# A Robust Hybrid Iterative Linear Detector for Massive MIMO Uplink Systems

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## Abstract

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## 1. Introduction

## 2. Background

## 3. Methods

#### 3.1. System Model

**G**) is ${\mathbf{H}}^{H}\mathbf{H}$. However, a direct computation of ${\mathbf{A}}^{-1}$ requires $\mathcal{O}\left({K}^{3}\right)$. However, literature is rich with methods to avoid the direct and exact matrix inversion which relieves a burden of high computational complexity.

#### 3.2. Neumann Series

#### 3.3. Gauss–Seidel

#### 3.4. Jacobi

#### 3.5. A Stair Matrix

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- ${\mathbf{S}}_{\left(i,i-1\right)}=0,{\mathbf{S}}_{\left(i,i+1\right)}=0$ where $i=2,4,\dots ,2\u230a\frac{K}{2}\u230b$
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- ${\mathbf{S}}_{\left(i,i-1\right)}=0,{\mathbf{S}}_{\left(i,i+1\right)}=0$ where $i=1,3,\dots ,2\u230a\frac{K-1}{2}\u230b+1$

#### 3.6. Proposed Method

Algorithm 1: Detection method based on joint JA, GS, and a stair matrix |

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: ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{S}}^{-1}{\mathbf{y}}_{MF}$ ${\widehat{\mathbf{x}}}_{\left(1\right)}={\mathbf{D}}^{-1}\left({\widehat{\mathbf{x}}}_{MF}+\left(\mathbf{D}-\mathbf{A}\right){\widehat{\mathbf{x}}}_{\left(0\right)}\right)$ Iteration:for j = 2:1:n Apply the GS method using (7) end Return $\widehat{\mathbf{x}}$. |

## 4. Complexity Analysis

## 5. Numerical Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Performance of the GS, JA, NS, and proposed algorithm at $30\times 160$ mMIMO size and $n=2$.

**Figure 4.**Performance of the GS, JA, NS, and proposed algorithm at $40\times 160$ mMIMO size and $n=2.$

**Figure 5.**Performance of the GS, JA, NS, and proposed algorithm at $50\times 160$ mMIMO size and $n=2$.

**Figure 6.**Performance of the GS, JA, NS, and proposed algorithm at $60\times 160$ mMIMO size and $n=2$.

**Figure 7.**Performance of the GS, JA, NS, and proposed algorithm at $60\times 160$ mMIMO size and $n=3$.

**Figure 8.**Performance of the GS, JA, NS, and proposed algorithm at $60\times 160$ mMIMO size and $n=4$.

**Figure 9.**A comparison between a detector based on the GS method and the proposed method in 40 mMIMO system to obtain BER = ${10}^{-3}$.

Method | Complexity |
---|---|

NS | $(n-2){K}^{3}+N{K}^{2}+NK$ |

GS | $4n{K}^{2}$ |

JA | $n(4{K}^{2}-2K)$ |

Proposed Algorithm | ${K}^{2}(1+4n)+K-3$ |

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**MDPI and ACS Style**

Albreem, M.A.; Alsharif, M.H.; Kim, S.
A Robust Hybrid Iterative Linear Detector for Massive MIMO Uplink Systems. *Symmetry* **2020**, *12*, 306.
https://doi.org/10.3390/sym12020306

**AMA Style**

Albreem MA, Alsharif MH, Kim S.
A Robust Hybrid Iterative Linear Detector for Massive MIMO Uplink Systems. *Symmetry*. 2020; 12(2):306.
https://doi.org/10.3390/sym12020306

**Chicago/Turabian Style**

Albreem, Mahmoud A., Mohammed H. Alsharif, and Sunghwan Kim.
2020. "A Robust Hybrid Iterative Linear Detector for Massive MIMO Uplink Systems" *Symmetry* 12, no. 2: 306.
https://doi.org/10.3390/sym12020306