Robust Secure Beamforming in MISO Full-Duplex Two ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TVT.2015.2394370, IEEE Transactions on Vehicular Technology IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY

1

Robust Secure Beamforming in MISO Full-Duplex Two-Way Secure Communications Renhai Feng, Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin

Abstract—Considering worst-case channel uncertainties, we investigate the robust secure beamforming design problem in multiple-input-single-output full-duplex two-way secure communications. Our objective is to maximize worst-case sum secrecy rate under weak secrecy conditions and individual transmit power constraints. Since the objective function of the optimization problem includes both convex and concave terms, we propose to transform convex terms into linear terms. We decouple the problem into four optimization problems and employ alternating optimization algorithm to obtain the locally optimal solution. Simulation results demonstrate that our proposed robust secure beamforming scheme outperforms the non-robust one. It is also found that when the regions of channel uncertainties and the individual transmit power constraints are sufficiently large, because of self-interference, the proposed two-way robust secure communication is proactively degraded to one-way communication. Index Terms—Full-duplex, multiple-input-single-output (MISO), secure communications, two-way, worst-case model.

I. I NTRODUCTION N full-duplex communications, a transceiver is able to simultaneously transmit and receive signals on the same frequency band by employing advanced technologies on antenna, electronics, and signal processing [1]–[4]. Because the transmit and receive antennas are close to one another, selfinterference is high. Thus, the main challenge of full-duplex communication systems is to suppress self-interference. Different self-interference suppression schemes, such as antenna isolation, time cancelation and spatial precoding, have been extensively investigated in the literature [1]–[4]. Because of openness of wireless transmission medium, wireless information is susceptible to eavesdropping [5], [6]. In [7]–[10], secure communication for two-way relay networks was investigated. Secure communication is also a critical issue for full-duplex communications. In [11], the one-way secure communication scheme was investigated where a full-duplex multi-antenna destination simultaneously acts as a cooperative

I

Copyright (c) 2013 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. This work was supported in part by the National Natural Science Foundation of China under Grants 61472458, 61202498, and 61173148, and in part by the Fundamental Research Funds for the Central Universities (15lgzd10 and 15lgpy15). R. Feng, 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], {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: liquanzhong2009@ gmail.com). R. Feng is also with the School of Information Engineering, Shenzhen University, Shenzhen 518060, Guangdong, China.

jammer and an information receiver. In [11], perfect channel state information (CSI) was considered. In practice, it is difficult to obtain perfect CSI because of channel estimation and quantization errors. The robust secrecy transmission in multiple-input-single-output (MISO) half-duplex one-way secure communication systems was extensively studied in the literature [12], [13] where the channel uncertainties are modeled by the worst-case model. In [14], considering worst-case channel uncertainties, the robust transmit beamforming design in MIMO full-duplex two-way communications systems was proposed where the existence of eavesdropper is not considered. In [15], Vishwakarma et al. studied the robust secure beamforming design in single-input-single-output (SISO) fullduplex two-way secure communication systems where the worst-case sum secrecy rate is maximized. However, to the best of our knowledge, the research on robust secure beamforming in MISO full-duplex two-way secure communication systems is missing. In this correspondence, considering worst-case channel uncertainties, we investigate the robust secure beamforming design problem in MISO full-duplex two-way secure communication systems where two full-duplex sources, with multiple transmit antennas and single receive antenna, exchange information with the existence of a single-antenna eavesdropper. The aforementioned scenario may be found in future wireless secure communications between vehicles and roadside infrastructure networks. This is because the short range wireless communication is suitable for full duplex communication since the self-interference is relatively weak. Furthermore, the vehicular network is a typical ad-hoc network, which suffers security problems. Here, our objective is to maximize worst-case sum secrecy rate under weak secrecy conditions and individual transmit power constraints. Since the objective function of the optimization problem includes both convex and concave terms, we propose to transform convex terms into linear terms. We decouple the problem into four optimization problems and employ alternating optimization algorithm to obtain the locally optimal solution. Furthermore, since rank relaxation is used to solve the aforementioned problem, we prove that the obtained solution is rank-one through analyzing Karush-Kuhn-Tucker (KKT) conditions. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The conjugate transpose, trace and rank of the matrix A are denoted as A† , tr(A), and rank(A), respectively. By A ≽ 0 or A ≻ 0, we mean that A is positive semidefinite or positive definite, respectively. CN (0, σ 2 ) denotes the distribution of a circularly symmetric complex Gaussian vector with mean 0 and variance σ 2 .

0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.


S1

2

S2 h21

h12

h11

h22 …

…

N

noises at S1 , S2 , and E, respectively. In this correspondence, we assume that the natural isolation self-interference scheme proposed in [1] is employed such that the receiving radio frequency (RF) chain is not saturated. Substituting (1) into (3)-(5), we know that y1 and y2 include ˆ 11 x1 and h ˆ 22 x2 , respectively. the self-interference terms h Since the sources, S1 and S2 , know their own transmitted signal vectors, they can subtract the resulting self-interference terms from the received signals [1]. Thus, the remaining received signals at S1 and S2 are

h1E

N

h2E E

y˜1 = ∆h†11 x1 + h†21 x2 + n1 ,

Fig. 1. The system model of the full-duplex two-way secure communication system.

II. S YSTEM M ODEL Consider a full-duplex two-way secure communication system, as shown in Fig. 1, which consists of two full-duplex sources, S1 and S2 , and an eavesdropper, E. The two fullduplex sources are equipped with N + 1 antennas, including N transmit antennas and one receive antenna. The eavesdropper is equipped with a single antenna. Denote the channel responses from the source, S1 , to itself and the source, S2 , as h11 ∈ CN ×1 and h12 ∈ CN ×1 , respectively. Denote the channel responses from S2 to itself and S1 as h22 ∈ CN ×1 and h21 ∈ CN ×1 , respectively. Denote the channel responses from S1 and S2 to an eavesdropper, E, as h1E ∈ CN ×1 and h2E ∈ CN ×1 , respectively. We assume that the sources only have imperfect CSI on hij , i ∈ {1, 2}, j ∈ {1, 2, E}. This assumption is valid when the eavesdropper is active [16]. The active eavesdropper may register in the network as a subscribed user [17]. The active eavesdropper may also act as either (both) a jammer or (and) a classical eavesdropper [18]. 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 [19]. In this correspondence, we model the channel uncertainties by the worst-case model [12], [13], ˆ ij + ∆hij hij = h

(1)

ˆ ij denotes the estimate of the channel hij , ∆hij where h denotes the channel uncertainty. In (1), ∆hij is bounded by the elliptical region [12], [13] Hi = {∆hi |∆h†ij Vij ∆hij ≤ 1}

(2)

where the matrices Vij ≻ 0, assumed to be known, determine the qualities of CSI. When the two sources, S1 and S2 , transmit signal vectors x1 ∈ CN ×1 and x2 ∈ CN ×1 , respectively, the received signals at S1 , S2 , and E are y1 = h†11 x1 + h†21 x2 + n1 , y2 = yE =

h†12 x1 + h†22 x2 + n2 , h†1E x1 + h†2E x2 + nE

(3) (4) (5)

where n1 ∼ CN (0, σ12 ), n2 ∼ CN (0, σ22 ), and nE ∼ 2 CN (0, σE ) are the circularly symmetric complex Gaussian

y˜2 =

h†12 x1

+

∆h†22 x2

+ n2 ,

(6) (7)

respectively. It is noted that the terms ∆h†11 x1 and ∆h†22 x2 in (6)-(7) include the self-interference which cannot be eliminated. Thus, the mutual information rates of x2 relative to y˜1 and x1 relative to y˜2 are ) ( h†21 x2 x†2 h21 , (8) I(x2 ; y˜1 ) = log2 1 + σ12 + ∆h†11 x1 x†1 ∆h11 ) ( h†12 x1 x†1 h12 , (9) I(x1 ; y˜2 ) = log2 1 + N2 + ∆h†22 x2 x†2 ∆h22 respectively. The upper bound of the eavesdropper information rate is [5], [6] ( ) h†1E x1 x†1 h1E + h†2E x2 x†2 h2E I(x1 , x2 ; yE ) = log2 1 + . 2 σE (10) We assume that X1 = x1 x†1 and X2 = x2 x†2 . Substituting X1 and X2 into (8)-(10), we have I(x2 ; y˜1 ) = ξ1 + ξ2 ,

(11)

I(x1 ; y˜2 ) = ξ3 + ξ4 ,

(12)

I(x1 , x2 ; yE ) = −ξ5 −

2 log2 σE

(13)

where

[ ( )] ξ1 = log2 σ12 + tr(∆h11 ∆h†11 X1 ) + tr h21 h†21 X2 , [ ] ξ2 = − log2 σ12 + tr(∆h11 ∆h†11 X1 ) , )] ( [ ξ3 = log2 σ22 + tr(∆h22 ∆h†22 X2 ) + tr h12 h†12 X1 , [ ] ξ4 = − log2 σ22 + tr(∆h22 ∆h†22 X2 ) , [ ] 2 ( ) ∑ † 2 ξ5 = − log2 σE + tr hiE hiE Xi .

(14) (15) (16) (17) (18)

i=1

In this correspondence, we investigate the robust secure beamforming problem which maximizes worst-case sum secrecy rate of the MISO full-duplex two-way secure communication system under individual transmit power constraints. The optimization problem is formulated as follows max

min R

X1 ≽0,X2 ≽0 ∆hij

s.t.

tr(Xi ) ≤ P, rank(Xi ) = 1, i ∈ {1, 2}


(19)


3

(n−1)

R = I(x2 ; y˜1 ) + I(x1 ; y˜2 ) − I(x1 , x2 ; yE ).

(20)

It is noted that R defined in (20) is the sum secrecy rate under weak secrecy conditions [5], [6]. Remark: The required signaling overhead for the practical system includes the exchange of the CSI, the qualities of CSI, and the obtained robust secure beamforming vectors. III. ROBUST S ECURE B EAMFORMING In the objective function of the problem (19), since ξ1 and ξ3 are concave and ξ2 , ξ4 and ξ5 are convex, R is non-convex. We have the following proposition. Proposition 1: Let a ∈ ℜ1×1 be a positive scalar and f (a) = − lnab2 + log2 a + ln12 . We have − log2 b =

max

a∈ℜ1×1 ,a>0

f (a)

(21)

and the optimal solution to the right-hand side of (21) is a = 1b . Proof : Since f (a) is convex, the optimal solution to the (a) = 0. right hand side of (21) is obtained by letting ∂f∂a We transform ξ2 , ξ4 and ξ5 into convex optimization problems ξ2 = ξ4 = ξ5 =

max 1×1

ζ1 (a1 ),

(22)

max

ζ2 (a2 ),

(23)

max

ζ2 (a3 )

(24)

a1 ∈ℜ

,a1 >0

a2 ∈ℜ1×1 ,a2 >0 a3 ∈ℜ1×1 ,a3 >0

where

) ai ( 2 1 ζi (ai ) = − σi + tr(∆hii ∆h†ii Xi ) + log2 ai + , ln 2 ln 2 i ∈ {1, 2}, (25) ( ) 2 ( ) ∑ a3 2 ζ3 (a3 ) = − σE + tr hiE h†iE Xi ln 2 i=1

1 . (26) ln 2 We propose to decouple (19) into four optimization problems and alternatively optimize a1 , a2 , a3 , X1 , and X2 . In the nth iteration, when the optimal X1 in the (n − 1)th iteration, (n−1) denoted as X1 , is obtained, we solve ( ( )) a1 (n) (n−1) a1 = arg max − σ12 + max tr ∆h11 ∆h†11 X1 a1 >0 ∆h11 ln 2 1 . (27) + log2 a1 + ln 2 Note that in the derivation from (19) to (27), we exchange the optimization sequence of a1 and ∆h11 because the objective function of problem (27) is convex with respect to a1 and ∆h11 . Similarly, when the optimal X2 in the (n − 1)th (n−1) iteration, denoted as X2 , is obtained, we solve ( ( )) a2 (n−1) (n) σ22 + max tr ∆h22 ∆h†22 X2 a2 = arg max − a2 >0 ∆h22 ln 2 1 . (28) + log2 a2 + ln 2 + log2 a3 +

(n−1)

When X1

where P is the individual transmit power constraint and

(n)

a3

and X2 are obtained, we solve ( ) 2 ( ) ∑ a3 (n−1) † 2 = arg max − σE + max tr hiE hiE Xi a3 >0 ∆hiE ln 2 i=1 + log2 a3 + (n)

(n)

(n)

When a1 , a2 , and a3 { } (n) (n) X1 , X2 = arg

1 . ln 2

(29)

are obtained, we solve

( ) ( ) (n) (n) min ζ1 a1 + ζ2 a2 X1 ≽0,X2 ≽0 ∆hij ( ) (n) + ζ3 a3 + (ξ1 + ξ3 ) max

tr(X1 ) ≤ P, tr(X2 ) ≤ P

s.t.

(30)

where the constraints rank(Xi ) = 1, i ∈ {1, 2} are omitted by employing semidefinite relaxation (SDR). In order to solve (27), we should solve the following optimization problem first ( ) (n−1) max tr ∆h11 ∆h†11 X1 ∆h11

s.t. ∆h†11 V11 ∆h11 ≤ 1.

(31)

It is noted that the optimal ∆h11 should satisfy that the inequality constraint is active. With the equality constraint ∆h†11 V11 ∆h11 = 1, we rewrite (31) as an unconstrained optimization problem, ∆h†11 X1

(n−1)

max

∆h11

∆h11 . † ∆h11 V11 ∆h11

(32)

The problem (32) is the maximization of a generalized Rayleigh quotient whose optimal objective value is the maxi−1 (n) mum eigenvalue of V11 X1 , denoted as φ1,max . Substituting the optimal objective value of problem (32), φ1,max , into (27), we have ) a1 ( 2 1 (n) a1 = arg max − σ1 + φ1,max + log2 a1 + . a1 >0 ln 2 ln 2 (33) The closed-form solution to problem (33) is ( )−1 (n) a1 = σ12 + φ1,max .

(34)

Similarly, we obtain the closed-form solution to problem (28) ( )−1 (n) a2 = σ22 + φ2,max (35) −1 where φ2,max is the maximum eigenvalue of V22 X2 . In order to solve (29), we should solve the following optimization problem first ( ) (n−1) † ˆ † (n−1) ˆ Ti = max tr xi (hiE + ∆hiE )(hiE + ∆hiE ) xi (n)

∆hiE

s.t.

∆h†iE ViE ∆hiE ≤ 1, (n−1)

i ∈ {1, 2}, where Xi function of Ti is (n−1) †

Li =xi

(n−1) (n−1) † xi .

= xi

(36) The Lagrangian

ˆ iE + ∆hiE )(h ˆ iE + ∆hiE )† x (h i

+ λi (∆h†iE ViE ∆hiE − 1)


(n−1)

(37)


4

where λi denotes the Lagrange multiplier. Since Li is convex in ∆hiE , the KKT conditions are sufficient for maximizing (36). Thus, we have [ ( )] (n−1) ˆ ˆ † + V−1 + 2√κi V−1 , (38) Ti = tr Xi hiE h iE iE iE

This problem is convex. Taking the partial derivative of g with respect to τ1 , we have

i ∈ {1, 2}, where

When ∂g/∂µ < 0, i.e., τ1 > 1/a1 − τ2 − σ12 , g is monotonically decreasing with the increase of τ1 . When ∂g/∂µ ≥ 0, (n) i.e., τ1 ≤ 1/a1 − τ2 − σ12 , g is monotonically increasing with the increase of τ1 . In (43), τ1 is function of X1 . Thus, it indicates that with the increase of transmit power of S1 , the worst-case sum secrecy rate decreases. In our simulations, it is found that when the regions of channel uncertainties and the individual transmit power constraints are sufficiently large, because of self-interference, the two-way robust secure communication is proactively degraded to one-way communication, i.e., tr(X1 ) = 0 or tr(X2 ) = 0. This means that the maximum worst-case sum secrecy rate of one-way communication is the same as that of two-way communication. By employing S-procedure [20] and introducing slack variables η1 ≥ 0, η2 ≥ 0, η3 ≥ 0, η4 ≥ 0, and η5 ≥ 0, we transform (41d)-(41i) into linear matrix inequalities (LMIs) [21] as follows [ ] η1 V11 − X1 0 E1 , ≽ 0, (47) 0 −η1 + τ1 [ ] ˆ 21 η2 V21 + X2 X2 h E2 , † † ˆ ˆ ˆ 21 − τ2 ≽ 0, (48) h21 X2 −η2 + h21 X2 h [ ] η3 V22 − X2 0 E3 , ≽ 0, (49) 0 −η3 + τ3 [ ] ˆ 12 η4 V12 + X1 X1 h E4 , † † ˆ ˆ ˆ 12 − τ4 ≽ 0, (50) h12 X1 −η4 + h12 X1 h [ ] ˆ 1E η5 V1E − X1 −X1 h E5 , ˆ † X1 −η5 − h ˆ † X1 h ˆ 1E + τ5 ≽ 0, (51) −h 1E 1E [ ] ˆ 2E η6 V2E − X2 −X2 h E6 , ˆ † X2 −η6 − h ˆ † X2 h ˆ 2E + τ6 ≽ 0. (52) −h 2E 2E

(46)

(n)

)

(

ˆ iE h ˆ † X(n−1) tr h iE i ( ) . κi = (n−1) −1 tr Xi ViE Thus, the closed-form solution to problem (29) is ( 2 )−1 (n) a3 = σE + T1 + T 2 .

(39)

(40)

To solve (30), we rewrite it as follows max

Ω

(41a)

s.t.

tr(X1 ) ≤ P, tr(X2 ) ≤ P, rank(X1 ) = 1, rank(X2 ) = 1,

(41b) (41c)

X1 ≽0,X2 ≽0

(n)

∂g 1 a = 2 − 1 . ∂τ1 (σ1 + τ1 + τ2 ) ln 2 ln 2

tr(∆h11 ∆h†11 X1 ) D τ1 , ∀ ∆h11 ∈ H11 , (41d) ) ( tr h21 h†21 X2 ≥ τ2 , ∀ ∆h21 ∈ H21 , (41e) tr(∆h22 ∆h†22 X2 ) D τ3 , ∀∆h22 ∈ H22 , (41f) ) ( (41g) tr h12 h†12 X1 ≥ τ4 , ∀ ∆h12 ∈ H12 , ( ) tr h1E h†1E X1 ≤ τ5 , ∀ ∆h1E ∈ H1E , (41h) ( ) tr h2E h†2E X2 ≤ τ6 , ∀ ∆h2E ∈ H2E (41i) where “D” represents either “≥” or “≤” and (n)

a1 (σ12 + τ1 ) (n) + log2 a1 ln 2 (n) a (σ12 + τ3 ) (n) + log2 (σ22 + τ3 + τ4 ) − 2 + log2 a2 ln 2 ) (n) ( 2 + τ5 + τ6 a3 σE 3 (n) − + log2 a3 + . (42) ln 2 ln 2 Note that in (41d) and (41f), the inequality signs are undetermined. They are determined as follows. From (8)(9), I(x2 ; y˜1 ) and I(x1 ; y˜2 ) decrease with the increases of tr(∆h11 ∆h†11 X1 ) and tr(∆h22 ∆h†22 X2 ), respectively. Since our objective is to maximize worst-case sum secrecy rate of the MISO full-duplex two-way secure communication system, the inequality signs in (41d) and (41f) are “≤”. Thus, we have Ω = log2 (σ12 + τ1 + τ2 ) −

τ1 = min tr(∆h†11 ∆h11 X1 ). ∆h11

(43)

Let g include all the terms in (41a) related with τ1 , (n)

a1 (σ12 + τ1 ) . (44) ln 2 Since the value of τ1 is related with the worst-case channel uncertainty, we should optimize τ1 such that g is in its worstcase, i.e., minimized. Thus, we have the optimization problem g = log2 (σ12 + τ1 + τ2 ) −

min g. τ1

(45)

The problem (41) is reformulated as follows max

X1 ≽0,X2 ≽0

s.t.

Ω tr(X1 ) ≤ P, tr(X2 ) ≤ P, Ek ≽ 0, ηk ≥ 0, k ∈ {1, · · · , 6}.

(53)

This problem is a semidefinite programming (SDP) with linear objective function and LMI constraints. We have the following proposition. Proposition 2: The solution of X1 to problem (53) has rank of less than or equal to one. Proof : See Appendix A. Similarly, we can prove that the solution of X2 to (53) has rank of less than or equal to one. We summarize the proposed alternating optimization algorithm in Algorithm 1. We have the following proposition to theoretically prove the convergence of Algorithm 1.



5

Algorithm 1 The proposed alternating optimization algorithm for robust secure beamforming 2:

3:

(0)

(0)

(0)

Initialize: n = 0, a1 = 1, a2 = 1, a3 = 1; (n) (n) Repeat: Solve the problem (53) to obtain X1 and X2 ; n := n + 1; (n) (n) (n) Obtain a1 , a2 , and a3 by (34), (35), and (40), respectively; Until: convergence.

Proposition 3: The sequence (n)

(n)

(n)

(n)

(n)

{(X1 , X2 , a1 , a2 , a3 )} generated by the Algorithm 1 has limit point Γ⋆ = (X⋆1 , X⋆2 , a⋆1 , a⋆2 , a⋆3 ) which is a KKT point of the problem (19). Proof : See Appendix B. Remark:(From [22], an SDP prob( the complexity of solving)) 1/2 · log(1/ϵ) lem is O nsdp · msdp n3sdp + m2sdp n2sdp + m3sdp where msdp denotes the number of semidefinite cone constraints, nsdp denotes the dimension of the semidefinite cone, and ϵ is the accuracy of solving the SDP. Compare the SDP (53) with the standard form in [22], we have msdp = 8 and nsdp = N + 1. Thus, the computational complexity of our proposed scheme is ( ) L · O 8(N + 1)7/2 + 64(N + 1)5/2 + 512(N + 1)1/2 · log(1/ϵ)

(54)

where L is the iteration number of alternating optimization. IV. S IMULATION R ESULTS ˆ 11 , In the simulations, all the entries of channel estimates, h ˆ 12 , h ˆ 22 , h ˆ 21 , h ˆ 1E , and h ˆ 2E , are independent and identically h distributed (i.i.d.) complex Gaussian random variables with zero mean and unit variance. All the channel uncertainties are bounded by (2) where Vij = δ −1 I/100, i ∈ {1, 2}, j ∈ {1, 2, E}. The variances of the Gaussian noises are 2 σ12 = σ22 = σE = σ 2 . We produce 1000 randomly generated channel realizations and compute the average worst-case sum secrecy rate. In Fig. 2 and Fig. 3, we present the average worst-case sum secrecy rate comparison of our proposed robust secure beamforming scheme (denoted as “Robust” in the legend) and the non-robust secure beamforming scheme (denoted as “Non-Robust”) where the sources, S1 and S2 , are equipped with N = 2 and N = 4 transmit antennas, respectively. In Fig. 4, we present the average worst-case sum secrecy rate comparison for different transmit antennas, N where P/σ 2 = 10 dB. The average worst-case secrecy rate of any source, S1 or S2 , is half of the average worst-case sum secrecy rate. Considering perfect CSI, the average worst-case sum secrecy rate of the optimal secure beamforming scheme (denoted as “Perfect CSI”) is also presented. When perfect CSI is available, the optimal secure beamforming scheme is derived using the similar method proposed in Section III where ∆hij = 0 for i ∈ {1, 2}, j ∈ {1, 2, E}. For the

Avergae Worst−Case Sum Serecy Rate (bps/Hz)

1:

7 Perfect CSI Robust, δ=0.01 Non−Robust, δ=0.01 Robust, δ=0.05 Non−Robust, δ=0.05 Robust, δ=0.1 Non−Robust, δ=0.1

6

5

4

3

2

1

0

0

2

4

6

8

10

12

14

P/σ2 (dB)

Fig. 2. Average wosrt-case sum secrecy rate versus P/σ 2 ; performance comparison of our proposed robust secure beamforming scheme and the nonrobust one where N = 2.

non-robust secure beamforming scheme, the channel estimates are directly used as the actual channel responses without considering channel uncertainties. From Fig. 2-Fig. 4, it is observed that our proposed robust secure beamforming scheme outperforms the non-robust one. It is also found that when δ = 0.1, with the increase of P/σ 2 from 12 dB to 14 dB, the worst-case sum secrecy rate decreases for the non-robust scheme whereas increases for our proposed robust scheme. This is because for the non-robust scheme, imperfect CSI is erroneously considered as perfect CSI. Thus, the selfinterference is not considered. With the increase of P/σ 2 , self-interference increases which causes that the worst-case sum secrecy rate decreases. However, with the increase of P/σ 2 , our proposed two-way robust secure communication is proactively degraded to the one-way communication, i.e., tr(X1 ) = 0 or tr(X2 ) = 0, to eliminate self-interference. Thus, the worst-case sum secrecy rate of our proposed robust scheme still increases with the increase of P/σ 2 . In Fig. 5, we present the average worst-case sum secrecy rate comparison for different δ where P/σ 2 = 10 dB. From Fig. 5, it is found that when δ approaches to 0, the performance of the proposed “Robust” scheme approaches to that of the “Perfect CSI” scheme.

V. C ONCLUSIONS In this correspondence, we have proposed a robust secure beamforming scheme in the MISO full-duplex two-way secure communication system. Simulation results demonstrate that our proposed robust scheme outperforms the non-robust one. It is also found that when the regions of channel uncertainties and the individual transmit power constraints are sufficiently large, because of self-interference, the two-way robust secure communication is proactively degraded to one-way communication.



6

8 Perfect CSI Robust, δ=0.01 Non−Robust, δ=0.01 Robust, δ=0.05 Non−Robust, δ=0.05 Robust, δ=0.1 Non−Robust, δ=0.1

9 8 7 6

Average Worst−Case Sum Serecy Rate (bps/Hz)

Average Worst−Case Sum Serecy Rate (bps/Hz)

10

5 4 3 2 1

0

2

4

6

8

10

12

14

P/σ2 (dB)

Fig. 3. Average wosrt-case sum secrecy rate versus P/σ 2 ; performance comparison of our proposed robust secure beamforming scheme and the nonrobust one where N = 4.

6 5 4 3 2 1

Perfect CSI, N=4 Robust, N=4 Non−Robust, N=4 Perfect CSI, N=2 Robust, N=2 Non−Robust, N=2

0 −4 10

−3

10

−2

10 δ

−1

10

0

10

Fig. 5. Average wosrt-case sum secrecy rate versus δ; performance comparison of our proposed robust secure beamforming scheme and the non-robust one where P/σ 2 = 10 dB.

we have rank(Q4 ) + rank(E4 ) ≤ N + 1. From S-procedure [20], we have λ4 ≥ 0. Furthermore, it can be verified that λ4 > 0. Thus, X1 + λ4 V21 ≻ 0 has full rank. Since rank(Q4 ) + rank(E4 ) ≤ N + 1, we know that rank(Q4 ) ≤ 1. For the constraint E1 ≽ 0, we have

10 Worst−Case Sum Serecy Rate (bps/Hz)

7

8

6

Q1 E1 = −Q1 G1 X1 G†1 + Q1 diag (η1 V11 , −η1 + τ1 ) (57) Perfect CSI Robust, δ=0.01 Non−Robust, δ=0.01 Robust, δ=0.05 Non−Robust, δ=0.05 Robust, δ=0.1 Non−Robust, δ=0.1

4

2

0

2

3

4

5

6

7

8

where G1 = [I, 0]† . Similarly, we have rank(Q1 ) ≤ 1. For the constraint E5 ≽ 0, we have Q5 E5 = −Q5 G5 X1 G†5 + Q5 diag (η5 V1E , −η5 + τ5 ) (58) 9

N

Fig. 4. Average wosrt-case sum secrecy rate versus N ; performance comparison of our proposed robust secure beamforming scheme and the nonrobust one where P/σ 2 = 10 dB.

ˆ 1E ]† . Similarly, we have rank(Q5 ) ≤ 1. where G5 = [I, h Furthermore, from KKT conditions, we have Q0 X1 = 0. Thus, we have rank(Q0 ) ≤ 1. ¯ 1 ) with respect to X1 and Taking partial derivative of L(X and applying KKT conditions, we have −Q0 + G†1 Q1 G1 − G†4 Q4 G4 + G†5 Q5 G5 + qI = 0. (59) Multiplying both sides of (59) with X1 , we have (G†4 Q4 G4 )X1 = (G†1 Q1 G1 + G†5 Q5 G5 + qI)X1 .

A PPENDIX A P ROOF OF P ROPOSITION 2 The Lagrangian dual function of (53) with respect to X1 is given by

(60)

Since G†1 Q1 G1 + G†5 Q5 G5 + qI ≻ 0 has full rank, we have rank((G†1 Q1 G1 + G†5 Q5 G5 + qI)X1 ) = rank(X1 ). (61)

¯ 1 ) = − tr(Q0 X1 ) − tr(Q1 E1 ) − tr(Q4 E4 ) − tr(Q5 E5 ) L(X + q(tr(X1 ) − P ) (55)

It is noted that rank(G†4 Q4 G4 ) = 1, we have rank(X1 ) ≤ 1.

where the dual variables Q0 ∈ H+ , Q1 ∈ H+ , Q4 ∈ H+ , Q5 ∈ H+ , and q ≥ 0 are corresponding to the constraints X1 ≽ 0, E1 ≽ 0, E4 ≽ 0, E5 ≽ 0, tr(X1 ) ≤ P in (53), respectively. For the constraint E4 ≽ 0, we have

A PPENDIX B P ROOF OF P ROPOSITION 3

Q4 E4 = Q4 G4 X1 G†4 + Q4 diag (η4 V21 , −η4 − τ4 )

We have the following lemma. Lemma [23, Corollary 2] Consider the problem min f (X, a) s.t. (X, a) ∈ X × A X,a

(56)

ˆ 21 ]† . From KKT conditions, we have where G4 = [I, h Q4 E4 = 0. Since the size of Q4 and E4 is (N +1)×(N +1),

(62)

where f (X, a) is a continuously differentiable function; X ⊆ Cm×n and A ⊆ R are closed, nonempty, and convex subsets. Suppose that the sequence {(X(n) , a(n) )} generated by



7

optimizing X and a alternatively has limit points. Every limit point of {(X(n) , a(n) )} is a stationary point of (62). It is noted that by Proposition 1, the problem (19) is transformed into max

min ζ1 (a1 ) + ζ2 (a2 ) + ζ3 (a3 ) + (ξ1 + ξ3 )

s.t.

tr(Xi ) ≤ P, rank(Xi ) = 1, i ∈ {1, 2}. (63)

X1 ≽0,X2 ≽0, ∆hij a1 >0,a2 >0,a3 >0

The objective function of (63) is continuously differentiable. The feasible set is closed, nonempty, and convex. Further(n) (n) (n) (n) (n) more, the sequence {(X1 , X2 , a1 , a2 , a3 } generated by optimizing (X1 , X2 ), a1 , a2 , and a3 alternatively has limit points because of the individual transmit power constraints in (63). By Bolzano-Weierstrass theorem, we know (n) (n) (n) (n) (n) that {(X1 , X2 , a1 , a2 , a3 } have limit points. Therefore, invoking Lemma 1, we conclude that every limit point Γ⋆ generated by Algorithm 1 is a stationary point of (63). In the following, we will prove that every stationary point of (63) is also a stationary point of (19). Note that from Proposition 2, the rank-one constraints on X1 and X2 in (63) and (19) are omitted. Denote the objectives of problems (63) and (19) as ϕ1 (Γ) and ϕ2 (X1 , X2 ), respectively, where Γ = (X1 , X2 , a1 , a2 , a3 ). Since Γ⋆ is a stationary point of (63), we have [ ] tr ∇Xi ϕ1 (Γ⋆ )† (Xi − X⋆i ) ≤ 0, ∀ tr(Xi ) ≤ P, i ∈ {1, 2}, (64) ∇ak ϕ1 (Γ⋆ )† (ak − a⋆k ) ≤ 0, ∀ ak > 0, k ∈ {1, 2, 3}.

(65)

From Proposition 1, (25)-(26), and (65), we have ( )−1 a⋆i = σi2 + tr(∆hii ∆h†ii X⋆i ) , i ∈ {1, 2} ) ( 2 ) −1 ( ∑ † ⋆ ⋆ 2 . a3 = σE + tr hiE hiE Xi

(66) (67)

i=1

Substituting (66)-(67) into (64), it can be verified that ∇Xi ϕ2 (X⋆1 , X⋆2 ) = ∇Xi ϕ1 (Γ⋆ ).

(68)

Therefore, we conclude from (64) and (68) that [ ] tr ∇Xi ϕ2 (X⋆1 , X⋆2 )† (Xi − X⋆i ) ≤ 0, ∀ tr(Xi ) ≤ P, i ∈ {1, 2},

(69)

i.e., (X⋆1 , X⋆2 ) is an optimal solution of the following problem [ ] max tr ∇Xi ϕ2 (X⋆1 , X⋆2 )† (Xi − X⋆i ) s.t. tr(Xi ) ≤ P. (70) Xi

Hence, (X⋆1 , X⋆2 ) must satisfy the KKT conditions of (70)

R EFERENCES [1] 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. [2] B. P. Day, A. R. Margetts, D. W. Bliss, and P. Schniter, “Full-duplex MIMO relaying: Achievable rates under limited dynamic range,” IEEE J. Sel. Areas Commun., vol. 30, no. 8, pp. 1541-1553, Sept. 2012. [3] M. Duarte, A. Sabharwal, V. Aggarwal, R. Jana, K. K. Ramakrishnan, C. Rice, and N. K. Shankaranarayanan, “Design and characterization of a full-duplex multi-antenna system for WiFi networks,” IEEE Trans. Veh. Tech., vol. 63, no. 3, pp. 1160-1177, Mar. 2014. [4] P. Lioliou, M. Viberg, M. Coldrey, and F. Athley, “Self-interference suppression in full-duplex MIMO relays,” in Proc. Asilomar 2007, pp. 658-662. [5] A. E. Gamal, O. O. Koyluoglu, M. Youssef, and H. E. Gamal, “Achievable secrecy rate regions for the two-way wiretap channel,” IEEE Trans. Inf. Theory, vol. 59, no. 12, pp. 8099-8114, Dec. 2013. [6] A. Somekh-Baruch, “On the secure interference channel,” in Proc. 2013 51st Annual Allerton Conf. Commun., Control, and Computing, pp. 770773. [7] Y. Yang, C. Sun, H. Zhao, H. Long, and W. Wang, “Algorithms for secrecy guarantee with null space beamforming in two-way relay networks,” IEEE Trans. Signal Process. , vol. 62, no. 8, pp. 2111-2126, Apr. 2014. [8] J. Mo, M. Tao, Y. Liu, and R. Wang, “Secure beamforming for MIMO two-way communications with an untrusted relay,” IEEE Trans. Signal Process., vol. 62, no. 9, pp. 2185-2199, May 2014. [9] H.-M. Wang, Q. Yin, and X.-G. Xia, “Distributed beamforming for physical-layer security of two-way relay networks,” IEEE Trans. Signal Process., vol. 61. no. 5. pp 3532-3545. July 2012. [10] H.-M. Wang, M. Luo, Q. Yin, and X.-G. Xia, “Hybrid cooperative beamforming and jamming for physical-layer security of two-way relay networks,” IEEE Trans. Inf. Forensics and Security, vol. 8, no. 12, pp. 2007-2020, Dec. 2013. [11] G. Zheng, I. Krikidis, J. Li, A. P. Petropulu, and B. Ottersten, “Improving physical layer secrecy using full-duplex jamming receivers,” IEEE Trans. Signal Process., vol. 61, no. 20, pp. 4962-4974, Oct. 2013. [12] Q. Li and W.-K. Ma, “Spatially selective artificial-noise aided transmit optimization for MISO multi-eves secrecy rate maximization,” IEEE Trans. Signal Process., vol. 61, no. 10, pp. 2704-2717, May 2013. [13] J. Huang and A. L. Swindlehurst, “Robust secure transmission in MISO channels based on worst-case optimization, IEEE Trans. Signal Process., vol. 60, no. 4, pp. 1696-1707, Apr. 2012. [14] J. Zhang, O. Taghizadeh, and M. Haardt, “Robust transmit beamforming dsign for full-duplex point-to-point MIMO systems,” in Proc. ISWCS 2013, pp. 1-5. [15] S. Vishwakarma and A. Chockalingam, “Secret full-duplex communication,” [Online]. Available: http://arxiv.org/abs/1311.3918. [16] 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. 46864698, Oct. 2008. [17] 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. [18] X. Zhou, B. Maham, and A. Hjorungnes, “Pilot contamination for active eavesdropping,” IEEE Trans. Wireless Commun., vol. 11, no. 3, pp. 903907, Mar. 2012. [19] A. Mukherjee and A. L. Swindlehurst, “Detecting passive eavesdroppers in the MIMO wiretap channel,” in Proc. IEEE ICASSP 2012, pp. 28092812. [20] Z.-Q. Luo, J. F. Sturm and S. Zhang “Multivariate nonnegative quadratic mappings,” SIAM J. Optimization, vol. 14, no. 4, pp. 1140-1162, 2004. [21] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [22] I. Polik and T. Terlaky, “Interior Point Methods for Nonlinear Optimization,” in Nonlinear Optimization, G. Di Pillo and F. Schoen, Eds., 1st ed. New York, NY, USA: Springer-Verlag, 2010, ch. 4. [23] L. Grippo and M. Sciandrone, “On the convergence of the block nonlinear Gauss-Seidel method under convex constraints,” Operation research letter, vol. 26, pp. 127-136, 2000.

ˆ i I + Di = 0, λ ˆ i (P − tr(X⋆ )) = 0, ∇Xi ϕ2 (X⋆1 , X⋆2 ) − λ i ˆ i ≥ 0, Di ≽ 0, i ∈ {1, 2} X⋆ Di = 0, λ (71) 1

ˆ i and Di are Lagrangian multipliers. The conditions where λ in (71) are exactly the KKT conditions of the problem (19). 0018-9545 (c) 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.