H 0 y.
y rej = y - d 0 d H 0 d H 0 d 0 y. (7.5)
In our case, equation (7.5) can be implemented by the subtraction of the expected value of y from y, as the denominator in equation (7.5) coincides to the number of pulses, and d 0 contains only ones.

Once the clutter rejection is performed on the range compressed data, the azimuth compression is performed through an IFFT. Figure 7.4 represents the range-Doppler maps for the three configurations after radar processing and zero-Doppler rejection. The first column illustrates the plots with a large-range colorbar in order to visualize the zero-Doppler rejection, whereas the plots of the second column provide the results but centered around the response of the scatterer and in a lower colorbar range in order to observe the sidelobe level of each configuration.

. Result Analysis

After applying the processing provided in subsection 7.2.1, we obtain a transceiver response on the range-Doppler map.

Figure 7.4 illustrates the resulting transceiver contribution, on which a Doppler shift of 313 Hz was applied, as observed on the Doppler axis. Sub-figure 7.4b displays the configuration where the chirp pulses are transmitted and compressed with the MF, whereas sub-figures 7.4d and 7.4f show the CPFSK compressed with MF and MMF, respectively. We can observe the classical sinuscardinals in both range and Doppler directions for the chirp pulse. For the CPFSK signals, the sinus-cardinal can be seen along the Doppler direction, whereas for the range axis, the sidelobe averaging created from the different embedded messages in the phase of each pulse can be remarked as in chapter 6. This effect provides lower sidelobes than using one identical signal for the image generation, but creates a higher noise level, especially for the case when the signals are compressed with matched filtering (sub-figure 7.4d). To numerically quantify this effect, we compute the average noise level over a small window away from the mainlobe the area of which is indicated by a red box on sub-figure 7.4b. Table 7.2 provides the values of the average noise floor over a small window located on the bottom right side of each range-Doppler map. The values of this table validates the visual remarks: the noise level of the chirp is the lowest as the energy is mainly restrained along the range and the Doppler axes, followed by the CPFSK firstly compressed with the MMF and then with the MF for which the sidelobe energy is diffused in the 2D plane.

More information on the point scatterer of the transceiver can be provided by plotting the cuts along the range and Doppler axis, as presented in figure 7.5, along with the spectrum after clutter suppression and radar processing. The cut providing the range compression is illustrated in sub-figure 7.5a, and a zoom on the compression peak is provided in sub-figure 7.5b. Observing these two figures, the curve created from chirp pulses shows lower sidelobe level away from the mainlobe peak than both cases of CPFSK codes, with performance closer to the configuration with the MMF. However, when zooming on the mainlobe peak, the sidelobes close to it are higher than the ones for the CPFSK signals, due to the absence of weighting window for the chirp. For the case of CPFSK signals compressed with MMF, the support of the compression output is larger than the rest of the results as the support of the mismatched filter q is three times larger than the transmitted signal. It can be remarked from both sides of the compression output in sub-figure thus less fluctuating spectrum for the point scatterer. We observe that the spectrum energy of CPFSK decreases at the extremities, but less than the chirp. Finally, sub-figure 7.5d displays the cut along the Doppler axis, which is very similar in all three configurations, as expected. The small peaks around the mainlobe are created by some defects of the transceiver. Since they are present for all three different configurations, it is not due to the code choice or the range processing. Finally, the classical PSLR and ISLR criteria are applied on the resulting compressed output in order to numerically evaluate the performance. They are displayed in table 7.3, along with an extra criterion, called the ISLR 2D. It is defined as the classical ISLR (the ratio between all sidelobe energy and the mainlobe energy) but it takes into account the 2D rectangular grid around the mainlobe, to better apprehend the 2D spreading created by the diversity of the transmitted signals.

Let us denote W the input image on which the ISLR 2D is computed. It is of size P r × P dop , where P r denotes the number of range cells, and P dop the number of Doppler cells. The ISLR 2D is then mathematically defined as where Ω ML,r and Ω ML,dop are the sets that contain the mainlobe position along the range and Doppler directions, respectively. The best results for the PSLR and the ISLR values are given by the configuration CPFSK MMF, as in chapter 6, when comparing the corner reflector of the SAR image, and the worst case is given by the chirp configuration, followed by the CPFSK MF configuration. These values are computed with respect to the 1D range axis of the image. However, when observing the area around the mainlobe, the configurations for which non-identical signals are transmitted present higher energy, and thus the practical motivation of computing the ISLR 2D. Observing table 7.3, the best value for the ISLR 2D is provided by the chirp pulse compressed with the MF. Such a result was expected, as the chirp pulse does not diffuse sidelobe energy away from the range and Doppler axis. On the contrary, for both CPFSK configurations, there is an off-axis sidelobe energy increase. The worst result is given when CPFSK signals are compressed with the MF. The difference between this case and the one compressed with the MMF is around 2.5 dB. Nevertheless, even if the chirp is the best with an ISLR 2D of -7.18 dB, the case of CPFSK MMF has only a difference of around 0.6 dB, proving thus a satisfactory performance for this configuration.

. Aircraft ISAR Imaging using CPFSK Codes

In a similar framework, data acquisitions were performed while HYCAM was transmitting signals in the direction of a flying airborne platform. The target of interest was an Airbus A320 landing at Orly airport, close to Paris. Figure 7.6 illustrates the distance between the radar and the airport, as well as the runway that indicates the landing direction of the platform, and figure 7.7 provides a photo of such an A320 aircraft.

The same signals as in the previous section were transmitted, alternating between a chirp and a CPFSK signal as illustrated in figure 7.2, and using the same parameters displayed in table 7.1. However, in this configuration the target was moving. Besides from a range Doppler map, this section will also provide an Inverse SAR (ISAR) image of the aircraft, and the processing applied on the data is slightly different from the SAR one after the range compression. A fundamental difference is that the clutter response in ISAR is much lower and will less pollute the image, allowing us to have a better vision of the signal behavior on the target, compared to a SAR image where all neighboring pixels usually contribute with their sidelobes in the under-consideration cell.

For both the range-Doppler and the ISAR image, a step of range migration needs to be applied.

For the former, we will apply the Keystone transform, and for the latter a classical ISAR range alignment method. The range compression step coincides in both cases. Thus, in the following, we first introduce the raw data and the corresponding range compression. Afterwards, the two processing methods are introduced, with their associated results comparing the CPFSK outputs with the chirp one.

. Raw Data Analysis

We present in figure 7.8 the amplitude and the spectrum of the chirp and CPFSK signals as in section 7.2.

The amplitude in sub-figure 7.8a has the same high energy block caused by the coupling as remarked for the transceiver case. The main difference lies in the intensity of the amplitude along the time axis, which is created because of the difference of the line of sight of the radar for the two different configurations. For the transceiver case, the radar was slightly tilted towards the ground, whereas in the aircraft configuration the radar is pointing towards the sky. For the transceiver, due to clutter contributions, high energy can be observed along the whole time axis, whereas in this setting the first block of energy after the coupling corresponds to the aircraft and the second to some unknown contribution further away from the plane, probably some clutter contribution.

To avoid polluting the final image with the transmitter-receiver coupling, it is removed from the beginning of the pulse. After cutting the coupling, the spectrum of the received signal is plotted in sub-figure 7.8b. We can identify the rectangular behavior of the chirp spectrum which is a bit distorted, and roughly the mainlobe of the spectrum for the CPFSK, which is also not very clean, as we cannot see clearly where the mainlobe of the spectrum ends.

. Range Compression

Afterwards, range compression is performed either with the matched or the mismatched filter. The same three configurations are considered as for the transceiver:

; Chirp acquisitions with matched filter, ; CPFSK acquisitions with matched filter, ; CPFSK acquisitions with mismatched filter.

After range compression, the position of the aircraft can be observed on the range axis. However, here there are two main different lines detected along the range axis, as it can be observed in sub-figure 7.9a, that represents the output of the data after range compression. It is more probable that the plane range position corresponds to the one that has the highest energy. To ensure the result, the aircraft should appear on a non-zero Doppler value on the range-Doppler map due to its motion. After performing Doppler processing via an IFFT, the range-Doppler map is displayed in sub-figure 7.9b. We can observe that the aircraft is located at a range of around 6530 m, whereas the other contribution probably corresponds to clutter elements since it is located at zero Doppler.

A problem that arises here is range migration. If the target was stationary or very slow, then its contribution would appear at the same range cell over the whole pulse train. Then by performing a classical Doppler compression, a well-formed, full-resolution compression peak would be generated in the azimuth direction, as for the transceiver point scatterer. However, here the aircraft has a non-negligible velocity with respect to the range resolution, creating range migration, affecting both range and azimuth resolution. These types of planes have a velocity of order of 100 m/s at landing, which results to a maximum range migration of 210m as the integration time is T int = 2.1s. Since the range resolution is equal to ρ r = c 2B = 0.98m, a maximum of 214 migrated cells could appear. However the radial velocity is inferior to the plane speed and in our context is small as the range migration is around 30 m, observed in sub-figure 7.10a. Indeed, by calculating the plane velocity we have

v plane = f d c 2 f c ≈ 15m/s, for f d = 300 Hz.
where the choice of f d = 300 Hz is observed in figure 7.10, and can be chosen in the interval [220,300] Hz. Thus, the distance due to migration is d migration = 15 × 2.1 = 31.5m, (7.7) that approximately validates the migration observed in sub-figure 7.10a coincides with the output. The contribution points of the platform thus clearly migrates over time, which are spread in both range and Doppler of the target response. This phenomenon can be observed in sub-figure 7.10b, which is a zoom of sub-figure 7.9b, where the plane contribution is spread over the range and Doppler axes. For this reason, the next two subsections deal with two different ways of realigning the range profile in order to provide a focused illustration of the aircraft.

. Keystone Transform

The Keystone method is a transformation performed on the range compressed data to correct the range migration [Richards, 2014]. The goal is to limit the spreading in range and Doppler due to the range migration, which means to get rid of the rectangular behavior observed in sub-figure 7.9b and obtained after processing due to the motion of the platform. This would have been the case if the mainlobe after pulse compression would have not depended on the range bin, and thus every peak would have been in the same range bin.

The objective of the Keystone transformation is hence to remove the coupling between the frequency of the range dimension and the velocity that is created after the range compression. To do so, a variable change of the slow-time variable can be performed in order to re-scale the slowtime axis with respect to the fast-time frequency [Richards, 2014]. Explanations on the Keystone transform are provided in Appendix B. After this variable change, the range compressed data are re-aligned, and thus the range migration is removed, as shown in figure 7.11. It slightly degrades the Doppler resolution. Figure 7.12 illustrates the Fourier Transform in fast-time and inverse Fourier transform in slow time data before and after providing the Keystone transformation. They show the change of grid in the range frequency space, as we can observe a different geometry after the transformation (difference at the border of sub-figure 7.12b).

The range-Doppler images for all the three configurations for different signals with the Keystone processing are provided in figure 7.13. The image created by the chirp pulse with MF, illustrated in sub-figure 7.13a, is the reference case to be compared with the CPFSK. Observing the chirp pulse in sub-figure 7.13a, the energy contribution is located in the range and Doppler directions of the highenergy points, but not on the 2D plane. However for the images created from the CPFSK signals, the sidelobe energy is spread in the 2D map, and especially along the Doppler axis. This remark The average energy level away from the plane is higher compared to the chirp image. To provide a better comparison, we compute the average noise level of each sub-figure of figure 7.13 over a small window located at the bottom right of each image, the area of which is illustrated in sub-figure 7.13a by a red box. Theses values are displayed in table 7.4, and the best output is for chirp as expected, followed by the case CPFSK MMF and then CPFSK MF, and each with a difference of around 10 dB from its previous one.

Moreover, a zoom on the plane is also provided in figure 7.14, for which the color contrast of the points is chosen in the range [-50, 0] in dB to emphasize the high energy points. In these subfigures, we can remark that the plane is easily noticeable and distinguishable in every configuration, with different sidelobe distribution for each one of them.

Moreover, the range cut of the highest energy point of the plane, illustrated in sub-figure 7.15a, has lower ISLR. Because of the typical decreasing behavior of the chirp sidelobes, it is particularly visible at range 6500m-6550m, whereas the CPFSK signals have lower sidelobes around the highenergy points. Away from the mainlobes, the chirp sidelobes meet the sidelobes of the CPFSK MF on the range axis. The sidelobes of the latter case are higher than when they are compressed with MMF, which shows the smallest ISLR compared to the other cases. Finally, the Doppler cuts are displayed in sub-figure 7.15b, that correspond to the highest energy peak along the Doppler direction of the aircraft. All the curves present a very similar behavior, but not identical, unlike the previous studied cases, as it can be observed in table 7.5, where the mean value of the Doppler cut for each curve is computed. We assume that the differences between the curves come from the Keystone algorithm and more precisely from the performed variable change.

Thanks to the Keystone transform, the range profiles have been correctly realigned, enabling to obtain a well-focused range-Doppler image. In the next subsection, we will analyze the data with a different algorithm related to the ISAR processing.

. Classical ISAR Algorithm

In this subsection, a classical ISAR algorithm is presented. The following description is not a complete literature review on the ISAR method, as this domain is not much explored in this thesis, but the main features are explained in order to accompany our results by a theoretical background. Firstly, a brief description of the framework is provided, and then the generated results are displayed.

. Brief Description of the ISAR framework

The term ISAR stands for Inverse Synthetic Aperture Radar, as it represents an opposite configuration from the classical SAR framework. Whereas the SAR framework consists in exploiting the movement of an airborne platform measures to create a synthetic antenna and improve the azimuth resolution, the ISAR framework is mainly introduced as a ground stationary radar that images moving targets. One could say that the difference between the two configurations lies in the definition of the reference point [START_REF] Chen | Inverse synthetic aperture radar[END_REF]. For example, in a SAR framework, if the reference point is taken from the radar to the target, then the framework resembles to an ISAR configuration. The substantial difference is the cooperation of the target, since in a SAR framework, the airborne motion is known, as opposed to the non-cooperative target in ISAR. If the relative velocity of the target is unknown to the radar, then the azimuth direction processing where g r (x, y) is the reflectivity function, and p(t) is the distance from the radar to the scatterer such that p(t) = R(t) + x cos θ (t)y sin θ (t) for a target positioned at (x, y) with range R(t) and rotation angle θ (t), as illustrated in figure 7.16, that changes with respect to time.

The main processing pillars of an ISAR framework consists in applying a range alignment and a phase correction on the range compressed data. The range alignment, also called range tracking, consists in keeping the scatterers energy in their range cells, and the phase correction, also called Doppler tracking, maintain a stable Doppler shift, as if the platform had a constant velocity. These methods can focus raw data into an ISAR image as long as the angular velocity Ω is quasi-stable such that θ (t) can be modeled as θ (t) = θ 0 + Ωt + γt 2 , with a small angular acceleration γ. We

Raw data

Range compression Range alignment Phase adjustment Polar reformatting ISAR image Figure 7.17: ISAR processing algorithm.

can remark that the two variables creating the ISAR image are the range and the rotation angle functions, instead of the range and azimuth ones for the SAR framework.

The resolution for the range direction is the same as in SAR, and is as usual dependent on the bandwidth such that ρ r = c 2B . The resolution on the other direction linked to the rotation angle is called cross-range resolution. It is expressed as [START_REF] Chen | Inverse synthetic aperture radar[END_REF] ρ cr = c 2 f 0 ΩT int , (7.9)

where T int is the integration time. Figure 7.17 illustrates the main steps of an ISAR processing. Here, this processing is briefly introduced to provide an overall description [START_REF] Chen | Inverse synthetic aperture radar[END_REF]:

• Once the raw data are retrieved from the antenna, range compression is applied. Usually the matched filter performs the pulse compression, but in our framework, we also use the MMF with the CPFSK signals.

• Afterwards, range alignment is performed, which compensates the transitional motion of the range profiles by aligning them (not-aligned profiles are illustrated in sub-figure 7.10a).

In such a way, the aircraft scatterers will remain in the same range bins. In order to do so, a reference range profile is chosen as the dominant scatterer by finding the maximum magnitude of each range profile. Then the number of cells that need to be shifted compared to the reference point is estimated, and each profile is aligned with respect to the reference one. The dominant scatterer algorithm is one way to deal with range misalignments.

• However because of this procedure, phase drifts may occur from one range profile to another, which create phase errors that result in a non-linear phase over the number of pulses. Thus the next step is the phase adjustment, and the most common way to perform phase adjustment is the minimum variance method. In this method, the scatterers with the smallest variance are determined, they are averaged and used as a phase correction to compensate the phase for all the range bins. After this step, the corrected phase function is linear, and the ISAR image is usually well focused.

• In the ISAR framework, Doppler shifts can also occur due to high rotation of the target, which lead to an unfocused image and the need for further processing. Such an effect occurs when the angular velocity is not constant, creating a Doppler shift that shifts the scatterers to the neighboring cells. Rotational motion compensation step deals with this problem by using the polar formatting algorithm from the tomography world. More precisely, if the rotation angle is large, then the data have a "circular" behavior [START_REF] Chen | Inverse synthetic aperture radar[END_REF] and can be expressed onto a polar grid with variables f 0 and θ . This algorithm seeks to remove this effect by mapping the polar coordinate data onto a rectangular grid in order to focus the scatterer points of the aircraft. In our configuration, the rotation induced from the movement and the velocity of the platform is small, but this algorithm is still applied to slightly improve the output result.

. Results from the HYCAM Acquisitions

After presenting the theoretical ISAR imaging framework, we provide in figure 7.18 the output images of the aircraft for the chirp with the MF and the CPFSK with the MF and the MMF.

The results have similar behavior with the previous images using Keystone during the processing, but the representation of the planes is not the same even if the scatterers are identical. The aircraft can be seen in all configurations even if the noise floor is variable among the three configurations. In terms of image representation, the ISAR images are represented with respect to the range and the frequency induced by the angular velocity Ω.

Comparing the different outputs, sub-figure 7.18a displays the chirp pulse compressed with the ). As mentioned earlier, the mismatched filter minimizes the sidelobe level on the range axis, but the off-axis sidelobes increase. This effect can be quantified by computing the average energy over a small window away from the scatterers on each configuration, the area of which is indicated by a red box in sub-figure 7.18a. Results are provided in table 7.6. As expected the best value is given by the chirp for which the noise floor level is as low as -86 dB, followed firstly by the CPFSK MMF case with 8 dB of difference and lastly by the CPFSK MF configuration with 9 dB of difference from the previous one. These results are close to the ones observed with Keystone. A zoomed version of the aircraft sub-figures is also provided in figure 7.19 with a reduced colorbar. For the chirp case, range sidelobes are higher close to the main energy points than for the CPFSK case. However in cross-range direction, the sidelobe level for the two CPFSK cases is higher compared to the chirp. Still, the aircraft can be easily spotted, especially after the ISAR processing.

In order to better visualize the range and cross-range sidelobes on the aircraft image, we provide the cuts of the highest energy point along the two dimensions in figure 7.20. Sub-figures 7.20a and 7.20b illustrate these cuts for range and cross-range axes respectively. For the range dimension, the sidelobes of the chirp look higher as they slowly decrease, whereas for the CPFSK cases, they are more uniform, with the lowest level for the CPFSK MMF case. On the other side, for the cross range axis, the chirp case provides lower sidelobes away from the main peaks followed by the CPFSK MMF and then the CPFSK MF case. We would expect to have a similar behavior on the cross-range axis as for the Doppler axis previously. A potential reason for this difference is maybe the phase adjustment step of the ISAR algorithm. Unfortunately, it is not possible to compute the evaluation criteria as we did for the transceiver, because the number of high-energy scatterers is more than one and the contribution of each one of them affects the neighboring points.

. Conclusion

In this last chapter, we provided some real data acquired by HYCAM using the CPFSK modulation and mismatched filtering, in order to evaluate their performance. To do so, both cases of a transceiver and an aircraft were considered in order to provide the performance of a single high scatterer and the behavior of a set of scatterer points. Overall, the chirp provides a better contrast, whereas the CPFSK MMF provides a more uniform energy level spread in the 2D axis and lower sidelobes close to the mainlobe in range direction. The use of MMF instead of MF enables to strongly improve the image quality in the case of the CPFSK code, and it seems robust to the deformation after hardware impact. Moreover, even if the MMF was computed for a zero-Doppler value, the Doppler of both the transceiver and the plane seem not to degrade the performance. Finally, we can conclude that the images using CPFSK signals and compressed with MMF have acceptable performance for a radar-communication system, at least from a radar perspective, and they can tackle the problem of the spectral congestion by transmitting information, and performing radar sensing at the same time, under the constraint that the SNR after mismatched filtering is lower than the SNR of the system.

CHAPTER 8

Conclusion and Future Work

. Conclusion

The work of this thesis allows us to respond to our initial question: Can we create a cooperative SAR-communication system using communication signals that provides an acceptable radar performance? We can now reply Yes! Our work is a small contribution to the large problem of electromagnetic spectral congestion, by creating a joint SAR-communication system. As we have seen, the main strategies that could have been followed to tackle this problem were co-existence, cooperation, and co-design. As our work is more radar-oriented, we chose to create a cooperative system and work with communication signals that are unusual to radar applications in order to ensure information transmission, and then to focus on the radar processing optimization. Through an extensive comparison, the CPFSK modulation has been chosen to be the waveform of the joint system, because of its constant envelop and spectral containment.

Performing matched filtering compression with such signals generates high sidelobes which can be problematic for the image contrast. For this reason, we have proposed to replace the MF by a MMF as it minimizes the sidelobe energy. It can be computed by solving a convex optimization problem. We have remarked that in cases where the received signals are degraded compared to the transmitted ones, the filter is no more optimal and it can result to deteriorated performance. Thus to limit a sidelobe increase on the output, we have defined optimization problems that can handle intra-pulse Doppler shifts or off-grid delay problems. The former one can be computed using any signal as it is signal-independent, whereas the latter one is directly computed in the special case of CPFSK. The choice of using a reinforced filter is dependent on the application. However, when considering a SAR-communication system, an interesting characteristic for the communications is the maximization of the data rate. In our framework, it can be achieved by transmitting a non-identical pulse at each acquisition. Choosing such framework implies that a mismatched filter

Titre:

  Traitement Optimisé des Formes d'Ondes à Modulation de Phase Continue pour les Systèmes Conjoints Radar-Communication : Application à l'Imagerie Radar Mots clés: Radar à Synthèse d'Ouverture, Communications, Filtre Désadapté, Modulation à Phase Continue, Forme d'Onde

  Framework and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Basics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.1.1 Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.1.2 Radar Signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.3 Linear Frequency Modulated Pulse . . . . . . . . . . . . . . . . . . . . . . . 13 2.1.4 Matched Filter and Range Resolution . . . . . . . . . . . . . . . . . . . . . 15 2.1.5 Ambiguity Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.6 Coherent Pulse Train and Doppler Resolution . . . . . . . . . . . . . . . . . 18 2.2 SAR Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.2.1 Synthetic Aperture Radar Principles . . . . . . . . . . . . . . . . . . . . . . 22 iii 14 Algorithme ISAR. Raw data = données brutes, range compression = compression distance, range alignment = alignement de distance, phase adjustment = ajustement de phase, polar reformatting = reformatage polaire, ISAR image = image ISAR. . xxxv 15 Images ISAR de l'avion par le radar HYCAM pour le chirp et les CPFSK avec filtrage désadapté Range = distance, frequency induced from the angular velocity = fréquence induite par la vitesse angulaire. . . . . . . . . . . . . . . . . . . . . . . xxxv 2.1 A simplified radar system diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 A Simple Example of a Radar Framework . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Representation of the transmitted and the received pulses and a configuration when the range values exceed the range ambiguity. In this case, the received echoes of the first transmitted signal arrive after the transmission of the second pulse. This could lead to localization problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Frequency, phase and spectrum plot of the chirp pulse centered at f 0 = 0. . . . . . 14 2.5 Matched filter output for a noiseless chirp for a target at τ = 0s. On the second sub-figure, the part of the curve in blue corresponds to the mainlobe and the rest of the curve in orange illustrates the sidelobes.T p = 1µs, so the resolution before matched filtering would have been 150m. . . . . . . . . . . . . . . . . . . . . . . . . 16 2.6 An example where a strong energy target masks a low-energy one after matched filtering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.7 Ability of target discrimination with respect to the range resolution. The first subfigure represents a configuration where two targets cannot be separated because their relative delay is lower than the range resolution. On the contrary, in the the second sub-figure, the two targets are separated by a distance higher than the resolution and thus they can be discriminated. . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.8 The ambiguity function of the chirp pulse is dispayed in the middle, on top the zero-Doppler cut and on the left the zero-delay cut. Here B = 352MHz and T p = 1.5µs. 19 2.9 Ambiguity function of a Gold code along with the zero-Doppler and zero-delay cuts. Here B = 352MHz and T p = 1.5µs. . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.10 Coherent train of pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.11 Range-Doppler processing for a simulated scatterer using a train of chirp pulses. Parameters: B = 352MHz, T p = 1.5µs, number of pulses 1500 and PRF= 5000. . . . 21 2.12 Zero-Doppler and zero-delay cuts after the range-Doppler processing on a chirp pulse. 21 2.13 Optical image from Google Maps and SAR image acquired with the airborne platform SETHI [Ruault du Plessis and Dreuillet, 2013]. . . . . . . . . . . . . . . . . . 23 2.14 SAR Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.15 Illustration of the Integrated Sidelobe Level Ratio (ISLR) and Peak-to-Sidelobe Level Ratio (PSLR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.1 A simple example of a digital communication system. . . . . . . . . . . . . . . . . 32 3.2 Binary constellation represented by 0 and 1 bits and embedded in a signal. . . . . 32 3.3 An 8-Phase Shift Keying circular constellation and a 16-Quadrature Amplitude Modulation constellation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.4 ITU Frequency Spectrum from [Manning, Catherine, 2018]. . . . . . . . . . . . . . 35 3.5 IEEE Frequency Bands. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 4.21 Phase, spectrum and amplitude of an Orthogonal Frequency Division Multiplexing (OFDM) pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.22 Ambiguity function of an OFDM code, with the zero-delay cut on the left, and the zero-Doppler cut on the top. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4.23 Zoom on the mainlobe of the OFDM ambiguity function. . . . . . . . . . . . . . . 75 4.24 Spectra and 99% Power Bandwidth for different codes. The 99% Power Bandwidth is delimited by the vertical lines with color associated to the curve colors of the different codes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.25 Auto-correlation of the filtered signals after 1.2B and 2B cuts with the initial code. The small windows on each sub-figure is a zoom at the peak of the mainlobe to visualize the loss after the compression. . . . . . . . . . . . . . . . . . . . . . . . . 77 4.26 Amplitude distortion of LFM-PSK, BPSK and CPFSK after filtering with spectral cut at 1.2B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.1 Different compression outputs for a Continuous Phase Frequency-Shift Keying (CPFSK) code. The matched filter, the considered mismatched filter (with β = 1 dB) and the analytic mismatched filter are displayed. . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Matched filter ('MF ') and mismatched filter ('MMF 0 ') that minimizes the PSLR criterion for a CPFSK code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.3 Matched and mismatched filter output for the transmitted signal and mismatched filter with a signal that undergoes a Doppler shift equal to 404000 Hz. The ISL value of the MF for a zero-Doppler shift signal is 1.46 dB, the ISL value of MMF for a zero-Doppler shift signal is -4.47 dB and the ISL value MMF for the signal with Doppler shift is -2.08 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 5.4 Integrated sidelobe level for a CPFSK code in dB over two different Doppler intervals and for different compression filters. . . . . . . . . . . . . . . . . . . . . . . . . . . 91 5.5 On grid and off-grid points on the time axis. . . . . . . . . . . . . . . . . . . . . . . 93 5.6 Matched filter output for an on-grid and an off-grid signal sampled at sampling frequency 1.5B. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.7 Mean ISL for CPFSK signals over the sampling cell Θ for 100 Monte Carlo simulations. The standard deviation is represented by the bars around the mean value of the ISL obtained on the Monte Carlo simulations. The chosen sampling rate is 2 samples per symbol. The symbol number is 100 per CPFSK signal. . . . . . . . . . 95 5.8 Mean and standard deviation values on 50 Monte Carlo iterations for the mean ISL of CPFSK signals over the sampling cell with respect to the sampling frequency factor α representing the number of samples per symbol. Here, the CPFSK signals contain 50 symbols. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.9 Mean and standard deviation values on 50 Monte Carlo iterations for the mean ISL of CPFSK signals over the sampling cell with respect to the symbol number. The sampling frequency factor is 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 5.10 Duality gap between the primal and the dual optimization problem. The number of variables indicated here correspond to our considered problem. . . . . . . . . . . . 100 6.1 Real-data SAR image acquired from SETHI using chirps. . . . . . . . . . . . . . . 110 6.2 SAR Image Reconstitution Algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.9 Range compressed data and range-Doppler image generation after IFFT for the case of the chirp pulse compressed with MF. . . . . . . . . . . . . . . . . . . . . . . . . 148 7.10 Range migration observed on the range compressed data, and the plane illustration after azimuth compression through IFFT, without range migration correction. . . . 148 7.11 Range compressed data after application of Keystone transformation. . . . . . . . 150 7.12 Range compressed data in the Fourier space for the fast time and in the time domain for slow-time before and after Keystone transformation. We can observe the interpolation effect at the border of (b). . . . . . . . . . . . . . . . . . . . . . . . . 150 7.13 Range-Doppler image of the aircraft for the three configurations presented before after applying the Keystone transform. The red box corresponds to the zone used for noise level estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 7.14 Zoom of the range-Doppler image on the aircraft for the three configurations. . . . 152 7.15 Range and Doppler cuts of the highest energy point on the aircraft. . . . . . . . . 153 7.16 Illustration of the two main variables R(t) and θ (t). This figure is taken from [Chen and Martorella, 2014]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.17 ISAR processing algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.18 ISAR images of the aircraft for three different configurations. The red box corresponds to the zone used for noise level estimation. . . . . . . . . . . . . . . . . . . 156 7.19 Zoom of the ISAR images on the aircraft for three different configurations. . . . . 157 7.20 Cuts along the range and the cross-range dimensions plotted for highest energy point of the aircraft. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 1 PSLR et ISLR pour le signal réfléchi sur le répondeur. Les valeurs surlignées indiquent la meilleure valeur de chaque colonne. . . . . . . . . . . . . . . . . . . . . . xxxiv 2.1 Chirp parameters for the generation of Figure 2.4. . . . . . . . . . . . . . . . . . . 15 2.2 ISLR and PSLR values for a compressed chirp pulse. . . . . . . . . . . . . . . . . . 29 3.1 Various radar and communication applications in different electromagnetic bands. 37 4.1 Parameters used for the generation of the different waveforms. . . . . . . . . . . . 67 4.2

  .24. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Energy loss of the mainlobe after matched filter (MF ) with the input non-filtered signal with spectral cuts at 1.2B and 2B. . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1 Parameters used in this chapter to generate the signals. In case of any change in one or more values, it is specified in the main body of the document. . . . . . . . . 82 5.2 Mean ISL values for the compression with the different filters. . . . . . . . . . . . . 96 5.3 Parameters of the algorithm 1 for the mismatched filter computation. . . . . . . . 103 5.4 Computational cost of the proposed algorithm and the CVX solver for one optimal mismatched filter, and the extrapolated computational time for 6522 filters. . . . . 103 6.1 Parameters and characteristics of the CPFSK codes and the real-data Synthetic Aperture Radar (SAR) images. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 6.2 PSLR and ISLR values in the range direction on the different impulse responses.

7. 1

 1 Parameters of the system and the code generation . . . . . . . . . . . . . . . . . . 7.2 Average value of the noise floor over a small window away from the mainlobe. . . . 7.3 PSLR, ISL, and ISLR 2D for the transceiver signal. The highlighted values indicate the best value for each column. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Average value of the noise floor over a small window away from the mainlobe on the bottom right of each sub-figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Mean value along the Doppler cut in order to observe the energy difference in dB between the outputs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Average noise level on the bottom right part of the plane sub-figures in figure 7.18.

Figure 1 :

 1 Figure 1: Allocation des fréquences aux États-Unis en 2016. Image récupérée du site https: //www.ntia.doc.gov/files/ntia/publications/2003-allochrt.pdf.

Figure 2 :

 2 Figure 2: Géométrie d'un radar aéroporté. Range = distance, azimuth = azimut, target = cible, ground range = distance au sol, near range = distance courte, far range = distance longue, slant range = distance radar-cible, beam footprint = emprunt du sol.

  Auto-corrélation d'un chirp, équivalente à la sortie de ce signal traité avec le filtre adapté. Time = temps. Traitement range-Doppler pour un point brillant simulé en utilisant des chirps. Range = distance.

Figure 3 :

 3 Figure 3: Résultats de la compression distance d'un chirp et la sortie du traitement range-Doppler.

Figure 4 :

 4 Figure 4: Un système radar-communication coopératif. Radar target = cible radar, communication transmitter = émetteur des communications, receiver = récepteur, mutual infromation = information mutuelle, radar transmitter = émetteur radar, radar-communication receiver = récepteur du système radar-communication.

⇝

  la modulation Phase Shift Keying (PSK), ⇝ la modulation PSK intégrée dans la phase d'un chirp LFM-PSK, Fonctions de densité spectrale théoriques pour plusieurs signaux CPFSK pour différents indices de modulation h et tailles de constellations M. L'axe horizontal correspond aux fréquences normalisées.

Figure 5 :Figure 6 :⇝

 56 Figure 5: Phase linéaire par morceaux d'un signal CPFSK et des densités spectrales de puissance théoriques pour plusieurs paramètres.

Figure 7 :Figure 8 :

 78 Figure 7: Points sur-grille et hors-grille sur l'axe temporel. On-grid samples = points sur-grille, off-grid values = valeurs hors-grille.

Figure 9 :

 9 Figure 9: ISL pour un code CPFSK en dB pour le problème des décalages Doppler intra-pulse inconnus et les décalages de retard hors-grille.

L

  Figure 11: Images re-synthétisées à partir d'un signal chirp (a) et de signaux CPFSK (b) avec filtrage désadapté. Range axis = axe distance, azimuth axis = axe azimut.

Figure 12 :

 12 Figure 12: Cartes range-Doppler pour le signal réémis par le répondeur, obtenues pour le signal chirp et les signaux CPFSK avec le MF et le MMF.

Figure 13 :Figure 14 :

 1314 Figure 13: Migration distance sur les données comprimées en distance. Range axis = axe distance, transmitted pulses = impulsion transmise.

Figure 15 :

 15 Figure 15: Images ISAR de l'avion par le radar HYCAM pour le chirp et les CPFSK avec filtrage désadapté Range = distance, frequency induced from the angular velocity = fréquence induite par la vitesse angulaire.

  Figure 2.1: A simplified radar system diagram.

  Figure 2.2: A Simple Example of a Radar Framework

Figure 2 . 3 :

 23 Figure 2.3: Representation of the transmitted and the received pulses and a configuration when the range values exceed the range ambiguity. In this case, the received echoes of the first transmitted signal arrive after the transmission of the second pulse. This could lead to localization problems.

Figure 2 . 4 :

 24 Figure 2.4: Frequency, phase and spectrum plot of the chirp pulse centered at f 0 = 0.

  Definition of the mainlobe and the sidelobes around it.

Figure 2 . 5 :

 25 Figure2.5: Matched filter output for a noiseless chirp for a target at τ = 0s. On the second sub-figure, the part of the curve in blue corresponds to the mainlobe and the rest of the curve in orange illustrates the sidelobes.T p = 1µs, so the resolution before matched filtering would have been 150m.

Figure 2 . 6 :

 26 Figure 2.6: An example where a strong energy target masks a low-energy one after matched filtering.

Figure 2 . 7 :

 27 Figure 2.7: Ability of target discrimination with respect to the range resolution. The first subfigure represents a configuration where two targets cannot be separated because their relative delay is lower than the range resolution. On the contrary, in the the second sub-figure, the two targets are separated by a distance higher than the resolution and thus they can be discriminated.

Figure 2

 2 Figure 2.8: The ambiguity function of the chirp pulse is dispayed in the middle, on top the zero-Doppler cut and on the left the zero-delay cut. Here B = 352MHz and T p = 1.5µs.

  is given by r(t) = au N (t -τ c )e j2π f d t + n(t).

Figure 2

 2 Figure 2.9: Ambiguity function of a Gold code along with the zero-Doppler and zero-delay cuts. Here B = 352MHz and T p = 1.5µs.

Figure 2

 2 Figure 2.10: Coherent train of pulses. Range-Doppler processing of a chirp pulse

Figure 2 . 12 :

 212 Figure 2.12: Zero-Doppler and zero-delay cuts after the range-Doppler processing on a chirp pulse.

Figure 2 .

 2 Figure 2.13: Optical image from Google Maps and SAR image acquired with the airborne platform SETHI [Ruault du Plessis and Dreuillet, 2013].

Figure 2 .

 2 Figure 2.14: SAR Geometry

Figure 2

 2 Figure 2.15: Illustration of the ISLR and PSLR.

Figure 3 Figure 3

 33 Figure 3.1: A simple example of a digital communication system.

  Representation of symbol using the PSK scheme.

  Representation of symbol using the QAM scheme.

Figure 3 . 3 :

 33 Figure 3.3: An 8-Phase Shift Keying circular constellation and a 16-Quadrature Amplitude Modulation constellation.

Figure 3 . 4 :

 34 Figure 3.4: ITU Frequency Spectrum from [Manning, Catherine, 2018].

Figure 3 .

 3 Figure 3.5: IEEE Frequency Bands.

Figure 3 . 7 :

 37 Figure 3.7: A co-existence radar-communication system.

Figure 3 .

 3 Figure 3.8: A cooperative radar-communication system using radar-based waveforms that are slightly modified to embed communication symbols.

Figure 3 Figure 3 .

 33 Figure 3.9: A cooperative radar-communication system using communication-based waveforms to perform both missions.

  Ambiguity function of Gold code in dB.

Figure 3 .

 3 Figure 3.11: Ambiguity functions of Kasami and Gold codes.

Figure 3 .

 3 Figure 3.12: A co-design radar-communication system. The transmitted waveform is optimized for both radar and communication missions.

Figure 4 .

 4 Figure 4.1: Power Spectral Density of CPM codes with different constellations with h = 0.5, L = 1, and a rectangular shaping function g for one code and for an average of 100 different PSDs.

Figure 4

 4 Figure 4.2: Power Spectral Density of CPM codes with different memory lengths L with M = 4, h = 0.5 and a rectangular shaping function g.

  phase smoothing functions u over time.

Figure 4 . 3 :Figure 4 . 4 :

 4344 Figure 4.3: Phase and Frequency Shaping Functions.

Figure 4 .Chapter 4 .Figure 4 . 5 :

 4445 Figure 4.3 illustrates the two different windows for two memories of length L equal to 1 and 3. More precisely, sub-figure 4.3a illustrates the frequency functions g, whereas sub-figure 4.3b provides the

Figure 4 . 6 :

 46 Figure 4.6: PSD of CPM codes with different modulation indices with M = 8, L = 1, and rectangular shaping function.

Figure 4 . 7 :

 47 Figure 4.7: Phase cylinder for a rectangular CPM code with modulation index h = 0.5 and a constellation composed of 4 symbols.

Figure 4 . 8 :

 48 Figure 4.8: Final phase states for modulation index h = 0.5.

Figure 4 . 9 :

 49 Figure 4.9: Phase states of a CPFSK signal for modulation index h = 0.5 and a binary constellation.

Figure 4 .

 4 Figure 4.9 illustrates the phase transitions of a CPFSK code for a binary constellation and a modulation index of h = 1 2 . It has linear transitions and each node provides two branches that link to the next states.

  (a) Theoretical Power Spectral Density Function for CPFSK signals for different modulation indices h and different constellation sizes M. (b) Simulated Power Spectral Density function as the Fourier Transform of the correlation function of the CPFSK signals for different modulation indices h and different constellation sizes M.

Figure 4 .

 4 Figure 4.10: CPFSK Power Spectral Density Functions.

Figure 4 .

 4 Figure 4.11: Illustration of an ideal rectangular spectrum.

Figure 4 . 12 :Figure 4 .

 4124 Figure 4.12: Ambiguity function of a CPFSK code, with the zero-delay cut on the left, and the zero-Doppler cut on the top. The parameters used for this example are h = 0.5, M = 8, T c = 10ns, B = 352MHz, and T p = 1.5µs.Ambiguity function of CPFSK

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.14: Phase and spectrum of a binary PSK pulse.

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.17: Phase and spectrum of a LFM-PSK pulse.

  17

  )

4. 3 .Figure 4 .

 34 Figure 4.20: Guard Interval of an OFDM code.

Figure 4 .Figure 4 .Figure 4 .

 444 Figure 4.21: Phase, spectrum and amplitude of an OFDM pulse.

Figure 4 .

 4 Figure 4.24: Spectra and 99% Power Bandwidth for different codes. The 99% Power Bandwidth is delimited by the vertical lines with color associated to the curve colors of the different codes.

Figure 4 .

 4 Figure 4.25: Auto-correlation of the filtered signals after 1.2B and 2B cuts with the initial code.The small windows on each sub-figure is a zoom at the peak of the mainlobe to visualize the loss after the compression.

Figure 4 .

 4 Figure 4.26: Amplitude distortion of LFM-PSK, BPSK and CPFSK after filtering with spectral cut at 1.2B.

Figure 5

 5 Figure 5.1: Different compression outputs for a CPFSK code. The matched filter, the considered mismatched filter (with β = 1 dB) and the analytic mismatched filter are displayed.

Figure 5

 5 Figure 5.2: Matched filter ('MF') and mismatched filter ('MMF 0 ') that minimizes the PSLR criterion for a CPFSK code.

Figure 5 . 3 :

 53 Figure 5.3: Matched and mismatched filter output for the transmitted signal and mismatched filter with a signal that undergoes a Doppler shift equal to 404000 Hz. The ISL value of the MF for a zero-Doppler shift signal is 1.46 dB, the ISL value of MMF for a zero-Doppler shift signal is -4.47 dB and the ISL value MMF for the signal with Doppler shift is -2.08 dB.

  Figure 5.4 shows the integrated sidelobe level in dB obtained for the different filters considering the two given Doppler intervals. The filters that are plotted in figure 5.4 are listed below: 'MF' : the classical matched filter.

Figure 5 . 4 :

 54 Figure 5.4: Integrated sidelobe level for a CPFSK code in dB over two different Doppler intervals and for different compression filters.

Figure 5

 5 Figure 5.5: On grid and off-grid points on the time axis.

Figure 5 . 6 :

 56 Figure 5.6: Matched filter output for an on-grid and an off-grid signal sampled at sampling frequency 1.5B.

Figure 5 . 7 :

 57 Figure 5.7: Mean ISL for CPFSK signals over the sampling cell Θ for 100 Monte Carlo simulations. The standard deviation is represented by the bars around the mean value of the ISL obtained on the Monte Carlo simulations. The chosen sampling rate is 2 samples per symbol. The symbol number is 100 per CPFSK signal.

  Figure 5.7 illustrates the average ISL obtained for the different filters, applied to 100 Monte Carlo simulations over CPFSK pulses with a randomly drawn sequence of information symbols, for which different delay values are applied. In addition, in figure 5.7, the standard deviation is

Figure 5

 5 Figure 5.8: Mean and standard deviation values on 50 Monte Carlo iterations for the mean ISL of CPFSK signals over the sampling cell with respect to the sampling frequency factor α representing the number of samples per symbol. Here, the CPFSK signals contain 50 symbols.

Figure 5

 5 Figure 5.9: Mean and standard deviation values on 50 Monte Carlo iterations for the mean ISL of CPFSK signals over the sampling cell with respect to the symbol number. The sampling frequency factor is 2.

Figure 5 .

 5 Figure 5.10: Duality gap between the primal and the dual optimization problem. The number of variables indicated here correspond to our considered problem.

Figure 6 .

 6 Figure 6.1: Real-data SAR image acquired from SETHI using chirps.

Figure 6

 6 Figure 6.2: SAR Image Reconstitution Algorithm.

Figure 6 . 3 :

 63 Figure 6.3: Impulse responses of the chirp and the CPFSK codes for one or different transmitted codes with their associated matched or mismatched filters. The abbreviation 'MF' corresponds to matched filter and 'MMF' to mismatched filter. The small window on the top right side of each figure represents a zoom around the mainlobe.

Figure 6 . 4 :

 64 Figure 6.4: Mismatched filter compression for 8 different codes, along with their averaged value.

Figure 6

 6 Figure 6.5: Range and Azimuth cuts of all the impulse responses illustrated in figure 6.3.

Figure 6 . 7 :Figure 6

 676 Figure 6.7: Re-synthesized SAR images obtained in the different considered configurations. Note that all five sub-figures have the same intensity.

Figure 6 .

 6 Figure 6.10: Two plane shadows on the SAR image.

Figure 6 .

 6 Figure 6.11: Auto-correlation of a chirp pulse with sidelobes hidden by a clutter contribution. The clutter has values up to -40 dB.

Figure 6 . 12 :

 612 Figure 6.12: Illustration of the same corner reflector for the four CPFSK configurations.

Figure 6 .

 6 Figure 6.13: Pearson correlation coefficients of the corner reflector for different signal configurations.

Figure 6 .

 6 Figure 6.14: Pearson correlation coefficients of the building for different signal configurations.

Figure 6 .

 6 Figure 6.16: Pearson correlation coefficients of the forest for different signal configurations.

Figure 6 .

 6 Figure 6.17: The red square area of the lake where the average value is considered.

  Figure 6.18: Two plane shadows on the re-synthesized SAR images for the chirp and CPFSK codes and their associated processing.

Figure 7 .

 7 Figure 7.1: HYCAM ground-based radar owned by ONERA with antennas in horizontal position https://www.onera.fr/en/demr/research-units.

Figure 7 . 2 :

 72 Figure 7.2: Framework of real acquisitions.

Figure 7 . 3 :

 73 Figure 7.3: Amplitude and Spectrum of a received chirp signal and one CPFSK as transmitted back from the transceiver.

Figure 7 . 4 :

 74 Figure 7.4: Range-Doppler maps of the transceiver after radar processing and zero-Doppler rejection for the chirp and the CPFSK with MF and MMF. The right column corresponds to zooms on the transceiver signatures. The red square delimits the zone used for noise level estimation.

Figure 7 . 5 :

 75 Figure 7.5: Range and Doppler axis cut of the transceiver point scatterer for the chirp and the CPFSK with MF and MMF in (a) and (d) respectively. For better visualization, (b) provides a zoom on the peak of the range axis. The spectrum after range and Doppler compression and clutter rejection for the chirp and the CPFSK for MF and MMF is plotted in (c).

Figure 7 . 6 :

 76 Figure 7.6: Illustration of the position of HYCAM, the ORLY Airport and the airport runway.

Figure 7 . 7 :

 77 Figure 7.7: An A320 aircraft. The figure is taken from https://fr.wikipedia.org/wiki/ Airbus_A320.

Figure 7 . 8 :

 78 Figure 7.8: Amplitude and Spectrum of a received chirp signal and one CPFSK reflected on the aircraft.

Figure 7 .

 7 Figure 7.9: Range compressed data and range-Doppler image generation after IFFT for the case of the chirp pulse compressed with MF.Range migration

Figure 7 .

 7 Figure 7.10: Range migration observed on the range compressed data, and the plane illustration after azimuth compression through IFFT, without range migration correction.

Figure 7 .

 7 Figure 7.11: Range compressed data after application of Keystone transformation.

Figure 7 . 12 :

 712 Figure 7.12: Range compressed data in the Fourier space for the fast time and in the time domain for slow-time before and after Keystone transformation. We can observe the interpolation effect at the border of (b).

Figure 7 .

 7 Figure 7.13: Range-Doppler image of the aircraft for the three configurations presented before after applying the Keystone transform. The red box corresponds to the zone used for noise level estimation.

Figure 7 .

 7 Figure 7.14: Zoom of the range-Doppler image on the aircraft for the three configurations.

4 :

 4 Average value of the noise floor over a small window away from the mainlobe on the bottom right of each sub-figure.

Figure 7 .

 7 Figure 7.15: Range and Doppler cuts of the highest energy point on the aircraft.

  Mean value along the Doppler cut in order to observe the energy difference in dB between the outputs.

Figure 7 .

 7 Figure 7.16: Illustration of the two main variables R(t) and θ (t). This figure is taken from [Chen and Martorella, 2014].

Figure 7 .

 7 Figure 7.18: ISAR images of the aircraft for three different configurations. The red box corresponds to the zone used for noise level estimation.

Figure 7 .

 7 Figure 7.19: Zoom of the ISAR images on the aircraft for three different configurations.

  highest p int al ng the cr ss-range axis Chirp CPFSK MF CPFSK MMF (b) Cross-range cut.

Figure 7 .

 7 Figure 7.20: Cuts along the range and the cross-range dimensions plotted for highest energy point of the aircraft.

  N+k-a ) exp -jπh T I n(N+k-a) τ F a,a s * (t N+l-a ) exp jπh N+k-a ) F a,a s * (t N+l-a )

Table 1 :

 1 Comme auparavant pour les réponses impulsionnelles des CPFSK et les filtres désadaptés associés, PSLR et ISLR pour le signal réfléchi sur le répondeur. Les valeurs surlignées indiquent la meilleure valeur de chaque colonne.

	xxxiv

  This chart is a graphic single-point-in-time portrayal of the Table of Frequency Allocations used by the FCC and NTIA. As such, it does not completely reflect all aspects, i.e., footnotes and recent changes made to the Table of Frequency Allocations. Therefore, for complete information, users should consult the Table to determine the current status of U.S. allocations.

																																			FX-SAT
																																			MOBILE	FX-SAT (S -E)	B-SAT. MOB**
																																			FIXED	BCST -SAT.
																																		Earth Expl Sat Space Res.	Radiolocation BCST SAT.	FIXED FIXED SATELLITE (S-E)	FIXED SATELLITE (S-E)	EARTH EXPL. SAT. FX SAT (S-E) SPACE RES.	FIXED SATELLITE (S-E)	FIXED SATELLITE (S-E)	FIXED SATELLITE (S-E) MOBILE SAT. (S-E)	FX SAT (S-E) MOBILE SATELLITE (S-E)	FX SAT (S-E) STD FREQ. & TIME MOBILE SAT (S-E)	EARTH EXPL. SAT. MOBILE FIXED SPACE RES.	FIXED MOBILE	MOBILE** FIXED	EARTH EXPL. SAT. FIXED MOBILE** RAD.AST SPACE RES.	FIXED MOBILE	INTER-SATELLITE	FIXED	RADIO ASTRONOMY SPACE RES. (Passive)	AMATEUR AMATEUR SATELLITE	Radio-Amateur location RADIO-LOCATION Earth Expl. Satellite (Active)	FIXED	RADIONAVIGATION INTER-SATELLITE	INTER-SATELLITE RADIOLOCATION SATELLITE (E-S)	RADIONAVIGATION FIXED SATELLITE	(E-S)	FIXED FIXED SATELLITE	(E-S)	std freq e-e-sat MOBILE INTER-SAT.	& time e-e-sat (s-s) FIXED MOBILE INTER-SAT.	Earth Exploration Satellite (S-S) FIXED MOBILE INTER-SATELLITE	FIXED SATELLITE (E-S) MOBILE SATELLITE (E-S)
	SPACE RESEARCH (deep space)	RADIONAVIGATION INTER-SAT SPACE RES.	FIXED MOBILE SPACE RESEARCH (space-to-Earth)	SPACE	RES.	FIXED MOBILE SAT. (S-E) FIXED	FIXED-SATELLITE	MOBILE FIXED FIXED SATELLITE MOBILE SAT.	FIXED SAT MOBILE SAT. EARTH EXPL SAT (E-S) Earth Expl. Sat (s -e) SPACE RES. (E-S)	FX-SAT (S-E)	FIXED MOBILE BROAD-CASTING BCST SAT.	RADIO FIXED MOBILE** ASTRONOMY FIXED SATELLITE (E-S)	MOBILE SATELLITE (E-S) FIXED SATELLITE (E-S)	MOBILE RADIONAV. SATELLITE MOBILE SAT (E-S).	FIXED MOBILE MOB. SAT(E-S) RADIONAV.SAT.	FIXED MOBILE FX SAT(E-S)	SPACE RESEARCH	MOBILE FIXED	INTER-SAT EARTH EXPL-SAT (Passive) SPACE RES.	INTER-SAT SPACE RES. EARTH-ES	INTER-SAT EARTH-ES SPACE RES. MOBILE FIXED	EARTH EXPLORATION SAT. (Passive) SPACE RES. MOBILE FIXED INTER -SAT	FIXED MOBILE	INTER-SAT RADIO-LOC. MOBILE FIXED EARTH EXPLORATION SAT. (Passive) SPACE RES..	MOBILE FIXED	INTER-SATELLITE FIXED MOBILE**	MOBILE** INTER-SATELLITE	MOBILE INTER-SATELLITE	RADIOLOC. Amateur	Amateur Sat. Amateur RADIOLOC.	AMATEUR SAT AMATEUR RADIOLOC.	EARTH EXPL SAT. (Passive) FIXED MOBILE INTER-SATELLITE SPACE RESEARCH (Passive)	Amatuer FIXED MO-BILE INTER-SAT. SPACE RES. E A R T H EXPL . SAT	EARTH EXPL SAT. (Passive) SPACE RESEARCH (Passive)

Figure 3.6: Frequency Allocation Chart in United States in 2016. Figure taken from https: //www.ntia.doc.gov/files/ntia/publications/2003-allochrt.pdf.

Table 3 .

 3 1: Various radar and communication applications in different electromagnetic bands.

Table 4 .

 4 1: Parameters used for the generation of the different waveforms.

	Parameter	Gold	Random BPSK LFM-PSK OFDM CPFSK
	B		352 MHz
	f s			4B
	T p	1.4e-6s		1.5e-06 s
	h		-		0.5
	N b	511 bits	529 bits	529 bits	529 bits 450 bits
	T c		2.8e-9 s		1.5e-06 s 1e-8 s
	M		2		8

Table 4 . 3

 43 .24.

	Signals	MF (cut at 1.2B) (cut at 2B) MF
	BPSK	-0.62 dB	-0.31 dB
	LFM-PSK	-1.14 dB	-0.35 dB
	CPFSK	-0.10 dB	-0.01 dB

: Energy loss of the mainlobe after matched filter (MF) with the input non-filtered signal with spectral cuts at 1.2B and 2B.

Table 5 .

 5 1: Parameters used in this chapter to generate the signals. In case of any change in one or more values, it is specified in the main body of the document.

	B	165.5 MHz
	Pulse Duration T p	1 µs
	Modulation index h	0.5
	Chip duration T c	10 ns
	Sampling Period T s	5 ns
	Constellation	4
	Symbol Number	100
	Allowed loss in processing gain β	2 dB

Table 5 .

 5 2: Mean ISL values for the compression with the different filters.

Table 5 .

 5 4: Computational cost of the proposed algorithm and the CVX solver for one optimal mismatched filter, and the extrapolated computational time for 6522 filters.

	Value

Table 6 .

 6 1: Parameters and characteristics of the CPFSK codes and the real-data SAR images.

	Parameter	Value
	CPFSK Characteristics
	Bandwidth B	305 MHz
	Pulse Duration T p	4.8 µs
	Chip Duration T c	19.2 ns
	Bit Number N b	1000
	Modulation Index h	0.4
	Constellation M	16
	Number of pulses N pulses	6522
	Real Image Characteristics
	Band	X
	Bandwidth B	500 MHz
	Pulse Duration T p	8 µs
	Azimuth Resolution	0.38 m
	Platform velocity	124 m/s
	PRF	5 kHz
	Waveform	Chirp

Table 6 .

 6 2: PSLR and ISLR values in the range direction on the different impulse responses.

  Re-Synthesized SAR Images

			127
	Signals	PSLR ISLR small (dB) (dB)
	Chirp with MF	-11.5	-8.6
	One CPFSK with MF	-12.9	-7.3
	Different CPFSK with MF	-13.0	-10.6
	One CPFSK with MMF	-13.7	-11.72
	Different CPFSK with MMF -13.2	-11.7

Table 6 .

 6 3: Range PSLR and ISLR for the corner reflector of every re-synthesized SAR image.

Table 6 .

 6 4: Mean values of the correlation coefficients for different areas presented in Figure

		Corner Building Lake Reflector	Forest
	One CPFSK with MF	0.7007	0.658	0.3682 0.655
	Different CPFSK with MF	0.642	0.711	0.464	0.761
	One CPFSK with MMF	0.908	0.8335 0.6035 0.882
	Different CPFSK with MMF 0.8887	0.8745	0.682 0.9116

Table 6 .

 6 5: MSE and SSIM values for the corner reflector and the building images.

		Reflector	Building
	Signals	MSE SSIM	MSE SSIM
	One CPFSK with MF	52.28 0.2876 72.56 0.256
	Different CPFSK with MF	62.6 0.1942 51.53 0.2998
	One CPFSK with MMF	19.16 0.6875 29.77 0.5366
	Different CPFSK with MMF 23.22 0.627 24.77 0.592

Table 6 .

 6 lower than 0.4). 6: MSE and SSIM values for the lake and the forest images.

			Lake	Forest
	Signals	MSE	SSIM	MSE SSIM
	One CPFSK with MF	227.5	0.0	69.9 0.1371
	Different CPFSK with MF 160.4 0.02261 46.5	0.322
	One CPFSK with MMF	76.5	0.213	24.33 0.5557
	Different CPFSK with MMF 54.66 0.3425 19.47 0.6294

Table 7 .

 7 1: Parameters of the system and the code generation

	Parameter	Value
	Radar System Characteristics
	Band	S
	PRI CPFSK	200 µs
	Integration Time	2.1 s
	Number of CPFSK Pulses N CPFSK pulses	10500
	Number of Non-Identical CPFSK Pulses N CPFSK s	500
	CPFSK Code Characteristics
	Bandwidth B	152 MHz
	Sampling Frequency	200 MHz
	Chip Duration T c	23 ns
	Pulse Duration T p	100 µs
	Bit Number N b	1299
	Modulation Index h	0.5
	Constellation C	8
	Loss-in-processing gain	2 dB

ground-based radar HYCAM, with which the acquisitions are performed.

Table 7 .

 7 2: Average value of the noise floor over a small window away from the mainlobe.

	Signal & Processing Average Noise Level (dB)
	Chirp MF	-71.6
	CPFSK MF	-60.3
	CPFSK MMF	-65.6

Table 7 .

 7 7.5a. Sub-figure7.5c provides the spectrum plot for the different configurations after the radar processing. It is plotted after range-Doppler compression, and thus it provides an averaged and 3: PSLR, ISL, and ISLR 2D for the transceiver signal. The highlighted values indicate the best value for each column.

	Signal & Processing PSLR (dB) ISLR (dB) ISLR 2D (dB)
	Chirp MF	-15.46	-10.47	-7.18
	CPFSK MF	-19.25	-11.97	-4.08
	CPFSK MMF	-20.43	-13.08	-6.54

Table 7 .

 7 6: Average noise level on the bottom right part of the plane sub-figures in figure7.18.

Toutes ces nouvelles technologies étant disponibles sur le marché, elles occupent davantage de bande passante et demandenent de plus en plus de bande spectrale. Par exemple, les opérateurs de communication britanniques ont payé 119,3 millions de livres pour 1800MHz de bande passante en 2015, l'Allemagne facture plus de 5 milliards d'euros pour 4 bandes de fréquences de communications et les États-Unis ont vendu 28GHz d'allocation spectrale pour les applications 5G pour 702 millions de dollars[START_REF] Liu | Joint radar and communication design: Applications, state-of-the-art, and the road ahead[END_REF]. Or, le spectre électromagnétique est une ressource limitée et il est également utilisé pour des applications autres que les communications (voir figure1). Par conséquent, le spectre électromagnétique est de plus en plus congestionné.

Dans cette thèse, on considère le cadre de l'imagerie radar synthétique (Synthetic Aperture Radar (SAR)), dont la géométrie est représentée en figure 2. La particularité du traitement SAR est l'utilisation d'une antenne virtuelle, en exploitant le mouvement du porteur pendant l'émission et la réception du train d'impulsions, améliorant la résolution azimutale de l'image de sortie. Dans une telle configuration, les émetteurs sont susceptibles d'utiliser une large bande passante, afin d'améliorer également les capacités de résolution distance. La combinaison de cette application avec les systèmes de communication devient alors encore plus pertinente. Il existe très peu de littérature sur la création de systèmes conjoints SAR-communication ayant pour objectif la génération d'une image radar de haute qualité et la transmission simultanée d'informations.Dans un cadre classique de SAR, les formes d'ondes utilisées sont principalement des chirps, et les signaux reçus sont généralement traités avec un filtre adapté (Matched Filter (MF)) afin de créer une image qui représente la réflectivité du sol. Mais malgré toutes ses qualités radar, le chirp ne permet pas la transmission d'information. On pourrait alors penser transmettre une forme d'onde de communication, que l'on traiterait de manière analogue au chirp pour obtenir une image. Cependant, le MF appliqué à de telles formes d'ondes génère généralement des lobes secondaires forts. L'objectif de cette thèse est alors double: il faut proposer une forme d'onde adéquante, et le traitement approprié, pour satisfaire au mieux les objectifs d'imagerie et des communications.

Pour en revenir à notre cadre de SAR-communication, nous considérons par ailleurs que les signaux émis par l'antenne de l'avion diffèrent les uns des autres, afin de maximiser le débit de données. Cela implique qu'un filtre désadapté différent doit être calculé pour chaque signal émis. Nous avons remarqué qu'il est impossible de calculer tous les filtres en un temps raisonnable, en utilisant l'algorithme d'optimisation convexe CVX comme nous le faisions auparavant. Nous avons donc développé un algorithme qui réduit considérablement le temps de calcul par résolution du problème d'optimisation dans l'espace dual. Nous avons montré que notre algorithme peut calculer les filtres 250 fois plus rapidement que les algorithmes génériques de Matlab en considérant les paramètres du cadre SAR. Ce travail a été publié dans une conférence internationale[START_REF] Chatzitheodoridi | Fast computation of the optimal integrated sidelobe level filter for large-scaled problems[END_REF].On liste ci-dessous les 3 publications dans des conférences internationales de ce chapitre, ainsi qu'une contribution dans une conférence nationale et un workshop:

https://www.radartutorial.eu/01.basics/Doppler%20Dilemma.en.html

(a) Optical Image (Screenshot from Google Maps). (b) SAR image.

This notation C N (0, R x ) indicates that the variable follows a complex normal distribution with 0 mean value and R x covariance matrix.

Briefly, a MIMO system is composed by multiple transmitters and receivers[Gomez, 2014]. During the transmission, each antenna transmits a different waveform compared to the other collocated ones.

The successive interference cancellation algorithm takes into account target range information in order to suppress its contribution, and thus perform communication decoding. Once the message is retrieved, the communication signal is removed and the radar compression is performed for detection through the matched filter.

Kasami and Gold codes are usually considered for their good cyclic cross-correlation properties. They are created through pairs of maximal length binary codes.

The authors vary the length of the frames using a variable data length.

This criterion depends on the spectral efficiency, and it can be also extended to the data rate.

DSSS is a multiple access technique used mainly in the communications field that shares the available bandwidth to avoid interference between the users, which have a unique identity[START_REF] Shaojian | Radar-communication integration based on dsss techniques[END_REF].

Wideband code division multiple access system is a system where the spread bandwidth of the codes is larger than the bandwidth capacity of the channel[Milstein,

2000]. It is mainly used for a 3G standard in order to provide high-speed data rate.

A maximal length binary sequence code is a binary pseudo-random code, created using polynomials in the Galois field theory[START_REF] Rife | Transfer-function measurement with maximum-length sequences[END_REF]. They are composed by M = 2 N -1 elements, and they are mainly used in communications for their good circular autocorrelation properties.

In the article that we published for this problem[START_REF] Chatzitheodoridi | A mismatched filter for integrated sidelobe level minimization over a continuous Doppler shift interval[END_REF], we used Kasami codes[Kasami, 1966] and Gold codes as at that point we had not yet considered the CPFSK codes.

Acknowledgments

I would like to express my heartfelt gratitude to the following people, without whom this research and manuscript would not have been possible ! I would like to thank the jury members, Sylvie MARCOS, Thierry CHONAVEL, Laurent FERRO-FAMIL, Pierfrancesco LOMBARDO, and Henri RUGGIERO (invité), for their interest on my PhD topic and their constructive questions and criticism.

is the first category of cooperation detailed in the following. Two other popular categories of cooperative systems are also explored; they correspond to having one system that conducts both missions using a specific waveform. The first one consists in using a radar-based waveform that maintains the radar processing and optimized the communication performance, and the second one using a communication-based code in order to perform again both tasks. In each case, the performance of the other system needs to be optimized, for instance, by modifying the waveform, rearranging the frequency distribution, or changing the compression technique. The case where communication codes are used both for radar and communication missions is the one that we will adapt for our work, and the chosen waveform will be detailed in chapter 4.

. Cooperation of two systems with physically separated antennas

The first presented category corresponds to cooperative but distinct systems with specific antennas dedicated to each mission. They can still be co-located, but then the radar and communication antennas would not be identical and they would transmit different waveforms. The aim is to provide a collaborative framework to improve the performance of both systems.

A smooth transition from a co-existence to a cooperation system is introduced in [START_REF] Chalise | Performance tradeoff in a unified passive radar and communications system[END_REF]. The authors use the previously introduced evaluation criteria to create a passive radar-communication system. Two different waveforms are used for each mission, and the aim is to find a trade-off between the maximization of the detection probability and the preservation of a desirable information rate. The corresponding optimization problem can be expressed as max P r ,P c P D , s.t. R ≥ r m , P r + P c ≤ P T , (3.5) where P D is the detection probability for the radar performance, P r and P c are the allocated power for the radar and the communication waveforms respectively, R is the data rate at the communication receiver, r m is the minimum tolerated data rate value, and P T is the total power of the system. This optimization problem thus intends to maximize the detection probability while ensuring a minimum data rate and respecting a maximum transmitted power.

A completely different approach is investigated in [START_REF] Chiriyath | Simultaneous radar detection and communications performance with clutter mitigation[END_REF], where the considered system performs simultaneously transmission for target detection and message decoding in cluttered environment. To do so, a given bandwidth is shared between the two systems, using a water-filling algorithm to adequately perform the splitting. The criterion to optimize is based on information theory. For the communications, this criterion is quite natural but for the radar, the authors derived a "radar information rate", that represents the estimation uncertainty of a parameter to be estimated. The difference lies in the method the signal is being processed at reception, which is called successive interference cancellation 5 , and improves the performance of both mis- In chapter 3, extensive information on joint radar-communication systems has been provided along with preliminary details on communication systems. We specified that our aim is to create a cooperative SAR-communication system that uses communication codes and optimizes the processing so as to fit the radar expectations.

In this chapter, we provide in section 4.1 an in-depth study of the waveform choice for our system, namely Continous Phase Modulation (CPM) codes. As we shall see, this waveform presents interesting spectral properties. Its behavior can change dramatically in terms of phase, spectrum, embedding symbols, etc, as its parameters are modified. By restraining some of their parameters, we decide to use a special case of CPM called Continuous Phase Frequency-Shift Keying (CPFSK). The main reason for motivating this choice lies in the ability of expressing the power spectral density and thus providing an expression for their bandwidth, which is not always that clear for communication codes. Once the desired code is well established, and in order to provide a complete evaluation, different communication codes are introduced and studied in section 4.2. They are chosen among the most cited ones in the literature. Finally the last section 4.3 provides a comparison of these codes with the CPFSK codes in terms of spectral quality.

53

Part III

SAR-Communication System:

Simulations and Results PSLR, the best value corresponds to the 1_CPFSK_MMF case, followed by the N_CPFSK_MMF case with only 0.6 dB of difference. These theoretical results should be confirmed by the study of the corner reflector, as the only difference between the corner reflector of the image and the set of impulse responses generated to compare them is that the former is embedded in clutter, and thus sidelobes with lower energy than the clutter cannot be measured.

6.1.5 . Generation of the Re-synthesized SAR Images and a First Visual Quantification

The impulse responses that we have just analyzed can now be used in the reconstruction algorithm, in order to provide re-synthesized SAR images. After providing the estimated ground reflectivity of the real-data SAR image in equation (6.3), it is convolved with each impulse response in order to provide re-synthesized range compressed data. Thereafter, azimuth compression is applied, and the re-synthesized SAR images are finally created from CPFSK communication codes. They are illustrated in figure 6.7. An optical image of the area of interest is displayed in figure 6.6, taken as a screenshot from Google Maps. It is a bit easier to observe the different elements of the scene on an optical image, and to recognize which objects reflect more energy back to the radar.

More precisely for the radar images in figure 6.7, the white color corresponds to high-energy points and the black one to low-energy areas. Sub-figure 6.7a illustrates the SAR image obtained with the chirp, and it is used as a reference for comparison. As mentioned before, this is also a re-synthesized SAR image, so that it shares the same range resolution as the other images. From this SAR image, we can observe the sinus-cardinal shaped sidelobes after compression, especially for high-energy points, for both range and azimuth axes. This can be seen for instance at the borders of a building.

In the case when only one CPFSK code is used (sub-figures 6.7b & 6.7c), some high sidelobe Chapter 6. Re-Synthesized SAR Images Generation and Analysis similar. On the contrary, when the output values are close to 0, then the images tend to be very different [START_REF] Hachemi | The interest of coherence from radar SAR images in the follow-up of urban extensions of focani city (romania)[END_REF]. In our framework, we want to compare the pixel energy of the generated images with respect to the configuration of a chirp compressed with MF in order to provide the best possible combination of signal transmission and processing procedure. The idea is that an image close to the reference one will be advantageous for SAR specialists, as it will preserve significant information such as shape, contrast, etc.

. Structural Similarity (SSIM) Index

The authors in [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF] propose a different comparative error estimator based on various characteristics. Their motivation stems from the human eye precision. They show that structural similarity information creates a satisfying error estimator when comparing similar images. The Structural Similarity Index (SSIM) definition is based on the computation of • the luminance l, which depends on the mean intensity of the images,

• the contrast c, which depends on the standard deviation of the images,

• the structure s, which is the inner product of the images after removing the luminance and the contrast.

Denoting again A, B the two images, the SSIM index can be mathematically expressed as follows [START_REF] Wang | Image quality assessment: from error visibility to structural similarity[END_REF], Avanaki, 2008]

where C 1 and C 2 are constants close to zero to avoid instability of the fraction, C 3 = C 2 2 , µ A and σ A are respectively the mean and the standard deviation of the image A and

The output of this formula is equal to 1 when the input images are identical. Thus, the less the compared images are similar, the smaller the SSIM index. According to [START_REF] Asamoah | Measuring the performance of image contrast enhancement technique[END_REF], the images are considered to be similar if the computed index is at least 0.4.

This method is compared to the mean square error (Mean Square Error (MSE)) [START_REF] Asamoah | Measuring the performance of image contrast enhancement technique[END_REF], James et al., 2013], which is computed as (6.10) with (N, M) the size of rows and columns, respectively, of the images A, B. The MSE output is 0 when the two inputs are identical.

. Average Value of Low-Energy Pixels of a SAR Image

The average value of low-energy pixels of the generated images gives an insight into the average thermal noise level or equivalently to the so-called Noise Equivalent Sigma Zero (NEσ 0 ) [START_REF] Gouveia | Pre-processing of Sentinel-1 C-band SAR images based on incidence angle correction for dark target detection[END_REF], as in this area the presence of signals is very low.

Let A be a sub-image of the SAR image that has low intensity pixels, then we define the normalized average value N 0 over a low intensity area as

When using different signals and compression techniques, N 0 gives an insight on the energy difference of the system between the average noise level and the potential artifacts that are created, especially due to the compression techniques. These differences can be linked to the sidelobe level observed in the 2D impulse responses or the loss-in-processing gain β of the optimization problem.

CHAPTER 7 ISAR using Real CPFSK Acquisitions from a Ground-based Radar

In chapter 6, we created re-synthesized SAR images using CPFSK signals in order to provide a semi-realistic framework, as we used chirp-acquired raw data but simulated impulse responses. This is the closest results to a realistic SAR framework that we could obtain without developing a dedicated airborne system. Although real airborne acquisitions could not be performed, measures using a ground-based radar was a possibility to verify that the CPFSK is a good candidate for radar imaging. As the radar is stationary, the considered application is ISAR, the principle of which is close to SAR one.

Firstly, in section 7.1, the framework and the data characteristics of the system are provided. In the following, two different configurations are considered. In the first approach, the transmitted communication codes are reflected on a transceiver, which allows to provide the real impulse response of the codes. It will permit to study the quality of the impulse responses of the codes after the transmission. These results are provided in section 7.2. In the second approach presented in section 7.3, an airborne platform is imaged.

. Framework and Data Characteristics

In this section 7.1, we describe the acquisition system along with the parameters used to generate the CPFSK.

The ground-based radar used for the transmission and the reception of the signals is called HYCAM. HYCAM is a surface radar owned by ONERA that mainly operates in S-band. It is used for various projects and studies that can either provide aid to other radar applications or test new radar concepts on real data. Some of the possible applications are multi-static configurations, MIMO, polarimetry, super-resolution, and others [START_REF] Brouard | Hycam: A new S band surface radar testbed[END_REF]. Figure 7.1 illustrates the Chapter 8. Conclusion and Future Work needs to be computed for every different signal, which results to an unfeasible computational time, when using generic convex algorithms from the literature. For this reason, we have proposed a fast algorithm that can compute 250 times faster all the needed filters, and thus makes it attainable to create a large amount of range compressed data with these filters. However even with this improvement, it is still not possible to compute them in real-time.

Moreover, in order to provide realistic results, we have generated SAR images that take into consideration an estimation of the integrated ground reflectivity acquired from real acquisitions. To do so, it has been convoluted with the simulated impulse responses of the CPFSK that are created from the range compressed signals with matched or mismatched filtering. These images have been compared to the classical chirp case through various evaluation criteria. We have concluded that the performance of CPFSK signals compressed with mismatched filtering is not very far from the chirp performance, even if the latter one is slightly better. However, we should not forget that one of the goals is to transmit information, and it is the price to pay to perform both tasks.

Finally, in the absence of an available system to provide real airborne acquisitions with CPFSK, we have used a ground-based radar in two different frameworks to validate our system to the most realistic extent. Firstly, range-Doppler maps have been created by illuminating a transceiver. This case has allowed us to visualize the impact of the communication signals on the transceiver, and at the same time to compute the classical radar evaluation criteria, in order to compare them with the reference chirp case. Secondly, we have created ISAR images by illuminating an airborne platform from the same ground-based radar. Only visual comparison could be performed, but nonetheless the results are satisfying when using CPFSK modulation, as they are also robust from a Doppler point of view and they are not deformed due to the hardware. These re-synthesized and real results conclude the work of this thesis, and show that satisfactory radar image performance can be obtained with communication signals.

. Future Work

The decisions that we made during this work determined the different tasks that we performed. In this section, we provide some ideas of future work, either by changing the path we chose or by complementing the current work.

Starting from the beginning, the choice of a cooperative system is a significant choice that we made, as our work is more radar-oriented. However the choice of the signal could be different compared to the criteria that we want to validate. For example, if the constant amplitude constraint was more flexible, then OFDM codes could be used for the system as in [START_REF] Liu | Design of integrated radar and communication system based on MIMO-OFDM waveform[END_REF], Dokhanchi et al., 2018], but in a SAR-communication framework, which is very less examined [START_REF] Baudais | MIMO-OFDM signal optimization for SAR imaging radar[END_REF]. Still if the CPM codes are chosen, waveform optimization with respect to the symbol choice could be considered for example for better auto-correlation and cross-correlation properties. Moreover, we could also study the noise level with respect to the CPM signals.

Afterwards, for a more complete examination of our system, the communication aspect could be explored more. The aspects that could be examined range from more waveform-based characteristics, like the BER, to more hardware-based specifications like synchronization. Also the channel could be estimated in order to take into consideration its impact on the signals, in a similar context 8.2. Future Work 163 to [START_REF] Liu | Toward dual-functional radar-communication systems: Optimal waveform design[END_REF]. Such considerations would provide a more detailed system for both radar and communication aspects. Moreover, as mentioned in [Feng et al., 2020, Han and[START_REF] Han | [END_REF], networking could be a future joint radar-communication solution as it can take into consideration multi-user configurations that would help with the massive number of devices.

From a compression point of view, the mismatched filter that minimizes the integrated sidelobe level was chosen because of the SAR framework, but other options could be considered. For example, a similar criterion choice could be the PSLR to minimize the largest sidelobe. Moreover if the ISL is preferred, then constraints could be added in the problem that deal with for example the SNR, or any other different window functions as the ones proposed in [START_REF] Coutino | Waveform-aware optimal window function design for mismatched filtering[END_REF]. Another perspective of this work could be to combine the two different optimization problems that we proposed (with intra-pulse Doppler shifts and off-grid delay shifts) into one common problem by optimizing both cases simultaneously.

Ideally, real-acquisitions from an airborne platform could be made in order to create CPFSK images in the SAR framework. The compromise between the data rate maximization and the mismatched filter computational time can be taken into consideration, meaning that reducing the non-similar pulses (less data rate) can decrease the computational time of the filters and create them in a more feasible time interval. Maybe even considering real-time acquisitions would be possible if the filters are not over-sampled. Additionally, for real-time acquisitions, deep learning could be used to learn the MMF behavior. We still have to provide an explicit expression for M Ξ , that defines our optimization problem through the following equation

The definition of M Ξ is given by

where

Let us first calculate the matrix M( f d ). An element of the correlation matrix Λ Λ Λ can be expressed as :

where m ∈ 1, K + N -1 and n ∈ 1, K and assuming that s i = 0 if i ≤ 0 or i > N .

Appendix A. Cost Function Calculations for the Mismatched Filter Problem

Using (A.2), the (k, l) element of M, with k, l ∈ 1, K , can be expressed as:

where M(0) = Λ Λ Λ T (0)FΛ Λ Λ * (0), which is close to the auto-correlation of the received signal for the zero-Doppler frequency and only depends on the transmitted signal s.

It follows that,

A.2 . Off-Grid Delay Configuration

Let us calculate the matrix ||M 1 2

Θ q|| 2 2 , that defines the cost function of the optimization problem as follows:

In order to find the expression of M Θ , we use the components of the correlation matrix defined as

Hence, M Θ can be now calculated. Let k, l ∈ 1, N , then,

Keystone Transform

Let s b be the baseband transmitted signal. The received signal that corresponds to the m th pulse can be expressed as

with τ m = 2R m c , R m = R re f -mνT r , ν is the relative radial velocity of the target with respect to the radar, R re f is the target range, and T r the PRI. Thus it can be written as

Applying the matched filter on the received signal, a peak appears where the delay is detected, that can be approximately modeled by a sinus-cardinal with mainlobe width equal to 1 B , as

The term -mνT r in the sinus-cardinal creates the spreading problem along both range and Doppler axis. The ideal output that keystone transformation tries to achieve is a sinus-cardinal expression independent of the velocity.

By taking the Fourier transform on the fast-time axis of equation (B.1), and since the Fourier Transform of a sinus-cardinal function is a single window function, we obtain

Now the explanation of the spreading problem can be easily observed, due to the second term of expression (B.2) that includes not only the slow time m and the target velocity, but also the fast-time frequency. Thus, the aim of the Keystone transformation is to discard the term exp j 4π c f t mνT r Appendix B. Keystone Transform from the fast-time Fourier transform data Y ( f t , m). To do so, let us write Y ( f t , m) in a continuous way also for the slow-time variable as

where t m ∈ T int is the continuous variable for the slow-time axis, and T int the integration time.

Observing the second term, the following variable change will allow to re-scale the slow-time domain and get rid of the f t contribution from this term:

Rewriting equation (B.3) using this variable change, we have (B.5) which corresponds to the desired output. This result shows that if the slow-time is re-scaled for the fast-time Fourier transform data, then this new function corresponds to the desired output of the fast-time Fourier transform slowtime inverse Fourier transform. An inverse Fourier transform in the range dimension and a Fourier transform in the pulse number dimension then provides the focused range-Doppler image that is no longer impacted from range migration neither in the range nor the Doppler axis. The last remaining step is to provide a sampling grid for the slow-time axis, which is provided through sinus-cardinal interpolation. It is performed through 1D interpolation on the slow-time data for each range bin [Richards, 2014].