Page 58 - Integrated Wireless Propagation Models

36 C h a p t e r O n e

where Ed is the diffracted filed and E; is the incident field, while F is a function of the
phase difference �<jl, which is given by
5+ 0.5
F = (1.9.2.2.1.3)
J2. sin( �<P + %)

and the phase difference �<P is

(1.9.2.2.1.4)

where C and 5 are the Fresnel integrals, expressed as
I
2
�
C = cos( x) dx
2
5 = I sin ( � x) dx

As we know, the diffraction loss is due to the knife edge, so the diffraction loss is
relative to the diffraction parameter v, and the expression of v with terms of geometrical
parameters is given by

- -h' 2(d� + d;)
V - (1.9 .2.2.1.5)
1
Nl'd' 2
1
where A is the wavelength and the other parameters could be found in Fig. . 9.2.2.1 . 1 .
h
o
In the real world, usually d1, d2 » , the expression f diffraction parameter v can be
simplified by
(1.9.2.2.1.6)

1
where A is the wavelength and the other parameters can be found in Fig. . 9.2.2.1 . 1 .
o
Notice how the integration limits n Eq. (1.9.2.2.1.4) indicate the nature f the sum­
i
mation of secondary sources from the top of the knife edge, with parameter v, up to
infinity. The eventual result L(v) is illustrated in Fig. . 9.2.2. . 2. It can be numerically
1
1
evaluated using standard routines for calculating Fresnel integrals or approximated for
v > 1 with accuracy better than 1 dB.
1 0.225
L(v) "' 20 log � = 20 log -- v < -2.4 (1. 9 .2.2.1. 7)
v
1tVv2
The approximate solutions for the different values of v can be found in Table 3.1.2.3.1
in Chap. 3.
1
Figure . 9.2.2.1.3 shows an example of the Fresnel zones. By the definition of an
ellipsoid, the radius of the nth zone rk must match the following condition:
(1.9 .2.2.1.8)