Open Collections

UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Resource allocation and performance analysis for wireless communication systems with radio frequency… Dong, Yanjie 2016

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
24-ubc_2016_september_dong_yanjie.pdf [ 957.6kB ]
Metadata
JSON: 24-1.0305736.json
JSON-LD: 24-1.0305736-ld.json
RDF/XML (Pretty): 24-1.0305736-rdf.xml
RDF/JSON: 24-1.0305736-rdf.json
Turtle: 24-1.0305736-turtle.txt
N-Triples: 24-1.0305736-rdf-ntriples.txt
Original Record: 24-1.0305736-source.json
Full Text
24-1.0305736-fulltext.txt
Citation
24-1.0305736.ris

Full Text

fysourwy Allowution unx dyrzormunwyAnulysis zor kirylyss Wommuniwutiongystyms kith fuxio FryquynwyEnyrgy HurvystingbyYanjie DongB.Eng., Xidian University, P.R. China, 2011A 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© Yanjie Dong, 2016iiAvstruwtRadio frequency energy harvesting (RF-EH) is a promising technology to increase thelifetime of the wireless nodes, and there are many use cases for this emerging technology.Despite a number of advantages of the RF-EH technology, several challenges remain to besolved in order to fully exploit its potential. For example, the impact of integrating the RF-EH technology into transmitters of the wireless communication systems remains unknown.Besides, heterogeneous quality-of-service (QoS) and the nonlinearity of the energy harvesterrequire fundamental investigation from both resource allocation and performance analysisperspective. Failing to address these issues can wipe out the advantages that the RF-EHtechnology brings. In this thesis, we consider some of these challenges and develop solutionsas described below.First, an energy efficiency (EE) maximization problem is studied in a distributed an-tenna system with RF-EH capability. The energy harvester on each radio remote head canscavenge energy over all frequency band for practical amount of energy. A low-complexitysemi-distributed algorithm is proposed to maximize the system EE via subchannel alloca-tion and power control.Next, an innovative optimization framework is proposed to formulate the long-termpower minimization problem in a simultaneous wireless information and power transferringsystem. The formulated problem contains both long-term and short-term QoS constraints,which is difficult to solve via standard optimization methods. Thus, the stochastic opti-mization theory is used to propose a dynamic power control and time switching algorithmfor suboptimal solution. By tuning a control parameter, the power consumption can ap-proach its optimal value at the expense of the delay of the wireless nodes with best efforttraffic.Finally, the performance of the wireless powered relaying systems with nonlinear en-ergy harvester is investigated. We derive an analytical expression of the complementarycumulative distribution function (CCDF) for the end-to-end signal-to-noise ratio. Com-pared with our analytical results, the linear energy harvester overestimates the CCDF ofthe end-to-end signal-to-noise ratio when the relay is placed closer to the source node.iiidryzuwyThis thesis is based on the research work conducted in the School of Engineering atThe University of British Columbia, Okanagan Campus, under the supervision of Prof. Md.Jahangir Hossain and Prof. Julian Cheng. Both published and submitted works are con-tained in this thesis.Chapter 3 of this thesis has been published in the drowyyxings oz ]EEE [lovul hylywomAmuniwutions Wonzyrynwy FDEI. The research idea, problem formulation and solution, andnumerical simulations related to this paper are the results of my own independent work.Dr. Zhang provided me with some suggestions to improve the system model. Prof. Hossain,Prof. Cheng and Prof. Leung helped me prepare the manuscripts for scholarly publicationby checking the validity of analytical and numerical results and proofreading.Chapters 4 and 5 of this thesis have been published in the ]EEE Wommuniwutions LytAtyrs. I am the principle contributor for these works. Prof. Hossain and Prof. Cheng helpedme check validity of the analytical and numerical results and proofread the manuscripts.The reuse of all the materials in this thesis is authorized by the co-authors of corre-sponding publication. A list of my papers published at The University of British Columbia,Okanagan Campus is provided below.Journul dupyr duvlishyx1. Y. Dong, M. J. Hossain, and J. Cheng, “Joint power control and time switching forSWIPT systems with heterogeneous QoS requirements,” ]EEE WommunB LyttB, vol.20, no. 2, pp. 328–331, Feb. 2016 (Part of Chapter 4).2. Y. Dong, M. J. Hossain, and J. Cheng, “Performance of wireless powered amplify andforward relaying over Nakagami-m fading channels with nonlinear energy harvester,”]EEE WommunB LyttB, vol. 20, no. 4, pp. 672–675, Apr. 2016 (Part of Chapter 5).Wonzyrynwy dupyr Awwyptyx1. Y. Dong, H. Zhang, M. J. Hossain, J. Cheng, and V. C. M. Leung, “Energy efficientresource allocation for OFDMA full duplex distributed antenna systems with energyivCeefTcerecycling,” in drowB oz ]EEE [lovywom, Dec. 2015, pp. 1-6 (Part of Chapter 3).Journul dupyr guvmittyx1. H. Zhang, Y. Dong, J. Cheng, M. J. Hossain, V. C. M. Lenug, “Fronthauling for 5GLTE-U ultra dense cloud small cell networks,” submitted to ]EEE kirylyss WommuAniwutions.vhuvly oz WontyntsAvstruwt B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B iiidryzuwy B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B ivhuvly oz Wontynts B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B viList oz huvlys B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B ifiList oz Figurys B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B fiList oz Awronyms B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B fiiiAwknowlyxgymynts B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B fiivWhuptyr EN Introxuwtion B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B E1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 DASs With RF-EH Capability . . . . . . . . . . . . . . . . . . . . . 21.1.2 SWIPT Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.3 WPR Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Literature Review and Motivation . . . . . . . . . . . . . . . . . . . . . . . 31.2.1 Energy Efficient Resource Allocation Algorithms in DASs With RF-EH Capability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2.2 Joint Power Control and Time Switching in SWIPT Systems WithHeterogeneous QoS Requirements . . . . . . . . . . . . . . . . . . . 41.2.3 Performance of Amplify and Forward Relaying With Nakagami-mFading and Nonlinear Energy Harvester . . . . . . . . . . . . . . . . 51.3 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . . . . . . 6Whuptyr FN Buwkgrounx B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B L2.1 RF-EH Receivers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.2 Stochastic Optimization Theory . . . . . . . . . . . . . . . . . . . . . . . . . 9viG45LE BF 6BAGEAGF2.2.1 Queue Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.2 Lyapunov Drift Function . . . . . . . . . . . . . . . . . . . . . . . . 112.2.3 One-slot Conditional Lyapunov Drift Function . . . . . . . . . . . . 112.3 Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.1 Rayleigh Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Rician Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Nakagami-m Distribution . . . . . . . . . . . . . . . . . . . . . . . . 13Whuptyr 3N Enyrgy Ewiynt fysourwy Allowution zor cFDaA Full DuplyfiDistrivutyx Antynnu gystyms kith Enyrgy Hurvysting B B B B B EI3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.1 Overall Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.1.2 System SE, Power Consumption and EE . . . . . . . . . . . . . . . . 163.2 Subchannel Allocation and Power Control . . . . . . . . . . . . . . . . . . . 183.2.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.2 Suboptimal Subchannel Allocation Algorithm . . . . . . . . . . . . . 193.2.3 Power Control Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 203.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24Whuptyr HN Joint dowyr Wontrol unx himy gwitwhing zor gkIdh gystymskith Hytyrogynyous eog fyquirymynts B B B B B B B B B B B B B B F74.1 System Model and Problem Formulation . . . . . . . . . . . . . . . . . . . . 274.1.1 Overall Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.1.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 304.2 Suboptimal Solution for JPCTS . . . . . . . . . . . . . . . . . . . . . . . . . 314.2.1 Virtual Throughput Queue . . . . . . . . . . . . . . . . . . . . . . . 314.2.2 Dynamic Power Control and Time Switching (DPCTS) Algorithm . 314.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Whuptyr IN EnxAtoAynx gbf oz kirylyss dowyryx Amplizy unx Forwurxfyluying kith bukugumiAm Fuxing Whunnyls unx bonlinyurEnyrgy Hurvystyr B B B B B B B B B B B B B B B B B B B B B B B B B B B HD5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.1.1 Overall Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.1.2 Nonlinear Energy Harvester Model . . . . . . . . . . . . . . . . . . . 425.2 End-to-End SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42viiG45LE BF 6BAGEAGF5.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Whuptyr JN Wonwlusions B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B ID6.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51Bivliogruphy B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B IFAppynxiwys B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B ILAppendix A: Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . 59Appendix B: Proof of the Convergence of PC-ADMM Algorithm . . . . . . . . . 60Appendix C: Proof of the Properties of DPCTS Algorithm . . . . . . . . . . . . . 62Appendix D: Proof of Theorem 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 64viiiList oz huvlysTable 3.1 Simulation Parameters Setting . . . . . . . . . . . . . . . . . . . . . 24ixList oz FigurysFigure 2.1 Block diagram of the antenna separating receiver. . . . . . . . . . . 9Figure 2.2 Block diagram of the time switching receiver. . . . . . . . . . . . . . 9Figure 2.3 Block diagram of the power splitting receiver. . . . . . . . . . . . . 10Figure 3.1 A full duplex distributed antenna system with b = 6 antennas andc = 2 WNs. The northeast of this figure illustrates a RRH whichis equipped with both information transmitter and energy harvester. 16Figure 3.2 Illustration of convergence of the proposed PC-ADMM algorithmand the JSPA algorithm with 5 WNs in the system. . . . . . . . . . 25Figure 3.3 Illustration of convergence of the proposed PC-ADMM algorithmand the JSPA algorithm with 15 WNs in the system. . . . . . . . . 25Figure 3.4 System EE versus SE requirement. . . . . . . . . . . . . . . . . . . . 26Figure 4.1 A SWIPT system with multiple WNs scavenging energy and detect-ing information from the same signal. The figure also shows theframe structure with WN i using /iPts (t) of tth time slot for EH and1− /iPts (t) for ID. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 4.2 The tradeoff of delay of WNs with the average power consumption. 38Figure 4.3 The performance variation against the proportion of BENs in thesystem with λj = 5 nats/sec/Hz and λk = 4 nats/sec/Hz, j ∈ B andk ∈ D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Figure 5.1 An illustration of a wireless powered relay system that consists ofsource, relay and destination. The relay is equipped with a nonlinearenergy harvester with input/output relation shown in the northwestof this figure. This figure also illustrates the three phase protocolused in the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 5.2 An illustration of the impact of the channel fading severity on thesystem CCDF with ysPr = 22m and yrPy = 4m. . . . . . . . . . . . . 44xLIFG BF FIGUEEFFigure 5.3 An illustration of the impact of the channel fading severity on thesystem CCDF with ysPr = 13m and yrPy = 13m. . . . . . . . . . . . . 45Figure 5.4 An illustration of the impact of the channel fading severity on thesystem CCDF with ysPr = 4m and yrPy = 22m. . . . . . . . . . . . . 46Figure 5.5 CCDF versus ysPr with ysPr + yrPy = 26 m. . . . . . . . . . . . . . . . 47Figure 5.6 Outage capacity versus time switching factor with ysPr = 2m andyrPy = 24m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 5.7 Outage capacity versus time switching factor with ysPr = 13m andyrPy = 13m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 5.8 Outage capacity versus time switching factor with ysPr = 24m andyrPy = 2m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49xiList oz AwronymsAwronyms DynitionsADMM Alternating direction method of multipliersBBU Base band unitBEN Best effort nodeCCDF Complementary cumulative distribution functionCSI Channel state informationDAS Distributed antenna systemsDPCTS Dynamic power control and time switchingDSN Delay sensitive nodeEE Energy efficiencyEH Energy harvestingFD Full duplexFDDAS Full duplex distributed antenna aystemsID Information DetectionJPCTS Joint power control and time switchingJSPA Joint subchannel and power allocationLOS Line-of-SightMINLP Mixed integer nonlinear programmingNP-hard Non-deterministic polynomial-time hardOFDMA Orthogonal frequency division multiple accessPC-ADMM Power control using ADMMPS Power splittingQoS Quality of serviceRF Radio frequencyRF-EH Radio frequency energy harvestingRF-EH-WCS Radio frequency energy harvesting wireless communication systemRRH Radio remote headSE Spectrum efficiencySNR Signal-to-noise ratioSWIPT Simultaneous wireless information and power transferxiiLifg bf 4cebalmfTS Time switchingWCS Wireless communication systemWN Wireless nodeWPR Wireless powered relayingxiiiAwknowlyxgymyntsI am deeply grateful to my supervisors Prof. Md. Jahangir Hossain and Prof. JulianCheng for their enthusiasm, guidance, advice, encouragement, and support. They grantedme a great flexibility and freedom in my research work. They taught me the academicknowledge and research skills. I will continue to be influenced by their rigorous scholarship,clarity in thinking, and professional integrity. It is my honor to study and do research undertheir supervision.I would like to express my thanks to Prof. Heinz Bauschke to serve as my externalexaminer. I would also like to thank Prof. Thomas Johnson for serving on the committee.I really appreciate their valuable time and constructive comments on my thesis.I owe many people for their generosity and support during my M.A.Sc. study atThe University of British Columbia, Okanagan Campus. I would like to thank my dearcolleagues Dr. Haijun Zhang and Dr. Samy Soliman for sharing their academic experiencesand constructive viewpoints generously with me during our discussions. Also, I need tothank Mr. Kun Yuan, a Ph.D. student at UCLA. It is his generous discussion that helpedme overcome several obstacles in my research. I would also like to thank my dear friends forsharing my excitement and encouraging me when I was in frustration during this journey.Finally, I would like to thank my parents for their patience, understanding and supportall these years. All my achievements would not have been possible without their constantencouragement and support.xivWhuptyr EIntroxuwtionEBE IntroxuwtionIn 1960s, a small helicopter hovering at a height of 50 feet was powered by a radiofrequency (RF) source with a direct current power supply of 270 W operating at 2.45 GHzon the ground [1]. A space-to-earth radio frequency energy harvesting (RF-EH) systemwas implemented using gigantic transmit antenna arrays on a satellite and receive antennaarrays at a ground station [2]. Due to the advancement in circuit design, low power RF-EH technology also finds its application in wireless communication systems [3], [4]. Theauthors in [3] proposed a system architecture for RF charging stations in an uplink cellularsystem. In [4], a harvest-then-transmit protocol was introduced for radio frequency energyharvesting wireless communication systems (RF-EH-WCSs). Moreover, various modernbeamforming techniques were employed to improve the energy harvesting efficiency forwireless nodes (WNs) [4], [5].Limited battery capacity has been a major obstacle for the WNs. For example, tradi-tional wireless sensor systems suffer from outage due to the limited battery lifetime of thesensors. Frequent battery replacement increases the operational expenditure. Sometimes,it is impossible to replace the batteries for those WNs that are embedded in the structureor implanted in the human body. On the other hand, the weather dependence and theindoor inaccessibility of the natural sources make the energy harvesting from the naturalsources less attractive. As a result, the RF-EH technology became popular (see [6–8] andreferences therein). This technology can provide controllable, constant, and predictableenergy supply for the WNs.In the RF-EH-WCSs, WNs can perform information detection (ID) and RF-EH fromthe RF signals. Hence, the RF-EH-WCSs have found their applications in various systemssuch as wireless sensor systems [9], [10], wireless body systems [11], and wireless chargingsystems [12], distributed antenna systems (DASs) [13] and wireless cooperative systems[14]. Commercial RF energy harvesters such as dowyrwustyr1 and Wotu systym2 havealready been introduced in the market.Ghttp:DDfififiCpofierxvstxoCxomDproyuxtsDpofierxvsterBtrvnsmittersDHhttp:DDfififiCossivinxCxomDxotvD11.1. IagebWhcgibaDespite a number of advantages of the RF-EH-WCSs, several challenges remain to besolved in order to fully exploit the potential of the RF-EH technology. Generally, thereare many use cases for the RF-EH-WCSs [15]. In this thesis, we consider the DASs withRF-EH capability, simultaneous information and power transferring (SWIPT) systems andwireless powered relaying (WPR) systems. In what follows, we provide a brief descriptionof these systems and describe major challenges associated with them.EBEBE DAgs kith fFAEH WupuvilityIt is reported that each year cellular operators spend more than 10 billion dollars onelectricity [16], and this amount keeps increasing at a rate of 10% on average every year [17].Cisco estimates that the wireless data traffic will continue to grow exponentially and reachover 24 exabytes per month in 2019 [18]. This trend translates to a tremendous bandwidthdemand and energy consumption. In order to reduce the operational expenditure, energyefficient protocol design has received a great deal of research attention.DASs have been introduced as a promising candidate architecture for future wirelesscommunication systems due to its enhancement in both spectrum efficiency (SE) and energyefficiency (EE) [19], [20]. In a typical DAS, the system functionalities are split betweena base band unit (BBU) and a set of low cost radio remote heads (RRHs). Specifically,the BBU performs computationally intensive signal processing tasks, while the RRHs areresponsible for basic computation and simple RF operations, e.g., up/down conversion andamplification. All the RRHs are distributed in the coverage area and connected to theBBU via optical fiber. Hence, the DASs can reduce the distance between transmitter andreceiver, which improve the performance of the WNs in the system.However, traditional energy efficient resource allocation algorithms only consider thechannel state information (CSI) between transmitters and receivers. Once each transmitteris equipped with an energy harvester, the resource allocation decision in the DAS willdepend not only on the CSI between transmitters and receivers, but also on the CSIbetween transmitters and energy harvesters. Thus, the energy efficient resource allocationalgorithms should be redesigned for the DASs with RF-EH capability.EBEBF gkIdh gystymsIn the SWIPT systems, a part of the received signal is used for ID, and the other partis used for RF-EH. The SWIPT systems can support reliable information reception andprolong the lifetime of energy constrained WNs. Based on the progression of different gen-erations of wireless communication systems, it can be predicted that the SWIPT systemswill be required to support various types of traffics having different quality-of-service (QoS)21.2. LigeeTghee Eeiiew TaW MbgiiTgibarequirements [21], [22]. On the other hand, to cope with the ever-increasing energy con-sumption, the SWIPT systems should minimize the energy consumption while maintaininga certain data rate. As a result, the resource allocation algorithms in the SWIPT systemsshould be carefully designed to support the traffics with heterogeneous QoS requirements,e.g., delay-sensitive traffic and best-effort traffic.EBEB3 kdf gystymsIn dual-hop wireless communication systems, an important use case of RF-EH technol-ogy is wireless relaying systems with energy constrained relay [14, 23, 24], which is alsoknown as WPR systems. The relay in the WPR systems can scavenge energy from the RFsignals of the source. As a result, the lifetime of the relays is prolonged, and the mainte-nance expenditure is reduced. On the other hand, the nonlinear input/output relation ofthe energy harvester makes the traditional linear energy harvester based analysis resultsinaccurate.To assist the future resource allocation, we first analyze the performance of the WPRsystems with nonlinear energy harvester. Due to the application of the RF-EH technologyat the relay, the first-hop and the second-hop signal-to-noise ratios (SNRs) are coupled.The coupled SNRs and the nonlinearity of the circuit components require a fundamentalanalysis on the performance of the WPR systems. Moreover, the analysis results will assistthe design of the resource allocation algorithms in the WPR systems.EBF Lityrutury fyviyw unx aotivutionEBFBE Enyrgy Ewiynt fysourwy Allowution Algorithms in DAgs kithfFAEH WupuvilityThe design of energy efficient resource allocation algorithms has been a topic underintense investigations [25]. With energy efficient resource allocation algorithm, both thegreenhouse gas emission and the operational expenditure of the wireless communicationsystems are reduced. Meanwhile, it has been demonstrated that the DASs outperform theco-located multiple antenna base station in EE [26]. Thus, several research efforts havebeen made on the EE issue of the DASs [27–31]. For examples, the authors in [27] studiedthe joint subchannel and power allocation problem in the DASs and proposed an optimalpower control algorithm that maximizes the system EE, which is defined as the ratio ofsystem SE over the total consumed power. However, the proposed algorithm does notguarantee the data rate requirements of users. In a related work [28], a power allocationalgorithm was developed to maximize the system EE in a fractional frequency reuse based31.2. LigeeTghee Eeiiew TaW MbgiiTgibamulticell communication systems. A statistical EE model was developed in [29] for theDASs, and a joint antenna selection and power control algorithm was proposed to optimizethe long term EE. In [30], the authors developed a subchannel allocation and power controlalgorithm that can enhance the EE of the DASs at expense of the SE. In [31], the authorsstudied the power control in the DASs, and an optimal power control algorithm was derivedby exploiting the mathematical property of the EE function.Recently, some progress has been made on the application of RF-EH technology [32–34] in wireless communication systems. In [32], the authors utilized a time switching (TS)energy harvester in a single relay network and studied the tradeoff between energy harvest-ing time and ergodic capacity. The authors in [33] proposed an energy efficient algorithmto jointly optimize TS factor of energy harvester, transmission power, and subchannelallocation for an orthogonal frequency division multiple access (OFDMA) based wirelesscommunication systems. The RF-EH issue at the base station was studied in [34] witha single relay scenario; however, the developed algorithm cannot be utilized in the DASswith multiple users due to the requirements of multiple beamforming vectors.To conclude, the system EE, which is always an important performance metric in thedesign stage of the wireless communication systems, has not been studied in the DASswith RF-EH capability. In Chapter 3, we consider an OFDMA based DASs with RF-EH capability, and we develop an energy efficient resource allocation algorithm for such asystem.EBFBF Joint dowyr Wontrol unx himy gwitwhing in gkIdh gystymskith Hytyrogynyous eog fyquirymyntsOne of the key features for the future SWIPT systems is to support heterogeneousQoS requirements. Several approaches such as scheduling [35], [36], admission control[37] and resource allocation [22, 33, 38] can be used to implement the heterogeneous QoSrequirements. In this thesis, we mainly focus on the resource allocation to support the het-erogeneous QoS requirements. The design of resource allocation algorithm for the SWIPTsystems has recently received a great deal of attention. The related works are brieflysummarized as follows. In [22], a joint beamforming and power splitting (PS) factor al-location algorithm was designed to minimize the consumed power under the constraintsof throughput and harvested energy. The authors in [38] investigated the optimal designof the SWIPT systems to maximize the weighted sum-rate for multiuser orthogonal fre-quency division multiplexing systems, where both TS and PS architectures were studied.The authors in [33] studied the EE of the SWIPT systems and proposed two algorithmsfor continuous and discrete PS factors. In [35], two scheduling schemes were proposed for41.2. LigeeTghee Eeiiew TaW MbgiiTgibathe SWIPT systems to control the tradeoff between the ergodic rate and the harvestedenergy. To this end, all resource allocation schemes in the SWIPT systems are based oninstantaneous CSI [21, 22, 33, 38]. Besides, the algorithms in [21], [22] are devised only forhomogeneous QoS requirements of the WNs.Motivated by the work in [21, 22, 33, 35, 38], we study the problem of joint power controland time switching (JPCTS) in SWIPT systems by considering the long-term power costand heterogeneous QoS requirements for WNs in Chapter 4.EBFB3 dyrzormunwy oz Amplizy unx Forwurx fyluying kith bukugumiAmFuxing unx bonlinyur Enyrgy HurvystyrThe recent research progress on the WPR systems are summarized as follows. Theend-to-end SNR outage probability was first studied for the WPR systems with amplify-and-forward protocol [14]. The work in [14] was extended to multiple source-destinationpairs scenario [23]. The performance of the WPR systems that can scavenge energy fromambient RF signals was studied in [24]. In [39], the outage and the diversity performances ofthe WPR systems were investigated using the theory of stochastic geometry. The scheme in[40] combined the conventional full-duplex relay with the RF-EH technology. By optimizingthe TS factor, the authors in [40] maximized the system throughput. However, the currentliterature [14, 23, 24, 39, 40] that studies the performance of WPR are based on conventionallinear energy harvesters. It was reported in [41] that the linear energy harvester is notpractical due to the nonlinearity of the diodes, inductors, and capacitors. To our bestknowledge, the performance of the WPR with nonlinear energy harvester has not beeninvestigated.While the aforementioned works provide a good understanding of the WPR systems[14, 23, 24, 39, 40], all of them assumed Rayleigh fading. However, the Nakagami-mfading is a generalized model that matches the various measurement data better thanthe Rayleigh fading [42]. Field test results showed that the Nakagami-m fading providedthe best matches to land mobile and indoor mobile multipath propagation [43]. On theother hand, a statistical analysis of the experiment data in [44] also showed that theNakagami-m distribution fits the urban multipath channel environment better than theother distributions such as Rayleigh, Rician, and log-normal distributions. Meanwhile,the beamforming technique qualifies the practical utilization of wireless power transfer[45]. Hence, a performance study of beamforming in the WPR with Nakagami-m fadingchannels and nonlinear energy harvester is a valuable step towards the practical applicationof the RF-EH technology. In Chapter 5, we study the performance of the WPR systemswith the Nakagami-m fading channels and a nonlinear energy harvester.51.3. Ghefif Bhgliae TaW 6bageiUhgibafEB3 hhysis cutliny unx WontrivutionsThe thesis is organized into six chapters. Chapter 1 presents the history and develop-ment of the RF-EH-WCSs. In addition, this chapter provides a detailed literature reviewrelated to the rest of this thesis.Chapter 2 provides detailed technical background for the entire thesis. First, thesystem-level architectures of the RF-EH receivers are presented and classified into threecategories based on the flow of the RF signals: antenna separating receiver, TS receiverand PS receiver. Second, the basic concepts in stochastic optimization theory are reviewed.Finally, the related channel fading models that will be adopted in this thesis are presented.In Chapter 3, an OFDMA-based full duplex distributed antenna system (FDDAS) withRF-EH capability is proposed. The energy efficient resource allocation problem in the FD-DAS with RF-EH capability is formulated to jointly allocate the transmission power andthe subchannel. The formulated problem is a mixed integer nonlinear problem (MINLP),which is NP-hard. To derive a low-complexity semi-distributed algorithm, we decouplethe formulated problem into two subproblems: subchannel allocation problem and powercontrol problem. Corresponding algorithms are proposed in this chapter to solve the twosubproblems. Compared with the joint subchannel allocation and power allocation (JSPA)algorithm developed in [30], the combination of the proposed algorithms has less compu-tational complexity.In Chapter 4, an innovative resource allocation framework is proposed to minimizethe long-term power consumption in the SWIPT systems with heterogeneous QoS require-ments. The long-term power minimization problem in the SWIPT systems with heteroge-neous QoS requirements is formulated to jointly allocate the transmission power and TSfactors. The formulated problem contains both long-term and short-term QoS constraints,which is difficult to solve via standard optimization methods. With the stochastic opti-mization theory, a low-complexity dynamic algorithm is proposed to solve the formulatedproblem suboptimally. By tuning an introduced control parameter, the suboptimal powerconsumption can approach its optimal value at the expense of the delay of best-effort traf-fic nodes (BENs). The proposed algorithm is easy to implement in practical systems as itonly requires the current CSI. Besides, the impact of continuous and discrete TS factors isstudied within the proposed resource allocation framework.In Chapter 5, a nonlinear energy harvester model is proposed for the WPR systems.The nonlinearity in the energy harvester makes the analysis results, which are based onthe linear energy harvester, inaccurate. Thus, the performance of the WPR systems withthe nonlinear energy harvester is studied. The beamforming technique and Nakagami-mfading model are used to make a step forward towards the practical application of the61.3. Ghefif Bhgliae TaW 6bageiUhgibafRF-EH technology. Numerical results show that the results based on the traditional linearenergy harvester overestimate the end-to-end SNR when the energy constrained relay nodeis placed closer to the source node. Moreover, the obtained analytical expressions willassist the design of the resource allocation algorithms in the WPR systems.Chapter 6 concludes the thesis. Future works are also suggested.7Whuptyr FVuwkgrounxIn this chapter, we review the system-level design on RF-EH receivers. We presentthe fundamental concepts of stochastic optimization theory, which is then followed by thebasics of some commonly used multipath fading channels.FBE fFAEH fywyivyrsElectromagnetic waves, such as infrared, X-rays and RF signals, convey energy. RFsignals are widely considered in the current RF-EH-WCSs. Typical frequency band for RFsignals ranges from 300MHz to 300GHz. To enable the receivers to harvest energy fromthe RF signals, an energy harvester should be incorporated into the traditional informa-tion receivers. This has motivated vast amount of research activities in the design of theRF-EH receiver architecture. For examples, the authors in [46], [47] studied the applicationof RF-EH technology in a point-to-point system and proposed an ideal receiver architec-ture. The proposed receiver architecture in [46], [47] is assumed to detect information andharvest energy from the same RF signal simultaneously. However, the proposed receiverarchitecture in [46], [47] is not realistic for circuit implementation because current RF-EHcircuits cannot decode the carried information from the same RF signals directly [48]. Asa result, researchers started to investigate the practical RF-EH receiver architecture. Wecan classify the current RF-EH receivers into three categories as follows:• Antynnu gypuruting fywyivyrN This type of RF-EH receiver is equipped withan energy harvester and information detector with independent antennas. Figure2.1 shows the block diagram of the antenna separating RF-EH receivers, where theantennas are divided into two groups, one for energy harvesting and the other forinformation receiving. Thus, the antenna separating receiver enables to perform theRF-EH and ID concurrently. Similar to the antenna separating receivers, one canalso combine an energy harvester with the information transmitter to enable thetransmitter to recycle energy from the ambient RF signals [13].• himy gwitwhing fywyivyrN Different from the previous antenna separating re-ceiver, the TS receiver harvests energy and detects information over orthogonal time82.2. FgbchTfgic BcgimimTgiba GhebelAnt #2Ant #1Energy HarvesterInformation DetectorFigure 2.1: Block diagram of the antenna separating receiver.slots as shown in Fig. 2.2. In this setting, the receiver separates each frame into twotime slots for RF-EH and ID, where /ts (/ts ∈ [0P 1]) and 1 − /ts are, respectively,the ratio of the RF-EH slot and the ID slot in a frame. The receiver can switch itsoperations periodically between energy harvesting and information detection. Hence,the receiver can trade the amount of harvested energy for the data rate by tuningthe ratio of RF-EH slot, /ts.Energy HarvesterInformation DetectorTime Switcherts1 r-tsrFigure 2.2: Block diagram of the time switching receiver.• dowyr gplitting fywyivyrN The PS receiver combines the energy harvester withthe information detector using a power splitter as shown in Fig. 2.3. The powersplitter separates the input power into two streams: one stream with power ratio0 ≤ /ps ≤ 1 is for RF-EH, and another stream, 1 − /ps, is used for ID. Differentfrom the TS setting, the power splitter introduces the processing noise zp to theinformation detector, which impairs the received SNR.FBF gtowhustiw cptimizution hhyoryStochastic optimization theory uses the Lyapunov drift function, whose analytical ex-pression will be defined later, to control the system performance dynamically [49]. Thestudied system operates in discrete time mode with the index of each frame t denoting92.2. FgbchTfgic BcgimimTgiba GhebelEnergy HarvesterInformation DetectorPower Splitterps1 r-psrpzFigure 2.3: Block diagram of the power splitting receiver.a unit time interval [tP t+ 1), t ∈ {0P 1P . . . P i}, where the term i can take an arbitrarylarge value. Now, we consider a system of c queues corresponding to c WNs, and letq (t) = [qi (t)]c×1 denote queue backlog vector. The Lyapunov function a (q (t)) is definedas a scalar function of the queue backlog vector q (t). The value of the Lyapunov functiona (q (t)) increases when total queue backlog moves toward unstable. Control actions aremade in each frame to optimize the performance metric Obj (t) while keeping the Lya-punov drift function finite. In the sequel, we present a brief review of several fundamentalconcepts in the stochastic optimization theory.FBFBE euyuy DynumiwsEach entry qi (t) of queue backlog vector q (t) evolves with the stochastic arrival processand the server process, i = 1P 2P . . . P c . The initial state of qi (0) <∞ is a random variableand takes a nonnegative value. With the arrival process λi (t) and the server process gi (t),we can define the queue dynamic function asqi (t+ 1) = max [qi (t)−gi (t) P 0] + λi (t) (2.1)where the arrival process λi (t) and the server process gi (t) are random variables at t-thframe and take nonnegative real values. We call qi (t) the backlog in frame t, as it canrepresent an amount of work that needs to be done. For example, the backlog of a trafficqueue qi (t) represents the amount of information bits in the traffic queue.Dynition FBEB (ayun futy gtuvly [49]) A discrete-time queue qi (t) is mean rate stableif limt→∞E[|qi(t)|]t = 0, where E [·] denotes the expectation operator.fymurk 2.2B If a discrete-time queue is mean rate stable, the time average expected arrivalis less than or equal to the time average expected departure such that the backlog of sucha queue is finite. This statement can be justified as follows.102.2. FgbchTfgic BcgimimTgiba GhebelFrom the dynamic function (2.1), we obtainqi (t+ 1)− qi (t) = max [λi (t)−gi (t) P λi (t)− qi (t)] . (2.2)Taking the expectation and the telescoping sums of (2.2) for 0 ≤ t ≤ i , we haveE [|qi (i )|]i≥ E [qi (i )]i− E [qi (0)]i≥i∑t=0E [λi (t)−gi (t)]t. (2.3)Letting i → ∞, we obtain limi→∞ 1i∑it=0 E [λi (t)] ≤ limi→∞ 1i∑it=0 E [gi (t)], wherethe terms limi→∞ 1i∑it=0 E [λi (t)] and limi→∞1i∑it=0 E [gi (t)] correspond to the timeaverage expected arrival rate and time average expected processing rate.A mean-rate-stable traffic queue indicates the information bits in such a queue willdepart this queue in finite time.FBFBF Lyupunov Drizt FunwtionTo measure of the queue backlog vector q (t), we define a scalar quadratic Lyapunovfunction a (q (t)) asa (q (t)) =12‖q (t)‖22 (2.4)where the operator ‖·‖2 denotes the u2-norm.The Lyapunov function defined in (2.4) is always nonnegative, and it is equal to zeroif and only if all components of q (t) are zero. Generally, there are some other forms ofLyapunov function according to different applications, such asa (q (t)) = qi (t) log (1 + q (t)) (2.5)anda (q (t)) =c∑i=1exp (−ϑ (κ− qi (t))) (2.6)where ϑ and κ are positive auxiliary variables.FBFB3 cnyAslot Wonxitionul Lyupunov Drizt FunwtionWith the defined Lyapunov function, we can calculate the one-frame conditional Lya-punov drift function as∆ (q (t)) , E [a (q (t+ 1))− a (q (t)) |q (t) ] . (2.7)112.3. 6hTaael MbWelfThis drift is the expected change in the Lyapunov function over one frame, given that thecurrent state in tth frame is q (t).By introducing a control parameter k S 0, we can define the Lyapunov drift-plus-penalty as∆ (q (t)) + k E [Obj (t) |q (t) ] (2.8)where Obj (t) represents a quantity that can be traded for queue backlog.FB3 Whunnyl aoxylsWireless channels can experience multipath fading and shadowing. Multipath fadingis due to the constructive and destructive combination of randomly delayed, reflected,scattered, and diffracted signal components. This type of fading is relatively fast, andtherefore is responsible for short-term signal variations, where both the signal envelopeand signal phase fluctuate over time.Depending on the symbol duration of the transmitted signal and the coherence time offading channels, fading can be classified into slow fading and fast fading. Coherence time isdefined as the time period over which we can consider the fading process to be correlated.Slow fading occurs when the symbol duration is less than the channel coherence time,and fast fading occurs when the symbol duration is greater than the channel coherencetime. Similarly, depending on the relation between the transmitted signal bandwidth andthe channel coherence bandwidth, fading can also be classified into frequency-nonselectivefading and frequency-selective fading. Coherence bandwidth is defined as the frequencyrange over which the fading process is correlated. If the transmitted signal bandwidth ismuch smaller than the channel coherence bandwidth, the fading is considered to be flat;otherwise, the fading is considered to be frequency selective.In this thesis, we only focus on slow and frequency-nonselective fading channels. It iscommon to use statistical distributions to describe the random behavior of the receivedsignal amplitude over the fading channels. Most widely used statistical models include theRayleigh, Rician, and Nakagami-m distributions.FB3BE fuylyigh DistrivutionThe Rayleigh distribution is frequently used to model the time varying characteristicsof the received signal amplitude in a wireless propagation scenario where there is no directline-of-sight (LOS) path between the transmitter and the receiver. The probability density122.3. 6hTaael MbWelffunction (PDF) of the Rayleigh distributed random variable m is given byfm (x) =xσ2exp(− x22σ2)=2xΩhayleighexp(− x2Ωhayleigh)P x ≥ 0 (2.9)where Ωhayleigh = 2σ2 is the mean square value of the received signal amplitude.FB3BF fiwiun DistrivutionWhen a LOS path exists between the transmitter and the receiver in addition to weakerrandom multipath signal components, the received signal amplitude is modeled as a Riciandistributed random variable m whose PDF is given byfm (x) =2x (J + 1)Ωhiyianexp(−J − (J + 1)x2Ωhiyian)I02x√J (J + 1)Ωhiyian P x ≥ 0 (2.10)where J = VH2σHis the Rician factor defined as the ratio of the LOS power V2 to the scatteredpower 2σ2, the average amplitude power is denoted by Ωhiyian = E[m2]= V2 + 2σ2,and Iv (·) is the ,th-order modified Bessel function of the first kind defined as I, (x) =∑∞m=0(x/2),+Hmm!Γ(,+m+1) . As the strength of the LOS signal diminishes to zero, i.e., J = 0, theRician distribution specializes to a Rayleigh distribution. As the value of J approachesinfinity, the fading effect tends to vanish.FB3B3 bukugumiAm DistrivutionIntroduced by Nakagami in the early 1940’s, the Nakagami-m distribution is a versatiledistribution used to model the multipath fading in wireless channels. Empirical data showthat the Nakagami-m fading model often gives the best fit to land-mobile and indoor-mobile multipath propagation. The PDF of the Nakagami-m distributed random variablem is given byfm (x) =2Γ (m)(mΩNakagami-m)mx2m−1 exp(− mx2ΩNakagami-m)P x ≥ 0Pm ≥ 12(2.11)where ΩNakagami-m is the mean square value of the amplitude, and Γ (·) is the Gamma functiondefined as Γ (x) =∫∞0 xx−1z−xyx [50, eq. (8.310.1)]. The fading severity parameter m isdefined asm = ΩHE[(mH−Ω)H] . Beyond its empirical justification, the Nakagami-m distributionis often used for the following reasons. First, the Nakagami-m distribution can be usedto model fading conditions more or less severe than the Rayleigh fading. When m = 1,the Nakagami-m distribution specializes to the Rayleigh distribution. When m = 0.5, it132.3. 6hTaael MbWelfbecomes the one-sided Gaussian distribution. As the value of the parameter m increases,the fading severity decreases. Second, the Rician distribution can be approximated by theNakagami distribution with J =√mH−mm−√mH−m and m =(J+1)H2J+1 for m S 1. The correspondingsquared Nakagami-m fading amplitude has a Gamma PDF asfmH (x) =(mΩNakagami-m)mΓ (m)xm−1 exp(− mxΩNakagami-m)P x ≥ 0Pm ≥ 12. (2.12)14Whuptyr 3Enyrgy Ewiynt fysourwyAllowution zor cFDaA FullDuplyfi Distrivutyx Antynnugystyms kith Enyrgy HurvystingIn this chapter, we consider an OFDMA based FDDAS with RF-EH capability anddevelop an energy efficient resource allocation scheme for such a system. In particular,in order to optimize the EE of FDDAS, we formulate the problem of joint subchannelallocation and power control as a MINLP problem. Since the formulated optimizationproblem is NP-hard whose computation complexity rises exponentially with larger systemscale, we develop a low complexity subchannel allocation and power control algorithm. Theconvergence property of the proposed algorithm is established. Simulation results show thatthe proposed algorithm for the FDDAS has a higher system EE than the existing algorithmfor the FDDAS without RF-EH capability.3BE gystym aoxyl3BEBE cvyrull DyswriptionWe consider the downlink transmission scenario of the FDDAS which consists of a BBU,c WNs, b RRHs, that use OFDMA technique with K subchannels assuming c ≪ K.Let N = {1P 2P . . . P c},M = {1P 2P . . . Pb} and K = {1P 2P . . . PK}, respectively, denote theset of WNs, RRHs and subchannels. As shown in Figure 1, all RRHs are distributed acrossthe coverage area and are connected to BBU via optical fiber backhaul links. All WNsare assumed to have single antenna for reception. It is assumed that all wireless channelsexperience block fading, which indicates the channel fading remains constant during aframe and varies for different frames. The CSI is perfectly known at the BBU as well as atthe RRHs. We assume that each RRH is equipped with two antennas: one for downlinkinformation transmission and one for RF-EH. Thus, RRHs are operating in full duplex153.1. Flfgem MbWelRRH 2RRH 3RRH 6RRH 4RRH 5BBU/RRH 1RechargablebatteryEnergyharvesterInformationtransmitterPowerPowerInformation SignalOptical FibreWN 1WN 2Figure 3.1: A full duplex distributed antenna system withb = 6 antennas andc = 2WNs.The northeast of this figure illustrates a RRH which is equipped with both informationtransmitter and energy harvester.(FD) mode and can transmit information and recycle energy simultaneously. The circuitdesign of RRHs depends on the specific type of RF-EH techniques, e.g., electromagneticinduction and electromagnetic radiation. In this chapter, we focus on recycling energy viaelectromagnetic radiation and each RRH is equipped with both information transmissionunit and energy harvesting unit which can recycle energy across all subchannels over thedownlink period. In addition, the recycled energy is used to replenish a rechargeablebattery of each RRH.3BEBF gystym gE, dowyr Wonsumption unx EEThe received signals at WN i and at RRH j can be respectively expressed asyiPmN =∑j∈M∑k∈KvkjPihkjPi√pkjPiskjPi + ziPmNP ∀i ∈ N (3.1)andyjPhhH =∑i∈N∑n∈M∑k∈KvknPigknPj√pknPisknPi + zjPhhHP∀j ∈M (3.2)where vkjPi (or vknPi) is a binary variable which equals to 1 if the subchannel k is allocatedto user i from RRH j (or n); otherwise, it equals to 0; pkjPi (or pknPi) denotes the transmit163.1. Flfgem MbWelpower from RRH j (or n) to WN i on subchannel k; skjPi (or sknPi) denotes the data symboltransmitted from RRH j (or n) to WN i on subchannel k. Both values of E[|skjPi|2] andE[|sknPi|2] are equal to 1. In (3.1), hkjPi denotes the complex fading coefficient between RRH jand WN i on subchannel k. The complex fading coefficient from RRH n to RRH j (n ̸= j),is denoted by gknPj , and gknPn is the complex coefficient of nth loop link. In (3.1)-(3.2), ziPmNand zjPhhH, respectively, denote the additive white Gaussian noise at WN i and RRH j. Weconsider all the channels (except the loop links) experience independent Rayleigh fadingand pathloss attenuation. The SE in nats/sec/Hz of WN i can be written asgi (P PO) =∑j∈M∑k∈KvkjPi log(1 + pkjPiHkjPi)(3.3)where P ,[pkjPi]b×c×Kis the power allocation matrix, O ,[vkjPi]b×c×Kis the sub-channel allocation matrix, HkjPi ,|hkj;i|HσHiwhere σ2i is the power of ziPmN.The net power consumption of the considered FDDAS can be denoted asjTP (P PO) = eBBk +behhH+1ψ∑i∈N∑j∈M∑k∈KvkjPipkjPi︸ ︷︷ ︸Power Concumed by Amplifiers− η∑i∈N∑j∈M∑n∈M∑k∈KvkjPipkjPi∣∣∣gknPj∣∣∣2︸ ︷︷ ︸Power Recycled by Energy Harvesters(3.4)where eBBk andbehhH respectively denote the circuit power consumed at BBU and RRHs;the term 1ψ∑i∈N∑j∈M∑k∈KvkjPipkjPi is the total power consumed by the power amplifiers of allRRHs with the power amplifier’s efficiency ψ; the term η∑j∈M∑i∈N∑n∈M∑k∈KvkjPipkjPi∣∣∣gkjPn∣∣∣2 isthe energy harvested by all RRHs with the energy harvesting efficiency3, η ∈ [0P 1].Following [29–31], we define the system EE of FDDAS as the ratio of the overall SEand the total power consumptionZZ (P PO) =j (P PO)jTP (P PO)(3.5)where j (P PO) =∑i∈Ngi (P PO) is the overall SE of FDDAS.3In this xhvpterA the energy hvrvesting exienxy is the rvtio of the output pofier fiith the input pofierof the energy hvrvesterC173.2. FhUchTaael 4llbcTgiba TaW Cbwee 6bagebl3BF guvwhunnyl Allowution unx dowyr WontrolIn this section, we formulate the problem of subchannel allocation and power control asan optimization problem that maximizes the system EE of the considered FDDAS. Then,we develop a low complexity suboptimal subchannel allocation and power control algorithmby decoupling the original optimization problem.3BFBE drovlym FormulutionThe goal is to obtain the subchannel allocation matrix O and the power allocationmatrix P to maximize ZZ (OPP ) in (3.5). Mathematically, we can formulate the EEmaximization problem asmax(P PA)ZZ (P PO) =j (P PO)jTP (P PO)(3.6a)s.t.∑k∈K∑i∈NvkjPipkjPi ≤ emaxP ∀j (3.6b)gi (P PO) ≥ gmini P ∀i (3.6c)∑j∈M∑i∈NvkjPi ≤ 1P ∀k (3.6d)vkjPi ∈ {0P 1} P ∀iP jP k (3.6e)0 ≤ pkjPi ≤ emaxP ∀iP jP k (3.6f)where the constraint (3.6b) is the power budget of a RRH; the constraint (3.6c) is toguarantee the minimum SE requirement gmini ; the constraint (3.6d) indicates that eachsubchannel can be assigned to at most one RRH-WN link; the constraint (3.6e) indicatesthat vkjPi is a binary variable which takes value either 0 or 1; and (3.6f) is the power rangeof each RRH-WN link in a given subchannel.The optimization problem in (3.6) contains two types of variables: the linear powerallocation variables and the binary subchannel allocation variables. The objective function(3.6a) is a quasi-concave function of the power allocation variables pkjPi; the throughput ofWN i, gi (P PO), is a concave function of pkjPi and a linear function of vkjPi. Due to thebinary variables vkjPi, the formulated optimization problem in (3.6) is a MINLP problem,which is NP-hard [51]. To obtain a near optimal solution to such a problem, a generalapproach is to relax the binary variable vkjPi, and subsequently solve the relaxed problemusing the so-called subgradient method [52]. However, the complexity of this approach isO (8b3c3K3) [53], which is prohibitively large even for moderate values of b , c , andK.183.2. FhUchTaael 4llbcTgiba TaW Cbwee 6bageblIn what follows, we decouple the original optimization problem in (3.6) into to subprob-lems, namely, the subchannel allocation problem and the power control problem. Then,we develop a suboptimal subchannel allocation algorithm and a power control algorithmseparately.3BFBF guvoptimul guvwhunnyl Allowution AlgorithmTo develop the subchannel allocation scheme, we define a new variable p˜kjPi = vkjPipkjPi,∀iP jP k such that P˜ =[p˜kjPi]b×c×K, and rewrite the original optimization problem in (3.6)asmaxAZZ (P PO) =j(P˜ PO)jTP(P˜ PO) (3.7a)s.t.∑i∈N∑k∈Kp˜kjPi ≤ emaxP ∀j (3.7b)∑j∈M∑i∈NvkjPi ≤ 1P ∀k (3.7c)vkjPi ∈ {0P 1} P (3.7d)0 ≤ p˜kjPi ≤ vkjPiemaxP ∀iP jP k (3.7e)∑j∈M∑k∈KvkjPi log(1 +p˜kjPiHkjPivkjPi)≥ gmini P ∀i. (3.7f)droposition 3BEB ]z thy wonstruint (3.7f) is ignoryx, givyn P˜ thy optimizution provlym in(3.7a)A(3.7e) wun vy solvyx vy thy zollowing suvwhunnyl ullowution ulgorithmvkj∗Pi∗ = 1P (j∗P i∗) = argmaxjPiHkjPi0P otherwise.(3.8)droozB See Appendix A.Since the subchannel allocation algorithm in (3.8) does not consider SE requirement ofWNs, additional steps are required to guarantee each WN’s SE requirement. The resultingsubchannel allocation algorithm is summarized in Algorithm 1. In Algorithm 1, Steps5-12 allocate subchannels to WN i, ∀i, until its minimum SE requirement is satisfied.Then Steps 13-21 allocate the remaining subchannels to the RRH-WN links using a greedyalgorithm [54]. According to the greedy algorithm, a particular subchannel in the remainingsubchannel set is allocated to the RRH-WN link that has the highest channel fading gain.193.2. FhUchTaael 4llbcTgiba TaW Cbwee 6bageblAlgorithm E Proposed Subchannel Allocation1: Initiulizution2: Set vkjPi = 0 and gi = 0, ∀iP jP k3: BBU equally divide K subchannel among b RRHs, i.e., the number of subchannelallocated to RRH j, Kj =Kb P ∀jI: RRH j divide its power equally on Kj subchannels, ∀j5: zor i = 1 : c xo6: whily gi < gmini xo7: vk∗j∗Pi = 1, where (j∗P k∗) = argmaxjPkHkjPiM: K := K \ k∗N: Kj := Kj − 11E: gi := gi + vk∗j∗Pi log(1 + bemaxK Hk∗j∗Pi)11: ynx whily12: ynx zor13: whily K ̸= ϕ xo C ϕ is the null set1I: vkj∗Pi∗ = 1, where (j∗P i∗) = argmaxjPiHkjPi15: iz Kj∗ S 0 thyn16: Kj∗ := Kj∗ − 117: K := K \ k1M: ylsy1N: M :=M\ j∗2E: ynx iz21: ynx whily22: The subchannel allocation matrix O∗ is acquiredAfter this procedure, all the subchannels are assigned to WNs.3BFB3 dowyr Wontrol AlgorithmTo derive the power control algorithm, we first transform the original fractional formobjective function in (3.7a) into a subtractive form function, which results in a convexoptimization problem (3.9). On the other hand, the ADMM algorithm [55] can solveconvex optimization problems by breaking them into several smaller subproblems, whichcan be solved in a parallel way. Solving all the subproblems has the same result of solvingthe original problem. Besides, each of the subproblem can be easier to handle comparedwith the original problem. Hence, we leverage the ADMM algorithm to solve the proposedoptimization problem (3.9).For the subchannel allocation matrix O∗ =[(vkjPi)∗]b×c×Kobtained using Algorithm203.2. FhUchTaael 4llbcTgiba TaW Cbwee 6bagebl1, we can rewrite the optimization problem in (3.7a)-(3.7f) in a subtractive form asmineP − U(P˜ PO∗)= qjTP(P˜ PO∗)− j(P˜ PO∗)(3.9a)s.t.∑k∈K∑i∈Np˜kjPi ≤ emaxP ∀j (3.9b)gi(P˜ PO∗)≥ gmini P ∀i (3.9c)0 ≤ p˜kjPi ≤ vk∗jPiemaxP ∀iP jP k (3.9d)where q is an auxiliary variable4 and the update procedure for q will be shown in Algorithm2. The optimization problem in (3.9) can be solved using ADMM, which is fast convergingand has the flexibility of distributing the computational load between the BBU and theRRHs. Generally, there are two basic forms of ADMM: unscaled form and scaled form [55].In this work, we select the unscaled form of ADMM for its concise expression.In order to solve the optimization problem in (3.9) using unscaled form ADMM, weintroduce two auxiliary vectors: u and v where u ∈ RbcK×1 consists of elements in powerallocation matrix P˜ , and v ∈ RbcK×1 is a global auxiliary vector with each elementcorresponding to that in u. Let uj ∈ RbcK×1 be the vector associated with RRH j andu =∑j∈M uj . Also, we define Φ1, Φ2 and Φ3 as the set of constraints with respect to(3.9b), (3.9c) and (3.9d) respectively. In addition, we also introduce an indicator functionδ (v) ={0P v ∈ Φ2∞P otherwise. (3.10)Using these notations, we can reformulate the optimization problem in (3.9) into a generalform consensus optimization with regularization [55] asmin(uPv)− U (uP O∗) + δ (v)s.t. u− v = Du ∈ Φ1 ∩ Φ3(3.11)where v is the global consensus variable and u− v = D is the consensus constraint. Here,the indicator function δ (v) is regarded as regularization to be handled by the BBU.4It hvve ween provey in p33A iheorem 1r thvt min eP V∗UTP( eP ;A∗)− U ( eP ;A∗) R V∗UTP ( eP ∗;A∗)−U( eP ∗;A∗) R EA fihere eP ∗ is the optimvl pofier vlloxvtion mvxtrixC213.3. FimhlTgiba EefhlgfThe augmented Lagrangian in scaled form of (3.11) can be derived asaφ = −U (uPO∗) + δ (v)− <2‖‖22 +<2‖u− v + ‖22 (3.12)where  ∈ RbcK×1 is the scaled dual variable vector and < S 0 is a constant calledpenalty parameter. The optimization problem in (3.11) can be solved with following stepsuτ+1 := arg minu∈ΦG∩Φ3{−U (uPO∗) + <2‖u− vτ + τ‖22}(3.13)vτ+1 := arg minv∈ΦH{∥∥uτ+1 − v + τ∥∥22}(3.14)τ+1 := τ +(uτ+1 − vτ+1) (3.15)where t denotes the iteration index.Expanding (3.13), we obtain (3.16) asuτ+1j := arg minuj∈θj{∑i∈N∑k∈KqukjPi(1ψ−∑n∈Mη∣∣∣gkjPn∣∣∣2)−∑i∈N∑k∈Kvk∗jPi log(1 +ukjPiHkjPivk∗jPi)+<2∑i∈N∑k∈K[ukjPi −(zkjPi)τ+(µkjPi)τ]2}P ∀j ∈M (3.16)which can be solved distributively at each RRH.According to [55], eq. (3.14) can be simplified tovτ+1 =∏ΦH(uτ+1 + τ)(3.17)where∏ΦHdenotes the Euclidean projection onto Φ2. Our proposed power control algo-rithm is summarized in Algorithm 2, whose convergence property is established in AppendixB.3B3 gimulution fysultsIn this section, we present simulation results to demonstrate the performances of ourproposed subchannel allocation and power control algorithm. The simulation parametersare specified in Table 3.1.The RRHs in the system are placed at the center and the four vertexes of a 8 meters×8 meters square area. The WNs are uniformly distributed over the square area. Thechannel power gain of each link consists of both pathloss and short-term fading. Theshort-term fading power gain is modeled using an exponentially distributed random variable223.3. FimhlTgiba EefhlgfAlgorithm F Power Control using ADMM (PC-ADMM)1: Initiulizution2: Each RRH j ∈M collects CSI and reports it to BBU3: BBU decides the subchannel allocation matrix O∗ through Algorithm 1I: BBU initializes u0 = D, v0 ∈ Φ2, 0 S 0, punishment factor q = 0, stop criteria ξ S 0and iteration index τ = 05: whily U(P˜ PO∗)S ξ xo6: Each RRH j ∈M updates uτ+1j by (3.16) and reports it to BBU simultaneously7: BBU combines uτ+1j into uτ+1 =∑j∈M uτ+1j , ∀j ∈MM: BBU updates vτ+1 through (3.17)N: BBU updates τ+1 through (3.15)1E: BBU updates qτ+1 :=j(v+GPA∗)jTP(v+GPA∗)and distributes qτ+1 to each RRH throughoptical fiber11: τ := τ + 112: ynx whily13: Optimal EE is achieved, i.e., ZZ∗ = qwith unit mean.Figures 3.2 and 3.3 depict the convergence behavior of our proposed PC-ADMM algo-rithm over several iterations with the number of users c = 5 and c = 15. In Figs. 3.2 and3.3, we compare our algorithm with the JSPA algorithm, which was developed in [30] tomaximize the EE in DAS without RF-EH capability. From Figs. 3.2 and 3.3, we observethat our proposed PC-ADMM algorithm converges after 3-5 iterations, and have the simi-lar number of iterations to that of the JSPA algorithm. However, compared with the JSPAalgorithm, the proposed PC-ADMM algorithm is not required to solve the optimizationproblem in (3.11) in each iteration. This key feature of Algorithm 2 can save computationexpenditure for the FDDAS. On the other hand, from Figs. 3.2 and 3.3, we also observethat the required number of iterations for convergence of our proposed algorithm doesnot change with the number of users in the system. This suggests that the PC-ADMMalgorithm can be utilized in a FDDAS with a large number of users. In addition, Figs.3.2 and 3.3 also illustrate that the proposed PC-ADMM algorithm designed for FDDASoutperforms the JSPA algorithm in DAS without RF-EH capability.Figure 3.4 plots the tradeoff between system EE and SE for the proposed PC-ADMMalgorithm and the JSPA algorithm. We observe that the proposed algorithm performsbetter in terms of EE than the JSPA algorithm in DAS without RF-EH capability. Wealso observe that the system EE decreases as the target SE increases. This phenomenoncan be explained as follows. As the SE requirement of WNs increases, more power isrequired to meet the higher SE requirement. Figure 3.4 also shows that the performance233.4. FhmmTelTable 3.1: Simulation Parameters SettingParameters taluesSimulation scenario 8 meters× 8 metersTotal bandwidth 5 MHzBandwidth per subchannel 180 KHzNoise power of unit bandwidth 10−G7:4 mW/HzQuantization noise per subchannel 10−4:7 mWCircuit power cost RRH: 10 mW, BBU: 100 mWTX power of RRH 200 mW (maximum)Antenna gain RRH: 9 dBi, WN: 0 dBiMinimum distance between WN and RRH 2 metersPathloss factor, α 3Harvester efficiency, η 0.8Amplifier efficiency, ψ 0.28Number of RRHs, b 5Number of WNs, c 5Number of Subchannels, K 15gain of our proposed PC-ADMM algorithm over the JSPA algorithm becomes larger asthe target SE increases. For example, the performance gain of the proposed PC-ADMM isabout 30% for a target SE of 24 nats/sec/Hz. This is because that to support the higherSE of each WN, the RRHs are required to transmit more power, which eventually leads tomore recycled energy.3BH gummuryIn this chapter, we developed an energy efficient resource allocation scheme for theOFDMA based FDDAS with RF-EH capability. We formulated the problem of subchannelallocation and power control as a MINLP problem, which is NP-hard. We proposed alow complexity semi-distributed algorithm by exploiting the ADMM and fractional pro-gramming. The convergence of the proposed algorithm was established. Simulation resultsshowed that our proposed algorithm outperforms the JSPA algorithm for the DASs withoutRF-EH capability in terms of EE.243.4. FhmmTel1 2 3 4 5 600.10.20.30.40.50.60.70.80.9Number of iterationEnergy Efficiency (nats/joule/Hz)  PC−ADMMJSPA wo Energy RecyclingFigure 3.2: Illustration of convergence of the proposed PC-ADMM algorithm and the JSPAalgorithm with 5 WNs in the system.1 2 3 4 5 600.10.20.30.40.50.60.70.8Number of iterationEnergy Efficiency (nats/joule/Hz)  PC−ADMMJSPA wo Energy RecyclingFigure 3.3: Illustration of convergence of the proposed PC-ADMM algorithm and the JSPAalgorithm with 15 WNs in the system.253.4. FhmmTel14 16 18 20 22 240.20.250.30.350.40.450.50.550.60.650.7Spectrum Efficiency (nats/sec/Hz)Energy Efficiency (nats/joule/Hz)PC−ADMMJSPA wo Energy RecyclingFigure 3.4: System EE versus SE requirement.26Whuptyr HJoint dowyr Wontrol unx himygwitwhing zor gkIdh gystymskith Hytyrogynyous eogfyquirymyntsThe JPCTS problem is investigated for SWIPT systems that support two types of traf-fic, namely, delay sensitive traffic and best effort traffic, simultaneously. Using a cross-layerdesign approach and the stochastic optimization theory, a dynamic algorithm is proposedto jointly allocate the transmission power and TS factor at the RF-EH receivers. As such,the QoS of the supported traffics is satisfied while the overall average power consumptionof the SWIPT system is minimized. By introducing a control parameter, the proposedalgorithm design can provide a tradeoff between the average power consumption and QoSrequirements of BENs. Simulation results depict that the proposed algorithm can tradethe average power consumption for the delay of BENs. Meanwhile, by setting differentminimum service rates of delay sensitive traffic nodes (DSNs), the algorithm can also tradethe delay of DSNs for the average power consumption.HBE gystym aoxyl unx drovlym FormulutionHBEBE cvyrull DyswriptionAs shown in Fig. 4.1, we consider the downlink transmission of a SWIPT systemwhich consists of a single antenna central node (CN) and c single antenna WNs. LetN = {1P 2P . . . P c} represent the set of WNs. Each WN is equipped with a TS energyharvester (c.f. Fig. 4.1). Based on the delay requirement, we classify the WNs into twosets: BENs and DSNs. The set of DSNs and BENs are, respectively, denoted by D and Bso that N = D ∪ B. We assume WNs are connected to the CN via orthogonal channels.The SWIPT system operates in discrete time mode with the index of each frame t denotinga unit time interval [tP t+ 1), t ∈ {0P 1P 2P . . .}; therefore, the terms “power” and “energy”274.1. Flfgem MbWel TaW CebUlem FbemhlTgibaCNWN 1WN 2. ..WN NEHIDInformation SignalEnergy Signal1l2l Nl. . .( ),tsi tr ( ), ts1 i tr-. . . . . .jK2K11jK+1KEH ID1Q2Q NQDL Traffic QueuesFigure 4.1: A SWIPT system with multiple WNs scavenging energy and detecting infor-mation from the same signal. The figure also shows the frame structure with WN i using/iPts (t) of tth time slot for EH and 1− /iPts (t) for ID.will be used interchangeably. To guarantee the energy causality5, all WNs operate RF-EHfirst and ID afterwards (c.f. Fig. 4.1). We assume the harvested energy from the RFsignals is used to compensate the energy consumed by the circuit at each WN because theamount of harvested energy is limited. Here, the nth WN uses /iPts (t) portion of the tthframe for EH. The value of /iPts (t) can be either discrete or continuous, i.e.,/iPts (t) ∈{1KP . . . PjKP . . . P 1}P ∀iP t (4.1)or/iPts (t) ∈ [0P 1] P ∀iP t. (4.2)At CN, c DL queues corresponding to c WNs are used to buffer the random arrivalinformation bits. The queue backlog and the arrival process at CN for WN n in the tthframe are, respectively, denoted by qi (t) and λi (t), where λi (t) is identical independent dis-tributed (i.i.d.) over all frames with E [λi (t)] = λ¯i. The complex channel coefficient of WNi in the tth frame is denoted by hi (t), which is i.i.d. over all frames. Hereinafter, we definethe output of the JPCTS scheme for WN i in the tth frame as θi (t) , {pi (t) P /iPts (t)},where pi (t) and /iPts (t) are, respectively, the transmission power and TS factor of WN5Energy xvusvlity mevns thvt the energy hvrvestey in the future xvnnot we usey for the present inforBmvtion yetextionC284.1. Flfgem MbWel TaW CebUlem FbemhlTgibai in the tth frame. For notational convenience, we denote the arrival process, the queuebacklog, the output of JPCTS scheme for the overall system and the complex channel co-efficient vector of the SWIPT system respectively by  (t) = [λi (t)]c×1, q (t) = [qi (t)]c×1, (t) = [θi (t)]c×1 and v (t) = [hi (t)]c×1, where the mean arrival rate of the SWIPTsystem is denoted as E [ (t)] =  with  ,[λ¯i]c×1. We also consider unit bandwidth.The transmission rate and harvested energy in the tth frame for WN i are, respectively,denoted bygi (t) = (1− /iPts (t)) log(1 + pi (t)|hi (t)|2σ2i)(4.3)andeiPrec (t) = η/iPts (t) pi (t) |hi (t)|2 (4.4)where σ2i is the noise power at WN i, and η is the energy harvesting efficiency of the energyharvester6, η ∈ [0P 1].Each traffic queue qi (t) for BEN i evolves asqi (t+ 1) = max [qi (t)−gi (t) P 0] + λi (t) P i ∈ B. (4.5)Without loss of generality, the terms gi (t), λi (t) and qi (t) in (4.5) have the unit ofnats/Hz. We define the total power consumption and the instantaneous throughput at theCN in the tth frame, respectively, aseTP (t) =∑i∈N pi (t) (4.6)andg (t) ,∑i∈N gi (t). (4.7)Hence, we can obtain the time average throughput and average power consumption of theSWIPT system, respectively, asg¯ = limi→∞1ii−1∑t=0E [g (t)] (4.8)ande¯TP = limi→∞1ii−1∑t=0E [eTP (t)]. (4.9)To quantify the delay of BENs, we first review the concept of stability of discrete-time6In this xhvpterA the energy hvrvesting exienxy is the rvtio of the output pofier fiith the input pofierof the energy hvrvesterC294.1. Flfgem MbWel TaW CebUlem FbemhlTgibaqueue. With Definition 2.1, we can characterize the QoS metric of BENs by guaranteeingthe BENs’ traffic queue mean rate stable. For DSNs, the QoS metric is assessed by theinstantaneous service rate. As in [21, 22, 38], the EH constraint requires that the outputpower of the energy harvester must be large enough to power ID modules, e.g., amplifierand decoder.HBEBF drovlym FormulutionOur objective is to minimize the time average power consumption and to maintain acertain amount of throughput in the SWIPT system. Mathematically, this objective canbe formulated as an optimization problem asmin(t)e¯TP (4.10a)s.t. g¯ ≥ gth (4.10b)qi (t) are mean rate stableP i ∈ BP ∀t (4.10c)gi (t) ≥ gmini P i ∈ DP ∀t (4.10d)eiPrec (t) ≥ emini P ∀iP t (4.10e)(4.1) or (4.2) (4.10f)where gth in the constraint (4.10b) is the average throughput requirement of the SWIPTsystem; the constraint (4.10c) indicates the information bits buffered in the traffic queuefor BENs depart in finite time; the constraint (4.10d) is used to guarantee the QoS ofDSNs; the constraint (4.10e) indicates the harvested energy should be above a thresholdin each frame in order to power its circuit for stable operation; the constraint (4.10f) is thefraction of each frame for WN i to harvest energy, i ∈ N . In an interference-free system,the problem formulation in [21, 22, 33, 38] is a special case of (4.10) when the constraints(4.10b) and (4.10c) are removed.The optimization problem formulated in (4.10a)-(4.10f) is related to time average;therefore, we cannot use the standard convex optimization method to solve this prob-lem. To tackle the time average throughput constraint in (4.10b), we will introduce avirtual throughput queue in Section 4.2.1. By using the stochastic optimization theory[49], we introduce a dynamic algorithm to solve the problem (4.10a)-(4.10f) suboptimally.304.2. FhUbcgimTl Fblhgiba fbe JC6GFHBF guvoptimul golution zor JdWhgHBFBE Virtuul hhroughput euyuyWe define the virtual throughput queue as Z (t), which evolves withZ (t+ 1) = max [Z (t) + n (t) P 0] (4.11)where n (t) , gth −g (t) and Z (0) = 0.droposition HBEB hhy wonstruint (4.10b) is myt iz thy virtuul throughput quyuy is myunruty stuvlyBdroozB Clearly, eq. (4.11) is equivalent toZ (t+ 1) ≥ Z (t) + n (t) . (4.12)Rearranging (4.12) and taking the expectation of telescoping sum for 0 ≤ t ≤ i − 1 of(4.12), we haveE [Z (i )] ≥i−1∑t=0E [n (t)] = igth −i−1∑t=0E [g (t)]. (4.13)Dividing (4.13) by i and setting i → ∞, we obtain limi→∞ 1i E [Z (i )] ≥ gth − g¯.Meanwhile, we have limi→∞ 1i E [|Z (i )|] = 0 because Z [t] is mean rate stable. In addition,the inequalities 0 ≤ limi→∞ 1i E [Z (i )] ≤ limi→∞ 1i E [|Z (i )|] always hold. Thus, weobtain gth− g¯ ≤ limi→∞ 1i E [Z (i )] = 0. Furthermore, the constraint (4.10b) is satisfied.HBFBF Dynumiw dowyr Wontrol unx himy gwitwhing (DdWhg) AlgorithmWe now develop the suboptimal algorithm based on some important inequalities de-scribed below.Importunt InyquulitiysThe feasible region under v (t) is Sh(t) = {Constraints (4.10b)− (4.10f)}. We assumethat the complex channel coefficient v (t) has a stationary distribution .h, and the bound-edness assumptions [49] hold. Then, the time average power consumption has both upperbound emaxTP and lower bound eminTP described aseminTP ≤ E[e˜TP ( (t) Pv (t))]≤ emaxTP (4.14)314.2. FhUbcgimTl Fblhgiba fbe JC6GFwhere e˜TP ( (t) Pv (t)) , eTP (t). The rationale of (4.14) is two folds: 1) a power quantitymust be consumed to guarantee constraints (4.10b)-(4.10e); 2) all physical quantities arebounded in practical systems.Due to the stationarity of u (t), we havePr{limi→∞1ii−1∑t=0g (t) = g¯}= 1andPr{limi→∞1ii−1∑t=0eTP (t) = e¯TP}= 1.Now we define a concatenated vector of all traffic queues and virtual throughput queue as! (t) , [q (t) P Z (t)]. Hence, the Lyapunov function of ! (t) is defined asa (! (t)) , 12∑n∈Bq2i (t) +12Z2(t). (4.15)The one frame conditional Lyapunov drift is defined as∆ (! (t)) , E [a (! (t+ 1))− a (! (t))|! (t)] . (4.16)Besides, we have the following two inequalitiesq2i (t+ 1)− q2i (t) = (max [qi (t)−gi (t) P 0] + λi (t))2 − q2i (t)≤ λ2i (t) +g2i (t) + 2qi (t) (λi (t)−gi (t))(4.17)andZ2(t+ 1)− Z2(t) = (max [Z (t) + n (t) P 0])2 − Z2(t)≤ n 2(t) + 2Z (t)n (t) .(4.18)Substituting (4.17), (4.18) and (4.15) into (4.16), the upper bound of the Lyapunov driftplus penalty of (4.10a)-(4.10f) is shown as∆ (! (t)) + k E [eTP (t)|! (t)] ≤ Ψ+ k E [eTP (t)|! (t)]+∑i∈Bqi (t)E [λi (t)−gi (t)|! (t)] + Z (t)E [n (t)|! (t)] (4.19)where Ψ ≥ 12∑i∈B E[λ2i (t) +g2i (t)]+ 12E[n 2(t)]is a positive constant. The right-hand-side (RHS) of (4.19) will be used to define DPCTS algorithm.324.2. FhUbcgimTl Fblhgiba fbe JC6GFAlgorithm DysignThe Lyapounov drift-plus-penalty minimization algorithm, which is designed for theoptimization of queueing networks and other stochastic systems [49], is presented here asAlgorithm 3 to solve (4.10a)-(4.10e). In this algorithm, we introduce a positive parameterk to control the queue backlog.Algorithm 3 DeCih Algorithm1: At the start of each frame, CN observes traffic queues’ backlog qi (t), i ∈ B, virtualthroughput queue’s state Z (t) and complex channel coefficients vector v (t) and dis-tributes those information to WNs over the control channel.2: For DSNs i ∈ D, the resource allocation scheme is obtained byminθi(t)L1 (pi) = k pi (t)− Z (t)gi (t)s.t. (4.10d), (4.10e) and (4.10f).(4.20)3: For BENs i ∈ B, the resource allocation scheme is obtained byminθi(t)L2 (pi) = k pi (t)− (qi (t) + Z (t))gi (t)s.t. (4.10e) and (4.10f).(4.21)I: CN updates the qi (t) and Z (t) based on (4.5) and (4.11), i ∈ N .We note that Algorithm 3 allows each WN to implement JPCTS in each frame with-out prior CSI knowledge, which suggests that the proposed Algorithm 3 can be easilyimplemented in a practical SWIPT system. The properties of the Algorithm 3 are threefold:1. The time average constraints (4.10b) and (4.10c) are satisfied.2. The total queue backlog for BENs are upper-bounded byΨ + k(e¯ ∗TP − eminTP)ϵ. (4.22)3. The power consumption e¯ c is bounded bye¯ ∗TP ≤ e¯ c ≤ e¯ ∗TP +Ψk. (4.23)where e¯ ∗TP denotes the optimal power consumption of the optimization problem(4.10). With a larger control parameter k , the actual power consumption e¯ c ap-proaches the optimal power consumption e¯ ∗TP.334.2. FhUbcgimTl Fblhgiba fbe JC6GFdroozB See Appendix C.When the control parameter k approaches infinity, Algorithm 3 approaches the optimalvalue of (4.10a)-(4.10f) with O(1k)7 at the expense of increasing the queue length of BENswith the rate of O (k ). According to the Little’s law, the queue length is proportional tothe delay. Hence, we conclude there exists a power-BENs’ delay tradeoff.golution zor Wontinuous hg FuwtorWe observe that the optimization problem (4.20) is feasible if and only ifeminiηpi (t) |hi (t)|2+gminilog(1 + pi (t)|hi(t)|HσHi) ≤ 1. (4.24)Since L1 in (4.20) increases monotonically with /iPts (t), eq. (4.20) is minimized when/iPts (t) =eminiηpi(t)|hi(t)|HH. Thus, the optimization problem (4.20) can be reformulated asminpi(t)L1 (pi (t)) = k pi (t) + Z (t)(eminiηpi (t) |hi (t)|2− 1)log(1 + pi (t)|hi (t)|2σ2i)(4.25a)s.t.eminnηpi (t) |hi (t)|2+gminilog(1 + pi (t)|hi(t)|HσHi) ≤ 1. (4.25b)The second-order derivative of L1 (pi (t)) is shown asU2aUp2i (t)=2Zeminiηp3i (t) |hi (t)|2(2 log(1 + pi (t)|hi (t)|2σ2i)− pi (t) |hi (t)|2σ2i + pi (t) |hi (t)|2)+ Z (t)(1− eminiηpi (t) |hi (t)|2) |hi (t)|4(σ2i + pi (t) |hi (t)|2)2 . (4.26)We note that the function 2 log(1 + pi (t)|hi(t)|HσHi)− pi(t)|hi(t)|HσHi+pi(t)|hi(t)|His a monotonically in-creasing function of pi (t). As a result, we obtain2 log(1 + pi (t)|hi (t)|2σ2i)− pi (t) |hi (t)|2σ2i + pi (t) |hi (t)|2≥ 2 log (1 + 0)− 0σ2i + 0= 0. (4.27)Together with 1− eminiηpi(t)|hi(t)|HH≥ 0, we conclude UHaUpHi (t)≥ 0. Besides, eq. (4.25b) is a convexconstraint. We conclude the optimization problem in (4.25a)-(4.25b) is convex and can be7O =x) mevns thvt the v quvntity vpprovxhes to its optimvl vvlue fiith the svme mvgnituye of xC344.3. FimhlTgiba Eefhlgfsolved by, for example, the bisection search method. Above procedures can also be usedto solve (4.21) because (4.21) shares a structure similar to (4.20).golution zor Diswryty hg FuwtorFor the optimization problem (4.20) with discrete TS factor, we can derive the lowerbound of pi (t) with a given /i (t) from (4.10d)-(4.10e) aspi (t) ≥ max{(zRminiG−i;ts(t) − 1)σ2i|hi (t)|2Peminiη/iPts (t) |hi (t)|2}. (4.28)The derivative of L1 (pi) isUL1 (t)Upi (t)< 0P pi (t) <Z(t)(1−/i;ts(t))k −σHi|hi(t)|H= 0P pi (t) =Z(t)(1−/i;ts(t))k −σHi|hi(t)|HS 0P pi (t) SZ(t)(1−/i;ts(t))k −σHi|hi(t)|H. (4.29)As a result, the optimal value of pn (t) for a given /n (t) is shown aspi (t) = max{1hi (t)(exp[gmini1− /iPts (t)]− 1)Peminiη/iPts (t) |hi (t)|22PZ (t) (1− /iPts (t))k− σ2i|hi (t)|2P 0}(4.30)which means there is an injection between /iPts (t) and pi (t) in each frame. Finally, eachWN can implement the JPCTS scheme θ∗i (t) ={/∗iPts (t) P p∗i (t)}via a brute force search.The complexity is O (c)8, which increases linearly with the number of time slots in eachframe. A similar conclusion can be reached for (4.21).HB3 gimulution fysultsWe consider a SWIPT system consisting of 10 WNs, of which there are 5 BENs and 5DSNs. We set the traffic arrival rate of DSNs as λ¯i = 4 nats/sec/Hz, i ∈ D. The valuesof c and η are 6 and 0.8, respectively; the minimum recycled power emini is 0.01 mW,i ∈ N ; the average throughput requirement gth is 50 nats/sec/Hz; the power of interferenceplus noise is 10−6 mW. For simplicity, we suppose |hi (t)|2 take an integer9 in the interval[−21P−30] dB uniformly.Nihe opervtion itervtion of the proposey vlgorithm hvs the svme mvgnituye fiith N in the fiorst xvseC9ihe proposey frvmefiork xvn vyvpt to vny xhvnnel moyels fiith iCiCyC yistriwution over yierent frvmesC354.4. FhmmTelFigure 4.2(a) shows the tradeoff between the average queue backlog of BENs and theaverage power consumption of the SWIPT system. We observe that the average powerconsumption e¯TP decreases monotonically with an increase of control parameter k , whilethe opposite trend is observed for the average queue backlog of BENs versus the controlparameter. From the Little’s law, the average queue backlog of BENs is proportional to theaverage delay of BENs. This indicates that by tuning k we can trade e¯TP with the averagedelay of BENs. Besides, we also observe that a larger traffic arrival rate of BENs leads toa larger power consumption and more delay of BENs under the same control parameter k .The reason is that a larger traffic arrival rate of BENs λBE will consume more transmissionpower in order to guarantee the stability of the BENs’ traffic queues.Figure 4.2(b) illustrates that the tradeoff between minimum service rate gmin andaverage power consumption e¯TP in the SWIPT system. When gmin increases, the averagepower consumption increases. Generally, given λDS, the minimum service rate is inverselycorrelated with the delay of DSNs. Hence, from Fig. 4.2(b), we can conclude that theaverage power consumption can be traded with the delay of DSNs. This is due to the factthe power consumption of DSNs is exponentially related to DSNs’ minimum service rateas shown in (4.30).Figure 4.3 shows the impact of portion of BENs on the system performance for bothcontinuous TS scheme and discrete TS scheme. For comparison, we devise a benchmarkscheme where there are 10 DSNs and 0 BEN in the system, and the constraints (4.10b) and(4.10c) are dropped. This benchmark scheme relates to the schemes in [21, 22, 33, 38] forthe interference-free systems. In Fig. 4.3(a), we observe that the continuous TS scheme hascomparable average power consumption to the discrete TS scheme. However, the averagedelay of BENs for the continuous TS scheme is strictly less than that of the discreteTS scheme, indicating that continuous TS scheme has a higher control precision thanthe discrete TS scheme. In Fig. 4.3(b), we observe that the average power consumptiondecreases faster when the number of BENs increases. Meanwhile, we also note that thepower consumption of the benchmark scheme is not sensitive to the control parameter. Thisis because there is no time average constraint in the formulated problem. On the otherhand, Fig. 4.3(b) shows the average delay per BEN deteriorates more severely when thenumber of BENs decreases. These two observations indicate that the proposed algorithmis more effective when there are more BENs in the system.HBH gummuryIn this chapter, a stochastic optimization framework was presented to investigate thetradeoff between power-delay tradeoff of the SWIPT systems. Using the stochastic op-364.4. FhmmTeltimization theory, the long-term power consumption can be traded for DSNs’ delay andBENs’ delay. The impacts of continuous and discrete TS metrics on the system perfor-mance were also studied. It was found that the discrete TS metric can achieve the sameperformance as the continuous TS metric by setting large number of time slots.374.4. FhmmTel1 2 3 4 5 6 7 8 9 10200210220230240250260270280Average Power Consumption (mW)Control Parameter, V  0481216202428Average Queue Length of BENs (nats/Hz)BENs’ Arrival Rate = 4 nats/sec/HzBENs’ Arrival Rate = 5 nats/sec/HzBENs’ Arrival Rate = 6 nats/sec/Hz=v) Avervge queue length of BEcs versus the vvervge pofierxonsumptionC1 2 3 4 5 6 7 8 9 10200250300350400450Control Parameter, VAverage Power Consumption (mW)  Rmin = 9 nats/sec/HzRmin = 9.5 nats/sec/HzRmin = 10 nats/sec/Hz=w) binimum servixe rvte of Dhcs versus the vvervge pofierxonsumptionCFigure 4.2: The tradeoff of delay of WNs with the average power consumption.384.4. FhmmTel1 2 3 4 5 6 7 8 9 10100110120130140150160Control Parameter, VAverage Power Consumption (mW)  K = 30, # DSN = 3, # BEN = 7K = 30, # DSN = 5, # BEN = 5K = 30, # DSN = 7, # BEN = 3K = 30, BenchmarkCont, # DSN = 3, # BEN = 7Cont, # DSN = 5, # BEN = 5Cont, # DSN = 7, # BEN = 3Cont, Benchmark=v) Avervge pofier xost versus V C1 2 3 4 5 6 7 8 9 1011.522.533.5Control Parameter, VAverage Delay per BEN (sec)  K = 30, # DSN = 3, # BEN = 7K = 30, # DSN = 5, # BEN = 5K = 30, # DSN = 7, # BEN = 3Cont, # DSN = 3, # BEN = 7Cont, # DSN = 5, # BEN = 5Cont, # DSN = 7, # BEN = 3=w) Avervge yelvy per BEc versus V CFigure 4.3: The performance variation against the proportion of BENs in the system withλj = 5 nats/sec/Hz and λk = 4 nats/sec/Hz, j ∈ B and k ∈ D.39Whuptyr IEnxAtoAynx gbf oz kirylyssdowyryx Amplizy unx Forwurxfyluying kith bukugumiAmFuxing Whunnyls unx bonlinyurEnyrgy HurvystyrThe end-to-end SNR of WPR systems with amplify-and-forward protocol is studiedfor the Nakagami-m fading channels. Different from the existing literature, we considerthe nonlinearity of the energy harvester. An analytical expression is derived for the com-plementary cumulative distribution function (CCDF) of the end-to-end SNR. Using theCCDF, the outage capacity is calculated.IBE gystym aoxylIBEBE cvyrull DyswriptionAs shown in Fig. 5.1, we consider a WPR system where a source h communicates witha destination Y through a relay g. The nodes h and Y are equipped with cs and cyantennas, respectively. We assume the relay is equipped with one antenna. Without lossof generality, we assume there is no correlation among these antennas10. Following [14],[40], we assume that the direct link h → Y does not exist due to severe link attenuation.The CSI is assumed to be available at nodes h, g and Y. The channels are assumed toexperience block fading with independent and non-identically Nakagami-m distribution, sothat they remains constant during a frame and varies for different frames.We separate each frame into three time slots for energy transmission, information trans-mission and information reception. Each frame has a unit duration. As shown in Fig. 5.1,G0ihis xorresponys to the vntennvs thvt vre physixvlly sepvrvtey wy vt levst hvlf of the fivvelengthC405.1. Flfgem MbWelS DEnergyReceptionInformationReceptionInformationTransmissiontsr ( )t s112r-RInput Power (mW)OutputPower(mW)thPthPh( )t s112r-Figure 5.1: An illustration of a wireless powered relay system that consists of source, relayand destination. The relay is equipped with a nonlinear energy harvester with input/outputrelation shown in the northwest of this figure. This figure also illustrates the three phaseprotocol used in the system.the first time slot is used for energy harvesting with the duration as /ts where /ts ∈ [0P 1].The remaining part of the frame is equally divided into two time slots for informationprocessing such that the relay uses the half of that, (1− /ts) R2, for information receptionand the remaining half, (1− /ts) R2, for information transmission.In the first and the second time slots, the source signal s is weighted with an cs × 1beamforming vector ws to form the transmission vector, where the term E[|s|2] = 1. Usingmaximal-ratio transmission, the received signal at g is denoted asyr = ‖vsPr‖2√pss+ zr (5.1)where zr ∼ CN(0P σ2r)is the noise vector; vsPr is the cs×1 channel matrix with Nakagami-m1 fading entries with fading parameter m1; ps is the transmit power of the source.In the third time slot, the relay amplifies the received signal and forwards it to the des-tination. Using the maximal-ratio combining, the received signal at Y after the combineris denoted by [34]yy = ζ‖vrPy‖2‖vsPr‖2√pss+ ζ‖vrPy‖2zr +v†rPy‖vrPy‖2zy (5.2)where zy ∼ CN(0P σ2yEcd)is the noise vector; vrPy is the cy × 1 channel matrix withNakagami-m2 fading entries with fading parameter m2; ζ =√pr√ps‖hs;r‖H is the relay gainwhich implies that the relay can invert the first-hop channel perfectly.415.2. EaW-gb-EaW FAEIBEBF bonlinyur Enyrgy Hurvystyr aoxylIn the literature, the total harvested energy at relay is usually formulated as a linearmodel [24, 39]pLinearout = ηps ‖vsPr‖22 (5.3)where the term ps ‖vsPr‖22 is the received signal power at the relay g. It has been notedthat the linear model is not practical [41], as an energy harvesting circuit usually com-prises diodes, inductors and capacitors. On the other hand, the nonlinear energy harvesterproposed in [41] is not analytically tractable. As an improvement of conventional linearenergy harvester, we propose a piece-wise linear energy harvester model that captures thesaturation character of practical circuit. As shown in Fig. 5.1, the energy harvester willoutput a constant power denoted by ηe th when the input power is beyond the thresholde th, where e th is the saturation threshold and η is the energy harvesting efficiency in thelinear region11. Hence, we can obtain the transmit power of the relay aspr ={2/tsηps1−/ts ‖vsPr‖22 P ps ‖vsPr‖22 ≤ e th2/tsηe th1−/ts P ps ‖vsPr‖22 S eth.(5.4)In the next section, we study performance of the wireless powered relay systems overthe Nakagami-m fading channels and derive the CCDF of the end-to-end SNR.IBF EnxAtoAEnx gbfPerforming some algebraic manipulations in (5.2), the end-to-end SNR can be expressedasγeq =γsPrγrPyγsPr + γrPy(5.5)where γsPr , ps‖hs;r‖HHσHrand γrPy ,pr‖hr;d‖HHσHd. The variables ‖vsPr‖22 and ‖vrPy‖22 are randomvariables following the distributions Gamma(csm1PmGΩs;r)and Gamma(cym2PmHΩr;d)re-spectively, where ΩsPr and ΩrPy are the pathloss attenuation for the h → g link and g→ Ylink respectively. Here, the probability density function of the Gamma random variableGamma (αP β) is defined as fGamma(αPβ) (x) , βx−G exp{−xβ}Γ(α) , for x S 0 and αP β S 0.GGIn this xhvpterA the energy hvrvesting exienxy is the rvtio of the output pofier fiith the input pofierof the energy hvrvester in the linevr regionC425.2. EaW-gb-EaW FAEWith the assistance of (5.4), we can show the analytical expression of γrPy asγrPy =2/tsηps1−/ts‖hs;r‖HH‖hr;d‖HHσHdP ps ‖vsPr‖22 ≤ e th2/tsηe th1−/ts‖hr;d‖HHσHdP ps ‖vsPr‖22 S e th.(5.6)hhyorym IBEB hhy WWXF oz ynxAtoAynx gbf oz thy stuxiyx kdf systym is xynotyx usPr {γeq S γ} = I1 + I2 whyry I1 unx I2 ury ryspywtivyly xynyx usI1 = ωcdmH−1∑m=0csmG−1∑n=0εmPn∫ d−10 zn−m exp{− mHvΩr;dbz −mGγzΩs;rv}yzP γ < y0P γ ≥ y(5.7)I2 =δcdmH−1∑m=0csmG−1∑n=0m∑k=0µmPnPk∫∞d−1 zn−k exp{− mHγΩr;dxz −mGγzΩs;rv}yzP γ < yδcdmH−1∑m=0csmG−1∑n=0m∑k=0vmPnPkKn−k+1(2γ√mGmHΩr;dΩs;rxv)P γ ≥ y(5.8)whyryω =1Γ (csm1)(m1γΩsPrv)csmGexp{− m1γΩsPrv}(5.9a)εmPn =1m!(m2vΩrPyb)m( csm1 − 1n)(5.9b)δ =(mGγΩs;rv)csmGΓ (csm1)exp{− m1γΩsPrv− m2γΩrPyx}(5.9c)µmPnPk =1m!(m2γΩrPyx)m( csm1 − 1n)(mk)(5.9d),mPnPk = 2µmPnPk(m2ΩsPrvm1ΩrPyx)n−k+GH(5.9e)with v = psσHr, b = 2/tsηps(1−/ts)σHd, x = 2/tsηeth(1−/ts)σHdunx y = ethσHrB Hyry, K, (·) xynotys thy vthAorxyrmoxiyx Vyssyl zunwtion oz sywonx kinx oID, yqB LBHGFBJq with thy unulytiwul yfipryssion usK, (x) ,(x2)µ ∫∞0z−t−xH4t2t,+GytBdroozB See Appendix D.fymurk 5.2B Let e th → ∞, we note that y = e thσHr→ ∞. Hence, the RHS of (5.8)approaches 0 when y→∞. On the other hand, we can obtain the closed form expression435.3. AhmeeicTl Eefhlgffor I1 by setting y→∞ asI1 = 2ωcdmH−1∑m=0csmG−1∑n=0εmPn(ΩsPrv2m2ΩrPybm1γ)n−m+GHKn−m+1(2√m1m2γΩsPrΩrPyb). (5.10)In other words, the CCDF reduces to (5.10) when y → ∞. Hence, we conclude that theconventional linear energy harvester is a special case of the proposed energy harvester whenthe saturation threshold is set to infinity.IB3 bumyriwul fysultsIn this section, numerical results are provided to illustrate the performance of thewireless powered relay with the proposed nonlinear energy harvester. Simulation resultsare also provided to verify the numerical results. Unless otherwise specified, the sourceand destination are equipped with 2 antennas. The relay has only one antenna. Thetransmission power of the source is 103 mW. The SNR threshold γ is 40 dB. The power ofnoise plus interference at both relay and destination are 10−6 mW. The pathloss exponentis 2.5. The efficiency, time switching factor and saturation threshold of the energy harvesterare assumed to be 0.8, 0.5 and 10 mW, respectively.35 40 45 50 55 6010−410−310−210−1100SNR Threshold, γ (dB)CCDF of γeq  m1=1, m2=2m1=2, m2=1m1=m2=2m1=m2=4SimulationFigure 5.2: An illustration of the impact of the channel fading severity on the systemCCDF with ysPr = 22m and yrPy = 4m.Figures 5.2-5.4 show the impact of channel fading severity on the CCDF of the end-to-445.3. AhmeeicTl Eefhlgf35 40 45 50 55 6010−410−310−210−1100SNR Threshold, γ (dB)CCDF of γeq  m1=1, m2=2m1=2, m2=1m1=m2=2m1=m2=4SimulationFigure 5.3: An illustration of the impact of the channel fading severity on the systemCCDF with ysPr = 13m and yrPy = 13m.end SNR. Fig. 5.2, Fig. 5.3 and Fig. 5.4 correspond to, Case 1: ysPr = 4m and yrPy = 22m;Case 2: ysPr = 13m and yrPy = 13m; Case 3: ysPr = 22m and yrPy = 4m, respectively.When the fading parameters m1 and m2 increase from 2 to 4, the CCDF of the end-to-endSNR deteriorates for these three cases. This observation indicates the more severe thechannel fading is, the worse the coverage of the wireless powered relay becomes. As therelay becomes closer to the destination, we also observe that the system CCDF becomesmore sensitive to the first-hop fading parameter. The reason is that the less severe fadingin the first-hop, the more energy is harvested by the relay and this in turn enhances theCCDF.Figure 5.5 shows the impact of the proposed energy harvester with different coveragethresholds. The CCDF of end-to-end SNR of traditional linear energy harvester, whichare shown in red curves, has a quasi-convex feature with increasing value of ysPr, whichis also referred as “doubly near-far phenomenon”12 [56]. However, in our settings, theenergy harvester has a saturation threshold eth. Due to the threshold eth, that the relayapproaches the source has no contribution on the system CCDF, which is shown in theblack curves. However, when the relay is placed closer to the destination, we also observean increase in the system CCDF for both red curves and black curves. The reason is asfollows. The increasing in ysPr lowers the chance of saturation for the energy harvester. AsGHihe vvvilvwle energy vny the rexeivey hcg vt the relvy is youwly yistvnxeByepenyent on the rstBhopsignvl vttenuvtionC455.3. AhmeeicTl Eefhlgf35 40 45 50 55 6010−410−310−210−1100SNR Threshold, γ (dB)CCDF of γeq  m1=1, m2=2m1=2, m2=1m1=m2=2m1=m2=4SimulationFigure 5.4: An illustration of the impact of the channel fading severity on the systemCCDF with ysPr = 4m and yrPy = 22m.ysPr increases, the second-hop link length yrPy decreases, which enhances the second-hopSNR. The enhancement of the second-hop SNR compensates the decrease of harvestedenergy.Figures 5.6-5.8 plot the outage capacity versus the time switching factor. When thetime switching factor /ts increases, the outage capacity increases first. After reachingthe peak, the outage capacity decreases, which indicates the existence of an optimal timeswitching factor for different source-relay distance. This observation can be understood asfollows. Generally, the outage capacity is influenced by two factors: information detectionratio 1−/ts2 and the transmission power of the relay. When the time switching factor /tsincreases, the relay has more energy to transmit information to the source, which enhancesthe outage capacity of the WPR system. After the peak point, increasing the availableenergy at the relay can have less impact on the outage capacity because the log-concavityof the Shannon capacity formula. Instead, after the peak point, reducing the time forinformation detection plays a dominant role in the outage capacity. Thus, the outagecapacity decreases after the peak point.We also note that the peak outage capacity attains the minimum value when the relayis located in the middle of the source and destination. Compare the peak outage capacityin Fig. 5.6, Fig. 5.7 and Fig. 5.8, we note that the outage capacity in Fig. 5.7 is the lowestone with 0.11 nats/sec/Hz for ps = 40 mW when compared with the other two figures withthe same transmission power ps. Fig 5.7 corresponds to the case where the relay locates at465.4. FhmmTel2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 2400.10.20.30.40.50.60.70.80.91ds,r (m)CCDF of γeq  γ = 40 dBγ = 43 dBγ = 46 dBγ = 49 dBSimulationFigure 5.5: CCDF versus ysPr with ysPr + yrPy = 26 m.the middle of the source and destination. This phenomenon can be understood as follows.Compared with the case with ysPr = 2 m, the case with ysPr = 13 m has less receivedenergy for transmission, and the distance advantage of the case with ysPr = 13 m cannotcompensate the disadvantage from the less received energy. On the other hand, comparedwith the case with ysPr = 2 m, the case with ysPr = 13 m has less end-to-end SNR and theenergy advantage of the case with ysPr = 13 m cannot compensate the disadvantage fromthe lower SNR. Thus, the case with ysPr = 13 m exhibits the lowest outage capacity.IBH gummuryIn this chapter, we studied the performance of the WPR over the Nakagami-m fad-ing channels with a nonlinear energy harvester which is piece-wise linear. We derived ananalytical expression of end-to-end SNR’s CCDF. With the analytical expression of theCCDF of the end-to-end SNR, we observe that the traditional performance analysis basedon linear energy harvester at the relay overestimates the end-to-end SNR. The aforemen-tioned situation is obvious when the relay gets closer to the source node. For differentcombination of ysPr and yrPy, there always exist an optimal value of /ts. This observationwill assist the design of the resource allocation algorithm in the WPR systems in the futurestudies.475.4. FhmmTel0 0.2 0.4 0.6 0.8 100.20.40.60.811.21.4Time Switching Ratio, ρtsOutage Capacity (Nats/sec/Hz)  ps = 10 mWps = 20 mWps = 30 mWps = 40 mWSimulationFigure 5.6: Outage capacity versus time switching factor with ysPr = 2m and yrPy = 24m.0 0.2 0.4 0.6 0.8 100.020.040.060.080.10.12Time Switching Ratio, ρtsOutage Capacity (Nats/sec/Hz)  ps = 10 mWps = 20 mWps = 30 mWps = 40 mWSimulationFigure 5.7: Outage capacity versus time switching factor with ysPr = 13m and yrPy = 13m.485.4. FhmmTel0 0.2 0.4 0.6 0.8 100.20.40.60.811.21.4Time Switching Ratio, ρtsOutage Capacity (Nats/sec/Hz)  ps = 10 mWps = 20 mWps = 30 mWps = 40 mWSimulationFigure 5.8: Outage capacity versus time switching factor with ysPr = 24m and yrPy = 2m.49Whuptyr JWonwlusionsThis chapter concludes the thesis with some comments on resource allocation and per-formance analysis in the RF-EH-WCSs. We also present a discussion on possible futureworks.JBE gummury oz WontrivutionsIn this thesis, we studied three categories of RF-EH-WCS systems. They are DASswith RF-EH capability, SWIPT systems and WPR systems. The technical contributionsof this thesis can be summarized as follows.• An EE maximization problem has been formulated for the OFDMA-based FDDASwith RF-EH capability. Due to the NP-hardness of the formulated problem, we de-couple the EE maximization problem into two subproblems: subchannel allocationproblem and power control problem, to investigate the low complex suboptimal al-gorithm. A heuristic algorithm and a PC-ADMM algorithm have been proposed tosolve the subchannel allocation problem and the power control problem, respectively.Simulation results show the combination of the two proposed algorithms, which isreferred to as suboptimal algorithm hereinafter, outperforms the JSPA algorithm in[30]. Compared with the JSPA algorithm, the proposed suboptimal algorithm haslower complexity; therefore, the proposed suboptimal algorithm is more suitable forpractical application.• A stochastic optimization framework has been presented to investigate the tradeoffbetween the transmission power and the delay of the WNs in the SWIPT systemswith heterogeneous QoS requirements. With a control parameter k , the proposedstochastic optimization framework can trade the long-term power consumption forthe delay of BENs. Because the proposed stochastic optimization framework onlyrequires current CSI to optimize the long-term power consumption, it is convenientfor practical implementation. Besides, we also study the impact of the continuousand discrete TS factors on the system performance. Simulation results are used toverify our work.506.2. Fhghee Jbekf• The performance of the WPR systems with the energy constrained relay has beeninvestigated over the Nakagami-m channels. In the WPR systems, the energy con-strained relay is equipped with a nonlinear energy harvester to scavenge energy fromthe RF signals of the source. We derive an analytical expression for the CCDF ofthe end-to-end SNR. Numerical results are presented to verify our analysis.JBF Futury korksThe proposed suboptimal algorithm for the FDDAS with RF-EH capability in Chap-ter 3 is a semi-distributed solution algorithm, which requires a central control node todistribute part of the global information. However, the future generation wireless commu-nication systems are envisioned to be fully distributed. Hence, the design of the decentral-ized algorithms without central control node becomes an attractive research topic. On theother hand, machine learning provides a theoretical framework to develop algorithms with-out central control node. It has been shown that the decentralized algorithms can offercomparable performance compared to the centralized solutions [57]. Yet, the decentral-ized solutions are more robust against the central control node failure than the traditionalsemi-distributed solutions.Traditional single antenna transmitter-receiver pair can harvest limited amount of en-ergy. The beamforming technology can be utilized in the SWIPT systems to have a rel-atively higher amount of energy delivery [45]. However, the application of beamformingtechnology also poses a new challenge in the downlink transmission period of the SWIPTsystems. Generally, there are two types of WNs: ID WNs and RF-EH WNs. The SWIPTbase station needs to balance its limited transmission power between conveying the infor-mation to the ID WNs and delivering energy to the RF-EH WNs. One can, therefore,design the beamforming vectors for the SWIPT systems in order to balance the SNR forthe ID WNs and the energy for the RF-EH WNs.The resource allocation in the WPR systems is also a valuable topic. For example,based on the analysis in Chapter 5, we observe that there is an optimal TS factor for theenergy constrained relay. The optimal TS factor varies for different source-relay link andrelay-destination link. When there are multiple energy constrained relays in the system,it is more complicated to obtain the optimal TS factors for the energy constrained relaysin a centralized way due to the requirement of signaling exchange. This situation is evenobvious when the energy constrained relays have no energy at the beginning of the frame.Thus, one can develop a distributed algorithm for the energy constrained relays to obtainthe optimal TS factors without signaling overhead.51Vivliogruphy[1] W. C. Brown, “Experiments involving a microwave beam to power and position ahelicopter,” ]EEE hrunsB UyrospB ElywtronB gystB, vol. AES-5, no. 5, pp. 692–702, Sep.1969. → pages 1[2] J. O. McSpadden and J. C. Mankins, “Space solar power programs and microwavewireless power transmission technology,” ]EEE aiwrowB augB, vol. 3, no. 4, pp. 46–57,Dec. 2002. → pages 1[3] K. Huang and V. K. N. Lau, “Enabling wireless power transfer in cellular networks:Architecture, modeling and deployment,” ]EEE hrunsB kirylyss WommunB, vol. 13,no. 2, pp. 902–912, Feb. 2014. → pages 1[4] L. Liu, R. Zhang, and K.-C. Chua, “Multi-antenna wireless powered communicationwith energy beamforming,” ]EEE hrunsB WommunB, vol. 62, no. 12, pp. 4349–4361,Dec. 2014. → pages 1[5] G. Yang, C. K. Ho, and Y. L. Guan, “Dynamic resource allocation for multiple-antennawireless power transfer,” ]EEE hrunsB gignul drowyssB, vol. 62, no. 14, pp. 3565–3577,Jul. 2014. → pages 1[6] H. J. Visser and R. J. M. Vullers, “RF energy harvesting and transport for wirelesssensor network applications: Principles and requirements,” drowB ]EEE, vol. 101, no. 6,pp. 1410–1423, Jun. 2013. → pages 1[7] Y. Mao, Y. Luo, J. Zhang, and K. B. Letaief, “Energy harvesting small cell networks:Feasibility, deployment and operation,” urliv pryprint urlivNEIDEBDFJFD, 2015. →pages[8] S. Bi, C. K. Ho, and R. Zhang, “Wireless powered communication: Opportunities andchallenges,” urlivN EHDLBFGGIvF, Dec. 2014. → pages 1[9] H. Nishimoto, Y. Kawahara, and T. Asami, “Prototype implementation of ambientRF energy harvesting wireless sensor networks,” in drowB ]EEE oz gynsors, Nov. 2010,pp. 1282–1287. → pages 152Bibliography[10] Y. Dong, M. J. Hossain, and J. Cheng, “Joint power control and time switching forSWIPT systems with heterogeneous QoS requirements,” ]EEE WommunB LyttB, vol. 20,no. 2, Feb. 2016. → pages 1[11] X. Zhang, H. Jiang, L. Zhang, C. Zhang, Z. Wang, and X. Chen, “An energy-efficientasic for wireless body sensor networks in medical applications,” ]EEE hrunsB ViomyxBWirwuits gystB, vol. 4, no. 1, pp. 11–18, Feb. 2010. → pages 1[12] X. Lu, D. Niyato, P. Wang, D. I. Kim, and Z. Han, “Wireless charger networkingfor mobile devices: Fundamentals, standards, and applications,” ]EEE kirylyss WomAmunB, vol. 22, no. 2, pp. 126–135, Apr. 2015. → pages 1[13] Y. Dong, H. Zhang, M. J. Hossain, J. Cheng, and V. C. M. Leung, “Energy efficientresource allocation for OFDMA full duplex distributed antenna systems with energyrecycling,” in drowB oz ]EEE [lovywom, Dec. 2015, pp. 1–6. → pages 1, 8[14] A. A. Nasir, X. Zhou, S. Durrani, and R. A. Kennedy, “Relaying protocols for wire-less energy harvesting and information processing,” ]EEE hrunsB kirylyss WommunB,vol. 12, no. 7, pp. 3622–3636, Jul. 2013. → pages 1, 3, 5, 40[15] L. Xiao, P. Wang, D. Niyato, D. I. Kim, and Z. Han, “Wireless networks with RFenergy harvesting: A contemporary survey,” ]EEE WommunB gurvyys : hutsB, vol. 17,no. 2, pp. 757–789, Nov. 2014. → pages 2[16] E. Oh, B. Krishnamachari, X. Liu, and Z. Niu, “Toward dynamic energy-efficientoperation of cellular network infrastructure,” ]EEE WommunB augB, vol. 49, no. 6,pp. 56–61, Jun. 2011. → pages 2[17] W. V. Heddeghem, S. Lambert, B. Lannoo, D. Colle, M. Pickavet, and P. Demeester,“Trends in worldwide ICT electricity consumption from 2007 to 2012,” WomputyrWommunB, vol. 50, no. 1, pp. 64–76, Sep. 2014. → pages 2[18] Wiswo Visuul bytworking ]nxyfiN [lovul aovily Xutu hruzAw Forywust ipxuty, FDEHAFDEM, Feb. 3, 2015. [Online].Available: http://www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/white paper c11-520862.pdf → pages 2[19] A. Attar, H. Li, and V. C. M. Leung, “Green last mile: how fiber-connected massivelydistributed antenna systems can save energy,” ]EEE kirylyss WommunB augB, vol. 18,no. 5, pp. 66–74, Oct. 2011. → pages 253Bibliography[20] H. Kim, S.-R. Lee, K.-J. Lee, and I. Lee, “Transmission schemes based on sum rateanalysis in distributed antenna systems,” ]EEE hrunsB kirylyss WommunB, vol. 11,no. 3, pp. 1201–1209, Mar. 2012. → pages 2[21] S. Timotheou, I. Krikidis, G. Zheng, and B. Ottersten, “Beamforming for MISO inter-ference channels with QoS and RF energy transfer,” ]EEE hrunsB kirylyss WommunB,vol. 13, no. 5, pp. 2646–2658, May 2014. → pages 3, 5, 30, 36[22] Q. Shi, L. Liu, W. Xu, and R. Zhang, “Joint transmit beamforming and receive powersplitting for MISO SWIPT systems,” ]EEE hrunsB kirylyss WommunB, vol. 13, no. 6,pp. 3269–3280, Jun. 2014. → pages 3, 4, 5, 30, 36[23] Z. Ding, S. M. Perlaza, I. Esnaola, and H. V. Poor, “Power allocation strategies inenergy harvesting wireless cooperative networks,” ]EEE hrunsB kirylyss WommunB,vol. 13, no. 2, pp. 846–860, Feb. 2014. → pages 3, 5[24] Y. Gu and S. Aı¨ssa, “RF-based energy harvesting in decode-and-forward relayingsystems: Ergodic and outage capacities,” ]EEE hrunsB kirylyss WommunB, to bepublished. → pages 3, 5, 42[25] F. Meshkati, H. V. Poor, and S. C. Schwartz, “Energy-efficient resource allocation inwireless networks,” ]EEE gignul drowyssB augB, vol. 24, no. 3, pp. 58–68, May 2007.→ pages 3[26] C. He, B. Sheng, P. Zhu, D. Wang, and X. You, “Energy efficiency comparisonbetween distributed and co-located MIMO systems,” ]ntB JB WommunB gystB, vol. 27,no. 1, pp. 81–94, 2014. [Online]. Available: http://dx.doi.org/10.1002/dac.2345 →pages 3[27] X. Chen, X. Xu, and X. Tao, “Energy efficient power allocation in generalized dis-tributed antenna system,” ]EEE WommunB LyttB, vol. 16, no. 7, pp. 1022–1025, Jul.2012. → pages 3[28] Y. Xu, X. Xu, Y. Li, X. Chen, and X. Tao, “Energy-efficient power control for frac-tional frequency reuse,” in drowB oz ]EEE d]afW, London, UK, Sep. 2013, pp. 2962–2966. → pages 3[29] Y. Li, M. Sheng, Y. Zhang, X. Wang, and J. Wen, “Energy-efficient antenna selectionand power allocation in downlink distributed antenna systems: A stochastic optimiza-tion approach,” in drowB oz ]EEE ]WW, Sydney, NSW, Jun. 2014, pp. 4963–4968. →pages 4, 1754Bibliography[30] C. He, G. Y. Li, F.-C. Zheng, and X. You, “Energy-efficient resource allocation inofdm systems with distributed antennas,” ]EEE hrunsB VyhB hywhnolB, vol. 63, no. 3,pp. 1223–1231, Mar. 2014. → pages 4, 6, 23, 50[31] H. Kim, S.-R. Lee, C. Song, K.-J. Lee, and I. Lee, “Optimal power allocation schemefor energy efficiency maximization in distributed antenna systems,” ]EEE hrunsB WomAmunB, vol. 63, no. 2, pp. 431–440, Feb. 2015. → pages 3, 4, 17[32] Y. Gu and S. Aı¨ssa, “Interference aided energy harvesting in decode-and-forwardrelaying systems,” in drowB oz ]EEE ]WW, Sydney, NSW, Jun. 2014, pp. 5378–5382.→ pages 4[33] D. W. K. Ng, E. S. Lo, and R. Schober, “Wireless information and power transfer:Energy efficiency optimization in OFDMA systems,” ]EEE hrunsB kirylyss WommunB,vol. 12, no. 12, pp. 6352–6370, Dec. 2013. → pages 4, 5, 21, 30, 36, 60[34] Y. Zeng and R. Zhang, “Full-duplex wireless-powered relay with self-energy recycling,”]EEE kirylyss WommunB LyttB, vol. 4, no. 2, pp. 201–204, Apr. 2015. → pages 4, 41[35] R. Morsi, D. S. Michalopoulos, and R. Schober, “Multiuser scheduling schemes forsimultaneous wireless information and power transfer over fading channels,” ]EEEhrunsB kirylyss WommunB, vol. 14, no. 4, pp. 1967–1982, Apr. 2015. → pages 4, 5[36] J. Li, N. Bao, W. Xia, and L. Shen, “Adaptive user scheduling and resource manage-ment for multiuser mimo downlink systems with heterogeneous delay requirements,”in drowB oz ]EEE kWbW, Apr. 2013, pp. 1351–1356. → pages 4[37] R. Balakrishnan and B. Canberk, “Traffic-aware QoS provisioning and admission con-trol in OFDMA hybrid small cells,” ]EEE hrunsB VyhB hywhnolB, vol. 63, no. 2, pp.802–810, Feb. 2014. → pages 4[38] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer in mul-tiuser OFDM systems,” ]EEE hrunsB kirylyss WommunB, vol. 13, no. 4, pp. 2282–2294,Apr. 2014. → pages 4, 5, 30, 36[39] Z. Ding, I. Krikidis, B. Sharif, and H. V. Poor, “Wireless information and powertransfer in cooperative networks with spatially random relays,” ]EEE hrunsB kirylyssWommunB, vol. 13, no. 8, pp. 4440–4453, Aug. 2014. → pages 5, 42[40] C. Zhong, H. A. Suraweera, G. Zheng, I. Krikidis, and Z. Zhang, “Wireless informationand power transfer with full duplex relaying,” ]EEE hrunsB WommunB, vol. 62, no. 10,pp. 3447–3461, Oct. 2014. → pages 5, 4055Bibliography[41] E. Boshkovska, D. W. K. Ng, N. Zlatanov, and R. Schober, “Practical non-linearenergy harvesting model and resource allocation for SWIPT systems,” ]EEE WommunBLyttB, vol. 19, no. 12, pp. 2082–2085, Dec. 2015. → pages 5, 42[42] M. Xia, C. Xing, Y.-C. Wu, and S. Aı¨ssa, “Exact performance analysis of dual-hop semi-blind af relaying over arbitrary Nakagami-m fading channels,” ]EEE hrunsBkirylyss WommunB, vol. 10, no. 10, pp. 3449–3459, Oct. 2011. → pages 5[43] T. Aulin, “Characteristics of a digital mobile radio channel,” ]EEE hrunsB VyhB hywhAnolB, vol. 30, no. 2, pp. 45–53, May 1981. → pages 5[44] G. L. Turin, F. D. Clapp, T. L. Johnston, S. B. Fine, and D. Lavry, “A statisticalmodel of urban multipath propagation,” ]EEE hrunsB WommunB, vol. 21, no. 1, pp.1–9, Feb. 1972. → pages 5[45] Z. Ding, C. Zhong, D. W. K. Ng, M. Peng, H. A. Suraweera, R. Schober, and H. V.Poor, “Application of smart antenna technologies in simultaneous wireless informationand power transfer,” ]EEE WommunB augB, vol. 53, no. 4, pp. 86–93, Apr. 2015. →pages 5, 51[46] L. R. Varshney, “Transporting information and energy simultaneously,” in drowB oz]EEE ]g]h, Toronto, ON, Jul. 2008, pp. 1612–1616. → pages 8[47] P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and powertransfer,” in drowB oz ]EEE ]g]h, Jun. 2010, pp. 2363–2367. → pages 8[48] X. Zhou, R. Zhang, and C. K. Ho, “Wireless information and power transfer: Archi-tecture design and rate-energy tradeoff,” ]EEE hrunsB WommunB, vol. 61, no. 11, pp.4754–4767, Nov. 2013. → pages 8[49] M. J. Neely, gtowhustiw bytwork cptimizution with Uppliwution to Wommuniwution unxeuyuying gystyms. Morgan & Claypool, 2010. → pages 9, 10, 30, 31, 33, 62[50] A. Jeffrey and D. Zwillinger, huvly oz ]ntygruls, gyriys, unx droxuwts. AcademicPress, 2007. → pages 13, 43[51] S. A. Vavasis, bonlinyur cptimizutionN Womplyfiity ]ssuys. New York, NY, USA:Oxford University Press, Inc., 1991. → pages 18[52] H. Zhang, C. Jiang, N. C. Beaulieu, X. Chu, X. Wen, and M. Tao, “Resource alloca-tion in spectrum-sharing ofdma femtocells with heterogeneous services,” ]EEE hrunsBWommunB, vol. 62, no. 7, pp. 2366–2377, Jul. 2014. → pages 1856Bibliography[53] S. P. Boyd and L. Vandenberghe, Wonvyfi cptimizution. Cambridge University Press,2004. → pages 18[54] G. Bansal, M. J. Hossain, V. K. Bhargava, and T. Le-Ngoc, “Subcarrier and powerallocation for OFDMA-based cognitive radio systems with joint overlay and underlayspectrum access mechanism,” ]EEE hrunsB VyhB hywhnolB, vol. 62, no. 3, pp. 1111–1122, Mar. 2013. → pages 19, 59[55] S. P. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimizationand statistical learning via the alternating direction method of multipliers,” FounxBhrynxs auwhB LyurnB, vol. 3, no. 1, pp. 1–122, Jan. 2011. → pages 20, 21, 22, 60[56] H. Ju and R. Zhang, “Throughput maximization in wireless powered communicationnetworks,” ]EEE hrunsB kirylyss WommunB, vol. 13, no. 1, pp. 418–428, Jan. 2014.→ pages 45[57] D. T. Hoang, D. Niyato, P. Wang, and D. I. Kim, “Performance optimization forcooperative multiuser cognitive radio networks with rf energy harvesting capability,”]EEE hrunsB kirylyss WommunB, vol. 14, no. 7, pp. 3614–3629, Jul. 2015. → pages 5157Appynxiwys58Appynxifi Adrooz oz droposition 3BEWe first prove that the optimization problem in (3.6a)-(3.6f) is equivalent to (3.7a)-(3.7f). After a substitution of (3.3) and (3.4) in (3.7a) with p˜kjPi = vkjPipkjPi, the first-orderderivative of (3.7a) with respect to vkjPi can be written asF , UUvkjPi j(P˜ PO)jTP(P˜ PO)=1jTP(log(1 +p˜kjPiHkjPivkjPi)− p˜kjPiHkjPip˜kjPiHkjPi + vkjPi).Hence, the second-order derivative of (3.7a) in terms of vkjPi becomesUFUvkjPi= − 1vkjPi(p˜kjPiHkjPip˜kjPiHkjPi + vkjPi)2≤ 0. (A.1)As a result, F is a monotonically decreasing function of vkjPi, and we have F S 0 due tolimvkj;i→∞ F = 0. Hence, we conclude thatj( eP PA)jTP( eP PA) is monotonically increasing in termsof vkjPi. Therefore, in order to maximize thej( eP PA)jTP( eP PA) , one needs to allocate the RRH-WNlink with the highest HkjPi, which can be proved by contradiction [54, Theorem 1].59Appynxifi Vdrooz oz thy Wonvyrgynwy ozdWAADaa AlgorithmWe first prove that the energy efficiency q increases in each iteration. Then, we proveAlgorithm 2 converges. It has been proved in [33] that U (vPO∗) S 0 . Thus, we haveU (vτ PO∗) = j (vτ PO∗)− qτjTP (vτ PO∗)= qτ+1jTP (vτ PO∗)− qτjTP (vτ PO∗)=(qτ+1 − qτ)jTP (vτ PO∗) S 0(B.1)where qτ+1 = j(v PA∗)jTP(v PA∗).Furthermore, we conclude qτ+1 S qτ S 0 due to jTP (vτ PO∗) S 0. With the aid of itera-tion steps in (3.13)-(3.15), the following two equations hold due to limτ→∞∥∥vτ+1 − vτ∥∥2=0 [55]limτ→∞∥∥j (vτ+1PO∗)− j (vτ PO∗)∥∥2= 0 (B.2)limτ→∞∥∥jTP (vτ+1PO∗)− jTP (vτ PO∗)∥∥2 = 0. (B.3)By substituting (B.2)-(B.3) into∥∥U (vτ+1PO∗)− U (vτ PO∗)∥∥, we have the following deriva-tion ∥∥U (vτ+1PO∗)− U (vτ PO∗)∥∥2=∥∥j (vτ+1PO∗)− qτ+1jTP (vτ+1PO∗)− j (vτ PO∗) + qτjTP (vτ PO∗)∥∥2≤∥∥j (vτ+1PO∗)− j (vτ PO∗)− qτ+1jTP (vτ+1PO∗)+ qτ+1jTP (vτ PO∗)∥∥2≤∥∥j (vτ+1PO∗)− j (vτ PO∗)∥∥2+∣∣qτ+1∣∣ ∥∥jTP (vτ+1PO∗)− jTP (vτ PO∗)∥∥2 .As a result, we can achievelimτ→∞∥∥U (vτ+1PO∗)− U (vτ PO∗)∥∥2= 0.On the other hand, we also note that the power budget constraints in (3.9b) give an upper604cceaWik 5. Cebbf bf ghe 6baieegeace bf C6-4DMM 4lgbeighmbound for j (vPO∗). Together with the argument that jTP (vτ PO∗) is lower bounded bythe circuit power of both the BBU and the b RRHs, we can conclude that the sequence{q1P q2P . . . P qτ P . . .}with each element qτ , j(v PA∗)jTP(v PA∗)is upper bounded. Using the factthat a bounded monotonically increasing sequence converges, we havelimτ→∞ qτ = q∗.61Appynxifi Wdrooz oz thy dropyrtiys oz DdWhgAlgorithmFuwt WBEB Vy sytting  striwtly within thy wupuwity rygion oz (4.10), thy womplyfi whunAnyl woywiynts v (t) hus u stutionury xistrivution .hB hhy quuntitiys n (t) unx ri (t) uryvounxyxB ky huvy E [eTP (t)] = E[e˜TP (∗ (t) Pv (t))]unx E [m (t)] = E[n˜ (∗ (t) Pv (t))]BUs suwh, ∗ (t) ∈ Sh(t) yfiists with thy zollowing inyquulitiys sutisyxBE[e˜TP (∗ (t) Pv (t))]≤ e¯ ∗TP + δ (C.1a)E[n˜ (∗ (t) Pv (t))]≤ δ (C.1b)E [λi (t)] ≤ E[g˜i (θ∗i (t) P hi (t))]− ϵP ∀i (C.1c)whyry ϵ S 0 unx δ is un urvitrury smull positivy wonstuntB e¯ ∗TP is thy optimul vuluy oz thyoptimizution provlym (4.10)BdroozB See [49, Appendix 4.A] for detailed proof.Now, we start the proof of the properties of Algorithm 3. Because the proposed DPCTSalgorithm minimizes the RHS of (4.10) under the constraints (4.10d), (4.10e) and (4.10f).As a result, we have∆ (! (t)) + k E [eTP (t)|! (t)] ≤ Ψ+ k E [e ∗TP (t)|! (t)]+∑i∈Bqi (t)E [λi (t)−g∗i (t)|! (t)] + Z (t)E [n ∗ (t)|! (t)] (C.2)where e ∗TP (t), n ∗ (t) and g∗i (t) are the returned values by the DPCTS algorithm in eachframe. Substituting (C.1a)-(C.1c) into (C.2) and letting δ → 0, we obtain∆ (! (t)) + k E [eTP (t)|! (t)] ≤ Ψ+ k e¯ ∗TP − ϵ∑i∈Bqi (t) (C.3)where Ψ ≥ 12∑i∈B E[λ2i (t) + r2i (t)]+ 12E[n 2(t)].624cceaWik 6. Cebbf bf ghe Cebceegief bf DC6GF 4lgbeighmE) Because qi (t) ≥ 0, we can obtain the upper bound for ∆ (! (t)) from (C.3) as∆ (! (t)) ≤ Ψ+ k (e¯ ∗TP − E [eTP (t)|! (t)]) . (C.4)Taking the iterated expectation and the telescoping sums of (C.4) for 0 ≤ t ≤ i − 1, weobtainE [a (! (i ))]− E [a (! (0))] ≤ iX (C.5)where the constant X = Ψ+ k(e¯ ∗TP − eminTP).Substituting (4.15) in (C.5) and rearranging (C.5), we have the following expression1i∑i∈BE[q2i (i )]+1iE[Z2(i )] ≤ 2X + 2E [a (! (0))]i.Hence, the traffic queue qi (i ) are mean rate stable due tolimi→∞E [|qi (i )|]i≤ limi→∞√2X +2E [a (! (0))]i= 0.Similarly, we attain limi→∞E[|Z(t)|]i = 0.F) Noting that E [a (! (i ))] ≥ 0, we can obtain the following expression by taking theiterated expectation and the telescoping sum of (C.3)ki∑t=0E [eTP (t)] ≤ iΨ+ k i e¯ ∗TP − ϵki∑t=0∑i∈BE [qi (t)] + E [a (! (0))] . (C.6)Dividing (C.6) by k i and letting i →∞, we obtain (4.22).G) Taking the iterated expectation and the telescoping sums of (C.3), we can get1ii−1∑t=0∑i∈BE [qi (t)] ≤Ψ+ k(e¯ ∗TP − 1ii−1∑t=0E [eTP (t)])ϵ. (C.7)Substituting (4.14) into (C.7) and letting i →∞, eq. (4.23) yields.63Appynxifi Ddrooz oz hhyorym IBEWe separate the CCDF into two parts based on the saturation threshold e th asI1 = Pr{γeq S γP ps ‖vsPr‖22 < e th}I2 = Pr{γeq S γP ps ‖vsPr‖22 ≥ e th}.Hereinafter, we derive the analytical expressions for I1 and I2.Dyrivution oz thy yfipryssion oz I1If γ ≥ y, we note that I1 = 0. For the case γ < y, we can let j , k , m and n , re-spectively, denote 1v‖hs;r‖HH, 1b‖hs;r‖HH‖hr;d‖HH, 1‖hs;r‖HHand 1‖hr;d‖HH. As both ‖vsPr‖22 and ‖vrPy‖22are Gamma random variables, the PDFs for m and n can be, respectively, written asfm (x) =1xHf‖hs;r‖HH(1x)and fn (y) =1yHf‖hr;d‖HH(1y).From (5.3), the relation between (jP k ) and (mPn ) is j = mv and k =mnb . Using aJacobian determinant, we obtain the joint PDF of (jP k ) as fjk (uP v) =bufm (vu) fn(bvvu)where fm (·) and fn (·) are the PDFs for m and n respectively. The derivation for theexpression of I1 is shown as follows.I1 =(mGΩs;r)cscrmGΓ (cscrm1)crcdmH−1∑m=0G∫Gdvm!(vu)cscrmG+1 m2ΩrPybv(1uγ − 1)m× exp− m2ΩrPy bv ( 1uγ − 1) −m1ΩsPrvu yu=(γmGvΩs;r)cscrmGΓ (cscrm1)crcdmH−1∑m=01m!(vm2ΩrPyb)m cscrmG−1∑n=0(cscrm1 − 1n)×d−1∫0zn−m exp{− m2vΩrPybz− m1γ (z + 1)ΩsPrv}yz.(D.1)644cceaWik D. Cebbf bf Ghebeem 5.1Dyrivution oz thy yfipryssion oz I2As γsPr ≥ y, the output of energy harvester is saturated. Hence, γsPr = ps‖hs;r‖HHσHrandγrPy =2/tsηe th1−/ts‖hr;d‖HHσHdare independent. Hence, the derivation for the expression of I2 isshown as follows.I2 =(mGvΩs;r)csmGγcsmG−1Γ (csm1)exp{−m2γΩrPyx}cdmH−1∑m=01m!(m2γΩrPyx)m×∞∫max(y−γP0)(1 +γz)m(1 +zγ)csmG−1exp{− m2γ2ΩrPyxz− m1 (z + γ)ΩsPrv}yz=exp{− mHγΩr;dx −mGγΩs;rv} ( mGaΩs;r)NsmGΓ(csmG)cdmH−1∑m=01m!(mHγΩr;dx)m csmG−1∑n=0(csm1 − 1n)×m∑k=0(mk)∞∫d−1zn−k exp{− mHγΩr;dxz −mGγzΩs;rv}yzP γ < y2 exp{− mHγΩr;dx −mGγΩs;rv} ( mGaΩs;r)NsmGΓ(cscrmG)cdmH−1∑m=01m!(mHγΩr;dx)m csmG−1∑n=0(csm1 − 1n)×m∑k=0(mk)(mHΩs;rvmGΩr;dx)n−k+GHKn−k+1(2γ√mGmHΩr;dΩs;rvx)P γ ≥ y.(D.2)We obtain the analytical expression of the CCDF by taking the summation over (D.1)and (D.2).65

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.24.1-0305736/manifest

Comment

Related Items