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To understand the in uence of the built-up area relief on the signal intensity, let us rst examine the height distribution function Ph z . The graph of this function versus height z of a built-up overlay is presented in Figure 5.17. For n ) 1 Ph z describes the case where the buildings are higher than h1 (this is a very rare case as most buildings are at the level of a minimal height h1 ). The case when all buildings
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n = 0.2 n = 0.5 n=1 n=2 n=5 n= 0 h1 h2 z
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FIGURE 5.17. Buildings height distribution function Ph z versus the current height z for various parameters n of built-up pro le.
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have heights close to h2 , (i.e., most buildings are tall), is given by n ( 1. For n close to zero, or n approaching in nity, most buildings have approximately the same height h2 or h1, respectively. For n 1 we have the case of building heights uniformly distributed in the range h1 to h2 . The same result is obtained from analyzing the built-up layer pro le F z1 ; z2 . For the case when the minimum antenna height is above the rooftop level, that is, z2 > h2 > h1 , then according to Reference [77] we get ! h2 h1 F z1 ; z2 H h1 z1 h1 z1 n 1 5:73a h2 z1 n 1 H z1 h1 H h2 z1 n 1 h2 h1 n and for the case where the minimum antenna height is below the rooftop level, that is z2 < h2 , we derived [77] " # F z1 ; z2 H h1 z1 h1 z1 h2 h1 n 1 h2 z2 n 1 H z1 h1 H h2 z1 n 1 h2 h1 n From Equation (5.73) we can determine the average building height as " h2 n h2 h1 = n 1 h which reduces to " h1 h2 =2 h 5:75 5:74 h2 h1 n 1 h2 z2 n 1 n 1 h2 h1 n
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for the case n 1 of a uniformly distributed pro le investigated in References [57,58].
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Point #4. Terrain Factor n = 1; Transmitter Antenna hT = 10, 40, 100 m 70
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Minimum to Maximum Heigh of the Building Level
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F(hR ), m
30 40 50 60 70 Receiver Antenna Heigh h , m
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FIGURE 5.18. Distribution of F hR versus the receiver antenna height hR for various heights of the transmitter antenna: hT 10; 40, and 100 m and h1 and h2 are the minimum and maximum of built-up relief.
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As there are many geometrical factors in the built-up layer pro le: the antenna heights z1 and z2 , the minimum and maximum building heights h1 and h2 , and the building relief that appear in formulas (5.73a) and (5.73b), we consider their effects on function F z1 ; z2 separately. In Figure 5.18, F hT ; hR ; n F z1 ; z2 , given by expressions (5.73a) or (5.73b) for z2 > h2 > h1 or z2 < h2, respectively, is depicted as a family of curves versus the receiving antenna height [77]. The discrete parameters are denoted by ; . These parameters are the transmitter antenna height, ranging between 10 m (bottom curve) to 100 m (top curve), and n. The minimum and maximum heights of the buildings overlay pro le are indicated by the dotted vertical lines. We have chosen n 1 which corresponds to a uniform distribution of building heights. By inspection of the displayed curves, it is obvious that for a constant transmitter antenna height, as the receiver antenna height increases, the value of F z1 ; z2 becomes smaller and the effect of the building layer on the path loss is reduced. Thus, for a transmitter antenna height hT 40 m, as the receiver antenna height hR increases from 40 m (at the bottom level of rooftops) to 50 m (some intermediate building height value), F hR decreases sharply from 15 m to zero. A more gradual decrease for F hR is evidenced for higher transmitter antennas, (e.g., see curve for hT 100 m).
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