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Physical-layer security for visible-light communication systems Mostafa, Ayman 2017

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Physical-Layer Security for Visible-LightCommunication SystemsbyAyman MostafaB.Sc., Alexandria University, 2006M.A.Sc., McMaster University, 2012A 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)April 2017c© Ayman Mostafa, 2017AbstractVisible-light communication (VLC) is an enabling technology that exploits the light-ing infrastructure to provide ubiquitous indoor broadband coverage via high-speedshort-range wireless communication links. On the other hand, physical-layer securityhas the potential to supplement conventional encryption methods with an additionalsecrecy measure that is provably unbreakable regardless of the computational powerof the eavesdropper.The lack of wave-guiding transmission media in VLC channels makes the commu-nication link inherently susceptible to eavesdropping by unauthorized users existingin areas illuminated by the data transmitters. In this thesis, we study transmissiontechniques that enhance the secrecy of VLC links within the framework of physical-layer security.Due to linearity limitations of typical light-emitting diodes (LEDs), the VLCchannel is more accurately modelled with amplitude constraints on the channel input,rather than the conventional average power constraint. Such amplitude constraintsrender the prevalent Gaussian input distribution infeasible for VLC channels, makingit difficult to obtain closed-form secrecy capacity expressions. Thus, we begin withderiving lower bounds on the secrecy capacity of the Gaussian wiretap channel subjectto amplitude constraints.We then consider the design of optimal beamformers for secrecy rate maximiza-tion in the multiple-input single-output (MISO) wiretap channel under amplitudeiiAbstractconstraints. We show that the design problem is nonconvex and difficult to solve,however it can be recast as a solvable quasiconvex line search problem. We alsoconsider the design of robust beamformers for worst-case secrecy rate maximizationwhen channel uncertainty is taken into account.Finally, we study the design of linear precoders for the two-user MISO broadcastchannel with confidential messages. We consider not only amplitude constraints, butalso total and per-antenna average power constraints. We formulate the design prob-lem as a nonconvex weighted secrecy sum rate maximization problem, and providean efficient search algorithm to obtain a solution for such a nonconvex problem. Weextend our approach to handle uncertainty in channel information.The design techniques developed throughout the thesis provide valuable tools fortackling real-world problems in which channel uncertainty is almost always inevitableand amplitude constraints are often necessary to accurately model hardware limita-tions.iiiPrefaceThis thesis is based on research work performed under the supervision of Profes-sor Lutz Lampe. For all the chapters, as well as the corresponding publications,I conducted the literature surveys, formulated the problems, proposed the solutions,performed the analyses, implemented the simulations, and wrote the manuscripts.Professor Lampe helped by guiding the direction of research, validating the analyses,and providing feedback to improve the manuscripts.The content of Chapter 2 has been published in the following papers:• A. Mostafa and L. Lampe, “Physical-Layer Security for MISO Visible LightCommunication Channels,” in IEEE Journal on Selected Areas in Communica-tions - Special Issue on Optical Wireless Communications, vol. 33, no. 9, pp.1806–1818, Sept. 2015.• A. Mostafa and L. Lampe, “Physical-Layer Security for Indoor Visible LightCommunications,” in Proceedings of 2014 IEEE International Conference onCommunications (ICC), Sydney, NSW, Australia, pp. 3342–3347, Jun. 2014.• A. Mostafa and L. Lampe, “Securing Visible Light Communications via FriendlyJamming,” in Proceedings of 2014 IEEE Globecom Workshops (GC Wkshps),Austin, TX, USA, pp. 524–529, Dec. 2014.ivPreface• A. Mostafa and L. Lampe, “Enhancing the Security of VLC Links: Physical-Layer Approaches,” in Proceedings of 2015 IEEE Summer Topicals MeetingSeries (SUM), Invited Presentation, Nassau, Bahamas, pp. 39–40, Jul. 2015.The content of Chapter 3 has been published in the following papers:• A. Mostafa and L. Lampe, “Optimal and Robust Beamforming for Secure Trans-mission in MISO Visible-Light Communication Links,” in IEEE Transactionson Signal Processing, vol. 64, no. 24, pp. 6501–6516, Dec. 2016.• A. Mostafa and L. Lampe, “Pattern Synthesis of Massive LED Arrays for Se-cure Visible Light Communication Links,” in Proceedings of 2015 IEEE Inter-national Conference on Communication Workshop (ICCW 2015), London, UK,pp. 1350–1355, Jun. 2015.The content of Chapter 4 has been submitted for publication:• A. Mostafa and L. Lampe, “On Linear Precoding for the Two-User MISO Broad-cast Channel with Confidential Messages and Per-Antenna Constraints,” sub-mitted in Jan. 2017.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Visible-Light Communication . . . . . . . . . . . . . . . . . . 11.1.2 Physical-Layer Security . . . . . . . . . . . . . . . . . . . . . . 31.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.2.1 Are VLC Links Secure? . . . . . . . . . . . . . . . . . . . . . 5viTABLE OF CONTENTS1.2.2 Amplitude Constraints . . . . . . . . . . . . . . . . . . . . . . 61.2.3 Uncertain Channel Information . . . . . . . . . . . . . . . . . 81.3 Preliminaries and Definitions . . . . . . . . . . . . . . . . . . . . . . . 101.3.1 The VLC Channel Model and Modulation Scheme . . . . . . . 111.3.2 The Optical Channel Gain . . . . . . . . . . . . . . . . . . . . 141.3.3 Beamforming for the MISO VLC Channel . . . . . . . . . . . 161.3.4 The Wiretap Channel and the Secrecy Capacity . . . . . . . . 181.3.5 The Two-User Broadcast Channel with Confidential Messages(BC-CM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.3.6 What Does “Secure Transmission” Mean? . . . . . . . . . . . . 211.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 221.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 252 Achievable Secrecy Rates for VLC Wiretap Channels . . . . . . . . 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 The Scalar VLC Wiretap Channel . . . . . . . . . . . . . . . . . . . . 282.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 292.2.2 Achievable Secrecy Rates . . . . . . . . . . . . . . . . . . . . . 302.2.3 Upper Bound on the Secrecy Capacity . . . . . . . . . . . . . 332.2.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 382.3 The MISO VLC Wiretap Channel . . . . . . . . . . . . . . . . . . . . 392.3.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 402.3.2 Achievable Secrecy Rates . . . . . . . . . . . . . . . . . . . . . 412.3.3 Numerical Example from a VLC Scenario . . . . . . . . . . . 432.4 The Scalar VLC Wiretap Channel Aided by a Friendly Jammer . . . 462.4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 48viiTABLE OF CONTENTS2.4.2 Achievable Secrecy Rate . . . . . . . . . . . . . . . . . . . . . 502.4.3 Numerical Example from a VLC Scenario . . . . . . . . . . . 522.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563 Optimal and Robust Beamforming for Secure MISO VLC Links . 573.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.3 Optimal and Robust Beamformer Design . . . . . . . . . . . . . . . . 623.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 623.3.2 Optimal Beamforming with Perfect Channel Information . . . 633.3.3 Robust Beamforming with Imperfect Channel Information . . 703.4 Uncertainty Sets for the Eavesdropper’s Channel in VLC Scenarios . 763.4.1 Uncertain Eavesdropper’s Location . . . . . . . . . . . . . . . 783.4.2 Uncertain Eavesdropper’s Orientation . . . . . . . . . . . . . . 823.4.3 Uncertain LEDs Half-Intensity Angle . . . . . . . . . . . . . . 853.4.4 Uncertain NLoS Components . . . . . . . . . . . . . . . . . . 873.4.5 Combined Uncertainties . . . . . . . . . . . . . . . . . . . . . 883.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 913.5.1 Performance Comparisons . . . . . . . . . . . . . . . . . . . . 913.5.2 Worst-Case Secrecy Rate Performance in VLC Scenarios . . . 963.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 Linear Precoding for the Two-User MISO BC-CM . . . . . . . . . 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1084.2.1 The Two-User MISO BC-CM . . . . . . . . . . . . . . . . . . 1094.2.2 Linear Precoding . . . . . . . . . . . . . . . . . . . . . . . . . 111viiiTABLE OF CONTENTS4.2.3 Transmit Constraints and Secrecy Rate Regions . . . . . . . . 1124.3 Precoder Design with Perfect Channel Information . . . . . . . . . . 1174.3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . 1174.3.2 The Outer Problem . . . . . . . . . . . . . . . . . . . . . . . . 1214.3.3 The Dual of the Inner Problem . . . . . . . . . . . . . . . . . 1244.3.4 The Search Algorithm . . . . . . . . . . . . . . . . . . . . . . 1284.3.5 Per-Antenna Amplitude Constraint . . . . . . . . . . . . . . . 1314.4 Robust Precoder Design with Imperfect Channel Information . . . . . 1324.4.1 Channel Uncertainty Model . . . . . . . . . . . . . . . . . . . 1334.4.2 Total and Per-Antenna Average Power Constraints . . . . . . 1344.4.3 Per-Antenna Amplitude Constraint . . . . . . . . . . . . . . . 1384.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1394.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445 Conclusions and Future Directions . . . . . . . . . . . . . . . . . . . . 1465.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1465.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160A The Trapezoidal Distribution . . . . . . . . . . . . . . . . . . . . . . . 160B Proofs and Derivations for Chapter 3 . . . . . . . . . . . . . . . . . . 163B.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.2 Components of h0 and J0 from (3.53) . . . . . . . . . . . . . . . . . . 163ixTABLE OF CONTENTSC Proofs and Derivations for Chapter 4 . . . . . . . . . . . . . . . . . . 165C.1 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 165C.2 Derivation of the Dual Problem (4.57) . . . . . . . . . . . . . . . . . 168xList of Tables2.1 Simulation parameters for the MISO wiretap channel. . . . . . . . . . 452.2 Simulation parameters for the scalar channel aided by a friendly jammer. 543.1 Bisection search to solve the maximization problem in (3.31). . . . . . 753.2 Simulation parameters for the VLC scenario. . . . . . . . . . . . . . . 974.1 Subgradient-based search algorithm to solve the maximization problemin (4.29). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130xiList of Figures1.1 An example VLC scenario in which physical-layer security is applicable. 61.2 Simplified block diagram of a SISO PAM VLC system. . . . . . . . . 111.3 Current-power response of a typical LED. . . . . . . . . . . . . . . . 121.4 Geometry of an LoS VLC link with arbitrary receiver orientation. . . 151.5 Beamforming in conjunction with PAM for the MISO VLC Channel. 161.6 A general wiretap channel. . . . . . . . . . . . . . . . . . . . . . . . . 181.7 A general two-user broadcast channel with confidential messages (BC-CM). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.1 Problem scenario for the SISO case. . . . . . . . . . . . . . . . . . . . 292.2 Lower and upper bounds on the secrecy capacity of the scalar Gaussianwiretap channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.3 Problem scenario for the MISO case. . . . . . . . . . . . . . . . . . . 402.4 Layout of the LEDs for the MISO case. . . . . . . . . . . . . . . . . . 442.5 Spatial distribution of the SNR at the receivers height (0.85 m abovethe floor level) without beamforming. . . . . . . . . . . . . . . . . . . 452.6 Achievable communication rate between Alice and Bob as a functionof Bob’s location without secrecy constraints. . . . . . . . . . . . . . 462.7 Secrecy rate obtained with the ZF beamformer (2.32) as a function ofEve’s location when Bob is located at (−0.9,−2.0). . . . . . . . . . . 47xiiLIST OF FIGURES2.8 Secrecy rate obtained with the ZF beamformer (2.32) as a function ofBob’s location when Eve is located at (1.6,−0.7). . . . . . . . . . . . 472.9 Problem scenario for the scalar channel aided by a friendly jammer. . 482.10 Achievable secrecy rates (2.36) for the scalar channel aided by a friendlyjammer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 532.11 Layout of the LEDs for the scalar channel aided by a friendly jammer. 542.12 Secrecy rate obtained with the jamming beamformer (2.41) as a func-tion of Eve’s location when Bob is located at (−0.7,−0.9). . . . . . . 552.13 Secrecy rate obtained with the jamming beamformer (2.41) as a func-tion of Bob’s location when Eve is located at (0.3,−1.5). . . . . . . . 553.1 (a) Secrecy rates (3.5) obtained with the optimal, GEV, and ZF beam-formers versusA/σ, subject to the constraint ‖w‖p ≤ 1, for p = 1, 2,∞.(b) α? corresponding to the optimal beamformer wα? . . . . . . . . . . 923.2 Secrecy rates (3.83) of the optimal, GEV, and ZF beamformers underdifferent lp-norm constraints, versus the number of eavesdroppers when20 log10(A/σ) = 20 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . 933.3 Worst-case secrecy rates (3.85) of the robust, non-robust, GEV, and ZFbeamformers versus hE with hB = 0, 0.2, 0.4. All the beamformers aresubject to the amplitude constraint ‖w‖∞ ≤ 1, and 20 log10(A/σ) =20 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.4 Layout of the LEDs on the ceiling. There exist 25 light fixtures. Eachfixture has 10 × 10 cm2 surface area and encloses 4 LEDs located atthe corners of the fixture. . . . . . . . . . . . . . . . . . . . . . . . . . 973.5 Worst-case secrecy rate (3.44) versus α with uncertain Eve’s location.θE = 0 and ΨE = 70◦. . . . . . . . . . . . . . . . . . . . . . . . . . . . 99xiiiLIST OF FIGURES3.6 Worst-case secrecy rate (3.44) versus α with uncertain Eve’s orienta-tion. xE = −1.25, yE = 0, zE = 0.85, θE ∈ [0, θmax], and φE ∈ [0, 360◦]. 1013.7 Worst-case secrecy rate (3.44) versus α with uncertain Eve’s locationand LEDs half-intensity angle ζ3-dBi , i = 1, . . . , N . xE ∈ [−2.25,−0.25],yE ∈ [−2.5, 2.5], zE = 0.85, θE = 0, and ΨE = 70◦. . . . . . . . . . . . 1023.8 Worst-case secrecy rate (3.44) versus α with uncertain Eve’s locationand NLoS components. xE ∈ [−2.25,−0.25], yE ∈ [−2.5, 2.5], zE =0.85, θE = 0, and ΨE = 70◦. . . . . . . . . . . . . . . . . . . . . . . . 1034.1 The secrecy capacity region obtained with optimal S-DPC along withthe secrecy rate regions of the GEV precoder, the precoder obtainedwith Algorithm 4.1, and the ZF precoder, subject to the total powerconstraint PdB = 10 log10 PTot. The number of antennas N ∈ {2, 4}. . 1404.2 Achievable secrecy rate regions of the proposed and ZF precoders sub-ject to total power constraint (TPC), per-antenna power constraint(PAPC), and amplitude constraint (AmC). PTot = NPi = NA2i , i =1, . . . , N , PdB = 10 log10 PTot, and N = 4. The secrecy capacity regionwith optimal S-DPC is included for the case of total power constraint. 1424.3 Worst-case secrecy rate regions with different channel uncertainty lev-els, 1 and 2, subject to total power constraint (TPC), per-antennapower constraint (PAPC), and amplitude constraint (AmC). PTot =NPi = NA2i , i = 1, . . . , N , PdB = 10 log10 PTot = 15 dB, and N = 4. . . 143A.1 The trapezoidal distribution in (A.1) with b < a. . . . . . . . . . . . . 161A.2 The differential entropy (A.2) as a function of b when a = 2. . . . . . 162xivList of Abbreviations5G 5th GenerationBC-CM Broadcast Channel with Confidential MessagesCDMA Code-Division Multiple AccessDAC Digital-to-Analog ConverterDC Direct CurrentDD Direct DetectionFoV Field-of-ViewGaN Gallium NitrideGEV Generalized Eigenvaluei.i.d. Independent and identically-distributedIEEE Institute of Electrical and Electronics EngineersIM Intensity ModulationIoT Internet of ThingsLD Laser DiodeLED Light-Emitting DiodeLoS Line-of-SightMC Multi-CarrierMIMO Multiple-Input Multiple-OutputMISO Multiple-Input Single-OutputMTC Machine-Type CommunicationxvList of AbbreviationsNLoS Non-Line-of-SightOOK On-Off KeyingPAM Pulse-Amplitude ModulationPD PhotodiodePDF Probability Density FunctionPLC Power-Line CommunicationQAM Quadrature Amplitude ModulationRF Radio FrequencyS-DPC Secret Dirty-Paper CodingSINR Signal-to-Interference-plus-Noise RatioSISO Single-Input Single-OutputSNR Signal-to-Noise RatioVLC Visible-Light CommunicationVPPM Variable Pulse-Position Modulationw.r.t. With respect toZF Zero-ForcingxviNotationAlice The transmitterBob The (intended) receiverEve The eavesdropperRN The set of N -dimensional real-valued numbersRN+ The set of N -dimensional nonnegative real-valued numbers0 The all-zero column vector1N The all-one column vector with length NIN The N -dimensional identity matrix(·)T Transpose[x]+ max {x, 0}|x| Absolute value of x‖x‖p Lp-norm of the vector x, p ≥ 1‖x‖∞ Chebyshev or l∞-norm of the vector x‖X‖F Frobenius norm of the matrix XTr(X) Trace of the square matrix XDiag(x1, . . . , xN) The diagonal matrix with diagonal elements x1, . . . , xN⊗ Kronecker productI(·) Indicator functiond·e Ceiling functionlogb(·) The logarithm to base b, b ∈ {2, 10}xviiNotationln(·) The natural logarithm (i.e., the logarithm to base e)p(x) Probability density function for the random variable XE{X} Expected value of the random variable Xvar{X} Variance of the random variable Xh(X) Differential entropy of the random variable XD(p(·)‖q(·)) Relative entropy between the distributions P and QI(X;Y ) Mutual information between the random variables X and YN (0, σ2) The Gaussian distribution with zero mean and variance σ2U [−a, a] The uniform distribution over the interval [−a, a]Q(·) The Q-function∇subf(x) A subgradient of the function f at x{·}B A quantity relevant to Bob, the intended receiver{·}E A quantity relevant to Eve, the eavesdropperxviiiAcknowledgmentsIt has been a true pleasure and an enjoyable learning experience to pursue my doctoraldegree under the supervision of Professor Lutz Lampe. Over the course of the pastfour years, I found his door always open, and I was extremely fortunate to work withsuch a distinguished researcher and amazing supervisor. I am sincerely grateful forhis guidance, patience, encouragement, and support in so many ways.I thank the members of my examining committee at UBC, Professors DavidMichelson, Julian Cheng (UBC Okanagan), Vikram Krishnamurthy, Sathish Gopalakr-ishnan, and Brian Marcus (Mathematics), for their time and insightful comments.Special thanks go to Professor Ashish Khisti at the University of Toronto for servingas the external examiner, and also for participating in the final exam.I thank the Natural Sciences and Engineering Research Council of Canada (NSERC).This thesis would not have been possible without their funding and support.I thank my colleagues in the Communications Lab at Kaiser 4090 for the wonderfultimes and interesting discussions we had over the years. Their companionship hassignificantly enriched my experience at UBC.I also take the opportunity to express my sincere gratitude to my Master’s super-visor at McMaster University, Professor Steve Hranilovic. He taught me how to be aresearcher, and with whom I wrote my very first research paper.I am deeply indebted to my wife, Shereen, for her continuous love and support.She shared with me the difficult journey of graduate studies right from the beginning,xixAcknowledgmentsand was always there whenever I needed her encouragement. I also thank my sons,Mostafa and Omar, who always cheered for me, thinking I am a smart person and agreat dad. Their excitement when I go back home was my biggest rewarding momenteveryday.Last, but definitely not least, I am forever indebted to my Parents. WhateverI have accomplished is because of them, and in fact my biggest motivation was toplease them. Words would fall short to express how much I am grateful for their care,sacrifice, encouragement, endless love, and unconditional support.Ayman MostafaApril 19, 2017Vancouver, BCxxDedicationTo my Parents, my wife Shereen, and my sons Mostafa and OmarxxiChapter 1Introduction1.1 Background1.1.1 Visible-Light CommunicationVisible-light communication (VLC) is an enabling technology that exploits illumina-tion devices, mostly high-brightness light-emitting diodes (LEDs), to establish high-speed short-range wireless communication links [1, 2, 3, 4, 5, 6, 7]. In typical VLCsystems, information is relayed by the means of modulating the output intensity ofthe LEDs, whereas at the receiver side, the data signal is recovered using simplephotodiodes (PDs). The use of laser diodes (LDs), rather than LEDs, can resultin higher data rates [8], however LDs are not popular for illumination purposes asthey are more expensive and have the potential to cause eye or skin injuries. Datarates can also be improved by using imaging receivers [8, 9], however this comes withincreased complexity and cost.VLC systems take advantage of the license-free light spectrum and immunity toradio frequency (RF) interference. In addition, VLC transmitters can exploit theexisting lighting infrastructure where legacy incandescent and fluorescent lamps arebeing replaced with LED-based luminaires that have longer lifespan, smaller size,lower power consumption, higher energy-conversion efficiency, and improved colorrendering without using toxic chemicals [2, 5, 10]. Thus, the integration between1Chapter 1. Introductionpower-line communication (PLC) and VLC systems has the potential to provideubiquitous indoor broadband coverage with seamless handover [11, 12]. Furthermore,since typical lighting systems utilize multiple luminaires that are sufficiently separatedto provide uniform illumination, VLC systems can readily benefit from multiple-antenna techniques to achieve higher data rates [9] and enhance the reliability [13]and security [14] of VLC networks. Moreover, due to line-of-sight (LoS) propagationand confinement of light waves by opaque surfaces, VLC links cause limited inter-cell interference. Such advantages qualify VLC systems for realizing small-size cells,termed as “LiFi attocells” [15], in fifth generation (5G) networks featuring cells withcoverage ranges on the order of a few meters [16].The IEEE 802.15.7 standard [17], released in 2011, was a big step towards thecommercialization and widespread deployment of VLC networks [18, 19]. It definesthree physical layer modes, with the second and third modes, PHY II and PHY III,respectively, supporting data rates up to 96 Mbit/sec [20, 21]. In fact, much higherdata rates have already been demonstrated in laboratory conditions. A prototypeVLC system utilizing high-power LEDs to achieve bidirectional real-time transmissionwith a total rate of 500 Mbit/sec over a 2 m distance was implemented in [22]. In [23],the authors demonstrated a 16-user multi-carrier code-division multiple access (MC-CDMA) VLC system that achieves 750 Mbit/sec sum rate over a 1.5 m distanceusing off-the-shelf LEDs. Furthermore, the use of µLEDs with smaller size (e.g.,on the order of 50 µm) and lower junction capacitance allows higher modulationbandwidth. The authors in [24] utilized a single gallium nitride (GaN) µLED toestablish a 3 Gbit/sec VLC link over a 5 cm distance.2Chapter 1. Introduction1.1.2 Physical-Layer SecurityWith the unprecedented increase in traffic volumes over wireless networks, data pri-vacy and secrecy are becoming a major concern for users, as well as for networkadministrators. Conventional security schemes are typically implemented at upperlayers of the network stack via access control, password protection, and end-to-endencryption. Such schemes are deemed secure as long as the computational power ofpotential eavesdroppers remains below certain limits. For example, the eavesdrop-pers do not have sufficient computational power to perform an exhaustive search forthe password, or determine the prime factors of a large integer (to obtain the secretkey and decrypt the encrypted message), within a reasonable amount of time. Dur-ing the past few years, however, physical-layer security has emerged as a promisingtechnique that can complement conventional encryption methods with an additionalsecrecy layer that is provably unbreakable regardless of the computational power ofthe eavesdroppers [25, 26, 27, 28, 29, 30, 31]. Moreover, physical-layer security has thepotential to provide lightweight standalone secrecy solutions in communication sys-tems functioning under severe hardware or energy constraints such as machine-typecommunication (MTC) devices in the Internet of Things (IoT) [32].Physical-layer security refers to transmission schemes that exploit dissimilaritiesamong the channels of different receivers in order to hide information from unautho-rized receivers, without reliance on upper-layer encryption techniques. The underly-ing idea behind such a secrecy scheme is to sacrifice a portion of the communicationrate, that otherwise would be used for useful data transmission, in order to confusepotential eavesdroppers and diminish their capability to infer information at anypositive rate, via carefully-designed signaling and coding schemes.The innovative idea of quantifying secrecy via information-theoretic measures can3Chapter 1. Introductionbe traced back to Shannon [33] who proposed equivocation1 as a quantitative measureof the secrecy level of encrypted messages [26, Section 3.1]. Almost three decadeslater, the foundations of information-theoretic security were laid down by Wyner inhis seminal paper [34] that studied the problem of secret communication over thedegraded broadcast channel. In that paper, Wyner introduced the so-called wiretapchannel model to describe the scenario in which the transmitter has one secret mes-sage intended for one receiver, while the other receiver, whose channel is degraded,acts as an eavesdropper. Wyner also introduced the notion of secrecy capacity asa performance measure that specifies the maximum communication rate that guar-antees reliable reception of the secret message by the intended receiver and entirehiddenness from the eavesdropper. The work of Wyner motivated the characteriza-tion of the secrecy capacity of the scalar, i.e., the single-input single-output (SISO),Gaussian wiretap channel2 [36]. Wyner’s model was then extended to the (nonde-graded) wiretap channel [37], where the eavesdropper’s channel need not be degraded.Such an extension has ultimately led to the characterization of the secrecy capacity ofthe multiple-input single-output (MISO) [38, 39] and multiple-input multiple-output(MIMO) Gaussian wiretap channels [40, 41, 42, 43]. Furthermore, when the eaves-dropper’s channel is not accurately known or entirely unknown to the transmitter,the works in [44, 45, 46] proposed the transmission of jamming signals, i.e., artificialnoise, in conjunction with the information-bearing signal, in order to increase theinterference seen by the eavesdropper and diminish its capability to decode the secretmessage.The wiretap channel model was further extended to the two-user broadcast chan-nel with confidential messages (BC-CM) [47]. Such a model studies the scenario in1As defined in his paper [33, Section 11], equivocation is the conditional entropy of the trans-mitted message after knowing the received signal.2Recall that the scalar Gaussian broadcast channel is a degraded channel [35, Section 15.1.3].4Chapter 1. Introductionwhich the transmitter has two independent secret messages, one intended for eachreceiver, and each message should be kept confidential from the other receiver. Thesecrecy capacity regions of the two-user MISO BC-CM and the two-user MIMO BC-CM were characterized in [48] and [49], respectively.1.2 Motivation1.2.1 Are VLC Links Secure?VLC links are often deemed eavesdropping-proof, however this is not necessarily true,especially in public areas or in multi-user scenarios. With the lack of optical fibers,or any sort of wave-guiding transmission media, the VLC channel has a broadcastnature. This makes VLC links inherently susceptible to eavesdropping by unintendedor unauthorized users having access to areas illuminated by the data transmitters.Typical scenarios include public spaces such as classrooms, meeting rooms, libraries,shopping centers, and aircrafts, to name a few.Accordingly, our research efforts in this thesis are directed towards enhancing thesecrecy of VLC networks within the framework of physical-layer security.Figure 1.1 depicts a typical VLC scenario in which physical-layer security is appli-cable. The figure shows, for example, a governmental office utilizing a VLC network.The shaded area (at the bottom of the figure) is open to the public, making sensitiveinformation vulnerable to overhearing by potential eavesdroppers. An interestingdesign problem is to devise a physical-layer security scheme that maintains reliablecommunication among the office personnel and prevents users located in the shadedarea from reliably decoding the transmitted messages.5Chapter 1. IntroductionFigure 1.1: An example VLC scenario in which physical-layer security is applicable.1.2.2 Amplitude ConstraintsTypical VLC systems utilize LEDs for data transmission whereby the input currentsignal modulates the output intensity of the LEDs. Typical LEDs, however, havelimited linear operation region beyond which electro-optical conversion becomes non-linear (see, e.g., Figure 1.3). Such a nonlinearity can be partially compensated viapredistortion of the input current signal [50]. However, predistortion can be effec-tive only within certain operation limits beyond which the output intensity saturates,leading to clipping distortion of the transmitted signal. Thus, the modulating currentsignal must satisfy certain amplitude constraints in order to maintain linear electro-optical conversion and avoid undesirable nonlinear effects. As a consequence, inten-sity modulation (IM) channels are typically modelled with amplitude constraints onthe channel input, rather than the conventional average power constraint [51, 52, 53].In fact, all modern digital transmitters experience amplitude constraints because6Chapter 1. Introductionof the digital-to-analog converters (DACs) incorporated at the transmitter front-end.Clearly, these DACs have finite ranges, and thus the transmitted signals are subjectto amplitude constraints. Therefore, taking amplitude constraints into account canbe crucial to model hardware limitations, not only in IM systems, but in fact in allpractical communication systems.Now, an amplitude constraint on the channel input will render the prevalent Gaus-sian input distribution infeasible. Unfortunately, this makes amplitude constraintsdifficult to handle (in terms of obtaining analytic capacity expressions), and thereforethey are often overlooked in favor of the more convenient average power constraint.Compared to the massive body of literature on the Gaussian wiretap channel un-der the average power constraint, works that considered the amplitude-constrainedGaussian wiretap channel are quite rare. Even in the absence of secrecy constraints,characterization of the capacity of amplitude-constrained Gaussian channels is quitechallenging. In his seminal paper [54, Section 26], Shannon referred to the difficultyof obtaining an analytic expression for the capacity of the peak-limited, i.e., theamplitude-constrained, Gaussian channel. Instead, he derived a lower bound and anasymptotic upper bound that is valid at high peak signal-to-noise ratio (SNR). Outof his Ph.D. work [55, 51], Smith came up with the rather surprising result that thecapacity-achieving input distribution for the amplitude-constrained Gaussian chan-nel is discrete with finite support, i.e., it has a finite number of mass points. Closed-form lower and upper bounds on the capacity of the amplitude-constrained Gaussianchannel were derived in [53]. For the Gaussian wiretap channel, the authors in [56]followed the approach devised in [51] and proved that the secrecy capacity-achievingdistribution under the amplitude constraint is also discrete with finite support.In this thesis, we shall characterize the performance of amplitude-constrained7Chapter 1. IntroductionGaussian wiretap channels via closed-form secrecy rate expressions. We will also con-sider the design of beamformers for the MISO wiretap channel and linear precodersfor the two-user MISO BC-CM when the beamformers or precoders are subject to am-plitude constraints. In fact, it is fair to say that the novelty of many of the problemsconsidered in this thesis comes from taking amplitude constraints into account.1.2.3 Uncertain Channel InformationCompared to conventional encryption techniques, the performance of physical-layersecurity schemes is inherently sensitive to channel conditions, that is the secrecyperformance can be severely degraded if the designed scheme is based on inaccuratechannel information. In fact, this is a major drawback that may hinder any effortto deploy practical physical-layer security systems as it is almost always unrealistic,in real-world scenarios, to assume that the channel gain of the intended receiver orthe eavesdropper is accurately known to the transmitter. On one hand, informationregarding the intended receiver’s channel may suffer from estimation errors, asidefrom inevitable quantization errors imposed by the finite rate of the feedback channel.On the other hand, there is probably no feedback from the eavesdropper if it is anunregistered user and shall remain silent to hide its presence. In such a case, thetransmitter may resort to less reliable information sources, such as possible locationsof the eavesdropper, in order to obtain an estimate of its channel gain. In all cases,channel information available to the transmitter will never be accurate, and adoptinga physical-layer security scheme based on such inaccurate information may lead to asecrecy outage with catastrophic consequences.Based on the above discussion, it becomes clear that any practical physical-layersecurity system must take channel uncertainty into account. In other words, we8Chapter 1. Introductionhave to adopt the so-called robust transmission schemes. Among various possibleapproaches to achieve robust secure transmission, we shall consider worst-case op-timization. In such an approach, one chooses some uncertainty sets that are be-lieved to contain all possible realizations of the channel gains for the receiver andthe eavesdropper. Then, the design problem is formulated to optimize the perfor-mance measure, i.e., the secrecy rate, corresponding to the worst-case realization ofthe uncertain channel gains. Now we have to face the question of how to choose rea-sonable uncertainty sets. In fact, the validity of the worst-case optimization approachdepends mostly on such a choice. On one hand, unreasonably large (i.e., too con-servative) uncertainty sets may render the design problem infeasible. On the otherhand, small uncertainty sets can lead to an overestimate of the achievable secrecyrate and, consequently, secrecy outage may occur.Typical works in the physical-layer security literature assume spherical uncertaintysets for both the receiver’s and eavesdropper’s channels. For example, uncertainty inthe eavesdropper’s channel is typically modelled byhE ∈{hˆE + e : ‖e‖2 ≤ },where hˆE is the transmitter’s erroneous estimate of the eavesdropper’s channel hE,e is an unknown (but norm-bounded) error vector, and  is some known constantthat quantifies the amount of uncertainty. This error model is well accepted to takeinto account channel uncertainty caused by limited, i.e., finite-rate, feedback from thereceiver [57, Lemma 1]. In wiretap scenarios, however, such an uncertainty modelmay become inapplicable if the eavesdropper is a passive receiver that remains silentto hide its presence from the transmitter, i.e., there is no feedback, hˆE, regarding theeavesdropper’s channel. Fortunately, in indoor VLC scenarios, it is often reasonable9Chapter 1. Introductionto assume that the transmitter has some uncertain information regarding the locationand/or orientation of the eavesdropper (recall, for example, the scenario in Figure 1.1,wherein potential eavesdroppers can only exist within areas of the room that are opento the public). Furthermore, the LoS path is typically dominant in VLC scenarios,and thus the channel gain can be accurately approximated by a deterministic functionof the location and orientation of the receiver, as well as the emission pattern of theLEDs (see Eq. (1.4) in Section 1.3.2). This is unlike the case of RF channels whereinrich scattering environments typically give rise to significant multipath components,which are usually unpredictable.Therefore, in this thesis we develop the idea of choosing uncertainty sets for theeavesdropper’s channel based on the uncertain parameters in the LoS channel gainequation, i.e., based on uncertain information regarding the location and orientationof the eavesdropper. Then, we use such uncertainty sets, along with spherical uncer-tainty sets for the intended receiver’s channel, in order to formulate the worst-casesecrecy rate maximization problem and obtain a robust transmission scheme.1.3 Preliminaries and DefinitionsIn this section, we present some of the key concepts and definitions used throughoutthe entire thesis. We begin with describing the VLC channel model and the mod-ulation scheme that we adopt. We then recall the generalized Lambert’s cosine lawused to model the emission pattern of typical LEDs. We also explain how transmitbeamforming can be implemented in IM channels. Furthermore, we review two fun-damental constructs in physical-layer security, namely, the wiretap channel and thetwo-user BC-CM, and recall the relevant definitions of achievable secrecy rates andsecrecy rate regions. Finally, we clarify what the term “secure transmission” precisely10Chapter 1. IntroductionDimmingcontrolBias-TVisible lightInput data x(t)y(t)IDCPTX(t)PAM encoderPRX(t)|x(t)|   μMIIDC     μMI    PredistorterSISO VLC ChannelFigure 1.2: Simplified block diagram of a SISO PAM VLC system.means in the context of physical-layer security.1.3.1 The VLC Channel Model and Modulation SchemeTypical VLC systems utilize illumination LEDs for data transmission. Such LEDsare incoherent light sources3, and thus IM is the only feasible transmission scheme.As a consequence, direct detection (DD) at the receiver using simple PDs is sufficientfor demodulation [5, 6, 58, 59].In this thesis, we adopt the DC-biased pulse-amplitude modulation (PAM) scheme4illustrated in Figure 1.2. The transmit element is an illumination LED driven by afixed bias current IDC ∈ R+ that sets the average radiated optical power Popt = ηIDC,where η (mW/mA) is the electro-optical conversion efficiency of the LED. Thecurrent-power response of a typical LED is depicted in Figure 1.3.The PAM scheme is described as follows. Information symbols from a single-stream data source are stochastically encoded5 into a zero-mean current signal x(t),3Unlike LDs, LEDs emit photons with random phases.4Note that PHY I and PHY II in the IEEE 802.15.7 standard use the OOK and (binary) VPPMschemes [20, Tables I and II]. However, restricting the transmitted signal to such binary schemeswould not allow much room for optimization and performance enhancement, especially when secrecyconstraints are taken into account.5Stochastic encoding adds randomization to confuse the eavesdropper. See, e.g., [26, Chapter 3].11Chapter 1. Introduction1.24623.74621.2462x(t)Current (mA)Optical power (mW)IDC   μMI)IDC (1+μMI)IDCPopt = hIDC(1+μMI)Popt   μMI)Popth x(t)Figure 1.3: Current-power response of a typical LED.t = 1, 2, . . .. The codewords are chosen such that E{X} = 0 and |x(t)| ≤ A ∀t,where X is the random variable counterpart of x(t), A , µMIIDC, and µMI ∈ [0, 1] istermed as the modulation index. The modulation index, in turn, is chosen such thatthe LED maintains linear electro-optical conversion over the input current range[IDC − A, IDC + A], as illustrated in Figure 1.3. If nonlinearity is severe, digitalpredistortion of the input current signal x(t) may become necessary to linearize theLED response around the DC bias point [50]. The codewords are then superimposedon the DC bias, via a bias-T circuit, to imperceptibly modulate the output intensityof the LED. Thus, the instantaneous emitted optical power PTX(t) can be expressedasPTX(t) = η(IDC + x(t)). (1.1)Since E{X} = 0, the data6 signal x(t) does not alter the average radiated optical6With slight abuse of notation, we shall use the term “data” to refer to the “codewords corre-sponding to the secret message”. However, it is essential to keep in mind that x(t) is a sequence ofsecrecy codewords rather than uncoded data symbols.12Chapter 1. Introductionpower and, consequently, it has no effect on the illumination level.We shall assume narrow-band transmission, that is the bandwidth of the trans-mitted signal is well below the modulation bandwidth (or the cutoff frequency) ofthe LED, and is also smaller than the inverse of the maximum excess delay of theVLC channel. In other words, we shall ignore possible low-pass filtering caused bythe LED characteristics or multipath propagation. Consider, for example, a VLCsystem in which phosphorus-coated blue LEDs are utilized for transmission, and bluefiltering is applied at the receiver. This setup allows 3-dB modulation bandwidth ofabout 20 MHz [60, Figure 3], and the excess delay in a medium-sized room is about10-20 nsec [61]. Thus, a transmitted signal whose bandwidth is limited to 10 MHz,for example, should not suffer noticeable distortion from the frequency response ofthe LEDs or the channel.Thus, under the assumption of narrow-band transmission, the frequency responseof the VLC channel is almost flat near DC [58], and it is sufficient to characterize theoptical channel by its DC gain given by the ratio of transmitted to received opticalpowers. From (1.1), the instantaneous received optical power isPRX(t) = hoptPTX(t)= hoptη(IDC + x(t)), (1.2)where hopt ∈ R+ is the DC optical channel gain that shall be specified in the nextsubsection. The received optical power, in turn, is converted by a PD into a propor-tional photocurrent RPDPRX(t), where RPD (µA/mW) is the responsivity of the PD.Then, the DC term RPDhoptηIDC is blocked, and the resulting signal is amplified by atransimpedance amplifier with gain Ta (mV/µA) to produce a voltage signal y(t) ∈ Rthat is a scaled, but noisy, version of the input signal x(t). Dominant noise sources in13Chapter 1. IntroductionVLC channels are the thermal noise in the receiver electronic circuits, i.e., the ampli-fier noise, and the shot noise caused by ambient illumination from sunlight or otherlight sources. Both noise processes are well modelled as signal-independent additivewhite Gaussian noise [52, 59]. Thus, the discrete-time VLC channel in Figure 1.2 canbe modelled byy(t) = hx(t) + n(t), t = 1, 2, . . . , (1.3)where h , ηhoptRPDTa is the DC channel gain, and n(t) denotes independent andidentically-distributed (i.i.d.) zero-mean Gaussian noise samples with variance σ2,i.e., N ∼ N (0, σ2), where N is the random variable counterpart of n(t). The chan-nel model in (1.3) is a scalar Gaussian channel whose input x(t) is subject to theamplitude constraint |x(t)| ≤ A ∀t = 1, 2, . . ..1.3.2 The Optical Channel GainFigure 1.4 illustrates the geometry of an LoS VLC link. The receiver is pointingtowards an arbitrary direction specified by the unit vectoru = [sin θ cosφ sin θ sinφ cos θ]T ,where θ ∈ [0, pi] is the zenith (or polar) angle, and φ ∈ [0, 2pi] is the azimuth angle.We shall refer to u as the orientation vector.We assume that the LED has an azimuth-symmetric generalized7 Lambertianemission pattern. We also assume that the LoS path is dominant over multipath com-ponents caused by diffuse reflections from nearby surfaces8. Under these assumptions,7In the case of (non-generalized) Lambertian emission pattern, the Lambertian order m is equalto 1, which corresponds to a half-intensity angle ζ3-dB = 60◦.8This assumption will be relaxed in Section 3.4.4 wherein non-line-of-sight (NLoS) componentsare taken into account.14Chapter 1. IntroductionufθxyzdLEDψζΨ Optical receiverFigure 1.4: Geometry of an LoS VLC link with arbitrary receiver orientation.the DC optical channel gain hopt can be accurately approximated by [58, Eq. (10)]hopt =(m+ 1)APD2pi‖d‖22(cos ζ)m Ts gc cosψ IΨ(ψ) (1.4a)=(m+ 1)APD2pi‖d‖m+32dmz Ts gc dTu IΨ(cos−1dTu‖d‖2), (1.4b)where m is the Lambertian order, APD is the area of the PD, d = [dx dy dz]T is thedisplacement vector between the PD and the LED, ζ is the angle of irradiance fromthe LED (measured w.r.t. the LED axis), Ts is the gain of the optical filter, gc isthe gain of the optical concentrator within its field-of-view (FoV), ψ is the angle ofincidence from the LED (measured w.r.t. the receiver axis), and IΨ(·) is an indicatorfunction defined asIΨ(ψ) ,1 |ψ| ≤ Ψ0 |ψ| > Ψ,15Chapter 1. Introductionw2w1 IDCwNx2(t)xN(t)x1(t)s(t)IDCIDCy(t)|s(t)|   μMIIDC     μMI    ||w||     Input data PAM encoderPredistorterAdditionMultiplicationPredistorterPredistorterPRX(t)PTX,1(t)PTX,2(t)PTX,N(t)MISO VLC ChannelFigure 1.5: Beamforming in conjunction with PAM for the MISO VLC Channel.where Ψ is the semi-angle FoV of the concentrator. Assuming an idealized non-imaging concentrator, the gain gc can be approximated by [58, Eq. (8)]gc =n2rsin2 Ψ, (1.5)where nr is the refractive index of the concentrator material. Furthermore, the Lam-bertian order m is determined bym =−1log2(cos ζ3-dB), (1.6)where ζ3-dB is the half-intensity angle of the LED.1.3.3 Beamforming for the MISO VLC ChannelWhen the transmitter has N > 1 LEDs that are sufficiently separated and can bemodulated independently of each other using separate drivers, we end up with aMISO channel having N transmit elements. Figure 1.5 illustrates a MISO VLC16Chapter 1. Introductionsystem utilizing transmit beamforming along with PAM. Similar to the SISO case,information symbols are stochastically encoded into codewords S such that E{S} = 0and |s(t)| ≤ A ∀t. Then, the codewords are multiplied by a fixed vector w ∈ RN ,‖w‖∞ ≤ 1, termed as the beamformer, resulting in the modulation current vectorx(t) = ws(t). (1.7)Thus, after adding the DC bias to each LED, the vector of instantaneous opticalpowers transmitted from the LEDs can be expressed asPTX(t) = η(IDC1N + x(t))= η(IDC1N +ws(t)). (1.8)With multiple-LED transmission, the total received optical power, PRX(t), is the sumof optical powers collected from individual LEDs, i.e., PRX(t) is given byPRX(t) = hToptPTX(t)= ηhTopt(IDC1N +ws(t)), (1.9)where hopt ∈ RN+ is the DC optical channel gain vector. Then, after removing theDC component from the output of the PD, the received signal y(t) can be expressedasy(t) = hTws(t) + n(t), t = 1, 2, . . . , (1.10)where h , ηhoptRPDTa is the DC channel gain vector, and n(t) denotes i.i.d. Gaus-sian noise samples with variance σ2. Equation (1.10) specifies a Gaussian MISOchannel with transmit beamforming, and the transmitted signal vector is subject to17Chapter 1. Introduction   X (n)YB(n)YE(n)MDecoderDecoder   AliceEvep(yB|x)p(yE|x)BobStochastic encoderFigure 1.6: A general wiretap channel.the amplitude constraint‖w‖∞ ≤ 1, (1.11a)|s(t)| ≤ A ∀t = 1, 2, . . . . (1.11b)1.3.4 The Wiretap Channel and the Secrecy CapacityThe wiretap channel is a broadcast channel model that was originally proposed byWyner [34], and later extended by Csiszár and Körner [37], to study the followingcommunication problem: The transmitter (Alice) aims to send a confidential mes-sage M ∈ {1, 2, · · · , 2nRs} to the receiver (Bob) and keep the message entirely secretfrom the eavesdropper (Eve) without using secret-key encryption. Figure 1.6 illus-trates such a scenario, and the individual channels to Bob and Eve are specified bythe marginal transition probability density functions (PDFs) p(yB|x) and p(yE|x),respectively.In order to send the secret message M , Alice will stochastically encode M into acodewordX(n) that is transmitted over the broadcast channel in n channel uses. Thus,the information rate is1nlog2(2nRs) = Rs bits/channel use. Both Bob and Eve willattempt decoding their received signals. Let Mˆ denote the message decoded by Bob,where Mˆ ∈ {1, 2, · · · , 2nRs}. Then, decoding error happens when Mˆ 6= M . Let P (n)e18Chapter 1. Introductiondenote the average probability of decoding error at Bob, then the communicationrate Rs is said to be achievable and secure, i.e., Rs is an achievable secrecy rate, ifthere exists a sequence of (2nRs , n) codes such thatlimn→∞P (n)e = 0, (1.12a)limn→∞1nI(M ;Y (n)E ) = 0. (1.12b)The condition in (1.12a) requires the transmission rate Rs to be reliable, i.e., can bereliably decoded by Bob. On the other hand, (1.12b) is the weak secrecy constraintwhich requires the rate of information leaked to Eve to vanish [26, Section 3.3].The secrecy capacity is defined as the maximum achievable secrecy rate. Bydefinition, any achievable secrecy rate is a lower bound on the secrecy capacity9.Csiszár and Körner [37] have shown that the secrecy capacity of the (nondegraded)wiretap channel illustrated in Figure 1.6 is [26, Corollary 3.4]Cs = maxp(u,x)(I(U ;YB)− I(U ;YE)), (1.13)where U is an auxiliary random variable that satisfies the Markov chain U → X →(YB, YE). Except for a few specials cases, the optimization problem in (1.13) is typi-cally difficult to solve, and usually it is unclear how to choose the auxiliary variable Uin an optimal way. For the special case of the degraded wiretap channel, i.e., whenX → YB → YE forms a Markov chain, it can be shown that the choice U = X isoptimal (see [26, Corollary 3.5]), and thus (1.13) simplifies toCs = maxp(x)(I(X;YB)− I(X;YE)). (1.14)9Therefore, we use the terms “achievable secrecy rate” and “lower bound on the secrecy capacity”interchangeably.19Chapter 1. Introduction  X (n)Y1(n)Y2(n)M1, M2DecoderDecoderTransmitterp(y1|x)p(y2|x)User 1Stochastic encoder              User 2Figure 1.7: A general two-user broadcast channel with confidential messages (BC-CM).1.3.5 The Two-User Broadcast Channel with ConfidentialMessages (BC-CM)The wiretap channel model was extended by Liu et al. [47] to the two-user BC-CMillustrated in Figure 1.7. In such a model, the transmitter has two independentconfidential messages: M1 ∈ {1, 2, · · · , 2nR1} is intended for User 1 and should bekept secret from User 2, and M2 ∈ {1, 2, · · · , 2nR2} is intended for User 2 and shouldbe kept secret from User 1.The transmitter encodes the pair (M1,M2) into a codeword X(n) that is trans-mitted in n channel uses. Similar to the wiretap channel, let P (n)e,1 denote the averageprobability of decoding error at User 1, i.e., the average probability that Mˆ1 6= M1,where Mˆ1 is the decoded message, and P(n)e,2 denote the average probability of decod-ing error at User 2. Then, the rate pair (R1, R2) is said to be achievable and secureif there exists a sequence of (2nR1 , 2nR2 , n) codes such thatlimn→∞P(n)e,1 = 0, limn→∞P(n)e,2 = 0, (1.15a)limn→∞1nI(M1;Y (n)2 ) = 0, limn→∞1nI(M2;Y (n)1 ) = 0, (1.15b)where (1.15a) specifies the reliability requirements for both users, and (1.15b) is themutual confidentiality constraint using the weak secrecy measure.20Chapter 1. IntroductionCompared to the wiretap channel, evaluating the secrecy performance of the two-user BC-CM is obviously more challenging as it requires the characterization of asecrecy capacity region rather than the secrecy capacity (which is just a scalar). LetU1 and U2 be auxiliary random variables such that (U1, U2) → X → (Y1, Y2) formsa Markov chain. Then, it was shown in [47, Theorem 4] that the secrecy rate pair(R1, R2) satisfying0 ≤ R1 ≤ I(U1;Y1)− I(U1;Y2|U2)− I(U1;U2), (1.16a)0 ≤ R2 ≤ I(U2;Y2)− I(U2;Y1|U1)− I(U1;U2) (1.16b)is achievable for the general two-user BC-CM illustrated in Figure 1.7.1.3.6 What Does “Secure Transmission” Mean?The term “secure transmission scheme” can be ambiguous to a reader not familiarwith the terminology used in the physical-layer security literature. Thus, it may beuseful to clarify what “secure transmission” literally means.In the context of physical-layer security, transmission schemes, such as the beam-formers proposed in Chapter 3 and precoders proposed in Chapter 4, are said to be“secure” when they lead to positive secrecy rates. Thus, a typical problem of inter-est is to find transmission schemes that maximize the achievable secrecy rate. Note,however, that having a positive secrecy rate is a necessary but not sufficient conditionto achieve secure transmission. In other words, applying a transmission scheme thatleads to a positive secrecy rate does not immediately render the communication linksecure. Instead, it makes secure transmission possible provided that an appropriatesecrecy codebook is constructed and used to encode the transmitted messages. The21Chapter 1. Introductionsecrecy codebook, which is revealed to all parties, should ensure reliable reception bythe receiver (like regular channel codes), and also have sufficient randomization to al-low stochastic encoding and confuse the eavesdropper. In other words, the codebookshould satisfy the reliability and secrecy constraints in (1.12) for the wiretap chan-nel, or the corresponding constraints in (1.15) for the two-user BC-CM. The designof secrecy codebooks is an involved subject that is beyond the scope of this thesis.The interested reader, however, can refer to [26, Chapter 6] or [31, Section VII] andthe references therein.1.4 Contributions of the ThesisWe claim that this thesis is the first to consider enhancing the secrecy of VLC systemswithin the framework of physical-layer security. By taking amplitude constraintsinto account, we encounter a novel category of design problems in which closed-formsolutions usually cease to be possible. Furthermore, by taking channel uncertaintyinto account, we help make physical-layer security schemes more applicable to real-world scenarios in which the assumption of perfect channel information is almostalways impractical. Our contributions in the entire thesis are summarized as follows.1. Achievable Secrecy Rates subject to Amplitude Constraints: With thelack of analytic expressions for the secrecy capacity of amplitude-constrainedGaussian wiretap channels, we resort to closed-form bounds. In Chapter 2,we begin with deriving lower and upper bounds on the secrecy capacity of thescalar channel under the amplitude constraint. We derive the lower boundsusing the uniform input distribution in conjunction with the entropy powerinequality. For the upper bound, we devise an approach to obtain upper boundson the secrecy capacity of degraded wiretap channels, and apply the devised22Chapter 1. Introductionapproach to the scalar Gaussian wiretap channel. We then exploit the lowerbound along with transmit beamforming in order to obtain an achievable secrecyrate for the MISO wiretap channel. This achievable rate will serve as thedesign equation, i.e., the objective function, in all the optimization problemsencountered in Chapter 3 wherein the design of the beamformer is studied indetail. We also consider in Chapter 2 the scenario in which the scalar channelbetween the transmitter and intended receiver is aided by a friendly jammercapable of sending jamming signals using multiple transmit elements. We derivea closed-form secrecy rate expression when both the data and jamming signalsare subject to amplitude constraints. Our contributions in Chapter 2 werepublished in [62, 14, 63, 64].2. Optimal and Robust Beamforming for the MISO VLCWiretap Chan-nel: In Chapter 3, we focus on the MISO VLC wiretap channel. In particular,we study the design of transmit beamformers that maximize the achievablesecrecy rate, subject to amplitude constraints. Such constraints render the de-sign problem nonconvex and difficult to solve. We show, however, that thisnonconvex problem can be transformed into a solvable quasiconvex line searchproblem. Our approach to solve the optimization problem is generic in the sensethat it can handle general lp-norm constraints on the beamforming vector, i.e.,for any p ≥ 1. We also consider the more realistic case of imperfect chan-nel information regarding the receiver’s and eavesdropper’s links. We tacklethe worst-case secrecy rate maximization problem, again subject to amplitudeconstraints. In our treatment, uncertainty in the receiver’s channel is due tolimited feedback, and is modelled by spherical uncertainty sets. On the otherhand, there is no feedback from the eavesdropper, and the transmitter shall uti-23Chapter 1. Introductionlize the LoS channel gain equation to map the eavesdropper’s nominal locationand orientation into an estimate of the channel gain. Thus, we derive channeluncertainty sets based on inaccurate information regarding the eavesdropper’slocation and orientation, as well as the emission pattern of the LEDs. We alsoconsider channel mismatches caused by the uncertain NLoS components. Thework in Chapter 3 was published in [65, 66].3. Linear Precoding for the Two-User MISO BC-CM: In Chapter 4, weturn our focus to the more general two-user MISO BC-CM communicationmodel. We study the design of linear precoders for secure transmission on sucha channel subject to total and per-antenna10 average power constraints, and alsosubject to amplitude constraints. In both cases, we tackle the design problemby formulating a weighted secrecy sum rate maximization problem. The for-mulated problem involves a fractional objective function, making it nonconvexand difficult to solve. Nevertheless, we show that this nonconvex problem canbe transformed into an equivalent, but more tractable, problem. We propose asubgradient-based search algorithm to obtain a solution, and characterize thecondition under which the obtained solution is guaranteed to be globally opti-mal. Furthermore, we show that our problem formulation and solution approachcan be easily extended to handle the robust version of the design problem withuncertain channel information regarding both receivers. Our work in Chapter 4was submitted for possible publication [67].10In Chapter 4, we generalize the channel model by considering different types of constraintson the channel input. Accordingly, in that chapter, we use the general term “antenna” to denotegeneral transmit and receive elements. In a VLC system, for example, the transmit antenna wouldbe an LED and the receive antenna would be a PD.24Chapter 1. Introduction1.5 Organization of the ThesisThe structure of the thesis reflects the list of contributions in the previous section,and is as follows.In Chapter 2, we derive closed-form secrecy rate expressions for the Gaussianwiretap channel subject to amplitude constraints. Three cases are considered, namely,the scalar wiretap channel, the MISO wiretap channel, and the scalar channel aided bya friendly jammer having multiple transmit elements. We provide numerical examplesfrom typical VLC scenarios in order to get insight into the secrecy performance ofVLC wiretap channels.In Chapter 3, we consider the design of beamformers for the MISO VLC wire-tap channel. The design equation is the secrecy rate expression derived in Chapter 2,and the design parameter is the beamformer subject to amplitude constraints. Underthe premise of perfect channel information, we show that the nonconvex secrecy ratemaximization problem can be optimally solved using a simple line search algorithm.We then extend our approach to the design of robust beamformers that maximizethe worst-case secrecy rate with imperfect channel information. In order to obtainreasonable uncertainty models for the eavesdropper’s channel, we derive uncertaintysets based on the uncertain parameters in the VLC channel gain equation. We usenumerical examples to compare the performance of the optimal and robust beam-formers with conventional beamforming schemes, and also to illustrate the secrecyperformance in typical VLC scenarios.In Chapter 4, we consider linear precoding for the two-user MISO BC-CM subjectto total and per-antenna average power constraints, and also subject to amplitudeconstraints. We begin with deriving closed-form secrecy rate pair expressions. Then,we provide a unified framework to tackle the design problem via weighted secrecy sum25Chapter 1. Introductionrate maximization. We also extend our approach to take channel uncertainty intoaccount. We use numerical examples to validate the solution method and compare theperformance of the proposed linear precoder with conventional precoding schemes.Finally, in Chapter 5, we summarize our contributions and findings in the thesis,and outline some topics for future research.Appendices A, B, and C contain proofs and derivations relevant to Chapters 2,3, and 4, respectively.26Chapter 2Achievable Secrecy Rates for VLCWiretap Channels2.1 IntroductionIntensity modulation (IM) is the only feasible transmission scheme for VLC systemsthat utilize LEDs. Due to linearity limitations of typical LEDs, the input current sig-nal, i.e., the intensity-modulating signal, must satisfy certain amplitude constraintsin order to maintain linear electro-optical conversion and avoid nonlinear or clippingdistortion (see Figure 1.3). Therefore, IM channels are typically modelled with am-plitude constraints on the channel input, rather than the conventional average powerconstraint [52, 53]. Consequently, a proper characterization of the secrecy perfor-mance of VLC links should involve the secrecy capacity of amplitude-constrainedGaussian wiretap channels. In [56], it was shown that the secrecy capacity of thescalar wiretap channel under the amplitude constraint is achieved by a discrete inputdistribution having a finite number of mass points. For sufficiently-small amplitudeconstraints, the symmetric binary input distribution has been shown to be opti-mal [56, Section IV]. For the general case, however, it is difficult to explicitly solvefor the maximizing distribution, and thus the secrecy capacity can be only found vianumerical methods. Since closed-form expressions are typically crucial for systemdesign purposes, one might resort to lower bounds on the secrecy capacity.27Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsAccordingly, in this chapter, we derive closed-form secrecy rate expressions forthe wiretap channel subject to amplitude constraints. Three scenarios are considered,namely, the scalar wiretap channel, the MISO wiretap channel, and the scalar wiretapchannel aided by a friendly jammer. In all scenarios, the data and jamming signals(when applicable) are subject to amplitude constraints. For the scalar channel, we usethe uniform input distribution in conjunction with the entropy power inequality toobtain lower bounds on the secrecy capacity. We also devise a technique to derive anupper bound. Next, we leverage beamforming to obtain a lower bound on the secrecycapacity of the MISO channel. We characterize the secrecy performance when simplezero-forcing (ZF) beamforming is applied. Finally, we consider the scalar channelwhen it is aided by a friendly jammer having multiple transmit elements, but doesnot know the message that is being transmitted. We derive a closed-form secrecy rateexpression after restricting the jamming signal such that it causes no interference tothe intended receiver.The remainder of this chapter is divided into three main sections, correspondingto the three scenarios we consider, besides the conclusions section. The scalar andMISO wiretap channels are considered in Sections 2.2 and 2.3, respectively, whereasthe scalar channel aided by a friendly jammer is considered in Section 2.4. In eachsection, we begin with describing the problem scenario and system model, then wederive closed-form secrecy rate expressions followed by a numerical example. Weconclude the chapter in Section 2.5.2.2 The Scalar VLC Wiretap ChannelIn this section, we consider the scalar VLC wiretap channel, i.e., the amplitude-constrained scalar Gaussian wiretap channel. Because of the amplitude constraint,28Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelshBhEEveBobAliceFigure 2.1: Problem scenario for the SISO case.there is no analytic expression for the secrecy capacity, and thus we derive closed-formlower and upper bounds.2.2.1 System ModelWe consider the simple VLC scenario illustrated in Figure 2.1. The service area, orsimply the room, is illuminated by a single light fixture that is also utilized by Alicefor data transmission. The fixture may have one LED, or multiple LEDs modulatedby the same current signal, e.g., all the LEDs are connected in series. The intendedreceiver (Bob) and the eavesdropper (Eve) have a single photodiode (PD), each.Utilizing the Gaussian channel model in (1.3), the signals received by Bob andEve, respectively, are given byyB(t) = hBx(t) + nB(t), (2.1a)yE(t) = hEx(t) + nE(t), (2.1b)where x(t) ∈ [−A,A] is the transmitted signal, hB ∈ R+ and hE ∈ R+ are Bob’sand Eve’s channel gains, respectively, and nB(t) and nE(t) are i.i.d. Gaussian noisesamples with variances σ2B and σ2E, respectively. For simplicity, and without loss of29Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsgenerality, we assume that σ2B = σ2E = σ2. Such an assumption can be simply fulfilledby properly scaling yB or yE.2.2.2 Achievable Secrecy RatesAssuming hB > hE, the secrecy capacity of the scalar wiretap channel in (2.1) is [36]CSISOs = maxp(x)(I(X;YB)− I(X;YE)) (2.2a)s.t. |X| ≤ A, (2.2b)where maximization is performed over all the input distributions p(x) that satisfy theamplitude constraint |X| ≤ A. Now, because of the amplitude constraint, obtaining aclosed-form solution for (2.2) is a formidable task, if not unfeasible [56]. Nevertheless,it was shown that the maximization problem in (2.2) is convex [56, Eq. (9)], andthe optimal distribution p?(x) that maximizes the difference I(X;YB)− I(X;YE) isdiscrete with a finite number of mass points. Thus, the problem in (2.2) can beefficiently solved via numerical methods. Nevertheless, closed-form expressions aretypically of great interest for system design purposes. Therefore, we provide closed-form lower bounds on the secrecy capacity of the wiretap channel in (2.1), as follows.Proposition 2.1. (Lower Bound on the Secrecy Capacity)The secrecy capacity of the scalar Gaussian wiretap channel in (2.1) subject to theamplitude constraint |x(t)| ≤ A ∀t is lower-bounded asCSISOs ≥12ln(1 +2A2h2Bpieσ2)−(1− 2Q(δ + AhEσ))ln2(AhE + δ)√2piσ2(1− 2Q ( δσ))−Q(δσ)− δ√2piσ2e−δ22σ2 +12, (2.3)30Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelswhere δ > 0 is a free parameter, and Q(·) is the Q-function.Proof: The secrecy capacity in (2.2) can be lower-bounded by the difference betweenthe capacities of Alice-Bob and Alice-Eve channels, as follows.CSISOs = maxp(x)(I(X;YB)− I(X;YE))≥ maxp(x)I(X;YB)−maxp(x)I(X;YE)= CB − CE, (2.4)where the inequality follows from the fact thatmaxu(f1(u)− f2(u)) ≥ maxuf1(u)−maxuf2(u)for arbitrary functions f1 and f2. Then, CB and CE, respectively, can be lower- andupper-bounded as [53, Theorem 5]CB ≥ 12ln(1 +2A2h2Bpieσ2), (2.5a)CE ≤(1− 2Q(δ + AhEσ))ln2(AhE + δ)√2piσ2(1− 2Q ( δσ)) +Q( δσ)+δ√2piσ2e−δ22σ2 − 12,(2.5b)where δ > 0 is a free parameter. Replacing CB and CE in (2.4) with the lower andupper bounds in (2.5a) and (2.5b), respectively, yields the lower bound in (2.3). Proposition 2.2. (Lower Bound on the Secrecy Capacity)The secrecy capacity of the scalar Gaussian wiretap channel in (2.1) subject to the31Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsamplitude constraint |x(t)| ≤ A ∀t is lower-bounded asCSISOs ≥12ln6A2h2B + 3pieσ2pieA2h2E + 3pieσ2. (2.6)Proof: The secrecy capacity in (2.2) can be lower-bounded using the entropy-maximizing uniform input distribution as follows.CSISOs(a)≥ I(X;YB)− I(X;YE)= h(YB)− h(YB|X)− h(YE) + h(YE|X)= h(YB)− h(YE)= h(hBX +NB)− h(YE)(b)≥ 12ln(e2h(hBX) + e2h(NB))− 12ln (2pievar{YE})(c)=12ln(4A2h2B + 2pieσ2)− 12ln(2pie(4A2h2E12+ σ2))=12ln6A2h2B + 3pieσ2pieA2h2E + 3pieσ2, (2.7)where (a) follows from dropping the maximization over p(x), (b) from lower-boundingh(hBX + NB) using the entropy power inequality [35, Theorem 17.7.3] and upper-bounding h(YE) by the differential entropy of a Gaussian random variable havingvariance var{YE}, and (c) from choosing X ∼ U [−A,A], i.e., p(x) is the uniformdistribution over the interval [−A,A], and substituting withh(hBX) = ln(2AhB),var{YE} = var{hEX}+ var{NE} = (2AhE)212+ σ2.32Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsNote that the uniform distribution p(x) = U [−A,A] is the maximum-entropydistribution over the input range [−A,A] subject to the constraint E{X} = 0. Sucha constraint is necessary to ensure that the average radiated optical power, andconsequently the illumination level, is not altered by E{X} (recall the modulationscheme described in Section 1.3.1).2.2.3 Upper Bound on the Secrecy CapacityIn [68, 69, 53], the authors used the dual channel capacity expression in [70, The-orem 8.4] to obtain upper bounds on the capacity of the Gaussian channel underamplitude constraints. Here, we follow a similar approach in order to derive an up-per bound on the secrecy capacity of the scalar Gaussian wiretap channel. First, wenote that, for the case hB > hE, the wiretap channel in (2.1) has the same secrecyperformance as that of the physically degraded wiretap channel characterized by [26,Section 5.1]yB(t) = x(t) + nB(t), (2.8a)yE(t) = yB(t) + nE(t), (2.8b)with NB ∼ N (0, σ2h2B ) and NE ∼ N (0,σ2h2E− σ2h2B). Note from (2.8) that X → YB → YEforms a Markov chain. Next, we introduce the following theorem.Theorem 2.1. (Upper Bound on Conditional Mutual Information)Let X, YB, and YE be three random variables with a joint distribution p(x, yB, yE) thatfactors as p(x)p(yB|x)p(yE|yB), i.e. X → YB → YE forms a Markov chain. Then, the33Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsconditional mutual information I(X;YB|YE) is upper-bounded asI(X;YB|YE) ≤ Ep(x){D(p(yB|X, yE)‖q(yB|yE))} , (2.9)where D(·‖·) denotes the relative entropy, p(yB|x, yE) = p(yB|x)p(yE|yB)p(yE|x) , and q(yB|yE)is an arbitrary conditional distribution.Proof: We begin with [71, Eq. (2.4.20)]I(X;YB|YE) =∫∫∫p(x, yB, yE) lnp(yB|x, yE)p(yB|yE) dx dyB dyE. (2.10)We also have [35, Eq. (2.65)]D(p(yB|yE)‖q(yB|yE)) =∫∫p(yB, yE) lnp(yB|yE)q(yB|yE) dyB dyE=∫∫∫p(x, yB, yE) lnp(yB|yE)q(yB|yE) dx dyB dyE. (2.11)Adding (2.10) to (2.11) yieldsI(X;YB|YE) + D(p(yB|yE)‖q(yB|yE))=∫∫∫p(x, yB, yE) lnp(yB|x, yE)q(yB|yE) dx dyB dyE= Ep(x){∫∫p(yB, yE|X) ln p(yB|X, yE)q(yB|yE) dyB dyE}= Ep(x){D(p(yB|X, yE)‖q(yB|yE))} . (2.12)Then, the inequality in (2.9) follows since the relative entropy D(p(yB|yE)‖q(yB|yE))is always nonnegative [35, Theorem 2.6.3]. Note from (2.12) that equality holds in (2.9) when D(p(yB|yE)‖q(yB|yE)) = 0, i.e.,34Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelswhen p(yB|yE) = q(yB|yE) ∀yB, yE. Note also that the inequality in (2.9) holds for anyinput distribution p(x). Now, consider the specific distribution p?(x) that achievesthe secrecy capacity, i.e.,p?(x) , argmaxp(x)I(X;YB|YE), (2.13)where maximization is over all the distributions that satisfy the constraints on thechannel input X. Using p?(x) in (2.9) results in the following upper bound on thesecrecy capacity.Corollary 2.1. (Upper Bound on The Secrecy Capacity of the Degraded WiretapChannel)An upper bound on the secrecy capacity of the degraded wiretap channel X → YB → YEis given byCs ≤ Ep?(x){D(p(yB|X, yE)‖q(yB|yE))} , (2.14)where p?(x) is as defined in (2.13), and q(yB|yE) is an arbitrary conditional distribu-tion.Now, we are ready to derive an upper bound on the secrecy capacity of the wiretapchannel in (2.1), as follows.Proposition 2.3. (Upper Bound on the Secrecy Capacity of the Scalar GaussianWiretap Channel)The secrecy capacity of the scalar Gaussian wiretap channel in (2.1) subject to theamplitude constraint |x(t)| ≤ A ∀t is upper-bounded asCSISOs ≤12lnA2h2B + σ2A2h2E + σ2. (2.15)35Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsProof: Substituting for D(·‖·) in (2.14) yieldsCs ≤ Ep?(x){∫∫p(yB, yE|X) ln p(yB|X, yE)q(yB|yE) dyB dyE}= Ep?(x){∫∫p(yB, yE|X) ln p(yB|X, yE) dyB dyE}︸ ︷︷ ︸I1−Ep?(x){∫∫p(yB, yE|X) ln q(yB|yE) dyB dyE}︸ ︷︷ ︸I2. (2.16)Now, we have to calculate the terms I1 and I2.The first term I1 can be written asI1 = Ep?(x){∫∫p(yB, yE|X) ln p(yB|X, yE) dyB dyE}= Ep?(x){−h(YB|X = x, YE)}= −h(YB|X, YE). (2.17)Recall that for a Markov chain X → YB → YE, we haveh(X, YB, YE) = h(X) + h(YB|X) + h(YE|YB). (2.18)In addition, for any random variables X, YB, and YE, we haveh(X, YB, YE) = h(X) + h(YE|X) + h(YB|X, YE). (2.19)From (2.17)–(2.19), we can see thatI1 = − (h(YB|X) + h(YE|YB)− h(YE|X)) . (2.20)36Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsFor notational convenience, define γ2B and γ2B, respectively, asγ2B ,σ2h2B, γ2E ,σ2h2E.Thus, we haveI1 = −12ln(2pieγ2B(γ2E − γ2B)γ2E). (2.21)In order to calculate I2 in (2.16), we choose the conditional distribution q(yB|yE) asq(yB|yE) = 1√2pis2e−(yB−µyE)22s2 , (2.22)where µ and s2 are constants to be determined in (2.25). Again, for a Markov chainX → YB → YE, we havep(yB, yE|x) = p(yB|x) p(yE|yB)=1√2piγ2Be− (yB−x)22γ2B1√2pi(γ2E − γ2B)e− (yE−yB)22(γ2E−γ2B) . (2.23)Using (2.22) and (2.23), we getI2 = −Ep?(x){∫∫p(yB, yE|X) ln q(yB|yE) dyB dyE}= −Ep?(x) 1√2piγ2B∞∫−∞e− (yB−X)22γ2B∞∫−∞1√2pi (γ2E − γ2B)e− (yE−yB)22(γ2E−γ2B)×(−12ln(2pis2)− (yB − µyE)22s2)dyB dyE=12ln(2pis2) + Ep?(x) 1√2piγ2B∞∫−∞e− (yB−X)22γ2B12s2(µ2(γ2E − γ2B)+ (µ− 1)2 y2B)dyB=12ln(2pis2) + Ep?(x){12s2(µ2(γ2E − γ2B)+ (µ− 1)2 (X2 + γ2B))}≤ 12ln(2pis2) +12s2(µ2(γ2E − γ2B)+ (µ− 1)2 (A2 + γ2B)) , (2.24)37Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelswhere the inequality follows from Ep?(x){X2} ≤ A2. In order to minimize the expres-sion in (2.24), we differentiate w.r.t. µ and s2. After setting the resulting partialderivatives to zero, we obtain the minimizersµ =A2 + γ2BA2 + γ2Eand s2 =(A2 + γ2B)(γ2E − γ2B)A2 + γ2E. (2.25)Substituting from (2.25) back into (2.24) and adding the result to (2.21), we getCs ≤ 12ln(A2 + γ2B)γ2E(A2 + γ2E)γ2B=12lnh2BA2 + σ2h2EA2 + σ2, (2.26)which is the upper bound in (2.15). It is worth mentioning that the upper bound in (2.26) can be simply obtainedby relaxing the amplitude constraint |X| ≤ A into the average power constraintE{X2} ≤ A2 and noting that (2.26) is the secrecy capacity of the Gaussian channelunder the average power constraint. Nevertheless, the framework we proposed viaTheorem 2.1 and Corollary 2.1 can be used to derive upper bounds on the secrecycapacity of degraded wiretap channels with arbitrary conditional distributions p(yB|x)and p(yE|yB), i.e., the main and degraded channels need not be Gaussian.2.2.4 Numerical ExampleFigure 2.2 depicts the bounds in (2.3), (2.6), and (2.15). Three groups of thesebounds are shown using 20 log10 (hB/hE) = 10, 20, and 30 dB. The lower boundin (2.3) is calculated using δ = σ ln (1 + 2AhE/σ) as proposed in [53]. As can beseen, both (2.3) and (2.6) along with (2.15) tightly bound the secrecy capacity atasymptotically low and high SNRB, where SNRB , h2BA2/σ2. Note that the lower38Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels-20 -10 0 10 20 30 40 50 60 70 80-0.500.511.522.533.54Figure 2.2: Lower and upper bounds on the secrecy capacity of the scalar Gaussianwiretap channel.bound in (2.6) incurs a fixed gap ln√pie/6 = 0.1765 nats/sec/Hz at asymptoticallyhigh SNRB. Nevertheless, since typical VLC links operate at SNR values below40 dB (see, e.g., Figure 2.5), the lower bound in (2.6) is more appropriate for VLCscenarios. Furthermore, (2.6) is more analytically-tractable, and therefore it will beused to obtain secrecy rate expressions for the MISO wiretap channel.2.3 The MISO VLC Wiretap ChannelIn this section, we utilize one of the lower bounds we derived in the previous sectionalong with beamforming to obtain a secrecy rate expression for the MISO wiretapchannel subject to amplitude constraints on the channel input vector.39Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelshBhEEveBobAliceFigure 2.3: Problem scenario for the MISO case.2.3.1 System ModelWe consider the MISO scenario illustrated in Figure 2.3. The room is illuminated byN identical light fixtures utilized for data transmission. Using the vectorized versionof the channel model in (1.3), the signals observed by Bob and Eve, respectively, areyB(t) = hTBx(t) + nB(t), (2.27a)yE(t) = hTEx(t) + nE(t), (2.27b)where x(t) ∈ RN is the transmitted signal vector subject to the amplitude constraint‖x(t)‖∞ ≤ A ∀t, hB ∈ RN+ and hE ∈ RN+ are fixed channel gain vectors, and nB(t) andnE(t) are i.i.d. Gaussian noise samples with variance σ2. Unlike the scalar wiretapchannel in (2.1), the MISO wiretap channel in (2.27) is nondegraded, provided thathB and hE are linearly independent.40Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels2.3.2 Achievable Secrecy RatesA single-letter characterization of the secrecy capacity of the nondegraded wiretapchannel in (2.27) was derived by Csiszár and Körner as [37]CMISOs = maxp(u,x)(I(U ;YB)− I(U ;YE)), (2.28)where U is an auxiliary random vector that satisfies the Markov chainU →X → (YB, YE).Unlike the scalar case, the optimization problem in (2.28) is nonconvex, in general.Furthermore, it is unclear how to choose U . For the Gaussian MISO channel undertotal average power constraint, it was shown in [39] that the secrecy capacity isachieved via beamforming, i.e., the choice U = X = wS is optimum, where w is thebeamformer, i.e., a fixed vector, and S is a Gaussian random variable. Accordingly,we propose the use of beamforming to obtain a lower bound on the secrecy capacityof the MISO wiretap channel in (2.27) under the amplitude constraint, as follows.Proposition 2.4. (Lower Bound on the Secrecy Capacity)The secrecy capacity of the MISO wiretap channel in (2.27) subject to the amplitudeconstraint ‖x(t)‖∞ ≤ A ∀t is lower-bounded asCMISOs ≥12ln6A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2, (2.29)where w ∈ RN is any beamforming vector that satisfies the constraint ‖w‖∞ ≤ 1.Proof: The proof follows directly from combining beamforming and the lower bound41Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsin (2.6), as follows.CMISOs = maxp(u,x)(I(U ;YB)− I(U ;YE))(a)≥ I(X;YB)− I(X;YE)(b)≥ I(wS;YB)− I(wS;YE)(c)≥ 12ln6A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2, (2.30)where (a) follows from dropping the maximization and setting U = X, (b) fromchoosingX = wS such that ‖w‖∞ ≤ 1 and |S| ≤ A, i.e., restricting the transmissionscheme to beamforming, and (c) from choosing p(s) = U [−A,A] and utilizing thelower bound in (2.6). Although suboptimal, beamforming is preferable as it is a linear scheme with lowimplementation complexity. Furthermore, beamforming reduces the vector channelinto a scalar version which enables the use of scalar channel codes. Note that thesecrecy rate expression in (2.30) provides a design equation for the beamformer w.2.3.2.1 Optimal BeamformingThe optimal beamformer w? that maximizes the secrecy rate in (2.30) isw? = argmax‖w‖∞≤112ln6A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2. (2.31)The optimization problem in (2.31) is nonconvex and difficult to solve, mainly becauseof the amplitude constraint ‖w‖∞ ≤ 1. In fact, we shall devote a considerable portionof Chapter 3 to solving (2.31).42Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels2.3.2.2 Zero-Forcing BeamformingThe secrecy rate expression in (2.30) can be simplified by restricting the beam-former w to Eve’s null space. Then, the best ZF beamformer wZF is obtained bywZF = argmax‖w‖∞≤1hTBw (2.32a)s.t. hTEw = 0, (2.32b)which yields the ZF secrecy rateRZFs =12ln(1 +2A2(hTBwZF)2pieσ2). (2.33)Unlike (2.31), the problem in (2.32) is a linear program, and thus can be solved withlower computational complexity. Furthermore, the ZF beamformer wZF makes itunnecessary to use secrecy codebooks, i.e., secure transmission can be achieved withregular channel codes.2.3.3 Numerical Example from a VLC ScenarioHere we provide some numerical results to get insight into the secrecy performanceof the ZF beamformer in a typical indoor VLC scenario. The problem geometry isillustrated in Figure 2.4, and the simulation parameters are provided in Table 2.1.There exist 16 down-facing light fixtures attached to the ceiling. Each fixture encloses4 LEDs, and each LED radiates 1 W optical power. The half-intensity angle is 60◦,and the modulation index is set to 10%. Bob and Eve are located at height 0.85 mabove the floor level, e.g., on desks, and their receivers have a 60◦ FoV (semi-angle).We use a Cartesian coordinate system (x, y) at the receivers height to specify their43Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels1 m1 m5 m5 m951 2 3 487610 11 1216151413xyFigure 2.4: Layout of the LEDs for the MISO case.locations. The origin (0, 0) corresponds to the room center, and all distances arespecified in meters. Noise power is calculated using [9, Eq. (6) and Table I] with70 MHz receiver bandwidth, and the result is averaged over the entire room area.The average electric noise power is −98.82 dBm.Figure 2.5 shows the spatial distribution of the SNR at the receivers height withoutbeamforming, i.e., w = 1N . As can be seen, the SNR reaches its maximum value,39.40 dB, at the room center, and decays to 24.97 dB at the corners.Figure 2.6 shows the achievable communication rate RB, between Alice and Bob,as a function of Bob’s location, without secrecy constraints. This rate is obtainedusing (2.33) after replacing wZF with w = 1N .In Figure 2.7, Bob’s location is fixed at (−0.9,−2.0) and the secrecy rate (2.33) isdepicted as a function of Eve’s location within the entire room area. As expected, thesecrecy rate significantly decreases when Eve is close to Bob. Once Eve is relativelyfaraway, e.g., more than about 2.5 m apart, the secrecy rate is almost independent ofEve’s exact location. It is also interesting to characterize the loss in communication44Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsTable 2.1: Simulation parameters for the MISO wiretap channel.Problem geometryRoom dimensions (W × L×H) 5× 5× 3 m3Light fixtures height (Alice) 3 mReceivers height (Bob and Eve) 0.85 mNumber of light fixtures N 16Transmitter characteristicsNumber of LEDs per fixture 4Average optical power per LED Popt 1 WModulation index µMI 10%LEDs half-intensity angle ζ3-dB 60◦Receiver characteristicsReceiver FoV Ψ 60◦Refractive index of the concentrator nr 1.5PD responsivity RPD 0.54 (A/W)PD surface area APD 1 cm2Average noise power σ2 −98.82 dBm242622830321 234361038040-1-1-2-226283032343638Figure 2.5: Spatial distribution of the SNR at the receivers height (0.85 m above thefloor level) without beamforming.45Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels222.531 23.51004-1-1-2-22.22.42.62.833.23.43.63.8Figure 2.6: Achievable communication rate between Alice and Bob as a function ofBob’s location without secrecy constraints.rate caused by the secrecy constraint, i.e., RB − Rs, by comparing the secrecy ratesin Figure 2.7 with RB(−0.9,−2.0) = 3.2256 nats/sec/Hz from Figure 2.6.Finally, in Figure 2.8, Eve’s location is fixed at (1.6,−0.7) and the secrecy rate(2.33) is shown as a function of Bob’s location. As can be seen, even when Bob isrelatively faraway from Eve, the secrecy rate Rs still depends on Bob’s location, i.e.,Rs exhibits stronger dependence on hB than hE.2.4 The Scalar VLC Wiretap Channel Aided by aFriendly JammerIn this section, we study the secrecy performance of the scalar VLC wiretap channelwhen it is aided by a friendly jammer having multiple transmit LEDs. A jamming46Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels00.5211.51 222.510303.5-1-1-2-2 00.511.522.53Figure 2.7: Secrecy rate obtained with the ZF beamformer (2.32) as a function ofEve’s location when Bob is located at (−0.9,−2.0).00.5211.521 22.53103.504-1-1-2-2 00.511.522.533.5Figure 2.8: Secrecy rate obtained with the ZF beamformer (2.32) as a function ofBob’s location when Eve is located at (1.6,−0.7).47Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelshJBhJEEveBobhAEJammingDataJammerAlicehABFigure 2.9: Problem scenario for the scalar channel aided by a friendly jammer.signal is transmitted to degrade Eve’s reception while causing no interference to Bob,which leads to an increase in the achievable secrecy rate between Alice and Bob.Both the data and jamming signals are subject to amplitude constraints.After a formal description of the system model, we derive a closed-form expressionfor the achievable secrecy rate. Then, we provide a numerical example to illustratethe performance in a typical VLC scenario.2.4.1 System ModelWe consider the VLC scenario illustrated in Figure 2.9. The room is illuminated byNJ + 1 identical light fixtures. Each fixture consists of a group of LEDs modulatedby the same current signal. Alice, the transmitter, sends her data via a single fixture.On the other hand, the jammer utilizes the remaining NJ fixtures, but it does notknow the data transmitted by Alice. Bob and Eve have a single PD, each.Without help from the jammer, securing the connection between Alice and Bobis not possible unless Bob is closer to Alice than Eve. On the other hand, a jammerequipped with multiple transmit elements, and without having access to the trans-mitted data, can help secure the connection by transmitting a carefully-designedjamming signal that increases the interference seen by Eve, i.e., degrades her signal-to-interference-plus-noise ratio (SINR), while causing no interference to Bob.48Chapter 2. Achievable Secrecy Rates for VLC Wiretap ChannelsUtilizing the channel model in (1.3), the signals received by Bob and Eve, respec-tively, areyB(t) = hABx(t) + hTJBxJ(t) + nB(t), (2.34a)yE(t) = hAEx(t) + hTJExJ(t) + nE(t), (2.34b)where hAB ∈ R+ and hAE ∈ R+ are the channel gains from Alice to Bob and Eve,respectively, hJB ∈ RNJ+ and hJE ∈ RNJ+ are the channel gain vectors from the jammerto Bob and Eve, respectively, x(t) ∈ R is the data signal, xJ(t) ∈ RNJ is the jammingsignal, and nB(t) and nE(t) are i.i.d. Gaussian noise samples with variance σ2. Thedata and jamming signals are subject to the amplitude constraints |x(t)| ≤ A ∀tand ‖xJ(t)‖∞ ≤ A ∀t, respectively. Furthermore, both x(t) and xJ(t) are designedsuch that E{X} = 0 and E{XJ} = 0. Thus, neither x(t) nor xJ(t) has an effect onillumination. Finally, we assume that hJB and hJE are linearly independent, and allthe channel gains are accurately known to all the terminals.In order to derive a secrecy rate expression for the wiretap channel in (2.34), wehave to simplify the expressions in (2.34) by imposing the following restrictions. First,the jamming signal xJ(t) shall cause no interference to Bob, i.e., hTJBxJ(t) = 0 ∀t.Such a restriction is not necessarily optimal as it might well be the case that allowingnonzero interference at Bob would permit higher interference at Eve and probablyhigher achievable secrecy rate. Second, the jammer shall adopt a beamforming strat-egy, i.e., the jamming signal is constructed as xJ(t) = wJj(t), where wJ ∈ RNJ ,‖wJ‖∞ ≤ 1, is the jamming beamformer, and j(t) ∈ [−A,A] is a zero-mean ran-dom jamming symbol. Beamforming is preferred as it allows simple implementation,however it might be an inappropriate jamming strategy if there are many eavesdrop-pers with probably orthogonal or near-orthogonal channels. Finally, to simplify the49Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsderivation of a closed-form secrecy rate expression, we assume that both X and Jhave uniform distributions over the interval [−A,A]. After applying such restrictions,the wiretap channel in (2.34) simplifies toyB(t) = hABx(t) + nB(t), (2.35a)yE(t) = hAEx(t) + hTJEwJj(t) + nE(t). (2.35b)We are now ready to derive an achievable secrecy rate expression for (2.35), whichwill also be achievable for (2.34).2.4.2 Achievable Secrecy RateProposition 2.5. (Achievable Secrecy Rate)An achievable secrecy rate, in (nats/sec/Hz), for the wiretap channel in (2.35) is[Rs]+, where Rs is given byRs =12ln(1 +2A2h2ABpieσ2)−lnhAE|hTJEwJ|+|hTJEwJ|2hAE|hTJEwJ| ≤ hAEhAE2|hTJEwJ|otherwise=12ln(1 +2A2h2ABpieσ2)−min{lnhAE|hTJEwJ|+|hTJEwJ|2hAE,hAE2|hTJEwJ|}, (2.36)where wJ ∈ RN is any jamming vector that satisfies the constraints hTJBwJ = 0 and‖wJ‖∞ ≤ 1.Proof: Without loss of generality, we assume in the following that hTJEwJ is non-negative. If hTJEwJ < 0, then wJ can be replaced with −wJ without violating theamplitude constraint or changing the secrecy rate results.First, we recall our assumption in the previous subsection that X ∼ U [−A,A]50Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsand J ∼ U [−A,A]. Thus, we haveh(hABX) = ln(2AhAB), (2.37a)h(hAEX) = ln(2AhAE), (2.37b)h(hTJEwJJ) = ln(2AhTJEwJ). (2.37c)Next, let the random variable VE be defined asVE , hAEX + hTJEwJJ. (2.38)Then, VE has a trapezoidal distribution (see Appendix A), and its differential entropyish(VE) = min{ln(2AhAE) +hTJEwJ2hAE, ln(2AhTJEwJ) +hAE2hTJEwJ}. (2.39)Furthermore, it is clear from (2.35b) and (2.38) that X → VE → YE forms a Markovchain. Now, the secrecy capacity of the wiretap channel in (2.35) can be lower-bounded as follows.Cs ≥ I(X;YB)− I(X;YE)(a)≥ I(X;YB)− I(X;VE)= h(YB)− h(YB|X)− h(VE) + h(VE|X)(b)≥ 12ln(e2h(hABX) + e2h(NB))− h(NB)− h(VE) + h(hTJEwJJ)(c)=12ln(4A2h2AB + 2pieσ2)− 12ln(2pieσ2)− h(VE) + ln(2AhTJEwJ)(d)=12ln(1 +2A2h2ABpieσ2)−min{lnhAEhTJEwJ+hTJEwJ2hAE,hAE2hTJEwJ}, (2.40)51Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelswhere (a) follows from the data-processing inequality [35, Theorem 2.8.1], (b) fromlower-bounding h(YB) using the entropy power inequality [35, Theorem 17.7.3], (c) bysubstituting from (2.37a) and (2.37c) for h(hABX) and h(hTJEwJJ), respectively, and(d) by substituting from (2.39) for h(VE). Figure 2.10 depicts Rs in (2.36) as a function of hTJEwJ for different values of hAE.Note that Rs is upper-bounded by12ln(1 +2A2h2ABpieσ2), which is the achievable ratebetween Alice and Bob, without secrecy constraints, subject to |x(t)| ≤ A ∀t [53,Theorem 5]. Note also that Rs is a nondecreasing function of hTJEwJ for hTJEwJ ≥ 0,and a nonincreasing function of hAE. Thus, under the assumption that hJB and hJEare perfectly known to the jammer, the optimal jamming beamformer that maximizesRs while causing no interference to Bob is obtained bymaximize‖wJ‖∞≤1hTJEwJ (2.41a)s.t. hTJBwJ = 0, (2.41b)which is a simple linear program and can be efficiently solved.2.4.3 Numerical Example from a VLC ScenarioIn this subsection, we provide a numerical example by simulating a typical indoorVLC scenario. The problem geometry is illustrated in Figure 2.11, and the simulationparameters are provided in Table 2.2. The room has a size of 5 × 5 × 3 m3, and isilluminated by 9 identical light fixtures. Each fixture has 7 LEDs, and each LEDradiates 1 W optical power. The fixture at the center is used by Alice for data trans-mission, while the remaining 8 fixtures are exploited for jamming. The modulationindex for all the LEDs is 10%. Bob and Eve are located at height 0.85 m above the52Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels0 1 2 3 4 5 6 7 8 9 1000.511.522.53Figure 2.10: Achievable secrecy rates (2.36) for the scalar channel aided by a friendlyjammer.floor level, and their receivers have a 70◦ FoV and a single PD, each. We use a two-dimensional coordinate system (x, y) to identify the receivers locations. The origin(0, 0) corresponds to the room center at the receivers level. Noise power is calculatedusing [9, Eq. (6) and Table I] with a receiver bandwidth of 70 MHz, and the result isaveraged over the entire room area. The average noise power is −98.39 dBm.In Figure 2.12, we plot the secrecy rate (2.36) as a function of Eve’s locationwhen Bob is located at (−0.7,−0.9), while in Figure 2.13, we fix Eve’s location at(0.3,−1.5) and plot (2.36) as a function of Bob’s location.In both figures, the jamming beamformer wJ is obtained with (2.41). We notethat, when Eve is sufficiently close to Bob, jamming is restrained by the null spaceof Bob, resulting in considerably reduced secrecy rates. On the other hand, whenBob and Eve are faraway, the jammer is able to significantly degrade Eve’s reception,53Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels87654321xy5 m1⅔  mAlice5 m1⅔  mFigure 2.11: Layout of the LEDs for the scalar channel aided by a friendly jammer.Table 2.2: Simulation parameters for the scalar channel aided by a friendly jammer.Problem geometryRoom dimensions (W × L×H) 5× 5× 3 m3Light fixtures height (Alice and the jammer) 3 mReceivers height (Bob and Eve) 0.85 mTotal number of light fixtures NJ + 1 9Transmitter characteristicsNumber of LEDs per fixture 7Average optical power per LED Popt 1 WModulation index µMI 10%LEDs half-intensity angle ζ3-dB 60◦Receiver characteristicsReceiver FoV Ψ 70◦Refractive index of the concentrator nr 1.5PD responsivity RPD 0.54 A/WPD surface area APD 1 cm2Average noise power σ2 −98.39 dBm54Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channels020.511 21.51002-1-1-2-2 00.20.40.60.811.21.41.6Figure 2.12: Secrecy rate obtained with the jamming beamformer (2.41) as a functionof Eve’s location when Bob is located at (−0.7,−0.9).00.5211 21.510202.5-1-1-2-2 00.20.40.60.811.21.41.61.82Figure 2.13: Secrecy rate obtained with the jamming beamformer (2.41) as a functionof Bob’s location when Eve is located at (0.3,−1.5).55Chapter 2. Achievable Secrecy Rates for VLC Wiretap Channelsand the resulting secrecy rate is almost independent of Eve’s channel, but is upper-bounded by the achievable rate between Alice and Bob.2.5 ConclusionsUnlike RF channels, the VLC channel is more accurately modelled with amplitudeconstraints on the channel input, making it difficult to obtain analytic secrecy ca-pacity expressions even for the simple SISO case. Therefore, we derived closed-formlower and upper bounds on the secrecy capacity of the amplitude-constrained scalarwiretap channel. Then, we utilized beamforming to obtain an achievable secrecy ratefor the MISO channel. The numerical results revealed that ZF is an appropriatestrategy for secure transmission in VLC scenarios, provided that the transmitter hasaccurate channel information. When feasible, ZF is a favorable transmission schemeas it eliminates the need to use secrecy codebooks.We also derived a closed-form secrecy rate expression for the scalar wiretapchannel when the signal received by the eavesdropper is degraded by amplitude-constrained jamming signals transmitted from a helper node. In addition, we formu-lated a simple linear program to optimize the jamming beamformer, assuming perfectchannel information.In the next chapter, we will focus on the MISO channel and study the design ofbeamformers for secrecy rate maximization subject to amplitude constraints.56Chapter 3Optimal and Robust Beamformingfor Secure MISO VLC Links3.1 IntroductionIn the previous chapter, we utilized the uniform input distribution to derive closed-form secrecy rate expressions for the scalar wiretap channel under the amplitudeconstraint. Then, we leveraged beamforming to obtain a closed-form secrecy rateexpression for the MISO wiretap channel. In this chapter, we focus on the design ofthe beamformer itself. In particular, we study the design of transmit beamformers forsecure downlink transmission in indoor MISO VLC links in the presence of a passiveeavesdropper (Eve) attempting to overhear the message conveyed by light waves tothe intended receiver (Bob). Assuming uniform input distribution, our performancemeasure is the secrecy rate expression (2.29) derived in the previous chapter for theamplitude-constrained MISO wiretap channel.Under the premise of perfect channel information, we first consider the designof optimal beamformers that maximize the achievable secrecy rate subject to ampli-tude constraints. Such constraints render the optimization problem nonconvex anddifficult to solve. Nevertheless, we show that this nonconvex problem can be recastas a solvable quasiconvex line search problem. We then consider the more generaland more realistic case in which the transmitter (Alice) has uncertain information57Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksregarding Bob’s and Eve’s channels. We study the design of robust beamformers thatmaximize the worst-case secrecy rate, again subject to amplitude constraints. Theresulting max-min optimization problem is more complex than its non-robust counter-part, but still can be reformulated as a quasiconvex line search problem. Tractabilityof the reformulated problem, however, depends on the geometries of the uncertaintysets. For Bob’s channel, we consider uncertainty arising from quantization errorsimposed by the finite rate of the feedback channel. Such uncertainty is well mod-elled with N -dimensional spherical sets centered at the nominal estimate availableto Alice, where N is the number of transmit elements. For Eve’s channel, however,we do not assume any feedback because Eve is a passive eavesdropper. Instead, wetake advantage of the fact that the line-of-sight (LoS) path is typically dominantin VLC channels. Moreover, the LoS channel gain can be accurately approximatedby a deterministic function of the receiver’s location and orientation, along with theemission pattern of the LEDs (recall the LoS channel gain expression in (1.4)). Intypical VLC scenarios, it is sensible to assume that Alice has some knowledge ofEve’s location and orientation (recall, for example, the scenario in Figure 1.1). Thus,a reasonable estimate of Eve’s channel can be obtained from such information. Ac-cordingly, we derive uncertainty sets that reflect Alice’s imprecise knowledge of Eve’slocation and orientation, as well as the emission pattern of the LEDs. We also con-sider possible channel mismatches caused by non-line-of-sight (NLoS) components.Such components are due to diffuse reflections from nearby surfaces, and they are nottaken into account by the channel gain equation in (1.4). All the derived uncertaintysets are well structured in the sense that they lead to solvable worst-case secrecy ratemaximization problems.The secrecy performance of the Gaussian MISO wiretap channel with perfect58Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkschannel information, subject to a total average power constraint, was studied in [72,38, 39, 43]. Lower bounds on the secrecy capacity were obtained in [72] and [38]. Inaddition, it was shown in [38] that beamforming is the optimal transmission strategyif the channel inputs are Gaussian. These results were generalized in [39] and [43]where it was shown that Gaussian signaling, along with beamforming, is in factoptimal, and closed-form secrecy capacity expressions were derived.The design of robust transmission schemes with imperfect channel information,based on worst-case secrecy rate maximization, was considered in [73, 74, 75, 76, 77,78]. In [73], the authors observed similarities between the cognitive radio and wiretapchannel models, and considered the design of robust beamformers in conjunction withspherical uncertainty sets for Eve’s channel. The authors in [74] studied robust beam-forming along with discrete uncertainty sets corresponding to inaccurate informationregarding Eve’s location, under the assumption of LoS propagation for RF channels.Worst-case secrecy rate maximization for the MISO channel wiretapped by multi-ple eavesdroppers having multiple antennas was considered in [75] using sphericaluncertainty sets for the receiver’s and eavesdroppers’ channels. In [76], the authorsconsidered the use of artificial noise generated by a friendly jammer and studied thedesign of robust data and jamming covariance matrices, under both individual andglobal power constraints. The work in [77] considered the design of robust transmitcovariance matrices for the MIMO wiretap channel in the low SNR regime using alinearized secrecy rate expression, i.e., the secrecy rate is approximated by a linearfunction of the covariance matrix. A similar approach was utilized in [78] wherethe data and jamming covariance matrices are alternatively optimized after lineariz-ing the nonconcave term in the secrecy rate expression based on Taylor’s first-orderapproximation.59Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksCompared to the previously mentioned works, our work in this chapter has thefollowing two key differences:1) We design the beamformer w subject to a per-transmit-element amplitudeconstraint, i.e., ‖w‖∞ ≤ 1. As mentioned in Section 1.2.2, amplitude con-straints explicitly arise in VLC systems because of limitations on the linearoperation region of the LEDs. Furthermore, as a side advantage, our ap-proach to solve the design problem is in fact applicable to general lp-normconstraints, i.e., ‖w‖p ≤ 1, for any p ≥ 1. On the other hand, the worksin [72, 38, 39, 43, 73, 74, 75, 76, 77, 78] consider a total power constraint PToton the transmitted signal vector, that is ‖w‖2 ≤√PTot, or, more generally,Tr(E{XXT}) ≤ PTot, where E{XXT} is the transmit covariance matrix.2) We do not assume feedback from Eve regarding her channel information. In-stead, we exploit Alice’s imprecise knowledge of Eve’s location and orientationto obtain an estimate of Eve’s channel gain. Specifically, we derive uncertaintysets for Eve’s channel based on the uncertain parameters in the LoS channelgain equation in (1.4). We also consider uncertainty caused by the NLoS com-ponents. On the other hand, the works in [73, 75, 76, 77, 78] assume sphericaluncertainty sets for Eve’s channel, that is ‖hE − hˆE‖2 ≤ hE , where hˆE is Alice’serroneous estimate of hE, and hE is some known constant. This model is wellaccepted to take into account channel uncertainty caused by limited feedbackfrom the receiver [57, Lemma 1]. In wiretap scenarios, however, the sphericaluncertainty model becomes inapplicable if Eve is a passive eavesdropper andnot part of the communication network.The remainder of this chapter is organized as follows. The system model is de-scribed in Section 3.2. In Section 3.3, we consider the design of optimal and robust60Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksbeamformers under the assumptions of perfect and imperfect channel information,respectively. In Section 3.4, we derive uncertainty sets for Eve’s channel based onthe uncertain parameters in the LoS channel gain equation. In Section 3.5, we pro-vide numerical examples to compare the performance of the proposed beamformerswith conventional schemes, and evaluate the worst-case secrecy rate performance ina typical VLC scenario. We conclude the chapter in Section 3.6.3.2 System ModelWe consider secure downlink transmission from Alice to Bob over an indoor VLC linkin the presence of a passive eavesdropper, Eve (recall the scenario in Figure 2.3). Theservice area is illuminated by NFix light fixtures attached to the ceiling. Each fixtureencloses NLED high-brightness LEDs that can be modulated independently of eachother using separate drivers. Thus, the total number of LEDs is N = NFix ×NLED.Next, we recall the beamforming scheme described in Section 1.3.3 whereby thetransmitted signal vector x(t) ∈ RN is constructed asx(t) = ws(t), (3.1)where w ∈ RN is the beamformer and s(t) ∈ R is the data symbol. Due to linearitylimitations of the LEDs, the transmitted signal vector x(t) must satisfy the amplitudeconstraint‖x(t)‖∞ ≤ A ∀t. (3.2)In order to satisfy (3.2), we let S ∼ U [−A,A], where S is the random variablecounterpart of the data symbol s(t), and choose the beamformer w such that it61Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkssatisfies the constraint‖w‖∞ ≤ 1. (3.3)Thus, utilizing the MISO channel model in (1.10), the signals received by Bob andEve, respectively, areyB(t) = hTBws(t) + nB(t), (3.4a)yE(t) = hTEws(t) + nE(t), (3.4b)where hB ∈ RN+ and hE ∈ RN+ are Bob’s and Eve’s channel gain vectors, respectively,and nB(t) and nE(t) are i.i.d. Gaussian noise samples with variance σ2.3.3 Optimal and Robust Beamformer Design3.3.1 Problem FormulationUtilizing the result of Proposition 2.4, an achievable secrecy rate, in (bits/sec/Hz),for the MISO wiretap channel in (3.4) isRs =[12log26A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2]+, (3.5)where the beamformer w is subject to the amplitude constraint ‖w‖∞ ≤ 1. A typ-ical problem of interest is to find the optimal beamformer w? that maximizes theachievable secrecy rate, i.e.,w? = argmax‖w‖∞≤1Rs. (3.6)In fact, our main goal in this chapter is to solve the design problem in (3.6). To thisend, we have to overcome two major difficulties. Firstly, the optimization problem62Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksin (3.6) is clearly nonconvex, and the amplitude constraint ‖w‖∞ ≤ 1 makes itdifferent from the well-known Rayleigh quotient maximization problem. In the nextsubsection, we introduce Proposition 3.1 to transform this nonconvex problem into asolvable quasiconvex line search problem. Secondly, it is unrealistic to assume thatthe channel gain vectors hB and hE are precisely known to Alice. Therefore, a moreappropriate design approach is to devise reasonable uncertainty sets, HB and HE,that enclose all possible realizations of hB and hE, respectively, and solve the robustcounterpart [79] of (3.6) to maximize the secrecy rate corresponding to the worst-caserealization of (hB,hE) ∈ HB ×HE. That is to solvemaximize‖w‖∞≤1Rs ∀(hB,hE) ∈ HB ×HE, (3.7a)or, equivalently,maximize‖w‖∞≤1minhB∈HB,hE∈HERs. (3.7b)We will tackle the robust design problem (3.7) in Section 3.3.3 via Proposition 3.2,whereas in Section 3.4, we shall discuss methods to model uncertainty in Eve’s chan-nel, in VLC scenarios, without feedback from Eve.3.3.2 Optimal Beamforming with Perfect ChannelInformationOur focus in this subsection is on solving the design problem in (3.6) under thepremise of perfect channel information. Although the constraint on the beamformeris specified by ‖w‖∞ ≤ 1, i.e., an amplitude or l∞-norm constraint, we shall in factsolve the problem subject to a general lp-norm constraint, i.e., for any p ≥ 1.63Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksProposition 3.1. (Certain hB and hE) Let ‖w‖p, p ≥ 1, denote the lp-norm of w,then the maximization problemmaximize‖w‖p≤16A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2(3.8)is equivalent to the quasiconvex optimization problem (or quasiconcave maximizationproblem)maximizeα∈[αmin,√6/pie]6A2(hTBwα)2 + 3pieσ2pieα2A2(hTBwα)2 + 3pieσ2, (3.9)where αmin, the lower bound on α, isαmin = minw,αα (3.10a)s.t. hTBw = 1, (3.10b)|hTEw| ≤ α, (3.10c)and, for each α ∈ [αmin,√6/pie], wα is obtained bywα = argmax‖w‖p≤1hTBw (3.11a)s.t. |hTEw| ≤ αhTBw. (3.11b)Proof: Our goal is to prove that the problem in (3.8) is equivalent to the line searchproblem in (3.9), and the objective function in (3.9) is quasiconcave w.r.t. the searchvariable α. The latter part, in particular, is not straightforward.64Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksUsing the auxiliary variable τ ≥ 3pieσ2, the problem in (3.8) can be expressed asmaximize‖w‖p≤1,τ6A2(hTBw)2 + 3pieσ2τ(3.12a)s.t. pieA2(hTEw)2 + 3pieσ2 ≤ τ, (3.12b)or, equivalently,maximizeτf(τ)τ, (3.13)where f(τ) is defined asf(τ) , max‖w‖p≤16A2(hTBw)2 + 3pieσ2 (3.14a)s.t. |hTEw| ≤√τ − 3pieσ2pieA2. (3.14b)Note that the constraints in (3.12b) and (3.14b) are equivalent. In the following, weshow that the objective function in (3.13) is quasiconcave w.r.t. τ by establishing theconcavity of f(τ). For notational convenience, we introduce a new variable ε ≥ 0,defined asε ,√τ − 3pieσ2pieA2. (3.15)Then, we define the perturbation function ϕ(ε) asϕ(ε) , max‖w‖p≤1hTBw (3.16a)s.t. |hTEw| ≤ ε. (3.16b)It is clear that ϕ(ε) is nonnegative and nondecreasing for all ε ≥ 0. Furthermore, theperturbed problem in (3.16) is convex, and thus ϕ(ε) is concave [80, Section 5.6.1].65Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksAs a consequence, ϕ(ε) is continuous and its right and left derivatives11, ϕ′+(ε) andϕ′−(ε), exist for all ε > 0. These derivatives are nonincreasing in the sense that, forany ε2 > ε1 > 0, we have [81, Theorem 1.6]ϕ′−(ε1) ≥ ϕ′+(ε1) ≥ ϕ′−(ε2) ≥ ϕ′+(ε2) ≥ 0, (3.17)where the last inequality holds since ϕ(ε) is nondecreasing. Moreover, for any ε0 ≥ 0and any ε ∈ {ε : ε > 0, ϕ′+(ε) = ϕ′−(ε)}, i.e., any ε at which ϕ(ε) is differentiable, wehave [80, Section 3.1.3]ϕ(ε0) ≤ ϕ(ε) + ϕ′(ε)(ε0 − ε). (3.18)Substituting with ε0 = 0 into (3.18), we getϕ(ε) ≥ ϕ(0) + εϕ′(ε) ≥ εϕ′(ε), (3.19)where the second inequality holds since ϕ(0) is nonnegative. We are now ready toprove that f(τ) ≡ 6A2(ϕ(ε))2+3pieσ2 is concave w.r.t. τ ≡ pieA2ε2+3pieσ2. The rightand left derivatives of f(τ) can be written in terms of ϕ′+(ε) and ϕ′−(ε), respectively,asf ′+(τ) =6pieϕ(ε)εϕ′+(ε), f′−(τ) =6pieϕ(ε)εϕ′−(ε). (3.20)From (3.17) and (3.20), it is clear thatf ′−(τ) ≥ f ′+(τ) for any τ > 3pieσ2. (3.21)11We resort to one-sided derivatives, rather than the ordinary two-sided derivative ϕ′(ε), becauseϕ(ε) is not necessarily smooth or differentiable over the whole interior of its domain. Particularly,there exist, in general, some ε > 0 at which ϕ′+(ε) 6= ϕ′−(ε). These are the points where ϕ′+(ε) andϕ′−(ε) have jump discontinuities. Nevertheless, since ϕ(ε) is concave, there are only countably manysuch jumps, i.e., the set {ε : ε > 0, ϕ′+(ε) 6= ϕ′−(ε)} has zero Lebesgue measure [81, Section 1.8].66Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksFurthermore, when ϕ(ε) is twice differentiable (and consequently f(τ) is twice differ-entiable), we havef ′′(τ) =3pieε2((ϕ′(ε)− ϕ(ε)ε)ϕ′(ε) + ϕ(ε)ϕ′′(ε))≤ 0, (3.22)where the inequality holds since ϕ′(ε) ≤ ϕ(ε)/ε, ϕ′(ε) ≥ 0, ϕ(ε) ≥ 0, and ϕ′′(ε) ≤ 0(the last inequality follows from (3.17) or the second-order condition of concavity [80,Section 3.1.4]). Combining (3.21) and (3.22) yieldsf ′−(τ1) ≥ f ′+(τ1) ≥ f ′−(τ2) ≥ f ′+(τ2), (3.23)for any τ2 > τ1 > 3pieσ2. Hence, f(τ) is concave [82, Theorem 24.2]. Then, itis straightforward to verify that f(τ)/τ is quasiconcave by noting that all the β-superlevel sets {τ : τ ≥ 3pieσ2, f(τ)/τ ≥ β}, for all β ∈ R, are convex, i.e., intervals,including the empty set and infinite intervals [80, Section 3.4.1].Next, we define the new variable α ≥ 0 asα , εϕ(ε)=√τ − 3pieσ2pieA2(ϕ(ε))2, ϕ(ε) 6= 0. (3.24)Alternatively, for some given α ≥ 0, τ can be expressed in terms of α asτ = g(α) , pieα2A2(hTBwα)2 + 3pieσ2, (3.25)67Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere wα is defined aswα , argmax‖w‖p≤1hTBw (3.26a)s.t. |hTEw| ≤ αhTBw. (3.26b)The problem in (3.26) is clearly equivalent to the perturbed problem in (3.16) when αand ε satisfy (3.24), or, equivalently, when α and τ satisfy (3.25). Thus, hTBwα ≡ ϕ(ε).Furthermore, we note from (3.26) that hTBwα is nondecreasing w.r.t. α (since increas-ing α relaxes the constraint in (3.26b)). Thus, g(α), as defined in (3.25), is a strictlyincreasing function of α. Substituting with τ = g(α) back into (3.13) and changingthe optimization variable into α, the problem in (3.13) can be written asmaximizeαf(g(α))g(α),or, equivalently,maximizeα6A2(hTBwα)2 + 3pieσ2pieα2A2(hTBwα)2 + 3pieσ2, (3.27)where wα is as defined in (3.26). Since f(τ)/τ is quasiconcave w.r.t. τ , and τ = g(α)is strictly increasing w.r.t. α, we conclude that f(g(α))/g(α) is quasiconcave w.r.t. α,and hence the problem in (3.27) is quasiconvex, i.e., a quasiconcave maximizationproblem.Finally, the search interval for optimal α can be lower-bounded by the smallest68Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksfeasible α, given byαmin = minw,αα (3.28a)s.t. hTBw = 1, (3.28b)|hTEw| ≤ α, (3.28c)and the upper bound αmax =√6/pie is simply obtained by noting that f(g(α)) ≥g(α), and thus Rs ≥ 0, only if α ≤√6/pie, which completes the proof. Remarks:• Proposition 3.1 has a practical interpretation. It states that the achievablesecrecy rate is a quasiconcave function of the parameter α, which is the ratioof the signal level at Eve to the signal level at Bob. This is provably true foran arbitrary lp-norm constraint on the beamformer w.• Setting α = 0 in (3.11) corresponds to ZF, i.e., wα=0 is the best ZF beamformer.• If N ≥ 2, and hB and hE are linearly independent, then αmin = 0 and ZF isfeasible.• The case of K > 1 colluding eavesdroppers, or, equivalently, a single eavesdrop-per having K receiving elements, can be also handled using Proposition 3.1after replacing the inequalities in (3.10c) and (3.11b) with ‖HTEw‖2 ≤ α and‖HTEw‖2 ≤ αhTBw, respectively, whereHE , [hE1 . . . hEK ] and hEk , k = 1, . . . , K,is the channel gain vector of the kth eavesdropper.Proposition 3.1 involves two optimization problems; the outer problem (3.9) andthe inner problem (3.11). The outer problem is a quasiconvex line search problemwhose globally optimal solution can be found by performing a bisection search on69Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksα ∈ [αmin,√6/pie], on a logarithmic scale. In the next subsection, we propose Algo-rithm 3.1 to solve (3.9), as well as the corresponding problem in the more generalcase of uncertain channel information. In each iteration of the bisection search, theinner problem (3.11) should be solved to obtain wα and calculate the objective func-tion in (3.9). The inner problem is clearly convex for any p ≥ 1, and thus it can beefficiently solved.Using (3.9), the achievable secrecy rate, as a function of α, isRs(α) =[12log26A2(hTBwα)2 + 3pieσ2pieα2A2(hTBwα)2 + 3pieσ2]+. (3.29)Let α? denote the global maximizer of (3.9), then the optimal beamformer w? isthe solution of (3.11) corresponding to α = α?, i.e., w? ≡ wα? , and the maximumachievable secrecy rate is Rs(α?).3.3.3 Robust Beamforming with Imperfect ChannelInformationIn this subsection, we extend Proposition 3.1 to take into account uncertainty inchannel information for both Bob and Eve.Proposition 3.2. (Uncertain hB and hE) Given a convex set HB and an arbitraryset HE, the max-min problemmaximize‖w‖p≤1minhB∈HB,hE∈HE6A2(hTBw)2 + 3pieσ2pieA2(hTEw)2 + 3pieσ2, (3.30)70Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksfor any p ≥ 1, is equivalent to the quasiconvex problemmaximizeα∈[αmin,√6/pie]6A2t2α + 3pieσ2pieα2A2t2α + 3pieσ2, (3.31)where αmin isαmin = minw,αα (3.32a)s.t. hTBw ≥ 1 ∀hB ∈ HB, (3.32b)|hTEw| ≤ α ∀hE ∈ HE, (3.32c)and, for each α ∈ [αmin,√6/pie], tα is obtained from(wα, tα) = argmax‖w‖p≤1,tt (3.33a)s.t. hTBw ≥ t ∀hB ∈ HB, (3.33b)|hTEw| ≤ αt ∀hE ∈ HE. (3.33c)Proof: The proof is mostly along the same line as that of Proposition 3.1. Themax-min problem in (3.30) can be expressed asmaximizeτf(τ)τ, (3.34)71Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere f(τ) is defined asf(τ) , max‖w‖p≤1minhB∈HB6A2(hTBw)2 + 3pieσ2 (3.35a)s.t. |hTEw| ≤√τ − 3pieσ2pieA2∀hE ∈ HE. (3.35b)Next, we define the perturbation function ϕ(ε) asϕ(ε) , max‖w‖p≤1,tt (3.36a)s.t. |hTBw| ≥ t ∀hB ∈ HB, (3.36b)|hTEw| ≤ ε ∀hE ∈ HE, (3.36c)where ε is defined as in (3.15). Note from (3.35) and (3.36) that f(τ) ≡ 6A2(ϕ(ε))2 +3pieσ2. Note also that, unlike (3.16), the perturbed problem in (3.36) is not convexbecause of the constraint in (3.36b). This nonconvexity can be eliminated by imposingthe additional constrainthTBw ≥ 0 ∀hB ∈ HB, (3.37)or, equivalently, replacing (3.36b) withhTBw ≥ t ∀hB ∈ HB. (3.38)The additional constraint, however, may render the solution suboptimal. Let ϕ(ε)72Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksbe defined asϕ(ε) , max‖w‖p≤1,tt (3.39a)s.t. hTBw ≥ t ∀hB ∈ HB, (3.39b)|hTEw| ≤ ε ∀hE ∈ HE. (3.39c)Then, ϕ(ε) ≤ ϕ(ε), i.e., a nonzero gap may exist between the two optimal values. Inthe sequel, we show that this gap actually disappears with an additional technicalassumption on HB.Lemma 3.1. If HB is a convex set, then ϕ(ε) = ϕ(ε), i.e., the problems in (3.36)and (3.39) are equivalent.Proof: The proof is provided in Appendix B.1.Following the same approach from the proof of Proposition 3.1, it can be shownthat f(τ) ≡ 6A2(ϕ(ε))2 + 3pieσ2 is concave w.r.t. τ , and thus f(τ)/τ is quasiconcave.Next, we introduce the variable α ≥ 0 via the substitutionτ = pieα2A2t2α + 3pieσ2, (3.40)where tα is obtained from(wα, tα) = argmax‖w‖p≤1,tt (3.41a)s.t. hTBw ≥ t ∀hB ∈ HB, (3.41b)|hTEw| ≤ αt ∀hE ∈ HE. (3.41c)73Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksNote from (3.39) and (3.41) that tα ≡ ϕ(ε) whenever α and τ satisfy (3.40). Substi-tuting (3.40) back into (3.34), the latter can be rewritten asmaximizeα6A2t2α + 3pieσ2pieα2A2t2α + 3pieσ2. (3.42)Similar to (3.27) in the proof of Proposition 3.1, we note from (3.40) and (3.41) that τis strictly increasing w.r.t. α. Thus, the objective function in (3.42) is quasiconcavew.r.t. α. Finally, αmin can be obtained by modifying the problem in (3.28) tominimizew,αα (3.43a)s.t. hTBw ≥ 1 ∀hB ∈ HB, (3.43b)|hTEw| ≤ α ∀hE ∈ HE, (3.43c)which completes the proof. Remarks:• αmin > 0 implies that ZF is not feasible.• αmin ≥√6/pie implies that the max-min problem is not feasible and the worst-case secrecy rate is zero (e.g., when HB ∩HE 6= ∅).Similar to (3.9) in Proposition 3.1, the outer problem (3.31) is quasiconvex, andthus it can be efficiently solved by performing a bisection search on α. We proposeAlgorithm 3.1, provided in Table 3.1, to obtain a solution α? with accuracy α (dB).Assuming α = 0.2 dB, Algorithm 3.1 shall converge in at most [80, Section 4.2.5]⌈log2(20 log10√6/pie10−10)− log2 α⌉= 10 iterations.74Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksTable 3.1: Bisection search to solve the maximization problem in (3.31).Algorithm 3.1 Bisection search to solve (3.31) in Proposition 3.21: Solve (3.32) to obtain αmin2: if αmin < 10−10, then αmin := 10−103: Initialize α = 20 log10√6/pie and α = 20 log10 αmin4: given the required accuracy α (dB), set the positive constant ∆α such that0 < 20 log10 ∆α < α5: while α− α ≥ α do6: α(dB) :=α + α27: Solve (3.33) with α to obtain tα, where α = 10α(dB)208: Calculate the objective in (3.31), f(α) =6A2t2α + 3pieσ2pieα2A2t2α + 3pieσ29: Solve (3.33) with α + ∆α to obtain tα+∆α10: Calculate f(α + ∆α) =6A2t2α+∆α + 3pieσ2pie(α + ∆α)2A2t2α+∆α + 3pieσ211: if f(α + ∆α)− f(α) > 0, then α := α(dB) else α := α(dB)12: end while13: return α? := α75Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksNote, however, that the inner problem (3.33) should be solved twice in each iteration.Thus, although Proposition 3.2 is valid in principle for any convex set HB and anarbitrary set HE, it is practically useful only when (3.33) is tractable, i.e., can be effi-ciently solved. The inner problem (3.33) is a robust convex program whose tractabilitydepends solely on the geometries of HB and HE [79, 83, 84]. In Section 3.4, we usea spherical set HB to accommodate quantization errors caused by limited feedbackfrom Bob. For Eve’s channel, we use discrete, interval, and ellipsoidal sets to modeldifferent uncertainty sources that cause inaccurate estimates of hE in VLC scenarios.Using a spherical set HB, and discrete, interval, or ellipsoidal sets HE, and assumingthat12 p ∈ {1, 2,∞}, the inner problem (3.33) can be expressed as a second-ordercone program, which can be efficiently solved.From (3.31), the worst-case secrecy rate, as a function of α, isRwcs (α) =[12log26A2t2α + 3pieσ2pieα2A2t2α + 3pieσ2]+. (3.44)The best worst-case secrecy rate is equal to Rwcs (α?), and is achieved by the robustbeamformer wα? .3.4 Uncertainty Sets for the Eavesdropper’sChannel in VLC ScenariosRecall from Sections 1.3.1 and 1.3.2 that the LoS DC channel gain from the ithtransmit LED can be accurately approximated by12We need the assumption p ∈ {1, 2,∞} merely to state that the resulting problem is a second-order cone program. However, the problem is still convex and equally solvable, e.g., via the CVXtoolbox [85], for any p ≥ 1.76Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkshi = ηRPDTa(m+ 1)APD2pi‖di‖22(cos ζi)m Ts gc cosψi IΨ(ψi) (3.45a)= ηRPDTa(m+ 1)APD2pi‖di‖m+32dmz Ts gc dTi u IΨ(cos−1dTi u‖di‖2), (3.45b)where all the terms in (3.45) are defined in Sections 1.3.1 and 1.3.2. Note that weassume equal heights for all the LEDs, i.e., the vertical distance between the PD atthe receiver and each LED is dz regardless of the LED index (dz is independent of i).Now, our focus in this section is on deriving uncertainty sets for Eve’s channelbased on the uncertain parameters in (3.45). Our motivation towards this approachis the lack of feedback from Eve regarding her channel when Eve is a passive ornon-cooperative receiver. In particular, we take advantage of the fact that hE can bepredicted from Eve’s location and orientation using (3.45) if the LoS path is dominantand the emission pattern of the LEDs is known. Such information can be mappedinto an estimate of hE surrounded by a reasonable uncertainty setHE. Unfortunately,the channel gain expression in (3.45) is quite complex, and mapping such uncertainparameters altogether into a useful HE that makes the inner problem (3.33) solvableis quite difficult. Thus, we begin with studying uncertainty sets corresponding to oneuncertain parameter at a time. We also consider uncertainty caused by the NLoScomponents in hE. Cases involving more than one uncertainty source will also bebriefly discussed.Throughout the entire section, we assume an amplitude constraint on w, i.e.,‖w‖∞ ≤ 1. Furthermore, we assume a spherical uncertainty set for Bob’s channel,i.e., hB ∈ HB,HB ={hˆB + ehB : ‖ehB‖2 ≤ hB}, (3.46)77Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere the nominal vector hˆB is known to Alice via limited feedback from Bob, andthe bounded error term ehB is due to quantization errors. Substituting (3.46) backinto (3.33b), the latter can be expressed ashˆTBw − hB‖w‖2 ≥ t. (3.47)3.4.1 Uncertain Eavesdropper’s LocationIn this subsection, we consider uncertainty caused by inaccurate information regard-ing Eve’s location. We assume that Eve is located inside a three-dimensional rectan-gular region (or box) B with dimensions (2lx, 2ly, 2lz). We also assume, without lossof generality, that B is centered at the origin, i.e.,B = {Lv : v ∈ R3, ‖v‖∞ ≤ 1} , (3.48)where L , Diag(lx, ly, lz). Furthermore, we choose the origin (or the center of B) asthe nominal location of Eve.Let δ = [δx δy δz]T, δ ∈ B, denote the deviation of the actual location of Eve fromthe origin. Using (3.45b), the channel gain hi, i = 1, . . . , N , anywhere inside B, as afunction of δ, ishi(δ) = ηRPDTa(m+ 1)APD2pi‖di − δ‖m+32(dz − δz)m Ts gc (di − δ)Tu IΨE(cos−1(di − δ)Tu‖di − δ‖2),(3.49)and the set of all possible channel realizations inside B can be written asHBE = {h(δ) : δ ∈ B} . (3.50)78Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksIf we substitute with HE = HBE back into (3.33c), we will end up with an intractablesemi-infinite optimization problem. Therefore, we shall discuss methods to approxi-mate HBE, based on the volume of B, in order to make (3.33) solvable.3.4.1.1 Small Uncertainty RegionFor sufficiently-small B, e.g., max {2lx, 2ly, 2lz} ≤ 0.5 m, we can assume that thesubset of LEDs seen by Eve’s receiver at a particular location δ,Iδ ={i : IΨE(cos−1(di − δ)Tu‖di − δ‖2)= 1, i ∈ {1, . . . , N}},is identical for all δ ∈ B. In other words, the output of the indicator function in(3.49) is independent of δ for all the LEDs and is solely determined by the nominallocation of Eve. Under this assumption, the channel gain in (3.49) can be written ashi(δ) = ci(dz − δz)m(di − δ)Tu‖di − δ‖m+32, (3.51)whereci , ηRPDTa(m+ 1)APD2piTs gc IΨE(cos−1dTi u‖di‖2). (3.52)Furthermore, with sufficiently-small B, h(δ) can be well approximated by its first-order approximation around the center of B, that ish(δ) ≈ h¯(δ) = h0 + J0δ, (3.53)79Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere h0 ≡ h(0), J ∈ RN×3 is the Jacobian matrix (or matrix of partial derivatives),defined asJ ,∂h1(δ)∂δx∂h1(δ)∂δy∂h1(δ)∂δz.........∂hN(δ)∂δx∂hN(δ)∂δy∂hN(δ)∂δz , (3.54)and J0 ≡ J(0). The entries of h0 and J0 are provided in Appendix B.2. Usingthe linearized channel gain expression in (3.53), the uncertainty set in (3.50) can beapproximated byH¯BE ={h0 + J0Lv : v ∈ R3, ‖v‖∞ ≤ 1}. (3.55)Substituting with H¯BE back into (3.33c), the inner problem (3.33) can be expressed asmaximize‖w‖∞≤1,tt (3.56a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.56b)|hT0w + vTLJT0w| ≤ αt ∀v : ‖v‖∞ ≤ 1, (3.56c)or, equivalently,maximize‖w‖∞≤1,tt (3.57a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.57b)hT0w + ‖LJT0w‖1 ≤ αt, (3.57c)hT0w − ‖LJT0w‖1 ≥ −αt, (3.57d)80Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhich is a second-order cone problem. Similarly, the problem in (3.32) can be ex-pressed asminimizew,αα (3.58a)s.t. hˆTBw − hB‖w‖2 ≥ 1, (3.58b)hT0w + ‖LJT0w‖1 ≤ α, (3.58c)hT0w − ‖LJT0w‖1 ≥ −α. (3.58d)3.4.1.2 Large Uncertainty RegionIf the uncertainty region B is relatively large, the first-order approximation in (3.53)may become poor. Nevertheless, B can first be divided into K non-overlapping boxes,Bk, k = 1, . . . , K, such that⋃Kk=1 Bk = B. Then, the first-order approximation isapplied inside each box, around its center, and (3.56) is solved with the correspondingK constraints.Alternatively, the region B can be discretized using a three-dimensional finegrid...B , and the inner problem (3.33) is approximated bymaximize‖w‖∞≤1,tt (3.59a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.59b)|hT(δ)w| ≤ αt ∀δ ∈ ...B , (3.59c)where the entries of h(δ) are obtained with (3.49). Although discretization is astraightforward approach that leads to linear constraints, the number of constraintsmay grow up very quickly with large uncertainty regions.81Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links3.4.2 Uncertain Eavesdropper’s OrientationIn this subsection, we assume that Eve has the freedom to adjust the direction ofher receiver, (θE, φE), θE ∈ [θmin, θmax], φE ∈ [φmin, φmax], to her advantage. In otherwords, the exact direction of Eve’s receiver is unknown to Alice. The uncertainty set Ucontaining all possible realizations of Eve’s orientation vector u (refer to Figure 1.4)can be written asU = {u = [sin θ cosφ sin θ sinφ cos θ]T : θ ∈ [θmin, θmax], φ ∈ [φmin, φmax]} , (3.60)and the channel gain hi, i = 1, . . . , N , as a function of u, is given byhi(u) = cidmz‖di‖m+32dTi u, (3.61)where ci is as defined in (3.52). For notational convenience, let D ∈ RN×3 be definedasD , dmz[c1d1‖d1‖m+32. . .cNdN‖dN‖m+32]T. (3.62)Then, h(u) can be expressed ash(u) = Du. (3.63)Note from (3.52) and (3.62) that D depends on u via the indicator function in thedefinition of ci, i = 1, . . . , N . Thus, the mapping from u to h in (3.63) is not linear,in general. The set of all possible channel gains for Eve is given byHUE = {Du : u ∈ U} . (3.64)82Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksSubstituting with HUE back into (3.33c), the inner problem (3.33) can be written asmaximize‖w‖∞≤1,tt (3.65a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.65b)maxu∈U|uTDTw| ≤ αt. (3.65c)In order to efficiently solve (3.65), we shall differentiate between two cases, as follows.3.4.2.1 Small Angle VariationsIn this case, we assume that Eve’s freedom to adjust her receiver’s orientation is lim-ited in the sense that the subset of LEDs inside Eve’s FoV at a particular direction u,Iu ={i : IΨE(cos−1dTi u‖di‖2)= 1, i ∈ {1, . . . , N}},remains unchanged for all u ∈ U . Perhaps the most practical case in which the aboveassumption may hold is when the permissible variations of the zenith angle θE isrelatively small and close to zero, i.e., θE ∈ [0, θmax], where θmax is relatively small(e.g., θmax ≤ 30◦). If Iu is fixed for all u ∈ U , then D is independent of u, and h,as given in (3.63), is a linear function of u. In this case, the left-hand side of theinequality in (3.65c) can be upper-bounded asmaxu∈U|uTDTw| ≤ max‖u‖2≤1uTDTw = ‖DTw‖2. (3.66)83Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksThen, the problem in (3.65) is replaced bymaximize‖w‖∞≤1,tt (3.67a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.67b)‖DTw‖2 ≤ αt, (3.67c)which is a second-order cone problem.3.4.2.2 Large Angle VariationsWith arbitrary zenith and/or azimuth angle variations for Eve’s receiver, D becomesdependent on u, and linearity between h and u no longer holds. In this case, itbecomes difficult to obtain a mathematically-convenient uncertainty set HUE over thecontinuum of θE and φE. Thus, we resort to sampling h(u) over U , and the innerproblem (3.33) is approximated bymaximize‖w‖∞≤1,tt (3.68a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.68b)|hT(θ, φ)w| ≤ αt ∀(θ, φ) ∈ ...Θ × ...Φ, (3.68c)where the components of h(θ, φ) are obtained with (3.45b), and...Θ and...Φ are finegrids on the intervals [θmin, θmax] and [φmin, φmax], respectively.84Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links3.4.3 Uncertain LEDs Half-Intensity AngleAssuming generalized Lambertian emission, the emission pattern of the LEDs is fullydetermined by the Lambertian orderm = −1/log2(cos ζ3-dB), (3.69)where ζ3-dB is the half-intensity angle of the LEDs. This angle is typically spec-ified by the LED manufacturer as a nominal value in the datasheet. In practice,however, the actual angle of each LED will deviate from the nominal value. In thissubsection, we study channel uncertainty caused by this deviation. In particular, weassume an interval uncertainty model in which ζ3-dB ∈ [ζ3-dBmin , ζ3-dBmax ], and allow inde-pendent realizations of ζ3-dB for each LED. Then, we map the interval [ζ3-dBmin , ζ3-dBmax ]into independent interval uncertainties for each entry of hE.We begin with rewriting the channel gain from (3.45a) ashi(mi) = κi(mi + 1)(cos ζi)mi , i = 1, . . . , N, (3.70a)wheremi = −1/log2(cos ζ3-dBi ), ζ3-dBi ∈[ζ3-dBmin , ζ3-dBmax], (3.70b)andκi , ηRPDTaAPD2pi‖di‖22Ts gc cosψi IΨE(ψi). (3.70c)Next, we define mmin and mmax, respectively, asmmin , −1/log2(cos ζ3-dBmax ), (3.71a)mmax , −1/log2(cos ζ3-dBmin ). (3.71b)85Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksThen, in order to map the interval [mmin,mmax] into [hmini , hmaxi ], i = 1, . . . , N , wefirst show that hi is a quasiconcave function of mi. Differentiating hi w.r.t. mi yieldsh′i(mi) = κi(cos ζi)mi(1 + (mi + 1) ln(cos ζi)). (3.72)From (3.72), for κi 6= 0 and i = 1, . . . , N , we note thath′i(mi) ≥ 0 for mi ≤ m?i ,h′i(mi) < 0 for mi > m?i ,where m?i , −(1 + 1/ ln(cos ζi)). Thus, hi(mi) is a quasiconcave function with globalmaximizerm?i . Consequently, the uncertainty setHζ3-dBE corresponding to the interval[ζ3-dBmin , ζ3-dBmax ] can be written asHζ3-dBE ={[h1 . . . hN ]T : hi ∈ [hmini , hmaxi ], i = 1, . . . , N}, (3.73a)where, for i = 1, . . . , N,hmini =hi(mmin) if m?i > mmax,min{hi(mmin), hi(mmax)} if m?i ∈ [mmin,mmax],hi(mmax) if m?i < mmin,(3.73b)hmaxi =hi(mmax) if m?i > mmax,hi(m?i ) if m?i ∈ [mmin,mmax],hi(mmin) if m?i < mmin.(3.73c)86Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC LinksDefine hˆ ∈ RN+ and Hˆ ∈ RN×N+ , respectively, ashˆ , 12[hmax1 + hmin1 . . . hmaxN + hminN ]T, (3.74a)Hˆ , 12Diag(hmax1 − hmin1 , . . . , hmaxN − hminN ). (3.74b)Then, Hζ3-dBE can be written asHζ3-dBE ={hˆ+ Hˆv : v ∈ RN , ‖v‖∞ ≤ 1}. (3.75)Similar to (3.55)–(3.57), substituting withHζ3-dBE into (3.33c), the inner problem (3.33)can be expressed asmaximize‖w‖∞≤1,tt (3.76a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.76b)hˆTw + ‖Hˆw‖1 ≤ αt, (3.76c)hˆTw − ‖Hˆw‖1 ≥ −αt. (3.76d)3.4.4 Uncertain NLoS ComponentsIn this subsection, we consider channel uncertainty arising from the NLoS compo-nents caused by diffuse reflections from nearby surfaces. Taking into account signalcontributions from both the LoS and NLoS paths, the channel gain can be writtenashi = hLoSi + hNLoSi , i = 1, . . . , N, (3.77)87Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere hLoSi is the LoS component obtained with (3.45), and hNLoSi is the unknownNLoS component. We shall consider a simple multiplicative uncertainty model inwhich hNLoSi is an uncertain fraction, γi, of hLoSi , that ishNLoSi = γihLoSi , 0 ≤ γi ≤ γmax, i = 1, . . . , N, (3.78)where γmax , maxiγi. The actual value of γmax depends mostly on the problemgeometry as well as the diffuse reflectivity of nearby surfaces. In practice, γmax canbe measured or predicted using numerical simulations. Simulation results reportedin [59] show γmax of about 12% (see the discussion after Figure 6 in [59]). Note,however, that the multiplicative model in (3.78) is applicable only when the LoSpath between the ith LED and the PD exists, i.e., hLoSi 6= 0, and is dominant. Inother words, (3.78) does not take into account the case in which the received signalconsists entirely of NLoS components, e.g., when the LoS path is blocked or outsidethe receiver FoV.From (3.77) and (3.78), the set of all possible channel gain vectors can be writtenasHγmaxE ={[h1 . . . hN ]T : hi ∈ [hLoSi , (1 + γmax)hLoSi ], i = 1, . . . , N}, (3.79)which is similar to Hζ3-dBE in (3.73a), and thus we can proceed with the same stepsfrom the previous subsection.3.4.5 Combined UncertaintiesSo far we have derived separate sets corresponding to uncertainties in location, ori-entation, half-intensity angle, and NLoS components. In practice, however, these88Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksuncertainties will mostly happen in combination with each other. Thus, more inclu-sive sets that take into account the aggregate uncertainty are required. Unfortunately,it is difficult, in general, to derive such sets or provide a unified treatment for differ-ent combinations of uncertainties because, as we mentioned earlier, the channel gainexpression in (3.45) is a complex function of the uncertainty sources. Nevertheless,one intuitive approach to circumvent such a difficulty is to sample the channel gainvector over the variables with lower dimension or smaller uncertainty size. Consider,for example, the case of uncertain location and LEDs half-intensity angle, that isHB×ζ3-dBE = HBE ×Hζ3-dBE={h(δ, ζ3-dB) : δ ∈ B, ζ3-dB ∈ [ζ3-dBmin , ζ3-dBmax ]N},where ζ3-dB = [ζ3-dB1 . . . ζ3-dBN ]T. If N > 3, i.e., the dimension of ζ3-dB is bigger thanthe dimension of δ, then B can be discretized using a three-dimensional K-point grid,...B = {δ1, . . . , δK}, and the problem in (3.76) is modified tomaximize‖w‖∞≤1,tts.t. hˆTBw − hB‖w‖2 ≥ t,hˆTkw + ‖Hˆkw‖1 ≤ αt, k = 1, . . . , K,hˆTkw − ‖Hˆkw‖1 ≥ −αt, k = 1, . . . , K,where hˆk and Hˆk are obtained as in (3.74) using the components of hmin(δk) andhmax(δk), for k = 1, . . . , K. The same idea can be applied to other combinations ofuncertainty sources.Furthermore, there exist specific cases of combined uncertainties in which dis-89Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkscretization may not be necessary. Consider, for example, the special, but practicallyrelevant, case of small location and angle uncertainties. With such a combination,the linear channel gain models considered in Sections 3.4.1.1 and 3.4.2.1 are bothapplicable, and an explicit formulation of the optimization problem can be obtainedas follows. First, we rewrite the linearized channel gain expression from (3.53) ash¯(δ,u) = h0 + J0δ= Du+G0(I3 ⊗ u)δ= Du+G0(δ ⊗ I3)u, (3.80)where D is as defined in (3.62), and the entires of G0, G0 ∈ RN×9, can be in-ferred from (B.2b)–(B.2d) in Appendix B.2. Then, the inner problem (3.33) can beexpressed asmaximize‖w‖∞≤1,tt (3.81a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.81b)maxδ∈B,‖u‖2≤1|uTDTw + uT(δT ⊗ I3)GT0w| ≤ αt. (3.81c)The constraint in (3.81c) can be replaced by a set of second-order cone constraints,given by‖DTw + (vT(k) ⊗ I3)GT0w‖2 ≤ αt, k = 1, . . . , 8, (3.82)where v(k) ∈ R3, k = 1, . . . , 8, are the vertices (or corners) of B.90Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links3.5 Numerical ExamplesIn this section, we provide numerical examples to verify the performance gains ofthe beamformers proposed in Section 3.3 compared to conventional beamformingschemes. We also demonstrate the design of robust beamformers in a typical VLCscenario and investigate the resulting worst-case secrecy rate performance in conjunc-tion with the uncertainty models devised in Section 3.4.3.5.1 Performance ComparisonsAll the results presented in this subsection are obtained under the following assump-tions. The number of transmit elements is N = 4. The entries of hB and hE aregenerated i.i.d. according to the uniform distribution over the interval [0, 1], and theresults are averaged over 1000 independent trials. The optimal and robust beam-formers are obtained via Algorithm 3.1, where the outer maximization problem issolved with accuracy α = 0.2 dB and the inner problem is solved using the CVXtoolbox [85] along with the MOSEK solver [86].3.5.1.1 Optimal versus Suboptimal Beamformers under Differentlp-norm ConstraintsIn this example, we compare the secrecy rate performance of the optimal beamformerwith the generalized eigenvalue (GEV) and ZF beamformers, under the premise ofperfect channel information.Figure 3.1(a) depicts the secrecy rates (3.5) versus A/σ. These secrecy rates areobtained with wα? , wGEV, and wα=0, corresponding to the optimal, GEV, and ZFbeamformers, respectively, subject to the constraint ‖w‖p ≤ 1, for p = 1, 2,∞. Theoptimal beamformer wα? is obtained with Proposition 3.1, and the corresponding α?91Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links0 2 4 6 8 10 12 14 16 18 2000.511.522.50 2 4 6 8 10 12 14 16 18 20-35-30-25-20-15-10-5Figure 3.1: (a) Secrecy rates (3.5) obtained with the optimal, GEV, and ZF beam-formers versus A/σ, subject to the constraint ‖w‖p ≤ 1, for p = 1, 2,∞. (b) α?corresponding to the optimal beamformer wα? .is shown in Figure 3.1(b). The beamformer wGEV is the generalized eigenvector of thematrix pair (6A2hBhTB + 3pieσ2IN , pieA2hEhTE + 3pieσ2IN) corresponding to its largestgeneralized eigenvalue, where wGEV is scaled such that ‖wGEV‖p = 1, for p = 1, 2,∞.The ZF beamformer wα=0 is obtained by solving (3.11) with α = 0.As expected, we note from Figure 3.1(a) that the optimal beamformer provides thebest performance for all p = 1, 2,∞, however at the cost of increased computationalcomplexity. We also note that the secrecy rates of the optimal and GEV beamformerscoincide when p = 2. This is because GEV beamforming is optimal under the l2-normconstraint [39]. Furthermore, we note that the ZF beamformer outperforms its GEVcounterpart under the l∞-norm constraint, and it approaches the performance of the92Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links1 2 3 4 5 600.511.522.5Figure 3.2: Secrecy rates (3.83) of the optimal, GEV, and ZF beamformers under dif-ferent lp-norm constraints, versus the number of eavesdroppers when 20 log10(A/σ) =20 dB.optimal beamformer as A/σ increases. Figure 3.1(b) shows that α? is nonincreasingw.r.t. A/σ for all p = 1, 2,∞. Thus, the ZF beamformer is in fact asymptoticallyoptimal at high A/σ for all p. Moreover, it can be observed that α? decreases rapidlyas p increases. This reveals that the performance gap between the ZF and optimalbeamformers narrows quickly with increasing p at high A/σ.Figure 3.2 shows the secrecy rate performance versus the number of eavesdrop-pers K when 20 log10(A/σ) = 20 dB. Each eavesdropper has a single receive element,and there is no collaboration among the eavesdroppers, i.e., centralized processing ofthe received signals is not permitted. The secrecy rates are obtained withRs(w) =[12log26A2(hTBw)2 + 3pieσ2pieA2‖HTEw‖2∞ + 3pieσ2]+, (3.83)93Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkswhere HE , [hE1 . . . hEK ]. The GEV beamformer is the generalized eigenvector ofthe matrix pair (6A2hBhTB + 3pieσ2IN , pieA2HEHTE + 3pieσ2IN) corresponding to thelargest generalized eigenvalue. The optimal beamformer is obtained with Proposi-tion 3.1 after replacing the constraint in (3.11b) with ‖HTEw‖∞ ≤ αhTBw, and the ZFbeamformer is obtained by setting α = 0.We note that the GEV beamformer is optimal when K = 1 and p = 2. We alsonote that, as K increases, the GEV beamformer outperforms the ZF scheme evenwhen p 6= 2. Obviously, ZF becomes infeasible once K ≥ N .3.5.1.2 Robust versus Non-Robust SchemesIn this example, we compare the worst-case secrecy rate performance of the robustbeamformer with non-robust schemes. We assume that the uncertainty sets for Bob’sand Eve’s channels, respectively, areHB ={hˆB + ehB : ‖ehB‖2 ≤ hB}, (3.84a)HE ={hˆE + ehE : ‖ehE‖∞ ≤ hE}. (3.84b)The entries of the nominal vectors hˆB and hˆE are generated at random, and theresults are averaged over 1000 trials.In Figure 3.3, we plot the worst-case secrecy rateRwcs (w) =12log2minhB∈HB6A2(hTBw)2 + 3pieσ2maxhE∈HEpieA2(hTEw)2 + 3pieσ2+ (3.85)versus hE , for hB = 0, 0.2, 0.4, and 20 log10(A/σ) = 20 dB. We compare the perfor-mance of the robust beamformer from Proposition 3.2 with its non-robust counterpartfrom Proposition 3.1, as well as the GEV and ZF beamformers. All the beamformers94Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.200.511.522.5Figure 3.3: Worst-case secrecy rates (3.85) of the robust, non-robust, GEV, and ZFbeamformers versus hE with hB = 0, 0.2, 0.4. All the beamformers are subject tothe amplitude constraint ‖w‖∞ ≤ 1, and 20 log10(A/σ) = 20 dB.are subject to the amplitude constraint ‖w‖∞ ≤ 1. Substituting from (3.84a) and(3.84b) into (3.33b) and (3.33c), respectively, the inner problem (3.33) is expressedasmaximize‖w‖∞≤1,tt (3.86a)s.t. hˆTBw − hB‖w‖2 ≥ t, (3.86b)hˆTEw + hE‖w‖1 ≤ αt, (3.86c)hˆTEw − hE‖w‖1 ≥ −αt. (3.86d)Then, the robust beamformer is obtained via Algorithm 3.1. On the other hand, thenon-robust, GEV, and ZF beamformers are obtained using the nominal vectors hˆB95Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksand hˆE.As expected, we note from Figure 3.3 that the robust beamformer outperformsits non-robust counterparts, and the performance gain becomes more evident withincreasing hB and hE .3.5.2 Worst-Case Secrecy Rate Performance in VLCScenariosIn this subsection, we investigate the worst-case secrecy rate performance in a typicalVLC scenario using the robust beamformer from Proposition 3.2 along with theuncertainty sets derived in Section 3.4.We consider a room of size 5 × 5 × 3 m3 illuminated by 25 square-shapedlight fixtures uniformly distributed over 4 × 4 m2 of the ceiling area, as depicted inFigure 3.4. Each fixture occupies 10× 10 cm2 and encloses 4 high-brightness 2.5-WLEDs located at the corners of the fixture. Each LED radiates 570 mW optical power(or radiant flux). Emitted light is “warm-white” (i.e., color temperature is between2700 and 3000 K) with luminous efficiency 284 lm/W [10, Table 3.2]. The resultingluminous flux is 0.570 × 284 ≈ 162 lm per LED. The nominal half-intensity angle(measured from the center) is 60◦, and the peak (or center) luminous intensity is51 cd. The resulting illuminance, averaged over a horizontal 4 × 4 m2 illuminationgrid at height 0.85 m above the floor level, is 339 Lux. For convenience, all simulationparameters are provided in Table 3.2.All the following results are generated with Bob and Eve having PDs of areaAPD = 1 cm2 and responsivity RPD = 100 µA/mW/cm2. The modulation index µMIis set at 10%. The noise power is assumed to be equal everywhere with 20 log10 σ =−114 dBm. This value is obtained with [9, Eq. (6)] using the average received DC96Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links-2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5-2.5-2-1.5-1-0.500.511.522.5Figure 3.4: Layout of the LEDs on the ceiling. There exist 25 light fixtures. Eachfixture has 10 × 10 cm2 surface area and encloses 4 LEDs located at the corners ofthe fixture.Table 3.2: Simulation parameters for the VLC scenario.Simulation setupRoom size 5× 5× 3 m3Number of fixtures NFix 25Fixture size 10× 10 cm2Number of LEDs per fixture NLED 4Total number of LEDs N 100LED electrical and optical characteristicsForward voltage 3.6 VForward current IDC 700 mAInput electric power 2.52 WOptical power / electric current η 813.6 µW/mAOutput optical power (or radiant flux) Popt 569.52 mWLuminous efficiency (warm-white color) 284 lm/WLuminous flux 161.74 lmLuminous efficacy 64.18 lm/WNominal half-intensity angle ζ3-dB 60◦Peak (or center) luminous intensity 51.48 cdModulation index µMI 10%Optical receiver characteristicsGain of the optical filter Ts 1Lens refractive index nr 1.5PD responsivity RPD 100 µA/mW/cm2PD surface area APD 1 cm297Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linksoptical power (averaged over the horizontal plane at height 0.85 m) with FoV Ψ = 70◦and receiver bandwidth of 10 MHz. All location coordinates are specified in metersw.r.t. the room center at the floor level.In all scenarios, we assume that Bob is located at (xB, yB, zB) = (1.7173, 0.7496, 0.85)with orientation (θB, φB) = (15◦, 240◦) and FoV ΨB = 70◦. Furthermore, we use thespherical set in (3.46) to model uncertainty in Bob’s channel, where the entries of hˆBare obtained with (3.45), i.e., hˆB = h(xB, yB, zB, θB, φB,ΨB), and hB = 0.1‖hˆB‖2.The nominal estimate hˆB is fixed and assumed to be known to Alice via feedbackfrom Bob. Parameters relevant to Eve are provided in the caption of each figure.In all cases, for the sake of illustration, we plot the worst-case secrecy rate versus αusing (3.44), where 20 log10 α = −50,−49, . . . , 0 dB. We also include the case ofcertain Eve’s channel for comparison purposes. For each α, we use the CVX tool-box [85], in conjunction with the MOSEK solver [86], to solve (3.33) using the relevantuncertainty set HE from Section 3.4.3.5.2.1 Uncertain Eavesdropper’s LocationFigure 3.5 shows the worst-case secrecy rate performance with uncertain Eve’s lo-cation. We include three groups of curves corresponding to three uncertainty re-gions, B, of different volumes. All the regions are rectangular and centered at(x, y, z) = (−1.25, 0, 0.85). Four curves are generated for each B corresponding tothe combinations of two methods to approximate HBE and two methods to modulatethe LEDs. We refer to the case in which the affine approximation (3.53) is usedas “Linearized”, and to the case in which B is discretized as “Discretized”. For the“Linearized” case, B is divided into identical boxes, each of volume 2lx × 2ly × 2lz =0.5×0.5×0.25 m3, then (3.53) is applied to each box and wα is obtained with (3.57).For the “Discretized” case, HBE is approximated by sampling the channel gain in the98Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 000.511.522.533.544.5Figure 3.5: Worst-case secrecy rate (3.44) versus α with uncertain Eve’s location.θE = 0 and ΨE = 70◦.three-dimensional space using a 10×10 ×10 cm3 grid, and wα is obtained with (3.59).Furthermore, we refer to the case in which each LED is modulated independently as“LEDs”, and to the case in which all LEDs in one fixture are modulated with thesame current signal as “Fixtures”.As expected, we note from Figure 3.5 that Rwcs decreases as the uncertainty aboutEve’s location increases. For the case of certain Eve’s location, we can see that the ZFbeamformer is practically optimal. In addition, Figure 3.5 reveals that independent-LED modulation does not provide much improvement, if any, compared to the morepractical and less expensive “Fixture” modulation scheme. This is also expected sinceLEDs in the same fixture are relatively close to each other and have almost identicalchannel gains. Figure 3.5 also validates the affine approximation in (3.53) when lx,ly, and lz are chosen properly.99Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links3.5.2.2 Uncertain Eavesdropper’s OrientationIn Figure 3.6, we plot the worst-case secrecy rate performance with uncertain Eve’sorientation. We also investigate the impact of Eve’s FoV on the secrecy rate. Thecurve “Small θmax” is generated withwα obtained from (3.67), whereas all other curvesare obtained with (3.68) after discretizing the intervals Θ = [0, θmax] and Φ = [0, 360◦]using ∆θ = ∆φ = 4◦.For the case θmax = 0, i.e., no uncertainty about Eve’s orientation, we can seethat ZF is essentially optimal. We also note that the curve “Small θmax” almostcoincides with the curve corresponding to ΨE = 90◦ and θmax = 30◦. Thus, thelinear channel gain model that leads to the problem in (3.67) is indeed valid for smallangle variations and wide FoV. Figure 3.6 also reveals that reducing Eve’s FoV has anegative impact on the secrecy rate performance, which can be explained as follows.First, we recall from (1.5) that reducing the FoV of the concentrator increases its gain(inside the FoV). Second, the limited FoV of Eve’s receiver, in conjunction with herability to adjust orientation, increases the space of her received signal as measuredby the number of nonzero singular values of the matrix H...UE whose columns are theelements of the discretized uncertainty setH...UE . Obviously, increasing the signal spaceavailable to Eve makes it more difficult for Alice to suppress Eve’s signal, i.e., moreof the degrees of freedom available to Alice are exploited, and thus the secrecy rateis reduced.100Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links-50 -40 -30 -20 -10 0 1000.511.522.533.544.5Figure 3.6: Worst-case secrecy rate (3.44) versus α with uncertain Eve’s orientation.xE = −1.25, yE = 0, zE = 0.85, θE ∈ [0, θmax], and φE ∈ [0, 360◦].3.5.2.3 Uncertain Eavesdropper’s Location and LEDs Half-IntensityAngleFigure 3.7 depicts the secrecy performance with uncertain Eve’s location and LEDshalf-intensity angle. We consider half-intensity angle uncertainties up to ±20◦ aroundthe nominal value of 60◦, and the location uncertainty region{(xE, yE) : xE ∈ [−2.25,−0.25], yE ∈ [−2.5, 2.5]}is discretized using ∆x = ∆y = 20 cm. As can be seen, even relatively small half-intensity angle deviations, e.g., ±5◦, can significantly reduce the worst-case secrecyrate.101Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 000.511.522.533.5Figure 3.7: Worst-case secrecy rate (3.44) versus α with uncertain Eve’s location andLEDs half-intensity angle ζ3-dBi , i = 1, . . . , N . xE ∈ [−2.25,−0.25], yE ∈ [−2.5, 2.5],zE = 0.85, θE = 0, and ΨE = 70◦.3.5.2.4 Uncertain Eavesdropper’s Location and NLoS ComponentsIn Figure 3.8, we show the worst-case secrecy rate performance with uncertain Eve’slocation and NLoS components. Similar to the results in Figure 3.7, the locationuncertainty region is discretized using ∆x = ∆y = 20 cm. We investigate the per-formance when the strength of the NLoS components can go up to γmax = 60% ofthe LoS path. Note that γmax = 60%, or even 40%, is a too pessimistic or too con-servative assumption. In typical scenarios with only diffuse reflections, i.e., no largewindows or mirrors, γmax will probably be less than 20% (see, e.g., Figure 4 in [87]or the discussion after Figure 6 in [59]).102Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Links-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 000.511.522.533.5Figure 3.8: Worst-case secrecy rate (3.44) versus α with uncertain Eve’s location andNLoS components. xE ∈ [−2.25,−0.25], yE ∈ [−2.5, 2.5], zE = 0.85, θE = 0, andΨE = 70◦.3.6 ConclusionsIn this chapter, we studied the design of transmit beamformers for secrecy rate max-imization in MISO wiretap channels subject to amplitude constraints. Such con-straints are typically difficult to handle and, because of that, they are often overlookedin favor of the more convenient total power constraint.We tackled the nonconvex secrecy rate maximization problem by transformingit into an equivalent quasiconvex line search problem. Our approach is conceptu-ally simple but provably optimal for general lp-norm constraints, and the equivalentproblem itself is practically meaningful. Furthermore, our approach proved helpful in103Chapter 3. Optimal and Robust Beamforming for Secure MISO VLC Linkstackling the more complex robust design problem with uncertain channel information.We used the VLC scenario as a practical example in which reasonable estimatesof the eavesdropper’s channel can be obtained without feedback from the (passive)eavesdropper. Numerical results show that the excess degrees of freedom provided bythe large number of LEDs in typical VLC transmitters can be effectively utilized tocompensate for the lack of accurate information regarding the eavesdropper’s channel.In the next chapter, we consider the more general two-user MISO BC-CM model.104Chapter 4Linear Precoding for the Two-UserMISO Broadcast Channel withConfidential Messages4.1 IntroductionIn the previous chapter, we studied the design of transmit beamformers for the MISOwiretap channel under the amplitude constraint. In such a scenario, the transmitterhad one secret message for the intended receiver (Bob), while the other receiver (Eve)acted only as an eavesdropper. In this chapter, we consider the more general scenarioin which the transmitter has two independent secret messages, one intended for eachuser, and each message should be kept confidential from the other user. Such a modelis referred to as the two-user broadcast channel with confidential messages (BC-CM).Note that the two-user BC-CM reduces to the wiretap channel when the informationrate to one of the users is set to zero.Extension of the wiretap channel to the two-user BC-CM was considered in [47]wherein the authors derived inner and outer bounds on the secrecy capacity regionof the discrete memoryless BC-CM. Achievability of the secrecy capacity region ofthe two-user MISO BC-CM was established in [48] using the so-called secret dirty-105Chapter 4. Linear Precoding for the Two-User MISO BC-CMpaper coding (S-DPC) scheme under the total (average) power constraint. This codingscheme was extended in [49] to the MIMO BC-CM, and it was shown that the secrecycapacity region is rectangular under the matrix power (or input covariance matrix)constraint. Under the total power constraint, however, the secrecy capacity region canbe only found by performing an exhaustive search over the set of all input covariancematrices that satisfy the total power constraint. Due to the complexity of S-DPCand the lack of a simple solution to the practical case of total power constraint,the authors in [88] proposed a low-complexity linear precoding scheme for the two-user MIMO BC-CM based on generalized singular value decomposition. The workin [89] also considered the secrecy rate region of the two-user MIMO BC-CM underthe total power constraint via formulating a nonconvex weighted secrecy sum ratemaximization problem. An iterative algorithm based on a block successive lower-bound maximization method was proposed to solve such a nonconvex problem.In this chapter, we study the design of linear precoders for the two-user MISOBC-CM. Our treatment will not be limited to VLC channels in the sense that wedesign the precoders not only subject to amplitude constraints, but also subject tototal and per-antenna power constraints. Note that, in this chapter, we use the term“antenna” to designate general transmit and receive elements. In a VLC system, thetransmit antenna is an LED and the receive antenna is a PD.Under amplitude constraints, the secrecy capacity region of the two-user MISOBC-CM is unknown, and thus our motivation to find achievable secrecy rate regionsbased on linear precoding is clear. However, we also consider linear precoding forthe cases of total and per-antenna power constraints, wherein S-DPC is known to beoptimal, for the following reasons:1) Our approach to formulate and solve the problem of linear precoder design106Chapter 4. Linear Precoding for the Two-User MISO BC-CMis equally applicable to all the aforementioned constraints. In other words,total and per-antenna power constraints can be considered without additionaldifficulty.2) More importantly, the case of total power constraint is the only case in whichthe secrecy capacity region is precisely known and has been characterized inclosed form. Therefore, it sets a benchmark that can be used to quantify theperformance loss incurred by using a suboptimal linear precoding scheme, andalso to validate our approach to solve the precoder design problem when thetotal power constraint is considered.3) Finally, for the case of per-antenna power constraint, it seems that the secrecycapacity region can be only found via an exhaustive search over the set of allinput covariance matrices that satisfy the per-antenna power constraint, whichentails high computational complexity. Even when it is found, the S-DPCscheme utilized to achieve the secrecy capacity region is difficult to implement.Therefore, our proposed linear precoding scheme provides a viable solution withlower implementation complexity for the case of per-antenna power constraint.After fixing the input distribution, our goal in this chapter is to find linear pre-coders that achieve the boundary points of the secrecy rate region. To this end, weformulate the precoder design problem as a weighted secrecy sum rate maximizationproblem, subject to any of the aforementioned constraints. The resulting problem,however, has a fractional objective function, making it nonconvex and difficult tosolve. To circumvent such a difficulty, we first simplify the objective function using alower bound on the weighted secrecy sum rate. Then, we transform the problem intoan equivalent one having only two optimization variables. We show that the equiv-alent problem is more tractable and can be solved iteratively using the subgradient107Chapter 4. Linear Precoding for the Two-User MISO BC-CMmethod. In each iteration, we solve a convex inner optimization problem to updatethe value of the outer problem, and also to obtain the subgradient vector that spec-ifies the search direction for the next iteration. We provide a condition under whichthe obtained solution is guaranteed to be globally optimal. Furthermore, we showthat the inner problem can be easily modified to take into account channel uncer-tainty caused by limited feedback from both receivers. This leads us to the design ofrobust linear precoders that guarantee a certain worst-case secrecy rate performance.To the best of our knowledge, the work in this chapter is the first to considerlinear precoding for the two-user MISO BC-CM, subject to per-antenna power oramplitude constraints. Furthermore, it is the first work to consider robust precoding,for the same channel, by taking channel uncertainty into account.The remainder of this chapter is organized as follows. The system model, pre-coding scheme, and transmit constraints are described in detail in Section 4.2. InSection 4.3, we solve the precoder design problem under the premise of perfect channelinformation. In Section 4.4, we extend the design problem to its robust counterpartby considering uncertainty in channel information. In Section 4.5, we provide ournumerical examples to illustrate the achievable secrecy rate regions of the proposedprecoder. We conclude the chapter in Section 4.6.4.2 System ModelIn this section, we describe the channel model, the linear precoding scheme, theachievable secrecy rate regions, and the constraints imposed on the transmitted signalvector.108Chapter 4. Linear Precoding for the Two-User MISO BC-CM4.2.1 The Two-User MISO BC-CMWe study the problem of secret communication between one transmitter and twoindependent receivers over the Gaussian MISO broadcast channel. The transmitterhas N ≥ 2 antennas, and each receiver has a single antenna. In each communicationsession, the transmitter has two independent confidential messages, one intended foreach receiver. The two messages are simultaneously broadcasted, and the transmittershall ensure that each message can be reliably decoded by its intended receiver, andis kept confidential from the other one.We assume narrow-band transmission over a quasi-static, i.e., non-fading, Gaus-sian broadcast channel. The transmitted and received baseband signals, as well asthe channel gain vectors, are real-valued, i.e., the carrier phase is not modulated.This model is typical for intensity-modulation (IM) channels, including VLC chan-nels, and is also applicable to RF systems utilizing amplitude modulation schemes,such as amplitude-shift keying (where the baseband data symbols are real-valued).Under these assumptions, the signals observed by the two receivers can be expressedasy1(t) = hT1 x(t) + n1(t), (4.1a)y2(t) = hT2 x(t) + n2(t), (4.1b)where x(t) ∈ RN is the transmitted signal vector, h1 ∈ RN and h2 ∈ RN are thechannel gain vectors, and n1(t) and n2(t) are i.i.d. Gaussian noise samples withvariance σ2. We assume that h1 and h2 are linearly independent to ensure that theMISO broadcast channel in (4.1) is nondegraded.Let X be an input random vector that satisfies the constraints on the channel109Chapter 4. Linear Precoding for the Two-User MISO BC-CMinput, and Y1 and Y2 be the output random variables. Also letU 1 andU 2 be auxiliaryrandom variables. Then, it was shown in [47, Theorem 4] (see also [48, Lemma 2])that for any joint PDF p(u1,u2,x, y1, y2) that can be written as13p(u1,u2) p(x|u1,u2) p(y1, y2|x),the secrecy rate pair (R1, R2) satisfying0 ≤ R1 ≤ I(U 1;Y1)− I(U 1;Y2|U 2)− I(U 1;U 2), (4.2a)0 ≤ R2 ≤ I(U 2;Y2)− I(U 2;Y1|U 1)− I(U 1;U 2) (4.2b)is achievable for the two-user MISO BC-CM in (4.1). Achievability of the rate pairin (4.2) was proved based on a double-binning scheme [48, Section IV]. Thus, givena joint PDF p(u1,u2,x), the achievable secrecy rate region can be determined us-ing (4.2). On the other hand, given a certain constraint on the channel input X,it remains unclear how to choose p(u1,u2,x) such that the secrecy rate region ismaximized. For the case of total power constraint, it was shown that the secrecycapacity region of the MISO BC-CM in (4.1) can be characterized in closed form [48,Theorem 1], and the boundary points are achievable with the S-DPC scheme. Thisscheme, however, is difficult to implement in practice [88]. In addition, with per-antenna power constraints, there is no closed-form characterization, and apparentlythe secrecy capacity region can be only found via an exhaustive search over all in-put covariance matrices that satisfy the per-antenna power constraint. Furthermore,the S-DPC scheme proposed in [48] does not seem to be applicable to the case withamplitude constraints. This motivates us to consider the linear precoding scheme13In other words, (U1,U2)→X → (Y1, Y2) forms a Markov chain.110Chapter 4. Linear Precoding for the Two-User MISO BC-CMdescribed in the next subsection.4.2.2 Linear PrecodingWe study the secrecy performance of the two-user MISO BC-CM in (4.1) when thetransmitted signal vector is constructed asx(t) = w1s1(t) +w2s2(t) = Ws(t), (4.3)where w1 ∈ RN and w2 ∈ RN are fixed beamformers, s1(t) ∈ R and s2(t) ∈ R areindependent symbols (i.e., secrecy codewords) intended for Users 1 and 2, respec-tively, W = [w1 w2] is termed as the precoding matrix, or simply the precoder, ands(t) = [s1(t) s2(t)]T is the vector of transmitted symbols. Although suboptimal, theprecoding scheme in (4.3) is simple to implement. Furthermore, it will enable us tohandle per-antenna power or amplitude constraints.Substituting (4.3) back into (4.1), the signals received at both users can be writtenasy1(t) = hT1w1s1(t) + hT1w2s2(t) + n1(t), (4.4a)y2(t) = hT2w1s1(t) + hT2w2s2(t) + n2(t). (4.4b)Let S1 and S2 denote the random variable counterparts of s1(t) and s2(t), respectively.Then, the transmission scheme in (4.3) corresponds to choosingU 1 = w1S1, U 2 = w2S2, and X = U 1 +U 2. (4.5)Substituting from (4.5) into (4.2), the achievable secrecy rate pair in (4.2) can be111Chapter 4. Linear Precoding for the Two-User MISO BC-CMwritten as0 ≤ R1 ≤ I(S1;Y1)− I(S1;Y2|S2), (4.6a)0 ≤ R2 ≤ I(S2;Y2)− I(S2;Y1|S1). (4.6b)Note that joint encoding is not utilized in (4.5), i.e., S1 and S2 are independent, andthus I(U 1;U 2) = I(S1;S2) = 0.4.2.3 Transmit Constraints and Secrecy Rate RegionsIn this subsection, we describe the transmit constraints considered in the chapter,and derive closed-form expressions for the corresponding secrecy rate pairs (R1, R2).4.2.3.1 Total Average Power ConstraintThe most common constraint imposed on the input of Gaussian channels is the total(or sum) average power constraint. It is mathematically convenient, and often leadsto closed-form solutions. Furthermore, it provides much insight into the performanceof the communication system for a given power budget. Mathematically, a totalaverage power constraint PTot requires the transmitted codewords X to satisfy theinequalityTr(E{XXT}) ≤ PTot, (4.7)where E{XXT} is the transmit covariance matrix. An obvious way to comply withthe transmission scheme in (4.3) and satisfy the constraint in (4.7) is to choose S1and S2 to be i.i.d. standard Gaussian random variables, that isS1 ∼ N (0, 1), S2 ∼ N (0, 1), (4.8a)112Chapter 4. Linear Precoding for the Two-User MISO BC-CMand to ensure that the precoder W satisfies the inequality‖W‖2F ≤ PTot. (4.8b)Note that our choice of equal variances for the distributions of S1 and S2 (both haveunity variance) involves no loss of generality because the power allocated to each usercan still be adjusted from the entries of the precoding matrix W.Now, for a given W, and with Gaussian codewords S1, S2 ∼ N (0, 1), the mutualinformation terms in (4.6a) are simply calculated asI(S1;Y1) =12log2(1 +(hT1w1)2(hT1w2)2 + σ2), (4.9a)I(S1;Y2|S2) = 12log2(1 +(hT2w1)2σ2), (4.9b)where information is measured in (bits/sec/Hz). Similar expressions can be obtainedfor the corresponding terms in (4.6b), and thus we end up with the achievable secrecyrate pairR1 =12[log2(1 +(hT1w1)2(hT1w2)2 + σ2)(σ2(hT2w1)2 + σ2)]+, (4.10a)R2 =12[log2(1 +(hT2w2)2(hT2w1)2 + σ2)(σ2(hT1w2)2 + σ2)]+. (4.10b)4.2.3.2 Per-Antenna Average Power ConstraintDespite its simplicity, the total average power constraint (4.7) is often not sufficientto capture practical limitations that arise from implementation constraints. For ex-113Chapter 4. Linear Precoding for the Two-User MISO BC-CMample, the so-called digital beamforming14 scheme requires a dedicated transmit RFchain for each antenna element15. Clearly, each of these chains has its own powerbudget. Thus, a more realistic approach to model power limitations at the transmit-ter is to impose an individual power constraint on each RF chain, or, equivalently,on each antenna element, in addition to the total power constraint. A per-antennaaverage power constraint Pi, i = 1, . . . , N, can be expressed asE{X2i } ≤ Pi, i = 1, . . . , N, (4.11)where Xi is the ith entry of X. Depending on the values of PTot and P1, . . . , PN , oneof the constraints in (4.7) and (4.11) may become redundant. In particular:i) If∑Ni=1 Pi ≤ PTot, the per-antenna power constraint (4.11) becomes dominantand (4.7) can be ignored.ii) If Pi ≥ PTot for all i ∈ {1, . . . , N}, then (4.11) is obviously redundant and thetotal power constraint (4.7) is sufficient.iii) If neither of the above two cases holds, both (4.7) and (4.11) can be active, andthus they should be taken into account.Similar to the case of dominant total average power constraint, we let the code-words S1 and S2 be i.i.d. standard Gaussian random variables. Then, in order tosatisfy the constraint in (4.11), the entires of W should be chosen such thatw21i + w22i ≤ Pi, i = 1, . . . , N, (4.12)14In fact, all the transmission schemes considered in this thesis fall into the category of digital (orbaseband) beamforming.15Such a constraint is relaxed in the so-called hybrid beamforming scheme where the number ofRF chains can be smaller than the number of antennas.114Chapter 4. Linear Precoding for the Two-User MISO BC-CMwhere w1i and w2i are the ith entries of w1 and w2, respectively. Since S1 and S2are Gaussian, the secrecy rate pair expressions in (4.10) remain valid for any W thatsatisfies (4.12).4.2.3.3 Per-Antenna Amplitude ConstraintBy now, we already know that amplitude constraints typically arise in the design ofIM systems. In such a case, the input signal must satisfy the amplitude constraint|Xi| ≤ Ai, i = 1, . . . , N. (4.13)This constraint obviously renders the Gaussian distribution infeasible for the channelinput. Nevertheless, (4.13) can be fulfilled by choosing the codewords S1 and S2according to the uniform distribution over the interval [−1, 1], i.e.,S1 ∼ U [−1, 1], S2 ∼ U [−1, 1], (4.14a)and choosing the entries of the precoder W such that they satisfy the constraint|w1i|+ |w2i| ≤ Ai, i = 1, . . . , N. (4.14b)Unlike the Gaussian input distribution in (4.8a), the uniform distribution in (4.14a),along with Gaussian noise, do not immediately lead to closed-form expressions forI(S1;Y1) − I(S1;Y2|S2) in (4.6a), or the similar terms in (4.6b). Nevertheless, wecan lower-bound these terms to obtain closed-form expressions for the secrecy ratepair (R1, R2), as follows.115Chapter 4. Linear Precoding for the Two-User MISO BC-CMFirst, we rewrite I(S1;Y1)− I(S1;Y2|S2) ash(Y1)− h(Y1|S1)− h(Y2|S2) + h(Y2|S1, S2). (4.15)Using the entropy power inequality [35, Theorem 17.7.3], the differential entropyh(Y1) can be lower-bounded ash(Y1) = h(hT1w1S1 + hT1w2S2 +N1)≥ 12log2(22h(hT1w1S1) + 22h(hT1w2S2) + 22h(N1))=12log2(4(hT1w1)2 + 4(hT1w2)2 + 2pieσ2). (4.16)On the other hand, the conditional differential entropy h(Y1|S1) can be upper boundedby the differential entropy of a Gaussian random variable having equal variance, thatish(Y1|S1) = h(hT1w2S2 +N1)≤ 12log2(2pie(13(hT1w2)2 + σ2)). (4.17)Similarly, we haveh(Y2|S2) ≤ 12log2(2pie(13(hT2w1)2 + σ2)). (4.18)We also haveh(Y2|S1, S2) = h(N2) = 12log2(2pieσ2). (4.19)116Chapter 4. Linear Precoding for the Two-User MISO BC-CMSubstituting (4.16)-(4.19) back into (4.15) yields the secrecy rate expressionR1 =[12log2(4(hT1w1)2 + 4(hT1w2)2 + 2pieσ2)σ22pie(13(hT1w2)2 + σ2) (13(hT2w1)2 + σ2)]+ . (4.20a)Similarly, we haveR2 =[12log2(4(hT2w2)2 + 4(hT2w1)2 + 2pieσ2)σ22pie(13(hT2w1)2 + σ2) (13(hT1w2)2 + σ2)]+ . (4.20b)4.3 Precoder Design with Perfect ChannelInformationIn this section, we focus on the design of the precoder W under the assumption ofperfect channel information. We begin with the case of total and per-antenna averagepower constraints. Then, we show in Section 4.3.5 that the problem formulation andsolution method can be easily modified to handle per-antenna amplitude constraints.4.3.1 Problem FormulationBy designing W we mean finding the set of precoding matrices that achieve theboundary of the secrecy rate region characterized by (R1, R2). Assuming total andper-antenna average power constraints, the design problem can be expressed by the117Chapter 4. Linear Precoding for the Two-User MISO BC-CMtwo-objective optimization problemmaximizeW(R1, R2) (4.21a)s.t. ‖W‖2F ≤ PTot, (4.21b)w21i + w22i ≤ Pi, i = 1, . . . , N, (4.21c)where partial ordering and maximization of the pair (R1, R2) are w.r.t. the nonnega-tive orthant R2+ [80, Section 4.7.5]. In the context of multi-objective optimization, afeasible matrix W that achieves a secrecy rate pair on the boundary of the set of allachievable rate pairs is referred to as Pareto optimal, and the corresponding secrecyrate pair (R1, R2) is a Pareto optimal pair. Thus, solving (4.21) means finding Paretooptimal matrices W.The standard approach towards solving (4.21) is to scalarize the objective via aweighted sum [80, Section 4.7.5], that is to replace (R1, R2) with ρ1R1 + ρ2R2, wherethe weights ρ1 ≥ 0 and ρ2 ≥ 0 are free parameters. Different Pareto optimal pointscan be obtained by adjusting the relative weight ρ1/ρ2 to different values between 0and ∞. This can be carried out by choosing16 ρ1 = ρ and ρ2 = 1− ρ, where ρ is afree parameter taking values in the interval [0, 1]. Thus, for any ρ ∈ [0, 1], we havethe weighted secrecy sum rate maximization problemmaximizeWRwsum(ρ) (4.22a)s.t. ‖W‖2F ≤ PTot, (4.22b)w21i + w22i ≤ Pi, i = 1, . . . , N, (4.22c)16Although constraining ρ1 and ρ2 to sum to 1 looks arbitrary here, we will need this restrictionin the proof of Proposition 4.2, particularly to ensure that the optimization problem in (C.6) isconvex.118Chapter 4. Linear Precoding for the Two-User MISO BC-CMwhereRwsum(ρ) , ρR1 + (1− ρ)R2 (4.22d)is the weighted secrecy sum rate. It is clear that solving (4.22) with ρ = 1 correspondsto finding the maximum achievable secrecy rate for User 1 when User 2 is treated asan eavesdropper, while ρ = 0 yields the maximum achievable secrecy rate for User 2.Ideally we would like to solve (4.22) with the objective Rwsum calculated using therate expressions in (4.10). Using these expressions, however, would make (4.22) verydifficult to solve, except for the special cases ρ = 0 and ρ = 1. In order to make theproblem tractable, we will simplify the objective of (4.22) by replacing Rwsum withthe lower bound Rˆwsum, given byRˆwsum(ρ) = ρRˆ1 + (1− ρ)Rˆ2, (4.23)where, for hT1w1 6= 0 and hT2w2 6= 0, Rˆ1 and Rˆ2, respectively, are given byRˆ1 = log2∣∣hT1w1∣∣σ((hT1w2)2 + σ2)12 ((hT2w1)2 + σ2)12, (4.24a)Rˆ2 = log2∣∣hT2w2∣∣σ((hT2w1)2 + σ2)12 ((hT1w2)2 + σ2)12. (4.24b)From (4.10) and (4.24), it is clear that17 Rˆ1 < R1 and Rˆ2 < R2. Thus, for anyρ ∈ [0, 1], we have the inequality Rˆwsum < Rwsum. Substituting from (4.24) into (4.23),we obtainRˆwsum(ρ) = log2∣∣hT1w1∣∣ρ ∣∣hT2w2∣∣1−ρ σ((hT2w1)2 + σ2)12 ((hT1w2)2 + σ2)12. (4.25)Note that Rˆwsum is a tight lower bound for Rwsum in (4.22d) when the SNR at both17The inequality Rˆ1 < R1 results from dropping the term 1 in the logarithm in (4.9a) and droppingthe operator [·]+ from the rate expression in (4.10a). In a similar way, it can be shown that Rˆ2 < R2.119Chapter 4. Linear Precoding for the Two-User MISO BC-CMreceivers is sufficiently high. However, unlike Rwsum, whose nonnegativity is ensuredby the [·]+ operators in (4.10), Rˆwsum can be negative since Rˆ1 and/or Rˆ2 can benegative when the corresponding SNR is sufficiently low. Nonetheless, maximizingRˆwsum is still beneficial even when its optimal value ends up to be negative becausethe maximization problem is only used for the design of W. The achievable secrecyrate pair, however, is obtained by substitutingW into (4.10), i.e., the achievable ratepair is guaranteed to be nonnegative. Now, we formulate our design problem as18maximizeWln(hT1w1)ρ(hT2w2)1−ρ((hT2w1)2 + σ2)12 ((hT1w2)2 + σ2)12(4.26a)s.t. ‖W‖2F ≤ PTot, (4.26b)w21i + w22i ≤ Pi, i = 1, . . . , N. (4.26c)Note that the formulation in (4.26) implicitly adds the two constraints hT1w1 ≥ 0and hT2w2 ≥ 0. These additional constraints cause no loss in performance becausean optimal w1 that results in negative hT1w1 can always be replaced with −w1 with-out reducing the optimal value or violating the constraints on W. In a similar way,the implicit constraint hT2w2 ≥ 0 can be justified. Note also that, unlike the ex-pressions in (4.24), the formulation in (4.26) does not exclude the cases hT1w1 = 0and hT2w2 = 0 as the objective function remains well defined even when optimal Wleads to hT1w1 = 0 or hT2w2 = 0. For example, the solution w1 = 0 (which results inhT1w1 = 0) would be optimal only when19 ρ = 0, resulting in (hT1w1)ρ = 00 = 1.In the next subsection, we shall explain in detail our approach to solve (4.26).18Using the natural logarithm in the objective of (4.26) (instead of the logarithm to base 2) willslightly simplify the notation when differentiation becomes involved.19This is true because we assume that h1 and h2 are linearly independent. On the other hand,if h1 and h2 are collinear and ‖h1‖2 ≤ ‖h2‖2, then w1 = 0 would be optimal for all ρ ∈ [0, 1],i.e., User 1 cannot achieve positive secrecy rates and should always be treated as an eavesdropperbecause its channel h1 is degraded.120Chapter 4. Linear Precoding for the Two-User MISO BC-CM4.3.2 The Outer ProblemUsing the auxiliary variables δ1 ≥ 0 and δ2 ≥ 0, the problem in (4.26) can be expressedasmaximizeW,δ1,δ2ln(hT1w1)ρ(hT2w2)1−ρ(δ21 + σ2)12 (δ22 + σ2)12(4.27a)s.t. |hT2w1| ≤ δ1, |hT1w2| ≤ δ2, (4.27b)‖W‖2F ≤ PTot, (4.27c)w21i + w22i ≤ Pi, i = 1, . . . , N. (4.27d)Let f(δ1, δ2) denote the optimal value of the perturbed problemmaximizeWρ ln(hT1w1) + (1− ρ) ln(hT2w2) (4.28a)s.t. |hT2w1| ≤ δ1, |hT1w2| ≤ δ2, (4.28b)‖W‖2F ≤ PTot, (4.28c)w21i + w22i ≤ Pi, i = 1, . . . , N. (4.28d)Then, the problem in (4.27) can be written asmaximizeδ1,δ2≥0ϕ(δ1, δ2), (4.29a)whereϕ(δ1, δ2) , f(δ1, δ2)− 12ln((δ21 + σ2)(δ22 + σ2)). (4.29b)Now, we can see that solving the design problem in (4.26) entails solving (4.28)and (4.29) iteratively. For obvious reasons, we shall refer to (4.29) as the outer121Chapter 4. Linear Precoding for the Two-User MISO BC-CMproblem, and to (4.28) as the inner problem.The inner problem is clearly convex, and thus can be efficiently solved usingstandard convex optimization packages. On the other hand, the outer problem isnonconvex because the objective function ϕ(δ1, δ2) is not concave, in general. Nev-ertheless, the following two propositions reveal that ϕ(δ1, δ2) has a special structurethat makes the outer problem solvable, i.e., its global maximum can be efficientlyobtained, when a certain condition is satisfied. Even when such a condition is notsatisfied, these propositions still give us guidelines for approaching the outer problem.Proposition 4.1. The objective function of the outer problem (4.29) is concave whenrestricted inside the region {(δ1, δ2) : 0 ≤ δ1 ≤ σ, 0 ≤ δ2 ≤ σ}.Proof: The proof is fairly straightforward. The first term in (4.29b), i.e., f(δ1, δ2),is concave for all δ1, δ2 ≥ 0 because the perturbed problem in (4.28) is convex [80,Section 5.6.1]. On the other hand, the second term −12ln ((δ21 + σ2)(δ22 + σ2)) isconcave only when 0 ≤ δ1 ≤ σ and 0 ≤ δ2 ≤ σ (this can be easily verified after writingdown the Hessian matrix). Thus, ϕ(δ1, δ2) is concave when δ1, δ2 ≤ σ. Proposition 4.2. The objective function of the outer problem (4.29) is quasiconcavewhen restricted to any line (in the nonnegative orthant R2+) passing through the origin.The proof, which is provided in Appendix C.1, is based on the observation thatthe first term in (4.29b) is nondecreasing (w.r.t. R2+), while the second term ismonotonically decreasing20. Note that the condition in Proposition 4.2 is weakerthan stating that ϕ(δ1, δ2) is quasiconcave, as the latter condition would require ϕ tobe quasiconcave when restricted to any line in R2+.Combining Propositions 4.1 and 4.2 immediately yields the following conclusion:20See [80, Section 3.6.1] for the notion of monotonicity w.r.t. a generalized inequality on thenonnegative orthant.122Chapter 4. Linear Precoding for the Two-User MISO BC-CMCorollary 4.1. For the outer problem (4.29), any local maximum inside the region{(δ1, δ2) : 0 ≤ δ1 ≤ σ, 0 ≤ δ2 ≤ σ} is a global maximum.Corollary 4.1 suggests that we start searching for the solution of (4.29) insidethe region δ1, δ2 ≤ σ. If the search algorithm terminates at δ? = (δ?1, δ?2) such thatδ?1, δ?2 ≤ σ, then δ? is guaranteed to be the (globally) optimal solution of (4.29). Onthe other hand, if δ?1 > σ or δ?2 > σ, then we will accept δ? as a (possibly) suboptimalsolution. It is worth to mention that the numerical results show that ϕ(δ1, δ2) is aunimodal function with only one maximum, for all δ1, δ2 ≥ 0, and no other stationarypoints. However, it is difficult, in general, to rigorously prove that a multivariablefunction is unimodal, beyond concavity or quasiconcavity. Therefore, we can onlyconjecture that ϕ(δ1, δ2) is unimodal (for all δ1, δ2 ≥ 0), and consequently any localmaximum is global.Now, we have to choose a reasonable search algorithm to solve (4.29). Since theobjective function ϕ(δ1, δ2) is differentiable almost everywhere21, a natural choicefor the search algorithm is the subgradient method in which the subgradient vectorsare used as the search directions [90, 91]. Let the vector ∇subf(δ1, δ2) ∈ R2+ be asubgradient22 of f at (δ1, δ2), where the two entries of ∇subf are both nonnegativesince f is nondecreasing w.r.t. δ1 and δ2. Then, the corresponding subgradient of ϕis given by∇subϕ(δ1, δ2) = ∇subf(δ1, δ2)−[δ1δ21 + σ2δ2δ22 + σ2]T. (4.30)Before we proceed to the details of the search algorithm, we need to find ∇subfin order to calculate the search direction ∇subϕ at any (δ1, δ2). This will be our goalin the next subsection.21This is because f(δ1, δ2) is not necessarily differentiable (everywhere). Nevertheless, sincef(δ1, δ2) is concave, it is differentiable almost everywhere.22The term “supergradient” is probably more appropriate here because f(δ1, δ2) is a concavefunction.123Chapter 4. Linear Precoding for the Two-User MISO BC-CM4.3.3 The Dual of the Inner ProblemThe inner problem (4.28) is a convex problem whose constraints satisfy Slater’s con-dition, and thus strong duality holds [80, Section 5.2.3]. As a consequence, theoptimal value of the inner problem, i.e., f(δ1, δ2), is identical to the optimal value ofits (Lagrange) dual. Furthermore, the optimal Lagrange multipliers associated withthe two constraints in (4.28b) provide a subgradient vector23 for f at (δ1, δ2) [92,Section 8.5.6]. Therefore, our next task is to derive the dual problem for (4.28).We begin with reformulating (4.28) asmaximizeW,z1,...,z4ρ ln z1 + (1− ρ) ln z2 (4.31a)s.t. |z3| ≤ δ1, |z4| ≤ δ2, (4.31b)‖w1‖22 + ‖w2‖22 ≤ PTot, (4.31c)w21i + w22i ≤ Pi, i = 1, . . . , N, (4.31d)hT1w1 = z1, hT2w2 = z2, (4.31e)hT2w1 = z3, hT1w2 = z4, (4.31f)where we have introduced four new variables, z1, . . . , z4, and four associated equalityconstraints (4.31e)-(4.31f). The Lagrangian associated with the reformulated problemin (4.31) isL(W, z1, . . . , z4, λ1, λ2, γ,µ, ν1, . . . , ν4)= ρ ln z1 + (1− ρ) ln z2 − λ1 (|z3| − δ1)− λ2 (|z4| − δ2)− γ (‖w1‖22 + ‖w2‖22 − PTot)− N∑i=1µi(w21i + w22i − Pi)− ν1(hT1w1 − z1)− ν2(hT2w2 − z2)− ν3(hT2w1 − z3)− ν4(hT1w2 − z4), (4.32)23Note that f has more than one subgradient at the points (δ1, δ2) where f is non-differentiable.124Chapter 4. Linear Precoding for the Two-User MISO BC-CMwhere λ1 ≥ 0 and λ2 ≥ 0 are the Lagrange multipliers associated with the perturbedconstraints in (4.31b), γ ≥ 0 is the Lagrange multiplier associated with the totalpower constraint (4.31c), µ = [µ1 . . . µN ]T, with entries µi ≥ 0, i = 1, . . . , N , is theLagrange multiplier vector associated with the per-antenna power constraint (4.31d),and ν1, . . . , ν4 are the Lagrange multipliers associated with the equality constraintsin (4.31e)-(4.31f). Upon rearranging the terms in the Lagrangian (4.32), the dualfunction g is obtained by maximization over the primary variables W, z1, . . . , z4,that isg(λ1, λ2, γ,µ, ν1, . . . , ν4) = λ1δ1 + λ2δ2 + γPTot +N∑i=1µiPi+N∑i=1maxw1i(− (ν1h1i + ν3h2i)w1i − (γ + µi)w21i)+N∑i=1maxw2i(− (ν2h2i + ν4h1i)w2i − (γ + µi)w22i)+ maxz1(ν1z1 + ρ ln z1) + maxz2(ν2z2 + (1− ρ) ln z2)+ maxz3(ν3z3 − λ1|z3|) + maxz4(ν4z4 − λ2|z4|) , (4.33)where h1i and h2i are the ith entries of h1 and h2, respectively. Now, we have tosolve all the maximization terms in (4.33) analytically. In fact, we havemaxw1i(− (ν1h1i + ν3h2i)w1i − (γ + µi)w21i)=(ν1h1i + ν3h2i)24(γ + µi), γ + µi > 0, i = 1, . . . , N, (4.34a)maxz1(ν1z1 + ρ ln z1) = −ρ ln −ν1ρ− ρ, ν1 < 0, (4.34b)125Chapter 4. Linear Precoding for the Two-User MISO BC-CMmaxz3(ν3z3 − λ1|z3|) =0 |ν3| ≤ λ1∞ otherwise, (4.34c)where (4.34a) is a simple unconstrained quadratic concave maximization problem,(4.34b) follows from the conjugate of the negative logarithm function (see [80, Ex-ample 3.21]), and (4.34c) follows from the conjugate of the absolute value function(see [80, Example 3.26]). Note that the condition γ + µi > 0 in (4.34a) is alwayssatisfied because, for each antenna, at least one of the constraints (i.e., the totalpower constraint or the per-antenna power constraint) must be active. Thus, γ + µiis strictly positive for all i = 1, . . . , N . Using the expressions in (4.34), the dualproblem can be formulated as24minimizeλ1,λ2,γ,µ,τ1,τ2,ν1,...,ν4 δ1λ1 + δ2λ2 + PTotγ +∑Ni=1(Piµi + τ1i + τ2i)−ρ ln −ν1ρ− (1− ρ) ln −ν21− ρ− 1 (4.35a)s.t. ν1, ν2 < 0, |ν3| ≤ λ1, |ν4| ≤ λ2, (4.35b)γ ≥ 0, µi ≥ 0, γ + µi > 0, i = 1, . . . , N, (4.35c) τ1i ν1h1i + ν3h2iν1h1i + ν3h2i 4(γ + µi)  0, i = 1, . . . , N, (4.35d) τ2i ν2h2i + ν4h1iν2h2i + ν4h1i 4(γ + µi)  0, i = 1, . . . , N, (4.35e)where we have used Schur complement, in conjunction with the auxiliary variables τ1iand τ2i, i = 1, . . . , N , to formulate the linear matrix inequality constraints in (4.35d)and (4.35e). Two special cases of the dual problem (4.35) are worth mentioning.24We maintain the fixed term −1 in the objective function in (4.35a) to have its optimal valueequal to the optimal value of the inner problem (4.28), i.e., equal to f(δ1, δ2).126Chapter 4. Linear Precoding for the Two-User MISO BC-CMFirst, at the corner point ρ = 0, the Lagrange multipliers λ1, ν1, and ν3 are set tozero, and the dual problem (4.35) simplifies tominimizeλ2,γ,µ,τ2,ν2,ν4(δ2λ2 + PTotγ +N∑i=1(Piµi + τ2i)− ln(−ν2))− 1 (4.36a)s.t. ν2 < 0, |ν4| ≤ λ2, (4.36b)γ ≥ 0, µi ≥ 0, γ + µi > 0, i = 1, . . . , N, (4.36c) τ2i ν2h2i + ν4h1iν2h2i + ν4h1i 4(γ + µi)  0, i = 1, . . . , N, (4.36d)where we have used the convention that 0 ln 00= 0 while simplifying the objective.A similar simplification can be obtained for the other corner point, i.e., at ρ = 1.Second, for the case in which there is only a total power constraint, i.e., whenthe per-antenna constraint in (4.31d) does not exist or is not active, the Lagrangemultiplier vector µ is set to 0, and (4.35) simplifies tominimizeλ1,λ2,γ,τ1,τ2,ν1,...,ν4 δ1λ1 + δ2λ2 + PTotγ + τ1 + τ2−ρ ln −ν1ρ− (1− ρ) ln −ν21− ρ− 1 (4.37a)s.t. ν1, ν2 < 0, |ν3| ≤ λ1, |ν4| ≤ λ2, (4.37b)γ > 0, (4.37c) τ1 (ν1h1 + ν3h2)Tν1h1 + ν3h2 4γIN  0, (4.37d) τ2 (ν2h2 + ν4h1)Tν2h2 + ν4h1 4γIN  0. (4.37e)127Chapter 4. Linear Precoding for the Two-User MISO BC-CMThe dual problem (4.35) is, of course, convex and thus can be efficiently solvedto obtain f(δ1, δ2), as well as ∇subf(δ1, δ2). Let {λ?1, λ?2, γ?,µ?, τ ?1, τ ?2, ν?1 , . . . , ν?4} bethe optimal solution of (4.35) for given δ1 and δ2. Then, f(δ1, δ2) is equal to theoptimal value of the objective, and the vector [λ?1 λ?2]T is a subgradient of f at (δ1, δ2).Consequently, the subgradient vector in (4.30) can be written as∇subϕ(δ1, δ2) =[λ?1 −δ1δ21 + σ2λ?2 −δ2δ22 + σ2]T. (4.38)Having obtained ∇subϕ(δ1, δ2), we are now ready to use the subgradient methodto solve (4.29).4.3.4 The Search AlgorithmIn this subsection, we turn our focus to the search algorithm used to find a solutionfor the outer problem (4.29), i.e., to find δ? = [δ?1 δ?2]T that maximizes ϕ(δ1, δ2).A typical subgradient method uses the iteration [91]δ(k+1) = δ(k) + α(k) ∇subϕ(δ(k)), k = 1, 2, . . . , (4.39)where δ(k) is the start point at the kth iteration (with δ(1) being the initial point),α(k) > 0 is the kth step size, and δ(k+1) is the end point after k iterations. Thenumerical results in Section 4.5 reveal that, when the noise variance σ2 is equal to 1,the values of δ?1 and δ?2 can be on the order of 10−4 up to 101. This several orders ofmagnitude difference suggests that the search is better carried out on a logarithmicscale, rather than the ordinary linear scale, in order to improve the accuracy andmaintain numerical stability (so convergence is achieved within a reasonable numberof iterations).128Chapter 4. Linear Precoding for the Two-User MISO BC-CMLet δdB be defined as δdB , [20 log10(δ1) 20 log10(δ2)]T. Then, the subgradient∇subϕ on the logarithmic scale, i.e., when differentiation is w.r.t. 20 log10(δ1) and20 log10(δ2), is given by∇subϕ(δdB) = ln 1020δ1(λ?1 −δ1δ21 + σ2)δ2(λ?2 −δ2δ22 + σ2) . (4.40)Now, we proceed with the search algorithm as follows. First, we choose an initialpoint δ(1)dB, such that δ(1)1 ≤ σ and δ(1)2 ≤ σ. This point is iteratively updated byδ(k+1)dB = δ(k)dB + αFixdB∇subϕ(δ(k)dB)‖∇subϕ(δ(k)dB)‖2, k = 1, 2, . . . , (4.41)where αFixdB is a fixed step size in dB. That is, for each iteration, we take a step αFixdBin the direction of the subgradient. This iteration shall continue until we overshootthe peak, i.e., when ϕ(δ) starts to decrease. Once the peak is encountered, we reducethe step size and use the iterationδ(K+l+1)dB = δ(K+l)dB +αFixdBl∇subϕ(δ(K+l)dB )‖∇subϕ(δ(K+l)dB )‖2, l = 1, . . . , L, (4.42)where K is the number of iterations using (4.41), i.e., with a fixed step size, and Lis the maximum number of iterations with a decreasing step size. Unlike K, L isdetermined in advance according to the required accuracy of the solution. Therefore,the search will terminate after K+L total iterations, and the solution δ?dB is obtainedwith accuracy αFixdB /L dB. For convenience, the algorithm is summarized in Table 4.1.Upon solving the outer problem (4.29), we solve the inner problem (4.28) using δ?to obtain the optimum precoding matrix W?. Then, the secrecy rate pair (R1, R2)129Chapter 4. Linear Precoding for the Two-User MISO BC-CMTable 4.1: Subgradient-based search algorithm to solve the maximization problemin (4.29).Algorithm 4.1 Subgradient-based method to solve (4.29)1: Set the initial (fixed) step size αFixdB and the maximum number of iterations withdecreasing step size L2: Set the binary switch REDUCE = false3: Set the indexes k = 0 and l = 14: Choose an initial point δ(1)dB such that δ(1)1 ≤ σ, δ(1)2 ≤ σ5: while l ≤ L do6: Solve (4.35) to obtain f(δ(k+l)dB ), λ?1(k+l), λ?2(k+l)7: Calculate ϕ(δ(k+l)dB ) using (4.29b)8: Calculate ∇subϕ(δ(k+l)dB ) using (4.40)9: if ϕ(δ(k+l)dB ) ≤ ϕ(δ(k+l−1)dB ), then10: REDUCE = true11: end if12: if REDUCE = false, then13: Update δ(k+l)dB using (4.41)14: k := k + 115: else16: Update δ(k+l)dB using (4.42)17: l := l + 118: end if19: end while20: return δ?dB = argmax {ϕ(δ(1)dB), . . . , ϕ(δ(k+L+1)dB )}130Chapter 4. Linear Precoding for the Two-User MISO BC-CMis calculated by substituting W? into (4.10). We repeat the entire procedure withdifferent values of ρ ∈ [0, 1] to obtain different points (R1, R2) on the boundary ofthe secrecy rate region.4.3.5 Per-Antenna Amplitude ConstraintIn this subsection, we design the precoding matrix W subject to the per-antennaamplitude constraint (4.14b). Fortunately, the problem formulation and solutiontechniques developed in the previous subsections are immediately applicable. In fact,we just need to modify the weighted secrecy sum rate expression (4.25) and the innerproblem (4.28), and consequently its dual (4.35), to take the amplitude constraintinto account.Similar to (4.25), we need a weighted secrecy sum rate expression that is amenableto optimization. From (4.20), the rate expressions R1 and R2, respectively, can belower-bounded byRˆ1 = log23√2∣∣hT1w1∣∣σ√pie ((hT1w2)2 + 3σ2)12 ((hT2w1)2 + 3σ2)12, (4.43a)Rˆ2 = log23√2∣∣hT2w2∣∣σ√pie ((hT2w1)2 + 3σ2)12 ((hT1w2)2 + 3σ2)12. (4.43b)Then, for any ρ ∈ [0, 1], we have the weighted secrecy sum rateRˆwsum(ρ) = log23√2∣∣hT1w1∣∣ρ ∣∣hT2w2∣∣1−ρ σ√pie((hT2w1)2 + 3σ2)12 ((hT1w2)2 + 3σ2)12. (4.44)131Chapter 4. Linear Precoding for the Two-User MISO BC-CMSimilar to (4.28), we formulate the inner problem asmaximizeWρ ln(hT1w1) + (1− ρ) ln(hT2w2) (4.45a)s.t. |hT2w1| ≤ δ1, |hT1w2| ≤ δ2, (4.45b)|w1i|+ |w2i| ≤ Ai, i = 1, . . . , N. (4.45c)Then, following the same procedure as in Section 4.3.3, it can be shown that the dualproblem for (4.45) isminimizeλ1,λ2,µ,ν1,...,ν4 δ1λ1 + δ2λ2 +∑Ni=1(Aiµi)−ρ ln −ν1ρ− (1− ρ) ln −ν21− ρ− 1 (4.46a)s.t. ν1, ν2 < 0, |ν3| ≤ λ1, |ν4| ≤ λ2, (4.46b)|ν1h1i + ν3h2i| ≤ µi, i = 1, . . . , N, (4.46c)|ν2h2i + ν4h1i| ≤ µi, i = 1, . . . , N, (4.46d)where the Lagrange multipliers λ1, λ2, ν1, . . . , ν4 are defined as in (4.35), and µ =[µ1 . . . µN ]T is the Lagrange multiplier vector associated with the amplitude con-straint (4.45c).4.4 Robust Precoder Design with ImperfectChannel InformationOur solutions in Section 4.3 were based on the assumption that the channel gainvectors h1 and h2 are precisely known to the transmitter. In this section, we capitalizeon our approach and tackle the more general design problem in which the transmitter132Chapter 4. Linear Precoding for the Two-User MISO BC-CMhas only uncertain estimates of h1 and h2. We will see that the problem formulationis very similar to its non-robust counterpart, and thus the solution approach will alsobe similar. Therefore, our pace in this section will be relatively fast, and much ofthe details and derivations encountered in the previous section will be omitted forbrevity.4.4.1 Channel Uncertainty ModelWe adopt the spherical uncertainty model (or norm-bounded error model) in whichthe actual channel gain vectors, h1 and h2, respectively, are modelled byh1 ∈ H1, H1={hˆ1 + e1 : ‖e1‖2 ≤ 1}, (4.47a)h2 ∈ H2, H2={hˆ2 + e2 : ‖e2‖2 ≤ 2}, (4.47b)where H1 and H2 are N -dimensional spherical sets, hˆ1 ∈ RN and hˆ2 ∈ RN are thechannel vector estimates available to the transmitter, e1 ∈ RN and e2 ∈ RN are un-known (but norm-bounded) error vectors, and 1 and 2 are known constants thatquantify the amount of uncertainty for each channel. This error model is well ac-cepted for representing channel uncertainty caused by quantization errors and finite-rate feedback from each receiver to the transmitter [57, Lemma 1].Given the uncertain channel information in (4.47), our goal in this section isto design the precoding matrix W in order to optimize the performance in termsof the worst-case secrecy rate pair (Rwc1 , Rwc2 ). That is to solve the two-objectiveoptimization problemmaximizeW(Rwc1 , Rwc2 ) (4.48)subject to power or amplitude constraints, where, for fixedW, the worst-case secrecy133Chapter 4. Linear Precoding for the Two-User MISO BC-CMrates Rwc1 and Rwc2 are determined byRwc1 = minh1∈H1,h2∈H2R1, (4.49a)Rwc2 = minh1∈H1,h2∈H2R2. (4.49b)Similar to our approach in the previous section, we shall tackle (4.48) by solving aweighted worst-case secrecy sum rate maximization problem, as we see in the followingtwo subsections.4.4.2 Total and Per-Antenna Average Power ConstraintsIn this subsection, we solve the weighted worst-case secrecy sum rate maximizationproblem subject to total and per-antenna power constraints. First, we need to sim-plify the worst-case secrecy rate expressions in order to obtain a weighted sum ratethat is amenable to optimization. Substituting from (4.10a) into (4.49a), we obtainRwc1 =[12log2 minh1∈H1(1 +(hT1w1)2(hT1w2)2 + σ2)+12log2 minh2∈H2(σ2(hT2w1)2 + σ2)]+≥ 12log21 + minh1∈H1(hT1w1)2maxh1∈H1(hT1w2)2 + σ2+ 12log2 σ2maxh2∈H2(hT2w1)2 + σ2 (4.50a)≥ log2minh1∈H1∣∣hT1w1∣∣σmaxh1∈H1((hT1w2)2 + σ2)12 maxh2∈H2((hT2w1)2 + σ2)12, (4.50b)where the first inequality follows from dropping the [·]+ operator and applying theinequalityminxf1(x)f2(x)≥minxf1(x)maxxf2(x),134Chapter 4. Linear Precoding for the Two-User MISO BC-CMwhich holds for arbitrary functions f1 and f2, and the second inequality follows fromdropping the term 1. We shall use (4.50b) to formulate the weighted secrecy sum ratefor the optimization problem, while we use the better bound in (4.50a) to calculatethe worst-case secrecy rate Rwc1 after obtaining W. Similarly, we haveRwc2 ≥12log21 + minh2∈H2(hT2w2)2maxh2∈H2(hT2w1)2 + σ2+ 12log2 σ2maxh1∈H1(hT1w2)2 + σ2 (4.51a)≥ log2minh2∈H2∣∣hT2w2∣∣σmaxh2∈H2((hT2w1)2 + σ2)12 maxh1∈H1((hT1w2)2 + σ2)12. (4.51b)Next, we combine the rate expressions in (4.50b) and (4.51b) using the weights ρ and1− ρ, for any ρ ∈ [0, 1], to formulate the robust design problemmaximizeWlnminh1∈H1(hT1w1)ρ minh2∈H2(hT2w2)1−ρmaxh2∈H2((hT2w1)2 + σ2)12 maxh1∈H1((hT1w2)2 + σ2)12(4.52a)s.t. ‖W‖2F ≤ PTot, (4.52b)w21i + w22i ≤ Pi, i = 1, . . . , N. (4.52c)135Chapter 4. Linear Precoding for the Two-User MISO BC-CMProblem (4.52), in turn, can be expressed asmaximizeW,z1,z2,δ1,δ2lnzρ1 z1−ρ2(δ21 + σ2)12 (δ22 + σ2)12(4.53a)s.t. hT1w1 ≥ z1 ∀h1 ∈ H1, (4.53b)hT2w2 ≥ z2 ∀h2 ∈ H2, (4.53c)|hT2w1| ≤ δ1 ∀h2 ∈ H2, (4.53d)|hT1w2| ≤ δ2 ∀h1 ∈ H1, (4.53e)‖W‖2F ≤ PTot, (4.53f)w21i + w22i ≤ Pi, i = 1, . . . , N. (4.53g)Utilizing the expressions of the uncertainty sets H1 and H2 in (4.47), the inequalitiesin (4.53b), (4.53c), (4.53d), and (4.53e), respectively, can be replaced byhˆT1w1 − 1‖w1‖2 ≥ z1, (4.54a)hˆT2w2 − 2‖w2‖2 ≥ z2, (4.54b)|hˆT2w1|+ 2‖w1‖2 ≤ δ1, (4.54c)|hˆT1w2|+ 1‖w2‖2 ≤ δ2. (4.54d)Similar to (4.28), let f(δ1, δ2) denote the optimal value of the perturbed problemmaximizeW,z1,z2ρ ln z1 + (1− ρ) ln z2 (4.55a)s.t. (4.54a), (4.54b), (4.54c), (4.54d), (4.53f), (4.53g). (4.55b)136Chapter 4. Linear Precoding for the Two-User MISO BC-CMThen, the robust design problem (4.53) can be expressed asmaximizeδ1,δ2≥0f(δ1, δ2)− 12ln((δ21 + σ2)(δ22 + σ2)). (4.56)Again, we shall refer to (4.56) as the outer problem, and to (4.55) as the inner prob-lem. It is clear that the inner problem (4.55) is convex, and the outer problem (4.56)is essentially identical to (4.29). Thus, it can be shown that Propositions 4.1 and 4.2hold for the objective of (4.56) as well. Consequently, (4.56) can be solved iterativelyusing Algorithm 4.1. In each iteration, the subgradient vector ∇subf(δ1, δ2) is ob-tained by solving the dual of the inner problem (4.55). Such a dual problem can beformulated asminimizeλ1,λ2,γ,µ,τ1,τ2,χ1,χ2,η1,η2,ν1,ν2 δ1λ1 + δ2λ2 + PTotγ +∑Ni=1(Piµi + τ1i + τ2i)−ρ ln χ1ρ− (1− ρ) ln χ21− ρ− 1 (4.57a)s.t. χ1, χ2 > 0, |ν1| ≤ λ1, |ν2| ≤ λ2, (4.57b)‖χ1hˆ1 − η1 − ν1hˆ2‖2 ≤ λ12 + χ11, (4.57c)‖χ2hˆ2 − η2 − ν2hˆ1‖2 ≤ λ21 + χ22, (4.57d)γ ≥ 0, µi ≥ 0, γ + µi > 0, i = 1, . . . , N, (4.57e)τ1i η1iη1i 4(γ + µi)  0, i = 1, . . . , N, (4.57f)τ2i η2iη2i 4(γ + µi)  0, i = 1, . . . , N, (4.57g)where λ1 and λ2 are the Lagrange multipliers associated with the constraints (4.54c)and (4.54d), respectively. Derivation of the dual problem (4.57) is provided in Ap-137Chapter 4. Linear Precoding for the Two-User MISO BC-CMpendix C.2.4.4.3 Per-Antenna Amplitude ConstraintWith amplitude constraints, we use the definitions in (4.49) to obtain the worst-case counterparts of the secrecy rate expressions in (4.20). Furthermore, the innerproblem (4.55) is modified tomaximizeW,z1,z2ρ ln z1 + (1− ρ) ln z2 (4.58a)s.t. (4.54a), (4.54b), (4.54c), (4.54d), (4.58b)|w1i|+ |w2i| ≤ Ai, i = 1, . . . , N, (4.58c)and it can be shown that its dual is given byminimizeλ1,λ2,µ,χ1,χ2,η1,η2,ν1,ν2 δ1λ1 + δ2λ2 +∑Ni=1(Aiµi)−ρ ln χ1ρ− (1− ρ) ln χ21− ρ− 1 (4.59a)s.t. χ1, χ2 > 0, |ν1| ≤ λ1, |ν2| ≤ λ2, (4.59b)‖χ1hˆ1 − η1 − ν1hˆ2‖2 ≤ λ12 + χ11, (4.59c)‖χ2hˆ2 − η2 − ν2hˆ1‖2 ≤ λ21 + χ22, (4.59d)|η1i| ≤ µi, |η2i| ≤ µi, i = 1, . . . , N. (4.59e)Then, we proceed with the same steps from the previous subsection and use Algo-rithm 4.1 to obtain the precoder W.138Chapter 4. Linear Precoding for the Two-User MISO BC-CM4.5 Numerical ExamplesIn this section, we provide numerical examples to demonstrate the performance of theproposed linear precoder, in terms of the achievable secrecy rate regions, subject topower or amplitude constraints. We also show the performance of the robust precoderunder different channel uncertainty levels along with different constraints.For simulation purposes, the elements of the channel gain vectors h1 and h2 (or hˆ1and hˆ2 for the robust case) are generated randomly (i.i.d. random variables) accordingto the standard normal distribution N (0, 1). To obtain the achievable secrecy rateregion, we generate 21 points, i.e., secrecy rate pairs (R1, R2), on the boundary ofthe region by solving the weighted secrecy sum rate maximization problem usingρ = 0, 0.05, 0.10, . . . , 1.00. The final results that we plot are obtained by averagingover 1000 realizations of the channel gain vectors. In all cases, the noise variance σ2is equal to 1 at both receivers.We begin with the case of perfect channel information and total power constraint.This case is particularly important as it is the only case for which the secrecy capacityregion is precisely known, and the boundary points can be calculated using a closed-form expression. This capacity region sets a benchmark that enables us to quantifythe loss incurred by using a suboptimal linear precoding scheme, and also to validatethe algorithm we use to obtain the linear precoder.In Figure 4.1, we plot the secrecy capacity region obtained with the optimalS-DPC scheme [48, Theorem 1], along with the secrecy rate region of the linearprecoder proposed in Section 4.3, subject to a total power constraint specified byPdB , 10 log10 PTot. We also include two other conventional linear precoding schemes,namely, the generalized eigenvalue (GEV) and the zero-forcing (ZF) schemes, forcomparison purposes.139Chapter 4. Linear Precoding for the Two-User MISO BC-CM0 0.5 1 1.5 2 2.500.511.522.5Figure 4.1: The secrecy capacity region obtained with optimal S-DPC along with thesecrecy rate regions of the GEV precoder, the precoder obtained with Algorithm 4.1,and the ZF precoder, subject to the total power constraint PdB = 10 log10 PTot. Thenumber of antennas N ∈ {2, 4}.The proposed linear precoder is obtained using Algorithm 4.1 along with the dualproblem (4.37). The dual problem is solved using the CVX toolbox [85] in conjunctionwith the MOSEK solver [86]. For Algorithm 4.1, we use δ(1) = (10−1, 10−1) or, equiv-alently, δ(1)dB = (−20 dB,−20 dB), as the initial point, and start searching with a fixedstep αFixdB = 1 dB. The maximum number of iterations after encountering a peak isL = 10, i.e., the final solution δ?dB is obtained with accuracy αFixdB /L = 0.1 dB. For theGEV precoder, the beamformers w1,GEV and w2,GEV are obtained as follows. Let v1be the generalized eigenvector of the matrix pair (σ2IN + PToth1hT1 , σ2IN + PToth2hT2 )140Chapter 4. Linear Precoding for the Two-User MISO BC-CMcorresponding to its largest generalized eigenvalue. Then,w1,GEV =√ρPTotv1‖v1‖2 .Similarly, we havew2,GEV =√(1− ρ)PTot v2‖v2‖2 ,where v2 is the generalized eigenvector of the matrix pair (σ2IN + PToth2hT2 , σ2IN +PToth1hT1 ) corresponding to its largest generalized eigenvalue. The ZF precoder isobtained by solving the inner problem (4.28), without the per-antenna power con-straint (4.28d), using δ1 = δ2 = 0. For all three linear precoders, the achievable ratepairs (R1, R2) are obtained by substituting with the precoder W into (4.10).Several interesting conclusions can be drawn from Figure 4.1. First, we notethat the GEV precoder yields slightly better performance than our precoder fromSection 4.3, especially at low power levels. This is due to the fact that we use thesimplified lower bound in (4.25) as the objective function of the weighted secrecy sumrate maximization problem, rather than the more complex expression in (4.22d). Thissuggests that the GEV is probably a good linear precoder when there is only a totalpower constraint (and channel information is accurately known to the transmitter).Note, however, that there is no counterpart of the GEV scheme for cases involvingper-antenna power or amplitude constraints. Particularly, unlike the cases in Sec-tion 3.5.1.1, where we simply scaled the beamformer wGEV to satisfy the lp-normconstraint for all p = 1, 2,∞, scaling the precoder WGEV , [w1,GEV w2,GEV] to sat-isfy the per-antenna power or amplitude constraints would significantly deterioratethe performance. Instead, w1,GEV and w2,GEV should be scaled by different factors,however it is unclear how to choose these factors in an optimal way. We also note141Chapter 4. Linear Precoding for the Two-User MISO BC-CM0 0.5 1 1.5 2 2.5 300.511.522.53Figure 4.2: Achievable secrecy rate regions of the proposed and ZF precoders subjectto total power constraint (TPC), per-antenna power constraint (PAPC), and ampli-tude constraint (AmC). PTot = NPi = NA2i , i = 1, . . . , N , PdB = 10 log10 PTot, andN = 4. The secrecy capacity region with optimal S-DPC is included for the case oftotal power constraint.from Figure 4.1 that the ZF precoder has the worst performance among all otherprecoders at all power levels. Performance gaps, however, significantly decrease asthe number of antennas or transmit power increases.In Figure 4.2, we show the achievable secrecy rate regions of the proposed lin-ear precoder, subject to the total power constraint (4.8b), the per-antenna powerconstraint (4.12), and the per-antenna amplitude constraint (4.14b). The secrecycapacity region obtained with optimal S-DPC (for the case of total power constraint)and the secrecy rate regions of the ZF precoder (for all constraints) are also included.The power level indicated in the figure specifies the total power constraint in dB,i.e., PdB = 10 log10 PTot. For comparison purposes, we choose the per-antenna power142Chapter 4. Linear Precoding for the Two-User MISO BC-CM0 0.5 1 1.5 2 2.5 3 3.500.511.522.533.5Figure 4.3: Worst-case secrecy rate regions with different channel uncertainty levels,1 and 2, subject to total power constraint (TPC), per-antenna power constraint(PAPC), and amplitude constraint (AmC). PTot = NPi = NA2i , i = 1, . . . , N , PdB =10 log10 PTot = 15 dB, and N = 4.constraint as Pi = PTot/N , and the amplitude constraint as Ai =√PTot/N , fori = 1, . . . , N . Thus, the per-antenna power constraint also implies the total powerconstraint, and the amplitude constraint implies the total and per-antenna powerconstraints. The number of antennas N is set to 4. As expected, the proposed linearprecoder outperforms the ZF precoder, under all constraints, however at the cost ofincreased computational complexity.Finally, in Figure 4.3, we plot the worst-case secrecy rate regions obtained withthe robust precoder considered in Section 4.4, subject to (4.8b), (4.12), and (4.14b),separately. Similar to the previous example, we choose Pi = PTot/N and Ai =√PTot/N , for all i = 1, . . . , N , where 10 log10 PTot = 15 dB and N = 4. The case1 = 2 = 0 designates perfect channel information, and is included for comparison143Chapter 4. Linear Precoding for the Two-User MISO BC-CMpurposes. As expected, we note from Figure 4.3 that increased uncertainty levelshave negative impact on the worst-case secrecy rate region.4.6 ConclusionsIn this chapter, we considered the design of linear precoders for the two-user MISOBC-CM subject to total and per-antenna power constraints, and also subject to am-plitude constraints. Per-antenna constraints are typically more difficult to handle, butthey are essential for modelling hardware limitations in practical systems employingmultiple transmit antennas. Although suboptimal, linear precoding is particularlyattractive because of low implementation complexity. On the other hand, the opti-mal S-DPC scheme is difficult to implement, and can be only found via an exhaustivesearch when per-antenna power constraints are taken into account. Furthermore, theoptimal scheme is unknown under amplitude constraints. Therefore, our proposedlinear precoding scheme provides a viable solution to an open problem that has notbeen addressed in the published literature.We formulated the linear precoder design problem as a weighted secrecy sumrate maximization problem that is transformed into a more tractable problem havingonly two optimization variables. We proposed a subgradient-based search algorithmto obtain a solution, and provided a condition under which the obtained solutionis guaranteed to be optimal. Our approach is applicable to any combination ofthe total power, per-antenna power, and per-antenna amplitude constraints. It isalso applicable to the robust design problem when channel uncertainty is taken intoaccount.We used the total power constraint case, in which the secrecy capacity region isprecisely known, to validate our approach and compare the performance of the linear144Chapter 4. Linear Precoding for the Two-User MISO BC-CMprecoder with the optimal S-DPC scheme. Numerical results show negligible losswhen the SNR is sufficiently high. Compared to the idealistic case of total powerconstraint and perfect channel information, the results show considerable reductionin the achievable secrecy rate region when per-antenna constraints and channel un-certainty are taken into account.145Chapter 5Conclusions and Future Directions5.1 ConclusionsPhysical-layer security has the potential to complement existing encryption tech-niques with an additional secrecy measure that is provably unbreakable regardlessof the computational power of the eavesdropper. It can also be a viable lightweightsecrecy solution under severe hardware or energy constraints. While physical-layersecurity has been an active research area for more than a decade, it still has notgot much attention from practical system designers. Perhaps the main reason forsuch a disregard is performance sensitivity to channel conditions. Particularly, theperformance of physical-layer security schemes can be severely degraded, and secrecyoutage may occur, if the design is based on inaccurate channel information.In this thesis, we proposed the use of physical-layer security techniques to enhancethe secrecy of visible-light communication (VLC) systems. We had a twofold purposefrom such a proposal. First, the broadcast nature of the VLC channel makes an addi-tional secrecy layer a sensible approach. Second, VLC links can be a reasonable plat-form for the deployment of physical-layer security prototypes as realistic assumptionsregarding channel information can be made in typical VLC scenarios with dominantLoS path. Furthermore, by adopting robust transmission schemes that take channeluncertainty into account, performance sensitivity to channel estimation errors can besignificantly alleviated. Although in this thesis we mainly focused on VLC systems146Chapter 5. Conclusions and Future Directionsfunctioning in indoor environments, the techniques we developed are also applicableto outdoor scenarios.Existing physical-layer security schemes assume Gaussian input distribution andtotal transmit power constraint, making them inapplicable to VLC channels whereinamplitude constraints on the channel input are inherent due to linearity limitationsof typical LEDs.Accordingly, in this thesis, we studied the design of physical-layer security schemesfor the Gaussian wiretap channel subject to amplitude constraints. Three majorcontributions have been presented:Firstly, with the lack of closed-form secrecy capacity expressions for the amplitude-constrained Gaussian wiretap channel, we utilized the maximum-entropy uniform in-put distribution, along with the entropy power inequality, to establish a closed-formlower bound on the secrecy capacity. We also developed a method to derive an upperbound on the secrecy capacity of degraded wiretap channels, and applied that methodto the scalar Gaussian wiretap channel. Then, we used the lower bound along withbeamforming to obtain a closed-form secrecy rate expression for the MISO wiretapchannel. We later used that expression as a performance metric for the beamformerdesign. We also derived a closed-form secrecy rate expression for the amplitude-constrained scalar wiretap channel when it is aided by a friendly jammer sendingartificial noise that is also subject to amplitude constraints.Secondly, we studied the design of beamformers for the MISO wiretap channelsubject to amplitude constraints. Unlike the case of total power constraint, whichis readily solvable as a Rayleigh quotient maximization problem, the design problemunder the amplitude constraint is more difficult to solve. Nevertheless, we trans-formed such a difficult problem into a quasiconvex line search problem that is easily147Chapter 5. Conclusions and Future Directionssolved with a bisection search. In addition, we showed that our solution techniqueis applicable to arbitrary lp-norm constraints on the beamformer. We also solvedthe worst-case secrecy rate maximization problem when channel uncertainties for thereceiver and eavesdropper are taken into account.Thirdly, we studied the design of linear precoders for the two-user MISO broadcastchannel with confidential messages (BC-CM). We developed a general approach thatcan handle the design problem subject to any combination of the total, per-antennapower, or amplitude constraints. Although suboptimal, our linear precoding schemeentails low implementation complexity. Furthermore, it provides a viable solution tothe cases of per-antenna power or amplitude constraints where there is no closed-formcharacterization of the secrecy capacity region. We formulated the design problem asa weighted secrecy sum rate maximization problem, then we transformed the probleminto a more tractable form that can be solved with an iterative search algorithm. Weused the case of total power constraint to quantify the performance loss incurredby using a suboptimal linear precoding scheme and also to validate our approach tosolving the design problem. We also considered the design of robust linear precodersto maximize the worst-case secrecy rate region when channel uncertainty is takeninto account.The numerical results revealed considerable decline in the achievable secrecy rateswhen channel uncertainty and amplitude constraints are taken into account as com-pared to the idealistic case of perfect channel information and total power constraint.Therefore, the design techniques we developed throughout the thesis provide valu-able tools for tackling real-world problems in which channel uncertainty is almostalways inevitable and per-antenna constraints are essential for accurate modelling ofhardware limitations.148Chapter 5. Conclusions and Future DirectionsFinally, it is worth mentioning that physical-layer security is a research area thathas its origins from information theory. On the other hand, the design of VLC sys-tems is mostly treated as a practical engineering problem. By combining these tworesearch areas in one thesis, we aim to serve both communities and help narrow thegap between information theory and practical system design. For example, amplitudeconstraints impose an inherent practical limitation that is difficult to treat mathe-matically, however we derived lower bounds on the secrecy capacity, subject to theseconstraints, in order to circumvent such a difficulty. Furthermore, by using realisticchannel gain models from VLC scenarios and linking the physical sources of chan-nel uncertainty (e.g., location or orientation uncertainty) with the uncertainty setsused in robust optimization problems, we help make the techniques from informationtheory and convex optimization more approachable to practical system designers.5.2 Future WorkSecure Transmission with Discrete Input Distribution:In Chapter 2, we derived closed-form secrecy rate expressions for the amplitude-constrained Gaussian wiretap channel based on the (continuous) uniform input dis-tribution. Then, we used the resulting expression for beamformer design in the MISOwiretap channel. Similarly, we used a secrecy sum rate expression based on the uni-form input distribution for linear precoder design in the two-user MISO BC-CM underamplitude constraints. We note, however, that the codewords or signals transmittedin practical communication systems cannot have a continuous distribution. Instead,they must be drawn from a discrete constellation, i.e., a discrete distribution withfinite support. One practical reason for such a limitation is the finite resolution of149Chapter 5. Conclusions and Future Directionsthe digital-to-analog converters (DACs) incorporated at the transmitter front-end.Although we know that the optimal input distribution for the amplitude-constrainedscalar wiretap channel is discrete (and we conjecture that this is also the case for theMISO wiretap channel and the two-user MISO BC-CM), there is no closed-form char-acterization of these optimal distributions and they can only be found via numericaltechniques. Since closed-form expressions are required for beamformer or precoderdesign, an interesting research direction is to find secrecy capacity-approaching dis-crete input distributions which yield closed-form secrecy rate expressions that areamenable to optimization.Linear Precoding for the MIMO Wiretap Channel underAmplitude Constraints:In Chapter 3, we considered the design of beamformers for the MISO wiretap channelunder amplitude constraints. Our approach in Propositions 3.1 and 3.2 took advan-tage of the fact that only one data stream is being transmitted (since the intendedreceiver has one antenna), and the signal term in the numerator of the secrecy rateexpression (3.5) is a squared linear function of the beamformer. This ultimately ledto convex formulations of the inner problems (3.11) and (3.33). If the intended re-ceiver, however, has multiple antennas, then simultaneous transmission of multipledata streams should be considered. Consequently, equalization may become neces-sary or desirable at the receiver as well as the eavesdropper. The resulting secrecyrate expression, however, will become more difficult to handle, and the inner prob-lems corresponding to (3.11) and (3.33) will no longer be convex. Thus, a naturalextension to the work presented in Chapter 3 is to consider linear precoding for theMIMO wiretap channel subject to amplitude constrains.150Chapter 5. Conclusions and Future DirectionsSimilar remarks can be made about the design of linear precoders for the two-userMIMO BC-CM when multiple data streams are simultaneously transmitted to eachuser.Linear Precoding for the Complex-Valued MISO WiretapChannel under Amplitude Constraints:Throughout the entire thesis, we assumed real-valued transmitted signals and channelgain vectors. Such an assumption is applicable to intensity modulation (IM) systems,as well as RF systems utilizing only amplitude modulation, i.e., the carrier phase isnot modulated. Extension to the more general case of complex-valued transmission,such as quadrature amplitude modulation (QAM), shall make the design problemsconsidered in Chapters 3 and 4 more difficult to handle. For example, with complex-valued channel gain and beamforming vectors, the inner problem (3.11), as well as itsrobust counterpart (3.33), would involve maximization of the magnitude of a complex-valued quantity. Obviously, this is a nonconvex problem, and thus the techniques wedeveloped via Propositions 3.1 and 3.2 will have to be modified in order to deal withnonconvexity of the inner problem.Finally, the problem of deriving achievable secrecy rate expressions for the complex-valued Gaussian wiretap channel subject to amplitude constraints can also be of greatinterest. Among various feasible input distributions that can be utilized, the circularuniform distribution sounds like a good candidate to begin with.151Bibliography[1] Z. Ghassemlooy, S. Arnon, M. Uysal, Z. Xu, and J. 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Then, the PDF of the sum Z , X + Yis obtained by the convolutionpZ(z) = (pX ∗ pY )(z)=z + a+ b4ab−a− b ≤ z ≤ −|a− b|,min{12a,12b}−|a− b| ≤ z ≤ |a− b|,−z + a+ b4ab|a− b| ≤ z ≤ a+ b,0 otherwise,(A.1)as depicted in Figure A.1 for the case b < a. For obvious reasons, the distribu-tion pZ(z) is referred to as the trapezoidal distribution. Note that the uniform distri-bution U [−a, a] is a special case of the trapezoidal distribution (A.1) when b→ 0.160Appendix A. The Trapezoidal Distribution a a1/2a pX(x)x b b1/2b pY(y)y  a+b)   a b) a b a+b1/2a pZ(z) = (pX*pY)(z)zFigure A.1: The trapezoidal distribution in (A.1) with b < a.The differential entropy of Z (in nats) ish(Z) = −∫pZ(z) ln pZ(z)dz=ln(2a) +b2ab ≤ aln(2b) +a2botherwise= min{ln(2a) +b2a, ln(2b) +a2b}. (A.2)In Figure A.2, we plot the differential entropy (A.2) as a function of b when a = 2.Note that h(Z) is differentiable for all b > 0, including the point b = a.161Appendix A. The Trapezoidal Distribution0 1 2 3 4 5 6 7 8 9 1000.511.522.533.544.55Figure A.2: The differential entropy (A.2) as a function of b when a = 2.162Appendix BProofs and Derivations for Chapter 3B.1 Proof of Lemma 3.1Let the pair (w?, t?) be an optimal solution of the nonconvex perturbed problemin (3.36), where t? ≡ ϕ(ε). If HB is convex, then the linear function fw?(hB) , hTBw?maps HB into an interval with three possible outcomes:i) If hTBw? ≥ 0 for all hB ∈ HB, then t? ≥ 0.ii) If hTBw? ≤ 0 for all hB ∈ HB, then t? ≥ 0. This also implies that −hTBw? ≥ 0for all hB ∈ HB. Note that if (w?, t?) is a solution to (3.36), then (−w?, t?) isalso a solution.iii) If there exist h1,h2 ∈ HB such that hT1w? > 0 > hT2w?, then t? = 0.From the above cases, we see that hTBw? ≥ 0, or hTBw? ≤ 0, for all hB ∈ HB, isa necessary condition to obtain nonzero t?. Thus, we lose nothing by imposing theconstraint in (3.37), provided that HB is a convex set. B.2 Components of h0 and J0 from (3.53)From (3.51), for i = 1, . . . , N , ci 6= 0, we have1cihi(δ) =(dz − δz)m(di − δ)Tu‖di − δ‖m+32, (B.1a)163Appendix B. Proofs and Derivations for Chapter 31ci∂hi(δ)∂δx=−(dz − δz)meT1 u‖di − δ‖m+32+(m+ 3)(dx,i − δx)(dz − δz)m(di − δ)Tu‖di − δ‖m+52, (B.1b)1ci∂hi(δ)∂δy=−(dz − δz)meT2 u‖di − δ‖m+32+(m+ 3)(dy,i − δy)(dz − δz)m(di − δ)Tu‖di − δ‖m+52, (B.1c)1ci∂hi(δ)∂δz=−m(dz − δz)m−1(di − δ)Tu− (dz − δz)meT3 u‖di − δ‖m+32+(m+ 3)(dz − δz)m+1(di − δ)Tu‖di − δ‖m+52, (B.1d)where ej, j = 1, 2, 3, is the jth column of the identity matrix I3. Substituting withδ = 0 back into (B.1), we obtainhi(0) = cidmz dTi u‖di‖m+32, (B.2a)∂hi(0)∂δx= ci( −dmz eT1‖di‖m+32+(m+ 3)dx,idmz dTi‖di‖m+52)u, (B.2b)∂hi(0)∂δy= ci( −dmz eT2‖di‖m+32+(m+ 3)dy,idmz dTi‖di‖m+52)u, (B.2c)∂hi(0)∂δz= ci(−mdm−1z dTi − dmz eT3‖di‖m+32+(m+ 3)dm+1z dTi‖di‖m+52)u. (B.2d)164Appendix CProofs and Derivations for Chapter 4C.1 Proof of Proposition 4.2Consider the unit vector u = [u1 u2]T, u1 ≥ 0, u2 ≥ 0, ‖u‖2 = 1, and let ϕu(t), t ≥ 0,denote the function ϕ from (4.29b) with its domain restricted to the line passingthrough the origin along the direction u, i.e.,ϕu(t) , ϕ(tu) = ϕ(tu1, tu2)= fu(t)− 12(ln(u21t2 + σ2) + ln(u22t2 + σ2)), (C.1)where fu(t) , f(tu). Our goal here is to prove that, for each u, there exists onepoint t? such that ϕu(t) is nondecreasing for t ∈ [0, t?] and nonincreasing for t ≥ t?,i.e., ϕu(t) is quasiconcave.Since f(δ1, δ2) is concave, its restriction to a line is also concave. As a consequence,fu(t) is continuous and twice differentiable almost everywhere, meaning that thereare only countably many points where f ′′u(t) may not exist [93, Chapter 13]. In orderto simplify the notation, we will first restrict ourselves to the points at which fu(t)is twice differentiable, then we will see that extension to all t > 0 is straightforward.Differentiating (C.1) w.r.t. t, we obtainϕ′u(t) = f′u(t)−(u21tu21t2 + σ2+u22tu22t2 + σ2). (C.2)165Appendix C. Proofs and Derivations for Chapter 4Further differentiation yieldsϕ′′u(t) = f′′u(t) + u21u21t2 − σ2(u21t2 + σ2)2+ u22u22t2 − σ2(u22t2 + σ2)2. (C.3)Let t? denote any point at which ϕ′u(t) = 0. Then, we need to show that there isonly one such point. Setting t = t? and substituting with ϕ′u(t?) = 0 in (C.2) yieldf ′u(t?) =u21t?u21t?2 + σ2+u22t?u22t?2 + σ2. (C.4)Using (C.3) and (C.4), ϕ′′u(t?) can be written asϕ′′u(t?) = f ′′u(t?) + (f ′u(t?))2 − 2u21u22t?2(u21t?2 + σ2)(u22t?2 + σ2)− u21σ2(u21t?2 + σ2)2− u22σ2(u22t?2 + σ2)2. (C.5)Now we will show that the sum f ′′u(t?) + (f ′u(t?))2 is always nonpositive, and thusϕ′′u(t?) is also nonpositive. To do this, we first need to show that efu(t) is a concavefunction. Let G(δ1, δ2) denote the optimal value of the perturbed problemmaximizeW(hT1w1)ρ(hT2w2)1−ρ (C.6a)s.t. |hT2w1| ≤ δ1, |hT1w2| ≤ δ2, (C.6b)‖W‖2F ≤ PTot, (C.6c)w21i + w22i ≤ Pi, i = 1, . . . , N. (C.6d)Since the objective function in (C.6a) is concave (see [80, Problem 3.16 (f)]), theperturbed problem (C.6) is convex, and thus G(δ1, δ2) is a concave function. Next,we note from (4.28) and (C.6) that G(δ1, δ2) = ef(δ1,δ2). Thus, Gu(t) , G(tu) = efu(t),166Appendix C. Proofs and Derivations for Chapter 4and we haveG′′u(t) = Gu(t)(f ′′u(t) + (f′u(t))2). (C.7)Since Gu(t) is concave, it holds that G′′u(t) ≤ 0 [80, Section 3.1.4]. Furthermore, sinceGu(t) is nonnegative, we must havef ′′u(t) + (f′u(t))2 ≤ 0. (C.8)Thus, f ′′u(t?) + (f ′u(t?))2 ≤ 0 and, consequently, ϕ′′u(t?) ≤ 0. The last inequality tellsus that ϕ′u(t) can experience zero-crossing only from positive to negative. Since thiscan happen only once, we conclude that there is only one point t? such thatϕ′u(t) ≥ 0 for t ≤ t?,ϕ′u(t) ≤ 0 for t ≥ t?.Hence ϕu(t) is quasiconcave.In order to extend the proof to include the points at which fu(t) is not differ-entiable, we just need to replace the derivative of fu(t) with any element from itssubdifferential. Specifically, since fu(t) is concave, it is continuous and has right andleft derivatives over the whole interior of its domain (i.e., for all t > 0) [81, Theo-rem 1.6]. Such derivatives are nonincreasing in the sense that, for any t2 > t1 > 0,we havef ′u(t−1 ) ≥ f ′u(t+1 ) ≥ f ′u(t−2 ) ≥ f ′u(t+2 ). (C.9)Now, at the points where f ′u(t+) 6= f ′u(t−), i.e., fu(t) is non-differentiable, we willallow f ′′u(t)→ −∞ and let f ′u(t) take any value in the interval [f ′u(t+), f ′u(t−)], mak-ing (C.8) hold for all t > 0. Thus, ϕ′′u(t?) is always nonpositive including, possibly,167Appendix C. Proofs and Derivations for Chapter 4ϕ′′u(t?)→ −∞. In other words, ϕ′u(t+) (or, equivalently, ϕ′u(t−)) can experience zero-crossing only from positive to negative, even if ϕ′u(t+) has jump discontinuity at thecrossing point. Following the same argument for the differentiable case, we concludethat ϕu(t) is quasiconcave for all t ≥ 0. C.2 Derivation of the Dual Problem (4.57)The problem in (4.55) can be reformulated asmaximizeW,M,z1,...,z4ρ ln z1 + (1− ρ) ln z2 (C.10a)s.t. hˆT1w1 − 1‖w1‖2 ≥ z1, (C.10b)hˆT2w2 − 2‖w2‖2 ≥ z2, (C.10c)|z3|+ 2‖w1‖2 ≤ δ1, (C.10d)|z4|+ 1‖w2‖2 ≤ δ2, (C.10e)‖m1‖22 + ‖m2‖22 ≤ PTot, (C.10f)m21i +m22i ≤ Pi, i = 1, . . . , N, (C.10g)w1 = m1, w2 = m2, (C.10h)hˆT2w1 = z3, hˆT1w2 = z4, (C.10i)where we have introduced the new variables M, z3, and z4, along with the equalityconstraints in (C.10h)-(C.10i).168Appendix C. Proofs and Derivations for Chapter 4The Lagrangian associated with (C.10) isL(W,M, z1, . . . , z4, χ1, χ2, λ1, λ2, γ,µ,η1,η2, ν1, ν2)= ρ ln z1 + (1− ρ) ln z2− χ1(−hˆT1w1 + 1‖w1‖2 + z1)− χ2(−hˆT2w2 + 2‖w2‖2 + z2)− λ1 (|z3|+ 2‖w1‖2 − δ1)− λ2 (|z4|+ 1‖w2‖2 − δ2)− γ (‖m1‖22 + ‖m2‖22 − PTot)− N∑i=1µi(m21i +m22i − Pi)− ηT1 (w1 −m1)− ηT2 (w2 −m2)− ν1(hˆT2w1 − z3)− ν2(hˆT1w2 − z4). (C.11)Rearranging the terms, and maximizing w.r.t. the primary variablesW,M, z1, . . . , z4,we obtain the dual functiong(χ1, χ2, λ1, λ2, γ,µ,η1,η2, ν1, ν2)= λ1δ1 + λ2δ2 + γPTot +N∑i=1µiPi+ maxw1((χ1hˆ1 − η1 − ν1hˆ2)Tw1 − (λ12 + χ11)‖w1‖2)+ maxw2((χ2hˆ2 − η2 − ν2hˆ1)Tw2 − (λ21 + χ22)‖w2‖2)+N∑i=1maxm1i(η1im1i − (γ + µi)m21i)+N∑i=1maxm2i(η2im2i − (γ + µi)m22i)+ maxz1(−χ1z1 + ρ ln z1) + maxz2(−χ2z2 + (1− ρ) ln z2)+ maxz3(ν1z3 − λ1|z3|) + maxz4(ν2z4 − λ2|z4|) . (C.12)The first maximization in the Lagrangian (C.12) is the conjugate of the l2-norm169Appendix C. Proofs and Derivations for Chapter 4function [80, Example 3.26], and is solved asmaxw1((χ1hˆ1 − η1 − ν1hˆ2)Tw1 − (λ12 + χ11)‖w1‖2)(C.13)=0 ‖χ1hˆ1 − η1 − ν1hˆ2‖2 ≤ λ12 + χ11∞ otherwise.(C.14)Then, after solving the other maximization terms in (C.12), which are similar tothose in (4.34), the dual problem (4.57) follows.170

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