# Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks

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## Abstract

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## 1. Introduction

## 2. System Model

#### 2.1. System Specifications

#### 2.2. Channel Model

#### 2.3. Objective Function

## 3. Proposed Algorithm

#### 3.1. Initial Digital Beamforming Matrix Design

#### 3.2. Algorithm Design

Algorithm 1 Alt-MMSE Algorithm. |

Input: $\mathit{H}$, ${\mathit{F}}_{RF}$, ${\mathit{W}}_{RF}$1: Determine ${\mathit{H}}_{eff}={\mathit{H}}^{\ast}{\mathit{W}}_{RF}$ 2: Calculate the SVD of ${\mathit{H}}_{eff}$ according to Equations (11) and (19) 3: Determine ${\mathit{W}}_{BB}$ according to Equations (12) and (13) 4: $for\mathit{i}\le {\mathit{N}}_{\mathit{t}}^{RF}$ 5: $do$ 6: Calculate ${\mathit{F}}_{e}$ according to Equation (16) 7: Obtain the SVD of ${\mathit{F}}_{\mathit{e}\mathit{t}}^{\ast}{\mathit{F}}_{RF}={\mathit{U}}_{et}{\mathit{S}}_{et}{\mathit{V}}_{et}$ 8: Determine ${\mathit{F}}_{BB}={\mathit{V}}_{et}{\mathit{U}}_{et}^{\ast}$ according to Equation (19) 9: Obtain the argument of $arg\left({\mathit{F}}_{RF}\right)=arg({\mathit{F}}_{opt}{({\mathit{F}}_{BB})}^{\u2020})$ using Equation (15) 10: Calculate ${\mathit{W}}_{er}=\mathit{H}{\mathit{F}}_{RF}{\mathit{F}}_{BB}$ 11: Calculate the SVD of ${\mathit{W}}_{er}^{\ast}{\mathit{W}}_{RF}={\mathit{U}}_{er}{\mathit{S}}_{er}{\mathit{V}}_{er}$ 12: Determine ${\mathit{W}}_{BB}={\mathit{V}}_{er}{\mathit{U}}_{er}^{\ast}$ using the method of Equation (19) 13: Obtain the argument of $arg\left({\mathit{W}}_{RF}\right)=arg({\mathit{W}}_{opt}{({\mathit{W}}_{BB})}^{\u2020})$ using the method of Equation (15) 14: $\mathit{i}=\mathit{i}+1$ 15: end16: Determine ${\mathit{F}}_{BB}={\mathit{N}}_{\mathit{s}}\frac{{\mathit{F}}_{BB}}{\Vert {\mathit{F}}_{RF}{\mathit{F}}_{BB}{\Vert}_{F}}$, ${\mathit{W}}_{BB}={\mathit{N}}_{\mathit{s}}\frac{{\mathit{W}}_{BB}}{\Vert {\mathit{W}}_{RF}{\mathit{W}}_{BB}{\Vert}_{F}}$ 17: return ${\mathit{F}}_{RF}$,${\mathit{F}}_{BB}$, ${\mathit{W}}_{RF}$, ${\mathit{W}}_{BB}$ |

## 4. Simulation Results

#### 4.1. Spectral Efficiency Analysis when ${\mathit{N}}_{\mathit{s}}\ne {\mathit{N}}_{\mathit{t}}^{RF}={N}_{r}^{RF}$

#### 4.2. Spectral Efficiency Analysis when ${\mathit{N}}_{\mathit{s}}={\mathit{N}}_{\mathit{t}}^{RF}={N}_{r}^{RF}$

#### 4.3. Impact of RF Chains

#### 4.4. Impact of Antennas

#### 4.5. Energy Efficiency Analysis

#### 4.6. Bit Error Rate (BER) Analysis

## 5. Complexity Analysis

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**System performance when number of RF links ${\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}={\mathit{N}}_{\mathit{r}}^{\mathit{R}\mathit{F}}=\mathbf{8}$, and number of streams ${\mathit{N}}_{\mathit{s}}=\left\{\mathbf{2},\mathbf{4},\mathbf{8}\right\}$.

**Figure 3.**System performance when number of data streams equals RF links (${\mathit{N}}_{\mathit{s}}={\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}={\mathit{N}}_{\mathit{r}}^{\mathit{R}\mathit{F}}$).

**Figure 4.**System performance when number of data streams is ${\mathit{N}}_{\mathit{s}}=\left\{\mathbf{2},\mathbf{4},\mathbf{8}\right\}$ and SNR $=$ 0 dB.

**Figure 5.**Comparison of spectral efficiency against number of BS antennas with 8 RF chains and SNR = 0 dB.

**Figure 6.**Comparison of spectral efficiency against the number of user antennas with 8 RF chains and SNR = 0 dB.

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

Number of transmitting antennas ${\mathit{N}}_{\mathit{t}}$ | 256 |

Number of receiving antennas ${\mathit{N}}_{\mathit{r}}$ | 64 |

Antenna arrangement | UPA |

Antenna spacing $\mathit{d}$ (mm) | λ/2 |

Number of clusters ${\mathit{N}}_{\mathit{c}\mathit{l}}$ | 5 |

Number of scatterers ray ${\mathit{N}}_{\mathit{r}\mathit{a}\mathit{y}}$ | 10 |

Angle of Arrival (AoA), Angle of Departure (AoD) distribution | Laplacian |

Number of data streams ${\mathit{N}}_{\mathit{s}}$ | 8 |

Signal-to-noise ratio (SNR) | 30 dB |

Algorithm | Complexity |
---|---|

PP [18] | $\mathcal{O}\left(\mathbf{40}{\mathit{N}}_{\mathit{t}}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}{\mathit{N}}_{\mathit{s}}\right)$ |

PC-SIC [19] | $\mathcal{O}\left(\mathbf{4}{\mathit{N}}_{\mathit{t}}^{\mathbf{2}}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}+\mathbf{16}{\mathit{N}}_{\mathit{t}}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}+\mathbf{4}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}\right)$ |

OMP [20] | $\mathcal{O}({\mathit{N}}_{\mathit{t}}^{\mathbf{2}}{\left({\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}\right)}^{\mathbf{2}}{\mathit{N}}_{\mathit{s}})$ |

MO-AltMin [24] | $\mathcal{O}\left(\mathbf{10}{\mathit{N}}_{\mathit{t}}^{\mathbf{2}}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}{\mathit{N}}_{\mathit{s}}\right)$ |

Proposed Alt-MMSE | $\mathcal{O}\left({\mathit{N}}_{\mathit{t}}{\mathit{N}}_{\mathit{t}}^{\mathit{R}\mathit{F}}{\mathit{N}}_{\mathit{s}}\right)$ |

© 2019 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/).

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**MDPI and ACS Style**

Mohammed, S.L.; Alsharif, M.H.; Gharghan, S.K.; Khan, I.; Albreem, M.
Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks. *Symmetry* **2019**, *11*, 1424.
https://doi.org/10.3390/sym11111424

**AMA Style**

Mohammed SL, Alsharif MH, Gharghan SK, Khan I, Albreem M.
Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks. *Symmetry*. 2019; 11(11):1424.
https://doi.org/10.3390/sym11111424

**Chicago/Turabian Style**

Mohammed, Saleem Latteef, Mohammed H. Alsharif, Sadik Kamel Gharghan, Imran Khan, and Mahmoud Albreem.
2019. "Robust Hybrid Beamforming Scheme for Millimeter-Wave Massive-MIMO 5G Wireless Networks" *Symmetry* 11, no. 11: 1424.
https://doi.org/10.3390/sym11111424