# Exact Performance Analysis of Amplify-and-Forward Bidirectional Relaying over Nakagami-m Fading Channels with Arbitrary Parameters

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## Abstract

**:**

## 1. Introduction

## 2. System Model

## 3. Performance Analysis

#### 3.1. Moment Generating Function

#### 3.2. Higher Order Moments

#### 3.3. Ergodic Capacity

#### 3.4. Average SEP

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Average signal error probability (SEP) of the differential phase shift keying (DPSK) modulation. SNR: signal-to-noise ratio.

Notation | Description |
---|---|

${\alpha}_{1}$ | Scaling factor of the total relay power assigned to ${S}_{1}$ |

${\alpha}_{2}$ | Scaling factor of the total relay power assigned to ${S}_{2}$ |

$B\left(\xb7,\xb7\right)$ | Beta function |

$erfc\left(\xb7\right)$ | Complementary error function |

${}_{2}{F}_{1}\left(\xb7\right)$ | Gaussian hypergeometric function |

$\mathsf{\Gamma}\left(\xb7\right)$ | Gamma function |

$G\left(\xb7\right)$ | Meijer’s G function |

$H\left(\xb7\right)$ | Fox’s function |

${K}_{\nu}\left(\xb7\right)$ | The $\nu $th order modified Bessel function of the second kind |

Relay Mode | Transmission Direction | a | ${\mathit{\gamma}}_{1}$ | ${\mathit{\gamma}}_{2}$ |
---|---|---|---|---|

Two time slot bidirectional relay | ${S}_{2}\to R\to {S}_{1}$ | $\frac{{p}_{r}}{{p}_{1}+{p}_{r}}$ | $\frac{{p}_{2}{\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ | $\frac{\left({p}_{r}+{p}_{1}\right){\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ |

${S}_{1}\to R\to {S}_{2}$ | $\frac{{p}_{r}}{{p}_{r}+{p}_{2}}$ | $\frac{{p}_{1}{\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ | $\frac{\left({p}_{r}+{p}_{2}\right){\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ | |

Three time slot bidirectional relay | ${S}_{2}\to R\to {S}_{1}$ | $\frac{{p}_{r}}{{p}_{r}+{\alpha}_{1}{p}_{1}}$ | $\frac{{\alpha}_{2}{p}_{2}{\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ | $\frac{\left({p}_{r}+{\alpha}_{1}{p}_{1}\right){\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ |

${S}_{1}\to R\to {S}_{2}$ | $\frac{{p}_{r}}{{p}_{r}+{\alpha}_{2}{p}_{2}}$ | $\frac{{\alpha}_{1}{p}_{1}{\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ | $\frac{\left({p}_{r}+{\alpha}_{2}{p}_{2}\right){\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ | |

Four time slot bidirectional relay | ${S}_{1}\to R\to {S}_{2}$ | 1 | $\frac{{p}_{1}{\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ | $\frac{{p}_{r}{\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ |

${S}_{2}\to R\to {S}_{1}$ | 1 | $\frac{{p}_{2}{\left|{h}_{2}\right|}^{2}}{{N}_{0}}$ | $\frac{{p}_{r}{\left|{h}_{1}\right|}^{2}}{{N}_{0}}$ |

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**MDPI and ACS Style**

Qin, D.; Wang, Y.; Zhou, T.
Exact Performance Analysis of Amplify-and-Forward Bidirectional Relaying over Nakagami-*m* Fading Channels with Arbitrary Parameters. *Energies* **2019**, *12*, 1277.
https://doi.org/10.3390/en12071277

**AMA Style**

Qin D, Wang Y, Zhou T.
Exact Performance Analysis of Amplify-and-Forward Bidirectional Relaying over Nakagami-*m* Fading Channels with Arbitrary Parameters. *Energies*. 2019; 12(7):1277.
https://doi.org/10.3390/en12071277

**Chicago/Turabian Style**

Qin, Dong, Yuhao Wang, and Tianqing Zhou.
2019. "Exact Performance Analysis of Amplify-and-Forward Bidirectional Relaying over Nakagami-*m* Fading Channels with Arbitrary Parameters" *Energies* 12, no. 7: 1277.
https://doi.org/10.3390/en12071277