Statistical Characterization and Modeling of Radio Frequency Signal Propagation in Mobile Broadband Cellular Next Generation Wireless Networks

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 fgure out how much signal propagation loss occurs along the propagation paths. However, there are issues in fnding 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. Tree 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. Te evaluated Maximum Likelihood model accurately estimates the relevant wireless channel properties based on observed feld data. Te 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.


Introduction
Adequate knowledge of spatial radio frequency channel parameters is critical to cellular network engineering [1][2][3][4]. Accurate estimation of the network parameters is necessary for estimating the location probability and shadow margin computations, aiding efective network planning and optimization processes [5][6][7][8][9]. Te work in [5] investigated macrocell path loss prediction employing artifcial intelligence techniques. On the measurements of radio feld 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 feld measurements at 1.9 GHz. In [8], the authors presented radio frequency measurements and capacity analysis for industrial indoor environments. Te work presented focuses on measurements campaign, including feld 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][11][12][13][14].
In order to address the problem of determining the most suitable model for a specifc environment, several parameter estimation approaches have been exploited recently [15][16][17][18][19][20]. Specifcally, 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]. Te authors applied the experimental fndings 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 fbre 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. Teir 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][19][20]. Specifcally, the work in [18] presented a study on empirical path loss models to accurately predict TV signals for secondary users. Te 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][24][25], the least square and absolute deviation regression methods were applied to estimate the parameters of the deployed radio frequency channel measurements from diferent wireless propagation environments. In particular, the work in [23] reported an experimental study of UMTS radio signal propagation characteristics, employing feld 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. Te current contribution exploited an efcient parameter-based ML estimation model combined with the lognormal distribution function to estimate spatial variations of wireless propagated signals. Te study focused on practical feld tests performed on a commercial mobile broadband network. Te fndings 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. Te main contributions of the paper are outlined as follows: (i) An efcient parameter-based ML estimation model combined with the lognormal distribution function to estimate spatial variations of wireless propagated signals is proposed (ii) Te performance of the proposed hybrid parameter estimation model compared with the least square regression method was examined (iii) Te cumulative hazard plots of propagation loss distribution of ML and LS models with the measurement obtained from diferent site locations were demonstrated (iv) Te mean prediction error with ML and LS estimated parameters on measured pathloss loss data were determined Te remainder of this paper is organized as follows: in Section 2, the preliminaries are highlighted briefy. 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.

Materials and Methods
Tis section briefs the measurement campaign, signal propagation model, and maximum-likelihood estimators.

Measurements Campaign and Signal Propagation
Modeling. Te measurement campaign was conducted in the built-up areas of Port-Harcourt, Nigeria. Te tested 4G LTE network operates at 1900 MHz. Field measurements were taken using drive test tools in and around the investigated environment [27][28][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 fles and processed similarly to earlier works [30][31][32]. According to Rappaport [33], the experimental received signal power and propagation loss are logarithmically related to the propagation distances, d i, and transmit power P TX is defned by the following equation: where X i and L of express the location-specifc fading and ofset parameters, respectively. Equation (1) describes the signal propagation loss model. Specifcally, it is assumed that L of can be precisely achieved using a small reference measurement number. In the model, the shadow fading parameter X i is assumed to be a specifc random variable such that X i ∼N(0, σ 2 ). Te key attenuation model parameters such as α and σ 2 are derived relative to their dependence on the actual wireless propagation environment [27,29,34,35].

Maximum Likelihood Estimators.
Te Maximum Likelihood (ML) estimation is an indispensable and efective channel parameter estimation method that fnds practical application in signal processing [36][37][38][39]. Te ML method can be deployed to examine the behaviour of channel data parameters. Tis study employs the likelihood function [40][41][42] to determine the ML estimation parameters in the measured pathloss data. Specifcally, the likelihood function of the lognormal distribution for P i (i � 1, 2, 3, . . ., n) dataset is achievable by considering the product of the probability densities expressed in equations (2) to (6): f(·) signifes the lognormal distribution with parameters: and Te lognormal distribution log-likelihood function for P i (i � 1, 2, 3, . . ., n) dataset can be obtained by exploring the natural log of the likelihood function (7) to (11): Te next step is to fnd μ and ω 2 , which maximize L(P/μ, ω 2 ). Tus, for μ, we have the following equation: Equation (11) also implies that equations (13) and (14) hold: nμ Similarly, to fnd ω 2 , which maximize L(P/μ, ω 2 ), according to (15) to (17): − n Equation (17) implies the defnitions in (18) and (19): and By applying the expression in equations (15) and (19) can also be written as follows: Terefore, the ML estimation model parameters are defned in (21):μ � n i�1 ln P i /n and

Results and Discussions
Te results of the characterized parameters and predictive analysis of the propagation loss data using the ML estimate approach are briefed. Te parameters of the pathloss data obtained via the least square (LS) regression estimation are provided for deductive comparison [15,16]. Te 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. Te 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 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. Te prediction error attained by engaging the ML-based and ML-based estimation approaches is quantitively defned in Table 3.

Conclusions
Tis study considers parameter estimation for spatial variations of a radio frequency channel based on experimental measurements derived from an operational 4G LTE broadband network. Te work developed a combined maximum-likelihood estimation model and a lognormal distribution function. Te explored ML-based model reliably estimates the specifed wireless channel parameters compared with measured feld data for the investigated environments. In order to test the validity of the proposed model, standard statistical metrics were employed for

Data Availability
Te data that support the fndings of this study are available from the corresponding author upon reasonable request.

Ethical Approval
Tis article does not contain any studies with human participants or animals performed by any of the authors.

Conflicts of Interest
Te authors declare that they have no conficts of interest.