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Energy efficient resource allocation for Non-Orthogonal Multiple Access (NOMA) systems Fang, Fang 2017

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Energy Efficient Resource Allocationfor Non-Orthogonal Multiple Access(NOMA) SystemsbyFang FangM.A.Sc., Lanzhou University, 2013B.Sc., Lanzhou University, 2010A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)December 2017c© Fang Fang, 2017The following individuals certify that they have read, and recommend to the Collegeof Graduate Studies for acceptance, a thesis/dissertation entitled: Energy EfficientResource Allocation for Non-Orthogonal Multiple Access (NOMA) Systemssubmitted by Fang Fang in partial fulfilment of the requirements of the degree of Doctorof PhilosophyDr. Julian Cheng, School of EngineeringSupervisorDr. Md. Hossain Jahangir, School of EngineeringSupervisory Committee MemberDr. Feng Chen, School of EngineeringSupervisory Committee MemberDr. Joshua Brinkerhoff, School of EngineeringUniversity ExaminerDr. Lin Cai, Electrical & Computer Engineering, University of VictoriaExternal ExamineriiAbstractNon-orthogonal multiple access (NOMA) is a promising technique for the fifth genera-tion (5G) mobile communication due to its capability of achieving high spectral efficiencyand high data rate. A popular NOMA scheme uses power domain to achieve multipleaccess. By applying successive interference cancellation (SIC) technique at the receivers,multiple users with different power levels can be multiplexed on the same frequency band,providing higher sum rate than that of conventional orthogonal multiple access (OMA)schemes. The energy consumption has increased rapidly in recent years. To save energyand meet the requirement of green communications in 5G, we focus on the energy efficientresource optimization for NOMA systems. Our research aim is to maximize the systemenergy efficiency in NOMA systems by considering perfect channel state information (CSI)and imperfect CSI via resource management.We first study the energy efficient resource allocation for a downlink single cell NOMAnetwork with perfect CSI. The energy efficient resource allocation is formulated as a non-convex problem. A low-complexity suboptimal algorithm based on matching theory isproposed to allocate users to subchannels. A novel power allocation is designed to furthermaximize the system energy efficiency. However, the perfect CSI is challenging to obtain inpractice. We subsequently investigate energy efficiency improvement for a downlink NOMAsingle cell network by considering imperfect CSI. To balance the system performance andcomputational complexity, we propose a new suboptimal user scheduling scheme, whichclosely attains the optimal performance. By utilizing Lagrangian approach, an iterativepower allocation algorithm is proposed to maximize the system energy efficiency.Implementing NOMA in Heterogeneous networks (HetNets) can alleviate the cross-iiiAbstracttier interference and highly improve the system throughput via resource optimization. Byconsidering the cochannel interference and cross-tier interference, an iterative algorithm isproposed to maximize the macro cell and small cells energy efficiency. Simulations resultsshow that the proposed algorithm can converge within ten iterations and can achieve highersystem energy efficiency than that of OMA schemes.ivPrefaceA list of my publications at The University of British Columbia is provided in thefollowing.Refereed Journal PublicationsJ1. Haijun Zhang, Fang Fang, Julian Cheng, Wei Wang, Keping Long, and VictorC. M. Leung,“Energy-efficient resource allocation in 5G NOMA heterogeneous net-works,”IEEE Wireless Communications Magazine, Feb. 2017. (Submitted)J2. Fang Fang, Haijun Zhang, Julian Cheng, Se´bastien Roy, and Victor C. M. Leung,“Joint user scheduling and power allocation optimization for energy efficient NOMAsystems with imperfect CSI,” Accepted for publication in IEEE Journals on SelectedAreas on Communications, 2017.J3. Fand Fang, Haijun Zhang, Julian Cheng, and Victor C. M. Leung, “Energy-efficientresource allocation for downlink non-orthogonal multiple access (NOMA) network,”IEEE Transactions on Communications, vol. 64, no. 9, pp. 3722–3732, July 2016.Refereed Conference PublicationsC1. Fang Fang, Haijun Zhang, Julian Cheng, and Victor C. M. Leung, “Energy-efficientresource scheduling for NOMA systems with imperfect channel state information,”Proceedings of IEEE International Conference on Communications (ICC 2017), Paris,France, May 21-25, 2017.vPrefaceC2. Fang Fang, Haijun Zhang, Julian Cheng, and Victor C. M. Leung, “Energy effi-ciency of resource scheduling for non-orthogonal multiple access (NOMA) wirelessnetwork,” Proceedings of IEEE International Conference on Communications (ICC2016), Kuala Lampur, Malaysia, May 23-27, 2016.viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Acronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 7Chapter 2: Background on Energy Efficient Non-Orthogonal Multiple Ac-cess (NOMA) Systems . . . . . . . . . . . . . . . . . . . . . . . . 102.1 Non-Orthogonal Multiple Access (NOMA) Systems . . . . . . . . . . . . . . 102.1.1 Successive Interference Cancelation (SIC) Technology in NOMA Sys-tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.1.2 Performance Analysis for NOMA Systems . . . . . . . . . . . . . . . 13viiTABLE OF CONTENTS2.2 Energy Efficiency in Communication Networks . . . . . . . . . . . . . . . . 162.2.1 Research Motivation of Energy Efficiency . . . . . . . . . . . . . . . 162.2.2 Energy Efficiency Definition in Wireless Communications Networks . 172.2.3 Resource Optimization and Convex Optimization Approaches . . . . 182.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Chapter 3: Energy Efficient Resource Allocation for Downlink NOMANetwork with Perfect CSI . . . . . . . . . . . . . . . . . . . . . . 203.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3 Subchannel Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.1 Subchannel Matching Problem Formulation . . . . . . . . . . . . . . 263.3.2 Suboptimal Matching for Subchannel Assignment Algorithm in NOMA 283.3.3 Power Ratio Factor Determination . . . . . . . . . . . . . . . . . . . 303.3.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.4 Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.1 DC Programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.4.2 Power Proportional Factor . . . . . . . . . . . . . . . . . . . . . . . . 333.4.3 Subchannel Power Allocation by DC Programming . . . . . . . . . . 343.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43Chapter 4: Energy Efficient Resource Allocation for Downlink NOMANetwork with Imperfect CSI . . . . . . . . . . . . . . . . . . . . . 474.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.1.1 Imperfect Channel Model . . . . . . . . . . . . . . . . . . . . . . . . 494.1.2 Scheduled Channel Capacity and Outage Capacity . . . . . . . . . . 494.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 514.3 Methodology and Problem Solution . . . . . . . . . . . . . . . . . . . . . . . 52viiiTABLE OF CONTENTS4.3.1 Optimization Problem Transformation . . . . . . . . . . . . . . . . . 524.4 Energy Efficient Resource Allocation Scheme . . . . . . . . . . . . . . . . . 564.4.1 User Scheduling Scheme Design . . . . . . . . . . . . . . . . . . . . . 574.4.2 Energy Efficient Power Allocation Algorithm . . . . . . . . . . . . . 594.4.3 Power Allocation Expression Derivation . . . . . . . . . . . . . . . . 624.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72Chapter 5: Energy Efficient Resource Allocation for NOMA Heteroge-neous Networks (HetNets) . . . . . . . . . . . . . . . . . . . . . . 735.1 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 745.1.1 NOMA HetNet System Model . . . . . . . . . . . . . . . . . . . . . 745.1.2 Channel Description . . . . . . . . . . . . . . . . . . . . . . . . . . . 755.1.3 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 775.2 Energy Efficient Resource Allocation for NOMA HetNets . . . . . . . . . . 805.2.1 Energy Efficiency Optimization for the Entire System Algorithm Design 805.2.2 Energy Efficiency Optimization for Small Cells . . . . . . . . . . . . 825.2.3 Algorithm Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.2.4 Macro Cell Energy Efficiency Maximization . . . . . . . . . . . . . . 895.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 935.4 Imperfect CSI Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100Chapter 6: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1016.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116ixTABLE OF CONTENTSAppendix A: Proof of Convergence of Algorithm 6 . . . . . . . . . . . . . . . . . 116Appendix B: Derivation of the Optimal Power Allocation Policy in Chapter 4 . . 118Appendix C: Derivation of the Optimal Power Allocation {ps,∗u,n} for SUEs in Chap-ter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120xList of FiguresFigure 1.1 Mobile data traffic growth predicted by Ericsson [7]. . . . . . . . . . 3Figure 2.1 OFDMA versus NOMA systems. . . . . . . . . . . . . . . . . . . . . 13Figure 3.1 System model of a downlink NOMA single-cell network. . . . . . . . 21Figure 3.2 Sum rate of the system versus different number of users. . . . . . . 38Figure 3.3 Energy efficiency of the system versus different number of users. . . 39Figure 3.4 Sum rate of the system versus BS power. . . . . . . . . . . . . . . . 40Figure 3.5 Energy efficiency of the system versus the maximum BS power Ps. . 41Figure 3.6 Energy efficiency of the system versus Pc/Ps. . . . . . . . . . . . . . 44Figure 3.7 Energy efficiency of the system versus different number of users. . . 45Figure 4.1 Energy efficiency performance versus the number of users . . . . . . 64Figure 4.2 Energy efficiency performance versus the number of iterations forAlgorithm 6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65Figure 4.3 Energy efficiency performance versus the number of users. . . . . . 67Figure 4.4 Energy efficiency performance versus the number of iterations forAlgorithm 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Figure 4.5 Energy efficiency performance versus BS maximum power. . . . . . 70Figure 4.6 Energy efficiency performance versus users. . . . . . . . . . . . . . . 71Figure 5.1 NOMA based heterogeneous networks. . . . . . . . . . . . . . . . . 74Figure 5.2 Energy efficiency versus the number of iterations for Algorithm 8. . 94Figure 5.3 Energy efficiency versus number of small cells with perfect CSI. . . 95xiLIST OF FIGURESFigure 5.4 Energy efficiency of the system versus the number of the small cellswith the estimation error variance 0.05. . . . . . . . . . . . . . . . . 98Figure 5.5 Energy efficiency of the system versus the number of the small cellwith different estimation error variances. . . . . . . . . . . . . . . . 99xiiList of AcronymsAcronyms Definitions1G First Generation2G Second Generation3G Third Generation4G Fourth Generation5G Fifth GenerationAMPS Advanced Mobile Phone SystemATSC Advanced Television Systems CommitteeAWGN Additive White Gaussian NoiseBS Base StationCRNN Channel Response Normalized by NoiseCSI Channel State InformationEE Energy EfficiencyFDE Frequency Domain EqualizationFTPA Fractional Transmit Power AllocationGPRS General Packet Radio ServiceGPS Global Positioning SystemGSM Global System for Mobile CommunicationsxiiiList of AcronymsHetNet Heterogeneous NetworkICT Information and Communication Technology (ICT)IMT International Mobile TelecommunicationsIoT Internet of ThingsIP Internet ProtocolLTE Long Term EvolutionLTE-Advanced Long Term Evolution-AdvancedITU International Telecommunication UnionITU-R International Telecommunications Union-RadioKKT Karush-Kuhn-TuckerLDM Layer Division MultiplexingMBS Macro Base StationMIMO Multiple-Input Multiple-OutputMUE Macro User EquipmentsMUST Multiple-User Superposition TransmissionNOMA Non-orthogonal Multiple AccessOFDMA Orthogonal Frequency Division Multiple AccessOMA Orthogonal Multiple AccessQ1 First QuarterQoS Quality of ServiceSBS Small Base StationSC SubchannelxivList of AcronymsSIC Successive Interference CancellationSINR Signal-to-Interference-plus-Noise RatioSNR Signal-to-Noise RatioSOMSA Suboptimal Matching Scheme for Subchannel AssignmentSUE Small User EquipmentUE User EquipmentUT User TerminalsxvList of SymbolsSymbols Definitionsarg max {·} Points of the domain of the function at which the functionvalues are maximizedlog2(·) The log function with base 2max {·} The maximum value of the functionmin {·} The minimum value of the functions.t. Subject toE[·] The statistical expectation operationQ1(·) The first-order Marcum Q-functionlog(·) The log function with base 10Pr[·] The probability of an event| · | The absolute value of the argumentxviAcknowledgementsI am deeply grateful to my supervisor Dr. Julian Cheng for his constant guidance,advice, encouragement and support for my PhD study. He granted me a great flexibilityand freedom in my research work. He taught me academic knowledge, research skills andwriting skills. I will continue to be deeply influenced by his work enthusiasm, rigorousscholarship, clarity in thinking and professional integrity. It is my honor to study and doresearch under his supervision.I would like to express my thanks to Dr. Lin Cai from University of Victoria to serveas my external examiner. I am honored to have her on my committee. I would also liketo thank Dr. Joshua Brinkerhoff for being my University Examiner and Dr. Md. JahangirHossian and Dr. Chen Feng for being my committee members. I really appreciate theirvaluable time. Besides, I would like to give special thanks to Dr. Huijun Zhang for insightfuldiscussions and valuable suggestions on my research work.I want to thank my love Tianlong Liu for his continuing encouragement and accom-panying on my PhD adventure. I would also like to thank my dear landlord family (Taijifamily) in Kelowna, Bruce Taiji, Jane Taiji, Sian Taiji, grandma and grandpa, for theircontinuing support and encouragement when I was in depression during my PhD journey.I would like to thank my dear colleagues Bingcheng Zhu, Md. Zoheb Hassan, HuiMa, Guanshan Ye and Fan Yang for sharing their academic experiences and constructiveviewpoints during my PhD study at The University of British Columbia. I also would liketo thank all my dear friends who helped me a lot when I was in Kelowna and I miss thefun we had together.Finally, I would like to thank my parents and my younger sister for their patience,xviiAcknowledgementsunderstanding, unconditional love and support over all these years. All my achievementswould not have been possible without their constant encouragement and support.xviiiChapter 1Introduction1.1 Background and MotivationThe evolution of communication systems began with use of drums, smoke signal andsemaphore in the early human history [1]. Since the invention of the telephone by Alexan-der Graham Bell in 1876, the instant communication across long distance has ignited therevolution of communication systems. The discovery of radio waves demonstrated commu-nication systems using radio signals. The time of wireless communication had begun sinceMarconi first demonstrated radio transmission in 1895 from the Isle of Wight to a tugboatthat was 18 miles away. Radio technology was developed rapidly to enable transmissionover longer distances with better quality, less power, and smaller, cheaper devices, therebyenabling public and private radio communications, television, and wireless networking [2].The researchers at AT&T Bell Laboratories developed the cellular concept to solve thecapacity problem emerged during the 1950s and 1960s [3].During the 1980s, the first generation (1G) mobile telecommunication systems wereinvented for commercial use. The first analogue cellular system, advanced mobile phonesystem (AMPS), was widely deployed in North America. With the increasing demand ofcapacity and high quality of communications, the development of digital cellular technologybecame significant. The second generation (2G) mobile telecommunication networks werecommercially launched in Finland by Radiolinja in 1991. This network used the global sys-tem for mobile communications (GSM) standard [2]. The 2G systems had higher spectrumefficiency and offered more mobile data services than the 1G systems. The introductionof the general packet radio service (GPRS) became the first major step in the evolution11.1. Background and Motivationof GSM networks towards the third generation (3G) telecommunication technology. Inthe early 1980s, the third generation telecommunication technology was developed by theInternational Telecommunication Union (ITU). Compered with the 2G networks, the 3Gnetworks offered higher data rate and greater security. By using the bandwidth and loca-tion information of 3G devices, global positioning system (GPS), location-based services,mobile Internet access, video calls and mobile TV were developed into applications. A newgeneration of cellular standards has appeared approximately every ten years since the 1Gsystems were introduced. In March 2008, requirements of the fourth generation mobiletelecommunication (4G) technology standards were specified by the International Telecom-munications Union-Radio communications sector (ITU-R) [4]. A major step from 3G to4G is that the 4G systems can support all-Internet Protocol (IP) based communication,such as IP telephony, instead of traditional circuit-switched telephony service. The spreadspectrum radio technology used in the 3G networks was abandoned in the 4G systems.The main technology in 4G systems were the orthogonal frequency division multiple ac-cess (OFDMA) multi-carrier transmission and other frequency domain equalization (FDE)schemes.In the fourth generation mobile communication systems such as long-term evolution(LTE) and LTE-Advanced [5], OFDMA has been widely adopted to achieve higher datarate. The demand for mobile traffic data volume is expected to be 500-1,000 times largerin 2020 than that in 2010 [6].Figure 1.1 shows total global monthly data (ExaBytes per month) and voice traffic fromQ1 2012 to Q1 2017 [7]. It depicts a continued strong growth in data traffic. This growth isdriven by increased smart phone subscriptions and average data volume per subscription,which has been fueled primarily by more viewing of video content. Data traffic grew around12% quarter-on-quarter and around 70% year-on-year.To further meet the overwhelming requirement of data rates, various new techniqueshave been proposed in recent years, and these techniques include massive multiple-inputmultiple output (MIMO) [8], millimeter wave communications [9], LTE-U[10], C-RAN [11],21.1. Background and MotivationFigure 1.1: Mobile data traffic growth predicted by Ericsson [7].31.2. Literature ReviewSON [12] and non-orthogonal multiple access (NOMA)[13]. Among them, NOMA takesadvantage of spectrum efficiency by allowing multiple users to occupy the same subchannel,which is different from the resource allocation in OFDM systems [14–16]. By applyingsuccessive interference cancellation (SIC) in the NOMA systems, superposition coded signalcan be correctly decoded and demodulated at the receiver [17–20]. Therefore, NOMA hasbeen well considered as a promising candidate for the next generation mobile communicationsystems.With the overwhelming increase of the traffic data and mobile devices, the energy costhas rapidly increased and become an important issue in the green cellular network because ofthe increasing amount of CO2 emission levels caused by energy consumption [22–25]. Thus,fast growing energy consumption and limited global energy resources are the importantmotivations for the research on energy efficient communications in NOMA networks. Inthis thesis, we mainly focus on the energy efficient resource allocation for NOMA networks.We aim to design different resource allocation schemes to maximize the energy efficiency ofdifferent NOMA systems.1.2 Literature ReviewSince the basic concept of NOMA was introduced and the cell-edge user throughputperformance improvement was demonstrated in [26], NOMA has attracted much researchattention. The NOMA system has also been envisioned as a key technology in the fifthgeneration mobile communication systems [27]. A popular NOMA scheme uses power do-main to achieve multiple access. More specifically, by utilizing the SIC technology, multipleusers can be multiplexed on the same frequency band with different power levels [28, 29].This is different from the conventional orthogonal multiple access (OMA) scheme. In thedownlink NOMA system, the user who has the higher channel state information (CSI) canremove the interference signal from the other users who has less CSI [30, 31]. This protocolenables NOMA to achieve higher spectrum efficiency than the OMA scheme. In addi-41.2. Literature Reviewtion to its spectral efficiency gain, academic and industrial research has also demonstratedthat NOMA can effectively support massive connectivity, which is important for ensuringthat the forthcoming 5G network can support the Internet of Things (IoT) functionalities[32–35].The multiuser superposition transmission scheme has been proposed in the third gen-eration partnership project long-term evolution advanced (3GPP-LTE-A) networks [36],where NOMA is referred to as multi-user superposition transmission (MUST). A variationof NOMA, termed layer division multiplexing (LDM) was proposed for next generationdigital TV standard advanced television systems committee (ATSC) 3.0. Extensive studiesfor NOMA system have been conducted. By considering a single frequency band in theNOMA system with uniformly deployed mobile users, the outage probability was first de-rived with perfect CSI, and the numerical results confirmed the derived outage probabilityexpressions [37]. Furthermore, the outage performance of a NOMA system with two typesof imperfect CSI was analyzed for the downlink single cell NOMA system, and numericalresults showed that the average sum rate matches well with the Monte Carlo simulations[38]. By exploiting the outage probability from [38], the problem of energy efficient userscheduling and power allocation was investigated for NOMA systems by considering onlytwo users multiplexed on one subchannel and imperfect CSI [39].Besides the performance analysis, the resource allocation was investigated in NOMAsystems. By using fractional transmit power allocation (FTPA) among users and equalpower allocation across subchannels, the authors in [40, 41] compared system-level per-formance of the NOMA system with the OMA system, and showed that the overall cellthroughput, cell-edge user throughput, and the degree of proportional fairness of NOMAare all superior to those of the OMA scheme. Though FTPA is simple to implement, itfails to optimally allocate power among multiplexed users on each subchannel. Thus, anew power allocation scheme based on water filling was proposed to achieve high spectralefficiency [42]. A cooperative relay system based on NOMA was proposed in [43], wherethe improvement of the spectral efficiency was presented by numerical results. A greedy51.2. Literature Reviewsubchannel and power allocation algorithm was proposed for the NOMA system [44], anda cooperative NOMA transmission scheme, where some users have prior information of theother users’ message, was proposed in [45] to improve spectrum efficiency. Driven by therapidly increasing of the energy cost, the energy-efficient power allocation was investigatedfor NOMA systems [46]. By using statistical channel state information at the transmit-ter, a near optimal power allocation scheme was proposed to maximize the system energyefficiency [46].The problem of joint subcarrier allocation and power allocation was investigated ina full-duplex NOMA system [47], where the proposed suboptimal iterative scheme withlow computation complexity can achieve close-to-optimal performance. The application ofcombining NOMA with MIMO technologies has attracted recent research attention. TheMIMO NOMA design for small packet transmission and the multi-user detection for uplinkNOMA systems were investigated in [48] and [49], respectively. The fairness clusteringproblem was solved in MIMO NOMA scenarios [50], where an algorithm was proposed toachieve a good tradeoff between the complexity and the throughput.Driven by the rapid increase of wireless terminal equipments and wide usage of mobileInternet, HetNets have emerged as one of the most promising network infrastructures toprovide high system throughput and large coverage of indoor and cell edge scenarios in the5G wireless communication systems. In such an architecture, a macro cell is overlaid byseveral small cells, e.g., microcell, picocell and femtocell, to significantly improve the systemthroughput and the spectral efficiency. For HetNets, frequency band sharing between macrocell and small cells is viable, and it is also more efficient to reuse the frequency bands withina macro cell. However, the cross-tier interference can severely degrade the quality of thewireless transmission. The advantage of the HetNet also comes with fundamental challengessuch as cross-tier interference mitigation and resource scheduling. In previous research,the cross-tier interference control and cancellation in HetNets have been investigated, e.g.precoding technique and resource management [15]. In the traditional OFDMA HetNets,the frequency band can be divided into several sub frequency bands, and users in the same61.3. Thesis Outline and Contributionssmall cell are assigned to different sub frequency bands in order to avoid the inter-cellinterference [15]. Spectrum sharing between macro cell and small cells is applicable inHetNets; however, the cross-tier interference and the co-channel interference can severelydegrade the communication quality in HetNet. Therefore, implementing NOMA in HetNetscan alleviate the cross-tier interference and significantly improve the system throughput viaresource optimization [51]. To maximize the sum rate of small cells, an iterative subchannelallocation and power allocation algorithm is proposed to closely approach the optimalsolution within limited iterations [51].To this end, most research works on NOMA systems have focused on the case that thebase station (BS) knows the perfect knowledge of the CSI [40, 41, 52]. Since the perfect CSIis challenging to obtain in practice, power-efficient resource allocation scheme was designedto minimize the total transmit power [53], where the proposed scheme can achieve close-to-optimal performance. We assume that a channel estimation error model where the BS onlyknows the estimated channel gain and a priori knowledge of the variance of the estimationerror [54, 55]. In this situation, the scheduled user data rate may exceed the maximumachievable data rate due to the estimated channel gain. Therefore, an outage probabilityrequirement should be considered for the resource allocation to maximize system energyefficiency.1.3 Thesis Outline and ContributionsIn this thesis, we present the research work conducted on the following three topics:− Energy efficient resource allocation for downlink NOMA networks with perfect CSI− Energy efficient resource allocation for downlink NOMA networks with imperfect CSI− Energy efficient resource allocation for downlink NOMA HetNets.The summary and contributions of each chapter are as follows.71.3. Thesis Outline and ContributionsChapter 1 presents background knowledge on the history and development of cellularcommunication systems. In addition, this chapter provides a detailed literature reviewrelated to the rest of the thesis.Chapter 2 presents the required technical background for the entire thesis. We first in-troduce the SIC technology and discuss practical issues associated with SIC implementation.After that, we introduce the NOMA system and analyze its performance gain comparedto the conventional OMA scheme. As the energy efficient design has an important rolein modern communications, a detailed introduction of energy efficiency in communicationis provided. Finally, we discuss the resource allocation types and optimization tools inwireless communication systems.Chapter 3 studies the energy efficient resource allocation for NOMA system with theperfect channel state information. Unlike most previous works focusing on resource allo-cation to maximize the throughput, we aim to optimize subchannel assignment and powerallocation to maximize the energy efficiency for the downlink NOMA network. Assumingperfect knowledge of the channel state information at the base station, we propose a low-complexity suboptimal algorithm which includes energy-efficient subchannel assignmentand power proportional factors determination for subchannel multiplexed users. We alsopropose a novel power allocation across subchannels to further maximize energy efficiency.Since both optimization problems are non-convex, difference of convex programming is usedto transform and approximate the original non-convex problems to convex optimizationproblems. Solutions to the resulting optimization problems can be obtained by iterativelysolving the convex subproblems.In Chapter 4, we investigate energy efficiency improvement for a downlink NOMA sin-gle cell network by considering imperfect CSI. The energy-efficient resource optimizationproblem is formulated as a non-convex optimization problem with the constraints of outageprobability limit, the maximum power of the system, the minimum user data rate and themaximum number of multiplexed users sharing the same subchannel. To efficiently solvethis problem, the probabilistic mixed problem is first transformed into a non-probabilistic81.3. Thesis Outline and Contributionsproblem. An iterative algorithm for user scheduling and power allocation is proposed tomaximize the system energy efficiency. The optimal user scheduling based on exhaustivesearch serves as a system performance benchmark, but it has high computational com-plexity. To balance the system performance and the computational complexity, a newsuboptimal user scheduling scheme is proposed to schedule users on different subchannels.Based on the user scheduling scheme, the optimal power allocation expression is derivedby the Lagrange approach. By transforming the fractional-form problem into an equivalentsubtractive-form optimization problem, an iterative power allocation algorithm is proposedto maximize the system energy efficiency. Simulation results demonstrate that the proposeduser scheduling algorithm closely attains the optimal performance. In this work, multipleusers can occupy the same subchannel, which is different from [39, 56, 57] where only twousers can be supported on the same subchannel.In Chapter 5, NOMA is extended and applied to HetNets. In this chapter, we design ajoint resource allocation scheme to maximize the system energy efficiency via the Lagrangianapproach. We maximize not only small cells energy efficiency but also macro cell energyefficiency. Simulation results show that our proposed resource allocation scheme for NOMAHetNets can achieve higher energy efficiency than that of the conventional OMA scheme.Chapter 6 summarizes the entire thesis and lists our contributions. In addition, somefuture works related to our current research are also suggested.9Chapter 2Background on Energy EfficientNon-Orthogonal Multiple Access(NOMA) SystemsIn this chapter, we provide a brief description of NOMA system and SIC, and SIC is themain technology applied at the receiver in NOMA systems. Then, the resource allocationbackground knowledge in wireless network is presented. Finally, we introduce the energyefficiency concept in wireless communications and present our research motivations.2.1 Non-Orthogonal Multiple Access (NOMA) Systems2.1.1 Successive Interference Cancelation (SIC) Technology in NOMASystemsNon-orthogonal multiple access has been recognized as a promising multiple accesstechnique for the fifth generation networks due to its superior spectral efficiency. In NOMAsystems, the main technology is SIC technology. Considering a downlink transmissionscenario where a single base station transmits superposition coded information to two users.The channel gains from the base station to these two users are h1 and h2, respectively.Assume that |h1| < |h2|, and both h1 and h2 are perfectly known to both the transmitter102.1. Non-Orthogonal Multiple Access (NOMA) Systemsand receivers. The transmit signal or the sum of two signals, can be expressed asx = x1 + x2 (2.1)where xk is the signal intended for user k, k = 1, 2. Therefore, the received signal at user kcan be written byyk = hkx+ wk, k = 1, 2 (2.2)where wk ∼ CN(0, σ2n)is independent and identically distributed (i.i.d.) complex additivewhite Gaussian noise (AWGN) with mean zero and variance σ2n. The transmit signal xhas an average power constraint of P = P1 + P2 where P1 and P2 are the transmit powersfor User 1 and User 2, respectively. The key idea of SIC technique is that users who havehigher channel gains can decode the data that has been successfully decoded by the userswho have less channel gains [58]. In this case, since User 2 has a higher channel gain thanUser 1, User 2 can decode the data that has been successfully decoded by User 1. Thus,the superposition coding scheme can be implemented by the following steps [59]:1. The transmit signal is superposition coded signals of the two users.2. At the receiver, User 1 treats User 2’s signal as noise and decodes its data from y1.3. User 2 with better channel performs SIC, i.e., it decodes the data of User 1 andsubtracts User 1’s signal from y2. After that, User 2 can decode its data.According to the Shannon’s capacity formula, the data rate for User 1 and User 2 withbandwidth B can be achieved byR1 = B log(1 +P1|h1|2P2|h1|2 + σ2n)bits/s (2.3)R2 = B log(1 +P2|h2|2σ2n)bits/s. (2.4)We consider K users are distributed in a downlink network with SIC at the receivers112.1. Non-Orthogonal Multiple Access (NOMA) Systemsand |hK | ≥ |hK−1| ≥ · · · ≥ |h1|. The boundary of the capacity region is given byRk = B log1 + Pk|hk|2σ2n + |hk|2K∑j=k+1Pj , k = 1, 2, · · · ,K (2.5)for all possible splits P =K∑k=1Pk of the total power at the base station. The optimal pointsare achieved by superposition coding at the transmitter and SIC at each of the receivers.The cancellation order at every receiver is always to decode the weaker users before decodingits own data.We have discussed the advantages of SIC technology in the downlink network. SIChas a significant performance gain over the conventional orthogonal multiple access (OMA)techniques. It takes advantage of the strong channel of the nearby user to give it high ratewhile providing the weak user with the best possible performance. Here we discuss severalpotential practical issues in applying SIC in a wireless system [59].− Complexity will increase when the number of the users increases: In the downlink,applying SIC at the mobile receivers means that the user needs to decode informationintended for some of the other users, which would not happen in the conventionalsystem. Thus the decoding complexity at each mobile user will increase when thenumber of users multiplexed on the same frequency band increases. However, we haveseen that superposition coding in conjunction with SIC has the largest performancegain when the users have large disparate channels from the base station. To avoidthe high complexity of SIC, user grouping can be a solution in practice. In order toreduce the complexity of decoding, it is suggested to separate users in the cell intogroups containing small number of users. Each group users can be multiplexed onthe same subchannel and superposition coding based SIC is performed. Therefore,the SIC can achieve the performance gain with low complexity.− Imperfect channel state information estimation: The interfering signal from the other122.1. Non-Orthogonal Multiple Access (NOMA) Systemsusers must be reconstructed before it is removed from the received signal. Thiscontribution depends on the estimation of channel state information. The imperfectestimation of CSI will lead to residual cancellation errors. One concern is that if thedifference of the users’ received powers is large, the residual error from cancelling thestronger user can still swamp the weaker users signal. On the other hand, it is alsoeasier to get an accurate channel estimate when the user has high CSI. It turns outthat these two effects compensate each other and the effect of residual errors does notgrow with the power disparity.− Analog-to-digital quantization error: When the difference of received powers of theusers is large, a large dynamic range of the analog-to-digital (A/D) converter is re-quired. For example, if the power disparity is 20 dB, even 1-bit accuracy for the weaksignal would require an 8-bit A/D converter. This may well pose an implementationconstraint on how much gain SIC can offer.2.1.2 Performance Analysis for NOMA SystemsUE2FrequencyPowerOFDMAUE1(better CSI)UE2(worse CSI)FrequencyPowerUE1NOMA: SICFigure 2.1: OFDMA versus NOMA systems.In this section, we discuss basic NOMA with SIC and analyze its performance gainover OFDMA schemes. A popular NOMA scheme uses power domain to achieve multipleaccess. Figure 2.1 presents the power domain comparison over frequency of the NOMAsystem and the OFDMA system. By applying SIC technique at the receivers, multiple132.1. Non-Orthogonal Multiple Access (NOMA) Systemsusers with different power levels can be multiplexed on the same frequency band, providinghigher sum rate than that of conventional orthogonal multiple access (OMA) schemes.In NOMA systems, SIC is applied at the receivers. Let us focus on the two-user case fora downlink NOMA system. Consider that two users are multiplexed on the same subchannelwith the channel gains |h1|2 ≥ |h2|2 shown in Fig. 2.1, where hm = gm ·PL−1(d),m = 1, 2,and where gm is assumed to be the Rayleigh fading channel gain and PL−1(d) is the pathloss function between the BS and UTm at distance d. Denote the assigned power on SCnby pn. The bandwidth of the subchannel is Bsc and the power proportional factors for User1 and User 2 are β1 and β2, respectively.For NOMA systems, SIC is applied at User 1 with a higher channel gain than User 2.According to the SIC protocol, User 1 can cancel the interference signal from User 2. Thedata rates of User 1 and User 2 in NOMA systems can be respectively represented asR1 = Bsclog2(1 +|h1|2β1pnσ2n)(2.6)andR2 = Bsclog2(1 +|h2|2β2pn|h2|2β1pn + σ2n). (2.7)The sum rate can be written byRNOMA = Bsclog2(1 +|h1|2β1pnσ2n)+Bsclog2(1 +|h2|2β2pn|h2|2β1pn + σ2n). (2.8)In OFDMA systems, we assume OFDMA with orthogonal user multiplexing. The totalbandwidth Bsc is occupied by these two users (assume that each user has half bandwidth).The data rates of User 1 and User 2 in OFDMA systems can be respectively represented asR1 =12Bsclog2(1 +|h1|2pnσ2n)(2.9)142.1. Non-Orthogonal Multiple Access (NOMA) SystemsandR2 =12Bsclog2(1 +|h2|2pnσ2n). (2.10)Therefore, the sum rate of these two users in OFDM system can be expressed asROFDM =12Bsclog2(1 +|h1|2pnσ2n)+12Bsclog2(1 +|h2|2pnσ2n). (2.11)Using the parameters values in [60], we set β1 = 1/5, β2 = 4/5, B = 1 Hz,|h1|2pnσ2n= 20dB and |h2|2pnσ2n= 0 dB . Assume each user has the same weighted bandwidth (Bsc = 1Hz). Therefore, ROFDM = 3.33 + 0.50 = 3.83 bits/sec and RNOMA = 4.39 + 0.74 = 4.53bits/sec. The gain of the NOMA system is 34% more than that of the OFDMA scheme.Now consider better channel conditions, which means we improve the channel gains bysetting |h1|2pnσ2n= 30 dB and |h2|2pnσ2n= 10 dB. Assume each user has the same weightedbandwidth (Bsc = 1 Hz), β1 = 1/5, β2 = 4/5. Therefore, ROFDM = 4.99 + 1.73 = 6.72bits/sec and RNOMA = 7.65 + 1.87 = 9.52 bits/sec. The gain of the NOMA system is 42%more than that of the OFDMA scheme.Based on the above numerical examples, it can be concluded that the sum data rate ofthe NOMA system will be improved when the channel gains increase, and the performancegain of NOMA over OFDMA increases when the channel conditions improve.The difference of two systems’s sum rate in high signal-to-noise ratio (SNR) region canbe calculated. Consider that two users are multiplexed on the same subchannel with the152.2. Energy Efficiency in Communication Networkschannel gains |h1|2 ≥ |h2|2. Therefore, the difference of sum rate can be written as [61]RNOMA −ROFDM=Bsclog2(1 +|h1|2β1pnσ2n)+Bsclog2(1 +|h2|2β2pn|h2|2β1pn + σ2n)− 12Bsclog2(1 +|h1|2pnσ2n)− 12Bsclog2(1 +|h2|2pnσ2n)→pnσ2n→∞log2(pnσ2n|h1|2β1)+ log2(1β1)− log2(pnσ2n|h1| |h2|)=log2 (|h1|)− log2 (|h2|)(2.12)which is not a function of SNR. From (2.12), it can be concluded that the sum rate gapbetween NOMA and OFDMA will be increased when the channel gain difference of the twousers is enlarged. Therefore, NOMA systems can outperform OFDMA systems only if thechannel gain difference exists.2.2 Energy Efficiency in Communication Networks2.2.1 Research Motivation of Energy EfficiencyFrom an operation point of view, approximately 600 TWh of the world wide electri-cal energy is consumed by the information and communication technology (ICT). By theend of 2030, this number is expected to grow to 1700 TWh [21]. This increasing energyconsumption becomes an important issue in the green cellular network because of the in-creasing amount of CO2 emission levels caused by energy consumption [22–25]. Therefore,the fast growing energy consumption and limited global energy resources are the importantmotivations for the research on energy efficient wireless communication systems.162.2. Energy Efficiency in Communication Networks2.2.2 Energy Efficiency Definition in Wireless CommunicationsNetworksDuring the past decades, many research works have been conducted to improve sys-tem throughput. However, with the exponential growth of wireless data traffic, energyconsumption of wireless networks has been rapidly increasing. Therefore, finding the trade-off of high data rate and energy saving is an urgent task in the next generation wirelesscommunication systems.Energy efficiency is commonly defined as the ratio of data rate to the power consump-tion. Bits per Joule is commonly used to measure energy efficiency performance in wirelessnetworks [62, 63]. For energy-efficient communication, it is desirable to send the maximumamount of data with a given amount of energy. With bandwidth B, the achievable data rateR = B log(1 + P |h|2σ2n), where P is the transmit power, σ2n is the AWGN power and |h|2 isthe channel power gain between transmitter and receiver. Given an amount of energy ∆Ethat is consumed in a duration ∆T , we have ∆E = P∆T . Therefore, the energy efficiency(EE) is defined asEE =R∆T∆E=RPbits/Joule. (2.13)The power consumption includes transmit power and circuit power consumption. Thecircuit power consumption is the additional device power consumption, which includessignal processing and active circuit blocks such as analog-to-digital converter, digital-to-analog converter, synthesizer, and mixer during the transmission [64]. Denote the additionaldevice power consumption, circuit power, as Pc. Thus, the overall power assumption isP + Pc. Energy efficiency needs to be redefined as data rate bits/s per unit energy, wherean additional circuit power factor, Pc, needs to be taken into consideration. Therefore, theenergy efficiency is defined asEE =R∆T∆E=RP + Pcbits/Joule. (2.14)172.2. Energy Efficiency in Communication NetworksNote that the circuit power consumption Pc is independent of the transmit power.In this thesis, we consider two definitions of system energy efficiency. In Chapter 3, thesystem energy efficiency is formulated as a summation of each subchannel energy efficiency.In Chapter 4, the system energy efficiency is formulated as the ratio of system sum rateto the total power consumption. Different resource allocation schemes are proposed toimprove the system energy efficiency. Regardless different definitions of system energyefficiency, our research conclusion will be the same. By resource allocation, the energyefficiency of a NOMA system can be made higher than that of an OFDMA system.2.2.3 Resource Optimization and Convex Optimization ApproachesResource management plays an important role to improve the energy efficiency in wire-less communication systems. The main resource management in wireless communicationsis frequency, time and power optimization. There are different mechanisms for resourcemanagement in wireless networks. The most important ones include congestion control,routing, subchannel allocation and power control [65]. In this thesis, we mainly focus onsubchannel allocation (user scheduling) and power allocation to maximize the system en-ergy efficiency in NOMA networks. Subchannel allocation means that the schedular needsto assign different users to different subchannels. Since different subchannels have differentgains, different subchannel allocation schemes can achieve different performance. Powerallocation means that the schedular needs to allocate different powers to the users, whichcan also achieve different performance gains. One of the most common and effective math-ematical tools to solve the resource allocation problem in wireless communication networksis the convex optimization method.182.3. SummaryLet us consider a standard form convex problem:minxf0(x)s.t. fi(x) ≤ 0, i = 1, 2, ...,mhi(x) = 0, i = 1, 2, ..., px ∈ C.(2.15)Equation (2.15) describes the problem of finding an x that minimizes f0(x) among all xvalues that satisfy the conditions fi(x) ≤ 0, i = 1, 2, ...,m, hi(x) = 0, i = 1, 2, ..., p andx ∈ C. We call x ∈ C the optimization variable and f0 the objective function or costfunction. fi(x) and hi(x) are the inequality and equality constraint functions, respectively,and C is the constraint set. The domain of the objective and constraint functions are definedasD =m⋂i=0dom fi ∩m⋂i=0dom hi ∩ C. (2.16)The convexity of this problem can be proved by the following conditions [66]. Firstthe objective function f0(x) should be convex. Second the inequality constraint functionsfi (i = 1, 2, ...,m) and equality constraint functions hi (i = 1, 2, ..., p) should be convex.Thus, it is proved that the problem (2.15) is convex, and we can find a global optimalsolution x∗ ∈ D to this problem by using standard algorithms from convex optimizationtheory [66], e.g, interior point method and sequential quadratic programming.2.3 SummaryIn this chapter, we presented the essential technical background knowledge on theNOMA system. Compared with the OFDMA system, the performance gain of the NOMAsystem is presented. Moreover, energy efficiency aspect in wireless communications wasprovided and the convex optimization approach was briefly introduced.19Chapter 3Energy Efficient ResourceAllocation for Downlink NOMANetwork with Perfect CSIIn this chapter, we aim to optimize subchannel assignment and power allocation to maxi-mize the energy efficiency for the downlink NOMA network. Assuming perfect knowledge ofthe channel state information at base station, we propose a low-complexity suboptimal algo-rithm that includes energy-efficient subchannel assignment and power proportional factorsdetermination for subchannel multiplexed users. We also propose a novel power allocationacross subchannels to further maximize energy efficiency. Since both optimization problemsare non-convex, difference of convex programming is used to transform and approximate theoriginal non-convex problems to convex optimization problems. Solutions to the resultingoptimization problems can be obtained by iteratively solving the convex subproblems. Sim-ulation results show that the NOMA system equipped with the proposed algorithms yieldsmuch better sum rate and energy efficiency performance than the conventional orthogonalfrequency division multiple access scheme.3.1 System ModelFigure 3.1 shows a downlink NOMA network. A BS transmits its signals to M userterminals (UTs) through N subchannels, and SIC is employed at the receiver of UTs.We denote n as index for the nth subchannel where n ∈ {1, 2, · · · , N} and denote m203.1. System ModelUTUTUTDLDLDLUTUTUTDLDLUTDLDLUTDLBSFigure 3.1: System model of a downlink NOMA single-cell network.as index for the mth mobile user where m ∈ {1, 2, · · · ,M}. In the cell, M users areuniformly distributed in a circular region with radius R. The total bandwidth of the system,BW , is equally divided into N subchannels where the bandwidth of each subchannel isBsc = BW/N . Let Mn ∈ {M1,M2, · · · ,MN} be the number of users allocated on thesubchannel n (SCn) and the power allocated to the lth user on SCn is denoted by pl,n.Then, the subchannel and BS power constraints are given byMn∑l=1pl,n = pn andN∑n=1pn = Ps,where pn and Ps are, respectively, the allocated power on SCn and the total transmittedpower of the BS. In NOMA systems, we assume that the BS has full knowledge of thechannel state information. According to the NOMA protocol [26], multiple users can beallocated to the same subchannel with SIC technique. A block fading channel is consideredin the system model, where the channel fading of each subchannel remains the same, but itvaries independently across different subchannels. Based on the parameters and constraintsof the system, the BS needs to assign multiple users (with different power levels) to differentsubchannels and allocate different powers across subchannels. Considering Mn users areallocated on SCn, the symbol transmitted by the BS on each subchannel SCn can beexpressed asxn =Mn∑i=1√pi,nsi (3.1)213.1. System Modelwhere si is the modulated symbol of the ith user on SCn, which is denoted by UTi,n1. Thereceived signal at the lth user on SCn isyl,n = hl,nxn + zl,n =√pl,nhl,nsl +Mn∑i=1,i 6=l√pi,nhl,nsi + zl,n (3.2)where hl,n = gl,n · PL−1(d) is the coefficient of SCn from the BS to UTl,n, and where gl,nis assumed to have Rayleigh fading channel gain, and PL−1(d) is the path loss functionbetween the BS and UTl,n at distance d. The impact of users’ channel conditions onthe performance gain of NOMA over OFDMA was studied in [61]. In this work, theauthors presented that the performance gain of the NOMA over OFDMA will increasewhen the difference of channel gain of users become larger. The authors in [37] showedthat the distances between BS and UTs will affect the performance of the NOMA system.In this paper, we assume these distances of different users are known by BS. Let zl,n ∼CN (0, σ2n) be the additive white Gaussian noise with mean zero and variance σ2n. In adownlink NOMA network, each subchannel can be shared by multiple users. Each user onSCn receives its signals as well as interference signals from the other users on the samesubchannel. Therefore, without SIC at receiver, the received signal-to-interference-plus-noise ratio (SINR) of the lth user on the SCn is written bySINRl,n =pl,n|hl,n|2σ2n +Mn∑i=1,i 6=lpi,n|hl,n|2=pl,nHl,n1 +Mn∑i=1,i 6=lpi,nHl,n(3.3)where σ2n = E[|zl,n|2] is the noise power on SCn and Hl,n , |hl,n|2/σ2n represents the channelresponse normalized by noise (CRNN) of the lth user on SCn. Based on the Shannon’scapacity formula, the achievable sum rate of SCn is written byRn = BscMn∑l=1log2 (1 + SINRl,n) = BscMn∑l=1log2(1 +pl,nHl,n1 + Il,n)(3.4)1Without causing notational confusion, UTi,n denotes the ith user on SCn, while UTm denotes the mthuser in the cell, where m ∈ {1, 2, · · · ,M}.223.1. System Modelwhere Il,n is the interference that UTl,n receives from the other users on the SCn, whichcan be expressed asIl,n =Mn∑i=1,i 6=lpi,nHl,n. (3.5)In NOMA systems, the SIC process is implemented at UT receiver to reduce the inter-ference from the other users on the same subchannel. The optimal decoding order for SIC isthe increasing order of CRNNs. Based on this order, any user can successfully and correctlydecode the signals of the other users with smaller CRNN values. Thus, the interferencefrom the users having poorer channel condition can be cancelled and removed by the userwho has better channel condition. In order to maximize the sum rate of SCn, the NOMAprotocol allocates higher power to the users with lower CRNN [26], i.e., for two users UTi,nand UTj,n sharing the same SCn with CRNNs |Hi,n| ≥ |Hj,n| 2, we always set pi,n ≤ pj,n toguarantee the weak user’s communication quality. This assumption is widely used in theNOMA scheme [40, 41]. Consider that Mn users are allocated on SCn with CRNNs order|H1,n| ≥ |H2,n| ≥ · · · ≥ |Hl,n| ≥ |Hl+1,n| ≥ · · · ≥ |HMn,n| . (3.6)According to the optimal SIC decoding order, User l can successfully decode and removethe interference symbols from users i > l. However, the interference symbol from User i(i < l) cannot be removed and will be treated as noise by User l. Therefore, the SINR ofUser l with SIC at receiver can be written asS˜INRl,n =pl,nHl,n1 +l−1∑i=1pi,nHl,n. (3.7)2Without causing notational confusion, hl,n is the channel gain of UTl,n, while Hl,n , |hl,n|2/σ2n denotesthe channel response normalized by noise of UTl,n233.2. Problem FormulationThen the data rate of the lth user on SCn can be expressed asRl,n (pl,n) = Bsclog21 + pl,nHl,n1 +l−1∑i=1pi,nHl,n . (3.8)Therefore, the overall sum rate of the NOMA system can be written asR =N∑n=1Mn∑l=1Rl,n (pl,n) =N∑n=1Rn (pn) . (3.9)3.2 Problem FormulationIn this section, we formulate the energy-efficient subchannel assignment and powerallocation as an optimization problem. For energy-efficient communication, it is desirableto maximize the amount of transmitted data bits with a unit energy, which can be measuredby energy efficiency. For each subchannel in the NOMA system, given assigned power pnon SCn and additional circuit power consumption Pc, the energy efficiency over SCn isdefined asEEn =RnPc + pn. (3.10)Then the overall energy efficiency of the system can be given byEE =N∑n=1EEn. (3.11)For the downlink NOMA network, SIC technique is well investigated in [17, 20]. Theimplementation complexity of SIC at the receiver increases with the maximum numberof the users allocated on the same subchannel. In order to keep the receiver complexitycomparatively low, we consider a simple case where only two users are allocated on the samesubchannel. This assumption is important because it also restricts the error propagation.243.3. Subchannel AllocationIn this case, given that the two users sharing SCn with CRNNs |H1,n| ≥ |H2,n|, the sumrate of SCn can be expressed asRn (pn) = Bsclog2 (1 + βnpnH1,n) +Bsclog2(1 +(1− βn) pnH2,n1 + βnpnH2,n)(3.12)where βn is the power proportional factor for the two users on SCn. Generally, βn is usedfor the user who performs SIC on SCn and βn ∈ (0, 1). The optimal power proportionalfactor can be decided within our proposed subchannel assignment scheme. To obtain anenergy-efficient resource allocation scheme for this system, we formulate the energy effi-ciency optimization problem asmaxpn>0N∑n=1Rn (pn)Pc + pn(3.13)subject to C1 : Rl,n(pn) ≥ RminC2 :N∑n=1pn = Ps(3.14)where C1 guarantees user minimum data rate constraint and Rmin is denoted as minimumdata rate determined by quality of service (QoS) requirement. The constraint C2 ensuresthe maximum BS power constraint. Since this optimization problem is non-convex andNP-hard, it is challenging to find the global optimal solution within polynomial time. Tosolve this problem efficiently, we will treat subchannel assignment and subchannel powerallocation separately.3.3 Subchannel AllocationIn this section, we investigate the energy-efficient matching algorithm for subchannelassignment in the NOMA network. For the optimization problem (3.13), it can be shownthat the subchannel assignment and power allocation for subchannels are coupled with eachother in terms of energy efficiency. Due to the considerable complexity of global optimumsolution, we decouple subchannel assignment and power allocation to obtain a suboptimal253.3. Subchannel Allocationsolution. We first propose a greedy subchannel-user matching algorithm by assuming equalpower is allocated on each subchannel, in which each power proportional factor βn is alsodetermined to allocate different powers to the multiplexed users on the same subchannel.We define the parameter βn as the proportional factor of assigned power to the user whoperforms SIC on SCn. By decomposing the objective function into difference of convexfunctions, the suboptimal matching scheme for subchannel assignment is decided by a DCprogramming approach.3.3.1 Subchannel Matching Problem FormulationTo describe the dynamic matching between the users and the subchannels, we considersubchannel assignment as a two-sided matching process between the set of M users andthe set of N subchannels. Considering only two users can be multiplexed on the samesubchannel due to the complexity of decoding, we assume M = 2N [26]. We say UTm andSCn are matched with each other if UTm is allocated on SCn. Based on the perfect channelstate information, the preference lists of the users and subchannels can be denoted byPF UT = [PF UT (1), · · · , PF UT (m), · · · , PF UT (M)]TPF SC = [PF SC(1), · · · , PF SC(n), · · · , PF SC(N)]T(3.15)where PF UT (m) and PF SC(n) are the preference lists of UTm and SCn, respectively.We say UTm prefers SCi to SCj if UTm has higher channel gain on SCi than that on SCj ,and it can be expressed asSCi (m)  SCj (m) . (3.16)As an example, we consider four users and two subchannels with the following channel gainmatrixH =[0.197, 0.778; 0.437, 0.143; 0.322, 0.545; 0.272, 0.478]263.3. Subchannel Allocationwhere row index denotes the users and column index denotes the subchannels. Therefore,we have the preference list of the users asPF UT (1) =[2 1]T, PF UT (2) =[1 2]TPF UT (3) =[2 1]T, PF UT (4) =[2 1]Tand the preference list of the subchannels asPF SC(1) =[2 3 4 1]TPF SC(2) =[1 3 4 2]T.We say SCn prefers user set qm to user set qn (qn, qm is denoted as subsets of {1, 2, · · · ,M})if the users in set qm can provide higher energy efficiency than users in set qn on SCn, andwe represent this scenario asEEn (qm) > EEn (qn) , qm, qn ⊂ {UT1, UT2, · · · , UTMn} . (3.17)Matching theory has been studied in [67, 68], where various properties and types of pref-erences have been discussed. Based on the preference lists of users and subchannels, thesubchannel assignment problem is formulated as a two-sided matching problem [67, 68].Definition 1 : (Two-sided Matching) Consider users and subchannels as two disjointsets, M = {1, 2, · · · ,M} and N = {1, 2, · · · , N}. A two-to-one, two-sided matching M isa mapping from all the subsets of users M into the subchannel set N satisfying UTm ∈Mand SCn ∈N1) M(UTm) ∈N .2) M−1(SCn) ⊆M .3) |M(UTm)| = 1, |M−1(SCn)| = 2.4) SCn ∈M(UTm)⇔ UTm ∈M−1(SCn).273.3. Subchannel AllocationCondition 1) states that each user matches with one subchannel, and Condition 2) repre-sents each subchannel can be matched with a subset of users. Condition 3) states that thenumber of users can be allocated on each subchannel is limited to two. Condition 4) meansthat UTm and SCn are matched with each other.Definition 2 : (Preferred Matched Pair) Given a matching M that UTm /∈ M−1(SCn)and SCn /∈ M(UTm). If EEn (Snew) > EEn(M−1(SCn)) where Snew ⊆ {UTm} ∪ S andS =M−1(SCn), where S is the user set that has been assigned to SCn, Snew becomes thepreferred users set for subchannel n and (UTm, SCn) is a preferred matched pair. Basedon the above definition, we will describe in Section 3.3.2 the matching action between theusers and the subchannels. If each subchannel has to select the best subset of users toallocate, it will cause considerable complexity especially when the number of users is large.Because the optimal solution requires to search all the possible combinations of the usersto maximize energy efficiency. To reduce the complexity, a suboptimal matching algorithmis proposed for subchannel assignment as follows.3.3.2 Suboptimal Matching for Subchannel Assignment Algorithm inNOMAIn this subsection, we propose a suboptimal matching algorithm for subchannel assign-ment. The main idea of this matching model is that each user sends the matching requestto its most preferred subchannel according to its preference list. This preferred subchannelhas the right to accept or reject the user according to energy efficiency that the all userscan provide on this subchannel. Based on the equal power allocation across subchannels,the user selection algorithm is a process of finding the preferred matching pair for each userand subchannel.Algorithm 1 describes the proposed low-complexity suboptimal matching scheme for asubchannel assignment (SOMSA) scheme to maximize the system energy efficiency. Thisalgorithm includes initialization and matching procedures. In the initialization step, prefer-ences lists of subchannels and users are decided according to the channel state information,283.3. Subchannel AllocationAlgorithm 1: Suboptimal Matching for Subchannel Assignment1: Initialize the matched list SMatch(n) to record users matched on SCn for all thesubchannels ∀n ∈ {1, 2, · · · , N}.2: Initialize preference lists PF UT (m) and PF SC (n) for all the users∀m ∈ {1, 2, · · · ,M} and all the subchannels ∀n ∈ {1, 2, · · · , N} according to CRNNs.3: Initialize the set of unmatched users SUnMatch to record users who has not beenallocated to any subchannel.4: while {SUnMatch} is not empty do5: for m = 1 to M do6: Each user sends matching request to its most preferred subchannel nˆ according toPL UT (m).7: if |SMatch (nˆ)| < 2 then8: Subchannel nˆ adds user m to SMatch (nˆ), and removes user m from {SUnMatch}9: end if10: if |SMatch (nˆ)| = 2 then11: a) Find power proportional factor βn for every two users in Sqm ,Sqm ⊂ {Smatch(nˆ),m} by using (3.18), or exhaustive search method or DCprogramming algorithm in Section 3.4.1.12: b) Subchannel nˆ selects a set of 2 users Sqm satisfying maximum energyefficiency Enˆ(qm) ≥ Enˆ(qn), qm, qn ⊂ {Smatch(nˆ),m}.13: c) Subchannel nˆ sets Smatch(nˆ) = qm, and rejects other users. Remove theallocated users from {SUnMatch}, add the unallocated user to {SUnMatch}.14: d) The rejected user removes subchannel from their preference lists.15: end if16: end for17: end while293.3. Subchannel Allocationand SMatch(n), ∀n ∈ {1, 2, · · · , N} and SUnMatch are initialized to record the allocatedusers on SCn and unallocated users of the system, respectively. In the matching procedure,at each round, each user sends the matching request to its most preferred subchannel. Ac-cording to the preferred list of each user (PF UT (m), ∀m ∈ {1, 2, · · · ,M}) which is a listof subchannels ordered by decreasing channel gains, the mth user will find the first non-zero entry in PF UT (m) and send matching request to the corresponding subchannel. Thesubchannel accepts the user directly if the number of allocated users on this subchannel isless than two. When the number of the allocated users equals to two, only the subset ofusers that can provide higher energy efficiency will be accepted or it will be rejected. Thismatching process will terminate when there is no user left to be matched. After that, theallocated user and the corresponding subchannels in the preference list are set to zero. Theproposed SOMSA converges to a stable matching after a limited number of iterations [68].3.3.3 Power Ratio Factor DeterminationIn Algorithm 1, it is required to determine the power proportional factor βn for everytwo subchannel users. In this section, we will first review the existing fractional transmitpower allocation scheme and the exhaustive searching method. Then we will introduce anew energy-efficient power allocation algorithm based on DC programming in Section 3.4.1.It will be shown in the simulation results that the new algorithm can result in improvedenergy efficiency.Fractional transmit power allocationAccording to the SINR expression in (3.7), the transmit power allocation to one useraffects the achievable sum rate as well as the energy efficiency on each subchannel. Due toits low computational complexity, FTPA is widely adopted in OFDMA systems and NOMAsystems [20, 26]. In the FTPA scheme, the transmit power of UTm on SCn is allocated303.3. Subchannel Allocationaccording to the channel gains of all the multiplexed users on SCn, which is given aspl,n = pnHl,n−αMn∑i=1Hi,n−α(3.18)where α (0 ≤ α ≤ 1) is a decay factor. In the case α = 0, it corresponds to equal powerallocation among the allocated users. From (3.18), it is clear that when α increases, morepower is allocated to the user with poorer CRNN. Note that the same decay factor shouldbe applied to all subchannels and transmission times.Exhaustive searching methodIn finding power proportional factor βn, the method of exhaustion can also be exploitedfor βn ∈ (0, 1). The optimal value can be found through searching all βn values in (0, 1)using a sufficiently small step size. Therefore, the optimal power proportional factors forthe multiplexed users can be obtained. However, the computational complexity of theexhaustion method is much higher than FTPA. Therefore, in the following, we considera suboptimal but efficient DC programming to allocate power among multiple users tomaximize the energy efficiency.3.3.4 Complexity AnalysisThe optimal subchannel assignment scheme can only be obtained by searching over allpossible combinations of the users and selecting the one that maximizes the system energyefficiency. If we have M users and N subchannels (M = 2N). The scheduler needs tosearch (2N)!2Ncombinations. The time complexity of exhaustive searching is O( (2N)!2N). Inorder to compare the complexity of different algorithms, we take natural logarithm of thecomplexity. The logarithm complexity is O(ln((2N)!) −N) = O(ln((2N)!)). By using theStirling’s formula, ln(n!) = n lnn−n+O(ln(n)), the logarithm complexity of the exhaustivesearching can be written as O(N lnN). In the SOMSA algorithm, the complexity of theworst case is O(N2). Taking natural logarithm of the complexity, the logarithm complexity313.4. Power Allocationis O(lnN). Since O(lnN) < O(N lnN) and actual complexity of SOMSA is much less thanthe complexity of the worst case, the complexity of SOMSA algorithm is much less thanthe optimal subchannel assignment scheme. It can be shown that for a small number ofusers (M = 4), the SOMSA will yield the identical results from the exhaustive search.3.4 Power AllocationAs mentioned in Section 3.3, equal power allocation is assumed across subchannels inSOMSA. In order to further improve the energy efficiency of the NOMA system, we considerto obtain the energy-efficient subchannel power allocation instead of equal power allocation.In this section, we introduce the DC programming approach and discuss its application infinding power proportional factors as well as power allocation across subchannels.3.4.1 DC ProgrammingDC programming approach has been studied recently to solve non-convex optimizationproblems [69]. It is shown that DC programming can be applied if the objective function canbe written as a minimization of a difference of two convex functions, which is representedasminx∈χ q (x) = f (x)− g (x) (3.19)where x = [x1, x2, · · ·xL]T and χ is a convex set; f (x) and g (x) are continuous, convex orquasi-convex [69]. In general, the problem defined in (3.19) is non-convex. However, it canbe solved suboptimally by using Algorithm 2. The key idea of Algorithm 2 is to convert anon-convex problem to convex subproblems by using successive convex approximations. Inthis algorithm, ε is the difference tolerance and the term −g (x) in the objective function(3.19) is replaced by −g (x(k))−∇gT (x(k)) (x− x(k)) in (3.20). The convex optimizationproblem in (3.20) can be solved by using standard algorithms from convex optimizationtheory [66, 70, 71], i.e., interior point method and sequential quadratic programming.323.4. Power AllocationAlgorithm 2: Iterative, Suboptimal Solution for DC Problems [70]Initialize x(0), set iteration number k = 0.while∣∣q (x(k+1))− q (x(k))∣∣ > ε doDefine convex approximation of q(k) (x) asqˆ(k) (x) = f (x)− g(x(k))−∇gT(x(k))(x− x(k))(3.20)Solve the convex problemx(k+1) = arg minx∈χqˆ(k) (x) (3.21)k ← k + 1end whileThe convergence of Algorithm 2 can be easily proved byq(x(k))= qˆ(k)(x(k))≥ qˆ(k)(x(k+1))≥ q(x(k+1))(3.22)where q(x(k))= qˆ(k)(x(k))is the kth iteration step, and qˆ(k)(x(k)) ≥ qˆ(k) (x(k+1)) can beobtained by (3.21). Therefore, q(x(k))monotonically decreases when k increases. Underan additional assumption that f (x) and g (x) are continuous and differentiable on theconstraint set. In this case, Algorithm 2 always returns a point of q (x), which may not bethe global optimal solution [69].3.4.2 Power Proportional FactorConsidering two users UT1 and UT2 who are to be multiplexed over SCn with CRNNsH1,n ≥ H2,n. According to the principle of SIC decoding sequences, UT1 can cancel theinterfering power term of UT2, whereas UT2 treats the symbol power UT1 as noise. Theproblem of finding βn to maximize energy efficiency of SCn can be formulated asmaxβn∈(0,1)Bsclog2 (1 + βnpnH1,n)Pc + pn+Bsclog2(1 +(1−βn)pnH2,n1+βnpnH2,n)Pc + pn(3.23)333.4. Power Allocationwhich can be rewritten asmaxβn∈(0,1)Bsclog2 (1 + βnpnH1,n) +Bsclog2(1+pnH2,n1+βnpnH2,n)Pc + pn. (3.24)In order to use the DC programming approach, we can convert (3.24) to DC representationminβn∈(0,1)− Bsclog2 (1 + βnpnH1,n)Pc + pn−Bsclog2(1+pnH2,n1+βnpnH2,n)Pc + pn(3.25)orminβn∈(0,1)(f (βn)− g (βn)) (3.26)where f (βn) = −Bsclog2(1+βnpnH1,n)Pc+pn and g (βn) =Bsclog2(1+pnH2,n1+βnpnH2,n)Pc+pn, and both terms areconvex functions with respect to βn because ∇2f (βn) > 0 and ∇2g (βn) > 0. Therefore,the DC programming approach can be used to find βn by replacing x with βn in Algorithm2.3.4.3 Subchannel Power Allocation by DC ProgrammingGiven the subchannel-user matching scheme and power proportional factors on differentsubchannels by Algorithm 1, the optimization problem in (3.13) can rewritten asmaxpn≥0N∑n=1Bsclog2 (1 + βnpnH1,n)Pc + pn +Bsclog2(1+pnH2,n1+βnpnH2,n)Pc + pn (3.27)subject to C1 : Rl,n(pn) ≥ Rmin; C2 :N∑n=1pn = Ps (3.28)where Rl,n(pl,n) is defined in (3.8). Since Rl,n(pl,n) is a linear function respect to theassigned power pn on SCn. The constraint C1 can be converted to pn > pn,min, wherepn,min is the minimum assigned power on SCn and it is determined by Rmin. Condition C2in (3.28) guarantees BS power constraint. Note that the optimization problem in (3.27) is343.4. Power Allocationnon-convex with respect to pn. However, the representation of (3.27) is similar to the DCproblem representation. Thus (3.27) can be rewritten asminpn≥0−N∑n=1Bsclog2 (1 + βnpnH1,n)Pc + pn +Bsclog2(1+pnH2,n1+βnpnH2,n)Pc + pnminpn≥0F (P)−G (P)(3.29)where P = [p1, p2, · · · , pn, · · · , pN ]T represents the allocated powers on the subchannels,andF (P) = −N∑n=1Bsclog2 (1 + βnpnH1,n)Pc + pn−N∑n=1Bsclog2 (1 + pnH2,n)Pc + pn;G (P) = −N∑n=1(Bsclog2 (1 + βnpnH2,n)Pc + pn).Problem (3.27) can be written asminP0Q (P) = minP0F (P)−G (P)subject to C1 : P  Pmin; C2 : ‖P‖1 = Ps (3.30)where Pmin = [p1,min, p2,min, · · · , pn,min, · · · , pN,min]T and P  Pmin means all the elementsin P are larger than the corresponding elements in Pmin, pn > pn,min. Proposition 1 provesconvexity of F (P) and G (P), Therefore, the DC programming approach can be appliedto realize energy-efficient power allocation using Algorithm 3. Once the power allocationover subchannels is obtained, we replace the equal power allocation with our new powerallocation scheme to achieve higher energy efficiency of the system.In Algorithm 3, ∇G(P(k))is the gradient of G(P) at the point P(k) and it is calculated353.4. Power AllocationAlgorithm 3: DC Programming Algorithm for Power Allocation across SubchannelsInitialize P(0), set iteration number k = 0. The Objective function Q (P), convexfunctions F (P) and G (P) .while∣∣∣Q(P(K+1))−Q(P(k))∣∣∣ > ε doDefine convex approximation of G(k) (P) at P(k) asQ(k) (P) = F (P)−G(P(k))−∇GT(P(k))(P−P(k))(3.31)Solve the convex problemP(k) = arg minpn≥pn,min, ‖P‖1=PsQ(k) (P) (3.32)k ← k + 1end whileby∇G(P(k))=N∑n=1Bsclog2 (1 + βnpnH2,n)− (Pc + pn) βnH2,n(1+βnpnH2,n) ln 2(Pc + pn)2 . (3.33)Since (3.32) and the power domain are convex, problem (3.33) can be solved by either theinterior point method or the sequential quadratic programming. In order to use the DCprogramming approach, the quasi-convexity of F (P) and G (P) needs to be established. Itis easy to show thatf (P) = −N∑n=1Bsclog2 (1 + βnpnH1,n)−N∑n=1Bsclog2 (1 + pnH2,n)andg (P) = −N∑n=1Bsclog2 (1 + βnpnH2,n)are convex since ∇2f (P) and ∇2g (P) are positive semi-definite matrices.Proposition 1 : If−f1 (pn) = Bsclog2 (1 + βnpnH1,n) +Bsclog2 (1 + pnH2,n)363.5. Simulation Resultsand−g1 (pn) = Bsclog2 (1 + βnpnH2,n)are strictly concave in pn, −F1 (pn) = −f1(pn)Pc+pn and −G1 (pn) =−g1(pn)Pc+pnare quasi-concavewith constant Pc. Inspired by [72], we can prove Proposition 1 as follows.Proof : Denote the α-sublevel sets of function −F1 (pn) asSα = {pn > 0| − F1 (pn) ≥ α} . (3.34)−F1 (pn) is strictly quasi-concave if and only if Sα is strictly convex for any α. In this case,when α < 0, there are no points satisfying −F1 (pn) = α. Therefore, Sα is strictly convexwhen α ≤ 0. When α > 0, we can rewrite Sα as Sα = {pn > 0|α (Pc + pn) + f1 (pn) ≤ 0}.Since f (pn) is strictly convex in pn, Sα is therefore also strictly convex. Hence, −F1 (pn)and −G1 (pn) are strictly quasi-concave. Therefore, F1 (pn) and G1 (pn) are quasi-convex.As a result, F (P) and G (P) are quasi-convex. 3.5 Simulation ResultsIn this section, Monte Carlo simulation results are presented to evaluate the performanceof the proposed resource allocation algorithms for NOMA systems. In the simulations, weconsider one base station located in the cell center and all the user terminals are uniformlydistributed in a circular range with radius of 500 m. We set the minimum distance amongall the users to be 40 m, and the minimum distance from users to BS is 50 m. Thebandwidth is 5 MHz. Let M users be randomly distributed in the cell. In NOMA systems,to reduce demodulating complexity of the SIC receiver, we consider each subchannel isonly allocated with two users. In OFDMA schemes, each user can only be assigned toone subchannel. In the simulations, we compare the performance of NOMA systems withOFDMA systems, both with resource allocation algorithms. For the subchannel power373.5. Simulation Results10 20 30 40 50 604850525456586062Number of users per BSTotal sum rate of system (Mbps)  NOMA−DCNOMA−EQOFDMAFigure 3.2: Sum rate of the system versus different number of users.383.5. Simulation Results10 15 20 25 30 35 40 45 50 55 6011.522.533.544.55x 107Number of UE per BSTotal Energy Efficiency of System (bits/Joule)  NOMA−DC−DC Pc=1wNOMA−DC−EQ Pc=1wOFDMAFigure 3.3: Energy efficiency of the system versus different number of users.allocation, we compare our proposed suboptimal algorithm with equal power allocationscheme based on our proposed subchannel assignment scheme. FTPA for multiplexed userson subchannel is also compared with our proposed algorithms. In our simulations, weset BS peak power, Ps, to be 41 dBm and circuit power consumption Pc = 1 W [73]. Themaximum number of users is 60 and σ2n =BWN N0, where N0 = −174 dBm/Hz is the AWGNpower spectral density. In the simulations, we set the value of α as 0.4 [40].In Fig. 3.2, the performance of the total sum rate is evaluated with the number of usersM (M varies from 10 to 60). We set difference tolerance ε = 0.01, and the bandwidth islimited to 5 MHz. It is shown that the total sum rate increases when the number of theusers grows. As the number of users grows larger, the sum rate continues to increase, butthe rate of growth becomes slower, as expected from the Shannon’s formula in calculating393.5. Simulation Results0 2 4 6 8 10 120102030405060Power of BS with 10 users (Watt)Total sum rate of system (Mbps)  10.6 10.75555.5  NOMA−DCNOMA−EQOFDMAFigure 3.4: Sum rate of the system versus BS power.403.5. Simulation Results0 2 4 6 8 10 1200.511.522.533.54x 107Power of BS with 10 users (Watt)Energy effiency of system (bits/Joule)  NOMA−DCNOMA−EQOFDMAFigure 3.5: Energy efficiency of the system versus the maximum BS power Ps.413.5. Simulation Resultsthe sum rate. From Fig. 3.2, we observe that the performance of NOMA system withthe proposed resource allocation algorithms, including subchannel assignment and powerallocation, is much better than the OFDMA scheme. For example, when the number ofusers is 30, the sum rate of the proposed algorithm (NOMA-DC) 3 is 12.5% more than thatof the OFDMA scheme, and the sum rate of equal power allocation (NOMA-EQ) is 11.9%more than that of the OFDMA scheme. That is because in the OFDMA scheme, eachsubchannel can only be used by one user. As a result, BS cannot fully use the spectrumresources. For different subchannel power allocation schemes, the sum rate of NOMA-DCis higher than that of NOMA-EQ.Figure 3.3 shows the energy efficiency versus the number of users with the same con-straints of Fig. 3.2. It can be observed that the energy efficiency also increases when thenumber of users grows. The trend of curve is similar to the sum rate curves due to theenergy efficiency expression. From this figure, the performance of our proposed subchan-nels and power allocation is much more energy-efficient than the OFDMA scheme. Ourproposed subchannel power allocation through the DC programming achieves better per-formance than the equal power allocation. When the number of users is 30, the energyefficiency of NOMA-DC is 33% more than that of the OFDMA scheme and 19% more thanNOMA-EQ.In Fig. 3.4, the performance of the total sum rate versus BS power with a fixed circuitpower of Pc = 1 W, the total number of users M = 10, and the BS power is from 1 W to12 W. In Fig. 3.4, the sum rate of the system increases as the BS power grows. In NOMAsystems, our proposed algorithm using DC for subchannel power allocation performs betterthan equal power allocation. Both algorithms outperform the OFDMA system.Figure 3.5 illustrates the total energy efficiency versus BS power with the fixed circuitpower of Pc = 1 W, and the number of users is M = 10, and the BS power rangesfrom 1 W to 12 W. It shows that the total energy efficiency first increases from 0 when BStransmit power increases. After the power reaches a certain level, the total energy efficiency3NOMA-DC uses DC programming to allocate power across subchannels.423.6. Summarybegins to decrease. That is because there is a tradeoff between transmission capacity andpower consumption for the energy-efficient power allocation. From Fig. 3.5, it is seen thatNOMA-DC can achieve better performance than NOMA-EQ and the OFDMA scheme. Forlarger BS power levels, NOMA-DC achieves much better performance than NOMA-EQ andOFDMA.Figure 3.6 shows the total energy efficiency versus circuit power to BS power ratioPc/Ps. The system energy efficiency decreases when the ratio Pc/Ps increases. With thefixed BS power of 12 W, the system achieves less energy efficiency when the circuit powerincreases. According to the definition of energy efficiency, its value will become smallerwhen Pc increases. However, the NOMA system equipped with the proposed resourceallocation algorithms still outperforms the OFDMA system.In Fig. 3.7, FTPA among multiplexed users with equal subchannel power scheme iscompared with the NOMA-DC, NOMA-EQ and OFDMA schemes. Fig. 3.7 shows thatthe energy efficiency increases as the number of users grows. NOMA-DC-DC4 performs thebest among those schemes. When user number is 20, the energy efficiency of NOMA-DC-DC is 35% more than that of the OFDMA scheme and 30% more than NOMA-FTPA-EQ5.NOMA-DC-EQ6achieves 12% more than the OFDMA scheme in terms of energy efficiency.3.6 SummaryBy assigning only two users to the same subchannel, we proposed energy-efficient re-source allocation algorithms for a downlink NOMA wireless network. These algorithmsinclude subchannel assignment, power proportional factors determination for multiplexedusers and power allocation across subchannels. By formulating the subchannel assignmentproblem as a two-sided matching problem, we proposed the SOMSA algorithm to maximize4NOMA-DC-DC uses DC programming approach to determine the power proportional factors and allo-cate different powers across the subchannels.5NOMA-FTPA-EQ uses FTPA to determine the power proportional factors, and equal power allocationacross the subchannels.6NOMA-DC-DC uses DC programming approach to determine the power proportional factors and equalpower allocation across the subchannels.433.6. Summary0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.511.522.53x 107Pc/Ps with 10 usersEnergy effiency of system (bits/Joule)  NOMA−DCNOMA−EQOFDMAFigure 3.6: Energy efficiency of the system versus Pc/Ps.443.6. Summary10 15 20 25 30 35 40 45 50 55 6011.522.533.544.55x 107Number of users per BSEnergy efficiency of system (bits/Joule)  NOMA−DC−DC Pc=1wNOMA−DC−EQ Pc=1wOFDMANOMA−FTPA−EQ Pc=1wFigure 3.7: Energy efficiency of the system versus different number of users.453.6. Summarythe system energy efficiency. Power proportional factors for the multiplexed users on eachsubchannel are determined by SOMSA. In the power allocation across subchannels scheme,since the objective function is non-convex, DC programming was utilized to approximatethe non-convex optimization problem into the convex subproblem. Therefore, a subopti-mal power allocation across subchannels was obtained by solving the convex subproblemsiteratively. Based on the resource scheduling from the proposed SOMSA algorithm, fur-ther improvement in the system energy efficiency was achieved by the proposed subchannelpower allocation scheme. Through extensive simulations, the performance of the proposedalgorithms for resource allocation was compared with the OFDMA system. It was shownthat the total sum rate and energy efficiency of NOMA system are both higher than theOFDMA scheme. The proposed power allocation for subchannel users outperforms theFTPA scheme. Moreover, the proposed subchannel power allocation achieves better per-formance than the equal power allocation scheme.46Chapter 4Energy Efficient ResourceAllocation for Downlink NOMANetwork with Imperfect CSIIn this chapter, we investigate energy efficiency improvement for a downlink NOMAsingle-cell network by considering imperfect CSI. The energy efficient resource schedulingproblem is formulated as a non-convex optimization problem with the constraints of out-age probability limit, the maximum power of the system, the minimum user data rateand the maximum number of multiplexed users sharing the same subchannel. Differentfrom previous works, the maximum number of multiplexed users can be greater than two,and the imperfect CSI is first studied for resource allocation in NOMA. The probabilis-tic mixed problem is first transformed into a non-probabilistic problem. To balance thesystem performance and the computational complexity, a new suboptimal user schedulingscheme is proposed to schedule users on different subchannels. Based on the user schedulingscheme, the optimal power allocation expression is derived by the Lagrangian approach. Bytransforming the fractional-form problem into an equivalent subtractive-form optimizationproblem, an iterative power allocation algorithm is proposed to maximize the system en-ergy efficiency. Finally, we present some selected numerical results to demonstrate that theproposed user scheduling algorithm closely attains the optimal performance.474.1. System Model4.1 System ModelWe consider a downlink single cell NOMA network where a BS is located in the cellcenter and M users are uniformly distributed within the cell. We define M and N asthe numbers of users and subchannels, respectively. The BS transmits its signals to Musers through N subchannels. We denote n as the index for the subchannel where n ∈{1, 2, · · · , N}. The total bandwidth B is shared by N subchannels and each subchanneloccupies a bandwidth of Bsc = B/N . According to the NOMA protocol, multiple userscan share the same subchannel. We denote the set of users on subchannel n (SCn) as Unand denote the number of users on SCn as Mn∆= |Un| , n ∈ {1, 2, · · · , N} where M = M1 +M2 + · · ·+MN . We denote m as the index of the mth user multiplexed on each subchannel.PT is denoted as the total transmitted power of the system and letN∑nMn∑mpm,n ≤ PT wherepm,n is the allocated power to the mth user (UEm) on SCn.We assume that Mn users are multiplexed on the subchannel n. The signal transmittedby the BS through SCn can be expressed asxn =Mn∑m=1√pm,nsm (4.1)where sm is the modulated symbol, which represents the transmitted symbol of UEm onSCn. We define hm,n as the channel gain from the BS to the mth user on SCn. Withoutloss of generality, we assume channel gains of the Mn users on SCn are perfectly knownand are sorted as |hMn,n| ≥ |hMn−1,n| ≥ · · · ≥ |hm,n| ≥ · · · ≥ |h1,n|. In the NOMA system,by exploiting the SIC technique at the receivers, the signal received by UEm on SCn canbe formulated asym,n = hm,nxn + zm,n= hm,n√pm,nsm + hm,nMn∑i=m+1√pi,nsi + zm,n(4.2)where hm,n = Dm,ngm,n, where Dm,n = d− 12m,n and gm,n ∼ CN (0, 1) respectively account for484.1. System Modelpath loss coefficient and the Rayleigh fading channel gain between the BS to the mth useron SCn, and where dm,n is the distance from the BS to the mth user on SCn. The noiseterm zm,n is a zero-mean complex AWGN random variable with variance σ2z .4.1.1 Imperfect Channel ModelMost previous works assumed that the BS has the whole knowledge of CSI. However,the perfect CSI is challenging to obtain in practice. In this paper, we investigate the energyefficiency optimization by assuming that the small scale fading channel is estimated at theBS. Since the path loss and shadowing are large scale fading factors and are slowly varying,we assume that the path loss and shadowing coefficients Dm,n can be estimated perfectlyby the BS [74, 75]. By using the minimum mean square error channel estimation errormodel [73, 76, 77], we can model the Rayleigh fading coefficient between the BS and themth user on SCn asgm,n = gˆm,n + em,n (4.3)where gm,n is the real Rayleigh fading channel coefficient between the BS to the mth useron SCn; gˆm,n ∼ CN (0, 1−σ2e) denotes the estimated channel gain and em,n is the estimatederror which follows a complex Gaussian distribution with mean zero and the variance σ2e .We assume that random variables gˆm,n and em,n are uncorrelated.4.1.2 Scheduled Channel Capacity and Outage CapacityIf the BS has the perfect CSI hm,n = Dm,ngm,n, according to the Shannon’s capacityformula, the maximum achievable data rate of the mth user on SCn can be written asCm,n = Bsclog2 (1 + Γm,n) (4.4)494.1. System ModelwhereΓm,n =pm,n|hm,n|2|hm,n|2Mn∑i=m+1pi,n + σ2z.(4.5)In (5), Γm,n is the signal-to-interference-plus-noise ratio for themth user on SCn; |hm,n|2Mn∑i=m+1pi,nis the interference from the users who have higher channel gains on SCn. However, in prac-tice, the BS only has the estimated fading channel coefficient gˆm,n. The data rate mayexceed the maximum achievable data rate. Therefore, in this paper, we adopt outage prob-ability as a metric to measure the performance of the case when the scheduled data rateexceeds the achievable data rate with imperfect CSI. The scheduled data rate of the mthuser on SCn can be written asrm,n = Bsclog2 (1 + Φm,n) (4.6)whereΦm,n =pm,n|hˆm,n|2|hˆm,n|2Mn∑i=m+1pi,n + σ2z(4.7)and where hˆm,n = Dm,ngˆm,n. Therefore, the system average outage sum rate is defined as[73]R(U ,P) =N∑n=1Mn∑m=1rm,n Pr [rm,n ≤ Cm,n|gˆm,n] (4.8)where Pr [rm,n ≤ Cm,n|gˆm,n] denotes the probability of the event when the scheduled datarate rm,n, which is based on the estimated channel gain gˆm,n, is less than or equal to themaximum data rate Cm,n, which is based on the real channel gain gm,n. The resource allo-cation includes user scheduling policy (U) and power allocation policy (P = {pm,n, ∀m,n}for pm,n).504.2. Problem Formulation4.2 Problem FormulationFor energy efficient communication, we aim to maximize the total communication datarate with the unit power cost. Therefore, we formulate the system energy efficiency as aratio of the system sum rate to the total power consumption. The energy efficiency of thesystem is formulated asEE(U ,P) = R(U ,P)Ps(U ,P) (4.9)where Ps(U ,P) = Pc + PT is the total power consumption of the system; Pc and PT =N∑n=1Mn∑m=1pm,n are the circuit power consumption and transmission power, respectively. Theenergy efficiency maximization problem can be formulated asmaxU ,PEE(U ,P) (4.10)s.t. C1 : Pr[Cm,n < rm,n|gˆm,n] ≤ εout, ∀m,nC2 :N∑n=1Mn∑m=1pm,n ≤ Pmax,C3 : rm,n ≥ Rmin, ∀m,nC4 : pm,n ≥ 0,∀m,nC5 : |Un| ≤ Umax,∀n(4.11)where C1 specifies the channel outage probability requirement εout; C2 is the transmissionpower limit constraint for the BS in the downlink single cell network and Pmax is themaximum transmitted power of the BS; C3 describes that each user data rate must be largerthan the minimum user data rate Rmin, which is determined by the QoS requirement; C4ensures that the power allocated to each user is positive, and C5 specifies the user numberallocated on each subchannel must be less than the maximum number of the users Umax,which can be greater than two.514.3. Methodology and Problem Solution4.3 Methodology and Problem SolutionThe objective function in (4.10) is a non-convex function with non-convex probabilisticconstraints. The global optimal resource allocation cannot easily be obtained in practicesince this problem cannot be optimally solved in polynomial time. In order to solve thisoptimization problem efficiently, we first deal with the non-convex probabilistic constraintC1 in (4.11) and transform the probabilistic mixed problem into a non-probabilistic prob-lem. An iterative resource allocation algorithm for user scheduling and power allocation isshown as Algorithm 4.4.3.1 Optimization Problem TransformationThe outage probability requirement in C1 is a complicated non-convex function of sched-uled data rate and powers. Therefore, the exact closed form expression for resource alloca-tion cannot be obtained. However, we can address this issue by the following approxima-tions. We first rewrite the scheduled data rate asrm,n = Bsclog2 (1 + Φm,n)= Bsclog2(1 +am,nbm,n) (4.12)where Φm,n =am,nbm,n= 2rm,nBsc − 1 and am,n = bm,n(2rm,nBsc − 1), and the achievable data rate isCm,n = Bsclog2 (1 + Γm,n)= Bsclog2(1 +cUm,ncDm,n)(4.13)524.3. Methodology and Problem Solutionwhere cUm,n = pm,nD2m,n|gm,n|2, cDm,n = D2m,n|gm,n|2Mn∑i=m+1pi,n + σ2z . The outage probabilitycondition can be bounded byPr[Cm,n < rm,n|gˆm,n] ≤ εout= Pr[Γm,n < Φm,n|gˆm,n] ≤ εout.(4.14)According to the total probability theorem, the outage probability can be written asPr[Γm,n < Φm,n|gˆm,n]= Pr[cUm,ncDm,n< 2rm,nBsc − 1|gˆm,n]= Pr[E1] · Pr [cUm,n ≤ am,n|gˆm,n]+ Pr[E2] · Pr [cUm,n > am,n|gˆm,n](4.15)wherePr[E1] = Pr[cUm,ncDm,n< 2rm,nBsc − 1|cUm,n ≤ am,n, gˆm,n]Pr[E2] = Pr[cUm,ncDm,n< 2rm,nBsc − 1|cUm,n > am,n, gˆm,n].(4.16)Following [73, 76, 77], we can show that the outage probability constraint C1 can be derivedfromPr[cDm,n ≥ bm,n|gˆm,n] ≤ εout/2 (4.17)andPr[cUm,n ≤ am,n|gˆm,n]= εout/2. (4.18)534.3. Methodology and Problem SolutionTo show this, we note that since bm,n = am,n/(2rm,nBsc − 1), we havePr[cDm,n ≥ bm,n|gˆm,n]= Pr[cDm,n ≥ am,n/(2rm,nBsc − 1)|gˆm,n]= Pr[am,ncDm,n≤ 2rm,nBsc − 1|gˆm,n].(4.19)Therefore, eq. (4.17) can be rewritten as Pr[am,ncDm,n≤ 2rm,nBsc − 1|gˆm,n]≤ εout2 . When cUm,n >am,n, we can always havePr[E2] = Pr[cUm,ncDm,n≤ 2rm,nBsc − 1|gˆm,n]≤ εout/2. (4.20)From (4.18), we have Pr[cUm,n > am,n|gˆm,n]= 1 − εout/2. Since Pr[E1] ≤ 1, we have from(4.15), (4.17) and (4.18)Pr[Γm,n < Φm,n|gˆm,n] ≤ εout/2 + (εout/2)(1− εout/2)= εout − ε2out/4 ≈ εout, for εout  1.(4.21)Therefore, the constraint C1 can be rewritten as (4.17) and (4.18).We now integrate the probabilistic constraints in (4.17) and (4.18) into (4.10) andtransform to a revised optimization problem as follows. By using the Markov inequality,we have [73, 77, 78]Pr[cDm,n ≥ bm,n|gˆm,n]= Pr[D2m,n|gm,n|2Mn∑i=m+1pi,n ≥ bm,n − σ2z |gˆm,n]≤E[D2m,n|gm,n|2Mn∑i=m+1pi,n]bm,n − σ2z=D2m,n|gm,n|2Mn∑i=m+1pi,nbm,n − σ2z.(4.22)544.3. Methodology and Problem SolutionAccording to (4.17), let the right side of (4.22) equals to εout/2, we haveD2m,n|gm,n|2Mn∑i=m+1pi,nbm,n − σ2z=εout2. (4.23)Since |gm,n|2 ∼ CN (gˆm,n, σ2e) is a non-central chi-squared distributed random variable withtwo degrees of freedom, the left side of (4.18) can be rewritten asPr[cUm,n ≤ am,n|gˆm,n]= Pr[pm,nD2m,n|gm,n|2 ≤ am,n|gˆm,n]= Pr[|gm,n|2 ≤ am,nD2m,npm,n|gˆm,n]=F|gm,n|2(am,nD2m,npm,n)=1−Q1√2|gˆm,n|2σ2e,√2σ2eam,nD2m,npm,n(4.24)whereQ1 (a, b) = exp(−a2 + b22) ∞∑k=0(ab)kIk (ab)is the first-order Marcum Q-function and Ik(·) is the kth order modified Bessel function ofthe first kind. Based on (4.19), let (4.24) equal to εout/2, then we haveam,n = F−1|gm,n|2 (εout/2) ·D2m,npm,n (4.25)where F−1|gm,n|2(·) is the inverse function of F|gm,n|2(·). Based on |gm,n|2 = |gˆm,n|2 + σ2e ,bm,n = am,n/(2rm,nBsc − 1), (4.23) and (4.24), we haveD2m,n|gm,n|2Mn∑i=m+1pi,nam,n/(2rm,nBsc − 1)− σ2z=Mn∑i=m+1pi,nD2m,n(|gˆm,n|2 + σ2e)F−1|gm,n|2(εout/2)·D2m,npm,n2rm,nBsc −1− σ2z=εout2.(4.26)554.4. Energy Efficient Resource Allocation SchemeTherefore, the data rate for the mth user on SCn (rm,n) with the outage probability con-straint can be written byR˜m,n = Bsclog2(1 + Φ˜m,n)(4.27)whereΦ˜m,n =εoutF−1|gm,n|2 (εout/2) ·D2m,npm,nεoutσ2z + 2D2m,n(|gˆm,n|2 + σ2e) Mn∑i=m+1pi,n. (4.28)Now, the transformed average sum rate for the entire system can be written by [59]R˜(U ,P) =N∑n=1Mn∑m=1(1− εout) R˜m,n. (4.29)The energy efficient optimization problem can be reformulated asmaxU ,PE˜E(U ,P) = R˜(U ,P)Ps(U ,P) (4.30)s.t. C1 :N∑n=1Mn∑m=1pm,n ≤ Pmax,C2 : R˜m,n ≥ Rmin, ∀m,nC3 : pm,n ≥ 0,∀m,nC4 : |Un| ≤ Umax, ∀n.(4.31)The transformed non-probabilistic optimization problem (4.30) is still non-convex. Theoptimal solution to this non-convex optimization problem (4.30) is challenging to obtainin practice. In order to solve (4.30) efficiently, we will design an iterative algorithm tomaximize the system energy efficiency.4.4 Energy Efficient Resource Allocation SchemeIn this section, we design an iterative algorithm for the energy efficient joint user schedul-ing and power allocation, as shown in Algorithm 4. This algorithm includes user scheduling564.4. Energy Efficient Resource Allocation Schemesubproblem and power allocation subproblem. In each iteration, user scheduling and powerallocation are updated iteratively until convergence. We first assume equal power allocationfor all the users. The user scheduling is optimized by the proposed Algorithm 5 which pro-vides better performance than the corresponding algorithm in [57]. Based on the obtaineduser scheduling scheme, the power allocation scheme can be updated by Algorithm 6, wherethe optimal power allocation policy can be obtained by the Lagrangian approach. An it-erative power allocation scheme is proposed and the optimal closed form power expressionis derived for each user. Due to the non-convexity of the problem in (4.30), the optimalsolution cannot be obtained in polynomial time. However, in this iterative algorithm, thesystem energy efficiency slightly improves at each iteration and it converges at the end ofthe procedure. It is observed that the constraints in (4.31) satisfy the necessary Karush-Kuhn-Tucker (KKT) conditions. Algorithm 4 finds at least one locally optimal solutionwith the potential of being a global optimum.Algorithm 4: Energy Efficient Resource Allocation Algorithm1: Initialize the power allocation for each user pm,n = Pmax/M .2: Initialize the maximum iterations Lmax, the index l = 1 and maximum tolerance ε.3: while∣∣EE(l) − EE(l−1)∣∣ > ε or l ≤ Lmax do4: 1. Given P (l), obtain user scheduling scheme U l by the proposed suboptimalalgorithm (Algorithm 5).5: 2. Update the power allocation scheme P (l+1) according to the propose powerallocation policy shown in Algorithm 6 of Section 4.4.2.6: 3. Set l = l + 1 and compute EE(l+1).7: end while4.4.1 User Scheduling Scheme DesignIn this section, we design a user scheduling scheme to assign users to different subchan-nels in order to maximize the system energy efficiency. Since the maximum number ofusers that can be multiplexed on the same subchannel is less than Umax, the global optimalsolution can only be obtained by the exhaustive search method, which has exponential com-plexity with respect to the number of subchannels. Thus we propose a novel suboptimal574.4. Energy Efficient Resource Allocation Schemeuser scheduling scheme to reduce the complexity. The user scheduling can be expressed asUoptimal = arg maxU EE(U) =R˜(U)Ps(U) . (4.32)Algorithm 5: A Novel Suboptimal User Scheduling Algorithm1: Initialize the power allocation for each user Pm,n.2: Construct the estimate channel gain H ∆= [|hˆm,n|]M×N .3: Initialize the sets Uun to record the unallocated user in the system.4: Initialize the lists for all the subchannels EE(n) to record the energy efficiency ofSCn.5: while Uunun 6= ∅ do6: Find the maximum value |hˆm,n| in H using|hˆm,n| = arg maxm∈Uun,n∈Hun(H) .7: EEn,possible=∅.8: EEn,i=∅.9: if the number of multiplexed users on this subchannel is less than Umax then10: a) Schedule the user m onto the subchannel n.11: b) Uun=Uun\UEm.12: end if13: if the number of multiplexed users on this subchannel equals Umax then14: a) Assume UEm is allocated on SCn and the user set is now Un,possible.15: b) Calculate the energy efficiency EEn possible of the Umax users from Un,possibleon SCn.16: c) Un = arg maxU∈Un,possible(EEn possible) and UEi /∈ Un.17: d) Uun=Uun\Un.18: Let the ith and nth row’s elements in H be zeros.19: Let the nth column’s elements in H be zeros.20: end if21: end whileAlgorithm 5 describes the proposed suboptimal user scheduling process. We first initial-ize the power allocation scheme. When Algorithm 5 is used for the first time, we assumeequal power allocation for each user pm,n =PmaxM . Uun is initialized to record the userswho have not been allocated to any subchannel. In the scheduling procedure, we needto find the user who has the maximum channel gain and allocate it to the corresponding584.4. Energy Efficient Resource Allocation Schemesubchannel if the number of users multiplexed on this subchannel is less than Umax. If theuser number is equal to Umax, the users should be selected from the user set Un,possible. Theusers who can provide the maximum energy efficiency on this subchannel will be allocatedon this subchannel. The user who has been selected will be returned to Uun. This processterminates if there is no user left to be allocated.The complexity of the proposed algorithm is less than the optimal exhaustive search.Since the user scheduling subproblem is non-convex, the optimal user scheduling algorithmcan only be obtained through exhaustive search. For a given power allocation scheme, thebest user scheduling scheme can be obtained to maximize the system energy efficiency bysearching all the combinations of users and subchannels. The exhaustive search dependson the maximum number of users multiplexed on each subchannel Umax. For the opti-mal solution, the BS needs to search N !((M1)+(M2)+ · · ·+ (MUmax)) combinations. Thetime complexity of exhaustive search is O(N !2M ). The performance of the proposed userscheduling algorithm is compared with the optimal solution and the existing scheme in [57].Simulation comparison is shown in Section 4.5. With the same complexity, the proposedalgorithm can achieve improved energy efficiency performance.4.4.2 Energy Efficient Power Allocation AlgorithmAfter user scheduling, the power allocation for all the users is still equal power allocationscheme. In order to further improve the system energy efficiency, a closed form optimalpower allocation expression is derived. Given the user scheduling scheme, the optimizationproblem in (4.30) is still a non-convex optimization problem with respect to pm,n. Theobjective function in (4.30) has a non-linear fractional form which makes the problemchallenging to solve. In order to reduce the complexity in solving the power allocationproblem, successive convex approximation and parameter transformation are exploited tosolve this problem. We design an iterative algorithm for power allocation in Algorithm 6.The lower bound of log2(1+q) can be expressed as αlog2q+β ≤ log2(1+q) for any q ≥ 0[79]. The bound will become tight when q = q¯, α = q¯1+q¯ and β = log2 (1 + q¯)− q¯1+q¯ log2q¯. By594.4. Energy Efficient Resource Allocation Schemeutilizing the lower bound, the lower bound of data rate for the mth user on the subchanneln can be written asR∗m,n = Bscαm,nlog2(Φ˜m,n)+ βm,n. (4.33)Therefore, the energy efficient optimization problem can be rewritten bymaxP0EE(P) =R∗ (P)Ps (P)(4.34)s.t. C1 :N∑n=1Mn∑m=1pm,n ≤ Pmax,C2 : R∗m,n ≥ Rmin, ∀m,n(4.35)where R∗ =N∑n=1Mn∑m=1(1− εout)R∗m,n, P ∆= [pm,n]Mn×N and P  0 means all elements of P arepositive. The objective function in (4.34) is non-convex. In order to avoid high complexity ofthe solution to this problem, we introduce the following parameter transformation. Withoutloss of generality, we define the maximum energy efficiency of the system ast∗ = maxP0R∗ (P)Ps (P)=R∗ (P∗)P ∗s (P∗). (4.36)Note that the problem in (4.34) is a concave-convex fractional optimization problemwhich is a nonlinear fractional program, and it can be transformed to an equivalent param-eterized non-fractional form as follows [66]maxP0R∗ (P)− tPs (P) (4.37)s.t. C1 :N∑n=1Mn∑m=1pm,n ≤ Pmax,C2 : Rm,n ≥ Rmin, ∀m,n(4.38)where t is a parameter introduced to scale the weight of Ps. For a given value t, the solutionto problem (4.37) is denoted by P and the optimal solution to (4.37) is defined as P∗. Let604.4. Energy Efficient Resource Allocation SchemeAlgorithm 6: An Iterative Power Allocation AlgorithmInitialize the maximum number of the iterations Lmax and the maximum tolerance ε.Initialize the energy efficiency t and the iteration index l = 0.while∣∣R∗ (Pl)− t∗Ps (Pl)∣∣ > ε or l ≤ Lmax do1. Given the energy efficiency t, solve the power allocation using (4.45).2. Set l = l + 1 and let tl =R∗(Pl−1)P ∗s (Pl−1).end whileus define the functionf(t)∆= maxP0{R∗ (P)− tPs (P)} . (4.39)From (4.37), f(t) is negative when t approaches infinity while f(t) is positive when t ap-proaches minus infinity. Obviously, f(t) is convex with respect to t. Hence, a proposedalgorithm can be exploited to determine the maximum energy efficiency. Therefore, solving(4.32) is equivalent to finding the maximum energy efficiency t∗, which can be achieved ifand only iff (t∗) = maxP{R∗ (P)− t∗Ps (P)}= R∗ (P∗)− t∗Ps (P∗)= 0(4.40)where P∗ is the optimal power allocation policy. Therefore, an iterative resource allocationalgorithm for power allocation can be proposed in Algorithm 6. Given the maximumiteration number and maximum tolerance, the energy efficiency improves for each iterationuntil the algorithm converges. In each iteration, the Lagrange multiplier approach is usedto solve the problem in (4.37). Appendix A proves the convergence of Algorithm 6. Giventhe energy efficiency t, the optimal power allocation for each user will be developed in thefollowing subsection.614.4. Energy Efficient Resource Allocation Scheme4.4.3 Power Allocation Expression DerivationIn Algorithm 6, given the energy efficiency t, the optimization in (4.37) is a concavemaximization problem with respect to power allocation policy P. This problem can besolved by its dual problem and the difference between the primal and dual solution is zerowhen strong duality holds [66, 80]. In this section, we solve the primal problem of (4.37)by solving its associated dual problem. The Lagrangian function can be written byL (P, λ,υ) =N∑n=1Mn∑m=1(1− εout)R∗m,n (P)− tPs (P)+λ(Pmax −N∑n=1Mn∑m=1pm,n)+N∑n=1Mn∑m=1υm,n(R∗m,n (P)−Rmin)(4.41)where λ and υ = [υm,n]Mn×N are the Lagrange multipliers corresponding to the constraintsC1 and C2 in (4.36). The constraints are KKT conditions [66] for optimizing power allo-cation, thus the dual problem of (4.37) isminλ,υg (λ,υ) (4.42)s.t. λ ≥ 0,υ  0 (4.43)whereg (λ,υ) = maxP0L (P, λ,υ) . (4.44)To solve (4.37), we decompose it into two layers: inner layer and outer layer. We first solvethe inner layer problem to obtain the power allocation policy, and then use the outer layerto compute the dual variables λ and υm,n iteratively.For a fixed Lagrange multiplier and a given energy efficiency t, the problem is a standardoptimization problem with the KKT conditions. The optimal power allocation policy for624.5. Simulation Resultsthe mth user on SCn can be derived in Appendix B aspm,n =αm,nBN (1− εout + υm,n)ln 2 (λ+ t) +m−1∑l=1A(pl,n)(4.45)whereA(pl,n) = (1− εout + υl,n)αl,n2(|gˆl,n|2 + σ2e)D2l,nΦ˜l,npl,nεoutF−1|gl,n|2 (εout/2). (4.46)Given the power allocation scheme in (4.43), the outer layer primal problem can be solvedby the gradient method since the objective function is differentiable. Therefore, the dualvariables can be updated with gradient descent asλ(l′+ 1)=[λ(l′)− ξ1 (l′)(Pmax − N∑n=1Mn∑m=1pm,n)]+(4.47)υm,n(l′+ 1)=[υm,n(l′)− ξ2 (l′) (R∗m,n −Rmin)]+,∀m,n(4.48)where l′is the iteration index. ξ1(l′)and ξ2(l′)are positive step sizes at iteration l′. Sincethe transformed problem (4.37) is concave, this guarantees that the iteration converges toan optimal solution to problem (4.37) based on appropriate step sizes.4.5 Simulation ResultsIn this section, we evaluate the performance of our proposed resource allocation algo-rithms for NOMA system through Monte Carlo simulations. The system parameters usedin our simulations are given as follows. We consider one BS located in the cell center andM users are uniformly distributed on the circular range with radius of 500 m. In the sim-ulations, we set the minimum distance among users as 40 m and the minimum distancebetween base station and users as 50 m. The total bandwidth in this system is 5 MHzthat is divided into N subchannels. In the NOMA and OFDMA systems, we assume the634.5. Simulation Results10 20 30 40 50 60 70 8022.533.544.55x 106Number of UE per BSTotal Energy Efficiency of System (bits/Joule)  EE−NOMA−DCEE−NOMA−ProposedEE−NOMA−ExistingEE−OFDMAFigure 4.1: Energy efficiency performance versus the number of users644.5. Simulation Results1 2 3 4 5 6 7 8 9 101.61.822.22.42.62.833.23.43.6x 107IterationsEnergy Efficiency (bits/Joule)  |U|=30, Algorithm 3|U|=45, Algorithm 3|U|=60, Algorithm 3Figure 4.2: Energy efficiency performance versus the number of iterations for Algorithm 6.654.5. Simulation Resultssmall-scale Rayleigh fading channels between the base station and users, and the 3GPPurban path loss model with a path loss factor of 3.76 [81] 7. In OFDMA systems, eachuser can only be assigned to one subchannel. In NOMA systems, the maximum userscan be multiplexed on the same subchannel is three. In the simulations, we compare ourproposed resource allocation algorithms for NOMA systems with a conventional OFDMAsystem. We set circuit power consumption Pc = 30 dBm and the BS peak power Pmax isfrom 10 dBm and 40 dBm. The maximum number of users is 75 and σ2z =BNN0, whereN0 = −174 dBm/Hz is the AWGN power spectral density. The maximum error tolerancefor the algorithms is set to  = 0.01.Figure 4.1 compares the proposed user scheduling scheme with the suboptimal sub-channel allocation in [57]. In this figure, we set Pmax = 41 dBm. The outage probability is0.05 and the variance of estimated error for channel gain is 0.1. Based on the equal powerallocation for users, the new proposed user scheme can achieve higher energy efficiency thanthe existing schemes in [44] and [57]. For example, when the number of users is 60, theenergy efficiency of our proposed resource allocation scheme for the NOMA system is 2%more than the subchannel algorithm in [57] and 30% more than that of the existing schemein [44] with equal power allocation.Figure 4.2 evaluates the energy efficiency performance versus the number of iterationsof Algorithm 6. We set the minimum normalized data rate normalized by bandwidth foreach user as 7 bits/s/Hz and the maximum transmit power as Pmax = 40 dBm. The outageprobability is 0.1 and the variance of estimated error for channel gain is 0.1. Based on theuser scheduling scheme shown in Algorithm 4, the convergence of the proposed iterativepower allocation algorithm (Algorithm 6) is presented with different number of users. Asobserved from Fig. 4.2, the system energy efficiency converges after 4 iterations. Thesystem with 60 users can achieve higher energy efficiency than the system with 45 and 30users.7Let D be the distance from the base station to the different users. The path loss model from the basestation to its users is PL dB = 15.3 + 37.6log10D.664.5. Simulation Results10 15 20 25 30 35 4012345678x 106Pmax (dBm)Total Energy Efficiency of System (bits/Joule)  EE−NOMA−Proposed−Var(e)=0.1EE−NOMA−Proposed−Var(e)=0.01EE−Exhaustive−Var(e)=0.1EE−Exhaustive−Var(e)=0.01Figure 4.3: Energy efficiency performance versus the number of users.674.5. Simulation Results1 2 3 4 5 6 7 8 9 101.81.922.12.22.32.42.5x 107IterationsEnergy Efficiency (bits/Joule)  |U|=60, Algorithm 1|U|=45, Algorithm 1|U|=30, Algorithm 1Figure 4.4: Energy efficiency performance versus the number of iterations for Algorithm 4.684.5. Simulation ResultsFigure 4.3 shows the proposed user scheduling comparison with the optimal user schedul-ing through exhaustive search. To evaluate the performance of the proposed user schedulingalgorithm, we adopt the equal power allocation scheme for each user and the total transmitpower for this system is set from 10 dBm to 40 dBm. Since the number of users is fixed,the energy efficiency decreases when the total transmit power consumption increases. Thisis because the transmit power consumption grows faster than the system sum rate. Asshown in this figure, the energy efficiency of the NOMA system with our proposed userscheduling scheme is close to the exhaustive search solution especially when the transmitpower is large. The complexity of the proposed user scheduling scheme is lower than theoptimal solution especially when the user number is large. When Pmax = 25 dBm, theoptimal algorithm can only achieve 1.1% more than the proposed user scheduling scheme.Figure 4.4 shows the convergence of the joint user scheduling and power allocationalgorithm for different number of users with Pmax = 40 dBm. It is observed that the overalliterative algorithm converges after 5 iterations. This result demonstrates that the energyefficiency increases when the user number increases.In Fig. 4.5, the performance of energy efficiency is evaluated versus the number of users.In our simulation, we set the channel estimation error variance as σ2e = 0.05 and the outageprobability εout as 0.1. It is observed that the system energy efficiency increases when thenumber of users increases. As the number of users grows, the energy efficiency continuesto increase, but the rate of growth becomes slower. The performance of our proposedresource allocation scheme for the NOMA system achieves higher energy efficiency thanthat of OFDMA scheme as well as NOMA system with equal power allocation scheme. Forexample, when the number of users is 30, the energy efficiency of the proposed resourceallocation scheme for the NOMA system is 38% more than that of the OFDMA schemewith equal power allocation.Figure 4.6 demonstrates the energy efficiency performance of NOMA systems with dif-ferent estimation error variances versus the number of users M . As observed in Fig. 6,the energy efficiency of the system deteriorates when the error variances increases. When694.5. Simulation Results10 20 30 40 50 60 70 8000.511.522.53x 107Number of UE per BSTotal Energy Efficiency of System (bits/Joule)  EE−NOMA−Proposed Pc=1wEE−NOMA−EQ Pc=1wEE−OFDM Pc=1wFigure 4.5: Energy efficiency performance versus BS maximum power.704.5. Simulation Results10 20 30 40 50 60 70 801.021.031.041.051.061.071.081.091.11.111.12x 108Number of UE per BSTotal Energy Efficiency of System (bits/Joule)  EE−NOMA Var(e)=0.1EE−NOMA Var(e)=0.05EE−NOMA Var(e)=0.01Figure 4.6: Energy efficiency performance versus users.714.6. Summarythe number of users is 60, the energy efficiency of the proposed resource allocation schemewith σ2e = 0.01 is 1.2% more than that with σ2e = 0.05 and is 2.2% more than that withσ2e = 0.1. Thus, as expected, the channel estimation error can degrade the energy efficiencyperformance.4.6 SummaryFor the NOMA system with imperfect CSI, we solved the resource allocation optimiza-tion problem by transforming a probabilistic mixed non-convex optimization problem to anon-probabilistic problem. A novel low-complexity suboptimal user scheduling algorithmwas proposed to maximize the system energy efficiency. Given the user scheduling scheme,we proposed an optimal power allocation scheme and derived a closed form power allocationexpression for users on each subchannel where the maximum user number can be greaterthan two. The effectiveness of the proposed scheme was verified by computer simulationsand compared to the existing scheme in terms of energy efficiency. It was shown that theenergy efficiency of the NOMA system with the proposed resource allocation scheme ishigher than the considered referential scheme as well as the OFDMA scheme.72Chapter 5Energy Efficient ResourceAllocation for NOMAHeterogeneous Networks(HetNets)Implementing NOMA in HetNets can alleviate the cross-tier interference and improvethe system throughput via resource optimization. In this chapter, we aim to maximizethe whole system energy efficiency in NOMA HetNet via subchannel allocation and powerallocation. The task of energy efficient subchannel and power allocation in both macrocell and small cells is formulated as an integer mixed nonconvex problem. An iterativealgorithm is proposed to maximize the macro cell and small cells energy efficiency. Byconsidering the cochannel interference and cross-tier interference, an iterative algorithm isproposed to solve this optimization problem. In the proposed algorithm, convex relaxationand dual decomposition approaches are exploited to find the closed form optimal powerallocation expression in each iteration. Finally, the complexity analysis and simulationresults are provided to evaluate the system energy efficiency performance.735.1. System Model and Problem FormulationMacro BSUEUEFemtocellPicocellSmall RRHOptical fiberWireless linkMUEUE7UE1UE3UE6UE2UE4UE5PowerOFDMANOMA:Superposition & power allocationFrequencyMUEMUEMUEMUEMUEMUEMUEFigure 5.1: NOMA based heterogeneous networks.5.1 System Model and Problem Formulation5.1.1 NOMA HetNet System ModelIn this system, we consider a downlink NOMA heterogeneous small cell network shownas Fig. 1, where one macro base station (MBS) is located in the center of macro cell. Weassume that M macro user equipments (MUEs) are uniformly distributed within the macrocell overlaid by S small cells, e.g., picocells and femtocells. The index for the MUE and smallcells are defined as m ∈ {1, 2, · · · ,M} and s ∈ {1, 2, · · · , S}, respectively. In each smallcell, one small base station (SBS) is located in the small cell center and U user equipments(SUEs) are uniformly distributed within the small cell. We define u ∈ {1, 2, · · · , U} asthe index of SUEs in each small cell. The MBS transmits signals to M MUEs through Nsubchannels and each small cell occupies one SC. The total bandwidth is B, which is dividedinto N subchannels and each subchannel occupies bandwidth Bsc = B/N .The maximumtransmit power for MBS is PMmax and the maximum transmit power of all small cells is PSmax.A block fading channel is adopted in this system model, where the channel fading of745.1. System Model and Problem Formulationeach subcarrier is assumed to be the same within a subchannel, but it varies independentlyacross different subchannels. In this system, we allow MUEs and SUEs to reuse these Nsubchannels in order to improve the system spectrum efficiency. The cross-tier interferencecaused by the macro cell will be effectively mitigated by applying the NOMA technique.Note that the interference between small cells is neglected due to the following two reasons:first, because different small cells will occupy different subchannels and the subchannels areindependent with each other; second, the inter-cell interference can be ignored due to thesevere wall penetration loss [16]. We assume each UE (MUE or SUE) is equipped with onesingle antenna and is connected to one BS (MBS or SBS). We assume user association to theMBS and SBSs are completed before the resource allocation, which means that the users inthe small cell are only connected to their corresponding SBS, and MUEs are connected tothe MBS. Both the MBS and SBSs have the full knowledge of the channel state informationobtained by the backhaul between the MBS and SBSs.5.1.2 Channel DescriptionAccording to the NOMA protocol, superposition coding and successive interferencecancelation are implemented in the BSs and UEs, respectively. In NOMA HetNets, eachsmall cell applies NOMA, which means that the users in the same cell can be multiplexedon the same subchannel. The MUEs share the same subchannels with the SUEs. On eachsubchannel, by applying the NOMA protocol, the users who have larger channel gain willdecode and remove the message from the users who have smaller channel gain. Denote psu,nand pm,n as the assigned power to the uth UE in the small cell s on the nth subchannel andtransmit power from MBS to MUE m on SC n. The SBS s sends messages to SUE u throughsubchannel n. Define gsu,n and hMSs,u,n, respectively, as the channel gain on subchannel n fromSBS s to SUE u and the channel gain on the link n from MBS to SUE u in SBS s. Denoteα = [αs,n]S×N as the subchannel indicator matrix for the small cells, where αs,n = 1 denotesthat subchannel n is assigned to the small cell s. Denote β = [βm,n]M×N as subchannelindicator matrix for the MUEs, where βm,n = 1 denotes that subchannel n is assigned to755.1. System Model and Problem Formulationthe MUE m. We assume that each small cell has U SUEs. Generally, the channel gains ofSUEs are sorted as |gsU,n| ≥ |gsU−1,n| ≥ · · · ≥ |gsu,n| ≥ · · · ≥ |gs1,n|. The received signal bySUE u by SBS s on subchannel n isysu,n = gsu,n√psu,nssu,n + gsu,nU∑l=u+1√psl,nssl,n +M∑m=1βm,nhMSs,u,n√pm,nsm,n + zsu,n. (5.1)The first term is the expected received signal from SBS s to SUE u; the second termis the interference from the users in the same small cell; the third term is the cross-tierinterference; zsu,n ∼ CN (0, σ2z) is the zero-mean complex additive white Gaussian noiserandom variable with variance σ2z . We assume that the MBS and SBS know the perfectCSI and are connected by wired links [83]. According to the Shannon’s capacity formula,the data rate from SBS s to SUE u on SC n isRsu,n = Bsc log2(1 + γsu,n) (5.2)whereγsu,n =∣∣gsu,n∣∣2psu,n∣∣gsu,n∣∣2 U∑l=u+1psl,n +M∑m=1βm,n∣∣hMSs,u,n∣∣2pm,n + σ2z (5.3)is the SINR of SUE u in small cell s. The total sum rate of small cells isRS =S∑s=1U∑u=1N∑n=1αs,nRsu,n. (5.4)The energy efficiency of small cell tier is defined as a ratio of the total sum rate of smallcells to the total power consumption of small cellsEES =RSPST + PSc(5.5)where PST =S∑s=1N∑n=1U∑u=1αs,npsu,n is the total transmit power consumption and PSc is thetotal circuit power consumption of the small cells.765.1. System Model and Problem FormulationIn the macro cell, we define H = [hMm,n]M×N as the channel gain on the link from MBSto MUE m on subchannel n. The interference from the small cell is also considered in thiswork. Denote fSMs,m as the channel gain between SBS s to MUE m on the same subchannel.Without loss of generality, we assume Mn users are multiplexed on the subchannel n andchannel gains are sorted as |hMMn,n| ≥ · · · ≥ |hMm,n| ≥ · · · ≥ |hM1,n|. According to theShannon’s capacity formula, the data rate of MUE m can be written asRm,n = Bsc log2(1 + γm,n) (5.6)whereγm,n =∣∣hMm,n∣∣2pm,nMn∑i=m+1βm,n∣∣∣hMi,n∣∣∣2pi,n + N∑n=1U∑u=1αs,npsu,n∣∣fSMs,m ∣∣2 + σ2z . (5.7)Therefore, the sum rate of macro cell isRM =M∑m=1N∑n=1βm,nRm,n. (5.8)The energy efficiency of the macro cell tier is defined as a ratio of the total sum rate tothe total power consumption in the macro cellEEM =RMPMT + PMc(5.9)where PMT =M∑m=1N∑n=1βm,npm,n is the total transmit power consumption of the macro celland PMc is the total circuit power consumption of the macro cell.5.1.3 Problem FormulationOur goal is to maximize the entire system energy efficiency including macro cell energyefficiency and small cell energy efficiency. For each tier, network’s energy efficiency isformulated as a ratio of system sum rate to the total power consumption. The objective775.1. System Model and Problem Formulationis to maximize summation of network tier energy efficiency. Assume the MBS and SBSshave the perfect CSI. The resource allocation is performed by the entire system under thefollowing definitions and constraints:− The total power constraint:PST =S∑s=1N∑n=1U∑u=1αs,npsu,n ≤ PSmax (5.10)PMT =M∑m=1N∑n=1βm,npm,n ≤ PMmax. (5.11)− Quality of service requirement: The SUE data rate should be guaranteed for theirbasic communication, which requires the following constraint:Rsu,n ≥ Rmin,∀s, u, n. (5.12)The MUE data rate should also be guaranteed for their basic communication, whichrequiresRm,n ≥ Rmin,∀m,n. (5.13)− Cross-tier interference constraints: The interference from SBS s to MUEs who arealso multiplexed on subchannel n. The cross-tier interference limit is constrained bya threshold Isn,thIsn =N∑n=1αs,nU∑u=1psu,n∣∣fSMs,m ∣∣2 ≤ Isn,th, ∀n. (5.14)The interference from MBS to SUE u in small cell s is also limited by a thresholdIMn,thIMn =M∑m=1βm,npm,n|hMSs,u,n|2 ≤ IMn,th, ∀n. (5.15)785.1. System Model and Problem FormulationThe energy efficiency of NOMA HetNets can be defined asEE = EES + EEM . (5.16)Therefore, the energy efficient resource allocation for a downlink NOMA HetNet systemcan be formulated asmax{αs,n},{βm,n},{psu,n},{pm,n}EE (5.17)s.t. C1 : PST ≤ PSmax;PMT ≤ PMmax,C2 : psu,n ≥ 0, ∀s, u, n; pm,n ≥ 0,∀m,nC3 : Rsu,n ≥ Rmin, ∀s, u, n;Rm,n ≥ Rmin, ∀m,nC4 : Isn ≤ Isn,th, ∀s; IMn ≤ IMn,th, ∀nC5 : αs,n ∈ {0, 1}∀s, n;βm,n ∈ {0, 1},∀m,nC6 :S∑s=1αs,n ≤ 1,∀nC7 :S∑s=1U∑u=1αs,n +M∑m=1βm,n ≤ Umax,∀n(5.18)where constraint C1 is the transmitted power limitation for all SBSs and MBS; ConstraintC2 demonstrates that the transmitted power of BS should be no less than zero; ConstraintC3 describes the heterogeneous QoS requirement that the data rata of each UE should beno less than the minimum user data rate Rmin. In constraint C4, the cross-tier interferencesfrom small cell and macro cell are limited by Isn,th and IMn,th, respectively; Constraints C5and C6 are imposed to guarantee that each subchannel can only be assigned to at most onesmall cell according to C5; Constraint C7 limits the user number on the same subchannel.795.2. Energy Efficient Resource Allocation for NOMA HetNets5.2 Energy Efficient Resource Allocation for NOMAHetNetsIt is challenging to find the global optimal solution to the problem (5.17) within poly-nomial time. To solve this problem efficiently, we first deal with macro cell subchannelallocation to MUEs with equal power allocation policy. Based on the value of βm,n, theenergy-efficient subchannel allocation and power allocation for SUEs can be iterativelysolved by Algorithm 8 where a closed form optimal power allocation expression is derivedby the Lagrangian approach. Finally, we update the power allocation for MUEs to furtherimprove the system energy efficiency shown in Algorithm 9.5.2.1 Energy Efficiency Optimization for the Entire System AlgorithmDesignAssume the MBS knows the entire knowledge of channel statement information H =[hMm,n]M×N . To reduce the complexity of the global optimal solution, we first decouple theproblem into macro cell energy efficiency maximization and small cell energy efficiency max-imization subproblems. A low-complexity suboptimal algorithm is designed to maximizethe system energy efficiency, as shown in Algorithm 7.In this algorithm, we first determine subchannel allocation for MUEs in macro cell.Equal power is allocated for all MUEs. We define a set as UMun to record the unallocatedMUEs and let it equal to 1, 2, · · · ,M . For each user, we will find subchannel n∗ who has themaximum channel gain among N subchannels. We need to check the number of MUEs onsubchannel n∗. Allocate this MUE on subchannel n∗ if the user number is less than UMmax.However, if the MUE number is equal to UMmax, this subchannel n∗ will choose the user setwho can provide the maximum energy efficiency [57]. The MUE who has not been chosenwill be put back into the set UMun. For this unallocated MUE, we will repeat the allocationprogress until it has been allocated on one subchannel. This subchannel allocation forMUEs procedure will terminate until all the MUEs has been allocated on subchannels.805.2. Energy Efficient Resource Allocation for NOMA HetNetsAlgorithm 7: An Iterative Energy Efficient Resource Allocation Algorithm forNOMA HetNets1: Initialize the power allocation for MUEs pm,n = Pmax/M .2: Initialize the sets UMun = 1, 2, · · · ,M to record the unallocated user in the system.3: while UMun is not empty do4: for m = 1 to M do5: 1. Find the subchannel n∗ who has the maximum channel gain in Hn∗ = maxnH6: if the number of MUEs on this subchannel n∗ is less than UMmax then7: a) Schedule the MUE m onto the subchannel n∗.8: b) UMun=UMun\m.9: end if10: if the number of multiplexed users on this subchannel equals Umax then11: a) Subchannel n∗ selects a set of UMmax users who can provide maximum energyefficiency.12: b) UMun=UMun\m.13: c) The unchosen MUE will go back to Step 1 and find the maximum channelgain among {1, 2, · · · , N}\n∗ repeat this step until it have been allocated onone subchannel.14: end if15: end for16: end while17: After all MUEs are allocated on subchannels, αs,n and Psu,n can be optimized for smallcells by Algorithm 8.18: Power allocation for MUEs can be updated by Algorithm 9.815.2. Energy Efficient Resource Allocation for NOMA HetNetsAfter all the MUEs have been assigned to different subchannels, we focus on subchannelallocation for small cells and power allocation for SUEs. An iterative algorithm is proposedto solve this problem, as shown in Algorithm 8. To further improve the entire system energyefficiency, another iterative algorithm is proposed to update the power allocation for MUEs,as shown in Algorithm 9 of Section 5.2.4. In both Algorithm 8 and Algorithm 9, closed formpower allocation expressions for SUEs and MUEs are derived by the Lagrangian approach.Details of the derivation can be found in Appendix C.5.2.2 Energy Efficiency Optimization for Small CellsIn this subsection, we design an iterative algorithm to allocate subchannels to smallcell and power allocation for SUEs in order to maximize the small cells energy efficiency.Based on the value of βm,n, the energy efficient resource allocation for small cells can beformulated asmax{αs,n},{psu,n}S∑sU∑u=1N∑n=1αs,nRsu,nS∑sU∑u=1N∑n=1αs,npsu,n + PSc(5.19)s.t. C1 :S∑s=1N∑n=1U∑u=1αs,npsu,n ≤ PSmax,C2 : psu,n ≥ 0, ∀s, u, nC3 : αs,nRsu,n ≥ Rmin,∀s, u, nC4 :S∑s=1U∑u=1αs,npsu,n∣∣fSMs,m ∣∣2 ≤ Isn,th,∀nC5 : αs,n ∈ {0, 1}, ∀s, nC6 :S∑s=1αs,n ≤ 1, ∀n.(5.20)This problem is mixed-integer programming due to constraint C5. The optimal solutionto this non-convex problem has an extremely high complexity. To efficiently solve this825.2. Energy Efficient Resource Allocation for NOMA HetNetsproblem, by using convex relaxation, we first relax the subchannel indication variable αs,nto be a continuous real variable in [0,1]. Since the range of αs,n is between zero and one,we could consider it as a time-sharing factor for subchannel n. It denotes the fraction oftime that small cell s occupies subchannel n during one block transmission. This relaxationwas first proposed to modify the integer mixed problem with the relaxed constraints forsubcarrier allocation in OFDM [84]. The duality gap of the relaxed problem is proved tobe zero [85].Since the problem is fractional nonlinear programming, and it can be transformed toan equivalent parameterized non-fractional form [66]. The equivalent subtractive problemwith relaxation can be formulated asmax{αs,n},{psu,n}S∑sU∑u=1N∑n=1αs,nRsu,n − t(S∑sU∑u=1N∑n=1αs,npsu,n + PSc)(5.21)s.t. C1 :S∑s=1N∑n=1U∑u=1αs,npsu,n ≤ PSmax,C2 : psu,n ≥ 0, ∀s, u, nC3 : αs,nRsu,n ≥ Rmin,∀s, u, nC4 :S∑s=1U∑u=1αs,npsu,n∣∣fSMs,m ∣∣2 ≤ Isn,th,∀nC5 : αs,n ∈ [0, 1],∀s, nC6 :S∑s=1αs,n ≤ 1, ∀n(5.22)where t is a parameter introduced to scale the weight of the total power consumption ofsmall cells. For a given value t, the solution to the problem can be denoted as {αs,n} and{psu,n}. We definef (t) , max{αs,n},{psu,n}S∑sU∑u=1N∑n=1αs,nRsu,n − t(S∑sU∑u=1N∑n=1αs,npsu,n + PSc). (5.23)835.2. Energy Efficient Resource Allocation for NOMA HetNetsIt is observed that f (t) is negative when t approaches infinity, while f (t) is positive whent approaches minus infinity. Therefore, f (t) is convex with respect to t. Define {α∗s,n} and{ps,∗u,n} are the optimal subchannel allocation policy and power allocation policy for problem(5.20). Therefore, the maximum energy efficiency t∗ can be achieved if and only iff (t∗) = RS({α∗s,n},{ps,∗u,n})− t∗ (PST ({α∗s,n},{ps,∗u,n})+ PSc ) = 0 (5.24)where the maximum energy efficiency of the small cells can be defined ast∗ =RS({α∗s,n},{ps,∗u,n})PST({α∗s,n},{ps,∗u,n})+ PSc . (5.25)For notational simplicity, we denote the actual power allocation to SUE u in small cells on subchannel n as p˜su,n = αs,npsu,n. Thus, the data rate of SUE u in small cell s onsubchannel n can be written byR˜su,n = Bsclog21 +∣∣gsu,n∣∣2p˜su,n∣∣gsu,n∣∣2 U∑l=u+1p˜sl,n + αs,nIMn + αs,nσ2z . (5.26)Then the problem can be rewritten asmax{αs,n},{psu,n}S∑sU∑u=1N∑n=1αs,nR˜su,n − t(S∑sU∑u=1N∑n=1p˜su,n + PSc)(5.27)845.2. Energy Efficient Resource Allocation for NOMA HetNetss.t. C1 :S∑s=1N∑n=1U∑u=1p˜su,n ≤ P smax,C2 : p˜su,n ≥ 0,∀s, u, nC3 : αs,nR˜su,n ≥ Rmin, ∀s, u, nC4 :S∑s=1U∑u=1p˜su,n∣∣fSMs,m ∣∣2 ≤ Isn,th,∀nC5 : αs,n ∈ [0, 1], ∀s, nC6 :S∑s=1αs,n ≤ 1, ∀n.(5.28)For a given value t, the Hessian matrix of the objective function in (5.27) with respect top˜su,n and αs,n is negative semi-definite. The objective function (5.27) is concave [66]. As theinequality constraints in (5.28) are convex, the feasible set of objective function is convex.Being a convex optimization problem, the transformed optimization problem in (5.27) hasa unique optimal solution, i.e., the local solution is the optimal solution, and it can beobtained in polynomial time. Therefore, the dual decomposition method can be used tosolve this problem. The Lagrangian function of the problem (5.27) can be written byL({αs,n} ,{psu,n} , t, λs,νs,µs,ηs)=S∑sU∑u=1N∑n=1αs,nR˜su,n − t(S∑sU∑u=1N∑n=1p˜su,n + PSc)+ λs(P smax −S∑s=1N∑n=1U∑u=1p˜su,n)+S∑sU∑u=1N∑n=1νsu,n(αs,nR˜su,n −Rmin)+N∑n=1µsn(Isn,th −S∑s=1U∑u=1p˜su,n∣∣fSMs,m ∣∣2)+N∑n=1ηsn(1−S∑s=1αs,n)(5.29)where λs ≥ 0, νs  0, µs  0, and ηs  0 are the Lagrange multipliers corresponding to855.2. Energy Efficient Resource Allocation for NOMA HetNetsthe power constraints. The Lagrangian function can be rewritten asL({αs,n} ,{psu,n} , t, λs,νs,µs,ηs)=S∑s=1N∑n=1Ls,n({αs,n} ,{psu,n} , λs,νs,µs,ηs)− tPSc + λs(PSmax)− S∑s=1N∑n=1U∑u=1νsu,nRmin +N∑n=1µsnIsn,th +N∑n=1ηn(5.30)whereLs,n({αs,n} ,{psu,n} , t, λs,νs,µs,ηs)=U∑u=1αs,nR˜su,n − tU∑u=1p˜su,n − λsU∑u=1p˜su,n +U∑u=1νsu,nαs,nR˜su,n− µsnU∑u=1p˜su,n∣∣fSMs,m ∣∣2 − ηsnαs,n.(5.31)Given by t, the dual problem of (5.27) isminλs,νs,µs,ηsg (λs,νs,µs,ηs) (5.32)s.t. λs ≥ 0,νs,µs,ηs  0 (5.33)whereg (λs,νs,µs,ηs) = max{αs,n},{psu,n}L({αs,n} ,{psu,n} , t, λs,νs,µs,ηs) . (5.34)We decompose the dual problem into two layers: inner layer and outer layer. We firstsolve the inner layer problem to obtain the subchannel allocation power allocation policyfor small cells, and then outer layer to compute the dual variables iteratively. To reducethe complexity of the optimal solutions, the lower bound is applied to achieve the optimalsolution iteratively. With lower bound [79], the data rate of SUE u in small cell s onsubchannel n can be rewritten byRˆsu,n = Bscasu,nlog2(γsu,n) + bsu,n. (5.35)865.2. Energy Efficient Resource Allocation for NOMA HetNetsAccording to (5.30), the Lagrangian dual function can be decomposed into S × Nsubproblems. We define the actual optimal power allocation to SUE u in small cell s onsubchannel n as p˜s,∗u,n = αs,nps,∗u,n. According to the KKT conditions, assume we have thechannel gains satisfying |gsU,n| ≥ |gsU−1,n| ≥ · · · ≥ |gs1,n|, the optimal power allocation forSUEs in small cell s can be derived asps,∗u,n =p˜s,∗u,nαs,n=Bscasu,n(1 + νsu,n)u−1∑l=1Bscasl,n(1 + νsl,n)(γsl,nps,∗l,n)+ ln 2(ηs + λs + µsn∣∣fSMs,m ∣∣2) (5.36)whereγsu,n =∣∣gsu,n∣∣2ps,∗u,n∣∣∣gs1,n∣∣∣2 U∑l=u+1ps,∗l,n + IMn + σ2z. (5.37)It can be observed from (5.36) and (5.37) that the power allocation policy is a fixed pointequation. Denote α = {psu,n} and P S = {psu,n} where s = 1, 2, · · · , S, u = 1, 2, · · · , U andn = 1, 2, · · · , N . According to the positivity, monotonicity and scalability of variable P S ,the power allocation can be updated by each iteration with (5.36) and (5.37).To obtain αs,n, the partial derivation of the Lagrangian can be expressed as∂Ls,n (· · · )∂αs,n= ∆s,n − ηsn< 0, αs,n = 0= 0, 0 < αs,n < 1> 0, αs,n = 1∀s, n (5.38)875.2. Energy Efficient Resource Allocation for NOMA HetNetswhere∆s,n =U∑u=1(1 + νsu,n)Bscasu,nlog2∣∣gsu,n∣∣2ps,∗u,n∣∣gsu,n∣∣2 U∑l=u+1ps,∗l,n + IMn + σ2z−U∑u=1(1 + νsu,n) Bscasu,nln 2 IMn + σ2z∣∣gsu,n∣∣2 U∑l=u+1ps,∗l,n + IMn + σ2z− (t+ λs)U∑u=1ps,∗u,n − µsnU∑u=1ps,∗u,n∣∣fSMs,m ∣∣2.(5.39)Subchannel n∗ is assigned to small cell s with the largest ∆s,n, that isα∗s,n = 1|n∗=maxn∆s,n ,∀s. (5.40)The outer layer primal problem can be solved by the gradient method since the objec-tive function is differentiable. Therefore, the dual variables can be updated with gradientdescent asλs (i+ 1) =[λs (i)− ξ1 (i)×(P smax −S∑s=1N∑n=1U∑u=1p˜su,n)]+, (5.41)νsu,n (i+ 1) =[νsu,n (i)− ξ2 (i)×(αs,nR˜su,n −Rmin)]+,∀s, u, n (5.42)µsn (i+ 1) =[µsn (i)− ξ3 (i)×(Isn,th −S∑s=1U∑u=1p˜su,n∣∣fSMs,m ∣∣2)]+, ∀n (5.43)ηsn (i+ 1) =[ηsn (i)− ξ4 (i)×(1−S∑s=1αs,n)]+,∀n (5.44)where i is the iteration index. ξ1 (i), ξ3 (i), ξ3 (i) and ξ4 (i) are positive step sizes at iterationi. Since the transformed problem (5.27) is concave, this guarantees that the iteration processconverges to an optimal solution to problem (5.27) based on appropriate step sizes.885.2. Energy Efficient Resource Allocation for NOMA HetNets5.2.3 Algorithm DesignIn this subsection, we design an iterative algorithm to obtain the energy efficient sub-channel allocation and power allocation in small cells, as shown in Algorithm 8. Note thatthe wire links between MBS and SBSs are assumed to help SBSs coordinate with MBS.In Algorithm 8, given the maximum iteration number and maximum tolerance, the en-ergy efficiency improves for each iteration until it converges. In each iteration, the Lagrangemultiplier approach is used to solve the problem (5.27). Given the energy efficiency t, theoptimal power allocation for each user and subchannel indication determination will bedeveloped iteratively until convergence.Algorithm 8: An Iterative Energy Efficient Resource Allocation Algorithm1: Initialize the maximum number of the iterations Imax and the maximum tolerance ε.2: Initialize the energy efficiency t and the iteration index i = 0.3: Initialize psu,n with a uniform power distribution among all subchannels4: Initialize subchannel allocation for small cells αs,n method in5: while |RS (α(i),P S(i))− t(i− 1) (PST (α(i),P S(i))+ PSc ) | > ε or i ≤ Imax do6: for n = 1 to N do7: for s = 1 to S do8: for u = 1 to U do9: 1. Given the energy efficiency t(i), update ps,∗u,n according to the optimalpower allocation policy eq.10: 2. Calculate ∆s,n according to eq. (5.36).11: 3. Update α∗s,n according to eq. (5.40).12: 4. Update λs, νs, µs, ηs by (5.41)-(5.44).13: end for14: end for15: end for16: Set i = i+ 1 and t(i) =RS(α(i−1),PS(i−1))PST (α(i−1),PS(i−1))+PSc.17: end while5.2.4 Macro Cell Energy Efficiency MaximizationBased on Algorithm 8, the power allocation and subchannel allocation have been de-termined for all the small cells. Now each small cell occupies one subchannel and has theinterference to the MUEs who are multiplexed on the same subchannel. However, the MUEs895.2. Energy Efficient Resource Allocation for NOMA HetNetson the same subchannel are allocated with equal power. To improve the energy efficiencyof the macro cell, we now update the power allocation of MUEs. We define the powerallocation policy of MUEs as PM = {pm,n}. The macro cell energy efficiency optimizationproblem can be formulated asmaxPMM∑m=1N∑n=1Rm,nM∑m=1N∑n=1pm,n + PMc(5.45)s.t. C1 :M∑m=1N∑n=1pm,n ≤ PMmaxC2 : pm,n ≥ 0,∀m,nC3 : Rm,n ≥ Rmin, ∀m,nC4 :M∑m=1βm,npm,n|hMSs,u,n|2 ≤ IMn,th.(5.46)For the power allocation, this optimization problem can be transformed into an equiv-alent subtractive problem with a parameter ηM [66], which is defined as the system energyefficiency based on given power allocation following by the similar steps (5.23)-(5.25). Wedefine a function of ηM function asf(ηM), max{pm,n}M∑m=1N∑n=1Rm,n − ηM(M∑m=1N∑n=1pm,n + PMc)(5.47)which is convex with respect to ηM . Therefore, the optimal energy efficiency for macro cellηM,∗ can be achieved whenf(ηM,∗)= RM({p∗m,n})− ηM,∗ (PMT (p∗m,n)+ PMc ) = 0 (5.48)with the optimal power allocation PM,∗ = {p∗m,n} for MUEs. Therefore, the transformed905.2. Energy Efficient Resource Allocation for NOMA HetNetssubtractive form problem can be written bymaxpMM∑m=1N∑n=1Rm,n − ηM(M∑m=1N∑n=1pm,n + PMc)(5.49)s.t. C1 :M∑m=1N∑n=1pm,n ≤ PMmax,C2 : pm,n ≥ 0,∀m,n,C3 : Rm,n ≥ Rmin,∀m,nC4 :M∑m=1pm,n|hMSs,u,n|2 ≤ IMu,n,th, ∀u, n.(5.50)To solve (5.49), an iterative algorithm is proposed to find the optimal power allocation forMUEs by iteratively solving the convex subproblems, as shown in Algorithm 9. By utilizingthe dual decomposition method, the Lagrangian function of problem (5.49) can be writtenbyL({pm,n} , ηM , λM ,νM ,µM)=Mn∑m=1N∑n=1Rm,n − ηM(M∑m=1N∑n=1pm,n + PMc)+ λM(PMmax −M∑m=1N∑n=1pm,n)+M∑m=1N∑n=1νMm,nRm,n −Rmin +N∑n=1U∑u=1µMu,n(IMu,n,th −M∑m=1pm,n|hMSu,n |2)=Mn∑m=1N∑n=1Lm,n({pM}, ηM , λM ,νM ,µM)− ηMPMc + λM (P smax)−M∑m=1N∑n=1νMm,nRm,n +N∑n=1µMn IMu,n,th(5.51)whereLm,n({pm,n} , ηM , λM ,νM ,µM)=(1 + νMm,n)Rm,n −(ηM + λM)pm,n −U∑u=1µMu,npm,n|hMSu,n |2(5.52)λM , νM and µM are the Lagrange multipliers corresponding to the power constraints.915.2. Energy Efficient Resource Allocation for NOMA HetNetsGiven ηM , the corresponding dual problem of (5.49) isminλM≥0,νM ,µM0max{pm,n}L({pm,n} , ηM , λM , νM , µM) . (5.53)The dual decomposition approach is exploited to solve the dual problem (5.53). Given bythe parameter ηM and fixed Lagrangian multipliers λM , νM and µM , the inner subproblemis a convex problem. Therefore, the optimal power allocation for MUEs on subchannel ncan be written bypM,∗m,n =BscaM2,n(1 + νM1,n)m−1∑l=1BscaMl,n(1 + νMl,n)(γMl,npM,∗l,n)+ ln 2(ηM + λM +U∑u=1µMu,n∣∣hMSu,n ∣∣2) (5.54)whereγMm,n =∣∣hMm,n∣∣2pm,nMn∑i=m+1∣∣hMm,n∣∣2pi,n + N∑n=1U∑u=1αs,npsu,n∣∣fSMs,m ∣∣2 + σ2z . (5.55)Given the power allocation scheme in (5.54), the outer layer primal problem can besolved by the gradient method since the objective function is differentiable. Therefore, thedual variables can be updated with gradient descent asλM (i+ 1) =[λM (i)− ζ1 (i)×(PMmax −M∑m=1N∑n=1pm,n)]+, (5.56)νMm,n (i+ 1) =[νsu,n (i)− ζ2 (i)× (Rm,n −Rmin)]+, ∀m,n, (5.57)µMu,n (i+ 1) =[µMu,n (i)− ζ3 (i)×(IMu,n,th −M∑m=1pm,n|hMSu,n |2)]+,∀u, n (5.58)where i is the iteration index; ζ1 (i), ζ2 (i) and ζ3 (i) are positive step sizes at iteration i.Based on (5.42)-(5.55), Algorithm 9 is proposed to update the power allocation forMUEs in macro cell. Since the transformed problem (5.49) is concave, this guaranteesthat the iteration process converges to an optimal solution to problem (5.49) based onappropriate step sizes.925.3. Simulation ResultsAlgorithm 9: An Iterative Energy Efficient Resource Allocation Algorithm1: Initialize the maximum number of the iterations Imax and the maximum tolerance ε.2: Initialize the energy efficiency ηM and the iteration index i = 0.3: Initialize pMm,n with a uniform power distribution among all subchannels.4: while |RM (PM (i))− ηM (i− 1) (PMT (PM (i))+ PMc ) | > ε or i ≤ Imax do5: 1. Given the energy efficiency ηM , update p∗m,n according to the optimal powerallocation policy (5.54) and (5.55).6: 2. Update ηM , λM , νM and µM by (5.55)-(5.58).7: 3. Set i = i+ 1 and ηM (i) =RM(PM (i−1))PMT (PM (i−1))+PMc.8: end while5.3 Simulation ResultsSimulation results are presented to demonstrate the effectiveness of the proposed algo-rithms. In our simulations, we assume that all users are uniformly distributed in each smallcell coverage area, and the small cells are uniformly distributed in the macro cell coveragearea. The radius of the macro cell is 300 m. The radius of each small cell is 10 m. Smallcell has a minimum distance of 50 m from the macro base station. The minimum distancebetween small cell base stations is 40 m. The path loss8 model is based on [81]. We as-sume that the shadowing standard deviation between base station and the users is 10 dB.The channel fading is composed of shadowing fading, path loss, and Rayleigh fading. Theadditive white Gaussian noise power is set as σ2=3.9811 × 10−14 W. We assume that themaximum transmit power is 40 dBm at the macro cell base station, the maximum transmitpower is 17 dBm in each small cell, and circuit power of each user is 20 dBm. The minimumdata rate of each user with unit bandwidth (Rmin) is 7 bits/s.Figure 5.1 evaluates the energy efficiency performance versus the number of iterationsof Algorithm 8. We set the minimum data rate with unit bandwidth for each user as 7bits/s and the maximum transmit power as Pmax = 40 dBm. From Fig. 5.1, we can observe8Let D be the distance from the corresponding base station to the different users, let RSBS be the radiusof each small cell, and the loss of wall L is 10 dB. The path loss models we use are listed as: 1. from smallbase station to its the small cell user, PL dB = 38.46 + 20log10D + 0.7D; 2. from small base station tomacro cell user, PL dB = max((15.3 + 37.6log10(D−RSBS)), (38.46 + 20log10(D−RSBS))) + 0.7RSBS +L;3. from macro cell base station to small base station, PL dB = 15.3 + 37.6log10D; 4. from macro cell basestation to small cell users, PL dB = 15.3 + 37.6log10D + 2L.935.3. Simulation Results1 2 3 4 5 6 7 8 9 10 11 120123456789x 109IterationsEnergy Efficiency (bits/Joule)  |SUE|=45, Algorithm 8|SUE|=30, Algorithm 8|SUE|=15, Algorithm 8Figure 5.2: Energy efficiency versus the number of iterations for Algorithm 8.945.3. Simulation Results10 20 30 40 50 60 70 802.62.833.23.43.63.844.24.4x 108Number of SBS Per MBSTotal Energy Efficiency of System (bits/Joule)   EE_NOMA_HetNet_Proposed EE_OFDM_HetNetFigure 5.3: Energy efficiency versus number of small cells with perfect CSI.955.4. Imperfect CSI Discussionthat the convergence of the proposed iterative resource allocation algorithm (Algorithm 8)converges within 10 iterations, which suggests that the proposed scheme is practical. Thesystem with 60 SUEs can achieve higher energy efficiency than the system with 45 and 15users.Figure 5.2 shows the performance of the system energy efficiency versus the numberof the small cells. The bandwidth is limited to 10 MHz, and we set the peak power ofthe entire system to be 20 W and circuit power to be 0.1 W on each subchannel. Figure5.2 indicates that the total energy efficiency increases when the number of the small cellsincreases. From Fig. 5.2, we can observe that the NOMA HetNet outperforms the OFDMheterogeneous network (OFDM HetNet) because OFDM HetNet cannot fully utilize thespectrum resources.5.4 Imperfect CSI DiscussionIn this subsection, we study the energy efficient resource allocation for the downlinkNOMA heterogeneous network with imperfect CSI. In practice, perfect CSI at the trans-mitter is difficult to achieve due to channel estimation errors, feedback and quantizationerrors. To maximize the energy efficiency, we can formulate the energy efficient resourceallocation as a probabilistic mixed non-convex optimization problem under the constraintsof outage probability limitation, the maximum transmitted power and the number of mul-tiplexed users on one subchannel.The resource allocation scheme is optimized based on the imperfect CSI. To solve thisproblem, we decouple the problem into subchannel allocation and power allocation sub-problems separately to maximize the system energy efficiency. We assume that the BSshave the estimated value of the CSI. In this situation, the user data rate may not meet theminimum data requirement determined by QoS. Therefore, an outage probability require-ment is considered for the resource scheduling to maximize the system energy efficiency.The energy efficient optimization problem is formulated as a probabilistic mixed non-convex965.4. Imperfect CSI Discussionoptimization problem. In order to efficiently solve the optimization problem, we can firsttransform the probabilistic mixed non-convex optimization problem to a non-probabilisticoptimization problem. The outage probability constraint can be transformed to minimumpower constraints for UEs sharing the same subchannel. In this transformed problem, wecan treat subchannel allocation and power allocation separately. The subchannel allocationstarts with assigning equal power allocation on subchannels. The optimal solution to theuser subchannel allocation subproblem is challenging to obtain in practice as it is requiredto search all the possible combinations. A suboptimal subchannel allocation algorithm canbe obtained by using the estimated CSI. In order to cancel the interference from the othersmall cells, we let each small cell only occupy one subchannel. MUEs will multiplex onthese subchannels, and hence the SUEs will not interfere with each other. The MUE withlarge channel gain will be chosen to multiplex on its corresponding subchannel. For theeach subchannel, we set the maximum number of multiplexed users to reduce the complex-ity of decoding at the receivers. The subchannel allocation will terminate if there is noMUE left to be allocated. During the subchannel allocation, the power allocation for themultiplexed users on one subchannel can be determined by FTPA. The complexity of theproposed suboptimal algorithm is less than the optimal solution that can only be obtainedby the exhaustive search.Given the user subchannel allocation scheme, the underlying optimization problem canbe shown to be a convex function respect to the power variable under the constraint of themaximum power of the system. Therefore, a unique optimal solution can be found by agradient assisted binary search algorithm. Therefore, the new power allocation scheme canreplace the equal power allocation to further improve the system energy efficiency.Figure 5.4 shows the energy efficiency versus the number of the small cells with thechannel gain estimation error variance 0.05. As shown, the energy efficiency increases whenthe number of the small cells grows. From Fig. 5.2, the NOMA HetNet outperforms OFDMHetNet in terms of energy efficiency.Fig. 5.5 shows the energy efficiency of the NOMA HetNet performance with different975.4. Imperfect CSI Discussion10 20 30 40 5001234567x 108Number of SBS Per MBSTotal Energy Efficiency of System (bits/Joule)   EE_OFDM_HetNet_EQ_ImperfectCSI EE_NOMA_HetNet_EQ_ImperfectCSI EE_NOMA_HetNet_FTPA_ImperfectCSIFigure 5.4: Energy efficiency of the system versus the number of the small cells with theestimation error variance 0.05.985.4. Imperfect CSI Discussion10 20 30 40 50 60 70 802.62.833.23.43.63.844.24.4x 108Number of SBS Per MBSTotal Energy Efficiency of System (bits/Joule)   EE_OFDM_HetNet_EQ_Var(error)=0.1 EE_OFDM_HetNet_EQ_Var(error)=0.01 EE_OFDM_HetNet_EQ_Var(error)=0 EE_NOMA_HetNet_FTPA_Var(error)=0.1  EE_NOMA_HetNet_FTPA_Var(error)=0.01  EE_NOMA_HetNet_FTPA_Var(error)=0Figure 5.5: Energy efficiency of the system versus the number of the small cell with differentestimation error variances.995.5. Summaryestimation error variances. When the number of small cells is 20, the energy efficiency ofNOMA HetNet with FTPA power allocation scheme is 27% more than that of the OFDMHetNet with the equal power (EQ) allocation scheme under the imperfect channel CSI.5.5 SummaryIn this chapter, we introduced the concept of NOMA HetNets and proposed an energyefficient subchannel allocation and power allocation scheme for a downlink NOMA HetNetwith perfect CSI. We solved the resource optimization problem with the help of convexoptimization. It is envisioned that a joint subchannel and power allocation approach canfurther enhance the energy efficiency of the overall system. We also discussed the energyefficiency maximization problem in the NOMA HetNet with imperfect CSI. The effective-ness of the proposed schemes was compared to the existing scheme and verified computersimulations to in terms of energy efficiency.100Chapter 6ConclusionsIn this chapter, we summarize the contributions of this thesis. Also, we present severalpotential future research topics that are related to our accomplished work.6.1 Summary of ContributionsIn this thesis, we investigated the energy efficient resource allocation design for downlinkNOMA networks. We first studied the energy efficient resource allocation for a single cellNOMA network with perfect CSI. Then we extended the energy efficient resource allocationto the imperfect CSI case. Furthermore, the entire system energy efficiency maximizationproblem was studied in a NOMA HetNet. In the end, we discussed the related future work.Here we summarize the results obtained in each chapter.Contribution 1: the energy-efficient resource allocation problem was formulated for adownlink NOMA wireless network by assigning only two users to the same subchannel. Toefficiently solve this problem, we first decoupled it into subchannel allocation and powerallocation subproblems. Matching theory was exploited by subchannel assignment subprob-lem formulation and a low-complexity suboptimal algorithm was proposed to maximize thesystem energy efficiency. To allocate powers across subchannels, DC programming wasutilized to approximate the non-convex optimization problem into the convex subproblem.Therefore, a suboptimal power allocation across subchannels was obtained by solving theconvex subproblems iteratively. The system energy efficiency was further improved by theproposed subchannel power allocation scheme. Numerical results showed that both sumrate and energy efficiency performance of the proposed resource allocation scheme outper-1016.2. Future Worksformed the OFDMA system.Contribution 2: For the NOMA system with imperfect CSI, the formulated probabilis-tic mixed non-convex resource optimization problem was first transformed into to a non-probabilistic problem. Based on the proposed low-complexity suboptimal user schedulingscheme, we proposed an iterative allocation algorithm for power allocation. In power allo-cation, we derived a closed form power allocation expression for users on each subchannelwhere the maximum user number can be greater than two. The performance of the pro-posed scheme was verified by computer simulations and compared to the existing schemein terms of energy efficiency. It was shown that the energy efficiency of the NOMA systemwith our proposed resource allocation scheme is higher than the existing scheme in [57] aswell as the OFDMA scheme.Contribution 3: the single-cell NOMA system model was extended into HetNets. Tomaximize the entire NOMA HetNet, an energy efficient subchannel allocation and powerallocation scheme was proposed for a downlink NOMA HetNet with perfect CSI. We solvedthe resource optimization problem with the help of convex optimization. The energy ef-ficiency maximization was considered on both small cells and marco cell. In small cell, ajoint subchannel and power allocation approach can enhance the system energy efficiency.The optimal power allocation scheme in macro cell was proposed to further improve theoverall system energy efficiency. Moreover, imperfect CSI case was also discussed. Simu-lation results show that our proposed resource allocation scheme for NOMA HetNets canachieve higher energy efficiency than that of conventional OMA systems. Finally, the en-ergy efficiency for NOMA HetNets with imperfect CSI was discussed, and the performancewas evaluated by the simulation results.6.2 Future WorksIn Chapter 5, we presented an energy efficient resource scheme for a downlink NOMAHetNet. Resource allocation is one of the methods to improve the energy efficiency of 5G1026.2. Future WorksNOMA networks. Small cell in HetNet can also be energy efficient because low power istypically required due to the short distance between transmitter and receiver. However,there are still many challenges and open issues in achieving energy-efficient 5G NOMAnetworks. 1) NOMA with MIMO: Not only user scheduling and power allocation, but alsobeamforming can be considered in the energy efficient resource allocation. Furthermore,different antenna configurations for macro cell and small cell in HetNet will make theproblem more complex. Finding effective ways to combine NOMA and massive MIMO is achallenging but important topic in future research works. 2) NOMA with energy harvesting:To save more energy, 5G NOMA HetNet is expected to harvest energy from solar, wind,thermoelectric, and so on. One open problem is how to manage those renewable energysource because they are intermittent over time and space. 3) NOMA with game theory:User pairing is a key research direction in energy-efficient 5G NOMA networks. Matchinggame theory can be an effective tool to optimize the user pairing of energy-efficient 5GNOMA networks. 4) NOMA with open/closed access small cells: There are three accessmodes in small cell networks: open access, closed access, and hybrid access. Different accessmodes of NOMA small cells require different energy saving methods. 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Dinkelbach, “On nonlinear fractional programming,” Management Science, vol. 13,pp. 492–498, Mar. 1967. → pages 116113Bibliography[83] V. Chandrasekhar and J. G. Andrews, “Femtocell networks: A survey,” IEEE Com-munications Magazine, vol. 46, no. 9, pp. 59–67, Sept. 2008. → pages 76[84] C. Y. Wong, R. Cheng, K. Lataief, and R. Murch, “Multiuser OFDM with adaptivesubcarrier, bit, power allocation,” IEEE Journal on Selected Areas in Communications,vol. 17, no. 10, pp. 1747–1758, Oct. 1999. → pages 83[85] W. Yu and R. Lui, “Dual methods for nonconvex spectrum optimization of multicarriersystems,” IEEE Transactions on Communications, vol. 54, no. 7, pp. 1310–1322, July2006. → pages 83114Appendix115Appendix AProof of Convergence of Algorithm6We will now prove the convergence of Algorithm 6 [82]. Based on (4.37) and (4.38), wefirst prove that f(t) is a strictly monotonic decreasing function of t, i.e., if t(1) < t(2), thenf(t(1)) > f(t(2)).Proof: Let P(1) and P(2) be the optimal power allocation schemes for f(t(1)) and f(t(2)),respectively. We havef(t(2))= maxP0{R∗ (P)− t(2)Ps (P)}= R∗(P(2))− t(2)Ps(P(2))> R∗(P(1))− t(2)Ps(P(1))> R∗(P(1))− t(1)Ps(P(1))= f(t(1)).(A.1)Let t(1) =R∗(P(1))Ps(P(1))and P be an arbitrary power allocation policy. We have f(t(1)) ≥ 0because f(t(1))= maxP0{R∗ (P)− t(1)Ps (P)} ≥ R∗ (P(1))− t(1)Ps (P(1)) = 0. Now we areready to prove the convergence of Algorithm 3. During each iteration, the energy efficiencyt increases. We assume t∗ is the maximum energy efficiency and its corresponding powerallocation scheme is P∗. Now we want to prove that the energy efficiency t is increasedto the optimal t∗ when f(t∗) = 0. First, let P(l) be the optimal power allocation schemesin the lth iteration. We assume that t(l) 6= t∗ and t(l+1) 6= t∗. Based on (38), we have116Appendix A. Proof of Convergence of Algorithm 6f(t(l))> 0 and f(t(l+1))> 0. In Algorithm 3, t(l+1) =R∗(P(l))Ps(P(l)), then we havef(t(l))= R∗(P(l))− t(l)Ps(P(l))= t(l+1)Ps(P(l))− t(l)Ps(P(l))= Ps(P(l))(t(l+1) − t(l))> 0.(A.2)Since Ps(P(l))> 0, we have t(l+1) − t(l). When the number of iterations is sufficientlylarge, we will have f(t(l))= 0 and find t∗.117Appendix BDerivation of the Optimal PowerAllocation Policy in Chapter 4For each subchannel, we assume that channel estimation has error and the users areassorted according to the estimated channel gains |hˆ1,n| ≤ · · · ≤ |hˆm,n| ≤ · · · ≤ |hˆMn,n|.When m = 1, decode the first user and let∂L (P, λ)∂p1,n=BN(1− εout + υ1,n)α1,n 1ln 21p1,n− t− λ = 0. (B.1)p1,n =(BN (1− εout + υ1,n))α1,nln 2 (t+ λ). (B.2)When m = 2, decode the second user and let∂L (P, λ)∂p2,n=BN(1− εout)α2,n 1ln 21p2,n+BN(1− εout)α1,n 1ln 2Φ˜1,n2(|gˆ1,n|2 + σ2e)D21,np1,nεoutF−1|g1,n|2 (εout/2)+BNυ2,nα2,n1ln 21p2,n+BNυ2,nα1,n 1ln 2Φ˜1,n2(|gˆ1,n|2 + σ2e)D21,np1,nεoutF−1|g1,n|2 (εout/2)− t− λ = 0.(B.3)Then we can havep2,n =α2,nBN (1− εout + υ2,n)ln 2 (t+ λ) +A(p1,n)(B.4)118Appendix B. Derivation of the Optimal Power Allocation Policy in Chapter 4whereA (p1,n) =BN(1− εout + υ1,n)α1,n2(|gˆ1,n|2 + σ2e)D21,nΦ˜1,np1,nεoutF−1|g1,n|2 (εout/2) (B.5)andΦ˜1,n =εoutF−1|g1,n|2 (εout/2) ·D21,np1,nεoutσ2z + 2D21,n(|gˆ1,n|2 + σ2e)Mn∑i=2pi,n. (B.6)When m = 3, decode the second user and let∂L (P, λ,υ)∂p3,n=BN(1− εout + υ3,n)α3,n 1ln 21p3,n+∂∂p3,n(BN(1− εout + υ2,n)α2,nlog2(Φ˜2,n))+∂∂p3,n(BN(1− εout + υ1,n)α1,nlog2(Φ˜1,n))− t− λ = 0.(B.7)We can obtainp3,n =α3,n(BN (1− εout + υ3,n))ln 2 (t+ λ) +A (p1,n) +A (p2,n). (B.8)Therefore, by deduction, the power of the mth users on SCn can be derived as (4.43).119Appendix CDerivation of the Optimal PowerAllocation {ps,∗u,n} for SUEs inChapter 5For any subchannel n, we assume that the users are assorted according to the channelgains |gs1,n| 6 |gs2,n| 6 · · · 6 |gsU,n|.To obtain p˜s1,n, we let∂Ls,n (· · · )∂p˜s1,n=αs,nBnas1,n(1 + νs1,n) 1ln 21p˜s1,n− t− λs − µsn∣∣fSMs,m ∣∣2=0.(C.1)Then we haveps,∗1,n =p˜s,∗1,nαs,n=Bnas1,n(1 + νs1,n)ln 2(t+ λs + µsn∣∣fSMs,m ∣∣2) . (C.2)To obtain ps,∗2,n, we let∂Ls,n (· · · )∂p˜s2,n=αs,nBnas1,n(1 + νs1,n) 1ln 2(−γs1,nps1,n)+ αs,nBnas2,n(1 + νs2,n) 1ln 2(1p˜s2,n)− t− λs − µn∣∣fSMs,m ∣∣2=0.(C.3)120Appendix C. Derivation of the Optimal Power Allocation {ps,∗u,n} for SUEs in Chapter 5Then we can haveps,∗2,n =p˜s,∗2,nαs,n=Bnas2,n(1 + νs2,n)Bnas1,n(1 + νs1,n)(γs1,nps,∗1,n)+ ln 2(t+ λs + µsn∣∣fSMs,m ∣∣2) (C.4)whereγs1,n =|gs1,n|2ps,∗1,n|gs1,n|2U∑l=u+1ps,∗l,n + IMn + σ2z.(C.5)Similarly, to obtain ps,∗3,n, we let∂Ls,n (· · · )∂p˜s3,n=αs,nBnas1,n(1 + νs1,n) 1ln 2(−γs1,nps3,n)+ αs,nBnas2,n(1 + νs2,n) 1ln 2(−γs2,np˜s3,n)+ αs,nBnas3,n(1 + νs3,n) 1ln 2(1p˜s3,n)− t− λs − µsn∣∣fSMs,m ∣∣2=0.(C.6)We can obtainps,∗3,n =p˜s,∗3,nαs,n=Bnas3,n(1 + νs3,n)Bnas1,n(1 + νs1,n)(γs1,nps,∗1,n)+Bnas2,n(1 + νs2,n)(γs2,nps,∗2,n)+ C(C.7)whereC = ln 2(t+ λs + µsn∣∣fSMs,m ∣∣2) . (C.8)Therefore, by deduction, the optimal power allocated on SUE u in small cell s on subchanneln can be derived as (5.35).121

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