# Directional Modulation Technique Using a Polarization Sensitive Array for Physical Layer Security Enhancement

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Our Contribution

## 2. System Model

## 3. Review of Polarized Beamforming

## 4. Principle of the Proposed Multi-Beam DM Technique Using a PSA

#### 4.1. From a Sampling Perspective

#### 4.2. From a Signal Processing Perspective

## 5. Security Performance Analysis for the Proposed DM Scheme

#### 5.1. Security Performance Analysis When Eves with Polarization Information

**y**is

**y**is a $K\times 1$ vector representing the received signals, $\mathbf{H}={[{\mathbf{s}}_{1},\cdots ,{\mathbf{s}}_{j},\cdots ,{\mathbf{s}}_{K}]}^{T}$ is a $K\times 2N$ matrix denoting the channel matrix from transmitter to receiver, ${\mathbf{s}}_{j}$ is a $2N\times 1$ vector denoting the steering vector. $\mathbf{W}=[{\mathbf{w}}_{1},\cdots ,{\mathbf{w}}_{j},\cdots ,{\mathbf{w}}_{K}]$ is a $2N\times K$ matrix denoting the antenna weights. The variable $\mathbf{\xi}$ with distribution $\mathcal{C}\mathcal{N}(0,{\sigma}^{2}{\mathbf{I}}_{K})$ is the normalized additive white Gauss noise (AWGN), where $\mathcal{C}\mathcal{N}$ denotes a complex and circularly symmetric random variable.

#### 5.2. Security Performance Analysis When Eves without Polarization Information

#### 5.3. Security Performance Analysis from a Signal Processing Perspective

#### 5.4. Metrics

## 6. Simulations and Discussions

#### 6.1. SER

#### 6.2. Secrecy Rate

#### 6.3. Robustness

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## References

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**Figure 4.**The numbers of the Lagrangian multiplier method computations for the proposed directional modulation (DM) scheme.

**Figure 6.**Beam patterns for broadside ${\theta}_{ML}={0}^{\circ}$ for single beam for symbol pairs (

**a**) “00,00”, (

**b**) “00,01”, (

**c**) “00,11”, (

**d**) “00,10”.

**Figure 7.**Phase patterns for broadside ${\theta}_{ML}={0}^{\circ}$ for single beam for symbol pairs (

**a**) “00,00”, (

**b**) “00,01”, (

**c**) “00,11”, (

**d**) “00,10”.

**Figure 8.**The resulting symbol error rate (SER) curve for broadside ${\theta}_{ML}={0}^{\circ}$ for two data streams.

**Figure 9.**Beam patterns for broadside ${\theta}_{ML}={30}^{\circ}$ for single beam for symbol pairs (

**a**) “00,00”, (

**b**) “00,01”, (

**c**) “00,11”, (

**d**) “00,10”.

**Figure 10.**Phase patterns for broadside ${\theta}_{ML}={30}^{\circ}$ for single beam for symbol pairs (

**a**) “00,00”, (

**b**) “00,01”, (

**c**) “00,11”, (

**d**) “00,10”.

**Figure 11.**The resulting SER curve for off-broadside ${\theta}_{ML}={30}^{\circ}$ for two data streams.

**Figure 13.**The resulting SER curve versus elevation angle for two data streams in two desired directions.

**Figure 14.**The simulated far-field (

**a**) magnitude patterns and (

**b**) phase patterns for 50 symbols with variable polarization information.

**Figure 15.**The SER simulation results versus elevation angle obtained for receivers using a polarization sensitive antenna or a single-polarized antenna.

**Figure 16.**The SER performance versus SNR for receivers at the desired direction ${0}^{\circ}$ utilizing a polarization sensitive antenna or a single-polarized antenna.

**Figure 17.**Secrecy rate performance of the proposed DM scheme and the conventional AD-aided DM schemes. (

**a**) Achievable rate of the LU versus SNR; (

**b**) Achievable rate of the Eve versus SNR; (

**c**) Secrecy rate of the system.

Symbols | Usage | Symbols | Usage |
---|---|---|---|

${(\xb7)}^{T}$ | Transpose operator | ${(\xb7)}^{H}$ | Complex conjugate transpose operator |

${(\xb7)}^{-1}$ | Inverse operator | ${(\xb7)}^{+}$ | Moore-Penrose pseudo inverse operator |

$\left|\xb7\right|$ | Modulus operator | ${\u2225\xb7\u2225}_{2}$ | ${l}_{2}$-norm operator |

⊗ | Kronecker product operator | $\mathsf{\Pi}$ | Quadrature operator |

$\mathsf{\Sigma}$ | Sum operator | $erfc(\xb7)$ | Complementary error function |

$\mathsf{\Xi}(\xb7)$ | Phase acquisition function | $\mathcal{C}\mathcal{N}(\xb7,\xb7)$ | Standard normal distribution |

$max(\xb7,\xb7)$ | Returns the largest element | ${[\xb7]}^{\u2020}$ | $max(\xb7,0)$ |

$\mathbb{R},\mathbb{C}$ | Real number, complex number | ${\mathbf{I}}_{K}$ | Identity matrix with size $K\times K$ |

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**MDPI and ACS Style**

Zhang, W.; Li, B.; Le, M.; Wang, J.; Peng, J. Directional Modulation Technique Using a Polarization Sensitive Array for Physical Layer Security Enhancement. *Sensors* **2019**, *19*, 5396.
https://doi.org/10.3390/s19245396

**AMA Style**

Zhang W, Li B, Le M, Wang J, Peng J. Directional Modulation Technique Using a Polarization Sensitive Array for Physical Layer Security Enhancement. *Sensors*. 2019; 19(24):5396.
https://doi.org/10.3390/s19245396

**Chicago/Turabian Style**

Zhang, Wei, Bin Li, Mingnan Le, Jun Wang, and Jinye Peng. 2019. "Directional Modulation Technique Using a Polarization Sensitive Array for Physical Layer Security Enhancement" *Sensors* 19, no. 24: 5396.
https://doi.org/10.3390/s19245396