# Floquet Spectral Almost-Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense-Massive-MIMO, Intelligent-Surfaces, 5G, and 6G Applications

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## Abstract

**:**

## 1. Introduction

## 2. Almost-Periodic Modulation to Study Strongly Coupled Arrays

#### 2.1. Modulation in Infinite Almost-Periodic Arrays

#### 2.2. Modulation in Finite Almost-Periodic Arrays

## 3. MoM-GeC Modelization Based on Floquet Analysis

## 4. Numerical Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Spectral representation of the interactions of a unit cell with its neighbors (infinite and finite cases) (valid for strong coupling interaction by using Floquet phases).

**Figure 6.**Variation of radiation pattern against Floquet states and application of superposition theorem for 10 elements of antenna array (uni-dimentionnal configuration) at 77 GHz: obtained by the MoM-GEC method.

**Figure 7.**Variation of radiation pattern against Floquet states and application of superposition theorem for 100 elements of antenna array (uni-dimentionnal configuration) at 77 GHz: obtained by the MoM-GEC method.

**Table 1.**Directivity versus some Floquet states (considering 100 antenna arrays) and the superposition theorem (or the modulation as explained in Formula (13), which transformed to study a finite array) for ${\varphi}_{s}=0,{\theta}_{s}={30}^{\circ}$ steering angles (used for 5G application).

Floquet States | Directivity Values (dB) at 24 GHz |
---|---|

$({\alpha}_{-49},\beta =0)$ | 23.0200 |

$({\alpha}_{-10},\beta =0)$ | 22.5953 |

$({\alpha}_{+20},\beta =0)$ | 22.8874 |

$({\alpha}_{+30},\beta =0)$ | 23.2055 |

Superposition | 23.3009 |

Floquet States | Directivity Values (dB) at 77 GHz |

$({\alpha}_{-49},\beta =0)$ | 22.8771 |

$({\alpha}_{-10},\beta =0)$ | 24.1083 |

$({\alpha}_{+20},\beta =0)$ | 23.4323 |

$({\alpha}_{+30},\beta =0)$ | 23.1353 |

Superposition | 23.5783 |

**Table 2.**Directivity versus Floquet states and the superposition theorem (or the modulation as explained in Formula (13), which transformed to study a finite array) for ${\varphi}_{s}=0,{\theta}_{s}={45}^{\circ}$ steering angles (FMCW radar application).

Floquet States | Directivity Values (dB) at 24 GHz |
---|---|

$({\alpha}_{-2},{\beta}_{-1})$ | 21.7249 |

$({\alpha}_{-1},{\beta}_{-1})$ | 26.7351 |

$({\alpha}_{0},{\beta}_{-1})$ | 21.1290 |

$\left({\alpha}_{+1}{\beta}_{-1}\right)$ | 20.8615 |

$({\alpha}_{-2},{\beta}_{0})$ | 20.6344 |

$({\alpha}_{-1},{\beta}_{0})$ | 21.0667 |

$({\alpha}_{0},{\beta}_{0})$ | 14.8990 |

$({\alpha}_{+1},{\beta}_{0})$ | 21.0737 |

Superposition | 21.0455 |

Floquet States | Directivity Values (dB) at 77 GHz |

$({\alpha}_{-2},{\beta}_{-1})$ | 22.3762 |

$({\alpha}_{-1},{\beta}_{-1})$ | 18.0573 |

$({\alpha}_{0},{\beta}_{-1})$ | 21.9229 |

$\left({\alpha}_{+1}{\beta}_{-1}\right)$ | 22.1662 |

$({\alpha}_{-2},{\beta}_{0})$ | 20.9345 |

$({\alpha}_{-1},{\beta}_{0})$ | 21.3088 |

$({\alpha}_{0},{\beta}_{0})$ | 13.7990 |

$({\alpha}_{+1},{\beta}_{0})$ | 21.3088 |

Superposition | 21.7022 |

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**MDPI and ACS Style**

Bilel, H.; Taoufik, A. Floquet Spectral Almost-Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense-Massive-MIMO, Intelligent-Surfaces, 5G, and 6G Applications. *Electronics* **2022**, *11*, 36.
https://doi.org/10.3390/electronics11010036

**AMA Style**

Bilel H, Taoufik A. Floquet Spectral Almost-Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense-Massive-MIMO, Intelligent-Surfaces, 5G, and 6G Applications. *Electronics*. 2022; 11(1):36.
https://doi.org/10.3390/electronics11010036

**Chicago/Turabian Style**

Bilel, Hamdi, and Aguili Taoufik. 2022. "Floquet Spectral Almost-Periodic Modulation of Massive Finite and Infinite Strongly Coupled Arrays: Dense-Massive-MIMO, Intelligent-Surfaces, 5G, and 6G Applications" *Electronics* 11, no. 1: 36.
https://doi.org/10.3390/electronics11010036