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From the presented illustration, it follows that the Loo s curve and the Abdi s curve, both are close to the measured data for different cases and channel conditions and cannot be distinguished. This is very important for link budget design. These empirical results indicate the utility of an Abdi s model [13] for LMS channels. 14.4. PHYSICAL STATISTICAL MODELS In pure statistical models, the input data and computational effort are quite simple, as the model parameters are tted to measured data. Because of the lack of physical background, such models only apply to environments that are very close to the one they have been inferred from. On the contrary, pure deterministic physical models provide high accuracy, but they require actual analytical path pro les and timeconsuming computations. A combination of both approaches has been developed by the authors. The general method relates any channel simulation to the statistical distribution of physical parameters, such as building height, width and spacing, street width or elevation and azimuth angles of the satellite link. This approach is henceforth referred to as the Physical-Statistical approach [2,14,15]. The main concept of such an approach is sketched in Figure 14.13. As for physical models, the input knowledge consists of electromagnetic theory and a full physical understanding of the propagation processes. However, this knowledge is then used to analyze a statistical input data set, yielding a distribution of the output predictions. The output predictions are not linked to speci c locations. Physical-statistical models therefore require only simple input data such as distribution parameters (e.g., mean building height and building height variance, as was done in 5 for land communication links). This modeling describes the geometry of mobile satellite propagation in built-up areas and proposes statistical distributions of building heights, which are used in the subsequent analysis. We will consider only two of them, which have been fully proved by numerous experiments for land land and land satellite communication links:
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FIGURE 14.13. Algorithm of the physical-statistical model.
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FIGURE 14.14. Geometry for mobile-satellite communication in built-up areas.
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 a model of shadowing based on the two-state channel Lutz model;  a multiparametric stochastic model. 14.4.1. The Model of Shadowing The geometry of the situation, which was analyzed in References [2,14,15] by Saunders and his colleagues, is illustrated in Figure 14.14. It describes a situation where a mobile is situated on a long straight street with the direct ray from the satellite impinging on the mobile from an arbitrary direction. The street is lined on both sides with buildings whose height varies randomly. In the presented model, the statistics of the building height in typical built-up areas will be used as input data. A suitable form was sought by comparing it with geographical data for the cities of Westminster and Guildford, UK [14,15]. The PDFs that were selected to t the data are the lognormal and Rayleigh distributions with unknown parameters of a mean value, m, and standard deviation, sb . The PDF for the lognormal distribution is [2,15]
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1 pb hb p e hb 2psb ln2 hb =m 2s2 b
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The PDF for the Rayleigh distribution is presented in 1. We will repeat it using notations made in References [2,15]: hb b2 pb hb 2 e 2sb sb
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TABLE 14.4. Best- t Parameters for the Theoretical PDFs Lognormal PDF City Westminster Guildford Mean (m) 20.6 7.1 Standard deviation 0.44 0.27 Rayleigh PDF Standard deviation 17.6 6.4
To nd the appropriate parameters for these functions in order to t the data measurements as accurately as possible, the probability density function was found by minimizing the maximum difference between the two cumulative distribution functions. The parameters for each PDF are quoted in Table 14.4 from References [14,15], where all parameters are in meters. The direct ray is judged to be shadowed when the building height hb exceeds some threshold height hT relative to the direct ray height hr (see Figure 14.14). The shadowing probability, Ps , can then be expressed in terms of the probability density function of the building height, pb hb as in References [2,15]: Ps Pr hb > hT