# Dual-Hop Cooperative Relaying with Beamforming Under Adaptive Transmission in κ–μ Shadowed Fading Environments

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Contributions

- We obtain new and exact results in an analytic form for the OP and average capacity using adaptive transmission techniques, such as optimal power and rate adaptation (OPRA), optimal rate adaptation with a constant transmit power (ORA), truncated channel inversion with a fixed rate (TIFR), and channel inversion with a fixed rate (CIFR).
- Using the obtained analytical results, we analyzed the system performance for various combinations of source and destination antennas and for different shadowing and fading parameters.
- It should be noted that all the obtained analytical expressions are general for $\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed fading scenario, and we can therefore easily transform these expressions into some special cases, namely, Nakagami-$\widehat{m}$/Nakagami-$\widehat{m}$, $\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $/$\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $, Rayleigh/Rayleigh, Rician-shadowed/Rician-shadowed, Rician/Rician, and mixed Rayleigh, $\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $, Rician-shadowed, Nakagami-$\widehat{m}$, and Rician fading links. These fading arrangements can occur in various applications including satellite, micro-/macro-cellular, and/or hybrid satellite/terrestrial communication systems.
- Our results can efficiently be used to investigate the behavior of various channels like the ones in land mobile satellite systems, underwater acoustic communications, body centric communications, and other different wireless communication applications.

## 2. System and Channel Models

#### 2.1. System Model

#### 2.2. $\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $ Shadowed Fading Model

## 3. Performance Analysis

#### 3.1. Outage Probability Analysis

#### 3.2. Capacity Analysis with Adaptive Transmission

#### 3.2.1. Optimal Power and Rate Adaptation (OPRA)

#### 3.2.2. Optimal Rate Adaptation with Constant Transmit Power (ORA)

#### 3.2.3. Channel Inversion with Fixed Rate (CIFR)

## 4. Special Cases

#### 4.1. Special Cases for Outage Probability Analysis

#### 4.2. Special Cases for Average Capacity Analysis

## 5. Numerical Results and Discussions

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Outage probability against average SNR per hop for different arrangements of antenna and fading parameter $\mu $ when ${\kappa}_{1}={\kappa}_{2}=1$, ${m}_{1}={m}_{2}=2$, and ${\gamma}_{th}=10$ dB.

**Figure 3.**Average channel capacity for four different adaptive transmission methods with respect to the average SNR per hop when ${m}_{1}={m}_{2}=0.5$, ${\mu}_{1}={\mu}_{2}=1$, ${\kappa}_{1}={\kappa}_{2}=1$, and ${N}_{1}={N}_{2}=2$.

**Figure 4.**Average channel capacity of OPRA scheme for some channel models, viz., the Rician (when ${K}_{1}={K}_{2}=5$), $\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $ (when ${\kappa}_{1}={\kappa}_{2}=4$ and ${\mu}_{1}={\mu}_{2}=2$), Rician-shadowed (when ${K}_{1}={K}_{2}=8$ and ${m}_{1}={m}_{2}=0.5$), Rayleigh fading, and Nakagami-$\widehat{m}$ (when the shaping factor of the Nakagami-$\widehat{m}$ is set to 2 for each link).

**Figure 5.**Average channel capacity of four adaptive transmission policies for different arrangements of antennas, N (${N}_{1},\phantom{\rule{0.277778em}{0ex}}{N}_{2}$), when ${\overline{\gamma}}_{1}={\overline{\gamma}}_{2}=8$ dB, ${\kappa}_{1}={\kappa}_{2}=1$, ${\mu}_{1}={\mu}_{2}=1$, and ${m}_{1}={m}_{2}=1$.

**Figure 6.**Average channel capacity under different adaptive transmission techniques for different shadowing parameter, $m\phantom{\rule{0.277778em}{0ex}}({m}_{1},\phantom{\rule{0.222222em}{0ex}}{m}_{2})$, when ${\overline{\gamma}}_{1}={\overline{\gamma}}_{2}=10$, ${N}_{1}={N}_{2}=2$, ${\kappa}_{1}={\kappa}_{2}=1$, and ${\mu}_{1}={\mu}_{2}=1$.

**Figure 7.**Average channel capacity of ORA technique with respect to the fading parameter $\kappa \phantom{\rule{0.277778em}{0ex}}({\kappa}_{1},\phantom{\rule{0.277778em}{0ex}}{\kappa}_{2})$ for different shadowing parameter m when $\overline{\gamma}=8$ dB, ${\mu}_{1}={\mu}_{2}=1$, and ${N}_{1}={N}_{2}=2$.

**Figure 8.**Average channel capacity using ORA technique of dual-hop relaying systems operating over $\kappa -\mu $ shadowed fading links and its equivalent Nakagami-q fading links for different number of antennas.

Fading Distribution | $\mathit{\kappa}$ | $\mathit{\mu}$ | m |
---|---|---|---|

Rician shadowed | $\kappa =\phantom{\rule{0.277778em}{0ex}}K$ | $\mu =\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1$ | $m=\phantom{\rule{0.277778em}{0ex}}m$ |

Nakagami-$\widehat{m}$ | $\kappa \phantom{\rule{0.277778em}{0ex}}\to 0$ | $\mu =\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\widehat{m}$ | $m\to \phantom{\rule{0.277778em}{0ex}}\infty $ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\mu $ | $\kappa =\phantom{\rule{0.277778em}{0ex}}\kappa $ | $\mu =\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mu $ | $m\to \phantom{\rule{0.277778em}{0ex}}\infty $ |

Rician | $\kappa =\phantom{\rule{0.277778em}{0ex}}K$ | $\mu =\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1$ | $m\to \phantom{\rule{0.277778em}{0ex}}\infty $ |

Rayleigh | $\kappa \phantom{\rule{0.277778em}{0ex}}\to 0$ | $\mu =\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}1$ | $m\to \phantom{\rule{0.277778em}{0ex}}\infty $ |

One-sided Gaussian | $\kappa \phantom{\rule{0.277778em}{0ex}}\to 0$ | $\mu =0.5$ | $m\to \phantom{\rule{0.277778em}{0ex}}\infty $ |

**Table 2.**Special cases obtained from general analyses for $\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed fading channels with different parameter settings [33].

First Hop/Second Hop | ${\mathit{\kappa}}_{1}$ | ${\mathit{\mu}}_{1}$ | ${\mathit{\kappa}}_{2}$ | ${\mathit{\mu}}_{2}$ | ${\mathit{m}}_{1}$ | ${\mathit{m}}_{2}$ |
---|---|---|---|---|---|---|

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${\kappa}_{2}$ | ${\mu}_{2}$ | ${m}_{1}$ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed/Rician-shadowed | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${K}_{2}$ | 1 | ${m}_{1}$ | ${m}_{2}$ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed/Rician | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${K}_{2}$ | 1 | ${m}_{1}$ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed/Nakagami-$\widehat{m}$ | ${\kappa}_{1}$ | ${\mu}_{1}$ | 0 | ${\widehat{m}}_{2}$ | ${m}_{1}$ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed/Rayleigh | ${\kappa}_{1}$ | ${\mu}_{1}$ | 0 | 1 | ${m}_{1}$ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ${m}_{2}$ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/Rician-shadowed | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${K}_{2}$ | 1 | ∞ | ${m}_{2}$ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/Rician | ${\kappa}_{1}$ | ${\mu}_{1}$ | ${K}_{2}$ | 1 | ∞ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/Nakagami-$\widehat{m}$ | ${\kappa}_{1}$ | ${\mu}_{1}$ | 0 | ${\widehat{m}}_{2}$ | ∞ | ∞ |

$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $/Rayleigh | ${\kappa}_{1}$ | ${\mu}_{1}$ | 0 | 1 | ∞ | ∞ |

Rician-shadowed/Rician-shadowed | ${K}_{1}$ | 1 | ${K}_{2}$ | 1 | ${m}_{1}$ | ${m}_{2}$ |

Rician-shadowed/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed | ${K}_{1}$ | 1 | ${\kappa}_{2}$ | ${\mu}_{1}$ | ${m}_{1}$ | ${m}_{2}$ |

Rician-shadowed/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | ${K}_{1}$ | 1 | ${\kappa}_{2}$ | ${\mu}_{2}$ | ${m}_{1}$ | ∞ |

Rician-shadowed/Rician | ${K}_{1}$ | 1 | ${K}_{2}$ | 1 | ${m}_{1}$ | ∞ |

Rician-shadowed/Nakagami-$\widehat{m}$ | ${K}_{1}$ | 1 | 0 | ${\widehat{m}}_{2}$ | ${m}_{1}$ | ∞ |

Rician-shadowed/Rayleigh | ${K}_{1}$ | 1 | 0 | 1 | ${m}_{1}$ | ∞ |

Rician/Rician | ${K}_{1}$ | 1 | ${K}_{2}$ | 1 | ∞ | ∞ |

Rician/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed | ${K}_{1}$ | 1 | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ${m}_{2}$ |

Rician/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | ${K}_{1}$ | 1 | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ∞ |

Rician/Rician-shadowed | ${K}_{1}$ | 1 | ${K}_{2}$ | 1 | ∞ | ${m}_{2}$ |

Rician/Nakagami-$\widehat{m}$ | ${K}_{1}$ | 1 | 0 | ${\widehat{m}}_{2}$ | ∞ | ∞ |

Rician/Rayleigh | ${K}_{1}$ | 1 | 0 | 1 | ∞ | ∞ |

Nakagami-$\widehat{m}$/Nakagami-$\widehat{m}$ | 0 | ${\widehat{m}}_{1}$ | 0 | ${\widehat{m}}_{2}$ | ∞ | ∞ |

Nakagami-$\widehat{m}$/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed | 0 | ${\widehat{m}}_{1}$ | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ${m}_{2}$ |

Nakagami-$\widehat{m}$/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | 0 | ${\widehat{m}}_{1}$ | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ∞ |

Nakagami-$\widehat{m}$/Rician-shadowed | 0 | ${\widehat{m}}_{1}$ | ${K}_{2}$ | 1 | ∞ | ${m}_{2}$ |

Nakagami-$\widehat{m}$/Rician | 0 | ${\widehat{m}}_{1}$ | ${K}_{2}$ | 1 | ∞ | ∞ |

Nakagami-$\widehat{m}$/Rayleigh | 0 | ${\widehat{m}}_{1}$ | 0 | 1 | ∞ | ∞ |

Rayleigh/Rayleigh | 0 | 1 | 0 | 1 | ∞ | ∞ |

Rayleigh/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ shadowed | 0 | 1 | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ${m}_{2}$ |

Rayleigh/$\kappa \phantom{\rule{-0.166667em}{0ex}}-\phantom{\rule{-0.166667em}{0ex}}\mu $ | 0 | 1 | ${\kappa}_{2}$ | ${\mu}_{2}$ | ∞ | ∞ |

Rayleigh/Rician-shadowed | 0 | 1 | ${K}_{2}$ | 1 | ∞ | ${m}_{2}$ |

Rayleigh/Rician | 0 | 1 | ${K}_{2}$ | 1 | ∞ | ∞ |

Rayleigh/Nakagami-$\widehat{m}$ | 0 | 1 | 0 | ${\widehat{m}}_{2}$ | ∞ | ∞ |

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**MDPI and ACS Style**

Bhutto, Z.; Yoon, W. Dual-Hop Cooperative Relaying with Beamforming Under Adaptive Transmission in *κ*–*μ* Shadowed Fading Environments. *Electronics* **2019**, *8*, 658.
https://doi.org/10.3390/electronics8060658

**AMA Style**

Bhutto Z, Yoon W. Dual-Hop Cooperative Relaying with Beamforming Under Adaptive Transmission in *κ*–*μ* Shadowed Fading Environments. *Electronics*. 2019; 8(6):658.
https://doi.org/10.3390/electronics8060658

**Chicago/Turabian Style**

Bhutto, Zuhaibuddin, and Wonyong Yoon. 2019. "Dual-Hop Cooperative Relaying with Beamforming Under Adaptive Transmission in *κ*–*μ* Shadowed Fading Environments" *Electronics* 8, no. 6: 658.
https://doi.org/10.3390/electronics8060658