Secure Beamforming for MIMO Two-Way Communications with an Untrusted Relay
aa r X i v : . [ c s . I T ] D ec Secure Beamforming for MIMO Two-WayCommunications with an Untrusted Relay
Jianhua Mo,
Student Member, IEEE , Meixia Tao,
Senior Member, IEEE ,Yuan Liu,
Member, IEEE , and Rui Wang
Abstract
This paper studies the secure beamforming design in a multiple-antenna three-node system wheretwo source nodes exchange messages with the help of an untrusted relay node. The relay acts as bothan essential signal forwarder and a potential eavesdropper. Both two-phase and three-phase two-wayrelay strategies are considered. Our goal is to jointly optimize the source and relay beamformers formaximizing the secrecy sum rate of the two-way communications. We first derive the optimal relaybeamformer structures. Then, iterative algorithms are proposed to find source and relay beamformersjointly based on alternating optimization. Furthermore, we conduct asymptotic analysis on the maximumsecrecy sum-rate. Our analysis shows that when all transmit powers approach infinity, the two-phase two-way relay scheme achieves the maximum secrecy sum rate if the source beamformers are designed suchthat the received signals at the relay align in the same direction. This reveals an important advantage ofsignal alignment technique in against eavesdropping. It is also shown that if the source powers approachzero the three-phase scheme performs the best while the two-phase scheme is even worse than directtransmission. Simulation results have verified the efficiency of the secure beamforming algorithms aswell as the analytical findings.
Index Terms
Two-way relaying, physical layer security, signal alignment, untrusted relay.
J. Mo is now with Wireless Networking and Communications Group, The University of Texas at Austin. M. Tao, Y. Liu and R.Wang are with the Department of Electronic Engineering, Shanghai Jiao Tong University, P. R. China. (Email: [email protected], { mxtao, yuanliu, liouxingrui } @sjtu.edu.cn).The material in this paper was partly presented at IEEE Wireless Communications and Networking Conference, Shanghai,China, April 2013 [1]. I. I
NTRODUCTION
A. Motivation
Cooperative relaying has been shown effective for power reduction, coverage extension andthroughput enhancement in wireless communications. Recently, with the advance of wirelessinformation-theoretic security at the physical layer, a new dimension arises for the design ofrelaying strategies. In specific, from a perspective of physical-layer security, a relay can befriendly and may help to keep the confidential message from being eavesdropped by others,while an untrusted relay may intentionally eavesdrop the signal when relaying. The case ofuntrusted relay exists in real life. For example, the relays and sources belong to different networkin today’s heterogenous network, where the nodes have different security clearances and thusdifferent levels of access to the information. It is therefore important to find out whether theuntrusted relay is still beneficial compared with direct transmission and if so what is the newrelay strategy.The goal of this work is to study the physical layer security in two-way relay systemswhere the relay is untrusted and each node is equipped with multiple antennas. Comparedwith traditional one-way relaying, the problem in two-way relaying is more interesting. This isbecause by applying physical layer network coding, the relay only needs to decode the network-coded message rather than each individual message and hence the network coding procedureitself also brings certain security. We will try to address three important questions. First, underwhat conditions, should we treat the two-way untrusted relay as a passive eavesdropper or seekhelp from it? This is a challenging problem because different power constraints and antennasconfigurations may result in different answers. Second, if help is necessary, how to jointlyoptimize the source and relay beamformers? Typically this would be a non-convex problemand very difficult to solve. Thirdly, would physical layer network coding, originally known forthroughput enhancement in two-way relay systems, bring new insights to the new performancemetric of information security?
B. Related Work
We first briefly review the existing works related to beamforming design in MIMO two-wayrelay systems without taking secrecy into account. Then according to the relay being trusted oruntrusted, we classify the related work on secure communication in relay systems.
1) Beamforming in MIMO Two-way Relay Systems:
When the source nodes are each equippedwith a single antenna, [2] proposed the optimal relay beamforming structure and a convexoptimization algorithm to find the capacity region. For the case of multi-antenna uses, the problemis much more difficult. The work [3] showed an optimal structure of the relay precoding matrixand proposed an alternating optimization method (optimize the relay precoding matrix and sourceprecoding matrices alternatively) to maximize the achievable weighted sum rate. Based on anothercriterion of mean-square-errer (MSE), [4] proposed an iterative method for the joint source andrelay precoding design.
2) Trusted Relay:
In this case, the relay is a legitimate user or acts like a legitimate user whowill help to counter external eavesdroppers and increase the security of the networks. Most ofthe work has focused on traditional one-way relaying secret communication (e.g., [5]–[11]).Only a few attempts have been made very recently to study two-way relaying secret communi-cation [12]–[16]. Specifically, [12] and [13] investigated the relay and jammer selection problemin the two-way relay networks. The authors in [14] studied beamforming design in MIMO two-way relaying for maximizing secrecy sum rate which is proven to be achievable in [17]. Jointdistributed beamforming and power allocation was considered in [15] for maximizing secrecysum rate in two-way relaying networks with multiple single-antenna relays. Several secret keyagreement schemes were proposed in [16].
3) Untrusted Relay:
Untrusted relay channels with confidential messages was first studiedin [18], where an achievable secrecy rate was obtained. A destination-based jamming (DBJ)technique was proposed in [19], [20] without source-destination link. The performance of DBJin fading channel and multi-relay scenarios was analyzed in [21]. When the source-destinationlink exists, authors in [22] discussed whether cooperating with the untrusted relay is betterthan treating it as a passive eavesdropper. A Stackelberg game between the two sources andthe external friendly jammers in a two-way relay system was formulated as a power controlscheme in [23]. In [24], the authors considered MIMO one-way amplify-and-forward (AF) relaysystems and jointly deigned the source and relay beamforming using alternating optimization.[25] examines the secrecy outage probability in one-way non-regenerative relay systems.From these existing literature, it is found that the problem of joint source and relay beam-forming for MIMO two-way untrusted relaying has not been considered yet.
C. Contribution
In this paper, we investigate physical layer security in MIMO two-way relay systems, wherethe two sources exchange confidential information with each other through an untrusted relay.The relay acts as both an essential helper and a potential eavesdropper, but does not makeany malicious attack. In our previous work [1], we considered the two-phase scheme. In thisextension, we study both two-phase and three-phase two-way relay schemes. In particular, weformulate the joint secure source and relay beamforming design for each two-way relay scheme.The objective is to maximize the secrecy sum rate of the bidirectional links subject to the sourceand relay power constraints. Furthermore, we conduct asymptotic analysis on maximum secrecysum rate of the different two-way relay schemes in comparison with direct transmission.The main contributions and results of this paper are summarized as follows: • The optimal structure of the relay beamforming matrix for fixed source beamformers isderived. With this structure, the number of unknowns in the relay beamformer is significantlyreduced and thus the joint source and relay beamformer design can be simplified. • Iterative algorithms based on alternating optimization are proposed to find a solution ofthe joint source and relay beamformers. These algorithms are convergent but cannot ensureglobal optimality due to the nonconvexity of the optimization problems. • Via asymptotic analysis, we show that when the powers of the source and relay nodesapproach infinity, the two-phase scheme achieves the maximum secrecy rate if the transceiverbeamformers are designed such that the received signals at the relay align in the samedirection. This reveals an important advantage of signal alignment techniques in againsteavesdropping. It gives a new perspective to achieve the physical layer security, and alsolowers the source antenna number requirement for ensuring security. • It is also shown via asymptotic analysis that when the power of the relay goes to infinityand that of the two sources approach zeros, the three-phase two-way relay scheme performsthe best while the two-phase performs even worse than direct transmission.
D. Organization and Notations
The rest of the paper is organized as follows. Section II describes the system model andproblem formulations. The optimal secure beamformers for two- and three-phase two-way relay schemes are presented in Section III. Asymptotical results are detailed in Section IV. Compre-hensive simulation results are given in Section V. Finally, we conclude this paper in SectionVI.Notations: Scalars, vectors and matrices are denoted by lower-case, lower-case bold-face andupper-case bold-face letters, respectively. [ x ] + denotes max (0 , x ) . Tr( A ) , A − , Rank( A ) , k A k F , A ∗ and A H denote the trace, inverse, rank, Frobenius norm, conjugate and Hermite of matrix A , respectively. span( A ) represents the column space (range space) of A and dim( A ) denotesthe dimension of A . The projection matrix onto the null space of A is denoted by A N . k q k denotes the norm of the vector q . σ max ( A ) is the largest singular value of A . λ max ( A ) is thelargest eigenvalue of the matrix A and ψψψ max ( A ) is the eigenvector of A corresponding to thelargest eigenvalue. λ max ( A , B ) is the largest generalized eigenvalue of the matrices A and B . ψψψ max ( A , B ) is the generalized eigenvector of ( A , B ) corresponding to the largest generalizedeigenvalue. We use P DTi , P Pi and P Pi to represent the transmit power of node i ∈ { A, B, R } in two-way direct transmission, two-phase two-way relaying and three-phase two-way relaying,respectively. Throughout this paper, n i denotes the zero mean circularly symmetric complexGaussian noise vector at node i ∈ { A, B, R } with n i ∼ CN ( , I ) .II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
We consider a two-way relay system as shown in Fig. 1, where two source nodes A and B exchange information with each other with the assistance of a relay node R . The relay acts as bothan essential helper and a potential eavesdropper but does not make any malicious attack. Notethat the decode-and-forward (DF) relay strategy is not applicable here since the relay is untrustedand not expected to decode the received signal from the source nodes. As such, we assume therelay adopts AF strategy, which also has low complexity. The number of antennas at nodes A , B and R are denoted as N A , N B and N R , respectively. As shown in Fig. 1, T A ∈ C N B × N A , T B ∈ C N A × N B , H A ∈ C N R × N A , G A ∈ C N A × N R , H B ∈ C N R × N B , G B ∈ C N B × N R denote thechannel matrices of link A → B , B → A , A → R , R → A , B → R and R → B , respectively. Ifthe system operates in time division duplex (TDD) mode and channel reciprocity holds, then wehave T A = T TB , H A = G TA , and H B = G TB . For simplicity, we only consider single data streamfor each source node in this paper. Denote the transmitted symbol at the source i as s i ∈ C with E ( | s i | ) = 1 , and the associated beamforming vector as q i ∈ C N i × , for i ∈ { A, B } . Different two-way relay schemes have been studied in the literature [26], [27]. In this paper,we study the two-phase and three-phase two-way relay schemes. For the purpose of comparison,the two-way direct transmission is also considered in Appendix A, wherein the relay node istreated as a pure eavesdropper [24], [28].
A. Two-Phase Two-Way Relay Scheme
In the first phase of the two-phase two-way relay scheme, A and B simultaneously transmitsignals to the relay node R . The received signal at relay is, y PR = H A q A s A + H B q B s B + n R . (1)In the second phase, the relay node amplifies y PR by multiplying it with a precoding matrix F and broadcasts it to both A and B. The transmit signal vector from the relay node is expressedas x PR = Fy PR . (2)After subtracting the back propagated self-interference, each source node i obtains the equiv-alent received signals, y Pi = G i FH ¯ i q ¯ i s ¯ i + G i Fn R + n i , i ∈ { A, B } , (3)where ¯ i = { A, B } \ i .The information rate from node i to node ¯ i is R Pi ¯ i = 12 log (cid:0) q Hi H Hi F H G H ¯ i K − i G ¯ i FH i q i (cid:1) , (4)where K ¯ i = G ¯ i FF H G H ¯ i + I . (5)If the untrusted relay wants to eavesdrop the signals from both source nodes, it may try to fullydecode the two messages s A and s B . Therefore, the achievable information rate at the untrustedrelay can be expressed as the maximum sum-rate of a two-user MIMO multiple-access channel, given by R PR , I (cid:0) y PR ; s A , s B (cid:1) = 12 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I + h H A q A H B q B i q HA H HA q HB H HB (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( a ) = 12 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) I + q HA H HA q HB H HB h H A q A H B q B i(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 log (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q HA H HA H A q A q HA H HA H B q B q HB H HB H A q A q HB H HB H B q B (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 12 log (cid:16) k H A q A k + k H B q B k + k H A q A k k H B q B k − k q HB H HB H A q A k (cid:17) . (6)where ( a ) is from the identity | I + AB | = | I + BA | .The achievable secrecy sum rate [17] of the two source nodes is thus given by, R Ps = (cid:2) R PAB + R PBA − R PR (cid:3) + (7)Our goal is to maximize the secrecy sum rate by jointly optimizing the relay and sourcebeamformers F , q A and q B . The problem can be formulated as max { F , q A , q B } R Ps (8a)s. t. k q i k ≤ P Pi , i ∈ { A, B } , (8b) Tr (cid:0) FH A q A q HA H HA F H + FH B q B q HB H HB F H + FF H (cid:1) ≤ P PR . (8c)where (8c) is the relay power constraint. B. Three-Phase Two-Way Relay Scheme
In the first phase of the three-phase two-way relay scheme, source node A transmits. Thereceived signals at the relay R and the source node B are respectively given by y PR = H A q A s A + n R , (9) y PB = T A q A s A + n B . (10) where n R and n B are the noises at the relay node and source node B in the first phase,respectively.In the second phase, node B transmits. The received signals at the relay and the source node A are respectively given by y PR = H B q B s B + n R , (11) y PA = T B q B s B + n A . (12)where n R and n A are the noises at the relay node and source node A in the second phase,respectively.In the third phase, the relay node R amplifies the received signals y PR and y PR by multiplyingthem with F A and F B , respectively. The broadcast signal from the relay is thus x PR = F A y PR + F B y PR . (13)After subtracting the self-interference, the two source nodes obtain the signals as y PA = G A F B H B q B s B + G A ( F A n R + F B n R ) + n A , (14) y PB = G B F A H A q A s A + G B ( F A n R + F B n R ) + n B . (15)Combining (12) and (14), we obtain the information rate from B to A as, R PBA , I ( y PA , y PA ; s B ) (16) = 13 log (cid:16) q HB T HB T B q B + q HB H HB F HB G HA · (cid:0) G A (cid:0) F A F HA + F B F HB (cid:1) G HA + I (cid:1) − G A F B H B q B (cid:17) . Likewise, the information rate from A to B is, R PAB = 13 log (cid:16) q HA T HA T A q A + q HA H HA F HA G HB · (cid:0) G B (cid:0) F A F HA + F B F HB (cid:1) G HB + I (cid:1) − G B F A H A q A (cid:17) . The information sum rate leaked to the untrusted relay can be obtained from (9) and (11): R PR , I ( y PR ; s A ) + I ( y PR ; s B ) (17) = 13 log (cid:0)(cid:0) q HA H HA H A q A (cid:1) (cid:0) q HB H HB H B q B (cid:1)(cid:1) Thus, the secrecy sum rate is given by R Ps = [ R PBA + R PAB − R PR ] + . (18)We can formulate the secrecy sum rate maximization problem for three-phase two-way relayscheme as max { F A , F B , q A , q B } R Ps (19a)s. t. k q i k ≤ P Pi , i ∈ { A, B } , (19b) Tr (cid:16) F A H A q A q HA H HA F HA + F B H B q B q HB H HB F HB + F A F HA + F B F HB (cid:17) ≤ P PR , (19c)where (19c) is the relay power constraint.In these two schemes, we assume that one of the source nodes, say A is responsible for the jointdesign of source and relay beamformers. After finishing the design, A sends the correspondingdesigned beamformer to B and the relay. Then, the two source nodes and the untrusted relaywill use their beamformers to process the transmit signals.III. S ECURE B EAMFOMRING D ESIGNS
After introducing the problem formulations in (8) and (19) for the two-phase and three-phase two-way relay schemes, respectively, we now present algorithms to design these securebeamformers in this section.
A. Secure Beamforming in Two-Phase Two-Way Relay Scheme
We first obtain the optimal structure of the secure relay beamforming matrix F . Then wepresent an iterative algorithm to find a local optimal solution for the joint secure source andrelay beamformers. Define the following two QR decompositions: [ G HA G HB ] = VR P , (20) [ H A q A H B q B ] = UR P . (21)where V ∈ C N R × min { N A + N B ,N R } , U ∈ C N R × are orthonormal matrices and R P , R P are uppertriangle matrices. Lemma 1.
In the two-phase two-way relay scheme, the optimal relay beamforming matrix F ∈ C N R × N R that maximizes the secrecy sum rate has the following structure: F = VAU H , (22) where A ∈ C min { N A + N B ,N R }× is an unknown matrix.Proof: Note that the relay beamforming matrix F only influences the information rate R PAB and R PBA . Therefore, the optimal F that maximize the secrecy sum rate is actually the same tothe F that maximizes the information sum rate R PAB + R PBA . Due to the rank-one precoding ateach source node, we have the equivalent channel H i q i from source node i to relay. Therefore,applying the results in [3], we readily have Lemma 1.According to Lemma 1, the number of unknowns in F is reduced from N R to { N R , N A + N B } , which reduces the computational complexity of the joint source and relay beamformingdesign.We note that it is not easy to find the optimal solution to the problem (8). Even after substitutingthe optimal structure of F (22) into (7), the problem is still nonconvex since the secrecy sumrate is not a convex function of q A , q B and A . Therefore, we optimize the source beamformingvectors q A , q B and the unknown matrix A in the relay beamforming matrix F in an alternatingmanner. Given q A and q B , we use the gradient method shown in Appendix B to search A . Given F and q i , we can find the optimal q ¯ i , where the optimization method is shown in AppendixC. Formally, we present the method in Algorithm 1 as follows. Here, the initial points of thecomplex vectors q A and q B can be randomly generated as long as they satisfy the given powerconstraint. Algorithm 1
Iterative algorithm for secure beamforming in two-phase two-way relay scheme Initialize A , q A and q B . Repeat (a) Optimize A given q A and q B based on gradient method given in Appendix B;(b) Optimize q B given A and q A according to Appendix C;(c) Optimize q A given A and q B according to Appendix C by swapping A and B ; Until the secrecy sum rate does not increase. Note that the algorithm always converges because the secrecy sum rate is finite and does notdecrease in every iteration.
B. Secure Beamforming in Three-Phase Two-Way Relay Scheme
Similar to the two-phase case, we define the following QR decomposition: [ G HA G HB ] = VR P , (23)where V ∈ C N R × min { N A + N B ,N R } is an orthonormal matrix and R P ∈ C min { N A + N B ,N R }× ( N A + N B ) is an upper triangle matrix. Then we give the optimal structure of the relay beamforming matrices F A and F B in the following lemma. Lemma 2.
In the three-phase two-way relay scheme, the optimal relay beamforming matrices F A , F B that maximize the secrecy sum rate have the following structure: F A = Va ( H A q A ) H k H A q A k , F B = Va ( H B q B ) H k H B q B k , (24) where a ∈ C min { N A + N B ,N R }× , a ∈ C min { N A + N B ,N R }× are unknown vectors.Proof: See Appendix D.Lemma 2 simplifies the design of two beamforming matrices F i to the design of two beam-forming vectors a i . Thus, the number of unknowns is reduced to { N R , N A + N B } . Note thatthe number of unknowns in the relay beamforming matrices is the same for two- and three-phaseschemes.Lemma 1 and 2 show that the optimal relay beamforming contains three parts: (i) matching tothe received signal; (ii) combination or other operation of the information-bearing signals; (iii)beamforming to the intended receiver. This structure is similar to the optimal relay beamformingstructure in other systems, for example, the two-way relaying system without secrecy constraintin [2], [3] and one-way relaying system with secrecy constraint in [24]. These structure are alsoused in our following asymptotical analysis.Since problem (19) is also nonconvex, we adopt the iterative method to obtain a solution where q A , q B , a and a are alternatively optimized until the secrecy sum rate does not increase. Thealgorithm, denoted as Algorithm 2, is very similar to the Algorithm 1 and omitted.Since the problems (8) and (19) are both nonconvex, the iterative algorithms presented inthe previous section cannot ensure global optimality. However, letting the transmit power on each node approach zero or infinity, we can derive interesting intuitions which lead to theasymptotically optimal solution for secure beamforming. In the next section, we present suchasymptotic analysis. IV. A SYMPTOTIC A NALYSIS
The goal of this section is to find the asymptotical optimal secure beamforming design whenthe relay power P R approaches infinity. We first present the analysis when the two source powersare also infinite in Subsection IV-A, followed by the analysis when the two source powersapproach zero in Subsection IV-B. Finally, we briefly discuss the case where the relay power P R approaches zero. For comparison purpose, the asymptotic result for the direct transmissionis presented in this section as well. A. The Case of High Relay and Source Powers
Proposition 1 (2P) . When P R → ∞ , P A → ∞ and P B → ∞ , the maximum secrecy sum rateof the two-phase two-way relay scheme is, R P max ≈ max ( q A , q B ) ∈S
12 log k H A q A k k H B q B k k H A q A k + k H B q B k , if N A + N B > N R ,
12 log − ( σ max ( U HA U B )) , if N A + N B ≤ N R , , (25) where set S is { ( q A , q B ) : ∃ β ∈ R , β H A q A = H B q B and k q A k ≤ P A , k q B k ≤ P B } , σ max ( U HA U B ) is the maximum singular value of matrix U HA U B , U A ∈ C N R × min { N A ,N R } and U B ∈ C N R × min { N B ,N R } are obtained from the QR de-composition of H A and H B , respectively, i.e., H i = U i R i , i ∈ { A, B } , (26) where R i ∈ C min { N R ,N i }× N i are upper triangle matrices.Proof: We first prove the following fact:When P R → ∞ , the information rate from i to ¯ i in two-phase two-way relay scheme is lim P R →∞ R Pi ¯ i = 12 log (cid:0) q Hi H Hi H i q i (cid:1) , (27) To prove (27), we first plug in the optimal structure of F to (4) and let F = t VAU H where t is a real number. When P R → ∞ , we just let t → ∞ . Thus, q Hi H Hi F H G H ¯ i K − i G ¯ i FH i q i = q Hi H Hi t UA H V H G H ¯ i (cid:0) t G ¯ i VAA H V H G H ¯ i + I (cid:1) − · G ¯ i t VAU H H i q i ( a ) = q Hi H Hi U (cid:16) I − (cid:0) I + t A H V H G H ¯ i G ¯ i VA (cid:1) − (cid:17) U H H i q i = q Hi H Hi UU H H i q i − q Hi H Hi U (cid:0) I + t A H V H G H ¯ i G ¯ i VA (cid:1) − · U H H i q i ( b ) = q Hi H Hi H i q i − q Hi H Hi U (cid:0) I + t A H V H G H ¯ i G ¯ i VA (cid:1) − U H H i q i where ( a ) is from the matrix inverse lemma and ( b ) is from QR decomposition (21). Sincenodes A , B and R all have multiple antennas, we have Rank ( G ¯ i ) ≥ with probability1 as every element of G ¯ i are drawn from continuous distribution. Therefore, it is alwayspossible to find A such that (cid:0) A H V H G H ¯ i G ¯ i VA (cid:1) ∈ C × is positive definite matrix. Hence,the eigenvalue of (cid:0) I + t A H V H G H ¯ i G ¯ i VA (cid:1) approaches positive infinity when t → ∞ . Asa result, the term q Hi H Hi U (cid:0) I + t A H V H G H ¯ i G ¯ i VA (cid:1) − U H H i q i approaches zero and we ob-tain that when P R → ∞ , R Pi ¯ i ≥ log (cid:0) q Hi H Hi H i q i (cid:1) . In addition, it is easy to see that lim P R →∞ R Pi ¯ i ≤ log (cid:0) q Hi H Hi H i q i (cid:1) . Therefore, we obtain (27).Substituting (6) and (27) into (7), we obtain the achievable sum-rate as lim P R →∞ R Ps = 12 log − f ( q A , q B ) (28)where f ( q A , q B ) , | q HA H HA H B q B | (cid:0) k H B q B k (cid:1) (cid:0) k H A q A k (cid:1) . (29)From (28), we see that to maximize lim P R →∞ R Ps , we should maximize f ( q A , q B ) . An upperbound of f ( q A , q B ) is, f ( q A , q B ) < | q HA H HA H B q B | k H B q B k k H A q A k ≤ , (30)and this upper bound can be approached when P A → ∞ and P B → ∞ , i.e., ¯ f ( q A , q B ) , lim P A →∞ ,P B →∞ f ( q A , q B ) = | q HA H HA H B q B | k H B q B k k H A q A k . (31) Therefore, the problem is transformed to maximize ¯ f ( q A , q B ) , which is to find two vectors withthe minimum angle from the column spaces of U A and U B .For the case N A + N B > N R , with probability one, we can find q A and q B such that β H A q A = H B q B (32)where β can be an arbitrary non-zero real number. Under this condition, ¯ f ( q A , q B ) can take itsmaximum value of in (30) . Therefore, substituting the condition (32) into (28), we obtain lim P R →∞ R Ps = 12 log − f ( q A , q B )= 12 log (cid:0) k H A q A k (cid:1) (cid:0) k H B q B k (cid:1) k H A q A k + | H B q B | ≈
12 log k H A q A k k H B q B k k H A q A k + k H B q B k if P A → ∞ , P B → ∞ . At last, we maximize over the possible alignment directions and obtain the first part of Proposition1. On the other hand, if N A + N B ≤ N R , we have, (cid:13)(cid:13) q HB H HB H A q A (cid:13)(cid:13) ( a ) = (cid:13)(cid:13) q HB R HB U HB U A R A q A (cid:13)(cid:13) ( b ) ≤ σ max (cid:0) U HB U A (cid:1) k R A q A k k R B q B k ( c ) = σ max (cid:0) U HB U A (cid:1) k U A R A q A k k U B R B q B k ( d ) = σ max (cid:0) U HB U A (cid:1) k H A q A k k H B q B k (33)where ( a ) and ( d ) are from (26), ( b ) is from the singular value decomposition of U HB U A and theequality can be achieved by letting q A = R − A ψ max (cid:0) U HA U B U HB U A (cid:1) and q B = R − B ψ max (cid:0) U HB U A U HA U B (cid:1) where the upper triangle matrices R i ∈ C N i × N i are invertible, and ( c ) is from q Hi R Hi R i q i = q Hi R Hi U Hi U i R i q i . Substituting (33) back to (28), we obtain the second part of Proposition 1.Notice that we always have σ max (cid:0) U HB U A (cid:1) < when N A + N B ≤ N R . The proof is as follows. An algorithm to find q A and q B was shown in [29, Lemma 1]. First, as (cid:13)(cid:13) q HB H HB H A q A (cid:13)(cid:13) ≤ k H A q A k k H B q B k and the equality in ( b ) can be achieved, weknow that σ max (cid:0) U HB U A (cid:1) ≤ . Second, if σ max (cid:0) U HB U A (cid:1) = 1 , there is an intersection subspacebetween the space span( H A ) and span( H B ) such that β H A q A = H A q A where β is a realnumber. However, according to dimension theorem [30] and because the entries of the channelmatrices are generated from continuous distribution, we have dim(span( H A ) ∩ span( H A ))= dim(span( H A )) + dim(span( H B )) − dim(span([ H A , H B ]))= N A + N B − ( N A + N B )= 0 . Consequently, there is no intersection subspace and we have σ max (cid:0) U HB H A (cid:1) < .Thus, the proof of Proposition 1 is completed.Proposition 1 is essentially similar as the so-called signal alignment. In [29], this technique wasfirst proposed to achieve the degrees of freedom of the MIMO Y channel which is a generalizedtwo-way relay channel with three users. The key idea of the signal alignment is to align thetwo desired signal vectors coming from two users at the receiver of the relay to jointly performdetection and encoding for network coding. Specifically, if N A + N B > N R , there is intersectionsubspace between the column spaces of H i with probability one and thus there exists β ∈ R such that (32) holds. As illustrated in Fig. 2, the secure beamformers at the two source nodes arechosen such that the two received signals align in the same direction at the relay node. Intuitively,aligning the signal vectors at the relay node will hinder the relay node decode the source messagesand make the system more secure. After self-interference cancellation, the two source nodes willobtain the desired signal. The maximum secrecy sum rate goes to infinity as the source powersapproach infinity. On the other hand, if N A + N B ≤ N R , there is no intersection subspace withprobability one and there is an upper bound for the maximum secrecy sum rate. Specifically, U i is the orthonormal basis of the column space of H i . Thus, arccos (cid:0) σ max (cid:0) U HA U B (cid:1)(cid:1) , is theminimum angle between any two vectors from the respect two column spaces. Actually, it iscalled the minimum principal angle of these two subspaces [31]. Proposition 2 (3P) . When P R → ∞ , P A → ∞ and P B → ∞ , the maximum secrecy sum rate of the three-phase two-way relay scheme is, R P max ≈ X i ∈{ A,B } Θ i , (34) where Θ i ∈ " (cid:2) log (cid:0) + λ max (cid:0) T Hi T i , H Hi H i (cid:1)(cid:1)(cid:3) + , (cid:2) log (cid:0) λ max (cid:0) T Hi T i , H Hi H i (cid:1)(cid:1)(cid:3) + , if N i ≤ N R and Θ i = 13 (cid:20) log (cid:18) P i (cid:19) + log (cid:0) λ max (cid:0) H N i T Hi T i H N i (cid:1)(cid:1)(cid:21) , if N i > N R .Proof: See Appendix F.Proposition 2 shows that the secrecy sum-rate of the three-phase scheme will reach a floor ifthe untrusted relay has more antennas than the two source nodes.
Proposition 3 (DT) . When P A → ∞ and P B → ∞ , the maximum secrecy sum rate of thetwo-way direct transmission scheme is R DT max ≈ X i ∈{ A,B } Ω i , (35) where Ω i = (cid:2) log λ max (cid:0) T Hi T i , H Hi H i (cid:1)(cid:3) + , if N i ≤ N R (cid:2) log P i + log λ max (cid:0) H N i T Hi T i H N i (cid:1)(cid:3) , if N i > N R , and the optimal beamforming q DTi is given in (47) .Proof:
This lemma is based on [24, Lemma 7]. Here, we assume that the entries of the chan-nel matrices are generated from continuous distribution. As a result,
Rank( H m × n ) ≥ min { m, n } with probability one. In addition, the condition H N i T Hi = in [24, Lemma 7] is also satisfiedwith probability one.As shown in Proposition 3, the secrecy sum-rate of the direct transmission scheme will alsoreach a floor if untrusted relay has more antennas than the source nodes. This is similar to thethree-phase case. From Proposition 1, 2, and 3, we find that the asymptotic comparison among these threeschemes depend not only on the antenna numbers N A , N B , N R but also on specific channelrealizations. In the following, we only present the comparison results in two cases. Corollary 1.
When P R → ∞ , and N A ≤ N R , N B ≤ N R , N A + N B > N R , the maximumof secrecy sum rate of the two-phase two-way relay scheme keeps increasing when the twosource powers P A and P B increase while the maximum of secrecy sum rates of two-way directtransmission and three-phase scheme both approach constants. Thus, we have R P max ≥ max (cid:8) R DT max , R P max (cid:9) . (36) Proof:
It can be easily verified from Proposition 1, 2 and 3.
Remark 1.
As shown in [24], [28] for two-way direct transmission, in the infinite power case,the infinite maximum secrecy sum rate needs N A > N R or N B > N R . Proposition 1 reveals thatwith the signal alignment techniques at the untrusted relay, the infinite maximum secrecy sumrate only needs N A + N B > N R , which lowers the requirement of the numbers of antennas atthe two sources. The result clearly demonstrates the benefits of signal alignment for physicallayer security, which is the unique feature in two-way relaying. Corollary 2.
When P R → ∞ , P A → ∞ , P B → ∞ and N A > N R , N B > N R , R DT max ≥ R P max . (37) Proof:
When N i > N R , we have R DT max = X i ∈{ A,B }
12 [log P i + O (log P i )] (38) R P max = X i ∈{ A,B }
13 [log P i + O (log P i )] (39)where the order notation O ( P ) means that O ( P ) / P → as P → ∞ . Thus, the Corollary 2follows.From this Corollary we see that when the number of antennas at each source node is largerthan the number of antennas at the relay node, direct transmission performs better than thethree-phase two-way relaying at high SNR. B. The Case of High Relay Power and Low Source Powers
Proposition 4 (2P) . When P R → ∞ , P A → and P B → , the optimal source beamformingvectors of the two-phase two-way relay scheme are q A = √ P A ψψψ max (cid:0) H HA H B H HB H A (cid:1) k ψψψ max ( H HA H B H HB H A ) k , (40) q B = √ P B ψψψ max (cid:0) H HB H A H HA H B (cid:1) k ψψψ max ( H HB H A H HA H B ) k , (41) and the maximum secrecy sum rate is R P max ≈
12 ln 2 P A P B λ max (cid:0) H HA H B H HB H A (cid:1) . (42) Proof:
See Appendix E.Note that q A and q B are determined by the concatenated channel H HA H B . Proposition 5 (3P) . When P R → ∞ , P A → and P B → , the maximum secrecy sum rate ofthe three-phase two-way relay scheme satisfies
12 ln 2 X i ∈{ A,B } (cid:20) P i λ max (cid:18) T Hi T i − H Hi H i (cid:19)(cid:21) + ≤ R P max ≤
12 ln 2 X i ∈{ A,B } P i λ max (cid:0) T Hi T i (cid:1) . Proof:
Substituting the above upper bound and lower bound of lim P R →∞ R Ci ¯ i given in (56)into the (18), we can easily prove Proposition 5. Proposition 6 (DT) . When P A → and P B → , the maximum secrecy sum rate of the two-waydirect transmission scheme is, R DT max ≈
12 ln 2 X i ∈{ A,B } (cid:2) P i λ max (cid:0) T Hi T i − H Hi H i (cid:1)(cid:3) + and the optimal beamforming q DTi are given in (47) .Proof:
It is easily obtained from (48) or [24, Lemma 6].We find that different from the two-phase scheme, the secrecy sum rates of the direct transmis-sion and the three-phase scheme are closely related to the term T Hi T i − α H Hi H i ( α = 0 , , ). Corollary 3.
When P R → ∞ , P A → and P B → , we have R P max ≥ R DT max ≥ R P max . Proof:
This corollary can be easily obtained from Proposition 4, 5 and 6. Since H Hi H i arepositive semidefinite matrices, λ max (cid:0) T Hi T i − H Hi H i (cid:1) ≥ λ max (cid:0) T Hi T i − H Hi H i (cid:1) . Therefore,the three-phase two-way relay scheme is better than direct transmission scheme. In addition, R P max approaches zero faster than the other two schemes. Thus, the proof of Corollary 3 iscompleted.This Corollary clearly suggests that when the two source powers are extremely low, it is thebest to apply the three-phase two-way relay scheme for secure transmission. C. The Case of Low Relay Power
In this subsection, we present the asymptotic secrecy sum rate when relay power approacheszero.First, we briefly show when the relay power P R → , the maximum secrecy sum rate of thetwo-phase two-way relay scheme R P max approaches zero. As the relay power approaches zero, theinformation rate through the relay link goes to zero, which means that R PAB + R PBA approacheszero. On the other hand, the information rate leaked to untrusted relay R PR is not related to therelay power and does not approach zero. Therefore, the secrecy sum rate is zero when P R → . Corollary 4.
When the relay power P R → , R DT max ≥ R P max ≥ R P max . (43) Proof:
See Appendix G.Corollary 4 shows that the direct transmission is the best when the relay power is low. In therelay system without secrecy constraint, the similar conclusion hold [32].we can now summarize the main comparison results in Table I. Utilizing Table I, we canchoose the best transmission scheme under different scenarios.Note that besides the three schemes we considered in this work, four-phase one-way relayscheme is also possible for secure bi-directional transmission. In this four-phase scheme, theconventional one-way relaying is used twice for communications as A → R → B and B → R → A . It can be shown that this four-phase scheme is the best when P R → ∞ , P A → and TABLE IT
HE COMPARISON OF THE THREE SCHEMES IN TERMS OF THE MAXIMUM SECRECY SUM RATE . (I
N THE TABLE , WE USE ‘DT’, ‘2P’, ‘3P’
TO DENOTE THE THREE SCHEMES . A ND , A > B MEANS THAT SCHEME A IS BETTER THAN SCHEME
B.)Conditions Comparison P R → ∞ P A → , P B → > DT >
2P (Corollary 3) P A → ∞ , P B → ∞ N A + N B > N R , N A ≤ N R , N B ≤ N R > DT and 2P >
3P (Corollary 1) N A > N R , N B > N R DT >
3P (Corollary 2)Other cases Channel dependent P R → DT > >
2P (Corollary 4) P B → . For the other cases, this scheme is either suboptimal or the comparison depends onthe channel realization. V. S IMULATION R ESULTS AND D ISCUSSIONS
In this section, we perform simulation for all the cases discussed in section IV and V. Inthe simulation, we assume that the channel reciprocity holds, i.e., H A = G TA , H B = G TB and T A = T TB . The following example of channel coefficients realization (every entry of the matricesis generated from CN (0 , distribution) is used to show the asymptotical performance. ¯H A = . − . i . − . i . . i . . i − . . i − . . i . − . i − . − . i − . . i − . . i − . − . i − . − . i − . . i − . − . i . . i ¯H B = . . i − . − . i . − . i . − . i . . i − . − . i − . − . i . . i − . . i − . . i − . . i − . . i − . . i . . i . − . i ¯T A = . . i . − . i − . − . i . − . i − . − . i − . − . i − . − . i − . . i . . i If the channel matrix we need is smaller than the dimension of the above matrices, we simplychoose the left upper part of the corresponding matrix. For instance, if N A = 2 , N R = 3 , wechoose H A = ¯H A (1 : 3 , .Note that Algorithm 1 and 2 are not guaranteed to find the optimal solution and the convergentpoint may be far from the optimal solution. A method to cope with this problem is to randomlygenerate multiple initializations and choose the one with the best performance. Fig. 3 illustratesthe convergence behavior of Algorithm 1 with different initializations. It is seen that whenthe initialization vectors q A and q B are chosen based on the signal alignment technique, thealgorithm converges faster than the case of random generated vectors. Thus, in the rest of oursimulation, we choose the asymptotic optimal beamforming vectors shown in Section IV as theinitial points of q A and q B . A. High Relay Power and High Source Powers
Fig. 4, 5 and 6 compare the secrecy sum rates obtained by different schemes. Here the relaypower is fixed at P R = 40 dB, but the source powers are changing. The results for the two-phaseand three-phase two-way relay scheme are obtained using the Algorithm 1 and 2 proposed inSection III. For the direct transmission, we use the closed-form expression 48 given in AppendixA.Case N A = 2 , N R = 3 , N B = 2 : This is an example of the case when N A < N R , N B < N R and N A + N B < N R . The curve for signal alignment of 2P is obtained by forcing β H A q A = H B q B . Fig. 4 clearly verifies the importance of signal alignment for security asanalyzed in Corollary 1. We see that in Fig.4 the maximum secrecy sum rate of two-phasescheme goes to infinity with the increase of the source powers, while that of the other twoschemes reach floors. Under this channel setup, the upper bound of the secrecy sum rate of thedirect transmission scheme is about . bps/Hz and that of three-phase scheme is . bps/Hz.Case N A = 3 , N R = 2 , N B = 3 : This is an example of the case when N A > N R and N B > N R . As shown in Fig. 5, the maximum secrecy sum rate for these schemes all approachto infinity as the powers increase. We find that the direct transmission scheme is the best. Thisagrees with our analysis in Corollary 2. Actually, as shown in (38) and (39), the degrees offreedom of the direct transmission scheme is one and the degrees of freedom of the three-phasescheme is . In this case, although the signal alignment of the two-phase scheme is feasible, the direct transmission scheme is better than the two-phase scheme.Case N A = 2 , N R = 5 , N B = 2 : This is the scenario when N A + N B ≤ N R . Underthis condition, all the schemes have upper bounds for their secrecy sum rates. The comparisonresults are shown in Fig. 6. It is shown that the two-phase scheme is the best. We also plot thecurve for two-phase scheme when P R = 50 dB. The curve can approach the upper bound moreclosely than the curve when P R = 40 dB. This implies that to approach the upper bound givenin (25), we need the powers of all the three nodes go to infinity and the relay power should bemuch larger than the source powers. In this case, although the signal alignment of the two-phasescheme cannot be achieved, the two-phase scheme is better than the direct transmission scheme.From Fig. 4 and Fig. 6, we can see that increasing the number of antennas at the relay reducesthe performance. This is in contrast to the relay system without secrecy constraints, where withmore antennas at the relay, the performance will be better. B. High Relay Power and Low Source Powers
Fig. 7 shows the performance of three schemes when P R = 40 dB and the source powers arelow. We find that the two-phase scheme is much worse than the other schemes and three-phasescheme is better than the direct transmission scheme, which verifies Corollary 3. By carefulobservation, we see that R P max decreases to zero as twice faster as R DT max and R P max when thesource powers tend to zero. Moreover, we also find that the asymptotical results are quite accuratewhen the source powers are low. C. Low Relay Power
In Fig. 8, we compare the three schemes when the relay power is as low as -20dB. We findthat the maximum secrecy sum rate of two-phase scheme is close to zero and direct transmissionis better than three-phase scheme, which verifies Corollary 4. The reason is that the only link A ⇆ R ⇆ B of the two-phase scheme is very weak while there are strong direct links in theother two schemes with high source powers. D. General Relay SNR and Fading Channels
We have considered the high relay power and low relay power case. In this subsection, weconsider the general relay power. For this case, this is no asymptotic results and we perform simulation with different channel realizations (every entry of the matrices is generated from CN (0 , distribution) and obtain average secrecy sum rate. For the two-phase and three-phasescheme, the simulation results are obtained by Algorithm 1 and 2.In Fig. 9, we compare the average secrecy sum rates of the three schemes with varying relaypower. The source powers are fixed at dB and N A = N B = 2 , N R = 3 . The average rate of thedirect transmission scheme does not change with the relay power as the relay does not transmitin this scheme. The average rate of the three-phase scheme increases with the relay power andhas similar performance with direct transmission at high relay power. For the two-phase scheme,as the relay power increases, the average rate rises from zero to as high as . bps/Hz. We cansee that the two-phase scheme is much better than the other two schemes when relay poweris high. The reason is that in this case, signal alignment can be achieved when P R is large as N A + N B > N R .In Fig. 10, we plot the average secrecy sum rate versus the relay antenna number. The sourcenode A and B both have three antennas. The relay power is dB and the source powers are dB.From the figure, we see that the average rate of the direct transmission scheme monotonicallydecreases with N R . The reason is that the untrusted relay will be more powerful to eavesdrop thedirect transmission signal as N R increases. For the two-phase transmission scheme, the averagerate achieves the largest value when N R = 4 . The reason is that when N R is small, the relaydoes not have enough abilities to help the two-way transmission and when N R is large, the relaywill be more powerful to decode the received signals. For the three-phase scheme, the averagerate also decreases with N R in this case.VI. C ONCLUSION
In this paper, we investigated a MIMO two-way AF relay system where the two source nodesexchange confidential information with an untrusted relay. For both two-phase and three-phasetwo-way relay schemes, we proposed efficient algorithms to jointly design the secure sourceand relay beamformers iteratively. Furthermore, we analyzed the asymptotical performance ofthe secure beamforming schemes in low and high power regimes of the sources and relay.Simulation results validate our asymptotical analysis.From these results, we can conclude that the conventional two-way direct transmission ispreferred when the relay power goes to zero. When the relay power approaches infinity and source powers approach zero, the three-phase two-way relay scheme performs best. Moreover,when all powers go to infinity, the two-phase two-way relay scheme has the best performance ifsignal alignment techniques are used, which also lowers the requirement of numbers of antennasat the source nodes for security. A PPENDIX AS ECURE B EAMFORMING OF T WO -W AY D IRECT T RANSMISSION S CHEME
For the two-way direct transmission scheme, the transmission consists of two time slots. Inthe first time slot, A transmits while B and R listen. During the second time slot, B transmitswhile A and R listens. The received signals at B and R in the first time slot are respectivelygiven by y DTB = T A q A s A + n B , (44) y DTR = H A q A s A + n R , (45)and the received signals in the second time slot are similar.An achievable secrecy sum rate of this two-way direct transmission scheme given by [28] is, R DTs = X i ∈{ A,B } (cid:20) log q Hi T Hi T i q i q Hi H Hi H i q i (cid:21) + . (46)We want to maximize the secrecy sum rate R DTs subject to the source power constraints. Ac-cording to [33], [28] and [24], the optimal beamforming q DTi of the two-way direct transmissionscheme is given by q DTi = p P DTi ψψψ max ( I + P DTi T Hi T i , I + P DTi H Hi H i ) k ψψψ max ( I + P DTi T Hi T i , I + P DTi H Hi H i ) k , i ∈ { A, B } , (47)and the maximum secrecy sum rate is given by R DT max ( P DTA , P
DTB )= X i ∈{ A,B } (cid:2) log λ max (cid:0) I + P DTi T Hi T i , I + P DTi H Hi H i (cid:1)(cid:3) + , (48)where the factor of is due to the use of two orthogonal phases. ∂B ( A , µ ) ∂ A ∗ = − X i ∈{ A,B } log e V H G Hi K − i G i FH ¯ i q ¯ i q H ¯ i H H ¯ i U − V H G Hi K − i G i FH ¯ i q ¯ i q H ¯ i H H ¯ i F H G Hi K − i G i FU (cid:0) q H ¯ i H H ¯ i F H G Hi K − i G i FH ¯ i q ¯ i (cid:1) + µ AU H H A q A q HA H HA U + AU H H B q B q HB H HB U + A P PR − Tr (cid:0) AU H H A q A q HA H HA UA H + AU H H B q B q HB H HB UA H + AA H (cid:1) (51)A PPENDIX BS EARCH A USING G RADIENT M ETHOD
Substituting (22) into (8), we obtain a subproblem of optimizing A given q A and q B asfollows, min A − R Ps (49a)s. t. Tr (cid:0) AU H H A q A q HA H HA UA H (49b) + AU H H B q B q HB H HB UA H + AA H (cid:1) ≤ P PR . (49c)The logarithmic barrier function associated with (49) is, B ( A , µ ) = −R Ps − µ ln (cid:16) P PR − Tr (cid:0) AU H H A q A q HA H HA UA H + AU H H B q B q HB H HB UA H + AA H (cid:1)(cid:17) (50)where µ > is the barrier parameter.The gradient of B ( A , µ ) with respect to A is given by (51) shown at the top of the next page,With this gradient, we use gradient descent method to search A .A PPENDIX CS EARCH O PTIMAL q B G IVEN F AND q A IN TWO - PHASE TWO - WAY RELAY S CHEME
First, we rewrite (8) in the homogenized form with respect to q B , as (52) shown at the top ofthe next page. Then, we can follow the same procedure in [34, Section III-B] or [24, AppendixA] to find the optimal q B . The basic idea is to first relax (52) into a fractional semidefiniteprogramming problem, which is then transformed to a SDP problem using Charnes-Coopervariable transformation. At last, the rank-one matrix decomposition theorem [35, Theorem 2.3]is used. Here we omit the details. max q B ,t Tr H HB F H G HA K − A G A FH B q B q HB q B t ∗ q HB t | t | Tr H HB (cid:0)(cid:0) k H A q A k (cid:1) I − H A q A q HA H HA (cid:1) H B k H A q A k q B q HB q B t ∗ q HB t | t | (52a) s.t. Tr I 00 q B q HB q B t ∗ q HB t | t | ≤ P B , (52b) Tr H HB F H FH B q B q HB q B t ∗ q HB t | t | ≤ P r − Tr (cid:8) FF H (cid:9) − Tr (cid:8) FH A q A q HA H HA F H (cid:9) , (52c) Tr q B q HB q B t ∗ q HB t | t | = 1 . (52d)A PPENDIX DP ROOF OF L EMMA N R > N A + N B .Without loss of generality, we can express F i as F i = h V V ⊥ i a i B i c i D i U Hi U ⊥ Hi (53)where V is from (23) , V ⊥ ∈ C N R × ( N R − N A − N B ) such that (cid:2) V V ⊥ (cid:3) is unitary , U i is H i q i k H i q i k , U ⊥ i ∈ C N R × ( N R − such that (cid:2) U i U ⊥ i (cid:3) is unitary, and a i ∈ C ( N A + N B ) × , c i ∈ C ( N R − N A − N B ) × , B i ∈ C ( N A + N B ) × ( N R − , D i ∈ C ( N R − N A − N B ) × ( N R − . Therefore, we obtain (54) shown at thetop of the next page. Therein, ( a ) is from the above property of F i (53), ( b ) is from that P i =1 G A U i B i B Hi U Hi G HA is positive semidefinite matrix. We see that the information rate from Bto A R PBA = log (1 + q HB H HB H B q B + x BA ) is not related to c i and D i and achieves a upperbound when B i = . Similarly, the information rate from A to B, R PAB , is also not related to c i and D i and achieves a upper bound when B i = . In addition, the power consumed by the x BA , q HB H HB F HB G HA (cid:0) G A (cid:0) F A F HA + F B F HB (cid:1) G HA + I (cid:1) − G A F B H B q B ( a ) = k H B q B k a H V H G HA X i =1 G A Va i a Hi V H G HA + X i =1 G A VB i B Hi V H G HA + I ! − G A Va b ) ≤ k H B q B k a H V H G HA X i =1 G A Va i a Hi V H G HA + I ! − G A Va (54)relay is Tr (cid:0) F A H A q A q HA H HA F HA + F B H B q B q HB H HB F HB + F A F HA + F B F HB (cid:1) = k H A q A k (cid:0) k a k + k c k (cid:1) + k H B q B k (cid:0) k a k + k c k (cid:1) + X i =1 k a i k + X i =1 k B i k F + X i =1 k c i k + X i =1 k D i k F We find that the relay power is increased when B i , c i , D i is not zero. Therefore, it leads to B i = , c i = and D i = .When N R ≤ N A + N B , we can express F i as F i = V h a i B i i U Hi U ⊥ Hi (55)where V is from (23) , U i is H i q i k H i q i k , U ⊥ i ∈ C N R × ( N R − such that (cid:2) U i U ⊥ i (cid:3) is unitary, and a i ∈ C N R × , B i ∈ C N R × ( N R − . Similar as the above case, we can prove that the optimal B i = . A PPENDIX EP ROOF OF P ROPOSITION P A → , P B → into (27), we have lim P R →∞ R Ps = 12 log − q HB H HB H A q A q HA H HA H B q B ( q HB H HB H B q B )( q HA H HA H A q A )= −
12 log (cid:18) − q HB H HB H A q A q HA H HA H B q B (1 + q HB H HB H B q B ) (1 + q HA H HA H A q A ) (cid:19) ≈ −
12 log (cid:0) − q HB H HB H A q A q HA H HA H B q B (cid:1) ≈
12 ln 2 (cid:13)(cid:13) q HB H HB H A q A (cid:13)(cid:13) . To maximize k q HB H HB H A q A k , we obtain Proposition 4.A PPENDIX FP ROOF OF P ROPOSITION q HB H HB F HB G HA (cid:0) G A (cid:0) F A F HA + F B F HB (cid:1) G HA + I (cid:1) − G A F B H B q B = k H B q B k a H V H G HA X i =1 G A Va i a Hi V H G HA + I ! − G A Va a ) ≤ k H B q B k a H V H G HA (cid:0) G A Va a H V H G HA + I (cid:1) − G A Va b ) = k H B q B k a H V H G HA (cid:16) I − G A Va (cid:0) a H V H G HA G A Va + 1 (cid:1) − · a H V H G HA (cid:17) G A Va = k H B q B k k G A Va k k G A Va k ≤ k H B q B k where ( a ) is from that G A Va a H V H G HA is positive semidefinite, ( b ) is from the matrix inverselemma. The above third term in (16) also has a lower bound by simply letting a = a = ¯a , q HB H HB F HB G HA (cid:0) G A (cid:0) F A F HA + F B F HB (cid:1) G HA + I (cid:1) − · G A F B H B q B = k H B q B k a H V H G HA (cid:0) G A Vaa H V H G HA + I (cid:1) − G A Va = 12 k H B q B k (cid:16) − (cid:0) a H V H G HA G A Va (cid:1) − (cid:17) = 12 k H B q B k k G A Va k k G A Va k → k H B q B k as P R → ∞ Therefore, we have
13 log (cid:18) q Hi T Hi T i q i + 12 q Hi H Hi H i q i (cid:19) ≤ lim P R →∞ R Pi ¯ i ≤
13 log (cid:0) q Hi T Hi T i q i + q Hi H Hi H i q i (cid:1) . (56)To prove Proposition 2, we first substitute the upper bound and lower bound into (18). Afterthat, the proof procedure of Proposition 2 is similar to the proof of [24, Lemma 7]. In addition,we assume that the entries of channel matrices are generated from continuous distribution.A PPENDIX GP ROOF OF C OROLLARY P i = P DTi = P Pi = P Pi , i ∈ { A, B } and P R = P PR = P PR .When the relay power P R → , there are only direct links between the two source nodes forthe three-phase scheme. Thus, the maximum secrecy sum rate of the three-phase two-way relayscheme R P max is R P max (57) ≈ max q A , q B X i ∈{ A,B } (cid:20) log q Hi T Hi T i q i q Hi H Hi H i q i (cid:21) + = 13 X i ∈{ A,B } (cid:2) log (cid:0) λ max (cid:0) I + P Pi T Hi T i , I + P Pi H Hi H i (cid:1)(cid:1)(cid:3) + = 13 X i ∈{ A,B } (cid:20) log (cid:18) λ max (cid:18) I + 32 P i T Hi T i , I + 32 P i H Hi H i (cid:19)(cid:19)(cid:21) + . In addition, we have λ max (cid:18) I + 32 P i T Hi T i , I + 32 P i H Hi H i (cid:19) ( a ) = λ max (cid:18) P i T Hi T i − P i H Hi H i , I + 32 P i H Hi H i (cid:19) + 1= max ψψψ ψψψ H (cid:0) P i T Hi T i − P i H Hi H i (cid:1) ψψψψψψ H (cid:0) I + P i H Hi H i (cid:1) ψψψ + 1 ≤ max ψψψ ψψψ H (cid:0) P i T Hi T i − P i H Hi H i (cid:1) ψψψψψψ H ( I + P i H Hi H i ) ψψψ + 1= 32 λ max (cid:0) P i T Hi T i − P i H Hi H i , I + P i H Hi H i (cid:1) + 1 ( b ) ≤ (cid:0) λ max (cid:0) P i T Hi T i − P i H Hi H i , I + P i H Hi H i (cid:1) + 1 (cid:1) ( c ) = (cid:0) λ max (cid:0) I + P i T Hi T i , I + P i H Hi H i (cid:1)(cid:1) , where ( a ) and ( c ) are from λ max ( A , B ) = λ max ( A − B , B ) + 1 , ( b ) is from x + 1 ≤ x when x is a nonnegative real number.Therefore, we obtain R DT max ≥ R P max when P R → . Together with R P max → when P R → ,we obtain Proposition 4. R EFERENCES [1] J. Mo, M. Tao, Y. Liu, B. Xia, and X. Ma, “Secure beamforming for mimo two-way transmission with an untrusted relay,”in
IEEE Wireless Communications and Networking Conference (WCNC) , 2013, pp. 4180–4185.[2] R. Zhang, Y.-C. Liang, C. C. Chai, and S. Cui, “Optimal beamforming for two-way multi-antenna relay channel withanalogue network coding,”
IEEE J. Sel. Areas Commun. , vol. 27, no. 5, pp. 699–712, 2009.[3] S. Xu and Y. Hua, “Optimal design of spatial source-and-relay matrices for a non-regenerative two-way MIMO relaysystem,”
IEEE Trans. Wireless Commun. , vol. 10, no. 5, pp. 1645 –1655, May 2011.[4] R. Wang and M. Tao, “Joint source and relay precoding designs for MIMO two-way relaying based on MSE criterion,”
IEEE Trans. Signal Process. , vol. 60, no. 3, pp. 1352 –1365, march 2012.[5] L. Lai and H. El Gamal, “The relay–eavesdropper channel: Cooperation for secrecy,”
IEEE Trans. Inf. Theory , vol. 54,no. 9, pp. 4005–4019, Sep. 2008.[6] C. Jeong and I.-M. Kim, “Optimal power allocation for secure multicarrier relay systems,”
IEEE Trans. Signal Process. ,vol. 59, no. 11, pp. 5428 –5442, Nov. 2011.[7] L. Dong, Z. Han, A. Petropulu, and H. Poor, “Improving wireless physical layer security via cooperating relays,”
IEEETrans. Signal Process. , vol. 58, no. 3, pp. 1875 –1888, Mar. 2010.[8] I. Krikidis, J. Thompson, and S. Mclaughlin, “Relay selection for secure cooperative networks with jamming,”
IEEE Trans.Wireless Commun. , vol. 8, no. 10, pp. 5003–5011, Oct. 2009. [9] D. Ng, E. 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] J. Huang and A. Swindlehurst, “Cooperative jamming for secure communications in MIMO relay networks,”
IEEE Trans.Signal Process. , vol. 59, no. 10, pp. 4871 –4884, Oct. 2011.[11] J. Mo, M. Tao, and Y. Liu, “Relay placement for physical layer security: A secure connection perspective,”
IEEE Commun.Lett. , vol. 16, no. 6, pp. 878 –881, june 2012.[12] J. Chen, R. Zhang, L. Song, Z. Han, and B. Jiao, “Joint relay and jammer selection for secure two-way relay networks,”
IEEE Trans. Inf. Forensics Security , vol. 7, no. 1, pp. 310 –320, Feb. 2012.[13] Z. Ding, M. Xu, J. Lu, and F. Liu, “Improving wireless security for bidirectional communication scenarios,”
IEEE Trans.Veh. Technol. , vol. 61, no. 6, pp. 2842 –2848, Jul. 2012.[14] A. Mukherjee and A. L. Swindlehurst, “Securing multi-antenna two-way relay channels with analog network coding againsteavesdroppers,” in
Proc. IEEE Eleventh Int Signal Processing Advances in Wireless Communications (SPAWC) Workshop ,2010, pp. 1–5.[15] 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. 60, no. 7, pp. 3532 –3545, Jul. 2012.[16] T. Shimizu, H. Iwai, and H. Sasaoka, “Physical-layer secret key agreement in two-way wireless relaying systems,”
IEEETrans. Inf. Forensics Security , vol. 6, no. 3, pp. 650 –660, Sep. 2011.[17] E. Tekin and A. Yener, “The general gaussian multiple-access and two-way wiretap channels: Achievable rates andcooperative jamming,”
IEEE Trans. Inf. Theory , vol. 54, no. 6, pp. 2735 –2751, Jun. 2008.[18] Y. Oohama, “Coding for relay channels with confidential messages,” in
Information Theory Workshop.
IEEE, 2001, pp.87–89.[19] X. He and A. Yener, “Two-hop secure communication using an untrusted relay: A case for cooperative jamming,” in
IEEEGlobal Telecommunications Conference, 2008 , 30 2008-dec. 4 2008, pp. 1 –5.[20] ——, “Two-hop secure communication using an untrusted relay,”
EURASIP J. Wirel. Commun. Netw. , vol. 2009, pp.9:1–9:10, May 2009. [Online]. Available: http://dx.doi.org/10.1155/2009/305146[21] L. Sun, T. Zhang, Y. Li, and H. Niu, “Performance study of two-hop amplify-and-forward systems with untrustworthyrelay nodes,”
IEEE Trans. Veh. Technol. , vol. PP, no. 99, p. 1, 2012.[22] X. He and A. Yener, “Cooperation with an untrusted relay: A secrecy perspective,”
IEEE Trans. Inf. Theory , vol. 56, no. 8,pp. 3807 –3827, Aug. 2010.[23] R. Zhang, L. Song, Z. Han, and B. Jiao, “Physical layer security for two-way untrusted relaying with friendly jammers,”
IEEE Trans. Veh. Technol. , vol. 61, no. 8, pp. 3693 –3704, Oct. 2012.[24] C. Jeong, I.-M. Kim, and D. I. Kim, “Joint secure beamforming design at the source and the relay for an amplify-and-forward MIMO untrusted relay system,”
IEEE Trans. Signal Process. , vol. 60, no. 1, pp. 310 –325, Jan. 2012.[25] J. Huang, A. Mukherjee, and A. Swindlehurst, “Secure communication via an untrusted non-regenerative relay in fadingchannels,”
IEEE Trans. Signal Process. , vol. 61, no. 10, pp. 2536–2550, 2013.[26] B. Rankov and A. Wittneben, “Spectral efficient protocols for half-duplex fading relay channels,”
IEEE J. Sel. AreasCommun. , vol. 25, no. 2, pp. 379 –389, Feb. 2007.[27] S. J. Kim, P. Mitran, and V. Tarokh, “Performance bounds for bidirectional coded cooperation protocols,”
IEEE Trans. Inf.Theory , vol. 54, no. 11, pp. 5253–5241, Aug. 2008. (cid:17)(cid:17)(cid:17) A (cid:17)(cid:17)(cid:17)(cid:17)(cid:17)(cid:17) B (cid:17)(cid:17)(cid:17) H A G A H B G B T A T B R(E) (cid:17)(cid:17)(cid:17)
Fig. 1. MIMO two-way relay model.
MAC phaseBC phaseSelf-interferencecancellation Source Node A Relay Node R Source Node B
Fig. 2. The signal vectors of the two-phase two-way relaying scheme.[28] A. Khisti and G. Wornell, “Secure transmission with multiple antennas I: The MISOME wiretap channel,”
IEEE Trans.Inf. Theory , vol. 56, no. 7, pp. 3088 –3104, Jul. 2010.[29] N. Lee, J.-B. Lim, and J. Chun, “Degrees of freedom of the mimo y channel: Signal space alignment for network coding,”
IEEE Trans. Inf. Theory , vol. 56, no. 7, pp. 3332 –3342, july 2010.[30] G. Strang,
Linear Algebra and Its Applications . Pacific Grove, CA, USA, 2004.[31] G. H. Golub and C. F. Van Loan,
Matrix computations . John Hopkins University Press, 2012.[32] M. Chen and A. Yener, “Power allocation for F/TDMA multiuser two-way relay networks,”
IEEE Trans. Wireless Commun. ,vol. 9, no. 2, pp. 546–551, 2010.[33] S. Shafiee and S. Ulukus, “Achievable rates in gaussian miso channels with secrecy constraints,” in
IEEE InternationalSymposium on Information Theory, 2007 , Jun. 2007, pp. 2466 –2470.[34] A. De Maio, Y. Huang, D. Palomar, S. Zhang, and A. Farina, “Fractional QCQP with applications in ML steering directionestimation for radar detection,”
IEEE Trans. Signal Process. , vol. 59, no. 1, pp. 172–185, 2011.[35] W. Ai, Y. Huang, and S. Zhang, “New results on hermitian matrix rank-one decomposition,”
Mathematical programming ,vol. 128, no. 1-2, pp. 253–283, 2011. A and q B S e c r e cy s u m r a t e ( bp s / H z ) q A ,q B from signal alignment and A from 4 random initial pointsq A , q B from signal alignment and A from 8 random initial pointsq A , q B and A from 4 random initial pointsq A , q B and A from 8 random initial points Fig. 3. Convergence behaviour comparison of different initialization methods for Algorithm 1. N A = N B = 2 , N R = 3 , P R = 30 dB and P A = P B = 10 dB. P A =P B (dB) S e c r e cy s u m r a t e ( bp s / H z ) DTUpper bound of DT2PSignal alignment of 2P3PUpper bound of 3P
Fig. 4. Comparison of the three schemes in high power regimes when N A = 2 , N R = 3 , N B = 2 and P R = 40 dB. A =P B (dB) S e c r e cy s u m r a t e ( bp s / H z ) DTDT, asymptotic (Prop. 3)2P2P, asymptotic (Prop. 1)3P3P, asymptotic (Prop. 2)
Fig. 5. Comparison of the three schemes in high power regimes when N A = 3 , N R = 2 , N B = 3 and P R = 40 dB. A =P B (dB) S e c r e cy s u m r a t e ( bp s / H z ) DTUpper bound of DT (Prop. 3)2P, P R =40dB2P, P R =50dBUpperbound of 2P (Prop. 1)3PUpperbound of 3P (Prop. 2) Fig. 6. Comparison of the three schemes in high power regimes when N A = 2 , N R = 5 , N B = 2 and P R = 40 dB. −20 −18 −16 −14 −12 −10 −8 −6 −4 −2 010 −4 −3 −2 −1 P A =P B (dB) S e c r e cy s u m r a t e ( bp s / H z ) DTDT, asymptotic (Prop. 6)2P2P, asymptotic (Prop. 4)3PUpper bound of 3P (Prop. 5)Lower bound of 3P (Prop. 5)
Fig. 7. Comparison of the three schemes with high relay power when N A = 2 , N R = 3 , N B = 2 and P R = 40 dB. −20 −15 −10 −5 0 5 10 15 2000.20.40.60.811.21.41.61.82 P A =P B (dB) S e c r e cy s u m r a t e ( bp s / H z ) DT2P3P3P, asymptotic (Eq. 57 )
Fig. 8. Comparison of the three schemes with low relay power. N A = 2 , N R = 3 , N B = 2 and P R = − dB. −10 −5 0 5 10 15 20 25 30 35 4000.511.522.5 P R (dB) A v e r age s e c r e cy s u m r a t e ( bp s / H z ) DT2P3P
Fig. 9. Comparison of the three schemes with varying relay power, P A = P B = 15 dB, N A = N B = 2 , N R = 3 . N R A v e r age s e c r e cy s u m r a t e DT2P3P
Fig. 10. Comparison of the three schemes with varying relay antenna number, P A = P B = 15 dB, P R = 25= 25