# Asymptotic Ergodic Capacity Analysis for FSO Communication between Mobile Platforms in Maritime Environments

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## Abstract

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## 1. Introduction

## 2. System and Channel Models

#### 2.1. Foggy Channel

#### 2.2. Pathloss

#### 2.3. Pointing Error

## 3. Performance Analysis: Ergodic Capacity

${\mathbf{I}}_{\mathbf{a}}$ | ${\mathbf{I}}_{\mathbf{p}}$ | |

Power of expectation | ${\left(\mathbb{E}\left[{I}_{a}\right]\right)}^{rn}={\left(\frac{z}{z+1}\right)}^{krn}$ | ${\left(\mathbb{E}\left[{I}_{p}\right]\right)}^{rn}={\left(\frac{{\xi}^{2}{A}_{0}}{{\xi}^{2}+1}\right)}^{rn}$ |

nth moment | $\mathbb{E}\left[{{I}_{a}}^{rn}\right]={\left(\frac{z}{rn+z}\right)}^{k}$ | $\mathbb{E}\left[{{I}_{p}}^{rn}\right]=\frac{{\xi}^{2}{{A}_{0}}^{rn}}{rn+{\xi}^{2}}$ |

Derivative of power of expectation | $\frac{\partial}{\partial n}{\left(\mathbb{E}\left[{I}_{a}\right]\right)}^{rn}=kr{\left(\frac{z}{z+1}\right)}^{krn}ln\left(\frac{z}{z+1}\right)$ | $\frac{\partial}{\partial n}{\left(\mathbb{E}\left[{I}_{p}\right]\right)}^{rn}=r{\left(\frac{{\xi}^{2}{A}_{0}}{{\xi}^{2}+1}\right)}^{rn}ln\left(\frac{{\xi}^{2}{A}_{0}}{{\xi}^{2}+1}\right)$ |

Derivative of nth moment | $\frac{\partial}{\partial n}\mathbb{E}\left[{{I}_{a}}^{rn}\right]=-krz\frac{1}{{\left(rn+z\right)}^{2}}{\left(\frac{z}{rn+z}\right)}^{k-1}$ | $\frac{\partial}{\partial n}\mathbb{E}\left[{{I}_{p}}^{rn}\right]=\frac{-{\xi}^{2}r{{A}_{0}}^{rn}}{{\left(rn+{\xi}^{2}\right)}^{2}}+\frac{{\xi}^{2}r{{A}_{0}}^{rn}}{rn+{\xi}^{2}}ln\left({A}_{0}\right)$ |

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

Notation | Definition |

FSO | Free-space optical |

Tx | Transmitter |

Rx | Receiver |

SNR | Signal-to-noise ratio |

BER | Bit error rate |

Probability density function | |

HD | Heterodyne detection |

IM/DD | Intensity modulation/direct detection |

PD | Photo-detector |

Parameters | Definition |

${I}_{l}$ | pathloss |

${I}_{a}$ | foggy channel |

${I}_{p}$ | pointing error |

l | the length of the propagation link (km) |

$\alpha $ | signal attenuation random variable (dB/km) |

$\beta $ | continuous scale parameter |

$\mathrm{\Gamma}(\xb7)$ | Gamma function |

k | continuous shape parameter of fog |

V | visibility |

$\lambda $ | wavelength |

q | size distribution of the scattering particles |

${\omega}_{z}$ | beamwidth of a Gaussian beam |

a | aperture radius |

s | the radial displacement between the centers of the beam and the detector |

${A}_{0}$ | fraction of the collected power at $s=0$ |

v | ratio between the beamewidth and aperture radius |

${\omega}_{{z}_{eq}}$ | the equivalent beamwidth |

$\mathrm{erf}\left(x\right)$ | error function |

$\xi $ | raio of the equivalent beam radius at the receiver |

$\sigma $ | standard deviation of the pointing error displacement at the receiver |

$\overline{C}$ | ergodic capacity |

${\gamma}^{\prime}$ | instantaneous SNR |

${\mu}^{\prime}$ | average electrical SNR |

## Appendix A. Derivative of Expectation of I_{a}

## Appendix B. Derivative of Expectation of I_{p}

## Appendix C. Derivative of the Moments of I_{a}

## Appendix D. Derivative of the Moments of I_{p}

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**Figure 1.**System and channel models of FSO communication between mobile platforms in maritime environments.

**Figure 2.**Ergodic capacity results for varying fog types under both HD and IM/DD techniques with $l=0.2$ km.

**Figure 3.**Ergodic capacity results for varying pointing errors under both HD and IM/DD techniques with $l=1$ km.

**Figure 4.**Ergodic capacity results for combined effects of foggy channel with pathloss and pointing error under both HD and IM/DD techniques with $l=0.5$ km.

**Figure 5.**Ergodic capacity results for varying both boresight (i.e., $\mu =0,110,180$) and jitter (i.e., $\xi =0.3216,\infty $) with dense fog and ${I}_{l}=5.7943\times {10}^{-22}$ under HD technique with $l=0.5$ km.

Fog Type | Dense | Thick | Moderate | Light |
---|---|---|---|---|

$V\left(m\right)$ | 0–50 | 50–200 | 200–500 | 500–1000 |

k | 36.05 | 6.00 | 5.49 | 2.32 |

$\beta $ | 11.91 | 23.00 | 12.06 | 13.12 |

Parameter | Symbol | Value |
---|---|---|

Wavelength | $\lambda $ | 1550 nm |

Receiver radius | a | 10 cm |

Beamwidth | ${\omega}_{z}$ | 150 cm |

Distance between Tx and Rx | l | 0.2–1 km |

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**MDPI and ACS Style**

Han, J.E.; Nam, S.S.; Yoon, C.; Hwang, D.D.; Alouini, M.-S.
Asymptotic Ergodic Capacity Analysis for FSO Communication between Mobile Platforms in Maritime Environments. *Appl. Sci.* **2023**, *13*, 6978.
https://doi.org/10.3390/app13126978

**AMA Style**

Han JE, Nam SS, Yoon C, Hwang DD, Alouini M-S.
Asymptotic Ergodic Capacity Analysis for FSO Communication between Mobile Platforms in Maritime Environments. *Applied Sciences*. 2023; 13(12):6978.
https://doi.org/10.3390/app13126978

**Chicago/Turabian Style**

Han, Jae Eun, Sung Sik Nam, Changseok Yoon, Duck Dong Hwang, and Mohamed-Slim Alouini.
2023. "Asymptotic Ergodic Capacity Analysis for FSO Communication between Mobile Platforms in Maritime Environments" *Applied Sciences* 13, no. 12: 6978.
https://doi.org/10.3390/app13126978