# Spatial-Temporal-DBSCAN-Based User Clustering and Power Allocation for Sum Rate Maximization in Millimeter-Wave NOMA Systems

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## Abstract

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## 1. Introduction

- mmWave can strongly correlate the channels of different users through its highly directional transmissions; thus, the necessary combination between NOMA and mmWave can be achieved; and,
- with the higher connectivity and the high capacity provided by NOMA, the benefits of mmWave-NOMA systems can be multiplied based on the aforementioned characteristics.

- The optimization problem was investigated to maximize the sum-rate of a downlink mmWave- NOMA-based system by jointly optimizing user clustering, PA, and HBF strategies. It is challenging to solve this problem, because it involves combinatorial complexity. Thus, we separated the underlying problem into sub-problems of the user clustering, PA, and HBF problems.
- To solve the user clustering problem, we propose a novel spatial–temporal density-based spatial clustering of applications with noise (ST-DBSCAN)-based algorithm for user clustering in downlink mmWave-NOMA systems. Our proposed ST-DBSCAN algorithm exploits the similarity between the position of users and the correlation between the channel gains of users to group users into clusters. Recent research has only focused on the correlations between user channel gains and has not considered the spatial distribution of users. In this study, we consider both user positions and channel gains to maximize the system sum-rate. When the users are in close proximity to each other, the BS can form a narrower beam in order to achieve a higher beam gain than that achievable with a wider beam [9]. By exploiting this characteristic, our ST-DBSCAN-based user clustering algorithm can enhance the performance of mmWave-NOMA systems.
- We obtain a sub-optimal inter-cluster PA under fixed and arbitrary HBF. To deal with the non-convexity of the PA problem, we separate this problem into two sub-problems, intra-cluster PA, and inter-cluster PA. Based on the intra-cluster PA that was solved in [10], we propose an inter-cluster PA algorithm. Moreover, we apply the boundary-compressed particle swarm optimization (BC-PSO) algorithm to eliminate inter-cluster interference, to boost the performance of the system.
- Simulation results show that the proposed ST-DBSCAN-based user clustering algorithm outperforms other user clustering algorithms. The simulation results also show that our PA and HBF solutions can improve the performance of mmWave-NOMA systems.

## 2. Related Work

#### 2.1. User Pairing

#### 2.2. User Clustering

## 3. System Model

#### 3.1. Network Model

#### 3.2. User Spatial Distribution Model

#### 3.3. Channel Model

#### 3.4. Signal Model

## 4. Problem Formulation

## 5. ST-DBSCAN-Based User Clustering for mmWave-NOMA Systems

#### 5.1. Overview and Preliminaries of ST-DBSCAN

**Definition 1**(Neighborhood)

**.**

**Definition 2**(Eps-neighborhood of a point)

**.**

**Definition 3**(Core object and border object)

**.**

**Definition 4**(Directly density-reachable)

**.**

**Definition 5**(Density-reachable)

**.**

**Definition 6**(Density-connected)

**.**

**Definition 7**(Cluster)

**.**

- 1.
- $\forall a,b$: (Maximality) If $a\in C$ and b is density-reachable from a with regard to minPts and Eps, then $b\in C$.
- 2.
- $\forall a,b\in C$: (Connectivity) A point a is density-connected to point b with regard to minPts and Eps.

**Definition 8**(Noise)

**.**

#### 5.2. ST-DBSCAN Based User Clustering

Algorithm 1 ST-DBSCAN-based user clustering algorithm |

Input: Set of user information $\mathcal{U}=\{{U}_{1},{U}_{2},\cdots ,{U}_{n}\}$.Input: Eps1 (${\u03f5}_{1}$): Radius between users in geographical coordinates.Input: Eps2 (${\u03f5}_{2}$): Maximum channel gain between users.Input: minPts: Density threshold (minimum number of users within ${\u03f5}_{1}$ and ${\u03f5}_{2}$ distance).Input: $\Delta \u03f5$: Threshold value of a cluster.
- 1:
- Cluster_Number = 0
- 2:
**for**$i=1\mathbf{to}$ n**do**- 3:
**if**${U}_{i}$ is not assigned in a cluster**then**- 4:
- $A=$ Retrieve_Neighbors(${U}_{i},{\u03f5}_{1},{\u03f5}_{2}$)
- 5:
**if**$\left|A\right|<minPts$**then**- 6:
- Mark ${U}_{i}$ as noise user
- 7:
**else**- 8:
- Cluster_Number = Cluster_Number + 1
- 9:
**for**$j=1$**to**$\left|A\right|$**do**- 10:
- Assigned all users in A with current Cluster_Number
- 11:
**end for**- 12:
- Push(all users in A)
- 13:
**while not**isEmpty()**do**- 14:
- Current_User = Pop()
- 15:
- B = Retrieve_Neighbors(Current_User, ${\u03f5}_{1}$, ${\u03f5}_{2}$)
- 16:
**if**$\left|B\right|\ge minPts$**then**- 17:
**for**all users u in B**do**- 18:
**if**(u is not noise user**or**not assigned in a cluster)**and**|Cluster_Avg() - u.value| $\le \Delta \u03f5$.**then**- 19:
- Mark u with current Cluster_Number
- 20:
- Push(u)
- 21:
**end if**- 22:
**end for**- 23:
**end if**- 24:
**end while**- 25:
**end if**- 26:
**end if**- 27:
**end for**
Output: Set of clusters $\mathcal{K}$ and set of users in cluster ${\mathcal{C}}_{k},wherek\in \mathcal{K}$. |

## 6. Solution of Power Allocation and Hybrid Beamforming

#### 6.1. Power Allocation with Fixed Beamforming

#### 6.1.1. Intra-Cluster Power Allocation

#### 6.1.2. Inter-Cluster Power Allocation

**Proposition**

**1.**

**Proof.**

Algorithm 2 Inter-cluster power allocation algorithm |

Input: $U,K,\left\{{C}_{k}\right\},{P}_{t},\left\{{\mathbf{h}}_{k}\right\},\left\{{\overline{SINR}}_{k}\right\},\mathbf{W},\mathrm{and}\phantom{\rule{4.pt}{0ex}}{T}_{\mathrm{max}}.$- 1:
- ${p}_{k}^{\u2606\left(0\right)}=\frac{{P}_{t}}{K}(1\le k\le K)$.
- 2:
**for**$t=1:{T}_{max}$**do**- 3:
**for**$k=1:K$**do**- 4:
- 5:
- Obtain ${p}_{k}$ in (30)
- 6:
- ${p}_{k}^{\u2606\left(t\right)}={p}_{k}$
- 7:
**end for**- 8:
**end for**- 9:
- ${p}_{k}^{\u2606}={p}_{k}^{\u2606\left({T}_{max}\right)}$
Output: Inter-cluster PA: $\left\{{p}_{k}^{\u2606}\right\}$. |

#### 6.2. HBF with BC-PSO

#### 6.2.1. DBF Using AZF

#### 6.2.2. BC-PSO Based ABF

## 7. Simulation Results

## 8. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Proof of Proposition 1

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**Figure 3.**Comparison of achievable sum-rate between different clustering algorithms under LoS and NLoS channel models with $U=14$, $K=2$, and $N=8$.

**Figure 4.**Comparison of achievable sum-rate between proposed mmWave-NOMA and mmWave-OMA, mmWave-NOMA Ideal.

**Figure 5.**Comparison of achievable sum-rate between proposed inter-cluster PA algorithm and no-inter-cluster PA scheme.

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**MDPI and ACS Style**

Hoang, H.-T.; Pham, Q.-V.; Hwang, W.-J.
Spatial-Temporal-DBSCAN-Based User Clustering and Power Allocation for Sum Rate Maximization in Millimeter-Wave NOMA Systems. *Symmetry* **2020**, *12*, 1854.
https://doi.org/10.3390/sym12111854

**AMA Style**

Hoang H-T, Pham Q-V, Hwang W-J.
Spatial-Temporal-DBSCAN-Based User Clustering and Power Allocation for Sum Rate Maximization in Millimeter-Wave NOMA Systems. *Symmetry*. 2020; 12(11):1854.
https://doi.org/10.3390/sym12111854

**Chicago/Turabian Style**

Hoang, Huu-Trung, Quoc-Viet Pham, and Won-Joo Hwang.
2020. "Spatial-Temporal-DBSCAN-Based User Clustering and Power Allocation for Sum Rate Maximization in Millimeter-Wave NOMA Systems" *Symmetry* 12, no. 11: 1854.
https://doi.org/10.3390/sym12111854