Exploiting Direct Link in Two-Way Half-Duplex Sensor Network over Block Rayleigh Fading Channel: Upper Bound Ergodic Capacity and Exact SER Analysis

Relay communication, in which the relay forwards the signal received by a source to a destination, has a massive consideration in research, due to its ability to expand the coverage, increase the capacity, and reduce the power consumption. In this paper, we proposed and investigated energy harvesting (EH) based two-way half-duplex (TWHD) relaying sensors network using selection combining (SC) over block Rayleigh fading channel. In this model, we proposed the direct link between two sources for improving the system performance. For the system performance analysis, we investigated and derived the closed-form of the exact and upper bound Ergodic capacity (EC) and the exact form of the symbol error ratio (SER). By using the Monte Carlo simulation, the correctness of the research results is verified in the influence of the main system parameters. From the discussions, we can see that the analytical and simulation agree well with each other.

the single antenna receiver station for energy and information via power splitting protocol studied in [14,15]. In [16], the energy harvesting and information transmission in the wireless multi-antenna communication systems are investigated via a simplified three-node setup. The new wireless RF powered network with the H-AP harvested energy in the downlink and transferred the information in the uplink by TDMA is proposed and demonstrated in [17].
Furthermore, the cooperative relaying network has been deeply studied in the last ten years for improving the system performance of the communication network. The source and destination nodes can be communicated by helping the intermediate relay when the direct link between them is weak or with the long distance. The authors in [18] considered and investigated the outage probability of the three-node cooperative relaying network with a fundamental switching between energy harvesting and data relaying. The relay selection problem in the AF relay network with QoS and harvested energy constraints is proposed in [19]. In this paper, the authors investigated the dependence of the system ergodic capacity and the outage probability on the amount of energy transferred to the RF energy harvesters. In [20], the authors investigated joint wireless information and energy transfer in a two-user MIMO interference channel, in which each receiver either decodes the incoming information data or harvests the RF energy. Further, joint wireless information and energy transfer methods in a general K-user MIMO interference channel was studied in [21]. A wireless cooperative network with multiple source-destination pairs communicate with each other via an energy harvesting relay is presented in [22], and authors investigated the relay's strategies to distribute the harvested energy among the multiple users and their impact on the system performance. In [23], the authors focused on the two-way amplify-and-forward relaying channels with an energy harvesting relay node. Here, the relay node harvests energy from signals from two sources and uses this energy for information transferring between them with amplify-and-forward mode. For multiple-input multiple-output relay channels [24], proposed a low complexity dynamic antenna switching between information decoding and energy harvesting based on the principles of the generalized selection combiner. The authors in [25] proposed and studied a cooperative multi-hop secured transmission protocol to underlay cognitive radio networks by formulating an effective signal-to-interference-plus-noise ratio (SINR) as well as secrecy capacity under the constraints of the maximum transmit power, the interference threshold, and the hardware impairment level. Furthermore [26], presented a partial relay selection (PRS) protocol to enhance the secrecy performance for cooperative cognitive radio networks (CRNs). For this purpose, the authors investigated the secrecy outage probability (SOP) and the probability of non-zero secrecy capacity (NSC) of the proposed network system. As shown in [27], the problem of designing a good strategy for EH in AF wireless relay systems is considered, and the combination between the wireless power transfer and cooperative jamming (CJ) is studied in [28] for enhancing the physical security in public transportation networks. Full duplex (FD) cognitive radio network, implementing the technique of energy harvesting (EH), is proposed and studied in [29]. And the authors in [30] investigated how to maximize the energy efficiency of the BS while making full use of the relay's renewable energy and meeting the average throughput requirement. From that point of view, the system performance analysis of the communication cooperative relaying network is the hot research direction in our time.
In this paper, we proposed and investigated Energy Harvesting (EH) based Two-Way Half-Duplex (TWHD) relaying cooperative sensor network using selection combining over block Rayleigh fading channel. Firstly, we proposed the system model with two sources S 1 , S 2, and one intermediate relay R. Then, we investigated the system performance in terms of the ergodic capacity (EC) and SER. Finally, all the mathematical analytical expressions are verified by Monte Carlo simulation, and the influence of some main system parameters on the system performance is demonstrated. From the discussions, we can see that the analytical and simulation agree well with each other. The main contribution of this research can be pointed out as the followings: (1) Energy harvesting based two-way half-duplex relaying cooperative network using selection combining over block Rayleigh fading channel is proposed and investigated (2) The closed-form of the upper bound EC and exact SER of the model system is derived.
(3) All the results are convinced by Monte Carlo simulation in connection with all primary system parameters.
The rest of this manuscript can be drawn as follows. We provide the system model, the energy and information transfer phases in Section 2. The closed-form expressions of system upper bound EC and SER are derived in Section 3. We introduce the results and some discussions in Section 4. In the last section, some conclusions are proposed.

Relaying Network Model
In this section, the energy harvesting based two-way half-duplex relaying sensor cooperative network is drawn in Figure 1. In this system model, sources are denoted as S 1 and S 2 , and the relay is R. We assume that all links between them are available and are block Rayleigh fading channels. The EH and information transformation (IT) for this proposed model system are illustrated in Figure 2. In this protocol, the transmission is divided into blocks of length T, which consists of three-time slots. In the first time slot T/3, the R harvests energy ρP 1 from the source node S 1, and the source uses the energy (1−ρ)P 1 for information transmission to R and S 2 (here 0 < ρ < 1: is the power splitting factor). In the second interval time T/3, the R harvests energy ρP 2 from the source node S 2, and the source S 2 uses the energy (1−ρ)P 2 for information transmission to R and S 1 . Finally, the remaining time slot T/3 is used for information transferring from the R to the source nodes S 1 and S 2 [30][31][32][33][34][35].
Sensors 2020, 20, x FOR PEER REVIEW 3 of 17 (2) The closed-form of the upper bound EC and exact SER of the model system is derived.
(3) All the results are convinced by Monte Carlo simulation in connection with all primary system parameters.
The rest of this manuscript can be drawn as follows. We provide the system model, the energy and information transfer phases in Section 2. The closed-form expressions of system upper bound EC and SER are derived in Section 3. We introduce the results and some discussions in Section 4. In the last section, some conclusions are proposed.

Relaying Network Model
In this section, the energy harvesting based two-way half-duplex relaying sensor cooperative network is drawn in Figure 1. In this system model, sources are denoted as S1 and S2, and the relay is R. We assume that all links between them are available and are block Rayleigh fading channels. The EH and information transformation (IT) for this proposed model system are illustrated in Figure 2. In this protocol, the transmission is divided into blocks of length T, which consists of three-time slots. In the first time slot T/3, the R harvests energy ρP1 from the source node S1, and the source uses the energy (1−ρ)P1 for information transmission to R and S2 (here 0 1    : is the power splitting factor).
In the second interval time T/3, the R harvests energy ρP2 from the source node S2, and the source S2 uses the energy (1−ρ)P2 for information transmission to R and S1. Finally, the remaining time slot T/3 is used for information transferring from the R to the source nodes S1 and S2 [30][31][32][33][34][35].     (2) The closed-form of the upper bound EC and exact SER of the model system is derived.
(3) All the results are convinced by Monte Carlo simulation in connection with all primary system parameters.
The rest of this manuscript can be drawn as follows. We provide the system model, the energy and information transfer phases in Section 2. The closed-form expressions of system upper bound EC and SER are derived in Section 3. We introduce the results and some discussions in Section 4. In the last section, some conclusions are proposed.

Relaying Network Model
In this section, the energy harvesting based two-way half-duplex relaying sensor cooperative network is drawn in Figure 1. In this system model, sources are denoted as S1 and S2, and the relay is R. We assume that all links between them are available and are block Rayleigh fading channels. The EH and information transformation (IT) for this proposed model system are illustrated in Figure 2. In this protocol, the transmission is divided into blocks of length T, which consists of three-time slots. In the first time slot T/3, the R harvests energy ρP1 from the source node S1, and the source uses the energy (1−ρ)P1 for information transmission to R and S2 (here 0 1    : is the power splitting factor).
In the second interval time T/3, the R harvests energy ρP2 from the source node S2, and the source S2 uses the energy (1−ρ)P2 for information transmission to R and S1. Finally, the remaining time slot T/3 is used for information transferring from the R to the source nodes S1 and S2 [30][31][32][33][34][35]. Figure 1. The proposed system model.

Energy Harvesting Phase
Let S 1 transmits the symbol x 1 in the first phase. The received signal at the relay node R and source node S 2 can be expressed, respectively, as where E |x 1 | 2 = P 1 , E{•} is expectation operator and P 1 represents the average transmit power at the S 1 . Further, n I r and n I 2 denote the zero-mean additive white Gaussian noise (AWGN) with variance N 0 and h 1,R , h 2,R are the channel gain of S 1 -R and S 2 -R links, respectively.
The harvested energy at the relay node can be given as where 0 < η ≤ 1 is energy conversion efficiency and 0 < ρ < 1 is the power splitting factor. In the second phase, the source node S 2 will transmit the symbol x 2 to the nodes R and S 1 . Therefore, the received signals at the R and S 1 can be expressed, respectively, as where E |x 2 | 2 = P 2 , P 2 represents the average transmit power at the S 2 . Further, we assume that n II r and n II 1 are the zero-mean additive white Gaussian noise (AWGN) with variance N 0 . Similar to the first phase, the total harvested energy at the relay node can be obtained as We assume that the average transmit power from source S 1 and S 2 is equal. So, the Equation (4) can be rewritten as where we denote P 1 = P 2 = P. Therefore, the average transmit power at the relay node can be given as

Information Transmission Phase
In the first phase, after doing EH, S 1 will broadcast the information to the R node and S 2 with remaining power (1 − ρ)P. Hence, the received signal at the R node and S 2 node can be expressed, respectively, as where h 1,2 is the channel gain of S 1 -S 2 link.
Similar to the first phase, the received signal at the R node and S 1 node can be given in the second phase, respectively, as where h 2,1 is the channel gain of S 2 -S 1 link. Finally, in the third phase, the received signal at the source S 1 and S 2 can be expressed, respectively, as where we denote E |x R | 2 = P R . And h R,1 , h R,2 are the channel gain of R-S 1 and R-S 2 links, respectively. In AF technique, in order to ensure that the transmission power at R is P R , the amplifying coefficient χ can be chosen as From Equation (9), the received signal at the source S 1 can be rewritten as where n III 1 denote the zero-mean additive white Gaussian noise (AWGN) with variance N 0 . Replace Equations (7) and (8) into (11); finally, we have where n r = n I r + n II r denote the total AWGN at the relay with variance N 0 . This signal contains both messages x 1 and x 2 , while the only x 2 is the desired signal at x 1 . Since node x 1 perfectly knows its transmitted symbol x 1 , it can eliminate the corresponding self-interference term χh R,1 h 1,R 1 − ρx 1 from y III 1 . Therefore, Equation (12) can be rewritten as From Equation (13), the signal to noise ratio (SNR) of S 2 -R-S 1 link can be calculated as Replace (10) into (14), and after doing some algebra, we have the final form where we denote ψ = P 2 N 0 = P N 0 , ϕ 1 = h R,1 2 , ϕ 2 = h 2,R 2 .
In the second phase, S 2 will transmit the data to S 1 directly, from Equation (8) the received signal destination can be given as where we denote ϕ 3 = h 2,1 2 .
Finally, using the selection combining (SC) at the receiver S 1 , the end to end SNR of AF mode at the source S 1 can be obtained as

Upper Bound Ergodic Capacity Analysis
The EC of the proposed system can be formulated as the following It is easy to observe that Equation (18) is tough to compute in the closed-form expression. So, we will calculate it in the UPPER bound analysis. From Equation (18), EC in UPPER bound can be given as the following equation where C UP = log 2 1 + E γ AF e2e . Here, we have In order to calculate Equation (20), we have to find F γ AF e2e (x). So, combined with Equation (17), (x) can be given as Firstly, we consider P 1 where λ 1 , λ 2 are the mean of the random variable (RV) ϕ 1 , ϕ 2 , respectively. Applying Equation (3.324.1) in [36], Equation (22) can be reformulated as Secondly, P 2 can be obtained by the following equation where λ 3 is the mean of RV ϕ 3 . Substituting Equations (23) and (24) into (21), we can obtain F γ AF e2e (x) as followings where K v (•) is the modified Bessel function of the second kind and vth order.

Derivate Equation (25) by using the formula
Substituting Equation (26) into (20), we have The first term of Equation (27) can be derived as where we demote And then applying Equation (6.643,3) in [36], A 2 can be reformulated as where Γ(•) is the Gamma function and W(•) is the Whittaker function.
The second term of Equation (27) can be derived as where we denote By applying Equation (6.643, 3) in [36], we can calculate B 1, B 2 , B 3, and B 4 , respectively, as follows: where W(•) is the Whittaker function which can be defined in [36].

SER Analysis
In this section, we obtain new expressions for the SER at the destination. We first consider the outage probability, which was obtained in [31]. Thus, we have ∞ t e −x 2 /2 dx is the Gaussian Q-function, while a and b are constants, which are specific for modulation type. (a, b) = (1, 1) for binary phase-shift keying (BPSK) and (a, b) = (1, 2) for Quadrature Phase Shift Keying (QPSK). As a result, before obtaining the SER performance, the distribution function of γ AF e2e is expected. Then, we begin rewriting the SER expression given in Equation (37) directly in terms of outage probability at the source S 1 by using integration, as follows Substituting Equation (25) into (38), we have where we denote Apply Equation (3.361,2) in [36], I 1 and I 2 can be obtained as, respectively and Moreover, I 3 and I 4 can be denoted as Applying Equation (6.614,4) in [36], I 3 and I 4 can be claimed as, respectively where Γ(•) is the Gamma function and W(•) is the Whittaker function.
In this section, we investigate the impact of ψ on the system performance of the proposed system in cases with and without the direct link between the S 1 and S 2 nodes. The Figure 3 shows the system EC versus ψ in the presence of the direct link between the S 1 and S 2 sources. In Figure 3, we set some primary system parameters as η = 0.8, λ 1 = 5, λ 2 = λ 4 = 10, λ 3 = 2, = 0.5, and 0.85. In this simulation, we consider both the exact and upper bond EC in the influence of ψ as shown in Figure 3. From Figure 3, we can state that both the exact and upper bond system EC rise while we vary ψ from 10 dB to 10 dB and the exact EC is higher than the upper bond system EC with all ψ values. Moreover, the analytical expression of the exact and upper bond system EC in the above section is verified by the simulation results using Monte Carlo Simulation. Furthermore, the comparison system EC in cases with and without a direct link between the sources S 1 and S 2 is illustrated in Figure 4 with the primary system parameters as η = 0.5 and 0.85, respectively. In the same way with the above case, the system EC significantly increases with rising ψ from 10 dB to 10 dB in both cases with and without a direct link between the sources S 1 and S 2 as in Figure 4. In addition, the system EC in the case with a direct link is better than in the case without a direct link between two sources. It can be observed that the direct link can lead to more useful information transmission in the proposed system. In the model system with the direct link, system has two way to transfer the information as via helping relay and direct link. With two way of information transmission via relay and direct link, this case is effective in information transferring in comparison with the case only with helping of the relay as in [45]. Further, the analytical and the simulation results match well for all possible values of ψ as shown in Figure 4.
In order to further observe the effect of power splitting coefficient on the exact and upper bond system EC, Figure 5 shows the exact and upper bond system EC versus with the main system parameters as the follows η = 0.8, λ 1 = 5, λ 2 = λ 4 = 10, λ 3 = 2, ψ = 5, and 10 dB, respectively. In this case, we vary the power splitting coefficient from 0 to 1. From the result, we can see that the exact and upper bond system EC increases significantly to the optimal values while increases to 0.4, and after that falls up from the optimal values with the rising of to 1. This is the case because more energy used for the harvesting energy process can lead to an increase of the system EC. Still, the over-harvesting energy process can cause less information transmission and lead to the falling of the system EC as shown in Figure 5. In addition, the difference of the upper and exact maximum ergodic capacity as shown in Figure 3 with = 0.5 is 0.4681 bps/Hz (≈9.5%) 0.4681 bps/Hz (≈9.5%) and = 0.85 is 0.3927 bps/Hz (≈9.1%); and in Figure 5, the difference of the upper and exact maximum ergodic capacity with ψ = 5dB is 0.4398 bps/Hz (≈12.7%) and with ψ = 10dB is 0.51 bps/Hz (≈10.2%).
Furthermore, the comparison of the system EC of the cases with and without a direct link between the sources S 1 and S 2 is drawn in Figure 6 with η = 0.5 and 0.85, respectively. As shown in Figure 6, the system EC in the case with the presence of the direct link is better than the case without in connection with the better information and energy transmission processes with the direct link between the sources. In both Figures 5 and 6, the simulation and the analytical values are the same with all values of to confirm the analytical analysis in the above section. Sensors 2020, 20, x FOR PEER REVIEW 13 of 17             Finally, Figures 7 and 8 show the effect of the ψ and ρ on the system SER, respectively. Here, we set the main system parameters as η = 0.8, λ1 = 5, λ2 = λ4 = 10, λ3 = 2, ψ = 10 dB for Figure 8 and ρ = 0.5 for Figure 7, respectively. Moreover, the system SER significantly falls while the ψ varies from −10 dB to 10 dB as shown in Figure 7. From the results in Figure 8, it can be seen that the system SER decreases with ρ varies from 0.1 to 0.5, and after that increases with ρ from 0.5 to 1. The optimal value of system SER can be obtained with ρ from 0.4 to 0.6. It can be observed from Figures 6 and 7 that the analytical results have an agreement to the simulations for both cases to verify the correctness of the above analytical section.  Finally, Figures 7 and 8 show the effect of the ψ and on the system SER, respectively. Here, we set the main system parameters as η = 0.8, λ 1 = 5, λ 2 = λ 4 = 10, λ 3 = 2, ψ = 10 dB for Figure 8 and = 0.5 for Figure 7, respectively. Moreover, the system SER significantly falls while the ψ varies from −10 dB to 10 dB as shown in Figure 7. From the results in Figure 8, it can be seen that the system SER decreases with varies from 0.1 to 0.5, and after that increases with from 0.5 to 1. The optimal value of system SER can be obtained with from 0.4 to 0.6. It can be observed from Figures 6 and 7 that the analytical results have an agreement to the simulations for both cases to verify the correctness of the above analytical section.
Finally, Figures 7 and 8 show the effect of the ψ and ρ on the system SER, respectively. Here, we set the main system parameters as η = 0.8, λ1 = 5, λ2 = λ4 = 10, λ3 = 2, ψ = 10 dB for Figure 8 and ρ = 0.5 for Figure 7, respectively. Moreover, the system SER significantly falls while the ψ varies from −10 dB to 10 dB as shown in Figure 7. From the results in Figure 8, it can be seen that the system SER decreases with ρ varies from 0.1 to 0.5, and after that increases with ρ from 0.5 to 1. The optimal value of system SER can be obtained with ρ from 0.4 to 0.6. It can be observed from Figures 6 and 7 that the analytical results have an agreement to the simulations for both cases to verify the correctness of the above analytical section.

Conclusions
In this paper, we investigated the EH Based TWHD relaying sensor network system using selection combining over block Rayleigh fading channel. In this model, we propose the direct link between two sources for improving the system performance. For the system performance analysis, we investigate and derived the closed-form of the upper bound EC and the exact form of the SER. By using the Monte Carlo simulation, the correctness of the research results is verified in the influence of the main system parameters. From the discussions, we can see that the analytical and simulation agree well with each other. This research can be considered as a novel recommendation for a communication cooperative relaying sensor network.