Abstract
An accurate assessment of the spatial and temporal radio frequency channel characteristics is essential for complex signal processing and cellular network optimization. Current research has employed numerous models to figure out how much signal propagation loss occurs along the propagation paths. However, there are issues in finding the right model for a particular terrain because these models are not universally applicable. By employing the lognormal function and the Maximum Likelihood model, a hybrid probabilistic statistical distribution model was evolved. Three LTE cell site locations in Port Harcourt, Nigeria, were used to create a hybrid model that describes the functional stochastic signal propagation loss in the area. The evaluated Maximum Likelihood model accurately estimates the relevant wireless channel properties based on observed field data. The minor square regression approach and the proposed hybrid parameter estimation methodology are compared. When it comes to estimating standard deviation errors as well as the root mean square errors, the ML-based approach consistently outperforms the least square regression model. Finally, the proposed hybrid probabilistic statistical distribution model would be useful for mobile broadband network planning in related wireless propagation conditions.
1. Introduction
Adequate knowledge of spatial radio frequency channel parameters is critical to cellular network engineering [1–4]. Accurate estimation of the network parameters is necessary for estimating the location probability and shadow margin computations, aiding effective network planning and optimization processes [5–9]. The work in [5] investigated macrocell path loss prediction employing artificial intelligence techniques. On the measurements of radio field strength and pathloss determination in UMTS networks, Isabona et al. [6] characterized the signal propagation loss in typical 3G wireless networks. In the built-up area of South-South Nigeria, Isabona and Peter [7] described signal propagation loss based on field measurements at 1.9 GHz. In [8], the authors presented radio frequency measurements and capacity analysis for industrial indoor environments. The work presented focuses on measurements campaign, including field testing, modeling, and a comparative analysis of multifrequency band propagation characteristics for cellular networks. By using experimental and simulated propagation data, estimating the spatial and temporal radio frequency channel parameters is key to addressing the proliferating issues in complex signal processing, cellular network systems design, and optimization [10–14].
In order to address the problem of determining the most suitable model for a specific environment, several parameter estimation approaches have been exploited recently [15–20]. Specifically, the work in [15] examined transmit power estimation focusing on the signal strength of the wireless network with cooperative receiver nodes using the Maximum Likelihood (ML) estimation [21, 22]. The authors applied the experimental findings to validate the explored ML estimation. In [16], the authors investigated the Maximum Likelihood estimation combined with signal statistics to determine the performance of intensity-modulated fibre optic links.
In related work, the authors in [17] reported realistic predictive modeling of stochastic path attenuation losses in wireless channels over microcellular urban, suburban, and rural terrains using probability distribution functions. Their study revealed that the normal distribution was most suitable for the statistical predictive modeling of signal path loss data. Similar predictive analyses have been reported [18–20]. Specifically, the work in [18] presented a study on empirical path loss models to accurately predict TV signals for secondary users. The authors of the work in [19] posed and answered a question on why is shadow fading lognormal. In [20], the authors investigated the fading characteristics of wireless channels on a high-speed railway in hilly terrain. In [23–25], the least square and absolute deviation regression methods were applied to estimate the parameters of the deployed radio frequency channel measurements from different wireless propagation environments. In particular, the work in [23] reported an experimental study of UMTS radio signal propagation characteristics, employing field measurements in the GSM band. In [24], the authors presented RF propagation measurement and modeling to facilitate network planning of outdoor wireless local area networks operating in the 2.4 GHz band.
Similarly, the work in [25] examined path loss propagation prediction and optimization, employing the popular Hata model at 800 MHz in an urban area. In a similar study, Gentile et al. [26] proposed a suitable methodology for benchmarking radio-frequency channel sounders through a system model. The current contribution exploited an efficient parameter-based ML estimation model combined with the lognormal distribution function to estimate spatial variations of wireless propagated signals. The study focused on practical field tests performed on a commercial mobile broadband network. The findings of this work demonstrated that the proposed ML-based model estimates the relevant wireless channel parameters for the tested environments, in comparison with the measured data, with minimal errors. The main contributions of the paper are outlined as follows:(i)An efficient parameter-based ML estimation model combined with the lognormal distribution function to estimate spatial variations of wireless propagated signals is proposed(ii)The performance of the proposed hybrid parameter estimation model compared with the least square regression method was examined(iii)The cumulative hazard plots of propagation loss distribution of ML and LS models with the measurement obtained from different site locations were demonstrated(iv)The mean prediction error with ML and LS estimated parameters on measured pathloss loss data were determined
The remainder of this paper is organized as follows: in Section 2, the preliminaries are highlighted briefly. Section 3 gives an overview of the simulated and experimental measurements and analyses. Section 4 presents the results and discussions. Finally, Section 5 provides a concise conclusion to the paper.
2. Materials and Methods
This section briefs the measurement campaign, signal propagation model, and maximum-likelihood estimators.
2.1. Measurements Campaign and Signal Propagation Modeling
The measurement campaign was conducted in the built-up areas of Port-Harcourt, Nigeria. The tested 4G LTE network operates at 1900 MHz. Field measurements were taken using drive test tools in and around the investigated environment [27–29]. Real-time 4G LTE signal strength obtained from the evolved base station (eNodeBs) was processed and analyzed in MATLAB. In particular, the Reference Signal Received Power (RSRP) was extracted from the logged files and processed similarly to earlier works [30–32]. According to Rappaport [33], the experimental received signal power and propagation loss are logarithmically related to the propagation distances, di, and transmit power is defined by the following equation:where and Loff express the location-specific fading and offset parameters, respectively. Equation (1) describes the signal propagation loss model. Specifically, it is assumed that Loff can be precisely achieved using a small reference measurement number. In the model, the shadow fading parameter Xi is assumed to be a specific random variable such that Xi∼N(0, σ2). The key attenuation model parameters such as α and σ2 are derived relative to their dependence on the actual wireless propagation environment [27, 29, 34, 35].
2.2. Maximum Likelihood Estimators
The Maximum Likelihood (ML) estimation is an indispensable and effective channel parameter estimation method that finds practical application in signal processing [36–39]. The ML method can be deployed to examine the behaviour of channel data parameters. This study employs the likelihood function [40–42] to determine the ML estimation parameters in the measured pathloss data. Specifically, the likelihood function of the lognormal distribution for (i = 1, 2, 3, …, n) dataset is achievable by considering the product of the probability densities expressed in equations (2) to (6):f(·) signifies the lognormal distribution with parameters:and
The lognormal distribution log-likelihood function for (i = 1, 2, 3, …, n) dataset can be obtained by exploring the natural log of the likelihood function (7) to (11):
The next step is to find and , which maximize . Thus, for , we have the following equation:
Equation (11) also implies that equations (13) and (14) hold:
So
Similarly, to find , which maximize , according to (15) to (17):
Equation (17) implies the definitions in (18) and (19):and
By applying the expression in equations (15) and (19) can also be written as follows:
Therefore, the ML estimation model parameters are defined in (21): and
3. Results and Discussions
The results of the characterized parameters and predictive analysis of the propagation loss data using the ML estimate approach are briefed. The parameters of the pathloss data obtained via the least square (LS) regression estimation are provided for deductive comparison [15, 16]. The cumulative hazard plots are presented in Figures 1–3. Table 1 shows the measured loss estimated parameters and their estimation accuracies using the two approaches. The cumulative hazard plots are employed to visually examine the ML and LS models and their distributive prediction and reliability on the measured propagation loss. From the plotted mean prediction graphs of Figures 4–9 and the summarized prediction results in Table 2, it is evident that the ML estimation is superior to the LS approach. In Table 2, for instance, employing the mean absolute error (MAE), mean percentage error (MAPE), root mean square error (RMSE), and standard deviation error (SDE) statistics, the ML model, attains 1.82, 3.97, 1.99, and 0.79, respectively, in site location 1. In contrast, the LS model achieved 2.70, 11.85, 3.44, and 2.13, respectively. The ML posed similar parameter estimation and prediction performance over the LS approach, as revealed in Table 2 for site locations 1 and 2.
Figures 10–12 show exponential CDF plots to demonstrate the accuracy attained by the ML approach in estimating (predicting) the measured path loss values acquired over three study locations. It can be found from the three graphs that the ML-based estimation closely maps the measured path loss values up to 70% each before deviations. In contrast, the LS-based approach could only accurately predict 30–50% of the measured path loss values sample. The prediction error attained by engaging the ML-based and ML-based estimation approaches is quantitively defined in Table 3.
4. Conclusions
This study considers parameter estimation for spatial variations of a radio frequency channel based on experimental measurements derived from an operational 4G LTE broadband network. The work developed a combined maximum-likelihood estimation model and a lognormal distribution function. The explored ML-based model reliably estimates the specified wireless channel parameters compared with measured field data for the investigated environments. In order to test the validity of the proposed model, standard statistical metrics were employed for benchmarking. Regarding the mean absolute error (MAE), Mean percentage error (MAPE), root mean square error (RMSE), and standard deviation error (SDE) statistics, the ML model approach attains 1.82, 3.97, 1.99, and 0.79 in site location 1. In contrast, the LS model achieved 2.70, 11.85, 3.44, and 2.13 values, respectively, for the same site location. Similar parameter estimation and prediction performance of the ML method over the LS approach are demonstrated for site locations 1 and 2. Future work would focus on optimizing the parameters of the proposed hybrid model for optimal performance in a related wireless propagation environment.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Ethical Approval
This article does not contain any studies with human participants or animals performed by any of the authors.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The work of Agbotiname Lucky Imoize was supported in part by the Nigerian Petroleum Technology Development Fund (PTDF) and in part by the German Academic Exchange Service (DAAD) through the Nigerian-German Postgraduate Program under Grant no. 57473408. The authors extend their appreciation to Taif University for funding the current work via Taif University Researchers Supporting Project number (TURSP-2020/119), Taif University, Taif, Saudi Arabia.