Artigo - PDF
Scientific Society Journal
ISSN: 2595-8402
Journal DOI: 10.61411/rsc31879
REVISTA SOCIEDADE CIENTÍFICA, VOLUME 7, NÚMERO 1, ANO 2024
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ARTIGO ORIGINAL
Modelo matemático para localização ótima de torres de transmissão de sinal celular
Cristiane Ruiz Gomes1; Igor Ruiz Gomes2; Herminio Simões Gomes3
Como Citar:
GOMES, Cristiane Ruiz; GOMES, Igor Ruiz; GOMES, Herminio Simões. Modelo matemático para localização Ótima de Torres de Transmissão de sinal celular. Revista Sociedade Científica, vol.7, n. 1, p.4768-4779, 2024.
https://doi.org/10.61411/rsc202477517
Área do conhecimento: Engenharias.
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Sub-área: Telecomunicações.
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Palavras-chaves: Mathematical Model; Optimization; Radio Propagation; KNN Classifier.
Publicado: 14 de outubro de 2024.
Resumo
Este artigo aplica um modelo matemático para localização ótima de torres de transmissão de sinal celular em um cenário distinto do cenário original para o qual o modelo foi desenvolvido. A otimização proposta tem como objetivo maximizar a quantidade de usuários atendidos restrito a um padrão mínimo de qualidade de sinal. Para validar o modelo foram feitas campanhas de medição em 44 pontos em um cenário residencial no município de Castanhal-PA, nas frequências de 1800 MHz e 2600 MHz. As saídas do modelo fornecem a potência recebida, a perda média e a localização ótima da nova torre a ser instalada.
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Mathematical model for optimal location of cellular transmission towers
Abstract
This paper applies a mathematical model for the optimal location of cellular signal transmission towers in a scenario different from the original scenario for which the model was developed. The proposed optimization aims to maximize the number of users served while maintaining a minimum signal quality standard. To validate the model, measurement campaigns were carried out at 44 points in a residential setting in the municipality of Castanhal-PA, at frequencies of 1800 MHz and 2600 MHz. The model outputs provide the received power, the average loss and the optimal location of the new tower to be installed.
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Modelo matemático para la ubicación óptima de torres de transmisión celular
Resumen
Este artículo aplica un modelo matemático para la ubicación óptima de torres de transmisión de señal celular en un escenario diferente al escenario original para el cual se desarrolló el modelo. La optimización propuesta tiene como objetivo maximizar el número de usuarios atendidos, restringidos a un estándar mínimo de calidad de señal. Para validar el modelo, se realizaron campañas de medición en 44 puntos de un escenario residencial en el municipio de Castanhal-PA, en las frecuencias de 1800 MHz y 2600 MHz. Las salidas del modelo proporcionan la potencia recibida, la pérdida promedio y la ubicación óptima de. la nueva torre a instalar.
Palabras-clave: Modelo Matemático; Optimización; Propagación de Radio; Clasificador KNN.
1. Introduction
The understanding of the electromagnetic phenomenon by Maxwell, in the nineteenth century allowed the development of many technological improvements to the humanity. Among these, wireless telecommunication services as radio, cell phone and mobile internet services. In this context, models that can estimate and/or predict the behavior of electromagnetic waves in various frequency ranges are fundamental to the improvement and application of wireless telecommunication services.
Validation of these models, in special the empirical models, is one of the main steps of the modelling process. This article aims to applicate the empirical model proposed in [1] in a different scenario than that of its first fitment. The model tested here is based in the K-Nearest-Neighborhood (KNN) classifier technic and in the discretization of the studied scenario in a map which each unit of mapping is a square. Work [1] showed that this model obtained good results, considering its initial purpose.
Some related works are [2], [3], [4], [5] and [6]. Work [2] shows a study of the influence of vegetation in the 700 MHz band for Outdoor-to-Indoor paths. Where there is more vegetation, the authors of work [2] conclude that a loss of signal of about 10 dB and a bigger mean delay occur. The excess of vegetation in the studied scenario, was pointed out by the authors as the main causes or spreading and absorption of the signal.
The authors present in [3] a three-layer deterministic model based on Dyadic Green’s Functions. They worked in the UHF frequency range and emphasized differences in the received power for densely wooded urban scenarios and different climatic conditions. The authors of work [4] calculate the value of the Signal to Interference Ratio (SIR) in LTE air-to-ground networks in low altitude flights. For low altitudes, the results show that the SIR is greater for macrocells than microcells. The situation is the opposite in higher altitudes.
Study [5] describes the main topics for planning the installation of new cellular technologies, in particular, 5G. The authors underline the importance of changing the large RBS (Radio Base Station) to a small cell format and the importance of optimizing the positioning of the new RBS for better coverage, as well as propagation models designed for city maps, refarming frequencies, etc. In [6], a model for planning LTE networks is put forward in maritime regions by means of combinations of transmitters when they are not further than 100 km from each other.
The next sections of the article are distributed as follows: Section 2 explains the methodology of the work; Section 3 presents the propagation model used here; Section 4, development and discussions, contains the description of the measurement campaign, the results obtained and the discussion about them; Finaly, section 5 includes the final considerations of this work.
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2. Methodology
This paper presents a quantitative research for the adaptation and validation of a mathematical model for the optimal location of cell phone towers. For this purpose, measured data on the received power of RBS were collected in a residential area of the city of Castanhal-PA. These data were then inserted into a mathematical model developed for predicting cell phone signal in the Amazon region. Finally, the results obtained by the model adjusted for the considered scenario were used for the optimal location of the RBS to be inserted.
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2.1 The Model Used
The proposed model in [1] analyzes pattern of frequencies to determining a polynomial behavior with the purpose of the model being used in frequencies that were not measured. The model allows adjustments for different scenarios including routes with or without vegetation, buildings or passage through fresh water (such as rivers or lakes). Propagation trend curves were designed for the frequencies 521 MHz, 2100 MHz and 2600MHz, that were measured.
First, for each measured frequency, a trend curve was generated, expressed in the form:
(1)
Where is y the power trendline received in a data frequency, x the distance to the reference-point and ai,bi, i=1,2,3 are parameters to be determined through parabolic fitting by the least squares method.
Subsequently, the discretization and categorization of the scenario under study is carried out. The KNN classifier was used to classify grid squares that did not have a previous vegetation and building classification.
To determine the class of an element that does not belong to the training set, the KNN classifier looks for the K elements of the training set that are closest to the unknown element, that is, have the smallest “distance”. It is verified which classes of these K elements are the most frequent class. This class is assigned to the unknown element [7]. Here, the most common metric was used, which is the Euclidean Distance, given by:
(2)
Where X=(x1 , x2 ,…, xn) and Y=( y1 , y2 , …, yn ) belong to Rn.
Lastly, the calculation of received signal strength at each point depends on the total sum of losses in each square. Each measured point is at a different distance from the RBS, which means there is a need to determine a) how many and b) what kinds of squares were crossed.
The loss function or received strength depends on the number of points crossed and hence, the distance. Thus, this function can be expressed by the formula in (3):
(3)
The Euclidian distance was used to measure the distance from the transmission tower to the measured points. Since the scenario consists of a map with a discretization mesh, a straight line between two arbitrary points on the map can have a jagged or serrated (i.e. aliasing) line, in the shape of a stairway. What matters for the purposes of this study is to count how many and what types of squares were crossed by the abovementioned line, which connects the transmitter to the measured point.
A linear system of equations was formed by means of the measurement data and the number of each type of square used. Then there is an equation of the system for each measured point. N measured points, implies there are N equations. The equations of the system are of the form of (4):
(4)
with: αj ; losses in each type of square to be calculated; qi,j number of each type j of the square crossed in Equation i ; bi : signal strength received at point i.
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3. Development and Discussions
Measurements were performed in the 1800 MHz and 2600 MHz bands, using the Android Network Cell Info application installed on the Motorola One mobile device, to monitor the received signal strength in dBm. This application is a signal measurement and monitoring tool that supports several cellular networks and makes it possible to measure the RSSI, RSRP, RSRQ, CQI and SINR parameters for the location and current network to which the device is connected, with proper visualization and tracking coordinates in the map area. The measurements were conducted in a private region (condominium), located in the city of Castanhal in the state of Pará-Brazil, with an area of approximately 0.33 km². Data were collected from 44 points. The mobile device used to obtain the signal was connected to different networks (GSM, EDGE, LTE, HSDPA and HSPA+). The closest transmitter tower in the region is located at (1.282137°S, 47.949870°W). Information regarding the tower can be obtained from [8]. Figure 1 displays the satellite image containing the measurement locations (green marker) and the nearest transmitting tower (red marker).
Figure 1 - Scenario under study. Transmission tower (green marker), measurement points (red markers)
The location optimization model for installing new towers, maximizing coverage and minimizing losses, was adjusted and applied in a different scenario from the original, for testing and validation.
The residential condominium area was divided into a grid of 30 by 16 square, each of them has 29.5 m on a side. Figure 2 shows the scenario under study with the grid. Each of the squares has been assigned a numerical characteristic. Number 1 if it is an open place, number 2 if it is a place with low buildings and 3 if it is a place with some afforestation.
Figure 2 - Scenario under study with grid for application of the mathematical model.
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Then, a map of received powers for each grid point was made using a generalized regressor (Newgrnn function from Matlab®). Thus, it was possible to estimate a power value for each square of the grid.
Assuming that the signal suffers interference from the environment, it was assumed that the amount of crossed squares and their characteristics are related to the losses up to the considered square. The loss at each point was calculated from the transmission tower to the grid point, counting the total number of each characteristic crossed to that square. This was done for all 480 points of the grid.
Using (4) for the case under study, a system of 44 equations with 3 unknowns was set up, which are:
In the present work four candidate sites were used. These were located approximately 590 m from the center of the condominium. Figure 3 shows the candidate site for the towers in relation to the condominium.
Figure 3 - Candidates Sites
With the calculated alphas, two coverage maps were obtained, one for each new tower placed. The values of
To determine the best position for an RBS, the average loss was calculated for each tower considered. Table 1 presents the results.
Table 1: Average loss calculated for each scenario considering each tower individually.
ERB Tower | Average Loss (dB) |
TX1 | 59.74 |
TX2 | 37.52 |
TX3 | 44.17 |
TX4 | 49.03 |
Figure 4 show the received power values, in the considered scenario, estimated by the model. In these figures, it is possible to observe that Fig. 4(b) referring to the placement of the TX2 tower presents a large yellow area representing a better received power, which corroborates with the lower average loss presented in Table 1.
Figure 4 - Estimated received power (dBm) inside the condominium considering: (a) TX1; (b) TX2; (c) TX3 and (d) TX4.
4. Final considerations
This work showed that the model proposed in [1] can be used in different scenarios. The model can be adapted to other environments with few measurements. The results also show that we can obtain different signal prediction depending on the candidate location, facilitating the final installation choice considering the objective of maximizing the signal coverage area.
For the scenario studied, the model indicates that the best possible position for installing a new tower, taking into account the candidate locations, is location number 2, as it has the lowest average loss. The figures showing the distribution of the received power validate the results found.
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5. Indication of future work
For future works, it is intended to carry out a new measurement campaign in coastal cities and apply the model to these scenarios, to verify the possible different influences on signal degradation.
6. Biographies(s)
Cristiane Ruiz Gomes, é Doutora em Engenharia Elétrica (com ênfase em Telecomunicações) pela Universidade Federal do Pará (2015). Professora de Ensino Superior desde 2004. Atuando nas seguintes Universidades Federais: UNIFAP, UFPA, UFC e UFRJ. Atualmente é professora Associada II lotada no Instituto de Matemática da Universidade Federal do Rio de Janeiro (DM-IM-UFRJ). Tem experiência em Ensino de Matemática, Educação Matemática, Matemática Aplicada às Telecomunicações e Sistemas Elétricos de Potência.
Igor Ruiz Gomes, possui Bacharelado em Ciência da Computação pelo Centro Universitário do Estado do Pará (2007). Mestrado (2010) e Doutorado (2018) em Engenharia Elétrica na área de Telecomunicações pela Universidade Federal do Pará. Atualmente é Professor Adjunto II da Faculdade de Computação da UFPA campus de Castanhal. Tem experiência na área de Ciência da Computação, com ênfase em Inteligência Computacional, atuando em: algoritmos adaptativos, matemática aplicada e telecomunicações.
Herminio Simões Gomes, Licenciado em Matemática pela Universidade Federal do Pará (1976), Mestre em Matemática Aplicada (1981) e Doutor em Engenharia Elétrica (1986) ambos pela Universidade Estadual de Campinas. Atualmente é professor Titular aposentado da Universidade Federal do Pará. Tem experiência na área de Matemática, com ênfase em Matemática Aplicada, atuando principalmente nos seguintes temas: Comunicações Móveis, Redes Bayesianas, Aproximantes de Padé, Testes de hipótese e Simulação.
http://lattes.cnpq.br/0696136066497209
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7. Declaração de direitos
O(s)/A(s) autor(s)/autora(s) declara(m) ser detentores dos direitos autorais da presente obra, que o artigo não foi publicado anteriormente e que não está sendo considerado por outra(o) Revista/Journal. Declara(m) que as imagens e textos publicados são de responsabilidade do(s) autor(s), e não possuem direitos autorais reservados à terceiros. Textos e/ou imagens de terceiros são devidamente citados ou devidamente autorizados com concessão de direitos para publicação quando necessário. Declara(m) respeitar os direitos de terceiros e de Instituições públicas e privadas. Declara(m) não cometer plágio ou auto plágio e não ter considerado/gerado conteúdos falsos e que a obra é original e de responsabilidade dos autores.
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8. References
GOMES, Cristiane R. et al. Optimum Positioning of Base Station for Cellular Service Devices Using Discrete Knowledge Model. Journal of Microwaves, Optoelectronics and Electromagnetic Applications, ISSN 21791074, v. 19, p. 428-443, 2020. https://doi.org/10.1590/2179-10742020v19i4941
RIBEIRO, Leonardo G. et al. Influence of Vegetation on the Outdoor-to-Indoor Mobile Radio Propagation in 700 MHz Band. Journal of Microwaves, Optoelectronics and Electromagnetic Applications, ISSN 21791074, v. 18, p. 427-438, 2019. https://doi.org/10.1590/2179-10742019v18i31628
GOMES, Cristiane R. et al. Radio-Wave Propagation Model for UHF Band in Different Climatic Conditions with Dyadic Green’s Function. Journal of Microwaves, Optoelectronics and Electromagnetic Applications, ISSN 21791074, v. 14, p. 60-72, 2015. https://doi.org/10.1590/2179-10742015v14i1427
CAI, Xuesong et al. Interference modeling for low-height air-to-ground channels in live LTE networks. IEEE Antennas and Wireless Propagation Letters, v. 18, n. 10, p. 2011-2015, 2019. DOI: 10.1109/LAWP.2019.2936264
TAUFIQUE, Azar et al. Planning wireless cellular networks of future: Outlook, challenges and opportunities. IEEE Access, v. 5, p. 4821-4845, 2017. DOI: 10.1109/ACCESS.2017.2680318
PARK, Minyoung et al. LTE maritime coverage solution and ocean propagation loss model. In: 2017 International Conference on Performance Evaluation and Modeling in Wired and Wireless Networks (PEMWN). IEEE, 2017. p. 1-5. DOI:10.23919/PEMWN.2017.8308033
MURPHY, Kevin P. Machine learning: a probabilistic perspective. MIT press, ISBN 978-0-262-01802-9, 2012.
Brazilian National Agency of Telecommunications (ANATEL). Disponível em: https://informacoes.anatel.gov.br/paineis/infraestrutura. Acesso em: 10 março 2021.
Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brasil.
Universidade Federal do Pará, Castanhal, Brasil.
Universidade Federal do Pará, Belém, Brasil.
