2788 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

Secret Channel Training to Enhance Physical LayerSecurity With a Full-Duplex Receiver

Shihao Yan , Member, IEEE, Xiangyun Zhou , Senior Member, IEEE, Nan Yang , Member, IEEE,

Thushara D. Abhayapala , Senior Member, IEEE, and A. Lee Swindlehurst , Fellow, IEEE

Abstract— This paper proposes a new channel training (CT)scheme for a full-duplex receiver to enhance physical layersecurity. Equipped with NB full-duplex antennas, the receiversimultaneously receives the information signal and transmitsartificial noise (AN). In order to reduce the non-cancellableself-interference due to the transmitted AN, the receiver hasto estimate the self-interference channel prior to the datacommunication phase. In the proposed CT scheme, the receivertransmits a limited number of pilot symbols that are known onlyto itself. Such a secret CT scheme prevents an eavesdropperfrom estimating the jamming channel from the receiver tothe eavesdropper, hence effectively degrading the eavesdroppingcapability. We analytically examine the connection probability(i.e., the probability of the data being successfully decoded by thereceiver) of the legitimate channel and the secrecy outage prob-ability due to eavesdropping for the proposed secret CT scheme.Based on our analysis, the optimal power allocation between CTand data/AN transmission at the legitimate transmitter/receiveris determined. Our examination shows that the newly proposedsecret CT scheme significantly outperforms the non-secret CTscheme that uses publicly known pilots when the number ofantennas at the eavesdropper is larger than one.

Index Terms— Physical layer security, channel training, fullduplex, artificial noise, power allocation.

I. INTRODUCTION

AS WIRELESS communications become increasinglyubiquitous, a growing amount of research has been

devoted to security issues pertaining to wireless data trans-mission. This is due to the fact that wireless communicationsare vulnerable to security threats, such as eavesdropping andjamming attacks, due to the open nature of the wirelessmedium [1], [2]. Against this background, physical layer

Manuscript received October 18, 2017; revised February 15, 2018 andApril 19, 2018; accepted April 25, 2018. Date of publication May 7, 2018;date of current version May 22, 2018. This work was supported by theAustralian Research Council’s Discovery Projects under Grant DP150103905.The associate editor coordinating the review of this manuscript and approvingit for publication was Dr. Lifeng Lai. (Corresponding author: Shihao Yan.)

S. Yan is with the School of Engineering, Macquarie University, Sydney,NSW 2109, Australia (e-mail: shihao.yan@mq.edu.au).

X. Zhou, N. Yang, and T. D. Abhayapala are with the ResearchSchool of Engineering, Australian National University, Canberra, ACT2601, Australia (e-mail: xiangyun.zhou@anu.edu.au; nan.yang@anu.edu.au;thushara.abhayapala@anu.edu.au).

A. L. Swindlehurst is with the Center for Pervasive Communications andComputing, University of California at Irvine, Irvine, CA 92697 USA (e-mail:swindle@uci.edu).

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

Digital Object Identifier 10.1109/TIFS.2018.2834301

security is emerging as a promising technique to realize andenhance the secrecy of wireless communications and is alsocompatible and complementary to traditional cryptographictechniques [3], [4]. For example, the inherent randomness ofa wireless channel can be utilized to extract cryptographickeys based on channel reciprocity [5], and physical-layer-based secure communications can be used to distribute keysfor the initialization of a wireless network in order to supportupper-layer cryptographical techniques.

In the pioneering studies of physical layer security (e.g., [6]and [7]), a wiretap channel was established as the funda-mental model to characterize physical layer security, in whichan eavesdropper (Eve) attempts to intercept the data trans-mission between a transmitter (Alice) and a legitimatereceiver (Bob). In the context of multiple-input multiple-output (MIMO) wiretap channels, artificial noise (AN)-aidedsecure transmission is of growing interest due to its robust-ness and desirable performance (e.g., [8]–[13]). AN wasinitially proposed to be transmitted in the null space of themain channel (i.e., the channel between Alice and Bob) byAlice to deliberately confuse Eve while avoiding interferenceto Bob [8]. Then, this AN-aided secure transmission wasstudied and extended in numerous scenarios. For example,Wang et al. [10] proposed an artificial fast fading schemein order to enhance physical layer security when Eve has moreantennas than Alice. The method of transmitting AN by acooperative jammer (i.e., an external helper) was exploited(e.g., [9]). However, this method suffers from some issueswith regard to mobility, synchronization, and trustworthi-ness [14], [15]. As a result of the full-duplex techniquescoming to reality [16]–[18], these issues are being addressedby replacing the external helper by a full-duplex receiver thatcan simultaneously receive information signals and transmitAN (e.g., [14], [15], and [19]-[22]). Meanwhile, the impactof full-duplex techniques employed by an active eavesdropperon physical layer security was examined within a hierarchicalgame framework in [23].

One of the key challenges faced in designing prac-tical full-duplex transceivers is self-interference and thusmany techniques have been developed in the literature tosuppress it [16]–[18]. Among the different types of self-interference cancellation techniques, the channel-aware tech-nique has attracted increasing research interest since it isnormally the last line of defense against self-interference

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YAN et al.: SECRET CT TO ENHANCE PHYSICAL LAYER SECURITY WITH A FULL-DUPLEX RECEIVER 2789

in the digital domain [17], [18]. In channel-aware self-interference cancellation, the channel state information (CSI)of the self-interference channel (i.e., the channel betweenthe transmit and receive antennas of Bob) is first esti-mated and then the self-interference is suppressed by beam-forming or subtraction. We note that the self-interferencechannel in this work refers to the channel from the transmitantenna to the receive antenna or the channel from the transmitRF chain to the receive RF chain when the full-duplex Bobis achieved by connecting two RF chains to a single antennathrough a circulator [17]. This is due to the fact that there isleakage in the circulator from the transmit RF chain to thereceive RF chain. As such, in this work we do not considerthe impact of ADC, DAC, and other hardware impairmentson the self-interference cancellation. In the literature, [24]examined a novel secure on-off transmission scheme with ANby considering a practical scenario where the channel trainingand feedback are limited. It is shown that considering thetraining and feedback cost there exists an optimal number oftransmit antennas that maximizes the net secrecy throughput.In addition, [25] investigated the role of AN in both thechannel training and data transmission in physical-layer secretcommunications, where the AN is used to prevent Eve fromestimating the eavesdropper’s channel in the training phase andthe AN is adopted to mask the transmission of the confidentialinformation in the data transmission phase. However, how toperform the self-interference channel estimation and how toallocate transmit power between the channel training (CT)and data transmission have not been addressed in the contextof physical layer security. These questions are of significantimportance in the design of practical communication systemsto achieve security. This is due to the fact that fewer resources(e.g., transmit power, time slots) are left for data transmissionif more resources are allocated to channel estimation (althoughmore accurate channel estimation is achieved). In addition, forchannel estimation we have to consider the amount of infor-mation about the channels that is leaked to Eve. The secrecyperformance of the wiretap channel with a full-duplex receiveris highly related to these questions. The assumption that theCSI of the self-interference channel is perfectly known iswidely adopted in the literature and thus the self-interferencecan be fully cancelled when the full-duplex Bob is equippedwith multiple transmit or receive antennas [14]. This assump-tion cannot be justified in many practical scenarios in whichthe self-interference channel consists of not only deterministicdirect paths but also random reflected paths from nearbyscatterers. This partially motivates this work, which, for thefirst time, examines CT in the wiretap channel with a full-duplex receiver.

In the aforementioned studies where AN is transmitted bya cooperative jammer, the jammer has to know the channelfrom the jammer to Bob (i.e., jammer-Bob channel) in orderto avoid interference to Bob. To this end, public pilots withknown transmit power have to be transmitted by the jammerin order to enable the estimation of the jammer-Bob channelat Bob (since they are two separate devices). Meanwhile, Evecan estimate the channel from the jammer to Eve (i.e., jammer-Eve channel) based on the known pilots and transmit power

of these pilots. As such, in these studies the jammer-Evechannel is normally assumed to be known by Eve (e.g., [9]).A similar assumption that Eve knows the jamming channel(i.e., the channel from Bob’s transmit antennas to Eve) in thewiretap channel with a full-duplex receiver is adopted in theliterature (e.g., [14] and [22]). This assumption ignores oneproperty of this wiretap channel, which is that Bob knowsexactly the transmitted signals and transmit power. This meansthat the pilots and transmit power used to estimate the self-interference channel are not required to be public. Ignoranceof this property in the literature has meant fact that thebenefits of transmitting AN by a full-duplex Bob rather than anexternal jammer have not been fully exploited. Recently, ourconference paper [26] proposed the secret CT design basedon this property of the wiretap channel with a full-duplexreceiver. However, we would like to clarify that the perfor-mance of this proposed design was only examined throughsimulations in [26], which lead to the fact that the powerallocation was also determined based on these simulations.These simulations are of high signal processing cost, whichsignificantly increases the implementation cost of the proposedsecret CT design. In order to fully exploit the benefits offeredby the secret CT design, in this work we theoretically analyzethe performance of this design, which leads to the followingspecific contributions.

• Following our conference paper [26], in this work wefurther examine the CT design problem in the contextof wiretap channels with a full-duplex receiver. Specifi-cally, we apply the non-secret CT scheme to the wiretapchannel with a full-duplex receiver as a benchmark, basedon which we further develop a new secret CT scheme byutilizing the fact that a full-duplex Bob knows exactly thesignal he transmits.

• In the secret CT scheme, secret pilots are utilized toestimate the self-interference channel with limited symbolperiods to prevent Eve from obtaining the CSI of thejamming channel, which is different from the non-secretCT scheme that utilizes publicly known pilots to esti-mate the self-interference channel. In order to fullyexplore and analyze the benefits offered by the proposedsecret CT scheme, we derive closed-form expressionsfor its connection probability (CP), which is the prob-ability that Bob successfully decodes Alice’s message,and its secrecy outage probability (SOP). Based onthe derived CP and SOP, the optimal transmit powerallocations between CT and data/AN transmission atAlice and Bob are determined under average powerconstraints.

• For the sake of comparison, we also examine the secrecyperformance of the non-secret CT scheme and theassociated power allocation between CT and data/ANtransmission in the wiretap channel with a full-duplexreceiver. Our examination shows that our proposed secretCT scheme significantly outperforms the non-secret CTscheme when NE > 1, where NE is the number ofantennas at Eve. We also find that the performanceadvantage of our proposed secret CT scheme increasesas NE increases.

2790 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

Fig. 1. The wiretap channel with a full-duplex receiver, where Aliceis equipped with a single antenna, Bob is equipped with NB full-duplexantennas, and Eve is equipped with NE antennas.

The rest of this paper is organized as follows. Section IIdetails the system model, the secret CT scheme, and thenon-secret CT scheme. Section III derives the CP and SOPof the secret CT. Section IV presents the optimal transmitpower allocations at Alice and Bob in the secret CT scheme.Section V provides numerical results to compare the proposedsecret and non-secret CT approaches. Finally, Section VImakes some concluding remarks.

Notation: Scalar variables are denoted by italic symbols.Vectors and matrices are denoted by lower-case and upper-case boldface symbols, respectively. Given a complex numberz, |z| denotes the modulus of z. Given a complex vector x,‖x‖ denotes the Euclidean norm and x† denotes the conjugatetranspose of x. The L × L identity matrix is referred to as IL

and E[·] denotes expectation.

II. SYSTEM MODEL

A. Channel Model

The wiretap channel of interest is illustrated in Fig. 1, whereAlice is equipped with a single antenna, Bob is equippedwith NB full-duplex antennas, and Eve is equipped with NE

antennas. We assume that Bob operates in full-duplex mode,i.e., all NB antennas are used for reception and transmissionsimultaneously. We denote h ∈ C

NB ×1 as the main channelvector, g ∈ C

NE ×1 as the channel vector between Alice andEve (referred to as the eavesdropper’s channel), G j ∈ C

NE ×NB

as the jamming channel matrix, and Hs ∈ CNB ×NB as the

self-interference channel matrix. We assume all the wirelesschannels within our system model are subject to indepen-dent quasi-static Rayleigh fading with equal block length.The self-interference channel considered throughout this workis the effective self-interference channel after channel-unawareinterference cancellation. Following [16], [18], we know thatthe deterministic components (e.g., line-of-sight components)in the self-interference channel can be removed throughchannel-unaware interference cancellation, and thus it isreasonable to assume that the residual self-interference channelafter channel-unaware cancellation is subject to independentquasi-static Rayleigh fading. We note that this assumption hasbeen widely adopted in the literature to examine the impactof self-interference in full-duplex systems (e.g., [27]–[29]).We further assume that the entries of h, g, G j , and Hs

are independent and identically distributed (i.i.d.) circularly

symmetric complex Gaussian random variables with zero-mean. We adopt the assumption that the variance of eachentry in h, g, and G j is normalized to one, but the varianceof each entry in Hs is σ 2

s . This assumption is to maintainthe generality of these channels, since the fading variances(including path loss) of h, g, and G j can be effectivelyabsorbed into the noise variance at Bob and the transmitpowers of Alice and Bob, while the fading variance of Hs isquantified by σ 2

s . This is due to the fact that only the statisticaldistributions of such channels affect the design of the differentCT schemes, which will be shown in our following analysis.The assumption that statistical information about G j is knowncan be justified as follows. Considering the independent quasi-static Rayleigh fading, in order to know the distribution of G j

we only have to know the corresponding large-scale pathlosses (e.g., the location of Eve). We note that Eve mayhave been an active transmitter or receiver in previous timeslots, where the corresponding large-scale path losses wereestimated. Considering the static system settings, we assumethe large-scale path losses are fixed or slowly varying, whichjustifies our assumption that the distribution of G j is publiclyknown. Finally, we note that this assumption is widely adoptedin the context of physical layer security (e.g., [30]) and evena stronger assumption that the instantaneous realization of G j

is known is also widely adopted (e.g., [8], [14], and [31]).We assume that the total duration of each block consists

of T symbol periods, including pilot and data symbols. In thepilot symbol periods, Alice and Bob send pilots to enablethe estimation of the main channel and the self-interferencechannel, respectively. The pilots used by Alice are publiclyknown. During the data symbol periods, Alice transmits confi-dential information to Bob while the full-duplex Bob sends ANto aid this secure transmission. We denote Alice’s transmitpowers for pilots and data by PAp and PAd , respectively.We also denote Bob’s transmit powers for pilots and ANby PBp and PBa , respectively. We consider an average powerconstraint over a fading block [32], in which the total energyfor a fading block at a transmitter (i.e., Alice or Bob) issubject to a fixed upper bound. We also consider the passiveeavesdropping scenario, in which Alice does not know the CSIof the eavesdropper’s channel.

B. Secret Channel Training Scheme

In the considered wiretap channel, the pilots sent by Bob toestimate the self-interference channel can be kept secret fromEve. This is due to the fact that Bob knows exactly what hetransmits and thus he does not have to share his pilots withother devices. As such, we next develop a specific CT strategydedicated to the wiretap channel with a full-duplex receiver,which is named as the secret CT scheme.

In this secret CT method, we first set TB = NB , where TB

is the number of symbol periods used to estimate the self-interference channel Hs . This assumption is to guarantee areliable estimate of Hs at Bob according to the principle ofthe Linear Minimum Mean Square Error (LMMSE) estimation(i.e., if TB < NB Bob cannot achieve a reliable estimateof Hs) [33]. We note that TB = NB is also a strict requirementfor the secret CT scheme, since when TB > NB , Eve can

YAN et al.: SECRET CT TO ENHANCE PHYSICAL LAYER SECURITY WITH A FULL-DUPLEX RECEIVER 2791

Fig. 2. A transmission block of T symbols with the secret CT scheme andthe non-secret CT scheme. (a) Secret channel training scheme. (b) Non-secretchannel training scheme.

obtain partial information about the jamming channel G j

through blind channel estimation [34]. This is due to thefact that once TB > NB , Eve will have more observationsthan unknown parameters to estimate and thus she can applysubspace-based channel estimation without knowing the pilotssent by Bob. Setting TB = NB guarantees that the estimationproblem of G j at Eve is ill-posed due to the unknown pilots,and thus Eve cannot achieve any information about G j in thesecret CT scheme.

To enable Bob to estimate the main channel, Alice transmitsits publicly known pilots. We note that Alice and Bob haveto transmit pilots during different symbol periods in order toachieve orthogonality between Alice’s and Bob’s pilots, dueto the constraint TB = NB . In this work, we set the number ofsymbol periods used to estimate the main channel to be 1 sinceAlice is equipped with a single antenna. We note that priorstudies on optimal training resource allocation have shownthat the optimal number of pilot equals the number of transmitantennas (which is NB for the self-interference channel and1 for the main channel in this work), under the average powerconstraint [32]. A transmission block of T symbols with thesecret CT scheme is illustrated in Fig. 2 (a).

When Alice transmits the pilot, the corresponding receivedsignal at Bob is given by

zB = √PAphsA + wB, (1)

where zB ∈ CNB×1, sA ∈ C1×1 is the pilot transmitted byAlice satisfying sAs†

A = 1, and wB ∈ CNB×1 is the noise atBob with i.i.d entries, each of which follows the distributionCN (0, σ 2

B). Considering the LMMSE estimator, based on theknown pilot Bob achieves the estimate of h as [33]

h =√PAp

PAp + σ 2B

zBs†A. (2)

Based on the properties of LMMSE estimation [33], the entriesof h are i.i.d and each follows the distribution CN (0, σ 2

h),

where

σ 2h

= PAp

PAp + σ 2B

. (3)

Again, due to the properties of LMMSE estimation, the estima-tion error h = h − h is independent of h and the entries of hare i.i.d, each of which follows the distribution CN (0, σ 2

h),

where

σ 2h

= σ 2B

PAp + σ 2B

. (4)

Since Alice’s pilot is publicly known, Eve can estimatethe eavesdropper’s channel g following a similar procedureas detailed above. Likewise, the entries of her estimate on g,denoted by g, are i.i.d and each of them follows the distributionCN (0, σ 2

g ), where

σ 2g = PAp

PAp + σ 2E

, (5)

and σ 2E is the receiver noise power at Eve. The estimation

error g = g − g is independent of g and the entries of g arei.i.d, each of which follows the distribution CN (0, σ 2

g ), where

σ 2g = σ 2

E

PAp + σ 2E

. (6)

When Bob transmits pilots over NB symbol periods withhis NB full-duplex antennas, the signal at his receive antennasis given by

ZB = √PBpHsSB + WB , (7)

where ZB ∈ CNB ×NB , SB ∈ CNB ×NB are the pilots transmittedby Bob satisfying SBS†

B = INB , and WB ∈ CNB ×NB isthe noise at Bob with i.i.d entries, each of which followsthe distribution CN (0, σ 2

B). Again, adopting the LMMSEestimator (based on the known SB and σ 2

s ) Bob obtains theestimate of Hs as

Hs =√PBpσ

2s

PBpσ 2s + σ 2

B

ZBS†B . (8)

Likewise, the estimation error Hs = Hs − Hs is indepen-dent of Hs and each of its entries follows the distributionCN (0, σ 2

Hs), where

σ 2Hs

= σ 2Bσ 2

s

PBpσ 2s + σ 2

B

. (9)

When Bob transmits the pilots SB , the received signalmatrix at Eve is given by

ZE = √PBpG j SB + WE , (10)

where WE ∈ CNE ×NB is the noise at Eve with i.i.d entries,each of which follows the distribution CN (0, σ 2

E ). Due toZE ∈ CNE ×NB and that Eve does not know the pilots SB ,Eve cannot achieve any information on G j . We note thatin order to prevent Eve from obtaining any information on G j ,the communication system should be carefully designed. Forexample, Eve could learn G j based on the control messages(used to establish the communication link, e.g., synchroniza-tion) or feedback (e.g., used to feed back the CSI of the mainchannel to Alice if Alice is equipped with multiple antennas)sent from Bob to Alice. To prevent this, different resources(e.g., frequencies) must be used for control messages or feed-back than those used for data communications. In this work,we assume that this independence can be guaranteed and Eve

2792 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

can only learn statistical information (e.g., the distribution)of G j based on pre-existing transmissions between Alice andBob.

We note that in the context of wireless energytransfer (WET), a channel learning method that requires onlyone feedback bit for each energy receiver was proposed in [37],which achieves a higher energy transfer efficiency. Relative tothis channel learning method, our proposed channel trainingscheme targets at improving physical layer security in thecontext of wiretap channels. We do not consider the costof feedback in the proposed channel training scheme, sincein the considered system the transmitter is only equippedwith one antenna and multiple antennas are only consideredin the full-duplex receiver. Our proposed scheme utilizes oneproperty of the full-duplex receiver to keep the pilot sequencesunknown to the eavesdropper (but known at the receiver)in order to prevent the eavesdropper from learning hercorresponding channels, which leads to an improved secrecyperformance.

C. Data Transmission With AN Following Secret CT

During the data symbol periods, Alice transmits a datastream while Bob transmits AN to confuse Eve. In additionto Eve, the AN also causes interference to Bob through theself-interference channel due to channel estimation errors.In general, Bob has two strategies to suppress such interferencebased on the estimated self-interference channel Hs . First, Bobcan subtract the known part of AN based on Hs at his receiveantennas since Bob knows the AN he transmits. Second, Bobcan transmit AN in the null space of Hs , based on the ideathat AN that lies in the null space of Hs does not cause anyinterference to Bob. We note that the second approach requiresthat the number of Bob’s transmit antennas is greater than thatof his receive antennas, which is not satisfied in our systemmodel. Therefore, in this work we assume that Bob adopts thefirst strategy to suppress the interference caused by the AN.

The received signal at Bob in each data symbol period isgiven by

yB = √PAd hx +

√PBa

NBHsn + vB, (11)

=√PAd (h + h)x +

√PBa

NB(Hs + Hs)n + vB, (12)

where x ∈ C1×1 denotes the transmitted signal satisfyingE[|x |2] = 1, n ∈ CNB×1 is the AN vector, whose entries arei.i.d circularly-symmetric complex normal random variableswith zero mean and unit variance, and vB ∈ CNB×1 is thenoise vector at Bob with i.i.d entries, each of which follows thedistribution CN (0, σ 2

B). Knowing Hs and n, Bob can removeHsn from yB by subtraction and obtain the effective receivedsignal as

y′B = √

PAd hx + √PAd hx +

√PBa

NBHsn + vB, (13)

where√PBa/NB Hsn is the residual self-interference.

Although Bob knows that his received signal is subject to

interference caused by channel estimation errors in h and Hs ,he cannot suppress such interference since he does not knowh and Hs . As such, the optimal combining technique thatmaximizes the signal-to-interference-plus-noise ratio (SINR)at Bob is maximum ratio combining (MRC) based on h. Then,focusing on the CP (similar to the outage probability) andfollowing [38], the instantaneous SINR at Bob for given h, h,and Hs is

γB = μB‖h‖2

μB |h†h|2‖h‖2

+ μS‖h†Hs‖2

‖h‖2+ 1

, (14)

where μB = PAd/σ 2B and μS = PBa/σ

2B/NB .

Likewise, the received signal at Eve in one data symbolperiod is given by

yE = √PAd gx + √

PAd gx +√PBa

NBG j n + vE , (15)

where vE ∈ CNE ×1 is the noise vector at Eve with i.i.d entries,each of which follows the distribution CN (0, σ 2

E ). We notethat, due to the known structure of the pilots SB , Eve maystill obtain some information on the jamming channel matrixG j , even though she does not know either SB or PBp. Forexample, she can approximate G j G

†j by ZE Z†

E . Due to theuncertainty in the information on G j leaked to Eve, it is verydifficult to determine the optimal combining strategy at Eve,and the optimal strategy will lead to mathematically intractableanalysis. As confirmed numerically, the MRC strategy canoutperform the MMSE combining strategy with G j G

†j ≈

ZE Z†E in terms of achieving higher SINRs at Eve. As such,

in this work we assume that Eve adopts MRC based on g tocombine the received signals. Then, following (15) the SINRat Eve for the secret CT scheme is given by

γE = μE‖g‖2

μE |g†g|2‖g‖2 + μJ ‖g†G j ‖2

‖g‖2 + 1, (16)

where μE = PAd/σ 2E and μJ = PBa/σ

2E/NB .

D. Traditional Channel Training Schemeas a Benchmark

In order to better illustrate the benefits of the secret CTscheme, we now consider the traditional CT scheme as abenchmark. Unlike the secret CT scheme, in the traditionalCT scheme the pilot transmitted by Bob (i.e., SB ) is publiclyknown, which can be jointly designed with the pilot trans-mitted by Alice (i.e., sA). As such, in the traditional CTscheme we do not need the constraint TB = NB becauseBob’s pilots are known by Eve anyway. Hence, Alice andBob can simultaneously transmit pilots over 1 + NB symbolperiods while still ensuring the orthogonality of their pilots.This setting also guarantees a fair comparison between thesecret CT scheme and the traditional CT scheme, since thetotal number of symbol periods allocated to CT is 1 + NB

in both schemes. A transmission block of T symbols with thetraditional CT scheme is illustrated in Fig. 2 (b). Therefore,σ 2

h, σ 2

h, and σ 2

Hs(which are given by (3), (4), (5), (6), and (9),

YAN et al.: SECRET CT TO ENHANCE PHYSICAL LAYER SECURITY WITH A FULL-DUPLEX RECEIVER 2793

respectively, in the secret CT scheme) for the traditional CTscheme are changed to

σ 2h

= PAp(1 + NB )

PAp(1 + NB ) + σ 2B

, (17)

σ 2h

= σ 2B

PAp(1 + NB ) + σ 2B

, (18)

σ 2g = PAp(1 + NB )

PAp(1 + NB ) + σ 2E

, (19)

σ 2g = σ 2

E

PAp(1 + NB ) + σ 2E

, (20)

σ 2Hs

= σ 2Bσ 2

s

PBpσ 2s (1 + NB)/NB + σ 2

B

. (21)

In the traditional CT scheme, the pilot SB is public and thusEve can estimate the jamming channel G j . The estimationerror of G j can be obtained in a similar format as givenin (21). Since Eve knows that her received signal is subjectto the interference caused by the channel estimation errorand AN, the optimal combining technique that maximizes theSINR at Eve is MMSE based on g and her estimate of G j

(denoted by G j ). Here, we note that MMSE outperforms MRCin terms of achieving a higher SINR at Eve in the traditionalCT scheme. This is due to the fact that MMSE utilizes Eve’sknowledge of G j (in addition to g) to suppress the interferencein order to maximize the SINR, while MRC ignores the avail-able G j . Following (15) and applying the MMSE combiner,for the traditional CT scheme the instantaneous SINR at Eveis

γE = PAd wgg†w†

w(PBa

NBG j G

†j + PBa

NBG j G

†j + PAd gg† + σ 2

E INE

)w†

,

(22)

where

w = g†(

G j G†j + σ 2

E INE

)−1. (23)

III. SECRECY PERFORMANCE ANALYSIS OF THE SECRET

CHANNEL TRAINING SCHEME

During the data symbol periods, Alice adopts a fixed-ratewiretap code that can be described by two rate parameters,namely, the codeword rate RB and the redundancy rate RE ,which are predetermined and fixed [40]–[42]. The actualinformation rate is given by Rs = RB − RE . For such acoding scheme, Bob cannot reliably decode the transmittedinformation when the capacity of the main channel (i.e., CB ) isless than RB , while perfect secrecy against Eve fails when thecapacity of the eavesdropper’s channel (i.e., CE ) is larger thanRE [40]–[42]. We refer to the probability of achieving reliabledecoding as CP and refer to the probability of failing to achieveperfect secrecy as SOP. We note that one conventional secrecyoutage probability in the literature is defined as the probabilityof the secrecy capacity Cs being less than the secrecy rate Rs ,where Cs = CB − CE . In this work, the reliability cannot beguaranteed due to the fact that the main channel suffers fromestimation errors, i.e., CB is not available. As such, if we adopt

the conventional secrecy outage probability in this work, it willbe a combination of the CP and SOP and cannot separate theCP and SOP. Therefore, in this work we adopt RB and RE asthe two interesting code parameters to derive the CP and SOP,respectively. Although the ergodic secrecy capacity can beformulated for a passive eavesdropping scenario where Aliceand Bob do not know the CSI of the eavesdropper”s channel,in this work we did not adopt it as a performance metric. First,in this work we assume that all wireless channels are subjectto independent quasi-static Rayleigh fading, for which outageprobability is more appropriate than ergodic channel capacity,regardless of whether secrecy or non-secrecy communicationis considered [43]. In addition, there exists a certain SOPassociated with the ergodic secrecy capacity. Thus, the ergodicsecrecy capacity cannot fully capture the secrecy performanceof the system. We also note that the secrecy rate Rs is differentfrom the secrecy capacity examined in [44] from the perspec-tive of information theory. In [44], the secrecy capacity isachieved by encryption over the channel. Without encryption,the secrecy capacity is achieved only when the instantaneousCSI of both the main channel and the eavesdropper’s channelis available at Alice. Due to channel estimation error on themain channel and the passive eavesdropping scenario for theeavesdropper’s channel, the secrecy transmission rate suffersfrom connection outages and secrecy outages. As such, in thiswork we focus on analyzing the CP and SOP.

A. Connection Probability

The CP, which is the probability that Bob can decode themessage for a given RB with a negligible decoding errorprobability, is given by [38] and [45]

Pc = Pr(log2(1 + γB) ≥ RB). (24)

We derive the CP of the secret CT scheme in the followingtheorem.

Theorem 1: The CP of the secret CT scheme is derived as

Pc = cNB e− c

μh

�(NB + 1)�(NB )(cμH + μh)NB

×NB −1∑

i=0

(NB −1i

)�(NB + i + 1)

(cμH + μh)i+1(μHμh)−i−1

×[

2 F1

(1, NB + i + 1; NB + 1; μh

cμH + μh

)

− 2 F1

(1, NB +i +1; NB+1; μh(μt −μH )

μt (cμH + μh)

)], (25)

where c = 2RB − 1, μh = μBσ 2h

, μH = μSσ 2H

, μt = μBσ 2h

,�(n) is the Gamma function given by �(n) = (n − 1)!for positive integer n, and 2 F1(·, ·; ·; ·) denotes the Gausshypergeometric function [46, eq. (9.100)].

Proof: The proof is provided in Appendix.We would like to clarify that the derived CP given in (25)

is indeed a function of all power parameters, i.e., PAp , PAd ,PBp, and PBa . This is due to the fact that following (3) and (4)both μh and μt are functions of PAp and PAd while μH is afunction of PBp and PBa as per (9).

2794 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

B. Secrecy Outage Probability

The SOP, which is the probability that the capacity of theeavesdropper’s channel is no less than RE , is given by [45]

Pso = Pr(log2(1 + γE ) ≥ RE

). (26)

We derive the SOP of the secret CT scheme for positive σ 2E

in the following theorem.Theorem 2: The SOP of the secret CT scheme for positive

σ 2E is derived as

Pso = d NE e− d

μg μNB −NEg

�(NB + 1)�(NE )(dμJ + μg)NB

×NE −1∑

i=0

(NE −1i

)�(NB + i + 1)

(dμJ + μg)i+1(μJ μg)−i−1

×[

2 F1

(1, NB + i + 1; NB + 1; μg

dμJ + μg

)

− 2 F1

(1, NB +i +1; NB +1; μg(μd − μJ )

μd(dμJ + μg)

)], (27)

where d = 2RE − 1, μg = μEσ 2g , and μd = μEσ 2

g .Proof: Comparing (16) with (14), we find that γE follows

the same type of distribution as γB . As such, the proof issimilar to the proof of Theorem 1 and thus omitted here.

IV. OPTIMAL TRANSMIT POWER ALLOCATION IN THE

SECRET CHANNEL TRAINING SCHEME

We have seen in the previous section that the powervalues are key design parameters affecting the connectionand secrecy performances. In this section, we determine theoptimal transmit power allocation between CT and data/ANtransmission at Alice and Bob in the secret CT scheme underaverage power constraints. We note that in the consideredpassive eavesdropping scenario, the instantaneous SINR at Eveis not available and thus the power allocation does not dependon the instantaneous SINR at Eve. In particular, we cannotachieve the optimal power allocation that maximizes thesecrecy capacity of the considered system.

A. Objective Function

As mentioned in Section III, data transmission in theconsidered wiretap channel may incur connection and secrecyoutages. Considering block fading channels, we adopt theeffective throughput subject to a given secrecy constraint asour key performance metric, which is given by [42]

η = T − NB − 1

T(RB − RE )Pc,

s.t. Pso ≤ ε, (28)

where ε is the maximum allowable SOP (i.e., the predeter-mined secrecy requirement of the system) and RB > RE

in order to ensure a positive information rate. In this work,we consider fixed-rate transmission, in which RB and RE area priori fixed. We note that T is fixed in our system model, andthus the maximization of η subject to Pso ≤ ε given in (28)is equivalent to the following optimization

max Pc s.t. Pso ≤ ε. (29)

We note that we cannot guarantee perfect secrecy (i.e., the SOPcannot be zero) in (28) due to the fact that Alice doesnot have perfect CSI of the eavesdropper’s channel due tothe passive eavesdropping scenario and/or channel estimationerrors. We also note that in this work we cannot considerthe case of NB → ∞ for a given finite T . This is due tothe fact that the number of time slots used to estimate theself-interference channel is at least the same as NB in orderto achieve a reliable channel estimate. In order to achieve apositive effective secrecy throughput, we have NB < T − 1 asper (28).

B. Power Allocation Under Average Power Constraints

In this work, we consider an average power constraint atboth Alice and Bob. Following (29) the power allocationoptimization for the secret CT scheme can be presented as

maxPAp ,PAd ,PBp,PBa

Pc, (30)

s.t. Pso ≤ ε, (31)

PAp + PAd (T − NB − 1) ≤ EA, (32)

PBp NB + PBa(T − NB − 1) ≤ EB , (33)

where EA and EB are respectively the total powers availableat Alice and Bob for each block of T symbol periods; hence,the average power constraints per symbol for Alice and Bobare EA/T and EB/T . We next detail how to determine thesolution to (30) (i.e., the optimal values of PAd , PAp , PBa,and PBp).

Theorem 3: The optimal value of PAd that maximizes Pc

subject to the constraints given in (31), (32), and (33) can beobtained through

P∗Ad = argmax

0<PAd <PmAd

Pc(P†Ap,PAd ,P†

Bp,P†Ba), (34)

where

PmAd = EA

T − NB − 1, (35)

P†Ap = EA − PAd (T − NB − 1), (36)

P†Bp = EB − P†

Ba(T − NB − 1)

NB, (37)

and P†Ba as a function of PAd can be obtained by solving the

following equation

Pso = ε. (38)

Proof: We first note that Pso and Pc are both monoton-ically decreasing functions of PBa . As such, Pso = ε isalways achieved in order to maximize Pc subject to the secrecyconstraint (31). Otherwise (i.e., if Pso < ε), we can decreasePBa to increase Pc. Following (27), we note that Pso onlydepends on PBa and PAd . As such, we can obtain P†

Ba as afunction of PAd by solving (38). We also note that Pso is nota function of PAp or PBp, while Pc monotonically increasesas PAp or PBp increases. Then, we can conclude that theequality in both (32) and (33) is always guaranteed, whichleads to (36) and (37), respectively. Finally, (35) is achieveddue to PAp > 0. This completes the proof of Theorem 3.

YAN et al.: SECRET CT TO ENHANCE PHYSICAL LAYER SECURITY WITH A FULL-DUPLEX RECEIVER 2795

Fig. 3. Connection probability Pc versus PBp for different values of PApwhere PAd = PBa = 20dB, RB = 5, NB = 3, σ 2

s = 1, and σ 2B = 1.

By substituting P∗Ad , P∗

Ap , P∗Ba, and P∗

Bp into (25), we canobtain the maximum CP of the secret CT scheme. We note thatthe optimization problem given in (34) can be solved by a one-dimensional numerical search. Specifically, we can adopt agrid search on PAd to solve this optimization problem, since itsvalue must lie in the interval (0,Pm

Ad ) as given in Theorem 3.We can pick a value of PAd within (0,Pm

Ad ), then determinethe values of P†

Ap , P†Ba , and P†

Bp as per Theorem 3, and finallyachieve a value of Pc following Theorem 1.

V. NUMERICAL RESULTS

In this section, we present numerical results to examine thesecrecy performance of the proposed secret CT scheme withthe non-secret CT scheme as the benchmark. We note that itis the ratio between the transmit power (i.e.,PAp , PAd , PBp,and PBa) and the receiver noise power (i.e., σ 2

B ) that affectsthe numerical results. We also note that the unit of the wiretapcode rates (i.e., RB and RE ) in this work is the bit, which isomitted as well.

In Fig. 3 we plot Pc with different choices of pilot transmitpower values. In this figure, we first observe that Pc increasesas PAp or PBp increases. This is due to the fact that as PAp

and PBp increase, the channel estimation errors for the mainself-interference channels decrease, respectively. Furthermore,we observe that the CP of the non-secret CT scheme is higherthan that of the secret CT scheme under the specific non-optimized and unconstrained settings. This can be explained bythe fact that in the non-secret CT scheme both Alice and Bobtransmit pilots in NB + 1 symbol periods while in the secretCT scheme Alice and Bob transmit pilots in 1 and NB symbolperiods, respectively, and thus for fixed PAp and PBp morepower is utilized in the non-secret CT scheme. As we willshow in Fig. 5, this observation does not hold when the powerallocation is optimized under the average power constraints aswell as the secrecy outage constraint.

Fig. 4. Secrecy outage probability Pso versus NE for different values of NB ,where PAd = 10dB, PBa = 20dB, PAp = 10dB, PBp = 20dB, σ 2

E = 1,and RE = 2.

In Fig. 4 we plot Pso versus the number of antennas atEve for different numbers of antennas at Bob. As expected,in this figure we first observe that when NE = 1 the SOP is thesame for the secret CT scheme and the non-secret CT scheme,due to the fact that knowing the CSI of the jamming channelin the non-secret CT scheme does not help Eve for NE = 1.For NE > 1, we observe that Pso for the secret CT scheme issignificantly lower than that for the non-secret CT scheme andthe gap increases as NE increases. This can be explained bythe fact that in the secret CT scheme Eve does not know thejamming channel while she does in the non-secret CT schemeand the CSI of the jamming channel offers more informationto Eve as NE increases. Finally, the figure confirms that Pso

decreases as NB increases.In Fig. 5 we plot the maximum CPs of the secret and non-

secret CT schemes versus the secrecy constraint indicator ε fordifferent values of NE . In this figure and the following figures,we set a small value −20dB for σ 2

E . In this figure, we firstobserve that as ε increases the maximum CP increases, whichdemonstrates the tradeoff between the effective throughput andthe secrecy constraint. For example, by comparing the valuesof the maximum CP for ε = 0.01 and ε = 0.15 we cansee a high cost for reducing the maximum CP to achievesecrecy. We also observe that the proposed secret CT schemesignificantly outperforms the non-secret CT scheme in terms ofachieving a much higher maximum CP. This is due to the factthat the secret CT scheme prevents Eve from obtaining the CSIof the jamming channel. This observation also demonstratesthe benefits of transmitting AN by a full-duplex Bob relativeto transmitting AN by an external helper. Finally, we observethat the performance gap between these two schemes increasesas NE increases.

Under the same settings as in Fig. 5, we plot Alice’s optimaltransmit power for data (i.e., P∗

Ad ) and Bob’s optimal transmitpower for AN (i.e., P∗

Ba) versus ε in Fig. 6 and Fig. 7,

2796 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

Fig. 5. The maximum connection probability of the secret and non-secretCT schemes versus the secrecy constraint ε for different values of NE , whereNB = 8, RB = 5, RE = 3, T = 300, EA/T = EB/T = 10dB, σ 2

s = 1,and σ 2

B = 0dB.

Fig. 6. Alice’s optimal data transmit power P∗Ad versus the secrecy constraint

ε for different values of NE , where NB = 8, RB = 5, RE = 3, T = 300,σ 2

s = 1, EA/T = EB/T = 10dB, and σ 2B = 0dB.

respectively. We first observe that P∗Ad increases as ε increases

in Fig. 6, which demonstrates that more transmit power isallocated to data transmission as the secrecy constraint isrelaxed. We also observe that P∗

Ba decreases as ε increasesin Fig. 7, which demonstrates that as the secrecy constraintis relaxed less transmit power is allocated to AN at Bob.In Fig. 6, we further observe that more transmit power isallocated to data transmission at Alice as NE decreases.In Fig. 7, we observe that less transmit power is allocatedto AN at Bob as NE decreases. As expected, in Fig. 6 wesee that P∗

Ad for the secret CT scheme is higher than that forthe non-secret CT scheme and in Fig. 7 we see that P∗

Ba forthe secret CT scheme is lower than that for the non-secret

Fig. 7. Bob’s optimal AN transmit power P∗Ba versus the secrecy constraint

ε for different values of NE , where NB = 8, RB = 5, RE = 3, T = 300,σ 2

s = 1, EA/T = EB/T = 10dB, and σ 2B = 0dB.

Fig. 8. The maximum connection probability and Bob’s optimal pilottransmit power for the secret CT scheme versus the fading power of theself-interference channel σ 2

s , where ε = 0.10, RB = 5, RE = 3, T = 300,NB = NE = 2, EA/T = 15dB, EB/T = 10dB, and σ 2

B = 0dB.

CT scheme. This is due to the fact that Eve obtains lessinformation regarding the jamming channel in the secret CTscheme.

In Fig. 8 we examine the impact of the fading power of theself-interference channel σ 2

s (i.e., the variance of each entryin Hs) on the secrecy performance and power allocation of thesecret CT scheme. As expected, we observe that the maximumconnection probability of the secret CT scheme decreases asσ 2

s increases in Fig. 8(a). This is due to the fact that, accordingto (9), the channel estimation error of Hs increases as σ 2

sincreases, which means that the connection probability forthe secret CT scheme is a monotonically decreasing functionof σ 2

s . Noting the limited value range of the y-axis in Fig. 8(a),

YAN et al.: SECRET CT TO ENHANCE PHYSICAL LAYER SECURITY WITH A FULL-DUPLEX RECEIVER 2797

we would like to highlight that the secrecy performance ofthe secret CT scheme is insensitive to σ 2

s , especially whenσ 2

s > 0dB. In Fig. 8(b), we observe that Bob’s optimal pilottransmit power P∗

Bp first increases and then remains constantas σ 2

s increases. The initial increase in P∗Bp is due to the

fact that as σ 2s increases the performance of the channel

estimation decreases and thus more power should be allocatedto the channel training to counteract this performance loss inthe estimate of the self-interference channel. Again, noting thelimited value range of the y-axis in Fig. 8(b), we can concludethat the power allocation of the secret CT scheme is insensitiveto σ 2

s as well.In the secret CT scheme Bob can transmit AN during the

training of the main channel to further enhance the physicallayer security of the considered system, since the transmittedAN can increase the estimation error of the eavesdropper’schannel. The role of AN in channel training in wiretapchannels has been examined in the literature (e.g., [25], [35],and [36]). When Bob transmits AN during the training of themain channel, the estimation error of the main channel willdepend on the realization of the self-interference channel andthe estimation error of the eavesdropper’s channel will dependon the realization of the jamming channel, which makes theclosed-form performance analysis mathematically intractable.As such, we numerically approximate the performance of theupdated secret CT scheme, in which the training of the self-interference channel is conducted before the training of themain channel and the transmission of AN by Bob is consid-ered during the training of the main channel. As expected,we confirm that the transmission of AN by Bob during thetraining of the main channel improves the performance of theproposed scheme.

VI. CONCLUSION

In this work, we devised a new secret CT scheme basedon the fact that in the wiretap channel with a full-duplexreceiver, Bob knows exactly what he transmits. To study theperformance of the proposed secret CT scheme, we derivedclosed-form expressions for the CP and SOP, based on whichthe power allocation between CT and data/AN transmission isoptimized. Our examination demonstrates that when NE > 1the secret CT scheme significantly outperforms the non-secret CT scheme in terms of achieving a much higher CPsubject to the same secrecy constraint, and when NE = 1they achieve the same secrecy performance. The secrecyperformance improvement of the secret CT scheme relativeto the non-secret CT scheme increases as NE increases.

APPENDIX

Following (14), we can rewrite γB as

γB = γBn

γBs + 1, (39)

where γBn = μB‖h‖2, γBs = γB1 + γB2 , γB1 =μB |h†h|2/‖h‖2, and γB2 = μS‖h†Hs‖2/‖h‖2. We note thathere ‖h‖2 is a random variable since we are interested in thechannel-independent CP, based on which the power allocation

between channel estimation and data transmission can bedetermined. We note that γBn and γBs are independent dueto the fact that h, h, and Hs are independent of each other,and thus following (24) we have

Pc = Pr(γB ≥ 2RB − 1

)

= Pr

(γBs ≤ γBn − c

c

)

a=∫ ∞

cFγBs

(x − c

c

)fγBn

(x)dx

=∫ ∞

0FγBs

( x

c

)fγBn

(x + c)dx, (40)

wherea= is achieved by noting γBs ≥ 0, FγBs

(·) is the cdfof γBs , fγBn

(·) is the probability density function (pdf) of γBn ,and as defined in Theorem 1 we have c = 2RB − 1. We recallthat the entries of h are i.i.d and each of them follows thedistribution CN (0, σ 2

h), thus fγBn

(·) is given by [43]

fγBn(x) = x NB −1

�(NB )μNBh

e− x

μh . (41)

We next derive an expression for FγBs(·) when μt �= μH ,

i.e., FγBs(x) = Pr(γB1 + γB2 ≤ x). Since h and h are

independent, the cdf of γB1 is given by [43]

FγB1(x) = 1 − e

xμt . (42)

Given that h is independent of Hs , the pdf of γB2 isgiven by [43]

fγB2(x) = x NB−1

�(NB )μNBH

e− x

μH . (43)

We note that γB1 and γB2 are independent. As such, for μt �=μH we can derive FγBs

(x) as

FγBs(x)

=∫ x

0FγB1

(x − y) fγB2(y)dy

=∫ x

0

[1 − e

x−yμt

]fγB2

(y)dy

=∫ x

0fγB2

(y)dy − e− xμt

�(NB )μNBH

∫ x

0y NB−1e

− (μt −μH )yμt μH dy

b=γ

(NB , x

μH

)

�(NB )−

μNBt e− x

μt γ(

NB , (μt−μH )xμt μH

)

�(NB ) (μt − μH )NB, (44)

whereb= is achieved with the aid of the following identity [46,

eq. (3.351.1)]∫ u

0xne−μx dx = μ−n−1γ (n + 1, μu), (45)

and γ (n, t) is the lower incomplete gamma function, whichcan be expanded for positive integer n as

γ (n, t) = (n − 1)![

1 − e−tn−1∑

m=0

tm

m!

]

. (46)

2798 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

Then, substituting (41) and (44) into (40), we have

Pc = μ−NBh

�(NB )2

∫ ∞

0(x + c)NB −1e

− x+cμh γ

(NB ,

x

cμH

)dx

−μNBt μ−NB

h

�(NB )2

∫ ∞

0

(x +c)NB−1

ex+cμh

+ xcμt

γ

(NB ,

(μt − μH )x

cμtμH

)dx

c= e− c

μh μ−NBh

�(NB )2

NB −1∑

i=0

(NB − 1

i

)cNB −i−1

×[ ∫ ∞

0xi e

− xμh γ

(NB ,

x

cμH

)dx

− μNBt

(μt −μH )NB

∫ ∞

0

xi

ecμt+μhcμt μh

xγ

(NB ,

(μt −μH )x

cμtμH

)dx

],

(47)

wherec= is obtained with the aid of the following

identity [46, eq. (1.111)]

(a + x)n =n∑

i=0

(n

i

)xi an−i . (48)

We then solve the integrals in (47) and obtain the resultsin (25) with the aid of the following identity [46, eq. (6.455.2)]

∫ ∞

0

xu−1

eβxγ (v, αx)dx

= αv�(u + v)

v(α + β)u+v 2 F1

(1, u + v; v + 1; α

α + β

),

[α + β > 0, β > 0, u + v > 0]. (49)

So far, we have proved (25) for μt �= μH . We next prove (25)for the special case of μt = μH . Noting (42), (43), and thath is independent of h, for μt = μH we have

FγBs(x) =

γ(

NB + 1, xμt

)

�(NB + 1)=

γ(

NB + 1, xμH

)

�(NB + 1)

d=γ

(NB , x

μH

)

�(NB )−

(x

μH

)NBe− x

μH

�(NB + 1), (50)

whered= is achieved with the aid of γ (N+1, x) = Nγ (N, x)−

x N e−x . Then, substituting (41) and (50) into (40), for μt =μH we have

Pc = μ−NBh

�(NB )2

∫ ∞

0(x + c)NB−1e

− x+cμh γ

(NB ,

x

cμH

)dx

− μ−NBh (cμH )−NB

�(NB )�(NB + 1)

∫ ∞

0x NB (x +c)NB−1e

− x+cμh

− xcμH dx

= e− c

μh μ−NBh

�(NB )2

NB −1∑

i=0

(NB − 1

i

)cNB−i−1

×[ ∫ ∞

0xi e

− xμh γ

(NB ,

x

cμH

)dx

− (cμH )−NB

NB

∫ ∞

0x NB +i e

− cμH +μhcμH μh

xdx

]

e= cNB e− c

μh

�(NB + 1)�(NB )(cμH + μh)NB

×NB −1∑

i=0

(NB −1i

)�(NB + i + 1)

(cμH + μh)i+1(μH μh)−i−1

×[

2 F1

(1, NB +i +1; NB+1; μh

cμH +μh

)−1

], (51)

wheree= is achieved with the aid of (49) and the following

identity [46, eq. (3.351.3)]∫ ∞

0xne−ux dx = n!u−n−1, u > 0. (52)

We note that for μt = μH in (25) we have

2 F1

(1, NB + i + 1; NB + 1; μh(μt − μH )

μt (cμH + μh)

)= 1. (53)

As such, comparing (51) with (25) we know that (25) isalso valid for the special case of μt = μH . Therefore,following (47) and (51) the proof of Theorem 1 is completed.

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Shihao Yan (S’11–M’15) received the B.S. degreein communication engineering and the M.S. degreein communication and information systems fromShandong University, Jinan, China, in 2009 and2012, respectively, and the Ph.D. degree in elec-trical engineering from the University of New SouthWales, Sydney, Australia, in 2015. From 2015 to2017, he was a Post-Doctoral Research Fellowwith the Research School of Engineering, AustraliaNational University, Canberra, Australia. He iscurrently a University Research Fellow with the

School of Engineering, Macquarie University, Sydney, Australia. His currentresearch interests are in the areas of wireless communications and statisticalsignal processing, including physical layer security, covert communications,and location spoofing detection.

Xiangyun Zhou (SM’17) received the Ph.D. degreefrom Australian National University (ANU) in 2010.He is currently a Senior Lecturer with ANU.His research interests are in the fields of commu-nication theory and wireless networks. He was arecipient of the Best Paper Award at ICC’11 andthe IEEE ComSoc Asia-Pacific Outstanding PaperAward in 2016. He was named the Best YoungResearcher in the Asia-Pacific Region in 2017 byIEEE ComSoc Asia-Pacific Board. He also served assymposium/track and workshop co-chairs for major

IEEE conferences. He was the Chair of the ACT Chapter of the IEEECommunications Society and Signal Processing Society from 2013 to 2014.He has been serving as an Editor for various IEEE journals, including the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE WIRELESSCOMMUNICATIONS LETTERS, and the IEEE COMMUNICATIONS LETTERS.He served as a Guest Editor for the IEEE Communications Magazine’s featuretopic on wireless physical layer security in 2015.

Nan Yang (S’09–M’11) received the B.S. degreein electronics from China Agricultural Universityin 2005, and the M.S. and Ph.D. degrees in electronicengineering from the Beijing Institute of Technologyin 2007 and 2011, respectively. He was a Post-Doctoral Research Fellow with the CommonwealthScientific and Industrial Research Organization from2010 to 2012 and the University of New South Walesfrom 2012 to 2014. He has been with the ResearchSchool of Engineering, Australian National Univer-sity, since 2014, where he is currently a Future

Engineering Research Leadership Fellow and a Senior Lecturer. His generalresearch interests include massive multi-antenna systems, millimeter-waveand terahertz communications, ultra-reliable low latency communications,cyber-physical security, and molecular communications. He received theTop Editor Award from the Transactions on Emerging TelecommunicationsTechnologies in 2017, the Exemplary Reviewer Award from the IEEE TRANS-ACTIONS ON COMMUNICATIONS in 2016 and 2015, the Top Reviewer Awardfrom the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY in 2015,the IEEE ComSoc Asia-Pacific Outstanding Young Researcher Award and theExemplary Reviewer Award from the IEEE WIRELESS COMMUNICATIONS

LETTERS in 2014, and the Exemplary Reviewer Award from the IEEECOMMUNICATIONS LETTERS in 2013 and 2012. He was also a co-recipientof the Best Paper Awards from IEEE GLOBECOM 2016 and IEEE VTC2013-Spring. He is currently serving in the Editorial Board of the IEEETRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONSON VEHICULAR TECHNOLOGY, and Transactions on Emerging Telecommu-nications Technologies.

2800 IEEE TRANSACTIONS ON INFORMATION FORENSICS AND SECURITY, VOL. 13, NO. 11, NOVEMBER 2018

Thushara D. Abhayapala (M’00–SM’08) receivedthe B.E. degree (Hons.) in engineering and thePh.D. degree in telecommunications engineeringfrom the Australian National University (ANU),Canberra, in 1994 and 1999, respectively. He wasthe Director of the Research School of Engineering,ANU, from 2010 to 2014, and also the Leaderof the Wireless Signal Processing Program withthe National ICT Australia from 2005 to 2007.He is currently the Deputy Dean of the College ofEngineering and Computer Science, ANU. He has

supervised over 30 Ph.D. students and co-authored over 200 peer reviewedpapers. His research interests are in the areas of spatial audio and acousticsignal processing, multi-channel signal processing, and spatial aspects ofwireless communications. He is a member of the Audio and Acoustic SignalProcessing Technical Committee of the IEEE Signal Processing Society from2011 to 2016. He is an Associate Editor of IEEE/ACM TRANSACTIONS ONAUDIO, SPEECH, AND LANGUAGE PROCESSING. He is a Fellow of EngineersAustralia.

A. Lee Swindlehurst (F’04) received the B.S. andM.S. degrees in electrical engineering from BrighamYoung University (BYU), in 1985 and 1986, respec-tively, and the Ph.D. degree in electrical engineeringfrom Stanford University in 1991. He was with theDepartment of Electrical and Computer Engineering,BYU, from 1990 to 2007, where he served asthe Department Chair from 2003 to 2006. From1996 to 1997, he held a visiting scholar positionwith Uppsala University and the Royal Institute ofTechnology, Sweden. From 2006 to 2007, he was

on leave, while he was a Vice President of Research for ArrayComm LLC,San Jose, CA, USA. Since 2007, he has been a Professor with the ElectricalEngineering and Computer Science Department, University of California atIrvine, Irvine, where he served as an Associate Dean for Research andGraduate Studies with the Samueli School of Engineering from 2013 to 2016.From 2014 to 2017, he was also a Hans Fischer Senior Fellow with theInstitute for Advanced Studies, Technical University of Munich. His researchfocuses on array signal processing for radar, wireless communications, andbiomedical applications, and he has over 300 publications in these areas.He was a recipient of the 2000 IEEE W. R. G. Baker Prize Paper Award,the 2006 IEEE Communications Society Stephen O. Rice Prize in the Field ofCommunication Theory, the 2006 and 2010 IEEE Signal Processing Society’sBest Paper Awards, and the 2017 IEEE Signal Processing Society DonaldG. Fink Overview Paper Award. He was the inaugural Editor-in-Chief of theIEEE JOURNAL OF SELECTED TOPICS IN SIGNAL PROCESSING.