Physical Layer Security in a Hybrid TPSR Two ‐ Way Half ‐ Duplex Relaying Network over a Rayleigh Fading Channel: Outage and Intercept Probability Analysis

: In this paper, the system performance of a hybrid time–power splitting relaying (TPSR) two ‐ way half ‐ duplex (HD) relaying network over a Rayleigh fading channel is investigated in terms of the outage probability (OP) and intercept probability (IP). The proposed model has two sources, A and B, which communicate with each other with the help of an intermediate relay (R) under the presence of an eavesdropper (E). The physical layer security (PLS) was considered in this case. Firstly, we derived the closed ‐ form expressions of the exact and asymptotic IP in two cases, using MRC (maximal ratio combining) and SC (selection combining) techniques. The closed ‐ form expressions of the system OP was then analyzed and derived. All the analytical expressions of the OP and IP of the system model were verified by a Monte Carlo simulation in connection with all the main system parameters. In the research results, the analytical and simulation values were in total agreement, demonstrating the correctness of the system performance analysis.

(2) The closed form expressions of the exact and asymptotic IP in two cases (with MRC (maximal ratio combining) techniques and SC (selection combining) techniques), and the closed form expression of the system OP, are derived.
(3) The correctness of the analytical expressions is verified using a Monte Carlo simulation. The rest of this paper is organized as follows. The hybrid TPSR two-way HD relaying communication network is presented in the second section. The system OP and IP are analyzed and derived in the third section. The research results and some discussion are provided in the fourth section. Finally, the conclusion is proposed in the last section of the paper.

Network Model
In this section, a hybrid TPSR two-way HD relaying network over a Rayleigh fading channel is described (Figure 1). Two sources, A and B, communicate with each other with the help of a relay (R), in the presence of an eavesdropper (E). In this system model, all the block-fading channels are Rayleigh fading channels. The Energy Harvesting (EH) and Information Transmission (IT) phases in the interval T are displayed in Figure 2. In the first half-interval, αT, R harvests energy ((1−β)PA) from source A, and the source transfers the information βPA to R, where α is the time-switching factor, β is the power splitting factor, and 0 0.5    and 0 1    . Actually, we can consider different values of β for A-to-R and B-to-R links, i.e., β1 for A-to-R and β2 for B-to-R. The analysis should be the same as it is here. However, the resulting formula may be more complex. Without loss of generality, we assume the same β for both A-to-R and B-to-R links, to make the result simpler and more readable.
A B R E Figure 1. The relaying network model.

Energy Harvesting Phase
In the first phase, T  , A transmits the message A x with the power A P and the signal received at R can be calculated as follows: where     is the expectation operator.
The EH relay employs a fixed power-splitting factor, , to split the received RF power into two parts: is used for information transmission, and the remaining power, The amount of harvested energy during the first and second phase can be formulated as The average transmitted power at the relay can be calculated by where

Information Transmission Phase
In the first phase, after completing EH, A will broadcast the information to the node R and to B with the remaining power Hence, the signal received at R can be given as the below equation: Similar to the first phase, the signal received at R in the second phase can be expressed as . In our proposed model, we considered the amplify and forward (AF) mode. In order to ensure that the transmission power at R is PR, the amplifying coefficient  was chosen as follows: Combining Equation (7) with Equations (5) and (6), the received signal at the A node can be rewritten as Substituting Equation (7) into Equation (10) Therefore, Equation (11) can be reformulated as The signal to noise ratio (SNR) at node A can be formulated as Substituting Equation (5) into Equation (9) and then doing some algebra. In this paper, we assume that the channels are reciprocal, so BR . Hence, the end-to-end SNR at A can be obtained by where In the total information transmission phase, while R will receive both legitimated messages, A x and B x , from source nodes A and B, E will also overhear the information transmitted by the A and B nodes. Thus, the signal received at E can be obtained by where and AE BE h h are the channel gain of the A-E and B-E links, respectively, and E n is AWGN with variance N0. During the broadcast signal phase, E also overhears the information from R. Hence, the received signal at E can be given by In our model, we analyzed the intercept (IP) at node A. Because our system model was symmetrical, we could analyze at either node A or node B. Therefore, using Equations (15) and (16), the SNR at E (when decoding a successful message from node A) in the two different phases can be determined, respectively, by

Outage Probability (OP)
The outage probability at node A can be defined by where Using Equation (14), Equation (19) can be rewritten as   Applying eq (3.324,1) to the table of the integral [35], Equation (20) can be reformulated as  (17) and (18). Hence, the endto-end SNR at E can be obtained by

Intercept Probability
From Equations (14) and (12), we can now claim the capacities at node E are The IP can be calculated as where th C is a predetermined threshold.
Combining Equations (22) and (23), Equation (24) can be rewritten as Comment: We assume that all of the channels are Rayleigh fading channels. Firstly, in order to find the IP in Equation (22), we have to determine the cumulative distribution function (CDF) and probability density function (PDF) of X, Y, respectively, as follows:

Lemma 1. Exact Analysis
Substituting Equations (26) and (28) We can now solve for 1 P and 2 P .
By using results from Equation (27), 1 P can be obtained as Next, we consider 2 P . Using Equation (29) where , , where 2 P  was defined in Equation (32).

Lemma 2. Asymptotic Analysis
At the high SNR regime      , the IP can be approximated by   It is easy to observe from Equations (26) and (27) that we can obtain the CDF of X and Y at a high SNR, respectively, as follows: The PDF of Y can be obtained from Equation (35): Substituting Equations (35) and (36) into Equation (34), the IP, in this case, can be determined by where , The integral in Equation (37) is not complicated and in easy to compute. Hence, we can obtain

Case 2: Instantaneous End-to-End SNR at E under SC Technique
With the SC technique, E will choose the max SNR from Equations (17) and (18). Hence, the endto-end SNR at E can be expressed by

Lemma3. Exact analysis
The IP can be given by   From Equations (26) and (27), we have Lemma4. Asymptotic analysis From Equation (31), the asym IP can be given by Remark. The position of the E is integrated into the channel gains , and Communication between A and B with the assistance of R does not require knowledge about the position of the E. The position of E only affects the IP of E.
In Figure 3, the system IP is considered as a function of the time switching factor α, with the main system parameters set as Cth = 0.5 bps/Hz, ψ = 10 dB and β = 0.5, 0.85, respectively. In this simulation stage, we vary the time switching factor α from 0 to 0.5, as shown in Figure 3. As shown in Figure 3, the system IP decreases to 0 as α changes from 0 to 0.3, and remains at 0 with further rises of α. In Figure 3, we considered both cases using SC and MRC techniques. When α increases, SNR threshold also increases, because  (23)). Therefore, IP decreases.
The IP versus the power splitting factor β, is also plotted in Figure 4. In this figure, we set the primary system parameters as Cth = 0.5 bps/Hz, α = 0.3, and ψ = 10 dB, 15 dB, respectively, and vary the power splitting factor β from 0 to 1. From Figure 4, we can see that the system IP has a critical increase with the rising of β. It can be observed that when β increases, the SNR at E in phase three (Equation (18)) increases. Therefore, IP increases. In addition, the system IP of the MRC technique is better than the system IP of the SC technique as shown in Figures 3 and 4. Moreover, the simulation produces an agreement between the analytical curves (drawn in Figures 3 and 4), demonstrating the correctness of the system performance analysis described in the above section.  In Figure 5, we investigated the influence of ψ on the system IP, with the main system parameters as Cth = 0.5 bps/Hz, β = 0.5 in both cases, using SC and MRC techniques. From the research results, we can see that the exact system IP significantly rises while ψ varies from 0 to 25, and then retains a constant value near the asymptotic system IP. When SNR increases (i.e., when the power of sources A and B increases), the harvested energy at R also increases. Therefore, the transmit power of all sources and relays increases, leading to an increase in IP. This can be verified using Equations (17) and (18). As in Figures 3 and 4, we can see that the simulation and analytical values are the same, and the values obtained with the MRC technique are better than those obtained with the SC technique for verifying the correctness of the system performance analysis described in the third section. Moreover, the system OP versus the time switching factors α and ψ is displayed in Figures 5 and  6, respectively. Here, the main system parameters are set at Cth = 0.5 bps/Hz, ψ = 10 dB, η = 0.8, and β = 0.5, 0.85 for Figure 6, and Cth = 0.5 bps/Hz, α = 0.3, β = 0.5 and η = 0.5, 1 for Figure 7, respectively. From Figure 6, it can be observed that the system OP decreases greatly with a rise of α from 0.05 to 0.25 and then increases greatly as α rises from 0.25 to 0.45. The optimal value for system OP is obtained with α near 0.2-0.25, as shown in Figure 6. In the same way, the system OP decreases greatly with rising ψ, as shown in Figure 7. In both Figures 6 and 7, we can state that the simulation and analytical curves do not differ, demonstrating the correctness of the analytical section of this paper.  Finally, the system OP versus β is presented in Figure 8, with the main system parameters set as Cth = 0.5 bps/Hz, ψ = 10 dB, η = 1, and α = 0.3, 0.15, respectively. In Figure 8, we can see that the system OP decreases greatly as α rises from 0 to 0.5 and then increases greatly as α rises from 0.5 to 1. The optimal value of the system OP is obtained with α near 0.5, as shown in Figure 8. Again, the simulation and analytical curves do not differ, demonstrating the correctness of the analysis.

Conclusions
In this paper, the system performance of a hybrid TPSR two-way HD relaying network over a Rayleigh fading channel was investigated, in terms of the system OP and IP. The closed form expressions of the exact and asymptotic IP in two cases (with MRC (maximal ratio combining) and SC (selecting combining) techniques) were derived to analyze the system performance. Moreover, the closed form expression of the system OP was analyzed and derived. The correctness of all the analytical expressions of OP and IP in the system model was verified using a Monte Carlo simulation, in connection with all main system parameters. The results show that the analytical and simulation values agree well with each other.