Sabine Mounir

Philippe Jean-Claude Lionel

Michèle Nadine Patrice

Annie Michel

Keywords: MU-MIMO, IEEE 802.11ac, capacity, antenna arrays, indoor propagation measurements, TGac channel models, transmit beamforming, EIRP

The deployment facility and flexibility of wireless technologies have allowed considerable evolution of IEEE 802.11 standards. The arrival of the optical fiber to the home has further motivated research studies to increase data rates and capacity. In this context, the latest IEEE 802.11ac standard ratified in 2014 proposes new technologies to achieve maximum data rates above 1 Gbit/s. In particular, 802.11ac includes the Multi-User Multiple Input Multiple Output (MU-MIMO) and Transmit Beamforming (TxBF). These smart antenna technologies assume a variable transmit antenna pattern in function of the users' position and the corresponding multipath propagation channel characteristics. However, the current radio planning tools give a fixed coverage map independent of any adaptive or smart antenna processing.

This thesis studies the impact of the MU-MIMO and TxBF techniques on radio engineering, system optimization as well as radio planning tools. Indeed, the first part of this thesis deals with refining the sum rate capacity gains by identifying interesting system configurations and optimal system parameters (antenna spacing, tradeoff between antenna number and number of spatial streams, antenna arrangement... ). The second part addresses the beamforming gains in terms of the Signal-to-Noise Ratio (SNR) gain and interference level to have realistic radio coverage modeling.

Hence, we have firstly analyzed the MU-MIMO to Single User MIMO (SU-MIMO) sum rate capacity gain based on the 802.11ac correlated MU-MIMO channel models. By analyzing the results, we have proposed system recommendations to optimize the MU-MIMO in a residential environment, mainly related to system configuration. We have also highlighted a relevant correlation coefficient which can be used to decide whether MU-MIMO or SU-MIMO is suitable and to select users in MU-MIMO group. Moreover, we have formulated the convex optimization problem of maximizing the capacity in a Multi-User Multiple Input Multiple Output (MU-MIMO) context with multiple receiving antennas taking into account the Equivalent Isotropically Radiated Power (EIRP) constraint for Wireless Local Area Network (WLAN) in Europe.

Secondly, we have conducted a 2-user Multiple-Input Multiple-Output (MIMO) propagation channel measurement campaign in a residential environment to confirm the obtained radio engineering and system optimization recommendations. We have been able to extend this study to other potential parameters which have influence on MU-MIMO to SU-MIMO capacity gain such as path loss and antenna array geometries.

Finally, we have evaluated the performance of TxBF by analyzing the Signal-to-Noise Ratio (SNR) gain as well as the interference reduction. This part identifies the system parameters which have the biggest influence on the realistic radio coverage simulation. It proposes a statistical modeling to predict the beamforming gains achieved by the antenna processing.

Dedication

To my family...

XIII

Résumé étendu en français 1 Introduction

Le contexte global de cette thèse est celui de la montée en débit au-delà de 500 Mbit/s, voire de 1 Gbit/s sur l'accès radio, notamment en environement résidentiel. Pour cela, différents standards sont en fin de normalisation, notamment le standard IEEE 802.11ac dans la bande 5 GHz. Pour répondre à ce besoin de montée en débit, de nouvelles techniques sont mises en oeuvre comme l'utilisation de largeurs de bande plus importantes (jusqu'à 160 MHz), la formation de faisceaux connue sous "transmit beamforming (TxBF)" , puis, dans un second temps, l'utilisation du MIMO multi-utilisateurs (MIMO-MU) permettant une réutilisation spatiale du spectre. En outre, l'arrivée de la fibre optique a davantage motivé l'exploration de nouvelles solutions pour augmenter le débit au sein des réseaux locaux domestiques.

Pour réaliser l'ingénierie radio, les outils d'ingénierie actuels supposent tous que le réseau d'antennes de bornes Wi-Fi a un diagramme de rayonnement indépendant des positions des stations clientes et constant dans le temps. Les nouveaux systèmes MIMO-MU, ou même MIMO mono-utilisateur notés MIMO-SU avec "transmit beamforming" remettent en cause cette modélisation dans la mesure où le diagramme d'antennes formé devient dépendant du canal multi-trajets et de chacun des terminaux clients servis à un instant donné. L'objet de la thèse est d'analyser l'impact de ces nouvelles techniques MIMO sur l'ingénierie radio et notamment sur la modélisation des couvertures et des interférences pour contribuer à la conception de la prochaine génération d'outils d'ingénierie.

Le premier objectif de la thèse consiste à affiner les gains de capacité et surtout préciser les domaines d'emploi en indoor du MIMO-MU par rapport au MIMO-SU (lien en vue directe (LOS) ou avec obstacle (NLOS), comment constituer les groupes d'utilisateurs. . . ). Les travaux s'appuient sur une modélisation réaliste du canal de propagation MIMO indoor, basée sur des canaux standardisés indoor dans un premier temps. Ensuite, une campagne de mesures a été réalisée au cours de cette thèse pour confirmer et compléter les résultats obtenus.

Le second objectif de la thèse consiste à identifier quelle modélisation de l'ensemble propagation, antennes et ses traitements, permettrait de simuler une couverture radio réaliste pour l'ingénierie de ces nouveaux systèmes (802.11ac) tant du critère de puissance reçue que des interférences.

XV 2 Techniques MIMO-MU pour les réseaux locaux sans fil

Les communications sans fil utilisent le rayonnement des ondes dans un environnement donné par un émetteur (Tx).

L'onde émise interagit avec l'environnement physique d'une manière complexe avant d'arriver au récepteur (Rx). En effet, l'onde se propage et est soumise à plusieurs phénomènes : réflexions, diffractions, transmissions... L'onde reçue dépend de l'environnement, de l'emplacement des objets et de l'activité humaine.

Cet environnement souvent dynamique exige une analyse approfondie du canal de propagation radio afin d'estimer avec précision les paramètres des signaux reçus.

Pour tirer profit de la propagation multi-trajets à l'intérieur des bâtiments, les techniques MIMO ont fait l'objet d'une attention particulière. Celles-ci ont été proposées pour augmenter la capacité, en exploitant les propriétés spatiales du canal à trajets multiples. Ces techniques MIMO ont d'abord été étudiées dans des scénarios mono-utilisateur, puis étendues à des scénarios multi-utilisateurs. Dans ce contexte, la nouvelle norme IEEE 802.11ac finalisée en janvier 2014 normalise le MIMO-MU pour augmenter la capacité pour le sens descendant. Ce chapitre rappelle d'abord les caractéristiques principales de la couche physique de la nouvelle norme 802.11ac avant d'énumérer en détail les phénomènes de propagation influant le canal de propagation dans les reséaux sans fil. Ensuite, nous décrivons la liaison descendante MIMO-MU, les techniques de précodage proposées dans la littérature pour permettre la réutilisation spatiale et les capacités liées.

Le standard 802.11ac

La norme IEEE 802.11ac permet d'atteindre des débits allant jusqu'à 6, 9 Gbit/s au niveau de la couche physique pour la bande de 5 GHz. Par conséquent, cette norme est destinée aux services qui requièrent des débits élevés, tels que la télévision haute définition, l'affichage sans fil (interface multimédia haute définition (HDMI) sans fil), les stations d'accueil sans fil (connexion sans fil avec des périphériques), et autres. Ci-après, nous passons en revue quelques-unes des principales caractéristiques de la norme IEEE 802.11ac (Tableau 1).

Les techniques MIMO-MU

Dans les systèmes MIMO-MU, le point d'accès communique simultanément avec plusieurs terminaux clients en exploitant la dimension spatiale. Ceci est possible en utilisant la connaissance de chacun des canaux reliant les antennes de l'émetteur aux différentes antennes des récepteurs en s'appuyant sur la technique de "feedback" explicite. Dans le contexte MIMO-MU, l'émetteur doit gérer l'interférence inter utilisateurs. Pour ce faire, plusieurs méthodes, nommées méthodes de précodage, sont proposées dans la littérature. La diagonalisation par blocs, la plus étudiée, consiste premièrement à annuler l'interférence inter-utilisateurs, et ensuite à optimiser la capacité de chacun des liens. Les qu'elle offre des performances assez proches de la borne théorique de capacité appliquant le principe du "dirty paper coding".

Simulation de capacités multi-utilisateurs basées sur des canaux normalisés (TGac)

Dans ce paragraphe, nous considérons la liaison descendante d'un scénario MIMO-MU dans laquelle un point d'accès IEEE 802.11ac est équipé de plusieurs antennes dipôles (jusqu'à 10 antennes) et transmet à deux récepteurs, chacun avec deux antennes dipôles. Ce chapitre propose des recommandations pour optimiser les performances MIMO-MU en se basant sur les canaux standardisés pour 802.11ac (TGac). Tout d'abord, nous décrivons brièvement ce modèle. Ensuite, nous donnons les résultats numériques pour l'impact du nombre d'antennes d'émission sur le gain de capacité MIMO-MU par rapport au MIMO-SU. Nous expliquons les résultats obtenus en mettant en évidence un coefficient de corrélation qui décrit la similarité entre les canaux de deux utilisateurs. Enfin, nous formulons le problème d'optimisation de la capacité au sens Shannon du MIMO-MU en tenant compte de la contrainte de Puissance Isotrope rayonnée équivalente (PIRE). Il s'agit de la puissance répondant aux exigences de la réglementation européenne pour les réseaux WLAN dans la bande de fréquence 5 GHz: P IRE ≤ 23 dBm pour la bande basse, P IRE ≤ 30 dBm pour la bande haute.

Modèle TGac

Les modèles normalisés ou standardisés sont un outil important pour la conception des nouveaux systèmes radio. Ils permettent d'évaluer les avantages et les performances de différentes techniques (modulation, codage...) d'une manière unifiée. C'est dans ce cadre que les canaux 802.11 sont proposés pour les milieux indoor. Le modèle IEEE 802.11 TGn a été développé pour les canaux WLAN MIMO aux fréquences de 2.4 GHz XVII

Nombre d'antennes d'émission et corrélation

Dans ce paragraphe, nous évaluons l'impact de l'augmentation du nombre d'antennes d'émission sur le gain de capacité MIMO-MU par rapport au MIMO-SU. Nous examinons ce gain de capacité en considérant que la répartition de la puissance totale d'émission est équitable entre les différents flux spatiaux et sous porteuses. Le SNR est défini comme le rapport entre puissance totale émise et la puissance du bruit, et est fixé ici à 20 dB. Le canal est normalisé pour avoir une atténuation moyenne de 0 dB. La figure 2 donne la moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction du nombre d'antennes d'émission. On considère deux canaux: le canal normalisé TGac-B dans les conditions NLOS et le canal i.i.d Rayleigh décorrélé. Le graphe affiche les valeurs moyennes et les quantiles à 10% et 90% (q 10 et q 90 ) servant à estimer l'intervalle de confiance. La première observation tirée de la figure 2 est que le gain de capacité MIMO-MU par rapport au MIMO-SU augmente lorsque le nombre d'antennes d'émission augmente. Le gain passe de 1, 2 à 1, 65 pour l'environnement résidentiel: jusqu'à 45% de gain de capacité. Nous observons également que le gain de capacité obtenu passant de n T = 4 à n T = 6 est plus élevé que celui observé en passant de n T = 6 à n T = 8. Ceci peut être expliqué par le fait que nous ne pouvons pas tirer profit du "transmit beamforming" lorsque n T = 4, c'est à dire lorsque le nombre d'antennes d'émission est le même que le nombre total de flux spatiaux. Une autre explication concernant le coefficient de corrélation est donnée ci-après. Dans cette partie, nous définissons un coefficient de corrélation qui décrit le degré de similarité entre les canaux de 2 utilisateurs. La moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction de la moyenne du coefficient de corrélation est donnée sur la figure 3. Le nombre d'antennes d'émission est égal à 6. La moyenne est calculée par rapport au temps, et chaque point représente l'un des 100 tirages de 2 utilisateurs. Lorsque le coefficient de corrélation augmente, le gain de capacité diminue: plus de 30% de capacité en moins lorsque le coefficient de corrélation passe de 0,05 à 0,35. En effet, lorsque les canaux des deux utilisateurs sont corrélés, l'algorithme de précodage se montre moins performant. Ce coefficient de corrélation a un autre avantage: il peut être utilisé comme critère pour regrouper les utilisateurs dans un scénario MIMO-MU. Enfin, une analyse basée sur la régression linéaire est effectuée: les résultats suggèrent une relation linéaire entre gain de capacité et corrélation. Le gain de capacité MIMO-MU par rapport à MIMO-SU estimé y ′ peut être exprimé en fonction de la corrélation x sous forme:

y ′ = a 1 x + a 0 (1)
où y ′ , a 1 et a 0 désignent respectivement le gain de capacité estimé, la pente de régression et l'ordonnée à l'origine. Dans le cas précis n T = 6, les valeurs numériques sont listées dans le tableau 2.

XX

Contrainte de PIRE dans un contexte multi-utilisateurs

Le gain de capacité MIMO-MU par rapport au MIMO-SU est étudié en appliquant la contrainte de limitation de PIRE cette fois-ci. Deux méthodes de répartition de la puissance totatle d'émission sont analysées: répartition équitable et non de la puissance parmi les flux spatiaux. Dans notre formulation, nous étudions le cas de 2 utilisateurs, mais les résultats et les algorithmes présentés dans cette section peuvent être généralisés à n'importe quel nombre d'utilisateurs.

Les techniques MIMO-MU consistent à appliquer un précodage linéaire transmis aux flux spatiaux. Par conséquent, le diagramme d'antenne ainsi que son gain sont modifiés en fonction de l'emplacement d'utilisateurs et les paramètres de propagation. La valeur de PIRE est ainsi modifiée. La régulation européenne a fixé la PIRE dans les bandes de fréquence 5 GHz à 200 mW (bande basse) ou 1 W (bande haute). Cette contrainte peut être différente dans d'autres pays où elle est plutôt basée sur la puissance totale émise.

Nous avons formulé le problème de maximisation de capacité MIMO-MU sous la contrainte de la limitation de PIRE. La figure 4 présente les résultats du gain de capacité MIMO-MU par rapport au MIMO-SU pour les systèmes tenant compte ou non de la contrainte de PIRE. Les valeurs moyennes et les quantiles à 10% et 90% (q 10 and q 90 ) servant à estimer l'intervalle de confiance sont représentés. Le cas où la contrainte de PIRE est considérée est noté eirp-égal. Nous notons le cas habituel où la puissance d'émission est directement liée au SNR par basic. Le rapport signal à bruit est fixé à 20 dB dans les deux cas. Le nombre d'antennes d'émission varie de 4 à 10. La figure 4 montre que le gain de capacité MIMO-MU augmente quand le nombre d'antennes d'émission augmente pour le modèle de canal TGac-B. Ce gain passe de 1,2 à 1,77 pour un système tenant compte de la contrainte de PIRE dans un milieu résidentiel, ce qui représente plus de 50% de gain de capacité. A noter que ce gain est égal à 45% pour un système ne tenant pas compte de la contrainte de PIRE (basic). Ces résultats montrent que l'augmentation du nombre d'antennes d'émission est également favorable au gain de capacité MIMO-MU par rapport au MIMO-SU pour des canaux corrélés en respectant la contrainte de PIRE.

Capacité MIMO-MU en environment réel de type résidentiel

Ce chapitre présente les résultats numériques basés sur une campagne de mesures menée au cours de la thèse dans un environnement résidentiel typique rencontré au sein des réseaux domestiques. Afin d'avoir une comparaison significative, nous avons effectué une comparaison entre le modèle de canal TGac-B et de l'environnement mesuré. Le premier objectif de cette campagne de mesures est de confirmer les résultats précédents et de valider les conclusions d'ingénierie radio basées sur des modèles de canaux, puis d'étendre les résultats à différentes géométries de réseaux d'antennes et à d'autres paramètres de propagation comme l'atténuation. Nous avons également évalué l'impact de l'augmentation du nombre de flux spatiaux et des valeurs de SNR. Enfin, nous avons élaboré une étude comparative entre les résultats simulés et mesurés.

Campagnes de mesures

La Figure 5 montre l'environnement où nous avons mené la campagne de mesure: un milieu résidentiel. Il s'agit d'un appartement réel de 12 m ×7 m formé de matériaux et meubles européens. Le plafond est à 2,53 m. Les deux scénarios LOS et NLOS ont XXII Figure 7 -Moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction de la moyenne du coefficient de corrélation. Tout d'abord, nous avons résumé les caractéristiques de la nouvelle norme 802.11ac ainsi que les mécanismes de base de la propagation radio. Les différents modèles de canaux MIMO ont également été présentés. En particulier, nous avons détaillé le canal TGac utilisés pour nos simulations. Ensuite, nous avons décrit un modèle de système MIMO-MU, les techniques de précodage et les méthodes de calcul de capacités.

Gain de beamforming

Ensuite, nous avons proposé des recommandations pour optimiser les performances MIMO-MU en se basant sur les canaux standardisés TGac. Ces recommandations sont données en termes de paramètres systèmes/antennes et de la disposition des utilisateurs: LOS, NLOS... Tout d'abord, nous avons détaillé le processus des de simulation, le modèle de canal évalué et ses paramètres. Ensuite, nous avons fourni les résultats de gain de capacité MIMO-MU par rapport au MIMO-SU en fonction des configurations d'antennes et des paramètres de canaux de propagation. Enfin, nous avons mené une étude sur ces gains de capacité en tenant compte de la contrainte de la PIRE. En particulier, nous avons démontré qu'une légère augmentation du nombre d'antennes d'émission par rapport au nombre total de flux spatiaux transmis améliore significativement le gain en capacité MIMO-MU sur MIMO-SU. Nous avons également mis en évidence une définition de la corrélation des canaux correspondant qui est utile pour décider si MIMO-MU apporte plus de capacité que le MIMO-SU. Ce paramètre permettrait de sélectionner les utilisateurs dans un groupe d'utilisateurs MIMO-MU, et également d'optimiser la conception des antennes MIMO. Ensuite, nous avons formulé le problème d'optimisation de la capacité MU-MIMO en tenant compte de la contrainte de la PIRE. Nous avons montré que la technique de précodage MIMO-MU tire profit de la contrainte de la PIRE.

Nous avons ensuite présenté les résultats d'une campagne de mesures réalisée au sein d'Orange Labs Belfort en utilisant un analyseur de réseau. Cette campagne a été menée au cours de la thèse dans un environnement résidentiel typique des réseaux domestiques. Le premier objectif de cette campagne de mesures était de confirmer les résultats précédents et ainsi valider les conclusions d'ingénierie radio basées sur des modèles de canaux, puis d'étendre ces résultats à différentes géométries de réseaux d'antennes et à d'autres paramètres potentiels tels que l'atténuation de propagation. Les résultats obtenus confirment les recommandations basées sur des simulations TGac. En outre, nous avons étendu ces conclusions en étudiant l'impact de la configuration des antennes de réception, le nombre de flux spatiaux et les valeurs de SNR. En particulier, l'augmentation du nombre d'antennes de réception augmente légèrement le gain en capacité du MIMO-MU par rapport au MIMO-SU. En outre, la sélection de différentes antennes de réception n'a pas d'impact significatif sur ce gain de capacité. Ensuite, nous avons évalué l'atténuation du canal et son impact sur le gain de capacité MU-MIMO en utilisant différentes méthodes d'accès au canal. Une différence d'atténuation entre récepteurs peut être bénéfique pour le gain de capacité MIMO-MU par rapport au MIMO-SU selon le type de la méthode d'accès.

Enfin, le dernier chapitre évalue la méthode du "transmit beamforming" en se basant sur les données de mesure dans un contexte MIMO-SU. Pour évaluer les XXIX performances de cette technique, nous avons défini deux métriques: le gain sur le SNR et la réduction d'interférence. Plusieurs configurations MIMO (nombre d'antennes émettrices et réceptrices, nombre de flux spatiaux, géométries de réseaux d'antennes) ont été comparées pour identifier celle qui fournit les meilleurs gains sur le SNR. En particulier, nous avons prouvé que l'augmentation du gain sur le SNR est possible lorsque nous augmentons le nombre d'antennes d'émission, diminuons le nombre d'antennes de réception, ou diminuons le nombre de flux spatiaux. Ces conclusions permettraient d'ajouter les gains de "beamforming" aux cartes de couverture simulées par les outils d'ingénierie actuels.

En outre, nous avons cherché à savoir si l'introduction de cette technique de "transmit beamforming" pourrait augmenter ou réduire l'effet des interférences générées par un lien interférant. Nous avons pu montrer qu'utiliser la technique de "transmit beamforming" peut réduire dans certaines directions les interférences.

Les travaux de cette thèse pourraient être poursuivis par le développement des points suivants:

-Cette thèse suppose que l'émetteur connaît parfaitement le canal.

Context

Wireless communication is one of the most vibrant, challenging and interesting areas in the communication field today.

Over the past two decades, wireless technologies have thrived and have even become standard for household and office appliances. The deployment facility and freedom of movement they enable have made them a very attractive solution for setting up local access networks.

The Enterprise Wireless Local Area Network (WLAN) market has become highly competitive and the vendors are proposing vast offerings and unique solutions. WLAN is affordable, easy to deploy and manage. That is why, recent trends in the industry have shown increase in the demand for enterprise WLAN. According to the International Data Corporation (IDC)1 , the Worldwide WLAN Market has shown solid growth (12.8%) in the third Quarter of 2013 [START_REF]IDC: Analyze the future[END_REF]. Also, the overall WLAN market revenues are forecast to exceed 11 billion in 2017 and nearly 50 percent greater than 2012 revenues, according to Dell'Oro Group [4](this growth includes deployment of the 802.11ac upgrade cycle). In home networks, more than 10 millions of residential gateways are deployed in France.

The major standards for WLANs are the Institute of Electrical and Electronics Engineers (IEEE) 802.11 family, commonly known as Wireless Fidelity (Wi-Fi). Since its 1997 initial specification, the IEEE 802.11 standard has gone through several order to evaluate radio wave propagation in an indoor environment, various techniques may be used such as ray tracing launching technique, channel measurements and channel models. The 2D and 3D ray tracing models are both used widely in the literature. However, these techniques have not been normalized for MU-MIMO. Moreover, they would need very high computation time. The task group of 802.11ac (TGac) has designed standardized channel models for indoor wireless local area networks by including MU-MIMO techniques. We have first based our research approach on these standardized channel models. Since MU-MIMO TGac channel models are mainly based on measurement campaigns in Line-Of-Sight (LOS) scenarios like in offices [START_REF] Breit | Multi-user MIMO channel measurements[END_REF][6] [START_REF] Kwak | Channel measurements in corridors for TGac[END_REF][8], we have conducted afterwards a measurement campaign to confirm and complete our results based on TGac simulations. However, since the measurement equipment was not available until the end of the 2 nd year of this Ph.D work, the measurement analysis began at the end of 2014 and was hence scheduled for less than one year. With this constraint, we could not go further in the measurement analysis.

Thereby, in the first part of this thesis, the objective is to refine the capacity gains for indoor MU-MIMO compared to the SU-MIMO systems by identifying interesting system configurations (deployment recommendations such as LOS or Non-Line-Of-Sight (NLOS), impact of distances or other environmental parameters ..) and optimal system parameters (like number and arrangement of antennas, number of spatial streams, ..). The results are based on both a standardized MIMO channel modeling and a measurement campaign conducted in an indoor environment. This part concludes system and deployment recommendations. In the second part of the thesis, the objective is to define how smart antenna affects radio coverage and interference. This characterization could be included in radio planning tools. This part identifies the propagation parameters which have the biggest influence on the realistic radio coverage simulation in terms of received power and interference level.

Main contributions

During this thesis, we have focused our study on identifying the deployment recommendations or optimal system configuration of MU-MIMO techniques compared to SU-MIMO from a physical point of view. We have studied the impact of antennas and propagation channels on the Block Diagonalization (BD) capacity gain in home networks under a total transmitted power constraint. Note that the BD is a precoding technique that cancels the interference between users in a MU-MIMO context. We have also highlighted a relevant channel correlation definition that is useful to decide whether MU-MIMO outperforms SU-MIMO and to select the users into a MU-MIMO user group. We have formulated the optimization problem for computing capacity satisfying the Equivalent Isotropically Radiated Power (EIRP) constraint in MU-MIMO context. We have confirmed the results based on measured channels in a residential environment. The measurement campaign we have conducted, extended the study to various antenna array geometries, as well as path loss impact. Finally, we have proposed a statistical model that describes the TxBF processing in terms of received power gain and interference reduction. Based on this model, we will be able to apply the beamforming gains in the upcoming radio planning tools.

Contents

The remainder of the report is divided as follows. In Chapter 2, we first introduce the fundamentals of WLAN. Afterwards, we give the new features implemented in the latest IEEE 802.11ac standard. We then present the basics of a wireless channel and an overview of the channel models. We finish by presenting the different MU-MIMO techniques and related capacity formulas. Chapter 3 analyzes and summarizes the recommendations based on IEEE 802.11ac standardized channel for a two-user case, taking or not into account the European regulation in terms of EIRP. In Chapter 4, we present the results based on our measurement campaign conducted in a residential environment. This environment is compared to the simulated TGac-B channel model. This chapter completes the findings of the simulation results in terms of system optimization. It also extends the study to reveal the impact of path loss on MU-MIMO performance. Chapter 5 investigates the modeling of radio coverage using smart antennas in terms of received power gain of the TxBF and interference level reduction. Finally, we conclude this thesis and give some potential further research directions in Chapter 6.

List of publications

Introduction

Radio propagation study is an essential step to perform appropriate design, deployment, and management strategies for any wireless network. The transmitted wave interacts with the physical environment in a complex way before arriving at the receiver (Rx). Indeed, radio propagation is heavily site-specific and can vary depending on the terrain, velocity of communicating devices, interacting objects and movement of people.

Under the suggested solutions to overcome the issues related to multipath propagation, MIMO communications have attracted considerable attention. They have demonstrated the potential for increased Shannon capacity by exploiting the spatial properties of the multipath channel [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF]. If the channel is rich scattering (independent and identically distributed (i.i.d.) Rayleigh channel) then a linear increase in capacity is possible by simultaneously increasing the number of receiving and transmitting antennas (the capacity increases according to the min of the number of transmitting and receiving antennas). MIMO techniques were first investigated in single-user scenarios, then extended to multi-user MIMO scenarios. The MU-MIMO technology uses adaptive antenna 1 arrays to provide high spectral efficiency with moderate receiver complexity. In this context, the new IEEE 802.11ac standard ratified in January 2014 normalizes the MU-MIMO for increased data rates (above 1 Gbit/s). The aim behind these techniques is to improve the system performance by increasing channel capacity and spectrum efficiency extending coverage range. These techniques also reduce multipath fading, co-channel interference, system complexity and cost, Bit Error Rate (BER), and outage probability [START_REF] Sarkar | A survey of various propagation models for mobile communication[END_REF].

This chapter first gives the physical features of the new standard 802.11ac in Section 2.2, before listing in details the basic propagation mechanisms which impact the propagation channel in wireless communication (Section 2.3). Afterwards, we describe the MU-MIMO downlink system, the precoding techniques in Section 2.5 and their related capacities in Section 2.6. Finally, the conclusion is drawn in Section 2.7.

The 802.11ac standard

The IEEE 802.11ac standard is an emerging Very High Throughput (VHT) WLAN standard that can achieve PHYsical layer (PHY) data rates of up to 6.9 Gbit/s for the 1. Adaptive antenna is as example a phased array where the gains and the phases of the signals induced in the various elements are weighted to adjust the gain of the array in a dynamic manner along a particular direction, while simultaneously placing nulls along undesired directions.

5 GHz frequency band [START_REF] Perahia | Gigabit wireless LANs: an overview of IEEE 802.11ac and 802.11ad[END_REF]. Note that the rates at the Internet Protocol (IP) layer are lower in practice.

The scope of IEEE 802.11ac includes single link throughput supporting at least 500 Mbit/s, multiple-stations throughput of at least 1 Gbit/s, and backward compatibility and coexistence with legacy IEEE 802.11n devices in the 5 GHz band. Consequently, this standard is targeted at higher data rate services such as high-definition television, wireless display (high-definition multimedia interface (HDMI) replacement), wireless docking (wireless connection with peripherals), and rapid sync-and-go (quick upload/download). The IEEE 802.11ac standardization process started in 2008 and finished in December 2014.

In general, IEEE 802.11ac could be schematized as a lateral extension of IEEE 802.11n in which the two basic notions of MIMO and wider channel bandwidth are enhanced. PHY and Multiple-Access Channel (MAC) layers modifications have been introduced with the VHT standard. In addition, channelization has been modified and MU-MIMO has been introduced for greater spectral efficiency.

Hereafter, we review some of the key features of IEEE 802.11ac such as larger bandwidth, higher order modulation, and MU-MIMO transmission mode.

Mandated use of the 5 GHz band

The WLAN market is now transitioning from IEEE 802.11n to IEEE 802.11ac [START_REF] Jones | Emerging technologies for WLAN[END_REF], due to the promise of higher throughput and more reliable performance available in the 5 GHz unlicensed band. IEEE 802.11ac mandates the use of the 5 GHz frequency band, a band with the availability of more significant spectrum compared to the commonly used 2.4 GHz band. Indeed, in Europe, there is approximately 400 MHz of spectrum available. In contrast, the 2.4 GHz band can accommodate only three non-overlapping 20 MHz wide channels. This has led to many competing devices per channel and heavy levels of interference. The larger spectrum availability in the 5 GHz band provides more network capacity, and leads to fewer competing devices per channel and thus reduced interference compared to traditional single-band 2.4 GHz IEEE 802.11n networks.

Maximum data rate

One key feature of IEEE 802.11ac is the increase in data rate compared to IEEE 802.11n. This is achieved through the use of expanded channel bandwidth and higherorder modulation. Figure 2.1 shows the peak data rate and the per-Spatial Stream (per-SS) data rate for various WLAN standards that have evolved over the years. Spatial streams are composed of independent data streams. 802.11b and 802.11g support peak data rates of 11 and 54 Mbit/s, respectively. IEEE 802.11n increased the peak data rate to 600 Mbit/s. IEEE 802.11ac has further increased the peak data rate to 6.9 Gbit/s, over ten times that of 802.11n.

A maximum data rate of 6.93 Gbit/s is thus obtained using: 160 MHz bandwidth, 8 Spatial Stream (SS), 256-Quadrature Amplitude Modulation (QAM) with coding rate 5/6, and short Guard Interval (GI) (400 ns) [START_REF] Jones | Emerging technologies for WLAN[END_REF]. coding rates are defined: 3/4-rate and 5/6-rate. For comparison, 802.11n supports up to 64-QAM with the same coding rates. Thus, 802.11ac achieves a 33% increase in peak data rate over 802.11n. The combination of higher-order modulation and increased channel bandwidth enables an 802.11ac device to support approximately three to six times higher data rate compared to an 802.11n device for the same number of antennas or SS [START_REF] Jones | Emerging technologies for WLAN[END_REF]. 802.11n achieves a maximum of 150 Mbit/s per-SS (108 data tones in 40 MHz of bandwidth with a maximum of 5.0 bits per tone). This results in a maximum data rate of 600 Mbit/s, assuming the maximum supported 4 SS MIMO transmission of 802.11n. 802.11ac reaches 433 Mbit/s per-SS using 80 MHz channel bandwidth (234 data tones with 6.67 bits per tone), and 867 Mbit/s per-SS using 160 or 80 + 80 MHz bandwidth. This results in a maximum data rate of 6.9 Gbit/s, assuming the maximum supported 8 SS MIMO transmission of 802.11ac. While 6.9 Gbit/s is an eye-catching maximum data rate, a more commonly cited advantage of 802.11ac is the ability to cross the 1 Gbit/s barrier with small form-factor devices [START_REF] Jones | Emerging technologies for WLAN[END_REF]. A two-antenna 802.11ac device (using a maximum of two SS) can support a maximum data rate of 1.73 Gbit/s. Furthermore, an 802.11ac device can surpass the data rate of an 802.11n device, with much lower complexity and cost but with an increased bandwidth range. For example, an 802.11n device requires three antennas (three spatial streams) to achieve a similar maximum data rate (450 Mbit/s) as a single-antenna 802.11ac device.

Rate-range

In addition to increasing the maximum data rate, these enhancements also lead to improved rate over-range performance of 802.11ac compared to 802.11n. Figure 2.2 shows simulated performance of both technologies [START_REF] Jones | Emerging technologies for WLAN[END_REF], using a path loss model validated with measurements within a large building. Both technologies support three spatial streams, and have three transmitting and receiving antennas (3 × 3). The figure shows the TCP/IP throughput vs. the distance between the wireless devices. It can be seen that the 802.11ac devices can connect at twice the range of the 802.11n devices, at the maximum TCP/IP throughput of the 802.11n device (approximately 280 Mbit/s). For an end user, this translates to 802.11ac devices experiencing higher throughput across most locations in a home/office environment. Another observation from Figure 2.2 is that the peak rate of the 802.11ac device is three times that of the 802.11n device. For the end user, this translates to 802.11ac-enabled devices experiencing much higher throughput for in room and peer-to-peer scenarios.

However, the use of 80 MHz is guaranteed in optimal cases only where 80 MHz bandwidth can be available. In the case of a reduced available bandwidth, as in dense environments with many AP, in order to increase the throughput, the standard increases the number of SS to eight, as well as it proposes the MU-MIMO processing as solution to enhance the spectral efficiency. Explicit compressed feedback (ECFB) is known to provide the most precise estimate of the channel, taking into account all the imperfections at the transmitter and the receiver. However, ECFB comes with a lot of overhead. For an AP with four antennas, the compressed feedback varies from 180 to 1800 bytes, depending on the number of client antennas and level of compression. For an 80 MHz client, sounding with just one single antenna takes about 250 microseconds.

MU-MIMO

In 802.11n, a transmission to a device happens using SU-MIMO modes, where the data rate to a device scales with the minimum number of antennas of each device and with the signal to noise ratio. An 802.11n AP must transmit data to different devices, attempting to divide up the network throughput between stations based on Carrier Sense Multiple Access/Collision Avoidance (CSMA/CA) which acts to prevent collisions before they happen. Unfortunately, 802.11n network capacity is then limited by lower-cost devices that have a smaller number of antennas [START_REF] Heusse | Performance anomaly of 802.11b[END_REF]. For instance, with single-antenna receivers, the 802.11n does not take benefit from MIMO advantages to improve throughput. In this context, 802.11ac MU-MIMO transmission modes allow simultaneous transmissions to multiple devices, as depicted in Figure 2.4. This significantly improves the spectral efficiency of a WLAN when there are stations with limited numbers of antennas. Essentially, MU-MIMO captures the maximum spatial capacity without requiring all individual stations to have a large number of antennas. The 802.11ac standard allows a MU-MIMO transmission to send up to four stations simultaneously. Each station may receive up to four SS, but the total number of SS in a MU-MIMO transmission is limited to eight SS summed across all stations.

Transmit beamforming and MU-MIMO require knowledge of the channel state to

Multipath propagation channel modeling

The channel refers to the medium between the transmitter and the receiver. We have seen in the previous section that the real wireless channel is complicated due to complex propagation environment. The transmitted signal may be reflected, diffracted and scattered by surrounding obstacles, or attenuated and absorbed by obstructions causing a fluctuation of signal power. Hence, multipath signals are received in a terrestrial environment, i.e., where different forms of propagation are present and the signals arrive at the receiver from transmitter via a variety of paths, causing multipath interference. This is known as multipath fading. Adding the effect of movement of either the Tx or Rx, the received overall signal amplitude or phase changes over a small amount of time.

Consider a single-input single-output system, the overall transmission channel (propagation channel as well as the transmitting and receiving antennas) can be represented as a linear filter with time varying impulse response in continuous time h(τ, t). The received signal y(t) at time t is expressed using complex baseband equivalent signals and filter as:

y(t) = x(t) * h(τ, t) + n(t) = +∞ -∞ h(τ, t)s(t -τ ) dτ + n(t) (2.1)
where x(t) is the transmitted signal, n(t) is the corrupting noise, and * denotes the convolution operation.

As channel can be time-varying, time t is needed, and τ is the multipath delay. The Channel Impulse Response (CIR) h(τ, t) for a single-input single-output system experiencing multipath fading with L paths is given by:

h(τ, t) = L-1 l=0 α l (t)δ(τ -τ l (t)) (2.2)
where α l (t) and τ l (t) are respectively the complex attenuation and the delay of the l th propagation path at time t. We denote L(t) by L for sake of simplicity. The characteristics of the channel can also be represented in the frequency domain by the Channel Frequency Response (CFR). CFR can be obtained by applying Fourier transform to the CIR in the time-delay domain τ :

H(f, t) = +∞ -∞ h(τ, t)exp{-j2πf τ } dτ = L-1 l=0 α l (t)exp{-j2πf τ l (t)} (2.3)
The continuous-time system model is generally sampled into an equivalent discrete time model. The received signal is given by:

y k = L-1 l=0 h l,k s k-l + n k (2.4)
where y k = y(kT sym ), k is an integer and T sym is the sampling period. L is the number of taps, h l,k is the l th channel tap. The frequency domain response becomes:

H(f, k) = L-1 l=0 h l,k exp{-j2πlf T sym } (2.5)

Propagation channel parameters

The average Power Delay Profile (PDP), referred to as E(P DP (τ, t)), gives the distribution of signal average power received over a multipath channel as a function of propagation delays [START_REF] Sizun | Collection technique et scientifique des telecommunications[END_REF]. The influence of particular obstacles can be noticed in the PDP, where the peaks, echoes, and other irregularities are displayed. The PDP can be thought of as a density function, of the form:

P DP (τ, t) = | h(τ, t) | 2 +∞ 0 | h(τ ′ , t) | 2 dτ ′ (2.6)
Mean Excess Delay, denoted τ e (t) is the first moment of the power-delay profile (as shown in Figure 2.7) [START_REF] Sizun | Collection technique et scientifique des telecommunications[END_REF]. It is expressed as:

τ e (t) = +∞ 0 τ P DP (τ, t) dτ (2.7)
The time reference 0 is given by the first path of the channel. Root Mean Square (RMS) Delay Spread (DS) τ RM S (t) is the square root of the second central moment of a power-delay profile [START_REF] Sizun | Collection technique et scientifique des telecommunications[END_REF], as seen in Figure 2.7. It is the standard deviation of the mean excess delay as expressed in Equation 2.8. The RMS delay is a good measure of the multipath delay spread. It gives an indication of the importance of the Inter-Symbol Interference (ISI). Strong echoes (relative to the shortest path) with long delays contribute significantly to τ RM S (t).

τ RM S (t) = +∞ 0 (τ -τ e ) 2 P DP (τ, t) dτ (2.8)
Maximum Excess Delay τ m is measured with respect to a specific power level, which is characterized as the threshold of the signal [START_REF] Sizun | Collection technique et scientifique des telecommunications[END_REF]. When the signal level is lower than the threshold, it is processed as noise. For example, the maximum excess delay can be specified as the excess delay τ m for which the average P DP (τ ) falls below -30 dB with respect to its peak value, as shown in Figure 2.7.

Figure 2.7 summarizes the delay dispersion parameters characterizing the propagation channel defined above.

The coherence bandwidth B coh is the statistical average bandwidth of the radio channel, over which signal propagation characteristics are frequency correlated at a given correlation level. It characterizes the range of frequencies over which the channel can be considered correlated or nearly flat. It is usually defined as the bandwidth over which The fast fading is modeled using either Rician distribution or Rayleigh fading. Rayleigh fading occurs when there are multiple indirect paths between the transmitter and receiver and no distinct dominant path, such as LOS path. Sometimes the dominant non fading signal due to line-of-sight in the channel superimposes itself on the random multipath components. The effect of the dominant signal over the weaker multipath signal gives rise to a Rician distribution. The Rician distribution is described by a parameter K, which is defined as the ratio of signal power in dominant component over the (local-mean) scattered power. The absence of direct line-of-sight signal K ⇒ ∞ degenerates Rician distribution into Rayleigh.

Otherwise, it is assumed to be a slow-fading channel if T sym < T coh , i.e. B sym > B d . It is important to note that the velocity of the mobile unit or the velocity of objects using the channel through a baseband signal determines whether a signal undergoes fast fading or slow fading.

MIMO propagation channel modeling for systems

The design guidelines and performance evaluation of radio communication require developing effective propagation models featuring the relevant characteristics of the radio propagation in an operating environment. MIMO channel propagation models have been developed as a suitable, low-cost, and convenient alternative, since the site-measurements are expensive and time consuming.

Power angular spectrum

Since the spatial (angular) distribution of the multi-path components is important in determining the MIMO system performance [START_REF] Zhijun | The impact of multi-path angular spread of MIMO system capacity[END_REF], this paragraph highlights the incorporate additional concepts such as Angular Spread (AS), Angle of Arrival (AoA), Angle of Departure (AoD), Power Angular Spectrum (PAS) and the antenna array correlation matrices for the Tx and Rx combinations, as illustrated in Figure 2.9.

Power angular spectrum defines the average received power as a function of Θ, where Θ represents either the angle-of arrival for receiving antenna or angle-of-departure for transmitting antenna. The AoD is defined to be the angle with which a departing ray's power is transmitted by the Tx array with respect to the broadside. Inversely, the AoA is defined to be the angle with which an arriving ray's power is received by the Rx array with respect to the broadside. The power angular spectrum value, denoted P AS, at an angle θ ∈] -π, π] is given by:

P AS(θ) = E( +∞ -∞ | h(τ, θ) | 2 dτ ) (2.10)
where h(τ, θ) defines the CIR as a function of the delay τ and the angle θ.

Similar to PDP, we can define the mean angle θ, and the RMS AS θ RM S . This determines how spread out multipath power is about the horizon. It is defined as the standard deviation of the PAS.

TGn channel models

Standardized models are an important tool for the development of new radio systems. They allow to assess and compare the benefits of different transmission techniques to enhance capacity and improve performance, in a manner that is unified for many parties.

In indoor area, radio propagation is influenced by the presence of building materials. The transmitted signal reaches the receiver through more than one path, due to reflection, refraction, and diffraction of the radio wave by objects such as walls, windows, and doors inside a building. The IEEE 802.11n task group proposes in [START_REF] Erceg | Tgn channel models[END_REF] a set of channel models applicable to indoor MIMO WLAN systems as an extension of the Single-Input Single-Output (SISO) channel models proposed by Medbo et al. These models are the TGn channel models and are classified to be stochastic in Figure 2.10. The newly developed MIMO models are based on the cluster model developed by Saleh and Valenzuela [START_REF] Erceg | Tgn channel models[END_REF].

The channel models comprise a set of 6 profiles, labeled A to F , which cover the scenarios of Rayleigh flat fading, residential, residential/small office, typical office, large office, and large space (indoors and outdoors). Each channel model has a path loss model including shadowing, and a MIMO multipath fading model, which describes the multipath delay profile, the spatial properties, the K-factor distribution, and the Doppler spectrum.

Each channel model has a certain number of taps (one for model A, and 9 to 18 for models B -F ). Each model further comprises a number of clusters, which correspond to overlapping subsets of the tap delays. Each tap is characterized by a relative delay (with respect to the first path delay). Table 2.1 summarizes the basic model parameters.

The average power of each tap in a particular cluster was determined so that the sum of the powers of overlapping taps corresponding to different clusters corresponds to the powers of the original average power delay profiles. values were assigned to each tap and cluster (using statistical methods) that agree with experimentally determined values reported in the literature.

Cluster AS was experimentally found to be in the 20 • to 40 • range, and the mean AoA was found to be random with a uniform distribution. With the knowledge of each tap power, AS, and AoA (AoD) and for a given antenna configuration, TGn generates the correlation matrix. The correlation matrix for each tap is based on the PAS with AS being square root of the second moment of PAS.

The various steps of the cluster modeling approach are briefly described below.

-Start with delay profiles of models, designed as A for Rayleigh, B, C, D, E and F for other specific environments. -Manually identify clusters in each of the five models.

-Extend clusters so that they overlap, determine tap powers (see Appendix A).

-Assume Laplacian PAS shape of each cluster and corresponding taps.

-Assign AS to each cluster and corresponding taps.

-Assign mean AoA and AoD to each cluster and corresponding taps.

-Assume antenna configuration.

-Calculate correlation matrices for each tap as in [START_REF] Erceg | Tgn channel models[END_REF]. Figure 2.11 and Figure 2.12 show model B and its delay profile with clusters. Clearly, two clusters can be identified: cluster 1 corresponds to tap delays 0 to 40 ns (in steps of 10 ns), while cluster 2 corresponds to tap delays 20 to 80 ns (also in steps of 10 ns). Hence, clusters 1 and 2 comprise 5 and 7 tap delays, respectively, and they overlap in 3 tap delays (20, 30 and 40 ns). Each cluster is assigned a set of spatial properties: a mean AoA, a mean AoD, an AS at the receiver, and an AS at the transmitter. These parameters assume the same values for all tap delays related to a given cluster. These parameters determine the transmit and receive correlation matrices associated with each tap delay.

The IEEE 802.11n channel models make the following assumptions:

-The PAS and the Power Delay Spectrum (PDS) are separable: each tap is modeled independently. -The PAS and the Doppler spectrum for each tap are separable: the spatial correlation (correlation matrices) and temporal correlation (Doppler spectrum) for each tap are modeled independently. -Each tap is modeled using the Kronecker model for Rician correlated channels,

TGac channel models

In order to support MAC throughput above 1 Gbit/s using one or more of the following technologies: higher order MIMO (≥ 4 × 4), or higher bandwidth (≥ 40 MHz), or the MU-MIMO technology with ≥ 4 transmitting antennas, the TGac task group proposes modifications to the basic TGn model [START_REF] Breit | TGac channel model addendum[END_REF].

The TGn channel models assume minimum tap spacing of 10 ns and were employed for system bandwidth of up to 40 MHz. TGac systems may have much larger bandwidth. Hence, the channel sampling rate is increased by reducing the PDP tap spacing by a factor defined in [START_REF] Breit | TGac channel model addendum[END_REF].

The TGn channel models were originally conceived for MIMO systems with a transmitter with four antennas and a receiver with four antennas, and are based on the Kronecker channel correlation model assumption. TGn proposes that it is sufficient for channel models to tightly bound and sweep the range of performance in real environments. Furthermore, it is desirable that the channel model is simple enough and builds on TGn channel models to allow a fair and efficient comparison of different standards proposals.

Finally, to support MU-MIMO scenarios, the TGac proposed modifications to the power angular spectrum. These modifications are summarized as follows [START_REF] Erceg | Tgn channel models[END_REF]:

-The defined TGn azimuth spread for each cluster remains the same for all users.

-For each user, independent random offsets between +/ -180 • are introduced for the AoA, the direct tap and the NLOS taps. -For each user, independent random offsets between +/ -30 • are introduced for the AoD, the direct tap and the NLOS taps. In fact, it is shown [START_REF] Breit | TGac channel model addendum[END_REF] that for the same transmitter location, different receiver locations lead to a different AoA at the receiver as depicted in Figure 2.13. Specifically, the measurements report that clusters AoA vary by 0 -20 degrees in NLOS scenario (classroom) and 0 -60 degrees in LOS scenario (great hall), depending on location. This is equivalent to a MU-MIMO scenario with a fixed transmitter location and receivers at different locations as depicted in Figure 2.13. Hence, the TGac conjectures that for the same transmitter location, different receiver locations lead to a different AoD at the transmitter. This conjecture is justified based on simulations evaluating the channel capacity [START_REF] Breit | TGac channel model addendum[END_REF].

MU-MIMO techniques

In this section, we give an overview about the MU-MIMO techniques. First, we present the assumed hypothesis throughout this thesis. Afterwards, we give the MU-MIMO system model as described in the literature. Finally, we synthesize the most explored MU-MIMO techniques in research distinguishing two cases: the single-antenna users and the multiple antennas users. 802.11ac, the standard specifies a single compressed beamforming method that relies on the use of explicit feedback to implement MU-MIMO [START_REF] Perahia | Gigabit wireless LANs: an overview of IEEE 802.11ac and 802.11ad[END_REF] [START_REF] Redieteab | Cross-layer optimization for next generation Wi-Fi[END_REF]. However, propagation channels change over time in actual radio environments due to moving people resulting in Doppler effect for example. Hence, CSI is not perfect. A simple channel prediction scheme to provide CSI is proposed in [START_REF] Bui | Performance evaluation of a multi-user MIMO system with prediction of time-varying indoor channels[END_REF], and its effectiveness is demonstrated through simulations of BER performance using a measurement campaign in a meeting room.

To mitigate the effects of intersymbol interference in frequency selective channels, OFDM is now a simple and well-accepted technique which converts a broadband frequency selective channel into a series of narrowband channels by transmitting data in parallel over many subcarriers. Combining OFDM with MIMO, producing so called MIMO-OFDM, significantly reduces receiver complexity in wireless multiuser broadband systems. Since OFDM uses multiple subcarriers, optimal linear precoding for MIMO-OFDM can be implemented by deriving linear precoders for each subcarrier independently. Hereafter, we assume either the channel experiences a flat fading so that the channel is frequency non-selective or we operate in narrowband subchannels guaranteed by the use of OFDM. Hence, the channel fading characteristic can be considered to be narrowband for each frequency bin. This assumption is satisfied in the case of the environment considered by TGac channel models. In fact, the subcarrier frequency spacing defined in the 802.11ac standard [START_REF]and Information Exchange Between Systems -Local and Metropolitan Area Networks -Specific Requirements: Part 11: Wireless LAN medium access control (MAC) and physical layer (PHY) specifications. Amendment 3: Specification for operation in additional regulatory domains. IEEE (std.)[END_REF] is ∆f = 312.5 kHz, and the measured coherence bandwidths are ≥ 1 MHz: B coh ≫ ∆f [START_REF] Kumar | Microwave indoor radio propagation measurements and modeling at 5 GHz for future wireless LAN systems[END_REF] in most of the indoor environments in the 5 GHz frequency band.

MU-MIMO system model

In a MU-MIMO transmission, there are two challenges in a MU-MIMO scenario: Uplink (UL) and Downlink (DL) channels as depicted in Figure 2.14. The UL channel, is used to model multiple users transmitting simultaneously to a single AP. The DL or Broadcast Channel (BC) is used to model the case where the AP is transmitting simultaneously to multiple users. The UL challenge is addressed using array processing and multi-user detection techniques by the AP in order to separate the signals transmitted by the users. The challenge for BC is that the interference cancellation is required at the AP to have cheap terminals with low power consumption. This report focuses on the capacity of the MU-MIMO DL channels. The UL MU-MIMO is not considered by the 802.11ac standard. Thereby, a presentation of the mathematical model can be quite useful. Indeed, techniques MU-MIMO processing techniques are based on such approaches.

In the following, we describe the typical system model considered in this thesis. The studied IEEE 802.11ac MU-MIMO system is composed of K users connected to one AP, as shown in Figure 2. [START_REF] Heusse | Performance anomaly of 802.11b[END_REF].

The AP has n T antennas and communicates with K users simultaneously. Each user k has n R k antennas. We define n R = K k=1 n R k , as the total number of the receiving antennas.

According to Equation 2.1, the channel from the AP to the k th user is represented by a n R k × n T channel matrix H k . It is the MIMO subchannel transfer function for a subcarrier. For the sake of simplification, the subcarrier index is not mentionned. We define the n R × n T channel matrix H as H = [H T 1 . . . H T K ] T , and its elements

H ij as H = H ij 1≤j≤n T 1≤i≤n R
. Note that each element H ij of H is the CFR presented in Subsection 2.3.2 between i th receiving antenna and the j th transmitting antenna and for a particular subcarrier. The received signal is written as:

y = HX + n (2.12)
where X is the transmitted signal.

Let s k represents the N ss,k × 1 transmit data symbol vector for user k, where N ss,k is the number of independent spatial streams transmitted simultaneously for the k th user. The total number of spatial streams is N ss = K k=1 N ss,k . We define the N ss ×1 transmit data symbol vector as:

s = [s 1 T s 2 T . . . s K T ] T .
Next section addresses how X can be designed based on s using precoding techniques for MU-MIMO.

Precoding techniques

Precoding technique is the process of using the information at the transmitter in a way that mitigates multi-user interference on the one hand, and maximizes the sum rate capacity on the other. The non-linear techniques are known to achieve optimum capacity. Actually, it has been proven that the capacity region of the DL MU-MIMO systems is achieved with Dirty Paper Coding (DPC) method [START_REF] Lee | Dirty paper coding vs. linear precoding for MIMO broadcast channels[END_REF] called also the capacity bound. This non-linear technique consists of precoding the data in order to avoid the effect caused by the interference without canceling it. However, it has high computational complexity.

The capacity bound can be approached by linear techniques. Linear precoding techniques are low-complexity transmission techniques compared to non-linear techniques [START_REF] Bui | Performance evaluation of a multi-user MIMO system with prediction of time-varying indoor channels[END_REF].

Linear precoding techniques, supported by the standard IEEE 802.11ac, assume that the transmitted signal X is generated by a linear combination of data symbols contained in a vector s. It has been shown that linear precoding techniques can achieve sum rate capacity close to the DPC capacity bound with lower complexity [START_REF] Lee | Dirty paper coding vs. linear precoding for MIMO broadcast channels[END_REF]. Next, we will present some precoding techniques proposed in the literature in order to identify a relevant method for our study in next chapters. First, the simple case of single antenna receivers is detailed in next subsection. Afterwards, we present the precoding techniques taking into account multiple antennas per user.

Linear precoding techniques for single antenna receivers

Each user has only one receive antenna (n R k = 1 for k = 1, .., K). In this case, we have n R = K. Various approaches exist to solve the problem of designing linearly X given s. Some of these techniques are Channel Inversion (CI) and regularized channel inversion methods [START_REF] Pan | Generalized multiuser orthogonal space-division multiplexing[END_REF].

Channel inversion

The channel inversion technique consists in canceling the effects of the channel by precoding s with the pseudo inverse H † of the channel matrix [START_REF] Garcia-Garcia | Multi-user MIMO capacity from measurements in indoor environment with in-and outdoor base stations[END_REF]:

X = H † s = H H (HH H ) -1 s f or : n T ≥ n R = K (2.13)
The inversion in the Equation 2.13 can be done only when n T ≥ n R = K. In [START_REF] Peel | A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: channel inversion and regularization[END_REF], it has been proven that the sum rate as K goes to infinity is constant when n T = K unlike the capacity bound of a classical SU-MIMO system [START_REF] Peel | Space-Time Processing for MIMO Communications (chapter:Linear and Dirty-Paper Techniques for the Multi-User MIMO Downlink[END_REF] which increases linearly with min(n R , n T ) [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF]. This poor capacity is explained by looking at the eigenvalues of the matrix (HH H ) -1 . This matrix is not well conditioned for random complex Gaussian matrices when H is square, resulting in a very low SNR at the receivers. The CI technique is then a good solution for low noise or high-power situations [START_REF] Peel | A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: channel inversion and regularization[END_REF].

Therefore, any approach to improve channel inversion must seek to reduce the effects of ill-conditioned channel matrices.

Regularized channel inversion

One technique used to regularize an inverse is to add a multiple of the identity matrix before inverting. This technique is called regularized channel inversion and is written:

X = H H ((HH H + ζI) -1 )s (2.14)
where ζ is the regularization parameter.

In practice, the regularization factor, as a function of the total transmitted power P T x , is commonly chosen as ζ = n T P T x [START_REF] Peel | A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: channel inversion and regularization[END_REF]. It is shown in [START_REF] Peel | A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: channel inversion and regularization[END_REF] that this expression approximately maximizes the Signal-to-Interference-plus-Noise Ratio (SINR) at each receiver, and leads to linear capacity growth with n T [START_REF] Kaltenberger | Correlation and capacity of measured multi-user MIMO channels[END_REF] [START_REF] Kamal | Multiuser MIMO downlink system capacity analysis in wireless communication for time varying channel[END_REF]. Using this technique, each user sees some interference. Actually, if ζ = 0 , the transmitter does not cancel out perfectly all interference. Furthermore, this method has less flexibility in adjusting the transmitted power to each user: changing the transmitted power to one user changes the interference for all other users.

Discussion

The achievable sum rates for channel inversion and regularized channel inversion methods are explicitly given in [START_REF] Kaltenberger | Capacity of linear multi-user MIMO precoding schemes with measured channel data[END_REF]. The authors in [START_REF] Peel | A vector-perturbation technique for near-capacity multiantenna multiuser communication-part I: channel inversion and regularization[END_REF] have discussed their performance versus the increase of the number of users as well as high SNR. The sum rate of regularized inversion has linear growth with K, although its slope is different from the sum capacity. The regularization channel inversion method shows a big improvement over channel inversion method, a gap to capacity bound however remains, especially at high SNR.

The channel inversion and regularized channel inversion methods could be used with multiple antennas (n R k > 1). However, system performance is not optimal. Several linear methods that take advantage of the presence of multiple antennas at the receivers have been proposed by [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF] [START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF]. The technique that was widely explored for multiple antenna receivers is block diagonalization which is discussed in Subsection 2.5.5.1. This technique relaxes the interference cancellation constraint between spatial streams to interference cancellation between users only.

The rate gap between the BD and CI techniques with perfect Channel State Information at the Transmitter (CSIT) is given in [START_REF] Lee | High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding[END_REF]. According to [START_REF] Morales Cespedes | Achievable throughput with block diagonalization on OFDM indoor demonstrator[END_REF], a difference of 3.2 bits/s/Hz is seen between CI and BD sum-rates for the 2-user equipped with 2 antennas case connected to a transmitter with 4 transmitting antennas in indoor scenarios.

Linear precoding techniques for multiple antenna receivers

Block diagonalization

Block diagonalization [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF], also referred to as block channel inversion, is a generalization of the channel inversion technique when there are multiple antennas at each user.

BD scheme is one popular alternative due its low implementation complexity and its capability of approaching the capacity bound at high SNR.

The AP employs a BD precoding matrix W k of size n T × N ss,k for each user, which transforms the data s k to the transmitted vector W k s k . The n T × N ss precoding matrix is hence defined as:

W = [W 1 . . . W K ].
For each IEEE 802.11ac OFDM subcarrier, the received signal vector at the k th user is given by:

y k = H k W k s k + H k K i=1,i =k W i s i + n k (2.15)
where n k = [n k,1 , . . . , n k,n R k ] T is the noise vector composed of complex Gaussian random variables for the k th user. The components n k,i are i.i.d. with zero mean and variance σ 2 n for k = 1, . . . , K and i = 1, . . . , n R k . We define the noise vector n as:

n = [n 1 T n 2 T . . . n K T ] T . The corresponding signals y = [y T 1 y T 2 . . . y T K ]
T at all users can be arranged as:

y = HWs + n (2.16)
According to Equation 2.15, the component H k W k s k denotes the desired signal for the k th user, and the other components are the interference values between the k th and the other K -1 users. Hereafter, noise components are ignored. Therefore, it is important for the MU-MIMO technique to eliminate these interference values. In this direction, the BD precoding matrix must satisfy for any H k and s i with i = k:

H k K i=1,i =k W i s i = 0 (2.17) It follows that H i W k = 0 for i = k and 1 ≤ i, k ≤ K.
Here, we define the following new MIMO channel matrix Hk except for the k th user as:

Hk = H T 1 . . . H T k-1 H T k+1 . . . H T K (2.18)
The condition in Equation 2.17 implies that W k needs to be in the nullspace of Hk of dimension dim(N ull( Hk )) = n T -rank( Hk ). Note that the precoding matrix W k should be a nonzero matrix, otherwise, no signal is transmitted. To guarantee the existence of a nonzero precoding matrix, a sufficient condition is that the number of the transmit antennas is larger than the sum of the receive antennas of any K -1 users [START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF] and is written as

n T > max( K i=1,i =k n R i , k = 1, 2, . . . , K) (2.

19)

The BD precoding matrix W k is a cascade of two precoding matrices A k and B k .

W k = A k B k
, where A k is computed by Singular Value Decomposition (SVD) in order to nullify the IUI and B k is designed using any classical method for optimizing SU-MIMO capacity.

The SVD of the matrix Hk is defined in Equation 2.20 where Ũk , Ãk , and Ṽk are the left singular vector matrix, the matrix of singular values of Hk and the right singular vector matrix respectively.

Hk = Ũk Ãk ṼH k (2.20) We set Ṽk = [ Ṽ(1) k Ṽ(0) k ] where Ṽ(1)
k denotes the right singular matrix corresponding to non-zero singular values, referred to as the single space of all users except for the k th user. Term Ṽ(0) k denotes the right singular matrix corresponding to zero singular values, i.e. singular vectors in the null space which does not interfere with the other K -1 users. In order to produce zero interference at the other users, the matrix A k is chosen by using the weight matrix A k as the null space Ṽ(0)

k : A k = Ṽ(0) k . We can see that the dimension of Ṽ(0) k is n T × N ss,k .
Under the sufficient condition given in Equation 2.19 and the assumption of i.i.d. channel, we can obtain than the dimension N ss,k of the null subspace is [START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF].

N ss,k = n T - K i=0,i =k n R i
By substituting into Equation 2.15, we obtain Equation 2.21 where the MU-MIMO system denoted by Equation 2.15 has been decoupled to K parallel SU-MIMO systems as depicted in Figure 2. [START_REF] Redieteab | Cross-layer optimization for next generation Wi-Fi[END_REF].

y k = H k Ṽ(0) k B k s k + n k (2.21)
A close observation from Equation 2.21 and Figure 2.16 shows that we can think of an equivalent SU-MIMO channel of user k as H k Ṽ(0) k and the equivalent transmit preprocessing for the equivalent SU-MIMO channel for user k can be represented as B k . The design of B k is the same as designing the transmit preprocessing for a SU-MIMO system. In this report, it is designed to optimize the channel capacity [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF] and based channel matrix of the BD precoding is a block diagonal matrix. Hence MIMO decoders can take advantage of the diversity present in the inter stream interference for each user.

The authors in [START_REF] Ravindran | Limited feedback-based block diagonalization for the MIMO broadcast channel. Selected Areas in Communications[END_REF] evaluate the BD technique using a limited feedback system. They have established the advantage of using BD technique compared to CI in terms of feedback load quantifying the loss based on the throughput as a function of the feedback level.

Based on experimental mobile WiMAX system with real implementation of BD technique, it has been shown in [START_REF] Yamaguchi | Transmission performance evaluation for downlink MU-MIMO system: Computer simulation with block diagonalization algorithm and field experiments based on mobile WiMAX system[END_REF] that BD throughput is almost the same using either measured or simulated channels.

The BD scheme is just one of many linear precoding techniques that can be used on the MU-MIMO broadcast channel with multiple user antennas. It is primarily limited when the number of receiving antennas is greater than the number of transmitting antennas.

To overcome the dimensionality constraint, [START_REF] Stankovic | Multi-user MIMO downlink precoding for users with multiple antennas[END_REF] have proposed successive Minimum Mean Square Error (MMSE) precoding (SMMSE). SMMSE provides higher antenna and diversity gain than MMSE by suppressing the interference only between the antennas located at the two different terminals.

In [START_REF] Stankovic | Generalized design of Multi-User MIMO precoding matrices[END_REF], the authors use a different approach by separating the problem of IUI mitigation and the optimization of the overall system performance with respect to the different optimization criteria. The proposed method is called Regularized block diagonalization (RBD).

Different modifications of the original RBD algorithm provide different performance improvements. RBD is capable to adapt to different levels of SNR in order to provide a higher information sum rate of the system. At high SNR and high IUI it is able to provide the high diversity. RBD represents a good compromise between capacity/BER performance and computational complexity.

In [START_REF] Boccardi | A near-optimum technique using linear precoding for the MIMO broadcast channel[END_REF], the authors propose a linear transmission technique based on the BD technique using joint coordinated transmit-receive processing using a receiver beamformer to select a subset of the eigenmodes of a given user, and a transmitter beamformer in order to guarantee the orthogonality between the different users. This method is evaluated based on numerical results. It has been shown how the proposals achieve a significant fraction of the DPC sum rate for practical systems with finite K, and outperform previous BD schemes.

To sum up, there are various linear precoding techniques and their performance evaluation are compared in the literature. They achieve almost the same capacity with relatively the same complexity. Since BD is the one that has been widely explored in the literature and is used as reference for further improvement or extensions, this report will focus on simulating the MU-MIMO processing based on its principle.

Note also that in the standard 802.11ac uses the feedback to obtain the CSI. Hence, the MU-MIMO is chosen when the SNR is sufficiently high to get an acceptable feedback.

Non-linear precoding techniques

Non-linear precoding techniques are known to be capacity achieving using the notion of DPC, which originates from [START_REF] Costa | Writing on dirty paper (corresp.). Information Theory[END_REF]. This reveals that any known interference at the transmitter can be subtracted without the penalty of radio resources if the optimal precoding scheme is applied on the transmit signal. In other words, it proposes to benefit from the known CSI at the Tx to transmit information, instead of fighting against this interference. Optimal performance can be achieved this way. But this technique is very difficult to be implemented in practice.

There are several suboptimal and simplified variants of dirty paper coding. Other non-linear precoding techniques are Tomlinson-Harashima Precoding (THP), Vector Perturbation (VP) and Lattice Reduction Aided (LRA) methods. However, these techniques have a very computational complexity and are not supported by IEEE 802.11ac.

We present in Appendix B a comparison of BD and DPC capacities based on the correlated TGac-B channel model.

Channel capacity of wireless channels

An important tool for characterizing the performance of any communication channel is the Shannon capacity. It is the maximum rate of communication for which arbitrarily small error probability can be achieved [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF].

A transmission scheme that maximizes the capacity for one user in the network might result in unacceptably high interference for the other users, rendering their links useless. If high throughput is the goal, a better approach might be to maximize the sum capacity of the network, or the maximum sum transmission rate, where the IUI is taken into consideration.

SU-MIMO capacity

The capacity of SU-MIMO additive white Gaussian noise channels with CSI known at the transmitter was first studied by Telatar [START_REF] Telatar | Capacity of multi-antenna gaussian channels[END_REF] and Foschini [START_REF] Foschini | On limits of wireless communications in a fading environment when using multiple antennas[END_REF]. It is the maximum amount of information that can be transmitted as a function of available bandwidth given a constraint power.

The considered SU-MIMO system applies a singular value decomposition and its capacity is computed for each OFDM subcarrier as detailed in [START_REF] Bouhlel | Transmit beamforming analysis for mimo systems in indoor residential environment based on 3d ray tracing[END_REF]:

C SU -M IM O , k = Nss ,k i=1 log 2 (1 + p ik σ 2 n σ 2 ik ) (2.24)
where p ik is the allocated power to the i th spatial stream for the k th user, σ ik are the singular values of the channel matrix for the k th user after the singular value decomposition, and σ 2 n is the noise power.

arithmetic mean of C 1 and C 2 as can be seen in Equation 2.25.

C SU,T DM

A = C 1 + C 2 2 (2.25)

CSMA/CA method

The IEEE 802.11ac standard uses for the channel access the CSMA/CA method, where each user verifies the absence of other co-channel signals before transmitting a frame. The data frames are supposed to have equal size for each user, denoted below L = L 1 = L 2 , which implies a variable transmission duration [START_REF] Heusse | Performance anomaly of 802.11b[END_REF]. According to Figure 2.17(b), the time duration of the SU-MIMO, denoted t SU , is as:

t SU = t 1 + t 2 .
Thus, the CSMA/CA sum capacity of the SU-MIMO system (2×L)

C SU,CSM A = L C 1 + L C 2 .
Hence, the CSMA/CA SU-MIMO sum capacity is then equal to the harmonic mean of C 1 and C 2 expressed as in Equation 2.26. We recall that ideal CSI is considered with no protocol or frame overhead in this ideal case.

C SU,CSM A = 2 1 C 1 + 1 C 2 (2.26)

MU-MIMO capacity

For a MU-MIMO system with K users and n R k receiving antennas for each user k, the sum rate using BD [START_REF] Kaltenberger | Capacity of linear multi-user MIMO precoding schemes with measured channel data[END_REF] for a particular propagation channel sample is expressed for each OFDM subcarrier by Equation 2.27.

C M U -M IM O = K k=1 Nss ,k i=1 log 2 (1 + p ik σ 2 n µ 2 ik ) (2.27)
where p ik is the power dedicated to the i th spatial stream for the k th user, µ ik are the singular values of the effective channel for the k th user after applying the IUI cancellation, and σ 2 n is the noise power. The methods of scaling the power will be analyzed in the following chapter.

Main existing results and future works

Compared to a SU-MIMO TDMA system, DPC can bring a theoretical performance gain in terms of channel capacity of up to max(min(n T /n R k , K), 1) in an i.i.d. Rayleigh fading channel [START_REF] Kaltenberger | Capacity of linear multi-user MIMO precoding schemes with measured channel data[END_REF]. The expected loss in Rayleigh fading due to block diagonalization compared to DPC is given in Appendix B.

BD and DPC techniques are shown to be equivalent at low and very high SNR, small number of users (low K), and high number of transmitting antennas [START_REF] Kaltenberger | Capacity of linear multi-user MIMO precoding schemes with measured channel data[END_REF] [START_REF] Lee | High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding[END_REF].

Most MU-MIMO studies are based on Rayleigh channels, and only few articles addresses the MU-MIMO challenges based on indoor measurements (see Appendix C).

Some measured channels show an increase of the sum rate capacity over SU-MIMO by a factor of 3.2 using DPC and by a factor of 2.7 using Regularized Channel Inversion (RCI) with four transmit and four users with one receive antenna each at high SNR [START_REF] Kaltenberger | Real-time multi-user MIMO channel sounding and capacity evaluations[END_REF]. These results encourage us to simulate the MU-MIMO processing based on BD technique since it outperfoms the RCI technique. The DPC gain compared to BD is however important with large number of users [START_REF] Lee | High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding[END_REF] which is not the case with 802.11ac standard. It actually limits the max number of users to 4 in a MU-MIMO group.

It has been shown in [START_REF] Kaltenberger | On the characterization of MU-MIMO channels[END_REF] that channel correlation impacts capacity gap. Still, this result is based on single antenna users. The authors in [START_REF] Nishimori | 16x16 multiuser mimo testbed employing simple adaptive modulation scheme[END_REF] prove that MU-MIMO capacity gain becomes insignificant comparing to SU-MIMO when n R increases with 16 transmitting antennas. Furthermore, the MU-MIMO is highly sensitive to Doppler [START_REF] Kakishima | Indoor experiments on 4-by-2 multi-user MIMO employing various transmitter antenna arrangements in LTE-advanced downlink[END_REF], [START_REF] Ogawa | An overview of actual SUand MU-MIMO performances in a measured indoor channel[END_REF].

Introduction

The first objective of this thesis is to refine the capacity gains by identifying deployment recommendations or optimal system configuration for indoor MU-MIMO compared to the SU-MIMO systems. Accordingly, the aim of this chapter is to give recommendations to optimize MU-MIMO performance based on TGac simulations in terms of channel propagation parameters and users' configuration: LOS, NLOS, distances... Consequently, this chapter is organized as follows. First, a detailed description of the simulation process, the evaluated channel model and its parameters are given in Section 3.2. Next, we provide the simulation results of the impact of antennas and propagation channels on the BD capacity gain over SU-MIMO in Section 3.3. Afterwards, we study in Section 3.4 the impact of transmit power control meeting the European regulation requirements for WLAN in the 5 GHz frequency band in terms of EIRP. Finally, we conclude in Section 3.5 by summarizing system recommendations for an optimal use of MU-MIMO compared to SU-MIMO.

TGac-B channel model

Simulation setup

Since we study the benefit of using MU-MIMO techniques in home networks in the context of the arrival of the optical fiber to the home, this chapter shows results according to the channel model TGac-B. Actually, the TGac-B channel model covers the scenarios of residential environments. It has a path loss model including shadowing, and a MIMO multipath fading model which describes the multipath power delay profile, the spatial properties, the Rician K-factor, the Doppler spectrum and the PAS.

Typically, the fading process is characterized by a Rayleigh distribution for the non line-of-sight paths and a Rician distribution for a line-of-sight path. The TGac-B channel model has 9 Rayleigh-fading taps, and each tap has a Bell Doppler spectrum to consider the random time variability of an indoor channel due to human activity for example. Each tap is characterized by a relative delay (with respect to the first path delay). It further comprises a number of clusters, which correspond to overlapping subsets of the tap delays. Each cluster is assigned a set of spatial properties: a mean AoA, a mean AoD, an AS at the receiver, and an AS at the transmitter. These parameters assume the same values for all tap delays pertaining to a given cluster. These parameters determine according to the procedure given in [START_REF] Erceg | Tgn channel models[END_REF] the transmit and correlation matrices associated with each tap delay. The relative motion between the transmitter and receiver causes Doppler shifts. Local scattering typically comes from many angles around the user. This scenario causes a range of Doppler shifts, known as the Doppler spectrum. The maximum Doppler shift corresponds to the local scattering components whose direction exactly opposes the user's trajectory.

A Matlab source code [START_REF] Erceg | Tgn channel models[END_REF] was used to compute the different 802.11ac channel samples based on the implementation of fading channels as objects. The main loop of the source code is designed to process Fading Number Of Iterations bursts of samples. In our case, its value is set in the Matlab channel model to 512. The length of a burst can be expressed in coherence times, where the coherence time is defined in the TGn channel models as the inverse of the Cut-off frequency f D Hz = 5.8333 Hz which is the ratio between the wavelength and the movement speed of the scattering environment v 0 . In our simulation, we use a simulation length equal to 55 coherence times of the MIMO channel to simulate the fast fading. The choice of the value 55 for coherence times is justified by the convergence of the MU-MIMO to SU-MIMO capacity gain as the value of coherence times is increasing as can be seen in Figure 3.1. The figure gives the MU-MIMO to SU-MIMO capacity gain versus the simulation length in coherence times for a system with two receivers with two antennas each, both in NLOS with the Tx equipped with 4 antennas. The antennas are spaced with half wavelength. The capacity is computed using an average SN R = 20 dB. The simulation values stabilize after the fictive threshold of 50 coherence times. Table 3.1 summarizes the input parameters used for our simulations. Thereafter, a uniform linear array of antennas at the AP is simulated with the propagation channel model TGac-B (15 ns RMS delay spread) for the 5.25 GHz frequency band. Note that linear array antenna is the only antenna array geometry supported by TGac channel model. The simulated system is composed of one access point equipped with multiple antennas (linear array of 0 dBi omnidirectionnal and vertically polarized antennas), and two receivers. Each receiver has two 0 dBi omnidirectionnal antennas. By default, the antenna spacing is set to 0.5 λ for both the transmitter and the receiver.

Normalization

It is demonstrated in [START_REF] Loyka | On physically-based normalization of MIMO channel matrices[END_REF] that the physics of antenna arrays and propagation channel should be taken into account when the normalization of the channel matrix is chosen, so that SNR has proper physical meaning. The conclusions are physical and correspond to realistic systems. The paper pointed out various normalizations and gave equivalent formulas for different parameters needed. The antenna array geometry and the transmission strategy (coherent/non-coherent) limit the choice of normalization and determine how the capacity and other performance metrics scale with the number of antennas, which is more pronounced for densely-populated antenna arrays. The commonly used normalization is E( H 2 F ) = n T n R . It corresponds to a 0 dB average path loss. This normalization implies the equality in Equation 3.1.

1 N samples N samples p=1 n R i=1 n T j=1 |H ij | 2 = n T n R (3.1)
For the sake of simplicity, we omit to write the components of the channel matrix as a function of the index of the channel sample p (N samples = 488). We denote the normalized channel matrix H ′ . We have H ′ = αH where α is the normalization constant. Thus, the normalized channel matrices H ′ 1 and H ′ 2 for each user are given in Equation 3.2.

                   H ′ 1 = n T n R 1 N samples N samples p=1 n R 1 i=1 n T j=1 |H 1ij | 2 H 1 H ′ 2 = n T n R 2 N samples N samples p=1 n R 2 i=1 n T j=1 |H 2ij | 2 H 2 (3.2)

TGac and Rayleigh channel

Rayleigh fading is exhibited for each tap (except for the LOS tap which follows a Rice fading with a 0 dB Rician factor), with the assumption that the real and imaginary parts of the taps are modeled by i.i.d. zero-mean Gaussian processes so that different taps are uncorrelated. We recall that for each IEEE 802.11ac OFDM subcarrier, the channel matrix is computed through a discrete Fourier transformation (N = 56 subcarriers for a 20 MHz channel) of the tap delay representation. To have comparisons with an ideal case, we also study a non-correlated Rayleigh channel.

Impact of antennas and spatially correlated propagation channel on BD capacity gain

The objective of this thesis it to know which among the MU-MIMO and SU-MIMO techniques is better in a residential environment. Thereby, most graphs in this chapter highlight the MU-MIMO capacity gain over SU-MIMO expressed as the average of MU-MIMO (simulated BD method) to SU-MIMO capacity ratio. For 2 users, the optimal value of MU-MIMO to SU-MIMO capacity gain is theoretically proven to be 2 for TDMA system based on a Rayleigh channel. Indeed, compared to a SU-MIMO TDMA system, DPC can bring a theoretical performance gain of up to max(min(n T /n R , K), 1) in an i.i.d. Rayleigh fading channel [START_REF] Jindal | Dirty-paper coding versus TDMA for MIMO broadcast channels[END_REF].

We first study in this section the impact of antennas and propagation channel on the BD capacity gain for IEEE 802.11ac MU-MIMO in home networks. No EIRP constraint is considered in this section. We examine the effect of the number of the transmitting antennas, their spacing, and the average SNR, defined as the ratio of P T x to the average noise power: SN R = P T x N σ 2 n , where N = 56 is the total number of subcarriers defined in IEEE 802.11ac standard for a 20 MHz bandwidth.

P T x is considered equally shared among the N subcarriers throughout this thesis. Other strategies could be considered as in [START_REF] Saavedra | Multidiffusion et diffusion dans les systemes OFDM sans fil[END_REF]. However, the impact of Peak-to-Average Power Ratio (PAPR) should be globally studied. We also draw for comparative study the results for i.i.d. Rayleigh channels. The aim of this study is to assess the weight of each parameter on the BD capacity gain over SU-MIMO and to give recommendations to optimize MU-MIMO performance.

Transmit antenna spacing effect

For the following, the number of transmitting antennas is fixed to n T = 6, and SN R = 20 dB. To study the impact of the transmitting antenna spacing, six values are used: 0.25 λ, 0.5 λ, 0.75 λ, 1 λ, 1.25 λ and 1.5 λ. No antenna coupling is considered [START_REF] Krusevac | Mutual coupling effect on thermal noise in multi-element antenna systems[END_REF][54] [START_REF] Ogawa | An overview of actual SUand MU-MIMO performances in a measured indoor channel[END_REF].

In Figure 3.3, the first value (0.25 λ) presents an isolated and very low gain (33%) compared to the other spacings. For a transmit antenna spacing of 0.5 λ and 0.75 λ, the capacity gain of MU-MIMO compared to SU-MIMO is around 50%. For a transmit antenna spacing of 1 λ and above, the capacity gain of MU-MIMO compared to SU-MIMO is around 53% and it almost attains the gain in a Rayleigh channel. Antenna spacing has no effect on the Rayleigh channel since its MU-MIMO channel matrix elements are complex Gaussian and independent. Considering a trade-off between the overall antenna size and the MU-MIMO capacity gain, we recommend an antenna spacing equal to 0.5 λ. This is set as the default value for the rest of this report. Nevertheless, for low SNR, SU-MIMO performs better than MU-MIMO in terms of capacity which has been described already in the limitations of BD scheme. Note that it is not practical to have too low SNR for MU-MIMO as it would not be possible to have CSI with no errors to apply the precoding.

SNR effect

The MU-MIMO to SU-MIMO capacity gain is 30% when the SNR increases from 0 to 10 dB, from 10 to 20 dB, or from 20 to 30 dB, and around 10% when it changes from 30 to 40 dB. To conclude about the SNR, the desired range and the desired bit rate are put forward. For the following results, a middle case is evaluated with SN R = 20 dB.

Number of transmitting antennas

In this subsection, we assess the impact of increasing the number of transmitting antennas on MU-MIMO to SU-MIMO capacity gain. Under the same total transmitted power, we have considered two transmitted power allocation strategies. We first share the allocated power equally among the spatial streams. Secondly, we optimize the power allocation using the Water-filling algorithm. In both cases, the SNR is set to SN R = 20 dB.

Equal power sharing

We assume here that the transmitted power is equally shared among the spatial streams. Figure 3.5 gives the average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas for the TGac-B NLOS and Rayleigh channels. Average values, 10% and 90% quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. The first observation drawn from Figure 3.5 is that the MU-MIMO capacity gain over SU-MIMO increases with the number of transmitting antennas. It changes from 1.2 to 1.65 for the residential environment, i.e. around 45% of capacity gain. We also observe that the gain from n T = 4 to n T = 6 is much higher than the one observed from n T = 6 to n T = 8 or the one observed from n T = 8 to n T = 10. This can be explained by the fact that we cannot take benefit of the transmit beamforming for n T = 4, since the number of transmitting antennas is the same as the total number of spatial streams. Another explanation concerning the channel correlation is given hereafter.

In order to optimize the MU-MIMO capacity gain and have a less congested system, we recommend using n T = 6, when we have a system with two receivers and two antennas each.

Optimized power sharing: Water-filling algorithm

The capacity of the MU-MIMO system can be further increased if we assign extra power at the transmitter by allocating the power according to the Water-filling algorithm to all the channels. The capacity optimization problem is presented in Equation 3.3

   max subcarrier K k=1 Nss ,k i=1 log 2 (1 + p ik σ 2 n µ 2 ik ) such that K k=1 Nss ,k i=1 p ik = P T x N (3.3)
The power p ik is allocated according to the Water-filling algorithm. The problem 3.3 is a constrained maximization problem solved based on Lagrange technique. This technique is characterized by the Lagrange multiplier λ Lagrange by maximizing the function F defined in Equation 3.4. In our case, we have considered a maximization for each subcarrier separately.

F = K k=1 Nss ,k i=1 log 2 (1 + p ik σ 2 n µ 2 ik ) -λ Lagrange ( P T x N - K k=1 Nss ,k i=1 p ik ) (3.4)
From Equation 3.4, we deduce the expression of p ik = (

1 λ Lagrange -σ 2 n µ 2 ik
) + , where (.) + is defined as: (x) + = x if x ≥ 0 and (x) + = 0 if x < 0. The Lagrange multiplier is found based on the constraint of the optimization problem. The process of Waterfilling algorithm is similar to pouring the water in the vessel. The unshaded portion of the graph in Figure 3.7 represents the inverse of the power gain of a specific channel. The shadow portion represents the power allocated or the water. In [START_REF] Palomar | Practical algorithms for a family of waterfilling solutions[END_REF], a practical algorithm is detailed to evaluate numerically a general Water-filling solution. Figure 3.8 analyzes the benefit of using Water-filling algorithm in optimizing the MU-MIMO capacity gain over SU-MIMO. It shows the average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas for TGac-B channel NLOS using or not the Water-filling algorithm.

Average values, 10% and 90% quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. The first observation drawn from Figure 3.8 is that the average MU-MIMO capacity gain over SU-MIMO is almost the same as the obtained gain using the Water-filling algorithm with slight gain in 90% quantiles: around 5%. This is explained by the fact that both MU-MIMO and SU-MIMO systems take benefit from the Water-filling algorithm as was proven widely in the literature. Hence, using this complex algorithm for increasing MU-MIMO capacity gain is not necessary. Next section will explore other ways in optimizing MU-MIMO system satisfying a given power or EIRP constraint. ) is considered to be the worst case for MU-MIMO [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF]. Actually, with BD, each user precoding matrix lies in the null space of all other users' channels. Hence, if H 1 = H 2 , then H 2 = Ṽ(0) 1 . These theoretical results for extreme cases and our simulations revealed that the definition in Equation 3.7 was the most relevant to explain MU-MIMO to SU-MIMO capacity gain [START_REF] Spencer | Channel allocation in multi-user MIMO wireless communications systems[END_REF].

ρ = H 1 H 2 H 2 F n R 1 n R 2 (3.7)
Let us define the rows of the channel matrix of the j th user as:

H j = (L j 1 , . . . , L j n R j ) T
where

L j i = (l j i,1 , . . . , l j i,n T ) represents the i th row of dimension 1 × n T . The applied channel normalization implies that || L j i ||= 1. The product H 1 H 2 H = (L 1 1 , . . . , L 1 n R 1 ) T (L 2 1 , . . . , L 2 n R 2
)

H is a matrix whose elements are:

L 1 i L 2 j H 1≤j≤n R 1 1≤i≤n R 2
. Hence, the correlation coefficient can be expressed as:

ρ = 1 n R 1 n R 2 n R 1 i=1 n R 2 j=1 || L i 1 L j 2 H || 2 (3.8)
The component || L i 1 L j 2 H || 2 represents the correlation between each single receiving antenna subsystem of the first user and each single receiving antenna subsystem of the second user. This is the more common correlation coefficient used in the case of single antenna receivers. E(ρ) represents an average value of the correlation coefficient between any single antenna receiver subsystem combination.

The impact of the number of transmitting antennas versus average correlation coefficient E(ρ) is presented in Figure 3.9. The averaging is performed over all the MU-MIMO channel samples. The average correlation coefficient decreases with the increase of the number of transmitting antennas. The two types of residential channel (LOS, NLOS) follow the same trend. The values are higher but remain relatively close to the ones obtained for i.i.d. Rayleigh channel. We can observe that even if the simulated Rayleigh channel has independent elements in H 1 and H 2 , the average correlation coefficient E(ρ) is not zero, which is not an intuitive result. Through analytic calculation (see next subsection), the average correlation coefficient ρ for an i.i.d. MIMO Rayleigh fading channel is proven to be:

E(ρ) = 1 n T .
Figure 3.9 -Average correlation coefficient versus the number of transmitting antennas.

In Figure 3.10, we highlight the effect of transmit antenna spacing on the defined correlation coefficient for a Tx with 6 transmitting antennas in NLOS conditions. The figure shows that increasing the antenna spacing decorrelates the channels to attain the optimum value 0.17 obtained for the Rayleigh channel. The recommended value of 0.5 λ spacing gives a correlation coefficient value of 0.2 which is very close to the optimal value. Figure 3.10 -Average correlation coefficient versus transmit antenna spacing.

Impact of correlation coefficient on capacity gain

The average of MU-MIMO to SU-MIMO capacity ratio versus the average of the correlation coefficient E(ρ) is presented in Figure 3.11 considering n T = 6. The average is computed here only over time, and each point represents one of the 100 samples of user couples. When the correlation coefficient increases, the capacity gain decreases: more than 30% of capacity loss when the correlation coefficient goes from 0.05 to 0.35. Actually, when both channels (H 1 and H 2 ) are correlated, close in other words, the BD algorithm does not show great performance since the existence of the precoding matrix is not guaranteed [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF]. Thus, the gain decreases. It could be the case when the two users are close. This correlation coefficient has another advantage: it can be computed to optimize the users grouping in a MU-MIMO scenario. For example, a system could be optimized by selecting the K users minimizing

n R 1 i=1 n R 2 j=1 ||H 1 H 2 H || 2 n R 1 n R 2
between any 2-user.

Since Figure 3.11 reveals a linear trend between correlation ρ and the capacity gain, an analysis based on the supervised linear regression is carried out. To formulate the problem, the linear regression will determine how the MU-MIMO to SU-MIMO capacity gain y is affected by the changes in the correlation coefficient x according to the regression line in Equation 3.9.

y ′ = a 1 x + a 0 (3.9)

where y ′ , a 1 and a 0 denote respectively the estimated capacity gain, the regression slope of the line and the intercept. The fitted line minimizes the error variance.

Besides, a coefficient of determination denoted r 2 is defined in Equation 3.10 as the ratio between the variance of y ′ and the variance of y. Note that r is referred to as correlation coefficient between x and y in the particular case of linear regression.

r 2 = V ar(y ′ ) V ar(y) (3.10)
Finally, the RMSE, also termed standard error of the regression is calculated by dividing the error variance by n -2 (n is the number of samples, in our case n = 100 number of random draws). In fact, linear regression removes two degrees of freedom from the data (by estimating two parameters a 0 and a 1 ). The RMSE is simplified as can be seen in Equation 3.11:

RM SE = V ar(y)(1 -r 2 )( n -1 n -2 ) (3.11)
The numerical values of the different parameters are summarized in Table 3 The y intercept for LOS scenario is slightly greater than the one in NLOS conditions: almost 2% of capacity gain.

The coefficient of determination r 2 (equivalently the correlation coefficient r between x and y) tends to 1, and the RMSE tends to 0. This confirms the validity of predicting the capacity gain based on the correlation coefficient.

These results will be confirmed next chapter based on measurements. Note also that this linear regression analysis could be performed for other transmitting antennas number.

Correlation coefficient for Rayleigh fading channel

For a MIMO i.i.d. Rayleigh fading channel, each element of the channel matrix follows a zero mean complex Gaussian process (with the same standard deviation σ) and all these elements are independent. Hence, the complex Gaussian coefficients L 1 1 (p) and L 2 1 (q) where 1 ≤ p, q ≤ n T can be written in terms of their amplitudes r 1 p , r 2 q , and phases φ 1 p , φ 2 q as:

L 1 1 (p) = r 1 p e jφ 1 p n T m=1 (r 1 m ) 2 and L 2 1 (q) = r 2 q e jφ 2 q n T m=1 (r 2 m ) 2 (3.12)
where r p 1 , r q 2 follow a Rayleigh law, and φ 1 p , φ 2 q a uniform low in [0, 2π]. Note that r p 1 , r q 2 , φ 1 p , φ 2 q are independent. The denominators stand for for channel normalization. The expression of the correlation coefficient in Equation 3.8 is thus simplified as:

E(ρ) = 1 n R 1 n R 2 n R 1 i=1 n R 2 j=1 E(|| L 1 i L 2 j H || 2 ) (3.13)
The components

E(|| L 1 i L 2 j H || 2 )
do not depend on the indexes i and j since the corresponding laws for the matrix elements L 1 i and L 2 j H do not. The equation in 3.13 is then simplified as:

E(ρ) = E(|| L 1 1 L 2 1 H || 2 ) (3.14) = E | n T p=1 L 1 1 (p)L(p) 2 1 H | 2 = E | n T p=1 r 1 p r 2 p n T m=1 (r 1 m ) 2 n T m=1 (r 2 m ) 2 e j(φ 1 p -φ 2 p ) | 2 (3.15)
Based on the statistical independence r 1 p , r 2 p , φ 1 p , φ 2 p , we can write:

E(ρ) = n T p=1 E (r 1 p r 2 p ) 2 n T m=1 (r 1 m ) 2 n T m=1 (r 2 m ) 2 + 0 (3.16)
Since all terms in the sum are identical, we can write:

E(ρ) = n T E (r 1 1 r 2 1 ) 2 n T m=1 (r 1 m ) 2 n T m=1 (r 2 m ) 2 (3.17)
We can deduce the equation below since r 1 1 and r 2 1 are independent:

E(ρ) = n T E (r 1 1 ) 2 n T m=1 (r 1 m ) 2 (r 2 1 ) 2 n T m=1 (r 2 m ) 2 (3.18) Let us define A as A = E (r 1 1 ) 2 n T m=1 (r 1 m ) 2
. We have:

A = E (r 1 1 ) 2 n T m=1 (r 1 m ) 2 (3.19) = E (r 1 1 ) 2 (r 1 1 ) 2 + (r 1 2 ) 2 + . . . + (r 1 n T ) 2 = E (r 1 1 ) 2 + (r 1 2 ) 2 + . . . + (r 1 n T ) 2 (r 1 1 ) 2 + (r 1 2 ) 2 + . . . + (r 1 n T ) 2 - (r 1 2 ) 2 (r 1 1 ) 2 + (r 1 2 ) 2 + . . . + (r 1 n T ) 2 -. . . - (r 1 n T ) 2 (r 1 1 ) 2 + (r 1 2 ) 2 + . . . + (r 1 n T ) 2 (3.20) = 1 -(n T -1)A (3.21)
The quantity A is consequently equal to A =

1 n T .
Similarly, we have

E (r 2 1 ) 2 n T m=1 (r 2 m ) 2 = 1 n T .
Finally, we prove that the average of the correlation coefficient defined in Equation 3.7 is the inverse of the total number of the transmitting antennas as can be seen in Equation 3.22. This result is validated through simulations (Figure 3.9).

E(ρ) = 1 n T (3.22)
In this section, we have studied the impact of antennas and propagation channel on the BD capacity gain based on a same total transmitted power. Based on simulations, we have given recommendations to optimize MU-MIMO capacity in terms of number of transmitting antennas, their spacing and SNR effect. In the case of the EIRP constraint, it may not be evident that TxBF and MU-MIMO linear precoding still improve the system performance. Therefore, next section evaluates the impact of the EIRP constraint on IEEE 802.11ac MU-MIMO capacity gain using simulations based on the IEEE 802.11ac channel models.

EIRP constraint in a multi-user context

In this section, we still consider the downlink MU-MIMO scenario, in which an IEEE 802.11ac access point with multiple antennas (up to 10) is transmitting to two receivers, each one with two antennas. The BD method is investigated under the EIRP constraint. Two different power allocation schemes of the spatial streams are analyzed to optimize MU-MIMO capacity: equal and unequal power sharing among the spatial streams under the same EIRP constraint. In our formulation, we study K = 2 but the results and algorithms presented in this section can be generalized to any number of users K.

Related work

The MU-MIMO techniques consist of applying a linear precoding to the transmitted spatial streams based on the channel matrices H 1 , H 2 . Consequently, the antenna array pattern and gain are modified as functions of the user location and their corresponding propagation channel properties. This directly impacts the EIRP. The European regulation sets the EIRP limit in the 5 GHz frequency bands to 200 mW or 1 W depending on the frequency channels. This constraint may differ in other countries where it can be rather based on the total transmitted power. The EIRP constraint is rarely evaluated in the literature for MIMO systems. In most of the MIMO performance results, the packet error rate or the capacity value is evaluated based on the same total transmitted power (denoted throughout this report as P T x ) which is related to the SNR. The used SNR is commonly defined as the ratio of P T x to the average noise power as in the previous section with 0 dB average path loss normalized channel. Few recent studies have focused on the capacity optimization problems under total transmitted power constraint [START_REF] Saavedra | Multidiffusion et diffusion dans les systemes OFDM sans fil[END_REF][START_REF] Zhao | Efficient power allocation strategy in multiuser MIMO broadcast channels[END_REF][START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF]. Sometimes, this optimization is performed on each subcarrier of the 802.11 OFDM signal [START_REF] Saavedra | Multidiffusion et diffusion dans les systemes OFDM sans fil[END_REF]. In [START_REF] Kuzminskiy | Downlink beamforming under EIRP constraint in WLAN OFDM systems[END_REF], a new EIRP-based solution for IEEE 802.11 power scaling is proposed. However, this study is dedicated to only one single user system with a single spatial stream and receiving antenna. MU-MIMO linear precoding, like BD [START_REF] Zhao | Efficient power allocation strategy in multiuser MIMO broadcast channels[END_REF][START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF], modifies dynamically the antenna array pattern and gain as function of the current MU-MIMO propagation channel. This may change the EIRP of the transmitting antenna array if transmitted power remains unchanged. MU-MIMO and TxBF are commonly associated with a large number of transmitting antennas used to improve the antenna array gain and performance, as stated for MIMO i.i.d. Rayleigh channel [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF][START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF].

Problem statement

EIRP in linear precoding

For any receiver location, i.e. for any H matrix, the transmitting antenna array pattern is modified by the W precoding matrix. We have used a linear array of n T omnidirectional 0 dBi gain antennas with a regular spacing ∆, typically ∆ = λ/2. The manifold of the transmitter antenna array a(Θ) can be expressed in function of the Θ angle with the antenna array axis, a n T × 1 vector as written in Equation 3.23.

a(Θ)

T = [1, e -2jπ∆ cos(Θ)/λ , ..., e -2jπ(n T -1)∆ cos(Θ)/λ ] (3.23)
In practice with real antennas, a 3D antenna pattern can be included in this processing. Since the used TGac channel model is only a 2D model, the antenna array pattern expression is simplified to a 2D problem. The average radiated power d(Θ) in any direction Θ relative to the antenna array direction is expressed in Equation 3.24 as a function of the antenna input signals defined as X = Ws.

d(Θ) = E(| a(Θ) T X | 2 ) (3.24)
The radiated power contribution of any subcarrier is simply expressed as a function of the transmit diagonal correlation matrix B = E(ss

T ) = E(diag( s1 s T 1 , . . . , sK s T K )) = diag(p ik ), 1 ≤ i ≤ N ss,k , 1 ≤ k ≤ K as: d(Θ) = a(Θ) H WBW T a(Θ) (3.25)
The total radiated power d total (Θ) of all the subcarriers of the IEEE 802.11ac system is expressed as:

d total (Θ) = subcarrier d(Θ) (3.26)
The EIRP is then deduced as follows:

EIRP = max Θ d total (Θ) (3.27)
In the particular case in which the total transmitted power P T x is equally shared among the spatial streams and subcarriers, we have p ik = P T x N Nss , where N ss denotes the total number of spatial streams. Thus, d(Θ) can be simplified as:

d(Θ) = P T x N N ss a(Θ) H WW T a(Θ) (3.28)

Optimization problems

The capacity optimization process consists of finding the optimal value of the transmitted power values p ik compatible with the EIRP constraint. Considering the spatial streams, two power allocation schemes are evaluated: equal power allocation and unequal power allocation.

The European regulation sets the maximum EIRP in an indoor area to 23 dBm for the lower band (5150 -5350 MHz) for the 5 GHz frequency. For practical reasons, such as the computation time, this section focuses on the case where each subcarrier has the same allocated total transmitted power. Furthermore, an unequal subcarrier power allocation may not have a favorable impact on the peak-to-average power ratio of the OFDM signal. The general optimization problem is thus expressed in this case as:

   max subcarrier K k=1 Nss ,k i=1 log 2 (1 + p ik σ 2 n µ 2 ik ) such that EIRP ≤ 23 dBm (3.29)
As we have a sum of logarithmic functions, our problem is a standard convex optimization problem with constraint and is resolved using the Matlab-based library for convex optimization namely CVX [START_REF] Grant | Matlab software for disciplined convex programming[END_REF] to find the optimal solution. A simpler optimization problem consists in solving the case with equally distributed powers, i.e. p ik = P T x N Nss . This optimization problem has only one unknown variable P T x . This problem is simplified by seeking the maximum antenna array gain d total (Θ) P T x and then scaling the power according to the EIRP limit. The case K = 1 uses the same optimization method for computing the SU-MIMO capacity for both equal and unequal power allocation under EIRP constraint.

Evaluated systems and SNR considerations

The 802.11ac MU-MIMO systems based on BD schemes are evaluated. The results are presented in Section 3.4.3 and are compared to SU-MIMO systems relying on the same antennas and total power or EIRP constraint. Three capacity optimization techniques are evaluated and compared.

The first one is the usual MIMO system (denoted basic), with a constant transmitted power P T x equally shared among the spatial streams. BD-basic and SU-basic denote the corresponding studied systems. For this case, the average signal to noise ratio SNR per subcarrier is defined as SN R = P T x N σ 2 n . This is the common SN R definition adopted in most of the published MIMO capacity studies.

The second optimization considers a 23 dBm EIRP constraint and a total transmitted power equally shared among the spatial streams. This optimization is labeled eirpequal. A dynamic power scaling is applied, as detailed above, as a function of each channel matrix snapshot H. SUeirp-equal and BDeirp-equal denote respectively the corresponding SU-MIMO and MU-MIMO systems applying this technique.

The third one (eirp-unequal) considers a 23 dBm EIRP constraint and a total transmitted power unequally and dynamically shared among the spatial streams. SUeirp-unequal and BDeirp-unequal denote the corresponding systems applying this technique.

For eirp-equal and eirp-unequal systems, the common SN R definition is biased since P T x is no more constant, and depends on each channel matrix computation. We subsequently define the average SN R EIRP under EIRP constraint as SN R EIRP = EIRP N σ 2 n for eirp-equal and eirp-unequal systems.

Note that the maximum antenna array gain is n T . Since SN R = P T x N σ 2 n for a basic system, it implies that its corresponding SN R EIRP value is upper bounded by n T P T x N σ 2 n . However, direct comparison between the two systems remains unfair.

Simulation results and analysis

Results for equal power allocation

In this section, we consider a system with an equal power allocation among the spatial streams. 3.13 present the simulation results in terms of the MU-MIMO to SU-MIMO capacity ratio for basic and eirp-equal systems. Average values, 10% and 90% quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. SN R EIRP is set to 20 dB for eirp-equal and SN R for basic is also set to 20 dB. n T varies from 4 to 10 transmitting antennas.

These figures show that the MU-MIMO to SU-MIMO capacity ratio increases with the number of transmitting antennas for TGac-B (Figure 3.12) and Rayleigh (Figure 3.13) channels. The capacity ratio changes from 1.2 to 1.77 for the eirp-equal system in a residential environment, which represents more than 50% of capacity gain. Note that the gain without the EIRP constraint is around 45% [START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. It has been shown in previous studies that increasing the number of transmitting antennas favorably impacts the capacity gain on an i.i.d. Rayleigh channel under SNR constraint [START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF][START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF]. We have been able to prove in this section that this result holds even under the EIRP constraint and with correlated channels as in TGac models. The difference between the 10% and 90% quantiles reduces as the number of antennas increases. This shows that fading has less impact on the capacity values which become less scattered.

The basic and eirp-equal comparisons are biased. In fact, a system relying on a total transmitted power does not satisfy a constant EIRP constraint since it may have an increasing EIRP as n T increases. Intuitively, we could expect that for a basic system, the MU-MIMO to SU-MIMO capacity ratio increases more rapidly in function of n T than for an eirp-equal system, but simulations prove the opposite. In fact, SU-MIMO takes 2 . This means that for a system under the same EIRP constraint, the allocated power tends to be lower for SU-MIMO than for MU-MIMO. For the basic system, the allocated power is the same for SU-MIMO and MU-MIMO. Figure 3.14 shows the average capacity values for MU-MIMO and SU-MIMO. It is well observed that the MU-MIMO capacity increases more rapidly with n T compared to SU-MIMO.

Impact of power allocation strategy

The previous results have been obtained under the EIRP constraint with equally shared power among the spatial streams. In this section, the impact of allocating an unequal and optimized power to each spatial stream according to Equation 3.29 is highlighted. Figure 3.15 compares the average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas for eirp-equal and eirp-unequal systems. As the number of the transmitting antennas increases, the capacity gain of the MU-MIMO to SU-MIMO of the eirp-unequal becomes greater than the one achieved of the eirp-equal system. However, this gain is not significant compared to the computational complexity: it is around 5% for 4 transmitting antennas, and it decreases to 1% for 10 transmitting antennas. To check whether the MU-MIMO capacity always outperforms the SU-MIMO, we have evaluated the probability when the MU-MIMO capacity is lower than the SU-MIMO capacity. This is illustrated in Figure 3.16 for eirp-equal and eirp-unequal schemes versus the number of transmitting antennas.

Figure 3.16 shows that there are some channel snapshots for which the MU-MIMO is less efficient than SU-MIMO, particularly when n T is equal to 4. The probability reaches 22% for equal power sharing between spatial streams. However, the unequal power sharing decreases this probability to 14%. We also observe on Figure 3.15 that the MU-MIMO capacity gain for unequal sharing is slightly greater than the one observed for a fair power distribution. Nevertheless, the gain is not significant: we have around 3% of capacity gain by contrast to high computational complexity. The probability is almost 0 (≤ 1%) for n T = 6 and is equal to 0 for n T ≥ 8.

These results can be explained by examining the overall system: for n T = 4, the MU-MIMO system is composed of 4 antennas in the transmit and the receive sides with 4 spatial streams. This gives no diversity possibilities. The number of spatial streams becomes strictly less than the number of transmitting antennas when n T increases. As a result, the system takes benefit from transmit diversity and shows probabilities which tend to 0.

Antenna spacing effect

For additional results, different transmitting antenna spacings are evaluated for a system in NLOS conditions composed of a Tx with n T = 6 antennas and SN R EIRP = 20 dB. The evaluated values are: 0.25 λ, 0.5 λ, 0.75 λ, 1 λ, 1.25 λ and 1.5 λ. Figure 3.17 shows an isolated and very low gain (37%) at 0.25 λ due, inter alia, to the high correlation between the fading of the channels caused by the small spacing between the transmitting antennas. As the number of the transmitting antenna spacing is increasing, the MU-MIMO to SU-MIMO capacity gain for the residential environment (TGac-B) is around the value of the capacity gain of Rayleigh channel (57%). Antenna spacing has no effect on the Rayleigh channel since its MU-MIMO channel matrix elements are complex Gaussian random variables and independent. Considering a trade-off between the antenna size and the MU-MIMO capacity gain, the recommendation is to have the antenna spacing equal to 0.5 λ.

Conclusion

The purpose of this chapter is to give recommendations for radio engineering and system optimization. In this chapter, we have investigated the impact of antennas and propagation channel on the BD capacity gain for the 802.11ac MU-MIMO in home networks.

Figure 3.17 -Average of MU-MIMO to SU-MIMO capacity ratio versus transmitting antenna spacings.

In particular, we have proved that a small increase of the number of transmitting antennas compared to the total number of the transmitted spatial streams improves significantly the user channel de-correlation and the MU-MIMO capacity gain over SU-MIMO: for example a MU-MIMO gain over SU-MIMO of 45% is achieved for a 20 dB SNR, 4 spatial streams and 6 transmitting antennas. We have also highlighted a relevant channel correlation definition that is useful to decide whether MU-MIMO outperforms SU-MIMO and to select the users into a MU-MIMO user group [START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. Afterwards, we have formulated the optimization problem of the sum capacity for IEEE 802.11ac MU-MIMO systems, considering multiple spatial streams and antennas for each user and taking into account the EIRP constraint [START_REF] Issiali | Impact of EIRP Constraint on MU-MIMO 802.11ac Capacity Gain in Home Networks[END_REF]. Then we have derived numerical results for a typical home network environment based on the spatially correlated IEEE 802.11ac channel models. Two transmit power allocation methods have been evaluated: equal and unequal sharing among the spatial streams under the same EIRP constraint. We have compared these two strategies with the common MU-MIMO BD linear under the total transmitted power constraint. We have shown that MU-MIMO precoding takes more benefit from the EIRP constraint. In particular, we have proved that under EIRP constraint, it is also recommended to have a number of transmitting antennas slightly greater than the total number of spatial streams to guarantee a MU-MIMO capacity gain over SU-MIMO with antenna spacing equal to 0.5 λ. These simulated results need to be confirmed in further work with real MU-MIMO measured propagation channels [START_REF] Issiali | IEEE 802.11ac multiuser MIMO capacity and impact of antenna array geometry based on indoor measurements[END_REF]. Table 3 No deployment conclusion could be drawn based on the relative positions of the AP and users since the TGac channel model is not geometrical. Next chapter will explore deployment recommendations. Moreover, the channel model is restricted to only simulate uniform linear antenna arrays. Other array geometries need to be evaluated based on measurements.

Introduction

This chapter presents the numerical results based on a measurement campaign conducted during the thesis in a residential environment typically encountered in home networks. In order to have a meaningful comparison, we have performed a comparison between the TGac channel models and the measured environment. The first objective of this measurement campaign was to confirm the previous results and validate the radio engineering conclusions based on channel models, then to extend the results to different antenna array geometries and other potential parameters such as the path loss. We have also evaluated the impact of increasing the number of spatial streams and the SNR values. Afterwards, we have evaluated the measured path loss impact on MU-MIMO capacity gain using different channel access methods. Finally, we have drawn a comparative study between the simulated and measured results.

Experiment

In this section, we describe the performed measurements for MU-MIMO channels, based on which we evaluate the MU-MIMO performance. First, we present a state of the art of the previous studies based on measured data. We further describe the measurement environment, as well as the studied scenarios. Finally, we introduce the the measurement setup and the post processing.

Related work

Few articles have studied MU-MIMO capacity based on measured indoor MU-MIMO propagation channels (see Appendix C). In a narrow indoor corridor environment, the authors in [START_REF] Rusek | Indoor multi-user MIMO: Measured user orthogonality and its impact on the choice of coding[END_REF] have analyzed DPC gain over linear processing for two single-antenna receivers and revealed that this gain is almost insignificant for low and high user channel orthogonality. Studies in [START_REF] Fushiki | Experimental evaluations of multiuser MIMO with two-dimensional antenna configuration[END_REF][START_REF] Kakishima | Indoor experiments on 4-by-2 multi-user MIMO employing various transmitter antenna arrangements in LTE-advanced downlink[END_REF][START_REF] Zheng | Performance examinations of multi-user MIMO systems with a compact antenna cube[END_REF] have focused on achieving capacity or throughput improvement through the use of various transmitting antenna arrangements, antenna designs and antenna configurations. It has been shown in [START_REF] Zheng | Performance examinations of multi-user MIMO systems with a compact antenna cube[END_REF] by evaluating channel capacity that a compact tri-polarization antenna cube combined with a simplified pattern circuit are suitable for MU-MIMO systems with antenna selection. It has been shown in [START_REF] Fushiki | Experimental evaluations of multiuser MIMO with two-dimensional antenna configuration[END_REF], using one transmitter with 8 antennas and four single antenna receivers, that constraining the antenna arrangement to 7 λ is beneficial (a gain of 12.8 % of spectral efficiency is achieved) in an indoor environment (room), where λ indicates the wavelength of carrier frequency. However, none of these articles highlights the deployment recommendations or optimal system configuration of the MU-MIMO compared to SU-MIMO, or studies in details MU-MIMO capacity gain over SU-MIMO with multiple antennas receivers based on measurements in home networks.

Measurement scenarios

Channel measurement setup

The MU-MIMO propagation channel is sounded using a Vector Network Analyzer (VNA) based on a frequency domain technique. We collect the S 21 parameter since the propagation channel is the device under test. The VNA, depicted in Figure 4.4(a), is connected by cables of 10 m to the Tx, 20 m and 5 m to the first and second user respectively. Hence, the maximum distance between the Tx and Rx is 30 m. The VNA has probed M = 2048 frequency tones between 5.15 GHz and 5.40 GHz stepped by ∆f = 122.07 kHz. This configuration permits a maximum propagation delay of 1 ∆f = 8192 ns. This value is well above the propagation delay that can be usually observed in such environment but it is useful to estimate the noise level of the measurement data for post-processing. The high number of frequency bins allows also to improve the dynamic (the difference between the peak power and noise level) of the CIR, which is between from the non physical delay area of the average PDP. We force to 0 the corresponding CIR complex samples with an average power below this noise level and also the sample corresponding to a dynamic greater than 30 dB in order to process measurements with PDP over a 100 MHz bandwidth as in Equation 4.2. To get 100 MHz, the applied Hanning window was reduced accordingly. This choice is justified by the fact that the average PDP of TGac channels are by default computed over 100 MHz. The average PDP is computed over the 480 virtual antennas. In the LOS case, the computation gives a mean value of τ RM S = 10.87 ns, and τ RM S = 17.93 ns for the NLOS case. This result is in accordance with the numerical values in [START_REF] Mcdonnell | RMS delay spread in indoor LOS environments at 5.2 GHz[END_REF]. The LOS-NLOS refers to the case where one of the users is on LOS with the Tx and the other one on NLOS with Tx. It shows a middle case between the LOS and NLOS scenarios with τ RM S = 13.5 ns.

τ RM S (k) =
Figure 4.6 shows the delay spread of all measured channels compared to TGac channels. Despite of the dispersion of measured τ RM S , the average value is equal to 15.433 ns. The TGac-B channel model is defined to have 15 ns. Thus, both environments show an average delay spread around 15 ns and can be considered to have a similar frequency selectivity .

Superdiagonal of transmit and receive covariances

We study the coefficients r T x and r Rx of the receive and transmit correlation matrices R T x and R Rx as defined in [START_REF] Ozcelik | What makes a good mimo channel model[END_REF][18] (Equation 4.3), of dimension n T × n T and n R × n R respectively, for NLOS scenario with n T = 8 transmitting antennas arranged in linear array and two receivers with two antennas each.

R T x = 1 n R E | H T H | R Rx = 1 n T E | HH H | (4.
3)

The transmit and receive correlation matrices give insight to what extent the signals leaving the different transmitting antennas and the signals arriving at the different receiving antennas respectively are correlated. According to [START_REF] Kaltenberger | Correlation and capacity of measured multi-user MIMO channels[END_REF], it has been shown that in indoor and outdoor scenarios, almost no correlation can be measured at the receive side between the different users. However, at the transmit side, there is slightly, but not significantly more correlation. This has been shown to badly impact the MU-MIMO performance.

The correlation coefficient is featured to have superdiagonal form. The superdiagonal entry of a matrix is one that is directly above and to the right of the main diagonal. Each component shows the correlation between the antenna corresponding to its row and its adjacent antenna. The antennas are spaced by λ 2 . We have based our computation on the second user channel matrix H 2 because it includes all the various positions shown in Figure 4.2. The n T × n T transmit correlation matrix has the superdiagonal coefficients

as r T x = 1 n R E( n R 2 i=1 H ij Hi(j+1) ) for j = 1, 2, . . . , 7. The superdiagonal of the n R 2 × n R 2 receive correlation is r Rx = 1 n T E( n T j=1 H 1j H2j ),
where the H ij are the elements of the channel matrix H.

Fading

In this subsection, we evaluate the narrowband fast fading distribution. We characterize the CDF of narrowband fading of the measured channel and we compare it to the Rayleigh distribution as illustrated in Figure 4.8. The figure shows that the fading in the measured environment fits exactly the Rayleigh distribution considered for the TGac-B channel model.

Analyzing the previous parameters, we have shown that the two environments meet the same trend. Hence, they are expected to give similar results.

Statistical results

The measured data allows the study of various types of transmitting antenna array geometries using the 480 virtual antennas and an increased number of receiving antennas. This section first presents MU-MIMO results based on normalized 0 dB average path loss for each user. Afterwards, we consider measured path loss (non-normalized propagation channels). In this section, all numerical results are computed under the EIRP constraint (EIRP = 23 dBm). Note also that the power is equally shared among the spatial streams as well as subcarriers.

Impact of transmitting antennas considering a normalized channel

In this section, the effect of transmitting antennas (number and geometry of antennas) on MU-MIMO system with two receivers with two antennas each is analyzed based on a normalized channel. The reason of using the normalization is to keep only fast-fading effects so that the average SNR at the receiving antennas is set to a fixed value and can be easily adjusted as a parameter. The applied channel normalization in this article implies that the average propagation loss is set to 0 dB for both users [START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. The average SNR is defined as SN R = EIRP/σ 2 n where EIRP = 23 dBm for this study, and is set to 20 dB. The aim is to assess the impact of transmitting antenna configuration on the BD capacity gain over SU-MIMO and to give recommendations to optimize MU-MIMO performance. To highlight the MU-MIMO capacity gain over SU-MIMO, most graphs below show the average of MU-MIMO to SU-MIMO capacity ratio.

Number of transmitting antennas

Figure 4.9 gives the average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas arranged in an ULA. It also includes 10% (q 10 ) and 90% (q 90 ) quantiles as a reference. The first observation drawn from Figure 4.9 is that the MU-MIMO capacity gain over SU-MIMO grows logarithmically with the number of transmitting antennas. It changes from 1.27 to 1.7 for the residential environment, i.e. around 43% of capacity gain. For 4 transmitting antennas, the quantile q 10 of capacity gain is less than 1. This can be explained by the fact that we cannot benefit from transmit beamforming gain since the number of transmit antennas is the same as the total number of spatial streams.

Figure 4.10 shows the average capacity values for MU-MIMO and SU-MIMO systems. The capacity value for MU-MIMO increases more rapidly than SU-MIMO. It achieves 24 bits/s/Hz versus 14 bits/s/Hz for SU-MIMO with 8 transmitting antennas. In order to optimize the MU-MIMO capacity gain and have a less congested system, we recommend using 6 transmitting antennas in a system with two receivers and two antennas for each receiver. If we aim at reaching higher capacities, using 8 transmitting antennas allows 2 bits/s/Hz of capacity increase.

Impact of receiving antennas considering a normalized channel

In this section, the effect of the number of receiving antennas and the impact of the selected receiving antennas on MU-MIMO system with two receivers with two antennas each are analyzed based on a normalized channel.

Impact of selecting different receiving antennas

As can be seen in Figure 4.16, each user is equipped with four antennas. However, in the previous simulations we have evaluated a Rx with two antennas. In this subsection, we study the effect of selecting different receiving antennas on the MU-MIMO to SU-MIMO capacity gain and correlation.

Figure 4.17 gives the average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient considering different receiving antenna arrangements. The In terms of channel capacity values, Figure 4.20 shows an increase of about 3 bits/s/Hz for Tx with 4 or 6 antennas. Using a third antenna in a receiver will enlarge its design. Regarding the obtained performance, we do not recommend to increase the number of receiving antennas excepted if the number of spatial streams per user is 3. In the case of 2 spatial streams per user, it is preferable to increase the number of transmitting antennas and keeping 2 antennas at the receiver.

Impact of increasing the number of spatial streams

In this subsection, we highlight the effect of the number of spatial streams in optimizing the MU-MIMO techniques. Before giving the statistical results, we present the simulated system depicted in

Impact of the path loss difference on MU-MIMO gain

In this section, we consider the propagation channel including its measured path loss on a 20 MHz bandwidth. The EIRP is equal to 23 dBm. The noise power is set to -93 dBm which corresponds to the noise temperature at 300 • K for a 20 MHz band with a noise factor of 8 dB. The number of transmitting antennas is set to 6 arranged in ULA geometry. The SU-MIMO capacity is expressed considering TDMA and CSMA/CA channel access methods. The transmitter communicates with two receivers. Each receiver is equipped with two antennas spaced by λ 2 . We consider the average received power at each user in dBm.

Figure 4.23 shows the average of MU-MIMO to SU-MIMO capacity ratio versus the absolute difference of the average received powers (in dB) denoted as ∆P by the two receivers Rx 1 and Rx 2 . We observe that when ∆P is below 15 dB, both channel access methods give almost the same results. Nevertheless, compared to SU-MIMO CSMA/CA method, it is advantageous to group users with larger ∆P and use MU-MIMO: the capacity gain can be greater than 2. Actually, if C 1 is very small compared to C 2 , then C SU,CSM A is penalized by C 1 [START_REF] Heusse | Performance anomaly of 802.11b[END_REF] which is not the case of C M U -M IM O . We also notice that the capacity gain in all cases is higher than 60% in a 12 m ×7 m apartment with a 23 dBm EIRP. This proves the benefit of using the MU-MIMO method rather than SU-MIMO. that high difference in the received powers means low level of similarity between the users' channels. Hence, the correlation coefficient is low.

Measured versus simulated environments: numerical results

This section gives a thorough comparison between the numerical results based on simulated and measured channels. Since the TGac channel models simulate antennas arranged in ULA, we will confine the comparisons on evaluating a Tx with antennas arranged in ULA. We have presented in previous sections global statistical results including both LOS and NLOS cases. In this section, we will distinguish the two cases in order to have meaningful comparison with TGac simulation results.

The used notation for measurements comprises three cases. First, we denote the case where two receivers are in LOS with the transmitter as "Meas. LOS". Second, the case where the two receivers are in NLOS with the transmitter is refereed to as "Meas. NLOS". Finally, the case where one of the two users is in LOS and the other one is NLOS conditions with the Tx is denoted as "Meas. LOS/NLOS".

Similarly, we denote the TGac-B channel model with users in LOS and NLOS with the Tx as "TGac-B LOS" and "TGac-B NLOS" respectively. The SNR is set to 20 dB and is based on the EIRP constraint. The expression of MU-MIMO to SU-MIMO capacity gain as a function of correlation coefficient motivates us to look for a simplified expression of the correlation coefficient as a function of the physical parameters such as users' relative position.

Figure 4.29 plots the correlation coefficient as a function of angles and distances for a Tx with n T = 4 and two receivers in NLOS with n R 1 = n R 2 = 2 spaced by λ 2 as defined in TGac-B channel models. Indeed, for each drawing, four angles are drawn randomly by the TGac channel models. However, the points are dispersed and no conclusion could be drawn except that the points are randomly distributed and with a nearly periodicity.

Similarly, Figure 4.30 plots the correlation coefficient versus the users' position for a transmitter with n T = 8 arranged in ULA based on measurements. We denote

Conclusion

A MU-MIMO measurement campaign has been conducted in a residential environment for two users with two antennas each. The obtained sum rate capacity under the EIRP constraint results confirm that 6 is the recommended number of transmitting antennas to keep a compact size of the transmitting antenna arranged in

Introduction

The ability to accurately predict radio-propagation for wireless communication systems, is becoming crucial for system design and radio planning. In this context, the Wireless local Area Network Design and Analysis (WANDA) software has been developed by Orange Labs Belfort to design indoor WLAN.

Transmit beamforming analysis and modeling

Related work

Transmit beamforming is a feature of 802.11ac standard. It shapes the effective radiation pattern of the antennas pointing towards the direction of the dominant multi-paths by controlling the power and the phase of the transmitted signals. Various techniques may be used to implement the TxBF. There are phased antenna arrays (direction based beamformers), codes book (for Long Term Evolution (LTE)), dominant eigen mode selection (single beamforming technique with one antenna weighting vector and that carries only one spatial stream [START_REF] Nguyen | On the performance of link adaptation techniques in MIMO systems[END_REF]), and the more general SVD-MIMO with one precoding matrix carrying all the spatial streams. This latter is selected in the 802.11ac standard and is studied in this chapter.

In the literature, TxBF has been studied in order to highlight its benefits compared to the use of a simple MIMO system using spatial multiplexing only. Most of the publications studied outdoor [START_REF] Desai | Field data showing the downlink adaptive beamforming gains in an experimental IEEE 802.16e-2005 OFDMA system[END_REF] or outdoor-indoor [START_REF] Hermosilla | Empirical comparison of MIMO and beamforming schemes for outdoor-indoor scenarios[END_REF] environments based either on measurements made with equipment that implements TxBF or on a purely theoretical approach. Most of the results concern the dominant eigen mode TxBF. There are few results concerning the more general SVD-MIMO systems with multiple spatial streams [START_REF] Tran | Performance analysis of MIMO-OFDM systems using indoor wideband mimo channel measurement data[END_REF][75], but they rarely concern typical European residential environments and European building materials.

In fact, studies have analyzed several MIMO configurations in order to conclude on the relevance of a possible use of beamforming. The TxBF requires the knowledge of the MIMO propagation channel between the transmitter and the receiver. This implies frame exchanges between transmitter and receiver and an overhead in the frame control. For instance, in [START_REF] Hermosilla | Empirical comparison of MIMO and beamforming schemes[END_REF], it was shown that for small values of SNR (or high interference level) the dominant eigen mode Beamforming (BF) schemes can perform close to the MIMO system with singular value decomposition in terms of spectral efficiency, and can even outperform a MIMO system with CSI only at the receiver.

Other studies have focused on the comparison between introducing the dominant eigen mode and transmit diversity [START_REF] Friedlander | Beamforming versus transmit diversity in the downlink of a cellular communications system[END_REF]. It was shown that, under ideal conditions (uncorrelated fading), transmit diversity has an advantage over a beamforming system. However, under handoff conditions, the beamforming system has both array gain and diversity, which improves its performance relative to transmit diversity.

The SVD, which is seen as one of the most relevant MIMO precoding techniques, is present in publications that address the topic of eigen beamforming with multiple streams. Indeed, the analysis in [START_REF] Busche | SVD-based MIMO precoding and equalization schemes for realistic channel knowledge: Design criteria and performance evaluation[END_REF][START_REF] Lebrun | MIMO transmission over a time-varying channel using svd[END_REF][START_REF] Morozov | A novel combined CSI feedback mechanism to support multi-user MIMO beamforming schemes in TDD-OFDMA systems[END_REF][START_REF] Sun | ICI/ISI-aware beamforming for MIMO-OFDM wireless systems[END_REF] have focused on evaluating the impact of channel estimation error on the performance of MIMO system that implements SVD-MIMO technique. It was proved that in case of non-ideal channel knowledge and a limited accuracy in the channel matrix estimation, a reduced number of eigenmodes in the precoding process becomes an optimum and leads to an improved BER. A typical system employing SVD shows capacity degradation when incorrect CSI is used to transmit data. Therefore, some papers propose a new linear processing architecture which reduces the effect of incorrect CSI at the transmitter.

Other approaches such as prototyping [START_REF] Quadri | Industrial prototyping: A common architecture and case study of IEEE 802.11n transmit beamforming[END_REF] have been also explored in order to assess the feasibility of an IEEE 802.11n transmit beamforming architecture before starting its mass production. However, it does not exploit the effect of the feedback delay to evaluate the added value of beamforming.

In the context of earlier studies, it was shown in [START_REF] Andersen | Array gain and capacity for known random channels with multiple element arrays at both ends. Selected Areas in Communications[END_REF] for example, that in the case of a single stream transported by a MIMO system with n T transmitting antennas and n R receiving antennas, a significant antenna array gain n T × n R can be achieved in LOS conditions i.e. in fully correlated received signals. For multiple streams and using the SVD-MIMO technique, a theoretical bound of the gain was defined in the case of a rich scattering environment (Rayleigh fading). This bound is (

√ n R + √ n T ) 2 .
Other works such as [START_REF] Sengul | Diversity analysis of single and multiple beamforming[END_REF] have also analyzed the diversity gain for such channels and SVD MIMO technique, and calculated that the achievable diversity order is (n T -N ss +1)(n R -N ss +1) for N ss spatial streams.

In [START_REF] Bouhlel | Transmit beamforming analysis for mimo systems in indoor residential environment based on 3d ray tracing[END_REF], the transmit beamforming is analyzed based on simulations performed in a typical residential environment. The obtained results show that enabling TxBF increases the received power when n T is important compared to N ss (if N ss = n T , no power gain is achieved). Moreover, it was shown, for a given N ss , that with TxBF enabled, it is more advantageous to increase the number of receiving antennas than the transmitting antennas number. Two optimal values of transmitting antenna spacing were proved to optimize the TxBF gain: around 0.5 λ and 1.25 λ. Accordingly, we have set the antenna spacing to 0.5 λ. Finally, it is proved that the TxBF can improve the transmission range especially when N ss is sufficiently small compared to n T . However, the published results are based on ray tracing tools, and no conclusion was carried out about the impact of the EIRP constraint on the numerical results.

In this chapter, we evaluate the TxBF performance under the EIRP constraint based on measured data in home environment. We reveal in particular the benefits of the introduction of this option compared to the use of a simple MIMO system with spatial multiplexing and spatial expansion. The comparisons have been performed using different metrics evaluated With Beamforming (W-BF) and Without Beamforming (Wo-BF) based on the SVD MIMO technique. This study addresses the impact of the overall MIMO configuration by varying the number of transmitting antennas, the number of receiving antennas and the number of spatial streams. Several transmitting antenna array geometries are compared to identify which one has the most important BF gain. The interference level reduction is then evaluated in order to investigate whether the introduction of TxBF can increase or reduce the interference generated by an AP with TxBF activated. Note that the TxBF as defined in 802.11ac standard is designed to optimize capacity but not to reduce or cancel interference to the undesired receivers. Moreover, the figure shows that even under the EIRP constraint, the transmit beamforming still achieves gain. It changes from 2 dB to 4 dB. The highest SNR gain 6 dB is achieved with a transmitter equipped with n T = 8 antennas and a receiver with n R = 1 antenna. Hence, we can fully benefit from the TxBF when the number of transmitting antennas is largely greater than the number of receiving antennas and spatial streams, whereas the SNR gain is null when n T = N ss .

Secondly, we evaluate two spatial streams N ss = 2 to assess its impact on TxBF. The number of receiving antennas varies in this case in n R = 2, n R = 3. Figure 5.7 gives the SNR gain as a function of the number of transmitting antennas for N ss = 2. The same conclusion about the number of the transmitting and receiving antennas is drawn from Figure 5.7. Moreover, Figure 5.7 shows that increasing the number of spatial streams decreases the SNR gain. Indeed, the highest value of SNR gain is achieved by

n T = 8, n R = 2.
Therefore, with TxBF, increasing the SNR gain is possible when we increase the number of transmitting antennas, decrease the number of receiving antennas, or decrease the number of spatial streams compared to the number of transmitting antennas, even under the EIRP constraint. The beamforming gains are in the range of 1 to 6 dB which is very interesting to extend the WLAN coverage.

Impact of the geometry of transmitting antenna arrays

In this subsection, we assess the impact of antenna array geometries on the SNR gain. We simulate different antenna array geometries as in previous chapter (Chapter 4,

Statistical results of interference level reduction

In this section, we give numerical results of interference reduction to see whether the introduction of TxBF increases or reduces the interference difference generated by the AP. The interference reduction is evaluated as a function of the number and the geometry of antennas as well as the number of spatial streams.

Interference level

In this section, we study the average value of the interference reduction for the various 2-user measurement configurations. The transmitter has the antennas arranged in ULA.

At first, we study the case with one spatial stream. Figure 5.9 gives the average of the interference reduction as a function of the average of SNR gain for one spatial stream. Increasing the number of transmitting antennas decreases the interference level. The interference level is reduced from 1.5 dB to almost 3 dB when the number of transmitting antennas increases from n T = 2 to n T = 8. However, increasing the number of receiving antennas does almost not impact the interference level. The interference reduction values slightly increase (less than 0.5 dB) passing from n R = 1 to n R = 3. The highest value of interference reduction 3.5 dB in this case (N ss = 1) is achieved by a system with n R = 3, n T = 8.

Secondly, we set the number of spatial streams to N ss = 2.

Conclusion

In the recent years, MIMO has attracted considerable attention for its potential to increase channel capacity. The MU-MIMO techniques have been proposed to further increase the capacity by spatial reuse. However, no study gives explicit answers to whether using MU-MIMO or SU-MIMO is the best choice in a residential environment, or gives optimal configurations to benefit fully from these techniques. In addition, current network planning tools need to be reviewed taking into account the beamforming gains. In this thesis, we have addressed these issues related to MU-MIMO and smart antennas.

Firstly, the physical features of the new standard 802.11ac as well as the basic propagation mechanisms have been summarized in Chapter 2. The MIMO channel models have also been presented. We have highlighted in particular the TGac channel models. The MU-MIMO system model, the precoding techniques and their related capacities have been next recalled. Indeed, we have given in this chapter a short state of the art of the MU-MIMO techniques. For our studies, we have selected BD technique for the MU-MIMO processing.

In Chapter 3, we have proposed recommendations to optimize MU-MIMO performance based on TGac simulations in terms of channel propagation parameters and users' configuration: LOS, NLOS, distances... A particular interest has been paid to study the power control required to meet the European regulation for WLAN in the 5 GHz frequency band in terms of EIRP. First, we have given a detailed description of the simulation process, the evaluated channel model and its parameters. Next, we have provided simulation results in function of antennas and propagation channels parameters on the BD capacity gain over SU-MIMO. In particular, we have proved that a small increase in the number of transmitting antennas compared to the total number of transmitted spatial streams improves significantly the user channel de-correlation and the MU-MIMO capacity gain over SU-MIMO. We have also highlighted a relevant channel correlation definition that is useful to decide whether MU-MIMO outperforms SU-MIMO and to select the users into a MU-MIMO user group [START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. Afterwards, we have formulated the optimization problem of maximizing the sum capacity under the EIRP constraint for the IEEE 802.11ac MU-MIMO. In our formulation, we have looked for the power value which maximizes the sum capacity for each spatial stream of each user [START_REF] Issiali | Impact of EIRP Constraint on MU-MIMO 802.11ac Capacity Gain in Home Networks[END_REF]. We have compared these two strategies with the common linear MU-MIMO BD under the total transmitted power constraint. We have shown that MU-MIMO precoding takes benefit from the EIRP constraint. The MU-MIMO to SU-MIMO capacity gain increases under the EIRP constraint.

In Chapter 4, we have presented the results of a measurement campaign using the VNA.

This campaign has been conducted during the thesis in a residential environment typically encountered in home networks at Orange Labs Belfort. The first objective of this measurement campaign was to confirm the previous results and validate the radio engineering conclusions based on channel models, then to extend the results to different antenna array geometries and other potential parameters such as the propagation path loss. Obtained results confirm the recommendations based on TGac simulations. Furthermore, we have extended the conclusions to study the impact of receiving antennas and the number of spatial streams and the SNR values. In particular, increasing the number of receiving antennas increases the MU-MIMO to SU-MIMO capacity gain by only 5%. Moreover, selecting receiving antennas spaced by 0.5λ or 0.5 √ 2λ has no significant impact on MU-MIMO to SU-MIMO capacity gain. Afterwards, we have compared the measured propagation path loss impact on MU-MIMO capacity gain using TDMA and CSMA/CA channel access methods.

Finally, in Chapter 5, we have evaluated the transmit beamforming gain based on measured data in a SU-MIMO context. The performance evaluation has been made based on the SNR gain and interference reduction. Several MIMO configurations (number of transmitting and receiving antennas, number of spatial streams, antenna array geometries) have been compared to identify which one has the most important impact on the TxBF quantified in terms of received power. In particular, we have proved that increasing the SNR gain is possible when we increase the number of transmitting antennas, decrease the number of receiving antennas, or decrease the number of spatial streams. These conclusions have been drawn to correct and update the beamforming gains of the coverage maps in the current planning tools. Moreover, we have investigated whether the introduction of TxBF can increase or reduce the effect of interference generated by an interfering link. We have been able to show that TxBF reduces the interference in the environment but not for the 100% of locations. The values of the cyclic shifts to be used are specified in Table E

Abstract

The deployment facility and flexibility of wireless technologies have allowed considerable evolution of IEEE 802.11 standards. The arrival of the optical fiber to the home has further motivated the research studies to increase data rates and capacity. In this context, the latest IEEE 802.11ac standard ratified in 2014 proposes new technologies to achieve maximum data rates above 1 Gbit/s. In particular, 802.11ac includes the Multi-User Multiple Input Multiple Output (MU-MIMO) and Transmit Beamforming (TxBF). These smart antenna technologies assume a variable transmit antenna pattern in function of the users' position and the corresponding multipath propagation channel characteristics. However, the actual radio planning tools give a fixed coverage map independent of any adaptive or smart antenna processing.

This thesis studies the impact of the MU-MIMO and TxBF techniques on radio engineering, system optimization as well as radio planning tools. Indeed, the first part of this thesis deals with refining the sum rate capacity gains by identifying interesting system configurations and optimal system parameters. The second part addresses the beamforming gains in terms of the Signal-to-Noise Ratio (SNR) gain and interference level to have realistic radio coverage modeling.

Hence, we have firstly analyzed the MU-MIMO to Single User MIMO (SU-MIMO) sum rate capacity gain based on the 802.11ac correlated MU-MIMO channel models. By analyzing the results, we have proposed system recommendations to optimize the MU-MIMO in a residential environment, mainly related to system configuration such as the number of transmitting antennas (antenna spacing, tradeoff between antenna number and number of spatial streams, antenna arrangement...). A correlation coefficient between channels has been highlighted to explain the obtained gains. Moreover, we have formulated the convex optimization problem of maximizing the capacity in a MU-MIMO context with multiple receiving antennas taking into account the Equivalent Isotropically Radiated Power (EIRP) constraint for Wireless Local Area Network (WLAN) in Europe.

Secondly, we have conducted a 2-user Multiple-Input Multiple-Output (MIMO) propagation channel measurement campaign in a residential environment to confirm the obtained radio engineering and system optimization recommendations. We have been able to extend this study to other potential parameters to have influence on MU-MIMO to SU-MIMO capacity gain such as path loss and antenna array geometries.

Finally, we have evaluated the performance of TxBF by analyzing the Signal-to-Noise Ratio (SNR) gain as well as the interference reduction. This part identifies the system parameters which have the biggest influence on the realistic radio coverage simulation. It proposes a statistical modeling to predict the beamforming gains achieved by the antenna processing.

C

  To the memory of my mother Rachida DARIF... A great women who has sacrificed so much for us, and whom I still miss every day... To my father Ramdane ISSIALI... To my brother Walid ISSIALI... This humble work is a sign of my love for you... Thank you... V 6.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A All detailed parameters of TGac-B channel XII B Comparison of TDMA SU-MIMO, DPC, and BD sum rate capacity using MU-MIMO and spatially correlated channels

Figure 2 -Figure 3 -

 23 Figure 2 -Moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction du nombre d'antennes d'émission.

Figure 4 -

 4 Figure 4 -La moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU pour IEEE TGac-B channel (residential).

  Nous présentons le gain sur le SNR comme métrique d'évaluation du gain de "beamforming". Nous évaluons la capacité MIMO en respectant la contrainte de la limitation de PIRE: P IRE = 23 dBm et en variant les valeurs d'atténuation de canal pour prendre en compte différentes valeurs de SNR. En effet, en utilisant les 67 configurations différentes de 2 utilisateurs, nous avons normalisé le canal pour avoir des atténuations comprises dans l'intervalle: 45 dB, 110 dB . Pour chaque valeur d'atténuation, on simule la valeur médiane de la capacité MIMO. En tenant compte de la tendance linéaire décroissante des valeurs médianes de capacités en fonction du path loss, nous avons appliqué une régression linéaire comme le montre l'exemple Figure 8. La figure montre les valeurs médianes de capacité MIMO avec et sans "transmit beamforming" pour un système avec deux antennes à l'émission, une antenne en réception et un flux spatial en fonction de l'atténuation moyenne de propagation. Le gain sur le SNR est défini comme la différence entre les deux droites de régression en dB comme montre la figure. Le gain sur le SNR a été évalué pour différentes configurations en fonction du nombre d'antennes d'émission et de réception. La Figure 9 donne le gain sur le SNR en fonction du nombre d'antennes d'émission pour un seul flux spatial. Le gain sur le SNR augmente avec l'augmentation du nombre d'antennes d'émission, alors qu'il diminue lorsque le nombre d'antennes de réception augmente. Outre, la figure 9 montre qu'il y a toujours XXV Dans cette thèse, nous avons abordé les problématiques ci-dessus liées aux techniques MIMO-MU et aux antennes intelligentes.

Figure 2 . 3 -

 23 Figure 2.3 -Beamforming basics.

Figure 2 .

 2 Figure 2.10 -MIMO channel models.

  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.2 TGac-B channel model . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Simulation setup . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.2 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.3 TGac and Rayleigh channel . . . . . . . . . . . . . . . . . . . . 46 3.3 Impact of antennas and spatially correlated propagation channel on BD capacity gain . . . . . . . . . . . . . . . . . . . . 47 3.3.1 Transmit antenna spacing effect . . . . . . . . . . . . . . . . . . 47 3.3.2 SNR effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.3.3 Number of transmitting antennas . . . . . . . . . . . . . . . . . 48 3.3.4 Correlation coefficient . . . . . . . . . . . . . . . . . . . . . . . 52 3.4 EIRP constraint in a multi-user context . . . . . . . . . . . . . 59 3.4.1 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.4.3 Simulation results and analysis . . . . . . . . . . . . . . . . . . 62 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Figure 3 . 1 - 1 -

 311 Figure 3.1 -MU-MIMO to SU-MIMO capacity gain versus the simulation length values.

  By setting the distance between the Tx and the Rx, the TGac-B channel model simulates either the LOS or NLOS. Indeed, LOS conditions are assumed only up to a breakpoint distance d BP = 5 m (d BP = 5 m for TGac-B channel). Since the Rician factor for LOS is set to 0 dB which is very close to a Rayleigh channel, almost similar results are obtained with the TGac-B LOS and NLOS scenarios. This 0 dB weak Rice factor of the model B clearly appears in the Cumulative Distribution Function (CDF) of the first tap of Figure 3.2. It almost merges with the CDF of the Rayleigh distributed amplitude of the other taps. In Figure 3.2, the curve in green corresponds to the reference values, whereas the blue curves are the outcomes of the simulation for the 9 taps. This chapter is focusing on the TGac-B NLOS channel model after giving an example of LOS numerical results.

Figure 3 . 2 -

 32 Figure 3.2 -CDF of the taps power of TGn Channel Model Case B (9 taps)

Figure 3 . 3 -

 33 Figure 3.3 -Average of MU-MIMO to SU-MIMO capacity ratio versus transmit antenna spacing.

Figure 3 .

 3 Figure 3.4 shows the MU-MIMO capacity gain over SU-MIMO versus SNR with 6 transmitting antennas spaced by 0.5 λ in NLOS conditions. For high SNR, MU-MIMO outperforms SU-MIMO in terms of capacity with gain changing from 10% till 70%.Nevertheless, for low SNR, SU-MIMO performs better than MU-MIMO in terms of capacity which has been described already in the limitations of BD scheme. Note that it is not practical to have too low SNR for MU-MIMO as it would not be possible to have CSI with no errors to apply the precoding.The MU-MIMO to SU-MIMO capacity gain is 30% when the SNR increases from 0 to 10 dB, from 10 to 20 dB, or from 20 to 30 dB, and around 10% when it changes from 30 to 40 dB. To conclude about the SNR, the desired range and the desired bit rate are put forward. For the following results, a middle case is evaluated with SN R = 20 dB.

Figure 3 . 4 -

 34 Figure 3.4 -Average of MU-MIMO to SU-MIMO capacity ratio versus the SNR with 6 transmitting antennas spaced by 0.5 λ in NLOS conditions.

Figure 3 .

 3 6 shows the average capacity value for MU-MIMO and SU-MIMO. The capacity value for MU-MIMO increases more rapidly than SU-MIMO. It achieves 27.5 bits/s/Hz versus 16.5 bits/s/Hz for SU-MIMO for n T = 10.

Figure 3 . 5 -

 35 Figure 3.5 -Average of MU-MIMO to SU-MIMO capacity ratio versus the versus the number of transmitting antennas.

Figure 3 . 6 -

 36 Figure 3.6 -Channel capacity values versus the number of transmitting antennas.

Figure 3 . 8 -

 38 Figure 3.8 -Average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas.

Figure 3 . 11 -

 311 Figure 3.11 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n T = 6).

Figure 3 .

 3 Figure 3.12 -MU-MIMO to SU-MIMO capacity ratio for an IEEE TGac-B channel (residential).

Figure 3 .

 3 Figure 3.13 -MU-MIMO to SU-MIMO capacity ratio for an i.i.d Rayleigh channel.

Figures 3. 12

 12 Figures 3.12 and 3.13 present the simulation results in terms of the MU-MIMO to SU-MIMO capacity ratio for basic and eirp-equal systems. Average values, 10% and 90% quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. SN R EIRP is set to 20 dB for eirp-equal and SN R for basic is also set to 20 dB. n T varies from 4 to 10 transmitting antennas.These figures show that the MU-MIMO to SU-MIMO capacity ratio increases with the number of transmitting antennas for TGac-B (Figure3.12) and Rayleigh (Figure3.13) channels. The capacity ratio changes from 1.2 to 1.77 for the eirp-equal system in a residential environment, which represents more than 50% of capacity gain. Note that the gain without the EIRP constraint is around 45%[START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. It has been shown in previous studies that increasing the number of transmitting antennas favorably impacts the capacity gain on an i.i.d. Rayleigh channel under SNR constraint[START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF][START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF]. We have been able to prove in this section that this result holds even under the EIRP constraint and with correlated channels as in TGac models. The difference between the 10% and 90% quantiles reduces as the number of antennas increases. This shows that fading has less impact on the capacity values which become less scattered.The basic and eirp-equal comparisons are biased. In fact, a system relying on a total transmitted power does not satisfy a constant EIRP constraint since it may have an increasing EIRP as n T increases. Intuitively, we could expect that for a basic system, the MU-MIMO to SU-MIMO capacity ratio increases more rapidly in function of n T than for an eirp-equal system, but simulations prove the opposite. In fact, SU-MIMO takes

  Figures 3.12 and 3.13 present the simulation results in terms of the MU-MIMO to SU-MIMO capacity ratio for basic and eirp-equal systems. Average values, 10% and 90% quantiles (q 10 and q 90 ) are represented to estimate the confidence intervals. SN R EIRP is set to 20 dB for eirp-equal and SN R for basic is also set to 20 dB. n T varies from 4 to 10 transmitting antennas.These figures show that the MU-MIMO to SU-MIMO capacity ratio increases with the number of transmitting antennas for TGac-B (Figure3.12) and Rayleigh (Figure3.13) channels. The capacity ratio changes from 1.2 to 1.77 for the eirp-equal system in a residential environment, which represents more than 50% of capacity gain. Note that the gain without the EIRP constraint is around 45%[START_REF] Issiali | Impact of antennas and correlated propagation channel on BD capacity gain for 802.11ac multi-user MIMO in home networks[END_REF]. It has been shown in previous studies that increasing the number of transmitting antennas favorably impacts the capacity gain on an i.i.d. Rayleigh channel under SNR constraint[START_REF] Spencer | Zero-forcing methods for downlink spatial multiplexing in multiuser MIMO channels[END_REF][START_REF] Choi | A transmit preprocessing technique for multiuser MIMO systems using a decomposition approach[END_REF]. We have been able to prove in this section that this result holds even under the EIRP constraint and with correlated channels as in TGac models. The difference between the 10% and 90% quantiles reduces as the number of antennas increases. This shows that fading has less impact on the capacity values which become less scattered.The basic and eirp-equal comparisons are biased. In fact, a system relying on a total transmitted power does not satisfy a constant EIRP constraint since it may have an increasing EIRP as n T increases. Intuitively, we could expect that for a basic system, the MU-MIMO to SU-MIMO capacity ratio increases more rapidly in function of n T than for an eirp-equal system, but simulations prove the opposite. In fact, SU-MIMO takes

Figure 3 . 14 -

 314 Figure 3.14 -Average of capacity values achieved by the basic and eirp-equal systems.

Figure 3 . 15 -

 315 Figure 3.15 -Average of MU-MIMO to SU-MIMO capacity ratio for eirp-equal and eirp-unequal systems [TGac-B channel model].

Figure 3 .

 3 Figure 3.16 -Proba (C M U -M IM O /C SU -M IM O < 1) versus the number of transmitting antennas.

Figure 4 .

 4 Figure 4.1 represents the environment of the experiment. It displays the 3D indoor residential scene used to perform measurements. It is a typical and real middle sized apartment with a 12 m ×7 m surface and European building materials and furniture. The ceiling is at 2.53 m. Both LOS and NLOS scenarios have been probed. Hereafter

Figure 4 . 1 -

 41 Figure 4.1 -Living room and corridor of the apartment. Two locations of the Tx are considered, denoted as Tx 1 and Tx 2 in Figure 4.2. For each Tx position, multiple positions of the two receivers are evaluated. We denote by Rx 1 and Rx 2 the positions of the first and the second user respectively. During the measurements, nothing moved in the environment of the experiment to keep the same measurement conditions. Finally, the obtained measurement data base corresponds to 67 various 2-users configurations by combining the Tx and Rx positions.

  Tx: 480 virtual antennas.

Figure 4 . 4 -

 44 Figure 4.4 -Measurement equipment.

Figure 4 . 6 -

 46 Figure 4.6 -Cumulative distribution function of the RMS delay spread [ns].

Figure 4 .

 4 7 shows the CDF of the average of the superdiagonal values of the transmit and receive correlation matrices. The first observation drawn from this figure is that the receive correlation shows lower values compared to the transmit correlation. The measured environment shows slightly less correlated values at the Tx and Rx side than the TGac-B channel model.

Figure 4 . 7 -

 47 Figure 4.7 -CDF of the average of the superdiagonal values.

Figure 4 . 8 -

 48 Figure 4.8 -Comparison of measured and Rayleigh fading CDFs [dB].

Figure 4 . 9 -

 49 Figure 4.9 -Average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas.

Figure 4 .

 4 Figure 4.10 -Average MU-MIMO to SU-MIMO capacity values versus the number of transmitting antennas.

Figure 4 . 12 -

 412 Figure 4.12 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient.

Figure 4 .

 4 Figure 4.13 -Average MU-MIMO capacity values versus the average correlation coefficient.

Figure 4 . 14 -

 414 Figure 4.14 -Random antenna array geometries.

17

 17 Figure 4.17 -MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n T = 6).

  Figure 4.17 -MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n T = 6).

Figure 4 .

 4 19 depicts the MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient. The first observation drawn from Figure 4.19 is that the number of receiving antennas does not impact the correlation coefficient. This result is expected by the formula of the correlation coefficient proved in Equation 3.22 for Rayleigh channel. Increasing the number of receiving antennas increases the MU-MIMO to SU-MIMO capacity gain of 5%.

Figure 4 . 21 .

 421 It is composed of one Tx with 4 transmitting antennas spaced with 0.5 λ and two receivers equipped with four antennas each: n R 1 = n R 2 = 4. As previously stated in Chapter 2, the MU-MIMO can

Figure 4 .

 4 Figure 4.18 -Average MU-MIMO capacity values versus the average correlation coefficient (n T = 6).

Figure 4 .

 4 Figure 4.19 -Average MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient.

Figure 4 .Figure 4 . 22 -

 4422 Figure 4.22 -Average MU-MIMO capacity values versus the SNR.

Figure 4 . 23 -

 423 Figure 4.23 -Average MU-MIMO to SU-MIMO capacity ratio versus ∆P .

Figure 4 .

 4 Figure 4.24 -MU-MIMO to SU-MIMO capacity ratio versus ∆P and correlation coefficient.

Figure 4 . 27 -

 427 Figure 4.27 -Correlation coefficient versus the number of transmitting antennas.

Figure 4 . 28 -

 428 Figure 4.28 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient.

Figure 4 . 29 -

 429 Figure 4.29 -Correlation coefficient versus the users' position [TGac-B channel].

Figure 4 . 30 -

 430 Figure 4.30 -Correlation coefficient versus the users' position [Measured channel].

Figure 5 .

 5 1 gives an example of Wi-Fi radio coverage proposed by WANDA. It has been simulated at the conclusion and propose modeling for radio planning tools in Section 5.5.

Figure 5 . 6 -

 56 Figure 5.6 -The SNR gain as a function of the number of transmitting antennas for N ss = 1.

Figure 5 . 7 -

 57 Figure 5.7 -The SNR gain as a function of the number of transmitting antennas for N ss = 2.

Figure 4 .

 4 [START_REF] Perahia | Gigabit wireless LANs: an overview of IEEE 802.11ac and 802.11ad[END_REF]. The transmitter is composed of n T = 8 transmitting antennas arranged in 5 different geometries: CCA with 0.5λ spacing, and ICA with different radiuses: ICA 0.5λ, ICA 1λ, ICA 2λ and ICA 3λ.We study the case of one spatial stream N ss = 1. The number of receiving antennas varies in n R = 1, n R = 2, n R = 3. Figure5.8 gives the SNR gain as a function of the receiving antennas for different antenna array geometries. The figure confirms that an increase in the number of receiving antennas decreases the SNR gain of almost 1 dB for all antenna array geometries except the ICA 1λ where it decreases by more than 1 dB. The ICA 1λ achieves the highest SNR gain 6 dB with a receiver equipped with n R = 1 receiving antenna. For n R = 2 and n R = 3 receiving antennas, the ICA 1λ shows the highest SNR gain with 4, 3 dB and 3, 6 dB respectively.All antenna array geometries show almost the same SNR gain values. The variation is less than 0.5 dB which is insignificant regarding the difference between SNR gain values when varying the number of transmitting antennas.

Figure 5 . 8 -

 58 Figure 5.8 -The SNR gain as a function of the number of receiving antennas for N ss = 1.

Figure 5 .

 5 10 gives the average of the interference level difference as a function of the average of SNR gain for . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Figure A. 3 -

 3 Figure A.3 -The PDP of IEEE 802.11 channel model case B (9 taps).

Figure A. 4 -

 4 Figure A.4 -The CDF of the taps of the IEEE 802.11 channel model case B (9 taps).

Figure A. 6 -

 6 Figure A.6 -Doppler spectra of the taps of the IEEE 802.11 channel model case B (9 taps).

[

Figure D. 2 -

 2 Figure D.2 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 0.5 λ).

Figure D. 3 -

 3 Figure D.3 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 1 λ).

Figure D. 4 -

 4 Figure D.4 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 2 λ).

Figure D. 5 -

 5 Figure D.5 -Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 3 λ).

-n T = 4 , N ss = 1 , D = 1 2 1 1 1 1 T

 411 -n T = 3, N ss = 2,

Enfin, à partir

  de ces mesures nous avons évalué les performances de la technique du TxBF en analysant le gain sur le SNR ainsi que la réduction d'interférence. Cette partie propose finalement une modélisation statistique pour intégrer la prise en compte des gains de beamforming dans la simulation des couvertures radio avec un outil d'ingénierie. N° d'ordre : 15ISAR 32 / D15 -32

Table 1 -

 1 résultats proposés dans cette thèse se basent principalement sur cette technique vu Caractéristiques du standard 802.11ac.

	XVI

Table 2 -

 2 Résultats numériques des paramètres de régression linéaire pour 6 antennes d'émission.

	Scenario	a 0	a 1	r	Root Mean Square Error (RMSE)
	LOS	1.7319 -1.1290 0.9425	0.0379
	NLOS	1.7153 -1.1046 0.9682	0.0226

Chapter 1 Introduction Contents 1.1 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.2 Motivations and objectives . . . . . . . . . . . . . . . . . . . . . 3 1.3 Main contributions . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.5 List of publications . . . . . . . . . . . . . . . . . . . . . . . . . 5

  Profil de puissance pour le modèle TGac-B en fontion du retard des taps. XVIII 2 Moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction du nombre d'antennes d'émission. . . . . . . . . . . . . . . . . . . XIX 3 La moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction de la moyenne du coefficient de correlation(n Classification of fading channels. . . . . . . . . . . . . . . . . . . . . . . . 21 2.9 The MIMO channel angular parameters at both Tx and Rx. . . . . . . . 23 2.10 MIMO channel models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.11 Cluster based channel model B (2 clusters). . . . . . . . . . . . . . . . . . 26 2.12 TGac-B model average power delay profile with cluster extension (overlapping clusters). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 XXXIII 2.13 AoA and AoD for two-users MU-MIMO system. . . . . . . . . . . . . . . . 2.14 Uplink and downlink MU-MIMO systems. . . . . . . . . . . . . . . . . . . 2.15 Diagram of MU-MIMO system. . . . . . . . . . . . . . . . . . . . . . . . . 2.16 K parallel SU-MIMO channels. . . . . . . . . . . . . . . . . . . . . . . . . 2.17 TDMA and CSMA/CA channel access methods. . . . . . . . . . . . . . . 3.1 MU-MIMO to SU-MIMO capacity gain versus the simulation length values. 3.2 CDF of the taps power of TGn Channel Model Case B (9 taps) . . . . . . 3.3 Average of MU-MIMO to SU-MIMO capacity ratio versus transmit antenna spacing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Average of MU-MIMO to SU-MIMO capacity ratio versus the SNR with 6 transmitting antennas spaced by 0.5 λ in NLOS conditions. . . . . . . . 3.5 Average of MU-MIMO to SU-MIMO capacity ratio versus the versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . 3.6 Channel capacity values versus the number of transmitting antennas. . . . 3.7 Water-filling model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Average correlation coefficient versus the number of transmitting antennas.3.10 Average correlation coefficient versus transmit antenna spacing. . . . . . . 3.11 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n Measurement equipment. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Average PDP for each receiving antenna. . . . . . . . . . . . . . . . . . . . 4.6 Cumulative distribution function of the RMS delay spread [ns]. . . . . . . 4.7 CDF of the average of the superdiagonal values. . . . . . . . . . . . . . . . 4.8 Comparison of measured and Rayleigh fading CDFs [dB]. . . . . . . . . . 4.9 Average of MU-MIMO to SU-MIMO capacity ratio versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXXIV 4.10 Average MU-MIMO to SU-MIMO capacity values versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.11 Antenna array geometries. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.12 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.13 Average MU-MIMO capacity values versus the average correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.14 Random antenna array geometries. . . . . . . . . . . . . . . . . . . . . . . 4.15 CDF of the correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . 4.16 The two users indices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.17 MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n T = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.18 Average MU-MIMO capacity values versus the average correlation coefficient (n T = 6). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.19 Average MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.20 Average MU-MIMO capacity values versus the average correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.21 MU-MIMO system model with n T = n R 1 = n R 2 = 4. . . . . . . . . . . . . 4.22 Average MU-MIMO capacity values versus the SNR. . . . . . . . . . . . . 4.23 Average MU-MIMO to SU-MIMO capacity ratio versus ∆P . . . . . . . . . 4.24 MU-MIMO to SU-MIMO capacity ratio versus ∆P and correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The SNR gain as a function of the number of receiving antennas for N ss = 1.108 5.9 The interference level as a function of the average of SNR gain for N ss = 1.109 5.10 The interference level as a function of the average of SNR gain for N ss = 2.109 5.11 Interference level reduction map for a MIMO system with n T = 4 arranged in ULA, n R = 3, and N ss = 1 [LOS scenario with Tx 1 ]. . . . . . . . . . . . 5.12 Interference level reduction map for a MIMO system with n T = 4 arranged in ULA, n R = 3, and N ss = 1 [NLOS scenario with Tx 2 ]. . . . . . . . . . . 5.13 Interference level reduction map for a MIMO system with n T = 4 arranged in ULA, n R = 3, and N ss = 1 [NLOS scenario with Tx 1 ]. . . . . . . . . . . 5.14 Interference level reduction map for a MIMO system with n T = 8 arranged in ULA, n R = 3, and N ss = 1 [LOS scenario with Tx 1 ]. . . . . . . . . . . . 5.15 Interference level reduction map for a MIMO system with n T = 8 arranged in CCA, n R = 3, and N ss = 1 [LOS scenario with Tx 1 ]. . . . . . . . . . . . 5.16 Interference level reduction map for a MIMO system with n T = 8 arranged in CCA, n R = 3, and N ss = 1 [NLOS scenario with Tx 2 ]. . . . . . . . . . .

	AoA AoD AP AS BC BD BER BF BS CSD CCA CDF CI CIR CFR CSI CSIT CSMA/CA Carrier Sense Multiple Access/Collision Avoidance Angle of Arrival Angle of Departure Access Point Angular Spread Broadcast Channel Block Diagonalization Bit Error Rate Beamforming Base Station Cyclic Shift Delay Crossed Circular Array Cumulative Distribution Function Channel Inversion Channel Impulse Response Channel Frequency Response Channel State Information Channel State Information at the Transmitter DL Downlink DS Delay Spread DPC Dirty Paper Coding EIRP Equivalent Isotropically Radiated Power GI Guard Interval IUI Inter User Interference ISI Inter-Symbol Interference LOS Line-Of-Sight MIMO Multiple-Input Multiple-Output MU-MIMO Multi-User Multiple Input Multiple Output MCS Modulation and Coding Schemes NLOS Non-Line-Of-Sight OFDM Orthogonal Frequency-Division Multiplexing PAS Power Angular Spectrum PAPR Peak-to-Average Power Ratio per-SS per-Spatial Stream PHY PHYsical layer PDP Power Delay Profile PDS Power Delay Spectrum QAM Quadrature Amplitude Modulation Rx receiver RMS Root Mean Square RMSE Root Mean Square Error RCI Regularized Channel Inversion SISO Single-Input Single-Output SINR Signal-to-Interference-plus-Noise Ratio SNR Signal-to-Noise Ratio SS Spatial Stream SU-MIMO Single User MIMO SVD Singular Value Decomposition Tx transmitter TxBF Transmit Beamforming UL Uplink ULA Uniform Linear Array TDMA Time Division Multiple Access THP Tomlinson-Harashima Precoding VNA Vector Network Analyzer VHT Very High Throughput VP Vector Perturbation List of Figures 1 4.25 Average MU-MIMO to SU-MIMO capacity ratio versus versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.26 Average MU-MIMO channel capacity versus the number of transmitting List of Tables 1 Caractéristiques du standard 802.11ac. . . . . . . . . . . . . . . . . . . . . XVII 2 Résultats numériques des paramètres de régression linéaire pour 6 antennes d'émission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXI 2.1 Basic channel model parameters. . . . . . . . . . . . . . . . . . . . . . . . 3.1 Global parameters set for TGac-B channel model. . . . . . . . . . . . . . -----------Notation We briefly summarize below the main notations used throughout this report. Wherever possible, we have tried to keep the same notations from one chapter to another. The matrices and vectors are indicated with boldface. t Time τ Delay f Frequency 5.8 A.1 Display example of simulation parameters at the start of a simulation. . . A.2 Amplitude of the channel taps versus times of the IEEE 802.11 channel model case B (9 taps). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Numerical results of linear regression parameters for 6 transmitting Channel matrix for user k (Transfer channel function at a given frequency) H k antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Precoding matrix for user k W k 3.3 Recommendations to optimize MU-MIMO. . . . . . . . . . . . . . . . . . diag(v) Diagonal matrix formed using the vector v A.3 The PDP of IEEE 802.11 channel model case B (9 taps). . . . . . . . . . A.4 The CDF of the taps of the IEEE 802.11 channel model case B (9 taps). . A.5 Spatial correlation of the taps of the IEEE 802.11 channel model case B (9 taps). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.6 Doppler spectra of the taps of the IEEE 802.11 channel model case B (9 taps). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1 Cumulative distributions of the sum rate capacity for n T = 6, n R 1 = diag(M) Diagonal matrix formed using the diagonal of matrix M 4.1 Numerical results of linear regression parameters for different transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Convolution * (.) T Transposition 4.2 Numerical results of linear regression parameters for different transmitting antennas [TGac-B channel model]. . . . . . . . . . . . . . . . . . . . . . . 4.3 Linear regression parameters for 8 transmitting antennas [Measured Vector norm-2 || . || Absolute value | . | (.) Complex conjugate channel]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (.) H Transpose conjugate antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.27 Correlation coefficient versus the number of transmitting antennas. . . . . 4.28 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.29 Correlation coefficient versus the users' position [TGac-B channel]. . . . . 4.30 Correlation coefficient versus the users' position [Measured channel]. . . . 4.31 Angles in degrees [Measured channel]. . . . . . . . . . . . . . . . . . . . . 5.1 Coverage map for a transmitter placed in the living room. . . . . . . . . . 5.2 Transmission scheme in case of an SVD based TxBF precoding. . . . . . . 5.3 Transmission scheme without TxBF (Wo-BF). . . . . . . . . . . . . . . . . 5.4 SU-MIMO capacity as a function of path loss. . . . . . . . . . . . . . . . . 5.5 Transmission scheme quantifying interference. . . . . . . . . . . . . . . . . n R 2 = 2 achieved by several MU-MIMO strategies. . . . . . . . . . . . . . D.1 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (n T = 8). . . . . . . . . . . . . . . . . . . . . . . . . C.1 Survey of measurement studies [Part I]. . . . . . . . . . . . . . . . . . . . C.2 Survey of measurement studies [Part II]. . . . . . . . . . . . . . . . . . . . E(.) Expectation || Mean Frobenius norm || . || F N Number of 802.11ac subcarriers C.3 Survey of measurement studies [Part III]. . . . . . . . . . . . . . . . . . . n T Number of transmitting antennas D.2 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 0.5 λ). . . . . . . . . . . . . . . . . . . . . . . D.1 Numerical results of linear regression parameters for different transmitting antennas for CCA with n T = 8. . . . . . . . . . . . . . . . . . . . . . . . . Number of receiving antennas n R n R k Number of receiving antennas for the k th user D.3 Average of MU-MIMO to SU-MIMO capacity ratio versus the average D.2 Numerical results of linear regression parameters for ICA with n T passing K Number of users correlation coefficient (ICA 1 λ). . . . . . . . . . . . . . . . . . . . . . . . from 4 to 10 and different radiuses. . . . . . . . . . . . . . . . . . . . . . . k User index running from 1 to K D.4 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 2 λ). . . . . . . . . . . . . . . . . . . . . . . . trace() Trace E.1 Cyclic shift values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H † Pseudo inverse D.5 Average of MU-MIMO to SU-MIMO capacity ratio versus the average correlation coefficient (ICA 3 λ). . . . . . . . . . . . . . . . . . . . . . . . I Identity matrix
	i.i.d. IP ICA IDC Wo-BF 5.6 The SNR gain as a function of the number of transmitting antennas for independent and identically distributed Internet Protocol Irregular Circular Array International Data Corporation Without Beamforming W-BF N ss = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . With Beamforming Wi-Fi 5.7 The SNR gain as a function of the number of transmitting antennas for Wireless Fidelity WLAN Wireless Local Area Network N ss = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	XXXI XXXII XXXV XXXVI XXXVII XXXIX

Il est donc pertinent d'évaluer l'impact d'une estimation imparfaite du canal sur les résultats obtenus. -Une analyse approfondie des caractéristiques spatiales du canal de propagation (direction du départ, direction d'arrivée) permettrait une comparaison plus poussée des résultats des modèles de canal MIMO et de la campagne de mesure. -Il serait intéressant de mener une campagne de mesures avec un réseau d'antennes compact incluant ainsi l'effet de couplage des antennes. Dans notre campagne de mesures, nous avons évalué le seul cas avec 8 antennes d'émission disposées en réseau linéaire uniforme, les autres géométries se basant sur des réseaux virtuels. -Il serait possible d'aller plus loin dans l'analyse des tirages aléatoires des 8 antennes d'émission pour proposer une géométrie de réseaux optimisée à 8 antennes et la tester dans d'autres environnements résidentiels pour généraliser les résultats. -Il est pertinent de développer un algorithme rapide simulant le traitement d'antenne pour l'associer à un canal simulé par tracé/lancer de rayons afin de prédire les gains de "transmit beamforming" en chaque point de la carte de couverture au lieu d'une correction statistique globale. XXX Acronyms T = 6). . . . . . . XX 4 La moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU pour IEEE TGac-B channel (residential). . . . . . . . . . . . . . . . . . . XXII 5 L'environnement de mesure indoor, les 2 positions de l'émetteur et les 12 positions de réception. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXIII 6 Les différentes géométries de réseaux d'antennes Tx. . . . . . . . . . . . . XXIV 7 Moyenne du gain de capacité MIMO-MU par rapport au MIMO-SU en fonction de la moyenne du coefficient de corrélation. . . . . . . . . . . . . XXV 8 Gain moyen de SNR obtenu en activant TxBF. . . . . . . . . . . . . . . . XXVI 9 Gain sur le SNR en fonction du nombre d'antennes d'émission pour N ss = 1.XXVII 10 Le schéma de caractérisation d'interférence. . . . . . . . . . . . . . . . . . XXVII 11 La réduction d'interférence en fonction de la moyenne du gain sur le SNR pour N ss = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XXVIII 1.1 MU-MIMO combined to TxBF techniques [1]. . . . . . . . . . . . . . . . . 3 2.1 Peak data rate and per-SS data rate of WLAN technologies [2]. . . . . . . 11 2.2 Rate-over-range of three spatial streams 802.11ac and 802.11n devices [2]. 13 2.3 Beamforming basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Illustration of MU-MIMO 802.11ac and SU-MIMO 802.11n [2]. . . . . . . 15 2.5 Multipath propagation in an indoor environment. . . . . . . . . . . . . . . 16 2.6 Propagation phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.7 An illustration of a typical power-delay profile and the definition of the delay parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.8 T = 6). . . . . . . . . . . . . . . . . . . . . . . . . 3.12 MU-MIMO to SU-MIMO capacity ratio for an IEEE TGac-B channel (residential). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 MU-MIMO to SU-MIMO capacity ratio for an i.i.d Rayleigh channel. . . 3.14 Average of capacity values achieved by the basic and eirp-equal systems. . 3.15 Average of MU-MIMO to SU-MIMO capacity ratio for eirp-equal and eirp-unequal systems [TGac-B channel model]. . . . . . . . . . . . . . . . 3.16 Proba (C M U -M IM O /C SU -M IM O < 1) versus the number of transmitting antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.17 Average of MU-MIMO to SU-MIMO capacity ratio versus transmitting antenna spacings. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Living room and corridor of the apartment. . . . . . . . . . . . . . . . . . 4.2 The indoor environment: 2 positions of Tx with 12 positions of Rx. . . . . 4.3 5 GHz transmitting antenna pattern -Energy pattern [Measurement made by Orange Labs ]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4

Chapter 2 MU-MIMO techniques for wireless local area networks Contents 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The 802.11ac standard . . . . . . . . . . . . . . . . . . . . . . .

	-K. Issiali, V. Guillet, G. El Zein and G. Zaharia. IEEE 802.11ac Multi-user MIMO
	capacity and impact of antenna array geometry based on indoor measurements.
	In Personal, Indoor and Mobile Radio Communications (PIMRC), August 2015.
	2.2.1 Mandated use of the 5 GHz band . . . . . . . . . . . . . . . . .
	2.2.2 Maximum data rate . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.3 Wider bandwidth and higher modulation order . . . . . . . . .
	2.2.4 Rate-range . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.5 Eight spatial streams . . . . . . . . . . . . . . . . . . . . . . . .
	2.2.6 Transmit beamforming . . . . . . . . . . . . . . . . . . . . . . . Journal papers 2.2.7 MU-MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
	-K. Issiali, V. Guillet, G. El Zein and G. Zaharia. Impact of Antennas and 2
	Correlated Propagation Channel on BD Capacity Gain for 802.11ac Multi-User
	MIMO in Home Networks. Mediterranean Telecommunication Journal (RMT),
	June 2015.
	-K. Issiali, V. Guillet, G. El Zein and G. Zaharia. MU-MIMO 802.11ac Capacity
	Gain under EIRP Constraint for a Typical Home Environment. Lecture Notes in
	Electrical Engineering (LNEE), April 2016.
	International conferences
	-K. Issiali, V. Guillet, G. El Zein and G. Zaharia. Impact of Antennas and
	Correlated Propagation Channel on BD Capacity Gain for 802.11ac Multi-User
	MIMO in Home Networks. In WIreless Technologies, embedded and Intelligent
	Systems (WITS), April 2015: selected paper for publication in RMT journal.
	-K. Issiali, V. Guillet, G. El Zein and G. Zaharia. Impact of EIRP Constraint on
	MU-MIMO 802.11ac Capacity Gain in Home Networks. In Mediterranean Conf.
	On Inf. and Com. Techn. (MedICT), May 2015: best selected paper for
	publication in extended version in MDPI journal.

.3 Basics of propagation channel in wireless networks . . . . . .

	2.3.1 Multipath propagation mechanisms . . . . . . . . . . . . . . . .
	2.3.

2 Multipath propagation channel modeling . . . . . . . . . . . . 2.3.3 Propagation channel parameters . . . . . . . . . . . . . . . . . 2.3.4 Classification of multipath channels . . . . . . . . . . . . . . .

2.4 MIMO propagation channel modeling for systems . . . . . . .

2.4.1 Power angular spectrum . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Classification of the MIMO channel models . . . . . . . . . . . 2.4.3 TGn channel models . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 TGac channel models . . . . . . . . . . . . . . . . . . . . . . . 2.5 MU-MIMO techniques . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 MU-MIMO system model . . . . . . . . . . . . . . . . . . . . . 2.5.3 Precoding techniques . . . . . . . . . . . . . . . . . . . . . . . . 2.5.4 Linear precoding techniques for single antenna receivers . . . . 2.5.5 Linear precoding techniques for multiple antenna receivers . . . 2.5.6 Non-linear precoding techniques . . . . . . . . . . . . . . . . . 2.6 Channel capacity of wireless channels . . . . . . . . . . . . . . 37 2.6.1 SU-MIMO capacity . . . . . . . . . . . . . . . . . . . . . . . . 37 2.6.2 MU-MIMO capacity . . . . . . . . . . . . . . . . . . . . . . . . 39 2.

7 Main existing results and future works . . . . . . . . . . . . . 39

Table 2 .

 2 Next, AS, AoA, and AoD

	Parameters/Environment	A	B	C	D	E	F
	Avg 1 st Wall Distance (m)	5	5	5	10	20	30
	RMS Delay Spread (ns)	0	15 30	50 100 150
	Maximum Delay (ns)	0	80 200 390 730 1050
	Number of Taps	1	9	14	18	18	18
	Number of Clusters	N/A 2 2	2	3	4	6

1 -Basic channel model parameters.

Table 3 .

 3 .2. It shows that setting the Rician factor to 0 dB for LOS scenarios in TGac-B channel models gives almost similar results with slightly high capacity gain for LOS: 73 % for LOS compared to 71 % for NLOS case. 2 -Numerical results of linear regression parameters for 6 transmitting antennas.

	Scenario	a 0	a 1	r	RMSE
	LOS	1.7319 -1.1290 0.9425 0.0379
	NLOS	1.7153 -1.1046 0.9682 0.0226

Table 3 .

 3 .3 summarizes the previous conclusions and ends in proposing system recommendations. 3 -Recommendations to optimize MU-MIMO.

	Parameters	MU-MIMO to SU-MIMO capacity gain	Recommendations for radio engineering
	SNR effect: 0 -40 dB	Till 80% of gain	The desired range is put forward
	Transmit antenna spacing: 0.5 λ -1.5 λ	Insignificant gain of 2 %	0.5 λ n T should be slightly greater
	Number of		than the number of spatial
	transmitting antenna	40% of gain	streams: 6 transmitting
	n T : 4 -8		antennas for two two-antenna users and two SS per user
		almost the same gain
	LOS or NLOS	(LOS gives a slightly	no recommendation
		high gain)
	MU-MIMO with Water-filling algorithm	the same gain as equal power sharing	not recommended excepted if n T = n R = N ss
			-Beneficial use
			-Power	allocation
	EIRP constraint	10% of gain with 6 transmitting antennas	strategy: power	sharing	equal for
			low complexity

Table 4 .

 4 [START_REF]IDC: Analyze the future[END_REF] gives the regression parameters for the studied configuration. The correlation coefficient is low. A better prediction would need other parameters to be included.

	n T	a 0	a 1	r	RMSE
	8	0.1200 0.0139 0.4496 0.0350

Table 4 . 3

 43

-Linear regression parameters for 8 transmitting antennas [Measured channel].

Table D .

 D Hz (0.87/1Gbps) when SNR is 31 and 36 dB, respectively for n T = 16, n R = 4 K = 4. The frequency utilization (K = 4): 2.4 to 3.3 times higher compared to the eigenvector beamforming method in SU-MIMO, when n T = 16 and n R changes from 2 to 4. Table C.3 -Survey of measurement studies [Part III]. 1 -Numerical results of linear regression parameters for different transmitting antennas for CCA with n T = 8. MU-MIMO to SU-MIMO capacity gain as a function of correlation coefficient for different number of transmitting antennas arranged in ICA with different radiuses: ICA 0.5 λ, ICA 1 λ, ICA 2 λ and ICA 3 λ. Table D.2 summarizes the related regression parameters.

	26] Indoor and indoor-outdoor scenarios with multiples base stations in the Royal Institute of Technology (KTH) using DSP DA Converter at 1766.6 MHz (BW: 9.6 kHz). [48] LOS scenario in a conference room at the NTT DOCOMO RD Center in Yokosuka, Japan using an implemented LTE-Advanced transceiver [47] indoor	4 Tx and 4 Rx. Rx: 4-monopole array. Tx: dual-polarized sectorial antennas with directive pattern in elevation with high gain (Outdoor), and dual-polarized patch antennas with low profile and low gain (Indoor). Different noise and interference levels K = 2, n T = 4, n R k = 2. 3.92625 GHz, BW: 90 MHz Antenna arrangements at Tx: CAA and DAA. Tx and Rx:dipole antennas with omni-directional beam patterns in azimuth with 2 dBi of gain. 4.85 GHz BW: 100 MHz MU-MIMO: n T =, K = 4, a 0 a 1 4 1.5525 -1.0471 0.8964 0.0292 Downlink with no CSI with close antenna elements, in both indoor and outdoor locations, performs very similar to a channel with CSIT (slightly higher in indoor). Spatially, separated locations for Base Station (BS) shows slightly lower average capacity and non-Gaussian distribution for the measured environment. Peak throughput greater than 1 Gbit/s is achieved for the CAA. Throughput of approximately 700 to 950 Mbps is achieved for DAA. MU-MIMO is robust against the antenna separation. moving speed, antenna separation (fading correlation), and transmitter antenna arrangement on the achievable throughput of MU-MIMO. Frequency utilization of over 43.5/50 r RMSE 5 1.6809 -0.9379 0.8838 0.0267 6 1.7458 -0.8899 0.9016 0.0213 bits/s/n T 8 1.8141 -0.8235 0.8863 0.0169
	environment	n R k = 4 Tx and Rx:
	(a room) using	Sleeve antennas are
	MU-MIMO	used for the
	testbed	transmitter and
	according to	receiver antennas
	802.11a	arranged in linear
	standard	arrays spaced by 1.0
		and 0.5 wavelengths,
		respectively.

Table D .

 D 2 -Numerical results of linear regression parameters for ICA with n T passing from 4 to 10 and different radiuses.

		1.503 1.509	-0.974	-0.89	-0.879	-0.9	0.03 0.02 0.017 0.015 0.853 0.829 0.755 0.782
	5	1.74 1.667 1.669 1.671	-1	-0.778	-0.85	-0.858	0.03 0.022 0.019 0.015 0.875 0.739 0.723 0.731
	6	1.757 1.726 1.726 1.722	-1.339	-0.88 -0.88	-0.753	0.04 0.021 0.021 0.017 0.897 0.852 0.852 0.687
	8	1.807 1.789 1.804 1.818	-0.853	-0.615	-0.779	-0.892	0.034 0.019 0.019 0.015 0.835 0.755 0.746 0.795
	10	1.849 1.841 1.828 1.828	-0.737	-0.689	-0.723	-0.736	0.033 0.026 0.018 0.015 0.807 0.716 0.723 0.698

Table E .

 E .1[START_REF]IEEE standard for information technology -telecommunications and information exchange between systems local and metropolitan area networks -specific requirements[END_REF]. 1 -Cyclic shift values. La facilité de déploiement et la flexibilité des technologies sans fil ont permis une évolution considérable des normes IEEE 802.11. L'arrivée de la fibre optique à la maison a également motivé des études de recherche pour accroître les débits et la capacité. Dans ce contexte, la dernière norme IEEE 802.11ac ratifiée en 2014 a normalisé de nouvelles technologies pour atteindre des débits maximaux au-dessus de 1 Gbit/s. En particulier, 802.11ac propose le Multiple Input Multiple Output Multi-Utilisateurs (MIMO-MU) et le Transmit Beamforming (TxBF). Ces technologies d'antennes intelligentes supposent un diagramme d'antenne d'émission qui dépend de la position des utilisateurs et des caractéristiques du canal de propagation multi-trajets correspondant. Cependant, les outils d'ingénierie et de planification radio actuels simulent une carte de couverture fixe et indépendante de tout traitement d'antennes adaptatives côté émission. Cette thèse étudie l'impact de ces techniques MIMO-MU et TxBF et propose des solutions adaptées pour l'ingénierie radio et l'optimisation du système ainsi que pour les outils de planification radio. En effet, la première partie de cette thèse consiste à affiner les gains de capacité en identifiant les configurations et les paramètres système optimaux. La deuxième partie aborde les gains de beamforming en termes du gain sur le rapport signal à bruit (SNR) et sur la réduction du niveau d'interférence pour avoir une modélisation réaliste de la couverture radio. Par conséquent, nous avons tout d'abord analysé le gain de capacité MIMO-MU par rapport au MIMO Single User (MIMO-SU) en se basant sur les modèles de canaux standardisés pour le contexte MIMO-MU corrélés 802.11ac. En analysant les résultats, nous avons proposé des optimisations du système pour maximiser les gains de capacité MIMO-MU dans un environnement résidentiel, principalement liées à la configuration du système, telles que le nombre d'antennes d'émission, l'espacement d'antennes, un compromis entre le nombre d'antennes et le nombre de flux spatiaux, la géométrie du réseau d'antennes... Un coefficient de corrélation entre canaux a été mis en évidence pour expliquer les gains obtenus. En outre, nous avons formulé le problème d'optimisation convexe de maximisation de capacité dans le cadre d'un système MIMO-MU équipé de plusieurs antennes de réception tout en tenant compte de la contrainte de la puissance isotrope rayonnée équivalente (PIRE) pour un réseau local sans fil (WLAN) en Europe.Nous avons ensuite mené une campagne de mesures de canaux de propagation avec deux utilisateurs dans un environnement résidentiel pour confirmer et compléter les résultats obtenus avec des canaux simulés. Nous avons été en mesure d'étendre cette étude à d'autres paramètres pouvant potentiellement influencer le gain de capacité MIMO-MU par rapport au MIMO-SU, tels que l'atténuation de propagation et la géométrie des réseaux d'antennes.

	Résumé
	Number of	Cyclic shift for	Cyclic shift for	Cyclic shift for	Cyclic shift for
	space-time	space-time	space-time	space-time	space-time
	streams	stream 1 [ns]	stream 2 (ns)	stream 3 (ns)	stream 4 (ns)
	N ss = 1 N ss = 2 N ss = 3 N ss = 4	0 0 0 0	---400 ns -400 ns -400 ns	-----200 ns -200 ns	-------600 ns

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Not Applicable

(a) The first six taps (b) The last three tapsFigure A.5 -Spatial correlation of the taps of the IEEE 802.11 channel model case B (9 taps).

Acknowledgements

Chapter 4 MU-MIMO capacity based on propagation measurements in indoor environment

Contents Chapter 5 Transmit beamforming gain analysis for network planning tools based on propagation measurements

Future work

In this thesis, many issues related to MU-MIMO system have been addressed. However, multiple suggestions have not been explored and can be the topics for future studies:

-This thesis assumes a perfect channel state of information at the transmitter. It is relevant to assess the impact of an imperfect CSI on the obtained results. -A deep analysis of spatial characteristics (AoA, AoD) of the propagation channel could be done to compare the results between TGac MIMO channel models and measurement campaign. -It would be interesting to conduct a measurement campaign with a compacted antenna array including the antenna coupling effect. In our measurement campaign, we have evaluated only the case with 8 transmitting antennas arranged in ULA. -It is possible to go further in analyzing the random draws of 8 transmitting antennas to propose an optimized array geometry and to test it in other residential environments. -It is relevant to develop rapid algorithm simulating the antenna processing in the case of ray tracing or ray launching models in order to predict the beamforming gains in each point of the coverage map instead of a global statistical correction. -An accurate prediction of the correlation coefficient as a function of the users' locations would be interesting to decide about the benefit of MU-MIMO.

Appendix A

All detailed parameters of TGac-B channel

The following figures (A.2, A.3 A.4 A.5 A.6) present the amplitude of the channel taps, the PDP, the CDF, the spatial correlation properties and the Doppler spectra respectively of a SU-MIMO set-up with n T = 2, n R = 2 working at 5.25 GHz carrier frequency, with λ 2 wavelength spacing at the transmitter and receiver, using IEEE 802.11 TGac channel model TGac-B. It is sampled in time domain at 2500 Hz to simulate fading with NLOS conditions. 512 FadingNumberOfIterations, where each iteration has a length of 55 CoherenceTimes, have been generated. In order to reduce the set of output samples, a downsampling factor of 48 has been applied. Per block, one sample has been stored. Dashed red curves/markers correspond to the reference values, whereas the blue curves/markers are the outcomes of the simulation. Moreover, in the Doppler plot, the green curve represents the Welch periodogram [START_REF] Schumacher | Description of a Matlab implementation of the indoor MIMO WLAN channel model[END_REF]. [START_REF] Schumacher | Description of a Matlab implementation of the indoor MIMO WLAN channel model[END_REF]. The addition of the LOS component causes the achieved PDP of models TGac-B to be shifted with respect to the reference one, up to the value of the K factor (in this case K is equal to 0). In Figure A.5, the spatial correlation coefficients of the simulated impulse responses match the coefficients computed during the initialization phase. The spatial correlation tends to be significant for certain taps. This is partly due to the small spacing (half a wavelength) between the two receive antenna elements. Finally, the simulated Doppler spectra shown in Figures A.6 reproduce the bell shape spectrum selected by IEEE 802.11 TGac Channel Model Special Committee. On these figures, the red vertical lines are drawn at +/ -f requency Doppler . The upper green line is set at the maximum of the Doppler spectrum, and the lower green line lies 10 dB below. Ideally, the Doppler spectrum should meet the crossing of the red and green lines. This would be the case, should the jitter be removed from the sampled spectra presented.

Appendix B

Comparison of TDMA SU-MIMO, DPC, and BD sum rate capacity using MU-MIMO and spatially correlated channels

We study the MU-MIMO broadcast channel and the achievable throughput for the optimal strategy of dirty paper coding. We compare its performance to that achieved with block diagonalization with lower complexity based on correlated channels.

It was found in [START_REF] Costa | Writing on dirty paper (corresp.). Information Theory[END_REF] that dirty-paper coding is optimum capacity achieving for the broadcast MU-MIMO channel. In [START_REF] Jindal | Sum power iterative water-filling for multi-antenna gaussian broadcast channels[END_REF], two iterative algorithms have been proposed to find the optimum sum capacity bound for the multiple-antenna broadcast channel. The authors have used the fact that the Gaussian broadcast and multiple-access channels are dual. This means that their capacity regions, and therefore their sum rate capacities, are equal. These algorithms compute the sum rate capacity achieving for the dual multiple-access channel. This can be converted to the equivalent optimal strategies for the broadcast channel. The algorithms employ a simple iterative Water-filling procedure that probably converges to the optimum. The two algorithms use the single-user Waterfilling procedure in each iteration. Based on this algorithm and Water-filling, we find the sum capacity of the multiple-antenna broadcast channel.

Figure B.1 illustrates the performance of the BD algorithm using or not Water-filling and the sum-rate bound computed with DPC for a case involving

and SN R = 20 dB. The SNR here is defined as in Chapter 3, Section 3.3 without the EIRP constraint. We have also plotted the SU-MIMO capacity to compare. The results are based on the correlated TGac-B channel model and the cumulative distribution function of the capacity achieved by each method is plotted. Block diagonalization with equal power sharing among the spatial streams is labeled MU-MIMO BD, whereas MU-MIMO BD WF refers to the BD algorithm employing the optimal power allocation via Water-filling. SU-MIMO refers to the case with equal power sharing with no precoding The expected loss in Rayleigh fading due to block diagonalization compared to DPC is given by [START_REF] Lee | High SNR analysis for MIMO broadcast channels: Dirty paper coding versus linear precoding[END_REF]:

In our case when K = 2, n T = 6 and n R 1 = n R 2 = 2, loss DP C-BD = 1.5 bits/s/Hz in Rayleigh channels.

Appendix C

Measurement campaigns synthesis

antennas arranged in two cross-polarized pairs. Rx: 3G Panorama Antennas A characterization of the (dis-)similarity of the channels of different users using: spatial separation of the user's, correlation matrices, co-linearity of the MIMO channel matrices and correlation of large scale fading. The structure of the MIMO channel matrices changes significantly with the inter-user distance [captured by the co-linearity measure]. The transmit and the full correlation matrix also show some dependence on the inter-user distance.

Receive correlation matrices are independent of the inter-user distance. The shadowing correlation was found to be very low in all cases even when the nodes are quite close.

[49] Indoor conference room LOS/NLOS VNA at 5 GHz. The performance of MU-MIMO precoding drops drastically when the users are close together in an outdoor scenario due to the strong correlation at the transmitter. MU-MIMO provides a higher throughput than SU-MIMO. In outdoor scenarios with a LOS, the capacity drops significantly when the users are close together, due to high correlation at the transmitter side of the channel. Outdoor: The results show that MU-MIMO provides a higher throughput than SU-MIMO. However, the throughput in the measured channels is by far worse than the one in channels without spatial correlation. Of all the evaluated linear precoding schemes, the RCI precoder performs best in the measured channels.

SU-MIMO: n

Appendix D

Capacity gain versus correlation coefficient

The

Spatial expansion precoding matrix

Cyclic Shift Delay (CSD) and spatial expansion versus antenna mapping are applied at the transmitter, when no beamforming scheme is considered. It is necessary because closely spaced antennas act as beamforming arrays without wide phase spacing, and it is possible to inadvertently create signal maxima and minima over receive antennas due to interference patterns. This is avoided by giving each transmit antenna's signal a large phase shift relative to the others. Below are examples of spatial mapping matrices that might be used [START_REF]IEEE standard for information technology -telecommunications and information exchange between systems local and metropolitan area networks -specific requirements[END_REF]. There are many other alternatives; implementation is not restricted to the spatial mapping matrices shown. The examples are:

-Direct mapping Q k is a diagonal matrix of unit magnitude complex values that takes one of two forms:

1. Q k = I, the identity matrix 2. A CSD matrix in which the diagonal elements represent cyclic shifts in the time domain.

-Indirect mapping: Q k may be the product of a CSD matrix and a unitary matrix such as the Hadamard matrix or the Fourier matrix. -Spatial expansion: Q k is the product of a CSD matrix and a square matrix formed of orthogonal columns. In our implementation, we have evaluated the spatial expansion scheme [START_REF]IEEE standard for information technology -telecommunications and information exchange between systems local and metropolitan area networks -specific requirements[END_REF].The spatial expansion may be performed by duplicating some of the N ss streams to form the n T streams, with each stream being scaled by the normalization factor Nss n T . The spatial expansion may be performed by using, for instance, one of the following matrices, denoted D, left-multiplied by a CSD matrix, denoted M CDS (k), and/or possibly multiplied by any unitary matrix. The resulting spatial mapping matrix is then Q k = M CDS (k).D, where D may take on one of the following values:

-n T = 2, N ss = 1, D = 1