Robust Energy Efficient Beamforming in MISOMESWIPT Systems With Proportional Secrecy Rate
Abstract
The joint design of beamforming vector and artificial noise covariance matrix is investigated for the multipleinputsingleoutputmultipleeavesdropper simultaneous wireless information and power transferring (MISOMESWIPT) systems. In the MISOMESWIPT system, the base station delivers information signals to the legitimate user equipments and broadcasts jamming signals to the eavesdroppers. A secrecy energy efficiency (SEE) maximization problem is formulated for the considered MISOMESWIPT system with imperfect channel state information, where the SEE is defined as the ratio of sum secrecy rate over total power consumption. Since the formulated SEE maximization problem is nonconvex, it is first recast into a series of convex problems in order to obtain the optimal solution with a reasonable computational complexity. Two suboptimal solutions are also proposed based on the heuristic beamforming techniques that trade performance for computational complexity. In addition, the analysis of computational complexity is performed for the optimal and suboptimal solutions. Numerical results are used to verify the performance of proposed algorithms and to reveal practical insights.
I Introduction
Energy harvesting technology, which enables devices to scavenge energy from ambient resources, has attracted significant attention from academia and industry in recent decade [2, 3]. Harvesting energy from natural resources requires the naturalresource energyharvesting (NREH) module to be directly exposed in natural resources, such as solar and wind [3, 4, 5]. Moreover, the NREH module has a large form factor [3], which is not suitable for small devices, e.g., implantable wireless sensors. Another drawback of NREH takes root in the volatility of natural resources. Since the radiofrequency (RF) signals deliver both information and energy as well as penetrate the obstacles, radiofrequency energyharvesting (RFEH) technology becomes an alternative to overcome the shortcomings of NREH technology. Since the energy of RF signals is controllable, the RFEH technology is more suitable for stable energy provision [6, 7, 8].
The current research on RFEH technology can be categorized into two branches. The first branch focuses on wireless powered communication systems (WPCSs), where the wireless devices harvest energy from RF signals and then use the harvested energy to transmit information [8, 9, 10]. For example, the authors in [9, 10] proposed algorithms to maximize the system throughput in the multiuser WPCSs, where the base station (BST) broadcasts the energy signals and receives information signals in halfduplex mode [9] or fullduplex mode [10]. The WPCSs also find the applications in devicetodevice communications [11], secure communications [11, 12], wireless relaying communications [13, 14, 15] and passive communications [16]. These works mainly focused on the usage of harvested energy to optimize certain performance metrics, such as throughput [9, 10], system secrecy [11, 12], outage probability [13, 14, 15], power consumption [16], and energy efficiency (EE) [17, 18]. Another branch of research on RFEH technology is named as simultaneous wireless information and power transferring (SWIPT) systems, where the energy and information signals are broadcast simultaneously [19, 20, 21, 22]. The concept of SWIPT systems was originally proposed by Varshney in [19], where he studied the fundamental tradeoff within the rateenergy region in pointtopoint (PtP) systems. Then, the authors in [20] proposed two practical receiver architectures, where the RF signals are split into two streams for energy harvesting module and information detecting module. Moreover, the authors in [20, 21] investigated the rateenergy region of PtP systems for linear energy harvester and nonlinear energy harvester, respectively. Since the application of multiple antenna technology can enable the transmission of practical amount of energy, the research on the WPCSs and SWIPT systems was extended to multipleinputsingleoutput (MISO) [23, 24, 25, 15, 26] and multipleinputmultipleoutput (MIMO) systems [27, 28, 29, 30, 31].
Ia Related Works and Contributions
The current research on SWIPT systems focus on three types of receivers: 1) timeswitching receiver [15, 30]; 2) powersplitting receiver [23, 24, 30]; and 3) antennaseparating receiver [25, 26, 29, 30, 31]. Based on the receiver setups, significant research efforts have been made to investigate the system power minimization [24], system throughput maximization [30, 31] and harvested energy maximization (HEM) [25, 26, 30, 31]. For examples, the authors in [25] and [26] studied the HEM for the antennaseparating receiver with linear energy harvester and nonlinear energy harvester, respectively.
Though the broadcast characteristic of wireless channels enables onetomany energy delivery, it also increases the probability of legitimate messages to be eavesdropped in the SWIPT systems [32, 33, 34, 35, 36]. Therefore, the combination of physical layer security with SWIPT systems becomes an emerging research topic. For examples, the authors in [32] investigated the secrecy rate maximization (SRM) and HEM in the PtP systems with multiple energyharvesting nodes (EHNs) colocated with eavesdroppers (EVEs). Combining CharnesCooper transformation [37] with onedimensional search method, the authors in [32] proposed optimal solutions to the SRM and HEM problems. Then, the authors in [33] studied the SRM in singlestream and multiplestream MIMOPtP systems with perfect channel state information (CSI). Using a semidefinite relaxation (SDR) technique and an inexact blockcoordinatedescent method, the optimal and nearoptimal solutions are obtained for the singlestream and multiplestream cases. The authors in [38] studied the secure communication in the renewable resource powered distributed antenna systems with SWIPT capability, which allows each access point to exchange energy with the central processor. By jointly designing the beamforming vector, artificial noise (AN) covariance matrix and energy exchange variables, the system power consumption is minimized in [38].
Due to the tremendous growth of energy consumption, the wireless operators require new approaches to reduce their energy bills or maximize the energy utilization efficiency. Hence, the EE optimization is one of the most important research issues for the future generation wireless communications [39, 40]. When multiple EVEs exist in the wireless communication systems, a natural extension to the traditional EE optimization becomes secrecy energy efficiency (SEE) optimization. The SEE optimization is more important than traditional EE optimization since the traditional EE optimization does not jointly consider the constructive and detrimental effects of artificial noise in energy efficiency. Although there a number of works considered EE optimization in SWIPT systems [23, 41], the SEE optimization in SWIPT systems has become an active research area recently [42]. Moreover, the SEE optimization induces a better energy utilization efficiency compared with the system throughput maximization or system power minimization. With perfect CSI, the authors in [42] investigated the SEE optimization problem in a MIMOPtP system with multiple EVEs. Using the Dinkelbach method, they proposed an iterative algorithm to obtain the optimal SEE. However, their proposed algorithm cannot be applied to the SWIPT systems with multiple legitimate users (LUEs). When multiple LUEs exist in the MISOSWIPT systems, the fairness issue among the LUEs becomes an important topic, which has yet been investigated.
Since the perfect CSI is challenging to obtain in MISO and MIMO systems^{1}^{1}1This situation is especially true when the users are passive or roaming to a new place and do not interchange information with the local BST., the authors in [34] studied the system power minimization in the multiuser multipleinputsingleoutputmultipleeavesdropper (MISOME)SWIPT systems with imperfect CSI. Specifically, they leveraged the energy signals and AN to secure the legitimate communication and satisfy the energy requirement. Then, the authors extended the work in [34] and jointly considered the transmission power, RFEH efficiency and interference power leakage as a multiobjective optimization problem with imperfect CSI [35]. In [36], the authors studied the impact of nonlinear energy harvester on system power consumption in a multiuser MISOMESWIPT system. To the authors’ best knowledge, the robust SEE optimization problem in the multiuser MISOMESWIPT systems with proportionalsecrecyrate (PSR) constraints has not been reported.
In this work, we consider a multiuser MISOMESWIPT system where a BST transmits information signals and jamming signals to the LUEs and EVEs, respectively. Meanwhile, the EHNs scavenge energy from the information signals and jamming signals. Different from [42], we study the SEE optimization via joint design of beamforming vector and AN covariance matrix under the imperfect CSI of EVEs and EHNs. Moreover, we include the RFEH constraints and PSR constraints to guarantee the harvested energy at the EHNs and ensure the fairness among LUEs as in [43, 44], respectively. The formulated SEE optimization problem is more complicated than the traditional EE optimization problem [23, 41] due to the nonconvex PSR constraints and nonconvex SEE function. In order to develop tractable algorithms, we first simplify the SEE optimization problem via exploiting the problem structure. Based on the introduced parameter, we leverage the SDR technique coupled with a onedimensional search method to address the simplified SEE optimization problem. The major contributions of this work can be summarized as follows.

We formulate the SEE maximization problem for a multiuser MISOMESWIPT system via the joint design of beamforming vector and AN covariance matrix. In order to guarantee the fairness among LUEs, we explicitly confine the secrecy rates of LUEs to a predefined ratio such that the secrecy rates of LUEs are proportional. Since the EVEs and EHNs are either roaming users or passive devices, they may not exchange CSI as frequently as that of LUEs. Hence, we consider the boundederror CSI of EVEs and EHNs.

The formulated SEE maximization problem in the MISOMESWIPT system is nonconvex in SEE function and PSR constraints. Therefore, it is difficult and ineffective to solve the nonconvex SEE maximization problem via standard methods. In order to develop effective algorithms, we exploit the structure of nonconvex PSR constraints such that a tractable form of SEE maximization problem is obtained where the strong duality holds. We theoretically prove that: 1) the nonconvex PSR constraints can be relaxed into a finite set of convex constraints; and 2) the relaxations are tight. Based on these facts, we propose a twostage algorithm such that an optimal solution is obtained via solving a series of convex optimization problems and a onedimensional search method.

In order to trade the performance for computational complexity, we propose two suboptimal algorithms based on several heuristic beamforming techniques, e.g., maximal ratio transmission (MRT) and zeroforcing beamforming (ZFBF). The computational complexity of the proposed optimal and suboptimal algorithms are quantitatively compared.
Numerical results are used to verify the performance of the proposed optimal and suboptimal algorithms.
IB Organization and Notations
The remainder of this paper is organized as follows. The system model and the SEE maximization problem are presented with imperfect CSI of EVEs and EHNs in Section II. The optimal and suboptimal algorithms are proposed in Section III and Section IV, respectively. Numerical results are used to verify the effectiveness of proposed solutions and provide several practical insights in Section V. Finally, Section VI concludes the work.
Notations: Vectors and matrices are shown in bold lowercase letters and bold uppercase letters, respectively. denotes the dimension complexvalue matrices. and are the Frobenius norm and absolute value, respectively. stands for “distributed as”. and denote, respectively, an dimensional identity matrix and a zero matrix with rows and columns. The expectation of a random variable is denoted as . obtains a vector by stacking the columns of under the other. denotes a diagonal matrix with the diagonal elements given by the matrices . represents the set made of , . For a square matrix , and denote its conjugate transpose and trace, respectively. and respectively denote that is a positive semidefinite and is a positive definite matrix.
Ii System Model and Problem Formulation
Iia Overall Description
We consider downlink transmission of a MISOMESWIPT system, which consists of one single BST, a set of LUEs, a set of EVEs and a set of EHNs. Let , and denote the set of LUEs, set of EVEs and set of EHNs, respectively. The BST is equipped with antennas for simultaneous transmission of information and artificial noise over the same single carrierfrequency band. Therefore, the LUEs and EVEs decode information via beamforming. Each LUE is equipped with one antenna to receive the legitimate information. Each EVE, which is also equipped with one single antenna, can be roaming user from other wireless communication systems and searching for the services from the local BST^{2}^{2}2This is due to the fact that the malicious roaming users have the potential to intercept signals intended for LUEs under the coverage of the same BST.. Therefore, we leverage the AN to interrupt EVEs and guarantee RFEH requirement of EHNs. Here, the EHNs are equipped with one antenna and can be the passive sensors to monitor the environment status of the MISOMESWIPT system.
IiB Signal Model
At the BST, the informationbearing signal for the th LUE is given as with . Therefore, the transmission signal for LUEs is denoted by
(1) 
where is the beamforming vector for the th LUE. In order to secure the BSTLUE links, the BST needs to transmit an AN vector to reduce information leakage to EVEs. Hence, the transmission signal at the BST is given as
(2) 
where the AN vector is modeled as a complex Gaussian random vector with mean zero and covariance . In particular, it is assumed that the informationbearing signals and the AN vector are statistically independent.
Remark 1
In order to guarantee the strongly secure communications, the informationbearing signals contain both main information and auxiliary information. Here, the main information must be reliably delivered for BSTLUE links, and the auxiliary information is used to increase randomness to mislead the EVEs [45].
The framebased frequency nonselective fading channels with unit duration for each frame is considered; therefore, the words “energy” and “power” can be used interchangeably. In each frame, the BST broadcasts informationbearing signals and jamming signals over the same single carrierfrequency band. Considering the Rayleigh fading channels, the channelcoefficient vector for the th BSTLUE link is circularly symmetric complex Gaussian (CSCG) distributed as . Here, captures the joint effect of link distance and carrier frequency for the th BSTLUE link [46, 47]. As a result, the received signals of the th LUE is expressed as
(3) 
where is the additive white Gaussian noise (AWGN) with mean zero and variance of the th LUE.
Based on the received signals in (3), the data rate of the th LUE is denoted as
(4) 
where
(5) 
with and , . In addition, the rank of each beamforming matrix is upper bounded as . The term denotes the bandwidth of the system.
Besides, the received signal at the th EVE is denoted as
(6) 
where is the AWGN with mean zero and variance at the th EVE, ; is the channelcoefficient vector for the th BSTEVE link.
Based on the signal model in (6), the information leakage to the th EVE of the th LUE by treating the interference as noise is denoted as [46]
(7) 
where
(8) 
where .
Denote the channelcoefficient vector of the th BSTEHN link as , the received baseband signal of the th EHN is denoted as
(9) 
where is the AWGN at the th EHN, and is associated with the pathloss and carrier frequency of the th BSTLUE link.
Ignoring the energy that can be harvested from the AWGN, the amount of harvested energy of the th EHN is denoted as
(10) 
where , and is the energy conversion efficiency.
IiC Secure Data Rate, Power Consumption and SEE
Since we consider strongly secure communication, the informationleakage rate of the th BSTLUE link is confined to be smaller than or equal to the rate of auxiliary information . Hence, the secrecy rate of the th BSTLUE link is written as [45, 48]
(11) 
with
(12) 
where .
Furthermore, the power consumption of BST in the MISOMESWIPT system is comprised of the power of downlink beamforming, power of AN and power of circuit. Therefore, the expression of power consumption is given as
(13) 
where the constant circuit power consumption is given as since the circuit power consumption is positively correlated to the number of antennas [49]. Here, the term denotes the baseband processing power consumption.
In this work, the SEE is defined as the ratio of the sum secrecy rates of BSTLUE links over the power consumption of BST. Based on (11) and (13), the SEE of MISOMESWIPT system is given as
(14) 
whose unit is Nats/joule.
Remark 2
Different from [23], we ignore the harvested energy in the power consumption and investigate the SEE maximization. This is due to the fact that the harvested energy is stored in the integrated battery of EHNs and cannot reduce the energy bills of the wireless operators.
IiD Channel State Information
We assume that the MISOMESWIPT system operates in timedivision duplex mode. At the beginning of each frame, each LUE sends a pilot signal to the BST. After receiving the pilot signals, the BST estimates the channels associated with each LUE that sends the pilot signal. Since each LUE facilitates each uplink transmission with pilot signals, the BST can exploit the uplink reciprocity to periodically update the downlink CSI for BSTLUE links. Therefore, we assume that the BST can perfectly estimate the downlink CSI for the BSTLUE links. However, the EVEs and EHNs can be the roaming users and/or lowpower nodes and do not exchange the pilot signals with the BST as frequently as the LUEs. The obtained CSI for the BSTEVE links can be imperfect subject to channel estimation errors and/or quantization errors. Hence, we are motivated to use the boundederror CSI model to formulate the CSI for the BSTEVE and BSTEHNs links as [50, 51]
(15)  
(16) 
where and are, respectively, the estimated channelcoefficient vector and channel uncertainty of the th BSTEVE link. Similarly, and are the estimated channelcoefficient vector and channel uncertainty vector of the th EHN. The positive constants and denote the radii of uncertainty region of and . Here, the vectors and are CSCG distributed having the terms and that take into account of the joint effect of pathloss and carrier frequency of the th BSTEVE link. In addition, the channel uncertainty vectors and capture the joint effect of estimation errors and timevarying characteristics of wireless channels.
Remark 3
When the roaming users are malicious and want to intercept the unauthorized services, they become the EVEs. However, the EVEs also need to exchange pilot signal with the BST to establish links to obtain their registered services. The lowpower nodes can be sensors in the coverage of the same BST. Although the lowpower nodes do not interact with the BST, they may still need to report messages to the data fusion. Thus, the BST can still estimate the CSI of BSTEHN links.
IiE Problem Formulation
Our objective is to maximize SEE via the joint design of beamforming vector and AN covariance matrix in the MISOMESWIPT system. At the beginning of each frame, the BST performs beamforming and jamming in a centralized way. The SEE maximization problem is formulated as
(17a)  
s.t.  (17b)  
(17c)  
(17d)  
(17e)  
(17f)  
(17g) 
where the constraints in (17b) are used to guarantee the proportional fairness on the secrecy rates of LUEs with and [43, 44]; and the informationleakage constraints and the harvestedpower constraints are respectively defined in (17c) and (17d). Here, the term in (17d) denotes the energy requirement of the th EHN; and the constant in the constraints in (17e) is the power budget of BST due to the circuit limitation.
Remark 4
Iii Optimal SEE Maximization
We observe that the finding the optimal solution to optimization problem (17) is challenging due to the nonconvex objective function (17a), the PSR constraints in (17b), the infinite amount of informationleakage constraints in (17c) and RFEH constraints in (17d) and the nonconvex rankone constraints in (17g). In order to solve the problem (17), we deal with nonconvexity in the objective function (17a) and constraints in (17b) and (17g) via a convex relaxation and SDR. Moreover, we obtain the equivalent convex forms in finite amount for the constraints in (17c) and (17d) via procedure [52].
Since the nonconvex objective function (17a) and PSR constraints in (17b) contain the secrecy rate of LUEs, we are motivated to handle the nonconvexity in (17a) and (17b) via convex relaxation. Introducing an auxiliary variable with , the objective function (17a) and PSR constraints (17b) are respectively relaxed as
(18) 
and
(19) 
where . Note that the sum secrecy rate of all BSTLUE links is lower bounded by , and the secrecy rate of the th BSTLUE link is lower bounded by , . Thus, the lower bounds of secrecy rates of BSTLUE links satisfy the predefined ratios.
Therefore, the convexrelaxation version of optimization problem (17) is obtained as
(20a)  
s.t.  (20b) 
In problem (20), we replace the objective function (17a) with its lower bound (18) and relax the PSR constraints in (17c) by (19). Before we proceed to justify the equivalence between (17) and (20), we verify the tightness of convex relaxations in (18) and (19) for problem (20) in Proposition 1.
Proposition 1
Suppose that the righthandside of (18) and the constraints in (19) are not inactive under the optimal beamforming matrices and AN covariance matrix to problem (20). Then, the inequalities in (18) and (19) are active via another optimal beamforming matrices and AN covariance matrix to (20), which can be constructed as
(21a)  
(21b)  
(21c) 
Proof:
See Appendix A. \qed
Based on Proposition 1, we conclude the equivalence between the problems (17) and (20), which is verified in Appendix B. Hence, we investigate the problem (20) to seek a low complexity algorithm for the problem (17).
In order to deal with the infinite amount of informationleakage constraints in (17c) and (17d), we first review the procedure [52] in Lemma 1.
Lemma 1 (procedure [52])
Denote the functions and as and with and . Then, the condition holds if and only if there is such that
(22) 
provided that there exists a point such that , .
With (7) and (8), we obtain an equivalent form of informationleakage constraints in (17c) as
(23) 
where with and .
Applying Lemma 1 to (23), we obtain an equivalent form of informationleakage constraints in (17c) as
(24) 
where , and is the introduced auxiliary variable, and .
Then, applying Lemma 1 to (25) and performing some algebraic manipulations, we obtain an equivalent form of RFEH constraints in (17d) as
(26) 
where , and is the introduced nonnegative auxiliary variable, .
Based on the previous manipulations in (19), (24) and (26), we can now convexify the feasible region of the problem (20) via SDR technique. By dropping the constraints in (17g), the feasible region of problem (20) is transformed into a convex one as
(27a)  
s.t.  (27b)  
(27c) 
Remark 5
The only challenge in solving the problem (27) is to deal with the nonconvex objective function. Since the objective function (27a) is joint convex with respect to the beamforming matrices , AN covariance matrix and auxiliary variables and , we propose a twostage algorithm to solve the problem (27): 1) solving the problem (27) with given ; 2) performing a onedimensional search to obtain the optimal .
Remark 6
The feasible region of problem (27) is a convex hull of the feasible region of problem (20). Hence, the optimal value of (27) serves as the lower bound of the optimal value of (20) with the optimal . Thus, we are motivated to show that the performed SDR in (27) given is tight such that the optimal solution to (27) given is the optimal value of (20).
Due to the power budget constraint in (17e), the strong duality of problem (27) may not hold. Hence, we are motivated to figure out an equivalent form of problem (27), whose strong duality holds. We observe that the power constraint in (17e) and objective function (27a) have a common term . Dropping the power constraint in (17e) and fixing the value of , we solve the problem (27) as
(28a)  
s.t.  (28b)  
(28c) 
The benefits of dropping the power constraint in (17e) are three folds: 1) the problem (28) is always feasible; 2) the strong duality of (28) always holds [52]; and 3) the computational complexity of solving (28) is lower than that of (27). Since the problem (27) is feasible if and only if the optimal value of (28) is less than or equal to , we exploit the monotonicity of objective function (28a) with respect to via the Proposition 2.
Proposition 2
Proof:
See Appendix C. \qed
With the Proposition 2, we can claim that there exists only one value of such that the optimal power consumption . The value of can be obtained via a onedimensional line search. Now, we investigate the optimality of SDR technique. Hereinafter, we ignore the superscript in order to keep the brevity of presentation.
Proposition 3
There exists an optimal solution to the problem (28) with the rank of beamforming matrices satisfying
(29) 
Proof:
See Appendix D. \qed
When the value of guarantees , the optimal solution to problem (27) is obtained via solving the problem (28) and performing the onedimensional search for the optimal based on Proposition 3. As a result, we propose Algorithm 1, which is named as SDP empowered twostage beamforming and artificial jamming (SDPTsBAJ) algorithm.
We observe that the generated augmented SEE sequence increases monotonically with the number of search points. Since the system SEE is upper bounded due to the maximum transmit power of BST, we conclude that the SDPTsBAJ algorithm in converges.
Iv Low Complexity Suboptimal SEE Maximization
Although the proposed SDPTsBAJ algorithm can significantly reduce the computational complexity compared with the brandandbound algorithm, we observe that the computational complexity of SDPTsBAJ algorithm increases fast with the number of LUEs. In order to reduce the computational complexity of the SDPTsBAJ algorithm, we design the beamforming vector based on heuristic beamforming techniques: ZFBF and MRT. Denote the matrices and as and , respectively. To remove the interference of AN, we perform the singular value decomposition to the matrix as with and construct the AN covariance matrix as
(30) 
where the matrix .
Similarly, we apply the ZFBF technique to the beamforming vectors of LUEs as
(31) 
where the matrix is obtained via singular value decomposition of the matrix with . Here, each entry of is a complex weight to each column vector of .
Substituting (30) and (31) into the problem (28), we obtain a simplified problem as
(32a)  
s.t.  (32b)  
(32c)  
(32d)  
(32e)  
(32f) 
where the matrices and are, respectively, denoted as
(33) 
and
(34) 
with .
The optimization problem (32) is convex and solvable in CVX. The rankone constraints for the matrices can be recovered following the procedure in [48]. Instead of solving (28), the proposed ZFBFTsBAJ algorithm solves (32) at the third step of the SDPTsBAJ algorithm.
In order to obtain the beamforming vector in ZFBFTsBAJ algorithm, we still need to calculate a complex weight vector . Thus, we leverage the MRT technique to reduce number of variables such that computational complexity of ZFBFTsBAJ algorithm is further reduced. Performing the MRT technique to the th ZFBF vector by setting , we obtain . Hence, the th MRTZFBF beamforming vector is given as
(35) 
where is the transmit power for the th BSTLUE link.
Substituting (30) and (35) into (19), we obtain the closedform transmit power for the th BSTLUE link as
(36) 
with given .
Substituting (30) and (35) into problem (28), we obtain a downgraded problem as
(37a)  
s.t.  (37b)  
(37c)  
(37d)  
(37e) 
where
(38) 
and
(39) 
Solving the optimization problem (37) in the third step of the SDPTsBAJ algorithm, we obtain the MRTZFBFTsBAJ algorithm.
Complexity Comparison: The optimization problem (28) has matrix variables and real variables, and each matrix variable is of size . Therefore, the number of decision variables of the SDPTsBAJ algorithm is at the order of . Moreover, the feasible region of problem (28) contains positivesemidefinite constraints with size , positivesemidefinite constraints with size and linear constraints. Hence, the number of iterations is at order of . The overall computational complexity is calculated as , where and . Here, the operator is the worst case computation bound [53]. Following the similar procedure, we summarize the computational complexities of the proposed SDPTsBAJ, ZFBFTsBAJ and MRTZFBFTsBAJ in Table I with as the number of onedimensional search.
Algorithms  Complexity Calculation 
SDPTsBAJ  
ZFBFTsBAJ  
MRTZFBFTsBAJ  
We observe that the computational complexity of the proposed SDPTsBAJ, ZFBFTsBAJ and MRTZFBFTsBAJ algorithms scale linearly with the number of onedimensional search points. The SDPTsBAJ algorithm has the highest computational complexity among the three proposed algorithms. Then, the computational complexity is followed by the proposed ZFBFTsBAJ algorithm. The computational complexity of the MRTZFBFTsBAJ algorithm is the lowest.
V Numerical Results
In this section, we present simulation results to demonstrate the performances of the proposed algorithms. The parameters , and follow the setup of indoor channels as dB, where and are respectively the carrier frequency and link distance [47]. The values of and are set as and , respectively. Unless otherwise specified, the simulation parameters are set in Table II.
Parameters  Symbols  Values 
Carrier frequency and system bandwidth  ,  900 MHz, 200 KHz 
Number of LUEs, EVEs and EHNs  , ,  3, 2, 2 
Number of antennas at BST  7  
Power of AWGN  , ,  30 dBm 
Maximum Tx power of BS  43 dBm  
Required harvested power of the th EHN  5 dBm  
Energy conversion efficiency of the th EHN  0.8  
Rate of auxiliary information of the th LUE 