# Variational Channel Estimation with Tempering: An Artificial Intelligence Algorithm for Wireless Intelligent Networks

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## Abstract

**:**

## 1. Introduction

#### 1.1. Problem Statement

#### 1.2. Previous Research and Limitations

#### 1.3. Objective

- Modeling of the channel estimation problem into a variational message passing algorithm;
- Evaluation of the performance error bound of our novel CEVTI algorithm;
- Use of a Monte Carlo simulation to verify the bit error rate (BER), convergence rate, and mutual information of the CEVTI approach; and
- Established the efficiency and superiority of our new algorithm, CEVTI.

#### 1.4. Contributions

- The modeling of the OFDM channel estimation problem into a new variational message passing algorithm;
- An evaluation of the performance error bound of our innovative variational tempering channel estimation algorithm; and
- A numerical simulation of the performance of CEVTI to show that in general cases, the proposed CEVTI algorithm performs better than other algorithms.

## 2. Background

## 3. Solution Framework

#### 3.1. Mean-Field CEVTI

#### 3.2. Tempered Joint

#### 3.3. Tempered ELBO

#### 3.4. Local Variational

## 4. The CEVTI Algorithm

#### Updates

## 5. Application of CEVTI

#### 5.1. Application of CEVTI for CDMA

#### 5.1.1. Channel Coefficient Estimation

#### 5.1.2. Noise Covariance Estimation

#### 5.1.3. Codeword Distribution Estimation

#### 5.2. Application of CEVTI in Massive MIMO

## 6. Simulations and Results

#### 6.1. Simulations

#### 6.2. Complexity Analysis

#### 6.3. Optimality Guarantee

**Theorem**

**1**

**.**Our CEVTI has optimality guarantee.

**Proof**

**of**

**Theorem**

**1.**

## 7. Conclusions and Future Directions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

WSNs | Wireless Sensor Networks |

VI | Variational Inference |

CEVTI | Channel Estimation Variational Tempering Inference |

CE | Channel Estimation |

CS | Compressive Sensing |

CSI | Channel State Information |

MP | Message Passing |

ELBO | Evidence Lower Bound |

MCMC | Monte Carlo Markov Chain |

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Parameter | Meaning | Value |
---|---|---|

${L}_{p}$ | pilot number | 32 |

N | number of users | 8 |

M | number of antennas | 4 |

${r}_{ij}$ | receiver j’s signal toward user i | |

$[{\alpha}_{1},{\beta}_{1}]$ | signal prior | [0.5,1] |

$[{\alpha}_{2},{\beta}_{2}]$ | user prior | [0.4,0.6] |

$\u03f5$ | convergence tolerance | ${10}^{-6}$ |

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**MDPI and ACS Style**

Liu, J.; Li, M.; Chen, Y.; Islam, S.M.N.; Crespi, N.
Variational Channel Estimation with Tempering: An Artificial Intelligence Algorithm for Wireless Intelligent Networks. *Sensors* **2020**, *20*, 5939.
https://doi.org/10.3390/s20205939

**AMA Style**

Liu J, Li M, Chen Y, Islam SMN, Crespi N.
Variational Channel Estimation with Tempering: An Artificial Intelligence Algorithm for Wireless Intelligent Networks. *Sensors*. 2020; 20(20):5939.
https://doi.org/10.3390/s20205939

**Chicago/Turabian Style**

Liu, Jia, Mingchu Li, Yuanfang Chen, Sardar M. N. Islam, and Noel Crespi.
2020. "Variational Channel Estimation with Tempering: An Artificial Intelligence Algorithm for Wireless Intelligent Networks" *Sensors* 20, no. 20: 5939.
https://doi.org/10.3390/s20205939