STUDY OF THE SLOW FADING IN
INDOOR ENVIRONMENT USING
PARABOLIC EQUATIONS
Fa ´ tima N. B. Magno, Joa ˜ o F. Souza, Zı ´nia A. Valente,
Jesse ´ C. Costa, and Gerva ´ sio P. S. Cavalcante
Universidade Federal do Para ´ , Bele ´ m, Para ´ , Brasil
Received 5 December 2006
ABSTRACT: This article compares slow fading component of the prop-
agation loss calculated through parabolic wave equation and some mod-
els of the literature with results obtained during measurement campaign,
in indoor environment. The frequency used was 850 MHz; a complex
refractive index was considered. The implicit finite difference scheme of
the Crank-Nicolson type was applied in order to get the solution of the
parabolic equation. The propagation was considered in 15° with direc-
tion paraxial. © 2007 Wiley Periodicals, Inc. Microwave Opt
Technol Lett 49: 1676–1679, 2007; Published online in Wiley Inter-
Science (www.interscience.wiley.com). DOI 10.1002/mop.22530
Key words: finite difference method; indoor; parabolic equation; prop-
agation loss; slow fading
1. INTRODUCTION
Traditionally, propagation models based on empiric and semiem-
piric models have been used in several environments. However, as
the traffic has increased, the cells sizes have reduced, these models
no longer supply good predictions. As a consequence, determin-
istic methods have been used [1].
Some methods proposed for calculation of indoor waves prop-
agation, such as the ray tracing and the numerical solution of
Maxwell’s equations, request quite a lot of computation time and
great capacity of memory. For this reason, for demanding less time
and amount of memory compared with the method totally elliptic,
the parabolic approach of the wave equation has been used to solve
scattering problems when the main interest is to determine the
characteristics of the channel fading [2].
The internal environment of a building is extremely complex
and there is the impact of several propagation mechanisms, such as
reflection, rough surface scattering, diffraction, and transmission
through walls and furniture. For this reason, it is necessary to know
the materials used in the construction, and their electrical proper-
ties [3].
The model presented in this paper is based on the method of the
parabolic equation using the finite difference scheme of Crank-
Nicolson to calculate the path loss in indoor environment. It aims
to calculate the received field for a building of five floors, the
Institute of Superior Studies of the Amazon (Bele ´m-Para ´). A
complex refractive index with differentiated values, depending on
the relative permittivity and conductivity, for the several materials
existing in the construction and objects inside the environment
studied. The frequency used was 850 MHz. With the value of the
field, the propagation loss was calculated for the proposed method,
separating the fast fading and slow fading components and com-
paring the curve obtained for the component of slow fading with
the results obtained during measurement campaign and with two
models existing in the literature.
2. PARABOLIC EQUATION METHOD
The radio wave propagation modeling requires the solution of
wave equation with proper boundary conditions [4]
2
x
2
+
2
z
2
+ k
2
r
2
= 0 (1)
where the field, , is considered y independent, k and r are the
propagation constant and refractive index, respectively. We define
u, representing the electric field, as follows
u x , z = exp - ikx x, z (2)
By substituting (2) in (1) and factored, the resulted equation is [4]
x
+ ik 1 - Q
x
+ ik 1 + Q
u = 0 (3)
being that these two terms correspond, respectively, progressive
and regressive waves and Q is the distinguishing pseudo-differen-
tial operator defined by Q = 1/k
2
2
/ z
2
+ r
2
x, z [4].
Disdaining the regressive waves, we have the formal solution
for the electric field u
u x + x , z = expikx - 1 + Qu x, z. (4)
Defining that Q=
1+q 1 + 0,5q, where q = r
2
+ (1/ k
2
)
2
/
z
2
- 1, that characterizes the approach for small angles, up to 15°
[5], in (4), doing first-order Taylor expansions of the exponential
term and also of the square-root, a standard parabolic equation is
found (SPE) [4]
2
u
z
2
x , z + 2ik
u
x
x , z + k
2
r
2
x , z - 1 u x , z = 0 (5)
3. FINITE DIFFERENCE METHOD
In this article was used the finite difference scheme of Crank-
Nicolson applied to the standard parabolic equation. The approach
of the central finite differences was calculated for the derivatives
of first and second order in x and z, respectively, and used in (5).
Putting
m
= x
m-1
+ x
m
/2 as the midpoint in the solution from
x
m-1
to x
m
range, u
j
m
= u x
m
, z
j
, b = 4ikz
2
/x and a
j
m
= k
2
r
2
m
, z
j
- 1z
2
is obtained [4]
u
j
m
- 2 + b + a
j
m
+ u
j+1
m
+ u
j-1
m
= u
j
m-1
2 + b - a
j
m
- u
j+1
m-1
- u
j-1
m-1
(6)
4. DESCRIPTION OF THE ENVIRONMENT
The model was developed to study the electromagnetic waves
propagation in indoor environments, taking in consideration the
different geometric structures, the many involved materials, be-
yond the great number of small objects, all intervening with the
electromagnetic wave propagation.
The Superior Studies of the Amazon was the building used for
the campaign of measurements. In this building, the one used to be
the test environment, there are corridors, computer science labo-
ratories, laboratories for experimental works, classrooms, and
bathrooms. There are different materials involved, such as brick
walls, doors, and teacher and student’s wooden tables, group of
laboratories benches made of formica, glass, and aluminum in the
windows, steel in the elevators.
5. PATH LOSS THROUGH PARABOLIC EQUATIONS AND
EXPERIMENTALLY
The simulation using the parabolic equation method (PE) was
made for the signal being transmitted by a plain wave, vertically
1676 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 49, No. 7, July 2007 DOI 10.1002/mop