Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Soliman, Nourhan Hesham
Doctor of Philosophy - PhD
University of British Columbia
The emergence of delay-constrained applications imposes stringent latency and reliability requirements in 5G and beyond 5G networks, necessitating a shift in the performance analysis of networks from the infinite blocklength assumption to finite blocklength. Since the blocklength in the finite blocklength regime (FBR) imposes a trade-off between delay, reliability, and information coding rate, the performance analysis under infinitely long codes would be less applicable in practical scenarios. This dissertation presents a novel mathematical framework to characterize the performance of large-scale networks employing short codewords providing tractable expressions, conducts comparisons with the performance in the infinite blocklength regime characterizing the gap between the two regimes which motivates the importance of the proposed work, and investigates the effect of network and transmission parameters (e.g. base-station density, power allocation, power control parameters) on the performance. In the analysis, we consider orthogonal multiple access (OMA) and non-orthogonal multiple access (NOMA) networks. For the large-scale OMA uplink and downlink (DL) networks, theoretical achievable rates, outage probability, reliability, and the coding rate meta distribution are derived using finite blocklength coding theory and stochastic geometry. Moreover, achievable rates under practical modulation schemes and multilevel polar coded modulation (MLPCM) are investigated. Numerical results provide theoretical performance benchmarks and highlight the potential of MLPCM in achieving close to optimal FBR performance. For large-scale DL NOMA networks with a two-user cluster, the work proposes inner and outer bounds on the achievable rate region in the FBR, characterizes the min-max rate, and investigates the impact of parameters such as code-length and network density on their performance. The MLPCM is also investigated and the results validated its compliance for future technologies. Furthermore, we compared the performance of large-scale OMA and NOMA networks in the FBR, showing that NOMA outperforms OMA by a gap that increases in the FBR compared to the asymptotic regime, which motivates the use of NOMA for future networks. Additionally, the potential of Convolutional Neural Network autoencoders (CNN-AE) in approaching the theoretical maximum achievable rate over a Gaussian channel in the FBR is investigated. The CNN-AE is compared to the theoretical maximum achievable rate and the achievable rates of other schemes from the literature, outperforming these benchmark schemes and approaching the theoretical maximum rate. This demonstrates the capability of CNN-AEs in learning good codes for delay-constrained applications.
Performance Analysis of LargeScale Networks in the FiniteBlocklength RegimebyNourhan Hesham SolimanB.Sc. Hons., Cairo University, 2016M.Sc., Cairo University, 2019A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2024© Nourhan Hesham Soliman, 2024The following individuals certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, a dissertation entitled:Performance Analysis of Large Scale Networks in the Finite Block-length Regimesubmitted by Nourhan Hesham Soliman in partial fulfilment of the requirementsof the degree of Doctor of PhilosophyDr. Anas Chaaban, School of EngineeringSupervisorDr. Md. Jahangir Hossain, School of EngineeringCo-SupervisorDr. Hesham Elsawy, Queen’s UniversitySupervisory Committee MemberDr. Chen Feng, School of EngineeringSupervisory Committee MemberDr. Suliman Gargoum, School of EngineeringUniversity ExaminerDr. Matthew Valenti, West Virginia UniversityExternal ExamineriiAbstractThe emergence of delay-constrained applications imposes stringent latency andreliability requirements in 5G and beyond 5G networks, necessitating a shift in theperformance analysis of networks from the infinite blocklength assumption to finiteblocklength. Since the blocklength in the finite blocklength regime (FBR) imposesa trade-off between delay, reliability, and information coding rate, the performanceanalysis under infinitely long codes would be less applicable in practical scenarios.This dissertation presents a novel mathematical framework to characterize theperformance of large-scale networks employing short codewords providing tractableexpressions, conducts comparisons with the performance in the infinite blocklengthregime characterizing the gap between the two regimes which motivates the impor-tance of the proposed work, and investigates the effect of network and transmissionparameters (e.g. base-station density, power allocation, power control parameters)on the performance. In the analysis, we consider orthogonal multiple access (OMA)and non-orthogonal multiple access (NOMA) networks. For the large-scale OMAuplink and downlink (DL) networks, theoretical achievable rates, outage probability,reliability, and the coding rate meta distribution are derived using finite blocklengthcoding theory and stochastic geometry. Moreover, achievable rates under practicalmodulation schemes and multilevel polar coded modulation (MLPCM) are investi-gated. Numerical results provide theoretical performance benchmarks and highlightthe potential of MLPCM in achieving close to optimal FBR performance. For large-scale DL NOMA networks with a two-user cluster, the work proposes inner and outerbounds on the achievable rate region in the FBR, characterizes the min-max rate,and investigates the impact of parameters such as code-length and network densityon their performance. The MLPCM is also investigated and the results validatedits compliance for future technologies. Furthermore, we compared the performanceof large-scale OMA and NOMA networks in the FBR, showing that NOMA outper-forms OMA by a gap that increases in the FBR compared to the asymptotic regime,which motivates the use of NOMA for future networks.Additionally, the potential of Convolutional Neural Network autoencoders (CNN-AE) in approaching the theoretical maximum achievable rate over a Gaussian channelin the FBR is investigated. The CNN-AE is compared to the theoretical maximumiiiAbstractachievable rate and the achievable rates of other schemes from the literature, outper-forming these benchmark schemes and approaching the theoretical maximum rate.This demonstrates the capability of CNN-AEs in learning good codes for delay-constrained applications.ivLay SummaryThe dissertation explores the impact of evolving 5G and beyond 5G networks inmeeting the demands of high-performance applications, which are subject to strin-gent delay constraints and high reliability. Works in the literature analyzed thenetworks without considering any delay constraints, providing impractical results.The study introduces a new way to look at network performance that takes intoaccount delay and error rates. It compares this method with the older, less practicalresults and checks how changes in the network can affect performance. Moreover,by looking at real-world situations, like mobile networks, it finds possible ways toget the best performance with shorter codes. One of these ways is a method calledmultilevel polar coded modulation, which performs very well. Focusing on the chal-lenges of these constraints, this dissertation introduces a new analytical frameworkto analyze network performance under low delay constraints. The study looks athow different variables can impact network performance. It uses real-world exam-ples such as mobile networks to find the best ways to characterize and improve theirperformance using practical schemes achieving a performance close to the theoreticalbounds. The dissertation also delves into advanced network setups and compares dif-ferent schemes, shedding light on their fundamental limits. Lastly, the study exploresthe use of intelligent algorithms to improve data transfer rates, showing a promisingperformance for applications with delay constraints.vPrefaceThis dissertation embodies research conducted under the supervision of Dr. AnasChaaban and the co-supervision of Dr. Md. Jahangir Hossain. I declare that I amthe first author of this thesis. I developed all the analytic models and performed allthe simulation results presented in this thesis. I used generative AI (ChatGPT) indrafting this thesis, mainly in text style to improving the efficiency and quality ofwriting only.Throughout the various chapters, I conducted comprehensive literature reviews,conceptualized research ideas, performed essential mathematical analyses, problemformulation, and optimization as well as executed simulations. Additionally, I au-thored the corresponding manuscripts.Dr. Chaaban offered valuable input on system models, solutions to problems,and their associated computational complexities, contributing to the refinement of allarticles for scholarly dissemination. Some collaborations on part of this dissertationare conducted with Prof. Md. Jahangir Hossain and Dr. Hesham ElSawy whooffered invaluable input and revisionsThe following catalog enumerates the journal and conference publications per-tinent to the dissertation chapters. This compilation encompasses both publishedworks and those currently under review or submission. Additionally, it includesthe research and academic awards that I have been honored with during my Ph.D.studies.− Published, Submitted, and In-preparation Journal PapersJ1. Nourhan Hesham, Anas Chaaban, Hesham ElSawy, and Jahanghir Hos-sain, “Finite Blocklength Regime Performance of Downlink Large ScaleNetworks”, at IEEE Transactions on Wireless Communications, vol. 23,no. 1, pp. 479-494, Jan. 2024 ([1], Chapter 2).J2. Nourhan Hesham, Hesham ElSawy, Jahanghir Hossain, and Anas Chaa-ban, “Rate Outage and Meta-Distribution for Uplink Networks in FiniteBlock-Length Regime”, Early Access, IEEE Transactions on VehicularTechnology (parts of Chapter 3).viPrefaceJ3. Nourhan Hesham, Hesham ElSawy, Jahanghir Hossain, and Anas Chaa-ban, “On the Performance of Large-Scale NOMA Networks in the FiniteBlocklength Regime”, submitted at IEEE Transactions on Wireless Com-munications, April 2024. (Chapter 4).− Referred Conference PublicationsC1. Nourhan Hesham and Anas Chaaban, “On the Performance of Large-Scale Wireless Networks in the Finite Block-Length Regime.”, IEEE In-ternational Conference on Communications, (ICC 2021), IEEE, Virtual,Montreal, Canada, July 2021 (Chapter 2).C2. Nourhan Hesham, Jahangir Hossain, and Anas Chaaban, “Transmis-sion Rate Analysis for Large Scale Uplink Networks in the Finite Block-Length Regime”, IEEE Wireless Communications and Networking Con-ference, (WCNC 2023), IEEE, Scotland, UK, March 2023 (parts of Chap-ter 3).C3. Nourhan Hesham, Mohamed Bouzid, Ahmed Abdelqader, and AnasChaaban, “Coding for the Gaussian Channel in the Finite BlocklengthRegime Using a CNN-Autoencoder”, IEEE International Black Sea Con-ference on Communications and Networking (BlackSeaCom), IEEE, Is-tanbul, Trkiye, June 2023 (Chapter 5).− Poster PresentationsP1. Nourhan Hesham and Anas Chaaban, “The Average Coding Rate ofLarge-Scale Wireless Networks in the Finite Block-Length Regime.”, IEEECommunications Theory Workshop, CTW 2021, Virtual Event, June 2021(Chapter 2).P2. Nourhan Hesham and Anas Chaaban, “The Average Coding Rate ofLarge-Scale Wireless Networks in the Finite Block-Length Regime.”, 2021IEEE North America School of Information Theory (NASIT 2021), Vir-tual Event, June 2021 (Chapter 2).− Research and Academic AwardsA1. UBC Okanagan Graduate Research Scholarship, University of BritishColumbia, 2023.A2. UBC Okanagan Graduate Research Scholarship, University of BritishColumbia, 2022.viiPrefaceA3. The University Graduate Fellowship (UGF), University of British Columbia,2021.A4. Special University of BC Okanagan Graduate Award, 2021.A5. The University Graduate Fellowship (UGF), University of British Columbia,2020.A6. International Four-Year Doctoral Partial Tuition Award (IDPT), Univer-sity of British Columbia, 2019.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2.1 Large-Scale Networks . . . . . . . . . . . . . . . . . . . . . . . 21.2.2 Stochastic Geometry . . . . . . . . . . . . . . . . . . . . . . . 51.2.3 Multiple Access Schemes . . . . . . . . . . . . . . . . . . . . . 81.2.4 Finite Blocklength Regime . . . . . . . . . . . . . . . . . . . . 91.2.5 Channel Coding for Large-scale Networks . . . . . . . . . . . . 131.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.1 Cellular Network using Orthogonal Multiple Access . . . . . . 171.3.2 Cellular Network using Non-orthogonal Multiple Access . . . . 19ixTABLE OF CONTENTS1.3.3 Coding Schemes for Large-Scale Networks in the FBR . . . . . 211.4 Research Gap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Thesis Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241.6 Contribution and Outline . . . . . . . . . . . . . . . . . . . . . . . . 24Chapter 2: Large-Scale Downlink OMA Networks in the Finite Block-length Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 282.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282.2 Average Rate Analysis in the Finite Blocklength Regime . . . . . . . 302.2.1 Conditional Gaussian Representation . . . . . . . . . . . . . . 312.2.2 Average Coding Rate . . . . . . . . . . . . . . . . . . . . . . . 322.2.3 Average Coding Rate Using Finite Constellations . . . . . . . 352.2.4 Multilevel Polar-Coded Modulation . . . . . . . . . . . . . . . 382.3 Rate Outage and Reliability Analysis in the Finite Blocklength Regime 402.3.1 Rate Outage Probability . . . . . . . . . . . . . . . . . . . . . 402.3.2 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . . 432.4 Coding Rate Meta Distribution in the Finite Blocklength Regime . . 442.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 462.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Chapter 3: Large-scale Uplink OMA Networks in the Finite Block-length Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 563.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Fractional Path-Loss Inversion Power Control . . . . . . . . . . . . . 593.2.1 Rate Outage Analysis . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Coding Rate Meta Distribution . . . . . . . . . . . . . . . . . 623.3 Special Cases: TCI and CIMP Power Control Schemes . . . . . . . . 643.3.1 TCI and CIMP Power Control Schemes . . . . . . . . . . . . 643.3.2 Average Coding Rate Analysis . . . . . . . . . . . . . . . . . . 663.3.3 Rate Outage Analysis . . . . . . . . . . . . . . . . . . . . . . 693.3.4 Reliability Analysis . . . . . . . . . . . . . . . . . . . . . . . 713.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.1 Numerical Results for Fractional Path-Loss Inversion PowerControl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723.4.2 Numerical Results for TCI and CIMP Power Control Schemes 763.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81Chapter 4: Large-Scale Downlink NOMA Networks in the FiniteBlocklength Regime . . . . . . . . . . . . . . . . . . . . . . 83xTABLE OF CONTENTS4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.1 Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . 854.1.2 NOMA System Model . . . . . . . . . . . . . . . . . . . . . . 854.1.3 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . 874.2 Average Rate in the FBR . . . . . . . . . . . . . . . . . . . . . . . . 884.2.1 Rate Region Inner Bound . . . . . . . . . . . . . . . . . . . . 894.2.2 Outer Bound Region . . . . . . . . . . . . . . . . . . . . . . . 914.2.3 Max-Min Average Rate . . . . . . . . . . . . . . . . . . . . . . 934.3 OMA Inner Bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4 Multi-Level Polar Coded Modulation Scheme . . . . . . . . . . . . . . 974.4.1 At the Transmitter . . . . . . . . . . . . . . . . . . . . . . . . 984.4.2 At the Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . 984.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106Chapter 5: Coding in the Finite Blocklength Regime Using a CNN-Autoencoder . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.2 Autoencoder for the Finite Blocklength Regime . . . . . . . . . . . . 1115.2.1 Neural Network structure . . . . . . . . . . . . . . . . . . . . 1115.2.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . 1125.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116Chapter 6: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.2 Current Limitations and Future Works . . . . . . . . . . . . . . . . . 120Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Appendix A: The Channel Dispersion for A Large-Scale Network Under aFinite Constellation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137Appendix B: Laplace Transform of the Aggregate Interference Power underFractional Path-Loss Inversion Power Control . . . . . . . . . . . . . 141Appendix C: Average Capacity and Average Square Root Dispersion Deriva-tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142Appendix D: The Average Sum Capacity . . . . . . . . . . . . . . . . . . . 145xiTABLE OF CONTENTSAppendix E: The Average Square Root Channel Dispersion for the AverageSum Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147xiiList of TablesTable 5.1 Parameters of the proposed CNN-AE. . . . . . . . . . . . . . . 113Table 5.2 Performance of the AE at different values of M1 and M2 andSNR = 10 dB. Here, ‘×’ means that the target FER was notachieved, whereas ‘X’ means it was achieved. . . . . . . . . . . 114xiiiList of FiguresFigure 1.1 Research Gap. The citations in the circles indicate sampleexisting works, the area marked with the black line indicatesthe research gap, and our contributions in the form of publica-tions where the labels (J=Journal, C=Conference) refer to thepapers in the preface on page vi and vii and are highlightedin red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Figure 2.1 A realization of a cellular network with exclusion region of r0between the user of interest and the serving BS, where BSsare randomly distributed with distances ri > r0 to the user. . 29Figure 2.2 CDF of the interference power B and its approximation atdifferent values ofP ,λ and r0 . . . . . . . . . . . . . . . . . . 36Figure 2.3 Multilevel Coded Modulation Block Diagram . . . . . . . . . 39Figure 2.4 K(Υ) versus Υ and the PDF of the SINR at r0 ∈ {150, 250} m 42Figure 2.5 Average coding rate analysis of a large-scale network with λ =1 BS/km2 for fixed r0 . . . . . . . . . . . . . . . . . . . . . . 47Figure 2.6 Average coding rate analysis for random r0 at λ = 1 BS/km2 48Figure 2.7 The average coding rate versus r−η0 α at λ = 1 BS/km2, r0 =150 m, and n = 128. . . . . . . . . . . . . . . . . . . . . . . . 49Figure 2.8 The outage probability versus r0 at λ ∈ {1, 9} BS/km2, ¯ ∈{10−2, 10−6}, and α = 0 dB. . . . . . . . . . . . . . . . . . . . 50Figure 2.9 The outage probability versus the rate threshold Rt at λ =1 BS/km2, n ∈ {128, 2048}, and Pσ2w= 0 dB, averaged withrespect to Rayleigh distributed r0. . . . . . . . . . . . . . . . 51Figure 2.10 Outage probability using M-QAM at λ = 1 BS/km2, ¯ = 10−2,n = 128 and Pσ2w= 0 dB . . . . . . . . . . . . . . . . . . . . . 52Figure 2.11 Reliability versus the FER threshold ¯ . . . . . . . . . . . . . 53Figure 2.12 The meta distribution of the coding rate versus pt . . . . . . 54xivLIST OF FIGURESFigure 3.1 The rate outage probability versus (Rt) for ρo = −60 dBm,and σ2w=−90 dBm . . . . . . . . . . . . . . . . . . . . . . . . 72Figure 3.2 Outage Probability versus Rt at n = 128 and  = 10−2 . . . . 74Figure 3.3 Outage Probability Analysis . . . . . . . . . . . . . . . . . . 75Figure 3.4 Rate meta distribution versus pt for n = 128, λ = 2 BS/Km2,¯ = 10−5, δ = 1, and p = 0.7 . . . . . . . . . . . . . . . . . . . 75Figure 3.5 The average coding rate vs. ρo/σ2w for n = {256, 2048},  ={10−2, 10−5}, ρo = −70 dBm for TCI and CIMP power controlschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Figure 3.6 The average coding rate versus λ at n = 128,  = 10−5, andρo ∈ {−50,−70,−90} dBm . . . . . . . . . . . . . . . . . . . 77Figure 3.7 The average coding rate vs. ρo/σ2w for n = 512,  = 10−2,ρo = {−90, −70, −50} dBm for TCI and CIMP power controlschemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78Figure 3.8 The rate outage probability versus rate threshold Rt for ρo=−50 dBm, n∈{128, 2048}, ¯∈{10−2, 10−5}, σ2w =−90 dBmfor TCI and CIMP power control schemes. . . . . . . . . . . 78Figure 3.9 The total outage probability versus λ forRt = 1, ρo = −70 dBm,n ∈ {128, 2048}, and ¯ ∈ {10−2, 10−5} . . . . . . . . . . . . . 80Figure 3.10 The total outage probability and the rate outage probabil-ity of the large-scale network versus blocklength n at λ =2 BS/km2, Rt = 0.1375, ρo = −70 dBm, σ2w = −90 dBm and¯ ∈ {10−5, 10−8}. . . . . . . . . . . . . . . . . . . . . . . . . . 80Figure 3.11 Reliability versus target FER ¯ at Rt = 0.1375 . . . . . . . . 81Figure 4.1 Multilevel Coded Modulation Block Diagram . . . . . . . . . 97Figure 4.2 Inner and Outer Bounds for n = {128, 256, 512, 2048} . . . . . 99Figure 4.3 Inner and Outer Bounds for  = {10−2, 10−3, 10−5} . . . . . 100Figure 4.4 Inner and Outer Bounds for λ = {0.1, 0.15, 1, 10} BS/km2.There is a negligible increase in the achievable average rateregion at the BS density increases beyond λ ≥ 1 BS/km2. . . 100Figure 4.5 Max-Min Rate for λ = {0.1, 0.15, 1} BS/km2 . . . . . . . . . 102Figure 4.6 Max-Min Rate versus BS density λ for n = 128 and  = 10−5 103Figure 4.7 Max-Min Rate versus BS density λ for OMA and NOMAschemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Figure 4.8 Relative Gain between OMA and NOMA max-min averagerate versus Blocklength n for λ ∈ {0.1, 1} BS/km2 and  ∈{10−5, 10−2} . . . . . . . . . . . . . . . . . . . . . . . . . . . 105xvLIST OF FIGURESFigure 4.9 MLPCM Performance . . . . . . . . . . . . . . . . . . . . . . 105Figure 5.1 A conventional communication system consisting of an en-coder, modulator, demodulator, and decoder. . . . . . . . . . 108Figure 5.2 Proposed CNN-AE. The ‘encoding’, ‘modulation’, ‘demodula-tion’, and ‘decoding’ layers are trained jointly. . . . . . . . . . 110Figure 5.3 The designed achievable rates of the CNN-AE compared withschemes from the literature and theoretical achievable rates . 116xviList of AcronymsAcronyms Definitions5G Fifth Generation6G Sixth GenerationAR Asymptotic RegimeAE AutoencoderARQ Automatic Repeat RequestAWGN Additive White Gaussian NoiseB5G Beyond Fifth GenerationBER Bit Error RateBSs Base stationsCAPEX Capital ExpenditureCCDF Complementary Cumulative Distribution FunctionCDF Cumulative Distribution FunctionCDMA Code Division Multiple AccessCF Characteristic FunctionCF-mMIMO Cell-Free Massive Multi-Input-Multi-OutputCIMP Channel Inversion with Maximum Power TransmissionCNN Convolutional Neural NetworksCNN-AE Convolutional Neural Networks AutoencoderCSI Channel-State InformationCSMA Carrier Sense Multiple AccessDL DownlinkDMC Discrete Memoryless ChannelE2E End-to-EndEE Energy EfficientEiD Equivalence in DistributionFBC Finite-Blocklength CodingFBR Finite Blocklength RegimeFDMA Frequency Division Multiple AccessFEP frame error probabilityFER Frame Error RatexviiList of AcronymsFFNN Feed-Forward Neural NetworkIoT Internet of thingsKPIs key performance indicatorsLDPC Low-Density Parity-CheckLoS Line-of-SightLTE Long Term EvolutionMBS Macro Base StationMIMO Multi-Input Multi-OutputMLPCM Multilevel Polar Coded ModulationNLoS Non-Line-of-SightNN Neural NetworksNOMA Non-orthogonal Multiple AccessOMA Orthogonal Multiple AccessPDAs Personal Digital AssistantsPGFL Probability Generating FunctionalPP Point ProcessPPP Poisson Point ProcessQAM Quadrature Amplitude ModulationRNN Recurrent Neural NetworkSE Spectral EfficiencySIC Successive Interference CancellationSIR Signal-to-Interference RatioSINR Signal-to-Interference-plus-Noise-RatioSNR Signal-to-Noise RatioTCI Truncated Channel InversionTDMA Time Division Multiple AccessUE User EquipmentUL UplinkURLLC Ultra-Reliable Low-Latency CommunicationsxviiiList of SymbolsSymbols Definitionsn Blocklength Frame error rateΨ The 2-dimensional BS point processλ The intensity of BSsr0 Intended distance between a UE and its serving BSri Distances between the intended UE and the interfering BSsy The received signalP Transmit power of BSsh0 Channel fading of the intended channelhi Channel fading of the ith interfering channelη Path loss exponents0 Transmitted codeword symbolsi Transmitted codeword symbol by the ith interfererIagg The aggregate interference from all other BSsw Circularly symmetric complex Gaussian noiseσ2w Noise varianceL Number of consecutive symbols determining the channelcoherence intervalα Signal to noise ratioRn,(·) The maximum coding rate for an AWGN channelV (·) Channel dispersion for an AWGN channelB Aggregate interference power of the downlink OMA networkLx(·) Laplace transformΥ Conditional signal to interference and noise ratioRDLn, (α0) Maximum average coding rate of the DL large-scale networkin the FBR at a distance r0C∞,0(α0) Average capacity of a DL large-scale network at a distance r0V(α0) Average square root channel dispersion of a DL large-scalenetwork at a distance r0R¯DLn, Maximum average coding rate of the downlink large-scalexixList of Symbolsnetwork in the FBRC¯∞,0 Average capacity in a DL large-scale networkV¯ Average square-root channel dispersion of a DL large-scale networkM Constellation sizeC(M)∞,0(α0) Average capacity of DL networks using M -QAM ata distance r0V(M)(α0) Average square-root channel dispersion of DL networks using M -QAM at a distance r0R(M)n, (α0) Average achievable coding rate of DL networks usingM -QAM at a distance r0R¯(M)n, Average achievable coding rate of DL networks using M -QAMC¯(M)∞,0 Average capacity of DL networks using M -QAMV¯(M) Average square-root channel dispersion of DL networksusing M -QAMdml Distances between symbols in the constellation setRc Coding rate of the Polar encoderk Number of information bitsZ Information bits fed into the MLPCM encoderQ Output code word of the Polar encoderZ˜ Decoded message from MLPCM¯ FER thresholdO(r0, Rt, n, ¯) Outage probability of a DL network in the FBRO(r0, Rt) Outage probability in the AR of a DL networkOl(r0, Rt, n, ¯) Outage probability lower bound of a DL network in the FBROu(r0, Rt, n, ¯) Outage probability upper bound of a DL network in the FBRO¯(Rt, n, ¯) Average outage probability of a DL network in the FBRTn,¯(r0, Rt) Reliability lower bound of a DL network in the FBRT¯n,¯(r0, Rt) Average reliability lower bound of a DL network in the FBRT∞,¯(r0, Rt) Reliability in the ARFRt(pt, n, ¯) Coding rate meta distribution of DL network in the FBRPs(Rt, n, ¯) Probability of successpt Probability of success thresholdΩ Signal to interference ratioMd The dth Moment of the probability of successPˆs(Rt, n, ¯) Approximated probability of successMˆd The dth Moment of the approximated probability of successxxList of SymbolsFˆRt(pt, n, ¯) Approximated coding rate meta distribution of DL networkin the FBRφ The UE point processui UEs distances from the intended BSP0 Transmit power of the intended UEPi Transmit power of UEixi Activity indicator for UEiΩU Signal to interference and noise ratio for Uplink networksρo Power control parameterδ Compensation factorIFI Aggregate interference for UL networks using fractionalpath-loss inversion power controlOUn,¯(Rt) Outage probability of UL network in the FBR usingfractional path-loss inversion power controlO¯Un,¯(Rt, υ) Outage probability upper bound of UL network in the FBRusing fractional path-loss inversion power controlOUn,¯(Rt,Λ) Outage probability lower bound of UL network in the FBRusing fractional path-loss inversion power controlPUs (Rt, n, ¯) Probability of success of UL networkF URt(pt, n, ¯) Coding rate meta distribution of UL network in the FBRMUd The dth Moment of the probability of successPˆUs (Rt, n, ¯) Approximated probability of success for an UL networkin the FBRMˆUd The dth Moment of the approximated probability of successfor an UL network in the FBRFˆ URt(pt, n, ¯) Approximated coding rate meta distribution for an ULnetwork in the FBRPu Maximum trasmit power of a UEP (t)i UEs transmit using TCI power controlΩ(t) SINR of UL networks using TCI power controlI (t) Aggregate interference of UL networks using TCI power controlP (m)i UEs transmit using CIMP power controlΩ(m) SINR of UL networks using CIMP power controlI (m) Aggregate interference of UL networks using CIMP power controlR¯(t)n, (·) Average coding rate an UL network with TCI power controlR¯n, (·) Average coding rate of active users an UL networkwith TCI power controlC¯(t) (·) Average capacity an UL network with TCI power controlxxiList of SymbolsV (t) (·) Average square-root channel dispersion an UL networkwith TCI power controlR¯(m)n, (·) Average coding rate an UL network with CIMP power controlC¯(m) (·) Average capacity an UL network with CIMP power controlV (m) (·) Average square-root channel dispersion an UL networkwith CIMP power controlC¯cu Average capacity for cell-center UEsC¯eu Average capacity for cell-edge UEsVcu Average square-root channel dispersion for cell-center UEsVeu Average square-root channel dispersion for cell-edge UEsO(t)(Rt, n, ¯) Rate outage probability of an UL network withTCI power controlO(t)total(Rt, n, ¯) Total outage probability of an UL network withTCI power controlOtr(Rt, n, ¯) Truncation outageO(m)(Rt, n, ¯) Rate outage probability of an UL network withCIMP power controlO(m)cu (Rt, n, ¯) Rate outage probability approximation for cell-center UEsO(m)eu (Rt, n, ¯) Rate outage probability approximation for cell-edge UEsT (t)(Rt, n, ¯) Average reliability for TCI power control in the FBRT (m)(Rt, n, ¯) Average reliability for CIMP power control in the FBRT¯ (i)(Rt, n, ¯) Average reliability for power control i ∈ {TCI,CIMP}in the ARκk Represents the coordinates of the kth BSλu User Equipments Intensityyi Received Signal of userihik Channel fading coefficient from BS k to useriχk Transmitted signal from BS kξ Power split factorsik Codeword symbol transmitted by BS k to userigik Distance between the interfering BS k and the useriIaggi Aggregate interference from all interfering BSs at useriγi SINR of useriIi Aggregate interference power at userii Per-user FER of useriS Network Statef Network policyRˆNOMAn, Set of all (n, )-achievable average rate pairs ∀f ∈ FxxiiList of SymbolsR¯NOMAn, (f) Set of all (n, )-achievable average rate pairs at policy fRˆNOMAn, Inner bound on the achievable average rate region for NOMAR¯NOMAn, (ξ, 1, 2) Achievable average rate region for a given (ξ, 1, 2)R¯NOMAi,n Average achievable rate of useri for NOMA networkR¯OBn, Outer bound on the achievable average rate regionR∗ Max-min rateξ¯opt Optimized power split factorR∗MC Achievable rate using the optimized power split factorRˆOMAn, Inner bound on the achievable average rate region for OMAR¯OMAn, (ξ, 1, 2) Achievable average rate region for a given (ξ, 1, 2)R¯OMAi,n Average achievable rate of useri for OMA networkni Blocklength assigned of useri for an OMA networkRMLPCMi MLPCM encoding rate for useriRcod Coding RateK Number of information bitsN Length of a codewordkmod Number of coded bitsR Overall rate of information transmissionxxiiiAcknowledgementsFirst and foremost, I offer boundless praise and gratitude to Allah, the All-Knowing and the Most-Generous. I am profoundly thankful for His countless bless-ings, which provided me with the strength, purpose, and patience to reach thissignificant milestone in my life.I extend my deepest appreciation to my Ph.D. supervisor, Dr. Anas Chaaban,for his exceptional guidance, unwavering encouragement, ethical commitment, andconstant support throughout my Ph.D. journey. His ethical conduct, kindness, fair-ness, and leadership qualities have been a beacon, shaping not just my academicpursuit, but my character as well. Under his mentorship, I have learned to hone myproblem-solving skills and cultivate a deeper, more critical thinking approach. Histechnical expertise has been indispensable in navigating through complex concepts,transforming them into coherent and manageable learning experiences. I am deeplygrateful for his unwavering faith in my potential, and his constant encouragement hassignificantly influenced my personal and academic growth. His exemplary leadershipand mentoring have made my PhD journey a fulfilling and enlightening experience.The lessons I’ve learned from him extend far beyond the confines of academia andwill continue to guide me in all my future paths and pursuits.I am sincerely grateful to my co-supervisor Prof. Jahangir Hossain. His invalu-able feedback, insightful discussions, and great experience have greatly enriched myresearch, providing me with a broader perspective and deeper understanding of myresearch area. Prof. Hossain’s meticulous attention to detail and his unwavering ded-ication towards fostering a nurturing academic environment have been instrumentalin advancing my research. His critical yet constructive critiques have challengedme to push my boundaries and strive for excellence in my work. His support andguidance have been a crucial part of my growth as a researcher, and for that, I amsincerely grateful.Additionally, I also extend my deepest gratitude to Dr. Hesham Elsawy for hissincere feedback, discerning critiques, and invaluable suggestions of my work. He hasalso provided me with the needed support on a technical scale and personal scale.His exceptional support on both a technical and personal scale has been a corner-stone of my progress. Dr. Elsawy’s vast experience and profound knowledge in ourxxivAcknowledgementsfield have significantly broadened my understanding and perspective, enriching thequality of my work. I also extend my thanks to Prof. Chen Feng for his sincere feed-back, discerning critiques, and invaluable suggestions during my research proposalexamination.I must express my deepest gratitude to my family, my beloved father Hesham,my incredible mother Nadia, my supportive brother & sister Amr and Riham, myaunt Nesrine & uncle Tamer, and my beloved uncle Ashraf, whose support has beenunwavering and who never stopped believing in me. You are the light of my life.Also, I am immensely thankful to my friends and labmates in Kelowna, BC whomade my journey easier. Each of you has contributed immeasurably to my journey,and for that, I am eternally grateful.xxvDedicationTo the days that I thought will never pass. To the people who were by my sidewhen these days were passing.xxviChapter 1Introduction1.1 MotivationWireless communication networks have been evolving to satisfy users’ demand,providing high data rates, more availability, high reliability, and low latency. Theproliferating low latency applications and diverse use-cases of beyond 5th generation(B5G) networks enforce stringent key performance indicators (KPIs) that cannot berealized via conventional coding schemes [2]. For instance, mission critical appli-cations require sub 1 msec latency that cannot be achieved with conventional longcodes [3]. The extreme low-power consumption profile of many Internet of Things(IoT) devices cannot support the complexity and long transmission intervals of con-ventional long codes [4]. Consequently, there is a paradigm shift towards short codesto comply with the stringent latency and power consumption constraints of the fore-seen B5G networks. Such reduced blocklength comes at the expense of inevitableerrors and lower information coding rates. The network tolerance to the inducederrors differs with the application, which can reach a frame error probability (FEP)lower than 10−9 for ultra-reliable low-latency communications (URLLC) [5]. Thismotivates the finite blocklength coding theory [6, 7], which characterizes the infor-mation coding rate as a function of the blocklength and frame error rate (FER). Fora context-aware short code design in B5G networks, it is of primary importance toextend finite blocklength coding theory to interference-prone scenarios that accountfor the intrinsic large-scale and dense deployments of future networks.To the best of the authors knowledge, the performance analysis of large-scalenetworks in the FBR was not been investigated yet. Hence, a significant contributionof this thesis is proposing a theoretical framework that characterizes the performanceof large-scale networks under the finite blocklength regime (FBR) assumptions. Weconsider large-scale cellular networks using orthogonal multiple access (OMA) andnon-orthogonal multiple access (NOMA) schemes. Our work encompasses a thoroughtheoretical analysis and introduces practical schemes, including a machine learningapproach designed to achieve performance levels close to the theoretical bounds.The thesis includes a comparative analysis, contrasting the performance of large-11.2. Backgroundscale OMA networks with that of large-scale NOMA networks and investigates theeffect of network parameters (e.g. base station density) and transmission parameters(e.g. power allocation) on the performance of the network.This chapter provides background knowledge on the topic, reviews existing work,identifies research gaps, and highlights the thesis contributions. Hence, this chapteris organized as follows. Sec. 1.2 discusses the unique challenges and opportuni-ties associated with communication networks operating on a massive scale and thendelves into the elucidation of the FBR, contextualizing its emergence within the tra-ditional framework of information theory and emphasizing its increasing relevance incontemporary communication systems. Subsequently, practical coding schemes usedin future networks are explored. Next, in Sec. 1.3, a literature review is provided,exploring the work done in this area. Then, in Sec. 1.4, we distill the key insightsfrom existing works, introducing the research gaps and persistent challenges. Weoutline the thesis objectives, contributions, and thesis outline in Sections 1.5 and1.6, respectively.1.2 Background1.2.1 Large-Scale NetworksThe global technological landscape has witnessed exponential growth in the use ofsmart devices such as smartphones, tablets, and personal digital assistants (PDAs),leading to a surge in the demand for robust wireless cellular infrastructure. By2050, it is projected that over 24 billion devices will be connected, necessitatingthe evolution of the cellular infrastructure to accommodate this growth [8]. Tradi-tional network expansion techniques like cell splitting are struggling to keep pacewith the rapid increase in the user population and their associated traffic. Further-more, deploying macro base stations (MBS) involves significant capital expenditure(CAPEX), which is challenging to recover with the decreasing service cost [9], [10].The industry is driving standardization bodies to develop novel solutions that canhandle the increased capacity demand. These include using small cells and densifyingnetworks, which have been added to the 3rd generation (3G), Long Term Evolution(LTE), and WiMAX standards. In this context, the term “large-scale wireless net-works” refers to networks with a high density of base stations (BSs) (or nodes)(> 1 BS/km2) covering a large area. These networks need to manage the big volumeof data traffic generated by a diverse range of wireless devices, from smartphonesto smart meters and appliances. However, the scarcity of the wireless spectrum andthe ever-increasing capacity demand require universal frequency reuse [10–14]. This21.2. Backgroundspectrum reuse increases spatial spectrum efficiency and network capacity, but atthe expense of greater interference. Moreover, the deployment and management of alarge number of base stations pose challenges including site acquisition, power sup-ply, backhaul connectivity, and interference management between neighboring cells.In this thesis, we will focus on modeling and management of interference and howthe performance analysis of large-scale networks is conducted.Interference is a critical challenge facing large-scale wireless networks. As wire-less communication systems evolve towards more aggressive utilization of spectralresources and efficient transmission systems, understanding the effects of interferenceon system performance becomes crucial. Interference in wireless networks primarilycomes from two sources: co-channel interference and adjacent channel interference.Co-channel interference occurs when the same frequency band is reused by mul-tiple communication links within the network’s coverage area and it comprises oftwo types of interference: intracell interference caused by users in the same cell,and intercell interference caused by users in another cell due to frequency reuse.Adjacent channel interference, on the other hand, is caused by signal spillage fromneighboring frequency bands. Several factors contribute to the complexity of man-aging and modeling interference in large-scale wireless networks. Firstly, the densedeployment of base stations in these networks increases the likelihood of interferencefrom neighboring cells. Secondly, the adoption of spectral resource reuse, while ben-eficial for enhancing network capacity, exacerbates co-channel interference. Thirdly,the proliferation of different wireless technologies and devices operating in the samefrequency bands further increases the potential for interference. Fourthly, the pres-ence of various stochastic variables necessitates their incorporation into the modelingprocess.Clearly, the interference phenomenon takes place at the physical layer of the re-ceiver, as an interfering (undesired) signal disturbing the reception of a given desiredsignal. However, the characteristics of the interfering signal and its disturbance ef-fects are determined by features of the interfering transmission at different layers ordomains. Therefore, an interference model can be viewed as the combination of thefollowing components (or sub-models) [15]: Propagation model, that describes theeffects of radio propagation on the received signal, such as deterministic path loss,small-scale and large-scale fading. Another factor specifically related to large-scalenetwork is the interferers (i.e., transmitters) spatial distribution model, that specifieshow terminals are distributed over the network area, ranging from a purely randomdistribution to an arbitrary terminal placement. Also the network operation model,that describes the interrelationship among terminals of a network. This interrela-tionship is mainly dictated by the multiple access control technique adopted in the31.2. Backgroundnetwork, which can be classified as random access (e.g., Carrier sense multiple ac-cess (CSMA) and Aloha) or deterministic access (e.g., time division multiple access(TDMA), code division multiple access (CDMA) and frequency division multiple ac-cess (FDMA)). The multiple access control technique determines when and whichterminals access the radio channel. In cellular networks, deterministic access schemesare used. Moreover, the traffic model, that describes the transmitter activity, is animportant parameter that needs to be taken into consideration while modeling theinterference. The effects listed above appear in any interference model either as a de-terministic process or as a random process, according to the scenario considered. Forinstance, if fading is a relevant effect of the radio propagation environment, then theinterference signal is a random process, regardless of the nature of all other effects.Therefore, an appropriate fading distribution must be adopted in the interferencemodel. Clearly, the nature of the effects (either deterministic or random) collectivelydetermines the nature of the interference modeled.Interference modeling in large-scale networks is typically categorized into deter-ministic and stochastic models, each with unique attributes and applications. De-terministic interference models offer a high level of precision, as they are predicatedon exact input parameters. These models calculate interference by factoring in thespecific locations of transmitters, receivers, and any present obstacles. Techniquessuch as ray-tracing, path loss models, and antenna radiation patterns are commonlyemployed in deterministic interference models to compute interference. However,despite their accuracy, deterministic models can be computationally arduous, par-ticularly in large-scale networks with a multitude of nodes. In contrast, stochas-tic interference models incorporate elements of randomness to reflect the inherentvariability in real-world networks. These models produce statistical descriptions ofinterference instead of exact values. They leverage probability distributions to repre-sent random variables, such as node locations, propagation conditions, and networktraffic. Renowned stochastic models include the Poisson Point Process (PPP) modeland the Mate´rn hard-core model, which are typically used in large-scale networkenvironments featuring numerous unpredictable variables. Each model type presentsits own strengths and weaknesses. While deterministic models provide detailed andaccurate interference estimations, their computational expense can be a limiting fac-tor. Stochastic models, on the other hand, offer more computational efficiency andeffectively handle uncertainty, but may fail to provide the detailed results achiev-able with deterministic models. In practice, a combination of both deterministicand stochastic models may be utilized by network engineers, contingent on the spe-cific requirements and constraints of the given network. Deterministic models areoften favored where high accuracy is paramount, while stochastic models are pre-41.2. Backgroundferred where computational efficiency and effective uncertainty management are ofgreater importance. In this dissertation, we employ a stochastic approach, providinga comprehensive performance analysis of the network. Network engineers strive tooptimize the network to enhance its performance, hence the performance of a spe-cific network configuration would likely surpass the average performance projectedthrough theoretical analysis. Despite this, the insights derived from such analysisprove to be very beneficial. These insights afford a deeper understanding of potentialnetwork performance and can assist network engineers in optimizing the network andpreempting possible issues. The average performance provided by stochastic models,although not always indicative of individual network performance, offers invaluabledata for overall network optimization.1.2.2 Stochastic GeometryStochastic geometry has emerged as a robust mathematical framework, capable ofmodeling various types of wireless networks, delineating their operation, and under-standing their behavior [16]. The primary strength of stochastic geometry analysiscomes from its capacity to encapsulate the inherent spatial randomness of wirelessnetworks. Moreover, stochastic geometry models can be naturally extended to incor-porate other sources of uncertainties such as fading, shadowing, power control and allthe other factors mentioned before. Also, in specific instances, stochastic geometryanalysis can yield closed-form expressions that dictate system behavior, enabling adeeper understanding of network operation and providing insightful design guidelinesoften elusive from computationally demanding simulations.In real-world scenarios, cellular networks are already established and, for anygiven city, the locations of BSs are fixed. However, stochastic geometry does notmodel the performance of a specific instantiation of the cellular network at a par-ticular geographical location. Instead, it provides a general analytical model that isapplicable, on average, across all cellular network instances, providing some insightson the performance of the network using a specific technology and recommendingsome informed design decisions. For example, when analyzing the impact of a specifictechnology on the performance in cellular networks, instead of repeating the analysisfor every geographical setup, we can derive general performance analysis, guidelines,and design insights that apply, on average, across all unique realizations. Follow-ing the observations in studies [17] and [18], BSs’ locations at various geographicallocations constitute random patterns. Therefore, a general analysis for cellular net-works should be predicated on the probabilistic spatial distribution of BSs ratherthan a specific network instantiation. By abstracting each BS location into a point51.2. Backgroundin Euclidean space, stochastic geometry models the probabilistic BS locations usinga stochastic point process (PP), which characterizes the random spatial patternsformed by points in space. We focus on the performance of a randomly selecteduser and/or the average performance of all users, which considers users in all loca-tions. From the network’s perspective, we are interested in the average performanceacross all cellular network instances. This average performance metric, referred toas spatially averaged performance, is the primary output from stochastic geometryanalysis.Examples of spatially averaged performance metrics of interest in cellular net-works include outage probability, ergodic capacity, symbol error probability, bit errorprobability, coverage probability, and the meta-distribution of coverage probability.Stochastic geometry gives expressions that relate the aforementioned performancemetrics to the cellular network parameters and design variables. Such expressionsare then used to understand the network behavior in response to the network pa-rameters and/or design variables. This helps to obtain insights into the networkoperation and extract design guidelines. Note that, by stochastic geometry analysis,the obtained design insights hold on average for all realizations of cellular networks.Hence, in light of the obtained expressions, communication system engineers canperform trade-off studies and take informed design decisions network planning.The initial phase of the analysis involves selecting an appropriate PP as an ab-straction of network elements, such as base stations (BSs) and users. The perfor-mance metrics under consideration are then represented as functions of the chosenPP. These functions can be evaluated employing stochastic geometry results. Thephrase “appropriate PP” is used to refer to a PP that strikes a balance betweenmathematical tractability and practical realism. A subsequent discussion will ad-dress the issue that an overly practical PP may hinder model tractability, leadingto the necessity of seeking approximations. Stochastic geometry offers two principaltechniques that convert a function involving all points in a PP into an integral overthe PP’s domain. These techniques are Campbell’s theorem and the probability gen-erating functional (PGFL). Campbell’s theorem, which requires an expectation overa random sum, can be directly applied to compute moments. Conversely, the PGFL,which involves an expectation over a random product, is well-suited to computingthe characteristic function (CF) of the aggregate interference [19]. The Campbell’stheorem is expressed asE[∑X∈Φh(X)] =∫h(x)λ(x) dx, (1.1)61.2. Backgroundwhere λ is the intensity of the point process Φ, X is a point in the point process Φ,and the PGFL is expressed as:E[∏X∈Φf(X)]= exp(−λ∫Rd(1− f(x))dx). (1.2)While the intensity of the PP is sufficient to determine the mean of aggregateinterference related to a general PP through Campbell’s theorem, obtaining tractableexpressions for higher-order moments is generally not feasible. Moreover, the PGFLmay not exist for all PPs to characterize the aggregate interference via its CF. Thus,we must rely on approximations to maintain tractability. The prevalent and widelyaccepted approximation for the interference associated with a general PP is theinterference generated from a PPP.It is important to highlight that the versatility of SG extends beyond modelinga 2-dimensional (2-D) network. Indeed, the capabilities of SG are not confined to asingle plane. It also has the potential to model a 3-dimensional (3-D) network includ-ing unmanned aerial vehicles (UAVs) and satellites, utilizing the principles of 3-Dstochastic geometry. This application of SG to 3-D networks showcases its adaptabil-ity and broad applicability. This is essential because 6G will exploit satellite, aerial,and terrestrial platforms jointly to improve radio access capability and to unlock thesupport of on-demand edge cloud services in the 3-D space by incorporating MobileEdge Computing (MEC) functionalities on aerial platforms and low orbit satellites[20].Addressing these interference challenges requires a multifaceted approach. Ad-vanced interference management techniques, such as interference cancellation, in-terference alignment, and coordinated multipoint transmission, are being explored.Additionally, innovative network architectures like heterogeneous networks [21] andmassive multiple-input multiple-output (MIMO) systems [22] are being consideredto improve network capacity and mitigate interference. Moreover, effective spectrummanagement is vital. Dynamic spectrum access techniques, which allow the oppor-tunistic use of underutilized spectrum bands, are being studied as a potential solution[23]. Cognitive radio technology, which enables devices to sense and adapt to theirradio environment, is another promising approach to improve spectrum utilizationand manage interference [24].71.2. Background1.2.3 Multiple Access SchemesSince interference is a fundamental challenge in wireless networks, and its miti-gation is a crucial aspect of network design. Multiple access schemes are designedto manage this interference by coordinating the simultaneous transmissions of mul-tiple users over shared resources. The core of wireless communication is centeredaround the effective management of the shared radio spectrum among numeroususers. With the ever-increasing demand for seamless and simultaneous communica-tion, it is paramount to have a system that can handle this complexity efficiently.This necessity gives rise to multiple access schemes. Multiple access schemes areintegral to wireless communication systems, offering a way for many users to concur-rently share a finite amount of radio spectrum, effectively mitigating or controlingpotential interference and maximizing the efficient use of available resources. Byemploying these schemes, a balance is struck between maximizing user access andminimizing mutual interference, thereby enhancing the overall network performance.These schemes are primarily bifurcated into two main categories that are con-sidered in this thesis: OMA and NOMA, each with their unique approach to in-terference mitigation and resource allocation. Traditional OMA protocols, whichinclude TDMA, FDMA, and CDMA, are traditional technologies in communicationsystems [25, 26]. These methods allocate distinct resources to users to prevent in-terference and maximize bandwidth efficiency. TDMA divides the communicationchannel into discrete time slots, assigning each user a specific slot to access the fullchannel bandwidth. However, this method requires precise timing synchronizationto avoid overlapping transmissions, which can be challenging to maintain [26, 27].FDMA assigns different frequency bands to users, separated by guard bands to avoidinterference, but this can lead to spectral inefficiency due to the under utilization ofthese guard bands [26, 27]. CDMA allows multiple users to transmit simultaneouslyover the same frequency bands using unique spreading codes to separate users, but itfaces the near-far problem where a strong nearby signal can overpower weaker ones[28]. Additionally, managing the orthogonality of codes in a dense user environmentis complex.NOMA, unlike OMA, permits multiple users to concurrently use the same time-frequency resources at varying power levels [29]. OMA’s exclusive allocation of re-sources such as time, frequency, and code to each user theoretically eliminates inter-ference, simplifying the detection of each user’s packet by a receiver [30]. However,this exclusive allocation limits the maximum number of supported users. Conversely,NOMA allocates the same two-dimensional time-frequency resource to multiple usersin the power domain, leveraging channel difference characteristics. This allows mul-tiple users with significant channel gain differences to share the same resource, thus81.2. Backgroundnot strictly limiting the number of supported users. Nevertheless, NOMA requires anintricate user signal detection process known as Successive Interference Cancellation(SIC) at the receiver node side, which enables each user to decode individual pack-ets from superposed packets in the power domain, achieving high cell capacity [29].Furthermore, NOMA can achieve higher spectral efficiency (SE) due to its superiorexploitation of channel diversity via SIC.Current requirements for high SE, energy efficiency (EE), user data-rate, andmassive connectivity in 5th generation (5G) wireless communications cannot be metsolely by implementing OMA [31]. NOMA, with its non-orthogonal resource alloca-tion, offers improved network performance compared to conventional OMA. NOMA’sability to allow numerous User Equipments (UEs) to share the same resource blocksimultaneously in the power domain leads to inter-UE interference in each resourceblock [29, 31]. However, if managed effectively, this can enhance SE. The SIC schemehas been developed to facilitate multi-user detection, assisting in decoding superim-posed signals. Various NOMA techniques with lower receiver complexity and signifi-cantly improved performance have been introduced, making them ideal for providingmassive connectivity for IoT applications [32]. NOMA’s advantage lies in its ca-pacity to allow a higher number of users to communicate simultaneously during ascheduling time than the number of sub-carriers, due to frequency reuse within a cell[31]. This could be instrumental in achieving massive simultaneous connectivity. Incontrast, in multi-carrier OMA transmission, all active users compete for sub-carrierallocation via the scheduling mechanism, increasing competition and latency andmaking massive connectivity infeasible with a limited number of sub-carriers. WithNOMA, many-to-one allocation can be achieved, considerably reducing competitionand latency. Additionally, since each transmitted symbol can carry multiple users’data with a single header, data overhead can be minimized [33].The choice between OMA and NOMA depends on the specific requirements andconstraints of the wireless network, such as the number of users, their data rates,and the characteristics of the wireless channels [30]. Both schemes have their uniquestrengths and potential drawbacks, and understanding these is key to designing effi-cient and robust wireless networks. The evolution from OMA to NOMA representsa shift from a paradigm of avoiding interference to one that embraces and managesit, paving the way for the next generation of wireless communications.1.2.4 Finite Blocklength RegimeAt this point, we have covered the technical definition of a large-scale network, itschallenges, and its mathematical definition. However, the proliferating applications91.2. Backgroundand diverse use-cases of 5G and B5G networks enforce stringent key performanceindicators that add more challenges to the network that needs to be studied. For thefuture cellular networks generation to support its use cases, it requires to support lowlatency and high reliability communications that cannot be realized via conventionalcoding schemes [2]. For example, mission-critical applications such as autonomousdriving, remote surgery, and industrial automation require ultra-low latency, typi-cally below 1 millisecond, which cannot be attained using conventional long codes [3].In the context of autonomous vehicles1, the necessity for URLLC becomes apparentas these vehicles must process and react to road conditions in real-time, requiring con-stant communication with other vehicles and infrastructure (V2X communications)[36]. Any delay, even milliseconds, can compromise safety and efficiency. Similarly,in remote surgery, where surgeons operate on patients from remote locations usingrobotic instruments, latency must be minimized to ensure precision and prevent de-lays that could lead to medical errors [37]. Additionally, Industrial automation alsorelies heavily on low latency communications to synchronize complex machinery andproduction lines, ensuring seamless operations and safety in factories [38]. In smartfactories, machinery and robots must communicate seamlessly and instantaneouslyto coordinate complex manufacturing processes. Any delay can disrupt operationsand lead to significant productivity losses or safety hazards. Similarly, augmentedreality and virtual reality applications, which are becoming increasingly prevalentin both entertainment and professional training scenarios, require low latency toprocess and update images in real time to provide immersive and interactive userexperiences [39, 40]. Furthermore, emergency response systems benefit greatly fromlow-latency communications. During disaster scenarios, coordinating rescue opera-tions and providing real-time data to first responders can save lives and mitigatedamage [41]. The ability to quickly process and relay information is crucial in thesehigh-stakes environments. Furthermore, the IoT devices, which are becoming ubiq-uitous in smart homes and cities, also demand low-power consumption profiles thatcannot support the complexity and long transmission intervals of conventional longcodes [4]. Indeed, short packets are the typical form of traffic generated by thesewireless technologies. This requires a fundamentally different design approach thanthe one used in current high-data rate systems, such as 4G (LTE) and WiFi. Con-1It is worth noting that autonomous vehicles are equipped with sensors and modules which aresimilar to connected vehicles. However, the main difference is their capability to make indepen-dent decisions and act accordingly (self-driving) [34]. Autonomous vehicles use two main wirelesstechnologies: dedicated short-range communications (DSRC) and vehicle-to-everything. With theadvancement of 5G, the URLLC capabilities of 5G networks along with cloud computing will beused [35].101.2. Backgroundsequently, there is a paradigm shift towards using short codes to comply with thestringent latency and power consumption constraints of the foreseen B5G networks.As these systems aim for URLLC, understanding the effect of using short codes ontheir performance becomes crucial for optimizing system performance. Hence, thissection delves into the FBR, exploring its significance, challenges, and implicationsfor information theory.In the realm of information theory, the traditional approach assumes infiniteblock lengths for communication systems. The classical channel coding theorem byClaude Shannon provides bounds on the achievable rate of reliable communicationfor infinite blocklength and vanishing frame error probability. From an informationtheory perspective, the channel code is defined as a map between the informationpayload and the signal transmitted over n channel uses. The task of a wirelessreceiver is to recover the information payload from a distorted and noisy version of thetransmitted signal. A fundamental result tells us that when n is large (long packets),there exist channel codes for which the information payload can be reconstructedwith high probability. However, when n is small (short packets), the dimensionalityof the code decreases which limits the ability to find an optimal code that can beconstructed with a vanishing error probability. In other words, such reduced codelength comes at the expense of inevitable errors and lower information coding rates.Thus, the code length of the FBR is a critical design parameter that imposes adelicate trade-off between FER and information coding rate.The deployment of short codes in large-scale networks introduces several tech-nical complexities. Primarily, the abbreviated code length inherently constrainstheir error-correcting capabilities, which is particularly detrimental in high-noiseor interference-laden environments where the robustness provided by longer codesis indispensable. This limitation significantly impacts the overall reliability of thecommunication system, as the shorter codes have less redundancy to correct the er-rors induced by the channel noise [42]. In addition, the design and optimization ofshort codes within the FBR necessitate a meticulous balance among several criticalperformance metrics: latency, reliability, and throughput. This balance is vital inscenarios such as real-time communication services or machine-type communicationsin IoT networks, where even minimal delays or slight errors can severely degradethe system performance and usability [6]. Achieving this balance involves complexoptimization problems that must account for both the theoretical limitations of shortcodes and the practical constraints of network operation. Moreover, the effective im-plementation of short codes across a heterogeneously conditioned large-scale networkrequires an advanced understanding of the varying channel states experienced by dif-ferent users and links. The disparate channel conditions can result in a spectrum111.2. Backgroundof performance outcomes, with some connections experiencing suboptimal reliabilityand others not fully leveraging the available coding gains. To mitigate these issues,network operators must employ dynamic resource allocation and adaptive codingtechniques. These strategies adjust the coding and transmission parameters in real-time based on the observed channel conditions, thereby optimizing the use of networkresources and enhancing overall communication efficacy [43]. Such adaptive systemsadd layers of complexity to network management, necessitating sophisticated algo-rithms capable of real-time analysis and decision-making to dynamically tailor thecoding strategies to the prevailing channel conditions. This approach ensures thatthe network can maintain high levels of reliability and efficiency, even under theconstraints imposed by the use of short codes. Overall, while short codes are es-sential for supporting the stringent requirements of modern communication systemsin terms of latency and reliability, they introduce significant challenges in terms ofcoding design, error correction efficacy, and network management. These challengesmust be carefully studied and addressed to fully harness the benefits of short codesin large-scale networks.During the last few years, significant progress has been made within the infor-mation theory community to address the problem of characterizing the performanceof the system using short packets. The theoretical principles governing short-packettransmission was used to study and characterize the performance metrics of point-to-point scenarios, which allowed them to assess their performance. Some existing workwas devoted to the assessment of the penalty incurred by finite blocklength [44–46].However, the work proposed does not provide tight bounds on the maximal achiev-able coding rate. Moreover, they are intractable expressions that are numericallyevaluated. A significant advancement was achieved in [6] where Polyanskiy et al.provided tight bound on the maximum achievable rate as the capacity under infiniteblocklength with a back-off term which is function of the code length n and FER ,which is easily evaluated. The FER play a crucial role in quantifying the reliabilityof communication systems and the blocklength n affects the latency of the system.Moreover, there is new term introduced which is the channel dispersion. The channeldispersion measures the stochastic variability of the channel relative to a determin-istic channel with the same capacity [6]. This concept helps in understanding howthe achievable rates deviate from the Shannon capacity when the number of chan-nel uses is finite. Then, the maximum coding rate for an additive white Gaussiannoise (AWGN) channel with signal-to-noise ratio (SNR) α, blocklength n, and FER121.2. Background ∈ (0, 0.5), is presented as follows [6]2Rn,(α) = C∞,0(α)−√V (α)Q−1()√n+12nlog2 n, (1.3)where C∞,0(α) = log2(1 + α) is the AWGN channel capacity in the AR, V (α) =α(α+2)(α+1)2log22(e) is known as the channel dispersion, and Q−1(·) is the inverse of the Q-function. Hence, the trade-off between reliability (error probability), latency (blocklength), and throughput (rate) is quantitatively analyzed, providing crucial insightsfor system design using (1.3).1.2.5 Channel Coding for Large-scale NetworksFrom a practical point of view, after studying the performance of the large-scalenetwork theoretically, the cellular networks require a coding scheme that can delivera performance closely aligned with theoretical results. Using information theory,the existence of a codebook that can achieve a target FER at a given codelengthis proven as long as the coding rate is below the theoretical maximum coding ratein the FBR characterized in [6]. The challenge lies in identifying this optimal code.Currently, two primary approaches exist: conventional coding schemes and artifi-cial intelligence-based schemes. Traditional communication systems are often engi-neered as a concatenation of independent blocks using an encoder/decoder and amodulator/demodulator to construct a code that is easy to analyze, optimize, anddesign. There are multiple coding schemes that have been demonstrated to performeffectively for various applications. For example, the 5G in 3GPP Release 16 Spec-ification # 38.212 [47] identifies polar codes [48, 49] for relatively long blocklengthand Reed-Muller for short blocklength, combined with quadrature amplitude mod-ulation (QAM). Polar codes are recognized for their capacity-achieving capabilities,making them highly reliable. However, their main disadvantage is the complexity ofthe successive cancellation decoding process. Turbo codes offer near-optimal perfor-mance, making them a high-performance error correction code option [50]. However,they require a high computational complexity for decoding. They are widely usedin 3G and 4G cellular networks due to their error correction capabilities, which helpensure data transmission reliability. Low-Density Parity-Check (LDPC) codes pro-vide near-Shannon limit performance, making them highly efficient. However, theirmain disadvantage is their decoding complexity, which can be a hindrance in someapplications. Despite this, LDPC codes are used in various high-speed data commu-nication standards, including some areas of 5G communications, due to their high2This approximation is shown to be tight at blocklengths (> 100) and FER > 10−6 in [6].131.2. Backgroundefficiency. Convolutional codes offer good performance and relatively low complexity,making them suitable for many wireless communication systems. One downside isthat they may not perform as well as more advanced coding schemes at high datarates. Nevertheless, they are widely used in communication systems due to theireffective error correction and decoding simplicity. Despite the effective performanceof these codes, the hard-detection process often acts as a bottleneck, limiting theirperformance. Hence, such design does not guarantee optimality compared to thetheoretical maximum in the FBR. To enhance the performance of conventional com-munication systems, joint design and optimization of coding and modulation in theFBR is necessary. Although this is often a tedious task, multilevel polar coded mod-ulation (MLPCM) was introduced as a joint coding and modulation scheme thatperforms better than polar codes for Gaussian channels [51, 52], but unfortunatelydoes not achieve the maximum achievable rates under short code-length. It is worthnoting that Reed-Muller codes are not capacity achieving codes, however, they per-form better than polar codes in the FBR. However, we will consider a MLPCMscheme since polar codes are already being used for moderate blocklengths in 5G.These limitations motivated the investigation of neural networks (NN) for code de-sign in communication systems, after their success in various fields. NNs have beenshown in various works [53–61], to be a convenient tool for code design in communi-cation systems, yielding better performance than conventional schemes in terms oferror rates. Work in this area started by replacing one or more blocks of the physicallayer with NNs for performance improvements and extended to replace the transmit-ter and receiver by introducing the autoencoders. We will delve into the work donein this area in the next section.In conclusion, the evolution of wireless networks and the increasing demand fordata traffic necessitates the development of novel solutions to handle the capacity.Large-scale wireless networks face significant challenges, particularly in managinginterference and the deployment of a large number of base stations. The use ofStochastic Geometry and point processes provides a mathematical framework forperformance analysis. However, the need for transition towards shorter codes due tothe support of delay constraint application in B5G networks, introduces additionalcomplexities due to inherently limited error-correcting capabilities of these shortercodes. Future research should focus on studying the performance of these systems,the trade-off between reliability and latency, and the effect of network parameterson the performance of the network; and proposing practical techniques to achieve aperformance closely aligned with theoretical results.141.3. Literature Review1.3 Literature ReviewCellular networks are now nearly universally deployed and are under ever-increasingpressure to increase the volume of data they can deliver to the users. Network den-sification is one of the strategies to provide better coverage, higher data rates, andlower latency. However, wireless cellular networks are prone to interference due tofrequency reuse, multi-user association, and environmental conditions, to name afew. With the densification of networks, the interference problem is exacerbated.For decades, the performance of these large-scale cellular networks has been thor-oughly studied using the classical Shannon capacity as a benchmark [16, 62–66].However, with network evolution, future networks are expected to support a multi-tude of use-cases that requires stringent delay requirements. In a network, there arefour dominant delays: transmission delay, propagation delay, processing delay, andqueuing delay. The transmission delay refers to the time needed to transmit all ofa packet’s bits into the channel. Propagation delay is the time a bit takes to travelfrom the sender to the receiver, related to the distance between them. Processingdelay involves the time routers need to process the packet header, check for errors,and determine the packet’s route. Lastly, queuing delay is the time a packet waits inrouting queues to be processed or transmitted, which can fluctuate based on networkcongestion. As the network becomes denser, the propagation delay decreases and thetransmission delay can dominate the propagation delay which motivates using shortcodes. For example, for a 4G network with macrocells with a size of 3 − 6.5 km[67], the propagation delay would reach up to 10 − 15 µsec. However, for small 5Gcells in the range of 500 m [68], the propagation delay can go up to 1.67 µsec. Forthe transmission delay, using a codeword of 215 bits which is a blocklength used in5G networks [69], with a transmission rate of 1 Gbps would result in a transmissiondelay of 32.76 µsec, which dominates the propagation delay. However, reducing thecodeword to 256 bits using the same transmission rate will result in a transmissiondelay of 0.256 µsec, which is a significant decrease in the transmission delay. Also,using short codes does not only reduce the transmission delay but it also reduces theprocessing delay and may reduce the queuing delay since the processing/bufferingdelay of short codewords is less than that of long codes. Since delay-constrainedapplications require the use of short codes, studying the achievable information ratein such applications using Shannon’s capacity expression becomes imprecise, as thisexpression is derived for infinitely long codes and vanishing error probability. Suchidealistic scenario is hereafter denoted as the asymptotic regime (AR). The AR anal-ysis in [62–66] is not applicable for networks operating in the FBR since short codes(e.g., 128 symbols) violate the AR assumptions and lead to inevitable errors as a151.3. Literature Reviewcost for constraints on delay. Instead, the coding rate under a codelength limitationshas to be used for such studies. Therefore, Polyanskiy et al.’s theorem in [6] is moreprecise to be used to characaterize the performance of the communication systemunder short codes.Motivated by the need to migrate towards the FBR, theory of coding in theFBR is used to characterize the performance several use cases such as relay channels[70, 71], NOMA [72], rate splitting [73], multi-user MIMO [74], massive-MIMO [75],cell-free massive MIMO [76], and URLLC [77–81]. For relaying channels, the overallerror probability of relaying is derived in [70], and the blocklength-limited capacityof relaying is investigated. Moreover, closed-form expressions for the decoding errorprobabilities, coding rates, and outage probabilities of direct and relay transmissionsare derived under the Nakagami-m fading channel in the FBR in [71]. For point-to-point NOMA, the maximum coding rate for each user was derived in [72] and theeffect of blocklength and reliability was studied. For rate splitting, the theorem ofcoding in the FBR provides a basis for determining the optimal division of a datastream into two or more streams, which helps increase the overall system capacitywhile maintaining a desired level of reliability. In [73], optimal linear precoders thatmaximize the sum rate with finite blocklength constraints and the relations betweenthe sum rate and blocklength of rate splitting multiple access were investigated fora wide range of network loads and user deployments. In multi-user MIMO systems,the maximum coding rate is characterized using the theory of coding in the FBR asa function of the number of BS antennas in [74]. This allows a better understandingof the system and optimization of resources to allow the simultaneous use of multipleantennas by multiple users, thereby enhancing data transmission rates and overallsystem maximum achievable rates. For Massive-MIMO and cell-free Massive MIMOsystems, bounds on the maximum coding rate are proposed for a given number ofantennas, given a finite block length [82–84]. This information aids in optimizingspectral and energy efficiency.For URLLC, the understanding of the trade-off between reliability, latency, andthroughput is crucial for the design of 5G and beyond systems, where URLLC is akey component. The theory of coding in the FBR and SG provide powerful toolsto characterize the performance of URLLC. URLLC is studied in [75, 78–81]. Theauthors in [77, 78] optimize the blocklength and other network parameters (e.g.,transmit power or transmitter location) to minimize the decoding error probabilitysubject to a latency constraint. Works have also been done in [79–81] to maximizethe achievable rate, improve latency, enhance reliability, and/or provide a securecommunication in the FBR. However, the models in [70, 72–81] are limited to smallscale networks. To the best of the authors’ knowledge, interference-prone charac-161.3. Literature Reviewterization of the performance of large-scale networks in the FBR is still an openproblem.In this thesis, we characterized the performance of large-scale cellular networksusing OMA and NOMA and we investigate the practical coding schemes that canachieve a performance close to the theoretical results proposed. In the next subsec-tions, we will present the work done for each network and the coding schemes usedfor practical networks.1.3.1 Cellular Network using Orthogonal Multiple AccessOMA cellular networks, as the prevailing technology in the sphere of networkdeployment, have established the foundational principles of modern cellular commu-nication. They have been extensively studied in the past decades. We will present asample of the research done in this area, starting with reviewing the downlink (DL)followed by the uplink (UL).For DL networks, the performance of large-scale OMA networks was studied in[62–66, 82, 85–87] using a stochastic geometry approach. In [63], the wireless net-work was modelled for the first time as a PPP using stochastic geometry, which led totractable results on the connectivity, the capacity, the outage probability, and otherfundamental limits of wireless networks. Using the PPP model for network modeling,the coverage probability of a cellular network in urban areas is provided in [64]. Theresults are compared to the hexagonal grid model to find that the stochastic geome-try model provides a more accurate upper bound on the coverage probabilities thanwhen using the hexagonal grid model. Moreover, in [65], the signal-to-interference-plus-noise-ratio (SINR) is characterized and is used to evaluate performance metricslike error probability, outage probability, and transmission rate. Additionally, in [62],a stochastic geometry model is used for the performance analysis and the planning ofdense IEEE 802.11 networks, and investigating the effect of network density on thelong term throughput. In [66], a comprehensive survey on the literature related tostochastic geometry models for single-tier as well as multi-tier and cognitive cellularwireless networks is presented with a taxonomy based on the target network model,the point process used, and the performance evaluation technique. All the workproposed in [62–66] are conducted under AR assumptions. However for the FBR,the network was studied in [82, 85–87]. Moreover, [82] discusses the anticipatedevolution of URLLC in future sixth-generation (6G) networks, utilizing MIMO tech-nology. The authors derive expressions for the expectation and variance of channeldispersion based on existing maximal achievable rate results. Then, they establishan upper bound for the average maximal achievable rate in massive MIMO systems.171.3. Literature ReviewThe study concludes that as the spatial degrees-of-freedom increases, MIMO systemsexhibit deterministic transmission characteristics. This finding suggests that latencyin 6G uRLLC could be reduced by shortening the blocklength in the time domainat the expense of spatial degrees-of-freedom, demonstrating an exchangeability ofspace and time. In [85], the design of Finite-Blocklength Coding (FBC) for achiev-ing extremely low latency in 6G systems is explored. The study proposes a jointFBC scheme in multi-user DL systems where data bits of multiple users are jointlyencoded over orthogonal channel resources, resulting in throughput gains comparedto expanded blocklength of FBC. The researchers present a matrix-based methodfor joint encoding, and formulate an Integer Programming problem to maximizeDL users’ throughput subject to a power constraint. By converting this probleminto a nonlinear bipartite matching problem, an algorithm is presented to achieveoptimal power-constrained throughput within the low latency requirement. Addi-tionally, in [86], the performance of point-to-point MIMO systems with large butfinite numbers of transmit and receive antennas is studied, particularly in scenar-ios with short packets. Analyzing the system using recent findings on achievablerates of finite-length codes, it provides closed-form expressions for error probabil-ity/throughput and identifies the minimum antenna count needed to meet specificrequirements. It also indicates that while the antenna count required under errorconstraints is sensitive to the length of short codewords, the effect is minimal formoderate/large length codewords. The work in [87] addresses the impact of shortblocklength codes, or FBR, on the reliability and capacity performances of ultra-reliable and low-latency services. The authors establish the joint convexity of theFER with respect to blocklength and transmit power within a specific region, whichis instrumental for designing systems for optimal performance. They apply this tomultiple user cases and prove its validity.For the UL large-scale networks, its performance was studied from many angles[88–93]. In [88], the UL of a single and a multi-tier network using a truncated channelinversion power control strategy is studied in terms of the rate outage probabilityand average spectral efficiency of transmission. However, in [89], the users’ powercontrol strategy is modified to allow edge users to send with maximum transmissionpower instead of being truncated. Moreover, in [90], the meta distribution analysisof the signal-to-interference ratio (SIR) is proposed for UL networks. Also, in [91],the delay-bound violation probability under a given end-to-end (E2E) latency boundis studied for UL in a multi-cell interference channel. The transmission reliabilityof URLLC is maximized in the FBR via optimizing the fixed transmission rate. In[92], the coverage probability and the area spectral efficiency are analyzed for the ULof dense small cell networks considering a practical path loss model incorporating181.3. Literature Reviewboth line-of-sight (LoS) and non-line-of-sight (NLoS) transmissions. Moreover, [93]analyzes a two-tier UL cellular network comprising femtocells and a macrocell, withcomponents spatially deployed based on an independent Poisson point processes.Through an open access assignment policy, analytical upper and lower bounds onthe outage probabilities of femto users and macro users are derived affected by fadingand path loss. At this point, once can see that OMA networks were studied fromdifferent perspectives, however, the performance of large-scale OMA networks in theFBR was not studied yet.1.3.2 Cellular Network using Non-orthogonal MultipleAccessNOMA is a promising method for enhancing the capacity of the network by re-source sharing, as it enables multiple users to transmit on the same time/frequencyresources with different power levels. Hence, it offers higher spectral efficiency, mas-sive connectivity by serving more users simultaneously, and lower latency due tosimultaneous transmission all the time rather than a dedicated scheduled time slot.This makes NOMA a good candidate for future technology. The performance of theNOMA scheme in the FBR and AR has been analyzed in many works [30, 72, 78, 94–103].For the AR, an analytical framework on system coverage and average user achiev-able rate in a DL NOMA system is proposed in [94] taking into consideration intercellinterference and comparing it to OMA. Also, SINR is characterized and the transmis-sion success probabilities is derived for two clustering scenarios in [95]. However, forthe FBR, the achievable effective capacity of two-user NOMA and its OMA counter-part is characterized under delay quality-of-service constraints for weak and strongusers under Rayleigh fading channels in [30]. Moreover, a closed-form expression forthe block error rates of NOMA users is derived and the near-optimal power allocationcoefficients and blocklength is obtained to ensure certain reliability in [96]. A closedform expression for the rate and outage probability for Gaussian and Rayleigh fad-ing channels is proposed in [72]. Furthermore, [72] studies the system performancein terms of reliability and throughput when applying the type-I Automatic repeatrequest (ARQ) protocol with limited number of retransmissions. Additionally, [97]derives the information-theoretic achievable rates for fading channel accurately up tothe second-order as a function of the blocklength under finite-alphabet inputs and anaverage total power constraint. The thesis also formulates rate and error-rate perfor-mance optimization problems in the FBR for finite-alphabet inputs from which theoptimal power allocation and constellation-rotation can be derived, and compares the191.3. Literature Reviewtheoretical results with the performance of polar codes and LDPC codes. In [78], theauthors explore power allocation strategies specifically tailored for URLLC in NOMAsystems, in the FBR. Moreover, [98] studies the effect of using NOMA in the UL ofa mobile network in reducing the queueing delay, and proposes closed-form approx-imations for the corresponding outage and error probabilities accounting for imper-fect channel state information and finite-length coding. The work in [99] addressesthe challenges of ultra-reliable low-latency tasks in NOMA systems, employing jointdecoding techniques within the constraints of finite blocklength transmissions. Itstudies the decoding error probability and power allocation optimization, aiming atmaximizing the effective throughput for a cell-center user under a minimum-requiredeffective throughput constraint at a cell-edge user. Moreover, in [100], the max-minfairness is studied for UL FBR-NOMA and a joint optimization of the power controland transmission rates is considered to improve the max-min effective throughputand simultaneously provide the quality-of-service guarantee of transmission reliabil-ity for each user. Lastly, [101] investigates the average block error rate in DL NOMAshort-packet communication systems, particularly in Nakagami-m fading channels.Also, [102] studies the combined utilization of NOMA and millimeter wave communi-cations in large-scale networks using stochastic geometry taking into considerationsfactors such as spatial randomness, blockages affecting channel characteristics, powerallocation, and imperfections in interference cancellation. The study concludes thatusing NOMA under millimeter wave channels enhances DL rates while minimallyimpacting individual users, and closest line-of-sight base station association provesadvantageous. Increased density of base station deployment is also found to furtherimprove performance. In [103], an analytical framework for assessing the performanceof large-scale co-channel UL and DL NOMA networks is introduced. The frameworkallows for deriving the meta distribution and moments of the conditional successprobability. By analyzing various network performance metrics, the study offers in-sights into success probability, mean local delay, and meta distribution. Moreover,the framework is applied to optimize power allocations for DL NOMA users to max-imize user success probability, considering latency constraints. The study concludesthat maximizing success probability with latency constraints can significantly affectoptimal power solutions and tends to favor orthogonal multiple access for low SIRthresholds. At this point, one can see that the NOMA scheme has been studiedfrom different perspectives, however, the performance characterization of NOMAlarge-scale networks in the FBR taking into consideration intercell and intracell in-terference is still an open problem.201.3. Literature Review1.3.3 Coding Schemes for Large-Scale Networks in theFBRIn practical communication networks, channel coding is used to increase the re-liability of the wireless communications system by adding extra bits in a controlledfashion and the modulation scheme is used to enhance the spectral efficiency of thecommunication system. There are two approaches to design these blocks: conven-tional approach and NN-based approach. The conventional approach designs eachblock separately using known coding schemes like convolutional codes, Turbo codes,LDPC codes, polar codes, and LDPC codes. Convolutional coding is commonly usedfor its ability to correct random bit errors and used in 2G (GSM) and 3G (UMTS)networks. Turbo coding and LDPC codes are used in 3G and 4G (LTE) systems[104]. More recently, the 5G in 3GPP Release 16 Specification # 38.212 [47] iden-tifies polar codes [48, 49] for relatively long blocklength and Reed-Muller for shortblocklength, with QAM. Such design does not guarantee optimality compared to thetheoretical maximum in the FBR. To enhance the performance of conventional com-munication systems, joint design and optimization of coding and modulation in theFBR is necessary. Although this is often a tedious task, MLPCM was introduced as ajoint coding and modulation scheme that performs better than polar codes for Gaus-sian channels [51, 52], but unfortunately does not achieve the maximum achievablerates under short code-length. As mentioned before, we will consider a MLPCMscheme with polar coding since polar codes are already being used for moderateblocklengths in 5G. These limitations motivated the investigation of NN for codedesign in communication systems, after their success in various fields.The intersection of machine learning and finite blocklength information theory isan area of active research. Techniques from machine learning, such as deep learn-ing and reinforcement learning, are being explored in [53–61] to design codes andmodulation schemes optimized for finite blocklength scenarios. NNs are found tobe a convenient tool for code design in communication systems, yielding better per-formance than conventional schemes in terms of error rates [53–61]. Work in thisarea started by replacing one or more blocks of the physical layer with NNs for per-formance improvements, such as neural decoders [53] and neural modulators anddemodulators [56, 57, 61]. Then, it was extended to propose an E2E autoencoder(AE) based communication system in [54, 55, 60]. More advanced NNs were pro-posed in [58] where a scheme using a Turbo-AE together with a feed-forward neuralnetwork (FFNN) for modulation is proposed. Yet, these methods only consideredeither specific coding rates (like a rate 1/3 turbo code in [58]), encoding a smallnumber of symbols (ranging from 1 to 15) [53–58, 60, 61], or a moderate number of211.4. Research Gapinformation bits due to the complexity of the NN. Moreover, only the bit error rate(BER) of these NN was investigated. As such, the following question arises: Canan NN be used to design a code that approaches/achieves the information-theoreticmaximum achievable rate?To answer this question, one has to study the achievable information rate undera constraint on the FER. In [59], the authors investigated the FER of an end-to-end communication system where the coding and modulation are designed usingan FFNN, with a limited number of symbols (up to 15). However, [59] did notinvestigate the maximum achievable rate of this architecture. This also applies to[53–58, 60, 61], it is currently unknown if any of the NNs proposed in these worksachieves good performance compared to the maximum rate in the FBR, since thisperspective was not studied in these papers. Complexity is also often a challengein this area, for instance, training an FFNN-AE like in [59] poses a challenge interms of complexity as the blocklength increases and makes it difficult to study theperformance under a larger number of symbols (say hundreds).1.4 Research GapAs can be observed from the previous discussion, the emergence of B5G networkshas inaugurated in a new era of connectivity, marked by an unprecedented diversityof applications and utilization scenarios. However, this proliferation of use-cases alsobrings with it a set of stringent requirements, particularly concerning latency, thattraditional coding schemes struggle to accommodate effectively [2]. As such, there’s adiscernible paradigm shift towards the adoption of short codes within the B5G frame-work, driven by the necessity to meet demanding latency and power consumptionthresholds. Yet, this transition to shorter code lengths inherently introduces errorsand diminishes information coding rates, thereby necessitating a delicate balance be-tween performance and efficiency. Notably, the tolerance for these errors varies acrossapplications, with URLLC demanding error rates as stringent as 10−9 [5]. There-fore, the study of finite blocklength coding theory has emerged as a critical avenuefor understanding and optimizing coding performance within B5G networks [6, 7].This theoretical framework offers a nuanced perspective, characterizing informationcoding rates as a function of code length and FER. Consequently, for the designand implementation of context-aware short codes in B5G networks, it becomes im-perative to extend finite blocklength coding theory into interference-prone scenarios,accounting for the intrinsic large-scale and dense deployments characteristic of futurenetworks.While existing literature has made notable strides in exploring coding schemes221.4. Research GapFigure 1.1: Research Gap. The citations in the circles indicate sample existing works,the area marked with the black line indicates the research gap, and our contributionsin the form of publications where the labels (J=Journal, C=Conference) refer to thepapers in the preface on page vi and vii and are highlighted in red.for smaller networks with large blocklengths, as depicted in Fig. 1.1, a discerniblegap persists concerning large-scale networks operating within the FBR. This is whereour thesis aims to make significant contributions, leveraging coding theory alongsidestochastic geometry tools to fill this void. Our approach entails a meticulous exami-nation of research gaps specific to each network type. In the case of OMA large-scalenetworks, the absence of a performance framework tailored to the FBR poses a no-table challenge, underscoring the need for tailored solutions to meet the evolvingdemands of B5G environments. Similarly, prevailing studies on NOMA large-scalenetworks predominantly focus on point-to-point transmission, overlooking the intri-cacies of intercell interference induced by neighboring NOMA clusters. Moreover,the nuanced characteristics and challenges of large-scale NOMA networks withinthe FBR remain largely unexplored in current literature. Our thesis endeavors tofill these critical research gaps, contributing significantly to the theoretical underpin-nings of wireless communication networks. By bridging these research gaps, our over-arching goal is to develop a comprehensive theoretical framework capable of preciselycharacterizing the performance of large-scale networks in the FBR. This frameworkrepresents a pivotal advancement in the domain of wireless communication networks,offering insights into network performance under latency and reliability constraints.231.5. Thesis ObjectivesThe accuracy provided by this framework promises to unlock significant opportu-nities for next-generation communication systems, thereby catalyzing the continuedevolution of wireless technology and its myriad applications.1.5 Thesis ObjectivesGiven the aforementioned background and identified research gaps, this disser-tation primarily focuses on developing a mathematical framework to analyze theperformance of various large-scale networks in the FBR. We focus on characterizingthe performance of the UL and DL OMA networks in terms of average coding rate,rate outage probability, reliability, and meta distribution of the coding rate and theDL NOMA networks in terms of average coding rate and average max-min rate. Theresearch focus can be narrowed down into the following research objectives.1. Analyzing the performance of a large-scale DL cellular networks using orthog-onal multiple access in the FBR.2. Analyzing the performance of a large-scale UL cellular networks using orthog-onal multiple access in the FBR using multiple power control schemes.3. Analyzing the performance of a large-scale DL cellular networks using non-orthogonal multiple access in the FBR by characterizing the achievable averagerate region and the max-min rate.4. Designing a machine learning based autoencoder that approaches the maximumcoding rate of an AWGN channel.1.6 Contribution and OutlineThe contributions of this thesis are strategically designed to address and alignwith some emerging 5G and beyond 5G use-cases which imposes stringent latencyconstraints. The key contribution of this thesis is to develop a framework to char-acterize the performance of large-scale networks in the FBR. Within this theoreticalframework, we defined the performance metrics considered in the FBR. Then usingstochastic geometry tools, we characterized these performance metrics to providetractable expressions. A key challenge posed was that not all performance metricscould be characterized in a tractable manner. For such intractable metrics, we pro-vided bounds and/or tight approximations on the performance metrics. Specifically,this thesis presents the following four main contributions.241.6. Contribution and Outline1. Analyzing the performance of a large-scale DL cellular networks using orthog-onal multiple access in the FBR. (Chapter 2)We developed a novel mathematical analysis to characterize the trade-off be-tween the code length, the FER, and the information coding rate in large-scaleDL OMA networks. We characterize the performance of a large-scale DL net-work in the FBR as a function of the blocklength and FER in terms of theaverage coding rate under a random distance between the user and its servingBS, under Gaussian codebooks, the average coding rate using QAM constella-tions, the rate outage probability, the reliability bounds and approximations,and the approximation of the coding rate meta distribution. Additionally,the thesis simulates the average coding rate achieved by MLPCM for differentQAM modulation orders in comparison with the obtained theoretical bench-marks. All results are validated via Monte Carlo simulations. It is shown thatMLPCM achieves rates close to the derived theoretical benchmarks. It is alsoshown that the derived outage probability bounds are tight and coincide withsimulation results. Additionally, it is shown that the proposed approximationof the coding rate meta-distribution is fairly tight under moderate and highSINR. Moreover, qualitative and quantitative comparisons are conducted withthe performance in the asymptotic regime to motivate the importance of ana-lyzing the performance in the FBR and demonstrate the discrepancy with ARresults. We also investigate the effect of various network parameters in theFBR.2. Analyzing the performance of a large-scale UL cellular networks using orthogo-nal multiple access in the FBR using multiple power control schemes. (Chapter3)The DL is operationally different from the UL in that uses power control. Inthe DL, the BSs are powerful units with large power sources, unlike the UEswhich are power-limited units. This necessitates the use of power control in ULnetworks, which in return affects the analysis. We proposed a theoretical anal-ysis of the performance of large-scale UL OMA networks in terms of averagecoding rate, rate outage probability, and coding rate meta distribution underfractional path-loss inversion power control. Furthermore, two special caseswill be studied which are truncated channel inversion (TCI) [88] and channelinversion with maximum power transmission (CIMP) [89]. Moreover, a com-prehensive discussion is provided, delving into the qualitative and quantitativenuances of the numerical findings, and examining the impact of network pa-rameters on the performance of the large-scale network in the FBR. Moreover,251.6. Contribution and Outlinea comparison between power control schemes is conducted, providing insightsinto their performance and leading to conclusive recommendations for the pre-ferred scheme.3. Analyzing the performance of a large-scale cellular networks using non-orthog-onal multiple access in the FBR by characterizing the achievable average rateregion and the max-min rate. (Chapter 4)While the prevailing understanding acknowledges NOMA’s superiority overOMA in terms of achieving higher data rates, our inquiry pivots on whetherthe NOMA’s out-performance persist in the realm of large-scale networks sub-ject to stringent reliability and latency constraints, and what quantifiable gapexists between these two multiple access techniques? To address this, we usedthe theory of coding in the FBR in conjunction with stochastic geometry toolsto characterize the average rate region for a large-scale NOMA network under aglobal reliability constraint and a short blocklength and then compare the per-formance of NOMA to OMA for a large-scale network in the FBR. We derivea theoretical achievable rate region for the large-scale NOMA network under aglobal FER and a finite blocklength requirement. This region serves as an innerbound for the maximum achievable rate region. We derive an outer bound onthe maximum achievable rate region for the large-scale NOMA network in theFBR, providing a benchmark for system performance. We express the max-min average rate and propose an algorithm to find an optimal max-min rate.Also, we propose a practical power allocation scheme to pursue a fair perfor-mance between the two NOMA users. We also investigate the performance ofa practical scheme to achieve rates close to the theoretical achievable bound.Specifically, we consider MLPCM and show that it can achieve rates close tothe theoretical achievable inner bound. We compare the performance of thelarge-scale OMA network with the performance of a large-scale NOMA net-work in the FBR. We study the behavior of the performance gap between theNOMA and OMA schemes with respect to network parameters and provide aqualitative and quantitative analysis, deriving some useful insights on the per-formance gain of NOMA in a large-scale network in the FBR. The comparisonshows that the performance gap between OMA and NOMA increases as wemove towards shorter blocklengths and stringent reliability constraints, whichpromotes the use of NOMA networks in future networks. Additionally, weexamine the impact of various network parameters on the large-scale NOMAnetwork performance. Specifically, we analyze the effect of BS density andreliability constraints.261.6. Contribution and Outline4. Designing a machine learning based autoencoder that approaches the maximumcoding rate of an AWGN channel. (Chapter 5)Exploring a practical scheme is essential to be investigated to provide a perfor-mance close to the theoretical performance. NNs have been a convenient toolfor code design in communication systems, yielding better performance thanconventional schemes in terms of error rates. In this thesis, we consider anAWGN channel as preliminary results which can be extended. Since Convo-lutional Neural Networks (CNNs) are known to be easier to train than manyother NNs, we investigates the potential of Convolutional Neural Networksautoencoders (CNN-AE) in approaching the theoretical maximum achievablerate over a Gaussian channel for a range of signal-to-noise ratios at a fixedblocklength and target FER, which is a different perspective compared to ex-isting works that explore the use of CNNs from bit-error and symbol-errorrate perspectives. Hence, we propose a CNN-AE architecture with few hyper-parameters and explain how to choose these parameters. Then, we jointlyoptimize the transmitter and receiver by training the AE, thus overcomingthe mathematical complexity of finding an optimal code. Then, we comparethe results with the performance of a scheme that combines polar codes andQAM, a scheme that combines Reed-Muller codes with QAM, a scheme thatuses MLPCM [52], and the turbo AE-based scheme proposed in [58]. Resultsshow that the proposed CNN-AE approaches the maximum coding rate in [6]and outperforms the benchmark schemes, which highlights the capability ofCNN-AEs in finding a good code that approaches the theoretical maximumrate in the FBR. Finally, we comment on the complexity and performance ofsimilar designs using recurrent neural networks (RNN) or an FFNN instead ofthe CNN.The rest of this thesis is organized as follows. In Chapter 2, we study the perfor-mance of cellular DL large-scale networks in the FBR using OMA. Then, in Chapter3, we extend the study to characterize the performance of cellular UL large-scalenetworks using OMA. In Chapter 4, we investigate the performance of cellular DLlarge-scale networks in the FBR using NOMA, and a performance comparison be-tween OMA and NOMA is conducted. Next, in Chapter 5, a neural network basedautoencoder is proposed.27Chapter 2Large-Scale Downlink OMANetworks in the FiniteBlocklength RegimeIn this chapter, novel mathematical analysis is developed to characterize thetrade-off between the code length, the FER, and the information coding rate inlarge-scale DL networks using OMA. To this end, achievable rates and outage inthe FBR are characterized for Gaussian codebooks as well as practical modulationschemes for a stationary and random distance between the UE and its serving BS.MLPCM [105–108], is presented as a practical validation for the obtained theoreticalachievable rates. All the analyses are validated using Monte Carlo simulations andinsights are drawn for each of the DL and UL.The chapter starts with presenting the system model and assumptions of theanalysis. Then, the average coding rate of a large-scale DL network in the FBR isderived, under Gaussian codebooks and under a constraint of QAM constellations,in Sec. 2.2. In Sec. 2.3 and Sec. 2.4, bounds on the rate outage probability arederived, followed by the characterization of the reliability, and the meta distributionof the coding rate. Next, in Sec. 2.5, the theoretical results are numerically evaluatedand compared with the performance of MLPCM as a practical scheme. Moreover, acomprehensive discussion is provided, delving into the qualitative and quantitativenuances of the numerical findings, and examining the impact of network parameterson performance. Finally, the chapter is concluded in Sec. 2.6.2.1 System ModelConsider a single-tier OMA large-scale DL network with universal frequency reuseand no intra-cell interference. The BSs are located according to a PPP3 Ψ ⊂ R2 with3The network is approximated as a PPP with interference exclusion region of radius r0, i.e., thedistance between a UE and its serving BS is r0 and there is no interfering BS within the regionwith a radius r0 from the UE [65].282.1. System Modelr0r1r2r3r4r5r6r7r8 r9InterferenceexclusionregionFigure 2.1: A realization of a cellular network with exclusion region of r0 betweenthe user of interest and the serving BS, where BSs are randomly distributed withdistances ri > r0 to the user.intensity λ BS/km2, as depicted in Fig. 2.1. The network applies universal frequencyreuse with one UE served per BS at a given resource block. Each UE is served byits geographically closest BS, where the distance between a UE and its serving BSis denoted as r0. The distances to the interfering BSs ordered with respect to theintended UE are denoted as r1, r2, . . . , ri, . . . such that ri+1 > ri. For the sake ofsimple presentation, the set Ψ˜ ∈ R is defined as the BSs distances to the desired UE.Hence, the received signal at the desired UE is given byy =√Ph0r−η/20 s0 +∑ri∈Ψ˜\{r0}√Phir−η/2i si︸ ︷︷ ︸Iagg+w, (2.1)where P is the transmit power of the BSs, h0 (resp. hi) represents the channelfading of the intended (resp. ith interfering) channel, η is the path loss exponent,s0 (resp. si) is a unit average power codeword symbol transmitted by the serving(resp. ith interfering) BS, Iagg is the aggregate interference from all other BSs, andw ∼ CN (0, σ2w) is circularly symmetric complex Gaussian noise with zero mean andvariance σ2w. We model h0 and hi to be circularly symmetric complex Gaussianwith zero mean and unit variance (modeling Rayleigh fading). An independent andidentically distributed block fading channel model is assumed where the channelcoefficients h0 and hi remain constant for a duration of L consecutive symbols andchange to independent realizations at the end of each symbol interval.The serving BS wants to send information to the UE using codewords from a codewith the rate R and length n symbols, where n can be defined following a latency292.2. Average Rate Analysis in the Finite Blocklength Regimeand/or energy consumption constraint. To ensure that the channel remains constantduring the transmission of n symbols, we require n = L/l for some integer l [7].This ensures that the channels remain constant during the transmission, but makesconsecutive transmission blocks have identical channels. Nonetheless, by consideringthe sequence of blocks consisting of the first block (of n symbols) in each coherenceinterval (of L symbols), we have independent channels and we can derive the averageperformance over all transmissions. This is because the remaining sequences of blocks(such as the second frame in each coherence interval) have identical statistics.To this end, two modes of operation are considered. The first mode assumesknowledge of channel-state information (CSI) at the transmitter and the receiverin the form of SINR availability.4 Under this mode, we investigate the averagecoding rate E{R} as a function of the FER and codelength (see Sec. 2.2). Thesecond mode assumes that the CSI is unknown at the transmitter but known atthe receiver. Under this scenario, we derive the outage probability and utilize thecoding-rate meta distribution to characterize the percentile of users achieving a giventransmission reliability (See Sec. 2.3).2.2 Average Rate Analysis in the FiniteBlocklength RegimeIn this section, the average coding rate for a large-scale network in the FBRis proposed, first without a constraint on the used constellation, and then underfinite constellations (QAM). Then, a practical MLPCM is presented to validate theachievability of the derived interference-prone theoretical benchmarks.Considering the first mode of operation, we assume that the SINR is availableat the transmitter and the receiver. Hence, the coding rate can be adapted basedon the SINR in each transmission block so that the transmission using codelength nis successful with a desired FER . This rate adaptation can be realized using thefollowing lemma, which characterizes the maximum achievable rate over an AWGNchannel in the FBR [6].Lemma 2.1. ([6]) For an AWGN channel with signal to noise power ratio (SNR)α, blocklength n, and FER  ∈ (0, 0.5), the maximum coding rate is approximatedas5Rn,(α) = C∞,0(α)−√V (α)Q−1()√n+12nlog2 n, (2.2)4Assuming SINR knowledge benchmarks the performance of the network.5This approximation is shown to be tight at blocklengths (> 100) and FER > 10−6 in [6].302.2. Average Rate Analysis in the Finite Blocklength Regimewhere C∞,0(α) = log2(1 + α) is the AWGN channel capacity in the AR, V (α) =α(α+2)(α+1)2log22(e) is known as the channel dispersion, and Q−1(·) is the inverse of theQ-function.In this mode, due to the random changes of the channels h0 and hi, we characterizeperformance by the average coding rate subject to a target FER  and codelength n.However, Lemma 2.1 is derived for an AWGN channel, and hence, cannot be directlyapplied to a large-scale network due to the additional non-Gaussian interferenceterm Iagg in (2.1). To overcome this, we utilize the equivalence in distribution (EiD)approach to express the interference as a conditionally Gaussian random variable [65,109, 110], as described next.2.2.1 Conditional Gaussian RepresentationUsing the EiD approach, we represent the aggregate interference for a given r0as followsIaggeid=√Bg, (2.3)whereeid= denotes the EiD, g ∼ CN (0, 1) and B > 0 is a positive random variableindependent of g with a probability density function (PDF) whose Laplace transform(LT) is given by [65],LB(u) = exp{ ∞∑k=1(−1)k2piλr20(Prη0)k E{|s|2k}uk(ηk − 2)k!}. (2.4)where s is the transmitted symbol from an arbitrary codebook by an arbitrary BSand follows the same distribution as s0 and si.The EiD in (2.3) is proved by showing that the product√Bg has the samecharacteristic function as Iagg [65]. Using (2.3), the equivalent system model isexpressed asy =√Ph0r−η/20 s0 +√Bg + w,where for a given B, the lumped term √Bg + w is Gaussian with zero mean andvariance B+ σ2w. Hence, the resulting conditional SINR, given B and h0, is given byΥ =|h0|2Pr−η0B + σ2w. (2.5)312.2. Average Rate Analysis in the Finite Blocklength RegimeThe random variable B in (2.5) augments the noise power with the aggregatenetwork interference power, for any constellation of s0 and si. If s0 and si is Gaussiandistributed (Gaussian codebooks), it can be shown that (2.4) simplifies to [65]LBg(u) = exp{−2piλuPr2−η0η − 2 2F1(1, 1− 2η; 2− 2η;−uPrη0)}. (2.6)Since at this point, interference is modeled as conditionally Gaussian, and sinceLemma 2.1 provides a tight approximation when noise is Gaussian, then the tight-ness of the approximation provided in Lemma 2.1 holds here too. Thus, from thispoint onward, we focus on analyzing performance using Rn, as a surrogate for themaximum achievable rate in the FBR given that this is a tight approximation. Underthis framework, the maximum average coding rate is defined next.Definition 2.2. Given a DL large-scale network with a distance r0 between thedesired UE and the serving BS as defined in (1), assuming the SINR knowledge isavailable at the transmitter and the receiver, we define the maximum average codingrate in the FBR with blocklength n and FER  with α0 =Pr−η0σ2w, as RDLn,(α0) =Eh0,B{Rn,(Υ)}, where Υ is the SINR defined in (2.5) and Rn,(Υ) is as defined inLemma 2.1.Next, we use Lemma 2.1 and Def. 2.2 to characterize the average coding rate fora large-scale network in the FBR, first using Gaussian codebooks, and then underfinite constellations (QAM). Then, we present a practical MLPCM to validate theachievability of the derived theoretical benchmarks.2.2.2 Average Coding RateBy virtue of the EiD, the AWGN results in Lemma 2.1 can be now extended tolarge-scale networks. For a given intended UE distance r0, the FBR average codingrate is characterized in the following theorem.Theorem 2.3. The maximum average coding rate of the large-scale network modeledby (2.1) with Gaussian codebooks, blocklength n, FER , and distance r0 between theUE and its serving BS is given byRDLn,(α0) = C∞,0(α0)−V(α0)Q−1()√n+12nlog2 n, (2.7)whereC∞,0(α0) =∫ ∞0exp(−2c − 1α0)LBg{2c − 1Pr−η0}dc,322.2. Average Rate Analysis in the Finite Blocklength RegimeV(α0) =∫ log2(e)0e−z(v)α0 LBg{rη0P z(v)}dv,where α0 =Pr−η0σ2w, z(v) =√11− v2log22(e)− 1, and LBg{·} is given in (2.6).Proof. By virtue of the EiD approach, Lemma 2.1 is applicable to large-scale net-works by replacing α by Υ where Υ is defined in (2.5), which leads to a max-imum coding rate Rn,(Υ) under a given h0 and B. The average rate is thenRDLn,(α0) = E{Rn,(Υ)}, where the averaging is over h0 and B. Thus, we need toaverage C∞,0(Υ) and√V (Υ) over B and h0. The term C∞,0(α0) = E{C∞,0(Υ)} isthe average capacity in a large-scale network in the AR with CSI available at theBSs, which was derived in [17]. It remains to find E{√V (Υ)} which is shown tobe equal to V(α0). From the definition of channel dispersion provided in (2.2), theaverage square root channel dispersion V(α0) can be written as followsV(α0) = E{√V (Υ)}= E{log2(e)√(1− 1(Υ + 1)2)}=∫ ∞0(1− F√V (v))dv, (2.8)where F√V (v) is the cumulative distribution function (CDF) of√V (Υ), and the laststep follows as an application of Fubini’s theorem [111]. The CDF of√V (Υ) for agiven Bg is given byF√V (v|Bg) = P(√V (Υ) < v|Bg)= P(|h0|2<(1α0+BgPr−η0)z(v)∣∣∣Bg)= 1− exp(−(1α0+BgPr−η0)z(v)), (2.9)where the last step follows from the exponential distribution of |h0|2. Averaging withrespect to the interference term Bg yields EBg{F√V (v|Bg)}=EBg{1− e−z(v)α0 e−Bgrη0P z(v)}.Hence, the CDF of the square root of the channel dispersion is given asF√V (v) =1− exp(−z(v)α0)LBg{rη0P z(v)}, 0 ≤ v < log2(e), (2.10)332.2. Average Rate Analysis in the Finite Blocklength Regimewhich is obtained using the definition of Laplace transform (Lx{u} = E {e−ux}). Bysubstituting (2.10) in (2.8), we obtainV(α0) =∫ log2(e)0exp(−z(v)α0)LBg{rη0P z(v)}dv,as defined aboveTheorem 1 governs the maximum average rate at a fixed distance r0 between theUE and its serving BS, considering the stochastic locations of the interfering BSsand the Rayleigh fading environment. The following corollary generalizes Theorem1 by accounting for the randomness of r0.Corollary 1. The average coding rate of the large-scale network modeled by (2.1)with blocklength n, FER , and Gaussian codebooks is given byR¯DLn, = C¯∞,0 −V¯Q−1()√n+12nlog2 n,where C¯∞,0 =∫∞0C∞,0(α0)fr0(r0)dr0 and V¯ =∫∞0V(α0)fr0(r0)dr0.Proof. The average capacity and average channel dispersion are derived by directlyaveraging (2.7) with respect to r0 which has the following probability density function[17]fr0(r0) = 2piλr0e−piλr20 , 0 < r0 <∞. (2.11)As can be seen in Theorem 2.3 and Corollary 1, the average coding rate of thelarge-scale network in the FBR is the average capacity in the AR, plus a penaltyterm (loss) which is a function of the blocklength n, and FER , in addition tothe average SNR α0 and the stochastic geometry of the network, manifested in theLaplace transform term. This penalty term implies a trade-off between the reliabilityand the average rate: the higher the reliability requirement (low FER), the lower therate. However, for a given average achievable rate, the reliability can be increased( decreased) by increasing the blocklength, converging to the AR performance as ngrows to infinity.Theorem 2.3 expresses the average coding rate under Gaussian codebooks, whichis of theoretical relevance. Characterizing the performance in the FBR when usinga finite constellation is of practical interest and is discussed next.342.2. Average Rate Analysis in the Finite Blocklength Regime2.2.3 Average Coding Rate Using Finite ConstellationsThis section focuses on achievable rates in the FBR under transmission usingstandard finite constellations. The maximum achievable rate using an M -ary con-stellation over an AWGN channel in the FBR was given in [112]. Following the samemethodology as in Sec. 2.2.2, we use the EiD approach to extend the results of [112]to large-scale networks.Consider an encoder that maps messages from a message set into a length nsequence of symbols chosen from an M -ary constellation consisting of M complex-valued symbols s1, s2, ..., sM . Using an M -ary QAM constellation, denote by C(M)∞,0(α0)the average capacity achieved in the AR, by V (M)(α0) the average square root chan-nel dispersion, and by R(M)n, (α0) the average coding rate under a blocklength n andFER . To characterize the average capacity of the M -ary QAM under FBR, theconditional mutual information I(s0; y|h0,B, r0) has to be averaged with respect tothe aggregate interference power B and the channel gain h0. However, the distribu-tion of the aggregate interference power B is unknown which leads to an intractableexpression. Although such an expression can be evaluated using Monte Carlo sim-ulations, an expression that is amenable to numerical integration is preferable. Asimilar argument applies for V (M)(α0). Hence, to obtain tractable integral expres-sions, we adopt an approximation for the distribution of the interference power Bproposed in [113, 114] which was shown to be a tight approximation. In particular,in [113, 114], the distribution of the interference power B was approximated by aGamma distributionf(x; q, θ1) =xq−1e−xθ1Γ(q)θq1, x > 0 (2.12)where q =4piλr20(η−1)(η−2)2 is the shape parameter, and θ1 =(η−2)P2(η−1)rη0 is the scale parameter.The approximation is based on a moment-matching approach. The first two momentsof B can be obtained from (2.4) asE{B} = dLB(u)du∣∣∣∣u=0=2piλr2−η0 P(η − 2) ,E{B2} = d2LB(u)du2∣∣∣∣u=0=(2piλr2−η0 P(η − 2))2+piλr2−2η0 P2(η − 1) .The scale and shape parameters of the gamma distribution are then obtained asq = E{B}2Var{B} and θ1 =Var{B}E{B} , where Var{B} = E{B2} − E{B}2.352.2. Average Rate Analysis in the Finite Blocklength RegimeFigure 2.2: CDF of the interference power B and its approximation at different valuesofP ,λ and r0Fig. 2.2 numerically demonstrates the accuracy of this approximation by plot-ting the approximate CDF of B (using (2.12)) and its exact CDF obtained from anumerical inversion of (2.4). The figure shows that the Gamma approximation in(2.12) provides a fairly tight approximation for the CDF of B. Note that the Gammaapproximation is crucial to maintain the tractability of the analysis. Using this ap-proximation, the average coding rate of a large-scale network in the FBR, at a givendistance r0 and using M -ary QAM constellation, is characterized.The maximum achievable rate for a finite constellation is provided in [112]. How-ever, the channel dispersion in [112] is defined as the variance of the informationdensity (V = V ar(i(x; y))). This definition is considered to be imprecise accordingto [6], where the channel dispersion is defined as the conditional variance of the in-formation density for finite constellation and is defined as the unconditional varianceof the information density in the case of using Gaussian signals. Therefore, in thischapter, we derive the maximum achievable rate using this definition for large-scalenetworks in the FBR in the following theorem.Theorem 2.4. For a DL large-scale network with BS density λ BS/km2, blocklengthn, FER , distance r0 between the UE and the serving BS, and an M-ary constellationwith symbols {sm}Mm=1 where sm = [Re{sm}, Im{sm}]T and Re{sm} and Im{sm}are the real and imaginary parts of sm, respectively, if the interference power Bfollows a Gamma distribution (2.12), then the average achievable coding rate can be362.2. Average Rate Analysis in the Finite Blocklength Regimeexpressed byR(M)n, (α0) = C(M)∞,0(α0)−V (M)(α0)Q−1()√n+12nlog2 n, (2.13)where α0 =Pr−η0σ2w,C(M)∞,0(α0) = log2(M)−1MpiM∑m=1∫ ∞0∫R2e−‖t‖2gm(t)fΥ(Υ|r0)dt dΥ (2.14)V (M)(α0) =∫ ∞0√VM(Υ)fΥ(Υ|r0) dΥ (2.15)VM(Υ) =M∑m=1∫R2e−‖t‖2Mpigm(t)2dt−(∫R2e−‖t‖2Mpigm(t)dt)2 (2.16)where gm(t) = log2(∑Ml=1 e−2√ΥtT (sm−sl)−Υ‖sm−sl‖2), t = [t1, t2]T is a 2-D real-valuedvector, Υ is the SINR given in (2.5), andfΥ(Υ|r0) =exp{−Υσ2wPr−η0}Pr−η0(1 + θ1rη0Υ/P)σ2w + qθ1(1 + θ1rη0Υ/P)q+1, 0 ≤ Υ ≤ ∞. (2.17)Proof. The average capacity of an M -ary constellation (C(M)∞,0(α0)) at a fixed r0 = ris derived by averaging the conditional mutual information between the transmittedsymbol s0 ∈ {sm}Mm=1 and the received signal y, i.e., I(s0; y|Υ, r0 = r), provided in[112], with respect to the SINR Υ which is a function of interference power B andfading channel h0 statistics. Similarly, the average square-root channel dispersion(V (M)(α0)) is derived by averaging the channel dispersion in (2.16) with respect to Υ.The derivation of (2.16) is provided in App. 6.2. However, instead of averaging overthe distribution of Υ which is unknown, we average over fΥ(Υ|r0) in (2.17) whichis obtained using the Gamma approximation provided in (2.12). The distribution ofthe SINR Υ at a fixed r0 is derived by defining the CDF as followsFΥ(Υ|r0) = P(|h0|2< Υ(B + σ2w)Pr−η0)= 1− E{e−ΥBPr−η0}e−Υσ2wPr−η0= 1− LB{ΥPr−η0}e−Υσ2wPr−η0 ,372.2. Average Rate Analysis in the Finite Blocklength Regimeusing the Gamma approximation provided in (2.12). The PDF of the SINR Υ at afixed r0 is given by differentiating the CDF as followsf(Υ|r0) = e−Υσ2wPr−η0Pr−η0(1 +θ1rη0 ΥP )σ2w + qθ1(1 +θ1rη0 ΥP )q+1, (2.18)this proves (2.17).The following corollary generalizes Theorem 2 by accounting for the randomnessof r0.Corollary 2. The average achievable coding rate of the large-scale network modeledby (2.1) with blocklength n, FER , and using an M -ary constellation with symbols{sm}Mm=1 is given byR¯(M)n, ≈ C¯(M)∞,0 −V¯ (M)Q−1()√n+12nlog2 n, (2.19)where C¯(M)∞,0 =∫∞0C(M)∞,0(α0)fr0(r0)dr0, V¯ (M) =∫∞0V (M)(α0)fr0(r0)dr0, and fr0(r0) as de-fined in (2.11).Proof. This is derived by averaging (2.13) with respect to r0.The average achievable rate depends on the distances between symbols in theconstellation set, i.e. ‖dml‖= ‖sm − sl‖ for m 6= l, where neighboring symbolscontribute more to error than far ones. Such distances are in turn affected by theconstellation choice. Moreover, the average achievable rate is affected by the networkparameters λ, r0 which will be investigated later in the numerical evaluations section.The achievable rates provided so far are theoretical. In the next section, weuse MLPCM, which was proposed in [115, 116] to be used in future generations, tovalidate the achievability of the obtained theoretical benchmarks.2.2.4 Multilevel Polar-Coded ModulationPolar codes are capacity-achieving codes for a binary-input channel, proposed byErdal Arkan in 2009 in [48]. The polar encoder is a nonsystematic encoder thatuses the SNR of an AWGN channel to calculate the Bhattacharyya parameter [48]that is then used in the construction of a polar code with rate k/n where k is thenumber of information bits and n is the code length. The code construction isbased on multiple recursive concatenations of a short kernel code which transforms382.2. Average Rate Analysis in the Finite Blocklength RegimeZDemultiplexerz1z2zlog2MEncoderEncoderEncoderEncoderM-aryModulatorq1q2qlog2Ms0ChannelyDecoderz˜1Delay Decoderz˜2Delay Decoderz˜log2M−1z˜log2M−2z˜1Delay Decoderz˜log2Mz˜1MultiplexerZ˜Figure 2.3: Multilevel Coded Modulation Block Diagramthe physical channel into virtual outer channels. When the number of recursionsbecomes large, the n virtual channels polarize into ≈ nC high-reliability channels(noiseless) and ≈ n(1−C) low-reliability channels (extremely noisy), where C is thecapacity of the underlying AWGN channel. Then, information bits are sent over themost reliable channels, and the inputs of the low-reliability channels are frozen tozero. Bits sent over the noiseless channels are received correctly, therefore achievinga rate close to capacity.To validate the theoretical results in Sec. 2.2.2 and 2.2.3, we use the MLPCMscheme shown in Fig. 2.3. The information bits (message) Z ∈ Fk log2 (M)2 are fed into ademultiplexer that splits them into log2(M) sub-messages {zi}log2 (M)i=1 where zi ∈ Fk2.Each sub-message zi is encoded using a polar encoder with rate Rc = k/n to obtaincodewords Q ∈ Flog2(M)×n2 , where qi ∈ Fn2 is the ith row of Q (ith codeword). Then,the transmitter collects one bit from each of the codewords qi to form a log2(M)-bit symbol which is then mapped to a symbol s0 from an M -ary constellation suchas the M -QAM constellation. The symbols are then scaled according to the powerconstraint in Sec. 2.1. The result of repeating this process for all n codewordsymbols of all log2 (M) sub-messages is one MLPCM codeword of length n, whichis then transmitted through the channel. At the receiver side, multistage decodingtakes place to efficiently recover the message. It breaks the decoding process intolog2(M) stages.In each stage i = 1, . . . , log2(M), the receiver feeds the demapper of the ithstage with the received signal y and the submessages recovered from previous stages(z˜1, . . . , z˜i−1), (which is defined as ∅ for i = 1), and the demapper demaps the sym-bols into bits to obtain q˜i, using the multilevel decoding criteria [105]. Then, thedemapped bits q˜i are decoded using a polar decoder to obtain z˜i using the successivecancellation algorithm provided in [51]. Finally, all sub-messages are fed to a multi-392.3. Rate Outage and Reliability Analysis in the Finite Blocklength Regimeplexer to form the message Z˜. This MLPCM will be incorporated in the large-scaleDL network Monte Carlo simulator to validate the achievability of the theoreticalrates obtained in Sec. 2.2.2 and 2.2.3.2.3 Rate Outage and Reliability Analysis in theFinite Blocklength RegimeIn the previous section, we studied the case where the SINR is known at thetransmitter. However, if the BS does not track the SINR, then the rate adaptationinvestigated in the previous section is not possible. Considering the second mode ofoperation, where the SINR is unknown at the transmitter but known at the receiver,the transmitter encodes at a constant (target) rate Rt and blocklength n. As a result,the resulting (conditional) FER conditioned on SINR Υ may or may not satisfy adesired design FER  due to the unknown stochastic variation of the intended channelfading and aggregate interference. In this case, it is important to keep this conditionalFER below a threshold. This is especially relevant bearing in mind that n = L/l,i.e., a large conditional FER means that there are coherence intervals that producelarge FERs for all blocks within these coherence intervals. This can lead to bursts offrame errors when the channel undergoes a period of coherence intervals with weakchannels, which is undesired.Thus, we define an FER threshold ¯, and it is desired to maintain the conditionalFER below ¯. If the conditional FER exceeds ¯, we say that we have an outage. Thisequivalently means that rate Rn,¯(Υ) supported by the channel at the given n andFER threshold ¯ decreases below Rt. This is defined formally as follows.Definition 2.5. Given a target rate Rt and a DL large-scale network with a distancer0 between the desired UE and the serving BS as defined in (2.1), we define the outageprobability in the FBR with blocklength n and FER threshold ¯ as O(r0, Rt, n, ¯) =P(Rn,¯(Υ) < Rt), where Rn,¯(Υ) is as defined in Lemma 2.1.This outage probability is studied in this section.2.3.1 Rate Outage ProbabilityWe start by reviewing the outage probability of a large-scale network in the AR.In this case, the outage is defined as the probability that the channel capacity (definedas the maximum rate such that the error probability vanishes as n → ∞) is lower402.3. Rate Outage and Reliability Analysis in the Finite Blocklength Regimethan a target rate Rt. The rate outage probability under a given r0 is given as [117]O(r0, Rt) = 1− e− (2Rt−1)σ2wPr−η0 LBg{2Rt − 1Pr−η0}. (2.20)Studying the rate outage probability under blocklength n and FER threshold ¯is generally difficult to simplify as it is a function of the fading distribution as wellas the network geometry. Instead, we provide bounds which are fairly tight in thefollowing theorem.Theorem 2.6. For an average SNR α0, the rate outage probability defined in Def.2.5 satisfies Ol(r0, Rt, n, ¯) ≤ O(r0, Rt, n, ¯) ≤ Ou(r0, Rt, n, ¯), whereOl(r0, Rt, n, ¯) = O(r0, Rt), (2.21)Ou(r0, Rt, n, ¯) = O(r0, Rt + an,¯−bn), (2.22)where O(r0, Rt) is defined in (2.20), an,¯ =√log22(e)nQ−1(¯), and bn = 12n log2 n.Proof. From Def. 2.5, we haveO(r0, Rt, n, ¯) = P(Rn,¯(Υ) < Rt) = P(log2(1 + Υ)− an,¯√1− 1(1 + Υ)2+bn < Rt).(2.23)Noting that K(Υ) =√1− 1(1+Υ)2is monotonically increasing in Υ as shown in Fig.2.4, satisfying K(Υ) ∈ [0, 1], we conclude that Ol(r0, Rt, n, ¯) ≤ O(r0, Rt, n, ¯) ≤Ou(r0, Rt, n, ¯), where the lower and upper bounds are obtained by setting K(Υ) to0 and 1, respectively.The outage probability lower bound confirms that the outage probability in theFBR is larger than that in the AR and the upper bound provides a guaranteedperformance. By observing Fig. 2.4, it can be seen that K(Υ) quickly approaches oneas Υ grows. The figure also shows that the density f(Υ|r0) in the interval of SINR,where the approximation K(Υ) ≈ 1 is tight (Υ > 5 e.g. ), increases as r0 decreases(which is the case for dense networks). Hence, the upper bound Ou(r0, Rt, n, ¯) canbe used to evaluate guaranteed outage performance in general and can be used as atight approximation for dense networks. This is verified in Fig. 2.8 and 2.9. Next,the spatially averaged outage probability is presented.412.3. Rate Outage and Reliability Analysis in the Finite Blocklength Regime0 5 10 15 20 25 30 350.10.20.30.40.50.60.70.80.9100.050.10.150.20.250.30.35Figure 2.4: K(Υ) versus Υ and the PDF of the SINR at r0 ∈ {150, 250} mCorollary 3. The average rate outage probability defined in Def. 2.5 with blocklengthn, FER ¯, and target rate Rt can be upper bounded asO¯(Rt, n, ¯) = 1−∫ ∞02piλr0e− 2Rt+an,¯−bn−1α0−piλr20LBg{2Rt+an,¯−bn − 1Pr−η0}dr0 (2.24)Proof. Using the law of iterated expectation, the average outage probability is theaverage of (2.22) with respect to r0, i.e., P(R > Rt) = Er0{P(R > Rt|r0)}. Theresult then follows since r0 follows a Rayleigh distribution.For a shadowed urban cellular network, the path-loss exponent η lies in the range3 − 5 [118], where η = 4 is a common value of practical relevance that is widelyutilized in the literature. For the case η = 4, the outage probability in (2.24) can besimplified as given next.Corollary 4. The average rate outage probability of the large-scale network modeledby (2.1) with blocklength n, FER ¯, target rate Rt, and path loss exponent η = 4,can be upper bounded asO¯(Rt, n, ¯) = 1− piλ√piPσ2wµ1eσ2wµ1µ224P Q(√µ22µ1σ2w2P)(2.25)where µ1 = 2Rt+an,¯−bn − 1 and µ2 = λpiPσ2wµ1(√µ1 arctan(√µ1) + 1).Proof. A closed-form expression for the upper bounded outage probability at η = 4422.3. Rate Outage and Reliability Analysis in the Finite Blocklength Regimeis obtained as followsO¯(Rt, n, ¯) = 1−∫ ∞0e−µ1r40σ2wP LBg{µ1r40P}fr0(r0) dr0 (2.26)(i)= 1−∫ ∞02piλr0e−σ2wµ1P (r40+µ2 r20)dr0 (2.27)(ii)= 1− piλ√pi Pσ2wµ1eµ22µ1σ2w4P Q(√µ22µ1σ2w2P)(2.28)where we used LBg{s}η=4 = exp{−piλ√sPtan−1(√sPr40)}, (i) is obtained using achange of variable µ2 =λpiPµ1σ2w(√µ1 arctan(√µ1) + 1), and (ii) is obtained using thechange of variables and the CDF of the Gaussian distribution with mean(µ22)andvariance(P2µ1σ2w).The rate outage probability of a large-scale network in the FBR is upper bounded,with guaranteed performance at high SINR, in a generic form in (2.22) and (2.24)and in a closed-form expression for η = 4 in (2.28), as a function of BS density λ,FER ¯, blocklength n, and average SNR α0. Next, the reliability of the networkunder the FBR.2.3.2 Reliability AnalysisWe define a metric to evaluate the overall network reliability for a fixed r0, definedas the probability of correctly decoding a codeword. A codeword is decoded correctlyin two cases in the FBR. The first case is when Rn,¯(Υ) > Rth, in which case thecodeword is decoded correctly with a probability larger than or equal to (1− ¯). Thesecond case is when Rn,¯(Υ) < Rth, in which case few codewords may still be decodedcorrectly. Since in practice, it is desirable to operate at low outage probability, hencethe second case does not contribute much to reliability. Thus, to simplify the analysis,we will lower bound the probability of decoding a codeword correctly in the secondcase by 0 (i.e., its contribution to reliability is neglected) and we will only considerthe first case. Hence, the reliability is lower bounded by the following expression,providing a guaranteed performance for a given r0,Tn,¯(r0, Rt) = (1−O(r0, Rt, n, ¯))(1− ¯). (2.29)The guaranteed reliability averaged over r0 is given byT¯n,¯(Rt) = (1− O¯(Rt, n, ¯))(1− ¯), (2.30)432.4. Coding Rate Meta Distribution in the Finite Blocklength RegimeAlso, for the sake of comparison, the reliability of the AR can be defined asT∞,0(r0, Rt) = 1−O(r0, Rt), (2.31)since the error probability is assumed to be vanishing as n → ∞, and hence failureoccurs when there is an outage. The reliability will be assessed numerically in Sec.2.5.Studying the rate outage probability and the overall system reliability gives moreinsight into the performance of the network. Moreover, studying the coding ratemeta distribution, which provides the percentiles of users achieving a rate Rt, willprovide a complete picture of the network performance. Hence, the meta distributionof the coding rate is derived next.2.4 Coding Rate Meta Distribution in the FiniteBlocklength RegimeThe meta distribution provides fine-grained information about network perfor-mance, in the form of the fraction of users that can achieve a minimum data rate Rtat a FER threshold ¯ with probability at least pt. The definition and analysis of thisquantity are provided next.Definition 2.7. Given a target rate Rt and a DL large-scale network with a distancer0 between the desired UE and the serving BS as defined in (2.1), we define thecoding rate meta distribution in the FBR with blocklength n and FER threshold ¯as FRt(pt, n, ¯) = P(Ps(Rt, n, ¯) < pt) where Ps(Rt, n, ¯) = P(Rn,¯(Ω) > Rt|Ψ˜) is theprobability of success, Ω is the signal-to-interference ratio (SIR) and Rn,¯(Ω) is asdefined in Lemma 2.1.For simplicity, we assume an interference-limited scenario where noise is neglectedand the SIR is given by Ω = Pr−ηo |h0|2∑ri∈Ψ˜/ro Pr−ηi |hi|2. Then, Ps(Rt, n, ¯) is given byPs(Rt, n, ¯) = P!(Rn,¯(Ω) > Rt|Ψ˜) = P!(log2(1 + Ω)−√V (Ω)Q−1(¯)√n+bn > Rt|Ψ˜)(2.32)where P!(·) is the reduced Palm measure of the point process [119],[120, Def. 8.8].Note that Ps(Rt, n, ¯) is a random variable which is a function of the stochasticityof the network. Its moments are defined asMd = E{Ps(Rt, n, ¯)d}, (2.33)442.4. Coding Rate Meta Distribution in the Finite Blocklength Regimeand are used to calculate the meta distribution as follows [119]FRt(pt, n, ¯) = P(Ps(Rt, n, ¯) > pt)=12+1pi∫ ∞0Im(e−t log(pt)Mjt)tdt, (2.34)where Im(·) is the imaginary part of a complex number and Mjt is obtained atd = jt in (2.33).In [119], a simple yet tight approximating distribution of Ps(Rt, n, ¯) is proposedfor the AR, which is the beta distribution given byf(Ps;Rt) ≈ Pϑ(β+1)−11−ϑs (1−Ps)β−1B(ϑβ1−ϑ , β) , (2.35)where B(·, ·) is the beta function, ϑ =M1 and the β = (M1−M2)(1−M1)M2−M21 . Hence, themeta distribution could be computed as the complementary cumulative distributionfunction (CCDF) of the beta distribution. However,M1 andM2 provided in (2.33)are difficult to evaluate in the FBR, and hence, an approximation is proposed in thischapter. At high SIR, where the term K(Υ)=√1− 1(1+Υ)2is approximately equalto 1, the approximated probability of success is given byPˆs(Rt, n, ¯) = P!(log2(1 + Ω)− an,¯+bn > Rt|Ψ˜).Therefore, the approximated probability of success can be expressed as a function ofthe SIR instead of the coding rate, as follows,Pˆs(Rt, n, ¯) = P!(Ω > 2Rt+an,¯−bn − 1|Ψ˜)=∏ri∈Ψ˜\{r0}11 + (2Rt+an,¯−bn − 1)(r0ri)η ,(2.36)The expression provided in (2.36) is obtained by averaging over the channel gains.Then, its moments are given byMˆd = E{Pˆs(Rt, n, ¯)d}= E∏ri∈Ψ˜\{r0}1(1 + (2Rt+an,¯−bn − 1)(r0ri)η)d= exp{−2piλ∫ ∞r0r(1− 1(1 + (2Rt+an,¯−bn − 1)(r0/r)η)d)dr}, (2.37)452.5. Numerical ResultsThus, the meta distribution can be evaluated using the approximated moments in(2.37) as followsFˆRt(pt, n, ¯) = P(Pˆs(Rt, n, ¯) > pt) =12+1pi∫ ∞0Im(e−t log(pt)Mˆjt)tdt. (2.38)Using the beta distribution approximation, the meta-distribution can be approxi-mated asFˆRt(pt, n, ¯) ≈ 1− Ipt(Mˆ1(Mˆ1 − Mˆ2)(Mˆ2 − Mˆ21),(1− Mˆ1)(Mˆ1 − Mˆ2)(Mˆ2 − Mˆ21)), (2.39)where Iz(x, y) =1B(x,y)∫ z0tx−1(1−t)y−1dt is the regularized incomplete beta function.2.5 Numerical ResultsThis subsection presents illustrative numerical results for the theoretical rateanalysis provided in Sec. 2.2.2 and 2.2.3, which are also validated via Monte Carlosimulation. The results presented investigate the effect of the blocklength n andthe FER , the tightness of the proposed theoretical approximations, and the per-formance comparison between the FBR and the AR. Then, the performance of theMLPCM is investigated in comparison with the theoretical average coding rate.Fig. 2.5 plots the maximum average coding rate and the average coding rate forM-QAM versus the average SNR α0 for a fixed r0. In Fig. 2.5(a), the average codingrate is plotted for n ∈ {128, 2048}, FER  ∈ {10−2, 10−5, 10−6}, and r0 = 250 m.Given a FER , a gap exists between the average coding rate in the FBR and theAR. The gap is around 1 dB for n = 2048 and is between 3 and 5 dB for n= 128at =10−5 and 10−6, respectively, which shows the severity of the SNR penalty inthe FBR compared to the AR. The figure shows the trade-off between reliability(represented by FER) and maximum average coding rate at a given n, where themaximum average coding rate decreases/increases as the FER decreases/increases.For instance, to achieve a given maximum average coding rate at n=128 and FER=10−5, we need around 2 dB more power compared with the same blocklength nunder the FER  = 10−2. Furthermore, the impact of small n becomes more severewhen the FER  is smaller, resulting in rate degradation. Overall, this shows that itis impractical to use results from the AR in the FBR.Second, the maximum achievable coding rate for different modulation schemes,presented in Sec. 2.2.3, is simulated for r0 = 150 m, λ = 1 BS/km2, blocklength462.5. Numerical Results(a) The maximum average coding rate atr0 = 250 m for different values of n and .(b) The average coding rate for M-QAM atr0 = 150 m, n = 128, and  = 10−2.Figure 2.5: Average coding rate analysis of a large-scale network with λ = 1 BS/km2for fixed r0n = 128, and FER  = 10−2 in Fig. 2.5(b). Fig. 2.5(b) demonstrates the ap-proximated maximum achievable coding rate using the Gamma approximation forvarious modulation orders, M = {2, 4, 8, 16}, (Theorem 2.4), along with the maxi-mum achievable rate under Gaussian codebooks (Theorem 2.3) versus α0. The MonteCarlo simulated maximum achievable coding rate for QAM is provided to show thetightness of the approximation. The approximated maximum achievable rate is veryclose to the simulated maximum achievable rate. Thus, the Gamma approximationin (2.12) leads to a fairly tight approximation. It is also observed that as the QAMmodulation order increases, the performance improves towards the maximum aver-age coding rate (without Gaussian codebooks) in a large-scale network in the FBR(Theorem 2.3).The spatially averaged coding rate is depicted in Fig 2.6(a) versus the transmitsignal to noise power ratio Pσ2w.6 The figure shows that for high SNR (above 0 dB),one cannot achieve the same rate and FER achieved at a given n by decreasing nand paying an SNR penalty. For instance, the rate achieved at n = 2048,  = 10−5,6The figure is plotted for a wide range of Pσ2w for illustrative purposes, bearing in mind that somevalues of Pσ2w in the plot may be impractical.472.5. Numerical Results(a) Maximum Average Coding Rate for dif-ferent values of n and .(b) Average coding rate for M-QAM at n =512, and  = 10−2.Figure 2.6: Average coding rate analysis for random r0 at λ = 1 BS/km2SNR = 10 dB cannot be achieved at n = 128 and  = 10−5. The only way toachieve such a rate at n = 128 is by paying a FER penalty, which is impractical forapplications that need high reliability. Hence, it is important to select n that balancesthe trade-off between reliability and coding rate. Similar to Fig. 2.5(a), Fig. 2.6(a)shows the importance of the results in this chapter for characterizing and optimizingthe performance of large-scale networks with latency-sensitive applications.Fig. 2.6(b) plots, the average achievable rate versus Pσ2wunder a Rayleigh dis-tributed r0, QAM constellation with M = 2, 4, 8, and 16, blocklenth n = 512, andFER  = 10−2, showing a similar trend as for fixed r0. It is observed that the averageachievable rate saturates at a rate of ≈ 2 bits/transmission for this choice of λ, n, and, which is low compared to the case r0 = 150 m. This performance is a consequenceof averaging the performance of nearby UEs which have relatively strong receiveddesired signals (high SINR) and far away UEs which suffer from more interference(low SINR).Finally, considering the same setting in Fig. 2.5(b), we simulate MLPCM witha QAM constellation for modulation orders M = 2, 4, 8, and 16 in Fig. 2.7. Thesimulations are performed by fixing the SNR and increasing the coding rate untilreaching the FER . The achievable rates are plotted along with the theoreticalexpressions proposed in (2.7) and (2.13). The figure shows that the achievable rates482.5. Numerical ResultsFigure 2.7: The average coding rate versus r−η0 α at λ = 1 BS/km2, r0 = 150 m, andn = 128.of the MLPCM scheme with 2-QAM or 4-QAM approaches the theoretical rate of thecorresponding QAM constellation. However, for 8-QAM and 16-QAM, a gap of ≈ 2.5dB exists between the theoretical results and the simulation results. Nonetheless, thisSNR gap translates to a small rate gap of fractions of bits/transmission due to theslow increase of the rate versus SNR in the large-scale network which is caused byinterference. Fig. 2.7 thus validates that the theoretical performance under finiteconstellations can be approached using MLPCM, which validates the potential ofMLPCM for future technologies.Now, we provide numerical results for the bounds and approximations of the rateoutage probability, reliability, and the coding rate meta-distribution. We investi-gate the tightness of these approximations and compare them with results from theliterature for the AR.First, in Fig. 2.8, the rate outage probability is plotted versus r0 with λ ∈{1, 9} BS/km2, η = 4, ¯ ∈ {10−2, 10−6}, Pσ2w= 0 dB, and Rt = 1 bit/transmission.The PDF of r0 is plotted to illustrate the probability of occurrence of the r0 values.The figure shows the Monte Carlo simulated outage probability, in addition to theupper bound in (2.22) and the outage probability in the AR (2.20) which serves as alower bound (cf. Thm. 2.6). It shows that the rate outage in the AR underestimatesthe rate outage probability in the FBR and the gap to the actual outage probabilityincreases as the FER threshold decreases. This gap can be as large as 30% in somecases, such as with r0 = 200 m, λ = 1 BS/km2, n = 128, and ¯ = 10−6, where theactual outage probability is 0.13, but (2.20) underestimates it as 0.1. Fig. 2.8 also492.5. Numerical Results(a) n = 128. (b) n = 2048.Figure 2.8: The outage probability versus r0 at λ ∈ {1, 9} BS/km2, ¯ ∈ {10−2, 10−6},and α = 0 dB.shows that the upper bound of the outage probability expression in (2.22) is fairlyaccurate to provide a convenient approximation for a broad range of n, λ, and ¯.The outage probability is seen to increase as the FER threshold ¯ decreases becausethe term√V (Υ)nQ−1(¯) in (2.7) increases as ¯ decreases. However, at a specific ¯, asn increases, the outage probability approaches the outage probability in the AR asexpected. Thus, while the results in [65] can be a good representative of the perfor-mance when n is relatively large (Fig. 2.8(a)), this is not the case when n is smallin which case our derived bounds and approximation are more accurate. Moreover,in Fig. 2.9, the rate outage probability approximation for a Rayleigh distributed r0,provided in (2.28), is plotted versus the target rate Rt and compared to the simulatedoutage probability for BS density λ = 1 BSs/km2, blocklengths n ∈ {128, 2048},FER threshold ¯ ∈ {10−2, 10−6}, and Pσ2w= 0 dB. This plot also proves the tightnessof the upper bound and the inaccuracy of the AR.In Fig. 2.10(a), the rate outage probability under QAM constellations is shown forvarious modulation orders versus r0, where the rate outage for a specific modulationscheme is defined as O(M)(r0, Rt, n, ¯) = P(R(M)n,¯ (Υ) < Rt|r0). This expression iscomputed using Monte Carlo simulation. The figure is plotted for n = 128, ¯ = 10−2,λ = 1, Pσ2w= 0 dB, and Rt ∈ {0.825, 1.85}. Similar to the previous plots, the rate502.5. Numerical Results(a) ¯ = 10−2. (b) ¯ = 10−6.Figure 2.9: The outage probability versus the rate threshold Rt at λ = 1 BS/km2,n ∈ {128, 2048}, and Pσ2w= 0 dB, averaged with respect to Rayleigh distributed r0.outage probability O(r0, Rt, n, ¯) acts as the lower bound of the outage probabilityof all constellation sets. Also, the higher order modulation (i.e. M = 16) yields anoutage probability lower than the lower order modulation (i.e. M = 2) at a fixed Rtas the modulation order limits the maximum transmission rate (log2(M)). However,at lower Rt as such Rt = 0.825, most of the high order modulation schemes provide aperformance close to the theoretical lower bound O(r0, Rt, n, ¯). Fig. 2.10(b) showsthe outage probability for random r0, i.e., O¯(M)(Rt, n, ¯) = P(R(M)n,¯ (Υ) < Rt) versusRt. As shown, the rate outage probability increases as the rate threshold Rt increases,and it jumps to 1 at a certain Rt for each modulation order, which is when Rt exceedsthe maximum rate achievable by the modulation order (log2(M)).The overall network reliability versus the FER threshold ¯ is evaluated in Fig.2.11 for fixed and random r0 for n ∈ {128, 2048} and Rt ∈ {0.1375, 1, 3.46}. It can beobserved that the reliability of the network increases as the FER threshold increasesuntil it reaches a maximum around ¯ = 10−2 then it rapidly decays. This is becausethe increase of (1 − O(r0, Rt, n, ¯)) with respect to ¯ is faster than the decrease of(1− ¯) for small ¯. However, for large ¯, (1−O(r0, Rt, n, ¯)) saturates to 1 and (1− ¯)dominates and contributes to the decay of the reliability at high ¯. In other words,at ¯ < 10−2, the rate of increase of (1 − O(r0, Rt, n, ¯)) is higher than the rate of512.5. Numerical Results(a) Outage probability versus r0, for Rt ∈{0.825, 1.85}.(b) Outage probability versus Rt.Figure 2.10: Outage probability using M-QAM at λ = 1 BS/km2, ¯ = 10−2, n = 128and Pσ2w= 0 dBdecay of (1 − ¯). However, at ¯ > 10−2, the rate of increase of (1 − O(r0, Rt, n, ¯))is nearly zero and the rate of decay of (1 − ¯) dominates. Therefore, the reliabilityrapidly decays. It is shown that for a fixed Rt, a gap exists between curves simulatedunder FBR and AR. This gap grows wider up to 0.15 as FER threshold ¯ decreasesbecause the short blocklengths (e.g. n = 128) are less reliable. Also, it is obviousthat the reliability of the network at random r0 is lower than that at fixed r0 asit averages the performance of users which are close and those that are far awayfrom the BS. Moreover, as Rt increases the approximated reliability becomes closeto the exact one. This is because the outage upper bound provided in Corollary 2.6is tight for high SINR. Hence, from Def. 2.5, as the Rt increases, only high SINRscan achieve the desired rate and FER threshold at the given n, which is where theapproximation becomes tighter. Whereas, for low Rt, low SINRs can also achievethe desired performance which loosens the approximation.Finally, we validate the coding rate meta distribution in Fig. 2.12(a). The codingrate meta distribution provided in (2.34) is simulated via Monte-Carlo simulationsat λ = 1 BSs/km2, r0 = 150 m, an area of 500 km2, a blocklength n= 128, FERthreshold ¯ = 10−5, and a rate threshold Rt = {0.1375, 0.3964, 1, 2.0574, 3.4594}.522.5. Numerical Results(a) Fixed r0: λ = 1 BS/km2, Pσ2w = 0 dB andr0 = 250 m.(b) Random r0: λ = 0.1 BS/km2 and Pσ2w =10 dB.Figure 2.11: Reliability versus the FER threshold ¯It is observed that there is a significant gap between the performance in the ARand the FBR, showing that the meta distribution of the coding rate in the ARoverestimates the performance of the network in the FBR. This gap can reach up to0.09 at Rt = 3.46 and pt = 0.9, which means 9% of the users predicted to achieve therequired performance in the AR will not achieve it in the FBR. It can also be observedthat the gap increases as Rt or pt increases. Thus, the AR analysis in [90, 119, 121]cannot be used to provide precise results in the FBR. For the sake of preciseness ofthe results, we have simulated the coding rate meta distribution including the noiseterm with noise power σ2w =−90 dBm [122]. The results show that the network isan interference-limited network and the noise can be neglected for the simplicity ofthe analysis. In Fig. 2.12(b), the approximation proposed in (2.36) is shown to bevery tight compared to the exact coding rate meta distribution in (2.34) for differentr0. The tightness of the approximation increases by decreasing r0 or alternativelyincreasing SIR as the term√1− 1(1+Ω)2converges to 1 as the SIR increases. Usingthis approximation it is easier to find an expression for the moments of the successprobability and use it to evaluate the meta distribution using the beta distributionprovided in (2.39). Hence, the beta approximation of the meta-distribution providesa close performance to the exact and approximated meta-distribution.532.6. Conclusion(a) The FBR versus the AR at n = 128 and¯ = 10−5.(b) Exact versus Approximation at r0 ∈{150, 250, 500}m at Rt = 1, n = 128, and¯ = 10−2.Figure 2.12: The meta distribution of the coding rate versus ptThus, we saw that our proposed expressions provide an accurate representationof the rate outage probability and the coding rate meta distribution in the FBR,which is more accurate than using AR expressions from the literature.2.6 ConclusionThis chapter investigates different aspects of the performance of a large-scale DLnetwork in the FBR using the theory of coding in the FBR in conjunction withstochastic geometric tools. We start with rate analysis where we derive the averagecoding rate using Gaussian codebooks and M -ary QAM constellation, and inves-tigate the practical scheme of MLPCM in the FBR showing that its performanceapproaches the theoretical benchmarks. We also study the rate outage probability,reliability, and the coding rate meta distribution, where we provide bounds and ap-proximations. From the results, we conclude that the performance analysis providedunder an AR provides a misleading (overestimated) performance and cannot be usedto characterize the performance in the FBR. The performance in the FBR providesaccurate rate and FER characterization, and approaches the AR performance at suf-542.6. Conclusionficiently large blocklengths > 2048 and high FER > 10−2. The developed expressionsfor the average coding rate, the rate outage probability, and the coding rate metadistribution are fairly tight as shown by our numerical results. In conclusion, the re-sults are relevant for the theoretical analysis of large-scale networks in the FBR, andthe theoretical performance can be approached using practical transmission schemessuch as MLPCM.55Chapter 3Large-scale Uplink OMA Networksin the Finite Blocklength RegimeIn this Chapter, we will present a theoretical performance analysis of the ULlarge-scale network. Chapter 2 studies a large-scale network in the FBR, however,it focuses on the DL which is operationally different from the UL. In the DL, theBSs are powerful units with large power sources, however, the UEs are power-limitedunits which necessitates the use of power control in the UL networks. Power controlis used to maintain an acceptable level of performance which adds a challenge to thesystem as it makes the transmitted power of the signal a function of the distance to itsserving BS, which exhibits a stochastic nature. Moreover, in the DL network, due tothe UE association technique, a deterministic interference exclusion region is definedwhere no interfering BS exists within this region. However, the UL network doesnot conform to this deterministic framework, where the interference exclusion regionexhibits a probabilistic nature. Additionally, the UL network is not homogeneous asthe DL network, which in return affects the analysis. Therefore, the performance ofa large-scale UL network in the FBR as a function of blocklength and FER becomesimperative.In this chapter, a theoretical analysis of the performance of large-scale UL OMAnetworks is proposed in terms of average coding rate, rate outage probability, andcoding rate meta distribution under fractional path-loss inversion power control.Furthermore, two special cases will be studied which are TCI [88] and CIMP [89].Moreover, a comprehensive discussion is provided, delving into the qualitative andquantitative nuances of the numerical findings, and examining the impact of networkparameters on the performance of the large-scale network in the FBR. Moreover, acomparison between power control schemes is conducted, providing insights into theirperformance and leading to conclusive recommendations for the preferred scheme.The chapter starts with presenting the system model in Sec. 3.1 and the assump-tions of the analysis. Then, the performance analysis of the network under fractionalpath-loss inversion power control, TCI and CIMP power control schemes are pre-sented in Sections 3.2 and 3.3, respectively. The theoretical analysis is presented,followed by the numerical results in Sec. 3.4, where the theoretical results are nu-563.1. System Modelmerically evaluated, and a comparison between power control schemes is conducted.Finally, the chapter is concluded in Sec. 3.5.3.1 System ModelWe consider a single-tier UL large-scale network utilizing universal frequencyreuse causing inter-cell interference, combined with OMA to avoid intra-cell interfer-ence.7 Hence, only one user per cell has access to the same resource block. However,its received signal at the BS is interfered by a large number of users from other cells.The BSs constitute a 2-dimensional (2-D) spatial PPP Ψ with intensity λ. The UEspoint process Φ = {ui; i ∈ N} is modeled by associating each UE with its geographi-cally closest BS. For this network, the BS receives a signal y from its connected UEthat lies at a distance r0. The distance r0 follows a Rayleigh distribution [17], i.e., itis distributed as fr0(r0) = 2piλr0 exp{−piλr20}, 0 < r0 < ∞, due to the nearest BSassociation assumption. The BS also receives interference from other UEs located atdistances u1, u2, ... , given byy =√P0h0r−η20 s0 +∑ui∈Φ\{r0}xi√Pihiu−η2i si + w, (3.1)where P0 (resp. Pi) is the transmit power of the intended UE (resp. UEi), h0 (resp.hi) is the channel coefficient from intended UE (resp. UEi) to the BS, η is the pathloss exponent, s0 (resp. si) is a unit average power codeword symbol transmitted bythe intended UE (resp. UEi), xi ∼ Bernoulli(p) is an activity indicator for UE i, andw is the AWGN noise with noise power σ2w.To mitigate interference, associating a UE with the closest BS combined withpower control strategy is used to enforcing the average received power at a givenBS from its associated UE to be strictly greater than the average interference powerreceived from any other UE. Hence, the average received signal power of UE0 at BS0is maintained above the average interfering signal power from any interfering UE.Utilizing a power control scheme, the transmit power Pi will be defined accordingto the used scheme which will be defined in each section. A block fading channelmodel is assumed to follow a Rayleigh distribution (h0 ∼ CN (0, 1), hi and h0 arei.i.d), where the channel coefficients remain constant for L consecutive symbols andchange to an independent realization in the next L symbols. To ensure that thechannel remains constant during the transmission of n symbols, we require n = L/l7OMA schemes avoid intra-cell interference through an orthogonal allocation of resources.573.1. System Modelfor some integer l ≥ 1.8 The SINR is given byΩU =P0|h0|2r−η0∑ui∈Φ\{r0}xiPi|hi|2u−ηi + σ2w(3.2)The intended UE sends information to its serving BS using codewords from a codewith rate R, a target FER ¯, and length n symbols. The value of n can reflect latencyand/or energy consumption constraints.9 Besides, the target FER ¯ can be definedfollowing a reliability constraint, so that the FER  of the network conditioned onthe SINR should be less than or equal ¯. Treating interference as noise and followingPolyanskiy et al. [6], the coding rate at SINR ΩU, blocklength n, and FER  can beapproximated byRn,(ΩU) = log2(1 + ΩU)−√V (ΩU)Q−1()√n+12nlog2(n), (3.3)where V (ΩU) = ΩU(ΩU+2) log22(e)(ΩU+1)2is the channel dispersion. Hence, assuming CSI knowl-edge in the form of SINR availability, the average coding rate is studied.However, in practical networks, the SINR is unknown at the transmitter andhence, the transmitter encodes at a constant (target) rate Rt and blocklength n. Asa result, the resulting FER  conditioned on SINR ΩU may or may not satisfy thedesired design FER ¯ due to stochastic fading and aggregate interference. In thiscase, it is important to keep this conditional FER below a threshold to avoid havingexcessive frame errors. Hence, the outage probability is necessary to be evaluatedunder this scenario.Next, the performance of the network under fractional path-loss inversion powercontrol is analyzed under the FBR.8l represents that number of frames that can be transmitted within one coherence interval. Theanalysis can be simulated by averaging over all coherence intervals and considering one codewordper coherence interval.9In a network, one of the dominant delays is the transmission delay, especially in dense networks.The transmission delay is a function of the blocklength, hence it should be designed to satisfy thelatency constraint. In applications with stringent energy consumption requirements, the averageenergy of a block should satisfy energy consumption constraints.583.2. Fractional Path-Loss Inversion Power Control3.2 Fractional Path-Loss Inversion PowerControlIn this section, the fractional path-loss inversion power control is utilized wherethe transmitted power is given asPi = ρorδηi (3.4)where δ is a compensation factor which determines the fraction of channel inversion(i.e., δ = 1 refers to full channel inversion whereas δ = 0 refers to no channelinversion), ρo is the power control parameter (at δ = 1, ρo is the average receivepower, and at δ = 0, ρo is the transmit power.), and ri is the distance between theUE and its serving BS. Utilizing the aforementioned power control scheme, the rateoutage probability and the coding rate meta distribution are characterized in thissection, providing a characterization of the network performance. We assume allusers in the network perform fractional path-loss inversion power control. Hence,the signal received by the serving BS from the intended UE indexed by 0 is distortedby noise σ2w and aggregate interference IFI received from all active UEs using thesame UL channel frequency in other cells. The Laplace transform of IFI is given byLIFI(s)= exp∞∫rη(1−δ)0ρo−2γ(1 + δ, piλ(ρom)2η(1−δ))psρ−2ηo (piλ)δ−1ηm1−2η (m+ s)dm , (3.5)where γ(a, b) =∫ b0ta−1e−t dt. Refer to Appendix 6.2 for the derivation of LIFI(s).Hence, the SINR is defined asΩUFI =ρo|h0|2r−η(1−δ)0IFI + σ2w, (3.6)Next, the rate outage probability and the coding rate meta distribution are charac-terized in this section, providing a characterization of the network performance.3.2.1 Rate Outage AnalysisWe start by the definition of the rate outage, which is provided as follows.Definition 1. Given a target rate Rt, the outage probability of large-scale UL networkfor a blocklength n and target FER ¯, is defined as OUn,¯(Rt) = P(Rn,¯(ΩUFI) < Rt).593.2. Fractional Path-Loss Inversion Power ControlDue to the added channel dispersion term V (ΩUFI), the rate outage probability isdifficult to evaluate. By studying the behavior of the channel dispersion as a functionof SINR ΩUFI, we have found that it increases from 0 to log22(e) as the SINR increasesfrom 0 to ∞. Hence, the outage probability can be bounded as follows.Proposition 1. The rate outage probability under the considered system model isupper bounded byO¯Un,¯(Rt, υ) = FΩUFI(max(υ, 2Rt+an,¯(∞)−bn − 1))+ FΩUFI(min(υ, 2Rt+an,¯(υ)−bn − 1))− FΩUFI(υ) (3.7)for any υ > 0, and lower bounded byOUn,¯(Rt,Λ) =FΩUFI(min(Λ, 2Rt−bn − 1))− FΩUFI(Λ)+ FΩUFI(max(Λ, 2Rt+an,¯(Λ)−bn − 1)) (3.8)for a Λ > 0, where an,¯(x) =√V (x)nQ−1(¯), bn = 12n log2(n), LI(·) is provided in(3.5), and FΩUFI(x) is the cumulative probability function (CDF) of the SINR ΩUFIgiven byFΩUFI(ΩUFI) = 1−∫ ∞02piλr0e−piλr20−ΩUFIrη(1−δ)0ρo/σ2w LIFI(ΩUFIρor−η(1−δ)0)dr0, (3.9)Proof. The rate outage probability is given byOUn,¯(Rt) = P(log2(1 + ΩUFI)−√V (ΩUFI)Q−1(¯)√n+ bn < Rt)To derive the upper bound, we define an,¯(x) =√V (x)nQ−1(¯) Noting that an,¯(x) isan increasing function of x, then for any υ > 0 we haveOUn,¯(Rt) =∫ υ0P(ΩUFI)1(Rn,¯(ΩUFI) < Rt) dΩUFI +∫ ∞υP(ΩUFI)1(Rn,¯(ΩUFI) < Rt) dΩUFI≤∫ υ0P(ΩUFI)1(log2(1 + ΩUFI)− an,¯(υ) + bn < Rt) dΩUFI+∫ ∞υP(ΩUFI)1(log2(1 + ΩUFI)− an,¯(∞) + bn < Rt) dΩUFI603.2. Fractional Path-Loss Inversion Power ControlHence, the rate outage probability is upper bounded byO¯Un,¯(Rt, υ) =∫ β10P(ΩUFI) dΩUFI +∫ β2υP(ΩUFI) dΩUFI= FΩUFI(β1) + FΩUFI(β2)− FΩUFI(υ)for any υ > 0, where β1 = min(υ, 2Rt+an,¯(υ)−bn−1) and β2 = max(υ, 2Rt+an,¯(∞)−bn−1). This concludes the derivation and leads to the result in (3.7).To derive the lower bound, we revise the rate outage probability definition whichis given as OUn,¯(Rt) = P (Rn,¯(ΩUFI) < Rt). Then, since an,¯(x) is increasing in x, forany Λ > 0 we haveOUn,¯(Rt)=∫ Λ0PΩUFI(ΩUFI)1(Rn,¯(ΩUFI) < Rt) dΩUFI +∫ ∞ΛPΩUFI(ΩUFI)1(Rn,¯(ΩUFI) < Rt) dΩUFI≥∫ Λ0PΩUFI(ΩUFI)1(log2(1 + ΩUFI)− an,¯(0) + bn < Rt) dΩUFI+∫ ∞ΛPΩUFI(ΩUFI)1(log2(1 + ΩUFI)− an,¯(Λ) + bn < Rt) dΩUFI.Define µ1 = min(Λ, 2Rt−bn−1) and µ2 = max(Λ, 2Rt+an,¯(Λ)−bn−1), and since an,¯(0) =0, the rate outage probability is lower bounded byOUn,¯(Rt) =∫ µ10PΩUFI(ΩUFI)dΩUFI +∫ µ2ΛPΩUFI(ΩUFI)dΩUFI= FΩUFI(µ1)− FΩUFI(Λ) + FΩUFI(µ2),for any Λ > 0.This concludes the derivation and leads to (3.8).In Proposition 1, the upper and lower bounds hold for arbitrary values of υ andΛ, respectively. However, to achieve a tight bound, these parameters have to beoptimized and this can be done numerically.To reduce the complexity of the expressions, simplified upper and lower boundsare provided in the following Corollary.Corollary 5. A simplified upper bound can be obtained by setting υ =∞ leading toO¯Un,¯(Rt,∞) = FΩUFI(2Rt+an,¯(∞)−bn − 1), (3.10)and a simplified lower bound can be obtained by setting Λ = 0 leading toOUn,¯(Rt, 0) = FΩUFI(2Rt−bn − 1), (3.11)613.2. Fractional Path-Loss Inversion Power ControlThe simplified upper bound is fairly tight at large Rt as we shall see in Fig. 3.1,whereas the simplified lower bound is approximately the rate outage probability inthe AR.3.2.2 Coding Rate Meta DistributionThe meta distribution provides more detailed information about the quality ofservice in terms of coding rate by providing the fraction of users achieving a minimumdata rate of Rt with probability at least pt at a blocklength n and a target FER ¯.Denote by PUs (Rt, n, ¯) the conditional success probability for an arbitrary, yet fixed,network realization Φ∪Ψ, defined as the achievability of a rate > Rt at a blocklengthn and FER ¯. Then, PUs (Rt, n, ¯) is given byPUs (Rt, n, ¯) = P!(Rn,¯(ΩUFI) > Rt|Φ,Ψ)= P!(log2(1 + ΩUFI)−√V (ΩUFI)nQ−1(¯) + bn > Rt|Φ,Ψ), (3.12)where P!(·) is the reduced Palm probability [120, Def. 8.8].Note that PUs (Rt, n, ¯) is a random variable which depends on the relative loca-tions between the UEs and the BSs within the network. Its moments are definedasMUd = E(PUs (Rt, n, ¯)d), (3.13)and can be used to evaluate the meta distribution via Gil-Pelaez theorem as fol-lows [119]F URt(pt, n, ¯) = P(PUs (Rt, n, ¯) > pt) (3.14)=12+1pi∫ ∞0Im(e−t log(pt)MUjt)tdt,where Im(·) denotes the imaginary part andMUjt is obtained by replacing b by jt in(3.13), where j is the imaginary unit.The expressions provided in (3.12), (3.13), and (3.14) can be evaluated numer-ically. To obtain an analytic expression, we utilize the moment matching methodwith a Beta distribution, proposed in [119], to approximate (3.14) to obtainfX(x) =xϑ(β+1)−11−ϑ (1− x)β−1B(ϑβ1−ϑ , β) , (3.15)623.2. Fractional Path-Loss Inversion Power Controlwhere B(·, ·) is the Beta function, ϑ =MU1 and the β = (MU1 −MU2 )(1−MU1 )MU2 −(MU1 )2. Hence, themeta distribution can be computed as the CCDF of the Beta distribution. However,MU1 and MU2 provided in (3.13) are difficult to evaluate due to the added channeldispersion term V (ΩUFI). Instead, a lower bound on the conditional success probabilityand an approximation of the meta-distribution are given in the following proposition.Proposition 2. The conditional success probability can be lower bounded byPˆUs (Rt, n, ¯) =∏ri∈Φ\{r0} p1 +θ2Pir−ηiρor−η(1−δ)0+(1− p) e −θ2σ2wρor−η(1−δ)0 , (3.16)where θ2 = 2Rt+an,¯(∞)−bn−1. Using (3.16) and the Beta distribution approximation,the meta distribution in (3.14) can be approximated asFˆ URt(pt, n, ¯) ≈ 1− Ipt(MˆU1 (MˆU1 − MˆU2 )(MˆU2 − (MˆU1 )2),(1− MˆU1 )(MˆU1 − MˆU2 )(MˆU2 − (MˆU1 )2)), (3.17)where Iz(x, y) =1B(x,y)∫ z0tx−1(1−t)y−1dt is the regularized incomplete Beta function,and the MˆU1 and MˆU2 are the moments of Pˆs(Rt, n, ¯) provided in (3.18) at the topof next page.Proof. Starting from (3.12), we upper bound the√V (ΩUFI) by log2(e) which is consid-ered a simple yet convenient bound (it quickly approaches log2(e) as ΩUFI increases),leading to Ps(Rt, n, ¯) ≥ Pˆs(Rt, n, ¯) wherePˆs(Rt, n, ¯) = P! (log2(1 + ΩUFI)− an,¯(∞) + bn > Rt|Φ) .Consequently,Pˆs(Rt, n, ¯) = P! (ΩUFI > θ2|Φ)= Ehi,xi ∏ri∈Φ\{r0}e−θ2xi|hi|2Pir−ηiρor−η(1−δ)0 e−θ2σ2wρor−η(1−δ)0 ,By averaging over the channel gains, we obtain (3.16). The moments of Pˆs(Rt, n, ¯)are given in (3.18). Substituting these moments in the Beta distribution (3.15) yields(3.17).633.3. Special Cases: TCI and CIMP Power Control SchemesMˆUd = E(Pˆs(Rt, n, ¯)d)=∫ ∞0e− ∫∞r02γ1+δ,piλ(ρom) 2η(1−δ)1− p1+θrη(1−δ)0ρom+(1−p)de−θdσ2wρor−η(1−δ)0(piλ)δ−1ρ−2ηo ηm1− 2ηdmf(r0) dr0, (3.18)3.3 Special Cases: TCI and CIMP PowerControl SchemesTo mitigate interference, associating a UE with the closest BS combined withpower control strategy is used to enforcing the average received power at a givenBS from its associated UE to be strictly greater than the average interference powerreceived from any other UE. In this section, we will study the performance of twopower control strategies, TCI and CIMP, for a large-scale network in the FBR. First,the two strategies are proposed followed by the performance in terms of averagecoding rate and rate outage probability.3.3.1 TCI and CIMP Power Control SchemesIn practical systems the UEs has a limited transmit power Pu, hence, not all theUEs can apply channel inversion without violating the maximum transmit powerconstraint (Pi ≤ Pu). It is worth noting that we assume the inversion of only thepath-loss component of the channel(r−η/20)for simplification [88, 89]. Taking intoconsideration the power limitations of the UEs, we adopt the following strategies.The first scheme is the TCI [88], where UEs that are able to transmit with powerless than the maximum trasmit power Pu and maintain an average receive powerthreshold ρo are allowed to transmit. Otherwise, the UE is disconnected. In otherwords, UEs lying at a distance > (Puρo)1η are disconnected. The active UEs transmitwith power(P(t)i = ρorηi)10. Hence, the SINR (Ω(t)) is given by:Ω(t) =ρo|h0|2σ2w + I (t)(3.19)10The superscript t stands for truncated channel inversion.643.3. Special Cases: TCI and CIMP Power Control Schemeswhere I (t) = ∑ri∈Φ P (t)i |hi|2r−ηi is the interference power. The interference powerdistribution has a LT, derived in [88], given by:LI(t)(s)=exp{−2piλ s 2η EP (t)(P (t)2η)∫ ∞(sρo)−1ηyyη + 1dy}(3.20)where EP (t)(P (t)2/η)is evaluated fromEP (t)(P (t)d)=ρdoγ(dη2+ 1, piλ(Puρo)2/η)(piλ)dη2(1− e−piλ(Puρo )2/η) (3.21)with d = 2/η and γ(a, b) =∫ b0ta−1e−t dt.The second scheme is the CIMP [89] where UEs are divided to two sets of UEs.UEs who lie within the region of radius(Puρo) 1ηfrom their serving BS, transmit withpower(P(m)i = ρorηi)11. However, UEs lying at the distance > (Pu/ρo)1η , transmitwith maximum power(P(m)i = Pu). In other words, All UEs are still being served butthe received signal is not guaranteed to satisfy the average receive power thresholdρo. The SINR (Ω(m)) is defined for the two sets of UEs according to their location asfollowsΩ(m) =ρo|h0|2σ2w+I(m) , r0 <(Puρo)1/ηPur−η0 |h0|2σ2w+I(m) , r0 ≥(Puρo)1/η (3.22)where I (m) =∑ui∈Φ P (m)i |hi|2u−ηi is the interference power. The interference powerdistribution has a laplace transform, derived in [89], given by:LI(m)(s)= exp{−2piλ s 2η EP (m)(P (m)2η)∫ ∞(sρo)−1ηyyη + 1dy}(3.23)where EP (m)(P (m)2/η)is evaluated fromEP (m)(P (m)d)=ρdoγ(dη2+ 1, piλ(Puρo) 2η)(piλ)dη2+ P due−piλ(Puρo )2η(3.24)11The superscript m stands for channel inversion with maximum power.653.3. Special Cases: TCI and CIMP Power Control Schemeswith d = 2/η.Next, the average coding rate analysis is provided for the large-scale OMA net-works in the FBR adopting the aforementioned power control strategies.3.3.2 Average Coding Rate AnalysisThe maximum achievable rate for a blocklength n and a FER  is proposed in [6],presented in (3.3). The average coding rate for the UL channel will be derived byaveraging over the randomness of the channel (h0, hi, ui, ri, r0). Next, we will startwith the average coding rate for TCI followed by CIMP power control.3.3.2.1 Average Coding Rate under Truncated Power ControlThe average coding rate of a large-scale UL network in the FBR under TCI powercontrol is given byR¯(t)n,(ρoσ2w)= R¯n,(ρoσ2w)(1− e−piλ(Puρo )2η)(3.25)where the rate of active users R¯n,(ρoσ2w)is given byR¯n,(ρoσ2w)≈ C¯(t)(ρoσ2w)−V(t)(ρoσ2w)Q−1()√n+log2(n)2n, (3.26)where the average capacity C¯(t)(·) is given by [88]C¯(t)(ρoσ2w)=∫ ∞0exp{−(2c − 1)σ2wρo}LI(t)(2c − 1ρo)dc (3.27)where the Laplace transform of the aggregate interference LI(t)(·) is given by (3.20),and the average square root of channel dispersion for an UL large-scale network isgiven byV(t)(ρoσ2w)= E(√V (Ω(t)))(3.28)where Ω(t) = ρo|h0|2σ2w+I(t) is the SINR. Active users always perform channel inversion,hence, r0 does not affect the performance. Hence, we average over (h0, I(t)). Byaveraging, the average square root of channel dispersion is given byV(t)(ρoσ2w)=∫ log2(e)0e−σ2wz(v)ρo LI(t)(z(v)ρo)dv (3.29)663.3. Special Cases: TCI and CIMP Power Control Schemeswhere z(v)=(√11− v2log22(e)− 1).This expression is derived similar to the derivation provided in Theorem 2.3.Next, the average coding rate under CIMP power control is derived.3.3.2.2 Average Coding Rate under Channel Inversion with MaximumPower TransmissionThe average coding rate of an UL large-scale network in the FBR under CIMPis given byR¯(m)n,(ρoσ2w)≈ C¯(m)(ρoσ2w)−V(m)(ρoσ2w)Q−1()√n+log2(n)2n, (3.30)where the average capacity C¯(m)(·) is given by [89]C¯(m)(ρoσ2w)= C¯cu(1−e−piλ(Puρo )2η)+ C¯eue−piλ(Puρo )2η(3.31)where C¯cu is the average capacity for cell-center users, and C¯eu is the average capacityfor cell-edge users and are given byC¯cu =∫ ∞0exp{−(2c − 1)σ2wρo}LI(m)(2c − 1ρo)dcandC¯eu =∫ ∞0∫ ∞(Puρo )1/η2piλr0 exp{−(2c − 1)σ2wρo}exp{−piλr20 + piλ(Puρo)2/η}LI(m)(2c − 1ρo)dr0 dc,where the Laplace transform of the aggregate interference power is given in (3.23),and the average square root of the channel dispersion V(m)(ρoσ2w)is provided in thefollowing Theorem 3.1Theorem 3.1. The average square root of the channel dispersion of a large-scale ULnetwork V(m)(ρoσ2w)= E(√V (Ω(m))), where the Ω(m) is the SINR provided in (3.22),is given by673.3. Special Cases: TCI and CIMP Power Control SchemesV(m)(ρoσ2w)= Vcu(1− e−piλ(Puρo )2η)+ Veue−piλ(Puρo)2η(3.32)whereVcu =∫ log2(e)0e−σ2wz(v)ρo LI(m)(z(v)ρo)dvVeu=∫ log2(e)0∫ ∞(Puρo)1η2piλr0LI(m)(z(v)ρo)e−σ2wz(v)ρo−piλr20+piλ(Puρo)2ηdr0dv,and z(v) is given in (3.29)Proof. The V(m)(ρoσ2w)is derived as follows. We start by expressing the CDF of√V (Ω(m)) for a given I(m) as followsF√V (v|I(m)) = P(√V (Ω(m)) < v|I(m))= PΩ(m) <√ 11− v2log22(e)− 1|I(m)= P(ρo|h0|2σ2w + I(m)< z(v)|I(m))P(r0 <(Puρo)1/η)+ P(Pur−η0 |h0|2σ2w + I(m)< z(v)|I(m))P(r0 >(Puρo)1/η)=(1− e−z(v)(σ2w+I(m))ρo)P(r0 <(Puρo)1/η)+(1− e−z(v)(σ2w+I(m))Pur−η0)P(r0 >(Puρo)1/η).By averaging on I(m), the F√V (v) is given byF√V (v) =(1−∫ ∞(Puρo )1/η2piλr0LI(m)(1ρoz(v))e−σ2wρoz(v)e−piλr20+piλ(Puρo)2/ηdr0)P(r0 >(Puρo)1/η)+(1− e−z(v)σ2wρo LI(m)(z(v)ρo))P(r0 <(Puρo) 1η)683.3. Special Cases: TCI and CIMP Power Control SchemesUsing an application of Fubinis theorem [111], the average square root of thechannel dispersion is evaluated as follows:V(m)(ρoσ2w)=∫ ∞0(1− F√V (v))dv=∫ log2(e)0e−z(v)σ2wρo LI(m)(z(v)ρo)P(r0 <(Puρo) 1η)dv+∫ log2(e)0∫ ∞(Puρo )1η2piλr0e−σ2wz(v)ρo−piλr20LI(m)(z(v)ρo)dr0 dvAs shown in (3.26) and (3.30) as n→∞, the back-off term vanishes and the resultsunder the infinite blocklength assumptions are obtained as a special case (asymptote)of the results in the FBR. Next, the rate outage probability is derived for the twopower control schemes.3.3.3 Rate Outage AnalysisThe rate outage probability is characterized in this section for a large-scale ULnetwork in a FBR under TCI and CIMP power control schemes, providing an abstractcharacterization of the network behaviour. From the basic definition, a simplifiedupper bound is provided in Sec. 3.2.1 for an UL channel, which its tightness is shownin the numerical results. Using the same analogy, the rate outage probability upperbound for the large-scale UL network in the FBR using TCI and CIMP power controlschemes.3.3.3.1 Rate Outage Probability under Truncated Power ControlUnder truncated power control, the rate outage is defined for the active UEs(cell center UEs) only. Since cell edge UEs that lie at a distance >(Puρo)1/ηareinactive due to truncation, they are also in outage. To capture both types of outage,we combine rate outage and inactivity outage into a quantity which we call ‘totaloutage’.The rate outage probability upper bound for active UEs is presented in the fol-lowing Corollary.693.3. Special Cases: TCI and CIMP Power Control SchemesCorollary 6. The rate outage probability of a large-scale UL network modeled by (3.1),with truncated channel inversion power control with blocklength n, target FER ¯,target rate Rt, average received power threshold ρo, maximum transmit power Pu,and noise power σ2w can be upper bounded asO(t)(Rt, n, ¯)=1−e−σ2w(2Rt+an,¯−bn−1)ρo LI(t)(2Rt+an,¯−bn − 1ρo)where an,¯ =√log22(e)nQ−1(¯), bn =log2(n)2n, and LI(t)(·) is provided in (3.20).The rate outage probability provided is a function of the network parameters,the designed average received power threshold ρo, the density of the BSs λ, the noisepower σ2w, the blocklength n and the target FER ¯ and this bound is fairly tightand can be used as an approximation and this is shown in the numerical results. Itis worth noting that the rate outage is significantly affected by ρo as it controls theamount of interference. In particular, high ρo ensures that edge UEs from neighboringcells are inactive, which reduces interference.However, the rate outage is not sufficient to characterize the outage probabilityof this network model as inactive UEs suffer from outage due to truncation. Hence,the total outage, denoted by OUtotal(Rt, n, ¯), is defined for this system as the sum ofrate outage for active UEs and truncation outage for inactive UEs as in [89].O(t)total(Rt, n, ¯) = O(t)(Rt, n, ¯)(1− e−piλ(Puρo )2η)+Otr(Rt, n, ¯), (3.33)where Otr(Rt, n, ¯) = e−piλ(Puρo)2ηdenotes the truncation outage.3.3.3.2 Rate Outage Probability under Channel Inversion withMaximum Power TransmissionIn this case, UEs that lie at a distance larger than(Puρo)1/ηtransmit with maxi-mum power Pu and cell-center UEs perform channel inversion power control. Thus,all UEs are active. We can approximate the rate outage probability in this case asgiven next.Corollary 7. The rate outage probability of a large-scale UL network modelledby (3.1) with channel inversion with maximum power transmission power controlwith blocklength n, target FER ¯, target rate Rt, average receive power threshold703.3. Special Cases: TCI and CIMP Power Control Schemesρo, maximum transmit power Pu, and noise power σ2w, can be approximated asO(m)(Rt, n, ¯)≈O(m)cu(1− e−piλ(Puρo )2η)+O(m)eu e−piλ(Puρo)2ηwhere O(m)cu and O(m)eu are the rate outage probability approximations for cell-centerand cell-edge UEs, respectively, given byO(m)cu = 1− e−σ2w(2Rt+an,¯−bn−1)ρo LI(m)(2Rt+an,¯−bn − 1ρo),O(m)eu = 1−∫ ∞(Puρo )1η2piλr0e−(2Rt+an,¯−bn−1)Pur−η0 /σ2w e−piλr20+piλ(Puρo)2ηLI(m)(2Rt+an,¯−bn − 1Pur−η0)dr0where an,¯ =√log22(e)nQ−1(¯), bn =log2(n)2n, and LI(m) is provided in (3.23).From the rate outage probability under this power control scheme, it can beobserved that the outage probability approaches the outage probability of cell-centerUEs as ρo decreases (for a given Pu), since the percentage of truncated cell-edge UEsdecreases. On the other hand, if ρo is large, then the outage probability behavesdifferently since it is dominated by cell-edge UEs3.3.4 Reliability AnalysisWe aim here to gauge the overall network reliability taking into consideration theoutage probability and the target FER ¯. A codeword is guaranteed to be receivedsuccessfully if there is no outage and no frame error. Thus, the network guaranteedperformance is measured in terms of the probability of successful reception of acodeword.For TCI power control, the reliability is given byT (t)(Rt, n, ¯) = (1−O(t)total(Rt, n, ¯))(1− ¯). (3.34)For CIMP power control, the reliability is given byT (m)(Rt, n, ¯) = (1−O(m)(Rt, n, ¯))(1− ¯). (3.35)Also, for the sake of comparison, the reliability for the AR is defined asT¯ (i)(Rt,∞, 0) = 1−O(i)(Rt,∞, 0), (3.36)where i ∈ {total,m} depending on the power control scheme. Since, the error prob-ability vanishes in the AR n→∞, and hence failure occurs when there is outage.Next, numerical simulations are conducted to analyze and compare the perfor-mance of the two schemes in the FBR.713.4. Numerical Results(a) n = 1280 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.410-310-210-1(b) n = 2048Figure 3.1: The rate outage probability versus (Rt) for ρo = −60 dBm, and σ2w =−90 dBm3.4 Numerical Results3.4.1 Numerical Results for Fractional Path-Loss InversionPower ControlWe consider a network with intensity λ = 2 BSs/km2, power control parameterρo = {−60} dBm, and path loss exponent η = 4. All the results are validated usingMonte Carlo simulations. In the simulation scenario, we set one PPP realization forthe BSs in an area of 30 × 30 km2 area. One UE is placed uniformly within thecoverage of each BS.For the outage probability, we vary the network and the fading realization andcalculate the instantaneous SINR ΩUFI and the instantaneous rate Rn,¯(ΩUFI). Then,we compare the instantaneous rate to the rate threshold Rt and average to evaluatethe outage probability. Further, we numerically optimize the bounds at each Rt tooptimally choose Λ and υ. In Fig. 3.1, the rate outage probability is plotted versusRt for n = {128, 2048} and target FER ¯∈{10−2, 10−5}. As shown in the figure, therate outage upper and lower bounds in (3.7) and (3.8) are tight bounds on the per-formance of the network especially at moderate to high Rt. It is worth noting thatthe proposed upper bound provides a guaranteed performance at all values of Rt and723.4. Numerical Resultscan be used as an approximation for moderate to high Rt. While it is evident thata gap exists between the rate outage probabilities in the AR and FBR, quantifyingand investigating the behavior of this gap is essential. This gap is due to the term√V (ΩUFI)nQ−1(), which is a function of SINR, which in turn depends on the networkscale and density. When the SINR is high/low the V (ΩUFI) increases/decreases, in-creasing/decreasing the gap between the FBR and the AR performance. However,the SINR is random and hence the gap needs to be characterized. This gap in-creases as the target FER ¯ becomes more stringent and the blocklength n becomesshorter, which makes the rate outage probability in the AR for short blocklengths(e.g. n = 128) misleading, which motivates the analysis in FBR. Moreover, it isnoticed that as the blocklength increases the rate outage decreases approaching theperformance in the AR.To investigate the performance of the simplified bound provided in Corollary5, the simplified upper bound versus Rt is plotted in Fig. 3.2 with the optimizedupper bound and the exact performance of the network. The simplified upper boundprovides a simple expression that can be easily evaluated but it is not tight forlow Rt. However, for moderate Rt, it is shown to be tight and can be used as anapproximation. The tightness of this bound at moderate to large Rt and the existenceof a gap between the exact outage probability and the bound is explained next.In Fig. 3.1 and 3.2, it is observed that a gap exists between the optimized boundsand the exact outage probability of the network. This gap increases at low Rt anddiminishes at moderate to high Rt. Hence, an analysis is conducted to gain insightsinto the behavior of this gap and to investigate the significance of the parametersυ and Λ, and their impact on the outage probability. We start with the outagedefinition, where OUn,¯(Rt) = P(Rn,¯(ΩUFI) < Rt) = P(C(ΩUFI) < Rt + an,¯(ΩUFI) − bn).The outage probability is calculated by integrating the the pdf of the SINR ΩUFI suchthat C(ΩUFI) < Rt+an,¯(ΩUFI)−bn. Hence, we plotted the capacity C(ΩUFI) and the termRt + an,¯(ΩUFI)− bn as functions of SINR ΩUFI in Fig. 3.3. The term Rt + an,¯(ΩUFI)− bnincreases from Rt − bn to Rt + an,¯(∞) − bn as SINR increases. To simplify ouranalysis, we upper bound this function by a step function that changes value fromRt+an,¯(υ)−bn to Rt+an,¯(∞)−bn at υ, which is also plotted in Fig. 3.3 for optimizedvalues of υ. It is worth noting that the optimal υ is chosen numerically such thatO¯Un,¯(Rt,∞) is minimized. For small Rt, we find that the approximating step functionoverestimates the region at the optimal choice of υ, leading to an integration overundesired region thus increasing the gap between the bound and the actual outageprobability. For moderate to large Rt, the point of intersection of the approximatedstep function and the capacity approximately matches with the point of intersectionof the two exact curves for any choice of υ. Hence, it does not overestimate the733.4. Numerical ResultsFigure 3.2: Outage Probability versus Rt at n = 128 and  = 10−2integration interval. The significance of υ is evident at low Rt. However, its impactis negligible at moderate to high Rt which justifies the tightness of the simplifiedbounds. For the lower bound, the same argument can be made.For the meta-distribution of the coding rate simulation, (3.17) is compared to asimulation with fixed BSs and UEs realizations, and the fading realization changesat each time slot and each user becomes active with probability p. The SINR isevaluated for the active users and the simulation runs until 20, 000 samples of SINRare collected for each UE, which are used to evaluate per UE coding rate. The meta-distribution of the coding rate is then evaluated across all UEs. In Fig. 3.4, the codingrate meta distribution is plotted versus pt for rate thresholds Rt ∈ {0.5, 1.2, 1.5}bits/transmission. It is shown that the approximation provided in (3.17) tightlycharacterizes the exact coding rate meta distribution for a broad range of Rt.12However, for small Rt, the approximation provides a lower bound to the codingrate meta-distribution. It is worth noting that, for all Rt, the approximated meta-distribution provides an achievable performance, meaning it is always guaranteedsince it is always below the exact meta distribution. Also, it is shown that thecoding rate meta distribution in the AR significantly overestimates the number ofusers achieving the rate threshold Rt, by 20% to 40%. Hence, our analysis is moreprecise in estimating the users’ performance.12It is worth noting that the exact meta-distribution and its lower bound, using the probabilityof success in (3.12) and (3.16), are numerically evaluated using Monte Carlo simulations.743.4. Numerical ResultsFigure 3.3: Outage Probability AnalysisFigure 3.4: Rate meta distribution versus pt for n = 128, λ = 2 BS/Km2, ¯ = 10−5,δ = 1, and p = 0.7753.4. Numerical Results0 5 10 15 20 25 30 35 4000.20.40.60.811.21.41.61.8Figure 3.5: The average coding rate vs. ρo/σ2w for n = {256, 2048},  = {10−2, 10−5},ρo = −70 dBm for TCI and CIMP power control schemes3.4.2 Numerical Results for TCI and CIMP Power ControlSchemesIn this subsection, numerical results are presented for a large-scale network withBSs modeled as a PPP with intensity λ = 2 BSs/Km2, at a blocklength n and FER .UEs are distributed according to PPP and the density of UEs is assumed to be highenough such that a UE is always connected. Simulations are performed for averagereceived power threshold ρo = {−90, −70, −50} dBm, the UEs’ maximum transmitpower Pu = 2 W , and path loss exponent η = 4.Fig. 3.5 illustrates the average coding rate of the large-scale UL network. Usingthe classical coding theory for studying networks with short blocklength transmissionis shown to be imprecise. However, it provides an upper bound on the average codingrate that can be achieved for large n. Using the theory of coding in the FBR is moreprecise as it demonstrates the effect of n and  on the performance of the network.For example, for a blocklength of n = 128, there is up to a 5 dB gap between anFER of 10−2 and 10−5. We also investigated the effect of network parameters onthe average coding rate in Fig. 3.6. The average coding rate is plotted versus BSdensity λ for different values of ρo. It is shown that at small λ, the number of activeusers under TCI power control is very small compared to the number of truncated763.4. Numerical Results0 1 2 3 4 5 6 7 8012345Figure 3.6: The average coding rate versus λ at n = 128,  = 10−5, and ρo ∈{−50,−70,−90} dBmusers. Hence, as λ increases the number of truncated users decreases as the cell sizedecreases and the average rate increases. On the contrary, for CIMP, the averagecoding rate is high at small λ and decreases as λ increases because as λ increases,the interference increases. It is also observed that as λ increases, both power controlschemes tend to perform the same.In Fig. 3.7, the average coding rate for a blocklength n = 512 and FER  = 10−2is plotted using the TCI and CIMP power control schemes. As shown in the figure,at high average receive power threshold ρo > −80 dBm, the TCI power controlexhibits better performance than the CIMP. As the TCI will aggressively truncateall the edge UEs who need transmit power P > Pu. Hence, all edge UEs are inactive,and the induced interference is less, consequently, it exhibits a better average rateperformance. However under CIMP, more interference is induced due to the activityof edge UEs. However, at low average receive power threshold ρo = −90 dBm, bothtechniques tend to perform equally as approximately all UEs will be able to performpath-loss inversion while meeting the average receive power threshold and the max.transmit power constraint.In Fig. 3.8, we compare the results provided in Corollaries 6 and 7 with the resultsfrom the literature in the AR [88, 89]. The rate outage probability is plotted versusthe rate threshold Rt for ρo = −50 dBm, a blocklength n∈{128, 2048} and an target773.4. Numerical ResultsFigure 3.7: The average coding rate vs. ρo/σ2w for n = 512,  = 10−2, ρo ={−90, −70, −50} dBm for TCI and CIMP power control schemes.Figure 3.8: The rate outage probability versus rate threshold Rt for ρo=−50 dBm,n∈{128, 2048}, ¯∈{10−2, 10−5}, σ2w =−90 dBm for TCI and CIMP power controlschemes.783.4. Numerical ResultsFER ¯∈{10−2, 10−5}. As shown in the figure, for short blocklengths, the rate outageprobability in the AR underestimates the actual rate outage probability under thegiven a blocklength n and target FER ¯. The relative gap can be significant in somecases, especially when the outage probability is small which is the practical regime ofoperation. Hence, it is more accurate to use the rate outage probability expressionsgiven in Corollaries 6 and 7 for applications with stringent delay and reliabilityrequirements. Also, it is observed that as the blocklength increases the rate outagedecreases approaching the performance in AR. It is worth noting that this figureis not provided to compare the performance under different power control schemes,since such a comparison would not be fair because the truncated channel inversionpower control deactivates some UEs, whereas channel inversion with maximum powertransmission does not.In Fig. 3.9, the total outage probability under the two power control schemes isinvestigated versus BS density λ for different values of n and ¯. It can be observedthat at low BS density (< 1 BS/Km2), the outage probability is high and decreasesas λ increases, because at low λ, cell size increases and the number of users lie ata distance >(Puρo) 1ηincreases. Hence, using the TCI, these users are truncated(truncation outage dominates) and using the CIMP, they will transmit with powerPu which does not guarantee reception at a power level > ρo which results in highoutage. However, at large λ > 1, the network becomes dense and the interferenceincreases, hence, the outage increases. We can also observe that the CIMP performsbetter than the TCI at low density networks because the users that lie at a distance>(Puρo)1/ηare truncated under the TCI and are being served with lower quality bythe CIMP. However, at large lambda (λ > 3), the truncation outage decreases as thenumber of users at a distance >(Puρo)1/ηdecreases and truncating some users willdecreases interference which makes the TCI performs better.Fig 3.10 illustrates the gap between the outage in the FBR and that in theAR. The total outage probability and rate outage probability are plotted versus theblocklength n for different values of target FER ¯ ∈ {10−5, 10−8}, at target SINRθ2 = −10 dB ρo = −70 dBm, and σ2w = −90 dBm. In Fig 3.10(a), the figure shows agap of> 0.25 at small n for TCI power control and a gap of> 0.1 at small n for CIMP.However, in Fig. 3.10(b), the gap the FBR and the AR for the truncated channelinversion power control increases to be > 0.2 at small blocklengths. However, theoutage probability decreases as n grows larger, and approaches the outage probabilityof the AR at sufficiently large n. Moreover, as the target FER becomes more tight( = 10−8) the gap increases.793.4. Numerical ResultsFigure 3.9: The total outage probability versus λ for Rt = 1, ρo = −70 dBm,n ∈ {128, 2048}, and ¯ ∈ {10−2, 10−5}(a) The total outage probability (b) The rate outage probabilityFigure 3.10: The total outage probability and the rate outage probability of the large-scale network versus blocklength n at λ = 2 BS/km2, Rt = 0.1375, ρo = −70 dBm,σ2w = −90 dBm and ¯ ∈ {10−5, 10−8}.803.5. ConclusionFigure 3.11: Reliability versus target FER ¯ at Rt = 0.1375The network reliability versus target FER ¯ is plotted in Fig. 3.11 under thetwo power control schemes at Rt ∈ {0.1375}. It is shown that the reliability of thenetwork increases as the target FER increases until it reaches a maximum around¯ = 10−2 then it rapidly decays. This is because the term (1−O(i)(Rt, n, ¯)) increasesexponentially with ¯ and the term (1−¯) decreases linearly with ¯. Hence, at ¯ < 10−2,the rate of increase of (1−O(i)(Rt, n, ¯)) is higher than the rate of decay of (1− ¯).However, at ¯ > 10−2, the increase of (1−O(i)(Rt, n, ¯)) slows down (where it becomesclose to 1) and the rate of decay of (1−¯) dominates. Therefore, the reliability rapidlydecays. Moreover, a gap exists between curves simulated under FBR and AR whichacts as an upper bound and it grows wider up to 0.4 as target FER ¯ decreasesbecause the short blocklengths (e.g. n = 128) are less reliable. It is also shown theTCI has lower (∼ 0.35) reliability of the CIMP due to truncation.3.5 ConclusionIn this chapter, we developed a framework to characterize the performance ofan large-scale UL OMA networks for short codes using the theory of coding in theFBR in conjunction with stochastic geometric tools. For the UL, we considered813.5. Conclusiona generic power control scheme which is the fractional path-loss inversion powercontrol then adopted two special cases which are TCI and CIMP power controlschemes. For the fractional path-loss inversion power control, we provided theoreticalexpressions for the rate outage probability and the coding rate meta-distributionwhere bounds and approximations for both of them are proposed. Numerical resultswere provided to highlight the tightness of the rate outage probability upper andlower bound. Moreover, we provided a qualitative analysis of the performance of thebounds. Also, the coding rate meta-distribution approximation tightly characterizesthe performance in the FBR for a large-scale network. For TCI and CIMP powercontrol schemes, we proposed closed-form expressions that characterizes the averagecoding rate, rate outage probability, and reliability. The numerical results showedthat the AR overestimates the performance of network and the gap between theperformance on the FBR and the AR is quantified. Moreover, a comparison betweenthe TCI and CIMP power control schemes is conducted under FBR assumptions.It is shown that the TCI power control provides better performance in terms ofoutage probability for dense networks. On the other hand, CIMP provides reliabilityand availability for low and high density network, and both schemes provide sameaverage coding rate for dense networks. In both analysis, it was shown that theperformance in the AR underestimates the rate outage probability and overestimatesthe coding rate meta distribution and the average coding rate, compared to therealistic performance of the network in the FBR. However, as the blocklength nincreases, the performance approaches the performance of the AR. In conclusion, ourresults can accurately depict the performance of large-scale networks in the FBR.82Chapter 4Large-Scale Downlink NOMANetworks in the FiniteBlocklength RegimeIn this Chapter, we will develop a theoretical performance analysis of a cellu-lar large-scale networks which employ non-orthogonal multiple access. NOMA is apromising method for enhancing the capacity of the network by resource sharing. Itenables multiple users to transmit on the same time/frequency resources with dif-ferent power levels. Hence, it offers higher spectral efficiency, massive connectivityby serving more users simultaneously, and lower latency due to simultaneous trans-mission all the time rather than dedicated scheduled time slot. Thus, its analysisis different from OMA considered in Chapter 2. This chapter provides the neces-sary framework for characterizing the performance of NOMA large-scale network inthe FBR. Extending the work to large-scale networks under latency and reliabilityconstraints is necessary. To the best of the authors knowledge, performance char-acterization of large-scale NOMA networks in the FBR is still an open problem.Furthermore, while the prevailing understanding acknowledges NOMAs superiorityover OMA in terms of achieving higher data rates, our inquiry pivots on the followingquestion: Does NOMAs out-performance persist in the realm of large-scale networkssubject to stringent reliability and latency constraints, and what is the gap betweenthese two multiple access techniques in this scenario? To address this, we utilizethe theory of coding in the FBR in conjunction with stochastic geometry tools withthe intention to cover to the aforementioned research gap. In particular, we aim tocharacterize the average rate region for a large-scale NOMA network under a globalreliability constraint and a short blocklength and then compare the performanceof NOMA to OMA for a large-scale network in the FBR. The contributions of thechapter can be summarized as follows:− Achievability: We derive a theoretical achievable rate region for the large-scale NOMA network under a global frame error rate and a finite blocklengthrequirement. This region serves as an inner bound for the maximum achievable83Chapter 4. Large-Scale Downlink NOMA Networks in the Finite Blocklength Regimerate region.− Converse: We derive an outer bound on the maximum achievable rate regionfor the large-scale NOMA network in the FBR, providing a benchmark forsystem performance.− Fairness: We express the max-min average rate and propose an algorithm tofind a suboptimal max-min rate. Also, we propose a practical power allocationscheme to pursue a fair performance between two NOMA users.− Practical scheme: We investigate the performance of a 5G compliant mod-ulation and coding scheme to achieve rates close to the theoretical achievablebound. Specifically, we consider MLPCM and show that it can achieve ratesclose to the theoretical achievable inner bound.We also benchmark the NOMA performance against OMA network in the FBR.We characterize the performance gap between the NOMA and OMA schemes withrespect to network parameters and provide a qualitative and quantitative analysis,deriving useful insights on the performance gain of NOMA in a large-scale networkin the FBR. The comparison shows that the performance gap between OMA andNOMA increases as we move towards shorter blocklengths and stringent reliabilityconstraints, which promotes the use of NOMA networks in future networks.In addition, we examine the impact of various network parameters on large-scale NOMA network performance. Specifically, we analyze the effect of BS densityand reliability constraints. The derived bounds and max-min rate provide a solidfoundation for advancing the design and implementation of high-performance wirelesscommunication systems, particularly in scenarios characterized by large-scale NOMAdeployments within the FBR.This chapter is organized as follows. In Sec. 4.1, the NOMA system modeland assumptions are presented, followed by the OMA system model which will beused for benchmarking. In Sec. 4.2, the rate performance analysis of the large-scaleNOMA network is presented where the inner bound, outer bound, and max-min rateare presented in sections 4.2.1, 4.2.2, and 4.2.3, respectively. Next, in Sec. 4.4, theMLPCM scheme is presented to be used as a practical validation of the theoreticalresults. The numerical results are presented in Sec. 4.5, including a comparisonbetween the theoretical results of NOMA and OMA as well as that of the MLPCMpractical scheme, followed by a detailed discussion. Finally, the chapter is concludedin Sec. 4.6.844.1. System Model4.1 System Model4.1.1 Network ModelWe consider a downlink cellular network, where a typical BS is located at theorigin, denoted BS0 and interfering BSs locations follow a 2-D homogeneous PPPΨ = {κk, k ∈ N} with a density λ, where κk ∈ R2 represents the coordinates of thekth BS. The network applies universal frequency reuse with one UE served per BS ata given resource block which results in intercell interference. It is assumed that allBSs are equipped with a single antenna and transmit with the same power P . TheUEs are modelled as a point process Φ with intensity λu, where λu λ. We assumethe nearest BS association assumption, hence users are associated with their nearestBS.Under this network setup, we define a user cluster to be the set of users usingthe same resource block and served by the same BS. Thus, there can be at leasttwo users in one cluster. Hence, the downlink channel is modeled as a broadcastchannel where the BS broadcasts the signals to all of its users that are served bythe same resource block. Depending on the multiple access scheme, the users withinthe clusters share the resource block, which can result in intracell interference. Fora generic user within the cluster served by BS0 is located at position u ∈ R2, thetransmission distance between this user and BS0 is denoted by r = ||u|| and itsdistribution is given byfr(r) = 2piλre−piλr2 , 0 < r <∞, (4.1)due to the nearest BS association assumption. Since the NOMA transmission requiresordering, we ordered the users within the cluster, served by BS0, with respect totheir ascending order of distance from the serving BS0 as (r1 < r2 < ...). Whilethe locations of the users are independent, the ordering done by the multiple accessscheme brings correlations which is considered. Under this model, we assume thateach BS transmits with a total power budget of P . In this work, we consider a clusterof two users. Next, NOMA and OMA multiple access schemes will be presented.4.1.2 NOMA System ModelIn NOMA, multiple users can share the same radio resources, such as time slotsand frequency bands, which results in intracell interference. The intracell interferencecan be mitigated using methods such as superposition coding and SIC [123]. Weassume that 2-users are associated with each BS, forming a NOMA cluster. Each BS854.1. System Modelserves its associated 2-users in one time-frequency resource block by multiplexing thesignals for each user with different power levels. The power assigned to each user isP1 = ξP and P2 = (1− ξ)P , where ξ ∈ [0, 1] is the power split factor. The receivedsignal at useri is given byyi = hi0r− η2i χ0 +∑κk∈Ψhikg−η/2ik χk︸ ︷︷ ︸Iaggi+w, i ∈ {1, 2} (4.2)where hik represents the channel fading coefficient from BS k to useri, η is the pathloss exponent, χk =√ξPs1k +√(1− ξ)Ps2k is the transmitted signal from BS kwhere sik is a unit average power codeword symbol transmitted by BS k to useri,gik = ||κk − ui|| is the distance between the interfering BSk located at κk and theuseri located at ui, and w ∼ CN (0, σ2w) is circularly symmetric complex Gaussiannoise with zero mean and variance σ2w. The term Iaggi is the aggregate interferencefrom all interfering BSs at useri. We model hi0 and hik to be circularly symmetriccomplex Gaussian with zero mean and unit variance (modeling Rayleigh fading). Anindependent and identically distributed block fading channel model is assumed wherethe channel coefficients hi0 and hik remain constant for a duration of L consecutivesymbols (one block) and change to independent realizations at the end of the block.From (4.2), it is clear that the users within the representative cluster are vul-nerable to two kinds of interference: intercell interference and intracell interference,unlike OMA which only suffers from inter-cell interference. However, to mitigatethe intra-cell interference, NOMA exploits superposition coding at the BS and SICat the receivers [72]. Specifically, in NOMA, user1 decodes the message of user2,then decodes its own message after subtracting the decoded message of user2, whileuser 2 only decodes its own message treating user1’s signal as noise. For the sake ofeasy reference, we will call user1 the near user and user2 the far user. The signal-to-interference-plus-noise ratio (SINR) is given byγ1 =ξP|h01|2r−η1I1 + σ2w, γ2 =(1− ξ)P|h02|2r−η2ξPr−η2 |h02|2+I2 + σ2w. (4.3)where Ii is the aggregate interference power at useri.The serving BS sends information to the two NOMA users using codewords fromcodes with the rates R1 and R2 and length n symbols, where n can be defined fol-lowing a latency constraint. To ensure that the channel remains constant duringthe transmission of n symbols, we require n = L/l for some integer l [7]. This en-sures that the channels remain constant during the transmission, but makes consec-utive transmission blocks within the same coherence interval have identical channels.864.1. System ModelNonetheless, by considering the sequence of blocks consisting of the first block (of nsymbols) in each coherence interval (of L symbols), we have independent channelsand we can derive the average performance over these blocks. This equals the overallaverage rate because the remaining sequences of blocks (such as the second frame ineach coherence interval) have identical statistics.4.1.3 Problem StatementThis thesis characterizes the average coding rate for a large-scale NOMA networkand compare it to the average coding rate of a large-scale OMA network. We assumethe knowledge of CSI at the BS and the user in the form of SINR availability.13Consequently, the BS can adapt its coding rate for each user based on the SINRin each transmission block so that the transmission using codelength n is successfulwith a maximum global FER , defined as follows.Definition 2. The global FER is the overall FER of the users in the network, de-fined as P(n)e = P((sˆ(n)01 , sˆ(n)02 ) 6= (s(n)01 , s(n)02 )), where sˆ(n)01 and sˆ(n)02 are the reconstructedversions of s(n)01 and s(n)02 , respectively, where s(n)0i is a frame of n-symbols.We shall use  to denote the largest allowed average FER, i.e., P(n)e < . Fora blocklength n and a maximum global FER , the achievability of a rate pair isdefined as follows.Definition 3. A non-negative rate pair (R1, R2) is said to be (n, )-achievable if thereexist a code with a blocklength n with a global FER P(n)e ≤ . let Rn, denote the setof all (n, )-achievable rate pairs.We define S as the network state, representing a network realization {(ri, gik)|i ∈{1, 2},∀k ∈ N} and channel gains {hik|i ∈ {1, 2},∀k ∈ N}. Denote(RNOMA1,n (S, ξ, 1), RNOMA2,n (S, ξ, 2))as the achievable rate pairs given state S and scheme parameters (ξ, 1, 2), where thepower split ξ, the per-user FERs are 1 = P(sˆ(n)01 6= s(n)01 ), and 2 = P(sˆ(n)02 6= s(n)02 ) whichsatisfy the global FER constraint 1−(1−1)(1−2) ≤ , achieved using superpositioncoding and SIC. Then, we define a policy f by which the BS selects transmissionscheme parameters to achieve an achievable rate pair based on the network state S,f(S). In other words, it maps the network state S to a choice of the power split13Assuming SINR knowledge benchmarks the performance of the network.874.2. Average Rate in the FBRfactor ξ and the per-user FER 1 and 2. Denote(RNOMA1,n (S, f(S)), RNOMA2,n (S, f(S)))as the achievable rate pair given state S and policy f(S).We are interested in the set of achievable average rate pairs(R¯NOMA1 (f), R¯NOMA2 (f))defined as (R¯NOMA1,n (f), R¯NOMA2,n (f))=(ES{RNOMA1,n (S, f(S))},ES{RNOMA2,n (S, f(S))}). (4.4)where f ∈ F , the set of policies that map from state S to scheme parameters{ξ, 1, 2}. Denote by RˆNOMAn, the set of all (n, )-achievable average rate pairs ∀f ∈ Fwhich satisfy the constraints of the network.RˆNOMAn, =⋃f∈FR¯NOMAn, (f) (4.5)whereR¯NOMAn, (f) ={(R¯1, R¯2)∈R2+ : R¯1 ≤ R¯NOMA1,n (f), R¯2 ≤ R¯NOMA2,n (f)}.This set describes the performance of the network, and is hence important tostudy. To this end, in what follows, we provide tractable expressions and bounds onRˆNOMAn, for two NOMA users in a large-scale network in the FBR.4.2 Average Rate in the FBRUsing the definitions provided in Sec. 4.1.1, we mathematically define the achiev-able rate region for large-scale NOMA network in the FBR in this section.In [124, Theorem 1], the achievable rate region for an AWGN broadcast channelattained by superposition coding with rate splitting was characterized. We followthe characterization in [124, Remark 7] obtained without rate splitting as given inthe following lemma.Lemma 4.1. Consider a two-user Gaussian broadcast channel with SNRs ψ1 andψ2, a maximum global FER , and blocklength n. The achievable rate region using884.2. Average Rate in the FBRsuperposition coding is given byRn, =⋃{ξ,1,2}∈[0,1]3{(R1, R2) ∈ R2+ :R2 ≤ C((1− ξ)ψ21 + ξψ2)−√V ′(ξψ2, ψ2)nQ−1(2) +log2(n)2n,R1 ≤ C (ξψ1)−√V (ξψ1)nQ−1(1) +log2(n)2n}(4.6)where C(x) = log2(1 + x), V (x) =x(2+x) log2(e)2(1+x)2is the point to point Gaussian dis-persion function, V (x, y) = x(2+y) log2(e)2(1+x)(1+y)is the cross dispersion function,V ′(x, y) =(y − x)(2xy + 3x+ y + 2) log2(e)2(1 + x)2(1 + y)2,ξ ∈ [0, 1] is the power split factor, and (1, 2) ∈ [0, 1]2 are the per-user FERs satis-fying(1− 1)(1− 2) ≥ 1− . (4.7)Proof. Complete details are given in [124].This lemma describes the rate region for an AWGN channel, which we utilizeto study the achievable average rate region of a NOMA large-scale network in theFBR. To this end, we need to include in the analysis the additional non-Gaussianinterference term Iaggi which captures the stochasticity of the network, in addition tothe random channel gains hik for i ∈ {1, 2}, k = {1, 2, ...}. Hence, in the upcomingsubsections, an inner bound and an outer bound will be derived for the large-scaleNOMA network’s achievable average rate in the FBR, followed by an analysis of themax-min rate.4.2.1 Rate Region Inner BoundIn this section, we provide an achievable inner bound that serves as a benchmarkfor achievable communication rates, shedding light on the fundamental performanceboundaries. The achievable rate region for a given network state S and a policy f(S)is characterized using Lemma 4.1 by replacing ψi by γi, presented in (4.3), during agiven transmission block, which leads to the following achievable region:RNOMAn, (S, f(S)) ={(R1, R2) ∈ R2+ : R1 ≤ RNOMA1,n (S, f(S)), R2 ≤ RNOMA2,n (S, f(S))}894.2. Average Rate in the FBRwhere f : S → (ξ, 1, 2),RNOMA1,n (S, f(S)) =C(γ1)−√V (γ1)nQ−1(1) +log2(n)2n, (4.8)RNOMA2,n (S, f(S)) = C (γ2)−√√√√V ′ ( ξP|h2|2r−ηI2+σ2w , P|h2|2r−ηI2+σ2w )nQ−1(2) +12nlog2(n) (4.9)This is an achievable rate region for a given state and a given policy. To calculatethe achievable average rate region, it is required to average over all the rate pairsat different network states, i.e., different network realization and different channelsand take the union over all policies f . Since this analysis becomes intractable, anachievable inner bound is proposed in the following theorem.Theorem 4.2. For a large-scale NOMA network, an inner bound on the achievableaverage rate region R¯NOMAn, for a blocklength n, a maximum global FER , a powersplit factor ξ, attained by superposition coding and SIC for NOMA, is given byRˆNOMAn, =⋃{ξ,1,2}∈[0,1]3(1−1)(1−2)≥1−R¯NOMAn, (ξ, 1, 2), (4.10)where the achievable average rate region for a given (ξ, 1, 2) is defined asR¯NOMAn, (ξ, 1, 2) ={(R¯1, R¯2) ∈ R2+ : R¯1 ≤ R¯NOMA1,n (ξ, 1, 2),R¯2 ≤ R¯NOMA2,n (ξ, 1, 2)}, (4.11)whereR¯NOMAi,n (ξ, 1, 2) = C¯i −V¯i√nQ−1(i) +12nlog2(n),the average capacities C¯1 and C¯2 are given byC¯1 =∫ ∞0∫ ∞0e−k(c)σ2wrη1ξP LI1{k(c)rη1ξP}fr1(r1)dc dr1, (4.12)C¯2 =∫ ∞0∫ log2 1ξ0e−k(c)σ2wrη2P−1(1−ξ−k(c)ξ) fr2(r2)LI2{k(c)rη2P−1(1− ξ − k(c)ξ)}dcdr2, (4.13)904.2. Average Rate in the FBRand the average square root channel dispersion V¯1 and V¯ ′2 are given byV¯1 =∫ ∞0∫ log2 e0e−z(v)σ2wrη1ξP LI1{z(v)rη1ξP}fr1(r1)dv dr1, (4.14)V¯2 =∫ √1−ξ2 log2 e0∫ ∞0e−z(v)σ2wrη2(1−ξ−z(v)ξ)PLI2{z(v)rη2P−1(1− ξ − z(v)ξ)}fr2(r2)dr2 dv (4.15)with k(c) = 2c − 1, z(v) =√11− v2log22(e)− 1, the PDF of r1 is fr1(r1) = 4piλr1e−2piλr21 ,and the PDF of r2 is fr2(r2) = 4piλr2e−piλr22(1− e−piλr22).Proof. The achievable average rate pairs for NOMA for a given policy are defined in(4.4) as(R¯NOMA1,n (f), R¯NOMA2,n (f))=(ES{RNOMA1,n (S, f(S))},ES{RNOMA2,n (S, f(S))}}, (4.16)and the achievable average rate region is defined as the set of all achievable rate pairsby taking the union over all policies as provided in (4.5).Let FNOMA, denote the set of policies that satisfy f(S1) = f(S2), i.e., the functionf is constant for all S. Note that FNOMA ⊆ F . As a result,RˆNOMAn, ⊆ RˆNOMAn, =⋃f∈FNOMAR¯NOMAn, (ξ, 1, 2), (4.17)where R¯NOMAn, (ξ, 1, 2) is provided in (4.11). This proves the inner bound. Forthe sake of coherent treatise, we moved the derivation details of the quantitiesES{RNOMA1,n (S, ξ, 1, 2)}and ES{RNOMA2,n (S, ξ, 1, 2)}to Appendix 6.2.The inner bound provides an achievable average rate region which can be evalu-ated for given n and . An outer bound on the maximum achievable region will beinvestigated next.4.2.2 Outer Bound RegionThe information-theoretic outer bound serves as an upper limit on the achievablerates, providing a benchmark for system performance. It also gauges the tightness ofthe inner bound providing insights into the fundamental trade-offs and constraintsinherent in NOMA networks. In [124], a cut-set bound for the achievable rate regionis proposed, which we repeat in the following lemma.914.2. Average Rate in the FBRLemma 4.3. Consider a two-user Gaussian broadcast channel with SNRs ψi, fora maximum global FER , and blocklength n. The achievable average rate region isouter bounded by ROBn,(S) given byROBn,(S) ={(R1, R2) ∈ R2+ : R1 ≤ ROB1,n(S, ξ, )R2 ≤ ROB2,n(S, ξ, )R1 +R2 ≤ ROBs,n(S, ξ, )}(4.18)whereROB1,n(S, ξ, ) = C(ψ1)−√V (ψ1)nQ−1() +log2(n)2n(4.19)ROB2,n(S, ξ, ) = C(ψ2)−√V (ψ2)nQ−1() +log2(n)2n(4.20)ROBs,n(S, ξ, ) = C(max(ψ1, ψ2))−√V (max(ψ1, ψ2))nQ−1(2) +log2(n)2n(4.21)Proof. Complete details are given in [124].In the following theorem, we apply Lemma 4.3 to obtain an information-theoreticouter bound for the large-scale NOMA network considered in this chapter.Theorem 4.4. The achievable average rate region for a large-scale network with amaximum global FER () and a blocklength n, is outer bounded by R¯OBn, given byR¯OBn, ={(R¯1, R¯2) ∈ R2+ : R¯1 ≤ R¯OB1,n(ξ, ),R¯2 ≤ R¯OB2,n(ξ, ),R¯1 + R¯2 ≤ R¯OBs,n(ξ, )}(4.22)whereR¯OBi,n(ξ, ) = C¯i −V¯i√nQ−1() +12nlog2(n), (4.23)R¯OBs,n(ξ, ) = C¯s −V¯s√nQ−1(2) +12nlog2(n), (4.24)924.2. Average Rate in the FBRwhereC¯i =∫ ∞0∫ ∞0e−k(c)σ2wrηiP LIi{k(c)rηiP}fri(ri)dc dri, (4.25)C¯s = C¯1 + C¯2 − C¯12, (4.26)C¯12 =∫ ∞0∫ ∞0∫ ∞0e−k(c)σ2w(rη1 +rη2 )P LI1{k(c)rη1P}LI2{k(c)rη2P}f(r1, r2) dr2 dr1 dc,(4.27)k(c) = 2c − 1, f(r1, r2) = 2fr(r2)fr(r1), 0 < r1 < r2, 0 < r2 < ∞ where fr(r) isdefined in (4.1), andV¯i =∫ ∞0∫ log2(e)0e−z(v)σ2wr−ηiP LIi{z(v)Pr−ηi}fri(ri)dv dri, (4.28)V¯s = V¯1 + V¯2 − V¯12, (4.29)V¯12 =∫ log2(e)0∫ ∞0∫ ∞0e−z(v)σ2w(rη1 +rη2 )P LI1{z(v)rη1P}LI2{z(v)rη2P}f(r1, r2) dr2 dr1 dv,(4.30)where z(v) =√11− v2log22(e)− 1Proof. At a given network state S, the region ROBn,(S) is proven to be the cutsetbound region in [124]. By averaging over all state, the region R¯OBn, is also an outerbound on the average achievable region for any multiple access scheme. It remainsto express the average capacity and the average square root channel dispersion foruser1 and user2, C¯i = E{C(γ¯i)} and V¯i = E{√V (γ¯i)}for i ∈ {1, 2}, respectively,where γ¯i =P|h0i|2r−ηiIi+σ2w . These can be derived similar to the derivation provided in Ch.2. For the sake of coherent treatise, we refrain the derivation details of the averagesum capacity C¯s and the average square root channel dispersion V¯s to Appendix 6.2and Appendix 6.2, respectively.Next, the max-min average rate is studied.4.2.3 Max-Min Average RateIn the previous sections, we provided an inner and an outer bound. Within theinner/outer bounds, a special rate pair of interest is (R1, R2) with R1 = R2 which934.2. Average Rate in the FBRAlgorithm 1: Optimal Power Split ξInitialize λ, n, , P , and N0Initialize ξ =E{r−η2 }E{r−η1 }Initialize the step size ∆ and the error tolerance ζEvaluate combinations of 1 and 2 that satisfies (1− 1)(1− 2) ≥ (1− )Set counter j = 1 for each value of (1, 2) dowhile |R¯1 − R¯2|≥ ζ doEvaluate R¯1 and R¯2 provided in (4.11)Update ξ = ξ − sign(R¯1 − R¯2)∆endR¯opt(j) = R¯1j = j + 1endEvaluate R∗ = max(R¯opt)ensures fairness. This can be found by deriving the max-min rate which is studiednext.For more proactive fairness in NOMA, we will consider the maximization of theminimum average rate as follows:R∗ =maxξ,1,2min{R¯NOMA1,n (ξ, 1, 2), R¯NOMA2,n (ξ, 1, 2)} (4.31)subject to 0 ≤ ξ ≤ 1,(1− 1)(1− 2) ≥ (1− ) (4.32)However, the terms R¯NOMAi,n (ξ, 1, 2) for i ∈ {1, 2} are integral expressions withunknown closed-form expressions which are a functions in ξ, 1, 2. Hence, the math-ematical analysis of the max-min average rate R∗ is not tractable. However, it couldbe computed numerically. To find an optimal max-min rate, an optimization algo-rithm is proposed in Algorithm 1, where we initialize the power split by the ratiobetween the average distance between the two users. Then, we utilize steepest de-scent optimization to find optimum max-min rate at each pair of (1, 2) that satisfiesthe maximum global FER constraint. Then, we choose the largest max-min rate overall per-user FER pairs.We also provide another approach, which proposes a sub-optimal solution thatguarantees instantaneous fairness in the network performance, unlike the previousapproach which guarantees fairness in the average network performance. This ap-proach depends on providing a fair distribution of resources such that both users are944.3. OMA Inner Boundprovided with the same quality of service. This can be achieved by designing thepower split factor ξ such that both users achieve the same rate, denoted as R∗MC. Forthe users to achieve the same performance in terms of coding rate for a blocklengthn and a maximum global FER , both users should have the same SINR γi. How-ever, adjusting the power split factor for both users such that both users have thesame SINR does not necessarily maximize the minimum rate. However, it providesa sub-optimal solution that straightforward to implement in practice. By equatingthe SINRs of both users provided in (4.3), we getξ|h1|2Pr−η1I1 + σ2w=(1− ξ)P|h2|2r−η2ξPr−η2 |h2|2+I2 + σ2w. (4.33)By solving for ξ, we obtainξ¯opt =12(−(γ¯1 + γ¯2) +√(γ¯1 + γ¯2)2 + 4γ¯1). (4.34)By using the optimized power split factor ξ¯opt, both users will achieve the sameperformance. However, this will restrain the performance of user1 such that it per-forms as user2 in each frame which degrades the average max-min rate in return ofstrict fairness between users at each network realization. Although this approach ismore restrictive than Algorithm 1, but it is easy to implement just by evaluating theξ¯opt based on the CSI and using it for transmission at each state. The instantaneousrates in (4.8) and (4.9) are evaluated using the SINRs γ1 and γ2 at ξ¯opt, and thenaveraging takes place to evaluate the average max-min rateThis concludes the analysis of NOMA. To benchmark the performance of NOMAagainst OMA, next, we provide the theoretical analysis for OMA.4.3 OMA Inner BoundWe compare NOMA with OMA where only one user is served by a BS in a singletime-frequency resource block. For a fair comparison with the NOMA system, inOMA, we consider two users being served by the same BS0. However, we use time-division multiple access (TDMA) to share the resource block between the two users,maintaining orthogonality. Using TDMA, we split the blocklength (time) betweenthe two users, maintaining the power constant through the transmission. Hence, theserving BS sends information to the two OMA users using codewords from codes withthe rates R1 and R2 and length n1 and n2 symbols for user1 and user2, respectively.The TDMA message is transmitted with a total blocklength n = n1 + n2.954.3. OMA Inner BoundIn this section, we provide an achievable average rate region for OMA large-scalenetwork that is derived from the results provided in [124] and Ch. 2. The innerbound is presented in the following proposition.Proposition 3. For a large-scale OMA network, an achievable average rate regionR¯OMAn, for a blocklength n, a maximum global FER , attained by time-division multipleaccess for a two-user OMA is given byR¯OMAn, =⋃{1,2}∈[0,1]2(1−1)(1−2)≥1−n1+n2=nR¯OMAn, (1, 2), (4.35)and the achievable average rate region for a given (ξ, 1, 2) is given byR¯OMAn, (1, 2) ={(R¯1, R¯2) ∈ R2+ : R¯1 ≤ R¯OMA1,n1 (1),R¯2 ≤ R¯OMA2,n2 (2)}(4.36)whereR¯OMAi,ni (i) =ninC¯i −√niV¯iQ−1(i)n+12nlog2(ni) (4.37)the average capacity C¯i and average square root channel dispersion V¯i of useri aregiven byC¯i =∫ ∞0∫ ∞0e−k(c)σ2wrηiP LIi{k(c)rηiP}fri(ri)dc dri, (4.38)V¯i =∫ ∞0∫ log2 e0e−z(v)σ2wrηiP LIi{z(v)rηiP}fri(ri)dv dri, (4.39)k(c) = 2c − 1, and z(v) =√11− v2log22(e)− 1Proof. For a single user, it is proved in Ch. 2 that a user given a blocklength n anda maximum error rate  can achieve an average maximum rateR¯OMA1,n () = C¯1 −V1√nQ−1() +12nlog2(n) (4.40)964.4. Multi-Level Polar Coded Modulation SchemeZ1Z2MLPCM EncoderMLPCM Encoders01s02√ξP√(1− ξ)P+y1y2MLPCM DecoderMLPCM DecoderZ˜2Z˜1ZDemultiplexerz1z2zlog2MEncoderEncoderEncoderEncoderM-aryModulatorq1q2qlog2Ms0 yDecoderz˜1Delay Decoderz˜2Delay Decoderz˜log2M−1z˜log2M−2z˜1Delay Decoderz˜log2Mz˜1MultiplexerZ˜Figure 4.1: Multilevel Coded Modulation Block DiagramBy considering two users, the blocklength is split between the two users and eachuser will transmit its codeword at blocklength ni < n and maximum FERs i < while the maximum global error rate constraint is satisfied. Hence, the maximumaverage achievable rate of each user is given asR¯OMAi,ni (i) =ninC¯i −√niV1nQ−1(i) +12nlog2(ni), (4.41)and the quantities C¯i and V¯i are derived in Ch. 2.The achievable region provided in the above proposition will be used to comparebetween the OMA and NOMA performance for a large-scale network in the FBR.Next, we will study a 5G compliant practical scheme that can be used to validatethe results and is readily suitable for future technologies.4.4 Multi-Level Polar Coded Modulation SchemeIn this section, we use MLPCM to validate the theoretical results in Sec. 4.2.Specifically, we implement the MLPCM scheme shown in Fig. 4.1. The BS encodesboth messages for both users using MLPCM at different rates RMLPCM1 and RMLPCM2 forusers 1 and 2, respectively. Then, the transmit power is divided between the two usersthrough the power split factor ξ and the two MLPCM codewords are superimposed,974.5. Numerical Resultsi.e., where the two codeword symbols s10 and s20 are added and transmitted. Eachuser then receives the received signal given in (4.2). User1 decodes the two messages,while user2 only decodes its dedicated message and treats the other message as noise.Details of the encoding and decoding are given next.4.4.1 At the TransmitterThe MLPCM encoder is implemented as shown in Fig. 4.1. For a modulationorder M , the information bits (message) Z ∈ Fk log2(M)2 are fed into a demultiplexerthat splits them into log2(M) sub-messages {zi}log2(M)i=1 where zi ∈ Fk2. Each sub-message zi is encoded using a polar encoder with rate Rc = k/n to obtain codewordsqi. Then, the transmitter collects one bit from each of the codewords qi to form alog2(M)-bit symbol which is then mapped to a symbol s0 from an M -ary constellationsuch as an M -QAM constellation. The symbols are then scaled according to thepower constraint in Sec. 4.1.2. The result of repeating this process for all n codewordsymbols of all log2(M) sub-messages is one MLPCM codeword of length n with arate RMLPCM = Rc log2(M), which is then transmitted through the channel.4.4.2 At the ReceiverMultistage decoding takes place to efficiently recover the message. It breaksthe decoding process into log2(M) stages, as shown in Fig. 4.1 such that in eachstage i = 1, . . . , log2(M), the receiver feeds the decoder of the ith stage with thereceived signal y and the submessages recovered from previous stages (z˜1, . . . , z˜i−1),(which is defined as ∅ for i = 1), and the decoder decodes the symbols into bitsto obtain q˜i, using the multilevel decoding criteria [105]. Then, the decoded bitsq˜i are decoded using a polar decoder to obtain z˜i using the successive cancellationalgorithm provided in [51]. Finally, all sub-messages are fed to a multiplexer to formthe message Z˜. This MLPCM decoding is implemented at each user. User1 willdecode all the codewords and user2 will only decode his own codeword.This process is done for a given code length n, and induces FERs 1 and 2defined as the per-user FERs where the coding rate of each user is tuned such thatthey should satisfy a maximum global FER  defined in Def. 2.4.5 Numerical ResultsThis section presents illustrative numerical results for the theoretical rate analysisprovided in Sec. 4.2, which are also compared to the results for large-scale OMA984.5. Numerical ResultsFigure 4.2: Inner and Outer Bounds for n = {128, 256, 512, 2048}network of Sec. 4.3. The results depict the effect of the BS density λ, blocklengthn, and the global FER  on performance. Then, it compares the performance of theMLPCM with the theoretical results. We consider a large-scale downlink NOMAnetwork with a BS transmit power P = 1 W, noise power σ2w = −110 dBm, BSdensity λ = 1 BS/km2, blocklength n = 128, and global FER  = 10−5, unlessotherwise stated.We start by addressing the main question of the thesis: Does NOMA still main-tain superiority over OMA in such scenarios with the presence of intercell inter-ference? The results in Fig. 4.2, 4.3, and 4.4 address this question and quantifythe gap between OMA and NOMA. In Fig. 4.2, the inner bound and outer boundof the rate region for the large-scale NOMA network are plotted for blocklengthn = {128, 256, 512, 2048} at  = 10−3 and the theoretical results are validated usingMonte-Carlo simulations. It is observed that as the blocklength decreases the averagerate region shrinks, but nonetheless, the superiority of NOMA over OMA persists.Moreover, it can be seen that the performance gain between NOMA and OMA in-creases as the blocklength n decreases. For example, to achieve an average rate of0.8 bits/transmission at user2, user1 can achieve an average rate of 0.855 and 1.39using NOMA and 0.366 and 0.825 using OMA for blocklengths n = 128 and 2048,respectively. Thus, the gain increases from 71.6% (relative to OMA) for n = 2048994.5. Numerical ResultsFigure 4.3: Inner and Outer Bounds for  = {10−2, 10−3, 10−5}Figure 4.4: Inner and Outer Bounds for λ = {0.1, 0.15, 1, 10} BS/km2. There is anegligible increase in the achievable average rate region at the BS density increasesbeyond λ ≥ 1 BS/km2.1004.5. Numerical Resultsto 133.5% for n = 128. This phenomenon can be rationalized by understanding thatas the blocklength diminishes, splitting the blocklength in the case of OMA withTDMA reduces the reliability of the codeword, linducing more errors. Therefore,NOMA users tend to exhibit superior performance with shorter blocklengths. Thishighlights the importance of NOMA in large-scale networks in the FBR.In Fig. 4.3, we investigate the effect of an ultra-high reliability constraint  on theaverage performance of the network using NOMA and OMA. The inner bound andouter bound of the rate region for the large-scale NOMA network, and an achievableaverage rate region for the large-scale OMA network are plotted for maximum globalFERs  = {10−2, 10−3, 10−5}. This figure shows the effect of decreasing  on thenetwork in the form of a shrinkage of the achievable rate region. This quantificationis important in the design and analysis of URLLC which is one of the use-cases of5G and beyond networks. Achieving low FER using short blocklength (n = 128)is challenging and requires the use of low-rate codes allowing better error detectionand correction. However, as observed in the figure, the decrease in rate can besignificant as the maximum global FER  decreases. For instance, R1 decreases by0.6 bits/transmission as  decrease from 10−2 to 10−5 for R2 = 0.6, and R2 decrease by0.2 bits/transmission as  decreases from 10−2 to 10−5 for R1 = 1 bits/transmission.At the same time, the performance gain of NOMA compared to OMA increases as decreases. For instance, the gain in R1 increases from 48.7% at  = 10−2 to 84.6%(relative to OMA) at  = 10−5 when R2 = 0.6 bits/transmission. This is directlyassociated with the division of blocklength, leading to shorter codewords, in the caseof OMA with TDMA. This division inherently limits the error-correcting abilities ofthe code, resulting in the need for lower rates to maintain the same FERs.In Fig. 4.4, the achievable average rate bound for the large-scale NOMA andOMA networks is plotted for different BS density λ = {10, 1, 0.15, 0.1} BS/km2.Here, we study the effect of densifying the network to achieve lower latency and betterperformance by reducing the propagation delay and the propagation loss which areboth a function of the distance between the serving BS and the user. The figureshows that increasing BS density initially expands the rate region, indicating betterperformance. However, beyond a certain point, adding more BSs does not boost thenetwork’s data rate further. To explain this point, we have to recall the factors thataffect the network’s performance and how they relate to BS density. The transmittedsignal by the serving BS is affected by three main factors: Fading, propagation loss,and interference. Both propagation loss and interference depend on the distancebetween the serving BS and the user. Increasing BS density reduces this distance,decreasing propagation loss but increasing interference from neighboring cells. Atlow BS density, propagation loss dominates, so increasing density initially helps in1014.5. Numerical ResultsFigure 4.5: Max-Min Rate for λ = {0.1, 0.15, 1} BS/km2enhancing the performance of the network by increasing the desired signal power.However, as density increases further, interference increases, until a point where itnegates the gain in the desired signal power, at which point increasing BS densitydoes not improve performance anymore.Now, we will study the performance of the large-scale network using NOMA andOMA in the FBR from another perspective which is fairness to ensure that every useror device in the network receives a reasonable share of the resources which improvesthe user satisfaction. We investigate the max-min average rate for NOMA and OMAfor different network parameters. In Fig. 4.5, the max-min rate is plotted versus themaximum global FER  for various network densities λ = {0.1, 0.15, 1} BS/km2.The figure shows the max-min rate computed using Algorithm 1 based on (4.31) andthe max-min rate obtained using the proposed power split in (4.34). It is clear thatAlgorithm 1 outperforms the proposed power allocation scheme ξ¯opt in Sec. 4.2.3. Itis also observed that the max-min rate monotonically increases with the maximumglobal FER  but the relative gain of increasing the BS density is relatively small(< 10%). This is because a higher tolerance of FER can provide a more liberalresource allocation scheme which leads to better performance in terms of achievablerate. Furthermore, we conduct an analysis of the max-min rate as a function of λ, as1024.5. Numerical ResultsFigure 4.6: Max-Min Rate versus BS density λ for n = 128 and  = 10−5depicted in Fig. 4.6. The figure also shows that the denser the network, the betterthe performance. However, the increase is rapid until it saturates at λ = 0.3 BS/km2.It is also observed that at low density networks the power split in (4.34) performsclose to the optimal performance.In Fig. 4.7 and Fig. 4.8, the max-min average rate of the large-scale NOMAnetwork is compared to the max-min average rate of the large-scale OMA network.In Fig. 4.7, the max-min average rate is plotted versus the maximum global FER for blocklengths n = {128, 1024} and BS densities λ = {0.1, 0.15, 1} BS/km2. It canbe observed that the NOMA scheme outperforms the OMA scheme with a gain upto 24% achieved at  = 10−5 and n = 128 and this gain increases with the decreaseof the maximum global FER  and blocklength n. Moreover, for a short blocklengthn = 128, as the maximum global FER  increases (increasing the network tolerance),a higher max-min average rate is achieved than the max-min average rate at largeblocklength. However, this behavior is observed for the OMA scheme due to dividingthe blocklength between users.In Fig. 4.8, the relative gain between the max-min average rate of NOMA andOMA is plotted versus the blocklength n. It is observed that as the blocklengthincreases the relative gain decreases, which concludes that NOMA outperforms OMA1034.5. Numerical ResultsFigure 4.7: Max-Min Rate versus BS density λ for OMA and NOMA schemesscheme with a larger gain in the FBR for large-scale networks. Also, it is shown thatthe gain increases as the network becomes denser in the FBR which promotes theNOMA scheme to be used for future technologies.In Fig. 4.9, the performance of the MLPCM scheme is compared with the the-oretical results. We simulated the MLPCM scheme with a QAM constellation formodulation order M = 4, a blocklength n = 128, and a maximum global FER . Thesimulations are performed by fixing the rate of user2 and increasing the coding rateof user1 until reaching the FER . The power split factor is also tuned to maximizethe coding rate of user1. The achievable rates are plotted along with the inner andouter bound on the achievable average rate region for the NOMA scheme proposedin (4.10) and (4.22), and the achievable average rate region for OMA provided in(4.35). The figure shows that the achievable rates of the MLPCM scheme with 4-QAM approach the theoretical rate region of NOMA. Thus, Fig. 4.9 validates thatthe theoretical performance can be approached using MLPCM, which validates thepotential of MLPCM for future technologies.1044.5. Numerical ResultsFigure 4.8: Relative Gain between OMA and NOMA max-min average rate versusBlocklength n for λ ∈ {0.1, 1} BS/km2 and  ∈ {10−5, 10−2}Figure 4.9: MLPCM Performance1054.6. Conclusion4.6 ConclusionThis chapter investigates different aspects of the performance of a large-scale DLNOMA network in the FBR using the theory of coding in the FBR in conjunctionwith stochastic geometry tools. Outer and inner bounds on the average achievablerate region are derived, providing insights into the achievable rates attained by thelarge-scale NOMA network in the FBR. Moreover, the effect of high-reliability con-straints on the performance of NOMA users is studied and the study concluded thatthe performance gain relative to OMA users escalates as the reliability constraintsbecomes more stringent and the blocklength decreases. This underscores the influ-ence of blocklength on the network’s performance and the challenges associated withshort blocklength and highlights the superiority of NOMA over OMA. Also, this workevaluates the impact of density, blocklength, and reliability on the performance of thelarge-scale network in the FBR, and the theoretical results are evaluated numericallyto underscore and draw insights into the superiority of NOMA over OMA in the FBRfor large-scale networks. Further, it was shown that the theoretical performance canbe approached using 5G compliant practical transmission schemes such as MLPCM.This work can be extended to investigate the performance of promising technologiessuch as cell-free massive MIMO, and comparing it to the performance of NOMA,aiming to derive insights essential for determining the most suitable technology for6G deployment.106Chapter 5Coding in the Finite BlocklengthRegime Using aCNN-AutoencoderIn the previous chapters, we provided a theoretical framework that characterizesthe performance of large-scale networks in the FBR. However, a practical scheme isessential to be investigated to provide a performance close to the theoretical perfor-mance. We have already proposed the MLPCM as a practical scheme, however, weare aiming at a more intelligent approach. It has been found that NNs have beena convenient tool for code design in communication systems, yielding better perfor-mance than conventional schemes in terms of error rates. However, as a preliminarystep, we will consider an AWGN channel rather than a large-scale network setup.Since CNNs are known to be easier to train than many other NNs, this chap-ter investigates the potential of CNN-AE in approaching the theoretical maximumachievable rate over a Gaussian channel for a range of signal-to-noise ratios at a fixedblocklength and target FER, which is a different perspective compared to existingworks that explore the use of CNNs from bit-error and symbol-error rate perspec-tives. Hence, we propose a CNN-AE architecture with few hyper-parameters andexplain how to choose these parameters. Then, we jointly optimize the transmitterand receiver by training the AE, thus overcoming the mathematical complexity offinding an optimal code. Then, we compare the results with the performance of ascheme that combines polar codes and QAM, a scheme that combines Reed-Mullercodes with QAM, a scheme that uses MLPCM [52], and the turbo AE-based schemeproposed in [58]. Results show that the proposed CNN-AE approaches the maxi-mum coding rate in [6] and outperforms the benchmark schemes, which highlightsthe capability of CNN-AEs in finding a good code that approaches the theoreticalmaximum rate in the FBR. Finally, we comment on the complexity and performanceof similar designs using RNN or an FFNN instead of the CNN.The chapter is organized as follows. We present the system model and reviewthe theoretical maximum achievable rates in the FBR. Then, in Sec. 5.2, we presentour proposed CNN-AE architecture. Numerical results are presented and compared1075.1. System ModelMessage sK bits EncoderRcod =KNN bits M -aryModulator x ∈ CnChannely ∈ CnDemodulatorN bitsDecoderEstimatedMessagesˆFigure 5.1: A conventional communication system consisting of an encoder, modu-lator, demodulator, and decoder.with results in [6, 49, 52, 58] in Sec. 5.3. Finally, the chapter is concluded and futurework directions are presented in Sec. 5.4.5.1 System ModelFor a general communication system, the transmitter feeds a message s withK information bits to an encoder that encodes the bits, with rate R, to x =(χ1, χ2, . . . , χn) ∈ Cn which must satisfy a power constraint1nn∑i=1|χi|2≤ P. (5.1)Then, x is sent through a Gaussian channel leading to the received signaly = x + wwhere w is an n-dimensional vector of independent and identically distributed cir-cularly symmetric complex Gaussian noise with zero mean and variance σ2w.At the receiver side, the decoder is designed to find an estimate sˆ of the transmit-ted message, using the conditional probability of the channel PY|X(y|x), such thatthe FER P(sˆ 6= s) does not exceed a predetermined value . The trade-off betweenthe blocklength n and the FER  forms a limitation on the performance of the systemin terms of the achievable information rate. Note that latency and reliability dependon n and , respectively.For a conventional communication system (Fig. 5.1), the encoding process isdivided into channel coding with coding rate Rcod =KNwhere the encoder maps the1085.1. System ModelK information bits to a codeword of length N bits, followed by an M -ary modulatorof order M = 2kmod which maps each kmod coded bits into a complex-valued symbol toobtain the vector x of length n, where n = N/kmod. The overall rate of informationtransmission is given by R = Rcodkmod bits per complex-valued transmission. Thismodel will serve as one of the benchmarks we use for comparison.The achievable rate of the conventional system described above usually deviatesfrom the channel capacity (the maximum achievable rate) due to the separation ofthe design of channel coding and modulation. This capacity was first characterizedby Shannon in [125], as the maximum rate of information transmission such that→ 0 as n→∞. For a Gaussian channel with SNRα =Pσ2w,the capacity is given byC = log2(1 + α) bits/transmission, (5.2)which can be achieved using a code that maps the K information bits s into thecomplex-valued codeword x without separating this mapping into binary coding andmodulation. However, for a delay-constrained application, short codes (small n) areused to ensure low latency. In this case, Shannon’s capacity which is defined for infi-nite blocklength n and vanishing error probability  becomes inaccurate. Polyanskiyet al. [6] derived a closed-form approximation of the maximum achievable rate forthe Gaussian channel in the FBR, for a blocklength n as short as 100 symbols andtarget FER 0 <  < 0.5. This maximum rate is characterized for a Gaussian channelwith SNR α as followsRn, = C −√VnQ−1() +O(log n)/n, (5.3)where C is defined in (5.2), Q−1(·) is inverse Q-function, O(log(n))/n is bounded in[6] by O(1) < O (log2(n)/n) <12nlog2(n) + O(1) and we neglect the term O(1) inour analysis and use the upper bound, andV =α(α + 2)(α + 1)2log22 eis the channel dispersion defined as the variance of the mutual information densitybetween xi and yi.The maximum achievable rate is defined as the capacity with a back-off termwhich is a function of the blocklength n, FER , and SNR. This theoretical result1095.1. System ModelInputK bitsEncodingConv1DConv1DConv1DL×M1L×M1ReshapeLayerL×N ′ModulationConv1DConv1DDemodulationn× kmodn×M2n complex-valued symbolsNormalizationLayerncomplex-valuedsymbolsChannelncomplex-valuedsymbolsConv1Dn×M2Conv1Dn× kmodReshapeLayerL×N ′DecodingL×M1L×M1Conv1DConv1DConv1DK bitsOutputFigure 5.2: Proposed CNN-AE. The ‘encoding’, ‘modulation’, ‘demodulation’, and‘decoding’ layers are trained jointly.will be used as another benchmark for assessing the performance of our proposedscheme.The separation of channel coding and modulation in the conventional communi-cation system shown in Fig. 5.1 leads to a sub-optimal system. This simplification isoften done because finding a coding scheme for the general system which approaches(5.3) is generally cumbersome. The objective of this chapter is to propose an NN tojointly optimize the transmitter and receiver functions to outperform conventionalmethods and approaches (5.3), as described next.1105.2. Autoencoder for the Finite Blocklength Regime5.2 Autoencoder for the Finite BlocklengthRegime5.2.1 Neural Network structureIn this section, a CNN-AE that is capable of approaching the maximum rate inthe FBR is proposed. The structure of our CNN-AE is shown in Fig. 5.2. Differentfrom [56], encoding and decoding blocks are added around the modulator and thedemodulator. In our approach, we model the 4 blocks (channel encoder, modulator,demodulator, and channel decoder) as NNs and train them jointly in an end-to-endmanner, while using K bits as an input at the transmitter (output at the receiver)and n complex-valued symbols as an output at the transmitter (input at the receiver).The AE’s objective is to find an n-dimensional code with rate R bits/transmissionthat allows the reconstruction of the message at the receiver side with an FER lessthan . Thus, the AE aims to learn the codebook in an end-to-end manner basedon the channel rather than the conventional method of using a careful mathematicalconstruction of encoding/decoding and modulation/demodulation.The CNN-AE architecture is designed to mimic the blocks of a conventionalcommunication system. Hence, each block is represented by a set of layers, andits dimensions are chosen depending on the function it performs. Let us supposethat we want to transmit a message with K information bits through an encodermapping these bits into N coded bits at rate Rcod = K/N , followed by a modulatorthat maps the N bits into n complex-valued symbols using a modulation order 2kmodwhere kmod = N/n. Since the code rate is Rcod = K/N , to simplify this encoding intoa code with a lower number of input/output bits, we write K = K ′L and N = N ′Lwhere L is the greatest common divisor of N and K. This allows us to interpretthe encoding of K bits into N bits at rate Rcod as the encoding of L sub-messagesof K ′ bits each into L sub-codewords of N ′ bits each at rate Rcod. Thus, the inputof the AE will be K = K ′L bits. Then to construct the codewords, we use three1-D convolutional (Conv1D) layers.14. The first two Conv1D layers will map eachsub-message of K ′ information bits (L sub-messages) into a higher dimensional spaceof M1 dimensions, leading to an output size of L ×M1. This will allow the AE tolearn a good placement of each of the L sub-codewords in an M1 dimensional space.Then, the third Conv1D layer is used to map the sub-codewords down to an N ′dimensional space (output size L × N ′). Finally, to prepare the LN ′ = N symbolsfor modulation into n complex-valued symbols, they are reshaped into an n × kmod14A larger number of layers can be used if needed, but it was noticed that 3 layers suffice for thepurpose of this chapter.1115.2. Autoencoder for the Finite Blocklength Regimematrix.To modulate the n symbols with kmod symbols each, we use two Conv1D layers.The first Conv1D layer maps the kmod symbols into a higher dimensional spaceof M2 dimensions using a Conv1D layer (output size n × M2). This allows theAE to learn a good placement of the modulation symbols in an M2-dimensionalspace. Then, the other Conv1D layer maps each of the n modulated M2-dimensionalsymbols into a complex-valued symbol which we represent using a 2-dimensional real-valued representation via another Conv1D layer (output size n×2). A (nontrainable)normalization layer is finally added to satisfy the average power constraint in (5.1) bynormalizing by the average power of the n symbols. Then, after the (nontrainable)channel layer, the receiver side including the demodulator and decoder are designedin the same manner to reconstruct the transmitted message. The parameters of AEare summarized in Table 5.1.We use Conv1D layers due to their lower complexity and better trainability thanFFNN and RNN [126]. Also, each layer is followed by an Exponential Linear Unit(ELU) activation function to allow for non-linear coding. However, we use a linearactivation function at the end of the mapper to allow for any positive or negativevalues at the output of the modulation. In the last layer at the receiver side, weuse a sigmoid activation function followed by thresholding at 0.5 to ensure that theoutput is a binary vector. Finally, we note that each trainable layer is followed by aBatch Normalization layer to help the model converge quickly.5.2.2 MethodologySince this work aims to demonstrate the existence of an AE that can achievea rate close to the maximum achievable rate in the FBL, we fix a blocklength nand a target FER . Then, we train the AE in an end-to-end manner at a certainSNR α and designed coding rate R, by feeding a dataset of information bits (Kbits in each data sample) and training via backpropagation. Then, we test the AEusing a different set of data and evaluate its FER. If the FER is larger than , wegradually decrease the designed coding rate R via choosing Rcod and kmod until theFER becomes smaller than . The CNN parameters are chosen accordingly, based onthe parameters presented in the previous section, where Rcod and kmod are selectedso that Rcodkmod = R, while the parameters M1 and M2 (whose impact is analyzedin Sec IV) are chosen as 200 and 100, respectively, since they turned out to workwell for the range of rates, code-length, and FER considered in this study.It is worth noting that works in the literature which study the BER achievedby an AE [53–56, 58] train the AE at a certain SNR with a fixed rate R and test1125.2. Autoencoder for the Finite Blocklength RegimeType of layer Activation function Output sizeEncoderInput None (LK ′ × 1)Conv 1D elu (L×M1)Conv 1D elu (L×M1)Conv 1D elu (L×N ′)Reshape None (n× kmod)ModulatorConv1D elu (n×M2)Conv1D linear (n× 2)Normalization Layer None (n× 2)ChannelGaussian channel None (n× 2)DemodulatorConv1D elu (n×M2)Conv1D linear (n× kmod)Reshape None (L×N ′)DecoderConv1D elu (L×M1)Conv1D elu (L×M1)Conv1D sigmoid (LK ′ × 1)Table 5.1: Parameters of the proposed CNN-AE.it at a wide range of SNRs where it can be observed that the BER decreases asSNR increases. This can also be done in our work. However, the goal of our workis to show that at any SNR, the proposed CNN-AE architecture can approach themaximum rate in the FBR. Since this maximum rate increases with SNR, so mustthe rate of the CNN-AE. Since increasing the rate of the CNN-AE requires changingthe dimensions of its input and output, this implies that a new training is neededwhen the SNR is increased. Note that this is not unique to our CNN-AE, sinceincreasing the rate of a conventional scheme requires changing the coding and/ormodulation parameters which is analogous to changing the architecture in our case.1135.3. Numerical ResultsM1,M2 (50, 20) (80, 50) (200, 100)Rate 3 bits/transm. × × XRate 1 bits/transm. X X XTable 5.2: Performance of the AE at different values of M1 and M2 and SNR = 10 dB.Here, ‘×’ means that the target FER was not achieved, whereas ‘X’ means it wasachieved.5.3 Numerical ResultsIn this section, simulations are performed to find the maximum rates that can beachieved at given FER  and blocklength n. Numerical results are presented basedon the methodology provided in Sec. 5.2.2 under the settings provided next. The AEperformance is compared to the performance of polar codes combined with QAM,Reed-Muller codes combined with QAM, MLPCM, and the TurboAE-MOD schemein [58] in addition to the theoretical maximum provided in (5.3).For simulations, we train and test the CNN-AE under a Gaussian channel atblocklength n = 128 and FER  = 10−2, using Google Colab notebook.15 Wegenerate 106 random binary vectors for each of the training and the testing sets. Itis worth noting that these results are tested using a data set of 106 messages. The setused is considered small for proper validation, and more extensive testing is requiredto obtain reliable results. The training is conducted using Adam optimizer [128] witha learning rate of 0.001 and binary cross entropy loss, while the parameters M1 andM2 are chosen as 200 and 100, respectively. The batch size is set to 500 while thenumber of epochs is 2000. We evaluate the accuracy and the FER as performancemetrics. Finally, the training SNR, a communication-related parameter, was fixedat a value that matches (equal or slightly lower than) the testing SNR. Simulationresults are discussed next.The achievable rates that satisfy an FER < 10−2 for a blocklength of n = 128are presented in Fig. 5.3, in addition to the maximum rate in the FBR, and theachievable rate polar and Reed-Muller coded QAM (with hard detection), MLPCM,and the AE-based scheme in [58]. While these schemes are conventional and em-ploy separate encoders and modulators, the MLPCM scheme uses joint coding andmodulation where the encoding and modulation are jointly designed so that eachsymbol carries information from different polar codes (layers) and the demodulation15Google Colab [127] is a service provided by Google which offers a cloud-based Jupyter notebook,as well as a virtual machine to run it. This virtual machine can be configured with a machinelearning GPU or TPU.1145.3. Numerical Resultsand decoding are done in a hierarchical manner (multi-stage decoding) [52]. Thebenchmark AE-based scheme is the TurboAE-MOD scheme provided in [58] whichjointly trains a Turbo-AE and a QAM modulator/demodulator, and whose perfor-mance is comparable to a classical turbo code combined with a QAM modulator asshown in [58].It can be seen from the figure that the CNN-AE outperforms the benchmarkschemes, and achieves rates close to the theoretical maximum provided in (5.3). Thisdemonstrates the capability of the proposed CNN-AE in learning a good code forthe channel in the FBR. It is worth noting that the TurboAE-MOD scheme, in [58]has the limitation of a fixed encoding rate of 1/3, which makes its architecture lessflexible. Thus, this TurboAE-MOD scheme provides rates that are multiples of 1/3depending on the selected modulation order.16 This rate control is coarse comparedto the proposed CNN-AE which has more flexibility in choosing the coding rate andmodulation order so that it approaches the theoretical bound.The dimensions of the encoder and modulator filter space (M1 and M2) werefurther investigated in Table. 5.2. The table records the ability of the network toreach the rates of 3 and 1 bits per transmission at SNR = 10 dB for different M1 andM2. At higher rates closer to the maximum achievable rate (R = 3), it was foundthat small values of M1 and M2 constrain the performance of the AE which makesit unable to reach the target FER. However, at lower rates (R = 1), it was foundthat smaller M1 and M2 can achieve the desired performance. This is because largerM1 and M2 give the AE more flexibility in placing the codewords and modulationsymbols in a higher dimensional space before mapping back to n symbols. However,it must be noted that larger M1 and M2 increase the training complexity.Finally, during our experiments, we investigated different types of AEs (FFNN,RNN, CNN) to examine their complexity. We found that to achieve a rate of R = 2for instance at n = 128, Rcod =12, kmod = 4, and  = 10−2 for SNR = 12 dB, theRNN needs 252, 427 trainable parameters while the CNN needs 103, 861 trainableparameters at M1 = 100 and M2 = 20. Moreover, the FFNN needed 723, 200trainable parameters and was still unable to achieve this rate making it inefficientand non-scalable to larger blocklengths. Comparing the RNN and the CNN, we foundthat they achieved similar performance in terms of achievable rates, but the CNN-AE was faster in terms of training time thanks to its parameter sharing technique.Hence, the proposed CNN-AE provides a better performance in terms of coding rateand complexity.16Note that the rate of the scheme in [129] can be improved using an AE architecture whichachieves a higher coding rate, which is worth additional investigation. However, our proposedCNN-AE demonstrates that there is no need to restrict attention to a TurboAE-MOD architecture.1155.4. ConclusionFigure 5.3: The designed achievable rates of the CNN-AE compared with schemesfrom the literature and theoretical achievable rates5.4 ConclusionWe proposed a CNN-AE as an attempt to provide a practical scheme that canprovide achievable rates over the Gaussian channel which approaches the theoreticalmaximum rate in the FBR. We tested the designed coding rate at each SNR andfound it achieved a FER less than . Also, we also showed that the proposed CNN-AE outperforms benchmark conventional schemes and another NN-based AE, theTurboAE-MOD scheme in [58]. The main conclusion of this work is that a CNN-AEcan be used to learn a code for the Gaussian channel in the FBR which approachesthe theoretical maximum rate. This work can be extended to cover more complexchannels that exist in real-world scenarios such as interference channels and Rayleighfading channels. Furthermore, with the increasing demand and the apparition of5G networks with dense deployment and massive connectivity, a promising researchdirection is to consider the performance of CNN-AE in large-scale wireless networksin the FBR.116Chapter 6ConclusionIn this chapter, we provide the concluding remarks on the accomplished works inthis thesis. In the preceding chapters, we embarked on a comprehensive explorationof critical research gaps within the domain of large-scale networks in the FBR. Ourinvestigation primarily focused on the significant tendency in prior works to concen-trate on analyzing the performance of the large-scale network under an asymptoticregime assumption, potentially leading to imprecise results that can not be used un-der FBR assumptions. To address this challenge, we advocated for a paradigm shifttowards a theoretical framework that analyzes the performance of the network underlatency and reliability constraints. Hence, we developed a framework to characterizethe performance of DL and UL of large-scale networks for short codes using the the-ory of coding in the FBR in conjunction with stochastic geometric tools. We haveconducted our analysis on OMA and NOMA networks in the FBR.Furthermore, we identified another significant challenge which is identifying apractical coding scheme that can achieve a performance closely aligned to the the-oretical results. To address this, we examined the performance of polar codes andMLPCM for large-scale networks using OMA and NOMA schemes. These codingschemes have shown promise in several communication systems, and analyzing theirperformance in large-scale networks provides valuable insights into their scalabilityand practicality. Additionally, we proposed an intelligent architecture based on aCNN-AE that is capable of approaching the theoretical maximum achievable rateover a Gaussian channel across a range of SNRs. The proposed CNN-AE archi-tecture, due to its inherent ability to learn complex patterns, can potentially findoptimal or near-optimal codes, thereby achieving a rate that is close to the theoret-ical maximum. Further, we conducted an extensive comparative analysis of variouscoding schemes and their performance. This included polar codes combined withQAM, Reed-Muller codes combined with QAM, MLPCM, and the Turbo AE-basedscheme. This comparison aimed to provide a comprehensive view of the performanceof these schemes under different conditions, showcasing their relative strengths andweaknesses. This analysis is crucial in identifying which coding scheme is best suitedto specific network conditions and requirements.The concluding remarks on the theoretical analyses in each chapter are provided1176.1. Concluding Remarksin the next section.6.1 Concluding RemarksSome concluding remarks on the accomplished works of this thesis are presentedas follows.− In Chapter 2, we presents a thorough investigation of the performance of alarge-scale DL network in the FBR using the theory of coding in the FBRand stochastic geometric tools. We developed a novel mathematical analysisto characterize the trade-off between the code length, the FER, and the in-formation coding rate in large-scale DL networks using OMA. The achievablerates and outage in the FBR are characterized for Gaussian codebooks andpractical modulation schemes. A practical scheme of MLPCM is validatedand shows that its performance approaches the theoretical benchmarks. Weprovide a comprehensive analysis of the average coding rate, the rate outageprobability, and the coding rate meta distribution, showing that these devel-oped expressions are fairly tight, as confirmed by our numerical results. Wealso demonstrate that the performance analysis under an AR overestimates theperformance and cannot be used to accurately characterize the performance inthe FBR. On the contrary, the FBR provides accurate rate and FER character-ization, approaching the AR performance at large blocklengths and high FER.Simulation results reveal that MLPCM achieves rates close to the derived the-oretical benchmarks and the derived outage probability bounds coincide withsimulation results, affirming the accuracy of our findings. In conclusion, ourwork is crucial for the theoretical analysis of large-scale networks in the FBRand proves that the theoretical performance can be approached using practicaltransmission schemes such as MLPCM.− In Chapter 3, we proposed a theoretical analysis of the performance of large-scale UL OMA networks, focusing on performance metrics such as the averagecoding rate, rate outage probability, and coding rate meta distribution underfractional path-loss inversion power control. Two special cases, TCI and CIMP,are also specifically examined. We developed a framework to characterize theperformance of large-scale UL OMA networks for short codes using the theoryof coding in the FBR with stochastic geometric tools. Theoretical expressionsfor rate outage probability and coding rate meta-distribution were providedunder the fractional path-loss inversion power control scheme, and bounds and1186.1. Concluding Remarksapproximations were proposed. Our numerical results highlighted the tight-ness of these propositions. For the special cases: TCI and CIMP power controlschemes, we proposed closed-form expressions characterizing key performanceindicators. It was shown that while TCI power control provides better perfor-mance for dense networks, CIMP ensures reliability and availability for bothlow and high-density networks. Our findings also revealed that the perfor-mance in the AR underestimates the rate outage probability and overestimatesthe coding rate meta distribution and the average coding rate, compared tothe realistic performance of the network in the FBR. However, with increasingblocklength, the performance approaches that of the AR. In conclusion, ourstudy accurately characterizes the performance of large-scale networks in theFBR and provides significant insights into power control schemes in the UL.− In Chapter 4, we studied the performance of NOMA networks in the FBR andinvestigated whether NOMA’s outperformance persists in large-scale networkssubject to strict reliability and latency constraints and what quantifiable gapexists between these two multiple access techniques. To this end, we employedthe theory of coding in the FBR in conjunction with stochastic geometry toolsto characterize the average rate region for a large-scale NOMA network un-der a global reliability constraint and a short blocklength. We derived boththe theoretical achievable rate region and an outer bound on the maximumachievable rate region for a large-scale NOMA network in the FBR, establish-ing benchmarks for system performance. Moreover, we proposed a practicalpower allocation scheme to achieve a balanced performance between the twoNOMA users and utilized MLPCM, which demonstrated close alignment withthe theoretical achievable inner bound. Furthermore, we conducted a com-parative analysis between the performances of large-scale OMA and NOMAnetworks in the FBR, studying the performance gap with respect to networkparameters. The comparison underlined NOMA’s superiority, especially undershorter blocklengths and stringent reliability constraints, propelling the use ofNOMA networks in future network deployments. Additionally, we scrutinizedthe impact of BS density and reliability constraints on large-scale NOMA net-work performance. In conclusion, our study offers valuable insights into theperformance of large-scale networks in the FBR, demonstrating that the theo-retical performance can be approached using 5G compliant practical transmis-sion schemes such as MLPCM.− In Chapter 5, we investigated the potential of CNN-AE as a practical schemein approaching the theoretical maximum achievable rate over a Gaussian chan-1196.2. Current Limitations and Future Worksnel for a range of signal-to-noise ratios at a fixed blocklength and target FER.We proposed a CNN-AE architecture with minimal hyper-parameters and pre-sented methods for parameter selection. The AE was trained to jointly optimizethe transmitter and receiver, thus overcoming the mathematical complexity ofidentifying an optimal code. The performance results were compared withthose of various schemes including polar codes with QAM, Reed-Muller codeswith QAM, MLPCM, and the Turbo AE-based scheme. Our results indicatedthat the proposed CNN-AE approaches the maximum coding rate and out-performs benchmark schemes. This underscores the capability of CNN-AEs inidentifying an effective code that approaches the theoretical maximum rate inthe FBR. Lastly, we discussed the complexity and performance of alternativedesigns using RNN or a FFNN instead of the CNN.6.2 Current Limitations and Future WorksDespite the significant advancements achieved in this thesis in the theoretical per-formance characterization of large-scale networks under the FBR, several challengespersist, warranting further research and innovation.5G networks employ NOMA schemes as they provide superior spectral efficiencyand support a larger number of users simultaneously compared to other multipleaccess schemes. Furthermore, 5G networks are designed to support use cases thatrequire URLLC, which makes the performance analysis of UL NOMA crucial. How-ever, studying the performance of UL NOMA presents several technical challenges,primarily due to the operational differences between UL and DL. One of the maindistinctions is the use of power control in UL. In UL NOMA networks, multipleUEs transmit data to a base station. These devices commonly have limited powerresources, making efficient power control essential. Implementing effective powercontrol strategies in UL NOMA is challenging, as it must balance the need for effi-cient power utilization, mitigate inter-user interference, and maintain high data rates.Moreover, in the FBR, these power control challenges become even more complex.The limited time for transmission in the FBR impact the reliability of communi-cation and hence, managing interference from multiple users becomes a significantdifficulty. From the previous challenges, we would find that the theoretical analysisof the achievable rate in large-scale UL NOMA networks involves complex math-ematical modeling and this complexity increases in the FBR due to the non-zeroprobability of error and the intricate relationship between blocklength, error prob-ability, and rate. Hence, the performance analysis of large-scale UL NOMA underlatency and reliability constraints should be studied especially given the deployment1206.2. Current Limitations and Future Worksof NOMA in 5G networks. This analysis could encompass the characterization ofthe average achievable rate region, rate outage probability, max-min rate for fair-ness, reliability, and the meta distribution of the coding rate. Each of these metricswould provide valuable insights into the feasibility and effectiveness of NOMA within5G networks. Furthermore, conducting a comprehensive examination of the influ-ence of network and transmission parameters on the network’s performance couldyield crucial understanding of the optimization strategies. Moreover, an exhaus-tive comparison between UL OMA, UL NOMA, and UL cell-free massive MIMOcould also be conducted. Such comparative analysis would offer significant insightsinto the most effective technology for future network deployment. This could guidedecision-making processes in the development and implementation of future networktechnologies.Another challenge involves analyzing the performance of potential technologies,such as cell-free massive MIMO, that are being considered for B5G networks. Cell-free massive MIMO is a promising technology for 6G wireless networks that aims toaddress the limitations of traditional cellular architectures. This technology evolvesfrom the Massive MIMO technology of 5G, with the primary distinction being theremoval of distinct cell boundaries, leading to the term “cell-free”. In a cell-freeMassive MIMO system, the network area is blanketed by a large number of accesspoints (APs) or antennas which serve all users in a cooperative manner, irrespectiveof their location. Key features and benefits of cell-free Massive MIMO include en-hanced connectivity, improved capacity, and spectral efficiency, robustness to fadingand interference, enhanced energy efficiency, and scalability. Enhanced connectivityis achieved as all APs cooperatively serve all users, ensuring uniform high-qualityservice, regardless of the user’s location. This is particularly beneficial in high userdensity or high mobility environments. The large number of APs serving users si-multaneously allows for very high spectral efficiency, leading to better frequencyspectrum utilization and higher overall network capacity. The system’s robustnessto channel fading and interference is improved due to the use of multiple APs for eachuser, leading to enhanced reliability and quality of service. By directing transmissionpower towards the intended users and minimizing power wasted in other directions,energy efficiency is enhanced, which is crucial for sustainable network operation.Lastly, the architecture of cell-free Massive MIMO is highly scalable, allowing forstraightforward network expansion by simply adding more APs. However, there aresome challenges for implementing cell-free Massive MIMO including the requirementfor a high-capacity backhaul network to coordinate between APs, complexity in sig-nal processing and resource management, and potential privacy issues due to thecooperative nature of the system. Additional research is required for addressing1216.2. Current Limitations and Future Worksthese concerns, positioning cell-free Massive MIMO as a promising technology for6G networks. Hence, the performance of cell-free massive MIMO network should becharacterized and the effect of network parameters should be studied. Consequently,a theoretical performance analysis of scalable cell-free massive MIMO networks couldprovide valuable insights into the viability of this technology for future mobile gen-erations. This analysis could encompass various user association schemes, such asdistance-based or SINR-based user association schemes. As with the research pro-posed in this thesis, the theory of coding in the FBR, supplemented with tools fromstochastic geometry, should be applied to characterize the performance of cell-freemassive MIMO networks. Key performance indicators such as the average codingrate, rate outage probability, reliability, and meta distribution should be thoroughlyanalyzed. Following the performance characterization, it is crucial to investigate theinfluence of various parameters on network performance. For instance, one couldexamine the effect of varying AP density, user distribution, or signal propagationconditions. Equally important is to conduct a comparison between the performanceof cell-free massive MIMO networks and current network configurations in differentregimes, such as the FBR and the AR. This comprehensive analysis would providea detailed understanding of the performance capabilities of cell-free massive MIMOnetworks, highlighting their strengths and potential limitations. 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Mag., vol. 57, no. 2, pp. 130–137, 2019. → pages 115136AppendicesAppendix A: The Channel Dispersion for ALarge-Scale Network Under a Finite ConstellationThe channel dispersion is defined as the conditional variance of the informationdensity conditioned on the input distribution as followsVM(Υ) = ES{Var{log2PY |S(y|s)PY (y)| S}}=1MM∑m=1E{log22PY |S(y|sm)PY (y)}− 1MM∑m=1E{log2PY |S(y|sm)PY (y)}2.where sm = [Re{sm}, Im{sm}]T is the 2-D real-valued vector form of the complex-valued symbol sm of an M -ary constellation.Define I1 = E{log22PY |S(y|sm)PY (y)}and I2 = E{log2PY |S(y|sm)PY (y)}2. The first term I1 isderived as followsI1 = E{log22 PY |S(y|sm)}− 2E{log2 PY |S(y|sm) log2 PY (y)}+ E{log22 PY (y)}= I11 − 2I12 + I13.The term I11 is given byI11 =∫R2e− ‖y−h0√Pr−η/20 sm‖2σ2w+Bpi(σ2w + B)log22e− ‖y−h0√Pr−η/20 sm‖2σ2w+Bpi(σ2w + B)dy,where y = [y1, y2]T . Define t =y−h0√Pr−η/20 sm√σ2w+B, hence by substitution, I11 is given byI11 =∫R2e−‖t‖2pi(log221pi(σ2w + B)+ 2‖t‖2log2 (e) log21pi(σ2w + B)+ log22 e‖t‖2)dt= log221pi(σ2w + B)− 2 log2(e) log21pi(σ2w + B)+ 3 log22(e).137The term I12 is given byI12 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log2 1MM∑l=1e− ‖y−h0√Prη/20 sl‖2(σ2w+B)pi(σ2w + B) log2e−‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B) dy,and is divided into four terms as I12 = I(1)12 + I(2)12 + I(3)12 + I(4)12 , whereI(1)12 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log21piM(σ2w + B)log21pi(σ2w + B)dy= log21piM(σ2w + B)log21pi(σ2w + B),I(2)12 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log21piM(σ2w + B)log2 e−‖y−h0√Prη/20 sm‖2(σ2w+B) dy= − log2(e) log21piM(σ2w + B),I(3)12 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log21pi(σ2w + B)log2M∑l=1e− ‖y−h0√Prη/20 sl‖2(σ2w+B) dy(i)=1pi∫R2e−‖t‖2log21pi(σ2w + B)log2M∑l=1e−‖t‖2−2tT (sm−sl)h0√Pr−η/20√σ2w+B− |h0|2Pr−η0σ2w+B‖sm−sl‖2dt(ii)=1pi∫R2e−‖t‖2log21pi(σ2w + B)log2M∑l=1e−(‖t‖2+2√ΥtT dml+Υ‖dml‖2)dt= − log2(e) log21pi(σ2w + B)+ E{log21pi(σ2w + B)log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2},where (i) follows from a change of variable t =y−h0√Pr−η/20 sm√σ2w+B, and (ii) follows byusing dml = sm − sl and Υ = |h0|2Pr−η0σ2w+B , and I(4)12 is given by138I(4)12 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log2 e−‖y−h0√Prη/20 sm‖2(σ2w+B) log2M∑l=1e− ‖y−h0√Prη/20 sl‖2(σ2w+B) dy(i)=∫R2e−‖t‖2pilog2 e−‖t‖2 log2M∑l=1e−(‖t‖2+2tT dml√Υ+Υ‖dml‖2)dt= 3 log22(e)− log2(e)Et{‖t‖2log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2},where (i) is obtained by the substitution of t, dml, and Υ.The term I13 is given byI13 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)(log21Mpi(σ2w + B)+log2M∑l=1e−‖y−h0√Prη/20 sl‖2(σ2w+B))2dy,and is divided into three terms as I13 = I(1)13 + I(2)13 + I(3)13 , whereI(1)13 =∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log221piM(σ2w + B)dy= log221piM(σ2w + B),I(2)13 = 2∫R2e− ‖y−h0√Prη/20 sm‖2(σ2w+B)pi(σ2w + B)log21piM(σ2w + B)log2M∑l=1e−‖y−h0√Prη/20 sl‖2(σ2w+B) dy= −2 log2(e) log21piM(σ2w + B)+ 2 log21piM(σ2w + B)E{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2},I(3)13 =∫R21pi(σ2w + B)e− ‖y−h0√Prη/20 sm‖2(σ2w+B) log22M∑l=1e−‖y−h0√Prη/20 sl‖2(σ2w+B) dy= 3 log22(e)− 2 log2(e) E{‖t‖2log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2}+ E{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2},139By adding I11, I12, and I13, I1 is given byI1 = log221M+ E{log22M∑l=1e−2√ΥtT dml−Υ‖dml‖2},+ 2 log21ME{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2}The second term I2 is the square of the relative entropy given asI2 = log22(M)− 2 log2(M)E{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2}+ E2{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2}.To compute the average square-root channel dispersion, I2 is subtracted from I1 andthen averaged over all constellation symbols, leading to the following expression asindicated in (2.16)VM(Υ) =1MM∑m=1Et{log22M∑l=1e−2√ΥtT dml−Υ‖dml‖2}− Et{log2M∑l=1e−2√ΥtT dml−Υ‖dml‖2}2 .140Appendix B: Laplace Transform of the AggregateInterference Power under Fractional Path-LossInversion Power ControlBy definition, the Laplace transform of IFI is given byLIFI(s) = Eexp−s ∑ui∈Φ\{u0}xiPi|hi|2u−ηi .The interference coming from the interfering UE point process {Φ \ u0} can be ap-proximated with the interference seen from a non-homogeneous PPP Φ˜ = {ti ; i ∈ N}with intensity function λUE(t) =(1− e−piλ||t||2)λ, where ti is the vector coordinatesof the UEs and λUE(t) models the intensity of UEs as a function of UE’s location t[90]. To simplify the analysis, we define m = ||t||ηP. Then, for an arbitrary realization,mi =||ti||ηPiwhere mi ∈ Φ˜o and follows a PPP with the following intensity function[90]λm(m) =2(piλ)1−αρ2ηoηm1−2ηγ(1 + α, piλ(ρom)2η(1−α)), (6.1)which is obtained by the Displacement and Mapping Theoreoms for the PPP [120,Theorems 2.33 & 2.34]. Hence, the Laplace transform of interference is given byLIFI(s) = Eexp ∑mi∈Φ˜\{r0}−sxi|hi|2mi= exp{−∫ ∞rη(1−α)0ρoλm(m)(1− E(e−sx|h|2m))dm}= exp{−∫ ∞rη(1−α)0ρoλm(m)(1− pmm+ s− (1− p))dm}= exp{−∫ ∞rη(1−α)0ρoλm(m)(psm+ s)dm}(6.2)By substituting (6.1) in (6.2), we get (3.5).141Appendix C: Average Capacity and AverageSquare Root Dispersion DerivationFor user1 located at r1, the average capacity C¯1 = E{C(γ1)} and the averagesquare root channel dispersion V¯1 = E{V (γ1)} are computed in a similar methodol-ogy as in [1]. The difference is that P is replaced by ξP and the PDF of r1 is differentsince r1 is defined as the distance of user1 to the serving BS. Since the nearest BSassociation strategy is used, the distance between a user and its serving BS followsRayleigh distribution (4.1). For two users placed at Rayleigh distributed distance rwithin the coverage of the BS, and given r1 < r2, the cummulative density function(cdf) of r1 is given asFr1(r1) = 1− (1− Fr(r1))(1− Fr(r1))and the PDF is given as fr1(r1) = 4piλr1e−2piλr21 .For user2 located at r2, the average capacity C¯2 and the average square rootchannel dispersion V¯2 are derived in the following subsections.Average Capacity for Far UserThe average capacity for the user2 is calculated from its cumulative distributionfunction (cdf) using Fubini’s theorem [111] as followsC¯2 = E {C(γ2)} =∫ ∞01− FC(c) dc. (6.3)The cdf of the capacity is evaluated as follows.FC(c) = E {P (log2(1 + γ2) < c|r2, I2)}= E{P((1− ξ)P|h2|2r−η2ξPr−η2 |h2|2+I2 + σ2w< 2c − 1∣∣∣r2, I2)}= E{P(|h2|2< k(c)(I2 + σ2w)rη2(1− ξ − ξk(c))P∣∣∣r2, I2)}= 1− Er2,I2{e−k(c)σ2wrη2(1−ξ−ξk(c))P e−k(c)I2rη2(1−ξ−ξk(c))P}= 1−∫ ∞0e−k(c)σ2wrη2P−1(1−ξ−ξk(c)) LI2{k(c)rη2P−1(1− ξ − ξk(c))}fr2(r2) dr2, (6.4)142where k(c) = 2c− 1, and the last step is obtained using the definition of the Laplacetransform (Lx{u} = E {e−ux}). The cdf of r2 isFr2(r2) = Fr(r2)2,and the PDF is given asfr2(r2) = 4piλr2e−piλr22(1− e−piλr22). (6.5)By substituting (6.4) in (6.3), the average capacity in (4.13) is obtained.Average Square Root Channel DispersionFor the average square root cross channel dispersion term for user2,E√V ′(ξP|h2|2r−η2I2 + σ2w,P|h2|2r−η2I2 + σ2w) ,we would find that it is hard to evaluate. Hence, since we have that [124]V ′(ξP|h2|2r−η2I2 + σ2w,P|h2|2r−η2I2 + σ2w)≤ V (γ2) (6.6)we will use this upper bound to obtain a lower bound on the rate constraint. Theaverage square root channel dispersion V¯ ′2 ≤ V¯2 = E {V (γ2)} in (4.15) is derived asfollows using Fubini’s theorem [111]V¯2 = E {V (γ2)} =∫ log2(e)01− FV (v) dv. (6.7)The cdf of the square root channel dispersion is evaluated as follows.FV (v) = E{P(√(1− 1(γ2 + 1)2)log22(e) < v∣∣∣r2, I2)}= EP (1− ξ)P|h2|2r−η2ξPr−η2 |h2|2+I2 + σ2w<√11− v2log22(e)−1∣∣∣r2, I2= E{P(|h2|2< z(v)(I2 + σ2w)rη2(1− ξ − ξz(v))P∣∣∣r2, I2)}= 1− Er2,I2{e−z(v)σ2wrη2(1−ξ−ξz(v))P e−z(v)I2rη2(1−ξ−ξz(v))P}= 1−∫ ∞0e−z(v)σ2wrη2P−1(1−ξ−ξz(v)) LI2{z(v)rη2P−1(1− ξ − ξz(v))}fr2(r2) dr2. (6.8)143By substituting (6.8) in (6.7) and adjusting the limits from 0 to√1− ξ2 log2(e).Since there exist a pole at√1− ξ2 log2(e), hence the function is undefined at values>√1− ξ2 log2(e), the average square root channel dispersion in (4.15) is obtained.144Appendix D: The Average Sum CapacityThe average sum capacity C¯s is given asC¯s = E {log2(1 + max(γ¯1, γ¯2)}=∫ ∞01− FCs(c) dc. (6.9)To evaluate the average sum capacity, the cdf of the average sum capacity is derivedas followsFCs(c) = E {P(log2(1 + max(γ¯1, γ¯2)) < c|r2, r1, I1, I2)}= E {P(max(γ¯1, γ¯2) < 2c − 1|r2, r1, I1, I2)}= E {P(γ¯1 < k(c), γ¯2 < k(c)|r2, r1, I1, I2)}= E {Fγ¯1(k(c)|r2, r1, I1)Fγ¯2(k(c)|r2, r1, I2)}= Er2,r1{(1− e−k(c)σ2wPr−η1 LI1{k(c)Pr−η1})(1− e−k(c)σ2wPr−η2 LI2{k(c)Pr−η2})}= Er2,r1{1− e−k(c)σ2wPr−η1 LI1{k(c)Pr−η1}− e−k(c)σ2wPr−η2 LI2{k(c)Pr−η2}+ e−k(c)(rη1 +rη2 )P/σ2w LI1{k(c)Pr−η1}LI2{k(c)Pr−η2}}= 1−∫ ∞0e−k(c)σ2wPr−η1 LI1{k(c)Pr−η1}fr1(r1) dr1−∫ ∞0e−k(c)σ2wPr−η2 LI2{k(c)Pr−η2}fr2(r2) dr2+∫ ∞0∫ ∞0e−k(c)(rη1 +rη2 )P/σ2w LI1{k(c)Pr−η1}LI2{k(c)Pr−η2}f(r2, r1) dr2 dr1,where f(r2, r1) = 2fr(r1)fr(r2). By substituting in (6.9), we obtainC¯s =∫ ∞0{∫ ∞0e−k(c)σ2wPr−η1 LI1{k(c)Pr−η1}fr1(r1) dr1145−∫ ∞0∫ ∞0e−k(c)(rη1 +rη2 )P/σ2w LI1{k(c)Pr−η1}LI2{k(c)Pr−η2}f(r2, r1) dr2 dr1+∫ ∞0e−k(c)σ2wPr−η2 LI2{k(c)Pr−η2}fr2(r2) dr2}dc= C¯1 + C¯2 − C¯12. (6.10)where C¯i =∫∞0∫∞0e−k(c)σ2wPr−ηi LIi{k(c)Pr−ηi}fri(ri) dri dc and the term C¯12 is defined asC¯12 =∫∞0∫∞0∫∞0e−k(c)(rη1 +rη2 )P/σ2w LI1{k(c)Pr−η1}LI2{k(c)Pr−η2}f(r2, r1) dr2 dr1dc.This proves (4.26).146Appendix E: The Average Square Root ChannelDispersion for the Average Sum RateThe average square root channel dispersion V OBs is derived using the same method-ology as the average sum capacity. Hence, we do not provide a detailed derivationhere, but highlight the main differences that in [1]. The average square root channeldispersion V OBs is given asV¯s = E{√(1− 1(max(γ¯1, γ¯2) + 1)2)log22(e)}=∫ log2(e)01− FVs(v) dv. (6.11)To evaluate the average square root channel dispersion, the cdf of the average squareroot channel dispersion is derived as followsFV¯s(v) = E{P(√(1− 1(max(γ¯1, γ¯2)+1)2)log22 e<v∣∣∣r2, r1, I1, I2)}= EP(max(γ¯1, γ¯2) <√11− v2log22(e)− 1∣∣∣r2, r1, I1, I2)= E{P(γ¯1 < z(v), γ¯2 < z(v)∣∣∣r2, r1, I1, I2)}= 1−∫ ∞0e−z(v)σ2wPr−η1 LI1{z(v)Pr−η1}fr1(r1) dr1−∫ ∞0e−z(v)σ2wPr−η2 LI2{z(v)Pr−η2}fr2(r2) dr2+∫ ∞0∫ ∞0e−z(v)(rη1 +rη2 )P/σ2w LI1{z(v)Pr−η1}LI2{z(v)Pr−η2}f(r2, r1) dr2 dr1. (6.12)By substituting in (6.11), we obtainV¯s =∫ log2(e)0{∫ ∞0e−z(v)σ2wPr−η1 LI1{z(v)Pr−η1}fr1(r1) dr1−∫ ∞0∫ ∞0e−z(v)(rη1 +rη2 )P/σ2w LI1{z(v)Pr−η1}LI2{z(v)Pr−η2}f(r2, r1)dr2dr1147+∫ ∞0e−z(v)σ2wPr−η2 LI2{z(v)Pr−η2}fr2(r2) dr2}dv= V OB1 + VOB2 − V OB12 , (6.13)where V¯i =∫ log2(e)0∫∞0e−z(v)σ2wPr−η1 LI1{z(v)Pr−η1}fr1(r1) dr1dv is defined in [1], and VOB12 =∫∞0∫∞0e−z(v)(rη1 +rη2 )P/σ2w LI1{z(v)Pr−η1}LI2{z(v)Pr−η2}f(r2, r1) dr2 dr1 dv. This proves (4.29).148
Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Chaaban, Anas
Hossain, Md. Jahangir
Performance analysis of large scale networks in the finite blocklength regime
Text