Optimization and Fundamental Insights inFull-Duplex Cellular Networks

JOSE MAIRTON BARROS DA SILVA JUNIOR

Doctoral ThesisStockholm, Sweden 2019

TRITA-EECS-AVL-2019:27ISBN 978-91-7873-147-3

KTH Royal Institute of TechnologySchool of Electrical Engineering and Computer Science

SE-100 44 StockholmSWEDEN

Akademisk avhandling som med tillstand av Kungl Tekniska hogskolan framlaggestill offentlig granskning for avlaggande av teknologie doktorsexamen i natverk ochsystemteknik fredag den 12 april 2019 klockan 10.15 i Kollegiesalen, KTH Campus,Brinellvagen 8, Stockholm.

c© 2019 Jose Mairton Barros da Silva Junior, unless otherwise stated.

Tryck: Universitetsservice US AB

AbstractThe next generations of cellular networks are expected to provide explosive data rate

transmissions and very low latencies. To meet such demands, one of the promising wirelesstransmissions candidates is in-band full-duplex communications, which enable wirelessdevices to simultaneously transmit and receive on the same frequency channel. Full-duplex communications have the potential to double the spectral efficiency and reduce thetransmission delays when compared to current half-duplex systems that either transmitor receive on the same frequency channel. Until recently, full-duplex communicationshave been hindered by the interference that leaks from the transmitter to its own receiver,the so-called self-interference. However, advances in digital and analog self-interferencesuppression techniques are making it possible to reduce the self-interference to manageablelevels, and thereby make full-duplex a realistic candidate for advanced wireless systems.

Although in-band full-duplex promises to double the data rates of existing wirelesstechnologies, its deployment in cellular networks must be gradual due to the largenumber of legacy devices operating in half-duplex mode. When half-duplex devices aredeployed in full-duplex cellular networks, the user-to-user interference may become theperformance bottleneck. In such new interference situation, the techniques such as userpairing, frequency channel assignment, power control, beamforming, and antenna splittingbecome even more important than before, because they are essential to mitigate both theuser-to-user interference and the residual self-interference. Moreover, introduction of full-duplex in cellular networks must comply with current multi-antenna systems and, possibly,transmissions in the millimeter-wave bands. In these new scenarios, no comprehensiveanalysis is available to understand the trade-offs in the performance of full-duplex cellularnetworks.

This thesis investigates the optimization and fundamental insights in the design ofspectral efficient and fair mechanisms in full-duplex cellular networks. The novel analysisproposed in this thesis suggests new solutions for maximizing full-duplex performance inthe sub-6 GHz and millimeter-wave bands. The investigations are based on an optimizationtheory approach that includes distributed and nonconvex optimization with mixed integer-continuous variables, and novel extensions of Fast-Lipschitz optimization. The analysissheds lights on fundamental questions such as which antenna architecture should be usedand whether full-duplex in the millimeter-wave band is feasible. The results establishfundamental insights in the role of user pairing, frequency assignment, power controland beamforming; reveal the special behaviour between the self-interference and user-to-user interference; analyse the trade-offs between antenna sharing and splitting foruplink/downlink signal separation; and investigate the role of practical beamformingdesign in full-duplex millimeter-wave systems. This thesis may provide input to futurestandardization process of full-duplex communications.

SammanfattningNasta generations mobilnat forvantas ha en kraftigt okad formaga for dataoverforing

och mycket laga latenser. For att ta tillvara pa dessa mojligheter ar en av de lovandeteknikerna tradlos overforing med full duplexkommunikation: tradlosa enheter som kansanda och ta emot pa samma frekvenskanal samtidigt. Full-duplexkommunikation har po-tential att dubbla spektraleffektiviteten och minska overforingsforseningarna jamfort mednuvarande halv-duplex system som antingen sander eller mottar pa samma frekvenskanal.Hittills har full-duplexkommunikation hindrats av storningar som lacker fran sandarentill sin egen mottagare, det som ofta kallas for sjalv-interferens. Forbattringar i digitalaoch analoga bekampningstekniker for sjalv-interferens gor det mojligt att minska sjalv-interferensen till hanterbara nivaer och darmed gora full-duplex en realistisk kandidat foravancerade tradlosa system.

Fastan in-band full-duplex ser ut att kunna dubblera datahastigheten i ett tradlostnatverk sa maste dess genomforande ske stegvis i cellulara natverk pa grund av det storaantalet av enheter som fungerar i ett halv-duplex satt. Nar halv-duplex enheter fordelas ifull-duplex cellulara natverk uppstar anvandare-till-anvandare interferens. I denna infer-enssituation som uppstar blir tekniker som anvandaranslutning, frekvenskanalstilldelning,kraftkontroll, beamforming och antennendelning annu viktigare an innan, eftersom dear absolut nodvandiga for att minska bada anvandare-till-anvandare interferens och enanvandares sjalvinterferens. Det finns annu ingen overgripande analys tillganglig somforstar avvagningarna som maste goras vid inforandet av full-duplex cellulara natverk.

Denna avhandling undersoker bade optimeringen och fundamentala insikter i designenav spektraleffektiva och rattvisa mekanismer i full-duplex cellulara natverk. Den nya analy-sen som presenteras i denna avhandling foreslar nya losningar for att maximera full-duplexprestanda bade i sub-6 GHz och millmetervags frekvensomraden. Studierna ar baseradepa ett optimeringsteoretiskt forhallningssatt som omfattar distribuerad och icke-konvexoptimering med kontinuerliga variabler blandat med heltalsvariabler, samt originella vari-anter av Fast-Lipschitz optimering. Analysen sprider ljus over fundamentala fragor sasomvilka antennarkitekturer som ska anvandas, och ifall full-duplex ar genomforbar i millime-tervagsfrekvensomrade. Resultaten etablerar fundamentala insikter nar det galler rollensom spelas av anvandarparning, frekvenstilldelning, effektreglering, samt stralformning.Resultaten visar ocksa det speciella forhallandet mellan sjalvinterferensen och anvandar-anvandarinterferensen. Dessutom analyserar resultaten avvagningarna mellan delning ochsplitting av antenner for upplank/nerlank signalseparation. Slutligen undersoks rollen avpraktisk design av stralformning i full-duplex millimetervagsystem. Denna avhandling kanbidra till framtida standardiseringar av full-duplexkommunikationer.

To the memory of my beloved grandfathers, Raimundo and Manoel

AcknowledgmentsMy PhD is coming to an end, and it is time to acknowledge the professors and everyone

that, maybe without knowing, helped in the journey.First of all, I would like to express my utmost gratitude towards my supervisor Prof.

Carlo Fischione and my co-supervisor Prof. Gabor Fodor, for their everlasting support,guidance, and encouragements in the last four years. You supported me since the beginningof my PhD journey and were always available to discuss everything, even during yourvacations and holidays. Thank you so much for helping me to think more concisely,technically, and to improve as a researcher and a person. Enjoying this journey by yourside was a remarkable and joyful experience that I will always remember.

I am also extremely grateful to Prof. Ashutosh Sabharwal at Rice University for givingme the opportunity of visiting his research group, for his kindness during my stay inHouston, and for the many joyful discussions. I would like to thank all friends at Ricewho made my stay joyful and fruitful, and special thanks to Dr. Niranjan Gowda, Xu Du,Xing Zhang, Gengshan Wang, and Nate Raymondi.

I would like to thank my co-authors Prof. Hadi Ghauch, Prof. Mikael Skoglund, Dr.Yuzhe Xu for the exciting discussions on various topics. With your help, I was able tograsp different viewpoints and learn a lot through our discussions.

I am also very grateful to Prof. Geoffrey Ye Li for acting as the opponent, Prof. IoannisKrikidis, Prof. Sofie Pollin, and Prof. Tharmalingam Ratnarajah for participating in thegrading committee, and Prof. Gyorgy Dan for acting as quality reviewer for this thesis.

I would like to offer my thanks to many people of my current and previous division,Network and Systems Engineering and Automatic Control, for building a warm andexciting environment. Special thanks go to Yuzhe Xu for the great discussions oncombinatorial problems and auction theory; Hadi Ghauch for the great discussions onnonconvex optimization, block coordinate descent methods, and always joining me forheavy metal concerts; Hossein Shokri, Rong Du, Xiaolin Jiang, Sindri Magnusson, andShashi Kant for technical discussions and suggestions in my research; Robert Mattila,Alexandros Nikou, Xinlei Yi, and Takuya Iwaki for the most diverse and joyful discussions;Carol Pascoal, Ezzeldin Zaki, Talita Rocha, Robert Mattila for helping with the Swedishtranslation of the abstract, and Hugo Costa for the Portuguese proofread; I also thank myBrazilian friends that are or were in Sweden, Benedito Neto, Bruna Parolo, Carol Pascoal,Daniel Arauujo, Icaro da Silva, Igor Guerreiro, Ivanes Lian, Laecio Ferreira, LeandroD’Andrea, Nıbia Bezerra, Pedro Batista, Rafael Guimaraes, Talita Rocha, Thiago Dantas,and Victor Farias. I would also like to thank my friends in Brazil for all the great discussionsand support, especially to Hugo Costa, Marciel Barros, Ridley Gadelha, and Thiago Moura.

I also thank the administrators of the Automatic Control and Network and SystemsEngineering divisions, Anneli Strom, Hanna Holmqvist, Connie Linell, and Eleni Nylenfor the assistance and support in the administrative process throughout these years.

I would also like to thank the Brazilian research-support agency CNPq (NationalScience and Technology Development Council) for funding and supporting my PhDstudies, the Lars Hierta Memorial Foundation for the grants I received, the Engblom

x

Foundation for supporting my research visit to Rice University, and the Swedish NationalInfrastructure for Computing at PDC Centre for High Performance Computing for thecomputational resources I used.

Last, but surely not least, I should thank all my family (in Portuguese). Eu soueternamente grato aos meus pais, Edna e Mairton, pelo amor, ensinamentos e compreensaodurante toda a minha vida. Sempre lembrarei do esforo que voces fizeram durante toda aminha vida e que me permitiu essa jornada maravilhosa. Eu tambem gostaria de agradeceras minhas irmas, Erika e Luana, e ao meu irmao, Kaio, pelas conversas e momentosfelizes. Por ultimo, mas definitivamente nao menos importante, gostaria de agradecer aminha amada esposa Taina por todo o carinho, felicidade, suporte e compreensao. Voce foifundamental durante toda minha caminhada e esse trabalho so foi possvel por estar ao seulado.

Jose Mairton B. da Silva Jr.Stockholm, April 2019

Contents

Contents xi

List of Figures xv

List of Tables xix

List of Acronyms xxi

I Thesis Overview 1

1 Introduction 31.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.1.1 Full-Duplex Communications . . . . . . . . . . . . . . . . . . 51.1.2 Self-Interference Cancellation for Full-Duplex Communications 71.1.3 Full-Duplex Applications in Cellular Networks . . . . . . . . . 101.1.4 Coordination Mechanisms for Single-Antenna Systems . . . . . 121.1.5 Coordination Mechanisms for Multi-Antenna Systems . . . . . 13

1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.2.1 Spectral Efficiency Maximization . . . . . . . . . . . . . . . . 161.2.2 Fairness Maximization . . . . . . . . . . . . . . . . . . . . . . 18

1.3 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 201.3.1 Fairness Maximization for Full-Duplex Cellular Networks . . . 211.3.2 Distributed Power Control for SE Maximization . . . . . . . . 221.3.3 Smart Antenna Assignment for SE Maximization . . . . . . . . 221.3.4 Low Resolution Phase Shifters for Practical Full-Duplex

Millimeter Wave . . . . . . . . . . . . . . . . . . . . . . . . . 231.3.5 Contributions not Covered in the Thesis . . . . . . . . . . . . . 23

2 Preliminaries 252.1 Hungarian Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2 Fast-Lipschitz Optimization . . . . . . . . . . . . . . . . . . . . . . . 262.3 Block Coordinate Descent . . . . . . . . . . . . . . . . . . . . . . . . 28

xi

xii Contents

2.4 Semidefinite Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . 302.5 Penalty Dual Decomposition . . . . . . . . . . . . . . . . . . . . . . . 32

II Included Papers 37

A Spectral Efficient and Fair User Pairing for Full-DuplexCommunication in Cellular Networks 39A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41A.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43A.3 System Model and Problem Formulation . . . . . . . . . . . . . . . . . 45A.4 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48A.5 Solution via Lagrange Dual Problem . . . . . . . . . . . . . . . . . . . 50A.6 Approximate Solution via Greedy Method . . . . . . . . . . . . . . . . 55A.7 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60A.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67A.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

B Fast-Lipschitz Power Control and User-Frequency Assignmentin Full-Duplex Cellular Networks 73B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75B.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78B.3 System Model and Problem Formulation . . . . . . . . . . . . . . . . . 80B.4 Power Control Analysis for JASEM . . . . . . . . . . . . . . . . . . . 82B.5 Fast-Lipschitz SINR Target Updates and Distributed Power Control . . 86B.6 Assignment Solutions for JASEM . . . . . . . . . . . . . . . . . . . . 91B.7 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 94B.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102B.9 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

C Smart Antenna Assignment is Essential in Full-Duplex Communications 109C.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111C.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114C.3 System Model and Problem Formulation . . . . . . . . . . . . . . . . . 116C.4 Solution Approach Based on Block Coordinate Descent . . . . . . . . . 121C.5 Convergence and Complexity Analysis . . . . . . . . . . . . . . . . . . 125C.6 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . . 128C.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134C.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

D 1-bit Phase Shifters Suffice for Large-AntennaFull-Duplex mmWave Communications 143D.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145D.2 Related Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147

Contents xiii

D.3 System Model and Problem Formulation . . . . . . . . . . . . . . . . . 148D.4 LowRes: Solution Approach using Penalty Dual Decomposition . . . . 152D.5 Convergence and Complexity Analysis . . . . . . . . . . . . . . . . . . 161D.6 Numerical Results and Discussions . . . . . . . . . . . . . . . . . . . . 162D.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169D.8 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

III Conclusions and Future Works 171

1 Conclusions 173

2 Future Works 175

Bibliography 177

List of Figures

1.1 The three main use cases in 5G [8]. Possible application of FD communi-cations could be in enhancing the spectral efficiency for eMBB, reducingthe latency for URLLC, and providing high connectivity density for mMTC. 4

1.2 We can divide in-band FD schemes in three configurations, bidirectionalfull-duplex, three-node full-duplex, relaying full-duplex, and the additionalHD mode. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 A range of areas and technologies in which FD has been envisioned,irrespective of the configuration in Figure 1.2. We show in purple someapplications for BFD, in blue for TNFD, and in green for RFD. In somecases, more than one configuration can be applied at the same time, suchas D2D communications, MIMO, mmWaves, and SWIPT. . . . . . . . . . 6

1.4 Separate and shared antenna architectures to enable signal separation inFD. Separate antennas use a part of the antennas to transmit (purple) andreceive (green), while shared antennas (color gradient between purple andgreen) use a circulator to transmit and receive simultaneously in the sameantenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5 Block diagram of transmission and reception using a separate antennaarchitecture, inspired by [10]. The general SI cancellation model is dividedin three domains: propagation, analog and digital. . . . . . . . . . . . . . . 8

1.6 Examples of TNFD employing FD with two UE pairs, for single- andmulti-antenna systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.7 A multi-antenna cellular network employing FD with two UE pairs. TheBS may use simultaneously on UL and DL all of its antennas, representedby the color gradient, which causes SI to all the antennas. . . . . . . . . . . 17

A.1 A full duplex cellular network employing TNFD with two UEs pairs. TheBS selects pairs of UE (pairing) and jointly schedules them for TNFDtransmission by allocating frequency channels in the UL and DL. Asthe figure illustrates, apart from SI, TNFD experiences the new UE-to-UEinterference that might limit the efficiency of FD communications. . . . . . 42

A.2 The 3 admissible areas for a user i in the UL and a user j in the DL to sharea frequency channel f that fulfil constraints (A.4b)-(A.4e). . . . . . . . . . 49

xv

xvi List of Figures

A.3 CDF of the minimum spectral efficiency among all users. We notice thatas we increase the number of frequency channels in the system, the gapbetween JAFM and P-JAFM diminishes, where in the 50th percentile thisrelative gap is approximately 1 %. . . . . . . . . . . . . . . . . . . . . . . 62

A.4 CDF of the relative optimality gap of JAFMA and D-JFMA with P-JFMA.We clearly see that the optimality gap diminishes when the number offrequency channels is increased, where in 57 % of the cases the gap isapproximately zero for JAFMA. . . . . . . . . . . . . . . . . . . . . . . . 62

A.5 CDF of the ratio of connected users in the system for different users’ load.Notice that JAFMA guarantees connection to at least 92 % in a system with19 UL and DL users and 82 % in a system with 25 UL and DL users. Thus,JAFMA is able to maintain a high ratio of connected users although thesystem is completely loaded. . . . . . . . . . . . . . . . . . . . . . . . . . 64

A.6 CDF of the modified Jain’s fairness index among all UL and DL usersfor different users’ load. We notice that as we increase the number of usersJAFMA increases its relative difference to AF-EPA and R-FMA, with 35 %at the 50th percentile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.7 CDF of the ratio of connected users in the system for different values ofβ. Notice that JAFMA guarantees connection to at least 82 % in a systemwith high SI level, i.e., JAFMA guarantees a high connection ratio to usersin system with severe SI. . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

A.8 CDF of the modified Jain’s index among all UL and DL users for differentSI cancelling levels. We notice that as β increases, the relative differencebetween JAFMA and AF-EPA also increases. . . . . . . . . . . . . . . . . 66

A.9 Illustrative plot of O(Pui , Pdmax) that shows the possible maxima of the

function and its transitions in the poles. . . . . . . . . . . . . . . . . . . . 68

B.1 A cellular network employing three node FD with two UEs pairs. Thebase station selects pairs of UEs, represented by the ellipses, and jointlyschedules them for FD transmission in the UL and DL. To mitigate UE-to-UE interference (red dotted line), it is advantageous to assign DL/ULusers to for FD transmission in the same frequency that are far apart, suchas UE1-UE2 and UE3-UE4. . . . . . . . . . . . . . . . . . . . . . . . . . . 76

B.2 Convergence of the FL power control algorithm 3. Notice the solutionconverges in approximately 12 iterations with an accuracy of 10−6. . . . . . 96

B.3 CDF of the sum spectral efficiency with reduced number of users. Theproposed G-FLIP achieves a performance close to the optimal P-OPT anda better than H-FLIP. Notice that H-FLIP has the lowest sum spectralefficiency regardless of the number of users. . . . . . . . . . . . . . . . . . 97

B.4 CDF of the relative optimality gap between P-OPT and the proposedG-FLIP and H-FLIP. The relative gap slowly increases with the number ofusers for G-FLIP. Conversely, for H-FLIP the relative gap almost doubleswhen increasing the number of users. . . . . . . . . . . . . . . . . . . . . . 98

List of Figures xvii

B.5 CDF of the sum spectral efficiency with β =−110 dB, i.e., the system islimited by the UE-to-UE interference. We notice that G-FLIP has a relativegain of approximately 16 % with respect to HD systems. In addition, mostof the gains can be achieved by a smart frequency assignment rather than asmart power control. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

B.6 CDF of the total power consumption with a system limited by the UE-to-UE interference. We notice that using FL power control solution we haveapproximately 48 % of energy saving gains. . . . . . . . . . . . . . . . . . 99

B.7 CDF of the sum spectral efficiency with β =−70 dB, i.e., the system islimited by the SI. We notice a performance degradation for full-duplexcommunications, but the relative difference between G-FLIP and HD isonly 5 %. Now, most of the gains can be achieved by a smart power controlinstead of a smart frequency assignment algorithm. . . . . . . . . . . . . . 100

B.8 CDF of the total power consumption when the system is limited by theSI. The proposed solution G-FLIP provides a relative energy savingof approximately 42 % with respect to HD and all other algorithmstransmitting with EPA. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

C.1 An example of separate (or split) [16,19,23,212], shared [17,22,213,214],and our proposed smart full-duplex architecture. . . . . . . . . . . . . . . . 112

C.2 An example of a multi-antenna cellular network employing FD with twoUE pairs. The BS may use simultaneously on UL and DL all of itsantennas, represented by the color gradient, which causes SI to all theantennas. To mitigate all interferences, it is advantageous to analysethe sharing/splitting of antennas between UL and DL, as well as devisetransceivers for UL/DL users. . . . . . . . . . . . . . . . . . . . . . . . . . 116

C.3 Convergence rate of the proposed solution A-SDP with 4 antennas and 2UL/DL users. Notice that the convergence is smooth is non-decreasing. . . 129

C.4 CDF of the sum spectral efficiency with reduced number of users. Theproposed A-SDP achieves a performance close to the exhaustive searchEXH and a better than SPLIT and HD. . . . . . . . . . . . . . . . . . . . . 130

C.5 CDF of the sum spectral efficiency with traffic asymmetry (10− 90 %) andreduced number of users. The proposed A-SDP achieves a performanceclose to the exhaustive search EXH, and SPLIT reaches A-SDP at the 50-th percentile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

C.6 Average sum spectral efficiency for different residual SI powers, assuming4 antennas at the BS and 2 UL/DL users. The proposed A-SDP has almostno loss of performance across different residual SI powers. Moreover,SPLIT decreases quickly and is outperformed by HD solution. . . . . . . . 132

C.7 Average weighted sum spectral efficiency for different residual SI powers,assuming 4 antennas at the BS and 2 UL/DL users. Once more, A-SDPmaintains the average performance but now SPLIT improves with anincrease in the residual SI power. . . . . . . . . . . . . . . . . . . . . . . . 133

xviii List of Figures

C.8 Average sum spectral efficiency for different residual SI powers, assuming4 antennas at the BS and 4 UL/DL users. With more users in the UL andDL, the antennas are no longer able to null more the self-interference, andthus the decrease with higher residual SI powers. . . . . . . . . . . . . . . 134

C.9 Average sum spectral efficiency for different Tx and Rx distortions in asystem with residual SI power of −70 dB, assuming 8 antennas at the BSand 4 UL/DL users. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

C.10 Average sum spectral efficiency for number of antennas at the BS whilemaintaining the ration between antennas and users equals to two. Assum-ing a residual SI power of −70 dB, the difference between A-SDP andSPLIT increases with the number of antennas, as well as the gains withrespect to HD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136

D.1 An example of a multi-antenna cellular network employing full-duplexmmWave with one user pair. . . . . . . . . . . . . . . . . . . . . . . . . . 146

D.2 Beamforming architectures for downlink and uplink, including the analogand digital beamformers. . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

D.3 Convergence of the sum spectral efficiency, assuming 64 antennas at basestation and digital self-interference cancellation of −25 dB. Fast andsmooth convergence, and 1-bit phase shifter has a similar sum spectralefficiency than other higher quantization bits solution. . . . . . . . . . . . . 164

D.4 The cumulative distribution function of the sum spectral efficiency. Noticethat 1-bit phase shifter is close to the infinite resolution phase shifter. . . . . 164

D.5 Histogram for the real and complex amplitudes of the analog precoder FRF. 165D.6 Average sum spectral efficiency for different self-interference residual

powers. For low residual self-interference power, 1 quantization bitoutperforms half-duplex. . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

D.7 Average spectral efficiency in the uplink and downlink. Notice that theimpact of the residual self-interference is much higher in the uplink than inthe downlink. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

D.8 Average sum spectral efficiency for a different number of radio-frequencychains at the base station. The performance gap between infinite and 1-bitphase shifter decreases with the number of radio-frequency chains due toan increase in the dimension of the digital precoder/combiner. . . . . . . . 167

D.9 Average sum spectral efficiency for different number of antennas at thebase station. The performance gap between infinite and 1-bit phase shifterincreases due to additional dimensions in the analog precoder matrix. . . . 168

List of Tables

1.1 Coordination mechanisms and performance objectives considered in thearticles and manuscripts reported in this thesis. . . . . . . . . . . . . . . . 21

A.1 Definition of sets, constants and variables . . . . . . . . . . . . . . . . . . 46A.2 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

B.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95B.2 Fast-Lipschitz qualifying conditions from [164]. Q3 implies the general

condition (GQC), but (Q3) is much easier to use from an analytical andcomputational point of view compared to (GQC). . . . . . . . . . . . . . . 104

C.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

D.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

xix

List of Acronyms

3-DAP 3-Dimensional Assignment Problem

3GPP 3rd Generation Partnership Project

5G 5th Generation

ADC Analog-to-Digital Converter

ADMM Alternating Direction Method of Multipliers

AWGN Additive White Gaussian Noise

BCD Block Coordinate Descent

BFD Bidirectional Full-Duplex

BSUM Block Successive Upper-Bound Minimization

BS Base Station

CDF Cumulative Distribution Function

CSI Channel State Information

DAC Digital-to-Analog Converter

D2D Device-to-Device

DL Downlink

DMRS Demodulation Reference Signal

eMBB Enhanced Mobile BroadBand

EPA Equal Power Allocation

FD Full-Duplex

FDD Frequency Division Duplex

FL Fast-Lipschitz

xxi

xxii List of Acronyms

HD Half-Duplex

HPA High Power Amplifier

JAFM Joint Assignment and Fairness Maximization

JAFMA Joint Assignment and Fairness Maximization Algorithm

JASEM Joint Assignment and Spectral Efficiency Maximization

KKT Karush-Kuhn-Tucker

LNA Low Noise Amplifier

LOS Line-of-Sight

LTE Long Term Evolution

MAC Medium Access Control

MIMO Multiple Input Multiple Output

MINLP Mixed Integer Nonlinear Programming

MISO Multiple Input Single Output

MMSE Minimum Mean Square Error

mMTC Massive Machine Type Communications

MSE Mean Squared Error

MU-MIMO Multi-User Multiple Input Multiple Output

NLOS Non-Line-of-Sight

NOMA Non-Orthogonal Multiple Access

NP Non-Deterministic Polynomial-Time

PC Power Control

PDD Penalty Dual Decomposition

QoS Quality of Service

rBSUM Random Block Successive Upper-Bound Minimization

RF Radio-Frequency

RFD Relaying Full-Duplex

RSRP Reference Signal Receive Power

List of Acronyms xxiii

RSSI Reference Signal Strength Indicator

SE Spectral Efficiency

SDR Semidefinite Relaxation

SDP Semidefinite Programming

SINR Signal-to-Interference-Plus-Noise Ratio

SI Self-Interference

SNR Signal-to-Noise Ratio

SRS Sounding Reference Signal

SWIPT Simultaneous Wireless Information and Power Transfer

TDD Time Division Duplex

TNFD Three-Node Full-Duplex

UE User Equipment

UL Uplink

URLLC Ultra-Reliable Low Latency Communications

WMMSE Weighted Minimum Mean Square Error

Part I

Thesis Overview

1

Chapter 1

Introduction

Dreams shape the world.Neil Gaiman

For the upcoming 5th generation (5G) of wireless communication systems, peaks ofdata rates in the order of tens of Gbit/s are expected [1, 2]. 5G networks will haveto feature different use cases [3], including enhanced mobile broadband (eMBB), ultra-reliable low latency communications (URLLC), and massive machine type communica-tions (mMTC). The eMBB extends the support of conventional mobile broadband servicesthrough improved peak/average/cell-edge data rates, capacity and coverage. The URLLCrepresents the requirement for network services with extreme demand on availability,latency and reliability. The mMTC supports the envisioned 5G scenarios with tens ofbillions of network-enable devices [4]. To meet such demands and cover different usecases, the research and standardization communities are currently studying physical layertechnologies, including massive multiple input multiple output (MIMO) systems, spectrumsharing in mmWave networks, non-orthogonal multiple access technologies, and full-duplex (FD) communications [5–8].

Recently, in-band FD has been proposed as a key enabling technology to increasethe spectral efficiency of conventional wireless transmission modes. FD communicationsovercome the assumption that it is not possible for radios to transmit and receivesimultaneously on the same time-frequency resource. In-band FD transceivers are expectedto improve the attainable spectral efficiency of traditional wireless networks operatingwith half-duplex (HD) transceivers by a factor close to two [9]. In addition to thespectral efficiency gains, full-duplex can provide gains in the medium access controllayer, in which problems such as the hidden/exposed nodes and collision detection can bemitigated [10–13]. Hence, FD communications provide relevant technology componentsto meet the requirements of the 5G use cases [12, 14], as shown in Figure 1.1.

Until recently, in-band FD was not considered as a solution for wireless networksdue to the inherent interference created from the transmitter to its own receiver, the socalled self-interference (SI). However, recent advancements in mitigating SI have beensuccessful [15–25]. Despite in-band FD promises to double the data capacity of existing

3

4 Introduction

Possible FD use cases

Figure 1.1: The three main use cases in 5G [8]. Possible application of FDcommunications could be in enhancing the spectral efficiency for eMBB, reducing thelatency for URLLC, and providing high connectivity density for mMTC.

technology, its deployment in wireless local area and cellular networks is challenging dueto the large number of legacy devices and wireless access points. A viable introductionof FD technology in cellular networks requires the wireless access points or base stations(BSs), typically equipped with multiple antennas and operating in sub-6 GHz or mmWavebands, to implement FD transceivers to support the simultaneous downlink (DL) and uplink(UL) communication with two distinct user equipments (UEs) on the same frequencychannel [11]. Specifically to the design of FD multi-antenna systems, the simultaneoustransmission and reception requires the BS to decide between FD architectures that eithersplit or share the antennas between UL and DL antennas. In addition, the users experiencethe UE-to-UE interference, which may become the performance bottleneck, especially asthe capability of FD transceivers to suppress SI improves. In FD cellular systems, it iscrucial to understand the trade-offs between UL and DL performance in the design ofefficient and fair coordination mechanisms to realize the FD potential for legacy UEs.

Hence, in this thesis we are interested in the design of such coordination mechanismsto optimize and provide fundamental insights into FD cellular networks. Specifically, ourwork seeks to answer the following research questions:

1. How can we ensure fairness levels to users in an operational FD cellular network?

1.1. Background 5

BS

UE1 UE2

HD

Node1 Node2

SI SI

BFD

BS

UE1 UE2

SI

TNFD

RelayNode1 Node2

SI

RFD

Freq. Channel 1

Freq. Channel 2

UE-to-UE Interf.

Self-Interf. (SI)

Figure 1.2: We can divide in-band FD schemes in three configurations, bidirectional full-duplex, three-node full-duplex, relaying full-duplex, and the additional HD mode.

2. What is the role of coordination mechanisms at the BS and at the user in achievinghigh spectral efficiency?

3. What are the fundamental trade-offs between sharing and splitting an antennabetween UL and DL in FD multi-antenna systems?

4. Are low resolution phase shifters sufficient for providing high spectral efficiency inFD mmWave systems?

1.1 Background

In this section we overview some fundamental aspects of FD communications, includingits history and recent developments in SI cancellation, cellular networks, and distributedsolutions for power control and assignment of users.

1.1.1 Full-Duplex Communications

In-band FD in wireless networks is not recent, the concept of transmission and reception inthe same frequency channel has been used since 1940 in radar systems [10]. SI was alreadya key challenge at that time, and the first circuits to null the SI were proposed and providedlow levels of cancellation. Such mild SI cancellation levels limited the transmit powerand provided a reduced range of detectable targets. In the last decade, wireless broadbandsystems, such as WiFi and cellular, started to experiment with in-band FD communications.Arguably, the first application of FD in such context was introduced for relaying purposes,where the relay could be used to improve the sum rate, area coverage or in difficult areasto implement an operational BS [10, 26].

With respect to the transmission configurations, we can categorize the in-band FD inthree application areas: bidirectional full-duplex (BFD), three-node full-duplex (TNFD),and relaying full-duplex (RFD) [11,27]. Figure 1.2 shows the standard half-duplex systemand the aforementioned configurations, as well as the corresponding interference scenarios.In BFD, two FD-capable nodes (either a UE or the BS) transmit and receive on the same

6 Introduction

IN-BAND FD

D2DCOMMU-

NICATIONS[28–36]

SMALLCELLS ANDHETNETS[37–54]

NOMA[55–63]

MIMO[64–76]

SWIPT[77–85]

COGNITIVERADIO

[86–94]

BACK-HAULING

[43, 95–102]MASSIVE

MIMO[103–114]

MMWAVES[115–127]

Figure 1.3: A range of areas and technologies in which FD has been envisioned,irrespective of the configuration in Figure 1.2. We show in purple some applications forBFD, in blue for TNFD, and in green for RFD. In some cases, more than one configurationcan be applied at the same time, such as D2D communications, MIMO, mmWaves, andSWIPT.

time-frequency resource, which creates SI for both nodes. For WiFi or IoT applications,this configuration is attractive due to distributed aspects of such networks. In contrast,TNFD involves three nodes, but only one requires FD capability. The FD-capable nodetransmits to its receiver node while receiving from another transmitter node on the samefrequency channel, in which SI is present only at the FD-capable node. For cellularnetworks, this configuration is attractive because instead of requiring all UEs to be FD,only the BS can be assumed FD-capable while UEs remain HD (Figure 1.2). In RFD, onenode transmits to a FD-capable relay and then retransmits the signal to the second node,where all transmissions occur in the same time-frequency resource. Since the relay is theonly FD-capable node, SI is present only at the relay. This configuration is appealing togeneral wireless networks because it can represent a master-worker architecture in IoT, aswell as a relay or backhaul architecture in cellular networks.

These new configurations extend the design options, and allow for many benefits suchas: higher spectral usage of the already available frequency resource, higher detection ofnodes, higher coverage area, and lower power consumption. For example, in BFD thenodes can ideally double the spectral efficiency, provided that an efficient SI cancellation

1.1. Background 7

Separate AntennaArchitecture

Txchain

Rxchain

Circulator

Shared AntennaArchitecture

Txchain

Rxchain

Figure 1.4: Separate and shared antenna architectures to enable signal separation in FD.Separate antennas use a part of the antennas to transmit (purple) and receive (green), whileshared antennas (color gradient between purple and green) use a circulator to transmit andreceive simultaneously in the same antenna.

is performed at each node, and also improve collision avoidance. The same ideas applyto the other configurations, but now with a broader application range. Figure 1.3 showssome FD applications in which all three configurations have been studied and envisioned,where the purple color represents BFD, the blue color represents TNFD, and the graycolor represents the RFD. Note that FD already spans different applications, ranging fromdevice-to-device (D2D) communications, to simultaneous wireless information and powertransfer (SWIPT) networks, and reaching backhauling and mmWaves, which show anevolution and vast potential of the technology. For many areas, more than one configurationis applicable at the same time, which is the case of MIMO – and massive MIMO –, SWIPT,non-orthogonal multiple access (NOMA), cognitive radio, and mmWaves. Despite SWIPTthat may use SI in a beneficial manner, the common drawback of FD in the technologiesand applications shown in Figure 1.3 remains the SI, which highlights that SI cancellationis crucial.

1.1.2 Self-Interference Cancellation for Full-Duplex Communications

The driving concept of FD communication is to allow simultaneous transmission andreception for a node. For this to happen, the FD nodes require specially designed transmitand receive radio-frequency (RF) chains. To separate these two RF chains, there are twoarchitectures available: separated and shared antennas [27] (as seen in Figure 1.4). Thefirst architecture consists of separating physically the RF chains, i.e., when the numberof antennas is greater than 2, use a part of the set of antennas to transmit and the otherpart to receive. In this situation, there is a loss in the degree of freedom in the spatialdomain. The second architecture shares all the antennas in the RF chains, and to isolatethe receiver from the transmitter, a special antenna interface such as duplexer or circulatoris used. Such interfaces represent a three-port device that prevents the transmitted signal

8 Introduction

CodingModulation

Interference CancellationDemodulation

Decoding

DAC ×∼

HPA

Transmit Chain

TransmitBits

ADC ×∼

LNA +

Receiver Chain

CancellationCircuit

ReceiveBits

Digital Domain Analog Domain Propagation Domain

Self-Interference

Transmit Signal

InterestReceived Signal

Figure 1.5: Block diagram of transmission and reception using a separate antennaarchitecture, inspired by [10]. The general SI cancellation model is divided in threedomains: propagation, analog and digital.

from one RF chain to leak into the receiver signal of the receiver RF chain, and they remaina subject of intensive research [25, 128, 129]. The most used interface is the circulator(shown in Figure 1.4), whose first applications come from radar technology [10, 27]. Withshared antennas, it is possible to have a single antenna to transmit and receive, which isadvantageous in single-antenna systems.

Regardless of the methods to separate the RF chains, the simultaneous transmissionand reception in the same frequency resource makes the transmitted signal to loop back tothe receiving antenna, causing SI. Depending on the transmitter power, the SI needs to becancelled by more than 100 dB to reduce its value to the noise floor [10, 17], which is anextremely high level to be cancelled. To have a better understanding of the impact of SI inthe system, we need to cancel a signal that is approximately 1 trillion times higher than thereceived signal [20].

In Figure 1.5, we present a general block diagram of transmission and receptionblocks to show a few steps used to cancel the SI signal when using a separate antennaarchitecture [10]. At the transmitter side, the bits are converted to digital baseband signalin the digital domain through coding and modulation schemes. These digital signals areconverted to analog using digital-to-analog converter (DAC), upconverted to the carrierfrequency, and amplified using a high power amplifier (HPA). Notice that this step ofconverting from digital to analog domain introduces several distortions in the transmitsignal, such as quantization noise due to DAC, phase noise due to frequency upconversion,and distortions due to HPA. After the conversion, the analog signal is finally transmittedthrough the radiating antennas. At the receiver side, the received analog signal contains theinteresting and self-interference signals. The signal is then converted from the analog to thedigital domain using a low noise amplifier (LNA), downconverted to the baseband, and thenconverted to digital through the analog-to-digital converter (ADC). In the digital domain,the signal is converted to bits through demodulation and decoding schemes. Notice that

1.1. Background 9

the digital signal is directly connected to the analog signal through the analog cancellationcircuit, while the analog cancellation circuit is directly connected to the receiver signalbefore the LNA and before the transmit signal is radiated through the transmit antenna. Inthe following, we explain the details of such connections.

From Figure 1.5, we observe that to cancel SI we need to rely on techniques thatcan be tackled from the propagation, analog, and digital domains [10, 129]. Usually, thefirst cancellation happens in the propagation domain, which includes the signal separationarchitectures mentioned above. Next, SI cancellation in the analog and/or digital domainsuppresses the residual SI. In addition to this classification, the SI cancellation can befurther categorized into passive and active cancellation to indicate in which manner thetechnique mitigates the SI channel. For simplicity, we briefly describe a few aspects of thesuppression techniques in each domain, and for an in-depth overview of such techniqueswe refer to [7,10,27,129] that provide tables with different methods of cancellation as wellas their advantages and disadvantages.

In the propagation domain, the goal is to isolate the transmit chain from the receiverchain by suppressing the SI signal before it impacts the receiver chain [10, 130, 131].Common propagation domain techniques include the physical separation of the antennasin the separate architecture, and the use of duplexers or circulators in the sharedarchitecture. Besides the physical separation between the transmitting and receivingantenna, the spatial separation, called directional SI suppression, can also be accomplishedby setting a directional beam towards the receiving antenna and a lobe to the transmittingantenna [105, 132–135]. Using propagation suppression, it was shown experimentally thatup to 65 dB of SI can be suppressed with omni-directional antennas [131], and up to 72 dBwith directional antennas [135]. The general weakness of different propagation domaincancellation methods is that while modifying the transmit and/or receive antenna patternsto mitigate the SI, the desired signal may be impacted. To avoid such problems, propagationdomain cancellation is usually implemented along with analog and/or digital propagationcancellation.

In the analog domain, SI cancellation is usually executed before the signal entersthe ADC, either before or after the downconverter and LNA [10, 136]. The goal of SIcancellation in the analog domain is to place a null for the SI channel at the carrierfrequency or at the analog baseband signal. Such cancellation techniques are performedat the cancellation circuit in Figure 1.5, and they usually subtract an estimate of the SIsignal after performing a propagation domain suppression [27]. Moreover, they can alsobe used to tap the transmit signal into the digital domain (to apply adjustments digitally),and then convert it back to the analog domain to cancel the SI [10, 15, 16, 131, 137]. Tofurther reduce the SI, propagation domain suppression is also used along with analogdomain cancellation, and some authors report up to 80 dB using both techniques [138].Nevertheless, it may be prohibitive to implement advanced SI cancellation techniques inthe analog domain. Hence, this suggests that sophisticated cancellation techniques can beimplemented in the digital domain.

In the digital domain, SI cancellation is performed using an estimate of the SI signalfrom before and after each cancellation step, including passive and/or analog, as seen withthe feedback control in Figure 1.5. However, due to the limited ADC dynamic range the

10 Introduction

cancellation provided by digital cancellation alone is not sufficient to bring the SI signalto noise floor levels. This suggests that the SI signal should be reduced before the ADC.Hence, digital domain cancellation can be understood as the last line of defense against SI,whose goal is to mitigate any residual SI signal left from propagation- and analog-domaintechniques [10]. Since it is easier to perform heavy calculations in the digital domain,digital cancellation can model linear and nonlinear distortions of the signal, and then cancelthese components in the main signal. Digital cancellation is used along with propagationand analog cancellation to further suppress the SI, in which authors have shown thatusing separated antennas along with analog and digital cancellation, the system is ableto suppress 74 dB of the SI [16]. However, it was noted that digital cancellation should notbe used when analog cancellation performs well [16]. Interestingly, this happens due to thephase noise [136], which is a distortion introduced in the transmit signal when convertingthe digital signal to analog and using power amplifiers.

The best SI suppression so far appears to have been achieved by Bharadia et al. [17],where 110 dB of SI suppression was obtained with analog and digital cancellation in Wi-Fi networks with bandwidth of 20 MHz. According to the authors, such performancecan also be achieved in current Long Term Evolution (LTE) systems regardless of thefrequency band. More recently, SI suppression has achieved high levels also for mobiledevices [22], FD MIMO relays [23], FD and D2D nodes [34], in mmWave bands [25,120],wideband communications [139], and real time cancellation [140]. As an example ofpractical success, researchers from Stanford University have founded a startup company,Kumu Networks, to develop practical FD radios that are currently being deployed in smallcell BSs [141].

With these high levels of SI cancellation circuits, many works assume that either theSI is fully cancelled [31, 142], or some residual value is left [16, 28, 37, 39, 143]. Providedthat some residual value is left, some authors consider three different types of residualSI: fixed value independent of the transmitter power [39], fixed value dependent of thetransmitter power in a linear function [37, 143], and a random variable following a Riciandistribution dependent on the line-of-sight (LOS) coefficient [16, 28]. Throughout thisthesis, we consider the latter two types; for single-antenna scenarios we assume the residualSI is fixed and depends on the transmitter power in a linear manner; whereas in multi-antenna scenarios we assume that the residual SI follows the Rician distribution.

We conclude that the historical drawback of FD, the SI interference, remains importantand needs to be mitigated, but the recent developments show the maturity and feasibilityof FD communications in a vast range of practical scenarios.

1.1.3 Full-Duplex Applications in Cellular Networks

Cellular networks today operate with universal frequency reuse, i.e., every neighbouringcell uses the same set of frequency resources [144]. Using multiple techniques to mitigatethe interference from neighbouring cells, cellular networks are able to provide high datarates. In addition to universal frequency reuse, cellular networks operate in the HDmode, in which the BS transmits in the DL in a time-frequency resource and receivesin the UL in other time-frequency resource. However, with the ever increasing demand

1.1. Background 11

BS

UE1

UE2

UE3

UE4

SI

Freq. Channel 1

Freq. Channel 2

UE-to-UE Interf.Self-Interf. (SI)

(a) Single-antenna TNFD.

BS

UEd1

UEu1

UEd2

UEu2

SI

Uplink

DownlinkUE-to-UE Interf.

Self-Interf. (SI)

(b) Multi-antenna TNFD.

Figure 1.6: Examples of TNFD employing FD with two UE pairs, for single- and multi-antenna systems.

for higher data rates with lower latencies, universal frequency reuse is not enough. Analternative to go beyond universal frequency reuse between cells is to fully reuse the time-frequency resource within cells, which means to substitute current HD transmissions forFD transmissions.

With the recent research in SI cancellation, and considering the different in-band FDconfigurations in Figure 1.2, a viable introduction of FD technology in cellular networksis offered by TNFD. This configuration requires that, at least the wireless access points orBSs, implement FD transceivers to support the simultaneous DL and UL communicationwith two distinct UEs on the same frequency channel [11]. Although recent research havepointed to practical FD mobile users [22, 120], its implementation in modern phones hasstill a long road to travel. Due to this reason, the most common assumption in TNFDcellular networks is that only the BS is full-duplex capable, whereas the majority of usersremain HD. As shown in Figure 1.2, TNFD also suffers from the UE-to-UE interference,which is present because the UL user is transmitting in the same frequency resource as theone the DL user is receiving from the BS. The UE-to-UE interference depends on the userslocation and propagation effects, as well as on the transmit powers of UL users.

For single-antenna cellular networks, there are inherent constraints of orthogonalitywithin the same transmission direction. That is, every UL user – as well as the BS inthe DL – must transmit in a different frequency channel. To cope with this orthogonality,another challenge appears, which is how to pair UL and DL such that the assignmentof both sets of users to frequency channels maximize the desired performance indicator,e.g., the spectral efficiency or fairness. Figure 1.6a highlights a situation with four usersand two frequency channels. Notice that it is advantageous to pair users far apart, such asUE1-UE4 and UE2-UE3, and then assign these pairs the same frequency channel. Hence,the pairing and assignment of users to frequency channels must be carefully performed.Such assignment problems are a new technical challenge that was not present in traditional

12 Introduction

wireless networks, and that we originally propose in this thesis.For multi-antenna cellular networks, the challenges are different due to the different

interference scenarios at the BS or UEs. Along with the complicated SI channel andUE-to-UE interference, the transmitter and receiver distortions already present in single-antenna may increase due to coupling losses with multiple antennas. Moreover, given themulti-antenna nature, the antenna architecture plays an important role in establishing thenumber of antennas to transmit in the DL and to receive in the UL. Figure 1.6b highlightsthis situation, in which the BS uses a subset of the antennas for UL reception (in green)while the remaining (in purple) antennas are used for DL transmission. To cope with suchproblems, beamforming design and antenna assignment– to split or share an antenna– areimportant to mitigate the interferences while providing high system performance. Suchsplitting-sharing problems are a new technical challenge that was not present in traditionalwireless networks, and that we originally propose in this thesis.

To deal with these new challenges, coordination mechanisms that take into accountpairing, frequency assignment – to which both are usually named assignment– powerallocation, beamforming design and antenna splitting and/or sharing are crucial for FDcellular networks. Coordination mechanisms are important to reduce the impacts of thedifferent sources of interference in TNFD cellular networks, and also to optimize a desiredperformance indicator such as spectral/energy efficiency, and fairness. However, theirapplication changes remarkably if applied in single- or multi-antenna scenarios, and assuch, each scenario requires a different analysis.

1.1.4 Coordination Mechanisms for Single-Antenna Systems

For single-antenna scenarios, coordination mechanisms – such as power allocation andassignment – are important to reduce the impacts of the different sources of interferencein TNFD cellular networks. Typical and natural objectives for many physical layerprocedures for FD cellular networks are maximizing the sum spectral efficiency andfairness. In order to achieve the two main goals, the coordination mechanisms can usepower allocation [145–147], assignment [37, 148, 149] or a joint mechanism that takesboth into account [39, 40, 46, 150–152]. The main motivation for considering joint aspectsis to further improve the desired performance indicator.

Most of the works in FD cellular networks aim to maximize the spectral efficiencyand analyse the theoretical doubling that FD provides. In [145, 146], the authors analysethe rate gain region of TNFD and BFD considering only power allocation, whereas in[147] the authors analyse power allocation only in the UL. On the other hand, in [37] theauthors use scheduling to improve the spectral efficiency in a cellular environment, as wellas the pairing UL and DL of users [148, 149]. Some works also take into account bothaspects, power allocation and assignment, to maximize the spectral efficiency. In [46], theauthors use this joint approach in a heterogeneous network, whereas [39, 40, 150–152]have a similar approach but towards small cellular networks. However, some of the aboveworks tackle the assignment problem from a subcarrier or scheduling perspective, makesimplifications to the model, or provide heuristic solutions.

Furthermore, there is a need to move from a fully centralized architecture towards a

1.1. Background 13

more decentralized architecture [3]. The objective is to use the infrastructure of the BSto help the UEs to communicate in a distributed manner, reducing the processing burdenat the BS and the latency. In TNFD, the burden of the BS is further increased by theSI cancellation circuits, which increases the need of distributed solutions. Although theTNFD configuration remains centralized, the functions – such as power allocation andassignment– may be distributed. Few works have developed distributed algorithms that areapplicable in TNFD networks [153, 154]. The authors of [153] have tackled the problemof the UE-to-UE interference from an information theoretic perspective, without relatingto resource allocation and power control. In [154], the authors have proposed a distributedpower control for general wireless networks using approximation techniques, but they havenot taken into account the specific aspects of TNFD in cellular networks. Therefore, thereis a need for distributed solutions in TNFD for cellular networks.

Another important objective is to improve the fairness and per-user quality of service(QoS) of FD cellular networks, but little has been done as emphasized in [11, 143].The work in [11] emphasizes the importance of fairness and that it may degrade by afactor of two compared with HD communications. However, the authors do not providepower allocation and assignment schemes that are developed with such objectives in mind.In [143], a QoS provisioning framework within bidirectional FD configuration is proposed,but without considering the implications of TNFD transmissions.

We notice herein research gaps that have not been tackled by the community–distributed and fair solutions for TNFD- that have great impact in the application of single-antenna FD in cellular networks.

1.1.5 Coordination Mechanisms for Multi-Antenna Systems

Different from single-antenna cases, in multi-antenna systems the spatial domain is alsoavailable to tackle the UE-to-UE interference as well as the SI through directionalsuppression [105, 132]. Recall that the directional suppression can be tackled throughpropagation domain cancellation, which suggests that the antenna architecture –separatedor split– in the system’s performance is important. Hence, coordination mechanismsfor multi-antenna scenarios– such as beamforming, and antenna splitting/sharing– areimportant to help mitigate the interferences, distortions, and preserve spatial multiplexing.Due to these interferences, there is an inherent coupling between UL and DL structures thatcan be translated as a coupling between UL and DL beamformers. Such couplings lead tothe design of joint UL and DL beamforming, which are necessary to overcome interferencechallenges and improve the desired performance indicator.

Using beamforming design to optimize the desired performance indicator, some worksaim for spectral efficiency maximization [66, 67, 72, 155], while others consider energyefficiency [65, 68] or fairness [156, 157]. In [66], the authors formulate spectral efficiencymaximization problem for single-antenna UEs in a single-cell TNFD topology, whilein [67] the authors aim to maximize the weighted sum spectral efficiency in a BFD andsingle-cell TNFD topology with multi-antenna UEs. In [72, 155], the authors considermulti-cell TNFD topology with (weighted) sum rate maximization with transmitter andreceiver distortions due to non-linearities in the analog-to-digital and digital-to-analog

14 Introduction

converters [155], and channel uncertainties [72]. For energy efficiency, the authors in [65]aim to maximize the spectral and energy efficiency of a single-cell TNFD while assumingmultiple-antenna users and without taking into account the UE-to-UE interference.Differently, the authors in [68] aim to minimize sum power consumption as well asmaximize energy efficiency for multi-antenna UEs in a BFD topology while taking intoaccount the transmitter and receiver distortions. For fairness, the authors in [156] aim tomaximize a proportional fair version of UEs rate while considering single-cell TNFDtopology. In a different manner, in [157] the objective is to maximize max-min sumrate through the harmonic sum of signal-to-interference-plus-noise ratios (SINRs), whiletaking into account transmitter and receiver distortions, and perfect/imperfect channel stateinformation for multi-cell TNFD topology. A common assumption throughout the worksmentioned above is that the separate antenna architecture has already been decided, whichdisregards the problem of splitting or sharing antennas between UL and DL.

However, some works in the literature have addressed the problem of which andhow to use each antenna architecture [75, 105, 111, 158]. In [158], the objective is tomaximize sum spectral efficiency or minimize symbol error using antenna selection fora BFD topology. In a different manner, the authors in [105] use digital beamformingfor SI cancellation in a TNFD topology with large-scale antennas, and their experimentsdemonstrate that the UL/DL antenna splitting has high impact in the SI cancellationand spectral efficiency. Similarly, in [111] the authors propose an antenna splitting andbeamforming to minimize the gap between demand and achievable rates for TNFDtopology, while assuming a number of shared and separated antennas and ignoring the UE-to-UE interference. Differently, authors in [75] consider a separate antenna architectureand propose an antenna splitting solution to minimize the sum mean squared error (MSE).Although these works consider the impact of the antenna architecture in the optimization ofthe desired performance indicator, they do not consider the fundamental trade-offs betweenthe architectures in practical scenarios with antenna distortions, UE-to-UE interference,and the transmitter and receiver distortions.

The aforementioned works consider multi-antenna FD in the sub-6 GHz band, andgiven the practical experimental developments for SI cancellation in mmWaves [120,121],a natural question is to investigate the performance of FD in the mmWave band. Thetheoretical investigation has started recently [116, 119, 122–127], which considered RFD,TNFD, and also the use of D2D communications. Some works do not take into accountthe use of hybrid beamforming [116, 119, 123, 124], which is crucial for the practicalapplication of mmWaves. In [116], the authors aim to maximize the energy efficiency–through power allocation solutions– for RFD topology operating in the mmWave band.The authors in [119] propose analog beamforming solutions to maximize the sum spectralefficiency in TNFD topology. In [123], the authors aim to maximize the sum spectralefficiency and minimize the sum power through relay selection and power allocationin a RFD topology with D2D communications. In a different manner, the authorsin [124] take into account UL power allocation and user association in their goal ofinvestigating the impact of TNFD topology in the mmWave band for cell-center andcell-edge UEs. Nevertheless, a few works take into account the challenges of hybridbeamforming [122, 125–127]. In [122], the authors consider digital beamforming (such as

1.2. Problem Formulation 15

zero forcing) and beam steering solutions to mitigate the interferences in TNFD topology.The authors in [126] propose hybrid beamforming solutions to mitigate the SI in BFDtopology, whereas in [125] the authors aim to minimize the total data queue buffer in anultradense TNFD topology. Differently, in [127] the authors aim to maximize the energyand spectral efficiency through hybrid beamforming designs in RFD topology. However,none of the works above takes into account the practical aspects of analog beamforming,such as the use of low resolution phase shifters.

Once more, we again notice research gaps that have not been tackled by the literature–trade-offs in antenna splitting/sharing architectures and practical design of TNFD mmWave–which are important for the application of multi-antenna FD in cellular networks.

1.2 Problem Formulation

In this thesis we take into account single- and multi-antenna scenarios in FD cellularnetworks. For the single-antenna scenarios, we propose joint formulations of powerallocation and assignment – pairing and/or frequency assignment– in order to maximizefairness and the spectral efficiency. For multi-antenna scenarios, we derive antennaassignment, power allocation and beamforming – digital and hybrid– in order to maximizethe spectral efficiency for sub-6 GHz and mmWave systems. We can pose such differentscenarios and performance indicators in a joint formulation of a mixed integer nonlinearprogramming (MINLP) problem as

maximizeX,pu,pd

f0(X,pu,pd) (1.1a)

subject to fi(X,pu,pd) ≤ bi, ∀i ∈ I, (1.1b)

hj(X) ≤ cj , ∀j ∈ J , (1.1c)gl(X) = 0, ∀l ∈ L, (1.1d)X ∈ S. (1.1e)

The main optimization variables are pu, pd and X, where pu, pd may be the transmitpower or beamforming vectors for a number I of UL and a number J of DL users, and X isa matrix with entries in the discrete domain, which may represent an assignment betweenUL and DL users or quantized elements in a phase shifter. Notice that the assignmentmatrix may vary greatly depending on the scenarios and problems being proposed. Forsingle-antenna, its size depends on the fading the environment is experiencing, where infrequency selective fading X has three dimensions: the first two for UL and DL usersand the third for the frequency channel. For multi-antenna, its size may depend on thenumber of UL/DL antennas and the number of RF chains. The objective function (1.1a)of the problem depends on the three variables, and it is nonconvex for the problems ofthis thesis. Constraint (1.1b) represents a series of inequality constraints that depend oneither two or three optimization variables. The inequality function vector fi(X,p

u,pd)is nonconvex with respect to two or more variables, but it is convex with respect tothe individual continuous variables pu or pd. This constraint may represent minimum

16 Introduction

QoS requirement per UL and DL user, as well as maximum transmit power for ULusers and the BS. The other constraint function vector in Eq. (1.1c) represents the binaryinequalities, and embed the orthogonality between UL/DL users and frequency channelsor the constant modulus. This constraint may represent that a UL user can be associated toonly one DL user and frequency channel, and it is applied for all users and frequencychannels. The scalar constraintsin Eq. (1.1d) represent the binary equality constraints,which are present only in the multi-antenna scenario with constant modulus constraintdue to the hybrid beamforming in mmWave systems. The last constraint (1.1e) requiresthe assignment to be discrete. Usually, we assume that matrix X represents a binaryassignment between UL and DL users to frequency channels or antenna usage in theUL and DL. For FD mmWave system, we assume that matrix X has quantized matrixelements that are complex and belong to the unit circle. In addition, the set of constraintsI, J , and L are complementary, the inequality function vectors fi(X,p

u,pd) and hj(X)represent different functions, and all inequalities in Eqs. (1.1b)-(1.1e) are component-wise.The optimization problem (1.1) is difficult to solve because it has discrete and continuousvariables intertwined in nonconvex functions. In fact, we show that for some applicationsthe problem is non-deterministic polynomial-time (NP)-hard, i.e., that no polynomial timesolution – in the sense of optimality – for the problem is known. With this in mind, wewill use different optimization techniques to provide an approximated or close-to-optimalsolution, either centralized or distributed, for the maximization of two practical objectives:spectral efficiency and fairness.

1.2.1 Spectral Efficiency Maximization

One of the main promises of FD is to theoretically double the spectral efficiency by thetransmission and reception in the same frequency channel. With this, one of the mostimportant performance objectives is the spectral efficiency. In the sequel, we provide anexample of a spectral maximization problem for multi-antenna systems in the sub-6 GHzband.

We are interested in spectral efficiency maximization while taking into accountbeamforming, power allocation, and antenna assignment. To this end, we need to definesome parameters. Let hui ∈ CM×1 denote the complex channel vector comprising slowfading, shadowing, and path-loss between transmitter UE i and the BS, hdj ∈ CM×1

denote the channel vector between the BS and receiving UE j, and gij ∈ C denote theinterfering channel gain between the UL transmitter UE i and the DL receiver UE j (as seenin Figure 1.7). Let HSI ∈ CM×M denote the SI channel matrix from the transmit antennasin DL to the receive antennas in the UL, which is modelled as Rician fading [16, 66].To account for non-ideal circuitry, we consider an additional additive white Gaussiandistortion signal at the transmitter and receiver [64], which are modelled in the UL ascuj ∈ C and eu ∈ CM×1, and in the DL as cd ∈ CM×1 and edj ∈ C, respectively. Thetransmitted power in the UL is denoted by qui ∈ R, whereas in the DL the beamformingvector is denoted by wd

j ∈ CM×1. For the antenna assignment, the M antennas at the BSmay transmit and receive simultaneously. Hence, an antenna operating in a single direction,that is either transmitting in DL or receiving in UL, is in the split or separate mode. In

1.2. Problem Formulation 17

BS

UEd1

UEu1 UEd

2

UEu2

hu1

hu2hd

1

hd2

g11

g12

g22

g21

HSI

Uplink

DownlinkUE-to-UE Interf.

Self-Interf. (SI)

Figure 1.7: A multi-antenna cellular network employing FD with two UE pairs. TheBS may use simultaneously on UL and DL all of its antennas, represented by the colorgradient, which causes SI to all the antennas.

contrast, an antenna operating in both directions simultaneously is in full-duplex mode, asseen by the gradient color in Figure 1.7. To model this behaviour, we define two binaryassignment vectors, xu,xd∈{0, 1}M×1, for UL and DL, respectively. In the UL, xui = 1if antenna i is used on UL and xui = 0 otherwise. Similarly, in the DL xdj = 1 if antennaj is used on DL, and xdj= 0 otherwise. However, it is useful to transform the assignmentvectors into diagonal assignment matrices, such that Xu=diag (xu) and Xd=diag

(xd).

We can apply Xu to the received UL symbol yu, creating the effective received symbolyu = Xuyu. Similarly, we can apply Xd to the transmitted signal

∑Jm=1 wd

msdm+cd,

creating the effective transmitted signal Xd(∑J

m=1 wdms

dm+cd

).

Treating SI as noise, the SINR at the BS of transmitting user i and the SINR at thereceiving user j of the BS are given by

γui =qui

∣∣∣rui Hhui

∣∣∣2rui

HΨui rui

, γdj =

∣∣∣∣rdjHhdj

H

wdj

∣∣∣∣2∣∣rdj ∣∣2 Ψdj

, (1.2)

where rui ∈ CM×1 and rdj ∈ C are the linear decoders at the BS and at the DLuser j, respectively; and Ψu

i and Ψdj are the covariance matrix and variance of the total

18 Introduction

interference plus noise in the UL and DL, respectively, and are defined as

Ψui =

I∑l 6=i

qul hul hul

H+κ

I∑l=1

qul hul hul

H+

J∑j=1

HSI

(wdjw

dj

H+κdiag

(wdjw

dj

H))

HHSI

+ β

I∑l=1

qul diag(hul h

ul

H)+ β

J∑j=1

diag(HSIw

djw

dj

HHH

SI

)+ σ2Xu, (1.3)

Ψdj =

J∑m6=j

hdmH

wdmwd

m

Hhdm+κ

J∑m=1

hdmH

diag(wdmwd

m

H)

hdm+

I∑i=1

|gij |2 qui (κ+β+1)

+ β

J∑m=1

hdmH

wdmwd

m

Hhdm + σ2, (1.4)

where ηu∼CN(0M , σ

2IM)

and ηdj ∼CN(0, σ2

)are additive white Gaussian noise at the

BS and at DL user j, respectively. Thus, the achievable sum spectral efficiency (in bps/Hz),for the UL and DL are given as Rui =

∑Ii=1 log2(1 + γui ) and Rdj =

∑Jj=1 log2(1 + γdj ),

respectively.Our goal is to explore the theoretical limits of a proposed smart antenna assignment

between UL and DL antennas, which allows every antenna to be either shared or separate,and show how good the proposed smart architecture is. Specifically, we formulate the jointantenna assignment and transceiver design problem

maximizexu,xd,{qui }

{wdj },{r

ui },{r

dj }

I∑i=1

αui Rui +

J∑j=1

αdjRdj (1.5a)

subject to qui ≤ Pumax, ∀i, (1.5b)∑J

j=1tr(wdjw

dj

H)≤ P dmax, (1.5c)

xu,xd ∈ {0, 1}M×1. (1.5d)

The optimization variables are xu,xd, {qui }, {rui } for all i, and {wdj }, {rdj } for all

j. Constraints (1.5b)-(1.5c) limit the transmit power per-user and the total DL power,while constraint (1.5d) ensures that the assignment variables are binary. The optimizationproblem (1.5) is a mixed integer nonlinear programming problem, which is known fortheir high complexity and computational intractability. In this thesis, we notice that theantenna assignment problem alone is NP-hard, and we provide a close-to-optimal solutionfor problem (1.5) in Chapter C.

1.2.2 Fairness Maximization

Provided that we have an operational usage of FD in cellular networks, we need to ensuresome level of fairness for all the users in the system. However, fairness can be defined in

1.2. Problem Formulation 19

many ways [159], such as a minimum QoS guarantee for all users, a high average spectralefficiency for all users, or we can guarantee a high spectral efficiency for the user withminimum spectral efficiency. As an example of formulation, we provide an example ofsuch formulation in single-antenna systems.

To this end, we need to first define some parameters. Let the number of UEs in the ULand DL be denoted by I and J , respectively, which are constrained by the total number offrequency channels in the system F , i.e., I ≤ F and J ≤ F . Let Gib denote the effectivepath gain between transmitter UE i and the BS,Gbjf denote the effective path gain betweenthe BS and the receiving UE j on frequency channel f , andGijf denote the interfering pathgain between the UL transmitter UE i and the DL receiver UE j on frequency channel f ,and β as the residual SI cancellation coefficient. The vector of transmit power levels inthe UL by UE i is denoted by pu = [Pu1 . . . P

uI ], whereas the DL transmit powers by

the BS is denoted by pd = [P d1 . . . PdJ ]. Accordingly, we define the assignment matrix,

X ∈ {0, 1}I×J×F , such that xijf = 1 if the UL UEi is paired with the DL UEj andassigned to frequency channel f , and xijf = 0 otherwise. The SINR at the BS of transmituser i and the SINR at the receiving user j on the frequency channel f of the BS are givenby

γuif =Pui Gibf

σ2 +∑Jj=1 xijP

dj β

, γdjf =P dj Gbjf

σ2 +∑Ii=1 xijP

ui Gijf

. (1.6)

The achievable spectral efficiency for each user is given by the Shannon equation (inbits/s/Hz) for the UL and DL asCui =

∑Ff=1 log2(1+γuif ) andCdj =

∑Ff=1 log2(1+γdjf ),

respectively.In this thesis, fairness maximization is understood as the maximization of the minimum

spectral efficiency of all users, i.e., max-min spectral efficiency. Therefore, the jointfrequency assignment of UEs in the UL and DL to frequency channels and power allocationproblem to increase the fairness of the system is formulated as

maximizeXu,Xd,pu,pd

min∀i,j{Cui , Cdj } (1.7a)

subject toF∑f=1

γuif ≥ γuth, ∀i, (1.7b)

F∑f=1

γdjf ≥ γdth, ∀j, (1.7c)

Pui ≤ Pumax, ∀i, (1.7d)

P dj ≤ P dmax, ∀j, (1.7e)I∑i=1

xuif ≤ 1, ∀f, (1.7f)

F∑f=1

xuif ≤ 1, ∀i, (1.7g)

20 Introduction

J∑j=1

xdjf ≤ 1, ∀f, (1.7h)

F∑f=1

xdjf ≤ 1, ∀j, (1.7i)

xuif , xdjf ∈ {0, 1}, ∀i, j, f. (1.7j)

The optimization variables are pu, pd, Xu, and Xd. Constraints (1.7b) and (1.7c) ensurea minimum SINR to be achieved in the DL and UL, respectively. Constraints (1.7d)and (1.7e) limit the transmit power and constraints (1.7f)-(1.7i) assure that only one UE inthe UL and DL can use frequency channel f and that any given frequency channel is usedby at most one UE in the UL and one in the DL. Differently from the spectral efficiencymaximization in Section 1.2.1, this problem is MINLP and belongs to the category ofmulti-level nonlinear bottleneck assignments, which is also extremely complex. In thisthesis, we provide an approximate solution for problem (1.7) in Chapter A. In addition,we provide a centralized solution to a multi-objective problem that takes into account bothspectral efficiency and fairness maximization.

1.3 Contributions of the Thesis

In this thesis we analyse key performance measures to optimize and provide fundamentalinsights to FD communications in cellular networks. In the second part of the thesis, eachchapter is based on a published or submitted manuscript. We present below the publicationseach chapter is based on:

[J1] Jose Mairton B. da Silva Jr., Gabor Fodor, Carlo Fischione, “Spectral Efficientand Fair User Pairing for Full-Duplex Communication in Cellular Networks”, IEEETransactions on Wireless Communications, Vol. 15, No. 11, pp. 7578-7593, Nov.2016.

[J2] Jose Mairton B. da Silva Jr., Gabor Fodor, Carlo Fischione, “Fast-Lipschitz PowerControl and User-Frequency Assignment in Full-Duplex Cellular Networks”, IEEETransactions on Wireless Communications, Vol. 16, No. 10, pp. 6672-6687, October2017.

[J3] Jose Mairton B. da Silva Jr., Hadi Ghauch, Gabor Fodor, Mikael Skoglundand Carlo Fischione, “Smart Antenna Assignment is Essential in Full-DuplexCommunications”, submitted to IEEE Transactions on Communications, 2019.

[J4] Jose Mairton B. da Silva Jr., Ashutosh Sabharwal, Gabor Fodor, and CarloFischione, “1-bit Phase Shifters Suffice for Large-Antenna Full-Duplex mmWaveCommunications”, submitted to IEEE Transactions on Wireless Communications,2019.

1.3. Contributions of the Thesis 21

Table 1.1: Coordination mechanisms and performance objectives considered in thearticles and manuscripts reported in this thesis.

[J1] [J2] [J3] [J4]Combinatorial Prob. User/Freq. User/Freq. Antenna QuantizationPower Alloc./Control X X - -Beamforming - - X X

Objective Minimum SE Sum SE Sum SE Sum SE

Table 1.1 shows the main coordination mechanisms, design aspects and performancemeasures present in the publications. We consider different combinatorial problems, powercontrol/allocation, and beamforming as coordination mechanisms, which are necessaryto mitigate the effect of the SI and UE-to-UE interference and to maximize the desiredperformance measure in the spectral efficiency of the system. All publications consider acombinatorial problem, in which [J1] and [J2] provide a user-frequency assignment, [J3]presents an antenna assignment, and [J4] yields a quantization problem. For the single-antenna systems in [J1] and [J2], power allocation and power control are provided; whereasfor the multi-antenna systems in [J3] and [J4] beamforming solutions are derived. The twoobjective functions considered in this thesis are the sum spectral efficiency (SE) (in [J2],[J3] and [J4]), either weighted or with weights equal to one, and the minimum spectralefficiency of the system, aiming at a fair allocation of resources (in [J1]).

1.3.1 Fairness Maximization for Full-Duplex Cellular Networks

Chapter A is based on [J1] and studies fairness in FD for cellular networks. We proposea joint problem of user-frequency channel assignment and transmit power allocation,formulated as a mixed integer nonlinear optimization. The objective of our proposedproblem is to maximize the spectral efficiency of the user with the lowest achievedspectral efficiency while respecting minimum QoS constraints, aiming at a fair and efficientcommunication. The proposed problem is NP-hard, implying that no optimal solutionin polynomial time can be obtained. Thus, we apply Lagrangian duality, allowing us toprovide the optimal power allocation and an initial solution for the assignment. However,the assignment cannot be solved efficiently, because this problem is also NP-hard, butnow with respect to the assignment variables. Then, we propose an approximated greedysolution to the assignment problem, and prove that the duality gap between the greedysolution with optimal power allocation and the primal solution is bounded and diminishesas the number of frequency channels increases. The numerical results show that ourproposed solution, named joint assignment and fairness maximization algorithm (JAFMA),improves the spectral efficiency of the users with low spectral efficiency in a wide rangeof scenarios with different users load and residual SI powers. The key finding is that theoptimization of the assignment and power allocation should be solved jointly; otherwise, arandom allocation with equal power allocation achieves a similar performance as the one

22 Introduction

provided by the treating the assignment and power allocation problems separately.

1.3.2 Distributed Power Control for Spectral Efficiency Maximization

Chapter B is based on [J2] and we study the sum spectral efficiency maximization problem,but aiming at a distributed power control solution. The focus of the problem is on jointuser-frequency channel assignment and power control, and the problem is formulated as aMINLP optimization. The joint problem is NP-hard, and to find an approximated solution,we decompose it into two parts that correspond to the user-frequency channel assignmentproblem and power control problem, respectively. However, the assignment problem isalso NP-hard. Then, we propose a greedy algorithm with guaranteed performance withrespect to the optimal solution. To solve the power control problem, we develop a novelpower control mechanism especially suited for FD networks. The proposed power controlscheme is a distributed algorithm that sets the SINR targets at each receiver such that theachieved sum rate is close-to-optimal. The sum rate maximization problem is non-convexand we therefore used Fast-Lipschitz (FL) optimization [160] to solve it in a distributedand fast manner. The numerical results show that our proposed solution, named G-FLIP,outperforms existing methods in the literature for interference-limited scenarios, providingsum spectral efficiency gains and savings in terms of power consumption. Our key findingis the trade-off between assignment and power control for different interference-limitedscenarios; a smart assignment solution provides more gains in interference-limited regimethan smart power control solution, whereas a power control solution is more important thanusing a smart assignment solution in the SI-limited regime.

1.3.3 Smart Antenna Assignment for Spectral Efficiency Maximization

Chapter C is based on [J3] and we study the fundamentals of antenna architectures thatallow FD communications to be realized in practice. To achieve simultaneous transmissionand reception in FD, there are two architectures for separating the signals which consist ofsplitting or sharing the available antennas, respectively. Then, we focus on understandingthe impact of the two antenna architectures for FD communications on key performancemetrics such as the achieved spectral efficiency. To this end, we propose a third architecturethat we call smart architecture, which is an intermediate between the two aforementionedarchitectures. We investigate the benefits of such architecture by a weighted sum spectralefficiency maximization problem, which includes an assignment problem to optimallyassign antennas, beamforming and power allocation. The proposed problem is MINLP andwith high complexity, and we use block coordinate descent to iteratively solve it. However,the antenna assignment problems are NP-hard and to obtain an approximate solution weresort to semidefinite relaxation. The numerical results indicate that the proposed smartarchitecture outperforms HD solutions and the equal splitting of the antennas betweenUL and DL. Moreover, results show that the proposed solution is also robust with respectto effects of the residual SI. Our key finding is that smart antenna assignment, includingsharing/splitting of antennas between UL and DL, is essential to FD communications.

1.3. Contributions of the Thesis 23

1.3.4 Low Resolution Phase Shifters for Practical Full-DuplexMillimeter Wave

Chapter D is based on [J4] and we study the practical aspects of FD mmWave systems,especially with the use of quantized phase shifters. In this chapter we inquire whether lowresolution phase shifters are sufficient for FD mmWave systems. To address this question,we formulate the problem of SI suppression and downlink beamforming as a MINLP. Dueto the coupling constraints introduced by hybrid beamforming and the constant modulusconstraint of analog precoders/combiners, we resort to penalty dual decomposition (PDD)to obtain a solution termed LowRes. Using PDD, we provide a near-optimal solution forlow resolution phase shifters and stationary solution for infinite resolution phase shifters.The numerical results show that the proposed LowRes using a single quantization bitoutperforms HD with infinite resolution for scenarios with high and low residual SI power.In addition, with an increasing number of RF chains the optimality gap between low andinfinite resolution phase shifters decreases; whereas with an increasing number of antennasthe optimality gap between low and infinite resolution phase shifters increases. Our keyfinding is that even single quantization bit phase shifters are sufficient for FD mmWavesystems.

1.3.5 Contributions not Covered in the Thesis

In addition the manuscripts listed above, I have worked in other topics and published othermanuscripts during my PhD studies. The following publications are not covered in thethesis to keep the consistency and length. The manuscripts are the following:

[BC1] Zexian Li, Fernando Sanchez Moya, Gabor Fodor, Jose Mairton B. da Silva Jr.,Konstantinos Koufos, “Device-to-Device (D2D) Communications”, in A. Osseiran,J.F. Monserrat, P. Marsch (eds.), “5G Mobile and Wireless Communications Tech-nology”, Cambridge University Press, pp. 107-136, 2016.

[J5] Jose Mairton B. da Silva Jr., Gabor Fodor, “A Binary Power Control Schemefor D2D Communications,” IEEE Wireless Communications Letters, Vol. 4, No.6, pp. 669-672, Dec. 2015. IEEE Best Readings Topics on Device-to-DeviceCommunications

[J6] Gabor Fodor, Sandra Roger, Nandana Rajatheva, Slimane Ben Slimane, TommySvensson, Petar Popovski, Jose Mairton B. da Silva Jr., Samad Ali, “An Overviewof Device-to-Device Communications Technology Components in METIS”, IEEEAccess, Vol. 4, pp. 3288-3299, 2016.

[J7] Rodrigo L. Batista, Carlos Filipe M. e Silva, Tarcisio F. Maciel, Jose Mairton B. daSilva Jr., Francisco R. P. Cavalcanti, “Joint Opportunistic Scheduling of Cellular andDevice-to-Device Communications,” Journal of Communication and InformationSystems, Vol. 32, no. 1, 2017.

24 Introduction

[C1] Jose Mairton B. da Silva Jr., Yuzhe Xu, Gabor Fodor, Carlo Fischione, “DistributedSpectral Efficiency Maximization in Full-Duplex Cellular Networks”, in Proc. IEEEInternational Conference on Communications Workshops (ICC’16), Kuala Lumpur,Malaysia, May 2016.

[C2] Jose Mairton B. da Silva Jr., Gabor Fodor, Carlo Fischione, “On the Spectral Effi-ciency and Fairness in Full-Duplex Cellular Networks”, in Proc. IEEE InternationalConference on Communications (ICC’17), Paris, France, May 2017.

[C3] Jose Mairton B. da Silva Jr., Hadi Ghauch, Gabor Fodor, Carlo Fischione, “Howto Split UL/DL Antennas in Full-Duplex Cellular Networks”, in Proc. IEEEInternational Conference on Communications Workshops (ICC’18), Kansas City,MO, USA, May 2018.

[C4] Jose Mairton B. da Silva Jr., Ashutosh Sabharwal, Gabor Fodor, Carlo Fischione,“Low Resolution Phase Shifters Suffice for Full-Duplex mmWave Communica-tions”, invited paper in Proc. IEEE International Conference on CommunicationsWorkshops (ICC’19), Shanghai, China, May 2019.

Chapter 2

Preliminaries

This chapter summarizes the background theory used in the thesis. In particular, Section 2.1introduces the Hungarian algorithm, a centralized solution to assignment problems.Section 2.2 we discuss the framework of Fast-Lipschitz optimization, a method to derivea distributed solution for nonconvex optimization problem. Section 2.3 presents thefundamentals of block coordinate descent, a method that provides solutions to nonconvexoptimization problems that have a block structure. In Section 2.4 we present semidefiniterelaxation, a relaxation method to provide approximated solutions for nonconvex quadraticproblems. Then, in Section 2.5 we discuss the framework of penalty dual decomposition, ageneralization of block coordinate descent for solving nonconvex problems with couplingconstraints.

2.1 Hungarian Algorithm

In this section, we summarize an efficient centralized method for solving assignmentproblems optimally and in polynomial time [161–163]. Let us suppose there are n agentsthat need to be assigned to n locations in a one-to-one basis. For each association, wedefine the costs aij to represent the cost expenses of assigning agent i to location j, whichare collected in the matrix A ∈ R. The objective of this problem is to minimize the overallcost of transportation, i.e., the association between agents to locations need to be performedsuch that the overall cost of transportation is minimized.

We can formulate this assignment problem as a linear optimization with binaryvariables as

minimizeX

∑I

i=1

∑J

j=1aijxij (2.1a)

subject to∑I

i=1xij = 1, ∀j, (2.1b)∑J

j=1xij = 1, ∀i, (2.1c)

xij ∈ {0, 1}, ∀i, j, (2.1d)

25

26 Preliminaries

where the assignment matrix X ∈ {0, 1}n×n is the optimization variable. This assignmentproblem can also be seen as a 2D assignment problem, which represents the twodimensions that we have to assign. The Hungarian algorithm is a polynomial timecentralized solution for this problem [161,162], and it was named by Kuhn [161] in honorof the work of Konig and Egervary on which it is based. The algorithm assigns in anoptimal manner the agents to the locations in O(n3) operations [163].

For the sake of simplicity, we present below the summarized steps of the Hungarianalgorithm in matrix form [163]:

Step 1 Subtract the smallest aijmin in each row of A from all the entries of its row.

Step 2 Similarly to Step 1, subtract the smallest entry aiminj in each column of A from allthe entries of its column.

Step 3 Cover rows and columns with the minimum number of lines so that all the zeroentries of matrix A are covered. For simplicity, let us denote this minimum numberas l.

Step 4 If l = n, we have an optimal assignment of zeros and we are finished. However,l < n, an optimal assignment of zeros is not possible yet, and we proceed to thenext step.

Step 5 Find the smallest entry aij not covered by any line. Subtract this entry from eachuncovered row, and then add it to each covered column. Go back to Step 3.

We use the Hungarian algorithm as a benchmarking solution for the pairing of UL and DLusers in Chapter B.

2.2 Fast-Lipschitz Optimization

In this section, we give an overview of the Fast-Lipschitz optimization framework, whichwas introduced in [160]. The use of this optimization framework is advantageous fornetwork applications that require distributed and fast solutions to problems that may not beconvex. Using fixed-point iterations, Fast-Lipschitz optimization converges to the optimalsolution while assuming that certain qualifying conditions– not related to the convexity ofthe problem– are met. For a thorough discussion of Fast-Lipschitz problems we refer theinterested reader to the following articles [160, 164, 165].

Definition 1. A problem is said to be in the Fast-Lipschitz form if it can be written as

maximizex

f0(x)

subject to xi ≤ fi(x) ∀i ∈ Axi = fi(x) ∀i ∈ B,

(2.2)

where

• f0(x) :Rn → Rm is a differentiable scalar (m=1) or vector valued (m≥2) function;

2.2. Fast-Lipschitz Optimization 27

• A and B are complementary subsets of {1, . . . , n}, i.e., A ∪ B = {1, . . . , n} andA ∩ B = ∅;

• The functions fi : f0(x) :Rn → R are all differentiable.

From the individual constraint functions, we form the vector valued function f:Rn→Rnas

f(x)=[f1(x) · · · fn(x)

]T.

One characteristic of Fast-Lipschitz optimization is that each variable xi is associated withone constraint fi(x), although this does not reduce the generality of the approach becauseone can always introduce redundant constraints satisfying such characteristic. The formx ≤ f(x) is general because any constraint on canonical form g(x) ≤ 0 can be written asx ≤ x− γg(x) for some γ > 0.

Definition 2. We restrict our attention to a bounding box D={x ∈ Rn |a ≤ x ≤ b} . Weassume D contains all candidates for optimality and that f maps D into D, f :D→D. Thisbox arises naturally in practice, since any real-world decision variable, such as transmittingpower, must be bounded.

Definition 3. A problem is said to be Fast-Lipschitz when it can be written on Fast-Lipschitz form and admits a unique Pareto optimal solution x∗, defined as the uniquesolution to the system of equations

x∗ = f(x∗). (2.3)

A problem written on FL form is not automatically Fast-Lipschitz. We present belowthe first qualifying conditions proposed in [160] that a problem in Fast-Lipschitz formneeds to fulfil. Many other qualifying conditions have been proved since [160], and for thisreason we will refer to the following conditions as old.

Old Qualifying Conditions. For all x ∈ D, f0(x) and f(x) must fulfil at least one of thefollowing cases, either (0) and (i) or (0) and (ii):

(0) ∇f0(x) > 0

AND (i. a) ∇f(x) ≥ 0(i. b) |||∇f(x)||| < 1

OR (ii. a) f0(x) = c1Tx

(ii. b) ∇f(x) ≤ 0, or more generally,∇f(x)2 ≥ 0(ii. c) |||∇f(x)|||∞ < 1

OR (iii. a) f0(x) ∈ R

(iii. b) |||∇f(x)|||∞ < δδ+∆

, where δ , mini minx∈D∇if0(x), and ∆ ,maxi maxx∈D∇if0(x).

28 Preliminaries

Theorem 1 (Fischione [160]). A problem in Fast-Lipschitz form that fulfils any pair of theOld Qualifying Conditions is Fast-Lipschitz, i.e., it has a unique Pareto optimal point givenby x∗ = f(x∗). Furthermore, x∗ can be found as the limit of the iterations xk+1 = f(xk).

Therefore, once we know the problem is Fast-Lipschitz, to obtain the solution we onlyneed to solve the system of equations (2.3). In general, solving this system of equationsis much easier than solving an optimization problem using standard Lagrangian duality.Specifically, the qualifying conditions assure that f(x) is contractive on D, which impliesthat the iterations x∗ := f(x∗) converge geometrically to the optimal point x∗ startingfrom any point x0 ∈ D.

We use the Fast-Lipschitz framework to develop a distributed power control solutionin Chapter B.

2.3 Block Coordinate Descent

In this section, we present a summary of block coordinate descent (BCD) methods,also known as nonlinear Gauss-Seidel methods, alternating optimization, or coordinatedescent [166]. BCD methods are represent solution approaches to solve optimizationproblems that have a block structure –groups of variables– which are frequent inapplications for signal processing, machine learning, and data analysis. For an in-depthdiscussion of BCD methods, we refer the interested reader to the following articles [166–168].

Let us consider the problem

minimizex

f(x) (2.4a)

subject to x ∈ X , (2.4b)

where

• f : Rn → R is a continuously differentiable function;

• x ∈ Rn is partitioned in m blocks x = (x1,x2, . . . ,xm) with n =∑mi=1 ni;

• X is the Cartesian product of m closed and nonempty subsets Xi ⊆ Rni ;

• Constraint (2.4b) is equivalent to xi ∈ Xi, for all i = 1, 2, . . . ,m.

The BCD solution consists of the following iterative approach:

xk+1i , arg min

zi∈Xi

f(xk1 ,xk2 , . . . ,x

ki−1, zi,x

ki+1, . . . ,x

km), (2.5)

which updates in a specified order the block components of x, and generates the sequenceof updates {xk} with xk = (xk1 ,x

k2 , . . . ,x

km). Hence, the solution using BCD consists of

obtaining a solution for a variable (xi) of the block, while the remaining are fixed, andthen continue the iterative process of obtaining a solution for the remaining blocks. Each

2.3. Block Coordinate Descent 29

iteration for the variables in the block represents a lower-dimensional version of the full-problem, and thus its solution may be easier than in the full-problem. Moreover, the ruleof updates are quite general, in which the most used ones are the following:

• Cyclic: Select i0 = 1 and ik+1 = [ik mod n], for k = 0, 1, 2, . . .

• Randomized: Select index ik with uniform probability from {1, 2, . . . , n}, with orwithout replacement

The convergence of BCD is the subject of intensive research, each with a specific set ofassumptions about the objective function f(x) or even constraint set X . Herein, we presentthe standard convergence results. Before presenting the convergence result, we first needsome definitions [167, 169, 170].

Definition 4. Assume that f : Rn → R is continuously differentiable. Then

i) f is quasiconvex if for every x,y ∈ Rn, λ ∈ [0, 1] we have

f(x + (1− λ)y) ≤ max{f(x), f(y)}, (2.6)

and if strict inequality holds with λ ∈ (0, 1), f is strictly quasiconvex;

ii) f is pseudoconxex if for every x,y ∈ Rn, we have

∇f(x)T(y − x) ≥ 0⇒ f(y) ≥ f(x). (2.7)

iii) A stationary point, or critical point, x ∈ X is the point that satisfies∇f(x)T

(y−x) ≥0, for every y ∈ X .

Definition 5. The level set L0X of a function f : Rn → R relative to X , corresponding to

a given point x0 ∈ X , is defined as

L0X , {x ∈ X |f(x) ≤ f(x0)}. (2.8)

The following theorem gives the basic convergence result for the BCD method.

Theorem 2 (Bertsekas [166]). Suppose that the function f is continuously differentiableover the convex set X in Eq. (2.4b). Furthermore, suppose that for each i and xi ∈ Xi, theminimum in Eq. (2.5) is uniquely attained. Let {xk} be the sequence generate by the BCDin Eq. (2.5). Then, every limit point of {xk} is a stationary point of Problem (2.4).

Note that Theorem 2 requires that each minimum in Eq. (2.5) be uniquely attained,which is fulfilled for strictly convex functions. However, even convexity may not beattainable for a block variable in the objective function. For these situations, the followingtheorems are suggested.

Theorem 3 (Grippo [167]). Suppose that the function f is strictly quasiconvex with respectto xi ∈ Xi, for each i = 1, . . . ,m − 2, and that the sequence {xk} generated by theBCD in Eq. (2.5) has limit points. Then, every limit point of {xk} is a stationary point ofProblem (2.4).

30 Preliminaries

Theorem 4 (Grippo [167]). Suppose that the function f is pseudoconvex on X and thatL0X is compact. Then, the sequence {xk} generated by the BCD in Eq. (2.5) has limit

points and every limit point of {xk} is a global minimizer of Problem (2.4).

Note that Theorems 3-4 are more general than Theorem 2 because they relax theassumption of strict convexity in all blocks to strictly quasiconvex for m − 2 blocks andpseudoconvexity of f , respectively.

We use the BCD method to develop a joint beamforming, power allocation, and antennaassignment solution in Chapter C.

2.4 Semidefinite Relaxation

In this section, we present a summary of the semidefinite relaxation (SDR) technique,which is a powerful, computationally efficient approximation technique to solve compli-cated, usually NP-hard, optimization problems. The SDR technique is applied to obtainapproximated solutions to many nonconvex quadratically constrained quadratic programs,i.e., problems with quadratic objective functions and constraints. Its application range isbroad, including beamforming design, speech recognition, and maximum cut problems ingraph theory. For a broader overview of SDR, we refer the interested reader to the followingarticles [171, 172].

Let us consider the real-valued homogeneous problem

minimizeX∈ Sn

xTCx (2.9a)

subject to xTAix Di bi, ∀i = 1, . . . ,m, (2.9b)

where Di can represent either ≥, =, or ≤ for each i; matrices C,A1, . . . ,Am ∈ Sn, inwhich Sn denotes the set of all real symmetric n×n matrices; and b1, b2 . . . , bm ∈ R. Theproblem definition above is for real-value x, but it can be generalized to complex-valuedx as well. Notice that we do not require the matrices C,A1, . . . ,Am ∈ Sn to be positivesemidefinite, which makes problem (2.9) nonconvex and usually NP-hard. An importantstep to derive the SDR are the following observations:

(i) xTCx = tr(CxxT

)along with xTAix = tr

(AixxT

);

(ii) X = xxT is equivalent to X � 0 and rank (X) = 1.

Using these observations, we can rewrite problem (2.9) in an equivalent form

minimizeX∈ Sn

tr (CX) (2.10a)

subject to tr (AiX) Di bi, ∀i = 1, . . . ,m, (2.10b)X � 0, rank (X) = 1. (2.10c)

With the problem formulation in (2.10), we identify that the fundamental difficulty insolving the problem is the rank constraint rank (X) = 1. If we drop the constraint, we

2.4. Semidefinite Relaxation 31

have the following relaxed version of problem (2.10)

minimizeX∈ Sn

tr (CX) (2.11a)

subject to tr (AiX) Di bi, ∀i = 1, . . . ,m, (2.11b)X � 0. (2.11c)

In fact, problem (2.11) is known as an SDR of problem (2.10), where the namederives from the fact that problem (2.11) is a semidefinite programming problem. Usingproblem formulation (2.11), we can obtain an efficient and convenient solution X?

using commonly available interior-point algorithms with a worst-case complexity ofO(max{m, n}4n1/2 log(1/ε)

)for a given solution accuracy ε > 0 [171].

However, there is still the fundamental issue of converting the globally optimal solutionX? to problem (2.11) into a feasible solution x to problem (2.9). If X? has rank one,then we can write X? = x?x?T and x is feasible and the globally optimal solution toproblem (2.9). If X? has rank larger than one, we need to obtain a rank one solution inan efficient manner such that vector x is feasible to problem (2.9). Notice that we retrievea feasible solution x from X?, which in general is not optimal to problem (2.9). To obtainthe rank-one approximate solution x, we detail two specific approaches that use eigen-decomposition and randomization.

For the eigen-decomposition, let us denote r = rank (X?) and write the eigen-decomposition of X? as

X? =∑r

i=1λiqiqi

T, (2.12)

where λ1≥λ2≥ . . .≥λr>0 are the eigenvalues and q1, . . . ,qr ∈ Rn are the respectiveeigenvectors. One straightforward approach is to retrieve a rank-one approximation fromX?

1 to X given by X?1 = λ1q1q1

T. Then, we define x =√λ1q1 as our candidate solution

to problem (2.9), and if x is not feasible, we project it into the feasible set of problem (2.9).The randomization approach consists of picking a random vector ξ ∈ Rn drawn

according to the Gaussian distribution with zero mean and covariance X, i.e. ξ ∼ N (0,X).The intuition of the randomization lies in the following stochastic quadratic problem:

minimizeX∈ Sn

Eξ∼N (0,X)ξTCξ (2.13a)

subject to Eξ∼N (0,X)ξTAiξ Di bi, ∀i = 1, . . . ,m. (2.13b)

Notice that by assumption X = E{ξξT}, which implies that the stochastic problem (2.13)is equivalent to the original quadratic problem (2.9). Hence, after obtaining an optimalsolution X? to problem (2.11), we can generate a random vector ξ ∼ N (0,X) anduse it to construct an approximate solution to problem (2.13). Usually, multiple randomsamplings are generated and we can pick only the best approximate solution among theones generated.

Although the randomization approach seems a simple heuristic, there are strongapproximation bounds for many applications. One of such examples is the Boolean

32 Preliminaries

quadratic maximization problem, which is defined as

vQP = maximizeX∈ Sn

xTCx (2.14a)

subject to x2i = 1, ∀i = 1, . . . ,m, (2.14b)

whose optimal objective value of the quadratic problem is vQP and assuming C � 0.Problem (2.14) is NP-hard in its current form and also for the minimization form. In [173],it is shown that when Cij ≤ 0 for all i 6= j, the following approximation holds

γvSDR ≤ vQP ≤ vSDR, (2.15)

where γ = 0.87856, and vSDR denotes the objective value of the optimal SDR, i.e.vSDR = tr (CX?). The result above tell us that using SDR with randomization, we obtaina solution whose expected objective value is at least 0.87856 times the optimal solutionof problem (2.14). Unfortunately, to the best of our knowledge there is no approximationbound for the Boolean quadratic minimization problem version.

In Chapter C in this thesis, we use SDR with randomization to obtain a feasible solutionfor the Boolean quadratic minimization problem.

2.5 Penalty Dual Decomposition

In this section, we present a brief summary of the PDD method, which is a generalization ofthe BCD method to optimization problems with nonconvex nonsmoooth objective functionsubject to nonconvex coupling constraints. Since we are interested in applications withcontinuously differentiable objective functions, we leave out the results of nonsmoothobjective functions for the interested reader. For a detailed presentation of the PDD method,we refer the reader to the following articles that proposed, derived its important results, andpresented interesting applications to signal processing and machine learning [174, 175].

Let us consider the following problem

minimizex

f(x) (2.16a)

subject to h(x) = 0, (2.16b)gi(xi) ≤ 0,∀i = 1, 2, . . . , n, (2.16c)x ∈ X , (2.16d)

where

• f : Rn → R is a scalar continuously differentiable function;

• h(x) : Rn → Rp is a vector continuously differentiable function;

• gi(xi) : Rn → Rqi is a vector continuously differentiable function with q=∑ni=1qi;

• x ∈ Rn is partitioned in m blocks x = (x1,x2, . . . ,xm) with n =∑mi=1 ni;

2.5. Penalty Dual Decomposition 33

Algorithm 1 PDD method1: Input: x0, δ1, λ1, η1, ε1, ν, θ1, θ2 and set k = 12: while

∥∥h(xk−1)∥∥∞ < ν do

3: xk = ORACLE(L(xk−1,λk, δk), εk

)4: if

∥∥h(xk)∥∥∞ < ηk then5: λk+1 ← λk + 1

δkh(xk)

6: δk+1 ← δk

7: ηk+1 ← θ1 min{ηk,∥∥h(xk)∥∥∞}

8: else9: λk+1 ← λk

10: δk+1 ← θ2δk

11: end if12: k ← k + 113: end while14: Output: xk

• X is the Cartesian product of m closed and nonempty subsets Xi ⊆ Rni ;

• Constraint (2.16d) is equivalent to xi ∈ Xi, for all i = 1, 2, . . . ,m.

Due to coupling constraint (2.16b), problem (2.16) cannot be solved using the BCDmethod. To overcome this difficulty, PDD uses the augmented Lagrangian approach tosolve a modified dual version of problem (2.16). Then, let us define the augmentedLagrange function L(x,λ, δ) as

L(x,λ, δ) , f(x) + λTh(x) +1

2δ‖h(x)‖22 , (2.17)

where λ is the dual variable, δ is the penalty parameter, and we assume xi ∈ Xi withXi , {xi|gi(xi) ≤ 0,xi ∈ Xi}. Using the definition of the augmented Lagrange function,we define the augmented Lagrangian problem as

minimizexi∈Xi

L(x,λ, δ). (2.18)

Using the augmented Lagrange function, the PDD method approximately solves theaugmented Lagrangian problem using an oracle in the inner loop, whereas in the outerloop it updates the dual variable λ or penalty parameter if necessary.

Hence, PDD method is double-loop iterative method whose algorithmic solution isshown in Algorithm 1. The initialization of the algorithm consists of a feasible solution x0,a dual variable λ > 0 (component-wise), penalty parameter δ1 > 0, constraint violation0 < η1 < 1, termination criteria 0 < ε1 < 1, and constant parameters 0 < θ1 < 1 and0 < θ2 < 1 to control the decreasing speed of the constraint violation and terminationcriteria. In line 3, PDD uses an oracle to obtain the solution with some accuracy εk

starting from the previous iteration xk−1 and using current values of the dual variableλk and penalty parameter δk. Then, if the constraint violation

∥∥h(xk)∥∥∞ is smaller than

a predefined threshold ηk, the algorithm updates the dual variable λk while it maintains

34 Preliminaries

the penalty parameter δk and diminishes the predefined threshold ηk using θ1 (see lines 5-7). Otherwise, the dual variable λk remains unchanged while the penalty parameter ηk isdiminished using θ2 (see lines 9-10). The stopping criteria for the PDD method is definedin line 2, and it states that the algorithm stops when the constraint violation is smaller thana predefined threshold ν with ν � η0.

An important part of PDD is the oracle in line 3, and this oracle can be the classicalBCD method presented in Section 2.3, or an inexact variant of BCD, such as the randomblock successive upper-bound minimization (rBSUM) [174, 176]. For the terminationcriteria of the oracle, we can use the an option based on updates in the block variablexk and another based on the progress of the augmented Lagrangian function. For updatesbased on the block variable, let us define

ek , PX {xk −∇xL(x,λ, δ)−∇g(xk)µk} − xk, (2.19)

where PX {y} denotes the projection of vector y to set X ; g(xk) , (gi(xi)) for all i =1, . . . , n; and µk , (µki )i for all i denotes the Lagrange multipliers associated with theconstraint (2.16c). Then, the two termination criteria are∥∥ek∥∥∞ ≤ εk,∀k, (2.20a)∣∣∣L(xk,λk, δk)− L(xk−1,λk, δk)

∣∣∣∣∣∣L(xk−1,λk, δk)∣∣∣ ≤ εk,∀k. (2.20b)

Before presenting the main convergence result of PDD, we need to describe theregularity conditions assumed for its convergence, which are Robinson’s and KKTconstraints qualification conditions [177]. Robinson’s condition is a general first-orderoptimality condition for nonlinear, not necessarily convex, problems [177]. However,Robinson’s condition is hard to check for many problems. In some situations, Robinson’scondition is equivalent to other constraint qualifications that may be easier to check.One of such cases is the Mangasarian-Framowitz constraint qualification [166], which isequivalent to Robinson’s condition if the domain is Rn. Such conditions require an in-depth knowledge of convex analysis, that is out of scope herein, so we refer the interestedreader to [166,177] for general results; and to [174] for specific results with both constraintqualifications that are relevant to PDD methods.

Hence, the convergence of the PDD method is established in the following theorem.

Theorem 5 (Shi [174]). Let {xk,µk}. The termination condition for the optimizationoracle involved in Algorithm 1 is ∥∥ek∥∥∞ ≤ εk,∀k, (2.21)

with εk, ηk, δk → 0 as k → ∞. Suppose that x? is a limit point of the sequence {xk}and at the limit point x? the Robinson’s condition holds for problem (2.16). Then, x?

satisfies h(x?) = 0, and it is KKT point of problem (2.16) that satisfies the KKT qualifyingcondition.

2.5. Penalty Dual Decomposition 35

Therefore, using the PDD method we are able to obtain a KKT point of problem (2.16),which is a stationary solution.

In Chapter D in the thesis, we use PDD method to obtain an optimal and suboptimalsolution for a nonconvex optimization problem with coupling constraints.

Part II

Included Papers

37

Part III

Conclusions and Future Works

171

Chapter 1

Conclusions

Everything created has a beginning, Destiny of theEndless... as everything created has an end.

Neil Gaiman

Our goal in this thesis was to optimize and provide fundamental insights to FD cellularnetworks. To this end, we formulated different research questions such as the ones inChapter 1; analysed key performance indicators, such as spectral efficiency and fairness;and proposed various solutions for different scenarios with FD communications, such assingle- and multi-antenna and for sub-6 GHz and mmWave bands. The challenges andapplications of FD are very different in single- and multi-antenna scenarios, and as suchwe derived results and provided insights for these two different categories.

In single-antenna systems, we faced the challenges of user-frequency channel assign-ment, and power allocation or control in order to maximize the desired performanceindicator. In the first two manuscripts [J1] and [J2], we considered a single-antennascenario in which the main questions were the feasibility of fairness and spectral/energyefficient solutions given UE-to-UE interference- and SI-limited scenarios. In [J1], wetackled fairness as performance indicator through an optimization problem to maximizethe minimum spectral efficiency achieved in the system while guaranteeing a minimumQoS to users. We proposed a solution to this problem, including user-frequency channelassignment and power allocation, that improved the minimum spectral efficiency andachieved the highest ratio of connected users in a wide range of scenarios. We obtainedfundamental insights related to the role of SI in the admission of users to share a frequencychannel, and the essential role of assignment and power allocation in achieving fairnessof the system. In [J2], we aimed spectral efficiency as performance indicator throughan optimization problem using distributed power control. We proposed a solution to thisproblem, including a centralized user-frequency channel assignment, that outperformedHD in terms of spectral efficiency in an interference-limited regime, and also obtainedhigh energy efficiency. In this work, the fundamental insights were related to the trade-offbetween a smart assignment and power control solutions in scenarios limited by UE-to-UEinterference or SI.

173

174 Conclusions

In multi-antenna systems, we faced the challenges of beamforming, antenna assign-ment, and in the mmWave band the additional complexity of hybrid beamforming withpractical constraints. In the last two manuscripts [J3] and [J4], we formulated naturalquestions to multi-antenna systems and the practical application to systems communicatingin the mmWave band. In [J3], we were interested in the fundamentals of architecturesthat allow signal separation in FD communications. To address such fundamentals weproposed a smart architecture that decides between splitting or sharing an specific antennain the UL and DL. We tackled the spectral efficiency as performance indicator throughan optimization problem with antenna assignment, beamforming, power allocation anddistortions at transmitter and receiver. We proposed a solution using the smart architecturethat outperformed simple antenna splitting and HD for scenarios with high and lowresidual self-interference (SI) power. We obtained fundamental insights about the essentialrole of smart antenna assignment in optimizing multi-antenna FD communications. In[J4], we investigated the feasibility of FD communications in the mmWave band withcurrently available hardware devices that use low quantization bits at the phase shifters.We considered the spectral efficiency as performance indicator through an optimizationproblem including hybrid precoding/combining. We proposed a solution to this problemthat using just 1 quantization bit was able to outperform HD for scenarios with differentresidual SI powers, number of antennas and RF chains. Our fundamental insight wasthe feasibility of FD mmWave communications using just 1 quantization bit at the phaseshifters.

Chapter 2

Future Works

That’s the funny thing about arriving somewhere.Once you’re there, the only thing you can really dois leave again.

Brandon Sanderson

In the second part of this thesis, we proposed some directions that are potential topicsof future research for the community and that we wish to pursue. Herein, we collect thesedirections and propose a few more that may be of interest. The key points are listed asfollows:

• Mobility in FD: In [J2] we analysed distributed power control with a centralized user-frequency assignment. In scenarios with mobility, the user-frequency assignmentand power control need to take into account the dynamics aspects of fast fading,arrival/departure of users in a frequency channel, and handover to maximize thedesired performance indicator (spectral or/and energy efficiency, fairness). Suchtopics are important to shed light into practical applications of FD cellular networks,and to the best of our knowledge, they have not addressed yet.

• Antenna splitting/sharing with multiple-antennas at user equipments (UEs): Anextension of [J3] is to consider multiple antennas at the UE side. Using moreantennas for transmitter/receiver beamforming may help to decrease the impact ofthe chosen antenna architecture at the base station (BS). Given that UEs are alreadystandardized with multiple antennas, there is a practical appeal to the topic.

• Channel estimation and pilot design for FD-MIMO: A natural extension is toconsider FD with multiple antennas at both transmitter and receiver, and this hasbeen studied recently. However, most of the works assume perfect or partial channelknowledge, and without taking into account a proper method for channel estimationand the design of pilots. The pilot-data trade-off is a well-studied topic in standardMIMO, but it has not been addressed in the context of FD communications.

175

176 Future Works

• Low ADC quantization: Low resolution phase shifters were analysed in [J4], butwe assumed infinite resolution for the ADCs. Using either a linear or nonlinearquantization model, the feasibility of full-duplex (FD) in the mmWave can beextended to situations with perhaps just 1-bit ADC at the BS. This topic has notbeen addressed yet in FD cellular networks.

• Practical FD mmWave with different antenna architectures: In [J4] we analysedpractical aspects of FD mmWave systems, i.e., when the phase shifters at the RFchains had low resolution. In different antenna architectures, the phase shifters canbe replaced by switches and a lens antenna array can be utilized to provide highercost-aware solutions than using phase shifters. The use of both architectures modifyentirely the problems dealt in [J4] and the study of these problems would providefurther guarantees in the applications of FD communications in mmWave networks.

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