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Effects of terrain features on wave propagation:
high-frequency techniques
M. Sarwar
Submitted for the Degree of Master of Science in Electrical Engineering
Department of Technology
University of Kalmar S-391 82 Kalmar SWEDEN
April 2008
M. Sarwar 2008
ii
Abstract
This Master thesis deals with wave propagation and starts with wave propagation
basics. It briefly presents the theory for the diffraction over terrain obstacles and describes
two different path loss models, the Hata model and a FFT-based model. The significance of
this paper is that it gives the simulation results for the models mentioned above and presents a
comparison between the results obtained from an empirical formula and the FFT-model. The
comparison shows that the approach based on Fast Fourier Transform is good enough for
prediction of the path loss and that it is a time efficient method.
Keywords: wave propagation, path loss, Hata model, half-screen, FFT-model.
Email: [email protected]
WWW: http://www.te.hik.se/
iii
Acknowledgments
I would like to thank my supervisor Sven-Erik Sandström, Lecturer in School of
Mathematics and Systems Engineering, Växjö University for his support throughout the
Master thesis work. I also appreciate to Magnus Nilsson and other lecturers in the Department
of Technology, University of Kalmar, where this thesis work was performed. And finally, I
am very grateful to my family and friends, who always help me whenever I need.
iv
TABLE OF CONTENTS
1. INTRODUCTION ................................................................................................................. 1
2. THEORETICAL BACKGROUND....................................................................................... 3
2.1 BASIC PROPAGATION THEORY ...................................................................... 3
2.1.1 Specular Reflection......................................................................................... 4
2.1.2 Scattering ........................................................................................................ 4
2.1.3 Diffraction....................................................................................................... 4
2.1.4 Multiple Diffraction ........................................................................................ 7
2.2 PATH LOSS MODELS.......................................................................................... 8
2.2.1 The Hata Model .............................................................................................. 8
2.2.2 The FFT based Model..................................................................................... 9
3. SIMULATION..................................................................................................................... 12
3.1 THE HATA MODEL ........................................................................................... 12
3.2 AN FFT BASED MODEL.................................................................................... 15
3.3 COMPARISON OF THE SIMULATION RESULTS FOR THE HATA MODEL
AND THE FFT BASED MODEL........................................................................ 17
4. CONCLUSION.................................................................................................................... 20
REFERENCES ............................................................................................................................. 21
APPENDIX A............................................................................................................................... 22
APPENDIX B ............................................................................................................................... 25
APPENDIX C ............................................................................................................................... 27
1
1. INTRODUCTION
The aim of this thesis work is to present the theory as well as the implementation of
different models for predicting the path loss between a base station antenna and mobile station
antennas. The prediction of path loss is a very important step in planning a mobile radio system
and accurate prediction methods are needed to determine the parameters of a radio system which
will provide efficient and reliable coverage of a specified service area [1]. In order to make these
predictions one should understand the factors that influence the signal strength. For instance, the
effects of buildings and other obstacles should be considered in an urban area. In rural places,
shadowing, scattering and absorption by trees and other vegetation can cause great path losses,
especially at higher frequencies.
One of the most common models for path loss prediction is the empirical model derived
by Hata [2]. Due to its reliability the Hata model serves as a check for the FFT-model discussed
in this work. The latter is based on a theoretical model called the half-screen model [3]. This
means that buildings are replaced by absorbing screens of vanishing thickness. The propagating
electric field is then treated by means of multiple diffraction, where reflections are not
considered.
Due to the complexity of exact prediction of the propagation of the electric field there are
some simplifications introduced in the current paper. The main limitation is that this thesis work
treats the scalar electric field and not the vector electric field.
The current thesis work is similar to [3] and [6]. The difference with [3] is that it presents
the affect of the various screen separation distances on the absolute path loss level. The
difference with [6] is that it treats the case when the base station antenna height is equal 50 m,
the screen separation distance is increased to 150 m and the simulation results are verified using
MATLAB.
The thesis first presents a brief discussion of the wave propagation and diffraction theory
in Chapter 2. It also contains two models for the propagation loss. In Chapter 3, these models are
simulated with the use of MATLAB. The simulation results are presented in the same section.
2
Chapter 4 presents some concluding remarks. The codes and scripts for simulating the models
are enclosed in appendices.
3
2. THEORETICAL BACKGROUND
2.1 BASIC PROPAGATION THEORY
The phenomena that influence radio wave propagation can generally be described by
three basic mechanisms: reflection, diffraction, and scattering. The propagation mechanisms
need to be studied for the development of propagation prediction models. The wave propagation
phenomena depend on the environment and differ whether one considers flat terrain covered
with grass, brick houses in a suburban area, or buildings in a modern city. Propagation models
are more efficient when only the most dominant phenomena are taken into account. Which radio
propagation phenomena to consider, and to what extent, depends on whether one is interested in
modelling the average signal strength, fading statistics, the delay spread or some other entity.
A number of factors complicate the investigation of propagation phenomena:
1. The distance between a base station and a mobile could range from some meters to
several kilometers.
2. Man-made objects and natural features have sizes ranging from much smaller than a
wavelength to much larger than a wavelength and this affects the propagation of radio
waves.
There are two approaches which can deal with these difficulties:
• Experimental investigations which are closer to reality but at the expense of weaker
control on the environment.
• Theoretical investigations which consider only simplified model of the reality but give
an excellent control of the environment.
The problem of designing the experiments, and the interpretation of the results, are the major
disadvantages of experimental investigations. Software simulation has one main advantage over
experimental investigations: the environment and the geometry are more easily described and
modified.
4
2.1.1 Specular Reflection
The specular reflection phenomenon is the mechanism by which a ray is reflected at an
angle equal to the incidence angle. The reflected wave fields are related to the incident wave
fields through a reflection coefficient. The most common expression for the reflection is the
Fresnel reflection coefficient which is valid for an infinite boundary between two media, for
example air and concrete. The Fresnel reflection coefficient depends on the polarization and the
wavelength of the incident wave field and on the permittivity and conductivity of each medium.
The use of the Fresnel reflection coefficient formulas is very popular in ray tracing software
tools.
In some books the reflection coefficients are considered to be constant to simplify the
computations. However the validity of a constant reflection coefficient is usually not
investigated.
Specular reflections are mainly used to model reflection from the ground surface and
from building walls.
2.1.2 Scattering
Rough surfaces and finite surfaces scatter the incident energy in all directions with a
radiation diagram that depends on the roughness and the size of the surface or volume. The
dispersion of energy through scattering entails a reduction of the energy reflected in the specular
direction. A simple method to account for the diffuse scattering is to reduce the coefficient by
multiplying with a factor smaller than one which depends exponentially on the standard
deviation of the surface roughness according to the Raleigh theory. This description does not
take into account the true dispersion of radio energy in various directions, but accounts for the
reduction of energy in the specular direction due to the diffuse components scattered in all other
directions.
2.1.3 Diffraction
Diffraction allows radio signals to propagate around the curved surface of the earth,
beyond the horizon, and to propagate behind obstructions. Although the received field strength
5
decreases rapidly as a receiver moves deeper into the obstructed (shadowed) region, the
diffraction field still exists and often has sufficient strength to produce a useful signal [4].
The phenomenon of diffraction is explained by Huygen’s principle, which states that all
points on a wavefront can be considered as point sources for the production of secondary
wavelets, and that these wavelets combine to produce a new wavefront in the direction of
propagation. Diffraction is caused by the propagation of secondary wavelets into a shadowed
region.
Consider a transmitter T and a receiver R separated by a distance 21 dd + as shown in
Figure 2.1.1 found in [1], page 35. The plane is normal to the line-of-sight path at a point
between T and R. On this plane we construct concentric circles of arbitrary radius and it is
apparent that any wave which has propagated from T to R via a point on any of these circles has
traversed a longer path TOR.
Figure 2.1.1 − Concentric circles defining the limits of the Fresnel zones at a given point on the propagation path.
The difference between the direct path and the diffracted path, called the excess path length (∆),
can be determined from the geometry of Figure 2.1.2 assuming that 21 ,ddh << and λ>>h
(where h is the height of the obstructing screen). Thus, the excess path length is,
( )
21
212
2 dd
ddh +⋅≈∆ . (2.1.1)
6
Figure 2.1.2 − The geometry of knife-edge diffraction.
The corresponding phase difference,
( )
21
21
2
2
22
dd
ddh +≈
∆=
λπ
λπ
φ . (2.1.2)
Equation (2.1.2) is often normalized using the dimensionless Fresnel-Kirchoff diffraction
parameter v which is given by,
( )
21
212
dd
ddh
λν
+= . (2.1.3)
Hence, the phase difference can be expressed as,
2
2v
πφ = . (2.1.4)
The total path loss from T to R can be computed as
(Ltot)dB = (Lfs)dB + (Lke)dB, (2.1.5)
where Lfs is the free space loss between T and R (if no obstacle had been present) and Lke is the
knife-edge diffraction loss. This loss can be obtained as a function of the diffraction parameter v,
as shown in Figure 2.1.3 found in [5], page 55.
7
Figure 2.1.3 − Diffraction loss as a function of the diffraction parameter v.
2.1.4 Multiple Diffraction
In reality, the propagation path may consist of more than one obstruction. In such cases
the total diffraction loss due to all obstacles must be computed. There are several approximate
methods available in literature [1]. Bullington suggested to replace the series of obstacles with a
single equivalent obstacle so that the path loss can be obtained using single knife-edge
diffraction model. Bullington’s method has the advantage of simplicity but important obstacles
below the paths of the horizon rays are sometimes ignored and this can cause large errors to
occur. The limitation of the Bullington method is overcome by the Epstein-Peterson method; this
method computes the attenuation due to each obstacle in turn and sums them to obtain the overall
loss. Another method is the Deygout method, in which the path loss is estimated by considering
the dominant knife-edge. The other losses, due to remaining knife-edges, are determined with
respect to the dominant edge.
Diffraction loss (dB)
30
25
15
20
10
5
0
-5
Diffraction parameter v (degrees)
-5 -4 -3 -2 -1 0 1 2 3 4 5
8
2.2 PATH LOSS MODELS
The proposed path loss prediction models are the Hata model and the FFT based model,
for which a brief theoretical description is given below.
2.2.1 The Hata Model
Path loss estimation is performed by using empirical models, if land cover is known only
roughly, so that the parameters required for semi-deterministic models cannot be determined.
Four parameters are used when estimating propagation loss with the Hata model [2]:
frequency cf , distance d, base station antenna height bh and the height of the mobile antenna mh .
In the Hata model, which is based on Okumura’s various correction functions, the basic
transmission loss in urban areas is given by,
( ) ( ) ( ) dhhahfdBL bmbcp loglog55.69.44log82.13log16.2655.69 −+−−+= . (2.3.1)
For a large sized city, the mobile antenna correction factor is given by,
( ) ( ) 97.475.11log2.32 −= mm hha . (2.3.2)
The practical range for the parameters is,
cf : 150 MHz to 1500 MHz
bh : 30 m to 200 m
mh : 1 m to 10 m
d : 1 km to 20 km
To obtain the path loss in suburban area, the standard Hata formula is modified as,
( ) ( ) ( )[ ] 4.528/log22 −−= cpps fdBLdBL . (2.3.3)
And for the path loss in open rural areas, the formula is modified as,
( ) ( ) ( ) 94.40log33.18log78.42 −+−= ccppo ffdBLdBL . (2.3.4)
9
2.2.2 The FFT based Model
As mentioned earlier the proposed model is based on the half-screen model, where the
buildings and trees are replaced by absorbing half-screens. The distance between the screens is a
parameter with which the path loss can be changed. The original terrain profile is shown in
Figure 2.3.1. The terrain replaced with the half-screens is presented in Figure 2.3.2. Figure 2.3.1
and Figure 2.3.2 are modified versions of the figures found in [6], page 195.
xi xi+1
Figure 2.3.2 – Simplified terrain profile
A wave is propagating from the base station antenna and its amplitude is set to zero
behind the screen due to absorption. Then the wave is propagating towards the next absorbing
screen with the field above the screen as a form of source. This is repeated along the entire
terrain profile.
Figure 2.3.1 – Original terrain profile
x
y (xi+1,yp)
10
The Helmholtz integral (see Appendix A) should be evaluated in order to determine the
diffracted field after an absorbing half-screen. Here, the electric field (E-field) is treated as a
complex scalar. The integration along the y-axis is fairly complicated but the Helmholtz integral
can be simplified to [6]:
( ) ∫∞
+ ⋅⋅∆
−=0
)2(
11 ),()(2
, dyyxErkHr
xikyxE ipi , (2.3.5)
where ii xxx −=∆ +1 is the distance between two neighbouring absorbing screens,
)2(
1H - the first order Hankel function of the second kind
k - the wave number
22 )()( yyxr p −+∆= - the distance between the points ),( yxi and ),( 1 pi yx +
When the scalar E-filed ),( yxE i is known, on and above the preceding half-screen, then with
(2.3.5) the scalar E-field on and above the following (next) screen, ),( 1 pi yxE + , can be
determined for any given position py .
To determine the scalar E-field above the first half-screen one can assume a line source,
)exp(1
ikrr
−⋅ . Because the calculated result is the field from the line source, then the final field
has to be transformed in order to get the corresponding point source field. An approximate
method is to multiply the calculated field with x
1, where x is the distance between the base
station antenna and the mobile antenna. This transformation is only performed at those locations
where one wants to determine the final path loss [6].
If we define a function W such that,
( ) )(2
)2(
1 rkHr
xikyyW p ⋅
∆−=− . (2.3.6)
then expression (2.3.5) can be written in the following way
( ) ∫∞
+ ⋅−=0
1 ),()(, dyyxEyyWyxE ippi , (2.3.7)
or
( ) ),()(,1 yxEyWyxE ipi ∗=+ . (2.3.8)
11
Thus, (2.3.7) can be interpreted as a convolution. This means that in order to solve the
integral one can determine the Fourier transform of the scalar E-field with respect to y and
multiply with the corresponding transformed W-propagator in the yk -space:
( ) ),(ˆ)(ˆ,ˆ1 yiyyi kxEkWkxE ⋅=+ , (2.3.9)
where yk is the y-component of λπ2
=k .
In order to determine W consider the wave equation in two dimensions:
02
2
2
2
2
=+∂
∂+
∂
∂Ek
x
E
y
E, (2.3.10)
with ),( yxE . Fourier transform of (2.3.10) with respect to y yields
0ˆ)(ˆ
22
2
2
=−+∂
∂Ekk
x
Ey , (2.3.11)
with ),(ˆˆykxEE = .
The solution for the one-dimensional Helmholtz equation (2.3.11) is:
)(22
)(),(ˆ iy xxkki
yy ekCkxE−−−
= . (2.3.12)
For 22kk y > the branch cut in equation (2.3.12) can be defined as follows,
)()()()( 2222222
)()()(),(ˆ iyiyiy xxkk
y
xxkki
y
xxkki
yy ekCekCekCkxE−−−−−−−−−
=== . (2.3.13)
The constant C is defined when ixx =
),(ˆ)( yiy kxEkC = , (2.3.14)
and the solution for (2.3.11) can be written as
)(22
),(ˆ),(ˆ iy xxkki
yiy ekxEkxE−−−
= . (2.3.15)
The scalar E-field at 1+ix can be determined by calculating the inverse Fourier transform of the
E -field. A Fast Fourier Transform (FFT) can be used in order to perform the Discrete Fourier
Transform (DFT). Using FFT instead of solving the integral directly reduces the computation
time considerably.
12
3. SIMULATION
3.1 THE HATA MODEL
The path loss in an urban area is calculated with the use of the Hata model for the base
station antenna heights H = 30m, 50m, 70m and the result is shown in the figure below. The
carrier frequency in this case is equal to 900 MHZ and the mobile antenna height is 1.5 m.
100
101
120
125
130
135
140
145
150
155
160
165
Distance [km]
Path loss [dB]
Figure 3.1.1 –The path loss in an urban area according to the Hata model.
Figure 3.1.2 shows the path loss in a suburban area at the same frequency as above and
for the same heights of base antenna and mobile antenna.
30m
50m
70m
13
100
101
110
115
120
125
130
135
140
145
150
155
Distance [km]
Path loss [dB]
Figure 3.1.2 - The path loss in a suburban area according to the Hata model.
In an open rural area one finds that the loss is considerably smaller, see Figure 3.1.3. The
conditions are the same as for the previous cases.
30m
50m
70m
14
100
101
-20
-15
-10
-5
0
5
10
15
20
25
Distance [km]
Path loss [dB]
Figure 3.1.3 - The path loss in an open rural area according to the Hata model.
If the frequency is varied instead one obtains the results shown in Figure 3.1.4. Here, the
base station antenna height is chosen to be 70m and the mobile antenna height is 1.5 m. The
MATLAB code for the Hata model simulation is presented in Appendix B.
30m
50m
70m
15
100
101
110
115
120
125
130
135
140
145
150
155
160
Distance [km]
Path loss [dB]
Figure 3.1.4 - The path loss in an urban area for the frequencies
450 MHz, 900 MHz, 1500MHz.
3.2 AN FFT BASED MODEL
For an FFT based model the base station antenna height is chosen to be H=50 m, the
mobile antenna height h=1.5 m and the height of the buildings G=15 m (the height of the
screens). The definitions of the model parameters are given in Figure 3.2.1, which is a modified
version of the figure found in [6], page 196.
The main parameters in the model are the distance between the screens w and the
distance w′ between the mobile antenna and the nearest screen. A reduction of the distance w
leads to an increase in the resulting path loss. The same is valid for the parameter w′.
1500 MHz
900 MHz
450 MHz
16
Figure 3.2.1 – Notation for the model parameters.
In the FFT simulation the following data was used:
- the carrier frequency 900=cf MHz
- the spatial resolution in the vertical y-direction 3
λ=∆y .
The path loss versus the distance between the transmitter and the receiver ( xS ∆⋅ ) for the FFT-
model with 100 screens is shown in Figure 3.2.2. The distance between screens is equal to 100m
and the total distance for the profile is then 10 km. The distance between the mobile antenna and
the nearest screen is w′=15 m.
103
104
115
120
125
130
135
140
145
150
155
160
Distance [m]
Path loss [dB]
Figure 3.2.2 – Path loss for the FFT-model.
17
For verification it is useful to compare the FFT-model to an empirical propagation model,
for instance the Hata model. The comparison is presented in the next subsection. The code for
the FFT-model is developed in MATLAB and given in Appendix C.
3.3 COMPARISON OF THE SIMULATION RESULTS FOR THE HATA MODEL
AND THE FFT BASED MODEL
The figures below present a comparison between the propagation loss computed by
means of the Hata model and the FFT-model, respectively. The propagation loss is plotted
versus the distance between base antenna and mobile antenna ( xS ∆⋅ ). In Figure 3.3.1 the
distance between screens is 100=∆x m and the distance between the mobile antenna and the
nearest screen w′=15 m. The carrier frequency is 900 MHz. The base station antenna height, the
mobile antenna height and the height of the buildings are the same as in a previous experiment,
i.e. H=50 m, h=1.5 m and G=15 m.
103
104
100
110
120
130
140
150
160
170
Distance [m]
Path loss [dB]
Figure 3.3.1 – Path loss according to the Hata model (dashed line)
and the FFT-model (solid line) with 100=∆x m.
18
The oscillations of the FFT result as well as the absolute level of the path loss depend on
the distance between screens. Thus, reducing x∆ down to 50 leads to more oscillations and to a
higher path loss, as shown in Figure 3.3.2.
103
104
100
110
120
130
140
150
160
170
Distance [m]
Path loss [dB]
Figure 3.3.2 – Path loss according to the Hata model (dashed line)
and the FFT-model (solid line) with 50=∆x m.
Increasing x∆ up to 125 m produces an FFT result with almost no oscillations and a lower path
loss, as shown in Figure 3.3.3.
19
103
104
100
110
120
130
140
150
160
170
Distance [m]
Path loss [dB]
Figure 3.3.3 – Path loss according to the Hata model (dashed line)
and the FFT-model (solid line) with 125=∆x m.
From the figures above it is obvious that the propagation loss slope of the FFT-model is
nearly the same as the slope of the Hata model, which means that the FFT-model can be used to
predict the propagation loss due to multiple diffraction.
20
4. CONCLUSION
The implementation of the FFT-model with multiple screens gives reasonable agreement
with the Hata model. This is verified numerically using MATLAB. The FFT method is
numerically efficient and also applicable to the cases where the Hata model is not valid, such as
equal antenna and building heights.
After performing different experiments, one finds that the shorter the distance between the
screens is, the higher the computed path loss becomes. Moreover, the ripple that appears in the
numerical results is reduced when the distance between screens is increased.
Consequently, wave propagation along a terrain profile can be treated as multiple
diffraction from absorbing half-screens and can be predicted with fair accuracy by means of the
FFT half-screen model.
A problem with the method is the heuristic choice of the model parameters and that
numerical convergence in the conventional sense cannot be separated from the actual modelling.
21
REFERENCES
[1] J.D. Parsons, The Mobile Radio Propagation Channel, Second Edition. John Wiley &
Sons Ltd, 2000.
[2] M. Hata, “Empirical Formula for Propagation Loss in Land Mobile Radio Services”,
IEEE VT, 29, p. 317-325, 1980.
[3] H. Holmquist, A DFT Half-Screen Propagation Model for Macrocellular Environments,
Master’s thesis, The Royal Institute of Technology, Sweden, 1992.
[4] T.S. Rappaport, Wireless Communications. Principles and Practice, Second Edition.
Prentice Hall PTR, 2002.
[5] L. Ahlin, J. Zander and B. Slimane, Principles of Wireless Communications, Third
Edition. Studentlitteratur, 2006.
[6] J-E. Berg and H. Holmquist, “An FFT Multiple Half-Screen Diffraction Model”, IEEE
VT, 1, p. 195-199, 1994.
22
APPENDIX A
The Helmholtz Integral
The wave equation:
022 =+∇ EkE . (A.1)
The Helmholtz integral, which gives the solution for the wave equation is shown below
∫
∂∂
−∂∂
=−−
S
ikrikr
dSn
E
r
e
r
e
nEPE
π41
)( . (A.2)
For the derivation of the Helmholtz integral (A.2) from Green’s theorem [3] ψ has been chosen
equal to r
e ikr−
, where r is the distance from a fixed point P to a variable point P′, both inside the
volume ν . However, the same result could have been obtained if one would choose
ξψ +=−
r
e ikr
, (A.3)
where ξ has no singularities inside and on surface S. Hence, by use of this extended ψ function,
(A.2) can be formulated as
∫
∂∂
+−
+
∂∂
=−−
S
ikrikr
dSn
E
r
e
r
e
nEPE ξξ
π41
)( , (A.4)
where 022 =+∇ ξξ k inside and on S. Suppose, that there is a function 1ξ such that
01 =+−
ξr
e ikr
, (A.5)
in every point on S. Then (A.4) can be written as
∫
+
∂∂
=−
S
ikr
dSr
e
nEPE 1
4
1)( ξ
π. (A.6)
This is an important result as E(P) can be computed by knowing E on S only. Below 1ξ is
determined for a plane surface.
23
The Rayleigh Integral
Choose for closed surface S the plane Sp, equal to 0=x , and a hemi-sphere Shsp with radius R in
the upper half space ( 0>x ). For this special choice of S expression (A.6) may be written as
∫∫
+
∂∂
+
+
∂∂
=−−
hspp S
ikr
S
ikr
dSr
e
nEdS
r
e
nEPE 11
4
1
4
1)( ξ
πξ
π. (A.7)
Assume that the electric field E in the upper half space ( 0>x ) is generated by causal sources in
the lower half space ( 0<x ), and that one wishes to compute E(P), the electric field E in a point
),,( ppp xzyP = corresponding to ),,( xxzzyyr ppp −−−= in the upper half space. Then, for a
finite time interval, say max0 Tt ≤≤ , R can always be chosen such that the contribution from Shsp
has not yet reached point P for times smaller than maxT . Hence, for a given maxT one can define
the radius max0 cTR = such that one may take
04
11 =
+
∂∂
∫−
hspS
ikr
dSr
e
nE ξ
π. (A.8)
for any 0RR > . Consequently, if one want to know E(P) for maxTt < , expression (A.7) may be
replaced by
∫
+
∂∂
=−
pS
ikr
dSr
e
nEPE 1
4
1)( ξ
π for maxcTR > . (A.9)
The function 1ξ has to be determined such that
01
2
1
2 =+∇ ξξ k inside and on Sp, (A.10)
and
01 =+−
ξr
e ikr
on Sp. (A.11)
If r
e rik
′=
′−
1ξ is chosen with ),,( zzyyxxr ppp −−−−=′ then 1ξ satisfies (A.10). Keeping in
mind that rr =′ on Sp, it can be seen directly that (A.11) is satisfied as well. Consequently, (A.9)
can be rewritten as
dSr
e
r
e
nEPE
pS
rikikr
∫
′
−∂∂
=′−−
π41
)( ,
24
or, using n
r
rn ∂∂
∂∂
=∂∂
and n
r
rn ∂
′∂′∂
∂=
∂∂
:
∫ −
∂
′∂−
∂∂+
−=pS
ikrdSen
r
n
r
r
ikrEPE
2
1
4
1)(
π. (A.12)
From the definitions of r and r ′
222 )()()( zzyyxxr ppp −+−+−= ,
222 )()()( zzyyxxr ppp −+−+−−=′ .
The gradient of these expressions
φcos−=⋅−=⋅∇≡∂∂
nenrn
rr , (A.13)
φ ′−=⋅−=⋅′∇≡∂
′∂′ cosnenr
n
rr , (A.14)
where re is the unit vector in the r direction and rx p /cos =φ . However, the definition of r and
r ′ states that φπφ −=′ , which implies φcos=∂
′∂n
r. Finally, using the derived expressions in
(A.12) the following is obtained
∫ −+=
pS
ikrdSer
ikrEPE φ
πcos
1
2
1)(
2. (A.15)
(A.15) is called the Rayleigh integral. It states that any electric field may be synthesized by a
dipole distribution on the plane pS .
For the two-dimensional version of the Rayleigh integral E is independent of z. Then for 0=x
(A.15) can be rewritten as
∫ ∫
+== −
y zl l
ikr dydzer
ikryxEPE φ
πcos
1),0(
2
1)(
2, (A.16)
or using the first order Hankel function of the second kind,
∫ =−=yl
dykrHyxEik
PE )(cos),0(2
)( )2(
1φ , (A.17)
where 22 )( yyxr pp −+= .
25
APPENDIX B
MATLAB CODE – HATA MODEL
Code #1
f_c=900;f_c=900;f_c=900;f_c=900; h_re=1.5;h_re=1.5;h_re=1.5;h_re=1.5; a_re=3.2*(a_re=3.2*(a_re=3.2*(a_re=3.2*(log10(11.75*h_re))^2log10(11.75*h_re))^2log10(11.75*h_re))^2log10(11.75*h_re))^2----4.97;4.97;4.97;4.97; d=1:1:10; d=1:1:10; d=1:1:10; d=1:1:10; % distances in km% distances in km% distances in km% distances in km h_te=30;h_te=30;h_te=30;h_te=30; Lu_30=69.55+26.16*log10(f_c)Lu_30=69.55+26.16*log10(f_c)Lu_30=69.55+26.16*log10(f_c)Lu_30=69.55+26.16*log10(f_c)----13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d); Lsu_30=Lu_30Lsu_30=Lu_30Lsu_30=Lu_30Lsu_30=Lu_30----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_30=Lu_30Lr_30=Lu_30Lr_30=Lu_30Lr_30=Lu_30----4.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; h_te=50;h_te=50;h_te=50;h_te=50; Lu_Lu_Lu_Lu_50=69.55+26.16*log10(f_c)50=69.55+26.16*log10(f_c)50=69.55+26.16*log10(f_c)50=69.55+26.16*log10(f_c)----13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d); Lsu_50=Lu_50Lsu_50=Lu_50Lsu_50=Lu_50Lsu_50=Lu_50----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_50=Lu_50Lr_50=Lu_50Lr_50=Lu_50Lr_50=Lu_50----4.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; h_te=70;h_te=70;h_te=70;h_te=70; Lu_70=69.55+26.16*log10(f_c)Lu_70=69.55+26.16*log10(f_c)Lu_70=69.55+26.16*log10(f_c)Lu_70=69.55+26.16*log10(f_c)----13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h6.55*log10(h6.55*log10(h6.55*log10(h_te))*log10(d);_te))*log10(d);_te))*log10(d);_te))*log10(d); Lsu_70=Lu_70Lsu_70=Lu_70Lsu_70=Lu_70Lsu_70=Lu_70----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_70=Lu_70Lr_70=Lu_70Lr_70=Lu_70Lr_70=Lu_70----4.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; % For urban area% For urban area% For urban area% For urban area figure(1);figure(1);figure(1);figure(1); semilogx(d,Lu_30);semilogx(d,Lu_30);semilogx(d,Lu_30);semilogx(d,Lu_30); hold onhold onhold onhold on semilogx(d,Lu_50,'semilogx(d,Lu_50,'semilogx(d,Lu_50,'semilogx(d,Lu_50,'----or');or');or');or'); hold onhold onhold onhold on semilogx(d,Lu_70,'gsemilogx(d,Lu_70,'gsemilogx(d,Lu_70,'gsemilogx(d,Lu_70,'g----*');*');*');*'); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel(''''Path loss'Path loss'Path loss'Path loss');););); grid on;grid on;grid on;grid on; % For suburban area% For suburban area% For suburban area% For suburban area figure(2);figure(2);figure(2);figure(2); semilogx(d,Lsu_30);semilogx(d,Lsu_30);semilogx(d,Lsu_30);semilogx(d,Lsu_30); hold onhold onhold onhold on semilogx(d,Lsu_50,semilogx(d,Lsu_50,semilogx(d,Lsu_50,semilogx(d,Lsu_50,''''----or'or'or'or');););); hold onhold onhold onhold on semilogx(d,Lsu_70,semilogx(d,Lsu_70,semilogx(d,Lsu_70,semilogx(d,Lsu_70,'g'g'g'g----*'*'*'*');););); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel('Path loss''Path loss''Path loss''Path loss');););); grid on;grid on;grid on;grid on; % For open rural area% For open rural area% For open rural area% For open rural area figure(3);figure(3);figure(3);figure(3); semilogx(d,Lr_30);semilogx(d,Lr_30);semilogx(d,Lr_30);semilogx(d,Lr_30); hold onhold onhold onhold on semilogx(d,Lr_50,semilogx(d,Lr_50,semilogx(d,Lr_50,semilogx(d,Lr_50,''''----or'or'or'or');););); hold onhold onhold onhold on semilogx(d,Lr_70,semilogx(d,Lr_70,semilogx(d,Lr_70,semilogx(d,Lr_70,'g'g'g'g----*'*'*'*');););); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel('Path loss''Path loss''Path loss''Path loss');););); grid on;grid on;grid on;grid on;
26
Code #2
h_te=70;h_te=70;h_te=70;h_te=70; h_re=1.5;h_re=1.5;h_re=1.5;h_re=1.5; a_re=3.2*(log10(11.75*h_re))^2a_re=3.2*(log10(11.75*h_re))^2a_re=3.2*(log10(11.75*h_re))^2a_re=3.2*(log10(11.75*h_re))^2----4.97;4.97;4.97;4.97; d=1:1:10; d=1:1:10; d=1:1:10; d=1:1:10; % distances in km% distances in km% distances in km% distances in km f_c=450;f_c=450;f_c=450;f_c=450; Lu_450=69.55+26.16*log10(f_c)Lu_450=69.55+26.16*log10(f_c)Lu_450=69.55+26.16*log10(f_c)Lu_450=69.55+26.16*log10(f_c)----13.82*l13.82*l13.82*l13.82*log10(h_te)og10(h_te)og10(h_te)og10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d); Lsu_450=Lu_450Lsu_450=Lu_450Lsu_450=Lu_450Lsu_450=Lu_450----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_450=Lu_450Lr_450=Lu_450Lr_450=Lu_450Lr_450=Lu_450----4.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; f_c=900;f_c=900;f_c=900;f_c=900; Lu_900=69.55+26.16*log10(f_c)Lu_900=69.55+26.16*log10(f_c)Lu_900=69.55+26.16*log10(f_c)Lu_900=69.55+26.16*log10(f_c)----13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d); Lsu_900=Lu_9Lsu_900=Lu_9Lsu_900=Lu_9Lsu_900=Lu_900000000----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_900=Lu_900Lr_900=Lu_900Lr_900=Lu_900Lr_900=Lu_900----4.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^24.78*(log10(f_c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; f_c=1500;f_c=1500;f_c=1500;f_c=1500; Lu_1500=69.55+26.16*log10(f_c)Lu_1500=69.55+26.16*log10(f_c)Lu_1500=69.55+26.16*log10(f_c)Lu_1500=69.55+26.16*log10(f_c)----13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)13.82*log10(h_te)----a_re+(44.9a_re+(44.9a_re+(44.9a_re+(44.9----6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d);6.55*log10(h_te))*log10(d); Lsu_1500=Lu_1500Lsu_1500=Lu_1500Lsu_1500=Lu_1500Lsu_1500=Lu_1500----2*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^22*(log10(f_c/28))^2----5.4;5.4;5.4;5.4; Lr_1500=Lu_1500Lr_1500=Lu_1500Lr_1500=Lu_1500Lr_1500=Lu_1500----4.78*(log10(f_4.78*(log10(f_4.78*(log10(f_4.78*(log10(f_c))^2c))^2c))^2c))^2----18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)18.33*log10(f_c)----40.98;40.98;40.98;40.98; % For urban area% For urban area% For urban area% For urban area figure(1);figure(1);figure(1);figure(1); semilogx(d,Lu_450);semilogx(d,Lu_450);semilogx(d,Lu_450);semilogx(d,Lu_450); hold onhold onhold onhold on semilogx(d,Lu_900,semilogx(d,Lu_900,semilogx(d,Lu_900,semilogx(d,Lu_900,''''----or'or'or'or');););); hold onhold onhold onhold on semilogx(d,Lu_1500,semilogx(d,Lu_1500,semilogx(d,Lu_1500,semilogx(d,Lu_1500,'g'g'g'g----*'*'*'*');););); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel('Path loss [dB]''Path loss [dB]''Path loss [dB]''Path loss [dB]');););); grid on;grid on;grid on;grid on; % For suburban area% For suburban area% For suburban area% For suburban area figure(2);figure(2);figure(2);figure(2); semilogx(d,Lsu_450semilogx(d,Lsu_450semilogx(d,Lsu_450semilogx(d,Lsu_450);););); hold onhold onhold onhold on semilogx(d,Lsu_900,semilogx(d,Lsu_900,semilogx(d,Lsu_900,semilogx(d,Lsu_900,''''----or'or'or'or');););); hold onhold onhold onhold on semilogx(d,Lsu_1500,semilogx(d,Lsu_1500,semilogx(d,Lsu_1500,semilogx(d,Lsu_1500,'g'g'g'g----*'*'*'*');););); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel('Path loss [dB]''Path loss [dB]''Path loss [dB]''Path loss [dB]');););); grid on;grid on;grid on;grid on; % For open rural area% For open rural area% For open rural area% For open rural area figure(3);figure(3);figure(3);figure(3); semilogx(d,Lr_450);semilogx(d,Lr_450);semilogx(d,Lr_450);semilogx(d,Lr_450); hold onhold onhold onhold on semilogx(d,Lr_900,semilogx(d,Lr_900,semilogx(d,Lr_900,semilogx(d,Lr_900,''''----or'or'or'or');););); hold onhold onhold onhold on semilogx(d,Lr_1500,semilogx(d,Lr_1500,semilogx(d,Lr_1500,semilogx(d,Lr_1500,'g'g'g'g----*'*'*'*');););); xlabel(xlabel(xlabel(xlabel('Distance [km]''Distance [km]''Distance [km]''Distance [km]');););); ylabel(ylabel(ylabel(ylabel('Path loss [dB]''Path loss [dB]''Path loss [dB]''Path loss [dB]');););); grid ongrid ongrid ongrid on
27
APPENDIX C
MATLAB CODE – FFT BASED MODEL
clear all;clear all;clear all;clear all; lambda=0.33;lambda=0.33;lambda=0.33;lambda=0.33; delta_x=100;delta_x=100;delta_x=100;delta_x=100; H=50;H=50;H=50;H=50; G=15;G=15;G=15;G=15; max_p=2022;max_p=2022;max_p=2022;max_p=2022; delta_yr=lambda/3;delta_yr=lambda/3;delta_yr=lambda/3;delta_yr=lambda/3; pppp=37;=37;=37;=37; y_p=0:max_py_p=0:max_py_p=0:max_py_p=0:max_p;;;; Efield=sqrt(delta_x^2+(y_pEfield=sqrt(delta_x^2+(y_pEfield=sqrt(delta_x^2+(y_pEfield=sqrt(delta_x^2+(y_p----H).^2*delta_yr^2);H).^2*delta_yr^2);H).^2*delta_yr^2);H).^2*delta_yr^2); Input_Efield=exp(i*2*pi/lInput_Efield=exp(i*2*pi/lInput_Efield=exp(i*2*pi/lInput_Efield=exp(i*2*pi/lambda*Efield)./Efield;ambda*Efield)./Efield;ambda*Efield)./Efield;ambda*Efield)./Efield; for y_p = 1:for y_p = 1:for y_p = 1:for y_p = 1:max_pmax_pmax_pmax_p+1,+1,+1,+1, if (y_p if (y_p if (y_p if (y_p----1) < G 1) < G 1) < G 1) < G New_Efield(y_p)=0; New_Efield(y_p)=0; New_Efield(y_p)=0; New_Efield(y_p)=0; elseif (y_p elseif (y_p elseif (y_p elseif (y_p----1) > G1) > G1) > G1) > G New_Efield(y_p)=Input_Efield(y_p); New_Efield(y_p)=Input_Efield(y_p); New_Efield(y_p)=Input_Efield(y_p); New_Efield(y_p)=Input_Efield(y_p); else else else else New_Efield(y_p)=Input_Efield(y_p)*0.5; New_Efield(y_p)=Input_Efield(y_p)*0.5; New_Efield(y_p)=Input_Efield(y_p)*0.5; New_Efield(y_p)=Input_Efield(y_p)*0.5; end end end end end end end end for y_p = 1:for y_p = 1:for y_p = 1:for y_p = 1:mmmmax_pax_pax_pax_p+1,+1,+1,+1, if (y_p if (y_p if (y_p if (y_p----1) > (1) > (1) > (1) > (mmmmax_pax_pax_pax_p)/2; )/2; )/2; )/2; temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p----mmmmax_pax_pax_pax_p*0.5*0.5*0.5*0.5----1)*pi/(1)*pi/(1)*pi/(1)*pi/(max_pmax_pmax_pmax_p*0.5))+1)/2; *0.5))+1)/2; *0.5))+1)/2; *0.5))+1)/2; New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); end end end end end end end end CFFT_New_Efield=fft(New_Efield)/(CFFT_New_Efield=fft(New_Efield)/(CFFT_New_Efield=fft(New_Efield)/(CFFT_New_Efield=fft(New_Efield)/(mmmmax_pax_pax_pax_p+1); +1); +1); +1); for y_p = 1:for y_p = 1:for y_p = 1:for y_p = 1:mmmmax_pax_pax_pax_p+1,+1,+1,+1, k_y1(y k_y1(y k_y1(y k_y1(y_p)=(y_p_p)=(y_p_p)=(y_p_p)=(y_p----1)/((1)/((1)/((1)/((mmmmax_pax_pax_pax_p+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi; k_y2(y_p)=((y_p k_y2(y_p)=((y_p k_y2(y_p)=((y_p k_y2(y_p)=((y_p----1)1)1)1)----((((mmmmax_pax_pax_pax_p+1))/((+1))/((+1))/((+1))/((max_pmax_pmax_pmax_p+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi; if (y_pif (y_pif (y_pif (y_p----1) < (1) < (1) < (1) < (max_pmax_pmax_pmax_p)/2; )/2; )/2; )/2; if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----i*sqrt((2*pi/lami*sqrt((2*pi/lami*sqrt((2*pi/lami*sqrt((2*pi/lambda)^2bda)^2bda)^2bda)^2----k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x); else else else else CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2----(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x); end end end end else else else else if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), CFFT_New_Efield(y_p)=CFFT_New_Efiel CFFT_New_Efield(y_p)=CFFT_New_Efiel CFFT_New_Efield(y_p)=CFFT_New_Efiel CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(d(y_p)*exp(d(y_p)*exp(d(y_p)*exp(----i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2----k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x); else else else else CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2----(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x); end end end end end end end end end end end end ICFFT_New_Efield=ifft(CFFT_New_Efield)*(ICFFT_New_Efield=ifft(CFFT_New_Efield)*(ICFFT_New_Efield=ifft(CFFT_New_Efield)*(ICFFT_New_Efield=ifft(CFFT_New_Efield)*(mmmmax_pax_pax_pax_p+1); +1); +1); +1); for sfor sfor sfor ss=1:100,s=1:100,s=1:100,s=1:100, TotFFTTotFFTTotFFTTotFFT(:,ss)=ICFFT_New_Efield;(:,ss)=ICFFT_New_Efield;(:,ss)=ICFFT_New_Efield;(:,ss)=ICFFT_New_Efield; for y_p = 1:for y_p = 1:for y_p = 1:for y_p = 1:max_pmax_pmax_pmax_p+1,+1,+1,+1, if (y_p if (y_p if (y_p if (y_p----1) < G 1) < G 1) < G 1) < G New_Efield(y_p)=0; New_Efield(y_p)=0; New_Efield(y_p)=0; New_Efield(y_p)=0; elseif (y_p elseif (y_p elseif (y_p elseif (y_p----1) > G1) > G1) > G1) > G New_Efield(y_p)=ICFFT_New_Efield(y_p); New_Efield(y_p)=ICFFT_New_Efield(y_p); New_Efield(y_p)=ICFFT_New_Efield(y_p); New_Efield(y_p)=ICFFT_New_Efield(y_p); else else else else New_Efield(y_p)=I New_Efield(y_p)=I New_Efield(y_p)=I New_Efield(y_p)=ICFFT_New_Efield(y_p)*0.5;CFFT_New_Efield(y_p)*0.5;CFFT_New_Efield(y_p)*0.5;CFFT_New_Efield(y_p)*0.5; end end end end end end end end for y_p = 1: for y_p = 1: for y_p = 1: for y_p = 1:mmmmax_pax_pax_pax_p+1,+1,+1,+1, if (y_p if (y_p if (y_p if (y_p----1) > (1) > (1) > (1) > (mmmmax_pax_pax_pax_p)/2; )/2; )/2; )/2; temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p temp3(y_p)=(cos((y_p----mmmmax_pax_pax_pax_p*0.5*0.5*0.5*0.5----1)*pi/(1)*pi/(1)*pi/(1)*pi/(max_pmax_pmax_pmax_p*0.5))+1)/2; *0.5))+1)/2; *0.5))+1)/2; *0.5))+1)/2; New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); New_Efield(y_p)=New_Efield(y_p)*temp3(y_p); end end end end
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end end end end CFFT_New_Efield=fft(New_Efield)/( CFFT_New_Efield=fft(New_Efield)/( CFFT_New_Efield=fft(New_Efield)/( CFFT_New_Efield=fft(New_Efield)/(mmmmax_pax_pax_pax_p+1); +1); +1); +1); for y_p = 1: for y_p = 1: for y_p = 1: for y_p = 1:mmmmax_pax_pax_pax_p+1,+1,+1,+1, k_y1(y_p)=(y_p k_y1(y_p)=(y_p k_y1(y_p)=(y_p k_y1(y_p)=(y_p----1)/((1)/((1)/((1)/((mmmmax_pax_pax_pax_p+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi; k_y2(y_p)=((y_p k_y2(y_p)=((y_p k_y2(y_p)=((y_p k_y2(y_p)=((y_p----1)1)1)1)----((((mmmmax_pax_pax_pax_p+1))/((+1))/((+1))/((+1))/((max_pmax_pmax_pmax_p+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi;+1)*delta_yr)*2*pi; if (y_pif (y_pif (y_pif (y_p----1) < (1) < (1) < (1) < (max_pmax_pmax_pmax_p)/2; )/2; )/2; )/2; if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), if (k_y1(y_p)^2<(2*pi/lambda)^2), CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2----k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x);k_y1(y_p)^2)*delta_x); else else else else CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2sqrt(k_y1(y_p)^2----(2*pi/lambda)(2*pi/lambda)(2*pi/lambda)(2*pi/lambda)^2)*delta_x);^2)*delta_x);^2)*delta_x);^2)*delta_x); end end end end else else else else if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), if (k_y2(y_p)^2<(2*pi/lambda)^2), CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp( CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(----i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2i*sqrt((2*pi/lambda)^2----k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x);k_y2(y_p)^2)*delta_x); else else else else CFFT_New_Efield(y_p)=CFFT CFFT_New_Efield(y_p)=CFFT CFFT_New_Efield(y_p)=CFFT CFFT_New_Efield(y_p)=CFFT_New_Efield(y_p)*exp(_New_Efield(y_p)*exp(_New_Efield(y_p)*exp(_New_Efield(y_p)*exp(----sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2sqrt(k_y2(y_p)^2----(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x);(2*pi/lambda)^2)*delta_x); end end end end end end end end end end end end ICFFT_New_Efield=ifft(CFFT_New_Efield)*( ICFFT_New_Efield=ifft(CFFT_New_Efield)*( ICFFT_New_Efield=ifft(CFFT_New_Efield)*( ICFFT_New_Efield=ifft(CFFT_New_Efield)*(mmmmax_pax_pax_pax_p+1);+1);+1);+1); end end end end fofofofor S=1:100,r S=1:100,r S=1:100,r S=1:100, tmp2(S)= tmp2(S)= tmp2(S)= tmp2(S)=----20*log10(abs(20*log10(abs(20*log10(abs(20*log10(abs(TotFFTTotFFTTotFFTTotFFT(51,S))/sqrt(S*delta_x))+(51,S))/sqrt(S*delta_x))+(51,S))/sqrt(S*delta_x))+(51,S))/sqrt(S*delta_x))+pppp;;;; endendendend x=[10:100]*delta_x; x=[10:100]*delta_x; x=[10:100]*delta_x; x=[10:100]*delta_x; AXIS([1000 10000 100 170])AXIS([1000 10000 100 170])AXIS([1000 10000 100 170])AXIS([1000 10000 100 170]) semilogx(x,tmp2(10:100)) semilogx(x,tmp2(10:100)) semilogx(x,tmp2(10:100)) semilogx(x,tmp2(10:100)) grid on;grid on;grid on;grid on; xlabel(xlabel(xlabel(xlabel('Distance [m]''Distance [m]''Distance [m]''Distance [m]');););); ylabel(ylabel(ylabel(ylabel('Path loss [dB]''Path loss [dB]''Path loss [dB]''Path loss [dB]'););););
