physical layer security for multi-user ... - wordpress.coma special thanks also goes to my...

Physical Layer Security forMulti-User MIMO Systems

Giovanni GeraciB.Sc. M.Sc.

Supervisor: Prof. Jinhong Yuan

A DISSERTATION SUBMITTED FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

March 2014

School of Electrical Engineering and Telecommunications

The University of New South Wales, Australia

When I was seventeen, I was working as an apprentice mechanic in myvillage, in Sicily. On a summer’s day, Mr L.G. walked in the workshopto get his car battery changed. While I was replacing the battery, heasked me how long I was going to work there. I said “until September,when school reopens”. Mr L.G. replied that I would have been betteroff quitting school, because I was wasting my time.

Mr L.G., this dissertation is dedicated to you.

iii

AcknowledgmentsBoarding a plane to Australia three years ago to start a Ph.D. was a choice I did not regret. That

flight marked the beginning of a journey whose destination is this thesis. For me, it has been

all about the journey, not the destination. Difficulties and challenges have made me stronger and

mature. I have learnt how to bounce back from failures. Travelling and coming across different

cultures has made me more open minded, flexible and creative. Since that long flight, I feel I have

grown as a researcher, but most importantly as a man. Therefore, even if one day I will become

Dr. Giovanni, I will always be Giovanni first, Doctor second.

Many people have helped me complete this Ph.D. successfully. None of them need to be named,

for if they know me, then they already know how grateful I am to them all, even if all they did was

just smile at me once.

Most of the things I did in the last three years would not have been possible without my supervisor.

He is a great person, and he always treated me as an adult, not just as a student. He probably

thinks I am smarter than I really am. A special thanks also goes to my co-supervisor at CSIRO

for providing many insightful suggestions and remarks throughout my Ph.D. Thanks to all the

sources of funding that have supported my studies, and allowed me to present my work around the

globe. Thanks to my advisors in France and United States, for offering me internships that greatly

contributed to my professional and personal growth. Thanks to all my research collaborators, not

just for their help in making my publications list longer, but more importantly for all I have learnt

from them. I also thank all the colleagues and staff in Sydney, Paris, and Austin. You know I

enjoy much more interacting with humans than with computers. Thanks to all the friends I am

lucky to have in every single continent of this amazing world. Thanks to my family for giving me

opportunities that not everyone was privileged to have.

v

Abstract

T his thesis studies physical layer security for multi-user multiple-input multiple-output (MIMO)

wireless systems. Due to the broadcast nature of the physical medium, wireless multi-user

communications are very susceptible to eavesdropping, and it is critical to protect the transmit-

ted information. Security of wireless communications has traditionally been achieved at the net-

work layer with cryptographic schemes. However, classical cryptography might not be suitable

in large dynamic networks, since it requires key distribution and management, and complex en-

cryption/decryption algorithms. A method that exploits the characteristics of wireless channels,

known as physical layer security, was proposed as an alternative to achieve perfect secrecy without

requiring encryption keys. The rates at which messages can be reliably transmitted to an intended

user while no information is leaked at the eavesdroppers, denoted as the secrecy rates, have been

studied for several network topologies. However, the secrecy rates achievable in generic multi-

user networks are still unknown. Hence in this thesis we study the secrecy rates achievable in

multi-user systems with practical transmission schemes.

We propose a linear precoder based on regularized channel inversion (RCI) for the broadcast

channel with confidential messages (BCC). In the BCC, a multi-antenna base station (BS) simul-

taneously transmits independent confidential messages to several spatially dispersed malicious

users that can eavesdrop on each other. We carry out a large-system analysis and obtain closed

form expressions for the achievable secrecy rates under Rayleigh fading, as well as the optimal

regularization parameter and the optimal network load. Simulations confirm that our analysis is

accurate even for finite systems. We compare the secrecy rate of the proposed precoder to two

upper bounds obtained without secrecy requirements and without interference, respectively, and

show that it has the same scaling factor as the two bounds in the high signal-to-noise ratio (SNR)

regime. We further extend our analysis to more practical scenarios where only imperfect chan-

nel state information is available at the BS, and where channel correlation is present among the

transmit antenna elements.

We then introduce the broadcast channel with confidential messages and external eavesdroppers

(BCCE). Unlike the BCC, in the BCCE not just malicious users, but also randomly located ex-

ternal nodes can act as eavesdroppers. We obtain the probability of secrecy outage and the mean

secrecy rate for the RCI precoder in the BCCE, for the two cases of non-colluding and colluding

eavesdroppers. We show that, irrespective of the collusion strategy at the external eavesdroppers,

a large number of transmit antennas drives both the probability of secrecy outage and the rate loss

due to the presence of external eavesdroppers to zero. Increasing the density of eavesdroppers by

a factor n, requires n2 as many antennas to meet a given probability of secrecy outage and a given

vii

mean secrecy rate. Our analysis demonstrates that the number of transmit antennas at the BS is a

key resource to secure communications against malicious users and external eavesdropping nodes.

We finally turn our attention to cellular networks where, unlike the case of isolated cells, multiple

BSs generate inter-cell interference, and malicious users of neighboring cells can cooperate to

eavesdrop. We characterize the probability of secrecy outage and the mean secrecy rate with

RCI precoding, accounting for the spatial distribution of BSs and users and the fluctuations of

their channels. We find that RCI precoding can achieve a non-zero secrecy rate with probability

of outage smaller than one. However we also find that unlike isolated cells, the secrecy rate in

a cellular network does not grow monotonically with the SNR, and the network tends to be in

secrecy outage if the transmit power grows unbounded. We further show that there is an optimal

value for the density of BSs that maximizes the secrecy rate, and this value is a decreasing function

of the SNR. Using the developed analysis, we clearly establish the importance of designing the

transmit power and the BS deployment density to make communications robust against malicious

users in other cells.

Contents

Acknowledgments v

Abstract vii

List of Acronyms xxi

Basic Notations xxiii

1 Introduction 11.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Thesis Overview and Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2 Overview of Physical Layer Security 132.1 Introduction to Physical Layer Security . . . . . . . . . . . . . . . . . . . . . . 142.2 The Wiretap Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3 Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.4 Multi-antenna Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.5 Multi-user Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3 Physical Layer Security in Isolated Cells: Achievable Rates 213.1 The Broadcast Channel with Confidential Messages . . . . . . . . . . . . . . . . 223.2 Achievable Secrecy Rates in the BCC . . . . . . . . . . . . . . . . . . . . . . . 23

3.2.1 Linear Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.2.2 Achievable Secrecy Rates with Linear Precoding . . . . . . . . . . . . . 253.2.3 Regularized Channel Inversion Precoding . . . . . . . . . . . . . . . . . 273.2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.3 Optimal Regularization Parameter and Power Allocation . . . . . . . . . . . . . 303.3.1 Regularized Channel Inversion Precoding with Power Allocation . . . . . 313.3.2 Optimal Power Allocation . . . . . . . . . . . . . . . . . . . . . . . . . 323.3.3 Joint Optimal Power Allocation and Regularization Parameter . . . . . . 34

3.4 Achievable Secrecy Rates in Practical Channels . . . . . . . . . . . . . . . . . . 363.4.1 Secrecy Rates in the Presence of Imperfect Channel State Information . . 36

xi

3.4.2 Secrecy Rates Under No CSI or Poisoned CSI . . . . . . . . . . . . . . . 393.4.3 Secrecy Rates under Transmit Channel Correlation . . . . . . . . . . . . 40

3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4 Physical Layer Security in Isolated Cells: A Large-System Analysis 434.1 Large-System Analysis of the Secrecy Rates . . . . . . . . . . . . . . . . . . . . 44

4.1.1 Large-System Secrecy Rates with RCI Precoding . . . . . . . . . . . . . 444.1.2 Secrecy Sum-Rate Maximizing Regularization Parameter . . . . . . . . . 454.1.3 Optimal Secrecy Sum-Rate . . . . . . . . . . . . . . . . . . . . . . . . . 474.1.4 Optimal Network Load . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

4.2 RCI Precoder with Power Reduction . . . . . . . . . . . . . . . . . . . . . . . . 524.2.1 Optimal Transmit SNR . . . . . . . . . . . . . . . . . . . . . . . . . . . 524.2.2 Power Reduction Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 52

4.3 Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.1 Secrecy Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.3.2 Multi-User Loss . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.3.3 Comparison to Other Linear Schemes . . . . . . . . . . . . . . . . . . . 58

4.4 Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.1 Minimum Required CSI . . . . . . . . . . . . . . . . . . . . . . . . . . 604.4.2 Channel Feedback in FDD Systems . . . . . . . . . . . . . . . . . . . . 634.4.3 Channel Training in TDD Systems . . . . . . . . . . . . . . . . . . . . . 63

4.5 Channel Correlation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5.1 Large-System Secrecy Sum-Rates . . . . . . . . . . . . . . . . . . . . . 664.5.2 Selection of the Optimal Regularization Parameter . . . . . . . . . . . . 674.5.3 Comparison to Other Linear Schemes . . . . . . . . . . . . . . . . . . . 684.5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5 Physical Layer Security in a Random Field of Eavesdroppers 735.1 Introduction to the BCCE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.2 Probability of Secrecy Outage . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2.1 Non-colluding Eavesdroppers . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 Colluding Eavesdroppers . . . . . . . . . . . . . . . . . . . . . . . . . . 79

5.3 Mean Secrecy Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805.3.1 Mean Secrecy Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 815.3.2 Secrecy Rate Loss due to the External Eavesdroppers . . . . . . . . . . . 825.3.3 Optimal Regularization Parameter . . . . . . . . . . . . . . . . . . . . . 83

5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6 Physical Layer Security in Cellular Networks 916.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.1.1 Network Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.1.2 RCI Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 936.1.3 Malicious Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

6.2 Achievable Secrecy Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 946.2.1 SINR at a Typical User . . . . . . . . . . . . . . . . . . . . . . . . . . . 956.2.2 SINR at the Malicious Users . . . . . . . . . . . . . . . . . . . . . . . . 956.2.3 Achievable Secrecy Rates . . . . . . . . . . . . . . . . . . . . . . . . . 97

6.3 Large-system Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 996.3.2 Characterization of Interference and Leakage . . . . . . . . . . . . . . . 1016.3.3 Probability of Secrecy Outage . . . . . . . . . . . . . . . . . . . . . . . 1036.3.4 Mean Secrecy Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

6.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1076.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7 Conclusions and Future Work 113

Appendices 117

A Appendix for Chapter 4 117A.1 Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117A.2 Proof of Theorem 4.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

B Appendix for Chapter 5 121B.1 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121B.2 Proof of Lemma 5.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122B.3 Proof of Lemma 5.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123B.4 Proof of Lemma 5.9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124B.5 Proof of Corollary 5.11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

C Appendix for Chapter 6 127C.1 Proof of Proposition 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127C.2 Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128C.3 Proof of Proposition 6.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129C.4 Proof of Theorem 6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130C.5 Proof of Corollary 6.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

Bibliography 132

List of Figures

2.1 The wiretap channel in WYNER [1975], where the eavesdropper’s channel is adegraded version of the main channel. . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 The general MIMOME wiretap channel. . . . . . . . . . . . . . . . . . . . . . . 17

3.1 The MISO broadcast channel with confidential messages (BCC). . . . . . . . . . 233.2 Simulated secrecy sum-rate SBCC achievable by RCI precoding in the BCC versus

SNR ρ, for various values of N = K. . . . . . . . . . . . . . . . . . . . . . . . 303.3 Comparison between the RCI precoder and the plain CI precoder, for various val-

ues of N = K. The secrecy loss is also shown as the gap between dashed andsolid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.4 Per-user secrecy rate vs. SNR ρ for K = 4 users: with separate (dashed) and joint(circle) optimization of ξ and p, and with equal power allocation (solid). The rateof the RCI-EP precoder without secrecy requirements (square) and the secrecycapacity of the MISOME channel (diamond) are also plotted. . . . . . . . . . . . 37

4.1 Comparison between the secrecy sum-rate with RCI precoding in the large-systemregime (4.8) and the simulated ergodic secrecy sum-rate for finite N . Three setsof curves are shown, each one corresponds to a different value of β. . . . . . . . 49

4.2 Mean normalized secrecy sum-rate difference between S?BCC (obtained using theoptimal ξ?BCC,N ) and SBCC(ξBCC) (obtained with ξBC from large-system analy-sis), for β = 0.8 and various values of the SNR. . . . . . . . . . . . . . . . . . . 50

4.3 Large-system secrecy sum-rate per transmit antenna as a function of β for RCIprecoding. The value of β? is indicated next to each curve. . . . . . . . . . . . . 51

4.4 Comparison betweenK? (obtained via simulations),K? (obtained via exhaustivesearch and large-system analysis), and the analytical approximation K? (obtainedvia large-system and large-SNR analysis). . . . . . . . . . . . . . . . . . . . . . 51

4.5 Comparison between the ergodic secrecy sum-rates SBCC and SrBCC achieved bythe RCI precoder and by the proposed RCI-PR precoder, respectively, for N = 10

transmit antennas. Three values of β are considered: 1.2, 1.4, and 1.6, correspond-ing to K = 12, 14, and 16 users. . . . . . . . . . . . . . . . . . . . . . . . . . . 54

xv

4.6 Comparison between the simulated ergodic per-user secrecy rate with RCI-PR(solid) and the two upper bounds: (i) per-user rate without secrecy requirements(dashed) and (ii) MISOME secrecy capacity (dotted), for K = 12 users. Threevalues of β are considered: 0.8, 1, and 1.2, corresponding to N = 15, 12, and 10

antennas. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.7 Comparison between the per-antenna secrecy sum-rate SiBCC/N with RCI pre-

coding in the large-system regime and the simulated ergodic secrecy sum-rateSiBCC/N , for finite N , and in the presence of a channel estimation error τ = 0.1.Three sets of curves are shown, each one corresponds to a different value of β. . . 62

4.8 Comparison between the ergodic per-user secrecy rates SBCC/K and SiBCC/K

with RCI-PR precoder, for N = 10, in the presence of perfect CSI and in thepresence of a channel estimation error τ2 = C

ρ , with C obtained from (4.37) forlog2 b = 1 bit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.9 Optimal relative amount of training Tt/T vs high-SNR approximation, for N =

K = 10 and rρ = ρ/ρul = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.10 Comparison between the simulated ergodic per-antenna secrecy sum-rate ScBCC/N

from (3.48) and the large-system approximation ScBCC/N from (4.50), for ν = 0.5

and various values of β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704.11 Relative secrecy rate loss (RBCC −RcBCC)/RBCC as a function of the correlation

coefficient ν, for β = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 714.12 CCDF of the normalized secrecy rate difference between using: (i) ξcBCC obtained

from (4.51) and (ii) ξc?BCC obtained by bi-sectional search for every channel real-ization, for β = 1, and ν = 0.5. . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1 Example of a BCCE with K = 5 malicious users and a density of external eaves-droppers λe = 0.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Comparison between the simulated probability of outage OBCCE,k and the large-system results OBCCE provided in Theorem 5.2 and Theorem 5.7, for a networkload β = 1, an SNR ρ = 10dB, and various values of λe. . . . . . . . . . . . . . 85

5.3 Comparison between the simulated ergodic per-antenna secrecy sum-rate undernon-colluding and colluding eavesdroppers, and the large-system results fromTheorem 5.10, for λe = 0.1, N = 10 transmit antennas, and various values ofthe network load β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.4 Comparison between the simulated ergodic per-user secrecy rate E[Rk] under non-colluding and colluding eavesdroppers, and the large-system results R from The-orem 5.10, for a network load β = 1, an SNR ρ = 10dB, and various values ofλe. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.5 Comparison between the simulated ergodic per-user secrecy rates of: (i) the BCCEwith non-colluding eavesdroppers, (ii) the BCCE with colluding eavesdroppers,and (iii) the BCC without external eavesdroppers, for a network load β = 1, anSNR ρ = 10dB, and various values of λe. . . . . . . . . . . . . . . . . . . . . . 87

5.6 Comparison between the large-system regularization parameter ξBCCE in (5.29)and the value ξBCCE that maximizes the average simulated secrecy sum-rate SBCCE

for a finite system with N = 10 transmit antennas, a network load β = 1, and anSNR ρ = 10dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.7 Normalized mean secrecy sum-rate difference between using ξ?BCCE, that maxi-mizes each realization of the secrecy sum-rate SBCCE, and ξBCCE, obtained fromlarge-system analysis in (5.29), under colluding eavesdroppers, for a network loadβ = 1, various values of the density of eavesdroppers λe, and various values ofthe SNR ρ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

6.1 Illustration of a cellular network. The star denotes a typical user. The circles,squares, and triangles denote BSs, out-of-cell users, and in-cell users, respectively,as discussed in Subsection 6.1.3. . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6.2 Example of percolation in a random plane network. Dots represent malicioususers, and discs represents the cooperation range of malicious users. Two mali-cious users can cooperate when their respective discs overlap. . . . . . . . . . . . 96

6.3 Comparison between the simulated ergodic secrecy rate RCELL in (6.11) and theapproximation RCELL in (6.23) versus the SNR, for N = 20 transmit antennas,an average of K = 20 users per BS, and η = 4. . . . . . . . . . . . . . . . . . . 101

6.4 Comparison between the simulated cumulative distribution functions (CDFs) of Iand L and the lognormal approximations in (6.32) and (6.33), for an SNR ρ =

10dB, N = 20 transmit antennas, K = 20 users per BS, ‖c‖ = r, and η = 4. . . 1036.5 Comparison between the simulated probability of secrecy outage OCELL and the

analytical result from Theorem 6.3, for N = 20 transmit antennas, K = 20 usersper BS, and three values of the density of BSs λb. . . . . . . . . . . . . . . . . . 107

6.6 Comparison between the simulated mean secrecy rate RCELL and the analyticalresult from Theorem 6.4, for N = 20 transmit antennas, K = 20 users per BS,and two values of the density of BSs λb. . . . . . . . . . . . . . . . . . . . . . . 108

6.7 Simulated probability of secrecy outage versus transmit SNR, for K = 10 usersper BS and various values of the number of transmit antennas N and density ofBSs λb. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

6.8 Simulated ergodic secrecy rate versus transmit SNR, forK = 10 users per BS andvarious values of the number of transmit antennas N and density of BSs λb. . . . 110

6.9 Simulated ergodic secrecy rate versus density of BSs, for N = 20 transmit anten-nas, K = 20 users per BS, and various values of the transmit SNR ρ. . . . . . . . 110

List of Tables

3.1 Algorithm for optimal power allocation. . . . . . . . . . . . . . . . . . . . . . . 343.2 Algorithm for joint optimal power allocation and regularization parameter. . . . . 35

6.1 Notation Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

xix

List of Acronymsa.s. Almost Surely.BC Broadcast Channel.BCC Broadcast Channel with Confidential messages.BCCE Broadcast Channel with Confidential Messages and External Eavesdroppers.BS Base Station.CCDF Complementary Cumulative Distribution Function.CDF Cumulative Distribution Function.CI Channel Inversion.CI-PR Channel Inversion with Power Reduction.CSI Channel State Information.DPC Dirty Paper Coding.FDD Frequency Division Duplex.GSVD Generalized Single Value Decomposition.i.i.d. Independent and Identically Distributed.MIMO Multiple-Input Multiple-Output.MISO Multiple-Input Single-Output.MISOME Multiple-Input Single-Output Multiple-Eavesdropper.MRC Maximal Ratio Combining.pdf Probability Density Function.PGFL Probability Generating Functional.PPP Poisson Point Process.RCI Regularized Channel Inversion.RCI-EP Regularized Channel Inversion with Equal Power Allocation.RCI-PR Regularized Channel Inversion with Power Reduction.RMT Random Matrix Theory.r.v. Random Variable.RVQ Random Vector Quantization.SG Stochastic Geometry.SINR Signal-to-Interference-Plus-Noise Ratio.SNR Signal-to-Noise Ratio.SUB Single User Beamforming.TDD Time Division Duplex.w.p. With Probability.

xxi

Basic NotationsR Real numbers.C Complex numbers.x Scalar in R or C.x Column vector.xk The kth element of x.X Matrix.I Identity matrix.xk The kth column of X.(·)∗ Complex conjugate.(·)T Transpose.(·)† Conjugate transpose.(·)−1 Inverse.tr · Trace.‖x‖ Euclidean norm.log Natural logarithm.log2 Logarithm in base 2.lim Limit.arg Argument.max Maximum.min Minimum.|·| Set cardinality.B Complement of a set B., Defined as.∀ For all.Ex [·] Mean with respect to random variable x.x ∼ CN (µ, σ2) Complex Gaussian random variable with mean µ and variance σ2.x ∼ Γ(k, θ) Gamma-distributed random variable with shape k and scale θ.x ∼ exp(λ) Exponential random variable with mean 1/λ.P Probability.1· Indicator function.[·]+ Maximum between the quantity in the brackets and zero.Lf Laplace transform of f .sgn (·) Sign function.erfc (·) Complementary error function.δ (·) Dirac’s delta.

xxiii

Chapter 1

Introduction

1.1 Background and Motivation

O ver the last two decades, wireless communications have advanced tremendously, becom-

ing an indispensable part of our lives. Wireless networks have become more and more

pervasive to guarantee global digital connectivity, and wireless devices have quickly evolved into

multimedia smartphones, which run applications that demand high-speed data connections. Multi-

user multiple-input multiple-output (MIMO) wireless techniques have received much attention as

a way to meet such demand by achieving high spectral efficiency LIM et al. [2013]. In a multi-user

MIMO wireless system, a central multi-antenna base station (BS) simultaneously communicates

to several users over the same frequency band. While it is known that the sum-capacity of multi-

user MIMO systems is achieved by using dirty paper coding (DPC) CAIRE AND SHAMAI [2003],

DPC requires high-complexity coding schemes that make it too complex to be implemented LI

et al. [2010].

Suboptimal precoding schemes have proven to be practical and effective in controlling inter-user

interference for the downlink of multi-user MIMO networks HOCHWALD et al. [2005]. Among

those, linear precoding schemes were proposed as a low-complexity alternative to DPC for multi-

user MIMO downlink implementations SPENCER et al. [2004a]. A popular and practical linear

precoding scheme to control inter-user interference is channel inversion (CI) precoding, some-

times known as zero forcing precoding YOO AND GOLDSMITH [2006]. To increase the sum-rate

1

2 Background and Motivation 1.1

performance of the CI precoder, the regularized channel inversion (RCI) precoder was proposed

to tradeoff the inter-user interference and the desired signal through a regularization parameter

PEEL et al. [2005]. The performance of these linear precoding schemes has been studied by a

large-system approach that employs random matrix theory (RMT) tools WAGNER et al. [2012].

As well as spectral efficiency, security is regarded as a critical concern in wireless multiuser net-

works, since users rely on these networks to transmit sensitive data. Due to the broadcast nature of

the physical medium, wireless multiuser communications are very susceptible to eavesdropping,

and it is essential to protect the transmitted information. Security of wireless communications has

traditionally been ensured by network layer key-based cryptography. However, classical cryptog-

raphy may not be suitable in the case of large dynamic wireless networks, since it raises issues like

key distribution and management (for symmetric cryptosystems) and high computational complex-

ity (for asymmetric cryptosystems). Moreover, classical cryptography is potentially vulnerable,

because it relies on the unproven assumption that certain mathematical functions are hard to invert

LIANG et al. [2009b]; LIU AND TRAPPE [2010]. To provide an additional level of protection and

to achieve perfect secrecy without requiring encryption keys, methods exploiting the randomness

inherent in noisy channels, known as physical layer security, have been proposed WYNER [1975];

CSISZAR AND KORNER [1978].

In the past few years, physical layer security has become a very active area of research, and has

witnessed significant growth MUKHERJEE et al. [2013]. The maximum rate at which a mes-

sage can be reliably transmitted to an intended user while the rate of information leakage at the

eavesdroppers vanishes asymptotically with the code length, denoted the secrecy capacity, has

been studied for several network topologies. The secrecy capacity of the wiretap channel, a three-

terminal network consisting of a transmitter, an intended user and an eavesdropper, was derived for

the MIMO case when all terminals have full channel state information (CSI) KHISTI AND WOR-

NELL [2010] . These results were then extended to the case of multiple users with a single external

eavesdropper, known as the multi-receiver wiretap channel EKREM AND ULUKUS [2011]. When

the eavesdropper’s channel is not known by the transmitter, it was shown that the transmission

of artificial noise is an effective method to reduce the eavesdropper’s signal-to-noise ratio GOEL

AND NEGI [2008]. Recently, the study of physical layer security was also extended to networks

where the intended users can act maliciously as eavesdroppers, but only two-user systems have

been considered in the literature LIU et al. [2013]. Determining the achievable secrecy rates for

1.1 Introduction 3

multi-user networks where any number of intended users are potentially eavesdropping remains

an open problem. Moreover, the performance of practical transmission schemes in such generic

multi-user networks is also unknown.

In this thesis, we study physical layer security for generic multi-user multi-antenna systems. We

propose a practical linear transmission scheme based on RCI precoding to simultaneously transmit

independent confidential messages to several spatially dispersed malicious users. We first consider

the case of an isolated cell, where all malicious users can eavesdrop on each other. We carry out a

large-system analysis and obtain closed form expressions for the achievable secrecy rates. These

expressions allow us to optimize several design parameters, like the regularization parameter of

the precoder, the power allocation, and the network load. We also extend our analysis to practical

scenarios where only imperfect CSI is available at the transmitter, and where channel correlation

is present among the transmit antenna elements.

We then consider confidential broadcasting in the presence of a random field of eavesdroppers.

In this case, not just malicious users, but also randomly located external nodes can act as eaves-

droppers. We study the performance of RCI precoding, and provide explicit expressions for the

probability of secrecy outage and the mean secrecy rate, for the two cases of non-colluding and

colluding eavesdroppers. We find that, as expected, the presence of external eavesdropping nodes

incurs a secrecy rate loss. However, irrespective of the collusion strategy at the external eaves-

droppers, a large number of transmit antennas drives this secrecy rate loss to zero. We also find

that increasing the density of eavesdroppers by a factor n, requires n2 as many antennas to meet

a given probability of secrecy outage. Our analysis demonstrates that the number of transmit an-

tennas at the BS is a key resource to secure communications against malicious users and external

eavesdropping nodes.

We finally turn our attention to cellular networks where, unlike the case of isolated cells, multiple

BSs simultaneously transmit confidential messages to several users, generating inter-cell inter-

ference. Moreover in cellular networks, malicious users of neighboring cells can cooperate to

eavesdrop. We obtain the probability of secrecy outage and the mean secrecy rate achievable with

RCI precoding, and we find that a non-zero secrecy rate can be achieved with probability of outage

smaller than one. Unlike isolated cells, we find that the secrecy rate in a cellular network does not

grow monotonically with the transmit power, and the network tends to be in secrecy outage if the

transmit power grows unbounded. Furthermore, we show that there is an optimal value for the

4 Thesis Overview and Contributions 1.2

density of BSs that maximizes the secrecy rate, and we thus establish the importance of designing

the BS deployment density to make communications robust against malicious users in other cells.

1.2 Thesis Overview and Contributions

In this section we outline the contributions of the thesis.

Chapter 3

Chapter 3 considers the multiple-input single-output (MISO) broadcast channel with confidential

messages (BCC), where a multi-antenna base station simultaneously transmits independent con-

fidential messages to several single-antenna users. First, we study the secrecy rates achievable

by a linear precoder based on regularized channel inversion (RCI). We then propose an algorithm

to jointly optimize the regularization parameter of the precoder and the power allocation vector.

Finally, we extend the secrecy rate analysis to the BCC with imperfect channel state information

(CSI) and transmit channel correlation.

The work in this chapter is based on the papers listed in Section 1.3 as J1 and C1.

New Contributions:

• Theorem 3.3: For a given channel matrix, we derive a secrecy sum-rate achievable by linear

precoding in the MISO BCC.

• Corollary 3.4: For a given channel matrix, we derive a secrecy sum-rate achievable by

regularized channel inversion precoding in the MISO BCC.

• Table 3.1: We propose an algorithm that optimizes the power allocation vector of the RCI

precoder, for a given regularization parameter.

• Table 3.2: We propose an algorithm that jointly optimizes the power allocation vector and

the regularization parameter of the RCI precoder.

• Subsection 3.4.1: We derive a secrecy sum-rate achievable by RCI precoding in the presence

of imperfect channel state information.

• Subsection 3.4.3: We derive a secrecy sum-rate achievable by RCI precoding in the presence

of transmit channel correlation.

1.2 Introduction 5

Chapter 4

In Chapter 4, we obtain deterministic approximations for the large-system secrecy rates achievable

by RCI precoding in the MISO BCC. We also derive the optimal regularization parameter and the

optimal network load. We then propose a precoder based on RCI and power reduction (RCI-

PR) that significantly increases the performance for large network loads and large signal-to-noise

ratio (SNR). Finally, we extend our large-system analysis to the case of imperfect CSI in both

frequency division duplex (FDD) and time division duplex (TDD) systems, and to the case of

transmit channel correlation.

The work in this chapter is based on the papers listed in Section 1.3 as J2, J3, C2, and C3.

New Contributions:

• Theorem 4.1: We obtain a deterministic large-system approximation for the secrecy rates

achievable by RCI precoding in the BCC.

• Theorem 4.2: We derive the optimal regularization parameter of the RCI precoder that max-

imizes the large-system secrecy sum-rate in the BCC.

• Theorem 4.3: We obtain a compact expression for the large-system optimal secrecy sum-

rate achievable by RCI precoding in the BCC under unitary network load.

• Proposition 4.1: We derive the optimal network load that maximizes the secrecy sum-rate

in the limit of large SNR.

• Proposition 4.2: We derive the value of the SNR that maximizes the secrecy sum-rate for

network loads larger than one, as well as the corresponding maximum secrecy sum-rate.

• Subsection 4.2.2: We propose a linear precoder based on RCI and power reduction (RCI-

PR), which increases the performance for large SNR and network loads larger than one.

• Theorem 4.4: We provide high-SNR approximations for the secrecy sum-rate achievable by

the proposed RCI-PR precoder.

• Section 4.3: We compare the performance of the proposed RCI-PR precoder to two upper

bounds obtained without secrecy requirements and without interference, respectively. We

also compare the RCI-PR precoder to other linear precoding schemes of similar complexity.

• Theorem 4.5: We provide a deterministic approximation for the secrecy rates achievable by

RCI precoding in the BCC under imperfect CSI.


• Corollary 4.6: We obtain the minimum number of feedback bits required in an FDD system

in order to meet a given rate gap with the case of perfect CSI.

• Theorem 4.7: We study the optimal value of the training interval that maximizes the high-

SNR secrecy sum-rate in a TDD system.

• Theorem 4.8: We obtain a deterministic approximation for the secrecy rate achievable by

RCI precoding in the BCC under transmit channel correlation.

• Theorem 4.10: We study the optimal regularization parameter that maximizes the secrecy

sum-rate of RCI precoding in the BCC under transmit channel correlation.

Chapter 5

In Chapter 5 we introduce the MISO broadcast channel with confidential messages and external

eavesdroppers (BCCE), where a multi-antenna base station simultaneously communicates to mul-

tiple malicious users, in the presence of randomly located external eavesdroppers. We find that,

irrespective of the collusion strategy at the external eavesdroppers, a large number of transmit an-

tennas drives the probability of secrecy outage to zero. We finally show that increasing the density

of eavesdroppers by a factor n, requires n2 as many antennas to meet a given probability of secrecy

outage and a given mean secrecy rate.


New Contributions:

• Lemma 5.1: We derive the probability of secrecy outage for the RCI precoder in the BCCE

under non-colluding eavesdroppers.

• Theorem 5.2: We obtain a large-system approximation for the probability of secrecy outage

with RCI precoding in the BCCE under non-colluding eavesdroppers.

• Lemma 5.4: We derive the probability of secrecy outage in the BCCE caused by the external

eavesdropper nearest to the base station.


in the BCCE caused by the external eavesdropper nearest to the base station.

• Lemma 5.6: We derive the probability of secrecy outage for the RCI precoder in the BCCE

under colluding eavesdroppers.

1.2 Introduction 7


with RCI precoding in the BCCE under colluding eavesdroppers.

• Remark 5.1: We show that the collusion among eavesdroppers does not significantly affect

the number of transmit antennas required to meet a given probability of secrecy outage.

Moreover we show that increasing the density of eavesdroppers by a factor n requires n2 as

many antennas in order to meet a given probability of secrecy outage.

• Lemma 5.9: We derive the mean secrecy rate achievable by RCI precoding in the BCCE.

• Theorem 5.10: We derive a large-system approximation for the mean secrecy rate achievable

by RCI precoding in the BCCE.

• Remark 5.2: We show that, irrespective of the collusion strategy at the external eavesdrop-

pers, a large number of transmit antennas drives the probability of secrecy outage to zero.

• Lemma 5.12: We study the optimal regularization parameter of the RCI precoder in the

BCCE in two extreme cases.

Chapter 6

Chapter 6 considers physical layer security for the downlink of cellular networks. For this scenario,

we show that RCI precoding can achieve a non-zero secrecy rate with probability of outage smaller

than one. However we also show that unlike isolated cells, the secrecy rate in a cellular network

does not grow monotonically with the signal-to-noise ratio (SNR), and the network tends to be in

secrecy outage if the transmit power grows unbounded. We finally show that there is an optimal

value for the density of BSs that maximizes the secrecy rate, and this value is a decreasing function

of the transmit SNR.


New Contributions:

• Proposition 6.1: We obtain an expression for the secrecy rate achievable by RCI precoding

in the downlink of a cellular network.

• Lemma 6.2: We derive the Laplace transform of the inter-cell information leakage in a

cellular network.

• Theorem 6.3: We obtain an approximation for the probability of secrecy outage with RCI

precoding in a cellular network.


• Remark 6.1: We find that in cellular networks, RCI precoding can achieve confidential

communication with probability of secrecy outage smaller than one. However unlike an

isolated cell, cellular networks tend to be in secrecy outage w.p. 1 if the transmit power

grows unbounded, irrespective of the number of transmit antennas.

• Theorem 6.4: We obtain an approximation for the mean secrecy rate achievable by RCI

precoding in a cellular network.

• Remark 6.2: We find that in cellular networks, RCI precoding can achieve a non-zero

secrecy rate. However unlike an isolated cell, the secrecy rate in a cellular network is

interference-and-leakage-limited, and it cannot grow unbounded with the SNR, irrespective

of the number of transmit antennas.

• Remark 6.3: We find that in a cellular network with a fixed load, there is an optimal value

for the deployment density of BSs that maximizes the mean secrecy rate, and this value is a

decreasing function of the SNR.

• Corollary 6.5: We obtain a lower bound on the secrecy rate in a cellular network, which can

be calculated without knowledge of the probability density functions of inter-cell interfer-

ence and information leakage.

1.3 Introduction 9

1.3 Publications

The following is a list of publications in books, refereed journals, and conference proceedings pro-

duced during this Ph.D. candidature. In some cases the journal papers contain materials partially

presented in the conference papers.

Book Chapter

GERACI, G. AND J. YUAN,. “Physical Layer Security for Multi-User MIMO Communica-

tions.” in Recent Trends in Multi-User MIMO Communications, InTech Publisher, 2013.

Journal Papers

J1 GERACI, G., M. EGAN, J. YUAN, A. RAZI, AND I. B. COLLINGS “Secrecy sum-rates for

multi-user MIMO regularized channel inversion precoding.” IEEE Trans. on Communica-

tions, vol. 60, no. 11, pp. 3472–3482, Nov. 2012.

J2 GERACI, G., A. Y. AL-NAHARI, J. YUAN, AND I. B. COLLINGS “Linear precoding

for broadcast channels with confidential messages under transmit-side channel correlation.”

IEEE Communications Letters, vol. 17, no. 6, pp. 1164–1167, June 2013.

J3 GERACI, G., R. COUILLET, J. YUAN, M. DEBBAH, AND I. B. COLLINGS “Large system

analysis of linear precoding in MISO broadcast channels with confidential messages.” IEEE

Journal on Selected Areas in Communications, vol. 31, no. 9, pp. 1660–1671, Sep. 2013.

(Second prize of the 2012-2013 IEEE Australia Council Student Paper Contest).

J4 GERACI, G., S. SINGH, J. G. ANDREWS, J. YUAN, AND I. B. COLLINGS “Secrecy rates

in the broadcast channel with confidential messages and external eavesdroppers.” IEEE

Trans. on Wireless Communications, accepted for publication.

J5 GERACI, G., H. S. DHILLON, J. G. ANDREWS, J. YUAN, AND I. B. COLLINGS “Physical

layer security in downlink multi-antenna cellular networks.” IEEE Trans. on Communica-

tions, under review.

10 Publications 1.3

Conference Papers

C1 GERACI, G., J. YUAN, A. RAZI, AND I. B. COLLINGS “Secrecy sum-rates for multi-

user MIMO linear precoding.” In Proc. of IEEE Int. Symp. on Wireless Commun. Systems

(ISWCS’11). Aachen, Germany, Nov. 2011.

C2 GERACI, G., J. YUAN, AND I. B. COLLINGS “Large system analysis of the secrecy sum-

rates with regularized channel inversion precoding.” In Proc. of IEEE Wireless Commun.

Networking Conference (WCNC’12). Paris, France, Apr. 2012.

C3 GERACI, G., R. COUILLET, J. YUAN, M. DEBBAH, AND I. B. COLLINGS “Secrecy sum-

rates with regularized channel inversion precoding under imperfect CSI at the transmitter.”

In Proc. of IEEE Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP’13). Van-

couver, Canada, May 2013.

C4 GERACI, G., S. SINGH, J. G. ANDREWS, J. YUAN, AND I. B. COLLINGS “MIMO Multi-

user secrecy rate analysis.” In Proc. of IEEE Int. Conf. on Commun. (ICC’14). Sydney,

Australia, June 2014, accepted for publication.

C5 GERACI, G., H. S. DHILLON, J. G. ANDREWS, J. YUAN, AND I. B. COLLINGS “A

new model for physical layer security in cellular networks.” In Proc. of IEEE Int. Conf. on

Commun. (ICC’14). Sydney, Australia, June 2014, accepted for publication.

During the course of these Ph.D. studies, several other joint contributory papers were produced.

They are not included in this thesis in order to maintain focus, however the respective details can

be found in the list below.

Journal Papers

J6 YANG, N., G. GERACI, J. YUAN, AND R. MALANEY “Confidential broadcasting via

linear precoding in non-homogeneous MIMO multiuser networks.” IEEE Trans. on Com-

munications, under review.

J7 YAN, S., G. GERACI, N. YANG, R. MALANEY, AND J. YUAN “Optimization of code rates

in SISOME wiretap channels.” IEEE Trans. on Wireless Communications, under review.

1.3 Introduction 11

Conference Papers

C6 KOBAYASHI, M., S. YANG, AND G. GERACI “Time-correlated MISO wiretap channel

with delayed CSIT.” In Proc. of IEEE Int. Workshop on Signal Processing Advances in

Wireless Commun. (SPAWC’13). Darmstadt, Germany, June 2013.

C7 LIU, C., G. GERACI, N. YANG, J. YUAN, AND R. MALANEY “Beamforming for MIMO

Gaussian wiretap channels with imperfect channel state information.” To appear in Proc. of

IEEE Global Communications Conf. (Globecom’13). Atlanta GA, USA, Dec. 2013.

C8 YAN, S., G. GERACI, N. YANG, R. MALANEY, AND J. YUAN “On the target secrecy rate

for SISOME wiretap channels.” In Proc. of IEEE Int. Conf. on Commun. (ICC’14). Sydney,

Australia, June 2014, accepted for publication.

C9 LIU, C., N. YANG, G. GERACI, J. YUAN, AND R. MALANEY “Secrecy in MIMOME

wiretap channels: Beamforming with imperfect CSI.” In Proc. of IEEE Int. Conf. on Com-

mun. (ICC’14). Sydney, Australia, June 2014, accepted for publication.

Chapter 2

Overview of Physical Layer Security

Summary This chapter provides a brief overview of physical layer security, which is a tech-

nique that enables the exchange of confidential messages over a wireless medium in the presence

of unauthorized receivers, without relying on key-based encryption. We first introduce the notion

of physical layer security for the simple wiretap channel, and then extend our discussion to fading

channels, multi-antenna channels, and multi-user channels. This chapter does not provide new

results, and it is intended to give some background information necessary for the understanding

of the rest of the thesis.

13

14 The Wiretap Channel 2.2

2.1 Introduction to Physical Layer Security

D ue to the broadcast nature of wireless communications, it is essential to protect the transmit-

ted signals from unintended receivers. Security of data transmission has traditionally been

achieved via key-based cryptographic techniques at the network layer MASSEY [1988]. However,

in dynamic wireless networks this raises issues such as key distribution (for symmetric cryptogra-

phy), and high computational complexity (for asymmetric cryptography). In addition, key-based

cryptography is based on the unproven assumption that it is computationally infeasible to decipher

a message without knowledge of the secret key SCHNEIER [1998]. These vulnerabilities motivated

a new information-theoretic approach to security at the physical layer. The fundamental principle

behind physical layer security is to exploit the features of wireless channels, i.e., fading and noise,

to control the amount of information that can be extracted by an unauthorized receiver. This is

possible by appropriately designing coding and transmit precoding schemes, and does not rely

on a secret key nor assumes any limitations for the eavesdropper’s computational power. While

the first studies on physical layer security for noisy channels date back to the ’70s, this field has

experienced a renewed interest only in the last decade.

This thesis deals with physical layer security for multi-user systems. Although this is meant as a

self-contained piece of work, we here assume that the reader is familiar with fading channels and

multiple-input multiple-output (MIMO) technologies FOSCHINI AND GANS [1998]; TELATAR

[1999]. Due to the advances in wireless communications, this knowledge is now readily available

in textbooks RAPPAPORT [1996]; GOLDSMITH [2005]; TSE AND VISWANATH [2005]; VUCETIC

AND YUAN [2003]. Rather than repeating this standard knowledge, the main focus of this chapter

is to review various aspects of physical layer security in modern wireless networks.

2.2 The Wiretap Channel

The foundations of information-theoretic security were laid in SHANNON [1949]. For a simple

three-terminal network comprising a transmitter, a legitimate receiver, and an eavesdropper, Shan-

non introduced the notion of perfect secrecy, by requiring that the a posteriori probability of the

secret message computed by the eavesdropper be equal to the a priori probability of the message.

Shannon showed that in order to achieve perfect secrecy with a non-reusable private key, the key

must be at least as long as the message.

2.2 Overview of Physical Layer Security 15

Source Encoder Main

channel Decoder

Wiretapchannel

𝑆𝑘 𝑋𝑛 𝑌𝑛

𝑍𝑛

𝑆 𝑘

Figure 2.1: The wiretap channel in WYNER [1975], where the eavesdropper’s channel is a de-

graded version of the main channel.

Physical layer security for noisy channels was first proposed by Wyner, who introduced the wiretap

channel WYNER [1975]. As shown in Fig. 2.1, in the wiretap channel a confidential message X

is transmitted to the legitimate receiver over a discrete memoryless channel (main channel). The

legitimate receiver observes Y , which then passes through an additional channel (wiretap channel)

before being received by the eavesdropper as Z. Unlike Shannon, whose notion of perfect secrecy

required that the mutual information between transmitter and eavesdropper be zero regardless of

the block length, Wyner considered the block-length-normalized mutual information, and defined

the equivocation rate of the eavesdropper as

Re ≤H(Sk|Zn

)n

, (2.1)

where the conditional entropyH(Sk|Zn

)represents the equivocation. If the equivocation rateRe

is arbitrarily close to the information rate R, then R is the secrecy capacity of the wiretap channel.

By assuming that the transmitter-eavesdropper link is a probabilistically degraded version of the

main channel, Wyner constructed a coding scheme to hide the information in the additional noise

impairing the eavesdropper, by mapping each confidential message to many codewords according

to an appropriate probability distribution. This way, one can maximize the transmission rate R in

the main channel while making negligible the amount of information leaked to the eavesdropper.

In LEUNG-YAN-CHEONG AND HELLMAN [1978], the authors considered the degraded wiretap

channel with additive Gaussian noise. They showed that the secrecy capacity Cs is given by

Cs = Cm − Cmw, (2.2)

16 Fading Channels 2.3

where Cm and Cmw are the capacities of the main channel and of the overall wiretap channel,

respectively. Therefore, a non-zero secrecy capacity can only be obtained if the eavesdropper’s

overall wiretap channel is worse than the legitimate receiver’s main channel.

A non-degraded version of Wyner’s wiretap channel was considered in CSISZAR AND KORNER

[1978], where the authors characterized the secrecy capacity as

Cs = maxV→X→Y Z

I(V ;Y )− I(V ;Z). (2.3)

The secrecy capacity is achieved by maximizing over all joint probability distributions such that a

Markov chain V , X , Y Z is formed, where V is an auxiliary input variable.

2.3 Fading Channels

Early work on physical layer security assumed non-fading channels and perfect channel state

information at the transmitter. More recently, the research community has considered channel

fading in wiretap channels as well as limited channel state information. It is usually assumed that at

least the statistics of the eavesdropper’s channel are known to the transmitter. This allows to define

secrecy rate outage metrics similarly to the conventional rate outage metrics. For instance, the

secrecy outage probability is the probability that the instantaneous secrecy rate falls below a given

threshold. The secrecy outage probability of slow fading channels was analyzed in BARROS AND

RODRIGUES [2006]; BLOCH et al. [2006], where the authors showed that secret communication is

possible even if the eavesdropper has a better average SNR than the one of the legitimate receiver.

In the special case when only the eavesdropper’s channel is affected by fading, and when this

fading is unknown, a transmission scheme was proposed in LI et al. [2007b] to achieve a positive

secrecy rate even when the main channel is arbitrarily worse than the eavesdropper’s average

channel. The case when no information at all is available about the eavesdropper’s channel was

studied, among others, in HE AND YENER [2010]; SWINDLEHURST [2009]; XIE AND ULUKUS

[2013]. For this case, a coding scheme that hides the secure message across different fading

states was proposed in GOPALA et al. [2008]. A different approach to this problem was taken

in LIANG et al. [2009a] by studying the compound wiretap channel, i.e., by characterizing the

eavesdropper’s channel with a finite set of states, and guaranteeing secure communication under

any state that may occur.


𝑁𝑡

Eavesdropper

Transmitter

𝑁𝑟

𝑁𝑒

Legitimate receiver

Figure 2.2: The general MIMOME wiretap channel.

2.4 Multi-antenna Channels

When the transmitter is equipped with multiple antennas, it is possible to exploit the available

spatial dimensions to increase the secrecy rates of wireless channels. In a multiple-input multiple-

output (MIMO) system where the transmitter, legitimate receiver, and eavesdropper are equipped

with Nt, Nr, and Ne antennas respectively as in Fig. 2.2, the signals received by the legitimate

receiver and eavesdropper can be written as

yb = Hbxa + nb

ye = Hexa + ne

(2.4)

where xa ∈ CNt×1 is the transmit signal with covariance matrix E[xax†a] = Qx and average

power constraint trQx ≤ P , Hb ∈ CNt×Nr and He ∈ CNe×Nr are the Gaussian channel

matrices at the legitimate receiver and eavesdropper, respectively, and nb and ne are the respective

Gaussian noise vectors. Physical layer security in MIMO channels was first considered in HERO

[2003], which examined the utility of space-time block coding.

The secrecy capacity of the degraded single-input multiple-output (SIMO) wiretap channel was

obtained in PARADA AND BLAHUT [2005], whereas LI et al. [2007a]; SHAFIEE AND ULUKUS

[2007] studied the multiple-input single-output (MISO) case. The general MIMO case was consid-

ered in KHISTI et al. [2007]; OGGIER AND HASSIBI [2008] which introduced the multiple-input

multiple-output multiple-eavesdropper (MIMOME) wiretap channel. In KHISTI et al. [2007] the

18 Multi-user Channels 2.5

authors showed that when the eavesdropper’s CSI is known, a transmit precoder based upon the

generalized singular value decomposition (GSVD) of the pencil (Hb,He) is optimal in the high-

SNR regime. For the special case of Nr = 1 (MISOME), the optimal transmit beamformer was

shown to be the generalized eigenvector ψm corresponding to the largest generalized eigenvalue

λm of

h†bhbψm = λmH†eHeψm. (2.5)

A few contributions in the literature have also studied the MIMO wiretap channel under a more

general matrix power-covariance constraint LIU AND SHAMAI [2009]; BUSTIN et al. [2009];

FAKOORIAN AND SWINDLEHURST [2013].

The transmission of artificial noise was proposed as strategy to achieve secrecy in the case when

only the statistics of He are known to the transmitter NEGI AND GOEL [2005]; GOEL AND NEGI

[2008]. The artificial noise is transmitted with the information signal, and it is usually designed

to be orthogonal to the legitimate receiver, i.e., in the nullspace of Hb, such that only the eaves-

dropper’s channel is degraded. Although this is a common design choice made for the sake of

simplicity, in general it is not the optimal approach in terms of secrecy rate. An optimal power

allocation and beamforming method for the artificial noise strategy were presented in ZHOU AND

MCKAY [2009].

2.5 Multi-user Channels

The principles of physical layer security can be extended to multi-user networks with more than

two receivers, i.e., broadcast channels. We can distinguish the two following cases: (i) all the users

are trusted, and the confidential messages must be protected only from external eavesdroppers, and

(ii) each message must be kept confidential from all other unintended users, i.e., each user is seen

as an eavesdropper for messages not intended for it, and it is therefore denoted a malicious user.

While in the first case one can still apply most schemes proposed for the single-user scenario, in

the second case new transmission techniques must be designed.

The first case, where one transmitter wants to communicate with several legitimate users in the

presence of an external eavesdropper, was considered, among others, in KHISTI et al. [2008];

BAGHERIKARAM et al. [2013]; EKREM AND ULUKUS [2011]; LIU et al. [2010b]. The second

case, where two confidential messages are sent to two receivers, and each receiver acts as an


eavesdropper for the other one, was studied in LIU et al. [2008]; LIU AND POOR [2009]. It

was shown in LIU et al. [2010a] that, under the matrix input power-covariance constraint, both

confidential messages can be simultaneously communicated at their respected maximum secrecy

rates by using dirty paper coding.

Although one can find several papers that study the MISOME channel with multiple eavesdrop-

pers, it should be noted that in the MISOME channel no messages are to be delivered to the

eavesdroppers, since they are not users. Other papers have studied the case of malicious users,

which are users acting as eavesdroppers. However, most of the prior art has focused on physical

layer security for systems with up to two malicious users only. Determining the achievable secrecy

rates for multi-user networks where any number of users are potentially eavesdropping remains

an open problem. Moreover, the performance of practical transmission schemes in such generic

multi-user networks is also unknown. Hence in this thesis we study physical layer security for

generic multi-user multi-antenna systems.

Chapter 3

Physical Layer Security in Isolated

Cells: Achievable Rates

Summary This chapter studies the multiple-input single-output (MISO) broadcast channel with

confidential messages (BCC). We consider the case where a multi-antenna base station simultane-

ously transmits independent confidential messages to several single-antenna users via regularized

channel inversion (RCI) precoding in the presence of Rayleigh fading. We derive expressions for

the secrecy rates achievable by RCI precoding, and we show that RCI precoding outperforms plain

channel inversion precoding. We then propose an algorithm to jointly optimize the regularization

parameter of the RCI precoder and the power allocation vector, and show that optimal power

allocation increases the secrecy sum-rate compared to equal power allocation. Finally, we derive

the achievable secrecy rates for more practical scenarios where only imperfect channel state in-

formation (CSI) is available at the transmitter, and where channel correlation is present among

the transmit antenna elements.

21

22 The Broadcast Channel with Confidential Messages 3.1

3.1 The Broadcast Channel with Confidential Messages

W e consider the downlink of a narrowband multi-user MISO system, consisting of a base

station (BS) with N antennas which simultaneously transmits K independent confidential

messages to K spatially dispersed single-antenna users, as depicted in Fig. 3.1. In this model,

denoted as the MISO broadcast channel with confidential messages (BCC), transmission takes

place over a block fading channel, and the transmitted signal is x = [x1, . . . , xN ]T ∈ CN×1. We

assume homogeneous users, i.e., each user experiences the same received signal power on average,

thus the model assumes that their distances from the transmitter are the same and unitary.

Note that despite this assumption, which is necessary to maintain tractability, the simplified model

captures all the key characteristics of the broadcast channels that affect physical layer security,

as discussed in the sequel. Our analysis can be extended to the scenario where the mobile users

have different distances from the serving BS. Results on this scenario can be found in YANG et al.

[2013].

The received signal at user k is given by

yk =N∑j=1

hk,jxj + nk (3.1)

where hk,j ∼ CN (0, 1) is the independent and identically distributed (i.i.d.) Rayleigh fading

channel between the jth transmit antenna element and the kth user, and nk ∼ CN (0, σ2) is the

noise seen at the kth receiver. The corresponding vector equation is

y = Hx + n (3.2)

where H = [h1, . . . ,hK ]† is the K ×N channel matrix. We assume E[nn†] = σ2IK , where IK

is the K × K identity matrix, define the transmit signal-to-noise ratio (SNR) as ρ , 1/σ2, and

impose the long-term power constraint E[‖x‖2] = 1.

It is required that the BS securely transmits each confidential message, ensuring that the unin-

tended users receive no information. This is performed at the secrecy rate R, defined as follows.

Let P(En) be the probability of error at the intended user, m be a confidential message, yne be the

vector of all signals received by the unintended users, and H(m|yne ) be the corresponding equivo-

cation. Then a (weak) secrecy rate R for the intended user is achievable if there exists a sequence

3.2 Physical Layer Security in Isolated Cells: Achievable Rates 23

𝑁

Transmitter

User 1

User 2

User K

Figure 3.1: The MISO broadcast channel with confidential messages (BCC).

of (2nR, n) codes such that P(En)→ 0 and 1nH(m|yne ) ≤ 1

nH(m)−εn with εn approaching zero

as n→∞ KHISTI AND WORNELL [2010].

For each user k, we denote byMk = 1, . . . , k − 1, k + 1, . . . ,K the set of remaining users. In

general, the behavior of the users cannot be determined by the BS. As a worst-case scenario, we

assume that for each user k, all users inMk can cooperate to jointly eavesdrop on the kth message.

This assumption is reasonable, because the confidentiality of the messages must be ensured in all

cases, including the worst case. Since the set of malicious usersMk can perform joint processing,

they can be seen as a single equivalent malicious user Mk with K − 1 receive antennas. Due

to the assumption of cooperating malicious users, interference cancellation can be performed at

Mk, which does not see any undesired signal term apart from the received noise. It will be shown

that despite this conservative but necessary assumption, a properly designed linear precoder can

achieve a per-user secrecy rate which is close to an upper bound on the secrecy capacity. Since

the average secrecy sum-rate is simply given by the average per-user secrecy rate multiplied by

number of users, our analysis is suitable for both individual secrecy rate and secrecy sum-rate, and

a properly designed linear precoder performs well also in terms of the secrecy sum-rate.

3.2 Achievable Secrecy Rates in the BCC

In this section, we derive achievable secrecy rates for the MISO BCC by using a linear precoder.

Although suboptimal, linear precoding schemes are of particular interest because of their low-

24 Achievable Secrecy Rates in the BCC 3.2

complexity implementations and because they can control the amount of crosstalk between the

users YOO AND GOLDSMITH [2006]; SPENCER et al. [2004b]; PEEL et al. [2005]; JOHAM et al.

[2005]. We then specialize and obtain the secrecy rates achievable by the regularized channel

inversion (RCI) precoder. RCI is a linear precoding scheme that was proposed to serve multiple

users in the MISO broadcast channel (BC). RCI precoding has better performance than plain

channel inversion, especially at low SNR PEEL et al. [2005].

3.2.1 Linear Precoding

In linear precoding, the transmitted vector x is derived from the vector containing the confidential

messages u = [u1, . . . , uK ]T through a deterministic linear transformation (precoding) YOO AND

GOLDSMITH [2006]; SPENCER et al. [2004b]; PEEL et al. [2005]; JOHAM et al. [2005]. We

assume that the entries of u are chosen independently, satisfying E[|uk|2] = 1, ∀k.

Let W = [w1, . . . ,wK ] be theN ×K precoding matrix, where wk is the kth column of W. Then

the transmitted signal and the power constraint are, respectively:

x = Wu =

K∑k=1

wkuk, (3.3)

E[‖x‖2

]= E

[‖Wu‖2

]=

K∑k=1

‖wk‖2 = 1. (3.4)

By employing linear precoding as in (3.3), the signals observed at the legitimate user k and at the

equivalent malicious user Mk are, respectively

yk = h†kwkuk +∑j 6=k

h†kwjuj + nk

yM,k =∑j

Hkwjuj + nk

(3.5)

where nk = [n1, . . . , nk−1, nk+1, . . . , nK ]T , h†k is the kth row of H, and Hk is a matrix obtained

from H by eliminating the kth row. The channel in (3.5) is a multi-input, single-output, multi-

eavesdropper (MISOME) wiretap channel KHISTI AND WORNELL [2010]. The transmitter, the

intended receiver and the eavesdropper of this MISOME wiretap channel are equipped with N , 1

andK−1 virtual antennas, respectively. Due to the simultaneous transmission of theK messages,

user k experiences noise and interference from all the uj , j 6= k.


3.2.2 Achievable Secrecy Rates with Linear Precoding

In the following, we derive an achievable secrecy sum-rate SBCC for the MISO BCC. Although

the design of codes for the MISO BCC is not the focus of this thesis, we prove the achievability of

SBCC with a code construction based on independent codebooks and linear precoding.

Lemma 3.1. An achievable secrecy sum-rate SBCC for the MISO BCC is given by

SBCC4=

K∑k=1

RBCC,k, (3.6)

where RBCC,k is an achievable secrecy rate for the kth MISOME wiretap channel in (3.5), k =

1, . . . ,K.

Proof. Assume that the BS uses independent codebooks for each user, where each codebook is

a code for the scalar wiretap channel KHISTI AND WORNELL [2010]. The confidential message

uk is obtained as a codeword independently drawn from the code Ck, corresponding to the kth

user. The rate RBCC,k of the code Ck is chosen according to the secrecy rate achievable for user

k in the presence of the equivalent malicious user Mk, i.e., by the secrecy rate achievable for the

MISOME wiretap channel in (3.5). The existence of such code is guaranteed by the definition

of secrecy rate CSISZAR AND KORNER [1978]. To construct the vector codeword for the MISO

BCC, the scalar codewords for each MISOME wiretap channel are stacked according to u =

[u1, . . . , uK ]T , and no additional binning is required. The vector u is then linearly precoded as in

(3.3), which means that each message uk is transmitted by beamforming, i.e., by signaling with

rank one covariance, along the direction wk. The secrecy sum-rate SBCC is then by definition the

sum of the simultaneously achievable secrecy rates RBCC,k.

Lemma 3.2. An achievable secrecy rate for the MISOME wiretap channel in (3.5) is given by

RBCC,k =[log2

(1 + γk

)− log2

(1 + γM,k

)]+, (3.7)

where γk and γM,k are the signal-to-interference-plus-noise ratios (SINR) for the message uk at

the legitimate receiver k and at the equivalent malicious userMk, respectively, and where we have

used the notation [x]+ , max (x, 0).


Proof. By noting that the MISOME wiretap channel in (3.5) is a nondegraded broadcast channel

KHISTI AND WORNELL [2010], the secrecy capacity is given by CSISZAR AND KORNER [1978]:

Cs = maxuk→wkuk→yk,yM,k

I(uk; yk

)− I(uk; yM,k

)(3.8)

where I(x; y) denotes mutual information between two random variables x and y. The secrecy

capacity Cs is given by the difference of the mutual informations at the intended user and at the

eavesdropper, respectively. Cs is achieved by maximizing over all joint probability distributions

such that a Markov chain uk → wkuk → yk,yM,k is formed, where uk is an auxiliary input

variable. By evaluating (3.8) with uk ∼ CN (0, 1) and with the linearly precoded data wkuk, we

obtain an achievable secrecy rate for the MISOME wiretap channel (3.5) given by

RBCC,k =[I(uk; yk

)− I(uk; yM,k

)]+(3.9)

(a)=[I(wkuk; yk

)− I(wkuk; yM,k

)]+, (3.10)

where (a) follows from wkuk being a deterministic function of uk KHISTI AND WORNELL

[2010]. Equation (3.7) then follows from (3.10) and from the statistics of uk.

From equation (3.7) it is clearly observed that for high-performance linear precoder design an

efficient tradeoff between maximizing γk and minimizing γM,k is required.

Theorem 3.3. A secrecy sum-rate achievable by linear precoding in the MISO BCC is given by

SBCC =K∑k=1

log2

1 +

∣∣∣h†kwk∣∣∣2σ2+

∑j 6=k

∣∣∣h†kwj∣∣∣21 + ‖Hkwk‖2

σ2

+

. (3.11)

Proof. By using Lemma 3.1 and Lemma 3.2, we have that an achievable secrecy sum-rate is

obtained as the sum of the secrecy rates RBCC,k in (3.7). A lower bound on the quantities RBCC,k

can be obtained by considering a genie-aided equivalent malicious user Mk which observes not

only the signals yM,k received by its K − 1 antennas, but also all the confidential messages

uj , j 6= k. Such channel clearly has an achievable secrecy rate smaller than the original channel.

The genie-aided equivalent malicious user Mk can perform interference cancellation, and it does

not see any undesired signal term apart from the received noise nk.


According to the previous considerations, the signals at the legitimate user and at the eavesdropper

of the kth equivalent MISOME wiretap channel become, respectively:

yk = h†kwkuk +∑j 6=k

h†kwjuj + nk,

yM,k = Hkwkuk + nk.

(3.12)

For the kth equivalent MISOME wiretap channel in (3.12), the SINRs at the legitimate user and

the eavesdropper are, respectively:

γk =

∣∣∣h†kwk

∣∣∣2σ2 +

∑j 6=k

∣∣∣h†kwj

∣∣∣2 , (3.13)

γM,k =‖Hkwk‖2

σ2. (3.14)

Since the noise in yM,k in (3.12) is spatially white, the optimal receive filter at Mk is the matched

filter (Hkwk)†. Equation (3.14) then follows. For a given channel H, substituting (3.13) and

(3.14) into (3.7) and then into (3.6) yields (3.11).

In the remainder of this chapter, we refer to equation (3.11) as the secrecy sum-rate. We note that

the secrecy sum-rate depends on the choice of the precoding matrix W, as well as on the channel

H and the noise variance σ2. A possible choice for W, based on regularized channel inversion, is

discussed in the following.

3.2.3 Regularized Channel Inversion Precoding

We now consider RCI precoding for the MISO BCC. Although plain channel inversion (CI) pre-

coding can achieve secrecy by canceling all signals leaked at the unintended users (provided that

N ≥ K), this comes at the cost of a poor sum-rate. The RCI precoder has better performance

than plain CI, particularly at low SNR PEEL et al. [2005]. For each message uk, RCI precod-

ing achieves a tradeoff between the signal power at the kth legitimate user and the crosstalk at the

other (K−1) unintended users for each signal. The crosstalk causes interference to the unintended

users. In the case when the unintended users are acting maliciously, the crosstalk also causes in-

formation leakage. Therefore, RCI achieves a tradeoff between signal power, interference, and

information leakage.


With RCI precoding, linear processing exploiting regularization is applied to the vector of mes-

sages u PEEL et al. [2005]. The RCI precoding matrix is given by

W =1√ζH†(HH† +NξIK

)−1=

1√ζ

(H†H +NξIN

)−1H† (3.15)

where

ζ = tr

H†H

(H†H +NξIK

)−2

(3.16)

is the power normalization constant. The transmitted signal x after RCI precoding can be written

as

x = Wu =1√ζH†(HH† +NξIK

)−1u (3.17)

=1√ζ

(H†H +NξIN

)−1H†u. (3.18)

The latter passes through the channel, producing the vector of received signals

y =1√ζH(H†H +NξIK

)−1H†u + n. (3.19)

The function of the regularization parameter ξ ∈ R is to improve the behavior of the inverse,

although it also produces non-zero crosstalk terms in (3.19). We note that when the regularization

parameter is zero, i.e., ξ = 0, the RCI precoder reduces to a CI precoder, which is therefore

a special case of RCI. As a result, an RCI precoder obtained by optimizing the regularization

parameter ξ will achieve a secrecy sum-rate greater or equal to the one achieved by the CI precoder,

i.e., in the special case of ξ = 0.

Using RCI precoding, the SINRs (3.13) and (3.14) at the legitimate user k and at the equivalent

malicious user Mk become, respectively

γk =

∣∣∣h†k (H†H +NξIK)−1

hk

∣∣∣2ζσ2 +

∑j 6=k

∣∣∣h†k (H†H +NξIK)−1

hj

∣∣∣2 , (3.20)

γM,k =

∥∥∥Hk

(H†H +NξIK

)−1hk

∥∥∥2

ζσ2. (3.21)

To rewrite (3.20) and (3.21) in a more compact way, we introduce the quantities

Ak = h†k

(H†kHk +NξIK

)−1hk (3.22)

and

Bk = h†k

(H†kHk +NξIK

)−1H†kHk

(H†kHk +NξIK

)−1hk. (3.23)


It is then possible to express (3.20) as NGUYEN et al. [2009]

γk =A2k

Bk + ζσ2 (1 +Ak)2 . (3.24)

In a similar fashion, we rewrite (3.21) as

γM,k =Bk

ζσ2 (1 +Ak)2 . (3.25)

We then obtain the following expression for the secrecy sum-rates achievable with RCI precoding.

Corollary 3.4. A secrecy sum-rate achievable by RCI precoding in the MISO BCC is given by

SBCC =

K∑k=1

RBCC,k (3.26)

where RBCC,k is the achievable secrecy rate at user k, given by

RBCC,k =

log2

1 +A2k

Bk+ζσ2(1+Ak)2

1 + Bkζσ2(1+Ak)2

+

. (3.27)

Proof. Corollary 3.4 follows by substituting (3.24) and (3.25) into (3.7) and then into (3.6).

3.2.4 Numerical Results

We now show the performance of RCI precoding in the BCC via simulations. The precoding

matrix W is normalized by√ζ, as in (3.15), in order to meet the power constraint in (3.4). This

corresponds to a long-term power constraint, which does not require the receivers to know the

instantaneous value of ζ PEEL et al. [2005].

Figure 3.2 shows the simulated secrecy sum-rate SBCC of the RCI precoder in the BCC, versus

the SNR ρ, for various values of the number of transmit antennas N and users K, with N = K.

The value of SBCC was averaged over 103 channels, and it was calculated by using the optimal

regularization parameter ξ that maximizes the average secrecy sum-rate. This optimal value of ξ

was obtained by linear search.

In Fig. 3.3 we compare the simulated secrecy sum-rate of the RCI precoder in the MISO BCC

to the one of the plain CI precoder YOO AND GOLDSMITH [2006]. The sum-rate SBC of the

30 Optimal Regularization Parameter and Power Allocation 3.3

0 5 10 15 20 250

10

20

30

40

50

60

70

80

SNR, ρ [dB]

Secrecy

sum-rate

N = K = 4

N = K = 8

N = K = 16

N = K = 32

Figure 3.2: Simulated secrecy sum-rate SBCC achievable by RCI precoding in the BCC versus

SNR ρ, for various values of N = K.

optimal RCI precoder in the MISO BC without secrecy requirements is also plotted. The figure

shows plots for K = 4, 8, 16, 32. We observe that CI precoding exhibits a large performance

loss compared to the secrecy sum-rate of the optimal RCI precoder, especially for large values of

K. We note that although CI precoding achieves secrecy in a simple way by completely canceling

the information leakage, this comes at the cost of a poor sum-rate. Secrecy can be achieved with

a significantly larger sum-rate by using the RCI precoder. We also observe that the secrecy loss,

i.e., the gap between the sum-rate of the RCI precoder without secrecy and the secrecy sum-rate

of the RCI precoder, is almost constant at high SNR for large K. This means that the secrecy

requirements do not affect the multiplexing gain.

3.3 Optimal Regularization Parameter and Power Allocation

In this section, we consider power allocation for the RCI precoder. We first propose a new algo-

rithm to obtain the power allocation vector p which achieves the optimal secrecy sum-rate with a

fixed regularization parameter ξ. We then extend our algorithm to jointly optimize p and ξ.


0 5 10 15 20 250

5

10

15

20

25

SNR, ρ [dB]

Secrecy

sum-rate

RCI without secrecy

RCI with secrecy

CI with secrecy

(a) K = 4

0 5 10 15 20 250

5

10

15

20

25

30

35

40

SNR, ρ [dB]

Secrecy

sum-rate

RCI without secrecy

RCI with secrecy

CI with secrecy

(b) K = 8

0 5 10 15 20 250

10

20

30

40

50

60

70

80

SNR, ρ [dB]

Secrecy

sum-rate

RCI without secrecy

RCI with secrecy

CI with secrecy

(c) K = 16

0 5 10 15 20 250

50

100

150

SNR, ρ [dB]

Secrecy

sum-rate

RCI without secrecy

RCI with secrecy

CI with secrecy

(d) K = 32

Figure 3.3: Comparison between the RCI precoder and the plain CI precoder, for various values

of N = K. The secrecy loss is also shown as the gap between dashed and solid lines.

3.3.1 Regularized Channel Inversion Precoding with Power Allocation

We consider the RCI precoding matrix with arbitrary power allocation given by

Wp = WD =1√ζH†(HH† +NξIK)−1D, (3.28)

where D = diag(√

p), and p = [p1, . . . , pK ]T is the power allocation vector. The vector p must

be chosen such that the power constraint tr

W†pWp

= 1 is met. Clearly, (3.28) generalizes the

RCI precoder W with equal power allocation (RCI-EP) in (3.15).

Clearly, (3.28) generalizes the RCI precoder W in (3.15). The columns of W will in general have

different norms, and hence each user will be allocated a different power level. However, in order

to make a distinction between the general case Wp, where the BS can choose the power allocation


matrix D, and the special case W, where it is implicitly assumed that D = I, in the following we

will denote W as the RCI precoder with equal power allocation (RCI-EP).

When the precoder Wp is used, the SINR at the kth legitimate user, given by (3.13), becomes

γk =pk|h†kwk|2∑

j 6=k pj |h†kwj |2 + σ2

, (3.29)

and the SINR at the equivalent malicious user Mk, given by (3.14), becomes

γM,k =pk‖Hkwk‖2

σ2=pk∑

j 6=k |h†jwk|2

σ2. (3.30)

From (3.29) and (3.30), we obtain the achievable secrecy sum-rate with power allocation

SpBCC =K∑k=1

[log2

(1 +

pk|h†kwk|2∑j 6=k pj |h

†kwj |2 + σ2

)− log2

(1 +

pk∑

j 6=k |h†jwk|2

σ2

)]+

. (3.31)

3.3.2 Optimal Power Allocation

To obtain the optimal power allocation vector p, we are required to solve the non-convex optimiz-

ation problem

maximizep

SpBCC(p)

subject to tr

W†pWp

≤ 1,

(3.32)

where SpBCC(p) is given by (3.31), Wp is given by (3.28), and the maximum total transmit power

over all antennas is one. In the following, we will ignore the notation [·]+ in (3.31) in the maxi-

mization problem. In fact, any negative term in the sum can be replaced by zero (thus increasing

the sum) by using pk = 0 which is always feasible.

We now reformulate the problem (3.32) by applying the transformation pk = log pk, k = 1 . . . ,K,

and obtain the optimization problem

maximizep

SpBCC(p)

subject to tr

W†pWp

≤ 1,

(3.33)

where p = [p1, . . . , pK ]T .

Lemma 3.5. The quantity l = − log2(1 + γM,k), with γM,k as in (3.30), is concave.


Proof. The quantity l = − log2(1 + γM,k) and its first and second derivatives are

l = − log2 (1 + γM,k) = − log2

(1 +

epk∑

j 6=k |h†jwk|2

σ2

),

∂l

∂pk=−∂ log2 (1 + γM,k)

∂pk=−(log2 e)e

pk∑

j 6=k |h†jwk|2

σ2 + epk∑

j 6=k |h†jwk|2

,

∂2l

∂p2k

=−∂2 log2 (1 + γM,k)

∂p2k

=−(log2 e)e

pk∑

j 6=k |h†jwk|2σ2(

σ2 + epk∑

j 6=k |h†jwk|2

)2 ≤ 0.

(3.34)

Hence by the second order condition, l is concave.

In order to solve the problem (3.33), we consider a modified version of the method as in PAPAN-

DRIOPOULOS et al. [2008] and SUNG AND COLLINGS [2010] which is based on a reformulation

of (3.33). This approach guarantees an improvement in the performance over the standard high-

SNR approximation in fading channels SUNG AND COLLINGS [2010]. In order to obtain the

reformulation, we use the following bound obtained in PAPANDRIOPOULOS et al. [2008]

a log z + b ≤ log(1 + z),

a =z0

1 + z0and b = log(1 + z0)− z0

1 + z0log z0,

(3.35)

for some z0 ≥ 0, with equality when z = z0.

Lemma 3.6. With the change of variables pk = log pk, k = 1 . . . ,K, the lower bound

aklog 2

log

(epk |h†kwk|2∑

j 6=k epj |h†kwj |2 + σ2

)+

bklog 2

≤ log2

(1 +

pk|h†kwk|2∑j 6=k pj |h

†kwj |2 + σ2

),

(3.36)

is concave in pk, k = 1, . . . ,K.

Proof. The result follows immediately using the method in Lemma 3.5.

We showed in Lemma 3.5 that the second term of the sum in (3.31) is concave by the second order

condition. By using the lower bound in (3.36) for the first term of (3.31), we obtain a concave

objective function. Since the constraints are affine, the optimization problem arising from (3.33)

and the bound (3.36) is a convex optimization problem. This convex optimization problem is given


by

maximizep

K∑k=1

[ak

log 2log

(epk |h†kwk|2∑

j 6=k epj |h†kwj |2 + σ2

)

+bk

log 2− log2

(1 +

epk∑

j 6=k |h†jwk|2

σ2

)]subject to tr

W†

pWp

≤ 1

(3.37)

and it can be solved by bisection method.

The power allocation vector can then be obtained using Algorithm 1 in Table 3.1. To show that

Algorithm 1 converges monotonically to a local optimum, we note that the constraint is the same

for both the tth and (t + 1)th subproblems. Hence, the solution of the tth subproblem (3.37) is

also feasible for the (t + 1)th subproblem (3.37). Moreover, by the bound in (3.35), the objective

function is monotonically increasing and converges to a local optimum.

Table 3.1: Algorithm for optimal power allocation.

Algorithm 1

Initialize iteration counter t = 0

Initialize all a(t)k = 1, b(t)k = 0

repeat

Solve (3.37) to obtain p(t)

Update a(t)k , b

(t)k at z0 = γk(p

(t))

Increment t

until convergence

Obtain pk = epk , k = 1, . . . ,K

3.3.3 Joint Optimal Power Allocation and Regularization Parameter

Having established an algorithm to determine the optimal power allocation vector p for a fixed ξ,

we now consider the joint optimization of ξ and p. The joint optimization problem can be written


as

maximizep,ξ

SpBCC(p, ξ)

subject to tr

W†pWp

≤ 1.

(3.38)

Even after using the transformation pk = log pk, k = 1, . . . ,K, the problem (3.38) is non-convex.

To solve this problem, we propose Algorithm 2 in Table 3.2.

At each iteration, Algorithm 2 optimizes the regularization parameter ξ and subsequently the

power allocation vector p. It is straightforward to prove that Algorithm 2 converges monoton-

ically and it thus provides a locally optimal pair (ξ,p) for the RCI precoder. We now show via

simulations that the RCI precoder with jointly optimal regularization parameter and power alloca-

tion vector outperforms RCI precoding with equal power allocation (RCI-EP).

Table 3.2: Algorithm for joint optimal power allocation and regularization parameter.

Algorithm 2

Initialize iteration counter t1 = 0, t2 = 0

Initialize pk = 1/ζ, and set pk = log pk, k = 1, . . . ,K

Initialize ξ0 = 1/ρ

repeat

Increment t1

Obtain the optimal ξ?t1 using steepest descent with ξt1−1 as initial point

Initialize all a(t2)k = 1, b(t2)

k = 0

repeat

Solve (3.37) to obtain p(t2)

Update a(t2)k , b

(t2)k at z0 = γk(ξ

?t1 , p

(t2))

Increment t2

until convergence

Set p = p(t2)

until convergence

Obtain pk = epk , k = 1, . . . ,K

36 Achievable Secrecy Rates in Practical Channels 3.4

In Fig. 3.4 we compare the simulated per-user secrecy rate of the RCI precoder with jointly op-

timized regularization parameter ξ and power allocation vector p to the RCI precoder with sep-

arately optimized ξ and p, and to the RCI precoder with equal power allocation (RCI-EP). We

observe that there is a negligible performance difference between the joint optimization and the

separate optimization. As a result, a low-complexity, near-optimal RCI precoder may be imple-

mented by optimizing the regularization parameter and the power allocation vector separately.

The figure shows that for K = 4, the RCI precoder with optimal power allocation outperforms

the RCI-EP precoder by up to 20%. Figure 3.4 also shows that optimal power allocation reduces

the sum-rate loss due to the secrecy requirements. For an SNR ρ ≥ 15dB, RCI with power al-

location achieves a per-user secrecy rate which is even higher than the per-user rate achieved by

the optimal RCI-EP without secrecy requirements. Furthermore, Fig. 3.4 shows the simulated se-

crecy capacity CMISOME of a MISOME channel with the same per-message transmitted power.

Although CMISOME is obtained in a single-user and interference-free system KHISTI AND WOR-

NELL [2010], at high SNR, RCI precoding with power allocation achieves a per-user secrecy rate

as large as CMISOME. It is not very surprising that the per-user secrecy rate of the RCI precoder

with optimal power allocation is close to the capacity of the MISOME channel, since the RCI

precoder with optimal power allocation should be close to the optimal linear precoder.

3.4 Achievable Secrecy Rates in Practical Channels

3.4.1 Secrecy Rates in the Presence of Imperfect Channel State Information

In the previous sections, we studied the secrecy rates achievable by RCI precoding for the case

when the transmitter has perfect channel state information (CSI). In this section, we consider a

more realistic scenario where only an estimation of the channel is available at the transmitter.

We model the relation between the channel H and the estimated channel H as

H = H + E (3.39)

where the matrix E represents the channel estimation error, and it is independent from H. The

entries of H and E are i.i.d. complex Gaussian random variables with zero mean and variances

1 − τ2 and τ2, respectively. The value of τ ∈ [0, 1] depends on the quality and technique used


0 5 10 15 20 250

1

2

3

4

5

6

SNR, ρ [dB]

Per-usersecrecyrate

CMISOME

RCI-EP w/o secrecy

RCI w joint optim.

RCI w separate optim.

RCI-EP

Figure 3.4: Per-user secrecy rate vs. SNR ρ for K = 4 users: with separate (dashed) and joint

(circle) optimization of ξ and p, and with equal power allocation (solid). The rate of the RCI-EP

precoder without secrecy requirements (square) and the secrecy capacity of the MISOME channel

(diamond) are also plotted.


for channel estimation, and it is the same for all users. When τ = 0 the CSI is perfectly known,

whereas τ = 1 corresponds to the case when no CSI is available at all.

The transmitter uses the knowledge of H to obtain the RCI precoding matrix W, given by

W =1√ζ

H†(HH† +NξIK

)−1

=1√ζ

(H†H +NξIN

)−1H† (3.40)

where ζ = tr(

H†H +NξIN

)−2H†H

is the power normalization constant in the presence of

CSI error.

An achievable secrecy sum-rate in the presence of imperfect channel state information is therefore

given by

SiBCC =K∑k=1

RiBCC,k (3.41)

where

RiBCC,k =[log2

(1 + γik

)− log2

(1 + γiM,k

)]+, (3.42)

with

γik =ρ∣∣∣h†kwk

∣∣∣21 + ρ

∑j 6=k

∣∣∣h†kwj

∣∣∣2 , (3.43)

γiM,k = ρ ‖Hkwk‖2 , (3.44)

and where wk is the kth column of W.

Frequency Division Duplex Systems

In the case of frequency division duplex (FDD) systems, we can assume that users quantize their

perfectly estimated channel vectors and send the quantization index back to the transmitter over a

limited-rate channel. We assume that the channel magnitude is perfectly known to the transmitter,

since it can be efficiently quantized, and that each channel direction is quantized using B bits

and random vector quantization (RVQ) JINDAL [2006]; RYAN et al. [2008]. In RVQ, each user

independently generates a random codebook with 2B vectors, isotropically distributed on the N -

dimensional unit sphere. RVQ generates a CSI that follows the model in (3.39), where the error


τ2 can be upper bounded as JINDAL [2006]

τ2 < 2−B

N−1 . (3.45)

Time Division Duplex Systems

In the case of time division duplex (TDD) systems, uplink and downlink transmissions alternate on

the same channel. The channel estimation at the transmitter is obtained from known pilot symbols

sent by the users. Let T be the channel coherence interval, i.e., the number of channel uses for

which the channel is constant. The interval T is divided into Tt uses for uplink training and T −Ttuses for the downlink transmission of data. The channel state information at the users is provided

by a training phase in the downlink. However, a minimal amount of training is sufficient for this

phase, and we can therefore neglect the overhead due to the downlink training MARZETTA et al.

[2009].

Each user transmits the same number Tt > K of orthogonal pilot symbols to the base station,

which estimates all the K channels simultaneously. The channel estimation error at the base

station depends on the number Tt as well as on the SNR ρul on the uplink channel, and it is given

by CAIRE et al. [2010]

τ2 =1

1 + Ttρul. (3.46)

3.4.2 Secrecy Rates Under No CSI or Poisoned CSI

This thesis considers the case when channel state information, either perfect or imperfect, is ob-

tained by the BS with the collaboration of users. In some cases, users might be in standby mode

without providing any CSI at all, or they might provide wrong (poisoned) CSI with the purpose

of improving their eavesdropping capability. In both cases, the secrecy performance of the system

would be affected, because without knowledge of the channel matrix, the BS would not be able to

effectively use linear precoding to control interference and information leakage. The case when

a subset of the users are in standby and do not provide CSI to the BS is discussed in GERACI

et al. [2013], whereas analyzing the achievable secrecy rates in the presence of poisoned CSI is

regarded as an interesting future research direction.


3.4.3 Secrecy Rates under Transmit Channel Correlation

In the previous sections, we assumed that the channels at the various users are independent and

identically distributed. In practice, channel correlation might be present due to local scatterers

around the BS and insufficient separation between antenna elements. We now consider correlation

amongst transmit antenna elements, and for tractability in the analysis, we assume a separable (or

Kronecker) correlation model MUHARAR AND EVANS [2009]; AL-NAFFOURI et al. [2009]. The

channel matrix can therefore be written as

H = HR12 (3.47)

where H ∈ CK×N contains i.i.d. circularly-symmetric complex Gaussian random variables, with

zero mean and unit variance, and where R ∈ CN×N is the non-singular transmit correlation ma-

trix, given by R = E[H†H]. We assume that each user experiences the same transmit correlation.

The channel vector for the kth user is h†k = h†kR12 ∈ CN×1.

An achievable secrecy sum-rate is therefore given by

ScBCC =K∑k=1

RcBCC,k (3.48)

with

RcBCC,k =[log2 (1 + γck)− log2

(1 + γcM,k

)]+, (3.49)

where

γck =ρA2

k

ρBk + ζ(

1 + Ak

)2 , (3.50)

γiM,k =ρBk

ζ(

1 + Ak

)2 , (3.51)

and where we have introduced the quantities NGUYEN AND EVANS [2008]

Qk =(H†kHk +NξR−1

)−1, (3.52)

Ak = h†kQkhk, (3.53)

Bk = h†kQkH†kHkQkhk, (3.54)

ζ = tr

QkHR−1

H†Qk

. (3.55)


3.5 Conclusion

In this chapter, we considered the MISO broadcast channel with confidential messages, where a

multi-antenna base station simultaneously transmits independent confidential messages to several

single-antenna users in the presence of Rayleigh fading. We studied the secrecy rates achievable

by a linear precoder based on regularized channel inversion (RCI). We showed that RCI precoding

outperforms plain channel inversion precoding, and that the secrecy requirements result in a loss

in terms of the sum-rate. We then proposed an algorithm to jointly optimize the regularization

parameter of the precoder and the power allocation vector, and showed that optimal power alloca-

tion increases the secrecy sum-rate compared to equal power allocation. Finally, we extended the

secrecy rate analysis to more practical scenarios where only imperfect channel state information is

available at the transmitter, and where channel correlation is present among the transmit antenna

elements.

Chapter 4

Physical Layer Security in Isolated

Cells: A Large-System Analysis

Summary In this chapter, we obtain deterministic approximations for the large-system secrecy

rate achievable by RCI precoding in the MISO BCC under Rayleigh fading. We also derive the

optimal regularization parameter and the optimal network load, and simulations confirm that our

analysis is accurate even for finite systems. We find that the RCI precoder performs poorly for

large network loads and large SNR. Therefore, we propose a precoder based on RCI and power

reduction (RCI-PR) that significantly increases the performance in such regime. We compare the

secrecy rate of our proposed RCI-PR precoder to two upper bounds obtained without secrecy re-

quirements and without interference, respectively, and show that it has the same high-SNR scaling

factor as the two bounds. We further study the secrecy rates achievable under imperfect CSI, and

determine the minimum CSI quality in order to maintain a given rate gap to the case with perfect

CSI. We finally extend our large-system analysis to the case of transmit-side channel correlation.

43

44 Large-System Analysis of the Secrecy Rates 4.1

4.1 Large-System Analysis of the Secrecy Rates

I n this chapter, we consider the MISO BCC as in Chapter 3. For this system model, we study the

secrecy rates achievable by the RCI precoder in the large-system regime, where both the number

of receivers K and the number of transmit antennas N approach infinity, with their ratio β =

K/N being held constant. We then derive the optimal regularization parameter ξ that maximizes

the secrecy sum-rate and an approximation for the optimal network load β. As in Chapter 3,

we assumed homogeneous users, i.e., their distances from the serving BS are equal. Our large-

system analysis can be extended to the general case where the mobile users have different distances

from the serving BS. In this case, the large-system deterministic equivalents of the useful signal,

interference, and information leakage, are given in the form of fixed-point equations, and can be

found in YANG et al. [2013].

4.1.1 Large-System Secrecy Rates with RCI Precoding

In the following we provide a deterministic approximation of the secrecy rates, which is almost

surely exact as N → ∞. The proof of this deterministic approximation can be found in Ap-

pendix A.1, and it holds only for values of ξ such that the minimum eigenvalue of(

1NHH† + ξI

)is bounded away from zero for all large N , almost surely. It can be shown that such condition is

verified only if ξ ∈ DN , where the set DN is defined as

DN = R\[−(

1 +√β)2− C

N12−ε,−(

1−√β)2

+C

N12−ε

], (4.1)

with constants C > 0 and ε > 0.

Theorem 4.1. Let ρ > 0 and β > 0. Let RBCC,k be the secrecy rate achievable by user k with

RCI precoding defined in (3.7). Then

supξ∈DN

|RBCC,k (ξ)−RBCC (ξ)| a.s.−→ 0, as N →∞, ∀k. (4.2)

RBCC denotes the secrecy rate in the large-system regime, given by

RBCC =

[log2

1 + γ

1 + γM

]+

, if ξ 6= 0, (4.3)

and

RBCC(0) = limξ→0

RBCC(ξ) =

log2

[1 + (1−β)ρ

β

]if β ≤ 1

log2β3[β+ρ(β−1)]

[β2+ρ(β−1)2]2

+

if β > 1(4.4)

4.1 Physical Layer Security in Isolated Cells: A Large-System Analysis 45

with

γ = g (β, ξ)ρ+ ρξ

β [1 + g (β, ξ)]2

ρ+ [1 + g (β, ξ)]2, (4.5)

γM =ρ

(1 + g (β, ξ))2 , (4.6)

and

g (β, ξ)=1

2

sgn(ξ) ·√

(1−β)2

ξ2+

2 (1+β)

ξ+1+

1−βξ−1

. (4.7)

Proof. See Appendix A.1.

The secrecy sum-rate SBCC can be therefore approximated by the large-system secrecy sum-rate

SBCC, given by

SBCC = KRBCC. (4.8)

4.1.2 Secrecy Sum-Rate Maximizing Regularization Parameter

The value of the regularization parameter ξ has a significant impact on the large-system secrecy

sum-rate SBCC. In the following, we derive the regularization parameter ξBCC that maximizes

SBCC.

Theorem 4.2. Let ρ > 0, β > 0. Let ξ?BCC,N = arg maxξ∈DN

SBCC(ξ) be the optimal regularization

parameter in DN , and denote S?BCC , SBCC(ξ?BCC,N ) the optimal secrecy sum-rate. Then

1

N[S?BCC − SBCC(ξBCC)]

a.s.−→ 0, as N →∞, (4.9)

where ξBCC ∈ DN is the optimal large-system regularization parameter, given, for N large

enough, by

ξBCC =−2ρ2 (1−β)2+6ρβ+2β2−2 [β (ρ+1)−ρ] ·

√β2 [ρ2+ρ+1]−β [2ρ (ρ−1)] + ρ2

6ρ2 (β + 2) + 6ρβ.

(4.10)


Proof. The value of ξBCC can be found by setting the derivative of SBCC to zero and studying its

maxima in each of the intervals which compose the set DN . Then we have

0(a)

≤ 1

N[S?BCC − SBCC(ξBCC)]

=1

N

[SBCC(ξ?BCC,N )− SBCC(ξ?BCC,N ) + SBCC(ξ?BCC,N )− SBCC(ξBCC)

+SBCC(ξBCC)− SBCC(ξBCC)]

(b)

≤ 1

N

[SBCC(ξ?BCC,N )− SBCC(ξ?BCC,N ) + SBCC(ξBCC)− SBCC(ξBCC)

] (c)−→ 0, (4.11)

where (a), resp. (b), follows from the definition of ξ?BCC,N , resp. ξBCC, and (c) follows from

Theorem 4.1.

When β = 1, the value of ξBCC in (4.10) reduces to

ξBCC =1

3ρ+ 1 +√

3ρ+ 1, for β = 1. (4.12)

We note that the value ξBCC that maximizes the secrecy sum-rate can be negative, and it differs

from the value ξBC = β/ρ that maximizes the sum-rate in the MISO BC without secrecy require-

ments NGUYEN et al. [2009]. Unlike ξBC, which grows unbounded as ρ → 0, ξBCC is upper

bounded by

ξBCC ≥ limρ→0

ξBCC = 1− β

2, ∀β > 0, (4.13)

although when β ≥ 2 it can be shown that SBCC = 0 irrespective of ξ and ρ. Similarly to ξBC, the

value of ξBCC decreases as the SNR increases. In the high-SNR regime, we have

limρ→∞

ξBCC − ξ∞BCC = 0, (4.14)

where ξ∞BCC approximates the high-SNR behavior of ξBCC and is given by

ξ∞BCC =

β2ρ for β < 1

13ρ for β = 1

−2(β−1)2

3(β+2) + β(2−β)2ρ(β+2) for β > 1

. (4.15)

We then have by the continuous mapping theorem

limρ→∞

S?BCC − S?∞BCC

S?BCC

= 0 (4.16)

with

S?BCC , SBCC (ξBCC) and S?∞BCC , limρ→∞

S?BCC. (4.17)


4.1.3 Optimal Secrecy Sum-Rate

By substituting ξBCC from (4.10) in (4.5) and (4.6), and then in (4.3) and (4.8), it is possible

to obtain the optimal secrecy sum-rate S?BCC achievable by RCI precoding in the large-system

regime. The secrecy sum-rate S?BCC is a function of N , β and ρ. For β = 1, we obtain the

following compact result.

Theorem 4.3. For β = 1, the optimal secrecy sum-rate S?BCC achievable by the RCI precoder in

the large-system regime is given by

S?BCC = K log2

9ρ+ 2 + (6ρ+ 2)√

3ρ+ 1

4 (4ρ+ 1). (4.18)

Proof. Equation (4.18) follows by substituting (4.12) in (4.5) and (4.6), and then in (4.3) and

(4.8).

The secrecy sum-rate in (4.18) satisfies

S?BCC > 0 ∀ρ > 0, for β = 1, (4.19)

and S?BCC is monotonically increasing with the SNR ρ. It can be shown that the same is true for

β < 1. However when β > 1, the secrecy sum-rate does not monotonically increase with ρ. It will

be shown in Section 4.2 that there is an optimal value of the SNR beyond which the achievable

secrecy sum-rate S?BCC starts decreasing, until it becomes zero for large SNR. When β ≥ 2 no

positive secrecy sum-rate is achievable at all.

These results can be explained as follows. In the worst-case scenario, the alliance of cooperating

malicious users can cancel the interference, and its received SINR is the ratio between the signal

leakage and the thermal noise. In the limit of large SNR, the thermal noise vanishes, and the

only means for the transmitter to limit the SINR at the malicious users is by reducing the signal

leakage to zero by inverting the channel matrix. This can only be accomplished when the number

of transmit antennas is larger than or equal to the number of users, hence only if β ≤ 1. When

β > 1 this is not possible, and no positive secrecy sum-rate can be achieved. When β ≥ 2,

the eavesdroppers are able to drive the secrecy sum-rate to zero irrespective of ρ. This result is


expected and consistent with the ones obtained for a single-user system in KHISTI AND WORNELL

[2010].

Since for β > 1 the RCI precoder performs poorly in the high-SNR regime, we have studied the

use of both user scheduling and artificial noise generation for possible secrecy enhancement. In the

presence of secrecy constraints, the strategy of reducing the network load β by simply selecting a

subset of the users to be served, commonly known as user scheduling (US), is not effective. User

scheduling would reduce the interference generated, thus increasing the SINR at the legitimate

receivers. However, US would not be effective to limit the number of malicious receivers in the

network, as the discarded users would still be able to eavesdrop. Moreover, a precoding matrix

designed by taking into account only the channels of the selected users would perform poorly,

generating large information leakage at the nearby discarded users. We have studied the use of

various US algorithms, including the optimal exhaustive search. We have omitted these results

since US does not perform well. We have also investigated the transmission of artificial noise

(AN) to limit the eavesdropping ability of the discarded users, while being harmless to selected

users. However, even combining US with AN seems not to achieve better performance than the

RCI-PR precoder proposed in this thesis, and we have omitted these results as well.

Remark 4.1. In order for Theorem 1 to hold with ξ = ξBCC, it is sufficient that ξBCC ∈ DN . Since

ξBCC in (4.10) depends on β and ρ, so does the accuracy of the deterministic approximation S?BCC

for finite N . We can distinguish the two following cases. (i) When β 6= 1, we have ξBCC ∈ DN∀ρ, and the approximation is accurate uniformly on ρ. (ii) When β = 1, ξBCC ∈ DN for all finite

ρ; if ρ→∞, then it is required that N = O(ρ2+ε), for some ε > 0, otherwise the approximation

gets weaker as ρ→∞ for N fixed. This means for instance that the approximation with N = 10

and ρ = 17dB is as accurate as the approximation with N = 40 and ρ = 20dB.

Figure 4.1 compares the secrecy sum-rate S?BCC of the RCI precoder from the large-system anal-

ysis to the simulated ergodic secrecy sum-rate SBCC with a finite number of users, for different

values of β. The value of S?BCC was obtained by (4.8) with ξBCC as in (4.10). The value of SBCC

was obtained by using the regularization parameter that maximizes the average secrecy sum-rate.

We observe that when β = 0.8 and when β = 1.2 the large-system analysis is accurate for all

values of N and SNR. When β = 1, the analysis is accurate at low SNR for all values of N , and

for high SNR larger values of N are required to increase the accuracy. The previous observations

are consistent with Remark 4.1.


0 5 10 15 200

1

2

3

4

SNR, ρ [dB]

Per-antennasecrecysum-rate

SBCC/N (N = 10)

SBCC/N (N = 20)

SBCC/N (N = 40)

S⋆BCC/N

β = 1.2

β = 0.8

β = 1

Figure 4.1: Comparison between the secrecy sum-rate with RCI precoding in the large-system

regime (4.8) and the simulated ergodic secrecy sum-rate for finite N . Three sets of curves are

shown, each one corresponds to a different value of β.

Figure 4.2 shows that using the regularization parameter ξBCC, obtained from large-system anal-

ysis, does not cause a significant loss in the secrecy sum-rate compared to using ξ?BCC,N , opti-

mized for each channel realization. The figure shows the normalized secrecy sum-rate difference

(S?BCC − SBCC(ξBCC)) /S?BCC, simulated for finite-size systems, β = 0.8 and various values of

the SNR. The value of S?BCC was obtained by using ξ?BCC,N , whereas SBCC(ξBCC) was obtained

by using ξBCC. We observe that the average normalized secrecy sum-rate difference is less than

2% for all values of N and ρ. As a result, one can avoid the calculation of ξ?BCC,N for every

channel realization, and ξBCC can be used with only a small loss of performance.

4.1.4 Optimal Network Load

Figure 4.3 depicts the per-antenna secrecy sum-rate S?BCC/N as a function of the network load

β, for several values of the SNR. We denote by β? the value of β ∈ R+ that maximizes the

per-antenna large-system secrecy sum-rate. It is possible to see from Fig. 4.3 that the value of β?

falls between 0 and 1, and that it is an increasing function of the SNR. An approximation for β?

in the large-SNR regime is given in the following.


5 10 15 20 25 30 35 4010

−5

10−4

10−3

10−2

10−1

Norm

alizedthroughputdifferen

ce

Number of transmit antennas, N

ρ = 0dBρ = 5dBρ = 10dBρ = 15dBρ = 20dB

Figure 4.2: Mean normalized secrecy sum-rate difference between S?BCC (obtained using the op-

timal ξ?BCC,N ) and SBCC(ξBCC) (obtained with ξBC from large-system analysis), for β = 0.8 and

various values of the SNR.

Proposition 4.1. In the limit of large SNR, the value β? of the optimal network load can be found

by solving the following fixed point equation

β? = ρ(

1− β?)e− 1

1−β? , (4.20)

and the network load β? tends to one for large SNR.

Proof. From (4.16), we have that S?∞BCC approximates S?BCC in the large-SNR regime. We then

obtain (4.20) by noticing that it must be β? ∈ [0, 1], and by setting ∂(S?∞BCC/N)/∂β = 0.

Figure 4.4 shows the optimal number of users K? obtained via simulations, for N = 10, 20, and

40 antennas. This is compared to K?, obtained from an exhaustive search on the large-system

secrecy sum-rate S?BCC, and to the closed-form approximation K? = Nβ?, obtained from (4.20)

in the high-SNR regime. We note that K? is accurate across the whole range of SNR, whereas

K? is accurate for medium-to-large values of the SNR.


0 0.5 1 1.5 20

1

2

3

4

5

Network load, β

S⋆

BCC/N

ρ = 5dBρ = 10dBρ = 15dBρ = 20dB

0.65

0.70

0.75

0.57

Figure 4.3: Large-system secrecy sum-rate per transmit antenna as a function of β for RCI pre-

coding. The value of β? is indicated next to each curve.

0 5 10 15 20 25 300

5

10

15

20

25

30

35

40

45

50

SNR, ρ [dB]

Optimalnumber

ofusers

K⋆ (simulations)

K⋆ (exhaustive search from S⋆BCC)

K⋆ = Nβ⋆ (analysis)

N = 10

N = 20

N = 40

Figure 4.4: Comparison between K? (obtained via simulations), K? (obtained via exhaustive

search and large-system analysis), and the analytical approximation K? (obtained via large-

system and large-SNR analysis).

52 RCI Precoder with Power Reduction 4.2

4.2 RCI Precoder with Power Reduction

We have found that for β > 1 the RCI precoder performs poorly in the high-SNR regime. In this

section, we first derive the optimal value of the SNR ρ? that maximizes the achievable secrecy

sum-rate S?BCC for β > 1. We then propose a linear precoder based on RCI and power reduction

which significantly increases the high-SNR secrecy sum-rate for 1 < β < 2.

4.2.1 Optimal Transmit SNR

When 1 < β < 2, there is an optimal value of the transmit SNR ρ?, provided in the following.

Proposition 4.2. The value of the SNR ρ? that maximizes the secrecy sum-rate S?BCC for 1 <

β < 2, and the corresponding maximum value of S?BCC are respectively given by

ρ? = arg maxρ

S?BCC (ρ) =β (2− β)

(β − 1)2 (4.21)

and

S?BCC (ρ?) = K log2

β2

4 (β − 1). (4.22)

Proof. If 1 < β < 2, then ρ? is the only stationary point of S?BCC, which can be found by setting

its derivative ∂S?BCC/∂ρ to zero. We note that limρ→ρ? ξBCC = 0. Therefore, S?BCC (ρ?) can be

obtained by considering ρ→ ρ? and ξ → 0 in (4.8) and after some algebraic manipulations.

4.2.2 Power Reduction Strategy

We now propose a power reduction strategy to prevent the secrecy sum-rate from decreasing at

high SNR, for 1 < β < 2. This is achieved by reducing the transmit power, and therefore

reducing the SNR to the value ρ? that maximizes the secrecy sum-rate. We denote this scheme as

the RCI precoder with power reduction (RCI-PR), whose precoding matrix WPR is given by

WPR =

1√ζH†(HH† +NξIK

)−1if β ≤ 1

1√κζ

(H†H +NξIN

)−1H† if 1 < β < 2

0 if β ≥ 2

(4.23)


where κ = max(

ρρ? , 1

)is the power reduction constant used for 1 < β < 2, and where ξ is

chosen from (4.10) evaluated with an SNR of min(ρ, ρ?). We note that (4.23) generalizes the

RCI precoder in (3.15) to the case when the power reduction strategy is employed.

Remark 4.2. We note from (4.10) that ξBCC(ρ?) = 0. Therefore when ρ ≥ ρ?, the optimal

value of ξ for the RCI-PR precoder is zero, and it reduces to a CI-PR precoder. Even if β > 1, it

is still possible to calculate WPR by expressing it as in (4.23) for 1 < β < 2.

We denote by RrBCC,k and RrBCC the secrecy rates achievable by the proposed RCI-PR pre-

coder (4.23) in finite systems and in the large-system regime, respectively. Similarly, SrBCC =∑Kk=1R

rBCC,k and SrBCC = KRrBCC denote the respective secrecy sum-rates. The following

theorem provides a high-SNR approximation of SrBCC.

Theorem 4.4. In the high-SNR regime, we have limρ→∞SrBCC−S

r∞BCC

SrBCC= 0, where Sr∞BCC approxi-

mates the large-system secrecy sum-rate SrBCC achieved by the RCI-PR precoder, and it is given

by

Sr∞BCC =

K log21−ββ +K log2 ρ for β < 1

K2 log2

2764 + K

2 log2 ρ for β = 1

K log2β2

4(β−1) for 1 < β < 2

0 for β ≥ 2

(4.24)

Proof. When β ≤ 1, the RCI-PR precoder reduces to the RCI precoder. Therefore, in this case

we have Sr∞BCC = S?∞BCC, with the latter defined in (4.17). The value of (4.24) for 1 < β < 2

is obtained by noting that for large SNR, RCI-PR forces ρ = ρ?, and by using Proposition 4.2.

The value for β ≥ 2 arises from the fact that no positive secrecy sum-rate is achievable in such a

condition, and the RCI-PR precoder (4.23) transmits zero power.

The abrupt change in the large-system secrecy rate as β = 1 is due to the fact that, in this case, the

minimum eigenvalue of the quantity 1NHH†, which affects the secrecy rate, becomes zero for all

large N , almost surely.

From (4.24) we can conclude that the behavior of our proposed RCI-PR precoder can be classified

into four regions. When β < 1, any secrecy sum-rate can be achieved, as long as the transmitter

54 RCI Precoder with Power Reduction 4.2

0 5 10 15 20 250

2

4

6

8

10

12

SNR, ρ [dB]

Secrecy

sum-rate

SBCC

SrBCC

β = 1.6

β = 1.2

β = 1.4

Figure 4.5: Comparison between the ergodic secrecy sum-rates SBCC and SrBCC achieved by the

RCI precoder and by the proposed RCI-PR precoder, respectively, for N = 10 transmit antennas.

Three values of β are considered: 1.2, 1.4, and 1.6, corresponding to K = 12, 14, and 16 users.

has enough power available, and the secrecy sum-rate scales linearly with the factor K. When

β = 1, the linear scaling factor reduces to K/2. When 1 < β < 2, the cooperating eavesdroppers

have more antennas than the transmitter, and thus they can limit the achievable secrecy sum-rate

regardless of how much power is available at the transmitter. When β ≥ 2, the eavesdroppers are

able to prevent secret communications, and the secrecy sum-rate is zero even if unlimited power

is available.

Figure 4.5 shows the simulated ergodic secrecy sum-rates with and without the power reduction

strategy for N = 10 transmit antennas and three values of β > 1. The figure shows that the

proposed RCI-PR precoder in (4.23) increases the secrecy sum-rate compared to the RCI precoder

in (3.15). By using the proposed power reduction strategy, it is possible to prevent the secrecy

sum-rate from decreasing at large values of the SNR ρ. For large ρ, the achieved secrecy sum-rate

equals the maximum across all values of ρ. Moreover, this is achieved by using a lower transmit

power, and the amount of power saved equals 10 log10 κ−1 dB.


4.3 Performance Comparison

In this section, we first compare the secrecy sum-rate SrBCC achieved by the proposed RCI-PR

precoder to the sum-rate S?BC achieved by the optimized RCI precoder in the BC without se-

crecy requirements, in the large-system regime. The gap between SrBCC and S?BC represents the

secrecy loss, i.e., how much the secrecy requirements cost in terms of the achievable sum-rate.

Furthermore, we compare the per-user secrecy rate RrBCC achieved by the proposed precoder to

the secrecy capacity CMISOME of a single-user MISOME wiretap channel KHISTI AND WOR-

NELL [2010]. The gap betweenRrBCC and CMISOME represents a multi-user loss, i.e., the loss due

to the requirement of serving multiple users at the same time.

4.3.1 Secrecy Loss

The secrecy sum-rate SrBCC for the MISO BCC is obtained by using the precoder in (4.23). The

optimal sum-rate S?BC in the MISO BC without secrecy requirements is obtained by using the

precoder in (3.15), and it is given by NGUYEN et al. [2009]

S?BC = K log2 [1 + g (β, ξBC)], (4.25)

with ξBC = β/ρ. Similarly to the secrecy sum-rate, there is an optimal value for the ratio β

that maximizes the per-antenna sum-rate S?BC/N without secrecy requirements NGUYEN et al.

[2009]; HOCHWALD AND VISHWANATH [2002]; WAGNER et al. [2012]. It is easy to show that

S?BC ≥ 0 for all values of β and ρ, with equality only for ρ = 0, and that S?BC tends to zero as

β → ∞. Hence, there is no limit to the number of users per transmit antenna β that the system

can accommodate with a non-zero sum-rate. However if we impose the secrecy requirements,

the secrecy sum-rate SrBCC is zero for β ≥ 2. Therefore, introducing the secrecy requirements

will limit the number of users that can be served with a non-zero rate to two times the number of

transmit antennas.

We now compare the secrecy sum-rate SrBCC to the sum-rate S?BC in the limit of large SNR. Again

by using the regularization parameter ξBC = β/ρ we obtain limρ→∞S?BC−S

?∞BC

S?BC= 0, with

S?∞BC =

K log2

1−ββ +K log2 ρ for β < 1

K2 log2 ρ for β = 1

K log2ββ−1 for β > 1

. (4.26)

56 Performance Comparison 4.3

By comparing (4.26) to (4.24), we can draw the following conclusions regarding the large-SNR

regime. If the number of transmit antennas N is larger than the number of users K, then Sr∞BCC =

S?∞BC and the secrecy requirements do not decrease the sum-rate of the network. Therefore by us-

ing the RCI-PR precoder in (4.23), one can achieve secrecy while maintaining the same sum-rate,

i.e., there is no secrecy loss. If N = K, then the secrecy loss is 12 log2(64

27) ≈ 0.62 bits per user,

but the linear scaling factorK/2 remains unchanged. Alternatively, one can achieve secrecy while

maintaining the same sum-rate, by increasing the transmit power by a factor 64/27 ≈ 3.75dB.

If N < K < 2N , then the secrecy loss is (2 − log2 β) bits per user, but the proposed precoder

transmits a lower power, which is always upper bounded by β(2−β)

(β−1)2 . Finally if K ≥ 2N , then

the secrecy requirements force the sum-rate to zero, whereas the sum-rate S?BC remains positive,

though it also tends to zero for large β.

We note that in the BC, i.e., when there are no secrecy requirements, user scheduling can be used to

achieve a higher multiplexing gain. This is not possible in the BCC, since discarding users does not

prevent them from eavesdropping. In the presence of secrecy constraints, the strategy of reducing

the network load β by simply selecting a subset of the users to be served, commonly known as

user scheduling (US), is not effective. User scheduling would reduce the interference generated,

thus increasing the SINR at the legitimate receivers. However, it would not be effective to limit

the number of malicious receivers in the network, as the discarded users would still be able to

eavesdrop. Moreover, a precoding matrix designed by taking into account only the channels of the

selected users would perform poorly, generating large information leakage at the nearby discarded

users. We have studied the use of various US algorithms, including the optimal exhaustive search.

We have omitted these results since US does not achieve better performance than the RCI-PR

precoder proposed in this thesis.

Since discarding users to reduce the network load β is not an effective strategy in the BCC, the

case β > 1 has practical significance, as it arises whenever the number of users is larger than

the number of transmit antennas. At high SNR, e.g. 20dB and above, when large interference

and information leakage limit the secrecy performance, the proposed RCI-PR precoder achieves

reasonably good performance by controlling the transmit power (see Fig. 4.5 and Fig. 4.6). At low

SNR, e.g. 5dB and below, when noise becomes a limiting factor, the performance is still partially

affected by the value of β, and RCI precoding takes it into account and achieves a nonnegative

secrecy rate (see Fig. 4.3 and Fig. 4.5).


4.3.2 Multi-User Loss

We now consider the multi-user loss, i.e., the loss due to the interference caused by the presence

of multiple users in the system. This is given by the gap between the secrecy rate RrBCC achieved

by the proposed RCI-PR precoder and the secrecy capacity CMISOME of the MISOME wiretap

channel, where one user is served at a time and the remaining users can eavesdrop KHISTI AND

WORNELL [2010]. We compare Rr∞BCC to CMISOME in the large-SNR regime. The former is

obtained by dividing (4.24) byK. The value ofCMISOME was obtained in KHISTI AND WORNELL

[2010], and for large SNR we have limρ→∞CMISOME−C∞MISOME

CMISOME= 0, where

C∞MISOME =

log2 ρ for β < 1

12 log2 ρ for β = 1

log21

(β−1) for 1 < β < 2

0 for β ≥ 2

. (4.27)

We remark that in CMISOME from KHISTI AND WORNELL [2010] a single-user system is con-

sidered. Therefore, only one message is transmitted to one legitimate user, and the user does not

experience any interference. By comparing (4.27) to Rr∞BCC, we can conclude that the multi-user

loss is log21−ββ and 0.62 bits per user for β < 1 and β = 1, respectively. Hence for β ≤ 1, the

proposed RCI-PR precoder achieves a secrecy rate which has the same linear scaling factor as the

secrecy capacity of a single-user system with no interference. When 1 < β < 2, the proposed pre-

coder suffers a multi-user loss of (2− 2 log2 β) bits, but again it has the advantage of transmitting

a limited power.

In Fig. 4.6 we compare the simulated per-user ergodic secrecy rate SrBCC/K of the RCI-PR pre-

coder to the rate SBC/K of the RCI precoder in the MISO BC without secrecy requirements.

These were obtained by using the regularization parameters ξBCC and ξBC, respectively. For

β < 1, the difference between SrBCC/K and SBC/K becomes negligible at large SNR, and se-

crecy can be achieved without additional costs. For β = 1, the two curves tend to have the same

slope at large SNR, but there is a residual gap between them. Therefore, secrecy can be achieved

at a lower rate. We note that in order to achieve secrecy without decreasing the rate, the required

additional power is less than 4dB at all SNRs. For 1 < β < 2, the sum-rate SBC tends to satu-

rate for large SNR, and so does the secrecy sum-rate SrBCC. In the simulations, for β = 1.2 and

ρ = 25dB, the gap is about 1.79 bits, close to 2− log2 β ≈ 1.74 bits. Moreover, we note that the

58 Performance Comparison 4.3

0 5 10 15 20 250

2

4

6

8

SNR, ρ [dB]

Per-userrate

SrBCC/K

SBC/K

CMISOME

β = 1.2

β = 0.8

β = 1

Figure 4.6: Comparison between the simulated ergodic per-user secrecy rate with RCI-PR (solid)

and the two upper bounds: (i) per-user rate without secrecy requirements (dashed) and (ii) MIS-

OME secrecy capacity (dotted), for K = 12 users. Three values of β are considered: 0.8, 1, and

1.2, corresponding to N = 15, 12, and 10 antennas.

proposed precoder saves 92% of the transmit power. The gap is smaller for smaller values of the

SNR, e.g., it reduces to about 0.72 bits when we set the transmit power to 10dB.

Figure 4.6 also shows the simulated secrecy capacity CMISOME of the MISOME wiretap channel.

For β ≤ 1, the RCI-PR precoder achieves a per-user secrecy rate which has the same linear

scaling factor as CMISOME. When 1 < β < 2, also CMISOME saturates at high SNR. In particular,

for β = 1.2 and ρ = 25dB, the gap with the RCI-PR precoder is about 1.47 ≈ 2−2 log2 β bits, but

the RCI-PR precoder saves 92% of the power. The gap is smaller for smaller values of the SNR,

e.g., it reduces to about 0.4 bits when we set the transmit power to 10dB. All these numerical

results confirm the ones obtained from the large-system analysis.

4.3.3 Comparison to Other Linear Schemes

We now compare the secrecy sum-rate achieved by the RCI precoder with ξBCC to the secrecy

sum-rates obtained from (4.8) by using: (i) ξ = 0 (CI precoder), (ii) ξ → ∞ (single user beam-


former) and (iii) ξ = ξBC (optimum RCI precoder in the BC without secrecy requirements). The

analytical results provided in this section are obtained as special cases of the RCI precoder, for

special values of the regularization parameter ξ. Although these results can be obtained for any

value of β, in the following we restrict ourselves to β = 1, which leads to compact expressions

that allow us to easily compare the performance of the different schemes.

The aim of the CI precoder is to cancel all the interference and information leakage, therefore

yielding to a secrecy sum-rate that coincides with the sum-rate when β ≤ 1. We note that for the

CI precoder it is ξ = 0, and the precoding matrix is given by

W =1

ζ0H†(HH†

)−1, (4.28)

with ζ0 = tr(

H†H)−1

.

The secrecy sum-rate achieved by CI precoding in the large-system regime grows at most sublin-

early with N →∞. In fact,

limξ→0

limN→∞

SBCC

N= 0, β = 1. (4.29)

This result is consistent with NGUYEN et al. [2009], where it was shown that the CI precoder

performs poorly in the large-system regime when the number of antennas equals the number of

users.

Similarly, we calculate the secrecy sum-rate achieved when ξ → ∞, i.e., for the single user

beamformer (SUB). Here, the transmitter beamforms in a direction such as to maximize the signal

strength of each user, without taking into account the interference it creates and the amount of

resulting information leakage. The secrecy sum-rate achieved by SUB in the large-system regime

is zero. In fact,

limξ→∞

limN→∞

SBCC

N=

[log2

2ρ+ 1

(ρ+ 1)2

]+

= 0, β = 1. (4.30)

Clearly, SUB performs poorly compared to the optimal RCI precoder. This is due to the intended

user suffering from a large amount of interference, while the malicious users may cancel the

interference by cooperating.

Finally, we consider ξ = ξBC = 1/ρ, which is the value that maximizes the sum-rate in a MISO

BC without secrecy requirements PEEL et al. [2005]. The secrecy sum-rate SBCC achieved by

60 Imperfect CSI 4.4

RCI with ξ = ξBC in the large-system regime is given by

SBCC(ξBC) = K log2

4ρ+ 1 + (2ρ+ 1)√

4ρ+ 1

2 (4ρ+ 1), β = 1. (4.31)

We observe that RCI precoding with ξ = ξBC outperforms CI precoding and SUB in the large-

system regime, but it is suboptimal compared to the use of ξBCC. For high SNR, the per-antenna

secrecy sum-rate gain provided by using ξ = ξBCC in place of ξ = ξBC is given by

limρ→∞

SBCC(ξBCC)− SBCC(ξBC)

N= log2

3√

3

4≈ 0.38 bits. (4.32)

4.4 Imperfect CSI

In this section, we study the large-system secrecy rates achievable by RCI precoding in the pres-

ence of imperfect CSI, and we determine how the CSI error must scale with the SNR in order to

ensure a constant high-SNR rate gap to the case with perfect CSI, i.e., to maintain the same multi-

plexing gain. We derive the minimum number of feedback bits required for FDD systems, and we

study the optimum amount of channel training for TDD systems. For simplicity, in the following

we will consider the RCI precoder without power reduction strategy, although the results obtained

can be easily extended to the case of RCI-PR precoding. We also focus on the case β ≤ 1, which

is the only one that yields a non-zero high-SNR multiplexing gain.

4.4.1 Minimum Required CSI

We now provide a deterministic approximation for the secrecy rate achievable by RCI precoding

in the presence of imperfect CSI. The following result is generic, and it holds irrespective of the

channel estimation technique, therefore it holds for both FDD and TDD systems.

Theorem 4.5. Let ρ > 0, β > 0, and ξ ∈ DN . Let RiBCC,k be the secrecy rate for user k in the

presence of channel estimation error with variance τ2, defined in (3.42). Define ρ ,ρ(1−τ2)ρτ2+1

and

ξ , ξ1−τ2 . Then ∣∣RiBCC,k −RiBCC

∣∣ a.s.−→ 0, as N →∞, ∀k (4.33)


where RiBCC is the large-system secrecy rate in the presence of imperfect CSI, given by

RiBCC =

log2

1 + g(β, ξ)ρ+ ξρ

β [1+g(β,ξ)]2

ρ+[1+g(β,ξ)]2

1 + ρ

[τ2 + 1−τ2

(1+g(β,ξ))2

]

+

. (4.34)

Proof. See Appendix A.2.

In the case of perfect CSI, i.e., τ = 0, eq. (4.34) reduces to (4.3). The large-system secrecy

sum-rate in the presence of imperfect CSI is given by

SiBCC = KRiBCC. (4.35)

We now determine how the CSI estimation error must scale with the SNR, to maintain a given rate

gap to the case with perfect CSI. In the following, we assume that the regularization parameter

ξBCC from (4.10) is used. We define the per-user gap ∆RBCC as the difference

∆RBCC , RBCC −RiBCC (4.36)

where RBCC and RiBCC are the large-system secrecy rates obtained by the RCI precoder under

perfect CSI and under CSI distortion τ2, respectively. We now derive the scaling of τ2 required to

maintain a constant secrecy rate gap for high SNR, so that the multiplexing gain is not affected.

Proposition 4.3. For β ≤ 1, b > 1, a CSI distortion τ2 = Cρ , with

C =

12

(√4b− 3− 1

)for β < 1

23

(√3b− 2− 1

)for β = 1

(4.37)

produces a high-SNR rate gap of log2 b bits.

Proof. For β ≤ 1, define

κτ , τ2ρ+τ4ρ(1 + g(β, ξBCC))2

1− τ2+τ2(1 + g(β, ξBCC))2

1− τ2(4.38)

with g(β, ξ) defined as in (4.7). We have

limρ→∞

∆RBCC =

limρ→∞

log2

[1 + β2

4ρ(1−β)2κτ

]= log2 b for β < 1

limρ→∞

log2

[1 + 1

4κτ]

= log2 b for β = 1(4.39)


0 5 10 15 200

1

2

3

4

SNR, ρ [dB]


SiBCC/N (N = 10)

SiBCC/N (N = 20)

SiBCC/N (N = 40)

SiBCC/N

β = 1.2

β = 1

τ = 0.1

β = 0.8

Figure 4.7: Comparison between the per-antenna secrecy sum-rate SiBCC/N with RCI precoding

in the large-system regime and the simulated ergodic secrecy sum-rate SiBCC/N , for finite N , and

in the presence of a channel estimation error τ = 0.1. Three sets of curves are shown, each one

corresponds to a different value of β.

where the logarithmic quantities arise since the rate gap ∆RBCC is given by the difference of

RBCC and RiBCC, which are both logarithmic functions.

Figure 4.7 compares the secrecy sum-rate SiBCC of the RCI precoder from the large-system anal-

ysis to the simulated ergodic secrecy sum-rate SiBCC for finite N , in the presence of a CSI error

τ = 0.1 and for different values of β. The purpose of this figure is to confirm the accuracy of our

large-system analysis, and the channel estimation error τ was chosen as a constant independent of

the SNR for the sake of simplicity. The values of SiBCC and SiBCC were obtained by (4.35) and

(3.41), respectively, with ξ = ξBCC. As expected, the accuracy of the large-system approximation

increases as N grows.

We note that it is possible to design the number of feedback bits or the training length to achieve

the desired value of τ , according to (3.45) and (3.46). However, the number of feedback bits will

impact the overhead on the uplink channel, and the training length will impact the fraction of time

used to transmit useful information.


4.4.2 Channel Feedback in FDD Systems

For the case of FDD systems, we obtain the following result on the minimum number of feedback

bits required in order to meet a given rate gap with the case of perfect CSI.

Corollary 4.6. In order to maintain a high-SNR secrecy rate offset of log2 b bits per user in the

large-system regime, it is sufficient to scale the number of feedback bits B per user as

B ≈

N−13 ρdB − (N − 1) [log2

(√4b− 3− 1

)− 1] if β < 1

N−13 ρdB − (N − 1) [log2

√3b−2−1

3 + 1] if β = 1. (4.40)

Proof. The result follows from (3.45) and Proposition 4.3.

Figure 4.8 shows the ergodic per-user secrecy rate SiBCC/K, achieved in the presence of a channel

estimation error that scales as τ2 = Cρ , with C obtained from Proposition 4.3 for log2 b = 1 bit.

This is compared to the ergodic rate SBCC/K, achieved in the presence of perfect CSI (τ = 0),

for N = 10. The simulations show a high-SNR gap of nearly 1 bit, which confirms the claims

made in Proposition 4.3.

4.4.3 Channel Training in TDD Systems

For the case of TDD systems, part of the transmission time is used for channel training. Therefore,

although a longer training interval improves the quality of the CSI, it also reduces the interval for

data transmission, and we cannot directly apply the result in Proposition 4.3. In the following, we

study the optimal value of the training interval Tt for high SNR, in the case when both the uplink

SNR ρul and the downlink SNR ρ grow with a finite ratio rρ , ρ/ρul.

Theorem 4.7. In the large-system regime, let ρ, ρul grow large with rρ = ρ/ρul constant. Then, an

approximation of the secrecy sum-rate maximizing amount of channel training Tt can be obtained

as a solution of the equations

T 3t q + T 2

t (rρq −Krρ) + Tt(r2ρq +KrρT − 2Kr2

ρ) + 2Kr2ρT = 0, (4.41)

T 3t 4q + T 2

t (4rρq − 4Krρ) + Tt(3r2ρq + 4KrρT − 6Kr2

ρ) + 6KTr2ρ = 0, (4.42)

for β < 1 and β = 1, respectively, and with q , −SBCC log 2.


0 5 10 15 200

1

2

3

4

5

6

SNR, ρ [dB]

Per-usersecrecyrate

SiBCC/K (τ2 = C/ρ)

SBCC/K (τ2 = 0)

β = 0.8

β = 1

Figure 4.8: Comparison between the ergodic per-user secrecy rates SBCC/K and SiBCC/K with

RCI-PR precoder, for N = 10, in the presence of perfect CSI and in the presence of a channel

estimation error τ2 = Cρ , with C obtained from (4.37) for log2 b = 1 bit.

Proof. By using (3.46), a large-system approximation for the secrecy sum-rate in a TDD system

is given by

SiBCC =T − TtT

K

log2

1+g(β, ξ)ρ+ ξρ

β [1+g(β,ξ)]2

ρ+[1+g(β,ξ)]2

1+ρ

[τ2 + 1−τ2

(1+g(β,ξ))2

]

+

(4.43)

with

ξ =ξ(1 + Ttρul)

Ttρuland ρ =

ρTtρulρ+ 1 + Ttρul

. (4.44)

Rewrite SiBCC = T−TtT (SBCC −K∆), where

limρ,ρul→∞

∆ =

log2

(1 +

r2ρ

T 2t

+rρTt

)for β < 1

log2

(1 +

3r2ρ

4T 2t

+rρTt

)for β = 1

. (4.45)

Then (4.41) and (4.42) can be obtained by setting ∂SiBCC/∂Tt = 0 and after further high-SNR

approximations.

Although it is difficult to formally prove that equations (4.41) and (4.42) always admit a solution

Tt ≤ T , in the following we show that a solution can be obtained in the high-SNR regime by


0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

SNR, ρ [dB]

Tt/T

Simulations

Analysis

T = 300

T = 100

T = 1000

Figure 4.9: Optimal relative amount of training Tt/T vs high-SNR approximation, for N = K =

10 and rρ = ρ/ρul = 10.

solving equations (4.41) and (4.42) numerically, and that this solution well approximates the values

obtained via simulation.

Figure 4.9 shows the simulated optimal relative amount of training Tt/T versus the downlink

SNR ρ, for a system with N = K = 10, and rρ = 10. This is compared to the high-SNR

approximations obtained from Theorem 4.7. Three sets of curves are shown, each for a different

coherence time T . The figure shows that for increasing SNR, the channel estimation becomes more

accurate, and less resources should be allocated to channel training. At high SNR, the optimal

amount of training converges to the value predicted by the analysis and provided in Theorem 4.7.

4.5 Channel Correlation

In this section, we analyze the performance of the RCI precoder under transmit-side channel cor-

relation in the large-system regime, and we determine the optimal regularization parameter. Al-

though we do not consider power reduction strategy, the results derived can be easily extended

to the RCI-PR precoder. Furthermore, we obtain the large-system secrecy rates achievable by

channel inversion precoding and single user beamforming under transmit channel correlation.

66 Channel Correlation 4.5

4.5.1 Large-System Secrecy Sum-Rates

The following theorem provides a deterministic approximationRcBCC for the per-user secrecy rates

RcBCC,k in (3.49), which is almost surely exact as N →∞. RcBCC is independent of user k.

Theorem 4.8. Let ρ > 0, β > 0, and ξ > 0. Then

∣∣RcBCC,k −RcBCC

∣∣ a.s.−→ 0, as N →∞, ∀k. (4.46)

RcBCC is the large-system secrecy rate under transmit channel correlation, given by

RcBCC =

log2

1 + ωρE22+ ξρ

β(1+ω)2E12

ρE22+(1+ω)2E12

1 + ρE22

(1+ω)2E12

+

, (4.47)

where

ω = E[

T(1 + ω)

ξ(1+ω)+βT

], Eij =E

[Ti

(ξ(1+ω)+βT)j

]. (4.48)

The expectations in (4.48) are taken over the random variable T, whose distribution function Λ(t)

is the limiting eigenvalue distribution of the correlation matrix R.

Proof. The secrecy rate in the presence of channel correlation is given in (3.49). As N,K →∞,

the quantities in (3.53), (3.54), and (3.55), respectively converge almost surely to MUHARAR AND

EVANS [2009]

A = ω, B =βE22

1− βE22, and ζ = − β

(1 + ω)2

∂ω

∂ξ, (4.49)

where ∂ω∂ξ = −(1+ω)2E12

1−βE22and ω, E12 and E22 are defined in (4.48). Then (4.46) follows from

(3.49), (4.49), by applying the continuous mapping theorem, the Markov inequality, and the Borel-

Cantelli lemma COUILLET AND DEBBAH [2011].

We note that (4.47) is a non-random function of ρ, β, and ξ. The large-system secrecy sum-rate

ScBCC in the presence of channel correlation is given by

ScBCC = KRBCC. (4.50)

Corollary 4.9. In the case of no channel correlation, RcBCC reduces to RBCC in (4.3).


Proof. When no channel correlation is present, we have R = I and Λ(t) = δ(t − 1), which

implies ω = g(β, ξ) and E12 = E22 = [ξ(1 + g(β, ξ)) + β]−2. Therefore A = g(β, ξ),

B = βg(β,ξ)ξ(1+g(β,ξ))2+β

, and ζ = βg(β,ξ)ξ(1+g(β,ξ))2+β

, and Corollary 4.9 follows.

4.5.2 Selection of the Optimal Regularization Parameter

The value of ξ affects the large-system secrecy rate RcBCC under transmit channel correlation in

(4.47). We now study the optimal regularization parameter ξcBCC that maximizes RcBCC under

transmit-side channel correlation.

Theorem 4.10. The regularization parameter ξcBCC that maximizes the secrecy rate RcBCC under

transmit-side channel correlation can be obtained by solving the fixed-point equation

ξ = β(ω2 − 1)E12 − ρE22

2ρω(1 + ω)E12. (4.51)

Proof. We obtain (4.51) by defining MUHARAR AND EVANS [2009]

c1 = (1 + ω)2E12, (4.52)

c2 = (βE22 + ξc1)(ρE22 + c1), (4.53)

c3 =βE22 + ξ(1 + ω)2E12

ρE22 + (1 + ω)2E12, (4.54)

c4 =ρ2c3ωc5

βc2, (4.55)

c5 =2ω′(1 + ω)E12E22 + (1 + ω)2(E′12E22 − E′22E12)

ω′, (4.56)

with

E′12E22 − E′22E12 = −2β(1 + ω + ξω′)(E13E33 − E223), (4.57)

and solving

∂RcBCC

∂ξ=

ω′

2RcBCCρc2

1β(1+ ρE22

c1)2 log 2

[c4(ρξ−β)(c2

1+ρc1E22)β + ρ2c5(β + ρωc3)]

= 0.

(4.58)


We note that ω, E12, and E22 in (4.51) depend on ξ. It is easy to verify that the value of the

regularization parameter ξcBCC that maximizes RcBCC differs from the value ξcBC = β/ρ that max-

imizes the rates in the MISO BC without secrecy requirements MUHARAR AND EVANS [2009].

Moreover in the presence of secrecy requirements, the channel correlation affects the optimal reg-

ularization parameter.

4.5.3 Comparison to Other Linear Schemes

We now provide large-system approximations for the secrecy rates achievable by channel inversion

precoding and single user beamforming in the presence of channel correlation. These approxima-

tions were obtained from some of the results given in MUHARAR AND EVANS [2009].

Channel Inversion Precoding

As the regularization parameter ξ → 0, we have the channel inversion (CI) precoder. The aim of

the CI precoder is to cancel all the interference and information leakage. With CI precoding the

transmitted signal is

x =

1√ζ0

H†(HH†

)−1u for β ≤ 1

1√ζ0

(H†H

)−1H†u for β > 1

(4.59)

where ζ0 is the power normalization constant. It can be shown that in the large-system regime, the

secrecy rate achievable by CI precoding under transmit-side channel correlation is given by

RcBCC =

log2

(1 + ρ

κ0

)for β ≤ 1

log2

1+ ρ

ρ(β−1)+β2ζ01+

ρ(β−1)

ζ0β2

+

for β > 1, for ξ → 0 (4.60)

with κ0 = β(∫ tdΛ(t)

1+c1t)−1 and ζ0 = (β − 1)(

∫ dΛ(t)t )−1.

Single User Beamforming

As the regularization parameter ξ → ∞, we have the single user beamformer (SUB). Here, the

transmitter beamforms in a direction such as to maximize the signal strength of each user, without


taking into account the interference it creates and the amount of resulting information leakage.

The SUB is given by x =(ζ∞

)− 12

H†u, where ζ∞ is the power normalization constant. It can be

shown that the secrecy rate achievable by SUB in the large-system regime and under transmit-side

channel correlation is

RcBCC =

log2

1 + ρE2[T]β(ρE[T2]+E[T])

1 + ρE[T2]E[T]

+

, for ξ →∞. (4.61)

We note that when β ≥ 1, SUB achieves RcBCC = 0 ∀ρ. When β < 1, there is always a value

of ρ beyond which RcBCC = 0. Such poor performance is due to the intended user suffering

from a large amount of interference, while the malicious users may cancel the interference by

cooperating.

4.5.4 Numerical Results

We now provide simulation results to confirm the accuracy of the analysis, and to show the per-

formance of RCI precoding in the presence of channel correlation. Calculating RcBCC in (4.47)

requires the limiting eigenvalue distribution of the correlation matrix R. In the following, we

consider the Toeplitz-exponential model GRAY [2005], where R has a Toeplitz structure and its

entries follow the distribution rij = ν|i−j|, governed by the correlation coefficient ν ∈ [0, 1].

It can be shown that under Toeplitz-exponential correlation, the large-system secrecy rate RcBCC

reduces to

RcBCC =

log2

1+ωρcTE+ ξρ

β(1+ω)2

ρcTE+(1+ω)2

1 + ρcTE

(1+ω)2

+

, (4.62)

with cTE = ξ(1+ω)(1+ν2)+β(1−ν2)ξ(1+ω)(1−ν2)+β(1+ν2)

.

Figure 4.10 compares the per-antenna secrecy sum-rate approximation ScBCC/N under transmit-

side correlation to the simulated ergodic per-antenna secrecy sum-rate ScBCC/N with a finite num-

ber of users, for ν = 0.5 and three values of β. The values of ScBCC and ScBCC were obtained from

(4.50) and (3.48), respectively, using ξ = ξcBCC obtained as a positive solution to (4.51). We note

that the accuracy of the analysis decreases with ρ. The loss of accuracy is due to the limitations of

the tools used to derive the deterministic approximations WAGNER et al. [2012]. One can increase

the accuracy at any given SNR ρ <∞ by increasing the dimension N .


0 5 10 15 200

1

2

3

4

SNR, ρ [dB]


ScBCC/N (N = 10)

ScBCC/N (N = 20)

ScBCC/N (N = 30)

ScBCC/N

β = 1.2

β = 0.8

β = 1

Figure 4.10: Comparison between the simulated ergodic per-antenna secrecy sum-rate ScBCC/N

from (3.48) and the large-system approximation ScBCC/N from (4.50), for ν = 0.5 and various

values of β.

Figure 4.11 shows the relative secrecy rate loss (RBCC−RcBCC)/RBCC due to channel correlation,

with RBCC and RcBCC from (4.3) and (4.47), respectively. We observe how low-to-moderate

correlation , i.e., ν < 0.4, does not affect the secrecy rate excessively, i.e., the loss is less than

10%. However, higher correlation can significantly degrade the performance of the RCI precoder,

especially at low SNR.

Figure 4.12 shows that using the large-system regularization parameter ξcBCC from (4.51) does not

cause a significant loss in the secrecy sum-rate compared to using a finite-system regularization

parameter ξ = ξc?BCC optimized by bi-sectional search for each channel realization. The comple-

mentary cumulative distribution function (CCDF) of the normalized secrecy sum-rate difference

(RcBCC(ξc?BCC)−RcBCC(ξcBCC))/RcBCC(ξc?BCC) between using ξc?BCC and ξcBCC depends on the val-

ues of SNR and β, where large SNR and β = 1 represent the worst case. The CCDFs in Fig. 4.12

were obtained for β = 1, and ν = 0.5. Since the average normalized secrecy sum-rate difference

is 2% percent or less for all values of N , the large-system regularization parameter ξcBCC from

(4.51) may be used instead of the finite-system regularization parameter with only a small loss of

performance. Such choice avoids the computation of ξc?BCC for each channel realization.


0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

Correlation coefficient, ν

Relative

secrecyrate

loss

ρ = 0dB

ρ = 5dB

ρ = 10dB

ρ = 15dB

ρ = 20dB

Figure 4.11: Relative secrecy rate loss (RBCC − RcBCC)/RBCC as a function of the correlation

coefficient ν, for β = 0.8.

4.6 Conclusion

In this chapter, we studied the large-system performance of RCI precoding in the MISO BCC.

We obtained deterministic approximations for the achievable secrecy rates under Rayleigh fading,

and we derived expressions for the optimal regularization parameter ξ and network load β. The

analysis proved to be accurate even for finite-size systems. We found that for β > 1 the RCI

precoder performs poorly in the high-SNR regime. We therefore proposed a linear precoder based

on RCI and power reduction (RCI-PR) to increase the high-SNR performance for network loads

in the range 1 < β < 2. The proposed RCI-PR precoder was showed to achieve a secrecy rate

with the same high-SNR scaling factor as both the following upper bounds: (i) the sum-rate of the

optimal RCI precoder in the MISO BC without secrecy requirements, and (ii) the secrecy capacity

of a single-user system without interference. We then studied the performance of RCI precoding

in the presence of CSI estimation error, and determined: (i) how the CSI error must scale with

the SNR in order to ensure a constant high-SNR rate gap to the case with perfect CSI, so that

the multiplexing gain is not affected, (ii) the minimum amount of channel feedback required to

the users in an FDD system , and (iii) the optimum amount of channel training in a TDD system.

We further extended the large-system analysis to the MISO BCC under transmit-side channel

72 Conclusion 4.6

0 0.01 0.02 0.03 0.04 0.050

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized throughput difference

CCDF

N = 5, ρ = 10dB, mean = 2.1 × 10−2

N = 10, ρ = 10dB, mean = 6.2 × 10−3

N = 20, ρ = 10dB, mean = 1.5 × 10−3

N = 40, ρ = 10dB, mean = 4.2 × 10−4

N = 40, ρ = 30dB, mean = 4.3 × 10−3

Figure 4.12: CCDF of the normalized secrecy rate difference between using: (i) ξcBCC obtained

from (4.51) and (ii) ξc?BCC obtained by bi-sectional search for every channel realization, for β = 1,

and ν = 0.5.

correlation. We found that low-to-moderate correlation only partially affects the secrecy rates.

However, high correlation degrades the performance of the RCI precoder, especially at low SNR.

Chapter 5

Physical Layer Security in a Random

Field of Eavesdroppers

Summary In this chapter, we introduce the MISO broadcast channel with confidential messages

and external eavesdroppers (BCCE), where a multi-antenna base station simultaneously commu-

nicates to multiple malicious users, in the presence of randomly located external eavesdroppers.

We study the performance of RCI precoding in the BCCE and provide explicit expressions for the

large-system probability of secrecy outage and mean secrecy rate with respect to the spatial distri-

bution of the nodes and to the fluctuations of their channels. Our analysis is shown to be tight via

simulations. We find that, irrespective of the collusion strategy at the external eavesdroppers, a

large number of transmit antennas drives both the probability of secrecy outage and the rate loss

due to the presence of external eavesdroppers to zero. Increasing the density of eavesdroppers by

a factor n, requires n2 as many antennas to meet a given probability of secrecy outage and a given

mean secrecy rate.

73

74 Introduction to the BCCE 5.1

5.1 Introduction to the BCCE

I n this section we introduce the MISO broadcast channel with confidential messages and ex-

ternal eavesdroppers (BCCE), where not only malicious users but also nodes external to the

network can act as eavesdroppers. This can be the case in a practical system, where external

nodes are randomly scattered in space. These nodes must be regarded as potential eavesdroppers,

otherwise the system would be vulnerable to secrecy outage. The BCCE therefore represents a

practical scenario that needs to be addressed. Similarly to the BCC, we assume homogeneous

users, i.e., each user experiences the same average received signal power. This models a scenario

where users are located at equal distances from the BS, or where the BS employs power control

to guarantee the same average received signal power. Without loss of generality, such power can

be assumed unitary. Although the results presented in this chapter can be easily extended to the

non-homogeneous case, the assumption of homogeneous users allows analytical tractability, and

it yields to closed form expressions for the SINRs at the legitimate user and at the malicious users

in the large-system regime.

The BCCE can be obtained from the BCC by including external single-antenna eavesdroppers in

the system. For the sake of tractability, in this thesis the external eavesdroppers are assumed to be

distributed on the two-dimensional plane according to a Poisson point process (PPP) Φe of density

λe STOYAN et al. [1996]. Figure 5.1 shows an example of BCCE, where the BS is at the origin,

and the users lie on a disc of radius 1. As a worst-case scenario, we assume that each eavesdropper

can cancel the interference caused by the remaining K − 1 messages. Assuming that the BS lies

at the origin, the SINR γe,k for the kth message at a generic eavesdropper located in e is then given

by

γe,k =

∣∣∣h†ewk

∣∣∣2‖e‖ησ2

(5.1)

where wk is the precoding vector for user k, h†e is the channel vector between the base station

and the eavesdropper in e, and it takes into account the Rayleigh fading, and η is the path loss

exponent. Some of the results provided in this chapter assume a path loss exponent η = 4. In this

special case, which is a reasonable value for η in a shadowed urban area RAPPAPORT [1996], it is

possible to obtain more compact expressions for quantities of interest, such as the probability of

secrecy outage and the mean secrecy rate.

5.1 Physical Layer Security in a Random Field of Eavesdroppers 75

−5 0 5−5

−4

−3

−2

−1

0

1

2

3

4

5

y

x

Multi-antenna base station

Malicious users

External eavesdroppers

Figure 5.1: Example of a BCCE with K = 5 malicious users and a density of external eavesdrop-

pers λe = 0.2.

The precoding vector wk is calculated independently of h†e, therefore they are independent isotropic

random vectors. The channel h†e has unit norm, whereas the precoding vector wk has norm 1√K

be-

cause it is obtained after the normalization ‖W‖2 =∑K

k=1 ‖wk‖2 = 1. The inner product h†ewk

is a linear combination of N complex normal random variables, therefore∣∣∣h†ewk

∣∣∣2 ∼ exp( 1K ).

In the following, we consider two types of external eavesdroppers, namely non-colluding eaves-

droppers and colluding eavesdroppers. In the non-colluding case, the eavesdroppers individually

overhear the communication without centralized processing. In the colluding eavesdroppers case,

all eavesdroppers are able to jointly process their received messages at a central data processing

unit. The secrecy rate RBCCE,k achievable by the kth user in the BCCE is given by

RBCCE,k =[log2

(1 + γk

)− log2

(1 + max (γM,k, γE,k)

)]+, (5.2)

where γE,k is the resulting SINR of the PPP of external eavesdroppers for the kth message. The

secrecy rate RBCCE,k is therefore affected by the maximum of the SINR γM,k at the alliance of

malicious users and the SINR γE,k at the external eavesdroppers. In the case of non-colluding

eavesdroppers, γE,k is the SINR at the strongest eavesdropper. In the case of colluding eavesdrop-

pers, all eavesdroppers can perform joint processing, and they can, therefore, be seen as a single

multi-antenna eavesdropper. After interference cancellation, each eavesdropper receives the use-

76 Probability of Secrecy Outage 5.2

ful signal embedded in noise, and the optimal receive strategy at the colluding eavesdroppers is

maximal ratio combining (MRC) which yields to an SINR γE,k =∑

e∈Φeγe,k given by the sum

of the SINRs γe,k at all eavesdroppers generated by the Poisson point process (PPP) Φe.

The achievable secrecy sum-rate is denoted by SBCCE and defined as

SBCCE =K∑k=1

RBCCE,k. (5.3)

5.2 Probability of Secrecy Outage

In this section, we derive the secrecy outage probability, i.e., the probability that the secrecy rate

RBCCE,k achievable by user k with RCI precoding in the BCCE is zero, for both cases of non-

colluding and colluding eavesdroppers. Then we study the secrecy outage probability in the large-

system regime, and determine how the number of antennas N must scale in order to guarantee a

given secrecy outage probability. The secrecy outage probability for user k is defined as

OBCCE,k , P(RBCCE,k = 0) =

1 if γk ≤ γM,k

P(γE,k ≥ γk | γk) otherwise(5.4)

As discussed in Chapter 4, RCI precoding ensures γk > γM,k in most cases. Therefore, the secrecy

outage probability is often given by the probability that RBCCE,k is driven to zero by the presence

of external eavesdroppers.

The results provided in this chapter can be extended to obtain the probability that the achievable

secrecy rate RBCCE,k is smaller than a target rate RT . In this case, the probability of outage

would be a function of RT . We note that for Gaussian fading channels, there is always a non-

zero probability that the legitimate user’s channel is better than that of the eavesdroppers, but

this probability is smaller than one, hence there is always a non-zero secrecy outage probability.

Moreover, we note that the analysis provided in this section leads to a secrecy outage probability

which is a function of the SINR at the legitimate user and at the malicious users. These SINRs can

be calculated at the BS since the channels at the users are known and deterministic.


5.2.1 Non-colluding Eavesdroppers

In the case of non-colluding eavesdroppers, γE,k is the SINR at the strongest eavesdropper E,

given by

γE,k = maxe∈Φe

γe,k = maxe∈Φe

∣∣∣h†ewk

∣∣∣2‖e‖ησ2

. (5.5)

In the case of non-colluding eavesdroppers, OBCCE,k is the probability that any eavesdropper has

an SINR greater than or equal to the SINR of the legitimate user k. We obtain the following result.

Lemma 5.1. The secrecy outage probability for user k in the presence of non-colluding eaves-

droppers is given by

OBCCE,k =

1 if γk ≤ γM,k

1− exp

[−

2πλeΓ(

2η

)η(Nβσ2γk)

2η

]otherwise

(5.6)

where Γ(·) is the gamma function defined as

Γ(z) ,∫ ∞

0tz−1e−tdt. (5.7)

Proof. See Appendix B.1.

By applying results from random matrix theory (RMT) COUILLET AND DEBBAH [2011], we

now obtain the large-system secrecy outage probability OBCCE in the presence of non-colluding

eavesdroppers. Like in Chapter 4, the large-system results are obtained for the case when both

the number of transmit antennas N and the number of users K grow to infinity in a fixed ratio

β = K/N . In order to simplify the notation in the rest of this chapter, we find it useful to define

the quantity µ , π32

2√βσ2

.

Theorem 5.2. The secrecy outage probability in the presence of non-colluding eavesdroppers

satisfies

|OBCCE,k −OBCCE|a.s.−→ 0, as N →∞, ∀k (5.8)

where

OBCCE =

1 if γ ≤ γM

1− exp

[−

2πλeΓ(

2η

)η(Nβσ2γ)

2η

]otherwise

(5.9)

and with γ and γM given by (4.5) and (4.6), respectively.

78 Probability of Secrecy Outage 5.2

Proof. Theorem 5.2 follows from Lemma 5.1, by noting that |γk−γ| a.s.−→ 0 and |γM,k−γM |a.s.−→

0 as N →∞, and by the continuous mapping theorem BILLINGSLEY [1995].

Corollary 5.3. If γ > γM and η = 4, then (i) the number of transmit antennas required in

order to guarantee a large-system secrecy outage probability OBCCE < ε in the presence of non-

colluding eavesdroppers is N >(µλeε√γ

)2, and (ii) the large-system secrecy outage probability

OBCCE decays as 1√N

.

Proof. The proof follows from Theorem 5.2, by noting that Γ(

12

)=√π, and that 1 − e−x > x

for 0 < x < 1.

A special case of the previous scenario is the one where only the eavesdropper which is nearest to

the base station attempts to eavesdrop. In this case we have

γE,k =

∣∣∣h†Ewk

∣∣∣2‖E‖ησ2

(5.10)

where

E = argmine∈Φe

‖e‖. (5.11)

Lemma 5.4. The secrecy outage probability for user k, caused by the external eavesdropper

nearest to the base station, under a path loss exponent η = 4, is given by

OBCCE,k =

1 if γk ≤ γM,k

2µλe√Nγk

exp(µ2λ2

eπNγk

)Q(µλe

√2

πNγk

)otherwise

(5.12)

where Q(·) is the Q-function defined as

Q(x) ,1√2π

∫ ∞x

exp

(−u

2

2

)du. (5.13)


By applying results from RMT, we now obtain the large-system secrecy outage probabilityOBCCE

caused by the eavesdropper which is nearest to the base station.

Theorem 5.5. The secrecy outage probability for user k, caused by the external eavesdropper

nearest to the base station, under a path loss exponent η = 4, satisfies



where

OBCCE =

1 if γ ≤ γMµλe√N

(1 + µ2λ2

eπN

)(1− 2µλe

π√N

)otherwise

(5.15)


Proof. Theorem 5.5 follows from Lemma 5.4, by first-order Taylor approximation of (5.12), by

noting that |γk − γ| a.s.−→ 0 and |γM,k − γM |a.s.−→ 0 as N → ∞, and by the continuous mapping

theorem BILLINGSLEY [1995].

5.2.2 Colluding Eavesdroppers

The colluding eavesdroppers case represents a worst-case scenario. In this case, all eavesdroppers

can perform joint processing, and they can therefore be seen as a single multi-antenna eavesdrop-

per. After interference cancellation, each eavesdropper receives the useful signal embedded in

noise, and the optimal receive strategy at the colluding eavesdroppers is maximal ratio combining

(MRC). This yields to an SINR γE,k at the colluding eavesdroppers given by

γE,k =1

σ2

∑e∈Φe

‖e‖−η∣∣∣h†ewk

∣∣∣2 . (5.16)

Lemma 5.6. The secrecy outage probability for user k in the presence of colluding eavesdroppers,

under a path loss exponent η = 4, is given by

OBCCE,k =

1 if γk ≤ γM,k

1− 2Q(µλe

√π

2Nγk

)otherwise

(5.17)


By applying results from RMT, we now obtain the large-system secrecy outage probabilityOBCCE

in the presence of colluding eavesdroppers.

Theorem 5.7. The secrecy outage probability in the presence of colluding eavesdroppers, under

a path loss exponent η = 4, satisfies


80 Mean Secrecy Rates 5.3

where

OBCCE =

1 if γ ≤ γM1− 2Q

(µλe

√π

2Nγ

)otherwise

(5.19)


Proof. Theorem 5.7 follows from Lemma 5.6, by noting that Γ(

12

)=√π, that |γk − γ| a.s.−→ 0

and |γM,k − γM |a.s.−→ 0 as N → ∞, and by the continuous mapping theorem BILLINGSLEY

[1995].

Corollary 5.8. Let γ > γM and η = 4, then (i) the number of transmit antennas required in order

to guarantee a large-system secrecy outage probability OBCCE < ε in the presence of colluding

eavesdroppers is N >(µλeε√γ

)2, and (ii) the large-system outage probability OBCCE decays as

1√N

.

Proof. The proof follows from Theorem 5.7 and by using 1− 2Q(x) <√

2πx for 0 < x < 1.

Remark 5.1. By comparing the results in Corollary 5.3 and Corollary 5.8, we can conclude

that (i) the collusion among eavesdroppers does not significantly affect the number of transmit

antennas N required to meet a given probability of secrecy outage in the large-system regime, and

(ii) increasing the density of eavesdroppers λe by a factor n requires increasing N by a factor n2

in order to meet a given probability of secrecy outage.

5.3 Mean Secrecy Rates

In this section, we derive the mean secrecy rates, averaged over the location of the external eaves-

droppers, achievable by RCI precoding in the BCCE, for both cases of non-colluding and collud-

ing eavesdroppers. We then study the mean secrecy rates in the large-system regime, and derive a

bound on the secrecy rate loss due to the presence of external eavesdroppers. Finally, we propose

a rule for the choice of the regularization parameter of the precoder that maximizes the mean of

the large-system secrecy rate.


5.3.1 Mean Secrecy Rate

We now obtain the following result for the mean secrecy rate at user k.

Lemma 5.9. The mean secrecy rate achievable at user k by RCI precoding in the BCCE is given

by

EΦe [RBCCE,k] =

0 if γk ≤ γM,k

log2(1+γk)

1−OBCCE,k

(1+γM,k)1−PBCCE,k

−∫ γkγM,k

log2(1 + y)fγE,k(y) dy otherwise

(5.20)

In (5.20), PBCCE,k is the probability that the SINR γE,k at the external eavesdroppers is greater

than or equal to the SINR γM,k at the malicious users, and for a path loss exponent η = 4 it is

given by

PBCCE,k , P(γE,k ≥ γM,k) =

1− exp

(− µλe√

NγM,k

)for non-colluding eavesdroppers

1− 2Q

(µλe

√π

2NγM,k

)for colluding eavesdroppers

(5.21)

and fγE,k(y) is the pdf of the SINR at the external eavesdroppers, given by

fγE,k(y) =

µλey

− 32

2√N

exp(− µλe√

Ny


µλey− 3

2

2√N

exp(−πµ2λ2

e4Ny


(5.22)


By applying results from RMT, we now obtain the large-system mean secrecy rate RBCCE achiev-

able by RCI precoding in the BCCE.

Theorem 5.10. The mean secrecy rate achievable for user k by RCI precoding in the BCCE

satisfies

|EΦe [RBCCE,k]−RBCCE|a.s.−→ 0, as N →∞, ∀k. (5.23)

RBCCE denotes the mean secrecy rate in the large-system regime, given by

RBCCE =

0 if γ ≤ γM

log2(1+γ)1−OBCCE

(1+γM)1−P

BCCE−∫ γγM

log2(1 + y)fγE,k(y) dy otherwise(5.24)

82 Mean Secrecy Rates 5.3

In (5.24), PBCCE is the probability that the SINR γE,k at the external eavesdroppers is greater

than or equal to the large-system SINR γM at the malicious users, and for η = 4 it is given by

PBCCE , P(γE,k ≥ γM ) =

1− exp

(− µλe√

NγM


1− 2Q

(µλe

√π

2NγM


(5.25)

Proof. Theorem 5.10 follows from Lemma 5.9, by replacing γk and γM,k with their respective de-

terministic approximations γ and γM , by applying the continuous mapping theorem, the Markov

inequality, and the Borel-Cantelli lemma BILLINGSLEY [1995].

5.3.2 Secrecy Rate Loss due to the External Eavesdroppers

By comparing the large-system mean secrecy rate of the MISO BCCE in (5.24) to the large-system

secrecy rate of the MISO BCC without external eavesdroppers in (4.3), for a given regularization

parameter ξ, we can evaluate the secrecy rate loss ∆e due to the presence of external eavesdrop-

pers, defined as

∆e , RBCC −RBCCE. (5.26)

We now obtain an upper bound on the secrecy rate loss ∆e.

Corollary 5.11. The secrecy rate loss ∆e due to the presence of external eavesdroppers satisfies

∆e ≤ ∆UBe ,

Cµλe√N, (5.27)

whereCµ is a constant independent ofN , λe, and of the cooperation strategy at the eavesdroppers,

given by

Cµ = µ

[RBCC√γ

+(√

γ −√γM

)+]. (5.28)

Proof. See Appendix B.5

Remark 5.2. It follows from Corollary 5.11 that, irrespective of the collusion strategy at the

external eavesdroppers, (i) as the number N of transmit antennas grows, the secrecy rate loss ∆e


tends to zero as 1√N

, and (ii) increasing the density of eavesdroppers λe by a factor n requires

increasing N by a factor n2 in order to meet a given value of ∆UBe .

5.3.3 Optimal Regularization Parameter

The value of the regularization parameter ξ has a significant impact on the secrecy rates. The

optimal large-system regularization parameters of the RCI precoder ξBC and ξBCC, for the MISO

BC and for the MISO BCC, respectively, were given in Chapter 4. In the MISO broadcast channel

with confidential messages and external eavesdroppers (BCCE), we denote by ξBCCE the regular-

ization parameter that maximizes the large-system mean secrecy rate. The value of ξBCCE can be

obtained by numerically solving the following equation

ξBCCE , arg maxξ

RBCCE (5.29)

with RBCCE given in (5.24). Since the secrecy rate of the MISO BCCE is affected by the SINR at

the external eavesdroppers, the optimal large-system regularization parameter ξBCCE is not just a

function of β and ρ, like ξBC and ξBCC, but it also depends on the number of transmit antennas N ,

the density of the eavesdroppers λe, and their collusion strategy. The value of ξBCCE should be

found as a compromise between: (i) maximizing the SINR γ at the legitimate user, and (ii) trading

off the SINR γM at the malicious users and the probability PBCCE that the external eavesdroppers

are more harmful than the malicious users. We have the following two extreme cases.

Lemma 5.12. The optimal large-system regularization parameter ξBCCE follows the trend:

ξBCCE → ξBCC as λe → 0

ξBCCE → ξBC as λe →∞(5.30)

Proof. For low densities λe, we have by Corollary 5.11 that RBCCE approaches RBCC, therefore

ξBCCE approaches ξBCC. For high densities λe, we have PBCCE,k = P(γE,k ≥ γM,k) → 1, and

the secrecy rateRBCCE,k in (5.2) is determined solely by γk and γE,k. Since γE,k does not depend

on ξ, maximizing the mean rate coincides with the rate maximization problem for the BC, and its

solution in the large-system regime is given by ξBC.

84 Numerical Results 5.4

5.4 Numerical Results

In this section, we provide numerical results to show the performance of RCI precoding in the

BCCE, under a path loss exponent η = 4. We consider finite-size systems, and simulate the

probability of secrecy outage, the secrecy rate, and the optimal regularization parameter of the

precoder, in different scenarios and under different system dimensions, network loads, SNRs, and

densities of eavesdroppers. The simulations show that many results obtained in Section 5.2 and

Section 5.3 by using random matrix theory and stochastic geometry tools hold even for networks

with a small number of users and transmit antennas and randomly located eavesdroppers.

In Fig. 5.2 we compare the simulated probability of outageOBCCE,k under non-colluding and col-

luding eavesdroppers, respectively, to the large-system results OBCCE provided in Theorem 5.2

and Theorem 5.7, respectively. In the simulations, the regularization parameter ξBCC in (4.10)

was used. We observe that for λe = 0.1 and small probabilities of secrecy outage, (i) N >(µλe

0.1√γ

)2= 34 yields to a secrecy outage probability smaller than 0.1, (ii) the secrecy outage

probability decays as 1√N

, and (iii) the collusion of eavesdroppers does not significantly affect

the probability of secrecy outage. All these observations are consistent with Corollary 5.3, Corol-

lary 5.8, and Remark 5.1.

In Fig. 5.3 we compare the simulated ergodic per-antenna secrecy sum-rate under non-colluding

and colluding eavesdroppers, to the large-system results from Theorem 5.10, for λe = 0.1, N =

10, ξ = ξBCC, and various values of β. We note that the accuracy of the large-system analysis

decreases with the SNR. The loss of accuracy is due to the limitations of the tools used from

RMT WAGNER et al. [2012]. Moreover, we note that the per-antenna secrecy sum-rate does not

monotonically increase with the SNR. This is due to the fact that in the worst-case scenario the

malicious users and the external eavesdroppers can cancel the interference, whereas the legitimate

user is interference-limited in the high-SNR regime. This is consistent with the case of β > 1 in

the BCC, studied in Chapter 4.

In Fig. 5.4 we compare the simulated ergodic per-user secrecy rate under non-colluding and col-

luding eavesdroppers, to the large-system results from Theorem 5.10, for β = 1, ρ = 10dB,

ξ = ξBCC, and various values of λe. We note that the accuracy of the large-system analysis in-

creases with N . Moreover, we observe that the expectation of the per-user secrecy rate increases

with N , and this benefit is more for larger values of λe. This happens because the mean received


5 10 15 20 25 30 35 400

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8


Probabilityofsecrecyoutage

Ok non-coll. (Simulations)O non-coll. (Analysis)Ok colluding (Simulations)O colluding (Analysis)

λe = 0.1, 0.2, 0.4

Figure 5.2: Comparison between the simulated probability of outage OBCCE,k and the large-

system results OBCCE provided in Theorem 5.2 and Theorem 5.7, for a network load β = 1, an

SNR ρ = 10dB, and various values of λe.

0 5 10 15 200

0.5

1

1.5

2

2.5

SNR, ρ [dB]


Simulations (non-coll.)Analysis (non-coll.)Simulations (colluding)Analysis (colluding)

β = 1.2

β = 0.8

β = 1

Figure 5.3: Comparison between the simulated ergodic per-antenna secrecy sum-rate under non-

colluding and colluding eavesdroppers, and the large-system results from Theorem 5.10, for λe =

0.1, N = 10 transmit antennas, and various values of the network load β.


5 10 15 20 25 30 35 400

0.5

1

1.5

2


Per-usersecrecyrate

Simulations (non-coll.)Analysis (non-coll.)Simulations (colluding)Analysis (colluding)

λe = 0.1, 0.2, 0.4

Figure 5.4: Comparison between the simulated ergodic per-user secrecy rate E[Rk] under non-

colluding and colluding eavesdroppers, and the large-system results R from Theorem 5.10, for a

network load β = 1, an SNR ρ = 10dB, and various values of λe.

power at each external eavesdropper scales as 1βN , hence having more transmit antennas makes

the system more robust against external eavesdroppers.

In Fig. 5.5 we compare the simulated per-user secrecy rate of (i) the BCCE with non-colluding

eavesdroppers, (ii) the BCCE with colluding eavesdroppers, and (iii) the BCC without external

eavesdroppers, for β = 1, ρ = 10dB, ξ = ξBCC, and various values of λe. We note that in the

BCC, the per-user secrecy rate is almost constant with N , for a fixed network load β. On the

other hand, the per-user secrecy rate of the BCCE increases with N . Again, this happens because

the mean received power at each external eavesdropper scales as 1βN , hence having more transmit

antennas makes the system more robust against external eavesdroppers. We also note that for

higher densities of eavesdroppers λe, larger values of N are required to achieve a given per-user

secrecy rate of the BCCE. More precisely, increasing λe by a factor 2, requires increasing N by a

factor 4. Moreover, the collusion of external eavesdroppers does not affect the scaling law of the

mean rate. These observations are consistent with Remark 5.2.

Figure 5.6 compares the large-system regularization parameter ξBCCE given by (5.29) to the value

ξBCCE that maximizes the average simulated secrecy sum-rate SBCCE, for a finite system with


5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


Per-usersecrecyrate

RBCC,k (λe = 0)

RBCCE,k (non-coll.)

RBCCE,k (colluding)

λe = 0.1, 0.2, 0.4

Figure 5.5: Comparison between the simulated ergodic per-user secrecy rates of: (i) the BCCE

with non-colluding eavesdroppers, (ii) the BCCE with colluding eavesdroppers, and (iii) the BCC

without external eavesdroppers, for a network load β = 1, an SNR ρ = 10dB, and various values

of λe.


10−2

10−1

100

101

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0.11

Density of external eavesdroppers, λe

Optimalregularizationparameter

ξBCCE (non-coll.)ξBCCE (non-coll.)

ξBCCE (colluding)ξBCCE (colluding)

ξBCC

ξBC

Figure 5.6: Comparison between the large-system regularization parameter ξBCCE in (5.29) and

the value ξBCCE that maximizes the average simulated secrecy sum-rate SBCCE for a finite system

with N = 10 transmit antennas, a network load β = 1, and an SNR ρ = 10dB.

N = 10, β = 1, and ρ = 10dB. The figure shows that for low densities of eavesdroppers λe,

ξBCCE tends to ξBCC = 0.0273, whereas for high densities λe, it tends to ξBC = 0.1. These ob-

servations are consistent with Lemma 5.12. The finite-system parameter ξBCCE follows a similar

trend. We note that both ξBCCE and ξBCCE are smaller in the case of non-colluding eavesdrop-

pers, and this can be explained as follows. A smaller value of ξ generates a smaller information

leakage to the malicious users. Therefore, it is especially desirable to have a smaller ξ when the

malicious users are the main concern, i.e., when their SINR is larger than the SINR at the ex-

ternal eavesdroppers, and this is more likely to happen when the external eavesdroppers are not

colluding.

Figure 5.7 shows that using the regularization parameter ξBCCE, obtained from large-system anal-

ysis, does not cause a significant loss compared to using the optimal parameter ξ?BCCE, optimized

for each realization of the channels and of the locations of the external eavesdroppers. The fig-

ure shows the mean secrecy sum-rate difference SBCCE(ξ?BCCE) − SBCCE(ξBCCE) normalized

by the mean optimal SBCCE(ξ?BCCE), simulated for finite-size systems, β = 1, various values of

the density of eavesdroppers λe, and various values of the SNR ρ. Figure 5.7 was obtained for

colluding eavesdroppers, but a similar trend was observed for non-colluding eavesdroppers. We


5 10 15 20 25 30 35 400

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08


Norm

alizedthroughputdifferen

ce

λe = 0.1, ρ = 0dBλe = 0.4, ρ = 0dBλe = 0.1, ρ = 20dBλe = 0.4, ρ = 20dB

Figure 5.7: Normalized mean secrecy sum-rate difference between using ξ?BCCE, that maximizes

each realization of the secrecy sum-rate SBCCE, and ξBCCE, obtained from large-system analysis

in (5.29), under colluding eavesdroppers, for a network load β = 1, various values of the density

of eavesdroppers λe, and various values of the SNR ρ.

note that calculating the optimal value ξ?BCCE requires the base station to know (i) the channels H

of all users, (ii) the realization of the PPP Φe, i.e., the locations of all external eavesdroppers, and

(iii) the channels h†e of all external eavesdroppers. On the other hand, calculating ξBCCE does not

require the knowledge of any of these quantities. We observe that the normalized mean secrecy

sum-rate difference is less than 7% for all values of N , λe, and ρ, and it decreases when N grows,

e.g., falling under 3% for N = 20. As a result, one can avoid the calculation of ξ?BCCE for every

realization of H, Φe, and h†e, and ξBCCE can be used with only a small loss of performance.

5.5 Conclusion

In this chapter, we considered the broadcast channel with confidential messages and external

eavesdroppers (BCCE), where a multi-antenna base station simultaneously communicates to mul-

tiple malicious users, in the presence of randomly located external eavesdroppers. We showed that,

irrespective of the collusion strategy at the external eavesdroppers, a large number of transmit an-

tennas N drives both the probability of secrecy outage and the rate loss due to the presence of

90 Conclusion 5.5

external eavesdroppers to zero. Increasing the density of eavesdroppers λe by a factor n, requires

n2 as many antennas to meet a given probability of secrecy outage and a given mean secrecy rate.

Using the developed analysis, we clearly established the importance of the number of transmit

antennas at the BS to make communications robust against malicious users and external eaves-

dropping nodes.

Chapter 6

Physical Layer Security in Cellular

Networks

Summary In this chapter, we study physical layer security for the downlink of cellular net-

works, where each BS simultaneously transmits confidential messages to several users, and where

the confidential messages transmitted to each user can be eavesdropped by both (i) other users in

the same cell and (ii) users in other cells. The locations of BSs and mobile users are modeled as

two independent two-dimensional Poisson point processes. Using the proposed model, we analyze

the secrecy rates achievable by RCI precoding by performing a large-system analysis that com-

bines results from stochastic geometry and random matrix theory. We obtain approximations for

the probability of secrecy outage and the mean secrecy rate, and characterize regimes where RCI

precoding achieves a non-zero secrecy rate. We find that unlike isolated cells, the secrecy rate in

a cellular network does not grow monotonically with the transmit power, and the network tends to

be in secrecy outage if the transmit power grows unbounded. Furthermore, we show that there is

an optimal value for the BS deployment density that maximizes the secrecy rate, and this value is

a decreasing function of the SNR.

91

92 System Model 6.1

Figure 6.1: Illustration of a cellular network. The star denotes a typical user. The circles, squares,

and triangles denote BSs, out-of-cell users, and in-cell users, respectively, as discussed in Subsec-

tion 6.1.3.

6.1 System Model

6.1.1 Network Topology

W e consider the downlink of a cellular network, as depicted in Fig. 6.1. Each BS transmits at

power P and is equipped with N antennas. For tractability, we assume that the locations

of the BSs are drawn from a homogeneous PPP Φb of density λb. We consider single-antenna

users, and assume that each user is connected to the closest BS. The locations of the users are

drawn from an independent PPP Φu of density λu. We denote by Kb and by Kb = |Kb| the set of

users and the number of users connected to the BS b, respectively. We denote by

Hb =[‖b− b1‖−ηhb,1, . . . , ‖b− bKb‖−ηhb,Kb

]† (6.1)

the Kb×N channel matrix for the BS b, where hb,j ∼ CN (0, I) is the normalized channel vector

that accounts only for the fading between the BS b and the user j ∈ Kb.

6.1 Physical Layer Security in Cellular Networks 93

6.1.2 RCI Precoding

Transmission takes place over a block fading channel. The signal transmitted by the generic BS

b is xb = [xb,1, . . . , xb,N ]T ∈ CN×1. The vector xb is obtained via RCI precoding from the vec-

tor of confidential messages mb = [mb,1, . . . ,mb,Kb ]T , whose entries are chosen independently,

satisfying E[|mb,j |2] = 1, ∀j. The transmitted signal xb after RCI precoding can be written as

xb =√PWbmb, where Wb = [wb,1, . . . ,wb,Kb ] is the N ×Kb RCI precoding matrix, given by

PEEL et al. [2005]

Wb =1√ζb

H†b

(HbH

†b +NξIKb

)−1, (6.2)

and where ζb = tr

H†bHb

(H†bHb +NξIN

)−2

is a long-term power normalization constant.

The function of the real regularization parameter ξ is to achieve a tradeoff between the signal power

at the legitimate user and the crosstalk at the other users served by the same BS. The optimal value

for the parameter ξ in cellular networks is unknown. Since the results obtained in this chapter

hold for any value of ξ, we will now assume that each BS sets ξ to the value that maximizes the

large-system secrecy rate in an isolated cell, given by ξBCC in (4.10).

6.1.3 Malicious Users

In general, the BSs cannot determine the behavior of the users, i.e., whether they act maliciously

as eavesdroppers or not. As a worst-case scenario, we assume that for each legitimate user, all

the remaining users in the network can act as eavesdroppers. For a user o connected to the BS

b, the set of Kb − 1 malicious users within the same cell is denoted by MIo = Kb\o, and the

set formed by the rest of the malicious users in the network is denoted by MEo = Φu\Kb. In

Fig. 6.1, the legitimate user o, the set of (intra-cell) malicious usersMIo, and the set of (external)

malicious users MEo are represented by star, triangles, and squares, respectively. The total set

of malicious users for the legitimate receiver o is denoted byMo = MIo ∪ME

o = Φu\o. It is

important to make such a distinction between the intra-cell malicious users inMIo and the external

malicious users inMEo . In fact, the BS b knows the channels of the intra-cell malicious users in

MIo ⊂ Kb, and exploits this information by choosing an RCI precoding matrix Wb which is a

function of these channels. The RCI precoding thus controls the amount of information leakage

at the malicious users inMIo. On the other hand, the BS b does not know the channels of all the

94 Achievable Secrecy Rates 6.2

other external malicious users inMEo , and Wb does not depend upon these channels. Therefore,

the signal received by the malicious users inMEo is not directly affected by RCI precoding.

Table 6.1: Notation Summary

Notation Description

Φb; λb A PPP modeling the locations of BSs; deployment density of BSs

Φu; λu An independent PPP modeling the locations of users; density of users

P ; ρ Downlink transmit power for each BS; transmit SNR ρ , Pσ2

N Number of transmit antennas for each BS

c BS which is closest to the origin o

Kb; Kb = |Kb| Set of users associated with BS b; number of users associated with BS b

MIo = Kc\o Set of the Kc − 1 malicious users in the same cell as the typical user

MEo = Φu\Kc Set formed by the malicious users in all the other cells

mb Confidential messages sent by BS b to its users

xb =√PWbmb Signal transmitted by BS b

hb,j ∼ CN (0, I) Channel vector between BS b and user j

η Path loss exponent

gb,o ∼ Γ(Kb, 1) Inter-cell interference power gain from BS b to the typical user in o

gc,e ∼ exp(1) Leakage power gain from BS c to the malicious user e ∈MEo

6.2 Achievable Secrecy Rates

In this section, we derive a secrecy rate achievable by RCI precoding for the typical user in the

downlink of a cellular network.


6.2.1 SINR at a Typical User

We consider a typical user o located at the origin, and connected to the closest BS, located in

c ∈ Φb. The distance between the typical user and the closest BS is given by ‖c‖. The typical user

receives self-interference caused by the other messages mc,u, u 6= o transmitted by the BS c, and

inter-cell interference caused by the signal transmitted by all the other BSs b ∈ Φb\c. The signal

received by the typical user is given by

yo =√P ‖c‖−η h†c,owc,omc,o +

√P ‖c‖−η

∑u∈Kc\o

h†c,owc,umc,u

+∑

b∈Φb\c

√P ‖b‖−η

Kb∑j=1

h†b,owb,jmb,j + no (6.3)

where ‖b‖ is the distance between the typical user and the generic BS b, and η is the path loss

exponent. The four terms in (6.3) represent the useful signal, the crosstalk (or self-interference),

the inter-cell interference, and the thermal noise seen at the typical user, respectively. The latter is

given by no ∼ CN (0, σ2), and we define the transmit SNR as ρ , P/σ2.

We assume that the legitimate receiver at o treats the interference power as noise. The SINR γo at

the legitimate receiver o is given by

γo =ρ‖c‖−η

∣∣∣h†c,owc,o

∣∣∣2ρ‖c‖−η∑u∈Kc\o

∣∣∣h†c,owc,u

∣∣∣2 + ρ∑

b∈Φb\cgb,oKb‖b‖−η + 1

, (6.4)

where we define wb,j ,√Kbwb,j and

gb,o ,Kb∑j=1

∣∣∣h†b,owb,j

∣∣∣2 . (6.5)

6.2.2 SINR at the Malicious Users

The cell where the typical user o is located is referred to as the tagged cell. For the typical user

o, the set of malicious users is denoted byMo = MIo ∪ME

o , whereMIo = Kc\o is the set of

remaining users in the tagged cell, andMEo = Φu\Kc is the set of all users in other cells.

We assume that each malicious user can communicate directly to any other malicious user within

a cooperation radius rc around it, i.e., cooperation is possible for distances smaller than rc. This


Figure 6.2: Example of percolation in a random plane network. Dots represent malicious users,

and discs represents the cooperation range of malicious users. Two malicious users can cooperate

when their respective discs overlap.

assumption comes from the following model: assume that malicious users can transmit at a certain

power PM and that their signal is attenuated over distance according to a deterministic decreasing

function l(d). Assume also that malicious users can successfully receive data if the signal is at

least t times stronger than the ambient noise, which has power Pn. Even under the condition that

the interference is perfectly canceled, there is a maximum distance beyond which the two users

cannot cooperate. This distance, referred to as the cooperation radius, is given by

rc , max

d :

PM l(d)

Pn≥ t. (6.6)

By connecting each pair of cooperating malicious users, it is possible to generate a random plane

network GILBERT [1961], which represents an infinite cooperation network of malicious users

with range rc. It is known that if the density of users satisfies λu > 8 log 2r2c

, the random plane

network contains an infinite cluster almost surely. An illustration of this phenomenon, known as

percolation, is provided in Fig. 6.2. As a result, if the density of users λu is large enough, then

percolation will occur, and there will be an infinite set of malicious users cooperating to eavesdrop

the message intended for the typical user.

Motivated by these observations, in the following we will consider the worst-case scenario where

all the malicious users inMo can cooperate to eavesdrop on the message intended for the typical

user in o. Since each malicious user is likely to decode its own message, it can indirectly pass this


information to all the other malicious users. In the worst-case scenario, all the malicious users in

Mo can therefore subtract the interference generated by all the messages mj , j 6= o.

After interference cancellation, the signal received at a malicious user i ∈ MIo in the tagged cell

is given by

yi =√P ‖i− c‖−η h†c,iwc,omc,o + ni (6.7)

where ‖i−c‖ is the distance between the BS c and the malicious user i ∈MIo. The signal received

at a malicious user e ∈MEo outside the tagged cell is given by

ye =√P ‖e− c‖−η h†c,ewc,omc,o + ne. (6.8)

We denote by γi and γe the SINRs at the malicious users i ∈MIo and e ∈ME

o , respectively.

Due to the cooperation among all malicious users inMo =MIo∪ME

o , the setMo can be seen as

a single equivalent multi-antenna malicious user, denoted by Mo. After interference cancellation,

Mo sees the useful signal embedded in noise, therefore applying maximal ratio combining is

optimal, and yields to an SINR given by

γM,o =∑i∈MI

o

γi +∑e∈ME

o

γe = ρ∑i∈MI

o

‖i− c‖−η∣∣∣h†c,iwc,o

∣∣∣2 +ρ

Kc

∑e∈ME

o

gc,e‖e− c‖−η, (6.9)

where

gc,e ,∣∣∣h†c,ewc,o

∣∣∣2 , (6.10)

with wc,o ,√Kcwc,o and ni, ne ∼ CN (0, σ2).

6.2.3 Achievable Secrecy Rates

We are now able to obtain an expression for the secrecy rate achievable by RCI precoding for the

typical user of a downlink cellular network.

Proposition 6.1. A secrecy rate achievable by RCI precoding for the typical user o is given by

RCELL ,

log2

(1 +

ρ‖c‖−η∣∣∣h†c,owc,o

∣∣∣2ρ‖c‖−η∑i∈MI

o

∣∣∣h†c,owc,i

∣∣∣2 + ρI + 1

)

− log2

(1 + ρ

∑i∈MI

o

‖i− c‖−η∣∣∣h†c,iwc,o

∣∣∣2 + ρL

)+

, (6.11)


where I and L represent the interference and leakage term, respectively, given by

I =∑

b∈Φb\c

gb,oKb‖b‖−η (6.12)

L =1

Kc

∑e∈ME

o

gc,e‖e− c‖−η. (6.13)

Proof. The BS c, the user o, and the equivalent malicious user Mo form an equivalent multi-input,

single-output, multi-eavesdropper (MISOME) wiretap channel KHISTI AND WORNELL [2010].

As a result, an achievable secrecy rate is given by

RCELL = log2 (1 + γo)− log2 (1 + γM,o)+ . (6.14)

Substituting (6.4) and (6.9) in (6.14) yields (6.11).

The statistics of the terms gb,o and gc,e in (6.5) and (6.10), respectively, can be characterized as

follows DHILLON et al. [2013].

Proposition 6.2. For regularized channel inversion precoding we have that (i) the inter-cell inter-

ference power gain at the typical legitimate user o is distributed as gb,o ∼ Γ(Kb, 1), and (ii) the

leakage power gain at the malicious user e ∈MEo is distributed as gc,e ∼ exp(1).

Proof. See Appendix C.1.

We now define the probability of secrecy outage and the mean secrecy rate for the typical user.

Definition 6.1. The probability of secrecy outage for the typical user o is defined as

OCELL , P(RCELL ≤ 0). (6.15)

The probability of secrecy outage also denotes the fraction of time for which a BS cannot transmit

to a typical user at a non-zero secrecy rate.

Definition 6.2. The mean secrecy rate for the typical user o is defined as

RCELL , E [RCELL] . (6.16)


6.3 Large-system Analysis

In this section, we derive approximations for (i) the secrecy outage probability, i.e., the probability

that the secrecy rate RCELL achievable by RCI precoding for the typical user o is zero, and (ii) the

mean secrecy rate achievable by RCI precoding in the downlink of a cellular network.

6.3.1 Preliminaries

Throughout the analysis, we make the following assumptions.

Assumption 6.1. For uniformity of notation, we assume Kc = Kb = K , λuλb

, ∀b, i.e., we

approximate the number of users served by each BS by its average value, given by the ratio between

the density of users and the density of BSs. In order for this equivalence to hold, we ignore a small

bias that makes the tagged cell bigger than a typical cell. This bias is a result of Feller’s paradox,

also known as waiting bus paradox in one dimension BACCELLI AND BREMAUD [2003].

Assumption 6.2. We assume ‖i− c‖ ≈ ‖c‖, ∀i ∈MIo, i.e., we approximate the distance between

the tagged BS c and each user connected to c by the distance between the BS c and the typical

user o. We then approximate the Voronoi region of the tagged BS c by a ball centered at c and with

radius r = 1√πλb

, i.e., B(c, r) ,m ∈ R2, ‖m− c‖ ≤ r

. For the sake of consistency, the value

of r is chosen to ensure that B(c, r) has the same area as the average cell.

Note that despite these assumptions, which are necessary to maintain tractability, our analysis

captures all the key characteristics of the cellular networks that affect physical layer security, as

discussed in the sequel. The simplified model also provides some fundamental insights into the

dependence of key performance metrics, such as secrecy outage and mean secrecy rate, on the

transmit power and BS deployment density.

Under Assumptions 6.1 and 6.2, we obtain the approximations MIo ≈ MI

o and MEo ≈ ME

o ,

where MIo is a set of K − 1 malicious users located at distance ‖c‖ from the BS c, and ME

o is a

set given by

MEo = e ∈ Φu ∩ B(c, r), (6.17)

100 Large-system Analysis 6.3

with B denoting the complement of the set B. We can then approximate the interference and

leakage terms in (6.12) and (6.13) as follows

I ≈ I =1

K

∑b∈Φb\c

gb,o‖b‖−η (6.18)

L ≈ L =1

K

∑e∈ME

o

gc,e‖e− c‖−η. (6.19)

We now carry out a large-system analysis by assuming that both (i) the average number of users

K in each cell, and (ii) the number of transmit antennas N at the each BS grow to infinity in a

fixed ratio β , KN . We can thus approximate the remaining random variables in (6.11) with the

following deterministic quantities NGUYEN AND EVANS [2008]; WAGNER et al. [2012]∣∣∣h†c,owc,o

∣∣∣2 ≈ α, ∑i∈MI

o

∣∣∣h†c,owc,i

∣∣∣2 ≈ χ, and∑i∈MI

o

∣∣∣h†c,iwc,o

∣∣∣2 ≈ χ, (6.20)

where

α =g (β, ξ)

1 + ξ

β [1 + g (β, ξ)]2

[1 + g (β, ξ)]2, χ =

1

[1 + g (β, ξ)]2, (6.21)

g (β, ξ) is given in (4.7), and where it follows from (4.10) that

limρ→∞

χ = 0, for β ≤ 1. (6.22)

An approximated secrecy rate can be therefore obtained as follows.

Definition 6.3. An approximation for the achievable secrecy rate RCELL is given by

RCELL ≈ RCELL ,

log2

(1+

ρα‖c‖−ηρχ‖c‖−η + ρI + 1

)−log2

(1+ρχ‖c‖−η+ρL

)+

. (6.23)

In Fig. 6.3 we compare the simulated ergodic secrecy rate RCELL in (6.11) to the approximation

RCELL in (6.23), obtained in the large-system regime under Assumptions 6.1 and 6.2. The secrecy

ratesRCELL and RCELL are plotted versus the transmit SNR ρ, for a system withN = 20 transmit

antennas, an average numberK = 20 of users per cell, a path loss exponent η = 4, and two values

of the density of BS λb. Figure 6.3 shows that RCELL and RCELL follow the same trend, and

that the approximation RCELL ≈ RCELL is reasonable. The figure also shows that in a cellular

network the secrecy rate does not monotonically increase with the transmit SNR. A more detailed

discussion on this phenomenon will be provided in Subsection 6.3.4.


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

SNR, ρ [dB]

Per-usersecrecyrate

RCELL

RCELL

λb = 0.01

λb = 0.1

Figure 6.3: Comparison between the simulated ergodic secrecy rate RCELL in (6.11) and the

approximation RCELL in (6.23) versus the SNR, for N = 20 transmit antennas, an average of

K = 20 users per BS, and η = 4.

6.3.2 Characterization of Interference and Leakage

We now provide some results on the Laplace transforms of the terms I and L which will be useful

in the remainder of the chapter.

Lemma 6.1. The Laplace transform of the interference term LI(s, ‖c‖) = E[e−sI ] is

LI(s, ‖c‖) = exp

(−( sK

) 2ηλbCη,K (s, ‖c‖)

)(6.24)

where

Cη,K (s, ‖c‖) =2π

η

K∑n=1

(K

n

)[B

(1;K − n+

2

η, n− 2

η

)−B

((1 +

sP

K‖c‖−η)−1;K − n+

2

η, n− 2

η

)], (6.25)

and B(x; y, z) =∫ x

0 ty−1(1− t)z−1dt is the incomplete Beta function.



Lemma 6.2. The Laplace transform of the leakage term LL(s) = E[e−sL] is

LL(s) = exp

(−λu

( sK

) 2ηDη(s)

)(6.26)

where

Dη(s) =2π

η

[B

(1;

2

η, 1− 2

η

)−B

(1

1 + sK r−η ;

2

η, 1− 2

η

)]. (6.27)

Proof. The proof is omitted since it is similar to the proof of Lemma 6.1.

The probability density functions (pdfs) fI and fL of I and L, respectively, can be obtained by

inverting the respective Laplace transforms LI and LL. We now propose simple approximations

for fI and fL, using the following well-known results BACCELLI AND BŁASZCZYSZYN [2009].

Proposition 6.3. The mean and the variance of the interference term I are respectively given by

µI =2πλb‖c‖−(η−2)

η − 2, (6.28)

σ2I

=πλb

(K +K2

)‖c‖−2(η−1)

K2 (η − 1), (6.29)

whereas the mean and the variance of the leakage term L are respectively given by

µL =2πλur

−(η−2)

K(η − 2), (6.30)

σ2L

=2πλur

−2(η−1)

K2 (η − 1). (6.31)


We then approximate the pdfs of I and L by lognormal distributions with the same respective

mean and variance, as follows.

Definition 6.4. The probability density functions of I and L can be approximated as follows

fI(x) ≈ 1

xσI,N√

2πexp

−(

log x− µI,N)2

2σ2I,N

, x > 0 (6.32)

fL(z) ≈ 1

zσL,N√

2πexp

−(

log z − µL,N)2

2σ2L,N

, z > 0 (6.33)


10−4

10−2

100

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

CDF

FI (sim.)

FI (approx.)

FL (sim.)

FI (approx.)

λb = 0.1

λb = 0.03

λb = 0.01

Figure 6.4: Comparison between the simulated cumulative distribution functions (CDFs) of I and

L and the lognormal approximations in (6.32) and (6.33), for an SNR ρ = 10dB,N = 20 transmit

antennas, K = 20 users per BS, ‖c‖ = r, and η = 4.

where

µI,N = logµI −1

2log

(1 +

σ2I

µ2I

), σ2

I,N= log

(1 +

σ2I

µ2I

)(6.34)

µL,N = logµL −1

2log

(1 +

σ2L

µ2L

), σ2

L,N= log

(1 +

σ2L

µ2L

). (6.35)

In Fig. 6.4 we compare the simulated cumulative distribution functions (CDFs) of I and L to the

lognormal approximations provided in (6.32) and (6.33). The CDFs are plotted for a transmit SNR

ρ = 10dB, N = 20 transmit antennas, an average of K = 20 users per BS, ‖c‖ = r, η = 4, and

three values of the density of BS λb. Figure 6.4 shows that the lognormal approximations provided

in Definition 6.4 are accurate for all values of λb.

6.3.3 Probability of Secrecy Outage

We now obtain an approximation for the probability of secrecy outage with RCI precoding.


Theorem 6.3. The probability of secrecy outage with RCI precoding can be approximated as

OCELL ≈ OCELL =

∫ ∞0

∫ ∞−∞

∫ ∞−∞

1(z≥τ(x,y)) fL(z) dz fI(x, y) dx 2λbπye−λbπy2

dy, (6.36)

where fI(x, y) is the probability density function of the interference I for ‖c‖ = y, fL(z) is the

probability density function of the leakage L, and where we have defined

τ(x, y) ,αy−η

ρχy−η + ρx+ 1− χy−η. (6.37)

Proof. By using approximation (6.23) in (6.15), we obtain

OCELL ≈ P(RCELL ≤ 0) = P

(ρχ‖c‖−η + ρL ≥ α‖c‖−η

χ‖c‖−η + I + 1ρ

)

= P

(L ≥ α‖c‖−η

ρχ‖c‖−η + ρI + 1− χ‖c‖−η

)(a)=

∫ ∞−∞

∫ ∞−∞

P(L ≥ τ(x, y)

)fI(x, y | ‖c‖ = y) f‖c‖(y) dx dy

=

∫ ∞0

∫ ∞−∞

E[1(L≥τ(x,y))

]fI(x, y) dx 2λbπye

−λbπy2dy

=

∫ ∞0

∫ ∞−∞

∫ ∞−∞

1(z≥τ(x,y)) fL(z) dz fI(x, y) dx 2λbπye−λbπy2

dy, (6.38)

where (a) holds by defining τ(x, y) as in (6.37), and by noting that the distance ‖c‖ between the

typical user and the nearest BS c has distribution HAENGGI [2005]

f‖c‖(y) = 2λbπy exp(−λbπy2), y > 0. (6.39)

The result provided in Theorem 6.3 allows to evaluate the probability of secrecy outage without

the need for Monte-Carlo simulations, which can be computationally expensive to account for all

users and all exact Voronoi cells. Moreover, Theorem 6.3 yields to the following asymptotic result

without the need to solve the integral.

In Chapter 5, we showed that for an isolated cell in a random field of eavesdroppers, a sufficient

number of transmit antennas allows the BS to cancel the intra-cell interference and leakage, and

to drive the probability of secrecy outage to zero. In a cellular network, the secrecy outage is also


caused by the inter-cell interference and leakage, which cannot be controlled by the BS. It is easy

to show that limρ→∞ τ(x, y) ≤ 0, which from Theorem 6.3 implies limρ→∞ OCELL = 1. We

therefore have the following result.

Remark 6.1. In cellular networks, RCI precoding can achieve confidential communication with

probability of secrecy outage OCELL < 1. However unlike an isolated cell, cellular networks tend

to be in secrecy outage w.p. 1 if the transmit power grows unbounded, irrespective of the number

of transmit antennas.

6.3.4 Mean Secrecy Rate

In the following, we derive the mean secrecy rate achievable by RCI precoding.

Theorem 6.4. The mean secrecy rate achievable by RCI precoding can be approximated as

RCELL ≈ RCELL, with

RCELL =

∫ ∞0

∫ αρχ− 1ρ−χy−η

−∞

log2

(1 +

ραy−η

ρχy−η + ρx+ 1

)∫ τ(x,y)

−∞fL(z)

−∫ τ(x,y)

−∞log2

(1 + ρχy−η + ρz

)fL(z) dz

fI(x, y) dx 2λbπye

−λbπy2dy. (6.40)


The result provided in Theorem 6.4 allows to evaluate the mean secrecy rate without the need

for computationally expensive Monte-Carlo simulations. Moreover, Theorem 6.4 yields to the

following asymptotic result without the need to solve the integral.

In Chapters 4 and 5 we showed that in an isolated cell, even in the presence of a random field

of eavesdroppers, a sufficient number of transmit antennas allows the BS to cancel the intra-cell

interference and leakage, and the secrecy rate increases monotonically with the SNR. In a cellular

network, the secrecy rate is also affected by the inter-cell interference and leakage, which cannot be

controlled by the BS. It is easy to show that limρ→∞αρχ− 1

ρ−χy−η ≤ 0, which from Theorem 6.4

implies limρ→∞ RCELL = 0. We therefore have the following result.


Remark 6.2. In cellular networks, RCI precoding can achieve a non-zero secrecy rate RCELL.

However unlike an isolated cell, the secrecy rate in a cellular network is interference-and-leakage-

limited, and it cannot grow unbounded with the transmit SNR, irrespective of the number of trans-

mit antennas.

Theorem 6.4 shows that an optimal value for the BS deployment density λb should be found as a

tradeoff between (i) increasing the useful power αy−η, and (ii) reducing the intra-cell interference

χy−η and leakage χy−η, and the inter-cell interference x and leakage z. We know from (6.22) that

χ vanishes at high SNR, thus the terms x and z become dominant in (6.40). For a given cell load

K = λuλb

, the terms x and z are minimized by small densities λb. We therefore have the following

result which we will validate by simulations in Section 6.4.

Remark 6.3. In a cellular network with a fixed load, i.e., average number of users per BS, there

is an optimal value for the deployment density of BSs that maximizes the mean secrecy rate, and

this value is a decreasing function of the SNR. The optimal value of λb can be found from (6.40)

by performing a linear search.

In order to calculate the mean secrecy rate in (6.40), one must obtain expressions for fI and fL

via Laplace anti-transform or via approximations, as discussed in Section 6.3.2. We now derive a

lower bound on RCELL which can be calculated without knowledge of fI and fL.

Corollary 6.5. The approximated mean secrecy rate RCELL can be lower bounded as RCELL ≥RLBCELL, with

RLBCELL =

∫ ∞0

∫ ∞−∞

[F∗1 (φ, y)LI(−i2πφ, y)−F∗2 (φ, y)LL(−i2πφ)

]dφ2λbπye

−λbπy2dy

+

,

(6.41)

and

F1(φ, y) =sgn(φ) e

2πi(χy−η+ 1ρ

)φ

2φ log 2

(1− e2πiαφy−η

), (6.42)

F2(φ, y) =−e2πi(χy−η+ 1

ρ)φ

log 2

[1

2 |φ| +γ

ρδ(φ)

]. (6.43)



0 5 10 15 20 25 300.5

0.6

0.7

0.8

0.9

1

SNR, ρ [dB]


OCELL (analysis)

OCELL (simulations)

λb = 0.01

λb = 0.03

λb = 0.1

Figure 6.5: Comparison between the simulated probability of secrecy outage OCELL and the ana-

lytical result from Theorem 6.3, for N = 20 transmit antennas, K = 20 users per BS, and three

values of the density of BSs λb.

6.4 Numerical Results

In Fig. 6.5 we compare the simulated probability of secrecy outage OCELL to the analytical result

given in Theorem 6.3, forN = 20 transmit antennas,K = 20 users per BS, and three values of the

density of BSs λb. The analytical curves were obtained by using lognormal approximations for the

pdfs fI(x, y) and fL(z). The figure shows that the result provided in Theorem 6.3 is accurate for

all values of λb at relatively low values of SNR. Due to the lognormal approximations, the result

is slightly less accurate at relatively high values of SNR, when the BS can cancel the intra-cell

interference and leakage, and the secrecy outage is mostly determined by I and L.

In Fig. 6.6 we compare the simulated mean secrecy rate RCELL to the analytical result given in

Theorem 6.4, for N = 20 transmit antennas, K = 20 users per BS, and two values of the den-

sity of BSs λb. The analytical curves were again obtained by using lognormal approximations for

the pdfs fI(x, y) and fL(z). The figure shows that the simulations and the analytical result from

Theorem 6.4 follow the same trend. Therefore, the analysis provides insights into the behavior

of the secrecy rate as a function of λb and the SNR. The result from Theorem 6.4 is accurate for

all values of λb at relatively low values of SNR. Again due to the lognormal approximations, the


0 5 10 15 20 25 300

0.05

0.1

0.15

0.2

SNR, ρ [dB]

Per-usersecrecyrate

RCELL (analysis)

RCELL (simulations)

λb = 0.01

λb = 0.1

Figure 6.6: Comparison between the simulated mean secrecy rate RCELL and the analytical result

from Theorem 6.4, for N = 20 transmit antennas, K = 20 users per BS, and two values of the

density of BSs λb.

analytical curve is less accurate at relatively high values of SNR, since χ vanishes as reported in

(6.22), and the secrecy rate in (6.40) is dominated by I and L. This inaccuracy could be avoided

by employing the exact pdfs of fI(x, y) and fL(z), obtained from their Laplace transforms in

Lemma 6.1 and Lemma 6.2, but the anti-transform operation is computationally expensive. Find-

ing better approximations for fI(x, y) and fL(z) is therefore identified as a promising research

problem.

In Fig. 6.7 we plot the simulated probability of secrecy outage versus the transmit SNR, for K =

10 users per BS and three values of the number of transmit antennasN . In this figure, two cases are

considered for the density of BSs λb, namely 0.01 and 0.1, while the density of users is given by

λu = Kλb. Figure 6.7 shows that RCI precoding achieves confidential communications in cellular

networks with probability of secrecy outage OCELL < 1, and that having more transmit antennas is

beneficial as it reduces the probability of secrecy outage. However unlike an isolated cell, cellular

networks tend to be in secrecy outage w.p. 1 if the transmit power grows unbounded, irrespective

of the number of transmit antennas. These observations are consistent with Remark 6.1.

In Fig. 6.8 we plot the simulated per-user ergodic secrecy rate versus the transmit SNR, forK = 10


−10 0 10 20 30 400.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR, ρ [dB]


N = 10

N = 12

N = 14

λb = 0.1

λb = 0.01

Figure 6.7: Simulated probability of secrecy outage versus transmit SNR, for K = 10 users per

BS and various values of the number of transmit antennas N and density of BSs λb.

users per BS and three values of the number of transmit antennasN . In this figure, again, two cases

are considered for the density of BSs λb, namely 0.01 and 0.1, while the density of users is given

by λu = Kλb. Figure 6.8 shows that in cellular networks RCI precoding can achieve a non-zero

secrecy rate, and that having more transmit antennas is beneficial as it increases the secrecy rate.

However unlike the secrecy rate in an isolated cell, the secrecy rate in a cellular scenario does not

grow unbounded with the SNR, even with a large number of transmit antennas. These observations

are consistent with Remark 6.2.

In Fig. 6.9 we plot the simulated per-user ergodic secrecy rate as a function of the density of BSs

λb, for N = 20 transmit antennas, K = 20 users per BS, and various values of the transmit

SNR. Figure 6.9 shows that there is an optimal value for the density of BSs λb that maximizes

the secrecy rate, and that such value is smaller for higher values of the SNR. This observation is

consistent with Remark 6.3.


−10 0 10 20 30 400

0.2

0.4

0.6

0.8

1

SNR, ρ [dB]

Per-usersecrecyrate

N = 10

N = 12

N = 14λb = 0.01

λb = 0.1

Figure 6.8: Simulated ergodic secrecy rate versus transmit SNR, for K = 10 users per BS and

various values of the number of transmit antennas N and density of BSs λb.

10−3

10−2

10−1

100

0

0.05

0.1

0.15

0.2

0.25

Per-usersecrecyrate

Density of BSs, λb

ρ = 0 dB

ρ = 10 dB

ρ = 20 dB

ρ = 30 dB

Figure 6.9: Simulated ergodic secrecy rate versus density of BSs, for N = 20 transmit antennas,

K = 20 users per BS, and various values of the transmit SNR ρ.


6.5 Conclusion

In this chapter, we considered physical layer security for the downlink of cellular networks, where

multiple base stations (BSs) generate inter-cell interference, and malicious users of neighboring

cells can cooperate to eavesdrop. We showed that RCI precoding can achieve a non-zero secrecy

rate with probability of outage smaller than one. However we also found that unlike isolated cells,

the secrecy rate in a cellular network does not grow monotonically with the transmit signal-to-

noise ratio (SNR), and the network tends to be in secrecy outage if the transmit power grows

unbounded. We further showed that there is an optimal value for the density of BSs that max-

imizes the secrecy rate, and this value is a decreasing function of the transmit SNR. Using the

developed analysis, we clearly established the importance of designing the transmit power and the

BS deployment density to make communications robust against malicious users in other cells.

Chapter 7

Conclusions and Future Work

In this thesis, we studied physical layer security for multi-user multi-antenna wireless systems.

Physical layer security was recently proposed as a method to protect confidential information

without requiring encryption keys.

We started by considering the MISO broadcast channel with confidential messages (BCC) under

Rayleigh fading, where a multi-antenna base station (BS) simultaneously transmits independent

confidential messages to several spatially dispersed malicious users that can eavesdrop on each

other. For this system set-up, we proposed a low-complexity linear precoder based on regularized

channel inversion (RCI), and we derived the achievable secrecy rates, i.e., the rates at which each

message can be reliably transmitted to the intended user while no information is leaked at the

other users. We showed that RCI precoding outperforms plain channel inversion precoding, and

that the secrecy requirements result in a loss in terms of the sum-rate. We then proposed an

algorithm to jointly optimize the regularization parameter of the precoder and the power allocation

vector, and showed that optimal power allocation increases the secrecy sum-rate compared to equal

power allocation by up to 20% at practical values of the signal-to-noise ratio (SNR). Finally, we

extended the secrecy rate analysis to more practical scenarios where only imperfect channel state

information is available at the transmitter, and where channel correlation is present among the

transmit antenna elements.

Fundamental tools from random matrix theory allowed us to carry out a large-system analysis and

obtain closed form approximations for the achievable secrecy rates in the MISO BCC. Simulations

113

114 Conclusions and Future Work 7.0

confirmed that the approximations obtained via our analysis are accurate even for finite systems

of practical size. We derived expressions for the optimal regularization parameter of the precoder

ξ and for the optimal network load β, which maximize the secrecy sum-rate in the MISO BCC.

These values are different to the respective ones that maximize the sum-rate in the absence of

secrecy requirements. We found that for β > 1, i.e., when the number of users K is larger than

the number of transmit antennas N , the RCI precoder performs poorly in the high-SNR regime.

This happens because an alliance of cooperating malicious users can cancel the interference, and

therefore has an unbounded signal-to-interference-plus-noise ratio (SINR), while the legitimate

user’s SINR is interference limited. The only means for the transmitter to limit the eavesdropper’s

SINR is by inverting the channel matrix, but this can only be accomplished when β ≤ 1. In order to

increase the high-SNR performance for network loads β > 1, we proposed a linear precoder based

on RCI and power reduction (RCI-PR). The proposed RCI-PR precoder was showed to achieve a

per-user secrecy rate with the same high-SNR scaling factor as both the following upper bounds:

(i) the per-user rate of the optimal RCI precoder in the MISO broadcast channel (BC) without

secrecy requirements, and (ii) the secrecy capacity of a single-user system without interference.

We then studied the large-system performance of RCI precoding in practical channels. We first

considered the presence of channel state information (CSI) error, and determined how the CSI error

must scale with the SNR in order to ensure a constant high-SNR rate gap to the case with perfect

CSI. Such constant rate gap ensures that the multiplexing gain is not affected. For frequency

division duplex (FDD) systems, where users quantize their estimated channel vectors and send

the quantization index back to the BS over a limited-rate channel, we determined the minimum

amount of feedback bits required in order to meet a given secrecy rate gap. For time division

duplex (TDD) systems, where the channel estimation at the transmitter is obtained from known

pilot symbols sent by the users, we determined the optimum amount of channel training that

maximizes the secrecy sum-rate. We then applied our analysis to the MISO BCC under transmit-

side channel correlation. We obtained large-system approximations for the secrecy rates and for

the optimal regularization parameter of the precoder. We found that low-to-moderate correlation

only partially affects the secrecy rates. However, high correlation degrades the performance of the

RCI precoder, especially at low SNR.

We then introduced the broadcast channel with confidential messages and external eavesdroppers

(BCCE), where a multi-antenna BS simultaneously communicates to multiple malicious users, in

7.0 Conclusions and Future Work 115

the presence of a PPP of external eavesdroppers. Unlike the BCC, in the BCCE not just malicious

users, but also randomly located external nodes can act as eavesdroppers. By using results from

stochastic geometry, we obtained the probability of secrecy outage and the mean secrecy rate

for the RCI precoder in the BCCE under Rayleigh fading, for the two cases of non-colluding

and colluding eavesdroppers. We showed that, irrespective of the collusion strategy at the external

eavesdroppers, a large number of transmit antennasN drives both the probability of secrecy outage

and the rate loss due to the presence of external eavesdroppers to zero. Increasing the density of

eavesdroppers λe by a factor n, requires n2 as many antennas to meet a given probability of

secrecy outage and a given mean secrecy rate. Our analysis demonstrated that the number of

transmit antennas at the BS is a key resource to secure communications against malicious users

and external eavesdropping nodes.

We finally turned our attention to cellular networks where, unlike the case of isolated cells, multi-

ple BSs generate inter-cell interference, and malicious users of neighboring cells can cooperate to

eavesdrop. We characterized the probability of secrecy outage and the mean secrecy rate with RCI

precoding, accounting for the spatial distribution of BSs and users (modeled as independent PPPs)

and the fluctuations of their channels. We found that RCI precoding can achieve a non-zero se-

crecy rate with probability of outage smaller than one. However we also found that unlike isolated

cells, the secrecy rate in a cellular network does not grow monotonically with the SNR, and the

network tends to be in secrecy outage if the transmit power grows unbounded. We further showed

that there is an optimal value for the density of BSs that maximizes the secrecy rate, and this value

is a decreasing function of the SNR. Using the developed analysis, we clearly established the im-

portance of designing the transmit power and the BS deployment density to make communications

robust against malicious users in other cells.

This thesis considered scenarios where mobile users are equipped with a single receive antenna. In

addition, the analysis focused on a model where the noise power at each legitimate user is the same

for all users, and where the legitimate users of each cell have similar distances from the serving

BS, i.e., similar path-loss. A direct extension of this work considers the broadcast of confidential

messages in multiuser networks with multi-antenna receivers, under unequal path loss YANG et al.

[2013]. Furthermore, this thesis studied the performance of RCI precoding in cellular networks

by using approximations for the probability density functions (pdfs) of the inter-cell interference

power I and leakage power L. Obtaining an exact characterization of the pdfs of I and L could

116 Conclusions and Future Work 7.0

be an interesting future research direction. Finally, investigating the secrecy rates achievable in

heterogeneous networks, where small BSs are overlaid within the macro cellular network based

on traffic and coverage demand, is identified as another promising research problem.

Appendix A

Appendix for Chapter 4

A.1 Proof of Theorem 4.1

Proof. By defining Ak as in (3.22) and Bk as in (3.23), the SINRs at the intended user and the

eavesdropper can be expressed as in (3.24) and (3.25). We rewrite Ak = NK−1v†NQNvN , where

Qk,N =

(1

K − 1H†kHk − zI

)−1

∈ CN×N , vk,N =1√N

hk ∈ CN , z = − Nξ

K − 1. (A.1)

By Bai and Silverstein’s Lemma PEACOCK et al. [2008]; COUILLET AND DEBBAH [2011];

COUILLET et al. [2012], we have

E[φk

∣∣∣∣v†k,NQk,Nvk,N −1

NtrQk,N

∣∣∣∣p] ≤ CpNp

(N

λN

) p2

=Cp

Np2 λ

p2N

= fN ∀p ≥ 1 (A.2)

where Cp is a constant depending only on p, and φk = 1|λ1(H†kHk)−Nξ|,...,|λN (H†kHk)−Nξ|>λN,

with λN → 0. Assume ξ ∈ D′N , withD′N = DN for β ≤ 1 andD′N = DN\[− C

N12−ε

,+ C

N12−ε

]for β > 1. Then we have N = O(λ−2−ε

N ), for some ε > 0, and mink≤Kφk a.s.−→ 1 COUILLET

et al. [2012]. It follows from the Markov inequality and the Borel-Cantelli lemma BILLINGSLEY

[1995] that maxk|v†k,NQk,Nvk,N − 1

N trQk,N| a.s.−→ 0, as N → ∞. The term 1N trQk,N is

by definition the Stieltjes transform mH†kHk,N

(z) of H†kHk,N . Similarly, it can be shown that

maxk|m

H†kHk,N(z) − m(z)| a.s.−→ 0, where m(z) can be obtained as the solution of m(z) =[

1− NK−1 − z − z N

K−1m(z)]−1

. This yields

Ak − g (β, ξ)a.s.−→ 0 (A.3)

117

118 Proof of Theorem 4.1 A.1

with

g (β, ξ) = β−1m(ξ) =1

2

±√

(1− β)2

ξ2+

2 (1 + β)

ξ+ 1 +

1− βξ− 1

, (A.4)

and where in order for m to be a Stieltjes transform, the sign of the square root must be chosen the

same as the sign of ξ SILVERSTEIN AND CHOI [1995].

We now rewrite Bk = Ak − N2ξ

(K−1)2 v†k,NQ2k,Nvk,N , and similarly we have

E[φk

∣∣∣∣v†k,NQ2k,Nvk,N −

1

NtrQ2

k,N

∣∣∣∣p] ≤ CpNp

(N

λ2N

) p2

=Cp

Np2 λpN

= gN ∀p ≥ 1. (A.5)

Again, if N = O(λ−2−εN ), for some ε > 0, we have max

k|v†k,NQ2

k,Nvk,N − 1N trQ2

k,N|a.s.−→ 0,

as N →∞. We note that 1N trQ2

k,N is the Stieltjes transform of Q2k,N , given by

1

NtrQ2

k,N =

∫dFN (λ)

(λRN− z)2 =

∂

∂z

∫dFN (λ)

λRN− z = m′RN

(z) (A.6)

where FN (λ) is the distribution of the eigenvalues of RN . Since both mRk,N(z) and m(z) are

analytic functions, we have maxk|m′Rk,N

(z)−m′(z)| a.s.−→ 0, as N →∞, and it follows that

Bk −[

1

βm(z)− N2ξ

(K − 1)2

∂

∂zm(z)

]= Bk −

[g (β, ξ) + ξ

∂

∂ξg (β, ξ)

]a.s.−→ 0. (A.7)

For the power normalization constant ζ we have

ζ = tr

(H†H +NξI

)−1−Nξtr

(H†H +MξI

)−2

=1

βm

HH†N(z′)− ξ

β2m′

HH†N(z′)

(A.8)

where z′ = −NξK . If ξ ∈ D′N , if follows that

ζ −[g (β, ξ) + ξ

∂

∂ξg (β, ξ)

]a.s.−→ 0. (A.9)

By the continuity of RBCC,k and RBCC, it follows that the previous convergence results also hold

for ξ ∈[− C

N12−ε

,+ C

N12−ε

]and β > 1. Equation (4.2) then follows from (3.27), (A.3), (A.7),

(A.9), and by applying the continuous mapping theorem, the Markov inequality, and the Borel-

Cantelli lemma.

A.2 Appendix for Chapter 4 119

A.2 Proof of Theorem 4.5

Proof. From NGUYEN et al. [2009], by defining ρ ,ρ(1−τ2)ρτ2+1

and ξ , ξ1−τ2 , a deterministic

approximation for γik is given by

γi = g(β, ξ)ρ+ ξρ

β

[1 + g(β, ξ)

]2

ρ+[1 + g(β, ξ)

]2 . (A.10)

By defining Ωk =(H†kHk +NξI

)−1and using (3.40), we can rewrite γiM,k in (3.44) as

γiM,k = ρBk + 2

(1 + Ak

)Qk +

(1 + Ak

)2Rk

ζ(

1 + Ak

)2 , (A.11)

with

Ak = h†kΩkhk, (A.12)

Bk = h†kΩkH†kHkΩkhk, (A.13)

Qk = h†kΩkH†kHk

(Ωk −

Ωkhkh†kΩk

1 + Ak

)ek, (A.14)

Rk = e†k

(Ωk −

Ωkhkh†kΩk

1 + Ak

)H†kHk

(Ωk −

Ωkhkh†kΩk

1 + Ak

)ek. (A.15)

If ξ ∈ DN , we have

Ak − g(β, ξ)

a.s.−→ 0, (A.16)

Bk −[g(β, ξ)

+ ξ∂

∂ξg(β, ξ)]

a.s.−→ 0, (A.17)

Qka.s.−→ 0, (A.18)

Rk −τ2

1− τ2

[g(β, ξ)

+ ξ∂

∂ξg(β, ξ)]

a.s.−→ 0, (A.19)

and

ζ − 1

1− τ2

[g(β, ξ)

+ ξ∂

∂ξg(β, ξ)]

a.s.−→ 0, (A.20)

hence a deterministic approximation for γiM,k is given by

γiM = ρ

τ2 +1− τ2(

1 + g(β, ξ))2

. (A.21)

Theorem 4.5 then follows from (3.42), (A.10), (A.21), and from the continuous mapping theorem

BILLINGSLEY [1995].

Appendix B


B.1 Proof of Lemma 5.1

Proof. If γk ≤ γM,k, then RBCCE,k in (5.2) is zero w.p. 1. If γk > γM,k, we have for non-

colluding eavesdroppers γE,k = maxe

γe,k, therefore

OBCCE,k = P(γE,k ≥ γk

∣∣∣ γk) = 1− EΦe

[ ∏x∈Φe

P(γx,k < γk

∣∣∣ γk)]

= 1− EΦe

[ ∏x∈Φe

[1− P

(γx,k ≥ γk

∣∣∣ γk)]]

(a)= 1− EΦe

[ ∏x∈Φe

[1− exp

(−Nβσ2γk‖x‖η

)]](b)= 1− exp

[−2πλe

∫ ∞0

y exp(−Nβσ2γk y

η)dy

](c)= 1− exp

[−πλe

∫ ∞0

exp(−Nβσ2γk u

η2

)du

](d)= 1− exp

[− 2πλe

η(Nβσ2γk)2η

∫ ∞0

e−tt2η−1dt

]

(e)= 1− exp

− 2πλeΓ(

2η

)η(Nβσ2γk)

2η

(B.1)

where (a) follows from the distribution of γe,k, and (b) follows by using ‖x‖ = y, by applying the

probability generating functional (PGFL) for the PPP Φe, given by STOYAN et al. [1996]

EΦe

[ ∏x∈Φe

f(x)

]= exp

−∫R2

[1− f(x)]λedx

(B.2)

121

122 Proof of Lemma 5.4 B.2

and by changing to polar coordinates. Moreover, in (c) we have used u = y2, in (d) we have used

t = Nβσ2γkuη2 , and (e) follows from the definition of the gamma function

Γ(z) ,∫ ∞

0tz−1e−tdt. (B.3)


Proof. If γk ≤ γM,k, then RBCCE,k in (5.2) is zero with probability one. If γk > γM,k, we have

for the eavesdropper nearest to the BS

OBCCE,k = P(γE,k ≥ γk

∣∣∣ γk)=

∫ ∞0

P(γE,k ≥ γk

∣∣∣ γk, ‖E‖ = x)f‖E‖(x)dx

=

∫ ∞0

P(x−ησ2

∣∣∣h†Ewk

∣∣∣2 ≥ γk ∣∣∣ γk, ‖E‖ = x)f‖E‖(x)dx

=

∫ ∞0

P( ∣∣∣h†Ewk

∣∣∣2 ≥ σ2γkxη∣∣∣ γk, ‖E‖ = x

)f‖E‖(x)dx

(a)=

∫ ∞0

exp(−Nβσ2γkx

η)f‖E‖(x)dx

(b)= 2πλe

∫ ∞0

x exp(−Nβσ2γk x

η − λeπx2)dx, (B.4)

where (a) holds because∣∣∣h†Ewk

∣∣∣2 ∼ exp( 1Nβ ), and (b) holds because the distance ‖E‖ between

the base station and the nearest eavesdropper E has distribution HAENGGI [2005]

f‖E‖(x) = 2λeπx exp(−λeπx2). (B.5)

For a path loss exponent η = 4, (B.4) reduces to

OBCCE,k = 2πλe

∫ ∞0

x exp(−Nβσ2γk x

4 − λeπx2)dx

(c)= πλe

∫ ∞0

exp(−Nβσ2γk u

2 − λeπu)du

(d)=

π32λe

2√Nβσ2γk

exp

[(πλe)

2

4Nβσ2γk

]erfc

(πλe

2√Nβσ2γk

)(B.6)

where in (c) we have used u = x2, and (d) follows from∫ ∞0

exp(−ax2 − bx)dx =1

2

√π

aexp

(b2

4a

)erfc

(b

2√a

). (B.7)

B.3 Appendix for Chapter 5 123


Proof. For the case of colluding eavesdroppers, the Laplace transform of the SINR is HAENGGI

et al. [2009]

LγE,k(s) = E

[exp

(− s

σ2

∑x∈Φe

‖x‖−η∣∣∣h†xwk

∣∣∣2)](a)= exp

−2πλe

∫R2

Eh

[1− exp

(− s

σ2

∣∣∣h†xwk

∣∣∣2 ‖x‖−η)]x dx(b)= exp

−πλe Eh

[∣∣∣∣ 1σh†xwk

∣∣∣∣ 4η

]Γ

(1− 2

η

)s

2η

(c)= exp

−πλe

(Nβσ2

)− 2η Γ

(1 +

2

η

)Γ

(1− 2

η

)s

2η

(B.8)

where (a) holds since Φe is a PPP HAENGGI et al. [2009], (b) follows since the fading is indepen-

dent of the point process, and (c) follows since∣∣∣h†xwk

∣∣∣2 ∼ exp( 1Nβ ). Under a path loss exponent

η = 4, (B.8) reduces to

LγE,k(s) = exp(−π

2λe2

√s

Nβσ2

). (B.9)

By inverse transform one can obtain the probability distribution function SOUSA AND SILVESTER

[1990]

fγE,k(y) =π

32λey

− 32

4√Nβσ2

exp(− π4λ2

e

16Nβσ2y

), (B.10)

which integrated yields the cumulative distribution function

FγE,k(y) = erfc

[π2λe

4√Nβσ2y

], (B.11)

from which the secrecy outage probability in (5.17) can be calculated as OBCCE,k = FγE,k(γk).

124 Proof of Lemma 5.9 B.4


Proof. We note from (5.2) that when γk ≤ γM,k, the secrecy rate RBCCE,k is zero ∀ γE,k. When

γk > γM,k, the mean secrecy rate is given by

EΦe [RBCCE,k|γk>γM,k]

= EΦe

[max

[log2

(1 + γk

)− log2

(1 + max (γE,k, γM,k)

), 0]]

= EΦe

[[log2

(1 + γk

)− log2


)]1(γE,k<γk)

]= EΦe

[log2

(1 + γk

)1(γE,k<γk) − log2


)1(γE,k<γk)

]= P (γE,k < γk) log2

(1 + γk

)− EΦe

[log2

(1+max (γE,k, γM,k)

)1(γE,k<γk)


(1 + γk

)− EΦe

[log2

(1 + γM,k

)1(γE,k<γM,k) + log2

(1 + γE,k

)1(γM,k<γE,k<γk)


(1 + γk

)− P (γE,k < γM,k) log2

(1 + γM,k

)−∫ γk

γM,k

log2(1 + y)fγE,k(y) dy

= log2

(1 + γk

)1−OBCCE,k−log2

(1 + γM,k

)1−PBCCE,k−∫ γk

γM,k


= log2

(1 + γk

)1−OBCCE,k

(1 + γM,k

)1−PBCCE,k−∫ γk

γM,k

log2(1 + y)fγE,k(y) dy (B.12)

where (i) 1(·) is the indicator function, (ii) OBCCE,k , P (γE,k ≥ γk) is given by the secrecy

outage probability; (iii) PBCCE,k , P (γE,k ≥ γM,k) is the probability that the SINR at the exter-

nal eavesdroppers is greater than or equal to the SINR at the malicious users, given in (5.21) and

obtained by calculations similar to the ones in Lemma 5.1 and Lemma 5.6; and (iv) fγE,k(y) is the

pdf of the SINR at the external eavesdroppers, given by (B.10) for colluding eavesdroppers, and

by

fγE,k(y) =∂P (γE,k < y)

∂y=π

32λey

− 32

4√Nβσ2

exp

(− π

32λe

2√Nβσ2y

)(B.13)

for non-colluding eavesdroppers.

B.5 Appendix for Chapter 5 125

B.5 Proof of Corollary 5.11

Proof. For γ ≤ γM , we have RBCC = 0 and RBCCE = 0, therefore ∆e = 0. For γ > γM and

fixed ξ, irrespective of the cooperation strategy at the eavesdroppers, we have

∆e = OBCCE log(1 + γ)− PBCCE log(1 + γM ) +

∫ γ

γM


(a)

≤ OBCCE ·RBCC +µλe

2√N

∫ γ

γM

y−12 dy

=

[1− exp

(− µλe√

Nγ

)]RBCC +

µλe√N

(√γ −

√γM

)≤ µλe√

NγRBCC +

µλe√N

(√γ −

√γM

)= µ

[RBCC√γ

+(√

γ −√γM

)] λe√N

(B.14)

where (a) holds because PBCCE > OBCCE, log2(1 + y) ≤ y, and fγE,k(y) ≤ µλey− 3

2

2√N

.

Appendix C


C.1 Proof of Proposition 6.2

Proof. Under RCI precoding, the BS b multiplies the confidential message mb,j destined for user

j, for 1 ≤ j ≤ Kb, by wb,j , so that the transmitted signal is a linear function of the message

mb,j , i.e., xb =√P∑Kb

j=1 wb,jmb,j . The inter-cell interference power gain at the typical user o

is given by gb,o =∑Kb

j=1

∣∣∣h†b,owb,j

∣∣∣2, with wb,j =√Kbwb,j . The normalized precoding vectors

wb,j have unit-norm on average, and they are calculated independently of h†b,o. Therefore, h†b,o and

wb,j are independent isotropic unit-norm random vectors, and∣∣∣h†b,owb,j

∣∣∣2 is a linear combination

of N complex normal random variables, i.e., exponentially distributed. As a result, we have that

gb,o ∼ Γ(Kb, 1), since it is the sum of Kb i.i.d. exponential r.v.

The leakage power gain at the malicious user e ∈ MEo is given by gc,e =

∣∣∣h†c,ewc,o

∣∣∣2, with

wc,o =√Kcwc,o. Similarly, we have that h†c,e and wc,o are independent isotropic unit-norm

random vectors. As a result, we have that gc,e ∼ exp(1) since it is a linear combination of N

complex normal r.v.

127

128 Proof of Lemma 6.1 C.2

C.2 Proof of Lemma 6.1

Proof. The Laplace transform of the interference term I can be derived as follows

E[e−sI

]= E

[e− sK

∑b∈Φb\c

gb,o‖b‖−η]

= E

∏b∈Φb\c

e−sKgb,o‖b‖−η

(a)= EΦb

∏b∈Φb\c

Lgb,o( sK‖b‖−η

)(b)= exp

−λb

∫R2∩B(o,‖c‖)

[1− Lgb,o

( sK‖b‖−η

)]db

(c)= exp

−λb

∫R2∩B(o,‖c‖)

[1− 1(

1 + sK ‖b‖−η

)K]

db

= exp

−λb

∫R2∩B(o,‖c‖)

(1 + s

K ‖b‖−η)K − 1(

1 + sK ‖b‖−η

)K db

(d)= exp

−λb∫R2∩B(o,‖c‖)

∑Kn=1

(Kn

) (sK ‖b‖−η

)n(1 + s

Kb‖b‖−η

)K db

= exp

−λb

K∑n=1

(K

n

)∫R2∩B(o,‖c‖)

(sK ‖b‖−η

)n(1 + s

K ‖b‖−η)K db

(e)= exp

−2πλb

( sK

) 2η

K∑n=1

(K

n

)∫ ∞‖c‖( s

K)− 1η

v−nη

(1 + v−η)Kvdv

(f)= exp

−λb

( sK

) 2ηCη,K (s, ‖c‖)

, (C.1)

where (a) follows since the channel powers gb,o are independent of the locations of the BSs, (b)

follows from the PGFL of a PPP STOYAN et al. [1996], (c) follows from the Laplace transform

of gb,o ∼ Γ(K, 1), (d) follows from the Binomial theorem, (e) follows by converting to polar

coordinates, and (f) follows by substituting (1 + ν−η)−1 = t and noting that the integral is the

difference of two incomplete Beta functions B(x; y, z) =∫ x

0 ty−1(1− t)z−1dt.

C.3 Appendix for Chapter 6 129

C.3 Proof of Proposition 6.3

Proof. The mean and variance of the interference can be obtained by applying Campbell’s theorem

and are given by BACCELLI AND BŁASZCZYSZYN [2009]

µI = E[I]

(a)= EΦb

∑b∈Φb\c

‖b‖−η = 2πλb

∫ ∞‖c‖

v−ηv dv =2πλb‖c‖−(η−2)

η − 2(C.2)

σ2I

= E[I2]− µ2

I

(b)=K +K2

K2E

∑b∈Φb\c

‖b‖−2η

=

2πλb(K +K2

)K2

∫ ∞‖c‖

v−2ηv dv =πλb

(K +K2

)‖c‖−2(η−1)

K2 (η − 1), (C.3)

where (a) follows from E[gb,o] = K, and (b) follows from E[g2b,o

]= K + K2. Similarly, the

mean and variance of the leakage are given by BACCELLI AND BŁASZCZYSZYN [2009]

µL = E[L]

(c)=

1

KEΦu

∑e∈ME

o

E [gc,e] ‖e− c‖−η =

2πλuK

∫ ∞r

v−ηv dv =2πλur

−(η−2)

K (η − 2)

(C.4)

σ2L

= E[L2]− µ2

L

(a)=

2

K2E

∑e∈ME

o

‖e− c‖−2η

=4πλuK2

∫ ∞r

v−2ηv dv =2πλur

−2(η−1)

K2 (η − 1),

(C.5)

where (c) follows from E[gc,e] = 1, and (d) follows from E[g2c,e

]= 2.

130 Proof of Corollary 6.5 C.5

C.4 Proof of Theorem 6.4

Proof. By using approximation (6.23) in (6.16), we obtain

RCELL ≈ E[RCELL

]= E

[log2

(1+


)−log2

(1+ρχ‖c‖−η+ρL

)+]

= E

[[log2

(1 +


)− log2

(1 + ρχ‖c‖−η + ρL

)]1(L<τ(I,‖c‖))

]

= EI,‖c‖

[log2

(1 +


)P(L < τ

(I , ‖c‖

))]

− EI,L,‖c‖

[log2

(1 + ρχ‖c‖−η + ρL

)1(L<τ(I,‖c‖))

](a)=

∫ ∞0

∫ αρχ− 1ρ−χy−η

−∞

log2

(1 +

ραy−η

ρχy−η + ρx+ 1

)∫ τ(x,y)

−∞fL(z)

−∫ τ(x,y)

−∞log2

(1 + ρχy−η + ρz

)fL(z) dz

fI(x, y) dx 2λbπye

−λbπy2dy, (C.6)

where 1(·) is the indicator function, and where the upper limit in the inner integration in (a) follows

from 0 ≤ L < τ(I , ‖c‖).

C.5 Proof of Corollary 6.5

Proof. The lower bound in (6.41) can be obtained as follows

RCELL

= E[RCELL

] (a)

≥E

[log2

(1 +


)− log2

(1 + ρχ‖c‖−η + ρL

)]+

=

∫ ∞0

∫ ∞−∞

[log2

(1 +

ραy−η

ρχy−η + ρx+ 1

)fI(x, y)

− log2

(1 + ρχy−η + ρx

)fL(x)

]dx 2λbπye

−λbπy2dy

+

(b)=

∫ ∞0

∫ ∞−∞

[F∗1 (φ, y)LI(−i2πφ, y)−F∗2 (φ, y)LL(−i2πφ)

]dφ 2λbπye

−λbπy2dy

+

.

(C.7)

C.5 Appendix for Chapter 6 131

Equation (a) follows from Jensen’s inequality E[x+] ≥ E[x]+. Equation (b) follows by Parse-

val’s theorem RUDIN [1987], and since

F1(φ, y) =sgn(φ) e

2πi(χy−η+ 1ρ

)φ

2φ log 2

(1− e2πiαφy−η

)(C.8)

F2(φ, y) =−e2πi(χy−η+ 1

ρ)φ

log 2

[1

2 |φ| +γ

ρδ(φ)

](C.9)

are the respective Fourier transforms of

f1(x, y) = log2

(1 +

ραy−η

ρχy−η + ρx+ 1

)(C.10)

f2(x, y) = log2

(1 + ρχy−η + ρx

), (C.11)

where γ = limn→∞(∑n

k=11k − log n

)is the Euler-Mascheroni constant. The functions F1(φ, y)

and F2(φ, y) can be obtained from the Fourier transforms of 1x and log |x|, and by applying the

differentiation and shift theorems.

Bibliography

AL-NAFFOURI, T., M. SHARIF, AND B. HASSIBI [2009]. “How much does transmit correlation

affect the sum-rate scaling of MIMO Gaussian broadcast channels?” IEEE Trans. Commun.,

vol. 57, no. 2, pp. 562–572.

BACCELLI, F. AND B. BŁASZCZYSZYN [2009]. Stochastic Geometry and Wireless Networks,

Volume I: Theory. Hanover, MA: Now Publishers Inc., 1st ed.

BACCELLI, F. AND P. BREMAUD [2003]. Elements of queueing theory: Palm martingale calculus

and stochastic recurrences. New York, NY: Springer-Verlag.

BAGHERIKARAM, G., A. S. MOTAHARI, AND A. K. KHANDANI [2013]. “The secrecy capacity

region of the Gaussian MIMO broadcast channel.” IEEE Trans. Inf. Theory, vol. 59, no. 5, pp.

2673–2682.

BARROS, J. AND M. RODRIGUES [2006]. “Secrecy capacity of wireless channels.” In Proc. IEEE

Int. Symp. on Inform. Theory (ISIT). pp. 356–360.

BILLINGSLEY, P. [1995]. Probability and measure, 3rd ed. Hoboken, NJ: John Wiley & Sons,

Inc.

BLOCH, M., J. BARROS, M. R. D. RODRIGUES, AND S. W. MCLAUGHLIN [2006]. “An oppor-

tunistic physical-layer approach to secure wireless communications.” In Proc. Allerton Conf.

on Commun., Control, and Computing. Monticello, IL.

BUSTIN, R., R. LIU, H. V. POOR, AND S. SHAMAI [2009]. “An MMSE approach to the se-

crecy capacity of the MIMO Gaussian wiretap channel.” EURASIP Journ. Wireless Commun.

Network.

133

134 Bibliography

CAIRE, G., N. JINDAL, M. KOBAYASHI, AND N. RAVINDRAN [2010]. “Multiuser MIMO

achievable rates with downlink training and channel state feedback.” IEEE Trans. Inf. Theory,

vol. 56, no. 6, pp. 2845–2866.

CAIRE, G. AND S. SHAMAI [2003]. “On the achievable throughput of a multiantenna Gaussian

broadcast channel.” IEEE Trans. Inf. Theory, vol. 49, no. 7, pp. 1691–1706.

COUILLET, R. AND M. DEBBAH [2011]. Random Matrix Theory Methods for Wireless Commu-

nications. Cambridge University Press.

COUILLET, R., F. PASCAL, AND J. W. SILVERSTEIN [2012]. “Robust M-estimation for array

processing: A random matrix approach.” IEEE Trans. Signal Process. Submitted. Available:

http://couillet.romain.perso.sfr.fr/docs/articles/robust est submitted.pdf.

CSISZAR, I. AND J. KORNER [1978]. “Broadcast channels with confidential messages.” IEEE

Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348.

DHILLON, H. S., M. KOUNTOURIS, AND J. G. ANDREWS [2013]. “Downlink MIMO Het-

Nets: Modeling, ordering results and performance analysis.” IEEE Trans. Wireless Commun.

Accepted for publication, available arXiv:1301.5034.

EKREM, E. AND S. ULUKUS [2011]. “The secrecy capacity region of the Gaussian MIMO multi-

receiver wiretap channel.” IEEE Trans. Inf. Theory, vol. 57, no. 4, pp. 2083–2114.

FAKOORIAN, S. A. A. AND A. L. SWINDLEHURST [2013]. “Full rank solutions for the MIMO

Gaussian wiretap channel with an average power constraint.” IEEE Trans. Signal Process.,

vol. 61, no. 10, pp. 2620–2631.

FOSCHINI, G. J. AND M. J. GANS [1998]. “On limits of wireless communications in a fading

environment when using multiple antennas.” Wirel. Pers. Commun., vol. 6, no. 3, pp. 311–335.

GERACI, G., H. S. DHILLON, J. G. ANDREWS, J. YUAN, AND I. B. COLLINGS [2013]. “Phys-

ical layer security in downlink multi-antenna cellular networks.” submitted to IEEE Trans.

Commun. Available arXiv:1307.7211.

GILBERT, E. N. [1961]. “Random plane networks.” J. Soc. Indust. Appl. Math., vol. 9, pp.

533–543.

GOEL, S. AND R. NEGI [2008]. “Guaranteeing secrecy using artificial noise.” IEEE Trans.

Wireless Commun., vol. 7, no. 6, pp. 2180–2189.

Bibliography 135

GOLDSMITH, A. [2005]. Wireless Communications. Cambridge University Press.

GOPALA, P. K., L. LAI, AND H. EL-GAMAL [2008]. “On the secrecy capacity of fading chan-

nels.” IEEE Trans. Inf. Theory, vol. 54, no. 10, pp. 4687–4698.

GRAY, R. M. [2005]. “Toeplitz and circulant matrices: A review.” Foundations and Trends in

Commun. and Inf. Theory, vol. 2, no. 3.

HAENGGI, M. [2005]. “On distances in uniformly random networks.” IEEE Trans. Inf. Theory,

vol. 51, no. 10, pp. 3584–3586.

HAENGGI, M., J. ANDREWS, F. BACCELLI, O. DOUSSE, AND M. FRANCESCHETTI [2009].

“Stochastic geometry and random graphs for the analysis and design of wireless networks.”

IEEE J. Sel. Areas Commun., vol. 27, no. 7, pp. 1029–1046.

HE, X. AND A. YENER [2010]. “Providing secrecy irrespective of eavesdropper’s channel state.”

In Proc. IEEE Global Commun. Conf. (Globecom). pp. 1–5.

HERO, A. O. [2003]. “Secure space-time communication.” IEEE Trans. Inf. Theory, vol. 49,

no. 12, pp. 3235–3249.

HOCHWALD, B. AND S. VISHWANATH [2002]. “Space-time multiple access: Linear growth in

the sum rate.” In Proc. Allerton Conf. on Commun., Control, and Computing. Monticello, IL.

HOCHWALD, B. M., C. B. PEEL, AND A. L. SWINDLEHURST [2005]. “A vector-perturbation

technique for near-capacity multiantenna multiuser communication - Part II: Perturbation.”

IEEE Trans. Commun., vol. 55, no. 5, pp. 537–544.

JINDAL, N. [2006]. “MIMO broadcast channels with finite-rate feedback.” IEEE Trans. Inf.

Theory, vol. 52, no. 11, pp. 5045–5060.

JOHAM, M., W. UTSCHICK, AND J. NOSSEK [2005]. “Linear transmit processing in MIMO

communications systems.” IEEE Trans. Signal Process., vol. 53, no. 8, pp. 2700–2712.

KHISTI, A., A. TCHAMKERTEN, AND G. WORNELL [2008]. “Secure broadcasting over fading

channels.” IEEE Trans. Inf. Theory, vol. 54, no. 6, pp. 2453–2469.

KHISTI, A. AND G. WORNELL [2010]. “Secure transmission with multiple antennas I: The

MISOME wiretap channel.” IEEE Trans. Inf. Theory, vol. 56, no. 7, pp. 3088–3104.

136 Bibliography

KHISTI, A., G. WORNELL, A. WIESEL, AND Y. ELDAR [2007]. “On the Gaussian MIMO

wiretap channel.” In Proc. IEEE Int. Symp. on Inform. Theory (ISIT). pp. 2471–2475.

LEUNG-YAN-CHEONG, S. AND M. HELLMAN [1978]. “The Gaussian wire-tap channel.” IEEE


LI, Q., G. LI, W. LEE, M. LEE, D. MAZZARESE, B. CLERCKX, AND Z. LI [2010]. “MIMO

techniques in WiMAX and LTE: a feature overview.” IEEE Comms. Mag., vol. 48, no. 5, pp.

86–92.

LI, Z., W. TRAPPE, AND R. YATES [2007a]. “Secret communication via multi-antenna transmis-

sion.” In Proc. IEEE Conf. on Inf. Sciences and Systems (CISS). pp. 905–910.

LI, Z., R. YATES, AND W. TRAPPE [2007b]. “Secret communication with a fading eavesdropper

channel.” In Proc. IEEE Int. Symp. on Inform. Theory (ISIT). pp. 1296–1300.

LIANG, Y., G. KRAMER, H. V. POOR, AND S. SHAMAI [2009a]. “Compound wiretap channels.”

EURASIP Journ. Wireless Commun. Network.

LIANG, Y., H. V. POOR, AND S. SHAMAI [2009b]. Information Theoretic Security. Dordrecht,

The Netherlands: Now Publisher.

LIM, C., T. YOO, B. CLERCKX, B. LEE, AND B. SHIM [2013]. “Recent trend of multiuser

MIMO in LTE-advanced.” IEEE Comms. Mag., vol. 51, no. 3, pp. 127–135.

LIU, R., T. LIU, H. V. POOR, AND S. SHAMAI [2010a]. “Multiple-input multiple-output Gauss-

ian broadcast channels with confidential messages.” IEEE Trans. Inf. Theory, vol. 56, no. 9, pp.

4215–4227.

—— [2010b]. “A vector generalization of Costa’s entropy-power inequality with applications.”

IEEE Trans. Inf. Theory, vol. 56, no. 4, pp. 1865–1879.

—— [2013]. “New results on multiple-input multiple-output broadcast channels with confidential

messages.” IEEE Trans. Inf. Theory, pp. 1346–1359.

LIU, R., I. MARIC, P. SPASOJEVIC, AND R. D. YATES [2008]. “Discrete memoryless interfer-

ence and broadcast channels with confidential messages: Secrecy rate regions.” IEEE Trans.

Inf. Theory, vol. 54, no. 6, pp. 2493–2507.

Bibliography 137

LIU, R. AND H. POOR [2009]. “Secrecy capacity region of a multiple-antenna Gaussian broadcast

channel with confidential messages.” IEEE Trans. Inf. Theory, vol. 55, no. 3, pp. 1235–1249.

LIU, R. AND W. TRAPPE [2010]. Securing Wireless Communications at the Physical Layer. New

York, NY: Springer Verlag.

LIU, T. AND S. SHAMAI [2009]. “A note on the secrecy capacity of the multiple-antenna wiretap

channel.” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2547–2553.

MARZETTA, T. L., N. JINDAL, AND A. LOZANO [2009]. “What is the value of joint processing

of pilots and data in block-fading channels?” In Proc. IEEE Int. Symp. on Inform. Theory

(ISIT). pp. 2189–2193.

MASSEY, J. [1988]. “An introduction to contemporary cryptology.” Proc. IEEE, vol. 76, no. 5,

pp. 533–549.

MUHARAR, R. AND J. EVANS [2009]. “Downlink beamforming with transmit-side channel cor-

relation: A large system analysis.” Technical Report. Available: http://www.cubinlab.

ee.unimelb.edu.au/˜rmuharar/doc/techRepCorr.pdf.

MUKHERJEE, A., S. A. A. FAKOORIAN, J. HUANG, AND A. L. SWINDLEHURST [2013]. “Prin-

ciples of physical layer security in multiuser wireless networks: A survey.” submitted to IEEE

Commun. Surveys and Tutorials. Available arXiv:1011.3754v2.

NEGI, R. AND S. GOEL [2005]. “Secret communication using artificial noise.” In Proc. of the

IEEE Veh. Tech. Conference (VTC), vol. 3. pp. 1906–1910.

NGUYEN, V. AND J. EVANS [2008]. “Multiuser transmit beamforming via regularized channel

inversion: A large system analysis.” In Proc. IEEE Global Commun. Conf. (Globecom). pp.

1–4.

NGUYEN, V. K., R. MUHARAR, AND J. S. EVANS [2009]. “Multiuser transmit

beamforming via regularized channel inversion: A large system analysis.” Technical

Report. Available: http://cubinlab.ee.unimelb.edu.au/˜rmuharar/doc/

manuscriptrevRusdha22-11.pdf.

OGGIER, F. AND B. HASSIBI [2008]. “The secrecy capacity of the MIMO wiretap channel.” In

Proc. IEEE Int. Symp. on Inform. Theory (ISIT). pp. 524–528.

http://www.cubinlab.ee.unimelb.edu.au/~rmuharar/doc/techRepCorr.pdf

http://www.cubinlab.ee.unimelb.edu.au/~rmuharar/doc/techRepCorr.pdf

http://cubinlab.ee.unimelb.edu.au/~rmuharar/doc/manuscriptrevRusdha22- 11.pdf

http://cubinlab.ee.unimelb.edu.au/~rmuharar/doc/manuscriptrevRusdha22- 11.pdf

138 Bibliography

PAPANDRIOPOULOS, J., S. DEY, AND J. EVANS [2008]. “Optimal and distributed protocols for

cross-layer design of physical and transport layers in MANETs.” IEEE/ACM Transactions on

Networking, vol. 16, no. 6, pp. 1392–1405.

PARADA, P. AND R. BLAHUT [2005]. “Secrecy capacity of SIMO and slow fading channels.” In

Proc. IEEE Int. Symp. on Inform. Theory (ISIT). pp. 2152–2155.

PEACOCK, M. J. M., I. B. COLLINGS, AND M. L. HONIG [2008]. “Eigenvalues distributions of

sums and products of large random matrices via incremental matrix expansions.” IEEE Trans.

Inf. Theory, vol. 54, no. 5, pp. 2123–2138.

PEEL, C. B., B. M. HOCHWALD, AND A. L. SWINDLEHURST [2005]. “A vector-perturbation

technique for near-capacity multiantenna multiuser communication - Part I: Channel inversion

and regularization.” IEEE Trans. Commun., vol. 53, no. 1, pp. 195–202.

RAPPAPORT, T. S. [1996]. Wireless Communications: Principles and Practice. IEEE Press, 1st

ed.

RUDIN, W. [1987]. Real and Complex Analysis. New York, NY: McGraw-Hill, 3rd ed.

RYAN, D., I. B. COLLINGS, I. V. L. CLARKSON, AND R. W. HEATH JR. [2008]. “Performance

of vector perturbation multiuser MIMO systems with limited feedback.” IEEE Trans. Commun.,

vol. 57, no. 9, pp. 2633–2644.

SCHNEIER, B. [1998]. “Cryptographic design vulnerabilities.” IEEE Computer, vol. 31, no. 9,

pp. 29–33.

SHAFIEE, S. AND S. ULUKUS [2007]. “Achievable rates in Gaussian MISO channels with secrecy

constraints.” In Proc. IEEE Int. Symp. on Inform. Theory (ISIT). pp. 2466–2470.

SHANNON, C. E. [1949]. “Communication theory of secrecy systems.” Bell System Tech. J.,

vol. 28, pp. 656–715.

SILVERSTEIN, J. W. AND S.-I. CHOI [1995]. “Analysis of the limiting spectral distribution of

large dimensional random matrices.” Journal of Multivariate Analysis, vol. 54, no. 2, pp. 295–

309.

SOUSA, E. AND J. SILVESTER [1990]. “Optimum transmission ranges in a direct-sequence

spread-spectrum multihop packet radio network.” IEEE J. Sel. Areas Commun., vol. 8, no. 5,

pp. 762–771.

Bibliography 139

SPENCER, Q. H., C. B. PEEL, A. L. SWINDLEHURST, AND M. HAARDT [2004a]. “An intro-

duction to the multi-user MIMO downlink.” IEEE Comms. Mag., vol. 42, no. 10, pp. 60–67.

SPENCER, Q. H., A. L. SWINDLEHURST, AND M. HAARDT [2004b]. “Zero-forcing methods

for downlink spatial multiplexing in multiuser MIMO channels.” IEEE Trans. Signal Process.,

vol. 52, no. 2, pp. 461–471.

STOYAN, D., W. KENDALL, AND J. MECKE [1996]. Stochastic geometry and its applications.

New York, NY: John Wiley & Sons Ltd., 2nd ed.

SUNG, C. K. AND I. COLLINGS [2010]. “Cooperative transmission with decode-and-forward

MIMO relaying in multiuser relay networks.” In Proc. IEEE Int. Conf. on Comm. (ICC). pp.

1–5.

SWINDLEHURST, A. L. [2009]. “Fixed SINR solutions for the MIMO wiretap channel.” In Proc.

Internat. Conf. Acoust. Speech Signal Process. pp. 2437–2440.

TELATAR, E. [1999]. “Capacity of multi-antenna Gaussian channels.” European Transactions on

Telecommunications, vol. 10, no. 6, pp. 585–595.

TSE, D. N. C. AND P. VISWANATH [2005]. Fundamentals of Wireless Communication. Cam-

bridge University Press.

VUCETIC, B. AND J. YUAN [2003]. Space-time Coding. John Wiley & Sons.

WAGNER, S., R. COUILLET, M. DEBBAH, AND D. T. M. SLOCK [2012]. “Large system analysis

of linear precoding in correlated MISO broadcast channels under limited feedback.” IEEE


WYNER, A. D. [1975]. “The wire-tap channel.” Bell System Tech. J., vol. 54, pp. 1355–1387.

XIE, J. AND S. ULUKUS [2013]. “Secure degrees of freedom of the gaussian wiretap channel

with helpers and no eavesdropper csi: Blind cooperative jamming.” In Proc. IEEE Conf. on Inf.

Sciences and Systems (CISS). pp. 1–5.

YANG, N., G. GERACI, J. YUAN, AND R. MALANEY [2013]. “Confidential broadcasting via

linear precoding in non-homogeneous MIMO multiuser networks.” submitted to IEEE Trans.

Commun.

140 Bibliography

YOO, T. AND A. GOLDSMITH [2006]. “On the optimality of multiantenna broadcast scheduling

using zero-forcing beamforming.” IEEE J. Sel. Areas Commun., vol. 24, no. 3, pp. 528–541.

ZHOU, X. AND M. MCKAY [2009]. “Physical layer security with artificial noise: Secrecy capac-

ity and optimal power allocation.” In Proc. IEEE Int. Conf. on Sig. Proc. and Commun. Syst.

(ICSPCS). pp. 1–5.

physical layer security for multi-user ... - wordpress.coma special thanks also goes to my...

Documents