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Physical layer security in MIMO power line communication networks Zhuang, Yifei 2014

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Physical Layer Security in MIMOPower Line Communication NetworksbyYifei ZhuangB.Eng., Tianjin University, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)August 2014c Yifei Zhuang 2014AbstractIt has been well established that multiple-input multiple-output (MIMO)transmission using multiple conductors can improve the data rate of powerline communication (PLC) systems. In this thesis, we investigate whetherthe presence of multiple conductors could also facilitate the communicationof confidential messages by means of physical layer security methods. In par-ticular, this thesis focuses on the secrecy capacity of MIMO PLC. Numericalexperiments show that multi-conductor PLC networks can enable a more se-cure communication compared to the single conductor case. On the otherhand, we demonstrate that the keyhole property of PLC channels generallydiminishes the secure communication capability compared to what wouldbe achieved in a similar wireless communications setting. Furthermore, weconsider the cases of unknown and partially known channel state informa-tion (CSI) about the eavesdropper channel. For this purpose, we providedeterministic channel uncertainty model parameters for PLC networks viathe bottom-up channel modelling method. Numerical results show how im-perfect CSI has a negative impact on secure communication, and enable usto analyze the tradeo↵ between choosing di↵erent transmission strategiesthat correspond to unknown CSI and partially known CSI.iiPrefaceThis thesis is based on the work conducted in the Department of Electricaland Computer Engineering at the University of British Columbia under theguidance and supervision of Professor Lutz Lampe.A version of Chapters 3 and 4 has previously been published in thefollowing conference paper.• Y. Zhuang, and L. Lampe, ”Physical layer security in MIMO powerline communication networks,” in IEEE Intl. Symp. Power Line Com-mun. (ISPLC), pp.272-277, Glasgow, UK, March/April 2014.I have transferred my copyright to the organizers of the conference above.However, I have retained my copyright for writing this thesis. I am theprimary author for the publication above. I have performed the majority ofthe work. Tasks include but are not limited to literature review, simulationdesign and coding, data analysis and manuscript editing. My supervisor Dr.Lampe provided invaluable guidance and help in a secondary role.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . x1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Multi-conductor Wiring . . . . . . . . . . . . . . . . . . . . . 11.2 Physical Layer Security . . . . . . . . . . . . . . . . . . . . . 41.3 Motivation and Scope . . . . . . . . . . . . . . . . . . . . . . 41.4 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 PLC Channel Characterization . . . . . . . . . . . . . . . . . 62.1 Wiring Code . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.1 Power Distribution . . . . . . . . . . . . . . . . . . . 72.1.2 Transmission Line Cables . . . . . . . . . . . . . . . . 82.2 PLC Channel Modelling . . . . . . . . . . . . . . . . . . . . . 82.2.1 Signal Propagation in PLC Channels . . . . . . . . . 102.2.2 P.u.l. Parameters . . . . . . . . . . . . . . . . . . . . 112.2.3 Transmission Line Basics . . . . . . . . . . . . . . . . 132.2.4 Bottom-up Channel Model Algorithms . . . . . . . . 142.2.5 MTL Channel Modelling . . . . . . . . . . . . . . . . 162.3 Noises in PLC . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19ivTable of Contents3 Physical Layer Security Solutions . . . . . . . . . . . . . . . 203.1 Physical Layer Security: State-of-the-art . . . . . . . . . . . 203.2 System Model - Secrecy Capacity of MIMO PLC . . . . . . . 233.3 Solutions to Secrecy Capacity – Perfect CSI . . . . . . . . . 253.3.1 The SISO Channel Case . . . . . . . . . . . . . . . . 263.3.2 The MISO Channel Case . . . . . . . . . . . . . . . . 263.3.3 The MIMO Channel Case . . . . . . . . . . . . . . . 273.4 Solutions to Secrecy Capacity - Unknown CSI . . . . . . . . 293.5 Solutions to Secrecy Capacity - Imperfect CSI . . . . . . . . 313.5.1 The MISO Channel Case . . . . . . . . . . . . . . . . 323.5.2 The MIMO Channel Case . . . . . . . . . . . . . . . 323.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 Physical Layer Security in MIMO PLC Networks . . . . . 354.1 Simulation Setting: Network Topology and Channel Model . 354.2 Comparing Secrecy Capacities in MIMO PLC Network . . . 384.3 Topology E↵ects on Secrecy Capacity . . . . . . . . . . . . . 394.4 A Comparison with Wireless Communication . . . . . . . . . 434.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455 Physical Layer Security in MIMO PLC with CSI Uncer-tainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.1 Capturing PLC Channel Uncertainty from PLC Topology . . 465.1.1 Mapping Unknown Eve’s Location to Channel Uncer-tainty . . . . . . . . . . . . . . . . . . . . . . . . . . . 475.1.2 Mapping Unknown Eve’s Impedance to Channel Un-certainty . . . . . . . . . . . . . . . . . . . . . . . . . 485.2 Impact of Partially Known CSI . . . . . . . . . . . . . . . . . 505.3 With Imperfect CSI - Choose The Best Transmission Strategy 515.4 With Imperfect CSI- Comparison of MISO and MIMO cases 525.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.1 Conclusions and Remarks . . . . . . . . . . . . . . . . . . . . 566.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 57Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58vList of Tables2.1 Cable parameters . . . . . . . . . . . . . . . . . . . . . . . . . 114.1 Transmission scheme . . . . . . . . . . . . . . . . . . . . . . . 39viList of Figures1.1 Outlet shapes around the world . . . . . . . . . . . . . . . . . 21.2 Illustration of MIMO transmission on multi-conductor wires. 32.1 Power line cable - symmetric cable . . . . . . . . . . . . . . . 92.2 Power line cable - ribbon cable . . . . . . . . . . . . . . . . . 92.3 Two-port network . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Transmission Line . . . . . . . . . . . . . . . . . . . . . . . . 132.5 Bottom-up channel modelling . . . . . . . . . . . . . . . . . . 162.6 An MTL segment . . . . . . . . . . . . . . . . . . . . . . . . . 173.1 Wyner’s wiretap channel . . . . . . . . . . . . . . . . . . . . . 213.2 Wiretap channels . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.1 Network topology . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Load circuit impedance . . . . . . . . . . . . . . . . . . . . . 374.3 Secrecy capacity in MIMO PLC network . . . . . . . . . . . . 404.4 Empirical CDF for secrecy capacity . . . . . . . . . . . . . . . 414.5 Secrecy capacity as function of wiretap position . . . . . . . . 424.6 Secrecy capacity for MIMOSE as function of impedance ofEve’s modem . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.7 A comparison for secrecy capacities in PLC and wireless net-works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 445.1 Mapping unknown eavesdropper’s location to PLC channeluncertainty, lm=0.2 m, at frequency 5MHz . . . . . . . . . 485.2 Mapping unknown eavesdropper’s location to PLC channeluncertainty, lm=1 m,at frequency 5MHz . . . . . . . . . . . 495.3 Channel uncertainty against Eve’s impedance . . . . . . . . . 505.4 Impact of partially known CSI on secrecy capacity . . . . . . 525.5 Secrecy capacity under unknown and partially known CSI, atfrequency 5MHz . . . . . . . . . . . . . . . . . . . . . . . . . 53viiList of Figures5.6 Secrecy capacity with perfect/unknown/partially known CSI 54viiiList of AbbreviationsPLC Power Line CommunicationMIMO Multiple Input Multiple OutputSISO Single Input Single OutputDC Direct CurrentAC Alternating CurrentLV Low VoltageMV Medium VoltageNEC National Eletrical CodeBER Bit Error RatePSD Power Spectral DensityCSI Channel State InformationSNR Signal to Noise RatioETSI European Telecommunications Standards InstituteITU-T International Telecommunication Union- Telecommunication Standardization SectorsMTL Multi Transmission LineCTF Channel Transfer FunctionOFDM Orthogonal Frequency Division MultiplexingBB-PLC BroadBand Power Line CommunicationAN Artificial NoiseixAcknowledgementsMy sincere gratitude goes to my supervisor Dr. Lutz Lampe, who has o↵eredme invaluable advice and strong support that have made this thesis possible.I have always been inspired by Dr. Lampe’s knowledgeable understandingof PLC. More importantly, his earnest attitude towards academic work hasspurred me to be serious and patient in not only research work but any othertasks in life.I feel lucky to have been exposed to the Communication Lab for twoyears, and surrounded by a group of talented colleagues, who have created afriendly and inspirational environment for me to study and exchange ideas.I would like to express my thanks to Mr. Florian Gruber, who has helpedme with the simulation of the channel modelling, to my colleagues Hao Ma,Ayman Mostafa, Jun Zhu, Ghasem Naddafzadeh Shirazi, who have sharedtheir research experiences and ideas with me. I would like to extend mygratitude to the people in Kaiser 4090; I enjoyed the journey a lot.I own many thanks to my parents, my aunt and my sister, who insistthat I should be happy and healthy no matter how busy and challenging lifeis.This work is funded by the Natural Sciences and Engineering ResearchCouncil Canada (NSERC).xChapter 1IntroductionPower Line Communication (PLC), which reuses power lines for the pur-pose of data communications, has a long history in serving electric power.Narrowband PLC applications, such as remote metering and control, voicecommunication, emerged very soon after the electrification of large parts ofNorth America and Europe [1, 2]. Nevertheless, it was only with the de-velopment of broadband PLC (BB-PLC) in the late 1990s, which enableshigh-speed in-home communications, that PLC technology started to at-tract attention in the general public. To support world wide deployment ofBB-PLC, international standards and specifications have been released. Forexample, the International Telecommunications Union-TelecommunicationStandardization Sector (ITU-T) and the Institute of Electrical and Electron-ics Engineers (IEEE) published ITU-T and IEEE 1901, respectively[3, 4]. In addition, the business association HomePlug Powerline Alliancecombined various PLC technologies to establish standards called HomePlug[5–7]. More recently, PLC has gained further momentum for its potential insupporting smart grid communications and smart home automation [8–10].However, the power grid was not originally designed for data transmissionand thus PLC has to cope with dicult signal propagation and noise condi-tions. Furthermore, and similar to wireless systems, PLC broadcasts infor-mation. Among other things, this makes PLC susceptible to eavesdropping.In this thesis, we focus on the analysis of counter-eavesdropping methodsfor BB-PLC for in-home network. Specifically, our research is based on thefollowing principles and methodologies.1.1 Multi-conductor WiringIn many countries, the so-called wiring code requires that the electricalwiring in residential and commercial buildings includes a third conductorprotective earth (PE) in addition to the phase (P) and neutral (N) wires.Figure 1.1 shows a collection of worldwide outlet shapes. It is noticeablethat the three conductor socket is very common globally. When it comes toexactly how common it is, the European Telecommunications Standards In-11.1. Multi-conductor WiringFigure 1.1: Multi-conductor wiring - Outlet shapes around the worldstitute (ETSI) did an investigation throughout the world [11], and reportedthat the PE conductor is installed in most of the outlets in western countries,all outlets in China and a small amount of outlets in Japan and Russia.While traditionally, PLC network has used the single “PN” conductorpair, forming a Single-Input Single-Output (SISO) channel, new PLC sys-tems make use of additional communication channels from the third conduc-tor pair, see Figure 1.2. According to Kirchho↵’s law, only two out of thethree transmission signals are independent, and thus the maximal spatialmultiplexing gain is two. Using three conductors to transmit two signalsopens a way to perform spatial processing, i.e. MIMO techniques, whichare already well established in wireless networks. Generally speaking, M21.1. Multi-conductor WiringFigure 1.2: Illustration of MIMO transmission on multi-conductor wires.conductors can facilitate M1 independent channels between a transmitterand a receiver. As a result, more than two wires for communication overpower lines, which is often referred to as MIMO PLC, should help to improvethe capacity and coverage of PLC networks.Analytical and experimental results have already demonstrated the sig-nificant benefits of MIMO PLC. For example, the calculation in [12] sug-gested that the in-home PLC channel capacity through the MIMO techniqueis increased almost 85% compared to the SISO case. The authors of [13]studied the e↵ect of noise correlation on the achievable rate and obtained asmuch as 40% rate increase for a scenario with two transmission and threereceiver ports. ETSI also launched large scale campaigns and measurementsto collect MIMO PLC channel characteristics and to demonstrate its com-munication ability [14, 15].The MIMO PLC technique has also been adopted by international stan-dards organizations in recent years. For example, ITU-T published theRecommendation G.9963 in 2011 [16] that includes MIMO transceiver ex-tension as an update to its first release in the ITU-T Meanwhile, theHomePlug Powerline Alliance also included MIMO signal processing as partof its HomePlug AV2 specification [7].31.2. Physical Layer Security1.2 Physical Layer SecurityConventional schemes for secure communication apply cryptography at theapplication layer. An alternative to this, providing secure communicationin an information-theoretic sense, was first presented by Wyner [17]. Thisso-called physical layer security is characterized by a secrecy capacity, whichis the maximal transmission rate that can be achieved without leaking in-formation to an eavesdropper, see e.g.[18].While it was assumed in Wyner’s wiretap channel model that the eaves-dropper channel is a degraded one compared to the legitimate channel, suc-cessive works studied non-degraded broadcast channel cases [19]. Thereare also results of physical layer security combined with a wide range oftechniques and applications [20–24]. For example, [20] investigated the roleof multiple antennas in secure communication. [21] evaluated Quality-of-Service (QoS) of secure transmission by means of physical layer security andan Artificial-Noise (AN) approach. [24] suggested that physical layer secu-rity is a promising solution for fast and robust wireless smart grid networks.Others probed into using physical layer security for wireless authentication[25], cognitive radio networks [26], ad-hoc networks [27], etc. More technicalbackground information about the state-of-the-art of physical layer securitywill be introduced in Chapter 3.The advantage of physical layer security lies in its relatively lower require-ments of the computation capacity at communication devices, and the strongability in continuously generating key srteams. Therefore, it is promising tostrengthen or even replace the higher layer cryptography methods. Particu-larly, in ad-hoc and decentralized networks where cryptographic protocol istoo complex, physical layer security technique becomes a potential candidatefor solving security problems.Recently, [28] presented first results on secure communication in PLC.The authors focused on the SISO case, and it has been shown that secrecycapacity in PLC is fairly low compared to wireless communications, espe-cially in the high signal-to-noise ratio (SNR) regime.1.3 Motivation and ScopeInspired by the benefit of MIMO PLC in terms of data rate, in this workwe investigate whether the availability of multiple conductors also improvessecure communication by means of physical layer security method. We referto this as physical layer security for MIMO PLC, bearing in mind that41.4. Organizationwe may not use all the signals for data transmission, i.e., we may not seeany multiplexing gain. For numerical examples, we concentrate on the caseof three conductor power cables, which are very common in in-home orindustry environments in many countries, see Section 1.1. To the best ofour knowledge, this is the first time that the MIMO PLC physical layersecurity is studied.The first evaluation is under the assumption that perfect channel stateinformation (CSI) is available at legitimate parties. We review known resultsand select the algorithms that best suit the situation in our research. Thenwe demonstrate secrecy capacity performance in MIMO PLC networks, an-alyze its unique features in PLC and provide a comparison to wireless cases.However, in practice, it is very likely that CSI about the eavesdropper’schannels is not available. Therefore, we do the further investigation underimperfect eavesdropper CSI.1.4 OrganizationThe remainder of this thesis is organized as follows. In Chapter 2 we intro-duce power line topology elements and PLC channel modelling methods. InChapter 3 we review secrecy capacity solutions that are transferrable to ourproblems. Chapter 4 applies the known algorithms to MIMO PLC networks.Chapter 5 considers the case of imperfect eavesdropper CSI and shows itsimpact on the system. Finally, the conclusion and suggested future work ofthis thesis are provided in Chapter 6.5Chapter 2PLC ChannelCharacterizationThe development of PLC technologies highly depends on a profound un-derstanding of the communication channel and its characteristics, includingsignal propagation, channel transfer function, noise, etc. In this chapter, wefirst introduce wiring practices in North America, including electric powerdistribution and the power line cable composition. This gives us funda-mental ideas about the power line topology. Next, we review PLC channelfeatures such as signal propagation and highlight the channel modellingmethods, which play a critical role in any further research of PLC networks.While a simple approach of PLC channel modelling used in early days isto obtain the channel transfer function (CTF) via field measurements, alot of recent work endeavours to express PLC CTF using network topologyelements, namely bottom-up channel modelling method. We compare thetwo methods, and then introduce a particular channel modelling methodfor MIMO PLC networks, which is based on Multi-conductor TransmissionLine (MTL) theory. We further explain why we adopt this channel mod-elling method as the underlying channel simulation algorithm in this thesis.Finally, we briefly introduce noise characteristics in PLC networks.2.1 Wiring CodeElectrical wiring follows regulations and standards. In the United States,the standard called National Electrical Code (NEC) is sponsored by theNational Fire Protection Association. In Canada, a similar electrical wiringpractice standard is used that complies with Canadian Electrical Code. Thewiring code provides basic rules for designing and installing electric wiresin buildings. For example, NEC gives the regulations of the colours ofphase, neutral and grounding wires, the calculation of circuit loading, thewiring methods and the wire types in a residential building. Furthermore,wiring code variations between di↵erent countries are still common today,62.1. Wiring Codebecause wiring practices were developed in parallel in di↵erent countriesdating back to early last century. For example, the installation and dis-tribution of grounding wire in the United State is a major di↵erence withthat in other nations. To better understand the signal propagation featuresand channel modelling methods in PLC, it is necessary to investigate basicwiring practices. In the following we particularly examine how electric poweris distributed to customers and what makes up the typical power cables.2.1.1 Power DistributionThe technique of using transformers to step-up the voltage and then convertit down in a few stages is regarded as one of the most innovative revolutionsin electrical history and has been in use since last generation [29].Multiple phase conductors are adopted in power grid systems, whichis often referred to as the polyphase system [30]. The three-phase electricpower system is the most common one, the first of which was built in 1891in Frankfurt [31]. Its prevalence was achieved in a more economical way:three-phase setting uses less conductor materials to transmit electrical powerthan single-phase or two-phase systems.At the final distribution transformer, however, not all of the phases,if there exist, will go into buildings. Usually only a single phase will bepresented at homes and oces. In North America, three phase circuitsare used in commercial constructions, but most household loads are onlyconnected as a single phase. In residential buildings, the three-phase powermight feed a multiple-unit apartment block, but each unit is only fed byone of the phases. In lower-density areas, only a single phase might be usedfor power distribution. An overview of the power delivery from high voltagepower grid to residential and commercial building is shown in Figures 2.1and 2.2 in [32].At an in-home environment, electric devices couple to what is called byNEC as outlets to obtain electric power. The phase wire, together withthe neutral and the protection earth (PE) wires forms a three-conductorstructure, which as stated in Section 1.1 is very common around the world.The electric functionality of the three conductors in an outlet is as follows.• N - Neutral wire provides the return current path of the circuit.• P - Phase wire, or Hot wire possesses electric potential relative to theelectrical ground or the neutral wire. It is the conductor that connectsto an electrical system.72.2. PLC Channel Modelling• PE - Protection Earth or Ground is a conductor with continuity tothe earth. In NEC, the PE wire keeps the exposed metal surface ofthe electric device at earth potential. NEC does not allow current inPE conductor under normal circumstances. In the event of a fault onthe hot wire, currents will flow to protect the circuit.2.1.2 Transmission Line CablesThe power line cable is made up of conductors in the central part of thecable to convey electricity, and insulators outside to support the conductorsand endure voltages at both normal conditions and possible outages.The insulation layer is usually very thick to o↵er a sound protection ofthe voltage. Sometimes in a high voltage application, a thin conductor shieldlayer is presented outside of the insulated layer that connects to the earthground and plays as an equalization to the voltage pressure in the insulationlayer. In practice, the following features are demonstrated:• Bundled conductor is a popular structure. It reduces power loss, es-pecially at high voltage range. In a bundled wire, multiple transmis-sion conductors placed in parallel that follows certain structures. Thestranded conductor deployment can be in symmetric or ribbon struc-tures.• The insulator materials are usually wet-process porcelain or toughenedglass. Typical types of insulator are pin-type or suspension type.Most North American residential and commercial buildings use non-metallicsheathes (NM) cables. Figures 2.1 and Figure 2.2 show typical three con-ductor cable architectures of modern power line cables, with the symmetricconductor structure and the ribbon conductor structure, respectively. Asmarked from the cable architecture pictures, the area with solid dark stu↵-ings in the central part of the cable presents the core conductors. The thinconductor layer outside of the cable is the copper sheath which plays therole of the high voltage equalizer. Other areas are dielectric materials.2.2 PLC Channel ModellingThere are two major categories in PLC channel modelling: top-down ap-proach and bottom-up approach. Top-down approach regards the channelas a black box and proposes a parametric model initially. It then launcheslarge scale measurements to find suitable parameters to fit the model, see82.2. PLC Channel ModellingFigure 2.1: Three conductor power line cable - symmetric cableFigure 2.2: Three conductor power line cable - ribbon cablee.g. [33, 34]. The top-down approach requires little computation but lacksof knowledge about inner relationships of the network, and thus is dicultto extend to other networks. Besides, measurement errors are unavoidable.Later on, as more PLC channel modelling results are carried out, re-vealing profound scientific phenomena behind the power line transmission,people turned to use the bottom-up channel modelling approach, which ex-presses the frequency response of the channel through the components ofthe network topology, such as loads, cable lengths and properties, etc. It ismore flexible and extendable to other networks in that the parameters arewell formulated. The network configuration is easy to be modified and thusa lot of di↵erent situations can be examined and even compared to eachother. The benefits of bottom-up channel modelling method are at the costthat the algorithm is hard to build up and it requires more computation.MIMO PLC networks have a lot more parameters and are potentiallymore complicated than SISO PLC networks. There is a lot of room to92.2. PLC Channel Modellingmanipulate the parameters and to investigate all the possible phenomena itis able to o↵er. Therefore, we use bottom-up approach to model the MIMOPLC channel in this thesis.In the following, we first examine the underlying signal propagationmechanism in PLC networks. Then we introduce basic knowledge of elec-tricity transmission: including per unit length parameters and transmissionline theory, based on which we go through the selected channel modellingalgorithm that we will be using in our later experiments.2.2.1 Signal Propagation in PLC ChannelsSimilar to wireless signal propagation, the power line signal also endures at-tenuation, which is mainly caused by signal reflection at terminals of thebranches. The forward-moving radio frequency signal waves will be re-flected due to the mismatch. This can happen between mismatched loadimpedances and the signal generator, or load impedances and the wires.On the other hand, signal loss also comes from radiation emissions, or theelectric field leakage that is caused by the skin e↵ect (see Section 2.2.2)or dielectric losses. Signal attenuation characteristics over the power linewere traditionally obtained through representative building measurementsbecause of the unknown specific wiring system in a building as well as theunavailable knowledge of the types and locations of the loads. For example,in 1980s the authors of [32] did the measurement of the power grid systemin a particular building over the frequency band 20-240 KHz. Later on,[35] extended the experimental frequency range up to 30MHz. [36] showedthrough measurement results that both the cable length and the branchnumber will cause the signal attenuation. But the number of the branch cir-cuit has a stronger impact on the signal attenuation compared to that of thecable length factor. Although the experimental setting is the same, signalattenuation varies in di↵erent measurements because of time and loads. Thecharacteristics of the signal attenuation therefore may largely depend on thefield of the measurement. Since the field measurement is neither able to cap-ture enough characteristics, nor extendable to the results of other cases, inrecent years people are working on constructing flexible signal attenuationmodels [37][36].PLC signals also have multi-path property. Signals transmitting throughthe power cable propagate along the shortest path. Additional paths comingfrom echoes merge at the receiver. Multi-path is one of the main reasons tocause bit errors in PLC.Without loss of generality, in modern PLC system analysis the coupled102.2. PLC Channel ModellingTable 2.1: Parameters for the three conductor symmetric cable, with rw:radius of the solid core conductor; d: distance between two conductors;ns: total number of strands in a conductor core; ne:number of strands;d: conductivity of the dielectric in conductors; c: conductivity of theconductor. From [38]Type rw / d (mm) ns / ne c / d (S/m)Symmetric 0.69 / 3.1 16 / 6 5.8 ⇥107 / 105energy into the wire is usually assumed to propagate in the transverse elec-tromagnetic (TEM) or quasi-TEM mode because the wavelength is longerthan the transversal size of the cable throughout the frequency range con-sidered.2.2.2 P.u.l. ParametersPower line per unit length (p.u.l.) parameters, i.e. the R,L,C,G parame-ters, can be analytically calculated through the geometry information andthe conductor material properties of the cable. Considering the TEM orquasi-TEM mode based signal propagation in a symmetric three conductorcable (see Figure 2.1) with the electrical constants presented in Table 2.1,further assuming homogeneous dielectric, we can calculate the p.u.l param-eters as follows.When current propagates along the power cable, more of them tends toflow on the outside of the wire instead of towards the central area due to theexistence of the self inductance. This phenomenon is called the skin e↵ect[39]. Skin depth  is written as: = 1p⇡µf (2.1)where f is the frequency, µ is magnetic permeability, and  is the conduc-tivity.Assuming that all the current is flowing within the skin depth of thecable transverse, we write the p.u.l resistance R parameter as [40]:R = 1⇡r2w when  >> rwR = 12⇡rw when  << rw (2.2)112.2. PLC Channel Modellingwhere the meaning of the parameter rw is explained in Table 2.1. It isnoted that the above calculation is based on a solid composition of a singlestrand conductor wire. When multiple stranded conductor is used, e.g. seeSection 2.1.2, the current flow area is actually less than the whole skin deptharea because of the gaps between the neighbouring conductor strands. [41]proposed a solution by introducing a coecient XR to the calculation of(2.2), which is the ratio of e↵ective area and the whole skin depth areawhen the conductor is solid:XR = e↵ective areatotal area = ne cos1 ⇣ rsrs ⌘ r2s  (rs  )qr2s  (rs  )22rw (2.3)see Table 2.1 for the description of parameters ne and rs. For simplicity, rsand rw has the relation: rs ⇡ rw/pns.While calculating the resistance of the p.u.l. parameter takes the cablematerial property and the cable geometry into account, the calculation ofthe L,C,G parameters depends more on the geometry factor. Sometimesthe values of L,C,G parameters are obtained through empirical results.Analytically, the p.u.l. inductance L can be calculated as [40]:L = µ0⇡ log✓ drw◆ (2.4)Based on the homogeneous dielectric insulator surroundings assumption,the relationship between the inductance L and the p.u.l. capacitance C, andG holds [42]: LC = µ"LG = µd (2.5)where " is the insulator material dielectric constant and can be expressedas the product of the relative dielectric constant of the material "r and thevacuum permittivity "0: " = "r"0.According to the transmission line theory and the TEM propagationmode assumption, the characteristic impedance Zc and the propagation con-stant  of each segment can be calculated based on the p.u.l. parametersobtained above: Zc = sR+ j!LG+ j!C = p(R+ j!L) (G+ j!C) (2.6)where ! is the frequency (rad/s).122.2. PLC Channel ModellingPort A Port B Transmission line + +   I1 I I I1 V1 V Figure 2.3: An illustration of two-port networkFigure 2.4: Transmission line element circuitFor a ribbon cable, see 2.2, it is important to define the reference con-ductor, based on which the p.u.l. parameters can be similarly calculatedanalytically [40].2.2.3 Transmission Line BasicsBased on Maxwell’s Equations, Oliver Heaviside [43] created the transmis-sion line model, and developed the telegrapher’s equations, which depict therelation of the voltage and current on an electric transmission line in termsof the distance and time. Oliver’s theory is applicable to the simplified twoconductor case. In the following we introduce the two conductor case and inSection 2.2.5 we use the extended multi-conductor version to help calculatethe multi-conductor CTF.In Heaviside’s theory, the transmission line is first considered as a seriesof two-port networks that are made up of small transmission line segments,see Figure 2.4. Two-port network model, shown in Figure 2.3, is the elemen-tary analytical unit in a black-box form and can often simplify the circuitcalculation process.132.2. PLC Channel ModellingIn Figure 2.4, the voltage V (x) and current I (x) of the line segmentcircuit(x) follow the expression [43]:@V (x)@x = (R+ j!L)I(x)@I(x)@x = (G+ j!C)V (x) (2.7)where R,L,C,G are cable parameters (see Section 2.2.2) for the segment,! is the frequency (rad/s). When the cable segment x is small enough, thetransmission line is considered in a lossless structure. Based on the cablep.u.l. parameters and the transmission line theory, the CTF of a circuit canbe calculated in the following..2.2.4 Bottom-up Channel Model AlgorithmsAs introduced in Section 2.2, the bottom-up channel modelling approachmakes use of the cable properties and the network topology, based on thep.u.l. parameters and the transmission line theory (see Sections 2.2.2 and2.2.3), to assemble the transfer function of the whole network. We intro-duce two algorithms here, one is the simple and classic ABCD model basedalgorithm developed by [44], the other algorithm called impedance carryback method was developed by [40]. The latter inherits the former method.While only the two-conductor case is considered here, it is easy to extendto the multi-conductor case in Section 2.2.5.ABCD ApproachABCD model is developed based on the two-port network, as shown inFigure 2.3, such that the transfer function can be calculated conveniently[44]. In Figure 2.3, the relation between the port A voltage and current pair[V1, I1] and the port B one [V2, I2] follows the equation:V1I1 = A BC D V2I2 (2.8)where A,B,C,D are constants chosen to satisfy the steady state of the circuit.The product of the ABCD matrices for a series of cable segments thenbecomes the CTF. Specifically, for a cable segment with small enough cablelength l, the elements in the ABCD matrix can be written as function ofp.u.l. characteristic impedance Zc propagation constant  (see (2.6)), and142.2. PLC Channel Modellingthe A BC D =  cosh(l) Zc sinh(l)1Zc sinh(l) cosh(l)  (2.9)Since Zc and  are frequency selective, ABCD constants are also function ofthe frequency. Therefore, at one frequency point, for a larger network witha few two-port networks chaining up, the overall ABCD matrix for the twoend networks is the product of ABCD matrices in between. It is done asfollows.From (2.8), it is easy to calculate the transfer function H (frequencyomitted): H = VLVs = ZLAZL +B + CZLZs +DZs (2.10)and the input impedance Z1Z1 = V1I1 = AZL +BCZL +D (2.11)where ZL is the network load impedance.When a two-port network segment with a bridge tap is presented, wecan calculate the equivalent impedanceZeq = ZcZbr + Zc tanh(brdbr)Zc + Zbr tanh(brdbr) (2.12)which can replace the particular segment as a corresponding ABCD ma-trix. Zbr and br are the characteristic impedance and the propagation con-stant parameters for the bridge tap circuit, respectively. The ABCD modelbased bottom-up channel modelling is simple and straightforward. However,the formulation becomes very complex when the network topology becomeslarge. This is why the impedance carry-back method was proposed.Impedance Carry-back MethodThe impedance carry-back channel modelling method [40] optimizes theABCD model based method (see Section 2.2.4). The basic idea is as follows:the network load is repeatedly updated by plugging in the equivalent bridgetap impedances (see (2.12)) to the current outlet, until the transmitter outletis arrived. The transfer function for each step is recorded and the overalltransfer function will be the product of these transfer functions in order.As illustrated in Figure 2.5, load impedance at node n1 is firstly updatedby adding equivalent impedance of outlet o1 and o2 to n1. The equivalent152.2. PLC Channel ModellingFigure 2.5: Bottom-up channel modelling method: impedance carry backmethod, from [40]circuit after first carry back will be Figure 2.5(b). Then n1 and o3 can alsobe carried back to the node n3. Finally, the backbone is reached.It is noted that the load impedance for the current circuit is the summa-tion of the equivalent impedances carried back from all of the outer branchesto the terminal of this circuit, plus the load impedance at this point.CTF as the insertion loss between the transmitter node and the receivernode is calculated by multiplying the voltage ratios between each unit (fre-quency omitted): H = V0VN+1 = ⇧N+1b=1 Hb (2.13)where Hb is the insertion loss for each step of the impedance carry back.The impedance carry-back method handles scaler elements rather than ma-trices. Therefore, it alleviates computational complexity compared withABCD based method.2.2.5 MTL Channel ModellingWe now extend the channel modelling for the two conductor case to theMTL case. In particular, we adopt the algorithm proposed by [38], which iscased on the TEM signal propagation mode, the MTL theory [42] and theimpedance carry-back method (see Section 2.2.4).We consider the three conductor cable segment in Figure 2.6. Thep.u.l. parameters for the multi-conductor transmission line are the matrixextensions of the scalar form R,L,C,G (see Section 2.2.2). For example,162.2. PLC Channel ModellingFigure 2.6: A Multi-transmission-line segment, from [38]R,L,C,G in Figure 2.6 can be written as:R = rg + r0 r0r0 rr + r0L =  lg lmlm lr C = cg + cm cmcm cr + cmG = gg + gm gmgm gr + gm (2.14)where r0, rg, rr, lr, lg, lm, cr, cg, cm, gr, gg, gm, are the p.u.l. parametersfor each conductor or between conductor pairs as indicated in Figure 2.6.Following the same rule of the impedance carry back method (see Section2.2.4), we can remap the complex layout, break it down to unit segments andcalculate the voltage ratio iteratively. The overall CTF will be the productof each segments’ CTFs from the root to the leaf segment. We include theresult as follows [38]:The impedance and admittance matrices of the cable regarding theRLCG parameters and the frequency point f are defined as (omitted indexf): Z = R+ j2⇡fLY = G+ j2⇡C (2.15)Applying the eigenvalue decomposition to the matrix Y Z, we get:Y Z = T 1 00 2T1 = T⇤T1 (2.16)172.3. Noises in PLCwith T as the eigenvector matrix and ⇤ as the eigenvalues. The char-acteristic impedance matrix is defined as ZC = Y 1TT1, where  =diag{1, 2} is the propagation constant matrix satisfying  = ⇤. Thenthe reflection coecient matrix can be expressed as:⇢LI = T1YC(YL +YC)1(YL YC)ZCT (2.17)where YC = Z1C . YL is the load admittance. If we write YR as theequivalent load admittance folded back to the coordinate x, we also haveY R(x) = T (ex + ex⇢LI )⇥ (ex  ex⇢LI)1T1YC (2.18)The MTL CTF for the k-th segment in the topology can be expressed as:Hk = ZCkT k(U  ⇢LI,k)⇥ (elk  elk⇢LI,k)1T1k Z1Ck (2.19)After obtaining the voltage ratio between each line segment, the overallCTF is calculated as the product of each voltage ratio.2.3 Noises in PLCPower line noises a↵ect the transmission quality of BB-PLC significantly.According to literatures, four types of noises are usually presented in atypical PLC network [1]:• Colored background noise: It comes from low power sources such asuniversal motors. Power spectral density (PSD) of the backgroundnoise is frequency dependent and increases as the frequency goes higher.• Periodic impulse noise: The generator of the periodic impulse noiseis usually the electric devices that create 50Hz or 100Hz harmonics.Periodic impulse noise can be both synchronous or asynchronous tothe electricity frequency.• Narrow band noise: It usually comes from periodic sinusoidal signalsthat are modulated over amplitude.• Asynchronous impulse noise: It is bursts and peaks that are obviouslygreater than the average amplitude of the whole wave. The majorsources of the impulse noise are usually external devices. Other factsthat create impulse noise include silicon-controller rectifiers, switchingdevices and electric motors.182.4. SummaryIt is verified that background noise in PLC is not white Gaussian, andspecific background noise models have been proposed [44, 45]. PLC noises,especially impulse noises, generate a hostile environment for data transmis-sion, and also propel the development of communication techniques. For ex-ample, orthogonal frequency-division multiplexing (OFDM) is considered tobe e↵ective in mitigating the influence of PLC impulse noises [46], becauseit spreads the interferences of impulse noises over several symbols as thediscrete Fourier transform proceeds. [45] found that the PLC backgroundnoise follows the Nakagami distribution, and proved that the performanceof OFDM system with Nakagami-distributed noise is the same as that withadditive white Gaussian noise. In this thesis, we adopt OFDM and considerthe noise being white Gaussian in each OFDM subband.2.4 SummaryIn this chapter, we briefly introduced what is regulated by the wiring code,including the power distribution and the electric power cables. We thenhighlighted the PLC channel modelling methods. It is usually assumed thatthe signal propagation in PLC follows the TEM mode, based on which twobottom-up channel modelling algorithms were introduced: the ABCD basedapproach and the impedance carry back approach. The latter was furtherdeveloped to the MTL case and is adopted for the channel simulation in thisthesis. Finally, we introduced noise features in PLC.19Chapter 3Physical Layer SecuritySolutionsIn this chapter, we review the origin and the development of physical layersecurity. We go through typical wiretap channels where physical layer se-curity are applied to, and discuss the applications where physical layersecurity has demonstrated its potential capability. Then we explain howphysical layer security is performed over MIMO PLC networks, and for-mulate the problems of this thesis. Based on the specific situation in ourproblem, we review literatures and select suitable solutions. The rate anal-ysis for secure SISO/MISO/MIMO transmission will be presented with per-fect/unknown/partially known CSI conditions.3.1 Physical Layer Security: State-of-the-artConventionally, information security or privacy is a characteristic of higherlayers of the OSI/ISO stack model. The security functionality is imple-mented by applying cryptographic protocols over the application layer as-suming that the lower layers have been well established and a perfect (errorfree) link is available. However, cryptography is expensive and complex, es-pecially in dynamic networks. Moreover, it is assumed that ciphers can notbe cracked, but this assumption has not been proved yet. As a consequenceof the fast growth of the computation ability in recent years, cyber attacksbecome more and more common [47].As a matter of fact, it is also technically possible to achieve or enhancesecure communication at physical layer, which is called physical layer se-curity. Continued and tremendous interests are demonstrated in physicallayer security because of new potentials it has brought into communicationnetworks: it requires little computation capability at the communicationdevices, and thus is applicable to devices that lack computational resources.Generating a shared secret key between the legitimate users based on thesecure information that is not presented at the eavesdropper is very ecient203.1. Physical Layer Security: State-of-the-artFigure 3.1: Wyner’s wiretap channel model [17]and useful. In particular, it enables the key exchange in a more frequentway, allows for one-time pad encryption, and does not depend on any otherparties to distribute the key [48]. Therefore, in some new wireless networkmodels such as the popular ad-hoc and the decentralized networks wherecryptographic protocol is too complex, physical layer security becomes apromising candidate for providing security in these networks.Physical layer encryption is supported by Shannon’s theory of perfectsecrecy. Wyner proved its feasibility in his work [17], where the wiretapchannel model was built, as shown in Figure 3.1. In Wyner’s wiretap chan-nel, the transmission takes place over the discrete memoryless channel withan eavesdropper, which is assumed to have a degraded channel version com-pared to that of the legitimate users. Based on such settings, Wyner provedthe possibility of secure communication between legitimate nodes with thepresence of the eavesdropper. The data rate at which the legitimate usercan exchange messages but the presented eavesdropper only obtains zeroinformation is defined as an achievable secrecy rate. Correspondingly, themaximum secrecy rate is termed as secrecy capacity. Wyner proved thatthere exists a positive secrecy rate Cs such that reliable transmission at arate up to Cs is possible.While Wyner constructed a degraded version of the wiretap channel,subsequent works proposed modified the wiretap channels to fit other appli-cation scenarios, see Figure 3.2, enumerated as follows.• Wyner wiretap channel: The type of wiretap channel proposed byWyner [17] is also known as degraded channel, see Figure 3.2a. Thethree parties are referred to as Alice, Bob and Eve, respectively. Aliceand Bob are communicating via the main channel (channel A block)while the eavesdropper Eve is wiretapping the channel through the213.1. Physical Layer Security: State-of-the-art(a) Wyner wiretap channel(b) Csisza´r and Ko¨rner wiretap channel(c) Keyhole wiretap channelFigure 3.2: Three typical wiretap channels, based on [28]other channel E after which the eavesdropper can reach the legitimatechannel. That is to say, Eve always has a worse channel condition,with more noise, or more degraded channel than Bob.• Csisza´r and Ko¨rner wiretap channel: [19] studied the broadcast chan-nel which is considered to be a general extension of Wyner’s degradedchannel, see Figure 3.2b. The legitimate channel (channel B) andthe eavesdropper’s channel (channel E) are independent of each other.This is suitable to model the star structure of PLC network where nowire segments are shared by other receivers. It can also present thewiretap channel in a typical wireless network.• Keyhole channel: This model, see Figure 3.2c, includes a commonsegment of the legitimate channel and the eavesdropper channel, bothof which also have their independent channel part (channel B block andchannel E block) afterwards. Signals first arrive at the keyhole positionbefore they branch o↵ to their receivers Bob and Eve, respectively.223.2. System Model - Secrecy Capacity of MIMO PLCThis model can be used to mimic the tree or bus structure of PLCnetworks, which is very common in an in-home environment.It is noted that the keyhole channel model can be a general scenariowhich includes the Wyner channel model and the Csisza´r and Ko¨rner channelmodel when the channel A block and the channel B block is reduced to zero,or assumed to be an error free link.In recent years, physical layer security has become a flourishing researcharea, and has found its path to various applications. For example, [49] firstexploited secure communication over a MIMO setting. After that, successiveworks investigated physical layer security over multiple-input single-outputmultiple-eavesdropper (MISOME) channel [20], multiple-input multiple-outputmultiple-eavesdropper (MIMOME) channel[20], and the special 2-2-1 [50]channel, etc. Besides, [51] applied cooperative relay method over physi-cal layer security. In addition, physical layer security is also used over themultiple-access channel (MAC) [52], the interference channel [53], and wire-less smart grid networks [24]. In our thesis, we continue the line of recentworks probing into the physical layer security performance in MIMO PLCnetworks.3.2 System Model - Secrecy Capacity of MIMOPLCWe now apply the physical layer security (see Sections 1.2 and 3.1) overMIMO PLC networks (see Section 1.1) in the form of a keyhole wiretapchannel, as shown in Figure 3.3. Figure 3.3 depicts the communicationbetween a transmitter and a legitimate receiver in the presence of an eaves-dropper. In the context of PLC, the “Power Line Network A” cloud inFigure 3.3 refers to the power line network from Alice to the point at whichthe “Power Line Network B” and “Power Line Network E” clouds brancho↵ to Bob and Eve, respectively. Each line represents multiple conductors.The clouds have their inputs and outputs connecting to Alice, Bob, Eve orthe keyhole position. Other branches and loads may also be included afterBob and Eve, i.e. they might be equivalently incorporated in “Power LineNetwork B” channel and “Power Line Network E” channel. This operationis allowed because we are using impedance carry back channel modellingmethod, as explained in Section 2.2.4. We would like to point out that theAlice-Bob and Alice-Eve channels share the segment “Power Line NetworkA” and the system fits in the keyhole wiretap channel, see Figure3.2c. Thisis di↵erent from typical wireless communication scenarios.233.2. System Model - Secrecy Capacity of MIMO PLCFigure 3.3: Illustration of communication with presence of an eavesdropperLet us denote the number of available conductor pairs at Alice, Bob, andEve as Nt, Nr and Ne, respectively. Furthermore, we assume the applica-tion of multicarrier modulation as it is a typical modern PLC system, suchthat the frequency-selective PLC channel is decomposed into a number ofparallel channels, e.g., the subcarriers of an OFDM system, e.g. [54]. Ateach subcarrier, we then can write the transmission model asyr = Hx+ zr ,ye = Gx+ ze , (3.1)where we omitted the subcarrier and time indexes for brevity. In (3.1),H 2 CNr⇥Nt and G 2 CNe⇥Nt is the MIMO channel between Alice andBob and between Alice and Eve, respectively, and zr and ze are additivenoise terms. Signal propagation in these MIMO PLC channels are assumedto follow the TEM mode, and the CTFs can be built based on the MTLbottom-up channel modelling method, see Section 2.2. Furthermore, weassume the noises to be white (in time) and Gaussian with covariance rINrand eINe (In denotes the n ⇥ n identity matrix), respectively. AlthoughPLC background noise is not white Gaussian, these assumptions are valid formulticarrier systems, see Section 2.3. Possible noise coloring in frequencydirection can be incorporated by considering subcarrier dependent SNRs.The vector x 2 CNt is the transmitted signal, whose covariance matrix S =E[xxH ] (E denotes the expectation operator) satisfies the power constrainttr(S)  P . (3.2)The secrecy capacity, i.e., the maximal rate that can be transmittedreliably between Alice and Bob while Eve cannot decode Alice’s messageat any positive rate, can be expressed in terms of the mutual informationI(u;yr) and I(u;ye) between auxiliary vector u and the outputs yr and ye,243.3. Solutions to Secrecy Capacity – Perfect CSIrespectively. More specifically, in its general form the secrecy capacity isgiven by [19] Cs = maxp(u,x)(I(u,yr) I(u,ye)) , (3.3)where u satisfies the Markov relation u ! x ! (yr,ye) and the powerconstraint tr(S)  P applies.We note that (3.3) is usually evaluated under the assumption that Aliceand Bob have perfect CSI, which for Alice means it also knows the Alice-Evechannel. In practice, this may or may not be a valid assumption, dependingon the nature of Eve. Our investigation in this thesis covers both the casesof perfect and imperfect CSI about the Alice-Eve channel available at Aliceand Bob. We assume that Alice and Bob always perfectly know the Alice-Bob channel. In terms of di↵erent knowledge of Alice-Eve CSI at Alice,detailed discussions are presented in Sections 3.4 and 3.5.The computation of the secrecy capacity (3.3) is generally not easy.In the PLC scenario (3.1) considered in this work, we are dealing with aGaussian channel. Depending on the number of conductors available in the“Power Line Network A”, “Power Line Network B” and “Power Line Net-work E” networks, di↵erent special cases of the general MIMO case apply[20]. There are known solutions/strategies for MIMO and its special casesunder perfect, imperfect, or unknown Alice-Eve CSI. In the following Sec-tions 3.3, 3.4, 3.5, we will look at known solutions that are transferrable toour problem with di↵erent Alice-Eve CSI conditions at Alice. We note thatonly one eavesdropper is considered in this thesis.3.3 Solutions to Secrecy Capacity – Perfect CSIWhen perfect CSI is available in the system, the secrecy capacity in MIMOsetting can be generally written as [18, 20, 55, 56]Cs,MIMO = maxS⌫0,tr(S)P ✓log det✓INr + 12r HSHH◆ log det✓INe + 12e GSGH◆◆ , (3.4)which is the di↵erence between mutual information of the legitimate (Alice-Bob) and the eavesdropper (Alice-Eve) channel.The problem, as expressed in (3.4), is to find the covariance matrix atthe transmitter side such that the secrecy rate arrives at a maximum value.253.3. Solutions to Secrecy Capacity – Perfect CSI3.3.1 The SISO Channel CaseIf PLC is performed over only two conductors, i.e., Nt = Nr = Ne = 1, wehave SISO Gaussian channels between all three participants. In this case,the eavesdropper’s channel is stochastically degraded with respect to theAlice-Bob channel, and we have the closed-form result [18]Cs,SISO = ⇥log(1 + kHk2P/2r ) log(1 + kGk2P/2e )⇤+ , (3.5)where H and G are scalars and [x]+ = max(x, 0). The expression (3.5)indicates that when Bob’s channel SNR is higher than Eve’s, the secrecycapacity is the di↵erence between the respective channel capacities. Other-wise, the secrecy capacity is zero.As noted, (3.5) is regarded as the instantaneous secrecy capacity expres-sion at a particular subcarrier frequency point. For communication over afrequency selective channel, the average secrecy capacity over the frequencyrange is often calculated. In wireless fading channels, sometimes outageanalysis is also used to characterize the actual secrecy capacity [18].3.3.2 The MISO Channel CaseIf Alice transmits over more than two conductors, but Bob only uses twofor reception, i.e., Nr = 1, then we have the multiple-input single-output(MISO) case. The capacity result for this case is general enough to covermultiple-conductor reception at Eve, i.e., it includes the MISOME scenario,of which MISOSE is a special case. The secrecy capacity can be written as[20] Cs,MISO = log max✓INt + P2r HHH, INt + P2e GHG◆+ , (3.6)where max(A,B) is the largest generalized eigenvalue of the pair (A,B).Generalized eigenvalue-eigenvector pair (, ) of matrix pair (A,B) isdefined to satisfy the expression:A = B (3.7)Obviously, when B is invertible, the generalized eigenvalue and eigenvec-tor decomposition becomes the regular eigenvalue-eigenvector of the matrixB1A. In our case, log max(B1A) is the closed-form solution since in ourcase B is always invertible.263.3. Solutions to Secrecy Capacity – Perfect CSIIt is also noted that in (3.6) the legitimate-user channel H is always a1⇥Nt vector, while matrix G is reduced to a vector if the eavesdropper onlyuses two conductors.The optimal transmission scheme for the MISO scenario is beam-forming(rank one covariance matrix S) solution. Specifically, the capacity is ob-tained by signaling along the direction of the generalized eigenvector thatcorresponds to the largest eigenvalue of the matrix B1A.Alternatively, the secrecy capacity for the MISOSE case can also befound by transforming (3.3) into a convex optimization problem, cf. [26].We first write down the equivalent form of the expression (3.6):maxS 1 +HSHH1 +GSGHs.t. tr(S)  PS ⌫ 0 (3.8)which is a quasi-convex optimization problem that can be reformulated againby transforming it into an equivalent epigraph form:maxS,⌧ ⌧s.t 1 +HSHH  ⌧ 1 +GSGHtr(S)  PS ⌫ 0, ⌧  0 (3.9)Problem (3.9) can be solved through bisection search over [0, ⌧¯ ] that includesa sequence of convex feasible problems.3.3.3 The MIMO Channel CaseWhen the legitimate transmission between Alice and Bob is performed overmultiple conductor pairs, i.e., Nt > 1 and Nr > 1, we are dealing with theMIMO case.The expression (3.4) is generally dicult to evaluate in the general MI-MOME case. [57] used an alternating optimization approach to tacklethe secrecy capacity maximization problem. The algorithm is complexand requires initialization of several variables. There are also results moreamenable for numerical evaluation available for important special cases. Inparticular, the work [50] presents the secrecy capacity for the (Nt = 2, Nr =2, Ne = 1) Gaussian MIMO channel asCs,MIMO,221 = log(1) , (3.10)273.3. Solutions to Secrecy Capacity – Perfect CSIwhere 1 is the largest eigenvalue of the matrixB 12AB 12 (3.11)with A = INt + P2r HHH , (3.12)B = INt + P2e GHG . (3.13)The secrecy capacity was found by first applying the proposed transmissionscheme and then proving that the result meets the tight upper bound. It isshown that Gaussian signalling beam-forming scheme is optimal.This solution supports the result of another work in [58]. The authors in[58] use the beam-forming approach to implement the secure transmissionover di↵erent MIMO channels and provide the sucient condition wherebeam-forming is optimal. They looked at the number of positive eigenvaluesof the matrixHHH GHG and find that if the matrix has one and onlyone positive eigenvalue, beam-forming becomes the optimal solution. SinceMISOME and MIMOSE cases both satisfy this “one positive eigenvalue”condition, the rank one beam-forming solution is tight. Therefore, in the221 channel which is a special case of MIMOSE channel, beam-formingmethod helps to achieve the secrecy capacity.In addition, there is also closed-form solution for the special case ofNt = 2 and general Nr and Ne. [59] obtainsCs,MIMO,2yz = log (max(⌧1, ⌧2)) , (3.14)where ⌧1 is the largest real root (if any) of the quadratic equation⌧2q3 + ⌧p3  q6 = 0. (3.15)and ⌧2 is the largest real root (if any) of the quartic equation⌧2q2 + ⌧p2  q52 4 ⌧2q1 + ⌧p1  q4 ⌧2q3 + ⌧p3  q6 = 0 (3.16)such that0 <  ⌧2q2 + ⌧p2  q52 (⌧2q1 + ⌧p1  q4) < 1, (3.17)283.4. Solutions to Secrecy Capacity - Unknown CSIThe coecients {pi}, {qi} are functions of elements in the known matri-ces: Sr = a1 b1b⇤1 c1Se = a2 b2b⇤2 c2 (3.18)with Sr = PrHHH,Se = PeGHG, which can be written in detail as:p1 = b⇤1b2  b1b⇤2  (1 + a1)(a2c2  |b2|2) (1 + a2)(a1c1  |b1|2),p2 =2b⇤1b2 + 2b1b⇤2 + (1 + a1)(a2  c2 + a2c2  |b2|2) + (1 + a2)(a1  c1 + a1c1  |b1|2),p3 =(1 + a1)(1 + c2) + (1 + a2)(1 + c1) b⇤1b2  b1b⇤2,q1 = a2(c2 + a2c2  |b2|2),q2 =a2  c2 + a22 + |b2|2 + a2(a2c2  |b2|2),q3 =1 + a2 + c2 + a2c2  |b2|2,q4 = a1(c1 + a1c1  |b1|2),q5 =a1  c1 + a21 + |b1|2 + a1(a1a1  |b1|2),q6 =1 + a1 + c1 + a1c1  |b1|2(3.19)The key of the algorithm is to express the covariance matrix asS = xe1eH1 + (1 x)uuH (3.20)where e1 = [1 0]T , 0  x < 1 and uHu  1.In MIMO PLC we often have Nt = 2 (see Section 3.2) and therefore Aliceis able to send two independent signals at the same time, see Section 1.1.As a result, the special algorithm with two transmit antennas is applicableto our problem for the MIMO case.3.4 Solutions to Secrecy Capacity - UnknownCSIPreviously we studied the cases where Alice is informed of CSI for boththe Alice-Bob channel and the Alice-Eve channel. However, such assump-tions are usually impractical, especially when the eavesdropper is passiveand does not transmit at all. Therefore, evaluating the secure transmissionperformance through physical layer security when CSI is not perfect is aninteresting and meaningful topic. We perform the investigation under the293.4. Solutions to Secrecy Capacity - Unknown CSIassumption that the Alice-Bob CSI is perfectly known at both Alice andBob, but the Alice-Eve CSI is not available. Depending on the eavesdrop-per’s nature, unknown or partially known CSI cases are studied respectively.In this section we introduce the secure transmission strategy under unknownAlice-Eve CSI. Imperfect Alice-Eve CSI case will be studied in Section 3.5.When CSI about the eavesdropper’s channel is completely unavailable,an e↵ective approach is to use artificial noise [60, 61], i.e. sending jammingsignals to the eavesdroppers while keeping Bob away from the interferenceof the jamming signal.In the artificial noise based transmission scheme, Alice isotropically sendsout a combination of two messages, one is the precoded information it needsto transmit to Bob, the other is the jamming signal designed in a waythat it lies in the null space of the desired signal at Bob but interferes theeavesdropper as much as possible:x = tz + z0 (3.21)where z is the scalar data stream, the Nt⇥ 1 vector t is the transmit beam-former for formulating the information message, and the Nt ⇥ 1 vector z0is the jamming signal. We assume tHt = 1. The artificial noise gets theallocated power p˜ that can be written as E(z0Hz0) = p˜.At the receiver end, Bob and Eve are using the following beam-formersto recover the original message:yˆr = wHr yr = wHr (Htz +Hz0 + nr)= wHr (Htz + nr) (3.22)yˆe = wHe ye = wHe (Gtz +Gz0 + ne) (3.23)The beam-former t is designed to be the right singular vector of H withthe largest singular value, and the receiving beam-former wr is chosen to bewr = Ht. This design allows Bob to safely recover the message informationfrom the transmitter. The artificial noise term is then chosen to be a linearcombination of the orthogonal basis of the null space of Ht, i.e. it signals atall other directions except for Bob’s receiving signal and thus protects theinformation.Although Alice does not know the eavesdropper’s channel, we can stillcalculate the secrecy capacity using the actual Alice-Eve channels bearingin mind that Alice can do the transmission regardless of the Alice-Eve CSI.However, it does not know the code rate without the knowledge of Alice-Eve channel. In practice, the selection of the transmission rate comes with303.5. Solutions to Secrecy Capacity - Imperfect CSIan insecurity zone at Alice. Legitimate communication is secure from theeavesdroppers outside of the zone, but not from the ones inside [20].It is noted that the artificial noise scheme can only be applied to the mul-tiple transmitter antenna case in wireless situation. Similarly, this schemeis only suitable for a PLC network with multiple feeding conductors, i.e.Nt > 1. Furthermore, [20] showed that this beam-forming based schememay achieve near-optimal performance in high SNR regime. This featurematches the PLC environment very well due to the high probability that thePLC network will o↵er high SNR. Another design parameter of the artificialnoise scheme is the power allocation between the information signal and thejamming signal. [62] proved through analytical and numerical results thatthe simple equal power allocation method is a near optimal strategy in thecase of noncolluding eavesdroppers. Since we are considering secure com-munication with one eavesdropper, the convenient equal power allocation isused between the information message and the jamming signal.3.5 Solutions to Secrecy Capacity - ImperfectCSIIn the condition of imperfect CSI, Alice can make use of the CSI estima-tion and error to design the worst-case robust transmission strategy. Twopopular channel uncertainty models exist in literatures:1. Deterministic model:G = {G : G = Gˆ+G, ||GHP1e G||F  "2} (3.24)where Gˆ is the estimated channel, G is the actual channel and P e is aconstant positive definite matrix that determines the size and shape ofthe uncertainty range. When P e is an identity matrix, the uncertaintyrange set is reduced to ||G||F  ", which describes the uncertaintyrange with the estimation point at the centre.2. Random Gaussian model:G =nG : vec (G) ⇠ CN ⇣vec(Gˆ),C⌘o (3.25)which assumes that the actual channel G follows the complex normaldistribution with the estimated channel Gˆ as mean, and the matrixC as the covariance matrix. A simplified version is available when thematrix C is reduced to an identity matrix "2cI.313.5. Solutions to Secrecy Capacity - Imperfect CSI3.5.1 The MISO Channel CaseFor secrecy capacity in MISO case with imperfect CSI, [63] tackled theproblem in a convex fashion, and proved that beam-forming is the optimaltransmit strategy. The spherical CSI uncertainty, i.e. model (3.24) with P eas an identity matrix is considered.The problem of secrecy capacity in MISO case with spherical CSI uncer-tainty at the eavesdropper’s channel is written as:maxS log(1+HSHH) log det(I+GSGH)s.t. S ⌫ 0tr(S)  PG = Gˆ+G||G||F  " (3.26)Since the optimal solution is proved to be rank one, the det(·) term canbe equivalently replaced by a trace item. Further using the Charnes-Coopertransformation, the above problem (3.26) can be written as the followingsemidefinite programming problem [63]:r⇤ =minZ,⇠,⌧ ⌧s.t. ⇠ +HZHH = 1M ⌫ 0tr(Z)  ⇠PZ ⌫ 0, ⇠  0,  0 (3.27)with M =  + eINe gˆgˆH "2  ⇠ + ⌧  gˆHgˆ  (3.28)where  = ZH N INe ,S = Z/⇠. The optimal secrecy rate can be solvedthrough R⇤ = log2( 1r⇤ ).3.5.2 The MIMO Channel CaseIn a general MIMOME situation with imperfect CSI, it is even harder tofind the optimal value.There are a few algorithms that endeavour to solve arelaxation of the problem. For example, [64] did the calculation in the low323.6. SummarySNR regime by taking an approximation of the logarithm term. Specifically,the key step in the relaxation is:log det✓I + 12r HQHH◆ ⇡ 12r tr(HQHH) (3.29)The algorithm then uses the approximation on the right side of (3.29) to forma solvable convex problem. This algorithm is applicable to our situation butmay not fit ideally because high SNRs are more common in a PLC network.Another relaxation in [65] that works on the whole SNR regime is toestimate the secrecy capacity with an outage probability. Channel uncer-tainty model (3.25) is used. Outage secrecy capacity in MIMO PLC withimperfect CSI can be solved through the following problem:minW ,R,S,x,yRs.t tr⇣C1/2 ST ⌦W C1/2⌘+p2 log ⇢ · x log ⇢ · y +R log |I +HHWH| + tr (S) log |S| + gˆH ST ⌦W  gˆ N e"vec⇣C1/2 ST ⌦W C1/2⌘p2C1/2 ST ⌦W  g¯ #2  xyINtNe C1/2 ST ⌦W C1/2 ⌫ 0y  0,S ⌫ 0tr (W )  P,W ⌫ 0 (3.30)where R is the target secrecy rate, g is the channel estimation, C is thechannel uncertainty model parameter with g ⇠ CN (gˆ,C), g = vec(G), gˆ =vec(Gˆ). It is usually assumed that C is an identity matrix. ⇢ is the secrecyoutage probability - the chance that a secrecy rate below the target rageR is not achievable. Therefore, (3.30) provides the secrecy rate R that thesystem can achieve with the probability 1 ⇢.3.6 SummaryIn this chapter, we discussed physical layer security, which provides a wayto achieve secure communication without cryptographic methods. We for-mulated the problems to find secrecy capacity in MIMO PLC with per-fect/unknown/partially known CSI about the eavesdropper channel at thelegitimate transceivers. Known methods have been reviewed and selectedto solve our problem. With perfect CSI, closed-form solutions are available333.6. Summaryfor SISO and MISO and some special MIMO cases. With unknown CSI,the artificial noise based transmission is an ecient scheme. With partiallyknown CSI, robust designs help to find worst-case secrecy capacity.34Chapter 4Physical Layer Security inMIMO PLC NetworksFollowing the discussion of the PLC channel modelling, the principles ofphysical layer security, and the computation of security rates, we are nowready to apply the information-theoretic approach to MIMO PLC networksand to examine the secure communication performance through numericalanalysis.According to Section 2.2, the bottom-up channel modelling method canexpress the MIMO PLC channel as function of network topologies (cablelengths, keyhole positions, conductor numbers, etc), coupling schemes andalso load impedances, etc. In this chapter, the basic network topology forthe simulation is first introduced. Then, through numerical experiments, wediscuss the benefits of MIMO transmission for secure communication, thee↵ects of network topology on secrecy capacity, and compare the character-istics of secrecy capacity for PLC and wireless networks.4.1 Simulation Setting: Network Topology andChannel ModelFigure 4.1 shows the basic network topology considered in the following. Itis in consistency with system model shown in Section 3.2, but simplified.The “Power Line Network” clouds are reduced to power line wires. Theunderlying power grid consists of three-conductor cables (e.g., P, PE, and Nin an in-house environment, see Sections 1.1 and 2.1).In Figure 4.1, Bob communicates to Alice via the grid segments PowerLine A and Power Line B, and Eve eavesdrops through a branch pointalong the Alice-Bob link, which we will refer to as wiretap position. Asdiscussed in Section 1.1, all three users can potentially use two conductorpairs, namely PN and PEN to transmit and receive signals, respectively.It is reiterated that an M -conductor cable provides (M  1) independentdi↵erential signals. Sometimes also the common-mode signal is considered354.1. Simulation Setting: Network Topology and Channel ModelFigure 4.1: Network topology considered for PLC secrecy capacity results.Three-conductor cables connect Alice, Bob and the eavesdropper Eve.additionally [66].For concreteness, we consider symmetric three conductor cables with thecircular cross section (see Figure 2.1), as discussed in Section 2.1.2. Assum-ing further homogeneous dielectric, the per-unit-length cable parameterscan be obtained in closed form. The resistance of the stranded conductor isobtained by calculating the solid resistance and then applying a correctionfactor that takes the e↵ective area into account. The specific derivation wasgiven in Section 2.2.2.To account for the e↵ect of other electrical devices in the PLC network,admittance matrices Y loads are included in the system model (Figure 4.1) torepresent the network beyond the experiment circuit. The admittances cap-ture power loads connected to the grid at Bob’s and Eve’s coupling points,as well as the e↵ect of grid elements located behind Bob’s and Eve’s PLCmodems. The elements of the admittance matrices are obtained throughplacing load impedances modelled as parallel RLC resonant circuits betweenconductor pairs. Hence, the impedances are frequency selective. The circuits364.1. Simulation Setting: Network Topology and Channel ModelFigure 4.2: Illustration of load circuit impedancesare described by (cf. [67])Z(f) = R1 + jQ⇣ ff0  f0f ⌘ . (4.1)The load impedance model is observed to have a frequency-selective prop-erty, which reflects the same characteristic shared by various load devicesin practice that is caused by inductance and capacitance impeding e↵ectsin DC circuits. For specific realizations of the impedances, the parame-ters resistance R, resonance frequency, f0, and quality factor Q are gener-ated independently and randomly from the uniform distribution on intervals[200, 1800] ⌦, [2, 28] MHz, and [5, 25] respectively. A few random realizationsof the amplitude of the impedance is presented in Fig. 4.2.The admittance matrices Y PLC in Figure 4.1 represent the e↵ect of thePLC receiver modems, which we assume to have an input impedance of 50 ⌦.We ignore coupling losses at the transmitter side for the capacity results inthis work.For MIMO PLC transmission in the following experiments, we considerthe frequency band from 2 to 28 MHz, which is the typical band for broad-band PLC. We compute the MIMO channel frequency responses for the374.2. Comparing Secrecy Capacities in MIMO PLC NetworkAlice-Bob and Alice-Eve channels, H and G, for the frequencies spaced24.4 kHz apart in the considered band. This mimics the use of OFDMfor broadband PLC, as introduced in Section 3.2. The computation of thechannel frequency responses is based on the multi-conductor transmissionline theory, as already introduced in Chapter 2. The noise is assumed to beGaussian and possible noise coloring in frequency direction can be incorpo-rated in SNR in subcarrier, see Section 3.2 for detailed discussion.Given the secrecy capacity Ci for the ith frequency, in the following wepresent the secrecy capacity averaged over all frequencies, i.e.,C = 1N NXi=1 Ci bps/Hz (4.2)where N is the number of frequency points considered. We note that thisprovides a value for the spectral eciency, in bits per second and Hertz(bps/Hz), under secrecy constraints. This capacity is shown as function ofthe equivalent transmitter side SNR, which is given by P/2r in the following.4.2 Comparing Secrecy Capacities in MIMOPLC NetworkFirst, we consider the PLC network from Figure 4.1 with fixed cable lengthsof 18 m for Power Line A, 12 m for Power Line B, and 15 m for PowerLine E, and 100 randomly generated sets of admittance matrices accordingto the description in the previous section using (4.1). We compare thedi↵erent combinations for transmission and reception of the PLC signals byAlice, Bob, and Eve as summarized in Table 4.1. It is noted that we refer“MIMOME” case as multiple-input multiple-output and one eavesdropperwith multiple conductor pairs.Figure 4.3 shows the secrecy capacities, averaged over the 100 channelrealizations, for di↵erent input/output combination scenarios as function ofthe SNR. As reference curves, we also include the channel capacities (SISO,MISO, MIMO) without any secrecy constraints (labelled “w/o Eve” in Fig-ure 4.3). First, we observe that the presence of Eve, or adding a securityconstraint, has an obviously negative influence on the achievable rate thatcan be transmitted to the legitimate receiver, Bob. In particular, SISOPLC su↵ers a drastic rate loss. However, MIMO PLC can greatly alleviatethe situation. That is, high secrecy rates can be achieved if Alice and Bobcommunicate through MIMO PLC, but Eve only has a single conductor384.3. Topology E↵ects on Secrecy CapacityTable 4.1: Combinations of signal transmission and reception considered forsecrecy capacity. See Figure 4.1.Alice Bob EveSISO P-N P-N P-NMISOSE1 P-N, PE-N P-N P-NMISOSE2 P-N, PE-N P-N PE-NMISOME P-N, PE-N P-N P-N, PE-NMIMOSE P-N, PE-N P-N, PE-N P-NMIMOME P-N, PE-N P-N, PE-N P-N, PE-Npair to eavesdrop (MIMOSE case). In fact, similar rates can be achieved ifonly Alice uses two conductor pairs. Intuitively, Alice can pre-distort themultiple-input PLC signal such that it is in the null-space of the Alice-Evechannel. The MISOSE2 case provides a somewhat higher capacity thanthe MISOSE1 case. This is expected since the correlation between the twoMISO channels is higher when the receivers are connected to the same con-ductor pair. Clearly, if Eve also uses two conductor pairs (MIMOME case),the secrecy rate drops compared to MIMOSE, but it is still significantlyhigher than in the SISOSE case. Finally, we note that sending over multipleconductor pairs helps, compared to the SISO case, even if the legitimatereceiver uses on a single conductor pair and the eavesdropper can receive ontwo pairs, which is the MISOME case.4.3 Topology E↵ects on Secrecy CapacityThe properties of the underlying power grid have a direct e↵ect on the PLCchannel frequency responses and thus on secrecy capacity. We next shedsome light on this by considering varying topology parameters, in particulardi↵erent lengths of the line segments and load impedances.First, the results from 100 network realizations used also for Figure 4.3are plotted in terms of the empirical cumulative density function (CDF) inFigure 4.4. The CDF plots support the statements based on the averageresults in Figure 4.3. Using Nt = 2 improves secrecy capacity over the SISOsignificantly, except when Nr = 1 and Ne = 2. Particularly beneficial is aMIMO or MISO link to Bob when Eve can connect to only one conductorpair. But also the MIMOME case shows notable improvements over theSISO case. For the MIMOSE scenarios, the question of what conductor pairBob and Eve are connected plays a role.394.3. Topology E↵ects on Secrecy Capacity0 5 10 15 20 25 30 35 40012345678SNR (dB)Secrecy Capacity (bps/Hz)  SISOSEMISOSE 1MISOSE 2MISOMEMIMOSEMIMOMEw/o Eve SISOw/o Eve MISOw/o Eve MIMOFigure 4.3: Secrecy capacity for SISO/MISO/MIMO PLC channels. Cablelengths are 18 m for Power Line A, 12 m for Power Line B, and 15 m forPower Line E. Averaged over 100 realizations of network load impedance.Next, we examine the e↵ect of eavesdropper’s wiretapping positions. Tothis end, we keep the cable length between Alice and the network loadsconstant at 30 m. The PLC modem for legitimate receiver Bob lies inthe middle of them at 15 m. Eve’s tapping location is varied in the range[0.3, 30] m with a step size of 0.3 m. The length of the Power Line E segmentis 13 m, and the network load impedances are one set of randomly generatedvalues.Figure 4.5 demonstrates Eve’s wiretapping position’s e↵ect on secrecycapacity. We observe that the secrecy capacity varies with the wiretappingposition. Even though the Alice-to-Eve distance and thus cable attenua-tion increases from left to right Figure 4.5, signal reflections along the lineslead to a di↵erent channel matrix G for di↵erent positions. Furthermore,and di↵erent from wireless transmission, also the Alice-Bob channel H is404.3. Topology E↵ects on Secrecy Capacity0 1 2 3 4 5 6 7 800. Capacity (bps/Hz)Empirical CDF (over network impedances)  SISOSEMISOSE 1MISOSE 2MISOMEMIMOSEMIMOMEFigure 4.4: Empirical CDF for secrecy capacity. Cable lengths are 18 mfor Power Line A, 12 m for Power Line B, and 15 m for Power Line E. 100realizations of network load impedances.SNR is 30 dB.a↵ected by the wiretap position. Hence, it is dicult to predict the exactbehaviour of secrecy capacity as a function of wiretap position. However,when the wiretap position goes beyond Bob, a significant increase in secrecycapacity is demonstrated in every case. This can explained by the changeof the two receiver’s channel conditions as this move happens, when Bobobtains an improvement in the channel condition that might be caused bythe elimination of a branch tap, whereas Eve su↵ers from a worse channelcondition in that a new branch circuit is added in to the Alice-Eve channel.The PLC branch taps between the transmission brings in signal reflectionsthat account for more signal distortion and thus deteriorates the channelcondition severely.Finally, we look at how an impedance applied at the position of Eve’sPLC modem can a↵ect the MIMO PLC secrecy capacity performance. We414.3. Topology E↵ects on Secrecy Capacity0 5 10 15 20 25 30012345678910wiretap position (m)Secrecy Capacity (bps/Hz)  SISOSEMISOSE 1MISOSE 2MISOMEMIMOSEMIMOMEFigure 4.5: Secrecy capacity as function of wiretap position. Link lengthfrom Alice to Bob is 15 m, from Bob to network loads is 15 m. Length ofPower Line E is 13 m, and one realization of network load impedances hasbeen generated randomly. SNR is 30 dB.consider the MIMOSE case, and place an additional impedance in parallel toEve’s modem. This impedance is varied over the range [2, 60]+j[28, 30] ⌦,for one set of randomly generated network impedances. Figure. 4.6 shows theMIMOSE capacity as function of the complex impedance. We observe thatsmall impedance magnitudes lead to higher secrecy capacities. This can beexplained by the fact that the voltage levels Ve (see Figure 4.1) are generallylower for smaller impedances. The exact shape of the surface will dependon all network components, i.e., cables and loads. We note that though thisexperiment is insightful in that it shows the possibility of a↵ecting secrecycapacity through changing the channel frequency responses G and H viaimpedance changes, it neglects that this will also have an e↵ect on the noiselevel at Eve’s receiver.424.4. A Comparison with Wireless Communication−40−2002040020406044.555.566.57imaginary part of impedance (Ω)real part of impedance (Ω)secrcy capacity(bps/Hz)Figure 4.6: Secrecy capacity for MIMOSE as function of impedance of Eve’smodem. Cable lengths are 18 m for Power Line A, 12 m for Power Line B,and 15 m for Power Line E, and one realization of network load impedanceshas been generated randomly. SNR is 30 dB.4.4 A Comparison with Wireless CommunicationIn this section, we provide a comparison between the numerical secrecy ca-pacity results for PLC and a wireless network with equivalent link qualities.That is, we consider the set of PLC networks used for the results in Fig-ures 4.3 and 4.5, and compute the average power gainsph,i,j = 1KN KXk=1 NXn=1 |h(k,n)i,j |2 ,pg,i,j = 1KN KXk=1 NXn=1 |g(k,n)i,j |2 , (4.3)434.4. A Comparison with Wireless Communication0 1 2 3 4 5 6 7 800. Capacity (bps/Hz)Empirical CDF   PLC SISO Wireless SISOPLC MISOSE 1Wireless MISOSE 1PLC MISOSE 2Wireless MISOSE 2PLC MIMOSEWireless MIMOSEPLC MIMOMEWireless MIMOMEFigure 4.7: A comparison for secrecy capacities in PLC and wireless net-works. Cable lengths are 18 m for Power Line A, 12 m for Power Line B, and15 m for Power Line E. 100 realizations of network load impedances.SNR is30 dB.where h(k,n)i,j and g(k,n)i,j are the (i, j) elements of H and G for the kth net-work realization and the n frequency value, respectively, and K and N arethe total number of PLC network realizations and frequencies considered.We use the average power gains ph,i,j and pg,i,j as the second moments ofRayleigh distributions to generate the channel gains of the correspondingwireless links. This allows us to compare the secrecy capacities for the PLCnetwork with those for a wireless network with complex Gaussian channelcoecients and the same average gains for all links.Figure 4.7 shows the CDFs for the di↵erent secrecy capacities for bothPLC and wireless networks. It can be seen that the secrecy capacities for thewireless system are generally higher than for the corresponding PLC case.This phenomenon can be explained by the physical properties of the PLC444.5. Summarynetwork, where the Alice-to-Bob and Alice-to-Eve signals are first trans-ported through the same segment of power line before they separate at thewiretap point. Hence, there are dependencies between H and G [68]. Inthe wireless system, we made the usual assumption of independent valuesfor H and G.Another observation that highlights the di↵erences between PLC andwireless networks is the MISOSE case. In the PLC network, there is adi↵erence between MISOSE1 and MISOSE2, as it matters whether Bob andEve connect to the same conductor pair. In the wireless system, there isno such notion and thus the capacity results for MISOSE1 and MISOSE2overlap, as the average link gains are the same for those cases.4.5 SummaryThis chapter investigated the secrecy capacity for MIMO PLC networksunder the assumption of perfect CSI at legitimate users.We built the MIMO PLC channel using the bottom-up channel mod-elling, as introduced in Chapter 2. and applied known capacity results (seeChapter 3, Section 3.3) to PLC settings.Our numerical results demonstrated that secrecy capacity performancein MIMO PLC is generally better than in the SISO case. In particular, se-crecy capacity is improved if the transmitter can use more conductor pairsthan the eavesdropper. We highlighted the e↵ects of the underlying powerline infrastructure on secrecy capacity, which is unique to PLC. For example,secrecy capacity depends on which wire pairs legitimate receiver and eaves-dropper couple to. The secrecy capacity is also a↵ected by the topologycomponents, such as the cable length, wiretap position, load impedances.Furthermore, the direct coupling of the legitimate and eavesdropper chan-nel due to the shared power line network leads to a generally lower secrecycapacity than in a corresponding wireless transmission scenario.45Chapter 5Physical Layer Security inMIMO PLC with CSIUncertaintySimilar to the evaluation process of secure communication under perfectCSI in Chapter 4, in this chapter we apply the known imperfect CSI se-crecy capacity solutions, as introduced in Sections 3.4 and 3.5, to MIMOPLC channel with imperfect CSI about the Alice-Eve channel. The MIMOPLC channel is also built by the bottom-up channel modelling method, seeSection 2.2.In the following, we first discuss how bottom-up channel modelling givesus insights into CSI uncertainty in PLC. Then, through numerical experi-ments, we show the secure transmission performance in MIMO PLC withimperfect CSI, including unknown and partially known eavesdropper’s CSI.5.1 Capturing PLC Channel Uncertainty fromPLC TopologyIn Chapter 3, we introduced two channel uncertainty models, including thedeterministic model (3.24) and the random Gaussian model (3.25). Whichone should we choose for PLC?In order to answer this question, knowing from Chapter 2 that the PLCchannel is expressed as function of the network topology elements as a re-sult of the bottom-up channel modelling method, we reexamine the PLCtopology as introduced in Chapter 2. As we see from Figures 2.1 and 2.2 in[32], in-home PLC infrastructure is usually fixed, including service panels,outlet locations, etc. Besides, regulated by wiring code, the circuit designsand cable types will not be too diverse, see Section 2.1.2. Therefore, it isreasonable to assume that all related elements are within a deterministicboundary of a fixed value. As a result, the deterministic model (3.24) isadopted in our research. The problem is how to determine the uncertainty465.1. Capturing PLC Channel Uncertainty from PLC Topologyrange. Conceiving that channel estimation error arises if the informationabout the physical settings is inaccurate, we can find the uncertainty rangefrom physical considerations via the bottom-up channel modelling method.Since the in-home PLC infrastructure is presumably fixed, we assume thatthe keyhole position is already known, and consider uncertain distance fromthe keyhole to the eavesdropper and uncertain PLC modem impedance atthe eavesdropper, i.e. we can map the unknown eavesdropper’s location andload parameters into channel uncertainty. To do so, we still use Figure 4.1in Chapter 4 as the system model. The same communication is happeningwith the same cable type, load circuits, frequency band and multi-carriermodulation. As noted, the keyhole position and the cable length from Aliceto Bob are known. However, the information of the eavesdropper’s distanceto the keyhole and the PLC modem load impedances at the eavesdropperis only limited to an estimation value and an error range. In the following,we do the mapping from the unknown eavesdropper’s location and from theunknown eavesdropper’s load to channel uncertainty one by one.5.1.1 Mapping Unknown Eve’s Location to ChannelUncertaintyIn the PLC network, as shown in Figure 4.1, both Power Line A and PowerLine B are fixed cable lengths of 15 m. Power Line E is estimated to bel0 =13 m with an error of l m. For one actual distance l = l0 +l, |l| <lm, we give random network loads, and calculate the actual CTF G andthe estimated CTF Gˆ for the Alice-Eve channel, based on which we calculatethe CTF error, in both ratio and absolute value forms:rG = ||G Gˆ||F||G||F (5.1)G = ||G Gˆ||F (5.2)We choose 200 realizations ofl uniformly distributed between [lm,lm].For each l, we repeat 200 random network loads. The results are shown inFigure 5.1, with the first row showing the error in the form of ratio and thesecond row showing the error in the form of absolute value. For each l,randomly picking one channel uncertainty over all network loads, we havefirst column of the result; averaging channel uncertainty over the networkloads, we have the second column; taking 90% maximum channel uncer-tainty over the network loads (referred to as 90% outage), we have the thirdcolumn.475.1. Capturing PLC Channel Uncertainty from PLC Topology12.85 12.9 12.95 13 13.05 13.1 13.1500. Line E (m)channel uncertainty (ratio) for a random load12.85 12.9 12.95 13 13.05 13.1 13.1500.0050.010.0150.020.0250.030.0350.04Power Line E (m)|channel uncertainty (ratio) average12.85 12.9 12.95 13 13.05 13.1 13.1500. Line E (m)channel uncertainty (ratio) 90% outage12.8 12.9 13 13.1 13.200.0050.010.0150.020.0250.030.0350.04Power Line E (m)channel uncertainty (absolute value) for a random load12.85 12.9 12.95 13 13.05 13.1 13.1500.0050.010.0150.02Power Line E (m)||channel uncertainty (absolute value) average12.85 12.9 12.95 13 13.05 13.1 13.1500.0050.010.0150.020.0250.030.0350.04Power Line E (m)|channel uncertainty (absolute value) 90% outageFigure 5.1: Mapping unknown eavesdropper’s location to PLC channel un-certainty, lm=0.2 m, at frequency 5MHzFigure 5.1 shows the result with lm=0.2 m, and Figure 5.2 shows theresult with a broader lm=1 m. Results reveal that as the estimation errorof eavesdropper’s location increases, the channel estimation error tends toincrease accordingly. The positive correlation is almost linear as seen fromthe result. From the result of Figure 5.2, on 90% outage probability, 13%channel estimation error is very likely to happen when 50cm estimationerror in the eavesdropper’s location is made, and 5% channel estimationerror arises corresponding to a 17cm estimation error in the eavesdropper’slocation. When the estimation error in the eavesdropper’s location becomesas large as 1 m, there are 90% of chance that the channel estimation erroris up to 30%.5.1.2 Mapping Unknown Eve’s Impedance to ChannelUncertaintyThe second channel uncertainty experiment focuses on the eavesdropper’sPLC modem impedance, again based on the PLC communication shown in485.1. Capturing PLC Channel Uncertainty from PLC Topology12.5 13 13.5 1400. Line E (m)channel uncertainty (ratio) for a random load12.5 13 13.5 1400. Line E (m)|channel uncertainty (ratio) average12.5 13 13.5 1400. Line E (m)channel uncertainty (ratio) 90% outage12 12.5 13 13.5 1400. Line E (m)channel uncertainty (absolute value) for a random load12.5 13 13.5 1400. Line E (m)||channel uncertainty (absolute value) average12.5 13 13.5 1400. Line E (m)|channel uncertainty (absolute value) 90% outageFigure 5.2: Mapping unknown eavesdropper’s location to PLC channel un-certainty, lm=1 m,at frequency 5MHzFigure 4.1, where fixed cable lengths for Power Line A, Power Line B andPower Line E are 15 m, 15 m, 13 m, respectively. A similar experiment asSection 5.1.1 is carried out with the eavesdropper’s PLC impedance estima-tion at 50 ⌦, and the actual impedance uniformly distributed in [45, 55]⌦and [5j, 5j]⌦ for real and imaginary part, respectively.For each actual impedance, 200 realizations of network loads are repeatedand the result is shown in Figure 5.3. The channel estimation error in ratioand absolute value forms are shown in the first and second row, respectively.For each actual impedance point, result of the random, average, and 90%outage channel uncertainty are shown in the first, second, third column,respectively.The results in Figure 5.3 support that the increase of the estimationerror in the eavesdropper’s load impedance will lead to the growth of theeavesdropper’s channel uncertainty. The symmetric curve implies that the495.2. Impact of Partially Known CSI46 48 5052 54−4−202400. partreal part|∆|/|He| for a random−4 −2 02 4464850525400.020.040.06imaginary partreal part|∆|/|He| mean−4 −2 02 4464850525400. partreal part|∆|/|He| 90% outage46 48 5052 54−4−202400. partreal part|∆| for a random−4 −2 02 4464850525400.020.04imaginary partreal part|∆| mean−4 −2 02 4464850525400. partreal part|∆| 90% outageFigure 5.3: Eve’s channel uncertainty range against Eve’s PLC loadimpedance, at frequency 5MHzestimation error in the real and imaginary part of the load impedance havethe same e↵ect on the channel uncertainty. For 5⌦ estimation error pre-sented at both the real part and the imaginary part of the eavesdropper’simpedance, up to 7% channel estimation error will happen at a 90% outageprobability. Comparing to the result in Section 5.1.1, we find that the esti-mation error of 17 cm in the location and 4⌦ in both real part and imaginarypart of the impedance have equal amount of influence on channel estimationerror, both at 5%.5.2 Impact of Partially Known CSIThis simulation operates on the same setting as in Section 5.1.1. Threelevels of estimation errors in the eavesdropper’s location are considered,with lm 2 (10cm, 20cm, 30cm), respectively. According to the simulationresults in Figure 5.2 in Section 5.1.1, the eavesdropper’s channel uncertaintycan be read out:G = Gˆ+G, ||G||F  ", " 2 (0.015, 0.03, 0.35) (5.3)We now apply the algorithm in Section 3.5.1 to MISOSE1 scenario (seeTable 4.1 ) with the above three levels of CSI uncertainty. Secrecy capacity505.3. With Imperfect CSI - Choose The Best Transmission Strategyis averaged over all the random realizations of network loads and the actualeavesdropper’s locations.Figure 5.4 shows the worst case secrecy capacity with di↵erent amountof CSI uncertainty, and reflects how imperfect eavesdropper’s CSI a↵ectsthe secrecy capacity. In Figure 5.4, the perfect secrecy capacity is includedas a reference. It is obvious that secrecy rate is reduced when imperfectCSI comes in and the secrecy capacity loss becomes higher as CSI uncer-tainty increases. This negative e↵ect is more obvious at the higher SNRrange, because transmit power becomes a less important constraint of thesecure communication ability at a higher SNR level. Also at the high SNRrange, we can observe and predict that, when SNR is no longer a con-straint, the secrecy capacity drop is smaller and smaller from perfect CSIlevel to lm=10cm, from lm=10cm to lm=20cm, and from lm=20cmto lm=30cm, i.e. CSI imperfection a↵ects the secrecy capacity more atsmaller CSI imperfection level. We think this is reasonable since the se-cure transmission system is tend to be more sensitive at a smaller channeluncertainty stage.5.3 With Imperfect CSI - Choose The BestTransmission StrategyThis experiment is conducted with exactly the same setting as in Section 5.2,but we add the artificial noise (AN) transmission scheme for each of the un-certainty degree, i.e. three di↵erent lm. The result is shown in Figure 5.5.From the result, we can observe that while the AN method provides worseperformance than robust design at lower SNR regime, it has significantlybetter secure transmission capability at higher SNR regime. In fact, whenSNR goes higher and higher, secrecy rate achieved by AN gets closer tothe secrecy capacity with perfect CSI. The result supports the statementthat AN strategy achieves near-optimal performance in high SNR regime,as discussed in Chapter 3. Moreover, although AN generally o↵ers lowersecrecy rate than the robust design at smaller imperfection case, i.e. atlm=10cm level, it gives close or better security than robust design at big-ger uncertainty range, i.e. at lm=20cm and 30cm levels. In conclusion,the experimental result suggests that AN should be used under high SNR,and/or high channel uncertainty condition.515.4. With Imperfect CSI- Comparison of MISO and MIMO cases0 5 10 15 20 25 30 35 4001234567SNR(dB)Secrecy Capacity(bps/Hz)  perfect∆lm=10cm imperfect∆lm=20cm imperfect∆lm=30cm imperfectFigure 5.4: Secrecy capacity under partially known CSI with di↵erentamount of imperfection, at frequency 5MHz5.4 With Imperfect CSI- Comparison of MISOand MIMO casesThis experiment uses the same setting as in Section 5.1.1. Three typesof CSI conditions are considered, including perfect, unknown and partiallyknown CSI, for the latter lm = 10cm is assumed. Both MISO and MIMOtransmission schemes are tested. Particularly, we consider MISOSE1 andMIMOME cases (see Table 4.1). For perfect and unknown CSI cases, al-gorithms in Section 3.3 and Section 3.4 are put to use, respectively. Forimperfect CSI with MISO and MIMO case, the strategies in Section 3.5.1and Section 3.5.2 are applied, respectively.The algorithm in the MIMO+imperfect CSI case (see Section 3.5.2 ) isbased on the random Gaussian channel uncertainty model (3.25). In orderto transfer the random Gaussian uncertainty model to the bounded uncer-tainty model, we did an approximation by making the probability that the525.4. With Imperfect CSI- Comparison of MISO and MIMO cases0 5 10 15 20 25 30 35 4001234567SNR(dB)Secrecy Capacity(bps/Hz)  perfect∆lm=10cm imperfect∆lm=20cm imperfect∆lm=30cm imperfect∆lm=10cm AN∆lm=20cm AN∆lm=30cm ANFigure 5.5: Secrecy capacity under unknown and partially known CSI, atfrequency 5MHzGaussian-distributed channel falls in the bounded area equal to 90%,i.e.there is 90% chance that ||GGˆ||F  0.015. As a result, we set "2c =0.0046.The secrecy capacity outage probability ⇢ (see (3.5.2)) is set at 0.01. Fig-ure 5.6 shows secrecy capacity performances under various CSI conditions.In Figure 5.6, the secure transmission capability in MISOSE1 is still bet-ter than the MIMOME case, since Alice has more conductor pairs than Evein the former case (see Section 4.2). Further, it is verified that unknown orpartially known CSI conditions pull down the secrecy capacity curve. Par-ticularly, in MISOSE1 case, the robust design with imperfect CSI helps torecover secrecy capacity better than the artificial noise strategy with un-known CSI, especially at the medium SNR range. This is probably becausewe have chosen a relatively smaller uncertainty range with lm = 10cm,and thus the robust design shows its advantage. But the situation is di↵er-ent in MIMOME case. In MIMOME, the artificial noise transmission actsas a preferred scheme compared to the robust design with imperfect CSI,535.5. Summary0 5 10 15 20 25 30 3501234567SNR(dB)Secrecy Capacity(bps/Hz)  MISOSE1 perfectMIMOME perfectMISOSE1 ANMIMOME ANMISOSE1 imperfectMIMOME imperfectFigure 5.6: Secrecy capacity under perfect/unknown/partially known CSI,at frequency 5MHzespecially in higher SNR regime. We think one of the main reasons for thisphenomenon is that the eavesdropper in MIMOME cases obtains more in-formation through two conductor pairs, and at the same time, it gets moreinterference from the jamming signal. In addition, as stated, the artificialnoise scheme possesses its advantage over higher SNR regime. Therefore,the result of the above two di↵erent strengths is that in MIMOME case, theartificial noise gives comparable secure transmission performance to MIS-OSE1 case at lower SNR regime, but a significantly better performance athigher SNR regime.5.5 SummaryIn this chapter, we evaluated secrecy capacity in MIMO PLC with imper-fect channel conditions, including unknown and partially known CSI forthe Alice-Eve channel. We investigated channel uncertainty in PLC from545.5. Summaryphysical considerations. By reexamining the special structure of the PLCtopology as introduced in Chapter 2, we selected deterministic channel un-certainty model from Chapter 3, and found channel uncertainty range forPLC networks via the bottom-up channel modelling method. Simulationresults have demonstrated the PLC channel uncertainty in both absolutevalue and ratio forms. Further, we incorporated the PLC channel uncer-tainty model to the known secrecy solutions, see Chapter 3, and analyzedthe imperfect secrecy capacity through numerical results. We verified thatthe secrecy capacity loss increases as the amount of channel uncertaintyincreases. By collectively showing the secure communication performancesunder perfect/unknown/partially known CSI, we investigated the impact ofimperfect CSI in a detailed way. The advice concluded from the simulationresults is that the artificial noise approach should be used in MIMO caseand larger channel uncertainty scenario.55Chapter 6Conclusion6.1 Conclusions and RemarksIn this thesis, we have investigated the secrecy capacity for MIMO PLCnetworks.We started with the introduction of wiring practices, including the elec-tric power distribution, the in-home power line infrastructures, and thepower cables. Further, we exploited PLC channel modelling approachesand selected the bottom-up channel modelling method to be the underly-ing channel simulation algorithm of this thesis. In addition, we reviewedexisted secrecy capacity solutions and strategies under both perfect and im-perfect CSI. Applying the known capacity results to PLC settings enablesus to capture the physical layer security performances and features in PLCnetworks.Under perfect CSI, our numerical results have demonstrated that secrecycapacity performance in MIMO PLC is generally better than in the SISOcase. In particular, secrecy capacity is improved if the transmitter can usemore conductor pairs than the eavesdropper. We have highlighted the ef-fects of the underlying power line infrastructure on secrecy capacity, whichis unique to PLC. For example, secrecy capacity depends on which wirepairs legitimate receiver and eavesdropper couple to. Furthermore, the di-rect coupling of the legitimate and eavesdropper channel due to the sharedpower line network leads to a generally lower secrecy capacity than in acorresponding wireless transmission scenario.Under imperfect CSI, we evaluated the secure transmission capabilityunder both unknown and partially known CSI, and analyzed the impact ofCSI imperfection. Considering that the PLC network usually has fixed in-frastructures due to wiring code regulations, we expressed the PLC channeluncertainty in a deterministic model and found the uncertainty range viathe bottom-up channel modelling method. Incorporating the PLC channeluncertainty into the known strategy or solutions for worst case secrecy ca-pacity, we evaluated the impact of CSI uncertainty. It is suggested thatAN approach should be adopted for higher uncertainty range, higher SNR566.2. Future Workregime and/or multiple conductors at the eavesdropper case.6.2 Future WorkWe suggest the following topics be the directions of potential future work asan extension of this thesis.• Extend the system model to a larger scale with more branches andterminals to account for secure transmission over a more complex net-work. This extended work investigates the capability of physical layersecurity in an in-building or community environment instead of in-home environment. In addition, we may consider two or more eaves-droppers presented in the communication. Further, it is also reason-able to look into the multiple phase wire case due to the fact thatthree-phase system is very common for North American residential orcommercial buildings.• Seek a PLC specific channel uncertainty model. As we were consid-ering PLC channel uncertainty model, see Section 5.1, we chose thedeterministic model (3.24), which can be written as:G1 = {G : G = Gˆ+G, ||G||F  "} (6.1)Considering G is a function of the eavesdropper’s cable length l viathe bottom-up channel modelling method, see Section 2.2, we expressthe channel uncertainty model in another way:G2 = {G : G = f(l), l 2 [l0 lm, l0 +m]} (6.2)where f(·) is the channel modelling(see Section 2.2.5), l0 is the esti-mated length and lm is the boundary of the estimated length.Although both of the models G1 and G2 describe the channel un-certainty that reflects unknown physical setting, i.e. unknown cablelength, they are di↵erent sets.In fact, G2 2 G1. Therefore, using channel uncertainty model G2,we may find better achievable secrecy capacity than the “worst casesecrecy capacity” as we have obtained in Figures 5.4 and 5.6. 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