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Energy-efficient power allocation and wireless backhaul design in heterogeneous small cell networks Liu, Hao 2016

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EnyrgyAEffiwiynt dowyr Allowution unxkirylyss Buwkhuul Dysign inHytyrogynyous gmull Wyll bytworksbyHao LiuB.Eng., Shandong University, P. R. China, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)July 2016c© Hao Liu, 2016The undersigned certify that they have read, and recommend to the College ofGraduate Studies for acceptance, a thesis entitled: Eaeegl-Efficieag Cowee Allo-cagioa aaW Wieeleff Backhahl Defiga ia Hegeeogeaeohf Small Cell Aeg-woekf submitted by Hao Lih in partial fulfilment of the requirements of the degreeof Master of Applied ScienceSupervisor, Professor (please print name and faculty/school above the line)Supervisory Committee Member, Professor (please print name and faculty/school abovethe line)Supervisory Committee Member, Professor (please print name and faculty/school abovethe line)University Examiner, Professor (please print name and faculty/school above the line)External Examiner, Professor (please print name and faculty/school above the line)(Date Submitted to Grad Studies)Additional Committee Members include:(please print name and faculty/school above the line)(please print name and faculty/school above the line)iiAvstruwtThe widespread applications of wireless services and dense devices access have trig-gered huge energy consumption. Due to the environmental and financial considerations,energy-efficient design in wireless networks has become an inevitable trend. Since themacrocell cannot satisfy the increasing data requirements of users, heterogeneous smallcell network is one of the promising techniques to provide wireless service. However,backhaul is the bottle neck in the deployment of heterogeneous small cell networks. Toaddress the challenges of backhaul design and energy efficiency, we study the energy-efficient power allocation and wireless backhaul bandwidth allocation in orthogonal fre-quency division multiple access heterogeneous small cell networks. Different from theexisting resource allocation schemes that maximize the throughput, the studied schememaximizes energy efficiency by allocating both transmit power of each small cell basestation to each user and unified bandwidth for backhauling, according to the channelstate information and the circuit power consumption. The problem is formulated as anon-convex nonlinear programming problem and then it is decomposed into two con-vex subproblems. A near optimal iterative resource allocation algorithm is designed tosolve the resource allocation problem. A suboptimal low-complexity approach is alsodeveloped by exploring the inherent structure and property of the energy-efficient de-sign. Simulation results demonstrate the effectiveness of the proposed algorithms whencompared with the existing schemes.iiidryzuwyThis thesis is based on [C1, SJ1]. My supervisor, Prof. Julian Cheng, co-authoredthe publication and supervised all my research work.dwxwrwwv Conxwrwnuw butliustionsCC. H. Liu, H. Zhang, J. Cheng, and V. C. M. Leung, “Energy efficient power allo-cation and backhaul design in heterogeneous small cell networks,” Ikhceediggl hfmae BEEE Bgmekgamihgae Chgfekegce hg Chffngicamihgl (BCC), Kuala Lumpur,Malaysia, May 23–27, 2016.dwxwrwwv Journsl butliustions :sutmittwv)eJC. H. Liu, H. Zhang, J. Cheng, and V. C. M. Leung, “Downlink energy efficiencyof power allocation and wireless backhaul bandwidth allocation in heterogeneoussmall cell networks,” submitted to BEEE Mkagl’ Pikeeell Chffng’ on February01, 2016.ivhuvly oz WontyntsStstrsut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiibrwxsuw . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivfstlw ox Contwnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vList ox Fiyurws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList ox Suronyms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList ox eymtols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiSuknowlwvywmwnts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDwviustion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiCzsptwr CL Introvuution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Organization and Contributions . . . . . . . . . . . . . . . . . . . 6Czsptwr DL Hwtwroywnwous emsll Cwll Nwtworks snv dwsouruw Msnsyw-mwnt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1 Overview of Heterogeneous Small Cell Networks . . . . . . . . . . . . . 92.2 Resource Management . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11vfSTLW aF CaNfWNfe2.2.1 Energy Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2.2 Backhaul . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.2.3 Convex Optimization based Resource Management . . . . . . . . 142.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Czsptwr EL dwsouruw Slloustion Movwliny . . . . . . . . . . . . . . . . . . C93.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Czsptwr FL Enwryy-Effiuiwnt dwsouruw Slloustion snv Tsukzsuliny . . . D54.1 Conditions of Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 Energy-Efficient Power Allocation . . . . . . . . . . . . . . . . . . . . . . 264.3 Energy-Efficient Wireless Backhaul Bandwidth Allocation . . . . . . . . 304.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Czsptwr 5L Slyoritzm Dwsiyn . . . . . . . . . . . . . . . . . . . . . . . . . EE5.1 Iterative Resource Allocation Algorithm . . . . . . . . . . . . . . . . . . 335.2 Low-Complexity Optimization Algorithm . . . . . . . . . . . . . . . . . 335.3 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Czsptwr HL Conulusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FJ6.1 Summary of Accomplished Work . . . . . . . . . . . . . . . . . . . . . . 486.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49Titlioyrspzy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5CSppwnviuws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59vifSTLW aF CaNfWNfeAppendix A: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Appendix B: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61Appendix C: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Appendix D: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66viiList oz FigurysFigure 2.1 Traffic demand in terabits for North America [1]. . . . . . . . . 10Figure 2.2 Heterogeneous small cell network. . . . . . . . . . . . . . . . . . 12Figure 2.3 Backhaul. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14Figure 2.4 Graph of a convex function. . . . . . . . . . . . . . . . . . . . . 16Figure 3.1 Topology of a heterogeneous small cell network. . . . . . . . . . 20Figure 5.1 The convergence in terms of energy efficiency of all small cellusers over the number of iterations. . . . . . . . . . . . . . . . . 40Figure 5.2 Energy efficiency versus the number of users per small cell. . . . 41Figure 5.3 Energy efficiency versus the number of small cells. . . . . . . . . 42Figure 5.4 Capacity versus the number of users per small cell. . . . . . . . 43Figure 5.5 Capacity versus the number of small cells. . . . . . . . . . . . . 44Figure 5.6 Energy efficiency versus the power constraint. . . . . . . . . . . 45Figure 5.7 Energy efficiency comparison for different algorithms. . . . . . . 46Figure 5.8 Energy efficiency comparison for the optimal solution and pro-posed algorithms. . . . . . . . . . . . . . . . . . . . . . . . . . . 47viiiList oz AwronymsAwronyms Dyfinitions1G First Generation2G Second Generation3G Third Generation3GPP 3rd Generation Partnership Project4G Fourth GenerationAMPS Advanced Mobile Phone SystemAWGN Additive White Gaussian NoiseBS Base StationCDMA Code Division Multiple AccessFDMA Frequency Division Multiple AccessGABS Gradient Assisted Binary SearchGSM Global System for Mobile CommunicationsLTE-Advanced Long Term Evolution-AdvancedMIMO Multiple-Input Multiple-OutputNMT Nordic Mobile TelephoneixList ox SuronymsOFDM Orthogonal Frequency Division MultiplexingOFDMA Orthogonal Frequency Division Multiple AccessQoS Quality-of-ServiceSINR Signal-to-Interference-plus-Noise RatioSIR Signal-to-Interference RatioSMS Short Massage ServiceSNR Signal-to-Noise RatioTD-SCDMA Time Division-Synchronous Code Division Multiple AccessWCDMA Wideband Code Division Multiple AccessWiMAX Worldwide Interoperability for Microwave AccessxList oz gymvolsgymvols Dyfinitionsargmax {·} Points of the domain of the function at which the functionvalues are maximizedlim The limit of the functionlog2(·) The log function with base 2max {·} The maximum value of the functionmin {·} The minimum value of the functionO (·) The time complexity of an algorithms.t. Subject toxiAwknowlyxgymyntsI am deeply grateful to my thesis supervisor Prof. Julian Cheng for his enthusi-asm, guidance, advice, encouragement, support, and friendship. I will continue to beinfluenced by his rigorous scholarship, clarity in thinking, and professional integrity.I would like to thank Dr. Haijun Zhang for his feedback and valuable suggestionson my research work. I really appreciate his valuable time and constructive commentson my thesis.I owe many people for their generosity and support during my study at The Uni-versity of British Columbia. I would like to thank my dear colleagues for sharing theiracademic experiences and constructive viewpoints generously with me during our dis-cussions. I would also like to thank my dear friends for sharing in my excitement andencouraging me when I was in frustration during this journey.Finally, I would like to thank my parents for their patience, understanding, support,and love over all these years. All my achievements would not have been possible withouttheir constant encouragement and support.xiiTo MM Loving ParentsxiiiWhuptyr EIntroxuwtionEBE Buwkgrounx unx aotivutionThe demand for information access promotes the development of communicationtechnology, and mobile communication technology enables users to get rid of the con-straints of wire communications. Therefore, wireless communication is playing an im-portant role in people’s everyday lives. Since Guglielmo Marconi’s research team suc-cessively demonstrated the first radio transmission from the Isle of Wight to a tugboat18 miles away in 1895 [2], the era of wireless communication had begun. The earliestmobile communication system can be traced back to 1920s. In 1928, students fromPurdue University invented the super heterodyne radio receiver working at 2 MHz, andDetroit police quickly installed that mobile receiver in police patrol cars for managingtraffic [3], which was the beginning of modern cellular mobile communication network.A cellular network or mobile network is a kind of communication network where the lastlink is wireless. A cellular network consists of cells and each cell is served by at least onefixed location transceiver which is known as a base station (BS). The BS serves the usersfor transmission of voice, data and others. The first generation (1G) mobile telecom-munication technology for commercial use was invented in 1980s and three main analogcommunication standards were introduced in those years. One standard is Nordic mobiletelephone (NMT), which was used in Nordic countries, Switzerland, the Netherlands,Eastern Europe and Russia. Another two standards are the advanced mobile phonesystem (AMPS) used in North America and Australia [4], and the total access commu-1C.C. Tsukyrounv snv Motivstionnications system (TACS) used in the United Kingdom. In 1991, the second generation(2G) cellular telecommunication networks were commercially launched on the globalsystem for mobile communications (GSM) standard in Finland by Radiolinja [5]. Whileradio signals on 1G networks are analog, radio signals on 2G networks are digital. Threeprimary benefits of 2G networks over their predecessors are that phone conversationswere digitally encrypted; 2G systems significantly improved spectrum efficiency, whichmeans more mobile users were allowed in 2G systems; and 2G introduced data servicesfor mobile, such as short massage service (SMS). In order to satisfy the growing demandfor data service, several telecommunications companies introduced wireless mobile Inter-net services in the third generation (3G) based on code division multiple access (CDMA)technique [6]. There were three main standards proposed for 3G including widebandcode division multiple access (WCDMA), CDMA2000 and time division-synchronouscode division multiple access (TD-SCDMA) to improve system capacity and data rate[7]. Due to the development of digital signal processing, integrated circuit technolo-gy and other new technologies, mobile communication technology has made a rapidprogress. In 2011, the fourth generation (4G) of mobile telecommunications technologywas proposed. As opposed to earlier generations, all 4G candidate systems replacedspread spectrum radio technology used in 3G systems by orthogonal frequency divisionmultiple access (OFDMA) multi-carrier transmission, making it possible to transmithigh bit rates despite extensive multi-path radio propagation. The peak data rate canbe further improved by smart antenna arrays for multiple-input multiple-output (MI-MO) communications. For 4G systems, the 3rd Generation Partnership Project (3GPP)proposed Long Term Evolution-Advanced (LTE-Advanced) [8] system and WorldwideInteroperability for Microwave Access (WiMAX) Forum proposed WiMAX-Advancedtechnology [9].Wireless communication networks have experienced tremendous growth in the pastfew decades, wireless service have migrated from the conventional voice-centric services2C.D. Litwrsturw dwviwwto data-centric services. It is obvious that one main target for modern wireless commu-nication is to provide higher capacity wireless links for end-users. Since it is well knownthat the achievable data rate is limited by transmit power and transmission bandwidth,one straightforward way to meet the quality-of-service (QoS) demands of end-users isto increase the transmit power and bandwidth. However, transmission bandwidth andpower are valuable and scarce resources in wireless mobile communication systems, andone could never use those resources as much as he desires. To overcome such con-tradiction, researchers have proposed techniques such as power allocation, subchannelallocation and the heterogeneous network. Heterogeneous small cell network is a typicalmulti-tier transmission scheme which can increase the mobile system capacity. OFDMAcan use the bandwidth more efficiently than those previous transmission schemes such asCDMA and frequency division multiple access (FDMA) [10]. Therefore, heterogeneoussmall cell network with OFDMA has been adopted by modern mobile communicationnetworks.There are several key challenging problems associated with heterogeneous small cellnetwork systems: energy efficiency, power allocation, user association, bandwidth al-location and backhauling. In this thesis, we will focus on the energy-efficient powerallocation and backhaul bandwidth allocation problem in heterogeneous small cell net-works and study the optimization techniques for resource allocation in heterogeneoussmall cell network with OFDMA.EBF Lityrutury fyviywWith the explosive growth of wireless communications, it is shown that higher ca-pacity wireless links are expected to meet the increasing QoS demands of multimediaapplications. These high data rate links also result in increasing device power consump-tion. The next generation communication systems need to provide higher data ratewith limited power and bandwidth due to the rapidly increasing demands for multime-3C.D. Litwrsturw dwviwwdia services and resource scarcity. Designing energy-efficient wireless communicationsystem becomes an emerging trend because of rapidly increasing system energy costsand rising requirements of communication capacity [11–13]. According to [14] and [15],the radio access part is a major energy consumer in conventional wireless cellular net-works, which accounts for up to more than 70 percent of the total energy consumption.It is reported that the total energy consumed by the infrastructure of cellular wirelessnetworks, wired communication networks, and Internet takes up more than 3 percentof the worldwide electric energy consumption and the portion is expected to increaserapidly in the future [16]. Therefore, increasing the energy efficiency of typical wirelessnetworks is important to overcome the challenges raised by the rising demands of energyconsumption. In recent years, energy-efficient system design has been received muchattention in academia. In [17], the impact of cell sizes on energy efficiency in cellularnetworks was studied. Several cross-layer approaches were also developed to obtainmore gain over the independent layer design for energy efficiency [18].With the exponential growth of mobile service requirements, macrocell networkcannot satisfy all users’ requirement for data service. In order to solve this problem,heterogeneous small cell network has been proposed to provide higher system capacityand data rates, and improve the system coverage with low infrastructure cost [19–21]. Many important problems related to heterogeneous small cell networks such asinterference mitigation, resource allocation, and QoS provisioning were addressed toreap the potential gains [22–24].Resource allocation, such as power allocation and bandwidth allocation, has beenwidely used to maximize the energy efficiency under power limit and QoS requirementsin heterogeneous small cell networks. Power allocation for energy efficiency has beenwidely studied in the literature. The distributed power control game was studied in[25] to maximize the energy efficiency of transmission for secondary users in cognitiveradio networks and an optimal power control problem was formulated as a repeated4C.D. Litwrsturw dwviwwgame. The authors in [26] studied energy-efficient power control and receiver designin cognitive radio networks and a non-cooperative power control game for maximizingenergy efficiency of secondary users was considered with a fairness constraint and inter-ference threshold. The authors of [27] formulated the energy-efficient spectrum sharingand power allocation in heterogeneous cognitive radio networks with femtocells as aStackelberg game and they proposed a gradient based iteration algorithm to obtainthe Stackelberg equilibrium solution to the energy-efficient resource allocation problem.Many works have been done to consider bandwidth allocation for energy efficiency. In[28], the authors studied the joint service pricing and bandwidth allocation for ener-gy and cost efficiency at the operator level in a multi-tier network where an operatordeploys heterogeneous small cell networks, and they formulated the problem as a Stack-elberg game. The problem of joint link selection, power and bandwidth allocation forenergy efficiency maximization for multi-homing networks was investigated in [29]. Anew energy-efficient scheme was presented in [30] to statistically meet the demands forQoS during the bandwidth allocation for wireless networks.Convex optimization is one of the most effective mathematical modeling tools to ex-plore resource allocation problem in wireless communication networks. The authors in[24] proposed a resource allocation scheme for cochannel femtocells to maximize systemcapacity under QoS and interference constraints. They formulated the power and sub-channel allocation problem as a mixed-integer programming problem and transformedit into a convex optimization problem, and the proposed problem was solved by the dualdecomposition method. In [31], the authors formulated the network resource allocationproblem as a convex optimization problem to maximize system throughput and mini-mize delay under a variety of realistic QoS and fairness constraints in wireless cellularand ad hoc networks. The globally optimal solutions were computed efficiently throughpolynomial time interior point methods. In [32], the authors analyzed power controlproblem in wireless cellular networks in high signal-to-interference ratio (SIR) and medi-5C.E. fzwsis arysnizstion snv Contritutionsum to low SIR regimes. In high SIR regime, the formulated non-convex problems weretransformed into convex optimization problems in the form of geometric programmingand were effectively solved for global optimality. In the medium to low SIR regime, theproblem could only be solved through the approach of successive convex approximation.In this thesis, we define backhaul as the connection between macro BS and smallcell BSs, and it is necessary to jointly consider the backhaul and radio access network.Several related works considered the backhaul to improve energy efficiency in wire-less networks. The authors of [33] studied energy efficiency of resource allocation inmulti-cell OFDMA downlink networks where the limited backhaul capacity, the circuitpower consumption and the minimum required data rate were considered. The resourceallocation problem for energy-efficient communication with limited backhaul capaci-ty was formulated. In [34], an energy-efficient model of small cell backhaul networkswith Gauss-Markov mobile models was proposed. In [35], the authors maximized sys-tem energy efficiency in OFDMA small cell networks by optimizing backhaul data rateand emission power, and they proposed a joint forward and backhaul link optimizationscheme by taking both the power consumption of forward links and the backhaul linksinto consideration.In this thesis, we study the energy-efficient power allocation and backhaul bandwidthallocation in heterogeneous small cell networks. A near optimal iterative resource allo-cation algorithm and a suboptimal low-complexity approach are proposed. Unlike theexisting works in the literature, we take power allocation for small cell BSs and band-width allocation for backhauling together into consideration in heterogeneous small cellnetworks to maximize energy efficiency of all small cell users.EB3 hhysis crgunizution unx WontrivutionsThis thesis consists of six chapters. Chapter 1 presents background knowledge ofdevelopment and technologies for wireless communications and cellular networks. In6C.E. fzwsis arysnizstion snv Contritutionsmodern mobile communications, increasing QoS demand is the main target of systemdesign, and therefore high transmit power and wider transmission bandwidth are de-sired. However, power and bandwidth are scarce resource and are usually limited inwireless communication system. Therefore, we focus on the energy-efficient resourceallocation in OFDMA based heterogeneous small cell network.Chapter 2 provides detailed technical and knowledge background for the entire the-sis. First, a heterogeneous small cell network is introduced and used to provide moreeffective service than macrocell network. Second, resource management techniques,such as energy efficiency and backhauling, are provided. Convex optimization was in-troduced for resource management since it is an effective tool to solve the resourceallocation problem.In Chapter 3, an energy-efficient OFDMA heterogeneous small cell optimizationframework is designed. The system model for power allocation and backhaul bandwidthallocation in heterogeneous small cell network is proposed to maximize the downlinkenergy efficiency for all small cell users. The corresponding problem is formulated as anonlinear programming problem, where maximum transmit power constraints of eachsmall cell BS to each small cell user, the downlink data rate constraint of small cell BSsand the minimum data rate between each small cell BS and each of its correspondingusers are considered to provide reliable and low energy consumed downlink transmissionfor small cell users.In Chapter 4, the conditions for optimization are provided and energy-efficient re-source allocation problems are solved. First, we show that the formulated problem inChapter 3 is a non-convex optimization problem and we can decompose it into twoconvex subproblems: one for power allocation and one for unified wireless backhaulbandwidth allocation. Second, we solve the subproblems of energy-efficient power allo-cation and energy-efficient backhaul bandwidth allocation.In Chapter 5, optimization algorithms are proposed and numerical results are pro-7C.E. fzwsis arysnizstion snv Contritutionsvided to demonstrate the effectiveness of the proposed algorithms. We first propose anear optimal iterative resource allocation algorithm and a suboptimal low-complexityapproach to solve the resource allocation problem. Then we analyze the complexity forthose two proposed algorithms. Finally, we use simulation results to demonstrate theeffectiveness of the proposed algorithms when compared with the existing schemes.Chapter 6 summarizes the entire thesis and lists our contributions in this thesis. Inaddition, some future works related to our current research are suggested.8Whuptyr FHytyrogynyous gmull Wyllbytworks unx fysourwyaunugymyntIn this chapter, we present background knowledge about energy-efficient resourceallocation in heterogeneous small cell networks. We first address the characteristic ofheterogeneous small cell network and the motivation that small cell network exits, andthen we introduce the basic concept of energy efficiency and backhaul. Finally, the basicconvex optimization knowledge related to resource allocation is presented.FBE cvyrviyw oz Hytyrogynyous gmull Wyll bytworksWith the development of mobile Internet and the explosive growth of wireless traffic,macrocell networks face a series of challenges.− The users’ demand of mobile service has a trend of exponential growth. With thedevelopment of mobile Internet and cloud computing technology, users’ demand ofdata service has risen rapidly. According to [36], the amount of global mobile datatraffic nearly tripled over three consecutive years from 2010 to 2012 and exceededthe traffic on the entire global Internet in 2000. The development of the smartphone technology promoted the mobile network services. Those services, such asInternet video, mobile data and mobile voice, lead to an exponential growth of the9D.C. avwrviww ox Hwtwroywnwous emsll Cwll NwtworksFigure 2.1: Traffic demand in terabits for North America [1].demand for mobile data, which are shown in Fig. 2.1 [1]. The traditional cellularnetwork cannot keep pace with the data explosion through the previous expensiveand incremental methods such as increasing the amount of spectrum or deployingmore macro base stations [37].− The demand for indoor communication service has increased dramatically, butmacrocell network has a limited coverage for indoor environment. According to[38], over 50 percent of the voice traffic and 70 percent of the data traffic occurin the indoor environment, and those figures seem to grow continually. Further-more, 3G and 4G mobile communication systems are typically deployed at highfrequencies and the penetration loss is huge when signals transmit between walls.Therefore, the data rate requirement of indoor users is a challenge for the coverageof macrocell.10D.D. dwsouruw MsnsywmwntTo offload the overloaded traffics in macrocells and to enhance the coverage andcapacity of the wireless networks, one method is to shorten the distance between themacro BS and user equipments. Small cells (e.g., picocells, femtocells and relay nodes)have been used to improve system capacity in hotspots for relieving the burden on over-loaded macrocells, which is considered as a promising technique to provide an effectivesolution for the challenges in current macrocells [19, 20]. Therefore, there is no doubtthat small cell has been paid much attention in recent years from academia and industrybecause it can help the system spatially reuse spectrum with low power consumptionand improve the system coverage with low infrastructure cost deployment [21]. Hetero-geneous small cell networks, where small cells are overlaid within a macrocell to improvecoverage and increase system capacity beyond the initial deployment of macrocells, havebeen regarded as a promising approach to meet the increasing data traffic demand andcoverage requirements, and to reduce energy consumption. A heterogeneous small cellnetwork is shown in Fig. 2.2.FBF fysourwy aunugymyntSince the demand for mobile service has an exponential growth and the scarcity ofresource, resource management has drawn much attention these days. In recent years,energy-efficient system and backhauling have been proposed to help saving energy andguarantee the QoS for multi-users. Convex optimization algorithms are used in resourceallocation problem for more efficient resource usage.FBFBE Enyrgy EffiwiynwyDuring the past decades, much effort has been made to enhance network throughput.However, high network throughput usually implies large energy consumption, which issometimes unaffordable for energy-aware networks or energy-limited devices. How to re-duce energy consumption while meeting throughput requirements in such networks and11D.D. dwsouruw Msnsywmwntsmall cellbase stationsmall cellusermacro cellusermacrocellbase station...small cellsmall cellsmall cellsmall cellmacrocellFigure 2.2: Heterogeneous small cell network.devices is an urgent task. Therefore, energy-efficient communication system becomesan inevitable trend.Over the past few decades, energy efficiency is commonly defined as informationbits per unit transmit energy, which has been studied from the information-theoreticperspective for various scenarios [39]. For an additive white Gaussian noise (AWGN)channel, it is well known that for a given transmit power, p, and system bandwidth, l ,the achievable transmission data rate is r =l log2(1+pg2() bits per second, where σ2( isAWGN power and g is channel power gain between transmitter and receiver. We canfurther write the transmission rate as r′ = log2(1 +pg2() bits per second per Hertz. Forenergy-efficient communication, it is desirable to send the maximum amount of datawith a given amount of energy. Given the energy ∆Z consumed in duration ∆i , theenergy can be rewritten as∆Z = p∆iO (2.1)12D.D. dwsouruw MsnsywmwntTherefore, we define the energy efficiency as the ratio of the amount of data r′∆itransmitted in duration ∆i to the amount of the given energy ∆Z, which is shown asηZZ =r′∆i∆Z=r′p(2.2)bits per Hertz per Joule.Except for transmit power, circuit power is also incurred by device electronics [40,41]. Circuit power represents the additional device power consumption of devices duringtransmissions [42], such as digital-to-analog converters, mixers and filters, and thisportion of energy consumption is independent of the transmission state. Denote thecircuit power as eC (typical value 0.1 W), thus the overall power assumption is eC + p.Taking circuit energy consumption into consideration, energy efficiency needs to beredefined as information bits per unit energy (not only transmit energy) [43], wherean additional circuit power factor, eC , needs to be added in the denominator of (2.2)written asηZZ =r′∆i∆Z=r′p+ eCO (2.3)FBFBF BuwkhuulIn a heterogeneous network, the backhaul portion of the network comprises theintermediate links between the core network or backbone network and the small cellnetworks. Backhaul has responsibility to carry packets to and from the core networkand it acts as a bandwidth provider which guarantees QoS to the subnetwork users.Generally, backhaul solutions can be roughly categorized into wired (leased lines, copperor fiber) and wireless (point-to-point or point-to-multipoint over high-capacity radiolinks). Wired solution is usually expensive and often impossible to be deployed in remoteareas, which makes wireless solution a more suitable and viable option. Heterogeneouswireless architecture can overcome the hurdles of wired solutions to create efficient13D.D. dwsouruw Msnsywmwntsmall cellbase stationsmall cellusermacrocellbase station...backhaulbackhaulbackhaulbackhaulsmall cellsmall cellsmall cellsmall cellmacrocellFigure 2.3: Backhaul.large coverage areas and high capacity with relatively lower deployment cost. There isa growing demand in emerging markets where cost is usually a major factor in decidingtechnologies, a wireless backhaul solution is able to offer ‘carrier-grade’ services, whereasthis is not easily feasible with wired backhaul connectivity [44]. In this thesis, we definethat backhaul as the connection between macro BS and small cell BSs as shown in Fig.2.3, and it is necessary to consider the joint design of backhaul and radio access network.FBFB3 Wonvyfi cptimizution vusyx fysourwy aunugymyntMobile wireless networks are invented and act as essential means of communicationsto provide reliable data transmission among many users. With the exponential increaseof users’ demand for mobile service, wireless communication system is hard to satisfyall requests due to the resource scarcity. Therefore, managing available communicationresources, such as power and bandwidth, has drawn much attention. Many research14D.D. dwsouruw Msnsywmwntefforts have been made in investigating effective methods to increase operation efficiencyand network capacity for the development of wireless communication systems. Convexoptimization is one effective mathematical modeling tool to explore resource allocationproblem in wireless communication networks. The convex optimization methods wereused extensively in modeling, analyzing and designing of communication systems [45,46]. Theoretically, convex optimization is appealing since a local optimum is also aglobal optimum for a convex problem.According to the definition of convex function in [45], a function f : dn → d isconvex if the domain of f , denoted by vom f , is a convex set) and if for any two pointsx)P x2 ∈ vom f , and θ with 0 ≤ θ ≤ 1, we havef(θx) + (1− θ)x2) ≤ θf(x)) + (1− θ)f(x2)O (2.4)Geometrically, this inequality means that the line segment between (x)P f(x))) and(x2P f(x2)) lies above the graph of f , which is shown in Fig. 2.4. We say f is concave if−f is convex. Convexity and concavity will be preserved under nonnegative weightedsummation, positive scaling, and pointwise maximum operation.An optimization problem with arbitrary equality and inequality constraints canalways be written in the following standard form [45]min f((x)s.t. fi(x) ≤ 0P i = 1P 2P OOOPmhi(x) = 0P i = 1P 2P OOOP px ∈ h(2.5)to find x that minimizes f((x) among all x values that satisfy the conditions fi(x) ≤ 0,i = 1P 2P OOOPm, hi(x) = 0, i = 1P 2P OOOP p and x ∈ h, where f( is called the objective)Ix s swt C is uonvwx, tzw linw swymwnt twtwwwn sny two points in C liws in C.15D.D. dwsouruw MsnsywmwntFigure 2.4: Graph of a convex function.16D.D. dwsouruw Msnsywmwntfunction or cost function; fi(x) and hi(x) are the inequality and equality constraintfunctions, respectively, and h is the constraint set. The domain of the objective andconstraint functions is defined asD =(m∩i=)vom fi)∩(m∩i=)vom hi)∩ hO (2.6)The problem in (2.5) is a convex optimization problem if the objective function andthe inequality constraint functions fi(x)(i = 1P 2P OOOPm) are convex; equality constraintshi(x)(i = 1P 2P OOOP p) are affine functions2; and the set h is convex. Violating any oneof those conditions will result in a non-convex problem. A feasible x∗ ∈ D is said tobe global optimal if f((x∗) ≤ f((x) for all x. Notice that when we want to find themaximum value of the objective function, we can rewrite the formulation asmax f((x)s.t. fi(x) ≤ 0P i = 1P 2P OOOPmhi(x) = 0P i = 1P 2P OOOP px ∈ hO(2.7)Problem (2.7) is still a convex optimization problem if the objective function is concaveand the other conditions are the same as problem (2.5).When formulating the resource allocation problems in wireless networks, it oftenhappens that the formulated objective and constraint functions are non-convex. Thus,the problem cannot be solved by a convex optimization method. Fortunately, manyoptimization problems have hidden convexity and can be equivalently transformed intoconvex problems. In this thesis, we first formulate the energy-efficient power allocationand backhaul bandwidth allocation problem in heterogeneous small cell networks as anon-convex problem, and then decompose it into two convex subproblems.2fzw sffinw xunution usn tw rwprwswntwv ty mstrix wqustion Ax = A, wzwrw A is s mstrix snv A iss vwutor ox sppropristw sizw.17D.E. eummsryFB3 gummuryIn this chapter, we presented the essential technical background knowledge for theentire thesis. A brief description of heterogeneous small cell networks and the motiva-tions of this kind of mobile network were provided. Then, the background knowledgeand basic concepts on energy efficiency and backhaul were introduced. Finally, the basicknowledge about convex optimization were provided.18Whuptyr 3fysourwy Allowution aoxylingIn this chapter, we design an energy-efficient OFDMA heterogeneous small cell op-timization framework and propose a system model for power allocation and backhaulbandwidth allocation to maximize the downlink energy efficiency for all small cell users.We formulate the problem as a nonlinear programming problem under QoS, transmitpower and backhaul data rate constraints.3BE gystym aoxylWe consider a heterogeneous small cell network as shown in Fig. 3.1 with a singlemacro BS, J small cells deployed within the macrocell range and K users randomlylocated in each small cell.The small cells share the same spectrum with macrocell. In this work, the uni-fied wireless backhaul bandwidth allocation is investigated. The unified bandwidthallocation factor β ∈ [0P 1], which is the fraction of bandwidth assigned for wirelessbackhauling at all small cell BSs within a macrocell range. For simplicity, all smallcells are assumed to have the same bandwidth allocation factor. We assume that themultiple antenna technology is used in the macro BS and each small cell corresponds toa beamforming group, so the interference for wireless backhaul between different smallcells can be neglected. The antenna array size at macro BS is c , which is much greaterthan the beamforming group size W and the number of small cells, c ≫ W and c ≫ J .In this work, we also assume that W ≥ J . Each small cell BS is equipped with single19E.C. eystwm Movwlsmall cellbase stationsmall cellusermacro cellusermacrocellbase station...small cellmacrocellsmall cellsmall cellsmall cellFigure 3.1: Topology of a heterogeneous small cell network.20E.D. brotlwm Formulstionantenna. OFDMA technology is used in each small cell to support the communicationbetween BS and users.3BF drovlym FormulutionOur objective is to maximize downlink energy efficiency of all small cell users throughtransmit power allocation and unified wireless backhaul bandwidth allocation in het-erogeneous small cell networks. Let gjPk be the channel power gain between the jthsmall cell BS and its kth user, and denote Gj as the channel power gain between macroBS and the jth small cell BS, where j ∈ {1P 2P OOOP J}, k ∈ {1P 2P OOOPK}. Let pjPk denotethe transmit power from the jth small cell BS to its kth user, and let b = [pjPk]J×Kdenote the power allocation matrix. Then the received signal-to-noise ratio (SNR) inthe wireless backhaul downlink of small cell j is given byγj =e(Gjσ2(3.1)where e( is the transmit power of the macro BS and σ2 is the AWGN power.We assume that different users in each small cell use different subchannels and co-channel interference between small cells as part of the thermal noise because of thesevere wall penetration loss and low power of small cell BSs [24]. The received signal-to-interference-plus-noise ratio (SINR) of small cell user k associated with small cell jis given byγjPk =pjPkgjPkσ2+IjPk(3.2)where IjPk is the interference power introduced by macro BS IjPk = e(GjPk, where GjPkis the channel power gain between macro BS and the kth user in the jth small cell.The achievable data transmission rate between the jth small cell BS and its kth user is21E.D. brotlwm Formulstiondetermined byrjPk =(1− βK)log2 (1 + γjPk) O (3.3)Therefore, we have the relation between rjPk and pjPk aspjPk = (2Krj;k)− − 1)σ2+IjPkgjPkrjPk =(1− βK)log2(1 +pjPkgjPkσ2+IjPk)O(3.4)Except for transmit power, circuit power is also incurred by device electronics insmall cell BSs [40, 41]. Circuit power represents the additional device power consump-tion of devices during transmissions [42], such as digital-to-analog converters, mixersand filters, and this portion of energy consumption is independent of the transmissionstate. If we denote the circuit power as eC , the overall power assumption of the jthsmall cell BS to the kth user is eC + pjPk.For energy-efficient communication, it is desirable to send the maximum amount ofdata with a given amount of energy for small cell BSs. Hence, given the amount of energy∆e consumed in a duration ∆t in each small cell BS to each user, ∆e = ∆t(eC + pjPk),the small cell BSs desire to send a maximum amount of data bits by choosing the powerallocation matrix and unified backhaul bandwidth allocation factor to maximizeJ∑j=)K∑k=)rjPk(βP pjPk)∆t∆e(3.5)which is equivalent to maximizingj(βPb) =J∑j=)K∑k=)jjPk(βP pjPk) (3.6)wherejjPk(βP pjPk) =rjPk(βP pjPk)eC + pjPkO (3.7)22E.D. brotlwm Formulstionj(βPb) is called energy efficiency for all small cell users and jjPk(βP pjPk) is the energyefficiency of the kth user in the jth small cell. The unit of the energy efficiency is bitsper Hertz per Joule, which has been frequently used in the literature for energy-efficientcommunications [39, 47–49].When the downlink channel state information is estimated by the small cell BSs, theresource allocation is performed by each small cell BS under the following constraints.− Mkaglfim ihpek chglmkaigm hf eaca lfaee ceee BL mh eaca nlek30 ≤ pjPk ≤ emaxP ∀jP k (3.8)where emax denotes the maximum transmit power of each small cell BS to eachuser.− Mae dhpgeigd dama kame chglmkaigm hf eaca lfaee ceee BL3 the throughput of thesmall cell is given bygj =K∑k=)rjPkO (3.9)According to [50], the capacity of the wireless backhaul downlink for small cell jisCj = βlog2(1 +c−W+1Wγj)O (3.10)The downlink wireless backhaul constraint requiresgj ≤ Cj (3.11)such that the downlink traffic of the jth small cell can be accommodated by itswireless backhaul.− Aemekhgegehnl JhL gnakagmee3 the QoS requirement gt should be guaranteed foreach user in each small cell to maintain the performance of the communication23E.E. eummsrysystemrjPk ≥ gtO (3.12)Our target is to maximize the energy efficiency of power allocation and unifiedbandwidth allocation for wireless backhauling in heterogeneous small cell networks un-der power constraint and data rate requirements. Thus, the corresponding problem forthe downlink can be formulated as the following nonlinear programming problemmaxPPj(βPb) = maxPpj;kJ∑j=)K∑k=)jjPk(βP pjPk) (3.13)s.t. C1 : 0 ≤ pjPk ≤ emaxC2 : gj ≤ CjC3 : rjPk ≥ gtC4 : 0 ≤ β ≤ 1O(3.14)3B3 gummuryIn this chapter, we proposed an energy-efficient OFDMA heterogeneous small celloptimization framework. We established a system model for power allocation and back-haul bandwidth allocation to maximize the downlink energy efficiency for all small cellusers. We formulated the problem as a nonlinear programming problem with the con-sideration of maximum transmit power constraints of each small cell BS to each smallcell user, the downlink data rate constraints of small cell BSs and the minimum datarate between each small cell BS and its corresponding users.24Whuptyr HEnyrgyAEffiwiynt fysourwyAllowution unx BuwkhuulingIn this Chapter, we present the conditions for proposed optimization problem anddesign the mathematical approaches for resource allocation. We first prove that theformulated problem in Chapter 3 is a non-convex optimization problem. We find thatthe problem is separable, so we decompose it into two convex subproblems: one for powerallocation and another for unified wireless backhaul bandwidth allocation. Then, wesolve the subproblems of energy-efficient power allocation and energy-efficient backhaulbandwidth allocation individually.HBE Wonxitions oz cptimulityWe can prove that the formulated objective function in (3.13) is not concave andwe notice that the continuous variable β and pjPk are separable in (3.13). A detailedproof is given in Appendix A. The constraint C2 in (3.14) is a nonlinear non-convexconstraint. The detailed proof is shown in Appendix B. Therefore, the optimizationproblem formulated in (3.13) and (3.14) is not convex. By fixing the transmit powerb, the constraint C2 in (3.14) becomes convex with the bandwidth allocation factor β.Therefore, we consider a decomposition approach to solve the energy-efficient resourceallocation problem. We decompose the non-convex optimization problem into two con-vex subproblems: one for energy-efficient power allocation and one for energy-efficient25F.D. Wnwryy-Wffiuiwnt bowwr Slloustionwireless backhaul bandwidth allocation.HBF EnyrgyAEffiwiynt dowyr AllowutionGiven a unique global bandwidth allocation factor β for wireless backhauling, wedemonstrate that optimal energy-efficient power allocation exists. The optimizationalgorithm begins with the power allocation subproblem P1 which is formulated asP1 : maxPj(b) = maxpj;kJ∑j=)K∑k=)jjPk(pjPk) (4.1)s.t. C1 : 0 ≤ pjPk ≤ emaxC2 : gj ≤ CjC3 : rjPk(pjPk) ≥ gt(4.2)whererjPk(pjPk) =(1− βK)log2(1 +pjPkgjPkσ2+IjPk)(4.3)is strictly concave and monotonically increasing with pjPk when rjPk(0) = 0 and pjPk = 0.According to the concept of quasiconcavity defined in [51], a function f that mapsfrom a convex set of real n-dimensional vectors h′ to a real number is called strictlyquasiconcave if for any x)P x2 ∈ h′ and x) ̸= x2, and λ with 0 Q λ Q 1, we havef(λx) + (1− λ)x2) S min{f(x))P f(x2)}O (4.4)The optimal energy-efficient power allocation achieves maximum energy efficiency,i.e.b∗ = argmaxPj(b)O (4.5)It is proved in Appendix C that j(b) has the following properties.26F.D. Wnwryy-Wffiuiwnt bowwr SlloustionEeffa 1’ If rjPk(pjPk) is strictly concave in pjPk, jjPk(pjPk) ∈ j(b) is strictly quasicon-cave. Furthermore, jjPk(pjPk) is first strictly increasing and then strictly decreasing inany pjPk, i.e., the local maximum of j(b) for each pjPk exists at a positive finite value.For strictly quasiconcave functions, if a local maximum exists, it is also globallyoptimal [51]. Hence, a unique globally optimal transmit power matrix always exists andits characteristics are summarized in Theorem 1 according to the proofs in AppendixC.Maehkef 1’ If rjPk(pjPk) is strictly concave, there exists a unique globally optimal trans-mission power matrix b∗ = {p∗jPk; (jP k) ∈ J × K} for b∗=argmaxPj(b), for eachelement in b∗, p∗jPk=argmaxpj;kjjPk(pjPk) where p∗jPk is given byUjj;k(pj;k)Upj;k∣∣∣pj;k=p∗j;k= 0P f(pjPk) = 0PiOeOP jjPk(p∗jPk) =rj;k(p∗j;k)PC+p∗j;k=Urj;k(pj;k)Upj;k∣∣∣pj;k=p∗j;kOIn order to solve the power allocation problem P1, we rewrite the objective functionin (4.1) asmaxpj;kjjPk(pjPk) = maxpj;krjPk(pjPk)eC + pjPkO (4.6)If each small cell user could reach the maximum energy efficiency, the whole small cellscould reach the maximum energy efficiency. The total data rate in each small cellcould not exceed the capacity of the wireless backhaul downlink for small cell j, thatis, gj ≤ Cj . We can approximate that the data rate for each user will be less than CjK ,rjPk(pjPk) ≤ CjK .Thus, P1 is equivalent toP1O1 : maxpj;kjjPk(pjPk) (4.7)27F.D. Wnwryy-Wffiuiwnt bowwr Slloustions.t. C1 : 0 ≤ pjPk ≤ emaxC2 : rjPk(pjPk) ≤ CjKC3 : rjPk(pjPk) ≥ gtO(4.8)We can rewrite C2 in (4.8) according to (3.4) aspjPk ≤(σ2+IjPkgjPk)(2()−)lgg2()+N−B+)BP(Gj2)−1)O (4.9)We can rewrite C3 in (4.8) according to (3.4) aspjPk ≥(σ2+IjPkgjPk)(2KRt)− −1)O (4.10)Therefore,ajPk ≤ pjPk ≤ HjPk (4.11)whereajPk =(σ2+IjPkgjPk)(2KRt)− −1)(4.12)HjPk = min{(σ2+IjPkgjPk)(2()−)lgg2()+N−B+)BP(Gj2)−1)P emax}(4.13)only if the following inequality is satisfiedajPk ≤ HjPkO (4.14)The energy-efficient power allocation is given bypˆ∗jPk = argmaxpj;krjPk(pjPk)eC + pjPk(4.15)subject toajPk ≤ pjPk ≤ HjPkO (4.16)28F.D. Wnwryy-Wffiuiwnt bowwr SlloustionWe can solve (4.6) by using Theorem 1 to find the optimal power allocation solution.We can also use the low-complexity iterative algorithms based on the gradient assistedbinary search (GABS) algorithm proposed in [52] to realize the energy-efficient powerallocation for the kth user in the jth small cell BS. The GABS algorithm is shown asfollows.Slyoritzm Gradient Assisted Binary Search (GABS) AlgorithmC: InitislizstionL Each small cell BS allocates the same transmit power to each user,pjPk S 0.2: Then do p())jPk = pjPk, h) ←Ujj;k(pj;k)Upj;k∣∣∣∣pj;k=p())j;k and x S 1 (let x = 2).3: ix h) Q 0 tzwn4: rwpwst5: p(2)jPk ← p())jPk , p())jPk ←p())j;kc , and h) ←Ujj;k(pj;k)Upj;k∣∣∣∣pj;k=p())j;k6: until h) ≥ 0I: wlsw8: p(2)jPk ← p())jPk × x and h2 ←Ujj;k(pj;k)Upj;k∣∣∣∣pj;k=p(2)j;k9: rwpwstC0: p())jPk ← p(2)jPk , p(2)jPk ← p(2)jPk × x and h2 ←Ujj(pj;k)Upj;k∣∣∣∣pj;k=p(2)j;kCC: until h2 ≤ 0C2: wnv ixC3: wzilw no convergence voC4: pˆ∗jPk ←p())j;k+p(2)j;k2 , h′ ← Ujj;k(pj;k)Upj;k∣∣∣pj;k=p^∗j;kC5: ix h′ S 0 tzwnC6: p())jPk = pˆ∗jPkCI: wlswC8: p(2)jPk = pˆ∗jPkC9: wnv ix20: wnv wzilw2C: autput pˆ∗jPk.If the output pˆ∗jPk satisfies the power constraint, i.e., pˆ∗jPk=p∗jPk. Otherwise, we canget the maximum jjPk(pjPk) byp∗jPk = ajPk (4.17)29F.E. Wnwryy-Wffiuiwnt iirwlwss Tsukzsul Tsnvwivtz Slloustionif pˆ∗jPk Q ajPk, or we can get the maximum jjPk(pjPk) byp∗jPk = HjPk (4.18)if pˆ∗jPk S HjPk, since jjPk(pjPk) is first strictly increasing and then strictly decreasing inany positive finite pjPk.HB3 EnyrgyAEffiwiynt kirylyss Buwkhuul BunxwixthAllowutionOnce the optimal solution b∗ = {p∗jPk; (jP k) ∈ J × K} is obtained for the convexsubproblem P1 parameterized by β, it can be used in the following subproblem P2 forthe unified wireless backhaul bandwidth allocationP2 : maxj(βPb∗) = maxJ∑j=)K∑k=)jjPk(βP p∗jPk) (4.19)s.t. C1 : 0 ≤ β ≤ 1C2 : gj(βPb∗) ≤ Cj(βPb∗)C3 : rjPk(βP p∗jPk) ≥ gt(4.20)In order to obtain the solution to the original problem in (3.13) and (3.14), the twosubproblems P1 and P2 are solved iteratively until convergence.Maximizing the objective function of P2 with respect to β is equivalent to maximiz-ing (1− β) only, because (4.19) is a monotonically decreasing function of β. ProblemP2 reduces to a feasibility problem whose solution is the smallest feasible value of βgiven constraints (4.20).30F.E. Wnwryy-Wffiuiwnt iirwlwss Tsukzsul Tsnvwivtz SlloustionAccording to C2 in (4.20), gj(βPb∗) ≤ Cj(βPb∗), we haveβ ≥K∑k=)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) O (4.21)According to C3 in (4.20), rjPk(βP p∗jPk) ≥ gt, we haveβ ≤ 1− Kgtlog2(1 +p∗j;kgj;k2+Ij;k) O (4.22)Therefore, we can get the optimal backhaul bandwidth allocation factor β writtenasβ = max {ϕj P j ∈ J} (4.23)whereϕj =K∑k=)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) (4.24)only if gt satisfies the following conditiongt ≤ min {<j P j ∈ J} (4.25)where<j =log2(1 + c−B+)BP(Gj2)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) O (4.26)A detailed derivation of (4.25) can be found in Appendix D.31F.F. eummsryHBH gummuryIn this chapter, we found the solutions to subproblems of energy-efficient powerallocation and energy-efficient wireless backhaul bandwidth allocation, respectively. Wefirst proved that the problem formulated in Chapter 3 is a non-convex optimizationproblem and we found that the problem was separable. Therefore, we decomposed thatproblem into two convex subproblems, which means we maximized energy efficiencyfor power allocation and wireless backhaul bandwidth allocation separately. Then,we designed mathematical approaches for energy-efficient power allocation and energy-efficient backhaul bandwidth allocation.32Whuptyr IAlgorithm DysignIn this chapter, we propose two optimization algorithms for resource managementand provide numerical results for the proposed algorithms. We first design a nearoptimal iterative resource allocation algorithm and a suboptimal but low-complexityapproach to solve the resource allocation problem, and then we analyze the complexityfor those two proposed algorithms. Finally, we use simulation results to demonstratethe effectiveness of the proposed algorithms when compared with the existing schemes.IBE Ityrutivy fysourwy Allowution AlgorithmAccording to the analysis of power allocation and wireless backhaul bandwidth allo-cation discussed in Chapter 4, we propose an iterative optimization algorithm as shownin Algorithm 1.In Algorithm 1, each small cell BS calculates ϕj according to (4.24) and then sendsϕj to macro BS. The macro BS chooses the maximum ϕj to be the optimal bandwidthallocation factor β and broadcasts β to all small cell BSs.IBF LowAWomplyfiity cptimizution AlgorithmTo reduce the complexity of Algorithm 1, we propose a low-complexity optimizationalgorithm where bandwidth allocation factor is calculated from the equal power alloca-tion and we fix β to calculate the power allocation according to the scheme proposedin Chapter 4. This low-complexity optimization algorithm is shown in Algorithm 2.335.E. Complwxity SnslysisSlyoritzm C Iterative Resource Allocation AlgorithmC: InitislizstionL Each small cell BS allocates the same transmit power to each user,pjPk S 0 and set l = 1.2: rwpwst3: Tsukzsul Tsnvwivtz Slloustion4: Compute optimum β according to (4.23).5: Macro BS broadcasts the updated wireless backhaul bandwidth allocation factorto all small cell BSs.6: xor each small cell BS voI: xor each small cell user vo8: bowwr Slloustion9: a) find pˆ∗jPk = argmaxjjPk (pjPk) according to GABS;C0: b) check power constraint;CC: ix ajPk ≤ pˆ∗jPk ≤ HjPk tzwnC2: p∗jPk = pˆ∗jPkC3: wnv ixC4: ix pˆ∗jPk Q ajPk tzwnC5: p∗jPk = ajPkC6: wnv ixCI: ix pˆ∗jPk S HjPk tzwnC8: p∗jPk = HjPkC9: wnv ix20: wnv xor2C: wnv xor22: l = l + 1.23: until total energy efficiency convergence or l = amax.IB3 Womplyfiity AnulysisSince the problem formulated in (3.13) and (3.14) is not convex, the only way toget the optimal solution is to use the method of exhaustion. If we assume that it costse operations to calculate rjPk and it costs f operations to calculate Cj , the complexityof checking C2 and C3 in (3.14) entails Ke +K +f operations and e + 1 operations,respectively. If we assume it costs h operations to calculate jjPk, the complexity ofobtaining the total energy efficiency of all small cell users entails JKh+(J − 1) (K − 1)operations. The total complexity of getting the value of objective function in (3.13)under the constraints in (3.14) entails Ke + K + e + 1 + JKh + (J − 1) (K − 1)345.E. Complwxity SnslysisSlyoritzm D Fixed β and Optimum Power Allocation AlgorithmC: InitislizstionL Each small cell BS allocates the same transmit power to each user,pjPk S 0.2: Tsukzsul Tsnvwivtz Slloustion3: Compute optimum β according to (4.23).4: Macro BS broadcasts the wireless backhaul bandwidth allocation factor to all smallcell BSs.5: xor each small cell BS vo6: xor each small cell user voI: bowwr Slloustion8: a) find pˆ∗jPk = argmaxjjPk (pjPk) according to GABS;9: b) check power constraint;C0: ix ajPk ≤ pˆ∗jPk ≤ HjPk tzwnCC: p∗jPk = pˆ∗jPkC2: wnv ixC3: ix pˆ∗jPk Q ajPk tzwnC4: p∗jPk = ajPkC5: wnv ixC6: ix pˆ∗jPk S HjPk tzwnCI: p∗jPk = HjPkC8: wnv ixC9: wnv xor20: wnv xoroperations for specific pjPk and β values. If we assume the value of the step size forpjPk is a and the value of the step size for β is b, there are)b(Pmaxa)JKchoices forthe values of pjPk and β. Therefore, the complexity for the method of exhaustion isO(JKSb(Pmaxa)JK).In Algorithm 1, the worst-case complexity of calculating bandwidth allocation fac-tor β from (4.23) entails J operations in each iteration. If we assume that it costs Ωoperations in each GABS to search the optimal power allocation without power con-straint, then the worst-case complexity of finding the power allocation for every user ineach small cell entails JK (Ω+4) operations in each iteration. Suppose the Algorithm 1requires ∆ iterations to converge, so the total complexity of Algorithm 1 is O (JKΩ∆).Since iteration is not applied in Algorithm 2, the total complexity of Algorithm 2 isO (JKΩ), which is less than that of Algorithm 1. In the simulation, the typical value355.F. Numwriusl dwsultsfor ∆ is around 16, the typical value for Ω is less than 500, and the typical values for)b andPmaxb are both 100. So the complexities of Algorithm 1 and Algorithm 2 arealways less than that of the method of exhaustion. When the number of small cells Jand the number of users in each small cell K increase, the complexity of the methodof exhaustion increases exponentially, so the complexity of the method of exhaustion ismuch larger than the complexities of proposed two algorithms.IBH bumyriwul fysultsSimulation results are presented in this section to evaluate the performance of theproposed power allocation and wireless backhaul bandwidth allocation algorithms. Inthe simulations, it was assumed that small cells are uniformly distributed in the macro-cell coverage area, and small cell users are uniformly distributed in the coverage areaof their serving small cell. AWGN power σ2=3O9811 × 10−)4 W. The coverage radiusof the macrocell is 500 m, and that of a small cell is 10 m. The small cell BS has aminimum distance of 50 m from the macro BS. The minimum distance between smallcell BSs is 40 m. We assume that the channel fading is composed of path loss, shad-owing fading, and Rayleigh fading. The pathloss model for small cell users is based on[53]. The shadowing between small cell BS and small cell users is 10 dB. At the macroBS, we assume that transmit power is 33 dBm, the antenna array size c = 100 andbeamforming group size W = 20. We consider that all the small cell users have the sameQoS requirement.Figure 5.1 shows the convergence in terms of the energy efficiency of all small cellusers for the proposed Algorithm 1 versus the number of iterations, where J = 5,gt = 0O01 bps/Hz, emax = 20 dBm. It can be observed that the proposed methodtakes nearly 16 iterations before converging to the stable solution. This result, togetherwith the previous analysis, ensures that the proposed Algorithm 1 is applicable inheterogeneous small cell networks.365.F. Numwriusl dwsultsFigure 5.2 shows the total energy efficiency of all small cell users when the numberof users per small cell is increased from 2 to 10. The energy efficiencies of Algorithm 2are shown when emax = 7 dBm, emax = 10 dBm and emax = 20 dBm, and the energyefficiency of Algorithm 1 is shown when emax = 20 dBm. The simulation parameters areset as J = 5, gt = 0O01 bps/Hz. Fig. 5.2 shows that the energy efficiency performanceof Algorithm 1 is 20% higher than that of Algorithm 2. It also can be seen from Fig.5.2 that the more number of users in small cell is, the better performance is obtainedbecause of the multi-user diversity.Figure 5.3 shows the total energy efficiency of all small cell users when the numberof small cells is increased from 3 to 15. The energy efficiencies of Algorithm 2 are shownwhen emax = 7 dBm, emax = 10 dBm and emax = 20 dBm, and the energy efficiencyof Algorithm 1 is shown when emax = 20 dBm. The simulation parameters are setas K = 5, gt = 0O01 bps/Hz. Fig. 5.3 indicates that more number of small cell is,the better performance is obtained. It can also be seen from Fig. 5.3 that the energyefficiency performance of Algorithm 1 is always better than that of Algorithm 2 and thegap between them becomes larger when the number of small cells increases. The energyefficiency performance of Algorithm 1 is 30% superior to that of Algorithm 2 when thenumber of small cells is 10.Figure 5.4 shows the total downlink capacity of all small cell users when the numberof users per small cell is increased from 2 to 10. The total downlink capacities ofAlgorithm 2 are shown when emax = 7 dBm, emax = 10 dBm and emax = 20 dBm,and the total downlink capacity of Algorithm 1 is shown when emax = 20 dBm. Thesimulation parameters are set as J = 5, gt = 0O01 bps/Hz. Fig. 5.4 shows that the totaldownlink capacity of Algorithm 1 is more than 3 bps/Hz higher than that of Algorithm2. It also can be seen from the Fig. 5.4 that the more number of users in small cell is,the better performance is obtained due to the multi-user diversity. The total downlinkcapacity of Algorithm 1 is 21% higher than that of Algorithm 2 when the number of375.F. Numwriusl dwsultsusers in each small cell is over 10.Figure 5.5 shows the total downlink capacity of all small cell users when the numberof small cells is increased from 3 to 15. The total downlink capacities of Algorithm 2 areshown when emax = 7 dBm, emax = 10 dBm and emax = 20 dBm, the total downlinkcapacity of Algorithm 1 is shown when emax = 20 dBm. The simulation parametersare set as K = 5, gt = 0O01 bps/Hz. Fig. 5.5 illustrates that Algorithm 1 is superiorto Algorithm 2 in terms of the total downlink capacity and the gap between thembecomes larger when the number of small cells increases. The total downlink capacityof Algorithm 1 is 29% larger than that of Algorithm 2 when there 14 small cells in theheterogeneous network.Figure 5.6 shows the total energy efficiency of all small cell users when using Algo-rithm 2 for power constraint emax ranging from 0 dBm to 12.79 dBm where the numberof users in each small cell is 3, 4, 5. The simulation parameters are set as J = 5,gt = 0O01 bps/Hz. Fig. 5.6 presents that the more users in each small cell, the highertotal energy efficiency can be obtained, which has already been shown in Fig. 5.2. Italso can be seen from the Fig. 5.6 that the larger power constraint is, the better per-formance is obtained. This is because the larger power constraint leads to the largerregion of the optimizing variable.Figure 5.7 shows the total energy efficiency of all small cell users when the number ofusers per small cell is increased from 2 to 10, for different algorithms. Algorithm 1 andAlgorithm 2 are the iterative optimization algorithm and the low-complexity optimiza-tion algorithm, respectively. Algorithm 3 is an existing energy efficiency optimizationalgorithm with equal power allocation and Algorithm 4 is an algorithm that uses theoptimal power allocation we proposed given a random β to optimize energy efficiency.All the algorithms are under the setting of emax = 20 dBm. Fig. 5.7 indicates thatthe more users in each small cell, the better performance can be obtained, which hasalready been shown in Fig. 5.2. It also can be seen from Fig. 5.7 that Algorithm 1 has385.5. eummsrythe best performance and then it follows by Algorithm 2, Algorithm 3 and Algorithm4. The energy efficiency performance of Algorithm 1 is 30.5% and 56.6% higher thanthat of Algorithm 3 and Algorithm 4, respectively.Figure 5.8 shows the total energy efficiency of all small cell users when the numberof small cells is increased from 2 to 5, for the optimal solution and those two proposedalgorithms. Since the complexity of the method of exhaustion is very high, we onlyconsider the situation with small dimension where there are two users located in eachsmall cell, K = 2. All the algorithms are under the setting of emax = 20 dBm andgt = 0O01 bps/Hz. From Fig. 5.8, we can observe that the difference between theoptimal solution and Algorithm 1 in terms of energy efficiency is very slight, whichensures the effectiveness of the proposed algorithms. The energy efficiency performanceof the optimal solution is only about 7% and 24% higher than that of Algorithm 1 andAlgorithm 2 when the number of small cells is 3, respectively.IBI gummuryIn this chapter, we designed two optimization algorithms for resource managementand provided numerical results for the proposed algorithms. We first proposed a nearoptimal iterative resource allocation algorithm and a suboptimal but low-complexityapproach to solve the resource allocation problem. Then we analyzed the complexity forthose two proposed algorithms. Finally, we used simulation results to demonstrate theeffectiveness of the proposed algorithms by comparing them with the existing schemes.395.5. eummsry0 2 4 6 8 10 12 14 16 18 20020406080100120140160180200Iteration indexEnergy efficiency of all small cell users (bits/Hz/Joule)  Pmax=20 dBm, K=7Pmax=20 dBm, K=5Figure 5.1: The convergence in terms of energy efficiency of all small cell users over thenumber of iterations.405.5. eummsry2 3 4 5 6 7 8 9 10130135140145150155160165170175180185Number of users per small cellEnergy efficiency of all small cell users (bits/Hz/Joule)  Algorithm 1, Pmax=20 dBmAlgorithm 2, Pmax=20 dBmAlgorithm 2, Pmax=10 dBmAlgorithm 2, Pmax=7 dBmFigure 5.2: Energy efficiency versus the number of users per small cell.415.5. eummsry2 4 6 8 10 12 14 1650100150200250300350400450500Number of small cellsEnergy efficiency of all small cell users (bits/Hz/Joule)  Algorithm 1, Pmax=20 dBmAlgorithm 2, Pmax=20 dBmAlgorithm 2, Pmax=10 dBmAlgorithm 2, Pmax=7 dBmFigure 5.3: Energy efficiency versus the number of small cells.425.5. eummsry2 3 4 5 6 7 8 9 1014151617181920Number of users per small cellTotal downlink capacity of all small cell users (bps/Hz)  Algorithm 1, Pmax=20 dBmAlgorithm 2, Pmax=20 dBmAlgorithm 2, Pmax=10 dBmAlgorithm 2, Pmax=7 dBmFigure 5.4: Capacity versus the number of users per small cell.435.5. eummsry2 4 6 8 10 12 14 160510152025303540455055Number of small cellsTotal downlink capacity of all small cell users (bps/Hz)  Algorithm 1, Pmax=20 dBmAlgorithm 2, Pmax=20 dBmAlgorithm 2, Pmax=10 dBmAlgorithm 2, Pmax=7 dBmFigure 5.5: Capacity versus the number of small cells.445.5. eummsry0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02115120125130135140145150Power constraint Pmax (W)Energy efficiency of all small cell users (bits/Hz/Joule)  Algorithm 2, K=5Algorithm 2, K=4Algorithm 2, K=3Figure 5.6: Energy efficiency versus the power constraint.455.5. eummsry2 3 4 5 6 7 8 9 10110120130140150160170180Number of users per small cellEnergy efficiency of all small cell users (bits/Hz/Joule)  Algorithm 1Algorithm 2Algorithm 3Algorithm 4Figure 5.7: Energy efficiency comparison for different algorithms.465.5. eummsry2 2.5 3 3.5 4 4.5 5406080100120140160180200220Number of small cellsEnergy Efficiency of all small cell users (bits/Hz/Joule)  Optimal solution, Pmax=20 dBmAlgorithm 1, Pmax=20 dBmAlgorithm 2, Pmax=20 dBmFigure 5.8: Energy efficiency comparison for the optimal solution and proposed algo-rithms.47Whuptyr 6WonwlusionsIn this chapter, we conclude the thesis by summarizing the accomplished work andsuggest some potential further works.6BE gummury oz Awwomplishyx korkIn this thesis, we developed two optimization algorithms for energy-efficient powerallocation and backhaul bandwidth allocation in heterogeneous small cell networks. Thenumerical results establish the effectiveness of proposed design when compared withthe existing schemes. To conclude the thesis, we summarize the accomplished work asfollows:− In Chapter 2, we provided detailed technical and knowledge background for theentire thesis. We first introduced heterogeneous small cell network which was usedto provide more effective service than macrocell network, and then we presentedsome resource management techniques such as energy efficiency and backhaul-ing. Convex optimization was introduced for resource management since it is aneffective tool to solve the resource allocation problem.− In Chapter 3, we provided energy-efficient OFDMA heterogeneous small cell opti-mization framework. The system model for power allocation and backhaul band-width allocation in heterogeneous small cell network was proposed to maximizethe downlink energy efficiency for all small cell users. The corresponding prob-lem was formulated as a nonlinear programming problem under the constraints of48H.D. Futurw iorkQoS, transmit power and backhaul date rate.− Chapter 4 provided the conditions for optimization and mathematical approach-es for transmit power allocation and backhaul bandwidth allocation. First, weshowed that the formulated problem in Chapter 3 is a non-convex optimizationproblem and we decomposed it into two convex subproblems: one for power al-location and another for unified wireless backhaul bandwidth allocation. Second,we solved the subproblems of energy-efficient power allocation and energy-efficientbackhaul bandwidth allocation.− In Chapter 5, suboptimal algorithms were designed and numerical results werepresented. We proposed a near optimal iterative resource allocation algorithm anda suboptimal but low-complexity approach to solve the energy-efficient resourceallocation problem. Then, we analyzed the complexity for those two proposedalgorithms. Finally, we used simulation results to demonstrate the effectivenessof the proposed algorithms by comparing them with the existing schemes.6BF Futury korkAlthough considerable research work on resource management have already beenproposed during the past few years, there are still some potential directions worthfurther investigation.− In this work, we considered the power allocation and backhaul bandwidth al-location for energy efficiency. However, spectral efficiency is also an importantsystem performance indictor to be studied. Therefore, resource allocation withbackhauling for spectral efficiency is worth investigating in the future.− For resource allocation, subchannel allocation is an important aspect. Therefore,the joint energy-efficient subchannel allocation and power allocation are worth49H.D. Futurw iorkstudying in the future.− In this work, we considered the small cell BS with single antenna. 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Bhargava, “Joint downlink cell association and band-width allocation for wireless backhauling in two-tier HetNets with large-scale an-tenna arrays,” acceimed, BEEE Mkaglacmihgl hg Pikeeell Chffngicamihgl, 2016.→ pages 23[51] E. Wolfstetter, Mhiicl ig Fickhechghficl3 Bgdnlmkiae Hkgagisamihg, :ncmihgl, agdBgcegmioel. Cambridge, UK: Cambridge University Press, 1999. → pages 26, 27,63[52] G. Miao, N. Himayat, and G. Y. Li, “Energy-efficient link adaptation in frequency-selective channels,” BEEE Mkaglacmihgl hg Chffngicamihgl, vol. 58, pp. 545–554,Feb. 2010. → pages 29[53] “Further advancements for E-UTRA, physical layer aspects,” 3GPP Std.TR 36.814v9.0.0, 2010. → pages 3657Appynxiwys58Appynxifi AIn order to prove that the objective function in (3.13),J∑j=)K∑k=)jjPk(βP pjPk), is concave,we first prove that function jjPk(βP pjPk) is concave.The energy efficiency function for each small cell user isjjPk(βP pjPk) =()−K)log2(1 +pj;kgj;k2+Ij;k)eC + pjPkO (A.1)The Hessian matrix for function jjPk(βP pjPk) can be written asHws(jjPk) = U2jj;kU2 U2jj;kUUpj;kU2jj;kUpj;kUU2jj;kUp2j;k (A.2)whereU2jjPkUβ2= 0 (A.3)U2jjPkUβUpjPk=(− )K ) PC+pj;k(ln 2)()+pj;kgj;k2+Ij;k)( gj;k2+Ij;k)+()K)log2(1 +pj;kgj;k2+Ij;k)(eC + pjPk)2 (A.4)U2jjPkUpjPkUβ=(− )K ) PC+pj;k(ln 2)()+pj;kgj;k2+Ij;k)( gj;k2+Ij;k)+()K)log2(1 +pj;kgj;k2+Ij;k)(eC + pjPk)2 (A.5)59Sppwnvix S.U2jj;kUp2j;k=( )−K )(gj;k2+Ij;k)−(PC+pj;k)(gj;k2+Ij;k)(ln 2)()+pj;kgj;k2+Ij;k)2(PC+pj;k)2−2( )−K )( )ln 2)( gj;k2+Ij;k) PC+pj;k)+pj;kgj;k2+Ij;k−lgg2()+ pj;kgj;k2+Ij;k)(PC+pj;k)+ O(A.6)For simplicity, we denote thatb) =U2jjPkUβUpjPk=U2jjPkUpjPkUβ(A.7)andb2 =U2jjPkUp2jPkO (A.8)Therefore, we can write the Hessian matrix for function jjPk(βP pjPk) asHws(jjPk) = 0 b)b) b2 O (A.9)It is obvious that the Hessian matrix is not negative semi-definite, so the functionjjPk(βP pjPk) is not concave and the objective function in (3.13),J∑j=)K∑k=)jjPk(βP pjPk), isnot concave. 60Appynxifi BIn this Appendix, we prove the C2 in (3.14) is not a convex constraint. The C2 in(3.14) can be written asK∑k=)(1− βK)log2(1 +pjPkgjPkσ2+IjPk)≤ βlog2(1 +c−W+1We(Gjσ2)O (B.0.1)We denote function Z(βP pjPk) asZ(βP pjPk) =K∑k=)(1− βK)log2(1 +pjPkgjPkσ2+IjPk)− βlog2(1 +c−W+1We(Gjσ2)≤ 0O(B.0.2)The Hessian matrix for function Z(βP pjPk) can be written asHws(Z) = U2ZU2 U2ZUUpj;kU2ZUpj;kUU2ZUp2j;k (B.0.3)whereU2ZUβ2= 0 (B.0.4)U2ZUβUpjPk=(− )K ) ( gj;k2+Ij;k)(ln 2)(1 +pj;kgj;k2+Ij;k) (B.0.5)U2ZUpjPkUβ=(− )K ) ( gj;k2+Ij;k)(ln 2)(1 +pj;kgj;k2+Ij;k) (B.0.6)U2ZUp2jPk=−()−K)(gj;k2+Ij;k)2(ln 2)(1 +pj;kgj;k2+Ij;k)2 O (B.0.7)61Sppwnvix T.For simplicity, we denote thatb ′) =U2ZUβUpjPk=U2ZUpjPkUβ(B.0.8)andb ′2 =U2ZUp2jPkO (B.0.9)Therefore, we can write the Hessian matrix for function Z(βP pjPk) asHws(Z) = 0 b ′)b ′) b ′2 O (B.0.10)It is obvious that Hessian matrix is not positive semi-definite, so the function Z(βP pjPk)is not convex and C2 in (3.14) is not a convex constraint. 62Appynxifi WIn order to prove the properties of j(b) in Lamma 1, we first focus on jjPk(pjPk)and then we can get the properties of j(b).According to [51], we denote the α–superlevel sets of jjPk(pjPk) ash = {pjPk ≥ 0|jjPk(pjPk) ≥ α} (C.0.1)where pjPk is nonnegative. Based on the propositions from [51], jjPk(pjPk) is strictlyquasiconcave if and only if h is strictly convex for any real number α. In this case, whenα Q 0, no points exist on the contour jjPk(pjPk) = α. When α = 0, only pjPk = 0 is on thecontour jjPk(0) = α. Hence, h is strictly convex when α ≤ 0. Now, we investigate thecase when α S 0. We can rewrite the h as h = {pjPk ≥ 0|αeC+αpjPk−rjPk(pjPk) ≤ 0}.Since rjPk(pjPk) is strictly concave with respect to pjPk, −rjPk(pjPk) is strictly convex withrespect to pjPk. Therefore, h is strictly convex. Hence, we have the strict quasiconcavityof jjPk(pjPk).Next, we can obtain the partial derivative of jjPk(pjPk) with pjPk asUjjPk(pjPk)UpjPk=(eC + pjPk)r′jPk(pjPk)− rjPk(pjPk)(eC + pjPk)2 =f(pjPk)(eC + pjPk)2 (C.0.2)where f(pjPk) = (eC + pjPk)r′jPk(pjPk) − rjPk(pjPk), r′jPk(pjPk) is the first partial derivativeof rjPk(pjPk) with respect to pjPk. If p∗jPk exists such thatUjj;k(pj;k)Upj;k∣∣∣pj;k=p∗j;k= 0, it isunique, i.e., if there is a p∗jPk such that f(p∗jPk) = 0. In the following, we investigate theconditions when p∗jPk exists.63Sppwnvix C.The derivative of f(pjPk) isf ′(pjPk) = (eC + pjPk)r′′jPk(pjPk) (C.0.3)where r′′jPk(pjPk) is the second partial derivative of rjPk(pjPk) with respect to pjPk. SincerjPk(pjPk) is strictly concave in pjPk, so r′′jPk(pjPk) Q 0, f′(pjPk) Q 0. Hence, f(pjPk) isstrictly decreasing.limpj;k→∞f(pjPk) = limpj;k→∞((eC + pjPk)r′jPk(pjPk)− rjPk(pjPk))= limpj;k→∞(eCr′jPk(pjPk) + pjPkr′jPk(pjPk)− rjPk(pjPk)) (C.0.4)wherer′jPk(pjPk) =(1− βK)(gjPkσ2 + IjPk)(1ln 2)(11 +pj;kgj;k2+Ij;k)(C.0.5)andlimpj;k→∞r′jPk(pjPk) = 0 (C.0.6)so we havelimpj;k→∞eCr′jPk(pjPk) = 0O (C.0.7)According to the L’Hopital’s rule, it is easy to show thatlimpj;k→∞pjPkr′jPk(pjPk) = limpj;k→∞(1− βK)(gjPkσ2 + IjPk)(1ln 2)(pjPk1 +pj;kgj;k2+Ij;k)= limpj;k→∞(1− βK)(gjPkσ2 + IjPk)(1ln 2)(1gj;k2+Ij;k)= limpj;k→∞(1− βK)(1ln 2)(C.0.8)limpj;k→∞(−rjPk(pjPk)) = limpj;k→∞[−(1− βK)log2(1 +pjPkgjPkσ2+IjPk)]= −∞ (C.0.9)64Sppwnvix C.solimpj;k→∞f(pjPk) Q 0O (C.0.10)Besides,limpj;k→(f(pjPk) = limpj;k→(((eC + pjPk)r′jPk(pjPk)− rjPk(pjPk))= eCr′jPk(p(()jPk)− rjPk(p(()jPk)(C.0.11)where p(()jPk denotes pjPk = 0r′jPk(p(()jPk) =(1− βK)(gjPkσ2 + IjPk)(1ln 2)(11 +pj;kgj;k2+Ij;k)∣∣∣∣∣pj;k=(=(1− βK)(gjPkσ2 + IjPk)(1ln 2) (C.0.12)rjPk(p(()jPk) = 0 (C.0.13)limpj;k→(f(pjPk) =(1− βK)(eCgjPkσ2 + IjPk)(1ln 2)S 0O (C.0.14)Therefore, together with limpj;k→∞f(pjPk) Q 0, we obtain that p∗jPk exists and jjPk(pjPk)is first strictly increasing and then strictly decreasing in pjPk. After achieving the max-imum energy efficiency of every user in each small cell, the total energy efficiency of allsmall cells users can be maximized.Lemma 1 is readily obtained. 65Appynxifi DFrom (4.21) and (4.22), we can get the interval for the wireless backhaul bandwidthallocation factor β. Problem P2 can be solved only if the upper bound of β larger orequal to its lower bound1− Kgtlog2(1 +p∗j;kgj;k2+Ij;k) ≥K∑k=)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) ≥ Kgtlog2(1 +p∗j;kgj;k2+Ij;k)gt ≤log2(1 + c−B+)BP(Gj2)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) O(D.0.1)Therefore, we have the condition for gt to guarantee that the problem P2 is solvablegt ≤ minlog2(1 + c−B+)BP(Gj2)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) O (D.0.2)In order to facilitate the representation, we denote <j as<j =log2(1 + c−B+)BP(Gj2)log2(1 +p∗j;kgj;k2+Ij;k)Klog2(1 + c−B+)BP(Gj2)+K∑k=)log2(1 +p∗j;kgj;k2+Ij;k) O (D.0.3)66Sppwnvix D.So (D.0.2) can be rewritten asgt ≤ min {<j P j ∈ J} O (D.0.4)67

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