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Interference mitigation and alignment for interference-limited communication systems Rahman, Md. Jahidur 2017

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Interference Mitigation and Alignment for Interference-LimitedCommunication SystemsbyMd. Jahidur RahmanM. E. Sc., The University of Western Ontario, 2010B. Sc. Engg., Khulna 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)December 2017c© Md. Jahidur Rahman, 2017AbstractWith limited availability of the communication spectrum and ever-increasing de-mands for high-data-rate services, it is natural to reuse the same time-frequency re-source to the greatest degree possible. Depending on the nature of transmission andreception of the users, this leads to different instances of interference, e.g., inter-userinterference in an interference network and self-interference in a Full-Duplex (FD)transmission. With a goal to mitigate such interference, in this thesis we investigateemerging interference-limited communication systems, such as FD, Device-to-Device(D2D), and Power Line Communication (PLC). To this end, we propose advancedsolutions, namely self-interference mitigation and Interference Alignment (IA).With an objective to reduce the power consumption, we study transceiver designfor FD multi-cell Multi-Input Multi-Output (MIMO) systems with guaranteed Qual-ity of Service (QoS). Considering realistic self-interference models and robustnessagainst Channel State Information (CSI) uncertainty, our numerical results revealtransmission scenarios and design parameters for which replacing half-duplex withFD systems is beneficial in terms of power minimization. If the system is not power-constrained, however, a natural objective is to optimize the total throughput givena power budget. Nonetheless, throughput maximization underserves the users thatexperience poor channels, which leads to QoS unfairness. Therefore, we propose afair transceiver design for FD multi-cell MIMO systems, which can be implementedin a distributed manner. We further extend our design to enforce robustness againstiiAbstractCSI uncertainty. As a second contribution within this design theme, the concept ofrobust fair transceiver design is also extended for D2D communications, where un-like the self-interference in FD transmission, the users suffer from strong inter-userinterference.Recognizing that simultaneous multiple connections in PLC contribute to (inter-user) interference-limited communication, we introduce IA techniques for PLC net-works, for which the results confirm a significant sum-rate improvement. To overcomethe implementation burden of CSI availability for IA techniques, we then study BlindInterference Alignment (BIA) for PLC X-network, and show that the characteristicsof the PLC channel thwart simple implementation of this technique via impedancemodulation. We therefore resort to a transmission scheme with multiple receivingports, which can achieve the maximum multiplexing gain for this network.iiiLay SummaryThe transmission and reception nature of the communication users may lead to dif-ferent interference scenarios, such as inter-user interference in an interference net-work and self-interference in a Full-Duplex (FD) transmission. As such, the systemperformance becomes interference-limited. In this thesis, we investigate emerginginterference-limited communication systems, such as FD, Device-to-Device (D2D),and Power Line Communication (PLC), where we exploit advanced solutions, namelyself-interference mitigation and Interference Alignment (IA).The general objective of this research is to devise robust, power-efficient, andfair resource allocation schemes for interference-limited communication systems. Tothis end, we consider design challenges, such as Channel State Information (CSI)uncertainty, power-efficiency, and Quality of Service (QoS) fairness, which are prac-tically relevant to these communication systems. The contributions of this thesiswill advance the next generation communication technologies to meet a variety ofcommunication needs and services.ivPrefaceThe material presented in this thesis is based on research performed by myself underthe supervision of Prof. Lutz Lampe in the Department of Electrical and ComputerEngineering at the University of British Columbia, Vancouver, Canada.A co-author of published contributions, Dr. Ali Cagatay Cirik, introduced me tothe topic of full-duplex communication, assisted me towards a problem formulationin a publication related to Chapter 3, and also provided feedback on the manuscriptsrelated to full-duplexing in Chapters 2 and 3.The contributions made by another co-author, Dr. Hamidreza EbrahimzadehSaffar, which studies diversity in power line communication reported in a publicationrelated to Chapter 4 is not part of this thesis.Below is a list of publications related to the work presented in this thesis.Publications Related to Chapter 2• Md. Jahidur Rahman, Ali Cagatay Cirik, and Lutz Lampe, “Power-EfficientTransceiver Design for Full-Duplex MIMO Multi-Cell Systems with CSI Uncer-tainty,” IEEE Access, vol. 5, pp. 22689-22703, 2017.vPrefacePublications Related to Chapter 3• Ali Cagatay Cirik*, Md. Jahidur Rahman*, and Lutz Lampe, “Robust FairnessTransceiver Design for a Full-Duplex MIMO Multi-Cell System,” in press, IEEETransactions on Communications, Nov. 2017. (*equal contribution, listed ac-cording to the authors’ last name)• Md. Jahidur Rahman and Lutz Lampe, “Robust Transceiver Optimization forUnderlay Device-to-Device Communications,” in Proceedings of IEEE Inter-national Conference on Communications (ICC), Track: Signal Processing forCommunications, pp. 7695–7700, Sept. 2015.Publications Related to Chapter 4• Md. Jahidur Rahman and Lutz Lampe, “Interference Alignment in Power LineCommunications,” in press, Invited contribution in Encyclopedia of WirelessNetworks (Section: Interference Characterization and Mitigation), S. Shen, X.Lin, and K. Zhang, Eds. Germany: Springer, 2017.• Md. Jahidur Rahman and Lutz Lampe, “Rate Improvement in MIMO PowerLine Communications Through Interference Alignment,” Submitted, 2017.• Md. Jahidur Rahman and Lutz Lampe, “Interference Alignment for MIMOPower Line Communications,” in Proceedings of IEEE International Symposiumon Power Line Communications and Its Applications (ISPLC), pp. 71–76, Apr.2015.• Lutz Lampe, Md. Jahidur Rahman, and Hamidreza Ebrahimzadeh Saffar,“Characteristics of Power Line Networks: Diversity and Interference Align-viPrefacement,” in Proceedings of IEEE International Symposium on Power Line Com-munications and Its Applications (ISPLC), pp. 1–6, Apr. 2017.(Best PaperAward)Additional publications from Ph.D. research (not included in this thesis) are pro-vided in Appendix A.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixNotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxivDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Interference and Noise . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Different Interference Mechanisms in Communication Systems . . . . 21.3 Emerging Interference-Limited Communication Systems . . . . . . . 31.3.1 Full-Duplex Transmission . . . . . . . . . . . . . . . . . . . . 3viiiTable of Contents1.3.2 Device-to-Device Communication . . . . . . . . . . . . . . . . 41.3.3 Power Line Communication . . . . . . . . . . . . . . . . . . . 51.4 Solutions to Mitigate Interference . . . . . . . . . . . . . . . . . . . . 71.4.1 Self-Interference Mitigation . . . . . . . . . . . . . . . . . . . 71.4.2 Interference Alignment . . . . . . . . . . . . . . . . . . . . . 91.5 Design Challenges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.5.1 Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . 121.5.2 Power-Efficiency . . . . . . . . . . . . . . . . . . . . . . . . . 131.5.3 Fairness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.6 Major Contributions and Thesis Outline . . . . . . . . . . . . . . . . 141.6.1 Major Contributions of the Thesis . . . . . . . . . . . . . . . 151.6.2 Organization of the Thesis . . . . . . . . . . . . . . . . . . . 182 Power Minimization in Full-Duplex Communication Systems . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Signal Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.3 Received Signals . . . . . . . . . . . . . . . . . . . . . . . . . 262.2.4 Decoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2.5 Uplink and Downlink SINRs . . . . . . . . . . . . . . . . . . 272.3 Sum-Power Minimization with Perfect CSI . . . . . . . . . . . . . . . 292.3.1 Approximated Problem . . . . . . . . . . . . . . . . . . . . . 302.3.2 Solving the Approximated Problem . . . . . . . . . . . . . . . 322.3.3 Convergence Analysis . . . . . . . . . . . . . . . . . . . . . . 342.3.4 Complexity Analysis . . . . . . . . . . . . . . . . . . . . . . . 35ixTable of Contents2.3.5 Run-Time Analysis . . . . . . . . . . . . . . . . . . . . . . . 352.4 Sum-Power Minimization with CSI Uncertainty . . . . . . . . . . . . 362.4.1 Sum-Power Minimization with Stochastic CSI Uncertainty . . 362.4.2 Sum-Power Minimization with Bounded CSI Uncertainty . . 402.5 Numerical Results And Discussions . . . . . . . . . . . . . . . . . . . 442.5.1 Perfect CSI Results . . . . . . . . . . . . . . . . . . . . . . . 482.5.2 Imperfect CSI Results . . . . . . . . . . . . . . . . . . . . . . 512.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 Fairness Considerations in Full-Duplex and Device-to-Device Com-munication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 Fairness in FD Communications . . . . . . . . . . . . . . . . . . . . 583.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 593.2.2 Fairness Design with Perfect CSI . . . . . . . . . . . . . . . . 613.2.2.1 Problem Formulation . . . . . . . . . . . . . . . . . 613.2.2.2 Harmonic-Sum Minimization . . . . . . . . . . . . . 623.2.2.3 Complexity Analysis . . . . . . . . . . . . . . . . . . 673.2.3 Fairness Design with Imperfect CSI . . . . . . . . . . . . . . 683.2.3.1 Receive Filter Design . . . . . . . . . . . . . . . . . 703.2.3.2 Precoder Design . . . . . . . . . . . . . . . . . . . . 713.2.4 Numerical Results and Discussions . . . . . . . . . . . . . . . 743.2.4.1 Perfect CSI Results . . . . . . . . . . . . . . . . . . 743.2.4.2 Imperfect CSI Results . . . . . . . . . . . . . . . . . 793.3 Fairness in D2D Communications . . . . . . . . . . . . . . . . . . . . 833.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 85xTable of Contents3.3.1.1 Precoding at the D2D Transmitter . . . . . . . . . . 863.3.1.2 Channel Uncertainty . . . . . . . . . . . . . . . . . 863.3.1.3 Interference Suppression at the D2D Receiver . . . . 873.3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . 883.3.2.1 Overview of the Optimization Problem . . . . . . . 893.3.3 Receive Filter Design . . . . . . . . . . . . . . . . . . . . . . 913.3.4 Precoder Design . . . . . . . . . . . . . . . . . . . . . . . . . 923.3.5 Numerical Results and Discussions . . . . . . . . . . . . . . . 953.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984 Interference Alignment for Power Line Communications . . . . . 1004.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1004.2 IA for MIMO PLC Interference Networks . . . . . . . . . . . . . . . 1034.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 1034.2.2 Precoding at the PLC Transmitter . . . . . . . . . . . . . . . 1054.2.3 Interference Suppression at the PLC Receiver . . . . . . . . . 1054.2.4 IA Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . 1074.2.4.1 Min-IL based IA . . . . . . . . . . . . . . . . . . . . 1074.2.4.2 Feasibility of IA . . . . . . . . . . . . . . . . . . . . 1094.2.4.3 Max-SINR based IA . . . . . . . . . . . . . . . . . . 1104.2.5 Numerical Results with AWGN . . . . . . . . . . . . . . . . . 1104.2.5.1 Comparison with Orthogonal Transmission . . . . . 1144.2.5.2 Comparison with Wireless Communication . . . . . 1154.2.6 Numerical Results with Practical Measured Noise . . . . . . . 1174.3 BIA for PLC X-Networks . . . . . . . . . . . . . . . . . . . . . . . . 1234.3.1 Blind Interference Alignment . . . . . . . . . . . . . . . . . . 123xiTable of Contents4.3.2 Feasibility of the BIA Through Impedance Modulation . . . . 1254.3.3 Achievability of the BIA for PLC X-Networks . . . . . . . . . 1304.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1325 Summary and Directions for Future Works . . . . . . . . . . . . . . 1345.1 Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . 1345.2 Directions for Future Works . . . . . . . . . . . . . . . . . . . . . . . 1365.2.1 Decentralized Algorithms for FD Communication Systems . . 1365.2.2 Robust Multi-cell D2D Communications . . . . . . . . . . . . 1375.2.3 Feedback Reduction for IA in PLC Networks . . . . . . . . . 1385.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . 138Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140A Appendix for Additional Publications . . . . . . . . . . . . . . . . . 153A.1 Additional Publications from Ph.D. Research (not included in thisthesis) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153B Appendix for Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . 154B.1 Replacement of the Unit Norm Constraint in (3.16) . . . . . . . . . . 154B.2 Proof of Convergence of Algorithm 1 . . . . . . . . . . . . . . . . . . 155C Appendix for Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . 156C.1 ABCD Matrix Representation . . . . . . . . . . . . . . . . . . . . . 156C.2 Property of the PLC Keyhole Channel . . . . . . . . . . . . . . . . . 157xiiList of Tables2.1 Comparison of the Run-Time (in seconds) for Algorithm 1 . . . . . . 352.2 Simulation Parameters and Corresponding Settings for an FD MIMOMulti-Cell System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 463.1 Comparison of Computational Complexity . . . . . . . . . . . . . . . 684.1 Rate gain of the matched over the mismatched IA design for differentnoise data sets and PLC MIMO configurations. . . . . . . . . . . . . 121xiiiList of Figures1.1 Illustrations of interference mechanisms in an interference network andfull-duplex transmission. . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 An illustration of the PLC interference network, where each transmit-ter (Tx 1 and Tx 2) communicate to its paired-receiver (Rx 1 and Rx2, respectively). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.3 An illustration of the PLC X-network, where each transmitter (Tx 1and Tx 2) has data to transmit to each of the two receivers (Rx 1 andRx 2) in the network. Solid and dashed arrows denote desired signaland interference, respectively. . . . . . . . . . . . . . . . . . . . . . . 71.4 An illustration of the self-interference mitigation through interferencecancellations at different stages of the FD communication system. . . 81.5 An illustration of the residual self-interference due to limited ADCdynamic range in a typical small-cell full-duplex communication. . . . 91.6 An illustration of the interference alignment for a 3-user MIMO wire-less interference network over two spatial signaling dimensions. . . . . 101.7 An illustration of the impact of imperfect CSI on the performance of IA. 121.8 A flow diagram of research problems addressed in this thesis (denotedby rounded rectangles) for interference-limited communication systemsconsidering practical design challenges (denoted by rectangles) throughthe interference alignment and self-interference mitigation techniques. 14xivList of Figures2.1 An illustration of a power-constrained FD MIMO multi-cell systemwith solar-powered BSs. Dashed arrows denote the self-interferenceand dash-dotted arrows denote the interference between different nodes.Solid lines denote the desired signals in the uplink and downlink trans-missions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2 Convergence of the objective function in (2.21) for perfect CSI designwith κ = β = −120 dB and a QoS constraint of 2.63 Mb/s. . . . . . . 472.3 Comparison of the average power required by FD and HD systems forthe perfect CSI design with a QoS constraint of 2.63 Mb/s and varyingtransceiver distortions. . . . . . . . . . . . . . . . . . . . . . . . . . . 482.4 Comparison of the average power required by FD and HD systemsfor the perfect CSI design with κ = β = −120 dB and varying QoSconstraints. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.5 Convergence of the objective function in (2.39) for a design with stochas-tic CSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63 Mb/s,η = 0.3, and λ = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . 502.6 Convergence of the objective function in (2.56) for a design with norm-bounded CSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63Mb/s, and s = 0.01. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512.7 Comparison of the average power required by robust FD and HD sys-tems with stochastic CSI uncertainty, κ = β = −120 dB, a QoS con-straint of 2.63 Mb/s, and η = 0.3. . . . . . . . . . . . . . . . . . . . . 522.8 Comparison of the average power required by robust FD and HD sys-tems with norm-bounded uncertainty, κ = β = −120 dB and a QoSconstraint of 2.63 Mb/s. . . . . . . . . . . . . . . . . . . . . . . . . . 53xvList of Figures2.9 Comparison of the average power required by robust and non-robustFD systems with stochastic CSI uncertainty, κ = β = −120 dB, a QoSconstraint of 2.63 Mb/s, and η = 0.3. . . . . . . . . . . . . . . . . . . 543.1 Full-duplex MIMO multi-cell system. Dashed arrows denote the self-interference and the dash-dotted arrows denote the interference be-tween different nodes. . . . . . . . . . . . . . . . . . . . . . . . . . . . 603.2 Convergence of the objective function in (3.18) for perfect CSI designwith κ = β = −120 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 753.3 Total sum-rates achieved for FD and HD setups with perfect CSI. . . 763.4 Total sum-rates achieved for FD and HD setups with perfect CSI andvarying distance between BSs. The parameters representing transceiverdistortion are chosen as κ = β = −110 dB and κ = β = −120 dB . . . 773.5 Comparison of CDFs of individual user rate among the proposed, Max-Min and sum-rate maximizing WMMSE designs with κ = β = −80dB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 783.6 Convergence of minimum SINR i.e., improved associated rate in thepresence of imperfect CSI with s = 0.02 and κ = β = −120 dB. . . . 793.7 Total sum-rates achieved for FD and HD setups with varying CSIuncertainties and κ = β = −120 dB. . . . . . . . . . . . . . . . . . . 813.8 Total sum-rates achieved for robust and non-robust FD setups withvarying CSI uncertainties and transceiver distortions. . . . . . . . . . 823.9 An illustration of underlay D2D communications in a cellular network.Solid and dashed lines indicate desired signals and interference, respec-tively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xviList of Figures3.10 Worst-case stream data rate with SNR for D2D users in a cellularinterference network, C = K = 2, NT = 2, NR = NpuT = NpuR = 3,dc1 = dc2 = dpu = 1,  = 0.15. . . . . . . . . . . . . . . . . . . . . . . . 963.11 Worst-case stream data rate with CSI error () for D2D users in acellular interference network, C = K = 2, NT = 2, NR = NpuT =NpuR = 3, dc1 = dc2 = dpu = 1. . . . . . . . . . . . . . . . . . . . . . . . 974.1 Illustration of a 3-conductor cables MIMO PLC interference networkwith 3 Tx-Rx pairs (setup-I). . . . . . . . . . . . . . . . . . . . . . . 1044.2 Illustration of a 3-conductor cables MIMO PLC interference networkwith 3 Tx-Rx pairs, a variation of setup-I (setup-II). . . . . . . . . . 1054.3 Percentage of feasible IA for K = 3 with 3-conductor cables i.e., Nt =Nr = 2 as shown in Fig. 4.1 (setup-I) and Fig. 4.2 (setup-II). . . . . . 1114.4 Percentage of feasible IA for K = 3 with 4-conductor cables i.e., Nt =Nr = 3. Tx-Rx pairs are positioned as in Fig. 4.1 and and Fig. 4.2 forsetups III and IV, respectively. . . . . . . . . . . . . . . . . . . . . . . 1124.5 Comparison of average sum-rates of IA and orthogonal transmissionsfor MIMO PLC network in Fig. 4.1. . . . . . . . . . . . . . . . . . . . 1144.6 Comparison of average sum-rates with IA algorithms for the MIMOPLC network in Fig. 4.1 and a wireless interference network withequivalent link qualities. . . . . . . . . . . . . . . . . . . . . . . . . . 1164.7 Subcarrier rate gain for a matched IA design with correlated noiseover a mismatched IA design (i.e., that ignores spatial noise correlationduring IA filter computation) for OL 3 of the dataset 1 across P-E/N-Eports in the network scenario 2. . . . . . . . . . . . . . . . . . . . . . 119xviiList of Figures4.8 Subcarrier rate gain for a matched IA design with correlated noise overa mismatched IA design across P-N/N-E ports with the same networkconfiguration as in Fig. 4.7. . . . . . . . . . . . . . . . . . . . . . . . 1204.9 Comparison of sum-rate performances among different IA designs andorthogonal transmissions for OL 3 across P-E/N-E ports of the dataset 1in the network scenario 2. . . . . . . . . . . . . . . . . . . . . . . . . 1224.10 A simple transmission scenario of an X-channel setting (cf. Fig. 1.3). 1244.11 PLC network with three communication nodes. . . . . . . . . . . . . 1264.12 An illustration of the transmission scheme for the achievability of blindIA for the PLC X-network using multiple receiving ports. . . . . . . . 1304.13 A comparison of multiplexing gains for the proposed BIA and an or-thogonal transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . 1315.1 An illustration of an underlay D2D communication in a cellular net-work with resulting interference from cellular users and neighboringmacrocells. Solid and dashed lines indicate desired signals and inter-ference, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . 138C.1 ABCD-matrix representation of a two-port network. . . . . . . . . . . 156xviiiList of Abbreviations3GPP Third Generation Partnership Project5G Fifth GenerationADC Analog-to-Digital ConverterAWGN Additive White Gaussian NoiseBIA Blind Interference AlignmentBS Base StationCDF Cumulative Distribution FunctionCPU Central Processing UnitCSI Channel State InformationD2D Device-to-DeviceDAC Digital-to-Analog ConverterDCP Difference of Convex Function ProgrammingDL DownlinkDoF Degrees of FreedomDR Dynamic RangeETSI European Telecommunications Standards InstituteFD Full-DuplexFDD Frequency-Division DuplexFDMA Frequency Division Multiple accessHD Half-DuplexxixList of AbbreviationsIA Interference AlignmentIEEE Institute of Electrical and Electronics EngineersLNA Low Noise AmplifierLO Local OscillatorLOS Line Of SightLTE Long Term EvolutionLTE-A LTE-AdvancedMax-SINR Maximum Signal-to-Interference-Plus-Noise RatioMIMO Multi-Input Multi-OutputMin-IL Minimum Interference LeakageMISO Multi-Input Single-OutputMMSE Minimum Mean-Squared ErrorMS Mobile StationMSE Mean Squared ErrorMSR Maximum Sum-RateMU-MIMO Multi-User MIMOMUI Multi-User InterferenceNLOS Non Line Of SightOFDM Orthogonal Frequency Division MultiplexingPA Power AmplifierPLC Power Line CommunicationPSD Power Spectral DensityPU Primary UserQoS Quality of ServiceRAM Random Access MemoryxxList of AbbreviationsRCP Remote Centralized ProcessorRF Radio FrequencySCA Successive Convex ApproximationSDP Semidefinite ProgrammingSDR Semidefinite RelaxationSIMO Single-Input Multi-OutputSINR Signal-to-Interference-Plus-Noise RatioSISO Single-Input Single-OutputSNR Signal-to-Noise RatioSOCP Second Order Cone ProgrammingSU Secondary UserTDD Time-Division DuplexTDMA Time Division Multiple AccessUL UplinkWMMSE Weighted Minimum Mean-Squared ErrorWSR Weighted Sum-RatexxiNotationsA Matrixa VectorIN N by N identity matrix[A]nn n-th row and n-th column of matrix A0N×M N by M zero matrix| · | Absolute value of a complex number(·)T Transpose(·)H Hermitian transposevec(·) VectorizationCN (µ, σ2) A complex Gaussian distribution with mean µ and variance σ2CN×M The set of complex matrices with a dimension of N by ME(·) Statistical expectation operator⊗ Kronecker product‖ · ‖2 l2-norm‖ · ‖F Frobenius normdiag (A) A diagonal matrix with the same diagonal elements as ATr(·) Trace of a matrixrank(·) Rank of a matrixd·e Upper bound⊥ Statistical independencexxiiNotations◦ Hadamard product or element-wise multiplication|S| The cardinality of set SxxiiiAcknowledgmentsFirst and foremost, I would like to thank Allah, the Almighty, for giving me thestrength, knowledge, and ability to undertake the research in this thesis.I would like to express my sincere gratitude to my advisor Prof. Lutz Lampefor the support during the course of my Ph.D. His knowledge, work ethic and per-severance are truly inspiring and his guidance has helped me in every stage of myresearch.I thank the members of my examining committee at UBC, Professors Vijay K.Bhargava, Vincent Wong, Md. Jahangir Hossain (UBC Okanagan), Karthik Pattabi-raman, and Ryozo Nagamune (Mechanical Engineering), for their time and insightfulcomments. Special thanks go to Professor Rodney Vaughan at the Simon FraserUniversity for serving as the external examiner, and also for physically participatingin the final exam.I am grateful to Natural Science and Engineering Research Council of Canadaand the University of British Columbia for supporting this research partially throughan Alexander Graham Bell Canada Graduate Scholarship and a Four Year DoctoralFellowship, respectively.My sincere thank goes to Dr. Ali Cagatay Cirik, who was a postdoctoral fellow inour research group, for his time and many fruitful discussions. I also thank my fellowcolleagues, to name a few, Naveen, Ayman, Gautham, Hao, in the CommunicationTheory Lab for numerous technical conversations. Finally, I would like to thank myxxivAcknowledgmentsmother and my wife, who have been the biggest support during the course of thisjourney.xxvDedicationTo my father (who left us on January 11, 2003), my mother and my wife.xxviChapter 1Introduction1.1 Interference and NoiseInterference and noise are two key performance-limiting factors in many communica-tions systems. Unlike noise, which can be generated internally as well as externallyof the communication systems, different instances of interference arise depending onthe nature of transmission and reception of the participating users over a given com-munication medium. One important difference between noise and interference is thefact that interference usually suffers from the same propagation disturbances as theuseful signal while the noise level is typically constant, at least over a short timeinterval [1].The nature of noise and interference have deep implications on the performanceof communication systems. In a noise-limited communication system, such as spacecommunications, performance degradation due to underlying noise cannot be avoided,since generally we do not have control over the noise sources. In contrast, for aninterference-limited communication system, as in cellular communications, the verynature of interference facilitate the mitigation and/or cancellation of the same phe-nomenon via intelligent manipulation over time, space, frequency, or combinationof these domains. In this thesis, we focus on intelligent manipulations of differentinstances of interference so as to improve the performance of interference-limitedcommunication systems.1Chapter 1. IntroductionRx 2Tx 2Rx 1Tx 1a) Pair-wise transmissions in an interference network impaired by the inter-user interferenceNode 1Node 2b) A full-duplex transmission impaired by the self-interferenceFigure 1.1: Illustrations of interference mechanisms in an interference network andfull-duplex transmission.1.2 Different Interference Mechanisms inCommunication SystemsContinuing the discussions from the previous section, here we provide two specificinstances of interference that arise due to the nature of transmission and receptionover a common communication medium. A commonly studied scenario is an interfer-ence network, where transmitters and receivers communicate on a pair basis over acommon communication medium. Naturally, the transmission of any given pair willinterfere the reception of the other pairs, hence they experience the inter-user inter-ference. As such, the system capacity of these networks is limited by this inter-userinterference [2].Interference may also arise due to simultaneous transmission and reception by2Chapter 1. Introductiona user over the same frequency band, e.g., in Full-Duplex (FD) transmission. Thisform of interference is known as the self-interference [3]. More specifically, the self-interference refers to the interference that a transmitting terminal causes to the re-ception of the desired signal by that terminal. Therefore, the system capacity islimited by the self-interference for this communications scenario. These are illus-trated in Figs. 1.1 (a) and (b), which highlight detrimental interference for the datatransmission in interference networks and FD communications, respectively.1.3 Emerging Interference-LimitedCommunication SystemsWith ever-increasing demands for high-data-rate services and limited spectrum, itis natural to aim at transmitting (and/or receiving) at the same time-frequency re-source. As presented above, this leads to communication scenarios where users ex-perience different forms of interference in the network. In what follows, we discussthree emerging communication techniques in the context of both wireless and wire-line communication systems, where users communicate in the same time-frequencyresource leading to inference-limited system performance.1.3.1 Full-Duplex TransmissionCommunication in cellular networks occurs in multipoint-to-point, i.e., users to BaseStation (BS), and point-to-multipoint, i.e., BS to users, which are commonly knownas uplink and downlink transmissions. This is usually achieved via orthogonalizingthe channel, i.e., the BS communicates with users in separate time-frequency re-sources. Obviously, higher rates could be achieved if the BS and users are able to3Chapter 1. Introductioncommunicate (i.e., transmit and receive) simultaneously in the same frequency band.This can be achieved through FD communication [4]. Conventional wireless com-munication systems that operate in Half-Duplex (HD) transmission mode–commonlyknown as Time-Division Duplex (TDD) or Frequency-Division Duplex (FDD), em-ploy two orthogonal channels to transmit and receive. Therefore, these transmissiontechniques cannot achieve the maximal spectral efficiency. FD transceivers have thecapability of transmission and reception at the same time over the same frequencyband [5, 6], and thus hold the promise to double the link capacity or increase thespectral efficiency due to more flexible access control and networking [7, 8].While FD communication has been known for many decades, only recently ithas attracted renewed attention. One key reason behind this that the traditional ap-proaches to increase spectral efficiency through advanced techniques, such as modula-tion, coding, multiplexing are thought to be exhausted [5]. Therefore, much researchefforts have been put into non-traditional approaches, such as FD communication.However, the major stumbling block for the exploitation of the FD ability of com-munication devices is the strong self-interference. This is particularly pronounced inconventional cellular systems with large cell sizes, which require a higher transmitpower (i.e., causing a stronger self-interference) to compensate for the higher cell-edgepath losses. Therefore, the exploitation of the FD ability requires sophisticated in-terference mitigation techniques to reap the potential benefit of doubling the spectralefficiency.1.3.2 Device-to-Device CommunicationDevice-to-Device (D2D) communication enables direct communication between twoor more users in cellular networks, with less intervention from the BS or the core4Chapter 1. Introductionnetwork. In a traditional cellular network, all communications between two usersmust go through the BS. This protocol suits the scenario in which users are notusually close enough to establish a direct communication between them. However,mobile users these days demand high-data-rate services, such as video sharing. Theseservices can be facilitated with low latency if they could be in range for D2D com-munications. Therefore, D2D communications in such scenarios can increase thespectral and energy efficiencies of the cellular network by offloading traffic from thecore network [9,10]. Furthermore, supported by the trend of proximity-based servicesfor commercial purposes and public safety needs, recently the D2D communicationhas gained considerable attention from the network operators and the research com-munity [11]. In D2D communications, the interference may result from simultaneoustransmissions of other D2D users within the same cell, and possibly from transmis-sions of D2D users in neighboring cells (e.g., in a multi-cell D2D communicationscenario), in addition to the interference originating from conventional cellular trans-missions [12]. It is apparent that the system performance of the D2D communicationis limited by the underlying inter-user interference in the network, which motivatesthe study of interference mitigation techniques for this new communication paradigm.1.3.3 Power Line CommunicationData transmission over power lines is an attractive solution for providing commu-nication services, even in hard-to-reach areas through the reuse of existing powergrid infrastructures [13, 14]. The technique permits the seamless implementation ofa communication system without the need for an additional wiring infrastructure.A closer look at the signal transmission over power lines would reveal that thedata communication is essentially an unintended broadcast transmission, since the5Chapter 1. IntroductionPower Line NetworkTx 2Tx 1Rx 1Rx 2Figure 1.2: An illustration of the PLC interference network, where each transmitter(Tx 1 and Tx 2) communicate to its paired-receiver (Rx 1 and Rx 2, respectively).communication signals can travel through the electricity grid. As it is illustrated inFig. 1.2, simultaneous pair-wise data communication (i.e., at the same time-frequencyresource) in a PLC network resembles data transmission and interference scenariosin wireless interference networks, similar to the illustration in Fig. 1.1(a). Anothercommunication protocol is illustrated in Fig. 1.3, where each transmit port may havesignals intended for each of the receive ports in the network. This is alike the datacommunication over the wireless X-network [15]. Due to the underlying inter-userinterference in the PLC network, the system performance (e.g., sum-rate) becomesinterference-limited. These inter-user interference has not been dealt with advancedtechniques for PLC networks, such as Interference Alignment (IA) [16–18], which weinvestigate in this thesis.6Chapter 1. IntroductionTx 1Tx 2Power Line NetworkRx 1Rx 2Figure 1.3: An illustration of the PLC X-network, where each transmitter (Tx 1and Tx 2) has data to transmit to each of the two receivers (Rx 1 and Rx 2) in thenetwork. Solid and dashed arrows denote desired signal and interference, respectively.1.4 Solutions to Mitigate InterferenceWhile in FD communication the interfering signal may be perfectly known from itsown transmission, this is not the case in an interference network, i.e, the other users’signal is not known. Despite knowing the interfering signal, the self-interference inFD communication cannot be canceled completely, mainly due to the channel esti-mation error and limited Dynamic Range (DR) of the associated components in thetransceiver. Therefore, in the case of FD, the (residual) self-interference is the perfor-mance limiting factor, while in the interference network it is the inter-user interferencethat limits the system performance. In the sequel, we discuss two solutions to dealwith the self-interference and the inter-user interference in communication networks.1.4.1 Self-Interference MitigationThe deployment of the FD communication is facilitated by the recent emergence ofshort-range communication systems (e.g., small-cell systems), which incurs a lowercell-edge path loss compared to traditional cellular systems. Therefore, the mitigationor reduction of the detrimental self-interference becomes much more manageable [5].7Chapter 1. IntroductionModulator DACDuplexer-ADCDemod -Digital CancellationAnalog CancellationLNAPALOIsolationTx dataRx dataDirect pathReflected pathSelf-InterferenceTx signalRx signalLOFigure 1.4: An illustration of the self-interference mitigation through interferencecancellations at different stages of the FD communication system.As illustrated in Fig. 1.4, the general idea of the self-interference mitigation is to sup-press (e.g., through isolation in the propagation domain) and subtract the dominantpart of the self-interference signal in the Radio Frequency (RF) analog domain, sothat the remaining signal can be processed for further interference reduction in thebaseband, i.e., digital domain [19–22].As shown in Fig. 1.5, for a small-cell BS which transmits at 24 dBm, and assuminga typical −100 dBm receiver noise floor with 15 dB isolation between the BS transmitand receive antennas, the BS’s self-interference will be 109 dB above the noise floor.Obviously, systems with larger cell sizes will suffer a higher self-interference leveldue to requirement of the increased transmit power. Considering a typical 14-bitAnalog-to-Digital Converter (ADC) which results in a DR of 54 dB [5], the systemsuffers from a residual self-interference of −45 dBm even with a perfect digital domaincancellation, which is 55 dB above the noise floor. This strong self-interference is amajor drawback for the exploitation of the FD ability of communication devices.In recent years, specialized self-interference cancellation techniques along with8Chapter 1. IntroductionBS transmit power (24 dBm) Rx noise floor (-100 dBm)Self-interference (9 dBm)Residual self-interference (-45 dBm)Digital domain cancellation of 54 dB(limited by ADC dynamic range)55 dB above receiver floorFigure 1.5: An illustration of the residual self-interference due to limited ADC dy-namic range in a typical small-cell full-duplex communication.promising results from experimental research have demonstrated adequate levels ofisolation between transmitting and receiving signals [23–25]. Nevertheless, such tech-niques are far from perfect owing to imperfections of associated radio components,such as non-ideal Power Amplifier (PA) and Low Noise Amplifier (LNA), Local Os-cillator (LO), ADC and Digital-to-Analog Converter (DAC) [26]. Further challengesarise due to inaccurate Channel State Information (CSI) in the interference paths,which makes complete cancellation of the self-interference unattainable [27].1.4.2 Interference AlignmentGenerally, there are three common ways to deal with the inter-user interference incommunication networks. First, if the interference is strong, the interfering signal canbe decoded, and hence the desired signal can be separated [28]. Second, when theinterference is weak, the interfering signal can be treated as an additional noise [29].On the other hand, when the strength of the interference is comparable to the desiredsignal, the conventional wisdom is to avoid the interference by orthogonalizationof the communication medium either in time or frequency domain. An extensive9Chapter 1. IntroductionTx 1Tx 2Tx 3Rx 1Rx 2Rx 3Figure 1.6: An illustration of the interference alignment for a 3-user MIMO wirelessinterference network over two spatial signaling dimensions.use of this technique is found in time or frequency division multiple access basedcommunication systems [30,31]. Such an orthogonal medium access makes sure thatmultiple users can access the channel without creating interference to each other.The access mechanism can be considered as a cake-cutting measure, where eachuser gets only a portion of the communication resources depending on the numberof participating users. For example, if there are K participating user-pairs in thenetwork, they can only receive 1Kportion of the total communication resources. Onthe other hand, if they communicate at the same time-frequency resource as in aninterference network, there will be K − 1 interfering signals. In general, K signalingdimensions will be needed to recover the desired signal. The fundamental question is,how many signaling dimensions are really needed to resolve the desired signal fromthe remaining K − 1 interfering signals? Alternatively, is it possible to recover thedesired signal within a reduced signaling dimensions?10Chapter 1. IntroductionIn order to circumvent the above under-determined problem, the notion of IAwas introduced as an approach to maximize interference-free signal space for thedesired user [16–18]. If the interference signals could be consolidated into a smallersubspace so that they do not span the entire signal space at the receiver and atthe same time, the desired signal could avoid falling into the interference space, itmay be possible to recover the desired signal interference-free. This is done via thetechnique of IA. In essence, IA allows interfering users to transmit simultaneouslyby consolidating the space spanned by the interference at receivers within a smallnumber of signaling dimensions, while keeping the desired signals separable frominterference so that they can be projected into the null space of interference, andrecovered interference-free. The alignment of the interference can be obtained intime domain (e.g., via symbol extension), in frequency domain (e.g., over multiplesubchannels), or in spatial domain (e.g., via multiple antennas) [32]. For example,Fig. 1.6 illustrates the IA for a 3-user Multi-Input Multi-Output (MIMO) wirelessinterference network over two spatial signaling dimensions. The idea is considered tobe a breakthrough since now the interference networks, such as D2D and power linecommunications considered in this thesis, can achieve a much higher rate employingthe technique of IA.1.5 Design ChallengesIn this section, we highlight relevant design challenges for emerging interference-limited communication systems that we investigate in this thesis. To this end, wefocus on design aspects that are pertinent to practical implementations of these com-munications systems.11Chapter 1. IntroductionIA with perfect CSI IA with imperfect CSIFigure 1.7: An illustration of the impact of imperfect CSI on the performance of IA.1.5.1 Imperfect CSIThe availability of CSI greatly simplifies the design of communication systems. Asfor IA, it is essential to precode the transmitted signals such that the interferingsignals are aligned at the corresponding receivers, while the desired signal can bedecoded interference-free. This necessitates the availability of the CSI, as such, theperformance of IA is limited by quality of the CSI. As it is illustrated on the leftside of Fig. 1.7, the IA design with perfect CSI would facilitate exact alignment oftwo interfering signals in one dimension, and therefore, the desired signal can berecovered inference-free on the second dimension (i.e., no projection of the interferingsignals onto the signal subspace). On the other hand, imperfect CSI would lead totransmit and receive filters not being designed properly. As shown on the right sideof Fig. 1.7, this will cause the interference signals projection onto the signal subspace,which results in a degradation of the Signal-to-Interference-Plus-Noise Ratio (SINR)[33]. Similarly, the performance gains offered by FD communication depend on theavailability of the CSI. In particular, the availability of the CSI will determine thequality of the self-interference cancellation, and hence the overall performance gainover an HD system [34].Depending on whether the quality of CSI is limited by estimation or quantization,12Chapter 1. Introductionone of two models of CSI uncertainty is usually adopted to develop robust transmis-sion techniques [35]. A Bayesian model assumes that the statistics of the CSI errordue to estimation noise are known. Different from this, bounded models considerthat the error belongs to a predefined bounded uncertainty region, with no furtherassumption on the statistical properties.1.5.2 Power-EfficiencyTransmit power is an important design parameter to evaluate the overall systemperformance, especially in power-constrained applications [36]. This has also beenemphasized for interference-limited communication systems (e.g., in cellular com-munications), since reducing transmit power has obvious benefits, such as reducedinterference for wireless networks and lower cost for wireless devices [37, 38]. Theseare even more pronounced for FD wireless devices as they consume more power dueto additional components and processing involved with the self-interference mitiga-tion [27, 39]. With increasing emphasis on incorporating energy awareness in FifthGeneration (5G) communication systems [40], in which the FD communication is alsoconsidered as a key technology [41], in this thesis we study power-efficient resourceallocation techniques for FD communications.1.5.3 FairnessFor cellular communications, the total throughput maximization is optimal if thegoal is to maximize combined sum-rates of the uplink and downlink transmissions.One giveaway of this approach is that users that experience good channels will beallocated all the resources [42–45]. For example, in an FD cellular system, as the self-interference power increases, it starts overwhelming the desired signals coming from13Chapter 1. IntroductionInterference-limited CommunicationsInterference AlignmentSelf-Interference MitigationSolutionSolutionRobust D2D Fairness DesignPower Line CommunicationsRobust FD Fairness DesignRobust FD Power-Efficient DesignFairnessCSI UncertaintyPower-EfficiencyFigure 1.8: A flow diagram of research problems addressed in this thesis (denotedby rounded rectangles) for interference-limited communication systems consideringpractical design challenges (denoted by rectangles) through the interference alignmentand self-interference mitigation techniques.the uplink users, which reduces the achievable rate in the uplink channel. Therefore,reducing the transmit power in the uplink channel and concentrating on the downlinkchannel is more beneficial to optimize the system sum-rate. In this case, uplink usersare not served, i.e., all the resources are devoted for the downlink transmission,which results in unfairness. The situation is compounded in multi-cell FD systems,where additional interference sources exist, which will degrade the performance ofthe users, especially for the ones at the cell-edge. Therefore, it is important to ensuresatisfactory performance among all the users in the network, which motivates us toconsider fairness design in emerging FD and D2D communications.1.6 Major Contributions and Thesis OutlineThe general objective of this research is to devise robust, power-efficient, and fairresource allocation schemes for interference-limited communication systems. As de-14Chapter 1. Introductionpicted in Fig. 1.8, we are concerned with interference-limited communications sys-tems, specifically FD, D2D, and power line communications. To this end, we considerdesign challenges, such as CSI uncertainty, power-efficiency, and fairness considera-tions, which are practically relevant to these communication systems. As for solu-tions, we concentrate on two sophisticated techniques, i.e., self-interference mitigationand interference alignment. Where possible, we strive to devise low-complexity solu-tions for optimizing the system performance.1.6.1 Major Contributions of the ThesisOur contributions in this thesis are summarized as follows.• Power-Efficient Transceiver Design for FD Communication Systems.Motivated by reducing power consumption for FD wireless devices, we explorepower-efficient transmit and receiver beamforming designs for an FD MIMOmulti-cell system. In particular, we assume that BSs operating in FD trans-mission mode serve multiple FD mobile users at the same time over the samefrequency band. To guarantee a certain Quality of Service (QoS), we enforcemaintenance of a minimum SINR for each user. Concerning these design con-straints together with realistic FD self-interference models, we investigate thetransmit and receive beamforming designs that minimize the joint transmissionpower of BSs and users. The non-convex precoder design problem is posed asa Difference of Convex Function Programming (DCP), which can be efficientlysolved via Successive Convex Approximation (SCA). Owing to the FD trans-mission both at BSs and users, our design approach considers a comprehensivesystem model that includes I) the self-interference at each FD BS and FD user,II) the interference among adjacent BSs, i.e., inter-BS interference, and III) the15Chapter 1. Introductioninterference among all the mobile users in all cells, where other cellular com-munication scenarios can be derived as a subset of this system model. In orderto account for practical design aspects, we consider modeling of the CSI uncer-tainties to propose both statistical and worst-case designs. Our contributionstoward this research theme has been published in [46].• Fairness Considerations in FD and D2D Communication Systems.Performance fairness is an important design consideration in interference-limitedcommunication systems. We realize that the existing works on FD cellular sys-tems focus on the maximization of overall throughput, which results in unfair-ness between uplink and downlink channels depending on the self-interferencepower and inter-user interference levels. Therefore, we consider the transmitand receive beamforming designs that maximize the SINRs in the uplink anddownlink channels to introduce fairness among the users in an FD MIMO multi-cell system. To this end, we formulate the fairness problem as a harmonic-summaximization approach, in an attempt to draw a balance between total userthroughput and fairness, and the solution allows for distributed computationsof the beamformers. In order to address practical design aspects, we considerthe transceiver design that enforces robustness against imperfect CSI while pro-viding fair performance among FD users. The robust fairness design problemis handled via a low-complexity iterative algorithm based on alternating opti-mization and the Semidefinite Relaxation (SDR) technique. Our contributiontoward this design theme has been accepted for publication in [47].As a second contribution within this design theme, we study robust fairnesstransceiver design for another emerging interference-limited communication sys-tem, such as D2D communications, where the main challenge lies in dealing with16Chapter 1. Introductionthe inter-user interference. To this end, we assume the D2D users employ IA tomitigate the inter-user interference. We consider that the multi-antenna D2Dusers in the cellular network operate as secondary users via underlay cognitivetransmission, where its transmission is constrained by the interference powerconstraint that is enforced by the primary network (i.e., macrocell). Similarto the previous design problem for FD communications, this design problem istackled via an alternating optimization and the SDR technique. We claim thatthis study is the first to investigate a robust transceiver design in the contextof underlay D2D communications. Our research work in this regard has beenpublished in [48].• Interference Alignment for Power Line Communications. With com-plete lack of advanced interference mitigation techniques in the context of PLC,we focus on investigating IA techniques for this interference-limited communi-cation system, where the system performance is limited by the underlying inter-user interference. We claim that this thesis is the first to consider such advancedinterference mitigation techniques in the context of PLC networks. To beginwith, we consider different MIMO PLC interference networks to study the feasi-bility of IA and evaluate the performance in terms of system sum-rate for thesesetups in the presence of Additive White Gaussian Noise (AWGN). However,unlike most communications, noise in MIMO PLC is often found to be spatiallycorrelated. Concerning this, we also study the performance of the IA for MIMOPLC networks in the presence of practical measured noise, and specifically in-vestigate the impact of spatial noise correlation. Another contribution towardthis research direction is to study the Blind Interference Alignment (BIA) forPLC X-networks, which works without requiring CSI at the transmitter. In17Chapter 1. Introductionparticular, we investigate network configurations, where seemingly simple re-alization of the BIA through impedance modulation is not achievable for thePLC X-network. This is followed by proposal of a transmission scheme thatenables the implementation of the BIA for the PLC X-networks. Our contribu-tion toward this research direction has been submitted for publication in [49]and accepted/published in [50–52].1.6.2 Organization of the ThesisThe thesis is structured around the list of contributions in the previous section andis organized as follows.In Chapter 1, we provide essential backgrounds on interference-limited communi-cation systems and also introduce a few emerging instances of such systems that areinvestigated in this thesis. We then discuss design challenges in the specific contextof these interference-limited communication systems to motivate our research. Thisis followed by introduction of relevant solutions so as to mitigate interference in thesecommunication systems.Chapter 2 focuses on the design of a power-efficient resource allocation techniquefor FD communication systems. To guarantee a certain QoS and considering realisticFD self-interference models, we investigate joint sum-power minimization of BSs andusers. Noting that the resulting optimization problem is NP-hard, we then divide thisoptimization problem into separate receive and transmit beamforming design steps,which can be solved iteratively. Practical design aspects are taken into account byway of stochastic and bounded uncertainties. Based on simulation parameters fromthe Third Generation Partnership Project (3GPP) standardization body, numericalresults suggest that FD systems generally outperform HD ones under a wide range18Chapter 1. Introductionof QoS constraints and transceiver distortions.Chapter 3 considers the fairness problem in both FD and D2D communications.We begin by considering an FD MIMO multi-cell system to provide fairness in theform of SINR maximization in the uplink and downlink channels. We then proposean alternating optimization algorithm to tackle the problem. Thereafter, we considerthe worst-case transceiver design under bounded CSI uncertainty. Numerical resultsverify the improved fairness performance when compared with other algorithms andconfirm the robustness against the CSI uncertainty.In the second part of this chapter, we study robust transceiver optimizationfor D2D communications that aims for SINR fairness among D2D users, similar tothe previous robust design problem for FD communications. Numerical simulationsdemonstrate the performance of the proposed transceiver compared to the benchmarkcase of an IA system without primary network/macrocell (non-cognitive). We ob-serve that at low SNR and high CSI error with relaxed interference power constraint,the worst-case stream data rate of the D2D users are close to that of the users innon-cognitive IA system but performance degrades significantly with stringent inter-ference power constraint.In Chapter 4, we study IA techniques for different PLC networks. AssumingAWGN, we first consider different MIMO PLC interference networks and exploit iter-ative Minimum Interference Leakage (Min-IL) and Maximum Signal-to-Interference-Plus-Noise Ratio (Max-SINR) algorithms to study the IA feasibility and sum-rateperformances. Our results show that the sum-rate of the PLC networks can be sig-nificantly improved through the exploitation of Max-SINR algorithm. In particular,it is found that at high Signal-to-Noise Ratio (SNR) the performance gain in terms ofsum-rate with IA over orthogonal transmission techniques is around 30% for a 3-user19Chapter 1. Introduction2 × 2 MIMO PLC network. We then study the performance of the IA for MIMOPLC networks with practical measured noise, and also investigate the impact of spa-tial noise correlation. To this end, we choose Max-SINR as a candidate algorithmwhich is known to offer the best performance in terms of sum-rate, but susceptibleto noise statistics. Unlike channel correlation in MIMO communications, our resultssuggest that noise correlation actually helps to improve the system performance. Wethen focus on the BIA to study its feasibility and propose a transmission scheme thatfacilitates the implementation of this technique for PLC X-networks. The results sug-gest that the maximum multiplexing gain can be achieved exploiting the proposedtransmission scheme.Finally, the conclusions and potential avenues for further research related to topicsstudied in this thesis are presented in Chapter 5. To this end, we stress on decentral-ized algorithm designs for FD communication systems. We then emphasize studyingmore inclusive communication scenarios for the D2D communication. Finally, in thischapter we discuss relevance and potential benefits of CSI feedback reduction for IAin PLC networks.20Chapter 2Power Minimization in Full-DuplexCommunication Systems2.1 IntroductionWhile the potential benefits of FD systems are easy to foresee, the implementation ofthese systems poses significant challenges. For example, since an FD system relies onsimultaneous transmission and reception, one major drawback for the exploitationof FD ability is the strong self-interference at the front-end of the receiver causedby the signal leakage from the transmit antennas to its receive antennas. Althoughsophisticated self-interference cancellation techniques may achieve certain levels ofisolation between transmitting and receiving signals [20,21], they are far from perfectdue to imperfections of radio components, such as amplifier non-linearity and oscilla-tor phase noise [26]. Furthermore, due to inherent inaccuracy in the CSI estimationof the associated interference paths, the complete cancellation of the self-interferencecannot be achieved [34]. Therefore, the system optimization in the context of FDcommunications under the residual self-interference were studied in [53–56], and thereferences therein.Owing to increased spectral efficiency and recent advances in hardware design, FDcommunication has been investigated for point-to-point MIMO systems in [34,55–59]and for single-cell systems in [60–63]. However, these studies assumed HD users and21Chapter 2. Power Minimization in Full-Duplex Communication Systemsdid not account for the signal distortion caused by non-ideal amplifiers, oscillators,ADCs, and DACs. FD communication has also been studied for multi-cell MIMOsystems [64–67]. The works [64–66] focused on the optimization of system sum-ratewhile [67] considered user selection with power control. The minimization of transmitpower has not been a design criterion, e.g., in [64–67].As we discussed in Chapter 1, the reduction of transmit power of the wirelessdevices is beneficial since it also translates to the reduction of associated interferencein wireless networks and lower cost of production for these devices [37, 38]. In thespecific context of FD communications, the mitigation of the self-interference requiresadditional components and processing power, hence FD devices consume relativelymore power [27, 39]. As there is an increasing emphasis on the design approachthat incorporates energy awareness [11], the design of power-efficient FD transceivershas recently been studied considering the sum-power minimization in the context ofinterference channels [58, 59] and relay networks [34]. In this chapter, we considerthe sum-power minimization design approach for FD MIMO multi-cell systems. Inparticular, we investigate the problem of minimizing the joint transmit power at BSsand users while meeting QoS requirements in the form of SINR. As illustrated inFig. 2.1, because of the FD transmission both at BSs and users, our design approachconsiders I) the self-interference at each FD BS and FD user, II) the interferenceamong adjacent BSs, i.e., inter-BS interference, and III) the interference among all themobile users in all cells. In addition, we consider transmitter and receiver distortionscaused by non-ideal amplifiers, oscillators, ADCs, and DACs in our study. Althoughthe resulting optimization problem is non-convex and NP-hard, we can represent itas a DCP, which can be solved via SCA [68–71]. While the global optimality cannotbe guaranteed, the objective value in SCA converges monotonically as it is improved22Chapter 2. Power Minimization in Full-Duplex Communication Systemswith each iteration. Within this context, in the first part of this chapter, our goalis to understand under what conditions replacing HD systems with FD ones may bebeneficial for the power minimization.This will be expanded in the second part of the chapter to consider the CSIuncertainty, similar to previous studies on FD communications in the context ofcognitive radio [72,73], physical layer security [74–76], point-to-point MIMO commu-nication [77], and single-cell multi-user system [78]. To this end, we present robustdesigns for the power-minimization FD operation considering both stochastic andbounded CSI uncertainties. Numerical results confirm that the proposed FD designsachieve power savings compared to an HD setup and a non-robust design, under awide range of QoS constraints and signal distortions at the transceiver.The rest of this chapter is organized as follows. The system model is presented inSection 2.2. The power-minimization FD transceiver design assuming perfect CSI isderived in Section 2.3. This is extended to the case of imperfect CSI in Section 2.4.Numerical results and discussions are provided in Section 2.5. Finally, we summarizeour findings in Section 2.6.2.2 System ModelIn this section, we discuss the system model for an FD MIMO multi-cell system.2.2.1 Signal ModelWe consider an FD cellular communication scenario having K cells, where cell khas one BS k, k = 1, . . . , K, as illustrated in Fig. 2.1. We assume that each BS,k is equipped with Mk transmit and Nk receive antennas, and serves Ik users inits cell. We denote ik to be the ith user in cell k, equipped with Mik transmit23Chapter 2. Power Minimization in Full-Duplex Communication SystemsBS k BS j......Mobile user ik Mobile user ljSolar cellBBkjHULjklHDLki jHUUk ji lHULkkiHDLki kHSIkHSIkiHFigure 2.1: An illustration of a power-constrained FD MIMO multi-cell system withsolar-powered BSs. Dashed arrows denote the self-interference and dash-dotted ar-rows denote the interference between different nodes. Solid lines denote the desiredsignals in the uplink and downlink transmissions.and Nik receive antennas. We define the set of BSs as K = {1, . . . , K} and usersas I = {ik | k ∈ {1, 2, . . . , K}, i ∈ {1, 2, . . . , Ik}}. We view HDLikj ∈ CNik×Mj as thechannel between BS j and user ik in the downlink transmission, HULklj∈ CNk×Mlj asthe channel between BS k and user lj in the uplink transmission, HUUiklj∈ CNik×Mljas the interference channel from the user lj to the user ik, HBBkj ∈ CNk×Mj as theinterference channel from the BS j to the BS k, HSIk ∈ CNk×Mk and HSIik ∈ CNik×Mikas the self-interference channel from the transmit antennas to the receive antennasfor the BS k and user ik, respectively.The availability of the CSI is crucial in maximizing the gains offered by the FDtransmission. We note that there are three types of CSI involved in the system24Chapter 2. Power Minimization in Full-Duplex Communication Systemsdesign, i.e., I) BSs to users (HDLikj ) or users to BSs (HULklj) channels, II) BSs to BSs(HBBkj ) channels, and III) users to users (HUUiklj) channels. Considering, for examplea 3GPP Long Term Evolution (LTE) system, each BS broadcasts the cell-specificreference signal, including its cell identity [79]. Therefore, BSs to users channelscan be estimated from the received reference signal at each user. Users then reportthe CSI via control and/or shared channels to the BSs [79]. Hence, type I channelscan be estimated, e.g., assuming channel reciprocity. The same cell-specific referencesignal can be used at other BSs to estimate the type II channels [67]. The type IIIchannels are difficult to obtain as there is no direct signaling between users. However,the channel estimation between users can be facilitated via neighbor discovery ateach user through the use of sounding reference signal in 3GPP LTE system [80].Similar mechanisms to estimate channels between users have been proposed for D2Dcommunications [81].2.2.2 PrecodingEach user ik in the uplink and downlink channels sends symbols, sULik∈ CdULik ×1 andsDLik ∈ CdDLik×1, respectively, where dULik and dDLikare the number of data streams in therespective direction. We assume that the symbols are independent and identically dis-tributed (i.i.d.) with unit power, i.e., E[sULik(sULik)H]= IdULikand E[sDLik(sDLik)H]=IdDLik, for the uplink and downlink transmissions, respectively. Together with thetransmit beamforming matrices VULik =[vULik,1, . . . ,vULik,dULik]∈ CMik×dULik and VDLik =[vDLik,1, . . . ,vDLik,dDLik]∈ CMk×dDLik in the uplink and downlink channels, the transmit-ted signals of the user ik and BS k can be written as, xULik= VULik sULikand xDLk =∑Iki=1 VDLiksDLik , respectively. For convenience, we collect all beamforming vectors,vXik,m, ik ∈ I, m ∈ M, X ∈ {UL,DL} in the stacked vector v, where M denotes25Chapter 2. Power Minimization in Full-Duplex Communication Systemsthe set of the data streams.2.2.3 Received SignalsAs mentioned earlier, we take into account the limited DR at FD nodes [26], whichhas also been applied in [53, 54, 56, 57, 65]. Essentially, non-ideal components, suchas amplifiers, oscillators, ADCs and DACs contribute to limited DR. To model thelimited receiver DR, an additive white Gaussian receiver distortion with a varianceequal to β times the power of the undistorted received signal is injected at eachreceive antenna [82]. Similarly, an additive white Gaussian transmitter noise witha variance equal to κ times the power of the intended transmit signal is injected ateach transmit antenna to model the limited transmitter DR [83]. Considering thelimited DR, the signals received by the BS k and that received by the user ik, can berespectively written asyULk =K∑j=1Ij∑l=1HULkljxULlj+ HSIk(xDLk + cDLk)+K∑j=1,j 6=kHBBkj xDLj + eULk + nULk , (2.1)yDLik =K∑j=1HDLikjxDLj + HSIik(xULik + cULik)+∑(l,j)6=(i,k)HUUikljxULlj+ eDLik + nDLik, (2.2)where nULk ∈ CNk×1 and nDLik ∈ CNik×1 denote the AWGN vector with zero mean andunit covariance matrix at the BS k and user ik, respectively. In (2.2), cULik∈ CMik×1 isthe signal distortion at the transmitter antennas of user ik, which models the effect oflimited DR to account for transmitter chain inaccuracies. Unlike the thermal noisecomponents, the covariance matrix of cULik depends on the power of the transmitantenna. It is modeled as cULik ∼ CN(0, κ diag(VULik(VULik)H)), cULik ⊥ xULik [26].In (2.2), eDLik ∈ CNik×1 is the additive receiver distortion at the receiver an-tennas of user ik. Similar to the transmitter side, it models the effect of lim-26Chapter 2. Power Minimization in Full-Duplex Communication Systemsited DR to account for the receiver chain inaccuracies, and modeled as eDLik ∼CN (0, βdiag (ΦDLik )) , eDLik ⊥ ΘDLik [26], where ΦDLik = Cov{ΘDLik } and ΘDLik is theundistorted received vector at the user ik, i.e., ΘDLik= yDLik − eDLik . In (2.1), cDLkand eULk are the transmitter and receiver distortion at the BS k, respectively, whichare modeled similarly. Furthermore, we note that as the FD nodes know their owntransmit signals and the self-interference channel, the terms HSIk xDLk and HSIikxULik canbe cancelled from the received signal yULk at the kth BS and yULikat the ikth user,respectively [26]. We denote the self-interference free received signals as y˜ULk and y˜ULik,respectively.2.2.4 DecodingWe make use of linear decoders, UULik =[uULik,1, . . . ,uULik,dULik]∈ CNk×dULik , and UDLik =[uDLik,1, . . . ,uDLik,dDLik]∈ CNik×dDLik , to process the received signals at the BS k and user ik,respectively. Then the estimates of data streams of user ik in the uplink and downlinkchannels are obtained as, sˆULik =(UULik)Hy˜ULk and sˆDLik=(UDLik)Hy˜DLik , respectively.Again, for convenience, we collect all decoding vectors, uXik,m, ik ∈ I, m ∈ M, X ∈{UL,DL} in the stacked vector u.2.2.5 Uplink and Downlink SINRsWith estimated data streams, the SINR values of the m-th stream associated with theuser ik in the uplink and downlink channels can be expressed as in (2.3) and (2.4),respectively, shown at the bottom of the following page. Here, ΣULik (v)(ΣDLik (v))denote the covariance matrix of the aggregate interference-plus-noise for the user ikin the uplink (downlink) channel, and can be approximated, under β  1 and κ 1,as in (2.5) and (2.6), respectively, given on the following page as well. We note that27Chapter 2. Power Minimization in Full-Duplex Communication SystemsΣULik (v) and ΣDLik(v) depend on non-local parameters, such as channel matrices andpre-coding matrices at other links. However, they can be determined locally providedthat there is a sufficient coherence time window within which all channel and pre-coding matrices do not change [84–86].We note that our system model considers the most general communication sce-nario, where the FD BSs communicate with the FD users in a multi-cell environment.The other communication scenarios, e.g., I) FD BSs and HD users in a multi-cell, II)FD BS and FD users in a single-cell, III) FD BS and HD users in a single cell, andIV) HD BSs and HD users in single and multi-cell environments, can be recovered asspecial cases.γULik,m(v,uULik,m)= ∣∣∣(uULik,m)H HULkikvULik,m∣∣∣2(uULik,m)HΣULik (v) +dULik∑n=1HULkikvULik,n(vULik,n)H (HULkik)H︸ ︷︷ ︸QULik(v)uULik,m −∣∣∣(uULik,m)H HULkikvULik,m∣∣∣2,(2.3)γDLik,m(v,uDLik,m)= ∣∣∣(uDLik,m)H HDLikkvDLik,m∣∣∣2(uDLik,m)HΣDLik (v) +dDLik∑n=1HDLikkvDLik,n(vDLik,n)H (HDLikk)H︸ ︷︷ ︸QDLik(v)uDLik,m −∣∣∣(uDLik,m)H HDLikkvDLik,m∣∣∣2.(2.4)28Chapter 2. Power Minimization in Full-Duplex Communication Systems2.3 Sum-Power Minimization with Perfect CSIIn this section, our goal is to study the joint sum-power minimization of the FDBSs and users assuming that perfect CSI is available at the FD transmitters whileenforcing an SINR constraint for each data stream. The corresponding optimizationΣULik (v) =∑(l,j) 6=(i,k)dULlj∑n=1HULkljvULlj ,n(vULlj ,n)H (HULklj)H+K∑j=1,j 6=kIj∑l=1dDLlj∑n=1HBBkj vDLlj ,n(vDLlj ,n)H(HBBkj)H+ INk +Ik∑l=1dDLlk∑n=1κHSIk diag(vDLlk,n(vDLlk,n)H) (HSIk)H+ βK∑j=1Ij∑l=1dULlj∑n=1diag(HULkljvULlj ,n(vULlj ,n)H (HULklj)H)+ βK∑j=1,j 6=kIj∑l=1dDLlj∑n=1diag(HBBkj vDLlj ,n(vDLlj ,n)H (HBBkj)H)+ βIk∑l=1dDLlk∑n=1diag(HSIk vDLlk,n(vDLlk,n)H (HSIk)H), (2.5)ΣDLik (v) =∑(l,j) 6=(i,k)dDLlj∑n=1HDLikjvDLlj ,n(vDLlj ,n)H (HDLikj)H+∑(l,j)6=(i,k)dULlj∑n=1HUUikljvULlj ,n(vULlj ,n)H(HUUiklj)H+ INik + κHSIikdULik∑n=1diag(vULik,n(vULik,n)H) (HSIik)H+ βK∑j=1Ij∑l=1dDLlj∑n=1diag(HDLikjvDLlj ,n(vDLlj ,n)H (HDLikj)H)+ β∑(l,j)6=(i,k)dULlj∑n=1diag(HUUikljvULlj ,n(vULlj ,n)H (HUUiklj)H)+ βdULik∑n=1diag(HSIikvULik,n(vULik,n)H (HSIik)H). (2.6)29Chapter 2. Power Minimization in Full-Duplex Communication Systemsproblem can be formulated asminv,uK∑k=1Ik∑i=1dULik∑m=1(vULik,m)HvULik,m +K∑k=1Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m (2.7)s.t. γXik,m(v,uXik,m) ≥ γXth, ik ∈ I, m ∈M, X ∈ {UL,DL}, (2.8)where γXth, X ∈ {UL,DL} is the QoS constraint.The above problem is not jointly convex in v and u, and it is known to be NP-hard [87]. In what follows, we show that an approximation of the original problemcan be solved efficiently.2.3.1 Approximated ProblemThe original problem in (2.7)-(2.8) can be solved (suboptimally) in an iterative man-ner where the sum-power converges. To this end, the problem is divided into sep-arate transmit and receive beamforming designs which can be solved alternatively.We note that with fixed transmit beamformers, the linear minimum mean-squarederror (MMSE) receiver is optimal in the sense that it maximizes the per-streamSINR [88, 89]. In this regard, the optimal MMSE receiver that maximizes the per-stream SINR can be expressed asuXik,m=(HXikVXik(VXik)H (HXik)H+ΣXik(v))−1HXikvXik,m. (2.9)When MMSE receivers are applied to recover the data streams, it leads to thefollowing well-known relationship [89](ΓXik,m)−1 = 1 + γXik,m, (2.10)30Chapter 2. Power Minimization in Full-Duplex Communication Systemswhere ΓXik,m and γXik,mare the Mean Squared Error (MSE) and SINR, respectively,and it is assumed that an MMSE receiver is applied to recover each stream of theuser, ik ∈ I, m ∈ M, X ∈ {UL,DL}. The corresponding MSE function can becomputed asΓXik,m(v,uXik,m)= |(uXik,m)HHXikvXik,m−1|2+(uXik,m)H{ΣXik(v)+dXik∑n=1,n6=mHXikvXik,n(vXik,n)H (HXik)H}uXik,m. (2.11)Assuming that the uplink and downlink transmissions employ certain QoS con-straints, γXth, the precoder design problem can be written asmin .v PBS+UE (2.12)s.t. (ΓXik,m)−1 (v,uXik,m) ≥ γXth + 1, (2.13)where PBS+UE is given byPBS+UE =∑Kk=1∑Iki=1∑dULikm=1(vULik,m)HvULik,m +∑Kk=1∑Iki=1∑dDLikm=1(vDLik,m)HvDLik,m.The above non-convex problem can be formulated as a DCP with the introductionof an upper bounding constraint for each MSE term, i.e., ΓXik,m ≤ f(θXik,m), wheref(θXik,m) is a monotonic log-concave function and θXik,mis an auxiliary variable, ik ∈I, m ∈M, X ∈ {UL,DL}. To that end, we assume f(θXik,m) = c−θXik,m , where c > 1.For later results, we choose c = 2 as suggested in [70].With this approximation, the problem in (2.12)-(2.13) can be reformulated asmin .v,θ PBS+UE (2.14)31Chapter 2. Power Minimization in Full-Duplex Communication Systemss.t. ΓXik,m(v,uXik,m) ≤ cθXik,m , (2.15)θXik,m ≥loge(γXth + 1)loge c, (2.16)Note that the above problem in still non-convex due to MSE upper boundingconstraint in (2.15).2.3.2 Solving the Approximated ProblemThe non-convex part of the above MSE constraint, f(θXik,m) = c−θXik,m , can be linearlyapproximated at a given point, θXτik,m, by the first-order Taylor seriesf(θXik,m, θXτik,m) = f(θXτik,m) + (θXik,m− θXτik,m)f ′(θXτik,m)= −aXτik,mθXik,m + bXτik,m, (2.17)where τ is the iteration index, aXτik,m and bXτik,mare the coefficients of the linear approx-imation, f ′(θXτik,m) is the first-order partial derivative with respect to θXik,m. Takingthe first-order partial derivative, we havef(θXik,m, θXτik,m) = c−θXτik,m + (θXik,m − θXτik,m)(−c−θXτik,m loge c)= −θXik,mc−θXτik,m loge c+ c−θXτik,m(1 + θXτik,m loge c). (2.18)Therefore, we arrive ataXτik,m = c−θXτik,m loge c, (2.19)bXτik,m = c−θXτik,m(1 + θXτik,m loge c). (2.20)At iteration τ with fixed θXτik,m, ik ∈ I, m ∈M, X ∈ {UL,DL}, the optimization32Chapter 2. Power Minimization in Full-Duplex Communication Systemsproblem in (2.14)-(2.16) can be written asmin .v,θ PBS+UE (2.21)s.t. ΓXik,m(v,uXik,m) ≤ −aXτik,mθXik,m + bXτik,m, (2.22)θXik,m ≥ logc(γXth + 1). (2.23)In the above optimization problem, all constraints are convex due to the linearapproximations. After solving the above optimization problem at each iteration, thenext point can be computed using an exact line search method i.e., θXτ+1ik,m= θXτik,m orbased on the equality of the MSE constraint, θXτ+1ik,m= − log2(ΓXτ+1ik,m ) [70].The problem is solved in a way that the sum-power converges by alternating be-tween the receive and transmit beamforming designs. The iterative transmit beam-formers are optimized by repeatedly computing the linear approximation and thensolving the above reformulated optimization problem in (2.21)-(2.23). The steps forsolving the optimization problem is summarized in Algorithm 1.Algorithm 1 Sum-Power Minimization with Perfect CSI.1: Initialize the transmit beamforming vectors vXik,m and θXτik,m, ik ∈ I, m ∈ M, X ∈{UL,DL}. Set τ = 0.2: repeat3: Calculate the MMSE receive beamforming vectors u from (2.9).4: repeat5: Calculate the linear approximation coefficients, aXτik,m and bXτik,m, ik ∈ I, m ∈M, X ∈ {UL,DL} from (2.19) and (2.20), respectively.6: Calculate v from (2.21)-(2.23).7: Update θXτik,m, ik ∈ I, m ∈M, X ∈ {UL,DL} using line search. Set τ = τ + 1.8: until Convergence (inner) of the objective function in (2.21) or a predefined numberof iterations is reached.9: until Convergence (outer) of the objective function in (2.21) or a predefined numberof iterations is reached.33Chapter 2. Power Minimization in Full-Duplex Communication Systems2.3.3 Convergence AnalysisThe Algorithm 1 is guaranteed to converge if it can be proved that the objective func-tion in (2.21) decreases monotonically at each optimization step and it is boundedbelow. We note that the original optimization problem in (2.7)-(2.8) and the ap-proximated optimization problem in (2.21)-(2.23) have the same objective function.Furthermore, the optimization variables in (2.21)-(2.23) satisfy the same QoS con-straint in (2.7)-(2.8). Therefore, it is sufficient to show that the sum-power convergesfollowing Algorithm 1 and the objective function is bounded below. It is apparentthat the sum-power is bounded below, i.e., PBS+UE > 0.We note the MMSE receive filter update at step 3 of Algorithm 1 minimizes theper-stream MSE, which means that less (or equal) transmit power will be needed tosatisfy that per-stream MSE. This leads to decreased required sum-power [88, 89].Furthermore, the SCA optimization related to the transmit beamformers convergesmonotonically due to the fact that the point of approximation is included in the ap-proximated convex problem via the update of θXτik,m, ik ∈ I, m ∈M, X ∈ {UL,DL},as presented in Section 2.3.2 [68,70,71]. Therefore, the objective function is guaran-teed to converge.We further note that we deal with a convex problem at each optimization step,which can be solved efficiently [90]. However, the above convergence proof onlyholds for the monotonically decreasing convergence to a limit point of the objectivefunction. Since the original problem in (2.7)-(2.8) is non-convex, generally the globaloptimality cannot be ensured.34Chapter 2. Power Minimization in Full-Duplex Communication SystemsTable 2.1: Comparison of the Run-Time (in seconds) for Algorithm 1Setting 1 Setting 2 Setting 316.63 30.91 39.952.3.4 Complexity AnalysisThe computational complexity is determined by the problem size, i.e, number andsize of the optimization variables and constraints. Assuming the same number oftransmit antennas (M), receive antennas (N) and same number of data streams (d)at each node, in this section, we analyze the computational complexity of the proposedalgorithm. Since the proposed algorithm relies on iterative update, we provide per-iteration complexity. We do so by omitting linear constraints since their impact onoverall complexity can be considered negligible.Given the number of users, |I| and assuming interior point method for convexquadratic programming [90], the total complexity of the each iteration of the opti-mization problem involves O(M3)+ |I| (MNd+M2d+N2d)+N (NM +Md+N2)calculations [84]. We note that calculations of some terms in the covariance matricescan be reused. For example, the second term in (2.5) can be reused to calculate thesixth term as it incurs the diagonalization of the same matrix.2.3.5 Run-Time AnalysisThe run-time of an algorithm depends on the problem size, the convergence accuracydesired from the optimization problem, and the machine on which the algorithm isrunning. In our case, we use a consumer grade computer machine with 1.6 GHzprocessor, 8 GB of Random Access Memory (RAM), and a 8-core Central ProcessingUnit (CPU). The algorithm was tested when the computer had some other activitiesrunning in the background. To this end, we consider three different antenna settings,35Chapter 2. Power Minimization in Full-Duplex Communication Systemssuch as {Mk = Nk = 4,Mik = Nik = 2, dXik = 1}, {Mk = Nk = 8,Mik = Nik =4, dXik = 2}, {Mk = Nk = 12,Mik = Nik = 6, dXik = 3} , ik ∈ I, X ∈ {UL,DL},termed as settings 1, 2, and 3, respectively. Table 2.1 presents the average run time(in seconds) with different antenna settings for a convergence accuracy of 10−5, whichis a reasonable time for the convergence.2.4 Sum-Power Minimization with CSIUncertaintyThe realization of the full potential for FD communications relies on the quality ofthe CSI available at the transceiver. The CSI can be obtained at each transmitter viachannel estimation through the pilot signals or it can be fed back to the transmitterusing quantized feedback signaling [35]. Nonetheless, due to inevitable estimationerror involved with the channel estimation process or limited capacity of the feed-back channel, the assumption of the perfect CSI availability is idealistic. In orderto account for this design challenge, in the following we study the sum-power mini-mization problem under the imperfect CSI scenario considering both stochastic andbounded uncertainties.2.4.1 Sum-Power Minimization with Stochastic CSIUncertaintyIn this section, we incorporate the stochastic CSI uncertainty into our design, wherethe uncertainty is usually modeled as a complex random matrix with normally dis-tributed elements, and the transmitter is assumed to know the distribution type andcorresponding parameters [91–94]. The statistical CSI uncertainty model is expressed36Chapter 2. Power Minimization in Full-Duplex Communication SystemsasH ∈ H ={H˜ + ∆ : ∆ ∼ CN (0, σδI)}, (2.24)where H˜, ∆, and σδ denote the estimated CSI, the channel error matrix, and thevariance of the CSI uncertainty, respectively. The estimated CSI and the channelerror matrix are assumed to be statistically independent.Applying the model (2.24) to the respective channels, HULklj , HDLikj, HUUiklj , HBBkj , HSIk ,and HSIik in the FD multi-cell system, we have the corresponding distributions for thechannel error matrices given by∆ULklj ∼ CN (0, σulδ I), (2.25)∆DLikj ∼ CN (0, σdlδ I), (2.26)∆UUiklj ∼ CN (0, σuuδ I), (2.27)∆BBkj ∼ CN (0, σbbδ I), (2.28)∆SIk ∼ CN (0, σsi,bδ I), (2.29)∆SIik ∼ CN (0, σsi,uδ I). (2.30)The above variance terms can be further modeled as [95]σδ = λρ−η. (2.31)The model captures the effect of the channel uncertainties pertaining to the es-timation accuracy. According to this model, the error variance for the associatedchannel depends on the nominal SNR, ρ, unless η = 0. The parameters λ > 0 andη ≥ 0 are meant to capture a variety of communication scenarios. For example,37Chapter 2. Power Minimization in Full-Duplex Communication Systemsthe case of perfect CSI can be obtained with λ = 0. Reciprocal channels and CSIfeedback can be captured with η = 1 and η = 0, respectively.Considering the CSI uncertainty, the optimization problem that we want to solvecan be written asmin .v,uK∑k=1Ik∑i=1dULik∑m=1(vULik,m)HvULik,m +K∑k=1Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m (2.32)s.t. γ˜Xik,m(v,uXik,m) ≥ γXth, ik ∈ I, m ∈M, X ∈ {UL,DL}, (2.33)where γ˜Xik,m(v,uXik,m), ik ∈ I, m ∈M, X ∈ {UL,DL} is the SINR that accounts forthe stochastic CSI uncertainty, and computed as in (2.35) and (2.36) for the uplinkand downlink transmissions, respectively. Under the assumption of the stochastic CSIuncertainty, it is intuitive that all nodes have access to H˜ and the statistics aboutthe CSI uncertainty, instead of H. With the above modeling of the stochastic CSIuncertainty, the interference plus noise covariance matrices in (2.5) and (2.6) canbe approximated as in (2.37) and (2.38), respectively. We obtain this approximationby omitting terms that involve multiplication of the CSI uncertainty associated witheach channel, since their products are negligibly small. Furthermore, Σ˜ULik(v) andΣ˜DLik(v) are obtained by directly replacing all instances of H by H˜ in (2.5) and (2.6)for the uplink and downlink transmissions, respectively.As in the previous section, at iteration τ and for a fixed θ˜Xτik,m, ik ∈ I, m ∈M, X ∈ {UL,DL}, the optimization problem in (2.32)-(2.33) can be cast as belowmin .v,θ P˜BS+UE (2.39)s.t. Γ˜Xik,m(v,uXik,m) ≤ −a˜Xτik,mθ˜Xik,m + b˜Xτik,m, (2.40)θ˜Xik,m ≥ logc(γXth + 1), (2.41)38Chapter 2. Power Minimization in Full-Duplex Communication Systemsγ˜ULik,m(v,uULik,m)= ∣∣∣(uULik,m)H H˜ULkikvULik,m∣∣∣2(uULik,m)H (Σ˜ULik,δ(v) +∑dULikn=1 H˜ULkikvULik,n(vULik,n)H (H˜ULkik)H)uULik,m −∣∣∣(uULik,m)H H˜ULkikvULik,m∣∣∣2 ,(2.35)γ˜DLik,m(v,uDLik,m)= ∣∣∣(uDLik,m)H H˜DLikkvDLik,m∣∣∣2(uDLik,m)H (Σ˜DLik,δ(v) +∑dDLikn=1 H˜DLikkvDLik,n(vDLik,n)H (H˜DLikk)H)uDLik,m −∣∣∣(uDLik,m)H H˜DLikkvDLik,m∣∣∣2 .(2.36)Σ˜ULik,δ(v) ≈ Σ˜ULik (v) +σulδK∑j=1Ij∑l=1dULlj∑n=1tr(vULlj ,n(vULlj ,n)H)+ σbbδK∑j=1Ij∑l=1dDLlj∑n=1tr(vDLlj ,n(vDLlj ,n)H)+ σsi,bδ κIk∑l=1dDLlk∑n=1tr(vDLlk,n(vDLlk,n)H)+ βσulδK∑j=1Ij∑l=1dULlj∑n=1tr(vULlj ,n(vULlj ,n)H)+ σbbδ βK∑j=1Ij∑l=1dDLlj∑n=1tr(vDLlj ,n(vDLlj ,n)H)+ σsi,bδ βIk∑l=1dDLlk∑n=1tr(vDLlk,n(vDLlk,n)H) INk ,(2.37)Σ˜DLik,δ(v) ≈ Σ˜DLik (v) +σdlδK∑j=1Ij∑l=1dDLlj∑n=1tr(vDLlj ,n(vDLlj ,n)H)+ σuuδK∑j=1Ij∑l=1dULlj∑n=1tr(vULlj ,n(vULlj ,n)H)+ σsi,uδ κdULik∑n=1tr(vULik,n(vULik,n)H)+ σdlδ βK∑j=1Ij∑l=1dDLlj∑n=1tr(vDLlj ,n(vDLlj ,n)H)+ σuuδ βK∑j=1Ij∑l=1dULlj∑n=1tr(vULlj ,n(vULlj ,n)H)+ σsi,uδ βdULik∑n=1tr(vULik,n(vULik,n)H) INik . (2.38)39Chapter 2. Power Minimization in Full-Duplex Communication Systemswhere parameters a˜Xτik,m and b˜Xτik,mare applicable for the design involving the stochasticCSI uncertainty, but follow the same construction as derived for the case of perfectCSI design as in (2.19) and (2.20), Γ˜Xik,m is the MSE with the stochastic CSI uncer-tainty, and P˜BS+UE is the total transmit power for the same, obtained as in (2.14).Alike the perfect CSI design, the above optimization problem is convex and can besolved iteratively following the similar steps as in Algorithm 1.2.4.2 Sum-Power Minimization with Bounded CSIUncertaintyNow we extend our design to deal with the bounded CSI uncertainty. To this end,we restrict the imperfect CSI within a norm-bounded deterministic (or worst-case)model, where the instantaneous CSI is assumed to be located in a known set ofpossible values [96–99]. The norm-bounded uncertainty model is expressed asH ∈ H ={H˜ + ∆ : ‖∆‖F ≤ }, (2.42)γˆULik,m(v,uULik,m)LB= ∣∣∣(uULik,m)H H˜ULkikvULik,m∣∣∣2⌈(uULik,m)HΣˆULik(v)uULik,m⌉+(uULik,m)H∑dULikn=1,n 6=m H˜ULkikvULik,n(vULik,n)H (H˜ULkik)HuULik,m,(2.45)γˆDLik,m(v,uDLik,m)LB= ∣∣∣(uDLik,m)H H˜DLikkvDLik,m∣∣∣2⌈(uDLik,m)HΣˆDLik(v) uDLik,m⌉+(uDLik,m)H∑dDLikn=1,n6=m H˜DLikkvDLik,n(vDLik,n)H (H˜DLikk)HuDLik,m,(2.46)40Chapter 2. Power Minimization in Full-Duplex Communication Systemswhere H˜, ∆, and  denote the nominal value of the CSI, the channel error matrix,and the uncertainty bound, respectively. The uncertainty sizes can be made relatedto the quality of the channels, where the radius of the uncertainty regions can be setto  = s‖H˜‖F , s ∈ [0, 1) [100]. Considering the CSI uncertainty, the optimizationproblem that is of interest is given as belowmin .v,uK∑k=1Ik∑i=1dULik∑m=1(vULik,m)HvULik,m +K∑k=1Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m (2.43)s.t. min .∆γXik,m(v,uXik,m) ≥ γXth,(uULik,m)HΣ˜ULik(v)uULik,m =∑(l,j)6=(i,k)dULlj∑n=1||(uULik,m)HH˜ULkljvULlj ,n||2F+K∑j=1,j 6=kIj∑l=1dDLlj∑n=1||(uULik,m)HH˜BBkj vDLlj ,n||2F+ ||uULik,m||2F + κIk∑l=1dDLlk∑n=1||(uULik,m)HH˜SIk diag(vDLlk,n(vDLlk,n)H) 12 ||2F + βK∑j=1Ij∑l=1dULlj∑n=1||(H˜ULkljvULlj ,n)Hdiag(uULik,m(uULik,m)H) 12 ||2F + βK∑j=1,j 6=kIj∑l=1dDLlj∑n=1||(H˜BBkj vDLlj ,n)Hdiag(uULik,m(uULik,m)H) 12 ||2F+βIk∑l=1dDLlk∑n=1||(H˜SIk vDLlk,n)Hdiag(uULik,m(uULik,m)H) 12||2F , (2.47)(uDLik,m)HΣ˜DLik(v) uDLik,m =∑(l,j)6=(i,k)dDLlj∑n=1||(uDLik,m)HH˜DLikjvDLlj ,n||2F+∑(l,j)6=(i,k)dULlj∑n=1||(uDLik,m)HH˜UUikljvULlj ,n||2F+ ||uDLik,m||2F + κdULik∑n=1||(uDLik,m)HH˜SIikdiag(vULik,n(vULik,n)H) 12 ||2F + β K∑j=1Ij∑l=1dDLlj∑n=1||(H˜DLikjvDLlj ,n)Hdiag(uDLik,m(uDLik,m)H) 12 ||2F + β∑(l,j)6=(i,k)dULlj∑n=1||(H˜UUikljvULlj ,n)Hdiag(uDLik,m(uDLik,m)H) 12 ||2F+ βdULik∑n=1||(H˜SIikvULik,n)Hdiag(uDLik,m(uDLik,m)H) 12 ||2F . (2.48)41Chapter 2. Power Minimization in Full-Duplex Communication Systemsik ∈ I, m ∈M, X ∈ {UL,DL}, ‖∆‖F ≤ . (2.44)Given the size of the CSI uncertainty, one way to guarantee the worst-case SINRis by obtaining its lower bound. The lower bound of the SINRs for the uplink anddownlink transmissions are given in (2.45) and (2.46), respectively, where ΣˆXik(v)includes both the estimated channel (H˜) and the CSI uncertainty (∆). Assumingthe same uncertainty bound for associated CSI uncertainties, tractable upper boundsfor the first terms in the denominators of (2.45) and (2.46), i.e., (uXik,m)HΣˆXik(v)(uXik,m)are given in (2.49) and (2.50), for the uplink and downlink transmissions, respectively.The bound follows from the properties that Tr(A1B1) = Tr(B1A1) for any A1 ∈CM¯X×N¯X , B1 ∈ CN¯X×M¯X and Tr(A2B2) ≤ Tr(A2)Tr(B2) for any positive definitematrices, A2, B2 ∈ CN¯X×N¯X [101], where(M¯X, N¯X)=(Nk, Mik) if X = UL,(Nik , Mk) if X = DL.(2.51)Finally, it exploits the bound on the CSI uncertainty from Tr(∆∆H) ≤ 2 to arriveat (2.49) and (2.50) for the uplink and downlink transmissions, respectively. A similartechnique that involves obtaining tractable forms related to norm-bounded CSI uncer-tainty can be found in [87,102]. With the lower bound of the SINR, γˆXik,m(v,uXik,m)LB,the optimization problem can be written as belowmin .v,uK∑k=1Ik∑i=1dULik∑m=1(vULik,m)HvULik,m +K∑k=1Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m (2.52)s.t. γˆXik,m(v,uXik,m)LB≥ γXth, ik ∈ I, m ∈M, X ∈ {UL,DL}. (2.53)By obtaining the lower bound of the SINR, we can write the following one-to-one42Chapter 2. Power Minimization in Full-Duplex Communication Systemsrelationship(ΓˆXik,m)UB =[1 + (γˆXik,m)LB]−1, (2.54)where (ΓˆXik,m)UB is the upper bound of the MSE, ik ∈ I, m ∈ M, X ∈ {UL,DL}.That is to say, we can exploit the upper bound (i.e., worst-case) of the MSE, (ΓˆXik,m)UB,to guarantee the lower bound (i.e., worst-case) of the SINR, (γˆXik,m)LB as a QoSd(uULik,m)HΣˆULik(v) uULik,me =(uULik,m)HΣ˜ULik(v)uULik,m︸ ︷︷ ︸From eqn. (2.47)+2K∑j=1Ij∑l=1dULlj∑n=1||vULlj ,n||2F ||uULik,m||2F+ 2K∑j=1Ij∑l=1dDLlj∑n=1||vDLlj ,n||2F ||uULik,m||2F + 2κIk∑l=1dDLlk∑n=1||diag (vDLlk,n(vDLlk,n)H) 12 ||2F ||uULik,m||2F+ 2βK∑j=1Ij∑l=1dULlj∑n=1||vULlj ,n||2F ||diag(uULik,m(uULik,m)H) 12 ||2F + 2βK∑j=1Ij∑l=1dDLlj∑n=1||vDLlj ,n||2F ||diag(uULik,m(uULik,m)H) 12 ||2F + 2βIk∑l=1dDLlk∑n=1||vDLlk,n||2F ||diag(uULik,m(uULik,m)H) 12 ||2F (2.49)d(uDLik,m)HΣˆDLik(v) uDLik,me =(uDLik,m)HΣ˜DLik(v) uDLik,m︸ ︷︷ ︸From eqn. (2.48)+2K∑j=1Ij∑l=1dDLlj∑n=1||vDLlj ,n||2F ||uDLik,m||2F+ 2K∑j=1Ij∑l=1dULlj∑n=1||vULlj ,n||2F ||uDLik,m||2F + 2κdULik∑n=1||uDLik,m||2F ||diag(vULik,n(vULik,n)H) 12 ||2F+ 2βK∑j=1Ij∑l=1dDLlj∑n=1||vDLlj ,n||2F ||diag(uDLik,m(uDLik,m)H) 12 ||2F + 2βK∑j=1Ij∑l=1dULlj∑n=1||vULlj ,n||2F ||diag(uDLik,m(uDLik,m)H) 12 ||2F + 2βdULik∑n=1||vULik,n||2F ||diag(uDLik,m(uDLik,m)H) 12 ||2F (2.50)43Chapter 2. Power Minimization in Full-Duplex Communication Systemsconstraint. For given H˜Xik , ik ∈ I, m ∈ M, X ∈ {UL,DL}, (ΓˆXik,m)UB can becomputed as(ΓˆXik,m)UB = |(uXik,m)HH˜XikVXik,m−1|2+d(uXik,m)HΣˆXik(v)uXik,me+ (uXik,m)HdXik∑n=1,n6=mH˜XikvXik,n(vXik,n)H (H˜Xik)HuXik,m. (2.55)At iteration τ and for a fixed θˆXτik,m, ik ∈ I, m ∈ M, the above optimizationproblem can be reformulated as belowmin .v,θ PˆBS+UE (2.56)s.t. (ΓˆXik,m)UB ≤ −aˆXτik,mθˆXik,m + bˆXτik,m, (2.57)θˆXik,m ≥ logc(γXth + 1), (2.58)where (ˆ·) denotes the related variables for the case of norm-bounded design, similar tothose in the stochastic uncertainty design. The optimization problem is also convexand can be solved similarly following the steps as in Algorithm 1.2.5 Numerical Results And DiscussionsConsidering both perfect and imperfect CSI, in this section we investigate the perfor-mance of the proposed sum-power minimization algorithms for an FD MIMO multi-cell system through numerical simulations. We choose the simulation parametersfrom the 3GPP LTE specifications for small-cell deployments [103]. As discussed inChapter 1, the small cells are suitable for deployment of the FD technology since thecell-edge path loss is less than that in conventional cellular systems, which makes the44Chapter 2. Power Minimization in Full-Duplex Communication Systemsproblem of self-interference much more manageable [5, 61,104].In particular, our simulation setup considers an outdoor multi-cell scenario withthree pico cells randomly dropped in a hexagonal macrocell. For simplicity, we assumethe same number of transmit and receive antennas at each BS, i.e., Mk = Nk =N, k ∈ K, and at each mobile user, i.e., Mik = Nik = M, ik ∈ I. We further assumethat there are two users in each cell, where each BS is equipped with N = 4 transmitand receive antennas, and each user is equipped with M = 2 transmit and receiveantennas1. Also, we consider that each user sends a single data stream in the uplink(UL) and downlink (DL) directions. We average our results over 500 independentchannel realizations. The stochastic CSI uncertainty is generated following the modelin (2.31), where the nominal SNR is calculated based on the standard transmit powerof the BS and mobile users specified in [103]. The path loss model for line-of-sight(LOS) and non-line-of-sight (NLOS) communications between the BS and users aregenerated according to the following probability [64]PLOS = 0.5−min(0.5, 5 exp(−0.156/d)) + min(0.5, 5 exp(−d/0.03)), (2.59)where d denotes the distance (km) between the BS and users.We view HULkik =√κULik H¯ULkikas the uplink channel between the user ik and the BSk. The same model has been used in [61, 65, 66]. Here, H¯ULkik denotes the small-scalefading following a complex Gaussian distribution with zero mean and unit variance,whereas the large-scale fading consisting of the path loss and shadowing is denotedby, κULik = 10(−Z/10), Z ∈ {LOS,NLOS}, where LOS and NLOS are calculated fromthe specific path loss model given in Table 2.2. In the same vein, we define the1Similar to studies in [26, 62], out of total Nk +Mk antennas at the BS k, we assume that onlyNk (Mk) antennas are used for HD transmission (reception). The same holds for the mobile users.45Chapter 2. Power Minimization in Full-Duplex Communication SystemsTable 2.2: Simulation Parameters and Corresponding Settings for an FD MIMOMulti-Cell SystemParameters SettingsCell Radius 40 mMinimum Distance 40 mbetween BSsCarrier Frequency 2 GHzBandwidth 10 MHzThermal Noise Density −174 dBm/HzNoise Figure BS: 13 dB, User: 9 dBPath Loss (dB) between LOS: 103.8 + 20.9 log10 dBS and users (d in km) NLOS: 145.4 + 37.5 log10 dPath Loss (dB) between 98.45 + 20 log10 d, d ≤ 50 musers (d in km) 175.78 + 40log10d, d>50 mPath Loss (dB) between LOS: 89.5 + 16.9 log10 d, d < 2/3 km,BSs (d in km) LOS: 101.9 + 40 log10 d, d ≥ 2/3 km,NLOS: 169.36 + 40 log10 dShadowing Standard Deviation 10 dBbetween BS and usersShadowing Standard Deviation 12 dBbetween usersShadowing Standard 6 dBDeviation between BSschannels between the BS and DL users, and that of between the UL and DL users.In order to simulate the self-interference channel, we adopt the model from [3].46Chapter 2. Power Minimization in Full-Duplex Communication Systems1 2 3 4 5 6 7 8 9 10−50−40−30−20−1001020Average sum−power (dB)Iterations  FDHDFigure 2.2: Convergence of the objective function in (2.21) for perfect CSI designwith κ = β = −120 dB and a QoS constraint of 2.63 Mb/s.Accordingly, at the BS k the self-interference channel is distributed asHSIk ∼ CN(√KR1 +KRH¯SIk ,11 +KRINk ⊗ IMk), (2.60)where KR denotes the Rician factor and H¯SIk is a deterministic matrix2. Table 2.2summarizes the simulation parameters and corresponding settings. Unless stated oth-erwise, in our experiments we stick to the above settings of the network parameters.In all our results, the rate is calculated as the spectral efficiency via log2(1 +ΓXik,m), ik ∈ I, m ∈M, X ∈ {UL,DL}. We also note that our results on the power2Without loss of generality, we assume KR = 1 and H¯SIk is a matrix of all ones [61].47Chapter 2. Power Minimization in Full-Duplex Communication Systems−120 −115 −110 −105 −100 −95 −90 −85 −80−45−40−35−30−25−20−15−10−50κ=β(dB)Average sum−power (dB)  FD−DLFD−ULFDHD−DLHD−ULHDFigure 2.3: Comparison of the average power required by FD and HD systems forthe perfect CSI design with a QoS constraint of 2.63 Mb/s and varying transceiverdistortions.efficiency do not account for the overhead owing to the execution of the optimizationalgorithms.2.5.1 Perfect CSI ResultsThe proposed sum-power minimization algorithm presented in Section 2.3 relies oniterative updates of the design parameters. The iterative nature of the algorithmensures a local optimal solution. To this end, it is desirable to observe the convergencebehavior of the presented algorithm. Fig. 2.2 shows the outer convergence of the sum-power function in (2.21) for FD and HD operations. As expected, we observe strictlynon-increasing behavior of the optimization objective at each iteration. Furthermore,48Chapter 2. Power Minimization in Full-Duplex Communication Systems0 0.5 1 1.5 2 2.5 3 3.5 4 4.5−90−80−70−60−50−40−30Average sum−power (dB)Rate (Mb/s)  FD−DLFD−ULFDHD−DLHD−ULHDFigure 2.4: Comparison of the average power required by FD and HD systems forthe perfect CSI design with κ = β = −120 dB and varying QoS constraints.we notice faster convergence of the HD setup. This is because the FD system needsto consider additional self-interference in the design process, which contributes to theslower convergence of the optimization algorithm.Fig. 2.3 provides a comparison of the average sum-power required by FD and HDsystems with respect to the transceiver distortion. For the HD operation, we assumethat each BS serves the same number of DL and UL users as in the FD system.As we observe from Fig. 2.3, FD and HD setups require the same average powerat around κ = β ≈ −90 dB. However, the FD transmission outperforms the HDone when κ = β < −90 dB. These levels of self-interference cancellation have beenachieved through a recent advanced technique reported in [25]. We also note that49Chapter 2. Power Minimization in Full-Duplex Communication Systems1 2 3 4 5 6 7 8 9 10−45−40−35−30−25−20−15−10−505Average sum−power (dB)Iterations  FDHDFigure 2.5: Convergence of the objective function in (2.39) for a design with stochasticCSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63 Mb/s, η = 0.3, andλ = 0.2.the power efficiency gain for the FD over that of the HD transmission varies withdifferent κ (β) values. This is due to the fact that the higher transmitter (receiver)distortion, represented by κ (β), corresponds to larger residual self-interference, whichnecessitates a higher required transmit power to maintain the same QoS constraint.Therefore, we generally achieve a higher power efficiency gain with smaller values ofκ (β). Furthermore, FD-DL transmission tends to consume more power at higherκ (β). This is mainly due to the fact that the user has a lower noise figure, whichalso gets compounded at higher κ (β), i.e., more interference.Fig. 2.4 demonstrates a comparison of the average sum-power required by FDand HD systems with varying QoS constraints. As we notice from Fig. 2.4, as the50Chapter 2. Power Minimization in Full-Duplex Communication Systems1 2 3 4 5 6 7 8 9 10−50−40−30−20−10010Average sum−power (dB)Iterations  FDHDFigure 2.6: Convergence of the objective function in (2.56) for a design with norm-bounded CSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63 Mb/s, ands = 0.01.QoS constraint increases, power requirements for both FD and HD systems increase.Furthermore, with increasing QoS constraints, the HD system requires increasinglymore power, i.e., the gap between the required power for FD and HD systems tendsto be larger. Also, at this level of κ (β), the power consumption is mainly determinedby the interference between the nodes, which is lower between the users due to higherpath losses.2.5.2 Imperfect CSI ResultsAfter observing the power efficiency gains offered by the FD transmission with perfectCSI, in this section we study the performance in the presence of imperfect CSI.51Chapter 2. Power Minimization in Full-Duplex Communication Systems0 0.03 0.06 0.09 0.12−50−40−30−20−10010203040Average sum−power (dB)λ  HD−DLHD−ULHDFD−DLFD−ULFDFigure 2.7: Comparison of the average power required by robust FD and HD systemswith stochastic CSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63 Mb/s,and η = 0.3.As discussed in Section 2.4, we consider both stochastic and norm-bounded CSIuncertainties, where either the statistics or the size of the CSI uncertainty is knownto the transmitter, respectively. For the sake of simplicity, in either case we assumethat the associated channel links experience the same CSI uncertainty.Similar to the algorithm presented in Section 2.3, each of the robust algorithmsin Section 2.4 also rely on iterative updates of the design parameters to arrive at alocal optimum. Figs. 2.5 and 2.6 show the outer convergences of the proposed robustsum-power minimization algorithms for FD and HD setups over multiple design pa-rameters considering the stochastic and the norm-bounded CSI uncertainties, respec-tively. In both cases, we observe strictly non-increasing behavior of the optimization52Chapter 2. Power Minimization in Full-Duplex Communication Systems0 0.005 0.01 0.015 0.02 0.025−50−48−46−44−42−40−38−36−34−32−30Average sum−power (dB)s  HD−DLHD−ULHDFD−DLFD−ULFDFigure 2.8: Comparison of the average power required by robust FD and HD systemswith norm-bounded uncertainty, κ = β = −120 dB and a QoS constraint of 2.63Mb/s.objectives, i.e., the reduction of the sum-power with each iteration. Furthermore, asin the case of perfect CSI, the HD setup generally converges in fewer optimizationiterations compared to its FD counterpart.In Fig. 2.7, we compare the average power requirements for the robust FD andHD setups in the presence of stochastic CSI uncertainty for a given QoS constraintand transceiver distortion, where the parameters λ and η are chosen similar to thosein [65,95]. In particular, we notice that for a given η, the required transmit power forFD and HD setups increase with larger λ. This is justified because the higher CSIuncertainty means increased noise; therefore they require more power to guaranteethe same QoS. We also note that the proposed FD design consumes less power to53Chapter 2. Power Minimization in Full-Duplex Communication Systems0 0.02 0.04 0.06 0.08 0.1−50−40−30−20−100102030λAverage sum−power (dB)  FD−DL (Non−robust)FD−UL (Non−robust)FD (Non−robust)FD−DLFD−ULFDFigure 2.9: Comparison of the average power required by robust and non-robust FDsystems with stochastic CSI uncertainty, κ = β = −120 dB, a QoS constraint of 2.63Mb/s, and η = 0.3.guarantee the same QoS than the HD one.Next we compare the average power requirements for the robust FD and HDsetups assuming norm-bounded CSI uncertainty for a given QoS and transceiverdistortion, as depicted in Fig. 2.8. The CSI uncertainty parameter is related to thequality of the associated channels through the parameter s, whose range is chosensimilar to settings in [65, 105]. We observe that the required transmit power for FDand HD setups increase with larger s. This corroborates the previous observationwith the stochastic CSI uncertainty, since the noise also increases with larger CSIuncertainty. It is also apparent from the robust results that the FD system is affectedslightly more by the increased CSI uncertainty than the HD one since the former54Chapter 2. Power Minimization in Full-Duplex Communication Systemsinvolves more channel links (i.e., associated CSI uncertainties).Finally, in Fig. 2.9 we present the average power required by the robust and non-robust FD setups against the stochastic CSI uncertainty, measured by λ and η. Thisresult exemplifies the benefits that can be harnessed through the robust design inthe presence of CSI uncertainty. As we see from this result, as λ increases the powerrequirements increase for both robust and non-robust FD systems. We also observethat the non-robust FD setup requires more power in order to guarantee the same QoSsince it does not consider the CSI uncertainty into the design process. Furthermore,as λ increases (i.e., with increased CSI uncertainty), the difference in required averagepower between the robust and non-robust FD systems tends to increases for the samereason. In particular, the robust design results in power savings that can be as highas 87% for the considered simulation scenario.2.6 ConclusionsIn this chapter, we have studied the transmit and receive beamforming designs inorder to minimize the transmit power for an FD MIMO multi-cell system whilemaintaining a certain QoS. In addition to the limited DR at the transceivers, such acommunication system also suffers from the self-interference as well as the co-channelinterference from other nodes. Due to the non-convex nature of the optimizationproblem, we approximate this as a DCP, which is then solved via SCA. Our resultsdemonstrate that the FD systems require less transmit power than the HD systemsunder low to moderate level of transceiver distortions and for increasing QoS con-straints. In dealing with both stochastic and norm-bounded CSI uncertainties, ourstudies suggest that the proposed robust FD designs similarly require less power whencompared with the HD setups. Moreover, we have quantified the transmit power that55Chapter 2. Power Minimization in Full-Duplex Communication Systemscan be saved through the robust design in comparison to a non-robust design. Theresults justify the importance of a robust design in power-constrained applications.56Chapter 3Fairness Consideration inFull-Duplex and Device-to-DeviceCommunication Systems3.1 IntroductionIn the previous chapter, we have addressed the problem of sum-power minimizationfor the power-constrained FD applications, with maintenance of a certain QoS. Inthis chapter, we address the QoS fairness problem, given a certain power constraint.There may be specific interference scenarios in communications networks, as we ad-dress in the sequel, which may skew the resource allocations in favor of users thatexperience good channels, i.e., weak interference, if one tries to maximize the totalsum-rate of the network. In this case, it is likely that some users in the networkwill be underserved, which leads to unfairness among the users. One possibility toachieve network-wide fairness is to maximize the minimum SINR among all the usersin the network. This is same as equalizing the SINR performance of all users, andthus it is a strategy for enforcing the desired level of fairness in the network. Theauthors in [106, 107] have shown that max-min problems can be solved by SecondOrder Cone Programming (SOCP) programs, which have a high computational com-57Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsplexity. In this chapter, our general objective is to design low complexity algorithmsthat achieve the desired fairness among the users. Furthermore, we also considerpractical design aspects, such as CSI uncertainty. To this end, in the sequel we ad-dress fairness considerations in the specific context of FD and D2D communications.As presented in the Chapter 1, the performance of both FD and D2D systems areinterference-limited. For the former, it is the (residual) self-interference that limitsthe performance, while the performance is limited by the inter-user interference forthe latter.The rest of this chapter is organized as follows. In Section 3.2, we investigate thefairness design in FD multi-cell MIMO communications. We study the same designcriteria for the D2D communications in Section 3.3. We summarize the findings ofthis chapter in Section 3.4.3.2 Fairness in FD CommunicationsIn Section 2.1 of Chapter 2, we discussed that the FD system suffers form the residualself-interference due to imperfections of radio components, such as amplifier non-linearity, oscillator phase noise. We also noted that the system optimization underthe influence of residual self-interference has been investigated in various studies, suchas [53–56].The beamformer design for the sum-rate maximization in FD multi-cell cellularsystems has been studied in [61, 65]. It has been reported that when the sum-ratemaximization problems are considered in FD systems, the increasing self-interferencepower overwhelms the desired signals of the uplink users, which leads to reductionin achievable rate in the uplink channel. Since the objective is to maximize the totalthroughput, a natural solution is to reduce the transmit power in the uplink channel.58Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsTherefore, the uplink users are underserved, since all the resources are devoted forthe downlink transmission. This leads to unfairness among the users. In the contextof multi-cell FD communications as it is illustrated in Fig. 3.1 and also discussedin Section 2.1 of Chapter 2, there will be additional interference owing to the FDtransmission both at BSs and users, i.e., I) the self-interference at each FD BS andFD user, II) the interference among adjacent BSs, i.e., inter-BS interference, and III)the interference among all the mobile users in all cells. This interference will furtherdegrade the performance of the cell-edge users. Therefore, in the first part of thissection, we propose a transceiver design that maximizes the harmonic-sum of SINRsfor MIMO systems, and derive an iterative low-complexity distributed algorithm.In the second part of this section, we extend our design to provide resilienceagainst inaccuracies of the CSI. To this end, the CSI errors are often modeled asGaussian random variables [94], and the robustness can be provided in the statisticalsense. Alternatively, another way to achieve robustness is by worst-case optimization,which designs the system to operate under the worst-case channel condition if theCSI uncertainty is bounded [96–98]. We adopt the second approach and propose alow complexity iterative algorithm based on SDR technique to achieve fairness.The numerical results show that the proposed FD system outperforms the HDsystem under self-interference levels that have been achieved recently. In addition,we verify the advantages of incorporating important practical issues, such as CSIuncertainty and fairness performance into the transceiver design.3.2.1 System ModelIn this section, we consider an FD multi-cell multi-user MIMO system as it is il-lustrated in Fig. 3.1. In particular, we consider a K cell FD system, where BS59Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsDLkxkULix ˆkDLiskULis kDLiUkULiVkDLiy jULlx ˆjDLlsjULls jDLlUjULlV jDLly1kDLVkDLiVkkDLIV1kDLskDLiskkDLIskULiU1kULUkkULIUˆ kkULIsˆkULis1ˆkULsULkyDLjx1 jDLVjDLlVjjDLIV1 jDLsjDLlsjjDLIsjULlU1 jULUjjULIUˆ jjULIsˆjULls1ˆjULsULjykULkiHkDLi kHkSIiHSIkHBBkjHjULklHk jUUi lHkDLi jHUser ki User jlBS k BS jFigure 3.1: Full-duplex MIMO multi-cell system. Dashed arrows denote the self-interference and the dash-dotted arrows denote the interference between differentnodes.k, k = 1, . . . , K is equipped with Mk transmit and Nk receive antennas, and servesIk users in cell k. We denote ik to be the ith user in cell k with Mik transmit andNik receive antennas. We define the set of BSs as K = {1, . . . , K} and users asI = {ik | k ∈ {1, 2, . . . , K}, i ∈ {1, 2, . . . , Ik}}. We adopt the same limited DR mod-els at the FD transmitters and at the FD receivers as in Chapter 2. Fortunately,the system descriptions from Section 2.2 of Chapter 2 are directly applicable for thisstudy. Hence, we do not repeat them in this section. Recall the signal received by60Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsthe BS k and that received by the user ik can be written, respectively, asyULk =K∑j=1Ij∑l=1HULkljxULlj+ HSIk(xDLk + cDLk)+K∑j=1,j 6=kHBBkj xDLj + eULk + nULk , (3.1)yDLik =K∑j=1HDLikjxDLj + HSIik(xULik + cULik)+∑(l,j)6=(i,k)HUUikljxULlj+ eDLik + nDLik. (3.2)For the convenience of presentation, we will use the following notation in the restof the this section:HXik =HULkik , if X = UL,HDLikk, if X = DL.(3.3)3.2.2 Fairness Design with Perfect CSI3.2.2.1 Problem FormulationThe problem that maximizes the minimum SINR of users can be formulated as:max .v,u min∀ik∈I,mX∈MγXik,m(v,uXik,m)(3.4)s.t.dULik∑m=1(vULik,m)HvULik,m ≤ Pik , ik ∈ I, (3.5)Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m ≤ Pk, k ∈ K, (3.6)where Pik in (3.5) is the transmit power constraint at the user ik, and Pk in (3.6)is the total power constraint at the BS k. Here, the optimization variable u de-notes the stacked vectors of all receive filters in the uplink and downlink chan-nel. Furthermore, M denotes the set of all uplink and downlink data streams, i.e.,61Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsM = {m = 1, . . . , dXik , ik ∈ I, X = {UL, DL}}. The problem formulation (3.4)-(3.6)has two critical shortcomings. First, it requires an iterative search for a SOCP feasi-bility test [107, 108], which has a high computational complexity and necessitates acentralized implementation. Second, it ignores the overall throughput of the cell forthe purpose of maintaining fairness. In the following, we address both those issuesthrough a reformulation of the original problem.3.2.2.2 Harmonic-Sum MinimizationIn order to balance both the throughput for cell-edge users and the overall through-put, we choose to maximize the harmonic-sum of the user throughput, which is thesum of reciprocals of the user SINRs [109]. The harmonic-sum has the followingproperties for any set of positive numbers:min {x1, x2, . . . , xN} ≥ 1∑Ni=11xi, (3.7)1NN∑i=1xi ≥ N∑Ni=11xi, (3.8)It is easy to see from (3.7) and (3.8) that maximizing the harmonic-sum will in-directly maximize the minimum value and the arithmetic average of SINRs, respec-tively. These properties allow the maximization of the harmonic-sum to indirectlybalance the overall throughput and the user fairness.The harmonic-sum objective function for multi-cell multistream multi-user MIMOsystems can be expressed asSINRH =1K∑k=1Ik∑i=1∑X∈{UL,DL}dXik∑m=11γXik,m. (3.9)62Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsMaximizing the harmonic-sum SINRH is equivalent to minimizing 1/SINRH. There-fore, using the uplink and downlink SINRs from (2.3) and (2.4) of Chapter 2, respec-tively, this problem can be written asminv,uK∑k=1Ik∑i=1∑X∈{UL,DL}dXik∑m=1(uXik,m)HQXik (v) uXik,m−∣∣∣(uXik,m)H HXikvXik,m∣∣∣2∣∣∣(uXik,m)H HXikvXik,m∣∣∣2 (3.10)s.t. (3.5), (3.6), (3.11)where QXik (v) , X ∈ {UL,DL}, is defined in (2.3) and (2.4). The problem (3.10)-(3.11)can be equivalently rewritten asminv,uK∑k=1Ik∑i=1∑X∈{UL,DL}dXik∑m=1((uXik,m)HQXik(v) uXik,m−1)(3.12)s.t.∣∣∣(uXik,m)H HXikvXik,m∣∣∣2 = 1, ik ∈ I, m ∈M, (3.13)(3.5), (3.6), (3.14)which can be further simplified asminV,UK∑k=1Ik∑i=1∑X∈{UL,DL}Tr((UXik)HQXik (V) UXik)(3.15)s.t.∣∣∣(uXik,m)H HXikvXik,m∣∣∣2 = 1, ik ∈ I, m ∈M, (3.16)(3.5), (3.6), (3.17)where the optimization variable V (U) denotes all transmit (receive) beamformingmatrices in the uplink and downlink channels and QXik (V) is a function of transmitbeamforming matrices, V. We also note that the phase rotation of the columnvectors of the transmit beamforming matrices, vXik,m, does not affect the unit norm63Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsconstraint (3.16). Therefore, we can replace the constraint∣∣∣(uXik,m)H HXikvXik,m∣∣∣2 = 1by(uXik,m)HHXikvXik,m= 1. It has been shown in Appendix B.1 that this replacementdoes not change the objective function and other constraints.The new optimization problem can be cast asminV,UK∑k=1Ik∑i=1∑X∈{UL,DL}Tr((UXik)HQXik (v) UXik)(3.18)s.t.((UXik)HHXikVXik)◦ IdXik = IdXik , ik ∈ I, ∀X, (3.19)Tr((VULik)HVULik)≤ Pik , ik ∈ I, (3.20)Ik∑i=1Tr((VDLik)HVDLik)≤ Pk, k ∈ K. (3.21)Note that the objective function (3.18) is not jointly convex over transmit beam-forming matrices in the set V and receive beamforming matrices in the set U (sincethey are coupled). Therefore, we cannot apply the standard convex optimizationmethods to obtain the optimal solution. However, as the objective function (3.18)is component-wise convex over the matrices in V and U, we employ an iterativealgorithm that finds the efficient solutions of V and U in an alternating fashion.Particularly, we update the transmit beamforming matrices in V when the receivebeamforming matrices in U are fixed. Thereafter, we update the receive beamform-ing matrices in U using V obtained at the previous step. The iterations continueuntil convergence or a pre-defined number of iterations is reached. The Lagrangianfunction of the problem (3.18)-(3.21) can be written asL (V,U,λ,∆) =K∑k=1Ik∑i=1∑X∈{UL,DL}Tr((UXik)HQXik (v) UXik)+∑ik∈IλULik(Tr((VULik)HVULik)− Pik)+64Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsK∑k=1λDLk(Ik∑i=1Tr((VDLik)HVDLik)− Pk)+K∑k=1Ik∑i=1∑X∈{UL,DL}Tr(∆Xik(IdXik− (UXik)H HXikVXik)) ,(3.22)where λULik and λDLk are dual variables associated with the power constraints (3.20)and (3.21), respectively, and ∆Xik is a diagonal matrix with dual variables for theequality constraint (3.19). Here, λ and ∆ are the set of all dual variables for thepower and equality constraints, respectively.The optimal transmit and receive beamforming matrices can be computed bytaking the derivative of the Lagrangian function L (V,U,λ,∆) with respect to Vand U, respectively. They can be expressed asUXik =(HXikVXik(VXik)H (HXik)H+ΣXik (v))−1HXikVXik∆˜Xik, (3.23)VXik =(λ¯XikIM¯Xik+ XXik (U))−1 (HXik)HUXik(∆Xik)H, (3.24)where XXik (U) is defined in (3.25)-(3.26), λ¯Xikin (3.27) given at the bottom of thenext page. Here, ∆Xik and ∆˜Xikare the diagonal matrices that scale the column vec-tors of the transmit and receive beamforming matrices, respectively so that equalityconstraint (3.19) is met. By plugging the optimal receive and transmit beamformingmatrices in (3.23) and (3.24) into (3.19), respectively, the optimal scaling matricescan be computed as∆Xik =(IdXik◦((UXik)HHXik(λ¯XikIM¯Xik+ XXik (U))−1 (HXik)HUXik))−1, (3.28)∆˜Xik=(IdXik◦((VXik)H (HXik)H(HXikVXik(VXik)H(HXik)H+ΣXik (v))−1HXikVXik))−1.(3.29)The values of the Lagrange multiplier λ¯Xik in (3.24) are calculated numerically by65Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsmaking sure that the power constraints in (3.20) and (3.21) are satisfied. If the valuesof the Lagrange multipliers λ¯Xik are negative, we assign λ¯Xikas zeros. The iterativealternating algorithm for solving the optimization problem (3.18)-(3.21) is given inAlgorithm 2. The proof of convergence for Algorithm 2 is given in Appendix B.2.XULik (U) =K∑j=1Ij∑l=1((HULjik)HUULlj(UULlj)HHULjik + β(HULjik)Hdiag(UULlj(UULlj)H)HULjik)+ κdiag((HSIik)HUDLik(UDLik)HHSIik)+ β(HSIik)Hdiag(UDLik(UDLik)H)HSIik+∑(l,j)6=(i,k)((HUUljik)HUDLlj(UDLlj)HHUUljik + β(HUUljik)Hdiag(UDLlj(UDLlj)H)HUUljik),(3.25)XDLik (U) =K∑j=1,j 6=kIj∑l=1((HBBjk)HUULlj(UULlj)HHBBjk + β(HBBjk)Hdiag(UULlj(UULlj)H)HBBjk)+Ik∑l=1(κdiag((HSIk)HUULlk(UULlk)HHSIk)+ β(HSIk)Hdiag(UULlk(UULlk)H)HSIk)+K∑j=1Ij∑l=1((HDLljk)HUDLlj(UDLlj)HHDLljk + β(HDLljk)Hdiag(UDLlj(UDLlj)H)HDLljk).(3.26)(λ¯Xik , N¯Xik, M¯Xik)={(λik , Nk, Mik) if X = UL,(λk, Nik , Mk) if X = DL.(3.27)66Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsAlgorithm 2 Harmonic-Sum Maximization Algorithm.1: Set the iteration number n = 0 and initialize the transmit beamforming matricesVX,[0]ik, ik ∈ I, X ∈ {UL,DL}.2: Compute the receive beamforming matrices UX,[n]ikfrom (3.23) and (3.29).3: repeat4: n← n+ 1.5: Configure λ¯Xik with an initial value.6: for l = 1, . . . do7: Update the transmit beamforming matrices VX,[n]ik, ∀ (i, k,X) using (3.24)and (3.28).8: Update λ¯Xik numerically using bisection search.9: end for10: Compute the receive matrices UX,[n]ik, ∀ (i, k,X) from (3.23) and (3.29).11: until convergence of the objective function in (3.18), or a predefined number of itera-tions is reached.The proposed algorithm can be computed in a distributed manner because thecomputation of transmit beamforming matrix of one user in (3.24) does not requireinformation about transmit beamforming matrices of other users. Therefore, thetransmit beamforming matrices can be computed for each user in parallel as long asthe receive beamforming matrices and CSI are shared among BSs. However, Max-MinSINR algorithm [107] requires computation of all transmit beamforming matrices ofall users without parallelism [110].3.2.2.3 Complexity AnalysisAssuming the same number of transmit antennas (M), receive antennas (N) andsame number of data streams (d) at each node, in this section, we will compare thecomputational complexity of the proposed algorithm with those of the weighted-sum-rate [84,111] and Max-Min SINR [107] algorithms, which also employ an alternatingiterative algorithm.The Max-Min SINR algorithm requires iterative searches for the highest minimumSINR value using the bisection algorithm, where each search requires to solve an67Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsTable 3.1: Comparison of Computational ComplexityProposed Algorithm Weighted-Sum-Rate Max-Min SINR|I|N (Nd+NM +M2) |I|N (d2 +Nd+NM +M2) M2d3 |I|3+ M3 +Nd2 +MNd+ d3 + d2M + M3 +MNd+ d3 + d2MSOCP problem. This can be solved using an iterative interior point method requiringO (M2d3 |I|3) calculations per iteration of the interior point method. The complexityof the weighted-sum-rate algorithm is analyzed in [84]. We summarize the complexityof these algorithms in Table 3.1. Given that the number of users, |I|, is expectedto be much greater than the number of transmit antennas M , receive antennas Nor the number of data streams per user, d, it can be easily seen from Table 3.1 thatoverall computational complexity of Max-Min SINR algorithm is much higher thanthe complexity of the proposed algorithm.3.2.3 Fairness Design with Imperfect CSIIn this section, we will study the fairness problem under imperfect CSI scenario. Wecharacterize the imperfect CSI using the norm-bounded deterministic (or worst-case)model, where the instantaneous channel lies in a known set of possible values [96–98].In particular, it is expressed asH ∈ H ={H˜ + Λ : ‖Λ‖F ≤ }, (3.30)where H˜, Λ, and  denote the nominal value of the CSI, the channel error matrix,and the uncertainty bound, respectively. Here, H represents all the channels in theFD multi-cell system.68Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsWith the imperfect CSI, the max-min optimization problem can be written as3max .v,u min∀ik∈I,mX∈Mmin‖Λ‖F≤γXik,m(v,uXik,m)(3.31)s.t.dULik∑m=1(vULik,m)HvULik,m ≤ Pik , ik ∈ I, (3.32)Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m ≤ Pk, k ∈ K. (3.33)To simplify the presentation, we will use the result in [112, Lemma 1], which hasalso been used in [45, Appendix A], [102] to express min‖Λ‖F≤ γXik,masγ˜Xik,m(v,uXik,m), min‖Λ‖F≤γXik,m(v,uXik,m)=(uXik,m)HEXik,m (v) uXik,m(uXik,m)HFXik,m (v) uXik,m, (3.34)where EXik,m and FXik,mare defined asEXik,m (v) = H˜XikvXik,m(H˜XikvXik,m)H− 2 ∥∥vXik,m∥∥2 IN¯Xik , (3.35)FXik,m (v) = Σ˜Xik(v) + H˜XikdXik∑l=1vXik,l(vXik,l)H (H˜Xik)H− H˜XikvXik,m(H˜XikvXik,m)H− 2 ∥∥vXik,m∥∥2 IN¯Xik + 2ΘXikIN¯Xik . (3.36)Here, Σ˜Xik(v) is obtained by replacing the channel matrices H in ΣXik (v) given in (2.5)3Note that under perfect CSI case, solving harmonic-sum problem instead of max-min problemresults in a distributed and a low complexity algorithm. On the other hand, for the imperfect CSIcase, the solution of both max-min and harmonic sum problems are centralized, which requires theuse of SDR technique, and thus harmonic-sum metric does not have much improvement over max-min metric in terms of complexity when there is a channel uncertainty. Therefore, in this section,we will only focus on the max-min problem.69Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsand (2.6) with the estimated ones H˜, and ΘXik is expressed asΘULik =K∑j=1Ij∑l=1(1 + β)∥∥∥VULlj ∥∥∥2F+Ik∑l=1(κ+ β)∥∥VDLlk ∥∥2F + K∑j=1,j 6=kIj∑l=1(1 + β)∥∥∥VDLlj ∥∥∥2F,(3.37)ΘDLik =K∑j=1Ij∑l=1(1 + β)∥∥∥VDLlj ∥∥∥2F+ (κ+ β)∥∥VULik ∥∥2F + ∑(l,j)6=(i,k)(1 + β)∥∥∥VULlj ∥∥∥2F.(3.38)Using the simplified SINR definition in (3.34) and epigraph form with the slackvariable γ, the problem (3.31)-(3.33) can be rewritten asminv,u,γ−γ (3.39)s.t.dULik∑m=1(vULik,m)HvULik,m ≤ Pik , ik ∈ I, (3.40)Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m ≤ Pk, k ∈ K, (3.41)γ˜Xik,m(v,uXik,m) ≥ γ, ik ∈ I, m ∈M. (3.42)Because of the minimum SINR constraints in (3.42), the optimization prob-lem (3.39)-(3.42) is non-convex. Thus we iteratively compute the transmit and receivebeamforming matrices to monotonically improve the minimum SINR.3.2.3.1 Receive Filter DesignThe receive beamforming matrices optimization problem to maximize the minimumSINR among all users’ data streams under fixed transmit beamforming matrices canbe solved independently, since the SINR terms in (2.3) and (2.4) depend on a single70Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsstream receive filter. Therefore, the optimal receiver beamforming matrices can becomputed by solving the following problem:max .uXik,mγ˜Xik,m(v,uXik,m). (3.43)Using [101, 113], and as derived in [45, Appendix B], the optimal solution of (3.43)can be given asuXik,m =(FXik,m (v))−1/2wXik,m∥∥∥(FXik,m (v))−1/2 wXik,m∥∥∥ , (3.44)where wXik,m is the principle eigenvector of(FXik,m (v))−1/2EXik,m (v)(FXik,m (v))−1/2,and EXik,m (v) and FXik,m(v) are defined in (3.35) and (3.36), respectively.3.2.3.2 Precoder DesignTo solve the transmit beamforming matrices design problem under fixed receivebeamforming matrices, we apply the inverse relationship between max-min fair-ness and power minimization problems proposed for broadcast and multicast chan-nels in [45], [106, Theorem 3], and [114, Claim 3], respectively. Denoting P˜ ={Pik , ik ∈ I, Pk, k ∈ K}, and ρik = Pik/P˜ , ik ∈ I and ρk = Pk/P˜ , k ∈ K, theproblem (3.39)-(3.42) can be rewritten asP(P˜)= minv,u,γ−γ (3.45)s.t.dULik∑m=1(vULik,m)HvULik,m ≤ ρikP˜ , ik ∈ I, (3.46)Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m ≤ ρkP˜ , k∈K, (3.47)71Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsγ˜Xik,m(v,uXik,m) ≥ γ, ik∈I,m∈M. (3.48)Now consider the power minimization problem below:Q (γ) = minv,u,ττ (3.49)s.t.dULik∑m=1(vULik,m)HvULik,m ≤ ρikτ, ik ∈ I, (3.50)Ik∑i=1dDLik∑m=1(vDLik,m)HvDLik,m ≤ ρkτ, k∈K, (3.51)γ˜Xik,m(v,uXik,m) ≥γ, ik ∈ I, m∈M. (3.52)Using a minimum SINR constraint γ∗ in (3.52), assume that the optimal solutionof the problem Q (γ∗) in (3.49)-(3.52) is v∗,u∗, τ ∗. It was shown in [106, Theorem 3]that v∗,u∗, γ∗ is the optimal solution of the problem P (τ ∗) in (3.45)-(3.48), and thuswe can solve the problemQ (γ) to solve the problem P(P˜), and vice versa. Under thefixed receiver beamforming matrices, the problemQ (γ) is a quadratically constrainedquadratic program (QCQP), which can be solved through SDR techniques [45]. LetV˜Xik,m = vXik,m(vXik,m)H, then the transmit beamforming matrices design problem canbe written asminV˜,ττ (3.53)s.t.dULik∑m=1Tr{V˜ULik,m}≤ ρikτ, ik ∈ I, (3.54)Ik∑i=1dDLik∑m=1Tr{V˜DLik,m}≤ ρkτ, k ∈ K, (3.55)γ˜Xik,m(V˜,uXik,m)≥ γ, ik ∈ I, m ∈M, (3.56)72Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsV˜Xik,m  0, ik ∈ I, m ∈M, (3.57)rank(V˜Xik,m)= 1, ik ∈ I, m ∈M, (3.58)where V˜ denotes all matrices V˜Xik,m, ik ∈ I, m ∈M, and γ˜Xik,m(V˜,uXik,m)is obtainedby replacing vXik,m(vXik,m)Hin (2.3) and (2.4) with V˜Xik,m. The problem (3.53)-(3.58) isstill non-convex because of the rank constraint in (3.58). By dropping this constraint,the problem (3.53)-(3.58) can be solved through semidefinite programming (SDP)techniques. If the optimal solution VXik,m has rank 1, then it is also an optimalsolution for (3.49)-(3.52), and the optimal vXik,m can be obtained through rank-onedecomposition. Otherwise, one can apply Gaussian randomization [42] to obtain theapproximate solution of the beamforming vectors. The steps of the proposed robustalgorithm are given in Algorithm 3. Since each step of the Algorithm 3 increases theminimum SINR, Algorithm 3 converges, as it has been shown in [45].Algorithm 3 Robust Fairness Algorithm.1: Initialize the transmit beamforming vectors vXik,m, ik ∈ I, m ∈M to ensure the powerconstraints are satisfied.2: repeat3: Compute the receive beamforming vectors uXik,m, ik ∈ I, m ∈M from (3.44).4: Update the target minimum SINR from (3.34).γ∗ = minik∈I, m∈Mγ˜Xik,m. (3.59)5: Compute the transmit beamforming vectors vXik,m, ik ∈ I, m ∈M by solvingQ (γ∗)through (3.53)-(3.58).6: Scale the transmit beamforming vectors vXik,m, ik ∈ I, m ∈M to ensure the powerconstraints are satisfied.7: Update the target minimum SINRγ∗ = minik∈I, m∈Mγ˜Xik,m. (3.60)8: until convergence of the minimum SINR, γ∗.73Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.2.4 Numerical Results and DiscussionsIn this section, we numerically investigate the proposed algorithms for an FD multi-cell multi-user MIMO system. As in Section 2.5 of Chapter 2, we choose the sim-ulation parameters from the 3GPP LTE specifications [103]. Therefore, the samesimulation settings in Section 2.5 are applicable that also includes Table 2.2 of Chap-ter 2, unless stated otherwise. The maximum transmit power for each BS and mobileuser is set to 24 dBm and 23 dBm, respectively. We average the results over 1000 in-dependent channel realizations. Similar to the previous chapter, the rate is calculatedas the spectral efficiency via log2(1 + ΓXik,m), , ik ∈ I, m ∈M, X ∈ {UL,DL}.3.2.4.1 Perfect CSI ResultsThe proposed fairness design presented in Section 3.2.2 is based on an iterative up-date of the design parameters. The iterative nature of the algorithm is to ensurethat a local optimal solution is obtained. Hence, it is of interest to observe theconvergence behavior of this algorithm. Fig. 3.2 shows the convergence of the ob-jective function in (3.18) for both FD and HD operations. As expected, the strictlynon-increasing behavior of the optimization objective is observed as the number ofiterations increases. Moreover, we observe that HD setup converges with relativelyfewer iterations with the penalty of worse performance. This is because the FD de-sign needs to consider additional interference terms in the design process, which maycontribute to the slower convergence of the algorithm.Fig. 3.3 provides a comparison of the sum rates achieved by FD and HD systems.The HD transmission design is a special case of our system model, which can beobtained by ignoring the additional interference for the UL and DL transmissions.Also, we assume that each BS in HD operation serves the same number of downlink74Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems0 2 4 6 8 10 12 14 16 18 2010152025IterationObjective Fucntion (dB)  FDHDFigure 3.2: Convergence of the objective function in (3.18) for perfect CSI designwith κ = β = −120 dB.and uplink users to guarantee fairness among them as in the FD system. As wesee from Fig. 3.3, both FD and HD transmissions achieve the same rates at aroundκ = β ≈ −92 dB. This is to say, FD outperforms HD transmission when κ = β <−92 dB, which has been achieved by recent advanced self-interference cancellationtechniques reported in [25]. We also note that the spectral efficiency gain for FDover HD transmission varies with different κ and β values. This is due to the factthat the higher transmitter (receiver) distortion, represented by κ (β) corresponds tolarger residual self-interference. Therefore, with smaller values of κ (β), we obtaina higher spectral efficiency gain. In particular, going from κ = β = −100 dB toκ = β = −120 dB, the spectral efficiency gain over HD operation improves from 25%75Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems−140 −135 −130 −125 −120 −115 −110 −105 −100 −95 −9020406080100120140κ=β (dB)Sum−rate (bits/s/Hz)  HD−DLHD−ULHDFD−DLFD−ULFDFigure 3.3: Total sum-rates achieved for FD and HD setups with perfect CSI.to 65%, respectively.In order to understand the impact of inter-cell interference, we study the systemsum-rate with respect to distance between BSs. Unlike the results presented above, inthis case we keep the positions of users fixed with respect to its BS location. We notethat when we increase the distance between the BSs, however, the inter-user distanceamong the users in different small-cells are also increased. Therefore, it essentiallycaptures both impacts of inter-cell and inter-user interference. As we see from Fig.3.4, as the distance between the BSs increases, the inter-cell as well as inter-userinterference decreases, which leads to an increase in total sum-rates and improvesthe spectral efficiency gains over the HD transmission, irrespective of transceiverdistortions. In particular, with an increase of BS distance from 40 m to 100 m, the76Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems20 30 40 50 60 70 80 90 10030405060708090100Spectral efficiency gain over HD (%)Distance between small cell BS (m)  DL(κ=β=−120)UL(κ=β=−120)FD(κ=β=−120)DL(κ=β=−110)UL(κ=β=−110)FD(κ=β=−110)Figure 3.4: Total sum-rates achieved for FD and HD setups with perfect CSI andvarying distance between BSs. The parameters representing transceiver distortionare chosen as κ = β = −110 dB and κ = β = −120 dBspectral efficiency gain over HD setup increases from from 65% to 76%, respectively.The relatively small increase in spectral efficiency over HD setup indicates that thesystem performance is largely dominated by residual self-interference inherent to FDsystem. We also note that as the FD setup experiences less transceiver distortion,it provides higher spectral efficiency gain over HD setup. This is apparent fromthe superior spectral efficiency gain obtained for κ = β = −120 dB over that ofκ = β = −110 dB.Fig. 3.5 shows the Cumulative Distribution Function (CDF) of three algorithms,where we compare the proposed algorithm with WMMSE and Max-Min SINR fair-ness algorithms. The individual user rate along the x-axis of this figure denotes the77Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems0 5 10 15 20 25 3000.10.20.30.40.50.60.70.80.91Rate (b/s/Hz)CDF  ProposedWMMSEMax−MinFigure 3.5: Comparison of CDFs of individual user rate among the proposed, Max-Min and sum-rate maximizing WMMSE designs with κ = β = −80 dBtotal rate obtained by each user in the uplink and downlink transmissions. It can beobserved from this result that the Max-Min fairness algorithm provides the highestfairness to the users at the lower individual rate, i.e., below 1 b/s/Hz, since it maxi-mizes the minimum SINR. Compared to WMMSE and proposed algorithms, the usersare less likely to be served by the Max-Min SINR design when the data rate increases,e.g., beyond 2 b/s/Hz. In contrast, it is more likely that a user with higher individ-ual rate will be served by WMMSE design since it maximizes the total throughput.Nonetheless, compared to the WMMSE design, a user is more likely to be served bythe proposed design if it experiences a rate, for instance below 9 b/s/Hz. Further-more, we note that the LTE user throughput requirement is specified at two points:78Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems2 4 6 8 10 12 14 16 18 201234567Sum−rate (bits/s/Hz)Iteration  FD−DLFD−ULFDHD−DLHD−ULHDFigure 3.6: Convergence of minimum SINR i.e., improved associated rate in thepresence of imperfect CSI with s = 0.02 and κ = β = −120 dB.at the average and at the fifth-percentile of the user distribution (where 95 percentof the users have better performance) [115]. The fifth-percentile users correspond toones operating on the cell edge [116]. It can be observed from this figure that theproposed algorithm almost surely provides a better fifth-percentile user throughputwhen compared to the WMMSE design, and also against the Max-Min SINR fairnessalgorithm when the individual rate is above 1 b/s/Hz.3.2.4.2 Imperfect CSI ResultsAfter observing the gains offered by FD transmission over HD setup with perfect CSI,in this section we show the performance with imperfect CSI. As discussed in Section79Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.2.3, we consider the norm-bounded CSI uncertainty to obtain worst-case fairnessamong mobile users. This is to say, knowing the boundary of the uncertainty regiondenoted by , we provide the worst-case transceiver design to obtain fairness amongmobile users.We first study the convergence behavior of the proposed algorithm with imperfectCSI, as presented in Section 3.2.3. Fig. 3.6 shows the convergence of the proposedalgorithm for both FD and HD setups over multiple design parameters. The re-sult is obtained by averaging the convergence behavior of the network, over severalchannel realizations. As expected, we observe strictly non-decreasing behavior of theoptimization objective, i.e., improved minimum SINR in terms of associated rate,as the number of iterations increase. Furthermore, as in the case of perfect CSI,it is observed that HD setup converges with relatively fewer optimization iterationscompared to its FD counterpart.In Fig. 3.7, we present the sum-rate performances for both FD and HD setupsagainst the measure of CSI uncertainty, s. As we see from this figure, as the boundaryof the CSI uncertainty, i.e., s increases, the sum-rate decreases for both FD and HDsystems. It is perceivable, since larger s means higher CSI uncertainty, hence lowerrate. Furthermore, we observe that for increasing CSI uncertainty, the performanceof FD setup falls more rapidly than its HD counterpart. For example, for the samedecrease in CSI quality from s = 0 to s = 0.2, the sum-rate difference between FDand HD setups goes from 28.5 b/s/Hz to 5 b/s/Hz. This is due to the fact thatthe FD system involves a larger number of channels, i.e., self-interference and inter-user interference channels; and thus increased CSI uncertainty with its transmission,which degrades its performance more than that of the HD transmission. Hence, theperformance of FD system is more susceptible to an increasing s in comparison to a80Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2020406080100120sSum−rate (bits/s/Hz)  HD−DLHD−ULHDFD−DLFD−ULFDFigure 3.7: Total sum-rates achieved for FD and HD setups with varying CSI uncer-tainties and κ = β = −120 dB.HD setup.Next we compare the performance of robust and non-robust FD setups with var-ious design parameters. The objective of this result is to show the performance gainthat can be harnessed through a robust design in the presence of CSI uncertainty.To this end, we make the following observations from Fig. 3.8. We observe that withsmall CSI uncertainty i.e., s = 0.1, as the κ and β increase, the sum-rate decreasessharply. This is to say, in this regime the transceiver distortion is a more limitingfactor on the sum-rate performance than the CSI uncertainty. On the other hand,with larger s, i.e., s = 0.2, the rate of decrease of total sum-rate with increasing κand β is relatively smaller, i.e., the sum-rate curves flatten out for the FD setup.81Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems−125 −120 −115 −110 −105 −100 −95 −90 −8581012141618202224Sum−rate (bits/s/Hz)κ=β (dB)  FD (non−robust), s=0.1FD (robust), s=0.1FD (non−robust), s=0.15FD (robust), s=0.15FD (non−robust), s=0.2FD (robust), s=0.2Figure 3.8: Total sum-rates achieved for robust and non-robust FD setups withvarying CSI uncertainties and transceiver distortions.This indicates that the CSI uncertainty is a more limiting factor on the sum-rateperformance than the transceiver distortion. In addition, we observe that with lowerκ and β, the difference between robust and non-robust design is noticeable. However,with increasing κ and β, the difference in performance becomes smaller as the systemperformance is dominated by transceiver distortion rather than CSI uncertainty inthis region. Finally, we observe that as the CSI uncertainty increases, the differ-ence between the sum-rate performance of the robust and non-robust designs alsoincreases.82Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.3 Fairness in D2D CommunicationsIn this section, we turn our focus on another interference-limited communication sys-tem, namely D2D communications. Different from the FD communication which isalso an interference-limited communication system, in this case, however, the sys-tem performance is limited by the inter-user interference, as in wireless interferencenetworks.The seeming spectrum scarcity and ever growing demands for new wireless ser-vices have encouraged the concept of opportunistic spectrum access through cognitiveradios so as to better utilize the available wireless spectrum [117]. In this communi-cation scenario, unlicensed Secondary Users (SUs) are allowed to dynamically accessthe spectrum of the Primary User (PU) provided that they do not degrade the per-formance of the latter [118]. For example, in an underlay cognitive concept, the SUsare only allowed to transmit if the resulting interference to the PU’s receiver doesnot exceed a certain threshold value (set by macrocells or the regulatory body). Bytreating macrocells as PUs and D2D users as SUs, the concept of (in-band) underlayD2D transmission is deemed to be a suitable technology in cellular networks [119]-[120]. To that end, controlling the interference power to the macrocell users is key tothe successful coexistence.The emergence of IA has shown that the sum-rate of the wireless interferencenetworks can scale linearly with the number of users in the network at high SINR.In the specific context of IA for D2D communications, recently there have beena few studies reported, such as [121–124]. In [121], the authors have consideredcellular communications with underlay D2D transmissions and have derived the totalDegrees of Freedom (DoF) achievable for the network. The authors in [122] haveconsidered IA for D2D communications, where several grouping mechanisms for D2D83Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsusers are considered. An opportunistic IA scheme has been proposed in [123], wherethe underlaying D2D users are selected only if they meet a certain scheduling metric.The study in [124] investigates IA schemes for D2D communications, where multipleD2D transmitters communicate to a single D2D receiver, as in a local area networksystem. However, none of these studies have taken into account SINR fairness amongthe D2D users. In addition, the IA techniques face the practical challenge of obtainingthe CSI at the transmitter [125,126]. In the context of MIMO interference networks,the technique of IA has been studied with the assumption that CSI is estimatedthrough channel estimation at the transmitter side [125], or it has been obtainedvia quantized feedback from the receiver [126]. To account for this practical designaspect, we also consider CSI uncertainty in our design, along with SINR fairness forD2D communications.To the best of our knowledge, this thesis reports the first study that considersthe transceiver optimization considering SINR fairness among the D2D users in thecontext of IA. We assume that users involved in the D2D communications coexistin an underlay fashion with the macrocell (i.e., PU) and form groups possibly basedon positions or specific services required by D2D users, as suggested in [122]. Tothis end, we aim at designing precoders and receiver filters so as to maximize theminimum SINR for D2D users in the presence of CSI uncertainty. Due to the non-convexity of the optimization problem, similar to the previous study for fairness inFD communications, we follow an alternating minimization algorithm that involvesthe SDR technique. It is observed that at low SNR and high CSI error with relaxedinterference power constraint, the D2D users perform very close to that of the usersin the non-cognitive system, but performance degrades considerably with stringentinterference power constraint.84Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsBase Station D2D TransmitterD2D ReceiverCellular UserD2D Group BD2D Group AFigure 3.9: An illustration of underlay D2D communications in a cellular network.Solid and dashed lines indicate desired signals and interference, respectively.3.3.1 System ModelLet us consider a cellular network, composed of one primary-user pair (one BS andone cellular user) denoted as user pair 0 and C D2D groups operating in underlaycognitive transmission mode, as shown in Fig. 3.9 for C = 2. Each D2D group c iscomposed of K secondary-user (D2D user) pairs. A D2D link in each group makes useof NT transmit and NR receive antennas. Also, each D2D user i in group c, encodesits messages, scm,i, onto dci ≤ min(NT , NR) independent data streams sent along thebeamforming vectors vcm,i, where m = 1, . . . , dci such that E[sci(sci)H ] = Idci i.e., datastreams are independent and have unit power. In D2D group c, the data streams areobserved at user j after passing through the channel, Hci,j. Furthermore, we assumethat the primary-link uses NpuT and NpuR antennas at the transmitter and receiver,respectively, to transmit dpu ≤ min(NpuT , NpuR ) independent parallel data streams.85Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.3.1.1 Precoding at the D2D TransmitterLet us consider V ci = [vc1,i, . . . , vcdci ,i] ∈ CNT×dci as the precoding matrix of user i inD2D group c. The resultant precoded signal, Sci , can be represented asSci =dci∑m=1vcm,iscm,i = Vci sci , (3.61)where sci ∈ Cdci×1 is transmitted symbol vector at user i in D2D group c. Note thateach data stream, scm,i, i.e., the m-th stream of user i in D2D group c is sent along thebeamforming vector, vcm,i. The maximum transmission power, Pci , for user i in D2Dgroup c, i.e., total power constraint is then given by,∑dcim=1(vcm,i)Hvcm,i = ‖V ci ‖2F ≤ P ci .3.3.1.2 Channel UncertaintyWe assume that the D2D link between transmitting user i and receiving user j ingroup c can be expressed as [96–98]Hci,j = Hˆci,j + ∆ci,j, (3.62)where Hˆci,j is the channel estimate available at the transmitter i and ∆ci,j is thecorresponding CSI error matrix. For the latter, we consider a norm-bounded modelin order to obtain a worst-case robust design where the CSI uncertainty is assumedto be lying in a spherical region within a radius, ci,j, asR,{∆ci,j :‖∆ci,j‖F ≤ci,j}={∆ci,j :Tr(∆ci,j(∆ci,j)H)≤ci,j2}. (3.63)This model is applicable where the CSI is obtained via feedback from the receiver(e.g., in FDD) or estimated at the transmitter through pilot signals (e.g., in TDD)86Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems[35].3.3.1.3 Interference Suppression at the D2D ReceiverThe signal received at user i in D2D group c, is given byyci = Hci,iVci sci +K∑j=1,j 6=iHcj,iVcj sj +C∑c′=1,c′ 6=cK∑j=1Hc′j,iVc′j sc′j + zci , (3.64)where Hci,i ∈ CNR×NT is the channel matrix and zci ∈ CNR×1 is the effective AWGNvector at user i of c-th D2D group with distribution, CN (0NR , σ2i INR). The firstterm in the above expression is the intended signal for user i in D2D group c, thesecond term refers to the interference from other users in D2D group c, and the thirdterm refers to the interference from users in the neighboring D2D groups. Note thatwe assume the inter-cell interference that results from the neighboring macrocells ismitigated through macrocell coordination and therefore neglected in our analysis.We further assume that interference from cellular users to receiving users in the D2Dgroups is negligible. This assumption is not specifically restrictive since we assumethat D2D group will be formed by nearby users. Using (3.62), the received signal canbe decomposed asyci =(Hˆci,i + ∆ci,i)Vci sci +K∑j=1,j 6=i(Hˆcj,i + ∆cj,i)Vcj scj +C∑c′=1,c′ 6=cK∑j=1(Hˆc′j,i + ∆c′j,i)Vc′j sc′j + zci=C∑c=1K∑j=1Hˆcj,iVcj scj +C∑c=1K∑j=1∆cj,iVcj scj + zci . (3.65)As with precoders, the transmitting D2D user also designs the receive filter, U ci =[uc1,i, . . . , ucdci ,i] based on available channel estimate Hˆci,i. Projecting the received signal87Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsonto the columns of U ci , on a per-stream basis it can be written as(ucm,i)Hyci =(ucm,i)H C∑c=1K∑j=1dcj∑l=1Hˆcj,ivcl,jscl,j+ (ucm,i)H C∑c=1K∑j=1dcj∑l=1∆cj,ivcl,jscl,j+ (ucm,i)Hzci , (3.66)where vcl,j is the precoder for l-th stream of user j in D2D group c. Therefore, theSINR of the m-th stream for user i in c-th D2D group, γcm,i, is given byγcm,i(ucm,i,V,Λ) =‖(ucm,i)H(Hˆci,i + ∆ci,i)vcm,i‖22Icm,i + I∆cm,i + ‖ucm,i‖22σ2i,c, (3.67)where Icm,i =∑(l,j,c′)6=(m,i,c) ‖(ucm,i)HHˆc′j,ivc′l,j‖22 accounts for the inter-stream, inter-userand inter-cell interferences from the neighboring D2D groups, I∆cm,i =∑(l,j,c′)6=(m,i,c)‖(ucm,i)H∆c′j,ivc′l,j‖22 is the residue interference due to the channel uncertainty at them-th stream of user i in D2D group c, V and Λ denote the sets of {vcm,i,∀m, i, c} and{∆ci,j,∀i, j, c}, respectively.3.3.2 Problem FormulationConsidering that the system performance is often limited by the worst stream, ourobjective is to find the precoders and receive filters that maximize the minimum SINRover all streams across all users of underlaying D2D groups in the cellular network.As for the interference power constraint, we follow a conservative approach whereeach D2D user is aware of a preset interference power constraint (set by the BS) thatis tolerable by the cellular user.88Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.3.2.1 Overview of the Optimization ProblemMaximizing the minimum SINR of the data streams considering the bounded channeluncertainty as well as an interference power constraint (that is tolerable by the macro-cell user) is an NP-hard problem. Following [45], [106, Theorem 3], and [114, Claim3], we consider the corresponding inverse problem, i.e., jointly minimizing the trans-mit power subject to a minimum SINR constraint. The inverse optimization problemcan be formulated asmin.αc,Uci ,Vciαc (3.68a)subject to Tr(∆ci,j∆ci,jH) ≤ ci,j2,∀i, j, c (3.68b)max ‖(U0)HHci,0V ci ‖2F ≤ P cth, ‖∆ci,0‖F ≤ ci,0,∀i, c (3.68c)γcm,i(ucm,i,V,Λ) ≥ γcth,∀m, i, c (3.68d)‖V ci ‖2F ≤ αcµci ,∀i, c (3.68e)where U0 is the receive filter for the cellular user (denoted as user 0), Hci,0 is the channelmatrix between user i in D2D group c to the cellular user, P cth is the preset interferencepower constraint that is tolerable by the cellular user, γcth is a certain minimum SINRthreshold, µci =P cimin(P c1 ,...,PcK)and αc is an optimization variable to ensure that D2Duser i is transmitting at a reduced power of αcµci ≤ Pi. In order to simplify the aboveoptimization problem, we can follow approaches in [45, 102, 112] so as to exploitthe bound on the channel uncertainty and derive the worst-case SINR expression asshown in (3.69), in the next page. Inequality a follows from the lower and upper boundproperties of triangle inequality applied at the numerator and denominator of (3.67),respectively. Inequality b follows from the properties that Tr(A1B1) = Tr(B1A1)for any A1 ∈ CM×N , B1 ∈ CN×M and Tr(A2B2) ≤ Tr(A2)Tr(B2) for any positive89Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsdefinite matrices, A2, B2 ∈ CN×N [101]. Considering this worst-case SINR, the aboveoptimization problem can be written asmin.αc,Uci ,Vciαc (3.70a)subject to max ‖(U0)HHci,0V ci ‖2F ≤P cth, ‖∆ci,0‖F ≤ ci,0,∀i, c (3.70b)γˆcm,i(ucm,i,V) ≥ γcth,∀i,m, c (3.70c)‖V ci ‖2F ≤ αcµci ,∀i, c (3.70d)Also, similar bounding can be applied to the interference power constraint involv-ing channel uncertainty to the cellular user. Using the property of triangle inequality,the interference power constraint can be decomposed asmax ‖UH0 Hci,0V ci ‖2F , ‖∆ci,0‖F ≤ ci,0≤ ‖UH0 Hˆci,0V ci ‖2F + ci,02dci∑m=1dpu∑r=1‖ur,0‖2‖vcm,i‖2, ∀i, c (3.71)inf∆ci,jγcm,i(ucm,i,V,Λ)=‖(ucm,i)H(Hˆci,i + ∆ci,i)vcm,i‖2∑(l,j,c′)6=(m,i,c) ‖(ucm,i)H(Hˆc′j,i + ∆c′j,i)vc′l,j‖2 + ‖(ucm,i)‖2σ2i,ca≥ ‖(ucm,i)HHˆci,ivcm,i‖2 − ‖(ucm,i)H∆ci,ivcm,i‖2∑l,j,c‖(ucm,i)HHˆcj,ivcl,j‖2+∑l,j,c‖(ucm,i)H∆cj,ivcl,j‖2−‖(ucm,i)HHˆci,ivcm,i‖2−‖(ucm,i)H∆ci,ivcm,i‖2+‖(ucm,i)H‖2σ2i,cb≥ ‖(ucm,i)HHˆci,ivcm,i‖2 − ci,i2‖(uci)‖2‖vcm,i‖2∑(l,j,c′) 6=(m,i,c) ‖(ucm,i)HHˆc′j,ivc′l,j‖2 + ‖(ucm,i)H‖2∑(l,j,c′) 6=(m,i,c) c′j,i2‖vc′ l,j‖2+ ‖(ucm,i)H‖2σ2i,c, γˆcm,i(ucm,i,V). (3.69)90Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemswhere ur,0 denotes the receive filter of r-th stream for the cellular user. Therefore,the overall optimization problem can be reformulated asmin.αc,Uci ,Vciαc (3.72a)subject to ‖UH0 Hˆci,0V ci ‖2F+ci,02dci∑m=1dpu∑r=1‖ur,0‖2‖vcm,i‖2 ≤ P cth,∀i, c (3.72b)γˆcm,i(ucm,i,V) ≥ γcth, ∀m, i, c (3.72c)||V ci ||2F ≤ αcµci ,∀i, c (3.72d)The SINR constraint in the above optimization problem is jointly non-convex inthe optimization variables U ci and Vci . Therefore, we follow an alternating minimiza-tion approach to solve this problem, as in the previous section. In particular, weoptimize the receive filters with fixed precoders and then minimum SINR over allstreams across all users in any given D2D group is obtained and the target SINRis updated as well. After that, the precoders are optimized with fixed receive fil-ters. Then the minimum SINR is determined with up-scaled transmit power andtarget SINR is updated accordingly. This process continues until the minimum SINRconverges. Each of these subproblems is discussed below.3.3.3 Receive Filter DesignFor the fixed precoders, the worst-case SINR depends on the receive filters. There-fore, the optimization of the receive filters can be formulated as an unconstrained91Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemssubproblemmax. γˆcm,i(ucm,i,V),∀m, i, c (3.73)Following [101, 113], and as derived in [45, Appendix B], the optimal solution,(ucm,i), of the above unconstrained optimization problem can be derived asucm,i ={Ψcm,i}−12wcm,i||{Ψcm,i}−12wcm,i||, (3.74)where wcm,i is the principal eigenvector of (Ψcm,i)− 12 Φcm,i(Ψcm,i)− 12 , andγˆcm,i(ucm,i,V) =(ucm,i)HΦcm,iucm,i(ucm,i)HΨcm,iucm,i,whereΨcm,i =∑l,j,cHˆcj,ivcl,j(vcl,j)H(Hˆcj,i)H +∑l,j,ccj,i2||vcl,j||2IMr− Hˆci,ivcm,i(vcm,i)H(Hˆci,i)H − ci,i2||vcm,i||2IMr + σ2i,cIMr ,andΦcm,i = Hˆci,ivcm,i(vcm,i)H(Hˆci,i)H − ci,i2||vcm,i||2IMr .3.3.4 Precoder DesignAs we have studied in the case of FD communications in Section 3.2.3.2, the problemof precoder design with QoS constraint is NP-hard in general [45,106,114]. Therefore,we apply the SDR technique to solve this problem, as in the previous section [45].92Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication SystemsLet us define the precoder of the m-th stream for user i in D2D group c as V cm,i =(vcm,i)(vcm,i)H , a semidefinite matrix. Following this definition, the first term on theright hand side of (3.71) can be expanded as below‖UH0 Hˆci,0Vi,c‖2F = ‖[UH0 Hˆci,0vc1,i, UH0 Hˆci,0vc2,i, . . . , UH0 Hˆci,0vcdci ,i]‖2F= Tr[(Hˆci,0)HU0UH0 Hˆci,0Vc1,i] + . . .+ Tr[(Hˆci,0)HU0UH0 Hˆci,0Vcdci ,i]≤dci∑m=1PHˆcm,ith , (3.75)where PHˆcm,ith is the interference power due to the transmission of the m-th stream fromuser i in D2D group c that is related to the estimated channel. We define acm,i ,vec(UH0 Hˆci,0Vcm,i) such that ‖acm,i‖2 ≤ PHˆcm,ith . Using the Schur complement [101], eachof the terms in the above equation can be written into a positive semidefinite formasΘHˆcm,ith =P Hˆcm,ith (acm,i)Hacm,i I  0. (3.76)Similar steps can be followed for the remaining streams of user i in D2D groupc as well. On the other hand, each of the components of the second term on theright hand side of (3.71) can be written as, Tr[V cm,i] ≤ P∆cm,ithci,02∑dpur=1 ‖ur,0‖2, such thatci,02∑dcim=1∑dpur=1 ‖ur,0‖2Tr[V cm,i] ≤∑dcim=1 P∆cm,ith , where P∆cm,ith is the interference powercomponent due to the transmission of the m-th stream from user i in D2D group cthat is related to the channel uncertainty. Note that with fixed receive filters andpositive semidefinite V cm,i, the SINR in (3.69) can be written as, γ˜cm,i, which is givenin the following page (3.77), where Vsd denotes the set of semidefinite precoders,{(V cm,i), ∀m, i, c}. Therefore, SINR constraints in the above optimization problemnow becomes convex inequalities. Hence, the problem of optimizing the precoders93Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemswith fixed receive filters and interference power constraint can be reformulated asmin .αc,V cm,iαc (3.78a)subject todci∑m=1PHˆcm,ith +dci∑m=1P∆cm,ith ≤ P cth, ∀i, c (3.78b)ΘHˆcm,ith  0, ∀m, i, c (3.78c)Tr[V cm,i] ≤P∆cm,ithci,02∑dpur=1 ‖ur,0‖2,∀m, i, c (3.78d)γ˜cm,i(ucm,i,Vsd) ≥ γˆcth,∀m, i, c (3.78e)dci∑m=1Tr[V cm,i] ≤ αcµci ,∀i, c (3.78f)V cm,i  0,∀m, i, c (3.78g)rank(V cm,i) = 1,∀m, i, c. (3.78h)Note that the optimal α∗c that is obtained from this minimum power precoderdesign problem will return the minimum SINR threshold for the original max-minfair SINR problem. The last two constraints in the above optimization problemensue from the definition of V cm,i i.e., positive semidefiniteness of Vcm,i. However,the problem is still non-convex due to rank-1 constraint. Therefore, we relax theproblem by dropping the above rank-1 constraint, and in turn, the problem becomesa semidefinite problem that can be efficiently solved using available optimizationγ˜cm,i(ucm,i,V)=Tr((Hˆci,i)Hucm,i(ucm,i)HHˆci,iVcm,i)− ci,i2‖(ucm,i)‖2Tr(V cm,i)∑(l,j,c′) 6=(m,i,c) Tr((Hˆc′j,i)Hucm,i(ucm,i)HHˆc′j,iVc′l,j) + ‖(ucm,i)‖2∑(l,j,c′)6=(m,i,c) c′j,i2Tr(V c′m,i)+ ‖(ucm,i)‖2σ2i,c, γ˜cm,i(ucm,i,Vsd). (3.77)94Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemstoolboxes.3.3.5 Numerical Results and DiscussionsIn this section, we illustrate the worst-case data rate performance of the D2D usersin a cellular network using numerical simulations. The simulations are performed inan underlay D2D-enabled cellular network, where there is one primary-user pair andtwo D2D groups (C = 2) that coexist with the PU. Each D2D group is composed ofK = 2 secondary-user pairs, where each D2D transmitter and receiver are equippedwith two and three antennas, respectively. We also assume that primary-user linkmakes use of three antennas both at the transmitter and receiver. We further considerthat both PU and SUs send one independent data stream. The channel model usedin the simulation is a quasi-static Rayleigh flat fading channel, which is obtained bygenerating independent and identically distributed (i.i.d) Gaussian random variableswith zero mean and unit variance. Furthermore, we assume that CSI uncertaintybounds are equal, i.e., i,j = ,∀i, j. We also assume that users in the D2D groupshave the same total transmit power i.e., P c1 = Pc2 , which gives µci = 1,∀i, c.As a benchmark, we have simulated a robust non-cognitive IA system (withoutthe macrocell) to compare the worst-case data rate that is achievable from a D2Denabled cellular interference network in the presence of channel uncertainty and aninterference power constraint from the BS. For the sake of simplicity, we neglectedthe path-loss in our simulation. Fig. 3.10 shows the comparison of the worst-casedata rate over all streams across all users in a D2D group underlaying in the cellularnetwork with that of the users in an IA system without the primary network (non-cognitive system). As it is seen from the figure, the users in the D2D group generallyperform worse relative to the users in the non-cognitive system since the former needs95Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems0 5 10 15 20 250.40.50.60.70.80.911.11.21.3SNR (dB)Worst-case data rate (b/s/Hz)  Pth=0.4, Robust D2DPth=0.6, Robust D2DPth=0.8, Robust D2DRobust conventionalFigure 3.10: Worst-case stream data rate with SNR for D2D users in a cellularinterference network, C = K = 2, NT = 2, NR = NpuT = NpuR = 3, dc1 = dc2 = dpu = 1, = 0.15.to satisfy an additional interference power constraint. In addition, it is observed thatas the interference power constraint becomes stringent, the worst-case stream datarate of the users in the D2D group degrades. That is to say, the worst-case streamdata rate of the users in the D2D group approach that of the users in the non-cognitive system when the interference power constraint is relaxed. However, at lowSNR with relaxed interference power constraint, e.g., at SNR=0 dB with Pth = 0.8,the worst-case stream data rate of the users in the D2D group is very close to that ofthe users in the non-cognitive system since this relaxed interference power constraintat low SNR has minimal impact on the optimization of precoders and receive filters.Fig. 3.11 shows the worst-case stream data rate with CSI error at an SNR of96Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems0.05 0.1 0.15 0.2 0.250.650.70.750.80.850.90.9511.051.11.15CSI errorWorst-case data rate (b/s/Hz)  Pth=0.4, Robust D2DPth=0.6, Robust D2DPth=0.8, Robust D2DRobust conventionalFigure 3.11: Worst-case stream data rate with CSI error () for D2D users in a cellularinterference network, C = K = 2, NT = 2, NR = NpuT = NpuR = 3, dc1 = dc2 = dpu = 1.10 dB. As expected, as the CSI error increases i.e., the radius of the uncertaintyregion increases, the worst-case stream data rate of the users in the D2D groupdegrades. Furthermore, the D2D users perform worse than that of the users in thenon-cognitive system due to an additional interference power constraint set by themacrocell BS or the primary network. However, as seen from this figure, with relaxedinterference power constraint, the performance gap of the worst-case stream data ratebetween the users involved in the D2D communication and that of the users in thenon-cognitive system becomes narrower in the high CSI error regime. The reasonbeing at the high CSI error with relaxed interference power constraint, the CSI errorbecomes the performance bottleneck.97Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systems3.4 ConclusionsIn this chapter, we have studied the design of precoders and receive filters for thefairness problems in FD and D2D communications. In the first part of this chapter,we have studied the transmit and receive beamforming designs in order to address thefairness problem in FD MIMO multi-cell systems under the assumption of limited-DR.In particular, we consider design approaches with both perfect and imperfect CSI.Since the globally optimal solution is difficult to obtain due to the non-convex natureof the problems, we resort to alternating minimization to obtain locally optimal solu-tions. The optimization objective is found to be converging in a few iterations. Thesimulation results suggest that the FD transmission provides better sum-rate perfor-mance when compared to that of HD transmission with low to moderate transceiverdistortion. Furthermore, we notice that the spectral efficiency gain of the FD trans-mission increases with increasing distance between the small-cell BSs due to reducedinter-cell interference. In comparison to the widely studied WMMSE and Max-MinSINR algorithms, our results reveal that the proposed algorithm offers a better fair-ness among the cell-edge users. In dealing with practical design issues, our studydemonstrates that the proposed robust design similarly provides improved sum-rateperformance when compared to HD transmission in the presence of bounded CSIuncertainty. In comparison to a non-robust design, the results demonstrates that ahigher sum-rate is achievable with the aid of a robust design in the presence of CSIuncertainty.In the second part of this chapter, we have studied the same problem of robustfairness transceiver design for D2D-enabled cellular network. For this research prob-lem, our objective was to maximize the minimum SINRs of the D2D users. In orderto protect the PU (i.e., macrocell users) from the cognitive transmission of the users98Chapter 3. Fairness Considerations in Full-Duplex and Device-to-Device Communication Systemsinvolved in the D2D communications, there is an interference power constraint whichis taken into account, in addition to the CSI uncertainty. Due to the non-convexityof the optimization problem, we have followed an iterative alternative minimizationalgorithm that involves the SDR technique. In order to evaluate the performance ofthe proposed transceiver, we have compared the worst-case stream data rate of usersinvolved in the D2D communications with that of users in an IA system without theprimary network (non-cognitive case). It is observed that at low SNR with relaxedinterference power constraint, the worst-case stream data rate of the users involvedin the D2D communications is very close to that of the users in the non-cognitivesystem. However, the performance degrades considerably with stringent interferencepower constraint. Further, in line with our expectation, the worst-case stream datarate of the D2D users degrades as the CSI error increases, while the performance gapbecomes narrower at high CSI error with relaxed interference power constraint.99Chapter 4Interference Alignment for PowerLine Communications4.1 IntroductionIn Chapters 2 and 3, we investigated interference mitigations techniques for interference-limited systems, such as FD and D2D communication systems. In this chapter, wefocus on another interference-limited system, namely PLC where the system perfor-mance is limited by the inter-user interference, as presented in Chapter 1. With theexploitation of existing large power grid infrastructures, data transmission over thepower grid, i.e., PLC is a cost-efficient solution for providing communication services,such as local area networking [14]. Besides the phase-neutral (P-N) port, which canenable Single-Input Single-Output (SISO) PLC technology, many countries deploymulticonductor cables, e.g., an additional earth (E) wire, that creates more feedingand receiving possibilities. Therefore, it becomes possible to transmit and receivedata over multiple conductors and opens a way to perform spatial processing viawell-established MIMO technology [127].The multiple simultaneous pair-wise communications in power line networks closelyresemble signal transmission over an interference channel in wireless networks, whereall users share the common communication medium (cf. Fig. 1.2). Furthermore,each feeding port may also intend to send signals to each of the receiving ports in100Chapter 4. Interference Alignment for Power Line Communicationsthe network, as in wireless X-network (cf. Fig. 1.3).As presented in Chapter 1, the IA is an optimal multiplexing gain transmissionstrategy. The benefit of achieving the maximum multiplexing gain makes IA anattractive spectral efficient transmission technique for PLC networks. The key ideaof the IA is to jointly design the precoders and receive filters such that each receivefilter creates both signal and interference subspaces, while each transmitter precodesits signals in a way to make sure that for any given receiver, the signals from undesiredtransmitters fall into the interference subspace of that receiver [32].Although IA has been extensively studied in the context of wireless networks,only recently has it been studied in the context of PLC networks, as part of contri-butions of this thesis. To present a thorough treatment, in this chapter, we start byexamining the feasibility of IA in the context of MIMO PLC interference networks.In particular, we investigate practicality of the Min-IL and Max-SINR algorithms toPLC networks that were originally proposed to achieve IA in the context of MIMOwireless interference networks [128]. Assuming AWGN, our initial results show thatIA is equally applicable in the context of MIMO PLC interference networks and thesum-rate can be significantly improved through the exploitation of the Max-SINRalgorithm. It is found that at high SNR, the performance gain in terms of sum-rateover orthogonal transmissions is around 30% for a 3-user 2× 2 MIMO PLC network.Furthermore, we compare the performance of a MIMO PLC interference networkwith that of a MIMO wireless interference network having equivalent link qualitiesof the former. Our results suggest that MIMO PLC offers a lower sum-rate than theequivalent MIMO wireless network.Since noise in PLC is dominated by disturbances induced into the grid from at-tached appliances and equipments, it is generally spectrally and spatially not white.101Chapter 4. Interference Alignment for Power Line CommunicationsThis has been substantiated in various measurement campaigns for MIMO PLC inindoor environments [129–132]. While spectral dependencies are of somewhat sec-ondary importance for the operation of linear IA over spatial MIMO channel, thespatial correlation is directly relevant for the IA filter design and performance. Con-cerning this, we investigate the impact of spatial noise correlation on the performanceof linear IA for MIMO PLC. We exploit the practical noise data collected by the spe-cial task force 410 of the European Telecommunications Standards Institute (ETSI)during a measurement campaign in several homes throughout Europe [131,132]. Theavailability of the measured noise data allows us to evaluate the IA performance withpractical levels of spatial noise correlation in real MIMO PLC networks. Since theprevious results reveal that the Max-SINR based IA design provides a better sum-rate performance than the Min-IL based design and also has the ability to take intoaccount the noise statistics [128], we only consider the former for this particular anal-ysis. Our specific objectives are twofold. First, we examine the impact of spatiallycorrelated noise on performance when the Max-SINR is designed assuming indepen-dent noise across different ports, which could be considered as a mismatched design.This is to say, we investigate the benefits of incorporating noise correlation into theIA design. Second, we show that assuming spatially independent noise at differentPLC receiver ports underestimates achievable rates. Overall, our results highlightthat proper exploitation of practical noise correlation levels in PLC leads to furthergains in sum-rate for transmission with IA. More specifically, with a proper IA designthat accounts for practical noise correlation levels, a rate gain as high as 34% over amismatched IA design is obtained for the considered PLC setup.The IA techniques generally require the availability of the CSI at the transmit-ter. The BIA scheme is an exception in that it works without having the CSI at the102Chapter 4. Interference Alignment for Power Line Communicationstransmitter [133, 134]. Rather, the technique requires that the channel seen by eachreceiving port changes following a specific pattern. Even though the load modula-tion changes the channel transfer characteristics over time slots, we show that it isineffective for the purpose of the blind IA in PLC X-network. Hence, we proposea transmission scheme that offers the desired channel variation and can achieve themaximum multiplexing gain with the proposed scheme for PLC X-networks.The rest of the chapter is organized as follows. Section 4.2 discusses the IA forthe MIMO PLC interference network with different IA techniques assuming bothAWGN and MIMO PLC measured noise. In Section 4.3, we present the BIA for PLCX-networks, where we investigate different network scenarios for the feasibility of thisIA scheme. We summarize this chapter in Section 4.4.4.2 IA for MIMO PLC Interference NetworksIn this section, we investigate the technique of IA for MIMO PLC interference net-works. In the first part of this section, we concentrate on the IA design assumingAWGN, while we present results with the practical measured noise in the second partof this section.4.2.1 System ModelLet us consider a K-user MIMO PLC network, composed of K transmitter-receiver(Tx-Rx) pairs as shown in Fig. 4.1 for K = 3. Similar to wireless interference net-works, we assume that all transmitters communicate simultaneously to its intendedreceiver on a pair basis, thereby creating interference-limited communication scenariofor the PLC network. The underlying PLC network consists of 3-conductor cables.i.e., P, E, and N. Due to Kirchhoff’s circuit law, with 3 conductors it is only possible103Chapter 4. Interference Alignment for Power Line CommunicationsTx 1 Tx 2 Tx 3Rx 2Rx 1 Rx 3PENLoadFigure 4.1: Illustration of a 3-conductor cables MIMO PLC interference network with3 Tx-Rx pairs (setup-I).to feed 2 transmit signals simultaneously over three possible ports, i.e., P–N, P–E orN–E ports. While on the receiving end all 3 ports are available, we restrict our at-tention to a symmetric case of equal number of feeding and receiving ports. Anothervariation of this network setup can be obtained by merely interchanging the positionsof the corresponding Tx-Rx pairs as shown in Fig. 4.2. Additionally, more conduc-tors can be used to create more feeding and receiving possibilities. In principle, withα conductors α− 1 communication channels are available between any given Tx-Rxpair.We assume that each Tx-Rx pair makes use of Nt and Nr ports at the transmittingand receiving end, respectively. Each transmitter, i, encodes its messages, smi , ontodi ≤ min(Nt, Nr) independent parallel data streams sent along the beamformingvectors, vmi . Note that each data stream is sent along the beamforming vector, vmi ,with a power, ρmi , for the m-th stream of i-th transmitter such that∑dim=1 ρmi ≤ Pi,where Pi is the maximum power of transmitter i.104Chapter 4. Interference Alignment for Power Line CommunicationsTx 1 Tx 2 Rx 3Rx 1Rx 2 Tx 3PENLoadFigure 4.2: Illustration of a 3-conductor cables MIMO PLC interference network with3 Tx-Rx pairs, a variation of setup-I (setup-II).4.2.2 Precoding at the PLC TransmitterFor this study, we assume that perfect CSI is available at the transmitters. Thiscan be assumed for broadband PLC, where transmitter side CSI is also used for bit-loading. Let us denote Vi = [v1i , v2i , . . . , vdii ] as the CNt×di precoding matrix which isdesigned based on the available channel estimate at PLC transmitter i. The precodedsignal, Si, at transmitter i can be represented asSi =di∑m=1vmi smi= Visi. (4.1)4.2.3 Interference Suppression at the PLC ReceiverThe received signal at the corresponding PLC receiver i is given byyi = Hi,iVisi +K∑j=1,j 6=iHi,jVjsj + wi, (4.2)105Chapter 4. Interference Alignment for Power Line Communicationswhere Hi,i ∈ CNr×Nt is the channel matrix between Tx-Rx pair i and wi ∈ CNr×1 is theeffective noise vector at PLC receiver i with distribution CN (0Nr , σ2i INr). Althoughwe assume AWGN for this initial study, later in this chapter we extend our resultsfor PLC-specific measured noise. We also assume the application of multicarriermodulation, such as Orthogonal Frequency Division Multiplexing (OFDM), bearingin mind the broadband PLC systems. It allows us to decompose frequency-selectivePLC channel into a number of parallel subchannels i.e., Hi,j,∀i, j, as in subcarriers ofan OFDM system. It is worthwhile to mention that compared to the wireless channelwhich is often modeled with the assumption of independent fading coefficients in thechannel matrix H, the MIMO PLC channel shows a rather high spatial correlation[135,136].As with the precoder, transmitter also designs the filter Ui = [u1i , u2i , . . . , udii ] ∈CNr×di , for receiver i, based on the perfect CSI available at its end. After projectingthe received signal onto the columns of Ui, per-stream components of the receivedsignal at each subcarrier can be written as(umi )Hyi =(umi )HHi,ivmi smi + (umi )H∑(j,l)6=(i,m)Hi,jvljslj + (umi )Hwi, (4.3)For brevity, the signal transmission model presented above does not show anysubcarrier index. Instead it is presented for an arbitrary subcarrier keeping in mindthat real system would perform signal processing per subcarrier basis. Following(4.3), the SINR for the m-th stream of i-th user, γmi is given byγmi =ρmi ‖(umi )HHi,ivmi ‖22(umi )HΥmi (umi ), (4.4)106Chapter 4. Interference Alignment for Power Line Communicationsand the corresponding interference plus noise covariance matrix, Υmi is given byΥmi =∑(j,l) 6=(i,m)ρljHi,jvlj(vlj)H(Hi,j)H + σ2i INr . (4.5)If the interference resulting from the simultaneous transmissions of all PLC trans-mitters are aligned into the null space of Ui at PLC receiver i, the following conditionshave to be satisfied for perfect IA [128],UHi Hi,jVj = 0,∀j 6= i (4.6)rank(UHi Hi,iVi) = di,∀i, (4.7)That is, the desired signals are received through a di × di full rank channel matrixwhile the interference is completely eliminated at receiver i.4.2.4 IA AlgorithmsIn this section, we present a brief background on two popular IA algorithms, namelyMin-IL and Max-SINR, that were proposed in the context of MIMO wireless inter-ference networks [128]. These iterative algorithms were devised based on the as-sumption that communication channel is reciprocal. We can assume that the PLCchannel transfer function is also reciprocal as suggested in [136], so these algorithmsare directly applicable for MIMO PLC systems as well.4.2.4.1 Min-IL based IAThe Min-IL algorithm is designed to iteratively minimize the leakage interferencepower at each receiver in order to find the optimum precoders and receive filters. Inparticular, the goal is to progressively achieve IA by reducing leakage interference107Chapter 4. Interference Alignment for Power Line Communicationsat each receiver. The quality of alignment is measured by the power of the leakageinterference at each receiver, i.e., the interference power that remains in the receivedsignal after the receive filter is applied. To this end, IA is said to be feasible ifthe leakage interference converges to zero. The total interference leakage power atreceiver i due to transmissions of all undesired transmitters j (j 6= i) is given byΘi = Tr(UHi ΦiUi), (4.8)And the interference covariance matrix, Φi is given asΦi =K∑j=1,j 6=idj∑l=1ρljHi,jvlj(vlj)HHHi,j. (4.9)The optimum receive filter, (Ui)opt for receiver i in the original network is thencalculated by solving(Ui)opt = arg minUiΘi. (4.10)As suggested in [128], the solution to the above problem is the eigenvectors cor-responding to di smallest eigenvalues of the interference covariance matrix Φi. Thenthe transmit precoding matrix in the reciprocal network would be the receive filterin the original network as derived above. Further, in the reciprocal network, receivefilter (←−U j)opt, for receiver j is obtained by,(←−U j)opt = (Vj)opt = arg min←−U j←−Θ j, (4.11)where←−Θ j is the total interference leakage at receiver j in the reciprocal network.108Chapter 4. Interference Alignment for Power Line CommunicationsSimilarly, (←−U j)opt is obtained by taking eigenvectors corresponding to dj smallesteigenvalues of the interference covariance matrix,←−Φ j which can be obtained as←−Φ j =K∑i=1,i 6=jdi∑m=1←−ρ mi←−H j,i←−v mi (←−v mi )H←−HHj,i. (4.12)The receive filters in the reciprocal network are then used as precoding matricesin the original network. The iteration continues until the algorithm converges.4.2.4.2 Feasibility of IAIn order to determine the feasibility of the IA for a given number of data streamstransmitted, we calculate the interference leakage in the desired signal space. If didenotes the number of streams transmitted from user i, it is necessary to meet thefollowing condition in order to achieve perfect IA,di∑m=1Λm[Φi] = 0, (4.13)where Λm[B] denotes them-th smallest eigenvalue of matrixB. Note that∑dim=1 Λm[Φi]basically indicates total interference power in the desired signal space. Furthermore,the fraction of interference power in the desired signal space is given byθi =∑dim=1 Λm[Φi]Tr(Φi). (4.14)When IA is feasible, the fraction of interference leakage in the desired signal spacewill be zero.109Chapter 4. Interference Alignment for Power Line Communications4.2.4.3 Max-SINR based IAThe Min-IL algorithm discussed above only seeks to achieve perfect IA. However,it makes no effort in maximizing the desired signal power. To be precise, Min-ILalgorithm does not consider the direct links Hi,i,∀i in its design at all. Keepingthis drawback in mind, Max-SINR algorithm was proposed to maximize SINR at thereceivers, instead of only minimizing the leakage interference. To this end, the receivefilters are chosen such that the SINR of stream m for user i is maximized as below(umi )opt = arg maxumiγmi . (4.15)The unit vector that maximizes γmi is given by(umi )opt =(Υmi )−1Hi,ivmi‖(Υmi )−1Hi,ivmi ‖. (4.16)As in the case of Min-IL algorithm, we can exploit channel reciprocity in order tocalculate the precoder matrices. One can refer to [128] for the convergence behaviorof these two algorithms.4.2.5 Numerical Results with AWGNIn this section, we illustrate the performance of IA in the context of MIMO PLCinterference networks assuming AWGN, where all Tx-Rx pairs communicate simul-taneously over the same MIMO PLC channel. We consider the broadband PLC fre-quency band and calculate the channel frequency response between different Tx-Rxpairs at a frequency separation of 24.4 kHz [51]. This imitates the use of multicarriermodulation such as OFDM for broadband PLC systems. The computation of channelfrequency response is based on MTL theory which was reported in [137].110Chapter 4. Interference Alignment for Power Line Communications2 3 40102030405060708090100Percentage of feasible IATotal no. of streams  Setup-ISetup-IIFigure 4.3: Percentage of feasible IA for K = 3 with 3-conductor cables i.e., Nt =Nr = 2 as shown in Fig. 4.1 (setup-I) and Fig. 4.2 (setup-II).At first, we consider 3-conductor cables PLC networks with 3 Tx-RX pairs. Twosuch setups are shown in Fig. 4.1 (setup-I) and Fig. 4.2 (setup-II), where K = 3 andNt = Nr = 2. Fig. 4.2 is a variation of Fig. 4.1 that is obtained by interchanging Tx-Rx pairs. Next, we consider another PLC network for K = 3 but with 4-conductorcables. To this end, we obtain a 3×3 MIMO PLC channel between any given Tx-Rxpair, where Tx-Rx pairs are positioned as in Fig. 4.1 and Fig. 4.2 for setups IIIand IV, respectively. A channel emulator for such PLC networks with branches andmultiple-conductor cables is available in [138]. In order to evaluate the feasibilityof IA, we consider different numbers of streams transmitted over the PLC networkand then calculate the leakage interference power in the desired signal space as in(4.14). When IA is feasible, the fraction of interference leakage power in the desired111Chapter 4. Interference Alignment for Power Line Communications2 3 4 50102030405060708090100Percentage of feasible IATotal no. of streams  Setup-IIISetup-IVFigure 4.4: Percentage of feasible IA for K = 3 with 4-conductor cables i.e., Nt =Nr = 3. Tx-Rx pairs are positioned as in Fig. 4.1 and and Fig. 4.2 for setups III andIV, respectively.signal space will be zero. We further assume that transmitters are provided withsame power, i.e., P1 = P2 = P3 as well as the streams are allocated equal poweri.e., ρmi = ρlj,∀i, j, l,m. Also, noise is kept fixed with variance, σ2i = 1,∀i. For eachnetwork realization, distance between any given Tx-Rx pair and load resistance aregenerated independently and randomly from uniform distributions on the intervals[5, 15] m and [5, 50] Ω, respectively. We generate 1000 such network realizations.Fig. 4.3 shows the percentage of feasible IA (in terms of interference leakagepower in the desired signal space calculated within nemerical errors) for MIMO PLCnetworks shown in Fig. 4.1 and Fig. 4.2. In particular, for these setups we considerthree instances, where∑Ki=1 di = 2, 3, 4 streams are transmitted over the networks in112Chapter 4. Interference Alignment for Power Line CommunicationsFigs. 4.1 and 4.2. Note that for K = 3 with Nt = Nr = 2, the number of interference-free streams or DoF for each user is upper bounded by di =Nt+NrK+1= 1,∀i [32].Therefore, the maximum achievable DoFs for both of these networks are the same,i.e,∑Ki=1 di =KNt2= 3. We observe from Fig. 4.3 that when∑Ki=1 di = 2, 3 streamsare transmitted over the setups in Fig. 4.1 and Fig. 4.2, IA is feasible in mostof the realizations. That is to say, interference leakage power in the desired signalspace is zero, and therefore feasible for these DoF allocations which is in line withMIMO wireless interference networks. On the other hand, transmitting more than3 streams i.e.,∑Ki=1 di = 4 streams, yields interference leakage to the desired signalspace, therefore IA is infeasible for this DoF allocation. It is important to note thatwhen the interference leakage in the desired signal space is not zero for some of therealizations as it is seen from Fig. 4.3 while∑Ki=1 di = 2, 3 are transmitted, it doesnot necessarily mean that IA is infeasible for these DoF allocations. That is becausethe iterative Min-IL algorithm may not converge to the global minimum, yieldingnon-zero interference leakage in the desired signal space.Fig. 4.4 shows the percentage of feasible IA for a PLC network with 4-conductorcables, i.e., Nt = Nr = 3 and K = 3, for setups III and IV as discussed in thebeginning of this section. The maximum achievable DoFs for these networks areupper bounded as∑Ki=1 di =KNt2= 4.5. To determine the feasibility of IA, wetransmit∑Ki=1 di = 2, 3, 4, 5 streams over these PLC networks. Similar to the previousfigure, when∑Ki=1 di = 2, 3 streams are transmitted over these setups, in most ofthe realizations the percentage of interference leakage in the desired signal space iszero, and therefore feasible for these DoF allocations. Although the total numberof DoFs for these setups are upper bound by 4.5, transmitting∑Ki=1 di = 4 streamsover these networks are infeasible since the total DoF allocation in this case is also113Chapter 4. Interference Alignment for Power Line Communications0 5 10 15 20 25 30 350.050.10.150.20.25SNR (dB)Average sum-rate (Mb/s)  IA (Max-SINR)IA (Min-IL)OrthogonalizationFigure 4.5: Comparison of average sum-rates of IA and orthogonal transmissions forMIMO PLC network in Fig. 4.1.limited by DoF upper bound of an individual user. As an example, a choice ofd1 = 2, d2 = d3 = 1 would be infeasible since user 1 violates the upper bound for itsDoF allocation. In the same context, d1 = d2 = 2, d3 = 0 would be infeasible. Lastly,a total DoF allocation of 5 streams is clearly infeasible, which is also confirmed bythe simulation results as shown in Fig. 4.4. Therefore, these results further validatethe feasibility of IA for PLC interference networks.4.2.5.1 Comparison with Orthogonal TransmissionIn this section, we compare the sum-rate performances of IA algorithms with trans-mission techniques that are based on channel orthogonalization, such as Time Di-vision Multiple Access (TDMA). With orthogonal transmission, it is only possible114Chapter 4. Interference Alignment for Power Line Communicationsto transmit 2 streams simultaneously at a given time-frequency resource over 2 an-tennas. To this end, we compute the singular value decomposition of MIMO PLCchannel and then 2 data streams are transmitted on the eigenmodes of the channel.To be fair with IA based precoding, orthogonal transmissions are provided with 3times transmission power i.e., Po = 3Pi,∀i for K = 3 at a given time/frequency slot.Fig. 4.5 compares the sum-rates for the Max-SINR, Min-IL and orthogonal trans-missions for the network setup in Fig. 4.1. As it is observed, the Max-SINR basedIA provides a higher sum-rate than the Min-IL algorithm based IA design. This isinline with the MIMO wireless networks as the Max-SINR algorithm considers directlinks Hi,i,∀i in the design of precoders and receive filters in an effort to maximizethe SINRs of direct links, thus provides higher sum-rate. Furthermore, the Max-SINR based IA outperforms channel orthogonalization technique, which is also inline with our expectation since channel orthogonalization only allows simultaneoustransmission of 2 data streams whereas the Max-SINR based IA allows simultaneoustransmission of 3 streams. To be precise, the sum-rate gain at high SNR for Max-SINR based IA over orthogonal transmission is around 30% for this network setup. Incontrast to the wireless communications, we see that the Min-IL algorithm performsworse than orthogonal transmission. This may be attributed to the fact that MIMOPLC channel exhibits a high level of spatial correlation, resulting a lower SINR forthe direct links Hi,i,∀i.4.2.5.2 Comparison with Wireless CommunicationIn this section, we provide a comparison between sum-rate performances of the MIMOPLC interference network and that of a MIMO wireless interference network withequivalent link qualities for IA transmission. In particular, we consider the setup in115Chapter 4. Interference Alignment for Power Line Communications0 5 10 15 20 25 30 350.050.10.150.20.250.3Average sum-rate (Mb/s)SNR (dB)  MIMO Wireless (Max-SINR)MIMO Wireless (Min-IL)MIMO PLC (Max-SINR)MIMO PLC (Min-IL)Figure 4.6: Comparison of average sum-rates with IA algorithms for the MIMO PLCnetwork in Fig. 4.1 and a wireless interference network with equivalent link qualities.Fig. 4.1 and compute the average power gains of the channel given as belowβi,j =1Nsim ×NsubNsim∑q=1Nsub∑f=1|Hq,fi,j |2, (4.17)where Hq,ri,j is the (i, j) elements in channel H for the q-th network realization and ffrequency value, Nsim and Nsub are the number of network realizations and frequenciesconsidered in the simulation. We then use these average power gains, βi,j, as thesecond order moments of Rayleigh distributions to generate the channel gains of thecorresponding wireless MIMO links. This way we can compare the sum-rate for theMIMO PLC interference network with a similar MIMO wireless interference networkhaving complex Gaussian channel coefficients with the same average gains for all116Chapter 4. Interference Alignment for Power Line Communicationsdirect and cross-channel coefficients.Fig. 4.6 compares the sum-rates for Max-SINR and Min-IL algorithms in thecontext of MIMO PLC setup in Fig. 4.1 and an equivalent wireless interferencenetwork, for which channel gains are obtained as discussed above. It is observed formthis result that the IA algorithms for MIMO PLC interference network provide lowersum-rates than that can be obtained from the equivalent MIMO wireless interferencenetwork, although we use the same IA algorithms for both communication scenarios.We note that this is due to the fact that MIMO PLC channel exhibits relativelyhigher spatial correlation than the wireless channel, which causes the degradation inthe sum-rate performance [136].4.2.6 Numerical Results with Practical Measured NoiseIn this section, we present our results with practical measure MIMO PLC noise. Tothis end, we restrict our analysis to Max-SINR based IA since it provides a bettersum-rate performance than the Min-IL based design and also has the ability to takeinto account the noise statistics. Different from previous section, we now assume thatthe noise wi is Gaussian distributed with the covariance matrix Ψi,Ψi =ψii ψijψji ψjj , (4.18)whose off-diagonal elements are non-zero for spatially correlated receiving ports [129]–[132].To quantify the performance of the Max-SINR based IA design in the presence ofspatially correlated noise, we simulate an interference-limited MIMO PLC networkas it is illustrated in Fig. 4.1. For this simulation, we set the distance between117Chapter 4. Interference Alignment for Power Line Communicationseach Tx-Rx pair at 100 and 200 meters to simulate two different network scenarios,i.e., network scenarios 1 and 2, respectively. The shorter distance between the Tx-RX pairs may replicate a power line wiring in a house, while the larger distancereplicates the same in a large office. In both network scenarios, PLC transmittersand receivers are terminated with load resistances chosen from the range between 10Ωto 50Ω. We consider the same frequency band from 2 to 30 MHz and calculate thechannel frequency responses between different Tx-Rx pairs at a frequency separationof 24.4 kHz. This imitates the use of OFDM for broadband PLC systems as in theprevious section. For the PLC MIMO transmission, we consider two configurations:transmission using the P-E/N-E and the P-N/N-E ports, respectively. The totaltransmit Power Spectral Density (PSD) over both ports is set to −55 dBm/Hz, whichis a commonly applied spectral mask for broadband PLC systems [127].The additive noise is taken from measured noise traces made available to us andreported in [131, 132]. Since the measurement probe applied a star-style receiver[131, 132] while we consider a delta-type receiver (see Fig. 4.2, cf. [127, Ch. 1]), wetake the difference of time domain measured noise between ports P, E, and N, toobtain equivalent noise in delta-type receiver configuration. We denote such possiblemodes as P-N, P-E, and N-E. We then obtain the noise covariance matrix of thenoise through the use of Welch’s spectral estimation technique. We assume that eachtransmitter sends a single data stream, i.e., di = 1, ∀i, and that Pi = P , ∀i, accordingto the above-mentioned PSD limit.We now investigate the rate performances for two IA designs: one that incor-porates the noise correlation in (4.18), i.e., a proper design and one that assumesuncorrelated noise in the IA filter computation (4.16), thus rendering a mismatcheddesign. The latter can be considered as a benchmark design, for the case that an esti-118Chapter 4. Interference Alignment for Power Line Communications0 200 400 600 800 1000050100150200250300350400Gain over mismatched IA design (%)Subcarriers  Individual rate gain across P−E/N−E portsAverage rate gain across P−E/N−E portsFigure 4.7: Subcarrier rate gain for a matched IA design with correlated noise overa mismatched IA design (i.e., that ignores spatial noise correlation during IA fil-ter computation) for OL 3 of the dataset 1 across P-E/N-E ports in the networkscenario 2.mation of the noise correlation is not attempted. We first evaluate the performancesof these designs for a noise dataset collected in Valencia, Spain (henceforth, denoted asdataset 1) at three different outlets (OLs) in the frequency range 2−30 MHz [131,132].While receiving ports may have roughly similar noise PSDs, the spatial correlationsacross these ports can be substantially different. For example, the average noisePSDs of the P-E and N-E receiving ports at OL 3 are about −67 dBm/Hz, while thespatial correlations across P-E/N-E and P-N/N-E receiving ports are 0.56 and 0.28,respectively.Figs. 4.7 and 4.8 illustrate the rate gains of the matched (proper) IA design overthe mismatched one as a function of the subcarrier index using the measured noise at119Chapter 4. Interference Alignment for Power Line Communications0 200 400 600 800 1000020406080100120Gain over mismatched IA design (%)Subcarriers  Individual rate gain across P−N/N−E portsAverage rate gain across P−N/N−E portsFigure 4.8: Subcarrier rate gain for a matched IA design with correlated noise overa mismatched IA design across P-N/N-E ports with the same network configurationas in Fig. 4.7.OL 3 and network scenario 2 across P-E/N-E and P-N/N-E ports, respectively. Wenotice that considering actual noise correlation in the IA design leads to significantimprovements in the rate performances. For example, the improvement for transmis-sion over P-E/N-E ports, as shown in Fig. 4.7, can be as high as 390% for individualsubcarriers and it is 34% on average. Comparing Figs. 4.7 and 4.8, we also observethat the (average) sum-rate gain is substantially larger for the port configurationwith the large noise correlation. This is not only due to a more pronounced receivermismatch but also due to fact that noise correlation generally improves the achiev-able rate. Finally, we note that for few subcarriers the rate gain is negative, i.e., themismatched IA design assuming uncorrelated noise performs better than the matcheddesign. We attribute this to the fact that the Max-SINR algorithm may be stuck in120Chapter 4. Interference Alignment for Power Line CommunicationsTable 4.1: Rate gain of the matched over the mismatched IA design for different noisedata sets and PLC MIMO configurations.Network Scenario 1 Network Scenario 2Noise Datasets Outlets Avg. Gain (P-E/N-E) Avg. Gain (P-N/N-E) Avg. Gain (P-E/N-E) Avg. Gain (P-N/N-E)1 1 16.92% 10.90% 28.17% 16.48%2 14.88% 5.23% 31.19% 8.96%3 16.80% 8.05% 34.38% 12.66%2 1 6.64% 5.49% 19.17% 15.78%2 11.93% 6.22% 30.33% 15.75%a local optimum as convergence to the global optimum point is not guaranteed dueto the non-convex nature of the optimization problem [128].Table 4.1 summarizes the rate gain results for noise data collected at the OLs fortwo different locations. Here, we also provide the results for a noise dataset fromPaiporta, Spain (henceforth, denoted as dataset 2) at two different OLs. The noisein this location has a relatively lower average PSD than that of the dataset 1 (e.g.,at OL 2, the average noise PSDs of P-E and N-E receiving ports are about −75dBm/Hz). Hence, the signal to noise ratio (SNR) is relatively higher than for thedataset 1. While the average correlation coefficient across P-E/N-E ports at this OLis 0.53 and thus similar to that in OL 3 of the dataset 1, the average correlationcoefficient across the P-N/N-E ports is significantly higher at 0.37.We observe that notable rate gains are achieved for all test cases. Although, theyare relatively lower for the higher SNR case, e.g., network scenario 1 vs. networkscenario 2 due to short transmission distances for the former, and OL 3 of the dataset 1vs. OL 1 of the dataset 2 due to lower noise PSD for the latter. Owing to the highercorrelation across P-N/N-E ports for the dataset 2, however, the rate gains for theseports are comparable in the network scenario 2 despite having lower noise PSD. Asimilar explanation holds across P-E/N-E ports, e.g., for OL 2 of the dataset 2 whencompared with OL 1 of the dataset 1 in the network scenario 2.121Chapter 4. Interference Alignment for Power Line Communications1 2 3 4 500.511.522.53x 107Different transmission schemesSum−rate (b/s)  IA, Matched designIA, Mismatched designIA, Uncorrelated noiseOrthogonal transmission, Po=POrthogonal transmission, Po=KPFigure 4.9: Comparison of sum-rate performances among different IA designs andorthogonal transmissions for OL 3 across P-E/N-E ports of the dataset 1 in thenetwork scenario 2.We finally compare the system sum-rate performances for the network scenario 2considering correlated noise from the dataset 1 at OL 3 for (1) a matched and (2)a mismatched IA design, as well as (3) an IA design assuming uncorrelated noisewith the same noise PSDs at the individual ports, i.e., the case that spatial noisecorrelation is absent. Fig. 4.9 shows the corresponding system sum-rate results. As abaseline, we also provide the results with conventional orthogonal transmission, suchas time division multiple access. While in one instance, labelled as case (4), we assumea transmit power, Po = Pi = P, ∀i, for orthogonal transmission, we also simulate thecase (5) where Po = KP , i.e., the system transmit power is identical to that inIA where all K users transmit simultaneously, as in previous section. Comparing122Chapter 4. Interference Alignment for Power Line Communicationscases (1) and (3) we observe that a channel with noise correlation is supporting anotably increased rate and that the matched IA design is able to reap those rate gains.Furthermore, comparing cases (1) to both (4) and (5), we conclude that the Max-SINR based IA design outperforms conventional channel orthogonalization techniquein terms of system sum-rate, even if the latter was allowed to transmit with a muchhigher per-user transmit power.4.3 BIA for PLC X-NetworksIn this section, we turn our focus on the PLC X-network, and specifically investigatethe BIA for these networks. In the first part of this section, we show that theseemingly simple realization of the channel variation through impedance modulation,which has been inspired by the known fact that load changes affect PLC channelfrequency responses, is not effective for the purpose of BIA for the PLC X-network.We then propose a transmission scheme through which one can realize the BIA forthe PLC X-network.4.3.1 Blind Interference AlignmentFig. 4.10 shows a simplified version of Fig. 1.3 in Chapter 1 for the X-channel scenario,in which two senders transmit messages to two users [133]. The X-channel is a 4-node network where two nodes ni, i ∈ {1, 2}, intend to communicate data uij to twodifferent nodes nj, j ∈ {3, 4}. It is shown in [133] that the maximum multiplexinggain of this channel can be achieved without CSI at the transmitters as follows.123Chapter 4. Interference Alignment for Power Line CommunicationsH23Sender n1Sender n2Receiver n3Receiver n4H24H13H14Figure 4.10: A simple transmission scenario of an X-channel setting (cf. Fig. 1.3).Nodes n1 and n2 transmitx1 =u13 + u14u13 + u140 (4.19)andx2 =u23 + u240u23 + u24 , (4.20)respectively, in three successive time slots. We also assume that channel frequency-selectivity is dealt with through the use of multicarrier modulation, such as OFDM.We therefore do not indicate this notation in the following. Denoting the channelgains from node ni to node nj in time slot k by Hij(k), node nj receivesyj =H1j(1)H1j(2)H1j(3) ◦ x1 +H2j(1)H2j(2)H2j(3) ◦ x2 +w. (4.21)Defining Hj(k) = [H1j(k)H2j(k)], uj = [u1j u2j]T and 0 = [0 0], the received124Chapter 4. Interference Alignment for Power Line Communicationssignal at node n3 can be rewritten asy3 =H3(1)H3(2)0︸ ︷︷ ︸M1u3 +H3(1)0H3(3)︸ ︷︷ ︸M2u4 +w . (4.22)For node n3 to recover u3 from y3 without interference from the signal intended fornode n4, the trick of interference alignment is to ensure thatH3(1) 6= κH3(2)H3(1) = H3(3)(4.23)for any constant κ. Then, matrices M 1 and M 2 in (4.22) have ranks 2 and 1,respectively, so that interference can completely be cancelled at node n3 withoutcancelling the desired signal u3. The conditions in (4.23) can be accomplished blindly,i.e., without CSI at the transmitters. Rewriting Eq. (4.21) for node n4, we find thatthe same interference cancellation can be achieved by enforcing H4(1) 6= κH4(3)and H4(1) = H4(2). Since four data symbols are transmitted in three time slots, amultiplexing gain of 4/3 is achieved, which is the maximum possible for this network[133].4.3.2 Feasibility of the BIA Through ImpedanceModulationIn this section, we make an attempt to achieve the BIA through impedance modula-tion, which would make it applicable to two-conductor e.g., SISO PLC networks. Aspresented in the previous section, the BIA scheme requires certain desired patterns125Chapter 4. Interference Alignment for Power Line CommunicationsZB2[A2 B2C2 D2][A3 B3C3 D3]ZB3Z2B[A1 B1C1 D1]Z1BZ3BZB1nBZ1 (Z1,S, Z1,L)n1Z2 (Z2,S, Z2,L)n2Z3 (Z3,S, Z3,L)n3Figure 4.11: PLC network with three communication nodes.of channel variations over time slots so as to facilitate the interference cancellation ata given receiver port. A seemingly possible way to achieve these channel variationsis by modulating the input impedance of the PLC receiver modem. Accordingly, westart by considering a two-conductor PLC setup so that the network componentsbetween the three PLC devices, labeled as n1, n2, and n3 in Fig. 4.11, and nodenB are represented through ABCD transmission line parameters. The correspondingABCD-matrices[Ai BiCi Di](see Appendix C.1 for details) relate the appropriatelyoriented voltages and currents at node ni and node nB with each other [139]. Wenote that they capture the aggregate effects of transmission lines, branches, loadsetc. including those components that are physically behind nodes ni from node nB’sperspective. For the following discussion it is irrelevant that the ABCD parametersare frequency-dependent, and we therefore do not indicate this dependency in ournotations.The benefit of the abstract model in Fig. 4.11 is that it fully captures the interde-pendencies of signals communicated between the PLC devices. First, as mentionedabove, all signals have to travel through node nB, which therefore is also referred126Chapter 4. Interference Alignment for Power Line Communicationsto as a keyhole [140]. Therefore, signals that are transmitted between, for exam-ple, node n1 and nodes n2 and n3 have part of their transmission path in common,namely from node n1 to node nB in this case. Secondly, changes of transmissionline parameters in one segment of the network, for example in the segment betweennodes nB and n3, have an effect on the signal propagation in other segments of thenetwork, for example in the segments between nodes n1 and nB and between n2 andnB. These features of multi-node transmission in PLC networks are different fromwireless communication and have deep implications as we present below.We now express the various relevant channel frequency responses and impedancesfor the network in Fig. 4.11. To this end, we denote the source impedance whenPLC device i is transmitting as Zi,S, Zi,L is the load impedance when PLC device i isreceiving, and when we do not need to specify whether a PLC device is transmittingor receiving, we refer to its impedance as Zi, i = 1, 2, 3. Furthermore, we denotethe impedance seen into/from node nB from/to node ni as ZiB and ZBi, respectively.Then, from (C.3) (derived in Appendix C.1) it follows thatZBi =DiZi +BiCiZi + Ai, (4.24)andZiB = ZBj‖ZBk = 11/ZBj + 1/ZBk, (4.25)where j 6= k, j 6= i, i, j, k ∈ {1, 2, 3}. Using (C.1) and (C.3), the voltage channelfrequency response from node ni to nB and from node nB and to ni can be expressedasHiB ,VBVi,S=ZiB(Ai + CiZi,S)ZiB +Bi +DiZi,S(4.26)127Chapter 4. Interference Alignment for Power Line CommunicationsandHBi ,Vi,LVB=Zi,LDiZi,L +Bi, (4.27)respectively, where Vi,S is the voltage of the source before its internal impedance Zi,S,Vi,L is the voltage at load impedance Zi,L, and VB is the voltage at node nB.Using (4.26) and (4.27), the channel frequency responses between the three nodesn1, n2, and n3 can be written asH12 = H1BHB2, (4.28)H13 = H1BHB3, (4.29)H23 = H2BHB3. (4.30)Now, we utilize these insights from the network model in Fig. 4.11. We notethat for the case of BIA in wireless communications [133], the requirement (4.23)has been met through staggered antenna switching at the receiving nodes. That is,a first receiver antenna configuration is chosen during slots k = 1 and k = 3, and asecond antenna configuration is applied during k = 2. In the case of PLC networks, areceiver modem connected to two conductors can change its input impedance. Sincethe frequency response to a device is dependent on this impedance, see e.g. expression(4.27), the BIA scheme as described above seems to be directly applicable, withoutthe need for an additional infrastructure, i.e., extra receiving ports.Let us consider nodes n1 and n2 in Fig. 4.11 as the two transmitters and node n3as one of the receivers. Then, we can write the channel vector from n1 and n2 to n3asH3(k) = [H13(k) H23(k)] (4.31)128Chapter 4. Interference Alignment for Power Line Communications= [H1B(k)HB3(k) H2B(k)HB3(k)] , (4.32)where time variation with k is accomplished through modifying Z3,L at node n3. Wenote that Z3,L directly affects HB3(k) via (4.27), but also H1B(k) and H2B(k) throughthe dependency chain:Z3,LEq.(4.24)−→ ZB3 Eq.(4.25)−→ (Z1B, Z2B) Eq.(4.26)−→ (H1B(k), H2B(k)) .On the other hand, we can rearrange (4.31) intoH3(k) = H23(k)[H13(k)/H23(k) 1] . (4.33)In Appendix C.2, we show that the ratio H13(k)/H23(k) is independent of ZB3. Hence,H13(k)/H23(k) = H13/H23 , c independent of k, and thus (see (4.22))M 1 =H3(1)H3(2)0 (4.34)=c ·H23(1) H23(1)c ·H23(2) H23(2)0 0 . (4.35)The rank of matrix M 1 is one, regardless of how H23(k) is changed due to reconfig-uration at node n3. Hence, spatial multiplexing is not achieved.We thus have shown that the BIA through receiver reconfiguration, a seeminglyattractive scheme for PLC networks, is not possible in principle due to the propertiesof the transmission-line signal propagation.129Chapter 4. Interference Alignment for Power Line CommunicationsPENTx 1Rx 1Tx 2Rx 2Figure 4.12: An illustration of the transmission scheme for the achievability of blindIA for the PLC X-network using multiple receiving ports.4.3.3 Achievability of the BIA for PLC X-NetworksIt is apparent from the above analysis that the ability to control the channel responseto a desired receiving port while keeping channel responses to other receiver portsunchanged at the same time is a daunting task. In this section, we show that this canbe achieved via a concept similar to antenna-staggering in the case of wireless com-munications. However, this requires multiple receiving ports, i.e., a multi-conductorstructure as in MIMO or Single-Input Multi-Output (SIMO) PLC networks.To prove this concept, we focus on a MIMO PLC X-network with switches at thereceiving end to alternate between the receiving ports, as it is illustrated in Fig. 4.12.The switching between the receiver ports essentially serves the same purpose of an-tenna switching in wireless communications [133]. Since switching at a given receivingport has negligible impact on the channel response to the other receiving port dueto the attached impedance in parallel as shown in Fig. 4.12, it can produce desiredchannel patterns for the realization of the blind IA.In order to simulate the blind IA, we consider a MIMO PLC X-network configu-ration with two transmitters and two receivers, as it is illustrated in Fig. 4.12. Note130Chapter 4. Interference Alignment for Power Line Communications0 20 40 60 80 10000.20.40.60.811.21.4SNR (dB)Multiplexing gain  Blind IAOrthogonalFigure 4.13: A comparison of multiplexing gains for the proposed BIA and an or-thogonal transmission.that we are not specifically restricted to a MIMO configuration, since a SIMO config-uration also offers multiple receiving ports. We assume a distance of 20 m betweenthe nodes, i.e., between Tx1 and Rx2, Rx2 and Tx2, Tx2 to Rx1. The positionsof transmitters and receivers are not limited to this configuration as one can obtaina different configuration by placing the transmitters and the receivers at differentlocations. The PLC transmitters and receivers are terminated with load resistanceschosen from the range between 25 Ω to 100 Ω. We choose the frequency band from 2to 30 MHz with a frequency separation of 24.4 kHz for the OFDM system. As inthe previous cases, the computation of channel frequency response is based on MTLtheory [137], via a simulator reported in [138]. We further assume that both trans-mitters are provided with same power, i.e., P1 = P2 and noise is kept fixed with131Chapter 4. Interference Alignment for Power Line Communicationsvariance equal to one. This permits the use of transmission power as the transmitterside SNR. We note that each transmitter sends a single data stream intended for eachof the receivers in the PLC X-network.To show the effectiveness of the proposed technique, we compute the multiplex-ing gain, which can be obtained by calculating the slope of total sum-rate curve [32],and plot with respect to the transmitter side SNR. The results in Fig. 4.13 suggestthat multiplexing gains of 4/3 and 1 can be achieved with BIA and an orthogonaltransmissions, respectively. Since an orthogonal transmission uses channel orthogo-nalization, the maximum multiplexing gain for this scheme is 1. We also note that forthe considered PLC X-network configuration in Fig. 4.12, the maximum multiplexinggain that is achievable is 4/3, as we show in Section 4.3.1 [133]. This confirms theachievability of the BIA with the proposed multiconductor transmission scheme.4.4 ConclusionsIn this chapter, we have addressed the IA design for PLC networks. As this is the firststudy considering IA for PLC networks, we have confirmed the feasibility of IA by in-vestigating different PLC network setups. It was found that the feasibility conditionsfor IA that exist in the context of MIMO wireless interference networks also hold forMIMO PLC interference network. For a 3-user 2×2 MIMO PLC network, our resultsshow that at high SNR, the Max-SINR based IA provides a significant performancegain of around 30% over the orthogonal transmission techniques. Furthermore, wecompared the sum-rate performance of a MIMO PLC interference network with thatof a MIMO wireless interference network having equivalent link qualities. It was ob-served that IA in the context of MIMO PLC provides lower sum-rate than its wirelesscounterpart. It can be attributed to the fact that MIMO PLC channel exhibits rela-132Chapter 4. Interference Alignment for Power Line Communicationstively higher spatial correlation than the MIMO wireless channel, which degrades thesum-rate. Moreover, we have evaluated the system sum-rate in the presence of thespatially correlated measured MIMO PLC noise. In the process, we have learnt thatthe spatial correlation actually helps to improve the system sum-rate, which justifiesits inclusion in the system design. Finally, we have considered the BIA for PLCX-networks. It is shown that even though the load modulation changes the channelresponses over time slots, it is ineffective for the purpose of the blind IA in PLCX-network. We then propose a multiconductor network configuration with multiplereceiving ports, essentially to serve the same purpose of antenna switching in wire-less communications. The transmission scheme is shown to achieve the maximummultiplexing gain with BIA for the PLC X-network.133Chapter 5Summary and Directions forFuture Works5.1 Summary of ContributionsThe research works presented in this thesis are focused on providing solutions forinterference-limited communication systems, such as FD, D2D, and power line com-munications. In the specific context of FD communications, we proposed robust,power-efficient, and fair transceiver designs for multi-cell MIMO FD communicationsystems. The concept of fair transceiver design is also studied in the context ofunderlay D2D networks. As a solution to mitigate interference in PLC networks,we investigated different IA techniques in the context of PLC interference and X-networks.In Chapter 2, we proposed a power-efficient design for multi-cell MIMO FD com-munications. To this end, the original non-convex and NP-hard problem is posedas a DCP and efficiently solved via SCA. In addition to assuming the availability ofperfect CSI, we considered both stochastic and bounded uncertainty in our designs.The results quantified potential power savings, with respect to an HD setup and anon-robust design, under a wide range of design parameters.Another design objective that has been investigated in Chapter 3 is the fairnessproblem in both FD and D2D communications. The sum-rate maximization approach134Chapter 5. Summary and Directions for Future Worksonly aims at optimizing the total network throughput, leaving users at the cell-edgeor the ones experiencing poor channel conditions unserved. Therefore, we proposed atransceiver design that guarantees performance fairness among the users. In additionto perfect CSI design, we considered imperfect CSI design by way of norm-boundeduncertainty to provide worst-case design formulation. The non-convex precoder op-timization problem is solved via a low-complexity iterative algorithm. The resultsconfirm a considerable improvement in fairness performance among the FD and D2Dusers.In Chapter 4, we studied IA techniques for rate improvement in PLC networks.Recognizing that multiple simultaneous connections in PLC lead to an interference-limited communication as in the wireless networks, we studied the applicability ofIA in the context of both PLC interference and X-networks. Initially, we focusedon studying the feasibility and evaluated the system sum-rate performance with theIA transmission assuming AWGN for MIMO PLC networks. As an extension of thiswork, we utilized measured noise from real MIMO PLC networks to evaluate thesystem performance. In particular, we investigated the impact of spatially correlatednoise with Max-SINR algorithm which takes into account noise statistics. Our resultsquantified improved sum-rate performance with spatially correlated noise, suggestingthat the noise correlation must be taken into account in the system design. Finally,we investigated the feasibility and achievability of the BIA for PLC X-networks. Inparticular, we explored network scenarios where the BIA is not achievable through theimpedance modulation. Therefore, we proposed a transmission scheme that facilitatesthe implementation of the BIA for the PLC X-network. The results confirm that theoptimal multiplexing gain can be obtained with the proposed transmission schemefor the PLC X-network.135Chapter 5. Summary and Directions for Future Works5.2 Directions for Future WorksThe complicated interference scenarios in FD, D2D and power line communicationspresent numerous challenging research problems. Some interesting avenues for futureresearch are summarized below.5.2.1 Decentralized Algorithms for FD CommunicationSystemsThe algorithms presented in Chapter 2 for the sum-power minimization in multi-cellMIMO FD communications rely on processing at the Remote Centralized Processor(RCP) unit. The RCP facilitates the computation and distribution of the precodersand the receiver filters among the BSs and users. Therefore, the implementationof the algorithm is limited by the backhaul capacity and computational power ofthe RCP. Furthermore, it necessitates the symbol level synchronization due to jointprocessing of the transmitted signals. In an effort to mitigate these implementationissues, an important future direction for this work would be investigating decentral-ized algorithms. A decentralized algorithm would allow the implementation withlimited backhaul capacity, less computation power, and flexible system requirements(e.g., strict carrier phase synchronization would no longer be needed at the BSs) [141].The decentralized algorithm can be enabled by controlling the inter-cell interfer-ence, while designing the precoders and receiver filters with minimal cooperation atthe BSs and the mobile users. Since the design problems for precoders are inher-ently coupled among the BSs, one way to decouple the precoder design problem is toexploit primal decomposition method so that each BS can independently design theprecoders for its own users [71].Although the beamformer design with perfect CSI in Chapter 3 can be imple-136Chapter 5. Summary and Directions for Future Worksmented in a distributed fashion, the robust fairness design relies on the centralizedprocessing. Therefore, a decentralized algorithm under constrained feedback capacityis also an interesting research problem for this design objective.5.2.2 Robust Multi-cell D2D CommunicationsIn Section 3.3 of the Chapter 3, we assumed that the inter-cell interference thatresults from transmissions of users in the neighboring macrocells is mitigated throughmacrocell coordination, and is therefore neglected in our analysis. As a next step, onecan consider such interference from neighboring macrocells as a colored noise in thereceived signal, as shown on the right side of Fig. 5.1. This will allow us to analyzethe impact of such interference on the performance of D2D users. Furthermore, inour previous analysis we assumed that the interference from cellular users to the D2Dreceivers are negligible. Although this is a valid assumption when the D2D users arelocated at a distance from the cellular users so that the interference can be neglected,a more inclusive communication scenario can be considered by assuming that thereare cellular users nearby that interfere the D2D reception in the uplink transmission.Another related future work for this topic would be considering the total interfer-ence power constraint from a given D2D group, rather than the individual interferencepower constraint from each D2D user. This assumption is less restrictive on the trans-mission power of D2D users, and hence it may improve the overall performance ofD2D communications.137Chapter 5. Summary and Directions for Future WorksBase Station D2D TransmitterD2D ReceiverCellular UserD2D Group BD2D Group AFigure 5.1: An illustration of an underlay D2D communication in a cellular networkwith resulting interference from cellular users and neighboring macrocells. Solid anddashed lines indicate desired signals and interference, respectively.5.2.3 Feedback Reduction for IA in PLC NetworksOne important characteristics of the PLC channel is that it is often correlated in thetime domain. As we discussed in Chapter 4, the implementation of the IA techniquerequires the availability of the CSI at the transmitter. In practice, the reduction of theCSI feedback is a key design aspect for practical implementation of IA algorithms.Therefore, a relevant future research direction for this work is to exploit the timecorrelation of the PLC channel in order to reduce the amount of the CSI feedback.There are a few studies in the context of wireless networks, which may be readilyapplicable in the context of PLC networks as well [142,143].5.3 Concluding RemarksThe performance benefits and implementation challenges of the IA techniques andthe FD communications have rekindled the interest in developing novel physical layersolutions. 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Gameiro, “Low-bit rate feed-back strategies for iterative IA-precoded MIMO-OFDM-based systems,” TheScientific World Journal, vol. 2014, Article ID 619454, 11 pages, 2014.doi:10.1155/2014/619454152Appendix AAppendix for AdditionalPublicationsA.1 Additional Publications from Ph.D.Research (not included in this thesis)• Md. Jahidur Rahman and Lutz Lampe, “Robust MSE-based Transceiver Opti-mization for Downlink Cellular Interference Alignment,” in Proceedings of IEEEInternational Conference on Communications (ICC), Track: Cognitive Radioand Networks, pp. 4624–4629, Sept. 2015.• Md. Jahidur Rahman, Moslem Noori, and Lutz Lampe, “A Low-ComplexityDesign for Robust SINR Fairness in MIMO Interference Networks,” in Proceed-ings of IEEE Wireless Communications and Networking Conference (WCNC),Track: PHY and Fundamentals, pp. 64–69, Mar. 2015.153Appendix BAppendix for Chapter 3B.1 Replacement of the Unit Norm Constraintin (3.16)We recognize that the new (restrictive) constraint contains the phase information,where the original constraint unit norm constraint does not contain any phase in-formation. However, we also note that the phase information in the new constraintdoes not change the objective function in (3.18) nor other constraints in (3.20) and(3.21). To show this, let us consider that (VXik,m)∗, ik ∈ I, m ∈ M, X ∈ {UL,DL}is the optimal solution of the problem in (3.18)–(3.21) obtained using the new restric-tive constraint. Since the new restrictive constraint involves a phase rotation, let usconsider that there is a phase rotation, ejθXik,m , ik ∈ I, m ∈ M, X ∈ {UL,DL} tothe optimal solution if one were to use the original unit norm constraint. If we canshow that the new optimal solution, (VXik,m)∗ejθXik,m , does not change the optimizationobjective or other constraints, we can claim that (VXik,m)∗ is also optimal solution ofthe original problem.Note that the optimization objective in (3.18), and constraints (3.20) and (3.21)involve quadratic forms of the optimization variables, VXik,m. Therefore, once wereplace them with (VXik,m)∗ejθXik,m , the phase information ejθXik,m that is involved with(VXik,m)∗ will vanish. Therefore, we conclude that the new constraint does not impact154Appendix B. Appendix for Chapter 3the optimization objective nor other constraints.B.2 Proof of Convergence of Algorithm 1Let f (V,U) be the optimization objective in (3.18). Then, for any feasible valueof V and U (i.e., constraints are satisfied), the Lagrangian L (V,U,λ,∆) in (3.18)is equal to f (V,U). Since L (V,U,λ,∆) is convex for V when all other variablesare fixed, a feasible optimal precoding matrix V∗,[n] at the nth iteration will be theminimum of the objective with respect to a given receive filter U[n], i.e.,L (V∗,[n],U[n],λ,∆) = minVL (V,U[n],λ,∆) . (B.1)The same observation can be made for the receive filter, U, i.e.,L (V[n],U∗,[n+1],λ,∆) = minUL (V[n],U,λ,∆) . (B.2)Combining observations made in (B.1) and (B.2), we can make the followinginequality statement:L (V∗,[n+1],U∗,[n+1],λ,∆) ≤ L (V∗,[n],U∗,[n+1],λ,∆)≤ L (V∗,[n],U[n],λ,∆) . (B.3)If we iterate between computing optimal precoding matrix and receive filter, (B.3)guarantees that the Lagrangian is always updated with an equal or smaller value. TheLagrangian with any feasible value of V and U is lower bounded by zero; therefore,the algorithm guarantees that the objective function converges to a limit point.155Appendix CAppendix for Chapter 4C.1 ABCD Matrix RepresentationI2V1 V2[A BC D]I1Figure C.1: ABCD-matrix representation of a two-port network.The ABCD-matrix representation of a two-port network relates the voltages andcurrents identified in Fig. C.1 as V1I1 = A BC D V2I2 . (C.1)The same ABCD parameters can be used when input and output are swapped, i.e.,transmission in the other direction is considered. Then we have V2−I2 = 1AD −BC D BC A V1−I1 . (C.2)If the reciprocity property holds, AD − BC = 1 is true. Hence, in this case (C.2)156Appendix C. Appendix for Chapter 4simplifies to  V2−I2 = D BC A V1−I1 . (C.3)We note that reciprocity can be assumed for power line networks, where the overallABCD-matrix is a cascade of reciprocal ABCD matrices, e.g. [139].C.2 Property of the PLC Keyhole ChannelWe consider the model in Fig. 4.11 and assume that either nodes n1 and n2 aretransmitting simultaneously, or that source and load impedance are identical forthose two nodes, so that Z1 = Z1,S and Z2 = Z2,S. In this appendix, we show thatthen the ratio of the channel frequency responses H13 and H23 is independent of thenetwork elements located between node nB and node n3.From (4.29) and (4.30) we can write the ratio asH13H23=H1BH2B. (C.4)Let us consider the numerator H1B first. Starting from (4.26) we obtainH1B =1A1 + C1Z1,S + (B1 +D1Z1,S)/Z1B(C.5)(a)=1A1 + C1Z1,S + (B1 +D1Z1,S)(1ZB2+ 1ZB3) (C.6)(b)=1(B1 +D1Z1,S)(1ZB1+ 1ZB2+ 1ZB3) , (C.7)where (a) follows from substituting Z1B using (4.25) and (b) from (4.24). Applying157Appendix C. Appendix for Chapter 4the same transformations to H2B leads toH2B =1(B2 +D2Z2,S)(1ZB2+ 1ZB1+ 1ZB3) . (C.8)Finally, substituting (C.7) and (C.8) into (C.4) gives usH13H23=H1BH2B=B2 +D2Z2,SB1 +D1Z1,S, (C.9)which only depends on the parameters of the network elements between nodes n1 andnB and nodes n2 and nB, respectively.158

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