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Secrecy Performance of Multi-user MISO VLC Broadcast Channels with Confidential Messages

September 2018
IEEE Transactions on Wireless Communications PP(99):1-1

September 2018
PP(99):1-1

DOI:10.1109/TWC.2018.2871055

Authors:

Mohamed Amine Arfaoui

Concordia University Montreal

Ali Ghrayeb

Texas A&M University at Qatar

Chadi Assi

Concordia University Montreal

We study in this paper the secrecy performance of a multi-user (MU) multiple-input single-output (MISO) visible light communication (VLC) broadcast channel with confidential messages. The underlying system model comprises K +1 nodes: a transmitter (Alice) equipped with N fixtures of LEDs and K spatially dispersed users, each equipped with a single photodiode (PD). The MU channel is modeled as deterministic and real-valued, and assumed to be perfectly known to Alice since all users are assumed to be active. We consider typical secrecy performance measures, namely, the max-min fairness, the harmonic mean, the proportional fairness and the weighted fairness. For each performance measure, we derive an achievable secrecy rate for the system as a function of the precoding matrix. As such, we propose algorithms that yield the best precoding matrix for the derived secrecy rates, where we analyze their convergence and computational complexity. In contrast, what has been considered in the literature so far is zero-forcing (ZF) precoding, which is suboptimal. We present several numerical examples through which we demonstrate the substantial improvements in the secrecy performance achieved by the proposed techniques compared to those achieved by the conventional ZF. However, this comes at a slight increase in the complexity of the proposed techniques as compared to that of ZF.

Average secrecy rate R * s for the max-min fairness (MMF), the harmonic mean (HM), the proportional fairness (PF) and the weighted fairness (WF) for the number of users K = 4 and K = 2.

…

Average execution time of the proposed scheme and of ZF precoding versus A 2 for the max-min fairness (MMF), the harmonic mean (HM), the proportional fairness (PF) and the weighted fairness (WF). The number of users is K = 4.

…

Convergence behavior of Algorithm 1 when K = 4 and A 2 = 0 dBm.

…

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IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS 1

Secrecy Performance of Multi-User MISO VLC

Broadcast Channels With Conﬁdential Messages

Mohamed Amine Arfaoui ,AliGhrayeb ,andChadiM.Assi

Abstract— We study, in this paper, the secrecy performance1

of a multi-user (MU) multiple-input single-output visible light2

communication broadcast channel with conﬁdential messages.3

The underlying system model comprises K+1 nodes: a transmit-4

ter (Alice) equipped with Nﬁxtures of LEDs and Kspatially5

dispersed users, each equipped with a single photo-diode. The6

MU channel is modeled as deterministic and real-valued and7

assumed to be perfectly known to Alice, since all users are8

assumed to be active. We consider typical secrecy performance9

measures, namely, the max–min fairness, the harmonic mean,10

the proportional fairness, and the weighted fairness. For each11

performance measure, we derive an achievable secrecy rate for12

the system as a function of the precoding matrix. As such,13

we propose algorithms that yield the best precoding matrix for14

the derived secrecy rates, where we analyze their convergence15

and computational complexity. In contrast, what has been con-16

sidered in the literature so far is zero-forcing (ZF) precoding,17

which is suboptimal. We present several numerical examples18

through which we demonstrate the substantial improvements in19

the secrecy performance achieved by the proposed techniques20

compared with those achieved by the conventional ZF. However,21

this comes at a slight increase in the complexity of the proposed22

techniques compared with ZF.23

Index Terms—Broadcast, MISO, secrecy performance, VLC.24

I. INTRODUCTION25

A. Motivation26 VISIBLE light communication (VLC) is a new27

communication technology that uses visible light28

as a transmission medium, i.e., the light emitted by light29

sources is used for illumination and data communication30

purposes simultaneously. VLC has gained signiﬁcant interest31

during the last decade, owing to its high speed and low32

deployment cost [1], robustness against interference and33

abundance in the available spectrum [2]. Various aspects34

of VLC systems have been studied in the literature. In [3],35

the authors proposed a VLC end to end architecture with36

suitable modulation schemes. In [4], the performance of VLC37

Manuscript received March 18, 2018; revised August 27, 2018; accepted

September 5, 2018. This work was supported in part by the Qatar National

Research Fund through NPRP under Grant NPRP8-052-2-029, in part by

FQRNT, and in part by Concordia University. The associate editor coordi-

nating the review of this paper and approving it for publication was N. Yang.

(Corresponding author: Ali Ghrayeb.)

M. A. Arfaoui and C. M. Assi are with the Concordia Institute for

Information Systems Engineering, Concordia University, Montreal, QC H3G

1M8, Canada (e-mail: m_arfaou@encs; assi@ciise.concordia.ca).

A. Ghrayeb is with the Electrical and Computer Engineering

Department, Texas A&M University at Qatar, Doha, Qatar (e-mail:

ali.ghrayeb@qatar.tamu.edu).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TWC.2018.2871055

systems was studied in terms of transmit data rates and signal- 38

to-interference-plus-noise (SINR) ratio. In [5]–[7], a thorough 39

review of the advantages of VLC was given, whereas, 40

in [8], the potential of VLC for indoor communications was 41

discussed. In [9] and [10], the authors studied the fundamental 42

limits of optical wireless channels. In [11], the viability of 43

VLC for 5G wireless networks was investigated. 44

Security issues arise naturally in VLC broadcast channels 45

due to its open nature. Each receiver is able to receive signals 46

that contain all information ﬂows from the transmitter. Hence, 47

some receivers may decode data that are not intended for them. 48

However, information ﬂows should be kept conﬁdential from 49

non-intended receivers [12]. This requires that the transmitter 50

employs security techniques to guarantee such conﬁdentiality 51

requirements. On the other hand, physical layer security (PLS) 52

has achieved great success in enhancing the security of wire- 53

less communications or complementing existing cryptographic 54

schemes for radio-frequency (RF) broadcast channels [13]. 55

Due to this, there have been recently many attempts to extend 56

the previous studies to VLC. The potential of PLS stems 57

from its ability of leveraging features of the surrounding 58

environments via sophisticated encoding techniques at the 59

physical layer [14]. Indeed, PLS schemes can be applied in 60

the same spirit to VLC systems. Furthermore, VLC systems 61

are characterized by many speciﬁcities that imply major differ- 62

ences compared to RF systems. Precisely, VLC channels are 63

quasi-static and real valued channels which seemingly simplify 64

the application of PLS techniques. However, due to the limited 65

dynamic range of the emitting LEDs, VLC systems impose a 66

peak-power constraint, i.e., an amplitude constraint, on the 67

channel input which makes unbounded inputs, like Gaussian 68

inputs, not admissible. As a result, the performance and the 69

optimization of PLS schemes must be revisited in the VLC 70

context due to its different operating constraints. 71

B. Related Work 72

Information theoretic approaches in PLS were developed 73

to achieve secure communication [15], [16]. Based on that, 74

many techniques were then introduced in PLS to enhance the 75

secrecy performance of VLC broadcast channels. The secrecy 76

performance of single-user (SU) MISO VLC wiretap systems 77

was investigated in [17]–[29]. Under perfect eavesdropper’s 78

channel state information (CSI), both beamforming and 79

zero-forcing beamforming were adopted [17], [18], whereas 80

artiﬁcial noise was considered in [19]–[21]. Under imperfect 81

eavesdropper’s CSI, robust beamfomring and artiﬁcial noise 82

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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were employed in [22] and [23], respectively. The randomly83

located terminals scenario with a single ﬁxed user and84

multiple randomly located eavesdroppers was investigated85

in [24]–[29], where the average secrecy performance was86

analyzed using stochastic geometry.87

The secrecy capacity of a Gaussian wiretap channel under88

an amplitude constraint has not been determined in closed89

form yet. In fact, from an information theoretic point-of-90

view, ﬁnding the signaling schemes that achieve the secrecy91

capacity of a Gaussian wiretap channel under an amplitude92

constraint is quite challenging and it is still an open problem.93

This is attributed to the fact that, when input distributions of94

unbounded support are not permissible, the optimal input dis-95

tribution is either unknown, or only known to be discrete [30]96

for the special case of a degraded Gaussian single-input single-97

output (SISO) wiretap channel. VLC falls in this category98

since amplitude constraints must be satisﬁed. Upper and lower99

bounds on the capacity of the free-space optical intensity100

channel under peak and average optical power constraint were101

derived in [31]. In addition prior works focused only on the102

uniform distribution [17], the truncated Gaussian [19] and the103

truncated generalized normal (TGN) [20], [32], where it was104

shown that TGN is the best choice of input signaling till now,105

since it encompasses several bounded input distributions and106

one can enhance the secrecy performance of the system by107

optimizing over its parameters.108

The secrecy performance of MU-MISO broadcast chan-109

nels has been studied in the literature [33]–[43]. However,110

adoption of techniques developed for RF channels for VLC111

channels may not be straightforward since RF signals are112

complex-valued, which is fundamentally different from the113

real and bounded VLC signals. Nevertheless, several studies114

on precoding designs for MU-MISO VLC broadcast channels115

were proposed in [44]–[48]. In [44] and [45], the system116

was considered without an external eavesdropper, whereas117

in [46] and [47], it was assumed that an external eavesdropper118

may exist within the same area. For both cases, the secrecy119

performance of the systems was investigated, where only the120

max-min fairness and the weighted fairness were used as121

secrecy performance measures. Moreover, zero-forcing (ZF)122

precoding was employed, to cancel the information leakage123

between users, in conjunction with uniform input signaling.124

However, although it is a simple precoding scheme, ZF is125

suboptimal in the sense that the secrecy performance of the126

system can be enhanced by searching for optimal precoding127

schemes. In [48], the same problem was considered for the128

two-user MISO broadcast channel with conﬁdential messages129

under per-antenna amplitude constraint, per-antenna power130

constraint and average power constraint were considered.131

However, assuming only two active users in a VLC system132

is not a realistic scenario especially for large geometric areas133

or dense networks.134

C. Contributions135

In this paper, we consider a MU-MISO VLC broadcast136

channel consisting of a transmitter (Alice), equipped with137

Nﬁxtures of LEDs, aiming to transmit Kconﬁdential mes-138

sages to Kspatially dispersed users, each equipped with 139

a single PD. The transmitted messages are assumed to be 140

conﬁdential such that each user is supposed to receive and 141

decode only his own message, i.e., users are ignorant about 142

messages not intended for them. The channel input is subject 143

to an amplitude constraint. We develop within this framework 144

linear precoding schemes that enhance typical secrecy per- 145

formance measures, namely: the max-min fairness (MMF), 146

the harmonic mean (HM), the proportional fairness (PF) and 147

the weighted fairness (WF). Speciﬁcally, we derive ﬁrst an 148

achievable secrecy rate of a single user existing within the 149

MU VLC network. Second, we formulate the problems of 150

linear precoding schemes that maximize the aforementioned 151

secrecy performance measures of the considered system. Then, 152

we propose iterative algorithms to ﬁnd the best linear pre- 153

coding schemes for all four ecrecy performance measures. 154

We also analyze the computational complexity of the proposed 155

scheme and compare it to that of ZF where we show that 156

the increase in complexity due to the proposed schemes 157

can be, in the worst case scenario, the square root of the 158

number of active users in the network. Finally, we compare the 159

performance of the proposed schemes to that of the con- 160

ventional ZF precoding [44]–[47] and we demonstrate that 161

substantial improvements can be achieved by the proposed 162

schemes. 163

D. Outline and Notations 164

The rest of the paper is organized as follows. Section II 165

presents the system model. Section III present the proposed 166

precoding schemes. Sections IV and V present the numerical 167

results and the conclusion, respectively. 168

The following notations are adopted throughout the paper. 169

Upper case bold characters denote matrices and lower case 170

bold characters denote column vectors. We use log(·), without 171

a base, to denote natural logarithms and information rates are 172

speciﬁed in (nats/s/Hz). All the mathematical operators and 173

parameters used in this paper are deﬁned in Table I. 174

II. SYSTEM MODEL 175

A. The VLC Channel Model 176

We consider a DC-biased intensity-modulation direct- 177

detection (IM-DD) scheme where the transmit element is 178

an illumination LED driven by a ﬁxed bias IDC ∈R+.179

The DC-offset sets the average radiated optical power and, 180

consequently, settles the illumination level. The data signal 181

s∈Ris a zero-mean current signal superimposed on IDC 182

to modulate the instantaneous optical power emitted from the 183

LED. In order to maintain linear current-light conversion and 184

avoid clipping distortion, the total current IDC +smust be 185

constrained within some range IDC ±νIDC where ν∈[0,1] 186

is the modulation index [17]. Consequently, smust satisfy 187

an amplitude constraint expressed as |s|νIDC. After that, 188

the total current IDC +sis converted into an optical power 189

and transmitted by the LED, in which the conversion factor is 190

denoted by η.191

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ARFAOUI et al.: SECRECY PERFORMANCE OF MU MISO VLC BROADCAST CHANNELS 3

TAB L E I

TABLE OF NOTATIONS

At the receiver side, the receiver’s PD, with a responsiv-192

ity Rp, converts the incident optical power into a propor-193

tional current. Finally the DC-offset IDC is removed and a194

transimpedance ampliﬁer, with gain T, is used to produce a195

voltage signal y∈R, which is a scaled, but noisy, version of196

the transmitted signal s. The noise process is well-modeled197

in VLC channels as signal-independent, zero-mean, additive198

white Gaussian noise (AWGN) with variance σ2

n,givenby199

σ2

n=σ2

sh +σ2

th,(1)200

where σ2

sh and σ2

th are the variances of the shot noise and201

thermal noise, respectively, [49]. The shot noise in an optical202

wireless channel is generated by the high rate physical photo-203

electronic conversion process204

σ2

sh =2qB (Pr+Ibg I2),(2)205

where qis the electronic charge, Bis the system bandwidth,206

Pr=RphIDC is the average received power, Ibg is the207

ambient current in the PD and I2is the noise bandwidth factor.208

The thermal noise is generated within the transimpedance209

receiver circuitry and its variance is given by210

σ2

th =8π¯

KTKARXB2cI2

G+2πΓFTA

RX I3Bc,(3)211

where ¯

Kis Boltzmann’s constant, TKis the absolute temper-212

ature, cis the ﬁxed capacitance of the PD per unit area, Gis213

the open-loop voltage gain, ΓFis the transimpedance channel214

noise factor and I3=0.0868 [50].215

Fig. 1. VLC path gain description.

Armed with the above description, the received signal is 216

expressed as 217

y=hs +n, (4) 218

where yrepresents the received signal, srepresents the zero- 219

mean transmitted signal subject to the amplitude constraint 220

|s|≤A,whereA=νIDC ,nrepresents AWGN N(0,σ

2)221

distributed and h=ηRTg ∈R+represents the channel gain, 222

in which gdenotes the path gain of the optical link. Assuming 223

that the considered LED has a Lambertian emission pattern, 224

the path gain is given as [51], [52] 225

g=⎧

⎨

⎩

2π(m+1)cosm(θ)ARX

d2cos(ψ)R|ψ|≤ψFoV

0|ψ|>ψ

FoV ,

(5) 226

where m=−log(2)

log(cos(φ1

)) is the order of the Lambertian 227

emission with half irradiance at semi-angle φ1

2(measured 228

from the optical axis of the LED). As shown in Fig 1, θ229

represents the angle of irradiance, dis the line-of-sight (LoS) 230

distance between the LED and the PD, ψis the angle of 231

incidence, ψFoV is the receiver ﬁeld of view (FoV) and 232

ARX =n2

sin2(ψFoV )APD is the receiver collection area, such 233

that ncis the refractive index of the optical concentrator and 234

APD is the PD area. 235

In most practical cases, the VLC channel is either constant 236

(e.g., indoors VLC with no mobility) or varies very slowly 237

compared to the transmission rate (mobility or outdoors VLC). 238

The channel coherence time is typically 0.1 to 10 ms whereas 239

the transmission rates are on the order of several tens of 240

Mpbs to several Gbps. Thus, the channel remains constant 241

over thousands up to millions of consecutive bits, and hence, 242

it is considered quasi-static in the scale of interest [53]. 243

Various VLC channel estimation methods were introduced 244

in the literature, especially the receiver’s location and the 245

channel parameters in downlink VLC, as described in (5). For 246

the estimation of the receiver’s location, [54] and references 247

therein proposed receiver positioning algorithms, whereas for 248

the channel estimation, [55] and [56] proposed estima- 249

tion methods using neural networks and statistical Bayesian 250

MMSE, respectively. 251

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B. The MU-MISO VLC Broadcast Channel252

As mentioned above, Alice is equipped with Nﬁxtures253

of LEDs and intends to transmit K(K≤N)conﬁdential254

messages to Kspatially dispersed users. For k∈[[ 1 ,K]] ,255

we denote by ukthe conﬁdential message relative to the256

kth user. The Kmessages are conﬁdential and Alice has257

to communicate each message to its intended user while258

keeping each user unaware of the other messages. The N×1259

transmitted signal is expressed as260

s=Wu =



k=1

wkuk,(6)261

where W=[w1,w2,...,wK]∈RN×Kis the precoding262

matrix of the system, such that for k∈[[ 1 ,K]] ,wk∈RN

263

is the precoding vector relative to the kth message uk,and264

u=[u1,u

2,...,u

K]Tis the zero-mean K×1vector of265

conﬁdential messages. The transmitted signal sis subject to a266

peak-power constraint, i.e., amplitude constraint, expressed as267

||s||∞≤A, (7)268

where A∈R∗

+. Without loss of generality, we assume that269

the Kmessages are independent and identically distributed270

(i.i.d) according to a generic continuous zero-mean random271

variable uthat satisﬁes |u|≤A. Consequently, ||u||∞≤A,272

E(u)=0and EuuT=σ2

uIK. On the other hand, in order273

to satisfy the amplitude constraint in (7), we impose the274

following constraint on the matrix W.275

||W||∞≤1.(8)276

Based on the above, the signal received at the kth user, for277

k∈[[ 1 ,K]] , is expressed as278

yk=hT

kwkuk+



i=1

i=k

kwiui+nk,(9)279

where hk∈RN

+is the channel gain vector of the kth user and280

nkis a Gaussian noise sample which is N(0,σ

2)distributed.281

As seen in (9), the ﬁrst term hT

kwkukis the desired signal of282

th kth user, while the second term K

i=1

i=k

kwiuiis the multi-283

user interference (MUI) and the third term nkis the Gaussian284

noise superimposed at the reception. Consequently, the SINR285

at the kth received is expressed as286

SINRk=hT

kwk2σ2

K

i=1

i=khT

kwi2σ2

u+σ2.(10)287

In this case, the system model of the MU-MISO VLC broad-288

cast channel is expressed as289

y=Hs +n=HWu +n,(11)290

where y=[y1,y

2,...,y

K]T,H=[h1,h2,...,hK]Tand291

n=[n1,n

2,...,n

K]T.292

The objective of this paper is to construct linear precoding293

schemes, represented by the precoding matrix W, that the294

secrecy performance of the MU-MISO VLC broadcast channel295

in (11) under the inﬁnity norm constraint in (8). This will be296

the focus of the next section.297

III. PROPOSED PRECODING SCHEMES 298

A. Single User Achievable Secrecy Rate 299

In this part, we derive an achievable secrecy rate of a single 300

user existing within the MU framework. Since the Kmessages 301

are conﬁdential, when Alice wants to communicate with a 302

certain user in the network, the remaining users are treated 303

as eavesdroppers to this communication link. Therefore, for 304

k∈[[ 1 ,K]] , the received signal at the kth user and at the 305

remaining K−1users are expressed as 306

yk=hT

kwkuk+hT

k¯

Wk¯

uk+nk

yk=¯

Hkwkuk+¯

Hk¯

Wk¯

uk+¯

nk,(12) 307

where ¯

yk,¯

ukand ¯

nkare the vectors y,uand nafter removing 308

the kth element, respectively, and ¯

Hkand ¯

Wkare the matrices 309

Hand Wafter removing the kth row and the kth column, 310

respectively.1In other words, the MISO VLC wiretap system 311

in (12) assumes that the remaining users are treated as a single 312

potential eavesdropper for the communication link between 313

Alice and the kth user. Based on this discussion, an achievable 314

secrecy rate of the kth Gaussian MISO VLC wiretap channel 315

in (12) is given in the following theorem. 316

Theorem 1: An achievable secrecy rate of the MISO VLC 317

Gaussian wiretap channel in (12) is equal to R+

s,k,where 318

Rs,k(pu,W)=1

2log ⎡

⎢

⎣

1+auK

i=1 hT

kwi2

1+buK

i=1

i=khT

kwi2⎤

⎥

⎦319

−1

2log ⎡

⎢

⎣1+bu



i=1

i=khT

iwk2⎤

⎥

⎦,(13) 320

where au=exp(2 hu)

2πeσ2and bu=σ2

σ2, such that hudenotes the 321

differential entropy of the random scalar variable uand σ2

u322

denotes its variance. 323

Proof: See Appendix. 324

Note that the achievable secrecy rate in (13) assumes that 325

all channels are perfectly known to Alice, which is a valid 326

assumption. In fact, since all users are active, Alice can 327

perfectly estimates the channel gain of each user through 328

feedbacks sent from each user. In addition, this result is valid 329

for any precoding matrix Wand any continuous random vari- 330

able u. In other words, the result of Theorem 1 is independent 331

from the inﬁnity constraint in (8) and from the choice of the 332

probability distribution pu, but under the assumption that it is a 333

continuous probability distribution. However, the objective of 334

the paper is developing well-structured designs of W, under 335

the inﬁnity norm constraint in (8), that enhance the secrecy 336

performance of the system. 337

1Note that ¯

ykrepresents the vector of received signals at the remaining

K−1users before performing any decoding. However, the remaining users

are assumed to be able to work in a collaborative manner to jointly remove the

interference caused for each other, which is in accordance with the worst-case

consideration in physical layer security studies.

IEEE Proof

ARFAOUI et al.: SECRECY PERFORMANCE OF MU MISO VLC BROADCAST CHANNELS 5

B. Secrecy Performance Measures338

Our objective here is to derive linear precoding schemes that339

maximize the secrecy performance with respect to certain per-340

formance measures under the amplitude constraint in (8). i.e.,341

max

WS(pu,W)342

s.t. ||W||∞≤1,(14)343

where f(pu,W)is the objective function that represents the344

secrecy performance measure of interest. Typical secrecy345

performance measures include [57]:346

i) Max-min fairness (MMF): S=min

1≤k≤KRs,k.347

ii) Harmonic mean (HM): S=KK

k=1 R−1

s,k−1.348

iii) Proportional fairness (PF): S=K

k=1 Rs,k1

K.349

iv) Weighted fairness (WF): S=K

k=1 dkRs,k,where350

(dk)1≤k≤K∈R+,351

with increasing order of achievable secrecy sum-rate and352

decreasing order of user fairness. In fact, for the case where353

dk=1for all k∈[[ 1 ,K]] ,wehave354

MMF ≤HM ≤PF ≤WF.(15)355

However, in terms of user fairness, MMF is the best secrecy356

performance measure [57]. In this subsection, we consider357

all the secrecy performance measures discussed above while358

assuming that the probability distribution puis ﬁxed and359

known.360

1) Max-Min Fairness: The max-min fairness measure aims361

to maximize the minimum achievable secrecy rate among the362

Kusers, which leads to the following optimization problem.363

P1:Rs(pu)=max

Wmin

k∈[[ 1 ,K]] Rs,k (pu,W)364

s.t. ||W||∞≤1.(16)365

Problem P1is a max-min problem which involves two opti-366

mization problems. The inner problem consists of ﬁnding the367

user with the lowest achievable secrecy rate for a ﬁxed precod-368

ing matrix W, whereas the outer problem involves ﬁnding the369

best precoding matrix that maximizes the achievable secrecy370

rate for a given user. Solving P1is difﬁcult due to the mutual371

dependence between the optimization parameters in the inner372

and outer problems and the non concavity of the achievable373

secrecy rate Rs,k. We reformulate problem P1as374

P1:Rs(pu)=−min

W,z −z375

s.t. z−Rs,k (pu,W)≤0,∀k∈[[ 1 ,K]] ,

||W||∞−1≤0,

376

(17)377

where zis simply an auxiliary variable. We perform the change378

of optimization variable expressed as379

W=H⊥√X=HTHHT−1√X,(18)380

where X=(xk,i)1≤k,i≤Kis a K×Kmatrix, such that for381

all (k, i)∈[[ 1 ,K]] 2,xk,i ∈R+.Letx=xT

1,xT

2,...,xT

KT

382

be the K2×1vector, such that for all k∈[[ 1 ,K]] ,383

xk=[xk,1,x

k,2,...,x

k,K ]T. Consequently, for k∈[[ 1 ,K]] ,384

the achievable secrecy rate Rs,k is a function of xand it is 385

re-expressed as 386

Rs,k (pu,x)=1

2log 1+au



i=1

xk,i

−1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xk,i

⎤

⎥

⎦387

−1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xi,k⎤

⎥

⎦.(19) 388

Furthermore, we impose the following inﬁnity norm constraint 389

on the matrix X.390

||√X||∞≤aH,(20) 391

where aH=min

||H||∞,1

||H⊥||∞. In this case, the inﬁnity 392

norm constraint in (20) and the inﬁnity norm constraint in (8) 393

are equivalent, i.e., if one is satisﬁed, the other is automatically 394

satisﬁed. In fact, if the constraint in (8) is satisﬁed, then 395

||√X|| =||HW||∞≤||H||∞||W||∞≤||H||∞≤aH,(21) 396

and if the constraint in (20) is satisﬁed, then 397

||W|| =||H⊥√X||∞≤||H⊥||∞||√X||∞398

≤min ||H⊥||∞||H||∞,1399

≤1.(22) 400

Based on the above, problem P1can be re-written as 401

P1:Rs(pu)=−min

x,z −z402

s.t. ⎧

⎪

⎨

⎪

⎩

k(pu,x,z)=z

−Rs,k (pu,x)≤0,∀k∈[[ 1 ,K]] ,

k(x)=||√xk||1

−aH≤0,∀k∈[[ 1 ,K]] .

403

(23) 404

The function (w,z)→−zis convex. However, the constraints 405

c1

k1≤k≤Kand c2

k1≤k≤Kare not convex and consequently 406

the optimization problem P1is not convex. However, one way 407

to solve problem P1is using the convex-concave procedure 408

(CCP) [58]. CCP is a heuristic method used to ﬁnd local 409

solutions to problems involving the difference of convex (DC) 410

functions. Note that, for all k∈[[ 1 ,K]] ,wehave 411

k(pu,x,z)=f1

k(pu,x,z)−g1

k(pu,x),(24) 412

where f1

k(w,z)and g1

k(w,z)are expressed, respectively, as 413

⎧

⎪

⎨

⎪

⎩

k(pu,x,z)=z−1

2log 1+au



i=1

xk,i,

k(pu,x)=−1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xk,i⎤

⎥

⎦

−1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xi,k⎤

⎥

⎦.

(25) 414

IEEE Proof

6IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

Clearly, the functions f1

kand g1

kare convex and, therefore,415

the constraint c1

kis a difference of two convex functions.416

To tackle this problem, we convexify the constraint c1

kthrough417

a simple linearization of the function g1

kby applying the ﬁrst-418

order Taylor series approximation around a given feasible419

point xl=[xT

l,1,xT

l,2,...,xT

l,K ]T, Consequently, the convex420

form of c1

k, denoted by ˜c1

k, is expressed as421

˜c1

k(pu,x,xl,z)=f1

k(pu,x,z)−˜g1

k(pu,x,xl),(26)422

where ˜g1

Kis the ﬁrst-order Taylor series approximation of g1

k,423

that is expressed as424

˜g1

k(pu,x,xl)=g1

k(pu,xl)+∇g1

k(xl)T(x−xl),(27)425

where ∇g1

kdenotes the gradient of the function g1

kwith respect426

to (w.r.t) x. In addition, ∇g1

k(x)=vT

1,vT

2,...,vT

KT,where427

for all i∈[[ 1 ,K]] ,vi=[vi,1,v

i,2,...,v

i,K ]T∈RKand it is428

expressed as429

vi,j =−δi,k (1 −δj,k )Ak−(1 −δi,k )δj,k Bk,(28)430

such that431

⎧

⎪

⎨

⎪

⎩

Ak=1

1+K

l=1

l=k

xk,l

Bk=1

1+K

l=1

l=k

xl,k

(29)

432

and δis the Kronecker delta function deﬁned, for all (i, j )∈433

N2,asδi,j =1,ifi=j,andδi,j =0,otherwise.434

In the same spirit, and for all k∈[[ 1 ,K]] , the constraint435

k(pu,x)can be written as the difference of two convex436

functions as437

k(pu,x)=f2

k(pu,x,z)−g2

k(pu,x),(30)438

where439

f2

k(pu,x,z)=−aH,

k(pu,x)=−||√xk||1.(31)

440

Clearly, the functions f2

kand g2

kare convex and, therefore,441

the constraint c2

kis a difference of two convex functions.442

To tackle this problem, we convexify the constraint c2

kthrough443

a simple linearization of the function g2

kby applying the ﬁrst-444

order Taylor series approximation around xl. Consequently,445

the convex form of c2

k, denoted by ˜c2

k, is expressed as446

˜c2

k(pu,x,xl)=−aH−˜g2

k(pu,x,xl),(32)447

where ˜g2

kis the ﬁrst-order Taylor series approximation of g2

k,448

that is expressed as449

˜g2

k(pu,x,xl)=g2

k(pu,xl)+∇g2

k(xl)T(x−xl),(33)450

where ∇g2

kdenotes the gradient of the function gkw.r.t x.451

In this context, ∇gk(x)=pT

1,pT

2,...,pT

KT, such that for452

all i∈[[ 1 ,K]] ,pi=[pi,1,p

i,2,...,p

i,K ]T∈RKand it is453

expressed as454

pi,j =δi,k

2√xk,j

.(34)455

Consequently, armed with the above, the convex form of 456

problem P1is given by 457

P

1(xl): Rs(pu)=−min

x,z −z458

s.t. ˜c1

k(pu,x,xl,z)≤0,∀k∈[[ 1 ,K]] ,

˜c2

k(pu,x,xl)≤0,∀k∈[[ 1 ,K]] .459

(35) 460

Problem P

1(xl)is a convex optimization problem that depends 461

on the linearization point xland can be solved efﬁciently using 462

standard optimization packages [59], [60]. Finally, the detailed 463

iterative algorithm for solving P1is given in Algorithm 1,464

where the initial point x0is a random feasible point that 465

satisﬁes the constraints of problem P1.466

Algorithm 1 Iterative Algorithm for Solving P1

1. Initialization:

i) Estimate Hand σ2.

ii) Choose an Initial feasible point x0.

2. Set:l=0.

3. Repeat

i) Solve P

1(xl).

ii) Assign the solution to xl+1 .

iii) Update iteration; l←l+1.

4. Termination: terminate step 3. when

i) |xl−xl−1|≤,or

ii) l=Lmax.

2) Harmonic Mean: When the optimization objective is to 467

maximize the harmonic mean of the overall system, the max- 468

imization problem becomes as follows. 469

P2:Rs(pu)=max

WKK



k=1

Rs,k(pu,W)−1−1

470

s.t. ||W||∞≤1.(36) 471

Problem P2is a non-linear non-convex optimization problem 472

that is difﬁcult to solve due to the structure of the objective 473

function and the expression of the achievable secrecy rate 474

Rs,k(pu,W). Therefore, we adopt a suboptimal approach in 475

solving P2, that is detailed as follows. In fact, since the 476

harmonic mean is lower than the arithmetic mean for any 477

positive valued set of reals, we have 478

KK



k=1

Rs,k(pu,W)−1−1

≤



k=1

dkRs,k(pu,W),(37) 479

where dk=1

K,forallk∈[[ 1 ,K]] . Consequently, a suboptimal 480

solution for problem P2can be given through the following 481

optimization problem. 482

P

2:W∗=argmax



k=1

dkRs,k(pu,W)483

s.t. ||W||∞≤1.(38) 484

IEEE Proof

ARFAOUI et al.: SECRECY PERFORMANCE OF MU MISO VLC BROADCAST CHANNELS 7

Adopting the change of variables used in the max-min fairness485

measure, problem P

2can be reformulated as486

P

2:x∗=argmin

xf(pu,x)487

s.t. c2

k(x)≤0,∀k∈[[ 1 ,K]] ,(39)488

where f(pu,x)=−K

k=1 dkRs,k(pu,x). The function489

x→ f(pu,x)can be expressed as the difference of two convex490

function as follows.491

f1(pu,x)=f3(pu,x)−f2(pu,x),(40)492

where f1(pu,x)and f2(pu,x)are expressed, respectively, as493

⎧

⎪

⎨

⎪

⎩

f1(pu,x)=−



k=1

2log 1+au



i=1

xk,i

f2(pu,x)=−



k=1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xk,i⎤

⎥

⎦

−



k=1

2log ⎡

⎢

⎣1+bu



i=1

i=k

xi,k⎤

⎥

⎦.

(41)

494

To overcome the problem of non-convexity of the objec-495

tive function, we apply the CCP method, described above.496

Consequently, we start by convexifying the function fthrough497

a simple linearization of the function f2by applying the ﬁrst-498

order Taylor series approximation around a given point xl.The499

convex form of f, denoted by ˜

f, is expressed as500

f(pu,x,xl)=f1(pu,x)−˜

f2(pu,x,xl),(42)501

where ˜

f2is the ﬁrst-order Taylor series approximation of f2,502

that is expressed as503

f2(pu,x,xl)=f2(xl)+∇f2(xl)T(x−xl),(43)504

in which ∇f2denotes the gradient of the function f2w.r.t x.505

In addition, ∇f2is expressed as506

∇f2=



k=1

dk∇g1

k,(44)507

where for all k∈[[ 1 ,K]] ,∇g1

kis given in (28). Consequently,508

the convex form of problem P

2, denoted by P

2(xl),isgiven509

by510

P

2(xl):Rs(pu)=−min

x˜

f(pu,x,xl)511

s.t. ˜c2

k(pu,x,xl)≤0,∀k∈[[ 1 ,K]] .512

(45)513

Problem P

2(xl)is a convex optimization problem that depends514

on the linearization point xland can be solved efﬁciently515

by using standard optimization packages [59], [60]. Finally,516

based on the above analysis, the detailed iterative algorithm517

for solving P

2is given in Algorithm 1, where it sufﬁces to518

replace P

1(xl)by P

2(xl)and the initial point x0is a random519

feasible point that satisﬁes the constraints of problem P

2.520

Finally, after determining a suboptimal solution x∗for 521

problem P

2, the harmonic mean of the system is expressed as 522

Rs(pu)=KK



k=1

Rs,k(pu,x∗)−1−1

.(46) 523

3) Proportional Fairness: For the proportional fairness 524

measure, the maximization problem becomes as follows. 525

P3:Rs(pu)=max

WK



k=1

Rs,k(pu,W)1

526

s.t. ||W||∞≤1.(47) 527

Similar to the previous secrecy performance measure, P3is 528

a non-linear non-convex optimization problem. To over- 529

come this, as was done before, we resort to a suboptimal 530

approach. Knowing that the geometric mean is lower than the 531

arithmetic mean for any positive valued set of reals, we have 532

K



k=1

Rs,k(pu,W)1

≤



k=1

dkRs,k(pu,W),(48) 533

where dk=1

K,forallk∈[[ 1 ,K]] . Consequently, a suboptimal 534

solution for problem P3can be given through the optimization 535

problem P

2discussed in the previous part. Finally, after 536

adopting the same approach and determining a suboptimal 537

solution x∗for problem P

2, the proportional fairness of the 538

system is expressed as 539

Rs(pu)=K



k=1

Rs,k(pu,x∗)1

.(49) 540

4) Weighted Fairness: When the optimization objective is to 541

maximize the weighted secrecy sum-rate of the overall system, 542

the maximization problem becomes as follows. 543

P4:Rs(pu)=max



k=1

dkRs,k(pu,W)544

s.t. ||W||∞≤1,(50) 545

where for all k∈[[ 1 ,K]] ,dk∈R+is an arbitrary weight 546

for the kth user. Problem P4is equivalent to P

2discussed in 547

the previous part, and therefore, the secrecy sum-rate of the 548

system for this case can be determined by following the same 549

approach developed for the harmonic mean measure. 550

C. Complexity Analysis 551

In this part, we evaluate the computational complexity of 552

the proposed precoding schemes and we compare it to that 553

of conventional ZF. In Algorithm 1, we employ the well 554

known interior point algorithm (IPA) in solving the invoked 555

convex problem. Therefore, we employ the number of Newton 556

steps, denoted by Ns, as a complexity measure. The number 557

of Newton steps denotes the number of recursive iterations 558

till convergence from a given starting point, i.e., the number 559

of required recursive steps to reach a local solution. Based 560

on [61], the worst-case Nsto reach a local solution in a non- 561

linear convex problem is expressed as 562

Ns∼problem size,(51) 563

IEEE Proof

8IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

TAB L E I I

NUMBER OF NEWTON STEPS OF ALGORITHM 1

where the problem size is the number of optimization564

scalar variables. In Algorithm 1, we are supposed to solve565

a non-linear convex problem at most L-times, and thus,566

we are employing the IPA at most L-times. Based on this,567

the worst-case complexity of our proposed schemes and of568

the ZF precoding are presented in Table II.569

Thus, for the adopted secrecy performance measures,570

we have571

Ns|proposed schemes

Ns|ZF ≈√K, (52)572

where Kis the number of active users. In other words,573

the computational complexity of our proposed schemes is574

approximately √Ktimes higher than of ZF.575

D. The Truncated Generalized Normal (TGN) Distribution576

The precoding schemes developed previously are valid for577

any continuous input distribution puwith support [−A, A].578

In our work, we adopt the TGN distribution as input signaling.579

The TGN distribution is a class of real parametric continuous580

probability distributions over a bounded interval, that adds a581

shape parameter to the truncated normal distribution. A TGN582

distribution over [−A, A],A∈R+, with a position parameter583

μ∈[−A, A], a scale parameter α∈R∗

+and a form584

parameter β∈R+is denoted by TGN(−A, A, μ, α, β)and585

its probability density function is given by586

pu(y)=⎧

⎪

⎨

⎪

⎩

αφ(y−μ

α)

Φ(A−μ

α)−Φ(−A−μ

α)−A≤y≤A

0otherwise,

(53)

587

where φis the standard generalized normal distribution that588

is deﬁned, ∀x∈R,asφ(y)= β

2Γ( 1

β)exp−|y|β,andΦis589

its cumulative distribution function that is deﬁned, ∀y∈R,590

as Φ(y)=1

2+sign(y)γ1

β,(y

β)β

2Γ( 1

β), where sign and γdenote591

the sign function and the incomplete gamma function, respec-592

tively. The expected value of a TGN(−A, A, μ, α, β)is equal593

to μ. Furthermore, according to the parameters μ,αand β,594

the TGN class over [−A, A]includes:595

•The truncated Laplace distribution when β=1,596

•The truncated normal distribution when β=2,597

•The uniform distribution when μ=0,α=Aand598

β→∞.599

Hence, through an optimal design of the parameters μ,α600

and β, the secrecy performance can be improved.601

In our scheme, we assume that random scalar variable u602

follows a TGN(−A, A, 0,α,β). The position parameter μis603

set to zero since uis zero-mean. In this case, the differ-604

ential entropy huand the variance σ2

uof uare expressed,605

Fig. 2. MU-MISO VLC system with N=16LEDs ﬁxtures and K=2

users.

respectively, as 606

⎧

⎪

⎨

⎪

⎩

hu=log2αΓ( 1

β)

β+η(α, β),

σ2

u=α2

Γ( 1

β)γ3

β,A

αβ,

(54) 607

where η(α, β)=logγ1

β,(A

α)β

Γ( 1

β)+γ1

β+1,(A

α)β

γ1

β,(A

α)β.608

In the above analysis, and since au=exp(2 hu)

2πeσ2and 609

bu=σ2

σ2, the secrecy rates Rs(pu)and RZF (pu)can be 610

enhanced by optimizing over the parameters αand βof 611

the probability distribution pu. However, since these maxi- 612

mizations are not straightforward, we adopt a discretization 613

approach as follows. Let D∈R+and M1,M2∈N614

such that M1=0and M2=0. Then we consider a 615

two dimensional grid G(D)=(Gi,j (B))1≤i≤M1

1≤j≤M2

,where 616

Gi,j (D)=(αi,β

j)such that for all i∈[[ 1 ,M

1]] ,αi=B×i

M1,617

and for all j∈[[ 1 ,M

2]] ,βj=B×j

M2. Based on this, for a given 618

secrecy performance measure, the highest secrecy sum-rate of 619

our proposed schemes and of ZF precoding are expressed, 620

respectively, as 621

⎧

⎨

⎩

R∗

s=max

(α,β)∈G(B)Rs(pu),

R∗

ZF =max

(α,β)∈G(B)RZF (pu).(55) 622

IV. SIMULATIONS RESULTS 623

A. Simulations Settings 624

To validate our proposed schemes, we consider a typical 625

VLC system consisting of a single room as shown in Fig. 2. 626

A Cartesian coordinate system, shown in Fig. 2, is used. 627

The parameters of the room, the transmitter and the users 628

are given in Table II. Alice is equipped with N=16 629

ﬁxture of LEDs, located at the ceiling of the room and at 630

IEEE Proof

ARFAOUI et al.: SECRECY PERFORMANCE OF MU MISO VLC BROADCAST CHANNELS 9

TABLE III

MU-MISO VLC SYSTEM PARAMETERS

(x, y)∈{1,2,3,4}×{1,2,3,4}. The users height measured631

from the room’s ﬂoor is 1m. Based on Table III and equations632

(1), (2) and (3), the average noise variance at the receivers633

is σ2=−98.82 dBm. The simulation results are obtained634

through 105independents Monte Carlo trials on the locations635

of users within the room. In addition, the central processing636

unit (CPU) of the machine on which all the simulations are637

performed is an Intel Core i5 from the second generation638

that has a dual-core, a basic frequency of 2.40 GHz and639

a maximum turbo frequency of 3.40 GHz. Moreover, We640

use =10

−3and Lmax =10as stopping criterion for641

Algorithm 1. Finally, The best input distribution used in all642

simulations is TGN(−A, A, 0,A,2).643

B. Secrecy Performance644

In this subsection we evaluate the secrecy performance of645

our proposed schemes. Fig. 3 presents the average secrecy rate646

R∗

sand R∗

ZF of the four secrecy performance measures con-647

sidered versus A2in dBm, where the number of active users648

is K=4. Fig. 3 shows that, for all the considered secrecy649

performance measures, the proposed precoding schemes out-650

perform the conventional ZF precoding. This result is expected651

since, as mentioned in subsection III-B, ZF precoding is a652

special case of our proposed scheme. On the other hand,653

Fig. 4 presents the average secrecy rate R∗

sof the proposed654

precoding schemes versus A2in dBm for the four secrecy655

performance measures when K=2and 4. As shown in the656

ﬁgure, the secrecy performance improves with the number of657

users.658

Fig. 3. Average secrecy rates R∗

sand R∗

ZF versus A2for the max-min

fairness (MMF), the harmonic mean (HM), the proportional fairness (PF) and

the weighted fairness (WF). The number of users is K=4.

Fig. 4. Average secrecy rate R∗

sfor the max-min fairness (MMF),

the harmonic mean (HM), the proportional fairness (PF) and the weighted

fairness (WF) for the number of users K=4and K=2.

C. Complexity Analysis 659

Another metric that we can use to evaluate the complexity 660

of the proposed schemes is the execution time, i.e., the amount 661

of time required to obtain the best precoding matrix. Fig. 5 662

presents the average execution time in seconds of the proposed 663

precoding schemes and of ZF precoding versus A2in dBm, 664

where the number of active users is K=4. This ﬁgure shows 665

that the complexity of our schemes is about two times that 666

of ZF. This result is also expected since for a given number K667

of active users, our proposed scheme consists of optimizing 668

over K2+1 variables whereas ZF precoding consists of 669

optimizing over K+1 variables only. 670

D. Convergence Behavior 671

Fig. 6 presents the convergence behavior of Algorithm 1 672

when applied to our precoding scheme, where the number 673

of active users is K=4. The amplitude constraint is 674

A2=0dBm. The convergence here is measured in terms 675

IEEE Proof

10 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS

Fig. 5. Average execution time of the proposed scheme and of ZF precoding

versus A2for the max-min fairness (MMF), the harmonic mean (HM),

the proportional fairness (PF) and the weighted fairness (WF). The number

of users is K=4.

Fig. 6. Convergence behavior of Algorithm 1 when K=4and

A2=0dBm.

of the relative error between consecutive iterations and that676

should be less than the adopted . This ﬁgure shows that677

on average, Algorithm 1 requires at most seven iterations to678

converge for the max-min fairness measure whereas it requires679

at most ﬁve iterations to converge for the other three measures.680

V. CONCLUSION681

In this paper, we considered the MU-MISO VLC Broadcast682

channel with conﬁdential messages. We proposed new precod-683

ing schemes aiming to maximize typical secrecy performance684

measures of the underlying system subject to a peak-amplitude685

constraint. Moreover, due to the amplitude constraint imposed686

to the channel input, we adopted the TGN distribution as input687

signaling and we optimized over its parameters to enhance688

the secrecy performance of the system. We compared the689

performance of our scheme to the conventional ZF precoding690

scheme and we showed through the numerical results that our691

techniques outperform the conventional ZF precoding scheme 692

in terms of secrecy performance. However, this improvement 693

comes at the expense of an increase of the execution time 694

of the proposed algorithms, which gives a trade-off between 695

complexity and performance improvement. 696

APPENDIX 697

PROOF OF THEOREM 1698

Based on the results of [33] and [34], a lower bound on 699

the secrecy capacity of the kth MISO VLC Gaussian wiretap 700

channel in (12) can be obtained as follows. 701

Ck≥max

p(uk)[I(uk;yk)−I(uk;¯

yk|¯

uk)−I(uk;¯

uk)]+702

≥⎡

⎣I(uk;yk)−I(uk;¯

yk|¯

uk)−I(uk;¯

uk)

 !

⎤

⎦

703

=[h(yk)−h(yk|uk)−h(¯

yk|¯

uk)+h(¯

yk|uk,¯

uk)]+,(56) 704

where inequality a holds by choosing any probability dis- 705

tribution p(uk),andI(uk;¯

uk)=0, since all the messages 706

are statistically independent. Now, we develop each term of 707

equation (47)-b. Since for all i∈[[ 1 ,K]] ,uiis continuous, 708

we can use the entropy power inequality (EPI) to determine 709

a lower bound on h(yk). Precisely, let (α)1≤i≤K∈R∗and 710

(x)1≤i≤Kbe Krandom scalar variables. Therefore, using the 711

EPI, we can obtain a lower bound on the differential entropy 712

of the random variable z=K

i=1 αixias follows. 713

h(z)=hK



i=1

αixi≥1

2log K



i=1

e2h(αixi)714

2log K



i=1

e2log(|αi|)+2h(xi)715

2log K



i=1

α2

ie2h(xi).(57) 716

Therefore, and based on (57), h(yk)can be lower bounded as 717

h(yk)≥1

2log K



i=1

e2h(hT

kwiui)+e2h(nk)718

2log K



i=1 hT

kwi2e2hu+2πeσ2719

2log 1+ e2hu

2πeσ2



i=1 hT

kwi2+1

2log 2πeσ2,720

(58) 721

where we used the fact that h(nk)=1

2log 2πeσ2.Onthe 722

other hand, h(yk|uk)=hK

i=k

i=1

kwi+nkand it can be 723

upper bounded by the differential entropy of random Gaussian 724

IEEE Proof

ARFAOUI et al.: SECRECY PERFORMANCE OF MU MISO VLC BROADCAST CHANNELS 11

variable with the same variance as725

h(yk|uk)≤1

2log ⎡

⎢

⎣2πe



i=k

i=1 hT

kwi2σ2

u+2πeσ2⎤

⎥

⎦

726

2log ⎡

⎢

⎣1+σ2

σ2



i=k

i=1 hT

kwi2⎤

⎥

⎦+1

2log 2πeσ2.

727

(59)728

In addition, h(¯

yk|¯

uk)=h¯

Hkwk+¯

nkcan be also upper

729

bounded by the differential entropy of random Gaussian vector730

with the same covariance matrix as731

h(¯

yk|¯

uk)732

≤1

2log "2πeσ2K−1####¯

Hkwk¯

HkwkTσ2

σ2+IK−1####$

733

2log "1+####¯

Hkwk####2

σ2

σ2$+K−1

2log 2πeσ2

734

2log ⎡

⎢

⎣1+σ2

σ2



i=k

i=1 hT

iwk2⎤

⎥

⎦+K−1

2log 2πeσ2.

735

(60)736

Furthermore, we have737

h(¯

yk|uk,¯

uk)=h(¯

yk|uk)=h(¯

nk)= K−1

2log 2πeσ2.(61)

738

Finally, by substituting the different terms of equation739

(56)-b by their expressions, we obtain the expression of the740

achievable secrecy rate Rs,k (pu)given in Theorem 1, which741

completes the proof.742

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Mohamed Amine Arfaoui received the B.E. degree 934

in electrical and computer engineering from the 935

École Polytechnique de Tunisie, Tunisia, in 2015, 936

and the M.Sc. degree in information systems engi- 937

neering from Concordia University, Montreal, QC, 938

Canada, in 2017, where he is currently pursuing the 939

Ph.D. degree in information systems engineering. 940

His current research interests include communication 941

theory, optical communications, and physical layer 942

security. 943

Ali Ghrayeb received the Ph.D. degree in electrical 944

engineering from The University of Arizona, 945

Tucson, AZ, USA, in 2000. He was with Concordia 946

University, Montreal, QC, Canada. He is currently 947

a Professor with the Department of Electrical and 948

Computer Engineering, Texas A&M University 949

at Qatar. He has co-authored the book Coding 950

for MIMO Communication Systems (Wiley, 2008). 951

His research interests include wireless and mobile 952

communications, physical layer security, massive 953

MIMO, wireless cooperative networks, and ICT 954

for health applications. He was a co-recipient of the IEEE GLOBECOM 955

2010 Best Paper Award. He served as an instructor or a co-instructor in 956

technical tutorials at several major IEEE conferences. He served as the 957

Executive Chair for the 2016 IEEE WCNC Conference and the TPC Co-Chair 958

for the Communications Theory Symposium at the 2011 IEEE GLOBECOM. 959

He has served on the editorial board of several IEEE and non-IEEE journals. 960

Chadi M. Assi received the Ph.D. degree from 961

The City University of New York (CUNY) in 2003. 962

He is currently a Full Professor with Concordia 963

University. His current research interests are in the 964

areas of network design and optimization, network 965

modeling, and network reliability. He was a recipient 966

of the prestigious Mina Rees Dissertation Award 967

from CUNY in 2002 for his research on wavelength- 968

division multiplexing optical networks. He is on the 969

Editorial Board of the IEEE COMMUNICATIONS 970

SURVEYS AND TUTORIALS,theIEEE TRANSAC-971

TIONS ON COMMUNICATIONS, and the IEEE TRANSACTIONS ON VEHICU-972

LAR TECHNOLOGIES.973