Acknowledgement

It is my pleasure to get the opportunity to express gratitude to all who contributed in the successful completion of my PhD thesis.

First of all, I would like to express my heartily thanks to my thesis supervisor Professor Xavier Lagrange for his constant guidance and supervision throughout the last three years. His confidence in my research work, and careful editing contributed enormously to the bringing up of this thesis. I would also like to express special thanks to my co-supervisors Dr. Philippe Mary and Dr. Jean Yves Baudais for their invaluable guidance, fruitful ideas, and generous advises throughout the whole period of my PhD. I got all kind of support from them in all difficulties I faced during my thesis.

It is also my pleasure to express my sincere thanks to the members of my thesis defence committees, Dr. Marco Di Renzo, Professor Jean-Marie Gorce. Professor Marie-Laure Boucheret and Professor Luc Deneire for reading and evaluating my manuscript that helped me to enrich my thesis.

I would like to thank all my colleagues in IETR for providing a friendly working environment and facilities to complete this thesis. I also express my gratitude to the officials and other staff members who rendered their help during the period of my thesis work. I express my gratitude particularly to Pascal Richard, Katell Kervella, Celine Laube and Aurore Gouin for their kindness help from the beginning of my thesis in all official and administrative steps.

I would like to express my thanks to my friends, and relatives including my in-laws family for their encouragement during my thesis.

I give all my love and thanks to my parents and my brother for their emotional and moral support during my thesis years.

I express my heartiest thanks to my wife for her continuous moral support during all my doctoral work. I express my love to my daughter for the joyful moments she gave me during my PhD thesis.

List of figures 3.1 Growth of mobile network traffic and revenue [START_REF] Fletcher | Green radio-sustainable wireless networks[END_REF] [START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF] [START_REF] Correia | Challenges and enabling technologies for energy aware mobile radio networks[END_REF] . . . . . [START_REF] Suarez | An overview and classification of research approaches in green wireless networks[END_REF].12 EE vs. SE for AWGN channel considering only the transmit power with N 0 = -168.83 dBm/Hz . .

Percentage of power consumption in a cellular network

Hexagonal cellular network

Poisson homogeneously distributed BSs

Components of a BS

Directions of the linear precoding vectors

Layered energy efficient

Distribution of power consumption in BSs

Daily data traffic profile for cellular systems in Europe

MIMO and CoMP

Heterogeneous network deployment with relay for coverage extension

Optimal -SE

List of tables

Parameters for the plots with and without

Coefficients of G(v

Mathematical notations and variables

Mathematical notations

||A|| 2
Spectral norm of matrix A lim sup M ||A|| 2 limit superior of ||A|| 2 , i.e. for every > 0, there exists M 0 ( ) such that ||A|| 2 ≤ lim sup M ||A|| 2 + for ∀M > M 0 ( )

B \ A In B but not in A B ⊂ R d B is a subset of R d

Chapitre 1 : introduction

Ce chapitre présente les motivations et le contexte des études de compromis EE-SE. Cela est ensuite suivi par une description de la structure et des contribution de la thèse. Le chapitre se termine par la liste des publications relative à ces travaux de thèse.

Chapitre 2 : préliminaires mathématiques

Les travaux exposés au chapitre 5 combinent quelques résultats fondamentaux de la théorie de matrices aléatoires (RMT) et des PPP, mobilisant les outils de géométrie stochastique et plus particulièrement les processus ponctuels. Ce chapitre résume les définitions, propriétés, lemmes ou théorèmes importants de la géométrie stochastique et RMT qui seront utilisés par la suite.

Ce chapitre illustre deux des plus importants théorèmes de la littérature relative aux PPP : les théorème de Slivnyak et de Campbell. Le théorème de Slivnyak établit que la statistique d'un PPP conditionnée à un point est la même que celle du PPP en entier. L'application de ce théorème se produit dans les réseaux sans fil lorsque le conditionnement est effectué relativement à la position de l'émetteur ou du récepteur. D'un autre côté, le théorème de Campbell est utilisé pour calculer la moyenne de la somme de fonctions évaluées à la position des point du processus ponctuel.

La transformée de Stieltjes est l'un des outils les plus importants utilisé pour résoudre les problèmes relatifs aux grandes matrices aléatoires. Dans le chapitre 5, les puissances des interférences et du signal utile sont calculées en utilisant la transformée de Stieltjes de la distribution spectrale limite (LSD) de matrices hermitiennes, qui est la transformée de Stieltjes de la distribution spectrale empirique (ESD) lorsque la taille des matrices croît. Le résultat obtenu pour la LSD des matrices de grande taille peut être utilisé pour approcher les résultats de l'ESD de matrice de taille finie. Ce chapitre présente les définitions de la transformée de Stieltjes, de la LSD et de l'ESD. Un théorème très utile de Silverstein et Bai sur la convergence de l'ESD vers la LSD est également présenté et qui sera particulièrement utile dans le chapitre 5. De plus, quelques résultats importants (lemme d'inversion de matrice, lemme de la trace, lemme de la perturbation de rang 1...) sont fournis.

Chapitre 3 : état de l'art

Dans ce chapitre, nous introduisons la SE qui est la métrique de performance conventionnelle des communications dans fil. La SE indique si le spectre est utilisé efficacement mais ne donne pas d'indication sur l'efficacité avec laquelle l'énergie est consommée. Un aperçu de la métrique la plus commune pour quantifié l'efficacité énergétique (EE) est ensuite présenté, incluant le bit par Joule qui mesure la quantité de bits distribués par le réseau par unité d'énergie consommée.

Cette mesure est utilisée pour évaluer l'efficacité énergétique d'un canal AWGN dans ce chapitre, et aussi pour un réseau hexagonal dans le chapitre 4. Cependant, la métrique est normalisée par la bande passante du système pour quantifier l'EE dans le chapitre 5.

Des modèles de consommation de puissance et de réseaux sont nécessaires pour calculer l'EE et la SE. Une description détaillée des modèles de consommation linéaires, qui sont les plus communément utilisés dans la littérature, sont présentés. Ces modèles sont simples et permettent de calculer l'EE avec une précision suffisante. Principalement, la puissance consommée par l'amplificateur de puissance (PA) est proportionnelle à la puissance de sortie et la consommation des autres composants est relativement constante. Cela justifie l'approximation linéaire de la consommation des BS en fonction de la charge. Selon le modèle de consommation de puissance (a) Réseau cellulaire hexagonal [START_REF] Rappaport | Wireless communications: principles and practice[END_REF] (b) Réseau PPP [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] FIGURE 1 -Modèles réseau linéaire, la consommation totale d'une BS peut être écrite comme

P total = aP t + b, (1)
Où a est déterminé par l'efficacité du PA, P t est la puissance d'émission et b est la consommation de puissance du circuit statique. Considérant en outre que la consommation d'énergie de l'unité d'alimentation en courant continu (PS), de l'unité de bande de base (BB) et du système de refroidissement n'est pas à l'échelle du nombre de unité de radiofréquence (RF), (1) peut être écrit comme [START_REF] Li | Throughput and Energy Efficiency Analysis of Small Cell Networks with Multi-Antenna Base Stations[END_REF] P total = aP t + M P RF + c, (

Où M est le nombre d'unité RF, P RF est la puissance consommée par unité RF et c représente la puissance de non transmission pour l'unité BB, l'unité PS et le refroidissement. Comme le montre la Fig. 1, deux types de modèles de réseau existent dans la littérature : le réseau régulier hexagonal avec des positions fixes des BS et le réseau PPP avec des positions aléatoires. Subsequently, the energy efficiency-area spectral efficiency (EE-ASE) region is investigated for multi-user multiple input single output (MU-MISO) Poisson point process (PPP) network when signal-to-leakage-and-noise ratio (SLNR) precoder is used.

η SE = sup E : P η SE < E ≤ (3)
η EE = η SE P totale (4

Background and motivation

The design and evolution of wireless networks is guided by the continuously increasing data rate demand of customers for high quality digital service. The exponentially increasing data rate requirement triggers the demand for high SE, which is a measure of how efficiently limited frequency spectrum is utilized. The deployment densification of wireless networks, i.e. micro and femto cell in long term evolution (LTE), multiple input multiple output (MIMO) technology, and reusing the frequency bandwidth over relatively small areas already appear as a suitable answer to meet that demand [START_REF] Bjornson | Energy-efficient future wireless networks: A marriage between massive MIMO and small cells[END_REF][START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF].

The roll-out densification leads to an elevation of the energy consumption of communication networks. The information and communication technology (ICT) industry currently contributes to 2% of total worldwide carbon emissions, and this contribution is expected to double over the next 10 years with exponential increase in ICT sector [START_REF] Association | Green Technologies: Concepts, Methodologies, Tools and Applications[END_REF]. Designing green wireless networks 1. 1. Background and motivation has become increasingly important in order to limit the electricity bill and the greenhouse gas emission. Therefore, one of the major concerns for the next generation of wireless networks is EE, which measures how efficiently energy is used to transmit information [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF][START_REF] Blume | Most Promising Tracks of Green Network Technologies[END_REF]. Obviously, the reduction of the energy consumption is not the only target to be pursued, and therefore the investigation of EE-SE tradeoff becomes increasingly important.

A large amount of works has been conducted to the problem of EE in wireless communications, see for instance [START_REF] Belmega | A Survey on Energy-Efficient Communications[END_REF] and references therein. The authors in [START_REF] Gorce | Energy-Capacity Trade-off Bounds in a Downlink Typical Cell[END_REF] Since the BS static power consumption cannot be neglected, several works addressed various techniques to turn off BS in low load scenarios [START_REF] Xu | Improving Network Energy Efficiency through Cooperative Idling in the Multi-Cell Systems[END_REF]. In [START_REF] Weng | Energy-Efficient Cellular Network Planning under Insufficient Cell Zooming[END_REF], the authors proposed cell zooming to save energy when the network is not fully loaded. The tradeoff between EE and SE considering a practical power consumption model has been a growing subject of interest in recent years [START_REF] Maaz | Energy efficiency analysis of hybrid-ARQ relay-assisted schemes in LTE-based systems[END_REF][START_REF] Imran | Most suitable efficiency metrics and utility functions[END_REF][START_REF] Sun | Energy efficiency analysis of one-way and two-way relay systems[END_REF][START_REF] Auer | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown (earth)[END_REF][START_REF] Desset | Flexible power modeling of LTE base stations[END_REF].

To describe the distribution of the interfering BSs, Kelif et al. have proposed a fluid model, which considers a continuum distribution of BS instead of a discrete summation of interfering signals [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF]. This work provides a closed form expression of the interference-to-signal ratio (ISR) allowing a tractable derivation of SE and EE in hexagonal cellular networks. However, their expression is limited to frequency reuse factor 1, where the whole system bandwidth is used in each cell. Since SE is inversely proportional to the frequency reuse factor, reusing the whole system bandwidth potentially increases SE and EE as well. But the major issue in this case is the interference caused by the transmission on the same frequencies. We use a semi-analytical method to obtain an expression of ISR allowing the derivation of the EE-SE tradeoff in a regular hexagonal network when frequency reuse factor is different from 1, which has never been done in literature to the best of our knowledge. Moreover, we have proposed -SE-EE tradeoff as an outage measure for performance evaluation when shadowing is considered.

On the other hand, MIMO enables space division multiple access (SDMA) that serves multiple users in the same time-frequency resources achieving higher SE compared to the conventional communication [START_REF] Ngo | Energy and spectral efficiency of very large multiuser MIMO systems[END_REF]. However, precoding must be used to mitigate the interference. Designing a precoder that maximizes the output signal-to-interference-plus-noise ratio (SINR) for each user is desired, but this is a challenging task. A suboptimal and more tractable precoder is zero-forcing (ZF), which is based on nulling the intra-cell interferences [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF][START_REF] Gupta | Downlink coverage probability in MIMO HetNets with flexible cell selection[END_REF][START_REF] Dhillon | Downlink MIMO HetNets: Modeling, ordering results and performance analysis[END_REF]. Although ZF performs well in high signal-to-noise ratio (SNR) regime, its performance deteriorates in low SNR regime.

Another drawback of this precoder is that it imposes a restriction on the minimum number of transmit antennas at BS [START_REF] Cheng | A new SLNR-based linear precoding for downlink multiuser multi-stream MIMO systems[END_REF]. Sadek et al. have proposed SLNR as an alternative optimization metric for designing precoder, which has a closed form solution [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF]. In contrast to zero-forcing solution, this precoder does not impose any restriction on the number of BS antennas and also considers the influence of noise [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF].

Although there have been numerous works based on SLNR precoder [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF][START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF][START_REF] Cheng | A new SLNR-based linear precoding for downlink multiuser multi-stream MIMO systems[END_REF][START_REF] Tarighat | A multi user beamforming scheme for downlink MIMO channels based on maximizing signal-to-leakage ratios[END_REF], these works did not consider multi-cell environment and also ignored the network geometry while computing the precoder. We investigate the performance of the SLNR precoder for cellular network taking into account non-homogeneous average SNR due to large scale fading, i.e. path loss, and also the interference created on the other-cell users. We consider a spatial PPP network model, which is more tractable compared to the hexagonal grid model. Although the PPP model is extensively used in literature, works in [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF][START_REF] Bjornson | Designing wireless broadband access for energy efficiency: Are small cells the only answer?[END_REF][START_REF] Gupta | Downlink coverage probability in MIMO HetNets with flexible cell selection[END_REF][START_REF] Dhillon | Downlink MIMO HetNets: Modeling, ordering results and performance analysis[END_REF] have used either the maximal ratio transmission (MRT) or ZF precoder for tractability. We develop the analytical expressions of ASE and EE in a PPP cellular network when SLNR precoder is used, using random matrix theory (RMT) and stochastic geometry. We review important definitions, properties and theorems related to point processes from [START_REF] Haenggi | Stochastic geometry for wireless networks[END_REF][START_REF] Haenggi | Interference in large wireless networks[END_REF].

Structure of the thesis and contributions

Point process

A point process is a random collection of points and plays a fundamental role in the description of the network geometry. A point process can be defined as Definition 2.1. (Point process) A point process is a countable random collection of points that reside in some measure space, usually the Euclidean space R d [START_REF] Haenggi | Stochastic geometry for wireless networks[END_REF]. The point process is alternatively defined as [START_REF] Haenggi | Interference in large wireless networks[END_REF]

Φ = ∞ i =1 δ x i ,
where δ x is the Dirac measure such that δ x (B ) = 1 B (x) for B ⊂ R d , where B is a Borel subset.

We will focus on the two dimensional Euclidean space R 2 since BSs and users in Chapter 5 are modelled by two point processes located in a two-dimensional circular disc.

Definition 2.2. (Intensity measure) [START_REF] Haenggi | Stochastic geometry for wireless networks[END_REF] The intensity measure of a point process Φ is equal to the average number of points in a set B ⊂ R d and is expressed as

∆(B ) = E[Φ(B )].
For example, intensity measure of the BS and user processes in a subset area of R

P(Φ ∈ B ) = P(Φ x ∈ B ).
The intensity measure of a homogeneous point process can be written as

∆(B ) = λ|B |,
where |B | is the Lebesgue measure and λ is the density of the point process Φ. A homogeneous point process has a constant density. Therefore, the intensity measure of such process is proportional to the Lebesgue measure, which assigns a measure to the subsets of d-dimensional

Euclidean space R d . For the two dimensional network considered in Chapter 5, Lebesgue measure corresponds to area, and densities of the BSs and users are defined as the average number of the processes per u.a. Both BSs and users are homogeneous point processes since their intensity measures are proportional to the area.

Two of the most important point processes are binomial point process (BPP) and PPP, which are descried in the following.

Binomial point process [1]

Φ = {x n } is a BPP if the number of points is fixed, and the points are identically and independently distributed on a compact set B ⊂ R d . Φ is a uniform BPP only if the points are uniformly distributed. The probability of having k < n nodes in A ⊂ B of a uniform BPP is

P Φ A = k = n k |A| |B | k 1 - |A| |B | n-k
, where |A| and |B | represent the Lebesgue measures corresponding to set A and B respectively.

Note that the number of points in Φ(A) and Φ(B ) are not independent since the total number of points is fixed. Due to this dependence, it is usually difficult to analyze the performance of cellular networks when the nodes are designed as points of BPP. However, PPP can be obtained as a limit of BPP when |B | → ∞ keeping n -1 |B | constant.

Poisson point process [1]

PPP is the most widely used point process. The characteristics of a homogeneous PPP of density λ are:

• the number of points in any B ⊂ R d is a Poisson random variable (r.v.) with mean λ|B | and can be characterized by the probability distribution function (PDF)

P Φ B = n = e -λ|B | (λ|B |) n n! , ( 2.1)
• the number of points in disjoint sets are independent r.vs.

PPP is more tractable compared to BPP due to the independence property.

P Φ A = k|Φ B = n = n k |A| |B | k 1 - |A| |B | n-k .
Proof.

P Φ A = k|Φ B = n = P Φ A = k, Φ B = n P Φ B = n P Φ A = k, Φ B \ A = n -k P Φ B = n . ( 2.2)
Since the number of points in the disjoint sets A and B \ A are independent according to the characteristic of PPP, (2.2) can be written as

P Φ A = k|Φ B = n = P Φ A = k P Φ B \ A = n -k P Φ B = n . ( 2.3)
Using (2.1), (2.3) can be expressed as

P Φ A = k|Φ B = n = e -λ|A| (λ|A|) k k! e -λ|B \A| (λ|B \A|) n-k (n-k)! e -λ|B | (λ|B |) n n! = n! k!(n -k)! |A| |B | k |B \ A| |B | n-k = n k |A| |B | k 1 - |A| |B | n-k . ( 2.4)
Since we consider an infinite network in Chapter 5, both the number of BSs and number of users converge to their intensity measures according to law of large number (LLN). Therefore, Bss and users can be characterized also by two BPPs according to the conditional property mentioned.

Function of point process

Mean of sum and product of functions evaluated at the points of a point process have wide applications in wireless communications, and they are described in the following.

Theorem 2.1. (Campbell's theorem) [START_REF] Haenggi | Stochastic geometry for wireless networks[END_REF] If Φ is a point process on R d with intensity measure ∆ and f : R d → R is a measurable function, then the random sum

S = x∈Φ f (x)
is a r.v. with mean

E[S] = R d f (x)∆(d x),
provided that the right hand side (RHS) is finite. If Φ has a density λ,

E[S] = R d f (x)λ(x)d x.
When Φ is homogeneous, Campbell's theorem can be written as

E[S] = λ R d f (x)d x.
In Chapter 5, Campbell's theorem is applied over the groups of interfering BS to calculate the mean of the aggregated inter-cell interference powers from each group. The measurable function in this context is the inter-cell interference power from a BS within the group.

Theorem 2.2. (Probability generating functional (PGFL)) [START_REF] Haenggi | Interference in large wireless networks[END_REF] The PGFL of the point process Φ is defined as

G f = E x∈Φ f (x) , where f (x) : R d → [0, ∞) is measurable. The PGFL of a PPP is G f = e -R d (1-f (x))∆(d x) .
PGFL is very useful to evaluate the Laplace transform of x∈Φ f (x), which can be expressed as

E e -s x∈Φ f (x) = E x∈Φ e -s f (x) = G e -s f (x) .
Most of the works in literature dealing with PPP networks consider exponential distribution for the desired signal power since it allows them to use PGFL and leads to simple analytical expressions for the coverage probability, i.e. complementary CDF (CCDF) of the SINR, and ergodic rate, i.e. rate averaged over the fast fading and the network geometry [START_REF] Novlan | A new model for coverage with fractional frequency reuse in OFDMA cellular networks[END_REF][START_REF] Andrews | A new tractable model for cellular coverage[END_REF][START_REF] Ganti | A new way of computing rate in cellular networks[END_REF][START_REF] Zhang | A stochastic geometry analysis of inter-cell interference coordination and intra-cell diversity[END_REF].

Since the distributions of the signal powers are unknown due to the consideration of SLNR precoding at BSs in Chapter 5, PGFL cannot be applied to calculate the ergodic rate. This makes the calculation of the ergodic rate of a typical user in a PPP network difficult when SLNR precoder is used. Therefore, we follow a different approach to calculate the ergodic rate in Chapter 5.

Slivnyak's theorem

Slivnyak's theorem is regarded as one of the striking properties of PPP. The theorem can be stated as Theorem 2.3. [START_REF] Haenggi | Stochastic geometry for wireless networks[END_REF] The reduced palm distribution equals the distribution of the PPP itself and can be written as

P !x = P,
where P is the distribution of the PPP, and P !x is the reduced Palm probability, defined as the distribution of the PPP conditioned to have a point at x ∈ R d .

The application of palm distribution arises in a wireless network when a condition is put on either the transmitter or the receiver's position. Slivnyak's theorem states that the law of Φδ x conditioned to have a point at x ∈ R d is the same as the law of Φ. For a homogeneous PPP, the theorem says

E y∈Φ\{x} f (y) = λ R d f (y)d y.
In Chapter 5, while Campbell's theorem is applied over the groups of interfering BS to calculate the mean of the aggregated inter-cell interference powers from each, we condition on the location of the typical connected BS, and the desired signal power is not taken into account in the calculation of the aggregated inter-cell interference power. This is allowed by Slivnyak's theorem.

Random matrix theory

Deriving the achievable rate region in multiple antenna systems is an intractable problem, and RMT has been introduced for analyzing such systems [START_REF] Couillet | Random matrix methods for wireless communications[END_REF]. When random matrices grow large with a given ratio between the number of rows and the number of columns, empirical spectral distribution (ESD) of the large dimensional matrices converges to deterministic functions. The results obtained for ESD of large dimensional matrices can be used to approximate the results for ESD of finite size matrices. There are a number of powerful and appealing theorems on the convergence of eigenvalue distribution of large dimensional matrices to deterministic functions.

This section provides the definitions and basic tools related to RMT used in Chapter 5, which are reproduced mostly from [START_REF] Couillet | Random matrix methods for wireless communications[END_REF] unless otherwise mentioned.

Definition 2.4. (Random matrix) An N × n matrix Y is a random matrix if it is a matrix valued r.v. on some probability space (Ω, F , P) with entries in some measurable space (R, G ), where F is a σ-field on Ω with probability measure P and G is a σ-field on R.

Here Ω is the sample space, i.e. set of all possible outcomes, F is a set of events and P is the probability of the events. In the context of our work, the space R can be either R or C since we encounter both real and complex r.vs.

µ X (A) = 1 N N i =1 1 A (λ i ).
This definition implies that empirical probability measure of eigenvalues of a matrix is the proportion of the realization of the eigenvalues in a set.

Definition 2.6. (Empirical spectral distribution) ESD of the matrix X ∈ C N ×N is defined as the empirical CDF of its eigenvalues, and is expressed as

F ∆ (x) = µ X -∞, x = 1 N N i =1 1 λ i ≤x (x),
where

∆ = diag(λ 1 , λ 2 , • • • , λ N )
represents a square diagonal matrix filled by the eigenvalues.

The definition implies that ESD is the proportion of the eigenvalues less than or equal to x.

Definition 2.7. (Limit spectral distribution) If F ∆ (x) converges in distribution towards a non random distribution function F ∆ l (x) as N → ∞, then the function F ∆ l (x) is called the limit spectral distribution (LSD) of X when F ∆ l (x) exists.
It is often sufficient to have convergence of F ∆ (x) in distribution to F ∆ l (x) for all x where F ∆ l (x) is continuous to obtain relevant results. In Chapter 5, eigen decomposition is applied to Hermitian matrices of size equal to the number of BS antennas. ESD of the matrix is random function for small number of BS antennas, but converges to a non-random function following Theorem 2.4 stated further below in the document, as the number of BS antennas tends to infinity.

The non-random function is called LSD.

Definition 2.8. (Moment of ESD) The k-th order moment of F ∆ can be expressed as

M k = ∞ -∞ λ k d F ∆ (λ) = 1 N N i =1 λ k i = 1 N tr X k ,
where tr (X k ) represents the trace of (X k ), i.e. the sum of the elements on the main diagonal of X k .

Stieltjes transform

Stieltjes transform is one of the most important tools used to solve the problems related to large random matrices. Marcěnko and Pastur first used Stieltjes transform approach to find the distribution of the eigenvalues of random matrices [START_REF] Marčenko | Distribution of eigenvalues for some sets of random matrices[END_REF].

Definition 2.9. (Stieltjes transform) Let F be a real-valued bounded measurable function with

support supp(F ) ⊂ R. Then the Stieltjes transform of F for z ∈ C \ supp(F ) is defined as m F (z) = supp(F ) 1 λ -z d F (λ).
Study of large dimensional random matrices is simplified due to the Stieltjes transform. For the Hermitian matrix X ∈ C N ×N , we can write

m F ∆ (z) = R 1 λ -z d F ∆ (λ) = 1 N tr ∆ -zI N -1 = 1 N tr X -zI N -1 .
Therefore, working with the Stieltjes transform simplifies to working with trace of the matrix (X -zI N ) -1 . Stieltjes transform has been used in communications engineering problems in a large extent [START_REF] Govindasamy | Spectral Efficiency in Single-Hop Ad-Hoc Wireless Networks with Interference Using Adaptive Antenna Arrays[END_REF][START_REF] Tse | Linear multiuser receivers: effective interference, effective bandwidth and user capacity[END_REF][START_REF] Silverstein | Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices[END_REF][START_REF] Li | On the distribution of SINR for the MMSE MIMO receiver and performance analysis[END_REF].

In Chapter 5, average quantities are calculated from the Stieltjes transform of LSD of positive semidefinite matrices, which are the limiting Stieltjes transform of ESD as the size of the matrices grow large. Since the eigenvalues of positive semidefinite matrices are non-negative, Stieltjes transforms are analytic over C \ R + .

Lemmas and theorem

Some basic lemmas, and a theorem on the convergence of the ESD of random matrices with asymptotically large dimensions to LSD are used in this thesis.

Lemma 2.1. (Matrix inversion lemma ) [START_REF] Silverstein | On the empirical distribution of eigenvalues of a class of large dimensional random matrices[END_REF] Let A be an invertible matrix of size M × M . Then, for any vector y ∈ C M and scalar τ ∈ C for which A + τyy H is invertible,

y H (A + τyy H ) -1 = y H A -1 1 + τy H A -1 y . Lemma 2.2. (Trace lemma) [50, 51] Let y = [y 1 , • • • , y M ]
T be an M × 1 vector where the y m are iid complex Gaussian r.vs. with unit variance. If A is an M × M matrix independent of y and

lim sup M ||A|| 2 ≤ ∞, then 1 M y H Ay -tr(A) M →+∞ -----→ 0,
where ||A|| 2 is the spectral norm of matrix A and lim sup M ||A|| 2 is the limit superior of ||A|| 2 , i.e.

for every > 0, there exists M 0 ( )

such that ||A|| 2 ≤ lim sup M ||A|| 2 + for ∀M > M 0 ( ). Lemma 2.3. (Rank-1 perturbation lemma) [52] Let, z ∈ C + , v = ℑ[z], A and B are N × N matrix
with B be Hermitian and r ∈ C N . Then

tr B -zI N -1 -B + rr H -zI N -1 A = r H B -zI N -1 A B -zI N -1 r 1 + r H B -zI N -1 r ≤ ||A|| 2 v for z < 0, tr B -zI N -1 -B + rr H -zI N -1 A ≤ ||A|| 2 |z| .
This lemma with z < 0 is used a number of times in Chapter 5. However, A is an identity matrix in the context of our work implying a spectral norm equal to one. We now state Theorem 2. The massive traffic increase is accompanied by increased energy consumption. Energy cost for running a network constitutes almost 50% of the operational expenditures (OPEX) of the operators [START_REF] Zeller | Sustainable wireless broadband access to the future Internet-The EARTH project[END_REF]. Mobile network traffic is envisioned to increase exponentially leading to an exponential increase in the energy cost in contrast to the increase of the revenue, as shown in Fig. 3.1. Reducing the energy consumption is critical to lowering the OPEX from an operator's perspective. Another important consequence of increasing energy consumption is the increase of carbon-dioxide (CO 2 ) emissions that has a devastating impact on climate change [START_REF] Bianzino | A survey of green networking research[END_REF][START_REF] Cox | Acceleration of global warming due to carbon-cycle feedbacks in a coupled climate model[END_REF].

denoted as F D , converges in distribution to F D l on [0, ∞) as n → ∞, and the ratio n N → γ au ∈ (0, ∞) as n, N → ∞. Then ESD of B surely converges in distribution to F B l such that for z ∈ C + , m F B l (z) is the unique solution of m F B l (z) = -z -γ au t 1 + t m F B l (z) d F D l (t ) -1 . ( 2
Since energy saving and environmental protection become inevitable needs, focus of research in wireless communications is shifting to EE oriented design. However, around 60% of the energy required for the operation of a cellular network is consumed at BSs, as can be seen in Fig. 3.2.

Therefore, the most energy-efficient designs are targeted to reduce the energy consumption in

Performance metrics

SE has driven the wireless system design in the past decades due to the high data rate demand of multimedia services. It indicates how efficiently frequency spectrum is used, but fails to provide any insight on how efficiently the energy is consumed. The fast growing data traffic volume and dramatic expansion of network infrastructures inevitably trigger tremendous escalation of energy consumption in wireless networks. Hence, EE is quickly becoming one of the key performance metrics to evaluate wireless communication systems together with SE that has been traditionally used.

Spectral efficiency

SE is defined as the achievable transmission rate per unit bandwidth and expressed in b/s/Hz as

η SE = R w , (3.1)
where R is the achievable transmission rate in b/s, also known as the capacity, over a communication channel with bandwidth w Hz. The achievable transmission rate of an AWGN channel has been computed by Shannon in [START_REF] Shannon | A mathematical theory of communication[END_REF]. We use η SE as the performance metric for the AWGN channel in this chapter, and also for the hexagonal network in Chapter 4 when only path loss is considered.

The Shannon capacity is not adapted to describe the performance of a system in shadowing environment when the transmitters have no channel state information (CSI). When shadowing is considered in Chapter 4, we define -SE, adapted from the -outage capacity definition in [START_REF] Tse | Fundamentals of Wireless Communication[END_REF].

The -SE is defined as the largest SE E such that the probability of SE being below E is less than or equal to and can be expressed as

η ( ) SE = sup E : P η SE < E ≤ . ( 3.2)
On the other hand, the performance metric used in Chapter 5 is ASE, i.e. average sum of the ergodic rate of users in b/s/Hz per u.a., since we are interested on the total sum rate, not the individual rate.

The capacity region of the multiple access and broadcast channel in a single cell network has been characterized in [START_REF] Telatar | Capacity of Multi-antenna Gaussian Channels[END_REF][START_REF] Tse | Multiaccess fading channels. I. Polymatroid structure, optimal resource allocation and throughput capacities[END_REF][START_REF] Yu | Iterative water-filling for Gaussian vector multipleaccess channels[END_REF] and [START_REF] Li | Capacity and optimal resource allocation for fading broadcast channels. I. Ergodic capacity[END_REF][START_REF] Yu | Trellis precoding for the broadcast channel[END_REF][START_REF] Vishwanath | Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels[END_REF][START_REF] Viswanath | Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality[END_REF][START_REF] Weingarten | The capacity region of the Gaussian multipleinput multiple-output broadcast channel[END_REF][START_REF] Yu | Uplink-downlink duality via minimax duality[END_REF] respectively. On the other hand, cellular network consists of multiple cells, and cells having the same set of frequencies interfere with each other. The common practice to obtain R in the interference-limited cellular networks is to consider the inter-cell interferences as noise [START_REF] Gilhousen | On the capacity of a cellular CDMA system[END_REF], as considered in Chapter 4 and Chapter 5.

Energy metrics

Energy metrics are introduced in order to assess and compare the energy consumption of different telecommunication networks and equipments [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF]. These metrics are useful in setting long term research target to achieve more energy efficient solutions. The energy consumption of the telecommunication networks can be attributed to the operational energy and to the embodied energy. The operational energy represents the energy consumed due to the actual run-time operation of the network and equipments. On the other hand, the embodied energy represents the energy consumed to manufacture and deploy the system [START_REF] Humar | Rethinking Energy Efficiency Models of Cellular Networks with Embodied Energy[END_REF]. Energy metrics given by various standardization bodies mainly quantify the operational energy. Therefore, in the following, an overview of the energy metrics is provided taking only the operational energy consumption into account.

Energy metrics can be divided into component, equipment and network level metrics [START_REF] Chen | Energy efficiency metrics for green wireless communications[END_REF].

While component and equipment level metrics are used to evaluate the energy consumption of a component within an equipment and the equipment respectively, network level metrics assess the consumption of a network as a whole. The efficiency of a PA is a component level metric, which is defined as [START_REF] Chen | Energy efficiency metrics for green wireless communications[END_REF],

η = P o P i , (3.3)
where P o is the effective output power and P i is the input power. Another component level metric corresponding to PA is peak-to-average-power ratio (PAPR), i.e. the ratio between the peak output power and the average output power, whose reduction ensures a better amplifier efficiency [START_REF] Jiang | An overview: peak-to-average power ratio reduction techniques for OFDM signals[END_REF].

Energy consumption rating (ECR) of an equipment is an equipment level metric and can be expressed in J/b as [START_REF]Network and telecom equipment-energy and performance assessment test procedures and measurement methodology[END_REF] ECR

= P f T f , ( 3.4)
where P f is the peak power in Watts (W) and T f is the maximum throughput in b/s. The lower ECR, the less energy is consumed to transport the same amount of data. ECR metric is developed with the aim to compare EE of the network equipments from different manufacturers. However, this metric can be used to measure EE of networks also [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF]. The definition of ECR fits only for full load and hence, variable load ECR (ECR-VL) is introduced to consider the dynamic nature of the traffic. ECR-VL can be expressed as [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF] ECR -VL = I β I P I I β I T I (

with β I , P I and T I be the weighting coefficient, power consumption and throughput respectively at I ∈ {0, 10, 30, 50, 100} percent of the load. The weighting coefficients are selected such that

I β I = 1.
Having the best peak efficiency assumed by ECR does not necessarily mean that the equipment has the best power management capabilities, as measured by ECR-VL. The telecommunication energy efficiency ratio (TEER) is another equipment level metric, which is defined as [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF] TEER = useful work power , (

where 'useful work' varies with the type and function of the equipment, and 'power' is the average power over the duration of the measurement test.

A network level metric is power per area, which is defined in W/m 2 as [74]

ρ = P C A C , ( 3.7)
where P C is the average power consumed in the network, and A C is the coverage area. Energy per unit area expressed in J/m 2 is another network level metric which relates the energy consumed in a network to the area covered. Both these network level metrics allow to compare various deployment strategies, e.g. heterogeneous networks with overlapping cells, cells with variables sizes. Since a network is coverage limited at low loads, these metrics are particularly relevant when the traffic is low. International telecommunication union (ITU) suggested a network level metric for wireless access networks that is defined in J/b/m 2 /subscriber as the power per number of subscribers, per throughput and per area [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF]. This metric is relevant for capacity limited systems since it takes the number of subscribers and the throughput into account.

Energy metrics are further classified into absolute metrics and relative metrics [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF]. While absolute metrics measure the actual EE of a component, equipment or a network, relative metrics quantify EE w.r.t. references. Absolute metrics can be further classified into EE metrics and energy consumption (EC) metrics [START_REF] Suarez | An overview and classification of research approaches in green wireless networks[END_REF]. While the EE metrics express the ratio of the performed work over the consumed energy, EC metrics are defined as the opposite. However, both of these metrics contain exactly the same information. Energy per bit is the most common EC metric in literature, particularly used for theoretical studies [START_REF] Verdú | Spectral efficiency of CDMA with random spreading[END_REF][START_REF] Sandhu | Non-ergodic power-bandwidth tradeoff in linear multi-hop networks[END_REF][START_REF] Oyman | A Shannon-theoretic perspective on fading multihop networks[END_REF][START_REF] Yao | On energy efficiency and optimum resource allocation of relay transmissions in the low-power regime[END_REF]. This metric is defined as the ratio between the consumed power and the bit rate, and can be measured in J/b as

E b = P R , (3.8)
where P is the consumed power. The bit per energy is an EE metric which measures the amount of bits delivered per unit energy consumed, and can be expressed in b/J as [START_REF] Chen | Characterizing energy efficiency and deployment efficiency relations for green architecture design[END_REF] b E = 1

E b = R P . ( 3.9)
This metric has been extensively used in literature, e.g. wireless sensor and ad-hoc networks [START_REF] Rodoplu | Bits-per-joule capacity of energy-limited wireless ad hoc networks[END_REF][START_REF] Meng | Growth of wireless ad hoc networks[END_REF][START_REF] Rodoplu | Bits-per-joule capacity of energy-limited wireless networks[END_REF], cooperating BSs [START_REF] Fehske | Bit per joule efficiency of cooperating base stations in cellular networks[END_REF], AWGN channels [START_REF] Kwon | Channel capacity in bits per joule[END_REF]. The metric b/J/Hz relates the number of transferred bits with the energy consumed in the network and the bandwidth [START_REF] Li | Throughput and energy efficiency analysis of small cell networks with multi-antenna base stations[END_REF][START_REF] Li | Energy efficiency analysis of small cell networks[END_REF]. This metric can be obtained from b E by normalizing w.r.t. bandwidth. An EE metric derived from ECR is the energy efficiency rate (EER), which is defined in b/J as [START_REF] Hamdoun | Survey and applications of standardized energy metrics to mobile networks[END_REF] EER = 1 ECR (3 .10) and fits only for full load.

On the other hand, energy consumption gain (ECG) and energy reduction gain (ERG) are the two most commonly used relative metrics. These metrics are useful only to compare systems with same characteristics. ECG is defined as [START_REF] Han | Green radio: radio techniques to enable energy-efficient wireless networks[END_REF] ECG

= E 1 E 2 , ( 3.11)
where E 1 and E 2 are the energy consumption of the reference system and the system under consideration respectively. On the other hand, ERG is defined as [START_REF] Han | Green radio: radio techniques to enable energy-efficient wireless networks[END_REF] ERG

= E 1 -E 2 E 2 .
To summarize, it is necessary to have reliable energy metrics to evaluate energy saving techniques and their performance in practical systems.

Cellular network models

Cellular network is based on the concept of replacing a single cell with high power BS by several cells with low power BSs for higher system capacity [START_REF] Rappaport | Wireless communications: principles and practice[END_REF]. The region over which the signal strength of a BS is above a minimum threshold is the coverage area of BS, also known as cell. On the side of the network models existing in literature, we can find two kind of works: the work considering regular hexagonal networks with fixed BS positions [START_REF] Anang | Cellular system information capacity change at higher frequencies due to propagation loss and system parameters[END_REF][START_REF] Pollakis | Base station selection for energy efficient network operation with the majorization-minimization algorithm[END_REF][START_REF] Gong | Base station sleeping and resource allocation in renewable energy powered cellular networks[END_REF][START_REF] Lee | Spectrum efficiency in cellular [radio[END_REF] and the researches dealing with PPP to model the BS distribution, e.g. [START_REF] Gorce | Energy-Capacity Trade-off Bounds in a Downlink Typical Cell[END_REF][START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF][START_REF] Gupta | Downlink coverage probability in MIMO HetNets with flexible cell selection[END_REF][START_REF] Dhillon | Downlink MIMO HetNets: Modeling, ordering results and performance analysis[END_REF] among others.

Hexagonal network

Among the shapes that could be considered for the geometry of a cell in regular networks, hexagon covers the maximum area and hence this pattern has been widely used to model the cellular deployment. In hexagonal network, the whole system bandwidth is divided and allocated to a number of cells. The set of cells among which the whole system bandwidth is divided constitutes a regular pattern. This regular pattern is called the cluster and this is repeated throughout the whole network. Fig. 3.3 is an example of hexagonal cellular network with three clusters.

K = i 2 + i j + j 2 , i 0, j 0, (3.12)
where i and j are non-negative integers. For example, K is 7 in Fig.

PPP network

Since the regular hexagonal network is highly idealized and not very tractable, Andrews et al.

have introduced a more tractable model in [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] by considering BSs as a PPP. In nearest-neighbour cell association model [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF][START_REF] Guo | Asymptotic deployment gain: A new approach to characterize coverage probability[END_REF], i.e. each user connects to its nearest BS, the coverage area of a BS is the set of points belonging to R 2 which are the closest from this BS than any other. This association rule leads to Voronoi tesselation [START_REF] Aurenhammer | Voronoi diagrams-a survey of a fundamental geometric data structure[END_REF], as shown in Fig. 3. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF]. The proposed model better captures the increasingly opportunistic and dense placement of BSs in urban cellular networks with highly variable coverage area. The main weakness is that BSs in some cases are located very close to each other but with a significant coverage area [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. However, the strength of this model can be stated as the independent positions of the BSs leveraging the tools from stochastic geometry, natural inclusion of different cell sizes and shapes, and the lack of edge effects [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].

The assumption that BSs are positioned completely at random has turned the PPP model more tractable compared to the hexagonal network model. PPP model is adopted for our works in Chapter 5.

BS power consumption models

Sophisticated power consumption models are required to calculate EE accurately in thereby.

Power consumption model of a BS must incorporate the power consumed by its components since considering only the transmit power leads to wrong conclusions for designing energy efficient systems [START_REF] Cui | Energy-constrained modulation optimization[END_REF][START_REF] Miao | Cross-layer optimization for energy-efficient wireless communications: a survey[END_REF].

Components of a BS

As shown in Fig. 3.5, a BS is comprised of a baseband (BB) unit and one or more transceivers.

Each transceiver consists of a radio frequency (RF) unit, a PA and an antenna connected through a feeder cable. In addition, BS contains a power supply (PS) unit that comprises the mains supply and the DC-DC unit. Moreover, a cooling system is also present to protect the telecommunications equipments from damage due to heat.

PA amplifies the low output power from the RF block into a signal with high enough power for transmission. It is one of the most power consuming components. Current wireless standards, i.e. RF unit comprises a receiver and a transmitter for uplink and downlink communications respectively. At the transmitter side, digital output signal from the BB unit is first converted to analog signal followed by an up-conversion in RF signal with sufficient signal amplification [START_REF] Imran | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF].

BB unit carries out digital signal processing, e.g. channel coding, decoding, signal detection, digital pre-distortion (only in the downlink for micro/macro BSs), modulation, demodulation, filtering, fast Fourier transform (FFT), inverse FFT for OFDM, etc. [START_REF] Imran | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]. The silicon technology, which is anticipated to be replaced by a new generation every two years, significantly affects the power consumption of the BB unit.

Mains supply unit performs the AC-DC conversion and connects to the power grid, while DC-DC unit works as a battery backup. On the other hand, cooling depends largely on environmental conditions. Power saving can be achieved by allowing as much natural cooling as possible to reduce the air conditioning. However, active cooling is relevant only for macro BSs, while smaller BSs can be cooled naturally [START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF].

Linear power consumption models

Since the RF output power per transmit antenna, P out is measured at the input of the antenna element, losses due to the antenna interface are usually not included in the calculation of the total power consumption of a BS. Traditionally PAs are located far from the antenna incurring significant power loss in the feeder cables. The effect of the feeder cable can be approximated by an efficiency term, i.e. ratio between the output power and input power, considering that the power consumption by the feeder cable scale with P out [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]. The power consumption of each PA is proportional to the power input to the feeder cable. Besides, the losses incurred by the cooling system, mains supply and DC-DC unit scale linearly with the power consumption of the other components, and the effects can be approximated by efficiency terms [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF].

Putting all pieces together, total power consumption of a BS at maximum load, i.e. P out = P max , can be expressed as [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]]

P total = M P max η PA η feed + P RF + P BB η DC η MS η cool , (3.14)
where M is the number of transceiver chains, P max is the maximum P out , P RF is the power consumed by RF unit, and P BB is the power consumed by BB unit when M is one. Moreover, η PA , η feed , η DC , η MS and η cool are the efficiencies of PA, feeder cable, DC-DC unit, mains supply and the active cooling system respectively. Note that the total power consumption scales with the number of transceiver chains, i.e. M .

Mainly the power consumption of PA scales with the output power, i.e. P out , and the con-3. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. Propagation model sumption of the other components is comparatively static, as shown by the authors in [START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF]. This justifies a linear approximation of the BS power consumption model at variable load [START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF]:

P total = M (aP out + P 0 ), (3.15)
where 0 < P out ≤ P max , a is the slope of the load-dependent power consumption, determined by the efficiency of PA and known as the PA factor, and P 0 represents the power consumption with a single transceiver at the minimum non-zero P out . Writing P t = M P out and b = M P 0 , (3.15) can be expressed as

P total = aP t + b. (3.16)
This model is extensively used in literature because of its simplicity [START_REF] Gong | Base station sleeping and resource allocation in renewable energy powered cellular networks[END_REF]. We use this model to calculate EE in Chapter 4. Considering further that the power consumption of the PS unit, BB unit and cooling system do not scale with the number of transceiver chains, (3.16) can be written as [START_REF] Li | Throughput and Energy Efficiency Analysis of Small Cell Networks with Multi-Antenna Base Stations[END_REF] P total = aP t + M P RF + c, (

where c represents the non-transmission power for BB unit, PS unit and cooling. This power consumption model is used in Chapter 5 to calculate EE. Since BSs in the hexagonal network considered in Chapter 4 have a single antenna, (3.16) and (3.17) are equivalent in that case.

Propagation model

Radio signal propagated through wireless channel is impaired by path loss, shadowing, multipath, fading, etc. An accurate but tractable propagation model is required to calculate the power at the receiver. Taking into account all the impairments of a wireless channel, power received by the typical k-th user at distance r i k from the BS i ≥ 0 can be written as [START_REF] Elsawy | Stochastic geometry for modeling, analysis, and design of multi-tier and cognitive cellular wireless networks: A survey[END_REF]

P i k = P t LX r -α i k , (3.18)
where P t is the transmit power, L is the propagation constant, X is a random variable accounting for the random channel power gain and r -α i k is the path loss with α being the path loss component. Due to the singularity at the origin, propagation model in (3.18) is an unbounded model, which is valid only for calculating the received power at the far field [START_REF] Inaltekin | On unbounded path-loss models: effects of singularity on wireless network performance[END_REF]. This model is widely used in literature due to its simplicity and mathematical tractability. We use this model throughout the whole thesis to calculate the received power. However, the parameters of the model vary for different scenario and will be mentioned when the model is applied.

Precoding techniques

Different precoding techniques are used at BSs with multiple antennas in order to increase the desired signal power, while reducing the interference or leakage to other users. This is done by sending the same data signal with different amplitudes and phases from different antennas such that they add coherently at the desired user, but destructively at the other users [START_REF] Björnson | Optimal resource allocation in coordinated multi-cell systems[END_REF]. Precoding enables global utilization of all spectral resources removing the need for frequency reuse technique. Dirty paper coding is based on non-linear interference pre-cancellation, and it is capacity achieving for a single cell [START_REF] Costa | Writing on dirty paper (corresp.)[END_REF]. However, deploying it in real-time systems is difficult due to high complexity of decoding and the required knowledge of all interfering signals at transmitter. On the other hand, optimal precoding technique is yet not known for multicell scenario [START_REF] Björnson | Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure[END_REF]. Although the linear precoders, e.g. MRT, ZF, SLNR, are suboptimal in the capacitysense, they are practically more feasible because of low complexity and robustness to channel uncertainty. MRT precoder maximizes the SNR by aligning the precoding vector with the channel vector of the desired user [START_REF] Lo | Maximum ratio transmission[END_REF]. This precoder does not consider the leakage created on other users, and is optimal in the low SNR regime. On the other hand, ZF technique is based on nulling the interference generated at non-intended users and can be achieved by selecting precoding vectors that are orthogonal to the channel vectors of the non-intended users [START_REF] Bjornson | Cooperative multicell precoding: Rate region characterization and distributed strategies with instantaneous and statistical CSI[END_REF][START_REF] Ginis | A multi-user precoding scheme achieving crosstalk cancellation with application to DSL systems[END_REF]. Therefore, the ZF precoder is available only when the number of BS antennas is greater than the number of users to whom no leakage is created. ZF is the optimal precoder in the high SNR scenarios where inter-user interference greatly dominates the noise term in the SINR expression [START_REF] Guthy | Low-complexity linear zero-forcing for the MIMO broadcast channel[END_REF].

SLNR precoder creates a balance between the two extremes, i.e. maximizing the desired signal power and minimizing the interference generated at other users, by maximizing the ratio between the desired signal power and the interference power that leaks to non-intended users plus the noise power at the desired user [START_REF] Bjornson | Cooperative multicell precoding: Rate region characterization and distributed strategies with instantaneous and statistical CSI[END_REF]. The principle of SLNR precoding has been used by different authors in literature and was first used by the authors in [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF]. An equivalent metric, known as the signal-to-generating-interference-plus-noise-ratio (SGINR), is used in [START_REF] Lee | Interference-aware decentralized precoding for multicell MIMO TDD systems[END_REF]. The maximization of the SLNR or SGINR is a generalized eigenvalue problem, and hence also named as the generalized eigenvalue based beamformer in [START_REF] Stridh | System evaluation of optimal downlink beamforming with congestion control in wireless communication[END_REF]. This precoder combines the benefit of MRT at low SNR with the benefit of ZF at high SNR. Furthermore, SLNR always exists while ZF is only possible under certain conditions. Fig. 3.6 illustrates the directions of MRT, ZF and SLNR precoding vectors. As illustrated in the figure, precoding vector for MRT follows the direction of the channel vector of the desired user, ZF precoding vector is orthogonal to the channel of non-intended users, and SLNR balances between these extremes and moves between them depending on the SNR. In Chapter 5, performance of the SLNR precoder in terms of EE-ASE tradeoff is compared with ZF considering a PPP network, and better performance is achieved for the SLNR precoder. However, the non-intended users for the precoders are not the same. While ZF considers only the intra-cell interference, the SLNR precoder takes into account leakage to the active users in the other cells also.

Energy efficient research approaches

Since EE is an important research topic, several researchers have proposed many solutions to design energy efficient networks. We present a brief survey of these methods in a three-layer stack where lower layers serve the upper layers to increase their attainable EE. The layers from the bottom are component layer, transmission techniques and radio resource management layer, and the cellular architecture layer as illustrated in Fig. in [START_REF] Correia | Challenges and enabling technologies for energy aware mobile radio networks[END_REF]. Among others, amplifiers with different architectures and features, e.g. class J amplifier, switched mode power amplifier, etc., are proposed in literature which have higher efficiencies compared to the conventional Doherty amplifier [START_REF] Han | Green radio: radio techniques to enable energy-efficient wireless networks[END_REF]. Authors in [START_REF] Claussen | Effects of joint macrocell and residential picocell deployment on the network energy efficiency[END_REF][START_REF] Han | Green radio: radio techniques to enable energy-efficient wireless networks[END_REF] have proposed a new BS architecture with the transceivers decoupled from the BB unit and mounted at the same physical location as the antenna element. Compared to the traditional architecture, the new one simplifies the installation, reduces the feeder loss and energy required for cooling due to more fresh air cooling. Switching off the transceivers when there is no traffic is a promising technique to improve EE [START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF], as considered in Chapter 5.

Transmission techniques and radio resource management layer

Cellular access networks have been traditionally designed targeting performance maximization at full load. However, the daily traffic shows high variations between peak and off-peak values with long periods of low loads. Moreover, only 20% of BSs carries 80% of traffic even during the busy hour resting the network under-utilized most of the time in most of the space [START_REF] Blume | Energy savings in mobile networks based on adaptation to traffic statistics[END_REF]. As Figure 3.9 -Daily data traffic profile for cellular systems in Europe [START_REF] Imran | Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF] illustrated in Fig. 3.9, a large difference exists between busy and quiet hours in the daily data traffic profile, i.e. percentage of active mobile broadband subscribers, in Europe. Moreover, only 16% of the mobile broadband subscribers are active even in peak hours. The large spatial and temporal traffic variation would not be issues if the power consumption of BSs was proportional to their load. But this is not the case due to the large static circuit power consumption into BS. The energy consumption of cellular networks can be reduced by adapting transmission techniques and resource allocation schemes to the load.

Unlike the conventional radio resource management (RRM) techniques aiming at achieving higher data rates, energy aware resource allocation schemes target to utilize the resources of the network, i.e. bandwidth, time, space, power, etc., to maximize EE without compromising the QoS. The insights on relation among these resources guide practical system designs towards a green evolution. Relation among the resources is often a tradeoff, i.e. one cannot be improved without reducing other. The authors in [START_REF] Chen | Fundamental trade-offs on green wireless networks[END_REF] (MU-MIMO) [START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF], as it has been summarized in Fig. 3. [START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF]. Despite the fact that these techniques have potential to increase the data rate, their EE is still an open issue because of the increase of circuit energy consumption [START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF]. EE of MIMO for short-range transmission has been found to be lower than single antenna when static circuit power consumption is considered by both and none of them adopt adaptive modulation techniques [START_REF] Cui | Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks[END_REF][START_REF] Liu | Energy efficiency of MIMO transmissions in wireless sensor networks with diversity and multiplexing gains[END_REF][START_REF] Li | Application of STBC-encoded cooperative transmissions in wireless sensor networks[END_REF][START_REF] Bougard | Smartmimo: Energy-aware adaptive mimo-ofdm radio link control for wireless local area networks[END_REF]. However, a balance between the energy used for transmission and circuit energy consumption can be achieved adapting the modulation order resulting in higher EE for MIMO [START_REF] Miao | Cross-layer optimization for energy-efficient wireless communications: a survey[END_REF]. Adapting the number of antennas to the traffic variation can further increase EE [START_REF] Liu | Energy efficiency of MIMO transmissions in wireless sensor networks with diversity and multiplexing gains[END_REF][START_REF] Bougard | Smartmimo: Energy-aware adaptive mimo-ofdm radio link control for wireless local area networks[END_REF].

Andrews et al. have studied EE in a PPP cellular network considering the ZF precoder and

shown that EE is an increasing function of the number of transmit antennas only when the different components of the BS power consumption satisfy certain conditions [START_REF] Li | Energy efficiency analysis of small cell networks[END_REF]. However, ZF precoder performs worse than the SLNR precoder in terms of EE-ASE tradeoff [START_REF] Alam | Energy Efficiency-Area Spectral Efficiency Tradeoff in PPP Network with SLNR Precoder[END_REF]. EE of CoMP is potentially better compared to the conventional single BS transmission scheme due to the spatial diversity induced by coordinated BSs [START_REF] Cao | Energy saving performance comparison of coordinated multi-point transmission and wireless relaying[END_REF]. Among other works, tradeoff between number of retransmissions and longer packets with forward error correction (FEC) is evaluated, and an energy-efficient algorithm combining FEC and automatic repeat request scheme is proposed in [START_REF] Lettieri | A QoS-aware, energy-efficient wireless node architecture[END_REF]. Furthermore, the authors in [START_REF] Miao | Energy efficient design in wireless OFDMA[END_REF] have shown how OFDM can improve EE when combined with adaptive modulation techniques.

Cellular architecture layer

To reduce the consumption of circuit energy, cellular networks should be dimensioned with as few nodes as possible while maintaining the desired coverage, capacity and other QoSs. Wireless access network is dimensioned for peak hour traffic as mentioned earlier, and deployment of new architectures, e.g. small cells, relays, repeaters, etc., has been recently considered as a way to meet the high data rate requirement. Although they save transmit power due to lower path loss, the circuit energy consumption is increased [START_REF] Gruber | EARTH-energy aware radio and network technologies[END_REF][START_REF] Maaz | Energy Minimization in HARQ-I Relay-Assisted Networks with Delay-limited Users[END_REF]. The authors of [START_REF] Ericson | Total network base station energy cost vs. deployment[END_REF] studied the tradeoff between transmit power and circuit power consumption by evaluating the area power consumption of different deployment scenarios for a minimum received power at the cell edge. They concluded that large cell deployments are more energy efficient than the small cell deployments due to the large circuit power consumption by the latter.

Since mobile traffic shows significant variations in spatial and temporal domain leading to significant energy loss in low load scenarios, one potential approach to save energy is to adapt the network to the dynamic load variation via switching off BSs and cell zooming, as investigated in [START_REF] De Domenico | Enabling green cellular networks: A survey and outlook[END_REF][START_REF] Niu | TANGO: traffic-aware network planning and green operation[END_REF][START_REF] Oh | Dynamic base station switching-on/off strategies for green cellular networks[END_REF][START_REF] Ashraf | Sleep mode techniques for small cell deployments[END_REF]. Cell zooming is a technique in which a cell under heavy load reduces its size Figure 3.11 -Heterogeneous network deployment with relay for coverage extension [START_REF] Correia | Challenges and enabling technologies for energy aware mobile radio networks[END_REF] through power control and the users are handed off to the neighbouring cells [START_REF] Niu | Cell zooming for cost-efficient green cellular networks[END_REF]. Deployment of heterogeneous network of small cells with overlay macro cell, as illustrated in Fig. 3.11, can save a huge amount of energy when combined with switching off BSs and cell zooming techniques [START_REF] Badic | Energy efficient radio access architectures for green radio: large versus small cell size deployment[END_REF][START_REF] Le | Possible power-saving gains by dividing a cell into tiers of smaller cells[END_REF]. Macro cell can provide a low bit rate but high coverage to users, while BSs with smaller cell sizes can be utilized to provide high bandwidth connections when needed. Lower amount of transmit power is required by the small cells to transmit at shorter propagation distances.

Besides, the small cells can be switched off during low traffic period leading to a huge energy saving. EE of different femto cell deployments has been studied in [START_REF] Khirallah | Energy and cost impacts of relay and femtocell deployments in long-term-evolution advanced[END_REF][START_REF] Calin | On femto deployment architectures and macrocell offloading benefits in joint macro-femto deployments[END_REF][START_REF] Dufková | Energy consumption comparison between macro-micro and public femto deployment in a plausible LTE network[END_REF][START_REF] Claussen | Effects of joint macrocell and residential picocell deployment on the network energy efficiency[END_REF][START_REF] Wang | System performance analysis of OFDMAbased femtocell networks[END_REF] and joint macro-femto cell deployment has been shown to consume more energy in contrast to traditional macro-only network for medium and high femtocell densities [START_REF] Khirallah | Energy and cost impacts of relay and femtocell deployments in long-term-evolution advanced[END_REF]. Relays provide coverage for areas that would otherwise be shaded or that would necessitate the use of extremely strong transmit powers. Enhanced concepts include relays that are switched on only when users are present in the area handled by them. EE of different relay schemes, i.e. amplify and forward, decode and forward, block Markov coding, are compared with that of direct transmission in [START_REF] Yao | On energy efficiency and optimum resource allocation of relay transmissions in the low-power regime[END_REF].

All these layers can be benefited through employing cognitive radio (CR) techniques, which can adapt the transmission parameters, resource allocation and also network layout to different channel and traffic conditions [START_REF] Amanna | Green Communications[END_REF]. CR can optimizes the usage of available spectrum causing less required power to achieve the same data rate, which can lead to better EE-SE tradeoff [START_REF] Hasan | Green cellular networks: A survey, some research issues and challenges[END_REF]. CR can also be used to mitigate interference resulting in energy saving. Several applications of CR for green radio, e.g. tuning the direction of power radiated from sector antenna, have been given in [START_REF] Palicot | Cognitive radio: an enabling technology for the green radio communications concept[END_REF].

International research projects

EE-SE tradeoff in AWGN channel

Fundamental EE-SE tradeoff without static power consumption

According to the Shannon's formula, the achievable transmission rate in b/s for AWGN channel is

R = w log 2 1 + P 0k w N 0 , (3.19)
where w is the channel bandwidth in Hz, P 0k is the power received by the typical k-th user from the typical 0-th transmitter expressed in W, and N 0 is the noise spectral density in W/Hz.

η SE = R w = log 2 1 + P t w N 0 . ( 3
η EE = R P t = η SE (2 η SE -1)N 0 . ( 3.21)
The EE-SE tradeoff for an AWGN channel considering only the transmit power is shown in Fig. 3.12 for N 0 = -168.

P total = aP t + b. ( 3.22)
When path loss is considered, the power received by the typical k-th user at distance r 0k from the typical 0-th transmitter is

P 0k = P t r 0k r 0 -α , ( 3.23)
where r α 0 is the propagation constant equal to L in (3.18) and random channel power gain X is 1. Using (3.23), (3.19) and (3.20), SE in b/s/Hz can be written as

η SE = log 2   1 + P t r 0k r 0 -α w N 0    . ( 3.24)
Using (3.22) and inserting the value of P t from (3.24), EE in b/J can be written as This large linear part is due to the large amount of energy spent in static power consumption, i.e. b, and is defined as circuit power dominated region. EE starts decreasing when SE is further increased resulting in transmission power dominated region. demonstrate that increasing the PA factor decreases EE implying the fact that PA is inefficient when a is large. Total power consumption increases as a becomes higher, but SE does not depend on it resulting in lower EE when a is increased. Besides, the transmission power dominated region starts for a smaller value of P t when a is large. This causes the optimal point to move towards a lower SE when a is higher.

η EE = wη SE aP t + b = wη SE a (2 η SE -1) w N 0 r 0k r 0 α + b . ( 3
Fig. 3.13b illustrates the EE-SE tradeoff for b ∈ {100W, 400W, 712W} when a is 14. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. The results demonstrate that increasing b decreases EE. This is due to the fact that the total power consumption increases as static power consumption becomes higher, but SE remains unchanged resulting in lower EE when b is increased. However, the circuit power dominates over a greater value of transmit power causing the optimal point to move towards a higher SE when b increases.

∂η EE ∂η SE = ∂ ∂η SE    wη SE a (2 η SE -1) w N 0 r 0k r 0 α + b    = w a 2 η SE -1 w N 0 r 0k r 0 α + b -1 +wη SE (-1) a 2 η SE -1 w N 0 r 0k r 0 α + b -2 aw N 0 r 0k r 0 α 2 η SE log e 2. ( 3
w a 2 η * SE -1 w N 0 r 0k r 0 α + b -1 = η * SE a 2 η * SE -1 w N 0 r 0k r 0 α + b -2 aw 2 N 0 r 0k r 0 α 2 η * SE log e 2. ( 3.27)
Dividing both sides of (3.27) by aw 2 N 0

r 0k r 0 α a 2 η * SE -1 w N 0 r 0k r 0 α + b -2
= 0, we can write

2 η * SE -1 + b aw N 0 r 0k r 0 α = η * SE 2 η * SE log e 2 =⇒ b aw N 0 r 0k r 0 α -1 = η * SE log e 2 -1 2 η * SE . ( 3.28)
The negative branch of the Lambert function 1 , denoted as W -1 , ranges from W -1 (-1/2) = -1 to

W -1 (0 -) = -∞.
On the other hand, positive branch of the Lambert function, denoted as W 0 (z), is ≥ -1 for z > 0 -. Solving (3.28) for η * SE , we obtain

η * SE = log 2 e   1 + W 0    1 e    b aw N 0 r 0k r 0 α -1          . (3. 29
)
W 0 is considered due to the fact that SE cannot be negative. With (3.25), maximum EE is

η * EE = wη * SE a(2 η * SE -1)w N 0 r 0k r 0 α + b . ( 3.30)
Using (3.24), the transmit power corresponding to maximum EE can be expressed as

P * t = (2 η * SE -1)w N 0 r 0k r 0 α . ( 3

Conclusion

In this chapter, we have first provided the state of the art performance metrics, cellular network and BS power consumption models. The general propagation model for the whole thesis is also described. Moreover, we illustrated the linear precoding techniques applied at multi-antenna BSs. A detailed survey of the research approaches and projects that aim at improving EE was also given. Finally, we have revisited the characteristic of EE-SE relation in AWGN channel first by considering only the transmit power and then using the linear power consumption model.

It is observed that EE is always decreasing with SE when only the transmit power is accounted.

However, there is also a large linear part where EE increases with SE when the static circuit power is taken into account. The results also illustrate that both optimal EE and SE decrease as the PA factor increases. Besides, although higher optimal SE is achieved when the static power consumption is increased, the optimal EE becomes lower.

.

Chapter 4 EE-SE tradeoff in a hexagonal cellular network 4.1 Introduction

This chapter deals with the EE-SE tradeoff in a downlink regular hexagonal cellular network. 1The CDF of SINR is higher in a regular network compared to PPP network and hence, it is better for an operator to be as close as possible to the hexagonal model [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF]. We assume a TDMA, FDMA or orthogonal FDMA (OFDMA) type of access leading to a perfect orthogonality between radio resources of a cell. Hence, there is no intra-cell interference. On the other hand, the decrease of signal power with distance allows to reuse the same frequency at spatially separated locations.

Reusing the frequency bandwidth over relatively small geographical areas is necessary for the efficient use of the available bandwidth, which potentially increases SE as well as EE. But the major issue in this case is CCI among the co-channel cells.

When the cell size and the BS transmit power are the same for all BSs, then CCI depends only on the ratio between the frequency reuse distance and cell radius, which is equal to 3K as obtained from (3.13). If K is increased, then the effective distance between the co-channel cells increases and CCI decreases. However, since the number of cells in a cluster increases, available bandwidth in a cell decreases. This has a negative effect on SE as well as on EE. It is not clear whether the increase of available bandwidth or the increase of interference has the dominating effect on the EE-SE tradeoff. The aim of this chapter is to study the effect of different frequency reuse factors, i.e. K , on the EE-SE tradeoff both with and without shadowing when the users are at different locations in the cell.

To describe the distribution of the interfering BSs, Kelif et al. have proposed a model, known as fluid model [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF]. A closed form expression of ISR is derived using the model allowing a tractable derivation of outage probability in hexagonal cellular networks. The expression for ISR provided by Kelif et al. is limited to the case where the system bandwidth is reused in each cell.

We obtain a parametric expression of ISR for various reuse cases with a curve fitting approach.

The parametric expressions of ISR allow us to derive the EE-SE tradeoff for several frequency reuse factors when non negligible static circuitry power consumption is assumed and path loss dependent channel modelling is considered. On the other hand, although SE is generally taken as equal to the Shannon capacity when only the path loss is taken into account, the capacity in Shannon's sense is not defined when shadowing is considered. We propose the -EE-SE tradeoff when shadowing is considered. The definition depends on the outage capacity achievable and its corresponding EE, and is particularly useful when long-term fading is present. We determine the CDF of SINR allowing to study the EE-SE tradeoff when shadowing is taken into account.

The results show that when static power consumption into the BS is considered, the EE-SE tradeoff has a large linear part before a sharp decreasing. Our results also illustrate that using the whole system bandwidth in each cell optimizes the EE-SE tradeoff only when a user is close to the BS. Moreover, better EE-SE tradeoff in the shadowing environment is achieved for higher implying higher outage in the system, and hence, should be set according to the desired QoS.

System model

A downlink regular hexagonal cellular network with different frequency reuse factors is considered. BSs are placed at the center of the cells and both BSs and users are equipped with a single antenna. Users are connected to the nearest BS. Transmit power of all BSs is the same that makes the network homogeneous causing the EE-SE study of a single cell valid for all the other cells in the network. The linear power consumption model introduced in (3. [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]) is considered for BSs :

P total = aP t + b. ( 4.1)
We assume that the parameters of the power consumption model corresponds to a resource block allocated to a user.

The power received by the typical k-th user at distance r i k from BS i ≥ 0 considering shadow-ing along with path loss is

P i k = P t r i k r 0 -α Y i k , (4.2)
where r α 0 is the propagation constant given by L in (3.18) which has the same value as for the AWGN channel, Y i k , given by X in (3.18), is a log-normal (LN) r.v. characterizing shadowing denoted as Y i k ∼ L N (0, σ), with zero mean and σ as standard deviation in dB. The factor

P t r i k r 0 -α
is the received power related with path loss and represents the average received power in shadowing condition. The user receives interference from all BSs having the same frequency sub-band as the connected BS. The connected BS is set for i = 0 without loss of generality, and the power received by the k-th user at distance r 0k from the connected BS is

P 0k = P t r H v r 0 -α Y 0k , (4.3)
where v = r 0k /r H is the normalized distance with 0 ≤ v ≤ 1. Let I = i ≥1 P i k the interference power aggregated from BS i (i ≥ 1) to the k-th user and σ 2 n the noise power on the total bandwidth available to the user. Then, the SINR of the user is

γ = P 0k σ 2 n + I = 1 w K N 0 P 0k + I P 0k , (4.4)
where w is the bandwidth of a resource block allocated to a user when frequency reuse factor is one and N 0 is the noise spectral density.

The ISR with shadowing depends on the r.v. Y i k and can be written as

f s (v, α) = I P 0k = i ≥1 r i k r H v -α Y i k Y 0k . ( 4.5)
We also study the EE-SE tradeoff without shadowing consideration, which is equivalent to work with the average received power in shadowing environment. The ISR without shadowing consideration, noted as f (v, α), is simply [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF] f (v, α)

= I P 0k = i ≥1 r i k r H v -α . ( 4.6)

EE-SE tradeoff without shadowing

Considering a frequency reuse factor K , the available frequency for a user is w/K and the Shannon SE in b/s/Hz is

η SE = 1 K log 2 1 + γ . ( 4.7)
Inserting the value of γ from (4.4), and using (4.6), (4.7) can be written as

η SE = 1 K log 2   1 + 1 w K N 0 P 0k + f (v, α)   . ( 4.8)
When shadowing is not considered, (4.3) can be written as

P 0k = P t r H v r 0 -α
. Inserting the value of P 0k in (4.8), we get

1 w K N 0 P t r H v ro -α + f (v, α) = 2 K η SE -1 ⇔ P t r H v r o -α w K N 0 = 1 1 2 K η SE -1 -f (v, α) ⇔ P t = r H v r o α w K N 0 (2 K η SE -1) 1 -f (v, α)(2 K η SE -1) . ( 4.9)
Since EE is defined as

η EE = wη SE aP t + b , ( 4.10)
inserting the value of P t from (4.9), (4.10) can be expressed as

η EE = wη SE a r H v r 0 α w K N 0( 2 K η SE -1) 1-f (v,α)(2 K η SE -1) + b . ( 4.11)

Parametric expression of f (v, α)

A pragmatic approach is adopted and a parametric expression of f (v, α) given in

f (v, α) = e i + j ≤5 A i , j (log e v) i α j . ( 4

.12)

The values of the coefficients A i , j for different K are provided in Table 4. [START_REF] Haenggi | Interference in large wireless networks[END_REF]. The simulated and the fitted values match very well, which can be observed in Fig. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF]. [START_REF] Haenggi | Interference in large wireless networks[END_REF]. The maximum value of the relative error, considering all the simulations, is 0.75% depicting a perfect match between the simulation and the parametric expression. It can also be observed that the ISR is higher for lower frequency reuse factor. Moreover, the ISR increases as the user moves away from the BS, and this is extremely significant for K =1, since it experiences more interference. The tractable closed-form of this semi-analytical expression enables us to avoid performing Monte Carlo simulations to study the EE-SE tradeoff for different frequency reuse factors.

Coefficients K = 1 K = 3 K = 4 K = 7 A 5,

Numerical results on EE-SE tradeoff without shadowing

In this subsection, we provide some numerical results to illustrate the effect of the frequency reuse factor on the EE-SE tradeoff. The parameters for all the plots remaining in this chapter are set according to Table 4.2 unless otherwise mentioned. Thus, there exists an optimal EE-SE tradeoff, which has to be obtained. However, since the network is homogeneous, i.e. all BSs have the same transmit power, and interference limited, SE converges towards the limit 1 K log 2 1 + 1 ISR with a small increase of P t after the optimal point is reached. This results in sharp decrease of EE when P t is further increased which is not observed in case of AWGN channel.

While the optimal frequency reuse factor in Fig. 4.

2 is K = 3 for v = 0.85, it is 1 for v = 0.5.
This behavior is based on the characteristic of (4.7). Since SE is inversely proportional to K , it increases for lower K resulting in higher EE for a particular P t . On the other hand, noise power increases for lower K because of higher available bandwidth. Moreover, ISR is also higher for lower K as can be observed from Fig. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF].1. Therefore, the SINR is decreasing for the given transmit power as K is decreasing. The ISR decrease is extremely significant for K = 1 compared to the other values of K when v = 0.85 as can be seen in Fig. 4. 1. This causes K = 3 to be the best choice at this distance. For a particular K , the decrease of SINR, due to high ISR as the user moves away from BS, results in lower SE and EE. We determine EE, SE and P t corresponding to the optimal point in the next section.

Determination of EE, SE and P t corresponding to optimal point

Derivative of EE in (4.11) w.r.t. SE is

∂η EE ∂η SE = ∂ ∂η SE wη SE a r H v r 0 α w K N 0( 2 K η SE -1) 1-f (v,α)(2 K η SE -1) + b = w a r H v r 0 α w K N 0( 2 K η SE -1) 1-f (v,α)(2 K η SE -1) + b -η SE a r H v r 0 α w K N 0 K log e 2 2 K η SE (1-f (v,α)(2 K η SE -1)) 2 a r H v r 0 α w K N 0( 2 K η SE -1) 1-f (v,α)(2 K η SE -1) + b 2 . ( 4
  a r H v r 0 α w K N 0 y 1-f (v,α)y + b   -1 K log 2 (y + 1) a r H v r 0 α w K N 0 K log e 2 (y+1) 1-f (v,α)y 2 a r H v r 0 α w K N 0 y 1-f (v,α)y + b 2 = 0 ⇔ a r H v r 0 α w K N 0 y 1 -f (v, α)y + b - 1 K log 2 (y + 1) a r H v r 0 α w K N 0 K log e 2 (y + 1) 1 -f (v, α)y 2 = 0 ⇔ y 1 -f (v, α)y -log 2 (y + 1) log e 2 (y + 1) 1 -f (v, α)y 2 = - b a r H v r 0 α w K N 0 . ( 4.14)
Multiplying both sides of (4.14

) by 1 -f (v, α)y = 0 , y - log 2 (y + 1)(y + 1) log e 2 1 -f (v, α)y = - b 1 -f (v, α)y a r H v r 0 α w K N 0 . ( 4.15)
There is no closed form expression for y. We obtained η * SE by solving (4.15) numerically. Using (4.9), the transmit power corresponding to η * SE can be written as

P * t = r H v r 0 α w K N 0 (2 K η * SE -1) 1 -f (v, α)(2 K η * SE -1) . ( 4.16)
Using (4.11), η * E E can be written as

η * E E = wη * SE a r H v r 0 α w K N 0 2 K η * SE -1 1-f (v,α) 2 K η * SE -1 + b . ( 4.17)
Fig. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF].3 illustrates the variation of η * SE , η * E E and P * t with frequency reuse factor. The results show that the optimal frequency reuse factor in terms of η * SE and η * E E is 1 for v = 0.5, while it is 3 for v = 0.85 and v = 1. Besides, K = 1 is the worst at v = 1, while this is not the case when v is equal to 0.85. This is due to fact that the increase of ISR becomes extremely significant for K = 1 when the user moves closer to the cell edge. On the other hand, ISR becomes lower as K is increased and requires higher P * t for the convergence of SE towards the limit, as it can be ). This results in higher P * t as K is increased.

-EE-SE tradeoff with long term shadowing

Since SE is an increasing function of γ, inserting (4.7) into (3.2), -SE can be expressed as

η ( ) SE = sup E : P 1 K log 2 1 + γ < E ≤ = sup E : P γ < 2 K E -1 ≤ = sup E : F γ 2 K E -1 ≤ , ( 4.18)
where F γ denotes the CDF of γ. Outage EE or -EE is then defined using (4.10) as

η ( ) EE = wη ( ) SE aP t + b . ( 4.19)
In order to characterize the EE-SE tradeoff, CDF of γ is needed to be computed first.

CDF of γ

The SINR of the k-th user at distance v from BS with shadowing consideration can be expressed using (4.4), (4.3), and (4.5) as

γ =    w K N 0 P t r H v r 0 -α Y 0k + f s (v, α)    -1 . ( 4.20)
Each term of the sum in the numerator of f s (v, α) is a LN r.v. Therefore, we can write log e (r

-α i k Y i k ) ∼ N (d µ i k , d 2 σ 2 )
, where d = log e 10/10 and µ i k = 1

d log e r -α i k . Numerator of f s (v, α) is sum of LN r.vs., and can be approximated by a LN r.v. according to the Fenton-Wilkinson method [START_REF] Fenton | The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems[END_REF].

Therefore, we can write log

e i ≥1 r -α i k Y i k ∼ N d µ F n , d 2 σ 2
F n , where [START_REF] Fenton | The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems[END_REF]

d µ F n = log e i ≥1 e d µ i k + d 2 σ 2 2 - d 2 σ 2 F n 2 = log e i ≥1 r -α i k + d 2 σ 2 2 - d 2 σ 2 F n 2 (4.21) 91 4.4.
-EE-SE tradeoff with long term shadowing and

d 2 σ 2 F n = log e e d 2 σ 2 -1 i ≥1 e 2d µ i k i ≥1 e d µ i k 2 + 1 = log e e d 2 σ 2 -1 i ≥1 r -2α i k i ≥1 r -α i k 2 + 1 . ( 4.22)
The ratio of two LN r.v. is a LN r.v. with mean and variance equal to the difference of the means and sum of the variances respectively. Therefore, f s (v, α) is a LN r.v. and is noted as

L N (µ F , σ F ) with d µ F = d µ F n -log e r -α 0k (4.23)
and

d 2 σ 2 F = d 2 σ 2 F n + d 2 σ 2 . ( 4.24)
Inserting (4.21) into (4.23), we get

d µ F = log e i ≥1 r -α i k + d 2 σ 2 2 - d 2 σ 2 F n 2 -log e r -α 0k . (4.25)
Inserting r 0k = r H v and (4.6) into (4.25), we can write

µ F = 1 d log e f (v, α) + d 2 σ 2 2 - d 2 σ 2 F n 2 . ( 4
µ F = 1 d log e f (v, α) + d 2 σ 2 2 - 1 2 log e e d 2 σ 2 -1 G(v, α) + 1 , (4.27)
where

G(v, α) = i ≥1 r -2α i k i ≥1 r -α i k 2 .
Inserting (4.22) into (4.24), we obtain

σ F = 1 d 2 log e e d 2 σ 2 -1 G(v, α) + 1 + σ 2 . ( 4.28)
Parametric expressions of G(v, α) are obtained for various frequency reuse factors in the same way as was done for f (v, α) and they have the same general form as follows:

G(v, α) = e i + j ≤5 B i , j (log e v) i α j . ( 4.29)
The values of the coefficients B i , j for different K are provided in Table 4. [START_REF] Suarez | An overview and classification of research approaches in green wireless networks[END_REF]. The maximum value of the relative error considering all the simulations is about 3%. This small relative error indicates a good match between the simulations and the parametric expressions, which can also be observed in Fig. We approximate

Table 4.3 -Coefficients of G(v, α) Coefficients K = 1 K = 3 K = 4 K = 7 B 5
w K N 0 P t r H v r 0 -α Y 0k ∼ L N (µ N , σ N ) with µ N = 10 log 10 w K N 0 -10 log 10 P t r H v r 0 -α σ N = σ (4.30)
Since the sum of two LN r.v. can be approximated by a LN r.v. and the inverse of a LN r.v. is also LN [START_REF] Cheikh | SIR Distribution Analysis in Cellular Networks Considering the Joint Impact of Path-Loss, Shadowing and Fast Fading[END_REF], we can write γ ∼ L N (µ γ , σ γ ). If µ N L , σ N L , µ F L and σ F L are the parameters in linear units corresponding to µ N , σ N , µ F and σ F respectively, then µ γ and σ γ can be written as [START_REF] Fenton | The Sum of Log-Normal Probability Distributions in Scatter Transmission Systems[END_REF]

       µ γ = 1 2d log e σ N L 2 +σ F L 2 (µNL+µFL) 2 + 1 -1 d log e µ N L + µ F L σ γ = 1 d log e σ N L 2 +σ F L 2 (µNL+µFL) 2 + 1 (4.31)
where

       µ N L = e d µ N +d 2 σ 2 /2 , µ F L = e d µ F +d 2 σ F 2 /2 σ 2 N L = e 2d µ N +d 2 σ 2 e d 2 σ 2 -1 σ 2 F L = e 2d µ F +d 2 σ F 2 e d 2 σ F 2 -1 (4.32)
The CDF of γ can now be expressed as

F γ (x) = 1 -P(γ > x) =
F γ (x) = 1 -Q 10 log 10 (x) -µ γ σ γ , ( 4.34)
where Q(x) is the Marcuum function that provides the CCDF of a standard normal r.v. X . Fig. 4.5 validates the approximations regarding CDF of SINR. In this figure, CDF of γ is plotted from Monte Carlo simulations performed for v = 0.5 and σ = 5 dB, and compared to the analytical approximation (4.34). An excellent match can be observed validating the approximations, and the maximum relative error is less than 2%.

η ( ) SE = sup E : 1 -Q 10 log 10 2 K E -1 -µ γ σ γ ≤ . (4.35) Solving 1 -Q 10 log 10 (2 K E -1)-µ γ σ γ = yields η ( ) SE providing E = 1 K log 2 10 σγQ -1 (1-)+µγ 10 + 1 , (4.36)
where Q -1 is the inverse of the Q function and E represents the -SE. Afterwards, η ( ) EE can be obtained by inserting the value of -SE into (4.19).

Numerical results on EE-SE tradeoff with shadowing

In this subsection, we provide some numerical results to illustrate the effect of frequency reuse factors and on the EE-SE tradeoff when shadowing is considered. Throughout this subsection, we set σ = 5 dB. Fig. 4.6 depicts the effects of K on the EE-SE tradeoff with shadowing. The EE-SE tradeoff has a large linear part before a sharp decreasing when η SE is increasing as also observed in the case without shadowing. For a particular P t and K , η SE increases as increases leading to higher values of η EE . However, high causes lot of outage in the system, which is not necessarily desirable, and hence the value should be set according to a certain QoS to meet. From Fig. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF].6a, it can be observed that K = 3 allows the highest EE and SE when v = 0.5 due to the significant SINR decrease in shadowing environment when K = 1. When the user moves further from the BS, e.g. v = 0.85, K = 7 achieves the best performance as observed in Fig. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF].6b. However, K = 1 performs the worst due to the extreme increase of ISR at this distance.

×10 4 K = 1 K = 3 K = 4 K = 7 (a) v = 0.

Conclusion

In this chapter, we have investigated the EE-SE tradeoff in homogeneous hexagonal interferencelimited network using a practical power consumption model considering frequency reuse factors both with and without shadowing. We obtained a parametric expression of f-parameter by curve fitting to analyze the EE-SE tradeoff without shadowing. Then we proposed approximations for the CDF of SINR to evaluate the EE-SE tradeoff with shadowing, which are validated by Monte Carlo simulations. Our results showed that the EE-SE curves have a large linear part, due to the static power consumption, followed by a sharp decreasing EE, since the network is homogeneous and interference-limited. Moreover, a frequency reuse equal to 1 for regions close to the BS and higher reuse factors in region closer to the cell edge optimize the EE-SE tradeoff. However, K = 1 is the best only for higher values of , i.e. i.e. higher outage in the system, even at a moderate distance due to significant SINR decrease when shadowing is taken into account. Moreover, better -EE-SE is achieved as increases, but the value should be set according to the QoS to meet. However, it should be noted that we do not consider feedback on channel state from the users to the BSs, and hence the BSs can not adapt their transmission to the channel state.

The BSs are assumed to transmit in best effort leading to equity in user access to the resources.

However, in reality users getting low throughput request again for resources since they are less served. This highlights the importance of resource management for cell-edge users which we did not take into account. Resource management based on the real traffic condition and user position will lead to different power allocation for the desired signal and the interfering signal. In this chapter, results are obtained assuming that the desired power and the interference power are same. The results may be different for different power allocation.

Chapter 5 EE-ASE tradeoff in a PPP network

Introduction

Higher ASE can be achieved when the communication between BS and users is in the same time-frequency resources [START_REF] Ngo | Energy and spectral efficiency of very large multiuser MIMO systems[END_REF]. Precoding must be used in downlink to mitigate the interference due to the communication in the same radio resources. Optimal precoder maximizing the network sum rate subject to a transmit power constraint is desired on the perspective of EE-ASE tradeoff. Achieving such precoder is a non-convex and non-trivial problem and therefore linear precoding techniques, e.g. MRT, ZF, SLNR, minimum mean square error (MMSE), are also of interest [START_REF] Nguyen | MMSE precoding for multiuser MISO downlink transmission with non-homogeneous user SNR conditions[END_REF][START_REF] Christensen | Weighted sum-rate maximization using weighted MMSE for MIMO-BC beamforming design[END_REF]. Although the ZF precoder nulls the intra-cell interference [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF][START_REF] Gupta | Downlink coverage probability in MIMO HetNets with flexible cell selection[END_REF][START_REF] Dhillon | Downlink MIMO HetNets: Modeling, ordering results and performance analysis[END_REF], it is not designed to limit the inter-cell interference. Moreover, the ZF precoder imposes a restriction on the minimum number of transmit antennas at BS [START_REF] Cheng | A new SLNR-based linear precoding for downlink multiuser multi-stream MIMO systems[END_REF]. A generalized MMSE precoder for downlink multicell MU-MISO systems considering different average SNRs of users has been proposed in [START_REF] Sohn | Generalized MMSE beamforming for multicell MIMO systems with random user geometry and channel feedback latency[END_REF]. The authors have shown that MMSE achieves higher throughput compared to other linear precoders by performing Monte Carlo simulations. However, evaluating the performance of MMSE precoder is generally difficult to handle theoretically.

Authors in [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF] have proposed SLNR as an optimization metric for designing precoder. SLNR does not impose any restriction on the number of BS antennas in contrast to ZF solution [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF] and moreover noise is considered. However, the previous works based on SLNR do not consider a multi-cell environment and also ignore the network geometry while computing the SLNR precoder solution [START_REF] Sadek | A leakage-based precoding scheme for downlink multi-user MIMO channels[END_REF][START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF][START_REF] Cheng | A new SLNR-based linear precoding for downlink multiuser multi-stream MIMO systems[END_REF][START_REF] Tarighat | A multi user beamforming scheme for downlink MIMO channels based on maximizing signal-to-leakage ratios[END_REF]. In this chapter, we investigate the performance of the SLNR precoder for cellular network taking into account non-homogeneous average SNR due to the large scale fading, i.e. path loss, and also the interference created on the other-cell users. The MMSE and SLNR precoders have been proved to be equivalent under symmetric scenario, i.e. all channels between BS and users have the same gain [START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF][START_REF] Björnson | Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure[END_REF]. Although the network geometry is non-symmetric in our work, we consider the SLNR precoder for its close-to-optimal performance and simplicity.

Hexagonal cellular network model has been extensively used by both academia and industry for long time. This is a useful model to represent well planned BS deployments. However, the locations of the BSs in cellular network are not so regular, and hexagonal grid cannot capture the randomness of the positions of the BSs. Besides, the model is not tractable when MU-MISO with precoding is considered. Although the positions of the BSs are not completely random, it is more tractable to consider that the BSs are positioned at random. PPP can capture such randomness completely leveraging the techniques from stochastic geometry. Hence, we consider a spatial PPP network model in this chapter.

Stochastic geometry has been used for the performance analysis of cellular networks intensively during the past decade with the work of Baccelli, Andrews or Haenggi among others [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF][START_REF] Haenggi | Interference in large wireless networks[END_REF]. The authors of [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF] have used the stochastic geometry tool to study the relation between ASE and different system parameters, e.g. BS density, number of BS antennas, considering a ZF precoder. The same authors have studied the relation between EE and the system parameters in [START_REF] Li | Energy efficiency analysis of small cell networks[END_REF] considering MRT precoder. Most of the previous works dealing with stochastic geometry to analyze the network performances have considered either a MRT or ZF precoder because of tractability [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF][START_REF] Bjornson | Designing wireless broadband access for energy efficiency: Are small cells the only answer?[END_REF][START_REF] Gupta | Downlink coverage probability in MIMO HetNets with flexible cell selection[END_REF][START_REF] Dhillon | Downlink MIMO HetNets: Modeling, ordering results and performance analysis[END_REF].

In this chapter, we derive approximations for ASE and EE in a MU-MISO cellular network with random topology when SLNR precoder is used, which has never been considered in literature to the best of our knowledge. The result is achieved by combining some fundamental results from RMT to PPP and tightness of the approximations for ASE and EE are validated by performing Monte Carlo simulations for a wide range of system parameters. 1 We also compare the EE-ASE tradeoff of the SLNR precoder with ZF precoder, which is commonly used in PPP networks. 2Finally, based on the derived expressions of ASE and EE, we illustrate the effect of number of BS antennas and BS density, with constant user density and constant BS-user density ratio, on the ASE and EE when SLNR precoder is used in PPP cellular networks.

System model

A downlink MU-MISO cellular network is considered where BSs are equipped with M transmit antennas and users have a single receive antenna. BSs and users are modelled by two independent PPPs with density λ b and λ u respectively and users are connected to the nearest BS. Some BSs do not transmit any signal, and are called 'inactive', since they do not have any user to serve due to the independent locations of BSs and users. All BSs have the same transmit power, which is equally divided among the active users in the cell. The whole system bandwidth is allocated to each user, and hence users experience both intra and inter-cell interferences. The linear power consumption model, introduced earlier in (3.17), is considered for BSs:

P total = aP t + M P RF + c.
(5.1)

Rayleigh flat fading channels between BS and users are considered and perfect CSI is assumed to be available at each BS. The signal received by the k-th user from the typical 0-th BS is given by

y k = P t r -α 0k u 0 h H 0k w 0k x 0k + (i , j )∈A P t r -α i k u i h H i k w i j x i j + n k , (5.2)
where Following the approach in [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF], BSs are divided into subsets denoted as Φ u , with u ∈ [0, u max ],

A = {i , j } ∈ N 2 |i = 0 ∧ j = k ∨ i = 0, ∀ j , r i
and each BS in the subset Φ u serves u users. The maximum number of users served simultaneously in a cell is set to u max in order to control several factors, i.e. spatial multiplexing gain, interference, beamforming gain, etc., that determine the SDMA gain. The number of users in different cells is assumed to be independent, and each BS group Φ u follows a homogeneous PPP distribution with density λ b p N (u), where p N (u) is the probability mass function (PMF) of the number of active users in a cell. The calculation of p N (u) requires the exact size distribution of the Poisson-Voronoi typical cell, which has been given in [START_REF] Calka | Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional poisson-voronoi tessellation and a poisson line process[END_REF]. However, since the expression is challenging to compute numerically, a curve-fitted equation has been proposed in [START_REF] Ferenc | On the size distribution of Poisson Voronoi cells[END_REF]. Using this equation, PMF of the number of active users in a cell can be expressed as in [START_REF] Li | Performance analysis of SDMA in multicell wireless networks[END_REF]:

p N (u) =        µ µ Γ(u+µ)ρ -u Γ(µ)u! 1 ρ +µ u+µ 0 ≤ u ≤ u max -1 ∞ n=u max µ µ Γ(n+µ)ρ -n Γ(µ)n! 1 ρ +µ n+µ u = u max (5.3)
where n is the number of users available in a cell, µ = 3.5 is a constant obtained through data fitting [START_REF] Ferenc | On the size distribution of Poisson Voronoi cells[END_REF], ρ = λ b λ u is the BS-user density ratio, and Γ(•) is the gamma function. When n ≤ u max , n users are served by BS, while BS randomly chooses u max users to serve when n > u max . Due to the limitation of active users to u max , their locations become correlated, which is very challenging to handle. In order to make the problem tractable, the simplifying assumption that the set of active users is the sum of independent PPPs with density λ b up N (u) with u ∈ [1, u max ] is made.

Since the sum of independent PPPs is another PPP, the set of active users is considered as a PPP with density λ au = u max u=1 λ b uP N (u). The network is considered to be a circular disc of radius R a , and total number of active users in this network is N au which is a r.v. in a given area. However, we have lim

R a →∞ N au (R a ) = lim R a →∞ λ au πR 2
a when the network size grows to infinity. Power received by the typical k-th user at distance r 0k from the 0-th BS is

P 0k = P t u 0 r -α 0k h H 0k w 0k 2 , (5.4)
where the propagation constant L in (3.18) is one and h H 0k w 0k 2 is the random channel power gain given by X . By grouping the interfering BSs into subsets, the SINR of the k-th user can be written as

γ(u 0 ) = P t r -α 0k u 0 h H 0k w 0k 2 j =k P t r -α 0k u 0 h H 0k w 0 j 2 + u max u=1 i ∈Φ u \{0} u j =1 P t r -α i k u h H i k w i j 2 + σ 2 n . (5.5)
According to Slivnyak's theorem illustrated in 2. 1.5, (Φ u \ {0}) has the same statistics as Φ u . At denominator, the first and second terms refer to the intra and inter-cell interferences respectively.

SLNR and ZF precoding schemes

In this section, expressions of the SLNR and ZF precoding vectors are presented in the context of a MU-MISO cellular network with random topology.

SLNR precoder

The SLNR is defined as the ratio of the received signal power at the desired user to the interference created by the desired signal on the other terminals, also known as leakage, plus the noise power of the desired user. For user k, SLNR can be expressed as

SLNR = P t r -α 0k u 0 w H 0k h 0k h H 0k w 0k j =k P t r -α 0 j u 0 w H 0k h 0 j h H 0 j w 0k + σ 2 n = r -α 0k w H 0k h 0k h H 0k w 0k w H 0k H0k D0k HH 0k w 0k + σ 2 n u 0 P t , (5.6)
where

H0k = h 01 , • • • , h 0(k-1) , h 0(k+1) , • • • , h 0N au and D0k = diag r -α 01 ,• • •, r -α 0(k-1) , r -α 0(k+1) ,• • •, r -α 0N au
represent the concatenated fading channels and a square diagonal matrix filled by the path losses from the 0-th BS to the active users in the network, except the k-th user. With average power constraint, the SLNR maximization for user k is:

w 0k =arg max w 0k ∈C M ×1 w H 0k h 0k h H 0k w 0k w H 0k H0k D0k HH 0k + σ 2 n u 0 P t w 0k 2 I M w 0k (5.7) subject to E w 0k 2 = 1
where • represents the L 2 vector norm. When M → ∞, the optimization problem in (5.7) can be proved to be equivalent to

w 0k =arg max w 0k ∈C M ×1 w H 0k h 0k h H 0k w 0k w H 0k H0k D0k HH 0k + σ 2 n u 0 P t I M w 0k (5.8)
subject to E w 0k 2 = 1.

Indeed, defining

w 0k =[w 0k1 , w 0k2 , • • • , w 0kM ] T , we can write E w 0k 2 = E M i w 2 0ki = M i E w 2 0ki = M E w 2 0ki = 1, (5.9)
where the final step in (5.9) is obtained considering that w 0ki are iid with the same second order moment. Let the random variable x i = M w 2 0ki with the mean value independent of M and equal to 1, i.e. the mean value of w 2 0ki decreases with M . Then, according to the law of large numbers (LLN),

w 0k 2 = 1 M M i x i LLN -→ E [x i ] = 1. (5.10)
For (5.10) be true, the variance of x i should be finite, which is assumed to be true and seems a reasonable assumption. From (5.9) and ( 5.10), we can write lim

M →∞ w 0k 2 = E[ w 0k 2
] and hence replace the problem in (5.7) by the problem in (5.8).

Using the generalized Rayleigh quotient theorem, the solution of (5.8) is the eigenvector corresponding to the maximum eigenvalue [START_REF] Golub | Matrix computations[END_REF]:

w 0k ∝ max eigenvect H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k h H 0k . (5.11)
The resulting SLNR is equal to the maximum eigenvalue λ max [START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF]. Any vector g 0k , which is in the eigenspace corresponding to λ max , satisfies the following eigenvector equation:

H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k h H 0k g 0k = λ max g 0k . (5.12)
Inserting a nonzero complex scalar δ 0k = h H 0k g 0k into (5.12), g 0k can be written as

g 0k = δ 0k λ max H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k . (5.13)
Imposing the power constraint, solution to (5.8) is

w 0k = H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k E H0k , D0k ,h 0k H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k 2 . (5.14)
Monte Carlo simulations have been performed on the CDF of SLNR when the precoding vector w 0k is as in (5.14) and when it is obtained by solving (5.8) and imposing an instantaneous power constraint, i.e. w 0k 2 = 1 which is equivalent to remove the expectation in the denominator of (5. A perfect match of the CDFs with the instantaneous and the average power constraint is observed in Fig. 5.1 when M = 10. This suggests that E[ w 0k 2 ] can be used as a good approximation of w 0k 2 even for M as small as 10.

The SLNR precoder causes a significant network overhead since each BS needs the CSI for all the active users in the network. However, considering the leakage to the users that are far away into the SLNR expression should have very little impact. Therefore, focus can be provided only on those users to whom much leakage is caused. It can be assumed that that each BS collects local CSI from the users in its own cell and non-local CSI from the users to whom a considerable leakage is created. This will results in a significant decrease in the network overhead due to the requirement of the CSI of all the active users in the network by each BS.

ZF precoder

ZF precoder is defined such that H H 0 G 0 = I u 0 [START_REF] Boccardi | Zero-forcing precoding for the MIMO broadcast channel under per-antenna power constraints[END_REF], where

H 0 = h 01 , • • • , h 0k , • • • , h 0u 0 ∈ C M ×u 0 and G 0 ∈ C M ×u 0
respectively represent the concatenated channels and non-normalized precoding vectors for the users in cell 0. Non-normalized precoding matrix G 0 can be chosen as the pseudo-inverse of H 0 , which for u 0 ≤ M can be written as [START_REF] Alam | Energy Efficiency-Area Spectral Efficiency Tradeoff in PPP Network with SLNR Precoder[END_REF]

G 0 = H 0 H H 0 H 0 -1 . ( 5.15)
Normalized precoding vector for the k-th user is given by

w 0k = [G 0 ] k [G 0 ] k , ( 5.16)
where [G 0 ] k is the k-th column of G 0 . In case of the ZF precoder, each BS requires CSI only from the users in the own cell.

Performance metrics

We provide the expressions of ASE and EE using stochastic system model.

Area spectral efficiency

Ergodic throughput of a cell with u 0 active users can be written as [START_REF] Wang | Average throughput analysis of downlink cellular networks with multi-antenna base stations[END_REF] R

BS (u 0 ) = u 0 R u (u 0 ), (5.17)
where R u (u 0 ) is the ergodic rate of a typical user when there are u 0 active users in the cell. The ergodic throughput of a typical BS averaged over the number of active users can be written as

R BS = E u 0 [u 0 R u (u 0 )] = u max u 0 =1 u 0 R u (u 0 )p N (u 0 ). (5.18)
Using (5.18), ASE in b/s/Hz/u.a. can be written as

η ASE = λ b u max u 0 =1 u 0 R u (u 0 )p N (u 0 ). (5.19)
Three different random processes, i.e. two independent PPPs governing users and BSs locations, and fading channels are taken into account in the system model. To calculate the ergodic rate of the k-th user, we first average the rate over the SINR conditioned on distance to the connected BS and then average over this random quantity. Therefore, the ergodic rate of the k-th user with u 0 users in the cell is

R u (u 0 ) = E r 0k E γ log 2 1 + γ(u 0 ) | r 0k , (5.20)
where E γ and E r 0k denote the expectations over SINR, and the distance of the k-th user from the connected BS respectively. Calculation of (5.20) can be done using coverage probability [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] or moment generating function approaches [START_REF] Di Renzo | Average rate of downlink heterogeneous cellular networks over generalized fading channels: A stochastic geometry approach[END_REF]. Both these methods necessitate finding the distribution of the terms h H 0k w 0k 2 , h H 0k w 0 j 2 and h H i k w i j 2 , which are difficult to obtain.

Using the Jensen's inequality, we first search for an upper bound of R u (u 0 ),

R u (u 0 ) ≤ E r 0k log 2 1 + E γ γ(u 0 )| r 0k = Ru (u 0 ). (5.21)
Secondly, E γ γ(u 0 )|r 0k is assumed to be close to the ratio of the average quantities in (5.5) as it has been done in [START_REF] Yu | SINR Analysis of the Subtraction-Based SMI Beamformer[END_REF] and is given by

E γ γ(u 0 )|r 0k (5.22)
E H0k , D0k ,h 0k

P t r -α 0k u 0 h H 0k w 0k 2 | r 0k E H0j , D0jk ,h 0 j j =k P t r -α 0k u 0 h H 0k w 0 j 2 | r 0k + E Hi j , Di j ,h i j u max u=1 i ∈Φ u \{0} P t r -α i k u u j =1 h H i k w i j 2 | r 0k + σ 2 n .
It has been verified by simulation that E γ γ(u 0 )|r 0k and the approximation in (5.22) are close to each other. While the numerator in (5.22) represents the average desired power, the first and second terms at denominator refer to the average intra and inter-cell interference powers respectively. Note that all these average powers are calculated conditioned on r 0k . Upper bound of R u (u 0 ) can be written as

Ru (u 0 )= ∞ r 0k ≥0 log 2 1+ E γ γ(u 0 )|r 0k f r 0k (r 0k )d r 0k , (5.23)
where f r 0k (r 0k ) is the PDF of the distance of the k-th user from its connected BS, and is given by [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] f r 0k (r 0k ) = e -λ b πr 2 0k 2πλ b r 0k .

(

With (5.19), an upper bound of ASE can be written as

ηASE = λ b u max u 0 =1
u 0 Ru (u 0 )p N (u 0 ).

(5.25)

Energy efficiency

EE is another important performance metric for cellular networks, which is defined as the ratio of ASE to the average power consumption per u.a. To determine the average power consumption per u.a., only non-transmission power consumption is considered for inactive BSs. Therefore, the average power consumption per u.a. can be written as

P A = λ b 1 -p N (0) aP t + M P RF + λ b c (5.26)
and hence EE is

η EE = η ASE P A , (5.27)
which is obtained using (5. [START_REF] Gorce | Energy-Capacity Trade-off Bounds in a Downlink Typical Cell[END_REF]) and (5.26). In a same way, upper bound of EE is ηEE = ηASE

P A , (5.28)
which is obtained using (5.25) and (5.26). The computation of ASE and EE requires to calculate an approximation of E γ γ(u 0 )|r 0k as suggested in (5.22). The simulation results in Section 5.6

will validate the approximation done on the average SINR.

Calculation of E γ γ(u 0 )|r 0k

In this section, we provide four theorems to evaluate E γ γ(u 0 )|r 0k . The first two theorems are used to calculate the average desired power and the last two ones are applied to calculate the inter-cell interference power. The third theorem is also used to determine the intra-cell interference power. All these theorems are obtained for the asymptotic regime, which is defined as follow ∼ in the rest of the thesis.

Average desired power conditioned on r 0k

Using (5.14), we can write

h H 0k w 0k 2 = h H 0k H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k 2 E H0k , D0k ,h 0k H0k D0k HH 0k + σ 2 n u 0 P t I M -1 h 0k 2 .
(

We manipulate (5.29) in order to apply Theorem 2.4 and obtain

h H 0k w 0k 2 = h H 0k 1 M H0k M α 2 D0k HH 0k + M α 2 -1 σ 2 n u 0 P t I M -1 h 0k 2 E H0k , D0k ,h 0k 1 M H0k M α 2 D0k HH 0k + M α 2 -1 σ 2 n u 0 P t I M -1 h 0k 2 .
(

The singular value decomposition (SVD) of the M ×(N au -1) matrix

M α 2 -1 H0k D 1 2 0k is Ū0k Σ0k VH 0k where Ū0k ∈ C M ×M and V0k ∈ C (N au -1)×(N au -1) are unitary matrices implying ŪH 0k Ū0k = Ū0k ŪH 0k = I M , VH 0k V0k = V0k VH 0k = I N au -1 .
Moreover, Σ0k is an M ×(N au -1) rectangular diagonal matrix with non-negative real numbers on the diagonal, which are the singular values of M (

The RHS of (5.31) is the eigen decomposition of 1 M H0k M α 2 D0k HH 0k and the non-zero elements of Σ0k are the square roots of the non-zero eigenvalues of 1 M H0k M α 2 D0k HH 0k . Considering this eigen decomposition and writing Λ0k = Σ0k ΣH 0k , (5.30) can be expressed as

h H 0k w 0k 2 = h H 0k Ū0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M ŪH 0k -1 h 0k 2 E Ū0k , Λ0k ,h 0k Ū0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M ŪH 0k -1 h 0k 2 = h H 0k Ū0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M -1 ŪH 0k h 0k 2 E Ū0k , Λ0k ,h 0k h H 0k Ū0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M -2 ŪH 0k h 0k . ( 5.32)
Writing t 0k = ŪH 0k h 0k , (5.32) can be expressed as

h H 0k w 0k 2 = t H 0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M -1 t 0k 2 E t 0k , Λ0k t H 0k Λ0k + M α 2 -1 σ 2 n u 0 P t I M -2 t 0k .
(

With

t 0k = [t 0k1 , t 0k2 , • • • , t 0kM ] T and Λ0k = diag λ0k1 , λ0k2 , • • • , λ0kM , (5.33
) can be written as

h H 0k w 0k 2 =   M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t   2 E t 0k , Λ0k M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t 2 .
(

Since w 0k does not include r 0k , the expectation of h H 0k w 0k 2 conditioned on r 0k is the same as its unconditional expectation. Therefore, using (5.34), we can write

E H0k , D0k ,h 0k h H 0k w 0k 2 | r 0k = E t 0k , Λ0k     M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t   2   E t 0k , Λ0k M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t 2 .
(

The expression of w 0k in (5.14) allows us to write (5.35) since the expectation of denominator is already present in (5.30). Numerator of ( 5.35) can be expressed as

E t 0k , Λ0k       M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t    2    =   E t 0k , Λ0k    M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t       2 + var t 0k , Λ0k    M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t    . (5.36)
We introduce two theorems to calculate (5.35). While Theorem 5.1 is used to calculate the first term in the RHS of (5.36), Theorem 5.2 is used to calculate the second term in the RHS of (5.36) and the denominator in (5.35).

Theorem 5.1. Considering the 0-th BS,

E t 0k , Λ0k    1 M M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t    a.r. ∼ m0k (z)
( 5.37) with m0k (z) is the unique, non-negative real solution of the following equation:

πcsc 2π α m-2 α 0k (z)α - z m0k (z) 2πλ au - m0k (z) α -2 πλ au γ au -α 2 + m0k (z) 2 α -1 × 2 F 1   1 - 2 α , 1 - 2 α , 2 - 2 α , πλ au γ au α 2 m0k (z) 1 + πλ au γ au α 2 m0k (z)    = 1 2πλ au , (5.38
)

where z = - M α 2 -1 σ 2 n u 0 P t
, csc is the cosecant function, and 2 F 1 (a, b, c, z) is the Gauss hypergeometric function [START_REF] Abramowitz | Handbook of mathematical functions: with formulas, graphs, and mathematical tables[END_REF].

Proof. See Appendix A.1. Theorem 5.2. Considering the 0-th BS,

var t 0k , Λ0k    1 M M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t    a.r. ∼ E t 0k , Λ0k      1 M M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t 2      a.r. ∼ m 0k (z),
where m 0k (z) is the differentiation of m0k (z) w.r.t. z and expressed as

m 0k (z) = m0k (z)   4π 2 λ au c sc 2π α m1-2 α 0k (z)α 2 -z +Q   -1 , ( 5.39
)

with z = - M α 2 -1 σ 2 n u 0 P t and Q = 2πλ au m0k (z) πλ au γ au α-1 2α -2 1 + m0k (z) πλ au γ au α 2 2 α -2 2 F 1   2 - 2 α , 1 - 2 α , 3 - 2 α , πλ au γ au α 2 m0k (z) 1 + πλ au γ au α 2 m0k (z)    - 2πλ au πλ au γ au α 2 -1 α -2 1 + m0k (z) πλ au γ au α 2 2 α -1 2 F 1   1 - 2 α , 1 - 2 α , 2 - 2 α , πλ au γ au α 2 m0k (z) 1 + πλ au γ au α 2 m0k (z)    .
Proof. See Appendix A. [START_REF] Fletcher | Green radio-sustainable wireless networks[END_REF].

Using (5.35), (5.36), and Theorems 5.1 and 5.2, the average desired signal power conditioned on r 0k can be expressed as

E H0k , D0k ,h 0k P t r -α 0k u 0 h H 0k w 0k 2 | r 0k a.r. ∼ P t r -α 0k u 0 1 + M m2 0k (z) m 0k (z)
.

(5.40)

Average inter-cell interference power conditioned on r 0k

We introduce Theorem 5.3 to calculate the average inter-cell interference power from the i -th BS conditioned on r i k and r 0k .

Theorem 5.3. Considering the i -th interfering BS,

E Hi j , Di j k ,h i j h H i k w i j 2 | r i k ,r 0k a.r. ∼ 1 1 + M α 2 r -α i k mi j k (z) 2 , (5.41)
where mi j k (z) is the unique, non-negative real solution of the following equation:

πcsc 2π α m-2 α i j k (z)α - z mi j k (z) 2πλ au - mi j k (z) α -2 πλ au γ au -α 2 + mi j k (z) 2 α -1 × 2 F 1   1 - 2 α , 1 - 2 α , 2 - 2 α , πλ au γ au α 2 mi j k (z) 1 + πλ au γ au α 2 mi j k (z)    = 1 2πλ au (5.42
)

with z = - M α 2 -1 σ 2 n u i P t . Proof. See Appendix A.3.
From Theorem 5.3 and by averaging over r i k , we state Theorem 5. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF]. The average inter-cell interference power conditioned on r 0k is

E Hi j , Di j ,h i j u max u=1 i ∈Φ u \{0} P t r -α i k u u j =1 h H i k w i j 2 | r 0k a.r. ∼ u max u=1 f 1 (u) -f 2 (u, r 0k ) .
where

f 1 (u) = 2πλ b p N (u)P t M 1-α 2 m α+2 α i j k (z) 2 F 1 2 α , α+2 α ; 2 + 2 α ; 1 (α + 2) m2 i j k (z) and f 2 (u, r 0k ) = 2πλ b p N (u)P t M 1-α 2 2 F 1 2 α , α+2 α ; 2 + 2 α ; 1 mi j k (z)M α 2 r -α 0k +1 1 mi j k (z) + M α 2 r -α 0k α+2 α (α + 2) m2 i j k (z)
.

Proof. See Appendix A. [START_REF] Rappaport | Wireless communications: principles and practice[END_REF].

Average intra-cell interference power conditioned on r 0k

Since h H 0k w 0 j has the same distribution as h H i k w i j , E H0j , D0jk ,h 0 j h H 0k w 0 j 2 | r 0k can be computed with (5.41) substituting r i k and mi j k (z) by r 0k and m0jk (z) respectively. Hence, we can write

E H0j , D0jk ,h 0 j h H 0k w 0 j 2 | r 0k a.r. ∼ 1 1 + M α 2 r -α 0k m0jk (z) 2 .
( 5.43) According to Lemma 2.3 for z < 0, m0jk (z) is equal to m0k (z) for large M , which is the unique, nonnegative real solution of (5.38). Since E H0j , D0jk ,h 0 j h H 0k w 0 j

2 | r 0k = E H0j , D0j k ,h 0 j h H 0k w 0 j 2 | r 0k
∀ j = j except k, inserting m0jk (z) = m0k (z) and using (5.43), the average intra-cell interference power conditioned on r 0k can be written as E H0j , D0jk ,h 0 j j =k

P t r -α 0k u 0 h H 0k w 0 j 2 | r 0k a.r. ∼ P t r -α 0k (u 0 -1) u 0 1 + M α 2 r -α 0k m0k (z) 2 .
(5.44)

Using (5.40), Theorem 5.4 and (5.44), (5.22) can be written as E γ γ(u 0 )|r 0k a.r. .45) Despite that the theorems presented above are derived in asymptotic regime, i.e. large N au and M , they provide accurate predictions on ASE and EE even for moderate values of M as it will be discussed in the next section.

∼ P t r -α 0k u 0 1 + M m2 0k (z) m 0k (z) P t r -α 0k (u 0 -1) u 0 1+M α 2 r -α 0k m0k (z) 2 + u max u=1 f 1 (u) -f 2 (u, r 0k ) + σ 2 n . ( 5

Evaluation of expressions of ASE and EE

In the section, tightness of the approximations for ASE and EE, i.e. (5.25) and (5.28), are verified by performing Monte Carlo simulations. The simulations are performed considering (5. [START_REF] Gorce | Energy-Capacity Trade-off Bounds in a Downlink Typical Cell[END_REF]) and

( 5.27), and also the instantaneous power constraint on the precoding vector, i.e. w 0k 2 = 1.

Throughout the rest of this chapter, α and σ 2 n are set to 4 and -97.5 dBm respectively. For the power consumption model, micro BSs are considered with P RF = 35 W, c = 34 W and a = 3.125 [START_REF] Li | Throughput and energy efficiency analysis of small cell networks with multi-antenna base stations[END_REF][START_REF] Li | Energy efficiency analysis of small cell networks[END_REF].

A circular area, whose radius is such that the average number of users in the network is N u , is considered for simulations. We set N u = 5000 except for the case ρ = 0.1, where N u = 2500 was used to reduce the simulation time. Besides, u max and λ u are equal to M and 5 • 10 -4 m -2 respectively. For the approximation of ASE, the number of active users in the network is determined as lim

R a →∞ N au (R a ) = lim R a →∞ λ au λ u N u (R a ) since lim R a →∞ N u (R a ) = lim R a →∞ λ u πR 2
a and lim

R a →∞ N au (R a ) = lim R a →∞ λ au πR 2 a .
The users are positioned uniformly in the area, and the typical user is considered to be located at center of the network. Figs. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].2a and 5.2b draw ASE vs. P t labelled on BS-user density ratio and number of BS antennas respectively. It is observed that ASE is increasing and converging towards a limit when P t exceeds 20 dBm. This is because the network is homogeneous (same transmit power) and interference limited, hence ASE saturates when noise power becomes negligible w.r.t. to the interference power. The results also demonstrate that higher ASE can be achieved by increasing the BS-user density ratio keeping user density constant or increasing the number of BS antennas. EE vs. P t is drawn on Figs. 5.3a and 5.3b for different BS-user density ratio and number of BS antennas respectively. It is observed that EE is first increasing when P t is increased. However, since ASE converges towards a limit for P t around 20 dBm, EE decreases for further increase Last but not least, our theoretical findings are very tight compared to the simulations results.

The maximum value of the relative error considering all the simulations is about 3%. This small relative error indicates a good match between the simulations and the analytical expressions suggesting that the analytical expressions of ASE and EE can be used as a good approximation of exact values even for M and N au as small as 10 and 2500 respectively. Moreover, despite the fact that the analytical expressions for ASE and EE developed from

Comparison with ZF precoder

We perform Monte Carlo simulations to obtain the EE-ASE tradeoff for the well known ZF precoder, i.e. (5.16), and compare the results with SLNR precoder, as shown in Figs. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].5a and 5.5b. The figures are labelled on BS-user density ratio, with M = 30, and number of antennas, with ρ = 0.1, respectively. The parameters for the plots are set as N u = 5000, u max = M and λ u = 5•10 -4 m -2 unless otherwise mentioned. It can be seen that SLNR precoder has a significant performance gain compared to ZF precoder in terms of achievable EE and ASE.

Higher ASE implies a better SINR for the SLNR precoder compared to ZF. This is due to the fact that although ZF precoder nulls the intra-cell interference, it does not account the leakage to other-cell users. Moreover, the total cancellation of the intra-cell interference by ZF precoding is done at the price of a decrease in the received desired signal power [START_REF] Cheikh | Multicellular zero forcing precoding performance in Rayleigh and shadow fading[END_REF]. On the other hand, the multi-cell SLNR precoder achieves a tradeoff between maximizing the received desired signal power of the intended user and minimizing the interference leakage to all other users in order to maximize the SLNR. This fact results in better SINR when SLNR precoder is used as it can also be seen in Fig. 5. [START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF].

However, the gain of SLNR over ZF precoder is less for u 0 = 20 compared to 10 and 30. The reason is that the maximum ASE for ZF precoder is achieved when the number of active usere in a cell is close to 3 5 M [START_REF] Wang | Average throughput analysis of downlink cellular networks with multi-antenna base stations[END_REF]. On the other hand, maximum ASE for SLNR precoder is achieved when the number of active usere in a cell is close to 3M, as will be shown in 5.8.1.

Effect of system parameters

Numerical results are provided in this section to investigate the influence of different system parameters on the performance of the system, i.e. ASE and EE, when SLNR precoder is used.

Effect of u max on ASE

Number of active users per cell plays a significant role on the performance of SDMA since it affects the spatial multiplexing gain, the aggregated interference, and the beamforming gain for each user. The BS-user density ratio, i.e. ρ, and the maximum number of active users allowed in a cell, i.e. u max , determine the number of active users per cell, and hence the performance gain n, when n ≤ u max , while BS randomly chooses u max users to serve when n > u max . Therefore, the number of active users does not change with u max when it is sufficiently large compared to n, and this leads to a constant ASE afterwards.

PMF of the number of active users, i.e. p N , with u max = ∞ or equivalently the PMF of the number of available users in a cell is plotted in Fig. 5.8. It is observed that the maximum of the PMF is shifted to higher value of u when ρ = 0.0077 compared to ρ = 0.1. This implies that the probability of having higher number of available users is higher when ρ is small. Hence, the value of u max after which the number of active users remain constant, is very large for ρ = 0.0077, while the value is very small for ρ = 0.1.

As observed in Figs. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].7a and 5.7b, ASE first increases rapidly with u max due to the spatial multiplexing gain obtained from serving more users. When ρ is small, as in Fig. 5.7a, the number of active users increases until a large value of u max resulting in higher interference and less beamforming gain for each user reducing the performance gain of SDMA. There is an optimal value of u max denoted as u * max after which ASE decreases slightly and reaches to a constant value. Moreover, the value of u * max is close to 3M . On the other hand, ASE becomes constant for a smaller u max when ρ is large without causing the decrease in ASE, as in Fig. 5.7b.

Effect of M on ASE and EE

We compare ASE for u max = u * max and u max = M when ρ = 0.0077 and 0.1, in Figs. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].9a and 5.9b respectively. As observed in Fig. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].9a, when BS density is much lower than the user density, i.e. ρ = 0.0077, ASE grows linearly with M both when u max = M and u * max . This is due to the fact that the number of active users per cell in this case is sufficiently large to allow higher spatial multiplexing gain provided by the higher number of antennas. Moreover, the gain for u max = u * On the other hand, when BS density is not much lower than the user density, i.e. ρ = 0.1, ASE does not increase linearly with M for both u max = u * max and u max = M . The reason is that the number of active users per cell becomes smaller compared to the number of BS antennas as M increases diminishing the spatial multiplexing gain provided by higher number of antennas.

Moreover, ASE is the same for u max = u * max and u max = M . In addition to ASE, EE depends on the average power consumption per u.a., i.e. P A , as can be seen from (5.28). The gain in terms of ASE for higher number of antennas is associated with more P A , which always increases linearly with M , as illustrated by (5.26) and observed in Fig. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].10a.

Since ASE does not increase linearly with M when ρ = 0.1, Fig 5.9b, but P A does, EE significantly decreases with M at ρ = 0.1, Fig. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].10b. Moreover, although ASE at ρ = 0.1 is much higher than ASE at ρ = 0.0077, P A is also much higher at ρ = 0.1. Hence, EE at ρ = 0.1 is lower than EE at ρ = 0.0077.

Effect of BS density with constant λ u on ASE and EE

We further plot ASE vs. ρ labelled on M in Fig. 5.11 with N u = 5000, λ u = 5 • 10 -4 m -2 , u max = M , P t = 40 dBm. Since the user density is constant, an increase in ρ implies that the BS density also The gain in terms of ASE for higher number of BSs is associated with more P A , which depends on 1p N (0) , as can be seen from ( 5 shown in Fig. 5. [START_REF] Li | Throughput and Energy Efficiency Analysis of Small Cell Networks with Multi-Antenna Base Stations[END_REF]. This leads to a linear increase of P A with ρ, as it can be observed in Fig. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].13a.

The decreasing rate of increase in ASE with ρ for M = 30 in Fig. 5.11 leads to a significant decrease in EE with ρ, as shown in Fig. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].13b.

Effect of BS density with constant ρ on ASE and EE

Figs. [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].14a and 5.14b draw the ASE vs. λ b labelled on number of BS antennas, i.e. M , and transmit power, i.e. P t , when BS-user density ratio, i.e. ρ, is 0.0077 and 1 respectively. The parameters for the plots are N u = 5000 and u max = M . We see that increasing λ b keeping ρ constant increases ASE. The results also demonstrate that while ASE increases linearly with λ b when P t = 40dBm, the rate of increase becomes higher slightly with λ b at P t = -20dBm. This is because ASE depends on upper bound of ergodic rate, i.e. Ru , and the PMF of the number of active users in a cell, i.e.

p N (u), as can be seen in (5.25). Since ρ is fixed, p N (u) is constant as can be seen from ( 5.3).

On the other hand, when P t is sufficiently large leading to interference limited scenario, i.e. be observed in Fig. 5. [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]. On the other hand, the increasing rate of ASE with BS density leads to an increasing EE with BS density at P t = -20 dBm. However, Ru converges towards a limit as λ b increases since the NSR becomes negligible compared to ISR. This will lead to a constant increase of ASE and a constant EE if λ b is further increased even for P t = -20 dBm.

It is also interesting to observe in Fig. 5.16b that higher number of BS antennas induces a loss in EE for ρ = 1, which is in contrast to ρ = 0.0077. The reason is that the spatial multiplexing gain provided by higher number of BS antennas cannot be achieved when ρ is large, i.e. ρ = 1. Therefore, the increase in P A , due to the larger amount of RF circuit power consumption, dominates over the increase of ASE resulting in lower EE for higher BS antennas at ρ = 1.

Conclusions

We have introduced a theoretical framework for approaching the upper bound of ASE in asymptotic regime for PPP networks when SLNR precoder is used by means of random matrix theory and stochastic geometry. The theoretical expression of the EE-ASE tradeoff has been found to be tight with the results obtained through Monte Carlo simulations, even for moderate values of the number of antennas and users in the network, and for a wide range of system parameters.

The results have shown that EE increases linearly as a function of ASE due to the important amount of power wasted in static circuitry. A sharp decrease of EE is observed when transmit power is increased beyond a certain level because of the saturation of ASE at this power. We also compared the performance of the SLNR precoder with ZF precoder. Our results have shown that there is a performance gain over ZF precoder in terms of EE-ASE tradeoff when SLNR precoder is used.

We also investigated the effect of the maximum number of active users allowed in a cell, number of BS antennas, and BS density with a constant user density and constant BS-user density ratio. Numerical results have shown that deploying more BSs or BS antennas increases ASE.

However, the performance gain depends on the BS-user density ratio and the number of BS antennas when user density is fixed. As long as the BS-user density ratio is small or equivalently the average number of users per cell is large compared to the number of antennas, ASE grows linearly with the number of antennas or BS density. The gain diminishes with the number of antennas or BS density if the BS-user density is large compared to the number of antennas.

Since the average power consumption per u.a. increases linearly with the number of antennas, the sub-linear increase in ASE for higher BS-user density ratio induces significant loss in EE when the number of antennas is increased. On the other hand, the BS activity probability decreases with the BS density when user density is fixed. However, the average area power consumption per u.a. increases linearly with BS density when the increased non-transmission power dominate over the transmission power. EE decreases significantly with BS density when ASE sub-linearly increases for higher number of antennas.

When BS density is increased keeping the BS-user density constant, ergodic rate of a user in the noise limited region, i.e. small transmit power, increases. This leads to a higher rate of increase in ASE and a sub-linear increase in EE with BS density. However, when the network is interference limited, the ergodic rate of a typical user converges to a constant value resulting in a linear increase in ASE and a constant EE with BS density. Therefore, ASE can be improved by deploying more BSs without sacrificing EE and also the ergodic rate of the users as the user density increases.

In this chapter, transmit power is assumed to be equally shared among the active users in a cell. Resource management based on the real traffic condition and user position will lead to different power allocation as already mentioned in Chapter 4. Joint distribution of the rates of the active users needs to be taken into account when the allocated resources are not equally shared among the active users.

Chapter 6

Conclusions and future works

Conclusions

Reusing the frequency bandwidth over relatively small geographical areas is one of the most promising strategies to increase SE and EE. The major concern in this case is CCI potentially decreasing SE and EE. In addition to the higher bandwidth, densification of the networks, by deploying small cells and multiple antennas at BSs, is also considered as a potential technique to increase ASE. Due to the large amount of circuit power consumption, the total energy consumption of the wireless networks increases causing a decrease in EE. In this thesis, we have characterized the achievable EE-SE region in a hexagonal cellular network considering several frequency reuse factors as well as shadowing. The EE-ASE region has been also studied in a MU-MISO cellular network with SLNR precoder, when BSs and users are modelled as PPP.

In the beginning, we provided the mathematical tools from stochastic geometry and random matrix theory which were later used in this thesis. We also conducted a brief survey on the performance metrics and energy-efficient research approaches from literature. Besides, the most commonly used models for network architecture, BS power consumption and propagation loss, which are also used in this thesis, have been described. Moreover, the linear precoders, e.g. MRT, ZF, SLNR, have been illustrated using a vector representation.

We have also revisited the EE-SE tradeoff for an AWGN channel. It was observed that EE decreases with SE when the static circuit power consumption is ignored. However, a linear part also exists in the EE-SE tradeoff curves where EE increases with SE when static circuit power consumption is taken into account. We have studied the effect of the PA factor and static circuit power consumption on the EE-SE tradeoff. The results demonstrated that both optimal EE and SE decrease as the PA factor increases. Besides, although higher optimal SE is achieved when the static power consumption is increased, the optimal EE becomes lower.

Although Kelif et al. derived a closed form expression of ISR using the fluid model allowing a tractable derivation of EE-SE tradeoff in hexagonal cellular networks [START_REF] Kelif | A Fluid Model for Performance Analysis in Cellular Networks[END_REF], the expression for ISR was limited to frequency reuse factor equal to one. We have characterized the EE-SE region in hexagonal cellular network for different frequency reuse factors both with and without shadowing. A downlink regular hexagonal cellular network with TDMA, FDMA or OFDMA type of access among the users in a cell has been considered in this work. Both BSs and users have been equipped with a single antenna.

We have obtained a parametric expression of ISR, as function of the normalized distance of the user and path loss exponent, for different frequency reuse factors with curve fitting approach.

The parametric expressions of ISR have allowed us to derive the EE-SE tradeoff for several frequency reuse factors when path loss dependent channel modelling is considered. On the other hand, we have proposed -SE and -EE to characterize the EE-SE tradeoff when shadowing is considered since the capacity in Shannon's sense is not defined with shadowing consideration.

We have obtained an analytical expression of CDF of SINR allowing to study the EE-SE tradeoff when shadowing is taken into account, and validated the expression of CDF by Monte Carlo simulations.

Consequently, we have investigated the EE-SE region with and without shadowing for different frequency reuse factors for a user at various distances. The effect of on the EE-SE tradeoff has also been investigated for the shadowing case. The EE-SE tradeoff curves have been observed

to have a large linear part as seen also for the AWGN channel due to the domination of the consumption of static power over the transmit power. However, EE later started decreasing with SE due to the transmit power domination leading to an optimal EE. Since SE converges towards a limit for a given transmit power in the interference limited network, there is little SE improvement beyond the optimal point and EE decreases sharply when the transmit power is further increased.

The results have also demonstrated that a frequency reuse factor equal to 1 for regions close to BS and higher reuse factors in region closer to the cell edge optimizes the EE-SE tradeoff.

Moreover, better EE-SE tradeoff can be achieved with higher or outage in the system when shadowing is considered. Besides, K = 1 allowed the best optimal EE-SE tradeoff only for higher values of even at a moderate distance since SINR decreases more significantly in shadowing environment for K = 1.

Then we have introduced a theoretical framework for obtaining the upper bound of ASE in asymptotic regime in a MU-MISO cellular network with random topology when SLNR precoder is used, and derived EE from a linear power consumption model afterwards. The results were achieved by applying some fundamental results from RMT to PPP. Although the PPP model is extensively used in literature, the previous works have considered either the MRT or the ZF precoder due to tractability. The theoretical expressions of ASE and EE for the SLNR precoder have been found to be tight with the exact results obtained through Monte Carlo simulations, even for moderate values of the number of antennas and users in the network, and for a wide range of system parameters. Hence, the expressions can be used for the performance analysis of a real network saving the time to run extensive Monte Carlo simulations in addition to providing useful insights.

The results have shown that EE increases linearly with ASE due to the significant amount of static circuit power consumption, and decreases sharply when transmit power is increased beyond a certain level because of the saturation of ASE in the interference-limited network. We have also compared the performance of SLNR and ZF precoders. Our results have shown that there is a performance gain over ZF precoder in terms of EE-ASE tradeoff due to better SINR for the SLNR precoder. The reason is that although ZF precoder nulls the intra-cell interference, it does not account the leakage to other-cell users. Moreover, it decreases the received desired signal power. On the other hand, the multi-cell SLNR precoder achieves a tradeoff between maximizing the received desired signal power of the intended user and minimizing the interference leakage to all other users. This fact results in better SINR when SLNR precoder is used.

We have also investigated the effect of maximum number of active users allowed in a cell on ASE. Numerical results have shown that ASE first increases rapidly due to the spatial multiplexing gain obtained from serving more users. However, ASE converges to a constant value when the maximum number of active users is higher than a particular value, which is higher for a smaller BS-user density ratio.

We also studied the effect of number of BS antennas, and BS density with a constant user density on ASE and EE. The results have illustrated that deploying more BSs or BS antennas increases ASE, but the performance gain depends on the BS-user density ratio and the number of BS antennas. ASE has been observed to grow linearly with number of antennas and BS density when the average number of users per cell is large enough to achieve the spatial multiplexing gain provided by MU-MISO. However, the gain diminishes with number of antennas and BS density if the BS-user density is large compared to the number of antennas.

On the other hand, although the average power consumption per u.a. always increases linearly with the number of antennas, it increases linearly with BS density only when the increased 131 6.2. Future works non-transmission power dominates over the transmission power. Sublinear increase in ASE for higher BS density and higher number of antennas when number of antennas and BS density increases respectively induces a significant loss in EE.

Finally, we illustrated the effect of BS density with constant BS-user density ratio on ASE and EE. In the noise limited regime, ergodic rate of a user increases, leading to a higher rate of increase in ASE and an increasing EE with BS density. However, when the network becomes interference limited, ergodic rate of a typical user converges to a constant value resulting in a linear increase in ASE and a constant EE with BS density. Therefore, ASE can be improved by deploying more BSs without sacrificing EE and also the ergodic rate of the users as the user density increases.

Future works

There are numerous things which can be addressed in the future:

-We have characterized the achievable EE-SE region in a hexagonal cellular network considering several frequency reuse factors. But the BSs have been equipped with a single antenna. Multiple antennas at BSs can increase the achievable data rate and more general results considering MU-MIMO can be derived in the future.

-Turning off BSs is believed to be a promising solution to save energy and taken into account in literature by several authors dealing with stochastic geometry. However, the switching off BSs is done randomly without considering the actual load in the cells, which is not desired. BSs can be turned off based on the actual load and its impact on the EE-ASE tradeoff can be studied afterwards when SLNR precoder is used in a PPP network.

-Deployment of heterogeneous network consisting of various small BSs, i.e. micro, pico, femto, etc., underlaid in a macro cellular network can save a huge amount of energy when combined with switching off BSs and hence gaining a lot of interest by the researchers recently. In a heterogeneous network, different amount of transmit power is required by the cells with different size. In addition, resource management based on the real traffic condition and user position leads to different power allocation for the desired signal and the interfering signal. In this thesis, results are obtained assuming that the desired power and the interference power are same. However, these powers will be different for different power allocation scheme that can be considered in the future.

F M α 2 r -α 0 j (τ) = P M α 2 r -α 0 j < τ = P r -α 0 j < τM -α 2 . (A.2)
Since r -α 0 j is a decreasing function of r 0 j , using (A.1), (A.2) can be written as

F M α 2 r -α 0 j (τ) = 1 -P r 0 j < τM -α 2 -1 α = 1 - τ -1 α M 1 2 0 2x R 2 a d x = 1 - τ -2 α M R 2 a . (A.3)

A.2 Proof of Theorem 5.2

Using the law of total variance [START_REF] Govindasamy | Spectral Efficiency in Single-Hop Ad-Hoc Wireless Networks with Interference Using Adaptive Antenna Arrays[END_REF][START_REF] Li | On the distribution of SINR for the MMSE MIMO receiver and performance analysis[END_REF], we can write

var t 0k , Λ0k    1 M M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t    = E Λ0k   var t 0k    1 M M l =1 |t 0kl | 2 λ0kl + M α 2 -1 σ 2 n u 0 P t | Λ0k       + var Λ0k   E t 0k    1 M M l =1 |t 0kl | 2

A.3 Proof of Theorem 5.3

We can write the precoder for the j -th user in the i -th cell as which is the concatenated fading channels from the i -th BS to all active users in the network except the j -th and the k-th user and

w i j =
Di j k = diag r -α i 1 , • • • , r -α i ( j -1) , r -α i ( j +1) , • • • , r -α i (k-1) , r -α i (k+1) , • • • , r -α i N au
which is the a square diagonal matrix filled by the path losses from the i -th BS to all active users in the network except the j -th and the k-th user. Since The CDF of the entries of M α 2 Di j k can be proved to converge almost surely to (A.4) in the asymptotic regime considering that the infinite network is centered around the i -th BS. Furthermore, Hi j k ∈ C M ×(N au -2) contains iid complex entries with E Hi j k11 -E Hi j k11 where t i k = [t i k1 , • • • , t i kM ], t N i j = t N i j 1 , • • • , t N i j M and Λi j k = λi j k1 , • • • , λi j kM . Both t i k and t N i j are complex Gaussian vectors with mean 0 and covariance matrix I M . Moreover, they are independent of Λi j k . Since h i k and h i j are independent, and t i k = ŪH i j k h i k , t N i j = ŪH i j k h i j , t N i j and t i k are also independent. Furthermore, the entries of t N i j and t i k are iid with mean 0. ∼

E Λi j k M -α M l =1 1 ( λi j kl -z) 2 M -α 2 + r -α i k mi j k (z) 2 E Λi j M l =1 1 λi j l -z 2 .
(A.

Abstract

One of the used strategies to increase the spectral efficiency (SE) of cellular network is to reuse the frequency bandwidth over relatively small areas. The major issue in this case is higher interference, decreasing the energy efficiency (EE). In addition to the higher bandwidth, densification of the networks (e. The EE-SE region in a hexagonal cellular network for different FRF, both with and without shadowing is first characterized. When shadowing is considered in addition to the path loss, the ε-SE-EE tradeoff is proposed as an outage measure for performance evaluation.

The EE-SE curves have a large linear part, due to the static power consumption, followed by a sharp decreasing EE, since the network is homogeneous and interference-limited. The results show that FRF of 1 for regions close to BS and higher FRF for regions closer to the cell edge improve the EE-SE optimal point. Moreover, better EE-SE tradeoff can be achieved with higher outage values. Besides, FRF of 1 is the best choice for very high outage value due to the significant signal-to-interference-plus-noise ratio (SINR) decrease.

In downlink, precoders are used in space division multiple access

3. 13 4 . 1 4 . 3

 134143 EE vs. SE for AWGN channel considering static circuit power consumption . . . . . 3.14 Variation of η * SE , η * EE and P * t with a for different b . . . . . . . . . . . . . . . . . . . . Fitting of the parametric expressions of f (v, α) with simulations for K = {1, 3, 4, 7} and α = {2.5, 3, 3.5, 4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 EE-SE tradeoff without shadowing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Variation of η * SE , η * E E and P * t with K . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Fitting of parametric expressions of G(v, α) with simulations for K = {1, 3, 4, 7} and α = {2.5, 3, 3.5, 4}

4. 5

 5 Comparison of the approximated CDF of the SINR with simulations . . . . . . . .

  .

4. 6

 6 EE-SE tradeoff with long term shadowing

3. 1

 1 Parameters for the AWGN channel with static power consumption . . . . . . . . . . 4.1 Coefficients of f (v,

lN (d µ i k , d 2 σ 2 ) 1 B∼α 5 blog e 10/ 10 ∆ 2 G 0 2 Fn 2 F 2 N 2 n 2 γ

 215102022222 0, I M ) Complex normal r.v. variable with mean 0 and covariance matrix I M δ x Dirac measure δ x (B ) ≡ 1 B (x) ∆(B ) Intensity measure of a point process in B ∂η EE ∂η SE Derivative of η EE w.r.t. η SE e Exponential function E[X ] Expectation of r.v. X E y [X ] Expectation of X over y E y [X |z] Expectation of X over y and conditioned on z F Real-valued bounded measurable function F ∆ Empirical cumulative distribution function of ∆, empirical spectral distribution F ∆Limiting empirical cumulative distribution function of ∆, limit spectral distributionf (x) Function of x 2 F 1 (a, b, c, z) Gauss hypergeometric function ∀ For all G[ f ] Probability generating functional for f [G 0 ] k k-th column of G 0 I M Identity matrix of size M × M ℑ[z] Imaginary part of z L N (0, σ) LN r.v.characterizing shadowing with zero mean and σ be the standard deviation in dB M k k-th order moment of F ∆ µ X (A) Empirical probability measure of the eigenvalues of X in A N The space of natural numbers Normal r.v. with mean d µ i k and standard deviation d 2 σ 2 in natural logarithm (Ω, F , P) Probability space Ω with σ-field F and measure P =⇒ Or Φ Point process Φ(B ) Number of points of Φ in B P !x Reduced palm measure of the point process Φ P[A] Probability of event A x∈Φ f (x) Product of function f evaluated at a point x of the point process Φ Q Marcuum function that provides the CCDF of a standard normal r.v. Q -1 Inverse of the Q function R The space of real numbers R d Real d -dimensional Euclidean space R +The space of non-negative real numbers(R, G ) G is a σ-field on R supp(F ) Support of F x∈Φ f (x) Sum of function f evaluated at a point x of Φ tr (X) Trace of X, sum of the elements on the main diagonal of X var[X ] Variance of r.v. X var y [X ] Variance of X over y var y [X |z]Variance of X over y and conditioned on z (x)Indicator function of x ∈ B • L 2 vector norm lim x→∞ f (x) = q f (x) approaches the limit q as x approaches ∞ a.r. Convergence in asymptotic regime Variables A, B Borel subsets of R d A i , j Coefficients of f (v, α) with i + j ≤ 5 Path loss exponent B i , j Coefficients of G(v, α) with i + j ≤Total power consumption at the minimum non-zero load b E Bit per energy β I Weighting coefficient for I % of the load c Non-transmission power for BB unit, PS unit and cooling D0k A square diagonal matrix filled by the path losses from the 0-th BS to the active users in the network, except the k-th user Di j k Square diagonal matrix filled by the path losses from the i -th BS to all active users in the network except the j -th and the k-th user d Square diagonal matrix filled by the eigenvalues δ 0kNon-zero complex scalar corresponding to the k-th user in the 0-th cellf s (v, α) ISR with shadowing f r 0k (r 0k ) PDF of r 0k G(v, α) i ≥1 r -2α i k i ≥1 r -α i k Concatenated non-normalizedprecoding vectors for the users in cell 0 g 0k Eigenvector in the eigenspace corresponding to λ max (for the k-th user in the 0-th cell) Γ(•) Gamma function γ SINR of a user in hexagonal network γ(u 0 ) SINR of a user in a cell with u 0 users in PPP network γ au lim N au ,M →+∞ N au M H 0 Concatenated channels for the users in cell 0 H0k Concatenated fading channels from the 0-th BS to the active users in the network, except the k-th user Hi j k Concatenated fading channels from the i -th BS to all active users in the network except the j -th and the k-th user h i k Vector representing the complex Gaussian distributed channel between the i -th BS and the k-th user I Mean of the signal received by the k-th user from the i -th BS (dB)u F n Mean of the numerator of f s (v, α) (dB) u F Mean of f s (v, α) (dB) u N Mean of NSR (dB) u γ Mean of γ (dB) n k AWGN for the k-th user µ N L , σ N L , µ F L , σ F L Linear number corresponding to µ N , σ N ,µ F and σ F µ Constant and the value is 3.5 N au Total number of active users in PPP network n Number of users available in a cell Power consumption at I % of the load P i Input power to an element P o Output power of an element P out Power input to an antenna P max P out at the maximum load P total Total power consumption of a BS P RF Power consumed by RF unit P BB Power consumed by BB unit when M = 1 P 0 power consumption at the minimum non-zero P out when M = 1 P i k Power received by the k-th user at distance r i k from the BS i ≥ 0 P t Total transmit power of a BS P A Average power consumption per u.a. Φ u Point process for the BS group with u users R Achievable transmission rate R a Radius of the PPP networkR u (u 0 ) Ergodic rate of a typical user when there are u 0 active users in the cell Ru (u 0 ) Upper bound of R u (u 0 ) R BS (u 0 ) Ergodic throughput of a BS with u 0 active users R BS Ergodic throughput of a typical BS averaged over the number of active users r H Radius of a hexagonal cell r i k Distance from the i -th BS to the k-th user ρ Ratio between BS density and user density σ 2 Variance of the signal for a user with shadowing (dB) σ Variance of the numerator of f s (v, α) (dB) σ Variance of f s (v, α) (dB) σ Variance of NSR with shadowing (dB) σ Noise power (dB) σ Variance of γ in dB Σ Singular value matrix T f Maximum throughput T I Throughput at I % of the load u i Number of users in the i -th cell U, V Unitary matrix v Normalized distance defined as r 0k /r H w Channel bandwidth when K = 1 w i j Precoding vector for the j -th user in the i -th cell x i j Transmitted symbol for the j -th user in the i -th cell X r.v. accounting for the random channel power gain Y i k LN r.v. characterizing shadowing for the k-th user from BS i ≥ 0 y k The signal received by the k-th user from the typical 0-th BS Résumé étendu Cette thèse caractérise la région d'efficacité énergétique et spectrale (EE-SE) d'un réseau cellulaire limité en interférences lorsque la consommation des circuits est prise en compte. Nous caractérisons en premier la région EE-SE d'un réseau cellulaire hexagonal avec un seul utilisateur par cellule, en considérant différentes facteurs de réutilisation de fréquence et avec ou sans masquage. La région d'efficacité énergétique et d'efficacité spectrale par unité de surface (EE-ASE) est ensuite étudiée avec des processus de Poisson ponctuels (PPP) et un réseau à entrée, sortie et utilisateur multiples (MU-MISO) utilisant un précodeur à rapport signal sur fuite plus bruit (SLNR).

5 FIGURE 3 -

 53 FIGURE 3 -EE vs. SE pour un canal AWGN en considérant la consommation d'énergie du circuit statique et dynamique

FIGURE 4 -Chapitre 5 : 5 FIGURE 5 -

 4555 FIGURE 4 -Validation de f (v, α) et du CDF approché du SINR avec des simulations .

FIGURE 6 -FIGURE 7 - 1 FIGURE 8 -Chapitre 6 :

 67186 FIGURE 6 -Variation de -SE et -EE optimales avec à v = 0.5

FIGURE 9 -FIGURE 10 -FIGURE 11 - 1 Introduction

 910111 FIGURE 9 -Effet du nombre d'antennes et ρ sur l'ASE avec N u = 5000, λ u = 5 • 10 -4 m -2 et P t = 40 dBm

Definition 2 . 5 .

 25 (Empirical probability measure of eigenvalues) Let X ∈ C N ×N be a Hermitian matrix with real eigenvalues λ 1 , λ 2 , • • • , λ N . The empirical probability measure of the eigenvalues of X is defined as

. 5 ) 3 State of the art 3 . 1 Introduction

 5331 In the context of Chapter 5, H represents the concatenated fading channels with iid entries of mean zero and variance one, N and n respectively represent the number of BS antennas and number of active users in the network which grow to infinity keeping a constant ratio between them, D represents a diagonal matrix filled by functions of path losses which are non-negative real entries and also independent of H.ChapterMobile communication has experienced a tremendous evolution since the introduction of global system for mobile communication (GSM) in 1980's. The major breakthrough in mobile communication took place in 2007 when the amount of data traffic exceeded the amount of voice traffic[START_REF] Aleksic | Energy efficiency of femtocell deployment in combined wireless/optical access networks[END_REF]. Number of mobile subscribers overtook the number of fixed broadband subscriptions in 2008[START_REF] Fehske | The global footprint of mobile communications: The ecological and economic perspective[END_REF]. The driving force behind this rapid growth was the introduction of mobile internet that caused a paradigm shift from low bandwidth services, e.g. voice and short message, to bandwidth hungry data services. Global mobile penetration rate is forecasted to reach 100% after 2020[START_REF] Fehske | The global footprint of mobile communications: The ecological and economic perspective[END_REF]. Moreover, internet of things will potentially increase the number of connected devices leading to a massive traffic increase in future.

Figure 3 . 1 -Figure 3 . 2 -

 3132 Figure 3.1 -Growth of mobile network traffic and revenue [2]

Figure 3 . 3 -

 33 Figure 3.3 -Hexagonal cellular network[START_REF] Rappaport | Wireless communications: principles and practice[END_REF]

  3.3 since there are seven cells in a cluster. The nearest co-channel neighbours of a particular cell can be obtained first by moving i cells along any chain of hexagons and secondly, turning 60 degrees counter-clockwise and moving j cells, as illustrated in the figure for cell A with i = 2 and j = 1. The distance between any two nearest co-channel cells is known as the frequency reuse distance. Since all the clusters in Fig.3.3 are neighbouring to each other, any two cells marked with the same letter are the nearest co-channel cells, and the distance between them is the frequency reuse distance. If r H is the radius of the hexagonal cell, then the distance between two adjacent cell centers is 3r H and the frequency reuse distance is l = 3K r H .(3.13) Interference between co-channel cells due to the transmission on the same frequency is known as the co-channel interference (CCI). Hexagonal network model is used in Chapter 4.

Figure 3 . 4 -

 34 Figure 3.4 -Poisson homogeneously distributed BSs [5]

Figure 3 . 5 -

 35 Figure 3.5 -Components of a BS[START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF]

Figure 3 . 6 -

 36 Figure 3.6 -Directions of the linear precoding vectors[START_REF] Björnson | Optimal resource allocation in coordinated multi-cell systems[END_REF]

  3.7.

Figure 3 . 7 -

 37 Figure 3.7 -Layered energy efficient approaches

Figure 3 . 8 -

 38 Figure 3.8 -Distribution of power consumption in BSs [8]

  have identified four fundamental tradeoffs of EE with network performance, i.e. deployment cost, SE, bandwidth, delay, for a network with AWGN channels. However, our focus is only on the EE-SE tradeoff and we revisit the relation for an AWGN channel at the end of this chapter. The aim is to differentiate the EE-SE characteristics achieved for interference limited multi-cell networks in Chapter 4 and Chapter 5 from noiselimited AWGN channel.

Figure 3 .

 3 Figure 3.10 -MIMO and CoMP[START_REF] Li | Energy-efficient wireless communications: tutorial, survey, and open issues[END_REF]

  EE has been an interesting research topic in recent years among the network operators, vendors, academia and the regulatory and standardization bodies, e.g. ITU, the 3rd generation partnership project (3GPP), European telecommunication standards institute (ETSI). Many projects have been initiated over the last decades in order to reduce the energy consumption. Majority of the works in literature have been concentrated on battery driven systems, e.g. mobile terminals, wireless ad-hoc and sensor networks[START_REF] Bae | End-to-end energy-bandwidth tradeoff in multihop wireless networks[END_REF][START_REF] Cui | Energy-efficiency of MIMO and cooperative MIMO techniques in sensor networks[END_REF][START_REF] Cui | Energy-constrained modulation optimization[END_REF]. However, many international research projects, e.g. Optimizing Power Efficiency in mobile RAdio Networks (OPERA-Net), mobile VCE Green-Radio, Energy Aware Radio and neTwork tecHnologies (EARTH), Green-touch, have been started over the last few years which have dealt with EE of cellular networks.EARTH project started in January 2010 with the aim to decrease the energy consumption of the 4 th Generation mobile network by half while sustaining the growth in data traffic[START_REF] Gruber | EARTH-energy aware radio and network technologies[END_REF]. EARTH follows a holistic approach targeting the whole system from an EE perspective. Energy efficiency evaluation framework (E 3 F) is an important contribution of EARTH that discusses the network modelling, EE performance metrics, energy aware RRM techniques, network architecture and coverage extension devices[START_REF] Auer | Framework for Energy Efficiency Analysis of Wireless Networks[END_REF]. GreenTouch is a consortium of leading ICT industry, academic and non-governmental research experts. The project has analyzed the fundamental limits of global communication systems with the target to deliver architecture, specifications and roadmaps to increase the network EE by a factor of 1000 by 2015 from the level in 2010[START_REF] Vetter | Towards energy efficient wireline networks, an update from GreenTouch[END_REF].The consortium concluded in 2015 and has been successful in its mission to improve the EE of communication networks by the factor it targeted [140]. The project GREAT from Huawei has identified, analysed and modelled the tradeoff among the fundamental resources, e.g. power, energy, spectrum and bandwidth, latency, deployment cost, for mobile networks[START_REF] Chih-Lin | Toward green and soft: a 5G perspective[END_REF]. Among the other projects, Mobile VCE Green Radio started in 2007 in the UK. This project has aimed at developing green radio architectures focusing mostly on BS design issues, and targeted to achieve 100 times energy reduction of the current wireless networks by 2020[START_REF] Olivier | Trends in global CO2 emissions 2012 report[END_REF]. OPERA-Net, a project led by France telecom, proposed energy saving solutions through cell-size breathing and sleep modes based on traffic loads[START_REF] Feng | A survey of energy-efficient wireless communications[END_REF]. This project started in June 2008 and ended on May 2011. The project TEPN, in which the thesis takes place, aims at making the network energy consumption proportional to the actual load, i.e. number of served users, requested bandwidth, by taking intelligent decisions, e.g. switching on and off network components[START_REF]Towards Energy proportional Network (TEPN)[END_REF]. This project started in 2013 and expected to end in 2017.

. 20 ) 19 Figure 3 . 12 -

 2019312 Figure 3.12 -EE vs. SE for AWGN channel considering only the transmit power with N 0 = -168.83 dBm/Hz

. 25 )Fig. 3 .

 253 Fig.[START_REF] Suarez | An overview and classification of research approaches in green wireless networks[END_REF].13a and Fig.3.13b respectively depict the effects of the PA factor, i.e. a, and static power consumption, i.e. b, on the EE-SE tradeoff. Parameters for all the remaining plots in this chapter are set according to Table3.1 unless otherwise mentioned. The value of r 0 is calculated

Fig. 3 .

 3 Fig. 3.13a shows the EE-SE tradeoff for a ∈ {5, 10, 14.5}[START_REF] Auer | Framework for Energy Efficiency Analysis of Wireless Networks[END_REF] when b is 712 W[START_REF] Auer | Framework for Energy Efficiency Analysis of Wireless Networks[END_REF]. The results

5 Figure 3 . 13 -

 5313 Figure 3.13 -EE vs. SE for AWGN channel considering static circuit power consumption

Figure 3 . 14 -

 314 Figure 3.14 -Variation of η * SE , η * EE and P * t with a for different b

  (4.6) is obtained for K equal to 1, 3, 4 and 7 for α ranging from 2.5 to 4 by fitting the Monte Carlo simulations

Figure 4 . 1 -

 41 Figure 4.1 -Fitting of the parametric expressions of f (v, α) with simulations for K = {1, 3, 4, 7} and α = {2.5, 3, 3.5, 4} .

Figure 4 . 3 -

 43 Figure 4.3 -Variation of η * SE , η * E E and P * t with K

4

 4

= 4 Figure 4 . 4 -

 444 Figure 4.4 -Fitting of parametric expressions of G(v, α) with simulations for K = {1, 3, 4, 7} and α = {2.5, 3, 3.5, 4} .

Figure 4 . 5 - 4 . 4 . 2

 45442 Figure 4.5 -Comparison of the approximated CDF of the SINR with simulations

85 Figure 4 . 6 -

 8546 Figure 4.6 -EE-SE tradeoff with long term shadowing

Figure 4 . 7 -

 47 Figure 4.7 -Optimal -SE vs.

25

 25

  v = 0.85

Figure 4 . 8 -

 48 Figure 4.8 -Optimal -EE vs.

variance σ 2 n

 2 k is the distance from the i -th BS to the k-th user, {i , k} ∈ N 2 , u i is the number of users in the i -th cell and n k is the AWGN with zero mean and for user k. Moreover, h i k ∼ C N (0, I M ) is an M × 1 vector representing the complex Gaussian distributed channel between the i -th BS and the k-th user where I M is an M × M identity matrix. In addition, w i j ∈ C M , x i j are the precoding vector and the transmitted symbol respectively for the j -th user in the i -th cell with E[|x i j | 2 ] = 1. Note that (•) H denotes the Hermitian (transpose-conjugate) operation.

Figure 5 . 1 -

 51 Figure 5.1 -CDF of the SLNR at M = 10

Definition 5 . 1 . 2 a= 1

 5121 (Asymptotic regime) Let R a be the radius of the circular area centered at BS of interest and γ au ∈ R + a constant. The asymptotic regime (a.r.) refers to the condition lim N au ,M →+∞N au M = γ au with lim R a →+∞ N au (R a )λ au πR and will be referred as a.r.

1 ( 2 Figure 5 . 2 -

 1252 Figure 5.2 -ASE vs. BS transmit power

Figs. 5 . 1 ( 2 Figure 5 . 3 - 1 ( 2 Figure 5 . 4 -

 512531254 Figs.[START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].4a and5.4b plot the EE-ASE tradeoff labelled on the BS-user density ratio and number of antennas respectively. The EE-ASE tradeoff has a large linear part before a sharp decrease when ASE is increased, as also observed in Chapter 4 for a hexagonal network. The linear behavior is due to the significant consumption of the RF circuit and non-transmission powers, i.e. P RF and c respectively. Moreover, since ASE converges towards a limit while EE decreases when P t is increased, a sharp decrease of EE is observed for a slight improvement of ASE, as it has been also observed for regular hexagonal network in Chapter 4. The results also demonstrate that ρ = 0.1 allows to achieve the best ASE without loosing too much EE compared to other values of ρ. In a same way, M = 30 achieves the best ASE without loosing EE compared to M = 10 or 20. However, optimal EE is not achieved always with the highest valued parameters. We study the effect of

1 ( 2 Figure 5 . 5 -

 1255 Figure 5.5 -Comparison of the EE-ASE tradeoff of SLNR and ZF precoder

Figure 5 . 6 -

 56 Figure 5.6 -Comparison of the mean received SINR of ZF and SLNR precoder for different u 0 with M = 30 and ρ = 0.04

1 Figure 5 . 7 -

 157 Figure 5.7 -ASE vs. u max at N u = 5000, λ u = 5 • 10 -4 m -2 and P t = 40 dBm

Figs. 5 .

 5 Figs.[START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF].7a and 5.7b illustrate ASE vs. u max labelled on M for ρ = 0.0077 and 0.1 respectively, with N u = 5000, λ u = 5 • 10 -4 m -2 and P t = 40 dBm. BSs serve all the users available in a cell, i.e.

Figure 5 . 8 - 1 Figure 5 . 9 -

 58159 Figure 5.8p N (u) vs. u with u max = ∞

Figure 5 . 10 -

 510 Figure 5.10 -Variation of average power consumption per u.a. and EE with number of BS antennas at N u = 5000, λ u = 5 • 10 -4 m -2 and P t = 40 dBm

Figure 5 .Figure 5 . 12 -

 5512 Figure 5.11 -ASE vs. BS-user density ratio at N u = 5000, λ u = 5 • 10 -4 m -2 , u max = M and P t = 40 dBm

Figure 5 . 13 -

 513 Figure 5.13 -Variation of average power consumption per u.a. and EE with BS-user density ratio at λ u = 5 • 10 -4 m -2 and P t = 40 dBm

P t = 40 = 1 Figure 5 . 14 -= 1 Figure 5 . 15 -= 1 Figure 5 . 16 -

 40151415151516 Figure 5.14 -ASE vs. BS density at N u = 5000 and u max = M

  Since |t 0kl | 2 are iid with both mean and variance equal to 1, (A.14) can be proved to be var t 0k ,

2 = 1 . 2 FFFF

 212 Therefore, according to the Theorem 2.4, ESD of 1 M Hi j k M α Di j k HH i j k , denoted as F Λi j k in the asymptotic regime converges to a non-random function F Λi j k l . Since the elements of Λi j k are non-negative, Λi j k → F Λi j k l implies m F Λi j k (z) → m Λi j k l (z) for negative values of z. Following the proof of Theorem 5.1, m Λi j k l (z) can be shown to be the unique, non-negative real solution of (5.42) where z =j k is used instead of m Λi j k l for notational simplicity. Therefore, inserting(A.30), (A.29) can be written as h H i k w i j 2 a.r.

M 2 Et 2 ,.

 22 -α |t i kl | 2 |t N i j l | 2 ( λi j kl -z) 2 M -α 2 + r -α i k mi j k (z) Di j , Λi j M l =1 |t Di j l | 2 λi j l -z Note that |t i kl | 2 , |t N i j l | 2 , ( λi j klz) -2 are independent to each other with E |t i kl | 2 = E |t N i j l | 2 = 1.Moreover, |t Di j l | 2 and ( λi j lz) -2 are independent to each other with E |t Di j l | 2 = 1. Therefore, (A.[START_REF] Tarighat | A multi user beamforming scheme for downlink MIMO channels based on maximizing signal-to-leakage ratios[END_REF]) can be written as E Hi j , Di j k ,h i j h H i k w i j 2 | r 0k ,r i k a.r.

Since λi j l and λi j kl are the eigenvalues of 1 M 20 ,

 120 Hi .39), (A.[START_REF] Bjornson | Designing wireless broadband access for energy efficiency: Are small cells the only answer?[END_REF]) can be expressed asE Hi j , Di j k ,h i j h H i k w i j 2 | r 0k ,r i k a.r. Avenue des Buttes de Coëmes CS 70839 F-35708 Rennes Cedex 7 Tel : 02 23 23 82 00 -Fax : 02 23 23 83 96 N° d'ordre : 17ISAR 06 / D17 -06 Résumé L'une des stratégies utilisée pour augmenter l'efficacité spectrale (ES) des réseaux cellulaires est de réutiliser la bande de fréquences sur des zones relativement petites. Le problème majeur dans ce cas est un plus grand niveau d'interférence, diminuant l'efficacité énergétique (EE). En plus d'une plus grande largeur de bande, la densification des réseaux (cellules de petite taille ou multi-utilisateur à entrées multiples et sortie unique, MU-EMSO), peut augmenter l'efficacité spectrale par unité de surface (ESuS). La consommation totale d'énergie des réseaux sans fil augmente en raison de la grande quantité de puissance de circuit consommée par les structures de réseau denses, réduisant l'EE. Dans cette thèse, la région EE-ES est caractérisé dans un réseau cellulaire hexagonal en considérant plusieurs facteurs de réutilisation de fréquences (FRF), ainsi que l'effet de masquage. La région EE-ESuS est étudiée avec des processus de Poisson ponctuels (PPP) pour modéliser un réseau MU-EMSO avec un précodeur à rapport signal sur fuite plus bruit (RSFB). Différentes densités de station de base (SB) et nombre d'antennes aux SB avec une consommation d'énergie statique sont considérées. Nous caractérisons d'abord la région EE-SE dans le réseau cellulaire hexagonal pour différentes FRF, avec et sans masquage. Avec le masquage, la mesure de coupure ε-EE-ES est proposée pour évaluer les performances. Les courbes EEES présentent une grande partie linéaire, due à la consommation de puissance statique, suivie d'une forte diminution de l'EE, puisque le réseau est homogène et limité par les interférences. Les résultats montrent qu'un FRF de 1 pour les régions proches de la SB et des FRF plus élevés sur le bord de la cellule améliorent le point optimal du EE-ES. En outre, un FRF de 1 est le meilleur choix pour une valeur élevée de coupure en raison d'une réduction du rapport signal sur interférence plus bruit (RSIB). Les précodeurs sont utilisés en liaison descendante des réseaux cellulaires MU-EMSO à accès multiple par division spatiale (AMDS) pour améliorer le RSIB. La géométrie stochastique a été utilisée intensivement pour analyser de tels systèmes complexes. Nous obtenons une expression analytique de l'ESuS en régime asymptotique, c.-à-d. nombre d'antennes et d'utilisateurs infinis, en utilisant des résultats de matrices aléatoires et de géométrie stochastique. Les SBs et les utilisateurs sont modélisés par deux PPP indépendants et le précodage RSFB est utilisé. L'EE est dérivée d'un modèle de consommation de puissance linéaire. Les simulations de Monte Carlo montrent que les expressions analytiques sont précises même pour un nombre faible d'antennes et d'utilisateurs. Les résultats montrent également que le précodeur RSFB offre de meilleurs performances que le précodeur forçage à zéro (FZ), qui est typiquement utilisé dans la literature. Les résultats numériques pour le précodeur RSFB montrent que déployer plus de SBs ou d'antennes aux BSs augmente l'ESuS, mais que le gain dépend du rapport des densités SB-utilisateurs et du nombre d'antennes lorsque la densité de l'utilisateur est fixe. L'EE augmente seulement lorsque l'augmentation de l'ESuS est plus importante que l'augmentation de la consommation d'énergie par unité de surface. D'autre part, lorsque la densité d'utilisateur augmente, l'ESuS dans la région limitée par les interférences peut être améliorée en déployant davantage de SB sans sacrifier l'EE et le débit ergodique des utilisateurs.

  g. small cells or multi-user multiple input single output, MUMISO) potentially increases the area spectral efficiency (ASE). The total energy consumption of the wireless networks increases due to the large amount of circuit power consumed by the dense network structures, leading to the decrease of EE. In this thesis, the EE-SE achievable region is characterized in a hexagonal cellular network considering several frequency reuse factors (FRF), as well as shadowing. The EE-ASE region is also studied using Poisson point processes (PPP) to model the MUMISO network with signalto-leakage-and-noise ratio (SLNR) precoder. Different base station (BS) densities and different number of BS antennas with static power consumption are considered.

(

  SDMA) MU-MISO cellular networks to improve the SINR. Stochastic geometry has been intensively used to analyse such a complex system.A closed-form expression for ASE in asymptotic regime, i.e. number of antennas and number of users grow to infinity, has been derived using random matrix theory and stochastic geometry. BSs and users are modeled by two independent PPP and SLNR precoder is used at BS. EE is then derived from a linear power consumption model.Monte Carlo simulations show that the analytical expressions are tight even for moderate number of antennas and users. Moreover, the EE-ASE curves have a large linear part before a sharply decreasing EE, as observed for hexagonal network. The results also show that SLNR outperforms the zero-foring (ZF) precoder, which is typically used in literature. Numerical results for SLNR show that deploying more BS or a large number of BS antennas increase ASE, but the gain depends on the BS-user density ratio and on the number of antennas when user density is fixed. EE increases only when the increase in ASE dominates the increase of the power consumption per unit area. On the other hand, when the user density increases, ASE can be improved by deploying more BS without sacrificing EE and the ergodic rate of the users.

  Le réseau hexagonal modélise grossièrement la réalité et pose des difficultés d'analyse qui peuvent être résolues avec les réseaux PPP. L'EE étant un sujet de recherche important, plusieurs solutions ont été proposées pour rendre les réseaux d'accès cellulaires efficacent en énergie. Un rapide survol des méthodes dans le cadre d'un modèle à 3 couches est présenté dans ce chapitre, et sont la couche des composants, la couche des techniques de transmission et de gestion de la ressource radio et enfin la couche d'architecture cellulaire. L'amélioration de l'EE des couches les plus basses peut également améliorer l'EE des couches supérieures. L'approche au niveau des composants consiste principalement à réduire la consommation du PA. Le réseau cellulaire montre une grande variation spatiale et temporelle du trafic. La consommation d'énergie peut être réduite en adaptant les techniques de transmission, les schémas d'allocation des ressources et le nombre de points d'accès tout en maintenant la couverture souhaitée, la capacité et d'autres mesures de la qualité de service (QoS).

	12
	10
	8
	6
	η
	4
	2
	0	0	1	2	3	4	5
				η SE (b/s/Hz)

EE (b/J)

×10

19

FIGURE 2 -EE vs. SE pour le canal AWGN avec P t seulement Nous terminons le chapitre en revisitant les caractéristiques de la relation EE-SE dans un canal AWGN avec un modèle de consommation d'énergie linéaire. Il est observé dans la Fig.

2

que l'EE décroit toujours avec l'augmentation de la SE quand seule la puissance transmise est considérée. Quand la puissance statique des circuits est prise en compte, il y a une large partie linéaire où l'EE augmente avec la SE avant de décroître, comme illustré dans la Fig.

3

. Cependant, lorsque la puissance transmise domine la puissance statique, l'EE décroît avec l'augmentation de la SE. On note également que l'EE comme la SE décroissent avec l'augmentation du facteur de puissance du PA. Cependant, même si une forte valeur de la SE optimale est obtenue lorsque la puissance statique augmente, l'EE optimale reste faible.

  provided fundamental EE-SE tradeoff using PPP for several multiplexing technologies, i.e. time division multiple access (TDMA), frequency division multiple access (FDMA) or superposition coding (SC). However, this work does not take into account the static power consumption part, which is the predominant part for base station (BS) power consumption. EE of cellular networks taking into account practi-

cal power consumption models for BS has been a growing subject of interest for the last few years, e.g.

[START_REF] Auer | How much Energy is Needed to Run a Wireless Network?[END_REF][START_REF] Humar | Rethinking Energy Efficiency Models of Cellular Networks with Embodied Energy[END_REF]

. Several works have studied the EE behavior according to the roll-out density

[START_REF] Li | Throughput and Energy Efficiency Analysis of Small Cell Networks with Multi-Antenna Base Stations[END_REF]

. It appears that EE is an increasing function of the BS density and the number of transmit antennas only when the different components of the BS power consumption satisfy certain conditions.

1.3 Publications 1.3.1 Journal paper

	1.2. Structure of the thesis and contributions
	description of these models is provided next. A detailed description of linear power consumption environment.
	models, which are the most commonly used in literature, is discussed afterwards. These models
	are simple and have the ability to calculate EE with reasonable accuracy. We also describe the
	In Chapter 5, EE-ASE region is investigated for a MU-MISO cellular network with random general propagation model for the whole thesis, and also an overview of the precoding techniques topology, i.e. BSs and users are modelled by two independent homogeneous PPP, when SLNR [J1] Ahmad Mahbubul Alam, Philippe Mary, Jean-Yves Baudais, and Xavier Lagrange. Asymptotic used for MU-MISO networks. Since EE is an important research topic, several researchers have precoder is used. The precoder is derived based on maximizing SLNR using the generalized Analysis of Area Spectral Efficiency and Energy Efficiency in PPP Network with SLNR Precoder. proposed many energy efficient solutions. A brief survey of these methods in a three-layer stack, and well-known research projects are also presented in this chapter. Finally, we revisit the EE-SE Rayleigh quotient theorem. Although BSs are not totally random in reality, they are modelled by Accepted in IEEE Transactions on Communications (TCOM).
	a PPP to facilitate the tractability. An expression of the mean SINR is derived for a typical user in tradeoff for an AWGN channel first taking into account only the transmit power. The study is then extended considering the static circuit power consumption. EE-SE tradeoff for different asymptotic regime, i.e. number of antennas and number of users grow to infinity, using RMT 1.3.2 Conference papers
	power amplifier (PA) factors and static circuit power consumptions are also analyzed. and stochastic geometry. ASE is calculated based on the expression of the mean SINR, and EE [C1] Ahmad Mahbubul Alam, Philippe Mary, Jean-Yves Baudais, and Xavier Lagrange. Energy
	In Chapter 4, EE-SE region in a hexagonal cellular network for different frequency reuse is derived afterwards using a linear power consumption model. The analytical expressions of Efficiency-Spectral Efficiency Tradeoff in Interference-Limited Wireless Networks with Shadow-
	factors both with and without shadowing consideration is characterized. A parametric expression ASE and EE provide useful insights in addition to saving the time to run extensive Monte Carlo ing. In IEEE 82nd Vehicular Technology Conference: VTC2015-Fall, volume 82, pp. 1-5, 2015.
	of ISR for different frequency reuse factors is obtained when only the path loss is considered simulations. EE-ASE tradeoff is then investigated for different number of BS antennas and BS
	allowing a tractable derivation of SE and EE. The expressions of ISR are obtained with a curve density with constant user density, using theoretical expressions and Monte Carlo simulations. [C2] Ahmad Mahbubul Alam, Philippe Mary, Jean-Yves Baudais, and Xavier Lagrange. Energy
	fitting approach, and are expressed as function of the normalized distance of the user and path The theoretical expressions of ASE and EE are found to be tight with the results obtained through Efficiency-Area Spectral Efficiency Tradeoff in PPP Network with SLNR Precoder. In IEEE 17th
	loss exponent. On the other hand, -SE and -EE are proposed to characterize the EE-SE tradeoff Monte Carlo simulations, even for moderate values of the number of antennas and users in International Workshop on Signal Processing Advances in Wireless Communications: SPAWC2016,
	when shadowing is considered along with the path loss. To allow the study of EE-SE tradeoff with the network, and for a wide range of system parameters. The results also show that the EE-ASE volume 17, pp. 1-6, 2016.
	shadowing, an analytical expression of the cumulative distribution function (CDF) of SINR is tradeoff curves have a large linear part before a sharply decreasing EE, as observed for hexagonal
	obtained and validated by Monte Carlo simulations. EE-SE region with and without shadowing network.
	for different frequency reuse factors is investigated when a user is at various distances. For the
	shadowing case, the effect of on the EE-SE tradeoff is also investigated. EE-ASE tradeoff with SLNR is then compared with ZF precoder. SLNR is observed to out-
	It is well known that for a single link AWGN channel SE and EE are two quantities varying perform the ZF precoder, which is typically used in literature when BS and users are modelled
	towards opposite directions when static power consumption is neglected or when it is not the by PPP. This implies a better SINR at user of interest for SLNR compared to ZF. In addition, the
	This thesis is a part of the project titled toward energy proportional networks (TEPN) of Cominlabs predominant factor [37]. However, a linear part where EE increases with SE also exists in the effect of system parameters, i.e. maximum number of allowed users in a cell, number of BS
	Labex, and is carried out at the institute of electronics and telecommunications of Rennes (IETR) EE vs. SE curves when static power consumption is taken into account. It is observed that antennas, BS density with constant user density and constant BS-user density ratio, on ASE and
	in Rennes, France. The manuscript is organized as follows : the EE-SE tradeoff curves for the hexagonal cellular network also have a large linear part as EE is investigated using the SLNR precoder. Numerical results show that deploying more BS or a
	Chapter 2 reviews important definitions, properties, lemmas, theorems, etc. from stochastic seen for the AWGN channel when static power consumption is considered. However, the EE large number of BS antennas increase ASE, but the gain depends on the BS-user density ratio
	geometry and RMT that are used later in Chapter 5, which combines some fundamental results decreases very sharply after reaching the optimal points, since the network is homogeneous and and on the number of antennas when user density is fixed. EE increases only when the increase
	from RMT to PPP leveraging the tools from stochastic geometry. interference-limited. It is shown that a frequency reuse equal to 1 for regions close to BS and in ASE dominates the increase of the power consumption per unit area (u.a.). On the other hand,
	Chapter 3 first defines SE, which is the traditional performance metric for wireless commu-higher reuse factors in region closer to the cell edge optimizes the EE-SE tradeoff. Moreover, when user density increases, ASE can be improved by deploying more BS without sacrificing EE
	nications. SE indicates how efficiently a frequency spectrum is used. An overview of the most better EE-SE tradeoff can be achieved with higher value of when shadowing is considered. But and the ergodic rate of the users.
	common energy metrics used to quantify EE is presented next. Energy metrics provide insight on higher causes a lot of outage in the system and therefore should be set according to a certain
	how efficiently the energy is consumed. Two kinds of network models exist in literature: regular quality of service (QoS) to meet. Moreover, frequency reuse factor 1 is the best choice for a very
	hexagonal networks with fixed BS positions and PPP networks with random BS positions. A short high value of outage, even at a moderate distance, due to significant SINR decrease in shadowing Chapter 6 provides the conclusions and future perspectives of our works.

Chapter 2 Mathematical preliminaries 2.1 Stochastic geometry

	Performance of cellular networks depends largely on the locations of BSs and users. The user
	locations are usually random. However, locations of the BSs can either be fixed as in Chapter 4
	or random as modelled in Chapter 5. Stochastic geometry is a tool used to capture the spatial
	randomness of the network nodes, and enables to study the behaviour of wireless networks
	averaged over random spatial realizations. Our work in Chapter 5 includes modelling the network
	nodes by PPP leveraging the tools from stochastic geometry, particularly the point process theory.

Definition 2.3. (Homogeneous

  2 in Chapter 5 are respectively the average number of BSs and users in that area. point process)[START_REF] Haenggi | Interference in large wireless networks[END_REF] A point process Φ = {x n } is called homogeneous if Φ x = {x n + x} has the same distribution as Φ for all x ∈ R d , i.e.,

Property 2.1. (Superposition) The superposition of the independent PPPs Φ k with density λ k is another PPP with density λ = k λ k . Property 2.2. (Independent thinning

	Property 2.3. (Conditional property) Conditioned on the number of points of Φ in a compact set
	B ⊂ R d , the distribution of Φ(A) for A ⊂ B is BPP characterized by
	Hence, PPP is more
	widely used to model the nodes in wireless networks, e.g. BSs and users are modelled by two
	independent homogeneous PPPs in Chapter 5. PPP has the following interesting properties:
	) Selecting a point of the PPP with probability p indepen-
	dently of the other points results in another PPP with density pλ. This is known as independent
	thinning.
	In Chapter 5, number of active users in different cells are considered to be independent and
	BSs are divided into subsets based on the number of active users in the cells. Hence, the BS
	groups are considered as thinned PPPs created by independent thinning from the process of
	all BSs. Moreover, the density of BSs is the sum of the densities of the BS groups following the
	superposition property.

  4, which is used in Chapter 5, and is very important result for us.

Theorem 2.

[START_REF] Rappaport | Wireless communications: principles and practice[END_REF]

.

[START_REF] Silverstein | Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices[END_REF]

Let B = 1 N HDH H , where H ∈ C N ×n contains iid complex entries with E H 11 -E H 11 2 = 1 where H 11 denotes the element of the first column and first row of the matrix H, D ∈ C n×n be an Hermitian positive semi-definite matrix and independent of H. Moreover, ESD of D,

  Whennoise is the only impairment, P 0k is equal to the transmit power P t , and other parameters in(3.18) are one. Inserting (3.19) with P 0k = P t into (3.1), SE in b/s/Hz can be expressed as

.2 EE-SE tradeoff with static power consumption

  83 dBm/Hz. As observed in this figure, EE vs. SE curve is a Pareto front tradeoff, i.e. EE decreases if SE increases, when only P t is considered. Moreover, maximum EE and minimum SE are obtained when the system is in bandwidth limited regime, i.e. w tends to

	infinity leading to lim w→∞	w log 2 1 + P t w N 0 = P t N 0 log 2 e.
	3.9

EE-SE region for the AWGN channel is characterized by considering the linear power consumption model introduced in (3.16):

Table 3

 3 .1 unless otherwise mentioned. The value of r 0 is calculated from COST-231 Hata model for suburban areas considering a carrier frequency at 1800 MHz, BS antenna effective height of 30m and mobile station antenna effective height of 3m[START_REF] Abhayawardhana | Comparison of empirical propagation path loss models for fixed wireless access systems[END_REF]. It can

	Parameter	Value
	w	10 MHz
	N 0	-168.83 dBm/Hz
	r 0k	0.5 km
	r 0	1.7 × 10 -4 km
	α	3.5

Table 3 .

 3

1 -Parameters for the AWGN channel with static power consumption be observed that the EE-SE tradeoff curves have large linear parts, where EE increases with SE before reaching to a optimal point beyond which SE cannot be increased without decreasing EE.

  labelled on static circuit power consumption, i.e. b. It is observed that η * SE , η * EE and P * t decreases when a increases. This implies that the lowest a is desirable to achieve the best EE-SE tradeoff, but this is achieved with higher transmit power. The results also show that increasing the static power consumption increases η

.31)

Figs.

3.14a, 3.14b

and 3.14c respectively illustrate the variation of η * SE , η * EE and P * t w.r.t. a * SE and P * t while decreasing η * EE .

Table 4 .

 4

	0	0.0043	0.0025	0.0015	0.0009
	A 0,5	0.0018	0.0006	0.0003	0.0002
	A 4,1	0.0098	0.0049	0.0031	0.0018
	A 1,4	0.0007 -0.0001 -0.0001 -0.00004
	A 3,2	0.0022	0.0022	0.0014	0.0009
	A 2,3	-0.0034 -0.0005 -0.0003 -0.0001
	A 4,0	0.0286	0.0169	0.0105	0.0064
	A 0,4	-0.0254 -0.009 -0.0054 -0.0031
	A 3,1	0.0961	0.0386	0.0251	0.0143
	A 1,3	-0.026	-0.001 -0.0003	0.0001
	A 2,2	0.0546	0.0226	0.0144	0.0079
	A 3,0	-0.0078 0.0166	0.0105	0.0075
	A 0,3	0.0944	0.0421	0.0236	0.0124
	A 2,1	0.1996	0.0675	0.0468	0.0276
	A 1,2	0.276	0.061	0.0379	0.0191
	A 2,0	-0.2363 -0.0434 -0.0282 -0.0131
	A 0,2	0.2685	0.0824	0.0843	0.0732
	A 1,1	0.7172	0.9695	0.9929	1.0046
	A 1,0	-0.0106 -0.0189 -0.0215 -0.0179
	A 0,1	-2.6125 -2.1043 -2.0724 -2.1864
	A 0,0	5.6798	4.1574	3.7802	3.4321

1 -Coefficients of f (v, α)

Table 4 .

 4 2 -Parameters for the plots with and without shadowing Fig. 4.2 depicts the effects of K on the EE-SE tradeoff without shadowing. Static circuit power dominates for low value of SE, and EE increases linearly with the SE as seen also for the AWGN channel. Therefore, EE can be increased without sacrificing the SE for this linear portion of the EE-SE tradeoff curves. Afterwards, EE decreases due to the transmission power domination.

	×10 4
	v=.5, K=1
	v=.5, K=3
	v=.5, K=4 v=.5, K=7	P t =1.53W
	v=.85, K=1
	v=.85, K=3
	v=.85, K=4
	v=.85, K=7
	P t =2.3W
	Figure 4.2 -EE-SE tradeoff without shadowing
	Parameter	Value
	w	10 MHz
	N 0	-168.83 dBm/Hz
	r 0	1.7 × 10 -4 km
	r H	0.5 km
	a	14.5 [74]
	b	712 W [74]
	α	3.5

  E Hi j , Di j ,h i j Hi j Di j HH i j + = h i 1 , • • • , h i ( j -1) , h i ( j +1) , • • • , h i N au and Di j = diag r -α i 1 , • • • , r -α i ( j -1) , r -α i ( j +1) , • • • , r -α i N aurespectively represent the concatenated fading channels and a square diagonal matrix filled by the path losses from the i -th BS to all active users in the network, except the j th user. Using (A.25), we can write Hi j , Di j ,h i j Hi j Di j HH i j +

				Hi j Di j	HH i j +	σ 2 n u i P t I M σ 2 n u i -1 P t I M h i j	-1	h i j	2	,	(A.25)
	where Hi j h H i k w i j	2 =	h H i k Hi j Di j E σ 2 HH i j + σ 2 n u i P t I M n u i P t I M -1	h i j -1	2 h i j	2
		=	h H i k E Hi j , Di j ,h i j 1 M Hi j k M	α 2 Di j k 1 M Hi j M HH i j k + M r α α 2 -1 i k α 2 Di j HH h i k h H i k + i j + M α 2 -1 σ 2 M P t n u i α 2 -1 σ 2 n u i P t I M -1 I M h i j -1 2 h i j	2	,	(A.26)

with Hi j k = h i 1 , • • • , h i ( j -1) , h i ( j +1) , • • • , h i (k-1) , h i (k+1) , • • • , h i N au

I

  M are invertible matrix of size M × M with M be a scalar, h i k ∈ C M be a vector, applying Lemma 2.1, (A.26) can be written as is independent of h i k and the elements of h i k are iid complex Gaussian with variance 1, applying Lemma 2.2 and using (A.27), we can write lim Di j , (A.28) can be written aslim Di j , Λi j t H Di j Λi j + Di j = [t Di j 1 , • • • , t Di j M ] and Λi j = [ λi j 1 , • • • , λi j M ].Therefore, the denominator of (A.29) can be expressed as E t Di j , Λi j t H Di j Λi j + Di j = E t Di j , Λi j

		Moreover, 1 M tr Λi j k +	M	α 2 -1 σ 2 n u i P t	I M	-1	can be represented by m	F Λi j k -	M	α 2 -1 σ 2 n u i P t	for large M .
												h H i k	1 M Hi j k M	α 2 Di j k	HH i j k +	M	α 2 -1 σ 2 n u i P t	I M	-1	h i j	2
			h H i k w i j	2 =	1+ M E Hi j , Di j ,h i j r α α 2 -1 i k	h H i k 1 M Hi j M 1 M Hi j k M α 2 Di j α 2 Di j k HH HH i j k + i j +	M M α 2 -1 σ 2 n u i P t α 2 -1 σ 2 P t n u i I M I M -1	h i k -1	h i j	2 .	(A.27)
	Since 1 M Hi j k M	α 2 Di j k	HH i j k +	M	α 2 -1 σ 2 P t n u i	I M	-1
												h H i k	1 M Hi j k M	α 2 Di j k	HH i j k +	M	α 2 -1 σ 2 n u i P t	I M	-1	h i j	2
	M →∞	h H i k w i j	2 = lim M →∞	1+ M E Hi j , Di j ,h i j	α 2 -1 r α i k	tr 1 M Hi j k M 1 M Hi j M α 2 Di j α 2 Di j k HH HH i j k + i j + M M	α 2 -1 σ 2 n u i P t α 2 -1 σ 2 P t n u i	-1 I M I M -1	h i j	2 .	(A.28)
	Applying the eigen decomposition: 1 M Hi j k M	α 2 Di j k	HH i j k = Ūi j k Λi j k	ŪH i j k and 1 M Hi j M	α 2 Di j	HH i j =
	Ūi j Λi j	ŪH i j , and inserting ŪH
												M -α 2 t H i k Λi j k +	M	α 2 -1 σ 2 n u i P t	I M	-1	t N i j	2
			M →∞	h H i k w i j	2 = lim M →∞	M -α 2 + E t M r -α i k M tr Λi j k + M α α 2 -1 σ 2 n u i P t 2 -1 σ 2 P t n u i	I M I M	-1 -2	t Di j	(A.29)
	where t M	α 2 -1 σ 2 n u i P t	I M	-2	t      l =1 M	1 M Hi j k M |t Di j l | 2 α 2 Di j k P t λi j l + M α 2 -1 σ 2 n u i	HH i j k +  2    	M	α 2 -1 σ 2 n u i P t (A.30)	I M and
		1 M Hi j k M	α 2 Di j k	HH i j k + M r α α 2 -1 i k	h i k h H i k +	M	α 2 -1 σ 2 n u i P t	α 2 -1 r α i k

i j k h i k = t i k , ŪH i j k h i j = t N i j and ŪH i j h i j = t

  Using (A.[START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF]) and (A.31), mean of h H i k w i j 2 conditioned on r 0k and r i k can be written asE Hi j , Di j k ,h i j h H i k w i j 2 | r 0k ,r i k

	Therefore, we can write
										
	M m=1	M n=1 n =m	E t ,λ	  	M -α t H i km t N i j m t H N i j n t i kn λi j km + M α 2 -1 σ 2 n u i P t λi j kn + M α 2 -1 σ 2 P t n u i	  
											
				=	M m=1	M n=1 n =m	E t ,λ	  	λi j km +	M -α t H i km t N i j m t H N i j n M α 2 -1 σ 2 n u i P t λi j kn + M	P t 2 -1 σ 2 α n u i	   E [t i kn ]
				= 0.				(A.33)
	Using (A.33) and (A.32), we can write
	E t i k ,t N i j , Λi j k	  M -α 2 t H i k Λi j k +	M	α 2 -1 σ 2 n u i P t	I M	-1	t N i j	2  
											
										= E t i k ,t N i j , Λi j k	    l =1 M	M	α 2 -1 σ 2 n u i	2	   	. (A.34)
										P t

M -α |t i kl | 2 |t N i j l | 2 λi j kl + a.r. ∼ E t i k ,t N i j , Λi j k

  -α i k h i k h H i k is a rank-1 matrix with h i k ∈ C M , 1 M Hi j M

	Since 1 M Hi j M	α 2 Di j	HH i j = 1 M Hi j k M	α 2 Di j k α 2 Di j HH i j k + M HH i j is the rank-1 perturbation of 1 α 2 -1 r -α i k h i k h H i k where M M Hi j k M	α 2 Di j k	HH i j k .
	Therefore, according to Lemma 2.3,
	lim M →∞
				36)

α 2 -1 r

Inverse of the function f (z) = ze z with z be any complex number can be written as z = f -1 (ze z ) = W (ze z ), where W denotes the Lambert function

This work has led to the publication [C1], cf.1.3.2

This work has been accepted for the publication [J1], cf.1.3.2

This work has led to the publication [C2], cf.1.3.2

List of acronyms ASE

-In this thesis, we have considered regular hexagonal network model with fixed BS positions and PPP model where BSs are positioned completely at random. Regular hexagonal network is highly idealized and not very tractable. On the other hand, the assumption that BSs are positioned completely at random has turned the PPP model more tractable compared to the hexagonal network model, but this model is far from reality. In real networks, position of BSs is neither fixed like regular hexagonal network nor totally random as considered in PPP model. Andrews et al. have shown in [START_REF] Andrews | A tractable approach to coverage and rate in cellular networks[END_REF] that the hexagonal network is optimistic and the PPP model is pessimistic in terms of coverage probability when compared with measurements from a real network. The works in this thesis can be extended in future by modelling the BSs by a point process that considers spatial correlation among the BSs without hampering the tractability.

-Deriving the theoretical expressions for the EE-ASE tradeoff in PPP networks using MMSE precoder is very challenging. The MMSE and SLNR precoders have been proved to be equivalent under symmetric scenario where all channels between BS and users have the same gain [START_REF] Patcharamaneepakorn | On the Equivalence Between SLNR and MMSE Precoding Schemes with Single-Antenna Receivers[END_REF][START_REF] Björnson | Optimal multiuser transmit beamforming: A difficult problem with a simple solution structure[END_REF].

However, finding the optimal parameters for the MMSE precoder in a non-symmetric scenario is generally hard. In future, theoretical expressions for the EE-ASE tradeoff in PPP networks can be developed using MMSE precoder and performance between the SLNR and MMSE precoder can be compared afterwards. Let us first derive the CDF of M α 2 r -α 0 j , where r 0 j denotes the distance from BS 0 to the j -th user. Note that M α 2 r -α 0 j , ∀ j = k are the diagonal components and also the eigenvalues of M α 2 D0k . Considering that BS 0 is positioned at the center of the network, the PDF of r 0 j is [START_REF] Ma | On capacity of wireless ad hoc networks with MIMO MMSE receivers[END_REF]:

The CDF of M α 2 r -α 0 j can be written as:

Since lim

λ au πR 2 a , using (A.3), we can write lim

The matrix (unitary) of eigenvectors Ū0k and the matrix of eigenvalues Λ0k are obtained from the eigen decomposition of where

is exponentially distributed with mean 1, and also independent of λ0kl , we have

According to Definition 2.6, ESD of 1 M H0k M α 2 D0k HH 0k can be expressed as

The Stieltjes transform of F Λ0k (x) on C\R + , according to Definition 2.9, is

Since the support of F Λ0k is on the nonnegative real axis, m F Λ0k is continuous in the neighborhood of the negative real line, and can be evaluated at negative real values of z [START_REF] Govindasamy | Spectral Efficiency in Single-Hop Ad-Hoc Wireless Networks with Interference Using Adaptive Antenna Arrays[END_REF][START_REF] Tse | Linear multiuser receivers: effective interference, effective bandwidth and user capacity[END_REF]. F Λ0k is random and depends on H0k ∈ C M ×(N au -1) and D0k ∈ R + (N au -1)×(N au -1) . The latter is independent of H0k .

Moreover, E H0k11 -E H0k11

Since the elements of Λ0k are non-negative, the integrand in (A. [START_REF] Hasan | Green cellular networks: A survey, some research issues and challenges[END_REF]) is bounded and positive for negative values of z. Hence,

Therefore, the term inside the sum in the RHS of (A.6) is m

in the asymptotic regime. Hence, (A.6) can be written as

Let us write m F Λ0k l = m0k for notational simplicity. By using (A.5), RHS of (A.9) can be expressed as .11) Applying lemma 1 from [START_REF] Govindasamy | Spectral Efficiency in Single-Hop Ad-Hoc Wireless Networks with Interference Using Adaptive Antenna Arrays[END_REF], (A.11) can be written as

Substituting (A.12) into (A.9) provides

After straightforward manipulations of (A.13), (5.38) is obtained and the proof is complete.

Using Bayes rule and inserting

, (A. [START_REF] Auer | D2. 3: Energy efficiency analysis of the reference systems, areas of improvements and target breakdown[END_REF]) can be written as

Applying Corollary 1.8 in [START_REF] Guionnet | Concentration of the spectral measure for large matrices[END_REF],

where δ 1 is the Sobolev inequality constant for the distribution of the entries of H0k ,

and (A.18) can be written as

Using (A. [START_REF] Gorce | Energy-Capacity Trade-off Bounds in a Downlink Typical Cell[END_REF]), (A.17) can be expressed as

As M → ∞, M S → 0 for α > 2. Hence, (A.15) can be expressed as

Using (A.22), we can write 1

∼ m 0k (z) , (A.23) where z = -

, and m F Λ0k l is denoted by m 0k for notational simplicity, which can be obtained as expressed in (5.39) by differentiating (A.13) w.r.t. z. On the other hand, using the fact .24) Combining (A.21), (A. [START_REF] Maaz | Energy efficiency analysis of hybrid-ARQ relay-assisted schemes in LTE-based systems[END_REF]) and (A.24) completes the proof.

A.4 Proof of Theorem 5.4

The mean inter-cell interference power conditioned on r 0k can be written as

∀i , ∀ j = j , and hence (A.41) can be written using Theorem 5.3 as .45)