# Hybrid TSR–PSR Alternate Energy Harvesting Relay Network over Rician Fading Channels: Outage Probability and SER Analysis

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## Abstract

**:**

## 1. Introduction

## 2. System Model

_{1}, R

_{2}and the destination D as shown in Figure 1. Each device works with a single antenna and in a half-duplex (HD) mode, and there is no direct link between R and D. In this model, S, and D have their own stable power supplies, while R1 and R2 operate with EH and alternately forward source data according to the AF protocol. We denote that h

_{1}and h

_{2}are the fading channel gains from the source to relays, g

_{1}and g

_{2}are the fading channel gain from the relays R

_{1}and R

_{2}to the destination D, and h

_{12}and h

_{21}are the gain factors between R

_{1}and R

_{2}, respectively [26,27,28,29].

_{1}and R

_{2}. After that, S transfers the part of energy $\rho {P}_{s}$ ($0\le \rho <1$) to R

_{1}in the next $(1-\alpha )T/2$ interval time and uses $(1-\rho ){P}_{s}$ energy to transfer information to R

_{1}. In the same interval time, R

_{2}harvests energy from S. In the final $(1-\alpha )T/2$ interval time, R

_{1}transfers information to D and R

_{2}harvests energy from R

_{1}[14,25,26,29].

_{1}forwards the source information data to D by using its energy harvested in the current T blocks and the previous T blocks. Please note that R

_{1}and R

_{2}always harvest energy from the received RF signals in all of the first T blocks. In the following T blocks, R

_{2}works as a helping relay, while R

_{1}harvests energy in all T blocks by overhearing the transmissions from S and R

_{2}. The EH and data relay of R

_{2}are performed similarly to the above procedure for R

_{1}. Thus, R

_{1}and R

_{2}will alternately forward source data in every T block. Compared with the TS-EH or PS-EH-based single-relay system, more energy can be harvested by relays in our protocol for the DT [14,26,29].

## 3. System Performance

_{1}and R

_{2}in $\alpha T$ and $(1-\alpha )T/2$ blocks. In the $(1-\alpha )T/2$ block, R

_{1}allocates $0\le \rho <1$ ($\rho $ is the power splitting factor) as part of the received source signal for the energy harvesting (EH). Therefore, the total harvested energy at R

_{1}can be given by

_{2}can be given by the equation below:

_{1}, after splitting the $\rho $ part of the received signal for the EH at the relays, the remaining signal at R

_{1}can be obtained as

_{0}at R

_{1}, and $E\left\{{\left|{x}_{s}\right|}^{2}\right\}={P}_{s}$ in which $E\{\u2022\}$ is expectation operator.

_{1}amplifies and forwards the signal to D in the next stage. The transmitted signal from R

_{1}can be expressed as

_{0}at D, $E\left\{{\left|{x}_{r1}\right|}^{2}\right\}={P}_{r1}$, and ${P}_{r1}$ is the average transmitted power of R

_{1}.

_{1}performs the delay-tolerant (DT) transmission mode, the end to end signal to noise ratio (SNR) at D can be calculated as

_{0}<< P

_{r}, the end to end SNR can be obtained:

_{2}can harvest energy from S in $\alpha T+(1-\alpha )T/2$ blocks, i.e., $\frac{T}{2}(1+\alpha )$ blocks, and R

_{2}can also harvest energy from R

_{1}in $(1-\alpha )T/2$ blocks. Therefore, the total harvested energy at R

_{2}when R

_{1}joins in the data transmission (DT) can be calculated by

_{2}, the total harvested energy at R

_{1}when R

_{2}joins in the DT can be obtained as

_{2}.

_{1}in the previous T blocks and current T blocks, the total harvested energy of R

_{1}for DT can be obtained as

_{1}, the total harvested energy of R

_{2}also can be obtained as the following:

_{1}can be calculated as

_{2}also can be obtained as

#### 3.1. Exact Outage Probability Analysis

**Theorem 1**(Exact Outage Probability)

**.**

#### 3.2. Asymptotic Outage Probability Analysis

#### 3.3. SER (Symbol Error Ratio) Analysis

**Theorem 2**(Exact SER)

**.**

**Theorem 3**(Asymptotic SER Analysis)

**.**

## 4. Numerical Results and Discussion

^{6}random samples of each channel gain, which are Rician distributed. The analytical curve and the simulation curve should match to verify the correctness of our analysis. All simulation parameters are listed in Table 2.

_{s}/N

_{0}= 10 dB, ρ = 0.5 and α = 0.5. From the results, we see that the outage probability decreases remarkably while η varies from 0 to 1. The research results show that the numerical results and simulation results match exactly with each other, validating the correctness of the theoretical analysis in the above section. Furthermore, the function of the outage probability to K is presented in Figure 4. Similarly, we set P

_{s}/N

_{0}= 10 dB, ρ = 0.5 and α = 0.5, and the outage probability decreases remarkably while K varies from 0 to 4. Once again, the simulation results and theoretical results agree well with each other.

_{s}/N

_{0}. In Figure 5, both the exact and asymptotic outage probability in cases TSR, PSR, and TSR–PSR are illustrated. The main parameters are set as R = 0.5, ρ = 0.3 and α = 0.3. From the results, the exact outage probability decreases and comes close to the asymptotic line when the ratio P

_{s}/N

_{0}increases from 0 to 20 dB. On another hand, the influence of R on the outage probability in three cases—TSR, PSR, TSR–PSR—is investigated in Figure 6 with P

_{s}/N

_{0}= 15 dB, ρ = 0.7 and α = 0.3. The outage probability significantly increases with R from 0 to 4. From Figure 5 and Figure 6, the analytical results and the simulation results match well with each other for all values of R and P

_{s}/N

_{0}.

_{s}/N

_{0}= 10 dB. It is clearly shown that the outage probability increases with increasing α and ρ, and the minimum outage probability can be obtained with α = 0 and ρ = 1. Moreover, SER versus the ratio P

_{s}/N

_{0}in three cases—TSR, PSR, and TSR–PSR—is shown in Figure 8. Furthermore, Figure 9 plots the comparison of the exact and asymptotic outage probability of three cases—TSR, PSR, and TSR–PSR—versus P

_{s}/N

_{0}. The results indicate that all the simulation and analytical values agreed well with each other.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Theorem 1—Exact Outage Probability

## Appendix B. Proof of Theorem 2—Exact SER Analysis

## Appendix C. Proof of Theorem 3—Asymptotic SER Analysis

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**Figure 1.**The system model. The green, red and black lines represent the first-hop and second-hop transmission, respectively. The green and dashed back lines represent the data transmission (DT) and energy harvesting (EH), respectively. (

**a**) R

_{1}-DT and R

_{2}-EH; (

**b**) R

_{2}-DT and R

_{1}-EH.

**Figure 2.**The information transmission and energy harvesting process. (

**a**) R

_{1}-DT and R

_{2}-EH; (

**b**) R

_{2}-DT and R

_{1}-EH.

Symbol | Definition |
---|---|

$0<\eta <1$ | Energy conversion efficiency |

$0\le \alpha <1$ | Time-switching factor |

$0\le \rho <1$ | Power-splitting factor |

P_{s}/N_{0} | Source-power-to-noise ratio |

K | Rician K-factor |

${\lambda}_{1}$ | Mean of ${\left|{h}_{1}\right|}^{2}$ |

${\lambda}_{2}$ | Mean of ${\left|{g}_{1}\right|}^{2}$ |

R | Source rate |

E_{r}_{1} | Harvested energy at relay 1 |

P_{r}_{1} | Average transmit power of relay 1 |

E_{r}_{2} | Harvested energy at relay 2 |

P_{r}_{2} | Average transmit power of relay 2 |

P_{out} | Outage probability |

${\gamma}_{e2e1}$ | End to end signal to noise ratio |

${K}_{v}(\u2022)$ | Modified Bessel function of the second kind and vth order |

$\mathsf{\Gamma}(\u2022)$ | Gamma function |

$F\left(\upsilon ,\beta ;\gamma ;z\right)$ | Hypergeometric function |

SER | Symbol error ratio |

$\beta $ | Amplifying factor |

$Q(t)$ | Gaussian Q-function |

P_{s} | Transmit power of the source |

T | Total time of processing |

Symbol | Values |
---|---|

$0<\eta \le 1$ | 0.7 |

${\lambda}_{1}$ | 1 |

${\lambda}_{2}$ | 1 |

P_{s}/N_{0} | 0:20 dB |

K | 3 |

R | 0.5 bps |

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## Share and Cite

**MDPI and ACS Style**

Nguyen, T.N.; Tran Tin, P.; Ha, D.H.; Voznak, M.; Tran, P.T.; Tran, M.; Nguyen, T.-L.
Hybrid TSR–PSR Alternate Energy Harvesting Relay Network over Rician Fading Channels: Outage Probability and SER Analysis. *Sensors* **2018**, *18*, 3839.
https://doi.org/10.3390/s18113839

**AMA Style**

Nguyen TN, Tran Tin P, Ha DH, Voznak M, Tran PT, Tran M, Nguyen T-L.
Hybrid TSR–PSR Alternate Energy Harvesting Relay Network over Rician Fading Channels: Outage Probability and SER Analysis. *Sensors*. 2018; 18(11):3839.
https://doi.org/10.3390/s18113839

**Chicago/Turabian Style**

Nguyen, Tan N., Phu Tran Tin, Duy Hung Ha, Miroslav Voznak, Phuong T. Tran, Minh Tran, and Thanh-Long Nguyen.
2018. "Hybrid TSR–PSR Alternate Energy Harvesting Relay Network over Rician Fading Channels: Outage Probability and SER Analysis" *Sensors* 18, no. 11: 3839.
https://doi.org/10.3390/s18113839