# Iterative Analog–Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems

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## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Contributions

#### 1.3. Notations

## 2. System Model Characterization

## 3. Transmitter Design

#### 3.1. Analog Precoder Design: No CSI at Users Terminals

#### 3.2. Analog Precoder Design: Average AoD knowledge at User Terminals

## 4. Analog–Digital Receiver Design

#### 4.1. Iterative Analog–Digital Equalizer

#### 4.1.1. Digital Feed-Forward Equalizer Design

#### 4.1.2. Analog Feed-Forward Equalizer Design

Algorithm 1 The proposed fully iterative hybrid space-frequency multi-user algorithm for broadband mmWave mMIMO systems | |||

1: | for$i=1,\dots ,{i}_{max}$do | ||

2: | Compute${\overline{W}}_{fd,k}^{(i)}$ accordingly to (20) | ||

3: | ${\overline{W}}_{res,k,0}^{(i)}={\overline{W}}_{fd,k}^{(i)}{\tilde{R}}_{k}^{(i-1)}$ | ||

4: | ${W}_{a,0}^{(i)}$= Empty Matrix | ||

5: | for$r=1,\dots ,{N}_{rx}^{RF}$do | ||

6: | ${\Pi}_{k,r}={\overline{W}}_{res,k,r-1}^{(i)}{A}_{rx}$; ${\Gamma}_{k,r}={({\tilde{R}}_{k}^{(i-1)})}^{1/2}{A}_{rx}$ | ||

7: | ${n}_{opt,r}=\mathrm{arg}{\mathrm{max}}_{l=1,\dots ,{N}_{CB}}{\displaystyle \sum _{k=1}^{S}\frac{{\left[{\Pi}_{k,r}^{H}{\Pi}_{k,r}\right]}_{l,l}}{{\left[{\Gamma}_{k,r}^{H}{\Gamma}_{k,r}\right]}_{l,l}^{}}}$ | ||

8: | ${W}_{a,r}^{(i)}=[{W}_{a,r-1}^{(i)}|{A}_{rx}^{\text{}({n}_{opt,r})}]$ | ||

9: | ${W}_{a,r}^{(i)}=[{W}_{a,r-1}^{(i)}|{A}_{rx}^{\text{}({n}_{opt,r})}]$ | ||

10: | ${W}_{d,k,r}^{(i)}={W}_{d,k}^{(i)}[{W}_{a,r}^{(i)}]$ | ||

11: | ${\overline{W}}_{res,k,r}^{(i)}=({\overline{W}}_{fd,k}^{(i)}-{W}_{ad,k,r}^{(i)}){\tilde{R}}_{k}^{(i-1)}$ | ||

12: | end for | ||

13: | ${B}_{d,k}^{(i)}=\left({W}_{d,k}^{(i)}{({W}_{a}^{(i)})}^{H}{H}_{k}-{I}_{U}\right){\left({\Psi}^{(i-1)}\right)}^{H}$ | ||

14: | ${\tilde{c}}_{k}^{(i)}={W}_{d,k}^{(i)}{({W}_{a}^{(i)})}^{H}{y}_{k}-{B}_{d,k}^{(i)}{\widehat{c}}_{k}^{(i-1)}$ | ||

15: | Compute${\widehat{c}}_{k}^{(i)}$and${\Psi}^{(i)}$ | ||

16: | end for |

#### 4.2. Complexity Comparison

## 5. Performance Results

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Derivation of (14)

## Appendix B. Equivalence between (28) and (29)

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**Figure 3.**Performance of the proposed hybrid iterative multi-user equalizer (Algorithm 1) for the NCSI precoder with ULA configuration and QPSK modulation.

**Figure 4.**Performance of the proposed hybrid iterative multi-user equalizer (Algorithm 1) for the PCSI precoder with ULA configuration and QPSK modulation.

**Figure 5.**Performance comparison between the Algorithm 1, the two-step, and GS/MMSE approaches for the NCSI precoder with ULA configuration and QPSK modulation.

**Figure 6.**Performance comparison between the Algorithm 1, the two-step, and GS/MMSE approaches for the PCSI precoder with ULA configuration and QPSK modulation.

**Figure 7.**Performance of the proposed hybrid iterative multi-user equalizer (Algorithm 1) for the PCSI precoder with ULA configuration and 16 QAM modulation.

**Figure 8.**Performance comparison between the Algorithm 1, the two-step and GS/MMSE approaches for the PCSI precoder with ULA configuration and 16 QAM modulation.

**Figure 9.**Performance of the proposed hybrid iterative multi-user equalizer (Algorithm 1) for the NCSI precoder with UPA configuration and QPSK modulation.

**Figure 10.**Performance of the proposed hybrid iterative multi-user equalizer (Algorithm 1) for the PCSI precoder with UPA configuration and QPSK modulation.

**Figure 11.**Performance comparison between the Algorithm 1 and the two-step approach for the NCSI precoder with ULA configuration, path loss, and shadowing and QPSK modulation.

**Figure 12.**Performance comparison between the Algorithm 1 and the two-step approach for the PCSI precoder with ULA configuration, path loss, and shadowing and QPSK modulation.

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

Carrier frequency | 72 GHz |

Antenna element spacing | Half-wavelength |

Array configuration | ULA or UPA |

${N}_{cl}$ | 4 |

${N}_{ray}$ | 5 |

${\sigma}_{{\varphi}_{q}^{rx}}^{},{\sigma}_{{\varphi}_{q}^{tx}}^{}$ | 10 degrees |

${N}_{c}$ | 512 |

$D$ | 128 |

$S$ | 128 |

$U$ | 8 |

${N}_{tx}$ | 16 |

${N}_{rx}$ | 32 |

${N}_{rx}^{RF}$ | 8 |

Modulation | QPSK or 16-QAM |

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## Share and Cite

**MDPI and ACS Style**

Magueta, R.; Castanheira, D.; Pedrosa, P.; Silva, A.; Dinis, R.; Gameiro, A.
Iterative Analog–Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems. *Sensors* **2020**, *20*, 575.
https://doi.org/10.3390/s20020575

**AMA Style**

Magueta R, Castanheira D, Pedrosa P, Silva A, Dinis R, Gameiro A.
Iterative Analog–Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems. *Sensors*. 2020; 20(2):575.
https://doi.org/10.3390/s20020575

**Chicago/Turabian Style**

Magueta, Roberto, Daniel Castanheira, Pedro Pedrosa, Adão Silva, Rui Dinis, and Atílio Gameiro.
2020. "Iterative Analog–Digital Multi-User Equalizer for Wideband Millimeter Wave Massive MIMO Systems" *Sensors* 20, no. 2: 575.
https://doi.org/10.3390/s20020575