# Energy Harvesting over Rician Fading Channel: A Performance Analysis for Half-Duplex Bidirectional Sensor Networks under Hardware Impairments

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## Abstract

**:**

## 1. Introduction

- The exact form expression of outage probability and achievable throughput at each destination node with imperfect hardware and in Rician fading environment are derived mathematically.
- We derive the exact-form cumulative distribution function (CDF) of the SNR at each destination node, and use this result to derive the integral exact-form of the SER at destination nodes.
- We also conduct the asymptotic analysis and provide the approximation of all performance factors mentioned above at high $P/{N}_{0}$ regime.
- The analytical results are all confirmed by Monte Carlo simulations. Using the simulation results, the effect of various system parameters on the system performance is carefully studied.

## 2. System Model

#### 2.1. System Model Description

#### 2.2. Energy Harvesting and Information Transfer Protocols

- ${\eta}_{i}$ denotes the hardware distortion noise at ${S}_{i}$ with zero mean and variance ${\kappa}^{2}P$. Here, $\kappa $ is sufficient to characterize the aggregate level of impairments of the channel [1].
- ${n}_{r}$ is the additive white Gaussian noise (AWGN) at R with zero mean and variance ${\sigma}_{r}^{2}$.

## 3. System Performance

#### 3.1. Outage Probability

**Theorem**

**1**(Outage probability—Exact form)

**.**

**Proof**

**of**

**Theorem**

**1.**

**Remark**

**1.**

#### 3.2. Achievable Throughput

#### 3.3. SER Analysis

**Theorem**

**2**(SER—Exact form)

**.**

**Proof**

**of**

**Theorem**

**2.**

#### 3.4. Asymptotic Analysis

#### 3.4.1. Outage Probability

**Theorem**

**3**(Outage probability—Asymptotic form)

**.**

**Proof**

**of**

**Theorem**

**3.**

#### 3.4.2. SER Analysis

**Theorem**

**4**(SER—Asymptotic form)

**.**

- If ${\xi}_{1}={\xi}_{2}$:$$\begin{array}{c}{SER}_{1}^{\infty}=\frac{\omega}{2}\hfill \\ -\frac{\omega \sqrt{\theta}}{2\sqrt{\pi}}{\zeta}_{1}{e}^{-2K}\sum _{l=0}^{\infty}\sum _{k=0}^{\infty}\sum _{m=0}^{l}\sum _{v=0}^{k+1}\left(\begin{array}{c}k+1\\ v\end{array}\right)\frac{{K}^{l+k}{\zeta}_{1}^{k}{\zeta}_{2}^{m}{a}^{m+v}{(-1)}^{v}\Gamma (m+k+1)}{l!m!{(k!)}^{2}{\zeta}_{1}^{m+1}{\theta}^{m+v+\frac{1}{2}}}\gamma \left(m+v+\frac{1}{2},\frac{\theta}{a}\right).\hfill \end{array}$$
- If ${\xi}_{1}\ne {\xi}_{2}$:$$\begin{array}{c}SE{R}_{1}^{\infty}=\frac{\omega}{2}-\frac{\omega \sqrt{\theta}}{2\sqrt{\pi}}{\zeta}_{1}{e}^{-2K}\sum _{l=0}^{\infty}\sum _{k=0}^{\infty}\sum _{m=0}^{l}\sum _{v=0}^{k+1}\sum _{p=0}^{\infty}{(m+k+1)}_{p}{\left[({\xi}_{1}-{\xi}_{2})\right]}^{p}\left(\begin{array}{c}k+1\\ v\end{array}\right)\hfill \\ \times \frac{{K}^{l+k}{\zeta}_{2}^{m}{a}^{m+v+p}{(-1)}^{v}\Gamma (m+k+1)}{l!m!{(k!)}^{2}p!{\zeta}_{1}^{m+p+1}{\theta}^{m+v+p+\frac{1}{2}}}\gamma \left(m+v+p+\frac{1}{2},\frac{\theta}{a}\right),\hfill \end{array}$$

**Proof**

**of**

**Theorem**

**4.**

#### 3.5. Optimal Time-Switching Factor

## 4. Numerical Results and Discussion

#### 4.1. Effects of Various Parameters on the System Performance

#### 4.2. Effect of Various Parameters on SER

#### 4.3. Optimal Time-Switching Factor

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AF | amplify-and-forward |

AWGN | additive white Gaussian noise |

BPSK | binary phase shift keying |

CDF | cumulative distribution function |

EH | energy harvesting |

I/Q | In-phase/Quadrature |

LOS | line-of-sight |

probability density function | |

PS | power splitting |

QPSK | quadrature phase shift keying |

RF | radio frequency |

RV | random variable |

SER | symbol-error-rate |

SNR | signal-to-noise ratio |

SWIPT | simultaneous wireless information and power transfer |

TS | time switching |

TWRN | two-way relay network |

WSN | wireless sensor network |

## Appendix A. Proof of Theorem 1

## Appendix B. Proof of Theorem 2

## Appendix C. Proof of Theorem 3

## Appendix D. Proof of Theorem 4

**Case 1:**ξ 2 =ξ 1 .

**Case 2:**ξ 2 <ξ 1 .

**Case 3:**ξ 2 >ξ 1 .

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Symbol | Parameter Names | Values |
---|---|---|

$\eta $ | Energy harvesting efficiency | 0.7 |

${\lambda}_{1}$ | Mean of $|{g}_{1}{|}^{2}$ | 0.5 |

${\lambda}_{2}$ | Mean of $|{g}_{2}{|}^{2}$ | 0.5 |

K | Rician K-factor | 3 |

$P/{N}_{0}$ | Source-power-to-noise ratio | 0–50 dB |

$\kappa ={\kappa}_{r}$ | Hardware impairment levels | 0, 0.15, 0.25 |

R | Source transmission rate | 1.5 bps/Hz |

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**MDPI and ACS Style**

Nguyen, T.N.; Quang Minh, T.H.; Tran, P.T.; Vozňák, M.
Energy Harvesting over Rician Fading Channel: A Performance Analysis for Half-Duplex Bidirectional Sensor Networks under Hardware Impairments. *Sensors* **2018**, *18*, 1781.
https://doi.org/10.3390/s18061781

**AMA Style**

Nguyen TN, Quang Minh TH, Tran PT, Vozňák M.
Energy Harvesting over Rician Fading Channel: A Performance Analysis for Half-Duplex Bidirectional Sensor Networks under Hardware Impairments. *Sensors*. 2018; 18(6):1781.
https://doi.org/10.3390/s18061781

**Chicago/Turabian Style**

Nguyen, Tan N., Tran Hoang Quang Minh, Phuong T. Tran, and Miroslav Vozňák.
2018. "Energy Harvesting over Rician Fading Channel: A Performance Analysis for Half-Duplex Bidirectional Sensor Networks under Hardware Impairments" *Sensors* 18, no. 6: 1781.
https://doi.org/10.3390/s18061781