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Resource allocation in wireless systems with conventional and energy harvesting nodes Ahmed, Imtiaz 2014

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RESOURCE ALLOCATION IN WIRELESS SYSTEMS WITHCONVENTIONAL AND ENERGY HARVESTING NODESbyImtiaz AhmedM.Sc. Eng., Bangladesh University of Engineering and Technology, 2008B.Sc. Eng., Bangladesh University of Engineering and Technology, 2006A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)October 2014c⃝ Imtiaz Ahmed, 2014AbstractHigh data rate, reliable communication, and low power consumption are the foremostdemands for next generation of wireless communication systems. The key challengeto the design of communication systems is to combat the detrimental effects of chan-nel fading, noise, and high power consumption. Wireless systems are often impairedby non–Gaussian noise, and the performance of systems designed for Gaussian noisecan degrade if non–Gaussian noises are present but are not taken into account. Thus,it is imperative to analyze systems that are impaired by non–Gaussian noise and tomanage their resources better to improve overall performance. Furthermore, there issignificant interest in using renewable energy for wireless systems. However, energyharvesting (EH) is a random process and the harvested energy should be expendedjudiciously to maximize aggregate system throughput. In this thesis, we considerwireless systems that are impaired by Gaussian and non–Gaussian noise and pow-ered by conventional energy sources and energy harvesters and propose appropriateresource allocation schemes for these systems.First, we propose optimal and fair power allocation schemes for a cooperativerelay network with amplify–and–forward relays that employs best and partial relayselections and is impaired by Gaussian and non–Gaussian noise. We derive closed–form expressions of asymptotic bit error rate and use this expression to allocatetransmit powers for different nodes with necessary energy consumption constraints.Second, we consider a network comprising a source, a relay, and a destination,iiAbstractwhere the source and the relay are EH nodes. We consider conventional and buffer–aided link adaptive relaying protocols, and propose offline and online resource allo-cation schemes that maximize the system throughput.Thirdly, we consider a multi–relay network with EH nodes and propose offline andonline joint relay selection and power allocation schemes that maximize the systemthroughput.Fourth, we consider a single source–destination link, where the source has a hybridenergy supply comprised of constant energy source and energy harvester. We proposeoffline and online power allocation schemes that minimize the energy consumptionfrom the constant energy source and thereby utilize the harvested energy effectively.iiiPrefaceThe contributions in this thesis have been published in different journals and sev-eral conferences. The works in Chapters 2–5 are performed under the supervision ofProf. Robert Schober and in collaboration with Dr. Aissa Ikhlef (former post-doctoralfellow in the department of Electrical and Computer Engineering (ECE) in the Uni-versity of British Columbia (UBC), Vancouver, Canada and currently with ToshibaResearch Europe Limited, Bristol, UK), Dr. Amir Nasri (former post–doctoral fel-low in ECE, UBC and currently with Fortinet Inc., Vancouver, Canada), Dr. Dio-midis S. Michalopoulos (former post–doctoral fellow in ECE, UBC and currently withthe department of ECE, University of Erlangen–Nuremberg, Germany), Dr. DerrickW. K. Ng (post–doctoral fellow in ECE, UBC and in the department of ECE, Univer-sity of Erlangen-Nuremberg, Germany), and Prof. Ranjan K. Mallik (Prof. in IndianInstitute of Technology (IIT), New Delhi, India). In all publications, I contributedthrough surveying the literature, proposing and developing the research idea, for-mulating and solving the problem, performing simulations, and writing the paperdraft.Two papers related to Chapter 2 have been published:• I. Ahmed, A. Nasri, D. S. Michalopoulos, R. Schober, and R. K. Mallik, “RelaySubset Selection and Fair Power Allocation for Best and Partial Relay Selectionin Generic Noise and Interference,” IEEE Trans. on Wireless Commun., vol.11, pp. 1828–1839, May 2012.ivPreface• I. Ahmed, A. Nasri, D. S. Michalopoulos, R. Schober, and R. K. Mallik, “Per-formance Analysis and Power Allocation of AF Best Relay Selection in GenericNoise,” in Proc. of IEEE WCNC, April 2012, Paris, France.Two papers related to Chapter 3 have been published:• I. Ahmed, A. Ikhlef, R. Schober, and R. K. Mallik, “Power Allocation for Con-ventional and Buffer-Aided Link Adaptive Relaying Systems with Energy Har-vesting Nodes,” IEEE Trans. on Wireless Commun., vol.13, pp. 1182–1195,Mar. 2014.• I. Ahmed, A. Ikhlef, R. Schober, and R. K. Mallik, “Power Allocation in EnergyHarvesting Relay Systems,” in Proc. of IEEE VTC, May 2012, Yokohama,Japan.One paper related to Chapter 4 has been published:• I. Ahmed, A. Ikhlef, R. Schober, and R. K. Mallik, “Joint Power Allocationand Relay Selection in Energy Harvesting AF Relay Systems,” IEEE WirelessCommun. Letters., vol. 2, pp. 239–242, Apr. 2013.Two papers related to Chapter 5 have been published:• I. Ahmed, A. Ikhlef, D. W. K. Ng, and R. Schober, “Power Allocation for aHybrid Energy Harvesting Transmitter,” IEEE Trans. on Wireless Commun.,vol. 12, pp. 6255–6267, Dec. 2013.• I. Ahmed, A. Ikhlef, D. W. K. Ng, and R. Schober, “Optimal Power Allocationfor a Hybrid Energy Harvesting Transmitter,” in Proc. of IEEE ICC, June2013, Hungary, Budapest.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Resource Allocation in Wireless Communication Systems . . . . . . . 21.1.1 Optimization Framework . . . . . . . . . . . . . . . . . . . . 31.1.2 Key Features of Resource Allocation Schemes . . . . . . . . . 51.2 Noise in Wireless Communication Systems . . . . . . . . . . . . . . . 81.3 Energy Harvesting in Wireless Communication Systems . . . . . . . 14viTable of Contents1.3.1 EH Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.3.2 EH Architectures . . . . . . . . . . . . . . . . . . . . . . . . . 191.4 Contributions and Results . . . . . . . . . . . . . . . . . . . . . . . . 221.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 242 Relay Subset Selection and Fair Power Allocation for Best and Par-tial Relay Selection in Generic Noise and Interference . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.3 Asymptotic Error Rate Analysis . . . . . . . . . . . . . . . . . . . . 332.3.1 Asymptotic PEP of Best Relay Selection . . . . . . . . . . . . 352.3.2 Asymptotic PEP for Partial Relay Selection . . . . . . . . . . 372.3.3 Diversity and SNR Gain . . . . . . . . . . . . . . . . . . . . . 392.4 Optimization of BRS and PRS in Generic Noise . . . . . . . . . . . . 392.4.1 Relay Subset Selection . . . . . . . . . . . . . . . . . . . . . . 392.4.2 Power Allocation with Energy Consumption Constraints . . . 402.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 Resource Allocation for Conventional and Buffer-Aided Link Adap-tive Relaying Systems with Energy Harvesting Nodes . . . . . . . 533.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.2.1 Conventional Relaying . . . . . . . . . . . . . . . . . . . . . . 583.2.2 Buffer–Aided Adaptive Link Selection . . . . . . . . . . . . . 613.3 Power Allocation and Link Selection for Buffer–Aided Relaying . . . 623.3.1 Offline Power Allocation . . . . . . . . . . . . . . . . . . . . . 62viiTable of Contents3.3.2 Online Power Allocation . . . . . . . . . . . . . . . . . . . . . 663.3.3 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 703.4 Power Allocation for Conventional Relaying . . . . . . . . . . . . . 713.4.1 Optimal Offline Power Allocation . . . . . . . . . . . . . . . . 713.4.2 Optimal Online Power Allocation by DP . . . . . . . . . . . . 733.4.3 Suboptimal Online Power Allocation . . . . . . . . . . . . . . 753.4.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 793.5.1 Performance of Different Power Allocation Schemes for Con-ventional Relaying . . . . . . . . . . . . . . . . . . . . . . . . 803.5.2 Performance of Different Power Allocation Schemes for LinkAdaptive Relaying . . . . . . . . . . . . . . . . . . . . . . . . 823.5.3 Comparison Between Conventional and Link Adaptive Relay-ing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 833.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924 Joint Power Allocation and Relay Selection in Energy HarvestingAF Relay Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 944.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.3 Joint Power Allocation and Relay Selection . . . . . . . . . . . . . . 974.3.1 Optimal Offline Optimization Algorithm . . . . . . . . . . . . 984.3.2 Suboptimal Online Optimization Algorithms . . . . . . . . . 1034.3.3 Extension to Multiple Relay Selection . . . . . . . . . . . . . 1064.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1064.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111viiiTable of Contents5 Power Allocation for an Energy Harvesting Transmitter with HybridEnergy Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1135.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3 Offline Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . 1185.3.1 Offline Power Allocation for Scenario 2 . . . . . . . . . . . . . 1195.3.2 Offline Power Allocation for Scenario 1 . . . . . . . . . . . . . 1235.3.3 Complexity of the Proposed Offline Power Allocation Schemes 1245.4 Online Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . 1245.4.1 Optimal Online Power Allocation . . . . . . . . . . . . . . . . 1255.4.2 Suboptimal Online Power Allocation . . . . . . . . . . . . . . 1305.4.3 Complexity of the Proposed Online Power Allocation Schemes 1355.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1355.5.1 Behavior of Offline Power Allocation Over Time Intervals k . 1365.5.2 Comparison of the Proposed Offline Schemes with the BaselineSchemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1385.5.3 Comparison of the Proposed Offline and Online Schemes . . . 1435.5.4 Effect of µ on the Proposed Schemes . . . . . . . . . . . . . . 1465.5.5 Impact of Various EH Profiles . . . . . . . . . . . . . . . . . 1485.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1486 Conclusions and Future Work . . . . . . . . . . . . . . . . . . . . . . 1516.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1516.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1536.2.1 Performance Analysis of Relay Selection Schemes Impaired byα–stable noise . . . . . . . . . . . . . . . . . . . . . . . . . . 153ixTable of Contents6.2.2 Intelligent Resource Allocation for EH Systems . . . . . . . . 1546.2.3 Wireless Information and Power Transfer in Relay Networks . 1556.2.4 Resource Allocation for a Multi–hop Conventional and Buffer–Aided Link Adaptive EH Relaying Protocols . . . . . . . . . 155Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157AppendicesA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175xList of Tables1.1 Examples of different EH sources and their performances; data col-lected from [1] and [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 Power and energy constraints for OPA, FPA–1, FPA–2, EPA, and ESP[3], cf. Fig. 2.6. For all schemes P0 + P1fX1 + P2fX2 + P3fX3 ≤ ET isvalid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.1 PH,Tot obtained for the parameters used for Fig. 5.9 for offline andsuboptimal online power allocation schemes. . . . . . . . . . . . . . . 148xiList of Figures1.1 The pdf of GGN. σ2 = 1 and different β have been considered. . . . 121.2 EH modules for wireless communication systems. Source: [4]. . . . . . 151.3 Solar EH system. Researchers at the University of California, LosAngeles (UCLA), used liquid-crystal display (LCD) screens to work asphotovoltaic cells for harvesting solar energy. Source: [5]. . . . . . . 161.4 EH roller sneaker which produces electricity from human motion. Source:[6]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171.5 EH architectures: (a) HSU architecture and (b) HU architecture. . . 202.1 Block diagram of the considered AF relay system with BRS and PRS.Solid lines indicate the direct S–D link and the selected relay link. . . 292.2 BER vs. average SNR, γ¯, for BRS with N = 4 AF relays and 4–PSK.I.i.d. Rayleigh fading and equal transmit powers for the source andthe relays. Scenario 1: All relays and the destination are corrupted byAWGN. Scenario 2: All relays and the destination are corrupted byϵ–mixture noise. Scenario 3: R1 and R3 are corrupted by ϵ–mixturenoise and unfaded synchronous CCI, respectively, and the other relaysand the destination are corrupted by AWGN. Solid lines with markers:Simulated BER. Dashed lines: Asymptotic BER obtained with (2.7)and (2.13). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46xiiList of Figures2.3 BER vs. average SNR, γ¯, for BRS with N = 3 AF and DF relays,respectively, and 4–PSK. The direct link is not used. γ¯11 = γ¯21 = γ¯,γ¯12 = γ¯22 = γ¯−10 dB, and γ¯13 = γ¯23 = γ¯−13 dB. Scenario 1: All relaysand the destination are corrupted by AWGN. Scenario 2: R2 and R3are corrupted by ϵ–mixture noise and Rician faded synchronous CCI,respectively, and the other relay and the destination are corruptedby AWGN. Solid lines with markers: Simulated BER. Dashed lines:Asymptotic BER obtained with (2.7) and (2.14). . . . . . . . . . . . . 472.4 BER vs. average SNR, γ, for 4–PSK PRS–S with N = 1, 2, 3. ForN = 1, γ¯0 = γ¯ and γ¯11 = γ¯21 = γ¯ − 10 dB. For N = 2, γ¯0 = γ¯,γ¯11 = γ¯21 = γ¯−3 dB, and γ¯12 = γ¯22 = γ¯− 10 dB. ForN = 3, γ¯0 = γ¯11 =γ¯21 = γ¯, γ¯12 = γ¯22 = γ¯ − 3 dB, and γ¯13 = γ¯23 = γ¯ − 10dB. Scenario 1:All relays and the destination are corrupted by AWGN. Scenario 2: Allrelays are subject to ϵ–mixture noise and the destination is corruptedby AWGN. Solid lines with markers: Simulated BER. Dashed lines:Asymptotic BER obtained with (2.7) and (2.17). . . . . . . . . . . . . 482.5 BER vs. average SNR, γ¯, for BRS with relay subset selection and 16–QAM. K = 2 of N = 3 relays are selected. γ¯0 = γ¯11 = γ¯21 = γ¯12 =γ¯22 = γ¯, γ¯13 = γ¯ − 17 dB, and γ¯23 = γ¯ − 10 dB. R2 is impaired byϵ–mixture noise and all other nodes are corrupted by AWGN. Solidlines with markers: Simulated BER. Dashed lines: Asymptotic BERobtained with (2.7) and (2.13). . . . . . . . . . . . . . . . . . . . . . 49xiiiList of Figures2.6 BER vs. ET/N0 for BRS with N = 3 AF relays and 16–QAM. Ω0 =Ω11 = Ω21 = 1, Ω12 = Ω22 = 0.1, Ω13 = Ω23 = 0.05. All noisevariances are equal to N0. R1 and R2 are impaired by GGN andunfaded synchronous CCI, respectively, and R3 and the destinationare impaired by AWGN. Solid lines with markers: Simulated BER.Dashed lines: Asymptotic BER obtained with (2.7) and (2.13). . . . . 513.1 System model for a single link S–R–D system where S and R are EHdevices. S and R have batteries with finite storage which can storethe energies harvested from the energy sources. The rectangle boxesin S and R represent the batteries and the hashed areas represent theamount of energy stored. . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Conventional relaying: Total number of transmitted bits vs. averagechannel SNR γ¯ for K = 10. . . . . . . . . . . . . . . . . . . . . . . . 803.3 Conventional relaying: Total number of transmitted bits vs. numberof time slots K for γ¯ = 25 dB. . . . . . . . . . . . . . . . . . . . . . 813.4 Link adaptive relaying: Total number of transmitted bits vs. averagechannel SNR γ¯ for K = 8 and K = 50. . . . . . . . . . . . . . . . . . 833.5 Comparison of conventional and link adaptive relaying: Total numberof transmitted bits vs. number of time slots K for γ¯S = γ¯R = γ¯ = 30dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 843.6 Comparison of online algorithms for conventional and link adaptiverelaying: Total number of transmitted bits vs. number of time slots Kfor γ¯S = 20 dB (Scenario 1) and γ¯S = 10 dB (Scenario 2) with γ¯R = 30dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86xivList of Figures3.7 Comparison of conventional and link adaptive relaying: Total numberof transmitted bits vs. HE for K = 60, γ¯S = 10 dB, and γ¯R = 30 dB. 873.8 Comparison of the HR assisted online schemes for conventional andlink adaptive relaying: Total number of transmitted bits vs. averageharvesting rate for K = 1000, γ¯S = 30 dB, and γ¯R = 15 dB. . . . . . 883.9 Total number of transmitted bits for offline and online power allocationschemes for conventional and link adaptive relaying when HR changesover the time of transmission for HS = 2 Joules, K = 50, and γ¯S =γ¯R = 30 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 893.10 Comparison of conventional and link adaptive relaying: Total numberof transmitted bits vs. Bmax for K = 100, γ¯S = 10 dB, and γ¯R = 30 dB. 903.11 Comparison of execution times of conventional and link adaptive re-laying: Execution time (in seconds) vs. number of time slots K forγ¯ = 30 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 914.1 System model for a multi–relay communication network. Source, S,and relays, Rn, n ∈ {1, 2, · · · , N} are EH nodes. Solid line indicatethe selected relay link. . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2 Total number of transmitted bits vs. number of time intervals K forγ¯ = 10 dB and average harvesting rate HE = 0.5 Joule. . . . . . . . . 1074.3 Convegence behavior of GBD vs. the number of iterations for K = 10,γ¯ = 10 dB, N = 5, and average harvesting rate HE = 0.5 Joule. . . . 1084.4 Total number of transmitted bits vs. average harvesting rate HE forK = 10 and γ¯ = 10 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 1104.5 Throughput (bits/sec) vs. number of selected relays for K = 30, γ¯ =10 dB, and N = 5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111xvList of Figures5.1 (a) System model for a communication link where the transmitter hasa hybrid energy supply. (b) Illustration of Scenario 1 with K trans-mission intervals, channel SNRs γk, and harvested energies Hk, wherek ∈ {1, 2, · · · ,K}. Here, RT bits arrive before transmission begins.(c) Illustration of Scenario 2. Here, Rk bits arrive just before timeinterval, k. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.2 Harvested energy Hk, fading level, water level, and optimal PE,k forScenario 1 vs. time interval k for K = 10 and HS = 2 Joules. . . . . . 1375.3 Consumed power for Scenario 1 vs. HS for K = 10 and RT = 75 bits.Only offline schemes are considered. . . . . . . . . . . . . . . . . . . . 1395.4 Consumed power for Scenario 1 vs. RT for K = 10 and HS = 6.75Joules. Only offline schemes are considered. . . . . . . . . . . . . . . 1415.5 Consumed power for Scenario 2 vs. RavgK for K = 10 and HS = 6.75Joules. Only offline schemes are considered. . . . . . . . . . . . . . . 1425.6 Consumed power for Scenario 1 vs. HS for K = 4 and RT = 30 bits. 1435.7 Consumed power for Scenario 1 vs. HS for K = 40 and RT = 300 bits. 1455.8 Consumed power for Scenarios 1 and 2 vs. HS for K = 100 andRT =K∑k=1Rk = 500 bits. . . . . . . . . . . . . . . . . . . . . . . . . . 1465.9 PH,k for Scenario 1 vs. k for K = 9 and HS = 2 Joules. . . . . . . . 1475.10 Consumed power for Scenario 1 vs. RT for K = 50 and two differentEH profiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149xviList of AbbreviationsAF Amplify–and–ForwardAWGN Additive White Gaussian NoiseBER Bit Error RateBICM Bit–Interleaved Coded ModulationBRS Best Relay SelectionCCI Co–Channel InterferenceCLT Central Limit TheoremCSI Channel State InformationDF Decode–and–ForwardDP Dynamic ProgrammingEH Energy HarvestingEPA Equal Power AllocationESP Equal Selection ProbabilityFH Frequency HoppingFPA Fair Power AllocationGGN Generalized Gaussian NoiseGMN Gaussian Mixture NoiseGP Geometric ProgramHR Harvesting RateHSU Harvest–Store–UsexviiList of AbbreviationsHU Harvest–Usei.i.d. Independent and Identically Distributedi.n.d. Independent and Non–Identically DistributedKKT Karush–Kuhn–TuckerLCD Light Emitting DiodeMGF Moment Generating FunctionMILP Mixed Integer Linear ProblemMINLP Mixed Integer Non–Linear ProgrammingNREL National Renewable Energy LaboratoryOPA Optimal Power Allocationpdf Probability Density FunctionPEP Pairwise Error ProbabilityPRS Partial Relay SelectionQAM Quadrature Amplitude ModulationQoS Quality of ServiceRF Radio FrequencyRFID Radio Frequency IdentificationsBB Spatial Branch and BoundSER Symbol Error RateSNR Signal–to–Noise RatioUCLA University of California, Los AngelesUWB Ultra–Wide BandxviiiNotationE(·) Statistical expectationA .= B A is asymptotically optimal to B (in the regime of high SNRunless otherwise stated)Pr{·} Probability of an event| · | Absolute value of a complex numbermx(p) 2pth order moment of xo(·) Order of a function, f(x) is o(x) if limx→0 f(x)/x = 0Γ(·) Gamma Function, Γ(x) !∞∫0e−ttx−1dtx! Factorial of x, x! ! 1. 2. · · · . xW (·) Lambert–W function[x]+ Non–negative value of x, [x]+ = max{x, 0}⌊x⌋ Floor operation on x, largest integer value not greater than xxixAcknowledgmentsFirst, I would like to express my invaluable gratitude to my supervisor, Prof. RobertSchober, for his relentless guidance, continuous support, high dedication, esteemedtechnical insights, constructive suggestions and comments, and effective advice. Hisenergy, drive, and enthusiasm have been a great source of motivation for me since Istarted working with him. This thesis would have been simply impossible withouthim. I am extraordinarily lucky to have had the chance to work with Prof. Schober,and will be grateful forever for his support and mentorship.I would like to thank my other supervisor, Prof. Lutz Lampe, for his overallguidance, constant support, and invaluable advice. It was very nice of him that heallowed me to join his group at the very last moment of my PhD. I am highly gratefulto him, especially for his exceptional support while I was doing a MITACS internship.I am highly grateful to Dr. Aissa Ikhlef and Dr. Derrick W. K. Ng for their con-structive suggestions and advice. Special thanks go to Dr. Diomidis S. Michalopoulos,Prof. Ranjan Mallik, Dr. Ali Nezampour, and Dr. Amir Nasri for their valuable com-ments and suggestions. I would like to thank my friend, Shankhanaad Mallick, forhis support and constructive suggestions. I am also grateful to my friends, Ghasemand Nikola and all other friends in communication lab in the department of ECE,UBC.Last but not the least, I would like to thank my wife Sonia, my father Prof.Farruk Ahmed, my mother, my brother Imran and his family, and my in–law familyxxAcknowledgmentsfor being very patient, caring, encouraging, and believing in me during all these years.Without their love and support I would have not been the person I am.This work was supported in part by grants from the Natural Sciences and Engi-neering Research Council of Canada (NSERC), the Institute for Computing, Infor-mation and Cognitive Systems (ICICS) at UBC, and TELUS.xxiDedicationTo My FamilyxxiiChapter 1IntroductionWireless communications and networking have undergone unprecedented develop-ment over the last few decades, becoming ubiquitous in a way that could hardly havebeen anticipated [7]. The rapid advancements of digital signal processing techniques,microprocessor technology, and radio frequency (RF) engineering have successfullyplaced wireless communication systems at the forefront of voice and data transmis-sion. In general, performance evaluation of a wireless communication system is basedpredominantly on its achievable data rates, i.e., how many data bits can be conveyedsuccessfully via a communication link in a given amount of time, and the bit (orsymbol or packet) error rate, i.e., how many data bits are detected incorrectly at thereceiver for a given number of transmitted bits. Traditionally, the error rate per-formance and/or throughput of wireless systems can be improved by increasing thetransmit power and/or channel bandwidth. However, increased power consumptionby wireless communication systems has been identified as a factor in the greenhouseeffect that is known to be detrimentally impacting the environment [8]. Moreover,high transmit power creates interference that requires receivers to have increasinglycomplex circuitry to mitigate its impact [9]. Furthermore, the amount of channelbandwidth available for wireless communications is limited and thus cannot be ex-ploited lavishly. A feasible response to these issues is much coveted by academics andindustry leaders. Ideally, a better system design would be obtained by optimizingthe available limited resources, e.g., the transmit power and the channel bandwidth.1Chapter 1. IntroductionBecause of the limited resources and the nature of the systems, an efficient systemdesign will require formulating and solving different optimization problems and, fromthere, proposing different resource allocation schemes. For instance, a transmit powerallocation scheme in a system with a constant energy source is significantly differentfrom a system whose available energy is a random variable. In this chapter, we willdescribe resource allocation schemes for wireless communication systems and providea brief overview on different communication systems powered by both constant andrandom energy sources, and impaired by Gaussian and non–Gaussian noises.Chapter 1 is organized as follows. Section 1.1 provides a brief overview of resourceallocation and corresponding optimization frameworks for wireless communicationsystems. Gaussian and different non–Gaussian noises present in wireless communi-cation systems are discussed in Section 1.2. Section 1.3 describes energy harvesting(EH) methods in communication systems. Section 1.4 summarizes the contributionsof this thesis, while Section 1.5 describes its structure.1.1 Resource Allocation in WirelessCommunication SystemsThe evolving next generation wireless communication systems are expected to over-come the fundamental challenges of existing systems and thus provide a dependableand precise quality of service (QoS) for users. This is quite a challenging task toaccomplish, especially for the emerging high data rate wireless applications. As afirst step, careful allocation of all available communication resources is required suchthat the system capacity is increased and the error rate performance is improved.Achieving certain specific goals for wireless communication systems demands that2Chapter 1. Introductionthe participating entities are managed efficiently and the communication resourcesare utilized effectively. One of the vital factors in improving the system through-put and operational efficiency is the development of appropriate resource allocationschemes.1.1.1 Optimization FrameworkIn general, resource allocation schemes are formulated as constrained optimizationproblems. A typical constrained optimization problem consists of a utility functionused as an objective/fitness function, a set of constraints used to confine a feasiblesolution set, and a set of optimization variables. The elements of an optimizationproblem are described in detail in the subsections that follow.Objective Function: An objective function in an optimization problem maps thesatisfaction of users into a real number and provides a tangible performance metric.Different objective functions are used in different communication systems as follows:• In most wireless communication systems, an aggregate end–to–end system through-put is a figure of merit integral to evaluation of the system’s performance [10].Considering a single link non–cooperative communication system with channelsignal–to–noise ratio (SNR) γ, bandwidth B, and transmit power P , the systemthroughput C is calculated by Shannon’s channel capacity formula [11],C = B log2(1 + γP ). (1.1)In general, an ergodic capacity/system throughput is used as the objective func-tion for an infinite horizon optimization (maximization) problem, whereas for afinite horizon problem, the system throughput aggregated over the considered3Chapter 1. Introductiontransmission time interval is adopted.• From a green communication standpoint, it is desirable that energy sourcesbe tapped into as little as possible to transmit a given number of data bits[12]. Therefore, optimization problems are often formulated with the aim ofminimizing the energy consumption [13].• Another important figure of merit is the average error rate or the outage prob-ability of a system. The quality of data transmission is often evaluated basedon closed–form expressions for the error rate and outage probability. Minimiz-ing the error rate (outage probability) while limiting the use of the availableresources (e.g., transmit power) improves the operational efficiency and ensuresa better quality of signal reception [14].Set of Constraints: In general, the QoS requirements are represented in an opti-mization problem by a set of constraints that are defined according to the physicallimitations of the system. For example, there is always a maximum limit on howmuch power is available for the purpose of transmission. This maximum limit de-pends on the type of the energy source. For a constant energy source, this limit isalways constant across the transmission time intervals. However, in case of a randomenergy source (e.g., EH node) the maximum limit on the available energy changesover time. Moreover, there is a maximum power budget for a cellular communica-tion system, to limit the amount of interference inflicted on other users [15]. On theother hand, in power minimization problems, we generally use a constraint to satisfythe minimum data rate requirement. The minimum data rate requirement is usuallyimposed in the problem formulation when there is a trade–off between the objectivefunction and the system throughput.4Chapter 1. Introduction1.1.2 Key Features of Resource Allocation SchemesIn real–world wireless communication systems, there are several important limita-tions on the transmitter that might vary according to the application. One of theselimitations arises from the energy source of the communication node, which is cap-tured by the long–term or short–term power constraints in the optimization problem.A long–term power constraint can be stated as follows:limK→∞1KK∑k=1Pk ≤ P¯ , (1.2)where K is the number of transmission time intervals, Pk is the transmit power ininterval k ∈ {1, 2, · · · ,K}, and P¯ is the available average power. This power con-straint is meaningful if the node is supplied by a constant energy source or equippedwith a battery that can be charged periodically at a constant rate. Furthermore, itis appropriate to use long–term utility functions if the performance is evaluated in aprobabilistic manner, e.g., based on the average error rate, ergodic capacity, averageoutage probability, and so on. In some cases, it is relevant to derive the analyti-cal expression of the average error probability and hence to adopt long–term utilityfunctions and constraints, cf. [16] to optimize the system’s performance.In some practical situations, we consider short–term utility functions and con-straints instead of the long–term utility functions and constraints. Batteries, forexample, might not have the capacity for getting charged fully in a periodic manner.Beside, in EH systems, the incoming energy is a random variable and therefore theavailability of energy supply is not guaranteed in every transmission interval. In suchsystems, the input power is subject to hard power constraints, i.e., the amount ofenergy consumed in each interval cannot exceed the amount of energy available in5Chapter 1. Introductionthe battery. In EH systems, the long–term power constraint in (1.2) cannot capturethe optimal use of the harvested energy in each time interval and hence short–termpower constraints are adopted as follows:Pk ≤ Bk, ∀k, (1.3)where Pk and Bk represent the transmit power and the available energy in the battery,respectively, in time interval k ∈ {1, 2, · · · ,K}. However, Bk is updated in eachinterval by1Bk = Bk−1 − Pk−1 +Hk, (1.4)where Hk is the amount of energy harvested from the renewable sources. As Pk de-pends on both the immediate and the past energy arrivals in (1.4), optimization prob-lems for EH communication systems cannot be formulated in probabilistic manner2 orover a single time interval, and hence we have to formulate the optimization problemfor EH systems with short–term utility function and constraints. For convenience,we denote the optimization problem with long–term utility function and constraintsas “long–term optimization” and the problem with short–term utility function andconstraints as “short–term optimization”.In this thesis, we consider both long–term and short–term optimization. In Chap-ter 2, we incorporate the effect of Gaussian and non–Gaussian noises in relay systemsand formulate a long–term optimization problem with the derived analytical expres-sion for average error rate. In Chapters 3–5, we examine EH systems and discuss1We use the linear model for the battery, as is well–established in the literature [17, 18].2By the term “probabilistic manner”, here we refer to an optimization problem with ergodicsystem throughput and average power consumption constraints.6Chapter 1. Introductionshort–term optimization for them.In general, finding resource allocation policies is more challenging for EH systemsthan for systems with constant energy supply because of the randomness of the arrivaltimes and the randomness of the amounts of harvested energy. Some of the existingliterature on wireless sensor networks with EH, e.g., [19, 20, 21, 22], considers long–term utility, analyzes the error rate performance, or proposes probabilistic powermanagement and scheduling policies. In particular, a large, possibly infinite numberof transmission time slots is considered, and thus the average error rate or the averagethroughput is calculated. This is in contrast to the contributions made in Chapters3–5, where we optimize power allocation as well as relay and link selection policiesover a finite number of time slots, which leads to more practical designs for realistictransmission scenarios. In this thesis, we consider two types of resource allocationscheme for EH systems, namely offline schemes and online schemes [17, 18].• In offline schemes, we assume that the non–causal information regarding thechannel SNR, the harvested energy, the number of incoming data bits, etc. areknown a priori. We formulate a single optimization problem considering thecausal and non–causal information over all the transmission time intervals, andsolve the optimization problem optimally. In general, offline schemes provideuseful performance upper bounds for the more practical online schemes. More-over, these schemes can provide design insights that can help us to design anefficient online scheme.• In practice, the amount of harvested energy, the channel SNR, and the numberof incoming data bits are random in nature and cannot be predicted in advance.Therefore, in this case, online resource allocation schemes have to be employedtaking into account the available information regarding the channel SNR, the7Chapter 1. Introductionharvested energy, and the incoming data bits. In general, an optimal onlineresource allocation scheme is formulated by computationally intensive stochas-tic dynamic programming (DP) [23]. To alleviate the computational cost ofthe optimal online scheme, we propose suboptimal but low–complexity onlineschemes.1.2 Noise in Wireless Communication SystemsMost contemporary wireless communication systems are designed with the assump-tion that the underlying noise is Gaussian distributed. This assumption is mainlybased on the analytical tractability and simple design of communication systems inGaussian noise and is often well justified by the central limit theorem (CLT). Nev-ertheless, in practice, wireless systems are also impaired by non–Gaussian noise andinterference3. The commonly observed non–Gaussian noises are man–made impulsivenoise, Gaussian mixture noise (GMN), generalized Gaussian noise (GGN), co–channelinterference (CCI), and ultra–wideband (UWB) interference, cf. [24]–[30] and refer-ences therein. For example, non–Gaussian platform noise is present if wireless chipsare implemented close to high–frequency central processors [31], and partial dischargeand switching effects cause impulsive noise in smart grid communication systems [32].It is acknowledged in the literature, cf. e.g. [29], that the performance of systems de-signed for Gaussian noise degrades alarmingly in non–Gaussian noise. Therefore,the inherent non–Gaussian noise in many wireless communication systems should betaken into account in the design process to achieve reliable performance and high datarates. However, another reason for assuming Gaussian noise in conventional system3To simplify our notation, in this thesis, “noise” refers to any additive impairment of the receivedsignal, i.e., our definition of noise also includes what is commonly referred to as “interference”.8Chapter 1. Introductiondesign is that the noise distribution usually changes rapidly and cannot be estimatedeasily. Nevertheless, it is important to investigate and study the performance ofwireless communication systems in non–Gaussian noise and interference.In general, the mathematical models for non–Gaussian noise can be divided intotwo broad categories, namely, noise models with exponentially–tailed probability den-sity functions (pdfs) and with algebraically–tailed pdfs [33]. A random variable X issaid to have an exponentially–tailed pdf, fX(x), if there exist c > 0 and a > 0 suchthatlimx→∞exp (xa) fX(x) = c. (1.5)Similarly, X is said to have an algebraically–tailed pdf, if there exist c > 0 and a > 0such thatlimx→∞xa fX(x) = c. (1.6)It can be easily shown from (1.6) that for noises with algebraically–tailed pdfs, thenth absolute moment Mn(X) ! E {|X|n}, for any n ≥ a does not exist, i.e., is in-finity. This is in contrast to the exponentially–tailed noises for which the momentsfor all values of n ≥ 0 are finite. Examples of noises with exponentially–tailed dis-tributions include Gaussian noise, GGN, Middleton’s Class A noise, GMN, and CCI.On the other hand, Middleton’s Class B and Class C [25], Generalized–t [34], andα–stable noise [33, 35] are noises with algebraically–tailed pdfs. In Chapter 2, weanalyze the error rate performance of a relay system that is impaired by Gaussianand non–Gaussian noise and interference. The analysis presented in Chapter 2 as-sumes that all noise moments are finite and hence it is applicable only in the case9Chapter 1. Introductionof exponentially–tailed type of noises. Note that exponentially–tailed noises includethe most common types of noise in communication systems and for this reason welimit our focus to them in Chapter 2. Analysis of the performance of a system im-paired by the noise with algebraically–tailed pdf is beyond the scope of this thesisand is presented instead in Chapter 6 as a potential subject for future research. It isworth mentioning that few algebraically–tailed noises, e.g., underwater acoustic noise,etc. can be approximated by exponentially–tailed noise (e.g., GGN), and hence wecan analyze their performance by our developed methods in Chapter 2. However,this kind of approximation and analysis show suboptimal performance, in general.Mathematical Modeling of Gaussian and Non–Gaussian Noise: In Chap-ters 3–5, we use Shannon’s capacity formula, which is valid for Gaussian noise andGaussian input signals4 to evaluate the system throughput [11]. However, Chap-ter 2 studies the performance for both Gaussian and different non–Gaussian noises.Therefore, we briefly review the properties of Gaussian and non–Gaussian noises inthe following:• Gaussian noise : As mentioned earlier, the Gaussianity property of noise is usu-ally justified by the CLT. Other important reasons for the popularity of Gaus-sian noise in communication system analysis are the preservation of the distri-bution under linear transformations and well–developed mathematical analysistools such as the Gaussian Q–function [36] and Turin’s formula [37]. The Gaus-sian pdf for a scalar (complex) random variable X with zero–mean and varianceσ2 is given byfX(x) =1πσ2exp(−|x|2σ2). (1.7)4In practice, the input signals do not follow Gaussian distribution. Instead, the input signalsare modulated into symbols having finite alphabet and also coded, if required. Note that thepractical modulation and coding schemes can be accommodated in the Shannon’s capacity formulaby multiplying all transmit powers by a constant.10Chapter 1. IntroductionNote that the maximum–likelihood (ML) receiver for Gaussian noise minimizesthe average Euclidean distance (ED) between the decoded symbol and the re-ceived signal. Unfortunately, the ED metric is not optimal for non–Gaussiannoise [33].• Laplacian noise : Laplacian noise is a non–Gaussian impulsive noise model.Generally, a noise is said to be impulsive if it is absent or possesses a lowamplitude most of the time but once in a while generates high amplitude spikes,i.e., impulses. The pdf of Laplacian noise is given byfX(x) =12σMexp(−|x|σM), (1.8)where σM is a scale parameter analogous to the variance in the Gaussian pdf. Itcan be shown that the ML receiver in Laplacian noise applies a median filter onthe received signal [33]. Note that there is no strong evidence for the existenceof Laplacian noise in practical communication systems [33].• GGN : A generalization of both the Gaussian and Laplacian noise models isknown as GGN, which is described by the pdffX(x) =β Γ(4/β)2π σ2 (Γ(2/β))2exp(−|x|βc), (1.9)where β > 0 is a parameter that controls the shape of the pdf, σ2 is the varianceof the noise, c !(σ2 Γ(2/β)Γ(4/β))β/2, and Γ(·) represents the Gamma function [38].Note that β = 1 and β = 2 correspond to the Laplacian and Gaussian pdfs,respectively. Fig. 1.1 shows the pdf of GGN for different values of β. It is worthmentioning that the lower the value of β, the more impulsive the noise will be.11Chapter 1. Introduction0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20.10.20.30.40.50.60.70.80.91Noise amplitude, |x|GGN pdf, fX(x)  β = 1.0 (Laplacian)β = 1.5β = 2.0 (Gaussian)β = 2.5Figure 1.1: The pdf of GGN. σ2 = 1 and different β have been considered.• GMN : This model is a generalized version of Middleton’s Class–A noise model[25], in which the number of interference sources is variable and their arrivaltime is not necessarily Poisson distributed [33]. The pdf of GMN is given byfX(x) =I∑i=1ciπσ2iexp(−|x|2σ2i), (1.10)where ci > 0,∑Ii=1 ci = 1, and σ2i , 1 ≤ i ≤ I, are noise parameters and thetotal variance is given by σ2 = ∑Ii=1 ciσ2i . The pdf in (1.10) indicates that atany time, a specific noise state i ∈ {1, 2, . . . , I} is present with probability ciand then a Gaussian random variable with variance σ2i is generated. Therefore,conditioned on the state of the noise, GMN is Gaussian distributed. GMNis a popular model for impulsive non–Gaussian noise in systems with receive12Chapter 1. Introductionantenna diversity [27], bit-interleaved coded modulation (BICM) [39, 40], andfor partial band interference in frequency hopping (FH) systems [41], etc.• ϵ–mixture noise : For I = 2, GMN with variance σ2 and c1 = 1− ϵ, c2 = ϵ, σ22 =κσ21 = κσ2/(1− ϵ + κϵ) simplifies to the well–known ϵ–mixture noise. In thisnoise model, the pdf of the noise can be characterized by only two parameters,namely, 0 < ϵ < 1 and κ ≫ 1. The physical interpretation of these parametersis as follows. The prevalent type of noise (i.e., the noise in (1 − ϵ) fractionof time) is a Gaussian background noise with variance σ21 = σ2/(1− ϵ + κϵ),whereas in ϵ fraction of the time (i.e., on average once every 1ϵ seconds), a strongGaussian interference with variance σ22 = κσ21 appears. Therefore, the smallerthe value of ϵ and the larger the value of κ, the more impulsive the noise willbe.• CCI : One of the most common types of interference is CCI, particularly incellular systems due to frequency re–use by different cells [42]–[43]. In fact,the capacity of cellular systems is mainly limited by CCI rather than the back-ground noise [44]. Mathematically, CCI from a single source of interference canbe expressed asx = a z, z !ku∑k=klg[k] i[k], (1.11)where a represents the fading gain of the channel between the interferer and thedesired user and kl, ku, g[k], and i[k] denote the lower limit, the upper limit,fixed (in general complex) coefficients, and the independent, identically dis-tributed (i.i.d.) interference symbols taken from an Mi–ary alphabet. Eq. (1.11)can be used to model multiple synchronous, a single asynchronous, or multipleasynchronous co–channel interferers. For example, for I synchronous interferers13Chapter 1. Introductionwe have kl = 1, ku = I, and g[k] and i[k] denote the gain and the transmittedsymbol of the kth interferer, respectively.Table 1.1: Examples of different EH sources and their performances; data collectedfrom [1] and [2].EH source PerformanceAmbient light (direct sunlight) 100mW/cm2Ambient light (illuminated office) 100µW/cm2Thermal (5 K gradient) 60µW/cm2Thermal (10 K gradient) 135µW/cm2Vibrational microgenerator (human motion) 4µW/cm3Vibrational microgenerator (machines) 800µW/cm3Ambient airflow 1mW/cm2Push buttons 50µJ/NHand generators 30W/KgIsotropic RF transmitter with 4W transmission (902–928 MHz) 5.5µW at 15mTX91501 Powercast transmitter with 3W transmission (915 MHz) 189µW at 5m1.3 Energy Harvesting in Wireless CommunicationSystemsIn wireless communication systems, the transceiver nodes expend their energy forprocessing and transmitting data. For some applications, connecting the transceivernodes to the power grid is cumbersome or may not even be possible. Pre–chargedbatteries can be a viable solution to this problem. In practice, the limited storagecapacity of batteries and high transmit powers may result in quick drainage of thebatteries. As a result, the batteries need to be replaced/recharged periodically, whichcan be sometimes impractical. An alternative solution is the deployment of EH nodes.EH nodes harvest energy from their surrounding environment, convert it to electricalenergy, and store the electrical energy in batteries to carry out their functions. In14Chapter 1. IntroductionFigure 1.2: EH modules for wireless communication systems. Source: [4].general, the energy can be harvested from unused ambient renewable energy sourcesusing solar, thermoelectric, and motion effects, or through other physical phenomena[19]. A list of various EH sources and their performances is provided in Table 1.1.EH nodes can be regarded as a promising option for deployment as they ensurea long system lifetime without the need for periodic battery replacements. In EHsystems, the energy can be independently harvested by the transceiver nodes duringthe course of data transmission at random times and in random amounts. For datatransmission (and for other signal processing tasks), EH nodes expend the energy fromtheir storage and only the unused energy remains in the batteries. In particular, ineach transmission time interval, the EH nodes are constrained to use, at most, onlythe energy available in their storage. These constraints necessitate the design of newtransmission strategies for the EH nodes, to ensure optimum performance in an EHenvironment.1.3.1 EH MethodsAs mentioned earlier, numerous EH methods are available in practice. Among them,a few EH modules are shown in Fig. 1.2. We describe some of these methods in the15Chapter 1. IntroductionFigure 1.3: Solar EH system. Researchers at the University of California, Los Angeles(UCLA), used liquid-crystal display (LCD) screens to work as photovoltaic cells forharvesting solar energy. Source: [5].following:• Solar EH systems : Solar energy is a convenient EH source and hence thereexist many implementations of solar EH nodes in the literature of sensor net-works and communication systems [45, 46]. The power density of solar energyat midday is approximately 100mW/cm2, i.e., 100mW/cm2 of eletrical powercan be harvested from the sun using polycrystalline solar cells, which have anefficiency of 16%–17%. Conversely, the power density in an indoor environment(e.g., in an illuminated office) is approximately 100µW/cm2. Solar energy iscaptured in a solar cell and then converted into electrical energy by exploitingthe photovoltaic effect. When the sunlight is incident upon a material surfaceon a solar panel, the electrons present in the material’s valence band absorbenergy and, being energized, jump to the conduction band of the material andbecome free. These highly energized, non–thermal electrons diffuse, and reach16Chapter 1. IntroductionFigure 1.4: EH roller sneaker which produces electricity from human motion. Source:[6].a junction where they are accelerated into a different material by a built–inpotential. This effect, referred to as the photovoltaic effect, generates an elec-tromotive force, and thus some of the light energy is converted into electricalenergy. In Fig. 1.3, we show the experimentation on a new technology thatturns LCD screens into photovoltaic cells, which would allow laptops and smartphones to harvest solar energy while they are in use [5].• Thermal EH systems : Several approaches, e.g., the Seebeck effect and thepiezothermal effect, among others, are exploited to convert the thermal energyinto electrical energy [47]. The efficiencies of these approaches are evaluatedby Carnot’s law. For example, for a thermal gradient of 5K with respect tothe normal ambient temperature of 300K, the thermal EH efficiency is approxi-mately 1.67% [1]. A commercial application of thermal EH systems is the Seiko17Chapter 1. IntroductionThermal wristwatch, which is powered by the heat generated by the humanbody [47].• Vibrational EH systems : Mechanical vibration is a potential EH source thatoccurs in human activities and industrial machinery, for example. Electric-ity can be generated from the vibration by many methods, e.g., piezoelectricmaterials, electromagnetic systems, and electrostatic systems. Piezoelectricmaterials exploit the ability of some materials, such as ceramics or crystals,to generate an electric potential difference between two nodes of a material inresponse to an implied mechanical stress. For electromagnetic converters, acoil oscillates in a static magnetic field and an electric potential is induced. Inthe case of electrostatic converters, electric charges on a set of flexible capac-itor plates create an electric potential once the plates are moved. Prominentexamples of vibrational EH sources include the vibration–based wristwatch,piezoelectric–powered radio–frequency identification (RFID) systems for shoes,and vibration–based micro–generators for intelligent sensor systems [47]. Fig.1.4 shows an example of exploitation of the piezoelectric effect from humanmovement in a roller sneaker.• Wind EH systems : Harvesting energy from wind is widely researched and ispopular for high power application. Large wind turbine–generators are used forsupplying power to remote loads, grids, and cellular base stations. According toa research conducted by the National Renewable Energy Laboratory (NREL),wind energy is considered the fastest growing electricity generation technologyin the world [48]. Expanding on the continuing success of large–scale wind EHsystems, attempts have been made to develop small–scale systems by consider-ing miniatured size and high portability.18Chapter 1. Introduction• RF EH systems : Recently, it has been shown in [49]–[52] that backgroundRF electromagnetic waves from ambient transmitters are also viable sources ofenergy. RF electromagnetic waves ranging from 3 KHz to 300 GHz are used asa medium to carry energy. RF EH can be regarded as a far–field energy transfertechnique. Therefore, the amount of energy that can be harvested depends onthe transmit power, the wavelength of the RF signal, and most importantly,the distance between the RF energy source and the harvesting node. Moreover,the RF–to–DC energy conversion efficiency has a vital impact on the amountof usable harvested energy. Although the development of RF EH systems isstill in its infancy, there are already a few companies that sell products forpractical implementations. The Powercast transmitter, which can transmitupto 3 W in the 915 MHz band, is an example of a commercial RF source [2]. Animportant advantage of RF EH systems is that RF signals can carry energy andinformation at the same time [49, 50]. Therefore, energy–constrained nodes canharvest energy and process the information simultaneously. This simultaneouswireless information and power transfer requires a careful design of the receiverso as to extract the correct information and harvest energy successfully [53, 54].Note that simultaneous information and power transfer is beyond the scope ofthis thesis and discussed in Chapter 6 as a potential future direction of research.1.3.2 EH ArchitecturesCommunication with the EH nodes strongly depends on the characteristics of thestorage capacity of the battery. For example, the storage capacity of a base stationis entirely different from the storage capacity of a small sensor node. A small sensornode may not have a battery at all, and hence can rely only on the energy har-19Chapter 1. IntroductionFigure 1.5: EH architectures: (a) HSU architecture and (b) HU architecture.vested immediately before its operation [55]. On the other hand, a base station canhave ample storage capacity of the battery in comparison to the sensor nodes [56].Therefore, the resource allocation policy for EH nodes highly depends on the batterymanagement algorithm to minimize the deleterious impact of energy outage5. Weclassify the EH architectures for wireless communications into two main categories,harvest–store–use (HSU) architecture and harvest–use (HU) architecture, as shownin Fig. 1.5.• HSU architecture : In this architecture, the energy is harvested from renewable5An energy outage is defined by a state when the transmitter is ready to transmit data bits, butcannot do so due to an insufficient amount of energy supply.20Chapter 1. Introductionsources and then stored in a battery for immediate and/or future use. EH is arandom process and there is no guarantee that the energy will be available infuture. Therefore, depending on the size of the battery, it is better to store asmuch extra energy as possible and thus reduce the randomness of the energyavailability. There is a large body of literature on resource allocation for EHsystems using the HSU architercture [17, 18, 57, 58, 59]. However, schedulingand optimizing the available resources for an HSU architecture is more involvedand strongly depends on the considered time horizon (finite or infinite). Forexample, the random energy arrival and management policy for EH systemsforms a Markov decision process [23], and hence the optimization problem forresource allocation cannot be formulated for a single time interval. Instead, ina given time interval, we have to incorporate the dependency of the resourceallocation policy on the factors of other time intervals. These considerationsresult in more complicated formulations of the optimization problem than theHU architecture, which is described next.• HU architecture : In an HU architecture, the harvested energy is not stored andhence powers the communication nodes directly [60]. Typically, a sensor nodeof miniature size does not contain a battery and thus its operation relies solelyon immediately–harvested energy. Examples of EH nodes with HU architectureinclude sensor nodes used for monitoring by bursty short message and push–button controllers [61]. The resource allocation policy for HU architecture isdifferent from that for HSU architecture. For example, it is sufficient to formu-late the optimization problem for the given time interval, and not to considerthe effect of the factors of other time intervals. There are a few contributions inthe literature on the design of EH systems with HU architecture, cf., [60, 62].21Chapter 1. Introduction1.4 Contributions and ResultsIn this thesis, we consider resource allocation algorithm designs for different wirelesscommunication systems with constant energy sources (Chapter 2), energy harvesters(Chapters 3 and 4), and a hybrid supply of energy (Chapter 5). A hybrid supplyconsists of a constant energy source and an energy harvester. The main contributionsof this thesis are as follows:1. We introduce a fair and flexible power allocation scheme for two amplify–and–forward (AF) relay selection systems, impaired by Gaussian and different non–Gaussian noises and powered by constant energy sources. The two AF relayselection schemes are denoted as best relay selection (BRS) and partial relayselection (PRS) in this thesis. To formulate the objective function for the opti-mization problem, we analyze the average error rate in the asymptotic regimeof high SNR for the considered systems. The derived analytical results arevalid for Gaussian and non–Gaussian noises with finite moments, independentand non–identically distributed (i.n.d.) Rayleigh fading, and arbitrary linearmodulation schemes. Our formulated power allocation schemes ensure the fair-ness with energy consumption constraints, which does not affect the achievablediversity gain. In order to reduce the signaling overhead required for channelstate information (CSI) acquisition, we propose a relay subset selection schemefor BRS and PRS. Simulation results confirm the accuracy of the proposed errorrate analysis, and optimal power allocation (OPA) was shown to outperformcompeting approaches to enforce fairness in energy consumption.2. We propose optimal resource allocation schemes for EH relay systems, where anEH source communicates with the destination via an EH decode–and–forward22Chapter 1. Introduction(DF) relay over fading channels. We consider two relaying protocols, namely,conventional relaying and buffer–aided link adaptive relaying protocols. Ourobjective is to maximize the system throughput over a finite number of trans-mission time slots for both relaying protocols. In the case of conventional re-laying, we propose optimal offline, optimal online, and several low–complexitybut suboptimal online joint source and relay transmit power allocation schemes.For buffer–aided link adaptive relaying, we propose an optimal offline and ef-ficient but suboptimal online schemes, which jointly optimize source and relaytransmit powers along with the link selection. Simulation results show thatbuffer–aided link adaptive relaying provides significant performance gains com-pared to conventional relaying but requires a higher complexity for computationof the resource allocation solution.3. We propose joint relay selection and power allocation schemes for maximizationof the throughput of an AF cooperative communication system, where thesource and the relays are EH nodes. We formulate an offline optimizationproblem which can be solved optimally by Generalized Bender’s Decomposition(GBD) method. For real–time implementation with low computational cost, wepropose two suboptimal online power allocation schemes. The performance ofthe proposed schemes is evaluated via simulations.4. We consider a point–to–point communication link, where the transmitter hasa hybrid energy supply comprised of a constant energy source and an energyharvester. We minimize the power consumed by the constant energy sourcefor transmission of a given amount of data in a given number of time inter-vals. Two scenarios are considered for packet arrival. In the first scenario, weassume that all data packets have arrived before transmission begins, whereas23Chapter 1. Introductionin the second scenario, we assume that data packets are arriving during thecourse of data transmission. For both scenarios, we propose optimal offline,optimal online, and low–complexity but suboptimal online transmit power al-location schemes. Simulation results reveal that the offline scheme performsbest among all considered schemes and the suboptimal online scheme providesa good performance–complexity tradeoff.1.5 Organization of the ThesisIn the following, we provide a brief overview of the remainder of this thesis:In Chapter 2, we derive accurate high SNR approximations for the error proba-bility and propose power allocation schemes for a cooperative network employing AFrelays and BRS or PRS. Since we do not assume that the noise distributions at thevarious network nodes are known at the destination, BRS and PRS are performedbased on the end–to–end and source–relay (relay–destination) SNRs, respectively,and the signals received via the relayed link and the direct link are combined withthe additive white Gaussian noise (AWGN) ML combining rule. We formulate anOPA problem for minimization of the asymptotic error probability under energy con-sumption constraints to enable fair energy consumption across relays. Furthermore,to reduce the amount of instantaneous CSI required for relay selection and the asso-ciated signaling overhead, we propose to pre–select a subset of relays based on thedeveloped error rate expressions which depend only on the average CSI.In Chapter 3, we propose offline and online power allocation schemes for conven-tional and buffer–aided link adaptive EH relay systems for a three–node cooperativecommunication system, where the source and the relay are assumed as EH nodes.In the case of conventional relaying, we formulate a convex optimization problem for24Chapter 1. Introductionthe offline power allocation, whereas for the online case, we propose a stochastic DPapproach to compute the optimal online transmit power. To alleviate the complexityinherent to DP, we also propose several suboptimal online power allocation schemes.For buffer–aided link adaptive relaying, we show that the joint offline optimization ofthe source and relay transmit powers along with the link selection results in a non–convex mixed integer non–linear problem (MINLP), which we solve optimally usingthe spatial branch–and–bound (sBB) method. However, we do not formulate an op-timal online scheme by DP for buffer–aided relaying, as otherwise, the formulationwould lead to a very high complexity and might not be implementable in practice.In Chapter 4, we consider AF relay selection where the source and the relays areEH nodes and thus the optimal relay selection depends on both the channel SNR andthe amount of harvested energy stored by the nodes. We propose offline and onlinejoint relay selection and source and relay transmit power allocation schemes thatmaximize the end–to–end system throughput over a finite number of transmissionintervals. The offline scheme results in a convex MINLP, which is solved by theGBD. However, as in Chapter 3, we do not formulate an optimal online scheme byDP, because such a formulation would lead to a very high complexity even greaterthan that of buffer–aided relaying for more than two relays. Instead, we formulatea suboptimal but low–complexity scheme, which exploits the statistical properties ofEH and relay selection.In Chapter 5, we consider a single communication link where the transmitter isequipped with a hybrid energy source, comprised of a constant energy source andan energy harvester. Our aim is to minimize the amount of energy drawn fromthe constant energy source, such that the harvested energy is efficiently utilized fortransmitting a given number of data packets over a finite number of transmission25Chapter 1. Introductionintervals. We propose optimal offline, optimal online, and low–complexity but sub-optimal online power allocation schemes for the considered system with two dataarrival scenarios. The solution of the optimization problem considered in Chapter 5provides insights regarding the optimal power allocation policy for communicationsystems with hybrid energy sources and thereby facilitates the design of reliable greencommunication systems.Finally, Chapter 6 summarizes the contributions of this thesis and provides direc-tions for future research.Appendices A and B contain the proofs of error rate expressions used in Chapter 2.In Appendix C, we enlist some related contributions that were accomplished duringmy PhD, but are not included in this thesis.26Chapter 2Relay Subset Selection and FairPower Allocation for Best and PartialRelay Selection in Generic Noise andInterference2.1 IntroductionBRS is an efficient approach to achieve full diversity in multi–relay systems [63, 64].Synchronization requirements are relaxed and spectral efficiency is increased com-pared to other cooperative diversity schemes where all relays transmit concurrently[65] and sequentially over orthogonal channels [66], respectively. Thus, the analy-sis of BRS has received considerable attention in the literature, cf. e.g. [63, 64, 67].However, since BRS is typically based on the instantaneous end–to–end SNR, a sig-nificant signaling overhead is required for CSI acquisition and feedback. To reducethe required amount of CSI for relay selection, PRS has been proposed. PRS usesonly the CSI of the source–relay or the relay–destination channels for relay selection,which reduces the estimation and signaling overhead at the expense of a loss in di-versity [68, 69]. In both BRS and PRS relays with favorable channel conditions are27Chapter 2. Fair Power Allocation in generic noise and interferenceselected more frequently, which drains their energy resources compared to relays withweaker channels. To overcome this problem, equal selection probability (ESP) BRShas been proposed in [3], where the end–to–end SNRs are biased such that all relaysare selected with equal probability. While this approach achieves fairness in energyconsumption among the relays, it may not yield high performance since relays withstrong channels and relays with weak channels transmit with identical powers andfor equal amounts of time.We mentioned in Chapter 1 that practical wireless communication systems areoften impaired not only by AWGN but also by non–Gaussian noise and interference.In this chapter, we derive asymptotic error probability of a cooperative networkemploying AF relays and BRS or PRS. The developed framework is applicable toall types of noise with finite moments and different network nodes may be impairedby different types of noise. Furthermore, we propose a relay subset selection schemeto reduce the requirement of instantaneous CSI and signaling overhead. BRS isperformed for the relays in the pre–selected subset, i.e., instantaneous CSI is requiredonly for the links of these relays.6 In addition, we exploit the derived error rateexpression and formulate an OPA problem with energy consumption constraints. Itis shown that BRS/PRS with OPA achieves significant performance gains comparedto BRS/PRS with ESP and equal power allocation (EPA), respectively, since differentrelays may transmit with different powers and for different amounts of time. However,the dependence of the energy consumption on the relay selection probability makesthe resulting OPA problem non–convex and more difficult to solve than the convexOPA problem for cooperative networks with orthogonal relay channels [16, 72, 73].The remainder of this chapter is organized as follows. In Section 2.2, the system6We note that rate–maximizing relay subset selection schemes have been proposed in [70, 71] forsystems impaired by AWGN. However, BRS and PRS are not considered in [70, 71] and the proposedrelay subset selection schemes are not motivated by reducing the CSI estimation overhead.28Chapter 2. Fair Power Allocation in generic noise and interferenceFigure 2.1: Block diagram of the considered AF relay system with BRS and PRS.Solid lines indicate the direct S–D link and the selected relay link.model for BRS and PRS is presented. The asymptotic bit error rate (BER) andsymbol error rate (SER) are analyzed in Section 2.3. In Section 2.4, the proposedschemes for relay subset selection and OPA with energy consumption constraints aredeveloped. The effectiveness of these schemes and the accuracy of the developedanalytical error rate expressions are evaluated based on computer simulations inSection 2.5. Section 2.6 concludes this chapter.2.2 System ModelIn this section, we introduce the considered signal and channel models.Channel Model: We consider a cooperative network with one source terminal,S, N half–duplex AF relays, Rk, k = 1, . . . , N , and one destination terminal, D,cf. Fig. 2.1. Transmission is organized in two phases. In the first phase, the source29Chapter 2. Fair Power Allocation in generic noise and interferencetransmits and the relays and the destination receive. In the second phase, one relay isselected for transmission and the destination receives. The fading gains of the S–D,S–Rk, and Rk–D links are denoted by h0, h1k, and h2k, respectively. We assumeindependent Rayleigh fading in all links, i.e., h0 ! a0ejθ0 , h1k ! a1kejθ1k , and h2k !a2kejθ2k are independent complex Gaussian random variables with zero mean andvariances Ω0 ! E {|h0|2}, Ω1k ! E {|h1k|2}, and Ω2k ! E {|h2k|2}, respectively, whereE{·} denotes statistical expectation. Here, the channel amplitudes, a0, a1k, and a2kare Rayleigh distributed, whereas the channel phases, θ0, θ1k, and θ2k are uniformlydistributed in [−π,π). The channel amplitudes are statistically independent of thechannel phases.n0, n1k, and n2k denote the (possibly) non–Gaussian noise samples at the destina-tion in the first phase of transmission, at relay k, and at the destination in the secondphase of transmission, respectively. Due to the distributed nature of relay networks,the noise samples at different relays are statistically independent and not necessarilyidentically distributed. For the proposed analysis we require all moments of all thenoises to exist. This is a mild assumption which is fulfilled by most types of noiseand interference of practical interest including AWGN, GMN, GGN, CCI, and UWBinterference (with α–stable noise being the only notable exception) [24].The instantaneous SNRs of the S–D, S–Rk, and Rk–D links are given by γ0 !P0a20/σ2n0 , γ1k ! P0a21k/σ2n1k , and γ2k ! Pka22k/σ2n2k , respectively, where σ2n0 ! E{|n0|2},σ2n1k ! E{|n1k|2}, σ2n2k ! E{|n2k|2}, and P0 and Pk are the average transmit powers ofthe source and relay k, respectively. In this chapter, we assume that S and Rk nodesare powered by constant energy sources. The corresponding average SNRs are givenby γ¯0 ! P0Ω0/σ2n0 , γ¯1k ! P0Ω1k/σ2n1k , and γ¯2k ! PkΩ2k/σ2n2k . For future reference,we also introduce the normalized noise samples as n¯0 ! n0/σn0 , n¯1k ! n1k/σn1k , and30Chapter 2. Fair Power Allocation in generic noise and interferencen¯2k ! n2k/σn2k .BRS: As customary in the literature, best relay selection is based on the maxi-mum bottleneck link SNR, which constitutes a tight upper bound for the end–to–endSNR [63]. The selected relay, Rks , is thus determined byks = argmaxk∈{1,...,N}{min{γ1k, γ2k}}. (2.1)For BRS, we assume that the destination acquires the CSI of all links (which requiresCSI feedback from the source or the relays), determines the best relay according to(2.1), and feeds ks back to the relays. Only the selected relay participates in thesecond phase of transmission.PRS: In PRS, relay selection is performed either by the source, based on thesource–relay SNRs or by the destination, based on the relay–destination SNRs. Thebest relay, Rks , is obtained as [68, 69]ks = argmaxk∈{1,...,N}γjk, (2.2)where j = 1 and j = 2 if the source and the destination select the relay, respectively.We refer to these relay selection schemes as PRS–S and PRS–D, respectively. In PRS–S and PRS–D, the source and the destination can directly estimate the required CSIbased on pilots sent by the relays, respectively. Thus, compared to BRS the signalingoverhead is reduced.Signal Model: The received signals at the destination in the first and secondphase of transmission are given byr0 =√P0h0x+ n0 (2.3)31Chapter 2. Fair Power Allocation in generic noise and interferenceandrks = Aksh2ksuks + n2ks = Aks√P0h2ksh1ksx+ n˜ks , (2.4)respectively, where uks =√P0h1ksx+ n1ks is the signal received at the selected relay,Aks is an amplification gain, and n˜ks ! Aksh2ksn1ks + n2ks is the effective noise. Thetransmitted symbol x is taken from an M–ary alphabet, A, with E{|x|2} = 1, suchas M–ary quadrature amplitude modulation (QAM) or M–ary phase–shift keying(PSK). The amplification gain is given by [74]Aks =√Pks/(P0a21ks + b), (2.5)where b = σ2n1ks is used for normalization of the average relay output power to Pks .Thus, for the simulation results shown in Section 2.5, b = σ2n1ks is adopted. However,in our performance analysis in Section 2.3, b = 0 is assumed to make the problemtractable as is customary in the literature [74]. Using b = 0 instead of b = σ2n1ks leadsto a very tight performance upper bound, especially at high SNR, cf. Section 2.5.Processing at Destination: We assume the destination does not know thedistribution of the noise at the different nodes of the network, which is a realisticassumption in practice. Therefore, the destination employs L2–norm combining [75](which is optimal in the maximum likelihood sense for AWGN but suboptimal fornon–Gaussian noise) to combine the signals received in the first and second phase oftransmission, yieldingL(x˜) =|r0 −√P0h0x˜|2σ2n0+|rks −√P0Aksh2ksh1ks x˜|2σ2n˜ks, (2.6)where x˜ ∈ A is a trial symbol and σ2n˜ks ! σ2n2ks +A2ksa22ksσ2n1ks . The detected symbol,32Chapter 2. Fair Power Allocation in generic noise and interferencexˆ ∈ A, is obtained from xˆ = argminx˜∈AL(x˜)7.2.3 Asymptotic Error Rate AnalysisIn this section, we derive the asymptotic error rate performances of BRS, PRS–S,and PRS–D in i.n.d. Rayleigh fading and generic, possibly non–Gaussian noise. Wefirst note that for sufficiently high SNR8, the SER and BER can be approximated bythose terms of the truncated union bound which dominate at high SNR [30, 36]. Forthe SER and BER of the considered linear modulation schemes thus leads toPXs.= ζ¯PXe (dmin) and PXb.=ζ¯log2MPXe (dmin), (2.7)respectively, where A .= B means that A and B are asymptotically (i.e., for highSNR) equal, PXe (d) denotes the asymptotic pairwise error probability (PEP) of twodistinct signal points xˆ, x ∈ A having an Euclidean distance of d ! |e|, e ! x − xˆ,from each other, dmin denotes the minimum Euclidean distance of constellation A, ζ¯is the average number of minimum–distance neighbors of each element of A. In (2.7),X is used to specify the considered relay selection scheme and X = B, X = PS, andX = PD stand for BRS, PRS–S, and PRS–D, respectively.Exploiting (2.3) and (2.6), the PEP of the considered relay selection schemes can7It is worth noting that an Lp–norm detector can be used to optimally detect the signals impairedby non–Gaussian noise [24]. For this detector, a tunable parameter p is adjusted offline for differentnoise distributions, and hence the detector can be implemented online. However, we need to knowthe noise distribution beforehand to implement the Lp–norm detector. This consideration is incontrast to the assumptions made in this chapter.8At what SNR values the analytical SER and BER approximations derived in this section becometight depends on the relay selection scheme, the number of relays, the type of fading, and the typeof noise. Typical SNR values where our analytical results become accurate lie between 20 dB and40 dB, cf. Section 2.5. For example, for BRS with N = 4 relays and independent and identicallydistributed Rayleigh fading, Fig. 2.2 reveals that the derived asymptotic BER expressions are tightfor SNRs larger than 25 dB.33Chapter 2. Fair Power Allocation in generic noise and interferencebe expressed asPXe (d) = Pr {L(xˆ) < L(x)} = Pr {∆0 +∆ks < 0} , (2.8)where Pr{A} denotes the probability of event A, x is the transmitted symbol, and∆0 ! (|√P0h0e+n0|2− |n0|2)/σ2n0 and ∆ks ! (|√P0Aksh2ksh1kse+n˜ks |2− |n˜ks |2)/σ2n˜ks .Furthermore, defining the moment generating functions (MGFs) of ∆0 and ∆ks asΦ∆0(s) ! Ea0,θ0{e−s∆0}and ΦX∆ks (s) ! Eaks ,θks{e−s∆ks}, respectively, (2.8) can berewritten asPXe (d) =12πjc+j∞∫c−j∞ΦX∆(s)dss, (2.9)where ΦX∆(s) ! Φ∆0(s)ΦX∆ks (s) and constant c > 0 lies in the region of convergence ofthe integral. Since Φ∆0(s) and ΦX∆ks (s) correspond to the direct S–D and the relayedlinks, respectively, the former MGF is identical for BRS and PRS, whereas the latterMGF depends on the selection scheme used. The asymptotic MGF of ∆0 can beobtained by averaging [16, Eq. (11)] with respect to the noiseΦ∆0(s) =1d2sγ¯0∞∑ξ=01ξsξm0(ξ) + o(γ¯−10), (2.10)where a function f(y) is o(y) if limy→0 f(y)/y = 0 and m0(ξ) ! E{|n¯0|2ξ} denotesthe 2ξ–th moment of n¯0. In the following subsections, we derive ΦX∆ks (s) and theasymptotic PEPs for the three considered relay selection schemes.34Chapter 2. Fair Power Allocation in generic noise and interference2.3.1 Asymptotic PEP of Best Relay SelectionFor BRS, we show in Appendix A that the asymptotic MGF of ∆ks can be expressedas ΦB∆ks (s) =N∑k=11N∏m=1,m ̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)(d2s)N∞∑ξ=0(ξ +N − 1)!sξ(ξ!)2(m1k(ξ)γ¯1k+m2k(ξ)γ¯2k)+o(γ¯−1c),(2.11)where m1k(ξ) ! E{|n¯1k|2ξ} and m2k(ξ) ! E{|n¯2k|2ξ} denote the noise moments of n¯1kand n¯2k, respectively, and γ¯c !∏Nk=1 min{γ¯1k, γ¯2k}. We note that the assumptionthat all noise moments exist is necessary for (2.10) and (2.11) to be valid since higherorder noise terms were absorbed into the respective o(·) terms. Combining (2.10) and(2.11) and assuming γ¯0, γ¯1k, γ¯2k →∞, we obtain the asymptotic MGFΦB∆(s).=N∑k=11N∏m=1,m ̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)(d2s)N+1γ¯0×∞∑ξ=0∑i+j=ξsξ(i+N−1)!m0(j)(i!)2j!(m1k(i)γ¯1k+ m2k(i)γ¯2k),(2.12)where ∑i+j=ξrepresents the summation over all possible combinations of non–negativeintegers i and j with i + j = ξ. Considering (2.9), the PEP, PBe (d), is given by thesum of the residues of ΦB∆(s)/s in the left hand side of the complex s–plain. Usingd’Alembert’s convergence test [76, 0.222] it can be shown that the right hand side of(2.12) is convergent for s ̸= 0, i.e., the only singularity of ΦB∆(s)/s is at s = 0. Thus,the asymptotic PEP is given by the residue of ΦB∆(s)/s at s = 0 or equivalently by thecoefficient associated with s0 in the series expansion of the right hand side of (2.12).Based on this observation, for high SNR, we only consider the term in the sum over35Chapter 2. Fair Power Allocation in generic noise and interferencei+ j with ξ = N + 1 in (2.12) to obtain the asymptotic PEP for BRS PBe (d).=1d2(N+1)N∑k=1∑i+j=N+1(i+N − 1)!m0(j)(i!)2j!γ¯0N∏m=1,m ̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)(m1k(i)γ¯1k+m2k(i)γ¯2k). (2.13)Combining (2.7) and (2.13) yields closed–form expressions for the SER and BERof AF BRS in generic, possibly non–Gaussian noise. The noise moments requiredin (2.13) can be found in [24, Tables I and II] for many types of noise of practicalinterest including AWGN, GGN, GMN, and CCI.No S–D Link: In some applications, the direct S–D link is negligible or cannotbe used. In this case, the PEP can still be obtained from (2.9) if we let Φ∆0(s) = 1.Thus, ΦB∆(s) = ΦB∆ks (s) holds and using a similar approach as before, we obtain forthe asymptotic PEPPBe (d).=N∑k=1Γ(2N)d2N(Γ(N + 1))2N∏m=1,m ̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)(m1k(N)γ¯1k+m2k(N)γ¯2k). (2.14)DF Relays: The asymptotic PEP of BRS with DF relays (without S–D link)in generic noise is also given by (2.14). A proof for this result can be found in [77,Appendix C] and corresponding numerical results are shown in Section 2.5.36Chapter 2. Fair Power Allocation in generic noise and interference2.3.2 Asymptotic PEP for Partial Relay SelectionWe show in Appendix B that the MGF of ∆ks for PRS–S can be written asΦPS∆ks (s) =N∑k=1∞∑ξ=01∑u1=0k ̸=1· · ·1∑uN=0k ̸=Nsξ(−1)N∑ν=1ν ̸=kuνξ!sd2×(m1k(ξ)γ¯1k+m2k(ξ)γ¯PS)+ o(γ¯−11k)+ o(γ¯−1PS), (2.15)where γ¯PS ! γ¯1kγ¯2k(1/γ¯1k+N∑i=1,i ̸=kui/γ¯1i). For high SNR, combining (2.10) and (2.15)we obtain the asymptotic MGF of ∆ for PRS–S asΦPS∆ (s).=N∑k=11(sd2)2γ¯0∞∑ξ=0∑i+j=ξ1∑u1=0k ̸=1· · ·1∑uN=0k ̸=Nsξm0(i)i!j!× (−1)N∑ν=1ν ̸=kuν (m1k(j)γ¯1k+m2k(j)γ¯PS). (2.16)Using again d’Alembert’s convergence test [76, 0.222], it can be shown that the righthand side of (2.16) is convergent for s ̸= 0. Thus, the asymptotic PEP can beexpressed by the residue of ΦPS∆ (s)/s at s = 0 or equivalently by the coefficientassociated with s0 in the series expansion of the right hand side of (2.16). Hence, forthe asymptotic PEP, in (2.16) only the term with ξ = 2 is relevant and we obtainP PSe (d).=1d4γ¯0N∑k=1∑i+j=2m0(i)i!j!1∑u1=0k ̸=11∑u2=0k ̸=2· · ·1∑uN=0k ̸=N(−1)N∑ν=1ν ̸=kuν×(m1k(j)γ¯1k+m2k(j)γ¯PS). (2.17)37Chapter 2. Fair Power Allocation in generic noise and interferenceCombining (2.7) and (2.17) yields closed–form expressions for the SER and BER ofPRS–S in generic, possibly non–Gaussian noise.The asymptotic PEP of PRS–D, P PDe , can be obtained in a similar fashion asthat of PRS–S and is given by P PDe (d).=1d4γ¯0N∑k=1∑i+j=2m0(i)i!j!1∑u1=0k ̸=11∑u2=0k ̸=2· · ·1∑uN=0k ̸=N(−1)N∑ν=1ν ̸=kuν (m1k(j)γ¯PD+m2k(j)γ¯2k), (2.18)where γ¯PD ! γ¯1kγ¯2k(1/γ¯2k +N∑i=1,i ̸=kui/γ¯2i).Remark 1: For N = 1 (no selection), BRS, PRS–S, and PRS–D are equivalent and(2.13), (2.17), and (2.18) give identical results. Furthermore, for N > 1, all terms in(2.17) involving noise moments m1k(j) cancel, i.e., the asymptotic PEP of PRS–S isindependent of the type of noise that impairs the relays. Similarly, for N > 1, allterms in (2.18) involving noise moments m2k(j) cancel and the asymptotic PEP ofPRS–D is independent of the type of noise that impairs the destination in the secondhop. In contrast, for the case N = 1 (no selection), the asymptotic PEPs of BRS andPRS depend on both the relay noise moments and the destination noise moments.Remark 2: For the special case of i.i.d. Rayleigh fading in the S–R and the R–D links,respectively, i.e., γ¯1k = γ¯1 and γ¯2k = γ¯2, 1 ≤ k ≤ K, for N > 1 the asymptotic errorrate of PRS–S (PRS–D) only depends on the relay–destination SNR, γ¯2 (source–relay SNR, γ¯1), but not on the source–relay SNR, γ¯1 (relay–destination SNR, γ¯2).In contrast, for N = 1 (no selection), the asymptotic error rate performance of PRSdepends on both γ¯1 and γ¯2. For the special case that all nodes are impaired by AWGN,similar observations have been made before in [68]. Here, we have extended this result38Chapter 2. Fair Power Allocation in generic noise and interferenceto non–Gaussian noise. However, we emphasize that for general i.n.d. Rayleigh fadingchannels (not considered in [68]), regardless of the type of noise, the error rate of PRSdepends on the SNRs of all links also for N > 1.2.3.3 Diversity and SNR GainIt is convenient to express the SER (or BER) in terms of the diversity gain Gd andthe SNR gain Gc, i.e., PXs.= (Gcγ¯)−Gd , where γ¯11 = . . . = γ¯1N = γ¯21 = . . . = γ¯2N =γ¯0 = γ¯ is assumed. Based on (2.7) and (2.13) we conclude that the diversity gain ofAF BRS is given by Gd = N + 1, irrespective of the type of noise. In contrast, thediversity gain of both PRS–S and PRS–D is Gd = 2, regardless of N and the typeof noise, cf. (2.7), (2.17) and (2.7), (2.18), i.e., full diversity is not achieved. On theother hand, the SNR gains of all considered relay selection schemes depend on thetype of noise via the noise moments.2.4 Optimization of BRS and PRS in Generic NoiseIn this section, we exploit the asymptotic error rate expressions from the previoussection for a reduction of the required signaling overhead via relay subset selectionand for optimization of the transmit powers of the source and the relays under energyconsumption constraints.2.4.1 Relay Subset SelectionIn systems with large numbers of relays, the overhead required for collection of theinstantaneous CSI of all links may be excessive. On the other hand, acquisition ofthe average CSI of all links in the form of average link SNRs and noise moments at39Chapter 2. Fair Power Allocation in generic noise and interferencethe relays and the destination may be possible since these parameters change muchmore slowly with time than the instantaneous CSI (i.e., the channel gains). The mainidea of relay subset selection is to pre–select K out of the N available relays basedon their average CSI and noise moments. Out of these K relays one relay is thenselected based on the corresponding instantaneous CSI of the involved links and theBRS criterion (2.1). The same idea can be applied to PRS.Let us denote all possible subsets of K out of N relays by Sk, 1 ≤ k ≤ N !/((N −K)!K!). We define the best subset for BRS, Sˆ, as that subset which minimizes thederived asymptotic PEP, PBe (d), in (2.14). As mentioned before, since the averageCSI changes very slowly with time, the best subset has to be updated relatively in-frequently. However, the best relay within the subset is updated relatively frequentlybased on the instantaneous CSI and (2.1).We note that subset selection reduces the achievable diversity order to Gd = K+1.However, if the average end–to–end SNRs and noise moments of the relays that arenot included in the selected subset are much worse than those of the relays includedin the subset, this loss in diversity will have a noticeable effect on performance onlyfor very high SNRs, cf. Section 2.5.2.4.2 Power Allocation with Energy ConsumptionConstraintsIn this subsection, we impose energy consumption constraints on the relays and thesource in order to limit the amount of energy consumed by each node. Throughoutthis section we assume that the average CSI, i.e., the link SNRs and the relevantnoise moments, is known for power allocation. However, knowledge of the exactdistribution of the noise is not required.40Chapter 2. Fair Power Allocation in generic noise and interferenceBased on the error rate expression developed in the previous section, we formulatethe following optimization problem for BRS, PRS–S, and PRS–D:minP≽0JX(P ) (2.19)s.t. P0 +N∑k=1PkfXk (P ) ≤ ET , (2.20)Emin,k ≤ fXk (P )Pk ≤ Emax,k, 0 ≤ k ≤ N, (2.21)where X ∈ {B,PS, PD}, JX(P ) is the objective function which is proportional tothe PEP PXe (d), P ! [P0 P1 . . . PN ], fX0 (P ) ! 1, fXk (P ) denotes the probability thatrelay k is selected, Emin,k and Emax,k denote, respectively, the normalized (to a certaininterval of time) minimum and maximum energy that can be allocated to node k, andET denotes the normalized total available energy. The normalized energy consumedby node k is given by fXk (P )Pk.We note that problem (2.19)–(2.21) encompasses many practical design problems.For example, in systems where the source and the relays are connected to the powergrid (constant energy source), we may only restrict the total consumed energy ETand set Emin,k = 0 and Emax,k = ∞, k = 0, 1, . . . , N . On the other hand, forapplications where the relays have their own power supply (e.g. a battery), we mayselect appropriate finite values for Emax,k, k = 0, 1, . . . , N , to avoid that relays withfavorable channel conditions drain their energy resources prematurely. We note that(2.19)–(2.21) is more difficult to solve than related power allocation problems in[16, 72, 73] for orthogonal relay channels, since fXk (P ) is in general not a posynomial9,and thus, (2.19) is in general not a geometric program (GP). In the following, we will9A posynomial is a function f(x1, · · · , xn) =N∑k=1ckxα1k1 · · ·xαnkn , where the variables xi andcoefficients ck are positive real numbers, and the exponents αik are real numbers [78].41Chapter 2. Fair Power Allocation in generic noise and interferencespecify JX(P ) and fXk (P ) for the considered relay selection schemes and discuss theresulting optimization problems.Power Allocation for BRSBased on the asymptotic PEP in (2.13), the objective function for BRS is defined asJB(P ) !N∑k=1∑i+j=N+1m0(j)(i+N − 1)!(i!)2j!P0N∏m=1,m ̸=k(P0ξ1mPmξ2mP0ξ1m+Pmξ2m)×(m1k(i)P0ξ1k+m2k(i)Pkξ2k), (2.22)where ξ1k ! Ω1k/σ2n1k and ξ2k ! Ω2k/σ2n2k . On the other hand, the selection proba-bility of relay k is given byfBk (P ) =∞∫0pzk(zk)N∏n=1n̸=kPzn(zk)dzk =1∑u1=0k ̸=1· · ·1∑uN=0k ̸=N(−1)N∑i=1i̸=kui(P0ξ1k + Pkξ2k)P0ξ1k + Pkξ2k + Pkξ1kξ2kN∑i=1i̸=kui(P0ξ1i+Piξ2i)Piξ1iξ2i, (2.23)where pzk(z) = γ¯1k+γ¯2kγ¯1kγ¯2k e− zγ¯1k γ¯2kγ¯1k+γ¯2k and Pzk(z) = 1 − e− zγ¯1k γ¯2kγ¯1k+γ¯2k denote the pdf andthe cumulative distribution function (cdf) of random variable zk = min{γ1k, γ2k},respectively. For any given N , fBk (P ) in (2.23) can be written as a single fractionwhere both the numerator and the denominator, λ(P ), are sums of positive terms.For example, for N = 2, we havefB1 (P ) = P1ξ11ξ21 (P0ξ12 + P2ξ22)/λ(P ) (2.24)42Chapter 2. Fair Power Allocation in generic noise and interferenceandfB2 (P ) = P2ξ12ξ22 (P0ξ11 + P1ξ21)/λ(P ), (2.25)where λ(P ) !P1ξ11ξ21 (P0ξ12 + P2ξ22) + P2ξ12ξ22 (P0ξ11 + P1ξ21). (2.26)However, since λ(P ) is a sum of products of the elements of P , fBk (P ) is not aposynomial and (2.22) is not a GP [78]. Thus, it is very difficult to solve (2.22)exactly. Hence, we follow the monomial fitting approach proposed in [78, Section 8]for this class of problems to find an approximate solution for (2.22) by solving a seriesof GPs. In particular, in the neighborhood of a given vector P i ! [P0i P1i . . . PNi],λ(P ) can be approximated as a monomialλ(P ) ≈ ciP0α0iP1α1i · · ·PNαNi , (2.27)where the monomial parameters are given by [78]αki =Pk(i)λ(P i)∂λ(P i)∂Pkiand ci =λ(P i)Pα0i0i Pα1i1i · · ·PαNiNi, (2.28)where k = 1, 2, . . . , N . Once we have replaced the denominator of fk(P ) by theapproximation in (2.27), fk(P ), JB(P ), and all polynomials in (2.19) are posynomials.Hence, (2.19) is a GP and can be efficiently solved using standard software [79]. Thisyields a new power allocation vector P i+1 and λ(P ) can be approximated in theneighborhood of P i+1 as shown in (2.27) but with i replaced by i + 1. This leadsto a new GP. This procedure is repeated until the power allocation vector P i doesnot change noticeably between iterations, i.e., ||P i+1 − P i|| < ϵ, where ϵ is a small43Chapter 2. Fair Power Allocation in generic noise and interferenceconstant (e.g. ϵ = 10−4). For initialization, a suitable P 0 which satisfies all constraintsin (2.19) is chosen.We note that for the adopted monomial fitting approach, convergence to theglobal optimal solution cannot be guaranteed in general [78]. However, in agreementwith the findings reported in [78], we were not able to find better power allocationsusing a (very complex and time consuming) numerical search, which suggests thatthe solution obtained with the proposed iterative algorithm is close to optimal.Power Allocation for PRSBased on the asymptotic PEPs in (2.17) and (2.18), it is straightforward to define forPRS–S and PRS–D objective functions JPS(P ) ∝ P PSe (d) and JPD(P ) ∝ P PDe (d),similar to the BRS case. We omit the expressions for JPS(P ) and JPD(P ) becauseof space limitation. The selection probabilities of relay k for PRS–S and PRS–D areobtained asfPSk (P ) =∞∫0pγ1k(x)N∏i=1,i ̸=kPγ1i(x)dx =1∑u1=0k ̸=1· · ·1∑uN=0k ̸=N(−1)N∑i=1,i ̸=kui1 +N∑i=1,i ̸=kuiξ1kξ1i(2.29)andfPDk (P ) =∞∫0pγ2k(x)N∏i=1,i ̸=kPγ2i(x)dx1∑u1=0k ̸=1· · ·1∑uN=0k ̸=N(−1)N∑i=1,i ̸=kui1 +N∑i=1,i ̸=kuiPkξ2kPiξ2i, (2.30)respectively, where pγjk(x) and Pγjk(x) denote the pdf and cdf of γjk, j = {1, 2},1 ≤ k ≤ N , respectively. Since for PRS–S the source–relay SNRs determine whichrelay is selected, fPSk (P ) is independent of Pi, i ∈ {0, 1, · · · , N}. Furthermore,44Chapter 2. Fair Power Allocation in generic noise and interferenceJPS(P ) is a posynomial. Therefore, for PRS–S the power optimization problem in(2.19) is a GP and can be solved exactly. In contrast, for PRS–D, neither fPDk (P )nor JPD(P ) are posynomials. Thus, for PRS–D, the resulting power optimizationproblem in (2.19) has to be solved using a monomial fitting approach similar to theone used for BRS.2.5 Numerical ResultsIn this section, we verify the analytical results derived in Section 2.3 with simulations.The asymptotic results shown in Figs. 2.2–2.6 were obtained by combining (2.7)with (2.13), (2.17), and (2.18). The considered noise models include ϵ–noise withparameters ϵ = 0.25 and κ = 50 (ϵ and κ denote the fraction of time where theimpulsive component is present and the ratio of the variances of the impulsive andthe Gaussian components, respectively), GGN with parameter β = 1 (i.e., Laplaciannoise), unfaded and Ricean (Ricean factor two) faded synchronous CCI, and AWGN.More details about the adopted noise models can be found in [24].In Fig. 2.2, we show the BER of AF BRS for 4–PSK and N = 4 relays. I.i.d. Rayleighfading is assumed, i.e., the average SNR in all links is γ¯. Three different noise sce-narios are considered, cf. caption of figure. For all three noise scenarios, the derivedasymptotic error rate expressions accurately predict the performance for sufficientlyhigh SNR. Furthermore, as expected from Section 2.3, for all noise scenarios, BRSyields the maximum diversity gain of Gd = N + 1 = 5. However, since the SNR gainis noise dependent, the BER curves have different SNR offsets for the different scenar-ios. For the considered example, Scenario 1, where all nodes are impaired by AWGN,yields the best performance since for the other two scenarios at least one node is af-fected by impulsive ϵ–mixture noise which has a negative effect on performance. For45Chapter 2. Fair Power Allocation in generic noise and interference0 5 10 15 20 2510-1010-910-810-710-610-510-410-310-210-1Average SNR (in dB)BER  Scenario 1 (sim)Scenario 2 (sim)Scenario 3 (sim)Asymptotic (th)Orthogonal RelayChannelsRelay SelectionFigure 2.2: BER vs. average SNR, γ¯, for BRS with N = 4 AF relays and 4–PSK. I.i.d.Rayleigh fading and equal transmit powers for the source and the relays. Scenario 1:All relays and the destination are corrupted by AWGN. Scenario 2: All relays and thedestination are corrupted by ϵ–mixture noise. Scenario 3: R1 and R3 are corruptedby ϵ–mixture noise and unfaded synchronous CCI, respectively, and the other relaysand the destination are corrupted by AWGN. Solid lines with markers: SimulatedBER. Dashed lines: Asymptotic BER obtained with (2.7) and (2.13).Scenario 3, we have also included the BER of cooperative relaying with orthogonalrelay channels, which was considered in [16]. As expected we notice that although thescheme with orthogonal relay channels outperforms BRS, both schemes achieve thesame diversity gain. More importantly, relay selection requires only two orthogonalchannels as compared to five for the scheme with orthogonal relay channels, i.e., relayselection is 2.5 times more bandwidth efficient.In Fig. 2.3, we compare BRS with AF and DF relays, respectively, assuming that46Chapter 2. Fair Power Allocation in generic noise and interference15 20 25 30 35 4010-810-710-610-510-410-310-210-1Average SNR (in dB)BER  AF (Scenario 1) (sim)DF (Scenario 1) (sim)AF (Scenario 2) (sim)DF (Scenario 2) (sim)Asymptotic (th)Scenario 2Scenario 1Figure 2.3: BER vs. average SNR, γ¯, for BRS with N = 3 AF and DF relays,respectively, and 4–PSK. The direct link is not used. γ¯11 = γ¯21 = γ¯, γ¯12 = γ¯22 =γ¯ − 10 dB, and γ¯13 = γ¯23 = γ¯ − 13 dB. Scenario 1: All relays and the destination arecorrupted by AWGN. Scenario 2: R2 and R3 are corrupted by ϵ–mixture noise andRician faded synchronous CCI, respectively, and the other relay and the destinationare corrupted by AWGN. Solid lines with markers: Simulated BER. Dashed lines:Asymptotic BER obtained with (2.7) and (2.14).the direct S–D link cannot be exploited. I.n.d. Rayleigh fading and two differentnoise scenarios are considered, cf. caption of figure. As expected from Section 2.3,for both noise scenarios AF and DF relays yield the same BER at sufficiently highSNR. This result was already known for the special case of impairment by AWGN,cf. [80], but is new for non–Gaussian types of noise and interference.Fig. 2.4 shows the performance of PRS–S under i.n.d. Rayleigh fading and twodifferent noise scenarios. For both scenarios, the destination is impaired by AWGN,whereas for Scenarios 1 and 2 the relays are corrupted by AWGN and ϵ–mixturenoise, respectively. For N = 1 (no selection) there exists a 1–dB performance gap47Chapter 2. Fair Power Allocation in generic noise and interference10 15 20 25 30 3510-810-710-610-510-410-310-2Average SNR (in dB)BER  N=1 (Scenario 1) (sim)N=1 (Scenario 2) (sim)N=2 (Scenario 1) (sim)N=2 (Scenario 2) (sim)N=3 (Scenario 1) (sim)N=3 (Scenario 2) (sim)Asymptotic (th)Figure 2.4: BER vs. average SNR, γ, for 4–PSK PRS–S with N = 1, 2, 3. For N = 1,γ¯0 = γ¯ and γ¯11 = γ¯21 = γ¯ − 10 dB. For N = 2, γ¯0 = γ¯, γ¯11 = γ¯21 = γ¯ − 3 dB, andγ¯12 = γ¯22 = γ¯ − 10 dB. For N = 3, γ¯0 = γ¯11 = γ¯21 = γ¯, γ¯12 = γ¯22 = γ¯ − 3 dB,and γ¯13 = γ¯23 = γ¯ − 10dB. Scenario 1: All relays and the destination are corruptedby AWGN. Scenario 2: All relays are subject to ϵ–mixture noise and the destinationis corrupted by AWGN. Solid lines with markers: Simulated BER. Dashed lines:Asymptotic BER obtained with (2.7) and (2.17).between both scenarios. On the contrary, for N > 1 both scenarios show identicalperformance as expected from Remark 1. Thus, for N > 1 the performance of PRS–Sis not affected by the type of noise that impairs the relays.In Fig. 2.5, we investigate the effectiveness of the considered relay subset selectionscheme. We assume i.n.d. Rayleigh fading channels and i.n.d. noise. Out of N = 3available relays, a subset of K = 2 relays is selected. From the two remaining relaysthe best one is selected using BRS. The BERs for all 3!/2! = 3 possible subsets areshown in Fig. 2.5. The subset consisting of relays R1 and R2 yields the best perfor-mance as accurately predicted by our asymptotic analysis. Thus, the subset selection48Chapter 2. Fair Power Allocation in generic noise and interference15 20 25 30 35 4010-910-810-710-610-510-410-310-2Average SNR (in dB)BER  R1 and R2 (sim)R1 and R3 (sim)R2 and R3 (sim)R1, R2, and R3 (sim)Asymptotic (th) No Subset SelectionSubsetSelectionFigure 2.5: BER vs. average SNR, γ¯, for BRS with relay subset selection and 16–QAM. K = 2 of N = 3 relays are selected. γ¯0 = γ¯11 = γ¯21 = γ¯12 = γ¯22 = γ¯,γ¯13 = γ¯ − 17 dB, and γ¯23 = γ¯ − 10 dB. R2 is impaired by ϵ–mixture noise and allother nodes are corrupted by AWGN. Solid lines with markers: Simulated BER.Dashed lines: Asymptotic BER obtained with (2.7) and (2.13).can be based on the asymptotic error rate expression in (2.13). For comparison, wehave also included the BER for BRS without subset selection. Although the lattercase results in a higher diversity gain, a noticeable performance gain compared tothe subset consisting of relays R1 and R2 is achieved only for SNRs higher than 30dB because of the comparably poor average channel conditions of R3. Thus, in thiscase, relay subset selection enables a significant reduction in signaling overhead atthe expense of a small performance loss.In Fig. 2.6, we investigate the performance of the proposed power allocationscheme for BRS with N = 3 AF relays and i.n.d. Rayleigh fading. In the con-sidered scenario, relays R1 and R3 experience the best and worst channel conditions,49Chapter 2. Fair Power Allocation in generic noise and interferenceTable 2.1: Power and energy constraints for OPA, FPA–1, FPA–2, EPA, and ESP[3], cf. Fig. 2.6. For all schemes P0 + P1fX1 + P2fX2 + P3fX3 ≤ ET is valid.OPA EPA FPA-1 FPA-2 ESP [3]P0 0 ≤ P0 <∞ P0 = ET /2 0 ≤ P0 ≤ ET /4 0 ≤ P0 <∞ P0 = ET /4P1 0 ≤ P1fX1 <∞ P1 = ET /2 0 ≤ P1fX1 ≤ ET /4 0 ≤ P1fX1 ≤ ET /3 P1fX1 = ET /4P2 0 ≤ P2fX2 <∞ P2 = ET /2 0 ≤ P2fX2 ≤ ET /4 0 ≤ P2fX2 ≤ ET /3 P2fX2 = ET /4P3 0 ≤ P3fX3 <∞ P3 = ET /2 0 ≤ P3fX3 ≤ ET /4 0 ≤ P3fX3 ≤ ET /3 P3fX3 = ET /4respectively. We consider five different power allocation/selection schemes: OPA, twofair power allocation (FPA) schemes, EPA, and ESP [3]. OPA, FPA–1, and FPA–2 are obtained by optimizing the transmit powers based on (2.19) under differentenergy consumption constraints, while EPA and ESP serve as benchmarks. EPA al-locates equal powers to the source and all relays, and ESP selects all relays with equalprobability while all nodes transmit with equal powers. The power allocation/energyconsumption of all considered schemes is summarized in Table 2.1. OPA and EPAdo not constrain the energy consumed by individual nodes and will drain the energyresources of the source and R1 more quickly than those of the other relays sincethe source always transmits and R1 is selected more often because of its favorablechannel conditions. Both FPA–1 and ESP force the source and all relays to consumethe same amounts of energy on average. For FPA–2, these stringent constraints arerelaxed. Fig. 2.6 reveals that OPA achieves a 1 dB performance gain compared toEPA. The price to be paid for the fairness enforced by FPA–1 is an SNR loss of 2.5dB compared to OPA. However, FPA–1 achieves a 2.5 dB gain compared to the fairbenchmark scheme ESP. The relatively poor performance of ESP is due to the factthat all relays are used for the same amounts of time and transmit with the samepowers, while FPA–1 is more flexible and uses relays with good channels more fre-50Chapter 2. Fair Power Allocation in generic noise and interference25 30 35 4010-910-810-710-610-510-410-310-2ET/N0 (in dB)BER  OPA (sim)EPA (sim)FPA-1 (sim)FPA-2 (sim)ESP  (sim)Asymptotic (th)Figure 2.6: BER vs. ET/N0 for BRS with N = 3 AF relays and 16–QAM. Ω0 = Ω11 =Ω21 = 1, Ω12 = Ω22 = 0.1, Ω13 = Ω23 = 0.05. All noise variances are equal to N0.R1 and R2 are impaired by GGN and unfaded synchronous CCI, respectively, andR3 and the destination are impaired by AWGN. Solid lines with markers: SimulatedBER. Dashed lines: Asymptotic BER obtained with (2.7) and (2.13).quently while allocating higher powers to relays with poor channels. FPA–2 achievespractically the same performance as OPA while still limiting the energy consumptionof the nodes.2.6 ConclusionsIn this chapter, we derived closed–form expressions for the asymptotic BER andSER of BRS and PRS with AF relays in i.n.d. Rayleigh fading and generic, possiblynon–Gaussian i.n.d. noise. The developed analytical results reveal that the diversity51Chapter 2. Fair Power Allocation in generic noise and interferencegains of BRS and PRS are independent of the type of noise but their SNR gainsdo depend on the type of noise via certain noise moments. With the derived errorrate expressions, we developed OPA scheme with energy consumption constraintsto avoid premature drainage of the energy resources of nodes with favorable channelconditions. Furthermore, to reduce the CSI feedback overhead, we considered schemesfor relay subset selection. Simulation results confirmed the accuracy of the proposedasymptotic analysis and OPA was shown to outperform competing approaches toenforce fairness in energy consumption.52Chapter 3Resource Allocation for Conventionaland Buffer-Aided Link AdaptiveRelaying Systems with EnergyHarvesting Nodes3.1 IntroductionAs mentioned in Chapter 1, EH nodes can operate autonomously over long periodsof time, as they are capable of harvesting energy from the surrounding environment.Therefore, EH nodes can be employed in different communication systems.Recently, transmission strategies for and performance analyses of EH nodes inwireless communication systems have been provided in [17, 18, 20, 81, 57, 58, 82].In [17], a single source–destination non–cooperative link with an EH source wasconsidered and an optimal offline along with an optimal and several sub–optimalonline transmission policies were provided for allocating transmit power to the sourceaccording to the random variations of the channel and the energy storage conditions.In [18], a similar system model was considered, where DP was employed to allocatethe source transmit power for the case when causal CSI was available. In [20], explicit53Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingthroughput–optimal and delay–optimal policies were developed for a single link point–to–point communication system. Several higher layer issues such as transmission timeminimization and transmission packet scheduling in EH systems were investigated in[57, 58, 82]. The use of EH relays in cooperative communication was introduced in[81], where a comprehensive performance analysis was performed for relay selectionin a cooperative network employing EH relays. A deterministic EH model (assuminga priori knowledge of the energy arrival times and the amount of harvested energy)for the Gaussian relay channel was considered in [59], [83], where delay and no–delayconstrained traffics were studied. In [84], cooperative communication with EH nodesusing an energy sharing strategy was studied. The concept of energy transfer in EHrelay systems was considered in [85], where an offline power allocation scheme wasproposed. The deployment of EH sensors in sensor networks has been extensivelydiscussed in the literature [19, 20, 21, 22].In this chapter, we consider a simple single link cooperative system where thesource communicates with the destination via a DF relay and both the source andthe relay are EH nodes. We assume that each node stores the harvested energy ina battery having finite capacity and the relay operates in the half–duplex mode. Inmost of the existing literature on half duplex relaying [81, 59, 86, 87], it is assumedthat relays receive a packet in one time slot from the source and forward it in thenext time slot to the destination. We refer to this approach as “conventional” relayingthroughout this chapter. Recently, it has been shown in [88, 89, 90] that equippingrelays with buffers can improve the performance of cooperative communication sys-tems. In fact, using buffers at the relays allows temporary storage of packets at therelay when the relay–destination channel quality is poor until the quality of the chan-nel has sufficiently improved. In [89], a buffer–aided adaptive link selection protocol54Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingwas proposed for the case where both source and relay have a constant energy supply.This protocol gives the relay the freedom to decide in which time slot to receive andin which time slot to transmit. We note that both conventional and buffer–aidedrelaying are important and deserve consideration since their applicability dependson the system requirements. For example, conventional relaying is suitable for delaysensitive systems, e.g., voice communications, because it entails the minimum possi-ble delay. On the other hand, buffer–aided relaying is suitable for delay non–sensitivesystems, e.g., data communications, where a certain amount of delay is tolerable, buta high average throughput is needed.In this chapter, we consider both conventional and buffer–aided link adaptiverelaying protocols. For both protocols, we propose offline and online (real–time)power allocation schemes that maximize the end–to–end system throughput over afinite number of transmission slots. For conventional relaying, we propose an optimalonline power allocation scheme which is based on a stochastic DP approach. Toavoid the high complexity inherent to DP, we also propose several sub–optimal onlinealgorithms. In case of buffer–aided link adaptive relaying, we formulate an offlineoptimization problem that jointly optimizes the source and the relay transmit powersalong with the link selection variable. Thereby, the link selection variable indicateswhether the source or the relay is selected for transmission in a given time slot. Theoptimization problem is shown to be a non–convex MINLP. We propose to use thesBB method to solve the offline MINLP problem optimally [91, 92, 93]. We alsopropose a practical online power allocation scheme for the buffer–aided link adaptiverelaying protocol.The buffer–aided link adaptive protocol considered in this chapter is differentfrom the no–delay constrained protocol in [59]. Firstly, [59] considers the Gaussian55Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingrelay channel whereas the source–relay and relay–destination links in this chapterare impaired by time–varying fading. Secondly, [59] only considers offline power al-location for delay and no–delay constrained relaying protocols, whereas we considerboth offline and online power allocation schemes for conventional and buffer–aidedlink adaptive relaying. Thirdly, although the no–delay constrained relaying protocolin [59] also employs a buffer, unlike the protocol in this chapter, it does not performadaptive link selection. Moreover, it was shown in [89] that optimizing power allo-cation and link selection for buffer–aided relaying over an infinite number of timeslots and for a constant energy supply always leads to an online solution, i.e., onlyknowledge of the current channel state is required and any additional knowledgeabout past or future channel states cannot further improve the solution obtained in[89]. In other words, for the problem considered in [89] the solution to the offlineoptimization problem can be implemented online. In contrast, since the optimizationin this chapter is performed over a finite number of time slots and the amount of en-ergy available for transmission is random, offline power allocation and link selectionpolicies outperform online policies, i.e., knowledge about past or future channel andEH states can further improve performance.The rest of this chapter is organized as follows. Section 3.2 describes the systemmodel. The proposed resource allocation schemes for buffer–aided link adaptive re-laying and conventional relaying are discussed in Sections 3.3 and 3.4, respectively.In Section 3.5, we show the simulation results and Section 3.6 concludes this chapter.3.2 System ModelWe consider an EH relay system, where the source, S, communicates with the des-tination, D, via a cooperative relay, R, as shown in Fig. 3.1. Both S and R are EH56Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingEnergy Sources: Solar, Wind, Biomechanical, Piezoelectric etc.  R D Battery Battery   Figure 3.1: System model for a single link S–R–D system where S and R are EHdevices. S and R have batteries with finite storage which can store the energiesharvested from the energy sources. The rectangle boxes in S and R represent thebatteries and the hashed areas represent the amount of energy stored.devices and are equipped with batteries, which have limited storage capacity andstore the harvested energy for future use. In particular, the batteries of S and Rcan store at most BS,max and BR,max Joules of energy, respectively. The participa-tion of S and R in signal transmission depends on the amount of harvested energystored in their batteries. The harvested energy can be of any form, e.g. solar, wind,or electro–mechanical energy. We assume that the data transmission is packet basedand organized in time slots of duration T . In the following, without loss of generality,we set T = 1s, and the system transmits for K time slots. Throughout this chapter,we assume that S is the central node. In particular, S acquires knowledge about thechannel SNRs and the harvested energies, executes the power allocation algorithms,and conveys the optimal transmit power for R and the link selection policy (for linkadaptive relaying) to R in each time slot. We assume that S has perfect knowledgeof the channel SNRs and the energy harvested at R10. In the next two subsections,10In practice, the channel SNRs may not be perfectly known to the source node due to differentsources of errors in the estimation process, e.g., background noise, quantization errors, and out-dated estimates. Furthermore, feedback errors and delay impair the quality of the estimates of theharvested energy at the relay node at the source node. However, studying the effect of imperfectchannel and energy knowledge is beyond the scope of this chapter and is an interesting topic for57Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingwe discuss the signal model, the system throughput, and the battery dynamics forconventional and buffer–aided link adaptive relaying.3.2.1 Conventional RelayingSignal Model: In conventional relaying, during the first time slot, S transmitsand R receives, and during the second time slot, R transmits and D receives. Thissequential process continues for K time slots, where K is assumed to be an evennumber. The received packet at R in the (2k − 1)th time slot is modelled asyR,2k−1 = hS,2k−1x2k−1 + nR,2k−1, k ∈ {1, 2, · · · ,K/2}, (3.1)where hS,2k−1 is the fading gain of the S–R link and nR,2k−1 denotes the noise sampleat R. The transmitted packet x2k−1 contains Gaussian–distributed symbols11. As-suming DF relaying, the detected packet, xˆ2k, is transmitted from R during time slot2k. Thus, the received packet at D is given byyD,2k = hR,2kxˆ2k + nD,2k, (3.2)where hR,2k and nD,2k denote the fading gain of the R–D link and the noise sample atD, respectively. hS,2k−1 and hR,2k can follow any fading distribution, e.g. Rayleigh,Rician, Nakagami–q, or Nakagami–m fading. nR,2k−1 and nD,2k are AWGN sampleshaving zero mean and unit variance. We assume the channels are quasi–static withineach time slot and the channel SNRs of the S–R and the R–D links are denoted byγS,2k−1 and γR,2k, respectively, where γS,2k−1 = |hS,2k−1|2 and γR,2k = |hR,2k|2. Wefuture work.11We note that practical modulation and coding schemes can be accommodated in the lateranalysis by multiplying all transmit powers by a constant χ > 1, which represents performance gapbetween the practical scheme and optimal one.58Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingassume γS,2k−1 and γR,2k are i.i.d. over the time slots. Furthermore, γS,2k−1 and γR,2kare mutually i.n.d. For future reference, we introduce the average SNRs of the S–Rand the R–D links as γ¯S and γ¯R, respectively.System Throughput: If x2k−1 is transmitted from S with transmit power PS,2k−1during time slot 2k − 1,ξS,2k−1 ! log2 (1 + γS,2k−1PS,2k−1) (3.3)bits of data can be received error–free at R. Similarly, if xˆ2k is transmitted from Rwith transmit power PR,2k,ξR,2k ! log2 (1 + γR,2kPR,2k) (3.4)bits of data can be received error–free at D. During the 2kth and the (2k−1)th timeslots, S and R, respectively, do not transmit any data, i.e., PS,2k = 0 and PR,2k−1 = 0.We assume that R ensures error–free detection by employing an appropriate errorcorrection coding scheme and hence xˆ2k = x2k−112. Therefore, the end–to–end (S–D) system throughput is given by 12 min{ξS,2k−1, ξR,2k} bits/s/Hz where the factor12 is due to the half–duplex constraint. Throughout this thesis, we assume thatthe number of transmitted packets is fixed in each time slot. On the contrary, thenumber of transmitted data bits (contained within data packets) per time slot, whichis calculated using Shannon’s capacity formula [11], is not fixed and changes over thetime slots depending on the channel SNR and the harvested energy. This type of datatransmission protocol is referred to as “adaptive rate transmission” in the literature[94].12To ensure an error–free detection, capacity achieving codes, e.g., turbo code and low–densityparity–check (LDPC) code are usually employed in practical wireless communication systems [11].59Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingBattery Dynamics: The energies stored in the batteries of S and R in time slotk are denoted by BS,k and BR,k, respectively. The transmit powers of S and R arelimited by the battery energies, i.e., 0 ≤ PS,2k−1 ≤ BS,2k−1 and 0 ≤ PR,2k ≤ BR,2k. Weassume throughout this chapter that the energy consumed by the internal circuitryof S and R is negligible compared to the transmit power [18]13. The energy harvesterat S harvests HS,m ≤ BS,max Joules of energy for transmission purpose during themth time slot, where m ∈ {1, 2, · · · ,K}. Similarly, the energy harvester at R collectsHR,m ≤ BR,max Joules of energy during the mth time slot. It is worth noting thatenergies are harvested during every time slot m at S and R. In general, the commonlyused model for the harvested energy is the Markovian model [18]. For example, solarenergy is usually modeled by a stationary Markov model. However, we did notuse any specific model for EH process to develop the resource allocation algorithmsthroughout this thesis. Rather, any stationary model can be accommodated for thedeveloped algorithms. Let HS ! E{HS,m} and HR ! E{HR,m} denote the averageEH rate of S and R over the time slots, respectively. Because of the spatial separationof S and R, we assume HS,m and HR,m are independent of each other and i.i.d. overthe time slots14. Similar to [18], we assume the stored energies at S and R increaseand decrease linearly provided the maximum storage capacities, BS,max and BR,max,are not exceeded, i.e.,BS,m+1 = min{(BS,m − PS,m +HS,m), BS,max}, ∀m (3.5)13If the energy consumed by the internal circuitry is not negligible, our problem formulations inSections 3.3 and 3.4 have to be changed accordingly. Problem formulations for EH systems account-ing for the energy consumed by the internal circuitry can be found in e.g. [95, 96]. Alternatively,the energy consumed by the internal circuitry may be supplied by a pre–charged battery and onlythe transmitted energy is generated via EH.14The EH process at different nodes can be correlated as well, e.g., when S and R are exposed tosunlight, their EH processes can be correlated. The algorithms developed in this chapter are alsovalid for the mutually correlated EH processes.60Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingBR,m+1 = min{(BR,m − PR,m +HR,m), BR,max}, ∀m. (3.6)Furthermore, BS,1 = HS,0 ≥ 0 and BR,1 = HR,0 ≥ 0, respectively, denote the availableenergies at S and R before transmission starts.3.2.2 Buffer–Aided Adaptive Link SelectionSignal Model: For buffer–aided link adaptive relaying, relay R is equipped with abuffer in which it can temporarily store the packets received from S. In this case,S decides whether S or R should transmit in a given time slot, k ∈ {1, 2, · · · ,K}[89]. Therefore, unlike conventional relaying, in any time slot k, S or R can transmitpackets. Let dk ∈ {0, 1} denote a binary link selection variable, where dk = 0 (dk = 1)if the S–R (R–D) link is selected for transmission. When dk = 0, the received packetat R is given byyR,k = hS,kxk + nR,k. (3.7)On the other hand, when dk = 1, the received packet at D is given byyD,k = hR,kxˆk + nD,k. (3.8)System Throughput: When dk = 0, S is selected for transmission and ξS,k bitsof data can be transmitted error–free via the S–R link. Hence, R receives ξS,k databits from S and appends them to the queue in its buffer. Therefore, the number ofbits in the buffer at R at the end of the kth time slot is denoted as Qk and givenby Qk = Qk−1 + ξS,k. However, when dk = 1, R transmits and the number of bitstransmitted via the R–D link is given by min{ξR,k, Qk−1}, i.e., the maximal number61Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingof bits that can be sent by R is limited by the number of bits in the buffer andthe instantaneous capacity of the R–D link [89]. The number of bits remaining inthe buffer at the end of the kth time slot is given by Qk = Qk−1 −min{ξR,k, Qk−1}.We assume that S has always data to transmit and the buffer at R has very large(possibly infinite) capacity to store them. Therefore, a total ofK∑k=1min{dkξR,k, Qk−1}bits are transmitted from S to D during the entire transmission time.Battery Dynamics: The battery dynamics for the link adaptive transmission pro-tocol are identical to those for conventional relaying.3.3 Power Allocation and Link Selection forBuffer–Aided RelayingIn this section, we propose offline and online joint link selection and power allocationschemes for EH systems employing buffer–aided relaying. Buffer–aided relaying ispreferable over conventional relaying for applications which can tolerate delays butrequire high throughput.3.3.1 Offline Power AllocationOur goal is to maximize the total number of transmitted bits (from S to D) deliveredby a deadline of K time slots for the link adaptive transmission protocol. The offline(prior) information about the full CSI and the energy arrivals at S and R in each timeslot are assumed to be known in advance. The resulting maximization problem issubject to a causality constraint on the harvested energy and the (maximum) storageconstraint for the batteries at both S and R. The offline optimization problem for62Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingthe link adaptive transmission protocol can be formulated as follows:maxT ≽0, {dk|∀k}K∑k=1dk log2(1 + γR,kPR,k) (3.9)s.t.q∑k=1((1− dk)PS,k + λS,k) ≤q−1∑k=0HS,k, ∀q (3.10)q∑k=1(dkPR,k + λR,k) ≤q−1∑k=0HR,k, ∀q (3.11)v∑k=0HS,k −v∑k=1((1− dk)PS,k + λS,k) ≤ BS,max, ∀v (3.12)v∑k=1HR,k −v∑k=1(dkPR,k + λR,k) ≤ BR,max, ∀v (3.13)q∑k=1(1− dk) log2 (1 + γS,kPS,k) ≥q∑k=1dk log2 (1 + γR,kPR,k) , ∀q (3.14)dk(1− dk) = 0, ∀k, (3.15)where T ! {PS,k, PR,k, λS,k, λR,k|k ∈ {1, 2, · · · ,K}}. Also, ∀q, ∀k, and ∀v standfor q ∈ {1, 2, · · · ,K}, k ∈ {1, 2, · · · ,K}, and v ∈ {1, 2, · · · ,K − 1}, respectively.The slack variables λS,k and λR,k ensure that constraints (3.12)–(3.14) can be metfor all realizations of γS,k, γR,k, HS,k, and HR,k. In particular, these slack variablesrepresent the power (possibly) wasted in each transmission interval15. Constraints(3.10) and (3.11) stem from the causality requirement on the energy harvested at Sand R, respectively. Moreover, (3.12) and (3.13) ensure that the harvested energydoes not exceed the limited storage capacity of the batteries at S and R, respectively.Constraint (3.14) ensures that R cannot transmit more bits than it has in its buffer.15For example, if HS,k and HR,k are large values (HS,k and HR,k are random variables and cannotbe controlled in the optimization problem) and BS,max and BR,max are small, then if λS,k and λR,kwere omitted, constraints (3.12) and (3.13) would not be satisfied and problem (3.9)–(3.15) wouldbecome infeasible. Therefore, slack variables λS,k and λR,k ensure that problem (3.9)–(3.15) isalways feasible.63Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingMoreover, (3.15) ensures that dk can only be 0 or 1, i.e., either S or R transmits ina given time slot, k ∈ {1, 2, · · · ,K}. We note that, in this optimization problem,although we are maximizing the throughput of the R–D link, using constraint (3.14)incorporates the effect of the throughput of the S–R link. As ξS,k and ξR,k areincreasing functions of PS,k and PR,k, respectively, the optimization problem in (3.9)–(3.15) can be restated as follows:maxT ′≥0, {dk|∀k}K∑k=1dkξR,k (3.16)s.t.q∑k=1((1− dk)(2ξS,k − 1)γS,k+ λS,k)≤q−1∑k=0HS,k, ∀q (3.17)q∑k=1(dk(2ξR,k − 1)γR,k+ λR,k)≤q−1∑k=0HR,k, ∀q (3.18)v∑k=0HS,k −v∑k=1((1− dk)(2ξS,k − 1)γS,k+ λS,k)≤ BS,max, ∀v (3.19)v∑k=1HR,k −v∑k=1(dk(2ξR,k − 1)γR,k+ λR,k)≤ BR,max, ∀v (3.20)q∑k=1(1− dk)ξS,k ≥q∑k=1dkξR,k, ∀q (3.21)dk(1− dk) = 0, ∀k, (3.22)where T ′ ! {ξS,k, ξR,k, λS,k, λR,k|k ∈ {1, 2, · · · ,K}}. The problem in (3.16)–(3.22)is a non–convex MINLP due to the binary variables dk and the non–convex and non–linear constraints (3.17)–(3.22). In the following, we propose two optimal methodsto solve the buffer–aided link adaptive offline optimization problem.Exhaustive SearchFor given dk, k ∈ {1, 2, · · · ,K}, the optimization problem in (3.16)–(3.22) is con-64Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingvex. Therefore, we can optimize ξS,k and ξR,k for given dk ∈ {0, 1} very efficiently.In this method, we optimize ξS,k and ξR,k for all possible combinations of dk, k ∈{1, 2, · · · ,K}, and select from all the solutions that combination of dk, k ∈ {1, 2, · · · ,K},which maximizes the cost function. This exhaustive search method provides theglobal optimal solution but with an exponential complexity. For instance, for K timeslots we have 2K−2 possible combinations of dk and hence to optimize ξS,k and ξR,k,we need to solve 2K−2 optimization problems. Therefore, in practice, this approachcannot be adopted in general, especially for large K. However, the exhaustive searchscheme can be effective for small K.Spatial Branch-and-Bound (sBB)As mentioned before, our problem is a non–convex MINLP. One of the recent advancesin (globally) solving MINLP problems is the sBB method [91, 92]. The sBB methodsequentially solves subproblems of problem (3.16)–(3.22). These subproblems areobtained by partitioning the original solution space. For each subproblem, the sBBmethod relies on the generation of rigorous lower and upper bounds of the problemover any given variable sub–domain. The feasible lower bounds are chosen to bethe local minimizers of the (sub)problems whereas the upper bounds are obtainedfrom convex relaxations. Interestingly, MINLP problems can be solved by using thewidely available open source solver Couenne [93, 92]. Couenne provides the globaloptimal solution for both convex and non–convex MINLP problems. It implementslinearization, bound reduction, and branching methods within a branch and boundframework. The convergence of the sBB method in a finite number of iterations isguaranteed [93]. However, the worst–case computational cost of sBB is exponentialin the size of the optimization variables [92].65Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying3.3.2 Online Power AllocationIn practice, only causal information about channels and harvested energies is avail-able for power allocation. Therefore, the offline power allocation scheme is not readilyapplicable as, at a given time slot, the future CSI and the upcoming harvested energyare not known in advance. To this end, we could formulate an optimal online schemeusing stochastic DP. Unfortunately, this approach leads to a very high computationalcost because of the adaptive link selection in every time slot and may not be imple-mentable in practice. Therefore, we propose two efficient suboptimal online schemeswhich have low complexity.Suboptimal Harvesting Rate (HR) Assisted Online Power AllocationWe propose an efficient online power allocation scheme referred to as “HR Assisted".To this end, we formulate an optimization problem which is based on the average datarate, the average energy causality constraints at S and R, and the average bufferingconstraint for K →∞ as follows:max{PS,k≥0, PR,k≥0, dk|∀k}1KK∑k=1dk log2(1 + γR,kPR,k) (3.23)s.t. 1KK∑k=1(1− dk)PS,k ≤1KK−1∑k=0HS,k (3.24)1KK∑k=1dkPR,k ≤1KK−1∑k=0HR,k (3.25)1KK∑k=1(1− dk) log2 (1 + γS,kPS,k) =1KK∑k=1dk log2 (1 + γR,kPR,k) (3.26)1Kdk(1− dk) = 0, ∀k (3.27)66Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingThe Lagrangian of (3.23)–(3.27) is given byL =1KK∑k=1dk log2(1 + γR,kPR,k)−ψSK(K∑k=1(1− dk)PS,k −HS,k−1)−ψRK(K∑k=1dkPR,k −HR,k−1)−1KK∑k=1βkdk(1− dk)−ϱK(K∑k=1dk log2(1 + γR,kPR,k)− (1− dk) log2(1 + γS,kPS,k)), (3.28)where ψS, ψR, ϱ, and βk are Lagrange multipliers. Differentiating (3.28) with respectto PS,k, PR,k, and dk and equating each of the differentiated expressions to zero leadsto the following optimum values of PS,k, PR,k, and dk:P ∗S,k =⎧⎪⎨⎪⎩ϑνS− 1γS,k , if γS,k >νSϑ AND dk = 0,0, otherwise,(3.29)P ∗R,k =⎧⎪⎨⎪⎩1νR− 1γR,k , if γR,k > νR AND dk = 1,0, otherwise,(3.30)d∗k =⎧⎪⎨⎪⎩1, if (CR > CS) OR(γR,k > νR AND γS,k < νSϑ),0, if (CR < CS)OR(γR,k < νR AND γS,k > νSϑ),(3.31)where CR = ln(γR,kνR)+ νRγR,k − 1, CS = ϑ ln(ϑγS,kνS)+ νSγS,k − ϑ, ϑ = ϱ/(1 − ϱ), νS =ψS ln(2)/(1− ϱ), and νR = ψR ln(2)/(1− ϱ). We observe that the optimal PS,k, PR,k,and dk depend on the instantaneous channel SNRs and Lagrange multipliers. TheLagrange multipliers can be solved efficiently, as shown in the next part of this section,without requiring any non–causal knowledge. Therefore, the optimal PS,k, PR,k, anddk are readily applicable in a real–time (online) environment with low implementationcomplexity.67Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingFinding the Lagrange Multipliers: Combining (3.29)–(3.31) and (3.24)–(3.26)yields the following conditions for K →∞:νR∫0[ ∞∫νSϑ(ϑνS−1γS,k)fγS,k(γS,k)dγS,k]fγR,k(γR,k)dγR,k+∞∫νR[ ∞∫L1(ϑνS−1γS,k)fγS,k(γS,k)dγS,k]fγR,k(γR,k)dγR,k = HS,E, (3.32)νSϑ∫0[ ∞∫νR(1νR−1γR,k)fγR,k(γR,k)dγR,k]fγS,k(γS,k)dγS,k+∞∫νSϑ[ ∞∫L2(1νR−1γR,k)fγR,k(γR,k)dγR,k]fγS,k(γS,k)dγS,k = HR,E, (3.33)νR∫0[ ∞∫νSϑlog2(ϑγS,kνS)fγS,k(γS,k)dγS,k]fγR,k(γR,k)dγR,k+∞∫νR[ ∞∫L1log2(ϑγS,kνS)fγS,k(γS,k)dγS,k]fγR,k(γR,k)dγR,k=νSϑ∫0[ ∞∫νRlog2(γR,kνR)fγR,k(γR,k)dγR,k]fγS,k(γS,k)dγS,k+∞∫νSϑ[ ∞∫L2log2(γR,kνR)fγR,k(γR,k)dγR,k]fγS,k(γS,k)dγS,k, (3.34)68Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingwhere L1 = − νSϑW⎛⎝−eγR,k−νRϑγR,k−1(νRγR,k) 1ϑ⎞⎠and L2 = − νRW(−eϑ−νSγS,k−1(νSϑγS,k)ϑ) . Here,W (·) is the Lambert W–function [97]. For Rayleigh fading, fγS,k(γS,2k−1) = (1/γ¯S)e−γS,2k−1/γ¯S and fγR,k(γR,2k) = (1/γ¯R)e−γR,2k/γ¯R . We need to solve (3.32)–(3.34) tofind the optimal νS, νR, and ϑ. The solution can be obtained by using the built–inroot–finding function in Mathematica. We note that the optimal νS, νR, and ϑ arecomputed offline before transmission starts. When transmission begins, P ∗S,k, P ∗R,k,and d∗k are calculated based on offline parameters ϑ, νS, and νR and online variablesγS,k and γR,k.Algorithm 3.1 HR Assisted Online Scheme For Buffer–Aided Link Adaptive Relay-ing1: Initialize the buffer status, Q0 = 0 bits;2: for k = 1 to K do3: Calculate BS,k and BR,k using (3.5) and (3.6), respectively.4: Calculate P ∗S,k, P ∗R,k, and d∗k using (3.29)–(3.31) and (3.24)–(3.26).5: if d∗k = 0 then6: if P ∗S,k > BS,k then7: P ∗S,k = BS,k;8: end if9: Qk = Qk−1 + log2(1 + γS,kP ∗S,k);10: else11: if P ∗R,k > BR,k then12: P ∗R,k = BR,k;13: end if14: if log2(1 + γR,kP ∗R,k) > Qk then15: P ∗R,k = 2Qk−1γR,k;16: end if17: Qk = Qk−1 − log2(1 + γR,kP ∗R,k);18: end if19: end for20: Obtain throughput =K∑k=1dk log2(1 + γR,kP ∗R,k).The solution of problem (3.23)–(3.27) provides an upper bound for the practicalcase where the storage capacity of the batteries is limited. Moreover, the problem69Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingmay yield PR,k ̸= 0 even if the buffer is empty at R. To avoid this undesirable behav-ior, we propose a practical but suboptimal online algorithm which is summarized inAlgorithm 3.1. At first, we calculate BS,k and BR,k using (3.5) and (3.6), respectively.We then calculate P ∗S,k, P ∗R,k, and d∗k from (3.29)–(3.31) and (3.24)–(3.26). To ensurethat P ∗S,k and P ∗R,k do not exceed the storage limits, we perform steps 6 to 8 and 11to 13, respectively. Steps 9 and 17 keep track of the arrival of data bits into and thedeparture of data bits out of the buffer, respectively. Steps 14 to 16 are adopted toensure that R transmits only if there is data in the buffer.Suboptimal Naive Online Power AllocationIn the suboptimal naive power allocation scheme for link adaptive relaying, at eachtime slot, k, S and R consider the amount of energy stored in their batteries as theirtransmit powers. Based on the transmit powers, S and R compute their capacities.Note that the buffer status should be taken into account in the computation of thecapacity of R. The S–R (R–D) link is selected if the capacity of S is greater (smaller)than that of R.3.3.3 ComplexityThe overhead of buffer–aided link adaptive relaying includes both the computationalcost and the required feedback of the corresponding power allocation schemes. Theworst–case computational cost of the sBB algorithm used to solve the offline powerallocation scheme for link adaptive relaying is exponential in K [92], whereas thecomputational cost of the exhaustive search algorithm is always exponential in K.Moreover, the worst–case complexity of the sBB algorithm is not likely to occur as isevident from the execution time results shown in Section V. Determining the exactand/or average complexity of the sBB algorithm is quite involved and beyond the70Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingscope of this thesis. The complexities of the proposed suboptimal online schemes forlink adaptive relaying are linear in K.Furthermore, for buffer–aided link adaptive relaying, S needs to know the instan-taneous values of γS,k, γR,k, HS,k, and HR,k along with the averages γ¯S, γ¯R, HS, andHR for execution of the online power allocation algorithms. Moreover, S also hasto track the status of the relay buffer. After calculating the optimal power and linkselection policies, S has to convey P ∗R,k as well as information regrading which link isselected to R.3.4 Power Allocation for Conventional RelayingIn this section, we propose an offline and several online power allocation schemes forthe considered EH system with conventional relaying. We note that conventional re-laying is preferable over buffer–aided relaying if small end–to–end delays are required.3.4.1 Optimal Offline Power AllocationLike for buffer–aided relaying, for conventional relaying, our objective is to maximizethe total number of transmitted bits (from S to D) delivered by a deadline of Ktime slots over a fading channel. We assume offline knowledge of the full CSI andthe energy arrivals at S and R in each time slot. The offline optimization problemfor maximization of the throughput of the considered system for K time slots can be71Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingformulated as follows:maxT ′′≽0K/2∑k=1min{ξS,2k−1, ξR,2k} (3.35)s.t. Constraints (3.10)− (3.13) with d2k−1 = 0 and d2k = 1, ∀k, (3.36)γS,2k−1PS,2k−1 = γR,2kPR,2k, ∀k, (3.37)where T ′′ ! {PS,2k−1, PR,2k,λS,2k−1,λR,2k|k ∈ {1, 2, · · · ,K/2}} and ∀k stands fork ∈ {1, 2, · · · ,K/2}. Like for buffer–aided relaying, constraints (3.36) ensure theenergy causality and limited energy conditions for conventional relaying. Constraint(3.37) ensures that the amount of information transmitted from S to R is identicalto that transmitted from R to D so as to avoid data loss at R. Constraint (3.37)is required since we assume individual power constraints for S and R. This is areasonable assumption since S and R have independent power supplies.Using (3.37) in (3.35) and (3.36), the considered offline optimization problem canbe rewritten asmaxT ′′′≽0K/2∑k=1ξS,2k−1 (3.38)s.t.l∑k=1(γS,2k−1PS,2k−1γR,2k+ λR,2k)≤2l−1∑k=0HR,k, ∀l (3.39)2m∑k=0HR,k −m∑k=1(γS,2k−1PS,2k−1γR,2k+ λR,2k)≤ BR,max, ∀m (3.40)Constraints (3.11) and (3.13) with d2k−1 = 0 and d2k = 1, ∀k, (3.41)where T ′′′ ! T ′′ \ {PR,2k|∀k}, and ∀l and ∀m stand for l ∈ {1, 2, · · · ,K/2} and m ∈{1, 2, · · · ,K/2−1}, respectively. Problem (3.38)–(3.41) forms a convex optimizationproblem and the optimum solution can be obtained either in closed form by using72Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingthe Karush–Kuhn–Tucker (KKT) conditions or by using any standard technique forsolving convex optimization problems [75], [79]. Let P ∗S,2k−1 denote the optimumsolution of the considered optimization problem. Then, the optimum PR,2k can beobtained asP ∗R,2k =γS,2k−1P ∗S,2k−1γR,2k. (3.42)3.4.2 Optimal Online Power Allocation by DPUnlike buffer–aided link adaptive relaying, the link selection policy is pre-defined forconventional relaying, i.e., d2k−1 = 0 and d2k = 1, k ∈ {1, 2, · · · ,K}. This featureof conventional relaying reduces the complexity of stochastic DP compared to linkadaptive relaying. Therefore, for conventional relaying, we consider a stochastic DPapproach for optimum online power allocation [18, 23]. Let c2k−1,2k ! (γS,2k−1,γR,2k, (HS,2(k−1) + HS,2k−3), (HR,2k−1 + HR,2(k−1)), BS,2k−1, BR,2k) denote the state fortime slots 2k − 1 and 2k. We note that HS,k = 0 for k < 0. Our aim is tomaximize the total throughput over K time slots. We assume the initial statec1,2 = (γS,1, γR,2, HS,0, (HR,0 +HR,1), BS,1, BR,2) is always known. We define a policyp = {(PS,2k−1(c2k−1,2k), PR,2k(c2k−1,2k)),∀c2k−1,2k, k = 1, 2, · · · ,K/2} as feasible if theEH constraints 0 ≤ PS,2k−1(c2k−1,2k) ≤ BS,2k−1 and 0 ≤ PR,2k(c2k−1,2k) ≤ BR,2k aresatisfied for all k. Hence, the objective function to be maximized can be reformulatedas [18]R(p) =K/2∑k=1E{min{ξ′S,2k−1, ξ′R,2k}|c1,2, p}, (3.43)73Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingwhere we used the definitions ξ′S,2k−1 ! log2 (1 + γS,2k−1PS,2k−1(c2k−1,2k)) and ξ′R,2k !log2(1 + γR,2kPR,2k (c2k−1,2k)). The expectation is with respect to the SNRs of thechannels and the harvested energies. In particular, for a given c1,2, the maximumthroughput can be obtained asR∗ = maxp∈PR(p), (3.44)where P denotes the space of all feasible policies.The maximum throughput during time slots 2k− 1 and 2k is denoted by J2k−1,2k(BS,2k−1, BR,2k). For a given c1,2, the maximum throughput, J1,2(BS,1, BR,2), canbe recursively obtained from JK−1,K(BS,K−1, BR,K), JK−3,K−2(BS,K−3, BR,K−2), · · · ,J3,4(BS,3, BR,4) [18]. For the last two time slots K − 1 and K, we haveJK−1,K(BS,K−1, BR,K) = max0≤PS,K−1≤BS,K−10≤PR,K≤BR,KγS,K−1PS,K−1=γR,KPR,K12min {ξS,K−1, ξR,K} (3.45)and for time slots 2k − 1 and 2k, we obtainJ2k−1,2k(BS,2k−1, BR,2k) = max0≤PS,2k−1≤BS,2k−10≤PR,2k≤BR,2kγS,2k−1PS,2k−1=γR,2kPR,2k12min {ξS,2k−1, ξR,2k}+J¯2k+1,2k+2(BS,2k−1 − PS,2k−1, BR,2k − PR,2k),(3.46)where J¯2k+1,2k+2(B′S,2k+1, B′R,2k+2) =Eγ˜S,2k+1,γ˜R,2k+2,H˜S,2k−1,H˜R,2k{J2k+1,2k+2(min{B′S,2k+1 + H˜S,2k−1, BS,max},min{B′R,2k+2 + H˜R,2k, BR,max})}. (3.47)74Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingHere, B′S,2k+1 ! BS,2k−1−PS,2k−1, B′R,2k+2 ! BR,2k−PR,2k, γ˜S,2k+1 (γ˜R,2k+2) representsthe SNR of the S–R (R–D) link in the (2k+1)th ((2k+2)th) slot given the SNR γS,2k−1(γR,2k) in the (2k−1)th (2kth) slot, and H˜S,2k−1 (H˜R,2k) denotes the harvested energyat S (R) in the (2k − 1)th (2kth) slot given the harvested energy HS,2k−2 (HR,2k−1)in the (2k − 2)th ((2k − 1)th) slot. It can be shown that the cost functions in (3.45)and (3.46) are concave in PS,2k−1 and PR,2k. Thus, (3.45) and (3.46) are convexoptimization problems and can be solved very efficiently [75]. Further simplificationof (3.45) yieldsJK−1,K(BS,K−1, BR,K) =12log2 (1 + γS,K−1ρK−1) , (3.48)where ρK−1 = min {BS,K−1, γR,KBR,K/γS,K−1}. Therefore, P ∗S,K−1 = min{BS,K−1,γR,KBR,KγS,K−1}, and P ∗R,K follows from (3.42). Similarly, (3.46) can be simplified asJ2k−1,2k(BS,2k−1, BR,2k) = max0≤PS,2k−1≤min{BS,2k−1,γR,2kBR,2k/γS,2k−1}12ξS,2k−1+J¯2k+1,2k+2(BS,2k−1 − PS,2k−1, BR,2k − γS,2k−1PS,2k−1/γR,2k). (3.49)Using (3.48) and (3.49), P ∗S,2k−1 and P ∗R,2k, k ∈ {1, 2, · · · ,K/2}, can be obtained fordifferent possible values of γS,k, γR,k, BS,k, and BR,k and can be stored in a look–uptable. This is done before transmission starts. When transmission starts, for givenrealizations of γS,2k−1, γR,2k, BS,2k−1, and BR,2k in time slots 2k−1 and 2k, the valuesof P ∗S,2k−1 and P ∗R,2k corresponding to these realizations are taken from the look–uptable.3.4.3 Suboptimal Online Power AllocationIn the proposed DP–based optimal online power allocation scheme, for a certain75Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingtransmission time slot, we consider the average effect of all succeeding time slots, c.f.(3.47). Due to the recursive nature of DP, the computational cost of this approachincreases alarmingly with increasing K. For this reason, in the following, we proposethree different suboptimal online power allocation schemes, which perform close tothe optimal DP approach but have reduced complexity.Suboptimal Simplified DP Power Allocation (“DP–I2” and “DP–I1”Schemes)In this scheme, we use the average effect of only 2 (or 4) following time slots toallocate the transmit power in each time slot. In particular, we assume for thecurrent time slot that all energies have to be spent over the following 2 (or 4) timeslots. Moreover, in the last two time slots, either S or R uses up all of its storedenergy. This scheme reduces the computational cost at the expense of a performancedegradation. We refer to the suboptimal DP schemes taking into account 4 and 2time slots as “DP–I2” and “DP–I1”, respectively.Suboptimal HR Assisted Power Allocation (“HR Assisted” Scheme)As for link adaptive relaying, we also propose an efficient “HR Assisted” online powerallocation scheme for conventional relaying. To this end, we first formulate an op-timization problem which is based on the average data rate and the average energycausality constraints at S and R for K →∞. Then, we revise the optimal solutionstaking into account the energy storage constraints in (3.5) and (3.6). The optimiza-76Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingtion problem for K →∞ can be formulated asmax{PS,2k−1≥0, PR,2k≥0,|∀k}1KK/2∑k=1min{log2(1 + γS,2k−1PS,2k−1), log2(1 + γR,2kPR,2k)}(3.50)s.t. Constraints (3.24) and (3.25) with d2k−1 = 0 and d2k = 1, ∀k, (3.51)γS,2k−1PS,2k−1 = γR,2kPR,2k. (3.52)Incorporating (3.52) into (3.50) and (3.51) yieldsmax{PS,2k−1≥0|∀k}1KK/2∑k=1log2(1 + γS,2k−1PS,2k−1) (3.53)s.t. 1KK/2∑k=1PS,2k−1 ≤1KK−1∑k=0HS,k (3.54)1KK/2∑k=1γS,2k−1γR,2kPS,2k−1 ≤1KK−1∑k=0HR,k. (3.55)Problem (3.53)–(3.55) is a convex optimization problem and using the optimizationtechniques described in subsection 3.3.2, the optimal P ∗S,2k−1 and P ∗R,2k can be ob-tained asP ∗S,2k−1 = min{[1ln(2)(ηS +ηRγS,2k−1γR,2k)−1γS,2k−1]+, BS,2k−1}(3.56)P ∗R,2k = min{γS,2k−1P ∗S,2k−1γR,2k, BR,2k}, (3.57)where ηS and ηR are Lagrange multipliers associated with (3.54) and (3.55), re-spectively. The optimal ηS and ηR are computed offline using the same method asdescribed in subsection 3.3.2 for link adaptive relaying before transmission starts.77Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingWhen transmission begins, P ∗S,2k−1 and P ∗R,2k are calculated using (3.56) and (3.57)with offline parameters ηS and ηR and online variables γS,2k−1, γR,2k, BS,2k−1, andBR,2k. For the (K − 1)th and the Kth time slots, we ensure that either S or R usesup all of the stored energy.Suboptimal Naive Power Allocation (“Naive” Scheme)In this suboptimal “naive” approach, for each time slot, only the stored energies athand determine the transmit power, i.e., this approach does not take into accountthe effect of the following time slots, and the transmitting node (S or R) uses up allof its available energy in each transmission interval. To be specific, for a particulartime slot k ∈ {1, 2, · · · ,K/2}, P ∗S,2k−1 = min{BS,2k−1,γR,2kBR,2kγS,2k−1}, and P ∗R,2k followsdirectly from (3.42).3.4.4 ComplexityIn this subsection, we discuss the overheads entailed by the power allocation schemesproposed for conventional relaying in terms of computational cost and the requiredfeedback. In the offline and the optimal online power allocation schemes, we solveconvex optimization problems where the number of constraints is a function of K.The required computational cost to solve a convex optimization problem dependson the method used (e.g. bisection method, interior–point–method, etc.) and ispolynomial in the size of the problem [75]. Therefore, the worst–case computationalcost of the offline power allocation scheme for conventional relaying is polynomialin the number of time slots K [75]. We observe from (3.46) and (3.47) that thecomplexity of the optimal online power allocation scheme by stochastic DP increasesexponentially with K. The complexities of the simplified versions of DP, i.e., DP–I2and DP–I1, are linear in K. Moreover, the complexities of the naive and HR assisted78Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingsuboptimal online schemes are linear in K as well. The complexities of the proposedoffline and online power allocation schemes are further investigated in terms of therequired average execution time in Section V.The required feedback information also contributes to the overhead of the pro-posed power allocation schemes. Like link adaptive relaying, S needs to know theinstantaneous values of γS,2k−1, γR,2k, HS,k, and HR,k along with the averages γ¯S, γ¯R,HS,E, and HR,E for execution of the online power allocation algorithms. Furthermore,S has to convey P ∗R,k to R.3.5 Simulation ResultsIn this section, we evaluate the performance of the proposed offline and online powerallocation schemes for the conventional and link adaptive relaying protocols. Weassume that the (overall) average harvesting rate is HS = HR = HE (except in Fig.3.9), and HS,k and HR,k independently take values from the set {0, HE, 2HE}, whereall elements of the set are equiprobable. For Figs. 3.2–3.6, 3.10, and 3.11, we assumeHE = 0.5 Joule. We adopt BS,max = BR,max = Bmax, where Bmax = 4 Joules forFigs. 3.2, 3.3, 3.7, Bmax = 10 Joules for Figs. 3.4–3.6, 3.9, 3.11, and Bmax = 100Joules for Fig. 3.8. We assume i.i.d. Rayleigh fading channels with γ¯S = γ¯R = γ¯for Figs. 3.2–3.5, 3.9, and 3.11 and i.n.d. Rayleigh fading channels for Figs. 3.6–3.8,and 3.10. We simulate 104 randomly generated realizations of the S–R and the R–Dchannels and the harvested energies at S and R to obtain the average throughput.79Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying0 5 10 15 20 25 30051015202530Average SNR (in dB)Total number of transmitted bits  Optimal OfflineOptimal OnlineDP-I 2DP-I 1NaiveHR AssistedFigure 3.2: Conventional relaying: Total number of transmitted bits vs. averagechannel SNR γ¯ for K = 10.3.5.1 Performance of Different Power Allocation Schemes forConventional RelayingIn this subsection, we show the performance of the proposed power allocation schemesfor conventional relaying. In particular, the impact of the average channel SNR andthe number of time slots on the total number of transmitted bits is studied.Fig. 3.2 shows the total number of transmitted bits for the power allocationschemes proposed for conventional relaying vs. the average channel SNR, γ¯, forK = 10. We observe that for all considered schemes, the total throughput increasesas γ¯ increases. We also notice that the offline scheme performs better than the onlinepower allocation schemes for all γ¯. This is due to the fact that in the optimal offline80Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying0 20 40 60 80 100 120050100150200250300350Number of time slotsTotal number of transmitted bits  Optimal OfflineDP-I 2DP-I 1NaiveHR AssistedFigure 3.3: Conventional relaying: Total number of transmitted bits vs. number oftime slots K for γ¯ = 25 dB.scheme we assume that both causal and non–causal information regarding the CSIand the harvested energy are available whereas the online schemes are based onlyon causal information regarding the CSI and the harvested energy. Moreover, asexpected, the optimal online scheme outperforms all considered suboptimal onlineschemes and performs close to the optimal offline scheme. The suboptimal onlineschemes DP–I2 and DP–I1 perform close to each other for all γ¯. We note that bothDP–I2 and DP–I1 outperform the HR assisted and the naive schemes.In Fig. 3.3, we show the total number of transmitted bits for the power alloca-tion schemes proposed for conventional relaying vs. the number of time slots K forγ¯ = 25 dB. We observe that the optimal offline method achieves the best perfor-81Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingmance. Among the different suboptimal online schemes, DP–I2 performs best. TheHR assisted scheme provides a similar performance as DP–I2 for large K. This ismainly due to the fact that the HR assisted scheme is based on the average harvestingrate which is more justified for large K. Moreover, we observe that the differencebetween the performances of DP–I2 and DP–I1 increases with increasing K. Thisshows that the consideration of the two next time slots instead of only the next timeslot for calculation of the optimal transmit powers becomes more important for largerK.3.5.2 Performance of Different Power Allocation Schemes forLink Adaptive RelayingIn this subsection, we show the impact of the average channel SNR and the numberof time slots on the total number of transmitted bits for the different power allocationschemes proposed for link adaptive relaying.Fig. 3.4 shows the total number of transmitted bits for link adaptive relayingvs. the average channel SNR γ¯ for K = 8 and K = 50. Here, we consider theexhaustive search offline algorithm for K = 8, and the HR assisted online, the naiveonline, and the sBB offline power allocation schemes for both values of K. Recallthat the optimal offline exhaustive search scheme is only effective for small K asthe complexity increases exponentially with K. For K = 8, we observe that theexhaustive search and the sBB schemes have exactly the same performance for allconsidered γ¯. This observation confirms that sBB scheme finds the global optimumsolution of non–convex MINLP problems [91, 92]. The performance gap between theoffline and the HR assisted online schemes is small at low γ¯ and large at high γ¯ forK = 8. Furthermore, the performance gap increases with γ¯ for K = 50. For K = 8,82Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying5 10 15 20 25 30050100150200250Average SNR (in dB)Total number of transmitted bits  Exhaustive SearchsBBHR Assisted OnlineNaive OnlineK=8K=50Figure 3.4: Link adaptive relaying: Total number of transmitted bits vs. averagechannel SNR γ¯ for K = 8 and K = 50.the naive online power allocation scheme has a small performance advantage over theHR assisted online algorithm for high γ¯, whereas for K = 50, the HR assisted onlinealgorithm shows better performance for all considered γ¯.3.5.3 Comparison Between Conventional and Link AdaptiveRelayingIn this subsection, we compare the power allocation schemes proposed for conven-tional and link adaptive relaying. For offline power allocation, we compare the optimalschemes and for online power allocation, we compare the suboptimal schemes for con-ventional relaying with the HR assisted online scheme for link adaptive relaying. Theoptimal DP approach (for conventional relaying) is not included in the comparison83Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying20 40 60 80 100 120 1400100200300400500600700Number of time slotsTotal number of transmitted bits  Optimal Offline (Link Adaptive)HR Assisted Online (Link Adaptive)Optimal Offline (Conventional)DP-I 2 Online (Conventional)HR Assisted Online (Conventional)Offline (No-delay constrained in [59])Figure 3.5: Comparison of conventional and link adaptive relaying: Total number oftransmitted bits vs. number of time slots K for γ¯S = γ¯R = γ¯ = 30 dB.because of its high complexity.Total number of transmitted bits vs. KFig. 3.5 shows the total number of transmitted bits vs. the number of time slots, Kfor the offline and online power allocation schemes for conventional and link adaptiverelaying. We assume symmetric S–R and R–D channels, i.e., γ¯S = γ¯R = γ¯. Weobserve that link adaptive relaying significantly outperforms conventional relayingfor offline power allocation. The performance gap increases with increasing K. Thisis mainly due to the fact that, for large K, we have more flexibility in selecting dk, i.e.,in selecting the S–R or R–D link for transmission to increase the system throughput.84Chapter 3. Resource Allocation for EH Conventional and Link Adaptive RelayingIn particular, for the offline case, the link adaptive relaying scheme can transmit 16and 55 additional bits compared to conventional relaying for K = 20 and K = 100,respectively. For small K, the HR assisted online scheme for link adaptive relayingdoes not show a better performance than the online schemes for conventional relaying.However, for relatively large K, e.g. K = 140, the HR assisted online scheme forlink adaptive relaying outperforms the DP–I2 and HR assisted online schemes forconventional relaying by 26 and 34 bits, respectively. We note that the gain of linkadaptive relaying over conventional relaying is the result of both power allocation andadaptive link selection since the power allocation affects the link selection and viceversa. Moreover, in Fig. 3.5, we have also included the performance of the no–delayconstrained protocol in [59]. We observe that, because we assume a direct S–D link isnot available, the performances of conventional relaying and the no–delay constrainedprotocol in [59] are identical for all considered values of K. This result was alreadyshown in [59, Fig. 4] for non–fading channels and we arrive at the same result forfading channels.In Fig. 3.6, we turn our attention to asymmetric links where we consider twoscenarios for the average channel SNRs. Scenarios 1 and 2 are valid for γ¯S = 20 dBand γ¯S = 10 dB, respectively, where γ¯R = 30 dB in both cases. We compare theperformance of the HR assisted online scheme for link adaptive relaying with that ofthe DP–I2 and HR assisted schemes for conventional relaying. In contrast to Fig. 3.5,in Fig. 3.6, we observe that the HR assisted online scheme for link adaptive relayingoutperforms the DP–I2 (HR assisted) schemes even for small numbers of time slots,e.g. K = 25. Moreover, Fig. 3.6 clearly shows that the performance gains of the HRassisted online scheme for link adaptive relaying over the DP–I2 (HR assisted) schemefor conventional relaying are significantly larger for asymmetric links compared to85Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying20 40 60 80 100 120 140050100150200250300350400450Number of time slotsTotal number of transmitted bits  HR Assisted Online (Link-Adaptive)DP-I 2 Online (Conventional)HR Assisted Online (Conventional)Scenario 1Scenario 2Figure 3.6: Comparison of online algorithms for conventional and link adaptive re-laying: Total number of transmitted bits vs. number of time slots K for γ¯S = 20 dB(Scenario 1) and γ¯S = 10 dB (Scenario 2) with γ¯R = 30 dB.symmetric links. The larger gains are caused by the flexibility introduced by thebuffer at the relay. In the link adaptive scheme, the stronger link can be used lessfrequently since relatively large amounts of information can be transferred every timethe link is used. Hence, the weaker link can be used more frequently to compensatefor its poor link quality. In contrast, in conventional relaying, both links are used forthe same amount of time regardless of their respective qualities.Number of transmitted bits vs. HEFig. 3.7 depicts the total number of transmitted bits for conventional and link adap-tive relaying vs. the average harvesting rate, HE, for γ¯S = 10 dB, γ¯R = 30 dB, andK = 60. We observe that the throughput increases with increasing HE for all con-86Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 120406080100120140160180200Average harvesting rate (Joules)Total number of transmitted bits  Optimal Offline (Link Adaptive)HR Assisted Online (Link Adaptive)Optimal Offline (Conventional)DP-I 2 (Conventional)HR Assisted Online (Conventional)Figure 3.7: Comparison of conventional and link adaptive relaying: Total number oftransmitted bits vs. HE for K = 60, γ¯S = 10 dB, and γ¯R = 30 dB.sidered power allocation schemes. We note that the slope of the throughput curvesis large for small HE and decreases with increasing HE. This behavior is partially(apart from the behavior of the log(·) function) due to the fact that the performanceof all schemes is limited by the finite storage capability of the batteries. For large HE,additional energy cannot be stored in the batteries and therefore the extra amountis wasted. We observe that the optimal offline and the HR assisted suboptimal on-line schemes for link adaptive relaying outperform the corresponding schemes forconventional relaying for all HE.In Fig. 3.8, we show the total number of transmitted bits vs. the average har-vesting rate HS = HR = HE for K = 1000. Here, we study the performance ofthe HR assisted schemes for conventional and buffer–aided relaying for large K. We87Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying0.5 1 1.5 2 2.5 3 3.5 42000250030003500400045005000Average harvesting rate (Joules)Total number of transmitted bits  Link AdaptiveConventionalFigure 3.8: Comparison of the HR assisted online schemes for conventional and linkadaptive relaying: Total number of transmitted bits vs. average harvesting rate forK = 1000, γ¯S = 30 dB, and γ¯R = 15 dB.observe that, like for small K, buffer–aided link adaptive relaying also significantlyoutperforms conventional relaying for large K.In Fig. 3.9, we show the performances of different power allocation schemes forboth conventional and buffer–aided link adaptive relaying assuming that HR changesover the time of transmission. In particular, we fix HS = 2 Joules and linearly changeHR, from one transmission time interval to the next from a high value (HR = 2 in t1)to a low value (HR = 0.25 in t5) and then back to a high value (HR = 2 in t9). Eachtransmission interval, t1, t2, · · · , t9, lasts for K = 50 time slots. At the beginning ofeach transmission interval, ti, the proposed optimization schemes are executed andthe total throughput during the K = 50 time slots is shown in Fig. 3.9. The changeof HR over time may model the impact that the change from cloudy weather (t5)88Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingt1 t2 t3 t4 t5 t6 t7 t8 t9160180200220240260280300Transmission timesTotal number of transmitted bits  ConventionalLink AdaptiveHR Assisted Online Optimal OfflineFigure 3.9: Total number of transmitted bits for offline and online power allocationschemes for conventional and link adaptive relaying when HR changes over the timeof transmission for HS = 2 Joules, K = 50, and γ¯S = γ¯R = 30 dB.to sunny weather (t1 and t9) has on a solar energy system. We observe that thethroughput increases (decreases) with increasing (decreasing) HR for both the offlineand online power allocation schemes. Most importantly, the change in throughputis large for conventional relaying and small for link adaptive relaying. In fact, forlink adaptive relaying, if the relay does not have enough energy to transmit data,the S–R link can be selected for transmission for several consecutive time slots andthe received data at R is stored in the buffer and is forwarded to the destination ata later time. On the other hand, for conventional relaying, in each time slot, thesource can only transmit the amount of data that the relay is able to forward to thedestination. For this reason, the link adaptive protocol is able to cope better withsudden changes in the EH profile. Note that this behaviour is observed for both the89Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying2 3 4 5 6 7 8 9 10100150200250Bmax (Joules)Total number of transmitted bits  Optimal Offline (Link Adaptive)HR Assisted Online (Link Adaptive)Optimal Offline (Conventional)DP-I 2 (Conventional)HR Assisted Online (Conventional)Figure 3.10: Comparison of conventional and link adaptive relaying: Total numberof transmitted bits vs. Bmax for K = 100, γ¯S = 10 dB, and γ¯R = 30 dB.offline and the online schemes.Total number of transmitted bits vs. BmaxIn Fig. 3.10, we show the total number of transmitted bits for conventional and linkadaptive relaying vs. Bmax for γ¯S = 10 dB, γ¯R = 30 dB, and K = 100. We observethat for all considered power allocation schemes, the throughput first increases withincreasing Bmax but starting at a certain value of Bmax, the throughput remainsunchanged. This can be explained by the fact that for HE = 0.5, small values ofBmax limit the performance since the extra amount of harvested energy cannot bestored in the batteries. However, the constant throughput of all the schemes for largeBmax indicates that, for the given parameters, increasing the storage capacities of the90Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relaying20 40 60 80 100 120 14010-410-310-210-1100101102103104Number of time slotsExecution time (in seconds)  Optimal Offline (Link Adaptive)HR Assisted Online (Link Adaptive)Optimal Offline (Conventional)DP-I 2 (Conventional)HR Assisted Online (Conventional)Figure 3.11: Comparison of execution times of conventional and link adaptive relay-ing: Execution time (in seconds) vs. number of time slots K for γ¯ = 30 dB.batteries beyond a certain value does not improve the performance of the system.Therefore, Fig. 3.10 provides an indication for the required storage capacities of thebatteries at S and R for different power allocation schemes to achieve a desiredperformance.Execution time vs. KIn Fig. 3.11, we show the average execution time (in seconds) vs. the number of timeslots, K, for all offline and online power allocation schemes for conventional and linkadaptive relaying. We ran all the algorithms on the same computer system. In par-ticular, all simulations were performed under MATLAB with the Intel(R) Core(TM)i7-2670QM (@2.20GHz 2.20GHz) processor. Therefore, it is reasonable to compare91Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingthe complexity of the algorithms based on their execution times. We observe thatfor both the offline and online schemes for both conventional and link adaptive relay-ing the execution time increases with K. Moreover, the required execution time forthe sBB algorithm is higher than that for the offline scheme for conventional relay-ing for all K. The complexity of forming the look–up tables for the DP–I2 schemeis not included in this analysis. Fig. 3.11 provides valuable information about thecomplexity–performance trade–off of the different proposed algorithms. For exam-ple, in case of conventional relaying, the HR assisted scheme is less complex than theDP–I2 scheme with only a small performance degradation for sufficiently high averagechannel SNR and large K, see Fig. 3.3. Therefore, it is preferable to apply the HRassisted scheme compared to the DP–I2 scheme. Moreover, although the worst casecomplexity of the sBB algorithm is exponential in K, from Fig. 3.11 we observe thatits average complexity is comparable to that of the offline power allocation schemefor conventional relaying. Thus, for link adaptive relaying, we can conclude that thesBB algorithm is preferable over the exhaustive search method (which is not shownin Fig. 3.11 due to its very high execution time).3.6 ConclusionsIn this chapter, we considered the problem of transmit power allocation for singlerelay networks, where the source and the relay harvest the energy needed for trans-mission from the surrounding environment. Two different transmission strategies,namely conventional relaying and buffer–aided link adaptive relaying, were consid-ered. We proposed optimal offline and optimal (conventional relaying) and severalsuboptimal online power allocation schemes maximizing the system throughput of theconsidered EH systems over a finite number of time slots. Simulation results showed92Chapter 3. Resource Allocation for EH Conventional and Link Adaptive Relayingthat the proposed suboptimal online schemes have a good complexity–performancetrade–off. Moreover, we showed that, for both offline and online optimization, adopt-ing the link adaptive protocol significantly improves the throughput compared toconventional relaying, especially for asymmetric link qualities, and buffer–aided linkadaptive relaying is more robust to the changes in the EH profile than conventionalrelaying.93Chapter 4Joint Power Allocation and RelaySelection in Energy Harvesting AFRelay Systems4.1 IntroductionIn Chapter 3, we have considered a single link EH DF relay system with two relayingprotocols. In this chapter, we turn our attention to AF relay system. We consider amulti–relay network, where the source and the AF relays are EH nodes. ConsideringEH AF relays in this chapter helps us to compare relay selection strategies for systemspowered by constant energy source, cf. Chapter 2 and energy harvester.As mentioned in the Introduction of Chapter 3, a comprehensive performanceanalysis was provided in [81] for relay selection in an EH network. However, [81]derives the error rate expressions in a probabilistic manner, possibly with the con-sideration of large number of transmission time intervals. Moreover, as stated inChapter 1, our focus in this thesis is to propose resource allocation schemes for afinite number of transmission time intervals for EH systems. The relay selectionstrategies described in [81] may not provide optimal results for finite number of timeintervals for an EH relay network. Moreover, conventional relay selection methods94Chapter 4. Joint Power Allocation and Relay Selectionsuitable for systems powered by constant energy sources [13, 98] cannot capture thefluctuating characteristics of the renewable energy over the time intervals, and hencecannot be applied in EH systems in a straight–forward manner. Therefore, in thischapter, we propose offline and online (real–time) joint relay selection and sourceand relay transmit power allocation schemes that maximize the end–to–end systemthroughput over a finite number of transmission intervals. The optimal relay selec-tion policy depends on both the channel SNR and the amount of harvested energiesstored at the batteries of the nodes.The rest of this chapter is organized as follows. In Section 4.2, we discuss theconsidered system model. Sections 4.3 and 4.4 provide joint power allocation andrelay selection schemes and simulation results, respectively. Section 4.5 concludesthis chapter.4.2 System ModelWe consider an EH AF relay system with one source, S, one destination, D, andN half–duplex relays, Rn, n ∈ {1, 2, · · · , N}. We assume that D is not an EHdevice and has continuous supply of power. S and Rn are EH devices and rely on theharvested energies to participate in signal transmission. The considered system modelcan be accommodated in different wireless communication systems, e.g., in wirelesssensor networks, where an EH sensor node transmits data to the fusion centre viaother EH sensor nodes [20]. In practice, a fusion centre can handle data from manysensor nodes and is powered by constant energy source. S and Rn are equipped withbatteries with limited storage capabilities that can store at most BS,max and BRn,maxJoules of energy, respectively. We assume the transmission is organized in equalduration time intervals and each interval comprises two time slots of duration T . In95Chapter 4. Joint Power Allocation and Relay SelectionFigure 4.1: System model for a multi–relay communication network. Source, S, andrelays, Rn, n ∈ {1, 2, · · · , N} are EH nodes. Solid line indicate the selected relaylink.the following, we set T = 1s. The total transmission time is equal to K intervals. Wealso assume that the transmitted packets contain Gaussian–distributed symbols andthe transmission is impaired by AWGN. We define γSRn,k and γRnD,k as the channelSNRs of the S–Rn and Rn–D links in the kth interval, respectively. The transmitpowers of S and Rn are denoted by PS,k and PRn,k, respectively. We assume thechannels are quasi–static within each interval (block fading), are independent of eachother, and may be non–identically distributed. However, γSRn,k (γRnD,k) is identicallydistributed over the time intervals. For future reference, we introduce the averageSNRs of the S–Rn and the Rn–D links as γ¯SRn and γ¯RnD, respectively. In each timeinterval k ∈ {1, 2, · · · ,K}, a single relay, Rζ , where ζ ∈ {1, 2, · · · , N}, is selected outof the N relays to forward the signals received from S to D. During the first timeslot, S transmits and Rζ receives, and during the second time slot, Rζ transmits and96Chapter 4. Joint Power Allocation and Relay SelectionD receives. The end–to–end SNR, SNReq,ζ,k, at D is given by [74]SNReq,ζ,k =PS,kγSRζ ,kPRζ ,kγRζD,kPS,kγSRζ ,k + PRζ ,kγRζD,k + 1(4.1)Thus, the end–to–end (S–D) system throughput in interval k is given by 12 log2(1 +SNReq,ζ,k) bits/s/Hz. We also define wn,k as the relay selection variable which canbe either 0 or 1. In particular, wn,k = 1 (wn,k = 0) indicates that Rn is selected (notselected) for transmission in time interval k.Battery Dynamics: The battery energy of a node N ∈ {S,R1, R2, · · · , RN} ininterval k is BN ,k. During transmission interval k, the transmit power of N is boundedby its battery energy, i.e., 0 ≤ PN ,k ≤ BN ,k. We assume that the stored energy at Nincreases and decreases linearly provided the maximum storage capacity, BN ,max, isnot exceeded, i.e.,BN ,k+1 = min{(BN ,k− PN ,k +HN ,k), BN ,max}, ∀k. (4.2)In time interval k, the energy harvester at node N collects HN ,k ≤ BN ,max Joules ofenergy. HN ,k is modelled as an ergodic random process with mean HN ! E{HN ,k}.BN ,1 = HN ,0 ≥ 0 denotes the available energy at N before transmission starts.4.3 Joint Power Allocation and Relay SelectionIn this section, we present the proposed offline and online joint relay selection andpower allocation algorithms.97Chapter 4. Joint Power Allocation and Relay Selection4.3.1 Optimal Offline Optimization AlgorithmOur goal is to maximize the total number of transmitted bits (from S to D) deliveredby a deadline of K intervals assuming offline knowledge of the full CSI and energyarrivals at S and Rn in each time interval. To this end, we formulate an offlineoptimization problem asmaxT ≽0, w1,1,··· ,wN,KK∑k=112log2 (1 + SNReq,k) (4.3)s.t.l∑k=1(PS,k + λS,k) ≤l−1∑k=0HS,k, ∀l (4.4)l∑k=1(wn,kPRn,k + λRn,k) ≤l−1∑k=0HRn,k, ∀l, ∀n (4.5)m∑k=0HS,k −m∑k=1(PS,k + λS,k) ≤ BS,max, ∀m (4.6)m∑k=0HRn,k−m∑k=1(wn,kPRn,k+λRn,k)≤BRn,max, ∀m, ∀n (4.7)N∑n=1wn,k = 1, ∀k (4.8)wn,k ∈ {0, 1}, ∀k, ∀n, (4.9)where SNReq,k !N∑n=1wn,kPS,kPRn,kγSRn,kγRnD,kPRn,kγRnD,k+PS,kγSRn,k+1and T ! {PS,k, PR1,k, · · · , PRN ,k, λS,k,λR1,k · · · ,λRN ,k|k ∈ {1, 2, · · · ,K}}. Also, ∀l, ∀m, ∀n, and ∀k stand for l ∈ {1, 2, · · · ,K}, m ∈ {1, 2, · · · ,K − 1}, n ∈ {1, 2, · · · , N}, and k ∈ {1, 2, · · · ,K}, respectively.The slack variables λS,k and λRn,k ensure that constraints (4.4)–(4.7) can be met forall realizations of γSRn,k, γRnD,k, HS,k, and HRn,k. In particular, these slack variablesrepresent the power (possibly) wasted in each transmission interval. Constraints (4.8)and (4.9) ensure that at a particular time interval k only one wn,k is equal to one,i.e., only one relay is selected. Constraints (4.4) and (4.5) ensure the causality of the98Chapter 4. Joint Power Allocation and Relay Selectionharvested energies at S and Rn, respectively. The (maximum) storage constraintsfor the batteries at both S and Rn are satisfied with constraints (4.6) and (4.7). Tomake the objective function of the optimization problem convex with respect to PS,kand PRn,k, we consider the following high SNR approximation:SNReq,k≈SNReq,k!N∑n=1wn,kPS,kPRn,kγSRn,kγRnD,kPRn,kγRnD,k + PS,kγSRn,k. (4.10)Problem (4.3)–(4.9) with SNReq,k replaced by SNReq,k is a non–convex MINLP. Thenon–convexity arises due to the multiplicative form of wn,k and the power allocationvariables of S and Rn in the objective function and the constraints. Due to the binarynature of wn,k, the optimization problem in (4.3)–(4.9) can be recast asmaxT ≽0, w1,1,··· ,wN,KK∑k=112log2(1 + SNReq,k)(4.11)s.t.l∑k=1(PRn,k + λRn,k) ≤l−1∑k=0HRn,k, ∀l, ∀n (4.12)m∑k=0HRn,k −m∑k=1(PRn,k + λRn,k) ≤ BRn,max, ∀m, ∀n (4.13)PRn,k ≤ wn,kk−1∑m=0HRn,m, ∀k, ∀n (4.14)Constraints (4.4), (4.6), (4.8), (4.9) (4.15)where SNReq,k !N∑n=1PS,kPRn,kγSRn,kγRnD,kPRn,kγRnD,k+PS,kγSRn,k. We note that constraint (4.14) relates therelay transmit power PRn,k to the relay selection variable wn,k. Specifically, whenwn,k = 1, i.e., relay Rn assists the source at time interval k, (4.12) and (4.13) aresatisfied because (4.14) serves as an upper bound for PRn,k. On the other hand,99Chapter 4. Joint Power Allocation and Relay Selectionif wn,k = 0, i.e., relay Rn is not selected to assist the source at time interval k,then constraint (4.14) automatically ensures that PRn,k = 0. Thus, the optimizationproblems in (4.3)–(4.9) with SNReq,k and (4.11)–(4.15) are equivalent.We observe that optimization problem (4.11)–(4.15) is a convex MINLP [75].Furthermore, the continuous optimization variables, T , can be linearly separatedfrom the integer (binary) optimization variables wn,k. Therefore, we can apply theGBD algorithm to solve this convex MINLP optimally [99, pp. 114–143]. GBDdecomposes problem (4.11)–(4.15) into two sub–problems: a primal problem anda master problem. PS,k, PRn,k, λS,k, and λRn,k are obtained by solving the primalproblem for fixed values of wn,k, which are obtained from the solution of the masterproblem. Solving the master problem gives wn,k for previously obtained PS,k, PRn,k,λS,k, and λRn,k along with the corresponding Lagrange multipliers. GBD iterativelysolves the primal problem and the master problem until their solutions converge. Forthe first iteration, a feasible initial value is adopted for the integer variables wn,k.In the following, we describe the primal and master problems for a given iterationi ∈ {1, 2, · · · , I}, where I is the number of iterations required for convergence ofGBD.Primal Problem (ith iteration): For the given optimal wn,k from iteration i− 1,w(i−1)∗n,k (or a given initial w(0)n,k), we formulate the primal problem as followsmaxT ≽0K∑k=112log2(1 + SNReq,k)(4.16)s.t. Constraints (4.4), (4.6), (4.12), (4.13) (4.17)PRn,k ≤ w(i−1)n,kk−1∑m=0HRn,m, ∀k and ∀n. (4.18)This is a convex optimization problem in PS,k, PRn,k, λS,k, and λRn,k [75] and can100Chapter 4. Joint Power Allocation and Relay Selectionbe solved optimally by any standard algorithm or solver, e.g., CVX [79]. Let P (i)∗S,k ,P (i)∗Rn,k, λ(i)∗S,k , and λ(i)∗Rn,k denote the optimal solutions at the ith iteration.Master Problem (ith iteration): To formulate the master problem, we need theLagrangian of the primal problem. The Lagrangian of the primal problem can bewritten asL(PS,k, PRn,k,λS,k,λRn,k, ξS,l, ξRn,l, ηS,m, ηRn,m,αn,k) =K∑k=112log2(1+N∑n=1PS,kPRn,kγSRn,kγRnD,kPRn,kγRnD,k + PS,kγSRn,k)−K∑l=1ξS,l(l∑k=1(PS,k + λS,k)−l−1∑k=0HS,k)−N∑n=1K∑k=1αn,k(PRn,k − wn,kk−1∑m=0HRn,m)−N∑n=1K∑l=1ξRn,l(l∑k=1(PRn,k + λRn,k)−l−1∑k=0HRn,k)−K∑m=1ηS,m(m∑k=0HS,k −m∑k=1(PS,k + λS,k)−BS,max)−N∑n=1K∑m=1ηRn,m(m∑k=0HRn,k−m∑k=1(PRn,k+λRn,k)−BRn,max), (4.19)where ξS,l, ξRn,l, ηS,m, ηRn,m, and αn,k represent the Lagrange multipliers associatedwith constraints (4.4), (4.12), (4.6), (4.13), and (4.14), respectively. Let the optimalsolution of the Lagrange multipliers of the primal problem be ξ∗S,l, ξ∗Rn,l, η∗S,m, η∗Rn,m,and α∗n,k. For a given P (i)∗S,k , P(i)∗Rn,k, λ(i)∗S,k , λ(i)∗Rn,k, ξ(i)∗S,l , ξ(i)∗Rn,l, η(i)∗S,m, η(i)∗Rn,m, and α(i)∗n,k , weformulate the master problem as followsmaxβM≥0, w1,1,··· ,wN,KβM (4.20)s.t. βM≤ L(P (j)∗S,k , P(j)∗Rn,k,λ(j)∗S,k ,λ(j)∗Rn,k, ξ(j)∗S,l , ξ(j)∗Rn,l,η(j)∗S,m, η(j)∗Rn,m,α(j)∗n,k ), j ∈{1,2,· · ·,i} (4.21)Constraints (4.8), (4.9). (4.22)101Chapter 4. Joint Power Allocation and Relay SelectionNow, problem (4.20)–(4.22) is a mixed integer linear problem (MILP) of wn,k and βMand therefore it can be solved optimally by any standard optimization toolbox, e.g.,MOSEK [100].GBD Algorithm: At iteration i, the optimal solution of master problem (4.20)–(4.22) is β(i)∗M , which is an upper bound for the optimum of problem (4.11)–(4.15).Moreover, at each iteration, the master problem has one additional constraint com-pared to those defined in the previous iteration. Thus, the newly obtained optimumof the master problem is always less than or equal to the previous one and thereforethis upper bound is always non–increasing. On the other hand, the primal problemin (4.16)–(4.18) provides the solution for fixed integer variables and therefore, theoptimum of the primal problem is always equal to or less than the optimum of theoriginal problem in (4.11)–(4.15). Thus, the solution of the primal problem alwaysgives a lower bound for the solution of the original problem. We set the lower boundat each iteration equal to the maximum of the lower bound of the previous iterationand the lower bound of the current iteration. At each iteration, we solve the primalproblem with the given solution of the master problem from the previous iteration.Then, we solve the master problem with the given solution of the primal problem.This process continues and within a finite number of iterations, the GBD algorithmconverges to the optimal solution due to the monotonicity of the upper and lowerbounds [90]. We summarize the GBD algorithm in Algorithm 4.1.We note that due its convexity, the primal problem can be solved in polynomialtime whereas the master problem has non–polynomial complexity as it requires solv-ing an integer problem. Nevertheless, since the GBD algorithm is executed offline,efficient commercial optimization software such as MOSEK can be used to solve themaster problem efficiently [100].102Chapter 4. Joint Power Allocation and Relay SelectionAlgorithm 4.1 : GBD Method1: Initialization: w(0)n,k, convergence parameter, ϵ, I ← φ and i ← 1.2: flag ← 1.3: while flag ̸= 0 do4: Solve primal problem (4.16)–(4.18) and obtain P (i)∗S,k ,P (i)∗Rn,k, λ(i)∗S,k , λ(i)∗Rn,k, ξ(i)∗S,l , ξ(i)∗Rn,l, η(i)∗S,m, η(i)∗Rn,m, α(i)∗n,kand the lower bound, LB(i).5: I ← I ∪ {i}.6: Solve master problem (4.20)–(4.22) and obtain w(i)∗n,k andthe upper bound, UB(i).7: if |UB(i) − LB(i)| ≤ ϵ then8: flag ← 0.9: end if10: Set i ← i+ 1.11: end while4.3.2 Suboptimal Online Optimization AlgorithmsIn practice, only causal information of channels and harvested energies is available.Therefore, the offline power allocation scheme is not readily applicable as the futureCSI and the upcoming harvested energies are not known in advance. In principle,the optimal online algorithm may be implemented with stochastic DP. However, DPentails a high complexity due to the joint relay selection and power allocation. Thus,to avoid the high complexity of DP, we propose suboptimal HR assisted and naiveonline schemes.HR Assisted SchemeIn this scheme, we jointly allocate the source and relay transmit powers and selectthe best relay based on the causal information of stored energies and channel SNRsas well as the average EH rates and the average channel SNRs. In particular, inorder to ensure that source and relays have (with high probability) sufficient energyfor transmission in their batteries without being overly conservative in the power103Chapter 4. Joint Power Allocation and Relay Selectionallocation, we limit the maximum transmit power based on the average energy HRHN , N ∈ {S,R1, · · · , RN}. This is straightforward for the source, where we imposePS,k ≤ HS, as the source transmits in every time interval. In contrast, only oneof the N available relays transmits in a given time interval. Thus, if we denotethe probability of selecting relay n by fn, the resulting relay power constraint iswn,kPRn,k ≤ HRn/fn. An analytical expression for fn is given by (2.23) in Chapter 2.Unfortunately, fn does not only depend on the average channel SNRs but also on thetransmit powers, which renders the overall problem highly non–convex. To avoid thecomplexity associated with this non–convexity, we approximate the source and relaytransmit powers by HS and NHRn , n ∈ {1, 2, · · · , N}, respectively, which makesfn independent of PS,k and PRn,k. Based on these considerations, the optimizationproblem for time interval k is formulated asmax[PR1,k,··· ,PRN,k]≽0, w1,k,··· ,wN,k12log2(1 + SNReq,k)(4.23)s.t. PS,k ≤ BS,k, PS,k ≤ HS, (4.24)wn,kPRn,k≤BRn,k, wn,kPRn,k≤τHRn/fn, ∀n (4.25)N∑n=1wn,k = 1, wn,k ∈ {0, 1}, ∀n, (4.26)where τ is a scaling factor that can be optimized to further improve performance.As SNReq,k in the objective function in (4.23) is an increasing function of PS,k, theoptimal PS,k can be obtained asP ∗S,k = min {BS,k, HS} . (4.27)104Chapter 4. Joint Power Allocation and Relay SelectionBased on (4.27) and a change of variables, optimization problem (4.23)–(4.26) canbe rewritten asmax[PR1,k,··· ,PRN,k]≽ 0, w1,k,··· ,wN,k12log2(1 + SNReq,k)(4.28)s.t. PRn,k ≤ wn,kΓn,k, ∀n, constraint (4.26), and PS,k = min {BS,k, HS} , (4.29)where Γn,k = min {BRn,k, τHRn/fn}. For a given τ , problem (4.28), (4.29) is a convexMINLP and therefore, to solve it, we can use GBD by following the same procedure asfor the offline optimization problem [90, pp. 114–143]. We note that problem (4.28),(4.29) is not a function of K, hence the complexity of the involved GBD algorithmis much lower than that of the GBD algorithm for the offline optimization problem.Moreover, for a given wn,k, problem (4.28), (4.29) is convex in PRn,k and the optimalPRn,k is obtained as P ∗Rn,k = wn,kΓn,k. Therefore, as an alternative to GBD, for eachk, we can calculate the objective function in (4.28) with P ∗Rn,k for all combinationsof wn,k, n ∈ {1, 2, · · · , N}, and select the combination that gives the best result. Forgiven channel and EH statistics, parameter τ can be optimized offline to maximizeperformance. However, we found that τ = 0.86 gives high performance for a widerange of channel and EH parameters.Naive SchemeIn this scheme, the source and the relays use their stored energies as their transmitpowers in each time interval. Using these transmit powers, we calculate the equivalentSNRs for all links. Then, we select the relay which provides the maximum equivalentSNR among all the relays. The complexity of this scheme is linear in K.105Chapter 4. Joint Power Allocation and Relay Selection4.3.3 Extension to Multiple Relay SelectionWe extend the idea of single relay selection (and power allocation) scheme to multiplerelay selection (and power) allocation scheme with necessary modifications in theoptimization problems. We assume that when more than one relay is selected, thetransmission happens over the orthogonal time slots. To select NS ≤ N relays, weadoptK∑k=1(1/(NS+1)) log2(1+SNReq,k) and (1/(NS+1)) log2(1+SNReq,k) in (4.11) and(4.28), respectively. Moreover, we incorporateN∑n=1wn,k = NS in (4.8) and (4.26). Theoffline and online optimization problems for joint multiple relay selection and powerallocation can also be solved by GBD optimally. Moreover, when NS = N , i.e., allrelays participate in transmission, the offline and online optimization problems leadto convex optimization problems and hence can be solved optimally and efficiently.4.4 Simulation ResultsIn this section, we evaluate the performance of the proposed offline and online jointrelay selection and power allocation schemes for the considered AF relay system. InFigs. (4.2)–(4.4), we assume that the average HR is HS = HR1 = · · · = HRN = HE =0.5 Joule, and HS,k and HRn,k independently take values from the set {0, HE, 2HE},where all elements of the set are equiprobable. We adopt BS,max = BR1,max =· · · = BRN ,max = Bmax = 100 Joules and set τ = 0.86. We consider exponentiallydistributed channel SNRs and show the performance of the proposed schemes forthree different scenarios for N and the channel SNRs. We simulate 104 realizations ofrandomly generated S–Rn and Rn–D channel SNRs and harvested energies at S andRn to obtain the average throughput. We consider three scenarios with three differentvalues of N . In Scenario 1, we assume N = 1 and γ¯SR1 = γ¯R1D = γ¯ where γ¯ = 10 dB.106Chapter 4. Joint Power Allocation and Relay Selection10 20 30 40 50 60 70 80 90 1000100200300400500600700Number of time intervals, KTotal number of transmitted bits  OfflineOnline-HROnline-NaiveScenario 2Scenario 3Scenario 1Figure 4.2: Total number of transmitted bits vs. number of time intervals K forγ¯ = 10 dB and average harvesting rate HE = 0.5 Joule.In Scenario 2, we assume N = 3 where the SNRs of the first link are the same as inScenario 1 and for links 2 and 3, we assume γ¯SR2 = γ¯SR3 = γ¯R2D = γ¯R3D = γ¯+10dB.In Scenario 3, we assume N = 5 where the first 3 links are identical to the links ofScenario 2 and the SNRs of links 4 and 5 are γ¯SR4 = γ¯SR5 = γ¯R4D = γ¯R5D = γ¯+20dB.In Figs. 4.2 and 4.4, we consider all three scenarios, whereas Figs. 4.3 and 4.5 includeonly Scenario 3.In Fig. 4.2, we show the total number of transmitted bits vs. the number of timeintervals, K. From Fig. 4.2, we observe that the throughput increases with increasingK and increasing N for all considered joint power allocation and relay selectionschemes. Furthermore, as expected, for each scenario, the optimal offline schemeis a performance upper bound for both online schemes. However, the HR assisted107Chapter 4. Joint Power Allocation and Relay Selection0 10 20 30 40 50 60 700100200300400500600Number of iterationsTotal number of transmitted bits  UBLBExhaustive SearchFigure 4.3: Convegence behavior of GBD vs. the number of iterations for K = 10,γ¯ = 10 dB, N = 5, and average harvesting rate HE = 0.5 Joule.scheme closely approaches the offline scheme and yields a significant gain comparedto the naive scheme. This gain is caused by the more efficient use of the harvestedenergy by the HR assisted scheme compared to the naive scheme. In particular, theHR assisted scheme limits the source and relay transmit powers taking into accountthe statistical properties of the channels and the harvested energies. In contrast, thenaive scheme spends in each time interval all energy stored in the batteries of thesource and the selected relay.In Fig. 4.3, we show the convergence behavior of GBD algorithm, which is adoptedin the offline scheme. We assume K = 10 and consider only one realization ofchannel SNRs and harvested energies. The upper bound and the lower bound comefrom the solutions of the master and the primal problems, respectively. We observe108Chapter 4. Joint Power Allocation and Relay Selectionthat with the number of iterations, the upper bound decreases and the lower boundincreases and converge at the optimal solution. For the considered example in thisfigure, we see the convergence is reached at the 66th iteration. We also include thethroughput performance of an exhaustive search method. In the exhaustive searchmethod, we search over all the combinations of the relay selection variables, wn,k andselect that combination which yields the best throughput. Note that for a given wn,k,n ∈ {1, 2, · · · , N} and k ∈ {1, 2, · · · ,K}, problem (4.11)–(4.15) is convex and hencecan be solved optimally and efficiently. We observe that the throughput obtained fromGBD algorithm exactly matches with the throughput obtained from the exhaustivesearch method. This finding confirms that GBD algorithm finds the global optimumsolution for convex MINLP [99].In Fig. 4.4, we compare our proposed HR assisted scheme with a baseline scheme.In the baseline scheme, we assume that the relay, ζ ∈ {1, 2, · · · , N} is selected ac-cording to the maximum bottleneck SNR of all the links as follows16:ζ = argmaxn∈{1,...,N}{min{γSRn,k, γRnD,k}}. (4.30)For the given relay selection policy in (4.30), we solve problem (4.28), (4.29) toobtain optimal power allocation policy. We observe that in Scenario 1, the proposedscheme and the baseline scheme show the identical results, as for N = 1, the relayis always selected for transmission. However, in Scenarios 2 and 3, we observe thatthe proposed scheme outperforms the baseline scheme. The baseline scheme does nottake into the account the energy state of the batteries at the relays and selects therelay according to the CSI. Therefore, it is likely for the baseline scheme to selecta relay that has a little amount of energy in its battery but has a better channel16Note that this is the BRS scheme considered in Chapter 2.109Chapter 4. Joint Power Allocation and Relay Selection0 5 10 15 20 25 30 35 40 45 50020406080100120Average harvesting rate (Joules)Total number of transmitted bits  Online - HRBaselineScenario 2Scenario 1Scenario 3Figure 4.4: Total number of transmitted bits vs. average harvesting rate HE forK = 10 and γ¯ = 10 dB.condition for transmission. This policy of relay selection for EH nodes degrades thethroughput performance. On the other hand, our proposed scheme considers boththe channel condition and the battery states to perform joint relay selection andpower allocation. This feature helps the proposed scheme to perform better than thebaseline scheme. Moreover, we observe that the performance gain of the proposedscheme over the baseline scheme is large for Scenario 3 compared to Scenario 2 dueto large number of relays participate in Scenario 3.Fig. 4.5 shows the throughput (bits/sec) of the single and multiple relay selectionschemes. Here, we assume HS = 1 Joule, HR1 = 0.5 Joule, HR2 = HR3 = 0.7Joule, HR4 = HR5 = 1 Joule and adopt Scenario 3. We observe that the throughputdegrades with the increasing number of selected relays. Selecting more than one relay110Chapter 4. Joint Power Allocation and Relay Selection1 2 3 4 511.522.533.54Number of selected relaysThroughput (bits/sec)  OfflineOnline-HROnline-NaiveFigure 4.5: Throughput (bits/sec) vs. number of selected relays for K = 30, γ¯ = 10dB, and N = 5.requires large number of time slots for transmission and hence degrades the spectralefficiency. This observation was already known for contant energy source [67], but isnew for EH systems.4.5 ConclusionIn this chapter, we proposed offline and online joint relay selection and transmit powerallocation schemes for EH AF relay system. Our offline optimization framework isbased on GBD algorithm, which provides optimal results for convex MINLP. Weproposed suboptimal but low–complexity HR assisted scheme, which performs closeto the offline scheme. We compared the HR assisted scheme with a baseline scheme111Chapter 4. Joint Power Allocation and Relay Selectionand showed by simulations that the relay selection policy for EH system should exploitnot only the CSI but also the energy state of the batteries for a better performance.112Chapter 5Power Allocation for an EnergyHarvesting Transmitter with HybridEnergy Sources5.1 IntroductionGreen communication has attracted significant attention in academia and industryas the rapidly increasing energy consumption of the equipment in wireless communi-cation systems has raised environmental concerns [8, 12]. In the literature, a numberof power allocation schemes, which aim to provide a balance between energy con-sumption and performance, have been reported for different wireless communicationsystems [101]–[104]. Most of these works assume that the energies are supplied bya constant energy source and/or a rechargeable battery. As mentioned in the ear-lier chapters, recently EH has attracted considerable interest as an environmentallyfriendlier supply of energy for communication nodes compared to traditional sourcesof energy. The harvested energy is practically free of cost and can ensure a perpetualsupply of energy.In Chapters 3 and 4, we have assumed that EH is the only source of energyfor the communication nodes. Furthermore, most of the works on communication113Chapter 5. Power Allocation for a Hybrid EH Transmittersystems with EH capability [17, 18, 58, 59, 105, 106] analyze wireless communicationsystems that are powered solely by the EH nodes. However, from a practical pointof view, to achieve both reliable and green communication, it is desirable to havea hybrid source of energy due to the intermittent nature of the harvested energy,cf. [107]. A hybrid energy source is a combination of a constant energy source, e.g.,power grid, diesel generator etc., and an EH source which harvests energy from solar,wind, thermal, or electromechanical effects. The concept of hybrid energy sources hasalso drawn interest from industry. For instance, Huawei has already developed basestations for rural areas which draw their energy from both solar panels and dieselgenerators [108]. Motivated by these considerations, in this chapter, we considera single communication link where the transmitter (e.g. a base station) is equippedwith a hybrid energy source, cf. Fig. 5.1(a), which comprises a constant energy sourceand an energy harvester. The constant energy source is assumed to be fed by a costlyand/or non–environmentally friendly generator, e.g., a diesel fuel power generator ina remote location or a nuclear power plant. In contrast, the harvested energy is greenand stems from a sustainable source of energy.In this chapter, our aim is to minimize the amount of energy drawn from theconstant energy source, such that the harvested energy is efficiently utilized for trans-mitting a given number of data packets over a finite number of transmission intervals.We assume that there is a battery in the hybrid energy source to store the harvestedenergy. We consider a non–ideal battery which may leak a fraction of the storedenergy over time. Thereby, the leakage depends on the charging and/or dischargingeffect, the chemical properties of the material, etc. [109, 110]. Note that our problemformulation is different from that in [17, 18, 58, 59, 105, 106] and [85, 109, 110], as[17, 18, 58, 59, 105, 106] and [85, 109, 110] consider throughput maximization and/or114Chapter 5. Power Allocation for a Hybrid EH Transmittertransmission time minimization for communication systems employing EH sourcesonly without exploiting a constant energy source. The solution of the optimizationproblem considered in this chapter provides insights regarding the optimal power al-location policy for communication systems with hybrid energy sources and therebyfacilitates the design of reliable green communication systems.We consider two scenarios for the arrival process of the data packets into thedata queue at the transmitter. In Scenario 1, the data packets that have to betransmitted arrive before the transmission begins and no packets arrive during thetransmission, cf. Fig. 5.1(b). In Scenario 2, the data packets may arrive duringthe course of transmission, cf. Fig. 5.1(c). For both scenarios, we derive offline andonline (real–time) power allocation schemes that minimize the total amount of energydrawn from the constant energy source. We propose optimal online power allocationschemes for both considered scenarios using a stochastic DP approach. To avoid thehigh complexity inherent to DP, we also propose suboptimal online algorithms.The remainder of this chapter is organized as follows. In Section 5.2, the systemmodel for the EH system is presented. Offline and online power allocation schemesfor Scenarios 1 and 2 are provided in Sections 5.3 and 5.4, respectively. In Section5.5, the effectiveness of the proposed power allocation schemes is evaluated based onsimulations. Section 5.6 concludes this chapter.5.2 System ModelSystem Description: We consider a single communication link, where a transmitter(source), S, communicates with a receiver (destination), D, as shown in Fig. 5.1(a).We assume that S has a data queue with infinite capacity which can store datapackets temporarily before their transmission. The energy required by S for signal115Chapter 5. Power Allocation for a Hybrid EH TransmitterFigure 5.1: (a) System model for a communication link where the transmitter has ahybrid energy supply. (b) Illustration of Scenario 1 with K transmission intervals,channel SNRs γk, and harvested energies Hk, where k ∈ {1, 2, · · · ,K}. Here, RT bitsarrive before transmission begins. (c) Illustration of Scenario 2. Here, Rk bits arrivejust before time interval, k.transmission and processing is supplied by a hybrid source of energy. The hybridsource includes a constant energy source, possibly connected through a cable to thepower grid, and an EH module which harvests energy from the surroundings. Theharvested energy is stored in a battery that can store at most Bmax Joules of energy.We consider a deadline of K transmission intervals and assume that data transmissionis packet based. The duration of each transmission interval is T and without loss ofgenerality, we assume T = 1s.We consider two scenarios for packet arrivals. In Scenario 1, we assume that RT116Chapter 5. Power Allocation for a Hybrid EH Transmitterbits have arrived at S before the transmission starts and have to be transmittedin K transmission intervals, cf. Fig. 5.1(b). We assume that additional bits do notarrive during the transmission. On the other hand, in Scenario 2, we assume thatRk bits arrive immediately before transmission interval k, where k ∈ {1, 2, · · · ,K},and all bits have to be transmitted by the end of the last transmission interval K,cf. Fig. 5.1(c).Channel Model: We assume that the transmitted packets contain Gaussian dis-tributed symbols and the transmission is impaired by AWGN. Let γk denote thechannel SNR of the S–D link, which is assumed to be i.i.d. over the time intervals.For future reference, we introduce the average SNR of the S–D link as γ¯. We denotetotal transmit power in interval k ∈ {1, 2, · · · ,K} by PE,k + PH,k, where PE,k andPH,k are supplied by the constant energy source and the energy harvester, respec-tively. Furthermore, the total powers drawn from the constant energy source andthe EH source are given by ρPE,k and ρPH,k, respectively. Here, ρ ≥ 1 is a constantthat accounts for the inefficiency of the non–ideal power amplifier. For instance, ifρ = 2, 100 Watts of power are consumed in the power amplifier for every 50 Wattsof power radiated in the radio frequency, and the efficiency of the power amplifier inthis case is 1ρ = 50%. We assume that the power required for signal processing atthe transmitter, which is constant in each time interval, is supplied by the constantenergy source and is excluded from the power allocation algorithm design.Hybrid Energy Model: We assume that Ek is the maximum energy that canbe drawn from the constant energy source in each interval, excluding the requiredconstant signal processing power17. On the other hand, the energy harvester atS collects Hk ≤ Bmax Joules of energy from its surroundings at the end of the17We consider the general case where Ek may change from one transmission interval to the next.However, for the simulation results shown in Section 5.5, we assume a constant energy supply, i.e.,Ek = E, ∀k.117Chapter 5. Power Allocation for a Hybrid EH Transmitterkth interval. Hk is modeled as an ergodic random process with average EH rateHS ! E{Hk}. Due to the inefficiency of the battery, a fraction of the stored harvestedenergy may be lost. We adopt the energy loss model from [111, 112] to incorporatethe imperfections of the battery. We assume that a factor of 1 − µ of the storedharvested energy is leaked per time interval, where 0 ≤ µ < 1 represents the efficiencyof the battery per time interval. Similar to [18], we assume that the harvested energystored in the battery increases and decreases linearly provided the maximum storagecapacity Bmax is not exceeded, i.e.,Bk+1 = min{µ(Bk − ρPH,k) +Hk, Bmax}, ∀k, (5.1)where B1 = H0 ≥ 0 denotes the available energy before transmission starts. Thus,Bk follows a first–order Markov process which depends only on the current state ofthe battery. Due to the finite storage capacity and the leakage of the battery, itis beneficial to draw the energy for packet transmission as quickly as possible fromthe battery so that more harvested energy can be stored in the future, and thus theamount of possibly wasted harvested energy is minimized.5.3 Offline Power AllocationIn this section, we develop offline power allocation strategies for Scenarios 1 and 2.For offline power allocation, it is assumed that both the causal and the non–causalinformation regarding the channel SNR and the harvested energy are available apriori. For offline power allocation, Scenario 1 may be viewed as a special case ofScenario 2 by setting R1 = RT and Rk = 0, where k = 2, 3, · · · ,K. Hence, we onlyformulate and describe the optimization problem for Scenario 2 in detail. We then118Chapter 5. Power Allocation for a Hybrid EH Transmitterobtain the solution for Scenario 1 by setting R1 = RT and Rk = 0, k = 2, 3, · · · ,K,see Section 5.3.2.5.3.1 Offline Power Allocation for Scenario 2We formulate the offline optimization problem for Scenario 2 as follows:minPE,k≥0, PH,k≥0, λH,k≥0K∑k=1ρPE,k (5.2)s.t.q∑k=1log2(1 + γk(PE,k + PH,k))≤q∑k=1Rk, ∀q (5.3)K∑k=1log2(1 + γk(PE,k + PH,k)) =K∑k=1Rk (5.4)l∑k=1ρµl−kPH,k ≤l−1∑k=0µl−k−1(Hk − λH,k), ∀l (5.5)q∑k=0µq−k(Hk − λH,k)−q∑k=1ρµq−k+1PH,k ≤ Bmax, ∀q (5.6)ρPE,k ≤ Ek, ∀k, (5.7)where l ∈ {1, 2, · · · ,K}, q ∈ {1, 2,· · · ,K−1}, and k ∈ {1, 2, · · · ,K}. Constraint (5.3)provides the flexibility to transmit the incoming data packets in future time intervals.Constraint (5.4) ensures that all the data packets are transmitted by a deadline ofK transmission intervals. Constraint (5.5) stems from the causality constraint onthe harvested energy and constraint (5.6) ensures that the harvested energy does notexceed the limited storage capacity of the battery. Thereby, λH,k is a slack variablethat ensures that problem (5.2)–(5.7) is always feasible. In particular, λH,k representsthe amount of harvested energy that is wasted in time interval k because of the limited119Chapter 5. Power Allocation for a Hybrid EH Transmitterstorage capacity of the battery18. The limitation on the amount of energy drawn fromthe constant energy source is reflected in constraint (5.7). Note that for a given timeinterval, for the constant energy supply, any extra amount of energy which is notused for transmission cannot be transferred to the next interval.Problem (5.2)–(5.7) is not a convex optimization problem because of the non–convexity of constraint (5.3) and the non–affinity of constraint (5.4). We combine(5.3) and (5.4) and transform problem (5.2)–(5.7) into the following problem:minPE,k≥0, PH,k≥0, λH,k≥0K∑k=1ρPE,k (5.8)s.t.K∑k=llog2(1 + γk(PE,k + PH,k))≥K∑k=lRk, ∀l (5.9)Constraints (5.5)− (5.7), (5.10)where l ∈ {1, 2, · · · ,K}. However, constraint (5.9) is an equivalent representation ofconstraints (5.3) and (5.4), and hence problem (5.8)–(5.10) is equivalent to problem(5.2)–(5.7), i.e., both problems have the same optimal solution. Problem (5.8)–(5.10)is a convex optimization problem and thus can be solved optimally and efficiently[75]. We note that problem (5.8)–(5.10) is not always feasible. Assuming Rk, channelSNRs, γk, and harvested energies, Hk, are given for all time intervals, i.e., k ∈{1, 2, · · · ,K}, a sufficient (but not necessary) condition for feasibility of problem(5.8)–(5.10) isK∑k=1log2(1 +γkρ(Ek +Hk)) ≥K∑k=1Rk. (5.11)18For example, if Hk is large (Hk is a random variable and cannot be controlled in the optimizationproblem) and Bmax is small, then if λH,k was omitted, constraint (5.6) would not be satisfied andproblem (5.2)–(5.7) would become infeasible.120Chapter 5. Power Allocation for a Hybrid EH TransmitterIf the problem is not feasible, we can extend the number of transmission intervals toK∗ > K by following similar steps as in [17] to avoid infeasibility. Note that duringtime intervals k′ ∈ {K+1,K+2, · · · ,K∗}, we have to assume that no additional datapackets arrive at S to avoid the possibility of facing further infeasibility. It is worthmentioning that problem (5.8)–(5.10) is always feasible if Ek →∞, ∀k, i.e., when theconstant energy supply is (practically) unlimited. In the following, we assume thatproblem (5.8)–(5.10) is feasible.As problem (5.8)–(5.10) satisfies Slater’s constraint qualification and is jointlyconvex in PE,k, PH,k, and λH,k, the duality gap between the dual optimum and theprimal optimum is zero [75]. Therefore, we solve the problem by solving its dual. Forthis purpose, we first provide the Lagrangian of problem (5.8)–(5.10) which can bewritten asL =K∑k=1ρPE,k +K∑l=1αl(l∑k=1ρµl−kPH,k −l−1∑k=0µl−k−1(Hk − λH,k))+K−1∑q=1ξq(q∑k=0µq−k(Hk − λH,k)−q∑k=1ρµq−k+1PH,k −Bmax)+K∑k=1βk (ρPE,k − Ek)−K∑l=1δl(K∑k=llog2(1 + γk(PE,k + PH,k))−K∑k=lRk)(5.12)where δl ≥ 0, αl ≥ 0, ξq ≥ 0, and βk ≥ 0 are the Lagrange multipliers associatedwith constraints (5.9), (5.5), (5.6), and (5.7), respectively. Note that the boundaryconditions PE,k ≥ 0, PH,k ≥ 0, and λH,k ≥ 0 are absorbed into the KKT conditionsfor deriving the optimal PE,k, PH,k, and λH,k. The dual of problem (5.8)–(5.10) canbe stated asmaxδl≥0, αl≥0, ξq≥0, βk≥0minPE,k≥0, PH,k≥0, λH,k≥0L. (5.13)121Chapter 5. Power Allocation for a Hybrid EH TransmitterUsing standard optimization techniques and the KKT optimality conditions, theoptimal PE,k, PH,k, and λH,k can be obtained asP ∗E,k =[ΞE,k −1γk− PH,k]+, (5.14)P ∗H,k =[ΞH,k−1γk−PE,k]+, and (5.15)λ∗H,k =[k−1∑i=0µk−i(Hi − λH,i)−k∑i=1µk−i+1ρPH,i +Hk −Bmax]+, (5.16)respectively, where [x]+ = max{x, 0} and λH,0 = 0. The power allocation solutionsin (5.14) and (5.15) can be interpreted as a form of water–filling, where ΞE,k =∑kj=1 δjρ ln(2)(1+βk)and ΞH,k =∑kj=1 δjρ ln(2)(∑Kj=k αjµj−k−∑K−1j=k ξjµj−k+1) are the water levels associatedwith P ∗E,k and P ∗H,k, respectively. P ∗E,k and P ∗H,k depend on each other due to constraint(5.9).From (5.15), we observe that when Bmax = ∞, we have ξq = 0, ∀q, and inthis case, the optimum water level for the EH source, ΞH,k, is monotonically non–decreasing. In this case, all harvested energy can be stored in the battery and thusP ∗H,k can be more efficiently distributed over the time intervals to minimize the useof the constant energy source. However, when Bmax is finite and if constraint (5.6)is satisfied with equality, i.e., at least one ξq ̸= 0, ∀q, then the monotonicity of theoptimum water level no longer holds. However, when constraint (5.6) is not satisfiedwith equality, the optimum water level is monotonically non–decreasing even forfinite Bmax. Furthermore, we observe from (5.14) that whenever constraint (5.7) isnot satisfied with equality, i.e., βk = 0, the optimum water level for the constantenergy source, ΞE,k, is monotonically non–decreasing for increasing k.We calculate the optimal Lagrange multipliers required in (5.14)–(5.16) via aniterative procedure [75]. We define t as the iteration index. For a given set of Lagrange122Chapter 5. Power Allocation for a Hybrid EH Transmittermultipliers (δ(t),αl(t), ξq(t),βk(t)) and a given value of P ∗H,k(t− 1), we obtain P ∗E,k(t)using (5.14) and then calculate P ∗H,k(t) based on (5.15) by using P ∗E,k(t) as PE,k. Wealso calculate λ∗H,k(t) based on (5.16) by using P ∗H,k(t) as PH,k. The initial set ofLagrange multipliers (δl(1),αl(1), ξq(1),βk(1)) are chosen from the feasible set, i.e.,δl(1) ≥ 0, αl(1) ≥ 0, ξq(1) ≥ 0, βk(1) ≥ 0. However, to calculate PE,k(1) for t = 1,PH,k(0) ≥ 0 is chosen such that (5.5) and (5.6) are satisfied. We update the Lagrangemultipliers as follows:δl(t+ 1) =[δl(t)−Υ1(t)(K∑k=llog2(1 + γk(PE,k + PH,k))−K∑k=lRk)]+, (5.17)αl(t+ 1) =[αl(t)+Υ2(t)(l∑k=1ρµl−kP ∗H,k(t)−l−1∑k=0µl−k−1(Hk − λ∗H,k(t)))]+, (5.18)ξq(t+ 1) =[ξq(t)+Υ3(t)(q∑k=0µq−k(Hk− λ∗H,k(t))−q∑k=1ρµq−k+1P ∗H,k(t)−Bmax)]+,(5.19)βk(t+ 1) =[βk(t) +Υ4(t)(ρP ∗E,k(t)− Ek)]+ , (5.20)where l ∈ {1, 2, · · · ,K}, q ∈ {1, 2, · · · ,K − 1}, and k ∈ {1, 2, · · · ,K}. Here, Υn(t),n ∈ {1, · · · , 4}, are positive step sizes. With the updated Lagrange multipliers,we solve P ∗E,k(t + 1) and P ∗H,k(t + 1) again and the same procedure continues untilconvergence. Note that due to the convexity of problem (5.8)–(5.10), the convergenceto the optimal solution is guaranteed as long as the step sizes satisfy the infinite travelcondition [75].5.3.2 Offline Power Allocation for Scenario 1For Scenario 1, P ∗E,k, P ∗H,k, and λ∗H,k are also given by (5.14), (5.15), and (5.16),respectively, but with δk = 0 for k ∈ {2, 3, · · · ,K}, i.e., the water levels are given123Chapter 5. Power Allocation for a Hybrid EH Transmitterby ΞE,k = δ1ρ ln(2)(1+βk) and ΞH,k =δ1ρ ln(2)(∑Kj=k αjµj−k−∑K−1j=k ξjµj−k+1) . Furthermore, in(5.17), we have to set R1 = RT and Rk = 0, k = 2, 3, · · · ,K. We note that thenumerators of the optimum water levels ΞE,k and ΞH,k for Scenario 1 contain onlyone Lagrange multiplier, δ1. Therefore, whenever constraint (5.7) is not satisfiedwith equality, i.e., βk = 0, ∀k, the optimum water level of PE,k for Scenario 1 remainsconstant. However, Bmax has the same impact on P ∗H,k for Scenario 1 as for Scenario2.5.3.3 Complexity of the Proposed Offline Power AllocationSchemesIn the offline power allocation scheme, we solve a convex optimization problem wherethe number of constraints is a function of K. The required computational cost tosolve a convex optimization problem is polynomial in the size of the problem [75].Therefore, for both considered scenarios, the worst–case computational cost of theproposed offline power allocation scheme is polynomial in the number of time intervalsK [75].5.4 Online Power AllocationIn practice, only causal information of channels and harvested energies is availablefor power allocation. Therefore, the offline power allocation scheme is not readilyapplicable as in a given time interval k, the future CSI and the upcoming harvestedenergy are not known in advance. In this section, for both considered scenarios, wepropose an optimal and a suboptimal, less complex online power allocation schemes.For the online schemes, the random data arrivals in Scenario 2 during transmission124Chapter 5. Power Allocation for a Hybrid EH Transmitterplay an important role19. Since this feature is not present in Scenario 1, to improve theclarity of our chapter, we describe the online power allocation schemes for Scenarios1 and 2 separately.5.4.1 Optimal Online Power AllocationFor optimal online power allocation, we employ a stochastic DP approach [18, 23]which exploits the causal information regarding the channel SNRs, the harvested en-ergies, and their pdfs. For Scenario 2, the causal information regarding the incomingdata bits and the pdf of the data bit arrival process also have to be known. Notethat the pdfs of the channel SNR, the harvested energy, and the incoming data bitscan be obtained via long–term measurements.Scenario 1Let c(1)k ! (γk, Hk−1, Bk, D(1)k−1, Tk) denote the state of the system in time interval kwhich includes channel SNR γk, incoming harvested energy Hk−1, stored harvestedenergy Bk, the total number of remaining bits to be transmitted over the follow-ing time intervals D(1)k−1, and the remaining number of time intervals Tk. D(1)k−1is calculated at the end of time interval (k − 1). Our aim is to minimize theamount of energy drawn from the constant energy source over K intervals and weassume the initial state c(1)1 = (γ1, H0, B1, D(1)0 , T1) is known. We define a policyp(1) = {PE,k(c(1)k ), PH,k(c(1)k ),∀c(1)k , k = 1, 2, · · · ,K}, as feasible if the constraints[PE,k(c(1)k ), PH,k(c(1)k )] ≽ 0 and ρPE,k(c(1)k ) ≤ Ek, ρPH,k(c(1)k ) ≤ Bk are satisfied for all19For example, for optimal power allocation, different system state definitions and Bellman’sequations result for Scenarios 1 and 2 [18]. Therefore, neither of the optimal online schemes bedirectly considered as a special case of the other.125Chapter 5. Power Allocation for a Hybrid EH Transmitterk. Moreover,D(1)k = D(1)k−1 − log2(1 + γk(PE,k + PH,k)) (5.21)for k ∈ {1, 2, · · · ,K} and D(1)0 = RT . The objective function to be minimized can bereformulated as [18]W (p(1)) =K∑k=1E{ρPE,k|c(1)1 , p(1)}, (5.22)where the expectation is with respect to the channel SNR and the harvested energy.In particular, for a given c(1)1 , the minimum amount of energy drawn from the constantenergy source can be obtained asW ∗ = minp(1)∈PW (p(1)), (5.23)where P denotes the space of all feasible policies. In general, the optimization of PE,kand PH,k cannot be performed independently in each time interval because of the EHconstraints. Therefore, to obtain W ∗, we adopt a stochastic DP approach by usingBellman’s equations [18].To this end, we denote the minimum energy drawn from the constant energysource in time interval k as J (1)k (Ek, Bk, D(1)k−1). For a given c(1)1 , the total minimumenergy W ∗ is given by J (1)1 (E1, B1, D(1)0 ), which can be recursively obtained fromJ (1)K (EK , BK , D(1)K−1), J(1)K−1(EK−1, BK−1, D(1)K−2), · · · , J(1)2 (E2, B2, D(1)1 ) [18]. For the126Chapter 5. Power Allocation for a Hybrid EH Transmitterlast time interval K, we haveJ (1)K (EK , BK , D(1)K−1) = minPE,K≥0, PH,K≥0ρPE,K≤EKρPH,K≤BKlog2(1+γK(PE,K+PH,K))≥D(1)K−1ρPE,K (5.24)and for time interval k ∈ {1, 2, · · · ,K − 1}, we have J (1)k (Ek, Bk, D(1)k−1) =minPE,k≥0, PH,k≥0ρPE,k≤EkρPH,k≤BkρPE,k+J¯(1)k+1(Ek+1, µ(Bk−ρPH,k), D(1)k−1 − log2(1 + γk(PE,k + PH,k))),(5.25)where J¯ (1)k+1(Ek+1, µ(Bk−ρPH,k), D(1)k−1 − log2(1 + γk(PE,k + PH,k))) =Eγ˜k+1,H˜k{J (1)k+1(Ek+1,min{µ(Bk−ρPH,k)+H˜k, Bmax}, D(1)k−1−log2(1+γk(PE,k+PH,k)))}.(5.26)Here, γ˜k+1 represents the random SNR in the (k + 1)th interval where the SNR γkin the kth interval is known. Similarly, H˜k denotes the random harvested energyin the kth time interval where the harvested energy Hk−1 in the (k − 1)th intervalis known and Ek is assumed to be known for all time intervals. The pdfs of thechannel SNR and the harvested energy have to be known for evaluation of (5.26).It can be shown that the cost functions in (5.24) and (5.25) are jointly convex inPE,k and PH,k. Therefore, (5.24) and (5.25) are convex optimization problems andcan be solved efficiently and optimally [75]. Note that (5.25) and (5.26) may notbe feasible for all γ˜k+1 and H˜k. We discard the results corresponding to those γ˜k+1and H˜k which provide infeasible solutions and only consider those γ˜k+1 and H˜k whichprovide feasible results in (5.25) and (5.26).127Chapter 5. Power Allocation for a Hybrid EH TransmitterUsing (5.24) and (5.25), P ∗E,k and P ∗H,k, k ∈ {1, 2, · · · ,K}, can be obtained fordifferent possible values of γk and Bk. The results are stored in a look–up table. Thisis done before transmission starts. When transmission starts, for a given realizationof γk and Bk, in time interval k, those values of P ∗E,k and P ∗H,k that correspond to thatrealization are taken from the look–up table. If (5.25) is not feasible for a given γk andBk in an interval k due to an insufficient available amount of energy for transmittingthe required number of data bits, the transmitter transmits as many bits as possibleusing the available power, i.e., P ∗H,k = Bkρ and P ∗E,k = Ekρ . If (5.24) is not feasible fora given γK and BK , then the transmitter extends the transmission deadline from Kto K∗ > K to ensure that all the bits are transmitted by the K∗th interval.Scenario 2We follow similar steps as for Scenario 1 and employ a stochastic DP approach alsofor Scenario 2. However, different from Scenario 1, the main challenge here is howto incorporate the random nature of the data bit arrival process to account for theeffect of data arrivals in future time intervals.Let c(2)k ! (γk, Hk−1, Bk, Rk, D(2)k−1, Tk) denote the state for time interval k, whereD(2)k = D(2)k−1 − log2(1 + γk(PE,k + PH,k)) + Rk+1 (5.27)represents the total number of remaining bits to be transmitted over the followingtime intervals which is calculated at the end of a given time interval (k − 1). Inparticular, D(2)0 = R1. Note that unlike Scenario 1, the number of bits Rk whicharrive immediately before interval k, is now considered as one of the state elementsfor Scenario 2. We assume the initial state c(2)1 = (γ1, H0, B1, R1, D(2)0 , T1) is known.We define a policy p(2) = {PE,k(c(2)k ), PH,k(c(2)k ),∀c(2)k , k = 1, 2, · · · ,K}, as feasible if128Chapter 5. Power Allocation for a Hybrid EH Transmitterthe constraints [PE,k(c(2)k ), PH,k(c(2)k )] ≽ 0 and ρPE,k(c(2)k ) ≤ Ek, ρPH,k(c(2)k ) ≤ Bk aresatisfied for all k. For a given c(2)1 , the minimum energy drawn from the constantenergy source can be obtained asW ∗ = minp(2)∈PK∑k=1E{ρPE,k|c(2)1 , p(2)}, (5.28)where the expectation is taken also with respect to the incoming data packets inaddition to the channel SNR and the harvested energy. We use again Bellman’sequations to obtain W ∗ for Scenario 2 and hence denote the minimum energy drawnfrom the constant energy source in time interval k as J (2)k (Ek, Bk, D(2)k−1) [18]. Fora given c(2)1 , the total minimum energy W ∗ = J (2)1 (E1, B1, D(2)0 ) can be recursivelyobtained from J (2)K (EK , BK , D(2)K−1), J(2)K−1(EK−1, BK−1, D(2)K−2), · · · , J(2)2 (E2, B2, D(2)1 ).For the last time interval K, we haveJ (2)K (EK , BK , D(2)K−1) = minPE,K≥0, PH,K≥0ρPE,K≤EKρPH,K≤BKlog2(1+γK(PE,K+PH,K))≥D(2)K−1ρPE,K (5.29)and for time interval k, we obtain J (2)k (Ek, Bk, D(2)k−1) =minPE,k≥0, PH,k≥0ρPE,k≤EkρPH,k≤BkρPE,k+J¯(2)k+1(Ek+1, µ(Bk−ρPH,k), D(2)k−1 − log2(1 + γk(PE,k + PH,k))),(5.30)where J¯ (2)k+1(Ek+1, µ(Bk−ρPH,k), D(2)k−1 − log2(1 + γk(PE,k + PH,k))) =Eγ˜k+1,H˜k,R˜k+2{J (2)k+1(Ek+1,min{µ(Bk−ρPH,k)+Hk,Bmax},D(1)k−1−log2(1+γk(PE,k+PH,k))+R˜k+2). (5.31)129Chapter 5. Power Allocation for a Hybrid EH TransmitterHere, R˜k+2 represents the random incoming data bits at the source just before the(k + 2)th interval. It can be shown that the cost functions in (5.29) and (5.30) arejointly convex in PE,k and PH,k [75]. A possible infeasibility of (5.30) and (5.31) canbe handled by following the same procedure as in case of Scenario 1. Furthermore,using (5.29) and (5.30), P ∗E,k and P ∗H,k, k ∈ {1, 2, · · · ,K}, are obtained for differentpossible values of γk, Bk, and Rk offline and the results are stored in look–up tablesand used during the course of transmission.5.4.2 Suboptimal Online Power AllocationIn the proposed DP–based optimal online power allocation algorithm, for a giventransmission interval k, we take into account the average effect of all succeeding timeintervals, cf. (5.26) and (5.31). Due to the recursive nature of DP, the computationalcost of this approach increases exponentially with increasing K. For this reason, inthe following, we propose a suboptimal but efficient online power allocation scheme,which performs close to the optimal DP approach with reduced complexity. To thisend, we assume that the average SNR γ¯ is known along with the causal informationof the channel SNRs, harvested energies, and the data arrivals (for Scenario 2). Asthe optimal offline algorithm results in water–filling solutions for P ∗E,k and P ∗H,k, ourobjective is to design an online algorithm which adaptively sets the required numberof data bits to be transmitted according to the current CSI. How the target numberof data bits to be transmitted is obtained is summarized in the following property.Property 1 Suppose P ′ is the power that a transmitter can use to send a requiredamount of data either in the current time interval k or in the next time interval k+1.Assuming that the transmitter only has causal information of the channel SNR, thegain of allocating P ′ in interval k instead of in interval k + 1 can be lower bounded130Chapter 5. Power Allocation for a Hybrid EH Transmitterby log2(γkγ¯).Proof 2 The expected gain in allocating P ′ in the current time interval k over futuretime interval k + 1 is given by∆G = log2(1 + γkP′)− Eγ˜k+1 {log2(1 + γ˜k+1P′)} . (5.32)Using Jensen’s inequality Eγ˜k+1 {log2(1 + γ˜k+1P ′)} ≤ log2(1 + γ¯P ′) in (5.32) yields∆G ≥ log2(1 + γkP′)− log2(1 + γ¯P′) ≈ log2(γkγ¯)(5.33)where the approximation holds for sufficiently large P ′.The significance of using the lower bound of ∆G in (5.33) for the proposed suboptimalonline power allocation algorithm will become apparent in the problem formulationsfor Scenarios 1 and 2 in the following. Note that exploiting ∆G in its original formin the suboptimal online power allocation would result in a non–convex optimizationproblem, which would be difficult to solve.Scenario 1Based on the findings in Property 1, we formulate the optimization problem for agiven time interval k ∈ {1, 2, · · · ,K − 1} as followsminPE,k≥0, PH,k≥0ρPE,k (5.34)s.t. log2(1 + γk(PE,k + PH,k)) ≥ ⌊ϑ(1)k ⌋ (5.35)ρPH,k ≤ Bk, (5.36)131Chapter 5. Power Allocation for a Hybrid EH TransmitterρPH,k ≥ min{(1− µ)Bk,2⌊ϑ(1)k ⌋ − 1γk}, (5.37)ρPE,k ≤ Ek, (5.38)where ϑ(1)k = max{0,min{D(1)k−1Tk+ log2(γkγ¯), D(1)k−1}}and ⌊·⌋ denotes a floor oper-ation. If problem (5.34)–(5.38) is not feasible in an interval k due to an insufficientavailable amount of energy for transmitting the required number of data bits, con-straint (5.35) is relaxed and the transmitter transmits as many bits as possible usingthe available power, i.e., P ∗H,k = Bkρ and P ∗E,k = Ekρ . Let us assume that optimizationproblem (5.34)–(5.38) is always feasible for all time intervals k ∈ {1, 2, · · · ,K − 1}for power allocation algorithm design. The right hand side of (5.35) incorporates theadaptive transmission of data packets based on the channel condition. In particular,based on Property 1, for a given interval k, if the channel SNR is better than theaverage channel SNR, i.e., γk > γ¯, in addition to the average data rate D(1)k−1Tk, weallow the transmission of an extra log2(γkγ¯)> 0 data bits. However, if the channelSNR is worse than the average channel SNR, i.e., γk < γ¯, we transmit less than theaverage data rate D(1)k−1Tksince log2(γkγ¯)< 0. We note that (5.35) implicitly includesa cost associated with missing the deadline as ϑ(1)k is inversely proportional to theremaining number of time intervals, Tk. However, for the Kth time interval, theright hand side of (5.35) is replaced by D(1)K−1. This means that in the last timeinterval, all the remaining data packets are transmitted irrespective of the channelcondition provided that a sufficient amount of energy can be drawn from the constantenergy source and the stored harvested energy. Furthermore, in the right hand sideof (5.37), the term (1−µ)Bk incorporates the effect of µ on PH,k in the optimizationproblem. For example, if µ is very small i.e., most of the stored harvested energyis lost due to leakage, (5.37) forces PH,k to use up all stored energy completely in132Chapter 5. Power Allocation for a Hybrid EH Transmitterthe current time interval so that the losses in future time intervals are minimized.Moreover, 2ϑ(1)k −1γkin (5.37) further avoids the possible waste of the harvested energyin the current time interval. For example, in some cases, using ρPH,k ≥ (1 − µ)Bkmight result in log2(1+γk(P ∗E,k+P ∗H,k)) > ϑ(1)k which would lead to a waste of energy.This is avoided by adopting min{(1− µ)Bk, 2ϑ(1)k −1γk}as the lower bound for PH,k.This choice efficiently reduces the loss of harvested energy in the current and futuretime intervals. If problem (5.34)–(5.38) is not feasible for time interval K, then thetransmitter extends the transmission deadline from K to K∗ > K to ensure that allthe bits are transmitted by the K∗th interval.Problem (5.34)–(5.38) is a convex optimization problem and can be solved opti-mally and efficiently [75]. The Lagrangian of problem (5.34)–(5.38) is given byL′1 =ρPE,k−δ′k(log2(1 + γk(PE,k + PH,k))−ϑ(1)k)−Bk) + ξ′k((1− µ)Bk − ρPH,k)+ α′k(ρPH,k + β′k(ρPE,k − Ek), (5.39)where δ′k, α′k, ξ′k, and β′k represent the Lagrange multipliers associated with (5.35),(5.36), (5.37), and (5.38), respectively. The dual of problem (5.34)–(5.38) can bestated asmaxδ′k≥0, α′k≥0, ξ′k≥0, β′k≥0minPE,k≥0, PH,k≥0L′1. (5.40)Using standard optimization procedures and KKT optimality conditions, the optimal133Chapter 5. Power Allocation for a Hybrid EH TransmitterPH,k and PE,k can be obtained asP ∗E,k =[δ′kρ ln(2)(1 + β′k)−1γk− PH,k]+and (5.41)P ∗H,k =[δ′kρ ln(2)(α′k − ξ′k)−1γk− PE,k]+, (5.42)respectively. We observe that the optimal solutions for PE,k and PH,k depend onlyon causal information regarding the instantaneous channel SNR and harvested en-ergy and also on the average SNR, γ¯, through the Lagrange multipliers. Moreover,(5.41) and (5.42) have a water–filling structure. Similar to the offline power alloca-tion algorithm, the optimal Lagrange multipliers in (5.41), (5.42) can be obtainediteratively.Scenario 2The suboptimal online power allocation for Scenario 2 can also be obtained fromproblem (5.34)–(5.38) after replacing ϑ(1)k by ϑ(2)k , where ϑ(2)k = max{0,min{(D(2)k−1+log2(γkγ¯)), D(2)k−1}}. Hence, for Scenario 2, P ∗E,k and P ∗H,k are also given by (5.41)and (5.42), respectively, but with δ′k and ξ′k being replaced by δ′′k and ξ′′k , respectively.δ′′k and ξ′′k represent the Lagrange multipliers associated with (5.35) and (5.37), whereϑ(1)k is replaced by ϑ(2)k . Similar to Scenario 1, the optimal solutions of PE,k andPH,k for Scenario 2 depend on the causal knowledge of the instantaneous channelSNR, harvested energy, and the average SNR, γ¯, through the Lagrange multipliers.Moreover, the optimal solutions of PE,k and PH,k for Scenario 2 are also influencedby the number of data bits that have just arrived through the Lagrange multipliers.134Chapter 5. Power Allocation for a Hybrid EH Transmitter5.4.3 Complexity of the Proposed Online Power AllocationSchemesThe complexity of the optimal DP based online power allocation scheme increasesexponentially with K. For the suboptimal online scheme, we solve a convex opti-mization problem for each time interval and the size of each convex problem does notdepend on K. Hence, the complexity of the suboptimal online scheme is linear in K.5.5 Simulation ResultsIn this section, we evaluate the performance of the proposed power allocation schemesfor Scenarios 1 and 2. We assume that in each time interval Hk, k ∈ {0, 1, · · · ,K−1},independently takes a value from the set {0, HS, 2HS}, where all elements of the setare equiprobable. For all presented simulation results, Bmax = 50 Joules and ρ = 2.5,which corresponds to a power amplifier efficiency of 40%20. In Figs. 5.3–5.10, weassume E1 = E2 = · · · = EK = 40 Joules whereas in Fig. 5.2, we assume E1 = E2 =· · · = EK = 8 Joules. As we consider a small duration for each time interval (T = 1s),the battery efficiency per interval is high [112]. Thus, we assume µ = 0.99 for allthe figures except for Fig. 5.9. The channel SNR follows an exponential distributionwith means γ¯ = 10 dB for Fig. 5.2, and γ¯ = 25 dB for Figs. 5.3–5.10. For Scenario2, Rk follows a uniform distribution with mean Ravg. For the simulation results inFigs. 5.3–5.8 and 5.10, 104 randomly generated realizations of the channel SNRs,harvested energies, and incoming data packets (for Scenario 2) are considered toobtain the average consumed powers. The total powers drawn from the constantenergy source and the harvested energy are denoted as PE,Tot =∑Kk=1 ρPE,k and20We consider a class A/B power amplifier with a moderate efficiency [113].135Chapter 5. Power Allocation for a Hybrid EH TransmitterPH,Tot =∑Kk=1 ρPH,k, respectively.For comparison, we consider two baseline schemes for offline optimization inFigs. 5.3–5.5. In baseline scheme 1, we minimize the total consumed power, i.e., thesum of the powers drawn from the constant energy source and the energy harvester.The offline optimization problems for baseline scheme 1 are obtained by adopting∑Kk=1 ρ(PH,k +PE,k) as objective function in (5.2) for Scenarios 1 and 2, respectively.The objective of baseline scheme 1 is to minimize the total consumed energy ratherthan the consumed non–renewable energy. In baseline scheme 2, we define the num-ber of data bits to be transmitted in each interval k as ⌊RT/K⌋ and Rk for Scenarios1 and 2, respectively. In each interval k, at first we draw power from the harvestedenergy and if the harvested energy cannot satisfy the bit rate requirement, then wedraw power from the constant energy source. It is worth mentioning that baselinescheme 1 follows a common approach for power minimization in the existing wirelesscommunication systems whereas baseline scheme 2 is a naive and basic policy thatprioritizes consuming renewable energy first.5.5.1 Behavior of Offline Power Allocation Over TimeIntervals kIn Fig. 5.2, we show the optimum power levels of PE,k and PH,k for the offline andsuboptimal online power allocation schemes for Scenario 1 as functions of the timeintervals k. In addition, we also show the inverse channel SNR 1γk and Hk as wellas the water levels of PE,k and PH,k for the offline scheme. HS = 2, RT = 40, andK = 10 are adopted in this figure.For the optimal offline power allocation scheme, we observe that the water levelof PE,k, ΞE,k, does not remain constant over the time intervals. As Ek is not large136Chapter 5. Power Allocation for a Hybrid EH Transmitter0 1 2 3 4 5 6 7 8 9 10024Time interval, k  0 1 2 3 4 5 6 7 8 9 103456Time interval, k  0 1 2 3 4 5 6 7 8 9 100246Time interval, k  Hk (Watts)1/γkWater level for PE,kPE,k (Watts) (Suboptimal Online)PE,k (Watts) (Oflline)Water level for PH,kPH,k (Watts) (Suboptimal Online)PH,k (Watts) (Offline)Figure 5.2: Harvested energy Hk, fading level, water level, and optimal PE,k forScenario 1 vs. time interval k for K = 10 and HS = 2 Joules.enough, Lagrange multiplier βk, associated with constraint (5.7), can be non–zeroand this causes ΞE,k to vary with k. For all time intervals, ΞE,k −P ∗H,k is larger thanthe inverse channel SNR and thus P ∗E,k > 0. We also observe that the water levelof PH,k, ΞH,k is monotonically non–decreasing. As Bmax is sufficiently large for theconsidered case, (5.6) is not met with equality and thus ξk = 0, for k ∈ {1, 2, · · · , 9}.This forces ΞH,k either to remain constant or to increase. In time interval k = 2,ΞH,k − P ∗E,k is less than the inverse channel SNR and therefore P ∗H,k = 0. In the restof the time intervals, ΞH,k−P ∗E,k is greater than the inverse channel SNR and therefore137Chapter 5. Power Allocation for a Hybrid EH TransmitterP ∗H,k > 0. We observe for the suboptimal online power allocation scheme that due tothe lack of non–causal information, this scheme cannot make full use of the harvestedenergy and thereby increases the power drawn from the constant energy source. Forinstance, we observe that in k = 2, although the inverse channel SNR is high, thesuboptimal online scheme draws more power from the harvested energy than theoptimal offline scheme and therefore cannot exploit the good channel condition (low1/γk) in k = 4 due to the lack of stored harvested energy in the battery. Moreover,compared to the offline scheme, the suboptimal online scheme consumes more energyfrom the constant energy source in k ∈ {4, 5}. Also, although no energy is harvestedat the end of interval k = 7, the offline scheme still draws energy from the batteryto transmit data bits in k = 8 to exploit the good channel condition. On the otherhand, suboptimal online scheme cannot draw harvested energy at the beginning ofk = 8, because it has already used up the stored harvested energy in k < 7. As theoffline scheme has non–causal knowledge about the channel SNR and the harvestedenergy, this scheme can adjust the power levels in all the intervals to minimize thetotal power consumption from the constant energy source.5.5.2 Comparison of the Proposed Offline Schemes with theBaseline SchemesFig. 5.3 shows PE,Tot and PH,Tot vs. the HR HS for Scenario 1, where RT = 75 bitsand K = 10. The upper and lower subfigures compare the proposed scheme withbaseline schemes 1 and 2, respectively. We observe that PE,Tot decreases with in-creasing HS for the proposed and the baseline schemes. Since we minimize the powerdrawn from the constant power source and the amount of data to be transmittedis constant, as the harvesting rate increases, more harvested energy is available for138Chapter 5. Power Allocation for a Hybrid EH Transmitter0 0.5 1 1.5 201020304050HS (Joules)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 1)PH,Tot (Baseline 1)0 0.5 1 1.5 201020304050607080HS (Joules)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 2)PH,Tot (Baseline 2)PE,Tot + PH,Tot  (Baseline 1)PE,Tot + PH,Tot  (Proposed)PE,Tot + PH,Tot  (Baseline 2) PE,Tot + PH,Tot  (Proposed)Figure 5.3: Consumed power for Scenario 1 vs. HS for K = 10 and RT = 75 bits.Only offline schemes are considered.use which results in increased consumption of harvested energy and decreased con-sumption from the constant power source. We observe that the PE,Tot (PH,Tot) curveof the proposed scheme always remains below (above) the PE,Tot (PH,Tot) curves ofboth baseline schemes, and that the gap between the schemes increases as the HRHS increases. As expected, for HS = 0 Joule, i.e., when there is no harvested energy,the proposed scheme and baseline scheme 1 yield the same PE,Tot and PH,Tot = 0.However, for HS = 0 Joule, the proposed scheme and baseline scheme 2 yield different139Chapter 5. Power Allocation for a Hybrid EH TransmitterPE,Tot. As baseline scheme 2 has to transmit a fixed number of data bits in each timeinterval irrespective of the channel conditions, it is less flexible and consumes a largeamount of power from the constant energy source. In particular, for HS = 1.5 Joules,we observe that adopting the proposed scheme allows to save 7 Watts and 55 Wattsof power drawn from the constant energy source compared to baseline schemes 1 and2, respectively.We have also included the total powers consumed by the proposed and the baselineschemes in Fig. 5.3. We observe that for baseline scheme 1, the increase of PH,Totwith increasing HS exactly compensates the decrease of PE,Tot with increasing HS,and therefore, the total consumed power does not change with HS. Intuitively, asbaseline scheme 1 minimizes the total consumed energy, HS has no impact on thetotal power consumption. On the other hand, for the proposed scheme, for largeHS, PH,Tot increases faster with HS than PE,Tot decreases with HS. As a result, thetotal consumed power increases with HS after a certain threshold. The reason forthe higher total power consumption is the increase in power drawn from the energyharvester to decrease the consumption from the constant energy source. Similarto the proposed scheme, the total power consumption of baseline scheme 2 is alsoinfluenced by HS and increases with increasing HS.In Figs. 5.4 and 5.5, we compare the performance of the proposed and the baselineoffline schemes for Scenarios 1 and 2 as functions of the amount of transmitted data.In particular, Fig. 5.4 shows PE,Tot and PH,Tot vs. RT for Scenario 1 whereas in Fig 5.5we present PE,Tot and PH,Tot vs. RavgK for Scenario 2 where Ravg = (1/K)K∑k=1Rk.For both scenarios, we adopt HS = 6.75 Joules and K = 10. We observe that thetransmit powers, PE,Tot and PH,Tot, increase with increasing RT (Ravg) because thelarger the amount of data that has to be transmitted, the higher the required transmit140Chapter 5. Power Allocation for a Hybrid EH Transmitter40 50 60 70 80 90 100050100150RT (bits)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 1)PH,Tot (Baseline 1)40 50 60 70 80 90 100050100150200250300RT (bits)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 2)PH,Tot (Baseline 2)PE,Tot when HS = 0Figure 5.4: Consumed power for Scenario 1 vs. RT for K = 10 and HS = 6.75 Joules.Only offline schemes are considered.power. However, the rate of increase is not the same for PE,Tot and PH,Tot. The ratewith which PE,Tot increases is small for low RT (Ravg), because for low RT (Ravg),the consumed power is mainly drawn from the energy harvester. On the other hand,for large RT (Ravg), PE,Tot increases rapidly as the harvested energy alone is notsufficient to supply the required power completely. Moreover, for large RT (Ravg),PH,Tot saturates because the maximum energy that the energy harvester can provideis limited byK∑k=1Hk. We observe again that in both scenarios our proposed schemesare more efficient in reducing PE,Tot than baseline schemes 1 and 2, respectively.For comparison, we also show PE,Tot for the proposed and baseline scheme 1 for141Chapter 5. Power Allocation for a Hybrid EH Transmitter40 50 60 70 80 90 100050100150RavgK (bits)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 1)PH,Tot (Baseline 1)40 50 60 70 80 90 100050100150200250300350400RavgK (bits)Consumed power (Watts)  PE,Tot (Proposed)PH,Tot (Proposed)PE,Tot (Baseline 2)PH,Tot (Baseline 2)PE,Tot when HR = 0Figure 5.5: Consumed power for Scenario 2 vs. RavgK for K = 10 and HS = 6.75Joules. Only offline schemes are considered.HS = 0 Joule (i.e., no energy harvester) and observe that both schemes yield identicalresults as expected. Besides, PE,Tot for HS = 0 Joule is always larger than PE,Tot forHS = 6.75 Joules as without the supplement of the energy harvester, the constantenergy source has to supply all the required power. Moreover, a comparison of thepowers consumed in Scenario 1 and Scenario 2 reveals that Scenario 1 requires a(slightly) lower PE,Tot than Scenario 2. This can be explained by the fact that knowingthe amount of data to be transmitted before the transmission starts provides moreflexibility to allocate the transmit powers over the transmission intervals than whenthe data packets arrive during the course of transmission.142Chapter 5. Power Allocation for a Hybrid EH Transmitter0 0.5 1 1.5 2 2.5 3 3.5 40246810121416HS (Joules)Consumed power (Watts)  PE,Tot (Offline)PH,Tot (Offline)PE,Tot (Optimal Online)PH,Tot (Optimal Online)PE,Tot (Suboptimal Online)PH,Tot (Suboptimal Online)Figure 5.6: Consumed power for Scenario 1 vs. HS for K = 4 and RT = 30 bits.5.5.3 Comparison of the Proposed Offline and OnlineSchemesFig. 5.6 shows PE,Tot and PH,Tot vs. the HR HS for Scenario 1 for all considered powerallocation schemes. We assume RT = 30 bits and K = 4. We observe that PE,Tot(PH,Tot) decreases (increases) with HS for all power allocation schemes. As expectedthe offline power allocation scheme performs better than the online power allocationschemes for all HS. Moreover, the optimal DP based online scheme outperforms thesuboptimal online scheme because DP makes optimal use of the statistical properties143Chapter 5. Power Allocation for a Hybrid EH Transmitterof the channel SNR and the harvested energy. We observe that the performancegaps between the offline scheme and the online schemes increase with increasing HS.Having non–causal knowledge regarding the channel SNR and the harvested energyhelps more in minimizing PE,Tot for high HS than for low HS. However, for lowHS, the gap between the offline and online schemes for PH,Tot is very small as all theharvested energies are used up for low HS regardless of the non–causal information (ofthe channel SNR and the harvested energy) for the considered RT . On the contrary,for high HS, the offline scheme makes efficient use of HS whereas the online schemesmay under–utilize the harvested energy and result in a lower PH,Tot. It is worthmentioning that a similar behavior can be observed for Scenario 2 (not shown here).Fig. 5.7 shows PE,Tot and PH,Tot vs. the HR HS for Scenario 1, for the offlineand suboptimal online power allocation schemes for K = 40 and RT = 300 bits. Theoptimal DP based online power allocation scheme has not been implemented herebecause of its inherent high computational cost for large K. We observe that theperformance gap between the offline and suboptimal online power allocation schemesincreases with increasing HS as also observed in Fig. 5.6. However, the performancegap between the offline PE,Tot and the suboptimal online PE,Tot is larger in Fig. 5.7compared to Fig. 5.6 for all HS. In fact, exploiting non–causal knowledge of thechannel SNR and the incoming harvested energy is more beneficial for large K thanfor small K. Note that a similar behavior can be observed for Scenario 2 (not shown).In Fig. 5.8, we show PE,Tot and PH,Tot vs. HS for Scenarios 1 and 2 for thesuboptimal online power allocation scheme. Here, we assume K = 100 and RT =K∑k=1Rk = 500 bits. We observe that Scenario 1 requires a lower PE,Tot than Scenario2 for all considered HS. The random arrival of data packets in Scenario 2 introducesadditional restrictions for power allocation compared to Scenario 1, where the amount144Chapter 5. Power Allocation for a Hybrid EH Transmitter0 0.5 1 1.5 2020406080100120HS (Joules)Consumed power (Watts)  PE,Tot (Offline)PH,Tot (Offline)PE,Tot (Suboptimal Online)PH,Tot (Suboptimal Online)PE,Tot + PH,Tot   (Offline)PE,Tot + PH,Tot(Suboptimal online)Figure 5.7: Consumed power for Scenario 1 vs. HS for K = 40 and RT = 300 bits.of data to be transmitted is known before transmission starts. Hence, the observationsmade for Scenarios 1 and 2 for the offline scheme in Figs. 5.4 and 5.5 also translate tothe suboptimal online scheme. On the other hand, since for the considered exampleRT =K∑k=1Rk = 500 bits have to be transmitted in only 100 time intervals, theharvested energy is completely used in both scenarios for all considered values of HS.Thus, we observe in Fig. 5.8 no significant difference for PH,Tot between Scenarios 1and 2.145Chapter 5. Power Allocation for a Hybrid EH Transmitter0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1050100150HS (Joules)Consumed power (Watts)  PE,Tot (Scenario 1)PH,Tot (Scenario 1)PE,Tot (Scenario 2)PH,Tot (Scenario 2)Figure 5.8: Consumed power for Scenarios 1 and 2 vs. HS for K = 100 and RT =K∑k=1Rk = 500 bits.5.5.4 Effect of µ on the Proposed SchemesIn Fig. 5.9, we show ρPH,k vs. time interval k for the optimal offline and subop-timal online power allocation schemes for K = 9, HS = 2 Joules, γ¯ = 25 dB,RT = 50 bits, and three different values of µ, namely µ = {0, 0.5, 1.0}. We assume[γ1 γ2 · · · γ9] = [104.02 23.86 219.11 145.77 190.44 107.76 119.56 623.05 102.54] forall the considered µ. We observe that when µ = 0, i.e., when the battery cannotstore the harvested energy at all, the offline and suboptimal online schemes use upall of the harvested energy immediately (right after its arrival) in the current timeinterval to avoid any possible loss of the harvested energy. Increasing µ helps the146Chapter 5. Power Allocation for a Hybrid EH Transmitter0 1 2 3 4 5 6 7 8 9024Time interval, k  0 1 2 3 4 5 6 7 8 9024Time interval, k  0 1 2 3 4 5 6 7 8 9024Time interval, k0 1 2 3 4 5 6 7 8 9024Time interval, kHk (Watts)PH,k (Suboptimal Online)PH,k (Offline)µ = 1.0µ = 0.5µ = 0Figure 5.9: PH,k for Scenario 1 vs. k for K = 9 and HS = 2 Joules.offline and suboptimal online schemes to store the harvested energy if necessary, andto use it when the channel is better. This property is observed for µ = 0.5 andµ = 1.0. In Table 5.1, we show PH,Tot obtained for the considered offline and sub-optimal online power allocation schemes for8∑k=0Hk = 10 Joules and the value of µassumed in Fig. 5.9. We observe that for µ = 0.5 and µ = 1, the offline schemeconsumes more harvested energy than the suboptimal online scheme. However, forµ = 0, all the harvested energy is used by both the considered schemes as any storedharvested energy would be completely lost.147Chapter 5. Power Allocation for a Hybrid EH TransmitterTable 5.1: PH,Tot obtained for the parameters used for Fig. 5.9 for offline and subop-timal online power allocation schemes.µ Offline (Joules) Suboptimal online (Joules)1.0 9.93 9.210.5 8.32 8.160 10 105.5.5 Impact of Various EH ProfilesIn Fig. 5.10, we show the impact of different EH profiles on PE,Tot and PH,Tot forScenario 1 as a function of RT . We assume K = 50 and consider offline powerallocation. Two cases, denoted as Case A and Case B, are considered here to showthe impact of two different EH profiles on the consumed powers. Case A assumes anEH profile, where the energy is harvested in every third time interval, whereas Case Bconsiders another EH profile, where the energy is harvested in every time interval. Inboth cases, the average EH rate is 5 Joules/interval and the channel SNR changes inevery time interval. We observe from Fig. 5.10 that the PE,Tot (PH,Tot) curve for CaseB is always below (above) the PE,tot (PH,Tot) curve for Case A. In Case A, the energyis harvested at a slower rate than in Case B, i.e., in Case A there is less flexibility forpower allocation. Thus, more energy is consumed from the constant energy sourcefor Case A to transmit the required amount of data bits.5.6 ConclusionsIn this chapter, we optimized the power allocation for a point–to–point communi-cation system with a hybrid energy source. The hybrid energy source includes aconstant energy source (non–renewable) and an energy harvester (renewable). The148Chapter 5. Power Allocation for a Hybrid EH Transmitter140 150 160 170 180 190 200 210 220 230 240050100150200250300350RTConsumed power (Watts)  PE,Tot (Case A)PH,Tot (Case A)PE,Tot (Case B)PH,Tot (Case B)Figure 5.10: Consumed power for Scenario 1 vs. RT for K = 50 and two differentEH profiles.harvested energy is stored in a battery which is modeled as a storage system withleakage. We proposed to minimize the amount of power drawn from the constantenergy source to make full use of the harvested energy. We presented optimal offline,optimal online, and suboptimal online power allocation schemes for two different dataarrival scenarios. We used stochastic DP to implement the optimal online power al-location scheme. Because of the inherently high complexity of DP, a low–complexitysuboptimal online power allocation scheme was also proposed. A comparison of theproposed power allocation schemes with baseline schemes revealed that the proposedschemes significantly reduce the power drawn from the constant energy source andutilize the harvested energy more efficiently. Moreover, simulation results revealedthat if the data to be transmitted arrives randomly over the transmission intervals,149Chapter 5. Power Allocation for a Hybrid EH Transmitterthe power consumption from the constant energy source is always higher than if thedata arrives before transmission starts.150Chapter 6Conclusions and Future WorkIn Section 6.1 of this final chapter, we briefly review our results and highlight thecontributions of this thesis to the literature. In Section 6.2, we propose directions forfuture research.6.1 Summary of ResultsThis thesis focused on resource allocation schemes for different cooperative and non–cooperative communication systems. We considered systems that are impaired byGaussian and non–Gaussian noises and powered by constant energy sources and en-ergy harvesters. The main results of each chapter are summarized as follows:In Chapter 2, we derived closed–form asymptotic error rate expressions for bestrelay selection (BRS) and partial relay selection (PRS) in an AF relay network. Ourderived expressions are valid for i.n.d. Rayleigh fading and noises with finite mo-ments. We used the derived error rate expressions to develop general power allocationframeworks for BRS and PRS with energy consumption constraints. Furthermore, weconsidered a relay subset selection scheme to reduce the CSI feedback overhead. Weused computer simulations to show the accuracy of the derived error rate expressionsand the effectiveness of the proposed power allocation schemes.In Chapter 3, we considered a single source–relay–destination link that used EHat the source and the relay. We considered conventional and buffer–aided link adap-151Chapter 6. Conclusions and Future Worktive relaying protocols and proposed optimal offline and optimal (for link adaptiverelaying) and suboptimal online resource allocation schemes for both protocols. Theoffline scheme for conventional relaying was formulated as a convex optimizationproblem and hence could be solved optimally and efficiently. On the other hand, theoffline scheme for link adaptive relaying results in a non–convex MINLP, which wasoptimally solved by the sBB method with relatively lower computational cost thanthe exhaustive search method. We formulated the optimal online scheme for conven-tional relaying by stochastic dynamic programming (DP). We did not formulate theoptimal online scheme for link adaptive relaying, because it would have incurred alarge computational cost. Our proposed suboptimal online schemes achieved a goodperformance–complexity trade–off. Simulation results revealed that link adaptive re-laying has significantly better throughput than conventional relaying, at the expenseof a higher complexity.In Chapter 4, we jointly optimized the relay selection and power allocation for anAF relay network that used EH nodes for the source and the relays. We proposed op-timal offline and suboptimal but low–complexity online resource allocation schemes.The offline scheme was initially formulated as a non–convex MINLP, which was thenconverted into a convex MINLP and optimally solved by the GBD algorithm. Weavoided formulating the optimal online scheme using DP due to its high computa-tional cost, but eventually proposed a low–complexity but suboptimal online scheme.Simulation results revealed that the suboptimal online scheme yields significantlymore throughput gain than a naive scheme and performs close to the offline scheme.In Chapter 5, we considered a point–to–point communication system that useda hybrid energy supply for its source. The hybrid energy supply consisted of aconstant energy source and an energy harvester. We considered two scenarios for152Chapter 6. Conclusions and Future Workthe data arrival process. In the first scenario, all the data packets arrived before thetransmission began. In the second scenario, the data packets may have arrived duringthe data transmission. Our objective was to minimize the amount of energy drawnfrom the constant energy source and thus efficiently utilize the harvested energy. Weproposed optimal offline, optimal online, and suboptimal but low–complexity onlineschemes for both data arrival processes. We compared our proposed schemes withbaseline schemes via simulations and observed that our proposed schemes performedmuch better than the baseline schemes in terms of reducing the energy consumptionfrom the constant energy source.6.2 Future WorksIn the following, we propose some ideas for future research that are based on theresults of this thesis.6.2.1 Performance Analysis of Relay Selection SchemesImpaired by α–stable noiseA broad class of non–Gaussian phenomena that is encountered in practice can becharacterized by α–stable distributions [114]. α–stable noise has heavier tails thanthe Gaussian distribution. Examples of α–stable noise include underwater acousticsignals, low–frequency atmospheric noise, and some types of man–made impulsivenoise. The characteristic component α, 0 ≤ α ≤ 2, defined in the pdf of α–stablenoise, controls the heaviness of the tails of the pdf. For instance, a small positive valueof α indicates severe impulsiveness, whereas α = 2 indicates the Gaussian distributionas its special case. The performance analysis in Chapter 2 is valid for the noises with153Chapter 6. Conclusions and Future Workfinite moments. However, our analysis in Chapter 2 cannot be applied to α–stablenoise, because it does not have finite moments. Developing new techniques to analyzeα–stable noise for AF relay selection systems would be an interesting direction forfuture research.6.2.2 Intelligent Resource Allocation for EH SystemsThe proposed offline and online resource allocation schemes for EH systems in Chap-ters 3–5 require non–causal or statistical information regarding the channel SNR,the harvested energy, and the data arrival process. Assuming non–causal informa-tion in the offline optimization framework is too optimistic in practice unless theunderlying processes are deterministic or highly predictable. These assumptions arerelaxed in the online optimization framework, which exploits statistical informationabout the system parameters. However, in many practical systems, the statisticalproperties may change over time or not be available. In both cases, learning basedalgorithms can be employed to identify the characteristics of the channel SNR andthe energy and data arrival processes. In [115], a single source–destination link isconsidered and the Q–learning method is used to identify the energy and data arrivalprocesses. The identified processes and collected information provided the bases forthe resource allocation schemes that were proposed to maximize the expected sumof the transmitted data over the lifetime of the source. In the future, researcherscould tackle the interesting challenge of exploiting machine–learning methods for EHcooperative communication systems in order to identify the EH characteristics of thecommunication nodes, their incoming data arrival process, and the channel SNRs ofall the involved links. Moreover, the performance and the complexity of differentmachine–learning methods applicable to EH systems could also be investigated.154Chapter 6. Conclusions and Future Work6.2.3 Wireless Information and Power Transfer in RelayNetworksResearchers have recently been very interested in the simultaneous transfer of in-formation and RF energy in different communication systems [54], [116]–[118]. Thesimultaneous transfer of information and energy ensures that RF signals are utilizedin transmitting both data and energy at the same time, and thus potentially offersgreat advantages to end users. However, as the information reception and RF EHhave different power sensitivities, it is vital that receivers are carefully designed tosuccessfully receive energy and information [119]. Time–sharing and power–splittingreceivers are the receivers most commonly discussed in the literature [53]. It wouldtherefore be interesting for researchers to study the behavior of wireless informationand power transfers in different relay networks, e.g., in conventional relaying, buffer–aided link adaptive relaying, multi–relay networks, etc. Analyzing the performanceof different receivers for different relay networks and comparing their performanceswould also be interesting.6.2.4 Resource Allocation for a Multi–hop Conventional andBuffer–Aided Link Adaptive EH Relaying ProtocolsMulti–hop21 communication provides better throughput and lower error rate perfor-mance compared to single–hop communication [120]–[123]. 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Commun.,vol. 58, pp. 2413–2424, Aug. 2010.[124] H. P. Hsu, Theory and Problems of Probability, Random Variables, and RandomProcesses. Schaum’s Outline Series.[125] “Wolfram Mathworld.” [Online]: http://functions.wolfram.com/Hypergeo/-metricFunctions/ Hypergeometric2F1/06/01/05/01/02.167Appendix AExploiting [16, Eq. (33) and (37)], for BRS the MGF of ∆ks can be expressed asΦB∆ks (s) = Eaks ,θks{e−s∆ks}=∞∑ξ=0s2ξ(2d)2ξ∑i+j=ξβiβj2i!2j!Ψijks , (A.1)where βi ! Γ(i+ 1/2)/(√πΓ(i+ 1)) andΨijks ! Eγ1ksγ2ks n¯1ks n¯2ks{Ωijks(d2s)|n¯1ks |2i|n¯2ks |2j}(A.2)withΩijks(s) !e−sγ1ksγ2ksγ1ks+γ2ks γ2j+i1ks γ2i+j2ks(γ1ks + γ2ks)2i+2j . (A.3)Since the channel gains and the noise samples are mutually independent, Ψijks can beexpanded asΨijks =N∑k=1Eγ1kγ2k{Ωijk (d2s)}En¯1kn¯2k{|n¯1k|2i|n¯2k|2j}. (A.4)Using the results in [124, Solved Problem 4.25], the joint probability density function(pdf) of γ1k and γ2k can be expressed asp(x, y) =⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩pγ1k(x)pγ2k(y)N∏m=1m̸=kPzm(x), x ≤ ypγ1k(x)pγ2k(y)N∏m=1m̸=kPzm(y), x > y(A.5)168Appendix A.where zm = min{γ1m, γ2m}, pγ1k(γ1k) = γ¯−11k e−γ1k/γ¯1k and pγ2k(γ2k) = γ¯−12k e−γ2k/γ¯2k areRayleigh pdfs, and Pzm(·) denotes the cdf of zm. Since we are interested in the highSNR behavior, we replace the cdf of zm with its first order Taylor series expansionPzm(z).= γ¯1m+γ¯2mγ¯1mγ¯2m z. Based on this approximation, Eγ1kγ2k{Ωijk (d2s)}in (A.4) can bewritten asEγ1kγ2k{Ωijk (s)}= ΨijAk(s) +ΨijBk(s), (A.6)where ΨijAk(s) =∞∫γ1k=0γ1k∫γ2k=0Ci,j,kγ,γ¯ γN−12k dγ1kdγ2k (A.7)and ΨijBk(s) =∞∫γ1k=0∞∫γ2k=γ1kCi,j,kγ,γ¯ γN−11k dγ1kdγ2k (A.8)withCi,j,kγ,γ¯ =e−sγ1kγ2kγ1k+γ2k γ2j+i1k γ2i+j2k e− γ1kγ¯1k e−γ2kγ¯2k(γ1k + γ2k)2i+2j γ¯1kγ¯2kN∏m=1,m ̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m) . (A.9)Applying the transformation of variables, γ1k = r2 cos2 φ and γ2k = r2 sin2 φ, the defi-nition Nγ¯ ! (γ¯1kγ¯2k∏Nm=1,m ̸=kγ¯1mγ¯2mγ¯1m+γ¯2m)−1, and the definition of the Gamma function,Γ(t) =∫∞0 xt−1e−xdx, in ΨijAk(s) of (A.8) leads toΨijAk(s) = 2Nγ¯Γ(i+ j +N + 1)π/4∫0(sin2φ)2i+j+N− 12 (cos2φ)2j+i+12(scos2φsin2φ + cos2φγ¯1k+ sin2φγ¯2k)i+j+N+1dφ. (A.10)Splitting the integration limit in (A.10) into two intervals [0, ϵ] and [ϵ,π/4] yieldsΨijAk(s) = 2Nγ¯Γ(i+ j +N + 1)(IAk + IIAk), (A.11)169Appendix A.where IAk and IIAk correspond to the integrals from 0 to ϵ and from ϵ to π/4, respec-tively. For ϵ → 0, sinφ → φ, and cosφ → 1, which leads toIAk =ϵ∫0(φ2)2i+j+N−12 γ¯i+j+N+11k((s+ 1γ¯2k)φ2γ¯1k + 1)i+j+N+1dφ. (A.12)Using [76, 3.194.1], (A.12) can be written asIAk =ϵ2(2i+j+N)γ¯i+j+N+11k2(2i+ j +N) 2F1(2i+ j +N, i + j +N + 1;2i+ j +N + 1;−(s+ 1/γ¯2k)γ¯1kϵ2), (A.13)where 2F1(·, ·; ·; ·) denotes the Gaussian hypergeometric function. Exploiting theasymptotic behavior of 2F1(·, ·; ·; z) for z →∞ [125], (A.13) can be rewritten asIAk =12(2i+ j +N)si+j+N+1[Γ(1− i)Γ(2i+ j +N + 1)(γ¯1ks)−(i−1)Γ(i+ j +N + 1)+(2i+ j +N)ϵ2(i−1)i− 1]+ o(γ¯−(i−1)+1k). (A.14)On the other hand, assuming ϵ → 0 and γ¯1k, γ¯2k →∞, IIAk can be stated asIIAk =1si+j+N+1π/4∫ϵ(sin2φ)i− 32 (cos2φ)j−N− 12dφ. (A.15)Due to the symmetry of ΨijAk(s) and ΨijBk(s) in (A.8), an asymptotic expression similarto (A.10) can also be developed for ΨijBk(s). This leads toΨijBk(s) = 2Nγ¯Γ(i+ j +N + 1)(IBk + IIBk), (A.16)170Appendix A.whereIBk =12(2j + i+N)si+j+N+1[Γ(1− j)Γ(2j + i+N + 1)(γ¯2ks)−(j−1)Γ(i+ j +N + 1)+(2j + i+N)ϵ2(j−1)j − 1]+ o(γ¯−(j−1)+2k)(A.17)and IIBk is independent of γ¯1k and γ¯2k.Based on (A.10) and (A.14)–(A.17), we evaluate now Eγ1kγ2k{Ωijk (s)} in (A.6) forfour different cases.Case 1: When i = 0 and j = 0,Eγ1kγ2k{Ωijk (s)}=(N − 1)!N∏m=1m̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)sN(1γ¯1k+1γ¯2k)+ o(γ¯−1c), (A.18)where γ¯c !∏Nk=1 min{γ¯1kγ¯2k}.Case 2: When i ̸= 0 and j = 0, we getEγ1kγ2k{Ωijk (s)}=(i+N − 1)γ¯1kN∏m=1m̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)si+N+ o(γ¯−1c). (A.19)Case 3: When i = 0 and j ̸= 0, we haveEγ1kγ2k{Ωijk (s)}=(j +N − 1)γ¯2kN∏m=1m̸=k(γ¯1mγ¯2mγ¯1m+γ¯2m)sj+N+ o(γ¯−1c). (A.20)Case 4: When i ̸= 0 and j ̸= 0, we obtainEγ1kγ2k{Ωijk (s)}= o(γ¯−2c). (A.21)171Appendix A.Combining (A.1), (A.4), (A.18)–(A.21), using En¯1kn¯2k{|n¯1k|2i|n¯2k|2j} = m1k(i)m2k(j),where m1k(i) ! E{|n¯1k|2i} and m2k(i) ! E{|n¯2k|2i}, and exploiting the identitys2ξβξ(ξ!)2 = 2ξ!, we obtain (2.11).172Appendix BFor PRS–S, relay selection depends only on the instantaneous S–R SNRs. Thus, thejoint pdf of γ1k and γ2k is given byp(x, y) = pγ1k(x)pγ2k(y)N∏i=1i̸=kPγ1i(x) = pγ1k(x)pγ2k(y)×1∑u1=0k ̸=11∑u2=0k ̸=2· · ·1∑uN=0k ̸=N(−1)N∑i=1,i ̸=kuie−γ1kN∑i=1,i ̸=kuiγ¯1i , (B.1)where Pγ1i(z) = 1 − e− zγ¯1i . Eqs. (A.1)–(A.4) are also valid for PRS–S and based on(B.1), Eγ1kγ2k{Ωijk (s)}in (A.4) can be written asEγ1kγ2k{Ωijk (s)}=1∑u1=0k ̸=1· · ·1∑uN=0k ̸=N(−1)N∑i=1i̸=kui ∞∫γ1k=0∞∫γ2k=0e−sγ1kγ2kγ1k+γ2k γ2j+i1k γ2i+j2k(γ1k + γ2k)2i+2j×e−γ1k(1γ¯1k+N∑i=1, i ̸=kuiγ¯1i)e−γ2kγ¯2kγ¯1kγ¯2kdγ1kdγ2k. (B.2)Applying γ1k = r2cos2φ, γ2k = r2sin2φ, and the definition of the Gamma function in(B.2) yieldsEγ1kγ2k{Ωijk (s)}=1∑u1=0k ̸=1· · ·1∑uN=0k ̸=N(−1)N∑i=1,i ̸=kui2Γ(i+ j + 2)γ¯1kγ¯2k(Ik + IIk + IIIk) , (B.3)173Appendix B.whereIk + IIk + IIIk =π/2∫0(sin2φ)2i+j+ 12 (cos2φ)2j+i+12(scos2φsin2φ + cos2φγ¯t+ sin2φγ¯2k)i+j+2dφ, (B.4)with γ¯t ! 1γ¯2k +N∑i=1,i ̸=kuiγ¯2i. Next, we spit the integral in (B.4) into the three parts Ik,IIk, and IIIk corresponding to the three integration intervals [0, ϵ), [ϵ,π/2 − ϵ), and[π/2− ϵ,π/2], respectively. For ϵ → 0 we can show thatIk =Γ(1− i)Γ(2i+ j + 2)γ¯1−it2(2i+ j + 1)Γ(i+ j + 2)s2i+j+1+ o(γ¯−(i−1)+t)(B.5)IIIk =Γ(1− j)Γ(2j + i+ 2)γ¯1−j2k2(2j + i+ 1)Γ(i+ j + 2)s2j+i+1+ o(γ¯−(j−1)+2k). (B.6)Furthermore, for γ¯1k, γ¯2k → ∞, IIk can be neglected due to its independence of γ¯1kand γ¯2k. ΦPS∆ks (s) in (2.15) can be obtained by evaluating Eγ1kγ2k{Ωijk (s)}for the fourdifferent cases (i = 0, j = 0), (i = 0, j ̸= 0), (i ̸= 0, j = 0), and (i ̸= 0, j ̸= 0) basedon (B.3), (B.5), and (B.6), and applying this result in (A.1) and (A.4), cf. AppendixA.174Appendix CSome research works that are not directly related to this dissertation but have beenpublished/submitted/under preparation during my time as a Ph.D. student at UBCare as follows:• I. Ahmed, R. Schober, and R. K. Mallik, “Asymptotic Performance of Lp–NormMIMO Detection,” in Proc. of IEEE VTC, September 2010, Ottawa, Canada.• I. Ahmed, A. Nasri, R. Schober, and R. K. Mallik, “Asymptotic Performance ofGeneralized Selection Combining in Generic Noise and Fading,” IEEE Trans.on Commun., vol. 60, pp. 916-922, May 2012.• I. Ahmed, A. Ikhlef, D. W. K. Ng, and R. Schober, “Optimal Resource Alloca-tion for Energy Harvesting Two-way Relay Systems with Channel Uncertainty,”in Proc. of IEEE GlobalSIP, December 2013, Austin, Texas, USA.• I. Ahmed, A. Ikhlef, D. W. K. Ng, and R. Schober, “Power Allocation for aHybrid Energy Harvesting Relay System with Imperfect Channel and EnergyState Information,” in Proc. of IEEE WCNC, 2014, Istanbul, Turkey.• I. Ahmed, D. W. K. Ng, A. Ikhlef, R. Schober, V. Wong, and L. Lampe, “Re-source Allocation and Wireless Energy Transfer for a Two-Way Relay Systemwith Channel and Energy State Uncertainty,” to be submitted.• I. Ahmed, A. Nezampour, R. Schober, and R. K. Mallik, “Asymptotic Perfor-mance Analysis of Lp–Norm MIMO Detection,” to be submitted.175Appendix C.• I. Ahmed, D. W. K. Ng, R. Schober, and L. Lampe, “Joint Relay Selection andPower Allocation for Energy Harvesting Relay System with Wireless EnergyTransfer,” under preparation.176

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