Page 57 - Integrated Wireless Propagation Models

I n t r o d u c t i o n t o M o d e l i n g M o b i l e S i g n a l s i n W i r e l e s s C o m m u n i c a t i o n s 35

1.9.2.1 . 1 Mathematical Presentation Consider the case of a point source located at a
point P0, oscillating at a frequency f The disturbance may be described by a complex
variable U known as the complex amplitude. It produces a spherical wave with wave­
0
length A and wave number p = 2rc/A. The complex amplitude of the primary wave
located at a distance r from P0 is given by

U(r) = Uo ei�r (1. 9 .2 1 .1)
.
r
The magnitude U decreases inversely proportional to the distance r where the wavelet
traveled, and the phase changes as p times the distance .
r
o
Using Huygens' theory and the principle f superposition f waves, the complex
o
amplitude at a further point i s found by summing the contributions from each point
P
on the sphere of radius r. The complex amplitude of the wavelet at P is then given by
iU(r)
U(P) = f � K(a) dS (1.9.2.1.2)
A r
5
where 5 describes the surface of the sphere and r is the distance between P0 and P. K(a)
is the inclination factor shown in Eq. (1.9.2.1.3), and a is the angle of incidence.
The various assumptions made by Fresnel emerge automatically in Kirchhoff's dif­
fraction formula,34 to which the Huygens-Fresnel principle can be considered an
approximation. Kirchoff gives the following expression for K(a):
i
=
K(a) - 2A. (1 + cos a) (1.9.2.1.3)
K has a maximum value at a = 0 as in the Huygens-Fresnel principle; K is equal to
0 at = n.
a
1.9.2.2 Knife Diffraction
1.9.2. . 1 Single-Knife Diffraction From Huygens' Principle, the fields in the shadow
2
area are not absolutely 0. There is still some energy to reach the shadow area via diffrac­
tion. Also, Huygens' Principle helps to analyze the diffraction caused by a knife edge in
a mathematical way.33·35.37
In electromagnetic theory, the field strength of a diffracted radio wave due to a knife
edge can be expressed as
�
= FeiM (1.9.2.2.1.1)
E o
where E 0 is the free-space electromagnetic field with no knife edge present, Ed is the dif­
fracted wave, F is the diffraction coefficient, and .:1<j> is the phase difference with respect
to the path of the direct wave.
We consider that the diffraction loss is a propagation loss, and thus it could be
expressed by the original power divided by the diffracted power in decibels, which is
o
given by ,32 -34

L(v) = 20 log��) = 20 logiFI (1.9.2.2.1.2)