# On the Performance of Secure Full-Duplex Relaying under Composite Fading Channels

Hirley Alves, Glauber Brante, Richard D. Souza, Daniel B. da Costa, Matti Latva-aho

aa r X i v : . [ c s . I T ] N ov On the Performance of Secure Full-DuplexRelaying under Composite Fading Channels

Hirley Alves, Glauber Brante, Richard D. Souza, Daniel B. da Costa, and Matti Latva-aho,

Abstract —We assume a full-duplex (FD) cooperative networksubject to hostile attacks and undergoing composite fadingchannels. We focus on two scenarios: a) the transmitter has fullCSI, for which we derive closed-form expressions for the averagesecrecy rate ; and b) the transmitter only knows the CSI of thelegitimate nodes, for which we obtain closed-form expressions forthe secrecy outage probability . We show that secure FD relaying isfeasible, even under strong self-interference and in the presenceof sophisticated multiple antenna eavesdropper. Index Terms —Full-duplex relaying, PHY security, secrecyergodic capacity, secrecy outage probability.

I. I

NTRODUCTION

Owing to the broadcast nature of the wireless medium,communication systems are vulnerable to the attack of eaves-droppers and security issues play an important role in systemdesign. Recently, cooperative communications has appeared asan alternative to enhance physical layer (PHY) security [1]–[3], where a trusted relay node helps the communication be-tween Alice and Bob through cooperation or by jamming Eve.Most works focus on half-duplex (HD) schemes. However,full-duplex (FD) communications have gained considerableattention and several promising signal processing techniquesfor self-interference – a power leakage between transmit andreceive signals – mitigation have been proposed [4]. Eventhough ideal FD operation is not yet attainable, practical FDcommunication is feasible via such self-interference mitiga-tion. This process results in residual self-interference that canbe modeled as a fading channel, which allows the emulationof various (non) line-of-sight conﬁgurations [4], [5].In this work, we assume a FD cooperative network in thepresence of a multi-antenna eavesdropper – named Eve. Alicecommunicates with Bob – which are the legitimate nodes– with the help of a trusted FD relay, which suffers self-interference. We also employ a general channel model whichencompasses Nakagami- m fading as well as Log-Normalshadowing, which is known as composite fading channel [6].We consider two scenarios according to the availability ofchannel state information (CSI) at Alice: a) Alice has theCSI of all users, thus perfect secrecy can be achieved [7]

H. Alves and M. Latva-aho are with the Centre for Wireless Communica-tions, University of Oulu, Oulu, Finland. (emails: {halves, matla}@ee.oulu.ﬁ).H. Alves, G. Brante and R. D. Souza are with Federal Univer-sity of Technology - Paraná (UTFPR), Paraná, Brazil (emails: {gbrante,richard}@utfpr.edu.br).D. B. da Costa is with the Federal University of Ceará (UFC), Ceará, Brazil(email: [email protected]).This work was supported by the Finnish Funding Agency for Technologyand Innovation (Tekes), University of Oulu Graduate School, Infotech OuluGraduate School, Academy of Finland, CAPES/CNPq (Brazil). and we focus on the average secrecy rate as the performancemetric, for which we present an accurate approximation inclosed-form; and b) Eve is a passive eavesdropper and Alicehas only CSI of the legitimate channels. Thus, perfect secrecycannot be guaranteed at all times [7], and we provide accurateapproximated closed-form expressions for the secrecy outageprobability. To the best of the authors’ knowledge, such analy-sis is still not available in the literature . Additionally, besidesthe developed mathematical framework, we show that secureFD relaying is feasible in both scenarios under consideration,even under the effect of self-interference.II. S YSTEM M ODEL

We consider a cooperative network with three legitimatesingle-antenna users: Alice (A), relay (R) and Bob (B), com-municating in the presence of Eve (E), which has N E antennasand employs maximal ratio combining (MRC). Moreover, weconsider that the relay operates in FD mode (with one antennadedicated to transmission and another to reception, in order toincrease the isolation of the self-interference [4]) and operatesunder the decode-and-forward (DF) protocol.The channels are affected by path-loss, shadowing and fad-ing which are assumed to be independent [6]. Then, in order toaccount for these channel impairments we adopt a compositefading distribution, where fading follows Nakagami- m distri-bution and shadowing is modeled as Log-Normal (LN) randomvariable (RV), whose squared-envelop follows a Gamma-LNdistribution [6]. Moreover, as proposed in [8], the compositesquared envelop – which represents the SNR of a given linkbetween nodes i and j – is well approximated by a singleLN RV, whose parameters depend on the actual distributionand are deﬁned as: shape µ dB “ ξ r ψ p m q ´ ln p m qs ` µ Ω p and log-scale σ “ ξ ζ p , m q ` σ p , where m is the shapeparameter of the Nakagami- m distribution, ξ “ ln p q { , Ω p is the mean squared-envelop, µ Ω p and σ Ω p is the meanand standard deviation of Ω p , respectively. A. Sum of LN RVs

As the density of Z “ ř k γ ij k – a RV representing thesum of k independent LN RVs – has no exact closed formexpression and its distribution is heavy-tailed and positivelyskewed, we resort to an approximation of the sum of LN RVsby a single LN RV [9] as follows. In this work we assume a multi-antenna Eve, composite fading channelsfor two distinct scenarios, whereas [3] only presents a preliminary inves-tigation on the secrecy outage probability of single antenna devices underRayleigh fading channels.

Deﬁnition 1 (

Cumulants and additivity property ): Let X and Y be two independent RVs. The cumulants of a RVcan be written as a function of the raw moments [10]. Forinstance, the ﬁrst and second cumulants of X are given as κ ,X “ E r X s and κ ,X “ E “ X ‰ ´ E r X s [10]. Then, theadditivity property of the cumulants gives that the cumulantsof X ` Y are the sum of the individual cumulants, therefore κ n,X ` Y ﬁ κ n,X ` κ n,Y , where κ n,X and κ n,Y are the n -thcumulants of X and Y , respectively [10].From Deﬁnition 1 and by approximating the sum of LN RVsby a single LN RV [9], we estimate the parameters of the sin-gle LN RV from the cumulants as µ “ ln ` κ ˘ ´ ln p a κ ` κ q and σ “ ln ` κ ` κ ˘ ´ ln p κ q , where µ is the mean and σ is the variance of the equivalent Normal p µ, σ q distribution inlogarithmic scale [10], [11, Ch. 26]. B. Legitimate Channel

The DF protocol can be decomposed into two phases:broadcast (BC) and multiple access (MAC). Differently fromHD relaying, the MAC phase starts simultaneously with theBC phase under the FD mode, so that the relay forwards toBob the message received from Alice in a previous phase atthe same time that Alice broadcasts a new message to the relayand Bob. Thus, the received signals at the relay and Bob are y R “ b P A d ´ ν AR h AR ¨ x ` a P R δ h RR ¨ ˜ x ` w R , (1) y B “ b P A d ´ ν AB h AB ¨ x ` b P R d ´ ν RB h RB ¨ ˜ x ` w B , (2)where h ij , i P t A , R u and j P t R , B u , denotes channelcoefﬁcient, P i is the transmit power, d ij is the distance betweennodes i and j, and ν is the path loss exponent. Additionally, w j is zero-mean complex Gaussian noise with unity variance, x is the unity energy transmitted symbol, while ˜ x is the unityenergy symbol forwarded by the relay. Let us remark that usea different codebook and x and ˜ x may be not identical.

1) FD Relaying:

The message is divided into L blocks, asshown in Fig. 1. Additionally, due to the inherent characteris-tics of the encoding/decoding scheme, ˜ x is delayed comparedto x , so that ˜ x r l s “ x r l ´ τ s , where ď l ď L and τ representsthe processing and block delay of τ ě blocks, which weassume hereafter to be τ “ . As pointed out in [12] thisdelay is large enough to guarantee that the received signalsare uncorrelated, and therefore can be jointly decoded. A ﬁrstanalysis on such decoding schemes for practical FD relaying isdone in [13], which is later extended in [14], where the authorsgeneralize the backward decoding scheme for any delay andnumber of blocks and show that performance is not affected forlarge L . Hereafter we assume regular encoding and backwarddecoding [1], [15] .Further, we consider all channels ( h ij ) as quasi-static inde-pendent and identically distributed (i.i.d.) according to com-posite Nakagami- m Log-Normal distribution. The relay suffersfrom self-interference, which we model as a composite fadingchannel denoted by h RR , and δ represents the overall self-interference attenuation factor. Additionally, since all RVs are Please refer to [15] for details on DF regular encoding and backwarddecoding, and to [1, Th. 2] for an analysis on the relay-eavesdropper channel. x [1] x [2] ··· x [ L − x [ L ] Alice ˜ x [1] ˜ x [2] ··· ˜ x [ L −

1] ˜ x [ L ] Relay

Fig. 1. Alice and relay block transmission. Notice that the frame is dividedinto L blocks and the relay’s transmission is delayed by one block ( τ “ ). i.i.d. the instantaneous SNRs of the legitimate links are LNdistributed as Γ ij ∼ LN p µ ij , σ ij q [8]. The SINR at the relayis Γ R “ Γ AR Γ RR , which is also a LN RV deﬁned as Γ R ∼ LN p µ R , σ R q , where µ R “ µ AR ´ µ RR and σ R “ p σ AR ` σ RR q [16]. The ﬁrst two cumulants of Γ ij can be readily attainedthrough Deﬁnition 1 as κ , Γ ij and κ , Γ ij . With that, and relyingon the additive property [16], we obtain the cumulants of theoverall SNR at Bob, Γ B “ Γ AB ` Γ RB , and then the sum oftwo LN RVs can be simply written as a single LN RV as Γ B ∼ LN p µ B , σ B q , where µ B “ ln ` κ , Γ B ˘ ´ ln p κ , Γ B ` κ , Γ B q and σ B “ ln ` κ , Γ B ` κ , Γ B ˘ ´ ln p κ , Γ B q . C. Eavesdropper Channel

We assume that Eve is equipped with N E antennas andapplies MRC to the received signals. Thus, the N E ˆ receivedsignal is y E “ b P A d ´ ν AE h AE ¨ x ` b P R d ´ ν RE h RE ¨ ˜x ` w E , (3)where h iE , i P t A , R u , denotes the channel coefﬁcients vectors( N E ˆ ) at the eavesdropper. Additionally, w E is zero-meancomplex Gaussian noise N E ˆ vector with unity variance.From Deﬁnition 1 we obtain the cumulants of γ iE k . More-over, once all RVs are i.i.d., we the cumulants of Γ iE “ ř k γ iE k are simply N E κ n, Γ iE , where i P t

A, R u and ď k ď N E . Then, since Γ AE and Γ RE are independent we resort againto Deﬁnition 1 and deﬁne the cumulants of Γ E “ Γ AE ` Γ RE as κ n “ N E p κ n, Γ AE ` κ n, Γ RE q , which allows us to write Γ E ∼ LN p µ E , σ E q , whose parameters are µ E “ ln ` κ , Γ E ˘ ´ ln p κ , Γ E ` κ , Γ E q and σ E “ ln ` κ , Γ E ` κ , Γ E ˘ ´ ln p κ , Γ E q .III. S CENARIO C VERAGE S ECRECY R ATE

First, let us suppose that Eve is part of the network and Aliceis able to acquire the CSI from it. For instance, this scenariomay represent Alice wanting to communicate privately toa certain user ( i.e. , Bob) without being overheard by otherlegitimate receivers. Thus, Alice can adapt its transmissionrate accordingly in order to achieve perfect secrecy. In such ascenario, the average secrecy capacity is an insightful metriconce it quantiﬁes the average secrecy rate [7].

Proposition 1:

The achievable rates of the DF protocolfor the relay-eavesdropper channel in the presence of self-interference is R s “ r log p ` Γ FD q ´ log p ` Γ E qs ` ,where r x s ` ﬁ max t x, u . Proof:

Based on the average rate derivation for traditionalrelaying in [17], plus given the cooperative secrecy rates from[1], and given that all RVs are independent we attain R s “ r R FD ´ R E s ` , “ r log p ` min t Γ R , Γ B uq ´ log p ` Γ E qs ` . (4) Let us introduce a new RV deﬁned as Γ FD “ min t Γ R , Γ B u ,whose CDF is introduced in Appendix A, then plugging it into(4) we conclude the proof. Theorem 1:

Assuming the non-negativity of the secrecy rateand the independence of RVs, the average secrecy rate is R s “ ż ż R s f Γ FD p γ FD q f Γ E p γ E q d γ FD d γ E , “ ż F Γ E p γ E q ` γ E r ´ F Γ FD p γ E qs d γ E . (5) Proof:

See [18].Next, since the CDFs of Γ FD and Γ E are known andwith help of Theorem 1 we attain the average secrecy rate(bits/s/Hz) as R s “ ż erfc p η E q erfc p η B q erfc p η R q p ` z q d z , (6)where η E “ µ E ´ ln p z q? σ E , η B “ ´ µ B ` ln p z q? σ E and η R “ ´ µ R ` ln p z q? σ R .Nevertheless, (6) does not have a closed form solution, there-fore we resort to Gauss-Laguerre quadrature [11, Ch. 25.4]. Theorem 2:

Under the assumption of perfect CSI and com-posite fading, the average secrecy rate in bits/s/Hz is R s » K ÿ k “ ω L k e χ L k ` η E ˘ erfc ` η B ˘ erfc ` η B ˘ , (7)where η i with i P t B , E , R u is written as in η i but replacing z by e χ L k ´ , and K is the order of the Laguerre polynomial. Proof:

Please see [18] and notice that we resort toGauss-Laguerre quadrature [11, Ch. 25.4] to attain (7), thus χ L k are the roots of the Laguerre polynomial while ω L k “ χ L k { ` p K ` q L K ` p χ L k q ˘ are the weights of the Gauss-Laguerre quadrature [11, Ch. 25.4]. The error can be ana-lytically estimated as indicated in [11, Ch. 25.4].IV. S CENARIO C ECRECY O UTAGE P ROBABILITY

Now let us suppose that Alice has no knowledge of Eve’sCSI except for channel statistics. Thus, in order to protectits transmission from a possible inimical attack, Alice com-municates with Bob with a constant secrecy rate R s ą ,which yields a certain secrecy outage probability . Note thatAlice assumes that Eve’ secrecy rate is R E “ R FD ´ R s ,thus security is compromised as soon as C E ą R E . Otherwise,perfect secrecy is assured. Then, secrecy outage probabilityis the appropriated metric to evaluate the performance of aquasi-static fading wiretap channel when the transmitter hasno CSI and the receivers have CSI of their own channelsonly [7]. Next, we determine the secrecy outage probability as O P “ Pr r R s ă R s s . Thus, an outage event occurs wheneverthe instantaneous secrecy rate R s falls below the target secrecyrate R s ﬁxed by Alice. Theorem 3:

Assuming the non-negativity of the secrecy rateand that only CSI of the legitimate channel is available at the In order to avoid confusion it is noteworthy that the secrecy capacity isdenoted by the calligraphic letter C and the secrecy achievable rate by letter R , while the attempted transmission rate is represented by the letter R . transmitter, we can deﬁne the secrecy outage probability inbits/s/Hz as O P » ´ K ÿ k “ ω H k erfc ´ ´ µ B ` ln p υ E q? σ B ¯ erfc ´ ´ µ R ` ln p υ E q? σ R ¯ ? π , (8)where υ E “ R s ` exp ` µ E ` ? σ E χ H k ˘ ` ˘ ´ . Proof:

Based on [1], [7], we can deﬁne the secrecy outageprobability under composite fading channels as O P “ Pr r R s ă R s s “ Pr r log p ` γ FD q ´ log p ` γ E q ă R s s“ ż F Γ FD p R s p ` z q ´ q f E p z q d z. (9)Note that the CDF of Γ FD is given in Appendix A. However,(9) does not have a closed-from solution. Therefore, we resortto semi-analytical solution based on Gauss-Hermite quadrature[11, Ch. 25.4]. The weights of the Hermite polynomial aregiven as ω H k “ ? π K ´ K ! { ` K H K ´ p χ H k q ˘ , where χ H k are its roots of order K [11, Ch. 25.4].V. N UMERICAL R ESULTS AND D ISCUSSIONS

Alice, the relay and Bob are assumed to be on a straight line,the relay is at the center and the distance between Alice andBob is m, unless stated otherwise. Additionally, we assumea path loss exponent of ν “ as well as unitary bandwidth. Weassume that all channels experience some LOS, and thereforeundergo fading with m “ , while the shadowing standarddeviation is σ “ dB . We consider equal power allocationthus total power is given as P A “ P R “ P .Fig. 2 depicts the average secrecy rate (Sc1) as a functionof transmit power P in dBm for different values of self-interference cancellation for N E P t , , u antennas. Eve’saverage SNR is set to µ E “ . and σ E “ . , whichmeans that µ Ω p “ ´ dB and σ Ω p “ dB so that Eveis closer to Alice and the relay than Bob. Notice that themore sophisticated is the self-interference cancellation (lowervalues of δ ), higher is the achievable average secrecy rate.For instance, at high SNR, the average secrecy rate can bedoubled with the reduction of the self-interference from -80dBto -90dB. Even though sophisticated interference mitigationschemes have been recently proposed, such cancellation levels– in the order of -90dB – are still a challenging task toachieve [4]. The average secrecy rate degrades as the num-ber of antennas grows. Nevertheless, it is still possible tocommunicate with perfect secrecy. Notice that in terms ofsecrecy capacity the impact of relatively poor self-interferencecancellation ( δ ą ´ dB) is much worse than that of havinga sophisticated eavesdropper (with several receive antennasapplying maximum ratio combining).Further, Fig. 3 shows the secrecy outage probability (Sc2)as a function of the transmit power for R s P t , u bits/s/Hz.Secrecy outage probability considerably increases with bet-ter self-interference attenuation and cancellation at the relay We assume that the polynomial order of K “ since it presents greataccuracy. We recall that the error can be analytically estimated which isanother advantage of such quadrature methods [11]. P (dBm)0 . . . . . . . . . . R s ( b i t s / s / H z ) Antennas N E = 2 N E = 4 N E = 8 SI Cancellation - δ −

60 dB −

75 dB −

90 dB

Fig. 2. Average secrecy rate R s as a function of the transmit power P indBm for different values of self-interference cancellation ( δ ) and number ofantennas N E . P (dBm)0.20.30.40.50.60.70.80.91.0 S ec r ec y O u t ag e P r o b a b ili t y Rate R s (bits/s/Hz) R s = 2 R s = 4 SI Cancellation - δ −

60 dB −

75 dB −

90 dB −

100 dB

Fig. 3. Outage probability, Pr r R s ă R s s , as a function of the transmitpower P in dBm with different values of self-interference cancellation ( δ ) aswell as target secrecy rate R s . (lower δ ). Notice also that as the target secrecy rate increases,the secrecy outage probability decreases since Alice imposesa more stringent secure rate requirement.VI. C ONCLUSIONS

We investigated the secrecy performance of FD relayingin the presence of a multiple antenna eavesdropper undercomposite fading. Two scenarios are considered based on theCSI availability at the transmitters: a) full CSI is available from both intended and unintended receivers; b) only thelegitimate channels are known to Alice. Our results showthat the self-interference at the relay considerably affectsperformance regardless of the scenario. However, even understrong self-interference and in the presence of a sophisticatedmultiple antenna eavesdropper, FD relaying is feasible.A PPENDIX

Let Γ R and Γ B be independent and LN distributed, thusCDF of Γ FD “ min t Γ R , Γ B u can be readily attained throughstandard statistical methods [16], such that the CDF is F Γ FD p z q “ ´

14 erfc ´ ´ µ R ` ln p z q? σ R ¯ erfc ´ ´ µ B ` ln p z q? σ B ¯ . (10)R EFERENCES[1] L. Lai and H. El Gamal, “The relay-eavesdropper channel: Cooperationfor secrecy,”

IEEE Trans. on Inf. Theory , vol. 54, no. 9, pp. 4005–4019,Sept 2008.[2] L. Dong, Z. Han, A. Petropulu, and H. Poor, “Improving wireless phys-ical layer security via cooperating relays,”

IEEE Trans. Sig. Process. ,vol. 58, no. 3, pp. 1875–1888, Mar. 2010.[3] H. Alves, G. Brante, R. Souza, D. da Costa, and M. Latva-aho, “On theperformance of full-duplex relaying under phy security constraints,” in

ICASSP , Italy, Sept. 2014, pp. 1–5.[4] M. Duarte, C. Dick, and A. Sabharwal, “Experiment-driven characteriza-tion of full-duplex wireless systems,”

IEEE Trans. on Wireless Commun. ,Dec. 2012.[5] H. Alves, D. da Costa, R. Souza, and M. Latva-aho, “Performance ofblock-markov full duplex relaying with self interference in nakagami-mfading,”

IEEE Wireless Commun. Lett. , vol. 2, no. 3, pp. 311–314, 2013.[6] J. Ilow and D. Hatzinakos, “Analytic alpha-stable noise modeling in apoisson ﬁeld of interferers or scatterers,”

IEEE Trans. Signal Process. ,vol. 46, no. 6, pp. 1601–1611, Jun. 1998.[7] M. Bloch, J. Barros, M. Rodrigues, and S. McLaughlin, “Wirelessinformation-theoretic security,”

IEEE Trans. on Inf. Theory , vol. 54,no. 6, pp. 2515–2534, 2008.[8] M.-J. Ho and G. L. Stüber, “Capacity and power control for CDMAmicrocells,”

ACM Journal on Wireless Net. , vol. 1, no. 3, pp. 355–363,Oct. 1995.[9] J. Wu, N. B. Mehta, and J. Zhang, “A ﬂexible lognormal sum approx-imation method,” in

Globecom , St. Louis, MO, 28 Nov.–2 Dec. 2005,pp. 3413–3417.[10] V. V. Petrov, “Sums of independent random variables,” 1975.[11] M. Abramowitz and I. A. Stegun,

Handbook of Mathematical Functions:with Formulas, Graphs, and Math. Tables . Dover, 1965.[12] T. Riihonen, S. Werner, and R. Wichman, “Hybrid full-duplex/half-duplex relaying with transmit power adaptation,”

IEEE Trans. on Wire-less Commun. , vol. 10, no. 9, pp. 3074–3085, Sept. 2011.[13] H. Alves, R. Souza, and G. Fraidenraich, “Outage, throughput andenergy efﬁciency analysis of some half and full duplex cooperativerelaying schemes,”

Trans. on Emerging Telecommun. Tech. , 2013.[14] M. Khafagy, A. Ismail, M. Alouini, and S. Aissa, “On the outage per-formance of full-duplex selective decode-and-forward relaying,”

IEEECommun. Letters , vol. 17, no. 6, pp. 1180–1183, 2013.[15] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative strategies andcapacity theorems for relay networks,”

IEEE Trans. Inf. Theory , vol. 51,no. 9, pp. 3037–3063, Sep. 2005.[16] A. Papoulis and S. U.Pillai,

Probability, Random Variables and Stochas-tic Processes . McGraw-Hill, 2001.[17] R. Nikjah and N. Beaulieu, “Exact closed-form expressions for theoutage probability and ergodic capacity of decode-and-forward oppor-tunistic relaying,” in

IEEE GLOBECOM’09 , Nov 2009, pp. 1–8.[18] L. Wang, N. Yang, M. Elkashlan, P. L. Yeoh, and J. Yuan, “Physicallayer security of maximal ratio combining in two-wave with diffusepower fading channels,”