# Robust Iterative Distributed Minimum Total MSE Algorithm for Secure Communications in the Internet of Things Using Relays

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## Abstract

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## 1. Introduction

^{H}to represent Hermitian transpose, Tr(.) to represent the trace of a matrix, E{.} to represent the expectation,

**I**to represent the identity matrix,

**0**to represent a matrix or vector whose all element are zeros, $\mathrm{Y}~\mathcal{C}\mathcal{N}\left(\mu ,{\mathsf{\sigma}}^{2}\right)$ to represent Y following the complex normal distribution with mean $\mu $ and variance ${\mathsf{\sigma}}^{2}$. $bd(.)$ to represent a block-diagonal matrix, $vec(.)$ to represent stack columns of a matrix on top of each other into a single vector, $\Vert .\Vert $ to represent 2-norm of a vector, $\u2a02$ to represent Kronecker product and $\u2102$ to represent the complex field.

## 2. System Model and Methods

## 3. The Iterative Distributed Algorithm of Solving Source, Relay, Destination and Eavesdropper Matrices

#### 3.1. Solution of Destination Matrices $\left\{{W}_{k}\right\}$

#### 3.2. Solution of Source Matrices $\left\{{U}_{k}\right\}$

#### 3.3. Solution of Relay Matrices $\left\{{V}_{m}\right\}$

## 4. The Convergence of the Proposed Algorithm

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Two-hop multiple input multiple output (MIMO) interference relay system model in the wireless Internet of Things (IoT).

Steps | Specific Progress |
---|---|

Step 1 | Set $n=0$, ${\mathrm{TMSE}}^{\left(n\right)}=0$ and initialize the $\left\{{U}_{k}^{\left(0\right)}\right\}$ and $\left\{{V}_{k}^{\left(0\right)}\right\}$ satisfying power constrains (11) and (12). |

Step 2 | Calculate $\left\{{W}_{k}^{\left(n+1\right)}\right\}$ and $\left\{{W}_{e,k}^{\left(n+1\right)}\right\}$ with $\left\{{U}_{k}^{\left(n\right)}\right\}$ and $\left\{{V}_{m}^{\left(n\right)}\right\}$ obtained from previous iteration. |

Step 3 | Update $\left\{{U}_{k}^{\left(n+1\right)}\right\}$ by solving the problem (25) with $\left\{{W}_{k}^{\left(n+1\right)}\right\}$, $\left\{{W}_{e,k}^{\left(n+1\right)}\right\}$ and $\left\{{V}_{m}^{\left(n\right)}\right\}$. |

Step 4 | Update $\left\{{V}_{m}^{\left(n+1\right)}\right\}$ by solving the problem (31) with $\left\{{W}_{k}^{\left(n+1\right)}\right\}$, $\left\{{W}_{e,k}^{\left(n+1\right)}\right\}$ and $\left\{{U}_{k}^{\left(n+1\right)}\right\},$ then calculate ${\mathrm{TMSE}}^{\left(n+1\right)}$. |

Step 5 | If ${\mathrm{TMSE}}^{\left(n+1\right)}-{\mathrm{TMSE}}^{\left(n\right)}\le \xi $, then end; otherwise set $n=n+1$, then go to step 2. |

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**MDPI and ACS Style**

Kong, Z.; Wang, D.; Li, Y.; Wang, C.
Robust Iterative Distributed Minimum Total MSE Algorithm for Secure Communications in the Internet of Things Using Relays. *Sensors* **2018**, *18*, 3914.
https://doi.org/10.3390/s18113914

**AMA Style**

Kong Z, Wang D, Li Y, Wang C.
Robust Iterative Distributed Minimum Total MSE Algorithm for Secure Communications in the Internet of Things Using Relays. *Sensors*. 2018; 18(11):3914.
https://doi.org/10.3390/s18113914

**Chicago/Turabian Style**

Kong, Zhengmin, Die Wang, Yunjuan Li, and Chao Wang.
2018. "Robust Iterative Distributed Minimum Total MSE Algorithm for Secure Communications in the Internet of Things Using Relays" *Sensors* 18, no. 11: 3914.
https://doi.org/10.3390/s18113914