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where X q, X 2 1; 1 ; q is the spatial wave vector, k 2p; shw and chw denote l k hyperbolic sine and cosine, respectively, and the parameter w accounts the density of the buildings and the range between the antennas 2 !1=2 3     1 1 2 5 10:35 1 1 w ln4 1 g0 d g0 d
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e e The dependence of W X; j0 =W 0; j0 as a function of the normalized parameter X for j0 p ; p ; p ; p (curves 1, 2, 3, 4, respectively) for g0 d 10 (built-up area with 2 3 4 6 high density of buildings around the receiver and transmitter, or the terminals are placed far from each other) is presented in gure 10.34a. The non symmetrical e spectrum W X; j0 is observed for the orientation angle j0 deviations from p (j0 is 2 the angle between the scatterer and radio path between terminal antennas). This phenomenon can be understood if we again return to the results of the analysis presented previously, where the PDF and the signal power angle-of-arrival distributions are uni ed and symmetrical relative to the wave path, which is directed strictly to the transmitter. If we consider that the position of a segment of the scatterer relative to radio path is not xed and that it can be oriented with equal probability anywhere in space, that is, the angle j0 is the angle regularly distributed within the angle interval [0, 2p], we immediately obtain the same case that is described in References [55,61]. According to these expressions, we get e hW X i 2 X 1 X2 1=2 10:36
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From this formula it follows that the spectrum of spatial frequencies for the case of h z1 " does not depend on the range d between the BS and MS antennas and on the building contours density g0 at the plane z 0. This result also follows from the e e dependence of hW X i=hW 0 i shown in Figure 10.34b. In the second case, when the transmitter/receiver is above the level of rooftop, " h, that is, z1 > " an increase of the illumination area ( r-area) surrounding the receiver/transmitter is observed. In this case the signal spectrum of normalized spatial frequencies X q 2 f 1; 1g can be determined as k e hW X i 2 1 X cos j0 X 1 X2 1=2 10:37
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PREDICTION OF SIGNAL DISTRIBUTION IN SPACE
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~ ~ FIGURE 10.34. The dependence W X; j0 =W 0; j0 as a function of the normalized p p p p parameter X (a) for j0 2 ; 3 ; 4 ; 6 (curves 1, 2, 3, 4) for g0 d 10. (b) The same dependence ~ ~ of hW X i=hW 0 i, but for j0 is regularly distributed from 0 to 2p. BS antenna is at the same level or below the rooftops.
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e e In Figure 10.35, the dependence W X; j0 =W 0; j0 as a function of normalized p p p p parameter X for j0 2 ; 3 ; 4 ; 6 (curves 1, 2, 3, 4, respectively) for g0 d 10 is shown. The deformation of the classical U-shape (shown in Fig. 10.346) is observed. 10.4.2. Signal Power Distribution in Doppler Shift Domain In built-up areas, the spatial distribution of signal strength fully determines the properties of temporal signal distribution obtained at the receiver. As indicated
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