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Optimal coordinated beamforming with artificial noise for secure SWIPT in multicell networks
EURASIP Journal on Wireless Communications and Networking volume 2018, Article number: 60 (2018)
Abstract
This paper investigates the multicell coordinated beamforming (MCBF) design for secure simultaneous wireless information and power transfer (SWIPT) in both centralized and distributed manners. In each cell, one transmitter serves multiple information receivers (IRs) and energy receivers (ERs) with the nonlinear energy harvesting (EH) model. Meanwhile, several eavesdroppers (Eves) intend to intercept the confidential information transmitted for IRs. To achieve a secure transmission, the artificial noise (AN) is embedded in the transmit signals of each transmitter. The proposed design is formulated into a powerminimization problem to guarantee the IRs’ information and ERs’ energy requirements while avoiding the information being intercepted by Eves. Since the problem is nonconvex and not easy to solve, a solution method based on semidefinition relaxation (SDR) is proposed and the global optimum is proved to be guaranteed with full channel state information (CSI). We further present a distributed ANaided MCBF for the system by using alternating direction method of multipliers (ADMM), with which each transmitter is able to calculate its own beamforming vectors and AN covariance matrix based on its local CSI. Simulation results show that our proposed distributed design converges to the global optimum obtained by the centralized one. It is also shown that by employing AN, the total required power of the system is reduced and the effect of AN on the system performance decreases with increment of transmit antennas. Compared with traditional linear EH model, optimizing the system under the nonlinear EH one avoids false output power at the ERs and saves power at the transmitter.
Introduction
Background
Recently, simultaneous wireless information and power transfer (SWIPT) has attracted increasing interests, where the same radio frequency (RF) signals are used for transmitting both energy and information. Thus, SWIPT is expected to prolong the lifetime of lowpower energyconstrained networks, such as wireless sensor networks (WSNs) and Internet of Things (IoT) [1–4]. Compared with traditional wireless communication system, SWIPT is equipped with the energy harvesting (EH) module, where the RFEH circuits enable energy receiver (ERs) to scavenge wireless energy by converting the received RF signals into output direct current (DC) power. As for the RFEH circuit, one of the most important performance measurement indexes is the RFtoDC conversion efficiency factor, which is the ratio of the input power in the RF signals and the output DC power. In most existing works, the RFtoDC conversion efficiency is regarded as a constant taking value in (0,1], referring to the linear EH model. However, according to very recent works, [5–7], the RFtoDC conversion efficiency turns out to depend on the level of the input power, which means RFtoDC conversion efficiency varies based on the input power and cannot be regarded as a constant anymore, referring to the nonlinear EH model. Therefore, to avoid the system performance loss, the transmit design for SWIPT should be redesigned based on the nonlinear EH model.
Meanwhile, green communications and high spectral efficiency (SE) are two important demands for future 5G networks. To achieve green communications, one way is designing energyefficient systems to achieve high energy efficiency (EE) [8], and the other way is designing energysaving systems to consume as less power as possible to meet system requirements [9]. To achieve high SE, many advanced communication technologies were proposed, such as cooperative relaying [10, 11], coordinated beamforming [12–14], and network coding [15]. Among them, multicell coordinated beamforming (MCBF) [12, 14] is regarded as one of the most promising technologies to achieve lowpower consumption and high SE, as well as guaranteeing quality of service (QoS) for receivers in multicell multiuser systems, since it is capable of exploiting the spatial degree of freedom (DoF) to migrate the intercell and intracell interference. Compared with the singlecell beamforming design, MCBF is more challenging since intercell interference cannot be neglected. Moreover, it is relatively easy for a singlecell transmitter to acquire channel state information (CSI) of all receivers, but for MCBF, a control center is required to gather all the intracell CSI and the intercell CSI for all transmitters, and the exchange of CSI between the control center and transmitters brings heavy burden and overhead to the system. As the future wireless systems prefer a flat Internet Protocol (IP) architecture [16] where all transmitters are directly connected with the core network rather than the control center, distributed MCBF is required to be developed where all transmitters work at their local CSI.
Due to the broadcast nature, another important issue in multicell multiuser systems is to avoid the confidential information leakage from the information receivers (IRs) to the eavesdropper (Eves). To achieve information security, physicallayer secure transmission has drawn increasing attention [17–19]. The main idea is to embed the artificial noise (AN) into the transmit signal to confuse Eves so that only the IRs can decode the confidential information correctly.
Motivation and contribution
To inherit the advantages of MCBF and SWIPT as well as achieving secure transmission, in this paper, we investigate ANaided MCBF designs for multicell multiuser SWIPT systems in both centralized and distributed manners. The contributions of our work is summarized as follows:

To realize green communication, we formulate an optimization problem to minimize the total required power of the whole system while satisfying the following system requirements: (1) minimal information rate requirement at each IR, (2) minimal EH requirement at each ER, and (3) avoiding any IR to be intercepted by any Eve.

With full CSI assumption, we optimize the system and solve the problem in a centralized manner. In this case, since the considered problem is nonconvex and cannot be solved directly, we solve it by a proposed solution method based on semidefinition relaxation (SDR) and the global optimal solution is proved to be always achieved.

With local CSI assumption, we also optimize the system and solve the problem in a distributed manner. In this case, each transmitter is able to calculate its own beamforming vectors and AN covariance matrix based on its local CSI. The proposed distributed design is derived based on alternating direction method of multipliers (ADMM).

Numerous simulation results are provided to demonstrate our analysis, which show that our proposed distributed design converges to the global optimum obtained by the centralized one. We also observe that by employing AN, the total required power of the system is reduced and the effect of AN on the system performance decreases with increment of transmit antennas. Compared with a traditional linear EH model, optimizing the system under the nonlinear EH model avoids a false output power at the ERs and also saves power at the transmitter.
Related work
Although MCBF and secure SWIPT has been investigated in the literature [19–31], they were separately discussed in various systems and only few works studied MCBF and secure SWIPT in a single system. In order to highlight the novelty of this paper, it is worth emphasizing the following differences between our work and existing ones [19–31].
Firstly, this is different from some existing works, see, e.g., [19, 20], where the system secrecy rate was maximized. In this paper, we consider the powerminimization design to cater for the requirement of green communications. The authors in [19, 20] formulated the problems as optimization (feasibility) problems and gave some approximate solutions based on the Bisection method. While we obtain the global optimal solution to our considered system and prove that AN also helps to save transmit power. Moreover, the intercell interference was not taken into consideration in these works.
Secondly, the powerminimization design was also investigated in some works, see, e.g., [21–27], but they all focused on singlecell SWIPT scenarios. Besides, they all adopted the ideal linear EH model, which mismatches the practical system and may lead to false optimization results of a system configuration. To avoid the performance loss, this paper considers the nonlinear EH model obtained by real data measurement [5–7].
Thirdly, although MCBF designs were investigated for various systems (see, e.g, [28–31]), most of them did not consider SWIPT and communication secrecy. For instance, the total required power was minimized under information rate constraints in [28, 29], the sum rate was maximized with zeroforcing beamforming in [30], and the energy efficiency was maximized with massive MIMO transition in [31]. As SWIPT plays a very important role in future wireless communication and secure transmission is an inevitable issue in broadcast SWIPT systems, it is of high importance to present suitable ANaided MCBF designs for multicell SWIPT systems. Therefore, in this paper, a secure SWIPT in multicell systems is investigated.
Fourthly, the existing works only designed the centralized ANaided MCBF for secure SWIPT system, while in our work, we also present an efficient distributed design, where each transmitter could generate its transmit beamforming using only a local CSI. In our work, multiple kinds of receivers, i.e., IRs, ERs and Eves, are considered, which is more general and more practical in the future system.
This paper is organized as follows. Section 2 gives the system model description and the problem statement. Section 3 gives the optimal centralized nonrobust design. The distributed design is given in Section 4. The simulation results are provided in Section 5. Finally, Section 6 concludes the paper.
Notations: Boldface lowercase and uppercase letters denote vectors and matrices, respectively. The set of all nbym complex matrixes is denoted by \({\mathbb {C}^{n \times m}}\). For a complex number a, a denotes the modulus. For a vector a, ∥w_{ n }∥_{2} denotes the Euclidean norm. The conjugate transpose, rank, trace,and determinant of the matrix A are denoted as A^{H}, rank(A), Tr(A), and det(A), respectively. A≽0 means A is a positive semidefinite (PSD) matrix. The symbol I denotes the identity matrix and 0 denotes a zero vector or matrix. The symbol \({{\mathbb E}}\left \{.\right \} \) represents the statistical expectation of the argument.
System model and problem formulation
Networks model
We consider a multicell multiuser downlink system with N_{c} cells as shown in Fig. 1. Each cell consists of one N_{t}antenna transmitter, K singleantenna IRs, L singleantenna ERs, and S singleantenna Eves. N_{c} transmitters transmit signals over a common frequency band, which means that the intercell and intracell interference coexist in the system. To prevent the information leakage, the energybearing spatially selective AN is embedded in the transmit signals of each transmitter.
For clarity, we use k, l, s, and n to denote the kth IR, the lth ER, sth Eve, and the nth cell/transmitter, respectively, where \(k \in \mathcal K\), \(l \in \mathcal L\), \(s \in \mathcal S\), and \(n \in \mathcal N_{\mathrm {c}}\). \(\mathcal K\buildrel \Delta \over = \left [1,2,\ldots,K\right ]\), \(\mathcal L\buildrel \Delta \over = \left [1,2,\ldots,L\right ]\), \(\mathcal S\buildrel \Delta \over =\left [1,2,\ldots,S\right ]\), and \(\mathcal N_{\mathrm {c}}\buildrel \Delta \over = \left [1,2,\ldots,N_{\mathrm {c}}\right ]\) denote the sets of IRs, ERs, Eves, and cells/transmitters, respectively. Thus, the transmit signal from the nth transmitter is
where \(\omega _{nk}\in \mathbb {C}\) with \({\mathbb E}\{\omega _{nk}^{2}\}=1\) is the data symbol for the kth IR transmitted in the nth cell. \(\mathbf {w}_{nk}\in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) is the corresponding transmit beamforming vector. \(\mathbf {z}_{n} \in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) denotes the AN vector which follows Gaussian distribution, i.e., \(\mathbf {z}_{n}\sim \mathcal {CN}(\mathbf {0},\mathbf {\Sigma }_{n})\) and Σ_{ n }≽0. Thus, the total required power of the system can be given by \( {\sum \nolimits }_{n = 1}^{{N_{\mathrm {c}}}} {\left ({{\sum \nolimits }_{k = 1}^{K} {\left \ {{\mathbf {w}_{nk}}} \right \_{2}^{2}} + {\text {Tr}}\left ({{\boldsymbol {\Sigma }_{n}}} \right)} \right)}\).
Then, the received signal at the kth IR and the lth ER in the nth cell can be, respectively, given by
and that at the sth Eve in cell n is
where h_{ mnk }, h_{ mnl }, and \(\mathbf {g}_{mns} \in \mathbb {C}^{{N_{\mathrm {t}}} \times 1}\) denote the channel vectors from the mth transmitter to the kth IR, the lth ER, and the sth Eve^{Footnote 1} in the nth cell, respectively. n_{ nk }(t), n_{ nl }(t), and v_{ ns }(t) are the Gaussian noises with variance \(\sigma _{nk}^{2}\), \(\sigma _{nl}^{2}\), and \(\sigma _{ns}^{2}\) at the kth IR, the lth ER, and the sth Eve, respectively. Following (2) and (4), the received SINR at the kth IR in cell n and that at the sth ER in cell n are given by (5) and (6), respectively. [!t]
[!t]
Nonlinear EH model
Each ER converts the received RF signals into output DC power by its RFEH circuits. At the lth ER in the nth cell, the input power of its RFEH circuits from the received RF signals is
In most existing works, the RFtoDC conversion efficiency ρ of the RFEH circuits is regarded as a constant in the interval (0,1], referring to the linear EH model, which indicates that the RFtoDC conversion efficiency is independent of the input power level. However, in practice, the RFEH circuits include various nonlinearities, such as the diode or diodeconnected transistor. As a result, the RFtoDC conversion efficiency depends on the input power level. To capture the dynamics of the RFtoDC conversion efficiency for different input power levels, in this paper, the nonlinear model is adopted [5–7]. The output DC power (harvested power) of the RFEH circuits at the sth ER is
with
where
Ψ_{ nl } is a logistic function of \(P_{nl}^{{\left (\text {ER}\right)}}\), M_{ nl } is a constant denoting the maximum output DC power, which indicates the saturation limitation of the RFEH circuits. a_{ nl } and b_{ nl } are constants representing some properties of the EH system, e.g., the resistance, the capacitance and the circuit sensitivity. In general, M_{ nl }, a_{ nl }, and b_{ nl } depend on the choice of hardware components for assembling the EH system and can be estimated through a standard curve fitting algorithm. Figure 2 provides an example of the nonlinear EH model, where the maximum output DC power M_{ nl } is set as 20 mW. One can observe that the output DC power increases with the increment of the input power at first, and then when it reaches the saturation region, the output DC power cannot surpass this saturation limitation, which is much different from the linear EH model, where the output DC power can always increases with the increment of the input power.
Problem formulation
Our goal is to minimize the total required power of the system by jointly optimizing the transmit beamforming vectors and covariance matrixes of the AN to meet the following two system requirements.

To guarantee the information rate requirement of each IR, its received SINR should be larger than a predefined threshold γ_{u}.

To prevent the information interception of each Eve, its received SINR should be lower than a predefined threshold γ_{e}.

To guarantee the EH requirement of each ER, its output DC power should be larger than a predefined threshold θ.
Then, our considered powerminimization ANaided MCBF design is mathematically formulated as
With (9b) and (9c), the secrecy capacity between each IR and its serving transmitter is guaranteed bounded below C_{sec}= log(1+γ_{u})− log(1+γ_{u}). Note that the value of γ_{u} and γ_{e} depends on the required QoS of IRs. A larger γ_{u} and a smaller γ_{e} indicate better system performance, but more power is required at transmitters.
Problem (9) is not convex due to the nonconvex constraint (9b)–(9d), which cannot be solved directly. Therefore, in Section 3, we will solve it by using SDR methods^{Footnote 2}.
Optimization in the power minimization design
To solve problem (9), one solution method based on SDR is proposed, where SDR means letting W_{ n k }=w_{ n k }w_{ n k }^{H} and removing the rankone constraint.
The EH constraint (9d) under the nonlinear EH model can be represented by
where τ_{ nl } indicates the required received power (input power of the RFEH circuits) under the nonlinear EH model, which can be given by
By employing SDR, the SDR form of (10) can be given by
Then, the SDR form of problem (9) can be given by
where
By doing so, problem (9) is expressed into a convex problem, i.e., problem (12). By using some offtheshelf solvers, e.g., SeduMi or CVX, the optimal solution, i.e., \(\mathbf {W}_{nk}^{\star }\) and \(\mathbf {\Sigma }_{n}^{\star }\), to problem (12) can be obtained. Note that, our goal is to obtain the optimal \(\mathbf {W}_{nk}^{\star }\) rather than \(\mathbf {W}_{nk}^{\star }\). Therefore, once we get \(\mathbf {W}_{nk}^{\star }\), we should recover \(\mathbf {W}_{nk}^{\star }\) from \(\mathbf {W}_{nk}^{\star }\). Generally, only when \(\mathbf {W}_{nk}^{\star }\) is rankone, problems (9) and (12) have the same optimal result and solution, which means that only in this case, \(\mathbf {W}_{nk}^{\star }\) can be recovered by rankone decomposition of \(\mathbf {W}_{nk}^{\star }\) without loss.
Proposition 1
The rankone solution to problem (12) always exists.
Proof
The proof of Proposition 1 can be found in the Appendix of this paper. □
Following Proposition 1, the optimal solution to problem (9) can be obtained by firstly solving problem (12) and then applying rankone decomposition to the optimal solution of Problem (12).
Proposition 2
Compared with the MCBF without AN (i.e., the non ANaided MCBF), the ANaided MCBF required less power under the same condition.
Proof
The non ANaided MCBF design can be formulated by setting the objective function of (9) as
and eliminating \(\left \{ {{\mathbf {\Sigma }_{m}}} \right \}_{m=1}^{N_{\mathrm {c}}}\) in (9b), (9c), and (9d). It also can be solved by our proposed SDRbased solution method. One can see that the optimal solution to the non ANaided MCBF is a feasible solution to problem (9) by setting \(\left \{ {{\mathbf {\Sigma }_{n}}} \right \}_{n=1}^{N_{\mathrm {c}}} = \mathbf {0}\). So the optimal result of problem (9) must be smaller than that of the non ANaided MCBF, which means the ANaided MCBF required less power than the non ANaided MCBF to satisfy the same system requirements. □
Proposition 2 indicates that by introducing the AN to the MCBF, the total required power of the system is reduced.
Distributed design with ADMM
Although we solve problem (9) efficiently in Section 3 by using SDR and obtain the global optimal solution, the proposed solution method is in a centralized manner, which means that to calculate the optimal beamforming vectors and AN covariance matrices, the CSI of all receivers in the system is required at each transmitter. Such a requirement yields heavy burden and overhead to the network, which blocks the deployment of the centralized design. Therefore, designing a distributed ANaided MCBF, where each transmitter only work with its local CSI, is of high importance. In this section, we shall apply ADMM to decentralize the proposed centralized ANaided MCBF.
The main idea of the proposed distributed algorithm is to decompose the primal centralized problem into N_{c}+1 subproblems, so that each transmitter can deal with its own beamforming vectors and AN covariance matrix based on local CSI at every iteration.
The first ingredient of the proposed distributed design is to find the penalty augmented problem of (12). For this, we define some slack variables as follows:
Then, problem (12) can be equally rewritten as
Since constraints (14), (15b), (15c), and (15d) are all convex w.r.t. the variables relevant to n of problem (15), the feasible set of problem (15) can be decomposed into N_{c}disjointed convex sets as (16).
For clarification, we define two new vector variables as follows:
where t collects all interference and AN variables and t_{ n } collects all interference and AN variables relevant to cell n (i.e., the interference received from and sent to neighboring cells). With the two vector variables, it is easy to find a linear mapping matrix Ω_{ n }∈{0,1}, such that t_{ n }=Ω_{ n }t.
As a result, the penalty augmented problem of (12) can be given by
where ρ_{ n }≥0 is a slack variable and c>0 is the penalty parameter. Note that problem (15) and problem (17) have the same optimal result and solution, and the added terms are required to ensure the convergence of ADMM algorithm.
The second ingredient of the proposed distributed design is dual decomposition so that each transmitter can generate its own beamforming vectors and AN covariance matrix with its local CSI. The dual problem of (17) can be given by
where v_{ n } and u_{ n } are associated dual variables. For given v_{ n } and u_{ n }, ({W_{ mk }}_{ k },{Σ_{ m }},t_{ m },p_{ m }) can be solved efficiently since the problem (18) is convex. So we can solve (18) by an iteration method. Particularly, at the (q+1)th iteration, the problem (18) can be decomposed into 2N_{c} convex subproblems, i.e., \(\forall n \in \mathcal N_{\mathrm {c}}\),
and
It is interesting to see that each subproblem in (19) is only relevant to the cell n. Thus, at the (q+1)th iteration of the distributed algorithm, each cell copies with one relevant subproblem in (19) to get (t_{ n }(q+1),p_{ n }(q+1)), and then exchanges (t_{ n }(q+1),p_{ n }(q+1)) with neighboring cells so that it can obtain (t(q+1),ρ_{ n }(q+1)) by solving (20). Then, according to ADMM, the dual variable can be updated by
For clarity, the proposed distrusted algorithm is summarized in Algorithm 1.
Simulation results and discussion
This section represents some simulation results to show the efficiency of the proposed centralized and distributed ANaided MCBF designs. For comparison, the non ANaided MCBF design is also simulated as a benchmark. We discuss the system performance in perspective of the SINR requirement at IRs, the allowable SINR threshold at Eves, the EH model at ERs, and the number of antennas at the transmitter. Besides, the centralized MCBF design and the distributed MCBF design are compared.
The simulation network scenario is shown in Fig. 1 where the number of cells N_{c} is 3 and the number of antennas N_{t} at each transmitter is 4. The intertransmitter distance D is 500 m. The number of IR, ER, and Eve in each cell is set as 2, 1, and 1, respectively. All IRs and ERs are located about 50 m from their serving transmitters and all ERs are located about 10 m from their serving transmitters, which is close enough that the lineofsight communication channel is expected. The channel mode adopted in our simulation is given by [29]
where \(\beta \left ({{d}} \right) = {10^{ \left ({128.1 + 37.6{{\log }_{10}}\left ({{d}} \right)} \right)/20}}\phantom {\dot {i}\!}\), d_{ mnk }, d_{ mnl }, and d_{ mns } denote the distance between the transmitter and the kth IR, lth ER, and sth Eve, respectively. ψ_{ mnk }, ψ_{ mnl }, and ψ_{ mns } represent the shadow fading, which follows the lognormal distribution with zero mean and standard deviation 8. φ denotes the transmitreceive antenna gain which is set to 15 dBi, and \({\hat {\mathbf {h}}_{mnk}}\), \({\hat {\mathbf {h}}_{mnl}}\), and \({\hat {\mathbf {g}}_{mns}}\) are the multipath fading which is modeled as Rayleigh fading for IRs and Eves and Rician fading for ERs. The noise power spectral density is − 162 dBm/Hz and the bandwidth B is 10 MHz. For each IR, the SINR requirement is set as 10 dB. For each Eve, the allowable SINR threshold is set as − 10 dB. For the nonlinear EH model of each ER, we set all M_{ l } as M=24 mW which corresponds to the maximum harvested power at each ER. Besides, we adopt a_{1}=...=a_{ L }=150 and b_{1}=...b_{ L }=0.024. The required EH power ς_{ s } at each ER is set as 10 mW. All parameters in the simulations are as described above unless specified. In the presented simulation results, each point of the curves is averaged over 1000 channel realizations.
Figure 3 shows the required power versus the SINR requirement at each IR γ_{u}. In the proposed ANaided MCBF design, the total required power contains two parts where one is associated with beamforming vectors \({\sum \nolimits }_{n = 1}^{{N_{\mathrm {c}}}} {{\sum \nolimits }_{k = 1}^{K} {\left \ {{\mathbf {w}_{nk}}} \right \_{2}^{2}} }\) and the other is associated with AN \({\sum \nolimits }_{n = 1}^{{N_{\mathrm {c}}}} {{\text {Tr}}\left ({{{\mathbf \Sigma }_{n}}} \right)}\). For clarity, we also plot these two parts in Fig. 3, where the power of beamforming part is marked by “Beamforming part (proposed)” and the power of AN part is marked by “AN part (proposed).” As γ_{ u } increases, more power is required in both beamforming part and AN part. Compared with the non ANaided MCBF, the proposed ANaide MCBF requires less power, which is consistent with Proposition 2.
Figure 4 compares the proposed ANaided MCBF under the nonlinear EH model and linear EH model, where for the linear EH model, the RFtoDC conversion efficiency ρ is set as 1 and 0.5, respectively. It is observed that for the linear EH model, the total required power increases linearly as each ER’s required output DC power θ increases. For the nonlinear EH model, the total required power also increases as θ increases, but there exists a saturation point on θ (e.g., M=24 mW in our example) due to the nonlinear EH circuit feature. Moreover, if the linear EH model with ρ=1 is adopted, it may result in false and deceptive output DC power at each ER. That is, when θ≤M, although less power is consumed by the linear EH model, the output DC power cannot meet the practical requirement (i.e., (9d) cannot be satisfied). When θ>M, although the transmit design can still be generated by the linear EH model, (9d) also cannot be satisfied. In contrast, for the nonlinear EH model, when θ≤M, (9d) can be guaranteed, and when θ>M, beamforming vectors and AN cannot be generated because of the saturation limitation of practical EH circuits. Thus, the false output DC power is avoided by employing the nonlinear EH model. Furthermore, if the linear EH model with ρ=0.5 is adopted, the design under the linear EH model is also feasible in practical system but more power is required compared with the design under the nonlinear EH model. This result demonstrates the advantage of employing the nonlinear EH model.
The relation between the required power and the allowable SINR threshold at each Eve γ_{ e } is investigated in Fig. 5. Since the small γ_{ e } may make (12) infeasible, we just show the results of γ_{ e } with the range from − 2 to − 10dB. It is observed that smaller γ_{ e } requires more total required power as well as the power in the AN part. Besides, the smaller γ_{ e }, the larger gap between the ANaided MCBF and the non ANaided MCBF, which indicates that to achieve highquality secure transmission, AN is required.
The total required power versus N_{t} is shown in Fig. 6, where N_{t} changes from 4 to 7. It is observed that the total required power of both the ANaided MCBF and the non ANaided MCBF decreases with the increment of N_{t}. The system performance gain between ANaided MCBF and non ANaided MCBF is more obvious for relative small N_{t}. As larger N_{t} indicates larger spatial DoF, it can be concluded that for system with limited spatial DoF, ANaided MCBF is a better choice to achieve lower power consumption. Besides, it is observed that with increment of transmit antennas, the effect of AN on the total required power decreases.
Figure 7 gives the comparison of total required power between the centralized algorithm and the distributed algorithm over 20 randomly generated channel realizations, and the iteration number is 20. It is seen that the distributed solution has a very close result to the centralized one, which means that the distributed algorithm can converge to the optimal solution to the centralized algorithm.
Conclusions
This paper studied the ANaided MCBF for multicell multiuser SWIPT, where the nonlinear EH model was adopted at each ER. We formulated an optimization problem to minimize the total required power by jointly optimizing the transmit beamforming vectors and the AN covariance matrixes at all transmitters, while guaranteeing IRs and ERs’ QoS and avoiding the information intercepted by any Eve. Since the problem is nonconvex, we solved it by applying SDR and proved that our proposed solution method guarantees the global optimal solution with full CSI in a centralized manner. We further presented a distributed ANaided MCBF design using ADMM, with which, each transmitter is able to calculate its own beamforming vectors and AN covariance matrix based on its local CSI. Simulation results showed that our proposed distributed algorithm converges to the optimal results obtained by the centralized one. Some insights are derived as follows. For the same secure transmission requirements, the proposed ANaided MCBF consumes less power than the non ANaided MCBF. With increment of transmit antennas, the total required power decreases and the effect of AN on the total required power decreases. Compared with the traditional linear EH model, employing the nonlinear EH model avoids false output power at the ERs and save power at the transmitter.
Appendix
Let \(\mathbf {W}_{nk}^{\star }\) represent the optimal solution to (12). Ynk⋆, \({\lambda _{nk}^{\star }}\), and \({\mu _{ns}^{\star } }\) are the corresponding optimal solutions to the dual problem of (12). The KKT equations corresponding to \(\mathbf {W}_{nk}^{\star }\) in (12) can be given by
where
which is a PSD matrix.
So, we have \(\text {rank}\left ({\mathbf {Y}_{nk}^{\star } \mathbf {W}_{nk}^{\star }} \right)=0\). Moreover, it is known that \(\text {rank}\left ({\mathbf {Y}_{nk}^{\star } \mathbf {W}_{nk}^ \star } \right) \ge \text {rank}\left ({\mathbf {Y}_{nk}^{\star }} \right) + \text {rank}\left ({\mathbf {W}_{nk}^{\star }} \right)  N_{\mathrm {t}}.\) Thus,
Since \({\mathbf {I}_{N{\mathrm {t}}}} + {\sum \nolimits }_{s = 1}^{S} {\frac {{\mu _{ns}^{\star }}}{{{\gamma _{\mathrm {e}}}}}} \left ({{\mathbf {g}_{nns}}\mathbf {g}_{nns}^{H}} \right)\) is positive definite, we have \({\mathbf {I}_{N{\mathrm {t}}}} + {\sum \nolimits }_{s = 1}^{S} {\frac {{\mu _{ns}^{\star }}}{{{\gamma _{\mathrm {e}}}}}} \left ({{\mathbf {g}_{nns}}\mathbf {g}_{nns}^{H}} \right) = \mathbf {A}{\mathbf {A}^{H}}\) where A is invertible, e.g., rank(A)=N_{t}. Therefore, \(\mathbf {Y}_{nk}^{\star } = \mathbf {A}\left ({{\mathbf {I}_{N{\mathrm {t}}}}  {\mathbf {A}^{ 1}}{\boldsymbol \Lambda }{\mathbf {A}^{ 1}}} \right)\mathbf {A},\) and thus,
Since h_{ nnk },h_{ nnl }≠0, we have
i.e., \({\text {rank}}\left ({\mathbf {Y}_{nk}^{\star }} \right) = {N_{\mathrm {t}}}  1\). Moreover, with Σ_{ m }≽0 and \(\sigma _{ns}^{2}>0\) and (12), we have \({\mathbf {W}_{nk}^{\star }}\ne 0\).
Following (22), if \({\text {rank}}\left ({\mathbf {Y}_{nk}^ \star } \right) = {N_{\mathrm {t}}}  1\), \({\text {rank}}\left ({\mathbf {W}_{nk}^ \star } \right)\ne 0\), or \({\text {rank}}\left ({\mathbf {W}_{nk}^ \star } \right)= 1\). Since \({\text {rank}}\left ({\mathbf {W}_{nk}^ \star } \right)\!\ne \! 0\), \({\text {rank}}\left ({\mathbf {W}_{nk}^ \star } \right)= 1\) always holds. As a result, \({\text {rank}}\left ({\mathbf {W}_{nk}^ \star } \right) = 1\).
Notes
 1.
The CSI of Eve can be estimated through the local oscillator power inadvertently leaked from the Eves’ receiver RF frontend [32].
 2.
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Acknowledgements
This work was supported in part by the Key Program of the National Natural Science Foundation of China (NSFC) under Grant U1334202, in part by the General Program of the NSFC under Grant 61671051, in part by the NFSC Outstanding Youth under Grant 61725101, in part by the Beijing Natural Science Foundation under Grant 4162049, and in part by the Fundamental Research Funds for the Central Universities Beijing Jiaotong University under Grants 2016JBM015 and 2017YJS063.
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Dr. YL and Dr. KX have equal contributions to the system modeling, algorithm design, performance analysis, and simulations. Dr. JL help to perform the experiments. Prof. PF and Prof. ZZ help to revise the manuscript. All authors read and approved the manuscript.
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Correspondence to Ke Xiong.
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Keywords
 Green communications
 SWIPT
 Beamforming
 Artificial noise
 ADMM