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The Fresnel integral in (4.91) gives the cumulative effect from several rst Fresnel zones covered by the obstruction. In Figure 4.19, the Fresnel (also called diffraction) parameter n in (4.91) is presented by the following formula [29 36]:
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FIGURE 4.19. Geometrical presentation of the Fresnel zones in terms of ellipsoids.
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From (4.92) and (4.93) one can obtain the physical meaning of the Fresnel Kirchhoff diffraction parameter n. Thus, the diffraction parameter v increases with 0 the number n of ellipsoids. All the above formulas are corrected for hn ( r0, r0 , that is, far from the terminal antennas. The volume enclosed by the ellipsoid and de ned by n 1 is known as the rst Fresnel zone. The volume between this ellipsoid and the one that is de ned by n 2 is the second Fresnel zone. The contributions to the total eld at the receiving point, from successive Fresnel zones, interfere by giving a very complicated interference picture at the receiver. If a virtual line OO0 is placed at the middle of the radio path TO0 R (i.e., TO0 O0 R, as shown in Fig. 4.19) then, if the height of the virtual point O0 (the virtual source of diffraction) h increases from h h1 (corresponding to the rst Fresnel zone) to h h2 (e.g., to the point O00 de ning the limit of the second Fresnel zone), then to h h3 (i.e., to the point O000 de ning the limit of the third Fresnel zone) and so on, the eld at the receiver R will oscillate. The amplitude of oscillations would essentially decrease as a smaller amount of wave energy penetrates into the outer zone relative to the inner zone. If, for example, some obstacles that we may model by a simple knife edge (with height above the line-of-sight line TOR, h, denoted in Fig. 4.20 as OO n ), lies 0 between the receiver and the transmitter at distances r0 and r0 , respectively, the Fresnel parameter can be presented as [1 8]
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and the phase difference F between the direct ray from the source placed at the point O (denoted TOR) and the diffracted ray from the point O n (denoted
O (n)
rn h
r0 '
FIGURE 4.20. Geometrical presentation of the knife-edge diffraction.
TO n R) can be obtained in the standard manner by use of a simple presentation of the path difference, r, and the phase difference, F, between these rays. From the geometry of the problem, shown in Figure 4.20, and using relationship (4.94) between r and v, the phase difference, F, can be presented as F 2p p r v2 l 2 4:95
From the above discussions, it is clear that any radio path in obstructive conditions requires a certain amount of a clearance around the central ray if free-space propagation is to occur. This effect can be understood by using the principle of Fresnel clearance, which is important in the design of point-to-point radio links, where communication is required along a single radio path. This clearance can be explained in terms of Fresnel zones. Thus, the rst Fresnel zone (for n 1 ) encloses all radio paths for which the additional path length r, de ned in (4.94), does not exceed l, and according to (4.95), a phase change is F1 p. The second Fresnel 2 zone (for n 2 ) encloses all paths for which the additional path length r does not exceed 2 l l, and correspondingly, F2 2p, and so on. The corresponding 2 radius of the rst Fresnel zone h1 can be derived by setting r l in (4.92). As a 2 result, h1 
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0 where f is measured in gigahertz and r r0 r0 in kilometer. The shape of the rst Fresnel zone and the effect of the obstruction on the clearance are clearly illustrated