IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, …Index Terms—Physical layer security, robust resource alloca-tion, full-duplex communications, active eavesdropping, jammer. I. INTRODUCTION

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017 885

Robust Resource Allocation to Enhance PhysicalLayer Security in Systems With Full-Duplex

Receivers: Active AdversaryMohammad Reza Abedi, Nader Mokari, Hamid Saeedi, Member, IEEE,

and Halim Yanikomeroglu, Fellow, IEEE

Abstract— We propose a robust resource allocation frameworkto improve the physical layer security in the presence of an activeeavesdropper. In the considered system, we assume that bothlegitimate receiver and eavesdropper are full-duplex (FD) whilemost works in the literature concentrate on passive eavesdroppersand half-duplex (HD) legitimate receivers. In this paper, theadversary intends to optimize its transmit and jamming signalparameters so as to minimize the secrecy data rate of thelegitimate transmission. In the literature, assuming that thereceiver operates in HD mode, secrecy data rate maximizationproblems subject to the power transmission constraint havebeen considered in which cooperating nodes act as jammers toconfound the eavesdropper. This paper investigates an alternativesolution in which we take advantage of FD capability of thereceiver to send jamming signals against the eavesdroppers. Theproposed self-protection scheme eliminates the need for externalhelpers. Moreover, we consider the channel state informationuncertainty on the links between the active eavesdropper andother legitimate nodes of the network. Optimal power allocationis then obtained based on the worst-case secrecy data ratemaximization, under a legitimate transmitter power constraintin the presence of the active eavesdropper. Numerical resultsconfirm the advantage of the proposed secrecy design and incertain conditions, demonstrate substantial performance gainover the conventional approaches.

Index Terms— Physical layer security, robust resource alloca-tion, full-duplex communications, active eavesdropping, jammer.

I. INTRODUCTION

A. State of the Art

PHYSICAL layer security, as a possible alternative to theuse of cryptographically secure schemes, has arisen as

a prominent frontier in wireless networks to maintain securedata transmission between the transmitter and the legitimate

Manuscript received December 23, 2015; revised June 10, 2016 andOctober 7, 2016; accepted November 15, 2016. Date of publication Novem-ber 29, 2016; date of current version February 9, 2017. This work wassupported by the Iran National Science Foundation under Grant 92028129.This paper was presented in part at the Vehicular Technology Conference,Boston, MA, USA. The associate editor coordinating the review of thispaper and approving it for publication was E. Koksal. (Corresponding author:Hamid Saeedi.)

M. R. Abedi, N. Mokari, and H. Saeedi are with the Department of Electricaland Computer Engineering, Tarbiat Modares University, Tehran, 14115, Iran(e-mail: [email protected]).

H. Yanikomeroglu is with the Department of Systems and ComputerEngineering, Carleton University, Ottawa, ON, K1S 5B6, Canada.

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TWC.2016.2633336

receiver to deal with the challenges in wireless distributionand management of secret keys together with the fact thatcomputational power is becoming easily available to usersnowadays [1]–[3].

In traditional communication systems, it is assumed thatterminals operate in half-duplex (HD) mode, i.e., they arenot able to receive and transmit data simultaneously. Recentadvances on electronics, antenna technology and signalprocessing allow the implementation of full-duplex (FD) ter-minals that can receive and transmit data at the same time andon the same frequency band as long as the self interference (SI)that leaks from the receiver output to the receiver input canbe dealt with [4], [5]. Antenna isolation, time cancellation andspatial precoding have been proposed in the literature for themitigation of SI [6]–[9].

An efficient way to increase the secrecy data rate in wirelesssystems is to degrade the decoding capability of the eaves-droppers by introducing controlled interference, or artificialnoise [10]. When the transmitter has multiple antennas, thesystem can be designed such that a subset of antennas areused to generate artificial noise in a manner not affectingthe legitimate receiver. When the transmitter is restricted tothe use of one antenna, a group of external relays can beemployed to collaboratively send jamming signals to degradethe eavesdropper channel. This approach is referred to ascooperative jamming (CJ). Note that a transmitter in a CJbased system can still have multiple antennas that are usedto increase the diversity order or, in general, the degrees offreedom of the system. This is in fact the case in this paper.

In the physical layer secure communication, eavesdroppingcan be classified into two cases called passive and activeeavesdropping. In the first case, the eavesdropper monitorscommunication and does not interfere with the communicationchannel [10]. In the second case, eavesdropper has the abilityof observing the communication medium as well as modifyingits contents [11].

In a system equipped with FD based legitimate terminals, anatural question is that whether one can take advantage of theFD capability to generate the required artificial noise, i.e, whilereceiving data whether it can simultaneously transmit friendlyjamming signals to degrade the eavesdropper channel. Thiseliminates the need to deploy relays which is very welcomefrom practical point of view. Deployment of either HD orFD receivers together with the fact that we have to deal with

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886 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 16, NO. 2, FEBRUARY 2017

passive or active eavesdroppers (PE or AE), leaves us withfour possible system categories: HD systems dealing with PE(abbreviated by HD-PE) or AE (HD-AE), and FD systemsdealing with PE (abbreviated by FD-PE) or AE (FD-AE).While many works have been devoted to analyse the firstcategory and to some extent the second category, only fewworks have analyzed the third category and basically no workhas focused on the last category, i.e., FD-AE.

This paper aims to shed light on the potential benefits ofusing a FD terminal to provide physical layer security inthe presence of an active eavesdropper (FD-AE category).We would like to see whether one can take advantage of theFD capability to generate the required artificial noise. In sucha system, a secondary question follows: How the performanceof such a system is compared with a traditional HD systemthat uses relays? To address this issue, we also consider HDsystems in the presence of AE as well.

B. Related Works

Majority of works in the literature assume systems withHD terminals in the presence of passive eavesdroppers(HD-PE category). In this regard, we can point out [10], [12],[13] in which transmitters with multiple antennas are assumedto generate artificial noise. On the other hand, CJ strategy isdeployed in [14]–[21] to generate artificial noise. Obtaining theoptimal CJ relay weights for maximizing the secrecy data rateis investigated in [14] and [15]. An opportunistic selection oftwo relays, where one relay re-forwards the transmitted signal,while the other uses the CJ strategy is discussed in [16] in thecontext of a multi-relay network. In [17], authors study thesecrecy outage probability using CJ for different levels of CSI.The optimal transmit beamforming together with artificialnoise design for minimizing the secrecy outage probabilityis addressed in [18] and [19]. The work in [20] combines CJwith interference alignment. A destination-assisted jammingscheme is used in [21] to prevent the system from becominginterference-limited. In [14]–[21], it is assumed that the perfectCSI between the transmitter and the legitimate receiver as wellas the eavesdropper is available at the transmitter which is nota practical assumption. Among few works in physical layersecurity that consider uncertainty on the CSI values, we canmention [22]–[24]. In these works, robust frameworks havebeen considered to take care of such uncertainty.

Few works have considered active eavesdropping in systemswith HD terminals (HD-AE category). In [11], secure commu-nications in the presence of malicious users which have theability of either eavesdropping or jamming (but not both) hasbeen considered. In this paper the optimal power allocationand the optimal strategy to alternate between jamming andeavesdropping modes is obtained. In [25], the authors formu-late the secure communication in the presence of an activeeavesdropper in the context of game theory. In [26], the authorsconsider the problem of power allocation of a user in thepresence of an active eavesdropper in an orthogonal frequencydivision multiplexing scheme. In [27], the authors investi-gate the resilience of wireless multiuser networks to passiveand active eavesdroppers under Rayleigh fading conditions.

In [28], the authors investigate a FD eavesdropper with SIthat optimizes its beamforming weights in order to minimizethe secrecy data rate of the system.

As for the FD-PE category, only few works can be men-tioned. A FD receiver that generates artificial noise is pro-posed in [29] to impair the eavesdropper’s channel and thesecrecy performance is evaluated based on the outage secrecyregion from a geometric perspective. In [30], a FD receiveris assumed to generate artificial noise to improve physicallayer security in a resource allocation framework. In thispaper, perfect CSI is assumed between the eavesdropper andother nodes of the considered network. In contrast, in [31],a robust secure resource allocation scheme is proposed inwhich, assuming imperfect CSI values, the performance ofa system with FD receiver is obtained and compared againsta HD system that uses relays in a CJ scheme. In [32], theauthors consider secure MIMO transmission in a wirelessenvironment, in which one legitimate transmitter, one legit-imate receiver and one eavesdropper are involved. In order tomaximize the secrecy data rate, a joint optimization scheme isproposed to assign the transmit/receive antennas for legitimatereceiver and to design the beamforming and power allocationfor the legitimate transmitter’s information and artificial noise.Finally, in a recent work [33], the authors provide a compre-hensive review on the current state of physical layer securitysystems. However, no works on the FD-AE category otherthan [34], the conference version of this paper, can be foundin the literature to the best of our knowledge.

C. Our Contributions

In this paper, we study the potential benefits of a FD receivernode simultaneously acting as a jammer and a receiver inthe presence of an active FD based eavesdropper (the FD-AEcategory), with the goal of improving the secrecy data ratefor the first time. We consider different scenarios in which weassume that the active eavesdropper is able to simultaneouslyoverhear and jam. We maximize the secrecy data rate in theproposed system models in which we assume that the CSIof eavesdropper links are imperfect. We take care of thisuncertainty through establishing a robust resource allocationframework. The performance is then compared against thetraditional CJ strategy (the HD-AE category). Such a com-parison is also missing from the literature even for passiveeavesdropping (i.e. HD-PE versus FD-PE category).

We consider a legitimate transmitter (LT) which acts as thesource, a legitimate receiver (LR) which acts as the destination,an active eavesdropper (AE), a jammer which are denoted bys, d , e, and j , respectively, when used as a subscript in ourformulations. The LT wants to transmit data to LR while theAE overhears. Also, the AE tries to degrade the reception ofthe information signal at the intended receiver.

To address the questions proposed in Subsection I-A, fourscenarios are studied, three with a HD receiver, which arecompared against a fourth scenario with a FD receiver. Thefirst scenario denoted by HD, represents a baseline scenariowhere the LT transmits data to LR. In the second scenario,denoted by HDJ, which in fact represents a simple typical

ABEDI et al.: ROBUST RESOURCE ALLOCATION TO ENHANCE PHYSICAL LAYER SECURITY 887

TABLE I

ABBREVIATIONS OF PROPOSED SCENARIOS

CJ scenario; the LT transmits to LR and a relay is also used asa jammer. One may ask if the deployed relay can be used as aconventional cooperative node instead of a jammer. To addressthis question, we consider the third scenario accordingly,denoted by HDR, where we deploy a decode-and-forward (DF)relay. Finally, we consider a scenario where the LT transmitsdata to the FD receiver without the help of any relays. Wealso consider extensions of HDR and HDJ scenarios wherewe let more than one realy/jammer node and we use a simplerelay selection strategy to obtain the best realy/jammer. Incases where FD scenario outperforms HDJ and HDR scenarioswith a single realy/jammer, we figure out the number theextra relays/jammers necessary to outperform the FD scenario.A summary of the proposed scenarios has been givenin Table I.

In all scenarios, we aim to maximize the achievable secrecydata rate by properly allocating the available resources. More-over, we assume CSI uncertainty between the AE and the net-work nodes and consider robust secrecy data rate optimizationproblems based on the worst-case secrecy data rate approach.By incorporating the channel uncertainties and exploiting theS-Procedure [35], we show that these robust optimizationproblems can be formulated into convex ones so as to obtaina closed form solution.

Simulation results indicate that in many realistic systemsettings, the FD scenario can demonstrate a comparable perfor-mance to that of the HDJ/HDR scenarios. This is an importantresult from practical point of view as it eliminates the need todeploy relays to deal with eavesdroppers. Nevertheless, if morethan one relay/jammer is allowed, the HDR/HDJ scenarios canalways outperform the FD scenario.

The organization of this paper is as follows. In the nextsection, some notations and assumptions are reviewed. InSection III, system models are illustrated for the four con-sidered scenarios. In Section IV, secrecy data rate max-imization problems are presented for the four consideredscenarios given in Table I. The performance of the proposedsecrecy transmission approaches is studied using several sim-ulation examples in Section VI, and conclusions are drawn inSection VII.

II. NOTATIONS AND ASSUMPTIONS

The following notations is used in the paper: E· denotesexpectation, (·)H the Hermitian transpose, ‖ · ‖ the Euclideannorm, (·)† the pseudo-inverse, Tr(·) is the trace operator, I isan identity matrix of appropriate dimension, H

N denotes theset of all N-by-N Hermitian matrices, C

M×N represents anM-by-N complex matrix set, and R (.) is the range space of

a matrix. Moreover, A � 0( A � 0) means A is a Hermitianpositive semidefinite (definite) matrix.

We assume the LT, jammer and eavesdropper have Ns ,N j and Ne transmit antennas, respectively. All HD receiversare assumed to have a single antenna.1 The legitimate FDreceiver is assumed to have Nd transmit antennas. We let gsdand gse denote the 1 × Ns vectors of channel gains betweenLT and destination as well as LT and AE, respectively. Inaddition, g j d and g j e denote the 1 × N j vector of channelgains between jammer and LR as well as jammer and AE,respectively. Accordingly, grd and gre denote the 1 × Nr

vectors of channel gains between relay and LR, and relay andAE, respectively. Moreover, ged denotes the 1 × Ne vectorsof channel gains between AE and LR. Finally for the scenariowith FD legitimate receiver (FD-LR), we let gde denote the1 × Nd vector of channel gains between FD-LR and AE.The FD-LR and AE transmit a jamming signal while theysimultaneously receive the LT transmitted signal. This createsa feedback loop channel between the input and output of theFD-LR as well as AE whose gains are denoted by vectorshd and he of dimension 1 × Nd and 1 × Ne , respectively.We assume that the naturally occurring noise at LR, relay andAE is zero-mean circular complex Gaussian with variance σ 2

d ,σ 2

r and σ 2e , respectively. To simplify the notations, we will

assume without loss of generality that σ 2d = σ 2

r = σ 2e = σ 2.

For all channel gains between the AE and different networknodes, it is assumed that only an estimated version of the gainis available. In particular, LT only has the knowledge of anestimated version of gse, i.e., gse and the channel error vectoris defined as egse = gse − gse. Moreover, the jammer onlyhas the knowledge of an estimated version of g j e, i.e., g j eand the channel error vector is defined as eg je

= g j e − g j e.Similarly, the relay only has the knowledge of an estimatedversion of gre, i.e., gre and the channel error vector is definedas egre

= gre− gre. Also, the AE only has the knowledge of anestimated version of ged , i.e., ged and the channel error vectorsare defined as eged

= ged − ged , respectively. Finally, only anestimated version of gde, i.e., gde, is available to the FD-LR.We define the channel error vector as egde

= gde− gde. For allcases, we assume that the channel mismatches lie in boundedsets [22], i.e., Egse

= {egse:‖ egse

‖2≤ ε2gse

}, Eg je= {eg je

:‖eg je

‖2≤ ε2g je

}, Egre= {egre

:‖ egre‖2≤ ε2

gre}, Eger

={eger

:‖ eger‖2≤ ε2

ger}, Eged

= {eged:‖ eged

‖2≤ ε2ged

},Egde

= {egde:‖ egde

‖2≤ ε2gde

}, where ε2gse

, ε2g je

, ε2ged

, ε2ger

and ε2gde

are known constants.In several optimization problems throughout the paper, we

indicate the variable that optimizes the utility function byadding a star to it as a superscript. For example for the problemmax

af (a), we set a∗ = argmax

af (a).

As far as self interference cancelation method is concerned,we remind that there are several methods in the literature suchas: 1) antenna isolation, 2) time cancellation and 3) spatialprecoding [6]–[9]. Time-domain cancellation and spatial null

1For simplicity, we assume that the eavesdropping is done by only onereceiving antenna. Note that this framework can be easily extended to thecase that adversary can exploit the multiple antennas for eavesdropping likejamming case.


Fig. 1. Schematic of the HD scenario.

space projection aim merely at minimizing the effect of loopinterference. However, the degrees of freedom in the spatialdomain allow for more sophisticated approach in which theuseful signal power is also improved.

Self-interference suppression precoding includes the follow-ing 2 schemes: 1) The zero forcing (ZF) precoder. 2) Theregularized channel inversion (RCI) precoder. The idea is toform the beams of the downlink signals to the legitimatereceiver, while simultaneously form beams to the legitimatetransmitter to send all-zero signals, which will avoid self-interference. The ZF precoder is designed to minimize themultiuser interference, while the RCI is designed to maximizethe signal-to-interference-plus-noise ratio (SINR). In particu-lar, [9] proposes to use a low-complexity ZF SI cancellationsolution. A similar framework based on ZF has also beenconsidered in [36] as well as in [37] and [38]. Althoughthe ZF precoder can perfectly avoid multi-user interference,the RCI precoder has better performance by maximizingthe SINR at each user at the expense of larger complexity.Meanwhile, simulation results in the [39] confirm that thesum-rate of the ZF precoder is very close to that of the RCIprecoder which makes the ZF precoder more attractive. Onthe other hand in the context of physical layer security, [30]proposes to use a low-complexity ZF SI cancellation solution.In this paper we also follow the ZF approach as it makesit possible to obtain closed form solutions for achievablesecrecy data rate.

III. SYSTEM MODELS

A. The HD Scenario

In the first proposed scenario, we consider one LT trans-mitting data to LR while one AE is presented2 as depictedin Fig. 1. In this model, LT sends private messages to LRin the presence of eavesdropper, who is able to eavesdropon the link between LT and LR and able to degrade thereception of the information signal at the LR by transmittingjamming signal. The achievable secrecy data rate is expressed

2Although we consider a single-carrier single user system, the extension toa multi-carrier multi-user system as in [40] and [41] is straightforward.

Fig. 2. Shcematic of the HDJ scenario.

as, RSe = max{0, RD

e − REe }, where

RDe = log2

(1 + gsd Qs gH

sd

σ 2 +�(Qe, eged )

), (1)

and

REe = log2

(1 + �(Qs , egse

)

σ 2 + he QehHe

), (2)

where �(Qs, egse) = ( gse + egse

)Qs( gse + egse)H ,

�(Qe, eged) = ( ged + eged

)Qe( ged + eged)H , [a]+ =

max{0, a} and Qs is the covariance matrix of the signaltransmitted by LT, xs , which is given by Qs = E{xs x H

s },and the power constraint is imposed such that Qs ∈ Q s ={ Qs : Qs � 0,Tr(Qs) ≤ Ps} where Ps is the maximumallowable transmission power on LT in the HD Scenario, Qeis the covariance matrix of the signal transmitted by AE, xe,and is given by Qe = E{xex H

e }, and the power constraint isimposed such that Qe ∈ Q e = { Qe : Qe � 0,Tr(Qe) ≤ Pe}where Pe is the maximum allowable transmit power on AE.We focus on optimizing the worst-case performance, where wemaximize the secrecy data rate for the worst channel mismatchegse

and egedin the bounded set Egse

and Eged, respectively.

B. The HDJ Scenario

In this subsection, we consider a cooperative jammingMISO communication system with an LT, a jammer, an LR,and an AE, as shown in Fig. 2. In this scenario, LT sendsprivate messages to the LR in the presence of an AE, who isable to eavesdrop on the link between LT and LR. The jammertransmits artificial interference signals to confuse the AE. Thedata rate at the destination can be written as

RDj e = log2

(1 + gsd Qs gH

sd

σ 2 + g j d Q j gHjd +�(Qe, eged

)

). (3)

The data rate of eavesdropper can be expressed as

REj e = log2

(1 + �(Qs, egse

)

σ 2 +�(Q j , eg je)+ he QehH

e

), (4)


Fig. 3. Schematic of the HDR scenario.

where �(Q j , eg je) = ( g j e + eg je

)Q j ( g j e + eg je)H .

Q j is the covariance matrix of the signal transmittedby jammer, x j , which is given by Q j = E{x j x H

j }, andthe power constraint is imposed such that Q j ∈ Q j ={ Q j : Q j � 0,Tr(Q j ) ≤ Pj } where Pj is the maxi-mum predefined transmit power on jammer. Therefore, thesecrecy data rate for the wiretap channel can be written as,RS

j e = max{0, RDj e − RE

j e}.

C. The HDR Scenario

Here we consider a scenario consisting of an LT node, aDF relay node, a LR node, and an AE (see Fig. 3). Thetransmission happens in two stages. In the first stage, the LTbroadcasts its encoded symbols to the relay. In the secondstage, relay that is assumed to have successfully decoded themessage, re-encodes the message and cooperatively transmitsthe re-encoded symbols to the LR. For a two-hop DF-basedrelay channel, the data rate through the relay link can bewritten as [42]

RDre = 1

2

[min

{log2

(1 + gsr Qs gH

sr

σ 2 + �(Qe2, eger)

),

log2

(1 + grd Qr gH

rd + gsd Qs gHsd

σ 2 +�(Qe1, eged )

)}], (5)

where �(Qe2, eger) = ( ger + eger

)Qe2( ger + eger)H and

�(Qe1, eged) = ( ged + eged

)Qe1( ged + eged)H . Note that we

have two covariance matrices for the signal transmitted by theeavesdropper: 1) Covariance matrix of the signal transmittedby the eavesdropper to destination, Qe1, and 2) Covariancematrix of the signal transmitted by the eavesdropper to relaystation, Qe2. Factor 1

2 appears because the relay transmissionis divided into two stages. Eavesdropper receives data duringboth hops, and the corresponding data rate can be expressedas

REre = 1

2log2

(1 + �(Qs , egse )+ �(Qr , egre )

σ 2 + he Qe1hHe + he Qe2hH

e

), (6)

where �(Qr , egre) = ( gre +egre

)Qr ( gre +egre)H . Therefore,

the secrecy data rate for the DF relaying wiretap channel canbe written as

RSre = max{0, RD

re − REre}

= 1

2

[min

{log2

(1 + gsr Qs gH

sr

σ 2 +�(Qe2, eger)

),

log2

(1 + grd Qr gH

rd + gsd Qs gHsd

σ 2 + �(Qe1, eged )

) }

− log2

(1 + �(Qs, egse

)+�(Qr , egre)


e

) ]+. (7)

Since the data rate of the relay link is limited by the SINRof the inferior hop, for a single data stream the transmitpower for source and the relay should be adjusted such that

gsr Qs g Hsr

σ 2+�(Qe2,eger )≤ grd Qr gH

rd +gsd Qs g Hsd

σ 2+�(Qe1,eged ).3 The data rate through

the relay link is equivalent to

RDre = 1

2log2

(1 + gsr Qs gH

sr


), (8)

Therefore, we can rewrite the secrecy data rate for a DFrelaying wiretap channel as

RSre = 1

2

[log2

(1 + gsr Qs gH

sr


)

− log2

(1 + �(Qs, egse

)+�(Qr , egre)


e

)]+.

(9)

D. The FD Scenario

Here, we consider one LT transmitting data to FD-LR whileone AE eavesdrops the FD-LR as depicted in Fig. 4. In otherwords, in this scenario, source sends private messages todestination in the presence of an eavesdropper, who is ableto eavesdrop and jam the link between source and distinction.The achievable secrecy data rate is expressed as follows:

RSe =

[log2

(1 + gsd Qs gH

sd

σ 2 + hd Qd hHd + �(Qe, eged

)

)

− log2

(1 + �(Qs , egse

)

σ 2 + he QehHe +�(Qd , egde

)

) ]+,

(10)

where �(Qd , egde) = ( gde + egde

)Qd ( gde + egde)H .

IV. PROBLEM FORMULATIONS AND SOLUTIONS

A. The HD Scenario

For the HD scenario, the optimization problem can bewritten as follows:

3Note that to solve OH D R , we can also assume gsr Qs gHsr

σ2+�(Qe2,eger )≥

grd Qr gHrd +gsd Qs gH

sdσ2+�(Qe1,eged )

. This does not change the solution method.


Fig. 4. Schematic of the FD scenario.

Problem O H D:

maxQs∈Q s

minQe∈Q e;egse ∈Egse ;

eged ∈Eged

RDe , (11a)

s.t. Tr(Qs) ≤ Ps , (11b)

Tr(Qe) ≤ Pe, (11c)

‖ egse ‖2≤ ε2gse, (11d)

‖ eged ‖2≤ ε2ged, (11e)

Qs � 0, (11f)

Qe � 0, (11g)

where (11b) and (11c) are the LT and AE power con-straints. At first, the covariance matrix of the signal trans-mitted by eavesdropper is obtained by solving the followingoptimization:

Problem O H D1 :

maxQe∈Q e

mineged ∈Eged

( ged + eged)Qe( ged + eged

)H , (12a)

s.t. he QehHe = 0,

(11c), (11e), (11g). (12b)

Note that Qe itself is an optimization variable and we find itfor the worst case of LT and the best case of eavesdropper.We consider constraint (12b) to cancel self-interference in fullduplex nodes. In this regard, we use a ZF constraint on theeavesdropping signal, which is equivalent to he QehH

e = 0.The maximin problem O H D

1 can be transformed toProblem O H D

2 :

maxQe∈Q e,ν

ν, (13a)

s.t. ( ged + eged)Qe( ged + eged

)H ≥ ν, (13b)

(11c), (11e), (11g), (12b),

where the constraints (13b) and (11e) can also be expressedas

egedQeeH

ged+ 2Re

(ged QeeH

ged

)+ ged Qe gH

ed − ν ≥ 0,

(14a)

−eged eHged

+ ε2ged

≥ 0. (14b)

To make the problem more tractable to solve and analyze,the next step is to turn (14a) and (14b) into linear matrixinequalities (LMIs), using the S-procedure [35]. Using theS-procedure, we know that there exists an eged

∈ CNe

satisfying both the above inequalities if and only if there existsa μ ≥ 0 such that[

μI Ne + Qe Qe gHed

ged Qe gHed Qe ged − με2

ged− ν

]� 0. (15)

Letting ψ = gHed Qe ged − με2

ged− ν, where ψ ≥ 0, O H D

2 canthen be expressed as

Problem O H D3 :

maxQe∈Q e;μ≥0;ψ≥0

− ψ + Tr(Qe gHed ged )− με2

ged, (16a)

s.t.

[μI Ne + Qe Qe gH

edged Qe ψ

]� 0,

(11c), (11g), (12b). (16b)

Problem O H D3 is a semidefinite program (SDP) that consists

of a linear objective function, together with a set of LMI con-straints. Therefore, we can solve this problem efficiently andobtain the optimal solution Q∗

e . Note that although eged doesnot explicitly appear in O H D

3 , the optimal robust covarianceQ∗

e is already based on the hidden worst-case e∗ged

that canbe expressed explicitly through the following problem:

Problem O H D4 :

mineged

( ged + eged )Q∗e( ged + eged )

H , (17a)

s.t. ‖ eged‖2≤ ε2

ged. (17b)

The above problem is a convex problem and thus strong dualityholds for (17a) and its dual.

Proposition 1: The worst-case channel mismatch for (17a)is given by

e∗ged

= − ged Q∗e(λI Ne + Q∗

e)−1, (18)

where λ is the solution of the following SDP problem:

maxλ≥0,γ

γ , (19a)

s.t.

[λI Ne + Q∗

e Q∗e gH

edged Q∗

e ged Q∗e gH

ed − λε2ged

− γ

]� 0,

(19b)

where γ is a non-negative auxiliary optimization variable.Proof: See Appendix. �With optimal values Q∗

e and e∗ged

, we can formulate theoptimization problem over Qs as

Problem O H D1 :

maxQs∈Q s

minegse ∈Egse

σ 4 + σ 2 gsd Qs gHsd + σ 2�(Q∗

e , e∗ged)[

σ 2 +�(Q∗e , e∗

ged)][σ 2 +�(Qs , egse )

] ,s.t. (11b), (11d), (11 f ). (20a)

The optimization problem O H D1 can be transformed to


Problem O H D2 :

maxQs∈Q s ,ν


e , e∗ged)

ν, (21a)

s.t.[σ 2 +�(Q∗

e , e∗ged)][σ 2 + �(Qs , egse )

]≤ ν,

(11b), (11d), (11 f ). (21b)

The constraints (21b) and (11d) can also be expressed as

− egseQs eH

gse− 2Re

(gse Qs eH

gse

)− gse Qs gH

se − σ 2

+ ν

σ 2 +�(Q∗e , e∗

ged)

≥ 0, (22a)

− egseeH

gse+ ε2

gse≥ 0. (22b)

Using the S-procedure [35], we know that there exists an egse

satisfying both the above inequalities if and only if there existsa μ ≥ 0 such that[

μI Ns − Qs − Qs gHse

− gse Qs ψ

]� 0, (23)

where I Ns is the identity matrix of dimension Ns and ψ =− gse Qs gH

se − με2gse

− σ 2 + νσ 2+�(Q∗

e ,e∗ged), then O H D

2 can be

converted intoProblem O H D

3 :

minQs∈Q s ;ψ≥0;μ≥0[

σ 2 +�(Q∗e , e∗

ged)][ψ + με2

gse+ σ 2 + Tr(Qs gH

se gse)]


e , e∗ged)

,

(24a)

s.t.

[μI Ns − Qs − Qs gH

se− gse Qs ψ

]� 0,

(11b), (11 f ), (24b)

Problem O H D3 consists of a linear fractional objective function

with a positive denominator, and thus is quasi-convex. Werewrite this problem by defining the non-negative variable t ,using the Charnes-Cooper transformation [43], [44] and lettingμ = μ/γ , ψ = ψ/γ and Qs = Qs/γ for some γ > 0 as

Problem O H D4 :

mint; Qs�0;μ≥0;

ψ≥0,γ

t,

s.t.[σ 2 + �(Q∗

e , e∗ged)]

(25a)[ψ + με2

gse+ γ σ 2 + Tr( Qs gH

se gse)]

≤ t, (25b)

γ σ 4 + σ 2 Tr( Qs gHsd gsd)+ γ σ 2�(Q∗

e , e∗ged) = 1,

(25c)[μI Ns − Qs − Qs gH

se

− gse Qs ψ

]� 0, (25d)

Tr( Qs) ≤ γ Ps . (25e)

Note that the solution for the optimal covariance, Q∗s , obtained

from O H D4 is already based on a hidden worst-case channel

Fig. 5. The pseudo code of Iterative algorithm for finding the LT’s transmittedpower and eavesdropper’s transmitted power and channel mismatches.

mismatch e∗gse

. Next, we will explicitly express e∗gse

under thenorm-bounded constraint. The problem is formulated as

Problem O H D5 :

minegse ∈Egse

( gse + egse )Q∗s ( gse + egse )

H , (26a)

s.t . ‖ egse‖2≤ ε2

gse. (26b)

The above problem is a non-convex problem since we wantto maximize a convex function. However, similar to Proposi-tion 1, we can still obtain the global optimum by solving itsdual problem, given by

e∗gse

= gse Q∗s (λI Ns − Q∗

s )†, (27)

where λ is the solution of the following problem:

maxλ≥0,γ

γ , (28a)

s.t.

[λI Ns − Q∗

s Q∗s gH

segse Q∗

s − gse Q∗s gH

se−λε2gse

− γ

]� 0. (28b)

The above optimization problem is a SDP and hence canbe solved efficiently using, for example, the interior-pointmethod [35].

It is important to note that using conventional con-vex optimization methods, the multi-variable optimizationproblems can be decoupled into multiple sub-problemsto simplify the solution and reduce the computationalcomplexity [22], [35], [45]. Thus, we use the the primaldecomposition method (for more details, refer to [46]–[48])by decomposing the original problem into several subproblemscontrolled by a master problem, and use an iterative methodto find the solution. In this case, from the previous section,we know that the subproblems involving Qs and Qe in (16)and (25) are both convex for a given p1 and p2. Hence, ourfirst step will be to estimate Qs and Qe for some initial p1and p2, and then we will find optimal values p1 and p2 for theresulting Qs and Qe. We then continue iteratively to find thechannel mismatches and power allocation. The steps for thejoint optimization that considers both the channel mismatchesand the power allocation between LT and eavesdropper areoutlined in a pseudo-code of Fig. 5.

B. The HDJ Scenario

For HDJ scenario, the optimization problem can be writtenas follows:


Problem O H D J :

maxQs∈Q s ;Q j ∈Q j


eg je ∈Eg je ;eged ∈Eged

RSj e, (29a)

s.t. Tr(Qs) ≤ Ps , (29b)

Tr(Q j ) ≤ Pj , (29c)

‖ eg je‖2≤ ε2

g je, (29d)

Q j � 0,

(11c), (11d), (11e), (11 f ), (11g), (29e)

where Ps is the maximum allowable transmission power onLT in the HDJ, HDR and FD scenarios. As the first step, thecovariance matrix of the signal transmitted by eavesdropper,Qe, is obtained. Hence, the corresponding optimization prob-lem is formulated as follows:

Problem O H D J1 :

maxQe∈Q e

mineged ∈Eged


)H , (30a)

s.t. he QehHe = 0,

(11c), (11e), (11g). (30b)

Problem O H D J1 can be solved in a similar way as that of

O H D1 . Then the covariance matrix of the signal transmitted by

jamming is obtained, and the optimization problem becomesProblem O H D J

1 :

maxQ j ∈Q j

mineg je ∈Eg je

( g j e + eg je )Q j ( g j e + eg je)H , (31a)

s.t. g j d Q j gHjd = 0,

(29c), (29d), (29e). (31b)

We can use a similar processing flow as O H D1 to solve O H D J

1and to obtain Q∗

j and e∗g je

. With optimal values for Q∗j and

e∗g je

, we can formulate the optimization problem over Qs as

Problem O H D J1 :

maxQs∈Q s

minegse ∈Egse

A

B,

s.t. (29b), (11d), (11 f ), (32)

where A = [σ 2 + �(Q∗

j , e∗g je)][σ 2 + gsd Qs gH

sd +�(Q∗

e , e∗ged)], B = [

σ 2 + �(Q∗e , e∗

ged)][σ 2 + �(Qs , egse

)+�(Q∗

j , e∗g je)]

and �(Q∗j , e∗

g je) = ( g j e + e∗

g je)Q∗

j ( g j e +e∗

g je)H . Solutions of O H D J can be easily obtained by follow-

ing the same line of arguments we made for O H D . Thus, wecan obtain Q∗

s and e∗gse

.

C. The HDR Scenario

For the HDR scenario, the optimization problem can bewritten as follows:

Problem O H D R:

maxQs∈Q s ;Qr ∈Q r

minQe1∈Q e; Qe2∈Q e;

egse ∈Egse ;egre ∈Egre ;eged ∈Eged ;eger ∈Eger

RSre, (33a)

s.t. Tr(Qr ) ≤ Pr , (33b)

Tr(Qe1) ≤ Pe, (33c)

Tr(Qe2) ≤ Pe, (33d)

‖ egre ‖2≤ ε2gre, (33e)

‖ eger‖2≤ ε2

ger, (33f)

Qe1 � 0, (33g)

Qe2 � 0, (33h)

Qr � 0,

(11d), (11e), (11 f ), (29b). (33i)

Similar to HD and HDJ scenarios, we initially obtain thecovariance matrix of the signal transmitted by the eavesdrop-per, Qe1 and Qe2. Now, by dropping the logarithm in (9), theoptimization problem O H D R is modified as follows:

Problem O H D R:

maxQs∈Q s ;Qr ∈Q r

minQe1∈Q e; Qe2∈Q e;

egse ∈Egse ;egre ∈Egre ;eged ∈Eged ;eger ∈Eger

1 + gsr Qs g Hsr

σ 2+�(Qe2,eger )

1 + �(Qs ,egse )+�(Qr ,egre )

σ 2+he Qe1hHe +he Qe2 hH

e

, (34a)

s.t.gsr Qs gH

sr

σ 2 + �(Qe2, eger)

≤ grd Qr gHrd + gsd Qs gH

sd

σ 2 + �(Qe1, eged)

(34b)

(11d), (11e), (11 f ), (29b), (33c), (33d), (33h),

(33i), (33b), (33e), (33 f ), (33g).

Qe1 and Qe2 are obtained by solving the following opti-mization problems:

Problem O H D R1 :

maxQe1∈Q e

mineged ∈Eged

( ged + eged)Qe1( ged + eged

)H , (35a)

s.t. he Qe1hHe = 0,

(11e), (33c), (33h), (35b)

and Problem O H D R1 :

maxQe2∈Q e

mineger ∈Eger

( ger + eger)Qe2( ger + eger

)H , (36a)

s.t. he Qe2hHe = 0,

(33 f ), (33d), (33i). (36b)

Problems O H D R1 and O H D R

1 are solved using a method similarto O H D

1 . As a result, e∗ged

and e∗ger

are also obtained. We canformulate the optimization problem over Qr as

Problem O H D R1

minQr ∈Q r

minegre ∈Egre

( gre + egre)Qr ( gre + egre

)H ,

s.t. (33b), (33e), (33g). (37)

We can solve this problem efficiently in a similar way asthat of O H D

1 and obtain the optimal solution Q∗r and e∗

gre.

With optimal values Q∗r , e∗

gre, Q∗

e1, Q∗e2, e∗

gerand e∗

ged,


we can formulate the optimization problem over Qs as Prob-lem O H D R

1 :

maxQs∈Q s

minegse ∈Egse

C

D,

s.t. (29b), (11d), (11 f ), (34b). (38)

where C = σ 2[σ 2+�(Q∗

e2, e∗ger)+gsr Qs gH

sr

]and D = [

σ 2+�(Qs, egse )+�(Q∗

r , e∗gre)][σ 2 +�(Q∗

e2, e∗ger)]. Solution of

O H D R5 can be easily obtained by following the same line of

argument we made for O H D5 .

D. The FD Scenario

In the FD scenario, the secrecy data rate maximizationproblem is given as

Problem OF D:

maxQs∈Q s;Qd∈Q d


egde ∈Egde ;eged ∈Eged

RSe , (39a)

s.t. Tr(Qd ) ≤ Pd , (39b)

‖ egde‖2≤ ε2

gde, (39c)

Qd � 0,

(29b), (11c), (11d), (11e), (11 f ), (11g). (39d)

Similar to previous scenarios, the covariance matrix of the sig-nal transmitted by eavesdropper is obtained first. Accordingly,the corresponding optimization problem is written as follows:

Problem OF D1 :

maxQe∈Q e

mineged ∈Eged


)H , (40a)

s.t. he QehHe = 0,

(11c), (11e), (11g). (40b)

We can solve this problem efficiently and obtain the optimalsolution Q∗

e and e∗ged

. Then, the covariance matrix of thesignal transmitted by jammer is obtained. Accordingly, thecorresponding optimization problem is written as follows:

Problem OF D1 :

maxQd∈Q d

minegde ∈Egde

( gde + egde)Qd( gde + egde

)H , (41a)

s.t. hd Qd hHd = 0, (41b)

(39b), (39c), (39d).

We can solve this problem efficiently in a similar way andobtain the optimal solution Q∗

d and e∗gde

. With optimal valuesQ∗

e , e∗ged

, Q∗d and e∗

gde, we can formulate the optimization

problem over Qs and egseas

Problem OF D1 :

maxQs∈Q s

minegse ∈Egse

E

F,

s.t. (29b), (11d), (11 f ). (42a)

where E =[σ 2 +�(Q∗

d , e∗gde)][�(Q∗

e , e∗ged)+ gsd Qs gH

sd +σ 2

]and F =

[σ 2 + �(Q∗

e , e∗ged)][σ 2 + �(Qs, egse ) +

�(Q∗d , e∗

gde)]. The optimization problem OF D

1 can be trans-formed to

Problem OF D2 :

maxQs∈Q s ,ν

[σ 2+�(Q∗

d , e∗gde)][�(Q∗

e , e∗ged)+ gsd Qs gH

sd +σ 2]

ν,

(43a)

s.t.[σ 2 +�(Q∗

e , e∗ged)]

[σ 2 +�(Qs , egse

)+�(Q∗d , e∗

gde)]

≤ ν, (43b)

‖ egse‖2≤ ε2

gse,

(29b), (11 f ). (43c)

The constraints (43b) and (43c) can also be expressed as

− egseQs eH

gse− 2Re

(gse Qs eH

gse

)− gse Qs gH

se − σ 2

−�(Q∗d , e∗

gde)+ ν

σ 2 +�(Q∗e , e∗

ged)

≥ 0, (44a)

− egseeH

gse+ ε2

gse≥ 0. (44b)

Using the S-procedure [35], we know that there exists an egse

satisfying both the above inequalities if and only if there existsa μ ≥ 0 such that


se− gse Qs ψ

]� 0. (45)

where ψ = − gse Qs gHse − με2

gse− σ 2 − �(Q∗

d , e∗gde) +

νσ 2+�(Q∗

e ,e∗ged), then OF D

2 can be converted into

Problem OF D3 :

minQs∈Q s ;ψ≥0;μ≥0

G

H, (46a)

s.t.


se− gse Qs ψ

]� 0, (46b)

(29b), (11 f ).

where G =[σ 2 + �(Q∗

e , e∗ged)][ψ + με2

gse+

σ 2 + Tr(Qs gHse gse) + �(Q∗

d , e∗gde)]

and H =[σ 2 + �(Q∗

d , e∗gde)][�(Q∗

e , e∗ged) + gsd Qs gH

sd + σ 2].

Problem OF D3 consists of a linear fractional objective function

with a positive denominator, and thus is quasi-convex. Weuse the Charnes-Cooper transformation, by letting μ = μ/γ ,ψ = ψ/γ and Qs = Qs/γ for some γ > 0, and rewritingOF D

3 as


Problem OF D4 :

mint; Qs�0;

μ≥0,ψ≥0,γ

t, (47a)

s.t.[σ 2 + �(Q∗

e , e∗ged)]

[ψ+με2

gse+γ σ 2+Tr( Qs gH

se gse)+γ�(Q∗d , e∗

gde)]

≤ t,

(47b)[σ 2 +�(Q∗

d , e∗gde)]

[γ�(Q∗

e , e∗ged)+ Tr( Qs gH

sd gsd)+ γ σ 2]

= 1, (47c)[μI Ns − Qs − Qs gH

se

− gse Qs ψ

]� 0, (47d)

Tr( Qs) ≤ γ Ps . (47e)

Note that the solution for the optimal covariance Q∗s

obtained from OF D4 is already based on a hidden worst-case

channel mismatch e∗gse

. Next, we will explicitly express e∗gse

under the norm-bounded constraint. The problem is formulatedas

Problem OF D5 :

minegse ∈Egse

( gse + egse)Q∗

s ( gse + egse)H , (48a)

s.t . ‖ egse‖2≤ ε2

gse. (48b)

OF D5 can be easily solved by following the same line of

argument we made for O H D5 .

V. MULTI NODE SCENARIOS

A. Multi Relay

We assume there are R relays in the HDJ scenario insteadof one. The set of relays is denoted by R = {1, 2, . . . , R}.Based on the optimal power allocation scheme proposed in theHDR scenario for each relay, the LT can calculate the secrecyrate for any relay by solving optimization problem for eachrelay. Therefore, the LT can select the relay corresponding tothe maximum secrecy rate. This can be done, via a simpleexhaustive search as, r∗ = arg maxr RS

re.

B. Multi Jammer

We assume there are J jammers in the HDJ scenario insteadof one. The set of jammers is denoted by J = {1, 2, . . . , J }.Based on the optimal power allocation scheme proposed inthe HDJ scenario for each jammer, the source can calculatethe secrecy rate for any jammer. Therefore, the source canselect the jammer corresponding to the maximum secrecy rate.This can be done, via a simple exhaustive search as, j∗ =arg max j RS

j e.

VI. SIMULATION RESULTS

This section presents the numerical results of the pro-posed scenarios. The proposed schemes have been evalu-ated in terms of secrecy data rate. We assume the LT,jammer, relay, AE and FD-LR have four transmit antennas,

Fig. 6. Typical positioning model of the network nodes.

i.e., Ns = Nd = Nr = N j = Ne = 4, while each HD receiverhas one. The channel matrices are assumed to be composed ofindependent, zero-mean Gaussian random variables with unitvariance. We perform Monte Carlo experiments consisting of1000 independent trials to obtain the average results. Similar to[22], the normalized background noise power is assumed tobe the same at LR and AE where σ 2

d = σ 2r = σ 2

e = 0 dBand for the transmit power of different nodes we assumePs = 2P, Ps = Pd = Pj = Pr = P = 5dB, Pe = 4dB,where Ps is the transmit power of the source for the HDscenario and Ps , Pd , Pj , and Pr are the transmit power forthe source, destination, jammer, and relay, respectively for allother scenarios. Pe represents the transmit power of the AE.Unless otherwise stated, we let ε2

gse= ε2

gde= ε2

g je= ε2

gre=

ε2ger

= ε2ged

= 0.5.

For simplicity, we consider a simple one-dimensionalsystem model, as illustrated in Fig. 6, in which the LT,jammer, relay, LR, and AE are placed along a line.The LT-relay/jammer distance is always assumed to be smallerthan the LT-RT or the LT-AE distance. Channels between anytwo nodes are simply modelled through distance-dependentattenuation. For example, gsd = d−c/2

sd where dsd is thedistance between the LT and LR where c is the path-lossexponent. We set c = 3.5 which is a typical value in theliterature, nevertheless, other values for c also lead to similarresults. The LT and LR distance is considered to be constant,in particular, we assume LT is located at the origin, i.e.,coordinates (0,0) and the LR stays at coordinates (50,0) (allthe distance units are in meters.).

A. Effect of Source-Eavesdropper Distance

Fig. VI-A shows secrecy data rate versus different posi-tions of the eavesdropper. The total transmit power is fixedat P = 5 dB [22]. We fix the jammer/relay location atcoordinates (25,0) (i.e., in an equal distance from LT and LR)and move the position of the AE from coordinates (30,0) to(90,0). As expected, the secrecy data rate for the HD scenariobecomes zero when the LR is at a farther position (to the LT)than the AE. As observed, when the AE moves away fromthe LT, the secrecy data rate increases for the HDR scenario,since the received signal power at the AE decreases. For theHDJ scenario, it is interesting to see that the secrecy data rateat first decreases, then increases, and eventually becomes equalto the secrecy data rate of the HD scenario. The reduction insecrecy data rate is because more jamming power is needed forcreating larger interference and less power is available for theLT to transmit the message signal, when the AE moves awayfrom the jammer. However, when the AE gets very far awayfrom the jammer and also the LT, we should spend most ofthe power on transmitting the message signal. In this situation,


Fig. 7. Secrecy data rate, R, vs. LT and eavesdropper distance, dse for HD,HDJ, and FD scenarios. The position of eavesdroppers varies from (30,0) to(90,0). The jammer/rely location is fixed at (25,0).

it is not worth spending a large amount of power on trans-mitting the jamming signal, since the received power of themessage signal at the AE is always small (regardless ofjamming) due to the large path loss. This explains why thesecrecy data rate could increase. In the FD scenario, when theAE moves away from the LT and gets close to LR, the secrecydata rate increases, since the received jammer signal power atthe AE increases.

B. Effect of Source-Jammer/Relay Distance

In Fig. 8(a), we fix the AE location at coordi-nates (70,0), and change the position of the jammer/relayfrom (5,0) to (45,0). All other parameters are the same asthose used in Fig. VI-A. As expected, the secrecy data rate ofthe HD and FD scenarios are independent of the jammer/relaylocations. When the jammer/relay moves away from the LT,the secrecy data rate for the HDR scenario first increasesand then decreases, and there is an optimal relay locationsomewhere between the LT and LR. The secrecy data rate ofthe HDJ scenario, on the other hand, monotonically increasesas the jammer gets closer to the AE since the received jammingpower at AE is larger for a smaller jammer-AE distance.

In Fig. 8(b), we fix the AE location at (30,0), and movethe position of the jammer/relay from (5,0) to (45,0). Allother parameters are the same as those used in Fig. VI-A.As expected, the secrecy data rate of HD and FD scenariosare independent of the jammer/relay locations. When thejammer/relay moves away from the LT, the secrecy data ratefor the HDJ scenario first increases and then decreases, andthere is an optimal location for jammer somewhere betweenLT and LR. In this case, the HDJ scenario produces a betterperformance than the HDR scenario.

Note that for both figures, the HDR scenario outperformsothers despite the fact that the whole time slot is divided intotwo equal slots; which would result in the reduction of secrecyrate intuitively. However, the effect of boosting the receivedSNR in HDR scheme is so pronounced that it compensatesfor this rate reduction.

Finally, in Figures 9(a) and 9(b), for different locations ofthe AE between (5,0) to (70,0), we investigate the conditions

Fig. 8. (a) Secrecy data rate R vs. LT and jammer/relay distance ds j forHD, HDJ, and FD scenarios. The position of jammer/relay varies from (5,0)to (45,0). The eavesdropper location is fixed at (70,0). (b) Secrecy data rateR vs. LT and jammer/relay distance ds j for HD, HDJ, and FD scenarios.The position of jammer/relay varies from (5,0) to (45,0). The eavesdropperlocation is fixed at (30,0).

on the realy/jammer location between (0,0) to (50,0) underwhich the FD scenario is preferred over the HDJ and HDRscenarios. As can be seen, when AE is close to the source,for the majority of the considered range, HDJ scenario out-performs FD. For example, if AE is located on (5,0), HDJscenario is preferred over FD on the whole considered range.However, when the AE is placed closer to the destination,FD scenario takes over in a broader range such that whenAE is located on (50,0), FD is preferred for the whole range.For the HDR scenario the observation is different. When theAE is located closer to the source, FD scenario outperformsHDR in a broader range. For example, if AE is located on(5,0), HDR outperforms FD only when the relay is locatedon (10,0) or closer to the source. Interestingly, as AE movestowards the destination, FD takes over in a more limited range.

C. Effect of Multi Jammer/Relay

For this part, we fix the AE location at coordinate (70,0) andinitially look for the conditions under which FD scenario out-performs HDR and HDJ with one relay/jammer. For exampleas can be seen in Fig. 8(a), FD scenario performs considerably


Fig. 9. (a) Jammer location ds j for which the FD and HDJ scenarios producethe same secrecy rate vs LT and eavesdropper distance dse for HDJ and FDscenarios. (b) Relay location dsr for which the FD and HDR scenarios producethe same secrecy rate vs LT and eavesdropper distance dse for HDR and FDscenarios.

better than HDR and HDJ scenarios when the relay andjammer are located at coordinates (45, 0) and (70,0), respec-tively. We now uniformly add some relays between coordinates(40,0) and (50,0) and apply the best relay selection strategy.Fig. 10(a) shows the secrecy rate for different numbers ofrelays in this case. As can be seen, for R ≥ 2, the secrecy rateof HDR scenario takes over that of the FD scenario. We repeatthe same experiment with jammers. This time, the jammerare uniformly place between coordinates (20,0) to (30,0).Fig. 10(b) shows the secrecy rate for different numbers ofjammers. As can be seen, for J ≥ 3, the secrecy rate of theHDJ scenario takes over that of the FD scenario.

D. Effect of CSI Uncertainty

Fig. 11(a) considers the same scheme as that of 8(a)but for different values of ε2. As expected, smallervalues of ε2 produce higher secrecy rates. Moreover, nospecific difference can be seen in the behaviour of theconsidered scenarios with respect to each other. Fig. 11(b)plots the secrecy data rate for different valuesof ε2

gse= ε2

gde= ε2

g je= ε2

gre= ε2

ger= ε2

ged= ε2

from 0 to 1. We assume the same locations as the last figure

Fig. 10. (a) Secrecy data rate R vs. number of relays for HD, HDJ, HDR, andFD scenarios. (b) Secrecy data rate R vs. number of jammers, for HD, HDJ,HDR, and FD scenarios. The locations of jammer/relay and eavesdropper arefixed at (45,0) and (70,0), respectively.

for the network nodes. It can be seen from Fig. 11(b) that thesecrecy data rate decreases as ε2 increases. This is due to thefact that when ε2 increases, a larger fraction of the transmitpower must be devoted to each transmitter in order to reacha higher secrecy data rate.

E. Summary

For the considered simulation setup, the following conclu-sions can be drawn:

• In all cases the HD scenario has the weakest performance.• In general, the performance of the HDJ and FD scenarios

highly depend on where the jammer/relay and AE arelocated.

• When the jammer/relay is equally distant from the LTand RT (Fig. VI-A) and if the AE is not too close tothe jammer/relay, the FD scenario outperforms the HDJscenario and performs only slightly inferior to the HDRscenario. This is a very attractive situation from practicalpoint of view as we can remove the need for an extranetwork node.

• In case the relay/jammer is considered to be portableand an estimate of the location of the eavesdropper is


Fig. 11. (a) Secrecy data rate vs. ds j for different values of ε2. (b) Secrecydata rate R vs. channel mismatch between the eavesdropper and network nodesε2

gse= ε2

gde= ε2

g je= ε2

gre= ε2

ger= ε2

ged= ε2 for HD, HDJ, HDR, and

FD scenarios. The positions of eavesdropper and jammer/relay are at fixed at(70,0) and (25,0), respectively.

at hand, the HDJ scenario may be preferred over the FDscenario if the jammer can be placed close enough tothe eavesdropper (Fig. 8(b)). A similar statement can bemade for the HDR scenario. Otherwise, the FD scenariostill outperforms the HDR as well as the HDJ scenarioor in fact the conventional CJ scheme (Fig. 8(a)).

• If mutiple realys/jammers are permitted, the HDR/HDJscenarios can take over FD most of the times.

VII. CONCLUSION

In this paper, we proposed a robust resource allocationframework to improve physical layer security in the presenceof an active eavesdropper. Four different scenarios, namely,HD, HDJ, HDR, and FD were analysed to assess the per-formance of a system with FD receiver as compared withconventional CJ schemes. To solve the proposed optimizationproblem, we obtained robust transmit covariance matrices forthe proposed scenarios based on worst-case secrecy data ratemaximization. We then transformed the resulting non-convexoptimization problems into quasi-convex problems.

Simulation results show that the preference of deploying theFD scenario over CJ or vice versa highly depends on where

the jammer/relay and eavesdropper are located. In general onecan conclude that if the jammer can be placed close enough tothe eavesdropper, a better performance is achieved comparedto the FD system. Otherwise, the FD scenario can generallytake over and if this is the case, it would be very favorablefrom practical point of view as we can remove the need foran extra network node.

APPENDIX

PROOF OF PROPOSITION 1

Problem O H D4 can be rewritten as

mineged

egedQ∗

e eHged

+ 2( ged Q∗e eH

ged)+ ged Q∗

e gHed , (A.1a)

s.t. ‖ eged‖2≤ ε2

ged, (A.1b)

with the Lagrangian

L(eged, λ) = eged

(λI Ne + Q∗e)e

Hged

+ 2( ged Q∗e eH

ged)

+ ged Q∗e gH

ed − λε2ged, (A.2)

where λ ≥ 0 and the dual function is given by

g(λ)= infeged

L(eged, λ)=

{O � 0, Q∗

e eHged

∈ R (λI Ne + Q∗e );

−∞, otherwise.

(A.3)

where O = ged Q∗e eH

ged−λε2

ged− ged Q∗

e(λI Ne + Q∗e)

† Q∗e gH

ed +λI Ne + Q∗

e . The unconstrained minimization of L(eged , λ) withrespect to eged

is achieved when eged= − ged Q∗

e(λI Ne +Q∗

e)†, and the dual problem is thus obtained as

maxλ

eged Q∗e gH

ed − λε2ged

− ged Q∗e(λI Ne + Q∗

e)† Q∗

e gHed ,

(A.4a)

s.t. λI Ne + Q∗e � 0, (A.4b)

Q∗e gH

ed ∈ R (λI Ne + Q∗e). (A.4c)

Using the Schur complement, the dual problem can be con-verted to the following SDP:

maxλ≥0,γ

γ , (A.5a)

s.t.

[λI + Q∗

e Q∗e gH

edged Q∗

e ged Q∗e gH

ed − λε2ged

− γ

]� 0, (A.5b)

which completes the proof.

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Mohammad Reza Abedi received the B.Sc. andM.Sc. degrees in electrical engineering from theAmirkabir University of Technology, Tehran, Iran.He is currently a Research Associate with TarbiatModares University, Tehran. His research interestsinclude different aspects of wireless communica-tions with concentration on radio resource allocationtechniques.


Nader Mokari received the Ph.D. degree inelectrical engineering from Tarbiat Modares Uni-versity, Tehran, Iran, in 2014. He joined theDepartment of Electrical and Computer Engineering,Tarbiat Modares University, as an Assistant Profes-sor, in 2015. He has been involved in a number oflarge scale network design and consulting projects inthe telecom industry. His research interests includedesign, analysis, and optimization of communicationnetworks.

Hamid Saeedi (S’01–M’08) received the B.Sc. andM.Sc. degrees from the Sharif University of Tech-nology, Tehran, Iran, in 1999 and 2001, respec-tively, and the Ph.D. degree from Carleton Uni-versity, Ottawa, ON, Canada, in 2007, all in elec-trical engineering. From 2008 to 2009, he was aPostdoctoral Fellow with the Department of Elec-trical and Computer Engineering, University ofMassachusetts, Amherst, MA, USA. In 2010, hejoined the Department of Electrical and ComputerEngineering, Tarbiat Modares University, Tehran.

His current research interests include coding and information theory, wirelesscommunications, and resource allocation in conventional, heterogeneous, andcognitive radio networks.

Dr. Saeedi has been a technical program committee member of someinternational conferences, including the Vehicular Technology Conference2012, the Wireless Communication and Networking Conference 2014, 2015,and 2016, and the International Conference on Communications 2015. He wasa recipient of some awards, including the Carleton University Senate Medal forOutstanding Academic Achievement, the Natural Sciences and EngineeringCouncil of Canada Industrial Research and Development Fellowship, andthe Ontario Graduate Scholarship. He is currently an Editor of the IEEECOMMUNICATIONS LETTERS.

Halim Yanikomeroglu (S’96–M’98–SM’12–F’17)was born in Giresun, Turkey, in 1968. He receivedthe B.Sc. degree in electrical and electronics engi-neering from the Middle East Technical University,Ankara, Turkey, in 1990, and the M.A.Sc. degreein electrical engineering and the Ph.D. degree inelectrical and computer engineering from the Uni-versity of Toronto, Toronto, ON, Canada, in 1992and 1998, respectively. From 1993 to 1994, hewas with the Research and Development Groupof Marconi Kominikasyon A. S., Ankara, Turkey.

Since 1998, he has been with the Department of Systems and ComputerEngineering, Carleton University, Ottawa, ON, Canada, where he is currentlya Full Professor. He was a Visiting Professor with the TOBB Universityof Economics and Technology, Ankara, Turkey, from 2011 to 2012. Hisresearch interests include wireless technologies with a special emphasis onwireless networks. In recent years, his research has been funded by Huawei,Telus, Allen Vanguard, Blackberry, Samsung, Communications ResearchCentre of Canada, and DragonWave. This collaborative research resultedin over 25 patents (granted and applied). He was a recipient of the IEEEOttawa Section Outstanding Educator Award in 2014, Carleton UniversityFaculty Graduate Mentoring Award in 2010, the Carleton University GraduateStudents Association Excellence Award in Graduate Teaching in 2010, andthe Carleton University Research Achievement Award in 2009. He is aRegistered Professional Engineer in Ontario, Canada. He has been involvedin the organization of the IEEE Wireless Communications and NetworkingConference (WCNC) from its inception, including serving as a SteeringCommittee Member and the Technical Program Chair or Co-Chair of theWCNC 2004, Atlanta, GA, USA, the WCNC 2008, Las Vegas, NV, USA,and the WCNC 2014, Istanbul, Turkey. He was the General Co-Chair ofthe IEEE Vehicular Technology Conference (VTC) 2010-Fall held in Ottawa,and he is serving as the General Chair of the IEEE VTC 2017-Fall whichwill be held in Toronto. He has served on the Editorial Boards of theIEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS

ON WIRELESS COMMUNICATIONS, and the IEEE Communications Surveysand Tutorials. He was the Chair of the IEEE Wireless Technical Committee.He is a Distinguished Lecturer of the IEEE Communications Society and aDistinguished Speaker of the IEEE Vehicular Technology Society.

Documents

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, …Index Terms—Physical layer security, robust resource alloca-tion, full-duplex communications, active eavesdropping, jammer. I. INTRODUCTION