# Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### Point Process Analysis

_{1}…s

_{n}, each based on an (x,y) coordinate in 2D space. Pattern analysis, based on the spatial associations between positions, is represented by the following types: 1. independent, 2. regular and 3. clustered (Figure 2) [7,8,9].

**Poisson random process**with distribution is given by Equation (1) [10]:

^{2}. The random variable N(A) is the number of events in the set A⊂X defined as a random process (Figure 3a).

**Z(s)**is called a

**determinantal**random point process if for a random subset A drawn according to Z(s), the process holds for every A⊂X (Equation (2)):

**inhibition process**[10] with distance δ, where δ is the threshold distance by deleting all pairs of events that are closer than δ (Figure 3b). Finally a point process with diversion called

**determinantal**

**point process [11,12,13]**is illustrated in Figure 3c. A determinantal point process is often used to induce diversity or repulsiveness among the points of a sample. Figure 3a illustrates simulated coordinates (x,y) of 100 positions of a

**Poisson point process**with estimated $\stackrel{\u2322}{\lambda}=2.018$ simulated on the unit square. Figure 3b illustrates simulated coordinates (x,y) of 100 points of an

**inhibition point process**with estimated $\stackrel{\u2322}{\lambda}=0.781$ and δ = 0.1 simulated on the unit square. Figure 3c illustrates simulated coordinates (x,y) of 100 points of a

**determinantal**

**point process**with estimated $\stackrel{\u2322}{\lambda}=3.01$ simulated on the unit square.

- The number of positions in a region A has a Poisson distribution with mean λN(A)
- The positions of these points are i.i.d. and uniformly distributed inside A
- The contents of two disjoint regions A and B are independent

_{i},s

_{j}) = γ(s

_{i}− s

_{j}) = γ(d) (depending only on direction and distance). For the isotropy case, the second-order intensity depends only on the distance between s

_{i}and s

_{j}. A brief presentation of the previous research involving point process analysis in antenna stations is given in Table 1 [19,20,21,22,23].

## 3. Results

^{−1}E [number of extra points within distance d of a randomly chosen arbitrary point]

^{2}, defining λ as value of the intensity under Poisson random point process; so K(d) = πd

^{2}. The value λK(d) could be defined as the expected number of points within distance d of a randomly chosen arbitrary point; hence (Equation (8)):

_{d}(d

_{ij}). Thus, estimation of the K-function is given by the form (Equation (9)):

_{ij}is the distance between the i-th and j-th observed event position, which can be viewed as the radius of a circle centered at event position i and passing through j; and w

_{ij}is a weighting value equal to this circle’s proportion of the entire area A. For specific distance value d, I

_{d}(d

_{ij}) is an indicator function which is 1 if d

_{ij}≤ d. It is clear that as d increases, w

_{ij}→∞. If $K(d)$ > πd

^{2}, the theoretical K-function, then there is evidence of clustering. Investigation for a clustering pattern could be achieved by plotting $\stackrel{\u2322}{K}(d)$ and $K(d)$, so spatial dependences could be illustrated. Figure 5b graphically compares Ripley’s K-function and the theoretical one, and clearly shows evidence of regularity, as well as the Poisson random point process with 1000 positions (Figure 5a) [28,29,30].

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 4.**Real data locations of base stations, antennas of the mobile network in Paris [14].

**Table 1.**Presentation of previous research into random point process involving antenna stations for telecommunications.

References | Brief Presentation |
---|---|

Aurélien Vasseur (2017) [14] | Focus on Poisson point process considering probabilistic modeling using data based on antennas of the mobile network in Paris |

Yingzhe Li, Franc¸ois Baccelli, Harpreet S. Dhillon, Jeffrey G. Andrews (2014) [19] | Poisson random point analysis using determinant point process |

Ezequiel Fattori, Pablo Groisman, and Carlos Sarraute (2016) [20] | Modeling the spatial distribution of cell phone antennas in the city of Buenos Aires (CABA) |

Benedikt Jahnel (2018) [21] | Probabilistic methods in Telecommunications |

A. Guo, Y. Zhong, M. Haenggi, and W. Zhang (2014) [22] | Modeling using Gauss–Poisson point process |

N. Deng, W. Zhou, and M. Haenggi (2015) [23] | Modeling using Ginibre point process |

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Zimeras, S.
Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques. *Telecom* **2022**, *3*, 541-547.
https://doi.org/10.3390/telecom3030030

**AMA Style**

Zimeras S.
Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques. *Telecom*. 2022; 3(3):541-547.
https://doi.org/10.3390/telecom3030030

**Chicago/Turabian Style**

Zimeras, Stelios.
2022. "Spatial Pattern Simulation of Antenna Base Station Positions Using Point Process Techniques" *Telecom* 3, no. 3: 541-547.
https://doi.org/10.3390/telecom3030030