# Impact of Stair and Diagonal Matrices in Iterative Linear Massive MIMO Uplink Detectors for 5G Wireless Networks

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. System Model

**w**is the $N\times 1$ additive white Gaussian noise (AWGN). In the MMSE detector, the transmitted signal can be estimated based on the equalization matrix ($\mathbf{A}$) as

#### 2.2. Definition of a Diagonal Matrix and a Stair Matrix

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

## 3. Diagonal and Stair Matrices in Iterative Linear M-MIMO UL Detectors

#### 3.1. Linear Detectors Based on Neumann Series and Newton Iteration

- ${\mathbf{A}}_{\left(0\right)}^{-1}={\mathbf{D}}^{-1}$, where $\mathbf{D}$ is the diagonal matrix,
- ${\mathbf{A}}_{\left(0\right)}^{-1}={\mathbf{S}}^{-1}$, where $\mathbf{S}$ is the stair matrix.

**D**and

**S**are extracted from each

**H**. Figure 1 shows that the condition in Equation (7) is always valid for both diagonal and stair matrices.

#### 3.2. Linear Detectors Based on Iterative Methods

**D**and

**S**and is expressed as one of the following solutions:

- Using a stair matrix: ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{S}}^{-1}{\mathbf{y}}_{MF}$,
- Using a diagonal matrix: ${\widehat{\mathbf{x}}}_{\left(0\right)}={\mathbf{D}}^{-1}{\mathbf{y}}_{MF}$.

## 4. Complexity Analysis

**D**and

**S**are exploited, respectively. In order to obtain a good performance, a detector based on

**S**requires a small number of iterations in comparison with a detector based on

**D**matrix. The required number of multiplications are listed in Table 1.

## 5. Numerical Results

**S**and $\mathbf{D}$. However, statistical values are smaller in the case of

**S**than the values in the case of

**D**. For instance, the mean values of the convergence condition are 0.7027 and 0.7255 when using

**S**and

**D**, respectively.

**S**and

**D**matrices. Detectors initialized by

**S**outperform detectors initialized by

**D**in each iteration. For example, at $n=1$, the BER $={10}^{-2}$ is obtained at SNR $=11$ dB and SNR $=13$ dB for the GS based detector using

**S**and

**D**, respectively. In the SOR based detector, the target performance is achieved at $n=1$ and $n=2$ using

**S**and

**D**, respectively. The worst performance is occurred when a detector based on the RI and the NS is initialized by

**D**. In other words, all detectors can achieve the target performance when the number of iterations is large (i.e., $n\ge 5$), but the best detector achieves the target performance with the smallest number of iterations (lowest complexity).

**S**when SNR $=14.5$ dB. In contrast, a detector based on the NS method and initialized by

**D**requires the highest computational complexity to achieve the target BER performance.

## 6. Conclusions

**D**and

**S**has been studied in M-MIMO UL detectors. It is shown that the initialization of a detector based on

**S**achieves a good balance between the performance and the computational complexity. A detector based on the GS method and initialized by

**S**obtained the best performance–complexity profile. However, a detector based on the NS method and

**D**requires a high number of iterations to achieve the target performance and hence it has the highest computational complexity.

**S**could be extended to nonlinear detectors such as the sphere decoding (SD) and successive interference cancellation (SIC) detectors. In addition, the performance-complexity profile of detectors based on local research and belief propagation (BP) could be developed by the usage of

**S**. However, the utilization of

**S**should be investigated in realistic radio channels such as the QUAsi Deterministic RadIo channel GenerAtor (QuaDRiGa) package.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Albreem, M.A.M. 5G wireless communication systems: Vision and challenges. In Proceedings of the 2015 International Conference on Computer, Communications, and Control Technology (I4CT) 2015, Kuching, Malaysia, 21–23 April 2015; pp. 493–497. [Google Scholar] [CrossRef]
- Mohammed, S.; Alsharif, M.; Gharghan, S.; Albreem, M. Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks. Symmetry
**2019**, 11, 1424. [Google Scholar] [CrossRef] [Green Version] - Alsharif, M.; Kim, S.; Kuruoğlu, N. Energy Harvesting Techniques for Wireless Sensor Networks/Radio-Frequency Identification: A Review. Symmetry
**2019**, 11, 865. [Google Scholar] [CrossRef] [Green Version] - Kim, H.; Kim, D.; Yang, S.; Son, Y.; Han, S. Mitigation of Inter-Cell Interference Utilizing Carrier Allocation in Visible Light Communication System. IEEE Commun. Lett.
**2012**, 16, 526–529. [Google Scholar] [CrossRef] - Albreem, M.A.; Juntti, M.; Shahabuddin, S. Massive MIMO Detection Techniques: A Survey. IEEE Commun. Surv. Tutor.
**2019**, 21, 3109–3132. [Google Scholar] [CrossRef] [Green Version] - Thian, B.S.; Goldsmith, A. Decoding for MIMO Systems with imperfect channel state information. In Proceedings of the 2010 IEEE Global Telecommunications Conference GLOBECOM, Miami, FL, USA, 6–10 December 2010; pp. 1–6. [Google Scholar] [CrossRef]
- Albreem, M.; Salleh, M. Lattice Sphere Decoding for Block Data Transmission Systems. Wirel. Person. Commun.
**2015**, 82, 1833–1850. [Google Scholar] [CrossRef] - Albreem, M.; Ismail, N. A Review: Detection Techniques For LTE System. Telecommun. Syst.
**2015**, 63, 153–168. [Google Scholar] [CrossRef] - Chaudhary, M.; Meena, N.K.; Kshetrimayum, R.S. Local search based near optimal low complexity detection for large MIMO System. In Proceedings of the 2016 IEEE International Conference on Advanced Networks and Telecommunications Systems (ANTS), Bangalore, India, 6–9 November 2016; pp. 1–5. [Google Scholar] [CrossRef]
- Takahashi, T.; Ibi, S.; Sampei, S. On Normalization of Matched Filter Belief in GaBP for Large MIMO Detection. In Proceedings of the 2016 IEEE 84th Vehicular Technology Conference (VTC-Fall), Montreal, QC, Canada, 18–21 September 2016; pp. 1–6. [Google Scholar] [CrossRef]
- Xie, T.; Han, Q.; Xu, H.; Qi, Z.; Shen, W. A Low-Complexity Linear Precoding Scheme Based on SOR Method for Massive MIMO Systems. In Proceedings of the IEEE Vehicular Technology Conference, Glasgow, Scotland, 11–14 May 2015; pp. 1–5. [Google Scholar] [CrossRef]
- Costa, H.; Roda, V. A Scalable Soft Richardson Method for Detection in a Massive MIMO System. Prz. Elektrotechniczny
**2016**, 92, 199–203. [Google Scholar] [CrossRef] - Minango, J.; de Almeida, C. Optimum and quasi-optimum relaxation parameters for low-complexity massive MIMO detector based on Richardson method. IEE Electron. Lett.
**2017**, 53, 1114–1115. [Google Scholar] [CrossRef] - Kong, B.Y.; Park, I.C. Low-complexity symbol detection for massive MIMO uplink based on Jacobi method. In Proceedings of the IEEE International Symposium on Personal, Indoor, and Mobile Radio Communications, Valencia, Spain, 4–8 September 2016; pp. 1–5. [Google Scholar] [CrossRef]
- Wang, F.; Zhang, C.; Liang, X.; Wu, Z.; Xu, S.; You, X. Efficient iterative soft detection based on polynomial approximation for massive MIMO. In Proceedings of the 2015 International Conference on Wireless Communications & Signal Processing, Nanjing, China, 15–17 October 2015; pp. 1–5. [Google Scholar] [CrossRef]
- Dai, L.; Gao, X.; Su, X.; Han, S.; I, C.L.; Wang, Z. Low-Complexity Soft-Output Signal Detection Based on Gauss Seidel Method for Uplink Multiuser Large-Scale MIMO Systems. IEEE Trans. Veh. Technol.
**2015**, 64, 4839–4845. [Google Scholar] [CrossRef] - Zhou, J.; Hu, J.; Chen, J.; He, S. Biased MMSE soft-output detection based on conjugate gradient in massive MIMO. In Proceedings of the 2015 IEEE 11th International Conference on ASIC, Chengdu, China, 3–6 November 2015; pp. 1–4. [Google Scholar] [CrossRef]
- Jiang, F.; Li, C.; Gong, Z.; Su, R. Stair Matrix and its Applications to Massive MIMO Uplink Data Detection. IEEE Trans. Commun.
**2018**, 66, 2437–2455. [Google Scholar] [CrossRef] - Prabhu, H.; Edfors, O.; Rodrigues, J.; Liu, L.; Rusek, F. Hardware efficient approximative matrix inversion for linear pre-coding in massive MIMO. In Proceedings of the 2014 IEEE International Symposium on Circuits and Systems, Melbourne, Australia, 1–5 June 2014; pp. 1700–1703. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**$\parallel \mathbf{I}-\mathbf{A}{\mathbf{A}}_{\left(0\right)}^{-1}\parallel $ in Equation (7) for ${10}^{4}$ channel realizations using (

**a**) a stair matrix and (

**b**) a diagonal matrix.

**Figure 2.**Performance of a detector based on iterative matrix inversion methods with

**D**and

**S**for $32\times 256$ M-MIMO system.

Method | Stair Matrix (S) | Diagonal Matrix (D) |
---|---|---|

NI | $2(n-1){K}^{3}+N{K}^{2}+K(N+3)-3$ | $2(n-1){K}^{3}+N{K}^{2}+NK$ |

RI | $4n{K}^{2}+K(2n+3)-3$ | $4n{K}^{2}+2nK$ |

SOR | $4n{K}^{2}+K(n+3)-3$ | $4n{K}^{2}+4nK$ |

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

JA | $2nK(2K-1)$ | $n(4{K}^{2}-2K)$ |

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

**Table 2.**Statistics of $\parallel \mathbf{I}-\mathbf{A}{\mathbf{A}}_{\left(0\right)}^{-1}\parallel $ in Equation (7) for ${10}^{4}$ channel realizations for $32\times 256$ M-MIMO system.

Matrix | Mean | Median | Standard Deviation | Prob. Quadratic Convergence |
---|---|---|---|---|

Stair (S) | 0.7027 | 0.7007 | 0.2776 | 1 |

Diagonal (D) | 0.7255 | 0.7233 | 0.2868 | 1 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Albreem, M.A.; Alsharif, M.H.; Kim, S.
Impact of Stair and Diagonal Matrices in Iterative Linear Massive MIMO Uplink Detectors for 5G Wireless Networks. *Symmetry* **2020**, *12*, 71.
https://doi.org/10.3390/sym12010071

**AMA Style**

Albreem MA, Alsharif MH, Kim S.
Impact of Stair and Diagonal Matrices in Iterative Linear Massive MIMO Uplink Detectors for 5G Wireless Networks. *Symmetry*. 2020; 12(1):71.
https://doi.org/10.3390/sym12010071

**Chicago/Turabian Style**

Albreem, Mahmoud A., Mohammed H. Alsharif, and Sunghwan Kim.
2020. "Impact of Stair and Diagonal Matrices in Iterative Linear Massive MIMO Uplink Detectors for 5G Wireless Networks" *Symmetry* 12, no. 1: 71.
https://doi.org/10.3390/sym12010071