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Physical layer security in massive MIMO systems Zhu, Jun 2016

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Physical Layer Security in Massive MIMO SystemsbyJun ZhuM. A. Sc., University of Victoria, 2011B. Sc., Southeast University, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)June 2016c© Jun Zhu, 2016AbstractMassive multiple-input multiple-output (MIMO) is one of the key technologies for theemerging fth generation (5G) wireless networks, and has the potential to tremen-dously improve spectral and energy eciency with low-cost implementations. Whilemassive MIMO systems have drawn great attention from both academia and industry,few eorts have been made on how the richness of the spatial dimensions oered bymassive MIMO aects wireless security. As security is crucial in all wireless systemsdue to the broadcast nature of the wireless medium, in this thesis, we study howmassive MIMO technology can be used to guarantee communication security in thepresence of a passive multi-antenna eavesdropper. Our proposed massive MIMO sys-tem model incorporates relevant design choices and constraints such as time-divisionduplex (TDD), uplink training, pilot contamination, low-complexity signal process-ing, and low-cost hardware components. The thesis consists of three main parts.We rst consider physical layer security for a massive MIMO system employingsimple articial noise (AN)-aided matched-lter (MF) precoding at the base station(BS). For both cases of perfect training and pilot contamination, we derive a tightanalytical lower bound for the achievable ergodic secrecy rate, and an upper bound forthe secrecy outage probability. Both bounds are expressed in closed form, providingan explicit relationship between all system parameters, oering signicant insightsfor system design.We then generalize the work by comparing dierent types of linear data and ANiiAbstractprecoders in a secure massive MIMO network. The system performance, in termsof the achievable ergodic secrecy rate is obtained in closed form. In addition, wepropose a novel low-complexity data and AN precoding strategy based on a matrixpolynomial expansion.Finally, we consider a more realistic system model by taking into account non-idealhardware components. Based on a general hardware impairment model, we derive alower bound for the ergodic secrecy rate achieved by each user when AN-aided MFprecoding is employed at the BS. By exploiting the derived analytical bound, weinvestigate the impact of various system parameters on the secrecy rate and optimizeboth the uplink training pilots and AN precoder to maximize the secrecy rate.iiiPrefaceChapters 24 are based on works under the supervision of Professor Robert Schoberand Professor Vijay K. Bhargava.For all chapters, I conducted the paper survey on related topics, formulated theproblems, proposed problem solutions, and performed the analysis and the simula-tions of the considered communication systems. I also wrote all paper drafts.Two papers related to Chapter 2 have been published:• J. Zhu, R. Schober, and V. K. Bhargava, Secure transmission in multicellmassive MIMO systems, IEEE Transactions on Wireless Communications, vol.13, no. 9, pp. 47664781, Sept. 2014.• J. Zhu, R. Schober, and V. K. Bhargava, Secure transmission in multi-cellmassive MIMO systems, in Proceedings of IEEE Global TelecommunicationsConference (Globecom 2013) Workshop, Atlanta, GA, Dec. 2013.Three papers related to Chapter 3 have been published:• J. Zhu, R. Schober, and V. K. Bhargava, Linear precoding of data and arti-cial noise in secure massive MIMO systems, IEEE Transactions on WirelessCommunications, vol. 15, no. 3, pp. 22452261, Mar. 2016.• J. Zhu, R. Schober, and V. K. Bhargava, Secrecy analysis of multi-cell massiveMIMO systems with RCI precoding and articial noise transmission, in Pro-ivPrefaceceedings of IEEE International Symposium on Communications, Control, andSignal Processing 2014 (ISCCSP'14), Athens, Greece, May 2014.• J. Zhu, R. Schober, and V. K. Bhargava, Secure downlink transmission inmassive MIMO system with zero-forcing precoding, in Proceedings of IEEEEuropean Wireless 2014 (EW'14), Barcelona, Spain, May 2014.Two papers related to Chapter 4 have been submitted:• J. Zhu, D. W. K. Ng, N. Wang, R. Schober, and V. K. Bhargava, Analy-sis and design of secure massive MIMO systems in the presence of hardwareimpairments, submitted to possible journal, Feb. 2016.• J. Zhu, R. Schober, and V. K. Bhargava, Physical layer security for mas-sive MIMO systems impaired by phase noise, in Proceedings of IEEE Inter-national workshop on Signal Processing advances in Wireless Communications2016 (SPAWC 2016), Edinburgh, UK, July 2016.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Massive MIMO Wireless Systems . . . . . . . . . . . . . . . . . . . . 21.1.1 Time-Division Duplex and Uplink Pilot Training . . . . . . . 21.1.2 Downlink Linear Precoding . . . . . . . . . . . . . . . . . . . 41.1.3 Multi-Cell Deployment and Pilot Contamination . . . . . . . 51.2 Hardware Impairments in Massive MIMO Systems . . . . . . . . . . 7viTable of Contents1.3 Physical Layer Security . . . . . . . . . . . . . . . . . . . . . . . . . 81.4 Physical Layer Security in Massive MIMO Systems . . . . . . . . . . 101.4.1 Is Massive MIMO Secure? . . . . . . . . . . . . . . . . . . . . 111.4.2 How to Improve Security for Massive MIMO? . . . . . . . . . 121.4.3 Prior Arts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.5 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 141.6 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 172 AN-Aided MF Precoding in Secure Massive MIMO Systems . . . 202.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.2.1 System and Channel Models . . . . . . . . . . . . . . . . . . 232.2.2 Uplink Training and Channel Estimation . . . . . . . . . . . 232.2.3 Downlink Data Transmission . . . . . . . . . . . . . . . . . . 262.2.4 Design of AN Precoding Matrix An . . . . . . . . . . . . . . 282.3 Achievable Ergodic Secrecy Rate Analysis . . . . . . . . . . . . . . . 292.3.1 Achievable Ergodic Secrecy Rate . . . . . . . . . . . . . . . . 292.3.2 Lower Bound on the Achievable User Rate . . . . . . . . . . . 312.3.3 Ergodic Capacity of the Eavesdropper . . . . . . . . . . . . . 322.3.4 Tight Upper Bound on the Ergodic Capacity of the Eavesdrop-per . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.4 Performance Analysis for Perfect Training . . . . . . . . . . . . . . . 352.4.1 Lower Bound on the Achievable Ergodic Rate . . . . . . . . . 362.4.2 Impact of System Parameters on Ergodic Secrecy Rate . . . . 382.4.3 Secrecy Outage Probability Analysis . . . . . . . . . . . . . . 412.5 Performance Analysis for Pilot Contamination . . . . . . . . . . . . 42viiTable of Contents2.5.1 Lower Bound on the Achievable Ergodic Rate . . . . . . . . . 422.5.2 Impact of System Parameters on Ergodic Secrecy Rate . . . 452.5.3 Secrecy Outage Probability Analysis . . . . . . . . . . . . . . 492.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 492.6.1 Ergodic Secrecy Rate and Secrecy Outage Probability . . . . 502.6.2 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . 522.6.3 Conditions for Non-zero Ergodic Secrecy Rate . . . . . . . . . 562.6.4 Optimization of the Net Ergodic Secrecy Rate . . . . . . . . . 582.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Linear Data and AN Precoding in Secure Massive MIMO Systems 613.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.2 System Model and Preliminaries . . . . . . . . . . . . . . . . . . . . 643.2.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . 643.2.2 Channel Estimation and Pilot Contamination . . . . . . . . . 663.2.3 Ergodic Secrecy Rate . . . . . . . . . . . . . . . . . . . . . . 683.3 Linear Data Precoders for Secure Massive MIMO . . . . . . . . . . . 703.3.1 Analysis of Existing Data Precoders . . . . . . . . . . . . . . 713.3.2 POLY Data Precoder . . . . . . . . . . . . . . . . . . . . . . 753.3.3 Computational Complexity of Data Precoding . . . . . . . . . 773.4 Linear AN Precoders for Secure Massive MIMO . . . . . . . . . . . . 783.4.1 Analysis of Existing AN Precoders . . . . . . . . . . . . . . . 783.4.2 POLY AN Precoder . . . . . . . . . . . . . . . . . . . . . . . 813.4.3 Computational Complexity of AN Precoding . . . . . . . . . 823.5 Comparison of Linear Data and AN Precoders . . . . . . . . . . . . 833.5.1 Comparison of SZF, CZF, and MF Data Precoders . . . . . . 85viiiTable of Contents3.5.2 Comparison of SNS, CNS, and Random AN Precoders . . . . 863.5.3 Ergodic Secrecy Rate Analysis . . . . . . . . . . . . . . . . . 873.6 Performance Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . 893.6.1 Ergodic Capacity of the Eavesdropper for Conventional LinearAN Precoders . . . . . . . . . . . . . . . . . . . . . . . . . . 903.6.2 Ergodic Secrecy Rate for Conventional Linear Data Precoders 903.6.3 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . 943.6.4 Conditions for Non-Zero Secrecy Rate . . . . . . . . . . . . . 973.6.5 Low-Complexity POLY Data and AN Precoders . . . . . . . 993.6.6 Complexity-Performance Tradeo . . . . . . . . . . . . . . . 1013.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1034 Hardware Impairments in Secure Massive MIMO Systems . . . . 1054.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.2 System and Channel Models . . . . . . . . . . . . . . . . . . . . . . 1084.2.1 Uplink Pilot Training under Hardware Impairments . . . . . 1094.2.2 Downlink Data Transmission and Linear Precoding . . . . . . 1134.2.3 Signal Model of the Eavesdropper . . . . . . . . . . . . . . . 1144.3 Achievable Ergodic Secrecy Rate in the Presence of Hardware Impair-ments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1164.3.1 Lower Bound on Achievable Ergodic Secrecy Rate . . . . . . 1164.3.2 Asymptotic Analysis of Achievable Rate for MF Precoding . . 1184.3.3 Generalized NS AN Precoding . . . . . . . . . . . . . . . . . 1214.3.4 Upper Bound on the Eavesdropper's Capacity . . . . . . . . . 1244.4 Guidelines for System Design . . . . . . . . . . . . . . . . . . . . . . 1254.4.1 Design of the Pilot Sequences . . . . . . . . . . . . . . . . . . 125ixTable of Contents4.4.2 Selection of bo for G-NS AN Precoding . . . . . . . . . . . . 1264.4.3 Secrecy in the Absence of AN . . . . . . . . . . . . . . . . . . 1264.4.4 Maximum Number of Eavesdropper Antennas . . . . . . . . . 1284.4.5 Number of LOs . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.4.6 Are hardware impairments Benecial for Security? . . . . . . 1304.5 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.5.1 Capacity of Eavesdropper for G-NS AN Precoding . . . . . . 1324.5.2 Achievable Ergodic Rate of MT for Dierent Pilot Designs . 1334.5.3 Optimal Power Allocation to Data and AN . . . . . . . . . . 1354.5.4 Achievable Ergodic Secrecy Rate for Non-Ideal Hardware Com-ponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1354.5.5 Maximum Tolerable Number of Eavesdropper Antennas . . . 1374.5.6 Is Additive Distortion Noise at the BS Benecial for Security? 1394.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1405 Summary of Thesis and Future Research Topics . . . . . . . . . . . 1425.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1425.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1445.2.1 Physical Layer Security in Massive MIMO Systems under Con-stant Envelope Precoding . . . . . . . . . . . . . . . . . . . . 1455.2.2 Physical Layer Security in Massive MIMO Systems with Lim-ited RF-Chain Constraints . . . . . . . . . . . . . . . . . . . 1465.2.3 Physical Layer Security in Massive MIMO Systems against Ac-tive Eavesdropping . . . . . . . . . . . . . . . . . . . . . . . . 147Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149xTable of ContentsAppendicesA Proofs in Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158A.1 Proof of Lemma 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 158A.2 Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 160A.3 Proof of Theorem 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 161B Proofs in Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163B.1 Derivation of hˆknm in Section 3.2.2 . . . . . . . . . . . . . . . . . . . 163B.2 Proof of Proposition 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . 164B.3 Derivation of 1Popt . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.4 Proof of Corollary 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 167B.5 Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 168B.6 Proof of Theorem 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 171C Proofs in Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C.1 Proof of Lemma 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C.2 Proof of Lemma 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 173C.3 Proof of Lemma 4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 174C.4 Proof of Lemma 4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 176C.5 Proof of Lemma 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177xiList of Tables3.1 SINR of the kth MT in the nth cell for linear data precoding and thesimplied path-loss model in (3.36). . . . . . . . . . . . . . . . . . . . 853.2 AN leakage for simplied path-loss model in (3.36). . . . . . . . . . . 85xiiList of Figures1.1 FDD mode versus TDD mode. . . . . . . . . . . . . . . . . . . . . . . 31.2 Pilot contamination in multi-cell massive MIMO systems. . . . . . . . 51.3 Transceiver hardware model. . . . . . . . . . . . . . . . . . . . . . . . 61.4 Physical layer security model. . . . . . . . . . . . . . . . . . . . . . . 82.1 Multi-cell massive MIMO system in the presence of a multi-antennaeavesdropper. The shaded cell is the local cell. The MTs in the lo-cal cell suer from the inter-cell interference caused by data and ANtransmission in the six adjacent cells. . . . . . . . . . . . . . . . . . . 222.2 Ergodic capacity of the eavesdropper seeking to decode the informationintended for the kth MT in the local cell vs. the normalized numberof MTs in the cell, , for a system with total transmit power ei = 10dB, b = 7, ϕ = 0:75, and ci = 100. . . . . . . . . . . . . . . . . . . 512.3 Ergodic secrecy rate and outage probability for perfect training, b =7, ei = 10 dB, K = 10, / = 0:3,  = 0:1, and ϕ = 0:75. . . . . . . . . 532.4 Ergodic secrecy rate and outage probability for pilot contamination,b = 7, ei = 10 dB, K = 10 MTs, / = 0:1,  = 0:1, ϕ = 0:75,  = K,and p = eiRK. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54xiiiList of Figures2.5 Ergodic secrecy rate vs. power allocation factor ϕ assuming perfecttraining, ci = 100, b = 7, ei = 10 dB, and / = 0:1. Black circlesdenote the optimal power allocation factor, ϕ∗, obtained with (2.33)and (2.34). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552.6 Ergodic secrecy rate vs. power allocation factor ϕ assuming pilot con-tamination, b = 7, ci = 100, ei = 20 dB,  = K, p = eiRK, and/ = 0:1. Black circles denote the optimal power allocation factor, ϕ∗,obtained with (2.51) and (2.52). . . . . . . . . . . . . . . . . . . . . . 562.7 Ergodic secrecy rate and optimal power allocation factor, ϕ∗, vs.  forb = 7, ei = 20 dB, ci = 100,  = 0:3, and / = 0:1. In case of pilotcontamination,  = K and p = eiRK. The ergodic secrecy rateswere obtained with (2.27), (2.28), (2.45), and (2.46). The optimalpower allocation factor was obtained with (2.33), (2.34), (2.51), and(2.52). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.8 sec vs.  and p for pilot contamination, total transmit power ei = 20dB, b = 7, ci = 100, / = 0:1, and  = K. . . . . . . . . . . . . . . . 582.9 Net ergodic secrecy rate vs.  for a system with optimal ϕ∗, ci = 100,b = 7,  = 0:1, ei = 10 dB, p = 0 dB, and / = 0:1. Black circlesdenote the maximum net ergodic secrecy rate. . . . . . . . . . . . . . 593.1 Ergodic capacity of the eavesdropper vs. the normalized number ofMTs in the cell, , for a system with ci = 200, ϕ = 0:75, ei = 10dB, / = 0:3, and b =  = 2. . . . . . . . . . . . . . . . . . . . . . . . 913.2 Analytical and simulation results for the ergodic secrecy rate vs. thenumber of BS antennas, ci , for a lightly loaded network with ϕ = 0:75,ei = 10 dB, p = eiR ,  = 0:1, K = 10, / = 0:1, and b =  = 2. . 92xivList of Figures3.3 Analytical and simulation results for the ergodic secrecy rate vs. thenumber of BS antennas, ci , for a dense network with ϕ = 0:75, ei =10 dB,  = 2, p = eiR ,  = 0:1, K = 40, / = 0:3, and b = 7. . . . 933.4 Ergodic secrecy rate vs. ϕ for dierent selsh data precoders for anetwork with ei = 10 dB, ci = 100,  = 2, p = eiRK,  = 0:1,/ = 0:1, and b = 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 943.5 Ergodic secrecy rate vs. ϕ for dierent data precoders for a networkwith ei = 10 dB, ci = 100,  = 2, p = eiR ,  = 0:1, and  = 0:1. 963.6 Ergodic secrecy rate vs. ϕ for dierent AN precoders for a networkwith ei = 10 dB, ci = 100,  = 2, p = eiR , b = 2, / = 0:1, and = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.7 s vs.  for dierent data and AN precoders for a network with ei =10 dB, ci = 100,  = 2, p = eiR , / = 0:3, and b = 2. . . . . . . 983.8 Ergodic secrecy rate for POLY and conventional selsh data precodersfor a network employing the optimal ϕ, ei = 10 dB,  = 1, p = eiR ,ci = 200, and  = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . 1003.9 Ergodic secrecy rate for POLY and SNS AN precoders for a networkemploying the optimal ϕ, ei = 10 dB,  = 1, p = eiR , ci = 200,and  = 0:1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1013.10 Ergodic secrecy rate (left hand side) and computational complexity(right hand side) of various linear data precoders for a network em-ploying ei = 10dB, ci = 1000, p = eiR , b =  = 2, / = 0:1,i −  = 100, and an SNS AN precoder. . . . . . . . . . . . . . . . . 102xvList of Figures3.11 Ergodic secrecy rate (left hand side) and computational complexity(right hand side) of various linear AN precoders for a network em-ploying ei = 10dB, ci = 1000, p = eiR , b =  = 2, / = 0:1,i −  = 100, and an SZF data precoder. . . . . . . . . . . . . . . . . 1034.1 Uplink training and downlink transmission phase. . . . . . . . . . . . 1094.2 Capacity of the eavesdropper vs. the normalized number of MTs  fora system with c = 128, co = 4, cE = 16, ei = 10 dB, ϕ = 0:25,BSt = 0:152, and G-NS AN precoding with bo = {1P 2P 4}. . . . . . . 1324.3 Achievable ergodic rate, k, and ak vs. phase noise standard deviation = ϕ for dierent pilot designs for a system with c = 128, co = 2,cE = 16, K = 16, p = eiRK, ei = 10 dB, ϕ = 0:5, and BSt = BSr =MTt = MTr = 0:052. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1344.4 Achievable ergodic secrecy rate vs. ϕ for SO and TO pilots and asystem with K = 4 , c = 128, co = bo = 2, cE = 4, p = eiRK,ei = 10 dB, and BSt = BSr = MTt = MTr = 0:152. . . . . . . . . . . 1364.5 Achievable ergodic secrecy rate vs. number of BS antennas for G-NSAN precoding and a system with K = 4, cE = 4, co = 16, Bo = 1,p = eiRK, ei = 10 dB, and BSt = BSr = MTt = MTr = 0:152. Theoptimal ϕ is adopted. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1374.6 AN and sec vs. the normalized number of MTs  for SO and TOpilots and a system with c = 128, bo = 2, p = eiRK, ei = 10 dB, = ϕ = 6◦, and BSt = BSr = MTt = MTr = 0:152. . . . . . . . . . . 1384.7 Achievable ergodic secrecy rate vs. BS distortion noise parameter BStfor a system with c = 128, K = 32, cE = 4, co = bo = 2, p =eiRK, ei = 10 dB, and BSr = MTt = MTr = 0:152. . . . . . . . . . . 139xviList of Abbreviations5G 5th Generation3GPP 3rd Generation Partnership ProjectLTE (3GPP) Long Term EvolutionLTE-A (3GPP) Long Term Evolution - AdvancedCDMA Code Division Multiple AccessMIMO MultipleInput MultipleOutputMISO MultipleInput SingleOutputSISO SingleInput SingleOutputAWGN Additive White Gaussian NoiseBS Base StationMT Mobile TerminalAN Articial NoiseNS Null SpacePHY Physical LayerMAC Media Access ControlCDF Cumulative Distribution FunctionPDF Probability Distribution Functionr.v. Random VariableCSI Channel State InformationDPC Dirty Paper CodingxviiList of AbbreviationsFDD Frequency Division Duplexi.i.d. Independent and Identically DistributedMAC Medium Access ControlMMSE Minimum Mean Squared ErrorZF Zero-ForcingRCI Regularized Channel-InversionMF Matched-FilterPOLY PolynomialMSE Mean Squared ErrorRF Radio FrequencySIC Successive Interference CancellationSINR SignaltoInterferenceplusNoise RatioSNR SignaltoNoise RatioTDD Time Division Duplexw.r.t. With Respect toPA Power AmplierCE Constant EnvelopeSO Spatially OrthogonalTO Temporally OrthogonalxviiiNotation(·)i Transpose(·)H Hermitian transpose0 Allzero column vector1 Allone column vectorI Identity matrixZ+ The set of positive integerC The set of complex numberCm×n The space of all m× n matrices with complex-valued elementsE[·] Expectation operatorvar(·) Variance operator|| · ||2 Euclidean norm operatordiag{x} A diagonal matrix with the elements of vector x on the main diagonaltr{·} Trace of a matrixrank{·} Rank of a matrixm Upper bound for mm Lower bound for mCN(0PΣ) A circularly symmetric complex Gaussian random variablewith zero mean and covariance matrix Σ2n A chi-square random variable with n degrees of freedom[ · ]klThe element in the kth row and lth column of a matrixxixNotation[ · ]+ max{0P x}P x ∈ R⌈x⌉ The smallest integer no smaller than x⌊x⌋ The largest integer no greater than x|S| The cardinality of set SxxAcknowledgmentsFirst and foremost I would like to express my deep and sincere gratitude to mysupervisors, Professor Robert Schober and Professor Vijay K. Bhargava, for theirpatient guidance, encouragement, and invaluable advice throughout my time as theirstudent. Both of them set great examples of being a distinguished researcher andexcellent teacher. The knowledge and attitude I learned from them benet me forever.I am thankful for the opportunity to work with them over the years. Without theirsupport and guidance, this thesis would not be possible.Also, I greatly thank the members of my doctoral committee, for their time andeort in evaluating my work and providing valuable feedback and suggestions.Many thanks go to my dear laboratory colleagues at Lab Kaiser 4090 in Vancouverand IDC in Erlangen, research collaborators, and friends, for their presences and fun-loving spirits that made the otherwise grueling experience enjoyable.Finally, I would like to thank the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC), the University of British Columbia, the German Aca-demic Exchange Service (DAAD), the Alexander von Humboldt Foundation, andthe China Scholarship Council (CSC), not only for providing the funding which al-lowed me to undertake the research, but also for giving me the opportunity to attendconferences and meet so many interesting people in the world.xxiDedicationTo My Parents, Grandma, and Girlfriend Miss Xie ZhangxxiiChapter 1IntroductionThe fth generation (5G) wireless system is expected to create a paradigm shift com-pared to the current Long Term Evolution (LTE)/LTE-Advanced systems in orderto meet the unprecedented demands for future wireless applications, including thetremendous throughput and massive connectivity. Massive multiple-input multiple-output (MIMO) [1]-[8], an architecture employing large-scale multiuser MIMO pro-cessing using the array of hundreds or even thousands of antennas, simultaneouslyserving tens or hundreds of mobile users, has been identied as a promising air in-terface technology to address a signicant portion of the above challenges. Besides,security is a vital issue in wireless networks due to the broadcast nature of the medium[10]. Despite the great eorts on massive MIMO from both academia and industry,the security paradigms guaranteeing the condentiality of wireless communicationsin 5G networks have scarcely been stated. These motivate us to consider the massiveMIMO system from the security perspective. This chapter provides an overview of aseries of fundamentals related to this thesis, including massive MIMO, physical layersecurity, and hardware impairments.The chapter is organized as follows. In Sections 1.1 and 1.2, we briey review thefundamentals of ideal and non-ideal hardware constrained massive MIMO systems,respectively. In Section 1.3, we introduce the concept of physical layer security. InSection 1.4, we motivate the thesis by illustrating why we consider physical layersecurity in massive MIMO systems. The contributions conducted in this thesis are1Chapter 1. Introductionsummarized in Section 1.5, and the thesis organization is provided in Section 1.6.1.1 Massive MIMO Wireless SystemsMassive MIMO systems, also known as large-scale antenna or very large MIMOsystems, equip base station (BS) antenna arrays with an order of magnitude moreelements than what is used in current systems, i.e., a hundred antennas or more, andsimultaneously serve low-complexity single-antenna mobile terminals (MTs) [1]-[8].Massive MIMO enjoys all the benets of conventional multiuser MIMO, such as im-proved data rate, reliability and reduced interference, but at a much larger scale andwith simple linear precoding/detection schemes [1]-[3]. Remarkable improvementsin rates as well as in spectral and power eciency can be achieved by focusing theradiating power onto the MTs with the very large antenna array [6]. Massive MIMOis therefore capable of achieving robust performance at low signal-to-interference-plus-noise ratio (SINR) with highly ecient and inexpensive implementations, as theeects of noise and interference vanish completely in the limit of an innite numberof antennas [5]. Other benets of massive MIMO include but are not limited to theextensive use of inexpensive low-power components, reduced latency, simplication ofthe media access control (MAC) layer, and robustness to intentional jamming [1]-[3].This section will review the fundamentals of massive MIMO systems from the follow-ing perspectives: uplink pilot training in Section 1.1.1, downlink linear precoding inSection 1.1.2, and pilot contamination in Section 1.1.3.1.1.1 Time-Division Duplex and Uplink Pilot TrainingIn this subsection, we review the operations for channel state information (CSI)acquisition applicable for massive MIMO systems. It is well understood that the2Chapter 1. IntroductionDownlink Pilot Training(prop. to # of BS antennas)Downlink Data Trans.Full/Reduced RateFeedbackTrainingPhaseData Trans.PhaseTrainingPhaseData Trans.PhaseUplink Pilot Training(prop. to # of MTs)Downlink Data Trans.FDD TDDFigure 1.1: FDD mode versus TDD mode.acquisition of CSI is essential for signal processing at the BS. Most current cellularsystems work on frequency-division duplex (FDD) mode, where the CSI is typicallyacquired via feedback (full or limited) [11], as shown in Fig. 1.1. However, whenthe BS is equipped with large excess of antennas compared with the number ofterminals, which is customary for massive MIMO systems, the time-division duplex(TDD) mode provides the only solution to acquire CSI. This is because the trainingburden for uplink pilots in a TDD system is proportional to the number of MTs,but independent of the number of BS antennas, while conversely the training burdenfor downlink pilots in an FDD system is proportional to the number of BS antennas[5]. The adoption of an FDD system imposes a severe limitation on the numberof antennas deployed at the BS. By exploiting the reciprocity between uplink anddownlink channels for TDD systems, the BS is able to eliminate the need for feedback,and uplink pilot training is sucient for providing the desired uplink and downlinkCSI.3Chapter 1. Introduction1.1.2 Downlink Linear PrecodingWith the desired downlink CSI available via uplink training by exploiting the channelreciprocity for TDD operation, the BS performs precoding in order to simultaneouslyserve multiple single-antenna MTs. Most precoding techniques are identical to thoseused for conventional multiuser MIMO schemes, but at a much larger scale. Thetheoretical sum-capacity optimal dirty paper coding (DPC) technique [12] is toocomplex to be implemented in practice even in a conventional MIMO system, andis thus not considered here. In contrast, linear precoding is typically adopted inmassive MIMO systems. The most popular scheme is matched-lter (MF) precoding,due to its simple processing and robustness to CSI error [2, 5, 8]. However, MFprecoding results in a performance degradation with increasing number of servingMTs. This is because when more MT channels exist, the near orthogonality betweenthe MT channels becomes weak, which increases the level of multiuser interference.In this case, zero-forcing (ZF)/regularized channel inversion (RCI) precoding arepreferable [13]-[15]. Like in the conventional MIMO system, the former suppressesthe multiuser interference, while the latter strikes a balance between MF and ZFprecoding. Unfortunately, they require high-dimensional matrix inversions, whichlead to a high computational complexity, especially when the number of BS antennasand MTs are both large.In order to reduce the computational complexity induced by conventional linearprecoding techniques, e.g., ZF/RCI precoding in massive MIMO systems, the re-lated literature has also investigated precoding schemes based on matrix polynomials,which avoids the need of large dimension matrix inversion calculations. The conceptwas originally conceived for code division multiple access (CDMA) uplink systems in[16] and later extended to MIMO systems in [17]. The main idea is to adopt matrix4Chapter 1. IntroductionUplink pilotsContaminated pilotsFigure 1.2: Pilot contamination in multi-cell massive MIMO systems.polynomials with several terms to approximate matrix inversion. Thereby, the coef-cients of the matrix polynomials need to be optimized in order to achieve a certainapproximation accuracy.1.1.3 Multi-Cell Deployment and Pilot ContaminationIn massive MIMO systems, each terminal is ideally assigned to an orthogonal uplinkpilot sequence. However, the maximum number of orthogonal pilot sequences arelimited by the coherence block length. When deployed in a multi-cell network theavailable orthogonal pilot sequences are quickly exhausted. As such, pilot sequenceshave to be re-used from one cell to another. The negative eect incurred by the pilotre-use is generally called pilot contamination [1, 8], as shown in Fig. 1.2. More pre-cisely, when the BS estimates the channel of a specic MT, it correlates the receivedpilot signal with the pilot sequence of that MT. In the case of pilot reuse between5Chapter 1. IntroductionFilterLow-NoiseAmplifierMixerA/DConverterFilterLow-NoiseAmplifierMixerA/DConverterDSPLocalOscillatorLocalOscillatorLocalOscillatorLocalOscillatorPowerAmplifierPowerAmplifierMixerMixerD/AConverterD/AConverterDSPReceiveAntenna 1ReceiveAntenna NTransmitAntenna 1TransmitAntenna NReceiverCircuitTransmitterCircuitFigure 1.3: Transceiver hardware model.cells, it actually acquires a channel estimate that is contaminated by a linear com-bination of channels associated with other MTs that share the same pilot sequence.The downlink precoding based on the contaminated channel estimates introducesinterference which is directed to the MTs that share the same pilot sequence. Thedirected interference grows with the number of BS antennas at the same speed as thedesired signal. Similar interference also exists for uplink data transmission.6Chapter 1. Introduction1.2 Hardware Impairments in Massive MIMOSystemsMassive MIMO, as reviewed in Section 1.1, is able to bring substantial improvementsin spectral and energy eciency to wireless systems, due to the high spatial resolu-tion and array gain provided by large-scale antenna arrays. Thus, massive MIMOhas been identied as the promising air interface technology for 5G wireless networks.When the large-scale antenna array is deployed in practice, the hardware cost is asignicant issue to address, as it scales linearly with the number of BS antennas [18],as shown in Fig. 1.3. In order to reduce the total expenditure, one solution is toadopt low-cost transceivers, in contrast to current expensive and high-quality circuitcomponents. Unfortunately, such transceivers are usually prone to hardware imper-fections, e.g., amplier non-linearities, I/Q-imbalance, phase noise, and quantizationerrors [18]. Although such impairments can be mitigated by analog and digital signalprocessing techniques [19], they cannot be removed completely, due to the random-ness introduced by dierent sources of imperfection. Transceiver implementationsconsist of various hardware components, including lters, converters, mixers, andampliers [18]. Each of them distorts the signals in a dierent manner. Generallyspeaking, non-ideal hardware components impair the transceiver by 1) introducingthe mismatch between the intended transmit signal and what is actually emitted and2) distorting the received signal in the reception processing. From the system perfor-mance perspective, instead of the individual behavior of each hardware component,the aggregate eects of all the residual hardware impairments is more important.Among the literature, several works have investigated the impact of hardwareimpairments on massive MIMO systems [18], [20, 21, 22]. The impact of phase noise7Chapter 1. IntroductionAliceBobEveAlice-Bob-Eve ModelMulti-User Multi-Antenna Network(Employing AN)BSEveMTDataANFigure 1.4: Physical layer security model.originating from free-running oscillators on the downlink performance of massiveMIMO systems was studied in [20] for dierent linear precoder designs. Constant en-velope precoding for massive MIMO was studied in [21] with the objective of avoidingdistortions caused by power amplier nonlinearities at the transmitter. The impactof the aggregate eects of several hardware impairments originating from dierentsources on massive MIMO systems was studied in [18] by modelling the residualimpairments remaining after compensation as additive distortion noises [19]. Theauthors in [22] presented closed-form expressions for the achievable user rates inuplink massive MIMO systems for a general residual hardware impairment modelincluding both multiplicative phase noise and additive distortion noise.1.3 Physical Layer SecuritySecurity is a vital issue in wireless networks due to the broadcast nature of themedium. Traditionally, security has been achieved through cryptographic encryptionimplemented at the application layer, which requires a certain form of information8Chapter 1. Introduction(e.g., key) shared between the legitimate entities [9, 10]. This approach ignores thebehavior of the communication channels and relies on the theoretical assumptionthat communication between the legitimate entities is error free. More importantly,all cryptographic measures assume that it is computationally infeasible for them tobe deciphered without knowledge of the secret key, which remains mathematicallyunproven. Ciphers that were considered potentially unbreakable in the past are con-tinually defeated due to the increasingly growth of computational power. Moreover,error free communication cannot be always guaranteed in non-deterministic wirelesschannels [10]. A novel approach for wireless security taking advantage of the char-acteristics of physical layer communication channels was proposed by Wyner in [23]and is referred to as physical layer security. The concept was originally developed forthe classical wire-tap channel [23], cf. Fig. 1.4 (left). Wyner showed that a source(Alice)-destination (Bob) pair can exchange perfectly secure messages with a positiverate if the desired receiver enjoys better channel conditions than the eavesdropper(Eve). However, this condition cannot always hold in practice, especially in wirelessfading channels. To make things worse, Eve enjoys a better average channel gainthan Bob as long as he/she is located closer to Alice than Bob. Therefore, perfectlysecure communication seems impossible, and techniques to enhance Bob's channelcondition while degrading Eve's are needed. One option is to utilize articial noise(AN) to perturb Eve's reception [24], as shown in Fig. 1.4 (right). Eves are typi-cally passive so as to hide their existence, and thus their CSI cannot be obtainedby Alice. In this case, multiple transmit antennas can be exploited to enhance se-crecy by simultaneously transmitting both the information-bearing signal and AN.Specically, precoding is used to make the AN invisible to Bob while degrading thedecoding performance of possibly present Eves [24, 25]. In [26], authors investigated9Chapter 1. Introductionthe secrecy outage probability for the AN-aided secrecy system, where only Alice hasmultiple antennas. When Eve is also equipped with multiple antennas, the work in[27] employs AN precoder to achieve a near-optimal performance in high signal-to-noise (SNR) regime. The contribution extends to a secrecy system where all nodeshave multiple antennas in [28].More recent studies have considered physical layer security provisioning in mul-tiuser networks [29]-[36]. Although the secrecy capacity region for multiuser networksremains an open problem, it is interesting to investigate the achievable secrecy ratesof such networks for certain practical transmission strategies. All aformentioned workgenerally assumed that Alice can acquire perfect CSI of Bob, which seems too ideal.Robust beamforming designs with estimated CSI were reported in [37]-[41]. Finally,the literature on physical layer security for the emerging massive MIMO systems willbe discussed in Section 1.4.3.1.4 Physical Layer Security in Massive MIMOSystemsThe emerging massive MIMO architecture oers tremendous performance gains interms of network throughput and energy eciency by employing simple coherentprocessing on the large-scale antenna array. However, very little attention has beengiven to the security issue in massive MIMO systems. In order to address this concern,we need rst to consider two fundamental questions: 1) Is massive MIMO secure? 2)If not, how can we improve security in massive MIMO systems? In this section, weillustrate the main motivation of this thesis by providing brief and general responsesto the two questions.10Chapter 1. Introduction1.4.1 Is Massive MIMO Secure?Compared with conventional MIMO, massive MIMO is inherently more secure, as thelarge-scale antenna array equipped at the transmitter (Alice) can accurately focus anarrow and directional information beam on the intended terminal (Bob), such thatthe received signal power at Bob is several orders of magnitude higher than thatat any incoherent passive eavesdropper (Eve) [42]. Unfortunately, this benet mayvanish if Eve also employs a massive antenna array for eavesdropping. The followingscenarios further deteriorate the security of the massive MIMO system:• As Eve is passive, it is able to move arbitrarily close to Alice without beingdetected by either Alice or Bob. In this case, the signal received by Eve can bestrong.• In a ultra-dense multi-cell network, Bob suers from severe multiuser inter-ference (both pilot contaminated and uncontaminated), while Eve may haveaccess to the information of all other MTs, e.g., by collaborating with them,and remove their interference when decoding Bob's information.• In practice, both Alice and Bob are equipped with low-cost transceivers toreduce the total expenditure, which are prone to hardware imperfections, whileEve has ideal hardware.In the aforementioned scenarios, unless additional measures to secure the communi-cation are taken by Alice, even a single passive Eve is able to intercept the signalintended for Bob [43]. Furthermore, we note that Eve could emit its own pilot sym-bols to impair the channel estimates obtained at Alice to improve his ability to decodeBob's signals during downlink transmission [44]. However, this would also increase11Chapter 1. Introductionthe chance that the presence of the eavesdropper is detected by Alice [45]. Therefore,in this thesis, we limit ourselves to passive eavesdropping.1.4.2 How to Improve Security for Massive MIMO?Massive MIMO systems oer an abundance of BS antennas, while multiple transmitantennas can be exploited for secrecy enhancement, e.g., by emitting AN. Therefore,the combination of both concepts seems natural and promising. There arise sev-eral challenges and open problems for physical layer security provisioning in massiveMIMO systems that are not present for conventional MIMO systems. We summarizethem as follows.• In a conventional massive MIMO system (without security), pilot contaminationconstitutes a limit on performance in terms of data throughput [5]. However, itseects on the AN design, as well as wireless security have not been considered.• One of the tremendous advantages of massive MIMO in the physical layer is thesimple processing, e.g., MF precoding. It remains unknown if more advancedand sophisticated signal processing techniques, e.g., ZF/RCI precoding and BScollaboration are benecial in terms of data throughput and security, in a pilotcontaminated environment.• In conventional MIMO systems, AN is transmitted in the null space (NS) of thechannel matrix [24]. The complexity associated with computing the NS maynot be aordable in case of massive MIMO and thus simpler AN precodingmethods are essential.• When deployed in practice, low-cost transceivers are equipped to reduce thetotal expenditure. Such components are usually prone to hardware imperfec-12Chapter 1. Introductiontions. The eects of the imperfections on the AN design, as well as the resultingsecurity performance remains an open problem.This thesis will provide detailed and insightful solutions to the aforementioned chal-lenges and problems. As massive MIMO will serve as an essential enabling technologyfor the emerging 5G wireless networks, it is expected that its design from physicallayer security perspective opens a new and promising research path. Related contri-butions will be summarized in Section 1.4.3.1.4.3 Prior ArtsIn this section, we summarize the related work on the topic of physical layer securityfor massive MIMO systems. In [46], the authors summarized the possible researchoptions for the design of physical layer security in the emerging massive MIMO sys-tems. Large system secrecy analysis of MIMO systems achieved by RCI precodingwas provided in [47]-[49]. In [50], the authors adopted the channel between Alice andBob as secrete key and showed that the complexity required by Eve to decode Alice'smessage is at least of the same order as a worst-case lattice problem. AN-aided jam-ming for Rician fading massive MIMO channels was investigated in [51], where thepower allocation is optimized between messages and AN for both uniform and direc-tional jamming. [52] investigated power scaling law for secure massive MIMO systemswithout the help of AN. The authors in [53] dened a new term secrecy area where alllegitimate MTs satisfy the secrecy outage probability requirements within this area.An optimal power allocation strategy was performed to maximize this area. In thecontext of massive MIMO relaying, the work presented in [54] and [55] comparedtwo classic relaying schemes, i.e., amplify-and forward (AF) and decode-and-forward(DF), for physical layer security with imperfect CSI at the massive MIMO relay. Au-13Chapter 1. Introductionthors in [56] provided a large system secrecy rate analysis for simultaneous wirelessinformation and power transfer (SWIPT) MIMO wiretap channels. Whereas [51]-[56]and the contributions in this thesis all assumed that Eve is passive, the so-called pilotcontamination attack [44], a form of active eavesdropping, was also considered in theliterature. In particular, several techniques for detection of the pilot contaminationattack were proposed in [42], including a detection scheme based on random pilotsand a cooperative detection scheme. Moreover, the authors in [57] developed a secretkey agreement protocol under the pilot contamination attack, and the authors in [58]proposed to encrypt the pilot sequence in order to hide it from the attacker. The en-cryption enables the MTs to achieve the performance as if they were under no attack.Methods for combating the pilot contamination attack in a multi-cell network wasreported in [59], which exploited the low-rank property of the transmit correlationmatrices of massive MIMO channels.1.5 Contributions of the ThesisThis is the rst thesis considering physical layer security in massive MIMO systems.In this thesis, we study secure downlink transmission in single/multi-cell massiveMIMO systems in the presence of a multi-antenna eavesdropper which attempts tointercept the signal intended for any one of the MTs. To arrive at an achievable se-crecy rate for this MT, we assume that the eavesdropper can acquire perfect knowl-edge of the CSI of all user data channels and is able to cancel all interfering MTsignals. Ergodic secrecy rate and secrecy outage probability are the two performanceevaluation metrics adopted in this thesis. The research work is divided into threechapters. The contributions in each chapter are as follows:1. AN-aided MF precoding: Chapter 2 presents the rst study of physical layer14Chapter 1. Introductionsecurity in pilot contaminated massive MIMO systems. In this chapter, we de-rive tight lower bounds for the ergodic secrecy rate and tight upper bounds forthe secrecy outage probability for both cases of perfect training and pilot con-tamination, when BS employs simple MF precoding and AN precoding. Thederived bounds are in closed form and provide signicant insight for systemdesign. In particular, the obtained results allow us to predict under what con-ditions (i.e., for what number of BS antennas, eavesdropper antennas, users,path-loss, number of cells, and pilot power) a positive secrecy rate is possible.Furthermore, we show that employing random AN precoding matrices is anattractive low-complexity option for massive MIMO systems. We also derivea closed-form expression for the fraction of transmit power that should be op-timally allocated to AN and show that, for a given number of BS antennas,this fraction increases with the number of eavesdropper antennas and decreaseswith the number of users in the system. The work in Chapter 2 was publishedin [43, 60].2. Linear data and AN precoding: Chapter 3 studies the performance-complexitytradeo of selsh and collaborative data and AN precoders. Selsh precodersrequire only the CSI of the MTs in the local cell but cause inter-cell interferenceand inter-cell AN leakage. In contrast, collaborative precoders require the CSIbetween the local BS and the MTs in all cells, but reduce inter-cell interferenceand inter-cell AN leakage. However, since the additional CSI required for thecollaborative precoders can be estimated directly by the local BS, the additionaloverhead and complexity incurred compared to selsh precoders is limited. Wethen derive novel closed-form expressions for the asymptotic ergodic secrecy ratewhich facilitate the performance comparison of dierent combinations of linear15Chapter 1. Introductiondata precoders (i.e., MF, selsh and collaborative ZF/RCI) and AN precoders(i.e., random, selsh and collaborative NS), and provide signicant insight forsystem design and optimization. In order to avoid the computational complex-ity and potential stability issues in xed point implementations entailed by thelarge-scale matrix inversions required for ZF and RCI data precoding and NSAN precoding, we propose polynomial (POLY) data and AN precoders andoptimize their coecients. We borrow the tools from free probability theory[61] to obtain the POLY coecients. This allows us to express the POLY coef-cients as simple functions of the channel and system parameters. Simulationresults reveal that these precoders are able to closely approach the performanceof selsh RCI data and NS AN precoders, respectively. The work in Chapter 3was published in [62, 63, 64].3. AN-aided MF precoding with hardware imperfections: Chapter 4 presentsthe rst study of physical layer security in hardware constrained massive MIMOsystems. For the adopted generic residual hardware impairment model, we de-rive a tight lower bound for the ergodic secrecy rate achieved by a downlink userwhen MF data precoding is employed at the massive MIMO BS. The derivedbound provides insight into the impact of various system and channel parame-ters, such as the phase noise variance, the additive distortion noise parameters,the AN precoder design, the amount of power allocated to the AN, the pilotsequence design, the number of deployed local oscillators (LOs), and the num-ber of users, on the ergodic secrecy rate. As conventional NS AN precoding issensitive to phase noise, we propose a novel generalized NS (G-NS) AN precod-ing design, which mitigates the AN leakage caused to the legitimate user in thepresence of phase noise at the expense of a reduction of the available spatial de-16Chapter 1. Introductiongrees of freedom. The proposed method leads to signicant performance gains,especially in systems with large numbers of antennas at the BS. Moreover, wegeneralize the spatially orthogonal (SO) and temporally orthogonal (TO) pilotsequence designs from [22] to orthogonal pilot sequences with arbitrary numbersof non-zero elements. Although SO sequences, which have no zero elements, arepreferable for small phase noise variances, sequence designs with zero elementsbecome benecial in the presence of strong phase noise. Our analytical and nu-merical results reveal that while hardware impairments in general degrade theachievable secrecy rate, the proposed countermeasures are eective in limitingthis degradation. Furthermore, surprisingly, there are cases when the additivedistortion noise at the BS is benecial for the secrecy performance as it canhave a similar eect as AN. The work in Chapter 4 was submitted in [65, 66].1.6 Organization of the ThesisIn the following, we provide a brief overview of the remainder of this thesis. Each ofthe Chapters 2-4 in this thesis is self-contained and included in separate journal orconference papers. The notations are dened separately for each chapter.In Chapter 2, we consider physical layer security provisioning in multi-cell massiveMIMO systems. Specically, we consider secure downlink transmission in a multi-cellmassive MIMO system with MF precoding and AN precoding at the BS in the pres-ence of a passive multi-antenna eavesdropper. We investigate the resulting achievableergodic secrecy rate and the secrecy outage probability for the cases of perfect trainingand pilot contamination. Thereby, we consider two dierent AN precoding matrices,namely, the conventional AN precoding matrix, where the AN is transmitted in thenull space of the matrix formed by all user channels, and a random AN precoding17Chapter 1. Introductionmatrix, which avoids the complexity associated with nding the null space of a largematrix.In Chapter 3, we consider that linear precoding of data and AN are employedfor secrecy enhancement. Four dierent data precoders (i.e., selsh ZF/RCI and col-laborative ZF/RCI precoders) and three dierent AN precoders (i.e., random, self-ish/collaborative null-space based precoders) are investigated and the correspondingachievable ergodic secrecy rates are analyzed. Our analysis includes the eects of up-link channel estimation, pilot contamination, multi-cell interference, and path-loss.Furthermore, to strike a balance between complexity and performance, linear pre-coders that are based on matrix polynomials are proposed for both data and ANprecoding. The polynomial coecients of the data and AN precoders are optimizedrespectively for minimization of the sum MSE of and the AN leakage to the mobileterminals in the cell of interest using tools from free probability and random matrixtheory.In Chapter 4, we investigate the impact of hardware impairments on the secrecyperformance of downlink massive MIMO systems in the presence of a passive multiple-antenna eavesdropper. Thereby, for the BS and the legitimate users, the joint eectsof multiplicative phase noise, additive distortion noise, and amplied receiver noiseare taken into account, whereas the eavesdropper is assumed to employ ideal hard-ware. We derive a lower bound for the ergodic secrecy rate of a given user when MFdata precoding and AN transmission are employed at the BS. Based on the derivedanalytical expression, we investigate the impact of the various system parameters onthe secrecy rate and optimize both the pilot sets used for uplink training and the ANprecoding.Finally, Chapter 5 summarizes the contributions of this thesis and outlines areas18Chapter 1. Introductionof future research.Appendices A-C contain the proofs of the propositions, corollaries, lemmas, andtheorems used in this thesis.19Chapter 2AN-Aided MF Precoding in SecureMassive MIMO Systems2.1 IntroductionMassive MIMO systems oer an abundance of BS antennas, while multiple transmitantennas can be exploited for secrecy enhancement. Therefore, the combination ofboth concepts seems natural and promising, which is the main motivation for thework presented in this chapter. Several new issues arise for physical layer securityprovisioning in multi-cell massive MIMO systems that are not present for conven-tional MIMO systems [10, 23, 27]-[38]. For example, pilot contamination is uniqueto massive MIMO systems and we study its eect on the ergodic secrecy rate andthe secrecy outage probability. Furthermore, for the user data, MF precoding is usu-ally adopted in massive MIMO systems [2, 5], since the matrix inversion needed forthe schemes used in conventional MIMO, such as RCI and minimum mean squarederror (MMSE) precoding, is considered to be computationally too expensive for thelarge matrices typical for massive MIMO. Similarly, whereas in conventional MIMOsystems, the AN is transmitted in the NS of the channel matrix [24], the complexityassociated with computing the NS may not be aordable in case of massive MIMOand simpler AN precoding methods may be needed. Finally, unlike most of the re-lated work [10, 23, 27]-[38], we consider a multi-cell setting where not only the data20Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemssignals cause inter-cell interference but also the AN, which has to be carefully takeninto account for system design.In this chapter, we study secure downlink transmission in multi-cell massiveMIMO systems in the presence of a multi-antenna eavesdropper, which attempts tointercept the signal intended for one of the users. To arrive at an achievable secrecyrate for this user, we assume that the eavesdropper can acquire perfect knowledgeof the CSI of all user data channels and is able to cancel all interfering user signals.Under this assumption, we derive tight lower bounds for the ergodic secrecy rateand tight upper bounds for the secrecy outage probability for the cases of perfecttraining and pilot contamination. The derived bounds are in closed form and providesignicant insight for system design. In particular, the obtained results allow us topredict under what conditions (i.e., for what number of BS antennas, eavesdropperantennas, users, path-loss, number of cells, and pilot powers) a positive secrecy rateis possible. Furthermore, we show that employing random AN precoding matricesis an attractive low-complexity option for massive MIMO systems. We also derive aclosed-form expression for the fraction of transmit power that should be optimallyallocated to AN and show that, for a given number of BS antennas, this fractionincreases with the number of eavesdropper antennas and decreases with the numberof users in the system.The remainder of this chapter is organized as follows. In Section 2.2, we describethe channel model, the channel estimation scheme, the transmission format, andtwo AN precoding matrix designs for the considered system. In Section 2.3, weprovide a simple lower bound on the achievable ergodic rate of the MT, a closed-form expression for the ergodic capacity of the eavesdropper, and a simple and tightupper bound for the ergodic capacity of the eavesdropper. In Sections 2.4 and 2.5, we21Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems/RFDO&HOO%6 (9( 07Figure 2.1: Multi-cell massive MIMO system in the presence of a multi-antennaeavesdropper. The shaded cell is the local cell. The MTs in the local cell suer fromthe inter-cell interference caused by data and AN transmission in the six adjacentcells.analyze the secrecy performance of the considered downlink multi-cell massive MIMOsystem for cases of perfect training and pilot contamination, repsectively. Analyticaland simulation results are presented in Section 2.6, and the chapter is concluded inSection 2.7.2.2 System ModelIn this section, we introduce the channel model, the channel estimation scheme, thetransmission format, and two AN precoding matrix designs for the considered securemulti-cell massive MIMO system.22Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.2.1 System and Channel ModelsIn this chapter, we consider a at-fading multi-cell system consisting of b cells, asdepicted in Fig. 2.1. Each cell comprises an ci -antenna BS and K single-antennaMTs1. The nth cell, n ∈ {1P : : : Pb}, is the local cell (the shaded area in Fig. 2.1).An eavesdropper equipped with cE antennas (equivalent to cE cooperative single-antenna eavesdroppers) is located in the local cell of the considered multi-cell region.The eavesdropper is passive and seeks to recover the information transmitted to thekth MT in the local cell. Let gkmn ∈ C1×cT and GmE ∈ CcE×cT denote the channelbetween the mth BS, m = 1P : : : Pb , and the kth MT in the local cell and the channelbetween the mth BS and the eavesdropper, respectively. gkmn =√kmnhkmn comprisesthe path-loss, kmn, and the small-scale fading vector, hkmn ∼ CN(0icT P IcT ). Similarly,we model the eavesdropper channel as GmE =√mEHmE, where mE and HmEdenote the path-loss and small-scale fading components, respectively. The elementsof HmE are modeled as independent and identically distributed (i.i.d.) Gaussianrandom variables (r.v.s) with zero mean and unit variance.2.2.2 Uplink Training and Channel EstimationWe assume that the BSs are perfectly synchronized and operate in the TDD modewith universal frequency reuse. Furthermore, we assume that the path-losses betweenall users in the system and the local BS, knm, m = 1P : : : Pb , k = 1P : : : K, areknown at the local BS, whereas the small-scale fading vectors hknm, m = 1P : : : Pb ,k = 1P : : : K, are not known and the local BS estimates only the small-scale fading1We note that the results derived in this chapter can be easily extended to multi-antenna MTs ifthe BS transmits one independent data stream per MT receive antenna and receive combining is notperformed at the MTs. In this case, each MT receive antenna can be treated as one (virtual) MT andthe results derived in this chapter are applicable. For example, the secrecy rate of a multi-antennaMT can be obtained by summing up the secrecy rates of its receive antennas.23Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsvectors of the MTs within the local cell. These assumptions are motivated by the factthat the path-losses change on a much slower time scale than the small-scale fadingvectors, and thus, their estimation creates a comparatively low overhead.The local BS estimates the downlink CSI of all MTs, hknn, k = 1P : : : P K, byexploiting reverse training and channel reciprocity [1]-[8]. We consider two scenarios:Perfect training and imperfect training which results in pilot contamination [8]. In theformer case, allbK MTs in the system emit orthogonal pilot sequences in the trainingphase having a suciently large pilot power p such that hˆknn = hknn, k = 1P : : : P ci ,can be assumed, where hˆknn denotes the estimated channel in the local cell. In thelatter case, the K pilot sequences used in a cell are still orthogonal but all cells usethe same pilot sequences. Let√!k ∈ C×1 denote the pilot sequence of length transmitted by the kth MT in each cell in the training phase, where !Hk !k = 1 and!Hk !j = 0, ∀P jP k = 1P : : : P K, k ̸= j. Assuming perfect synchronization, the trainingsignal received at the local BS, Ypilotn ∈ C×cT , can be expressed asYpilotn =b∑m=1K∑k=1√pknm!khknm +NnP (2.1)where Nn ∈ C×cT is a Gaussian noise matrix having zero mean, unit varianceelements. Assuming MMSE channel estimation [7, 8], the estimate of hknn givenYpilotn is obtained ashˆknn =√pknn!Hk(I + !k(pb∑m=1knm)!Hk)−1Ypilotn=√pknn1 + p∑bm=1 knm!Hk Ypilotn : (2.2)For MMSE estimation, we can express the channel as hknn = hˆknn + h˜knn, where24Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsthe estimate hˆknn and the estimation error h˜knn ∈ C1×cT are mutually indepen-dent. Hence, considering (2.2) we can statistically characterize hˆknn and h˜knn ashˆknn ∼ CN(0icT Pp knn1+p ∑Mm=1 knmIcT)and h˜knn ∼ CN(0icT P1+p ∑m ̸=n knm1+p ∑Mm=1 knmIcT), re-spectively. Still from (2.2), we also observe that !Hk Ypilotn is proportional to theMMSE estimate of hknm for any m, i.e.,hˆknm‖hˆknm‖=!Hk Ypilotn‖!Hk Ypilotn ‖P∀m: (2.3)Eq. (2.3) implies that the estimate of the kth MT in each cell is simply a scaledversion of the same vector !Hk Ypilotn . Hence, the BS is not able to distinguish be-tween the channel to its kth MT and to the kth MT in other cells [8]. In the samemanner, we also expand the channel hkmn = hˆkmn + h˜kmn,2where hˆkmn and h˜kmnare mutually independent. We also have hˆkmn ∼ CN(0icT Pp kmn1+p ∑Ml=1 kmlIcT)andh˜kmn ∼ CN(0icT P1+p ∑l̸=n kml1+p ∑Ml=1 kmlIcT), respectively.In order to be able to nd the required numbers of orthogonal pilot sequences, pilotsequence lengths of  ≥bK and  ≥ K are required for the cases of perfect trainingand pilot contamination, respectively. Furthermore, we note that the eavesdroppercould emit his own pilot symbols to impair the channel estimates obtained at the BSto improve his ability to decode the MTs' signals during downlink transmission [44].However, this would also increase the chance that the presence of the eavesdropperis detected by the BS [45]. Therefore, in this chapter, we assume the eavesdropper ispurely passive and leave the study of active eavesdroppers in massive MIMO systemsfor future work.2In this chapter, the local BS only needs to estimate hknn. The role of this expansion is tofacilitate a mathematical simplication in deriving the achievable rate in Section 2.5 for the case ofpilot contamination, by decomposing the inter-cell interference/AN leakage from the mth cell intocorrelated terms h^kmn and uncorrelated terms~hkmn with respect to (w.r.t.) the desired MT's channelestimate.25Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.2.3 Downlink Data TransmissionIn the local cell, the BS intends to transmit a condential signal snk to the kthMT.The signal vector for the K MTs is denoted by sn =[sn1P : : : P snK]i ∈ CK×1 withE[snsHn ] = IK . Each signal vector sn is multiplied by a transmit beamforming ma-trix, Fn = [fn1P : : : P fnkP : : : P fnK ] ∈ CcT×K , before transmission. As typical for massiveMIMO systems, we adopt simple MF precoding, i.e., fnk = (hˆknn)HR‖hˆknn‖ [5],[8], sincethe matrix inversion required for ZF and MMSE precoding is computationally too ex-pensive for the large number of users and antenna elements that are typical for massiveMIMO systems. Furthermore, we assume that the eavesdropper's CSI is not availableat the local BS. Hence, assuming that there are K Q ci MTs, the BS may use the re-maining ci −K degrees of freedom oered by the ci transmit antennas for emissionof AN to degrade the eavesdropper's ability to decode the data intended for the MTs[24, 37, 38]. The AN vector, zn = [zn1P : : : P zn(cT−K)]i ∼ CN(0cT−K P IcT−K), is mul-tiplied by an AN precoding matrix An = [an1P : : : P aniP : : : P an(cT−K)] ∈ CcT×(cT−K)with ‖ani‖ = 1, i = 1P : : : P ci − K. The considered choices for the AN precodingmatrix will be discussed in the next subsection. The signal vector transmitted by thelocal BS is given byxn =√pFnsn +√qAnzn =K∑k=1√pfnksnk +cT−K∑i=1√qanizniP (2.4)where p and q denote the transmit power allocated to each MT and each AN signal,respectively, i.e., for simplicity, we assume uniform power allocation across users andAN signals, respectively. Let the total transmit power be denoted by ei . Then, pand q can be represented as p = ϕeTKand q = (1−ϕ)eTcT−K , respectively, where the powerallocation factor ϕ, 0 Q ϕ ≤ 1, strikes a power balance between the information-26Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsbearing signal and the AN.The b − 1 cells adjacent to the local cell transmit their own signals and AN. Inthis work, in order to be able to gain some fundamental insights, we assume thatall cells employ identical values for p and q as well as ϕ. Accordingly, the receivedsignals at the kth MT in the local cell, ynk, and at the eavesdropper, yE, are given byynk =√phknnfnksnk +∑{mPl}̸={nPk}√phkmnfmlsml +b∑m=1√qhkmnAmzm + nnk (2.5)andyE =√pb∑m=1HmEFmsm +√qb∑m=1HmEAmzm + nEP (2.6)respectively, where nnk ∼ CN(0P 2nk) and nE ∼ CN(0cE P 2EIcE) are the Gaussiannoises at the kth MT and at the eavesdropper, respectively. The rst term on theright hand side of (2.5) is the signal intended for the kth MT in the local cell witheective channel gain√phknnfnk, which is assumed to be perfectly known at the kthMT in the local cell. The second and the third terms on the right hand side of(2.5) represent intra-cell/inter-cell interference and AN leakage, respectively. On theother hand, the eavesdropper observes an bci ×cE MIMO channel comprising Klocal user signals, (b − 1)K out-of-cell user signals, ci − K local cell AN signals,and (ci −K)(b − 1) out-of-cell AN signals. In order to obtain a lower bound onthe achievable secrecy rate, we assume that the eavesdropper can acquire perfectknowledge of the eective channels of all MTs, i.e., HmEfmkP∀mP k. We note howeverthat this is a quite pessimistic assumption because the uplink training performed inmassive MIMO [8] makes it dicult for the eavesdropper to perform accurate channelestimation.27Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.2.4 Design of AN Precoding Matrix AnIn this chapter, we consider two dierent designs for the AN precoding matrix An.NS method: For conventional (non-massive) MIMO, An is usually chosen to liein the null space of the estimated channel, hˆknn, i.e., hˆknnAn = 0icT−K , k = 1P : : : P K,which is possible as long as ci > K holds [24]. We refer to this method as N inthe following. If perfect CSI is available, i.e., hˆknn = hknnP the N -method preventsimpairment of the users in the local cell by AN generated by the local BS. In case ofpilot contamination, intra-cell AN leakage to the users in the local cell is unavoidable,but inter-cell AN leakage is suppressed due to pilot contamination, which is an extrabenet, and will be discussed in Section 2.5 in details. Unfortunately, for the largevalues of ci and K typical for massive MIMO systems, computation of the NS ofhˆknn, k = 1P : : : P K, is computationally expensive. This motivates the introduction ofa simpler method for generation of the AN precoding matrix.Random method: In this case, the columns of An are mutually independentrandom vectors. We refer to this method as R in the following. Here, we constructthe columns of An as ani = a˜niR‖a˜ni‖, where the a˜ni, i = 1P : : : P ci −K, are mutu-ally independent Gaussian random vectors. Note that the R-method does not evenattempt to avoid AN leakage to the users in the local cell. However, it may still im-prove the ergodic secrecy rate as the precoding vector for the desired user signal, fnk,is correlated with the user channel, hknn, whereas the columns of the AN precodingmatrix are not correlated with the user channel.Our results in Sections 2.4-2.6 reveal that although the N -method always achievesa better performance than the R-method, if pilot contamination and inter-cell inter-ference are signicant, the performance dierences between both schemes are small.This makes the R-method an attractive alternative for massive MIMO systems due28Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsto its simplicity.2.3 Achievable Ergodic Secrecy Rate AnalysisIn this section, we rst show that the achievable ergodic secrecy rate of the kth MTin the local cell can be expressed as the dierence between the achievable ergodicrate of the MT and the ergodic capacity of the eavesdropper. Subsequently, weprovide a simple lower bound on the achievable ergodic rate of the MT, a closed-form expression for the ergodic capacity of the eavesdropper, and a simple and tightupper bound for the ergodic capacity of the eavesdropper. The results derived in thissection are valid for both perfect training and pilot contamination as well as for bothAN precoding matrix designs. For convenience, we dene the ratio of the numberof eavesdropper antennas and the number of BS antennas as  = cERci , and theratio of the number of users and the number of BS antennas as  = KRci . In thefollowing, we are interested in the asymptotic regime where ci → ∞ but  and are constant.2.3.1 Achievable Ergodic Secrecy RateThe ergodic secrecy rate is an appropriate performance measure if delays can beaorded and coding over many independent channel realizations (i.e., over manycoherence intervals) is possible [25]. Considering the kth MT in the local cell, theconsidered channel is an instance of a multiple-input, single-output, multiple eaves-dropper (MISOME) wiretap channel [27]. In the following lemma, we provide anexpression for an achievable ergodic secrecy rate of the kth MT in the local cell.Lemma 2.1. An achievable ergodic secrecy rate of the kth MT in the local cell is29Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsgiven bygsecnk = [gnk − Xevenk ]+P (2.7)where [x]+ = max{0P x}, gnk is an achievable ergodic rate of the kth MT in the localcell, and Xevenk is the ergodic capacity between the local BS and the eavesdropper seekingto decode the information of the kth MT in the local cell. Thereby, it is assumed thatthe eavesdropper is able to cancel the received signals of all in-cell and out-of-cell MTsexcept the signal intended for the MT of interest, i.e.,Xevenk = E[log2(1 + pfHnkGHnEX−1GnEfnk) ]P (2.8)where X = q∑bm=1AHmGHmEGmEAm denotes the noise correlation matrix at theeavesdropper under the worst-case assumption that the receiver noise is negligible,i.e., 2E → 0.Proof. Please refer to Appendix A.1.Eq. (2.7) reveals that the achievable ergodic secrecy rate of the kth MT in thelocal cell has the subtractive form typical for many wiretap channels [10, 23, 27]-[38],i.e., it is the dierence of an achievable ergodic rate of the user of interest and thecapacity of the eavesdropper. Before we analyze (2.7) for perfect training and pilotcontamination in Sections 2.4 and 2.5, respectively, we derive general expressions forgnk and Xevenk , which apply to both cases.30Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.3.2 Lower Bound on the Achievable User RateBased on (2.5) an achievable ergodic rate of the kth MT in the local cell is given bygnk =E[log2(1 +|√pgknnfnk|2∑bm=1∑cT−Ki=1 |√qgkmnami|2 +∑{mPl}̸={nPk} |√pgkmnfml|2 + 2nk)]:(2.9)Unfortunately, evaluating the expected value in (2.9) analytically is cumbersome.Therefore, we derive a lower bound on the achievable ergodic rate of the kth MT inthe local cell by following the same approach as in [8]. In particular, we rewrite thereceived signal at the kth MT in the local cell asynk = E[√pgknnfnk]snk + n′nkP (2.10)where n′nk represents an eective noise, which is given by n′nk =(√pgknnfnk − E[√pgknnfnk])snk +b∑m=1gkmn√qAmzm +∑{mPl}≠{nPk}√pgkmnfmlsml + nnk:(2.11)Eq. (2.10) can be interpreted as an equivalent single-input single-output channel withconstant gain E[√pgknnfnk] and AWGN n′nk. Hence, we can apply Theorem 1 in [8]to obtain a computable lower bound for the achievable rate of the kth MT in thelocal cell as gnk = log2(1 + nk) ≤ gnk, where nk denotes the received signal-to-31Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsinterference-plus-noise ratio (SINR), given by nk =desired signal︷ ︸︸ ︷|E[√pgknnfnk]|2var[√pgknnfnk]︸ ︷︷ ︸signal leakage+b∑m=1cT−K∑i=1E[|√qgkmnami|2]︸ ︷︷ ︸AN leakage+∑{mPl}≠{nPk}E[|√pgkmnfml|2]︸ ︷︷ ︸intra- and inter-cell interference+ 2nk(2.12)with var[√pgknnfnk] = E[|√pgknnfnk − E[√pgknnfnk]|2]. We note that the derived lowerbound on the achievable rate is applicable to both AN precoding matrix designs andthe cases of perfect training and pilot contamination, respectively, cf. Sections 2.4and 2.5. The tightness of the lower bound will be conrmed by our results in Section2.6.2.3.3 Ergodic Capacity of the EavesdropperIn this section, we provide a closed-form expression for the ergodic capacity of theeavesdropper valid for both perfect training and pilot contamination. To gain moreinsight, we adopt a simplied path-loss model for the eavesdropper, i.e., the path-losses between the BSs and the eavesdropper are given by mE = 1 if n = m andmE = / if n ̸= m, i.e., the path-loss between the local BS and the eavesdropperis 1 and the path-loss between the BSs of the other cells and the eavesdropper is/ ∈ [0P 1].3 A similar simplied path-loss model was used in [7] for the user channels.The resulting ergodic secrecy capacity is summarized in the following theorem.Theorem 2.1. For ci →∞ and both the N and the R AN precoding matrix designs,3We note that the simplied path-loss model is only adopted to reduce the number of parameters.The ergodic capacity and the ergodic secrecy rate can also be derived for the original path-loss modelin closed form. However, the resulting equations are more cumbersome and less insightful comparedto those for the simplied model.32Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsthe ergodic capacity of the eavesdropper in (2.8) can be written asXevenk =1ln 2cE−1∑i=0i × 102∑j=1bj∑l=2!jlI(1RjP l)P (2.13)where i =(b(cT−K)i), 0 =∏2j=1 bjj ,(jP bj) =(Pci −K)P j = 1(/P (b − 1)(ci −K))P j = 2P(2.14) = qRp,!jl =1(bj − l)!ybj−lyxbj−l(xi∏s̸=j(x+1s)bs)∣∣∣∣x=− 1jP (2.15)and I(aP n) =∫∞01(x+1)(x+a)nyxP aP n > 0. A closed-form expression for I(·P ·) is givenin [67, Lemma 3].Proof. Please refer to Appendix A.2.A lower bound on the achievable ergodic secrecy rate of the kth MT in the local cellfor the N /R methods is obtained by combining (2.7), (2.12), and (2.13). However,the expression for the ergodic capacity of the eavesdropper in (2.13) is somewhatcumbersome and oers little insight into the impact of the various system parameters.Hence, in the next subsection, we derive a simple and tight upper bound for Xevenk .2.3.4 Tight Upper Bound on the Ergodic Capacity of theEavesdropperIn the following theorem, we provide a tight upper bound for the ergodic capacity ofthe eavesdropper.33Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO SystemsTheorem 2.2. For ci →∞ and both the N and the R AN precoding matrix gener-ation methods, the ergodic capacity of the eavesdropper in (2.8) is upper bounded by4Xevenk Q Xevenk ≈ log2(1 +a(1− )− xRa)= log2((1− )ϕ+ −ϕ+ )P (2.16)if  Q 1−xRa2, where we introduce the denitions a = 1+/(b−1), x = 1+/2(b−1),and  = a− xa(1−) .Proof. Please refer to Appendix A.3.Remark 2.1. We note that a nite eavesdropper capacity results only if matrix Xin (2.8) is invertible. Since GmE, m = 1P : : : Pb , are independent matrices withi.i.d. entries, X is invertible if b(ci − K) ≤ cE or equivalently  ≤ 1 − Rb .Regardless of the values of b and /, we have1− R[1 + /2(b − 1)] ≤ 1− xRa2 ≤ 1− Rb: (2.17)For b = 1 or / = 1, equality holds in (2.17). For b > 1 and / Q 1, the conditionfor  in Theorem 2.2 is in general stricter than the invertibility condition for X.Nevertheless, the typical operating region for a massive MIMO system is  ≪ 1[2, 5], where the upper bound in Theorem 2.2 is applicable.Eq. (2.16) reveals that Xevenk is monotonically increasing in , i.e., as expected, theeavesdropper can enhance his eavesdropping capability by deploying more antennas.Furthermore, in the relevant parameter range, 0 Q  Q 1 − xRa2, Xevenk is not4We note that, strictly speaking, we have not proved that (2.16) is a bound since we used anapproximation for its derivation, see Appendix C. However, this approximation is known to bevery accurate [69] and comparisons of (2.16) with simulation results for various system parameterssuggest that (2.16) is indeed an upper bound.34Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsmonotonic in  but a decreasing function for  ∈ (0P 1 − √xRa) and an increasingfunction for  ∈ (1 − √xRaP 1 − xRa2). Hence, Xevenk has a minimum at  =1 − √xRa. Assuming ci and cE are xed, this behaviour can be explained asfollows. For small K (corresponding to small ), the capacity of the eavesdropper islarge because the amount of power allocated to the intercepted MT, ϕeiRK, is large.As K increases, the power allocated to the MT decreases which leads to a decreasein the capacity. However, if K is increased beyond a certain point, X becomesincreasingly ill-conditioned which leads to an increase in the eavesdropper capacity.Combining now (2.7), (2.12), and (2.16) gives a tight lower bound on the ergodicsecrecy rate of the kth MT in the local cell for both the N and the R methods.To gain more insight, in the next two sections, we specialize the tight lower boundon the ergodic secrecy rate to the cases of perfect training and pilot contamination,respectively. This will allow us to further simplify the SINR expression of the kthMT in the local cell and the resulting ergodic secrecy rate expression.2.4 Performance Analysis for Perfect TrainingIn this section, we analyze the secrecy performance of the considered downlink multi-cell massive MIMO system under the assumption of perfect CSI, i.e., hˆknn = hknn,k = 1P : : : P K. To this end, for both considered AN precoding methods, we rstsimplify the lower bound on the achievable ergodic rate expression derived in Section2.3.2 by taking into account the perfect CSI assumption. Subsequently, exploitingthis result, we derive simple and insightful lower bounds on the achievable ergodicsecrecy rate. Finally, we obtain an upper bound on the secrecy outage probability.35Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.4.1 Lower Bound on the Achievable Ergodic RateWe rst characterize some of the terms in (2.12) for the case of perfect training inthe following lemma.Lemma 2.2. The received signal and interference powers at the kth MT in the localcell can be expressed asE[hknnfnk]2 = E2[x] and E[|hknnfmk|2] = E[|hknnami|2] = E[y2]P ∀n ̸= m (2.18)respectively, where x2 =∑cTl=1 |ul|2 ∼ 22cT , y2 = |ul|2 ∼ 22, ul are i.i.d. complexGaussian r.v.s with zero mean and unit variance, and E[y2] = 1.Proof. Since each element of hknn follows a Gaussian distribution with zero mean andunit variance and fnk =(hknn)H‖hknn‖ =(hknn)H‖hknn‖ , |hknnfnk|2 is a (scaled) chi-square r.v. with2ci degrees of freedom and statistically equivalent to x2. On the other hand, sincefml, ∀{mP l} ̸= {nP k}, and ami are unit-norm vectors and independent of the small-scale fading vector hknn, the normalized interference terms, |hknnfmk|2 and |hknnami|2,are (scaled) chi-square r.v.s with 2 degrees of freedom and statistically equivalent toy2.Introducing x and y in (2.12) and dividing both numerator and denominator byp, we obtain the SINRs for the N and R AN precoding matrices asNnk =knnE2[x]knnvar[x] + ∑bm̸=n kmn∑cT−Ki=1 E[y2] +∑{mPl}̸={nPk} kmnE[y2] + KϕeT(2.19)andRnk =knnE2[x]knnvar[x] + ∑bm=1 kmn∑cT−Ki=1 E[y2] +∑{mPl}≠{nPk} kmnE[y2] + KϕeTP (2.20)36Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsrespectively. The right hand sides of (2.19) and (2.20) dier only in the second termof the denominator, where Rnk contains an additional term knn∑cT−Ki=1 E[y2], whichis due to the AN leakage caused in the local cell. This term is absent in Nnk as, forperfect CSI, the N -method avoids AN leakage in the local cell. Hence, Nnk > Rnkalways holds. Since for large ci we have [8]limcT→∞E2[x]ci= 1 and limcT→∞var[x]ci= 0P (2.21)we obtain from (2.19) and (2.20)limcT→∞Nnk =knnci∑bm̸=n kmn(ci −K) +∑{mPl}≠{nPk} kmn +KϕeT(2.22)andlimcT→∞Rnk =knnci∑bm=1 kmn(ci −K) +∑{mPl}≠{nPk} kmn +KϕeTP (2.23)respectively. In order to obtain simple yet insightful results, we adopt in the followinga simplied path-loss model [7], similar to the simplied model introduced for theeavesdropper in Section 2.3.3. In particular, we model the path-losses as kmn = 1 ifn = m and kmn = / if n ̸= m, i.e., the path-loss between the local BS and the MTsin the local cell is 1 and the path-loss between the BSs of the other cells and the MTsin the local cell is /. Hence, (2.22) and (2.23) simplify tolimcT→∞Nnk =1(b − 1)/(1− ) + (b − 1)/+  + ϕeT(2.24)andlimcT→∞Rnk =1((b − 1)/+ 1)(1− ) + (b − 1)/+  + ϕeTP (2.25)respectively. The ergodic rate for the two considered AN precoding matrix generation37Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsmethods is lower bounded by gΨnk = log2(1 + Ψnk), where Ψ ∈ {N PR}. We note thatfor systems with few users, i.e.,  → 0, and ci → ∞, the lower bounds on theergodic rate reduce togNnk ≈ log2(1 +1(b − 1)/)and gRnk ≈ log2(1 +1((b − 1)/+ 1))P (2.26)i.e., performance is limited by AN leakage. This is in contrast to massive MIMOsystems without AN precoding, whose performance in the considered regime ( → 0)is only limited by pilot contamination [2, 5], which is not considered in this sectionbut will be addressed in Section V. Moreover, (2.26) suggests that the performancedierence between the N -method and the R-method diminishes if the AN leakagefrom adjacent cells, which is proportional to (b − 1)/ for both methods, dominatesthe AN leakage for the R-method in the local cell, which is proportional to .Closed-form expressions for the lower bound on the achievable ergodic secrecyrate of the kth MT in the local cell for the N /R methods are obtained by combining(2.7), (2.13), and (2.24)/(2.25). The tightness of the proposed lower bounds will beconrmed in Section 2.6 via simulations.2.4.2 Impact of System Parameters on Ergodic Secrecy RateIn this subsection, we provide insight into the inuence of the various system pa-rameters on the ergodic secrecy rate. Combining (2.7), (2.24)/(2.25), and the upperbound on the ergodic secrecy capacity in (2.16), simple lower bounds for the ergodic38Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemssecrecy rate valid for ci →∞ are obtained asgsecPNnk =[log2(b + ( + 1− b)ϕ− ( + 1)ϕ2b + [(1− ) + b]ϕ+ (1− )ϕ2)]+P (2.27)gsecPRnk =[log2((b+ 1) + [1− (b+ 1)]ϕ− ϕ2(b+ 1) + (b+ 1)(1− )ϕ)]+P (2.28)where b = (b − 1)/ + 1Rei and  = qRp = (1Rϕ − 1)R(1 − ) was used. In thefollowing, we rst investigate for what values of  a non-zero ergodic secrecy rate canbe achieved.Impact of : Let us denote the upper limit for  such that a positive secrecyrate can be achieved as sec. For the N -method and the R-method, we obtain from(2.27) and (2.28), respectively, positive secrecy rates if  Q Ψsec, Ψ ∈ {N PR}, withNsec =a2(1− )ab(1− ) + x→0=ab+ xRa=1 + /(b − 1)1Rei + /(b − 1) + xRa (2.29)andRsec =a2(1− )a(b+ 1)(1− ) + x→0=ab+ 1 + xRa=11 + 1R[ei (/(b − 1) + 1)] + xRa2 :(2.30)In both cases, Ψsec is obtained for ϕ → 0, i.e., almost the entire transmit poweris allocated to AN precoding. For both methods, sec is monotonically decreasingin . Furthermore, we always have Rsec Q Nsec, i.e., the N -method can toleratea larger number of eavesdropper antennas than the R-method at the expense of ahigher complexity in calculating the AN precoding matrix. The robustness of bothAN precoding matrix designs can be improved by increasing the transmit powerei . However, based on (2.29) and (2.30) it can be shown that even for ei → ∞,the maximum values of  that yield a non-zero ergodic secrecy rate are limited as39Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO SystemsNsec ≤ 4R3 and Rsec ≤ 1 regardless of the choice of b and /. We note that fora single-cell system with a single user, it was shown in [27] that the N -method canachieve non-zero secrecy rate for  Q 2. The smaller number of tolerable eavesdropperantennas in the considered massive MIMO system are caused by the suboptimal MFprecoding at the BS, which was chosen for complexity reasons. More sophiscatedprecoding techniques will be considered in Chapter 3.Impact of ϕ: Eqs. (2.27) and (2.28) reveal that zero secrecy rate results forϕ = ϕ0 = 0 and for a second value ϕ = ϕΨ1 , 0 Q ϕΨ1 Q 1, where Ψ ∈ {N PR}.Specically, ϕΨ1 is given byϕN1 = 1−a(1− )(b+ 1)a2(1− )(1 + Ra)− x (2.31)ϕR1 = 1−a(1− )(b+ 1)a2(1− )− x (2.32)where ϕΨ1 Q 1 follows from the condition  Q 1 − xRa2 which is required for thevalidity of the upper bound on the ergodic secrecy capacity in (2.16). For ϕ = 0, allpower is allocated to AN precoding and no power is left for information transmission.On the other hand, for ϕ = ϕΨ1 , the amount of AN generated is not sucient toprevent the eavesdropper from decoding the transmitted signal. This suggests thatfor  Q Ψsec, Ψ ∈ {N PR}, there exists an optimal ϕ, 0 Q ϕ Q ϕΨ1 , which maximizesthe achievable ergodic secrecy rate. The values of the optimal ϕ can be obtained from(2.27) and (2.28) asϕ∗N =−(b + b) +√b(b+ 1)( − b +  + b)1 + b+  − b P (2.33)ϕ∗R =− +√ −  − b +  + b1−  : (2.34)40Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO SystemsImpact of : It can be shown from (2.33) and (2.34) that for both the N and Rmethods the optimal ϕ is a monotonically increasing function of  ∈ (0P 1 − xRa2).Thus, as the number of MTs in the cell increase, the amount of power allocated toAN precoding decreases. This can be explained by the fact that as  increases, thetransmit power per MT used for information transmission, ϕeiRK, decreases. Tocompensate for this eect, a larger ϕ is necessary. On the other hand, the ergodicsecrecy rates for both the N and R methods are decreasing functions of  ∈ (0P 1−xRa2), cf. (2.27), (2.28), i.e., as expected, for a given number of users the ergodicsecrecy rates increase with increasing number of BS antennas. Surprisingly, thisproperty does not necessarily hold in case of pilot contamination, cf. Section 2.5.2.4.3 Secrecy Outage Probability AnalysisIn delay limited scenarios, where one codeword spans only one channel realization,outages are unavoidable since Alice does not have the CSI of the eavesdropper channeland the secrecy outage probability has to be used to characterize the performance ofthe system instead of the ergodic rate. For the considered multi-cell massive MIMOsystem, the rate of the desired user, gnk, becomes deterministic as ci → ∞, butthe instantaneous capacity of the eavesdropper channel remains a random variable.A secrecy outage occurs whenever the target secrecy rate g0 exceeds the actualinstantaneous secrecy rate. Thus, the secrecy outage probability of the kth MT inthe local cell is given by"out = Pr{gnk− log2(1+E) ≤ g0} = Pr{E ≥ 2gnk−g0−1} = 1−FE(2gnk−g0−1)P(2.35)41Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemswhere E = pfHnkGHnEX−1GnEfnk and FE(x) is given in Appendix A.2. A closed-formupper bound on the secrecy outage probability is obtained by replacing gnk withgΨnk = log2(1 + Ψnk) with Ψnk given in (2.24)/(2.25).2.5 Performance Analysis for Pilot ContaminationIn this section, we analyze the performance of the considered multi-cell massiveMIMO system for the case of pilot contamination. To this end, we simplify thelower bound on the achievable ergodic rate expression derived in Section 2.3.2 for thecase of pilot contamination, derive insightful and tight lower bounds on the ergodicsecrecy rate, and provide a closed-form expression for the secrecy outage probability.2.5.1 Lower Bound on the Achievable Ergodic RateThe lower bound on the achievable ergodic rate of the users derived in Section 2.3.2is also applicable in case of pilot contamination. Thus, in a rst step, we characterizethe four expectations/variances in the SINR expression in (2.12).Expressing the small-scale fading vector as hknn = hˆknn + h˜knn, cf. Section 2.2, thedenominator of (2.12) can be rewritten as (we omit the path-loss for the moment)E[hknnfnk] = E[‖hˆknn‖+h˜knn(hˆknn)H‖hˆknn‖]= E[‖hˆknn‖] =√pknn1 + p∑bm=1 knmE[x]P (2.36)where x2 ∼ 22cT , cf. Lemma 2.2. Furthermore, we observe from (2.2) that, at thelocal BS, the channel estimate for the kth MT in the local cell involves the sum ofall channel vectors between the local BS and the kth MTs in all cells weighted withscaling factors√p knm1+p ∑Ml=1 knl. Thus, the transmit beamforming vector for the kth MTin the local cell is also aected by the channel vectors between the local BS and42Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsthe kth MTs in all other cells. This is the fundamental problem introduced by pilotcontamination. Using this observation, the interference caused by the kth MT in themth cell to the local cell (i.e., the component of the third term of the denominator in(2.12) with l = k) is given byE[|hkmnfmk|2] = E[‖hˆkmn‖2]+ E[ hˆkmn‖hˆkmn‖(h˜kmn)Hh˜kmn (hˆkmn)H‖hˆkmn‖]=pkmn1 + p∑bl=1 kmlE[x2] +1 + p∑l ̸=n kml1 + p∑bl=1 kmlE[y2]P (2.37)where y2 ∼ 22, cf. Lemma 2.1. Exploiting now (2.36) and (2.37) and the denition ofvariance, i.e., var[x] = E[x2]− E2[x], we obtain for the signal leakage term in (2.12)var[hknnfnk] =pknn1 + p∑bm=1 knmvar[x] +1 + p∑m̸=n knm1 + p∑bm=1 knmE[y2]: (2.38)Furthermore, the interference from the lth MT, where l ̸= k, in the adjacent (i.e.,non-local) cells is given byE[|hkmnfml|2] = E[y2]P (2.39)as each fml, ∀l ̸= k, has unit norm and is independent of hkmn. While the termscalculated in (2.36)-(2.39) are identical for the N and R methods, the AN leakagedepends on the AN precoding matrix design. In particular, for the N -method, theAN is designed to lie in the NS of the estimated channels from each BS to all K MTsin its own cell, which is also a scaled version of the estimated channels from each BSto all K MTs in the local cell due to pilot contamination, cf. (2.3). This implies thatall hˆkmlP∀l are aligned, cf. Section 2.2.2. Hence, the AN leakage is obtained asE[|hkmnami|2] = E[h˜kmnamiaHmi(h˜kmn)H ] =1 + p∑l ̸=n kml1 + p∑bl=1 kmlE[y2]P∀mP (2.40)43Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsby exploiting E[|hˆkmmami|2] = 0 for N -method and the independence of ami, ∀i, andh˜kmn. On the other hand, for the R-method, the AN is generated randomly, such thatE[|hkmnami|2] = E[y2], since the ami, ∀i, have unit norm and are independent of hkmn.Plugging all intermediate results derived in this section so far into (2.12), weobtain Nnk =nkE2[x]nkvar[x] +∑bm=1(mk + ∑cT−Ki=1 mk +∑l ̸=k kmn)E[y2] +∑m̸=n mkE[x2] +KϕeT(2.41)and Rnk =nkE2[x]nkvar[x] +∑bm=1(mk + ∑cT−Ki=1 kmn +∑l ̸=k kmn)E[y2] +∑m̸=n mkE[x2] +KϕeTP(2.42)where mk = kmnp kmn1+p ∑Ml=1 kmland mk = kmn1+p ∑l̸=n kml1+p ∑Ml=1 kml. Adopting now the samesimplied interference model as in Section 2.4, the term∑bm=1 mk in (2.41) and(2.42) can be simplied as 1−  + (b − 1)/(1− /) = a− x. Other terms can besimplied in the same way. By combining all terms together, (2.41) and (2.42) canbe further simplied for large ci , the corresponding lower bound on the achievableergodic rates are given bygNnk = log2(1 +(a− x)(1− ) + a + (b − 1)/2+ ϕeT)(2.43)andgRnk = log2(1 +a(1− ) + a + (b − 1)/2+ ϕeT)P (2.44)where  = p 1+p a. From (2.43) and (2.44) we observe that gNnk > gRnk always holdsbut the performance dierence diminishes if aRx ≫ . We note that for both AN44Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsprecoding matrix designs the powers of the inter-cell interference are proportionalto a − 1 = (b − 1)/. Furthermore, for the N -method and the R-method, theAN leakage is proportional to (1 − xRa) and , respectively. Therefore, aRx ≫ implies that the inter-cell interference are much stronger (but with a weaker / tohave a ≫ x) than the AN leakage and/or the pilot power p is not suciently largeto prevent AN leakage for the N -method. Furthermore, for  → 0, we obtain gNnk =log2(1+R[(a− x)+(b −1)/2]) and gRnk = log2(1+R[a+(b −1)/2]), i.e., inthe asymptotic regime where the number of users is constant but the number of BSantennas increases without bound, the performance for both AN precoding matrixdesigns is limited by both AN leakage and pilot contamination.Since the ergodic capacity of the eavesdropper is not aected by the imperfectCSI at the local BS, a lower bound on the ergodic secrecy rate for pilot contaminationcan be calculated from (2.7), (2.8), and (2.43)/(2.44).2.5.2 Impact of System Parameters on Ergodic Secrecy RateTo gain more insight, we employ again the upper bound on the ergodic capacity of theeavesdropper provided in Theorem 2.2. Combining (2.7), (2.16), (2.43), and (2.44),we obtain simple lower bounds for the ergodic secrecy rate for the N and R methodsas gsecPNnk =[log2((b+ 1− x) + [( + 1)x− (b+ 1− x)]ϕ− ( + 1)xϕ2(b+ 1− x) + [(x+ x− 1) + (b+ 1− x)(1− )]ϕ+ (1− )(x+ x− 1)ϕ2)]+P(2.45)45Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO SystemsandgsecPRnk =[log2((b+ 1) + [x− (b+ 1)]ϕ− xϕ2(b+ 1) + [(x− 1) + (b+ 1)(1− )]ϕ+ (1− )(x− 1)ϕ2)]+P(2.46)respectively.In the following, we investigate the impact of the system parameters on the ergodicsecrecy rate in detail.Impact of : Similar to the perfect training case we investigate in the followingthe upper limit for  such that a positive secrecy rate can be achieved. We observefrom (2.45) and (2.46) that a non-zero secrecy rate can be achieved as long as  Q Ψsecholds whereNsec =a2(1− )a(1− )(1 + b− x) + x→0=a2a(1 + b− x) + xP (2.47)Rsec =a2(1− )a(1− )(1 + b) + x→0=a2a(1 + b) + x: (2.48)Eqs. (2.47) and (2.48) reveal that the robustness of the considered multi-cell MIMOsystem to eavesdropping is monotonically decreasing with increasing number of MTsin the system. On the other hand, allocating more resources to training, i.e., increas-ing  by increasing the pilot power, p , or the pilot sequence duration,  , leads to ahigher robustness against eavesdropping, i.e., a larger number of eavesdropper anten-nas can be tolerated. Furthermore, as expected, Nsec > Rsec, i.e., the more complexN -method is more robust to eavesdropping than the simpleR method. However, Rsecapproaches Nsec if both x and  are small, i.e., both methods have a similar robustnessto eavesdropping in case of strong pilot contamination but a small value of /, since,in this case, the N -method can no longer avoid AN leakage. We also note that, asexpected, since  Q 1 always holds, for R-method, the maximum tolerable number46Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsof eavesdropper antennas in case of pilot contamination is always smaller than thatin case of perfect training. Surprisingly, on the other hand, for the N -method, themaximum tolerable number of eavesdropper antennas for pilot contamination is pos-sible to be larger than that for perfect training, if  > b+1b+x, cf. (2.29), (2.30), and(2.47), (2.48). This mainly attributes to the inter-cell AN leakage suppression due topilot contamination, cf. Lemma 2.2 and (2.40). In particular, the channel estimatesin adjacent cells also involve the inter-cell interference channel vector between theadjacent BS and MTs in the local cell. Therefore, the AN emitted in adjacent cellsis aected by the inter-cell interference channels. In this regard, pilot contaminationis benecial for improving the system performance.Impact of ϕ: Similar to the case of perfect training, the ergodic secrecy rate forboth AN precoding matrix designs becomes zero for ϕ = ϕ0 = 0 also for the case ofpilot contamination, cf. (2.45) and (2.46), since zero power is allocated to informationtransmission in this case. A second zero of the ergodic secrecy rate occurs for ϕ = ϕΨ1 ,0 Q ϕΨ1 Q 1, where Ψ ∈ {N PR}. ϕΨ1 is obtained from (2.45) and (2.46) asϕN1 = 1−a( − 1)((b+ 1) + (x− 1))(a(a+ x)2 + (−a2 + a+ x) − a(x− 1)) (2.49)ϕR1 = 1−a( − 1)((b+ 1) + (x− 1))(a22 + (−a2 + a(x− 1) + x) − a(x− 1)) : (2.50)Furthermore, assuming  Q Ψsec and taking the derivatives of (2.45) and (2.46) withrespect to ϕ and setting them to zero, we obtain the optimal power allocation factorsfor the N and R methods asϕ∗N =−√(b+ 1− x)((−1 + x)+ (b+ 1))((x+ x)+ (−1 + )(b+ 1))(−x22 + ((2− 2x− )+ (−1 + )(b+ 1)) − x(−1 + x))+(−x22 + (b+ 1))2 + ((−x−  + 1)x2 + (( − 1 + x)b+  − 1 + x))(−x22 + ((2− 2x− )+ (−1 + )(b+ 1)) − x(−1 + x)) (2.51)47Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsandϕ∗R =−√((−1 + x)+ (b+ 1))(b+ 1)(x+ (−1 + )(b+ 1))((−1 + )(b+ 1)− x(−1 + x))+(( − 1 + x)b+  − 1 + x)((−1 + )(b+ 1)− x(−1 + x)) : (2.52)Impact of : Based on (2.51) and (2.52) it can be shown that, similar to the casefor perfect training, for pilot contamination, the optimal ϕ∗N and ϕ∗R are monotonicallyincreasing in . Furthermore, in Section 2.4, we found that, for perfect training, theergodic secrecy rate is monotonically increasing for decreasing . However, for agiven ϕ, it can be shown based on (2.45) and (2.46) that this is no longer true in caseof pilot contamination. In other words, if ϕ and the number of users K are xed,in case of pilot contamination, the ergodic secrecy rate is not maximized by makingthe number of BS antennas, ci , exceedingly large (i.e., ci ≫ K such that  → 0).Instead, there is an optimal nite number of BS antennas. We will investigate thisissue numerically in Section 2.6.Impact of : Pilot contamination impacts the ergodic secrecy rate via , wheresmaller values of  imply that the MTs expend less resources for uplink training (i.e.,they employ a smaller pilot power p and/or a shorter pilot sequence length, ). First,we observe from (2.45) and (2.46) that both gsecPNnk and gsecPRnk are increasing functionsof , i.e., as expected, if the MTs employ a higher pilot power and/or a longer pilotsequence for channel estimation, the ergodic secrecy rate improves. Furthermore,sec is an increasing function of , i.e., a higher uplink training power and/or longerpilot sequence lengths increase the operating region of the system where a non-zerosecrecy rate can be achieved.On the other hand, for a given coherence interval i , xed transmit power ei ,48Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsand xed pilot power p , the fraction of time allocated for training Ri (and as aconsequence ) can be optimized for maximization of the net ergodic secrecy rategiven by (1 − Ri )gsecPΨnk , Ψ ∈ {N PR}. We assume that the channels are constantwithin one coherence interval but change from one coherence interval to the next.We also emphasize that by using the (net) ergodic secrecy rate as a performancemeasure, we implicitly assume coding over many coherence intervals. For small  ,the factor (1− Ri ) is large but the ergodic secrecy rate, gsecPΨnk , is small because ofthe unreliable channel estimation. On the other hand, for large  , the factor (1−Ri )is small but the ergodic secrecy rate, gsecPΨnk , is large because of the more accuratechannel estimation. Hence,  can be optimized for optimal performance [68]. Theoptimization of  will be studied numerically in Fig. 2.9 in Section 2.6.2.5.3 Secrecy Outage Probability AnalysisPlugging (2.43) and (2.44) into the secrecy outage probability expression derived in(2.35), we obtain an upper bound for the secrecy outage probability for the case ofpilot contamination as"Ψout = 1− FE(2gΨnk−g0 − 1)P (2.53)where Ψ ∈ {N PR}.2.6 Numerical ExamplesIn this section, we evaluate the secrecy performance of the considered multi-cellmassive MIMO systems based on the analytical expressions derived in Sections 2.2-2.5 and via Monte-Carlo simulation. We consider a system with b = 7 hexagonalcells and adopt the simplied path-loss model, i.e., the severeness of the inter-cell49Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsinterference is characterized by parameter / only. The Monte-Carlo simulation resultsfor the ergodic secrecy rate of the kth MT in the local cell are based on (2.7) where theachievable ergodic rate gnk is obtained from (2.9) and the ergodic secrecy capacityof the eavesdropper is obtained from (2.8). Thereby, the expected values in (2.9)and (2.8) were evaluated by averaging over 5000 random channel realizations. TheMonte-Carlo simulation results for the outage probability are obtained from "out =Pr{gnk − log2(1 + E) ≤ g0}, which was evaluated again based on 5000 randomchannel realizations. The values of all relevant system parameters are provided inthe captions of the gures.2.6.1 Ergodic Secrecy Rate and Secrecy Outage ProbabilityFor the results shown in this section, we adopt a xed power allocation factor ofϕ = 0:75. The optimization of ϕ will be addressed in the next subsection.In Fig. 2.2, we verify the derived analytical expressions for the ergodic capacityof the eavesdropper which seeks to decode the information intended for the kth MTin the local cell. The analytical results were generated with (2.13) while the upperbound results were computed with (2.16). The vertical dashed lines denote  =1− xRa2. Fig. 2.2 reveals that for  Q 1− xRa2, the upper bound is very tight. For1− xRa2 Q  Q 1− Rb , the upper bound is not applicable, although the ergodiccapacity of the eavesdropper is still nite, cf. Theorem 2.2 and Remark 2.1. For  →1−Rb , the ergodic capacity of the eavesdropper tends to innity since X becomessingular. Furthermore, we observe from Fig. 2.2 that increasing inter-cell interference(i.e., larger inter-cell interference factors, /) has a negative eect on the ergodiccapacity of the eavesdropper, whereas as expected, the eavesdropper can improvehis performance by adding more antennas, cE (i.e., by increasing ). Moreover,50Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0 0.2 0.4 0.6 0.8 101234567β (K/NT )Ergodic capacity of eavesdropper (bps/Hz)  AnalysisSimulationUpper boundρ = 0.1, α = 0.4ρ = 0.3, α = 0.4ρ = 0.1, α = 0.1ρ = 0.3, α = 0.1Figure 2.2: Ergodic capacity of the eavesdropper seeking to decode the informationintended for the kth MT in the local cell vs. the normalized number of MTs in thecell, , for a system with total transmit power ei = 10 dB, b = 7, ϕ = 0:75, andci = 100.Fig. 2.2 conrms that the ergodic capacity of the eavesdropper is monotonicallydecreasing in  in the interval (0P 1 − √xRa) and monotonically increasing in  inthe interval (1−√xRaP 1− xRa2). The resulting minimum of the ergodic capacityof the eavesdropper at  = 1−√xRa is denoted by a black circle in Fig. 2.2.In Fig. 2.3, for the case of perfect training, we show the ergodic secrecy ratevs. the number of BS antennas (subgure (a)) and the secrecy outage probabilityvs. the target secrecy rate g0 (subgure (b)) for the kthMT in the local cell. Results51Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systemsfor both considered AN precoding matrix designs are shown. In subgure (a), lowerbound I was obtained based on (2.7), (2.13), (2.24), and (2.25) and lower bound IIwas obtained with (2.27) and (2.28). In subgure (b), the upper bound was obtainedwith (2.35). Fig. 2.3 reveals that the derived bounds for the ergodic secrecy rateand the secrecy outage probability are accurate. As expected, for the ergodic secrecyrate, lower bound I is somewhat tighter than lower bound II. Furthermore, increasingthe number of BS antennas ci improves both the ergodic secrecy rate as well as thesecrecy outage probability. Moreover, as expected, the N -method for generation ofthe AN precoding matrix always outperforms the R-method as the N -method avoidsintra-cell AN leakage.In Fig. 2.4, we show the same performance metrics as in Fig. 2.3, however, now forthe case of pilot contamination. In subgure (a), lower bound I was obtained basedon (2.7), (2.13), (2.43), and (2.44), whereas lower bound II was obtained with (2.45)and (2.46). In subgure (b), the upper bound was obtained with (2.53). Similar tothe case of perfect training, the derived bounds on the ergodic secrecy rate and thesecrecy outage probability are very tight. A comparison of Figs. 2.3 and 2.4 revealsthat pilot contamination causes a signicant performance degradation in terms ofboth ergodic secrecy rate and secrecy outage probability. Furthermore, unlike forthe case of perfect training, for pilot contamination, the ergodic secrecy rate is notmonotonically increasing in ci but has a unique maximum for both AN precodingmatrix designs.2.6.2 Optimal Power AllocationIn this subsection, we investigate the optimization of power allocation factor ϕ andillustrate its impact on the ergodic secrecy rate.52Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems80 100 120 140 1600.10.20.30.40.50.60.70.8Number of BS antennas (NT )Ergodic secrecy rate (bps/Hz)(a) Ergodic secrecy rate  SimulationLower bound ILower bound II0 0.5 100.10.20.30.40.50.60.70.80.91R0 (bps/Hz)Secrecy Outage Probability(b) Secrecy outage probability  Upper bound NSimulation NUpper bound RSimulation RN -methodR-methodNT = 80NT = 100NT = 120Figure 2.3: Ergodic secrecy rate and outage probability for perfect training, b = 7,ei = 10 dB, K = 10, / = 0:3,  = 0:1, and ϕ = 0:75.Figs. 2.5 and 2.6 show the ergodic secrecy rates of the kth MT in the local cell asfunctions of ϕ for the cases of perfect training and pilot contamination, respectively.The ergodic secrecy rate curves were obtained via Monte Carlo simulation and variousvalues of  and  are considered. The optimal values for ϕ obtained with (2.33)/(2.34)(for perfect training) and (2.51)/(2.52) (for pilot contamination) are denoted by blackcircles. As expected from our discussions in Sections 2.4 and 2.5, Figs. 2.5 and 2.6show that, for both the N and the R AN precoding matrix desigs, the optimal ϕ∗ isdecreasing in , i.e., the system should allocate more power to AN if the eavesdropper53Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems80 100 120 140 1600.10.120.140.160.180.20.220.240.26Number of BS antennas (NT )Ergodic secrecy rate (bps/Hz)(a) Ergodic secrecy rate  SimulationLower bound ILower bound II0 0.5 100.10.20.30.40.50.60.70.80.91R0 (bps/Hz)Secrecy Outage Probability(b) Secrecy outage probability  Upper bound NSimulation NUpper bound RSimulation RN -methodR-methodNT = 80NT = 100NT = 120Figure 2.4: Ergodic secrecy rate and outage probability for pilot contamination,b = 7, ei = 10 dB, K = 10 MTs, / = 0:1,  = 0:1, ϕ = 0:75,  = K, andp = eiRK.is becoming stronger, and increasing in , i.e., less power should be allocated to ANif the number of users increases. For  = 0:4, no results are shown for the case ofpilot contamination in Fig. 2.6 since the corresponding ergodic secrecy rates are zerofor all choices of ϕ, i.e.,  > sec holds in this case.In Fig. 2.7, we depict the ergodic secrecy rate and the optimal power allocationfactor, ϕ∗, as functions of the normalized number of MTs in each cell, . Thereby,the ergodic secrecy rate is calculated using the optimal ϕ∗, which was obtained basedon the analytical results in Sections 2.4 and 2.5 for the case of perfect training and54Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0 0.2 0.4 0.6 0.8 100.511.522.5φErgodic secrecy rate (bps/Hz)  N -methodR-methodα = 0.1, β = 0.5α = 0.1, β = 0.05α = 0.4, β = 0.05α = 0.4, β = 0.5Figure 2.5: Ergodic secrecy rate vs. power allocation factor ϕ assuming perfect train-ing, ci = 100, b = 7, ei = 10 dB, and / = 0:1. Black circles denote the optimalpower allocation factor, ϕ∗, obtained with (2.33) and (2.34).pilot contamination, respectively. We observe that, unlike the case when ϕ is xed,if ϕ is optimized, the ergodic secrecy rate is a non-increasing function of  also incase of pilot contamination, i.e., for a given number of users, increasing the numberof BS antennas is always benecial. On the other hand, for all considered cases, theoptimal value of ϕ is a monotonically increasing function of , i.e., as the numberof users in the system increases relative to the number of BS antennas, less poweris allocated to AN. Also, the performance gap between both AN precoding matrixdesign methods decreases with increasing .55Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.700.10.20.30.40.50.60.70.80.9φErgodic secrecy rate (bps/Hz)  N -methodR-methodα = 0.1, β = 0.05α = 0.1, β = 0.5Figure 2.6: Ergodic secrecy rate vs. power allocation factor ϕ assuming pilot con-tamination, b = 7, ci = 100, ei = 20 dB,  = K, p = eiRK, and / = 0:1.Black circles denote the optimal power allocation factor, ϕ∗, obtained with (2.51)and (2.52).2.6.3 Conditions for Non-zero Ergodic Secrecy RateIn Fig. 2.8, we illustrate for both AN precoding matrix designs under what conditionsa non-zero ergodic secrecy rate is possible. To this end, we plot sec as dened in(2.29), (2.30), (2.47), and (2.48) as functions of  for p = eiRK (subgure onleft hand side) and the amount of power, p , spent by the MTs for training for = 0:05P 0:5 (subgure on right hand side). For  ≥ sec, the ergodic secrecy rateis zero regardless of the amount of power allocated to AN. On the other hand, for56Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0 0.2 0.4 0.6 0.800.511.522.533.5β (K/NT )Ergodic secrecy rate (bps/Hz)  N -methodR-method0 0.2 0.4 0.6 0.800.10.20.30.40.50.60.7β (K/NT )φ∗Perfect trainingPerfect trainingPilot contaminationPilot contaminationFigure 2.7: Ergodic secrecy rate and optimal power allocation factor, ϕ∗, vs.  forb = 7, ei = 20 dB, ci = 100,  = 0:3, and / = 0:1. In case of pilot contamination, = K and p = eiRK. The ergodic secrecy rates were obtained with (2.27), (2.28),(2.45), and (2.46). The optimal power allocation factor was obtained with (2.33),(2.34), (2.51), and (2.52). Q sec, a positive ergodic secrecy rate can be achieved. We observe from Fig. 2.8that for both AN precoding matrix designs sec is a decreasing function of , whereasit is an increasing function of p , i.e., the more reliable the channel estimates, themore eavesdropper antennas can be tolerated before the ergodic secrecy rate drops tozero. However, sec saturates for large values of p . We note that the values of secare smaller for the R-method than for the N -method because of the larger intra-cell57Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0 0.5 100.20.40.60.811.21.4βαsec  N -methodR-method0 10 20 300.260.280.30.320.340.360.380.40.420.44pτ in dBαsec  β = 0.05β = 0.5Perfect TrainingPilot ContaminationN -methodR-methodFigure 2.8: sec vs.  and p for pilot contamination, total transmit power ei = 20dB, b = 7, ci = 100, / = 0:1, and  = K.AN leakage caused by the R-method.2.6.4 Optimization of the Net Ergodic Secrecy RateFig. 2.9 depicts the net ergodic secrecy rate, (1− Ri )gsecnk , as a function of , wherethe lower bounds in (2.45) and (2.46) were used to approximate gsecnk . The cases ofi = 100 and i = 500 are considered for K = 5 and K = 20 MTs. We assume thatp = 0 dB and  is varied by changing  and the optimal power allocation factor ϕ∗is employed. Thereby, the range of possible  is [KPi ), which directly translates into58Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems0.55 0.56 0.57 0.58 0.59 0.6 0.61 0.62 0.6300.20.40.60.811.21.4λNet ergodic secrecy rate (bps/Hz)  N -methodR-methodT = 500, K = 5T = 100, K = 5T = 500, K = 20Figure 2.9: Net ergodic secrecy rate vs.  for a system with optimal ϕ∗, ci = 100,b = 7,  = 0:1, ei = 10 dB, p = 0 dB, and / = 0:1. Black circles denote themaximum net ergodic secrecy rate.the range of possible  as  = p 1+p a. Fig. 2.9 reveals that the optimal  is (slightly)increasing in i since for larger values of i , more time for allocation to uplink trainingis available, i.e.,  can be increased resulting in a larger value for the optimal . ForK = 20, the lower limit of the permissible interval for  given by  = K yields themaximum net secrecy rate. In this case, increasing  beyond  = K does not improvegsecnk suciently to compensate for the decrease of the term 1− Ri .59Chapter 2. AN-Aided MF Precoding in Secure Massive MIMO Systems2.7 ConclusionsIn this chapter, we considered a multi-cell massive MIMO system with MF precodingand AN precoding at the BS for secure downlink transmission in the presence ofa multi-antenna passive eavesdropper. For AN precoding, we considered both theconventional NS AN precoding matrix design and a novel random AN precodingmatrix design. For both perfect training and pilot contamination, we derived twotight lower bounds on the ergodic secrecy rate and a tight upper bound on thesecrecy outage probability. The analytical expressions allowed us to optimize theamount of power allocated to AN precoding and to gain signicant insight into theimpact of the system parameters on performance. In particular, our results revealthat for the considered multi-cell massive MIMO system with MF precoding (1) ANprecoding is necessary to achieve a non-zero ergodic secrecy rate if the user and theeavesdropper experience the same path-loss, (2) secrecy cannot be guaranteed if theeavesdropper has too many antennas, (3) for the case of pilot contamination, theergodic secrecy rate is only an increasing function of the number of BS antennas ifthe amount of power allocated to AN precoding is optimized, and (4) the proposedrandom AN precoding matrix design is a promising low-complexity alternative to theconventional NS AN precoding matrix design.60Chapter 3Linear Data and AN Precoding inSecure Massive MIMO Systems3.1 IntroductionSince secrecy and privacy are critical concerns for the design of future communicationsystems [10], it is of interest to investigate how the large number of spatial degrees offreedom in massive MIMO systems can be exploited for secrecy enhancement [27, 30].If the eavesdropper (Eve) remains passive to hide its existence, neither the transmit-ter (Alice) nor the legitimate receiver (Bob) will be able to learn Eve's CSI. In thissituation, it is advantageous to inject AN at the transmitter to degrade Eve's channeland to use linear precoding to avoid impairment to Bob's channel as was shown in[24, 27]-[38] and [70], for single user and single-cell multiuser systems, respectively.However, in multi-cell massive MIMO systems, multi-cell interference and pilot con-tamination will hamper Alice's ability to degrade Eve's channel and to protect Bob'schannel. This problem was studied rst in Chapter 2 for simple MF data precodingand NS and random AN precoding. However, it is well known that MF data pre-coding suers from a large loss in the achievable information rate compared to otherlinear data precoders such as ZF and RCI precoders as the number of MTs increases[7]. Since it is expected that this loss in information rate also translates into a lossin secrecy rate, studying the secrecy performance of ZF and RCI data precoders in61Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsmassive MIMO systems is of interest. Furthermore, while NS AN precoding wasshown to achieve a better performance compared to random AN precoding [43], italso entails a much higher complexity. Similarly, the improved performance of ZFand RCI data precoding compared to MF data precoding comes at the expense of ahigher complexity. Hence, the design of novel data and AN precoders which allowa exible tradeo between complexity and secrecy performance is desirable. In theliterature, ZF and RCI data precoding were analyzed in the large system limit in[14, 15]. However, neither pilot contamination nor AN were taken into account andthe secrecy rate was not analyzed. Using a concept that was originally conceivedfor CDMA uplink systems in [16] and later extended to MIMO systems in [17], re-duced complexity linear data precoders that are based on matrix polynomials wereinvestigated for use in massive MIMO systems in [72, 73]. However, [72, 73] did nottake into account the eect of AN leakage for precoder design and did not study thesecrecy rate. Hence, the results presented in [72, 73], as well as the related workdiscussed in Chapter 1.3 [47]-[59], are not directly applicable to the system studiedin this chapter.In this chapter, we consider secure downlink transmission in a multi-cell massiveMIMO system employing linear data and AN precoding in the presence of a passivemulti-antenna eavesdropper. We study the achievable ergodic secrecy rate of suchsystems for dierent linear precoding schemes taking into account the eects of uplinkchannel estimation, pilot contamination, multi-cell interference, and path-loss. Themain contributions of this chapter are summarized as follows:• To address the impairments incurred by inter-cell interference as well as inter-cell AN leakage, we study both selsh and collaborative precoders. The formerrequires only the CSI of the MTs in the local cell but cause inter-cell interference62Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsand inter-cell AN leakage, whereas the latter requires the CSI between the localBS and the MTs in all cells, but reduce inter-cell interference and inter-cell ANleakage. Nevertheless, since the additional CSI required for the collaborativeprecoders can be estimated directly by the local BS, the additional overheadand complexity incurred compared to selsh precoders is limited.• We derive novel closed-form expressions for the asymptotic ergodic secrecy ratewhich facilitate the performance comparison of dierent combinations of lineardata precoders (i.e., MF, selsh and collaborative ZF/RCI) and AN precoders(i.e., random, selsh and collaborative NS), and provide signicant insight forsystem design and optimization.• In order to avoid the computational complexity and potential stability issuesin xed point implementations entailed by the large-scale matrix inversionsrequired for ZF and RCI data precoding and NS AN precoding, we proposePOLY data and AN precoders and optimize their coecients. Unlike [71] and[72], which considered polynomial data precoders for massive MIMO systemswithout AN generation, we use free probability theory [61] to obtain the POLYcoecients, which allows us to express the coecients as simple functions of thechannel and system parameters. Simulation results reveal that these precodersare able to closely approach the performance of selsh RCI data and NS ANprecoders, respectively.The remainder of this chapter is organized as follows. In Section 3.2, we outlinethe considered system model and review some basic results from Chapter 2. InSections 3.3 and 3.4, the considered linear data and AN precoders are investigated,respectively. In Section 3.5, the ergodic secrecy rates of dierent linear precoders are63Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemscompared analytically for a simple path-loss model. Simulation and numerical resultsare presented in Section 3.6, and some conclusions are drawn in Section 3.7.3.2 System Model and PreliminariesIn this section, we introduce the considered system model as well as the adoptedchannel estimation scheme, and review some ergodic secrecy rate results.3.2.1 System ModelWe consider the downlink of a multi-cell massive MIMO system with cell set M ={1P : : : Pb} and a frequency reuse factor of one, i.e., all BSs use the same spectrum.Each cell includes one ci -antenna BS, K ≤ ci single-antenna MTs, and potentiallyan cE-antenna eavesdropper. The eavesdroppers try to hide their existence andhence remain passive. As a result, the BSs cannot estimate the eavesdroppers' CSI.To overcome this limitation, each BS generates AN to mask its information-carryingsignal and to prevent eavesdropping [24]. In the following, the kth MT, k = 1P : : : P K,in the nth cell, n = 1P : : : Pb , is the MT of interest and we assume that an eavesdrop-per tries to decode the signal intended for this MT. We note that neither the BSsnor the MTs are assumed to know which MT is targeted by the eavesdropper. Thesignal vector, xn ∈ CcT×1, transmitted by the BS in the nth cell (also referred to asthe nth BS in the following) is given byxn =√pFnsn +√qAnznP (3.1)where sn ∼ CN(0K P IK) and zn ∼ CN(0cT P IcT ) denote the data and AN vectorsfor the K MTs in the nth cell, respectively. Fn = [fn1P · · · P fnK ] ∈ CcT×K and64Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsAn = [an1P · · · P ancT ] ∈ CcT×cT are the data and AN precoding matrices, respec-tively, and the ecient design of these matrices is the main scope of this chap-ter. Thereby, the structure of both types of precoding matrices does not depend onwhich MT is targeted by the eavesdropper. The AN precoding matrix An has ranka = rank{An} ≤ ci , i.e., a dimensions of the ci -dimensional signal space spannedby the ci BS antennas are exploited for jamming of the eavesdropper. The dataand AN precoding matrices are normalized as tr{FHn Fn} = K and tr{AHnAn} = a,i.e., their average power per dimension is one. The average powers p and q allocatedto the information-carrying signal for each MT and each AN signal, respectively, canbe written as p = ϕeTKand q = (1−ϕ)eTa, where ei is the total transmit power andϕ ∈ (0P 1] is a power allocation factor which can be optimized. For the sake of clarity,in this chapter, we assume that all cells utilize the same value of ϕ.The vectors collecting the received signals at the K MTs and the cE antennas ofthe eavesdropper in the nth cell are given byyn =b∑m=1Gmnxm + nn and yE =b∑m=1GmExm + nEP (3.2)respectively, with Gaussian noise vectors nn ∈ CN(0K P 2nIK) and nE ∈ CN(0cE P 2EIcE),where 2n and 2E denote the noise variances at one MT and one eavesdropper re-ceive antenna, respectively. Furthermore, Gmn = D1R2mnHmn ∈ CK×cT and GmE =√mEHmE ∈ CcE×cT are the matrices modeling the channels from the mth BS tothe K MTs and the eavesdropper in the nth cell, respectively. Thereby, Dmn =diag{1mnP : : : P Kmn} and mE represent the path-losses from the mth BS to the KMTs and the eavesdropper in the nth cell, respectively. Matrix Hmn ∈ CK×cT , withrow vector hkmn ∈ C1×cT in the kth row, and matrixHmE ∈ CcE×cT represent the cor-responding small-scale fading components. Their elements are modeled as mutually65Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsindependent and identically distributed (i.i.d.) complex Gaussian random variables(r.v.s) with zero mean and unit variance.For the design of the data and noise precoders, we consider two dierent ap-proaches: Selsh designs and collaborative designs. For the selsh designs, each BSdesigns its precoders only based on the estimate of the CSI in its own cell, Gnn, andwithout regard for the interference and the AN it causes to other cells. In contrast,for the collaborative designs, each BS designs its precoders based on the estimates ofthe CSI to the MTs in all cells, Gmn, m = 1P : : : Pb , in an eort to avoid excessiveinterference and AN to other cells. Although collaborative designs introduce morechannel estimation overhead at the BS, they may not always outperform selsh de-signs because of the imperfection of the CSI and the limited number of spatial degreesof freedom available for precoder design.3.2.2 Channel Estimation and Pilot ContaminationAs is customary for massive MIMO systems, we assume that the downlink and uplinkchannels are reciprocal and the CSI is estimated in an uplink training phase [2]-[8].To this end, all MTs emit pilot sequences of length  = KP  ∈ M and with pilotsymbol power p . We assume that the pilot sequences are mutually orthogonal, andthus can be assigned to at most  cells without mutual pollution. When  Q b , thisgives rise to so-called pilot contamination [2]-[8], because at least one pilot sequenceis shared between more than one MTs in a b -cell network. Furthermore, we assumethat the path-loss information changes on a much slower time scale than the small-scale fading. Hence, the path-loss matrices Dnm, m = 1P : : : Pb , can be estimatedperfectly and are assumed to be known at the BS for MMSE estimation of the small-scale fading gains [8]. At the nth BS, the estimate of the small-scale fading vector to66Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsthe kth MT in the mth cell, hˆknm, is obtained in Appendix B.1. For MMSE estimation,we havehknm = hˆknm + h˜knmP (3.3)where the estimate hˆknm and the estimation error h˜knm are mutually independentand can be statistically characterized as hˆknm ∼ CN(0cT P p knmknm+p knmIcT ) and h˜knm ∼CN(0cT Pknmknm+p knmIcT ), respectively, cf. [43], where knm = 1+p∑l∈Mm⊆M\{m} knlandMm denotes the set of cells sharing the same set of pilot sequences with the mthcell, which is a subset ofM excluding the mth cell, where⌊bR⌋ − 1 ≤ |Mm| ≤ ⌈bR⌉ − 1P ∀m: (3.4)Example: For b = 7,  = 2, we have 2 ≤ |Mm| ≤ 3, i.e., there are 2 or 3 cellssharing the same set of pilot sequences with the mth cell 5.To further clarify, when  = b , i.e., all bK MTs use their respective pilots, nopilot contamination exists, knm reduces to 1 as |Mm| = 0 in this case. For  = 1,i.e., the pilot sequences in one cell are orthogonal, but reused in all other cells, wesimply have |Mm| = b − 1, and the distributions of both hˆknm and h˜knm are identi-cal with those in Chapter 2. For future reference, we collect the estimates and theestimation errors at the nth BS corresponding to all K MTs in the mth cell in ma-trices Hˆnm = [(hˆ1nm)i P : : : P (hˆKnm)i ]i ∈ CK×cT and H˜nm = [(h˜1nm)i P : : : P (h˜Knm)i ]i ∈CK×cT , respectively.5For the results shown in Section 3.6, without loss of generality, we always assume |Mn| = 3 forcalculating the secrecy rate achieved by the MT in the nth cell when the parameters in this exampleare adopted.67Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems3.2.3 Ergodic Secrecy RateThe performance metric adopted in this chapter is the ergodic secrecy rate [30]. Inthis section, we review some results for the ergodic secrecy rate in multi-cell massiveMIMO systems employing linear data and AN precoding from [43], as these resultswill be needed throughout this chapter. Combining (3.1) and (3.2) we observe thatthe downlink channel comprising the BS, the kth MT, and the eavesdropper in the nthcell is an instance of a MISOME wiretap channel [27]. Hence, the achievable secrecyrate of the kth MT in the nth cell is bounded by the dierence of the capacities ofthe channel between the BS and the MT and the channel between the BS and theeavesdropper, see [43, Lemma 1], [47, Lemma 2]. Thus, a lower bound on the ergodicsecrecy rate of the kth MT in the nth cell is given by [43]gsecnk = [gnk − Xevenk ]+P k = 1P : : : P KP (3.5)where gnk denotes an achievable rate of the kthMT in the nth cell and Xevenk denotesthe ergodic capacity of the channel between the BS and the eavesdropper. In orderto obtain a tractable lower bound on the ergodic secrecy rate, we lower bound theachievable rate of the MT as gnk = log2(1 + nk) with SINR [43, Eq. (10)] nk =|E[√knnphknnfnk]|2var[√knnphknnfnk] +b∑m=1ct∑i=1E[|√kmnqhkmnami|2] + ∑{mPl}≠{nPk}E[|√kmnphkmnfml|2] + 1 :(3.6)Furthermore, we make the pessimistic assumption that the eavesdropper is able tocancel the received signals of all in-cell and out-of-cell MTs except the signal intendedfor the MT of interest. This leads to an upper bound for the eavesdropper's capacity,68Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsand consequently, to a lower bound for the ergodic secrecy rate.6Hence, the ergodiccapacity of the eavesdropper is given by [43, Eq. (7)]Xevenk = E[log2(1 + pfHnkGHnEX−1GnEfnk) ]P (3.7)where X = q∑bm=1GmEAmAHmGHmE ∈ CcT×cT denotes the noise correlation ma-trix at the eavesdropper under the worst-case assumption that the receiver noise atthe eavesdropper is negligible, i.e., 2E → 0. Denoting the normalized number ofeavesdropper antennas by  = cERci , a necessary condition for the invertibility ofmatrix X is  ≤ baRci . Hence, a non-zero secrecy rate can only be achieved ifthis condition is met. Consequently, a larger a implies that the BS is able to toleratemore eavesdropper antennas.IfHnEfnk and matrix X are statistically independent, which in turn means for thedata and AN precoders that vector fnk and the subspace spanned by the columns ofAn are mutually orthogonal, a simple and tight upper bound on (3.7) can be obtained.Since any ecient data/AN precoder pair has to keep the AN self-interference at thedesired MT small, this orthogonality condition holds at least approximately in prac-tice. In this case, for  Q a2aR(xci ) and ci →∞, where a = 1 +∑bm ̸=n mERnEand x = 1 +∑bm̸=n(mERnE)2, a simple and tight upper bound for Xevenk is given by[43, Theorem 1]Xevenk ≤ log2(1 +paqaRci − xqRa)= log2(1 +ϕ(1− ϕ)(a− xciR(aa))):(3.8)For b = 1, we have a2Rx = b = 1, i.e., the bound in (3.8) is applicable in the6This lower bound is achievable if the eavesdropper has access to the data of all interfering in-cell and out-of-cell MTs, which might be the case e.g. if the interfering MTs cooperate with theeavesdropper.69Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsentire range of  where Xevenk in (3.7) is nite. Forb > 1, we have a2Rx ≤b , i.e., thebound is not applicable for aa2R(xci ) ≤  ≤baRci . However, for strong inter-cellinterference, we have mE ≈ nE and a2Rx ≈ b , i.e., the bound is applicable for all for which Xevenk in (3.7) is nite. On the other hand, for weak inter-cell interference,we have mE ≪ nE, and matrix X will be ill-conditioned for aRci ≤  ≤ baRciand Xevenk will become very large. Hence, the bound is again applicable for the valuesof  (i.e., 0 ≤  ≤ aRci ), for which Xevenk in (3.7) assumes practically relevant values.More generally, [43, Figs. 2-4] and Section 3.6 suggest that (3.8) is applicable andtight for all values of  which permit a non-vanishing secrecy rate.Combining (3.5), (3.6), and (3.8), we obtain a tight and tractable lower boundon the secrecy rate [43]. It is noteworthy that the upper bound on the capacity ofthe eavesdropper in (3.8) is only aected by the dimensionality of the AN precoder,a, but not on the exact structures of An and Fn, as long as fnk and the subspacespanned by the columns of An are orthogonal. On the other hand, the achievablerate of the MT in (3.6) is aected by both the data and the AN precoders. In thefollowing two sections, we analyze the impact of the most important existing dataand AN precoder designs on the achievable rate gnk as ci → ∞, respectively, andpropose novel low-complexity data and AN precoders that are based on a polynomialmatrix expansion.3.3 Linear Data Precoders for Secure MassiveMIMOIn this section, we analyze the achievable rate of selsh and collaborative ZF/RCIdata precoding, respectively, and develop a novel POLY data precoder. In contrast70Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsto existing analysis and designs of data precoders for massive MIMO, e.g. [14, 15],[72, 73], the results presented in this section account for the eect of AN leakage,which is only present if AN is injected at the BS for secrecy enhancement. Weare interested in the asymptotic regime where KPci → ∞ but  = KRci and = cERci are nite.3.3.1 Analysis of Existing Data PrecodersFor ci → ∞, analyzing the achievable rate is equivalent to analyzing the SINR in(3.6). Thereby, the eect of the AN precoder can be captured by the termQ =b∑m=1ct∑i=1E[|√kmnhkmnami|2] =b∑m=1kmnE[hkmnAmAHm(hkmn)H ] (3.9)in the denominator of (3.6), which represents the inter-cell and intra-cell AN leakage.This term is assumed to be given in this section and will be analyzed in detail fordierent AN precoders in Section 3.4.Selsh ZF/RCI Data PrecodingThe selsh RCI (SRCI) data precoder for the nth cell is given byFn = 1LnnHˆHnnP (3.10)where Lnn = (HˆHnnHˆnn + 1IcT )−1, 1 is a scalar normalization constant, and 1 is aregularization constant. In the following proposition, we provide the resulting SINRof the kth MT in the nth cell.Proposition 3.1. For SRCI data precoding, the received SINR at the kth MT in the71Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsnth cell is given bySRCInk =1∑m∈Mn∪{n} ΓˆmSRCI+(1+G(P1))2G(P1)(ΓˆnSRCI+ΓˆnSRCI1(1+G(P1))2) +∑m∈Mn mkRnk P (3.11)where the setMn is dened in Section 3.2,G(P 1) = 12[√(1− )221+2(1 + )1+ 1 +1− 1− 1]P (3.12)andΓˆmSRCI =ΓSRCImkΓSRCI∑m∈Mn∪{n} mk + 1(3.13)withΓSRCI =K∑mR∈Mn∪{n}∑Kl=1 kmn + Q+ KϕeT P (3.14)mk = kmnp kmnkmn+p kmn, mk = kmnkmnkmn+p kmn, and  = qRp.Proof. Please refer to Appendix B.2.Regularization constant 1 can be optimized for maximization of the lower boundon the secrecy rate in (3.5), which is equivalent to maximizing the SINR in (3.11).Setting the derivative of SRCInk with respect to 1 to zero, the optimal regulariza-tion parameter is found as 1Popt = R∑m∈Mn∪{n} ΓˆmSRCI in Appendix B.3, and thecorresponding maximum SINR is given bySRCInk =1ΓˆnSRCIR∑m∈Mn∪{n} ΓˆmSRCIG(P 1Popt) +∑m∈Mn mkRnk: (3.15)On the other hand, for 1 → 0, the SRCI data precoder in (3.10) reduces to theselsh ZF (SZF) data precoder. The corresponding received SINR is provided in the72Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsfollowing corollary.Corollory 3.1. Assuming  Q 1, for SZF data precoding, the received SINR at thekth MT in the nth cell is given bySZFnk =1(1−)ΓˆSRCI +∑m∈Mn mkRnk: (3.16)Proof. Please refer to Appendix B.4.Collaborative ZF/RCI PrecodingThe collaborative RCI (CRCI) precoder for the nth cell is given byFn = 2LnHˆHnnP (3.17)where Ln = (HˆHn Hˆn + 2IcT )−1with Hˆn = [Hˆin1 : : : Hˆinb ]i ∈ CbK×cT , 2 is anormalization constant, and 2 is a regularization constant. The corresponding SINRof the kth MT in the nth cell is provided in the following proposition.Proposition 3.2. For CRCI data precoding, the received SINR at the kth MT in thenth cell is given byCRCInk =1∑Mm=1 ΓˆmCRCI+(1+G(P2))2G(P2)(ΓˆnCRCI+ΓˆnCRCI2(1+G(P2))2) +∑m∈Mn mkRnk P (3.18)where ΓˆmCRCI =ΓCRCImkΓCRCI∑Mm=1 mk+1with ΓCRCI =KQ+ KϕPT.Proof. The proof is similar to that for the SINR for the SRCI data precoder given inAppendix B.2.73Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsFurthermore, the optimal regularization constant maximizing the SINR (and thusthe secrecy rate) in (3.18) is obtained as 2Popt = R∑bm=1 ΓˆmCRCI, and the corre-sponding maximum SINR is given byCRCInk =1ΓˆnCRCIR∑bm=1 ΓˆmCRCIG(P 2Popt) +∑m∈Mn mkRnk: (3.19)On the other hand, for 2 → 0, the CRCI precoder in (3.17) reduces to thecollaborative ZF (CZF) precoder. The corresponding received SINR is provided inthe following corollary.Corollory 3.2. Assuming  Q 1Rb , for CZF data precoding, the received SINR atthe kth MT in the nth cell is given byCZFnk =1(1−)ΓˆnCRCI+∑m∈Mn mkRnk: (3.20)Proof. CZFnk in (3.20) is obtained by letting 2 → 0 in (3.18). The proof is similar tothat for the SINR for the SZF data precoder given in Appendix B.4.Remark 3.1. By comparing Propositions 3.1 and 3.2, we observe that SRCInk andCRCInk are identical for  = 1. In this scenario, the estimate of inter-cell CSI at theBS is nothing but a scaled version of that of the in-cell CSI, cf. Chapter 2, and bothschemes are equivalent. Therefore, we will focus more on the scenario of  > 1 in thesequel.Remark 3.2. Selsh data precoders require estimation of in-cell CSI, i.e., Hˆnn, only.In contrast, collaborative data precoders require estimation of both in-cell and inter-cell CSI at the BS, i.e., Hˆn. Furthermore, since collaborative data precoders attemptto avoid interference not only to in-cell users but also to out-of-cell users, more BS74Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsantennas are needed to achieve high performance. This is evident from Corollaries 3.1and 3.2, which reveal that ci > K and ci > K are necessary for SZF and CZFdata precoding, respectively. On the other hand, if successful, trying to avoid out-of-cell interference is benecial for the overall performance. Hence, whether selshor collaborative precoders are preferable depends on the parameters of the consideredsystem, cf. Sections 3.5 and 3.6.3.3.2 POLY Data PrecoderThe RCI and ZF data precoders introduced in the previous section achieve a higherperformance than simple MF data precoding [43]. However, they require a ma-trix inversion which entails a high computational complexity for the large valuesof K and ci desired in massive MIMO. Hence, in this section, we propose a low-complexity POLY data precoder which avoids the matrix inversion. As the goal is alow-complexity design, we focus on selsh POLY precoders, although the extensionto collaborative designs is possible.The proposed POLY precoder, Fn, for the nthBS can be expressed asFn =1√ciHˆHnnI∑i=0i(HˆnnHˆHnn)iP (3.21)where Hˆnn =1√cTHˆnn, and  = [0P : : : P I ]i are the real-valued coecients of theprecoder matrix polynomial, which have to be optimized. In the following, we showthat, for KPci →∞, the optimum coecients  do not depend on the instantaneouschannel estimates but are constant and can be determined by exploiting results fromfree probability [61] and random matrix theory [93]. To this end, we dene theasymptotic average MSE of the users in the nth cell as msen = limK→∞ 1KE [‖en‖2]75Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemswith error vectoren = &yn − sn = &(Gnn(√pFnsn +√qAnzn) + n˜n)− snP (3.22)where n˜n =∑m̸=nGmnxm + nn includes Gaussian noise, inter-cell interference, andinter-cell AN leakage. Furthermore, & is a normalization constant at the receiver,which does not impact detection performance. The optimal coecient vector  min-imizes msen for a given power budget ϕei for the information-carrying signal, i.e.,minP& msen s:t: : Tr{FHn Fn} = 1P (3.23)where we use the notation Tr {·} = limK→∞ tr {·} RK. The optimal coecient vector,opt, is provided in the following theorem.Theorem 3.1. For KPci →∞, the optimal coecient vector minimizing the asymp-totic average MSE of the users in the nth cell for the POLY precoder in (3.21) is givenbyopt = 3Π−1 P (3.24)where  = [P 2P : : : P I+1]i , [Π]iPj = Tr {Dnn}  i+j+(Tr {Dnn∆n}+ Tr{n}+eANcT p) i+j−1,Σn = E[n˜nn˜Hn ], ∆n = diag{1nn1nn+p 1nnP · · · P KnnKnn+p Knn}, and eAN =qE[Tr{GnnAnAHnGHnn}]. Furthermore,  l denotes the lth-order moment of the sumof the eigenvalues of HˆnnHˆHnn, i.e., l = limK→∞ 1K∑Kk=1 lk, which converges to l =∑l−1i=0(li)(li+1)ilfor K → ∞ [73, Theorem 2]. Finally, 3 is chosen such thatTr{FHn Fn} = 1 holds.Proof. Please refer to Appendix B.5.We note that opt does not depend on instantaneous channel estimates, and hence,76Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemscan be computed oine.3.3.3 Computational Complexity of Data PrecodingWe compare the computational complexity of the considered data precoders in termsof the number of oating point operations (FLOPs) [74]. Each FLOP represents onescalar complex addition or multiplication. We assume that the coherence time ofthe channel is i symbol intervals of which  are used for training and i −  areused for data transmission. Hence, the complexity required for precoding in onecoherence interval, consist of the complexity required for generating one precodingmatrix and i −  precoded vectors. A similar complexity analysis was conducted in[73, Section IV] for various selsh data precoders without AN injection at the BS.Since the AN injection does not aect the structure of the data precoders, we candirectly adapt the results from [73, Section IV] to the case at hand. In particular, theselsh MF, the SZF/SRCI, and the CZF/CRCI precoders require (2K−1)ci (i−),0:5(K2 + K)(2ci − 1) + K3 + K2 + K + ciK(2K − 1) + (2K − 1)ci (i − ),and 0:5(2K2 + K)(2ci − 1) + 3K3 + 2K2 + K + ci K(2K − 1) + (2K −1)ci (i − ) FLOPs per coherence interval, see [73, Section IV]. In contrast, forthe POLY data precoder, we obtain for the overall computational complexity (i −) ((I + 1)(2K − 1)ci + I(2ci − 1)K) FLOPs, which assumes implementation ofthe precoding operation by Horner's rule [73, Section IV].The above complexity expressions reveal that the additional complexity intro-duced by collaborative data precoders compared to selsh data precoders is at mosta factor of 3. In addition, the complexity savings achieved with the POLY dataprecoder compared to the SZF/SRCI data precoders increase with increasing K fora given i . We note however that, regardless of their complexity, POLY data pre-77Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemscoders are attractive as they avoid the stability issues that may arise in xed pointimplementations of large matrix inverses.3.4 Linear AN Precoders for Secure MassiveMIMOIn this section, we investigate the performance of selsh and collaborative NS (S/CNS)and random AN precoders. In addition, a novel POLY AN precoder is derived. Tothe best of the authors' knowledge, POLY AN precoding has not been considered inthe literature before.3.4.1 Analysis of Existing AN PrecodersFor a given dimensionality of the AN precoder, a, the secrecy rate depends on theAN precoder only via the AN leakage, Q, given in (3.9), which aects the SINR of theMT. Furthermore, the optimal POLY data precoder coecients in (3.24) are aectedby the AN precoder via the leakage term eAN. In this subsection, for ci → ∞, wewill provide closed-form expressions for Q and eAN for the SNS, CNS, and randomAN precoders.SNS AN PrecoderThe SNS AN precoder of the nth BS is given by [24]An = IcT − HˆHnn(HˆnnHˆHnn)−1HˆnnP (3.25)78Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemswhich has rank a = ci −K and exists only if  Q 1. We divide the correspondingAN leakage QSNS into an inter-cell AN leakage QSNSo and an intra-cell AN leakageQSNSi , where QSNS = QSNSo +QSNSi . For the SNS AN precoder, QSNSo is obtained asQSNSo =∑m∈MnkmnE[hkmnAmAHm(hkmn)H]+∑mR∈Mn∪{n} kmnE[hkmnAmAHm(hkmn)H]= E[tr{AmAHm} ] ∑m∈Mnmk +∑mR∈Mn∪{n} kmn= (ci −K) ∑m∈Mnmk +∑mR∈Mn∪{n} kmn P (3.26)where we exploited [71, Lemma 11] and the independence ofAm and h˜kmn (for contam-inated cells, i.e., m ∈Mn) and hkmn (for non-contaminated cells, i.e., m R∈Mn⋃{n}).In contrast, the intra-cell AN leakage power is given byQSNSi = knnE[hknnAnAHn (hknn)H]= knnE[h˜knnAnAHn (h˜knn)H]= (ci −K)nkP(3.27)as the SNS AN precoder matrix lies in the NS of the estimated channels of all KMTs in the nth cell. Similarly, the AN leakage relevant for computation of the POLYdata precoder is obtained ase SNSAN = (1− ϕ)ei limK→∞1KK∑k=1nk: (3.28)79Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsCNS AN PrecoderFor the CNS AN precoder at the nth BS, the AN is designed to lie in the NS of theestimated channels between all bK MTs and the BS, i.e.,An = IcT − HˆHn(HˆnHˆHn)−1HˆnP (3.29)which has rank a = ci − K and exists only if  Q 1R. The corresponding ANleakage to the kth MT in the nth cell is given byQCNS =b∑m=1kmnE[hkmnAmAHm(hkmn)H]= (ci − K)b∑m=1mk: (3.30)Furthermore, the CNS AN precoder results in the same eAN as the SNS AN precoder,cf. (3.28).Random AN PrecoderFor the random precoder, all elements of An are i.i.d. r.v.s independent of the channel[43], i.e., An has rank a = ci . Hence, hkmn and Am, ∀m, are mutually independent,and we obtainQrandom =b∑m=1kmnE[hkmnAmAHm(hkmn)H]= cib∑m=1kmn: (3.31)Furthermore, we obtain e randomAN = (1− ϕ)ei limK→∞ 1K∑Kk=1 knn.Remark 3.3. If the power and time allocated to channel estimation are very small,i.e., p → 0, the S/CNS AN precoders yield the same qQ and eAN as the randomAN precoder. This suggests that in this regime all considered AN precoders achievea similar SINR performance for a given MT. However, for p > 0, the S/CNS AN80Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsprecoders cause less AN leakage resulting in an improved SINR performance comparedto the random precoder at the expense of a higher complexity.3.4.2 POLY AN PrecoderTo mitigate the high computational complexity imposed by the matrix inversionrequired for the S/CNS AN precoders, while achieving an improved performancecompared to the random AN precoder, we propose a POLY AN precoder. Similar tothe POLY data precoder, we concentrate on the selsh design because of the desiredlow complexity, and hence, set a = ci − K. The proposed POLY AN precoder isgiven byAn = IcT − HˆHnn( J∑i=0,j(HˆnnHˆHnn)j)HˆnnP (3.32)where  = [,0P : : : P ,J ]i contains the real-valued coecients of the AN precoder poly-nomial, which have to be optimized. In particular,  is optimized for minimizationof the asymptotic average AN leakage caused to all MTs in the nth cell eAN. Thecorresponding optimization problem is formulated asmin eAN = qE[Tr{GnnAnAHnGHnn}]s:t: :Tr{AHnAn} = 1R − 1: (3.33)The solution of (3.33) is provided in the following theorem.Theorem 3.2. For KPci →∞, the optimal coecient vector minimizing the asymp-totic average AN leakage caused to the users in the nth cell for the AN precoderstructure in (3.32) is given byopt = Σ−1!P (3.34)where [Σ]iPj = i+j+1 + ϵ i+j and ! = [2 + ϵP : : : P J+2 + ϵJ+1]. Here,  l de-81Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsnotes again the lth order moment of the sum of the eigenvalues of matrix HˆnnHˆHnn,cf. Theorem 3.1. ϵ is chosen such that Tr{AHnAn} = 1R − 1.Proof. Please refer to Appendix B.6.3.4.3 Computational Complexity of AN PrecodingSimilarly to the data precoders, the complexity of the AN precoders is evaluated interms of the number of ops required per coherence interval i . For the SNS ANprecoder, the computation of An in (3.25) requires the computation and inversion ofa K×K positive denite matrix, which entails 0:5(K2+K)(2ci −1)+K3+K2+KFLOPs [74], and the multiplication of an ci ×K, an K×K, and an K×ci matrix,which entails ci (ci+K)(2K−1) FLOPs [74]. Furthermore, the i− vector-matrixmultiplications required for AN precoding entail a complexity of (2ci −1)ci FLOPs[74], respectively. Hence, the overall complexity is 0:5(K2+K)(2ci −1)+K3+K2+K + ci (ci + K)(2K − 1) + (2ci − 1)ci (i − ) FLOPs. Similarly, for the CNSAN precoder, we obtain a complexity of 0:5(2K2 + K(2ci − 1) + 3K3 + 2K2 +K + ci (ci + K)(2K − 1) + (2ci − 1)ci (i − ) FLOPs, whereas the randomAN precoder entails a complexity of (2ci − 1)ci (i − ) FLOPs as only the ANvector-matrix multiplications are required.Similar to the precoded data vector [73, Section IV], the POLY precoded ANvector can be generated using Horner's rule. Hence, based on (3.32), the transmittedAN vector in the nth cell can be obtained asAnzn = zn −(,0HˆHnnHˆnn(zn +,1,0HˆHnnHˆnn (zn + : : :))): (3.35)82Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsHence, Anzn can be computed eciently by rst multiplying Hnn with zn, whichrequires (2ci − 1)K FLOPs, then multiplying HˆHnn with the resulting vector, whichrequires (2K−1)ci FLOPs, adding zn to the resulting vector, and repeating similaroperations (J +1) times, see [16, 73] for details on Horner's rule. Overall, this leadsto a complexity of (J + 1) ((2K − 1)ci + (2ci − 1)K) (i − ) FLOPs.3.5 Comparison of Linear Data and AN PrecodersIn this subsection, we compare the secrecy performances of the considered data andAN precoders. Thereby, in order to get tractable results, we focus on the relativeperformances of SZF, CZF, and MF, cf. Chapter 2 data precoders and SNS, CNS, andrandom AN precoders. The performances of SRCI, CRCI, and POLY data precodersand the POLY AN precoder will be investigated via numerical and simulation resultsin Section 3.6.In order to gain some insight for system design and analysis, we adopt a simpliedpath-loss model. In particular, we assume the path losses are given bykmn =1P m = n/P otherwise(3.36)where / ∈ [0P 1] denotes the inter-cell interference factor. For this simplied model,a and x in (3.8) simplify to a = 1 + (b − 1)/ and x = 1 + (b − 1)/2. Furthermore,the SINR expressions of the linear data precoders considered in Section 3.3.1 and theMF precoder considered in Chapter 2 can be simplied considerably and are providedin Table 3.1, where we use the normalized AN leakage Q˜ = QRa. The expressionsfor the normalized AN leakage Q˜, the asymptotic average AN leakage eAN, and the83Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsdimensionality a of the considered linear AN precoders are given in Table 3.2. Underthe simplied model, mk dened in Proposition 3.1 simplies to1 =p 1+(1+|Mn|/)p  =p 1+bp P for m = n/21 =/2p 1+bp P for m ∈Mn/22 =/2p 1+(|Mn|+1)/p  P for m R∈Mn⋃{n}(3.37)with b = 1 + |Mn|/. Accordingly, the term∑m∈Mn∪{n} mk and∑bm=1 mk inPropositions 3.1 and 3.2 simplify to b − y1 and a − y1 − (x − y)2, respectively,where y = 1 + |Mn|/2. By combining all above intermediate results, ΓˆmSRCI andΓˆmCRCIP 1 ≤ m ≤b , simplify toΓˆmSRCI =ΓˆSRCIP for m = n/2ΓˆSRCIP for m ∈MnP ΓˆmCRCI =ΓˆCRCIP for m = n/2ΓˆSRCIP for m ∈Mn/22R1ΓˆSRCIP for m R∈Mn ∪ {n}P(3.38)respectively, whereΓˆSRCI =ΓSRCI1ΓSRCI(b− y1) + 1 P ΓˆCRCI =ΓCRCI1ΓCRCI (a− y1 − (x− y)2) + 1 P (3.39)andΓSRCI =ϕϕ(a− b) + (1− ϕ)Q˜+ eTP ΓCRCI =ϕ(1− ϕ)Q˜+ eT: (3.40)84Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsTable 3.1: SINR of the kth MT in the nth cell for linear data precoding and thesimplied path-loss model in (3.36).Data Precoder nkSZF1ϕ(1−)(1−ϕ)Q˜+ϕ(a−y1)+(y−1)1ϕ(1−)+ReTSRCI11RyG(PRyΓˆSRCI)+y−1CZF1ϕ(1−)(1−ϕ)Q˜+ϕ(a−y1−(x−y)2)+(y−1)1ϕ(1−)+ReTCRCI11R(y+(x−y)2R1)G(PR(y+(x−y)2R1)ΓˆCRCI)+y−1MF1ϕ(1−ϕ)Q˜+ϕa+(y−1)1ϕ+ReTTable 3.2: AN leakage for simplied path-loss model in (3.36).AN Precoder Q˜ eAN aSNS a− y1 (1− ϕ)ei (1− 1) ci −KCNS a− y1 − (x− y)2 (1− ϕ)ei (1− 1) ci − KRandom a (1− ϕ)ei ci3.5.1 Comparison of SZF, CZF, and MF Data PrecodersIn this subsection, we compare the performances achieved with SZF, CZF, and MFdata precoders for a given AN precoder, i.e., a and Q˜ are xed. Since the upperbound on the capacity of the eavesdropper channel is independent of the adopteddata precoder, cf. Section 3.2.3, we compare the considered data precoders based ontheir SINRs. Exploiting the results in Table 3.1, we obtain the following relationsbetween SZFnk , CZFnk , and MFnk :SZFnkMFnk= 1 + ((y+ |Mn|/2)SZFnk − 1)CZFnkSZFnk=1− 1−  +[(x− y)2R1 + (y− 1)( − 1)]1−  CZFnk : (3.41)Hence, for SZFnk > MFnk , we require SZFnk > 1R(y + |Mn|/2) = 1R(1 + 2/2|Mn|), andfor CZFnk > SZFnk , we need CZFnk > (−1)R((x−y)2R1+(y−1)(−1)). As expected,(3.41) suggests that for a lightly loaded system, i.e.,  → 0, all three precoders have85Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsa similar performance, i.e., CZFnk ≈ SZFnk ≈ MFnk . Moreover, when  = 1, we simplyhave CZFnk = SZFnk , as SZF and CZF are equivalent, cf. Remark 3.1. In the following,we investigate the impact of the number of MTs and the pilot power on the relativeperformances of the considered data precoders.Number of MTs : From (3.41), we nd that for SZFnk > MFnk and CZFnk > SZFnk tohold, the number of MTs has to meet K Q KSZF>MF and K Q KCZF>SZF, whereKSZF>MF =y1ϕci(1− ϕ)Q˜+ aϕ+ 1ReiKCZF>SZF =ϕ(x− y)2ci(1− ϕ)( − 1)Q˜+ ((a− y1)( − 1) + (x− y)2)ϕ+ ( − 1)ReiP(3.42)for  > 1, respectively. Interestingly, both the maximum numbers of MTs for whichthe SZF data precoder is advantageous compared to the MF data precoder, KSZF>MF,and the maximum number of MTs for which the CZF data precoder is advantageouscompared to the SZF data precoder, KCZF>SZF, decrease with increasing AN leakage,Q˜, the number of cells b and the number of contaminated neighboring cells |Mn|(via y), but increase with the amount of resources dedicated to channel estimation,p (via 1 and 2), and consequently with the channel estimation quality. However,while KSZF>MF decreases with increasing inter-cell interference factor / (via a, x, andy), KCZF>SZF increases.3.5.2 Comparison of SNS, CNS, and Random AN PrecodersIn this subsection, we analyze the impact of the AN precoders on the secrecy rate.AN precoders aect the ergodic capacity of the eavesdropper via a and the achievablerate of the MT via the leakage, Q˜. Since the upper bound on the ergodic secrecy rate86Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsof the eavesdropper in (3.8) is a decreasing function in a, we haveXevenk |random ≤ Xevenk |SNS ≤ Xevenk |CNS: (3.43)On the other hand, from Table 3.2, we observe Q˜random ≥ Q˜SNS ≥ Q˜CNS. Sinceaccording to Table 3.1 the SINRs for all data precoders are decreasing functions ofQ˜, for a given data precoder, we obtain for the lower bound on the ergodic rate ofthe kth MT in the nth cellgnk|random ≤ gnk|SNS ≤ gnk|CNS: (3.44)Considering (3.43), (3.44), and the expression for the ergodic secrecy rate, gsecnk =[gnk − Xevenk ]+, it is not a priori clear which AN precoder has the best performance.In fact, our numerical results in Section 3.6 conrm that it depends on the systemparameters (e.g. , , b , , p , and /) which AN precoder is preferable.3.5.3 Ergodic Secrecy Rate AnalysisIn this subsection, we provide closed-form results for the ergodic secrecy rate for SZF,CZF, and MF data precoding for the simplied path-loss model in (3.36). Thereby,the simplied path-loss model is extended also to the eavesdropper, i.e., nE = 1 andmE = /, m ̸= n, is assumed.Combining (3.5), (3.8), and the results in Table 3.1, we obtain the following lower87Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsbounds for the ergodic secrecy rate of the kth MT in the nth cell:gsecnk ≥[log2((Q˜+1ReT )+(a−Q˜)ϕ+y1ϕ(Q˜+1ReT )+(a−Q˜)ϕ+(y−1)1ϕ ·−ϕ+(1−)ϕ+)]+MFP[log2((Q˜+1ReT )+(a−y1−Q˜)ϕ+y1(1−)ϕ(Q˜+1ReT )+(a−y1−Q˜)ϕ+(y−1)1(1−)ϕ ·−ϕ+(1−)ϕ+)]+SZFP[log2((Q˜+1ReT )+(a−y1−(x−y)2−Q˜)ϕ+y1(1−)ϕ(Q˜+1ReT )+(a−y1−(x−y)2−Q˜)ϕ+(y−1)1(1−)ϕ ·−ϕ+(1−)ϕ+)]+CZFP(3.45)where  = a− xcTaa, and Q˜ and a are given in Table 3.2 for the considered ANprecoders. Eq. (3.45) is easy to evaluate and reveals how the ergodic secrecy rate ofthe three considered data precoders depends on the various system parameters. Togain more insight, we determine the maximum value of  which admits a non-zerosecrecy rate. This value is denoted by s in the following, and can be shown to bea decreasing function of ϕ for all conidered data precoders. Hence, we nd s bysetting gsecnk = 0 in (3.45) and letting ϕ→ 0. This leads tos =a21Q˜a+y1cT Ra+aReTfor MF(1−)a21Q˜a+y1(1−)cT Ra+aReT for SZFP(1−)a21Q˜a+y1(1−)cT Ra+aReT for CZF:(3.46)Eq. (3.46) reveals that for a given AN precoder, independent of the system param-eters, the MF data precoder can always tolerate a larger number of eavesdropperantennas than the SZF data precoder, which in turn can always tolerate a largernumber of eavesdropper antennas than the CZF data precoder. This can be explainedby the fact that the high AN transmit power required to combat a large number ofeavesdropper antennas drives the receiver of the desired MT into the noise-limitedregime, where the MF data precoder has a superior performance compared to the88Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO SystemsS/CZF data precoders. On the other hand, since s depends on both Q˜ and a, it isnot a priori clear which AN precoder can tolerate the largest number of eavesdropperantennas. For a lightly loaded network with small  and smallb , according to Table3.2, we have a ≈ ci for all three AN precoders. Hence, in this case, we expect theCNS AN precoder to outperform the SNS and random AN precoders as it achievesa smaller Q˜. On the other hand, for a heavily loaded network with large  and b ,the value of s of the CNS AN precoder is compromised by its small value of a andSNS and even random AN precoders are expected to achieve a larger s.3.6 Performance EvaluationIn this section, we evaluate the performance of the considered secure multi-cell mas-sive MIMO system. We consider cellular systems with b = 2 and b = 7 hexagonalcells, respectively, and to gain insight for system design, we adopt the simpliedpath-loss model introduced in Section 3.5, i.e., the severeness of the inter-cell inter-ference is only characterized by the parameter / ∈ (0P 1]. Various pilot contaminationpatterns are considered by having dierent pilot length  = KP  ∈ M. The sim-ulation results for the ergodic secrecy rate of the kth MT in the nth cell are basedon (3.5), (3.7), and the expression for the ergodic rate of the MT [43, Eq. (8)] andare averaged over 5P 000 random channel realizations. Note that, in this chapter, weconsider the ergodic secrecy rate of a certain MT, i.e., the kth MT in the nth cell. Thecell sum secrecy rate can be obtained by multiplying the secrecy rate of the kth MTby the number of MTs, K, as for the considered channel model, all MTs in the nthcell achieve the same secrecy rate. The values of all relevant system parameters areprovided in the captions of the gures. To enable a fair comparison, throughout thissection, we adopted the SNS AN precoder when we compare dierent data precoders89Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsand the SZF data precoder when we compare dierent AN precoders.3.6.1 Ergodic Capacity of the Eavesdropper for ConventionalLinear AN PrecodersIn Fig. 3.1, we show the ergodic capacity of the eavesdropper for the consideredconventional AN precoders. First, we note that the upper bound in (3.8) is verytight for all AN precoders and all consider values of  and . Furthermore, as increases, the ergodic capacity of all AN precoders decreases since the power allocatedto the information-carrying signal of the user that the eavesdropper tries to interceptdecreases with increasing  as the total power allocated to the information-carryingsignals of all users is xed. As expected, the eavesdropper's capacity benets fromlarger values of . Furthermore, as predicted in (3.43), because of their dierentvalues of a, the CNS AN precoder yields the largest eavesdropper capacity, whilethe random AN precoder yields the lowest. The performance dierences between thedierent AN precoders diminish for small values of  and  as the dependence of theeavesdropper capacity on a becomes negligible for small , cf. (3.8), and a ≈ ciholds for all precoders for small , cf. Table 3.2.3.6.2 Ergodic Secrecy Rate for Conventional Linear DataPrecodersIn Figs. 3.2 and 3.3, we show the ergodic secrecy rates of the kth MT in the nthcell vs. the number of BS antennas for the MF, SZF, CZF, SRCI, and CRCI dataprecoders for a lightly loaded and a dense network, respectively, and a xed powerallocation factor of ϕ = 0:75. In both gures, the analytical results were obtainedfrom (3.5), (3.7), and (3.15) for the SRCI data precoder, (3.19) for the CRCI data90Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.241.522.533.544.5βCapacity of Eavesdropper (bps/Hz)  CNS Upper boundCNS SimulationSNS Upper boundSNS SimulationRandom Upper boundRandom Simulationα = 0.3α = 0.2α = 0.1Figure 3.1: Ergodic capacity of the eavesdropper vs. the normalized number of MTsin the cell, , for a system with ci = 200, ϕ = 0:75, ei = 10 dB, / = 0:3, andb =  = 2.precoder, and (3.45) for the MF, SZF, and CZF data precoders. For all consideredprecoders, the analytical results provide a tight lower bound for the ergodic secrecyrates obtained by simulations. Furthermore, as expected, the RCI data precodersoutperform the ZF data precoders for both the selsh and the collaborative strategies,but the performance gap diminishes with increasing number of BS antennas.For the lightly loaded network in Fig. 3.2, we assume b = 2 cells with no pilotcontamination, i.e.,  = 2, and K = 10 users with a small inter-cell interference factorof / = 0:1. For this scenario, the collaborative designs outperform the selsh designs91Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems200 250 300 350 40033.544.555.56Number of BS antennas NTErgodic secrecy rate (bps/Hz)  CRCI sim.CRCI ana.CZF sim.CZF ana.SRCI sim.SRCI ana.SZF sim.SZF ana.MF ana.Figure 3.2: Analytical and simulation results for the ergodic secrecy rate vs. thenumber of BS antennas, ci , for a lightly loaded network with ϕ = 0:75, ei = 10 dB,p = eiR ,  = 0:1, K = 10, / = 0:1, and b =  = 2.and C/SZF precoding yield a large performance gain compared to MF precoding.This is expected from our analysis in Section 3.5.1 as for the parameters valid forFig. 3.2, we obtain from (3.42), KSZF>MF ≈ 280 and KCZF>SZF ≈ 46 for ci = 400.Intuitively, as the network is only lightly loaded and without pilot contamination,the collaborative data precoder can eciently reduce interference to the other celldespite the expense of spatial degrees of freedom.For the dense network in Fig. 3.3, we assume b = 7 cells,  = 2, K = 40 users,and a larger inter-cell interference factor of / = 0:3. In this case, for the consideredrange of ci , the collaborative precoder designs are not able to suppress inter-cell92Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems200 250 300 350 4000.20.30.40.50.60.70.80.9Number of BS antennas NTErgodic secrecy rate (bps/Hz)  CRCI sim.CRCI ana.CZF sim.CZF ana.SRCI sim.SRCI ana.SZF sim.SZF ana.MF ana.Figure 3.3: Analytical and simulation results for the ergodic secrecy rate vs. thenumber of BS antennas, ci , for a dense network with ϕ = 0:75, ei = 10 dB,  = 2,p = eiR ,  = 0:1, K = 40, / = 0:3, and b = 7.interference and AN leakage to other cells suciently well to outperform the selshprecoder designs. In fact, for ci = 400, we obtain from (3.42) KCZF>SZF ≈ 26, i.e.,our analytical results suggest that the SZF precoder outperforms the CZF precoderfor K = 40 which is conrmed by Fig. 3.3. Nevertheless, for ci > 400, the ergodicsecrecy rate for the CZF data precoder will eventually surpass that for the SZF dataprecoder.93Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.20.40.60.811.21.4φErgodic Secrecy Rate (bps/Hz)  SRCI sim.SRCI ana.SZF sim.SZF ana.MF sim.MF ana.β = 0.1β = 0.5Figure 3.4: Ergodic secrecy rate vs. ϕ for dierent selsh data precoders for a networkwith ei = 10 dB, ci = 100,  = 2, p = eiRK,  = 0:1, / = 0:1, and b = 7.3.6.3 Optimal Power AllocationIn this subsection, we investigate the dependence of the ergodic secrecy rate on thepower allocation factor ϕ and study the impact of system parameters such as , b ,and / on the optimal ϕ that maximizes the ergodic secrecy rate. The results in thissubsection were generated based on the analytical expressions in (3.5), (3.7), and(3.15) for the SRCI data precoder, (3.19) for the CRCI data precoder, and (3.45) forthe MF, SZF, and CZF data precoders.Fig. 3.4 depicts the ergodic secrecy rate of the kth MT in the nth cell for the selshdata precoders SRCI, SZF, and MF as a function of the power allocation factor ϕ. All94Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemscurves are concave and have a single maximum. For ϕ = 0 only AN is transmitted,hence gsecnk = 0 results since no data can be transmitted. For ϕ = 1, no AN istransmitted, hence gsecnk = 0 results since the capacity of the eavesdropper becomesunbounded (recall that we make the worst-case assumption that the eavesdropper canreceive noise-free). For 0 Q ϕ Q 1, a positive secrecy rate may result depending onthe system parameters and the precoding schemes. Since we keep the total transmitpower xed, the transmit power per MT decreases with increasing . To compensatefor this eect, the portion of the total transmit power allocated to data transmissionshould increase. This is conrmed by Fig. 3.4 where the optimal value of ϕ for = 0:5 is larger than that for  = 0:1. Furthermore, for a given , the optimal ϕis the larger, the better the performance of the adopted data precoder is, i.e., for amore eective data precoder, transmitting the data signal with higher power is morebenecial, whereas for a less eective data precoder impairing the eavesdropper witha higher AN power is more benecial.In Fig. 3.5, we show the ergodic secrecy rate vs. ϕ for the CRCI, CZF, and SZFprecoders. Similar to our observations in Fig. 3.4, for given system parameters,the optimal ϕ tends to be larger for more eective precoders that achieve a betterperformance. For the system withb = 7P / = 0:1, this can be observed by comparingthe optimal ϕ for the SZF and CZF precoders. Furthermore, while for the pilotcontamination free system with b = 2P / = 0:3, collaborative precoding is alwayspreferable, forb = 7P / = 0:1, SZF precoding outperforms CZF and CRCI precodingfor most considered values of ϕ, as in this scenario, suppressing the interferenceand AN leakage to the K = 20 MTs in other cells with the available ci = 100antennas is not worth at the expense of sacricing extra spatial degrees of freedomfor collaborative designs. In particular, from (3.42), we obtain KCZF>SZF ≤ 6 for95Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.800.511.522.53φErgodic Secrecy Rate (bps/Hz)  CRCI sim.CRCI ana.CZF sim.CZF ana.SZF sim.SZF ana.M = 2, ρ = 0.3M = 7, ρ = 0.1Figure 3.5: Ergodic secrecy rate vs. ϕ for dierent data precoders for a network withei = 10 dB, ci = 100,  = 2, p = eiR ,  = 0:1, and  = 0:1.b = 7P / = 0:1 and KCZF>SZF ≤ 30 for b = 2P / = 0:3, which conrms the resultsshown in Fig. 3.5.Fig. 3.6 depicts the ergodic secrecy rate vs. ϕ for the considered conventionalAN precoder structures. We consider a lightly loaded network with  = 0:1 anda moderately loaded network with  = 0:4. For  = 0:1, the CNS AN precoderoutperforms the SNS AN precoder since, in this case, for the CNS AN precoder,the negative impact of having (slightly) fewer dimensions available for degrading theeavesdropper's channel (smaller value of a) is outweighed by the positive impact ofcausing less AN leakage (smaller value of Q˜). On the other hand, for  = 0:4, the96Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.511.522.533.5φErgodic Secrecy Rate (bps/Hz)  CNS sim.CNS ana.SNS sim.SNS ana.Random sim.Random ana.β = 0.1β = 0.4Figure 3.6: Ergodic secrecy rate vs. ϕ for dierent AN precoders for a network withei = 10 dB, ci = 100,  = 2, p = eiR , b = 2, / = 0:1, and  = 0:1.CNS AN precoder has a substantially smaller a than the SNS precoder which cannotbe compensated by its larger Q˜. Despite having the largest value of a, the randomAN precoder has the worst performance for both considered cases because of its largeAN leakage.3.6.4 Conditions for Non-Zero Secrecy RateIn Section 3.5.3, we showed that a positive ergodic secrecy rate is possible only if Q s. In Fig. 3.7, using (3.46), we plot s as a function of . In the left hand sidesubgure, we compare MF, SZF, and CZF data precoding for SNS AN precoding, and97Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems0 0.05 0.1 0.150.540.560.580.60.620.640.660.680.70.72βαs  MFSZFCZF0 0.5 100.10.20.30.40.50.60.70.80.9βαs  RandomSNSCNSFigure 3.7: s vs.  for dierent data and AN precoders for a network with ei = 10dB, ci = 100,  = 2, p = eiR , / = 0:3, and b = 2.in the right hand side subgure, we compare random, SNS, and CNS AN precodingfor SZF data precoding. The comparison of the data precoders reveals that althoughSZF and CZF entail a much higher complexity, MF precoding achieves a larger s.Therefore, if the eavesdropper has a large number of antennas and small ergodicsecrecy rates are targeted, simple MF precoding is always preferable. On the otherhand, whether SNS or CNS AN precoder is preferable depends on the system load.For small values of , CNS AN precoding can tolerate more eavesdropper antennas,whereas for large values of , SNS AN precoding is preferable. Random AN precodingis outperformed by SNS and/or CNS AN preceding for any value of . A closer98Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsexamination of (3.46) reveals that this is always true if S/CZF data precoders areemployed. However, for the MF data precoder, there are parameter combination forwhich random AN precoding outperforms SNS and CNS AN precoding.3.6.5 Low-Complexity POLY Data and AN PrecodersIn this subsection, we evaluate the ergodic secrecy rates of the proposed low-complexityPOLY data and AN precoders. To this end, we consider again a lightly loaded net-work with little inter-cell interference (b = 2,  = 0:1, / = 0:1) and a dense networkwith more inter-cell interference (b = 7,  = 0:15, / = 0:3). All results shownin this section were obtained by simulation. For each simulation point, the optimalvalue of ϕ was found numerically and applied. In Figs. 3.8 and 3.9, we show theergodic secrecy rate of the kth MT in the nth cell as a function of the pilot energy,p . As expected, for all considered schemes, the ergodic secrecy rate is monoton-ically increasing in the pilot energy since more accurate channel estimates improveperformance and the total power used for data and AN transmission, ei , is assumedto be xed.In Fig. 3.8, we depict the ergodic secrecy rates for the proposed POLY dataprecoder for dierent values of I and compare them to those of conventional selshdata precoders. For the sake of comparison, all data precoders are combined withthe SNS AN precoder. As the number of terms of the polynomial I increase, theperformance of the POLY data precoder quickly improves and approaches that of theSRCI data precoder. The convergence is faster for the dense network considered in theright hand side subgure, where the performance dierence between all precoders issmaller in general since interference cannot be as eciently avoided as for the lightlyloaded network.99Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems−10 −5 0 5 10 1500.511.522.533.5Pilot power τpτ (dB)Ergodic secrecy rate (bps/Hz)M = 2, ρ = 0.1, β = 0.1−5 0 5 10 150.10.120.140.160.180.20.220.240.260.28Pilot power τpτ (dB)Ergodic secrecy rate (bps/Hz)M = 7, ρ = 0.3, β = 0.15  SRCISZFPOLYMFI = 1, 3, 5I = 1, 3, 5Figure 3.8: Ergodic secrecy rate for POLY and conventional selsh data precodersfor a network employing the optimal ϕ, ei = 10 dB,  = 1, p = eiR , ci = 200,and  = 0:1.In Fig. 3.9, we show the ergodic secrecy rates for the proposed POLY AN precoderfor dierent values of J and compare them to those of the random and SNS ANprecoders. For the sake of comparison, all AN precoders are combined with SZFdata precoding. The POLY AN precoder quickly approaches the performance of theSNS AN precoder as the polynomial order J increases. Similar to the POLY dataprecoders, the convergence is faster for the dense network where the performancedierences between dierent AN precoders are also smaller. For the denser network,even the random AN precoder is a viable option and suers only from a small loss inperformance compared to the SNS AN precoder.100Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems−10 −5 0 5 10 1500.511.522.533.5Pilot power τpτ (dB)Ergodic secrecy rate (bps/Hz)M = 2, ρ = 0.1, β = 0.1  SNSPOLYRandom−5 0 5 10 150.060.080.10.120.140.160.180.20.220.240.26Pilot power τpτ (dB)Ergodic secrecy rate (bps/Hz)M = 7, ρ = 0.3, β = 0.15J = 1, 3, 5J = 1, 3, 5Figure 3.9: Ergodic secrecy rate for POLY and SNS AN precoders for a networkemploying the optimal ϕ, ei = 10 dB,  = 1, p = eiR , ci = 200, and  = 0:1.3.6.6 Complexity-Performance TradeoIn this subsection, we investigate the tradeo between the ergodic secrecy rate per-formance and the computational complexity for the proposed data and AN precodersin Figs. 3.10 and 3.11, respectively. In particular, Figs. 3.10 and 3.11 depict theergodic secrecy rate on the right hand side and the computational complexity (inGiga FLOP) on the left hand side, both as a function of the numbers of users in acell. For the considered setting, the performance gains of collaborative data and ANprecoding compared to selsh strategies are moderate, but the associated increase incomplexity is substantial, especially for large K.101Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems100 200 300 40011.522.533.544.55Number of MTs in each cell, KErgodic secrecy rate (bps/Hz)Ergodic secrecy rate100 200 300 40000.511.522.533.5Number of MTs in each cell, KComputational complexity (Giga−FLOP)Computational complexity  CRCISRCIPOLYMFI = 1, 3, 5I = 1, 3, 5Figure 3.10: Ergodic secrecy rate (left hand side) and computational complexity(right hand side) of various linear data precoders for a network employing ei = 10dB,ci = 1000, p = eiR , b =  = 2, / = 0:1, i −  = 100, and an SNS AN precoder.Fig. 3.10 illustrates that for the considered setting a POLY data precoder withI = 1 achieves a better performance than the MF precoder but has substantiallylower complexity than the SRCI precoder. For large I, the POLY data precoder hasa lower complexity than the SRCI precoder for large K. However, even for small K,the POLY precoder may be preferable as it does not incur the stability issues thatmay arise in the implementation of the large-scale matrix inversions required for theSRCI precoder.Fig. 3.11 shows that for the considered setting the proposed POLY AN precoderwith J = 1 outperforms the Random AN precoder, and with J = 5 achieves al-102Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systems100 200 30011.522.533.5Number of MTs in each cell, KErgodic secrecy rate (bps/Hz)Ergodic secrecy rate100 200 30000.511.522.53Number of MTs in each cell, KComputational complexity (Giga−FLOP)Computational complexity  CNSSNSPOLYRandomJ = 1, 3, 5J = 1, 3, 5Figure 3.11: Ergodic secrecy rate (left hand side) and computational complexity(right hand side) of various linear AN precoders for a network employing ei = 10dB,ci = 1000, p = eiR , b =  = 2, / = 0:1, i − = 100, and an SZF data precoder.most the same performance as the SNS AN precoder but with a substantially lowercomplexity. We further observe that for small K, the proposed POLY AN precoderrequires even lower complexity than the Random AN precoder, owing to the ecientstructure given in (3.35) operated by Horner's rule.3.7 ConclusionsIn this chapter, we considered downlink multi-cell massive MIMO systems employinglinear data and AN precoding for physical layer security provisioning. We analyzed103Chapter 3. Linear Data and AN Precoding in Secure Massive MIMO Systemsand compared the achievable ergodic secrecy rate of various conventional data andAN precoders in the presence of pilot contamination. To this end, we also opti-mized the regularization constants of the selsh and collaborative RCI precoders inthe presence of AN and multi-cell interference. In addition, we derived linear POLYdata and AN precoders which oer a good compromise between complexity and per-formance in massive MIMO systems. Interesting ndings of this chapter include: 1)Collaborative data precoders outperform selsh designs only in lightly loaded systemswhere a sucient number of degrees of freedom for suppressing inter-cell interferenceand sucient resources for training are available. 2) Similarly, CNS AN precoding ispreferable over SNS AN precoding in lightly loaded systems as it causes less AN leak-age to the information-carrying signal, whereas in more heavily loaded systems, CNSAN precoding does not have sucient degrees of freedom for eectively degradingthe eavesdropper channel and SNS AN precoding is preferable. 3) For a large num-ber of eavesdropper antennas, where only small positive secrecy rates are achievable,MF data precoding is always preferable compared to SZF and CZF data precoding.4) The proposed POLY data and AN precoders approach the performances of theSRCI data and SNS AN precoders with only a few terms in the respective matrixpolynomials and are attractive options for practical implementation.104Chapter 4Hardware Impairments in SecureMassive MIMO Systems4.1 IntroductionSince security is a critical concern for future communication systems, facilitatingsecrecy at the physical layer of massive MIMO systems has received signicant atten-tion recently. All aforementioned works on secure massive MIMO systems includingChapters 2 and 3 are based on the assumption that the transceivers of the legitimateusers are equipped with perfect hardware components, i.e., the eects of hardwareimpairments were not taken into account. Nevertheless, all practical implementa-tions do suer from hardware impairments such as phase noise, quantization errors,amplication noise, and nonlinearities [18]. These impairments are expected to beparticularly pronounced in massive MIMO systems as the excessive number of BSantennas makes the use of low-cost components desirable to keep the overall capi-tal expenditures for operators manageable. Although hardware impairments can bemitigated by analog and digital signal processing techniques [19], they cannot beremoved completely, due to the randomness introduced by the dierent sources ofimperfection. The remaining residual hardware impairments can be modelled by acombination of phase noise and additive distortion noises at the transmitter and thereceiver [19]. Several works have investigated the impact of hardware impairments105Chapter 4. Hardware Impairments in Secure Massive MIMO Systemson massive MIMO systems [18], [20, 21, 22]. They all demonstrated that hardwareimpairments can severely limit the performance of massive MIMO systems. Thereby,a crucial role is played by the degradation caused by phase noise to the quality ofthe CSI estimates needed for precoder design. On the one hand, phase noise causesthe CSI estimates to become outdated more quickly, and on the other hand, it maycause a loss of orthogonality of the pilot sequences employed by the dierent users ina cell for uplink training. To overcome the latter eect, so-called TO and SO pilotsequences were investigated in [22]. Furthermore, the impact of the number of LOsemployed at the massive MIMO BS on the performance in the presence of phase noisewas studied in [20, 22].All aforementioned works [18], [20, 21, 22] studied the impact of hardware impair-ments in the context of conventional massive MIMO system design without regardfor communication secrecy. However, if communication secrecy is considered, anadditional challenge arises: Whereas the legitimate user of the system will likely em-ploy low-cost equipment giving rise to hardware impairments, the eavesdropper isexpected to employ high-quality equipment which can compensate for all hardwareimpairments except for the additive distortion noise at the BS. This disparity inequipment quality was not considered in the related work on physical layer security[43]-[64] nor in the related work on hardware impairments [18], [20, 21, 22] and ne-cessitates the development of a new analysis and design framework. For example,NS AN precoding, which was widely used to enhance the achievable secrecy rateof massive MIMO systems [25, 43, 64], becomes ineective in the presence of phasenoise.Motivated by the above considerations, in this chapter, we present the rst studyof physical layer security in hardware constrained massive MIMO systems. Thereby,106Chapter 4. Hardware Impairments in Secure Massive MIMO Systemswe focus on the downlink and adopt for the legitimate links the generic residualhardware impairment model from [19, 22], which includes the eects of multiplicativephase noise and additive distortion noise at the BS and the users. As a worst-case scenario, the eavesdropper is assumed to employ ideal hardware. Our maincontributions are summarized as follows.• For the adopted generic residual hardware impairment model, we derive a tightlower bound for the ergodic secrecy rate achieved by a downlink user whenMF data precoding is employed at the massive MIMO BS. The derived boundprovides insight into the impact of various system and channel parameters, suchas the phase noise variance, the additive distortion noise parameters, the ANprecoder design, the amount of power allocated to the AN, the pilot sequencedesign, the number of deployed LOs, and the number of users, on the ergodicsecrecy rate.• As conventional NS AN precoding is sensitive to phase noise, we propose anovel G-NS AN precoding design, which mitigates the AN leakage caused to thelegitimate user in the presence of phase noise at the expense of a reduction of theavailable spatial degrees of freedom. The proposed method leads to signicantperformance gains, especially in systems with large numbers of antennas at theBS.• We generalize the SO and TO pilot sequence designs from [22] to orthogonalpilot sequences with arbitrary numbers of non-zero elements. Although SOsequences, which have no zero elements, are preferable for small phase noisevariances, sequence designs with zero elements become benecial in the presenceof strong phase noise.107Chapter 4. Hardware Impairments in Secure Massive MIMO Systems• Our analytical and numerical results reveal that while hardware impairmentsin general degrade the achievable secrecy rate, the proposed countermeasuresare eective in limiting this degradation. Furthermore, surprisingly, there arecases when the additive distortion noise at the BS is benecial for the secrecyperformance as it can have a similar eect as AN.The remainder of this chapter is organized as follows. In Section 4.2, the mod-els for uplink training and downlink data transmission in the considered massiveMIMO system with imperfect hardware are presented. In Section 4.3, we derive alower bound on achievable ergodic secrecy rate and introduce the proposed G-NS ANprecoder design. The impact of the various system and channel parameters on thesecrecy performance is investigated based on the derived lower bound in Section 4.4.In Section 4.5, the achievable secrecy rate is studied via simulation and numericalresults, and conclusions are drawn in Section 4.6.4.2 System and Channel ModelsThe considered massive MIMO system model comprises an c -antenna BS, K single-antenna MTs, and an cE-antenna eavesdropper. The eavesdropper is passive in orderto hide its existence from the BS and the MTs. Similar to [18, 22], we assume thatafter proper compensation the residual hardware impairments manifest themselves atthe BS and the MTs in the form of 1) multiplicative phase noises at transmitter andreceiver, 2) transmit and receive power dependent distortion noises at transmitterand receiver, respectively, and 3) amplied thermal noise at the receiver. The impactof this general hardware impairment model on uplink training and downlink datatransmission is investigated in Sections 4.2.1 and 4.2.2, respectively, and the signalmodel for the eavesdropper is presented in Section 4.2.3.108Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsCoherence Block TUplink TrainingPhaseBDownlink DataTransmission PhaseT-BSub-Phase 1Sub-Phase 2Sub-Phase BoFigure 4.1: Uplink training and downlink transmission phase.4.2.1 Uplink Pilot Training under Hardware ImpairmentsIn massive MIMO systems, the CSI is usually acquired via uplink training by ex-ploiting the channel reciprocity between uplink and downlink [5, 8]. Here, we assumethat the rst B symbol intervals of the coherence time, which comprises i symbolintervals, are used for uplink training. Thereby, we split the training phase into Bosub-phases of lengths Bb, 1 ≤ b ≤ Bo, where∑Bob=1Bb = B, cf. Fig. 4.1. Furthermore,the K MTs are assigned to Bo disjunct sets Sb, 1 ≤ b ≤ Bo, with |Sb| ≤ Bb and∑Bob=1 |Sb| = K. In training sub-phase b, the MTs in set Sb emit mutually orthogo-nal pilot sequences !k = [!k(1)P !k(2)P : : : P !k(Bb)]i ∈ CBb×1P k ∈ Sb, for which weassume a per-pilot power constraint |!k(t)|2 = p P∀kP t, whereas all MTs k R∈ Sb aresilent. For larger values of Bb, the total energy of the pilot sequences is larger but, aswill be shown later, the loss of orthogonality caused by phase noise becomes also morepronounced. Hence, Bb or equivalently Bo (assuming a xed B) should be optimizedfor maximization of the secrecy rate. We note that the proposed pilot design is ageneralization of the SO and TO pilot designs proposed in [22, 20] which result asspecial cases for Bo = 1 and Bo = B, respectively.In symbol interval t ∈ Tb, where Tb denotes the set of symbol intervals in trainingsub-phase b, 1 ≤ b ≤ Bo, the received uplink vector yUL(t) ∈ Cc×1 at the BS is given109Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsbyyUL(t) =∑k∈SbΘk(t)gk(!k(t) + MTtPk (t)) + BSr (t) + UL(t): (4.1)Here, the channel vector of the kth MT, gk ∼ CN(0c P kIc), is modelled as blockRayleigh fading, where k denotes the path-loss. Thereby, gk is assumed to be con-stant during coherence time i and change independently afterwards. In (4.1), theterms Θk(t), MTtPk (t), BSr (t), and UL(t) characterize the hardware impairments af-fecting the uplink training phase and are explained in detail in the following:1) Phase noise: MatrixΘk(t) = diag(zj1k(t)11×cRco P : : : P zjbok (t)11×cRco)∈ Cc×c (4.2)models the phase noise originating from the free-running LOs equipped at the BSand the MTs [20]. Thereby, we assume that at the BS each group of cRco ∈ Zantennas is connected to one free-running LO. lk(t) =  l(t)+ϕk(t) is the phase noisethat distorts the link between the lth LO at the BS and the kth MT. Adopting thediscrete-time Wiener phase noise model [20], in time interval t, the phase noises atthe lth LO of the BS and the kth MT are modelled as  l(t) ∼ CN( l(t − 1)P 2 ),1 ≤ l ≤ co, and ϕk(t) ∼ CN(ϕk(t − 1)P 2ϕ), 1 ≤ k ≤ K, where 2 and 2ϕ are thephase noise (increment) variances at the BS and the MTs, respectively.2) Distortion noise: MTtPk (t) ∈ C and BSr (t) ∈ Cc×1 model the additive distortionnoise at the kth MT and the BS, respectively, which originates from the residual eectsafter compensation of hardware impairments such as power amplier non-linearitiesat the transmitter, quantization noise in the analog-to-digital converters (ADCs) atthe receiver, etc. [18]. Distortion noise is modeled as a Gaussian distributed randomprocess in the literature [18, 19]. This model has been experimentally veried in110Chapter 4. Hardware Impairments in Secure Massive MIMO Systems[76]. Furthermore, at each antenna, the distortion noise power is proportional to thecorresponding signal power, i.e., MTtPk (t) ∼ CN(0P MTtPk ) and BSr (t) ∼ CN(0c PΥBSr ),whereMTtPk = MTt E[|!k(t)|2] and ΥBSr = BSrK∑k=1E[|!k(t)|2]Rdiagk : (4.3)Here, Rdiagk = diag(|g1k|2P : : : P |gck |2), where gik denotes the ith element of gk, andparameters MTt P BSr > 0 denote the ratio between the additive distortion noise vari-ance and the signal power and are measures for the severity of the residual hardwareimpairments.3) Amplied thermal noise: UL(t) ∼ CN(0c P ULIc) models the thermal noiseamplied by the low noise amplier and other components such as mixers at thereceiver [22]. Therefore, the variance of this noise is generally larger than that of theactual thermal noise 2n, i.e., UL > 2n.For channel estimation, we collect the signal vectors received during the bthtraining phase in vector  b = [(yUL(Bb−1 + 1))i P : : : P (yUL(Bb))i ]i ∈ CBbc×1,b = 1P : : : P Bo, where Bb ,∑bi=1Bi and B0 = 0, and dene the eective channelvector at time t as gk(t) = Θk(t)gk. With these denitions, the MMSE estimateof the channel of MT k ∈ Sb at time t ∈ {B + 1P : : : P i} (i.e., during the datatransmission phase) can be written as [22]gˆk(t) = E[gk(t) Hb ](E[ b Hb ])−1 b =(k!Hk Θb(t)Σ−1b ⊗ Ic) bP (4.4)whereΘb(t) = diag(z−2 +2ϕ2|t−Bb−1−1|P : : : P z−2 +2ϕ2|t−Bb|)(4.5)111Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsandΣb =∑k∈Sbk(Wbk +Ubk)+ ULIBb : (4.6)Here, we adopted the denitions [Wbk]iPj = !k(i)!∗k(j)z−2 +2ϕ2|i−j|, iP j ∈ {1P : : : Bb},and Ubk = (MTt + BSr )pIBb .Considering the properties of MMSE estimation, the channel can be decomposedas gk(t) = gˆk(t)+e(t), t = 1P : : : P B, where gˆk(t) denotes the MMSE channel estimategiven in (4.4) and ek(t) represents the estimation error. gˆk(t) and e(t) are mutuallyuncorrelated and have zero mean [18, Theorem 1]. The error covariance matrix isgiven byE[ek(t)eHk (t)] = k(1− k!Hk Θb(t)Σ−1b Θb(t)!k)Ic : (4.7)Eqs. (4.4)-(4.7) reveal that for |Sb| > 1 and 2 P 2ϕ > 0, the channel estimate of thekth MT contains contributions from channels of other MTs emitting their pilots inthe same training sub-phase, i.e., pilot contamination occurs although the emittedpilots are orthogonal. This loss of orthogonality at the receiver is introduced by thephase noise via matrices Θb(t) andWbk, and can be avoided only by enforcing that inany sub-phase only one MT emits its pilots, i.e., |Sb| = 1, 1 ≤ b ≤ Bo. In particular,for the case |Sb| = Bb = 1, 1 ≤ b ≤ Bo = B, for symbol interval t ∈ {B + 1P : : : i},the MMSE channel estimate of MT k ∈ Sb can be simplied togˆk(t) =kz−2 +2ϕ2|t−b|pk(1 + MTt + BSr ) + ULyUL(b)P (4.8)with yUL(t) given in (4.1), i.e., gˆk(t) is not aected by the channels of other MTs112Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsdespite the phase noise. The corresponding error covariance matrix simplies toE[e(t)eH(t)] = k(1− pkpk(1 + MTt + BSr ) + UL)Ic : (4.9)Eqs. (4.4) and (4.8) reveal that the channel estimate depends on time t. As a conse-quence, ideally, the channel-dependent data and AN precoders employed for downlinktransmission should be recomputed for every symbol interval of the data transmissionphase, which entails a high computational complexity. Therefore, in the following,we assume that data and AN precoders are computed based on the channel estimatefor one symbol interval t0 (e.g., t0 = B + 1) and are then employed for precodingduring the entire data transmission phase, i.e., for t ∈ {B + 1P : : : P i}. For nota-tional conciseness, we denote the corresponding channel estimate by gˆk = gˆk(t0),k = {1P : : : P K}.4.2.2 Downlink Data Transmission and Linear PrecodingAssuming channel reciprocity, during the downlink data transmission phase, the re-ceived signal at the kth MT in time interval t ∈ {B + 1P : : : P i} is given byyDLk (t) = gHk ΘHk (t)(x+ BSt (t)) + MTrPk (t) + DLk (t): (4.10)In (4.10), similar to the uplink, BSt (t) ∼ CN(0c PΥBSt ) and MTrPk (t) ∼ CN(0P MTrPk (t))denote the downlink distortion noise [18] at the BS and the kth MT, respectively,whereΥBSt = BSt diag (m11P : : : P mcc) and MTrPk (t) = MTr gHk (t)Xgk(t) (4.11)113Chapter 4. Hardware Impairments in Secure Massive MIMO Systemswith X = E[xxH ] and mii = [X]iiP i = 1P : : : P c . Furthermore, DLk (t) ∼ CN(0P DL)represents the amplied thermal noise at the kth MT. For simplicity of presentation,we assume that parameters BSt , MTr , and DLare identical for all MTs.The downlink transmit signal x ∈ Cc×1 in (4.10) is modeled asx =√pFs+√qAz ∈ Cc×1P (4.12)where the data symbol vector s ∈ CK×1 and the AN vector z ∈ Ca×1, a ≤ c , aremultiplied by data precoder F ∈ Cc×K and AN precoder A ∈ Cc×a, respectively.As we assume that the eavesdropper's CSI is not available at the BS, AN is injectedto degrade the eavesdropper's ability to decode the data intended for the MTs [25,43, 64]. Thereby, it is assumed that the components of s and z are independentand identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG)random variables, i.e., s ∼ CN(0K P IK) and z ∼ CN(0aP Ia). In (4.12), p = ϕeiRKand q = (1− ϕ)eiRa denote the power assigned to each MT and each column of theAN, where ei is the total power budget and ϕ ∈ (0P 1] can be used to strike a balancebetween data transmission and AN emission. Combining (4.12) and (4.10) we obtainyDLk (t) =√pgHk (t)fksk+K∑l ̸=k√pgHk (t)flsl+√qgHk (t)Az+gHk (t)BSt (t)+MTrPk (t)+DLk (t)P(4.13)where sk and fk denote the kthelement of s and the kth column of matrix F, respec-tively.4.2.3 Signal Model of the EavesdropperWe assume that the eavesdropper is silent during the training phase, i.e., for t ∈{1P : : : P B}, and eavesdrops the signal intended for MT k during the data transmis-114Chapter 4. Hardware Impairments in Secure Massive MIMO Systemssion phase, i.e., for t ∈ {B + 1P : : : P i}. Let GE denote the channel matrix betweenthe BS and the eavesdropper with i.i.d. zero-mean complex Gaussian elements havingvariance E, where E is the path-loss between the BS and the eavesdropper. Sincethe capabilities of the eavesdropper are not known at the BS, we make worst-caseassumptions regarding the hardware and signal processing capabilities of the eaves-dropper with respect to communication secrecy. In particular, we assume the receivedsignal at the eavesdropper at time t ∈ {B + 1P : : : P i} can be modelled asyE(t) = GHEΨH(t)(x+ BSt (t)) ∈ CcE×1P (4.14)where Ψ(t) = diag(zj 1(t)1i1×cRco P : : : P zj bo (t)1i1×cRco). Thereby, we assumed thatthe eavesdropper employs high-quality hardware such that the only hardware impair-ments are the phase noise and the additive distortion noise at the BS. Eq. (4.14) alsoimplies that the thermal noise at the eavesdropper is negligibly small [25, 43, 64]. Fur-thermore, we assume that the eavesdropper has perfect CSI, i.e., it perfectly knowsthe eective eavesdropper channel matrix GHEΨH(t), and can perfectly decode andcancel the interference caused by all MTs except for the MT of interest [25, 43, 64].These worst-case assumptions lead to an upper bound on the ergodic capacity of theeavesdropper given byXE = E[log2(1 + E)] (4.15)whereE = pgkE(GHE (qAAH +ΥBSt )GE)−1(gkE)H(4.16)and gkE = fHk GE. We note that since we assumed that the thermal noise at thereceiver of the eavesdropper is negligible, E, and consequently XE, are independent ofthe path-loss of the eavesdropper, E. Furthermore, since perfect channel estimation115Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsat the eavesdropper was assumed, the phase noise can be compensated and the onlyremaining hardware impairment aecting the performance of the eavesdropper isthe additive distortion noise at the BS, which impacts the ergodic capacity of theeavesdropper in a similar manner as the AN injected at the BS, cf. (4.16).4.3 Achievable Ergodic Secrecy Rate in thePresence of Hardware ImpairmentsIn this section, we analyze the achievable ergodic secrecy rate of a massive MIMOsystem employing non-ideal hardware. To this end, we derive a lower bound on theachievable ergodic secrecy rate in Section 4.3.1, and present an asymptotic analysisfor the downlink data rate of the legitimate MTs when MF data precoding is adoptedby the BS in Section 4.3.2. In Section 4.3.3, a generalized NS AN precoder is proposedto avoid the AN leakage caused by phase noise for conventional NS AN precoding.Finally, in Section 4.3.4, a simple closed-form upper bound for the eavesdropper'scapacity for the new AN precoder is presented.4.3.1 Lower Bound on Achievable Ergodic Secrecy RateIn this chapter, we assume that communication delay is tolerable and coding overmany independent channel realizations is possible. Hence, we adopt the ergodicsecrecy rate achieved by a given MT as performance metric [25].Before analyzing the secrecy rate, we rst employ [22, Lemma 1] to obtain a lowerbound on the achievable rate for the multiple-input single-output (MISO) phase noisechannel given by (4.10). In particular, the achievable rate of the kth MT, 1 ≤ k ≤ K,116Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsin symbol interval t ∈ {B + 1P : : : P i} is lower bounded bygk(t) ≥ gk(t) = log2(1 + k(t))P (4.17)with SINR k(t) =p∣∣E [gHk (t)fk]∣∣2K∑l=1pE[|gHk (t)fl|2]− p |E [gHk (t)fk]|2 + E[gHk (t)(qAAH +ΥBSt )gk(t)]+ E[MTkPr (t)]+ DL:(4.18)The expectation operator in (4.18) is taken with respect to channel vectors, gk, aswell as the phase noise processes,  l(t) and ϕk(t). The SINR in (4.18) is obtainedby employing the average eective channel gain∣∣E [gHk (t)fk]∣∣ for signal detection,while treating the deviation from the average eective channel gain as Gaussiannoise having variance E[∣∣gHk (t)fk∣∣2]−|E [gHk (t)fk] |2, cf. [8]. Moreover, following [22,Lemma 1] we treated the multiuser interference and distortion noises as independentGaussian noises, which is a worst-case assumption for the calculation of the mutualinformation. Based on (4.17), we provide a lower bound on the achievable ergodicsecrecy rate of the kth MT, 1 ≤ k ≤ K, in the following Lemma.Lemma 4.1. : The achievable ergodic secrecy rate of the kth MT, 1 ≤ k ≤ K, isbounded below bygseck ≥ gseck =1i∑t∈{B+1P:::Pi}[gk(t)− XE]+ P (4.19)where gk(t), 1 ≤ k ≤ K, is the lower bound of the achievable ergodic rate of the kthMT given in (4.17) and XE is the ergodic capacity between the BS and the eavesdrop-per given in (4.15).117Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsProof. Please refer to Appendix C.1.XE in (4.19) is constant for all t ∈ {B + 1P : : : P i} as we made the worst-caseassumptions that the eavesdropper employs ideal hardware and has perfect CSI. Thesum in (4.19) is over the i−B time slots used for data transmission. Motivated by thecoding scheme for the non-secrecy case in [77], a similar coding scheme that supportsthe secrecy rate given in (4.19) is described as follows. For a given t ∈ {B+1P : : : P i},the statistics of gk(t) in (4.18) given the estimate gˆk are identical across all coherenceintervals and the corresponding channel realizations are i.i.d. Hence, we employ i−Bparallel channel codes for each MT; one code for each time t ∈ {B+1P : : : P i}, i.e., thetth channel code is employed across the tth time slots of multiple coherence intervals.Then, at each MT, the tth received symbols across the multiple coherence intervalsare jointly decoded [77]. With this coding strategy the ergodic secrecy rate givenin (4.19) is achieved provided the parallel codes span suciently many (ideally aninnite number) of independent channel realizations gk and phase noise samples  l(t)and ϕk(t).4.3.2 Asymptotic Analysis of Achievable Rate for MFPrecodingIn this subsection, we analyze the lower bound on the achievable ergodic rate of thekth MT, 1 ≤ k ≤ K, in (4.17) in the asymptotic limit cPK → ∞ for xed ratio = KRc . Thereby, we adopt MF precoding at the BS, i.e., fk = gˆkR‖gˆk‖, as iscommonly done for massive MIMO systems because of complexity concerns for moresophisticated precoder designs. In the following Lemma, we provide a closed-formexpression for the gain of the desired signal.118Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsLemma 4.2. : For MF precoding at the BS, the numerator of (4.18) reecting thegain of the desired signal at MT k ∈ Sb, 1 ≤ b ≤ Bo, can be expressed asE[gHk ΘHk (t)fk]=√kck · z−2 +2ϕ2|t−t0|P where k = k!Hk Θb(t0)Σ−1b Θb(t0)!k:(4.20)Proof. Please refer to Appendix C.2.The term z−2 +2ϕ2|t−t0|in (4.20) reveals the impact of the accumulated phase noisefrom the time of channel estimation, t0, to the time of data transmission, t, on thereceived signal strength at MT k. On the other hand, the phase noise within thetraining phase aects k, and consequently the received signal strength, via Θb(t0)and Σb, cf. (4.5), when multiple pilot sequences are simultaneously emitted in a giventraining sub-phase. In contrast, when TO pilots are adopted, i.e., only a single useremits pilots in each training sub-phase and Bb = 1, 1 ≤ b ≤ B, k in (4.20) reducesto k =pkpk(1+MTt +BSr )+UL and is not aected by the phase noise.Next, an expression for the multiuser interference power in the rst term of thedenominator of (4.18) is derived.Lemma 4.3. : When MF precoding is adopted at the BS, the power of the multiuserinterference caused by the signal intended for the lth MT, l ̸= k, at MT k ∈ Sb,1 ≤ b ≤ Bo, is given byE[∣∣gHk ΘHk (t)fl∣∣2] = (k + (m(1)kPl +m(2)kPl +m(3)kPl )(1− ϵco + ϵ))P if l ∈ Sb (4.21)and by k otherwise. Here, ϵ = z−2 |t−t0|, m(1)kPl =2k!Hl b(t0)−1b Ubk−1b b(t0)!l!Hl b(t0)−1b b(t0)!l, m(2)kPl =cco· 2k!Hl b(t0)−1b Wbk−1b b(t0)!l!Hl b(t0)−1b b(t0)!l, and m(3)kPl = c(1− 1co)·∣∣∣∣k!Hk b(t0)−1b b(t0)!l∣∣∣∣2!Hl b(t0)−1b b(t0)!l.119Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsProof. Please refer to Appendix C.3.Lemma 4.3 conrms that when the number of BS antennas is suciently large,i.e., c →∞, as long as l R∈ Sb, the impact of the multiuser interference from the lthMT vanishes, as is commonly assumed in the massive MIMO literature, e.g. [5, 6].However, the same is not true for MTs that emit pilots in the same training sub-phaseas MT k, i.e., MTs l ∈ Sb. Because of the impairment incurred by the phase noiseduring the training phase, the interference power of these MTs grows linearly with cand does not vanish compared to the strength of the desired signal in (4.20) in thelimit of c →∞.Furthermore, for the summand with l = k in the sum in the rst term of thedenominator of (4.18), we obtain E[∣∣gHk ΘHk (t)fk∣∣2] =E[tr(gk(t0)gHk (t0)ΨHt0(t)gˆkgˆHk‖gˆk‖2Ψt0(t))]= k + k(c − 1)k(1− ϵco+ ϵ)P (4.22)where k ∈ Sb and Ψt0(t) is dened in Appendix C.2. The last equality in (4.22) isobtained by applying [66, Theorem 1] [61]. The variance of the gain of the desiredsignal, gHk ΘHk (t)fk, is obtained by subtracting the right hand side of (4.22) from thesquare of the right hand side of (4.20).The two terms in the denominator of (4.18) originating from the hardware im-pairments at the BS and the kth MT, i.e., BSt (t) and MTrPk (t), respectively, can becalculated asE[∣∣gHk ΘHk (t)ΥBSt Θk(t)gk∣∣] = kBSt ei and E [MTrPk (t)] = kMTr ei : (4.23)Substituting the results in (4.20)-(4.23) into (4.18), we obtain the received SINR at120Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsMT k ∈ Sb in symbol interval t ask(t) =pckkpk(ak + xk) + qkakAN + k(BSt + MTr )ei + DLP (4.24)withak =∑l∈Sb(1 +(m(1)kPl +m(2)kPl +m(3)kPl)(1− ϵco+ ϵ)Rk)+ (K − |Sb|)P (4.25)xk =(1− 1co)(1− ϵ) + [(c − 1)k + 1](1− ϵco+ ϵ)−ckP (4.26)where k = kz−(2 +2ϕ)|t−t0|. Furthermore, ak and xk represent the multiuser inter-ference received at the kth MT and the variance of the gain of the desired signal,respectively. Moreover, the term akAN = E[gHk ΘHk (t)AAHΘk(t)gk]in (4.24) repre-sents the AN leakage in the received signal of the kth MT in time slot t. This termwill be characterized in detail for the considered AN precoders in Section 4.3.3.4.3.3 Generalized NS AN PrecodingThe AN leakage term akAN in (4.24) depends on the particular AN precoder used.Therefore, in this subsection, we rst evaluate akAN for the conventional NS precoder,where A is designed to lie in the NS of the estimated channel vectors of all MTs,gˆk, 1 ≤ k ≤ K, which is the most common design used in the literature [25, 43, 64].Subsequently, we propose and analyze the G-NS AN precoder design which is lesssensitive to hardware impairments than the conventional NS design.The AN leakage incurred by the conventional NS AN precoder is given in thefollowing Lemma.Lemma 4.4. : For the conventional NS AN precoder, where a = c−K [25, 43, 64],121Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsthe AN leakage power received at MT k ∈ Sb in time interval t is given byakAN = k(c −K)((1− 1co)(1− ϵ) + 1− k): (4.27)Proof. Please refer to Appendix C.4.In Lemma 4.4, the terms ϵ and k reect the negative impact of the hardwareimpairments on the AN power leakage. If only one LO is employed, i.e., co = 1, theimpact of ϵ is eliminated. However, the negative eect of ϵ increases as the number ofLOs, co, increases since the phase noise processes of dierent LOs are independentdestroying the orthogonality of the columns of A and gk(t), 1 ≤ k ≤ K.This problem can be mitigated by employing bo NS AN precoders where eachprecoder encodes the data signals intended for the antennas connected to coRboLOs. Thereby, co is assumed to be a multiple of bo, i.e., coRbo ∈ Z. The resultingAN preorder is referred to as G-NS AN precoder. More in detail, for the G-NSAN precoder, we divide each channel estimation vector, gˆk, 1 ≤ k ≤ K, into bosub-vectorsgˆk =[(gˆ(1)k)iP(gˆ(2)k)iP : : : P(gˆ(bo)k)i ]iP (4.28)where gˆ(m)k ∈ CcRbo×1, which contains the ((m − 1)cRbo + 1)th to the (mcRbo)thelements of gˆk for 1 ≤ m ≤ bo. Correspondingly, we split matrix A into bo sub-matrices as followsA =[Ai(1)PAi(2) : : : PAi(bo)]iP (4.29)with A(m) ∈ CcRbo×(cRbo−K), 1 ≤ m ≤ bo, i.e., we have a = cRbo − K. Now,matrixA(m) is designed to lie in the null-space of gˆ(m)k , 1 ≤ k ≤ K, i.e., A(m)gˆ(m)k = 0,1 ≤ k ≤ K, 1 ≤ m ≤ bo. For bo = 1, the G-NS precoder simplies to theconventional NS precoder. On the other hand, for bo = co, the antennas connected122Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsto each LO have their own NS AN precoder.The AN leakage of the G-NS precoder is analyzed in the following Lemma.Lemma 4.5. : For the G-NS AN precoder, where a = cRbo−K and 1 ≤bo ≤ co,the AN leakage power received at MT k ∈ Sb in time interval t is given byakAN = k(cbo−K)((1− boco)(1− ϵ) + 1− k): (4.30)Proof. Please refer to Appendix C.5.Several observations can be made from (4.30). First, we note that, as expected,for bo = 1, (4.30) reduces to (4.27). Second, the negative impact of the phasenoise via ϵ on the AN leakage can be completely eliminated by choosing bo = co.Third, the G-NS precoder requires the calculation of bo null spaces of dimensioncRbo ×K. Hence, computational complexity increases with bo. We will elaborateon the optimal choice of bo in Sections 4.4 and 4.5.The achievable rates of MT k ∈ Sb in time slot t with conventional NS and G-NSprecoding are obtained by inserting (4.27) and (4.30) into (4.24), respectively. Hence,for the proposed G-NS precoder, we obtaingk(t) = log2(1 +kϕc(ak + xk − k)ϕ+ k + k)P (4.31)where k = (cbo−K)((1− boco)(1− ϵ) + 1− k), k = (MTr +BSt +DLR(kei )),and  = KRc > 0.123Chapter 4. Hardware Impairments in Secure Massive MIMO Systems4.3.4 Upper Bound on the Eavesdropper's CapacityIn the following Proposition, we provide a tight and tractable upper bound on eaves-dropper's capacity.Proposition 4.1. : For c → ∞ and (G-)NS AN precoding, the eavesdropper'scapacity in (4.15) can be upper bounded asXE ≤ XE = log2(1 +pcEqa+ BSt ei − cE)P with  =(1 + BSt )2q2a+ (BSt )2p2K(1 + BSt )qa+ BSt pKP(4.32)for qa+BSt ei > cE, and where a = c−K and a = cRbo−K for the conventionalNS and the G-NS precoders, respectively.Proof. The term GHEΥBSt GE in (4.16) converges to a deterministic diagonal matrixfor c → ∞, and is therefore independent of gkE. Hence, similar steps as in [43,Appendix B, C] can be used to arrive at (4.32).We observe from (4.32) that, as expected, the capacity of the eavesdropper isincreasing in the number of its equipped antennas, cE. Interestingly, when no AN isinjected, i.e., q = 0, (4.32) reduces toXE∣∣∣∣q=0= log2(1 +cEBSt (K −cE))P (4.33)for K > cE. For perfect BS hardware, we have BSt → 0 and XE →∞ making securecommunication impossible. Hence, without AN injection, hardware impairments mayin fact be benecial for secure communication as the distortion noise at the BS actslike AN and may facilitate secrecy. This surprising insight will be studied morecarefully in the next section. Furthermore, the number of independent distortionnoise processes at the BS is equal to the number of users, K. Hence, K > cE is124Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsneeded to prevent the eavesdropper from nulling out the distortion noise and forachieving secrecy.4.4 Guidelines for System DesignIn this section, we exploit the analytical results derived in the previous section togain some insight into the impact of the various system and hardware impairmentparameters on system design. To this end, we carefully study the closed-form lowerbound on the achievable ergodic secrecy rate obtained by combining (4.19), (4.31),and (4.32).4.4.1 Design of the Pilot SequencesAssuming that we assign the maximum number of users to each training sub-phase,i.e., |Sb| = Bb, the relevant design parameter for the pilot sequences is the numberof training sub-phases Bo, or equivalently, the size of the training sub-phases Bb as∑Bob=1Bb = B. In particular, Bb aects the lower bound on the achievable ergodicrate of MT k in (4.31) via k, ak, and xk, where xk becomes proportional to k forc →∞, cf. (4.26). Thereby, close inspection of (4.20) reveals that k, which reectsthe power of the received useful signal, is not monotonic in Bb. This can be explainedas follows. On the one hand, since the power of each pilot symbol is constrained,i.e., |!k(t)|2 = p , ∀kP t, the sum power of the pilot sequence per MT increaseswith Bb. On the other hand, for larger Bb, more MTs are allowed to emit pilots intraining sub-phase b introducing more contamination due to phase noise. This has anadverse eect on the quality of the channel estimate and consequently on the powerof the received useful signal. Similarly, close inspection of (4.25) reveals that ak,which reects the multiuser interference incurred to the kth MT, is a monotonically125Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsincreasing function of Bb, as a lower channel estimation accuracy gives rise to moremultiuser interference. Considering the behaviour of k, ak, and xk and their impacton the achievable ergodic rate of MT k in (4.31), we conclude that Bb, 1 ≤ b ≤ Bo,should be optimized and the optimal value depends on the channel and hardwareimpairment parameters. Thereby, the optimal Bb is decreasing in the phase noisevariances, 2 and 2ϕ, as the degradation introduced by concurrent pilot emission bymultiple MTs is increasing in these parameters. This conclusion will be veried inSection 4.5.2 by numerically evaluating (4.19).4.4.2 Selection of bo for G-NS AN PrecodingThe number of G-NS AN precoding sub-matrices, bo, 1 ≤ bo ≤ co, employedaects the achievable ergodic secrecy rate via the AN leakage akAN in (4.30) andvia the (bound on the) eavesdropper capacity XE in (4.32). The AN leakage is adecreasing function with respect to bo, i.e., as far as the AN leakage is concerned,bo = co is preferable. On the other hand, since the dimensionality of the G-NSAN precoder is given by a = cRbo −K, the eavesdropper capacity is an increasingfunction of bo, cf. (4.32), which has a negative eect on the ergodic secrecy rate.Hence, bo has to be optimized. Since the eavesdropper capacity does not depend onthe phase noise, we expect that the optimal bo increases with increasing BS phasenoise variance, 2 , as 2 aects the AN leakage via ϵ in (4.30). This conjecture willbe numerically veried in Section 4.5.4.4.4.3 Secrecy in the Absence of ANIn [43, 64] it was shown that if perfect hardware is employed, injection of AN isnecessary to achieve secrecy. In particular, without AN generation, under worst-case126Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsassumptions regarding the noise at the eavesdropper, the eavesdropper capacity isunbounded. On the other hand, we showed in Section 4.3.4 that in the presenceof hardware impairments the eavesdropper capacity is bounded since the distortionnoise generated at the BS has a similar eect as AN. Motivated by this observation,in this section, we calculate the maximum number of eavesdropper antennas cE thatcan be tolerated if a positive secrecy rate is desired without AN emission.If AN is not emitted, we have ϕ = 1 or q = 0. In this case, the proposed lowerbound on the ergodic secrecy rate of the kth MT in time interval t simplies togseck (t)∣∣∣∣q=0=[log2(1 +kcak + xk + k)− log2(1 +BSt ( − ))]+: (4.34)where  = cERc denotes the normalized number of eavesdropper antennas. Inthe following Proposition, we provide a condition for the number of eavesdropperantennas that has to be met for secure communication to be possible.Proposition 4.2. : If AN is not generated, the maximum number of eavesdropperantennas that the system can tolerate while ensuring a positive ergodic secrecy rate iscE = ⌊ANc⌋, whereAN =kcBSt kcBSt + ak + xk + k∣∣∣t=B+1: (4.35)Proof. First, we note that gk(t) is a decreasing function of t. Hence, considering(4.19), it is sucient to ensure gk(B + 1) > XE for achieving a positive ergodicsecrecy rate. Eq. (4.35) is obtained by setting (4.34) to zero and observing thatgseck (t)∣∣∣∣q=0is a decreasing function of .Eq. (4.35) clearly shows that the additive distortion noise at the BS is essential forachieving a positive secrecy rate if AN is not injected as AN = 0 results if BSt = 0.127Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsOn the other hand, AN is a decreasing function of all other hardware impairmentparameters, i.e., BSr , MTt , MTr , DL, 2 , and 2 , as the corresponding hardwareimpairments aect only the achievable ergodic rate of the MT but not the ergodiccapacity of the eavesdropper. We note that AN is an increasing function of  sincethe dimensionality of the additive distortion noise at the BS is proportional to .4.4.4 Maximum Number of Eavesdropper AntennasNow, we consider the maximum number of eavesdropper antennas that can be toler-ated if a positive ergodic secrecy rate is desired and AN injection is possible. Com-bining (4.19), (4.31), and (4.32), the lower bound on the ergodic secrecy rate in timeinterval t can be expressed asgseck (t) =[log2(1 +kϕc(ak + xk)ϕ+ k(1− ϕ) + k)−log2(1 +ϕ(1− ϕ+ BSt − ′))]+P(4.36)where ′ = (1+BSt )2(1−ϕ)2cRa+(BSt )2ϕ2R1−ϕ+BSt.Proposition 4.3. : If AN injection is possible, a positive secrecy rate can be achievedby the kth MT if the number of eavesdropper antennas does not exceed cE = ⌊secc⌋,wheresec =(1 + BSt )kaaRc(k + MTr + BSt + DLR(kei )) + kc(1 + BSt )∣∣∣t=B+1(4.37)and ϕ→ 0, i.e., almost all transmit power is employed for AN generation.Proof. Exploiting again that gk(t) is a decreasing function of t it suces to considerthe ergodic secrecy rate for t = B + 1. Then, an expression for sec is obtained bysetting gseck (t) in (4.36) to zero. This expression is monotonically decreasing in ϕ and128Chapter 4. Hardware Impairments in Secure Massive MIMO Systemshence can be further simplied by letting ϕ→ 0 which yields (4.37).Proposition 4.3 reveals that, as expected, the number of eavesdropper antennasthat can be tolerated increases with the channel estimation accuracy (i.e., k) andthe number of spatial dimensions available for AN (i.e., a). Furthermore, similar toAN, sec is a decreasing function of the hardware impairment parameters BSr , MTt ,MTr , DL, 2 , and 2 , and an increasing function of BSt . However, unlike AN, secis independent of .4.4.5 Number of LOsThe number of LOs, co, aects the ergodic secrecy rate via the terms ak, xk, andk in the achievable ergodic rate in (4.31). For c → ∞, ak and xk are decreasingfunctions of co, i.e., less multiple access interference is caused if more LOs are em-ployed, whereas the AN leakage term k is an increasing function in co. Therefore,considering the specic form of the denominator of the fraction inside the logarithmin (4.31), the optimal value of co, which maximizes the ergodic secrecy rate, dependson ϕ. In particular, for a given bo, for ϕ = 1 no AN is injected and k cancels in theexpression for the achievable ergodic rate in (4.31). Hence, in this case, the ergodicsecrecy rate is a monotonically increasing function of co, i.e., increasing the numberof LOs is benecial. On the other hand, for a given bo, for ϕ Q 1, the optimal comaximizing the ergodic secrecy rate can be found by performing a numerical searchbased on (4.31).We note that by employing G-NS AN generation and enforcing bo = co, wecan avoid the harmful eect of the multiple LOs on the AN leakage term k. Inthis case, the achievable ergodic rate of the MT becomes an increasing function ofbo = co. However, at the same time, the number of dimensions available for AN129Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsinjection, a = cRbo − K, is a decreasing function of bo = co. Therefore, theoptimal bo = co maximizing the ergodic secrecy rate has to be found again by anumerical search.4.4.6 Are hardware impairments Benecial for Security?Since the hardware impairment parameters BSr , MTt , MTr , DL, 2 , and 2 onlyaect the legitimate user but not the eavesdropper, the corresponding hardware im-pairments are always detrimental to the ergodic secrecy rate. However, the additivedistortion noise at the BS aects both the achievable ergodic rate of the MT andthe capacity of the eavesdropper. Hence, it is not a priori clear if this hardwareimpairment is benecial or detrimental to the ergodic secrecy rate. The followingProposition provides a criterion for judging the benets of the additive BS distortionnoise.Proposition 4.4. : For time interval t, non-zero additive BS distortion noise withsmall BSt > 0, BSt → 0, is benecial for the achievable ergodic secrecy rate of thekth MT if and only if(1− ϕ)[1−cERa− (1−cERa−cERK)ϕ]× 1−cERa1− (1− 2ϕ)cERa Q(ckϕ+ )2kcP(4.38)where  = (ak + xk)ϕ+ k(1− ϕ) + (MTr + DLR(kei )).Proof. For additive BS distortion noise to be benecial for a given time interval t andsmall BSt > 0, the derivative @gseck (t)R@BSt at BSt = 0 has to be positive. Assuminggseck (t) > 0, this condition leads to @gk(t)R@BSt |BSt =0 > @XER@BSt |BSt =0, which canbe further simplied to (4.38).130Chapter 4. Hardware Impairments in Secure Massive MIMO SystemsRemark 4.1. : We note that the criterion in Proposition 4.4 only guarantees thatadditive BS distortion noise with small positive BSt is benecial. The ergodic secrecyrate, gseck (t), is in general not monotonic in BSt and larger BSt may be harmful evenif small BSt are benecial, see Section 4.5.5. Furthermore, since the right hand sideof (4.38) is always positive, we conclude that additive BS distortion noise with smallBSt is always benecial when ϕ = 1, i.e., when AN is not injected.4.5 Numerical ExamplesIn this section, we provide numerical and simulation results to verify the analysispresented in Sections 4.3 and 4.4 and to illustrate the impact of hardware impairmentson the ergodic secrecy rate. For the numerical results, we numerically evaluate theanalytical expression for the lower bound on the ergodic secrecy rate obtained bycombining (4.19), (4.31), and (4.32). For the simulation results, we employ MonteCarlo simulation and evaluate (4.19) using gseck (t) = log2(1+k(t)) and XE = log2(1+E) with k(t) and E given by (4.18) and (4.16), respectively, for 5P 000 independentchannel realizations. For simplicity, in this section, we assume that the path-lossfor all MTs is identical, i.e., k = 1, 1 ≤ k ≤ K, and the coherence block length isequal to i = 500 time slots. Typical values for the phase noise increment standarddeviations,  , ϕ, used include 0:06◦, which was adopted in the long-term evolution(LTE) specications [78], and 6◦, which corresponds to strong phase noise accordingto [79, 80]. Furthermore, typical values for the additive distortion noise MTt = BSr =BSt = MTr include {0P 0:052P 0:152} [18], whereas the amplied receiver noise was setto UL = DL = 1:582n [22], with 2n = 1. The specic values of the adopted systemand hardware impairment parameters are provided in the captions of the gures.131Chapter 4. Hardware Impairments in Secure Massive MIMO Systems0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.400.511.522.533.54βCapacity of Eavesdropper (bps/Hz)  SimulationUpper BoundMo = 1, 2, 4, L = N/Mo −KFigure 4.2: Capacity of the eavesdropper vs. the normalized number of MTs  for asystem with c = 128, co = 4, cE = 16, ei = 10 dB, ϕ = 0:25, BSt = 0:152, andG-NS AN precoding with bo = {1P 2P 4}.4.5.1 Capacity of Eavesdropper for G-NS AN PrecodingFig. 4.2 depicts the eavesdropper's ergodic capacity, XE, as a function of  for G-NSAN precoding with bo = {1P 2P 4}. Besides results for the analytical upper bound,XE, from (4.32), we also show simulation results for XE by averaging log2(1+E) over5P 000 independent channel realizations, where E is given by (4.16). From Fig. 4.2we observe that the proposed upper bound on the capacity of the eavesdropper isvery tight. Furthermore, as expected, the ergodic capacity of the eavesdropper is an132Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsincreasing function ofbo since the number of dimensions available for AN generation,a = cRbo−K, is a decreasing function ofbo. In fact, since a = cRbo−K > cE isneeded for successfully jamming the eavesdropper, for bo = 4, we depict the ergodiccapacity of the eavesdropper only for  Q 0:125. Nevertheless, as will be shownbelow, choosing bo > 1 may still be benecial as far as the ergodic secrecy rate isconcerned as the achievable ergodic rate of the MT is an increasing function of bo.4.5.2 Achievable Ergodic Rate of MT for Dierent PilotDesignsNext, we investigate the impact of the general pilot designs introduced in Section4.2.1 on the lower bound of the achievable ergodic rate of the considered MT givenin (4.31)7. Note that the capacity of the eavesdropper is not aected by the pilotdesign. For simplicity, we assume equal duration for all training sub-phases, Bb =BRBo, b ∈ {1P : : : P Bo}, and B = K. The same number of users is assigned to eachtraining sub-phase. In Fig. 4.3, we show the achievable ergodic rate of a MT intraining set SBo as well as the corresponding k, which reects the power of thereceived useful signal, and ak, which reects the multiuser interference. Results forBo = 1 (SO pilots), Bo = 2, and Bo = 16 (TO pilots) are shown. As predictedin Section 4.4.1, the multiuser interference, ak, is monotonically decreasing in Bo aslarger Bo improve the robustness against phase noise during the channel estimationphase, which allows better suppression of multiuser interference via MF precoding.Somewhat surprisingly, for  = ϕ ≤ 5◦, ak is a decreasing function of the phase noisevariance. This may be attributed to the fact that phase noise prevents the coherent7We note that all results obtained by numerically evaluating the analytical expressions derivedin this chapter were veried by simulations. However, the simulation results are not included in allgures for clarity of presentation.133Chapter 4. Hardware Impairments in Secure Massive MIMO Systems0 5 10 15 20 2500.511.522.5σψ = σφ(◦)Achievable Rate per MT (bps/Hz)0 5 10 15 20 2500.20.40.60.81σψ = σφ(◦)λk0 5 10 15 20 250.050.10.150.20.250.3σψ = σφ(◦)ak  Bo = 16(TO)Bo = 2Bo = 1(SO)Figure 4.3: Achievable ergodic rate, k, and ak vs. phase noise standard deviation = ϕ for dierent pilot designs for a system with c = 128, co = 2, cE = 16,K = 16, p = eiRK, ei = 10 dB, ϕ = 0:5, and BSt = BSr = MTt = MTr = 0:052.superposition of the multiuser interference generated by dierent MTs such that largeinterference values are avoided. On the other hand, for  = ϕ > 5◦, the detrimentaleects of the pilot contamination caused by the loss of orthogonality for Bo Q 16outweigh this positive eect and ak increases with the phase noise variance. For k,i.e., the received signal power, we observe from Fig. 4.3 that the optimal Bo dependson the phase noise variance. In particular, for small phase noise variances, smallBo are preferable since the increased pilot power outweighs the loss of orthogonalityduring training. On the other hand, for large phase noise variances, eventually TO134Chapter 4. Hardware Impairments in Secure Massive MIMO Systemspilots become optimal as the preserved orthogonality during training becomes crucial.The behaviour of k and ak is also reected in the behaviour of the achievable rateof the considered MT. In particular, for the considered system parameters, Bo = 1,Bo = 2, and Bo = 16 are optimal for  = ϕ ≤ 6◦, 6◦ Q  = ϕ ≤ 21◦, and = ϕ > 21◦(which is not a practical range), respectively. Hence, in practice, theoptimal Bo can be found by evaluating (4.31).4.5.3 Optimal Power Allocation to Data and ANFig. 4.4 shows the achievable ergodic secrecy rate as a function of the power allocationparameter ϕ for SO and TO pilots and dierent phase noise variances. G-NS ANprecoding with bo = co = 2 is adopted. The curve for ideal hardware components,i.e., BSt = BSr = MTt = MTr =  = ϕ = 0, is also provided for reference. Weinvestigate the optimal power allocation between data transmission and AN emissionfor the maximization of the ergodic secrecy rate achieved for dierent phase noiselevels. When the phase noise variance is small, i.e.,  = ϕ = 0:6◦, SO pilotsoutperforms TO pilots for all values of ϕ. However, this is not true for stronger phasenoise. We also observe that the optimal value for ϕ maximizing the ergodic secrecyrate is only weakly dependent on the phase noise variance.4.5.4 Achievable Ergodic Secrecy Rate for Non-IdealHardware ComponentsIn Fig. 4.5, we show the ergodic secrecy rate achieved with G-NS AN precoding fordierent values of bo as a function of the number of BS antennas. The cases of weak( = ϕ = 0:6◦) and strong ( = ϕ = 6◦) phase noise are considered. For weakphase noise, using large values of bo becomes benecial only for large numbers of135Chapter 4. Hardware Impairments in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.900.511.522.533.544.5φAchievable Secrecy Rate (bps/Hz)  IdealSO PilotsTO Pilotsσψ = σφ = 0.6◦σψ = σφ = 6◦σψ = σφ = 12◦Figure 4.4: Achievable ergodic secrecy rate vs. ϕ for SO and TO pilots and a systemwith K = 4 , c = 128, co = bo = 2, cE = 4, p = eiRK, ei = 10 dB, andBSt = BSr = MTt = MTr = 0:152.antennas, i.e., c > 200, as for smaller numbers of antennas the positive eect of largervalues of bo on the AN leakage is outweighed by their negative eect on the numberof spatial dimensions available for AN precoding. On the other hand, for strongphase noise, the AN leakage is larger and its mitigation by choosing bo = co = 16is benecial already for c > 150. These observations are in line with our theoreticalconsiderations in Section 4.4.2. Fig. 4.5 also conrms the accuracy of the derivedanalytical expressions for the ergodic secrecy rate.136Chapter 4. Hardware Impairments in Secure Massive MIMO Systems150 200 25033.23.43.63.844.24.44.64.8Number of BS Antennas, NAchievable Ergodic Secrecy Rate (bps/Hz)σψ = σφ = 0.6◦150 200 25000.20.40.60.811.21.4Number of BS Antennas, NAchievable Ergodic Secrecy Rate (bps/Hz)σψ = σφ = 6◦  SimulationLower BoundMo = 16, 8, 4, 2, 1Mo = 16, 8, 4, 2, 1Figure 4.5: Achievable ergodic secrecy rate vs. number of BS antennas for G-NS ANprecoding and a system with K = 4, cE = 4, co = 16, Bo = 1, p = eiRK, ei = 10dB, and BSt = BSr = MTt = MTr = 0:152. The optimal ϕ is adopted.4.5.5 Maximum Tolerable Number of EavesdropperAntennasFig. 4.6 depicts the (normalized) maximum tolerable number of eavesdropper anten-nas for achieving a positive ergodic secrecy rate for the case without AN generation,AN, and the case with AN generation, sec, as a function of the (normalized) num-ber of users, . Results for channel estimation based on SO and TO pilots as wellas the case of no phase noise ( = ϕ = 0◦) are shown for co = 2 and co = 4137Chapter 4. Hardware Impairments in Secure Massive MIMO Systems0 0.1 0.2 0.3 0.400.020.040.060.08No = 4βαAN0 0.1 0.2 0.3 0.40.250.30.350.40.45βαsecNo = 2  No phase noiseSO PilotsTO Pilots0 0.05 0.1 0.15 0.200.020.040.060.08βαANNo = 20 0.05 0.1 0.15 0.20.160.180.20.220.24βαsecNo = 4Figure 4.6: AN and sec vs. the normalized number of MTs  for SO and TO pilotsand a system with c = 128, bo = 2, p = eiRK, ei = 10 dB,  = ϕ = 6◦, andBSt = BSr = MTt = MTr = 0:152.LOs. First, we note that, as expected from our considerations in Section 4.4.5, forthe case without AN (ϕ = 1), increasing co from 2 to 4 is benecial, i.e., the numberof tolerable eavesdropper antennas increases. In contrast, if AN is injected, co = 2 ispreferable. Second, AN generation is benecial and improves the robustness againsteavesdropping, i.e., sec > AN. Third, as expected from Sections 4.4.3 and 4.4.4,AN is a monotonically increasing function of  whereas sec is independent of .Fourth, for the considered example of weak phase noise, SO pilots outperform theTO pilots for all considered cases.138Chapter 4. Hardware Impairments in Secure Massive MIMO Systems0 0.5 11.791.7951.81.8051.811.815κBStErgodic Secrecy Rate (bps/Hz) φ = 0.25, σψ = σφ = 0.06◦  κBSt = 0κBSt > 00 0.5 11.21.221.241.261.281.3κBStErgodic Secrecy Rate (bps/Hz) φ = 0.05, σψ = σφ = 0.06◦0 0.5 11.221.2251.231.2351.241.245κBStErgodic Secrecy Rate (bps/Hz) φ = 0.25, σψ = σφ = 6◦0 0.5 10.70.710.720.73κBStErgodic Secrecy Rate (bps/Hz) φ = 0.05, σψ = σφ = 6◦Figure 4.7: Achievable ergodic secrecy rate vs. BS distortion noise parameter BSt fora system with c = 128, K = 32, cE = 4, co = bo = 2, p = eiRK, ei = 10 dB,and BSr = MTt = MTr = 0:152.4.5.6 Is Additive Distortion Noise at the BS Benecial forSecurity?In Fig. 4.7, we show the achievable ergodic secrecy rate as a function of the BSdistortion noise parameter, BSt , for dierent phase noise variances and dierent powerallocation factors ϕ. For comparison, the achievable ergodic secrecy rates without BSdistortion noise (i.e., BSt = 0) are also shown. Fig. 4.7 shows that if the powerallocated to AN is substantial (e.g., ϕ = 0:05), the additional distortion noise has a139Chapter 4. Hardware Impairments in Secure Massive MIMO Systemsnegative eect on the ergodic secrecy rate. On the other hand, if the power assignedfor AN is not sucient (e.g., ϕ = 0:25), non-zero additive distortion noise at theBS is benecial as the distortion noise acts like additional AN. In particular, forϕ = 0:25,  = 0:06◦, we obtain for the left hand side and right hand side of (4.38)0:52 and 1:66, respectively, which we represent as (0:52P 1:66). Correspondingly, weobtain for ϕ = 0:25,  = 6◦and ϕ = 0:05,  = 0:06◦and ϕ = 0:05,  = 6◦thetupels (0:52P 2:53) and (0:80P 0:16) and (0:80P 0:35), respectively. These values andthe results in Fig. 4.7 suggest that (4.38) can indeed be used to predict whether ornot BS distortion noise is benecial.4.6 ConclusionsIn this chapter, we have investigated the impact of hardware impairments such as mul-tiplicative phase noise, additive distortion noise, and amplied receiver noise on thesecrecy performance of massive MIMO systems employing MF precoding for downlinkdata transmission. To mitigate the loss of pilot orthogonality during uplink trainingif multiple MTs emit pilots concurrently, a generalized pilot design was proposed.Furthermore, to avoid the AN leakage caused by the loss of orthogonality betweenthe user channels and the NS AN precoder if multiple noisy LOs are employed at theBS, a novel G-NS AN precoding scheme was introduced. For the considered system,a lower bound on the achievable ergodic secrecy rate of the users was derived. Thisbound was used to obtain important insights for system design, including the impactof the pilot sequence design, the AN precoder design, the number of LOs, and thevarious hardware impairment parameters. The following general conclusions can bedrawn: 1) Additive distortion noise at the BS may be benecial for the secrecy per-formance especially if little or no AN is injected; 2) all other hardware impairments140Chapter 4. Hardware Impairments in Secure Massive MIMO Systemshave a negative impact on the ergodic secrecy rate; 3) despite their susceptibility topilot contamination in the presence of phase noise, SO pilots are preferable exceptfor the case when the phase noise is very strong; 4) if the number of BS antennas issuciently large, the proposed G-NS AN precoder outperforms the conventional NSAN precoder in the presence of phase noise.141Chapter 5Summary of Thesis and FutureResearch TopicsIn this nal chapter, in Section 5.1, we summarize our results and highlight thecontributions of this thesis. In Section 5.2, we also propose ideas for future relatedresearch.5.1 Summary of ResultsThis thesis as a whole has focused on physical layer security for massive MIMOsystems. In the following, we briey review the main results of each chapter.In Chapter 2, we considered a multi-cell massive MIMO system with MF precodingand AN precoding at the BS for secure downlink transmission in the presence ofa multi-antenna passive eavesdropper. For AN precoding, we considered both theconventional NS AN precoding matrix design and a novel random AN precodingmatrix design. For both perfect training and pilot contamination, we derived twotight lower bounds on the ergodic secrecy rate and a tight upper bound on thesecrecy outage probability. The analytical expressions allowed us to optimize theamount of power allocated to AN precoding and to gain signicant insight into theimpact of the system parameters on performance. In particular, our results revealthat for the considered multi-cell massive MIMO system with MF precoding (1) AN142Chapter 5. Summary of Thesis and Future Research Topicsprecoding is necessary to achieve a non-zero ergodic secrecy rate if the user and theeavesdropper experience the same path-loss, (2) secrecy cannot be guaranteed if theeavesdropper has too many antennas, (3) for the case of pilot contamination, theergodic secrecy rate is only an increasing function of the number of BS antennas ifthe amount of power allocated to AN precoding is optimized, and (4) the proposedrandom AN precoding matrix design is a promising low-complexity alternative to theconventional NS AN precoding matrix design.In Chapter 3, we considered downlink multi-cell massive MIMO systems employ-ing linear data and AN precoding for physical layer security provisioning. We ana-lyzed and compared the achievable ergodic secrecy rate of various conventional dataand AN precoders in the presence of pilot contamination. To this end, we also opti-mized the regularization constants of the selsh and collaborative RCI precoders inthe presence of AN and multi-cell interference. In addition, we derived linear POLYdata and AN precoders which oer a good compromise between complexity and per-formance in massive MIMO systems. Interesting ndings of this chapter include: 1)Collaborative data precoders outperform selsh designs only in lightly loaded systemswhere a sucient number of degrees of freedom for suppressing inter-cell interferenceand sucient resources for training are available. 2) Similarly, CNS AN precoding ispreferable over SNS AN precoding in lightly loaded systems as it causes less AN leak-age to the information-carrying signal, whereas in more heavily loaded systems, CNSAN precoding does not have sucient degrees of freedom for eectively degradingthe eavesdropper channel and SNS AN precoding is preferable. 3) For a large num-ber of eavesdropper antennas, where only small positive secrecy rates are achievable,MF data precoding is always preferable compared to SZF and CZF data precoding.4) The proposed POLY data and AN precoders approach the performances of the143Chapter 5. Summary of Thesis and Future Research TopicsSRCI data and SNS AN precoders with only a few terms in the respective matrixpolynomials and are attractive options for practical implementation.In Chapter 4, we have investigated the impact of hardware impairments suchas multiplicative phase noise, additive distortion noise, and amplied receiver noiseon the secrecy performance of massive MIMO systems employing MF precoding fordownlink data transmission. To mitigate the loss of pilot orthogonality during up-link training if multiple MTs emit pilots concurrently, a generalized pilot design wasproposed. Furthermore, to avoid the AN leakage caused by the loss of orthogonal-ity between the user channels and the NS AN precoder if multiple noisy LOs areemployed at the BS, a novel G-NS AN precoding scheme was introduced. For theconsidered system, a lower bound on the achievable ergodic secrecy rate of the userswas derived. This bound was used to obtain important insights for system design,including the impact of the pilot sequence design, the AN precoder design, the num-ber of LOs, and the various hardware impairment parameters. The following generalconclusions can be drawn: 1) Additive distortion noise at the BS may be benecial forthe secrecy performance especially if inadequate AN is emitted; 2) all other hardwareimpairments have a negative impact on the ergodic secrecy rate; 3) despite theirsusceptibility to pilot contamination in the presence of phase noise, SO pilots arepreferable except for the case when the phase noise is very strong; 4) if the numberof BS antennas is suciently large, the proposed G-NS AN precoder outperforms theconventional NS AN precoder in the presence of phase noise.5.2 Future WorkIn the following, we propose some ideas for further research that are similar to or canbe based on the work in this thesis.144Chapter 5. Summary of Thesis and Future Research Topics5.2.1 Physical Layer Security in Massive MIMO Systemsunder Constant Envelope PrecodingEquipping large antenna array in massive MIMO systems requires each antenna ele-ment and its associated radio-frequency (RF) electronics, e.g. power ampliers (PAs),to be inexpensive and power-ecient. However, cheaply manufactured PAs are in gen-eral non-linear devices, which suer from linearity issues when processing signals withlarge amplitude-variations. A per-antenna constant envelope (CE) nonlinear precod-ing was considered in single-user massive MIMO systems in [21]. It was shown thatunder the per-antenna CE constraint at the BS transmitter, an equivalent single-inputsingle-output (SISO) model over additive white Gaussian noise (AWGN) is obtainedfor MISO system where we have a single-user equipped with a single-antenna [21].When a suciently large number of antennas is used, the corresponding achievablerate under a per-antenna CE constraint is close to the capacity of the multiple-inputsingle-output (MISO) channel under an average power constraint in the high-powerregime. More recently, the idea of per-antenna CE precoding has been extendedto multi-user massive MIMO systems over at and frequency-selective fading chan-nels [75] [81]. To the best of our knowledge, there is no work considering securetransmission under CE precoding in the presence of multi-antenna eavesdroppers.Therefore, one promising research option is to investigate novel data and AN pre-coding methods, which jointly satisfy the per-antenna CE constraints, and comparetheir performance with the secrecy capacity achieving scheme for the average totaltransmit power constrained channel. Some of the results have been reported in [82].145Chapter 5. Summary of Thesis and Future Research Topics5.2.2 Physical Layer Security in Massive MIMO Systemswith Limited RF-Chain ConstraintsWhen multiple antennas are deployed at the BS in a conventional manner, the com-plex baseband symbols are tuned for both amplitude and phase. The baseband sym-bols are then upconverted to the carrier frequency after passing through RF chains,whose outputs are coupled with antenna elements. This implies that each antennaelement is supported by one dedicated RF chain, which is far too expensive to deployin massive MIMO systems due to the large number of antenna elements. On theother hand, the rapid development of circuitry technology enables the high dimen-sional phase-only RF (or analog) processing. In [83] and [84], analog precoding wasconsidered to achieve full diversity order and near-optimal beamforming performancevia iterative algorithms. The authors in [85] have taken into account more practi-cal constraints, including only quantized phase control and nite-precision analog-to-digital (A/D) conversion. In order to further enhance the system performance,related literature [86]-[88] have considered a hybrid approach combining digital andanalog preocoding together. More precisely, a low dimensional (limited to the num-ber of RF chains) baseband precoding is employed based on the equivalent channelacquired from the product of the analog RF precoder and the actual channel matrix[87]. However, the problems of how the limited RF-chain constraints will aect thesystem security and how to design the transmission strategy to enhance the systemsecurity with such practical constraints have not been studied before. One foreseeablechallenge is that the BS may not possess sucient spatial dimensions for emittingAN due to the limited number of RF chains. This motivates future research in thisdirection, and some of the results have been reported in [89].146Chapter 5. Summary of Thesis and Future Research Topics5.2.3 Physical Layer Security in Massive MIMO Systemsagainst Active EavesdroppingThe contributions covered in this thesis, including the aforementioned two future re-search options, are based on the assumption that the eavesdroppers always remainpassive to hide their existence. Another promising topic is to investigate how massiveMIMO systems can combat active eavesdropping. Among multiple active eavesdrop-ping techniques, the pilot contamination attack [44] poses the most serious secrecythreat to the TDD based massive MIMO systems. For such attacks, the eavesdropperis able to acquire any training sequences assigned to legitimate MTs, as they are xedand repeatedly adopted for uplink training. In the training phase, the eavesdropperemits the pilots while all legitimate MTs transmit. As such, the estimates at theBS align with both the legitimate MT's and the eavesdropper's channel. The attacknot only reduces the estimation accuracy at the BS, but enhances the eavesdropper'scapability to detect his/her desired data signals. It is foreseeable that if the emittingpower for the eavesdropper is suciently large, the achievable secrecy rate eventuallyapproaches zero. In this scenario, emitting conventional NS based AN is no longerecient, as the designed AN also lies in the NS of the eavesdropper's channel due tothe pilot contamination. In the literature, the authors in [42] proposed several tech-niques to detect the attack by taking advantage of massive MIMO, while the authorsin [57] developed a secret key agreement protocol under pilot contamination attack.Methods for combating such attack in a multi-cell network (pilot contaminated) wasreported in [59], based on the assumption that the channel covariance matrix of theeavesdropper is low-rank. A more general combating strategy is essential and of greatimportance. 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We rst derive an expression for the secrecy ratefor given realizations of hmk and Hevem , k = 1P : : : P K, m = 1P : : : Pb . Since theMISOME channel in (2.5) and (2.6) is a non-degraded broadcast channel [27], thesecrecy capacity is given by [36], [90]Xsecnk (h) = maxsnk→wnksnk→ynkPyeveI (snk; ynk|h)− I (snk;yeve|h) P (A.1)where vector h contains the CSI of all user and eavesdropper channels and I(x; y|h)is the mutual information between two r.v.s x and y conditioned on the CSI vector.Xsecnk (h) is achieved by maximizing over all joint distributions such that a Markovchain snk → wnksnk → ynkPyeve results, where snk is an arbitrary input variable [36].Specically, for snk ∼ CN(0P 1) an achievable secrecy rate for the kth MT in the local158Appendix A. Proofs in Chapter 2cell, gsecnk (h), is given bygsecnk (h) =[I (snk; ynk|h)− I (snk;yeve|h)]+(a)=[I (wnksnk; ynk|h)− I (wnksnk;yeve|h)]+(b)≥[gnk (h)− Xevenk (h)]+(A.2)where (a) follows since wnksnk is a deterministic function of snk. Furthermore,gnk(h) ≤ max I (wnksnk; ynk|h) is an achievable rate of the kth MT in the localcell and Xevenk (h) = log2(1 + pwHnkHeveHn X−1Heven wnk) ≥ I (wnksnk;yeve|h) is an up-per bound on the mutual information I (wnksnk;yeve|h). Thus, follows (b). We notethat for computation of Xevenk (h) we made the worst-case assumption that the eaves-dropper can decode and cancel the signals of all MTs except the signal intended forthe MT of interest [91, Chapter 10.2].Finally, to arrive at the ergodic secrecy rate, we average gsecnk (h) over all channelrealizations, which results in [25]E[gsecnk (h)]= E[[gnk (h)− Xevenk (h)]+]≥[E [gnk (h)]− E [Xevenk (h)]]+= gsecnk : (A.3)Introducing the denitions of the achievable ergodic secrecy rate, gnk = E [gnk (h)],and the ergodic eavesdropper capacity, Xevenk = E [Xevenk (h)], completes the proof.159Appendix A. Proofs in Chapter 2A.2 Proof of Theorem 2.1We rst recall that the entries of Hevem , m = 1P : : : Pb , are mutually independentcomplex Gaussian r.v.s. On the other hand, for ct → ∞ and both AN shapingmatrix designs, the vectors vmlP l = 1P : : : P ct−K, form an orthonormal basis. Hence,Hevem Vm, m = 1P : : : Pb , also has independent complex Gaussian entries, which areindependent from the complex Gaussian entries of Heven wnk. Thus, the term eve =pwHnkHeveHn X−1Heven wnk in (2.8) is equivalent to the SINR of an ce-branch MMSEdiversity combiner withb(ct−K) interferers [25, 92]. As a result, for the consideredsimplied path-loss model, the cumulative density function (CDF) of the receivedSINR, eve, at the eavesdropper is given by [92]Feve(x) =∑ce−1i=0 ixi∏2j=1(1 + jx)bjP (A.4)where i, j, and bj are dened in [66, Theorem 1]. Exploiting (A.4), we can rewrite(2.8) asXeve(a)=1ln 2∫ ∞0(1 + x)−1Feve(x)yx=1ln 2ce−1∑i=0i ×∫ ∞0xi(1 + x)∏2j=1(1 + jx)bjyx(b)=1ln 2ce−1∑i=0i × 102∑j=1bj∑l=1∫ ∞0!jl(x+ 1)(x+ 1j)lyx(x)=1ln 2ce−1∑i=0i × 102∑j=1bj∑l=2!jlI(1RjP l)P (A.5)where 0, !jl, and I(·P ·) are dened in Theorem 2.1. Here, (a) is obtained usingintegration by parts, (b) holds if the order of x in the denominator of (A.4) is not160Appendix A. Proofs in Chapter 2smaller than that in the numerator, i.e., ct−K ≥ ceRb or equivalently 1− ≥ Rb ,which is also the condition to ensure invertibility of X in (2.8), and (c) is obtainedusing the denition of I(·P ·) given in Theorem 2.1. This completes the proof.A.3 Proof of Theorem 2.2Using Jensen's inequality and the mutual independence of w˜nk = Heven wnk andHevem Vm, m = 1P : : : Pb (cf. Appendix B), Xevenk in (2.8) is upper bounded byXevenk ≤ log2(1 + Ew˜nk[pw˜HnkE[X−1]w˜nk]): (A.6)Let us rst focus on the term E [X−1] in (A.6) and note that X is statistically equiv-alent to a weighted sum of two scaled Wishart matrices [93]. Specically, we haveX = qX1+/qX2 with X1 ∼ Wce(ct−KP Ice) and X2 ∼ Wce((b−1)(ct−K)P Ice),where WA(BP IA) denotes an A × A Wishart matrix with B degrees of freedom.Strictly speaking, X is not a Wishart matrix, and the exact distribution of X seemsintractable. However, X may be accurately approximated as a single scaled Wishartmatrix, X ∼ Wce(<P Ice), where parameters  and < are chosen such that the rsttwo moments of X and qX1 + /qX2 are identical [69, 94]. Equating the rst twomoments of the traces of these matrices yields [94]< = q(ct −K) + /q(b − 1)(ct −K)P (A.7)and2< = q2(ct −K) + /2q2(b − 1)(ct −K): (A.8)161Appendix A. Proofs in Chapter 2By exploiting the expectation of an inverse Wishart matrix given in [94, Eq. (12)],we obtain E[X−1] = 1(φ−ce−1)Ice with  = xqRa if < − ce > 1 or equivalently if Q 1− xRa2 for ct →∞. Plugging this result and E[w˜Hnkw˜nk] = ce into (A.6), wenally obtain the result in (2.16). This completes the proof.162Appendix BProofs in Chapter 3Appendix B provides the proofs of Propositions, Corollaries, and Theorems in Chap-ter 3.B.1 Derivation of h^knm in Section 3.2.2Let√!mk ∈ C×1 be the pilot sequence of length  transmitted by the kth MT inthemth cell in the training phase, where !Hlj!mk = 1, if l ∈Mm∪{m} and j = k, andequals zero otherwise, where set MmP ∀m is dened in Section 3.2.2. The trainingsignal received at the nth BS, Ypilotn ∈ C×cT is given in (2.1), with !mk instead of!k. Assuming MMSE channel estimation [7, 8], the estimate of hknm given Ypilotn canbe derived ashˆknm =√pknm!Hmk(I + pb∑l=1K∑j=1!ljjnl!Hlj)−1Ypilotn=√pknm!Hmk (I +Amk +Bmk)−1Ypilotn(a)=√pknm!Hmk ((I +Amk) (I +Bmk))−1Ypilotn=√pknm!Hmk (I +Bmk)−1 (I +Amk)−1Ypilotn=√pknm!Hmk(I −Bmk (I +Bmk)−1)(I +Amk)−1Ypilotn(b)=√pknm!Hmk (I +Amk)−1Ypilotn=√pknm1 + pknm + p∑l∈Mm knl!HmkYpilotn : (B.1)163Appendix B. Proofs in Chapter 3whereAmk = p∑l∈Mm∪{m}!lkknl!Hlk ∈ C× P (B.2)andBmk = p∑l∈Mm∪{m}∑j ̸=k!ljjnl!Hlj + p∑l R∈Mm∪{m}K∑j=1!ljjnl!Hlj ∈ C× : (B.3)In (B.1), (a) is due to AmkBmk = 0, while (b) uses !HmkBmk = 0. For the special caseof Mm = M\{m}, (B.1) reduces to (2.2) in Chapter 2 with n instead of m whenestimating the in-cell CSI.B.2 Proof of Proposition 3.1Considering (3.3) and (3.10), the eective signal power, i.e., the numerator in (3.6),can be expressed as [15]E2[hknnfnk] = 21E2[hknnLnn(hˆknn)H ] = 21E2[hknnLnPk(hˆknn)H1 + hˆknnLnPk(hˆknn)H]=21nk(mnk + Ank)2knn(1 +mnk)2P(B.4)where LnPk = (HˆnnHˆHnn − (hˆknn)Hhˆknn + 1IcT )−1, mnk = E[hˆknnLnPk(hˆknn)H ], andAnk = E[h˜knnLnPk(hˆknn)H ]. On the other hand, the intra-cell interference term inthe denominator of (3.6) can be expressed asE[∑l ̸=k|hknnfnl|2]= 21E[hknnLnPkHˆHnPkHˆnPkLnPk(hknn)H(1 + hˆknnLnPk(hˆknn)H)2 ] = 21nk(nnk +Bnk)knn(1 +m)2 P (B.5)164Appendix B. Proofs in Chapter 3where HˆnPk is equal to Hˆnn with the kthrow removed, andnnk = E[hˆknnLnPkHˆHnPkHˆnPkLnPk(hˆknn)H ]P Bnk = E[h˜knnLnPkHˆHnPkHˆnPkLnPk(h˜knn)H ]:(B.6)Due to pilot contamination, the data precoding matrix of the mth BS is a function ofthe channel vectors between the mth BS and the MTs in all cells with reused pilots.Hence, the inter-cell interference from the the mth BS (if m ∈Mn) is obtained asE[|hkmnfmk|2] =21mk(mmk + Amk)2kmn(1 +mmk)2+kmnkmn + pkmn(B.7)andE[∑l ̸=k|hkmnfml|2]= 21E[hkmnLmPkHˆHmPkHˆmPkLmPk(hkmn)H(1 + hˆkmmLmPk(hˆkmm)H)2 ] = 21mk(nmk +Bmk)kmn(1 +mmk)2 P(B.8)respectively. Meanwhile, by exploiting (B.4), (B.7), and the denition of the variance,i.e., var[x] = E[x2]− E2[x], we obtain for the rst term of the denominator of (3.6)var[hknnfnk] =knnknn + pknn: (B.9)According to [15, Eq. (16)] and [95, Theorem 7], for ci → ∞ and constant ,mmk converges to G(P 1) dened in (3.12) and Amk → 0. Similarly, nmk and BmkapproachnmkcT→∞= G(P 1) + 1 @@1G(P 1) (B.10)andBmkcT→∞=mkmk(1 + G(P 1))2(G(P 1) + 1 @@1G(P 1))P (B.11)165Appendix B. Proofs in Chapter 3respectively, where@@1G(P 1) = −G(P1)(1+G(P1))2+1(1+G(P1))2 .Moreover, the inter-cell interference from other non-contaminated cells (i.e., m R∈Mn⋃{n}) is calculated asE[hkmnFmPkFHmPk(hkmn)H]= E[tr{FmPkFHmPk} ]= K − 1P (B.12)where FmPk is equal to Fm with the kthcolumn removed. The rst equality in (B.12)is due to the fact that the precoding matrix for the other MTs (i.e., not the kthMTs) in adjacent cells are independent of hkmn and [71, Lemma 11], while the secondequality holds for ci →∞.On the other hand, the constant scaling factor 1 for SRCI precoding is given by[15, Eq. (22)]21 =ϕeG(P 1) + 1 @@1G(P 1): (B.13)Hence, employing (B.4)-(B.13) in (3.6), the received SINR in (3.11) is obtained asSRCInk=21nkm2nk(1+mnk)221∑m∈Mn∪{n} mk(nmk+Bmk)(1+mmk)2+∑m∈Mn21mkm2mk(1+mmk)2+ p∑mR∈Mn∪{n}∑Kl=1 kmn + qQ+ 1=Knkg+1TgT1g2(1+g)2K∑m∈Mn∪{n} mk(1+mkmk(1+g)2(1+g)2)+∑m∈Mn21mkg2(1+g)2p+ 21=ΓˆnSRCIg2(g + 1@g@1)(∑m∈Mn∪{n} ΓˆmSRCI + (1 + g)2)+∑m∈Mn ΓˆmSRCIg2=1(g+1TT1g)(∑m∈Mn∪{n} ΓˆmSRCI+(1+g)2)ΓˆnSRCIg2+∑m∈Mn ΓˆmSRCIRΓˆnSRCI=1∑m∈Mn∪{n} ΓˆmSRCI+(1+g)2g(ΓˆnSRCI+ΓˆnSRCI1(1+g)2) +∑m∈Mn mkRnk P (B.14)166Appendix B. Proofs in Chapter 3where we denote g = G(P 1) for notational simplicity, 21 =∑mR∈Mn∪{n}∑Kl=1 kmn+Q+ KϕeTand ΓˆmSRCI is dened in Proposition 3.1. This completes the proof.B.3 Derivation of 1PoptWe rst denote SRCInk =11RΓ+∑m∈Mn mkRnkin (3.11), whereΓ =ΓˆnSRCI· G(P 1) ·  + 1(1 + G(P 1))2Υ+ (1 + G(P 1))2 : (B.15)with Υ =∑m∈Mn∪{n} ΓˆmSRCI. From (B.15), it is obvious that the optimal 1 tomaximize SRCInk is equivalent to the one that maximizes Γ.In order to obtain the optimal 1Popt, we need the following steps:@Γ@1=ΓˆnSRCI(@g@1·  + (1 + g)2Υ+ (1 + g)2+ g · @@1( + (1 + g)2Υ+ (1 + g)2))=ΓˆnSRCIg·  + (1 + g)2Υ+ (1 + g)2(21(1 + g)@gR@1 + 1(1 + g)2+2(1 + g)@gR@1Υ+ (1 + g)2)=2Υ2g(1 + g)2 (Υ + (1 + g)2)2@g@1(1 − Υ)= 0P (B.16)where we denote g = G(P 1). This nally gives 1Popt = RΥ, which completes thederivation.B.4 Proof of Corollary 3.1SZFnk in (3.16) can be obtained from (3.11) as SZFnk = lim1→0 SRCInk . In particular,when 1 → 0, G(P 1) in (3.12) can be rewritten asG(P 1) = 121(√(1− )2 + 2(1 + )1 + 21 + (1− )− 1): (B.17)167Appendix B. Proofs in Chapter 3Plugging this into (B.15), we haveΓ =ΓˆnSRCI2(√(1− )2 + 2(1 + )1 + 21 + (1− )− 1)4∑m∈Mn∪{n} ΓˆmSRCI21 +(√(1− )2 + 2(1 + )1 + 21 + (1− ) + 1)2×[41 +(√(1− )2 + 2(1 + )1 + 21 + (1− )− 1)2 ]P (B.18)For 1 → 0, we simply havelim1→0SRCInk =1(1−)ΓˆnSRCI+∑m∈Mn mkRnk: (B.19)This completes the proof of Corollary 3.1.B.5 Proof of Theorem 3.1The objective function in (3.23) can be rewritten asmsen = &2pE[Tr{ I∑i=0i(HˆnnHˆHnn)i+1DnnI∑i=0i(HˆnnHˆHnn)i+1}]+ &2pE[Tr{ I∑i=0i(HˆnnHˆHnn)iHˆnnH˜HnnDnnH˜nnHˆHnnI∑i=0i(HˆnnHˆHnn)i}]− 2&√pE[Tr{D1R2nnI∑i=0i(HˆnnHˆHnn)i+1}]+ 1 + &2eAN + &2Tr {Σn} P(B.20)where we exploited E[snsHn ] = IK , the denition of eAN given in Theorem 3.1, thedenition of Fn in (3.21), the denition1√cTHnn = Hˆnn+ H˜nn, and H˜nn =1√cTH˜nn.In the following, we simplify the right hand side (RHS) of (B.20) term by term.To this end, we denote the rst three terms on the RHS of (B.20) by t1, t2, and t3,168Appendix B. Proofs in Chapter 3respectively. Using a result from free probability theory [61], the rst term convergesto [66, Theorem 1]t1 = &2pTr {Dnn}E[Tr{( I∑i=0i(HˆnnHˆHnn)i+1)2}]P (B.21)as matrixDnn is free from∑Ii=0 i(HˆnnHˆHnn)i+1. Similarly, the third term convergestot3 = −2&√pTr{D1R2nn}E[Tr{ I∑i=0i(HˆnnHˆHnn)i+1}]: (B.22)Furthermore, the second term can be rewritten ast2(a)= &2pE[Tr{H˜HnnDnnH˜nn}Tr{ I∑i=0i(HˆnnHˆHnn)iHˆnnHˆHnnI∑i=0i(HˆnnHˆHnn)i}](b)= &2pciTr {Dnn∆n} P (B.23)where (a) follows again from [66, Theorem 1] and (b) results from E[Tr{H˜HnnDnnH˜nn}] =Tr {Dnn∆n}, where∆n is dened in Theorem 3.1, (3.21), and the constraint in (3.23).Exploiting (B.21)-(B.23) and the eigen-decomposition of matrix HˆnnHˆHnn = TΛTH,where diagonal matrix Λ = diag (1P : : : P K) contains all eigenvalues and unitary ma-trix T contains the corresponding eigenvectors, the asymptotic average MSE becomesmsen = E[&2pTr {Dnn}Tr{Λ2( I∑i=0iΛi)2}− 2&√pTr{D1R2nn }Tr{ I∑i=0iΛi+1}]+1 + &2eAN + &2Tr {Σn}+ &2pciTr {Dnn∆n} : (B.24)Next, we introduce the Vandermonde matrix C1 ∈ RK×(I+1), where [C1]iPj = j−1i ,169Appendix B. Proofs in Chapter 3and  = [1P : : : P K ]i, which allows us to rewrite (B.24) in compact form asmsen = limK→∞1KE[&2pTr {Dnn}iCi1Λ2C1− 2&√pTr{D1R2nn}iCi1]+1 + &2eAN + &2Tr {Σn}+ &2pciTr {Dnn∆n} : (B.25)Similarly, the constraint in (3.23) can be expressed aslimK→∞1KE[iCi1ΛC1]= ci : (B.26)Thus, the Lagrangian function of primal problem (3.23) can be expressed as L1(P &) =msen + ϵ1(limK→∞ 1KE[iCi1ΛC1] − ci ), where ϵ1 is the Lagrangian multiplier.Taking the gradient of the Lagrangian function with respect to , and setting theresult to zero, we obtain for the optimal coecient vector opt:limK→∞1KE[Ci1Λ(Λ+ϵ1Tr {Dnn} &2pIK)C1] =Tr{D1R2nn}&√pTr {Dnn} limK→∞1KE[Ci1]:(B.27)Furthermore, taking the derivative of L1(P &) with respect to & and equating it tozero, and multiplying both sides of (B.27) by i and applying (B.26), we obtainϵ1&2p= Tr {Dnn∆n}+ eAN + Tr {Σn}cip: (B.28)The expressions involving C1, Λ, and  in (B.27) can be further simplied. For ex-ample, we obtain limK→∞ E[1K[Ci1ΛC1]mPn]= limK→∞ E[1K∑Kk=1 m+n−1k]. Sim-plifying the other terms in (B.27) in a similar manner and inserting (B.28) into (B.27)we obtain the result in Theorem 3.1.170Appendix B. Proofs in Chapter 3B.6 Proof of Theorem 3.2Exploiting E[znzHn ] = IcT , the constraint in (3.33), and a similar approach as wasused to arrive at (3.28), the objective function in (3.33) can be rewritten as eAN =qE[Tr{GnnAnAHnGHnn} ]= qE[Tr{DnnHˆnnAnAHn HˆHnn}]+(1−ϕ)eiTr{Dnn∆n}:(B.29)Using (3.32) and a similar approach as in Appendix B.5, (B.29) can be rewrittenaseAN = (1− ϕ)eiTr{Dnn∆n}+ qciTr {Dnn}E[− 2Tr{ J∑j=0,jΛj+2}+ Tr {Λ}+ Tr{Λ( J∑i=0,jΛj+1)2}](B.30)Dening Vandermode matrix C2 ∈ RK×(J+1), where [C2]iPj = j−1i , we can rewrite(B.30) in compact form as eAN =qciTr {Dnn} limK→∞1KE[−2iCi2Λ+1i+iCi2Λ3C2]+(1−ϕ)eiTr{Dnn∆n}P(B.31)where 1 denotes the all-ones column vector. Taking into account the constraint in(3.33), we can formulate the Lagrangian asL2() = eAN + ϵ2( limK→∞1KE[iCi2Λ2C2 − 2iCi2] + 1) (B.32)with Lagrangian multiplier ϵ2. The optimal coecient vector opt is then obtainedby taking the gradient of the Lagrangian function with respect to  and setting it to171Appendix B. Proofs in Chapter 3zero:limK→∞E[Ci2Λ2 (Λ+ ϵIK)C2] = limK→∞E[Ci2 (Λ+ ϵIK)]P (B.33)where we used ϵ = ϵ2qcTTr{Dnn} . Simplifying the terms in (B.33) by exploiting a similarapproach as in Appendix B.5, we obtain the result in Theorem 3.2.172Appendix CProofs in Chapter 4Appendix C provides the proofs of Lemmas in Chapter 4.C.1 Proof of Lemma 4.1The ergodic secrecy rate achieved by the kth MT in symbol interval t ∈ {B+1P : : : P i}is given by [43, Lemma 1]gseck (t) = E[[gk(t)− log2(1 + E)]+]≥ [E[gk(t)]− XE]+ (a)≥ [gk(t)− XE]+ = gseck (t)P(C.1)where gseck (t) is an achievable lower bound for gseck (t), and (a) uses (4.17). By aver-aging gseck (t) over all symbol intervals t ∈ {B+1P : : : P i} we obtain Lemma 4.1. Thiscompletes the proof.C.2 Proof of Lemma 4.2The expectation given in (4.20) for k ∈ Sb is calculated asE[gHk ΘHk (t)fk](a)= E[gˆHk ΨHt0(t)gˆk‖gˆk‖ zj(ϕk(t)−ϕk(t0))](b)= tr(E[gˆkgˆHk‖gˆk‖]E[ΨHt0(t)])E[zj(ϕk(t)−ϕk(t0))]=√kck · z−2 +2ϕ2|t−t0|P (C.2)173Appendix C. Proofs in Chapter 4where Ψt0(t) = diag(zj( 1(t)− 1(t0))1i1×cRco P : : : P zj( bo (t)− bo (t0))1i1×cRco)and k isdened in Lemma 4.2. In (C.2), (a) exploits that the channel estimate and theestimation error are uncorrelated [18], and (b) exploits the mutually independence ofgˆkgˆHk , ΨHt0(t), and zj(ϕk(t)−ϕk(t0)). This completes the proof.C.3 Proof of Lemma 4.3In (4.18), the term reecting the interference caused by the signal intended for MTl ∈ Sb to MT k ∈ Sb can be expanded asE[∣∣gHk ΘHk (t)fl∣∣2] = E[∣∣∣∣gHk (t0)ΨHt0(t) gˆl‖gˆl‖zj(ϕk(t)−ϕk(t0))∣∣∣∣2]= E[tr(gk(t0)gHk (t0)ΨHt0(t)gˆlgˆHl‖gˆl‖2Ψt0(t))](a)= k +(I2l !Hl Θb(t0)Σ−1b ΘH(t0)!lc− k)× E [(1ctr(ΨHt0(t)))2 ]P (C.3)where Xl = l!Hl Θb(t0)Σ−1b ⊗ Ic and I = E[tr(XHl gk(t0)gHk (t0)Xl b Hb)]. (a)exploits [66, Theorem 1] from free probability theory, since the phase drift matricesΨt0(t) and ΨHt0(t) are free from gk(t0)gHk (t0) andgˆlgˆHl‖gˆl‖2 . The further step is to expandI asI = E[tr(YHlkgkgHk YlkgkgHk)]+ tr(kXHl Xl(Σb − k(Wbk +Ubk))⊗ Ic) +E[tr(XHl gkgHk Xl(Ubk ⊗ diag(g(1)k P : : : P g(c)k)))]P (C.4)174Appendix C. Proofs in Chapter 4whereYlk = ΘHk (t0)Xl[ΘHk (Bb−1 + 1)!k(Bb−1 + 1)P : : : PΘHk (t0)!k(t0)]i: (C.5)Denoting the tth column of Ic by ect ∈ Cc×1, the rst term on the right hand sideof (C.4), denoted by I1, can be expanded asI1 =∑n1Pn2Pb1Pb2[kXleBbb1⊗ Ic ]n1n1 [kXleBbb2 ⊗ Ic ]Hn2n2 × !k(b1)!∗k(b2)Θ(n1P n2P b1P b2P t0)=∣∣tr (kXl(Θb(t0)!k ⊗ Ic))∣∣2 + tr (2kXHl Xl(Wbk ⊗ Ic))+c∑|n1−n2|≤ bb02k(ecn1)HXl((Wbk −Θb(t0)!k!Hk Θb(t0))⊗ ecn1(ecn2)H)XHl ecn2P (C.6)where the expectation with respect to the phase drift, Θ(n1P n2P b1P b2P t0), depends onthe number of LOs, co, and is given by Θ(n1P n2P b1P b2P t0) =E[zn1k (b1)−n1k (t0)−n2k (b2)+n2k (t0)]=z−2 +2ϕ2|b1−b2| |n1 − n2| ≤ cco Pz−2 +2ϕ2|t0−b1|z−2 +2ϕ2|t0−b2| |n1 − n2| > cco :(C.7)Furthermore, we rewriteUbk = (MTt +BSr )p∑Bbt=1 eBbt (eBbt )Hand diag(g(1)k P : : : P g(c)k)=∑cn=1 |(ecn )Hgk|2ecn (ecn )H . Using these results in the third term on the right handside of (C.4), denoted by I2, we obtainI2 = 2ktr(XHl Xl(Ubk ⊗ Ic))+c∑n=12k(ecn )HXl(Ubk ⊗ ecn (ecn )H)Xlecn : (C.8)Applying (C.6) and (C.8) in (C.3) and exploiting E[ (1ctr(ΨHt0(t)))2 ]= 1−ϵco+ ϵ, weobtain the result in Lemma 4.3 for kP l ∈ Sb.175Appendix C. Proofs in Chapter 4For the case of l R∈ Sb, the multiuser interference term simplies toE[∣∣gHk ΘHk (t)fl∣∣2] = E[∣∣∣∣gHk (t0)ΨHt0(t) gˆl‖gˆl‖zj(ϕk(t)−ϕk(t0))∣∣∣∣2]= kP (C.9)where the last equality follows from the independence of gk, gˆl, l R∈ Sb, and ΨHt0(t).This completes the proof.C.4 Proof of Lemma 4.4The AN leakage power received at the kth MT in time slot t can be expanded asakAN(t) = E[tr(gˆkgˆHk ΨHt0(t)AAHΨt0(t))]+ E[eHk (t0)ΨHt0(t)AAHΨHt0(t)ek(t0)]:(C.10)By using [66, Theorem 1], the rst term in (C.10) can be further expanded aska+(E[tr(gˆkgˆHk AAH)]− ka)E [( 1ctr (Ψt0(t)))2]= ka(1− 1co)(1− ϵ) P(C.11)since phase drift matrices Ψt0(t) and ΨHt0(t) are free from gˆkgˆHk and AAH. Further-more, we exploited gˆHk A = 0, 1 ≤ k ≤ K, which holds for the NS AN precoder.The second term in (C.10) is equal to ka(1− k), with k as dened in Lemma4.2, due to the mutual independence of the estimation error vector ek(t0), the phasedrift matrix Ψt0(t), and the AN precoder A. Combining these two terms completesthe proof.176Appendix C. Proofs in Chapter 4C.5 Proof of Lemma 4.5For the G-NS AN precoder, we rewrite the leakage power received at the kth MT intime slot t asakAN =bo∑m=1E[(g(m)k)H (Θ(m)k (t))HA(m)AH(m)Θ(m)k (t)g(m)k]P (C.12)where g(m)k ∈ CcRbo×1 contains the ((m− 1)cRbo+1)th to the (mcRbo)th elementsof vector gk, 1 ≤ m ≤ bo, and Θ(m)k (t) ∈ CcRbo×cRbo is a diagonal matrix withthe ((m − 1)cRbo + 1)th to the (mcRbo)th elements of matrix Θk(t) on its maindiagonal. Using similar steps as in Appendix C.4 but with coRbo substituted by cofor calculation of the expectation terms in (C.12), we obtain (4.30). This completesthe proof.177

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