# Low-Complexity Multi-User Parameterized Beamforming in Massive MIMO Systems

^{*}

## Abstract

**:**

## 1. Introduction

**Notations**: Boldface letter

**X**and

**x**denote a matrix and a column vector, respectively. ${X}^{H}$ and $\mathrm{tr}\left(X\right)$ denote the conjugate transpose and the trace of $X$, respectively. $\Vert x\Vert $ denotes the two-norm of

**x**and ${I}_{M}$ denotes an $\left(M\times M\right)$ identity matrix. $x~CN\left(m,C\right)$ refers to that

**x**is a circularly-symmetric complex Gaussian random vector with mean

**m**and covariance

**C**. ${\u2102}^{M\times 1}$ denotes all sets of M dimensional complex column vectors. The notation “$\underset{M\to \infty}{\overset{\mathrm{a}.\mathrm{s}.}{\to}}$” refers to almost sure convergence.

## 2. System Model

## 3. Proposed Beamforming Scheme

#### 3.1. Proposed Multi-User Parameterized Beamforming (MUPB)

**Lemma**

**1.**

**A**is uniformly bounded and

**x**is independent of

**A**, it can be shown that [26]

**Lemma**

**2.**

**A**be in Lemma 1, and

**x**and $y\sim CN\left(0,\frac{1}{M}{I}_{M}\right)$. When

**x**and

**y**are independent of

**A**, it can be shown that [26]

**Lemma**

**3.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Proof.**

#### 3.2. Computational Complexity

## 4. Performance Verification

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Lemma**

**4.**

## Appendix B

**Proof**

**of**

**Lemma**

**5.**

## Appendix C

**Proof**

**of**

**strict**

**quasi-concavity**

**of**${\overline{\mathrm{SLNR}}}_{k}$

**.**

**Theorem**

**A1.**

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**Figure 3.**Performance of multi-user parameterized beamforming (MUPB) according to ${\delta}_{\mathrm{th}}$.

Scheme | Computational Complexity |
---|---|

MRT | $4{K}_{\mathrm{total}}M$ |

ZF | $B{K}_{\mathrm{total}}{M}^{2}+\left(8B{K}_{\mathrm{total}}+4\right){K}_{\mathrm{total}}M+\frac{4}{3}B{K}_{\mathrm{total}}^{3}$ |

CEA-ZF | $\sum _{i=1}^{B}\left[\left({K}_{i}+{K}_{i}^{\mathrm{E}}\right){M}^{2}+\left(8{\left({K}_{i}+{K}_{i}^{\mathrm{E}}\right)}^{2}+4{K}_{i}\right)M+\frac{4}{3}{\left({K}_{i}+{K}_{i}^{\mathrm{E}}\right)}^{3}\right]$ |

Max SLNR | $\frac{4}{3}{K}_{\mathrm{total}}{M}^{3}+\left(\left(B+8\right){K}_{\mathrm{total}}+2\left(B-1\right)\right){M}^{2}+6{K}_{\mathrm{total}}M$ |

PB | $B{K}_{\mathrm{total}}{M}^{2}+\left(8B{K}_{\mathrm{total}}+18\right){K}_{\mathrm{total}}M+\frac{4}{3}B{K}_{\mathrm{total}}^{3}$ |

MUPB | $\sum _{i=1}^{B}\left({\psi}_{i}^{\mathrm{MRT}}+{\psi}_{i}^{\mathrm{PZF}}\right)}+\left[B\left(B-1\right)+10M\right]{K}_{\mathrm{total}$ |

Simulation Parameters | Value |
---|---|

Number of BSs B | 3 |

Cell radius ${R}_{\mathrm{max}}$ | 300 m |

Cell center radius ${R}_{\mathrm{c}}$ | 200 m |

Path loss ${\alpha}_{i,k},\text{}\forall i\text{}\mathrm{and}\text{}k$ | $148.1+37.6{\mathrm{log}}_{10}\left({d}_{i,k}\right)\text{}\mathrm{dB}$, where ${d}_{i,k}$ is the distance between BS i and user k in km |

Maximum transmit power ${P}_{\mathrm{T}}$ | 30 dBm |

Power allocation ${p}_{k},\text{}\forall k$ | ${P}_{\mathrm{T}}/K$ |

Noise power ${\sigma}_{n}^{2}$ | −92 dBm |

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**MDPI and ACS Style**

Jung, G.-W.; Lee, Y.-H.
Low-Complexity Multi-User Parameterized Beamforming in Massive MIMO Systems. *Electronics* **2020**, *9*, 882.
https://doi.org/10.3390/electronics9060882

**AMA Style**

Jung G-W, Lee Y-H.
Low-Complexity Multi-User Parameterized Beamforming in Massive MIMO Systems. *Electronics*. 2020; 9(6):882.
https://doi.org/10.3390/electronics9060882

**Chicago/Turabian Style**

Jung, Geon-Woong, and Yong-Hwan Lee.
2020. "Low-Complexity Multi-User Parameterized Beamforming in Massive MIMO Systems" *Electronics* 9, no. 6: 882.
https://doi.org/10.3390/electronics9060882