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

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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][2][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][5][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][9][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. erefore, 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]. e 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][13][14][15]. destination with jamming is investigated. e source, destination, and eavesdropper are considered to be under the uninterruptible power supplies. e 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. irdly, 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]. e 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. e 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 signalto-noise ratios (SNRs).
(iii) e 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. e simulation results imply that the optimal secrecy performance of the system can be achieved by optimizing the four metrics.
e rest of this paper is structured as follows. Section 2 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. Figure 1, the SWIPTenabled relay network is composed of four nodes, i.e., one source S, one AF relay R, one destination D, and one eavesdropper E. e 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. e 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].

Channel Description.
e 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]. e channel power gain between the nodes i and j is defined as |h ij | 2 , which is the exponential distribution with mean λ ij � d − L ij , where i and j ∈ S, R, D, E { }, i ≠ j, d ij is the distance between nodes i and j, and L is the path-loss exponent. e channel coefficients for S − R, R − E, R − D, and D − E links are represented as h SR , h RE , h RD , and h DE , respectively. e residual self-interference (RSI) channel coefficient at the destination is denoted as h DD , 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 σ 2 i -variance. e variance is formulated as σ 2 i � kP v r , where the two constants, i.e., k > 0 and v ∈ [0, 1], depend on the interference cancellation scheme at the relay [29].

Secure Transmission.
e communication process of the system includes energy harvesting, signal processing, and information transmitting, which are shown in Figure 1. 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].

Power-Splitting-Based Relaying.
e PS-based scheme for the secure communication in the SWIPT system with FD jamming is illustrated in Figure 2, where the source-todestination communication occurs in a slot of duration T. e slot is divided into two phases equally. e source transmits the information signal through power P S toward the relay, and the destination sends the jamming signal through power P D to the relay, where P S � ξP total and P D � (1 − ξ)P total are the power from the source and destination separately, ξ ∈ (0, 1) is the power allocation coefficient, and P total is the total system power. e 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.
For the energy harvesting at the relay based on the PSbased protocol, the linear EH and nonlinear EH (NLEH) models at R are discussed as following. Normally, the most existing research studies are based on the LEH model for it is tractable [21,31]. e linear energy collection at the relay is where η(0 < η < 1) represents the energy conversion efficiency. e forwarding power at the relay for transmitting signals is written as where E r � P S |h SR | 2 + P D |h RD | 2 . e energy conversion efficiency actually relies on the input power, which means that the energy collected is nonlinear with the input power [32]. e 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 where p 1 � c 1 − c 2 /c 3 , p 2 � c 3 , and E · { } is the expectation operator and c 1 , c 2 , and c 3 are constants related to the detailed circuit specifications such as the resistance and capacitance [20]. e specific values of c 1 , c 2 , and c 3 are 0.3929, 0.01675, and 0.04401, respectively [33]. e nonlinear energy harvesting at the relay can be expressed as Hence, the power to forward signals at R is given by After energy harvesting, the information processing is discussed next. e received signal at the relay in the first phase is expressed as where n R stands for the additive white Gaussian noise (AWGN) at the relay with zero mean and variance σ 2 .
Next, the AF relay transmits the amplified version of the received signal which is given by where l ∈ L, NL { } represents the LEH and NLEH, respectively.
e received signals at D and E with destination-aided jamming can be, respectively, expressed as where n D and n E denote the AWGNs at the destination and eavesdropper with zero mean and variance σ 2 , respectively. It is supposed that the destination can eliminate the AN signal by obtaining the exact information of the channel gain of the R − 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 where c S � P S /σ 2 , c l R � P l R /σ 2 , c D � P D /σ 2 , c Θ � E r /σ 2 , and kc v D is the residual self-interference. Similarly, the received SNR at E is obtained by the following equation:

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). e 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 − α)T/2 period, and the relay sends the information signal to the destination at the remaining time (1 − α)T/2 in the third phase. Both the LEH and NLEH models for the TS scheme are discussed in the following part.
For the aforementioned TS policy, the harvested energy during the period of αT in the first phase is given by E L H � ηαTE r . us, the power at R sending the amplified signal to the destination is written as P L R � 2ηαE r /(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 E NL H � αTp 1 E r /(E r + p 2 ), and the relay's transmit power forwarding the information to destination is given as In the second phase, the relay performs information processing after energy collections, where the signal received by the relay becomes en, the signal sent by the relay in the third phase is e received signals at D 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 Likewise, the received SNR at E is represented by

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.

Power-Splitting Protocol
e destination fails to recover the source message from the relay when the channel capacity of the R − D link is lower than the minimum transmission rate R t (bps/Hz), which means a connection interruption has occured. e channel capacity is defined as C M � 1/2log 2 (1 + Γ D ) in Shannon's coding theorem. e factor 1/2 denotes the effective time of information transmission between the source and destination.
Mathematically, the COP is defined as [21] where c t th � 2 2R t − 1 is the minimum transmission threshold.

Proposition 1. e closed-form expression of the COP for the SWIPT system with FD destination-aided jamming under the PS scheme is given by
where is the modified Bessel functions of the second kind (Eq. 8.407.2 in [35]).
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., P total ⟶ ∞, the analytical expression of COP in the SWIPT-enabled relay system adopting the LEH model is given by where 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.
e SOP is an indispensable indicator to measure the confidential performance, concerning the relationship between the channel capacity of the R − E link and the eavesdropping rate R e � R t − R s , where R s is the secrecy rate and R s ≤ R t [36]. e SOP reflects in the security outage when the standard channel capacity of the R − E link is higher than R e . e SOP is defined as where c e th � 2 2R e − 1 is the secrecy rate threshold.

Proposition 2.
e closed-form expression of SOP in the SWIPT system with FD destination-aided jamming under the PS scheme can be obtained as Energy harvesting Signal processing Information transmitting where 2 is the weight, and N is the nonnegative integer determining the accuracy and correctness of the calculation result.
Proof. See Appendix B.
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.
e theoretical expression of the SOP under the high SNRs in the LEH model is where Under the NLEH model, the expression of SOP in the high SNRs is P NL,c total⟶∞ SO � 0. Refer to Appendix B for the derivation process of asymptotic expressions of SOP accepting the PS policy.
rough 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. e TOP as a new outage metric is introduced in [21] to measure both the security and reliability of information transmission, which is defined as From the previous analysis, it is known that Γ l D and Γ l E 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 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: e 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. e 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. e ESC can be stated as where 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 Proposition 3.

e closed expression of the lower bound of ESC in the SWIPT network with PS-based relaying and FD jamming can be written as
where where m j � b 2 m y + 1 and m q � m w + a 2 .
Proof. See Appendix C. e 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 where where For the NLEH model, the asymptotic expression of ESC in the four-node SWIPT-enabled network is R sec NL,c total⟶∞ � 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. erefore, 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.  e closed-form expression for the COP of the SWIPT system with cooperative jamming under the TS scheme can be obtained as (1/2) where Proof. e proof obeys the same steps which are used to deduce the expression of COP under the PS strategy in Appendix A. us, we skip the proof for the TS policy for brevity.
Likewise, in the asymptotic case, the expression of COP under the LEH model is where e asymptotic-form expression of COP under the NLEH mode in the SWIPT system with FD jamming is P NL,c total⟶∞ CO � 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 c t th , which means meeting the requirements of secure communication between source and destination nodes is Security and Communication Networks more difficult; hence, interruptions are more likely to happen.

Proposition 5. e SOP for the SWIPTsystem with TS-based relaying and FD destination-aided jamming is obtained as
where Proof. Similar to the derivation of SOP under the PS strategy.
In the high SNR regime with P total ⟶ ∞, the expression for SOP employing the TS scheme under the LEH model is where m z � λ RE , m w � λ DE , and k 3 � ξ/ (1 − ξ). Under the NLEH model, the asymptotic expression of SOP adopting the TS policy is written as P NL,c total⟶∞ SO � 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. e expression for the TOP in the SWIPT with FD destination-aided jamming system utilizing the TS scheme is given by respectively. e 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 selfjamming signals can protect confidential information from wiretapping.

ESC.
In the TS policy, the instantaneous secrecy rate , which shows that the effective time is related to the TS ratio α.

Proposition 6.
e analytical expression of ESC for the SWIPT system with FD destination-aided jamming under the TS scheme is given by where where m j � b 2 m y + 1 and m q � m w + a 2 .
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 where where For the NLEH model, the analytical expression for ESC in the high SNR regime under the TS policy is R sec NL,c total⟶∞ � 0. e 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 ratio α not only affects ESC by influencing the instantaneous SNR at the destination but also directly determines ESC as an effective time factor. erefore, 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.

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, d ij � 5m, L = 2.7, ρ = 0.4, α = 0.4, σ 2 � − 90 dBm, R t � 3 bps/Hz, and R s � 2.5 bps/Hz. In the legend of the following figures, "Sim." and "Ana." stand for the Monte Carlo simulations and results of analytical expressions, respectively. Figure 4 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. e 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 5 examines that the SOP under different EH models varies quite significantly. e 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 Security and Communication Networks results as c total ⟶ ∞ when the transmit SNR is high enough. Due to the addition of the self-interference signal, the overall SOP is relatively small. Figure 6 shows the TOP versus the transmit SNR under the LEH and NLEH models adopting both the PS-and TSbased schemes. e 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 PSbased scheme under the same condition. e relationship between the ESC and transmit SNR is shown in Figure 7. e 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. is 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.

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. Figure 8 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.

Power-Splitting Ratio (ρ)/Time-Switching Ratio (α).
is 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. e 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 9 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. is 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. e 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 10 plots the regularities of the TOP in the PS ratio ρ and TS ratio α. e curves of TOP are similar to that of the COP under the corresponding transmit SNR. is is because the TOP is the resultant of the COP and SOP. Security and Communication Networks Figure 11 illustrates the achievable ESC for both the PS and TS schemes versus the ρ and α, respectively. e 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.

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 energyconstrained relay. e 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. e 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.
Appendix A e proof for the analytical expressions of COP under the PS-based scheme in general and asymptotic cases is in the following.
We let X � (1 − ρ)c S |h SR | 2 and Y � |h RD | 2 . According to the properties of CDF and the probability density function (PDF), X and Y follow the exponential distribution, where E X { } � m x and E Y { } � m y . Hence, the COP in (17) is rewritten as In this way, the issue of obtaining the closed-form of COP is turned to a mathematical problem of two-dimensional probability calculation. Letting μ � c t th a 1 /m x b 1 and ] � 1/m y and utilizing Eq. 3.324.1 in [35], the analytical expression of COP is represented as To get the asymptotic expression of COP, the instantaneous SNR at the destination with P total ⟶ ∞ is written as where c total � P total /σ 2 . Based on equations (2) and (5), one can get According to (9), P L,c total⟶∞ CO is described as where X and Y follow the exponential distributions, E X � m x , and E Y � m y . e tilde ∼ is attached on the variables in the asymptotic analysis to distinguish from those in the precise analysis.

B
e proof for the analytical expressions of SOP under the PSbased scheme in general case and asymptotic case is in the following.
Similar to the way of deriving the closed-form expression of COP, the following exponential distributions Z � c l R |h RE | 2 and W � a 2 c D |h DE | 2 are introduced, where the means of Z and W are c l R λ RE and a 2 c D λ DE , respectively. us, the SOP in (20) is rewritten as After further mathematical manipulations, P SO is expressed as

(B.2)
To compute the complicated integrals in (B.2), the Gauss-Laguerre quadrature is utilized, which is an extension of the Gaussian quadrature method for approximating the value of integrals and defined as where x i is the i-th root of Laguerre polynomial L n (x) as L n (x) � e x d n /dx n (x n e − x ); the weight ω i is given by Abramowitz and Stegun [38] as ω i � x i /(n + 1) 2 [L n+1 (x i ) 2 ], and N determines the precision and accuracy of the calculation result [39]. Based on equation (B.3), the closed-form expression of P SO is approximated as To certify asymptotic-form expressions of the SOP, the end-to-end SNR at the eavesdropper when the transmit SNR goes to infinity is obtained as  According to Jensen's inequality, E ln(1 + Γ l ε ) ⩽ ln(1 + E(Γ l E )) is obtained since ln(1 + x) is a concave function with respect to x. Substituting X, Y, Z, and W into E ln(1 + Γ l E ) , it is formulated as For the TS-based protocol, as c total tends to infinity, the end-to-end SNR at the destination is

Data Availability
No data were used to support the findings of the study.

Conflicts of Interest
e authors declare that they have no conflicts of interest.