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FIGURE 5.22. The region G and its boundaries change with changes in height factor & from 0 to 1.
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zero at the segment [AB] because there are some intersections of this segment with one of the arbitrary buildings (screens) that cross the path AB (see above Fig. 5.16). When the height of point B increases with respect to the rooftop height (i.e., when h), z2 > " the region G, where these re ections are observed, is formed mainly around the transmitting point A. That means there is no re ection in the neighborhood of the receiving point B. Also, for & 0:2 " 20 m, z2 25 m), the region that is h prohibited for re ections has the shape of a circle, the center and the boundary of h which are determined by (5.84). Moreover, an increase in the height factor & z2 > " spreads this region (for & 0:4; h 20 m; z2 33 m) to occupy the entire left halfplane (for & 0:5, h 20 m, z2 40 m). Any further increase in & & ! 1) limits the re ections to the neighborhood of point A. (see in Fig. 5.22 the circles and arcs for & 0:6, z2 50 m and & 0:8; z2 100 m . Effects of Multiple Scattering form Obstructions. To analyze the multiple scattering phenomena caused by the buildings (e.g., nontransparent screens), we assume, as in References [76,77], that the distribution of all obstructions placed above the rough terrain is satis ed by Poisson s distribution law. Consequently, the probability of the event for at least one ray being received after n-time scattering from the randomly distributed screens is Pn 1 expf hNn r ig : 5:86
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Here, the average number of n-time scattered rays from the screens can be obtained from the probability of the scattered points distribution mn r0 jr1 ; r2 ; . . . rn : hNn r; r0 i . . . m1 r0 jr1 ; r2 ; . . . ; rn . . . mn r0 jr1 ; r2 ; . . . ; rn drn drn 1 . . . dr1
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mi r0 jr1 ; r2 ; . . . ; rn exp g0 (
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n Y i 1
0:5g0 n jri 1 ri j jri ri 1 j sin2 ai =2 :
Here, the angle ai is an angle between vectors (ri 1 ri ) and (ri ri 1 ) for all i 1; 2; . . . ; n; r0 ; r1 ; r2 ; . . . ; rn are the radius-vectors of points A, C1 , C2 ; . . . ; Cn ; B, respectively (see Fig. 5.23). The examples of average values for once-, twice- and three-times-scattered rays from the randomly distributed buildings can be presented by using the MacDonald functions Kn w of the order n 1; 2; 3, respectively: hN1 r i 0:25pnr 2 K2 g0 r hN2 r i 9 pnr fK1 g0 r =8! 0:5pg0 r
2 2 1=2
5:88 K7=2 g0 r =7!g K11=2 g0 r =11!g 5:89 5:90
hN3 r i 8 pnr 2 3fK5 g0 r =10! 0:5pg0 r
n rn+1 - rn n Cn
l n
i i 2 C2 1 1 l 1 C1 2 Ci l 2 l i
r2 - r1
r1 - r0 A A
FIGURE 5.23. Geometry of multiple scattering by n randomly distributed buildings, as nontransparent screens.
The probability of occurrence for a single scattered wave (curve 1), a double scattered wave (curve 2), and a three-time scattered wave, calculated according to (5.86) (5.90) and observed at the range of 1 2 km from the source, is presented in Figure 5.24. In microcellular conditions (r < 1 2 km) the probability of observing these rays at the receiver for single-to-three-times scattered waves is equal to the unity. For short ranges from the transmitter, only single scattered waves can be observed. On the other hand, in the far eld, the effect of multiscattering becomes stronger than the single-scattering effect. All of the above mentioned probability formulas were substituted in the corresponding integral (5.54) instead of the shadow functions Zab for the signal eld intensity evaluation.