Beamforming in Non-Regenerative Two-Way Multi ... - IEEE Xplore

IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 10, OCTOBER 2014

5509

Beamforming in Non-Regenerative Two-Way Multi-Antenna Relay Networks for Simultaneous Wireless Information and Power Transfer Quanzhong Li, Qi Zhang, Member, IEEE, and Jiayin Qin

Abstract—Simultaneous wireless information and power transfer (SWIPT) is able to prolong the lifetime of energy constrained wireless networks. In this paper, we consider the relay beamforming design problem for SWIPT scheme in a non-regenerative two-way multi-antenna relay network. Our objective is to maximize the sum rate of two-way relay network under the transmit power constraint at relay and the energy harvesting (EH) constraint at EH receiver. For the non-convex EH-constrained relay beamforming optimization problem, we propose an iterative algorithm to find the global optimal solution based on semidefinite programming and rank-one decomposition theorem. To reduce computational complexity of global optimal solution, we transform the EH-constrained optimization problem to a difference of convex programming and propose a constrained concave convex procedure based iterative algorithm to find a local optimum. To further reduce the complexity, we propose a suboptimal solution based on the generalized eigenvectors method. When the case of multi-antenna sources is considered, we propose the alternating optimization based iterative algorithms. It is shown from simulations that considering the EH constraint, our proposed schemes outperform conventional relay beamforming schemes in the literature. Index Terms—Beamforming, energy harvesting (EH), nonregenerative, simultaneous wireless information and power transfer (SWIPT), two-way relay network.

I. I NTRODUCTION

S

IMULTANEOUS wireless information and power transfer (SWIPT) is a promising energy harvesting (EH) technique to solve the energy scarcity problem in energy constrained wireless communications [1]–[11]. The initial researches on SWIPT focus on point-to-point single-antenna wireless communication systems [1], [2]. Later on, employing SWIPT schemes in the multi-antenna wireless communication systems was extensively studied [3]–[11]. The SWIPT schemes for multiple-input-multiple-output (MIMO) channel and multipleinput-single-output (MISO) broadcast channel with a single information decoding (ID) receiver were investigated in [3]–[5]. For multiple ID receivers, the transmit beamforming design was

Manuscript received September 27, 2013; revised January 11, 2014 and March 21, 2014; accepted April 29, 2014. Date of publication May 5, 2014; date of current version October 8, 2014. This work was supported in part by the National Natural Science Foundation of China under Grant 61173148 and Grant 61202498 and in part by the Scientific and Technological Project of Guangzhou City under Grant 12C42051578. The associate editor coordinating the review of this paper and approving it for publication was C.-B. Chae. The authors are with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou 510006, China (e-mail: liquanzhong2009@ gmail.com; [email protected]; [email protected]). Digital Object Identifier 10.1109/TWC.2014.2321763

studied for SWIPT in MISO broadcast channel [6], [7]. In [8], [9], the SWIPT in a two-user MISO interference channel was considered. For the SWIPT scheme in non-regenerative oneway MIMO relay networks, rate-energy tradeoff was investigated in [10], [11]. In relay networks, two-way multi-antenna relaying scheme is able to enlarge wireless coverage and enhance spectral efficiency. Two-way multi-antenna relay networks have been extensively investigated in the literature [12]–[15]. In [12], the capacity region of the two-way relay channel with linear beamforming at the multi-antenna relay was analyzed where the single-antenna sources were considered. In [13], the beamforming design for multi-antenna sources and relay was considered to maximize the weighted sum rate by employing alternating optimization (AO). In [14], the algebraic normmaximizing (ANOMAX) transmission scheme which maximizes the weighted sum of the Frobenius norms of the effective channels was derived. In [15], the sum rate of two-way multi-antenna relay network with two single-antenna sources was maximized by employing the polynomial time difference of convex (POTDC) algorithm and generalized eigenvectors (GES) method. In [16], the linear precoder was designed for cooperative cognitive two-way relay systems to enhance the performance of the secondary user. In [17], the multiuser twoway relay system was investigated with the objective to maximize the sum rate or to minimize the sum mean-square-error (MSE) by designing the linear precoders for the base station and the relay. To the best of our knowledge, the research on beamforming design for SWIPT in non-regenerative two-way multi-antenna relay networks is missing. In this paper, we consider the relay beamforming design for SWIPT in the non-regenerative two-way multi-antenna relay network. Our objective is to maximize the sum rate of two-way relay network under the transmit power constraint at relay and the EH constraint at EH receiver. Because the EH constraint is non-convex, the considered relay beamforming design is much more challenging than the conventional relay beamforming design in [12]–[15]. The conventional schemes proposed in [12]–[15] cannot be applied directly for SWIPT in the two-way relay network. For the non-convex EH-constrained relay beamforming optimization problem, we propose an iterative algorithm to find the global optimal solution based on semidefinite programming (SDP) [18] and rank-one decomposition theorem [19]. The proposed iterative algorithm includes two loops, i.e., the outer-loop and inner-loop. The outer-loop is one-dimensional (1-D) linear

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search and the inner-loop is 1-D bisection search. To reduce the computational complexity, we transform the EH-constrained relay beamforming optimization problem to a difference of convex (DC) programming [20]. To solve this DC programming, we propose a constrained concave convex procedure (CCCP) based iterative algorithm [21] to find a local optimum. During each iteration of CCCP based iterative algorithm, a second-order cone programming (SOCP) [22] is solved. To further reduce the computational complexity, we propose a suboptimal solution based on the GES method [15]. We convert the EH-constrained relay beamforming optimization problem to an unconstrained optimization problem by restricting the beamforming vector to lie on the subspace of the original constraint set. Furthermore, we employ the conclusion in [15] to simplify the two-dimensional (2-D) search to 1-D bisection search to obtain the suboptimal solution. When the case of multi-antenna sources is considered, we also propose the AO based iterative algorithm. The rest of this paper is organized as follows. In Section II, we describe the system model and problem formulation. In Sections III–V, we propose the global optimal solution, the local optimal solution, and the low-complexity suboptimal solution, respectively. In Section VI, we propose the AO based iterative algorithm for the case of multi-antenna sources. Simulation results are provided in Section VII. We conclude our paper in Section VIII. Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The A∗ , AT , A† , A, and tr(A) denote the conjugate, transpose, conjugate transpose, Frobenius norm and trace of the matrix A, respectively. The ⊗ denotes the Kronecker product. The vec(A) denotes to stack the columns of a matrix A into a single vector a. The Re{a} denotes the real part of a. a denotes the Euclidean norm of a. The λmax (A) and λmin (A) denote the maximum and minimum eigenvalues of A, respectively. The ψ(A) denotes the eigenvector of A associating with the maximum eigenvalue. By A 0 or A 0, we mean that the matrix A is positive semidefinite or positive definite, respectively. a denotes the integer ceil of the scalar a.

Fig. 1. A non-regenerative two-way multi-antenna relay network for simultaneous wireless information and power transfer.

where hi,f ∈ CN ×1 , i ∈ {1, 2}, denotes the froward channel response vector from source i to relay, nr ∈ CN ×1 is the additive Gaussian noise vector at the relay which has zero mean and the covariance matrix σ 2 I. The harvested energy at the EH receiver is given by [3] E1 = P1 g1 2 + P2 g2 2

(2)

where gi ∈ CL×1 denotes the channel response from source i to the EH receiver, Pi = E[|xi |2 ] is the average transmit power of source i. In the second phase, the relay multiplies the received signal by a beamforming matrix, denoted as W ∈ CN ×N , and forwards the multiplication result to two sources. Thus, the transmit signal from the relay is ˜r = Wr

(3)

By using the equality vec(A1 A2 A3 ) = (AT3 ⊗ A1 )vec(A2 ), the transmit power at the relay is expressed as [15] E ˜r2 = w† Cw

(4)

where II. S YSTEM M ODEL AND P ROBLEM F ORMULATION A. System Model Consider a non-regenerative two-way relay network, shown in Fig. 1, which consists of two sources, a relay, and an energy harvesting (EH) receiver. As in [12], [15], each of the sources is equipped with a single antenna. The relay is equipped with N antennas and the EH receiver is equipped with L antennas. We assume that the direct link between the sources is sufficiently weak to be ignored. This occurs when the direct link is blocked due to long-distance path loss or obstacles [12], [14], [15]. The exchange of information symbols between source 1 and source 2 is divided into two phases. In the first phase, source 1 and source 2 simultaneously transmit the symbol x1 and x2 , respectively, to the relay. The received signal at the relay is expressed as r = h1,f x1 + h2,f x2 + nr

(1)

w = vec(W)

(5)

T C = P1 h1,f h†1,f + P2 h2,f h†2,f + σ 2 I ⊗ I.

(6)

The received signals at source 1 and source 2, denoted as y1 and y2 , respectively, are given by y1 = hT1,b Wh2,f x2 + hT1,b Wh1,f x1 + hT1,b Wnr + n1 ,

(7)

y2 = hT2,b Wh1,f x1 + hT2,b Wh2,f x2 + hT2,b Wnr + n2 ,

(8)

where hi,b ∈ CN ×1 denotes the backward channel response vector from relay to source i, ni is the additive Gaussian noise at source i which has zero mean and the variance σ 2 . Since the source i knows its own transmitted signal, it subtracts the resulting self-interference term hTi,b Whi,f xi from the received

LI et al.: BEAMFORMING IN NON-REGENERATIVE TWO-WAY MULTI-ANTENNA RELAY NETWORKS

signal. Thus, the remaining received signals at source 1 and source 2, denoted as y˜1 and y˜2 , respectively, are y˜1 = hT1,b Wh2,f x2 + hT1,b Wnr + n1 ,

(9)

y˜2 = hT2,b Wh1,f x1 + hT2,b Wnr + n2 .

(10)

From (9) and (10), the received signal-to-noise ratio (SNR) at source 1 and source 2, denoted as γ1 and γ2 , respectively, are γ1 =

w† Q2,1 w , w † R1 w + σ 2

(11)

γ2 =

w† Q1,2 w w † R2 w + σ 2

(12)

where Q2,1 = P2 Q1,2 = P1 Ri =

T , h2,f h†2,f ⊗ h1,b h†1,b

(13)

T h1,f h†1,f ⊗ h2,b h†2,b ,

(14)

σ I ⊗ 2

hi,b h†i,b

T (15)

in which the equalities vec(A1 A2 A3 ) = (AT3 ⊗ A1 )vec(A2 ) and (A1 ⊗ A2 )(A3 ⊗ A4 ) = (A1 A3 ) ⊗ (A2 A4 ) have been employed. In the second phase, the harvested energy at the EH receiver is given by [10] E2 = w† Dw

(16)

In this paper, we should investigate the optimization problem of beamforming vector w which maximizes the sum rate of two-way relay network under the transmit power constraint at relay and the EH constraint at EH receiver. The optimization problem is formulated as follows:

w† Q2,1 w w† Q1,2 w 1+ † · 1 + max w w R1 w + σ 2 w † R2 w + σ 2 ¯ s.t. w† Cw ≤ Pr , w† Dw ≥ Q (22) where (1/2) log2 (·) in (18) is omitted due to its monotonicity ¯ = Q − E1 . It is noted that the sum rate optimization and Q problem for non-regenerative two-way multi-antenna relay network without EH constraint has been investigated in [15]. Because the additional EH constraint is non-convex, the proposed scheme in [15] cannot be employed to solve problem (22). In the following sections, we will propose algorithms to solve problem (22). III. G LOBAL O PTIMAL S OLUTION It is observed that the objective function and the EH constraint in (22) are non-convex which causes problem (22) a non-convex optimization problem. In general, it is difficult or even intractable to obtain the global optimal solution to a nonconvex problem. We will obtain the global optimal solution to (22) by using semidefinite programming (SDP) and rank-one decomposition theorem. By introducing two slack variables t1 and t2 such that

where

D = P1 h1,f h†1,f + P2 h2,f h†2,f + σ 2 I

T

⊗ G†r Gr (17)

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1+

w† Q2,1 w ≥ t1 , w † R1 w + σ 2

(23)

1+

w† Q1,2 w ≥ t2 , w † R2 w + σ 2

(24)

and

in which Gr ∈ CL×N denotes the channel response matrix from relay to EH receiver.

we equivalently rewrite problem (22) as B. Problem Formulation

max

From (11) and (12), the sum rate of two-way relay network is expressed as

1 w† Q2,1 w w† Q1,2 w log2 1 + † 1 + . (18) 2 w R1 w + σ 2 w † R2 w + σ 2 The transmit power at relay and the harvested energy at EH receiver should be constrained as w† Cw ≤ Pr ,

(19)

E1 + E2 ≥ Q

(20)

w,t1 >0,t2 >0

s.t.

Qmax = E1 +

max

w|w† Cw≤Pr

E2

= E1 + λmax (C−1 D)Pr .

(21)

w† (Q2,1 − (t1 − 1)R1 )w ≥ σ 2 (t1 − 1), w† (Q1,2 − (t2 − 1)R2 ) w ≥ σ 2 (t2 − 1), ¯ (25) w† Cw ≥ Pr , w† Dw ≥ Q.

The problem (25) is a quadratically constrained quadratic problem (QCQP), which is still non-convex and difficult to solve. Employing the semidefinite relaxation method [18], we obtain the rank-one relaxation of (25) as follows max

X0,t1 >0,t2 >0

where Pr is the maximum allowable transmit power at relay and Q is the threshold of the harvested energy at EH receiver. According to [3], [4], the threshold Q should be chosen such that 0 ≤ Q ≤ Qmax where

t1 t2

s.t.

t1 t2 tr [(Q2,1 − (t1 − 1)R1 )X] ≥ σ 2 (t1 − 1), tr [(Q1,2 − (t2 − 1)R2 ) X] ≥ σ 2 (t2 − 1), ¯ (26) tr(CX) ≤ Pr , tr(DX) ≥ Q.

It is noted that the sufficient and necessary condition for the equivalence of problems (25) and (26) is that the optimal X of problem (26) is rank-one. We have the following lemma.

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Lemma 1: The problem (26) always has a rank-one optimal solution X. Proof: We have the following rank-one decomposition theorem to prove Lemma 1. Theorem 1 ([19], Theorem 2.3): Let Bi ∈ Cn×n , i ∈ I = {1, 2, 3, 4}, be a Hermitian matrix and Z ∈ Cn×n be a nonzero Hermitian positive semidefinite matrix. Suppose that n ≥ 3 and [tr(B1 Z), tr(B2 Z), tr(B3 Z), tr(B4 Z)] = [0, 0, 0, 0]

(27)

for any nonzero Hermitian positive semidefinite matrix Z ∈ Cn×n . If rank(Z) ≥ 2, we can find a rank-one matrix zz† in polynomial-time such that tr(Bi zz† ) = tr(Bi Z), i ∈ I. Suppose that X is the optimal solution to problem (26). Since C 0, we have [tr(F1 X), tr(F2 X), tr(CX), tr(DX)] = [0, 0, 0, 0]

(28)

where F1 = Q2,1 − (t1 − 1)R1 ,

(29)

F2 = Q1,2 − (t2 − 1)R2 .

(30)

If rank(X) ≥ 2, according to Theorem 1, we can find a rankone matrix xx† such that †

tr(F1 xx ) = tr(F1 X), †

(32)

tr(Cxx† ) = tr(CX),

(33)

tr(Dxx ) = tr(DX).

(34)

Thus, the problem (26) always has a rank-one optimal X = xx† . In the following, we will find the optimal solution to problem (26). We have the following lemma. Lemma 2: Given t1 , the problem (26) is a quasi-convex optimization problem with respect to (X, t2 ). Proof: See Appendix A. Thus, we propose an iterative algorithm to compute the optimal solution of the problem (26) which includes two loops, i.e., the outer-loop and inner-loop. The outer-loop is linear search over t1 while the inner-loop is bisection search over t2 to find the optimal (X, t2 ) given t1 . The search over t1 and t2 requires the upper and lower bounds of t1 and t2 . From (23), the lower bound of t1 , denoted as t1,l , is t1,l = 1. The upper bound of t1 , denoted as t1,u , is t1,u =

max

w|w† Cw≤Pr

1+

t2,l = 1 +

¯ † Q1,2 w ¯ w . ¯ † (R2 + σ 2 C/Pr )w ¯ w

(37)

In the outer-loop of the proposed iterative algorithm, we linearly search the optimal t1 over the interval [t1,l , t1,u ] from t1,u to t1,l . Denote the temporary optimal solution of t1 and t2 as t1 and t2 . From the derivation of (35)–(37), we know that (t1,u , t2,l ) is achievable. Thus, we initialize t1 and t2 as t1,u and t2,l . In the inner-loop of the proposed iterative algorithm, the bisection search interval of t2 is [t1 t2 /t1 , t2,u ] instead of [t2,l , t2,u ] because our task is to find the achievable t1 and t2 whose product is larger than t1 t2 . The bisection search of t2 begins with the feasibility check of t1 and the most likely feasible value of t2 over the interval [t1 t2 /t1 , t2,u ], i.e., t1 t2 /t1 , given by ﬁnd s.t.

X0 tr ((Q2,1 − (t1 − 1)R1 )X) ≥ σ 2 (t1 − 1), tr ((Q1,2 − (t2 − 1)R2 ) X) ≥ σ 2 (t2 − 1), ¯ tr(CX) ≤ Pr , tr(DX) ≥ Q.

(38)

(31)

tr(F2 xx ) = tr(F2 X), †

The lower bound of t2 , denoted as t2,l , is obtained by submit¯ which achieves the t1,u into (24) ting w

w† Q2,1 w w † R1 w + σ 2

w† Q2,1 w w w† (R1 + σ 2 C/Pr ) w −1 = 1 + λmax (R1 + σ 2 C/Pr ) Q2,1 .

(35)

Similarly, the upper bound of t2 , denoted as t2,u , is −1 t2,u = 1 + λmax (R2 + σ 2 C/Pr ) Q1,2 .

(36)

= max 1 +

If the problem (38) is feasible, the bisection search of t2 is performed and we obtain the new temporary optimal solution of t1 and t2 to replace the original ones, t1 and t2 . The proposed iterative algorithm to obtain the global optimal solution to problem (22) is summarized in Algorithm 1. Algorithm 1 Find the global optimal solution to problem (22) 1: Choose some large M and define Δt = (t1,u − t1,l )/M . Initialize t1 = t1,u , t2 = t2,l and the index j = M . 2: Set t1 = t1,l + jΔt. If t1 t2,u < t1 t2 Go to Step 3; Else Let t2 = t1 t2 /t1 . Solve the problem (38). If the problem (38) is infeasible Set j = j − 1 and go to Step 2; Else Solve problem (26) with the bisection search using the initial interval [t1 t2 /t1 , t2,u ] for t2 , and denote the optimal solution as (Xo , to2 ); Update t1 = t1 , t2 = to2 , j = j − 1 and go to Step 2. 3: If rank(Xo ) = 1 The optimal solution to the problem (22) is w = xo where Xo = xo (xo )† ; Else Employ Theorem 1 to find a rank-one matrix zo (zo )† such that (31)–(34) are satisfied. The optimal solution to the problem (22) is w = zo .


Remark (Complexity of Algorithm 1): The computation burden of Algorithm 1 is the computation of SDP (38). From [24], it is known √ that a rational SDP is solved, within a tolerance , in O( n log(1/)) iterations, where n is the dimension of the matrix. While the complexity per iteration of the interiorpoint method (IPM) for solving an SDP problem is O(mn3 + m2 n2 + m3 ) where m is the number of linear constraints. This means that for a given accuracy , IPM in total requires at most O((mn3.5 + m2 n2.5 + m3 n0.5 ) · log(1/)) arithmetic operations. For the SDP problem (38), n = N 2 and m = 4. Thus, the computational complexity of SDP (38) is O(N 7 log(1/)), which leads to the complexity of Algorithm 1 is O(T M N 7 · log(1/)) where T is the average iteration number of the bisection search. Given ε, the upper bound of T is given by 2 log2 ((t2,u − t2,l )/ε) [22]. IV. L OCAL O PTIMAL S OLUTION From Algorithm 1, to obtain global optimal solution is computationally expensive. In practice, developing a lower complexity algorithm to compute the local optimum of problem (22) is meaningful. In the following, we will propose an iterative algorithm based on the CCCP to find a local optimal solution. To employ the CCCP, we transform problem (22) to a DC programming. For proceeding, it is noted that the optimal beamforming vector w for problem (22) should satisfy that the transmit power constraint at relay is active, i.e., w† Cw = Pr .

(39)

Substituting (39) into problem (22), we obtain max w

s.t.

w † A1 w w † A2 w · w † B1 w w † B2 w w † B 3 w − w † A3 w ≤ 0

f˜(y, t1 ) = f (Uy, t1 ) =

N λj |yj |2 j=1

t1

gi (w, ti ) = w† Ai w/ti , i ∈ {1, 2, 3}

A3 = Pr D, Bi = Ri + σ 2 C/Pr , i ∈ {1, 2}, ¯ B3 = QC.

(43) (44) (45)

In (40), we have removed the equality constraint of relay transmit power (39). This is because the objective function and the constraint in (40) is homogeneous in w. An arbitrary positive scaling of w has no effect on the value of the objective function. Furthermore, if w satisfies the constraint, an arbitrary positive scaling of w also satisfies the constraint. By introducing slack variables t1 and t2 which are defined in (23) and (24), respectively, we equivalently rewrite problem (40) as √ max t1 t2 w,t1 >0,t2 >0

w† B1 w − w† A1 w/t1 ≤ 0, w† B2 w − w† A2 w/t2 ≤ 0,

(48)

where t3 = 1. The first-order Taylor expansion of gi (w, ti ) ˜ t˜i ) is [23] around the point (w, ˜ i /t˜2i . ˜ t˜i ) = 2Re{w ˜ † Ai w}/t˜i − w ˜ † Ai wt gi (w, ti , w,

(49)

In the (n + 1)th iteration of the proposed CCCP based iterative algorithm, we solve the following convex optimization problem, √ max t1 t2 (50a) (n) ˜ (n) , t˜i ≤ 0, i ∈ {1, 2, 3} w† Bi w−gi w, ti , w

(40)

(41) (42)

(47)

where y = [y1 , . . . , yN ]T . The function f˜(y, t1 ) is convex because it is the sum of convex functions [22]. Thus, the function f (w, t1 ) is convex because it is the affine transformation of f˜(y, t1 ) [22]. In the following, we will employ the CCCP proposed in [21] to find a local optimum of the DC programming (46). Define

s.t.

A1 = Q2,1 + R1 + σ 2 C/Pr , A2 = Q1,2 + R2 + σ 2 C/Pr ,

w † B 3 w − w † A3 w ≤ 0

√ where the replacement of the objective function t1 t2 with t1 t2 has no effect on the optimal solution. The problem (46) is a DC programming because we have√ the following lemma. Lemma 3: The functions − t1 t2 , w† Aw, and w† Aw/t1 , where t1 > 0, t2 > 0, A 0, are convex. √ Proof: From [22], the functions − t1 t2 and w† Aw are convex. Denote the singular value decomposition (SVD) of A as A = Udiag(λ1 , · · · , λN )U† . Let f (w, t1 ) = w† Aw/t1 . We have

w,t1 >0,t2 >0

where

s.t.

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(50b) ˜ (n) , ˜t(n) ) denotes the solution to problem (50) in where (w the nth iteration. The problem (50) can be transformed to a second-order cone programming (SOCP). By introducing a slack variable z, we convert (50a) to √ (51) max z s.t. t1 t2 ≥ z. It can be further converted to a conic quadratic-representable function as follows max z s.t. [2z, (t1 − t2 )] ≤ t1 + t2 .

(52)

The equivalence between (51) and (52) is because that the constraint of (52) is equivalent to 4z 2 + (t1 − t2 )2 ≤ (t1 + t2 )2 which is equivalent to the constraint of (51). By letting

† 2 (n) (n) (n) ˜ ˜ ai = w Ai w , t˜i

(53)

(54)

(46)

˜ bi = −2Ai w

(n)

(n) t˜i ,

(55)

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(50b) is rewritten as

w† Bi w + Re b†i w + ai ti ≤ 0

(56)

which can be converted to a second-order cone constraint, 1 2Bi2 w ≤ −Re b† w −ai ti +1. (57) i † −Re bi w −ai ti −1

number for the convergence of Algorithm 2. It is noted that compared with Algorithm 1, Algorithm 2 has much lower computational complexity. Remark (Initial Point for Algorithm 2): Any w0 whichsatisfies w0† B3 w0 − w0† A3 w0 ≤ 0 can be chosen as the initial point for Algorithm 2. Accordingly, we have w0† A1 w0 w0† A2 w0 [t1,0 , t2,0 ] = , . (60) w0† B1 w0 w0† B2 w0

The equivalence between (56) and (57) is because that the constraint of (57) is equivalent to 2 4w† Bi w + −Re b†i w − ai ti − 1

2 ≤ −Re b†i w − ai ti + 1

(58)

which is equivalent to the constraint of (56). Using (52) and (57), the problem (50) is converted to the following SOCP, max

w,t1 >0,t2 >0,z

s.t.

z [2z, (t1 − t2 )] ≤ t1 + t2 , 1 2Bi2 w † −Re bi w − ai ti − 1 ≤ −Re b†i w − ai ti + 1, i ∈ {1, 2, 3}. (59)

The proposed CCCP based algorithm to obtain the local optimal solution to problem (22) is summarized in Algorithm 2.

V. L OW-C OMPLEXITY S UBOPTIMAL S OLUTION Although Algorithm 2 has much lower complexity than Algorithm 1, it requires to solve a sequence of SOCPs. To further reduce the computational complexity, we propose a suboptimal solution which avoids solving SOCPs. We employ the generalized eigenvectors (GES) method proposed in [15] to solve problem (40). We convert the problem (40) to an unconstrained optimization problem by restricting the beamforming vector w to lie on the subspace of the constraint set {w|w† (B3 − A3 )w ≤ 0}. 2 Let V ∈ CN ×M consists of the M (1 ≤ M < N 2 ) eigenvectors of the matrix B3 − A3 associating with the eigenvalues being no greater than zero. Let ˜ w = Vw ˜ ∈ CM ×1 is an arbitrary vector, satisfying where w ˜ ≤ 0. ˜ † V† (B3 − A3 )Vw w

n := n + 1; 3: Until: convergence. Remark (Complexity of Algorithm 2): According to [25], the complexity of solving an SOCP problem is O(k 0.5 (m3 + m2 ki=1 ni + ki=1 n2i ) · log(1/)), where k denotes the number of second-order cone (SOC) constraints, m denotes the dimension of the optimization vector, and ni denotes the dimension of the ith SOC. For the SOCP (59), k = 4, m = N 2 and n1 = 3, n2 = · · · = nk = N 2 + 1. Thus, the computational complexity of solving the SOCP (59) is O(N 6 · log(1/)), which leads to the complexity of Algorithm 2 is O(Nmax N 6 log(1/)), where Nmax is the maximum iteration

(62)

Substituting (61) into the problem (40), we have max

Algorithm 2 Find the local optimal solution to problem (22) ˜ (n) , ˜t(n) ) = (w0 , t0 ); 1: Initialization: n = 0, (w 2: Repeat: ˜ (n) , ˜t(n) ) and denote the Solve the SOCP (59) with (w optimal solution in the (n + 1)th iteration as w and t ; Update ˜ (n+1) , ˜t(n+1) = (w , t ) w

(61)

˜ w

˜ 1w ˜ 2w ˜ †A ˜ w ˜ ˜ †A w · † † ˜ ˜ ˜ B1 w ˜ w ˜ B2 w ˜ w

(63)

where ˜ i = V† Ai V, A

(64)

˜ i = V† Bi V, B

(65)

i ∈ {1, 2}. According to [15], the global optimum to the prob˜ for which the gradient of lem (63) must be one of the vectors w objective function is zero. Using the condition of zero gradient, we obtain PA,1 ˜ ˜ 1 + ρA A ˜2 w ˜2 w ˜ = ˜ A B1 + ρB B (66) PB,1 where ˜ i w, ˜ †A ˜ PA,i = w

(67)

˜ i w, ˜ ˜ †B PB,i = w

(68)

ρA = PA,1 /PA,2 ,

(69)

ρB = PB,1 /PB,2 .

(70)


˜ is a generalized eigenvector of the From (66) the optimal w ˜ 1 + ρB B ˜ 1 + ρA A ˜ 2 ) and (B ˜ 2 ). However, we matrix pair (A ˜ directly from the generalized eigenvector cannot compute w because the unknown parameters ρA and ρB are correlated with ˜ Therefore, two-dimension (2-D) search over (ρA , ρB ) is w. ˜ It has been shown in [15] that required to find the optimal w. for every value of ρB , the corresponding maximization over ρA yields the maximal value which depends on ρB only very weakly. Thus, the 2-D search over (ρA , ρB ) can be replaced essentially without any loss by a one-dimension (1-D) bisection search over ρA only for one fixed value of ρB (e.g., the geometric mean of upper and lower bounds) [15]. For 1-D search over ρA , the required upper and lower bounds of ρA , denoted as ρA,u and ρA,l , respectively, are ρA,u = max ˜ w

ρA,l

˜ 1w ˜ †A ˜ w ˜ 2 )−1 A ˜1 , = λmax (A ˜ 2w ˜ †A ˜ w

˜ 1w ˜ †A ˜ w ˜ 2 )−1 A ˜1 . = λmin (A = min ˜ 2w ˜ w w ˜ †A ˜

(71)

tions. Thus, the complexity of Algorithm 3 is about O(T¯M 3 ) where T¯ is the average iterative number of bisection search. From [22], the average iterative number of bisection search is about 2 log2 ((ρA,u − ρA,l )/ε). VI. E XTENSION TO THE C ASE OF M ULTI -A NTENNA S OURCES In this section, we consider the case where two sources are equipped with multi-antennas. We assume that the source i, i ∈ {1, 2}, is equipped with Mi antennas. Denote the channel response matrices from source i to relay, from source i to EH receiver, and from relay to source i by Hi,f ∈ CN ×Mi , Gi ∈ CL×Mi , and Hi,b ∈ CMi ×N , respectively. Following the derivation in Section II, the received signal at source i is expressed as y˘i = r†i Hi,b WH¯i,f t¯i s¯i + r†i Hi,b Wnr + r†i ni

(72)

Similarly, the upper and lower bounds for ρB , denoted as ρB,u and ρB,l , respectively, are given by ˜ 2 )−1 B ˜1 , (73) ρB,u = λmax (B ˜ 2 )−1 B ˜1 . ρB,l = λmin (B

(74)

(76)

where si denotes the transmit symbol at source i with E[|si |2 ] = 1, ti ∈ CMi ×1 and ri ∈ CMi ×1 denote the transmit and receive beamforming vector at source i, respectively, and ¯i = 2; i = 1, (77) 1; i = 2. From (76), the received SNR at source i is given by

Defining the function

˜ 1w ˜ 2w ˜ †A ˜ w ˜ ˜ †A w ζ(ρA ) = · ˜ 1w ˜ 2w ˜ †B ˜ w ˜ †B ˜ w=φ( w ˜ ˜ A

5515

γi =

|r†i Hi,b WH¯i,f t¯i |2 . 2 σ 2 r†i Hi,b W + σ 2 ri 2

(78)

(75) ˜

˜

˜

1 +ρA A2 ,B1 +ρB B2 )

where φ(A, B) denotes the generalized eigenvector of the matrix pair (A, B) with respect to the largest eigenvalue, the proposed 1-D bisection search based algorithm to obtain the suboptimal solution to problem (22) is summarized in Algorithm 3.

The sum transmit power of sources and relay is expressed as P (ti , W) =

2

tr ti t†i + σ 2 tr(WW† )

i=1

+

2

tr WHi,f ti t†i H†i,f W† .

(79)

i=1

Algorithm 3 Find the low-complexity suboptimal solution to problem (22) by 1-D bisection search √ 1: Initialization: ρB = ρB,u ρB,l , and a small positive number, ε. 2: Repeat: Compute ρA = (ρA,u +ρA,l )/2, ρ1 = ρA − ε, ρ2 = ρA + ε. Compute ζ(ρ1 ) and ζ(ρ2 ) according to (75); If ζ(ρ2 ) < ζ(ρ1 ) ρA,u = ρ2 ; Else ρA,l = ρ1 ; 3: Until: ρA,u − ρA,l < ε. Remark (Complexity of Algorithm 3): The main computation burden of Algorithm 3 is to compute the function (75). The computation of the generalized eigenvector of matrix pair ˜ 1 + ρB B ˜ 1 + ρA A ˜ 2, B ˜ 2 ) requires O(M 3 ) arithmetic opera(A

The harvested energy at the EH receiver is given by E(ti , W) =

2

tr Gi ti t†i G†i + σ 2 tr Gr WW† G†r

i=1

+

2

tr Gr WHi,f ti t†i H†i,f W† G†r .

(80)

i=1

Our objective is to find the optimal ti , ri , i ∈ {1, 2}, and W which maximize the sum rate under the sum transmit power constraint at sources and relay and the EH constraint at EH receiver. The optimization problem is formulated as follows max

1 (log2 (1 + γ1 ) + log2 (1 + γ2 )) 2

s.t.

P (ti , W) ≤ PT , E(ti , W) ≥ Q

ti ,ri ,W

where PT is the maximum allowable sum transmit power.

(81)

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We propose to employ alternating optimization (AO) to solve the problem (81). It is observed that in the problem (81), when ti and ri are fixed, the proposed algorithms in Sections III–V are applicable to the optimization of W where hi,f is replaced by Hi,f ti , hTi,b is replaced by r†i Hi,b and gi is replaced by Gi ti . When ti and W are fixed, the optimization problem of ri is expressed as

It can be further converted to a conic quadratic-representable function as follows

roi = arg max γi

4z 2 + (γ1 − γ2 )2 ≤ (γ1 + γ2 + 2)2

ri

for i ∈ {1, 2} whose close-form solution is −1 roi = σ 2 Hi,b WW† H†i,b + σ 2 I Hi,b WH¯i,f t¯i .

(82)

max ti

s.t.

(83)

1 † H W† H†i,b ri r†i Hi,b WH¯i,f , ξi ¯i,f

(84)

(85) (86)

Υi = H†i,f W† G†r Gr WHi,f + G†i Gi , 2 ξi = σ 2 r†i Hi,b W + σ 2 ri 2

(87) (88)

for i ∈ {1, 2}, and (89) (90)

Define Ti = ti t†i and consider the rank-one relaxation of (84) as follows: Δ

max

Ti 0,γi

s.t.

(1 + γ1 )(1 + γ2 )

(91a)

tr(Φ1 T2 ) ≥ γ1 , tr(Φ2 T1 ) ≥ γ2 ,

(91b)

tr(Λ1 T1 ) + tr(Λ2 T2 ) ≤ P¯T ,

(91c)

¯ tr(Υ1 T1 ) + tr(Υ2 T2 ) ≥ Q

(91d)

where (1/2) log2 (·) in the objective function of (84) is omitted since the logarithm is monotonically increasing. By introducing the slack variable z > 0, we convert (91a) to max z s.t. (1 + γ1 )(1 + γ2 ) ≥ z 2 .

max

Ti 0,γi ,z>0

(94)

s.t.

z [2z, (γ1 − γ2 )] ≤ γ1 + γ2 + 2, tr(Φ1 T2 ) ≥ γ1 , tr(Φ2 T1 ) ≥ γ2 ,

¯ tr(Υ1 T1 ) + tr(Υ2 T2 ) ≥ Q.

Λi = H†i,f W† WHi,f + I,

P¯T = PT − σ 2 tr(WW† ), ¯ = Q − σ 2 tr Gr WW† G† . Q r

The equivalence between (92) and (93) is because that the constraint of (93) is equivalent to

tr(Λ1 T1 ) + tr(Λ2 T2 ) ≤ P¯T ,

where Φi =

(93)

which is equivalent to the constraint of (92). Thus, we equivalently transform the problem (91) into an SDP as follow:

It is found that roi is actually the minimum-mean-square-error (MMSE) receiver at source i. When ri and W are fixed, the optimization problem of ti is expressed as 1 (log2 (1 + γ1 ) + log2 (1 + γ2 )) 2 γ1 = tr Φ1 t2 t†2 , γ2 = tr Φ2 t1 t†1 tr Λ1 t1 t†1 + tr Λ2 t2 t†2 ≤ P¯T , ¯ tr Υ1 t1 t†1 + tr Υ2 t2 t†2 ≥ Q

max z s.t. [2z, (γ1 − γ2 )] ≤ γ1 + γ2 + 2.

(92)

(95)

It is noted that if the problem (95) has a rank-one optimal solution of Ti , i.e., the problem (91) has a rank-one optimal solution of Ti , the problem (91) is equivalent to the problem (84). Fortunately, we have the following theorem. Theorem 2: The rank-one optimal solution of the problem (91) always exists. Proof: See Appendix B. The proposed AO based iterative algorithm is summarized in Algorithm 4. Algorithm 4 The proposed AO based iterative algorithm for the case of multi-antenna sources (0)

(0)

1: Initialization: ti = ti , ri = ri , and k = 0. 2: Repeat: Compute W(k+1) using the Algorithm 1 or Algorithm 2 or (k) (k) Algorithm 3 with ti = ti and ri = ri ; (k+1) (k) Compute ri using (83) with ti = ti and W = (k+1) ; W (k+1) (k+1) by solving (95) with ri = ri and Compute ti (k+1) W=W ; Update k = k + 1; 3: Until: convergence. Remark (Complexity of Algorithm 4): The complexity of solving the SDP (95) is O(M 3.5 log(1/)) where M = max{M1 , M2 }[24]. Thus, the complexity of Algorithm 4 is K · O(M 3.5 log(1/) plus the K times of the complexity of Algorithm 1 or Algorithm 2 or Algorithm 3, where K is the average iteration number when the algorithm stops. (0) Remark (Initial Point for Algorithm 4): The initial point ti † in Algorithm 4 can be chosen as ψ(Gi Gi ), which maximizes the harvested energy, tr(Gi ti t†i G†i ), at the EH receiver. The (0) initial point ri can be chosen as ψ(Hi,b H†i,b ).


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VII. S IMULATION R ESULTS In this section, we present the computer simulation results of our proposed beamforming schemes. The sources and the EH receiver, if not specified, are equipped with single antenna. We assume that in the two-way multi-antenna relay network, all the entries in the channel response vectors are independent and identically distributed (i.i.d) complex Gaussian random variables with zero mean and unit variance. The channel reciprocity holds as in [15], i.e., hi,f = hi,b for i ∈ {1, 2}. To solve the SDPs and SOCPs, we use the Matlab-based CVX optimization software [26]. In all simulations, the average sum rate of the relay network is computed by using 1000 randomly generated channel realizations. The transmit power to noise power ratios of two sources, if not specified, are P1 /σ 2 = P2 /σ 2 = 10 dB. In the simulations, different relay beamforming schemes are considered including the proposed three schemes, i.e., global optimal solution (denoted as “Global” in the legend), local optimal solution (denoted as “Local”) and low-complexity suboptimal solution (denoted as “Suboptimal”), and three schemes in the literature, i.e., the algebraic norm-maximizing (denoted as “ANOMAX”) scheme [14], maximal-ratio reception and maximal-ratio transmission (denoted as “MRR-MRT”) scheme [12] and zero-forcing reception and zero-forcing transmission (denoted as “ZFR-ZFT”) scheme [12]. It is noted that the ANOMAX, MRR-MRT, and ZFR-ZFT schemes consider no EH constraint. The solutions obtained by aforementioned schemes may not be feasible solutions for EH-constrained relaying. When the solutions are not feasible, the sum rate of twoway relay network is assumed to be zero. It is noted that the sum rate computation in the aforementioned method only provides a performance metric for system comparison where both the wireless information transmission and the power transfer are important for the system design. It does not mean that when the EH constraint is not satisfied, the relay transmission is suspended.

Fig. 2. Average sum rate versus the iteration times when the threshold of EH constraint is Q = 0.5Qmax and the number of antenna at relay is N = 4.

Fig. 3. Average sum rate versus the iteration times when the transmit power to noise power ratio of relay is Pr /σ 2 = 15 dB and the threshold of EH constraint is Q = 0.5Qmax . TABLE I AVERAGE RUNNING T IME ( IN S ECONDS ) C OMPARISON OF A LGORITHMS

A. Convergence Performances of the Local Optimal Solution Before we compare the aforementioned schemes, we investigate the convergence performance of Algorithm 2. The initial point for Algorithm 2 is chosen by using w0 = Vx0 [see (61)] where x0 is a randomly generated vector. Accordingly, we obtain t1,0 and t2,0 by using (60). In Fig. 2, we present the average sum rate achieved by the local optimal solution for different iteration times and different transmit powers at relay. The threshold of EH constraint is Q = 0.5Qmax . The number of antenna at relay is N = 4. It is observed from Fig. 2 that after only 3 ∼ 4 iterations, the steady average sum rate is achieved regardless of Pr /σ 2 . This indicates that Algorithm 2 has fast convergence rate. In Fig. 3, we present the average sum rate for different iteration times and different number of antenna at relay when the threshold of EH constraint is Q = 0.5Qmax . In Fig. 3, the transmit power to noise power ratio of relay is Pr /σ 2 = 15 dB. It is found from Fig. 3 that after only 4 ∼ 5 iterations, the steady average sum rate is achieved.

B. Running Time Comparison of Proposed Algorithms In Table I, we compare the average running time of the proposed algorithms for one channel realization under different number of relay antenna, where the transmit power to noise power ratio of relay is Pr /σ 2 = 15 dB and the threshold of EH constraint is Q = 0.5Qmax . The Central Processing Unit (CPU) is Intel Core E7400 2.8 GHz. The size of Random Access Memory (RAM) is 4GB. The version of Matlab is R2008b and the version of SDPT3 solver we use in CVX is 4.0. From Table I, it is observed that to obtain the global optimal solution requires the running time about 7 ∼ 12 times more than to obtain the local optimal solution and about 3600 ∼ 4500 times more than to obtain the suboptimal solution. It is also found that the average running time increases with the increase of the number of antenna at relay, N , for all the algorithms.

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Fig. 4. Average sum rate versus Pr /σ 2 ; performance comparison of different schemes when the threshold of EH constraint is Q = 0.5Qmax and the number of antenna at relay is N = 4.

Fig. 5. Average sum rate versus τ ; performance comparison of different schemes when the transmit power to noise power ratio of relay is Pr /σ 2 = 20 dB and the number of antenna at relay is N = 4.

C. Average Sum Rate Versus the Transmit Power at Relay In Fig. 4, we present the average sum rate comparison of different schemes for various transmit power to noise power ratios at relay, Pr /σ 2 . The threshold of EH constraint is Q = 0.5Qmax . The number of antenna at relay is N = 4. It is observed from Fig. 4 that the proposed global optimal solution, local optimal solution and low-complexity suboptimal solution perform better than the ANOMAX, MRR-MRT, and ZFRZFT schemes. This is because the latter three schemes do not consider the SWIPT scheme in the two-way relay network. It is also found that the performance of proposed local optimal solution is close to that of global optimal solution, especially when the relay has high transmit power. D. Effect of the EH Threshold on Average Sum Rate Let Q = τ Qmax . In Fig. 5, we present the average sum rate comparison of different schemes for various EH thresholds, i.e. τ , when the transmit power to noise power ratio of relay is Pr /σ 2 = 20 dB and the number of antenna at relay is N = 4. From Fig. 5, it is found that the proposed global optimal solution and local optimal solution outperform the ANOMAX, MRR-MRT and ZFR-ZFT schemes for different EH thresholds. When τ ≥ 0.17, the proposed low-complexity suboptimal solution is better than the ANOMAX, MRR-MRT, and ZFR-ZFT schemes. It is also observed that with the increase of τ , the average sum rates achieved by all the schemes decrease. When τ is small, i.e., τ ≤ 0.5, the proposed local optimal solution performs almost the same as the global optimal solution. E. Average Sum Rate Versus the Number of Antenna at Relay In Fig. 6, we present the average sum rate comparison of different schemes for different number of antenna at relay when the transmit power to noise power ratio of relay is Pr /σ 2 = 20 dB and the threshold of EH constraint is Q = 0.5Qmax . It is observed from Fig. 6 that our proposed three schemes outperform the ANOMAX, MRR-MRT, and ZFR-ZFT schemes

Fig. 6. Average sum rate versus N ; performance comparison of different schemes when the transmit power to noise power ratio of relay is Pr /σ 2 = 20 dB and the threshold of EH constraint is Q = 0.5Qmax .

for different number of antenna at relay. With the increase of N , the average sum rates of our proposed three schemes increase while those of the ANOMAX, MRR-MRT, and ZFRZFT schemes decrease. This is because with the increase of N , the infeasible probabilities of the ANOMAX, MRR-MRT, and ZFR-ZFT schemes increase. It is also found that the proposed local optimal solution performs almost the same as the global optimal solution for different number of antennas at relay. F. Case of Multi-Antenna Sources In Fig. 7, we present the average sum rate comparison of different beamforming schemes for the case of multi-antenna sources. The numbers of antennas at sources, relay, and EH receiver are M1 = M2 = 3, N = 4, and L = 2, respectively. The threshold of EH constraint is assumed to be Q/σ 2 = 10 dB. The proposed AO based iterative algorithms where the relay bemforming matrix W is updated by using Algorithms 1, 2, and 3 are denoted as “AO with Alg. 1”, “AO with Alg. 2”, and “AO with Alg. 3”, respectively. The proposed AO based


5519

quasiconvex according to [22]. Therefore, the problem (96) is quasiconvex over X. By introducing the slack variable t2 into (96) such that 1 + (tr(Q1,2 X)/(tr(R2 X) + σ 2 )) ≥ t2 , we obtain (26). Thus, given t1 , the problem (26) is a quasiconvex optimization problem with respect to (X, t2 ). A PPENDIX B P ROOF OF T HEOREM 2 Suppose (Toi , γio ) is an optimal solution to the problem (91). Consider the following power minimization problem min

Ti 0

s.t. Fig. 7. Average sum rate versus PT /σ 2 for multi-antenna transceivers; performance comparison of different schemes. The threshold of EH constraint is Q/σ 2 = 10 dB. The number of antenna at sources, relay, EH receiver is M1 = M2 = 3, N = 4, and L = 2, respectively.

iterative algorithms are compared with the ANOMAX scheme. It is observed from Fig. 7 that the AO based iterative algorithms outperform the ANOMAX scheme. It is also found that when the sum transmit power, PT /σ 2 , is high, the AO with Alg. 2 scheme performs close to the AO with Alg. 1 scheme. VIII. C ONCLUSION In this paper, we have proposed the global optimal solution, local optimal solution and low-complexity suboptimal solution to the EH-constrained relay beamforming optimization problem for SWIPT in a non-regenerative two-way multi-antenna relay network. It is found from simulations that the global optimal solution achieves the global optimum while has the highest computational complexity. The local optimal solution performs a little bit worse than the global optimal solution while has lower complexity. The low-complexity suboptimal solution has much lower complexity than the global and local optimal solutions. Simulations have also shown that our proposed schemes outperforms the ANOMAX, MRR-MRT, and ZFRZFT schemes in the literature. We have also proposed the AO based iterative algorithms for the case of multi-antenna sources. Simulation results have shown that our proposed algorithms outperform the ANOMAX scheme. A PPENDIX A P ROOF OF L EMMA 2 Given t1 , the problem (26) is equivalently written as

tr(Q1,2 X) max t1 1 + X0 tr(R2 X) + σ 2 s.t.

tr [(Q2,1 − (t1 − 1)R1 ) X] ≥ σ 2 (t1 − 1), ¯ tr(CX) ≤ Pr , tr(DX) ≥ Q.

(96)

Since the function tr(Q1,2 X)/(tr(R2 X) + σ 2 ) is a linearfractional function with tr(R2 X) + σ 2 > 0, the objective is

tr(Λ1 T1 ) + tr(Λ2 T2 ) tr(Φ1 T2 ) ≥ γ1o , tr(Φ2 T1 ) ≥ γ2o , ¯ tr(Υ1 T1 ) + tr(Υ2 T2 ) ≥ Q.

(97)

Suppose Ti is an optimal solution to the problem (97). We will prove that (Ti , γio ) is also an optimal solution to problem (91). Since Toi is feasible for the problem (91), it is also feasible for the problem (97). Thus, we have tr (Λ1 T1 )+tr (Λ2 T2 ) ≤ tr (Λ1 To1 )+tr (Λ2 To2 ) ≤ PT

(98)

which indicates that (Ti , γio ) is feasible for the problem (91). Furthermore, since (Ti , γio ) and (Toi , γio ) has the same objective value in (91), (Ti , γio ) is indeed an optimal solution to the problem (91). According to Lemma 3.1 in [27], there exists an optimal solution Ti for the problem (97) such that (rank(T1 ))2 + (rank (T2 ))2 ≤ 3.

(99)

It can be verified that Ti = 0. Thus, from (99), we have rank(T1 ) = rank(T2 ) = 1. R EFERENCES [1] L. R. Varshney, “Transporting information and energy simultaneously,” in Proc. ISIT, 2008, pp. 1612–1616. [2] P. Grover and A. Sahai, “Shannon meets Tesla: Wireless information and power transfer,” in Proc. ISIT, 2010, pp. 2363–2367. [3] R. Zhang and C. K. Ho, “MIMO broadcasting for simultaneous wireless information and power transfer,” IEEE Trans. Wireless Commun., vol. 12, no. 5, pp. 1989–2001, May 2013. [4] C. Xing, N. Wang, J. Ni, Z. Fei, and J. Kuang, “MIMO beamforming designs with partial CSI under energy harvesting constraints,” IEEE Signal Process. Lett., vol. 20, no. 4, pp. 363–366, Apr. 2013. [5] Z. Xiang and M. Tao, “Robust beamforming for wireless information and power transmission,” IEEE Wireless Commun. Lett., vol. 1, no. 4, pp. 372– 375, Aug. 2012. [6] J. Xu, L. Liu, and R. Zhang, “Multiuser MISO beamforming for simultaneous wireless information and power transfer,” in Proc. IEEE ICASSP, 2013, pp. 4754–4758. [7] Q. Shi, L. Liu, W. Xu, and R. Zhang, “Joint transmit beamforming and receive power splitting for MISO SWIPT systems,” IEEE Trans. Wireless Commun, to be published. [8] C. Shen, W.-C Li, and T.-H Chang, “Simultaneous information and energy transfer: A two-sser MISO interference channel case,” in Proc. IEEE Signal Process. Commun. Symp., 2012, pp. 1–6. [9] J. Park and B. Clerckx, “Joint wireless information and energy transfer in a two-user MIMO interference channel,” IEEE Trans. Wireless Commun., vol. 12, no. 8, pp. 4210–4221, Aug. 2013. [10] B. K. Chalise, Y. D. Zhang, and M. G. Amin, “Energy Harvesting in an OSTBC based amplify-and-forward MIMO relay system,” in Proc. ICASSP, 2012, pp. 3201–3204.

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[11] B. K. Chalise, W.-K. Ma, Y. D. Zhang, H. A. Suraweera, and M. G. Amin, “Optimum performance boundaries of OSTBC based AF-MIMO relay system with energy harvesting receiver,” IEEE Trans. Signal Process., vol. 61, no. 17, pp. 4199–4213, Sep. 2013. [12] R. Zhang, Y.-C. Liang, C. C. Chai, and S. Cui, “Optimal beamforming for two-way multi-antenna relay channel with analogue network coding,” IEEE J. Sel. Areas Commun., vol. 27, no. 5, pp. 699–712, Jun. 2009. [13] S. Xu and Y. Hua, “Optimal design of dpatial dource-and-relay matrices for a non-regenerative two-way MIMO relay system,” IEEE Trans. Wireless Commun., vol. 10, no. 5, pp. 1645–1655, May 2011. [14] F. Roemer and M. Haardt, “Algebraic norm-maximizing (ANOMAX) transmit strategy for two-way relaying with MIMO amplify and forward relays,” IEEE Signal Process. Lett., vol. 16, no. 10, pp. 909–912, Oct. 2009. [15] A. Khabbazibasmenj, F. Roemer, S. A. Vorobyov, and M. Haardt, “Sumrate maximization in two-way AF MIMO relaying: Polynomial time solutions to a class of DC programming problems,” IEEE Trans. Signal Process., vol. 60, no. 10, pp. 5478–5493, Oct. 2012. [16] R. Wang, M. Tao, and Y. Liu, “Optimal linear transceiver designs for cognitive two-way relay networks,” IEEE Trans. Signal Process., vol. 61, no. 4, pp. 992–1005, Feb. 2013. [17] R. Wang, M. Tao, and Y. Huang, “Linear precoding designs for amplifyand-forward multiuser two-way relay systems,” IEEE Trans. Wireless Commun., vol. 11, no. 12, pp. 4457–4469, Dec. 2012. [18] Z.-Q. Luo, W.-K. Ma, A. M.-C. So, Y. Ye, and S. Zhang, “Semidefinite relaxation of quadratic optimization problems,” IEEE Signal Process. Mag., vol. 27, no. 3, pp. 20–34, May 2010. [19] W. Ai, Y. Huang, and S. Zhang, “New results on Hermitian matrix rankone decompostion,” Math. Programm., vol. 128, no. 1/2, pp. 253–283, 2011. [20] R. Horst and N. V. Thoai, “DC programming: Overview,” J. Optim. Theory Appl., vol. 103, no. 1, pp. 1–43, Oct. 1999. [21] A. J. Smola, S. V. N. Vishwanathan, and T. Hofmann, “Kernel methods for missing variables,” in Proc. 10th Int. Workshop Artif. Intell. Stat., 2005, pp. 325–332. [22] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge Univ. Press, 2004. [23] J. Magnus and H. Neudecker, Matrix Differential Calculus With Applications in Statistics and Econometrics. Hoboken, NJ, USA: Wiley, 2007. [24] A. Ben-Tal and A. Nemirovski, Lectures on Modern Convex Optimization: Analysis, Algorithms, Engineering Applications. Philadelphia, PA, USA: SIAM, 2001, ser. MPS-SIAM Series on Optimization. [25] 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. [26] M. Grant and S. Boyd, CVX: Matlab Software for Disciplined Convex Programming. [Online]. Available: http://cvxr.com/cvx [27] Y. Huang and D. P. Palomar, “Rank-constrained separable semidefinite programming with applications to optimal beamforming,” IEEE Trans. Signal Process., vol. 58, no. 2, pp. 664–678, Feb. 2010.

Quanzhong Li received the B.S. degree from Sun Yat-Sen University (SYSU), Guangzhou, China, in 2009. He is currently working toward the Ph.D. degree at the School of Information Science and Technology, SYSU. His research interests are in wireless communications powered by energy harvesting, cognitive radio, cooperative communications, and multipleinput-multiple-output (MIMO) communications.

Qi Zhang (S’04–M’11) received the B.Eng. (Hons.) and the M.S. degrees from the University of Electronic Science and Technology of China (UESTC), Chengdu, China, in 1999 and 2002, respectively, and the Ph.D. degree in electrical and computer engineering from the National University of Singapore (NUS), Singapore, in 2007. He is currently an Associate Professor with the School of Information Science and Technology, Sun Yat-Sen University, Guangzhou, China. From 2007 to 2008, he was a Research Fellow in the Communications Lab, Department of Electrical and Computer Engineering, NUS. From 2008 to 2011, he was at the Center for Integrated Electronics, Shenzhen Institutes of Advanced Technology, Chinese Academy of Sciences and The Chinese University of Hong Kong. His research interests are in wireless communications powered by energy harvesting, cooperative communications, ultra-wideband (UWB) communications.

Jiayin Qin received the M.S. degree in radio physics from Huazhong Normal University, Wuhan, China, in 1992 and the Ph.D. degree in electronics from Sun Yat-Sen University (SYSU), Guangzhou, China, in 1997. Since 1999, he has been a Professor with the School of Information Science and Technology, SYSU, China. From 2002 to 2004, he was the Head of the Department of Electronics and Communication Engineering, SYSU, China. From 2003 to 2008, he was the Vice Dean of the School of Information Science and Technology, SYSU, China. His research areas include wireless communication and submillimeter wave technology. Dr. Qin is the recipient of the IEEE Communications Society Heinrich Hertz Award for Best Communications Letter in 2014, the Second Young Teacher Award of Higher Education Institutions, Ministry of Education (MOE), China in 2001, the Seventh Science and Technology Award for Chinese Youth in 2001, the New Century Excellent Talent, MOE, China in 1999.