Weighted ABG: A general framework for optimal combination of ABG path-loss propagation models

Date of publication xxxx 00, 0000, date of current version xxxx 00, 0000. Digital Object Identifier 10.1109/ACCESS.2017.DOI Weighted ABG: A general framework for optimal combination of ABG Path-Loss Propagation Models D. CASILLAS-PEREZ1, C. CAMACHO-GÓMEZ1, S. JIMÉNEZ-FERNANDEZ1, A. PORTILLA-FIGUERAS1and S. SALCEDO-SANZ1 1Universidad de Alcalá, Spain. Corresponding author: S. Salcedo-Sanz (e-mail: [email protected]). This work is supported by the Universidad de Alcalá - ISDEFE Chair of Research in ICT and Digital Progress. Special mentions to Juan Antonio Carabaña, José Luis Ruiz-Mendoza and Sonia Castillo for their support and assessment. We would also like to thank the Secretaría de Estado de Telecomunicaciones e Infraestructuras Digitales, specifically D. Antonio Fernández-Paniagua and Pedro Luis Alonso for their guidance and problem definition. ABSTRACT In this paper we propose a novel path-loss model, the Weighted ABG (WABG), which suitably allows integrating or combine different available datasets, or previously proposed 5G propagation pathloss models from the state-of-the-art. Our proposal is therefore a new ABG-based approach which integrates other existing models, leading to the best possible approximation in least-square sense, considering different weighting policies. We evaluate the performance of the WABG in several 5G scenarios, and we carry out a complete comparison of the proposed method against several recently published ABG models, showing that the WABG obtains the best results in terms of model’s accuracy. INDEX TERMS 5G, path-loss, mmWave, ABG propagation models, models combination. I. INTRODUCTION The fifth generation of mobile communications systems (5G) will bring along a complete revolution in applications and experiences [1]. Many sectors such as health, industry, vehicles, videogames or business, among many others, will completely change thanks to this technology. Characteristics of the new mobile network include a 10 Gbps downlink throughput, 1 ms latency, 99.999% reliability and a large number of devices connected simultaneously to the access network [2], [3]. Wirelessly providing these high data rates and quality parameters demands new innovative technologies, not implemented in the fourth generation of mobile communications (4G), such as, massive multiple-input multipleoutput (mMIMO) [4]–[6] or non-orthogonal multiple access (NOMA) [7]–[9]. However, even with these new techniques, the frequency bands where legacy mobile communication systems are placed will not fulfill the requirements of the 5G technology. This is the reason why the Third Generation Partnership Project (3GPP [3]) has pointed out the need to use mmWave frequency bands for hosting future 5G services [10]. Among the different issues related to the use of mmWave frequencies, the extremely high path-loss occurring at these frequencies is a major technological issue [11]–[13]. Like in previous generations, radio propagation has been a key aspect on mobile communications design. Propagation models are used to estimate the path loss and, consequently, to determine the maximum coverage range of a given radio base station. Generally speaking, there are three types of radio propagation models [14]: ray tracing or optical models [15], [16], dominant path models [17] and statistical models [18]– [20]. Ray tracing-optical models are based on the addition of the individual contribution of multiple rays. These models require specifying the area under study including geographic information (buildings, streets, elevations), as well as building construction materials, in order to calculate the scattering, reflection and diffraction for each individual ray. Consequently, the ray-tracing models present a high computational cost, and need to be implemented on large simulators. Dominant path models only consider the main contributing ray between the transmitter and receiver, and specific information regarding the area and the environment’s materials [21]. They reduce the computational cost compared with the raytracing models, but also require the area’s materials database. Finally, statistical models (aka empirical models) provide VOLUME 4, 2016 1 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS fair estimations of the propagation path-losses by using few environment parameters, therefore they present a very low computational cost. The statistical strategy establishes an optimization problem based on different statistic criteria, such as, Maximum Likelihood (ML), Maximum a Posterior (MAP) criteria or Least-Squares (LS), and looks for the model which best adjusts the path-loss samples in this sense. In fact, radio propagation models implement mathematical formulation, which depending on the model’s complexity, may be used for either site specific and/or general site calculations [22]. These models are very useful when the available area information is reduced or when the computation time requirements are limited, for instance, in the estimation of the maximum cell range around the 360◦directions (note that in ray-tracing only one direction is obtained). Focusing on the statistic models, several studies propose the use of different propagation path-loss models in 5G systems [3], [12], [18], [19], [23]–[26]. However, the large-scale path-loss propagation model named the Alpha-Beta-Gamma (ABG) model [3], [12], [19] is currently the most used pathloss model in 5G applications, and it is also the model established by the 3GPP as the standard model for 5G [3]. There are specific ABG models for different 5G scenarios, such as, Urban Micro and Macro cells (UMi and UMa, respectively), in different environments such as Line-of-Sight (LOS) and Non-Line-of-Sight (NLOS). The implementations of these models cover a wide working range of distances and frequencies, many of which overlap in terms of distance or frequency [12], [19]. To name some examples, [27] studied the wideband channels at 9.6,28.8and 57.6 GHz in LOS and NLOS. In [28], [29] a measurement campaign and path-loss modeling was carried out in the 60 GHz band. And in [30] Aalto University studied frequencies in the Eband, specifically the 71 to 76 GHz and the 81 to 86 GHz bands. Although all of these models estimate similar pathloss values, especially in LOS (where propagation path-loss modelling is close to the free-space modeling), dissimilarities occur, and policies for balancing the path-loss approaches are needed to create an integral model. In the literature, there are not many integral approaches that deal with the existent ABG models and cover all the established distance-frequency ranges in 5G. The most recent approach in [19] proposes to create a database by combining different 5G path-loss databases. This new database leads to a new set of ABG coefficients and, therefore, we have named this procedure as the basic and integrated ABG model. To construct the new ABG model, a large dataset is needed to lead to reliable ABG coefficients. Nevertheless, the first issue of this approach is that merging is done without a balancing or weighting policy. All input models weight the same when creating the integrated (output) model, regardless of their accuracy. Hence, tending to overestimate those models made with a larger number of samples. The second issue is that, in order to create the new database, it requires that all input models are defined similarly (i.e. input models are defined by a path-loss dataset). This is, in general, an extremely hard requirement since we do not often count on public databases. Many published studies (summarized in [19]) only provide the resulting ABG coefficients of a model, its ranges and a noise figure indicating how accurate the model is with respect to its database. Input models and/or the provided raw path-loss datasets may cover different frequency or distance ranges, may be defined using ABG coefficients or field measurements, may have a larger or a smaller number of points, etc. Therefore, it is paramount to address correctly the importance of the former baseline models. The main objective of the present paper is to obtain a single (output) ABG-based model that is able to combine different (input) ABG models and/or (input) raw path-loss datasets provided by different studies. To perform the integration, the novelty of this research is that these inputs are weighted using a balancing policy that allows to surpass the aforementioned drawbacks of the state-of-the-art approaches. In this research, this model is named as Weighted ABG (WABG), and its main characteristic is that it preserves the low number of parameters characteristic to the ABG models. Figure 1 shows an illustrative flowchart of our proposal. We have tested the WABG in several 5G scenarios, including UMi and UMa scenarios, in both LOS and NLOS environments. We also compare the proposed WABG model with several recently published alternative ABG models, including the basic integrated ABG model, in the different 5G scenarios considered, obtaining excellent results in terms of path-loss standard deviation. The remainder of the paper is structured as follows: Section II reviews the theoretical aspects of the ABG propagation model. Section III describes in detail the proposed WABG model, including the weighting policies which can be applied to generate it. Section IV first shows an instructive example of how to build a WABG model from the beginning, then it details an study of the effect of the different weightings, and finally it presents a complete set of experiments in LOS and NLOS environments for the different 5G considered scenarios. This section also shows a direct comparison with the state-of-the-art ABG models. Section V closes the paper with some final conclusions and remarks. II. THE ABG MODEL The Alpha Beta Gamma (ABG) model [3] is a large-scale path-loss propagation model for all generic frequencies. The model is not highly constrained, so it is frequently used for specific scenarios, distinguishing LOS and NLOS environments; or finding specifically calculated ABG model for different 5G scenarios, such as UMi and its variants (UMi Street Canyon, UMiSC, and UMi Open Square, UMiOS) and UMa. In NLOS, for instance, generally ABG models divide the distance range in different partitions to gain accuracy in specific applications [3]. The ABG models calculate the path-loss as follows: PABG(d, f)[dB] = 10αlog10(d)+β+10γlog10(f)+ξ, (1) 2VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS FIGURE1:WABG Flowchart. The proposed WABG is an ABG-based model which integrates raw datasets and/or other ABG models from different studies using a weighting policy. where PABG is the path-loss in decibels (dB) logarithmically dependent on the distance dand frequency f, measure in meters (m) and Gigahertz (GHz) respectively. The involved distance dis the Euclidean 3D distance between the transmitter and the receiver, and the frequency frefers to the carrier frequency. The coefficients α, β, γ define the ABG model. Specifically, αand γare the coefficients related to the log-distance and log-frequency respectively, and βis an offset term. They all define a plane in the 3D space where the path-loss, log-distance and log-frequency form the Cartesian axes. The random variable ξmodels the large-scale signal fluctuation produced along the transmissions. A particular statistical distribution for the noise ξis not assumed, but a zero mean and a finite variance σ2is required. The ABG model discovers the α,βand γvalues by setting an ordinary linear least-squares problem, which consists of adjusting the coefficients to those which fit the best to a data set [19]. Let D={(xi, yi)|1≤i≤n}be the set of navailable pairs of input-output observations, where xi= (di, fi)is an input pair of distance-frequency, also referred as the independent variable; and yiits corresponding pathloss measure, the dependent variable. The established ABG model relates xiand yiby the Equation (1). The fit of a model to a data point (xi, yi)is measured by its residual, defined as the difference between the measured path-loss and the value predicted by the ABG model: ri=yi−PABG(di, fi).(2) The optimal parameter values are obtained by minimizing the sum of the squared residuals R: R= n X i=1 r2 i= n X i=1 (yi−PABG(di, fi))2.(3) Just for linear models, which is the case of the ABG (see Equation (1)) there exist a unique solution to the problem of finding the best parameters (ˆα, ˆ β, ˆγ)with following closedform expression:   ˆα ˆ β ˆγ = (XTX)−1XTY, (4) where Xis a matrix whose ijth-element is the ithobservation of the jth independent variable: X=         log10(d1) 1 log10(f1) . . .. . .. . . log10(di) 1 log10(fi) . . .. . .. . . log10(dn) 1 log10(fn)         (5) and Y= (y1,· · · , yi,· · · , yn)Tits corresponding pathloss vector whose ith-element is the path-loss of the ithobservation. Note that (XTX)−1XTis the Moore-Penrose inverse of X. The Gauss-Markov theorem guarantees that the obtained linear regressor is the Best Linear Unbiased Estimator (BLUE), in terms of minimum variance. The following assumptions concerning the set of error random variables {ξi|1≤i≤n}have been taken: 1) The noise variables ξiare zero mean: E[ξi]=0. 2) They are homoscedastic, that is, all they have the same finite variance Var[ξi] = σ2<∞. 3) Distinct error terms are uncorrelated Cov[ξi, ξj] = 0,∀i6=j. The ABG model can manage a single carrier frequency by fixing its γcoefficient to 0or 2[18], [20], [31], becoming a two-parameters model (αand β). The mathematical treatment to obtain the best coefficients (ˆα, ˆ β)is similar to the one shown above. III. PROPOSED WEIGHTED ABG MODEL Multiple ABG coefficients for the different state-of-the-art path-loss databases can been obtained for several 5G scenarios. Particularized for a single carrier frequency or a near band of carrier frequencies, ABG models have shown in general good accuracy, especially when they are computed from one specific database. However, ABG models computed from multiple databases, with different distance-frequency ranges, do not achieve such accurate results: Most of them collect all data from the multiple state-of-the-art datasets to create a larger one, obtaining the best parameters following VOLUME 4, 2016 3 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS the steps of the previous Section II (see [19]). These integrated models do not contemplate unbalanced data from the different database or the residual values associated with each range of distance or frequencies. In addition, note that assuming the homoscedastic preservation along the different distance-frequency ranges leads to inaccurate and unbalance integrated models. In order to improve these previous issues with integrated ABG models, we propose a weighted model which considers the heteroscedastic property along the different data from the dataset for managing multiple range of distances and frequencies. The proposed Weighted ABG model (WABG), which surpasses the mentioned inconveniences of integrated path-loss approaches, is following defined. We observe three main scenarios or cases for proposing the WABG path-loss model: 1) The first scenario is defined by a family of propagation path-loss datasets {Sj|1≤j≤m}. Each dataset Sjcontains a set of input-output observations Sj={(xi, yi)|1≤i≤nj}in its corresponding range of distances and frequencies. Some of them may contain significant distance or frequency gaps which may need to be filled using interpolation algorithms. Usually the different datasets contain observations from a single carrier frequencies or a near frequency band, which leads to very accurate individual ABG models. Consequently, each dataset Sjhas associated its own ABG model defined by its parameters ABG(Sj) = (αj, βj, γj), computed following the steps of Section II. Associated with the ABG model, the standard deviation of the error σcan be easily calculated, once the optimal ABG model is obtained, by applying Equation (6): σj=sRj nj =sPnj i=1 (yi−PABG(di, fi))2 nj (6) Note that the standard deviation σjexpressed in Equation (6) determines the model’s quality. 2) The second scenario for defining the WABG, is the case where we do not have access to the complete databases, but only to the already calculated ABG coefficients {(αj, βj, γj) = ABG(Sj)|1≤j≤m} and its corresponding standard deviation σjvalid for a specific distance-frequency range. This scenario corresponds, for example, to a case where the employed datasets are not public. 3) The third scenario is a mixture of the two previous ones, the general case, where we can have measurements for some distances and frequencies and also ABG coefficients for other, and we would like to collect all this information into a new integrated ABG path-model. In any of these scenarios, the proposed WABG model works by integrating (combining) several ABG models for different ranges of distances and frequencies. Thus, after obtaining the ABG model and its associated standard deviation in the case of scenario 1, directly from the given models in scenario 2, or a mixture of both in scenario 3, we create a synthetic dataset ˜ Sjfor each corresponding ABG model, and form a collection of observation datasets {˜ Sj|1≤j≤m}. In order to do this, we equally vary the distance and frequency covering all the range of each given ABG model, and annotate the associated standard deviation. As a practical rule, we vary distances and frequencies in steps of 1 m and 1 GHz respectively. In case of ABG models computed for a single frequency, we only vary the distance in its corresponding range. At the end of this process, we have msimulated databases with observations ˜ Sj={(xj i, yj i)|1≤i≤nj}, which belong to a specific model and have the same standard deviation σj. Observations from different databases may have different standard deviations, therefore, collecting all the observations in a unique set and calculating an ABG model leads to wrong path-loss estimations (such as the basic integrated ABG model). WABG manages these variability of the deviations, avoiding this important issue to construct a robust integrated model. The WABG establishes a weighted least-squares problem, a special case of generalized least-squares. It occurs when all the off-diagonal entries of the correlation matrix of the residuals are zero. The variances of the observations, which are the unique non-null elements placed along the covariance matrix diagonal, may still be unequal. It is the so-called heteroscedasticity property which differs from the basic integrated ABG models. In WABG, the optimal parameter values are obtained by minimizing a weighted sum of squared residuals, which involves the heterogeneous variance of the observations: S= n X i=1 Wiir2 i= n X i=1 Wii (yi−PABG(di, fi))2,(7) where Wis the inverse of the correlation matrix, see Equation (8). W=          1 σ2 1 · · · 0· · · 0 . . ..... . ..... . . 0· · · 1 σ2 i · · · 0 . . ..... . ..... . . 0· · · 0· · · 1 σ2 n          (8) Aitaken [32] proved that the coefficients calculated in this way (ˆα, ˆ β, ˆγ)are the BLUE if each weight is equal to the reciprocal of the variance of the measurement, see Equation (7), and provides the following closed-form solution:   ˆα ˆ β ˆγ = (XTW X)−1XTW Y. (9) where Xis a matrix whose ijth-element is the ithobservation of the jth independent variable, and Wis repre4VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS sented in Equation (8). Observe how the computation complexity to solve Equation (9) is reduced to invert matrix XTW X, similarly to the ABG models, where complexity is reduced to invert XTX. Particularized for our WABG model, Xis a block matrix defined by stacking all the observations of the different databases ˜ Sj: X=         X1 . . . Xj . . . Xm         , Xj=         log10(d1) 1 log10(f1) . . .. . .. . . log10(di) 1 log10(fi) . . .. . .. . . log10(dnj) 1 log10(fnj)         . (10) Note that the block Xjthat corresponded to an ABG model valid for a single frequency has its third column constant. The inverse of the correlation matrix Wis also a partitioned matrix, where the observations that belong to the same dataset Sjhave the same variance, as we expressed in Equation (11). W=         W1· · · 0· · · 0 . . ..... . ..... . . 0· · · Wj· · · 0 . . ..... . ..... . . 0· · · 0· · · Wm         , Wj=           1 σ2 j · · · 0· · · 0 . . ..... . ..... . . 0· · · 1 σ2 j · · · 0 . . ..... . ..... . . 0· · · 0· · · 1 σ2 j          nj×nj (11) Now the corresponding standard deviation, considering the weighting, is computed as depicted in Equation (12). σ=v u u tPN i=1 Wiir2 i PN i=1 Wii =v u u tPN i=1 Wii (yi−PABG(di, fi))2 PN i=1 Wii . (12) We will see in the next section different policies to form the inverse of the correlation matrix Win order to correctly balance the WABG model, attending to different criteria. As the inverse of the correlation matrix Wis diagonal, it can be expressed through the so-called weighting vector ω. A. WEIGHTING POLICIES We propose three main policies of weightings applied to the WABG model when it is built by integrating several ABG models from different datasets: 1) Equalizing by the Number of Points: The ABG models integrated by the WABG may cover different range of distance and frequencies, and be built from datasets with different number of points. Each ABG model given for computing a WABG should be considered with the same importance independently of the number of points that were used to create them. Consequently, equalizing the WABG providing same grade of importance requires the following weighting vector: ωN=M    n1times z}|{ 1 n1 ,· · · , nitimes z}|{ 1 ni ,· · · , nmtimes z}|{ 1 nm    ,(13) where niis the number of points using in each model and Mis the least common multiple of the set {ni|1≤i≤n}. 2) Weighting by the Standard Deviation: The initial ABG models used to create the WABG may have different standard deviation which should be consider to weight. ABG models with large standard deviation should have less weighting than others with a small one. A good weight vector ωfor the WABG uses the inverse of the standard deviation as follows: ωσ=    n1times z}|{ 1 σ2 1 ,· · · , nitimes z}|{ 1 σ2 i ,· · · , nmtimes z}|{ 1 σ2 m    (14) 3) Mixture of the previous policies: Computing the Hadamard product of both vectors: ω=ωNωσ(15) The corresponding weighting matrix Wis a matrix whose diagonal matches its weighting vector ω. Note that using a weighting vector of ones ω= (1,· · · ,1), the obtained WABG model becomes an ABG whose weighting matrix is the identity W=I. IV. COMPUTATIONAL EXPERIMENTS AND RESULTS In this section, we compare the WABG model with the stateof-the-art ABG models computed from several 5G datasets, specifically, from those one provided by the companies Nokia, Qualcomm, and the Universities of Aalborg (AAU), New York (NYU) and Aalto (ALU). We distinguish between three different kinds of 5G scenarios: Urban Microcellular Street Canyon (UMiSC), Urban Micro-cellular Open Street (UMiOS) and Urban Macro-cellular (UMa). We also distinguish between two different 5G environments: LOS and NLOS. The LOS environment presents lower standard deviation σcompared to the NLOS cases, a consequence of the non-existence of a direct Line-of-Sight. The ABG models for each 5G scenario in the NLOS environment have been obtained from [19]. The corresponding ABG models for each 5G in LOS were computed from the works published in [11], [12]. According to this, we will provide a WABG model covering a wide range of frequencies. In NLOS, where the number of available models is high, we will also provide a model for low and high frequencies. The division between low and high frequencies was established by the state-of-the-art methods [11], [12]. Consequently, we VOLUME 4, 2016 5 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS have followed this division to allow comparison among them. Note that this division is not a requirement of the proposed model and we can design a WABG model in any other frequency range. Moreover, the metric error used is the weighting standard deviation σ, as expressed in Equation (12). Note that the weighting standard deviation directly leads to the standard deviation assuming as weighting matrix the identity W=I. This section is then structured into three subsections: we first give an end-to-end real computation example of a WABG departing from both ABG models and datasets, the mixture scenario (see Section III). Then, we analyze and compare the proposed weighting policies of Section III-A. Finally, we give the WABG coefficients in LOS and NLOS for the studied 5G scenarios UMiSC, UMiOS and UMa, and we carry out a comparison with the available large-scale propagation path-loss models in the literature. A. REAL WABG COMPUTATION EXAMPLE This section presents the calculation of the WABG model in the third scenario, the most general scenario as it includes the first and second scenarios (all of them described in Section III). Figure 2 shows an example of the construction of the WABG model in this case. We have elaborated a low band WABG model using three different datasets corresponding to the 5G UMiSC scenario in NLOS environment. These three models correspond to the first three rows presented in Table 5. Namely, these are the two datasets provided in the Nokia/AAU ABG model [30] for 2and 18 GHz, and the Qualcomm dataset for 2.9 GHz [19]. Specifically, we consider 103evenly spaced points which emulated the captured real ones according to the ABG models (see Figure 2). We first estimate its corresponding ABG coefficients from the given set of points following the steps described in Section II. As we see in Figure 2, the obtained estimation is very accurate, see Table 1 fourth and last columns for numerical values. It can be seen that the provided number of points is enough to obtain a good estimation of the ABG coefficients. Once we have an ABG model for all the datasets, we evenly sample each ABG along its distance range, by creating sets of pairs Sj={(xi, yi)|1≤i≤nj}where xiis the distance dand yithe path-loss predicted by the model. Observe that the number of points njmay not be the same, consequently, weighting policies for balancing the models are necessary, as explained in Section III-A. For the following example, the Xmatrix is computed as follows: X=  X1 X2 X3 =                  log10(109) 1 log10(2.9) . . .. . .. . . log10(235) 1 log10(2.9) log10(19) 1 log10(2.0) . . .. . .. . . log10(272) 1 log10(2.0) log10(19) 1 log10(18) . . .. . .. . . log10(272) 1 log10(18)                  .(16) Note that the WABG models the datasets with different standard deviations, and we give larger weights to those models whose standard deviation is larger, as explained in Section III-A. As a consequence, the weighting vector ωobtained expanding Equation (15) and using weighting policy 3(see Section III-A for more detail) is presented in Equation (17). ω=M   n1times z}| { (n1σ2 1)−1,· · · ,(n1σ2 1)−1, n2times z }| { (n2σ2 2)−1,· · · ,(n2σ2 2)−1, n3times z }| { (n3σ2 3)−1,· · · ,(n3σ2 3)−1,  , (17) which leads to the corresponding weighting matrix presented in Equation (18), as deduced in Equation (11) and using policy 3. W=         W1· · · 0· · · 0 . . ..... . ..... . . 0· · · Wj· · · 0 . . ..... . ..... . . 0· · · 0· · · Wm         , Wj=M           1 njσ2 j · · · 0· · · 0 . . ..... . ..... . . 0· · · 1 njσ2 j · · · 0 . . ..... . ..... . . 0· · · 0· · · 1 njσ2 nj          nj×nj . (18) Collecting all the corresponding path-loss together in a single vector Y: Y= (y1,· · · , yn1, yn1+1,· · · , yn1+n2, yn1+n2+1,· · · , yn1+n2+n3)T,(19) and solving Equation (9), it leads to the sought WABG coefficients for this example:   ˆα ˆ β ˆγ = (XTW X)−1XTW Y =  3.2 25.3 2.3 (20) 6VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS FIGURE2:Example of WABG construction. A low band WABG is developed in the distance range from 19 to 272 m and frequency range from 2to 18 GHz. ABG models of both Nokia/AAU datasets [30] at 2.9and 18 GHz are given as input models. Simulated points of the Qualcomm dataset [19] at 2.9 GHz are also considered. We compute its associated standard deviation σ= 5.93 using Equation (12). Note how the calculated value is closer to the standard deviation of Qualcomm’s ABG model whose value is the smallest one. Weighting by the model’s standard deviation, it leads to approaches which are close to the most accurate models, as shown in Figure 2. The number of points of the given datasets has a strong influence on the ABG estimation. Obviously, the larger the number of points available for estimation, the more accurate is the estimation of the ABG coefficients. We show an experiment varying the number of points from 101to 104with an increment of 10, see Figure 3. These orders of magnitude are the most frequently found in the 5G databases. Table 1 summarizes the estimated ABG coefficients for the Qualcomm dataset, when varying the available number of points. As can be seen, the quality of the ABG model obtained is better when a larger number of points is available. Note that this is a classical result, well known in estimation theory. Nº Points 101102103104G.T. α3.963 3.937 3.915 3.896 3.9 β7.11 9.19 9.81 10.28 10.2 γ2.0 2.0 2.0 2.0 2.0 σ3.71 3.10 3.22 3.19 3.2 TABLE1:Coefficients of the estimated ABG models obtained varying the number of available points. Acronym G.T. refers to the Ground Truth, the real coefficients we want to estimate using the samples. FIGURE3:Estimation of the ABG model of the simulated UMiSC dataset (Qualcomm). Sub-figures a),b),c) and d) show evenly spaced simulated datasets with 101,102,103 and 104points. VOLUME 4, 2016 7 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS B. COMPARISON AMONG DIFFERENT WEIGHTINGS SCHEMES This experiment shows the influence of the weighting in the final WABG model. As inputs to construct the WABG model, we use three existing ABG models in the range of 2 to 18 GHz in a 5G UMiSC scenario with a NLOS environment from [19]. We extract points evenly spaced along its corresponding distance ranges and create the corresponding WABG as we did in the previous example using different weighting policies. Figure 4 shows four WABG models with different weighting policies. Figure 4.a) does not use weighting, so W=I, which means it is the basic integrated ABG model. Figure 4.b) and Figure 4.c) show WABG models which only use the number of points or the individual standard deviation respectively for weighting. Finally, Figure 4.d) shows a WABG model which uses the number of points and the standard deviation of each individual model to carry out the weighting process. We also provide different views of the previous figures in Figure 4.e), Figure 4.f), Figure 4.g) and Figure 4.h). Observe in Figure 4.e) the distance between the generated ABG model (blue) and the Qualcomm dataset (green), in comparison with the WABG model (blue) in Figure 4.h). In the first one there is no weighting, and the second one is weighted both by the number of points and by the accuracy of each individual model. The difference in the number of points between the databases causes the model to move away from those models made up of fewer points, in this case, the Qualcomm dataset (green). In Figure 4.f), weighting is done by the number of points. This configuration leads to a WABG model closer to the dataset that has fewer points, that is the Qualcomm dataset (green). In Figure 4.g), weighting is done by the accuracy of each model (σ) and it leads to a fair estimation when comparing it with the ABG model (blue) in Figure 4.e). Table 2 shows the coefficients generated for each weighting configuration and the standard deviation of the associated WABG model (computed using Equation (7)). It can be observed that the sensitivity of the standard deviation σwhen compared to the αcoefficient is lower than the sensitivity of the standard deviation σwhen compared to the γcoefficient. This is due to the fact that the three integrated models present quite similar behaviour along the distance, but this is not a generalized rule. Thus, the sensitivity of the standard deviation σwhen compared to the αcoefficient may be higher than the sensitivity of the standard deviation σwhen compared to the γcoefficient, as we can see in the low band model generated for UMiOS in NLOS, see Table 6. From now on, all the provided WABG models will be calculated considering weightings schemes by both number of points and standard deviation. C. LOS In order to further evaluate the performance of the proposed WABG, we create WABG models for LOS environments, in different 5G scenarios, specifically in UMiSC and UMa. WABG Number of Points - Standard Dev. Weighting F-F F-T T-F T-T α3.34 3.26 3.18 3.20 β26.47 26.87 26.89 25.39 γ2.03 2.11 2.23 2.29 σ7.92 7.71 6.78 5.89 TABLE2:WABG coefficients obtained from the different weighting schemes and associated WABG model standard deviation (σ). Scripts F and T refer to a False or True value of weighting by number of points and standard deviation. Table 3 shows the available ABG models for the mentioned 5G scenarios. Specifically, we have models for single frequencies, such as 28 or 73 GHz, or for several frequencies ranges, 0.8to 60 or 6to 100 GHz, in UMiSC scenario. In UMa, we have two ABG approaches that cover different frequency ranges. We do not have enough UMiOS path-loss models in order to create an integrate WABG with them, and then to compare it with the existent ones in the literature. Note the differences in distance ranges between UMiSC and UMa scenarios. We use the path-loss models shown in Table 3 to create an individual WABG for each 5G scenario, following the steps given in Section III. We have a mixture of models defined for single and range frequencies. Table 4 shows the coefficients of our WABG together with the ones given by Sun [19] and 3GPPT [3]. All the models cover a wide frequency range. The following equation has been used for specifying the maximum distance for the 3GP P : dM= 4 ·hBS ·hUT ·fc c(21) In UMiSC scenario, 3GPP maximum distance dMhas been computed assuming a transmitter height hBS = 10 m and a receiver height hUT = 1.5 m respectively. Carrier frequency fcis fixed to 0.5 GHz and cis the speed of light in vacuum 3×108ms−1. In UMa scenario, the standard transmitter and receiver heights are fixed to 25 and 1.5 m respectively. The carrier frequency is fixed to 1 GHz. These parameters represent suitable standard conditions. However, other restrictive conditions would drive to lower maximum distances. In UMiSC, our WABG model obtains very good accuracy, with a standard deviation of 3.2(that is 0.3 dB below the result presented in [19], but with a wider frequency application range: between 0.5to 100 GHz. The WABG achieves a standard deviation 1 dB smaller than the one provided by the 3GPP16 [3] with similar distance-frequency range. Observe how the maximum working range of the WABG (the maximum and minimum of the distance-frequency range) is bounded to the model ranges of the input models considered. Note that we could also expand the working range of our WABG model by integrating the 3GPP model, but in this case, it would prevent any comparison between the proposed model and the model that has been used as Ground Truth. In UMa scenario, our WABG model obtains the best value of standard deviation, 1 dB below the one obtained by Sun16 8VOLUME 4, 2016 Author et al.: Preparation of Papers for IEEE TRANSACTIONS and JOURNALS 10 1 90 d (m) 10 2 100 110 120 10 1 90 d (m) 10 2 100 110 120 10 1 90 d (m) 100 10 2 110 120 WABG Nokia/AAU 2 Qualcomm 2.9 Nokia/AAU 18 f (GHz) 100 102 101 102 Pabg (dB) 103 101 70 90 100 110 120 130 140 80 d (m) d (m) 70 90 100 110 120 130 140 80 100 102 102103 101 f (G ) d (m) 70 90 100 110 120 130 140 80 100 102 102103 101 f (G ) d (m) 70 90 100 110 120 130 140 80 100 102 102103 101 f (G ) 10 1 90 d (m) 100 10 2 Pabg (dB) 110 120 f (GHz) f (GHz) f (GHz) FIGURE4:Study on the effect of different weighting schemes in the construction of WABG models. a) Basic integrated ABG model (without weighting); b) and c) WABG models in which the number of points or the individual standard deviation (respectively) are used for weighting; d) WABG model which considers both the number of points and the standard deviation of each individual model for weighting. The second row (sub-figures e) to h)) shows different views of the previous figures. Env. Sce. Freq. (GHz) Reference Organization Dist. Range (m) αβ (dB) γσ (dB) LOS UMiSC 28 Sun16 [11] NYU 27-54 1.1 46.8 2.1 4.3 73 Sun16 [11] NYU 27-54 1.1 46.8 2.1 4.3 6-100 Rap17 [12] 5GCM [33] 1-121 1.9 32.9 2.1 2.0 0.8-60 Rap17 [12] METIS [34] 1-121 2.2 28.0 2.0 3.1 UMa 2-38 Sun16 [11] NYU 60-930 1.9 35.8 1.9 2.4 6-100 Rap17 [12] 5GCM [33] 1-930 2.0 32.4 2.0 4.1 TABLE3:Available ABG models used as WABG inputs for UMiSC and UMa 5G scenarios in LOS environment. The considered models cover different distance-frequency ranges. Env. Sce. Freq. Range (GHz) Method Dist. Range (m) αβ (dB) γσ (dB) LOS UMiSC 0.5-100 3GPP16 [3] 10-200 2.1 32.4 2.0 4.0 2-73.5Sun16 [19] 5-121 2.0 31.4 2.12.9 0.8-100 WABG 1-121 2.0 31.1 2.1 3.2 UMa 0.5-100 3GPP16 [3] 10-1000 2.2 28.0 42.0 4.0 2-73.5Sun16 [19] 58-930 2.8 11.4 42.3 4.1 2-100 WABG 1-930 1.9 34.2 42.03.1 TABLE4:Resulting WABG model for UMiSC and UMa scenario in LOS. We compare our WABG with the ABG model computed by Sun16 [19] and the 3GPP16 [3]. and 0.9 dB below the one by the 3GPP16 model. In addition, WABG covers more frequency range than that provided by Sun16, between 2to 100 GHz against 2and 73 GHz, respectively. The model in 3GPP16 expands our frequency range in low frequencies but with less accuracy. Note that the results presented in Table 4 do not show a significant reduction of the standard deviation σ. Nevertheless, it is lower than the rest of the evaluated state-of-the-art methods, and it is remarkable how the proposed WABG model catches loss modeling from the large frequency ranges, while keeping a reduced standard deviation value. D. NLOS The standard deviation values observed in the LOS environment are much lower than in the counterpart NLOS. In NLOS, the fading phenomenon is more pronounced, doubling the standard deviation figures in dBs. Similar to the procedure followed for the LOS case, we provide WABG approaches from the existing ABG models for the UMiSC, UMiOS and UMa in NLOS environment. Table 5 shows the provided ABG coefficients computed from the state-ofthe-art datasets in these environments [19]. We have 6,5 and 7different ABG models for UMiSC, UMiOS and UMa, respectively, considerably more than in LOS environment. VOLUME 4, 2016 9