# Analysis of SWIPT-Enabled Relay Networks with Full-Duplex Destination-Aided Jamming

Authors: Jinglan Ou; Hangchuan Shi; Liubin Wang; Rui Ma; Haowei Wu
Journal: Security and Communication Networks
 (2021)

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## Abstract

Simultaneous wireless information and power transfer (SWIPT) is a major breakthrough in the field of low-power wireless information transmissions. In this paper, the secrecy performance of the SWIPT-enabled relay network with full-duplex destination-aided jamming is assessed, where both the power-splitting (PS) and time-switching (TS) schemes at the relay are considered with the linear and nonlinear energy harvesting models. The relay harvests energy from the confidential signal and artificial noise sent by the source and destination, respectively, and forwards the amplified signal to the destination, in the presence of an eavesdropper. The analytical closed-form expressions of the connection outage probability (COP), secrecy outage probability (SOP), and transmission outage probability (TOP) for PS- and TS-based schemes are derived, and the closed-form expression of the lower bound of ergodic secrecy capacity (ESC) is calculated. The asymptotic-form expressions of the COP, SOP, TOP, and ESC are further analyzed to capture the valuable information in the high SNR regime. Numerical results verify the correctness of analytical results, reveal the effects of the PS/TS ratio, and transmit the signal-to-noise ratio on secrecy performance.


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## Body

## 1. Introduction

Wireless communication networks greatly facilitate our daily lives with the explosive growth of smart phones, wireless laptops, and tablet computers. Stable and continuous energy supply is a fundamental requirement for wireless devices, which are usually charged by wired power, batteries, or other natural resources such as wind and solar from the surrounding environment. As an emerging technique, wireless power transmission (WPT) technique harvesting energy from electromagnetic radiation of the radio frequency (RF) signal is a promising solution to provide sustainable energy for wireless devices [1–3]. Recently, RF power harvesting has been widely studied and applied in wireless systems which are hard to acquire regular electrical supply, especially under conditions of frequent movement and wide coverage of wireless terminals [4–6].As a combination of wireless communication and WPT, simultaneous wireless information and power transfer (SWIPT) was proposed in [7]. With the advantages of high spectral efficiency and low energy consumption, SWIPT has drawn extensive attention on account of its potential ability to support wireless sensors and medical implants without frequent battery charging [8–10]. However, signals in SWIPT systems are particularly vulnerable to malicious attacks due to the strong signal power as well as the open nature of the wireless channel, which results in concerns about the security issue. Therefore, a lot of studies have been performed to prevent the leakage of private information, and physical layer security (PLS) has been proved to be one of the most effective solutions to improve the security of SWIPT systems [11]. The pivotal character of PLS is to utilize the intrinsic secrecy properties of wireless channels, which can help to realize the secure communication in SWIPT [12–15].Cooperative jamming is regarded as a sensible approach to prevent information leakage for SWIPT. Generally, to degrade the eavesdropper, one helper node is deployed in the network to transmit a jamming signal such as the artificial noise (AN). The helper is also termed as a cooperative interference source which might be any node involved in the network, e.g., the source, relay, and destination.As the self-cooperative jamming, the source can radiate the energy signal and the information signal superposed with the AN. In [13], the PLS issues in a multiuser SWIPT system were studied, where one multi-antenna source sends energy and information with the AN signal simultaneously to the information receiver and energy receivers. This deliberate interference is added into the source signal in order to degrade the information eavesdropping at the energy receivers. To obtain the maximum secrecy rate by optimizing the power allocation and AN-aided transmission, a SWIPT system, composing of an eavesdropper, a source, and several users, was investigated in [16], where the source acts as a cooperative jammer as well.Legitimate communication partners, such as the relay, can also act as a jammer in the SWIPT network. In [17], a relay adopting the power-splitting (PS) scheme in the wiretap system was introduced to forward information and interfere the illegal eavesdropper, and an iterative algorithm was proposed to handle the problem of maximizing the secrecy rate in the worst case. A two-level strategy was proposed in [18] to achieve the minimization relay power, within the limit of secrecy rate and energy harvesting (EH) requirements through jointly optimizing the relay beamforming matrix, AN covariance and PS ratio.The destination node can also assist the network in sending interference as well as receiving the intelligence signal. Two-hop cooperative communication with the destination transmitting the jamming signal was considered, and both adaptive time conversion and power cutting schemes were presented in [19], where the maximum secrecy capacity and its relationship with the time conversion ratio and power cutting rate are studied. A SWIPT-enabled system by full-duplex (FD) jamming at the intended receiver is proposed to enhance secrecy performance and the secrecy throughput of the system is analyzed in [20]. The FD destination node with the ability of emitting artificial noise is exploited to achieve better secrecy transmission in the three-node eavesdropping scenario of SWIPT, where the destination employed time-switching (TS) protocol to harvest energy, and the closed-form expressions of the connection outage probability (COP), the secrecy outage probability (SOP), the transmission outage probability (TOP), and the secrecy energy efficiency are derived [21]. In our previous study [22], an FD destination-assisted jamming scheme was adopted in a SWIPT-enabled trusted relay network, and the optimization problem was designed to achieve the optimal secrecy capacity, where only the PS scheme with the linear EH (LEH) model was considered.Motivated by above observations, in this paper, the secrecy performance of a SWIPT-enabled four-node wiretap network with an amplify-and-forward (AF) relay and a destination with jamming is investigated. The source, destination, and eavesdropper are considered to be under the uninterruptible power supplies. The relay replenishes energy from the RF signals including the information signal from the source and the AN signal from the destination, after which the relay amplifies the signal and then sends the correct information signal to the destination.Our work is different from the following related papers: firstly, the source acted as a jammer in [13, 16], and the relay was the jammer in [17, 18]. Unlike these papers, the destination sent the jamming signal with FD to protect the classified information in our network, which is the fundamental difference between our and their works. Secondly, only the LEH model was utilized in [19, 21, 22], while both the linear and nonlinear EH models are considered in this treatise. Thirdly, the destination node was self-interference in our network, instead of disturbing the relay, which is unlike the network in [19]. In contrast to [20, 21], the author investigated the three-node system, while the network in this paper consists of four nodes. Fourthly, both the PS- and TS-based schemes are employed, while only the PS scheme was applied in [19, 22] and the TS in [22]. In addition, the COP, SOP, TOP, and ESC are analyzed in our research to provide a comprehensive and thorough guidance for practical application, while only SC was focused in [19, 22]. The contributions of our work are summarized as follows:(i)
A SWIPT-enabled relay network is studied, where an FD destination receives the information, while transmitting the AN jamming at the same time. The secrecy performance of the network is analyzed for both the PS- and TS-based schemes under both the nonlinear and linear EH models at the relay.(ii)
For both the PS and TS schemes, the closed-form expressions of the COP, SOP, TOP, and lower bound of the ergodic secrecy capacity (ESC) are presented under the linear and nonlinear EH models, and the asymptotic-form expressions of these metrics are given in the region of high signal-to-noise ratios (SNRs).(iii)
The correctness of analytical expressions is verified by simulations, and the impacts of different system parameters on the COP, SOP, TOP, and lower bound of ESC are revealed. The simulation results imply that the optimal secrecy performance of the system can be achieved by optimizing the four metrics.The rest of this paper is structured as follows. Section2 describes the system model for the secure communication via the relay under both the linear and nonlinear EH models. In Section 3, the analytical expressions of the COP, SOP, TOP, and ESC for the PS- and TS-based relaying are derived. Numerical results are presented in Section 4, where effects of different system parameters on the secrecy performance of the SWIPT system with FD destination-assisted jamming are offered. Finally, conclusions are presented in Section 5.


## 2. System Model

### 2.1. Network Description

As shown in Figure1, the SWIPT-enabled relay network is composed of four nodes, i.e., one source S, one AF relay ℛ, one destination D, and one eavesdropper E. The relay depends on the wireless power supplies from the source and destination, while the source, destination, and eavesdropper are supplied by the stable power source. The FD destination with two antennas can transmit the AN signal and receive information signal from the relay at the same time, and other nodes equipped with only one antenna operate in the half-duplex mode. In addition, the source S only has a direct link with the relay ℛ, while the direct links of S−D and S−E are blocked by the physical obstacles, which is a common assumption in [23, 24].Figure 1
SWIPT-enabled relay network.


#### 2.1.1. Channel Description

The quasi-static frequency nonselective channel is considered in this model, which indicates that all the channel coefficients remain constant within a transmission block, but vary independently from one packet to another [25, 26]. Without loss of generality, it is assumed that the instantaneous channel state information of both the main channel and eavesdropping channel are available [27]. The channel power gain between the nodes i and j is defined as hij2, which is the exponential distribution with mean λij=dij−ℒ, where iandj∈S,ℛ,D,E, i≠j, dij is the distance between nodes i and j, and ℒ is the path-loss exponent. The channel coefficients for S−ℛ, ℛ−E, ℛ−D, and D−E links are represented as hSR, hℛℰ, hRD, and hDE, respectively. The residual self-interference (RSI) channel coefficient at the destination is denoted as hDD, which is regarded as the independent complex Gaussian random variable [20]. According to [28], after cancellation, the destination-aided jamming signal is regarded as the RSI, which is hypothesised to be independent of other signals and yields to Gaussian distribution with zero mean and σi2-variance. The variance is formulated as σi2=kPrv, where the two constants, i.e., k>0 and v∈0,1, depend on the interference cancellation scheme at the relay [29].


#### 2.1.2. Secure Transmission

The communication process of the system includes energy harvesting, signal processing, and information transmitting, which are shown in Figure1. Firstly, the source transmits confidential information signal toward the relay, and the destination sends the AN signal to the relay simultaneously, and then, the relay collects energy from the received signal, after which the relay amplifies the received signal. Finally, the relay with the full energy forwards the confidential information to the destination under the protection of the jamming signal.In the following content, the PS- and TS-based schemes are introduced in the SWIPT system, where both the linear and nonlinear EH architectures are also taken into consideration at the relay. For the sake of simplicity, the energy costs for data processing are ignored in all nodes, which means that most of the energy is used for data delivery [30].


### 2.2. Relay Protocol

#### 2.2.1. Power-Splitting-Based Relaying

The PS-based scheme for the secure communication in the SWIPT system with FD jamming is illustrated in Figure2, where the source-to-destination communication occurs in a slot of duration T. The slot is divided into two phases equally. The source transmits the information signal through power PS toward the relay, and the destination sends the jamming signal through power PD to the relay, where PS=ξPtotal and PD=1−ξPtotal are the power from the source and destination separately, ξ∈0,1 is the power allocation coefficient, and Ptotal is the total system power. The received energy from the RF signal is divided into two parts in the first phase, which depends on the PS ratio ρ, as ρ for the energy harvesting and 1−ρ for the signal processing. In the second phase, the relay transmits the information to the destination.Figure 2
PS scheme for secure communication in the SWIPT system.For the energy harvesting at the relay based on the PS-based protocol, the linear EH and nonlinear EH (NLEH) models atℛ are discussed as following. Normally, the most existing research studies are based on the LEH model for it is tractable [21, 31]. The linear energy collection at the relay is(1)EHL=ηρTPShSR2+PDhRD22,where η0<η<1 represents the energy conversion efficiency.The forwarding power at the relay for transmitting signals is written as(2)PℛL=ηρEr,where Er=PShSR2+PDhRD2.The energy conversion efficiency actually relies on the input power, which means that the energy collected is nonlinear with the input power [32]. The NLEH model is also considered in this paper since it matches practical EH circuits better than the linear one [20]. To ensure the accuracy, practicability, and ease of processing for the analysis, the simplified NLEH model is exploited and represented as(3)PℛNL=p1EPℛLEPℛL+p2,where p1=c1−c2/c3, p2=c3, and E⋅ is the expectation operator and c1, c2, and c3 are constants related to the detailed circuit specifications such as the resistance and capacitance [20]. The specific values of c1, c2, and c3 are 0.3929, 0.01675, and 0.04401, respectively [33].The nonlinear energy harvesting at the relay can be expressed as(4)EHNL=p1ρErρEr+p2T2.Hence, the power to forward signals atℛ is given by(5)PℛNL=p1ρErρEr+p2.After energy harvesting, the information processing is discussed next. The received signal at the relay in the first phase is expressed as(6)yℛ=1−ρPShSRxS+1−ρPDhRDxAN+nℛ,where nℛ stands for the additive white Gaussian noise (AWGN) at the relay with zero mean and variance σ2.Next, the AF relay transmits the amplified version of the received signal which is given by(7)xℛl=Pℛlyℛ1−ρPShSR2+PDhRD2+σ2,where l∈L,NL represents the LEH and NLEH, respectively.The received signals atD and E with destination-aided jamming can be, respectively, expressed as(8)yDl=hRDxℛl+PDhDDxAN+nD,(9)yEl=hExℛl+PDhDExAN+nE,where nD and nE denote the AWGNs at the destination and eavesdropper with zero mean and variance σ2, respectively.It is supposed that the destination can eliminate the AN signal by obtaining the exact information of the channel gain of theℛ−D link, while the eavesdropper is interfered by the AN on the account of lacking channel information [25]. Plugging (7) into (8), the received instantaneous end-to-end SNR at D is given by(10)ΓDl=1−ργSγℛlhSR2hRD2γℛlhRD2+kγDv+11−ργΘ+1,where γS=PS/σ2, γℛl=Pℛl/σ2, γD=PD/σ2, γΘ=Er/σ2, and kγDv is the residual self-interference.Similarly, the received SNR atE is obtained by the following equation:(11)ΓEl=1−ργSγℛlhSR2hℛℰ21−ργℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+11−ργΘ+1.


#### 2.2.2. Time-Switching-Based Relaying

From the perspective of receiver’s complexity, the TS is simpler than the PS as energy harvesting and information processing are separate in commercial circuits [34]. In the TS-based solution, the energy acquisition and information processing of the RF signal at the relay are performed separately during different time periods, depending on α∈0,1. The TS-based scheme of the SWIPT system with destination-aided FD jamming is shown in Figure 3, and the time period T is split into three phases. In the first phase, the relay takes αT to harvest energy from the received RF signals. During the second phase, the relay amplifies the signal from the source within 1−αT/2 period, and the relay sends the information signal to the destination at the remaining time 1−αT/2 in the third phase. Both the LEH and NLEH models for the TS scheme are discussed in the following part.Figure 3
TS scheme for secure communication in the SWIPT system.For the aforementioned TS policy, the harvested energy during the period ofαT in the first phase is given by EHL=ηαTEr. Thus, the power at ℛ sending the amplified signal to the destination is written as PℛL=2ηαEr/1−α. Similar to the nonlinear derivation of the PS-based scheme in section 2.2.1, the expression of NLEH for the TS-based scheme is EHNL=αTp1Er/Er+p2, and the relay’s transmit power forwarding the information to destination is given as(12)PℛNL=2αp1Er1−αEr+p2.In the second phase, the relay performs information processing after energy collections, where the signal received by the relay becomes(13)yℛ=PShSRxS+PDhRDxAN+nℛ.Then, the signal sent by the relay in the third phase is(14)xℛl=PℛlyℛPShSR2+PDhRD2+σℛ2.The received signals atD and E are the same as (8) and (9), respectively, which are not repeated here. Embedding (13) and (14) into (8), the received instantaneous end-to-end SNR at D is(15)ΓDl=γSγℛlhSR2hRD2γℛlhRD2+kγDv+1γΘ+1.Likewise, the received SNR atE is represented by(16)ΓEl=γSγℛlhSR2hℛℰ2γℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+1γΘ+1.


## 2.1. Network Description

As shown in Figure1, the SWIPT-enabled relay network is composed of four nodes, i.e., one source S, one AF relay ℛ, one destination D, and one eavesdropper E. The relay depends on the wireless power supplies from the source and destination, while the source, destination, and eavesdropper are supplied by the stable power source. The FD destination with two antennas can transmit the AN signal and receive information signal from the relay at the same time, and other nodes equipped with only one antenna operate in the half-duplex mode. In addition, the source S only has a direct link with the relay ℛ, while the direct links of S−D and S−E are blocked by the physical obstacles, which is a common assumption in [23, 24].Figure 1
SWIPT-enabled relay network.


### 2.1.1. Channel Description

The quasi-static frequency nonselective channel is considered in this model, which indicates that all the channel coefficients remain constant within a transmission block, but vary independently from one packet to another [25, 26]. Without loss of generality, it is assumed that the instantaneous channel state information of both the main channel and eavesdropping channel are available [27]. The channel power gain between the nodes i and j is defined as hij2, which is the exponential distribution with mean λij=dij−ℒ, where iandj∈S,ℛ,D,E, i≠j, dij is the distance between nodes i and j, and ℒ is the path-loss exponent. The channel coefficients for S−ℛ, ℛ−E, ℛ−D, and D−E links are represented as hSR, hℛℰ, hRD, and hDE, respectively. The residual self-interference (RSI) channel coefficient at the destination is denoted as hDD, which is regarded as the independent complex Gaussian random variable [20]. According to [28], after cancellation, the destination-aided jamming signal is regarded as the RSI, which is hypothesised to be independent of other signals and yields to Gaussian distribution with zero mean and σi2-variance. The variance is formulated as σi2=kPrv, where the two constants, i.e., k>0 and v∈0,1, depend on the interference cancellation scheme at the relay [29].


### 2.1.2. Secure Transmission

The communication process of the system includes energy harvesting, signal processing, and information transmitting, which are shown in Figure1. Firstly, the source transmits confidential information signal toward the relay, and the destination sends the AN signal to the relay simultaneously, and then, the relay collects energy from the received signal, after which the relay amplifies the received signal. Finally, the relay with the full energy forwards the confidential information to the destination under the protection of the jamming signal.In the following content, the PS- and TS-based schemes are introduced in the SWIPT system, where both the linear and nonlinear EH architectures are also taken into consideration at the relay. For the sake of simplicity, the energy costs for data processing are ignored in all nodes, which means that most of the energy is used for data delivery [30].


## 2.1.1. Channel Description

The quasi-static frequency nonselective channel is considered in this model, which indicates that all the channel coefficients remain constant within a transmission block, but vary independently from one packet to another [25, 26]. Without loss of generality, it is assumed that the instantaneous channel state information of both the main channel and eavesdropping channel are available [27]. The channel power gain between the nodes i and j is defined as hij2, which is the exponential distribution with mean λij=dij−ℒ, where iandj∈S,ℛ,D,E, i≠j, dij is the distance between nodes i and j, and ℒ is the path-loss exponent. The channel coefficients for S−ℛ, ℛ−E, ℛ−D, and D−E links are represented as hSR, hℛℰ, hRD, and hDE, respectively. The residual self-interference (RSI) channel coefficient at the destination is denoted as hDD, which is regarded as the independent complex Gaussian random variable [20]. According to [28], after cancellation, the destination-aided jamming signal is regarded as the RSI, which is hypothesised to be independent of other signals and yields to Gaussian distribution with zero mean and σi2-variance. The variance is formulated as σi2=kPrv, where the two constants, i.e., k>0 and v∈0,1, depend on the interference cancellation scheme at the relay [29].


## 2.1.2. Secure Transmission

The communication process of the system includes energy harvesting, signal processing, and information transmitting, which are shown in Figure1. Firstly, the source transmits confidential information signal toward the relay, and the destination sends the AN signal to the relay simultaneously, and then, the relay collects energy from the received signal, after which the relay amplifies the received signal. Finally, the relay with the full energy forwards the confidential information to the destination under the protection of the jamming signal.In the following content, the PS- and TS-based schemes are introduced in the SWIPT system, where both the linear and nonlinear EH architectures are also taken into consideration at the relay. For the sake of simplicity, the energy costs for data processing are ignored in all nodes, which means that most of the energy is used for data delivery [30].


## 2.2. Relay Protocol

### 2.2.1. Power-Splitting-Based Relaying

The PS-based scheme for the secure communication in the SWIPT system with FD jamming is illustrated in Figure2, where the source-to-destination communication occurs in a slot of duration T. The slot is divided into two phases equally. The source transmits the information signal through power PS toward the relay, and the destination sends the jamming signal through power PD to the relay, where PS=ξPtotal and PD=1−ξPtotal are the power from the source and destination separately, ξ∈0,1 is the power allocation coefficient, and Ptotal is the total system power. The received energy from the RF signal is divided into two parts in the first phase, which depends on the PS ratio ρ, as ρ for the energy harvesting and 1−ρ for the signal processing. In the second phase, the relay transmits the information to the destination.Figure 2
PS scheme for secure communication in the SWIPT system.For the energy harvesting at the relay based on the PS-based protocol, the linear EH and nonlinear EH (NLEH) models atℛ are discussed as following. Normally, the most existing research studies are based on the LEH model for it is tractable [21, 31]. The linear energy collection at the relay is(1)EHL=ηρTPShSR2+PDhRD22,where η0<η<1 represents the energy conversion efficiency.The forwarding power at the relay for transmitting signals is written as(2)PℛL=ηρEr,where Er=PShSR2+PDhRD2.The energy conversion efficiency actually relies on the input power, which means that the energy collected is nonlinear with the input power [32]. The NLEH model is also considered in this paper since it matches practical EH circuits better than the linear one [20]. To ensure the accuracy, practicability, and ease of processing for the analysis, the simplified NLEH model is exploited and represented as(3)PℛNL=p1EPℛLEPℛL+p2,where p1=c1−c2/c3, p2=c3, and E⋅ is the expectation operator and c1, c2, and c3 are constants related to the detailed circuit specifications such as the resistance and capacitance [20]. The specific values of c1, c2, and c3 are 0.3929, 0.01675, and 0.04401, respectively [33].The nonlinear energy harvesting at the relay can be expressed as(4)EHNL=p1ρErρEr+p2T2.Hence, the power to forward signals atℛ is given by(5)PℛNL=p1ρErρEr+p2.After energy harvesting, the information processing is discussed next. The received signal at the relay in the first phase is expressed as(6)yℛ=1−ρPShSRxS+1−ρPDhRDxAN+nℛ,where nℛ stands for the additive white Gaussian noise (AWGN) at the relay with zero mean and variance σ2.Next, the AF relay transmits the amplified version of the received signal which is given by(7)xℛl=Pℛlyℛ1−ρPShSR2+PDhRD2+σ2,where l∈L,NL represents the LEH and NLEH, respectively.The received signals atD and E with destination-aided jamming can be, respectively, expressed as(8)yDl=hRDxℛl+PDhDDxAN+nD,(9)yEl=hExℛl+PDhDExAN+nE,where nD and nE denote the AWGNs at the destination and eavesdropper with zero mean and variance σ2, respectively.It is supposed that the destination can eliminate the AN signal by obtaining the exact information of the channel gain of theℛ−D link, while the eavesdropper is interfered by the AN on the account of lacking channel information [25]. Plugging (7) into (8), the received instantaneous end-to-end SNR at D is given by(10)ΓDl=1−ργSγℛlhSR2hRD2γℛlhRD2+kγDv+11−ργΘ+1,where γS=PS/σ2, γℛl=Pℛl/σ2, γD=PD/σ2, γΘ=Er/σ2, and kγDv is the residual self-interference.Similarly, the received SNR atE is obtained by the following equation:(11)ΓEl=1−ργSγℛlhSR2hℛℰ21−ργℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+11−ργΘ+1.


### 2.2.2. Time-Switching-Based Relaying

From the perspective of receiver’s complexity, the TS is simpler than the PS as energy harvesting and information processing are separate in commercial circuits [34]. In the TS-based solution, the energy acquisition and information processing of the RF signal at the relay are performed separately during different time periods, depending on α∈0,1. The TS-based scheme of the SWIPT system with destination-aided FD jamming is shown in Figure 3, and the time period T is split into three phases. In the first phase, the relay takes αT to harvest energy from the received RF signals. During the second phase, the relay amplifies the signal from the source within 1−αT/2 period, and the relay sends the information signal to the destination at the remaining time 1−αT/2 in the third phase. Both the LEH and NLEH models for the TS scheme are discussed in the following part.Figure 3
TS scheme for secure communication in the SWIPT system.For the aforementioned TS policy, the harvested energy during the period ofαT in the first phase is given by EHL=ηαTEr. Thus, the power at ℛ sending the amplified signal to the destination is written as PℛL=2ηαEr/1−α. Similar to the nonlinear derivation of the PS-based scheme in section 2.2.1, the expression of NLEH for the TS-based scheme is EHNL=αTp1Er/Er+p2, and the relay’s transmit power forwarding the information to destination is given as(12)PℛNL=2αp1Er1−αEr+p2.In the second phase, the relay performs information processing after energy collections, where the signal received by the relay becomes(13)yℛ=PShSRxS+PDhRDxAN+nℛ.Then, the signal sent by the relay in the third phase is(14)xℛl=PℛlyℛPShSR2+PDhRD2+σℛ2.The received signals atD and E are the same as (8) and (9), respectively, which are not repeated here. Embedding (13) and (14) into (8), the received instantaneous end-to-end SNR at D is(15)ΓDl=γSγℛlhSR2hRD2γℛlhRD2+kγDv+1γΘ+1.Likewise, the received SNR atE is represented by(16)ΓEl=γSγℛlhSR2hℛℰ2γℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+1γΘ+1.


## 2.2.1. Power-Splitting-Based Relaying

The PS-based scheme for the secure communication in the SWIPT system with FD jamming is illustrated in Figure2, where the source-to-destination communication occurs in a slot of duration T. The slot is divided into two phases equally. The source transmits the information signal through power PS toward the relay, and the destination sends the jamming signal through power PD to the relay, where PS=ξPtotal and PD=1−ξPtotal are the power from the source and destination separately, ξ∈0,1 is the power allocation coefficient, and Ptotal is the total system power. The received energy from the RF signal is divided into two parts in the first phase, which depends on the PS ratio ρ, as ρ for the energy harvesting and 1−ρ for the signal processing. In the second phase, the relay transmits the information to the destination.Figure 2
PS scheme for secure communication in the SWIPT system.For the energy harvesting at the relay based on the PS-based protocol, the linear EH and nonlinear EH (NLEH) models atℛ are discussed as following. Normally, the most existing research studies are based on the LEH model for it is tractable [21, 31]. The linear energy collection at the relay is(1)EHL=ηρTPShSR2+PDhRD22,where η0<η<1 represents the energy conversion efficiency.The forwarding power at the relay for transmitting signals is written as(2)PℛL=ηρEr,where Er=PShSR2+PDhRD2.The energy conversion efficiency actually relies on the input power, which means that the energy collected is nonlinear with the input power [32]. The NLEH model is also considered in this paper since it matches practical EH circuits better than the linear one [20]. To ensure the accuracy, practicability, and ease of processing for the analysis, the simplified NLEH model is exploited and represented as(3)PℛNL=p1EPℛLEPℛL+p2,where p1=c1−c2/c3, p2=c3, and E⋅ is the expectation operator and c1, c2, and c3 are constants related to the detailed circuit specifications such as the resistance and capacitance [20]. The specific values of c1, c2, and c3 are 0.3929, 0.01675, and 0.04401, respectively [33].The nonlinear energy harvesting at the relay can be expressed as(4)EHNL=p1ρErρEr+p2T2.Hence, the power to forward signals atℛ is given by(5)PℛNL=p1ρErρEr+p2.After energy harvesting, the information processing is discussed next. The received signal at the relay in the first phase is expressed as(6)yℛ=1−ρPShSRxS+1−ρPDhRDxAN+nℛ,where nℛ stands for the additive white Gaussian noise (AWGN) at the relay with zero mean and variance σ2.Next, the AF relay transmits the amplified version of the received signal which is given by(7)xℛl=Pℛlyℛ1−ρPShSR2+PDhRD2+σ2,where l∈L,NL represents the LEH and NLEH, respectively.The received signals atD and E with destination-aided jamming can be, respectively, expressed as(8)yDl=hRDxℛl+PDhDDxAN+nD,(9)yEl=hExℛl+PDhDExAN+nE,where nD and nE denote the AWGNs at the destination and eavesdropper with zero mean and variance σ2, respectively.It is supposed that the destination can eliminate the AN signal by obtaining the exact information of the channel gain of theℛ−D link, while the eavesdropper is interfered by the AN on the account of lacking channel information [25]. Plugging (7) into (8), the received instantaneous end-to-end SNR at D is given by(10)ΓDl=1−ργSγℛlhSR2hRD2γℛlhRD2+kγDv+11−ργΘ+1,where γS=PS/σ2, γℛl=Pℛl/σ2, γD=PD/σ2, γΘ=Er/σ2, and kγDv is the residual self-interference.Similarly, the received SNR atE is obtained by the following equation:(11)ΓEl=1−ργSγℛlhSR2hℛℰ21−ργℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+11−ργΘ+1.


## 2.2.2. Time-Switching-Based Relaying

From the perspective of receiver’s complexity, the TS is simpler than the PS as energy harvesting and information processing are separate in commercial circuits [34]. In the TS-based solution, the energy acquisition and information processing of the RF signal at the relay are performed separately during different time periods, depending on α∈0,1. The TS-based scheme of the SWIPT system with destination-aided FD jamming is shown in Figure 3, and the time period T is split into three phases. In the first phase, the relay takes αT to harvest energy from the received RF signals. During the second phase, the relay amplifies the signal from the source within 1−αT/2 period, and the relay sends the information signal to the destination at the remaining time 1−αT/2 in the third phase. Both the LEH and NLEH models for the TS scheme are discussed in the following part.Figure 3
TS scheme for secure communication in the SWIPT system.For the aforementioned TS policy, the harvested energy during the period ofαT in the first phase is given by EHL=ηαTEr. Thus, the power at ℛ sending the amplified signal to the destination is written as PℛL=2ηαEr/1−α. Similar to the nonlinear derivation of the PS-based scheme in section 2.2.1, the expression of NLEH for the TS-based scheme is EHNL=αTp1Er/Er+p2, and the relay’s transmit power forwarding the information to destination is given as(12)PℛNL=2αp1Er1−αEr+p2.In the second phase, the relay performs information processing after energy collections, where the signal received by the relay becomes(13)yℛ=PShSRxS+PDhRDxAN+nℛ.Then, the signal sent by the relay in the third phase is(14)xℛl=PℛlyℛPShSR2+PDhRD2+σℛ2.The received signals atD and E are the same as (8) and (9), respectively, which are not repeated here. Embedding (13) and (14) into (8), the received instantaneous end-to-end SNR at D is(15)ΓDl=γSγℛlhSR2hRD2γℛlhRD2+kγDv+1γΘ+1.Likewise, the received SNR atE is represented by(16)ΓEl=γSγℛlhSR2hℛℰ2γℛlγDhRD2hℛℰ2+γℛlhℛℰ2+γDhDE2+1γΘ+1.


## 3. Performance Analysis

In this section, we proceed to derive the closed-form expressions of the COP, SOP, TOP, and lower bound of ESC for both PS and TS schemes to comprehensively and directly analyze the secrecy performance of the system.


### 3.1. Power-Splitting Protocol

#### 3.1.1. COP

The destination fails to recover the source message from the relay when the channel capacity of theℛ−D link is lower than the minimum transmission rate Rtbps/Hz, which means a connection interruption has occured. The channel capacity is defined as CM=1/2log21+ΓD in Shannon’s coding theorem. The factor 1/2 denotes the effective time of information transmission between the source and destination.Mathematically, the COP is defined as [21](17)PCOl=PrΓDl<γtht,where γtht=22Rt−1 is the minimum transmission threshold.Proposition 1.
The closed-form expression of the COP for the SWIPT system with FD destination-aided jamming under the PS scheme is given by(18)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, mx=1−ργSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+11−ργΘ+1, and b1=γℛl. Here, K1⋅ is the modified Bessel functions of the second kind (Eq. 8.407.2 in [35]).Proof.
See AppendixA.
To further capture the useful information for the secrecy performance, the asymptotic analysis of COPs is conducted in terms of LEH and NLEH models.
In the high SNR regime, i.e.,Ptotal⟶∞, the analytical expression of COP in the SWIPT-enabled relay system adopting the LEH model is given by(19)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜x=λSR, m˜y=λRD, k1=ξd¯, k2=λdd1−ξλ, u˜=γthtk2/m˜xk1, v˜=1/m˜y, d¯=ηρλ, and λ=λSR=λRD=λℛℰ=λDE as the channel fading between nodes is independent and identical by default.
The asymptotic formula of COP adopting the PS scheme under the NLEH model isPCONL,γtotal⟶∞=1. Please refer to Appendix A for the derivation of asymptotic expressions of COP concerning the PS policy.
It can be found from the above analytical expressions that the COP eventually remains unchanged under the LEH model as the transmit SNR goes to infinity, while the COP is not reduced, but rises to reaching one instead under the NLEH model.


#### 3.1.2. SOP

The SOP is an indispensable indicator to measure the confidential performance, concerning the relationship between the channel capacity of theℛ−E link and the eavesdropping rate Re=Rt−Rs, where Rs is the secrecy rate and Rs≤Rt [36]. The SOP reflects in the security outage when the standard channel capacity of the ℛ−E link is higher than Re. The SOP is defined as(20)PSOl=PrΓEl⩾γthe,where γthe=22Re−1 is the secrecy rate threshold.Proposition 2.
The closed-form expression of SOP in the SWIPT system with FD destination-aided jamming under the PS scheme can be obtained as(21)PSOl=1mzmx2mx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where mz=γℛlλℛℰ, mw=a2γDλDE, a2=1−ργΘ+1, and b2=1−ργD. Here, xk is the kth root of Laguerre polynomial Lnx, wk=xk/N+12LN+1xk2 is the weight, and N is the nonnegative integer determining the accuracy and correctness of the calculation result.Proof.
See AppendixB.
Equation (21) is sophisticated and hard to understand, so the asymptotic-form expressions of SOP under both LEH and NLEH models are discussed as well. The theoretical expression of the SOP under the high SNRs in the LEH model is(22)PSOL,γtotal⟶∞=m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ. Here, Eix is the exponential integral, defined as Eix=∫−∞xet/tdt (Eq. 8.211.1 in [35]).
Under the NLEH model, the expression of SOP in the high SNRs isPSONL,γtotal⟶∞=0. Refer to Appendix B for the derivation process of asymptotic expressions of SOP accepting the PS policy.
Through the above work, the confidentiality interruption of the system can be analyzed accurately. Apparently, by rising the transmit SNR, SOP can achieve zero under the NLEH model, whereas the SOP in the LEH one stays nonnegative constant.
The TOP as a new outage metric is introduced in [21] to measure both the security and reliability of information transmission, which is defined as(23)PTOl=1−PrΓDl⩾γtht,ΓEl<γthe.
From the previous analysis, it is known thatΓDl and ΓEl are not independent of each other. By following the same derivation steps as COP and using the property of the cumulative distribution function (CDF), the analytical expression of TOP is(24)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
Equations (25) and (26) indicate the asymptotic-form expressions of the TOP of the SWIPT system with FD destination-aided jamming under the LEH and NLEH, respectively:(25)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,(26)PTONL,γtotal⟶∞=1.
The proof for the asymptotic TOP is the same as that of the asymptotic SOP, thus, the authors skip the proof for brevity. As stated above, the COP, SOP, and TOP can describe the statistical properties of the accessible secrecy performance, and the TOP plays a decisive role when considering both security and reliability.


#### 3.1.3. ESC

The ESC is another important index for measuring and analyzing the secrecy performance, which describes the average of the attainable secrecy rate for all possible channel conditions and stands for the maximum transmission rate when an eavesdropper cannot decode the confidential information that is being transmitted. The ESC can be stated as(27)R¯secl=ERsecl,where Rsecl=1/2log21+ΓDl−log21+ΓEl is the instantaneous secrecy rate and 1/2 is the valid time for the transmission of messages.According to Kalamkar and Banerjee [25], the expression in (27) has no closed-form and is intractable; inspired by it, the lower-bounded form of ESC is given by(28)R¯secl≥12ln2Eln1+ΓDl−Eln1+Γεl.Proposition 3.
The closed expression of the lower bound of ESC in the SWIPT network with PS-based relaying and FD jamming can be written as(29)R¯secl⩾12ln2K1−K2,where(30)K1=−∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzzk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.Proof.
See AppendixC.
The lower bound expression of ESC is too complicated to obtain intuitive information. Consequently, the asymptotic ESC when the transmit SNR goes to infinity is necessary. For the LEH model, the asymptotic form of ESC is(31)R¯secL,γtotal⟶∞⩾12ln2H1−H2,where (32)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xkH2=ln1+∑k=1Nwkm˜zxkd¯m˜zm˜zxk−m˜wλln1+d¯m˜zm˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the asymptotic expression of ESC in the four-node SWIPT-enabled network isR¯secNL,γtotal⟶∞=0. Refer to Appendix C for the derivation of asymptotic expressions of ESC under the PS policy.
From the asymptotic ESC under the LEH and NLEH models, the ESC in the NLEH mode is worsened to zero when the transmit SNR is extremely large. Therefore, it is irrational to increase the transmit SNR blindly to obtain high ESC. Although the closed expression of the lower bound of ESC presented in Proposition 3 is intricate, the closed-form one is able to achieve rapid calculation in popular mathematical software such as MATLAB, thereby providing an efficient method to acquire the ESC of the system and intuitive analysis of the effects about system parameters whereas avoiding the time-consuming Monte Carlo simulations.


### 3.2. Time-Switching Protocol

#### 3.2.1. COP

For the TS policy, the channel capacity isCM=1−α/2log21+ΓD, and the factor 1−α/2 stands stands for the effective time of the information transmission.Proposition 4.
The closed-form expression for the COP of the SWIPT system with cooperative jamming under the TS scheme can be obtained as(33)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, γtht=22Rt/1−α−1, mx=γSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+1γΘ+1, and b1=γℛl.Proof.
The proof obeys the same steps which are used to deduce the expression of COP under the PS strategy in Appendix A. Thus, we skip the proof for the TS policy for brevity.
Likewise, in the asymptotic case, the expression of COP under the LEH model is(34)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜y=λRD, u˜=γtht=k2/m˜xk1, v˜=1/m˜y, m˜x=λSR, k1=ξd¯, k2=λdd1−ξλ, and d¯=2ηαλ/1−α.
The asymptotic-form expression of COP under the NLEH mode in the SWIPT system with FD jamming isPCONL,γtotal⟶∞=1. Refer to Appendix D for the derivation of asymptotic expressions of COP under the TS policy.
It can be clearly found from (33) that the system adopting the TS strategy is more prone to cause a break between the source and destination than that adopting the PS strategy due to the higher threshold in the case of the same minimum transmission rate, which inspires us to choose the two schemes (TS and PS) according to the actual information transfer rate. In the actual parameter setting, increasing α is capable of raising the threshold γtht, which means meeting the requirements of secure communication between source and destination nodes is more difficult; hence, interruptions are more likely to happen.


#### 3.2.2. SOP

Proposition 5.
The SOP for the SWIPT system with TS-based relaying and FD destination-aided jamming is obtained as(35)PSOl=mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where γthe=22Re/1−α−1, b2=γD, a2=γΘ+1, mz=γℛlλℛℰ, and mw=a2γDλDE.Proof.
Similar to the derivation of SOP under the PS strategy.
In the high SNR regime withPtotal⟶∞, the expression for SOP employing the TS scheme under the LEH model is(36)PSOL,γtotal⟶∞=1m˜zm˜xk3γthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ.
Under the NLEH model, the asymptotic expression of SOP adopting the TS policy is written asPSONL,γtotal⟶∞=0. Refer to Appendix D for the derivation of asymptotic expressions of SOP under the TS scheme.
For the TS policy, the SOP under the NLEH model drops to zero as the transmit SNR goes to infinity, while SOP under the LEH converges to a deterministic value in the high SNR regime, which reminds us that increasing the transmit SNR appropriately can reduce the SOP in the practical nonlinear circuit design.
The expression for the TOP in the SWIPT with FD destination-aided jamming system utilizing the TS scheme is given by(37)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
The asymptotic form expressions for the TOP of the SWIPT-enabled network with the TS protocol under the LEH and NLEH models are(38)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,PTONL,γtotal⟶∞=1,respectively.
The above analysis of the TS policy alerts us to quantify both the reliability and security at both the legitimate and eavesdropper separately, that is, allocating appropriate time spending on gathering energy to ensure a reliable communication and enough transmit SNR to guarantee that self-jamming signals can protect confidential information from wiretapping.


#### 3.2.3. ESC

In the TS policy, the instantaneous secrecy rateRsec=1−α/2log21+ΓDl−log21+ΓEl, which shows that the effective time is related to the TS ratio α.Proposition 6.
The analytical expression of ESC for the SWIPT system with FD destination-aided jamming under the TS scheme is given by(39)R¯sec⩾1−α2ln2K1−K2,where(40)K1=−1my∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzxk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.
As the proof is similar to the derivation for the lower bound of ESC under the PS strategy, it is omitted for the sake of brevity.
Now, the case of the transmit SNR going to infinity is considered, and the lower bound of ESC under the TS policy is approximated as(41)R¯secL,γtotal⟶∞⩾1−α2ln2H1−H2,where(42)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xk,H2=ln1+∑k=1Nwkm˜zx˜jm˜zd¯m˜zxj−m˜wλln1+m˜zd¯m˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the analytical expression for ESC in the high SNR regime under the TS policy isR¯secNL,γtotal⟶∞=0. The proof is similar to the derivation for asymptotic expression ESC under the PS strategy, which is omitted here.
It can be clearly found from analytical expressions that the TS ratioα not only affects ESC by influencing the instantaneous SNR at the destination but also directly determines ESC as an effective time factor. Therefore, comparing with the PS strategy, the fluctuation in ESC under the TS scheme with α is more obvious. Hence, the TS ratio α must be carefully designed based on the actual environment to achieve the optimal ESC.


## 3.1. Power-Splitting Protocol

### 3.1.1. COP

The destination fails to recover the source message from the relay when the channel capacity of theℛ−D link is lower than the minimum transmission rate Rtbps/Hz, which means a connection interruption has occured. The channel capacity is defined as CM=1/2log21+ΓD in Shannon’s coding theorem. The factor 1/2 denotes the effective time of information transmission between the source and destination.Mathematically, the COP is defined as [21](17)PCOl=PrΓDl<γtht,where γtht=22Rt−1 is the minimum transmission threshold.Proposition 1.
The closed-form expression of the COP for the SWIPT system with FD destination-aided jamming under the PS scheme is given by(18)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, mx=1−ργSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+11−ργΘ+1, and b1=γℛl. Here, K1⋅ is the modified Bessel functions of the second kind (Eq. 8.407.2 in [35]).Proof.
See AppendixA.
To further capture the useful information for the secrecy performance, the asymptotic analysis of COPs is conducted in terms of LEH and NLEH models.
In the high SNR regime, i.e.,Ptotal⟶∞, the analytical expression of COP in the SWIPT-enabled relay system adopting the LEH model is given by(19)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜x=λSR, m˜y=λRD, k1=ξd¯, k2=λdd1−ξλ, u˜=γthtk2/m˜xk1, v˜=1/m˜y, d¯=ηρλ, and λ=λSR=λRD=λℛℰ=λDE as the channel fading between nodes is independent and identical by default.
The asymptotic formula of COP adopting the PS scheme under the NLEH model isPCONL,γtotal⟶∞=1. Please refer to Appendix A for the derivation of asymptotic expressions of COP concerning the PS policy.
It can be found from the above analytical expressions that the COP eventually remains unchanged under the LEH model as the transmit SNR goes to infinity, while the COP is not reduced, but rises to reaching one instead under the NLEH model.


### 3.1.2. SOP

The SOP is an indispensable indicator to measure the confidential performance, concerning the relationship between the channel capacity of theℛ−E link and the eavesdropping rate Re=Rt−Rs, where Rs is the secrecy rate and Rs≤Rt [36]. The SOP reflects in the security outage when the standard channel capacity of the ℛ−E link is higher than Re. The SOP is defined as(20)PSOl=PrΓEl⩾γthe,where γthe=22Re−1 is the secrecy rate threshold.Proposition 2.
The closed-form expression of SOP in the SWIPT system with FD destination-aided jamming under the PS scheme can be obtained as(21)PSOl=1mzmx2mx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where mz=γℛlλℛℰ, mw=a2γDλDE, a2=1−ργΘ+1, and b2=1−ργD. Here, xk is the kth root of Laguerre polynomial Lnx, wk=xk/N+12LN+1xk2 is the weight, and N is the nonnegative integer determining the accuracy and correctness of the calculation result.Proof.
See AppendixB.
Equation (21) is sophisticated and hard to understand, so the asymptotic-form expressions of SOP under both LEH and NLEH models are discussed as well. The theoretical expression of the SOP under the high SNRs in the LEH model is(22)PSOL,γtotal⟶∞=m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ. Here, Eix is the exponential integral, defined as Eix=∫−∞xet/tdt (Eq. 8.211.1 in [35]).
Under the NLEH model, the expression of SOP in the high SNRs isPSONL,γtotal⟶∞=0. Refer to Appendix B for the derivation process of asymptotic expressions of SOP accepting the PS policy.
Through the above work, the confidentiality interruption of the system can be analyzed accurately. Apparently, by rising the transmit SNR, SOP can achieve zero under the NLEH model, whereas the SOP in the LEH one stays nonnegative constant.
The TOP as a new outage metric is introduced in [21] to measure both the security and reliability of information transmission, which is defined as(23)PTOl=1−PrΓDl⩾γtht,ΓEl<γthe.
From the previous analysis, it is known thatΓDl and ΓEl are not independent of each other. By following the same derivation steps as COP and using the property of the cumulative distribution function (CDF), the analytical expression of TOP is(24)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
Equations (25) and (26) indicate the asymptotic-form expressions of the TOP of the SWIPT system with FD destination-aided jamming under the LEH and NLEH, respectively:(25)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,(26)PTONL,γtotal⟶∞=1.
The proof for the asymptotic TOP is the same as that of the asymptotic SOP, thus, the authors skip the proof for brevity. As stated above, the COP, SOP, and TOP can describe the statistical properties of the accessible secrecy performance, and the TOP plays a decisive role when considering both security and reliability.


### 3.1.3. ESC

The ESC is another important index for measuring and analyzing the secrecy performance, which describes the average of the attainable secrecy rate for all possible channel conditions and stands for the maximum transmission rate when an eavesdropper cannot decode the confidential information that is being transmitted. The ESC can be stated as(27)R¯secl=ERsecl,where Rsecl=1/2log21+ΓDl−log21+ΓEl is the instantaneous secrecy rate and 1/2 is the valid time for the transmission of messages.According to Kalamkar and Banerjee [25], the expression in (27) has no closed-form and is intractable; inspired by it, the lower-bounded form of ESC is given by(28)R¯secl≥12ln2Eln1+ΓDl−Eln1+Γεl.Proposition 3.
The closed expression of the lower bound of ESC in the SWIPT network with PS-based relaying and FD jamming can be written as(29)R¯secl⩾12ln2K1−K2,where(30)K1=−∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzzk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.Proof.
See AppendixC.
The lower bound expression of ESC is too complicated to obtain intuitive information. Consequently, the asymptotic ESC when the transmit SNR goes to infinity is necessary. For the LEH model, the asymptotic form of ESC is(31)R¯secL,γtotal⟶∞⩾12ln2H1−H2,where (32)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xkH2=ln1+∑k=1Nwkm˜zxkd¯m˜zm˜zxk−m˜wλln1+d¯m˜zm˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the asymptotic expression of ESC in the four-node SWIPT-enabled network isR¯secNL,γtotal⟶∞=0. Refer to Appendix C for the derivation of asymptotic expressions of ESC under the PS policy.
From the asymptotic ESC under the LEH and NLEH models, the ESC in the NLEH mode is worsened to zero when the transmit SNR is extremely large. Therefore, it is irrational to increase the transmit SNR blindly to obtain high ESC. Although the closed expression of the lower bound of ESC presented in Proposition 3 is intricate, the closed-form one is able to achieve rapid calculation in popular mathematical software such as MATLAB, thereby providing an efficient method to acquire the ESC of the system and intuitive analysis of the effects about system parameters whereas avoiding the time-consuming Monte Carlo simulations.


## 3.1.1. COP

The destination fails to recover the source message from the relay when the channel capacity of theℛ−D link is lower than the minimum transmission rate Rtbps/Hz, which means a connection interruption has occured. The channel capacity is defined as CM=1/2log21+ΓD in Shannon’s coding theorem. The factor 1/2 denotes the effective time of information transmission between the source and destination.Mathematically, the COP is defined as [21](17)PCOl=PrΓDl<γtht,where γtht=22Rt−1 is the minimum transmission threshold.Proposition 1.
The closed-form expression of the COP for the SWIPT system with FD destination-aided jamming under the PS scheme is given by(18)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, mx=1−ργSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+11−ργΘ+1, and b1=γℛl. Here, K1⋅ is the modified Bessel functions of the second kind (Eq. 8.407.2 in [35]).Proof.
See AppendixA.
To further capture the useful information for the secrecy performance, the asymptotic analysis of COPs is conducted in terms of LEH and NLEH models.
In the high SNR regime, i.e.,Ptotal⟶∞, the analytical expression of COP in the SWIPT-enabled relay system adopting the LEH model is given by(19)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜x=λSR, m˜y=λRD, k1=ξd¯, k2=λdd1−ξλ, u˜=γthtk2/m˜xk1, v˜=1/m˜y, d¯=ηρλ, and λ=λSR=λRD=λℛℰ=λDE as the channel fading between nodes is independent and identical by default.
The asymptotic formula of COP adopting the PS scheme under the NLEH model isPCONL,γtotal⟶∞=1. Please refer to Appendix A for the derivation of asymptotic expressions of COP concerning the PS policy.
It can be found from the above analytical expressions that the COP eventually remains unchanged under the LEH model as the transmit SNR goes to infinity, while the COP is not reduced, but rises to reaching one instead under the NLEH model.


## 3.1.2. SOP

The SOP is an indispensable indicator to measure the confidential performance, concerning the relationship between the channel capacity of theℛ−E link and the eavesdropping rate Re=Rt−Rs, where Rs is the secrecy rate and Rs≤Rt [36]. The SOP reflects in the security outage when the standard channel capacity of the ℛ−E link is higher than Re. The SOP is defined as(20)PSOl=PrΓEl⩾γthe,where γthe=22Re−1 is the secrecy rate threshold.Proposition 2.
The closed-form expression of SOP in the SWIPT system with FD destination-aided jamming under the PS scheme can be obtained as(21)PSOl=1mzmx2mx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where mz=γℛlλℛℰ, mw=a2γDλDE, a2=1−ργΘ+1, and b2=1−ργD. Here, xk is the kth root of Laguerre polynomial Lnx, wk=xk/N+12LN+1xk2 is the weight, and N is the nonnegative integer determining the accuracy and correctness of the calculation result.Proof.
See AppendixB.
Equation (21) is sophisticated and hard to understand, so the asymptotic-form expressions of SOP under both LEH and NLEH models are discussed as well. The theoretical expression of the SOP under the high SNRs in the LEH model is(22)PSOL,γtotal⟶∞=m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ. Here, Eix is the exponential integral, defined as Eix=∫−∞xet/tdt (Eq. 8.211.1 in [35]).
Under the NLEH model, the expression of SOP in the high SNRs isPSONL,γtotal⟶∞=0. Refer to Appendix B for the derivation process of asymptotic expressions of SOP accepting the PS policy.
Through the above work, the confidentiality interruption of the system can be analyzed accurately. Apparently, by rising the transmit SNR, SOP can achieve zero under the NLEH model, whereas the SOP in the LEH one stays nonnegative constant.
The TOP as a new outage metric is introduced in [21] to measure both the security and reliability of information transmission, which is defined as(23)PTOl=1−PrΓDl⩾γtht,ΓEl<γthe.
From the previous analysis, it is known thatΓDl and ΓEl are not independent of each other. By following the same derivation steps as COP and using the property of the cumulative distribution function (CDF), the analytical expression of TOP is(24)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
Equations (25) and (26) indicate the asymptotic-form expressions of the TOP of the SWIPT system with FD destination-aided jamming under the LEH and NLEH, respectively:(25)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,(26)PTONL,γtotal⟶∞=1.
The proof for the asymptotic TOP is the same as that of the asymptotic SOP, thus, the authors skip the proof for brevity. As stated above, the COP, SOP, and TOP can describe the statistical properties of the accessible secrecy performance, and the TOP plays a decisive role when considering both security and reliability.


## 3.1.3. ESC

The ESC is another important index for measuring and analyzing the secrecy performance, which describes the average of the attainable secrecy rate for all possible channel conditions and stands for the maximum transmission rate when an eavesdropper cannot decode the confidential information that is being transmitted. The ESC can be stated as(27)R¯secl=ERsecl,where Rsecl=1/2log21+ΓDl−log21+ΓEl is the instantaneous secrecy rate and 1/2 is the valid time for the transmission of messages.According to Kalamkar and Banerjee [25], the expression in (27) has no closed-form and is intractable; inspired by it, the lower-bounded form of ESC is given by(28)R¯secl≥12ln2Eln1+ΓDl−Eln1+Γεl.Proposition 3.
The closed expression of the lower bound of ESC in the SWIPT network with PS-based relaying and FD jamming can be written as(29)R¯secl⩾12ln2K1−K2,where(30)K1=−∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzzk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.Proof.
See AppendixC.
The lower bound expression of ESC is too complicated to obtain intuitive information. Consequently, the asymptotic ESC when the transmit SNR goes to infinity is necessary. For the LEH model, the asymptotic form of ESC is(31)R¯secL,γtotal⟶∞⩾12ln2H1−H2,where (32)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xkH2=ln1+∑k=1Nwkm˜zxkd¯m˜zm˜zxk−m˜wλln1+d¯m˜zm˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the asymptotic expression of ESC in the four-node SWIPT-enabled network isR¯secNL,γtotal⟶∞=0. Refer to Appendix C for the derivation of asymptotic expressions of ESC under the PS policy.
From the asymptotic ESC under the LEH and NLEH models, the ESC in the NLEH mode is worsened to zero when the transmit SNR is extremely large. Therefore, it is irrational to increase the transmit SNR blindly to obtain high ESC. Although the closed expression of the lower bound of ESC presented in Proposition 3 is intricate, the closed-form one is able to achieve rapid calculation in popular mathematical software such as MATLAB, thereby providing an efficient method to acquire the ESC of the system and intuitive analysis of the effects about system parameters whereas avoiding the time-consuming Monte Carlo simulations.


## 3.2. Time-Switching Protocol

### 3.2.1. COP

For the TS policy, the channel capacity isCM=1−α/2log21+ΓD, and the factor 1−α/2 stands stands for the effective time of the information transmission.Proposition 4.
The closed-form expression for the COP of the SWIPT system with cooperative jamming under the TS scheme can be obtained as(33)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, γtht=22Rt/1−α−1, mx=γSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+1γΘ+1, and b1=γℛl.Proof.
The proof obeys the same steps which are used to deduce the expression of COP under the PS strategy in Appendix A. Thus, we skip the proof for the TS policy for brevity.
Likewise, in the asymptotic case, the expression of COP under the LEH model is(34)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜y=λRD, u˜=γtht=k2/m˜xk1, v˜=1/m˜y, m˜x=λSR, k1=ξd¯, k2=λdd1−ξλ, and d¯=2ηαλ/1−α.
The asymptotic-form expression of COP under the NLEH mode in the SWIPT system with FD jamming isPCONL,γtotal⟶∞=1. Refer to Appendix D for the derivation of asymptotic expressions of COP under the TS policy.
It can be clearly found from (33) that the system adopting the TS strategy is more prone to cause a break between the source and destination than that adopting the PS strategy due to the higher threshold in the case of the same minimum transmission rate, which inspires us to choose the two schemes (TS and PS) according to the actual information transfer rate. In the actual parameter setting, increasing α is capable of raising the threshold γtht, which means meeting the requirements of secure communication between source and destination nodes is more difficult; hence, interruptions are more likely to happen.


### 3.2.2. SOP

Proposition 5.
The SOP for the SWIPT system with TS-based relaying and FD destination-aided jamming is obtained as(35)PSOl=mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where γthe=22Re/1−α−1, b2=γD, a2=γΘ+1, mz=γℛlλℛℰ, and mw=a2γDλDE.Proof.
Similar to the derivation of SOP under the PS strategy.
In the high SNR regime withPtotal⟶∞, the expression for SOP employing the TS scheme under the LEH model is(36)PSOL,γtotal⟶∞=1m˜zm˜xk3γthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ.
Under the NLEH model, the asymptotic expression of SOP adopting the TS policy is written asPSONL,γtotal⟶∞=0. Refer to Appendix D for the derivation of asymptotic expressions of SOP under the TS scheme.
For the TS policy, the SOP under the NLEH model drops to zero as the transmit SNR goes to infinity, while SOP under the LEH converges to a deterministic value in the high SNR regime, which reminds us that increasing the transmit SNR appropriately can reduce the SOP in the practical nonlinear circuit design.
The expression for the TOP in the SWIPT with FD destination-aided jamming system utilizing the TS scheme is given by(37)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
The asymptotic form expressions for the TOP of the SWIPT-enabled network with the TS protocol under the LEH and NLEH models are(38)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,PTONL,γtotal⟶∞=1,respectively.
The above analysis of the TS policy alerts us to quantify both the reliability and security at both the legitimate and eavesdropper separately, that is, allocating appropriate time spending on gathering energy to ensure a reliable communication and enough transmit SNR to guarantee that self-jamming signals can protect confidential information from wiretapping.


### 3.2.3. ESC

In the TS policy, the instantaneous secrecy rateRsec=1−α/2log21+ΓDl−log21+ΓEl, which shows that the effective time is related to the TS ratio α.Proposition 6.
The analytical expression of ESC for the SWIPT system with FD destination-aided jamming under the TS scheme is given by(39)R¯sec⩾1−α2ln2K1−K2,where(40)K1=−1my∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzxk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.
As the proof is similar to the derivation for the lower bound of ESC under the PS strategy, it is omitted for the sake of brevity.
Now, the case of the transmit SNR going to infinity is considered, and the lower bound of ESC under the TS policy is approximated as(41)R¯secL,γtotal⟶∞⩾1−α2ln2H1−H2,where(42)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xk,H2=ln1+∑k=1Nwkm˜zx˜jm˜zd¯m˜zxj−m˜wλln1+m˜zd¯m˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the analytical expression for ESC in the high SNR regime under the TS policy isR¯secNL,γtotal⟶∞=0. The proof is similar to the derivation for asymptotic expression ESC under the PS strategy, which is omitted here.
It can be clearly found from analytical expressions that the TS ratioα not only affects ESC by influencing the instantaneous SNR at the destination but also directly determines ESC as an effective time factor. Therefore, comparing with the PS strategy, the fluctuation in ESC under the TS scheme with α is more obvious. Hence, the TS ratio α must be carefully designed based on the actual environment to achieve the optimal ESC.


## 3.2.1. COP

For the TS policy, the channel capacity isCM=1−α/2log21+ΓD, and the factor 1−α/2 stands stands for the effective time of the information transmission.Proposition 4.
The closed-form expression for the COP of the SWIPT system with cooperative jamming under the TS scheme can be obtained as(33)PCOl=1−1mye−γtht/mx2μν1/2K12μν,where my=λRD, γtht=22Rt/1−α−1, mx=γSλSR, μ=γthta1/mxb1, ν=1/my, a1=kγDv+1γΘ+1, and b1=γℛl.Proof.
The proof obeys the same steps which are used to deduce the expression of COP under the PS strategy in Appendix A. Thus, we skip the proof for the TS policy for brevity.
Likewise, in the asymptotic case, the expression of COP under the LEH model is(34)PCOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜,where m˜y=λRD, u˜=γtht=k2/m˜xk1, v˜=1/m˜y, m˜x=λSR, k1=ξd¯, k2=λdd1−ξλ, and d¯=2ηαλ/1−α.
The asymptotic-form expression of COP under the NLEH mode in the SWIPT system with FD jamming isPCONL,γtotal⟶∞=1. Refer to Appendix D for the derivation of asymptotic expressions of COP under the TS policy.
It can be clearly found from (33) that the system adopting the TS strategy is more prone to cause a break between the source and destination than that adopting the PS strategy due to the higher threshold in the case of the same minimum transmission rate, which inspires us to choose the two schemes (TS and PS) according to the actual information transfer rate. In the actual parameter setting, increasing α is capable of raising the threshold γtht, which means meeting the requirements of secure communication between source and destination nodes is more difficult; hence, interruptions are more likely to happen.


## 3.2.2. SOP

Proposition 5.
The SOP for the SWIPT system with TS-based relaying and FD destination-aided jamming is obtained as(35)PSOl=mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk,where γthe=22Re/1−α−1, b2=γD, a2=γΘ+1, mz=γℛlλℛℰ, and mw=a2γDλDE.Proof.
Similar to the derivation of SOP under the PS strategy.
In the high SNR regime withPtotal⟶∞, the expression for SOP employing the TS scheme under the LEH model is(36)PSOL,γtotal⟶∞=1m˜zm˜xk3γthem˜y+m˜xk3⋅γtheλm˜wm˜xd¯k3eγtheλm˜w/m˜xm˜zd¯k3Ei−γtheλm˜wm˜xm˜zd¯k3+m˜z,where m˜z=λℛℰ, m˜w=λDE, and k3=ξ/1−ξ.
Under the NLEH model, the asymptotic expression of SOP adopting the TS policy is written asPSONL,γtotal⟶∞=0. Refer to Appendix D for the derivation of asymptotic expressions of SOP under the TS scheme.
For the TS policy, the SOP under the NLEH model drops to zero as the transmit SNR goes to infinity, while SOP under the LEH converges to a deterministic value in the high SNR regime, which reminds us that increasing the transmit SNR appropriately can reduce the SOP in the practical nonlinear circuit design.
The expression for the TOP in the SWIPT with FD destination-aided jamming system utilizing the TS scheme is given by(37)PTOl=1−1mye−γtht/mx2μν1/2K12μν+mx2mzmx+b2myγthee−γthe/mx∑k=1Nwkmz2xkγthemw+mxmzxke−γthea2/mxmzxk.
The asymptotic form expressions for the TOP of the SWIPT-enabled network with the TS protocol under the LEH and NLEH models are(38)PTOL,γtotal⟶∞=1−1m˜y2u˜v˜1/2K12u˜v˜+m˜xk3m˜zγthem˜y+m˜xk3⋅γtheλm˜wm˜xdk3eγtheλm˜w/m˜xm˜zdk3Ei−γtheλm˜wm˜xm˜zdk3+m˜z,PTONL,γtotal⟶∞=1,respectively.
The above analysis of the TS policy alerts us to quantify both the reliability and security at both the legitimate and eavesdropper separately, that is, allocating appropriate time spending on gathering energy to ensure a reliable communication and enough transmit SNR to guarantee that self-jamming signals can protect confidential information from wiretapping.


## 3.2.3. ESC

In the TS policy, the instantaneous secrecy rateRsec=1−α/2log21+ΓDl−log21+ΓEl, which shows that the effective time is related to the TS ratio α.Proposition 6.
The analytical expression of ESC for the SWIPT system with FD destination-aided jamming under the TS scheme is given by(39)R¯sec⩾1−α2ln2K1−K2,where(40)K1=−1my∑k=1Nwke1/mx1+a1/b1myxkEi−1mx1+a1b1myxkK2=ln1+mx∑k=1Nwkmzxkmjmzxk−mqlnmjmzxkmq,where mj=b2my+1 and mq=mw+a2.
As the proof is similar to the derivation for the lower bound of ESC under the PS strategy, it is omitted for the sake of brevity.
Now, the case of the transmit SNR going to infinity is considered, and the lower bound of ESC under the TS policy is approximated as(41)R¯secL,γtotal⟶∞⩾1−α2ln2H1−H2,where(42)H1=−∑k=1Nwke1/m˜xm˜yk4xkEi−1m˜xm˜yk4xk,H2=ln1+∑k=1Nwkm˜zx˜jm˜zd¯m˜zxj−m˜wλln1+m˜zd¯m˜zxk−m˜wλm˜wλ,where k4=k1/k2.
For the NLEH model, the analytical expression for ESC in the high SNR regime under the TS policy isR¯secNL,γtotal⟶∞=0. The proof is similar to the derivation for asymptotic expression ESC under the PS strategy, which is omitted here.
It can be clearly found from analytical expressions that the TS ratioα not only affects ESC by influencing the instantaneous SNR at the destination but also directly determines ESC as an effective time factor. Therefore, comparing with the PS strategy, the fluctuation in ESC under the TS scheme with α is more obvious. Hence, the TS ratio α must be carefully designed based on the actual environment to achieve the optimal ESC.


## 4. Discussion and Results

In this section, the correctness of the analytical expressions is demonstrated by Monte Carlo simulations and the impacts with respect to the PS/TS ratio and transmit SNR on the secrecy performance of the SWIPT-enabled system with FD destination-aided jamming are represented. As depicted in the following figures, theoretical results are in exact agreement with the numerical simulations, which shows that the closed expressions are correct. Unless otherwise specified, we set the default parameters according to [25, 37] as follows: η=0.7, ξ=0.5, dij=5m, ℒ = 2.7, ρ = 0.4, α = 0.4, σ2=−90dBm, Rt=3bps/Hz, and Rs=2.5bps/Hz. In the legend of the following figures, “Sim.” and “Ana.” stand for the Monte Carlo simulations and results of analytical expressions, respectively.


### 4.1. Transmit SNR

Figure4 plots the COP versus the transmit SNR under the LEH and NLEH models adopting both the PS- and TS-based schemes. For the NLEH model, the initial value of the COP decreases to the minimum at the beginning as the transmit SNR grows and then gives enlarged returns with further increase in the transmit SNR. For the LEH one, the original COP keeps decreasing with the SNR ascending until it converges to an exact value. The reason for the different variations of COP under the two EH models is that the information signal has a greater positive effect on the destination at first and benefits the reliability, while with the further rising of transmit SNR, the negative effect of the interference signal becomes more pronounced, which is reflected in the fact that the curve either reverses change or keeps equilibrium, after reaching the extreme value. One can also note that the COP under the TS scheme is larger than that under the PS.Figure 4
COP versus the transmit SNR.Figure5 examines that the SOP under different EH models varies quite significantly. The SOP in the LEH model increases rapidly with the ascending of the transmit SNR, attaining its maximum and keeping constant. By contrast, in the NLEH model, SOP gradually decreases to zero after reaching the maximum value. It is also discovered that the analytical results of SOPs can tally well with the asymptotic results as γtotal⟶∞ when the transmit SNR is high enough. Due to the addition of the self-interference signal, the overall SOP is relatively small.Figure 5
SOP versus the transmit SNR.Figure6 shows the TOP versus the transmit SNR under the LEH and NLEH models adopting both the PS- and TS-based schemes. The TOP is determined jointly by both the COP and SOP, and the trend of the TOP curve is similar to that of the COP in Figure 4, which inspires us to weigh the SOP and COP according to the actual situation to get desirable safety. One interesting observation is that the result of the TS-based scheme achieves lower than that of the PS-based scheme under the same condition.Figure 6
TOP versus the transmit SNR.The relationship between the ESC and transmit SNR is shown in Figure7. The ESC curve shows a positive growth before reaching the extreme value. After accomplishing the peak value, the ESC in the LEH model hardly changes, while the ESC of the NLEH one drops to zero gradually. This is because the negative influence of the interference signal on the ESC, and the positive influence of the source signal cancel out each other into a saturation region eventually under the LEH model. However, when the SNR is high, the effect of useful signals is slight, while the effect of noisy signals is dominant, resulting in a decrease in the ESC under the NLEH model.Figure 7
ESC versus the transmit SNR.From the above analysis, for different models, the higher transmit SNR does not signify better performance. Apparently, increasing the transmit SNR blindly may cause a waste of resources as well as performance degradation.


### 4.2. Power-Splitting Ratio (ρ)/Time-Switching Ratio (α)

Figure8 shows the impacts of the PS ratio ρ and TS ratio α on the COP. It can be clearly found from Figure 8(a) that the COPs under the LEH model are more sensitive to the PS ratio or TS ratio. Figure 8(b) shows that, (1) for the TS, the COP first goes down and then climbs up as α becomes large. This is because increasing α enhances transmission power at the relay, while cutting down the effective communication time. (2) For the PS, increasing ρ can better suppress the occurrence of connection interruption, especially, under the linear model. The reason is that a greater PS ratio ρ means the relay can obtain more energy and the effective communication time is unimpeded by ρ, which makes the end-to-end SNR at the destination higher.Figure 8
COP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure9 is provided to observe the impacts of the PS ratio ρ and TS ratio α on the SOP. In Figure 9(a), it can be obviously found that, (1) for the PS, the SOP has positive correlation with ρ. This is because increasing the PS ratio ρ means more power for energy harvesting and less power for information processing which continuously reduces the ability of the relay to receive signals, thereby making the SOP rapidly increase. (2) For the TS, once α crosses a fixed value, the SOP begins to degrade as α rises. The reason is that increasing α degrades the strength of the received signal at the relay and makes the eavesdropping channel of the relay worsen, which results in the SOP reducing. In Figure 9(b), the variation of SOP with α is nonlinear as the TS ratio α influences the whole communication processes.Figure 9
SOP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure10 plots the regularities of the TOP in the PS ratio ρ and TS ratio α. The curves of TOP are similar to that of the COP under the corresponding transmit SNR. This is because the TOP is the resultant of the COP and SOP.Figure 10
TOP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure11 illustrates the achievable ESC for both the PS and TS schemes versus the ρ and α, respectively. The ESC is limited by a secrecy capacity ceiling when ρ or α goes beyond a specific threshold. It should be noted that properly increasing α or ρ can effectively improve the ESC of the SWIPT system with FD destination-aided jamming. However, for the TS scheme, a high time conversion rate leads to short effective time of information transmission, and the ESC drops sharply.Figure 11
ESC versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.


## 4.1. Transmit SNR

Figure4 plots the COP versus the transmit SNR under the LEH and NLEH models adopting both the PS- and TS-based schemes. For the NLEH model, the initial value of the COP decreases to the minimum at the beginning as the transmit SNR grows and then gives enlarged returns with further increase in the transmit SNR. For the LEH one, the original COP keeps decreasing with the SNR ascending until it converges to an exact value. The reason for the different variations of COP under the two EH models is that the information signal has a greater positive effect on the destination at first and benefits the reliability, while with the further rising of transmit SNR, the negative effect of the interference signal becomes more pronounced, which is reflected in the fact that the curve either reverses change or keeps equilibrium, after reaching the extreme value. One can also note that the COP under the TS scheme is larger than that under the PS.Figure 4
COP versus the transmit SNR.Figure5 examines that the SOP under different EH models varies quite significantly. The SOP in the LEH model increases rapidly with the ascending of the transmit SNR, attaining its maximum and keeping constant. By contrast, in the NLEH model, SOP gradually decreases to zero after reaching the maximum value. It is also discovered that the analytical results of SOPs can tally well with the asymptotic results as γtotal⟶∞ when the transmit SNR is high enough. Due to the addition of the self-interference signal, the overall SOP is relatively small.Figure 5
SOP versus the transmit SNR.Figure6 shows the TOP versus the transmit SNR under the LEH and NLEH models adopting both the PS- and TS-based schemes. The TOP is determined jointly by both the COP and SOP, and the trend of the TOP curve is similar to that of the COP in Figure 4, which inspires us to weigh the SOP and COP according to the actual situation to get desirable safety. One interesting observation is that the result of the TS-based scheme achieves lower than that of the PS-based scheme under the same condition.Figure 6
TOP versus the transmit SNR.The relationship between the ESC and transmit SNR is shown in Figure7. The ESC curve shows a positive growth before reaching the extreme value. After accomplishing the peak value, the ESC in the LEH model hardly changes, while the ESC of the NLEH one drops to zero gradually. This is because the negative influence of the interference signal on the ESC, and the positive influence of the source signal cancel out each other into a saturation region eventually under the LEH model. However, when the SNR is high, the effect of useful signals is slight, while the effect of noisy signals is dominant, resulting in a decrease in the ESC under the NLEH model.Figure 7
ESC versus the transmit SNR.From the above analysis, for different models, the higher transmit SNR does not signify better performance. Apparently, increasing the transmit SNR blindly may cause a waste of resources as well as performance degradation.


## 4.2. Power-Splitting Ratio (ρ)/Time-Switching Ratio (α)

Figure8 shows the impacts of the PS ratio ρ and TS ratio α on the COP. It can be clearly found from Figure 8(a) that the COPs under the LEH model are more sensitive to the PS ratio or TS ratio. Figure 8(b) shows that, (1) for the TS, the COP first goes down and then climbs up as α becomes large. This is because increasing α enhances transmission power at the relay, while cutting down the effective communication time. (2) For the PS, increasing ρ can better suppress the occurrence of connection interruption, especially, under the linear model. The reason is that a greater PS ratio ρ means the relay can obtain more energy and the effective communication time is unimpeded by ρ, which makes the end-to-end SNR at the destination higher.Figure 8
COP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure9 is provided to observe the impacts of the PS ratio ρ and TS ratio α on the SOP. In Figure 9(a), it can be obviously found that, (1) for the PS, the SOP has positive correlation with ρ. This is because increasing the PS ratio ρ means more power for energy harvesting and less power for information processing which continuously reduces the ability of the relay to receive signals, thereby making the SOP rapidly increase. (2) For the TS, once α crosses a fixed value, the SOP begins to degrade as α rises. The reason is that increasing α degrades the strength of the received signal at the relay and makes the eavesdropping channel of the relay worsen, which results in the SOP reducing. In Figure 9(b), the variation of SOP with α is nonlinear as the TS ratio α influences the whole communication processes.Figure 9
SOP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure10 plots the regularities of the TOP in the PS ratio ρ and TS ratio α. The curves of TOP are similar to that of the COP under the corresponding transmit SNR. This is because the TOP is the resultant of the COP and SOP.Figure 10
TOP versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.Figure11 illustrates the achievable ESC for both the PS and TS schemes versus the ρ and α, respectively. The ESC is limited by a secrecy capacity ceiling when ρ or α goes beyond a specific threshold. It should be noted that properly increasing α or ρ can effectively improve the ESC of the SWIPT system with FD destination-aided jamming. However, for the TS scheme, a high time conversion rate leads to short effective time of information transmission, and the ESC drops sharply.Figure 11
ESC versusρ/α. (a) Transmit SNR = 60 dB. (b) Transmit SNR = 110 dB.


## 5. Conclusion

We have investigated the secrecy performance of the relay system with FD destination-aided jamming, where both the LEH and NLEH models have been considered in the energy-constrained relay. The closed-form expressions for the COP, SOP, TOP, and lower bound of ESC have been derived selecting both the PS and TS protocols. Monte Carlo simulations confirmed the correctness of the derived expression, and analytical results reveal the influence of network parameters on the secrecy performance, which provides some insights in the actual design. For instance, increasing the transmission SNR does not necessarily improve the security and reliability of the system and may even worsen them. Enhancing the intensity of the AN signal at the destination can benefit the security of the system, but can erode the reliability. The energy harvesting model adopted by the relay has a great influence on the secrecy performance of the system. For the future work, we would introduce the intelligent reflecting surface into our network to design a security scheme and achieve the optimal power transfer efficiency and secrecy capacity. In addition, it would be interesting to analyze the secrecy performance in the context of multi-users scenarios with the intelligent reflecting surface.



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*Source: 6193012-2021-07-08.xml*