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## State of the Art in GUS in the MIMO BC MISO BC

The Gram-Schmidt channel orthogonalization with pivoting (DPC-style GUS) was introduced in [52]. In [53], a proper BF-style GUS, a large K analysis for DPCstyle GUS and simulations were presented and the matrix inversion lemma for bordered matrices was used in order to lower the complexity of BF-style GUS. The BF with pseudo-BF-style GUS: SUS (semi-orthogonal) i.e., DPC-style GUS with inner product constraints limiting the size of pool of users for selection is analyzed in [51]. It is also shown that for BF-SUS, as for DPC-SUS, In [54] a small refinement is proposed but with more constraints. A simplified at finite SNR, but otherwise exact, sum rate expression for MISO BF (regularized ZF style) can be found in [55]. A suboptimal user selection with complexity of order K2 and an interesting power loading algorithm, equating the correct sum rate gradient with that of an equivalent virtual parallel channel and performing WF on the virtual parallel channel are also proposed in [55].

The transformation of the MIMO channel into a MISO channel, similarly to [54], is done in [15]. Pseudo-BF-style GUS (SUS) and analysis for use in DPC and in BF are carried out, the analysis only shows effect of antennas in higher-order terms. Single-stream MIMO BC and the use of receive antennas to minimize quantization error (for feedback) on resulting virtual channel particularly for partial CSIT (and CSIR) with (G)US are considered in [56]. In [57] the authors obtain the high SNR sum rate offset between BF and DPC without user selection. They extend the analysis of [58] from MISO to MIMO. It is also done in [59]. In [60] SESAM is introduced: proper DPC-style GUS for MIMO case (extension of [52] from MISO to MIMO). In [61] a BF-style GUS for MIMO-BC-BF is proposed. In the style of predecessors, only the receiver of the new stream to be added is adapted. They replace the proper geometric average of the stream channel powers by its harmonic average: 1/tr{diag((HiHHi )−1)}, which leads to a generalized eigenvector solution for the receive filter, the min Frob algorithm. It can be simplified to a classical eigenvector problem: the LISA algorithm is equivalent to the SESAM algorithm. In [62] the same greedy approaches are proposed now for max WSR, without user selection. In [63] the authors prove that working per stream is equivalent to working per user.

### A MIMO IBC Decomposition Scheme

The idea of pairing complementary channel realizations, ergodic interference alignment, was first proposed by Nazer et al. in [16]. The scheme allows each user of an IC to achieve half of his interference free rate, i.e., half of the rate he would achieve if he had the channel for himself. It thereby reaches the optimal DoF G/2 of the G-user SISO IC that was first achieved by asymptotic IA [4]. Some improvements have been made to the original ergodic IA scheme, for instance the channel coefficient distribution does not need to be symmetric [46], the sum of channel matrices does not need to be the identity matrix but can be relaxed to an arbitrary diagonal matrix [46], and simple strategies can be deployed to reduce latency [64]. Other efforts were made to generalize the ergodic IA scheme to different networks, for instance for relay networks in [65]. Ergodic IA was also adapted to secrecy scenarios, in which the information leakage is to be minimized in [66]. A variant of ergodic IA for delayed feedback is proposed in Chapter 8, it shows that the full sum DoF G/2 of the SISO IC can be preserved for feedback delay as long as half the channel coherence time. Another variant, for completely outdated feedback, is developed in [45] and achieves larger DoF than retrospective alignment [43].

However, to the best of our knowledge, the ergodic alignment scheme and its variants do not cover the general symmetric MIMO IC. Indeed, both IA and ergodic IA schemes are also directly applicable to the MIMO symmetric square case, by decomposing each multi-antenna node in single-antenna nodes, but only asymptotic IA was also extended to SIMO and MISO symmetric configuration in [22] whereas only the MISO setting is covered by ergodic IA with the variant for « recovering more messages » proposed in [16]. We extend ergodic IA techniques to the SIMO IC and achieve the same DoF as asymptotic IA. Together with the existing MISO result we can also cover MIMO configurations with M transmit antennas and N receive antennas for the cases where either M/N or N/M is an integer R, yielding DoF = min(M,N)GR/(R + 1).

#### A MIMO IBC proper scheme

Our focus in this section is on linear IA solutions that aim at approaching the proper bound. Limited symbol extensions are considered because it allows to reach decimal DoF and can facilitate the enforcement of the alignment while remaining realistic. We consider the IBC because it is realistic to assume multiple receivers in a cell and because of the potential DoF gains compared to the IC [24]. Interesting results on linear IA were first on the DoF level in the IC [21, 68– 71] where the authors try to settle the question of IA feasibility depending on the number of transmitter-receiver pairs, antennas and assuming generic channel realizations. Mainly, they indicate that it is sufficient to find a certain invertible Jacobian matrix, implicitly or explicitly, to prove the existence of an IA solution to a certain stream assignment, thereby proving the achievability of a certain sum DoF. Actual filter design was then studied to approach these DoF. At first, using only the spatial capacity of the nodes, i.e., multiple antennas: [18, 72, 73], and minimizing different cost functions. Minimizing the interference leakage is useful to determine whether alignment is possible or not. Minimizing mean square error is likely to give better performance especially at low SNR. Then, in order to attain decimal DoF, filter design with the help of symbol extension, i.e., using supersymbols over extension in time and/or frequency were investigated. The main issue with symbol extensions is that channel matrices become structured, diagonal or block diagonal, and the previously mentioned algorithms may converge toward rank deficient solutions. In [19, 20] the authors tried to overcome this issue by incorporating the rank of the direct link in the optimization problem but the proposed algorithms still do not always provide acceptable solutions and sometimes suffer from numerical errors. To the best of our knowledge, the IA algorithm for structured channel that achieves the best results is described in [29]. By adding two constraints to the original interference leakage minimization problem, the authors obtain an algorithm that minimizes the interference leakage while preserving the direct links. Here, we consider the interference minimization problem in the IBC with symbol extension and we add the same constraints as the authors in [29] because they proved to be efficient to yield the good results in the IC.

**Table of contents :**

Abstract

Résumé

Acknowledgements

Contents

List of Figures

Acronyms

**1 Résumé Long [FR] **

1.1 Motivation et Modèles

1.1.1 Introduction

1.1.2 Résumé des contributions

1.1.3 Modèle du Système et Notations

1.2 Plus d’utilisateurs

1.2.1 GUS MISO BF-style GUS critère

1.2.2 IBC

1.2.3 IBC Partiellement Connectés

1.3 DCSIT

1.3.1 FRoI

1.3.2 DCSIT dans le BC

1.3.3 DCSIT dans le IC

Résultat Principal

1.4 Conclusion et Perspectives

1.4.1 Conclusion

1.4.2 Perspectives

**2 Motivation and Models **

2.1 Introduction

2.2 Summary of Contributions

2.3 System Model and Notations

**I Benefits of Having Many/Too Many Users in a Cell **

**3 Too many users in a BC: Multi-User Diversity **

3.1 Introduction

3.1.1 State of the Art in GUS in the MIMO BC MISO BC MIMO BC

3.2 GUS in the MISO BC

3.2.1 MISO DPC style GUS

3.2.2 MISO BF style GUS

3.2.3 BF rate offset approximation

3.2.4 MISO BF-style GUS Criterion

3.2.5 Complexity

3.2.6 Simulation Results

3.3 GUS in the MIMO BC

3.3.1 New MIMO BF-style GUS Criterion

3.3.2 Simulation Results

3.4 Conclusion

**4 Multiple users in interfering cells: Interfering Broadcast Channels **

4.1 Introduction

4.2 A MIMO IBC Decomposition Scheme

4.2.1 Motivation

4.2.2 System Model and Background Ergodic IA

4.2.3 Main Results

4.2.4 SIMO ergodic IA

Example

Proof of Theorem

4.2.5 Discussions

Decomposability

Delay

Improvements

4.3 A MIMO IBC proper scheme

4.3.1 Motivation

4.3.2 System Model and Problem Setup

4.3.3 Main Result

Algorithm

4.3.4 Comparison with IA in the IC

4.3.5 Numerical Results

4.4 Conclusion

**5 Partially Connected Interfering Broadcast Channels **

5.1 Motivation

5.2 IBC System Model and Background

5.2.1 System Model

5.2.2 Relevance

Cell Edge and Cell Center Users

Dense Small Cells

Dual Heterogeneous Networks

DoF analysis

5.3 Transmission Strategy

5.3.1 Separation

5.3.2 DoF After Separation

5.3.3 Simple Strategies

5.3.4 Different Configurations

5.4 Dual Heterogeneous Networks

5.5 Finite SNR Performance Evaluation

5.5.1 Naive Method

5.5.2 Separation

5.5.3 Numerical Results

5.5.4 Comparison

5.6 Conclusion

**II Cost and Delay of CSIT Acquisition **

**6 Finite Rate of Innovation and Foresighted Channel Feedback **

6.1 Introduction

6.2 General Delayed CSIT State of the Art

6.3 Some Channel Model State of the Art

6.4 The Bandlimited Doppler Spectrum Case

6.4.1 The Noiseless bandlimited Case: two-time scale model

6.4.2 Back to the Noisy bandlimited Case

6.4.3 No exact bandlimited model anywhere

6.5 Linear Finite Rate of Innovation (FRoI) Channel Models

6.6 Foresighted Channel Feedback

6.7 Conclusion

**7 Delayed CSIT in the BC **

7.1 Introduction

7.2 System Model

7.3 CSI Acquisition Overhead

7.3.1 Training and Feedback

7.3.2 MAT CSIR distribution

7.4 Net DoF Characterization

7.4.1 ZF

7.4.2 TDMA-ZF

7.4.3 MAT

7.4.4 MAT-ZF

7.4.5 ST-ZF

7.4.6 ZF with FCFB

7.5 Numerical Results and Discussion

7.5.1 Optimization of the number of users

7.5.2 MAT

7.6 Multi-antennas receivers

7.6.1 STIA-MIMO Scheme for the MIMO BC

7.6.2 Longer Feedback delays

7.7 Discussion

**8 Delayed CSIT in the IC **

8.1 Introduction

8.2 System Model and Assumptions

8.3 Main Result

8.3.1 Ergodic IA

8.3.2 Ergodic IA with delayed CSIT

8.4 Feedback delay-DoF tradeoff

8.4.1 Time Sharing

8.4.2 Partial Optimality

8.5 MIMO IC or IBC Configurations

8.5.1 Square MIMO Configurations

8.5.2 Rectangular MIMO Configurations

8.5.3 IBC Configurations

8.6 Net DoF Characterization

8.6.1 CSI Acquisition Overheads

8.6.2 Asymptotic IA

8.6.3 Ergodic IA

8.6.4 TDMA-IA

8.6.5 IA with FCFB

8.6.6 Classic IA

8.6.7 TDMA-IA

8.6.8 TDMA

8.7 Numerical Results

8.7.1 Decomposition IA schemes

8.7.2 Decomposition and proper IA schemes

8.8 Conclusion

**9 Conclusion and Perspectives **

9.1 Conclusion

9.2 Perspectives

Appendices

**A More users **

A.1 Proof of Proposition

A.2 GUS MIMO algorithm

A.3 IA algorithm for IBC

A.4 Separation algorithm

**B Delayed CSIT **

B.1 Basis Function Optimization

B.1.1 Single Basis Function Case

B.1.2 Approach 1: FRoI model based Analysis

B.1.3 Approach 2: Biorthogonal Approach with decoupled Analysis

and Synthesis filters

Optimization with respect to g for a given f

Optimization with respect to f for a given g

B.1.4 Multiple Basis Functions

B.2 STIA for MIMO BC

List of Publications

**Bibliography**