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

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## Abstract

**:**

## 1. Introduction

_{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.

## 2. Relaying Network Model

_{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 $0<\rho <1$: 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].

#### 2.1. Energy Harvesting Phase

_{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

_{1}represents the average transmit power at the S

_{1}. Further, ${n}_{r}^{I}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{n}_{2}^{I}$ 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.

_{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

_{2}represents the average transmit power at the S

_{2}. Further, we assume that ${n}_{r}^{II}\text{\hspace{0.17em}}\mathrm{and}\text{\hspace{0.17em}}{n}_{1}^{II}$ are the zero-mean additive white Gaussian noise (AWGN) with variance N

_{0}.

_{1}and S

_{2}is equal. So, the Equation (4) can be rewritten as

#### 2.2. Information Transmission Phase

_{1}will broadcast the information to the R node and S

_{2}with remaining power $(1-\rho )P$. Hence, the received signal at the R node and S

_{2}node can be expressed, respectively, as

_{1}-S

_{2}link.

_{1}node can be given in the second phase, respectively, as

_{2}-S

_{1}link.

_{1}and S

_{2}can be expressed, respectively, as

_{1}and R-S

_{2}links, respectively.

_{R}, the amplifying coefficient $\chi $ can be chosen as

_{1}can be rewritten as

_{0}.

_{0}.

_{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 $\chi {h}_{R,1}{h}_{1,R}\sqrt{1-\rho}{x}_{1}$ from ${y}_{1}^{III}$. Therefore, Equation (12) can be rewritten as

_{2}-R-S

_{1}link can be calculated as

_{2}will transmit the data to S

_{1}directly, from Equation (8) the received signal destination can be given as

_{1}, the end to end SNR of AF mode at the source S

_{1}can be obtained as

## 3. Upper Bound Ergodic Capacity and Exact SER Analysis

#### 3.1. Upper Bound Ergodic Capacity Analysis

_{1}

_{2}can be obtained by the following equation

_{2}can be reformulated as

_{1,}B

_{2}, B

_{3,}and B

_{4}, respectively, as follows:

#### 3.2. SER Analysis

_{1}by using integration, as follows

_{1}and I

_{2}can be obtained as, respectively

_{3}and I

_{4}can be denoted as

_{3}and I

_{4}can be claimed as, respectively

_{1}can be obtained as

## 4. Numerical Results and Discussion

_{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, ρ = 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.

_{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 ρ 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 ρ increases to 0.4, and after that falls up from the optimal values with the rising of ρ 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 ρ = 0.5 is 0.4681 bps/Hz ($\approx $9.5%) 0.4681 bps/Hz ($\approx $9.5%) and ρ = 0.85 is 0.3927 bps/Hz ($\approx $9.1%); and in Figure 5, the difference of the upper and exact maximum ergodic capacity with $\psi =5\mathrm{dB}$ is 0.4398 bps/Hz ($\approx $12.7%) and with $\psi =10\mathrm{dB}$ is 0.51 bps/Hz ($\approx $10.2%).

_{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 Figure 5 and Figure 6, the simulation and the analytical values are the same with all values of ρ to confirm the analytical analysis in the above section.

_{1}= 5, λ

_{2}= λ

_{4}= 10, λ

_{3}= 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 Figure 6 and Figure 7 that the analytical results have an agreement to the simulations for both cases to verify the correctness of the above analytical section.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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## Share and Cite

**MDPI and ACS Style**

Tin, P.T.; Nguyen, T.N.; Tran, M.; Trang, T.T.; Sevcik, L.
Exploiting Direct Link in Two-Way Half-Duplex Sensor Network over Block Rayleigh Fading Channel: Upper Bound Ergodic Capacity and Exact SER Analysis. *Sensors* **2020**, *20*, 1165.
https://doi.org/10.3390/s20041165

**AMA Style**

Tin PT, Nguyen TN, Tran M, Trang TT, Sevcik L.
Exploiting Direct Link in Two-Way Half-Duplex Sensor Network over Block Rayleigh Fading Channel: Upper Bound Ergodic Capacity and Exact SER Analysis. *Sensors*. 2020; 20(4):1165.
https://doi.org/10.3390/s20041165

**Chicago/Turabian Style**

Tin, Phu Tran, Tan N. Nguyen, Minh Tran, Tran Thanh Trang, and Lukas Sevcik.
2020. "Exploiting Direct Link in Two-Way Half-Duplex Sensor Network over Block Rayleigh Fading Channel: Upper Bound Ergodic Capacity and Exact SER Analysis" *Sensors* 20, no. 4: 1165.
https://doi.org/10.3390/s20041165