# Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems

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## Abstract

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## 1. Introduction

## 2. Background

**w**is the noise vector. (1) is very popular in detection techniques to estimate the transmitted signal ($\mathbf{x}$). However, a perfect channel estimation is assumed at the BS. The free-matrix-inversion methods depend on the equalization matrix ($\mathbf{A}$), which is expressed as

#### 2.1. Successive over Relaxation

#### 2.2. Gauss–Seidel Detector

#### 2.3. Jacobi Method

#### 2.4. Approximate Message Passing

## 3. Proposed Methods

- Step 1: Compute the initial solution ${\widehat{\mathbf{x}}}_{\left(0\right)}$ as$${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{D}}^{-1}{\mathbf{y}}_{MF}.$$
- Step 2: Apply the SOR, GS, or JA methods where $n=0$, as
- -
- If SOR:$${\widehat{\mathbf{x}}}_{\left(1\right)}={(\mathbf{D}-\frac{1}{\omega}\mathbf{L})}^{-1}({\mathbf{y}}_{MF}+((1-\frac{1}{\omega})\mathbf{D}+\frac{1}{\omega}\mathbf{U}){\widehat{\mathbf{x}}}_{\left(0\right)}).$$
- -
- If GS:$${\widehat{\mathbf{x}}}_{\left(1\right)}={(\mathbf{D}-\mathbf{L})}^{-1}({\mathbf{y}}_{MF}+\mathbf{U}{\widehat{\mathbf{x}}}_{\left(0\right)}).$$
- -
- If JA:$${\widehat{\mathbf{x}}}_{\left(1\right)}={\mathbf{D}}^{-1}({\widehat{\mathbf{x}}}_{MF}+(\mathbf{D}-\mathbf{A}){\widehat{\mathbf{x}}}_{\left(0\right)}).$$

- Step 3: Apply the AMP algorithm as shown in (8), where $n=1$.

Algorithm 1:Initialize the AMP using the SOR method |

Input:$\mathbf{y},\mathbf{H},{\sigma}^{2},n,\omega $Output: Estimated signal $\widehat{\mathbf{x}}$Initialization:$\mathbf{A}={\mathbf{H}}^{H}\mathbf{H}+{\sigma}^{2}{\mathbf{I}}_{K}$ $\mathbf{D}=diag\left(\mathbf{A}\right)$, $\mathbf{U}=-triu\left(\mathbf{A}\right)$, $\mathbf{L}=-tril\left(\mathbf{A}\right)$ ${\mathbf{y}}_{MF}={\mathbf{H}}^{H}\mathbf{y}$ Initial estimations: ${\mathbf{z}}_{\left(0\right)}$ $=[0\cdots 0]$ ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{D}}^{-1}{\mathbf{y}}_{MF}$ ${\widehat{\mathbf{x}}}_{\left(1\right)}={\left(\right)}^{\mathbf{D}}-1\mathbf{D}+\frac{1}{\omega}\mathbf{U}$ Iteration:for j = 2 : 1 : n Apply the AMP algorithm using (6)–(8) end Return $\widehat{\mathbf{x}}$. |

Algorithm 2:Initialize the AMP detector using the GS method |

Input:$\mathbf{y},\mathbf{H},{\sigma}^{2},n,\omega $Output: Estimated signal $\widehat{\mathbf{x}}$Initialization:$\mathbf{A}={\mathbf{H}}^{H}\mathbf{H}+{\sigma}^{2}{\mathbf{I}}_{K}$ $\mathbf{D}=diag\left(\mathbf{A}\right)$, $\mathbf{U}=-triu\left(\mathbf{A}\right)$, $\mathbf{L}=-tril\left(\mathbf{A}\right)$ ${\mathbf{y}}_{MF}={\mathbf{H}}^{H}\mathbf{y}$ Initial estimations: ${\mathbf{z}}_{\left(0\right)}$ $=[0\cdots 0]$ ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{D}}^{-1}{\mathbf{y}}_{MF}$ ${\widehat{\mathbf{x}}}_{\left(1\right)}={\left(\right)}^{\mathbf{D}}-1$ Iteration:for j = 2 : 1 : n Apply the AMP algorithm using (6)–(8) end Return $\widehat{\mathbf{x}}$. |

Algorithm 3:Initialize the AMP detector using the JA method |

Input:$\mathbf{y},\mathbf{H},{\sigma}^{2},n,\omega $Output: Estimated signal $\widehat{\mathbf{x}}$Initialization:$\mathbf{A}={\mathbf{H}}^{H}\mathbf{H}+{\sigma}^{2}{\mathbf{I}}_{K}$ $\mathbf{D}=diag\left(\mathbf{A}\right)$, $\mathbf{U}=-triu\left(\mathbf{A}\right)$, $\mathbf{L}=-tril\left(\mathbf{A}\right)$ ${\mathbf{y}}_{MF}={\mathbf{H}}^{H}\mathbf{y}$ Initial estimations: ${\mathbf{z}}_{\left(0\right)}$ $=\left(\right)open="["\; close="]">0\cdots 0$ ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{D}}^{-1}{\mathbf{y}}_{MF}$ ${\widehat{\mathbf{x}}}_{\left(1\right)}={\mathbf{D}}^{-1}\left(\right)open="("\; close=")">{\widehat{\mathbf{x}}}_{MF}+\left(\right)open="("\; close=")">\mathbf{D}-\mathbf{A}$ Iteration:for j = 2 : 1 : n Apply the AMP algorithm using (6)–(8) end Return $\widehat{\mathbf{x}}$. |

## 4. Complexity Analysis

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Deployment map of 5G networks [3].

**Figure 3.**Performance of initialized approximate message passing (AMP)-based detector for $32\times 128$ massive multiple-input multiple-output (MIMO) system, 64QAM, and $n=2$.

**Figure 4.**Performance of initialized AMP-based detector for $32\times 128$ massive MIMO system, 64QAM, and $n=3$.

**Figure 5.**Performance of initialized AMP-based detector for $32\times 256$ massive MIMO system, 64QAM, and $n=2$.

Method | Number of Multiplications |
---|---|

AMP | $2KNn$ |

AMP-SOR | $2KNn+4{K}^{2}+4K$ |

AMP-GS | $2KNn+4{K}^{2}$ |

AMP-JA | $2KNn+4{K}^{2}-2K$ |

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**MDPI and ACS Style**

Albreem, M.A.; Kumar, A.; Alsharif, M.H.; Khan, I.; Choi, B.J.
Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems. *Sustainability* **2020**, *12*, 9281.
https://doi.org/10.3390/su12219281

**AMA Style**

Albreem MA, Kumar A, Alsharif MH, Khan I, Choi BJ.
Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems. *Sustainability*. 2020; 12(21):9281.
https://doi.org/10.3390/su12219281

**Chicago/Turabian Style**

Albreem, Mahmoud A., Arun Kumar, Mohammed H. Alsharif, Imran Khan, and Bong Jun Choi.
2020. "Comparative Analysis of Data Detection Techniques for 5G Massive MIMO Systems" *Sustainability* 12, no. 21: 9281.
https://doi.org/10.3390/su12219281