# Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders

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## Abstract

**:**

## 1. Introduction

## 2. Problem Formulation

**Definition**

**1.**

**Remark**

**1.**

- (i)
- Complete synchronization for $\left(\mathbf{F}\left(.\right),\mathbf{M}\right)=\left(I,X\left(t\right)\right)$.
- (ii)
- Anti–synchronization for $\left(\mathbf{F}\left(.\right),\mathbf{M}\right)=\left(I,-X\left(t\right)\right)$.
- (iii)
- Matrix projective synchronization for $\left(\mathbf{F}\left(.\right),\mathbf{M}\right)=\left(I,\mathbf{M}\left(t\right)\right)$.
- (iv)
- Inverse generalized synchronization for $\left(\mathbf{F}\left(.\right),\mathbf{M}\right)=\left(\mathbf{F}\left(Y\left(t\right)\right),I\right)$.

## 3. $\mathbf{F}$–$\mathbf{M}$ Synchronization

#### 3.1. Case 1: $d=m$

**Theorem**

**1.**

**Proof.**

#### 3.2. Case 2: $d<m$

**Theorem**

**2.**

**Proof.**

## 4. Numerical Example

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

## 5. Application to MIMO Secure Communications

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Chaotic attractors of the master system (29) when $\left(a,b\right)=\left(100,10\right)$ and $p=0.95$.

**Figure 2.**Attractors of the slave system (30) when $q=0.94$ and ${u}_{1}={u}_{2}={u}_{3}={u}_{4}=0$.

**Figure 6.**The original messages ${s}_{i}\left(t\right)$ to be transmitted by n transmit antennas in a MIMO secure communication system.

**Figure 7.**The transmitted signals ${\tilde{s}}_{i}\left(t\right)$ as well as the equalized signals ${\tilde{r}}_{i}\left(t\right)$ for an SNR of 40 dB.

**Figure 9.**The recovered messages ${\widehat{s}}_{i}\left(t\right)$ from the n receive antennas after a 50–tap low–pass filter.

Synchronization Schemes |
---|

Inverse full state hybrid projective synchronization [3] |

Matrix projective synchronization [4]. |

Generalized synchronization [5]. |

Inverse generalized synchronization [6]. |

Hybrid synchronization [7]. |

$\mathsf{\Phi}$–$\mathsf{\Theta}$ synchronization [8]. |

Q–S synchronization [9]. |

Reduced order synchronization [10]. |

Increased order generalized synchronization [11]. |

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**MDPI and ACS Style**

Ouannas, A.; Debbouche, N.; Wang, X.; Pham, V.-T.; Zehrour, O.
Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders. *Appl. Sci.* **2018**, *8*, 1746.
https://doi.org/10.3390/app8101746

**AMA Style**

Ouannas A, Debbouche N, Wang X, Pham V-T, Zehrour O.
Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders. *Applied Sciences*. 2018; 8(10):1746.
https://doi.org/10.3390/app8101746

**Chicago/Turabian Style**

Ouannas, Adel, Nadjette Debbouche, Xiong Wang, Viet-Thanh Pham, and Okba Zehrour.
2018. "Secure Multiple-Input Multiple-Output Communications Based on F–M Synchronization of Fractional-Order Chaotic Systems with Non-Identical Dimensions and Orders" *Applied Sciences* 8, no. 10: 1746.
https://doi.org/10.3390/app8101746