Prof Marie-Laure Boucheret

Prof Mohamed- Fadhel Ghannouchi

Prof Jean-Pierre Cances

Prof Geneviève Baudoin

Prof. Mérouane Debbah

Prof Carlos-Faouzi Bader

Mylène Pischellla

Anne-Laure Billabert

Catherine Algani

Michel Terré

Christophe Alexandre

Didier Pascal Chevalier

Le Ruyet

Christian Pautot

Salim Faci

Ali Dziri

Yahia Medjahdi

Tarek Raissi

Hayfa, Xinying, Meryem Hanen

Keywords:

Writing this thesis is a very interesting and very fruitful experience, which allowed me to realize the diculty of the synthesis exercise, especially when it relates to diversied activities ranging from teaching to research, including administrative responsibilities. After a brief overview of the research activities carried out during my PhD thesis, this synthesis exercise

Short introduction

This report draws my career as associate professor at Conservatoire National des Arts et Métiers (CNAM) since 2011. This document is structured in two parts. The rst part consists of one chapter: chapter 1. This chapter is an extended CV presenting in addition to some personal information, my education, my professional experience, my administrative responsibilities, and my teaching activities. In this chapter, I will also present various information about my research activities, namely the supervision, publications, collaborations as well as the research projects in which I have been involved. The second part of this report corresponds to my research thesis. This part is structured in six chapters:

• In chapter 2, I will present the research activities carried out since 2003. First, I will review my contributions in the eld of acoustics and digital audio to optimize the performance of high-delity loudspeaker systems by digital signal processing. In the second part of chapter 2, I will give an overview of the research activities carried at CNAM since 2011. During this decade I have been working on post-Orthogonal Frequency Division Multiplexing (OFDM) Multi-Carrier Waveforms (MWFs) for 5G and beyond systems, with important eorts to characterize and mitigate power amplier (PA) distortions in Single Input Single Output (SISO) and massive Multiple Input Multiple Output (MIMO) scenarios. For each research activity, I will present the projects under which the work has been carried, along with the dierent collaborations carried out, the main achievements and related publications.

• The third chapter introduces my contribution to the eorts made to nd the best candidate post-OFDM waveform for 5G. I will present a comparative study of many MWFs based on dierent key performance indicators (KPI), that suits well to 5G and beyond requirements. The interest of Filter Band Multi Carrier with Oset Quadrature Amplitude Modulation (FBMC-OQAM), in terms of spectral localization and robustness to asynchronous user access, has prompted us to work on advanced equalization structures for this MWF in the presence of frequency selective channels and mixed numerology.

• The time domain signals generated by the vast majority of the MWFs candidate for 5G exhibit signicant uctuations in instantaneous power, characterized by the Peak to Average Power Ratio (PAPR). This characteristic makes these MWFs very sensitive to the non-linearities induced by the PA. In Chapter 4, I will present the studies carried out to characterize the In-Band (IB) and Out-Of-Band (OOB) distortions induced by memoryless measured or modeled PA. A great originality of this contribution is its applicability to most of the MWFs candidate for 5G systems in presence of modeled or measured memoryless PAs.

• The high PAPR of MWFs strongly compromises the energy eciency of 4G, 5G and beyond transceivers. In chapter 5, I will present the solutions that we proposed to improve the energy eciency of several post-OFDM MWFs based transceivers. The rst family of the proposed solutions aims to reduce the PAPR and/or pre-distort the signal before amplication in a dissociated or joint manner. The second approach that will be presented modies the architecture of the PA and splits the multi-carrier signal into several sub-signals with lower PAPRs which can be amplied with better energy eciency.

• Among the prominent technologies proposed to reach the 5G New Radio (NR) requirements, Massive MIMO and mixed numerology have been retained to support various congurations on top of OFDM waveform. In chapter 6, I will present two contributions to massive-MIMO systems. The rst one is related to the analysis and cancellation of the interference caused by users using dierent numerologies in massive MIMO OFDM systems. The second contribution is the proposal of low complexity energy ecient emitter architecture combining MIMO precoding, PAPR reduction and Digital Pre-Distortion (DPD).

• Chapter 7: the last chapter of this report, will be dedicated to present the prospects and future directions of my research activities.

• 1 European FP7-STREP project N • 318362 EMPhAtiC (2013-2015), 3 ANR projects: ACCENT5 (2014-2017) WONG5 (2015-2019) and MAESANGES (2021-2024), with a strong involvement in the setting up and coordination of the WONG5 project, 2 PHC projects : Utique C3 (2012-2014) and Tassili ATOME5+ (2018-2021), with very active participation in the setting up, administrative and scientic coordination of these projects.

• 17 publications in international peer-reviewed journals, 1 book chapter, 35 publications in international conferences.

• 3 patent lings with 1 extended to many countries.

• 8 projects deliverables.

• 617 citations, h-index 1 : 13, i10-index 2 : 17.

• Participation to 3 in thesis committees as co-supervisor. • Several reviews of articles in international journals (IEEE Transactions on Signal Processing, IEEE Transactions on Vehicular Technology, IEEE Wireless Communication Letters, IEEE Communication Letters, IEEE Access).

• Participation to dierent teaching units, ranging from the rst year of B.Sc. to the last year of master's degree. Modernization of some teaching units, given in French and in English.

• In charge of some teaching modules related to electronics and signal processing.

• Proposal, in collaboration with Pr. Daniel Roviras, of a Massive Online Open Access Course (MOOC) on "introduction to signal processing".

• In charge of 3 degrees at CNAM: 2 engineering degrees and 1 bachelor's degree.

• Doctoral supervision and research grant (PEDR) holder since September 2017. In 2011, I joined CNAM, as an Associate Professor in electronics and signal processing. My research activities, carried within the CEDRIC laboratory, are related to the physical layer of radiocommunications systems with a special focus on performances analysis and comparison of OFDM and post-OFDM waveforms in 5G and beyond communications. In this context power amplier nonlinearities were shown to severely impact the whole system performance in both SISO and MIMO scenarios. After deep analysis of power amplier impairments, over many OFDM and post-OFDM waveforms, we have proposed dierent solutions based either on reducing the transmitted signal PAPR or on linearizing the power amplier characteristics.

1.5.2 International projects 2020-2023 ERASMUS+ program with Tunisian universities.

CNAM submitted in January 2019 an ERASMUS+ exchange project, in partnership with several Tunisian universities, including the University of Carthage. This project was accepted in September 2020. As part of this project, hosting of master 2 and doctoral students are planned during the years 2022 and 2023. Also, I will be asked to spend a stay in Tunisia, to provide a seminar or a conference, related to my research activities.

2018-2021 PHC-Tassili-ATOME5+ Project

I contributed to the setting up of the Hubert Curien Partnership project (Opti-misAtion de l'ecacité énergéTique des formes d'Ondes Multi-portEuses 5G+ en présence d'imperfections RF dans un contexte MIMO et massive MIMO) between the CEDRIC/LAETITIA team and the telecommunications laboratory at the University of Tlemcen, in Algeria. The objective of this project is to reduce the PAPR of post-OFDM multi-carrier modulations and compensate for power amplier imperfections in MIMO and massive MIMO context, by using modern Deep Learning techniques.

A joint supervision of a PhD thesis has been initiated within the framework of this project: thesis of Meryem Benosman.

2013-2015 FP7-STREP European project N • 318362 EMPhAtiC 3The aim of this project, involving thirteen partners, was to demonstrate the advantages of using lter bank based multi-carrier techniques: FBMC-OQAM, for allowing the coexistence between broad-band 4G/5G services with narrow-band legacy services dedicated to Professional Mobile Radio (PMR) applications. I worked on the reduction of FBMC-OQAM signals PAPR in the presence of nonlinear power amplier.

Our main contribution to this project was the adaptation of some PAPR reduction techniques proposed for OFDM: SLM and TR, by considering the overlapping between FBMC-OQAM symbols. The PhD of Krishna Bulusu was part of this project.

2011-2014 PHC-Utique-C3 Project

I contributed to the setting up of the Hubert-Curien Partnerships project (Communications Cognitives et Coopératives dans les réseaux ad-hoc sans l) between the CEDRIC/LAETITIA team, the signals and systems laboratory of CentraleSupélec, the higher school of communications of Tunis (SUP'COM) and the national school of engineers of Tunis (ENIT). This project focuses on cognitive and cooperative communications in ad-hoc wireless networks. One aspect of this project was to study the impact of power amplier non-linearities on multi-carrier systems (OFDM and FBMC-OQAM). The PhD of Hanen Bouhadda was carried out within the framework of this collaborative project.

National projects 2021-2024 ANR MESANGES Project

The MESANGES (Metasurface-based Electronically Steerable Antenna arrays for Next Generation EHF Systems) ANR project focuses on the use of mmWave frequencies for 5G+/6G/Wi-Fi systems. Our partners in this project are Orange, IEMN, CEA-Leti and INSA-IETR. More specically, most of the partner's eorts will be made on demonstrating the benets of coupling smart metasurface-based electronically steerable antenna arrays with dedicated hybrid multi-user beamforming algorithms. At CNAM, we will study beamforming solutions for the steering of the transmit and reect arrays for both single and multi-user cases. I will participate to the supervision of the PhD student that will be recruited to work on these tasks.

2015-2019 ANR WONG5 Project 4

I contributed to the setup and management of this ANR project. The objectives of the WONG5 project (Waveforms MOdels for Machine Type Communication inteGrating 5G Networks) are the study and proposal of the best suited waveform for the physical layer of Critical Machine Type Communication (C-MTC). Our partners in this project were THALES, CEA-Leti and CentraleSupélec-Rennes. My contributions to this project were as follows:

• Joint administrative and technical coordination of the project with Pr. Daniel ROVIRAS,

• Comparative study of more than fteen post-OFDM waveform in a C-MTC context, in terms of complexity, frequency localization, spectral eciency, latency, etc,

• Improvement of the energy eciency of the best suited waveforms to C-MTC systems, by reducing the PAPR,

Rak Zayani

Within the framework of the WONG5 project, I supervised Dr. Rak Zayani who was in charge of realizing a testbed to compare the performance of the selected post-OFDM waveforms against users asynchronism, frequency oset and power amplier nonlinearities. The proposed testbed was based on recongurable platforms (USRP NI-USRP-2942R device). The work carried out was highlighted by one journal paper and one conference paper.

Oct. 2016 -Sept. 2017

Yahia Medjahdi

I supervised Dr. Yahia Medjahdi who carried an exhaustive and comparative study of dierent waveforms candidates for C-MTC communications. This work was done within the framework of the WONG5 project. We carried out a critical study of fteen post-OFDM waveforms based on dierent criterion, such as frequency localization, robustness to time asynchronism, frequency oset, complexity, latency, spectral eciency, and sensitivity to power amplier eects. This research has resulted in two journal papers and six conference papers.

1.6.2 PhD students Since Jan. 2019

Meryem Benosman

• Title: Optimization of the energy eciency of post-OFDM waveforms, in the presence of power ampliers, by reduction of the PAPR and digital pre-distortion.

• Supervision rate: 40%.

• Defense date: scheduled for January 2022.

• Co-supervised with: Tlemcen university, Algeria.

• Funded by: PHC-Tassili ATOME5+ project.

• Publications: 1 conference paper.

Since

Oct. 2017 Khaled Tahkoubit

• Title: Study and techniques for reducing imperfections in the RF chain for post 5G multi-carrier waveforms.

• Supervision rate: 60%.

• Defense date: scheduled for November 2021.

• Co-supervised with: Oran university, Algeria.

• Funded by: Algerian government.

• Publications: 1 journal paper, 2 conference papers.

Since

Oct. 2018

Xinying Cheng

• Title: Energy Eciency enhancement for massive MIMO-OFDM systems with RF impairments.

• Supervision rate: 40%.

• Defense date: October 11 th , 2021.

• Committee: Reviewers: Pr. Martine LIENARD (Lille University) and Pr. Luc DENEIRE (Côte d'Azur University). Examiners: Pr. Yves LOUET (Centrale-Supélec) and Pr. Mohamed Slim ALOUINI (King Abdullah University of Science and Technology). Supervisors : Pr. Daniel Roviras (CNAM), Dr. Hmaied Shaiek (CNAM) and Dr. Rak Zayani (ISI-Tunis).

• Funded by: French ministerial grant.

• Publications: 2 journal papers.

2015-2019 Hayfa Fhima

• Title: Contributions of Widely Linear ltering to multi-carrier modulations based on lter banks (FBMC-OQAM).

• Supervision rate: 40%.

• Defense date: December 12 th , 2019.

• Committee: Reviewers: Pr. Tahar Ezzedine (INSAT) and Pr. Raoof Kosai (ENSIM). Examiners: Pr. Geneviève Baudoin (ESIEE) and Pr. Jamel Belhadj Taher (ENISO). Supervisors : Pr. Daniel Roviras (CNAM), Pr. Ridha Bouallegue (Supcom), Dr. Hmaied Shaiek (CNAM) and Dr. Rak Zayani (ISI-Tunis).

• Co-supervised with: SUP'COM, Tunisia.

• Funded by: Tunisian government and WONG5 ANR project.

• Publications: 1 journal paper, 4 conference papers.

• Actual position: Associate professor at ISIT'COM, Hammem Sousse, Tunisia.

2013-2016 Krishna Bulusu

• Title: Performance analysis and PAPR reduction techniques for Filter Bank based Multi-Carrier systems (FBMC-OQAM) with Non Linear Power Ampliers.

• Supervision rate: 60%.

• Defense date: April 29 th , 2016.

• Committee: Reviewers: Pr. Geneviève Baudoin (ESIEE) and Pr. Yves Louet (CentraleSupélec). Examiners: Ing. Laurent Martinod (CASSIDIAN), Pr. Jean-François Hélard (INSA-Rennes) and Dr. Rak Zayani (ISI-Tunis). Supervisors: Pr. Daniel Roviras (CNAM) and Dr. Hmaied Shaiek (CNAM).

• Funded by: EMPhAtiC European project ICT N • 318362 and ACCENT5 ANR project.

• Publications: 1 book chapter, 1 patent, 2 journal papers, 6 conference papers.

• Actual position: Associate professor at Mahindra-Ecole Centrale, Hayderabad, India.

2013-2016 Hanen Bouhadda

• Title: Study of the impacts of non-linearities in FBMC-OQAM systems.

• Supervision rate: 40%.

• Defense date: March 22 nd , 2016.

• Committee: Reviewers: Pr. Noureddine Hamdi (INSAT) and Pr. Raoof Kosai (ENSIM). Examiners: Pr. Geneviève Baudoin (ESIEE) and Pr. Ammar Bouallegue (ENIT). Supervisors: Pr. Daniel Roviras (CNAM), Pr. Ridha Bouallegue (Supcom), Dr. Hmaied Shaiek (CNAM) and Dr. Rak Zayani (ISI-Tunis).

• Co-supervised with: SUP'COM school, Tunisia.

• Funded by: PHC-Utique C3 project.

• Publications: 1 journal paper, 4 conference papers.

• Actual position: Associate professor at SUP'COM Tunis. This collaboration started in May 2011, with the setting up of the PHC-Utique C3 project, which made it possible to fund the stays in France of Hanen Bouhadda, of which I was the PhD co-supervisor. As part of this collaboration, Dr. Rak Zayani: member of the Innov'Com laboratory of the SUP'COM school in Tunis, was invited to CNAM in 2013 and 2015, for a period of 6 months and we worked on the characterization and correction of non-linearities introduced by power ampliers in FBMC-OQAM systems. Between April 2015 and January 2016, I coordinated, with the help of several members of the CEDRIC and Innov'Com laboratories, the organization of the ISW-5G winter school. In 2016, we started a second thesis jointly supervised by the two laboratories: thesis of Hayfa Fhima, on the application of Widely-linear ltering to quasi-rectilinear modulations of the FBMC-OQAM type.

2014-2016

Tampere University of Technology, Finland.

As part of the European EMPhAtiC project, I worked closely with the team of Pr. Markku Renfors from Tampere University of Technology (TUT) on the reduction of PAPR and the eects of nonlinearities for FBMC-OQAM systems. I oversaw the 3.3 deliverable of this project. We continued this collaboration with a publication in Transactions on Emerging Telecommunications Technology, on the coexistence of narrowband PMR systems with FBMC-OQAM based broadband systems. This collaboration was crowned with the editing of Chapter 13 of the book "Orthogonal Waveforms and Filter Banks for Future Communications Systems", entitled "Power Amplier Effects and Peak-to-Average Power Ratio Mitigation". In January 2016, I invited Pr. Marrku Renfors at ISW-5G Winter School.

2017-2019

Federal University of Technology Paraná, Curitiba, Brazil. Telecommunications laboratory of Tlemcen University, Algeria.

I contributed to the elaboration of partnership projects, for funding the stays of PhD students in joint supervision with the university of Tlemcen. In January 2018, we won a PHC funded project named "Optimization of the energy eciency of 5G Multicarrier waveforms (ATOME5+). This collaboration started in January 2019, with the PhD of Meryem Benosman. I am co-supervising this PhD dealing with the application of AI-based techniques to reduce PAPR and correct RF impairments.

2016-2018

CentraleSupélec/IETR, Rennes.

• I worked with Pr. Carlos BADER and several partners of the EMPhAtiC FP7 and ACCENT5 ANR projects on the writing of deliverables. Specically, we co-worked on the study of candidate waveforms for 5G and beyond wireless communications in terms of coexistence capabilities.

• I worked with Pr. Yves Louet as part of task 3 of the ANR WONG5 project, on techniques for improving the energy eciency of OFDM and post-OFDM waveforms. Within the framework of deliverable 3.3 of this project, we established a realistic power budget of a transmitter by taking into account OFDM and post-OFDM waveforms, PAPR reduction techniques complexities and realistic power amplier DC power.

2016-2018

CEA-Leti, Grenoble.

This collaboration began with the ANR WONG5 project in October 2015. I contributed to the critical comparative analysis of candidate waveforms for C-MTC communications. At CNAM, we worked on the implementation of many waveforms (CP-OFDM, WOLA-OFDM, UF-OFDM, FBMC-OQAM, FBMC-QAM, FFT-FBMC, WCP-COQAM) which were compared to the waveform proposed by CEA-Leti: the BF-OFDM. The comparison concerned several metrics, among which I can cite: the PSD, the complexity, the latency, the robustness against timing-oset and frequency-oset. In November 2020, I contributed, with Dr. Jean-Baptiste Doré and Dr. David Demmer, to the proposal of an ANR project for investigating the design of multi-user precoding schemes with partial channel state information (CSI) in cell free MIMO systems.

2016-2018

Thales Communications and Security (TCS), Gennevilliers.

This collaboration, with Dr. Sylvain Traverso began in January 2016 and focused on a comparative analysis of candidate post-OFDM waveforms for C-MTC. TCS contributed to the implementation of the waveforms: f-OFDM, N-Continous OFDM and FMT. Then, we compared these waveforms with those studied by CNAM and also by the other partners involved in the ANR WONG5 project.

Since

Sept. 2017

University of Science and Technology, Oran, Algeria.

Since July 2017, Khaled Tahkoubit: PhD student from the university of sciences and technologies of Oran, contacted us to work together on proposing new techniques for reducing the RF impairments for 5G and beyond systems. We also had the opportunity to welcome his PhD supervisor, Pr. Ali Pacha, in our research laboratory. We then decided to begin a co-supervision of Khaled Tahkoubit '

Dec. 2006

Second intellectual property awards winner (organized by "la compagnie nationale des conseils en propriété industrielle"), for the project "Four-way coaxial loudspeaker driven by active digital signal processor", December 2016, Paris. • I organized, with Pr. Daniel Roviras, a special session on Post-OFDM waveforms for 5G radio access technology. This special session was organized within the framework of the ISWCS 2017 conference in Bologna, http://iswcs2017.radionetworksgroup.org/.

• I organized, with Pr. Daniel Roviras, within the framework of the IEEE ISWCS 2016 conference in Pozna«, http://iswcs2016.radiokomunikacja.edu.pl/ a special session on PAPR reduction and power amplier pre-distortion for post-OFDM waveforms to be adopted for beyond 5G context.

• I organized, as general co-chair: ISW-5G http://www.supcom.mincom.tn/isw5g2016/: a winter school on post-waveforms OFDM for 5G. This school took place in Tunis from January 21 st to 22 nd , 2016 (150 participants).

• I was part of the organizing committee of the IEEE ISWCS 2012 conference https://web1.see. asso.fr/iswcs2012/images/call_for_paper_ISWCS2012_v5.pdf. This conference was organized by the CEDRIC/LAETITIA team from August 28 th to 31 st , 2012 at CNAM (320 participants).

Seminars

• "On the eciency of post-OFDM waveforms in the presence of power ampliers", university of Tlemcen, November 2019.

• "Presentation of WONG5 project", DGA-MI seminar on data processing and communications of drone systems, Bruz, May 2016.

1.8.6 Participation to PhD dissertation committees I participated, as a co-supervisor, to the following PhD dissertation committees:

• Hayfa Fhima: in December 2019, "Contributions of Widely Linear ltering to multi-carrier modulations based on lter banks (FBMC-OQAM)".

• Krishna Bulusu: in April 2016, "Performance analysis and PAPR reduction techniques for Filter Bank based Multi-Carrier systems (FBMC-OQAM) with Non Linear Power Ampliers".

• Hanen Bouhadda: in March 2016, "Study of the impacts of non-linearities in FBMC-OQAM systems".

Membership in selection committees

• In April 2021 and May 2020, I was part of the selection committees for a research and teaching assistant positions (ATER) in electronics and telecommunications at CNAM.

• In April 2014, I was a member of the selection committee for the recruitment of an associate professor in Communication, Electronics and Instrumentation at the University of Le Mans.

1.9 Teaching activities 1.9.1 Teaching activities at CNAM Description of the provided teaching: At CNAM, I give courses ranging from the rst year Bachelor (1 st year of Diplôme Universitaire de Technologies) to the last year of Master degrees. The provided teaching are courses, tutorials, practical works and tutored projects in electronics, signal processing and digital communications. Most of these courses are given in French and part of them are given in English.

Part of these courses are also available online with video resources on Moodle. The students concerned belong to the following elds:

• Master's degree in radiocommunication techniques,

• Master's degree in mobile telecommunications systems,

• Master's degree in sciences, technologies, health with specialization in networks and telecommunications,

• Engineering degree in electronic systems, telecommunications, and computer science,

• Engineering degree in electronic systems and railway signaling,

• Bachelor's degree in telecommunications, in partnership with DongGuan University of Technology (DGUT),

• Bachelor's degree in electrical engineering and computer science.

Tab.

Advanced digital communications

This course focuses on mobile radio channels with a study of the main models used to characterize large scale and small-scale fading eects. Much of the course is dedicated to modeling mobile radio channels and studying fading laws such as Rayleigh, Rice and Nakagami. Theoretical analysis of the performance over Rayleigh fading channel with diversity is also carried. Chapter 1. Curriculum Vitae

Microcontrollers

The course starts with an introduction to the general organization of a microprocessor system architecture and internal operation (main buses, UALs, registers, memories, etc.). The practical work is based on the STM32F429 (ARM Cortex-M4) microcontroller. The evaluation board used is the Nucleo-144. This platform integrates some peripherals (push buttons, LEDs) as well as Arduino Uno and STMicroelectronics Morpho connectors.

Teaching related responsibilities:

Since my arrival at CNAM, I have been responsible for the following teaching units:

• ELE008: Introduction to microcontrollers. I was actively involved in the modernization of this course, which was based on a 16-bit microcontroller: the 68HCS12 from Freescale. The new version of this course, proposed since September 2019, uses the STM32F429 (ARM Cortex-M4) microcontroller. The latter is equipped with a lot of features, in terms of input and output, and especially in terms of the variety of available evaluation platforms.

• ELE109: Advanced techniques in analog and digital electronics. I am responsible for the scientic coordination of this labs based unit, involving many colleagues from the electrical engineering department. I have actively participated in the evolution of the content of this unit intended for master's students. ESIEE-Paris: Since April 2017, I have been responsible for the course entitled "Mathematical tools for signal processing", intended for students in the rst year, specializing in telecommunications and networks at ESIEE-Paris. This course represents 30 hours of lessons, tutorials, and practical work. The main objective of this teaching unit is to introduce the tools and methods of analysis and processing of deterministic continuous-time and discrete-time signals.

CentraleSupélec: Since November 2018, I have been involved in a training session given by Cen-traleSupélec Paris, entitled digital signal processing: methods and techniques. This training is oered twice a year for experienced engineers working in electronics and data processing. My contribution lasts 1.5 hour for introducing adaptive lters, with a focus on the stochastic gradient algorithm (LMS) and the recursive least square algorithm (RLS).

Massive Online Open Access Course

In January 2015, I proposed, in collaboration with Pr. Daniel Roviras, the rst session of a Massive Online Open Access Course (MOOC) on "introduction to signal processing" https://www.fun-mooc. fr/courses/CNAM/01008S02/session02/about. This MOOC is intended for students who wish to have an overview of graduate studies in the eld of signal processing. This MOOC lasts six weeks. Each week consists of 5 to 6 video sequences followed by quizzes which allow learners to evaluate their achievements. At the end of each week, as well as at the end of the course, a graded assessment tests are oered. For

Teaching materials

Since 2011, I have been strongly involved in the educational life of the electrical engineering department at CNAM. I managed the development of two teaching units "ELE109: Advanced techniques in analog and digital electronics" and "ELE008: Introduction to microcontrollers", of which I am the manager and scientic coordinator. I have also prepared and edited materials, for courses, tutorials and practical works corresponding to the following topics:

• Advanced techniques in analog and digital electronics,

• Analog signal processing,

• Digital signal processing,

• Electronics,

• Introduction to signal processing,

• Microcontrollers.

Projects supervision and apprentices monitoring

Third-year engineering students in telecommunications and railway signaling at CNAM, prepare tutored projects for the study and development of prototypes meeting given specications. These specications are drawn up by the students and validated by the supervising teacher, who follows them up through progress meetings. Between 2017 and 2019, I followed two groups of engineering students from the previously mentioned CNAM engineering specializations.

Much of my teaching at CNAM is aimed at dual education system, combining apprenticeships in a company and vocational education at CNAM. This status requires the designation of a referent teacher who monitors the student's education, upon entering CNAM and until obtaining his diploma. The role of the referent teacher is to ensure the smooth running of the student's training, both academically and in terms of the adequacy between the missions carried out in this company and the diploma objectives. Since 2011, I follow an average of eight apprentices per year, between engineering and bachelor's degrees. I oversee CNAM professional bachelor's degree in sciences, technologies, health with a specialization in home automation and smart objects. Among my dierent tasks, I supervise the academic evolution of the syllabus, I preside the diplomas committees and I am the corresponding professor for the certication with French ministry of education (AERES).

Since

Sept.

I oversee CNAM degree in mechatronics, in agreement with the university of Poitiers. Among my dierent tasks, I supervise the academic evolution of the syllabus with my colleagues from the university of Poitiers, I preside the diplomas committees and I am the corresponding professor for the certication with the engineer title commission (CTI).

Since

Sept.

I oversee CNAM engineering degree in automation and robotics, with specializing in mechatronics. Among my dierent tasks, I supervise the academic evolution of the syllabus, I preside all national examinations (admission to the engineering school, diplomas committees) and I am the corresponding professor for the certication with the engineer title commission (CTI).

Elective mandates

Since May 2018

Elected member of the scientic council of the electrical engineering department.

Since

Jan. 2018

Elected member of the scientic council of the CEDRIC laboratory.

Part II

Part II: Research activities and prospects

Chapter 2

Overview of my research activities

Introduction

In this chapter, I will give an overview of my research activities, before and after being associate professor.

The rst part will be dedicated to the research activities carried, between 2003 and 2010, as PhD student and as a research engineer in the eld of acoustics and audio signal processing. The main contributions achieved, as well as related publications, made during this period will be presented.

The second part of this chapter will focus on all the research activities carried as associate professor at CNAM since February 2011. A general context of my research studies will be rst introduced. Then, I will present the four topics over which I have been working during this decade. These topics are: [START_REF] Chen | A 2.6 GHz Class-AB GaN Power Amplier with Maximum Output Power of 56 W Achieving 70% Power Added Eciency[END_REF] Comparative study of post-OFDM MWFs for 5G and beyond systems, (2) Theoretical analysis of IB and OOB PA distortions, (3) Proposal of signal processing and hybrid techniques to enhance PA eciency and (4) Enhancement of the energy eciency of OFDM based Massive-MIMO networks with mixed numerology. For each research topic, I will present the projects under which the work has been carried, along with the dierent collaborations carried out. I will also give an overview on the main achievements and related publications.

It's important to underline here that scientic and technical contributions for each of the previously mentioned research topics will be detailed in chapters 3, 4, 5 and 6.

PhD related research activities 2.2.1 Context

From October 2003 to July 2007, I prepared my PhD thesis with the Signal and Communications Department of TELECOM Bretagne, in collaboration with Cabasse Acoustic Center. This thesis dealt with the optimization of high-end multi-way loudspeaker systems radiation by using dedicated digital signal processing techniques. From April 2007 to February 2009, I worked as research engineer with Canon Inc. group. A summary of the scientic contributions achieved during the two periods is summarized in the following subsections.

Contributions

Free space optimization of the sound eld radiated by a multi-way loudspeaker system:

To reach this objective, we combined Cabasse knowhow in terms of electro-acoustical conception and advances in digital ltering applied to audio. Indeed, with more than 50 years' experience, Cabasse R&D engineers proposed in 2005, the rst full range coaxial loudspeaker in the word. This system can reproduce the entire audio band extending from 20 Hz to 20 kHz with a well-controlled directivity. We have rst studied the theoretical and measured sound eld radiated by this loudspeaker system, with respect to a classical one, whose radiating surfaces are spatially spread-distributed. For this study, I have considered the two acoustic systems, given by Fig. 2.1. Each system is made up of three piston-like radiating surfaces embedded in an innite at bae. System 1 is a source consisting of three coaxially mounted loudspeakers. System 2 is a source consisting of three radiating surfaces, which are spatially spread-distributed. The near elds radiated by the two acoustic systems at the level of two crossover frequencies: f 1 = 800 Hz and f 2 = 4000 Hz, are given by Fig. 2.2. In the case of system 1 and for f 1 = 800 Hz, the radiated waves are quasi-spherical with a decrease in sound pressure in 1/r. At f 2 = 4000 Hz, this system becomes more directional in its main axis while keeping a homogeneous radiation even in very close eld (r ≤ 0.5m). On the other hand, system 2 presents an asymmetric radiation with marked interference zones. The radiated eld irregularities are more important in the near eld region between 0.5 m and 1 m, where a shifting of 15 cm of the receiver can cause a variation of 25 dB in the amplitude of the radiated eld. The accuracy of this theoretical study has been validated through measurements on real systems and published in a conference paper [START_REF] Debail | Radiated sound eld analysis of loudspeaker systems: discrete geometrical distribution of circular membranes versus co-incident annular rings[END_REF]. The reference coaxial loudspeaker system proposed by Cabasse still suers from residual problems, such as imperfect phase alignment, frequency bands overlapping and non-at on-axis frequency response. One of the main contributions of my PhD was to propose a dedicated digital lter bank to correct the residual imperfections of the coaxial loudspeaker system and enhance its performances in order to approach as well as possible the theoretical point source. The system model of the digitally controlled multi-way loudspeaker is given by Fig. 2.3.

The originality of the digital lter bank I developed, and validated on real systems, is that it ensures an enhanced control of the system directivity making it possible to correct possible radiation shifts over transducers' frequency overlap bands. The proposed algorithm is based on a complex weighting of the crossover frequency responses, to reach a given target in terms of radiation pattern or directivity index. The proposed algorithm was rst patented [START_REF] Shaiek | Activation signal ltering device for e.g. professional listening system, has digital ltering unit with channel separation unit under constraint, where separation unit has separator lters covering frequency band between channels[END_REF] and then published in a conference and a journal papers [START_REF] Shaiek | Enhanced control of sound eld radiated by co-axial loudspeaker systems using digital signal processing techniques[END_REF][START_REF] Shaiek | Optimizing the Directivity of Multiway Loudspeaker Systems[END_REF]. By comparing the radiation diagrams of Figs. 2.4 (a) and 2.4 (b), we note that the crossover lter optimization makes it possible to ensure a more homogeneous and progressive directivity.

Optimization of the sound eld radiated by a multi-way loudspeaker system in a listening room: The second part of the optimization procedure is to enhance the sound eld radiated by the multi-way loudspeaker system is related to the transfer brought by the listening room: equalization of the Room Transfer Functions (RTFs). Indeed, many researchers were concerned with the correction of multiple room reections and acoustic reverberation. However, no universal solution has been adopted. In the context of my PhD the considered acoustic sources stimulate the very low frequencies room modes. I decided, rst, to work on the design of adequate digital lters to compensate for room mode, in the low frequency part of the audio spectrum. This choice is based on the fact that the resonant modes of a listening room are well localized in low frequencies. The identication and correction of these modes, in a privileged listening area in the room, would therefore allow a signicant improvement in the quality of the reproduced sound. The experimental protocol used for the measurement of RTF is given by Fig. 2.5. The proposed solution is a multiple-point low frequency compensation technique of the room modes. This approach consists in averaging the RTFs measured at dierent listening positions and to systematically detect the room modes, to nally correct them using cascaded second-order notch lters. After xing the total number of resonances to be corrected P , each resonance i is controlled by a biquadratic cell C i (z). The equalizer C(z) is so the cascade of P biquadratic cell: C(z) = P i=1 C i (z). where C i (z) is given by:

C i (z) = b i 0 + b i 1 z -1 + b i 2 z -2 1 + a i 1 z -1 + a i 2 z -2 .
(2.1)

The design of these lters is based on the method proposed by S. White [START_REF] White | Design of a biquadratic peaking or notch lter for audio equalization[END_REF], allowing to establish direct relations between a given room mode parameter and the transfer function coecients a i and b j , Chapter 2. Overview of my research activities (i = 1..2 and j = 0..2) of a biquadratic cell. The originality of the proposed method, for multi-point RTF equalization, is that it allows an automated design of digital cascaded notch lters giving the possibility to correct the peaks that are common to all the room responses measured over a given listening area. From Fig. 2.6, we can see the results of an example compensating for the eects of room modes at the three selected positions on the sofa shown in Fig. 2.5. Indeed, the cascade of biquadratic cells reduces on average the gains of the resonant peaks common to the dierent positions over the selected listening zone. The results corresponding to the proposed algorithms where published in two conference papers [START_REF] Ndo | An hybrid approach of low frequency room equalization: Notch lters based on common acoustical pole modeling[END_REF][START_REF] Lakhdhar | Iterative equalization of room transfer function using biquadratic lters[END_REF].

Implementation on DSP and evaluations tests: The goal being to lead to an industrial model, I carried out subjective tests to evaluate the proposed digital ltering over Cabasse coaxial loudspeakers. I started by characterizing the dispersion at the level of the loudspeaker's responses. Then, I implemented on a DSP1 (Analog Devices DSP, ADSP-21369) all the digital ltering algorithms proposed (delays, crossover, equalization and enhanced directivity control). Finally, I carried out blind subjective evaluation tests to access the advantages of coupling the coaxial loudspeaker with the proposed digital processing proposed. This work was published in two conference papers [START_REF] Shaiek | Design and real-time implementation of a digital lter bank dedicated to co-axial loudspeaker systems[END_REF][START_REF] Shaiek | An optimized full-bandwidth 20Hz-20kHz digitally controlled co-axial source[END_REF]. Then, I focused on the validation of an automatic room acoustic transfer function correction algorithm. A subjective evaluation of this algorithm, under dierent listening conditions and congurations, made it possible to conrm the robustness of the proposed method, patented in October 2012 [START_REF] Kerneis | Method for developing correction lters for correcting the acoustic modes of a room[END_REF]. Fig. 2.7 shows the prototype of "La Sphère": the rst full range and digitally controlled coaxial loudspeaker system exhibited in Radio France studio 103.

Research activities at CNAM

Context

Each generation of mobile communication system has had great social and economic impacts on peoples and societies. Particularly, it is expected that the new generation: 5G, will increase actual users experience, in terms of data rates, reliability and 0s-like latency, to connect anyone/anything anywhere and at any time. This motivated researchers and telecommunications manufacturers, all over the word, during the last decade to intensify their research & development activities on 5G system. Preliminary studies, carried-out during the exploratory research phase of 5G (2012-2015), announced a peak data rate of 20 Gbps, corresponding to two order increase with respect to the existing LTE networks [START_REF] Dahlman | 5G wireless access: requirements and realization[END_REF]. However, 5G is not only a technology that would provide enhanced Mobile Broad Band (eMBB) communications to increase the data rate. Indeed, we expect from 5G to serve new use cases such as machine-type communication (MTC), exhibiting dierent operating requirements in terms of availability, reliability, latency, throughput, and devices density. One of the rst contributions on 5G wireless networks requirements [START_REF] Dahlman | 5G wireless access: requirements and realization[END_REF], classied MTC applications into two main categories: massive MTC (mMTC) and mission critical MTC (cMTC). mMTC, also labeled as Internet of Things (IoT), exhibits high devices densities ( 1 M device/km 2 ), short transmitted packets and high energy constraints (long battery life). As applications related to mMTC, we can cite sensors and actuators deployment in smart homes and smart cities over a wide area for surveillance and area-covering measurements. In the other side cMTC principally concern communications between machines involved in applications like Industrial manufacturing processes, observation of critical environments, remote medical surgery and vehicle-to-vehicle/infrastructure (V2X). . . In cMTC context, latency, reliability, and high data rates are prominent. The maximum latencies targeted, in the cMTC scenario, is of the order of a millisecond. Fig. 2.8, gives an overview of the dierent scenarios and some applications envisaged in the context of 5G. From this gure, we can see that 5G will be designed to support diverse services with dierent data trac proles: throughput, latency and density of connected devices. One of the major challenges of 5G was the introduction of a New Radio (NR) access technology, which oers the exibility needed to support these very dierent types of services.

From another side, due to permanent debates on climate change, every emerging technology, including 5G, is scrupulously evaluated on its carbon footprint. As part of the information and communication technologies (ICT), 5G should keep mastered energy consumption. In 2018 the ICT sector's carbon footprint was evaluated to be 730 M t CO 2 -equivalents or 1.4% of overall global emissions, and the sector uses 800 T W h or 3.6% of the worldwide global electricity [START_REF]Exponential data growth -constant ict footprints[END_REF]. By 2025 these emissions are expected to triple with the deployment of 5G networks, if no serious eorts are made to reduce this footprint [START_REF] Andrae | Total consumer power consumption forecast[END_REF].

A signicant part of these emissions, about the half, is attributed to user's devices whereas the sixth is dedicated to xed and mobiles network infrastructures [START_REF] Vereecken | Power consumption in telecommunication networks: overview and reduction strategies[END_REF]. By user's devices, we mean older phones, xed cordless phones mobile phones, smartphones, tablets Routers, modems, CPE (customer premises equipment), desktop and laptop, PCs, public displays, connected devices, (IoT)... Device's densication, emphasized by 5G use cases, should not increase drastically the total energy consumption. From the infrastructure side, the highest power consumer is the base station (BS), for which the radiated power is function of devices density in the covered area. BS can account for approximately 50-80% of total energy consumption, with ∼ 60% of them reserved to the power amplier (PA) [START_REF] Jens | The Energy and Carbon Footprint of the Global ICT and E&M Sectors 20102015[END_REF]. International Telecommunication Union (ITU) has been leading international eorts to enable mobile broadband communications (3G, 4G and now 5G). In January 2012, ITU dened the 4G wireless cellular technology: International Mobile Telecommunications (IMT)-Advanced and this is now being deployed worldwide. This technology uses Orthogonal Frequency Division Multiplexing (OFDM) Multicarrier Waveform (MWF) for downlink (DL) communications. OFDM has demonstrated its supremacy for high data rate transmissions over frequency selective channels at low implementation cost. However, for uplink (UL), a single carrier Discrete Fourier Transform (DFT) pre-coded OFDM waveform has been adopted. The main reason behind the use of a single carrier waveform for UL is that OFDM modulated time domain signal exhibits high uctuations in terms of instantaneous power, with respect to the average one. The high Peak to Average Power Ratio (PAPR) of OFDM signals, make them very sensitive to PA nonlinear (NL) behavior. Indeed, PA is a crucial component of the transmitter, which input-output behavior is commonly dened by the Amplitude Modulation to Amplitude Modulation (AM/AM) and Amplitude Modulation to Phase Modulation (AM/PM) conversion characteristics. These characteristics are always nonlinear, in the vicinity of the saturation region, where PA eciency is maximized. The standardization of 5G NR started in Chapter 2. Overview of my research activities 2016. The rst release appeared in December 2017 [START_REF]Technical Specication Group Radio Access Network; Study on New Radio (NR) access technology[END_REF] and conrmed the appearance of specic features compared to previous releases. An important feature of 5G NR is that this technology will support scalable numerologies to cover dierent use case requirements [START_REF] Wg1 | Numerology requirements[END_REF]. Indeed, 5G will oer three main use cases: eMMB, mMTC and cMTC (labeled rather as Ultra Reliable and Low Latency services in release 15). Additionally, 5G should came with cost-eective deployment topologies, ranging from macro cells to small cells across sub 6 GHz and mmWave (∼ 52.6 GHz) spectrum. Another important feature of 5G NR is that it will exploit the full potential of Multiple Input Multiple Output (MIMO) antenna technology.

Particularly, in 5G NR the number of antenna elements integrating BS units may reaches hundreds, while being higher compared to the number of user's devices. This concept is dened as Massive MIMO 2 .

With massive MIMO and the large number of BS active antenna elements we can address simultaneously many users through beamforming, achieving thus higher throughput for the system. The beams directivities are adapted to users' needs and allows to limit emissions in directions where there is no need for communications.

As highlighted previously, several areas of research emerged during the exploratory and experimental research phases on 5G. Those related to the denition and development of 5G NR were of great interest to me and were well in line with the research activities carried out within the LAETITIA team of the CEDRIC laboratory. This team has strong expertise in the eld of advanced MWFs and particularly on Filter Bank Multi-Carrier (FBMC) systems. Members of the LAETIITIA team had strongly contributed to the European FP7 PHYDYAS project3 , involving 13 partners and led by Pr. Maurice Bellanger. The research activities that I have developed since my arrival at CNAM are centered around the following four axes:

• Advanced post-OFDM MWFs for 5G and beyond systems: One of the rst questions raised during the exploratory research phase of the 5G systems was: shall we maintain OFDM as the MWF for 5G NR? Otherwise, which MWF, among those proposed in the literature, would better meet the requirements of 5G use cases? This question motivated us to carry out, within the framework of the WONG5 project, an exhaustive comparative study of the most promising MWFs suited for cMTC/URLLC use case. Several Key Performance Indicators (KPIs) have been considered to compare the selected MWFs: frequency localization, spectral eciency, End-to-End Physical layer latency, robustness to time and frequency synchronization errors, power uctuation and transceiver complexity. Among all the MWFs candidate for 5G NR, FBMC-OQAM: the best frequency localized one is characterized by real domain orthogonality, that could be lost under frequency selective channels. In this context and based on the cyclo-stationarity of the OQAM signal, we have proposed advanced equalization structures based on Widely linear and Frequency Shift (FRESH) processing, that have been applied to a FBMC-OQAM system using dierent 5G numerologies.

• In-band and Out-of-Band distortions analysis: The primary source of Radio Frequency (RF) imperfections in a transceiver is the NL PA. For MWFs based transmissions, such as those candidates to reach the requirements of 5G use cases, the envelope of the signal to be amplied is far from being constant, presaging signicant non-linear distortions, if the PA is operated at low input Back O (IBO) (i.e. close to its saturation region). We characterized analytically, for dierent types of memoryless PA (models or measured), and for dierent MWFs, the IB distortions in terms of bit error rate or symbol error rate. For the characterization of OOB emissions, we proposed analytical prediction of spectral regrowth based on cumulants.

• PA eciency enhancement solutions for 5G and beyond MWFs: OFDM, as well as most MWFs candidate for 5G NR, exhibit high PAPR exceeding 10 dB. This property implies, to reduce the distortions induced by the PA, to work at large IBO values. However, this scenario is not attractive in terms of energy eciency. Dierent approaches can be proposed to improve the energy eciency. The rst proposed solutions are based on digital signal processing (DSP) algorithms to reduce the PAPR and to compensate for the NL PA characteristics by using Digital Pre-distortion (DPD). These processing are oered either separately or jointly. A signicant eort has been put on MWFs with specic signal structures, such as FBMC-OQAM. Another solution that we have proposed is based on an hybrid approach which consists of modifying the architecture of the RF transmitter by splitting the MWF time domain signal into several (2 or more) low PAPR sub-signals which can be amplied via PAs at higher eciency.

• Enhanced energy ecient Massive MIMO networks with mixed numerology and RF impairments: The diverse use cases requirements is an important challenge for the 5G NR, where the coupling of OFDM-based massive MIMO and mixed numerologies can be envisaged. We have investigated the use of spatial multiplexing of users with dierent numerologies and sharing the same bandwidth. We have also analyzed the Inter-Numerology Interference (INI) and derived the theoretical expressions for DL and UL scenarios, along with corresponding cancellation schemes. As last contribution in this direction, we considered energy-ecient DL transmission in OFDM based massive MIMO in presence of PA. We designed a low complexity transmission scheme taking into consideration massive MIMO precoding, PAPR reduction and DPD.

In what follows, I will give a summary of my scientic contributions presenting the context where results have been obtained (international or national projects) and the researchers involved in the publications (other collaborating laboratories and industrials, supervised postdoctoral researchers, PhD, and master students). A detailed presentation of the scientic contributions will be given in chapters 3, 4, 5 and 6.

MWFs for 5G and beyond systems

The strong constraint on the latency of cMTC systems requires minimizing network access times by allowing asynchronous communications between users. The same problem can also be found for mMTC applications, were the transmitted/received data packet sizes are usually quite small. The overhead associated with orthogonal transmission, in terms of time alignment signaling becomes so very large, when large numbers of devices are connected. In [START_REF] Medjahdi | Performance analysis in the downlink of asynchronous OFDM/FBMC based multi-cellular networks[END_REF] it has been demonstrated that OFDM is not adapted for asynchronous communications. Another drawback of OFDM is the spectral eciency loss inherent to the Cyclic Prex (CP) insertion. My rst research focus, carried out within the framework of the WONG5 project, concentrated on the study of post-OFDM MWFs, that would better meet the requirements of 5G cMTC/URLLC use case. This project involved CNAM, CEA-leti, THALES and CentraleSupélec. Beyond the relaxation of the constraint on time synchronisation, other criteria must be considered during the comparative study of post-OFDM MWFs. As Key Performance Indicators (KPIs), chosen with the WONG5 project partners, we considered: Power Spectral Density (PSD) localization, Spectral Eciency (SE), End-to-End Physical layer latency, PAPR, transceiver complexity, adaptability to MIMO transmissions and performances over frequency-selective and/or time varying channels. Fig. 2.9 gives an overview on this research activity.

Several post-OFDM MWFs have been intensively studied in the literature in the last decade. Those considered for comparison by WONG5 project were:

• WOLA-OFDM: for Weighted Overlap and Add OFDM [START_REF]R1-162199 -waveform candidates[END_REF],

• UF-OFDM: for Universal Filtered OFDM (or also UFMC) [START_REF] Vakilian | Universal-ltered multi-carrier technique for wireless systems beyond LTE[END_REF],

• f-OFDM: for ltered OFDM [START_REF] Abdoli | Filtered OFDM: A new waveform for future wireless systems[END_REF],

• N-Continous OFDM: [START_REF] Van De Beek | N-continuous OFDM[END_REF],

• FMT: for Filtered MultiTone [START_REF] Cherubini | Filter bank modulation techniques for very high speed digital subscriber lines[END_REF],

• FBMC-OQAM: for Filter Bank MultiCarrier with Oset Quadrature Amplitude Modulation [START_REF] Bellanger | FBMC physical layer: a primer[END_REF][START_REF] Bellanger | Lapped-OFDM as an alternative to CP-OFDM for 5G asynchronous access and cognitive radio[END_REF],

• WCP-COQAM: for Windowed Cyclic Prex-based Circular-OQAM [START_REF] Lin | Multi-carrier modulation analysis and WCP-COQAM proposal[END_REF],

• FBMC-QAM: for Filter Bank MultiCarrier with Quadrature Amplitude Modulation [START_REF] Abdoli | Weighted circularly convolved ltering in OFDM/OQAM[END_REF][START_REF] Yun | A new waveform enabling enhanced QAM-FBMC systems[END_REF], Figure 2.9: Comparative study of OFDM and post-OFDM MWFs for 5G and beyond systems.

• GFDM: for Generalized Frequency Division Multiplexing [START_REF] Fettweis | GFDM -generalized frequency division multiplexing[END_REF],

• FFT-FBMC: for Fast Fourier Transform FBMC [START_REF] Zakaria | A novel lter-bank multicarrier scheme to mitigate the intrinsic interference: Application to MIMO systems[END_REF],

• BF-OFDM: for Block-Filtered OFDM [START_REF] Gerzaguet | The 5G candidate waveform race: a comparison of complexity and performance[END_REF].

To meet the 5G requirements, each of these MWFs try to provide solutions to the limitations of CP-OFDM, via windowing and/or ltering at the transmitter (Tx) and/or receiver (Rx), where the ltering could be done at subcarrier level or at resource block (RB) level.

The study of the above-mentioned waveforms, considering CP-OFDM as a basis for comparison, was distributed among the various partners of the WONG5 project. At CNAM, we were in charge of the study of CP-OFDM, WOLA-OFDM, UFMC, FMT, FBMC-OQAM, FBMC-QAM and WCP-COQAM.

As rst contribution, we have demonstrated that WOLA-OFDM provides better PSD localization than classical CP-OFDM, thanks to the smooth time domain windowing applied at the transmitter side. Additionally, the receiver side windowing, leaded to reduced users' interference in an asynchronous context. This work, co supervised with Pr. Daniel Roviras and achieved by a postdoctoral researcher under my responsibility: Dr. Rak Zayani, was published in a conference paper [START_REF] Zayani | WOLA-OFDM: A potential candidate for asynchronous 5G[END_REF]. Inspired from [START_REF] Lin | Multi-carrier modulation analysis and WCP-COQAM proposal[END_REF], we extended the previous contribution, during Yahia Medjahdi's post-doc, to combine WOLA processing and circular pulse shaping lter. The proposed MWF called WOLA based circular oset quadrature amplitude modulation (WOLA-COQAM) has shown improved robustness, compared to classical WOLA-OFDM, against time synchronization errors. This contribution has been published in a conference paper [START_REF] Medjahdi | WOLA processing: A useful tool for windowed waveforms in 5G with relaxed synchronicity[END_REF].

The other MWFs: UFMC, FMT, FBMC-OQAM, FBMC-QAM and WCP-COQAM were studied as part of Dr. Yahia Medjahdi's post-doc, which I supervised with Pr. Daniel Roviras. The investigated MWFs have been classied based on their orthogonality criterion, which is a very important feature when adapting them to MIMO communications. The performances of these MWFs, and those studied by the others WONG5 project partners, have been deeply analyzed in a review journal paper [START_REF] Medjahdi | On the road to 5G: Comparative study of physical layer in MTC context[END_REF] and a conference paper [START_REF] Medjahdi | WOLA processing: A useful tool for windowed waveforms in 5G with relaxed synchronicity[END_REF]. The main conclusions of this exhaustive comparative study are the following:

• PSD localization: windowed/ltered MWFs oered better frequency localization than CP-OFDM. The best frequency localization was achieved by FBMC-OQAM, thanks to the steep slopes of its prototype lter,

• Spectral eciency: in short bursts case, all MWFs gave almost the same spectral eciency, except FBMC-OQAM, FMT and FBMC-QAM, due to long prototype lters. With long bursts scenario, all the MWFs share the same spectral eciency,

• Latency: for very short bursts, CP-OFDM-inspired MWFs generated the lowest latencies compared to the other MWFs requiring ltering operation. In the case of very long bursts, MWFs using CP or zero padding (ZP) gave higher latencies due to the number of CP/ZP added to the transmitted signal,

• Robustness against time and frequency synchronization errors: thanks to their wellfrequency localized transmit/receive lters, FMT, FBMC-OQAM, FFT-FBMC, FBMC-QAM and f-OFDM provided good performances compared to CP-OFDM,

• Complexity: Regarding the transmitter: CP-OFDM and WOLA-OFDM achieved the lowest complexities. However, N-continuous OFDM and UFMC required huge computational resources reaching up to 200 times CP-OFDM complexity. At the receiver side, WOLA-OFDM, N-continuous OFDM and BF-OFDM receivers showed a complexity like the CP-OFDM one,

• PAPR: as time domain signals, at the output of all the considered MWFs, have complex Gaussian distribution, we have conrmed quite same PAPR for all the MWFs,

• Adaptability to MIMO: The investigated MWFs have been classied based on their ability to keep the orthogonality: (1) in the complex domain (e.g. most of the OFDM-inspired MWFs), (2) in the real domain (OQAM based techniques) and (3) non-orthogonal MWFs (like GFDM and FBMC-QAM). Among these three groups, the rst one is directly adaptable to MIMO transmissions.

For the performance of the studied MWFs over frequency-selective and time-varying channels, we have considered, for WONG5 project scenario, the E model of the HIPERLAN/2 channel [START_REF] Medbo | Channel models for hiperlan/2 in dierent indoor scenarios[END_REF]. This channel model is designed for the 5 GHz band, for typical large open space outdoor environments and with Non-LOS (NLOS). From this study, it appeared that UF-OFDM is very sensitive to channel delay spread, whereas all other MWFs are much more robust. If for low mobility case (5 km/h), all MWFs have the same performance, at 150 km/h, WOLA-COQAM, FFT-FBMC and BF-OFDM experience an error oor ten times greater than the one obtained with the other MWFs. These results have been published in a conference paper [START_REF] Medjahdi | Impact of selective channels on post-OFDM waveforms for 5G machine type communications[END_REF].

Among the MWFs candidate for 5G NR, we have clearly identied FBMC-OQAM as the MWF exhibiting the best frequency localization, thanks to its prototype lter [START_REF] Medjahdi | Inter-cell interference analysis for ofdm/fbmc systems[END_REF], providing then the best performance in supporting asynchronous users access. However, the use of the OQAM modulation, break the orthogonality under frequency selective channels. As part of Hayfa Fhima's PhD, I have been interested on the study and implementation of advanced equalization schemes applied to FBMC-OQAM. As OQAM is known to be a cyclo-stationary (CS) and non-second order circular (NSOC) modulation, it was relevant to study advanced equalization techniques based on widely linear and Fresh approaches [START_REF] Chauvat | Etude de liaisons SISO, SIMO, MISO et MIMO à base de formes d'ondes FBMC-OQAM et de récepteurs Widely Linear[END_REF]. We developed three minimum mean squared error (MMSE) based equalizing structures: a linear equalizer (LE), a widely linear equalizer (WLE) and a widely linear FRESH equalizer (WL FRESH). For the proposed equalizers, both symbol-spaced (SSE) and fractionally spaced (FSE) processing have been evaluated, in synchronous DL/UL and asynchronous UL scenarios, with mixed numerologies transmissions. The results showed the supremacy of the proposed WL FRESH equalizer to suppress INI and inter symbol interferences (ISI) over frequency selective channels. The work carried out within the framework of this thesis was highlighted by a journal paper [START_REF] Fhima | Comparison of Linear, Widely Linear and Fresh equalizers for FBMC-OQAM Systems with dierent numerologies[END_REF] and four conference papers [START_REF] Fhima | Widely linear equalizer performance with multiple independent interferences[END_REF][START_REF] Chang | On the performance of a widely linear SC-FDE system under multiple independent interferences[END_REF][START_REF] Fhima | Performance of Linear and Widely Linear equalizers for FBMC/OQAM modulation[END_REF][START_REF] Fhima | Analysis of Widely Linear equalization over frequency selective channels with multiple interferences[END_REF].

More insights on the contributions related to the study of post-OFDM MWFs, for 5G and beyond systems, will be given in chapter 3.

Power amplier distortions analysis

MWFs are suering from a major drawback related to their high PAPR. The PAPR measures the variation of the instantaneous power compared to the mean power of the transmitted multicarrier signal. Low values of PAPR correspond to a transmitted signal which power is always very close to its mean power, while high PAPR values indicate that the instantaneous power can be very high compared to the mean. The distribution of the transmitted power is not very important as far as linear systems are used in the transmission chain. Unfortunately, some elements of the chain are nonlinear (NL). In particular, the power (PA) has a NL behavior. This device can be quasi linear for a certain part of the input powers and NL for high input powers. Furthermore, the output power of a PA is limited to a saturation power. For small input powers, the PA is operated in its quasi-linear region and the high PAPR of the multicarrier signal will not have any inuence on the transmitted signal. Nevertheless, this situation has a high cost in terms of energy eciency. Especially, for mobile applications with batteries, it is thus necessary to increase the power eciency. Real PA communications systems will then be operated as close as possible to the saturation power to increase the power eciency. By doing that, two negative eects will appear:

• Out-Of-Band (OOB) distortions: the output signal spectrum will be broadened compared to the input one. This is also known as spectral regrowth,

• In-Band (IB) distortions: the transmitted signal will be distorted causing transmission errors.

From what has been said previously, it appears clearly that the performances of CP-OFDM and post-OFDM waveforms are strongly aected by PA, especially if energy eciency optimization is targeted. This observation motivated us to develop deep studies of the two types of distortions induced by memoryless PA. These studies were carried out in terms of theoretical analysis of IB and OOB distortions and completed by simulations, using modelled or real PA. Fig. 2.10 illustrates the scenario and the context of the studies envisaged within the framework of this research axis.

In-Band distortions analysis

With all the advantages shown by the FBMC-OQAM MWF, we were interested within the framework of Hanen Bouhadda's PhD, in the analytical characterization of the IB distortions undergone by this MWF over Additive White Gaussian Noise (AWGN) and at fading Rayleigh channels. As a rst contribution, we proposed a theoretical approach to evaluate the Bit Error Rate (BER) performance for FBMC-OQAM as well as CP-OFDM MWFs in presence of Saleh PA model, exhibiting both AM/AM and AM/PM conversion characteristics. The proposed approach is based on Bussgang's theorem [START_REF] Bussgang | Crosscorrelation functions of amplitude-distorted gaussian signals[END_REF], for modelling the IB distortion as a complex gain and an uncorrelated additive noise. Through this study, we have demonstrated a very important feature of FBMC-OQAM. Indeed, when only the amplitude of the modulated signals is distorted by the PA, FBMC-OQAM exhibited the same performance, in terms of BER, as CP-OFDM. However, FBMC-OQAM is shown to be more sensitive to phase distortions than CP-OFDM. This sensitivity is related to the intrinsic interference term introduced by this modulation. This nding have been published in a conference paper [START_REF] Bouhadda | Sensitivity analysis of FBMC signals to non linear phase distortion[END_REF].

For FBMC-OQAM, as well as any other MWFs, we carried analytical study of the BER, in presence of memoryless PA. This study required theoretical expressions of the complex gain and the variance of the NL distortion noise term computed by Busgang theorem. For simple PA models, such as the Soft Envelope Limiter (SEL), Dardari et al. [START_REF] Dardari | A theoretical characterization of nonlinear distortion eects in ofdm systems[END_REF] proposed analytical expressions to compute these parameters. However, for other PA, such as Rapp, Saleh, or polynomial models, or for real measured PA, BER derivation was generally intractable. Another original contribution we proposed in this direction is the proposal of a generalized method based on a polynomial tting of the PA characteristics allowing to compute theoretical expressions of the BER for any PA model or measurement. The proposed approach has been applied to CP-OFDM and FBMC-OQAM MWFs for Additive White Gaussian Noise (AWGN) and Rayleigh channels and the obtained results have been published in a journal paper [START_REF] Bouhadda | Theoretical analysis of BER performance of non-linearly amplied FBMC/OQAM and OFDM signals[END_REF]. For Rayleigh at fading channels, the BER analytical expressions developed in [START_REF] Bouhadda | Theoretical analysis of BER performance of non-linearly amplied FBMC/OQAM and OFDM signals[END_REF], for CP-OFDM and FBMC-OQAM MWFs, required numerical integration. We investigated, in the journal paper [START_REF] Zayani | Closed-form BER expression for (QAM or OQAM)-based OFDM system with HPA nonlinearity over Rayleigh fading channel[END_REF], closed form expressions of the BER for both MWFs and showed that FBMC-OQAM signal has the same sensitivity to the PA nonlinearity as the CP-OFDM signal when the channel is perfectly equalized.

The work proposed in [START_REF] Bouhadda | Theoretical analysis of BER performance of non-linearly amplied FBMC/OQAM and OFDM signals[END_REF], has been extended, within the framework of the WOWG5 project, to study the Symbol Error Rate (SER) of other MWFs. The considered post-OFDM MWFs are: WOLA-OFDM, UFMC, f-OFDM and FBMC-OQAM. Two PA models have been considered in this work: (1) a 4 GHz long term evolution (LTE) user equipment PA provided by [START_REF]Realistic power amplier model for the new radio evaluation[END_REF] and modeled using a polynomial model [56], (2) a Rapp model. With this study, we developed a universal approach to compute theoretically the SER for any measured or modelled memoryless PA and for each MWF. Theoretical results validate the simulated ones and show that PA distortions strongly aect the performance of the studied MWFs. It was shown that the selected MWFs almost perform as the classical CP-OFDM when PA is operated near its saturation region. These results were part of deliverable [START_REF] Shaiek | Performance of the candidate waveforms in the presence of power amplier[END_REF] of WONG5 project and have been published in a journal paper [START_REF] Shaiek | Analytical analysis of SER for beyond 5G post-OFDM waveforms in presence of high-power ampliers[END_REF].

Out-Of-Band distortions analysis

Within the framework of the FP7 EMPHAtiC project, we have been interested on the prediction the spectral regrowth of CP-OFDM and FBMC-OQAM MWFs. Theoretical studies, using Price theorem and cumulants, have been conducted to predict theoretically the PSD at the output of dierent PA models.

From the comparative analysis of CP-OFDM and post-OFDM MWFs [START_REF] Medjahdi | On the road to 5G: Comparative study of physical layer in MTC context[END_REF], we have shown that FBMC-OQAM exhibits the best frequency localization. This advantage is maintained if the PA is linear. However, when the PA is operated near to its saturation region, there will be a spectrum regrowth and the advantage of this MWF, compared to CP-OFDM, will be reduced. In this case, it can be useful to predict the spectral regrowth induced by the NL PA to have a quite reliant information about the unused sub-carriers.

Two popular methods are commonly used for the prediction of the spectral regrowth of MWFs in presence of NL-PA: The rst method is based on Price's theorem [START_REF] Stern | Méthodes pratiques d'étude des fonctions aléatoires[END_REF] and the second one is based on cumulants [START_REF] Nikias | Higher-order spectral analysis[END_REF]. These methods are based on deriving a closed form expression for the auto-covariance function of the PA output, whose Fourier Transform (FT) yields the output PSD. With Price theorem, the estimation of the PSD regrowth is generally based on a (Solid State Power amplier) SSPA model approached by a given order polynomial model [START_REF] Khodjet-Kesba | Multicarrier interference evaluation with jointly non-linear amplication and timing errors[END_REF]. This latter can fail in predicting accurately the PSD regrowth, especially at the vicinity of the PA saturation region. During the master training of Saidou SALL, we proposed a suitable polynomial modelling for the PA conversion characteristics, to obtain a more reliable analytical estimation of the spectral regrowth of FBMC-OQAM. These results have been published on a conference paper [START_REF] Sall | Analysis of the nonlinear spectral re-growth in FBMC systems for cognitive radio context[END_REF].

However, Price theorem is generally applied to memoryless PA models, exhibiting only AM/AM distortion. Also, it involves intensive integral calculus for high polynomial orders which will impact the closed-form expression derivation. For these reasons, we have switched to methods based on cumulants [START_REF] Nikias | Higher-order spectral analysis[END_REF]. Within the rst part of Krishna Bulusu's PhD, we have extended the work proposed by Zhou et. al [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power ampliers[END_REF] to predict the spectral regrowth of FBMC-OQAM signals by using cumulants. In this work, we considered memoryless PA models exhibiting both AM/AM and AM/PM conversion characteristics and tted with polynomial function. Simulation carried for FBMC-OQAM conrmed the good accuracy of the derived closed form expressions of the predicted PSD. We also demonstrated that the Non-linearity order and the IBO play vital role in spectral regrowth prediction accuracy. These results have been published in a conference paper [START_REF] Bulusu | Prediction of spectral regrowth for FBMC-OQAM system using cumulants[END_REF].

It's worth to point that the approaches used for the prediction of the spectral regrowth for CP-OFDM and FBMC-OQAM can be applied to most of the post-OFDM MWFs candidate for 5G. Indeed, provided that the PA input signal can be viewed as complex Gaussian, prediction of the spectral regrowth, using Price theorem or cumulants, remains valid.

The scientic contributions related to this research topic will be given in chapter 4.

Enhanced PA eciency for 5G and beyond MWFs

According to several studies [START_REF] Dahlman | 5G wireless access: requirements and realization[END_REF][START_REF] Andrae | Total consumer power consumption forecast[END_REF], the ICT sector accounts for nearly 3% of the worldwide energy consumption. For mobile communications systems, the power consumption helps to dene the battery life. From the base station side, the estimated consumed power is about 80% of the total energy of cellular infrastructure [START_REF] Jens | The Energy and Carbon Footprint of the Global ICT and E&M Sectors 20102015[END_REF]. In fact, in today's macro base stations, the PA eciency plays a key role in the energy eciency of the whole transmitter chain as this device is the most power-consuming component. From another side, CP-OFDM as well as most of the MWFs candidate for 5G and beyond systems suer from high PAPR. If the PA is operated in its quasi-linear region, the high PAPR of the multicarrier signal will not have any inuence on the quality of the transmission. Nevertheless, this situation has a high cost in terms of energy eciency, especially for mobile applications with batteries. In order to increase power eciency, PA should be operated as close as possible to its saturation region, but this would introduce broadening of the amplied signal spectrum, as well as distortion over the transmitted signal itself. Many solutions have been proposed to increase the eciency of PA in 4G-5G networks. These solutions can be broadly classied into three categories:

• Signal processing-based approaches: These solutions do not require any change in the transmitter architecture. The PA eciency is increased with the help of PAPR reduction techniques and/or linearization techniques. These processing can be also cascaded to get an enhanced control of the PA eciency. Otherwise, PAPR reduction and PA linearization can be designed to work jointly by seeing the two pre-processing as signal adding techniques. In few words, all the techniques, related to this approach, intend to reduce the IBO by making the PA operating as close as possible to its saturation region.

• Modied architecture transmitter-based approaches: They aim at modifying the transmitter architecture. Several solutions can be used: envelope tracking, Doherty ampliers, Kahn techniques, parallel architectures, linear amplication with nonlinear component (LINC) also known as Out phasing technique.

• Network based approaches: The power eciency is increased while using smaller cells and massive MIMO technology. Adding to this, the idea to reduce the power amplier activation as well as on/o techniques. In this way the operating point of the power amplier can be updated according to the load of the cell (night and day trac changes).

Increasing the PA eciency could also be a mix of all the aforementioned techniques. In [START_REF] Brandon | Joint optimization of linearization methods at the transmitter for multi-carrier modulations[END_REF], the authors combined envelope tracking, PAPR reduction and on/o techniques for multi-carrier based transmissions. A third topic of my research activities, focused on studying and proposing dierent solutions to enhance PA eciency in 5G context, by using either signal processing-based solutions or modied architecture-based ones. An illustration of this research topic is given in Fig. 2.11. Regarding networkbased approaches and particularly those based on massive MIMO technology, some contributions will be presented in the next section. by using a trellis-based scheme involving, for the optimization of the PAPR, more than one symbol at a time. The proposed algorithms named Trellis base SLM (TSLM) scheme have been published in a journal and a conference papers [START_REF] Bulusu | Enhanced multicarrier techniques for narrowband and broadband PMR coexistence[END_REF][START_REF] Bulusu | Potency of trellis-based SLM over symbol-by-symbol approach in reducing PAPR for FBMC-OQAM signals[END_REF].

We have then proposed a third solution for reducing the PAPR of FBMC-OQAM signals, named as Dispersive Tone Reservation (DTR). This algorithm has been published in a conference paper [START_REF] Bulusu | Reduction of PAPR of FBMC-OQAM systems by dispersive tone reservation technique[END_REF].

PA linearization techniques for FBMC-OQAM: In the case of pre-distortion, we have demonstrated the need to consider the overlapping nature of FBMC-OQAM symbols. A novel DPD technique, named Overlapped Recursive Error Correcting (OREC) DPD has been proposed for FBMC-OQAM [START_REF] Bulusu | PA linearization of FBMC-OQAM signals with overlapped recursive error correcting predistortion[END_REF].

The OREC-DPD algorithm relies on a constant convergence rate, thereby, leading to slow convergence. Recently, we have enhanced this algorithm to a very fast converging one [START_REF] Bulusu | HPA linearization for FBMC-OQAM signals with fast convergence-digital predistortion[END_REF][START_REF] Bulusu | HPA linearization for next generation broadcasting systems with fast convergence-digital predistortion[END_REF] requiring few iterations.

Co-existence capabilities of a FBMC-OQAM system working with NL PA: One of the main objectives of the FP7 European EMPHAtiC project was to investigate the coexistence of narrow band and broadband systems for Professional Mobile Radio (PMR) communication systems like Public Protection and Disaster Relief (PPDR). In [START_REF] Medjahdi | Integrating LTE broadband system in PMR band: OFDM vs. FBMC coexistence capabilities and performances[END_REF], it has been shown that classical CP-OFDM based LTE system needs sucient guard-band or additional ltering in order to coexist in a PMR band, even without considering the RF impairments of the transmitter. In practice, it is imperative to consider the RF impairments especially those related to NL PA. During Krishna's Bulusu PhD, we investigated the coexistence of narrow band and broadband PMR systems that can support asynchronism, have low latency and are able to be operated with NL devices. We demonstrated that a LTE-like FBMC-OQAM broadband system working with NL PA devices can still co-exist with legacy PMR systems with high energy eciency, when proper PA linearization techniques are applied jointly with PAPR reduction methods. This work was carried out jointly with Pr. Markku Renfors from Tampere University of Technology (Finland). The obtained results have been published in a journal paper [START_REF] Bulusu | Enhanced multicarrier techniques for narrowband and broadband PMR coexistence[END_REF] and then completed by a conference paper [START_REF] Bulusu | Impact of HPA non-linearity on coexistence of FBMC-OQAM systems with PMR/PPDR systems[END_REF].

All of the contributions made to characterize and mitigate the IB and OOB eects of NL PA on FBMC-OQAM as well as the associated techniques to improve the eciency of PAs were published in chapter 13 of the book [START_REF] Renfors | Contributors. In Orthogonal Waveforms and Filter Banks for Future Communication Systems[END_REF]. I managed the coordination of this work, carried out jointly with colleagues from Tampere University of Technology.

Case of other MWFs: WOLA-OFDM, UF-OFDM, F-OFDM and BF-OFDM PAPR reduction techniques: Within the framework of the WONG5 ANR project, we have focused on applying PAPR reduction techniques for many MWFs candidate for 5G NR. As stated before, all the time domain signals generated with most post-OFDM modulations exhibit high PAPR levels, like those observed with regular CP-OFDM signal. There are many PAPR reduction techniques whose performance could be very dierent depending on several criteria: complexity, receiver modication (known as downward compatibility), IB distortions, OOB regrowth, etc. We focused on PAPR reduction techniques like: TR, SLM, pre-coding and applied them to CP-OFDM, WOLA-OFDM, UF-OFDM, f-OFDM and BF-OFDM.

For WOLA-OFDM, UF-OFDM and f-OFDM, the application of SLM and TR techniques were straightforward. The evaluation of the PAPR reduction techniques was performed considering the PAPR reduction gain and the complexity of the algorithms. Through this evaluation, we demonstrated that: (1) for all the considered post-OFDM MWFs, both SLM and TR remain as ecient as in CP-OFDM. (2) for UF-OFDM, both algorithms require huge complexity resources. These results give rise to a conference paper publication [START_REF] Tani | PAPR reduction of post-OFDM waveforms contenders for 5G Beyond using SLM and TR algorithms[END_REF].

Within the framework of the 3 rd work package of WONG5 project, I carried, jointly with colleagues from CentraleSupélec Rennes, a study to estimate the overall power budget of transmitters integrating PA DC power and consumption related to the PAPR reduction algorithms. The considered MWFs were: CP-OFDM, WOLA-OFDM, f-OFDM and BF-OFDM, while the PAPR reduction method is the SLM one. The obtained results have been published in a conference paper [START_REF] Louet | Global power amplier eciency evaluation with PAPR reduction method for Post-OFDM waveforms[END_REF].

As part of Khaled Tahkoubit's PhD we have adapted a DFT spreading technique, used for CP-OFDM, to BF-OFDM for UL transmissions. We have shown that SC-FDMA and DFT spread BF-OFDM achieve quite same PAPR reduction performance of around 2 dB. This contribution has been published in a conference paper [START_REF] Tahkoubit | PAPR reduction of BF-OFDM waveform using DFT-Spread technique[END_REF].

Testbed for demonstrating post-OFDM MWFs capabilities in cMTC context: The selection of the best post-OFDM adapted for cMTC scenario, has been carried out during the WONG5 project. Regarding sub-band ltering-based MWFs: f-OFDM and BF-OFDM won the race, where UF-OFDM lost it due to its important implementation complexity. In another hand WOLA-OFDM is also an interesting candidate for 5G cMTC applications thanks to its reduced complexity and good performance regarding the targeted KPIs.

We developed at CNAM a Software Dened Radio (SDR) testbed to demonstrate, in real world, the feasibility of the proposed DSP algorithms proposed to enhance the energy eciency of the proposed transceivers based on selected MWFs, while taking CP-OFDM as basis of comparison. This work, cosupervised with Pr. Daniel Roviras and Dr. Christophe Alexandre, has been carried by a postdoctoral researcher: Dr. Rak Zayani and two master students: Mrs. Xinying Cheng and Mrs. Xiaotian Fu.

Through this testbed, we have conrmed the supremacy of WOLA-OFDM and BF-OFDM in improving the energy eciency and the robustness to asynchronism. The PA eciency enhancement relied on PAPR reduction and DPD algorithms to operate the device at low IBO values. Furthermore, the developed testbed has been dedicated to evaluating the capability of the selected MWFs in handling multi-user signals when there is imperfect synchronization in time domain. A detailed description of the proposed testbed along with a selection of some obtained results have been published in a conference paper [START_REF] Zayani | A testbed for experimental performance evaluation of multicarrier waveforms in presence of RF PA[END_REF] and a journal paper [START_REF] Zayani | Experimental testbed of Post-OFDM waveforms toward future wireless networks[END_REF].

Joint PAPR reduction & DPD

Conventionally, PAPR reduction and DPD techniques are optimized separately and applied independently. Some studies focused on their association aiming at avoiding mutual eects. Among the proposed solutions, we can cite [START_REF] Gouba | A joint approach for PAPR reduction and predistortion by adding signal in cognitive radio[END_REF][START_REF] Landin | Peak-power controlled digital predistorters for RF power ampliers[END_REF]. From another side, the work carried during Krishna Bulusu's PhD showed the feasibility to view the DPD as an adding signal technique. This statement is also true for some PAPR reduction techniques such as clipping and TR. Thereby, an interesting idea has been raised saying that both PAPR reduction and DPD can be combined and made through a unique adding signal operation. This means that both processing can be done at one stretch by adding a signal, which not only reduces PAPR but also linearizes the PA, which yields to a joint optimization of the two processing.

With Dr. Rak Zayani, we took up this idea and apply it to CP-OFDM. We proposed a new technique called "Ping-Pong Joint Optimization" (P2JO) to optimize jointly PAPR reduction and predistortion for CP-OFDM signals. The proposed approach synthesizes, iteratively, one correction signal for reducing the PAPR and linearizing the PA conversion characteristics to reach a given performance in terms of Mean Square Error (MSE). We have shown that the proposed solution allows a better trade-o between PA eciency and linearity, compared to classical combinations of the two-processing. As the idea seemed very interesting to us, we started by patenting it via a European patent [START_REF] Zayani | Method for conditioning a signal for amplication thereof, method for amplication, associated conditioning device and amplication device[END_REF]. We published then the proposed algorithm and the related results in a journal article [START_REF] Zayani | Ping-pong joint optimization of PAPR reduction and HPA linearization in OFDM systems[END_REF].

Modied architecture transmitter-based approaches

The algorithmic solutions proposed previously, to improve the energy eciency of MWFs-based transceivers, would not require a modication of the architecture of the RF chain. These solutions aim to reduce uctuations in peak signal power and/or linearize the conversion characteristics of the PA. Another approach, to enhance PA eciency, would rather consist in changing the design of the RF chain. This can be done by amplifying the multi-carrier signal via several PAs, ensuring at least equal performance to classical schemes, but with lower overall power consumption. This is the key idea, for example, for the LINC technique. As part of Khaled Tahkoubit's thesis, we focused on answering this question: how could we break down a signal with a strong PAPR, coming from a given MWF, into signals with weaker PAPR, so that they can be amplied with better power eciency? This reection led us to propose a new method called Iterative Dichotomy PAPR (IDP) reduction. The proposed IDP algorithm is based on decomposing the MWF signal into a sum of M (M ≥ 2) signals having constant envelope and a residual error signal carrying all the original signal peaks. A judiciously weighted version of the residual signal is then added to each of the M constant envelope signals to give a lowered PAPR signal that could be amplied by M PA, at better eciency. We have carried out theoretical analysis for computing the M signals and conrmed their validity by simulation results. The rst part of this study concentrated on a second order decomposition of CP-OFDM signal (M = 2). At this order, a PAPR gain of 2.2 dB was achieved by the IDP algorithm compared to classical CP-OFDM transmitter. These results have been published in a journal paper [START_REF] Tahkoubit | Iterative dichotomy PAPR reduction method for multicarrier waveforms[END_REF].

As the transmitter architecture related to the IDP method is very similar to that of the LINC (also known as Out Phasing in the literature) one, we have compared the performances of these techniques. The comparison has been done following the three main points: (1) comparing the output power for the same DC consumed power and the same BER performance, (2) studying the inuence of timing errors in the combination process and (3) examining the eect of a mismatch between the PAs conversion characteristics. The results of this comparative study have been published in a conference paper [START_REF] Tahkoubit | Energy eciency comparison of Outphasing and IDP techniques with non-linear power ampliers[END_REF]. We have recently generalized the results published in [START_REF] Tahkoubit | Iterative dichotomy PAPR reduction method for multicarrier waveforms[END_REF] to any dichotomy order. These results were submitted to IEEE Transactions on Wireless Communications.

In chapter 5, I will present some of the results related to the enhancements of MWFs based transceivers in presence of NL PAs.

Massive MIMO networks with mixed numerology and nonlinear PAs

3GPP consortium converged, in December 2017 [START_REF]Technical Specication Group Radio Access Network; Study on New Radio (NR) access technology[END_REF], towards the requirements of NR: dened as the new radio interface for 5G systems. As expected, NR will enable new use cases, improve data rates, reduce latency, improve coverage capacity and reliability. These requirements should be fullled with enhanced network energy eciency, while allowing to enlarge the sub-6GHz spectrum to mmWave. Among the prominent technologies proposed to reach the 5G NR requirements, Massive MIMO and mixed numerology have been retained to support various congurations on top of OFDM waveform.

The major innovation of release 15 of 3GPP with respect to former standards is the support of various numerologies [START_REF] Wg1 | Numerology requirements[END_REF] which allows scalable Sub Carrier Spacing (SCS) and symbol duration. By doing so, challenges of 5G can be properly addressed. Indeed, shortening symbols reduces the latency required for some URLLC/cMTC. Moreover, enlarging the SCS increases the robustness against the Doppler eect occurring in mobility scenarios.

Massive MIMO is an emerging technology that uses many antennas at the BS serving many tens of terminals in the same frequency resources. Massive MIMO relies on spatial multiplexing with high number of antennas helping to focus energy into ever smaller regions of space. This beamforming brings huge improvements in throughput and radiated energy eciency.

As the power of the signal to be transmitted by each individual antenna is suciently reduced, massive MIMO includes extensive use of inexpensive low-power components. PA with very relaxed linearity requirements will necessitate PAPR reduction and/or predistortion signaling, which can exploit the massive MIMO system extra degrees of freedom, equal to the dierence between the number of BS antennas and the number of user's terminals.

As part of Xinying Cheng's PhD, I have been interested, since October 2018, in the study of two issues linked to massive-MIMO systems. The rst one is related to the interference caused by users using dierent numerologies and the other one concerns energy eciency with emitters architectures with joint MIMO precoding, PAPR reduction and DPD. A simplied illustration of these research topics is given by Fig. 2.12.

Analysis and cancellation of mixed-numerologies interference for massive MIMO-OFDM

The extremely diverse service requirements are an important challenge for the upcoming 5G wireless communication technologies, where OFDM-based massive MIMO and mixed numerologies transmission are proposed as solutions. During the rst year of Xinying Cheng's PhD we have introduced a new transceiver for massive MIMO-OFDM DL systems where we have investigated the use of spatial multiplexing of users, sharing the same bandwidth, whose associated numerologies are dierent. We rst introduced a precoding design that aims to manage the mixed numerologies spectrum sharing (SS) transmission. Then, we analyzed the Inter Numerology Interference (INI) and derived the theoretical expressions in massive MIMO-OFDM DL. We demonstrated that by using the proposed precoding scheme and considering two

Enhanced energy eciency of Massive-MIMO system

As said previously, the use of massive MIMO would signicantly improve the energy eciency of the 5G network, thanks to spatial multiplexing gain. However, the signals at the output of the multiuser (MU) precoder in single carrier or multi carrier mode exhibit high PAPR. From another side, the use of extensive and inexpensive low-power PA, could dramatically impacts the overall system performance if nothing is done to reduce the PAPR and/or increase the PA linearity region. During the second year of Xinying Cheng's PhD, we have been interested on this topic. We have designed an OFDM massive MIMO transceiver, taking into consideration MU precoding, PAPR reduction and DPD, with special care to the required computational complexity. The proposed precoder is solution of a convex optimization problem solved via steepest gradient descent (GD) approach. The proposed algorithm is compared to other equivalent transmission schemes, including dierent combinations of MU precoding, PAPR reduction and DPD processing. The main conclusion of this work is that the proposed transmission scheme provides low computational complexity than the other existing schemes while closely approaching their performance. A journal paper describing the proposed algorithm and its comparison with the existing solutions have been submitted to IEEE Transactions on Vehicular Technology.

Chapter 2. Overview of my research activities

The scientic contributions corresponding to this last research topic will be presented in chapter 6.

Conclusion

This chapter presented an overview of all my research activities. After a quick look to the main contributions made between 2003 and 2009 in the eld of acoustics and digital audio processing, I introduced the general context of my activities at CNAM. Then, I presented the four research topics developed as associate professor at CNAM, as well as the contributions and collaborations developed since 2011. As the research axes are now clearly identied and introduced, the main related scientic contributions will be developed in chapters 3, 4, 5 and 6.

Chapter 3

MWF for 5G and beyond systems

Introduction

URLLC communications require minimizing network access times by allowing possible asynchronous communications between users. Indeed, the overhead required by orthogonal transmission to ensure synchronization becomes large and brings large latencies concerning network access. Beyond the relaxation of the constraint on the time asynchronism, other criteria must be considered during the comparative study of post-OFDM MWFs. As Key Performance Indicators (KPIs), we can consider: Power Spectral Density (PSD) localization, Spectral Eciency (SE), End-to-End Physical layer latency, PAPR, transceiver complexity, adaptability to MIMO transmissions and performances over frequency-selective and/or time varying channels. Among the studied MWFs, FBMC-OQAM was shown to be very attractive regarding frequency localization and robustness to users asynchronism. However, the high level of intrinsic interference makes it very sensitive to channel selectivity. This raised the need for advanced equalization techniques, exploiting the statistical properties of OQAM signals.

Between 2015 and 2019, I have been interested, within the framework of the WONG5 ANR project, on the study and comparison of post-OFDM MWFs, that would better meet the requirements of 5G cMTC/URLLC use case. This project involved CNAM, CEA-leti, THALES and CentraleSupélec. The results presented in this chapter have been carried with two post-doctoral researchers: Dr. Yahia Medjahdi and Dr. Rak Zayani and a PhD student: Mrs. Hayfa Fhima.

Post-OFDM MWFs

MWFs classication

Before presenting the dierent MWF candidates for the physical layer of the future 5G URLLC applications, it should be interesting to classify them based on some crucial aspects. A possible classication of these MWFs can be achieved with respect to the orthogonality criterion. A given MWF is orthogonal when the transmitted symbols are separable at the receiver after propagation over distortion free channels. Based on this denition, we can distinguish three categories:

• Orthogonality in the complex domain C: were complex-valued data symbols are transmitted every symbol period T with a subcarrier spacing F , where the symbol density T F ≥ 1,

• Orthogonality in the real domain R: due to the use of time-frequency well-localized prototype function, the orthogonality is restricted to R by transmitting real-valued data symbols [START_REF] Saltzberg | Performance of an ecient parallel data transmission system[END_REF]. To maintain the symbol density at T F = 1, two real-valued symbols are transmitted per unit timefrequency lattice area. In such a case, we nd oset-QAM (OQAM)-based MWFs,

• Non-orthogonality: the concept of transmitting real data instead of complex one, makes more complex the MIMO adaptation of the OQAM based techniques due to the high level of inherent interference brought by the prototype lter. Other solutions have been proposed to optimize the prototype lter response and transmitting complex QAM symbols rather than real ones. In this category we can cite FBMC-QAM and GFDM proposals.

In addition to the orthogonality-based classication, MWF can be clustered by considering the possible usage of ltering/windowing at transmitter (Tx) and/or receiver (Rx) and the granularity: subcarrier

MWFs

Orthogonality Filtering granularity Guard interval

CP-OFDM C n/a CP WOLA-OFDM C n/a CP UF-OFDM C RB ZP f-OFDM C sub-band CP N-Cont.-OFDM C n/a CP FMT C subcarrier n/a FFT-FBMC C RB CP BF-OFDM C RB CP FBMC-OQAM R subcarrier n/a Lapped-OFDM R subcarrier n/a WCP-COQAM R subcarrier CP WOLA-COQAM R subcarrier CP FBMC-QAM Non Orthogonal subcarrier n/a GFDM Non Orthogonal subcarrier CP Table 3.1: MWFs classication.
wise vs resource block (RB) wise. Another classication can be based on wether a given MWF uses or not cyclic extensions to absorb lters transient response and/or channel delay spread. In Tab. 3.1, I give dierent possibilities to classify the MWFs that will be studied and compared in this chapter. CP-OFDM and Filtered/windowed MWFs which are orthogonal in the complex domain (WOLA-OFDM, UF-OFDM, N-Continuous OFDM, FMT, BF-OFDM, FFT-FBMC) are in fact orthogonal only for propagation channels which are not too frequency selective. For such MWFs, the use of MIMO technology is straightforward. However, ltered MWFs which are not orthogonal in the complex domain (FBMC-OQAM, Lapped-OFDM, WCP-COQAM, WOLA-COQAM, FBMC-QAM, GFDM) generate both ISI and ICI at reception. These interferences explain the strong performance degradation of most of the standard MIMO schemes used for CP-OFDM, even for propagation channels which are relatively weakly frequency selective. For this reason, advanced MIMO schemes, aiming at mitigating the eects of ISI and ICI, are required for non-complex orthogonal ltered MWFs.

MWFs description

CP-OFDM CP-OFDM is widely used in several wireless standards (e.g., IEEE 802.11 and 3GPP-LTE). It consists of splitting up a stream of complex symbols at high-rate into several lower-rate streams transmitted on a set of orthogonal subcarriers which are implemented using the inverse fast Fourier transform (IFFT). The CP-OFDM receiver can be implemented using the fast Fourier transform (FFT). In order to keep the orthogonality between subcarriers, a cyclic prex (CP) is usually inserted. Therefore, after the FFT operation, the channel equalization becomes trivial through a single coecient per subcarrier. This modulation is used for 4G/LTE system and it is also the one chosen for 5G NR. In the following, this MWF will be considered as basis of comparison.

WOLA-OFDM To maintain the benets of CP-OFDM while reducing the Out-Of-Band (OOB) emissions, straightforward enhancements can be made to smooth the time-domain discontinuities between adjacent symbols. This approach named WOLA-OFDM have been introduced by Qualcomm [START_REF]R1-162199 -waveform candidates[END_REF].

In WOLA-OFDM Tx, the soft edges are added to the cyclic extensions of a given symbol by a time domain windowing. More specically, the windowing operation can be performed on both the original CP and the newly added cyclic sux as depicted by Fig. 3.1 (a). To create this latter, we copy and append the rst W Tx samples of a given symbol to its end. Adjacent symbols could overlap with each other in the edge transition regions, which leads to a similar overhead as in the CP-OFDM.

In addition to the transmit windowing, an advanced receive windowing is applied to suppress the asynchronous inter-user interference (i.e., adjacent non-orthogonal signals). In the rst step, the receiver takes N s + 2W Rx samples (N s = N FFT denotes the FFT size), which correspond to the samples of one (a) Tx.

(b) Rx. Figure 3.1: WOLA processing.

WOLA-OFDM symbol. Then, these samples are windowed. In the second step, an Overlap and Add processing is applied to create the useful N s samples from the N s + 2W Rx ones, as depicted by 3.1 (b).

This MWF has been studied and implemented at CNAM, during the postdoctoral stay of Dr. Rak Zayani. The performance of this MWF in asynchronous conditions has been published in a well cited conference paper [START_REF] Zayani | WOLA-OFDM: A potential candidate for asynchronous 5G[END_REF].

UF-OFDM UF-OFDM has been proposed by Alcatel-Lucent Bell Laboratories [START_REF] Vakilian | Universal-ltered multi-carrier technique for wireless systems beyond LTE[END_REF], and it is also referred to UFMC in the literature [START_REF] Wild | 5G air interface design based on Universal Filtered (UF-)OFDM[END_REF]. UF-OFDM is a combination of ZP-OFDM (traditional CP-OFDM where the CP is replaced by a Zero Padding (ZP) [START_REF] Muquet | Cyclic prexing or zero padding for wireless multicarrier transmissions[END_REF]) and ltered-OFDM which will be presented in the next paragraph: each OFDM symbol at the output of the IFFT is ltered and the ZP is used to absorb the lter transient response. When the channel is perfect, UF-OFDM holds the orthogonality of the subcarriers. Nevertheless, the orthogonality is no longer sustained as the time spreading of the channel increases. At the reception, the multiuser interferences coming from time and frequency asynchronism are rst reduced by applying a window on the received UF-OFDM block symbols [START_REF] Schaich | Relaxed synchronization support of universal ltered multi-carrier including autonomous timing advance[END_REF]. Finally, a FFT of size two times greater than the IFFT used at the transmitter is applied to the received UF-OFDM block symbols and only even subcarrier indexes are kept. With Dr. Yahia Medjahdi, we have studied and implemented this MWF.

Filtered-OFDM f-OFDM was proposed as a 5G candidate at the 3GPP RAN1 workgroup [START_REF] Hisilicon | f-OFDM Scheme and lter design[END_REF]. At the transmitter side, the poor OOB radiation of traditional CP-OFDM is improved thanks to a lter applied to the CP-OFDM symbol. At the receiver side, the interferences coming from time and frequency asynchronous adjacent users are reduced using a similar lter at the input of a CP-OFDM receiver. The ltering process required by f-OFDM generates inter block interferences, but if the lter is properly designed the impact of these interferences on the link performance are negligible [START_REF] Hisilicon | f-OFDM Scheme and lter design[END_REF]. It must be noted that the ramp-up and the ramp down generated by the lter increase the burst length and consequently the latency, reducing so the spectral eciency.

N-continous OFDM The N-continuous OFDM scheme has been introduced in [START_REF] Van De Beek | N-continuous OFDM[END_REF]. The idea of this MWF consists in creating consecutive adjacent CP-OFDM symbols which are continuous in the time domain to improve the poor OOB radiation of traditional CP-OFDM. The construction of OFDM symbols will render the transmitted signal and its rst N derivatives continuous using a precoding matrix which is placed between the symbol mapping and the IFFT. One advantage of N-continuous OFDM scheme is to have traditional CP-OFDM as its core MWF at both transmitter and receiver sides. Nevertheless, in its basic form, this scheme requires the transmission of side information (precoding matrix) to the receiver to recover the data. A solution to cope with this problem has been proposed in [START_REF] Van De Beek | EVM-constrained OFDM precoding for reduction of out-of-band emission[END_REF] and consists in using a systematic precoding matrix.

FMT FMT is a multicarrier modulation technique that has been specically developed for DSL applications [START_REF] Cherubini | Filter bank modulation techniques for very high speed digital subscriber lines[END_REF]. In FMT, a conventional method of frequency division multiplexing is used, i.e., the subcarrier bands are not overlapping. Each subcarrier is a traditional single carrier modulation ltered with square-root Nyquist lter. This lter limits the eective transmission bandwidth of each band to B = (1 + α) • D s , where D s is the symbol rate and α is the roll o factor, which should be judiciously chosen to avoid large square root Nyquist lter length which drastically increases the latency and the overhead. Some recent works have proposed new lters [START_REF] Traverso | A family of square-root nyquist lter with low group delay and high stopband attenuation[END_REF] in order to cope as much as possible with this problem providing a good compromise between OOB radiation and group delay.

FBMC-OQAM & Lapped-OFDM One of the drawbacks of the CP-OFDM and some of the windowing/ltering-based OFDM MWFs previously introduced is the loss of spectral eciency due to the use of CP or specic guard band. This leads to a symbol density greater than 1. Thus, to avoid this drawback, we have to choose a MWF with symbol density of one. However, it is theoretically proven, based on the Balian Low theorem [START_REF] Saltzberg | Performance of an ecient parallel data transmission system[END_REF], that it is not possible to fulll simultaneously (1) a prototype lter which is well-localized in both time and frequency domain, (2) the transmission of data symbols satisfying complex orthogonality and (3) a symbol density equal to 1. To achieve these objectives, we can restrict the orthogonality condition to the real domain. In this case and over a distortion-free propagation channel, it is possible to recover the data at the receiver side when real-valued (or purely imaginary) Pulse Amplitude Modulation (PAM) is used instead of Quadrature Amplitude Modulation (QAM). With this solution and to maintain a symbol density equal to 1, we should transmit two real-valued symbols per unit area of the time-frequency lattice. One of the pioneering solutions, fullling the Balian Low theorem, and called FBMC-OQAM (FBMC based on transmitting Oset-QAM symbols), was proposed by Saltzberg [START_REF] Saltzberg | Performance of an ecient parallel data transmission system[END_REF] in the mid-1960s.

The key-idea of FBMC-OQAM is to use well-frequency localized prototype lters, providing thus improved adjacent channel leakage performance compared to CP-OFDM. To ensure orthogonality between adjacent symbols and adjacent subcarriers, while keeping maximum spectral eciency, Nyquist constraints on the prototype lter combined with OQAM are used, bringing so the orthogonality from the complex domain to the real one. Indeed, in OQAM, the in-phase and the quadrature components of a given QAM symbol are time staggered by half a symbol period. The duration of the prototype lters is usually a multiple of the FFT size (KN f f t ), where K is called the overlapping factor and N F F T the total number of subcarriers. With Dr. Yahia Medjahdi, we considered two prototype lters: (1) FBMC-OQAM based on the most used PHYDYAS lter [START_REF] Viholainen | Prototype lter design for lter bank based multicarrier transmission[END_REF] with K = 4 and (2) Lapped-OFDM using the sine prototype lter with K = 2 [START_REF] Bellanger | Lapped-OFDM as an alternative to CP-OFDM for 5G asynchronous access and cognitive radio[END_REF].

The baseband of the FBMC-OQAM transmitted signal can be dened as follows [START_REF] Bouhadda | Sensitivity analysis of FBMC signals to non linear phase distortion[END_REF] :

x(t) = N F F T -1 m=0 +∞ n=-∞ a m,n h(t -nT /2)e j 2π
T mt e jϕm,n , (

where h(t) is the prototype lter impulse response, a m,n are real-valued symbols and ϕ m,n is a phase term equal to π 2 (m + n) -πmn. Considering the shifted versions of h(t) in time and frequency noted:

γ m,n (t) = h(t - nT /2)e j 2π
T mt e jϕm,n , in a distortion-free noise-less channel, the demodulated signal y m0,n0 at time instant n 0 and subcarrier m 0 is given by:

y m0,n0 = x(t), γ m0,n0 (t) = +∞ -∞ i(t)γ * m0,n0 (t)dt = +∞ n=-∞ N F F T -1 m=0 a m,n +∞ -∞ γ m,n (t)γ * m0,n0 (t)dt = a m0,n0 + m =m0 n =n0 a m,n +∞ -∞ γ m,n (t)γ * m0,n0 (t)dt Ψm 0 ,n 0 = a m0,n0 + ju m0,n0 , (3.2)
where γ * m0,n0 (t) is the complex conjugate of γ m0,n0 (t) and ., . stands for the inner product. According to [START_REF] Bellanger | Spicication and design of prototype lter for lter bank based multicarrier transmission[END_REF], the prototype lter is designed such that the intrinsic interference term is orthogonal to the useful symbol i.e. is purely imaginary. Considering the PHYDYAS prototype lter proposed in [START_REF] Bellanger | Spicication and design of prototype lter for lter bank based multicarrier transmission[END_REF], WCP-COQAM and WOLA-COQAM Despite the various advantages of FBMC-OQAM systems, the long prototype lters could be questionable for URLLC communications [START_REF] Schaich | Waveform contenders for 5G -suitability for short packet and low latency transmissions[END_REF]. Besides, FBMC-OQAM signals are not suitable to short packet transmission due to long ramp-up/down of the prototype lter leading thus to a non-negligible loss in spectral eciency. In order to overcome this situation, burst truncation can reduce this loss but it has detrimental eects like additional interference and signicant OOB radiation [START_REF] Bellanger | Eciency of lter bank multicarrier techniques in burst radio transmission[END_REF]. Circular convolution with time-windowing was proposed in [START_REF] Abdoli | Weighted circularly convolved ltering in OFDM/OQAM[END_REF], [START_REF] Lin | Multi-carrier modulation analysis and WCP-COQAM proposal[END_REF] to remove the overhead signal while maintaining smooth transition at the burst edges. This solution is known as Windowed Cyclic Prex-based Circular-OQAM (WCP-COQAM). In order to avoid multipath channel interference, a CP can easily be inserted since COQAM corresponds to a block transform [START_REF] Lin | Multi-carrier modulation analysis and WCP-COQAM proposal[END_REF]. Thanks to circular convolution, the continuity of CP-COQAM signal is maintained inside a given CP-COQAM block. However, since signal discontinuities can be observed between dierent CP-COQAM blocks, there is no remarkable dierence between the CP-COQAM spectrum and CP-OFDM one [START_REF] Farhang-Boroujeny | OFDM versus lter bank multicarrier[END_REF]. Note that this behavior is independent of how well is localized the prototype lter frequency response. At the receiver side, the CP is removed, windowed samples are compensated and the receive circular convolution is applied afterwards. The OQAM decision is then made to recover the desired useful data symbols.

n 0 -3 n 0 -2 n 0 -1 n 0 n 0 + 1 n 0 + 2 n 0 + 3 m 0 -1 0.043j 0.125j 0.206j 0.239j 0.206j 0.125j 0.043j m 0 -0.067j 0 -0.564j 1 0.564j 0 0.067j m 0 + 1 0.043j -0.125j 0.206j -0.239j -0.206j -0.125j 0.043j
In [START_REF] Zayani | WOLA-OFDM: A potential candidate for asynchronous 5G[END_REF], we compared WOLA-OFDM and UF-OFDM in relaxed synchronization scenario. The obtained results show that WOLA-OFDM outperforms UF-OFDM in various asynchronous congurations. Motivated by this result, we investigated, during the postdoctoral stay of Dr. Yahia Medjahdi, the eectiveness of WOLA processing on WCP-COQAM. The modied MWF is called WOLA-COQAM, where the WOLA transmitter performs a time domain windowing to produce soft edges at the beginning and the end of the original transmitted block, which length is cyclically extended from N s = KN FFT samples to N s + CP + W T x (see Fig. 3.1 (a)). In addition to the transmit windowing, the WOLA processing is also applied at the receiver side in order to enhance asynchronous inter-user interference suppression, as illustrated in Fig. 3.1 (b). Note that the applied receive window is independent of the transmit one and its length is equal to N s + 2W Rx . This windowing is followed by Overlap and Add processing which minimizes the eects of windowing on the useful data.

FFT-FBMC In order to overcome the FBMC-OQAM intrinsic interference issue, a precoded FBMC scheme has been proposed in [START_REF] Zakaria | A novel FBMC scheme for spatial multiplexing with maximum likelihood detection[END_REF][START_REF] Zakaria | A novel lter-bank multicarrier scheme to mitigate the intrinsic interference: Application to MIMO systems[END_REF]. This scheme proceeds by precoding the data in a subcarrier-wise manner using an IFFT. In FFT-FBMC proposal, blocks of N/2 data complex symbols in each subcarrier k go through a N -IFFT operation. The N/2 data symbols are alternately fed to the rst and last N/2 bins of the N -IFFT. When the subcarrier index k is odd (resp. even), the symbols are fed to the rst (resp. last) N/2 bins. After that, the N -IFFT outputs are extended with a CP of size L, and fed to the FBMC modulator of M carriers.

Thus, the interference coming from others subcarrier is removed by a simple equalization thanks to the subcarrier-wise IFFT/FFT precoding/decoding and CP insertion. Whereas the interference coming from the adjacent carriers can be avoided by a special data transmission strategy and a good frequencylocalized prototype lter. Fig. 3.2 (a) depicts the scheme of the FFT-FBMC Tx proposed in [START_REF] Zakaria | A novel FBMC scheme for spatial multiplexing with maximum likelihood detection[END_REF].

At the output of the FBMC demodulator, the serial symbols in each subcarrier q are reshaped into blocks of size N + L to only keep N symbols in each block. This operation is referred to as "S/P + CP removal" in Fig. 3.2 (b) . After that, N symbols of each block are fed to a N -FFT whose only N/2 (a) Tx.

(b) Rx. Figure 3.2: FFT-FBMC transeiver. output symbols are kept for detection. Again, the rst N/2 output symbols are kept when the subcarrier index q is odd, and the last N/2 symbols are kept when q is even.

The intercarrier interference is avoided thanks to the N/2 zeros inserted in the N -IFFT in each subband, ensuring the orthogonality in the complex domain [START_REF] Zakaria | Theoretical analysis of the power spectral density for FFT-FBMC signals[END_REF] and the eciency of single-tap equalization [START_REF] Zakaria | FFT-FBMC equalization in selective channels[END_REF].

BF-OFDM Block-Filtered OFDM (BF-OFDM) is another precoded lter-bank multi-carrier modulation that has been introduced by CEA-leti [START_REF] Demmer | Block-Filtered OFDM: an exhaustive waveform to overcome the stakes of future wireless technologies[END_REF][START_REF] Gerzaguet | The 5G candidate waveform race: a comparison of complexity and performance[END_REF]. The precoding scheme is performed by means of CP-OFDM modulators (of size N ) and the ltering operation is applied with a Poly Phase network (PPN) (of size M ) as for FFT-FBMC. However, the main dierence with respect to this latter is the insertion of a lter pre-distortion stage at the transmitter side. The extra stage aims at compensating the distortion induced by the lter to atten the transmitted signal spectrum inside the carrier bandwidth. As a consequence, no ltering stage is required at the receiver side to properly recover the signal and the receiver scheme can be reduced to a simple M N 2 -FFT preceded by a CP removal.

FBMC-QAM FBMC-OQAM uses a per-subcarrier ltering, reducing thus OOB emission and providing more exibility to meet 5G NR requirements. Such enhancements are at the price of orthogonality condition that only holds in the real domain and is no longer valid in the presence of realistic channels. Indeed, the self-interference inherent to OQAM-based schemes is a major problem when considering MIMO schemes [START_REF] Lélé | The Alamouti scheme with CDMA-OFDM/OQAM[END_REF], [START_REF] Zakaria | Intrinsic interference reduction in a lter bank-based multicarrier using QAM modulation[END_REF]. Some solution schemes were proposed based on iterative interference estimation and cancellation. However, such solutions are limited by error propagation induced by the residual interference since the interference power is as strong as the useful signal power [START_REF] Bellanger | FBMC physical layer: a primer[END_REF], [START_REF] Zakaria | On ISI cancellation in MIMO-ML detection using FBMC/QAM modulation[END_REF]. In order to overcome this problem, it has been demonstrated, in [START_REF] Agazzi | On the use of tentative decisions to cancel intersymbol interference and nonlinear distortion (with application to magnetic recording channels)[END_REF] and [START_REF] Van Beneden | Cancellation of linear intersymbol interference for two-dimensional storage systems[END_REF], that the interference power must be small and should be kept under a certain threshold, in order to counteract the error propagation phenomenon and consequently make more ecient the interference cancellation scheme. In order to achieve this objective and reduce the interference level, the authors in [START_REF] Zakaria | On ISI cancellation in MIMO-ML detection using FBMC/QAM modulation[END_REF] proposed the use of QAM modulation, instead of OQAM one, in FBMC systems. Actually, a signicant part of self-interference is avoided by only transmitting QAM symbols every signalling period nT, n ∈ Z. Such a combination is called FBMC-QAM systems. In order to improve the performance of FBMC-QAM symbols, new prototype lters have been designed, optimizing simultaneously spectrum localization, self-interference level, and overall spectral eciency [START_REF] Yun | A new waveform enabling enhanced QAM-FBMC systems[END_REF].

GFDM Generalized Frequency Division Multiplexing (GFDM) is a MWF performing a time-frequency ltering over a data block [START_REF] Fettweis | GFDM -generalized frequency division multiplexing[END_REF]. This MWF is therefore exible but also non-orthogonal. A data block corresponds to the set of symbols transmitted over a group of N F F T consecutive subcarriers over N B time slots and thus is composed of N F F T × N B symbols. The sub-carrier wise ltering is performed by means of circular convolution. However, as the symbols overlap in both time and frequency, interference (inter and intra data blocks) is generated. It is worth noticing that inter data block interference can be avoided by proper dimensioning of the CP.

In the literature two receiver schemes have prevailed: the Matched Filter (MF) and the ZF architectures [START_REF] Michailow | Generalized frequency division multiplexing: Analysis of an alternative multi-carrier technique for next generation cellular systems[END_REF]. With the MF approach, the received blocks are ltered by the transmission matched lters. This scheme provides low complexity but poor performance due to ISI. When it comes to the ZF approach, the signal is demodulated with the Moore pseudo-inverse of the transmitter matrix. This scheme suers from noise enhancement and the provided performance depends on the properties of the transmitter matrix [START_REF] Michailow | Bit error rate performance of generalized frequency division multiplexing[END_REF].

MWFs comparison

In this section, I will present a comparison of the MWFs introduced previously. The studies, carried under my supervision, have been achieved during the postdoctoral stay of Dr. Yahia Medjahdi.

I would like to stress that the results presented in this section are a selection of the work published in the review paper [START_REF] Medjahdi | On the road to 5G: Comparative study of physical layer in MTC context[END_REF]. Also, to simplify the presentation, some results corresponding for example to the comparison of MWFs in terms of PAPR and robustness over frequency selective channels will not be developed in this section. Interested readers may refer to [START_REF] Medjahdi | On the road to 5G: Comparative study of physical layer in MTC context[END_REF][START_REF] Medjahdi | Impact of selective channels on post-OFDM waveforms for 5G machine type communications[END_REF], for a more detailed description of the results presented here.

Comparison scenario

Except for BF-OFDM and FFT-FBMC, the total number of active sub-carriers for all the other MWFs is equal to 1024 and the corresponding Sub Carrier Spacing (SCS) is equal to 15 kHz. Among these waveforms, the size of the guard interval for those using a CP is N CP = 72, corresponding to a maximum delay spread of 4.68 µs. For BF-OFDM and FFT-FBMC, we considered 64 carriers, with an SCS equal to 180 kHz (1 LTE-RB bandwidth.). The number of active subcarriers has been set at N/2 = 32. For these two waveforms, we have considered a CP of size N CP = 4.

The comparison scenario, depicted in Fig. 3.3, corresponds to two coexisting users sharing the available frequency band, where the blue colored area and the red colored one correspond to the time/frequency resources allocated to the user of interest (UI) and to the interfering one, respectively. The useful signal occupies 3 LTE RB corresponding to 540 kHz, while the interfering bandwidth is equal to 1.62 M Hz (i.e., 9 LTE-RB) equally portioned on the two sides of the useful signal. A guard-band of δ kHz, illustrated by the gray colored area, is separating the frequency bands of both users.

The receiver of interest is assumed to be perfectly synchronized, in both time and frequency domains (i.e. neither Timing Oset (TO) nor Carrier Frequency Oset (CFO) are considered), and is located at equal distance from both transmitters1 . However, as illustrated in Fig. 3.3, a time/frequency synchronization misalignment (τ and ε denote timing and carrier frequency osets, respectively) can occur between the receiver of interest and the interfering user. Note that we consider a TO distributed between -T /2 and +T /2, where T is the OFDM symbol duration (T = 66.66 µs). Due to this synchronization mismatch, the receiver of interest potentially suers from the interference inducing thus performance degradation. To highlight the impact of this interference, we consider perfect and noiseless channels between both transmitters on one side and the victim receiver on the other side.

Power Spectral Density

It is well established that traditional CP-OFDM has poor frequency domain localization. In LTE, 10% of the system bandwidth is used for guard bands. These large guard bands located at both edges of the spectrum are necessary to reach enough attenuation to meet LTE spectrum mask requirement. It is expected that 5G and beyond systems use more eciently the allocated bandwidth and improve the overall spectral eciency of the system. Thus, good, or excellent spectral containment will be a key parameter for future 5G and beyond MWF in order to support neighboring signals.

We present in Fig. 3.4 the PSD comparison of the considered MWFs. We choose to plot only the contribution of the interference users (IUs) so that we can observe at the same time the level of OOB emission and the level of emission within a spectral hole corresponding to the UI. As expected, the worst PSD performance is given by CP-OFDM MWF. The far-end PSD is dominated by the MWFs adopting subcarrier wise ltering, namely Lapped-OFDM, FBMC-QAM, FBMC-OQAM and FMT. BF-OFDM and FFT-FBMC provide also excellent PSD performances thanks to the use of the lter bank approach in the signal construction. N-continuous OFDM has a relatively slow decaying but provides good farend PSD. UF-OFDM and f-OFDM apply their lter to a group of subcarriers, and we can observe that their performances are in the same order of magnitude. WCP-COQAM and WOLA-COQAM present moderate far-end PSD performance due to time domain transition between successive blocks. The time domain windowing applied by WOLA-OFDM improves by about 20 dB the spectral containment of CP-OFDM, but its far-end PSD performance remains moderate. Like WCP-COQAM and WOLA-COQAM, GFDM is not very well localized in the frequency domain due to the block construction of GFDM signal generating time domain transitions between blocks.

Spectral eciency and latency

Spectral eciency (SE), given in bits/s/Hz, is a key parameter for high data rate systems since it gives a clear idea of the achievable data rates for a given bandwidth. In Tab. 3.3, we present the SE according to the number of transmitted parallel vector symbols S, and also its asymptotic version called Asymptotic Spectral Eciency (ASE) where S tends toward innity. The required number of parallel vectors is dierent for each MWF and depends on the number of complex QAM symbols N QAM to be transmitted, but also on the way a block symbol is built. S is given by: WF Spectral Asymptotic End-to-End Eciency Spectral Eciency Physical layer latency CP-OFDM

N FFT N FFT +N CP N FFT N FFT +N CP S × N FFT +N CP Fs WOLA-OFDM S×N FFT S×(N FFT +N CP )+W Tx N FFT N FFT +N CP S×(N FFT +N CP )+W Tx Fs UF-OFDM N FFT N FFT +ZP N FFT N FFT +ZP S × N FFT +L ZP UF-OFDM Fs f-OFDM S×N FFT S×(N FFT +N CP )+L Trunc N FFT N FFT +N CP S×(N FFT +N CP )+L Trunc Fs N-Cont OFDM N FFT N FFT +N CP N FFT N FFT +N CP S × N FFT +N CP Fs FMT S (S+K-1)(1+α) 1 1+α (S-1+K)N FFT Fs FFT-FBMC S× N 2 K+ 1 2 [S×(N +N CP )-1] N N +N CP M × N 2 ×S+M × N CP 2 ×(S-1)+[(N CP -3)× M 4 +K× M 2 ] Fs BF-OFDM S× N 2 K+ 1 2 [S×(N +N CP )-1] N N +N CP M 2 S×N +N CP (S-1)+ N CP -3 2 +K Fs FBMC-OQAM S S+K-1/2 1 N FFT S-1 2 +K Fs Lapped-OFDM S S+3/2 1 S+ 3 2 Fs WCP-COQAM KN FFT KN FFT +N CP KN FFT KN FFT +N CP S × KN FFT +N CP Fs WOLA-COQAM S×K×N FFT S×(K×N FFT +N CP )+W Tx K×N FFT K×N FFT +N CP S×(K×N FFT +N CP )+W Tx Fs FBMC-QAM S S+K-1 1 N FFT S-1+K Fs GFDM N B ×N FFT N B ×N FFT +N CP N B ×N FFT N B ×N FFT +N CP S N B ×N FFT +N CP

Fs

Table 3.3: Spectral eciency and End-to-End Physical layer latency comparisons.

S =            N QAM K×N F F T
for WCP-OQAM and WOLA-COQAM

N QAM N B ×N F F T for GFDM N QAM M × N 2
for FFT-FBMC and BF-OFDM

N QAM N F F T otherwise (3.3)
where . refers to the ceiling operation, N B is the number of time slots (for GFDM), N F F T is the total number of subcarriers for all the MWFs excepting FFT-OFDM and BF-OFDM. For these last ones, N and M are respectively the number of used subcarriers and the number of active carriers. We can observe that for small values of S, the MWFs which have their lter applied to each subcarrier have a spectral eciency penalty due to their longer impulse response. This is especially true for FMT since the overlap factor K is usually much longer: for instance, we usually consider K = 16 for a roll o factor of 0.25, instead of K = 2 for Lapped-OFDM or K = 4 for FBMC-OQAM and FBMC-QAM. All the MWFs which require a guard interval (CP or ZP) suer from the fact that this guard interval does not transmit any useful information. When S tends toward innity, only FBMC derivatives (Lapped-OFDM, FBMC-OQAM and QAM) achieve full capacity, i.e., ASE = 1.

The latency of a MWF is a crucial key parameter in URLLC communications. In our analysis, we use the End-to-End (E2E) physical layer latency criterion dened as the time delay from which the forward error correction is capable to decode the bits corresponding to the N QAM transmitted symbols. E2E latency comparison is provided by table 3.3. In this table, F s is the sampling frequency equal to 15.36 M Hz for all the MWFs, excepting FFT-FBMC and BF-OFDM, for which the sampling frequency is equal to 11.52 M Hz. In order to better assess the performance of the other MWFs, we present in Fig. 3.5 the E2E with respect to traditional CP-OFDM scheme. In this gure, we can observe that the large group delay of the FMT prototype lter and the loss of spectral eciency due to the roll o factor drastically increases the latency of FMT scheme. All other MWFs latencies are in the same order of magnitude as CP-OFDM. We can also observe that for small N QAM values, the latency of WOLA-OFDM is very similar to the one of CP-OFDM while MWFs with long impulse response lters suer and yield to higher latencies. When N QAM increases, FBMC based MWFs become a little bit better than OFDM.

Inuence of timing oset

In this section, we discuss the performance of the considered MWFs in multi-user asynchronous access in time. The asynchronous interference impact on the performance of the various MWF schemes will be assessed using the normalized mean square error (MSE) on the decoded symbols (of the UI) in ideal noiseless channel. Per-subcarrier MSE can provide a meaningful information about the distribution of asynchronous interference across useful subcarriers. Two cases of guard-bands are examined: δ = 0 and 75 kHz.

To distinguish the degradation induced by timing synchronization errors from the one caused by CFO, we consider in this section that there is no CFO between the interfering signal and the useful one. The timing misalignment τ varies from -T /2 = -33.33 µs to T /2 = 33.33 µs. The per-subcarrier MSEs of MWFs are depicted in Fig. 3.6. In CP-OFDM case, we can see that the orthogonality between subcarriers is maintained if the delay error τ does not exceed the CP duration. When τ is outside the CP interval, the orthogonality is no longer ensured. This loss of orthogonality gives rise to a strong level of asynchronous interference even when increasing the guard band. Such a behavior is due to the poor frequency localization of the rectangular transmit/receive OFDM lters.

For WOLA-OFDM, we can see that the interference level in the middle of the bandwidth becomes lower (approx. -35 dB) compared to CP-OFDM scheme. We can also see that blue colored area (MSE less than -30 dB) becomes larger when increasing the guard band. This can be explained by the fact that WOLA processing applied at the receiver is able to suppress inter-user interference as well.

Thanks to the per-RB ltering, the UF-OFDM scheme shows better performance compared to CP-OFDM. Indeed, thanks to additional windowing at the receiver side, UF-OFDM oers interesting gain when the TO is outside the ZP region. Like WOLA-OFDM, the ZP region oering the lowest MSE is signicantly reduced in UF-OFDM because the ZP length is originally used to absorb the transmit lter response. Furthermore, when increasing the spectral distance of the victim subcarrier from the interfering ones, the interference level decay is more important compared to CP-OFDM but less signicant when compared to WOLA-OFDM.

In f-OFDM scheme, since ltering is applied at both transmitter and receiver sides, the inner subcarriers are more protected compared to the previous schemes. In fact, the long lters used in this MWF oers a better frequency localization of the transmitted signals and a better protection against inter-user interference compared to WOLA-OFDM and UF-OFDM.

Concerning N-cont. OFDM, one can see that the MSE is close to that of UF-OFDM when TO exceeds the CP region. Inside this region, the orthogonality between subcarriers is maintained as in CP-OFDM case leading thus to the absence of asynchronous interference.

In FMT case, we can observe that there is no (or negligible) asynchronous interference. Such a result is due to no (or negligible) interaction between subcarriers. Indeed, each FMT subcarrier can be seen as a traditional single carrier modulation which respects the Nyquist criteria thanks to the long transmit/receive FMT lters. Note that this excellent robustness against asynchronous interference is obtained to the detriment of latency which is very high in such a case (see section 3.3.3).

Regarding FFT-FBMC case, one can observe that the MSE is almost between -30 dB and -38 dB except two regions:

• Inner subcarrier located at the edges of the useful RBs frequency bands, where the MSE is about -28 dB. Such a result can be explained by the fact that subcarrier gain at the RB edges is slightly lower than the gain of subcarriers located at the middle of the RB.

• Edge subcarriers (in the vicinity of interfering subcarriers), where the MSE varies from more than -10 dB when δ = 0 Hz to -30 dB when δ = 75 kHz. As in lter-bank MWFs the edge subcarriers are highly impacted by interference, but this distortion is spread over more than one subcarrier because of the non-uniform gain over RB subcarriers.

Although the fact that BF-OFDM transmitter is like FFT-FBMC one, the performances are not the same. Indeed, BF-OFDM MSE is higher than FFT-FBMC one when the timing errors are outside the CP region. This is a direct consequence of the BF-OFDM receiver which is no more than the classical CP-OFDM receiver (i.e a simple FFT). However, the FFT-FBMC receiver is more ecient in asynchronous case thanks to the ltering performed by the analysis lter-bank.

For FBMC-OQAM (using PHYDYAS prototype lter bank) and Lapped-OFDM, we can observe that a very small number of edge subcarriers are aected by interference thanks to the good spectral containment of FBMC-OQAM and Lapped-OFDM signals. Such a behavior is directly linked to the design of the prototype lter which is a key-property of FBMC based MWFs. Indeed, FBMC prototype lters are commonly designed in order to minimize the interaction between subcarriers (e.g. in PHYDYAS case, a subcarrier overlaps at most with a single subcarrier on each side). When increasing the guardband size, FBMC can reach the FMT performance by oering a MSE below -40 dB for the entire useful frequency band.

For WCP-COQAM, the windowing protection against asynchronous inter-user interference is less ecient compared to ltering one. This explains the poor WCP-COQAM performance in fully asynchronous case compared to FBMC-OQAM and Lapped-OFDM. By applying WOLA processing to WCP-COQAM, we can see signicant improvement in terms of normalized MSE. Indeed, lower interference level in the middle of the bandwidth is achieved by WOLA-COQAM. This can be explained by the fact that the WOLA processing applied at the receiver is able to reduce inter-user interference as well.

Since FBMC-QAM and GFDM are non-orthogonal MWFs, they suer from a high level of selfinterference which makes us unable to distinguish the asynchronous interference from the self-distortion. To overcome this limitation, we introduced interference cancellation schemes for both MWFs. With interference cancellation, the performances of FBMC-QAM are almost like FBMC-OQAM where only a single subcarrier at each edge is impacted by the asynchronous interference. Such a result is due to the well-frequency localization of Samsung-Type-I prototype lter where a given subcarrier only interacts with its immediate adjacent subcarriers. Inserting a guard-band δ > 15kHz (75 kHz in this case) makes all useful subcarriers free from asynchronous interference. Regarding GFDM, one can see that asynchronous interference is more important on the edges of the useful frequency band. Thanks to transmit/receive ltering, the asynchronous interference decay becomes important when increasing the spectral distance between UI and IU. Moreover, we can observe that the best performance is obtained when the TO is inside the CP interval except for the edge subcarriers when δ = 0 Hz.

It should be mentioned that the scenario considered is a noiseless case. These results are so a bound on the achievable performance. The interference cancellation scheme will suer from noise and performance will be worst in case of Additive White Gaussian Noise (AWGN) channel.

Inuence of carrier frequency oset

In this section, we assume that both users (UI and IU) are perfectly synchronized in time domain (τ = 0) but there is an oset between their respective carrier frequencies. The objective here is to examine the impact of CFO-induced inter-user interference on the performances of the various MWFs. The CFO ε considered here varies from -1.5 kHz to +1.5 kHz. No guard-band is considered between the UI subcarriers and the IU ones δ = 0 Hz.

In Fig. 3.7, we have the per-subcarrier MSE of the studied MWFs. In CP-OFDM and N-cont. OFDM cases, the edges subcarriers are more sensitive to CFO compared to inner ones. In fact, the MSE at the edges becomes important even for negligible CFO (from 150 Hz) while inner subcarriers keep best performances (MSE< -30 dB) even when ε = 1.5 kHz.

When it comes to WOLA-OFDM, UF-OFDM and f-OFDM, the same behavior can be reported. Indeed, except the sensitivity of a few number of edge subcarriers to CFO when there is no guardband, WOLA-OFDM and UF-OFDM provide good performance with a MSE below -35 dB for any subcarrier/CFO point.

In FMT case, the best performance is achieved where a negligible MSE (dark-blue color: MSE less than -40 dB) is shown for all useful subcarriers. This can be explained by the fact that the FMT prototype lter is extremely well frequency localized.

For FFT-FBMC, the MSE at the edges of each RB is about -30 dB, whereas it is less than -35 dB in the other subcarriers. As we have previously explained, this phenomenon is due to the lter shape in frequency domain. It is also worth noticing that except in each subcarrier edges the MSE is almost invariant with respect to CFO. In BF-OFDM case, the MSE is below -30 dB in a larger region around the 0 Hz CFO.

As previously discussed, the robustness against time asynchronism of OQAM-based MWFs is ensured thanks to transmit/receive ltering that limits the interaction between a given subcarrier and its neighborhood. Indeed, one can see that, like timing asynchronism, only a small number of subcarriers are suering from asynchronous inter-user interference (e.g., one subcarrier on each side in FBMC-OQAM case).

Due to the block-based structure which is built-in property of WCP-COQAM and WOLA-COQAM signals, these schemes are more sensitive to CFO-induced inter-user interference compared to FBMC-OQAM and Lapped-OFDM systems. In fact, one can observe that the asynchronous interference caused by IU is more important, impacting thus a higher number of useful subcarriers compared to other OQAMbased MWFs. Moreover, this interference is slowly decreasing w.r.t. to the spectral distance between a given victim subcarrier and the interfering signal.

As stated in the previous section, the performance of both FBMC-QAM and GFDM, have been implemented with interference cancellation based-receivers. In FBMC-QAM, since the considered CFO does not exceed the subcarrier spacing (i.e., |ε max | = 1.5 kHz), we can see that except the two subcarriers of the edges, the rest of useful subcarriers are completely protected against the asynchronous interference (it becomes almost negligible compared to the residual self-interference). In GFDM case, the asynchronous interference aects more than one subcarrier at each edge, but the subcarriers located at the middle remain almost free of asynchronous interference.

Complexity

This section aims at estimating the complexity of the transmitter and receiver schemes for the considered MWFs. The complexity will be assessed by counting the number of real multiplications per unit of time to perform both the modulation and demodulation process. It has been preferred to assess the number of multiplications per unit of time to compare as fairly as possible the schemes that do not share the same sampling frequency (FFT-FBMC and BF-OFDM). To do so, a burst of S symbol vectors (as dened in equation (3.3)) is considered. For the schemes that exhibit symbol overlapping, the complexity will be benchmarked when S tends to innity. In this analysis, we will assume that one complex multiplication can be carried out with three real multiplications [START_REF] Krantz | Handbook of complex variables[END_REF]. Moreover, the Cooley-Tukey implementation will be considered for the FFT implementation. The relative complexities of the dierent transmission and reception schemes, with respect to that of CP-OFDM, are given in Tab. 3.4.

Due to its precoding stage, N-Cont. OFDM is the most complex waveform with more than 205 times of the CP-OFDM complexity. Moreover, we can observe that waveforms relying on time convolution (UF-OFDM, f-OFDM) require higher computation resources for the transmitter scheme than the others. Note that, WOLA-COQAM, WCP-COQAM and GFDM are also more computation hungry.

On the contrary, waveforms based on polyphase network (FMT, FBMC-QAM, Lapped-OFDM and FBMC-QAM) provide an ecient hardware implementation. FFT-FBMC and BF-OFDM are more complex because of their precoding stage. As expected, the OFDM and the WOLA-OFDM are the least complex.

When it comes to the receiver schemes, waveforms relying on a simple FFT (OFDM, BF-OFDM, UF-OFDM, N-cont. OFDM) with an eventual light receiver processing (WOLA-OFDM) perform well. UF-OFDM is slightly more complex because of its double size FFT while BF-OFDM is slightly more ecient thanks to its longer sampling period. Polyphase-network based receivers still perform well while the f-OFDM is still the more complex because of the time-convolution. WOLA-COQAM, WCP-COQAM and GFDM receivers are highly complex respectively because the oversampled-FFT and the interference cancellation stage.

Evaluation and discussions

In this section, I will evaluate and discuss the simulation results obtained previously. It is not straightforward to rank the dierent criteria that have been analyzed in the light of a URLLC communications. In this scenario, some criteria are of prime importance such as: latency, asynchronous capabilities, reliability, energy eciency and spectral eciency. Therefore, to give an overview on all of the considered MWFs, we introduce in Fig. 3.8 a radar plot where each corner corresponds to a given criterion. The criteria, taken for comparison, are summarized in table 3.5. From Fig. 3 MWFs with complex orthogonality such as WOLA-OFDM, UF-OFDM and f-OFDM have the CP-OFDM latency level, a more or less moderated added complexity and good performance concerning TOs. For MWFs with real orthogonality and ltering applied at subcarrier level, the FBMC-OQAM, in comparison with CP-OFDM, exhibits higher latency, better performance concerning TO and higher complexity. Nevertheless, COQAM bases MWFs have poor performance concerning TO. Finally, for MWFs without orthogonality, the FBMC-QAM has higher latency, better TO performance and higher complexity than CP-OFDM. Dierently, the GFDM has a very high complexity at the receiver side if interference cancellation is considered and poor performance in terms of TO.

3.4 Advanced equalization structures for FBMC-OQAM

Context and motivation

As discussed in the previous section, FBMC-OQAM showed some interesting features. This MWF has the best frequency localization, providing then the best performance in supporting asynchronous access while ensuring high spectrum eciency. However, FBMC-OQAM loses rapidly its good properties under frequency selective channels and requires advanced equalization techniques [START_REF] Renfors | Contributors. In Orthogonal Waveforms and Filter Banks for Future Communication Systems[END_REF].

Since FBMC-OQAM system employs OQAM, which is a non-second order circular modulation (NSOC) of quasi-rectilinear (QR) type, classical linear equalization [START_REF] Chauvat | Etude de liaisons SISO, SIMO, MISO et MIMO à base de formes d'ondes FBMC-OQAM et de récepteurs Widely Linear[END_REF] is not an optimal solution to recover eciently the transmitted data. Indeed, in [START_REF] Gerstacker | Receivers with widely linear processing for frequencyselective channels[END_REF], authors demonstrated that for OQAM based MWFs, widely linear equalization (WLE) outperforms classical linear equalization (LE). Furthermore, FBMC-OQAM signal is also considered to be cyclo-stationary [START_REF] Dayana | Co-operative cyclo-stationary feature detection with universal ltered multi-carrier spectrum sensing for cognitive radio network[END_REF][START_REF] Lazov | Cyclo-stationary process analysis within telecom applications[END_REF], for which frequency shift ltering (Fresh) processing [START_REF] Chevalier | Reception lter impact on widely linear fresh receiver performance for SAIC/MAIC with frequency osets[END_REF][START_REF] Chauvat | Widely linear fresh receiver for saic/maic with frequency osets[END_REF][START_REF] Elgenedy | Cyclostationary noise mitigation for simo powerline communications[END_REF] has demonstrated its outstanding performance, with regards to LE and WLE approaches.

Motivated by the performance provided by FBMC-OQAM in asynchronous transmissions and the capability of WLE and WL Fresh equalizers in dealing with frequency selective channels when considering NSOC and cyclo stationary (CS) signals [START_REF] Chevalier | Reception lter impact on widely linear fresh receiver performance for SAIC/MAIC with frequency osets[END_REF][START_REF] Chauvat | Widely linear fresh receiver for saic/maic with frequency osets[END_REF][START_REF] Ankarali | Flexible radio access beyond 5G: a future projection on waveform, numerology, and frame design principles[END_REF], we have focused, within the Hayfa Fhima's PhD, on the design of new WL based equalization schemes for FBMC-OQAM.

Equalization structures

We consider the system model given in Fig. 3.3 with two users: a UI transmitting over frequency selective channels, is interfered by an IU which is located at δ Hz guard band from the UI. Both users don't use the same numerology, i.e., dierent parameter settings for the MWF, such as SCS, symbol duration and CP length, as dened in [START_REF] Wg1 | Numerology requirements[END_REF].

From Tab. 3.2, we can notice that, for a given FBMC-OQAM subcarrier, the main interferences come from the couple of the immediately adjacent sub-carriers. Additionally, the OQAM modulation can be seen as the transmission of two M -PAM (pulse amplitude modulation) streams on the same subcarrier. Hence, and without loss of generality, a given sub-carrier of an FBMC-OQAM transmission can be seen as the equivalent model shown in Fig. 3.9. In this gure, x(n) is the useful signal and i(n) is the interfering one. p 1 (m) and p 2 (m) are the prototype lters impulse responses. The signal at the matched lter output is down sampled with factor P to allow symbol spaced equalization (SSE) (for P = 1) or fractionally spaced equalization (FSE) (for P > 1). We denote y p (l) as the equalizer input signal. L e is the equalizer length (L e = L S for SSE and L e = P.L s for FSE) and γ p (l) is the equalizer output with P times the symbol rate. Finally, we consider the vector y p (l) of length L e , dened as Linear Equalization "LE"

The classical LE uses only the information carried by the auto-correlation matrix of the signal y p (l).

Hence, the LE is a vector ∈ C 1×Le where the coecients are given by [START_REF] Ikhlef | An enhanced mmse per subchannel equalizer for highly frequency selective channels for fbmc/oqam systems[END_REF]:

w LE = r xyp R -1 yp , (3.4)
where

r xyp = E[x(n)y H p (l)
] is a vector ∈ C 1×Le which represents the inter-correlation vector between the transmitted symbol and the equalizer input samples. R yp = E[y p (l)y H p (l)] represents the auto-correlation matrix ∈ C Le×Le of the equalizer input signal.

Widely Linear Equalization "WLE"

In WL processing [START_REF] Picinbono | Widely linear estimation with complex data[END_REF] the equalizer uses the information contained in the second order properties of the signal, i.e the information carried by both the auto-correlation and pseudo-autocorrelation matrix. Let us consider ỹp (l) = [y T p (l), y * p (l)] T to be a vector of length 2L e , which represents the WL equalizer input. The scheme of the WL processing is given by Fig. 3.10 (a). The number of the WLE input branches are doubled, compared to the classical linear processing. Over each branch, the signal and its conjugate version, will be followed by a sub-equalizer having the same number of taps as the LE (i.e., L e ). The WL equalizer ∈ C 1×2Le , is given by

wW L = r x ỹp R -1 ỹp , (3.5)
where r x ỹp is a vector ∈ C 1×2Le and represents the inter-correlation vector between the transmitted symbol and the equalizer input in its widely linear version (i.e.: ỹp (l)). Besides, R ỹp is the autocorrelation matrix ∈ C 2Le×2Le of the signal ỹp (l). This matrix can be represented by:

R ỹp = E[ỹ p (l)ỹ H p (l)] = R yp C yp C * yp R * yp , (3.6)
where C yp = E[y p (l)y T p (l)] represents the pseudo-autocorrelation matrix (∈ C Le×Le ) of the signal y p .

Widely Linear Fresh Equalization "WL Fresh"

When the NSOC signal is stationary, there is no correlation between the spectral components of the signal [START_REF] Gardner | Cyclic wiener ltering: theory and method[END_REF]. This is because the whole information is carried by the auto-correlation and the pseudoautocorrelation matrices. However, when the NSOC signal is non-stationary, this is no longer true.

In particular, when the signal is characterized as cyclo-stationary [START_REF] Chevalier | Reception lter impact on widely linear fresh receiver performance for SAIC/MAIC with frequency osets[END_REF][START_REF] Chauvat | Widely linear fresh receiver for saic/maic with frequency osets[END_REF][START_REF] Dayana | Co-operative cyclo-stationary feature detection with universal ltered multi-carrier spectrum sensing for cognitive radio network[END_REF][START_REF] Lazov | Cyclo-stationary process analysis within telecom applications[END_REF], there is correlation between the spectral components of the signal. Moreover, the auto-correlation and the pseudoautocorrelation functions of the interferer signal, composed by the IU signal plus the noise and called I(l), are periodic functions of time [START_REF] Chauvat | Widely linear fresh receiver for saic/maic with frequency osets[END_REF]. Hence, they can be represented as Fourier series expansions, given as follows

R I (t, ξ) = αi R αi I (ξ)e j2παit , (3.7)
C I (t, ξ) = βi C βi I (ξ)e j2πβit . (3.8)
where R αi I (ξ) and C βi I (ξ) are dened, respectively, as the non-conjugate and conjugate cyclic correlation functions. Likewise, α i and β i are called the Second Order (SO) non conjugate and conjugate cyclic frequencies of the IU signal. For OQAM signal, the SO non conjugate and conjugate cyclic frequencies are given as follows [START_REF] Chauvat | Etude de liaisons SISO, SIMO, MISO et MIMO à base de formes d'ondes FBMC-OQAM et de récepteurs Widely Linear[END_REF]:

     α i = i T 1 for i ∈ Z, β i = ±2δ + i T 1 for i ∈ Z,
where δ stands for the guard band between the UI and the IU.

The WLE exploits only the information contained in the couple (α i , β i ) = (0, 0), for any value of i.

For OQAM signals, the most two energetic conjugate cyclic frequencies are β 0 = 2δ and

β 1 = 2δ - 1 T 1 , [116, 117]. Hence, if δ = i 2T 1
, the WLE gives poor performance and becomes equivalent to the LE if there is no energy at the null conjugate cyclic frequency. To better consider the cyclo-stationarity of the signal, another structure of equalizer has been introduced, which exploits better the correlation between the spectral components of the signal. This structure is called Frequency Shift (Fresh) [START_REF] Chauvat | Widely linear fresh receiver for saic/maic with frequency osets[END_REF] equalizer. Without loss of generality, we will limit the WL Fresh equalizer to the use of only one set of couple of cyclic frequencies (±α 1 , ±β 1 ), which correspond to the most energetic ones. The structure of the used WL Fresh equalizer is given in Fig. 3.10 (b). With the use of the cyclic frequencies in addition to the null couple (α i = 0, β i = 0), the Fresh equalizer has 6 branches, resulting to an equalizer with 6 times more taps than the classical one.

Let us consider the observation vector zp

(l) = [z T 1 (l), z T 2 (l), z T 3 (l), z T 4 (l), z T 5 (l), z T 6 (l)] T
, where z i (l) = [z i (lT 0 ), z i (lT 0 -T e ), ..., z i (lT 0 -(L e -1)T e )] T , for i = 1, .., 6. z i (l), i ∈ {1, 3} is the frequency shifted version of the received signal y p (l); this shift is done via the use of the cyclic frequencies +α 1 , 0 and -α 1 , respectively. Likewise, z i (l), i ∈ {4, 6} represents the frequency shifted version of the received signal complex conjugate y * p (l) and that is done by the use of the conjugate cyclic frequencies +β 1 , 0 and -β 1 . Therefore, the WL Fresh equalizer is given by the following expression

wW LF resh = r xzp R -1 zp , (3.9)
where r xzp is a vector ∈ C 1×6Le and represents the inter-correlation vector between the transmitted symbols and the equalizer input in its Fresh version (i.e., zp (l)). Besides,

R zp = E[z p (l)z H p (l)] is a matrix ∈ C 6Le×6Le ,
and it is dened as the auto-correlation matrix of the signal zp (l).

Performance analysis

In this section, we evaluate the performance of the previously described equalization schemes (LE, WLE and WL Fresh) for a FBMC-OQAM system in dierent scenarios: UL/DL, synchronous/asynchronous and with mixed numerologies transmissions. We assume that UI and IU are using dierent numerologies with symbol duration ratios equal to a power of 2, T 1 = 2 a T 0 , where a is positive integer.

The UI and IU use dierent numerologies, with respectively M 1 = 64 and M 2 = 2 × M 1 = 128 subcarriers and transmitting 64-OQAM symbols. We recall that τ is the time shift describing the asynchronism between the UI and IU. δ denotes the guard band between the two users. When using the WL Fresh equalizer, the SO non conjugate and conjugate cyclic frequencies used are given by: α 1 = 1 T 1

and

β 1 = 2δ + 1 T 1 .
For both scenarios: DL and UL, we activate only 5 SCs for the UI (using numerology 1) and 8 SCs for the IU (using numerology 2).

DL case

In this scenario, we assume a perfect synchronization between UI and IU (τ = 0). The guard band δ is chosen to be equal to one UI sub-carrier spacing (δ = 1 M 1 ). Since only the SCs in the edge of the band are aected, we have focused on the impact of one edge IU subcarrier on the BER over the neighboring UI subcarrier, with δ equal to one UI sub-carrier spacing. The channel is a frequency selective 8 taps nite impulse response (FIR) lter whose coecients are randomly chosen. The BER performance of the SSE equalizers as well as the FSE equalizers are presented in Fig. 3.11. It is clear from this gure that the WL Fresh equalizer out-performs both the LE and WLE, whereas the worst behavior is provided by the LE. This is true for both SSE and FSE cases. Eventually, when P increases, the WLE and WL Fresh equalizers performance increase, and the WL Fresh gives always the best performance.

UL case For this case, users are not necessarily synchronous in time. Thus, we will study the impact of this asynchronism as well as the guard band on the system performance for LE, WLE and WL Fresh equalizers in their FSE version with P = 8. ).

The per-subcarrier MSE performance, when the two users are synchronous, is given by (Fig. 3.12). Whereas, Fig. 3.13 depicts the performance when the UI and IU are transmitting asynchronously. In the two contexts, an ideal channel is considered to see the impact of mixed numerologies on the dierent equalizer behaviors.

From Figs. 3.12 and 3.13, we can deduce that for all equalizer cases, all the SCs are aected by the interferer. This remark remains true regardless the guard interval δ size as well as the asynchronism time. Moreover, the WLE and the WL Fresh provide the same MSE performance when considering the SCs in the middle of UI bandwidth. When focusing on the SCs in the edge, we can see from the color degradation that the LE gives the worst performance in the two scenarios when compared to the WLE and WL Fresh.

Furthermore, an interesting and new nding can be deduced from Fig. 3.12: when the UI and IU are synchronous. In this scenario, we can note a notch when the δ = 3 2T 1 , where the performance of WLE and WL Fresh equalizers are very close. This can be explained by the fact that when δ = 3 2T 1 , the conjugate cyclic frequency β 3 is null and since the WLE uses the information contained in all (β i = 0, i ∈ Z), the WLE gives the same performance as the WL Fresh. Besides, when the guard band is out of the notch, the WL Fresh outperforms the WL equalizer over the edge SC. It is noticeable that when the guard band δ is large, the two users are suciently spaced in the frequency domain, meaning while, the IU does not impact the UI anymore, so, the two widely linear equalizers give the same performance. This remains valid in the two contexts (synchronous as well as for the asynchronous cases).

3.5. Conclusion 83

Conclusion

In this chapter, I presented the activities carried out as part of a rst line of research developed since my arrival at CNAM. This axis focuses on the study and proposal of MWFs adapted to cMTC/URLLC communications. Most of this work was carried out as part of the ANR WONG5 project leaded by CNAM between 2015 and 2019. This work has been carried out while keeping a watch on the recommendations of the standardization bodies for 5G, particularly in terms of MWFs candidates for 5G NR and of requirements for URLLC transmissions. The CP-OFDM, which was chosen for 5G NR, did not however seem to us to be the best MWFs in view of the studies that we carried out. Indeed, this MWF has signicant limitations in terms of frequency localization and spectral eciency. In my opinion, other MWFs such as WOLA-OFDM, f-OFDM or BF-OFDM seemed more ecient for a cMTC scenario. Beyond these complex orthogonal waveforms, we have taken a particular interest in the FBMC-OQAM which presented an excellent frequency localization thanks to a very selective ltering at sub-carrier level. However, the real domain orthogonality of this MWF made it very sensitive to the eects of frequency selective channels. In this sense, we have developed advanced equalization techniques exploiting the higher order statistical properties of OQAM signals.

It should be noted that all the work carried out within the framework of this chapter is done using perfect RF equipments (RF impairments induced by power ampliers). The aim of the next chapter will be to study the impact of non-linear power ampliers on the performance of a selection of MWFs introduced in this chapter. Chapter 4 PA distortions analysis

Introduction

Despite the discussed advantages of the MWFs presented in chapter 3, they may lose rapidly their good properties when non-linear (NL) radio frequency (RF) power ampliers (PAs) are used. Indeed, as these MWFs suer from high Peak to Average Power Ratio (PAPR) of the modulated signal, amplifying them at sucient PA eciency may induce undesirable eects.

The PA nonlinearities phenomena lead to in-band performance degradation and spectral regrowth causing adjacent channel interference. PA harmful eects may appear for both UL and DL transmissions. From another side, memory eects in PAs can introduce asymmetry in sidebands Adjacent Channel Leakage Ratio (ACLR) [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to rf power ampliers[END_REF][START_REF] Vuolevi | Measurement technique for characterizing memory eects in the rf power ampliers[END_REF][START_REF] Bosch | Measurement and simulation of memory eects in predistortion linearizers[END_REF].

In this regard, we have been interested on analytical characterization of the in-band (IB) and Out-of-Band (OOB) non-linear distortions, induced by memoryless PAs, on the performance of some post-OFDM MWFs. In particular, we focused, within the Hanen Bouhadda's PhD, on the NL PA eects on system BER and SER performance over AWGN and Rayleigh channels. The originality of this work, carried within the FP7 EMPhAtiC project and continued during the WONG5 project, is the proposal of a universal approach to make this analysis feasible for any measured or modelled memoryless PA and for many MWFs candidate for 5G NR. Regarding OOB eects, we have applied some approaches proposed in the literature: Price theorem and cumulants, in order to predict the spectral regrowth induced by any memoryless PA exhibiting both amplitude and phase distortions.

The objectives of this second research axis developed at CNAM was the study of the eects of PA impairments. Two PhD students: Hanen Bouhadda and Krishna Bulusu, co-supervised with Pr. Daniel Roviras, contributed strongly to the results presented in this chapter.

In a rst section of this chapter, I will present the context and objectives of the carried studies. In a second section, I will describe the PA characteristics and models that are used in the following. Next, section 4.4 is devoted to the analysis of in-band eects while section 4.5 is devoted to OOB eects and spectral regrowth analysis.

Context and motivation

MWFs suer from a major drawback related to their high PAPR. To characterize power uctuations, the Complementary Cumulative Distribution Function (CCDF) of the PAPR is a widely used performance criterion. The CCDF of the PAPR is dened as the probability that the PAPR per block of consecutive symbols exceeds a given level P 0 :

CCDF(P 0 ) = Prob [P AP R (i) > P 0 ] , (4.1)
where

P AP R(i) = max[|x(n)| 2 ] E[|x(n)| 2 ]
, i is a the block index, x(n) is the time domain signal for block number i

with n = [0; N Block -1], N Block is the considered block size 1 .
The CCDF of the PAPR of some of the MWFs described in chapter 3 are presented in Fig. 4.1, for a block size N Block = 1024. We can observe, from this gure, that all MWFs have almost equivalent PAPR performances.

The distribution of the transmitted power given by the CCDF of the PAPR is not very important as far as linear systems are used in the transmission chain. Unfortunately, some elements of the chain are NL, among them the PA was shown to be the most problematic. Indeed, PA at the emitter side can be quasi linear for a certain part of the input powers P in (see Fig. 4.2) and NL for input powers which are close to the 1dB compression point (P 1dB ). Furthermore, the output power P out , of a PA saturates for high values of the input power.

If the input power is always far from P 1dB , the PA is operated in its quasi linear region and the high PAPR of the MWF signal will not have any inuence on the quality of the transmission. This scenario is illustrated in Fig. 4.2 by an input signal with a large IBO. Nevertheless, this situation has a high cost in terms of energy eciency. The concept of energy eciency can be understood by dening the power added eciency (PAE) of a PA, as the ratio between the RF added power and the consumed one.

P AE = P out -P in P DC , (4.2)
where P DC is the power consumed by the PA.

Operating a PA with a very low mean input power gives a low mean output power compared to the consumed power P DC . Especially, for mobile applications with batteries it is necessary to increase the power eciency and operate the PA as close as possible to its saturation region. By doing that, two negative eects (distortions) will appear: (1) distortions will occur in the transmitted signal causing transmission errors, (2) output signal spectrum will be broadened compared to the input one. Within the framework of this research axis, I focused on the following two topics. 1. Firstly, I have focused on the study of the IB distortions induced by NL PAs. In this part, I investigated the SER/BER performance of some MWF based transmissions when memoryless NL PA are used. Theoretical and experimental analysis will be carried to show a very good agreement between theory and simulations for a selection of MWFs operating with some modelled or measured PAs.

2. Secondly, I have been concentrated on the prediction of the OOB spectral regrowth induced by NL memoryless PA. A theoretical study using Price theorem and cumulants will be presented to compute theoretically the spectral regrowth of a post-OFDM MWF signal when amplied by NL PAs. The 1dB compression point refers to the input power level where the transfer characteristics of the amplier have dropped by 1 dB from the ideal linear characteristics. The complex envelope z(t) of the signal at the input of the PA can be written as

PA models and characteristics

z(t) = ρ(t)e jϕ(t) , (4.4)
where: ρ(t) is the input signal modulus, and ϕ(t) is the input signal phase. Broadly, the nonlinear PA models can be classied into two categories: memoryless nonlinear models and nonlinear models with memory. Memoryless PA are characterized by AM/AM (amplitude to amplitude) and AM/PM (amplitude to phase) conversions. Nonlinear models with memory aim also to capture the eects that are manifested not only by the instantaneous input signal magnitude but also the frequency dependent characteristics of the signal envelope e.g., making AM/AM and AM/PM functions non-static and depend on the past input signal. Memory eects in PAs are attributed to various sources, thermal and electric [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to rf power ampliers[END_REF][START_REF] Vuolevi | Measurement technique for characterizing memory eects in the rf power ampliers[END_REF]. These eects are more important in high power PAs, but also appear as the signal bandwidth is increased. The studies presented in this chapter were carried out with PAs without memory. The extension to PAs with memory is a possible prospect with the ever-increasing bands of transmitted signals.

In our study, we assumed that the PA frequency response is constant over the useful signal frequency band, which allows neglecting the memory eects of the PA. Thus, any change in the signal input occurs instantaneously at the output. As a general formulation, the amplied signal u(t) can be written as

u(t) = F a (ρ(t))e j(Fp(ρ(t))+ϕ(t)) = S(ρ(t))e jϕ(t) , (4.5)
where: F a (ρ(t)) and F p (ρ(t)) are respectively the AM/AM and AM/PM characteristics of the PA. S(ρ(t)) = F a (ρ(t))e jFp(ρ(t)) is called the complex soft envelope of the amplied signal u(t).

Many memoryless PA models have been proposed and studied in literature. Among them we can cite the following models • The Soft Envelop Limiter (SEL), used for modeling a PA with a perfect linearization system. The global transfer function can be described by the following AM/AM and AM/PM functions

F a (ρ(t)) = ρ(t), ρ(t) ≤ v sat , v sat , ρ(t) > v sat , (4.6)
F p (ρ(t)) = 0,
where v sat is the PA input saturation level.

• The Rapp model, commonly used for modeling Solid State Power Ampliers (SSPA), exhibits only AM/AM conversion [START_REF] Rapp | Eects of HPA nonlinearity on 4-DPSK-OFDM signal for digital sound broadcasting systems[END_REF]. It can be expressed as

F a (ρ(t)) = G 0 ρ(t) (1 + ( G0ρ(t) vsat ) 2p ) 1 2p
, (4.7)

F p (ρ(t)) = 0,
where G 0 is the linear gain and p is a smoothness positive factor that controls the transition from the linear region to the saturation region. When p → ∞, the Rapp model converges towards a SEL model.

• The Saleh model, generally used for modeling Travelling Wave Tube Ampliers (TWTA) [START_REF] Saleh | Frequency-independent and frequency-dependent nonlinear models of TWT ampliers[END_REF].

According to this model, the AM/AM and AM/PM conversion characteristics can be expressed as follows

F a (ρ(t)) = v 2 sat ρ(t) ρ(t) 2 +v 2 sat , (4.8)
F p (ρ(t)) = φ 0 ρ(t) 2 ρ(t) 2 +v 2 sat , (4.9)
where φ 0 controls the maximum phase distortion introduced by this PA model.

• The polynomial model, widely used for purpose of theoretical analysis. The complex soft envelope of the signal at the output of this PA model can be written as

S(ρ(t)) = F a (ρ(t))e jFp(ρ(t)) = L l=1 a l ρ(t) l , (4.10)
where L is the polynomial order, and a l are the complex coecients of the polynomial function.

In band distortions analysis 4.4.1 Nonlinear distortion modeling

When considering a large number of subcarriers, due to central limit theorem, the PA input signal z(t) is assumed to be a zero mean, complex Gaussian random process. According to the Bussgang theorem [START_REF] Bussgang | Crosscorrelation functions of amplitude-distorted gaussian signals[END_REF], the NL PA output u(t) is related to the input z(t) by the following equation

u(t) = Kz(t) + d(t), (4.11)
where d(t) is a zero-mean noise, which is uncorrelated with z(t) and K is a complex gain with modulus |K| and phase φ K .

According to [START_REF] Dardari | A Theoretical characterization of nonlinear distortion eects in OFDM systems[END_REF], K can be computed analytically by

K = 1 2 E ∂S(ρ(t)) ∂ρ(t) + S(ρ(t)) ρ(t) , (4.12)
where E is the expectation operator. We recall that S(ρ) = F a (ρ(t)) exp(jF p (ρ(t))) is the complex soft envelop of the amplied signal u(t).

The variance σ 2 d of the NL noise d(t) is given by the following equation

σ 2 d = E(|d(t)| 2 ) = E |S(ρ(t))| 2 -|K| 2 E(ρ(t) 2 ). (4.13)
Analytical computation of K and σ 2 d

The analytical computation of the NL distortion parameters K and σ 2 d depends on the complexity of the expression of S(ρ(t)). Indeed and based on equations (4.12) and (4.13), the authors in [START_REF] Dardari | A Theoretical characterization of nonlinear distortion eects in OFDM systems[END_REF] computed analytically K and σ 2 d in the case of the SEL PA model. In this case, K and σ 2 d are expressed as

K = (1 -e - v 2 sat σ 2 z ) + 1 2 π v 2 sat σ 2 z erfc v 2 sat σ 2 z . (4.14) σ 2 d = σ 2 z (1 -e - v 2 sat σ 2 z -K 2 ), (4.15)
where σ 2 z is the variance of the PA signal input. Nevertheless, for more complicated expressions of S(ρ(t)), such as Rapp, Saleh or measured memoryless PA models, the derivation of analytical expressions for the parameters K and σ 2 d is generally intractable. In order to simplify the computation and obtain analytical expressions for these parameters, we proposed a polynomial decomposition of S(ρ(t)). By doing this, we can analytically compute the NL distortion parameters of equations (4.12) and (4.13) for any memoryless PA conversion characteristics after polynomial tting.

Proposed method

The approach proposed, during the Hanen Bouhadda's PhD, aims to approximate the PA conversion characteristics by a polynomial model as given by equation (4.10).

In classical polynomial models existing in literature, only odd coecients are generally used for tting the AM/AM curve [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power ampliers[END_REF]. In order to better t the complex soft envelope S(ρ(t)) of the amplied signal, we have chosen to use a full rank order polynomial with odd and even coecients.

Using a polynomial decomposition of S(ρ(t)), the computation of K and σ 2 d given by equations (4.12) and (4.13) can be done, giving closed form analytical expressions. This will be achieved by the computation of the expectation of a power of Rayleigh random variable, E[ρ(t) l ], where, l = 1......L. By combining equations (4.12) and (4.10), we can write

K = 1 2 L l=1 (l + 1)a l E ρ(t) l-1 . (4.16)
The variance σ 2 d of the NL noise distortion d(t), given in equation (4.13), becomes

σ 2 d = L l=1 |a l | 2 E ρ(t) 2l + 2 L n,l=1,n =l [a n a l ]E ρ(t) n+l -|K| 2 E ρ(t) 2 , (4.17)
where [.] stands for the real part.

The above theoretical expressions of K and σ 2 d involve the computation of the expectation of ρ(t) l (l is a positive integer). This expectation is equivalent to calculate the l th derivation of the Moment Generating Function (MGF). We can write E[ρ(t) l ] as

E[ρ(t) l ] = ∂ l M (t) ∂t l t=0 (4.18)
where l is a positive integer, and M (t) is the MGF: M (t) = e ρt . A generic expression for the computation of E[ρ(t) l ] is given in [START_REF] Raich | Orthogonal polynomials for power amplier modeling and predistorter design[END_REF]. It is expressed as follows

• For odd values of l, we have • For even values of l, we have

E ρ(t) l = ∂ l M (t) ∂t l t=0 = π 2 σ l z l-1 2 i=0 (2i + 1). ( 4
E ρ(t) l = ∂ l M (t) ∂t l t=0 = ( √ 2σ z ) l ( l 2 )!, (4.20)
where ! stands for the factorial operator.

Using equations (4.16), (4. [START_REF] Andrae | Total consumer power consumption forecast[END_REF], and (4.20), we obtain nally the generic expression for computing the complex gain K as

K = a 1 + π 8 L l=2, l even (l + 1)a l σ l-1 z l-2 2 i=0 (2i + 1) + 1 2 L l=3, l odd (l + 1)a l ( √ 2σ z ) l-1 ( l -1 2 )!, (4.21)
where:

σ z = √ P z = √ α 2 P x = P 1dB 10 IBO 10
is the standard deviation of the PA input signal.

The analytical expression of variance σ 2 d of the NL noise d(t) is given by

σ 2 d = L l=1 |a l | 2 2 l σ 2l z l! -2 |K| 2 σ 2 z + 4π 2 L l,n=1,l =n, (l+n) odd [a l a * n ]σ l+n z l+n-1 2 i=0 (2i + 1) + 2 L l,n=1,l =n, (l+n) even [a l a * n ]( √ 2σ z ) l+n ( l + n 2 )!. (4.22)
We should note that both K and σ 2 d are function of the signal IBO through σ 2 z that is given by σ 2 z = P 1dB

10 IBO/10 .

Validation of the analytical expressions of K and σ 2 d

In order to validate our analytical analysis of K and σ 2 d (equations (4.21) and (4.22), at rst, we will compute by simulation the values of K and σ 2 d obtained, respectively, by equations (4.12) and (4.13) for a given PA model (Rapp and Saleh). After approximation of each PA model with the polynomial model of order L = 10, the parameters K and σ 2 d can be theoretically computed based on equations (4.21) and (4.22). Tab. 4.1, which is extracted from [START_REF] Bouhadda | Theoretical analysis of BER performance of non-linearly amplied FBMC/OQAM and OFDM signals[END_REF], compares the simulated and theoretically computed values for dierent PA models and dierent scenarios. The results are given for 10 6 4QAM symbols modulated over 64 subcarriers. For the Saleh model, we have v sat = 1 V , and for the Rapp one, we used the same saturation voltage with p = 1 and G 0 = 1 (equation (4.7)).

According to table 4.1, we observe, with a polynomial approximation of order L = 10, a very good agreement between simulation and theoretical results for both K and σ 2 d , for both Rapp and Saleh PA models.

Sensitivity analysis of OQAM based MWFs to phase errors

This study was carried during Hanen Bouhadda's PhD, where the main objective was to characterize and compensate the IB distortions induced by NL PA over FBMC-OQAM signals. As a rst part of studies, we have been interested on the sensitivity of QAM and OQAM based MWF in the case where there is a phase oset at the receiving side or, equivalently, the NL PA complex gain K is not properly corrected at the receiver side. In this case the received signal after PA and channel ltering can be expressed as

y(t) = z(t) * [Kh c (t)] + d(t) * h c (t) + b(t), (4.23)
where h c (t) is the channel impulse response and * stands for the convolution product operator. We remind here that K is a complex gain that will be written |K| e jφ K . When considering CP-OFDM, an illustrative example of the output 4QAM constellation after PA is presented in Fig. 4.4 (a).

Let d OF DM be the distance projected on real axis of the received signal points aected by the phase rotation induced by the PA complex gain K. This distance is equal to

d OF DM = Re (a I m,n + ja Q m,n ) |K| e jφ K , (4.24
) where a I m,n and a Q m,n denote respectively the in-phase and the quadrature components of the transmitted complex symbol. Depending on the sign of a Q m,n , the decision distance is rather d 1 OF DM (a I m,n > 0) or d 2 OF DM (a I m,n < 0). As explained in section 3.2.2, FBMC-OQAM exhibits high intrinsic interference. Indeed, for this MWF, the transmission of a real symbol at time n 0 and on subcarrier m 0 generates a pure imaginary intrinsic interference u m0,n0 (equation (3.2)). However, in the presence of a phase oset (φ K , this interference is no longer imaginary. Consequently, by taking the real part of the received signal, we obtain a part of the useful signal distorted by the interference signal (u m0,n0 ). Fig. 4.4 (b) shows the 4QAM constellation aected by the PA complex gain and the distance representation in the FBMC-OQAM case. We note that this constellation is considered before the OQAM demodulation. The FBMC-OQAM distance d F BM C is given by

d F BM C = |K|(a m0,n0 cos(φ K ) -u m0,n0 sin(φ K )), (4.25)
where u m0,n0 is given by equation (3.2) and it corresponds to all possible combinations of adjacent symbols.

If no correction is made at the receiver side for the complex gain K, the 4QAM BER of CP-OFDM over AWGN channel is dened by

BER OFDM 4QAM = 1 2 erfc d 1 OF DM 2(σ 2 b + σ 2 d ) + 1 2 erfc d 2 OF DM 2 (σ 2 b + σ 2 d ) , (4.26)
where erf c(.) represents the Gauss complementary error function and σ 2 b is the variance of white Gaussian noise.

For the FBMC-OQAM case, the decision distance d F BM C can take a set of values corresponding to all possible values of u m0,n0 . We have then

BER FBMC 4QAM = um 0 ,n 0 P (u m0,n0 )erfc d F BM C 2 (σ 2 b + σ 2 d ) , (4.27)
where P (u m0,n0 ) is the distribution probability of u m0,n0 .

In the case where we assume a perfect estimation of the NL distortion parameter K (K is jointly estimated with the channel response), we do the correction and the decision distances d

BER OFDM 4QAM = BER FBMC 4QAM = erfc |K| 2 (σ 2 b + σ 2 d )
.

(4.28)

Using equation (4.26) and equation (4.27), we compare theoretical and simulation results for both CP-OFDM and FBMC-OQAM systems when a Saleh PA model is used. Fig. 4.5 depicts the CP-OFDM and FBMC-OQAM BER performance considering an IBO of 6dB and a 4QAM modulation scheme. Four scenarios are investigated: a) The linear case (no PA). b) Only the AM/AM characteristic of Saleh PA model is considered (i.e, φ 0 = 0 of equation (4.9)), c) Both AM/AM and AM/PM conversion characteristics are considered (with φ 0 = π/3) without complex gain correction at the receiver side, d) Both AM/AM and AM/PM conversion characteristics are considered (with φ 0 = π/3) with phase error correction. This gure shows that the analytical and simulation results are in good mach in the dierent scenarios. It can also be noted that in the cases b) [φ 0 = 0 (K is a real gain)] and d) [φ 0 = π 3 (with correction of the NL factor K)], FBMC-OQAM and CP-OFDM reach the same BER performance and that theoretical result of equation (4.28) is in very good agreement with simulation results. Dierence between cases b) and d) is only related to the increase of the NL noise variance σ 2 d when AM/PM conversion is considered with φ 0 = π 3 . For case c), a signicant degradation is present for FBMC-OQAM system compared to CP-OFDM. Even if this case is not very realistic (we have said earlier that correction of the complex gain K will be made jointly with the equalization process), it shows that FBMC-OQAM is more sensitive to phase estimation errors than CP-OFDM. This is mainly due to the distribution of the intrinsic interference in this MWF.

In [START_REF] Zayani | BER analysis of FBMC-OQAM systems with phase estimation error[END_REF], we have extended this work to the case where a phase estimation error occurred during the channel estimation process. The analysis carried in this paper proposed a BER analytical expressions for OQAM based MWF in AWGN and Raleigh at fading channels. The analysis are based on approximating the intrinsic interference of OQAM MWF with a Gaussian probability distribution. For interested readers please refer to this paper to get more information about the proposed analysis.

SER analysis of MWFs in presence of realistic NL PAs

We used the method proposed in section 4.4.1 to compute theoretically the gain K and σ 2 d to evaluate the symbol error rate (SER) of some MWF candidate for 5G NR. This analysis will be done for two PA models: PA1 and PA2, which will be described briey here after.

PA1: Polynomial model

This model is based on a measured commercial 4 GHz long term evolution (LTE) user equipment PA provided by [START_REF]Realistic power amplier model for the new radio evaluation[END_REF], exhibiting both AM/AM and AM/PM distortions. The measured AM/AM and AM/PM curves of this PA model are approximated using a full rank 9 th order (L = 9 in equation (4.10)) polynomial model.

PA2: Rapp model

The second power amplier model PA2, is based on a Rapp model and exhibits only amplitude distortion. The AM/AM conversion function: F a (•), of the Rapp model is given by equation (4.7). As expected from a Rapp model, PA2 AM/AM conversion characteristic shows a linear performance for low amplitudes of the input signal. The parameters G 0 and v sat , are computed to make the linear gain of this PA model equal to that of PA1. The knee factor p is chosen to be equal to 1.1 to have a dierent 1dB compression point from that of PA2 model. For purpose of theoretical analysis of PA2 eects on transmitted signals, the AM/AM characteristic of this PA model is approximated with a full rank polynomial model of order L = 15. The conversion characteristics of the two PA models are depicted by Fig. 4.6.

Case of AWGN channel

According to equation (4.23) and to the system model shown in Fig. 4.3, we can write the received signal y(t), in presence of AWGN channel, and after perfect compensation of the complex gain K at the receiver side as follows

y(t) = αx(t) + d(t) + b(t) K (4.29)
From equation (4.29), it is straightforward to write, that, over AWGN channel, the SER of MQAM modulated symbols for any MWF [START_REF] Proakis | Digital Communications[END_REF], as:

SER AW GN M QAM = 2(1 - 1 √ M )erf c   3α 2 |K| 2 log 2 (M ) 2(M -1)(σ 2 d + σ 2 b )   , (4.30)
where σ 2 b = σ 2 u /(log 2 (M ).E b /N 0 ) is the thermal noise variance corresponding to E b /N 0 and σ 2 u is the variance of the PA output signal.

From equation (4.3), it can be seen that IBO coecient α depends on the considered MWF scheme. Indeed, the average power of the transmitted signal x(t): P x can be inuenced by the windowing and/or the ltering processes used in the MWF transceiver. In [START_REF] Shaiek | Analytical analysis of SER for beyond 5G post-OFDM waveforms in presence of high-power ampliers[END_REF], we developed deep analysis on how we can compute theoretically IBO coecients for some post-OFDM MWF with respect to CP-OFDM one. distortion. As in the case of PA2, FBMC/OQAM and CP-OFDM show the same performance. However, WOLA-OFDM, UFMC and f-OFDM have lower performances than CP-OFDM. This behavior can be explained by the fact that the IBO coecient α is impacted by the windowing/ltering operations related to these MWFs. However, the mean power of signals at the output of CP-OFDM and FBMC-OQAM modulators are the same. Fig. 4.7 (b) presents the SER of the selected MWFs for the Rapp model PA (PA2) with an IBO equal to 4dB and a 64QAM modulation. As noticed for PA1, FBMC/OQAM and CP-OFDM have the same performance. WOLA-OFDM, UFMC and f-OFDM have lower performances than CP-OFDM as explained before.

Case of Rayleigh fading channel

In [START_REF] Zayani | Closed-form BER expression for (QAM or OQAM)-based OFDM system with HPA nonlinearity over Rayleigh fading channel[END_REF], we analyzed the eect of nonlinearities, with emphasis on amplitude and phase distortions, on the performance degradation of MWFs over frequency-at Rayleigh fading channel. We derived a closed-form BER expression for M-array (QAM or OQAM) based OFDM system, which is valid for any PA model, even for real measured one.

We assume a slowly varying at fading Rayleigh channel with a coherence time larger than the symbol duration. The channel response h c = νe jψ is characterised by a Rayleigh distributed gain ν and a uniformly distributed phase ψ ∈ [0, 2π]. Assuming a perfect knowledge of the channel state information (CSI) and the complex gain K at the receiver, the ZF equalized signal is given by

h * c |h c | 2 K y(t) = z(t) + d(t) K + h * c |h c | 2 K b(t) (4.31)
The signal model of equation (4.31) is valid for both QAM and OQAM signals. For a given average power P u at the PA output, the instantaneous Signal to Noise Ratio (SNR) of the detected signal can be derived as

γ = |K| 2 ν 2 σ 2 z ν 2 σ d 2 + σ b 2 (4.32)
In [START_REF] Zayani | Closed-form BER expression for (QAM or OQAM)-based OFDM system with HPA nonlinearity over Rayleigh fading channel[END_REF], we developed a closed-form BER expression of uncoded (QAM or OQAM) based OFDM over quasi-static frequency-at Rayleigh channel. This BER is given by the following equation

BER = a 1 - 2 √ π +∞ n=0 +∞ k=0 (-1) k+n k!n!(2n + 1) ( σ 2 b Γ ) k+1 ( 1 σ 2 d ) n+k+3/2 (b|K| 2 σ 2 x ) n+1/2 k + n + 3/2 (4.33) × 2 F 1 (k + 2, n + k + 3/2; n + k + 5/2; 1)
where Γ = E[ν 2 ] is the average fading power, 2 F 1 (., .; .; .) denotes the Hypergeometric function and (a,b) are modulation-specic constants with a = 2

√ M -1 √ M log2(M ) and b = 3log2(M ) M -1
for MQAM. To nd the number of terms required in the innite series of equation (4.33), we have checked many values of n and k and have veried their tightness with the simulations for dierent values of M . For all the results shown in Fig. 4.8, the innite series have been truncated to n = 10 and k = 50 terms with negligible loss in precision.

We considered FBMC-OQAM and CP-OFDM systems with N F F T = 64 subcarriers. For the PA model, we consider the Saleh's one operating at an IBO = 6 dB with a nominal phase distortion φ 0 = π/3. With these PA model parameters, the nonlinear parameters K and σ 2 d are respectively equal to 0.6624 + 0.1954j and 0.0074. It should be interesting to underline that this study is valid for any memoryless PA model because gain K and σ 2 d can be computed for any PA using equations (4.21) and (4.22). Fig. 4.8 shows an evaluation of the closed form BER quasi-static frequency-at Rayleigh fading channel. For each value of M , we can see a very good agreement between the Monte-Carlo simulation results and those obtained analytically by using equation (4.33). We can also underline, for a given value of M , the perfect matching between the BER achieved by the FBMC-OQAM system and the CP-OFDM based one.

Out of band distortions analysis

The frequency localization of most MWFs is compromised when realistic RF impairments are taken into account [START_REF] Ihalainen | Channel equalization in lter bank based multicarrier modulation for wireless communications[END_REF]. When a signal with time-varying envelope passes through a NL PA, spectral regrowth appears at the PA output. Spectral regrowth causes adjacent channel interference, which is very tightly controlled by regulatory bodies for many over-the-air applications. Prediction of spectral regrowth for a prescribed level of PA non-linearity can be very helpful for designing communication systems. Given the PA conversion characteristics, it is desirable to be capable to predict theoretically, whether the PA can be used to amplify certain types of signals. Through polynomial model of PA, we can derive a closed-form expression for the output auto-covariance function, whose Fourier transform (FT) yields to the PSD, for any stationary input.

There are two popular methods for spectral regrowth analysis, where we can derive a closed-form expression for the auto-covariance function of the PA output. The rst method is based on Price's theorem [START_REF] Skrzypczak | Power Spectral Density and Cubic Metric for the OFDM/OQAM Modulation[END_REF] and the later one is based on cumulants [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to rf power ampliers[END_REF]. In a scenario where AM/PM conversion is to be considered, it becomes complicated to predict the PSD using the Price theorem. Additionally, the computation of integration variables required by this method becomes complicated when high order polynomial PA models are used. Due to these two reasons, the cumulant-based method becomes more adequate for PSD prediction.

We recall that z(t) = ρ(t)e jϕ(t) and u(t) are the base-band equivalent for the PA input and output signals respectively (Fig. 4.3). For any measured or modeled PA, we can approximate its input output conversion characteristics by a NL polynomial as

u(t) = z(t) L l=0 a 2l+1 ρ(t) 2l = L l=0 a 2l+1 ρ(t) 2l+1 e jϕ(t) , (4.34)
where, 2L+1 is the order of the polynomial model and a 2l+1 are the complex coecients of the polynomial model.

In equation (4.34), only the odd order harmonics are taken into account by the polynomial tting process [START_REF] Benedetto | Principles of digital transmission with wireless applications[END_REF]. In [START_REF] Zhou | On baseband representation of a bandpass nonlinearity[END_REF], this approximation was compared to full ranking polynomial and the conclusion is that both approaches give the same performance in terms of base-band modeling of PA nonlinearity.

As a rst contribution, we proposed in [START_REF] Sall | Analysis of the nonlinear spectral re-growth in FBMC systems for cognitive radio context[END_REF], the prediction of the spectral regrowth of OQAM based MWF with Rapp PA model and using Price theorem. Then, during the Krishna Bulusu's PhD, we extended our analysis to any measured or modelled PA, using cumulants [START_REF] Bulusu | Prediction of spectral regrowth for FBMC-OQAM system using cumulants[END_REF]. I want to underline that the analysis presented here after are not restricted to CP-OFDM and FBMC-OQAM. Indeed, the proposed methods (based on Price theorem and on cumulants) remain valid provided that the signal at the output of the MWF Tx is Gaussian.

Prediction of spectral regrowth with Price theorem

Using the AM/AM odd order polynomial model with real coecient, the soft complex envelope of the amplied signal (see equation (4.34)) can be written as following

S(ρ(t)) = L l=0 a 2l+1 ρ(t) 2l+1 . (4.35)
The analytical estimation of the PSD re-growth, based on Price's theorem, which is applicable to any memoryless nonlinearity is presented in [START_REF] Skrzypczak | Power Spectral Density and Cubic Metric for the OFDM/OQAM Modulation[END_REF]. This theorem is summarized by the following formula

∂ n R u ∂R n z = E {S n (ρ(t)).S n (ρ(t -τ ))} , (4.36)
where n is the order of the derivative. R z and R u are respectively the cross-correlation functions of the PA input and output signals.

The derivation of equation (4.36), determines the relationship between R u and R z which can be written as

R u (τ ) = L l=0 η 2l+1 R z (τ ) 2l+1 , (4.37)
where η 2l+1 are integration coecients which are function of both the polynomial tting coecients (a 2l+1 with l = 1..L) and σ 2 z , which is the variance of the signal at the PA input. For more details about the way to compute the integration coecients η 2l+1 , please refer to [START_REF] Sall | Analysis of the nonlinear spectral re-growth in FBMC systems for cognitive radio context[END_REF].

By applying FT to equation (4.37), we can deduce the estimated PSD of the output signal Γ u (f ) given by

Γ u (f ) = L l=0 η 2l+1 Γ z (f ) * ... * Γ z (f ) 2l+1 , (4.38)
where * denotes the convolution product and Γ z (f ) the PSD of the PA input signal z(t).

To validate the accuracy of the spectral re-growth prediction given by equation (4.38), we will use a polynomial tting of Rapp PA model. For this model we used a saturation voltage v sat = 1 V and a knee factor p = 2.25, according to measurements corresponding to a real WIMAX amplier used in PHYDYAS European Project [START_REF]Application of the FBMC physical layer in a cognitive radio scenario[END_REF].

Two scenarios will be examined. In the rst one, a single user is occupying a given number of subcarriers. The second scenario correspond to a fragmented spectrum between 3 users. The FBMC-OQAM system transmits signals over 1024 sub-carriers. The modulated signals are gain-scaled to achieve a suitable IBO for the signals at the input of users PAs. Fig. 4.9 (a) gives the results concerning the PSD of the signal at the output of the Rapp model approximated by 3 rd and 5 th order odd coecients. The IBO is xed 1dB in this rst scenario. As shown in Fig. 4.9 (a), the 5 th order polynomial PA model is able to estimate the spectral re-growth of the amplied signal with a higher accuracy. It is straightforward that increasing the polynomial order model of the AM/AM characteristic, leads to a more accurate PSD estimation of the non-linearly amplied signals. We can see from Fig. 4.9 (b) that the estimated PSD at the output of a 5 th order polynomial ts well the simulated PSD corresponding to Rapp model. Indeed, a gap of about 25dB, between the PSD estimated by a 3 rd order polynomial and the simulated one, is observed at the overlapping bands between the users. In the case where the estimation of the PSD is based on the use of a polynomial model of order 3, we could see free bands between the three operating users, which is not really the case. For FBMC-OQAM, as for any other MWF systems, it is more appropriate to increase the polynomial model approximation for the PA characteristic. This leads to more accurate analytical estimation of the spectral re-growth of the nonlinearly amplied signals, but at an extent of augmented complexity related to the computation of the integration parameters η 2l+1 required by equation (4.38).

Prediction of spectral regrowth with cumulants

Price theorem involves integral calculus. It may sometime involve diculty in determining the integration constant which will impact the closed-form expression derivation. Additionally, in a scenario where AM/PM conversion is to be considered, it becomes complicated to predict the PSD using the Price theorem. Due to these two reasons, the cumulant-based method seems to be more adequate for PSD prediction [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to rf power ampliers[END_REF]. We carried this study during the Krishna Bulusu's PhD and within the framework of the FP7 EMPhAtiC project.

Because of their interesting properties, cumulants are used as an alternative to the moments of a distribution. Cumulants were rst introduced by Thorvald N. Thiele, who called them as semi-invariants [START_REF] Cramer | Mathematical methods of statistics[END_REF]. They were rst called cumulants in a paper by Ronald Fisher and John Wishart [START_REF] Fisher | The derivation of the pattern formulae of two-way partitions from those of simpler patterns[END_REF]. The n th order cumulant of a random variable z is dened as

c nz = ∂ n ∂t n C(t) t=0 , (4.39)
where C(t) is the cumulant generating function given by the logarithm of the MGF M (t) = E(e tz ).

C(t) = ln(E(e tz )) = ∞ n=1 c nz t n n! . (4.40)
For high number of sub-carriers, we can assume most of MWFs signals to be Gaussian process as evident from Central limit theorem. If z(t) is zero-mean, Gaussian and i.i.d, the auto-covariance function of the PA output u(t) is the same as its auto-correlation function: c 2u (τ ). Closed form expression of this function can be derived by using Cumulants. According to [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to rf power ampliers[END_REF], the auto-correlation function for a (2L + 1) th polynomial order nonlinearity is given as

c 2u (τ ) = L m=0 ξ 2l+1 |c 2z (τ )| 2l c 2z (τ ), (4.41)
where c 2z (τ ) is the auto-correlation function of the PA input signal and ξ 2l+1 are constant coecients given by

ξ 2l+1 = 1 l + 1 L k=l a 2k+1 k l (k + 1)![c 2z (0)] 2l 2 , (4.42)
where n m = n!

(n-m)!m! , n ≥ m. The FT of c 2u (τ ), obtained from equation (4.41), yields to the PSD of the amplied signal Γ u (f ) and is given by

Γ u (f ) = +∞ -∞ c 2u (τ )e -j2πf τ dτ = +∞ -∞ L l=0 ξ 2l+1 |c 2z (τ )| 2l c 2z (τ )e -j2πf τ dτ = L l=0 ξ 2l+1 Γ z (f ) Γ z (f ) . . . Γ z (f ) l+1 Γ z (-f ) Γ z (-f ) . . . Γ z (-f ) l , (4.43)
where denotes convolution. It has to be noted that, as Γ z (f ) is symmetric, i.e. Γ z (f ) = Γ z (-f ), the Γ u (f ) is also symmetric as well.

As from equation (4.38), we call these approachs as spectral prediction, because the knowledge of input PSD Γ z (f ) and the polynomial coecients a 2l+1 allows the prediction of the Γ u (f ).

Note that for many MWF, it is straightforward to have analytical expression of Γ z (f ). In [START_REF] Bulusu | Performance analysis and PAPR reduction techniques for lter-bank based multicarrier systems with non-linear power ampliers[END_REF], you can nd those corresponding to CP-OFDM and FBMC-OQAM.

To validate the PSD prediction given by equation (4.43), we carried simulation with a FBMC-OQAM system with 128 SCs. PHYDYAS prototype lter and a Saleh PA model (with φ 0 = π/3 and v sat = 1V ) has been used in the simulations. Fig. 4.10 (a) presents the simulations results with a signal user activating 10 active sub-carriers and operating at an IBO equal to 3dB. We can observe that the 11 th order polynomial approximation gives theoretical result very close to simulated PSD.

In a second scenario, we considered three users operating on closely spaced sub-bands. The three users are amplied at an IBO of 1dB. The details of active sub-bands for the three users are given below

• User 1: 30 th to 39 th sub-carrier,

• User 2: 45 th to 49 th sub-carrier,

• User 2: 52 nd to 56 th sub-carrier.

Simulation results for PSD prediction when 3 users share neighboring sub-carriers are shown in Fig. 4.10 (b). In this typical scenario, we can notice that despite operating the PA at an IBO = 1dB, the prediction of the PSD is well achieved up to -20dB, -25dB and -33dB with polynomial ttings of orders 3 rd , 7 th and 11 th respectively. This multiuser scenario conrms that when choosing thinner sub-bands for the dierent users, and as we are operating the PAs close to their saturation region, we need higher-order polynomial tting to predict the spectral regrowth accurately.

Conclusion

In this chapter, I presented a theoretical approach to characterize the IB eects of memoryless NL.

Based on Bussgang's theorem, we modeled the in-band distortion by a complex gain and an additive Gaussian noise which is uncorrelated with the signal at the PA input. These distortion parameters can be computed theoretically from polynomial approximation of any memoryless PA conversion characteristics.

The developed theoretical performances were found to agree completely with simulations, and FBMC-OQAM, BF-OFDM, WOLA-OFDM and f-OFDM showed quite similar performance with CP-OFDM in terms of BER/SER over AWGN and Rayleigh fading channels.

Polynomial models were also used to characterize the spectral regrowth of the transmitted MWFs introduced by NL PAs with dierent back-o values. Spectral regrowth prediction for CP-OFDM has been extensively studied in the literature, by using Price's theorem or by cumulants. Here, we have extended the analysis to FBMC-OQAM and we conrm the applicability of the proposed approaches to other MWFs, given that the signals emitted from all MWFs Tx are Gaussian processes.

In-Band distortion level along with prediction of spectral regrowth for a prescribed level of PA nonlinearity can be very helpful for designing communication systems. It gives very good understanding about the severity of the PA non-linearity and its impact. Thereby, one can perform PAPR reduction and PA linearization to enhance the system energy eciency and decrease the spectral regrowth to an acceptable level by regulatory standards. Some contributions on PAPR reduction and PA pre-distortion will be presented in the forthcoming chapter.

Chapter 5

Enhanced PA eciency for 5G and beyond MWFs

Introduction

PA linearity and energy eciency are two vital parameters in the context of signals with strong uctuations, as it is the case of most of MWF techniques. To get rid of the amplied signal distortion, the PA can be made to operate at high IBO, which leads to very poor energy eciency. On the other hand, increasing the PA linearity together with PAPR minimization allows to increase power eciency by increasing the mean power of the input signal very close to input saturation power. Furthermore, if the input signal is always in the linear region of the PA, in-band and OOB eects will be limited. PAPR reduction and predistortion are pure signal processing techniques, where the MWFs signals is pre-processed before amplication to ensure enhanced energy eciency. Other approaches can be also envisaged to increase the PA eciency, among them we can cite those based on modifying transmitter architecture.

As a third research activity axis, I have been interested, in the context of EMPhaTiC European project, Accent5 and WONG5 national projects, on enhancing the energy eciency of CP-OFDM and some post-OFDM MWFs transceivers by using both signal processing-based techniques as well as modied architecture-based ones. An important part of these studies has been carried with two PhD students: Krishna Bulusu and Khaled Tahkoubit. Also, I have collaborated with many researchers among them, I can cite colleagues from Tampere University of Technology, CentraleSupélec and AIRBUS.

In this chapter, section 5.2 will be devoted to present some signal processing based algorithms and proposed to enhance PA eciency by reducing the PAPR and/or linearizing PA conversion characteristics. In section 5.3, I will introduce an hybrid approach for enhancing PA eciency which is based on signal transformation along with modied PA architecture.

Signal processing based solutions

As explained and studied in the previous chapter, the presence of high peaks causes IB as well as OOB distortions when the MWF signal is passed through a PA, which does not have enough linear range. Looking at Fig. 5.1 (a), we can identify two signal processing approaches for mitigating the NL eects of the PA at the transmitter side:

• We can decrease the distribution of high values of the instantaneous power of the transmitted signal. This is equivalent to lowering the PAPR of z(t), as illustrated in Fig. 5.1 (b),

• We can increase the PA linear range by using PA linearization techniques. This is equivalent to increasing P z towards v 2 sat (see Fig. In multi-carrier-based communication systems, the PAPR reduction remains to be one of the most crucial issues that need to be solved eectively with a reasonable complexity. Existing popular PAPR schemes for OFDM signals, include clipping [START_REF] Ryu | PAPR reduction using soft clipping and ACI rejection in OFDM system[END_REF], Tone Injection (TI) [START_REF] Tellado | Multicarrier Modulation with Low PAR: Applications to DSL and Wirelesss[END_REF], Tone Reservation (TR) [START_REF] Tellado | Peak to average ratio Rreduction for multi-carrier modulation[END_REF], Active Constellation Extension (ACE) [START_REF] Yang | ACE with frame interleaving scheme to reduce peak-toaverage power ratio in OFDM systems[END_REF], Partial Transmit Sequence (PTS) [START_REF] Muller | OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences[END_REF], Selective Mapping (SLM) [START_REF] Bauml | Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping[END_REF] and block coding [START_REF] Wilkinson | Minimisation of the peak-to-mean envelope power ratio of multicarrier transmission schemes by block coding[END_REF]. Comparison of most of these aforesaid schemes can be found in [START_REF] Hee | An overview of peak-to-average power ratio reduction techniques for multicarrier transmission[END_REF] and [START_REF] Jiang | An overview: Peak-to-average power ratio of OFDM signals[END_REF]. In general, PAPR reduction techniques can be broadly classied into three categories namely clip eect transformations (clipping), block coding techniques and probabilistic approaches. The adaptation of most of the above mentioned PAPR reduction techniques is straightforward to post-OFDM MWF whose signals structures is like that of the CP-OFDM, i.e., no overlapping between symbols. However, this is not valid for lter bank based MWF like FBMC-OQAM, FBMC-QAM or BF-OFDM, where special care should be taken to take into account the overlapping nature of the symbols at the MWF Tx.

From another side, PA linearization aims to mitigate the IB and OOB emissions below the limits determined by standardization and regulatory bodies. These emission limits depend on technologies and standards and can be dierent between UL and DL. For example the Adjacent Channel Power Ratio (ACPR) limit in LTE uplink transmission is -30 dBc, which is usually obtainable without any PA linearization, while for the downlink it is generally -45 dBc, which usually cannot be met without some kind of PA linearization [START_REF] Renfors | Contributors. In Orthogonal Waveforms and Filter Banks for Future Communication Systems[END_REF].

PA linearization techniques include feedback linearization, feedforward linearization, RF predistortion, and Digital predistortion (DPD) [START_REF] Kenington | High-linearity RF amplier design[END_REF][START_REF] Katz | The evolution of PA linearization[END_REF]. Feedback techniques rely on a closed-loop feedback from the PA output to its input to reduce the emissions and are limited in terms of bandwidth. Feedforward linearization is an inherently wide band technique but requires considerable additional RF hardware. The accuracy, exibility, as well as practical bandwidth of feedforward are all limited by the analog components. RF predistortion is a wide band linearization technique, but its delity is rather limited due to the analog implementation. Digital predistortion, on the other hand, is probably the most eective and cost ecient among all linearization techniques. It has the greatest exibility, recongurability, and performance in most cases. Its bandwidth is mostly limited by the digital hardware and will therefore benet from the scaling of CMOS technology. The focus in the rest of this chapter will be on DPD, while for more information on other linearization techniques, the reader is referred to [START_REF] Kenington | High-linearity RF amplier design[END_REF][START_REF] Katz | The evolution of PA linearization[END_REF]. In the following, I will present the contributions related to dierent MWFs, while considering memoryless measured or modelled PAs. The proposed solutions focus either on PAPR reduction, DPD or on a combination of this two processing.

Case of OQAM based MWFs

During the Krishna Bulusu's PhD, I have been working on PAPR reduction and DPD for OQAM based MWF. I will present in the following some of the proposed algorithms. u=0 . After MWF modulation, the symbol achieving the least PAPR is selected for transmission.

From Fig. 5.2, it is evident that unlike CP-OFDM, the FBMC-OQAM symbol energy spreads into its succeeding symbol period intervals rather than in its own one, which means that almost all of its energy lies in the two succeeding symbols period intervals rather than its own one. So, to nd the optimal rotation for any m th FBMC-OQAM symbol, the overlapping of its previous symbols must be taken into consideration. This is the main idea behind the DSLM algorithm. This algorithm involves the following steps:

Step 1 -Initialization: We generate U phase rotation vectors {φ (u) } U -1 u=0 of length N F F T and initialize the counter m. As long as the condition 0 ≤ m ≤ M -1 (M is the total transmitted FBMC-OQAM symbols.) is satised, we perform the following Steps 2, 3, 4, 5, 6 and 7.

Step 2 -Phase rotation: The m th input symbol vector X m is phase rotated with U dierent phase

rotation vectors {φ (u) } U -1 u=0 , giving {X (u) m } U -1 u=0 .
Step 3 -FBMC-OQAM modulation: For all patterns of m th input symbol vector, we compute

x (u) m (t) = 2m-1 m =0 N -1 n=0 a (u m min ) m ,n γ m ,n (t)
overlapping past symbols

+ 2m+1 m =2m N -1 n=0 a (u) m ,n γ m ,n (t) current symbol , (5.1)
where x Step 4 -Partial PAPR Calculation: Then, we compute the partial PAPR of x (u) m (t) on a certain interval T 0 = [mT + T a , mT + T b ), which is any arbitrary interval within the [mT, mT + 4.5T ] interval.

P AP R (u) (T0) = max t∈T0 |x (u) m (t)| 2 1 T0 T0 |x (u) m (t)| 2 .dt , (5.2)
where T a ≥ 0 and T b < 4.5T . The choice of interval T 0 during this step plays vital role in increasing the peak reduction performance.

Step 5 -Selection: Among P AP R (u) T0 , the index u is chosen for the signal with least PAPR as per the below criterion

u m min = arg min 0≤u≤U -1 P AP R x (u) m (t) , mT + T a ≤ t < mT + T b . (5.3)
Step 6 -Updation: We update the current input symbol vector:

X (u m min ) m = X m . φ (u m min ) .
(5.4)

The index u m min will be stored in a vector U SI in order to be transmitted as Side Information (SI) for perfect recovery of the signal at the receiver.

Step 7 -Incrementation: Increment the value of m by 1 and go to Step 2. We published this algorithm in the conference paper [START_REF] Bulusu | Reduction of PAPR for FBMC-OQAM systems using dispersive SLM technique[END_REF], which is one of the most cited references for reducing the PAPR of FBMC-OQAM signals.

Fig. 5.3 shows the performance of DSLM for dierent values of U , when the partial PAPR of step 4 is computed over T 0 = [(m + 1)T, (m + 3)T ). From the simulation results, we can notice that the FBMC-OQAM with DSLM is trailing to that of OFDM with SLM in PAPR reduction performance. We can also perceive that the trail gap in terms of PAPR reduction performance, is increasing with the size of U . The reason for sub-optimality is that, in DSLM, the phase rotations of the past symbols are xed or already chosen since it is based on symbol-by-symbol optimization. Whatever peak reduction we may obtain for the current symbol, can probably be hampered by its immediate next symbol. This motivated us to investigate this sub-optimality and probe for optimal solutions.

Trellis SLM

Optimal approach would be to test all possible phase rotation vectors to nd the lowest PAPR in the overall signal composed of M FBMC-OQAM symbols. This would have huge complexity, as there is a need to perform an exhaustive search over U M possible rotation vectors. A quasi-optimal solution would be a trellis-based dynamic programming, where we perform a trellis-based search to nd the optimal phase rotations, considering several successive symbols at a time. This idea gave rise to the Trellis SLM (TSLM) algorithm published in [START_REF] Bulusu | Potency of trellis-based SLM over symbol-by-symbol approach in reducing PAPR for FBMC-OQAM signals[END_REF][START_REF] Bulusu | Reducing the PAPR in FBMC-OQAM systems with Low Latency Trellis-based SLM Technique[END_REF].

If we have to transmit M input symbol vectors {X 0 , X 1 , . . . , X M -1 }, then we need to nd Θ, which is the optimal set of M dierent phase rotation vectors that give the best PAPR.

Θ = {φ (u 0 min ) , φ (u 1 min ) , ..., φ (u M -1 min ) }, (5.5)
where, {u 0 min , u 1 min , ..., u M -1 min } are the indices of optimal phase rotation vectors for the M input symbol vectors, which are to be sent to the receiver as SI. With M FBMC-OQAM symbols and U phase rotation vectors, we need to nd the best path in the trellis of Fig. 5.4 that gives the lowest PAPR.

For 0 ≤ m ≤ M -1, every m th FBMC-OQAM symbol x m (t), obtained from modulation of input symbol vector X m , is represented as the m th stage in the trellis that will reach at time instant mT . At each stage, there will be U dierent states, representing the rotated FBMC-OQAM symbols. Among these states, any i th trellis state indicates rotation by phase rotation vector φ (i) . Between every two stages, there exist U 2 number of possible paths. The joint FBMC-OQAM modulation of m th and (m + 1) th rotated input symbol vectors X The TSLM algorithm involves the following steps:

Step 1 -Initialization: Firstly, we generate M complex input symbol vectors {X m } M -1 m=0 and U phase rotation vectors {φ (u) } U -1 u=0 of length N F F T . We initialize the counter m (m = 0) and the state metrics for all states of the rst stage (Ψ (u,0) for u = 0..U -1). As long as the condition 0 ≤ m ≤ M -2 is satised, we perform the following Steps 2, 3, 4, 5 and 6 in a repeated manner.

Step 2 -Phase rotation: Two input symbol vectors X m , X m+1 are phase rotated with U dierent phase rotation vectors {φ (u) } U -1 u=0 and {φ

(v) } U -1 v=0 , giving {X (u) m } U -1 u=0 and {X (v) m+1 } U -1
v=0 respectively. The symbols, a Step 3 -FBMC-OQAM modulation: FBMC-OQAM modulation is done jointly for all combination of the patterns of m th and (m + 1) th input symbols, is written by

x (u) m (t) = 2m+1 m =2m N -1 n=0 a (u) m ,n γ m ,n (t), x (v) m+1 (t) = 2m+3 m =2m+2 N -1 n=0 a (v) m ,n γ m ,n (t),
(5.6)

x (u,v) m,m+1 (t) = x (u) m (t -T ) + x (v) m+1 (t), 0 ≤ u, v ≤ U -1.
Step 4 -Path metric calculation: For each of the U 2 patterns of the modulated FBMC-OQAM signal x given by:

(u,v) m,m+1 (
Γ (u,v) (m,m+1) = f P AP R (u,v) m,m+1
(5.7) where f (.) is any convex function.

Step 5 -Survivor path identication: The states for stage m that are related to the survivor paths leading to stage m + 1, are stored in a state matrix λ(v, m) as given below

λ(v, m) = min u∈[0,U -1] Ψ (u,m) + Γ (u,v) (m,m+1) , v = 0, ..., U -1.
(5.8)

Step 6 -State metric updating: The state metric Ψ (v,m+1) , for the stage m + 1, can be updated as follows

Ψ (v,m+1) = Ψ (λ(v,m),m) + Γ (λ(v,m),v) (m,m+1) , v = 0, ..., U -1.
(5.9)

Step 7 -Incrementation: Increment the value of m by 1 and if 0 ≤ m ≤ M -2, then go to Step 2; Else, for 0 ≤ m = M -1, go to Steps 8.

Step 8 -Traceback: Once state metrics for all the M stages has been computed, then identify the state that has the least state metric as shown below

Θ(M -1) = min u∈[0,U -1]
Ψ (u,M -1) .

(5.10)

Then start tracing back from last stage to the rst one in order to nd the unique survivor path Θ by identifying the optimal states at each stage as below

Θ(k) = λ(Θ(k + 1), k), (5.11)
where, k = M -2, M -3, ..., 1, 0. This survivor path Θ is the set of optimal phase rotation vectors, that is obtained after solving the optimization problem by dynamic programming (DP) [START_REF] Bellamen | Applied dynamic programming[END_REF], which can help in reducing substantially, the number of paths one need to pick. The performance of TSLM for a FBMC-OQAM signal with 64 subcarriers and 4QAM modulation is shown in Fig. 5.3 in terms of CCDF plots. We can notice that FBMC-OQAM with TSLM is not only superior to FBMC-OQAM with DSLM scheme in PAPR reduction but also outperforms OFDM with classical SLM. This superior performance implies that when the overlapping nature of the FBMC-OQAM signals is well exploited then it can signicantly impact the PAPR reduction and such proper exploitation can be possible with trellis-based approach instead of symbol-by-symbol optimization. Another observation is that the lead gap between CCDF curves of OFDM and FBMC-OQAM gets narrowed as U increases.

Dispersive TR

We have come to conclusion that probabilistic PAPR reduction schemes provide optimal results, with TSLM algorithm. This has prompted us to investigate new category of PAPR reduction techniques. As a next step during the Krishna Bulusu's thesis, we studied an adding signal technique and we have chosen TR. The idea behind TR is to isolate energy used to cancel large peaks to a predened set of R tones, namely peak reserved tones (PRTs). Such a PRT does not carry any useful information, i.e., PRTs are disjoint from the data tones (DTs). Stated mathematically, the signal corresponding to the m th transmitted symbol will be

x m (t) = d m (t) + c m (t), 0 ≤ t < ∞, (5.12)
where, c m (t) is the peak cancellation signal and d m (t) is the data signal (i.e. related to data only). x m (t) can be represented in frequency domain as X given by

X m = D m + C m (5.13)
where the n-th component of the vector X m is given by

X m,n = D m,n , n ∈ B c C m,n , n ∈ B (5.14)
where,

• n = 1..N F F T is the subcarrier index,

• B is the set of PRTs indices, which is of length R,

• B c is the complement set of B, containing the DTs indices and is of length

N F F T -R, • D m,n is DT set and D m,n = 0, for n ∈ B, • C m,n is PRT set and C m,n = 0, for n ∈ B c .
The aim of TR scheme is to compute the optimal values of PRTs subject to

c m (t) = arg min Cm,n n∈B P AP R(M OD{D m + C m }) , 0 ≤ t < ∞, (5.15)
where MOD{.} is a generic modulation operation, corresponding to any MWF. The optimal c(t) can be obtained by solving equation (5.15) using convex optimization algorithms such as (Quadratically constrained quadratic programs) QCQP [START_REF] Tellado | Peak power reduction for multicarrier transmission[END_REF], (projection onto convex sets) POCS [START_REF] Gatherer | Controlling clipping probability in DMT transmission[END_REF], gradient search [START_REF] Tellado | Peak power reduction for multicarrier transmission[END_REF], etc. The QCQP has a computational complexity of O(RN 2 F F T ) and yields to the optimal result. The sub-optimal approaches such as POCS and gradient search have a computational complexity of O(N F F T log N F F T ) and O(N F F T ) respectively.

For FBMC-OQAM, we have proposed a dispersive version of the TR technique. This algorithm, named dispersive TR (DTR) [START_REF] Bulusu | Reduction of PAPR of FBMC-OQAM systems by dispersive tone reservation technique[END_REF], considers the overlapped past symbols, while calculating signal c m (t). The idea is closely related to the one used in DSLM and is deeply described in [START_REF] Bulusu | Reduction of PAPR of FBMC-OQAM systems by dispersive tone reservation technique[END_REF].

To analyse the impact of R on the performance of the proposed DTR algorithm, we have considered dierent scenarios where we have reserved 6.25%, 12.5% and 25% of the total sub-carriers (N F F T = 64). We can see from Fig. 5.5 that the exploitation of the overlapping nature of the FBMC-OQAM signals can signicantly impact the PAPR reduction. Indeed, the PAPR reduction performance of FBMC-OQAM with classical TR is outperformed by the one with DTR scheme by 3.2dB, for R = 8. Also, in the same gure, we can notice that the CCDF of PAPR for FBMC-OQAM with DTR is closely trailing that of OFDM with classical TR by roughly 1dB. Another important observation is that, as R increases, the trail gap between CCDF curves of OFDM and FBMC-OQAM is getting increased. The reason for performance lag of DTR scheme is that the PAPR reduction done for the current symbol can be hampered by the future symbols, which is again related to schemes that opt for symbol-by-symbol optimization.

PA linearization techniques

PA linearization has seen a large investment of research and development funding in both the industrial and academic sectors. With PA linearization the PA linearity is improved and extended so that the PA can be operated at lower IBO. This means that a lower-power lower-cost linearized PA can be used in place of a higher-power higher-cost one. Furthermore, the linearized PA operates more eciently since it is operated closer to saturation. Among all linearization techniques, DPD is one of the most costeective techniques. A predistorter, which (ideally) has the inverse characteristic of the PA, is used to compensate for the PA non-linearity. The benets of DPD are even more pronounced for CP-OFDM and post-OFDM MWFs where the PAPR is large. Details about predistortion principle and benets can be found in [START_REF] Kenington | High-linearity RF amplier design[END_REF][START_REF] Ghannouchi | Behavioral modeling and predistortion[END_REF][START_REF] Boumaiza | Adaptive digital/RF predistortion using a nonuniform LUT indexing function with built-in dependence on the amplier nonlinearity[END_REF]. In [START_REF] Zayani | Adaptive predistortion techniques for non-linearly amplied FBMC-OQAM signals[END_REF], we proposed an adaptive DPD based on neural networks. Because of the intrinsic interference of FBMC-OQAM system, careful implementation must be done, especially for the phase (AM/PM) correction. This method will not be detailed here and interested readers could refer to the conference paper [START_REF] Zayani | Adaptive predistortion techniques for non-linearly amplied FBMC-OQAM signals[END_REF], to get more insights on the proposed method.

In this section, I have chosen to present another DPD technique developed during the Krishna Bulusu's PhD thesis. This method is based on a signal adding approach. Indeed, If the memory eects of the PA are negligible, the predistortion can be seen as a memoryless non-linear function. In such case, it has been shown in [START_REF] Gouba | Digital predistortion expressed as an adding signal technique in OFDM context[END_REF], that the Bussgang's theorem allows the formulation of predistortion as an adding signal technique for CP-OFDM signals. The idea behind any adding signal DPD technique is that we add a predistortion signal p(k) to the signal x(k) to compensate the PA non-linearity. The predistorted signal x pd (k) can be expressed as

x pd (k) = P{x(k)} = x(k) + p(k), (5.16)
where P{.} is the predistortion function. The aim of a DPD technique is to obtain a function P such that

A P{x(k)} = A 0 x(k), (5.17)
where A 0 is a linear gain and A{.} is the PA amplication function.

The proposed DPD technique, to linearize FBMC-OQAM signals is termed as overlapped-recursive error correcting-digital predistortion (OREC-DPD) as it takes into account the overlapping nature of the FBMC-OQAM symbols. The manner in which the overlapping of previous symbols are taken into consideration is similar to that of the proposed PAPR reduction techniques and can be understood from Fig. 5.2. The DPD is done in baseband, which implies that rst we model the real PA and then we perform linearization. Illustration of a FBMC-OQAM system along with OREC-DPD scheme is shown in Fig. 5.6. The OREC-DPD algorithm is given below

Step 1: Step 1a -FBMC-OQAM signal generation -Firstly, we generate M complex input symbol vectors and oversample them by L times to obtain X 0 , X 1 , ..., X M -1 . Then, for 0 ≤ k < (M + 3.5)LN F F T -1, we perform discrete-time FBMC-OQAM modulation to obtain the FBMC-OQAM signal x(k).

Step 1b -Predistortion signal initialization -Then, we initialize m = 0 and the predistortion signal 108 Chapter 5. Enhanced PA eciency for 5G and beyond MWFs

p(k) = 0
Step 2: For mLN F F T ≤ k < (m + 4.5)L F F T -1, we initialize iteration counter q = 1 and then, initialize the accumulative partial predistortion signal p (0) m (k) that is related to the m th signal x m (k) to be zero.

p (0) m (k) = 0, mLN F F T ≤ k < (m + 4.5)LN F F T -1. (5.18)
Step 3: Then, we perform the following substeps:

Step 3a -Amplication -The predistortion signal p(k) should be added to the FBMC-OQAM signal x(k)

along with the current accumulative predistortion signal up to q th iteration p

m (k) and amplied by the PA model as follows

y (q) (k) = A x(k) + p(k) + p (q) m (k) . (5.19)
Step 3b -Recursive error computation -The recursive error signal e (q) (k) at iteration q, is the dierence between amplied signal y (q) (k) and the input signal x(k)

e (q) (k) = y (q) (k) -x(k). (5.20)
Step 3c -Recursive error correction -The accumulative partial predistortion signal p (q) m (k), is updated through recursive accumulation as below

p (q) m (k) = p (q-1) m (k) + λe (q) (k), (5.21)
where, λ is the convergence rate.

Step 4: If q ≤ Q, we increment q and go to Step 3. Else, we update the predistortion signal p(k) by accumulating with the partial predistortion signal p

(Q) m (k) p(k) = p(k) + p (Q) m (k). (5.22)
Step 5: If m ≤ M -1, we increment m and go to Step 2. Else, we proceed to next step.

Step 6: Add the obtained predistortion signal p(k) to the FBMC-OQAM signal x(k) to obtain the pre-distorted input signal x pd (k), which shall be fed into the real PA.

x pd (k) = x(k) + p(k) (5.23)
In the analysis, we considered transmission over 64 sub-carriers, with an oversampling factor L = 4. We used a Rapp model with p = 2.25 and xed λ = 0.1. The impact of IBO on the OREC-DPD scheme has been simulated for values of IBO = 1 dB, 2 dB, 5 dB with Q = 30 and presented in Fig. 5.7. It can noticed that there is a spectral de-growth for all the IBO values w.r.t. to the original NL spectral regrowth without any predistortion. This relative decrease is found to be around 18.35 dB, 31.12 dB and 34.62 dB respectively, at normalized frequency of 2.

The OREC-DPD algorithm described here and published in [START_REF] Bulusu | PA linearization of FBMC-OQAM signals with overlapped recursive error correcting predistortion[END_REF], is based on a constant convergence rate λ, thereby, leading to slow convergence. However, there are enhanced methods, in the literature to update λ at each iteration, according to knee factor p and/or IBO values. In [START_REF] Bulusu | HPA linearization for FBMC-OQAM signals with fast convergence-digital predistortion[END_REF][START_REF] Bulusu | HPA linearization for next generation broadcasting systems with fast convergence-digital predistortion[END_REF], we extended this work, where we propose a faster converging DPD technique requiring few iterations compared to OREC-DPD.

Co-existence capabilities of a FBMC-OQAM system working with NL PA

Professional mobile radio (PMR) and public protection and disaster relief (PPDR) communications (TETRA, TETRAPOL, TETRA/TEDS [6], APCO 25 in the US and others) are undergoing signicant transformation towards supporting broadband data communications services in their future generations. The terrestrial trunked radio (TETRA), as one of the earliest critical communication systems, has evolved from low-speed data in its rst release and integrates today the wide band TETRA enhanced data service (TEDS). TEDS supports higher data rates, up to several hundred kbps, and ts very well into the existing PMR frequency landscape. However, the real broadband evolution of wireless PMR communications is progressing towards enhanced technical solutions oering high speed data services, while still providing the legacy PMR services. PMR communications must support data-rich broadband services and connect people instantly.

To respond eectively to the wide variety of incidents taking place, professional users demand much faster data and real-time multimedia applications to share, such as video streaming on-the-move, access to remote databases (identity checks, CCTV cameras, etc.), le transfer (maps, pictures, satellite les, etc.), and location-based services.

The new required capacity can be achieved in two complementary ways: by obtaining new frequency bands for PMR/PPDR data services or by tting a novel broadband data service within the scarcely available spectrum devoted to PMR systems. During the Krishna Bulusu's PhD thesis and within the framework of the EMPhAtiC European project, we focused on the latter approach, which can be seen as a very challenging evolution path, but also the most realistic opportunity to benet from the advantages of latest developments in MWFs and related signal processing techniques. One of the major issues is being able to introduce new broadband data services within the current frequency allocation, in coexistence with current narrowband PMR/PPDR systems. Having in mind the widely expected dominance of 3GPP long-term evolution (LTE), it is natural to consider the LTE system as a reference basis in the PMR/PPDR system development due to its modularity and wide adoption in the civil world.

In [START_REF] Medjahdi | Integrating LTE broadband system in PMR band: OFDM vs. FBMC coexistence capabilities and performances[END_REF], it has been shown that a LTE-like broadband system based on CP-OFDM lacks coexisting capability, even in the absence of RF impairments, because of the high side lobes of this MWF. This implies the need in the context of 5G PMR systems, to probe for new MWFs waveforms that can: 1) support user asynchronism and low latency, 2) permit coexistence with narrowband systems and 3) be capable to be operated with NL devices with high energy eciency. In this regard, we have made a genuine attempt, to address the hitherto mentioned issues. Although, CP-OFDM have been a natural choice for next generation communications; still it is not considered the optimal MWF technique for PMR coexistence. The principal reason is the lack of sub-channels' spectral connement, making it especially inapplicable to non-synchronized mobile networks. In this contribution, we investigated the coexistence capabilities of CP-OFDM and FBMC-OQAM systems with a PMR system that operates on licensed narrowband located in PMR UHF bands in Europe and which are around 400 M Hz.

The deployed broadband system should satisfy the constraints imposed by the already existing and operational PMR communication devices. The PMR system considered for the analysis is dened by the European Telecommunications Standards Institute (ETSI) TETRA standard [6] which is currently widely used by public safety organizations, both private and governmental, and by around two million of business users over the world. Fig. 5.8, illustrates the considered coexistence scenario where a LTE-like broadband system is deployed between two TEDS systems of 25 kHz bandwidth each. ∆f 1 and ∆f 2 5.1: Blocking levels of the 25 kHz-TEDS receiver dened in [6] and reception mask .

refer to the frequency distances between the nearest adjacent interfering broadband LTE system and the nominal carrier frequencies of the existing active narrowband users. To guarantee such cohabitation, the OOB radiation of the deployed LTE system should not exceed the levels dened by the spectrum mask of the TEDS standard. The 25 kHz-TEDS reception mask parameters are depicted in Tab. 5.1.

We have tested the capability of FBMC-OQAM and CP-OFDM to fulll the TEDS mask, rst assuming a linear PA. The PSD of each MWFs is integrated over a 25 kHz bandwidth and compared with the TEDS mask in Fig. 5.9. As demonstrated in [START_REF] Medjahdi | Integrating LTE broadband system in PMR band: OFDM vs. FBMC coexistence capabilities and performances[END_REF], we conrmed that with linear PA, only FBMC-OQAM is able to satisfy the TEDS mask.

When using a NL PA, which is operated in its nonlinear region, spectral re-growth increases the interference and endangers the fullment of the TEDS mask. We assumed a perfectly linearized PA. The NL PA chosen model is thus a SEL. To respect the TEDS mask, we used PAPR reduction techniques together with adjusted IBOs. Nevertheless, as higher IBO results in energy wastage, increasing IBO to respect the mask is not an optimal approach and PAPR reduction techniques must be jointly applied. In this study, we considered the DSLM technique to reduce the PAPR for FBMC-OQAM. It is clear from Fig. 5.9, that with a symbiotic combination of IBO and U (number of complex vectors used in DSLM algorithm), the interference of FBMC-OQAM fulls to the TEDS spectrum mask. In the same gure, it can also be seen that as IBO is increased, the interference level is getting decreased. With an IBO = 8dB and with a DSLM technique (U = 8), the interference level is well reduced below the TEDS spectrum mask. Without PAPR reduction, the input PA signal needs a 12 dB of IBO to full the TEDS spectrum mask.

This work have been jointly carried with two colleagues from Tampere university of Technology and leaded to a joint journal paper in Transactions on Emerging Telecommunications Technologies [START_REF] Bulusu | Enhanced multicarrier techniques for narrowband and broadband PMR coexistence[END_REF]. A short version of this paper, restricted to CNAM contributions, has been published in a conference paper [START_REF] Bulusu | Impact of HPA non-linearity on coexistence of FBMC-OQAM systems with PMR/PPDR systems[END_REF]. (b) TR with 20 iterations and 8, 4% of PRT. Figure 5.10: CCDF of PAPR for CP-OFDM and some post-OFDM MWFs (N F F T = 256 and 16QAM).

Case of WOLA-OFDM, UF-OFDM and f-OFDM

As part of the WONG5 national project, we focused on PAPR reduction techniques for OFDM-inspired waveforms: WOLA-OFDM, UF-OFDM and f-OFDM which have almost PAPR levels similar to those observed with CP-OFDM signals. This work has been carried during the Master internship of Khaled Tani. It is worth to mention that, in contrast to CP-OFDM, there are only few works on PAPR reduction in the previously mentioned post-OFDM MWFs. To this end, we have chosen two classical PAPR mitigation techniques: SLM which is a probabilistic method and TR which is an adding signal method.

PAPR performances analysis

The PAPR performances for CP-OFDM, WOLA-OFDM, UF-OFDM and f-OFDM are plotted in Fig. 5.10 (a). First, we can see that without PAPR reduction, the three post-OFDM MWFs have PAPRs slightly higher than CP-OFDM one. This dierence is related to the fact that the average power of these MWFs, is lower than the CP-OFDM one, due to the windowing and/or ltering applied at the Tx side. A detailed explanation of this problem is given in [START_REF] Medjahdi | On the road to 5G: Comparative study of physical layer in MTC context[END_REF]. Using SLM method, the PAPR decreases by the same amount for all the MWFs. Fig. 5.10 (b), shows the performance of CP-OFDM, WOLA-OFDM, UF-OFDM and f-OFDM without and with an iteration TR algorithm with a PRT ratio of 8.4%, after a total number of 20 iterations. We can notice that like the SLM case, the PAPR is reduced by the same amount for all MWFs.

Complexity analysis and overall power budget

From the performances given in Fig. 5.10, it appears that both PAPR reduction algorithms achieved similar performance for the chosen simulations parameters. However, the energy budget required to achieve that 3.5 dB reduction of the PAPR could be very dierent between SLM and TR. Also, the implementation cost for each of the considered methods may be very dierent from one MWF to the other. Thus, we focused on assessing, for each MWF, the complexity of both PAPR reduction techniques, by computing the number of multiplications required to process a complex vector of a size equal to N F F T . The complexity analysis are summarized in Tab. 5.2.

We remind that, for UF-OFDM, the data is processed at the RB level (B active RBs out of all the available ones). For each RB, rst there is the predistortion stage with N F F T < N F F T complex multiplications. Then, there is the transposition to the time domain with only N F F T active subcarriers out of N F F T . The IFFT is therefore mainly fed by null elements and its complexity can be reduced to

N F F T + N F F T
2N F F T + N F F T 2 log 2 (N F F T ) 2N F F T + N F F T 2 log 2 (N F F T ) + 4WTx N F F T (BN F F T + B(2N F F T + N F F T 2 log 2 (N F F T )) + (B + 1)(N F F T + L F IR -1) + BN F F T L F IR 2 ) N F F T 2 log 2 (N F F T ) + (N F F T + N CP ) L 2 + (2N F F T + N CP + 2L -2) Normalized complexity 1 1, 01 242, 93 1, 67

TR

Per iteration complexity

N F F T . log 2 (N F F T ) N F F T log 2 (N F F T ) + 2WTx + 2WRx BN F F T + B(N F F T + N F F T 2 log 2 (N F F T )) + B(N F F T + L F IR -1) + BN F F T L F IR 2 + N F F T log 2 (2N F F T ) 2( N F F T 2 log 2 (N F F T ) + (N F F T + N CP ) L 2 + (N F F T + N CP + L -1))
Normalized complexity 1 As an illustration of the previous analysis, we computed in Tab. 5.2, the normalized complexity, with respect to CP-OFDM, of the SLM and TR algorithms applied to WOLA-OFDM, UF-OFDM and f-OFDM. This analysis is based on the simulation parameters given in [START_REF] Tani | PAPR reduction of post-OFDM waveforms contenders for 5G Beyond using SLM and TR algorithms[END_REF] and is obviously independent from the number of complex phase rotation vectors U and the number of iterations K iter . It appears clearly that SLM applied to UF-OFDM exhibits the highest complexity as compared to WOLA-OFDM, which requires mainly the same number of complex multiplications as CP-OFDM. One can so predict that to achieve a given PAPR reduction gain, SLM is almost less complex than TR. Also, the complexity required by UF-OFDM is huge compared to those required by the other MWFs. As a corollary we can disqualify UF-OFDM from a competition to access the overall power budget in which the consumption corresponding to PAPR reduction method is taken into account.

As a second step of our study, we targeted to estimate the power budget of transmission systems based on dierent MWFs when considering the consumption of (i) the modulation operation (ii) the PAPR reduction method used (in this analysis only SLM will be considered) and (iii) the PA DC power. According to a given IBO and a given value of U , the PAE, given by equation (4.2) can be redened as:

P AE(IBO(U )) = P out (IBO(U )) -P in (IBO(U )) P DC (IBO(U )) + P SLM (U ) , (5.24)
where P in , P out and P DC are respectively the input, output and DC powers of the PA for a given IBO value. P SLM (U ) is the power consumption related to the SLM PAPR reduction algorithm for a given number of rotation vectors U . First, it has to be mentioned that IBO(U ) depends on U and decreases with the increase of the PAPR reduction gain given by Fig. 5.10. Besides, the SLM PAPR reduction method consumption was provided by a system generator tool which generates the VHDL code from a high level language of the processing (Matlab). The targeted board was a FPGA board from Xilinx Zedboard (FPGA Zynq-7000 AP SoC XC7Z020-CLG484). The PA used in this study is a typical 1 W , 3 GHz LTE user equipment power amplier [161]. The AM/AM and AM/PM conversions characteristics of this PA, as well as its DC power are given in [START_REF] Louet | Global power amplier eciency evaluation with PAPR reduction method for Post-OFDM waveforms[END_REF].

Eciency results are sketched in Fig. 5.11. It is worth to point that as CP -OF DM and W OLA -OF DM exhibit the same complexity, the corresponding performances are quasi-identical. For each waveform, three initial IBO were considered: 8, 10 and 12 dB. This means that starting from these IBO values, the SLM method was performed (for U = 1, 2, 4, 8, 16 and 32) and the IBO has been decreased in order to achieve the same level of EVM as that obtained with the chain not integrating PAPR reduction. For each value of U , the power consumption of the SLM was evaluated and included based on the expression given in Tab. 6.5. First, as expected the larger the initial IBO value, the lower the eciency. Second, it appears that eciencies exhibit a maximum value which refers to the trade-o between the PAPR gain and the complexity of the associated SLM method. This remark holds for all waveforms. Before this maximum (U low), the PAPR gain provides a large PA eciency versus the complexity of SLM (U less than 10). But after this maximum, even though the PAPR gain is larger, as U increases, the associated SLM complexity and consumption is too large what penalizes the PA eciency. Eciency results are very similar between CP-OFDM and WOLA-OFDM as their respective complexities are of same order. The eciency shows a maximum value of about 10% when the initial IBO is set to 8 dB. For the f-OFDM waveform, the eciency is less whatever the initial IBO value and the maximum eciency value is below 6%. For the UF-OFDM waveform, the eciency results are not relevant at all for the used PA as the values are very low due to the high complexity as seen in Tab. 6.5.

It is important to note that for the 1 W PA considered in this study, the consumption of the FPGA is of the same order of magnitude as the power consumed by the PA, which greatly reduces the PAE. If a PA with higher output power was used, the PAE would be higher with a maximum for values of U greater than those observed in Fig. 5.11.

This work was done, jointly with colleagues from CentraleSupélec and resulted in the publication of a conference paper [START_REF] Louet | Global power amplier eciency evaluation with PAPR reduction method for Post-OFDM waveforms[END_REF].

Joint PAPR reduction & DPD

Conventionally, PAPR reduction and DPD techniques are optimized separately and applied independently. Some researchers focused on their association aiming at avoiding mutual eects. Among the proposed solutions, we can cite [START_REF] Gouba | A joint approach for PAPR reduction and predistortion by adding signal in cognitive radio[END_REF][START_REF] Landin | Peak-power controlled digital predistorters for RF power ampliers[END_REF]. From another side, the work carried during Krishna Bulusu's PhD showed the feasibility to view the predistortion as an adding signal technique. This statement is also true for some PAPR reduction techniques such as clipping and TR. Thereby, an interesting idea has been raised saying that both PAPR reduction and DPD can be combined and made through a unique adding signal operation. This means that both processing can be done at one stretch by adding a signal, which not only reduces PAPR but also linearizes the PA, which yields to a joint optimization of the two processing. With Dr. Rak Zayani, we took up this idea and apply it to CP-OFDM. We proposed a new technique called "Ping-Pong Joint Optimization" (P2JO). The proposed approach synthesizes, iteratively, one correction signal for reducing the PAPR reduction and linearizing the PA conversion characteristics to reach a given MSE performance.

The block diagram of P2JO technique is given in Fig. 5.12 and the corresponding algorithm is described in the following 1. Set the minimum EV M min , the reserved tone R, the initial clipping level A clip and the maximal iteration number M ax Iter . A clip is closely related with the mean power of the CP-OFDM signal σ 2

x and the percentage of the Peak Reserved Tones R/N F F T . This threshold is given by σ 2

x ln(N F F T /R) [START_REF] Jiang | Curve tting based tone reservation method with low complexity for papr reduction in ofdm systems[END_REF],

2. Set i = 0 and the initial time-domain signal y 0 = γ 0 x, where x is the original CP-OFDM symbol and γ 0 is a scalar gain to achieve the targeted IBO value, Figure 5.12: Ping-Pong Joint Optimization by adding signal.

3. TR : compute the clipping noise w i = ȳi -y i and the peak-cancelling signal c papr,i = IF F T (S r F F T (w i )), were S r is an identity matrix of order N F F T with zeros at data positions, 4. DPD : compute the predistortion signal c dpd,i = y dpd,iy i , where y dpd,i is the predistorted version of y i signal, 5. Update the correction signal y i+1 = y i + µ 1 c papr,i + µ 2 c dpd,i , 6. Compute the amplied signal z i+1 = A(y i+1 ), where A is the amplication operation,

7. Calculate X i+1 = S d F F T (z i+1
) where S d is an identity matrix of order N F F T with zeros at peak cancelling tones positions, 8. Calculate the error vector magnitude

EV M i+1 = E{|X i+1 -X 0 | 2 } E{|X 0 | 2 } , if EV M min < EV M i+1 < EV M i & i < M ax Iter , set i = i + 1
and go to step 3). otherwise, transmit y i+1 and go to step 9), 9. End.

In our analysis, we set µ 1 to 1, to keep a maximal performance in terms of power eciency and push the predistortion to oer its best performance. µ 2 can be modied to clearly evaluate how we can take into consideration the mutual eect between TR PAPR reduction and DPD. It is noteworthy that when µ 2 = 1 the P2JO converges to the same performance obtained by the classical approach, in which we cascade PAPR reduction and DPD. In the following analysis, it's worth to point that "µ%P 2JO" refers to a scenario where µ 2 = µ × 10 -2 . Otherwise, I underline that Output-Back-O (OBO) is used rather than IBO in the following analysis. Fig. 5.13 (a) shows BER performance of 1%P 2JO and 5%P 2JO, in presence of Saleh's PA model (equations (4.8) and (4.9)) compared to the classical approach (where PAPR and DPD processing are cascaded without any collaboration). We can see from this gure that with an increase in the number of reserved tones, the capability of the proposed P2JO is improved. Fig. 5.13 (b), shows the OOB emission reduction in the neighbouring channels based on µ%P 2JO, where the proposed algorithm clearly outperforms the classical approach.

The proposed algorithm allows a better trade-o between PA eciency and linearity while exhibiting reduced implementation complexity, compared to classical cascading of the two-processing. As the idea seemed very interesting to us, we started by patenting it via a European patent [START_REF] Zayani | Method for conditioning a signal for amplication thereof, method for amplication, associated conditioning device and amplication device[END_REF]. We published then the proposed P2JO algorithm and the related results in the journal article [START_REF] Zayani | Ping-pong joint optimization of PAPR reduction and HPA linearization in OFDM systems[END_REF]. For detailed description of this algorithms, please refer to [START_REF] Zayani | Ping-pong joint optimization of PAPR reduction and HPA linearization in OFDM systems[END_REF].

Testbed for demonstrating post-OFDM MWFs capabilities in cMTC context

As an objective of the last task of the WONG5 project, we developed, at CNAM, a software dened radio (SDR) based testbed. We aimed to demonstrate, in real world, the feasibility of some of the DSP based algorithms presented previously to enhance the energy eciency of a selection of MWFs transceivers in The SDR testbed is realized with two parts, Transmitter (Tx) and Receiver (Rx), to evaluate the performance of several MWFs. The testbed overall architecture is presented in Fig. 5.14. The baseband unit is software-based and implemented using Matlab and it realizes the digital algorithm, e.g., waveform generation, QAM modulation, channel equalization, demodulation, etc. It is worth mentioning that DPD and PAPR reduction methods are also implemented using Matlab.

For SDR hardware, two separate Universal Software Radio Peripherals (USRPs) devices, NI-USRP-2942R are used as transmitter and receiver, integrating digital/RF units. The USRP has tunable carrier frequencies in the range of 400 M Hz to 4.4 GHz and tunable transmission rates to 200 M sps [163]. For synchronization of USRP modules, an external clock is used.

In order to ensure a real-time SDR implementation, we have considered two computers. Computer 1 has two solid state disk (SSD), where the rst SSD is dedicated to data storage and is shared with computer 2 using a gigabit ethernet switch and the second SSD is dedicated to the operating system. Matlab on computer 1 realizes the digital algorithms and commands the GNU-radio that congures and uploads signal to USRP Tx. Also, a C++ program is implemented in computer 1 to get the received signal from the USRP Rx. This received signal is divided into small packets that will be stored in (a) SLM (U = 8) and DPD.

8 PRT and DPD. Figure 5.15: Measured PSD performance of dierent MWFs.

the shared SSD and will serve for the analyses in the baseband unit. Computer 2 realizes the analysis of received signals. It uses shared received packets and gives some visual performance results such as PSD, constellation, NMSE per subcarrier, average NMSE and BER. Two graphical interfaces have been developed to make the testbed more exible. On the transmitter side interface, we can choose dierent congurations, for example, we can select the waveform (CP-OFDM, WOLA-OFDM or BF-OFDM), the input back-o IBO (0, 3, 6, 9 or 12 dB), the PAPR reduction method (TR or SLM), the DPD and we can enable or disable interfering user.

We have used an AWGN channel without any frequency selectivity to clearly see the impact of the RF PA on the performance of the selected MWFs and their robustness to asynchronous transmissions. The PA used is a SSPA PE15A4017 from Pasternack with a bandwidth of 20 M Hz to 3 GHz and 27 dB Gain [161].

Using the testbed presented previously, extensive measurements were performed in a realistic laboratory-like environment. For the measurements, two dierent scenarios were performed to evaluate the performances of the selected MWFs. In scenario 1, the robustness of these MWFs against PA distortions is evaluated when considering DSP based techniques like PAPR reduction and DPD. Scenario 2 is dedicated to evaluating the capability of the selected MWFs to support asynchronous transmissions. The system model used for comparison is the same as that used in section 3. In order to study the in-band error with dierent E b /N 0 regimes, Fig. 5.16 plots measured BER when PAs are operated at an IBO of 3 and 6 dB . We can note that WOLA-OFDM and BF-OFDM provide almost the same performance compared to the classical CP-OFDM when SLM based PAPR reduction is performed along with DPD. The same behavioural has been noted when TR is used. At an IBO of 6 dB, BER performances provided by all MWFs are very close to the one performed in the linear case.

In scenario 2, as mentioned previously, we evaluate the robustness of the considered MWFs in multiuser asynchronous access. To well assess the performance of these MWFs, we measure the NMSE on the decoded symbols of the user of interest (UI). Per-subcarrier NMSE is assessed in Fig. 5.17, for guard band δ = 4, 883kHz (corresponding to 1 subcarriers spacing) and timing oset τ = 106 µs (corresponding to 1/2 symbol duration). We can see from Fig. 5.17 that the interference level decreases slowly as the spectral distance between the victim subcarrier and the interferer ones increases. The interference level achieved by WOLA-OFDM in the middle of the bandwidth is lower (approximately -34 dB) compared to CP-OFDM scheme. These good results are related to the WOLA processing applied at the receiver that can suppress inter-user interference resulting from the mismatched FFT capture window.

We move now to BF-OFDM, where additional remarks can be made. Thanks to per-RB ltering, this MWF shows better performance, at the emitter side, compared to CP-OFDM, when both BT-Gaussian and PHYDYAS lters are used. However, the gain of BF-OFDM for the inner subcarriers, located at the middle of the bandwidth, is marginal compared to CP-OFDM. This is a direct consequence of the BF-OFDM receiver which is no more than the classical CP-OFDM receiver (i.e., a simple FFT). Further, at τ = 106 µs, WOLA-OFDM performs better than BF-OFDM, with both prototype lters, as we note a gain of more than 3 dB. From another side, BF-OFDM provides better protection to the edge subcarriers (in the vicinity of interferer subcarriers) compared to both CP-OFDM and WOLA-OFDM. When comparing the two prototype lters considered for BF-OFDM, one can note that a small gain is performed by BT-Gaussian lter compared to the PHYDYAS one. This is directly related to the lter prototype behavioural.

A more detailed description of the proposed testbed along with additional results have been published in the conference paper [START_REF] Zayani | A testbed for experimental performance evaluation of multicarrier waveforms in presence of RF PA[END_REF] and the journal paper [START_REF] Zayani | Experimental testbed of Post-OFDM waveforms toward future wireless networks[END_REF].

Modied architecture transmitter-based approaches

5G wireless communication systems are expected to support a wide range of services with diverse requirements. One of the main concerns for 5G systems is the improvement of energy eciency. Furthermore, Multiple-Input Multiple-Output (MIMO) and massive MIMO technologies will be generalized in 5G. Multiplying power ampliers in MIMO transmitting chains, together with the constraint of an increased energy eciency and low cost are antagonist constraints. With MIMO transmitters, low-cost PA are needed but could have eventually quite poor properties concerning the linearity.

During Khaled Tahkoubit's PhD, we proposed a hybrid approach for enhancing PA eciency by modifying its structure. The main idea behind the proposed method is to decompose the multicarrier signal into M signals with much lower PAPR, by using an algorithm called Iterative Dichotomy PAPR (IDP). The M signals are amplied by M PAs, combined with a suitable combiner and then fed to the antenna. When M = 2, the IDP technique has an architecture similar to that of linear amplication using nonlinear components (LINC) [START_REF] Garcia-Dúcar | Design and experimental evaluation of a LINC transmitter for OFDM systems[END_REF]. In the LINC method, the multicarrier signal is converted into

Iterative Dichotomy PAPR reduction method (IDP)

The IDP method is downward compatible, i.e., there is no side information to be sent to the receiver. As shown by Fig. 5.19 (b), IDP decomposes the MWF signal x(n) into M sub-signals, x m (n), (m = 1, . . . , M ). We can write the signal at the output of the IDP processing as follows

x(n) = M m=1
x m (n).

(5.25)

The sub-signals x m (n), m = 1, . . . , M , are then amplied in each branch by a suitable PA. Before amplication, each sub-signal is scaled by an adequate scalar gain α m to ensure the targeted IBO m . As signals x m (n), have a very low correlation, we can write P x = M m=1 P xm , where P x is the power of the original signal x(n) and P xm is the power of the sub-signal x m (n).

The amplied M sub-signals are recombined by a suitable circuit and the constructed signal is transmitted by the antenna. We assume a perfect synchronized recombination of the M sub-signals at the output of the ampliers. In addition, if perfect linear amplication was used, the signal y(n) should be equal to an amplied version of x(n).

The general idea of IDP method is to decompose the original signal x(n) as in equation (5.25), where signal x m (n) is given by This ratio is related to the mean absolute value of v m (n). By doing so, we ensure that the variable part of the error signal ε M (n), to be added to the signal v m (n), is proportional to its power. The IDP algorithm steps are given in Algorithm 1.

x m (n) = v m (n) + R m M p=1 R p ε M (n), (5.26
(n) = x(n), m = 0, 2: R m+1 = E {|ε m (n)|}, 3: v m+1 (n) = R m+1 .e jϕm(n) , ϕ m (n): the phase of ε m (n), 4: ε m+1 (n) = ε m (n) -v m+1 (n),
In [START_REF] Tahkoubit | Iterative dichotomy PAPR reduction method for multicarrier waveforms[END_REF], we proposed a method to compute analytically R m , for M = 2. Recently, we generalized this computation to any order of the dichotomy decomposition M . We also derived relationship between the PAPR of the original signal x(n) and the one of the nal signals x m (n). After straightforward derivations, we obtain

P AP R xm = P x P xm R m M p=1 R p 2 P AP R x . (5.27)
For M = 2, we have shown, in [START_REF] Tahkoubit | Iterative dichotomy PAPR reduction method for multicarrier waveforms[END_REF], that the two nal signals have quite similar PAPRs equal, approximately to 0.6 times the linear PAPR of the original signal.

The PAPR CCDF of the IDP method with M equal to 2, 3 and 4 and the original CP-OFDM signal are shown in Fig. 5.20. At CCDF = 10 -3 , the PAPR of the CP-OFDM is about 10.8 dB. At this PAPR CCDF, there is a gain of 2.2 dB, in terms of PAPR, at a dichotomy order M = 2. For a higher dichotomy order M , we can reach a gain of 3.4dB and 4dB with M = 3 and M = 4, respectively. We can also notice a good agreement between numerical results and theoretical expectations which proves the accuracy of equation (5.27).

As pointed out in chapter 4, the nonlinear amplication causes OOB radiation over adjacent channel. However, the use of IDP technique minimizes this spectral re-growth. In Fig. 5.21, we show the normalized PSD of the input CP-OFDM signal and the output signal in the classical and IDP based congurations of Fig. 5.19 with IBO = 3 dB. We can notice the improvement achieved by the IDP method for three dichotomy values: M = 2, 3, 4. From this same gure, it can be clearly seen, that performance, in terms of spectral de-growth, increases with the IDP dichotomy order M .

Energy eciency analysis

We have studied the energy eciency of the IDP method, by comparing an IDP based OFDM transmitter to a classical one. For the amplication process we use a single power amplier P A 0 to amplify the CP-OFDM signal z(n)., while we use power ampliers P A 1 , P A 2 , ... , P A M to amplify signals z 1 (n), z 2 (n), ... , z M (n), in the IDP based transmitter, as depicted by 5.19 (b). The P A 0 model is based on the 2.6 GHz GaN amplier characteristics with 19 dB power gain and a PAE up to 70% (Fig 5.18). The other ampliers used in the IDP based scheme are derived from the P A 0 model by adjusting the power compression points. Because signals, at the output of IDP based and classical OFDM methods, have dierent powers, the output power curves of the P A 1 and P A 2 , used in IDP, are respectively scaled by a factors equal to P x1 /P x = 0.8642 and P x2 /P x = 0.1356. Fig. 5.22 shows the output power characteristics for P A 0 , P A 1 and P A 2 .

For a given dichotomy order M , all the P A m , m = 1..M have the same power gain and the same PAE as P A 0 . Nevertheless, because input signals z m (t) have dierent powers the AM/AM curve of the m th P A are scaled by factors equal to P xm /P x for both input voltage z n and output voltage u n . In the following, we give more insights on the approach used to set the parameters of the two schemes of Fig. 5.19, in order to compare their energy eciency.

From equation (4.2), the consumed DC powers by the classical OFDM and IDP based CP-OFDM schemes are written as

P DC OF DM = P u -P z P AE , (5.28)
P DC IDP = M m=1 P um -P zm P AE m . (5.29)
In order to compare the energy eciency of the two schemes given by Fig. 5.19, we xed both consumed DC power and the EVM at the same values and we compare the output signal powers P y and P u . To do that, we follow two steps:

1. [Step 1:] in order to get the same EVM for signals u(n) and y(n) we x a common IBO for all power ampliers P A 0 , P A 1 to P A M . Because the PAPR of x m (n) signals is lower than the one of the CP-OFDM signal x(n), the expected EVM with IDP method is lower than the EVM of a classical CP-OFDM. Then, we gradually reduce the IBO m (by increasing α m ) for each P A m until reaching the same EVM, IBO for the two schemes and we compare the output powers P y and P u .

Tab. 5.3 shows the ratio between the power delivered by the IDP system P y and the output power of the CP-OFDM system P u for the same consumed DC power. We can observe from these results that the output power is higher with IDP system. With M = 2, the output power P y is 1.26 times the power P u , that give an improvement of 26% for IBO = 3 dB. Furthermore, the higher is the dichotomy order M , the higher is the ratio between output powers. For M = 3 and M = 4 there is an improvement of 53% and 78% at IBO = 3 dB.

To complete this analysis and conrm the advantages of the proposed IDP method, we have compared it, in a conference paper [START_REF] Tahkoubit | Energy eciency comparison of Outphasing and IDP techniques with non-linear power ampliers[END_REF], to LINC method. Indeed, both techniques have the same transmitter architecture consisting in the generation of two signals that must be amplied by two PA and then combined for transmission. The comparison has been done following the three main points: (1) comparing the output power for the same DC consumed power and the same BER performance, (2) studying the inuence of timing errors in the combination process and (3) analyzing the eect of a mismatch between the ampliers. Concerning the output emitted power with same DC power and same BER performance, the IDP method outperforms the LINC one, with an emitted power which is twice that of LINC. Concerning timing errors, the two methods are quite impacted. However, the IDP technique is much more tolerant than LINC. Finally, with PA mismatch between the two branches, the IDP method shows low degradations compared to the LINC one.

Conclusion

In this chapter, I presented some contributions on techniques to improve the energy eciency of PA in the context of multi-carrier transmissions. Indeed, like CP-OFDM, all the post-OFDM MWFs presented in chapter 3 suer from high PAPR and are therefore very sensitive to PAs conversion characteristics. Initially, we proposed and adapted probabilistic and signal-adding-based solutions to reduce the PAPR and/or pre-distort the signals to be amplied. The originality of the proposed solutions was based on the fact that they take into account the structures of the signals to be amplied, such as FBMC-OQAM and BF-OFDM, where the overlap between the transmitted symbols must be taken into account in order to achieve acceptable performance. In a second part, I presented an approach to jointly achieve the reduction of PAPR and DPD. This approach, patented in 2017, is based on an iterative and ping-pong synthesis of a single signal that is added to the input of the PA to improve its eciency and linearity.

Another original contribution that we have proposed consists in posing the problem of improving the energy eciency of the PA, by modifying the architecture of the latter. This solution is based on splitting the signal to be amplied into several sub-signals with reduced PAPRs and that can be amplied by parallel stages with energy ecient PAs.

In all these contributions, we have kept a close watch on the complexity of the implementation required by the proposed algorithms. This enabled to identify those presenting a real gain in terms of energy eciency if the corresponding consumption is considered when establishing an overall power budget.

Chapter 6

Massive MIMO networks with mixed numerologies and nonlinear PAs

Introduction

5G NR requirements should be fullled with enhanced network energy eciency, while allowing to enlarge the sub-6GHz spectrum to mmWaves. Among the prominent technologies proposed to reach the 5G NR requirements, Massive MIMO-OFDM and mixed numerologies have been retained to support various congurations.

The major innovation of release 15 of 3GPP with respect to former standards is the support of various numerologies [START_REF] Wg1 | Numerology requirements[END_REF] which allows scalable Sub Carrier Spacing (SCS) and symbol duration. By doing so, challenges of 5G can be properly addressed. Indeed, shortening symbols reduces the latency required for URLLC/cMTC use cases. Moreover, enlarging the SCS increases the robustness against the Doppler eect occurring in mobility scenarios.

Massive MIMO is an emerging technology that uses many antennas at the BS serving many tens of terminals in the same frequency resources. Massive MIMO relies on spatial multiplexing with high number of antennas helping to focus energy into ever smaller regions of space. As the power of the signal to be transmitted by each individual antenna is suciently reduced, massive MIMO includes extensive use of inexpensive low-power components. PA with very relaxed linearity requirements will require energy eciency enhancement solutions, which can exploit the massive MIMO system extra degrees of freedom.

As part of Xinying Cheng's PhD, I have been interested in the study of two issues related to massive-MIMO systems. The rst one, which will be presented in section 6.2, is related to the analysis and cancellation of interferences caused by users using dierent numerologies. In section 6.3, I will present the second topic related to the enhancement of the energy eciency of dedicated emitters architectures, including joint MIMO precoding, PAPR reduction and DPD.

Analysis and cancellation of mixed-mumerologies interference

for massive MIMO-OFDM

Motivaion

Many studies focused on the analysis of mixed numerologies in SISO, MIMO and OFDM systems. Among these studies, we can cite [START_REF] Sahin | Multi-User Aware Frame Structure for OFDMA Based System[END_REF][START_REF] Lien | 5G New Radio: Waveform, frame Structure, multiple access, and initial access[END_REF][START_REF] Guan | 5G eld trials: Ofdm-based waveforms and mixed numerologies[END_REF][START_REF] Zhang | Subband ltered multi-carrier systems for multi-service wireless communications[END_REF][START_REF] Zhang | Multi-service system: An enabler of exible 5G Air Interface[END_REF][START_REF] Yazar | Reliability enhancement in multi-numerology-based 5G new radio using INI-aware scheduling[END_REF][START_REF] Zhang | Mixed numerologies interference analysis and inter-numerology interference cancellation for windowed ofdm systems[END_REF][START_REF] Kihero | Inter-numerology interference analysis for 5G and beyond[END_REF]. In these papers, the approach proposed for supporting diverse services is to introduce exible numerology allocation over adjacent sub bands. Some recent papers [START_REF] Rahman | Multi-user MIMO with exible numerology for 5G[END_REF] [174] focused on mixed numerologies spectrum sharing (SS). Unlike nonoverlapping mixed numerologies systems, it is dicult to prevent interference using windowed/ltered waveforms in mixed numerologies SS systems (i.e., users share the same time/frequency resources). In [START_REF] Choi | A transceiver design for spectrum sharing in mixed numerology environments[END_REF], the authors proposed a new transceiver design for classical MIMO-OFDM using a mixed numerologies SS scheme.

In the rst part of Xinying Cheng's PhD, we focused on the use of spatial multiplexing to link users who share the same bandwidth but have dierent numerologies. We considered a massive MIMO-OFDM system, where we carried, for both DL and UL scenarios, theoretical analysis of internumerology interference (INI) and proposed ecient INI cancellation schemes.

INI theoretical analysis for Massive MIMO-OFDM systems 6.2.2.1 DL scenario

We consider a massive MIMO-OFDM DL scheme with various numerologies. The BS has M t antennas that serve M r single-antenna users with dierent numerologies over a frequency-selective channel, where M t is far larger than M r . N U M numerologies, represented by index num, where num = 1, ..., N U M , can be used to divide the M r users into N U M groups. N num and CP num denote the IFFT/FFT size and CP size of group num, respectively. The proposed transceiver can suppress Intra-NI (multi-user interference (MUI) between users using the same numerology) by using several precoders as will be explained later. The BS sends data, over the m t -th antenna, to the m r -th user via channel √ α mr ht mr,mt , where α mr is the large-scale fading, ht mr,mt ∈ C 1×D is the channel impulse response between transmitting antenna m t and user m r , m t = 1...M t , m r = 1...M r and D is the number of channel impulse response taps. For a massive MIMO-OFDM system, the channel frequency response is denoted by H(num) ∈ C Mr×Mt×Nnum , where H(num) (m r , m t , :) = hf (num) mr,mt = F F T ( √ α mr ht mr,mt , N num ), (6.1)

were F F T (ht mr,mt , N num ) represents the channel frequency response corresponding to the impulse response ht mr,mt and which is computed with a FFT size equal N num . For clarity's sake and without loss of generality, we consider two users (M r = 2) using two separate numerologies, as shown in Fig. 6.1. For users 1 and 2, s (1) , s (2) ∈ C 1×N1 are signals resulting from an M-array QAM. We assume that N 1 = N × N 2 , CP 1 = N × CP 2 , where N = 2 i and i is an integer. This generalized synchronized scenario has also been considered in previous works [168] [171]. x It's worth noting that users in the proposed system share the same bandwidth, resulting in high spectrum performance. Inter-symbol interference (ISI) is avoided at the receiver side if the length of CP num is greater than the length of the multi-path channel.

The analysis of INI between users using dierent numerologies, we perform in the next paragraphs, is only valid for massive MIMO-OFDM systems, i.e. M t M r , assuming that the ZF precoders works well and can fully suppress Intra-numerology MUI. In the case of INI between users who use dierent numerologies, the received symbol after FFT can be written as follows

y 1 [n 1 ] = s 1 [n 1 ] + ini (2,1) [n 1 ] + b 1 [n 1 ], n 1 = 1, ..., N 1 , , (6.2)
y 2,n [n 2 ] = s 2,n [n 2 ] + ini (1,2) n [n 2 ] + b 2,n [n 2 ], n 2 = 1, ..., N 2 ., (6.3)
where y 1 is the received signal by user 1, y 2,n is the n-th received signal by user 2. INI from numerology 1 with N 1 to numerology 2 with N 2 = N 1 /N .

In this scenario, we consider that user 2, using numerology 2 with smaller IFFT/FFT size N 2 is interfered by user 1, which is using N 1 > N 2 . User 2 receives an all-zero signal if only the dash-line block in Fig. 6.1 is enabled. Since the ZF precoder removes all MUI, we only get INI from user 1 due to numerology 1. ini (1,2) n [n 2 ] is derived from the IFFT/FFT window sizes being unequal, as seen in Fig. 6.2 (a). Based on the analysis carried in, [START_REF] Cheng | Inter-numerology interference analysis and cancellation for massive MIMO-OFDM downlink systems[END_REF], the INI, from numerology 1 to numerology 2, is given by ini (1,2)

n = G (1) W (1) n N1 n1=1 Mt mt=1 hf (1) 2,mt [n 1 ]v (1) mt [n 1 ], (6.4)
where

G (1) = DFT N2 ∈ C N2×N2 , hf (1)
2,mt ∈ C 1×N1 is the MIMO channel frequency response between m t -th transmitting antenna and user 2. W [START_REF] Chen | A 2.6 GHz Class-AB GaN Power Amplier with Maximum Output Power of 56 W Achieving 70% Power Added Eciency[END_REF] n ∈ C N2×N1 is the n-th N 2 × N 1 part of matrix W (1) , where W (1) ∈ C N1×N1 is a rotated version of matrix of the inverse discrete Fourier transform (IDFT) [START_REF] Cheng | Inter-numerology interference analysis and cancellation for massive MIMO-OFDM downlink systems[END_REF]. v perfectly eliminate all the Intra-NI, which is part of the total MUI. In this case, we can note that

Mt mt=1 hf (1) 2,mt [n 1 ]v (1)
mt [n 1 ] = 0, then, we can get ini (1,2) n = 0 n = 1, ..., N .

Remark 1: Performance of users using small numerology (IFFT/FFT size) are perfect independently of channel selectivity and power allocation.

As a result, unlike SISO/classical MIMO, where INI exists on small numerology (IFFT/FFT size) [START_REF] Zhang | Mixed numerologies interference analysis and inter-numerology interference cancellation for windowed ofdm systems[END_REF] [START_REF] Rahman | Multi-user MIMO with exible numerology for 5G[END_REF], the INI of users with small IFFT/FFT size is completely suppressed when using the proposed transceiver design for massive MIMO-OFDM.

INI from numerology 2 with N 2 to numerology 1 with

N 1 = N × N 2 .
In comparison to user 2, user 1's transmission is not properly protected by the ZF precoder, Indeed, when only the solid-line block of 6.1 is enabled, user 1 receives zeros. In this case, where zeros are prepared for user 1, the received signal on user 1 is the INI induced by signal from user 2. This interference comes from N symbols of numerology 2, each of them of length N 2 = N 1 /N . The INI is induced by an uneven FFT window at the receiver side, as seen in Fig. 6.2 (b) and is given by the sum of INI from each symbol modulated with numerology 2. The total INI, can be expressed as ini (2,1)

= N n=1 ini (2,1) n , (6.5)
where n = 1, .., N , ini (2,1) n represents the interference from the n-th symbol with numerology 2.

126 Chapter 6. Massive MIMO networks with mixed numerologies and nonlinear PAs

The INI on the n 1 -th subcarrier with numerology 1 from the n-th symbol with numerology 2 is given by ini (2,1) n

[n 1 ] = N1 n 1 =1 N2 n2=1 E n (n 1 , n 1 )Z(n 1 , n 2 ) Mt mt=1 hf (1) 1,mt [n 1 ]v (2,n) mt [n 2 ], (6.6)
where

• n 1 = 1, ..., N 1 , n 1 = 1, ..., N 1 , n 2 = 1, ..., N 2 and m t = 1, ..., M t ,

• Z = G (2) 2 W (2) 2 ∈ C N1×N2 , • E n ∈ C N1×N1 =      G (2)
1 W

(2) 1

f or n = 1, G (2)
2 ID (2) n W

(2) 3

f or n = 2, ..., N. ,

• G

(2)

1 = 0 N1×CP1 DFT N1 ∈ C N1×(N1+CP1) , G (2)
2 = DFT N1 ∈ C N1×N1 , • W (2) 1 = IDFT N1 0 CP1×N1 ∈ C (N1+CP1)×N1 , W (2)
3 = IDFT N1 ∈ C N1×N1 , • ID (2) n =   0 T1×(N1/2) 0 T1×(N1/2) I T2 0 T2×(N1-T2) 0 T3×(N1/2) 0 T3×(N1/2)   ∈ C N1×N1 , •                  T 1 = N 1 -(N -n + 1)(N 2 + CP 2 ), T 2 = N 2 + CP 2 + D f or n = 2, ..., N -1 N 2 + CP 2 f or n = N, T 3 = (N -n)(N 2 + CP 2 ) -D f or n = 2, ..., N -1 (N -n)(N 2 + CP 2 ) f or n = N.
Depending on the frequency selectivity of the channel, we can have hf

(1) 1,mt [n 1 ] = hf (2) 1,mt [n 2 ], implying that ini (2,1) n = 0.
Impact of the channel response selectivity and N 1 /N 2 ratio.

In a constant channel, ini (2,1) n is zero due to the channel frequency response characteristic. Indeed, in this case, and regardless the N 1 /N 2 ratio, we have hf

(1) 1,mt [n 1 ] = hf (2) 1,mt [n 2 ]
. Unlike SISO and classical MIMO systems, the proposed massive MIMO scheme provides improved performance in presence of mixed numerologies when the channel is at. When the channel is frequency-selective, vector hf

(1) 1,mt [(n 2 -1)N + 1] = hf (2) 1,mt [n 2 ], (6.7)
where

n 2 = 1, ...N 2 .
We can begin by considering the n 2 -th subcarrier of the n-th symbol with numerology 2. Its impact on the n 1 -th subcarrier with numerology 1 is derived in equation (6.6). According to equation (6.7), the inuence of n 2 -th subcarrier is zero only when n 1 = (n 2 -1)N + 1, because under this situation, we could obtain hf 1,mr [n 2 ] is closely related to the channel selectivity and the dierence between N 1 and N 2 . Greater selectivity and a larger dierence between N 1 and N 2 result in a larger dierence in the frequency responses of the two channels, resulting so in higher INI.

Remark 2: In a constant channel, ini (2,1) is zero due to the channel frequency response characteristic. However this INI is increased by channel selectivity and dierence between N 1 and N 2 .

Impact of user's power allocation. We chose a basic power allocation scheme for users and focused on its impact of INI. We assume that the received signal power by the m r -th user (p mr ) is proportional to the inverse of its path-loss α mr . The power allocation for the M r users is given by

P = ρ M t A -1 , (6.8)
where ρ is a normalization factor that guarantees that tr(P) = P T x , with P T x , being the total emitted power. P = Diag(p 1 , ..., p Mr ) and A = Diag(α 1 , ..., α mr ). Then, as user m r 's large-scale fading increases, the corresponding emitted power increases. In our system, having two users with dierent path-loss, the generated INI power from user 1 on user 2 (p (1,2) ) and that from user 2 on user 1 (p (2,1) ) satisfy

p (1,2) p (2,1) = α 2 α 1 . (6.9)
As an example, if user 2 is farther away from the BS than user 1, the INI power seen by user 1 is proportional to the power of the signal at the output of the ZF precoder. That is to say, when α 2 > α 1 , user 1 will suer from higher INI as a result of user 2's improved transmitting power.

Remark 3: ini (2,1) n is directly inuenced by power allocated for dierent users.

UL scenario

In a second step we focused on the UL system. We consider a single-cell massive MIMO-OFDM UL system, where M t = 2 single-antenna users that are using dierent numerologies N 1 and N 2 , transmit signals to a BS equipped with M r antennas (with M t M r ), over a frequency-selective channel. The system model corresponding to this scenario is given in Fig. 6.3.

As in the UL scenario, we consider a generalized synchronized scenario (N 1 = N ×N 2 , CP 1 = N ×CP 2 , where N = 2 i and i is an integer), at the BS side. For the analysis of INI, we assume a noise free transmission so that the NMSE detection becomes ZF detection. The received signal is separated into two branches and the data stream are then detected through linear processing by two ZF detectors P(1) ∈ C 2×Mr×N1 and P(2) ∈ C 2×Mr×N2 . Due to the ZF detection characteristics, the following equations are valid [START_REF] Cheng | Analysis and Cancellation of Mixed-Numerologies Interference for Massive MIMO-OFDM UL[END_REF] Mr mr=1 P(1) (2, m r , n 1 )hf [START_REF] Chen | A 2.6 GHz Class-AB GaN Power Amplier with Maximum Output Power of 56 W Achieving 70% Power Added Eciency[END_REF] mr,1 [n 1 ]s (1) [n 1 ] = 0, (6.10)

Mr mr=1 P(2) (1, m r , n 2 )hf (2)
mr,2 [n 2 ]s (2,n) [n 2 ] = 0. (6.11) MUI within the same numerology is eliminated by the detection matrices. However, in a mixednumerologies scheme, INI exists between users sharing the same band.

Simulation results

In this section, we will check the accuracy of the derived analytical expressions for INI in a mixed numerologies SS massive MIMO-OFDM scheme. We consider two single-antenna users who use two dierent numerologies but share the same band in the same cell and the BS is equipped with one hundred antennas. A 16QAM with Gray mapping is considered. Two pairs of numerologies are taken into consideration, with the rst pair N 1 = 1024, N 2 = 512, CP 1 = 72, CP 2 = 36 and the second pair Tab. 6.1 shows the NMSE values in dB for the user of interest, based on the analysis carried in section 6.2.2.1. In this table, we compared dierent delay spread scenarios with users which are at equal distances from the BS. Tab. 6.1 shows that regardless the N 1 /N 2 ratio and the channel selectivity, transmissions for user 2 with small IFFT/FFT size still have the highest quality (NMSE value of about -300 dB). It should be noted that the value of -300 dB is related to Matlab's oating point precision.

These ndings are in full accordance with the study conducted in section 6.2.2.1, which demonstrates that numerology 2 does not suer from the INI from numerology 1 when the IFFT/FFT size is small. When the N 1 /N 2 ratio increases in a frequency-selective channel, user 1 suers from higher INI coming from numerology 2. User 1's reliability is worse when interference is from N 2 = 256 than when interference is from N 2 = 512 for the same frequency-selective channel. From another side, when considering the performance of a given pair of N 1 and N 2 under various channel selectivity, we can see that the more selective is the channel, the worse is the user 1 performance. The NMSE performances given in Tab. 6.1 assume that the user of interest and the interfering one are located at the same distance for the BS (users with the same path-loss: α 1 /α 2 = 0 dB). If we consider a more realistic scenario, we can use equation ( 6 where d k is the distance from the BS in km. User 2 performs better than user 1 regardless the channel selectivity and/or path-loss. For this reason, we will concentrate only on user 1 performance when realistic path-loss is considered. The impact of user's power allocation on system performance is shown in Tab. 6.2, with the number of channel taps D = 8.

User of

As the path-loss on user 2 grows, user 1 experiences more INI. When N 1 = 1024, N 2 = 512 and D = 8, there is a 20dB degradation when α 1 /α 2 = 20dB compared to α 1 /α 2 = 0dB. It can also be noticed for Tab. 6.2, that once INI cancellation is applied at the BS side, INI on user 1 falls to -300dB for all path-loss and N 1 /N 2 values.

For DL scenario, Fig. 6.5 (a) shows the symbol error rate (SER) vs. E b /N 0 using N 1 = 1024, N 2 = 512 and D = 18 to validate the INI analysis on user 1. We can see from this gure that when the path-loss of user 2 increases, user 1's SER suers signicantly. As an example, we can notice at SER = 10 -3 , that there is a SNR loss of ∼ 3 dB when we move from α 1 /α 2 = 0 dB to α 1 /α 2 = 26 dB. Tab. 6.3 shows the NMSE values in dB for the user of interest for the UL system described in section 6.2.2.2 under various channel conditions, with α 1 = α 2 . As expected the results given in Tab. 6.3 show that INIs are increasing with channel selectivity except for the at fading channel case. These ndings are consistent with the analysis presented in section 6.2.2.2, demonstrating that channel selectivity and the increase of N 1 /N 2 ratio, raise INIs. For example, under channel D = 8, there is 9 dB of degradation on N M SE compared to channel D = 2 and 13 dB under channel D = 18 for the same pair of numerologies, N 1 = 1024, N 2 = 512. Meanwhile, when comparing performance under the same channel selectivity, dierent pairs of numerologies behave dierently, with users suering more INI as the N 1 /N 2 ratio grows. a low or a high modulation order is used [START_REF] Guerreiro | Massive MIMO with nonlinear amplication: Signal characterization and performance evaluation[END_REF]. As discussed in chapter 4, the non-linearity of RF PA, is the main hardware impairment in the system. The eect of non-linear PA on the energy-ecient design of massive MIMO has been studied in [START_REF] Khan | Energy ecient design of massive mimo by considering the eects of nonlinear ampliers[END_REF] and the spatial characteristic of the non-linear distortion radiated from antenna arrays is shown in [START_REF] Mollén | Spatial characteristics of distortion radiated from antenna arrays with transceiver nonlinearities[END_REF]. Many research eorts have been recently conducted in multi-user (MU)-MIMO precoding based PAPR reduction. Low-PAPR Tomlinson-Harashima precoding schemes suitable for MU-MISO and MU-MIMO DL were initially described in [START_REF] Mohammed | A low-complexity precoder for large multiuser miso systems[END_REF] and [START_REF] Siegl | Selected basis for par reduction in multi-user downlink scenarios using lattice-reduction-aided precoding[END_REF], respectively. These schemes, however, require specic signal processing at the receiver side (i.e., in the mobile terminal) making them less attractive. Prabhu et al. [START_REF] Prabhu | A low-complex peak-to-average power reduction scheme for OFDM based massive MIMO systems[END_REF] proposed a peak-clipping scheme for PAPR reduction in OFDM-based massive MIMO where some of the antennas at the BS are reserved to compensate for peak-clipping signals. Another method has been introduced in [START_REF] Ni | A novel adaptive tone reservation scheme for PAPR reduction in largescale multi-user MIMO-OFDM systems[END_REF]] that aims at reserving some tones to reduce the PAPR in large-scale MU-MIMO-OFDM systems. However, these methods lack eciency since they sacrice spectral eciency by reserving some antennas or some tones, which may be not suitable for future wireless networks.

Recently, some research aimed to jointly perform MU precoding and PAPR reduction for massive MIMO-OFDM, like [START_REF] Bao | An ecient Bayesian PAPR reduction method for OFDM-based massive MIMO systems[END_REF] [186] [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF], where algorithm EM-TGM-GAMP, PROXIINF-ADMM and MU-PP-GDm were proposed, respectively. These recent algorithms showed great performance both in MUI and OOB radiation while ensuring acceptable PAPR performance. However, all these solutions, in addition to the fact that they do not consider PA nonlinearities, still have a quite high computational complexity. In [START_REF] Zayani | Ecient precoding for massive MIMO downlink under PA nonlinearities[END_REF], we have proposed the MU-PNL-GDm algorithm performing jointly MU precoding, PAPR reduction and digital-predistortion. This algorithm provides good performance, but its complexity is still challenging.

Motivated by all these considerations, we have focused, during Xinying Cheng's PhD, on the development of an ecient DL transmission scheme taking into consideration MU precoding, PAPR reduction and PA linearization. The proposed iterative algorithm, referred to as MU-2P, exploits the high-dimensional degrees of freedom, oered by equipping the BS by many antennas. During the design of this algorithm, we have paid special attention to the computational complexity.

System model

We consider a typical massive MIMO scenario, where the BS is equipped with M t antennas serving M r single-antenna users and M t is signicantly larger than M r . The BS transmits, via the m t -th antenna, to the m r -th user over a frequency-selective channel. Here, each antenna is equipped with a PA. The signal vector s n ∈ C Mr×1 contains the symbols associated with the n-th OFDM subcarrier for M r users, where n = 1, ..., N and it is chosen from a complex-valued constellations. In practice, OFDM systems typically specify certain unused subcarriers, which are used for guard-band (e.g., at both ends of the spectrum). Hence, the set of subcarriers available are divided into two sets χ and χ c , where the subcarriers in set χ are used for data transmission and the subcarriers in its complementary set χ c are used for guard band.

Moreover, we set s n = 0 Mr×1 for n ∈ χ c so that no signal is transmitted on the guard-band.

To suppress the MUI at the receivers, linear precoding scheme is implemented at the BS. Thus, the signal matrices c n ∈ C Mt×1 are linearly coded as

c n = 1 √ ζ w W n s n , n = 1, ..., N, (6.17)
where ζ w is the power normalization factor designed to obtain an average or instantaneous transmit power equal to σ 2 s (power of signal s). W n ∈ C Mt×Mr and c n ∈ C Mt×1 denote respectively the ZF precoding matrices for n-th OFDM subcarrier and the precoded signal on the n-th subcarrier through the M t antennas.

After precoding, the M t -dimension vector c n is reordered to construct x n , according to the following mapping

x t 1 . . . x t Mt = [c 1 . . . c N ] T . (6.18)
The N -dimensional vector x mt denotes the frequency domain signal to be transmitted from m t -th antenna. Then, the time-domain signal a t mt is obtained by applying an IFFT to x mt and a CP is then added to the time-domain samples of each antenna. At the receiver side, assuming that the length of the CP is larger than the length of the multipath channel, ISI and ICI are eliminated.

In this study, we will compare our proposed low complexity algorithm, with two reference schemes noted as (Ref. 1) and (Ref. 2) as well as existing schemes proposed in the literature. Fig. 6.8, depicts the considered schemes along with the proposed one. A brief description of the existing schemes, along with the presentation of the proposed algorithm will be given in the next sections.

For clarity sake and without loss of generality, the input-output relationships of the wireless propagation channel will be written in the frequency domain. Then, the signal received by M r users at n-th Scheme (1) : ZF precoding and PAPR reduction using null-space + DPD.

The key idea of this scheme is to use ZF precoding method to achieve perfect MUI cancellation and to reduce the PAPR by adding the peak-cancelling signals using the null-space of the channel. The PA nonlinearities are compensated by an external DPD for each PA/antenna. PAPR reduction is achieved by adding peak-canceling signals {p t mt } to precoded signals x t mt . The N -dimensional vector p t mt is the peak-cancelling signal added to the precoded signal of the m t -th transmit antenna, while p n ∈ C Mt×1 is the frequency-domain peak-canceling signal added to the precoded signal of the n-th subcarrier. To avoid MUI, the peak-canceling signals must be projected onto the null-space of the MIMO channel, satisfying

H n p n = 0 Mr×1 , n ∈ χ. (6.20)
There are two methods to compute the null-space V n using the singular value decomposition (SVD) [START_REF] Sun | Improved singular value decomposition (TopSVD) for source number estimation of low SNR in blind source separation[END_REF] or by computing

V n = I n -H H n H n H H n H
. Therefore

H n V n = 0 Mr×Mt-Mr , ∀n.

(6.21)

The key point is to design a peak cancelling signal in the frequency domain p n in such a way that it fulls p n = V n e n .

(6.22)

After applying the clipping method to reduce the PAPR, the original frequency-domain clipping noise at the n-th subcarrier is d n ∈ C Mt×1 . Then, the aim of this scheme is to solve the following convex optimization problem

minimize ėn G (e n ) = V n e n -d n 2 2 , n ∈ χ, subject to ėn = 0 (Mt-Mr)×1 , n ∈ χ c . (6.23)
The above formulation yields to an iterative algorithm with the peak-canceling signals constrained to be in the null-space of the corresponding MIMO channel matrices. This algorithm is referred to as ZF-PAPRnullspace. It is worth noticing that this algorithm can separate the PAPR reduction problem from the MU precoding by initializing precoded signals x n according to a pre-xed precoder. Scheme (2) : Joint precoding and PAPR reduction using null-space [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF].

In scheme (1) the PAPR reduction using null-space is performed, while applying ZF precoding before. However, scheme (2) corresponds to the joint precoding and PAPR reduction solution MU-PP-GDm proposed in [START_REF] Zayani | PAPR-Aware Massive MIMO-OFDM Downlink[END_REF]. In this scheme the joint precoding and PAPR reduction are done iteratively to remove 6.3.4 Scheme (4) : Proposed scheme for joint ZF precoding and PAPR reduction without null-space.

Compared to the previously presented schemes, we propose a new scheme which can jointly apply precoding and PAPR reduction without using the high-complexity required by the null-space matrix computation. The new scheme corresponds to scheme (4) on Fig. 6.8. The key idea is to iteratively reduce jointly the MUI and the clipping noise {d t mt }, which is obtained by clipping the time-domain signals {a t mt }. Given the clipping threshold λ, the clipped signal a n at the n-th subcarrier can be obtained by By minimizing the MSE of intended symbols s n through the channel H, the problem can be formulated as

āmt (k) = a mt (k), if |a mt (k)| < λ, λe jφ(k) , if |a mt (k)| > λ, (6.30
minimize J(x n ) = H n x n + H n d n -s n 2 2 , n ∈ χ, subject to x n = 0 Mt×1 , n ∈ χ c . (6.33)
A minimization strategy can be used to solve (6.33), in which we iteratively minimize the objective function. Note that the clipping noise used in the (l + 1)-th iteration is calculated from the previous l-th iteration. The procedure can be expressed as

x (l+1) n = argmin {xn} J x (l) n , d (l) n . (6.34)
The search directions of the steepest gradient descent (GD) method at l + 1 is determined by the negative gradient of the cost function J at iteration l, which is given by

∇ x J x (l) n , d (l) n = 2µH H n H n x (l) n + H n d (l) n -s n , (6.35)
where µ is the adaptation step size. The details of the proposed processing, referred to as MU-2P (for Multi-User for joint Precoding and PAPR reduction), are summarized in algorithm 2.

Complexity analysis

We adopt the number of complex multiplications to compare the complexity of the proposed algorithm to the one of the existing scheme (1), scheme (2) and scheme (3). For the proposed algorithm MU-2P, the computational cost is dominated by the gradient computation ∆x (l) n (Line 3 of algorithm 2) and the N-point IFFT/FFT (line 5 and 6). The gradient in line 3 involves O (3|χ|M t M r ) complex multiplications. The IFFT/FFT costs O (M t N log(N )) complex multiplications. Therefore, the proposed MU-2P requires a total of O (3|χ|M t M r + 2M t N log(N )) complex multiplications for each iteration.

For scheme (1):

ZF-PAPRnullspace, it needs O N (2M t M 2 r + M 3 r ) , O (N (M t M r )) and O 2M 2 t M 2 r + M r M 2 t + M 3 r
for ZF procoding matrix computation, ZF implementation and null-space computation. This leads to O 2|χ|M t + 2M t N log(N ) per iteration. The complexities of the dierent schemes are summarized in Tab. 6.5. For scheme (1), ( 2) and (4) (the proposed scheme), a dedicated DPD block per antenna branch is used. The DPD is based on the well-known multi-layer perceptron (MLP) neural network (NN) model. The NN predistorter has two layers, one input and one output with 52 weights. The complexity of the NN DPD is the same for all three algorithms.

l) n = 2µH H n H n x (l) n + H n d (l) n -s n , ∀n ∈ χ, 4: x (l+1) n = x (l) n -∆x (l) n , ∀n ∈ χ, 5: a t(l+1) mt = IF F T x t(l+1) mt , ∀m t = 1 . . . M t , 6: d t(l+1) mt = F F T a t(l+1) mt -a t(l+1) (
N (2M t M 2 r +M 3 r ) - - - ZF imple- mentation N (M t M r ) - - - Null-space computation 2M 2 t M 2 r + M r M 2 t + M 3 r 2M 2 t M 2 r + M r M 2 t + M 3 r - - Per iterarion 2|χ|M 2 t + 2M t N log(N ) 2|χ|M t M r + 2|χ|M 2 t + 2M t N log(N ) 3|χ|M t M r + 2|χ|M 2 t + 2M t N log(N ) 3|χ|M t M r + 2M t N log(N ) DPD 13N M t 13N M t - 13N M t

Simulation results

The performance of the previously presented schemes in multiuser massive MIMO system is evaluated by simulations. The BS is equipped with M t = 100 antennas serving M r = 10 single-antenna users. We consider OFDM with N = 512 subcarriers and use a spectral map χ, in which |χ| = 128 subcarriers are used for data transmission. A 16QAM Gray mapping is considered. We use the CCDF to evaluate the PAPR reduction performance.

The wireless channel is assumed to be frequency-selective and modeled as a tap delay line with D = 8 taps. The time-domain channel response matrices have i.i.d. circularly symmetric Gaussian distributed entries with zero mean and unit variance. The equivalent frequency domain response H n is obtained by FFT.

In order to evaluate the eciency of our proposed scheme in term of PAPR reduction, Fig. 6.9 shows the CCDF of the PAPR achieved by the dierent schemes for a same MUI level equal to -40 dB 1 . Note that PAPRs associated with all M t antennas are considered in computing the empirical CCDF. From Fig. 6.9, we can see that the PAPR gain is of 5.4 dB, 5.2 dB, 2.6 dB and 5.1 dB, when, respectively, scheme 1, scheme 2, scheme 3, scheme 4 are considered (at CCDF of 10 -3 ). The SER vs. SNR of all the schemes is given in Fig. 6.10, where the SNR is dened as SN R = E{ x n 2 2 }/N 0 and the PA is operated at IBO = 3 dB. All schemes reduce the in-band distortion compared with ZF scheme without PAPR reduction referred as (Ref .2). We can also observe that there are little SNR performance (less than 0.5 dB) losses between all the schemes compared with the ideal scheme (Ref .1) at the level of 10 -4 . This performance dierence is primarily because we add dierent power on the transmitter side with the dierent schemes.

We compare the proposed scheme with the existing three schemes and the reference schemes. In each algorithm, the adaptation step size µ is automatically adjusted to oer the best convergence rate to the optimal PAPR. For each scheme, the value of maxIter is chosen to reach a MUI of -40 dB to create a fair comparisons between all the algorithms. Fig. 6.11 shows the PSD at the receiver side for the dierent schemes at IBO = 3 dB. The PSD spectrum is the mean on all M r receiving antennas. From the PSD given by Fig. 6.11, we can observe that all schemes perform at nearly the same level, with a gain of 5 dB compared with (ref.

2).

According to the mentioned closed-form expressions in Tab. 6.5, it is possible to numerically assess First, we examine the eciency of our proposed scheme with M t = 100, M r = 10, N = 512 and | χ |= 256. Tab. 6.6 shows the complexity of dierent schemes. From this table we can observe, rst, that the complexity of the proposed scheme ( 4) is lower than the other schemes. Indeed, the complexity of the proposed scheme requires 29.31%, 31.08% and 68.47% of the complexity needed by scheme (1), scheme (2) and scheme (3). This is because ZF-PAPRnullspace and MU-PP-GDm have the initialization of null-space or ZF matrix computation and for MU-PNL-GDm, there is PA implementation in each iteration.

We move now to the complexity comparison considering channel variations. Since ZF precoding matrix computation, null-space computation, the iterative algorithms and NN DPD are all sensitive to channel variations, the coherence time of the channel will play an important role in the total complexity. Fig. 6.12 shows the number of complex multiplication versus channel coherence time with M t = 100 and M r = 10, where τ represents the number of OFDM symbols within which the channel stays stationary. Like the previous gure, the proposed scheme has the best performance, in terms of complexity, compared with all the other schemes while schemes (1) and ( 2) are the most sensitive among all.

The results presented in section 6.3, have been submitted to IEEE Transaction on Vehicular Technology, in a journal paper which is under review.

Conclusion

In comparison to mixed numerologies with SISO and classical MIMO systems, where users use adjacent frequency bands, in this chapter, we rst introduced new transmission schemes designed for MU massive MIMO-OFDM based 5G system that supports dierent services using dierent numerologies, while sharing the same band. The eciency of these mixed numerologies in massive MIMO-OFDM DL & UL systems was then investigated. We derived theoretical expressions of the INI occurring in the mixed numerologies scenario. We showed that, in the massive MIMO-OFDM DL scenario, INI is only created in frequency selective channels and only when users with large sub-carriers spacing communicate with users with small SCS. However, in uplink scenario, all users suer from INI in frequency selective channels.

Furthermore, we have veried that the theoretical INI expressions agrees with simulation ndings, suggesting that INI is inuenced by SCS gaps, channel selectivity, and power allocation. Last but not least, we proposed INI cancellation schemes based on the developed closed form INI expressions that can suppress the INI at the BS side without raising the users' complexity. These results demonstrated that massive MIMO-OFDM systems can accept mixed numerology transmissions while all users share the same band, indicating that this method can increase the spectrum eciency as well as meet the needs of future wireless communication.

In the second part of this chapter, I focused on energy-ecient DL schemes for massive OFDM-MIMO, taking into account PA nonlinearities. I have presented the existing schemes in the literature and proposed a new one. The main idea of the proposed scheme is to nd a trade-o between the clipping noise and the MUI which enables to transmit signals with low PAPR while maintaining acceptable transmission performance. The joint MU precoding and PAPR reduction method was formulated as a simple convex optimization problem for which an algorithm based on steepest gradient descent method was designed. The simulation results showed that the proposed algorithm allows a satisfying performance, while requiring lower computational complexities compared with the existing solutions. requirements of 5G, new requirements that were not considered in 5G have been added and expanded more widely. Moreover, as with 5G, not all requirements need to be or can be met at the same time, but new combinations of requirements will be expected by the future new use cases.

In order to meet the new requirements of 6G, new approaches and technologies have been proposed in the literature or are being studied and developed by manufacturers or as part of collaborative research programs. Fig. 7.3 illustrates some of the 6G possible technological approaches, which are listed below:

• The extension to sub-THz and THz bands,

• A further advancement of wireless transmission technologies,

• The use of AI and ML to optimize the performance of all network elements, including the physical layer,

• The control of propagation through recongurable intelligent surfaces (RIS).

As perspectives of my research activities, I envision the continuation of my actual research topics, which I will extend to 6G systems whose specications and, as seen from Fig. 7.1, will be even more demanding than those of 5G. Additionally, I intend to develop new research topics such as the introduction of Machine and Deep learning techniques to propose energy ecient transceivers which are partially or entirely based on AI and ML. Also, as another new research axis, I will be interested on RIS, which are electronically controllable for transmitting data to several users.

Mitigation of RF impairments in higher frequency massive MIMO systems

The rapid increase of connected mobile terminals in the 4G era pushed data rate requirements of 5G systems to new levels [START_REF] Wang | Cellular architecture and key technologies for 5g wireless communication networks[END_REF]. As the sub-6 GHz spectrum is congested, moving towards other ranges, such as millimeter waves (mmWaves) for 5G and beyond systems [START_REF] Wei | Key elements to enable millimeter wave communications for 5g wireless systems[END_REF] as well as Sub-Tera Hertz bands [START_REF] Bicaïs | Design of digital communications for strong phase noise channels[END_REF][START_REF] Saad | Mimo techniques for wireless terabits systems under sub-thz channel with rf impairments[END_REF] for 6G systems, are and will be the main solutions for achieving these requirements. In 5G NR, frequency bands up to 52.6 GHz are supported, and extension to approximately 100 GHz is examined for future releases. In addition, the U.S. Federal Communication Commission (FCC) recommends that frequencies higher than 5G allocated frequencies, such as 95 GHz to 3 T Hz, be considered for 6G [194]. In such high frequency bands from the upper part of the millimeter wave band to the Tera Hertz band, a remarkably wide frequency bandwidth can be utilized and is under investigation 13 to achieve extreme high data rates exceeding 100 Gbps [4].

The use of higher frequency bands, as those shown by Fig. 7.5, is quite regulated and is mostly available, allowing for mobile communication systems to operate on large bandwidths. However, the propagation characteristics on this elevated frequency ranges poses many engineering challenges. Indeed, according to Friis' Law, the isotropic path loss in free-space propagation is inversely proportional to the wavelength squared and implies that path loss at higher frequency ranges is more severe than in lower ones. Nevertheless, the antenna directional gain is also inversely proportional to the wavelength squared, and, thus, employing an array of highly directional antennas more than compensates for the free-space path loss [START_REF] Heath | An overview of signal processing techniques for millimeter wave mimo systems[END_REF]. In 5G related studies, such technology has been referred to as mmWave massive MIMO [START_REF] Larsson | Massive mimo for next generation wireless systems[END_REF].

Massive MIMO and high frequency bands are two attractive ideas considered to full the high capacity demands of 5G and 6G communication systems. Although these technologies allow huge capacity gains, the RF transceivers employed in these frequency ranges induce several hardware impairments as noise 7.2. Mitigation of RF impairments in higher frequency massive MIMO systems 147 Figure 7.5: Sub-THz 14 and THz spectrum [5]. quantization (due to low resolution DAC/ADC converters [START_REF] Zhang | On Low-resolution ADCs in practical 5G millimeterwave massive MIMO systems[END_REF]), nonlinear PA [START_REF] Zhao | A 60-GHz dual-mode Class AB power amplier in 40-nm CMOS[END_REF], phase noise [START_REF] Bicais | Phase noise model selection for Sub-THz Communications[END_REF] and IQ imbalance [START_REF] Fan | Joint IQ imbalance and PA nonlinearity pre-distortion for highly integrated millimeter-wave transmitters[END_REF]. All these RF impairments lead to a strong nonlinear distortion in the transmitted signal. From these technological challenges, and as a mid term research activity, I plan to work on the following topics:

• Impact of the memory eects of power ampliers: Given the large bandwidths involved in mmWaves, sub-THz and THz communications (500 M Hz [START_REF] Yu | Digital predistortion of Ultra-Broadband mmWave power ampliers with limited Tx/Feedback loop/baseband bandwidth[END_REF] in mmWaves bands and up to several GHz for sub-THz and THz links), the memory eects of power ampliers can no more be neglected [START_REF] Hammi | Metrics and methods for benchmarking of RF transmitter behavioral models with application to the development of a hybrid memory polynomial model[END_REF]. These eects must be considered when studying the In-band and Out-O-Band distortions. For spectral regrowth analysis, Zhou et. al proposed in [START_REF] Zhou | Spectral analysis of polynomial nonlinearity with applications to RF power ampliers[END_REF] theoretical analysis of the spectral regrowth induced by PA models which are based on polynomial models with memory. However, to the best of my knowledge, few contributions are made for theoretically characterizing the In-band distortions in presence of memory eects and over frequency selective channels. I aim to investigate this axis, based on some existing studies generalizing the application of Bussgang's theorem to power ampliers exhibiting memory eects [START_REF] Ermolova | Theoretical characterization of memory polynomial models with gaussian inputs[END_REF],

• Improvement of the energy eciency of transmitters/receivers: With the rise in frequency, the envisaged waveform wavelengths are more and more smaller, where we move from 10 cm for communications at 3 GHz, to respectively 10 mm and 5 mm, for communications around 30 GHz and 60 GHz. For these wavelengths, the validity of some approaches proposed to improve the energy eciency of power ampliers such as LINC method or IDP becomes questionable. Indeed. for IDP method, that we have developed as part of the Khaled Tahkoubit's PhD [START_REF] Tahkoubit | Iterative dichotomy PAPR reduction method for multicarrier waveforms[END_REF], it should be interesting to investigate the problem related to the combination of amplied millimeter wave signals. On the other hand, I plan to study the interest of using IDP method in a massive MIMO context, where signicant improvement is expected in terms of energy eciency compared to classical or LINC based schemes.

7.3 Waveform study and design for 6G communications CP-OFDM has been selected as the reference waveform in the 4 th generation of wireless communication systems. This MWF has been recently conrmed, in the 15 th 3GPP release, to be the waveform for 5G NR. As explained in chapter 6, the main novelty in 5G NR is the association of CP-OFDM to dierent numerologies to addresses several users' requirements/scenarios in terms of data rates, latencies and mobilities. Indeed, 5G will address three dierent services named as: (1) eMBB, with data rates reaching 100 M bps, (2) URLLC, with End-to-End latency lower than 1 ms and (3) mMTC, also named as Internet of Things (IoT). The increased heterogeneity of applications envisaged in beyond 5G generations, bring up to date the need to study and design advanced MWFs compared to CP-OFDM.

The waveform for 6G wireless systems should be exible and compliant with any required bandwidth for both sub 6 GHz and higher frequency carriers. This MWF should cope with the time/frequency channels selectivities, be spectral ecient, exhibit reduced complexity and latency, be robust against time and frequency asynchronous access and present limited PAPR, enhancing so the energy eciency of the whole transceiver. Dierent waveforms have been proposed in the last decades to partially reach those requirements. In addition to those presented in chapter 3, we can cite faster than Nyquist (FTN) signaling [START_REF] Mazo | Faster-than-nyquist signaling[END_REF], Orthogonal Time Frequency Space (OTFS) [START_REF] Hadani | OTFS: A new generation of modulation addressing the challenges of 5G[END_REF] and Lagrange Vandermonde Division Multiplexing (LVDM) [START_REF] Tourki | Lagrange vandermonde division multiplexing[END_REF] to name a few. FTN, is a non-orthogonal waveform that was rstly proposed by Mazo in 1975. This MWF achieves better spectrum eciency by increasing the symbol transmission rate faster than Nyquist limit in time domain, so higher SE can be achieved compared with classical Nyquist signaling MWFs. Regarding OTFS modulation, it has been introduced to cope with high mobility scenarios. When it turns to LVDM, this modulation generalizes ZP-OFDM while guaranteeing a perfect recovery of the transmitted complex data symbols. I plan to work on 6G MWFs, through the following directions:

Design of new waveforms. Firstly, I will continue working on the design of new waveforms satisfying exibility (bandwidth, MIMO compatibility), high users densities, asynchronous communications, low complexity/low latency and reduced PAPR. The proposed MWFs will be compared to CP-OFDM based on several key performance indicator (KPI). The considered KPIs that will be considered are: the spectrum eciency, the PAPR, the MIMO compatibility, the robustness to asynchronism and the overall complexity. Along with this activity, I will consider precoded ltered multicarrier schemes obtained from FFT-FBMC and BF-OFDM to reduce the PAPR while keeping the good time-frequency localization, Study of a FTN signaling approach. Secondly, in regard to the radio access technology almost reaching the Nyquist limit in the OFDM-based technology, FTN signaling, which compresses and transmits signals non-orthogonally using a sampling rate faster than the Nyquist limit, is more and more studied. In [START_REF] Lucciardi | Trade-o between spectral eciency increase and PAPR reduction when using FTN signaling: Impact of non linearities[END_REF], it has been shown that FTN based MWF may provide benets in terms of PAPR reduction. As an envisaged research direction, I will investigate new PAPR reduction techniques for FTN MWF, Optimization of existing waveforms. Thirdly, I will focus on the optimization of existing waveforms to enhance their capabilities based on the KPI dened previously. This optimization will also benet from major advances in the eld of articial intelligence. I plan to apply advanced deep learning techniques, as well as new approaches such as reinforcement learning to optimize existing waveforms in terms of latency, PAPR... Within this task, I will also concentrate on the so-called LVDM [START_REF] Tourki | Lagrange vandermonde division multiplexing[END_REF] to improve its robustness in asynchronous communication scenarios. Another possible contribution could also the be proposal of advanced receivers (equalization/detection) for non-orthogonal waveforms such as FTN, FBMC-QAM and GFDM in the context of high/very high 6G mobility scenarios.

Deep learning for energy ecient future communication systems

Meeting all the challenging 6G requirements identied in section 7.1 requires a hyper-exible network with congurable radios. In most of the prospective studies for 6G [2,3,4], it is proposed that AI and machine learning (ML) will be used in concert with radio sensing and positioning to learn about the static and dynamic components of the radio environment. This will be used to predict link loss events at high frequencies, to proactively decide on optimal handover instances in dense city networks and to determine optimal radio resource allocations for base stations and users. An important question is whether AI could be used to design optimal air interfaces on the y for a given environment and set of specic requirements. This suggests AI inspired air interfaces. However, their true performance, in particular, power and energy eciency in real use cases is an open research problem. Indeed, existing communication systems exhibit inherent limitations in translating theory to practice when handling the complexity of optimization for emerging wireless applications with high degrees of freedom. Deep learning has a strong potential to overcome this challenge via data-driven solutions and it can improve the performance of wireless systems. As a third research perspective, I will investigate deep learning approaches to enhance the energy eciency of future communication systems. The envisaged research topics are presented in the following:

• AI-based pre-encoder integrating pre-distortion and PAPR reduction: As part of Xinying Cheng's thesis, we proposed algorithms for OFDM massive MIMO context, to achieve joint multiuser pre-encoding with PAPR reduction and digital pre-distortion. These purely algorithmic solutions can present signicant implementation complexities. As a short-term perspective, we have started working on solutions based on deep learning and Meta learning approaches, to reduce the complexity of the joint precoding stage integrating Multi-User Interference (MUI) cancellation along with RF impairments mitigation. An interesting idea to reach this objective is to design a meta learning neural network [START_REF] Vanschoren | Meta-learning : A survey[END_REF], which allows a reduced complexity and complements standard machine learning approaches. The goal is to create a meta-learning model that can be extended to a new conguration that was never learnt during the learning process. We propose to employ two neural networks. The rst one, the NN Precoder, creates a precoded data vector, for a certain channel conguration, to compensate the PA nonlinearities or other hardware imperfection. The weights of the NN-Precoder associated to a channel conguration are generated by the second NN, the Meta-NN, which may be extended to any channel conguration. By doing so, we eliminate the NN Precoder's adaption and the associated computing complexity when the channel changes.

• Communication chain based on auto-encoders: Recent studies have shown the possibility of interpreting a communication system as an auto-encoder [START_REF] Shea | An introduction to deep learning for the physical layer[END_REF]. This approach makes it possible to develop a new way of thinking the design of the communication system as an end-to-end reconstruction task which seeks to jointly optimize the various functions of the transmitter and the receiver in a single process. An auto-encoder can therefore be seen as a neural network used for encoding and decoding a series of data. The simplest form of an auto-encoder is a multi-layered perceptron-based neural network. As shown in Fig. 7.6, the architecture is composed by an input layer, several hidden layers, and an output layer, which has the same number of nodes as the input layer. The aim is to reconstruct, at the output, the series of data presented at the input. Beyond the work published in the literature, few contributions have been made for energy eciency purposes [START_REF] Vahdat | PAPR reduction scheme for deep learning-based communication systems using autoencoders[END_REF]. We started investigating these aspects as part of Meryem Benosamn's PhD and also as part of a master internship, which I am currently supervising. As a rst result of this internship, we proposed an improvement of the classical autoencoder oering Gray coding capabilities and showing better BER performance than the classical M-QAM conguration. The proposed autoencoder architecture is given by Fig. 7.6. This architecture is a modied version of the autoencoder model proposed in [START_REF] Shea | An introduction to deep learning for the physical layer[END_REF] by adding pre and post-processing layers in order to map constellation points on a Gray like map, while modifying the learning loss from categorical cross-entropy to binary cross-entropy. I intend to continue this activity in the short and mid term by taking into account the PA induced RF impairments. Beamforming architectures for mmWaves in massive MIMO settings are a key technology for 5G [START_REF] Ribeiro | Energy eciency of mmwave massive mimo precoding with low-resolution dacs[END_REF].

On the other hand, recongurable intelligent surfaces (or RIS for Recongurable Intelligent Surface) using reect and transmit arrays are envisioned as low energy consumption architectures to meet future high speed communication systems [START_REF] Di Renzo | Smart radio environments empowered by recongurable intelligent surfaces: How it works, state of research, and the road ahead[END_REF]. This technology will allow to concentrate energy in certain directions of space, to achieve very high data rates with better energy eciency. Compared to conventional planar antenna arrays, the radiation pattern is adjusted by activating the meta-surface diode grid, as depicted by Fig. 7.7.

Driven by a revolution in electromagnetically tunable surfaces (e.g., based on metamaterials), 6G technology plan to control signal reections and refractions by using RIS. Open research problems range from the optimized deployment of passive reectors and metamaterial-coated smart surfaces to AI-powered operation of recongurable RIS 15 . As a rst step of studies, fundamental analysis to understand the performance of RIS and smart surfaces, in terms of the rate, latency, reliability, and coverage is required. Another important research direction is environmental AI whereby smart surfaces learn and autonomously recongure their material parameters. Challenges include how to focus signals with different angles of incidence in large metamaterial surfaces, requiring controllability of reection/refraction coecients. ML-driven smart surfaces in mobile environments may require continuous retraining, in which the access to sucient training data, high computational capabilities, and guaranteed low training convergence are needed. Holographic radio could be made possible with 6G by using RIS and similar structures [3]. Holographic RF allows the control of the entire physical space and the full closed loop of the electromagnetic eld through spatial spectral holography and spatial wave eld synthesis.

In 2020, CNAM, Orange, CEA-Leti, IEMN and INSA-IETR have won a national ANR research project which started in February 2021. This project is entitled MESANGES for Metasurface-based Electronically Steerable Antenna arrays for Next Generation EHF Systems. This project will investigate and implement two main types of metasurfaces based systems: reectarrays [START_REF] Dahri | A Review of high gain and high eciency reectarrays for 5G communications[END_REF] and transmitarrays [START_REF] Clemente | Electronicallysteerable transmitarray antennas for Ka-Band[END_REF]. Reect and transmitarray antennas, thanks to the spatial feeding mechanism and the possibility to integrate tunable device to control the aperture phase distribution, are an extremely attractive technology for the future high-performance and smart communication networks operating up to sub-THz frequencies. As stated previously, reect-or a transmitarray is composed of a focal system illuminating a planar or conformal array of elements also called unit-cells. PIN diodes or other technologies of RF switches are integrated on the unit-cells to implement beamforming functions. Both structures are used generally to deviate and to focus beams on desired directions or generate shaped patterns and multi beams. The main dierence between the two solutions is that, in the case of reectarrays, the electromagnetic wave does not pass through the surface as for transmitarrays, but it is reected onto it. In both cases challenges will lie in the design of the surface's elements for mmWaves and sub-THz bands, then on the control of the numerous PIN diodes connected. Optimization algorithms are thus necessary to eciently create beams towards the desired direction.

As part of this ANR project, I will focus on the design of hybrid beamforming schemes applied to both reectarrays and transmitarrays. The main dierence with respect to classical massive MIMO is that only one or few sources are active and that the radiation pattern diagram for both reectarray and transmitarray is adjusted by the activation of the grid of diodes of the metasurface [START_REF] Sawyer | Review of numerical optimization techniques for meta-device design[END_REF]. Another open issue is the beam management and the limited exchange of information between the receivers and the source. I envision the study of dierent approaches for beam management. A rst approach could be to perform two or more phases (large beams and then beam renement) [START_REF] Giordani | A Tutorial on beam management for 3GPP NR at mmWave frequencies[END_REF]. Another solution is to exploit the time/space sparsity of the channel in mmWaves and see the beam management problem as a sparse signal recovery problem. The problem can be solved using compressed sensing based algorithms where only the strongest channel coecients are fed back to the transmitter in order to compute a precoding matrix and select the grid of diodes of the metasurface.

Conclusion

This chapter concludes my "Habilitation à diriger les Recherches" (HDR) thesis. In this chapter, I have presented the main research perspectives in the short, medium, and long term. These perspectives are centered around four topics, the rst two being a continuation of the activities carried out over the last years. These topics will focus on the analysis/synthesis of waveforms for 6G systems, as well as the proposal of solutions to improve the energy eciency of transmitters/receivers in mmWaves, sub-THz and THz bands in the presence of power ampliers and/or other dirty RF phenomenon. As a third topic of my future research, I will investigate techniques based on machine learning to improve the energy eciency of transceivers when interpreting the communication system as an auto-encoder. Least but not last, I plan to develop, over the next four-ve years, a new research direction on Recongurable Intelligent Surfaces (RIS). Specically, I will focus on the beamforming algorithms for the steering of the transmit and reect arrays for both single and multi-user scenarios. We are currently in the process of recruiting a PhD student who will work on this topic.

Multi-carrier waveforms for 5G and beyond systems: from PA impairments to energy eciency enhancement Abstract: This report draws my professional career by focusing on the activities carried since 2011. The rst part is an extended Curriculum Vitae presenting various information about my research activities, namely the supervision, publications, collaborations as well as the research projects in which I have been involved.

In the second part of this report, which is structured in six chapters, I presented the research activities carried as associate professor at CNAM. During this period I have been working on post-Orthogonal Frequency Division Multiplexing (OFDM) Multi-Carrier Waveforms (MWFs) for 5G and beyond systems, with important eorts to characterize and mitigate power amplier (PA) distortions in Single Input Single Output (SISO) and massive Multiple Input Multiple Output (MIMO) scenarios. For each research activity, I have presented the projects under which the work has been carried, along with the dierent collaborations carried out, the main achievements and related publications.

As a rst research direction, I worked, within the framework of the ANR WONG5 national research project, on the study and proposal of the most suitable post-OFDM waveforms for critical machine type communications. This use case is one of the scenarios envisioned in future 5G communications and requires extra reduced latency, below 1 ms, as well as relaxed constraint on users synchronization. As this later constraint can diculty be fullled with conventional 4G/5G CP-OFDM waveform, I contributed to the study and proposal of alternative solutions based on windowing/lterinfg the transmitted/received signals. In parallel to this study, I focused on advanced equalization structures for FBMC-OQAM over frequency selective channels.

The time domain signals from the vast majority of the MWFs candidate for 5G exhibit signicant uctuations in instantaneous power compared to the average power, characterized by the Peak to Average Power ratio (PAPR). This characteristic makes these MWFs very sensitive to the non-linearities induced by the Radio Frequency (RF) power ampliers. As a second research topic, I focused on the characterization of the In-Band (IB) and Out-Of-Band (OOB) distortions induced by memoryless measured or modeled PA. The carried studies made it possible to characterize the transmission errors as well as the induced spectral regrowth. This information is essential for engineers working on RF chain specications for 5G and beyond systems.

The high PAPR of MWFs strongly compromises the energy eciency of 4G, 5G and beyond systems. As a third research topic, I proposed solutions to improve the energy eciency of several post-OFDM MWFs based transceivers. The rst family of the proposed solutions aims to reduce the PAPR and/or pre-distort the signal before amplication in a dissociated or joint manner. The second approach that we have proposed, to improve the energy eciency of post-OFDM MWFs, modies the architecture of the PA and splits the multi-carrier signal into several sub-signals with lower PAPRs which can be amplied with better energy eciency.

Among the prominent technologies proposed to reach the 5G New Radio (NR) requirements, massive MIMO and mixed numerology have been retained to support various congurations on top of OFDM waveform. In this direction, we proposed a rst contribution related to the analysis and cancellation of the interference caused by users using dierent numerologies in massive MIMO OFDM systems. Another contribution that we made, in this eld, is the proposal of low complexity energy ecient emitters architectures combining MIMO precoding, PAPR reduction and Digital Pre-Distortion (DPD).

As future works, I intend to focus on the design of new waveforms for 6G systems, as well as the proposal of solutions to improve the energy eciency of transceivers in mmWaves, sub-THz and THz bands. These solutions can be based on advanced learning techniques or on Recongurable Intelligent Surfaces (RIS).

Keywords

22 Chapter 1 .

 221 1 ref. Google Scholar, July 7 th 2021 2 ref. Google Scholar, June 15 th 2021 Curriculum Vitae • Participation to 3 selection committees for the recruitment of 1 associate professor and 2 research and teaching assistant positions (ATER).

1. 5

 5 Contribution to research and scientic projects 1.5.1 General context

  Dr. Bruno Sens Chang: Associate Professor at the Federal University of Technology Paraná, Curitiba in Brazil is a former PhD student of the LAETITIA team. Dr. Sens Chang's thesis focused on the proposal of new precoding and equalization techniques for FBMC-OQAM system. I worked with Dr Chang, as part of Hayfa Fhima's PhD, to propose equalization structures suited to rectilinear or quasi-rectilinear signals such as OQAM signals. I invited Dr. Sens Chang for two months (February and March 2018), where we worked jointly on writing a paper to ICT'2018 conference. Since Jan. 2018

1. 9 . 2

 92 Teaching activities outside CNAM ENSTA-ParisTech: Since 2013, I have been involved, with Pr. Michel Terré, to taught a signal processing course for rst year engineering students at ENSTA-ParisTech. Each year, I give 22 hours of tutorials and labs on: sampling, introduction to random signal processing, Fourier transform, FIR/IIR ltering and linear prediction.ISEP: Since September 2017, I have been managing the "Signal acquisition and processing" course, intended for third-year engineering students in Wireless systems at ISEP. This course is taught in English for a volume equal to 45 hours/year. Topics covered in this course range from signal sampling to analysis of the correlation of stationary random signals, including Fourier transform and ltering. Particular emphasis is placed on the supervision and monitoring of students in tutored project mode.

Figure 2 . 1 :

 21 Figure 2.1: Three-way loudspeaker system.

  (a) f1 = 800 Hz.(b) f2 = 4000 Hz.

Figure 2 . 2 :

 22 Figure 2.2: Fields radiated by the two models of acoustic systems.

Figure 2 . 3 :

 23 Figure 2.3: Digitally controlled co-axial loudspeaker system.

  Without optimization. (b) With optimization.

Figure 2 . 4 :

 24 Figure 2.4: Directivity characteristics of the co-axial loudspeaker system after crossover lters optimization.

Figure 2 . 5 :

 25 Figure 2.5: Experimental protocol for RTF measurements using a subwoofer with pass-band [20 Hz, 200 Hz].

  RTF at position(2, 4). (b) RTF at position(3, 4).

  RTF at position(4, 4).

Figure 2 . 6 :

 26 Figure 2.6: Multi-point low frequency equalization of RTF based on automated design of notch lters (P = 11).

Figure 2 . 7 :

 27 Figure 2.7: La Sphère prototype exhibited in Radio France studio 103.

Figure 2 . 8 :

 28 Figure 2.8: Scenarios and applications considered in the context of 5G during the exploratory phase.

Figure 2 .

 2 Figure 2.10: In-band and Out-of-Band nonlinear PA eects analysis.

Figure 2 . 11 :

 211 Figure 2.11: Energy ecient transmitters for 5G MWFs.

Figure 2 . 12 :

 212 Figure 2.12: Massive MIMO -mixed numerology and enhanced energy eciency.

Figure 3 . 3 :

 33 Figure 3.3: Coexistence scenario: two asynchronous users with τ [s] TO, ε [kHz] CFO and free guardbands of δ [kHz].

Figure 3 . 4 :

 34 Figure 3.4: Interference users PSD comparison.

  Figure 3.5: End-to-End Physical layer latency ratio with respect to traditional CP-OFDM scheme for a user which used 3 RB.

Figure 3 . 6 :

 36 Figure 3.6: Per-subcarrier NMSE against TO, δ = 0 and 75kHz (73.125 kHz for FFT-FBMC and BF-OFDM.)

Figure 3 . 7 :

 37 Figure 3.7: Per-subcarrier NMSE against CFO, δ = 0 kHz.

Figure 3 . 8 :

 38 Figure 3.8: Performance overview.

Figure 3 . 9 :

 39 Figure 3.9: System model with two users.

Figure 3 . 10 :

 310 Figure 3.10: Equalization schemes.

3. 4 .

 4 Advanced equalization structures for FBMC-

Figure 3 . 11 :

 311 Figure 3.11: BER performance over a frequency selective channel.

Chapter 3 .

 3 MWF for 5G and beyond systems

Figure 3 . 12 :

 312 Figure 3.12: MSE performances for synchronous UI and IU.

Figure 3 . 13 :

 313 Figure 3.13: MSE performances for asynchronous UI and IU (τ = T 1 2).

Figure 4 . 1 :

 41 Figure 4.1: CCDF of the PAPR of some MWFs calculated over a block length of 1024 samples.

Figure 4 . 2 :

 42 Figure 4.2: PA nonlinear behavior.

Fig. 4 . 3 ,

 43 Fig. 4.3, shows a generic transmission scheme of MWFs in presence of a NL PA.

4. 3 . PA models and characteristics 87 Figure 4 . 3 :P 1dB 10 IBO 10

 387431010 Figure 4.3: Basic system model for MWF transceiver.

Figure 4 . 4 :

 44 Figure 4.4: Example of 4QAM constellation before and after PA.

Figure 4 . 5 :

 45 Figure 4.5: CP-OFDM and FBMC-OQAM performance comparison, 4QAM, 64 subcarriers, Saleh's PA model.

Fig. 4 .

 4 7 (a) presents the SER of the selected MWFs for the 3GPP PA model (PA1) with an IBO equal to 4dB and a 16QAM modulation. We recall that this PA model exhibits both AM/AM and AM/PM Chapter 4. PA distortions analysis Volts Output signal in degrees (a) AM/AM. (b) AM/PM (for PA1 only).

Figure 4 . 6 :

 46 Figure 4.6: Conversion characteristics of PA1 and PA2.

Figure 4 . 7 :

 47 Figure 4.7: SER performance of several MWFs, IBO=4dB.

Figure 4 . 8 :

 48 Figure 4.8: BER vs E b /N 0 for M-ary (QAM or OQAM) based OFDM, N F F T = 64, Saleh PA model.

  model 5th order polynomial model (a) Single user case. (b) Multi user case.

Figure 4 . 9 :

 49 Figure 4.9: Spectral regrowth prediction using Price theorem with polynomial models of order 3 and 5. In a second scenario, we study the impact of the polynomial model order on the estimation of the PSD re-growth in the case where three users are operating on closely spaced sub-bands. The three users have dierent bandwidths and are amplied with the same PA model, but at dierent IBO values • User 1: 120 subcarriers with IBO = 1dB, • User 2: 50 subcarriers with IBO = 2dB,

  (a) Single user case (IBO=3dB).(b) Multiuser case (IBO=1dB).

Figure 4 . 10 : 100 Chapter 4 .

 4101004 Figure 4.10: Spectral regrowth prediction using cumulants with dierent polynomial model orders.

5 . 1 )

 51 (b).

Figure 5 . 1 :

 51 Figure 5.1: Linearized PA and input signal with lowered PAPR.

5. 2

 2 .1.1 PAPR reduction techniques Dispersive SLM Similar to the classical SLM scheme, we proposed a Dispersive SLM (DSLM) algorithm to reduce the PAPR of OQAM based MWFs. With DSLM, each input symbol vector X m is phase rotated with U 5.2. Signal processing based solutions 103 dierent input vectors giving {X (u) m } U -1

m

  (t) = 0 from t = [0, (2m + 1) T 2 + 4T ), a (u m min ) m ,n are OQAM symbols corresponding to previously rotated symbols X (u m min ) m and γ m,n (t) is the shifted version of the prototype lter impulse response in time and frequency noted (as per equation (3.2)).

Figure 5 . 2 :

 52 Figure 5.2: Power prole of two FBMC-OQAM symbols in the duration of [T, 4T ] w.r.t the current symbol.

Figure 5 . 3 :

 53 Figure 5.3: CCDF of PAPR for FBMC-OQAM symbols with DSLM and TSLM and OFDM with SLM.

Figure 5 . 4 :

 54 Figure 5.4: Illustration of the trellis diagram between M stages composed of U states.

Figure 5 . 5 :

 55 Figure 5.5: CCDF of PAPR plot for FBMC-OQAM symbols with PAPR calculated over [T, 3T ) with size of PRT set, R = {4, 8, 16}.

  m+1 respectively is represented in the trellis by the path ζ(u,v) (m⇒m+1)between u th state in m th stage and v th state in (m + 1) th stage. For a given arriving state v at stage m + 1, the survivor state in stage m is the one giving the lowest PAPR. over T 0 ∈ [mT + T, mT + 4.5T ].

  t), we compute partial PAPR over a duration T 0 as per equation (5.2), where T 0 ∈ [mT + T, mT + 4.5T ] interval since almost all of the energy of the FBMC-OQAM symbol x (u,v) m,m+1 (t), lies within that interval as shown in Fig. 5.2. For the path ζ (u,v) (m⇒m+1) , we calculate its path metric Γ (u,v) (m,m+1)

Figure 5 . 6 :

 56 Figure 5.6: Detailed illustration of OREC-DPD block.

Figure 5 . 7 :

 57 Figure 5.7: Illustration of IBO on OREC-DPD scheme with p = 2.25.

110 Chapter 5 .

 1105 Enhanced PA eciency for 5G and beyond MWFs oset from nominal Rx freq. blocking level reception level 50 kHz to 100 kHz -40 dBm -81.4 dBc 100 kHz to 200 kHz -35 dBm -86.4 dBc 200 kHz to 500 kHz -30 dBm -91.4 dBc > 500 kHz -25 dBm -96.4 dBc Table

Figure 5 . 8 :Figure 5 . 9 :

 5859 Figure 5.8: LTE broadband in coexistence with narrowband PMR systems.

  SLM with U = 16.

2 log 2

 22 (N F F T ) complex multiplications. The convolution with the baseband real lter (of length L F IR ) adds M L F IR 2 multiplications (neglecting the rise and fall time of the convolution). Finally the up-conversion to the carrier frequencies counts for (N F F T + L F IR -1) multiplications.

112 Chapter 5 .

 1125 Enhanced PA eciency for 5G and beyond MWFs

  (a) CP-OFDM and WOLA-OFDM. (b) f-OFDM. Figure 5.11: Global eciency integrating PAPR reduction.

Figure 5 . 13 :

 513 Performance of an OFDM system with R = 32, 16, 16QAM , OBO = 3dB, Saleh PA model.

Figure 5 . 14 :

 514 Figure 5.14: Overall architecture of the testbed.

  3.1. The User of Interest (UI) occupies 7 RBs, about 1.1 M Hz bandwidth from 2.0020 to 2.0031 GHz. As shown by Fig. 3.3, on each side of the Interfering User (IU), there are 7 RBs, occupying 1.1 M Hz bandwidth as IU. A guard-band δ is separating the frequency bands of both UI and IU. Also, an oset τ is introduced to create timing asynchronism. For each MWF (CP-OFDM, WOLA-OFDM and BF-OFDM), PAPR reduction is based on SLM or TR algorithms. Whereas, DPD is based on the memoryless polynomial model. Figs. 5.15shows measured PSD performance comparison of all MWFs when SLM and TR are, respectively, performed with DPD. We can clearly see the signicant gain performed with WOLA-OFDM and BF-OFDM compared to CP-OFDM especially for IBO of 6 dB. We can also note that BF-OFDM outperforms WOLA-OFDM due to the better spectrum containment provided by the sub-band ltering. For an IBO of 6 dB, the CP-OFDM reaches the PSD in the linear case. Nevertheless, because of the rectangular shaping, the PSD localization is poor compared to WOLA-OFDM and BF-OFDM.

5. 3 . 117 Figure 5 . 16 :

 3117516 Figure 5.16: BER performance of dierent MWFs using SLM and DPD, 16QAM.

Figure 5 .

 5 Figure 5.17: NMSE performance of dierent MWFs when τ = 106 µs and δ = 4, 883 kHz.

Figure 5 . 18 :

 518 Figure 5.18: Output power and PAE vs. input power of the used PA [1].

Figure 5 . 19 :

 519 Figure 5.19: OFDM transmitter in presence of PA: (a)-Classical case, (b)-IDP based OFDM.

  ) where v m (n) is a signal with constant envelope and ε M (n) is an error signal. R m , m = 1..M are real positive scalar compute by algorithm 1. Each sub-signals x m (n) is the sum of a constant envelope signal v m (n) with an additional high PAPR signal. To control the PAPR of the M sub-signals, only a certain percentage of ε M (n) is added to v m (n).

5. 3 .

 3 Modied architecture transmitter-based approaches 119 Algorithm 1 IDP Algorithm 1: Initialization : ε 0

Figure 5 . 20 :

 520 Figure 5.20: CCDF of the PAPR for OFDM and IDP schemes, 16 -QAM , N F F T = 256 and L = 4.

Figure 5 . 21 :

 521 Figure 5.21: PSD before and after PA (IBO = 3 dB), for OFDM system and IDP based system for M = 2, 3, 4.

120 Chapter 5 . 2 Figure 5 . 22 :

 12052522 Figure 5.22: Output powers for PAs used by IDP-based and classical OFDM based schemes.

Fig. 5 .

 5 [START_REF] Andrae | Total consumer power consumption forecast[END_REF] shows the transmitters corresponding to the classical OFDM scheme (Fig. 5.19 (a)) and the IDP based one (Fig 5.19 (b)).

( 1 ) 1 )

 11 mt is the modulated OFDM symbol on m th t transmitting antenna for user 1 and x (2,n) mt is user 2's n th modulated OFDM symbol on the m th t transmitting antenna (n = 1, ...N ). x (mt has a total symbol length of N times the symbol length of x (2,n) mt , and all of the symbols are aligned x mt (2) = [x mt (2, 1), x mt (2, 2), ...x mt (2, N )].

Figure 6 . 1 :

 61 Figure 6.1: System model of the massive MIMO OFDM DL system.

  s 1 [n 1 ] represents the original symbol on the n 1 -th subcarrier with numerology 1 and s 2,n [n 2 ] is the original symbol of the 6.2. Analysis and cancellation of mixed-mumerologies interference for massive MIMO-OFDM 125 (a) From user 1 to user 2. (b) From user 2 to user 1.

Figure 6 . 2 :

 62 Figure 6.2: FFT windows and INI on the receiver side.

=

  t ] . . . u (1) N1 [m t ] ∈ C 1×N1 is the reordered version of precoded signals as shown by Fig. 6I M r , the ZF precoder given by P

  . In this case, we can write hf

  [n 1 ] = hf

  [n 2 ]. For other values of n 1 (n 1 = (n 2 -1)N + 1), the dierence between hf

  [n 1 ] and hf

6. 2 . 127 Figure 6 . 3 :

 212763 Figure 6.3: System model of the massive MIMO OFDM UL with two dierent numerologies.

N 1 =

 1 1024, N 2 = 256, CP 1 = 72, CP 2 = 18. The time-domain channel responses ht mr,mt [d] = c d have circularly symmetric Gaussian distributed values of zero mean and unit variance, where d = 1, ..., D. Four wireless channels are represented by a tap delay line with D = 1, 2, 8, 18 taps. It is worth noticing that the CP duration is always adequate to suppress the ISI and ICI induced by the channel.

6 . 2 .

 62 .16) [177] for the LTE path-loss model. Pathloss(dB) = α = 128.1 + 37.6log 10 (d k ) (6.16) Analysis and cancellation of mixed-mumerologies interference for massive MIMOα 1 /α 2 = -26 dB w/o INI cancellation with INI cancellation (a) DL with α 1 /α 2 = 0, 20, 23 and 26 dB (b) UL with α 1 /α 2 = 0, -20 and -26 dB..

Figure 6 . 5 :

 65 Figure 6.5: SER performance with and without INI cancellation on user 1.

Figure 6 . 6 :Figure 6 . 7 :

 6667 Figure 6.6: User distribution in a ring where BS is in the center.

Figure 6 . 8 :

 68 Figure 6.8: Massive MIMO-OFDM schemes.

6. 3 .

 3 Enhanced energy eciency of Massive-MIMO systems 135 OFDM subcarrier can be expressed as r n = H n y n + b n , (6.19) where H n ∈ C Mr×Mt is the MIMO channel matrix, y n is the transmitted signal and b n is the receiver noise whose entries are i.i.d circularly-symmetric complex Gaussian distribution with zero-mean and σ 2 b variance. 6.3.3 Existing schemes Schemes (Ref .1) & (Ref. 2) As shown in Fig. 6.8, there are two reference schemes. (Ref. 1) is the classical ideal massive MIMO-OFDM scheme with ZF precoding and without any PA nonlinearities, which represents the best quality of transmission. On the contrary, (Ref. 2) represents the worst scenario among all the schemes, with only DPD processing before PAs in each branch.

  )where a mt (k) = |a mt (k)| e jφ(k) and φ(k) is the phase of a mt (k). The original frequency-domain clipping noise associated to the m t -th transmit antenna is d t mt = F F T (a t mt -a t mt ). Then, the eective transmission signal xn at the n-th subcarrier in the frequency domain is represented asxn = x n + d n , n ∈ χ. (6.31)Compared with scheme (3), here, d n stands for the clipping noise instead of PA distortion noise, which can be obtained with more simplicity. To ensure desirable spectral properties of the transmitted OFDM signals, the inactivated OFDM subcarrier (indexed by χ c ) must satisfy the following shaping constraints:xn = 0 Mt×1 , n ∈ χ c . (6.32)Consequently, the total MUI energy can be evaluated by H n xns n 2 2 .

mt 7 :

 7 scheme scheme 1 scheme 2 scheme 3 scheme 4 ZF computation

4 Figure 6 . 9 :Figure 6 . 10 :Figure 6 . 11 :Figure 6 . 12 :

 469610611612 scheme 1 scheme 2 scheme 3 scheme 4

Figure 7 . 1 :

 71 Figure 7.1: Overview on 6G requirements [2, 3, 4, 5].

Fig. 7 .

 7 4 summarizes the research topics that I intend to develop in the short, medium, and long term.

Figure 7 . 2 :

 72 Figure 7.2: Expected 6G applications [2, 3, 4, 5].

Figure 7 . 3 :

 73 Figure 7.3: Some 6G enabling technologies.

Figure 7 . 4 :

 74 Figure 7.4: Summary of future research directions.

Figure 7 . 6 :

 76 Figure 7.6: A communications system over an AWGN channel represented as an autoencoder.

7. 5

 5 Beamforming techniques for recongurable intelligent surfaces (RIS)

Figure 7 . 7 :

 77 Figure 7.7: Typical RIS based communication scenario.

  1.7.6 White papers [W1] M. Renfors, L. G. Baltar, N. Bartzoudis, D. Gregoratti, E. Kodis, L. Martinod, A. Myrvoll, S. Nedic, D. Petrov, D. Raki, V. Ringset, and H. Shaiek, "Flexible and Spectrally Localized Waveform Processing for Next Generation Wireless Communications". White paper of the 3 rd work package of FP7 EMPHATIC project, April 2015.

	1.7.7 PhD dissertations
	[PHD1] H. Shaiek. "Optimizing wide band co-axial loudspeaker systems using digital signal processing
	techniques". TELECOM Bretagne, Brest, France, July 2007.
	1.8 Scientic outreach
	1.8.1 Collaborations

Since 2012, I participated to several research projects, which have enabled me to diversify my scientic collaborations with French and international partners. Since Sept. 2011 Innov'Com laboratory, SUP'COM, Tunisia.

  s PhD. The defense of this thesis is scheduled in November 2021.

	1.8.2 Distinctions
	Since	Doctoral supervision and research grant: Prime d'Encadrement Doctoral et de
	Sept. 2017	Recherche (PEDR), since September 2017.
	Since	Mahindra-Ecole Centrale, Haydarabad, India.
	Sept. 2018

Dr. Krishna Bulusu: my second PhD student, joined Mahindra-Ecole Centrale engineering school in Hayderabad, India in September 2017, as an associate professor. He contacted us in January 2018, with a view to submitting a Franco-Indian partnership project, within the framework of the CEFIPRA program. The collaboration with Dr. Krishna Bulusu continued with further work on fast convergence PA linearization algorithms for FBMC-OQAM signals.

  Journals: IEEE Transactions on Signal Processing, IEEE Transactions on Vehicular Technology, IEEE Wireless Communication Letters, IEEE Communication Letters, IEEE Access, International Journal of Distributed Sensor Networks, Physical Communications.

	1.8.3 Reviewing activities
	• Conferences: IEEE VTC-Spring, IEEE ICC, IEEE ICCE and IEEE ISWCS, IEEE WCNC, IEEE
	PIMRC, ICT.
	• Technical Program Committee in conferences:
	IEEE IWCMC 2021,
	ICT 2018-2020,
	IEEE ICC 2017,
	IEEE ISWCS 2012, 2016-2021,
	IEEE VTC-Spring 2020.
	• Session chair:
	IEEE ISWCS 2014, Barcelona,
	IEEE ISWCS 2016, Poznan,
	1.8.4 Organization of scientic events

• • I am a research topic editor of "Waveform Design for Beyond 5G Networks" in Frontiers in Communications and Networks Journal. The call was announced in October 2020 and the submission deadline is scheduled for June 22 nd 2021. https://www.frontiersin.org/research-topics/ 16638/waveform-design-for-beyond-5g-networks,

Table 1 .

 1

	Topic		Language	Level	Classes	Tutorials	Projects
	Advanced	digital	FR	Master's	55h	48h	30h
	communications
	Advanced	tech-	FR	Master's	-	-	143h
	niques in analog
	and digital elec-
	tronics
	Analog signal pro-	FR	B.Sc.	120h	170h	77h
	cessing
	Digital signal pro-	FR+ENG	Master's	193h	135h	90h
	cessing
	Electronics		FR	B.Sc.	216h	333h	89h
	Introduction to sig-	FR	B.Sc.	81h	85h	95h
	nal processing
	Microcontrollers	FR+ENG	B.Sc.	186h	206h	222h

1.1 summarizes the teaching activities carried out at CNAM, since February 2011. My average teaching load is about 245 hours/year, corresponding to 33% of classes, 38% of tutorials and 29% of labs and projects. 1: Summary of my teaching activities at CNAM.

Table 1 .

 1 2: Information about the dierent MOOC sessions. each week discussions are open on a dedicated forum. The weekly feedbacks and the main questions asked on the forums are addressed during a live hangout session.Starting from an introduction to linear systems and ltering, we present time and frequency representation of signals with Fourier transform. Spectral analysis is introduced together with random signals. Next, we come to digital signal processing and its applications.This MOOC was oered in four sessions, between January 2015 and June 2018. Tab. 1.2 summarizes some information about this MOOC. Based on data extracted from FUN 6 platform, it appeared that 40% of the participants to this MOOC are master 1 or 2 students, 20% are preparing Bachelor degrees and 10% are PhD students.

	Session	Period	Participants
	2015	from January 17 th to February 27 th	3388
	2016	from February 15 th to March 27 th	4446
	2017	from May 2 nd to June 12 th	4256
	2018	from April 24 th to June 4 th	4004
	Total		16294

Table 3 .

 3 2: Transmultiplexer impulse response. the coecients Ψ m0,n0 are given in table 3.2. Consequently, a perfect reconstruction of the transmitted real symbols a m,n is obtained by taking the real part (OQAM decision) of the demodulated signal y m0,n0 .

Table 3 .

 3 .8, one can notice that some 3.4. Advanced equalization structures for FBMC-OQAM 4: Tx/Rx complexity normalized with respect to OFDM.

	77

Table 3 .

 3

5: Comparison criteria specication

Table 4 .

 4 1: Comparison between numerical and theoretical values of K and σ 2

d

.

Table 5 .

 5 2: Complexity analysis of SLM and TR for dierent MWFs.

Table 5 .

 5 3: Ratio between emitted power for the same DC consumption power and EVM.2. [Step 2:] when both schemes of 5.19 have the same EVM, we enforce a same consumed power P DC

	for OFDM PA [dB]	3	4	5	6	7	8	9	10
	M = 2	1.25 1.26 1.27 1.28 1.29 1.30 1.34 1.47
	M = 3	1.52 1.53 1.54 1.54 1.54 1.54 1.58 1.69
	M = 4	1.78 1.78 1.79 1.80 1.81 1.82 1.89	2

Table 6 .

 6

	6.2.4.1 Single user DL scenario
	User of Interest	user1 (N 1 = 1024)	user2	user2
				(N 2 = 512)	(N 2 = 256)
	Interfering User	user2(N 2 =	user2(N 2 =	user1(N 1 = 1024)
		512)	256)
	D = 1	-300	-300	-300	-300
	D = 2	-52	-48	-300	-300
	D = 8	-42	-37	-300	-300
	D = 18	-38	-33	-300	-300

1: NMSE(dB) of user of interest with interfering numerology and dierent channels for massive MIMO-OFDM DL.

Table 6 .

 6 2: NMSE(dB) of user of interest before and after INI cancellation for massive MIMO-OFDM UL.

	Interest		user 1 (N 1 = 1024)
	Interfering User	user 2(N 2 = 512)	user 2(N 2 = 256)
	α 1 /α 2 (dB)	original	corrected	original	corrected
		(dB)	(dB)	(dB)	(dB)
	20	-22	-300	-18	-300
	23	-19	-300	-14	-300
	26	-16	-300	-11	-300

Table 6 .

 6 3: NMSE(dB) of user of interest with interfering numerology and dierent channels for massive MIMO-OFDM UL.

	6.2.4.2 Single user UL case
	User of Interest	user1 (N 1 = 1024)	user2(N 2 =	user2(N 2 =
				512)	256)
	Interfering User	user2(N 2 =	user2(N 2 =	user1(N 1 = 1024)
		512)	256)
	D = 1	-300	-300	-300	-300
	D = 2	-56	-52	-52	-49
	D = 8	-47	-45	-44	-40
	D = 18	-43	-40	-39	-36

Table 6 .

 6 5: Complexity comparison of the massive MIMO-OFDM schemes.

	For scheme (2): MU-PP-GDm needs O 2M 2 t M 2 r + M r M 2 t + M 3 r for the null-space computation as initialization step and O 2|χ|M t M r + 2|χ|M 2 t + 2M t N log(N ) multiplications for each iteration.
	When	it	comes	to	scheme	(3),	the	complexity	of	MU-PNL-GDm	is
	O 3|χ|M t M r + 2|χ|M

Table 6 .

 6 Massive MIMO networks with mixed numerologies and nonlinear PAs the complexity of all schemes. The complexity of each scheme is computed assuming the same minimum MUI of -40 dB at the receiver side, i.e., including all the initialization stages, the iterations and the NN DPDs. To achieve a MUI of -40 dB, the ZF-PAPRnullspace, MU-PP-GDm, MU-PNL-GDm and the proposed MU-2P require 35, 32, 14 and 44 iterations, respectively. 6: Complexity comparison for each scheme with M t = 100, M r = 10, N = 512 and | χ |= 256.

	scheme	Complexity to
		achieve MUI=-40
		dB
	scheme 1	2.1349 × 10 8
	scheme 2	2.0133 × 10 8
	scheme 3	9.1375 × 10 7
	scheme 4	6.2565 × 10 7
	scheme 4/scheme 1	29.31 %
	scheme 4/scheme 2	29.5 %
	scheme 4/scheme 3	68.47 %

1

A MUI lower than -40 dB has been chosen and corresponds to a very low penalty concerning 16 and 64-QAM symbols 140 Chapter 6.

https://www.ict-emphatic.eu/Europeanproject

www.wong5.fr

https://anr.fr/Project-ANR-14-CE28-0026

https://www.fun-mooc.fr/fr/

Digital Signal Processor

also known as large-scale antenna systems, very large MIMO, hyper MIMO, full-dimension MIMO or ARGOS

http://www.ict-phydyas.org/

2.3.4.1 Signal processing-based solutionsThe classication of post-OFDM MWFs candidates for 5G NR, can be based on the way in which the subcarriers are ltered at the modulator level: ltering by sub-carrier or by group of sub-carriers. Depending on the spectral localization of the analysis lter, the length of the impulse response may spread out causing an overlap between the symbols. This phenomenon must be considered in the pre-processing algorithms such as those dedicated to the reduction of the PAPR or to the DPD.FBMC-OQAM caseBeing a MWF technique, FBMC-OQAM suers from high PAPR and its good frequency localization property is no longer valid in real communication systems and in presence of NL PAs. As a consequence, spectral regrowth may appear over the amplied signal, hampering much of its attractive appeal to be the ideal candidate for 5G NR. As part of the FP7 European project EMPHAtiC, the ANR project ACCENT5 and the PHC-C3 project, we have focused on improving the energy eciency of FBMC-OQAM transceivers by reducing the PAPR and designing dedicated DPD schemes.PAPR reduction techniques for FBMC-OQAM: Classical probabilistic schemes proposed for CP-OFDM systems cannot be directly applied to FBMC-OQAM. By taking overlapping signal structure of FBMC-OQAM into account, we have initially proposed a Dispersive SeLective Mapping (DSLM) scheme[START_REF] Bulusu | Reduction of PAPR for FBMC-OQAM systems using dispersive SLM technique[END_REF], which is essentially a symbol by-symbol approach. However, DSLM leaded to a sub-optimal performance w.r.t the classical SLM scheme applied to CP-OFDM as the peak reduction we may obtain for the current symbol, can possibly be destroyed by the future symbol. We have improved this technique

Note that in this work, we assumed the same transmit power per subcarrier for both useful and interfering users

Note that for CP-OFDM, N Block = N FFT

IBO is computed from the 1dB compression point.

http://www.brave-beyond5g.com/

https://5g-ppp.eu/rise-6g/

Acknowledgements

I have co-supervised two postdoctoral researchers and four master students.

2014-2017

ANR ACCENT5 Project 5 project (Advanced Waveforms, MAC Design and Dynamic Radio Resource project with THALES, CentraleSupélec-Rennes and Teamcast. I worked on workpackage 2 of this project for improving D2D Green aspects. contribution to this project was on the optimization of power amplier performance for FBMC-OQAM waveforms. The optimization of power amplier eciency was carried out by reducing the FBMC-OQAM PAPR and/or linearizing the power amplier characteristics. These studies were carried in the framework of Krishna's Bulusu PhD.

Projects submitted but not accepted for funding

Because PhD and Postdoctoral researchers are mostly funded by collaborative projects, since my arrival in CNAM, I had a great activity in proposing new academic and industrial projects. In addition to the above projects accepted and funded by national and international organizations, many proposals were prepared and submitted but not accepted. Among the collaborative submitted projects I d'Ondes AVancées• Funded by: ENIT-TELECOM Bretagne-Cabasse. ENIT-TELECOM Bretagne-Cabasse.

Chapter 6. Massive MIMO networks with mixed numerologies and nonlinear PAs INI from numerology 1 with N 1 to numerology 2 with N 2 = N 1 /N .

We assume a noise-free transmission channel and s (2) = 0 to simply focus on the INI caused by user 1. The BS uses P(2) for the detection of user 2's signal and the interference of each subcarrier in n-th symbol can be written as ini (1,2) n

G (1) (n 2 , k 2 )W (1) n (k 2 , n 1 )

mr,1 [n 1 ]s (1) [n 1 ], equals to 0 in constant channel (6.10) (6.12) where the matrix denition is the same as equation (6.6).

From equation (6.12), the INI obtained by user 2 is caused by all subcarriers from user 1.

INI from numerology 2 with N 2 to numerology 1 with

Similarly, we consider in this scenario s (1) = 0, and a noise-free transmission channel. The BS uses P(1) for the detection of user1's signal and the interference of each subcarrier can be written as ini (2,1)

mr,2 [k 1 ]s (2,n) [n 2 ]. equals to 0 in constant channel (6.11) (6.13) From (6.13), we can see that ini (2,1) [n 1 ] is caused by all subcarriers of user 2.

Impact of the channel response selectivity and N 1 /N 2 ratio.

Over at fading channels, dierent numerologies between any m t = 1...M t transmit antenna and any m r = 1...M r receive antenna, give

The INI on each subcarrier of user 1 and user 2 is expressed in equations (6.12) and (6.13), respectively. From equation (6.14), in a constant channel, elements in vector hf (1) mr,2 and vector hf (2) mr,2 are the same, which will leads to

With these substitutions and equations (6.10), (6.11), it is clear that ini (1,2) n = 0 and ini (2,1) = 0, for n = 1...N . As a rst conclusion, we can say that over frequency at channels, the proposed transceiver scheme for massive MIMO-OFDM based mixed numerologies SS UL transmissions does not produce any INI for both user 1 and user 2 (i.e. ini (1,2)

When the channel is frequency-selective, we have hf (1) mr,mt [(n 2 -1)N + 1] = hf (2) mr,mt

where

Equation (6.15) shows that there are only a few same-values within the two channel frequency responses for the same time-domain channel between user m t and receiving antenna m r . For other values (i.e. n 1 = (n 2 -1)N + 1), the dierence between hf (1) mr,mt [n 1 ] and hf (2) 1,mr [n 2 ] is closely related to the channel selectivity and to the N 1 /N 2 ratio. Greater selectivity and a larger ratio of N 1 /N 2 result in a larger dierence in the frequency responses of the two channels, inducing higher INI.

Remark 4: Due to the channel frequency response characteristics, in a constant channel, the proposed transceiver scheme for massive MIMO-OFDM based mixed numerologies SS UL transmissions does not produce any INI for both user 1 and user 2 (i.e. ini (1,2) n = 0, ini (2,1) = 0 ). The INIs for both user 1 and user 2 increases with channel selectivity and with N 1 /N 2 ratio.

Impact of user's power distribution.

Power allocation strategy has a similar impact as in the DL scenario. For interested reader please refer to [START_REF] Cheng | Analysis and Cancellation of Mixed-Numerologies Interference for Massive MIMO-OFDM UL[END_REF].

Remark 5: The INIs at user 1 and user 2 are directly aected by the power allocation for dierent users.

INI cancellation schemes

Based on the analytical expressions of INI developed in the previous sections, we will propose here after some INI cancellation schemes for both DL and UL scenarios. The main idea is to reduce the complexity of the single-antenna users by doing all the processing at the BS side.

DL INI cancellation

Only numerologies with large IFFT/FFT sizes suer from INI as compared to numerologies with smaller IFFT/FFT sizes, as seen in section 6.2.2.1. One of the key ideas in massive MIMO DL transmission is to keep all processes at the BS [START_REF] Larsson | Massive MIMO for next generation wireless systems[END_REF], which reduces receiver complexity while maintaining good transmission eciency. In addition, only the UL transmission estimates channel state in massive MIMO systems. The UL channel estimation is used for precoding in the DL transmission, but discrepancies between the UL and DL channels can be adjusted by a reciprocity adjustment method [START_REF] Vieira | Reciprocity calibration for massive MIMO: Proposal, modeling, and validation[END_REF].

On the BS side, Fig. 6.4 (a) indicates the proposed INI cancellation method's scheme. The key concept is to measure the INI from numerology 2 to numerology 1 using knowledge of the MIMO channel response and signals following user 2's precoding. Then, instead of transmitting s 1 , we transmit s 1 = s 1 -ini (2,1) . As we demonstrated in section 6.2.2.1, this adjustment does not introduce any INI at user 2, and the proposed transmission scheme perfectly protects the transmission for user 2. The transmit power in INI cancellation is p T x = p u + p ini , where p u is the power allocated to users and p ini is the power allocated to INI cancellation. It's worth noting that in equation (6.8), ρ is adjusted to fulll the constraint imposed by the total transmit power P T x when taking the p ini into account. According to simulations, the p ini is negligible compared to the power allocated to users (p u ), and it is worth noting that this would result in a minor SNR loss.

UL INI cancellation

As analyzed before, in section 6.2.2.2, INIs are created for each user in a frequency-selective channel. With the support of the INI analytical results presented previously, we implement an INI cancellation scheme by performing all processing at the BS side [START_REF] Larsson | Massive MIMO for next generation wireless systems[END_REF]. The full channel state is also calculated on the UL in massive MIMO, which can be used in ZF detection and INI cancellation. The proposed INI cancellation scheme for UL is shown in Fig. 6.4 (b), and it does not add any complexity to the users in this regard.

INI is estimated after the symbol estimation, then s(1) = ŝ(1) -ini (2,1) and s(2,n) = ŝ(2,n) -ini (1,2) n is applied in order to suppress the INI.

Furthermore, the proposed INI cancellation scheme can be extended to any method that employs more than two numerologies. The INI cancellation can be applied eectively for a given numerology by measuring and extracting the INIs from all other users that use dierent numerologies. It's worth noting, however, that the proposed INI cancellation approach is just as sensitive to imperfect CSI as ZF detections.

Chapter 6. Massive MIMO networks with mixed numerologies and nonlinear PAs Similar to the massive MIMO-OFDM DL system, we consider some more realistic scenarios with the LTE path-loss model given by equation (6.16) [177]. The impact of realistic path-loss over system performance is shown in Tab. 6.4, with D = 8 as the number of channel taps. As path-loss increases, users experience more INI. When N 1 = 1024, N 2 = 512 and D = 8, there is a 21 dB degradation on N M SE when α 1 /α 2 = 20 dB compared to α 1 /α 2 = 0 dB, and this degradation increases to 28 dB when α 1 /α 2 = 26 dB. These ndings match our study in section 6.2.2.2. It can be also noticed from this table that after the INI cancellation at the BS side, transmission of user 1 and user 2 reached about -70 dB for all path-loss values. Using N 1 = 1024, N 2 = 512, D = 8 and three separate pairs of locations, Fig. 6.5 (b) shows the SER vs. SNR of dierent users with/without INI cancellation. We can see that when the path-loss increases, user 1's performance suers from INI signicantly. As noticed in the DL scenario, the SNR loss is ∼ 3 dB for SER = 10 -2 , when we move from α 1 /α 2 = 0 dB to α 1 /α 2 = -26 dB.

Multi-user DL scenario

The simulations carried in the previous sections assume a simple cell including two devices, which are using two dierent numerologies. In a more realistic scenario, a cell includes many users that can be divided into several groups, each of them is using a given numerology. We will, consider a DL scenario as an example. Users are evenly distributed in the ring around the BS, with inner radius r 1 = 200 m and outer radius r 2 = 1000 m, as shown in Fig. 6.6 with M t = 100 and M r = 30. The thirty users are divided into three groups: Group 1 includes users 1 to 10, Group 2 includes users 11 to 20, and Group 3 includes users 21 to 30, using three dierent numerologies of IFFT/FFT sizes equal to 1024, 512, and 256, respectively. The work presented in this section have been published in two journal papers [START_REF] Bulusu | Prediction of spectral regrowth for FBMC-OQAM system using cumulants[END_REF] and [START_REF] Bulusu | Potency of trellis-based SLM over symbol-by-symbol approach in reducing PAPR for FBMC-OQAM signals[END_REF] for respectively DL and UL scenarios.

Enhanced energy eciency of Massive-MIMO systems 6.3.1 Motivation

As far as a large number of antennas is used at the BS side, massive MIMO precoders exhibit transmit signals with high PAPR, regardless of whether single-carrier or multi-carrier transmission and of whether Chapter 6. Massive MIMO networks with mixed numerologies and nonlinear PAs the MUI and reduce the PAPR using null-space MIMO channels and performing MU precoding. Then, the new optimization problem becomes minimize {ên, ẋn} (6.24) where

, n ∈ χ is the cost function of MUI elimination, while G (e n ) is given by equation (6.23) to obtain an optimal PAPR.

An alternating minimization strategy can be used to solve (6.24), in which we alternatively minimize the objective function with respect to x n and e n . In the (l + 1)-th iteration, the alternating procedure can be expressed as

Scheme (3) : Joint ZF precoding, PAPR reduction and DPD [START_REF] Zayani | Ecient precoding for massive MIMO downlink under PA nonlinearities[END_REF].

The above schemes ( 1) and ( 2) use clipping method for PAPR reduction and after that, a separate DPD for PA linearization. The key idea of scheme ( 3) is to jointly perform MU precoding and PA non-linearity mitigation, including PAPR reduction and DPD, using the PA model directly at each transmit antenna. By using the well-known Bussgang theorem [START_REF] Shaiek | Analytical analysis of SER for beyond 5G post-OFDM waveforms in presence of high-power ampliers[END_REF], one can decompose the nonlinear signal at the PA output into a linear function of the PA input and an uncorrelated distortion term. At each branch, the amplied signal y t mt = q mt x t mt + d t mt , where q mt is the m t -th PA complex gain and d t mt stands for the added zero-mean distortion noise with variance σ 2 d . Then, we can write the amplied signal in (6.19) as

where Q n = diag ([q 1 , q 2 , . . . , q Mt ]) is the M t × M t square diagonal matrix with elements {q mt } on the diagonal. Substituting (6.26) in (6.19), we have:

By minimizing the MSE of intended symbols and PA outputs through the channel, the optimization problem can be formulated as

subject to x n 2 2 ≤ P T x , (6.28) where P T x is the total transmitted power.

Then, scheme (3) aims to search, iteratively, for the symbols {x t mt } which can guarantee excellent transmission quality after the channel. The alternating procedure, in the (l + 1)-th iteration, can be expressed as

where P(.) denotes the amplication operation. The proposed algorithm is referred as MU-PNL-GDm for Multi-User (MU) precoding -Power Non-Linearities compensation (PNL) -using Gradient Descent with momentum (GDm) algorithm.

The three existing schemes introduced before proposed several solutions for PA non-linearity mitigation. However, they have high computational complexity. For scheme [START_REF] Chen | A 2.6 GHz Class-AB GaN Power Amplier with Maximum Output Power of 56 W Achieving 70% Power Added Eciency[END_REF], the high complexity comes from the matrix inversion of ZF precoding and the computation of the channel null-space. For scheme (2), although the ZF precoding matrix computation is removed, the computation of null-space remains high. What is more, in scheme (3), is that there is a PA model used in each iteration, which brings high complexity for the processing. The research activities I have carried during the last decade have mainly focused on topics related to improving the energy eciency of multicarrier based transceivers in presence of NL memoryless PAs. Also, I contributed to the proposal of schemes for massive MIMO transmissions including algorithms to mitigate the interferences induced by mixed numerologies, as well as the distortions introduced by RF PAs. Most of these studies considered 5G and beyond 5G systems and scenarios. Now that the rst releases of 3GPP on 5G have been published and that the deployment of this technology is progressing almost everywhere in the world, most of the researchers and operators eorts are focused on dening the issues, specications and scenarios for 6G in the 2030 era. Several consortia, collaborative projects and scientic events have emerged since 2018, the aim of which is to imagine and harmonize the societal and technical expectations of 6G. From these initiatives we can cite:

• Research projects: ICT-52 Hexa-X 4 , EC Smart Networks and Services partnership 5 , Japan 6G

• Forums, symposiums and conferences: IWON Global forum 10 , 6G Symposium 11 , 6G Wireless Summit 12 , In addition to these initiatives, there is a growing number of white papers and publications:

• White papers: Virginia Tech [2], 6G Flagship [3], NTT DOCOMO 6G [4]... • Publications: An example of the increasing number of IEEE 6G related publications is given in table 7.1.

From the preliminary studies and prospects on 6G [2,3,4], I summarized in Fig. 7.1 and Fig. 7.2, some expected requirements and applications for 6G wireless technologies. In addition to the higher