# Distributed Beamforming Design for Nonregenerative Two-Way Relay Networks with Simultaneous Wireless Information and Power Transfer

Authors: Keyun Liao; Sai Zhao; Yusi Long; Gaofei Huang; Dong Tang
Journal: Wireless Communications and Mobile Computing
 (2019)

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## Abstract

This paper considers the distributed beamforming design for a simultaneous wireless information and power transfer (SWIPT) in two-way relay network, which consists of two sources,K relay nodes and one energy harvesting (EH) node. For such a network, assuming perfect channel state information (CSI) is available, and we study two different beamforming design schemes. As the first scheme, we design the beamformer through minimization of the average mean squared error (MSE) subject to the total transmit power constraint at the relays and the energy harvesting constraint at the EH receiver. Due to the intractable expression of the objective function, an upper bound of MSE is derived via the approximation of the signal-to-noise ratio (SNR). Based on the minimization of this upper bound, this problem can be turned into a convex feasibility semidefinite programming (SDP) and, therefore, can be efficiently solved using interior point method. To reduce the computational complexity, a suboptimal beamforming scheme is proposed in the second scheme, for which the optimization problem could be recast to the form of the Rayleigh–Ritz ratio and a closed-form solution is obtained. Numerical results are provided and analyzed to demonstrate the efficiency of our proposed beamforming schemes.


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## Body

## 1. Introduction

Harvesting energy from environment has become a preferred strategy to overcome the key challenge of limited lifetime of devices equipped with battery in wireless networks, e.g., wireless sensor network (WSN) or Bluetooth low-energy mesh (BLE Mesh) network. As an efficient energy harvesting technique, simultaneous wireless information and power transfer (SWIPT) has gained considerable research attention in the academic field [1–6]. The concept of SWIPT was first introduced and a capacity-energy function was first proposed to characterize the fundamental tradeoff between simultaneous energy and information transfer in [1]. Considering SWIPT in multiple-input single-output (MISO) multicasting systems [2], the authors minimize the base station (BS) transmit power by jointly optimizing the beamforming vector at BS and the power splitting parameters of mobile station. In [3], considering a multiple-input multiple-output (MIMO) wireless broadcast scenario, where one receiver harvested energy and another receiver decoded information from the signals sent by a transmitter, two practical schemes, namely, time switching (TS) and power splitting (PS), were proposed. Motivated by the requirement of fifth generation, Xu et al. [4] investigated the cooperative SWIPT NOMA protocol design in single-input single-output (SISO) and multiple-input single-output (MISO). Meanwhile, the security of information transmission also became more and more important. Zhao et al. [5] considered the security of wireless network for SWIPT, which maximize the achievable secrecy rate subject to sum transmit power constraint at relays and energy harvesting (EH) constraint at the EH receiver. Aforementioned papers were investigated under perfect channel state information (CSI). In [6], the MIMO SWIPT system under the condition of imperfect CSI was studied.Considering its advantage in enhancing spectral efficiency, two-way relay (TWR) network is extensively studied [7–10]. In the TWR network, two source nodes exchanged information through a relay node, and the number of required time slots is reduced from four to two with the aid of analog network coding [7]. In [8], a cooperative relay scheme for distributed amplify-and-forward (AF) relays working under the TWR protocol was considered, and a closed-form solution was obtained. Considering the user fairness, a linear precoding design for relay in multiple-pair two-way MIMO relay systems was investigated [9]. Considering security, cooperative beamforming in cognitive two-way relay networks was studied in [10]. For SWIPT two-way relay channels, the authors of [11–15] considered the relay beamforming design for SWIPT in a nonregenerative TWR network with the objective to maximize the sum rate of TWR. Wen et al. [12] studied the joint source and relay design for MIMO two-way relay networks with SWIPT, which minimized the total mean square error guaranteeing sufficient energy harvested at the sources. The authors of [14, 15] developed a novel distributed energy beamforming scheme for realizing SWIPT in the TWR network, where the former considered two source node exchanged information via an EH relay node and the latter considered multiple-user exchanged messages with the help of an energy-constrained relay node. Considering imperfect CSI and energy efficiency , the robust optimization schemes of MIMO two-way relay networks were considered under the constraint of energy harvesting [16]. In the IoT networks or sensor networks, most of the nodes may have limited volume and limited computation capacity with limited energy supply, some of the relay nodes are providing data communication service, while other idle node(s) could harvest energy to prolong its/their life cycle. That is why we consider multiple relays in two-way transmission, while at the same time, an EH is placed to harvest energy. And to our best knowledge, there is little work on this system model by now.The main contributions of our paper are summarized as follows:(1)
We study the distributed beamforming design in SWIPT-based two-way multiple-relay networks. Differing from [14, 15], we consider the two-way SWIPT system with multiple relays, while only single relay is considered in [14, 15]. Our goal is to minimize the upper bound of average MSE while satisfying the harvested energy requirements of the EH receiver and the total power budgets of the multiple relays.(2)
For the case of SWIPT-based two-way multiple-relay networks, the initial problem is a nonconvex problem. Through appropriate mathematical transformation, we could recast the initial problem into the semidefinite programming (SDP) problem. Then, we can obtain a quasi-optimal solution by applying semidefinite relaxation (SDR).(3)
To avoid the high computational complexity resulting from solving the SDP problem, we find the suboptimal solution by converting the initial problem into the form of the Rayleigh–Ritz quotient, which is of low complexity.The rest of this paper is organized as follows: in Section2, the system model and the problem formulation are illustrated. Section 3 presents our proposed methods to obtain the solutions to the problem of beamforming design. Section 4 constructs complexity analysis of two schemes and provides simulation results. Finally, Section 5 has a conclusion about the paper.Notations: boldface lowercase and uppercase letters denote vectors and matrices, respectively. The transpose, conjugate, conjugate transpose, Frobenius norm, and trace of matrixA are denoted as AT,A∗,AH,A, and tr(A), respectively. vec(A) denotes to stack the columns of a matrix A into a single vector a. ⊗ denotes the Kronecker product. λmaxA and λminA denote the maximum and minimum eigenvalues of A, respectively. νA denotes the eigenvector of A associating with the maximum eigenvalue. By A≽0, we mean that A is positive semidefinite. I is the identity matrix. CN0,1 denotes the distribution of a circularly symmetric complex Gaussian vector with mean vector 0 and variance 1.


## 2. System Model

We consider a collaborative network consisting of two source nodes,K AF relays using the two-way protocol, and one EH receiver as shown in Figure 1.Figure 1
Illustration of the two-way relay network with SWIPT.Each node in the whole network is equipped with a single antenna. All nodes are subject to the half-duplex mode. The two source nodes complete the information exchange with the help ofK distributed relay nodes within two time slots. In the first time slot, both S1 and S2 transmit symbols to all relays in the same frequency band simultaneously. The symbol received by the kth relay can be describe as(1)rk=P1h1k,fs1+P2h2k,fs2+zk,where Pi, i∈1,2, are the transmit power at S1 and S2, respectively; hik,f are the channel fading coefficient from Si to the kth relay, where k∈K=1,2,3,…,K; si are the transmitted symbols from Si, and Esi2=1; and zk∼CN0,σk2 is the additive white Gaussian noise (AWGN) at the kth relay in the first time slot. We assume perfect channel state information (CSI) of all channels is known to both source nodes. The relay will perform the channel estimation from both sources to multiple relays and feedback the CSI to both source nodes. The EH nodes perform channel estimation from sources to the EH node in the first slot and perform channel estimation from relay to EH nodes in the second slot, then feedback to the sources. The synchronisation is performed by synchronisation signals sent from sources to relays and EH nodes so that all relays and EH can be perfectly synchronized.The symbol to be retransmitted by thekth relay is given by(2)tk=wkP1h1k,fs1+P2h2k,fs2+zk,where wk is the complex beamforming weight at the kth relay.In the second time slot, the processed symboltk is transmitted from the kth relay to both S1 and S2 simultaneously, and the symbol received by S1 and S2 can be, respectively, written as(3)y1=∑k=1Kh1k,btk+n21,(4)y2=∑k=1Kh2k,btk+n22,where n2i∈CN0,σi2 is AWGN at Si, respectively, in the second time slot and hik,b are the channel fading coefficient from the kth relay to Si, k∈K=1,2,3,…,K.Assume the channels are reciprocal during two time slots (e.g., under TDD mode), i.e.,hik,f=hik,b=hik for i∈1,2. Since each source node knows both the channels associated itself with the relay nodes and the weighted coefficients wk, it can subtract the self-interference from itself which is similar to [11]. After this operation, (3) and (4) become(5)y˜1=P2∑k=1Kh1kwkh2ks2+∑k=1Kh1kwkzk+n21,(6)y˜2=P1∑k=1Kh2kwkh1ks1+∑k=1Kh2kwkzk+n22.The signal received by the EH receiver within two time slots can be, respectively, written as(7)e1=P1g1s1+P2g2s2+nEH,e2=P1∑k=1Kg3kwkh1ks1+P2∑k=1Kg3kwkh2ks2+∑k=1Kg3kwkzk+nEH,where gi denotes the channel fading coefficient from source node S1 to the EH receiver, and g3k, k∈K, denotes the channel fading coefficient from the kth relay to the EH receiver, and nEH∼CN0,σEH2 denotes AWGN at EH receiver. For simplicity, the noise powers at two sources, K relays, and EH receiver, if not specified, are σ12=σ22=σk2=σEH2=σ2.The harvested energy at the EH receiver during two time slots is given, respectively:(8)E1=P1g12+P2g22,E2=P1∑k=1Kg3kwkh1k2+P2∑k=1Kg3kwkh2k2+σ2∑k=1Kg3kwk2.In this paper, we will design a distributed relay beamforming to minimize the average MSE of bidirectional transmissions of two source nodes subject to the EH constraint and the total relay power constraint.The MSE of each received symbol in traditional multiple-input multiple-output (MIMO) systems can be expressed by its signal-to-noise ratio (SNR) as MSE = 1/(SNR + 1) if the Wiener filter is used at the receiver [17]. In two-way relay networks, the average MSE of the received symbols for bidirectional transmissions of two source nodes in (5) and (6) can be described as(9)MSE=0.51+SNR1−1+0.51+SNR2−1,where SNRi are the SNR for the received symbols for S1 and S2, respectively.To simplify equation (9), the average MSE can be tightly approximated in the high SNR regime as(10)MSE˜=0.5SNR1−1+0.5SNR2−1≥MSE,where(11)SNR1=P2∑k=1Kh1kwkh2k2σ21+∑k=1Kh1k2wk2,SNR2=P1∑k=1Kh2kwkh1k2σ21+∑k=1Kh2k2wk2.It is noted thatMSE˜ is the upper bound of MSE. Based on MSE˜, the upper bound of MSE minimization problem can be formulated as(12a)minwkMSE˜,(12b)s.t∑k=1Kwk2P1h1k2+P2h2k2+σ2≤P3,(12c)E1+E2≥Q.It is noted that the optimization problem of the upper bound of MSE is a nonconvex problem due to the nonconvex of objective function and EH constraint. In the following section, we propose schemes to solve problem (12a)–(12c).


## 3. Beamforming Design for Multiple Relays

Firstly, the objective function in problem (12a)–(12c) can be rewritten as(13)MSE˜=0.5σ2+σ2wHB1HB1wP1wHge∗geTw+0.5σ2+σ2wHB2HB2wP2wHge∗geTw,where(14)w≜w1,…,wk,…,wKT,B1≜diagh21,…,h2k,…,h2KT,B2≜diagh11,…,h1k,…,h1KT,ge≜h11h21,…,h1kh2k,…,h1Kh2KT.Secondly, it is noted that the optimal beamforming vectorw for problem (12a)–(12c) should guarantee that the total transmit power constraint (12b) at relays is active, i.e., (12b) can be written equivalently as(15)wHB3P3w=1,where B3≜diagP1h112+P2h212+σ2,…,P1h1k2+P2h2k2+σ2,…,P1h1K2+P2h2K2+σ2T.Thirdly, substituting (15) into (12c), the constraint in (12c) can be recast as(16)wHP1gh1∗gh1T+P2gh2∗gh2T+σ2g3g3H−Q˜P3B3w≥0,where(17)gh1≜g31h11,…,g3Kh1KT,gh2≜g31h21,…,g3Kh2KT,Q˜≜Q−E1,g3≜diagg31,…g3k,…g3KT,and Q denotes the threshold of the harvested energy at the EH receiver.We choose0≤Q≤Qmax, where(18)Qmax=E1+λmaxB3−1B4P3,where B4=P1gh1∗gh1T+P2gh2∗gh2T+σ2g3g3H. And λmax⋅ indicates the maximum eigenvalue of a matrix. Substituting (15) into (13), ignoring the coefficient 0.5, problem (12a)–(12c) can be reformulated as(19a)minwwHσ2B3/P1P3+σ2B1HB1/P1+σ2B3/P2P3+σ2B2HB2/P2wwHge∗geTw,(19b)s.twHB3P3w=1,(19c)wHB4−Q˜P3B3w≥0.In optimization problem (19a)–(19c), we have transformed the inequality constraint of total relay transmit power into an equality constraint. This is because the objective function and the constraint in (15) 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.


### 3.1. The Local Optimal Beamforming Scheme

The optimization problem (19a)–(19c) is a fractional quadratically constrained quadratic problem (QCQP) which is intractable. We transform the problem (19a)–(19c) into the following fractional semidefinite programming (SDP) problem [18]:(20a)minWTrC1WTrC2W,(20b)s.tTrC3W=1,(20c)TrC4W≥0,where(21)C1=σ2B3P1P3+B1HB1P1+B3P2P3+B2HB2P2,C2=ge∗geT,C3=B3P3,C4=B4−Q˜P3B3,W=wwH.It is noted that the rank one constraint Rank(W) = 1 has been removed within the problem (20a)–(20c), and the problem (20a)–(20c) is a quasi-convex optimization problem.We have the following proposition.Proposition 1.
The fractional SDP problem (20a)–(20c) can be written equivalent as(22a)minWTrC1W,(22b)s.tTrC2W=1,(22c)TrC3W=1,(22d)TrC4W≥0.Proof of Proposition 1.
Suppose thatt1,W1o and t2,W2o are the optimal value and optimal solution to the problems (20a)–(20c) and (22a)–(22d), respectively. It is obvious that W2o is feasible to problem (20a)–(20c). When W=W2o, the objective value of (20a)–(20c) is t2. Therefore, t2≥t1.
On the contrary, it is easy to prove thatW1o/TrC2W1o is the feasible to problem (22a)–(22d). When W=W1o/TrC2W1o, the objective value of (22a)–(22d) is t1. Therefore, t1≥t2. That means, problems (20a)–(20c) and (22a)–(22d) have the same optimal values, e.g., t1=t2. Therefore, problem (20a)–(20c) is equivalent to (22a)–(22d).
Suppose thatWo is the optimal solution to the problem (22a)–(22d). If RankWo=1, the optimal solution to problem (20a)–(20c) is W1o which can be obtained by the eigenvalue decomposition technique, i.e., W1o=cu, where c and u denote the eigenvalue and eigenvector of Wo, respectively. In the case of RankWo≥2, we apply the Gaussian randomization technique [9] to obtain the final solution W1o of problem (19a)–(19c) from the solution of problem (20a)–(20c). To obtain the final solution of initial problem (12a)–(12c), we also need to scale W1o to satisfy the relay power constraint.


### 3.2. The Closed-Form Beamforming Scheme

To reduce the computational complexity, we propose a suboptimal closed-form solution to the problem (19a)–(19c). Define U as a matrix which consists of the a1≤a≤K eigenvectors of matrix C4 which are associated with the eigenvalues being greater than zero. We have(23)w=Ux,where x is an arbitrary vector such that(24)wHC4w≥0.Substituting (24) into problem (19a)–(19c), we have the equivalent maximization problem:(25)maxxxHD2xxHD1xs.txHD3x=1,where(26)D1=UHC1U,D2=UHC2U,D3=UHC3U.Introducex˜=D31/2x so that x=D3−1/2x˜. Substituting x into (25), we are able to transform problem (25) into the following form:(27a)maxx˜x˜HG2x˜x˜HG1x˜,(27b)s.tx˜Hx˜=1,where(28)G1=D3−1/2HD1D3−1/2,G2=D3−1/2HD2D3−1/2.It is noted that the problem (27a) and (27b) is a generalized Rayleigh quotient problem [16]. Defining q=G11/2x˜, we can rewrite (27a) and (27b) as(29)maxqqHG1H1/2G2G11/2qqHq.Thus, the optimal solution to (29) can be obtained as(30)q=vmaxG1H1/2G2G11/2=G1H1/2D3H−1/2UHge∗,where vmaxA denotes the eigenvector corresponding to the largest eigenvalue of matrix A. Using (30), we have(31)x˜=G11/2q=κG11/2G1H1/2D3H−1/2UHge∗,where κ=1/G11/2G1H1/2D3H−1/2UHge∗ is used to normalize x̃ that satisfies the constraint (27b). Substituting (31) into x=D3−1/2x̃ and combining (23), we have(32)w=κUD3−1/2G11/2G1H1/2D3H−1/2UHge∗=κUD3−1/2G11/2UD3−1/2G11/2Hge∗.


## 3.1. The Local Optimal Beamforming Scheme

The optimization problem (19a)–(19c) is a fractional quadratically constrained quadratic problem (QCQP) which is intractable. We transform the problem (19a)–(19c) into the following fractional semidefinite programming (SDP) problem [18]:(20a)minWTrC1WTrC2W,(20b)s.tTrC3W=1,(20c)TrC4W≥0,where(21)C1=σ2B3P1P3+B1HB1P1+B3P2P3+B2HB2P2,C2=ge∗geT,C3=B3P3,C4=B4−Q˜P3B3,W=wwH.It is noted that the rank one constraint Rank(W) = 1 has been removed within the problem (20a)–(20c), and the problem (20a)–(20c) is a quasi-convex optimization problem.We have the following proposition.Proposition 1.
The fractional SDP problem (20a)–(20c) can be written equivalent as(22a)minWTrC1W,(22b)s.tTrC2W=1,(22c)TrC3W=1,(22d)TrC4W≥0.Proof of Proposition 1.
Suppose thatt1,W1o and t2,W2o are the optimal value and optimal solution to the problems (20a)–(20c) and (22a)–(22d), respectively. It is obvious that W2o is feasible to problem (20a)–(20c). When W=W2o, the objective value of (20a)–(20c) is t2. Therefore, t2≥t1.
On the contrary, it is easy to prove thatW1o/TrC2W1o is the feasible to problem (22a)–(22d). When W=W1o/TrC2W1o, the objective value of (22a)–(22d) is t1. Therefore, t1≥t2. That means, problems (20a)–(20c) and (22a)–(22d) have the same optimal values, e.g., t1=t2. Therefore, problem (20a)–(20c) is equivalent to (22a)–(22d).
Suppose thatWo is the optimal solution to the problem (22a)–(22d). If RankWo=1, the optimal solution to problem (20a)–(20c) is W1o which can be obtained by the eigenvalue decomposition technique, i.e., W1o=cu, where c and u denote the eigenvalue and eigenvector of Wo, respectively. In the case of RankWo≥2, we apply the Gaussian randomization technique [9] to obtain the final solution W1o of problem (19a)–(19c) from the solution of problem (20a)–(20c). To obtain the final solution of initial problem (12a)–(12c), we also need to scale W1o to satisfy the relay power constraint.


## 3.2. The Closed-Form Beamforming Scheme

To reduce the computational complexity, we propose a suboptimal closed-form solution to the problem (19a)–(19c). Define U as a matrix which consists of the a1≤a≤K eigenvectors of matrix C4 which are associated with the eigenvalues being greater than zero. We have(23)w=Ux,where x is an arbitrary vector such that(24)wHC4w≥0.Substituting (24) into problem (19a)–(19c), we have the equivalent maximization problem:(25)maxxxHD2xxHD1xs.txHD3x=1,where(26)D1=UHC1U,D2=UHC2U,D3=UHC3U.Introducex˜=D31/2x so that x=D3−1/2x˜. Substituting x into (25), we are able to transform problem (25) into the following form:(27a)maxx˜x˜HG2x˜x˜HG1x˜,(27b)s.tx˜Hx˜=1,where(28)G1=D3−1/2HD1D3−1/2,G2=D3−1/2HD2D3−1/2.It is noted that the problem (27a) and (27b) is a generalized Rayleigh quotient problem [16]. Defining q=G11/2x˜, we can rewrite (27a) and (27b) as(29)maxqqHG1H1/2G2G11/2qqHq.Thus, the optimal solution to (29) can be obtained as(30)q=vmaxG1H1/2G2G11/2=G1H1/2D3H−1/2UHge∗,where vmaxA denotes the eigenvector corresponding to the largest eigenvalue of matrix A. Using (30), we have(31)x˜=G11/2q=κG11/2G1H1/2D3H−1/2UHge∗,where κ=1/G11/2G1H1/2D3H−1/2UHge∗ is used to normalize x̃ that satisfies the constraint (27b). Substituting (31) into x=D3−1/2x̃ and combining (23), we have(32)w=κUD3−1/2G11/2G1H1/2D3H−1/2UHge∗=κUD3−1/2G11/2UD3−1/2G11/2Hge∗.


## 4. Simulation Results

In the section, we present the simulation results of our proposed beamforming schemes. We assume that the channel from the source nodeSi to the kth relay is modeled as hik=di,k−χ/2h˜ik and the channel from the source node Si to the EH receiver is modeled as gi=di,EH−χ/2g˜i. The channel from the kth relay to the EH receiver is modeled as g3k=dk,EH−χ/2g˜3k, where di,k is the distance between Si and the kth relay; di,EH is the distance between Si and the EH receiver; dk,EH is the distance between the kth relay and the EH receiver; χ=3 is the path loss exponent; and h˜ik,g˜i, and g˜3k are the independent identically distributed (i.i.d.) Rayleigh fading channels, where i∈1,2, k∈K. We consider a symmetric network with d1,EH=d2,EH=8m, and K relays are symmetrically placed at the two sides of the EH receiver with d2j+1,EH = d2j,EH = j ∗ 6 m, where j∈1,…,K/2. Through all the simulations, the transmit power of two sources, if not specified, are P1=P2=10dB, and the noise variance is set to be σ2 = −25 dB. The number of Monte Carlo simulations is set to 1000.In the simulation, different beamforming schemes are considered including the proposed two schemes, i.e., local optimal scheme (denoted as “Local-optimal”), closed-form scheme (denoted as “Closed-form”), and the global optimal scheme(denoted as “Global-optimal”) in [11]. The “Global-optimal” scheme obtains the global optimal solution based on the two-dimensional search and semidefinite programming.Figure2 shows the average sum rate of both S1-to-S2- and S2-to-S1-directional transmissions versus P3 in the underlying networks, comparing the “Global-optimal” scheme, “Local-optimal” scheme, and “Closed-form” scheme. Here, we consider the threshold of the EH constraint is Q=0.5Qmax and the number of relay is K = 4. We observe that the “Global-optimal” scheme outperforms the proposed schemes “Local-optimal” scheme and “Closed-form” scheme. Meanwhile, the “Local-optimal” scheme has better performance than the “Closed-form” scheme. It is also noted that as P3 increases, the average sum rates increase for three schemes.Figure 2
Average sum rate versusP3 when the EH threshold Q=0.5Qmax and K = 4.In Figure3, we compare three schemes in terms of average sum rate versus various EH thresholds, where Q=τQmax, K=4, and P3 = 25 dB. It is noted that, for different EH thresholds, the “Global-optimal” scheme has the best performance, and the “Local-optimal” scheme outperforms the “Closed-form” scheme. It is also noted that as τ increases, the average sum rates decrease for all three schemes.Figure 3
Average sum rate versusτ when the transmit power to noise power ratio of relay P3 = 25 dB and K = 4.In Figure4, we present the average sum rate comparison of three schemes for different number of relays when the transmit power of relay P3  = 40 dB and the threshold of EH constraint Q=0.1Qmax. It is found that the “Global-optimal” scheme has the best performance and the “Closed-form” scheme has the worst performance. With the increase of the number of relays, the average sum rate of the three schemes increases.Figure 4
Average sum rate versus different number of relays when the EH thresholdQ=0.1Qmax and the transmit power at relays P3 = 40 dB.Here, we provide the complexity analysis of the “Global-optimal” scheme, “Local-optimal” scheme, and “Closed-form” scheme. According to [20], the complexity of obtaining an optimal solution for an SDP problem can be approximated as Omaxm,n4nlog1/ɛ, where ɛ denotes the solution accuracy and m and n are the number of constraints and the dimension of the matrix, respectively.(i)
For the “Global-optimal” scheme,m=4 and n=K. Thus, the computational complexity of obtaining an global optimal solution is OTmaxm,K4nlog1/ɛ+nrd where T is the average iteration number of the two-dimensional search and nrd is the complexity of the Gaussian randomization.(ii)
For the “Local-optimal” scheme, the computational complexity is from the SDP problem (22a)–(22d), m=3 and n=K. Therefore, the overall computational complexity of (22a)–(22d) is Omaxm,K4nlog1/ɛ+nrd.(iii)
For the “Closed-form” scheme, the main computational complexity is to solve the function of the problem (27a) and (27b). The computation of the generalized eigenvector of matrix pair G2,G1 requires Oa3 arithmetic operations. Thus, the total computational complexity of the “Closed-form” scheme is Oa3.Obviously, the computational complexity of the “Global-optimal” scheme is very high due to the two-dimensional search. The “Local-optimal” scheme has comparatively much lower computational complexity than the “Global-optimal” scheme, and the “Closed-form” scheme has the lowest computational complexity.


## 5. Conclusion

In this paper, we investigate the low complexity distributed beamforming scheme for the two-way multiple relays with SWIPT system. Though the “Global-optimal” scheme in [11] outperforms the proposed schemes, its computational complexity is very high compared to the two proposed schemes. The “Local-optimal” beamforming scheme has better performance than the “Closed-form” beamforming scheme; however, the computational complexity of the former is higher than that of the latter. Simulation results have shown the effectiveness of the proposed schemes. The multiple antennas equipped with each node in the system and more sophisticated EH model will be considered as an extension research of this paper.



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