AN-Aided Secure Beamforming in Power-Splitting-Enabled SWIPT MIMO Heterogeneous Wireless Sensor Networks

Abstract

In this paper, we investigate the physical layer security in a two-tier heterogeneous wireless sensor network (HWSN) depending on simultaneous wireless information and power transfer (SWIPT) approach for multiuser multiple-input multiple-output wiretap channels with artificial noise (AN) transmission, where a more general system framework of HWSN only includes a macrocell and a femtocell. For the sake of implementing security enhancement and green communications, the joint optimization problem of the secure beamforming vector at the macrocell and femtocell, the AN vector, and the power splitting ratio is modeled to maximize the minimal secrecy capacity of the wiretapped macrocell sensor nodes (M-SNs) while considering the fairness among multiple M-SNs. To reduce the performance loss of the rank relaxation from the SDR technique while solving the non-convex max–min program, we apply successive convex approximation (SCA) technique, first-order Taylor series expansion and sequential parametric convex approximation (SPCA) approach to transform the max–min program to a second order cone programming (SOCP) problem to iterate to a near-optimal solution. In addition, we propose a novel SCA-SPCA-based iterative algorithm while its convergence property is proved. The simulation shows that our SCA-SPCA-based method outperforms the conventional methods.

1. Introduction

Recently, Internet-of-Things (IoT) has emerged to be considered as one of the key technological concepts as a basic precondition implementing Industry 4.0 [1]. The recent developments in the IoT application, wireless sensor networks (WSNs), have played a significant role in connecting tens of thousands of wireless devices over the Internet in many applications, especially healthcare services, electronic instrumentation, transportation, and environmental assessment [2]. A WSN generally contains many sensor nodes (SNs) that is capable of perceiving their external environment, which can perform communicate and computations [3]. Unfortunately, because SN are battery-powered, the working lifetime of the entire WSNs is limited by energy consumption due to the energy restriction at SN [4]. In order to overcome these limitation, there are many investigations on enabling technologies for energy sustainablility in WSNs [5].

The emergence of mobile devices has resulted in a continuously growing requirement for wireless applications, such as an ultra-wide radio coverage, an ultra-high data service, and an ultra-large number of smart devices. Recently, a promising solution to cater for this requirement has been the fifth generation (5G) wireless technology [6,7]. Therefore, the deployment of low-cost femtocell base stations (FBSs) underlaid in the macrocell can provide ubiquitous coverage and extremely high data rate [8]. In this regard, to accommodate the explosive cell density and growing data traffic, heterogeneous networks (HetNets) have emerged as one of the most promising approaches in the 5G wireless systems by increasing the off-load portion of their traffic [9,10].

In recent years, by integrating the WPT with traditional wireless information transmission (WIT), simultaneous wireless information and power transfer (SWIPT) have been recognised as among the main candidate technologies for green communications [11,12,13,14,15], which has been shown to be in a position to improve the WPT performance in HetNets [16,17,18]. The idea of using SWIPT to implement wireless recharging of the batteries in HetNets was first studied in [16], where two integer linear programming methods were proposed to help wireless nodes to scavenge ambient energy. Tang et al. [17] provided a essential study of the QoS-constrained energy efficiency optimization problem for coordinated multi-point (CoMP)-SWIPT HetNets, a decoupled approach was proposed in which the power allocation procedure and beamforming design were separated. Considering multiple Power splitting (PS)-based heterogeneous users in [18], Jiang et al. investigated the optimal transmit power minimization design problem of SWIPT systems to achieve green communication design under non-linear energy harvesting model, where the semidefinite relaxation (SDR) are employed to solve and the globe optimal solution was theoretically guaranteed.

On the other hand, depending on the presence of the broadcasting nature of wireless channel and the multi-tier dynamic topology in SWIPT HetNets, the useful messages which intended to the information users may be entirely possible to be eavesdropped by the energy users, which brings huge technical challenges to secure SWIPT in HetNets [19]. Recently, physical-layer security (PLS) has emerged as a prominent technique to guarantee the secrecy communication by exploiting the characteristics of physical wireless channels, while defending against the eavesdroppers [20,21]. Various PLS techniques in SWIPT networks have been studied in MISO and MIMO systems [22,23,24,25,26,27,28,29]. More specifically, Ren et al. [29] studied jointly optimize design of the transmit beamforming and AN of the macrocell base station (MBS) and the femtocell base stations (FBS), the secrecy rate maximization problem of the eavesdropped macrocell users was reformulated by utilizing the one-dimensional line search and successive convex approximation.

In addition, there are some related works integrating the SWIPT in WSNs [30,31,32,33,34,35,36,37]. Guo et al. [30] applied the SWIPT technique to cooperative clustered WSN, a distributed iteration algorithm was proposed by exploiting fractional programming and dual decomposition. Considering different PS abilities of relays in a mobile wireless sensor networks with SWIPT, a resource allocation design (RAD) problem was formulated as a non-convex energy efficiency maximization problem, Guo et al. [31] proposed a cross-layer algorithm for rate control, PS, and power allocation. A unified framework was developed to study the impact of SWIPT on the performance with both time splitting (TS) and power splitting (PS) schemes [32] in a WSN over Nakagami-m fading channels. In the MIMO wireless powered underground sensor network, the optimal RAD problem was considered for throughput the system throughput maximization with quality of service (QoS) assurance in regard to communication reliability and diverse data traffic demands [33]. Focusing on two relaying strategies, a downlink (DL) multiple-input single-output (MISO) WSN with the nonorthogonal multiple access technique (NOMA) and SWIPT was investigated over Nakagami-m fading [34], where three antenna-relay-destination selection algorithms were proposed through analysis on downlink (DL) and UL transmissions. Assuming the sensor nodes always work in harvesting–transmitting mode, Hu et al. [35] studied the PLS of uplink (UL) transmission in SWIPT deployed in WSNs, in which the energy/secrecy outage probabilities were derived through comprehensive analysis on DL and UL transmissions, respectively. As for IoT sensor networks with a PS-enabled SWIPT technology, Chae et al. [36] proposed a novel SWIPT strategy where the access point have a choice over either sending a private/common message, and a hybrid beamforming design algorithm was proposed by considering the total transmit power minimization problem. By exploiting a fountain codes in the SWIPT WSNs, Yi et al. [37] proposed a new PS mechanism to adaptively divide data packets into the information data packets and the energy data packets. To the best of our knowledge, secure beamforming design to achieve secrecy capacity, particularly assuming practical power splitting (PS)-based heterogeneous wireless sensor networks (HWSN) with SWIPT transmission, has not been considered previously.

Motivated by the above analysis, in this paper, we consider a SWIPT-enabled two-tier HWSN with MIMO system, where a macrocell sink node and a femtocell sink node transmit co-channel information to the corresponding macrocell sensor nodes (M-SNs), high-priority sensor nodes (HP-SNs) and low-priority sensor nodes (LP-SNs), respectively. In order to harvest more energy from the ambient environment, the HP-SNs can use a power splitter. According to the ambient interference signals, the LP-SNs not only eavesdrop the confidential messages from the M-SNs, but also harvest wireless energy. By applying the interference, an AN-aided secrecy transmission is utilized at the femtocell sink node. In addition, we study the jointly optimization design of the transmit beamforming vectors at the macrocell sink node and femtocell sink node, the AN and PS ratio to maximize the minimal secrecy capacity (MSC) of the wiretapped M-SNs under the secrecy capacity constraint at the HP-SNs, the harvested energy constraint at the HP-SNs and LP-SNs, the transmit power constraint at the macrocell sink node and femtocell sink node. More explicitly, the main contributions of this paper can be summarized as follows:

• Firstly, we establish the system model of a two-tier MIMO HWSN, where SWIPT technique is performed at the femtocell sink node to recharge the HP-SNs and LP-SNs. To be specific, each HP-SN is equipped with power splitter. We consider a general system case where both the M-SNs and HP-SNs are faced under threat of secrecy leakage from the LP-SNs, where each LP-SN act as a malicious user.
• To promote the security performance and harvest more power, the AN is injected designedly into the transmit beamforming at the femtocell sink node. To achieve secrecy enhancement, a joint design problem is formulated for the MSC maximization while considering the fairness among multiple M-SNs. The original problem is non-convex and hard to solve directly.
• In order to reduce the performance loss of the rank relaxation from the SDR technique while solving the non-convex max–min program [38], we apply SCA technique, first-order Taylor series expansion and SPCA method, and then the recast problem can be converted to an SOCP problem [39] which can be directly solved to achieve a near-optimal solution. Moreover, we propose a novel SCA-SPCA-based iterative algorithm, while its convergence property of the proposed algorithm is proved.

Compared with our previous papers [23,24], major additional works and results incorporated in these papers are summarized in the following. (1) This paper has extended the problem of AN power maximization to both perfect and imperfect CSI cases, which introduces substantial changes in the analyses. (2) An SPCA-based iterative algorithm has been proposed to solve the problem such that the computational complexity is largely reduced compared with the 1-D search method used in the previous work.

The rest of this paper is organized as follows: In Section 2, we presents the HWSN system model and formulates the MSC maximization problem. Section 3 solves the non-convex problem and provide the proposed SCA-SPCA-based iterative algorithm. Section 4 outlines the simulation results. Finally, Section 5 concludes this paper.

Notation: Lower-case letters mean scalars, bold-face lower-case letters are denoted by vectors, and boldface upper-case letters are used for matrices. ${\mathbb{C}}^{M\times L}$ indicates the space of $M\times L$ complex matrices. $|·|$ defines the absolute value of a complex scalar. ${\parallel \mathbf{A}\parallel }_F$ describes the Frobenius norm of the matrix $\mathbf{A}$, $\mathrm{tr}(\mathbf{A})$ indicates the trace of $\mathbf{A}$. $\mathrm{E}\{·\}$ describes the mathematical expectation. ${[x]}^{+}$ equals $max\{x,0\}$.

2. System Model and Problem Formulation

In this section, a two-tier downlink HWSN is considered with MIMO-SWIPT, where a femtocell sink node is deployed with a macrocell sink node, as shown in Figure 1. Following [40], the smart antenna technology is applied for the sink nodes. Moreover, the femtocell sink node serves K HP-SNs and L LP-SNs, whereas the macrocell sink node serves M M-SNs. For the sake of notational simplicity, it is assumed that the m-th M-SN, the k-th HP-SN and the l-th LP-SN are denoted by M-SN${}_m$, HP-SN${}_k$, and LP-SN${}_l$, respectively. The system parameters that we have considered are defined in

Table 1. System Parameters.

Variables	Definition
M-SN${}_m$	the m-th M-SN
HP-SN${}_k$	the k-th HP-SN
LP-SN${}_l$	the l-th LP-SN
${\mathbf{s}}_m$	the transmit signal from macrocell sink node to M-SN${}_m$
${\mathbf{x}}_s$	the transmit signal from femtocell sink node
${\mathbf{v}}_m$	the transmit beamforming vector from macrocell sink node aiming at M-SN${}_m$
${\mathbf{w}}_k$	the transmit beamforming vector from femtocell sink node aiming at HP-SN${}_k$
$s_{p,m}$, $s_{s,k}$	the information-bearing signal intended for M-SN${}_m$ and HP-SN${}_k$
$\mathbf{z}$	the energy-carrying AN
${\mathbf{g}}_m$	the channel between macrocell sink node and M-SN${}_m$
${\mathbf{g}}_{c,k}$	the channel between macrocell sink node and HP-SN${}_k$
${\mathbf{G}}_{e,l}$	the channel between macrocell sink node and LP-SN${}_l$
${\mathbf{h}}_m$	the channel between femtocell sink node and M-SN${}_m$
${\mathbf{h}}_{c,k}$	the channel between femtocell sink node and HP-SN${}_k$
${\mathbf{H}}_{e,l}$	the channel between femtocell sink node and LP-SN${}_l$
$n_m$, $n_{c,k}$, $n_{e,l}$	the complex Gaussian noise at the M-SN${}_m$, the HP-SN${}_k$ and the LP-SN${}_l$
$n_{a,k}$	the additional noise introduced from the RF-to-signal conversion in the ID part at the HP-SN${}_k$

. In the smallcell of this HWSN system, two different types of receivers are considered: (i) one cluster of HP-SNs (such as low-power receivers), denoted by $\mathbb{K}$ = {HP-SN${}_1$,…, HP-SN${}_K$}, which is located far from the femtocell sink node, (ii) one cluster of LP-SNs (such as sensors), denoted by $\mathbb{L}$ = {LP-SN${}_1$,…,LP-SN${}_L$}, which are located close to the femtocell sink node. It is assumed that each M-SN and HP-SN have single receive antenna, LP-SN is equipped with $N_E$ receive antenna, whereas the macrocell sink node and femtocell sink node are equipped with $N_M\ge M$ antennas and $N_F\ge K+L{N}_E$ transmit antennas, respectively. This system operates as follows: the macrocells share their frequency spectrum with the femtocell to improve the spectral efficiency, whereas the confidential messages from the macrocell sink node and the femtocell sink node may be intercepted by LP-SNs. Additionally, all HP-SNs employ the PS scheme to perform information decoding (ID) and energy harvesting (EH) simultaneously, as shown in Figure 2.

As an active user in the HWSN, we assume that LP-SN can harvest energy to extend its battery life from the ambient radio frequency (RF) signals. However, LP-SNs may be potential eavesdroppers due to intercepting the information from the femtocell sink node to HP-SNs. Thus, to provide secure transmission in this HWSN, the femtocell sink node applies the AN to transmit beamforming, which not only acts as interference to the LP-SNs, but also supplies energy to the HP-SNs. The secure signal vector $\mathbf{s}$ from the macrocell sink node and ${\mathbf{x}}_s$ from the femtocell sink node can be written as

$$\begin{array}{ccc}\hfill \mathbf{s}& =& {\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m{s}_m,m=1,\dots ,M,\hfill \\ \hfill {\mathbf{x}}_s& =& {\displaystyle \sum _{k=1}^K}{\mathbf{w}}_k{s}_{s,k}+\mathbf{z},k=1,\dots ,K,\hfill \end{array}$$

where ${\mathbf{v}}_m\in {\mathbb{C}}^{N_P}$ and ${\mathbf{w}}_k\in {\mathbb{C}}^{N_s}$ define the transmit beamforming vector from the macrocell sink node aiming at M-SN${}_m$ and from the femtocell sink node aiming at HP-SN${}_k$, $s_m$ and $s_{s,k}$ are the information-bearing signal intended for M-SN${}_m$ and HP-SN${}_k$ with satisfying $\mathbb{E}\{s_m^2\}=1$ and $\mathbb{E}\{s_{s,k}^2\}=1$, respectively, and $\mathbf{z}$ represents the energy-carrying AN.

Therefore, the received signal at the M-SN${}_m$, the HP-SN${}_k$, and the LP-SN${}_l$ can be given by, respectively,

$$\begin{array}{ccc}\hfill y_m& =& \underset{desiredsignal}{\underbrace{{\mathbf{g}}_m^H{\mathbf{v}}_m{s}_m}}+\underset{intra-tierinterference}{\underbrace{{\mathbf{g}}_m^H{\sum }_{i=1,i\ne m}^M({\mathbf{v}}_i{s}_{p,i})}}+\underset{inter-tierinterference}{\underbrace{{\mathbf{h}}_m^H({\sum }_{k=1}^K{\mathbf{w}}_k{s}_{s,k}+\mathbf{z})}}+n_m,m=1,\dots ,M,\hfill \\ \hfill y_{c,k}& =& \underset{desiredsignal}{\underbrace{{\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{s}_{s,k}}}+\underset{intra-tierinterference}{\underbrace{{\mathbf{h}}_{c,k}^H\mathbf{z}+{\mathbf{h}}_{c,k}^H{\sum }_{i=1,i\ne k}^K({\mathbf{w}}_i{s}_{s,i})}}+\underset{inter-tierinterference}{\underbrace{{\mathbf{g}}_{c,k}^H{\sum }_{m=1}^M{\mathbf{v}}_m{s}_m}}+n_{c,k},k=1,\dots ,K,\hfill \\ \hfill {\mathbf{y}}_{e,l}& =& \underset{desiredsignal}{\underbrace{{\mathbf{H}}_{e,l}^H{\mathbf{w}}_l{s}_{s,l}}}+\underset{intra-tierinterference}{\underbrace{{\mathbf{H}}_{e,l}^H\mathbf{z}+{\mathbf{H}}_{e,l}^H{\sum }_{i=1,i\ne l}^K({\mathbf{w}}_i{s}_{s,i})}}+\underset{inter-tierinterference}{\underbrace{{\mathbf{G}}_{e,l}^H{\sum }_{m=1}^M{\mathbf{v}}_m{s}_m}}+{\mathbf{n}}_{e,l},l=1,\dots ,L,\hfill \end{array}$$

where ${\mathbf{g}}_m\in \mathbb{C}$ and ${\mathbf{h}}_m\in {\mathbb{C}}^{N_S}$ are denoted by the channel between macrocell sink node and M-SN${}_m$ as well as that between femtocell sink node and M-SN${}_m$, ${\mathbf{g}}_{c,k}\in {\mathbb{C}}^{N_P\times N_C}$ and ${\mathbf{h}}_{c,k}\in {\mathbb{C}}^{N_S\times N_C}$ indicate the channel vectors between macrocell sink node and HP-SN${}_k$ as well as that between the femtocell sink node and the HP-SN${}_k$, ${\mathbf{G}}_{e,l}\in {\mathbb{C}}^{N_P\times N_E}$ and ${\mathbf{H}}_{e,l}\in {\mathbb{C}}^{N_S\times N_E}$ are the channel vectors between the macrocell sink node and LP-SN${}_l$ as well as that between the femtocell sink node and LP-SN${}_l$, $s_m$ is confidential information-bearing signal for M-SN${}_m$ from the macrocell sink node satisfying $\mathbb{E}\{s_m^2\}=1$. In addition, $n_m\sim \mathcal{CN}(0,{\sigma }_m^2)$, $n_{a,k}\sim \mathcal{CN}(0,{\sigma }_{a,k}^2)$ and ${\mathbf{n}}_{e,l}\sim \mathcal{CN}(0,{\sigma }_{e,l}^2\mathbf{I})$ denote the complex AWGN at the M-SN${}_m$, the HP-SN${}_k$ and the LP-SN${}_l$, respectively.

Since the HP-SN utilizes PS scheme to simultaneously implement ID and EH, the received signal at the HP-SN${}_k$ is divided into ID portion and EH portion by the PS ratio ${\rho }_k\in (0,1]$. Hence, the split signal for ID part and EH part at the HP-SN${}_k$ can be expressed as, respectively,

$$\begin{array}{ccc}\hfill y_{c,k}^{ID}& =& \sqrt{{\rho }_k}({\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{s}_{s,k}+{\mathbf{h}}_{c,k}^H\mathbf{z}+{\mathbf{h}}_{c,k}^H{\displaystyle \sum _{i=1,i\ne k}^K}({\mathbf{w}}_i{s}_{s,i})+{\mathbf{g}}_{c,k}^H{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m{s}_m+n_{a,k})+n_{p,k},\hfill \\ \hfill y_{c,k}^{EH}& =& \sqrt{1-{\rho }_k}({\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{s}_{s,k}+{\mathbf{h}}_{c,k}^H\mathbf{z}+{\mathbf{h}}_{c,k}^H{\displaystyle \sum _{i=1,i\ne k}^K}({\mathbf{w}}_i{s}_{s,i})+{\mathbf{g}}_{c,k}^H{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m{s}_m+n_{a,k}),\hfill \end{array}$$

where $n_{p,k}\sim \mathcal{CN}(0,{\delta }_{p,k}^2)$ is the additional Gaussian noise resulted in the RF-to-signal conversion for the ID part at the HP-SN${}_k$.

Thus, the channel capacity of the HP-SN${}_k$ is expressed as

$$\begin{array}{c}{R}_{c,k}=log\left(1+\frac{{\rho }_k{\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{h}}_{c,k}}{{\rho }_k\left({\mathbf{h}}_{c,k}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k}+{\mathbf{g}}_{c,k}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k}+{\sigma }_{a,k}^2\right)+{\delta }_{p,k}^2}\right).\hfill \end{array}$$

(1)

The channel capacity of the LP-SN${}_l$ for decoding the desired signal of the HP-SN${}_k$ can be represented as

$$R_{c,lk}=log|\mathbf{I}+{\left({\sigma }_{e,l}^2\mathbf{I}+({\mathbf{H}}_{e,l}^H(\sum _{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l})\right)}^{-1}{\mathbf{H}}_{e,l}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{H}}_{e,l}|.$$

(2)

Thus, the secrecy capacity of the HP-SN${}_k$ can be written as

$$\begin{array}{c}\hfill C_{c,k}={\left[R_{c,k}-\underset{l}{max}{R}_{c,lk}\right]}^{+},\forall k,l.\end{array}$$

(3)

Additionally, the channel capacity of the M-SN${}_m$ can be expressed as

$$\begin{array}{c}\hfill R_m=log\left(1+\frac{{\mathbf{g}}_m^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{g}}_m}{{\mathbf{h}}_m^H\left({\sum }_{j=1}^K{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H\right){\mathbf{h}}_m+{\sigma }_m^2+{\mathbf{g}}_m^H{\sum }_{j\ne m}^M({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{g}}_m}\right).\end{array}$$

(4)

Suppose that the LP-SN${}_l$ is an eavesdropper to decode the message for the M-SN${}_m$. The channel capacity of the LP-SN${}_l$ for decoding the M-SN${}_m$ is given as

$$\begin{array}{c}\hfill R_{m,l}=log\left|\mathbf{I}+{({\mathbf{H}}_{e,l}^H({\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H\sum _{j\ne m}({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{G}}_{e,l}+{\sigma }_{e,l}^2\mathbf{I})}^{-1}{\mathbf{G}}_{e,l}^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{G}}_{e,l}\right|.\end{array}$$

(5)

Hence, following the concept of secrecy capacity in [20,21,22], the secrecy capacity of the M-SN${}_m$ can be written as

$$\begin{array}{c}\hfill C_m={\left[R_m-\underset{l}{max}{R}_{m,l}\right]}^{+},\forall l.\end{array}$$

(6)

where ${[a]}^{+}$ indicates the $max(0,a)$.

Moreover, the HP-SNs employs the PS ratio to harvest the power carried from the RF signal. In order to harvest the carried energy, the LP-SN receiver does not need to transform the received signal to the baseband. However, due to the law of energy conservation, we assume that the total harvested energy at the HP-SN${}_k$ and the LP-SN${}_l$, i.e., the RF-band power normalized by the baseband symbol, denoted by $E_{c,k}$ and $E_{e,l}$, respectively, from all $N_E$ receive antennas at the LP-SN receiver is proportional to that of the received RF baseband signal, i.e.,

$$\begin{array}{}\mathrm{(7a)}& \hfill E_{c,k}& =& {\eta }_{c,k}(1-{\rho }_k)\left({\mathbf{h}}_{c,k}^H({\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k}+{\mathbf{g}}_{c,k}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k}+{\sigma }_{a,k}^2\right),\forall k,\hfill \mathrm{(7b)}& \hfill E_{e,l}& =& {\eta }_{e,l}\left(\mathrm{tr}({\mathbf{H}}_{e,l}^H({\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l})+\mathrm{tr}({\mathbf{G}}_{e,l}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l})+N_E{\sigma }_{e,l}^2\right),\forall l,\hfill \end{array}$$

where $0\le {\eta }_{c,k}\le 1$ and $0\le {\eta }_{e,l}\le 1$ denote the energy conversion efficiency at the HP-SN${}_k$ and LP-SN${}_l$, respectively.

In this paper, our objective is to secure the WIT to legitimated users (i.e., M-SNs and HP-SNs) in the presence of potential eavesdroppers (i.e., LP-SNs) by jointly designing the AN-aided transmit beamforming. Moverover, we aim to promote the MSC performance by guaranteeing all M-SNs can achieve a certain secrecy capacity level. To this end, we maximize the MSC of the M-SN${}_m$ under the secrecy capacity constraint at the HP-SNs, the harvested energy constraint at the HP-SNs and LP-SNs, as well as the transmit power constraint at the macrocell sink node and femtocell sink node. Based on the system models, the MSC maximization (MSCM) problem is formulated as

$$\begin{array}{}& \hfill \underset{{\mathbf{w}}_k,\mathbf{z},{\mathbf{v}}_m,{\rho }_k}{max}& \underset{m=1,\dots ,M}{min}{C}_m\hfill \mathrm{(8a)}& \hfill s.t.& \underset{k}{min}{C}_{c,k}\ge {\overline{R}}^c,\forall k,\hfill \mathrm{(8b)}& & \underset{k}{min}{E}_{c,k}\ge {\overline{E}}^c,\forall k,\hfill \mathrm{(8c)}& & \underset{l}{min}{E}_{e,l}\ge {\overline{E}}^e,\forall l,\hfill \mathrm{(8d)}& & {\displaystyle \sum _{k=1}^K}\parallel {\mathbf{w}}_k{\parallel }^2+{\parallel \mathbf{z}\parallel }^2\le P,\forall k,{\displaystyle \sum _{k=1}^K}\parallel {\mathbf{w}}_k{\parallel }^2\le P_1,{\displaystyle \sum _{m=1}^M}{\parallel {\mathbf{v}}_m\parallel }^2\le P_2,\hfill \mathrm{(8e)}& & 0\le {\rho }_k\le 1,\hfill \end{array}$$

where ${\overline{R}}^c$ denotes the secrecy capacity requirement of the HP-SN${}_k$, ${\overline{E}}^c$ and ${\overline{E}}^e$ mean the harvested power requirement of the HP-SN${}_k$ and the LP-SN${}_l$, respectively, P represents the total transmit power at the femtocell sink node, $P_1$ indicates the transmit power about the information signal part at the femtocell sink node, $P_2$ stands for the total transmit power at the macrocell sink node. The constraints (8a) guarantees that the minimum secrecy rate should be achieved at the HP-SN${}_k$. The constraints and (8b) and (8c) guarantee that the minimum harvested power at the HP-SN${}_k$ and the LP-SN${}_l$ are no less than ${\overline{E}}^c$ and ${\overline{E}}^e$, respectively. The constraint (8d) limits the transmit power. Obviously, it is seen that problem (8) is non-convex due to the existence of coupling among variables ${\rho }_k$ and ${\mathbf{w}}_k$, which is difficult to solved directly.

3. Proposed Algorithm

In this section, we will propose an iterative algorithm to obtain near-optimal solution by applying the SCA method and SPCA technology [41]. To tackle problem (8) we firstly introduce an equivalent reformulation about problem (8). To circumvent this issue, problem (8) can be rewritten as

$$\begin{array}{}& \underset{{\mathbf{w}}_k,\mathbf{z},{\mathbf{v}}_m,{\rho }_k}{max}\underset{m=1,\dots ,M}{min}log\left(1+\frac{{\mathbf{g}}_m^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{g}}_m}{{\mathbf{h}}_m^H({\sum }_{k=1}^K{\mathbf{w}}_k{\mathbf{w}}_k^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_m+{\mathbf{g}}_m^H{\sum }_{j\ne m}({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{g}}_m+{\sigma }_m^2}\right)\hfill \mathrm{(9a)}& -log\left|\mathbf{I}+{({\mathbf{H}}_{e,l}^H({\displaystyle \sum _{k=1}^K}{\mathbf{w}}_k{\mathbf{w}}_k^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H\sum _{j\ne m}({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{G}}_{e,l}+{\sigma }_{e,l}^2\mathbf{I})}^{-1}{\mathbf{G}}_{e,l}^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{G}}_{e,l}\right|\hfill & s.t.log\left(1+\frac{{\rho }_k{\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{h}}_{c,k}}{{\rho }_k({\mathbf{h}}_{c,k}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k}+{\mathbf{g}}_{c,k}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k}+{\sigma }_{a,k}^2)+{\delta }_{p,k}^2}\right)\hfill \mathrm{(9b)}& -log\left|\mathbf{I}+{({\sigma }_{e,l}^2\mathbf{I}+{\mathbf{H}}_{e,l}^H(\sum _{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l})}^{-1}{\mathbf{H}}_{e,l}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{H}}_{e,l}\right|\ge {\overline{R}}^c,\hfill \mathrm{(9c)}& {\eta }_k(1-{\rho }_k)(\mathrm{tr}({\mathbf{h}}_{c,k}^H({\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k})+\mathrm{tr}({\mathbf{g}}_{c,k}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k})+N_C{\sigma }_{a,k}^2)\ge {\overline{E}}^c,\forall k,\hfill \mathrm{(9d)}& {\eta }_l(\mathrm{tr}({\mathbf{H}}_{e,l}^H({\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l})+\mathrm{tr}({\mathbf{G}}_{e,l}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l})+{\sigma }_{e,l}^2)\ge {\overline{E}}^e,\forall l,\hfill & (8\mathrm{d}),(8\mathrm{e}).\hfill \end{array}$$

It is observed that the determinant functions is in the $R_{c,lk}$ of the objective function (9a). We consider the following Lemma 1, which will be applied to provide a convex approximation form to $C_{l,k}$.

Lemma 1.

([42]) For any matrix $\mathit{A}$, which $\mathit{A}$ is positive semi-definite, i.e., $\mathit{A}⪰\mathit{0}$. It holds true that

$$|\mathit{I}+\mathit{A}|\ge 1+\mathrm{tr}(\mathit{A}),$$

(10)

where the equality holds if and only if $rank(\mathit{A})\le 1$.

Proof. 

See Appendix A. □

We introduce two real-valued slack variables ${\gamma }_c$ and ${\gamma }_e$. By applying the Lemma 1, problem (9) can be rewritten as

$$\begin{array}{}\mathrm{(11a)}& \underset{{\mathbf{w}}_k,\mathbf{z},{\mathbf{v}}_m,{\rho }_k,{\gamma }_c,{\gamma }_e}{max}{\gamma }_c-{\gamma }_e\hfill \mathrm{(11b)}& s.t.\frac{{\mathbf{g}}_m^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{g}}_m}{{\mathbf{h}}_m^H\left({\sum }_{k=1}^K{\mathbf{w}}_k{\mathbf{w}}_k^H+\mathbf{z}{\mathbf{z}}^H\right){\mathbf{h}}_m+{\mathbf{g}}_m^H{\sum }_{j\ne m}({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{g}}_m+{\sigma }_m^2}\ge 2^{{\gamma }_c}-1,\hfill \mathrm{(11c)}& \frac{\mathrm{tr}({\mathbf{G}}_{e,l}^H{\mathbf{v}}_m{\mathbf{v}}_m^H{\mathbf{G}}_{e,l})}{\mathrm{tr}\left({\mathbf{H}}_{e,l}^H\left({\sum }_{k=1}^K{\mathbf{w}}_k{\mathbf{w}}_k^H+\mathbf{z}{\mathbf{z}}^H\right){\mathbf{H}}_{e,l}\right)+{\mathbf{G}}_{e,l}^H{\sum }_{j\ne m}({\mathbf{v}}_j{\mathbf{v}}_j^H){\mathbf{G}}_{e,l}+{\sigma }_{e,l}^2}\le 2^{{\gamma }_e}-1,\hfill & (9\mathrm{b}),(9\mathrm{c}),(9\mathrm{d}),(8\mathrm{d}),(8\mathrm{e}).\hfill \end{array}$$

In order to make the problem (11) tractable, by introducing the auxiliary variables $v_m$, $w_m$, $u_m$, $t_l$, $e_l$, and $q_l$, we define new matrices ${\mathbf{G}}_m={\mathbf{g}}_m{\mathbf{g}}_m^H$, ${\mathbf{H}}_m={\mathbf{h}}_m{\mathbf{h}}_m^H$, ${\mathbf{G}}_{c,k}={\mathbf{g}}_{c,k}{\mathbf{g}}_{c,k}^H$, ${\mathbf{H}}_{c,k}={\mathbf{h}}_{c,k}{\mathbf{h}}_{c,k}^H$, ${\overline{\mathbf{G}}}_{e,l}={\mathbf{G}}_{e,l}{\mathbf{G}}_{e,l}^H$, and ${\overline{\mathbf{H}}}_{e,l}={\mathbf{H}}_{e,l}{\mathbf{H}}_{e,l}^H$. The constraints (11b) and (11c) can be written as, respectively,

$$\begin{array}{}\mathrm{(12a)}& {\mathbf{v}}_m^H{\mathbf{G}}_m{\mathbf{v}}_m\ge e^{v_m},\hfill \mathrm{(12b)}& e^{w_m}\ge {\displaystyle \sum _{k=1}^K}{\mathbf{w}}_k^H{\mathbf{H}}_m{\mathbf{w}}_k+{\mathbf{z}}^H{\mathbf{H}}_m\mathbf{z}+\sum _{j\ne m}({\mathbf{v}}_j^H{\mathbf{G}}_m{\mathbf{v}}_j)+{\sigma }_m^2,\hfill \mathrm{(12c)}& e^{u_m}\ge 2^{{\gamma }_c}-1,\hfill \mathrm{(12d}& v_m-w_m\ge u_m.\hfill \end{array}$$

$$\begin{array}{}\mathrm{(13a)}& {\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m\le e^{t_l},\forall m,\forall l,\hfill \mathrm{(13b)}& e^{e_l}\le {\displaystyle \sum _{k=1}^K}{\mathbf{w}}_k^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_k+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+\sum _{j\ne m}({\mathbf{v}}_j^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_j)+{\sigma }_{e,l}^2,\hfill \mathrm{(13c)}& e^{q_l}\le 2^{{\gamma }_e}-1,\hfill \mathrm{(13d)}& t_l-e_l\le q_l.\hfill \end{array}$$

According to [43], it is well-known that a constraint is such that a concave function is greater than or equal to a convex function, then the constraint is a convex form. Therefore, it is easily seen that (12a) and (13b) are convex now. However, (12b), (12c), (13a) and (13c) are still non-convex. Nevertheless, the functions $e^{w_m}$ and $e^{u_m}$ on the right-hand side (RHS) of (12b) and (12c) are convex, which makes (12) easier to deal with than (11b). Similarly, the functions $e^{t_l}$ and $2^{{\gamma }_e}-1$ on the left-hand sides (LHS) of (13a) and (13c) are also convex, we obtain that (13) is easier to handle than (11c).

Following the idea of [23,44], we apply the SCA method to deal with (12b), (12c), (13a) and (13c). To this end, let us define ${\overline{w}}_m^{(n)}$, ${\overline{u}}_m^{(n)}$, ${\overline{t}}_l^{(n)}$ and ${\overline{\gamma }}_e^{(n)}$ as the variables ${\overline{w}}_m$, ${\overline{u}}_m$, ${\overline{t}}_l$ and ${\overline{\gamma }}_e$ at the n-th iteration for a novel SCA-SPCA-based iterative algorithm given below. By employing first-order Taylor series expansions on $e^{w_m}$, $e^{u_m}$, $e^{t_l}$ and $2^{{\gamma }_e}$ i.e., $e^{{\overline{w}}_m^{(n)}}(w_m-{\overline{w}}_m^{(n)}+1)$, $e^{{\overline{u}}_m^{(n)}}(u_m-{\overline{u}}_m^{(n)}+1)$, $e^{{\overline{t}}_l^{(n)}}(t_l-{\overline{t}}_l^{(n)}+1)$ and $2^{{\overline{\gamma }}_e^{(n)}}\left(({\gamma }_e-{\overline{\gamma }}_e^{(n)})ln2+1\right)$, the linearization of the non-convex constraints in (12b), (12c), (13a) and (13c) are given by

$$\begin{array}{}\mathrm{(14a)}& e^{{\overline{w}}_m^{(n)}}(w_m-{\overline{w}}_m^{(n)}+1)\ge {\displaystyle \sum _{k=1}^K}{\mathbf{w}}_k^H{\mathbf{H}}_m{\mathbf{w}}_k+{\mathbf{z}}^H{\mathbf{H}}_m\mathbf{z}+\sum _{j\ne m}({\mathbf{v}}_j^H{\mathbf{G}}_m{\mathbf{v}}_j)+{\sigma }_m^2,\hfill \mathrm{(14b)}& e^{{\overline{u}}_m^{(n)}}(u_m-{\overline{u}}_m^{(n)}+1)\ge 2^{{\gamma }_c}-1,\hfill \mathrm{(14c)}& e^{{\overline{t}}_l^{(n)}}(t_l-{\overline{t}}_l^{(n)}+1)\ge {\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m,\forall m,\forall l,\hfill \mathrm{(14d)}& 2^{{\overline{\gamma }}_e^{(n)}}\left(({\gamma }_e-{\overline{\gamma }}_e^{(n)})ln2+1\right)-1\ge e^{q_l}.\hfill \end{array}$$

In addition, for the constraint (9b), we will reformulate it by using SPCA method. By introducing two slack variables $s_1>0$ and $s_2>0$, (9b) can be recast as

$$\begin{array}{}\mathrm{(15a)}& log(s_1{s}_2)\ge {\overline{R}}^c,\forall l,\hfill \mathrm{(15b)}& 1+\frac{{\rho }_k{\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{h}}_{c,k}}{{\rho }_k\left({\mathbf{h}}_{c,k}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k}+{\mathbf{g}}_{c,k}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k}+{\sigma }_{a,k}^2\right)+{\delta }_{p,k}^2}\ge s_1,\hfill \mathrm{(15c)}& |\mathbf{I}+{({\sigma }_{e,l}^2\mathbf{I}+{\mathbf{H}}_{e,l}^H(\sum _{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l})}^{-1}{\mathbf{H}}_{e,l}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{H}}_{e,l}|\le \frac{1}{s_2}.\hfill \end{array}$$

By apply the Lemma 1, the constraint (15c) can be recast, respectively, as

$$\begin{array}{c}\hfill 1+\frac{\mathrm{tr}\left({\mathbf{H}}_{e,l}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{H}}_{e,l}\right)}{{\sigma }_{e,l}^2+\mathrm{tr}\left({\mathbf{H}}_{e,l}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l}\right)}\le \frac{1}{s_2}.\end{array}$$

(16)

Then, (15) can be further simplified as

$$\begin{array}{}\mathrm{(17a)}& s_1{s}_2\ge 2^{{\overline{R}}^c},\forall l,\hfill \mathrm{(17b)}& \frac{{\mathbf{h}}_{c,k}^H{\mathbf{w}}_k{\mathbf{w}}_k^H{\mathbf{h}}_{c,k}}{{\mathbf{h}}_{c,k}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{h}}_{c,k}+{\mathbf{g}}_{c,k}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{g}}_{c,k}+{\sigma }_{a,k}^2+\frac{{\delta }_{p,k}^2}{{\rho }_k}}\ge s_1-1,\hfill \mathrm{(17c)}& \frac{{\sigma }_{e,l}^2+\mathrm{tr}\left({\mathbf{H}}_{e,l}^H({\sum }_{j\ne k}{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l}\right)}{{\sigma }_{e,l}^2+\mathrm{tr}\left({\mathbf{H}}_{e,l}^H({\sum }_{j=1}^K{\mathbf{w}}_j{\mathbf{w}}_j^H+\mathbf{z}{\mathbf{z}}^H){\mathbf{H}}_{e,l}+{\mathbf{G}}_{e,l}^H{\sum }_{m=1}^M({\mathbf{v}}_m{\mathbf{v}}_m^H){\mathbf{G}}_{e,l}\right)}\ge s_2.\hfill \end{array}$$

The inequality constraint (17a) is equivalent to $2^{{\overline{R}}^c+2}+{(s_1-s_2)}^2\le {(s_1+s_2)}^2$. Then, we can transform it to a conic quadratic-representable function as (18)

$$\begin{array}{c}\hfill ∥\left[\sqrt{2^{{\overline{R}}^c+2}}{s}_1-s_2\right]∥\le s_1+s_2,\forall l.\end{array}$$

(18)

And then, we transform the inequality constraints (17b) and (17c) into the following form as

$$\begin{array}{}\mathrm{(19a)}& \sum _{j\ne k}{\mathbf{w}}_j^H{\mathbf{H}}_{c,k}{\mathbf{w}}_j+{\mathbf{z}}^H{\mathbf{H}}_{c,k}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\mathbf{G}}_{c,k}{\mathbf{v}}_m+{\sigma }_{a,k}^2+\frac{{\delta }_{p,k}^2}{{\rho }_k}\le \frac{{\mathbf{w}}_k^H{\mathbf{H}}_{c,k}{\mathbf{w}}_k}{s_1-1},\hfill & {\sigma }_{e,l}^2+{\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_j+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m\hfill \mathrm{(19b)}& \le \frac{{\sigma }_{e,l}^2+{\sum }_{j\ne k}{\mathbf{w}}_j^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_j+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+{\sum }_{m=1}^M{\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m}{s_2},\hfill \end{array}$$

It is observed that the constraints (19a) and (19b) are non-convex. Fortunately, the RHS of both constraints is the quadratic-over-linear (QoL) function, which are convex form [43]. According to the idea of the constrained convex procedure [45], these QoL functions can be replaced by their first-order Taylor expansions, which can be converted into a convex programming problem. Specifically, we firstly define a function as

$$\begin{array}{c}\hfill f_{\mathbf{B},b}(\mathbf{w},y)=\frac{{\mathbf{u}}^H\mathbf{B}\mathbf{u}}{y-b},\end{array}$$

(20)

where $y\ge b$ and $\mathbf{B}⪰\mathbf{0}$. Then, the first-order Taylor expansion of (20) at a certain point $(\tilde{\mathbf{u}},\tilde{b})$ is written by

$$\begin{array}{c}\hfill F_{\mathbf{B},b}(\mathbf{u},y,\tilde{\mathbf{u}},\tilde{y})=\frac{2\Re \{{\tilde{\mathbf{u}}}^H\mathbf{B}\mathbf{u}\}}{\tilde{y}-b}-\frac{{\tilde{\mathbf{u}}}^H\mathbf{B}\tilde{\mathbf{u}}}{{(\tilde{y}-b)}^2}(y-b).\end{array}$$

(21)

By utilizing the above results of first-order Taylor expansion, for the certain points $({\mathbf{w}}_k,{\tilde{s}}_1)$, $(\mathbf{z},{\tilde{s}}_2)$ and $({\mathbf{v}}_m,{\tilde{s}}_2)$, we can respectively convert the constraints (19a) and (19b) into the following convex forms as

$$\begin{array}{}\mathrm{(22a)}& \sum _{j\ne k}{\mathbf{w}}_j^H{\mathbf{H}}_{c,k}{\mathbf{w}}_j+{\mathbf{z}}^H{\mathbf{H}}_{c,k}\mathbf{z}+{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m^H{\mathbf{G}}_{c,k}{\mathbf{v}}_m)+{\sigma }_{a,k}^2+\frac{{\delta }_{p,k}^2}{{\rho }_k}\le F_{{\mathbf{H}}_{c,k},1}({\mathbf{w}}_k,s_1,{\tilde{\mathbf{w}}}_k,{\tilde{s}}_1),\hfill & {\sigma }_{e,l}^2+{\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_j+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+{\displaystyle \sum _{m=1}^M}({\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m)\le {\sigma }_{e,l}^2(\frac{2}{{\tilde{s}}_2}-\frac{s_2}{{\tilde{s}}_2^2})\hfill \mathrm{(22b)}& +\sum _{j\ne k}{F}_{{\overline{\mathbf{H}}}_{e,l},0}({\mathbf{w}}_j,s_2,{\tilde{\mathbf{w}}}_j,{\tilde{s}}_2)+F_{{\overline{\mathbf{H}}}_{e,l},0}(\mathbf{z},s_2,\tilde{\mathbf{z}},{\tilde{s}}_2)+{\displaystyle \sum _{m=1}^M}{F}_{{\overline{\mathbf{G}}}_{e,l},0}({\mathbf{v}}_m,s_2,{\tilde{\mathbf{v}}}_m,{\tilde{s}}_2),\forall k.\hfill \end{array}$$

Denoting $g_{s_1,k}=F_{{\mathbf{H}}_{c,k},1}({\mathbf{w}}_k,s_1,{\tilde{\mathbf{w}}}_k,{\tilde{s}}_1)-{\sigma }_{a,k}^2-\frac{{\delta }_{p,k}^2}{{\rho }_k}$ and $g_{s_2,l}={\sigma }_{e,l}^2(\frac{2}{{\tilde{s}}_2}-\frac{s_2}{{\tilde{s}}_2^2})+{\sum }_{j\ne k}{F}_{{\overline{\mathbf{H}}}_{e,l},0}({\mathbf{w}}_j,s_2,{\tilde{\mathbf{w}}}_j,{\tilde{s}}_2)+F_{{\overline{\mathbf{H}}}_{e,l},0}(\mathbf{z},s_2,\tilde{\mathbf{z}},{\tilde{s}}_2)+{\sum }_{m=1}^M{F}_{{\overline{\mathbf{G}}}_{e,l},0}({\mathbf{v}}_m,s_2,{\tilde{\mathbf{v}}}_m,{\tilde{s}}_2)-{\sigma }_{e,l}^2$, (22a) and (22b) can be rewritten as the second-order cone (SOC) constraints as

$$\begin{array}{}& \parallel {[2{\mathbf{w}}_1^H{\mathbf{H}}_{c,k},\dots ,2{\mathbf{w}}_{k-1}^H{\mathbf{H}}_{c,k},2{\mathbf{w}}_{k+1}^H{\mathbf{H}}_{c,k},\dots ,2{\mathbf{w}}_K^H{\mathbf{H}}_{c,k},2{\mathbf{z}}^H{\mathbf{H}}_{c,k},2{\mathbf{v}}_1^H{\mathbf{G}}_{c,k},\dots ,2{\mathbf{v}}_M^H{\mathbf{G}}_{c,k},g_{s_1,k}-1]}^T\parallel \hfill \mathrm{(23a)}& \le g_{s_1,k}+1,\hfill \mathrm{(23b)}& \parallel {[2{\mathbf{w}}_1^H{\overline{\mathbf{H}}}_{e,l},\dots ,2{\mathbf{w}}_K^H{\overline{\mathbf{H}}}_{e,l},2{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l},2{\mathbf{v}}_1^H{\overline{\mathbf{G}}}_{e,l},\dots ,2{\mathbf{v}}_M^H{\overline{\mathbf{G}}}_{e,l},g_{s_2,l}-1]}^T\parallel \le g_{s_2,l}+1.\hfill \end{array}$$

Moreover, the constraints (9c) and (9d) can be reformulated as

$${\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\mathbf{H}}_{c,k}{\mathbf{w}}_j+{\mathbf{z}}^H{\mathbf{H}}_{c,k}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\mathbf{G}}_{c,k}{\mathbf{v}}_m\ge \frac{{\overline{E}}^c}{{\eta }_k(1-{\rho }_k)}-{\sigma }_{a,k}^2.$$

(24)

$${\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_j+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m\ge \frac{{\overline{E}}^e}{{\eta }_l}-N_E{\sigma }_{e,l}^2,\forall l.$$

(25)

Next, we will apply the SPCA method to obtain convex approximations of (24) and (25). By substituting ${\mathbf{w}}_j\triangleq {\tilde{\mathbf{w}}}_j+\Delta {\mathbf{w}}_j$, $\mathbf{z}\triangleq \tilde{\mathbf{z}}+\Delta \mathbf{z}$ and ${\mathbf{v}}_m\triangleq {\tilde{\mathbf{v}}}_m+\Delta {\mathbf{v}}_m$ into the LHS of (24) and (25), we have

$$\begin{array}{cc}& {\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\mathbf{H}}_{c,k}{\mathbf{w}}_j+{\mathbf{z}}^H{\mathbf{H}}_{c,k}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\mathbf{G}}_{c,k}{\mathbf{v}}_m\hfill \\ =& {\displaystyle \sum _{j=1}^K}{({\tilde{\mathbf{w}}}_j+\Delta {\mathbf{w}}_j)}^H{\mathbf{H}}_{c,k}({\tilde{\mathbf{w}}}_j+\Delta {\mathbf{w}}_j)+{(\tilde{\mathbf{z}}+\Delta \mathbf{z})}^H{\mathbf{H}}_{c,k}(\tilde{\mathbf{z}}+\Delta \mathbf{z})+{\displaystyle \sum _{m=1}^M}{({\tilde{\mathbf{v}}}_m+\Delta {\mathbf{v}}_m)}^H{\mathbf{G}}_{c,k}({\tilde{\mathbf{v}}}_m+\Delta {\mathbf{v}}_m)\hfill \\ \ge & {\displaystyle \sum _{j=1}^K}\left({{\tilde{\mathbf{w}}}_j}^H{\mathbf{H}}_{c,k}{\tilde{\mathbf{w}}}_j+2\Re \{{{\tilde{\mathbf{w}}}_j}^H{\mathbf{H}}_{c,k}\Delta {\mathbf{w}}_j\}\right)+{\tilde{\mathbf{z}}}^H{\mathbf{H}}_{c,k}\tilde{\mathbf{z}}+2\Re \{{\tilde{\mathbf{z}}}^H{\mathbf{H}}_{c,k}\Delta \mathbf{z}\}\hfill \\ & +{\displaystyle \sum _{m=1}^M}{\tilde{\mathbf{v}}}_m^H{\mathbf{G}}_{c,k}{\tilde{\mathbf{v}}}_m+2\Re \{{\tilde{\mathbf{v}}}_m^H{\mathbf{G}}_{c,k}\Delta {\mathbf{v}}_m\},\hfill \end{array}$$

(26)

where the inequality is achieved by removing the quadratic terms ${\sum }_{j=1}^K\Delta {\mathbf{w}}_j^H{\mathbf{H}}_k\Delta {\mathbf{w}}_j$, $\Delta {\mathbf{z}}^H{\mathbf{H}}_k\Delta \mathbf{z}$ and ${\sum }_{m=1}^M\Delta {\mathbf{v}}_m^H{\mathbf{G}}_{c,k}\Delta {\mathbf{v}}_m$.

Similarly, for the LHS of (25), it follows

$$\begin{array}{cc}& {\displaystyle \sum _{j=1}^K}{\mathbf{w}}_j^H{\overline{\mathbf{H}}}_{e,l}{\mathbf{w}}_j+{\mathbf{z}}^H{\overline{\mathbf{H}}}_{e,l}\mathbf{z}+{\displaystyle \sum _{m=1}^M}{\mathbf{v}}_m^H{\overline{\mathbf{G}}}_{e,l}{\mathbf{v}}_m\hfill \\ \hfill =& {\displaystyle \sum _{j=1}^K}{({\tilde{\mathbf{w}}}_j+\Delta {\mathbf{w}}_j)}^H{\overline{\mathbf{H}}}_{e,l}({\tilde{\mathbf{w}}}_j+\Delta {\mathbf{w}}_j)+{(\tilde{\mathbf{z}}+\Delta \mathbf{z})}^H{\overline{\mathbf{H}}}_{e,l}(\tilde{\mathbf{z}}+\Delta \mathbf{z})+{\displaystyle \sum _{m=1}^M}{({\tilde{\mathbf{v}}}_{\mathbf{m}}+\Delta {\mathbf{v}}_m)}^H{\overline{\mathbf{G}}}_{e,l}({\tilde{\mathbf{v}}}_m+\Delta {\mathbf{v}}_m)\hfill \\ \hfill \ge & {\displaystyle \sum _{j=1}^K}\left({{\tilde{\mathbf{w}}}_j}^H{\overline{\mathbf{H}}}_{e,l}{\tilde{\mathbf{w}}}_j+2\Re \{{{\tilde{\mathbf{w}}}_j}^H{\overline{\mathbf{H}}}_l\Delta {\mathbf{w}}_j\}\right)+{\tilde{\mathbf{z}}}^H{\overline{\mathbf{H}}}_{e,l}\tilde{\mathbf{z}}+2\Re \{{\tilde{\mathbf{z}}}^H{\overline{\mathbf{H}}}_{e,l}\Delta \mathbf{z}\}\hfill \\ & +{\displaystyle \sum _{m=1}^M}({{\tilde{\mathbf{v}}}_m}^H{\overline{\mathbf{G}}}_{e,l}{\tilde{\mathbf{v}}}_m+2\Re \{{{\tilde{\mathbf{v}}}_m}^H{\overline{\mathbf{G}}}_{e,l}\Delta {\mathbf{v}}_m\}).\hfill \end{array}$$

(27)

Based on (26) and (27), the linear approximations of the concave constraints (24) and (25) can be obtained as

$$\begin{array}{c}{\displaystyle \sum _{j=1}^K}\left({{\tilde{\mathbf{w}}}_j}^H{\mathbf{H}}_{c,k}{\tilde{\mathbf{w}}}_j+2\Re \{{{\tilde{\mathbf{w}}}_j}^H{\mathbf{H}}_{c,k}\Delta {\mathbf{w}}_j\}\right)+{\tilde{\mathbf{z}}}^H{\mathbf{H}}_{c,k}\tilde{\mathbf{z}}+2\Re \{{\tilde{\mathbf{z}}}^H{\mathbf{H}}_{c,k}\Delta \mathbf{z}\}\hfill \\ +{\displaystyle \sum _{m=1}^M}({\tilde{\mathbf{v}}}_m^H{\mathbf{G}}_{c,k}{\tilde{\mathbf{v}}}_m+2\Re \{{{\tilde{\mathbf{v}}}_m}^H{\mathbf{G}}_{c,k}\Delta {\mathbf{v}}_m\})\ge \frac{{\overline{E}}^c}{{\eta }_k(1-{\rho }_k)}-{\sigma }_{a,k}^2,\forall k.\hfill \end{array}$$

(28)

$$\begin{array}{c}{\displaystyle \sum _{j=1}^K}\left({{\tilde{\mathbf{w}}}_j}^H{\overline{\mathbf{H}}}_{e,l}{\tilde{\mathbf{w}}}_j+2\Re \{{{\tilde{\mathbf{w}}}_j}^H{\overline{\mathbf{H}}}_l\Delta {\mathbf{w}}_j\}\right)+{\tilde{\mathbf{z}}}^H{\overline{\mathbf{H}}}_{e,l}\tilde{\mathbf{z}}+2\Re \{{\tilde{\mathbf{z}}}^H{\overline{\mathbf{H}}}_{e,l}\Delta \mathbf{z}\}\hfill \\ +{\displaystyle \sum _{m=1}^M}({{\tilde{\mathbf{v}}}_m}^H{\overline{\mathbf{G}}}_{e,l}{\tilde{\mathbf{v}}}_m+2\Re \{{{\tilde{\mathbf{v}}}_m}^H{\overline{\mathbf{G}}}_{e,l}\Delta {\mathbf{v}}_m\})\ge \frac{{\overline{E}}^e}{{\eta }_l}-N_E{\sigma }_{e,l}^2,\forall l.\hfill \end{array}$$

(29)

Finally, by rearranging (8d) as

$$\begin{array}{c}\parallel [{\mathbf{w}}_1^T,\dots ,{\mathbf{w}}_K^T,{\mathbf{z}}^T,{{\mathbf{v}}_1}^T,\dots ,{\mathbf{v}}_M^T]\parallel \le \sqrt{P},\hfill \\ \parallel [{\mathbf{w}}_1^T,\dots ,{\mathbf{w}}_K^T]\parallel \le \sqrt{P_1},\hfill \\ \parallel [{{\mathbf{v}}_1}^T,\dots ,{\mathbf{v}}_M^T]\parallel \le \sqrt{P_2}.\hfill \end{array}$$

(30)

Eventually, the original MSCM problem (8) is transformed into following convex form as

$$\begin{array}{c}\underset{{\mathbf{w}}_k,\mathbf{z},{\mathbf{v}}_m,{\rho }_k,{\gamma }_c,{\gamma }_e,v_m,w_m,u_m,t_l,e_l,q_l,s_1,s_2}{max}{\gamma }_c-{\gamma }_e\hfill \\ s.t.(12\mathrm{a}),(12\mathrm{d}),(13\mathrm{b}),(13\mathrm{d}),(14\mathrm{a}),(14\mathrm{b}),(14\mathrm{c}),(14\mathrm{d}),(18),(23),(28),(29),(30),\hfill \\ 0<{\rho }_{c,l}\le 1,\forall l,\Delta {\mathbf{w}}_k={\mathbf{w}}_k-{\tilde{\mathbf{w}}}_k,\Delta \mathbf{z}=\mathbf{z}-\tilde{\mathbf{z}},\Delta {\mathbf{v}}_m={\mathbf{v}}_m-{\tilde{\mathbf{v}}}_m.\hfill \end{array}$$

(31)

Given $\Omega \triangleq \{{\tilde{\mathbf{w}}}_k,\tilde{\mathbf{z}},{\tilde{\mathbf{v}}}_m,{\tilde{w}}_m,{\tilde{u}}_m,{\tilde{t}}_l,{\tilde{\gamma }}_e,{\tilde{s}}_1,{\tilde{s}}_2\}$, problem (31) is a SOCP problem [43], which can be efficiently solved by convex optimization software such as CVX tool [46]. According to the SCA method and the SPCA technology, a convex approximation with the optimal solution can be updated iteratively, which implies that problem (8) can be optimally solved. We summarize the proposed SCA-SPCA-based algorithm in Algorithm 1. Note that due to the power constraints (30) at the femtocell sink node and the macrocell sink node, the objective function of the recast problem (31) always has an upper bound. Hence, the proposed SCA-SPCA-based algorithm can be guaranteed to iteratively converge to a global optimal solution, whose the convergence property is proved in the following theorem.

Algorithm 1 Proposed SCA-SPCA-based iterative Algorithm

1.  Given the tolerance of accuracy $\varphi$ (very small value), and the maximum iteration number $N^{max}$. 2.  Set $n=0$ and initialize ${\Omega }^{(0)}=\left\{{\tilde{\mathbf{w}}}_k^{(0)},{\tilde{\mathbf{v}}}_m^{(0)},{\tilde{\mathbf{z}}}^{(0)},{\overline{w}}_m^{(0)},{\overline{u}}_m^{(0)},{\overline{t}}_l^{(0)},{\overline{\gamma }}_e^{(0)},{\tilde{s}}_1^{(n)},{\tilde{s}}_2^{(n)}\right\}$. 3.  For $n=1:N^{max}$ do 4.          Solve problem (31) with ${\Omega }^{(n)}=\left\{{\tilde{\mathbf{w}}}_k^{(n)},{\tilde{\mathbf{v}}}_m^{(n)},{\tilde{\mathbf{z}}}^{(n)},{\overline{w}}_m^{(n)},{\overline{u}}_m^{(n)},{\overline{t}}_l^{(n)},{\overline{\gamma }}_e^{(n)},{\tilde{s}}_1^{(n)},{\tilde{s}}_2^{(n)}\right\}$ and denote the solution $\Omega$ by ${\Omega }^{*}$. 5.            Set ${\Omega }^{(n+1)}={\Omega }^{*}$. 6.            Update the iteration number $n\leftarrow n+1$. 7.              The objective function converges, i.e., $\left|\left({\gamma }_c^{(n+1)}-{\gamma }_e^{(n+1)}\right)-\left({\gamma }_c^{(n)}-{\gamma }_e^{(n)}\right)\right|\le \varphi$, or the maximum number of iterations is reached, i.e., $n\ge N^{max}$. 8.  end for 9.  Obtain${\{{\mathbf{w}}_k\}}_{k=1}^K$, ${\{{\mathbf{v}}_m\}}_{m=1}^M$ and $\mathbf{z}$.

Theorem 1.

Suppose that ${\gamma }_c^{(n)}$ and ${\gamma }_e^{(n)}$ are the optimal solution to problem (31) at the n-th iteration. Denoting $f_n^{*}({\gamma }_c^{(n)},{\gamma }_e^{(n)})$ as the optimal objective function at the q-th iteration, we have $f_{n+1}^{*}({\gamma }_c^{(n+1)},{\gamma }_e^{(n+1)})\ge f_n^{*}({\gamma }_c^{(n)},{\gamma }_e^{(n)})$. Thus, with the increase of n, $\parallel \Delta {\mathbf{w}}_k\parallel$, $\parallel \Delta \mathbf{z}\parallel$ and $\parallel \Delta {\mathbf{v}}_m\parallel$ approach 0. Hence, Algorithm 1 will converge.

Proof. 

See Appendix B. □

4. Simulation Results

In this section, the simulation results are provided to validate the performance of our proposed SCA-SPCA-based iterative algorithm. The system simulation parameters that we have considered for the simulations are summarized in

Table 2. System Simulation Parameters.

Number of M-SN, M	3
Number of HP-SN, K	2
Number of LP-SN, L	2
Number of LP-SN antenna, $N_E$	2
Number of transmit antenna at macrocell sink node, $N_M$	4
Number of transmit antenna at femtocell sink node, $N_F$	16
Distance between the macrocell sink node and the M-SN, $g_m$	10 m
Distance between the macrocell sink node and the HP-SN, $g_{c,k}$	5 m
Distance between the macrocell sink node and the LP-SN, $g_{e,l}$	3 m
Distance between the femtocell sink node and the M-SN, $h_m$	6 m
Distance between the femtocell sink node and the HP-SN, $h_{c,k}$	5 m
Distance between the femtocell sink node and the LP-SN, $h_{e,l}$	2 m
Speed of light, c	$3\times {10}^8\mathrm{m}{\mathrm{s}}^{-1}$
Carrier frequency, $f_c$	900 MHz
Path loss exponent, $\kappa$	$2.7$
Gaussian noise power at all the M-SN, ${\sigma }_m^2$	$-90\mathrm{dBm}$
Gaussian noise power at all the HP-SN, ${\sigma }_{c,k}^2$	$-50\mathrm{dBm}$
Gaussian noise power of all the LP-SNs, ${\sigma }_{e,l}^2$	$-90\mathrm{dBm}$
Additional noise power of all the HP-SNs, ${\delta }_{p,k}^2$	$-90\mathrm{dBm}$
Energy conversion efficiency at all the HP-SN and LP-SN, ${\eta }_{c,k}={\eta }_{e,l}$	0.3

. We assume that one smallcell is in the coverage area of a macrocell, where we set the radius of the macrocell and the smallcell to 20 m and 10 m, respectively [17]. Thus, the channel ${\mathbf{g}}_m$, ${\mathbf{g}}_{c,k}$, ${\mathbf{G}}_{e,l}$${\mathbf{h}}_m$, ${\mathbf{h}}_{c,k}$, and ${\mathbf{H}}_{e,l}$ are modelled as ${\mathbf{g}}_m=H(g_m){\overline{\mathbf{g}}}_I$, ${\mathbf{g}}_{c,k}=H(g_{c,k}){\widehat{\mathbf{g}}}_I$, ${\mathbf{G}}_{e,l}=H(g_{e,l}){\overline{\mathbf{G}}}_I$, ${\mathbf{h}}_m=H(h_m){\overline{\mathbf{h}}}_I$, ${\mathbf{h}}_{c,k}=H(h_{c,k}){\widehat{\mathbf{h}}}_I$, and ${\mathbf{H}}_{e,l}=H(h_{e,l}){\overline{\mathbf{H}}}_I$, where $H(d)=\frac{c}{4\pi f_c}{(\frac{1}{d})}^{\frac{\kappa }{2}}$, ${\overline{\mathbf{g}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, ${\widehat{\mathbf{g}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, ${\overline{\mathbf{G}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, ${\overline{\mathbf{h}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, ${\widehat{\mathbf{h}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, and ${\overline{\mathbf{H}}}_I\sim \mathcal{CN}(0,\mathbf{I})$, respectively. In the simulation results, we will compare the following optimization design schemes: the proposed algorithm, the MSCM scheme with fixed ${\rho }_k=0.5$, the non-AN MSCM scheme which is obtained by setting $\mathbf{z}=\mathbf{0}$, the null-AN MSCM scheme which means the AN vector lie in the null space of the HP-SNs.

In Figure 3, we illustrate the convergence performance of the proposed SCA-SPCA-based iterative algorithm with respect to (w.r.t.) iteration numbers. We set ${\overline{R}}^c=1$ bps/Hz, ${\overline{E}}^c=10$ dBm, ${\overline{E}}^e=10$ dBm, $P=60$ dBm and $P_1=40$ dBm. It is easily seen from Figure 3 that the convergence of all cases can be quickly achieved within three iterations. It is observed that the average minimal secrecy capacity (MSC) at the M-SN becomes great as $P_2$ increases.

Figure 4 illustrates the average MSC at the M-SN in terms of different target transmit power at macrocell sink node with ${\overline{R}}^c=1.5$ bps/Hz, ${\overline{E}}^c=5$ dBm, ${\overline{E}}^e=5$ dBm, $P=60$ dBm and $P_1=40$ dBm over 500 feasible channel realizations. It is observed that the average MSC increases with the target transmit power at macrocell sink node. In addition, the proposed SCA-SPCA-based algorithm outperforms all the conventional baseline schemes. Moreover, we can check that the proposed SCA-SPCA-based algorithm performs better than the non-AN MSCM, null- AN MSCM and the MSCM scheme with ${\rho }_k=0.5$, and the performance gap is always gradually increasing regardless of the target transmit power at macrocell sink node. This reveals that optimizing the PS ratio ${\rho }_k$ is important.

In Figure 5, we compare the average MSC at the M-SN w.r.t. the different target-harvested power at HP-SN by fixing ${\overline{R}}^c=1$ bps/Hz, ${\overline{E}}^e=10$ dBm, $P=70$ dBm, $P_1=50$ dBm and $P_2=60$ dBm. In this figure, we can observe that all schemes decrease with the increase of ${\overline{E}}^c$. Moreover, the proposed SCA-SPCA-based algorithm always perform better in average MSC than all three conventional baseline schemes. Compared with the non-AN MSCM and the MSCM scheme with ${\rho }_k=0.5$, the averageMSC of the proposed AN-aidedalgorithm is 1.2 bps/Hz and 0.8 bps/Hz higher at 0 dBm. It is observed that the proposed algorithm outperforms the null-AN MSCM.

In Figure 6, we plot the average MSC at the M-SN in terms of different number of transmit antennas at the macrocell sink node with ${\overline{R}}^c=0.5$ bps/Hz, ${\overline{E}}^c=5$ dBm, ${\overline{E}}^e=10$ dBm, $P=60$ dBm, $P_1=40$ dBm and $P_2=60$ dBm. We can check that the curves of the proposed SCA-SPCA-based algorithm and the MSCM scheme with fixed ${\rho }_k=0.5$ increase with the same slope. Moveover, when $N_M$ increases, the performance gap between the non-AN MSCM scheme and the proposed algorithm becomes wider. This indicates that the AN technology is essential in obtaining the performance gains. Furthermore, the proposed algorithm outperforms the null-AN MSCM scheme by 0.8 bps/Hz.

In Figure 7, in terms of the average central processing unit (CPU) running time, the computational complexities of the proposed method and the conventional one-dimensional line search (1-D) method are compared for a channel realization under various MF on a computer (The CPU is Intel Core i7-8500U 1.8GHz, and the size of random access memory (RAM) is 8 GB). In this simulation, we set ${\overline{R}}^c=1$ bps/Hz, ${\overline{E}}^c=50$ dBm, ${\overline{E}}^e=0$ dBm, $P=80$ dBm, $P_1=40$ dBm and $P_2=60$ dBm. We can see that our proposed algorithm has much lower complexity than the conventional 1-D method in [29].

5. Conclusions

Considering the security transmission with PS-enabled SWIPT approach, we have investigated the AN-based beamforming design problem in a two-tier MIMO HWSN where a femtocell is underlaying a macrocell. The jointly design problem of the transmit beamforming vectors, the AN and PS ratio has been modeled to maximize the MSC of the M-SNs subject to the secrecy capacity constraint, the harvested energy constraint and the transmit power constraint. To reduce the performance loss from the rank relaxation, an novel SCA-SPCA-based iterative algorithm has been proposed and its convergence property is proved. Numerical results have demonstrated that the proposed SCA-SPCA-based algorithm always outperforms the conventional non-AN, null-AN and “fixed PS ratio” schemes. As a future work, the proposed algorithm can be extended to a multi femtocell scenario in a macrocell. In that case, interference between femtocells needs to be considered. In addition, the dependency of non-linear power-splitting EH model and EH efficiency on the received user characteristic can be an extension of this work.