The past decade has witnessed the rapid development of Internet of Things (IoT). It is forecast that by 2025 about 30 billion IoT devices will be used worldwide[1, 2]. As conventional battery is not convenient for such a huge number of devices, simultaneous wireless information and power transfer (SWIPT) is recognized as a promising technology to prolong the operation time of wireless devices [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The separated information receiver (IR) and energy receiver (ER) are considered in [3, 4, 5]. The authors in [3, 15]
considered a multi-user wireless information and power transfer system, where the beamforming vector was designed by the zero-forcing (ZF) algorithm and updated by maximizing the energy harvested. In, the optimal beamforming scheme was proposed for achieving the maximum secrecy rate, while meeting the minimum energy requirement at the ER. In [5, 16], the authors designed the robust information and energy beamforming vectors for maximizing the energy harvested by the ER under specific constraints on the signal-to-interference plus noise ratio (SINR) at the IR. A power splitting (PS) scheme was utilized to divide the received signal into two parts in order to simultaneously harvest energy and to decode information [6, 7, 9, 10]. The authors of [11, 12] investigated both PS and time switching (TS) schemes and compared the performance of these two schemes.
As an important technique of expanding the coverage of networks, relaying can also beneficially enhance the communication security, whilst simultaneously enhancing energy harvesting [17, 18, 19, 20, 21]. For the case of perfect channel state information (CSI) situations, secure SWIPT invoked in relaying networks has been investigated [22, 23, 24]. Literature , proposed a constrained concave convex procedure (CCCP)-based iterative algorithm for designing the beamforming vector of multi-antenna aided non-regenerative relay networks. While in , the analytical expressions of the ergodic secrecy capacity were derived separately based on TS, PS and on ideal relaying protocols. The beamforming vectors of SWIPT were designed for amplify and forward (AF) two-way relay networks through a sequential parametric convex approximation (SPCA)-based iterative algorithm to find its locally optimal solution in .
By contrast, for the imperfect CSI scenarios, the channel estimation uncertainty model was considered[25, 26, 27, 28, 29]. In [25, 26, 27], the robust information beamformer and artificial noise (AN) covariance matrix were designed with the objective of maximizing the secrecy rate under the constraint of a certain maximum transmit power. The secrecy rate maximization (SRM) problem was a non-convex problem in [25, 26, 27], the critical process is, how to transform it into a tractable convex optimization problem by using the S-Procedure. In  and , the authors formulate the power minimization problem under a specific secrecy rate constraint and minimum energy requirement at the energy harvester (EH), which was solved in a similar manner. In general, for the imperfect CSI situations, the channel estimation error is usually modeled obeying the ellipsoid bound constraint, and then be transformed into a convex constraint by using the S-procedure.
Recently, a promising physical layer security technique, known as directional modulation (DM), has attracted a lot of attention. In contrast to conventional information beamforming techniques, DM has the ability to directly transmit the confidential messages in desired directions to guarantee the security of information transmission, while distorting the signals leaking out in other directions[30, 31, 32, 33, 34]. In , the authors proposed a DM technique that employed a phased array to generate the modulation. By controlling the phase shift for each array element, the magnitude and phase of each symbol can be adjusted in the desired direction. The authors in  proposed a method of orthogonal vectors and introduced the concept of AN into DM systems and synthesis. Since the AN contaminates the undesired receiver, the security of the DM systems is greatly improved. Subsequently, the orthogonal vector method was applied to the synthesis of multi-beam DM systems . The proposed methods in  and  achieve better performance at the perfect direction angle, but it is very sensitive to the estimation error of the direction angle. The authors in 
modeled the error of angle estimation as uniform distribution and proposed a robust synthesis method for the DM system to reduce the effect of estimation error. In, the authors also considered the estimation error of the direction angle and proposed a robust beamforming scheme in the DM broadcast scenario.
However, none of these contributions consider DM-based relaying techniques. For example, if the desired user is beyond the coverage of the transmitter or there is no direct link between the transmitter and the desired user, the above methods are not applicable. Moreover, in [33, 34], the proposed robust methods only designed the normalized confidential messages beamforming and AN projection matrix without considering the power allocation problem. In fact, the power allocation of confidential messages and AN has a great impact on the security of DM systems. To the best of our knowledge, there exists no DM-based scheme considering secure SWIPT, which thus motivates this work.
, instead of channel estimation error modeled obeying the ellipsoid bound constraint, we model the estimation error of direction angle as the truncated Gaussian distribution which is more practical in our DM scenario. The main contributions of this paper are summarized as follows.
1) We formulate the SRM problem subject to the total power constraint at an AF relay and to the minimum energy requirement at the ER. Since the secrecy rate expression is the difference of two logarithmic functions, it is noncovex and difficult to tackle directly. Additionally, the estimates of the eavesdropper directions are usually biased. To solve this problem and to find the robust information beamforming matrix as well as the AN covariance matrix, we convert the original problem into a twin-level optimization problem, which can be solved by a one-dimensional (1D) search and the classic semidefinite relaxation (SDR) technique. The 1D search range is bounded into a feasible interval. Furthermore, the SDR is proved tight by invoking the Karush-Kuhn-Tucker (KKT) condition.
2) To reduce the the search complexity, we propose a suboptimal solution for maximizing the signal-to-leakage-AN-noise-ratio (Max-SLANR) subject to the total power constraint of the relay and to the minimum energy required at the ER. Due to the existence of multiple eavesdroppers, we consider the sum-power of the confidential messages leaked out to all the eavesdroppers. This optimization problem is also shown to be nonconvex, but it can be transformed into a semidefinite programming (SDP) problem and then solved by the SDR technique. Its tightness is also quantified. To further reduce the computational complexity, we propose an algorithm based on successive convex approximation (SCA). Specifically, we first formulate the SRM optimization problem and then transform it into a second-order cone programming (SOCP) which is finally solved by the SCA method. Furthermore, we analyse and compare the complexity of the aforementioned three schemes.
3) The formulated optimization problems include random variables corresponding to the estimation error of the direction angles, which makes the optimization problems very difficult to tackle directly. To facilitate solving this problem, we derive the analytical expression of the covariance matrix of each eavesdroppers’ steering vector and substitute it into the optimization problems to replace the random variable. Moveover, we add relay and energy harvesting node to the DM-based secure systems, which further expand the application of DM technology. Simulation results demonstrate that the bit error rate (BER) performance of all our schemes in the desired direction is significantly better than that in other directions, while the BER is poor in the vicinity of the eavesdroppers’ directions, showing the advantages of our DM technology in the field of physical layer security.
The rest of this paper is organized as follows. Section II introduces the system model. In Section III, three algorithms are proposed to design the robust secure beamforming. Section IV provides our simulation results. while, Section V concludes the paper.
Notation: Boldface lowercase and uppercase letters represent vectors and matrices, respectively, , , , , , and denote conjugate, transpose, conjugate transpose, trace, rank, positiveness and semidefiniteness of matrix , respectively, , , and denote the statistical expectation, pure imaginary number, and Euclidean norm, respectively, and denotes the Kronecker product.
Ii System Model
As shown in Fig. 1, we consider AF-aided secure SWIPT, where the source transmitter sends confidential messages to an IR with the aid of an AF relay in the presence of an ER and eavesdroppers (). It is assumed that the AF relay is equipped with an -element antenna array, while all other nodes have a single antenna.
Similar to the literature on DM [33, 34], this paper adopts the free-space path loss model which is practical for some scenarios such as communication in the air and rural areas. The steering vector between node and node can be expressed as 
where is the path loss between node and node . The function can be expressed as
where denotes the angle of direction between node and node , denotes the distance between two adjacent antenna elements, and is the wavelength.
We assume that there is no direct link from the source to the IR, ER or to any of the eavesdroppers. Thus the relay helps the source to transmit the confidential message to IR. The relay node is assumed to operate in an AF half-duplex mode. Simultaneously, ER intends to harvest energy, while the eavesdroppers try to intercept the confidential message. The power of the signal is normalized to, . In the first time slot, the source transmits the signal to the relay, and the signal received at the relay is given by
where is the transmission power of the source, denotes the steering vector between the source and the relay, is a circularly symmetric complex Gaussian (CSCG) noise vector, and is the angle of direction between the source and the relay. In the second time slot, the relay amplifies and forwards the received signal to IR. The signal transmitted from the relay is given by
where is the beamforming matrix, and is the AN vector assumed to obey a (CSCG) distribution with . In general, the relay has a total transmit power constraint , therefore we have
The signal received at the IR, ER, and the -th eavesdropper can be expressed as
respectively, where , , and denote the steering vectors from the relay to IR, ER, and the -th eavesdropper respectively. Furthermore, , , and represent the CSCG noise at IR, ER, and the -th eavesdropper, respectively, while , , and . Without loss of generality, we assume that , , , and are all equal to .
Similar to the considerations in  and , namely that the perfect CSI of the destination is available at the relay, here we assume that the relay has the perfect knowledge of direction angles to the IR. However, there is an estimation error of the direction angles of eavesdroppers at the relay, and we assume that the relay has the statistical information about these estimation errors. Therefore, the -th eavesdropper’s direction angle to the relay can be modeled as
where is the estimate of the -th eavesdropper’s direction angle at the relay, and denotes the estimation error, while is assumed to follow a truncated Gaussian distribution spread over the interval
with zero mean and variance
. The probability density function ofcan be expressed as
where is the normalization factor defined as
Iii Robust Secure SWIPT Design
In this section, three algorithms are proposed to design the robust secure beamforming under the assumption that an estimation error of the direction angles of eavesdroppers exists at the relay. To design the robust beamforming matrix and AN covariance matrix, we first define
and . Let denote the -th row and -th column entry of , and can be written as
where and can be found in and , respectively. The specific derivation procedure is detailed in Appendix A.
According to , the energy harvested at the ER is given by 
where denotes the energy transfer efficiency of the ER.
From , the SINR at the IR can be expressed as
According to , the -th eavesdropper’s is given by
Thus, the achievable secrecy rate at the IR can be expressed as 
where the scaling factor is due to the fact that two time slots are required to transmit one message. By invoking Jensen’s inequality, the worst-case secrecy rate is given by
Iii-a Secrecy Rate Maximization based on One-Dimensional search Scheme (SRM-1D)
In this subsection, the robust information beamforming matrix and AN covariance matrix are designed by our SRM-1D scheme. Specifically, according to , , and , we maximize the worst-case secrecy rate subject to the total transmit power and the harvested energy constraints. Then the optimization problem can be formulated as
where denotes total power constraint at the relay, and the first term in denotes the minimum power required by the ER. We employ a 1D search and a SDR-based algorithm to solve problem (P1). Observe that is the difference of two logarithmic functions, which is non-convex and untractable. Similar to , we decompose into two sub-problems, yielding:
where is a slack variable. The main steps to solve the problem (P1) are as follows. First, for each inside the interval , we can obtain a corresponding by solving the problem . Second, upon substituting and into the objective function of , we obtain the secrecy rate corresponding to the given . Thirdly, we perform a 1D search for , compare all the secrecy rates obtained and then finally we find the optimal value for .
As for the above procedure of solving the problem (P1), the most important and complex part is to solve the problem to obtain . This are illustrated as follows. Upon defining , we can rewrite as
With the above vectorization, we show problem can be transformed into a standard SDP problem. Upon defining , can be rewritten as
Note that the rank constraint in is non-convex. By dropping the rank-one constraint in , the SDR of problem can be expressed as
It can be observed that constitutes a quasi-convex problem, which can be transformed into a convex optimization problem by using the Charnes-Cooper transformation . Upon introducing slack variable , problem can be equivalently rewritten as
where and . Since problem is a standard SDP problem , its optimal solution can be found by using SDP solvers, such as CVX. If the optimal solution of problem is , then will be the optimal solution of problem .
Since we have dropped the rank-one constraint in the problem and reformulated it as a SDR problem , the optimal solution of may not be rank-one and thus the optimal objective value of generally serves an upper bound of . Next, we show that the above SDR is in fact tight. We consider the power minimization problem as follows
where is the optimal value of problem . Observe that the optimal solution of problem is also an optimal solution of . The proof is similar to that in  and thus omitted here for brevity. In order to obtain the optimal solution of , we should first obtain the optimal solution and the optimal value of problem by solving . If , then we get the optimal solution of . Otherwise, the rank-one solution can be found by solving .
Lemma 1: The optimal solution in satisfies .
Proof: See Appendix B.
Since is a rank-one matrix, we can write
by using eigenvalue decomposition. Thus, the SDR is tight and the optimal solution ofis and . Up to now, we have solved the problem .
Let us now return to the procedure used for the problem . The maximum of should be found by a 1D search. According to the fact that the secrecy rate is always higher than or equal to zero, we get
From the transmit power constraint in , we have , hence
Observe that can be recast as
where . Therefore, we have . According to and , the upper bound of is given by
The proposed SRM-1D scheme is summarized in Algorithm 1.
Iii-B Maximization of Signal-to-Leakage-AN-Noise-Ratio (Max-SLANR) Scheme
In the previous subsection, we employed a 1D search and a SDR-based algorithm to solve problem (P1). Although we have already derived , to limit the range of the 1D search, the complexity of the 1D search still remains high since for each , a SDP with needs to be solved. In order to avoid employing the 1D search, we propose an alternative algorithm for the suboptimal solution of (P1). Specifically, we propose an algorithm to maximize the SLANR rather than secrecy rate, subject to the total power and to the harvested energy constraints. Based on the concept of leakage , from and , the optimization problem (P1) can be reformulated as (III-B) at the top of the next page. The numerator of the objective function in represents the received confidential message power at the IR, and the first term in the denominator denotes the sum of confidential message power leaked to all eavesdroppers.
Following similar steps as in Section III-A and dropping the rank-one constraint, the related SDR problem can be formulated as show in at the top of the page, where and . Note that all constraints in are convex. However, the objective function is a linear fractional function, which is quasi-convex. Similar to , we transform into a convex optimization problem by using the Charnes-Cooper transformation. Problem can then be equivalently rewritten as
where is a slack variable, and . To prove that the relaxation is tight, we consider the associated power minimization problem, which is similar to that in Section III-A, yielding
where is the optimal value of . Problem is a standard SDP problem.
Lemma 2: The optimal solution in satisfies .
Proof: See Appendix C.
Iii-C Low-complexity SCA Scheme
In the III-A and III-B, we have proposed the SRM-1D and the Max-SLANR schemes to obtain the information beamforming matrix and the AN covariance matrix. Both of the two schemes have high computational complexity because their optimization variables are matrices. To facilitate implementation in practice, we propose a low complexity scheme based on SCA in this subsection. Specifically, we first formulate the optimization problem, then convert it into the SOCP problem, and use the SCA method to solve the problem iteratively. Different from designing the AN covariance matrix in the previous two subsections, here we are devoted to designing the AN beamforming vector , where .
The optimization problem (20) can be rewritten as