Secure Green-Oriented Multiuser Scheduling for Wireless-Powered Internet of Things

Abstract

In this paper, we consider the secure green-oriented multiuser scheduling for the wireless-powered Internet of Things (IoT) scenario, in which multiple source sensors communicate with a controller assisted by an intermediate sensor with the existence of a passive tapping device. Due to the limited energy, all sensors must acquire energy from external power beacons (PBs). Specifically, for the security improvement, we introduce two multiuser scheduling schemes possessing the optimal PB chosen by the relay, i.e., the best source sensor is scheduled in a random way (BSR), while the best source sensor is decided by the best PB (BSBP). Furthermore, for every scheme, we derive the analytical expressions for the secrecy outage probability (SOP) and investigate the secure energy efficiency (SEE) optimization problem with constricted transmission power in PBs. Simulation results reveal that the BSBP scheme provides better secrecy performance, and elevating the PBs quantity or reducing both the ratio of distance from PBs to source users and the total communication distance to some extent is helpful for improving SEE. In addition, the time-switching factor shows an important effect upon secrecy performance of the considered system.

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1. Introduction

As the key architecture of the fifth generation (5G) mobile communication system, the Internet of Things (IoT) has attracted more and more attention [1]. The primary driving thought for the prospective IoT has relation with smart sensors, and wireless sensor networks (WSNs) have been observed as key enablers of IoT applications recently, where multiple sensor nodes are caused by instant or periodic data acquisition in multiple environments [2]. However, among various types of IoT sensors, most of the objects are resource-constrained, battery-powered, and characterized by both of the low energy and poor computation capacity, resulting in the one of the biggest barrier impeding generalization of IoT in the future which is the limitation of energy [3]. Meanwhile, the rapid development of IoT will give rise to the massive deployment of sensor nodes and a vast amount of information exchange, making it unfeasible to flexibly recharge and control the power-constricted devices. Due to the rising power costs, green-oriented methods have inevitably become the dominating design consideration in the IoT system. Fortunately, far-field wireless energy transfer (WET), which enables wireless devices to acquire energy from the broadcast signals for their operation and information exchange, has considered as an appealing method to provide consistent and stable energy to power-constrained users of IoT, particularly when traditional energy harvesting (EH) techniques from reproducible energy sources are inapplicable [4]. Such a new communication type is known as wireless-powered communication (WPC), where the wireless users of IoT are powered through surrounding electromagnetic signals, which is likely to be dedicated wireless transmitters utilized to charge wireless terminals. As such, it can completely eliminate the burden of battery renewal and/or recharging, avoid the interruption caused by the running out of power, and provide networks with theoretically perpetual lifespans [5].

Generally, a basic hurdle of wireless transmission upon the basis of WPC rest with the fast attenuation of radio frequency (RF) signals with distance changing. Moreover, energy amount acquired at wireless-powered devices is fairly restricted, which limits the coverage severely and has become a bottleneck for the widespread application of WPC. Consequently, the design of energy-efficient transmission is of great significance for transforming the green concept into future wireless-powered IoT. Although the harvested energy from RF signals appears to be inefficient and the WPC-based wireless communication seems to be not widely available at the current stage, practical applications for power-constrained devices of IoT have already been investigated, and more feasible applications will be implemented in the near future [6–8]. Until now, a lot of research efforts have also been devoted to explore the advantages of WPC in most applications of IoT since it could be easily combined with wireless terminals [9–13]. Specifically, typical wireless-powered communication networks (WPCN) were investigated in [9], where introduction of a dynamic time division multiple access (TDMA) protocol was performed for a multiuser WPCN, and the best time allocation for the downlink WPC and the uplink wireless information transmission (WIT) had been analyzed. Until now, the studies on WPCN have been extended to various communication systems. In particular, a joint wireless power transfer (WPT) and relay selection method were proposed for WPCN in [10], where the source and relays utilize the time-switching-based RF-EH method to acquire energy from a power beacon (PB) with multiple antennas. Differently, secrecy performance discussion of EH wireless sensor networks was examined in [11], where the authors proposed an optimization method, in which a wireless-powered friendly jammer was utilized for improving the secrecy performance of the considered model. Furthermore, the authors in [12] analyzed the secrecy performance of the EH sensor system and proposed a best-relay-and-best-jammer protocol for enhancing the secrecy of a system with the source user and multiple sensor relays harvesting energy from diverse PBs. Recently, inspired by abovementioned works, a secure energy-efficient transmission design for wireless-powered IoT possessing various PBs was investigated in [13], where the authors introduce different relay selection schemes possessing the optimal PB under the selection of the single source and solve the optimization problem of secure energy efficiency (SEE). It is worth mentioning that above works mostly focused on the secrecy performance improvement contributed by relay selection, while ignored the practical scenario with multiusers, which can be considered as the typical application of IoT.

1.1. Related Works

Remarkably, in an effort to reduce the complexity and the costs of resource-constricted wireless-powered IoT, the multiuser scheduling has drawn wide attention due to the significant potential performance improvement [14–19]. Particularly, the authors in [14] explore the largest energy efficiency for multiuser WPCN by joint power control and time allocation while taking account of the initial battery energy level of all the users. Furthermore, the authors in [15] applied the notion of proportional fair scheduling to WPCN and overcome the doubly near fair problem by an opportunistic scheme. Meanwhile, in a multiuser system, the authors in [16] consider multiuser scheduling criteria taking into account cochannel interference in a multicell environment and propose an adaptive scheduling scheme. Then, in a WPCN-based multi-input multioutput (MU-MIMO) multiuser system, [17] proposes a zero-forcing-based transmission method by a downlink energy beamformer to optimize the related parameters. Afterwards, the authors in [18] concentrate on an MIMO-WPCN, in which a dedicated multiantenna PB transfers RF energy to some starve users, and then these users send required information to the destination. And, last but not the least, [19] focuses on the cumulative transmission framework of multiuser scheduling in full-duplex wireless-powered IoT system and designs a novel throughput-oriented scheduling strategy, which models the dynamic charging and discharging procedures for all the devices in IoT as a finite-state Markov chain.

On the contrary, the broadcast characteristics of wireless transmission make the WPT and WIT more vulnerable to malicious attacks, turning secure transmission into a burdensome task [20]. Typically, the upper layer cryptographic techniques are explored for securing the privacy information against intercepting in traditional wireless transmissions [21]. However, conventional cryptographic technology is limited in wireless-powered IoT because of the requirements of high hardware complicacy and massive energy [22]. Furthermore, an eavesdropper (Eve) possessing unlimited computing capacity is likely to make such technology compromised [23]. Fortunately, physical layer security (PLS), as a promising approach to make sure the safety of wireless communication, has been a high-effectiveness supplement to current solutions. As far as wireless PLS is concerned, the fundamental thought is to utilize the features of wireless channels for transmitting information in a reliable manner from the source to the intended receiver and to make sure the privacy of the information, to put in different way, not to be intercepted or eavesdropped [24]. Currently, PLS has been broadly deployed for ensuring security of future wireless networks [25, 26] because it can provide the security of new network architectures such as the IoT. In recent years, some literatures have explored secure communications in multiuser-scheduling-aided WPCN [27, 28]. Specifically, [27] investigates the secrecy performance of dual-hop multiuser relay system by exploiting the maximal ratio transmission (MRT) scheme and proposes a threshold-based multiuser scheduling method. On the contrary, the authors in [28] consider a multiantenna wireless network, in which the base station and the users have been given with multiple antennas. However, it is worth noting that few works have considered the SEE in wireless-powered IoT scenario.

1.2. Paper Contributions and Organization

Enlightened by the aforementioned observations, this paper focuses on the secrecy performance analysis of a typical wireless-powered IoT, in which multiple source sensors that perform monitoring or operating tasks in a localized group and an intermediate node performs as relay are powered by multiple dedicated PBs. Furthermore, we propose two multiuser scheduling schemes and compare their secrecy performance for providing the secure energy-efficient transmissions. The main contributions of our work are listed as follows:

(i)

We explore the PLS in the wireless-powered IoT and propose two green-oriented multiuser scheduling schemes, in which the optimal PB is decided by the relay; meanwhile, the best source is scheduled in a random way (BSR) or chosen by the best PB (BSBP), respectively.

(ii)

For the two proposed schemes, we obtain the closed-form expressions of the secrecy outage probability (SOP) and solve the SEE optimization problem with constrained transmission power in PB by resorting to the searching method. Compared with the BSR scheme characterized by lower complexity and simpler application, the BSBP scheme can make full use of the power transfer links contributed by diverse PBs and present the better secrecy performance.

(iii)

Simulation results demonstrate that increasing the number of PBs is favorable to improve the SEE of the studied scenario. Meanwhile, decreasing both the ratio of distance from PBs to the sources and the total communication distance to some extent is advantageous for SEE. Additionally, the time-switching factor has an important effect upon the secrecy performance of the considered system, which is worth designing and optimizing carefully.

The remaining part of this paper is summarized as follows. Section 2 provides details concerning the system model, the process of WPT, signal analysis, and two proposed multiuser scheduling schemes. Section 3 derives the exact SOP of BSR and BSBP schemes, respectively. Then, in Section 4, the SEE optimization research is given. The simulation results are presented in Section 5. Eventually, conclusions of this paper are drawn in Section 6.

2. System and Channel Model

The system model, the process of WPT, signal analysis, and two multiuser scheduling strategies are presented in this section.

2.1. System Model

Consider a dual-hop multiuser uplink WPCN for the IoT application as illustrated in Figure 1, where multiple source users $S_n$, $n\in \mathcal{M}=\left\{1,\dots ,N\right\}$ communicate with the controller D aided by the decode-and-forward (DF) relay, and are overheard by the passive Eve E. Considering the constricted coverage of sensors, we assume the direct $S_n⟶D$ link is not available [10]. Meanwhile, owing to the limitation of energy in the IoT system, $S_n$ and R have to gain energy by WPT from a selected $P_m$, $m\in ℳ=\left\{1,\dots ,M\right\}$ to support data transmission. And, the controller is powered by on-grid power. Apart from that, considering the size and cost limitations, it is assumed that, in each sensor, the destination D and the Eve E are single-antenna and half-duplex devices [12]. It is worth highlighting that the above configurations have numerous practical applications, such as in IoT, where the multiple source sensors upload information via a certain sensor (performs as relay) that is limited to a single antenna due to size limitations and cost.

[figure omitted; refer to PDF]

Furthermore, we consider that each link will be affected by Rayleigh fading. Thus, the power gains of the channel are subject to exponential distribution with parameter ${\lambda }_{XY}$, where $X\in \left\{P_m,S_n,R\right\}$ and $Y\in \left\{S_n,R,E,D\right\}$. Meanwhile, the additive white Gaussian noise (AWGN) at R and D has zero mean and variance $N_0$. Compared with the full channel state information (CSI) assumption in [12, 27], where perfect CSI is considered in order to investigate the performance bound, we consider only the statistic CSI can be acquired, which is more practical due to the weak computation ability and small memory of sensors. The specific estimation method for obtaining CSI is shown in [29]. In practice, when the eavesdropper is a member of the network and wants to interpret information that is not passed on to him, the partial CSI of the wiretap link is available.

In order to facilitate mathematical modeling, the channel coefficients of the $P_m⟶S_n$, $P_m⟶R$, $S_n⟶R$, $S_n⟶E$, $R⟶E$, and $R⟶D$ transmission channels are represented by $h_{P_m{S}_n}$, $h_{P_{m}R}$, $h_{S_{n}R}$, $h_{S_{n}E}$, $h_{RE}$, and $h_{RD}$, respectively, which are independent and distributed in an identical manner (i.i.d.) from one block to next. Meanwhile, the distances of the $P_m⟶S_n$, $P_m⟶R$, $S_n⟶R$, $S_n⟶E$, $R⟶E$, and $R⟶D$ transmission links are represented by $d_{P_m{S}_n}$, $d_{P_{m}R}$, $d_{S_{n}R}$, $d_{S_{n}E}$, $d_{RE}$, and $d_{RD}$, respectively. Furthermore, it is assumed that the multiple PBs and multiple source users are close in proximity, i.e., a certain clustering protocol in place. This assumption is generally used in the WPCN [12], which brings about the equivalent mean channel power gains of the channels $P_m⟶S_n$, $P_m⟶R$, $S_n⟶R$, and $S_n⟶E$, respectively. For convenience, we define ${\lambda }_{P_m{S}_n}={\lambda }_{PS}$, ${\lambda }_{P_{m}R}={\lambda }_{PR}$, ${\lambda }_{S_{n}R}={\lambda }_{SR}$, and ${\lambda }_{S_{n}E}={\lambda }_{SE}$ for any $m\in ℳ$ and $n\in \mathcal{N}$.

2.2. Wireless Power Transfer

In the WPT process, the receiver adopts the EH model based on rectangular antenna structure. For the rectangular antenna, the received signal can be converted into a direct current (DC) signal by a rectifier consisting of a passive low-pass filter (LPF) and a Schottky diode [30]. Then, it is considered that the harvested energy by all users at the stage of WPT is entirely used to transmit message in WIT, which is known as the harvest-use (HU) mode as in [31]. This consideration is practical for devices of IoT because they are limited by the size and cost, which leads to the smaller batteries. As for the relaying strategy, the time-switching-based receiver (TSR) protocol is applied in the data transmission, thanks to its high throughput compared with power splitting-based receiver (PSR) protocol [32]. In particular, the duration between two successive data transmission is defined as a transfer time slot T, $\alpha T$ denotes the time of WPT, and $\left(1-\alpha \right)T$ represents the duration of WIT as illustrated in Figure 2, where $\alpha \in \left(\mathrm{0,1}\right)$ is the time-switching coefficient. Based on the dual-hop communication, the time window of WIT is divided into two parts, i.e., $\left(1-\alpha \right)T/2$ is for $S_n⟶R$ and the remaining $\left(1-\alpha \right)T/2$ is provided for $R⟶D$.

[figure omitted; refer to PDF]

Then, we consider the scenario that a particular PB is activated, while other PBs remain silent in order to reduce the computational complexity and energy consumption, because PB selection is an green-oriented WPT scheme [12]. To be more specific, the PB with the best link for the $P_m⟶R$ channel is chosen to implement WPT for $S_n$ and R. The index of certain PB can be expressed as$$\begin{array}{}\text{(1)}& m^{\ast }=\arg \underset{m\in ℳ}{\max }\left\{{\left|h_{P_{m}R}\right|}^2\right\},\end{array}$$where ${\left|h_{P_{m}R}\right|}^2$ is power gain of the link from the selected $P_m$ to R. Furthermore, the harvested energy at $S_n$ and R can be expressed as follows [33]:$$\begin{array}{c}\text{(2)}& E_{S_n}=\eta P_B\alpha T\frac{{\left|h_{P_{m^{\ast }}{S}_n}\right|}^2}{d_{P_{m^{\ast }}{S}_n}^{\theta }}=\eta P_B\alpha T{\gamma }_{P_{m^{\ast }}{S}_n},E_R=\eta P_B\alpha T\frac{{\left|h_{P_{m^{\ast }}R}\right|}^2}{d_{P_{m^{\ast }}R}^{\theta }}=\eta P_B\alpha T{\gamma }_{P_{m^{\ast }}R},\end{array}$$where $10\%<\eta <80\%$ denotes the EH efficiency factor, which is mainly determined by the EH circuitry and frequencies (e.g., 15 MHz–2.5 GHz) [34]; $P_B$ represents the transmit power of the PBs; θ is the path loss exponent; ${\left|h_{P_{m^{\ast }}{S}_n}\right|}^2$ and ${\left|h_{P_{m^{\ast }}R}\right|}^2$ are power gains of the channels from the selected $P_m$ to $S_n$ and R, respectively; ${\gamma }_{P_{m^{\ast }}{S}_n}={\left|h_{P_{m^{\ast }}{S}_n}\right|}^2/d_{P_{m^{\ast }}{S}_n}^{\theta }$ and ${\gamma }_{P_{m^{\ast }}R}={\left|h_{P_{m^{\ast }}R}\right|}^2/d_{P_{m^{\ast }}R}^{\theta }$. Note that we neglect the harvested energy from the noise since the source users and the relay sensor are passive, and their received noise powers are much smaller than the received powers contributed by the PBs [35].

Lemma 1.

If $X_k$, $k\in \mathcal{K}=\left\{1,\dots ,K\right\}$, is the random variable subject to the exponential i.i.d, the probability density function (PDF) and the cumulative distribution function (CDF) of $X=\underset{k\in \mathcal{K}}{\max }\left\{X_k\right\}$ can be expressed by$$\begin{array}{}\text{(3)}& f_X\left(x\right)=K{\lambda }_X{e}^{-{\lambda }_{X}x}{\left(1-e^{-{\lambda }_{X}x}\right)}^{K-1},\text{(4)}& F_X\left(x\right)={\left(1-e^{-{\lambda }_{X}x}\right)}^K,\end{array}$$in which x denotes the independent variable, and $1/{\lambda }_X$ is the average channel gain.

According to Lemma 1, the PDF of ${\gamma }_{P_{m^{\ast }}R}$ can be derived as$$\begin{array}{}\text{(5)}& f_{{\gamma }_{P_{m^{\ast }}R}}\left(x\right)=M{\lambda }_{PR}{e}^{-x{\lambda }_{PR}}{\left(1-e^{-x{\lambda }_{PR}}\right)}^{M-1},\end{array}$$where $1/{\lambda }_{PR}=E\left[{\left|h_{P_{m^{\ast }}R}\right|}^2\right]/d_{P_{m^{\ast }}R}^{\theta }=E\left[{\gamma }_{P_{m^{\ast }}R}\right]$ and $E\left[\cdot \right]$ is an expectation operator.

Assuming that the channel fading factors are still unchanged within a transfer time slot, the transmit power of $S_n$ and R are represented as follows [36]:$$\begin{array}{}\text{(6)}& P_{S_n}=\frac{E_{S_n}}{\left(1-\alpha \right)T/2}=\frac{2\eta P_B{\gamma }_{P_{m^{\ast }}{S}_n}\alpha }{1-\alpha },\text{(7)}& P_R=\frac{E_R}{\left(1-\alpha \right)T/2}=\frac{2\eta P_B{\gamma }_{P_{m^{\ast }}R}\alpha }{1-\alpha }.\end{array}$$

From (6) and (7), it is assumed that the SNRs at R and E in the first hop are denoted as ${\gamma }_{SR}$ and ${\gamma }_{SE}$, while the SNRs at D and E in the latter hop are represented as ${\gamma }_{RD}$ and ${\gamma }_{RE}$. And the SNRs can be given as$$\begin{array}{c}\text{(8)}& {\gamma }_{SR}=\frac{P_{S_n}{\left|h_{S_{n^{\ast }}R}\right|}^2}{N_0{d}_{S_{n^{\ast }}R}^{\theta }}=\frac{2\eta \alpha P_B{\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\left|h_{S_{n^{\ast }}R}\right|}^2}{N_0\left(1-\alpha \right)d_{S_{n^{\ast }}R}^{\theta }}\\ ={\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R},\end{array}$$where $n^{\ast }$ is the index of the chosen source sensor (i.e., scheduled user), ${\gamma }_B=P_B/N_0$, $\xi =2\eta \alpha /1-\alpha$, and ${\chi }_{S_{n^{\ast }}R}={\left|h_{S_{n^{\ast }}R}\right|}^2/d_{S_{n^{\ast }}R}^{\theta }$. Comparably, ${\gamma }_{SE}$, ${\gamma }_{RD}$, and ${\gamma }_{RE}$ are given as $$\begin{array}{c}\text{(9)}& {\gamma }_{SE}={\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E},{\gamma }_{RD}={\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD},\\ {\gamma }_{RE}={\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE},\end{array}$$where ${\gamma }_E=P_B/N_E$, $N_E$ denotes the variance of AWGN at E, and ${\chi }_{S_{n^{\ast }}E}={\left|h_{S_{n^{\ast }}E}\right|}^2/d_{S_{n^{\ast }}E}^{\theta }$, ${\chi }_{RD}={\left|h_{RD}\right|}^2/d_{RD}^{\theta }$, and ${\chi }_{RE}={\left|h_{RE}\right|}^2/d_{RE}^{\theta }$.

2.3. Multiuser Scheduling Scheme

Then, when the best PB is determined, we pay attention to the two multiuser scheduling schemes, one is a straightforward scheme with the lower complexity, in which the best source is scheduled randomly from multiple users (BSR), and another is a joint PB and source user selection scheme, in which the best source is chosen by the optimal PB (BSBP).

2.3.1. The BSR Scheme

For reduction of the complexity and costs of considered networks, the source user is scheduled randomly among the candidates in the BSR scheme. It is worth noting that ${\gamma }_{P_{m^{\ast }}{S}_n}$, ${\chi }_{S_{n}R}$, ${\chi }_{S_{n}E}$, ${\chi }_{RD}$, and ${\chi }_{RE}$ are exponentially distributed with parameters $1/{\lambda }_{PS}$, $1/{\lambda }_{SR}$, $1/{\lambda }_{SE}$, $1/{\lambda }_{RD}$, and $1/{\lambda }_{RE}$, respectively.

Remark 1.

The BSR scheme is applicable for the delay-sensitive and resource-constricted scenarios owing to its lower complexity of computation. However, the BSR scheme fails to have the diversity gain contributed by diverse users, which hinders the secrecy improvement of communication systems.

2.3.2. The BSBP Scheme

Aiming to obtain the diversity gain, the best user is chosen from the perspective of $P_{m^{\ast }}$ in this scheme. Specifically, based on the CSI of the $P_{m^{\ast }}⟶S_n$ channels, the index of the scheduled source $n^{\ast }$ can be drawn as$$\begin{array}{}\text{(10)}& n^{\ast }=\arg \underset{n\in \mathcal{N}}{\max }\left({\left|h_{P_{m^{\ast }}{S}_n}\right|}^2\right).\end{array}$$

Therefore, ${\chi }_{S_{n^{\ast }}R}$, ${\chi }_{S_{n^{\ast }}E}$, ${\chi }_{RD}$, and ${\chi }_{RE}$ are exponentially distributed with parameters $1/{\lambda }_{SR}$, $1/{\lambda }_{SE}$, $1/{\lambda }_{RD}$, and $1/{\lambda }_{RE}$, respectively, while the PDF of ${\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}$ can be drawn according to Lemma 1:$$\begin{array}{}\text{(11)}& f_{{\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}}\left(x\right)=N{\lambda }_{PS}{e}^{-x{\lambda }_{PS}}{\left(1-e^{-x{\lambda }_{PS}}\right)}^{N-1},\end{array}$$where $1/{\lambda }_{PS}=E\left[{\left|h_{P_{m^{\ast }}{S}_{n^{\ast }}}\right|}^2\right]/d_{P_{m^{\ast }}{S}_{n^{\ast }}}^{\theta }=E\left[{\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}\right]$.

Remark 2.

According to the (10) and (11), we find that the BSBP scheme is able to obtain the diversity gain contributed by multiple users. Note that this scheme schedules the source user, taking into account the selection of PB adequately. As a result, this scheme can decline the interruption probability of information communication for source sensors in an effective way, which is beneficial to energy-limited IoT applications.

3. Secrecy Outage Probability Analysis

In the system, we consider that $S_n$ and R use the different codebooks to improve secrecy performance. In line with [37], the secrecy capacity of the scenario can be indicated as$$\begin{array}{}\text{(12)}& C_s=\min \left(C_{s1},C_{s2}\right),\end{array}$$where $C_{s1}$ and $C_{s2}$ denote the achievable secrecy capacity of the dual-hop, respectively, which is presented to be$$\begin{array}{c}\text{(13)}& C_{s1}=\epsilon {\left[{\log }_2\left(\frac{1+{\gamma }_{SR}}{1+{\gamma }_{SE}}\right)\right]}^{+},C_{s2}=\epsilon {\left[{\log }_2\left(\frac{1+{\gamma }_{RD}}{1+{\gamma }_{RE}}\right)\right]}^{+},\end{array}$$where the reason why the factor $\epsilon =\left(1-\alpha \right)/2$ is that, the communication time of every hop is $\left(1-\alpha \right)T/2$ during a transmission slot, ${\left[x\right]}^{+}=\max \left(x,0\right)$. Therefore, the secrecy capacity $C_s$ can be updated as$$\begin{array}{}\text{(14)}& C_s=\epsilon \left[{\log }_2\min \left(\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}},\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}\right)\right].\end{array}$$

Regarding evaluation of the secrecy performance, the SOP is used as the figure of merit, which is regarded as an important indicator of PLS generally. From the perspective of information theory, the transmission incurs secrecy outage if $C_s$ is less than a predetermined secrecy rate threshold $R_{\text{th}}$. Specifically, the SOP of every scheme $P_{\text{sop}}^{\left(\text{sch}\right)}$ can be shown as $$\begin{array}{}\text{(15)}& P_{\text{sop}}^{\left(\text{sch}\right)}=\mathrm{Pr}\left(C_s^{\left(\text{sch}\right)}<R_{\text{th}}\right)=\mathrm{Pr}\left({\gamma }_{\text{sec}}^{\left(\text{sch}\right)}<\beta \right),\end{array}$$where $\text{sch}\in \left\{\text{BSR},\text{BSBP}\right\}$, $C_s^{\left(\text{sch}\right)}$ is the secrecy capacity of each scheme, and $\mathrm{Pr}\left\{\cdot \right\}$ is the probability.$$\begin{array}{c}\text{(16)}& {\gamma }_{\sec }^{\left(\text{sch}\right)}=\min \left(\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}},\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}\right),\beta =2^{R_{\text{th}}/\epsilon }.\end{array}$$

3.1. Derivation for the BSR Scheme

According to the BSR scheme, each source user of the system has the same opportunity to participate in the transmission. Therefore, the exact SOP of the BSR scheme should be formulated as$$\begin{array}{}\text{(17)}& P_{\text{sop}}^{\left(\text{BSR}\right)}=\frac{1}{N}\sum _{n=1}^N{P}_{sop,S_{n}R}=1-\sum _{m=1}^M\left(\begin{array}{c}M\\ m\end{array}\right){\left(-1\right)}^{m+1}\times \frac{4{\gamma }_B\left(\beta -1\right){\lambda }_{SE}{\lambda }_{RE}\sqrt{m{\lambda }_{PS}{\lambda }_{PR}{\lambda }_{SR}{\lambda }_{RD}}}{\xi \left({\gamma }_B{\lambda }_{SE}+\beta {\gamma }_E{\lambda }_{SR}\right)\left({\gamma }_B{\lambda }_{RE}+\beta {\gamma }_E{\lambda }_{RD}\right)}\times K_1\left(2\sqrt{\frac{m{\lambda }_{PR}{\lambda }_{RD}\left(\beta -1\right)}{{\gamma }_B\xi }}\right)\times K_1\left(2\sqrt{\frac{{\lambda }_{SR}{\lambda }_{PS}\left(\beta -1\right)}{{\gamma }_B\xi }}\right),\end{array}$$where $P_{\text{sop},S_{n}R}$ is the SOP when the source user $S_n$ is decided and $K_1\left(\cdot \right)$ represents the modified Bessel function of the second kind [38].

Proof.

See Appendix A.

3.2. Derivation for the BSBP Scheme

Furthermore, according to (16), the closed-form expression of SOP for BSBP scheme is calculated as$$\begin{array}{}\text{(18)}& P_{\text{sop}}^{\left(\text{BSBP}\right)}=1-\sum _{n=1}^N\sum _{m=1}^M\left(\begin{array}{c}N\\ n\end{array}\right)\left(\begin{array}{c}M\\ m\end{array}\right){\left(-1\right)}^{m+n}\times \frac{4{\gamma }_B\left(\beta -1\right){\lambda }_{SE}{\lambda }_{RE}\sqrt{mn{\lambda }_{PS}{\lambda }_{PR}{\lambda }_{SR}{\lambda }_{RD}}}{\xi \left({\gamma }_B{\lambda }_{SE}+\beta {\gamma }_E{\lambda }_{SR}\right)\left({\gamma }_B{\lambda }_{RE}+\beta {\gamma }_E{\lambda }_{RD}\right)}\times K_1\left(2\sqrt{\frac{m{\lambda }_{PR}{\lambda }_{RD}\left(\beta -1\right)}{{\gamma }_B\xi }}\right)\times K_1\left(2\sqrt{\frac{n{\lambda }_{SR}{\lambda }_{PS}\left(\beta -1\right)}{{\gamma }_B\xi }}\right).\end{array}$$

Proof.

See Appendix B.

4. Secure Energy Efficiency Maximization

Generally, security improvement comes at the expense of more energy consumption frequently. With regard to energy-limited IoT applications, recklessly pursuing secrecy improvement has a negative effect on the performance of networks. Consequently, it is of great significance to make sure the safe communication in the application of IoT with low energy cost. From the above, the SEE is utilized as the proper metric for evaluation of the secrecy performance [39]. Mathematically, the SEE of above discussed schemes can be given as$$\begin{array}{}\text{(19)}& {\eta }_s^{\left(\text{sch}\right)}=\frac{R_{\text{th}}\left(1-P_{\text{sop}}^{\left(\text{sch}\right)}\right)}{P_{\text{total}}},\end{array}$$where ${\eta }_s^{\left(\text{sch}\right)}$ denotes the SEE of each scheme, $P_{\text{total}}=\kappa P_B+P_c$ represents the total power cost at PBs, κ denotes the power factor, and $P_c$ and $P_B$ stand for the fixed power and transmit power at PBs, respectively. Consider that the harvested energy by the sensors is fully used for data transmission, while the power consumption of the circuity is ignored.

To find the best transmit power of PBs and the time-switching factor, the SEE maximization problem can be considered as$$\begin{array}{}\text{(20)}& \begin{array}{cc}\underset{P_B,\alpha }{\max }& {\eta }_s^{\left(\text{sch}\right)}=\frac{R_{\text{th}}\left(1-P_{\text{sop}}^{\left(\text{sch}\right)}\right)}{P_{\text{total}}}\\ \text{s.t.}& 0<P_B\le P_{\max }\\ & 0<\alpha <1,\end{array}\end{array}$$where $P_{\max }$ denotes the maximum transmit power of PBs. Obviously, the solving process of the exact expressions for $P_B$ and α is rather tedious. Instead, by using the searching method, the optimal $P_B$ and α can be derived on the basis of simulation and numerical analysis. It ought to be highlighted that equation (20) is of more practical importance to the IoT scenarios.

5. Numerical Results and Discussion

In this section, some numerical outcomes are provided for validation of the abovementioned secrecy analysis and for discussion of the joint effect of corresponding parameters upon the secrecy performance of the two proposed multiuser scheduling schemes. Consistent with [40], the simulation is conducted on a linear topology, in which the multiuser sensors in a localized group as well as multiple PBs, relay R, and controller D are arranged horizontally. Unless otherwise stated, we use the following parameters in accordance with [10] throughout this section. In particular, we set ${\gamma }_E=20$ dB, the predetermined secrecy rate $R_{\text{th}}=0.2$ bits/s/Hz, the energy conversion efficiency $\eta =0.6$, the distances are set to $d_{RD}=d_{SE}=d_{RE}=3$ m, $d_{BS}=\left(1/2\right)d_{SR}$, and $d_{BR}=d_{SR}-d_{BS}$. Moreover, we set the number of the users $N=3$, the power coefficient $\kappa =2.63$, the path loss exponent $\theta =2$, and $P_c=112.2$ mW. It is obvious that the theoretical results agree exactly with the simulation results, which validates the correctness of our derivations. It is worth noting that all the numerical results in this section come from the simulation environment of MATLAB, which are all true experimental results. Combining the parameter setting in the article with the program code, all the results can be reproduced.

Figure 3 describes the SOP of the proposed multiuser scheduling schemes versus ${\gamma }_B$ for different M with $\alpha =0.5$, $N=3$. Overall, it can be seen in this figure that the proposed BSBP scheme can achieve better secrecy performance than the BSR scheme in the whole range of ${\gamma }_B$ with different M, which indicates that the BSBP scheme is more effective to improve the secrecy performance of the considered system. Furthermore, the number of PBs contributes to the improvement of SOP. This is attributed to the fact that more PBs are able to provide larger diversity gain for improving SOP.

[figure omitted; refer to PDF]

Figure 4 shows the SEE of the system with BSR/BSBP scheme versus ${\gamma }_B$ for different M with $\alpha =0.5$ and $N=3$. It is illustrated in this figure that the function of SEE and ${\gamma }_B$ is unimodal function. The reason is that increasing ${\gamma }_B$ brings about the improvement of SEE in low energy consumption, but it has a negative impact on the SEE in high energy consumption. Furthermore, as expected, the SEE of BSBP scheme outperforms that of the BSR scheme in the whole region, which demonstrates the advantage of BSBP in improving the secrecy performance. In addition, by increasing the number of PBs, the better security can be achieved, which can be understood through the following discussion. Furthermore, Figure 5 depicts the SEE of two multiuser scheduling strategies versus α with $N=3$ and ${\gamma }_B=20$ dB in consideration of various M. Obviously, the function of SEE and α is also unimodal function. This is due to the fact that when α is small, the harvested energy is commonly insufficient for the operation of multiple source sensors and relay sensor, while if α is large excessively, the duration of information communication will be restricted seriously, which will result in high transmission interruption probability. On the contrary, similar to Figure 4, the BSBP scheme always provides the best SEE performance with varying α and M, which validates the advantage of proposed BSBP scheme again.

[figure omitted; refer to PDF]

[figure omitted; refer to PDF]

Figure 6 presents the SEE of the two proposed strategies versus $d_{PS}$ and M with $\alpha =0.5$ and ${\gamma }_B=20$ dB. It can be observed that when the group of PBs is closer to multiple source users and the number of PBs is much more relative, the two proposed strategies achieve the best efficiency. Meanwhile, the BSBP scheme is more effective when M is larger, which can be explained as the abovementioned discussion.

[figure omitted; refer to PDF]

Figure 7 plots the impact of $d_{SD}$ and $d_{PS}$ on the optimization of SEE by the searching method with BSR/BSBP scheme for ${\gamma }_B=20$ dB. As shown clearly in the figure, the proposed multiuser scheduling schemes are more effective when $d_{SD}$ and $d_{PS}$ are smaller. As a matter of fact, the condition makes it possible for the sensors to acquire plenty of energy more readily, of which effect is equivalent to increasing $P_B$ or expand α.

[figure omitted; refer to PDF]

6. Conclusion

In this paper, the secrecy performance analysis of wireless-powered IoT possessing diverse PBs has been investigated. Specifically, the two green-oriented multiuser scheduling schemes to promote the secrecy performance of the networks are proposed. For each scheme, the analytical closed-form expression of SOP is obtained, while the optimization problem of SEE is solved by resorting to the searching method. To shed light on future applications of wireless-powered IoT, simulation results are presented to demonstrate the accuracy of our analysis, and the secrecy performance of the two proposed schemes is discussed, subject to various important parameters of the system.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under Grant numbers 61501508 and 61671476.

Appendix

Appendix

A. Proof of Formula (17)

According to (16), ${\gamma }_{\text{sec}}^{\left(\text{BSR}\right)}$ is given by$$\begin{array}{c}\text{(A.1)}& {\gamma }_{\text{sec}}^{\left(\text{BSR}\right)}=\min \left(\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}},\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}\right)=\min \left({\gamma }_{\sec 1}^{\left(\text{BSR}\right)},{\gamma }_{\sec 2}^{\left(\text{BSR}\right)}\right).\end{array}$$

Then, in line with (15) and (16), we have$$\begin{array}{}\text{(A.2)}& P_{\text{sop}}^{\left(\text{BSR}\right)}=P_{\text{sop},S_{n}R}=\mathrm{Pr}\left({\gamma }_{\text{sec}}^{\left(\text{BSR}\right)}<\beta \right)=F_{{\gamma }_{\sec }^{\left(\text{BSR}\right)}}\left(\beta \right),\end{array}$$where $F_{{\gamma }_{\sec }^{\left(\text{BSR}\right)}}\left(\beta \right)$ is the CDF of ${\gamma }_{\text{sec}}^{\left(\text{BSR}\right)}$, which can be given with the help of (A.1) by$$\begin{array}{c}\text{(A.3)}& F_{{\gamma }_{\sec }^{\left(\text{BSR}\right)}}\left(\beta \right)=\mathrm{Pr}\left\{{\gamma }_{\sec }^{\left(\text{BSR}\right)}<\beta \right\}=\mathrm{Pr}\left\{\min \left({\gamma }_{\sec 1}^{\left(\text{BSR}\right)},{\gamma }_{\sec 2}^{\left(\text{BSR}\right)}\right)<\beta \right\}\\ =1-\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSR}\right)}>\beta \right\}\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSR}\right)}>\beta \right\}\\ =1-\left[1-\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSR}\right)}<\beta \right\}\right]\times \left[1-\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSR}\right)}<\beta \right\}\right].\end{array}$$

Furthermore, $\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSR}\right)}<\beta \right\}$ and $\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSR}\right)}<\beta \right\}$ can be derived aided by the Eq. (3.351.3) in [38], and the expression is listed as follows:$$\begin{array}{c}\text{(A.4)}& \text{Pr}\left\{{\gamma }_{\text{sec}1}^{\left(\text{BSR}\right)}<\beta \right\}=\text{Pr}\left\{\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}}<\beta \right\}={\int }_0^{\infty }{\int }_0^{\infty }{F}_{{\chi }_{S_{n^{\ast }}R}}\left[\frac{\beta \left(1+{\gamma }_E\xi xy\right)-1}{{\gamma }_B\xi x}\right]\times f_{{\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}}\left(x\right)f_{{\chi }_{S_{n^{\ast }}E}}\left(y\right)\text{d}x\text{d}y\\ =1-\frac{{\gamma }_B{\lambda }_{PS}{\lambda }_{SE}}{{\lambda }_{SE}{\gamma }_B+\beta {\gamma }_E{\lambda }_{SR}}\times 2\sqrt{\frac{{\lambda }_{SR}\left(\beta -1\right)}{{\gamma }_B\xi {\lambda }_{PS}}}{K}_1\left(2\sqrt{\frac{{\lambda }_{SR}{\lambda }_{PS}\left(\beta -1\right)}{{\gamma }_B\xi }}\right),\\ \text{(A.5)}& \mathrm{Pr}\left\{{\gamma }_{\text{sec}2}^{\left(\text{BSR}\right)}<\beta \right\}=\mathrm{Pr}\left\{\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}<\beta \right\}={\int }_0^{\infty }{\int }_0^{\infty }{F}_{{\chi }_{RD}}\left[\frac{\beta \left(1+{\gamma }_E\xi xy\right)-1}{{\gamma }_B\xi x}\right]\times f_{{\gamma }_{P_{m^{\ast }}R}}\left(x\right)f_{{\chi }_{RE}}\left(y\right)\text{d}x\text{d}y\\ =1-\sum _{m=1}^M\left(\begin{array}{c}M\\ m\end{array}\right){\left(-1\right)}^{m+1}\frac{m{\gamma }_B{\lambda }_{PR}{\lambda }_{RE}}{{\lambda }_{RE}{\gamma }_B+\beta {\gamma }_E{\lambda }_{RD}}\times 2\sqrt{\frac{{\lambda }_{RD}\left(\beta -1\right)}{m{\gamma }_B\xi {\lambda }_{PR}}}{K}_1\left(2\sqrt{\frac{m{\lambda }_{PR}{\lambda }_{RD}\left(\beta -1\right)}{{\gamma }_B\xi }}\right).\end{array}$$

Finally, by substituting (A.4) and (A.5) into (A.3) and performing some mathematical manipulations, (17) can be derived.

B. Proof of Formula (18)

Similar with ${\gamma }_{\sec }^{\left(\text{BSR}\right)}$ in (A.1), ${\gamma }_{\sec }^{\left(\text{BSBP}\right)}$ can be shown as$$\begin{array}{c}\text{(B.1)}& {\gamma }_{\text{sec}}^{\left(\text{BSBP}\right)}=\min \left(\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}},\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}\right)=\min \left({\gamma }_{\text{sec}1}^{\left(\text{BSBP}\right)},{\gamma }_{\text{sec}2}^{\left(\text{BSBP}\right)}\right).\end{array}$$

Meanwhile, according to (15) and (16), we find$$\begin{array}{}\text{(B.2)}& P_{\text{sop}}^{\left(\text{BSBP}\right)}=\mathrm{Pr}\left({\gamma }_{\text{sec}}^{\left(\text{BSBP}\right)}<\beta \right)=F_{{\gamma }_{\sec }^{\left(\text{BSBP}\right)}}\left(\beta \right),\end{array}$$where $F_{{\gamma }_{\sec }^{\left(\text{BSBP}\right)}}\left(\beta \right)$ is the CDF of ${\gamma }_{\sec }^{\left(\text{BSBP}\right)}$, and it can be expressed with the help of (A.1) as$$\begin{array}{c}\text{(B.3)}& F_{{\gamma }_{\text{sec}}^{\left(\text{BSBP}\right)}}\left(\beta \right)=\mathrm{Pr}\left\{{\gamma }_{\text{sec}}^{\left(\text{BSBP}\right)}<\beta \right\}=\mathrm{Pr}\left\{\min \left({\gamma }_{\sec 1}^{\left(\text{BSBP}\right)},{\gamma }_{\text{sec}2}^{\left(\text{BSBP}\right)}\right)<\beta \right\}\\ =1-\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSBP}\right)}>\beta \right\}\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSBP}\right)}>\beta \right\}\\ =1-\left[1-\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSBP}\right)}<\beta \right\}\right]\times \left[1-\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSBP}\right)}<\beta \right\}\right].\end{array}$$

After that, $\mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSBP}\right)}<\beta \right\}$ and $\mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSBP}\right)}<\beta \right\}$ can be obtained using the Eq. (3.351.3) in [38] and the expression listed below:$$\begin{array}{c}\text{(B.4)}& \mathrm{Pr}\left\{{\gamma }_{\sec 1}^{\left(\text{BSBP}\right)}<\beta \right\}=\mathrm{Pr}\left\{\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}R}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}{\chi }_{S_{n^{\ast }}E}}<\beta \right\}={\int }_0^{\infty }{\int }_0^{\infty }{F}_{{\chi }_{S_{n^{\ast }}R}}\left[\frac{\beta \left(1+{\gamma }_E\xi xy\right)-1}{{\gamma }_B\xi x}\right]\times f_{{\gamma }_{P_{m^{\ast }}{S}_{n^{\ast }}}}\left(x\right)f_{{\chi }_{S_{n^{\ast }E}}}\left(y\right)\text{d}x\text{d}y\\ =1-\sum _{n=1}^N\left(\begin{array}{c}N\\ n\end{array}\right){\left(-1\right)}^{n+1}\frac{n{\gamma }_B{\lambda }_{PS}{\lambda }_{SE}}{{\lambda }_{SE}{\gamma }_B+\beta {\gamma }_E{\lambda }_{SR}}\times 2\sqrt{\frac{{\lambda }_{SR}\left(\beta -1\right)}{n{\gamma }_B\xi {\lambda }_{PS}}}{K}_1\left(2\sqrt{\frac{n{\lambda }_{SR}{\lambda }_{PS}\left(\beta -1\right)}{{\gamma }_B\xi }}\right),\\ \text{(B.5)}& \mathrm{Pr}\left\{{\gamma }_{\sec 2}^{\left(\text{BSBP}\right)}<\beta \right\}=\mathrm{Pr}\left\{\frac{1+{\gamma }_B\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RD}}{1+{\gamma }_E\xi {\gamma }_{P_{m^{\ast }}R}{\chi }_{RE}}<\beta \right\}={\int }_0^{\infty }{\int }_0^{\infty }{F}_{{\chi }_{RD}}\left[\frac{\beta \left(1+{\gamma }_E\xi xy\right)-1}{{\gamma }_B\xi x}\right]\times f_{{\gamma }_{P_{m^{\ast }}R}}\left(x\right)f_{{\chi }_{RE}}\left(y\right)\text{d}x\text{d}y\\ =1-\sum _{m=1}^M\left(\begin{array}{c}M\\ m\end{array}\right){\left(-1\right)}^{m+1}\frac{m{\gamma }_B{\lambda }_{PR}{\lambda }_{RE}}{{\lambda }_{RE}{\gamma }_B+\beta {\gamma }_E{\lambda }_{RD}}\times 2\sqrt{\frac{{\lambda }_{RD}\left(\beta -1\right)}{m{\gamma }_B\xi {\lambda }_{PR}}}{K}_1\left(2\sqrt{\frac{m{\lambda }_{PR}{\lambda }_{RD}\left(\beta -1\right)}{{\gamma }_B\xi }}\right).\end{array}$$

Finally, by substituting (B.4) and (B.5) into (B.3) and performing some mathematical manipulations, (18) can be derived.

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