GeometryBased Statistical Model for the Temporal, Spectral, and Spatial Characteristics of the Land Mobile Channel
Abstract
The main drawback of channel models presented in the literature is the lack of physically justified integration of all basic phenomena such as fluctuations, channel dispersion, and selective fading that occur in the actual radio channels. Based on physical premises, presented in this paper, the developed channel model reproduces all basic phenomena that affect the temporal, correlational–spectral, and spatial characteristics of the modelled radio channels. This effect is achieved by the structure of the model, which includes both the geometric channel model and statistical models of the received signal parameters. The geometry used in the model is based on the Parsons–Bajwa multielliptical model and the relationship, which describes the Doppler frequency as a function of the spatial position of object. Therefore, this model is called the Doppler multielliptical channel model (DMCM). The source of input data that define the geometric and statistical parameters of the model is the power delay profile or the power delay spectrum. This makes sure that DMCM characteristics depend on the properties of the modelled propagation environment. A comparative analysis of the simulation results and the measurements that DMCM can correctly capture the actual transmission properties of real channels.
Keywords
Wireless communications Mobile channel Geometric channel model (GCM) Multipath channel Rayleigh fading Scattering Doppler multielliptical channel model (DMCM)1 Introduction
Model of the communication channel is one of the basic elements of the algorithms used in simulation studies of wireless systems. Thus, a faithful representation of the transmission channel characteristics substantially affects the correctness of the test results.
The main group consists of models of the flat fading channels, which are applicable only to narrowband systems. In this case, the modelling procedure is reduced to generating multiplicative interferences with certain statistical properties that are described by probability density function (PDF) of envelope. This group should include the models of: Rayleigh [1], Rice [2, 3], Suzuki type I, and Suzuki type II [4, 5]. In addition to the abovementioned models, extended models [6] are developed, which also take into account the relationship between the real and imaginary parts of the signal envelope. The differentiation of parameters of these parts leads to extended models of fast fading, examples of which are the following distributions: Weibull [7], Hoyt [8], Beckman [9], mNakagami [10], α–μ [11], κ–μ, and η–μ [12].
Modelling of the influence of the channel on wideband signals additionally requires considering differentiation in delays of particular components of the received signal or the channel dispersion, which results in the intersymbol interference (ISI) [2, 13]. In this case, the ISI channel model uses tapped delay line (TDL) [6], its coefficients are calculated on the basis of the channel impulse response. However, based on the TDL, the ISI channel models do not map the spatial properties of the multipath propagation, which cause selective fading. Therefore, in recent times, it is possible to notice the progressive development of geometric channel models (GCMs). These models are defined by spatial location of the transmitter, receiver, and scatterers that are described by the area form and the spatial density of their occurrence. The most commonly used forms of scattering areas are the circle [14, 15, 16, 17], ellipse [16, 18, 19], ring [20], and semiellipse [21]. The most frequently used PDF of scattering density are the following distributions: uniform [15, 16, 17, 18, 19, 20, 21], Gaussian [18, 22, 23], hyperbolic [24], and inverted parabolic [14], as well as exponential and Rayleigh [25]. GCMs are used primarily to determine the characteristics describing the spatial properties of the signals, such as the PDF of the angle of arrival (AOA) and the power azimuth spectrum (PAS). The main difficulty in the practical application of these models is the matching problem of parameters to transmission properties of the environment. This is due to the lack of physical premises to select both shape and dimension areas as well the PDF of scatterer occurrence.
A brief overview of the radio channel models shows that obtaining temporal, correlationalspectral, and spatial characteristics of the received signals requires the use of different models with different input database. An additional difficulty is the matching problem of parameters to the transmission properties of the modelled propagation environment. Thus, we can observe that there is no model that can integrate all the phenomena occurring in the actual channel and provide a complete mapping of the channel characteristics based on the transmission properties of the environment. To meet this problem, the suggestion of such a channel model is presented in this study.
The developed model uses the statistical properties of the received signal parameters, the Parsons–Bajwa multielliptical model [26], and the relationship describing the Doppler frequency as a function of the spatial position of objects [27]. Therefore, this model is called the Doppler multielliptical channel model (DMCM) and is an extension of the ISI channel models. This extension takes into account the Doppler effect, which is conditioned by the current position and motion parameters of the objects. The input data for the model are: the spatial position and motion parameters of objects (transmitter/receiver), the power delay profile (PDP), or the power delay spectrum (PDS) that are related to the type of modelled propagation environment. In this study, the authors tried to present wide DMCM possibilities to represent the impact of the channel on the received signal characteristics in the time domain (the envelope as a function of time), the statistics of signal [cumulative density function (CDF), PDF of envelope], correlational properties [autocorrelation function (ACF) of envelope], frequency [power spectral density (PSD)], and space [power azimuth spectrum (PAS)]. Based only on the PDP or PDS and motion parameters of the transmitter/receiver, the developed model could provide all basic characteristics of the channel, which depend on the relative position of the objects. This fact proves the originality of DMCM, when compared with models presented in the literature.
The paper is organized as follows. Section 2 presents the origin and basic assumptions of DMCM. Section 3 describes in detail the GCM as well as the methodology for Doppler frequency shift (DFS) and AOA for delayed components of the received signal. Section 4 presents the statistical models of the received signal component parameters. For a simple scenario, wide DMCM possibilities for channel modelling are shown in Sect. 5. Verification of the model by comparative evaluation of the selected measurement results against the results obtained using DMCM is presented in Sect. 6. Lastly, Sect. 7 provides a summary that highlights the usefulness of the model for simulation tests.
2 Origin and Basic Assumptions of DMCM
As shown in Fig. 1, a fundamental difference between DMCM and models, which are based on the TDL, consists of introducing GCM to the structure of the model. In this way, DMCM provides a mapping of the properties of the received signal not only in the time domain and the spectrum, but also in space. The structure of the model shows that for the modelled propagation environment, the statistical properties of \( r_{lk} ,\varphi_{lk} \), and f _{ Dlk } have a significant impact on the accuracy of the estimation of channel characteristics. Therefore, the statistical models of these parameters and geometric model are defined on the basis of the PDP or PDS, which provide the mapping of actual channel properties.
3 GCM for Delayed Components
3.1 Origin and Basic Assumptions of the GCM
From relation (4), it can be noted that τ _{ l } is one of the basic DMCM parameters, because it determines the sizes of particular ellipses. The values of τ _{ l } and their amount (the number L) are determined on the basis of the PDP or PDS for which the base could provide the measurements or parameters of the standard channel models such as COST 207 [31] and WINNER [32]. When using the results of the measurements, L represents the amount of τ for which the PDP or PDS achieve local extremes.
It should also be noted that any object trajectory can be approximated by straight sections, along which the object moves at constant speed. In this case, the parameters of the received signal are determined for each segment of movement in a coordinate system, the beginning of which defines one end of the segment and the direction of the coordinate x indicates the direction of velocity vector.
 1.
Propagation phenomena are considered only in a plane stretched on the velocity vector of the object and the line passing through the points where Tx and Rx are located (see Fig. 2).
 2.
The radiation characteristics of the transmitting and receiving antennas are omnidirectional.
 3.
The probability of propagation path is the same for each of the direction of departure wave, i.e. the probability of occurrence of a scatterer is the same for each direction when viewed from Tx.
 4.
Each propagation path from Tx to Rx consists of scatterers on exactly one scattering element.
 5.
Each scattering element is a reradiating omnidirectional element with the same scattering coefficient and uniform phase distribution.
3.2 Method of Determining AOA and DFS for Delayed Components

the transformation of coordinate system;

the appointment of the coordinates of scattering element for the specified AOD;

the appointment of the coordinates of the apparent source;

the appointment of the AOA and DFS for the specified AOD.
In the new OXYZ system, the coordinates of Rx and Tx are \( \left( {x_{\text{R}} ,y_{\text{R}} ,z_{\text{R}} } \right) = \left( {{\text{v}}t,0,0} \right) \) and \( \left( {x_{\text{T}} ,y_{\text{T}} ,z_{\text{T}} } \right) = \left( {x_{{ 0 {\text{T}}}} ,y_{{ 0 {\text{T}}}} \cos \beta_{0} + z_{{ 0 {\text{T}}}} \sin \beta_{0} ,0} \right) \), respectively. Due to the abovementioned coordinate system transformation, the problem of propagation modelling of the O_{0}X_{0}Y_{0}Z_{0} space is reduced to modelling propagation on the OXY plane.
The transformation of the coordinates to the O′X′Y′ system with its beginning in point O′ = E consists of a shift of the OXY coordinate system by the vector \( {\mathbf{p}} = \left[ {x_{E} ,\,y_{E} } \right] \) and rotation by an angle β, as shown in Fig. 3. The angle β is determined by the direction of the \( {\mathbf{v}} \) of the Rx and a line defined by the position of the Rx and Tx, i.e. \( \begin{array}{*{20}c} {\beta = \arcsin \left( {{{\left( {y_{\text{T}}  y_{\text{R}} } \right)} \mathord{\left/ {\vphantom {{\left( {y_{\text{T}}  y_{\text{R}} } \right)} d}} \right. \kern0pt} d}} \right)} & \wedge & {\beta = \arccos \left( {{{\left( {x_{\text{T}}  x_{\text{R}} } \right)} \mathord{\left/ {\vphantom {{\left( {x_{\text{T}}  x_{\text{R}} } \right)} d}} \right. \kern0pt} d}} \right)} \\ \end{array} \) where \( d = \sqrt {\left( {x_{\text{T}}  x_{\text{R}} } \right)^{2} + \left( {y_{\text{T}}  y_{\text{R}} } \right)^{2} } \) is the distance between Tx and Rx.
The apparent source method was used to determine the Doppler frequency f _{ Dlk } of the component of a signal coming from point Q_{ lk }. In Fig. 3, the source is designated as U_{ lk }. The determination of its coordinates consists of establishing the tangent l _{ H } of the ellipse at point Q_{ lk }, and then the perpendicular line l _{ V } to the tangent that passes through point Tx. These lines intersect at point V_{ lk }, which is located at the same distance from both Tx and U_{ lk }. This constitutes the basis for determining the coordinates \( \left( {x_{{{\text{U}}lk}}^{\prime } ,y_{{{\text{U}}lk}}^{\prime } } \right) \) of the apparent source.
4 Statistical Models of Received Signal Parameters
The choice of γ value depends on the environmental conditions present in the surroundings of transmitting and receiving antenna and the direction of signal transmission. As shown in [34], for large scattering that occur in the vicinity of the omnidirectional receiving antenna, i.e. in an urban environment with low positioning of the antenna, γ is smaller than 3. In mobile radio systems, this case corresponds to downlink signal transmission. For uplink transmission, the angular scattering intensity of the received components is much smaller; hence, we assumed that γ > 10.
5 Signal Characteristics of DMCM for a Sample Scenario
A wide range of mapped phenomena, which occur in the real radio channels, determines the originality of the presented model. Therefore, in this section, using DMCM, the authors present the effects of channel on the transmitted signals in the range of temporal and their statistical properties, correlationalspectral, and spatial properties for a sample scenario. It should be noted that the purpose of this investigation is not to analyse the phenomena occurring in the radio channels, but to show a wide range of DMCM possibilities.
This section presents a set of characteristics obtained by DMCM, which is shown to be compatible with the characteristics of the signals occurring at the output of the real channel. This indicates that the developed DMCM maps a wide range of phenomena that are closely related to the specific propagation scenarios.
5.1 Simulation Scenario
Values of P _{ l } and τ _{ l } determined on the basis of measured PDP for test scenario [29]
l  τ _{ l }  P _{ l }  

(μs)  (dB)  (l)  
0  0.00  0.0  1.0000 
1  0.25  −2.8  0.5248 
2  0.50  −5.6  0.2754 
3  0.75  −6.4  0.2291 
4  2.00  −24.7  0.0034 
5  2.90  −27.0  0.0020 
Analysis of the results was carried out for a route with a length S = 50 m. On this route, the measurement sections (M = 8) were designated. For these sections, we assumed that the received signal parameters have fixed values. Based on [35], the length of each section was adopted to be 40λ, where \( \lambda \cong 0.16\;{\text{m}} \) is the wavelength, which results from the frequency of the transmitted signal (\( f_{ 0} = 1860\;{\text{MHz}} \)). Moreover, in simulation, the number of paths in each cluster was adopted to be K _{ l } = 10 (l = 0, 1, 2, …, L) and the sampling frequency of the signal was \( f_{s} \cong 200\,f_{D\hbox{max} } \cong 17.24\;{\text{kHz}} \), where \( f_{D\hbox{max} } =  f_{D\hbox{min} } \cong 86.1\;{\text{Hz}} \).
5.2 Analysis of the Signal Envelope
For comparison of the simulation and theoretical results, in Fig. 5, theoretical Rayleigh distribution, which is indicated by dashed line, is also presented. Figure 5 indicates a good fit of results obtained by using DMCM for the analysed example of data.
5.3 ACF and PSD
The abovementioned graphs have been obtained for the scenario parameters, taking into account changes in β. As expected, we can observe that change in the PSD concentration is associated with β change, which follows from relation (19).
5.4 PAS
These characteristics describe the PASs of the received signals and can be used for the assessment of the spatial compatibility of devices and networks operating in a specified propagation environment. In DMCM, PAS methodology depends on the sizes of the ellipses, i.e. PDS, but not movement of the objects (Tx/Rx).
6 Verification of DMCM
A DMCM accuracy assessment was realized in relation to theoretical models available in the literature and empirical research carried out in urban environments. Statistical properties of the signal envelope (CDF and PDF), its ACF, PSD, and PAS were the basis for accuracy assessment of the developed model.
6.1 PDF and CDF of the Envelope
For the scenario described in Sect. 5, the obtained CDF and PDF of the signal envelope were used for accuracy evaluation of the mapping of Rayleigh fading in DMCM. The mean square error (MSE), δ ^{2}, was adopted as an accuracy measure of the estimation of the characteristics. For the data presented in Fig. 4, the MSEs of CDF and PDF are \( \delta_{CDF}^{ 2} = 0.0 9\times 10^{  4} \) and \( \delta_{PDF}^{ 2} = 1. 1 9 \times 10^{  4} \), respectively. It can be observed that \( \delta_{CDF}^{ 2} \) is an order of magnitude smaller, when compared with \( \delta_{PDF}^{ 2} \). Therefore, the graphical comparison of the simulation and theoretical results is limited to the PDF because only in this case, we can observe clear differences in the graphs. The average values of \( \delta_{CDF}^{ 2} \) and \( \delta_{PDF}^{ 2} \) determined on the basis of averaging the results of a hundred simulation procedures are \( \overline{{\delta_{CDF}^{2} }} = \left( {0.25 \pm 0.2} \right) \times 10^{  4} \) and \( \overline{{\delta_{PDF}^{2} }} = \left( {2.31 \pm 1.29} \right) \times 10^{  4} \), respectively. Small values of these errors indicate correct mapping of the typical Rayleigh fading by DMCM.
6.2 PSD
6.3 PAS
Values of P _{ l } and τ _{ l } determined on the basis of measured PDS for TU test scenario [28]
l  τ _{ l }  P _{ l }  

(μs)  (dB)  (l)  
0  0.00  0.0  1.000 
1  0.14  −1.6  0.692 
2  0.32  −3.2  0.479 
3  0.83  −10.0  0.100 
4  1.28  −17.7  0.017 
5  1.95  −23.9  0.004 
Values of P _{ l } and τ _{ l } determined on the basis of measured PDS for BU test scenario [28]
l  τ _{ l }  P _{ l }  

(μs)  (dB)  (l)  
0  0.00  0.0  1.000 
1  0.25  −1.4  0.724 
2  0.85  −5.0  0.316 
3  1.15  −6.2  0.240 
4  1.81  −10.9  0.081 
5  2.44  −13.9  0.041 
Comparison of the results showed that for different environmental conditions, DMCM provides diverse PAS and gives results that are consistent with the measurement results obtained in realworld conditions. A complete verification of DMCM requires tests in a wide range of empirical data.
In order to access the possibility of DMCM the authors have developed a software implementation in MATLAB (see Appendix).
7 Conclusions
This paper presented the radio channel model, which structure consists of a geometric channel model and statistic models of the received signal parameters. This model was named the Doppler multielliptical channel model (DMCM) due to the possibility of mapping the effects of moving objects and the dispersive nature of the modelled channels. The input data for the model were the spatial location and motion parameters of objects (transmitter/receiver), the PDP, or PDS, which are closely related to the transmission properties of propagation environment. As a result, the developed model allowed us to obtain all the basic characteristics of the channel, including those related to the spatial position of the objects. In contrast to the previously presented models in the literature, DMCM provides integration of all basic phenomena, such as fluctuations and delay spread, as well as the phenomena that stem from the spatial nature in real channels. In the present study, this fact was shown using the example of selected propagation scenarios.
The main difficulty in the practical application of existing models is the problem of matching of their parameters to modelled scenarios. The PDP and PDS are characteristics that, in the measurement practice, are the basis for the assessment of the transmission properties of a channel. Therefore, DMCM ensures evaluation of the impact of the channel on the signal characteristics within a wide scope, including time domain (envelope vs. time), value (PDF, CDF, and the ACF of the envelope), frequency (PDS), and space (PAS). Accordingly, the developed model provides a good mapping of the transmission properties of the modelled propagation environment. This fact significantly distinguishes DMCM from most of the models previously presented in the literature.
In contrast to standard models, such as COST 207 or WINNER, DMCM also provided mapping of the impact of the spatial position and motion parameters of the objects on the channel characteristics that determines its originality. In COST 207 and WINNER models, it is not possible to test of each scenario, because the channel transmission characteristics (e.g. PDS) are precisely defined and they do not depend on TxRx distance. In the empirical scenarios [28, 29], PDSs are significant difference in relation to PDSs of COST 207 and WINNER.
The use of the ellipsoid to 3D modeling and phase differences in the multiantenna system to MIMO modeling will be the part of next extended work. Lastly, the simplicity of the practical implementation of DMCM, when compared with most of the models presented in the literature, is also worth mentioning.
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