With the proliferation of smart devices and data-hungry applications, establishing ubiquitous, high-throughput and secure communications is gaining increased importance in next-generation systems [1, 2, 3]. The traditional macrocells generally have poor performance in terms of indoor coverage and cell edge rate. To tackle this issue, heterogenous networks (HetNets) have emerged as a promising next-generation architecture, which are generally supported by heterogenous base stations (BSs) having different service coverages [4, 5]. Specifically, the macrocell base station (MBS) can provide open access and wide coverage up to dozens of kilometers, while the low-power femtocell base station (FBS) and picocell base station (PBS) are typically deployed in indoor environments and near to femtocell users (FUs) and picocell users (PUs), respectively. As pointed out in , the ultra-dense deployment of femtocells is recognized as an effecient technique to realize 1000 times increase in wireless data rate for. However, due to the high spatial spectrum reuse in HetNets and the dense deployment of FBSs and PBSs, cross-tier interference is usually unavoidable in HetNets. Fortunately, according to [7, 8], the interference can be re-utilized as an effective radio-frequency (RF) energy source for wireless energy harvesting (WEH), which thus contributes to the green and self-sustainable communications. WEH has many advantages over conventional energy supply methods . For example, WEH is more reliable than natural energy supply, such as solar, wind and tide, which are significantly affected by climate and terrain. Also, it is more cost-effective compared to the widely adopted batteries recharge/replacement technique. Generally, the densely deployed HetNets are favorable from the perspective of improving efficiency of WEH, since the distances from energy harvesters to energy stations are substantially shortened.
Recently, WEH-based HetNets have received extension attention, in which the power beacons (PBs) provide power for other nodes via wireless energy transfer [10, 11, 12, 13, 14, 15, 16]. In , H. Tabassum and E. Hossain studied the optimal deployment of PBs in wirelessly powered cellular networks. In 
, the downlink resource allocation problem was investigated by S. Lohani et al. under both time-switching and power-splitting based simultaneous wireless information and power transfer (SWIPT) strategies for two-tier HetNets. The comprehensive analysis of the outage probability and the average ergodic rate in both downlink and uplink stages of wirelessly powered HetNets with different cell associations were presented by S. Akbar et al. in. Y. Zhu et al. in  further extended the above work into the Massive MIMO aided HetNets with WEH, where different user association schemes are investigated in terms of the achievable average uplink rate. From the view of green communications, the energy efficient beamforming designs for SWIPT HetNets was studied by M. Sheng et al. and H. Zhang et al. in  and , respectively. To increase energy harvesting efficiency multi-antenna PBs and users, J. Kim et al. in  considered sum throughput maximization under different cooperative protocols of two-tier wireless powered cognitive networks.
Furthermore, owing to the open network architectures of HetNets, the security issue faced by wireless powered HetNets has also drawn extensive attention . As a mature technique to guarantee secure communications from the information-theoretical perspective, physical-layer security (PLS) has been widely researched in both academia and industry [18, 19]. There have been some works considering applying PLS techniques to HetNets with WEH. In  and , the artificial noise based secrecy rate maximization was studied for secure HetNets with SWIPT. The authors in  proposed the max-min secrecy energy efficiency optimization for wireless powered HetNets, and a distributed ADMM approach was applied to reduce the information exchange overhead. Considering the more practical scenarios where the transmitter only has imperfect eavesdropper’s CSI, a secrecy SWIPT strategy for two-tier cognitive radio networks was investigated in .
Most of the existing works on HetNets with WEH focus on SWIPT HetNets, it is still an open challenging issue on how to design the optimal harvest-then-transmit strategies for HetNets. In this paper, we investigate the secrecy beamforming design for a wirelessly powered HetNet, where the wirelessly powered FBS transmits the confidential information to a single-antenna FU in the presence of a multi-antenna eavesdropper (Eve). The FBS can harvest energy from the PB and the MBS. Moreover, there is no cooperation among the MBS, the PB and the FBS, thus the resultant cross-tier interference is taken into account. In this wirelessly powered HetNet, the energy and information covariance matrices as well as the time splitting factor are jointly optimized to maximize the secrecy rate under different levels of FBS-EVE CSI. Our main contribution is summarized as :
Firstly, we study secrecy rate maximization (SRM) of the wirelessly powered HetNet having perfect global CSI. In order to address this non-convex perfect SRM problem, a relaxed problem using the matrix trace inequality is studied and proved to be tight, since it always provides a rank-1 optimal solution. Considering the joint-convexity and quasi-convexity of the relaxed problem on different variables, a convexity-based linear search is proposed for optimally solving the perfect SRM problem. In particular, the closed-form solution of this problem is derived when the cross-tier interference at the MU is negligible.
Secondly, the imperfect FBS-EVE CSI with deterministic and Gaussian random CSI errors are considered, respectively. For the deterministic CSI error, the worst-case SRM problem is studied, which can be addressed similarly to the perfect SRM problem using the S-procedure. For the Gaussian random CSI error, the outage-constrained SRM problem subject to the probabilistic secrecy rate constraint is studied by applying the Bernstein-type inequality (BTI)  and then an alternating optimization procedure is proposed. For both the worst-case and the outage-constrained SRM problems, the rank-1 property of the optimal solutions is also validated.
Finally, we consider the realistic scenario with unknown FBS-EVE CSI, in which artificial noise (AN) is utilized for improving secrecy performance. We design AN aided secrecy beamforming by maximizing the average AN power subject to the legitimate rate requirement at the FU. This robust design can be reformulated as a concave one and its optimal rank-1 solution is demonstrated. Similarly, for the inactive cross-interference constraint, the closed-form solution to this AN aided secrecy beamforming design is available.
In fact, the studied SRM problems belong to the nonconvex difference of convex functions (DC) programming, which are more challenging than that in  focusing on the sum-throughput maximization of cognitive WPCNs. Compared to the SRM problem of  where the single-antenna PB and Eve are assumed, these SRM problems are also more intractable due to the additional energy and interference constraints. Fortunately, we validate that the optimal energy and information beamformings are of rank-1 in the secrecy wirelessly powered HetNet, regardless of the availability of eavesdropper’s CSI. This conclusion also provides important insights for practical engineering applications.
The bold-faced lower-case and upper-case letters stand for vectors and matrices, respectively. The operators, and denote the transpose, Hermitian and inverse of a matrix, respectively. and represent the trace and determinant of , respectively. denotes the matrix spectral norm and indicates that the square matrix is positive semidefinite. and denote the rank of , respectively. Also, is defined. The words ‘independent and identically distributed’ and ‘with respect to’ are abbreviated as ‘i.i.d.’ and ‘w.r.t.’, respectively.
Ii System model and Problem Formulation
As shown in Fig. 1, we consider secure communications of the wirelessly powered HetNet, in which an -antenna MBS used for information transmission coexists with an -antenna PB deployed for wireless energy transfer and an -antenna FBS aiming for energy harvesting. Note that the FBS is
energy-limited and harvests energy for its communications from the RF signals transmitted by the PB and the MBS. The MBS and the PB transmit the information-bearing signal and energy-bearing signal to a single-antenna MU and an -antenna FBS, respectively. Then the FBS transmits the confidential signal to a single-antenna FU, while a multi-antenna Eve aims for intercepting the signal of the FBS. We focus our attention on the security of FBS and there is no cooperation between the MBS and the FBS. Hence, the signals transmitted from the MBS and the FBS actually impose interference on the FU and the MU, respectively. All wireless channels are assumed to be quasi-static flat-fading and remain constant during a whole time slot .
In the initial subslot, where denotes the time splitting factor, the FBS harvests energy from both the PB energy signal and MBS interfering signal . Let’s define as the covariance matrix of the PB energy signal subject to the maximum transmit power , i.e. , and for simplicity with being the MBS maximum transmit power. By neglecting the contribution of thermal noise to the total harvested energy at the FBS, the amount of energy harvested at the FBS is expressed as
where is the energy harvesting efficiency factor. and denote the PB-FBS channel and MBS-FBS channel, respectively.
Next, in the second subslot, the FBS transmits the confidential signal to the FU by utilizing the harvested energy in (1). The signals received at the FU and the Eve are then expressed as
where and denote the FBS-FU channel and the MBS-FU channel, respectively. and denote the FBS-EVE channel and the MBS-EVE channel, respectively. and are i.i.d circularly symmetric Gaussian noises at the FU and the EVE, respectively. Additionally, we define as the covariance matrix of the FBS information signal , the achievable rates (in bps/Hz) at the FU and the EVE are then given by
According to , the achievable secrecy rate of the wirelessly powered HetNet is actually the data rate at which the desired information is correctly decoded by the FU, while no information is wiretapped by the EVE. Mathematically, we have
In our work, we jointly optimize the PB and FBS transmit covariance matrices and the time splitting factor for maximizing the achievable secrecy rate in (4). The resultant SRM problem for the wirelessly powered HetNet is then formulated as
In problem (II), the constraint CR1 comes from the fact that the PB transmit power has a maximum threshold, and CR2 denotes the energy causality constraint of the wirelessly powered FBS. While CR3 models the average interference constraint of the secrecy wirelessly powered HetNet. Specifically, by defining and as the PB-MU channel and the FBS-MU channel, respectively, the terms and actually denote the total interference at the MU originating from the PB and the FBS, respectively. Since the PB energy transfer and the FBS information transfer are separated by the time splitting factor , we consider the average interference power constraint at the MU as shown in CR3, where denotes the minimum tolerable interference. It is readily inferred from CR3 that problem (II) is feasible for an arbitrary . However, due to the highly coupled variables , the SRM problem (II) is generally non-convex and challenging to address.
In the sequel, we will investigate the SRM under three different levels of the FBS-EVE CSI. In the first case, the global CSI of the wirelessly powered HetNet is available at the FBS via channel feedback and high-SNR training techniques. In the second case, by assuming imperfect FBS-EVE CSI at the FBS, a pair of robust SRM problems are investigated under deterministic and Gaussian random CSI errors, respectively. In the third case, we consider the more practical scenario that the FBS is not aware of the existence of Eve. In other words, the FBS-EVE CSI is completely unknown to the FBS.
Iii Perfect SRM under global CSI of the wirelessly powered HetNet
In this section, a convexity-based one dimensional search is proposed for optimally solving the SRM problem (II) under the global CSI of the wirelessly powered HetNet.
Iii-a Transformation of Problem (Ii)
Firstly, by introducing an auxiliary variable , the SRM problem (II) can be reformulated as
where and . Unfortunately, problem (III-A) is still difficult to address because of the nonconvex constraint CR4. In order to tackle this issue, we firstly consider a relaxation of CR4 based on the following lemma.
 For any positive semi-definite matrix , we have , where the equality holds if and only if .
where denotes the achievable maximum secrecy rate by solving problem (III-A). Based on Lemma 1, problem (III-A) clearly has a larger feasible region than problem (III-A), so that holds. We then introduce new variables and to equivalently transform problem (III-A) into
It is concluded from problem (III-A) that for any fixed the objective function is the perspective of a strictly concave matrix function , which is also strictly concave [27, p. 39]. Moreover, all constraints in problem (III-A) are convex. Therefore, it is inferred that problem (III-A) is jointly concave w.r.t. given any , and can be globally solved by the interior point method. To further investigate the tightness of the constraint when varying , we firstly consider solving problem (III-A) without
Let’s define and as the optimal solution to problem (9). The corresponding interference level is then expressed as . When , it is readily inferred that the constraint in problem (III-A) will be automatically satisfied using the optimal solution to problem (9), implying that in this context has no effect on problem (III-A) and can be neglected without loss of optimality. While for the case of , we provide an interesting insight in the following Theorem.
Theorem 1 is proved by contradiction as follows. Firstly, we consider an interference threshold satisfying and denote the obtained by solving problem (III-A) as , where is the optimal solution to problem (III-A) with and is assumed. It is then easily found that there is another interference threshold satisfying , based on which the optimal solution actually becomes a feasible solution to problem (III-A). In other words, we have , where is the optimal solution to problem (III-A) with . On the other hand, since , a smaller feasible region is observed for problem (III-A) with , which thus yields By combining the above two inequalities, it is concluded that Similarly, for an arbitrary threshold , the same maximum objective value of problem (III-A) is observed. This phenomenon hints that the constraint actually has no effect on problem (III-A) and thus can be ignored without loss of optimality. As discussed before, this happens only when , which contradicts to the original condition . Therefore, the initial assumption of is actually invalid, and we must have the optimal solution at the boundary of for problem (III-A) when . ∎
Let’s rewrite problem (III-A) by introducing an auxiliary variable as
Given any , problem (10) is jointly and strictly concave w.r.t. and thus the unqiue optimal solution exists. Based on this, it is readily inferred that the value of (10a) is continuous on . We observe that for a sufficiently small , the value of (10a) is strongly dominated by the active constraint (10c). Upon increasing , the feasible region specified by (10c) expands and thus the value of (10a) increases. However, when becomes large enough, the constraint (10d) with the small actually dominates the value of (10a). In this context, we find that the value of (10a) decreases with increasing . According to the above analysis, it can be inferred that there must exist a turning point for problem (III-A). Specifically, with the increase of small , the value of (10a) firstly increases until reaches , and then decreases. This property hints that problem (III-A) is unimodal (quasi-concave) w.r.t. , of which the globally optimal value can be found by GSS. ∎
According to Theorem 2, we firstly determine the one-dimensional search interval of as follows. For achieving a nonzero secrecy rate, the maximum value of actually corresponds to the maximum legitimate rate of the FU, which is derived by solving the following problem
It is clear that problem (11) is also jointly concave w.r.t. . With the optimal solution to problem (11), we readily infer that the value range of is . Overall, given any , by combining the joint concavity of problem (III-A) with the GSS for finding the globally-optimal , the problem (III-A) can be optimally solved. More importantly, we further prove that holds for the PSRM problem (II) and the relaxed problem (III-A), as shown in Theorem 3.
Please refer to Appendix A. ∎
Following the proof of Theorem 3, we propose a convexity-based linear search for globally solving the perfect SRM problem (II). To be specific, given any , the relaxed convex problem (III-A) is firstly addressed for obtaining the optimal solution (via variable substitution) and the resultant achievable secrecy rate. Then the GSS is applied to find the globally-optimal . Theorem 3 reveals that the optimal energy and information covariance matrices, i.e. and , to problem (II) satisfy . Physically, this property means that single-stream transmission of both PB and FBS are optimal for secrecy performance of the wirelessly powered HetNet.
Iv Robust SRMs under Imperfect FBS-EVE CSI
In this section, two types of robust SRM problems are investigated in depth for the secrecy wirelessly powered HetNet. Specifically, one is the worst-case SRM associated with deterministic FBS-EVE CSI error. In this case, we propose a convexity-based linear search method for finding the optimal worst-case solution. The other is the outage-constrained SRM subject to Gaussian random FBS-EVE CSI error, for which the convex reformulation is realized by introducing an auxiliary variable.
Iv-a The Proposed Worst-Case SRM
Recall the system model in Section II, when considering deterministically imperfect FBS-EVE channel, we have , where
denotes the estimated FBS-EVE channel andis the norm bounded CSI error, i.e, . Based on this, the achievable worst-case secrecy rate of the wirelessly powered HetNet is given by , and the resultant worst-case SRM problem is formulated as
Compared to the perfect SRM problem (II), problem (IV-A) is more challenging since the semi-infinite norm bounded CSI error is included in CR5. To make problem (IV-A) tractable, we firstly relax it using the above Lemma 1 to
where denotes the maximum achievable worst-case secrecy rate by solving problem (IV-A) and satisfies due to larger feasible region of problem (IV-A). For solving this non-convex problem effectively, both the equality and the identity are utilized to rewrite the constraint CR5 as
(S-procedure)  For the equation , in which , , and is a constant, the following equality holds
where a scalar variable is introduced. In contrast to problem (III-A), an additional SDP constraint is included in problem (17). For any fixed , we easily find that is a convex linear matrix inequality (LMI) w.r.t. , so problem (17) becomes jointly concave w.r.t. . Similar to Theorem 2, we can also prove that problem (17) is quasi-concave w.r.t , since the semi-infinite CSI error in the equivalent problem (IV-A) is independent of . Based on the above discussions, it is readily inferred that the proposed convexity-based linear search for problem (III-A) can be directly extended to problem (17). More importantly, an interesting insight is provided in the following Theorem, namely, problem (17) also admits the rank-1 optimal solution.
Please refer to Appendix C. ∎
Generally, the proof of Theorem 4 subject to the complicated LMI constraint is more difficult than that of Theorem 3. Based on Theorem 4, the globally optimal solution to the worst-case SRM problem (12) can also be obtained by successively solving the relaxed problem (17), for which the proposed convexity-based linear search in Section III still works.
Iv-B The Proposed Outage-Constrained SRM
It is widely recognized that the worst-case optimization is the most conservative robust design, which is only encountered in practical systems with a low probability. Hence, in this subsection, we consider the more general case of statistically imperfect CSI, in which the FBS-EVE CSI error is assumed to be complex Gaussian distributed, i.e.,, where denotes the positive semi-definite error covariance matrix. Inspired by the fact that under the unbounded Gaussian error , an absolutely safe beamforming design cannot be guaranteed, we instead consider the outage-constrained SRM to implement secure communications in the wirelessly powered HetNet. More specifically, by defining the maximum secrecy rate outage probability , a -safe design of our wirelessly powered HetNet is investigated. Mathematically, the outage-constrained SRM problem is formulated as
Clearly, the secrecy outage constraint CR6 indicates that the probability of the achievable secrecy rate being over should be higher than , and problem (18) aims for maximizing this -outage secrecy rate threshold . However, problem (18) is computationally intractable since the constraint CR6 does not have an explicit expression. Therefore, we consider replacing the function in CR6 by an easy-to-handle function via the following Lemmas.
 For an arbitrary positive-definite matrix , we have , where the optimal is derived as .
(Bernstein-type inequality (BTI) ) For an arbitrary vector , we assume , where , and is a constant. Then for any , the following convex approximation holds, i.e.
where and are a pair of slack variables.
Firstly, by invoking Lemma 3, the wiretap rate of the EVE can be rewritten as
where Since , we can re-express as with . Furthermore, through the vectorization of (21), we have
where . To make the probabilistic constraint (22) tractable, we adopt a popular conservative approximation, the so-called BTI in Lemma 4, for transforming it into a series of tractable convex constraints. Then the outage-constrained SRM problem (18) is reformulated as
where . Although problem (23) is still not jointly concave w.r.t , it is more tractable than the outage-constrained SRM problem (18). Specifically, by utilizing , and , we easily find that problem (23) is jointly concave w.r.t.