1 Introduction

In multipath propagation environments, the reception angle spread is one of the main causes of received signal distortions. Statistical properties of the angle of arrival (AOA) are critical to select both the antenna system pattern and signal-processing method. Therefore, the probability density function (PDF) models of AOA are important for the design of antenna systems and theoretical and simulation studies of signals in wireless systems.

In practice, the theoretical (geometrical) and empirical models of PDF are used to reproduce the statistical properties of AOA. The first group of models is defined by geometrical structures that describe the spatial location of the scattering areas in 2D or 3D. The most commonly used geometrical structures are: circle [1–4], ellipse [3, 5–7], ring [8], hemisphere [9], cutting hemisphere [10], and cylinder [11–13]. Distribution of scatterers in propagation environment is an additional characteristic that defines each geometrical model. For these models, the following distributions are used: uniform [2, 4, 5, 8, 14], Gaussian [15, 16], Raleigh and exponential [2], hyperbolic [1], conical [17], parabolic [3, 18], and inverted parabolic [19]. The choice of the geometrical structure (shape, position, size) and scattering distribution determines the accuracy of the mapping of the actual propagation conditions.

The empirical models are based on the standard PDFs that are used in the calculus of probability. The adaptation of empirical model to different environment conditions involves selecting PDF parameter that minimizes the approximation error for measurement data. The advantage of these models is simplicity of the analytical form in which the spread of AOA is described by a single parameter.

In [20], the comparative analysis of the 2D geometrical models with respect to measurement data for different propagation scenarios is presented. However, this analysis applies only to the geometrical models. Therefore, in order to assess the effectiveness of PDF of AOA mapping by the various models, extending the comparative analysis on the empirical models it is reasoned. The results allow for a comparison of PDF approximation errors for the simple empirical and analytically complex geometrical models. The simplicity of the analytical description of PDF significantly simplifies the correlation and spectral analysis of the received signal. Therefore, the purpose of the comparative analysis is to assess and identify those empirical models that minimize PDF of AOA approximation error for the different propagation scenarios.

This paper focuses on statistical properties of the azimuthal AOA (AAOA). The presented analysis refers to wireless access systems, for which the propagation has a dominant role in azimuth plane. In practice, this means that the power pattern width is no more than a dozen degrees in elevation plane. In the first part of this paper, the main purpose is a comparative analysis and approximation error evaluation of the empirical models for different propagation environments. In [20], the obtained results show that none of the geometrical models provides minimization of the approximation error for all analyzed measurement scenarios. With regard to the empirical models, certain attempts to compare them are shown in [21]. However, the presented results do not provide a generalization of the conclusions because the single propagation scenario and only two models (truncated Laplacian and truncated Gaussian) are analyzed. In this paper, a wider analysis is presented. Here, the following models are considered: modified Gaussian, modified Laplacian, modified logistic, and von Mises. For each model, the approximation accuracy is evaluated with respect to measurement data from seven different propagation scenarios. In the literature, there are also other models such as the truncated cosine and uniform distribution in a limited angle range [22]. These models are not considered because their graphical representations are significantly different from the measurement data.

In this part of the paper, the comparative analysis of statistical AAOA models is based on the approximation error evaluation relative to measurement data. This error is determined by the classic measure that is the least-square error (LSE), and the statistical measures such as the standard deviation, Kolmogorov–Smirnov and Cramer–von Mises statistic. Thus, the obtained results provide a reliable assessment of the suitability of the different empirical models to represent the statistical properties of the signal reception angle in an azimuth plane. Comparative analysis of four models, a variety of propagation environments, and multi-criteria evaluation of approximation error largely fill the gaps in the existing analyses that are presented in the open literature.

In this paper, the presented results have a practical significance. They are the basis to select model and to evaluate its approximation error for different types of environments. Angular dispersion of the received signals depends on the type of propagation environment. Therefore, analytical and simulation studies need to clearly identify the model parameter. For each type of environment, the selection method of the model parameter is presented in Part II of this paper [23].

Part I of the paper is organized as follows. For description of the angle statistical properties, the method for modification of the standard PDF is presented in Sect. 2. Descriptions of the scenarios and the results that are reference data for empirical models are contained in Sect. 3. For particular models, the criteria and the evaluation of approximation errors are presented in Sect. 4. As a result of a comparative analysis, the empirical models, which accurately map the measurement data for the analyzed scenarios, are indicated in Sect. 5.

2 Empirical Models of AAOA

In the open literature, the statistical AAOA analysis shows that the empirical data approximation is based on PDF such as the Laplacian [15, 21, 22, 24–29], Gaussian [22, 24, 30, 31], and von Mises [22, 32–36]. In this paper, the logistic distribution [37–40] is also taken into consideration because its graph is very similar to the measurement results. The range of AAOA limits the support of PDF to \(\left( {\left. { -\uppi,\uppi} \right\rangle } \right.\) interval. For creating PDF with limited support, the commonly used method consists of the truncated standard PDF and the introduction of normalizing constant. However, for asymmetrical position of PDF extreme relative to the interval center (\(\theta_{0} \ne 0\), where \(\theta_{0}\) is the average AAOA), the truncated PDF does not provide a mapping of AAOA real properties. In the polar coordinate system, this function is not continuity. Therefore, in the analytical description of the truncated distribution, the introduction of the additional modification is necessary. For example, the differences between the truncated and modified Laplacian distributions are presented in the Cartesian (Fig. 1a) and polar (Fig. 1b) coordinate systems.

Fig. 1
figure 1

Differences between the truncated and modified Laplacian distributions in a Cartesian and b polar coordinate systems

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The analytical relationships that define the empirical models of PDF such as the modified Gaussian, \(f_{G} \left( \theta \right)\), modified Laplacian, \(f_{L} \left( \theta \right)\), modified logistic, \(f_{S} \left( \theta \right)\), and von Mises, \(f_{M} \left( \theta \right)\) are shown in Table 1.

Table 1 Empirical models of PDF of AAOA (\(\theta_{0} \in \left. {\left( { -\uppi , { }\uppi} \right.} \right\rangle\))
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Minimization of the approximation error is a criterion to fit the model to measurement data. The ranges of the model parameters are determined by the boundary conditions of reception angle. These conditions are defined by the standard deviation, \(\sigma_{\theta }\), which is a measure of real angle dispersion. For the case of maximum reception angle concentration (receiving direct path), PDF approaches delta distribution. In practice, a finite accuracy of measurements is the basis to adopt \(\sigma_{\theta } = 1^{ \circ }\). For the von Mises distribution, this value is obtained for \(\kappa \to \infty\). Because of numerical calculation complexity of the Bessel function, the following approximation is used [41]:

$$\mathop {\lim }\limits_{\kappa \to \infty } {\text{I}}_{0} \left( \kappa \right) \cong \frac{\exp \left( \kappa \right)}{{\sqrt {2\uppi \kappa } }}$$
(5)

For this case, the von Mises PDF is

$$\mathop {\lim}\limits_{\kappa \to \infty} f_{M} \left(\theta \right) \cong \sqrt {\frac{\kappa}{{2{\uppi}}}} \exp \left({\kappa \left({\cos \left(\theta \right) - 1} \right)} \right)$$
(6)

For maximum spread of the reception angle, PDF of AAOA tends to uniform distribution, that is \(\sigma_{\theta } = 103^{ \circ }\). For the reception boundary conditions, the ranges of parameters, \(\sigma\), \(\lambda\), \(s\), \(\kappa\) for the modified Gaussian, modified Laplacian, modified logistic, and von Mises distributions are contained in Table 2.

Table 2 The ranges of the parameters for empirical models
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The modified Gaussian, Laplacian, and logistic PDF require the introduction of the normalized factors, \(C_{G} \left( \sigma \right)\), \(C_{L} \left( \lambda \right)\), \(C_{S} \left( s \right)\), respectively

$$C_{G} = \frac{1}{{\int\nolimits_{{ -\uppi}}^{\uppi} {f_{G} \left( \theta \right){\text{ d}}\theta } }} = \frac{1}{{{\text{erf}}\left( {\frac{\uppi}{\sqrt 2 \sigma }} \right)}}$$
(7)
$$C_{L} = \frac{1}{{\int\nolimits_{{ -\uppi}}^{\uppi} {f_{L} \left( \theta \right){\text{ d}}\theta } }} = \frac{1}{{1 - \exp \left( { - \lambda\uppi} \right)}}$$
(8)
$$C_{S} = \frac{1}{{\int\nolimits_{{ -\uppi}}^{\uppi} {f_{S} \left( \theta \right){\text{ d}}\theta } }} = \frac{{\left( {1 + \exp \left( {\frac{\uppi}{s}} \right)} \right) \, \left( {1 + \exp \left( { - \frac{\uppi}{s}} \right)} \right)}}{{\exp \left( {\frac{\uppi}{s}} \right) - \exp \left( { - \frac{\uppi}{s}} \right)}}$$
(9)

where erf(·) is the error function. These factors depend on parameters of particular PDFs. The normalized factors versus parameters of analyzed PDFs are contained in Table 1, whereas their graphical presentations are shown in Fig. 2.

Fig. 2
figure 2

Normalized factor versus model parameter for the modified distributions: a Gaussian, b Laplacian, c logistic

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The graphs show that the normalized factors are approximately equal to 1 for specified range of parameters of analyzed PDFs. For particular models, \(C_{G} \cong 1.000\) for \(\sigma \le 44^\circ\), \(C_{L} \cong 1.000\) for \(\lambda \ge 0.056/{^\circ }\), and \(C_{S} \cong 1.000\) for \(s \le 23{^\circ }\).

The angle spread of the received signals depends on the type of propagation environment. The empirical model adaptation to different environment conditions consists of such matching of its parameter that provides the best fit to the real PDF of AOA. For analyzed models, the graphical representations of parameter influence on the mapping of the reception angle spread are presented in Fig. 3.

Fig. 3
figure 3

Influence of the model parameter on the mapping of reception angle spread for the following PDFs: a modified Gaussian, b modified Laplacian, c) modified logistic, d von Mises

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The graphs show that the appropriate choice of PDF parameter enables the model fit to the measurement data.

3 Measurement Scenarios

Presented in [20], a review and evaluation of the theoretical models of PDF of AAOA are the basis for the selection of the measurement scenarios for comparative analysis of empirical models. In this part of the paper, the comparative analysis is focused on scenarios that concern the homogeneous propagation environments. With respect to empirical PDF, it means a unimodal function type. Compared to [20], here the set of measurement results is limited to seven scenarios that are described in [21, 42–46]. Out of all unimodal, empirical PDFs that are included in [20], the measurement data from [47] are not contained in this paper. For this case, the correction changing position of transmitter (Tx) is not included in AAOA averaging procedure.

The choice of the measurement scenarios provides the differentiation of the analyzed propagation environments. Reference data are derived from the measurements that have been made in both rural environment [43] as well as urban environment with different density of buildings [21, 42, 44–46]. The type of propagation environment is determined by the height of the receiving antenna location relative to the mean height of buildings. For [42, 45, 46] and [21] (Aarhus) scenarios, the location height of antennas significantly exceed the mean height of buildings. As a result, the propagation conditions correspond to the typical urban environment type. Some important factors that affect the wave propagation are the distance between Tx and receiver (Rx), and a carrier frequency of the sounding signal. For particular scenarios, these data are included in Table 3.

Table 3 Parameters of measurement scenarios
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For the macro- and microcell, the measure that defines the type of propagation environment is the rms delay spread (DS)

$$\sigma_{\tau } = \sqrt {\frac{{\int\nolimits_{0}^{\infty } {\tau^{2} P\left( \tau \right){\text{ d}}\tau } }}{{\int\nolimits_{0}^{\infty } {P\left( \tau \right){\text{ d}}\tau } }} - \left( {\frac{{\int\nolimits_{0}^{\infty } {\tau P\left( \tau \right){\text{ d}}\tau } }}{{\int\nolimits_{0}^{\infty } {P\left( \tau \right){\text{ d}}\tau } }}} \right)^{2} }$$
(10)

where \(P\left( \tau \right)\) is the power delay profile (PDP) or power delay spectrum (PDS).

PDP and PDS are the basis for numerical calculation of DS. In Table 3, these functions are presented in the form of a discrete set of \(\tau_{l}\) and \(P\left( {\tau_{l} } \right)\).

The diversity of DSs shows the different types of propagation environments such as rural, suburban, and urban areas with sparse and dense buildings. The antenna heights of Tx and Rx relative to the average height of the buildings is the main factor, which decides about diversity propagation conditions. In [43] (scenario 2), time characteristics such as PDS or PDP are not included in measurement data set. In this case, the descriptions of the propagation environment and measurement conditions are utilized. Hence, to assess the type of propagation environment, \(\tau_{l}\) and \(P\left( {\tau_{l} } \right)\) are adopted according to COST 207 for rural area [48].

4 Comparative Analysis of Empirical Models

The assessment of the approximation error is the basis for comparative analysis of empirical models of PDF of AAOA for different environmental conditions. Here, as in [20], LSE is used as a measure of goodness of fit of the model to the measurement data

$$LSE = \frac{1}{K}\sum\limits_{k = 1}^{K} {\left[ {f_{E} \left( {\theta_{k} } \right) - f\left( {\theta_{k} } \right)} \right]^{2} }$$
(11)

where \(f_{E} \left( {\theta_{k} } \right)\) (\(k = 1, \ldots ,K\)) denotes the normalized values from empirical dataset, \(K\) refers to the cardinality of the set of measurement data, and \(f\left( {\theta_{k} } \right)\) represents the values for the analyzed empirical model of PDF.

The application of this measure gives the opportunity to use the results from [20] to compare the empirical models and the best geometrical models against the measured data. LSE minimize is also used as a criterion for matching model parameters to measurement data for the particular propagation scenarios. The empirical model parameters that minimize the approximation errors of measurement data are included in Table 4, whereas the graphical representations of PDFs are shown in Figs. 4, 5, 6, 7, 8, 9 and 10. For the empirical PDFs, the standard deviation is marked as \(\sigma_{E}\).

Table 4 Optimal parameters of empirical models for individual scenarios
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Fig. 4
figure 4

Scenario 1. Matthews et al. [42]—Leeds

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Fig. 5
figure 5

Scenario 2. Pedersen et al. [43]—Bristol

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Fig. 6
figure 6

Scenario 3. Takada et al. [44]—Yokosuka

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Fig. 7
figure 7

Scenario 4. Fleury et al. [45]—Aalborg

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Fig. 8
figure 8

Scenario 5. Mogensen et al. [46]—Aalborg

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Fig. 9
figure 9

Scenario 6. Pedersen et al. [21]—Aarhus

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Fig. 10
figure 10

Scenario 7. Pedersen et al. [21]—Stockholm

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The empirical data are obtained on the basis of measurement data graphs that are presented in [20]. To extract the numeric data, the software WebPlotDigitizer is applied [49].

For the analyzed scenarios, LSE is the basis for the fit accuracy evaluation of the empirical models to measurement data. LSE is calculated for optimal parameters, and the obtained results are included in Table 5. In [20], the same measure is used to assess the approximation accuracy of the geometrical models. This gives us the opportunity to compare the approximation errors for empirical and theoretical models of PDF of AAOA. Therefore, the smallest approximation errors for the geometrical models from [20] are also included in Table 5.

Table 5 LSE for geometrical and empirical models
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For different measuring scenarios, the obtained results show that the uniform elliptical model (Rx outside) provides the smallest LSE in a group of the geometrical models. But a more accurate mapping of measurement data can be obtained by making use of the modified Laplacian model, which belongs to the simple empirical models. In general, the results in the Table 5 show that in addition to the uniform elliptical model (Rx outside), all the geometrical models give greater approximation errors as compared to the empirical models.

In this paper, the presented comparative analysis concerns the statistical functions that described the properties of AAOA. Therefore, the statistical measures should be used to assess the accuracy of PDF approximation.

The basic measure, \(\Delta \sigma\), is based on a comparison of the standard deviations for model and measurement data

$$\Delta \sigma = \left| {\sigma_{E} - \sigma_{\theta } } \right|$$
(12)

where \(\sigma_{E}\), \(\sigma_{\theta }\) are the standard deviations for measurement data and PDF model, respectively. These parameters are also called the rms azimuth angle spread, when they are determined on the basis of the power azimuth spectrum.

For the analyzed scenarios and PDF models, the comparative analysis results based on \(\Delta \sigma\) are included in Table 6.

Table 6 Results of the comparative analysis based on Δσ
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However, the application of \(\Delta \sigma\) is limited only to assess the dispersion degree of PDFs. Therefore, the similarity assessment of PDFs requires the use of dedicated statistical measures such as the Cramer–von Mises and Kolmogorov–Smirnov statistics [55]. These statistics are described by, respectively

$$W^{2} = \int\limits_{{ -\uppi}}^{\uppi} {\left[ {F_{E} \left( \theta \right) - F\left( \theta \right)} \right]^{ \, 2} {\text{d}}F\left( \theta \right)}$$
(13)
$$D_{n} = \mathop {\sup }\limits_{\theta } \left| {F_{E} \left( \theta \right) - F\left( \theta \right)} \right|$$
(14)

where \(F_{E} \left( \theta \right) = \int\limits_{{ -\uppi}}^{\theta } {f_{E} \left( \varphi \right){\text{d}}\varphi }\) is the cumulative distribution function (CDF) for the PDF based on the empirical data, and \(F\left( \theta \right) = \int\limits_{{ -\uppi}}^{\theta } {f\left( \varphi \right){\text{d}}\varphi }\) is CDF for the analyzed empirical PDF model.

Based on (13) and (14), the obtained results of the numerical calculations are included in Tables 7 and 8, respectively.

Table 7 Results of the comparative analysis based on the Cramer–von Mises statistic
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Table 8 Results of the comparative analysis based on the Kolmogorov–Smirnov statistic
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For each PDF model, to assess the accuracy of the approximation, the average values of the utilized measures are included in the last column of each table. Table 5 shows that the empirical models generally provide a better fit to the measurement data than the geometrical models. Only the uniform elliptical (Rx outside) and Gaussian models provide the results similar to the empirical models. Furthermore, for empirical models, the simplicity of analytical description is a significant prerequisite for the use of these models to assess AAOA statistical properties in analytical and simulation studies. Tables 6, 7 and 8 show that for all statistical measures, Laplacian and logistic models provide the smallest approximation error for large number of measuring scenarios. When choosing the best model, ambiguity occurs only for scenarios 1 and 3. Unlike other scenarios, the measurement results were obtained for the fixed positions of Tx and Rx. Thus, the empirical data from Leeds [42] and Yokosuka [44] do not account for the spatial averaging of measurements. This means that the evaluation of PDF of AAOA requires not only defining the model, type of environment, but also the choice of an appropriate measure for measurement campaign. The obtained results show that for a pair of the modified Laplacian–modified logistic and the modified Gaussian–von Mises models, convergence of the approximation errors occurs. Thus, these models can be used interchangeably for each pair.

5 Conclusion

In this paper, the comparative analysis of PDF empirical models has been performed based on approximation accuracy evaluation of the reception angle distribution to the measurement data for different propagation environments. The obtained results are an extension of comparative analysis that is presented in [20] and focus only on geometrical models.

Here, the classical PDFs are used to determine the empirical models of AAOA distributions. However, the mapping of the statistical properties of AAOA requires appropriate modification of these PDFs. The modifications, which are introduced into the classical models such as Gaussian, Laplacian, and logistic ensure the continuity of PDFs in polar coordinates. In physical interpretation, it provides the convergence of the measurement data and empirical models. For empirical and geometrical models, the comparison of approximation errors shows that simple empirical models fit better with the measurement data, whereas for most of the analyzed scenarios, the modified Laplacian and modified logistic models are best adopted to the empirical data. The obtained results show that minimizing the approximation error requires the selection and fitting of the model to the propagation environment type. The problem of the model adaptation to research scenario, that is, matching of model parameter to the type of propagation environment is essential for theoretical and simulation studies. This issue is examined in Part II of this paper [23].