IEEE COMMUNICATIONS LETTERS

1

Optimal Power Allocation for Multicarrier Secure Communications in Full-Duplex Decode-and-Forward Relay Networks Xiaobin Huang, Jing He, Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin

Abstract—For full-duplex (FD) relay networks, secure communication is an important issue. In this letter, considering decodeand-forward scheme, we investigate the optimal power allocation for multicarrier secure communication in FD relay networks. Our objective is to maximize the achievable secrecy rate of FD relay network subject to sum power constraint of the source and FD relay. The formulated non-convex optimization problem is equivalently transformed into a convex problem. Employing Karush-Kuhn-Tucker conditions, we obtain closed-form solution to the optimization problem. Simulation results have shown that the proposed optimal power allocation scheme for FD relaying outperforms that for conventional half-duplex relaying. Index Terms—Achievable secrecy rate, decode-and-forward (DF), full-duplex (FD), power allocation, relay network.

I. I NTRODUCTION N relay networks, the half-duplex (HD) relaying and the full-duplex (FD) relaying are two relaying modes [1]. In the conventional relay networks, the HD relaying is usually employed where the relay receives and transmits signals over two orthogonal frequency slots or time slots. Employing the FD relaying, the relay receives and transmits signals simultaneously on the same frequency, which potentially achieves the higher information rate than the conventional HD relaying. The main challenge to employ the FD relaying is the loopinterference (LI), which is that the transmitted signals from the FD relay may interfere the received signals at the relay. In [2], several LI mitigation schemes were proposed, such as natural isolation, time-domain cancelation, and spatial suppression. In [3], [4], the FD transmission schemes for the orthogonal frequency division multiplexing modulated (OFDM) signals were investigated. The aforementioned works do not consider the secure communications. The wireless information is susceptible to eavesdropping because of the openness of wireless transmission medium [5]. Thus, the secure communication is a critical issue for the relay networks. In [6], [7], the secure transmission schemes for the conventional HD relay networks were proposed. In [8], [9], the multicarrier secure communication for the OFDM signals in the HD relay networks was studied.

I

This work was supported in part by the National Natural Science Foundation of China under Grant 61472458, Grant 61202498 and Grant 61173148. X. Huang, J. He, Q. Zhang, and J. Qin are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail: [email protected], [email protected], {zhqi26, issqjy}@mail.sysu.edu.cn). Q. Li is with the School of Advanced Computing, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail: [email protected]).

In [10], [11], the secure communication for the FD relay networks was investigated. To the best of our knowledge, the research on the multicarrier secure communication for the OFDM signals in the FD relay networks is missing. In this letter, considering the decode-and-forward (DF) scheme, we investigate the optimal power allocation for multicarrier secure communication in the FD relay networks. Our objective is to maximize the achievable secrecy rate of the FD relay network subject to the sum power constraint of the source and the FD relay. The formulated optimization problem is non-convex because of the non-convex objective function. We equivalently transform the non-convex optimization problem into a convex problem, which is solved by the Karush-KuhnTucker (KKT) conditions. After derivations, we obtain the closed-form solution to the optimization problem. II. S YSTEM M ODEL Consider an FD relay network, which consists of one source, one FD relay, one legitimate destination, one eavesdropper, as shown in Fig. 1. Each node except the relay is equipped with a single antenna. The FD relay is equipped with a single transmit antenna and a single receive antenna which are isolated [1]. We assume that there is no direct link between the source and the legitimate destination. The reliable communication from the source to the legitimate destination is established by the FD relay. The FD relay employs the DF scheme to forward signals. When the FD relay forwards signals to the legitimate destination, the eavesdropper tries to overhear the signals from the relay. In this letter, we assume that the eavesdropper cannot overhear the signals from the source. This is reasonable because if the eavesdropper is close to the FD relay, since no LI cancelation scheme is employed, the signals from the source are much weaker than those from the FD relay. In the FD relay network, the total bandwidth is divided into N orthogonal subcarriers. The source and the FD relay transmit the OFDM signals in the frequency-selective channels where each subcarrier experiences flat fading. Over the nth subcarrier, n ∈ N = {1, 2, · · · , N }, the network comprises four wireless links, i.e., the source-relay (SR) link, the relaydestination (RD) link, the LI link, and relay-eavesdropper (RE) link, whose responses are denoted as hSR (n), hRD (n), hLI (n), and hRE (n), respectively. We assume the channel state information (CSI) of the whole network is known at the source and the FD relay. This assumption is valid when the eavesdropper is active [5]. The active eavesdropper may

IEEE COMMUNICATIONS LETTERS

LI Link

2

where [x]+ denotes max{x, 0} for a real random variable x and

RD Link

SR Link

Destination Source

FD Relay

RE Link

γSR (n) = |hSR (n)|2 /σR2 (n),

(6)

|hRD (n)| /σD2 (n), ˜ LI (n)|2 /σ 2 (n), |h R |hRE (n)|2 /σE2 (n).

(7)

γRD (n) = γLI (n) = Eavesdropper

Fig. 1. The system model of multicarrier secure communications in the FD relay network.

register in the network as a subscribed user [12]. The active eavesdropper may also act as either (both) a jammer or (and) a classical eavesdropper [13]. Furthermore, even for a passive eavesdropper, there is a possibility for one to estimate the CSI through the local oscillator power inadvertently leaked from the eavesdropper’s receiver radio frequency frontend [14]. Unlike the conventional HD relay, the FD relay receives and forwards simultaneously on the same frequency. Over the nth subcarrier, denoting the transmitted signals from the source and the FD relay as xS (n) and xR (n), respectively, the received signal at the relay is expressed as yR (n) = hSR (n)xS (n) + hLI (n)xR (n) + nR (n)

(1)

where nR (n) ∼ CN (0, σR2 (n)) denotes the additive Gaussian noise at the relay. In (1), hSR (n)xS (n) is the desired signal and hLI (n)xR (n) is the LI. In practice, the LI is much stronger than the desired signal [3]. Thus, the LI cancelation scheme [1] should be employed, including estimation of the LI link and substraction of a replicated interference. However, it is difficult to eliminate the LI because of the non-ideal channel estimation and signal processing [1]. After the imperfect LI cancelation, the received signal at the relay is ˜ LI (n)xR (n) + nR (n) y˜R (n) = hSR (n)xS (n) + h

(2)

˜ LI (n)xR (n) denotes the residual LI over the nth where h ˜ LI (n) denotes the response of residual LI link subcarrier and h over the nth subcarrier. The received signals at the legitimate destination and the eavesdropper are yD (n) = hRD (n)xR (n) + nD (n), yE (n) = hRE (n)xR (n) + nE (n),

(3) (4)

respectively, where nD (n) ∼ CN (0, σD2 (n)) and nE (n) ∼ CN (0, σE2 (n)) denote the additive Gaussian noises at the legitimate destination and the eavesdropper, respectively. Denoting the transmit powers of the source and the FD relay over the nth subcarrier as PS (n) = E[|xS (n)|2 ] and PR (n) = E[|xR (n)|2 ], respectively, the achievable secrecy rate over the nth subcarrier is given by [8] R(n) (PS (n), PR (n)) = { } + PS (n)γSR (n) 1 + min 1+P , P (n)γ (n) R RD R (n)γLI (n) log2 (5) 1 + PR (n)γRE (n)

γRE (n) =

2

(8) (9)

It is noted that in (5), it is assumed that the FD relay is only responsible for symbol-by-symbol decoding and forwarding as in [8], [15]. The source rather than the FD relay encodes the secrete information into symbols. Only the legitimate destination performs full decoding [8], [15]. In this letter, our objective is to derive the optimal power allocation scheme which maximizes the achievable secrecy rate of the FD relay network subject to the sum power constraint of the source and the FD relay. The optimization problem is formulated as max

N ∑

{PS (n),PR (n),n∈N }

s.t.

n=1 N ∑

R(n) (PS (n), PR (n)) (PS (n) + PR (n)) ≤ PT

(10)

n=1

where PT is the sum power constraint of the source and the FD relay. The problem (10) is non-convex because of its nonconvex objective function. In the following, we will transform the problem (10) into a convex problem, which can be solved efficiently using the KKT conditions [16]. III. ACHIEVABLE S ECRECY R ATE M AXIMIZATION It is noted that if γRD (n) ≤ γRE (n), we have { } PS (n)γSR (n) min , PR (n)γRD (n) ≤ PR (n)γRE (n) 1 + PR (n)γLI (n) (11) which indicates that the achievable secrecy rate R(n) (PS (n), PR (n)) over the nth subcarrier is zero. Thus, we should not allocate any transmit power over the nth subcarrier, i.e., PS (n) = PR (n) = 0. Denote Ω to be Ω = {n|γRD (n) > γRE (n), n ∈ N }.

(12)

We recast the problem (10) as follows ∑ max R(n) (PS (n), PR (n)) {PS (n),PR (n),n∈Ω} n∈Ω

s.t.

∑

(PS (n) + PR (n)) ≤ PT .

(13)

n∈Ω

For a given P (n) where P (n) = PS (n) + PR (n), we have the following lemma. Lemma 1: Over the nth subcarrier, n ∈ Ω, given P (n), the optimal transmit power allocation at the source and the FD relay, denoted as PSo (n) and PRo (n), respectively, should satisfy PSo (n)γSR (n) = PRo (n)γRD (n). (14) 1 + PRo (n)γLI (n) Proof : We prove Lemma 1 by reduction ad absurdum. Assume that given P (n), the optimal transmit power allocation

IEEE COMMUNICATIONS LETTERS

3

at the source and the FD relay, denoted as PS∗ (n) and PR∗ (n), which do not satisfy (14). Thus, we consider the following two conditions, PS∗ (n)γSR (n) < PR∗ (n)γRD (n), 1 + PR∗ (n)γLI (n) PS∗ (n)γSR (n) > PR∗ (n)γRD (n). 1 + PR∗ (n)γLI (n)

(15) (16)

where R(n) (PR∗ (n)) is a monotone decreasing function on PR∗ (n). From (14) and (15), we have PR∗ (n) > PRo (n). Therefore, R(n) (PR∗ (n)) < R(n) (PRo (n)), which is contradictory with the assumption that PS∗ (n) and PR∗ (n) are optimal. If we have the condition (16), the achievable secrecy rate over the nth subcarrier is ) ( 1 + PR∗ (n)γRD (n) R(n) (PR∗ (n)) = log2 (18) 1 + PR∗ (n)γRE (n) where R(n) (PR∗ (n)) is a monotone increasing function on PR∗ (n) because γRD (n) > γRE (n) for n ∈ Ω. From (14) and (16), we have PR∗ (n) < PRo (n). Therefore, R(n) (PR∗ (n)) < R(n) (PRo (n)), which is contradictory with the assumption that PS∗ (n) and PR∗ (n) are optimal. From Lemma 1, over the nth subcarrier, n ∈ Ω, given P (n), if PS (n) and PR (n) are optimal transmit power allocation at the source and the FD relay, we have γLI (n)γRD (n) 2 γRD (n) PR (n) + PR (n). γSR (n) γSR (n)

(19)

Substituting (19) into problem (13), we equivalently transform the problem (13) into ∑ ˆ max R(n) s.t. ξ ≤ PT (20) {PR (n),n∈Ω}

n∈Ω

where

( ) 1 + PR (n)γRD (n) ˆ R(n) = log2 , (21) 1 + PR (n)γRE (n) [ ( ) ] ∑ γLI (n)γRD (n) γRD (n) PR2 (n) + 1 + PR (n) . ξ= γSR (n) γSR (n) n∈Ω

(22) It is noted that the objective function of the problem (20) is concave and the constraint of the problem (20) is convex. Thus, the optimization problem (20) is convex, which can be solved by the KKT conditions. The Lagrangian associated with the problem (20) is ∑ ˆ L(PR (n), λ) = R(n) − λ (ξ − PT )

γRD (n) γRE (n) ∂L = − ∂PR (n) (1 + PR (n)γRD (n)) ln 2 (1 + PR (n)γRE (n)) ln 2 ( ) 2λγLI (n)γRD (n) γRD (n) − PR (n) − λ +1 =0 (24) γSR (n) γSR (n) and λ(ξ − PT ) = 0.

If we have the condition (15), the achievable secrecy rate over the nth subcarrier is ∗ (P (n)−PR (n))γSR (n) 1 + 1+P ∗ (n)γ LI (n) R R(n) (PR∗ (n)) = log2 (17) ∗ 1 + PR (n)γRE (n)

PS (n) =

associated with (20) are given by

(23)

n∈Ω

where λ ≥ 0 is the Lagrange multiplier. The KKT conditions

(25)

We have the following lemma. Lemma 2: The solution to (20) should satisfy ξ = PT . Proof : From (24)-(25), if λ = 0, we have 1/(ln 2) 1/(ln 2) ∂L = − . (26) ∂PR (n) 1/γRD (n) + PR (n) 1/γRE (n) + PR (n) Because γRD (n) > γRE (n) for n ∈ Ω, we have ∂P∂L(n) > 0. It R is contradictory with the KKT condition (24). Thus, we have λ > 0. From (25), the solution to the problem (20) should satisfy ξ = PT . It is noted that (24) can be transformed into a standard cubic equation on PR (n), PR3 (n) + aPR2 (n) + bPR (n) + c = 0

(27)

where a =(λγRD (n) ln 2/d) · [2γLI (n) · (γRD (n) + γRE (n)) + γRE (n) · (γSR (n) + γRD (n))], b =(λ ln 2/d) · [2γLI (n)γRD (n)

(28)

+ (γSR (n) + γRD (n)) · (γRD (n) + γRE (n))], c =(1/d) · [λ(γSR (n) + γRD (n)) ln 2 + γSR (n)(γRE (n) − γRD (n))],

(29)

2 d =2λγLI (n)γRD (n)γRE (n) ln 2.

(31)

(30)

The real closed-form solution to (27), given by Cardano’s formula [17], is √ √ PRo (n) = ej∠y1 3 |y1 | + ej∠y2 3 |y2 | − a/3 (32) where ∠x denotes the phase angle of an complex random variable x and √ √ y1 = −q/2 + ∆, y2 = −q/2 − ∆, (33) p = −a2 /3 + b, q = 2a3 /27 − ab/3 + c, 3

2

∆ = p /27 + q /4.

(34) (35)

It is noted that the closed-form solution (32) includes an unknown parameter λ which is determined by substituting (32) into ξ − PT = 0 and solving it. IV. S IMULATION R ESULTS In this section, we evaluate the performance of proposed optimal power allocation scheme for the FD relay network through computer simulations. In the FD relay network, the total bandwidth is divided into N = 16 orthogonal subcarriers. Each response of wireless links over the N subcarriers except the residual LI link, i.e., hSR (n), hRD (n), and hRE (n), n ∈ N , is independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance.

IEEE COMMUNICATIONS LETTERS

4

0

12

10 FD Opt, τ/σ2=0 dB FD Opt, τ/σ2=10 dB HD Opt

−1

8

Packet Error Rate

Achievable Secrecy Rate (bps/Hz)

10

6

4

10

−2

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FD Uniform, τ/σ2=10 dB FD Uniform, τ/σ2=0 dB FD Opt, τ/σ2=10 dB

2

FD Opt, τ/σ2=0 dB 0

−3

5

10

15

20

25

30

2

PT/σ (dB)

Fig. 2. Achievable secrecy rate versus PT /σ 2 , performance comparison of the proposed optimal power allocation scheme for FD relaying and that for conventional HD relaying proposed in [8].

The response of the residual LI link over the nth subcarrier, ˜ LI (n), is given by h ˜ LI (n) ∼ CN (0, τ ), n ∈ N , where i.e., h τ denotes the variance of the residual LI link response. The noise variances at the relay, the legitimate destination and the eavesdropper are σR2 (n) = σD2 (n) = σE2 (n) = σ 2 , n ∈ N . In Fig. 2, we present the achievable secrecy rate comparison of the proposed optimal power allocation scheme for FD relaying (denoted as “FD Opt” in the legend) and that for conventional HD relaying proposed in [8] (denoted as “HD Opt” in the legend). The variance of the residual LI link response to noise power ratio, τ /σ 2 is 0 dB or 10 dB. The sum transmit power constraint to noise power ratio, PT /σ 2 , sweeps from 5 dB to 30 dB. From Fig. 2, it is observed that our proposed “FD Opt” scheme outperforms the “HD Opt” scheme. From Fig. 2, it is also found that with the larger τ , the proposed “FD Opt” scheme performs worse. In Fig. 3, we present the packet error rate comparison of the proposed optimal power allocation scheme for FD relaying and the uniform power allocation scheme for FD relaying (denoted as “FD Uniform”) where PT /σ 2 = 15 dB. In our simulations, when the achievable secrecy rate is below the minimum packet data rate, the packet is received with errors. From Fig. 3, it is found that the packet error rate of the proposed “FD Opt” scheme is much lower than that of the “FD Uniform” scheme. V. C ONCLUSIONS In this letter, considering the DF scheme, we propose the optimal power allocation scheme for multicarrier secure communication in FD relay networks. Simulation results have shown that the proposed optimal power allocation scheme for FD relaying outperforms that for conventional half-duplex relaying. It is also found that the residual LI at the FD relay has significant effect on the performance of FD relaying. R EFERENCES [1] T. Riihonen, S. Werner, and R. Wichman, “Hybrid full-duplex/half-duplex relaying with transmit power adaptation,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp. 3074-3085, Sept. 2011.

10

1

2

3 4 5 6 Minimum Packet Data Rate (bps/Hz)

7

8

Fig. 3. Packet error rate versus minimum packet data rate, performance comparison of the proposed optimal power allocation scheme and the uniform power allocation scheme for FD relaying where PT /σ 2 = 15 dB.

[2] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback selfinterference in full-duplex MIMO relays,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5983-5993, Dec. 2011. [3] M. Jain, J. I. Choi, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, “Pratical, real-time, full duplex wireless,” in Proc. of ACM MobiCom 2011, pp. 301-312. [4] J.-H. Lee, “Self-interference cancelation using phase rotation in fullduplex wireless,” IEEE Trans. Veh. Technol., vol. 62, no. 9, pp. 44214429, Nov. 2013. [5] P. K. Gopala, L. Lai, and H. El Gamal, “On the secrecy capacity of fading channels,” IEEE Trans. Inf. Theory, vol. 54, no. 10, pp. 4686-4698, Oct. 2008. [6] L. Lai and H. E. Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005-4019, Sept. 2008. [7] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1875-1888, Mar. 2010. [8] C. Jeong and I.-M. Kim, “Optimal power allocation for secure multicarrier relay systems,” IEEE Trans. Signal Process., vol. 59, no. 11, pp. 54285442, Nov. 2011. [9] D. W. K. Ng, E. S. Lo, and R. Schober, “Secure resource allocation and scheduling for OFDMA decode-and-forward relay networks,” IEEE Trans. Wireless Commun., vol. 10, no. 10, pp. 3528-3540, Oct. 2011. [10] R. Bassily and S. Ulukus, “Secure communication in multiple relay networks through decode-and-forward strategies,” J. Commun. Netw., vol. 14, no. 4, pp. 352-363, Aug. 2012. [11] H. Alves, G. Brante, R. D. Souza, D. B. da Costa and M. Latva-aho, “On the performance of full-duplex relaying under phy security constraints,” in Proc. IEEE ICASSP 2014, pp. 3978-3981. [12] A. Chorti, S. M. Perlaza, Z. Han, and H. V. Poor, “On the resilience of wireless multiuser networks to passive and active eavesdroppers,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1850-1863, Aug. 2012. [13] X. Zhou, B. Maham, and A. Hjorungnes, “Pilot contamination for active eavesdropping,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 903907, Mar. 2012. [14] A. Mukherjee and A. L. Swindlehurst, “Detecting passive eavesdroppers in the MIMO wiretap channel,” in Proc. ICASSP 2012, pp. 2809-2812. [15] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [16] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [17] W. Dunham, “Cardano and the solution of the cubic,” Ch. 6 in Journey through Genius: The Great Theorems of Mathematics, pp. 133-154. John Wiley & Sons, Inc., 1990.

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Optimal Power Allocation for Multicarrier Secure Communications in Full-Duplex Decode-and-Forward Relay Networks Xiaobin Huang, Jing He, Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin

Abstract—For full-duplex (FD) relay networks, secure communication is an important issue. In this letter, considering decodeand-forward scheme, we investigate the optimal power allocation for multicarrier secure communication in FD relay networks. Our objective is to maximize the achievable secrecy rate of FD relay network subject to sum power constraint of the source and FD relay. The formulated non-convex optimization problem is equivalently transformed into a convex problem. Employing Karush-Kuhn-Tucker conditions, we obtain closed-form solution to the optimization problem. Simulation results have shown that the proposed optimal power allocation scheme for FD relaying outperforms that for conventional half-duplex relaying. Index Terms—Achievable secrecy rate, decode-and-forward (DF), full-duplex (FD), power allocation, relay network.

I. I NTRODUCTION N relay networks, the half-duplex (HD) relaying and the full-duplex (FD) relaying are two relaying modes [1]. In the conventional relay networks, the HD relaying is usually employed where the relay receives and transmits signals over two orthogonal frequency slots or time slots. Employing the FD relaying, the relay receives and transmits signals simultaneously on the same frequency, which potentially achieves the higher information rate than the conventional HD relaying. The main challenge to employ the FD relaying is the loopinterference (LI), which is that the transmitted signals from the FD relay may interfere the received signals at the relay. In [2], several LI mitigation schemes were proposed, such as natural isolation, time-domain cancelation, and spatial suppression. In [3], [4], the FD transmission schemes for the orthogonal frequency division multiplexing modulated (OFDM) signals were investigated. The aforementioned works do not consider the secure communications. The wireless information is susceptible to eavesdropping because of the openness of wireless transmission medium [5]. Thus, the secure communication is a critical issue for the relay networks. In [6], [7], the secure transmission schemes for the conventional HD relay networks were proposed. In [8], [9], the multicarrier secure communication for the OFDM signals in the HD relay networks was studied.

I

This work was supported in part by the National Natural Science Foundation of China under Grant 61472458, Grant 61202498 and Grant 61173148. X. Huang, J. He, Q. Zhang, and J. Qin are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail: [email protected], [email protected], {zhqi26, issqjy}@mail.sysu.edu.cn). Q. Li is with the School of Advanced Computing, Sun Yat-Sen University, Guangzhou 510006, Guangdong, China (e-mail: [email protected]).

In [10], [11], the secure communication for the FD relay networks was investigated. To the best of our knowledge, the research on the multicarrier secure communication for the OFDM signals in the FD relay networks is missing. In this letter, considering the decode-and-forward (DF) scheme, we investigate the optimal power allocation for multicarrier secure communication in the FD relay networks. Our objective is to maximize the achievable secrecy rate of the FD relay network subject to the sum power constraint of the source and the FD relay. The formulated optimization problem is non-convex because of the non-convex objective function. We equivalently transform the non-convex optimization problem into a convex problem, which is solved by the Karush-KuhnTucker (KKT) conditions. After derivations, we obtain the closed-form solution to the optimization problem. II. S YSTEM M ODEL Consider an FD relay network, which consists of one source, one FD relay, one legitimate destination, one eavesdropper, as shown in Fig. 1. Each node except the relay is equipped with a single antenna. The FD relay is equipped with a single transmit antenna and a single receive antenna which are isolated [1]. We assume that there is no direct link between the source and the legitimate destination. The reliable communication from the source to the legitimate destination is established by the FD relay. The FD relay employs the DF scheme to forward signals. When the FD relay forwards signals to the legitimate destination, the eavesdropper tries to overhear the signals from the relay. In this letter, we assume that the eavesdropper cannot overhear the signals from the source. This is reasonable because if the eavesdropper is close to the FD relay, since no LI cancelation scheme is employed, the signals from the source are much weaker than those from the FD relay. In the FD relay network, the total bandwidth is divided into N orthogonal subcarriers. The source and the FD relay transmit the OFDM signals in the frequency-selective channels where each subcarrier experiences flat fading. Over the nth subcarrier, n ∈ N = {1, 2, · · · , N }, the network comprises four wireless links, i.e., the source-relay (SR) link, the relaydestination (RD) link, the LI link, and relay-eavesdropper (RE) link, whose responses are denoted as hSR (n), hRD (n), hLI (n), and hRE (n), respectively. We assume the channel state information (CSI) of the whole network is known at the source and the FD relay. This assumption is valid when the eavesdropper is active [5]. The active eavesdropper may

IEEE COMMUNICATIONS LETTERS

LI Link

2

where [x]+ denotes max{x, 0} for a real random variable x and

RD Link

SR Link

Destination Source

FD Relay

RE Link

γSR (n) = |hSR (n)|2 /σR2 (n),

(6)

|hRD (n)| /σD2 (n), ˜ LI (n)|2 /σ 2 (n), |h R |hRE (n)|2 /σE2 (n).

(7)

γRD (n) = γLI (n) = Eavesdropper

Fig. 1. The system model of multicarrier secure communications in the FD relay network.

register in the network as a subscribed user [12]. The active eavesdropper may also act as either (both) a jammer or (and) a classical eavesdropper [13]. Furthermore, even for a passive eavesdropper, there is a possibility for one to estimate the CSI through the local oscillator power inadvertently leaked from the eavesdropper’s receiver radio frequency frontend [14]. Unlike the conventional HD relay, the FD relay receives and forwards simultaneously on the same frequency. Over the nth subcarrier, denoting the transmitted signals from the source and the FD relay as xS (n) and xR (n), respectively, the received signal at the relay is expressed as yR (n) = hSR (n)xS (n) + hLI (n)xR (n) + nR (n)

(1)

where nR (n) ∼ CN (0, σR2 (n)) denotes the additive Gaussian noise at the relay. In (1), hSR (n)xS (n) is the desired signal and hLI (n)xR (n) is the LI. In practice, the LI is much stronger than the desired signal [3]. Thus, the LI cancelation scheme [1] should be employed, including estimation of the LI link and substraction of a replicated interference. However, it is difficult to eliminate the LI because of the non-ideal channel estimation and signal processing [1]. After the imperfect LI cancelation, the received signal at the relay is ˜ LI (n)xR (n) + nR (n) y˜R (n) = hSR (n)xS (n) + h

(2)

˜ LI (n)xR (n) denotes the residual LI over the nth where h ˜ LI (n) denotes the response of residual LI link subcarrier and h over the nth subcarrier. The received signals at the legitimate destination and the eavesdropper are yD (n) = hRD (n)xR (n) + nD (n), yE (n) = hRE (n)xR (n) + nE (n),

(3) (4)

respectively, where nD (n) ∼ CN (0, σD2 (n)) and nE (n) ∼ CN (0, σE2 (n)) denote the additive Gaussian noises at the legitimate destination and the eavesdropper, respectively. Denoting the transmit powers of the source and the FD relay over the nth subcarrier as PS (n) = E[|xS (n)|2 ] and PR (n) = E[|xR (n)|2 ], respectively, the achievable secrecy rate over the nth subcarrier is given by [8] R(n) (PS (n), PR (n)) = { } + PS (n)γSR (n) 1 + min 1+P , P (n)γ (n) R RD R (n)γLI (n) log2 (5) 1 + PR (n)γRE (n)

γRE (n) =

2

(8) (9)

It is noted that in (5), it is assumed that the FD relay is only responsible for symbol-by-symbol decoding and forwarding as in [8], [15]. The source rather than the FD relay encodes the secrete information into symbols. Only the legitimate destination performs full decoding [8], [15]. In this letter, our objective is to derive the optimal power allocation scheme which maximizes the achievable secrecy rate of the FD relay network subject to the sum power constraint of the source and the FD relay. The optimization problem is formulated as max

N ∑

{PS (n),PR (n),n∈N }

s.t.

n=1 N ∑

R(n) (PS (n), PR (n)) (PS (n) + PR (n)) ≤ PT

(10)

n=1

where PT is the sum power constraint of the source and the FD relay. The problem (10) is non-convex because of its nonconvex objective function. In the following, we will transform the problem (10) into a convex problem, which can be solved efficiently using the KKT conditions [16]. III. ACHIEVABLE S ECRECY R ATE M AXIMIZATION It is noted that if γRD (n) ≤ γRE (n), we have { } PS (n)γSR (n) min , PR (n)γRD (n) ≤ PR (n)γRE (n) 1 + PR (n)γLI (n) (11) which indicates that the achievable secrecy rate R(n) (PS (n), PR (n)) over the nth subcarrier is zero. Thus, we should not allocate any transmit power over the nth subcarrier, i.e., PS (n) = PR (n) = 0. Denote Ω to be Ω = {n|γRD (n) > γRE (n), n ∈ N }.

(12)

We recast the problem (10) as follows ∑ max R(n) (PS (n), PR (n)) {PS (n),PR (n),n∈Ω} n∈Ω

s.t.

∑

(PS (n) + PR (n)) ≤ PT .

(13)

n∈Ω

For a given P (n) where P (n) = PS (n) + PR (n), we have the following lemma. Lemma 1: Over the nth subcarrier, n ∈ Ω, given P (n), the optimal transmit power allocation at the source and the FD relay, denoted as PSo (n) and PRo (n), respectively, should satisfy PSo (n)γSR (n) = PRo (n)γRD (n). (14) 1 + PRo (n)γLI (n) Proof : We prove Lemma 1 by reduction ad absurdum. Assume that given P (n), the optimal transmit power allocation

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at the source and the FD relay, denoted as PS∗ (n) and PR∗ (n), which do not satisfy (14). Thus, we consider the following two conditions, PS∗ (n)γSR (n) < PR∗ (n)γRD (n), 1 + PR∗ (n)γLI (n) PS∗ (n)γSR (n) > PR∗ (n)γRD (n). 1 + PR∗ (n)γLI (n)

(15) (16)

where R(n) (PR∗ (n)) is a monotone decreasing function on PR∗ (n). From (14) and (15), we have PR∗ (n) > PRo (n). Therefore, R(n) (PR∗ (n)) < R(n) (PRo (n)), which is contradictory with the assumption that PS∗ (n) and PR∗ (n) are optimal. If we have the condition (16), the achievable secrecy rate over the nth subcarrier is ) ( 1 + PR∗ (n)γRD (n) R(n) (PR∗ (n)) = log2 (18) 1 + PR∗ (n)γRE (n) where R(n) (PR∗ (n)) is a monotone increasing function on PR∗ (n) because γRD (n) > γRE (n) for n ∈ Ω. From (14) and (16), we have PR∗ (n) < PRo (n). Therefore, R(n) (PR∗ (n)) < R(n) (PRo (n)), which is contradictory with the assumption that PS∗ (n) and PR∗ (n) are optimal. From Lemma 1, over the nth subcarrier, n ∈ Ω, given P (n), if PS (n) and PR (n) are optimal transmit power allocation at the source and the FD relay, we have γLI (n)γRD (n) 2 γRD (n) PR (n) + PR (n). γSR (n) γSR (n)

(19)

Substituting (19) into problem (13), we equivalently transform the problem (13) into ∑ ˆ max R(n) s.t. ξ ≤ PT (20) {PR (n),n∈Ω}

n∈Ω

where

( ) 1 + PR (n)γRD (n) ˆ R(n) = log2 , (21) 1 + PR (n)γRE (n) [ ( ) ] ∑ γLI (n)γRD (n) γRD (n) PR2 (n) + 1 + PR (n) . ξ= γSR (n) γSR (n) n∈Ω

(22) It is noted that the objective function of the problem (20) is concave and the constraint of the problem (20) is convex. Thus, the optimization problem (20) is convex, which can be solved by the KKT conditions. The Lagrangian associated with the problem (20) is ∑ ˆ L(PR (n), λ) = R(n) − λ (ξ − PT )

γRD (n) γRE (n) ∂L = − ∂PR (n) (1 + PR (n)γRD (n)) ln 2 (1 + PR (n)γRE (n)) ln 2 ( ) 2λγLI (n)γRD (n) γRD (n) − PR (n) − λ +1 =0 (24) γSR (n) γSR (n) and λ(ξ − PT ) = 0.

If we have the condition (15), the achievable secrecy rate over the nth subcarrier is ∗ (P (n)−PR (n))γSR (n) 1 + 1+P ∗ (n)γ LI (n) R R(n) (PR∗ (n)) = log2 (17) ∗ 1 + PR (n)γRE (n)

PS (n) =

associated with (20) are given by

(23)

n∈Ω

where λ ≥ 0 is the Lagrange multiplier. The KKT conditions

(25)

We have the following lemma. Lemma 2: The solution to (20) should satisfy ξ = PT . Proof : From (24)-(25), if λ = 0, we have 1/(ln 2) 1/(ln 2) ∂L = − . (26) ∂PR (n) 1/γRD (n) + PR (n) 1/γRE (n) + PR (n) Because γRD (n) > γRE (n) for n ∈ Ω, we have ∂P∂L(n) > 0. It R is contradictory with the KKT condition (24). Thus, we have λ > 0. From (25), the solution to the problem (20) should satisfy ξ = PT . It is noted that (24) can be transformed into a standard cubic equation on PR (n), PR3 (n) + aPR2 (n) + bPR (n) + c = 0

(27)

where a =(λγRD (n) ln 2/d) · [2γLI (n) · (γRD (n) + γRE (n)) + γRE (n) · (γSR (n) + γRD (n))], b =(λ ln 2/d) · [2γLI (n)γRD (n)

(28)

+ (γSR (n) + γRD (n)) · (γRD (n) + γRE (n))], c =(1/d) · [λ(γSR (n) + γRD (n)) ln 2 + γSR (n)(γRE (n) − γRD (n))],

(29)

2 d =2λγLI (n)γRD (n)γRE (n) ln 2.

(31)

(30)

The real closed-form solution to (27), given by Cardano’s formula [17], is √ √ PRo (n) = ej∠y1 3 |y1 | + ej∠y2 3 |y2 | − a/3 (32) where ∠x denotes the phase angle of an complex random variable x and √ √ y1 = −q/2 + ∆, y2 = −q/2 − ∆, (33) p = −a2 /3 + b, q = 2a3 /27 − ab/3 + c, 3

2

∆ = p /27 + q /4.

(34) (35)

It is noted that the closed-form solution (32) includes an unknown parameter λ which is determined by substituting (32) into ξ − PT = 0 and solving it. IV. S IMULATION R ESULTS In this section, we evaluate the performance of proposed optimal power allocation scheme for the FD relay network through computer simulations. In the FD relay network, the total bandwidth is divided into N = 16 orthogonal subcarriers. Each response of wireless links over the N subcarriers except the residual LI link, i.e., hSR (n), hRD (n), and hRE (n), n ∈ N , is independent and identically distributed (i.i.d.) complex Gaussian random variable with zero mean and unit variance.

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0

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10 FD Opt, τ/σ2=0 dB FD Opt, τ/σ2=10 dB HD Opt

−1

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Packet Error Rate

Achievable Secrecy Rate (bps/Hz)

10

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FD Uniform, τ/σ2=10 dB FD Uniform, τ/σ2=0 dB FD Opt, τ/σ2=10 dB

2

FD Opt, τ/σ2=0 dB 0

−3

5

10

15

20

25

30

2

PT/σ (dB)

Fig. 2. Achievable secrecy rate versus PT /σ 2 , performance comparison of the proposed optimal power allocation scheme for FD relaying and that for conventional HD relaying proposed in [8].

The response of the residual LI link over the nth subcarrier, ˜ LI (n), is given by h ˜ LI (n) ∼ CN (0, τ ), n ∈ N , where i.e., h τ denotes the variance of the residual LI link response. The noise variances at the relay, the legitimate destination and the eavesdropper are σR2 (n) = σD2 (n) = σE2 (n) = σ 2 , n ∈ N . In Fig. 2, we present the achievable secrecy rate comparison of the proposed optimal power allocation scheme for FD relaying (denoted as “FD Opt” in the legend) and that for conventional HD relaying proposed in [8] (denoted as “HD Opt” in the legend). The variance of the residual LI link response to noise power ratio, τ /σ 2 is 0 dB or 10 dB. The sum transmit power constraint to noise power ratio, PT /σ 2 , sweeps from 5 dB to 30 dB. From Fig. 2, it is observed that our proposed “FD Opt” scheme outperforms the “HD Opt” scheme. From Fig. 2, it is also found that with the larger τ , the proposed “FD Opt” scheme performs worse. In Fig. 3, we present the packet error rate comparison of the proposed optimal power allocation scheme for FD relaying and the uniform power allocation scheme for FD relaying (denoted as “FD Uniform”) where PT /σ 2 = 15 dB. In our simulations, when the achievable secrecy rate is below the minimum packet data rate, the packet is received with errors. From Fig. 3, it is found that the packet error rate of the proposed “FD Opt” scheme is much lower than that of the “FD Uniform” scheme. V. C ONCLUSIONS In this letter, considering the DF scheme, we propose the optimal power allocation scheme for multicarrier secure communication in FD relay networks. Simulation results have shown that the proposed optimal power allocation scheme for FD relaying outperforms that for conventional half-duplex relaying. It is also found that the residual LI at the FD relay has significant effect on the performance of FD relaying. R EFERENCES [1] T. Riihonen, S. Werner, and R. Wichman, “Hybrid full-duplex/half-duplex relaying with transmit power adaptation,” IEEE Trans. Wireless Commun., vol. 10, no. 9, pp. 3074-3085, Sept. 2011.

10

1

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3 4 5 6 Minimum Packet Data Rate (bps/Hz)

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Fig. 3. Packet error rate versus minimum packet data rate, performance comparison of the proposed optimal power allocation scheme and the uniform power allocation scheme for FD relaying where PT /σ 2 = 15 dB.

[2] T. Riihonen, S. Werner, and R. Wichman, “Mitigation of loopback selfinterference in full-duplex MIMO relays,” IEEE Trans. Signal Process., vol. 59, no. 12, pp. 5983-5993, Dec. 2011. [3] M. Jain, J. I. Choi, D. Bharadia, S. Seth, K. Srinivasan, P. Levis, S. Katti, and P. Sinha, “Pratical, real-time, full duplex wireless,” in Proc. of ACM MobiCom 2011, pp. 301-312. [4] J.-H. Lee, “Self-interference cancelation using phase rotation in fullduplex wireless,” IEEE Trans. Veh. Technol., vol. 62, no. 9, pp. 44214429, Nov. 2013. [5] P. K. Gopala, L. Lai, and H. El Gamal, “On the secrecy capacity of fading channels,” IEEE Trans. Inf. Theory, vol. 54, no. 10, pp. 4686-4698, Oct. 2008. [6] L. Lai and H. E. Gamal, “The relay-eavesdropper channel: Cooperation for secrecy,” IEEE Trans. Inf. Theory, vol. 54, no. 9, pp. 4005-4019, Sept. 2008. [7] L. Dong, Z. Han, A. P. Petropulu, and H. V. Poor, “Improving wireless physical layer security via cooperating relays,” IEEE Trans. Signal Process., vol. 58, no. 3, pp. 1875-1888, Mar. 2010. [8] C. Jeong and I.-M. Kim, “Optimal power allocation for secure multicarrier relay systems,” IEEE Trans. Signal Process., vol. 59, no. 11, pp. 54285442, Nov. 2011. [9] D. W. K. Ng, E. S. Lo, and R. Schober, “Secure resource allocation and scheduling for OFDMA decode-and-forward relay networks,” IEEE Trans. Wireless Commun., vol. 10, no. 10, pp. 3528-3540, Oct. 2011. [10] R. Bassily and S. Ulukus, “Secure communication in multiple relay networks through decode-and-forward strategies,” J. Commun. Netw., vol. 14, no. 4, pp. 352-363, Aug. 2012. [11] H. Alves, G. Brante, R. D. Souza, D. B. da Costa and M. Latva-aho, “On the performance of full-duplex relaying under phy security constraints,” in Proc. IEEE ICASSP 2014, pp. 3978-3981. [12] A. Chorti, S. M. Perlaza, Z. Han, and H. V. Poor, “On the resilience of wireless multiuser networks to passive and active eavesdroppers,” IEEE J. Sel. Areas Commun., vol. 31, no. 9, pp. 1850-1863, Aug. 2012. [13] X. Zhou, B. Maham, and A. Hjorungnes, “Pilot contamination for active eavesdropping,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 903907, Mar. 2012. [14] A. Mukherjee and A. L. Swindlehurst, “Detecting passive eavesdroppers in the MIMO wiretap channel,” in Proc. ICASSP 2012, pp. 2809-2812. [15] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062-3080, Dec. 2004. [16] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [17] W. Dunham, “Cardano and the solution of the cubic,” Ch. 6 in Journey through Genius: The Great Theorems of Mathematics, pp. 133-154. John Wiley & Sons, Inc., 1990.