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Modeling and performance of microwave radio links in rain

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McGill University
Information Networks and Systems Laboratory
M odeling and performance of
microwave radio links in rain
Joe Nader
Department of Electrical Engineering,
McGill University
Montreal, Canada
August, 1998
A thesis submitted to the Faculty o f Graduate Studies and Research in partial ful­
fillment of the requirements of the degree of Master o f Engineering.
Copyright © Joe Nader, 1998
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ABSTRACT
Microwave radio links, operating in the millimeter wave region, must account for
the effects o f rain when considering transmission loss. In this work, a theoretical
model is used to generate the specific attenuations based on perturbation theory
with spheroidal or Pruppacher-Pitter raindrop shapes, and Marshal 1-Palmer or
Weibull drop size distributions. The specific attenuation is fitted to the power law
relation with rain rate and the parameters are used in a two-component rain rate
model in order to estimate the attenuation along the path.
The theoretical model is simulated and compared to the ITU and Crane prediction
methods. Both moderate and tropical climates are considered. A simple line-ofsight radio system is then simulated and evaluated by incorporating the rain attenu­
ation in the channel. Finally, three basic network blocks are discussed and ana­
lyzed for links affected by rain.
• • •
ill
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SOMMAIKE
Les liaisons radio, fonctionnant dans les longueurs d’ondes millimetriques, doivent
inclure les effets de la pluie en considerant les pertes de signal dues a la transmis­
sion. Base sur la methode de perturbation, un modele theorique est employe pour
produire 1’attenuation
specifique, utilisant des gouttes de pluie de forme
spheroi'dale ou Pruppacher-Pitter et une distribution de grandeur de goutte selon
Marshall-Palmer ou Weibull. Afin d ’estimer 1’attenuation sur le parcours, 1’attenu­
ation specifique est repesentee avec une relation exponentielle incluant Ie taux de
pluie, et sont, par ia suite, utilises dans un modele de taux de pluie a deux composantes.
Le modele theorique est simule et compare aux methodes de prediction de rUTT et
de Crane. Des climats moderns et tropicaux sont consideres. En incorporant
1’attenuation due a la pluie dans la transmission, un systeme de communication
terrestre simple est alors simule et evalue. Enfin, trois blocs elementaires de reseau
sont discutes et analyses pour les liaisons radio affectees par la pluie.
v
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ACKNOWLEDGMENTS
I would like to express my deepest gratitude to my supervisor Dr. Salvatore D.
Morgera for his guidance throughout my graduate studies at M cGill University.
I would like to thank Harris Corporation, Farinon division who financially sup­
ported the research.
I would like to thank Carolina Dieguez for her contribution to the world rain region
figures and her patience.
Finally, I would like to thank my parents for their continuous support and encour­
agement.
• •
V ll
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CONTENTS
1
ABSTRACT
iii
SOMMAIRE
v
ACKNOWLEDGMENTS
vii
CONTENTS
ix
LIST OF FIGURES
xiii
LIST OF TABLES
xvii
INTRODUCTION
1
1.1
1.2
1.3
2
Overview of Attenuation Prediction
Motivation o f our Research
Organization of Thesis
PREDICTION OF RELIABILITY
2.1
2.2
2.3
Rain-Rate Climate Models
2.1.1 ITU-R837 Rainfall Rate Statistics
2.1.2 Crane Rain Rate Climate Zone
Specific Attenuation
2.2.1 ITU-R838 Specific Attenuation Model for Rain
Path Attenuation Formulas
2
2
4
5
7
7
9
11
11
13
ix
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X
CONTENTS
2.4
2.3.1 ITU-R530 Prediction Model
2.3.2 Crane Prediction Model
System Performance
3 MODELING ATTENUATION AT A POINT
3.1
19
21
24
24
26
28
29
31
34
35
35
36
36
37
41
42
42
43
MODELING POINT-TO-PATH ATTENUATION
45
3.3
4.1
4.2
4.3
5
17
Derivation of the Total Cross-Section
3.1.1 Vector-eigenfiinction expansion
3.1.2 Perturbation method solution
3.1.2.1 The Zeroth-Order Approximation
3.1.2.2 The First-Order Approximation
3.1.3 Expansion Coefficients
3.1.3.1 Axisymmetry contribution
3.1.3.2 Nonaxisymmetry contribution
3.1.4 Total Cross Section
Raindrop Characteristics
3.2.1 Raindrop shape
3.2.1.1 Spherical
3.2.1.2 Spheroidal
3.2.1.3 Pruppacher-and-Pitter
3.2.2 Drop Size Distribution
3.2.2.1 M arshall-Palmer dsd
3.2.2.2 Weibull dsd
Theoretical Derivation
3.2
4
13
14
15
Spatial Correlation of Rain
Two-Component Model
4.2.1 Model for Volume Cell Component
4.2.2 Model for Debris Component
Application to Networks
4.3.1 Parallel Diversity
4.3.1.1
Model for Volume Cell Component
4.3.1.2
Model for Debris Component
4.3.2 Node Diversity
4.3.2.1Model for Volume Cell Component
4.3.2.2 Model for Debris Component
4.3.3 Relay
PERFORMANCE ANALYSIS
5.1
Specific Attenuation
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47
48
50
52
55
55
56
57
59
59
61
62
65
66
CONTENTS
5.2
5.3
5.4
Path Links
5.2.1 Temperate Continental (Montreal)
5.2.2 Tropical Climate (Singapore)
System Analysis
5.3.1 Temperate Continental (Montreal)
5.3.2 Tropical Climate (Singapore)
Network Links
5.4.1 Parallel Diversity
5.4.2 Node diversity
5.4.3 Relay
6 FINAL REMARKS AND FUTURE WORK
6.1
6.2
Summary of our work
Future Considerations
id
74
75
80
85
85
88
91
91
93
93
95
95
98
REFERENCES
101
BIBLIOGRAPHY
107
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•
LIST OF FIGURES
Figure 2 .1
Rain attenuation prediction procedure.
6
Figure 2.2
ITU world rain regions (source [ITU837]).
7
Figure 2.3
Crane world rain regions (source [Crane80]).
10
Figure 2.4
Crane U.S. rain regions (source [Crane80]).
10
Figure 2.5
Block diagram of a communication system.
15
Figure 3.1
Geometry of plane waves scattered by a distorted raindrop
19
Figure 3.2
The raindrop models with cquivoiumetric drop radius 3.25mm.
35
Figure 3.3
The two spherical coordinates system superimposed.
39
Figure 4.1
One-dimensional idealized spatial spectrum for ln(rain rate)
47
Figure 4.2
Spatial correlation function for ln(rain rate) and for rain rate (Climate D2)48
Figure 4.3
Geometry for a parallel topology
55
Figure 4.4
Geometry for cell contribution calculations for a parallel diversity
56
Figure 4.5
Geometry for debris contribution calculations for parallel diversity
57
Figure 4.6
Geometry for a node topology.
59
Figure 4.7
Geometry for cell contribution calculations for a node topology.
60
Figure 4.8
Geometry for debris contribution calculations for node topology.
61
Figure 4.9
Geometry for relay topology.
62
Figure 5.1
Total cross section comparing spheroidal and P-P raindrop model for 15(top)
and 38(bottom) GHz.
67
Figure 5.2
M-P and Weibull drop size distribution for different rain rates.
68
• • •
m i
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x iv
LIST OF FIGURES
Figure S.3
Specific attenuation varying with frequency for a 50mm/h rain rate.
(Spheroidal raindrops)
69
Figure 5.4
Specific attenuation comparing drop size distribution for spheroidal
raindrops and frequencies 15(top) and 38(bottom) GHz.
70
Figure 5.5
Specific attenuation comparing drop size distribution for P-P raindrops and
frequencies 15(top) and 38(bottom) GHz.
71
Figure 5.6
Specific attenuation with different frequencies
Weibull(bottom) dsd, assuming P-P raindrop model.
Figure 5.7
Specific attenuation comparing theoretical spheroidal and P-P raindrop
models with the ITU-R838 at 38GHz, for M-P(top) and Weibull(bottom)
dsd.
73
Figure 5.8
Path attenuation in Montreal using ITU-R530 prediction method for a
38GHz system and fTU-R838 specific attenuation model.
76
Figure 5.9
Path attenuation in Montreal using Crane prediction method for a 38GHz
system and ITU-R838 specific attenuation model.
76
Figure 5.10
Path attenuation in Montreal using the two-component model with
spheroidal based specific attenuation at 38GHz with M-P(top) and
Weibull(bottom) dsd.
77
Figure 5.11
Path attenuation in Montreal using the two-component model with P-P based
specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd. 78
Figure 5.12
Path attenuation comparison in Montreal for different prediction methods at
38 GHz and unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical
derivation using M-P dsd.
79
Figure 5.13
Path attenuation comparison in Montreal for different prediction methods
and unavailabilities o f 0.01 (top) and 0.001 %(bottom), theoretical derivation
using Weibull dsd.
79
Figure 5.14
Path attenuation in Singapore using ITU-R530 prediction method for a
38GHz system and ITU-R838 specific attenuation model.
81
Figure 5.15
Path attenuation in Singapore using Crane prediction method for a 38GHz
system and ITU-R838 specific attenuation model.
81
Figure 5.16
Path attenuation in Singapore using the two-component model with
spheroidal based specific attenuation at 38GHz with M-P(top) and
82
Weibull(bottom) dsd.
Figure 5.17
Path attenuation in Singapore using the two-component model with P-P
based specific attenuation at 38GHz with M-P(top) and Weibull(bottom)
dsd.
83
Figure 5.18
Path attenuation comparison in Singapore for different prediction methods at
38 GHz and unavailabilities of 0.01(top) and 0.001 %(bottom), theoretical
derivation using M-P dsd.
84
Figure 5.19
Path attenuation comparison in Singapore for different prediction methods
and unavailabilities of 0.01(top) and 0.001%(bottom), theoretical derivation
using Weibull dsd.
84
Figure 5.20
Fade margin comparison in Montreal for different prediction methods at 38
GHz for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical
derivation using M-P dsd.
86
for
M-P(top)
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and
72
LIST OF FIGURES
XV
Figure 5.21
Fade margin comparison in Montreal for different prediction methods for
unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation
using Weibull dsd.
86
Figure S.22
Maximum path length comparison in Montreal for different prediction
methods at 38 GHz for unavailabilities of 0.01 (top) and 0.001 %(bottom),
theoretical derivation using M-P dsd.
87
Figure 5.23
Maximum path length comparison in Montreal for different prediction
methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical
derivation using Weibull dsd.
87
Figure 5.24
Fade margin comparison in Singapore for different prediction methods at 38
GHz for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical
derivation using M-P dsd.
89
Figure 5.25
Fade margin comparison in Singapore for different prediction methods for
unavailabilities of 0.01 (top) and 0.001 %(bottom), theoretical derivation
using Weibull dsd.
89
Figure 5.26
Maximum path length comparison in Singapore for different prediction
methods at 38 GHz for unavailabilities of 0.01(top) and 0.001 %(bottom),
theoretical derivation using M-P dsd.
90
Figure 5.27
Maximum path length comparison in Singapore for different prediction
methods for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical
derivation using Weibull dsd.
90
Figure 5.28
Diversity improvement factor and ratio of the cell to debris component as a
function of baseline length for jointly exceeding an attenuation of 3dB.
Calculations are for a pair of terrestrial paths of 10km length with an
orientation angle of 45° between the baseline and path direction.
92
Figure 5.29
Diversity improvement factor and ratio of the cell to debris component as a
function of orientation angle for jointly exceeding an attenuation of 3dB.
Calculations are for a pair of terrestrial paths of 10km length with a 3km
baseline.
92
Figure 5.30
Diversity improvement factor and ratio of the cell to debris component as a
function of separation angle for jointly exceeding an attenuation of 3dB.
Calculations are for a node with paths of 10km length.
93
Figure 5 .31
Diversity improvement factor and ratio of the cell to debris component as a
function of separation angle for exceeding an attenuation of 3dB.
Calculations are for a relay with paths of 10km length.
94
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LIST OF TABLES
TABLE 1.1
Rain attenuation assumptions
4
TABLE 2.1
ITU-R837 Rainfall intensity exceeded (mm/h) (source [ITU837])
8
TABLE 2.2
Crane Rain-Rate Distributions (mm/h) (source [Crane80])
9
TABLE 2.3
Regression coefficients for estimating specific attenuation in (EQ. 2.1)
12
TABLE 2.4
Link availability
16
TABLE 3.1
Calculated Coefficients for different raindrop sizes (Spheroidal raindrops)37
TABLE 3.2
Computed deformation coefficients of raindrops (source [Oguchi77])
38
TABLE 3.3
Calculated coefficients for different raindrop sizes (P-P raindrops)
40
TABLE 4.1
Parameters for the Two-Component model by region (source [1]).
50
TABLE 5.1
Parameters for the empirical specific attenuation evaluated at 38GHz.
74
TABLE 5.2
Rainfall Rate Statistics (mm/h) for Montreal
75
TABLE 5.3
Rainfall Rate Statistics (mm/h) for Singapore
80
TABLE 5.4
Parameters for the microwave system
85
x v ii
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1
INTRODUCTION
Millimeter wave links offer large bandwidth and high speed communication for
integrated multimedia services. Such links are also quick to deploy, as compared to
cable and fibre optic wireline connections. The issue of greatest importance in the
study of the performance o f millimeter links is rainfall. In the design o f such sys­
tems, the attenuation due to rain must be accurately accounted for in order to
ensure system reliability and availability. Systems that are poorly designed lead to
an increase in transmission errors, or worst, to an outage in the received signal.
Unfortunately, the designer has to rely on rain process statistics which could con­
tain an insufficient amount o f data for long-term predictions. This is particularly
true for tropical climates which are vaguely understood, as compared to climate
regions in North-America or Europe where the largest number of observing sta­
tions is located [1].
The primary goal of a rain attenuation prediction method is to achieve acceptable
estimates of the attenuation incurred on the signal due to rain, given the system
requirements such as frequency, path length, polarization, path geometry, rain rate
distributions, and availability. The International Telecommunication Union, or ITU
prediction model [ITU530] is a first point o f reference for the engineer. Improve­
ments to the models come from measurement programs which provide further
insight into the physics o f precipitation systems and also data against which their
accuracy can be evaluated. These can be obtained from experimental links, which
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2
INTRODUCTION
provide accurate data, but, generally for only one specific frequency and path
geometry.
1.1 OVERVIEW OF ATTENUATION PREDICTION
A power law empirical relation between the specific attenuation and the rain rate
has been found to be a good approximation [01sen78]. The derivation of the spe­
cific attenuation is a known result from scattering theory [4], but the computation
is complicated using the perturbation method [Oguchi60], because it involves the
summation of spherical Bessel functions and associated Legendre functions.
Therefore, the empirical relation for evaluating the specific attenuation is
extremely practical, since the parameters are given in a tabulated form and calcula­
tions can be carried out in seconds.
The path attenuation was first calculated [Lin77] by directly applying the line rain
rate in the empirical relation previously discussed. Later, Crane [Crane80] found a
power law relation between the line rain rate and point rain rate from which he
evaluated the instantaneous rain profile. He then utilized the empirical relation
between the attenuation and rain rate to compute an exponentially fitted effective
path length. The attenuation along the path is then obtained by multiplying the spe­
cific attenuation by the computed effective path length.
By making the rain rate the main parameter of the attenuation prediction model,
there is no need to evaluate attenuation at different frequencies and path geome­
tries.
1.2 MOTIVATION OF OUR RESEARCH
The evolution of computers and the advancement o f programming tools has made
it possible to simulate complex models in a reasonable amount of time. The pur­
pose of this thesis is to model the attenuation due to rain and to analyze microwave
link performance when degraded by rain.
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1.2 MOTIVATION OF OUR RESEARCH
3
The modeling o f the attenuation will be divided into two parts. First, we carry out
the evaluation o f the specific attenuation according to the solution o f the total cross
section provided by the perturbation method used by Oguchi for spheroidal rain­
drops [Oguchi60]. The theory will be applied to the Pruppacher-and-Pitter shaped
raindrops [Pruppacher71], which was made available through the simplification of
the raindrop shape expression [Li94]. The general expression for the P-P shaped
raindrops will be used to represent the spherical and spheroidal shape as well. The
new expression obtained resulted in better approximation of the spheroidal shape
when compared with the one used by Oguchi [Oguchi60] or M orrison and Cross
[Morrison74]. This could explain why differences in the total cross section were
noted between the Morrison and Cross least squares technique and the Oguchi per­
turbation method. Finally, the total cross section is integrated with the drop size
distribution in order to obtain the specific attenuation due to rain.
In the second part, the modeling of the rain rate process as a combination of two
components [Crane89] is carried out, one with an exponential distribution attrib­
uted to volume cells and the second with a lognormal probability density function
for the debris component. The model estimates the probability of exceeding a cer­
tain attenuation threshold. A trial and error approach results in the path attenuation
for a specified time percentage. The two-component model for rain rate assumes
an empirical relation for the specific attenuation. Theoretically derived specific
attenuations can therefore be fitted by the least squares method with the parameters
then fed into the two-component model. The choice of the two-component model
was encouraged because o f its applicability to network analysis. Once the theoreti­
cal model was ready, the rain effect could be included in the channel o f a micro­
wave system and then analyzed. Two prediction methods are chosen to validate the
simulated rain attenuation. The first one is the ITU Recommendation 530
[ITU530], and the second one is the Crane model as described in [Crane80].
We will limit ourselves to the study of terrestrial line-of-sight microwave links.
The Earth-space links differ only by an adjustment of the path length that is
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4
INTRODUCTION
affected by rain. Table 1.1 summarizes the different assumptions for modeling the
rain attenuation.
TABLE 1.1 Rain attenuation assumptions
Specific Attenuation
Rain Rate Distribution
Raindrop Shapes
Spheroidal or P-P
Raindrop Size Distributions
M-P or Weibull
Cell Component
Exponential pdf
Debris Component
Lognormal pdf
1.3 ORGANIZATION OF THESIS
The intent of this thesis is to study the effect of different assumptions in the com ­
putation o f microwave radio system rain attenuation and to compare the resulting
approaches to the standards. Chapter 2 describes the ITU and Crane prediction
methods along with their climate models. The empirical specific attenuation pro­
vided by ITU-R838 will be used for the calculation of the attenuation. A simple
system is presented, which will be simulated with the rain attenuation modeled in
the channel to complete the analysis of a microwave single link affected by rain.
Chapter 3 models the specific attenuation, by computing the total cross section for
a generalized expression of the raindrop shape using the perturbation method. The
spherical, spheroidal, and Pruppacher-and-Pitter raindrop shapes are then fitted to
this general expression. A knowledge o f the drop size distribution is needed to
finalize the calculation of the specific attenuation. Chapter 4 describes the twocomponent model for modeling the rain rate distributions. The rain correlation
function, which is needed for the model, is then derived from its power spectrum.
Finally, the two-component model is applied on three simple network blocks.
Chapter 5 describes the implementation o f the above mentioned models and
includes simulation results and performance evaluations. Chapter 6 concludes the
thesis with a summary of our work and suggestions for future investigation.
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2
PREDICTION OF
RELIABILITY
The effects that hydrorneteor, such as cloud, raindrops, or snowflakes have on
communication systems are dependent both on the system frequency and the type
of particle present. The rain attenuation for centimetric (3-30GHz) or submillimetric (300-3000GHz) and, especially, millimetric (30-300GHz) microwave links
becomes dominant and must be accounted for in a system design. The latter region
of the spectrum is now being developed as new technology becomes available.
Unfortunately, there are not enough data to completely describe the rain process as
it affects the propagation of electromagnetic waves. Models are prepared to pro­
vide the best possible estimates given the information currently available. By tak­
ing the available statistical information into account, statistical predictions are
made for the occurrences of events that affect the availability of communication
systems.
Due to the observed relation between specific attenuation and the precipitation
rate, o f the form
kR“
[01sen78], prediction methods are constructed with the rain
rate as the primary parameter. Starting with a specification for a microwave sys­
tem, as can be seen in Figure 2.1, the path attenuation is calculated in three stages.
The first stage, which estimates the rain rate at availability (1-p), requires the
knowledge of rain rate distributions which characterize the geographical location
of the link. Rain rate climate models with sufficient long-term data are readily
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6
PREDICTION OF RELIABILITY
available for an estimate o f these rain rate distributions. In the second stage, the
attenuation at a point, expressed in decibels per kilometer, is computed using the
above mentioned formula. The parameters, fitted to theoretical derivations, are pre­
sented in tabular form. At the last stage, an effective path length is estimated to
account for the rain inhomogeneous characteristic in the horizontal. Finally, the
path attenuation is derived. The effective path attenuation depends on the rain rate
due to the complexity o f measuring the path attenuation for different frequencies
and distances.
Specification
Stage i
---------------- 1 I------------
1
r
Stage 2
Stage 3
~\ r
d-p)
Frequency
Effective
Path Length
Path
Attenuation
Path Length
F ig u re 2.1
Rain attenuation prediction procedure.
Once the rain attenuation is known for some design specification, the performance
of that system can be analyzed by incorporating the rain attenuation in the trans­
mission loss of the medium. A backward approach is also useful when the path
length between the antennas is to be determined.
Two prediction procedures will be considered because of their acceptability and
wide use in microwave system design. The first one is the ITU-Recommendation
530 [ITU530], which is considered the standard in predicting the path attenuation
based on the ITU-R837 [ITU837] climate regions and the ITU-R838 (ITU838]
parameters for the specific attenuation. The second one is the Crane attenuation
prediction model [Crane80], based on the Crane climate model [Crane80]. This
model has an important interest in North-America.
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7
2.1 RAIN-RATE CLIMATE MODELS
2.1 RAIN-RATE CLIMATE MODELS
The starting point for estimating rain attenuation is a knowledge o f the statistics of
rainfall rate. Direct measurements o f rain attenuation are not very practical
because they can only be obtained for a given frequency and path geometry and
need to span several years to be statistically accurate, therefore smoothing the
year-to-year variability. The preferred option is to use locally derived statistics of
rainfall rate with an integration time o f 1 min. However, this data might not be
available, in which case global distributions o f rainfall rate are provided for various
percentages according to specified zones.
There are two important climate regions that need to be considered. First, the ITUR837 “Rainfall Rate Statistics” (1TU837] is utilized in the ITU-R530 prediction
model [ITU530]. Second, the Crane “Rain-Rate Climate M odel” [Crane80] is uti­
lized in the Crane Attenuation Model [Crane80].
2.1.1 ITU-R837 Rainfall Rate Statistics
180
180
Figure 2.2
ISO
120
tO
00
ISO
120
90
80
30
30
60
90
120
ISO
180
120
ISO
180
ITU world rain regions (source [ITU837]).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
8
PREDICTION OF RELIABILITY
The climate regions used by ITU are shown in Figure 2.2, and the rain rate distri­
butions are given in Table 2.1. The distribution gives the rain rate that is exceeded
at a certain percentage o f time or year. The climate regions are labeled with letters
starting with least precipitations in region A, and finishing with the letter P for
higher rain rates.
TABLE 2.1 ITU-R837 Rainfall intensity exceeded (nun/h) (source [ITU837])
Percentage of time (%)
1.0
0.3
0.1
0.03
0.01
0.003
A
<0.1
0.8
2
5
8
14
22
B
0.5
2
3
6
12
21
32
C
0.7
2.8
5
9
15
26
42
D
2.1
4.5
8
13
19
29
42
E
0.6
2.4
6
12
22
41
70
F
1.7
4.5
8
15
28
54
78
G
3
7
12
20
30
45
65
H
2
4
10
18
32
55
83
J
8
13
20
28
35
45
55
23
42
70
100
105
150
K
1.5
4.2
12
0.001
L
2
7
15
33
60
M
4
11
22
40
63
95
120
N
5
15
35
65
95
140
180
P
12
34
65
105
145
200
250
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
9
2.1 RAIN-RATE CLIMATE MODELS
2.1.2 Crane Rain Rate Climate Zone
The Crane climate regions have different boundaries than the ITU regions as can
be seen from Figure 2.3 for the world regions and Figure 2.4 for the U.S. details.
The corresponding rain rate distributions, in Table 2.2, are, however, similar in
general.
TABLE 2 .2 Crane Rain-Rate Distributions (nun/h) (source [Crane80])
Percentage of time (%)
0.01
0.005
0.001
1.0
0.5
0.1
0.05
A
1.7
2.5
5.5
8.0
15.0
19.0
28.0
B
1.8
2.7
6.8
9.5
19.0
26.0
54.0
C
1.9
2.8
7.2
11.0
28.0
41.0
80.0
DI
2.2
4.0
11.0
16.0
37.0
50.0
90.0
D2
3.0
5.2
15.0
22.0
49.0
64.0
102.0
D3
4.0
7.0
22.0
31.0
63.0
81.0
127.0
E
4.0
8.5
35.0
52.0
98.0
117.0
164.0
F
0.8
1.2
5.5
8.0
23.0
34.0
66.0
G
3.7
7.0
22.0
33.0
67.0
85.0
129.0
H
6.4
13.0
51.0
77.0
147.0
178.0
251.0
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10
PREDICTION OF RELIABILITY
100
150
<0
120
00
30
120
190
79
00
00
30
30
45
ISO
150
120
90
00
30
0
30
Figure 2.3
Crane world rain regions (source [Crane801).
Figure 2.4
Crane U.S. rain regions (source [Crane80]).
90
90
ISO
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100
2.2 SPECIFIC ATTENUATION
11
2.2 SPECIFIC ATTENUATION
The fundamental quantity in the calculation o f rain attenuation statistics for terres­
trial paths is the specific attenuation y, representing the rain attenuation per unit
distance. R. L. Olsen [01sen78] extensively analyzed the relation y =
aRb
between
specific attenuation and rain rate from both theoretical and numerical viewpoints
and concluded that this relation is an approximation of a more general series rela­
tion in frequency and rain rate, which reduces to the simpler form at the frequen­
cies and rain rates o f practical interest.
In order to obtain the parameters for the formula, specific attenuations at different
rain rates, computed from theoretical models, were fitted by the least squares
method for a number o f frequency values. Then, the formula was tested against
available data in order to verify its correctness.
The ITU-Recommendation 838 [ITU838] has become the standard for the specific
attenuation estimation.
2.2.1 ITU-R838 Specific Attenuation Model for Rain
The specific attenuation is calculated using the following power-law relationship:
YR = k R a
(EQ 2.1)
where yR is the specific attenuation in dB/km and R the rain rate in mm/h.
The frequency-dependent coefficients k and a are given in Table 2.3 for vertical
and horizontal linear polarizations, and terrestrial paths.
For all other path geometries and polarization, the coefficients in (EQ. 2.1) can be
calculated from the values in Table 2.3 using the following equations:
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12
PREDICTION OF RELIABILITY
, _
+
+ ( * / / - ly)COS20COs2T]
2
(EQ 2J2)
[ k Ha H + k v a v +• ( k Ha H - k v a v ) c o s 20 c o s 2 t ]
—
a
where 9 is the path elevation angle and x is the polarization tilt angle relative to the
horizontal, e.g., for circular polarization x = 45°.
TABLE 2.3 Regression coefficients for estimating specific attenuation in (EQ. 2.1)
r
(GHz)
1
kv
0.0000352
2
kH
«H
0.0000387
«v
0.880
0.912
0.000138
0.000154
0.923
0.963
4
0.000591
0.000650
1.075
1.121
6
0.00155
0.00175
1.265
1.308
7
1.332
0.00265
0.00301
1.312
8
0.00395
0.00454
1.310
1.327
10
0.00887
0.0101
1.264
1.276
12
0.0168
0.0188
1.200
1.217
15
0.0335
0.0367
1.128
1.154
20
0.0691
0.0751
1.065
1.099
25
0.113
0.124
1.030
1.061
30
0.167
0.187
1.000
1.021
35
0.233
0.263
0.963
0.979
40
0.310
0.350
0.929
0.939
45
0.393
0.442
0.897
0.903
50
0.479
0.536
0.868
0.873
60
0.642
0.707
0.824
0.826
70
0.784
0.851
0.793
0.793
80
0.906
0.975
0.769
0.769
90
0.999
1.06
0.754
0.753
100
1.06
1.12
0.744
0.743
120
1.13
1.18
0.732
0.731
150
1.27
1.31
0.711
0.710
200
1.42
1.45
0.690
0.689
300
1.35
1.36
0.689
0.688
400
1.31
1.32
0.684
0.683
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2.3 PATH ATTENUATION FORMULAS
13
2.3 PATH ATTENUATION FORMULAS
The general prediction procedure proceeds by choosing an availability requirement
for the system. The availability is expressed in percentage of time per year that the
signal should be available.
In this section, a description of the two methods employed to model the rain atten­
uation is given for the ITU-R Recommendation 530 [ITU530], which will be con­
sidered the reference, and the Crane model [Crane80], which is largely used in the
industry.
2.3.1 ITU-R530 Prediction Model
The present ITU-R530 [ITU530] method predicts the effects of rain as an attenua­
tion on the signal and is considered to be valid in all parts o f the world for frequen­
cies up to at least 40GHz and path lengths up to 60km. The method is as follows:
1. Obtain the rain rate Ro.oi exceeded for 0.01% of the time. If this information is
not available from local sources of long-term measurements, an estimate can be
obtained from the information given in Recommendation ITU-R837 IITU837].
2. Compute the specific attenuation Yr (dB/km) for the frequency, polarization and
rain rate of interest using Recommendation ITU-R838 [ITU838].
3. Compute the effective path length deff from the path length d as follows:
d0 = 35e"°'°15*001
d '!f ~
(EQ 2.3)
For Ro.oi > lOOmm/h, use the value of 100 mm/h in place of Ro.oi4. An estimate of the path attenuation exceeded for 0.01 % of the time is given by:
*o.oi = Ygd'/f
(EQ 2.4)
5. Attenuation exceeded for other percentages of time p in the range 0.001% to 1%
may be deduced from the following power law:
p =
_ 0.12p~<0546+0043log|0,’)
‘o.oi
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(EQ 2.5)
14
PREDICTION OF RELIABILITY
2.3.2 Crane Prediction Model
Crane has extensively studied the effect of rain attenuation in the millimeter range.
His step-by-step application of the attenuation prediction model [Crane80] is as
fo llo w s:
1. Determine the rain rate distribution Rp. Locate path endpoints on Crane’s map
and determine the rain climate region. From the rain climate region, obtain the
rain rate distribution also supplied by the Crane model.
2. Determine the specific attenuation parameters k and a for the frequency of inter­
est.
3. Calculate the attenuation value Ap from R and the distance d as
\
=
(EQ2.6)
yR<*'ff
i,v5 I____ _.cS
- ------ - + - ----— e«*,
y
z
yd
8 < d < 22.5
I
(EQ 2.7)
0<d<5
y
with the remaining coefficients computed in the following manner
B = 0 .8 3 -0 .1 7 1 ^/?,,)
c = 0.026 - 0.03 In (Rp)
6 = 3.8 —0.61n(Rp)
B
u - —+ c
o
y = CLu
Z = 0LC
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(EQ 2.8)
15
2.4 SYSTEM PERFORMANCE
2.4 SYSTEM PERFORMANCE
In the evaluation of the system performance, the attenuation due to rain is included
in the channel model as illustrated in Figure 2.5.
Channel
J
Transmitting
antenna sain
Ji
Signal
in
4
~
t
Rr,
-
g at
~
Free space
loss
L FS
Rain
loss
- »»
1
b—
(
Signal
\
1-------------------------------------------------------------------------
Transmitter
F igure 2.5
Receiving
antenna gain
►
*
Receiver
Block diagram of a communication system.
The available signal power at the receiver may be determined as follows:
=
P tx + g a t ~
L f s - A p + G ar
(EQ 2.9)
where,
PRx
=
receive power level, in dBm
P fx
=
transmit power, in dBm
G at
=
gain of transmitter antenna, in dB
G ar
-
8a' n ° f receiver antenna, in dB
Lpg
=
free-space path loss, in dB
Ap
=
rain attenuation exceeded for p% of the time on a path length d, in dB
The transmitting and receiving antenna gains can be computed, assuming a para­
bolic shape, as follows:
g at
=
g ar
= 17.88 + 201og/ + 20 log Z)
(EQ2.10)
The free-space loss is given by:
L fs = 92.44 + 2 0 lo g / + 20Iog</
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(EQ2.11)
16
PREDICTION O F RELIABILITY
f
= frequency, in GHz
D
= antenna diameter, in m
d
= distance between antennas, in km
Finally, the fade margin M, which must be positive in order to meet the specifica­
tion o f the design, is given by:
M = P Rx - T Rx
(EQ 2.12)
where,
When
M
=
fade margin, in dBm
T rx
=
receive threshold for a specified bit error rate, in dBm
the fade margin
is negative, the signal is received under the
specified bit
error rate, or worst, an
outage could occur. When the availability o f
the signalis
not met, widely due to rain loss, the following adjustment can be made to increase
the fade margin:
1.
2.
3.
4.
Decrease the frequency
Decrease the path length between the transmitting and receiving antennas
Increase the antenna size
Increase the transmit power
The following table summarizes the outage time per year allowed for given avail­
ability percentages:
TABLE 2 .4 Link availability
Availability (1-p)
Unavailability (p)
Outage per year
99%
1%
3.7 days
99.9%
0.1%
8.8 hours
99.99%
0.01%
S2.6 min
99.999%
0.001%
S.3 min
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3
MODELING ATTENUATION
AT A POINT
In order to evaluate the specific attenuation of electromagnetic waves due to rain,
the total cross-section (sum of the absorption cross-section and the scattering
cross-section) of raindrops must be evaluated. The scattering of a plane electro­
magnetic wave by spherical particles of any material was treated by Mie and is
known as Mie scattering theory. Later, Stratton [4] reformulated M ie’s work using
a method o f expanding the scattered fields in a series o f spherical vector wave
functions. When the raindrop was found to be deformed, it was represented by a
spheroid, which was considered as a perturbation of a sphere. Mushiake
[Mushiake56] developed the theory of scattering of a plane electromagnetic wave
by
perfectly
conducting
spheroids. Oguchi
[Oguchi60],
[Oguchi64]
and
[Oguchi73] extended the work of Mushiake to the case o f spheroids of any mate­
rial. Morrison and Cross [Morrison74] modified the perturbation calculation by
using an equivolumic spherical drop with an appropriate perturbation parameter,
rather than the perturbation about an inscribed spherical drop. This modification
improved the results considerably and gave results closer to the least squares fitting
method that they were analyzing.
At present, the most realistic and best-accepted distorted raindrop model is the
Pruppacher-and-Pitter raindrop model [Pruppacher71], which is expressed as a
cosine series. Li et al. [Li94], realizing the difficulty with a cosine series in the cal­
culations of the scattered fields, introduced a simplified expression for the Prup17
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18
MODELING ATTENUATION AT A POINT
pacher-and-Pitter raindrop model. Using the simplified expression, they were able
to derive the total cross-section [Li95a], [Li95b] for the Pruppacher-and-Pitter
raindrop model using the perturbation method theory previously used for spheroi­
dal raindrops. In the following, a general expression for raindrops will be used to
fit the three raindrop models: spherical, spheroidal, and PP; thus, the same equa­
tions can be reused in calculating the total cross-section for the three raindrops by
using the proper param eter values.
Oguchi studied the Pruppacher-and-Pitter raindrops using the point-matching tech­
nique and least squares fitting that solve the boundary-value problem [Oguchi77].
The cosine series approximation to raindrops was used in his study. In other work,
Oguchi presented a review [Oguchi81] of different techniques used in the determ i­
nation of the scattering properties and references to papers written concerning the
methods. He has also contributed [Oguchi83] to the propagation and scattering of
electromagnetic wave in different hydrometeors.
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19
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
Consider an electromagnetic wave passing through a raindrop as shown in
Figure 3.1. Assume the raindrop shape is represented by the general simplified
expression considered by Li et al. [Li94]:
r = a ( l - v 1) [ / o( 0 ) + T^ - / I( 0 ) ]
O < 0 £ jc
(EQ3.1)
0 < $ < 2 jc
where
/o<e, = 1
j)
(EQ 3.2)
/l( e ) = [, + ^ w
H (Q )
( e - ! ] si„*e
represents the step function, 6 the incident angle, a the horizontal radius of
the raindrop, and Vj and v2 the upper and lower vertical deformation, respectively.
HI
►y
X
►y
X
Case I
F igure 3.1
Case II
Geometry of plane waves scattered by a distorted raindrop
The electromagnetic fields E and H inside and outside the raindrop are governed
by M axwell’s equations expressed as follows:
V x V x E ( r ) - k 2E ( r ) = 0
(EQ 3.3)
V x V x / f ( r ) - * 2t f ( r ) = 0
(EQ 3.4)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
20
where
MODELING ATTENUATION AT A POINT
k2 =
to|i0(a )e -ia ). (Iq is the free space permeability, e and
a
are the perm it­
tivity and the conductivity o f the raindrop, respectively. The permittivity and con­
ductivity for water were derived from Ray [Ray72].
A time dependence
is assumed throughout the derivations.
e iwl
The boundary conditions associated with the raindrop surface can be written as:
n x E in = n x E oul
(EQ 3.5)
h x H in = n x H ou,
(EQ 3.6)
where
and
H in ou,
represent the EM fields inside and outside the raindrop
scatterers, respectively, and
n
denotes an outward unit vector normal to the rain­
drop surface.
The vertical and horizontal polarization is denoted by the subscripts I and II,
respectively. The two pairs o f incident EM waves are expressed by:
^ ^
_ (rfcoCrsinO + zcosO)]
E' i = E ^ c o s B x — s m 0 z ) e
Hi =
* < ,£ /-
ojpo
ye
[<iO(-tS in 0 + 'CO50)l
(EQ 3.7)
(EQ 3.8)
and
E n = Eu ye
...
H i, =
(EQ 3.9)
k0E „ ,
copo
.
(c o s 0 x -s in 6 z )e
| i t 0(xsin8 + ;c o s e ) |
_
(EQ3.10)
The scattered fields due to the raindrop scatterers illuminated by the plane waves
shown in (EQ. 3.7-3.10) can be obtained by solving the Maxwell equations in (EQ.
3.3-3.4). The total cross section is then calculated from the scattered EM fields due
to the raindrop.
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21
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
3.1.1 Vector-eigenfunction expansion
Using a technique similar to that employed by Oguchi [Oguchi60] and Stratton [4],
the incident, scattered, and transmitted EM fields are expanded in term s of the fol­
lowing vector wave eigenfunctions:
M^
k)
=
•nzA kr)
e ) |s,n( ^ ) l e
* ^ k r p z (cos
[cos
J
(EQ3.11)
3P*(cos0) fcos
- z „ ( k r ) — n-K
.
ou
n(n
= -i
+
[sm
]-
J
l)z.(k r)
^ _ Jp -(c o s
0)jsin*m<*>)}^
3['-2B(Arr)]a/,J,(cos0)fcos/
(m<t>ne
30
[sin
j
sin
(EQ 3.12)
+- k ? T r
m
where
zn(k r )
fcos
d [rz n(kr)]
I-
represents the spherical Bessel functions of n-order, and />*(cos0) is
the associated Legendre function.
Considering the two polarizations of the incident waves illustrated in Figure 3.1,
the incident electromagnetic fields are expanded in the following forms:
(EQ 3.13)
'k
Hl,
=
i|r0I I
n
“t
^
^
(EQ 3.14)
+ b ^ A f^ A k o )
n = 1m = 0
where the spherical Bessel functions of the first kind, i.e.,
z n{kQr )
used in the above vector wave functions, the orthogonal properties o f
Wf,mn (*0)
and
=
j„ (k Qr ) ,
M r mn (*0)
are
and
are considered, and the coefficients o f the expanded incident EM fields,
b'
are given by:
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22
MODELING ATTENUATION AT A POINT
m P” (cosQ)^
_ ( 2 - 5 w)(2n + l ) ( / , - w )! ..
sin 6
n(n + 1 )
(n + m)!*
(EQ 3.15)
dP” ( cos 9)
'//
ae
d P ” (cosQ )i
=
ae
(2 - 8° )(2„
n(/i + 1)
(n + /n)!
m P*(cos0)
sin0
(EQ 3.16)
-//
where 5" is the Kronecker symbol:
m =n
(EQ 3.17)
m *n
The method of vector wave eigenfunction expansion is also applied to obtain the
scattered EM fields
and
E\
h
and the transmitted EM fields
H s.
U
E\
h
and
H ‘. .
ti
Because the EM fields outside the raindrop must satisfy the Sommerfeld radiation
conditions, the scattered EM fields can be expanded as follows, in terms of the
spherical Hankel functions of the second kind //*2> :
a : mnM £ \ ( k 0) + b'CmN g A k Q)
= £ o .y
y
a:,mnN $ A k 0) + b\u m M $ n ( k Q)
(EQ 3.18)
(EQ 3.19)
n= 1m=0
while the transmitted field inside the raindrop can be expanded into the following
forms:
a'smnM timA k ) + b ^ nN rinn( k )
(EQ 3.20)
n = Im = 0
Hj,
=—
oiUo y y
n =IfflsO
(k) + b'SmnM > n (k)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ3.21)
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
where
a * n,
,
a'
, and
boundary conditions, and
k
b‘
23
are the coefficients to be determined from the
denotes the propagation constant determined by the
raindrop medium.
The boundary conditions satisfied by the EM fields indicate the continuity o f the
tangential components o f the total EM fields across the surface of the raindrop.
The total EM fields outside the raindrop can be considered as the sum of the inci­
dent EM fields and the scattered fields:
E out — E
Hn
E*
(EQ 3.22)
= W + Hs
while those inside the raindrop are the transmitted fields:
E in = E ‘
(EQ 3.23)
= H‘
Substituting the expanded electromagnetic fields into Eq. 3.5-3.6, the expansion
coefficients are solved by matching the boundary conditions. The boundary condi­
tions can therefore be rewritten as:
E\
+ E*.
= Ef.
//'+ //;
= h\
h*
//♦
//♦
(EQ 3.24)
h*
//♦
//♦
1 d f ,( 0 )
1( 6 )
= e!
dB
7/«
+Z
(6 )
dB
(EQ3.25)
H'
iie
+ H S,.
if8
I
d f t (Q)
/ KB)
dB
H‘ + H S,
fir
hr
1 d f t(B)
= h!
7/9 + J 1(9, dB li>
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24
MODELING ATTENUATION AT A POINT
3.1.2 Perturbation method solution
When the deformation of the raindrop from a sphere is small, variation of the fields
from those for a sphere can be evaluated by a first-order perturbation theory. Sup­
posing that the parameter
is small, the scattering and transmission coeffi­
cients of the EM fields can be considered approximately the sum of the first two
terms of the expanded series with respect to the parameter
~Q%mn
S
"<j j, 0 “
f/tlfl
a ?rm n
g mn
■<35. 1 ■
(BIB
g mn
+
where
a p -«
g mn
b f*r/n n
Bi-°
5 mn
b *'emn
B'.o
and
V'
1 - v , B*'1
f,mn
()/nn
(p
=
(EQ 3.26)
fl'1
S mn
Bp i
:
s
or
t, q
= 0 or 1) are the expansion coefficients to be
determined. The required coefficients can be derived directly, but approximately,
from this series of equations under the zeroth-order and the first-order approxima­
tions.
3.1.2.1
The Zeroth-Order Approximation
Under the zeroth-order approximation, the solution of the unknown coefficients of
the scattered and transmitted EM fields can be obtained. The four coefficient equa­
tions are obtained [Li95a] and given in the matrix forms as follows:
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3.1 DERIVATION OF THE TOTAL CROSS-SECTION
25
'< * > P n l
p> -
Vn\
n = I
+ A £2,A(«(p)
© J3
A=
1----------- 1
©£
=o
I
* U 3 '«<p >-
p) +
P'(P)
(EQ 3.27)
A= 1
=0
^ P n ]
-i *,° „J
*U*oA,(P> - *&tf.(CP> * B & * o * S ° ( P )
n = 1
where the parameters 3„(x) and
p„ (x)
are defined as follows:
3 .< * » - ^
h ? \ k ’r)
* ' « r)
3'
°
P
= ^FT-
(EQ 3.28)
- 3 [(^ )y .(y r)]
j ~
itva(ifcV)
= a[(rr)AW(yr)]
}
k 'r d ( k 'r )
( k ' r)
"(
with the symbol
k’
denoting
k
and
k0 .
The inter-parameter operators <t>,°n and VF,°
satisfy:
<*>°,/r(0) = j'r(e)/»(e» [ S X ^ 5 ) +( ^ I ^ :]lsi"e‘i9
°
n r m
(EQ 3.29)
-
0
for / 0(6) and /,(6 ) defined in (EQ. 3.2); and other parameters p, £ are defined as
P = *oa O ~ vi)
5= F
Ko
= M l -v ,)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ 3.30)
26
MODELING ATTENUATION AT A POINT
3.1.2.2
The First-Order Approximation
Using a method similar to the zeroth-order approximation, four equations are
obtained [Li95a] for first-order approximate solutions. The results are given as fol­
lows in matrix form:
W
n.
m L^'n.
a L „ p [y „ ( P ) ]'-^ '0 Cp[;„(?p)l' + A z ° p [ h ? K p ) V
* U p 3 ' - (p )- a£ i 5 p 3 v c p ) + « y i 1p p '( p )
nr/„n = I
A'n.
* U 3 - (P ) " BSmn3 '>(^ P) + *£«« * « (P)
= 0
m'
fl£--3,-<?P )- fiS-»p,(P)
(EQ3.31)
ro f
a?n..P23'- ( P ) - Ag«,(?P)23,r-(?P) + ^ - « P 2^ '( P )
rn -
^ U p2[^ (P)1' - fls l ^ P)2^ ^ P)]' + flSm
°np2^ 2)(P)l'
ro f
n = I
-
A£ j« & P )
+ ^ ^ ° „ ^ 2)(P)
=0
. f In .
.^PnJ
r e
A y.[?PA (5P)]'-A *|.[pJi«)(p)]'
B ^ k j n^ p ) - B ^ k 0h ^ K p )
*a.
where the parameters 3„(p) and p ”(p) are defined as
3 „ (P )
Ipy'n(p)]]
■f p J
(2).
:«p . -
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ 3.32)
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
27
and the parameter operators are given by:
= j F (e,/ l ( e ) [ g ] ( ^ r ) + ( ^ : ) ( ^ : ] | Sin9d8
0
0
r,'„F(9) = J f < 9 F ^ [ / « -
(EO 3.33)
l < g |) ] s i „ 9 , »
0
a{BF(9) = j F ( 0 ) ^ g ^ [ / ( / + l ) / > T ^ ] s m 0 d B
o
The associated Legendre functions have the orthogonality property in the variable
range [-1,1] or [0,tc]. The solution o f the first-order scattered EM fields by the axisymmetric raindrops, i.e., spherical or spheroidal, can be easily derived. In nonaxisymmetric raindrops, i.e., Pruppacher-and-Pitter, the upper part, corresponding to
0 < 0 < ^ , and the lower part, corresponding to ^ < 0 < n
,
are no longer the same.
Thus, the integration range changes to [-1,0] instead of [-1,1], and the associated
Legendre functions are no longer orthogonal. Therefore, the solutions of the inte­
grals in (EQ. 3.33) are no longer available in the closed forms, so that the coeffi­
cients of the first-order expanded EM fields must be solved for numerically.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
28
MODELING ATTENUATION AT A POINT
3.1.3 Expansion Coefficients
To solve for the first-oder approximate coefficients of the EM fields, the zerothorder and the first-order coefficients of the expanded EM fields are expressed
[Li95a] in terms o f the axisymmetric and nonaxisymmetric contribution of the
raindrop as follows:
A s- °
$mn
A * '° °
jfmn
4 5 .0 1
f/mn
A '- 0
Jfm/i
A 1- 00
%mn
A t . 01
jtmn
B l-°
£mn
B s. 00
£mn
B'.o
B ‘.o°
S'""
_
- fU T- V', ]J
f,mn
B'.oi
£ mn
_
-
A s , 10
A °m n
A?;
1
ffmn
A t , 10
A »mn
A !%mn
;1
Vl
B £m
s-1n
1“
B >. 1
Vl
B l -10
omn
B t.\o
Smn
£mn
A?"
+ fV2-VO
U -v ,
A %mn
£"
J B l -11
c„ mn
B t.ll
£m n
_
.
Assuming that
(EQ 3.34)
B*. ° i
and
y
are very small, the high-order terms can be
neglected.
Only the scattered EM fields will be used in the calculation of the total cross sec­
tion, so that the first-order coefficients of the transmitted EM fields would not be
needed.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
3.1.3.1
29
Axlsymmetry contribution
By substituting (EQ. 3.34) into (EQ. 3.27), the expansion coefficients o f the EM
fields under the zeroth-order approximation can be derived as follows:
„ tnn
y„(^p)[py„(p)i, - y „ ( p ) [^ p y „ (? p ) r
^
= --------------------- rr.----------------— ----------------------------- a '
j n ( & n p h ? \ p ) r - h n \ p ) i & j aw
;
(EQ 3.36)
= SZ ■a ‘„
gmn
n
fm n
r
, ( 2),
,v
yn( p ) [ p ^ ( P ) i>- ^ ( P ) [ p ; n(P )r
o o -
a ,
yn( C p ) [ p ^ 2)( p ) r - A l 2,(p)[Cpy„ccp)i'
B s.o o _ -
°
^ ( p ) ^ p y "( *>p ) * - U ( ^ P ) tP l.|P ) l
^ )(p)[;py„(^p)], - ? y „ (;p ) tp ^ 2)(p )r
_ g ^ 2>(p)fpy«cp>r- cy,cp)tp*l2>( p )r
Aj,2,(p )tc p y « (5p ) i' - ? 2y«(5p)[p*«2>c p ) r
=
r
gmn
"
. b,
bl
. a ,.
?mfl
_ 5 * . b'e
(EQ 3.38)
= r * .fcl-
Smn
n
(EQ 3.39)
5
The above expansion coefficients of the EM fields represent those of the Mie scat­
tered EM fields corresponding to the spherical raindrop scatterers.
Applying a method similar to that of Oguchi [Oguchi60], the first term of
(EQ. 3.35) is expressed as follows:
AS.
rt + 2
1
10 _
y„(£p)
A >’mn ~
+ W „ A K m/
l = m ax{m , n —2, I )
(EQ 3.40)
n
+2
-[?py«c?p)i'
In A
/ = mux{m, n —2. !)
+ x f V nA)b
Sm 1
n+2
B l - 10 =
J,mn
«(« +
s
LCpy„(^p>r
?p
I — max(m, n -
2, i )
(EQ 3.41)
-?pyn(CP)
X
(X/«W„a + X } n U ) b ‘m l ± X } ' V l Aai1m l
/ = m ax (m . n - 2. I )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30
MODELING ATTENUATION AT A POINT
where the inter-parameters
r \ } - \ x } - 2, X/-2, and
A,J-2 are defined as:
= p2^ L ^ _ 7-|.(^p)2^ £ > i + 5f p2‘/[/,'2)(P>1
dp
dp
P)
zrt^pyKCp)] i
zrtpy<(p)]i
fpy'i(p)] j
Z;1 = Pp
-T ft
(EQ 3.42)
(EQ 3.43)
'[p */2,(p )]‘
p f Cp /KSp H i + S,*p
Sp J
(EQ 3.44)
(EQ 3.45)
(EO 3.46)
d[y,(p)]
=
Ai =
p
dp
d (£ p )
'
K
(EQ 3.47)
dp
j n{ Z , P ) [ p h ? \ p ) Y - h * \ p ) & p j n& p ) Y
(EQ 3.48)
A2 = A^2)(p)[Cpy„(CP)]' - C2y„(?p)[p*^2)(p)]'
(EQ 3.49)
a n d th e parameters <PfnA, vF 1nA, T,ln<t, an d , ft|nA are represented by:
fm 2+ /i~(/i
—
m + 1)(/i + m + 1)
-----------------— ------------------------- — -—
L
(2 n + l)(2n + 3)
. ( n + l ) 2(/i - m ) ((njf
n + m)
m)~I .
(2n — 1)(2rt
*U =
+ 1)
J
(EQ 3.50)
n(n + 3 ) (n + m + l)(n + m + 2 ) ,
(2n + 3)(2/i + 5)
B’
(/i —2)(/i + 1)(/t - m)(n - m - 1).
(2/i - 3)(2/« - 1)
0,
vui
‘ InA
_
-
I = n
2m ( n - m )
2n-l
2m(n + m + 1)
2/t + 3
I = n+2
/ = n —2
otherwise
I = n- 1
/ = n+ 1
0,
otherwise
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ 3.51)
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
r*
1 In* ~
2 m ( n - 1)n(n —m)
2/i - I
2m(n + 1)(n + 2)(/i + m + 1)
2n + 3
31
I = n-l
(EQ 3.52)
I - n+ 1
0.
otherwise
2n(n + 1)rn(n —m + 1)(« + m + 1)
2/i + I L
2 /j
+3
I = n
(n + l)(n —m)(n + m)~
2n —1
(EQ 3.53)
2n(n + 2)(« + 3)(/i + m + l)(n + m + 2)
(2/i + 3)(2/i + 5)
I = n +2
2(/»-2)(/i2- I )(/i - m - I ) ( / i - m ) ,
/■
•>„ - 3)(2/i - 1)
n’
(2n
0,
I = n-2
otherwise
with
2n+l
(n + m)!
(n —m)!
(EQ 3.54)
and the subscript A denoting axisymmetry of the raindrops.
3.1.3.2
N onaxisym m etry contribution
When the raindrops are deformed axisymmetrically, the zeroth-order approximate
coefficients correspond exactly to the Mie-scattering coefficients. There is an addi­
tional contribution, however, due to such a nonaxisymmetrical distortion. Under
the sphere-based approximation, the upper and lower parts o f the sphere have dif­
ferent radii, a (l-v ,) and a(l-v 2), respectively. Therefore, the function U(p) such as
j n(p), /*„2)(p),
3'„(p), p '„(p ), etc. can be expanded due to the nonaxisymmetry into
, where H(0) denotes the step function. Substituting
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
32
MODELING ATTENUATION AT A POINT
(EQ. 3.34) into (EQ. 3.27) and applying the above expansion, the additional scat­
tering coefficients can be obtained [Li95a] and written as follows:
-
^.v.OI
l
__
n(n + 1)/„A,j /«(£P)
ISp/.CSp)]'
/
N
X
/ = maxim. 1) \
/ + n se odd
/
\
t f ® l n N a omi
X
)
/ = maxim, 1) \
1+ n = odd
(
•
\
/ = maxim , I ) ^
I + n = even
J
(EQ 3.55)
04
h
+ 5*5-00
2 ?mn
2
J / =
1)
/ ♦ n s even
40
B s.oi
f. mn
[?py-(? p)i'
*
n(n + l)/„A2
?p
/
JL
X
♦/
s m u x (m . 1)
/♦/ i = even
40
1+
M
40
/ = m a xim . I)
^
+ X
J
/ = m axim . 1)
/ + /t = even
n/4>?n/v^ m/
2
= m a x (m , I )
/ + n s odd
,
.
Z /1<t , 2.Ar6 c m /
^
'
2 Smn
l + n s odd
(EQ 3.56)
The parameters <!>{{,* and Tp,* are defined by:
n(n + !)(/> + m)l
In + 1 (n —m)l'
* ? aN =
l(l+ D
P f ( 0)
n(n + I ) -1(1+ 1)' 1
n = I
dP”( 0)
dx
(EQ 3.57)
dP?(0)
n(n+I)
P"( 0)
rn(n+ ! ) - / ( / + 1)
dx
•Fg,* = -mP?( 0)Pr(0)
(EQ 3.58)
In the above derivation, the following inter-parameters have been used:
2 / + I (l-m )l
=0
(EQ 3.59)
(EQ 3.60)
The coefficients of the first-order scattered electromagnetic fields due to the nonax­
isymmetry of the raindrop scatterers are expressed as [Li95a]:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.1 DERIVATION OF THE TOTAL CROSS-SECTION
41.
II
°m n
=
33
I AS, 1 0 _____________! ___________
2 “mn n(n + I)/„A j
OB
y„(Cp)
X
sm l
ma.r(m, 1)
/ + n = odd
(EQ 3.61)
I = max(m, I )
I s
s even
-Kpyn(Cp)]'
/ =
/ = m a i(m , I ) ^
1)
/ + n = even
/ ♦ n = odd
U = ig.t. 10______^___ .
I,mn
2
£»»»
n(«
+
I )/„A ;
[Cpy,(?p)]'
*
/ = /nax(m . 1)
I + n = odd
X
o i/^ iU + w u
/ = mcix(m, l)L
l + n - even
(x/ &lnN + X l ^ l a N ^ e mi
-?py„(?p)
I = m ax(m , 1)1u l + n = odd
gm /
/s
I)
I + n = e v en
(EQ 3.62)
The symbols <t»/nA/ and T,1^ (the subscript N denotes nonaxisymmetry) are repre­
sented by:
1
**»]IniV =
[ n l ( n - m + 1 ) 0 - m + 1)/,+
(2 n + I )( 2 /+ 1)
-/(/i + l)(n + m ) 0 ~ m + I )//+ i.„_ i
m
Il.n +
,.„ + 1
(EQ 3.63)
- « ( / + l)(/ + m ) ( n - m + l)/ ;_i. n+i
+('« + 1 ) ( / + l)(/i + / n ) ( / + / n ) / ,
T,n/V “
L
2/ + I
//+ ,-n
, « ( n - m + 1),
+
2n + 1
T,U =
^ £ r T T 1 Un - m+
( / + l ) ( I + m) ,
27T1
(EQ 3.64)
( n + l ) (/ i + m ) ,
"|
2n + 1
''•"-•J
(EQ 3.65)
I
“ '■» ' ^ T w 7 T T ) ["<" - m + 1)(' - m +1)/' - 1” '
(EQ 3.66)
- ( n + 1)(n + m ) ( / - m + I ) / , + i . n - i
+ n ( /i- m + l)(/ + m )/,_, n + , - ( n + I)(/i+ /n ) ( /+
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
]
34
MODELING ATTENUATION AT A POINT
and the parameter
, solved by Li et al. [Li95a] for the range [-1,0] is given by:
p? (x)p nx)dx
0
I* n
1 (n + m)\
2/i + 1 (/i —m)\
_______I
n +l
even
I —n
dPp( 0)
r
dP?(0 )-|
(EQ 3.67)
n +l
odd
(*-/)(*
3.1.4 Total Cross Section
The total cross section is defined [Oguchi60] as the ratio o f the sum of absorbed
and scattered energies to the mean energy flow of the incident waves. Using a
/
method similar to that o f Oguchi’s [Oguchi60], the total cross section " q t under
the first-order perturbation approximation is as follows:
»Q r
.
(EQ 3.68)
where,
(EQ 3.69)
n
\
rMi-01A
” Q0
t'
471™.
!,Q't °
i
11£
nl Tl 1/
"
Ko
m P ”( cos0)
A s. 10
sin6
Qmn
n
dP”( cos 6)
n = Im=0 4 5. 11
de
%mn
\
J
rBs-0l\
f,mn
B Z,mn
i 10
B sU
hmn
V
/
'
9P*(cos9)
de
mP”( cos0)
sin0
►
1
n —1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ 3.70)
3.2 RAINDROP CHARACTERISTICS
35
3.2 RAINDROP CHARACTERISTICS
Assumptions made on the raindrop have a great impact on the resulting specific
attenuation. First, the shape of the raindrop affects the calculations of the total
cross-section depending on the parameters given to the raindrop shape equation
given in (EQ. 3.1) and (EQ. 3.2). The parameters a, v t, and v2 can be fitted for a
spherical, spheroidal, or Pruppacher-and-Pitter raindrop shape. Second, the rain­
drop size distribution will affect the specific attenuation when it is integrated with
the total cross-section.
3.2.1 Raindrop shape
There are three different raindrop shapes that will be considered. The spherical and
spheroidal [Oguchi77] shapes are axisymmetrical about the horizontal axis, while
the Pruppacher-and-Pitter [Oguchi77] shape is distorted. The Pruppacher-and-Pit­
ter model is more realistic than the first two mentioned. The raindrop starts almost
spherical, then becomes spheroidal and axisymmetric for midsize raindrops, and
finally becomes distorted and nonaxisymmetric. This leads to the use of this model
in heavy rain climates due to the increased presence of larger raindrops. Figure 3.2
shows the raindrop as seen by the three models. The raindrop effective radius is
taken to be 3.25mm, which is considered a large drop. The spheroidal and the
Pruppacher-and-Pitter consider an equivolumic raindrop, i.e., they have the same
volume as the spherical case. This turned out to be an important issue when deriv­
ing the total cross sections.
Spherical
Figure 3.2
Spheroidal
The raindrop models with equivolumetric drop radius 3.25mm.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
36
3.2.1.1
MODELING ATTENUATION AT A POINT
Spherical
For the spherical shape, the parameters to (EQ. 3.1) are straightforward:
a = aQ
(EQ 3.71)
v, = v2 = 0
where ag is the equivolumetric drop radius.
W hen these values are substituted into (EQ. 3.68), the resulting total cross-section
is identical to the Mie scattering.
3.2.1.2
Spheroidal
The spheroidal shape has been extensively used in the derivation of total cross-sec­
tion, notably [Oguchi60] and [Morrison74]. It is approximated with a relation
between axial ratio dependent on the equivolumetric drop radius ag and is approxi­
mated by l —0. l a0, when ag is expressed in mm. This value was used by
[Morrison74] and later by [Oguchi83]. Equating both volumes of sphere and
spheroid for the above constraint, the parameters in (EQ. 3.68) are derived:
mm
a
(EQ 3.72)
( I —0 . 1 a 0 ) ,/3
v, = v2 = O .Ia0
The axisymmetry is maintained by equating v t and v2, the upper and lower defor­
mation coefficients, respectively. Substituting the parameters into (EQ. 3.1) results
in a total cross-section with only
and
Q l°
components, which corresponds to
the axisymmetric case.
The raindrop shape (EQ. 3.1) along with the parameters in (EQ. 3.72) results in
less error when compared to the spheroidal model used by Oguchi [Oguchi60] and
Morrison [Morrison74].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.2 RAINDROP CHARACTERISTICS
37
The following table summarizes the values for different equivolumetric drop
radius.
TABLE 3.1 Calculated Coefficients for different raindrop sizes (Spheroidal raindrops)
3.2.1.3
r«
>
II
>
ag (m m )
a (mm)
0.25
0.252119
0.025
0.50
0.508622
0.050
0.75
0.769746
0.075
1.00
1.035744
0.100
1.25
1.306895
0.125
1.50
1.583501
0.150
1.75
1.865893
0.175
2.00
2.154435
0.200
2.25
2.449525
0.225
2.50
2.751606
0.250
2.75
3.061165
0.275
3.00
3.378744
0.300
3.25
3.704948
0.325
Pruppacher-and-Pitter
Pruppacher and Pitter [Pruppacher71] established an equation to describe the
shape of w ater drops falling at their terminal velocity in terms of the balance o f the
internal and external pressure at the surfaces o f the drops. To simplify the expres­
sion, they introduced a cosine series,
,
r'(G') =
a,
«o
1+
cncos(«0')
(EQ 3.73)
fl =0
to replace the nonlinear equation and evaluated the first ten coefficients cn (n = 0,
1,2, ..., 9) o f the series for different sphere-based radii ao- This model is now well
accepted and used by researchers for the calculation of microwave attenuation by
rain [Oguchi77], [Oguchi81].
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
38
MODELING ATTENUATION AT A POINT
TABLE 3.2 Computed deformation coefficients o f raindrops (source [Oguchi77]>
*0
(mm)
co
c2
Cj
«4
«5
0.25
-0.386680xl0"3
-0.115996x10*2
-0.151420x10"3
0.629391x10*®
0.406408x1O'5
0.50
-0.265873xI0'2
-0.795387x1 O'2
-0.162839xl0*2
-0.104182x1O'3
0.844674x10*4
0.75
-0.693210x1O'2
-0.207063x10-'
-0.474328xl0'2
-0.428598x10"3
0.267023xl0*3
1.00
-0.115073x10-'
-0.342597x10-'
-0.885078x10"2
-0.138068x10"2
0.593452x1O'3
1.25
-0.187815x10-'
-0.559112x10*'
-0.146408x10*
-0.228007x10"2
0.964343x1 O'3
1.50
-0.267192x10'*
-0.795319x10"'
-0.211670x10-'
-0.329143xI0*2
0.136429xI0*2
1.75
-0.350003x10-'
-0.104166
-0.282497x10*
-0.438808xl0*2
0.177493x1O'2
2.00
-0.429905x10-'
-0.127923
-0.354365x10-'
-0.550270x10*2
0.216212xl0'2
2.25
-0.504202x10-'
-0.15000
-0.425377x10-'
-0.661037x1O'2
0.251072xl0’2
2.50
-0.573820x10-'
-0.170669
-0.496486x10*'
-0.773188x1O'2
0.282382xl0'2
2.75
-0.639783x10-'
-0.190235
-0.568738x10"'
-0.889090xl0*2
0.3l0464xl0'2
3.00
-0.703284x10*'
-0.209049
-0.643374x10*'
-0.101157x10*'
0.335670x10"2
3.25
-0.766642x10-'
-0.227796
-0.722772x10*'
-0.114546x10-'
0.358806xl0'2
*0
(mm)
C6
c7
c*
0.25
-0.852069x1 O'6
0.125856x10"^
-0.853347x 10‘7
0.36571 lxlO '7
0.50
-0.152054x1 O'4
0.353713x1 O'5
-0.412700x10*5
0.249148x1 O'5
0.75
-0.398766x10"4
0.107211x10“*
-0.167668x10“*
0.877456x1 O*5
1.00
0.221543x1 O'3
0.545847x10“*
-0.107608x1O’3
-0.573183x10“*
1.25
0.362000x1 O'3
0.891036x10“*
-O.I75351xlO'3
-0.934761x10“*
1.50
0.515668x1 O'3
0.126781xl0"3
-0.248960x 10"3
-0.132844x10"3
1.75
0.676448x1 O'3
0.166094x1 O'3
-0.325320x10‘3
-0.173789x1O*3
2.00
0.832096x1O'3
0.204025x10"3
-0.398414xl0"3
-0.213118xlO"3
2.25
0.977315x10°
0.239278x10"3
-0.465649x10 3
-0.249457x1O'3
2.50
0.111374x10*2
0.272273x1 O'3
-0.527797x10"3
-0.283225x10"3
2.75
0.124315x1 O'2
0.303469x10*3
-0.585708x10‘3
-0.314883xl0"3
3.00
0.136761x1 O'2
0.333400xl0-3
-0.640364x10"3
-0.344966x1O*3
3.25
0.149126x10"2
0.363121xl0'3
-0.693735x1O*3
-0.374542x1O'3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.2 RAINDROP CHARACTERISTICS
39
In order to analytically derive the scattered held by the rain, a simple mathematical
formula given by (EQ. 3.1) is assumed.
Assuming 0'max is the angle at which the horizontal distance
r'(Q 'max) s in 6 'mux
in
(EQ. 3.73) reaches the maximum, the translated vertical distance is equal to:
(EQ 3.74)
Figure 3.3 shows the two coordinate systems. Under the new coordinate, the PP
raindrop model is fitted to the simplified model in (EQ. 3.1). The maximum and
minimum vertical distances from the horizontal plane, on which the points with
maximum horizontal distances are located, should remain unchanged in the two
coordinate systems.
8=K
8= 0
8=0
8 = x
F ig u re 3.3
The two spherical coordinates system superimposed.
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40
MODELING ATTENUATION AT A POINT
The parameters to (EQ. 3.1) are finally given by:
(EQ 3.75)
n ss 0
V|
1
sin0_
- C O S0.
(EQ 3.76)
cos0.
(EQ 3.77)
1 + ^ c ncos(nemux)
n=0
V, =
1 - -7
sin0_
1 + X CnCOs(/,0'"“ )
n=0
Numerically, the coefficients for different raindrop sizes are com puted and shown
in Table 3.3.
TABLE 3.3 Calculated coefficients for different raindrop sizes (P-P raindrops)
ao (mm)
a (nun)
0.25
0.250194
vt
0.001649
v2
0.002991
0.50
0.502608
0.009103
0.022635
0.75
0.760110
0.022384
0.059717
1.00
1.021407
0.033650
0.100362
1.25
1.294050
0.055970
0.160456
1.50
1.575835
0.080877
0.223404
1.75
1.866817
0.106975
0.286660
2.00
2.165145
0.132021
0.345647
2.25
2.469269
0.154547
0.399411
2.50
2.778873
0.174757
0.449140
2.75
3.093901
0.193242
0.495616
3.00
3.414610
0.210211
0.540041
3.25
3.742279
0.225862
0.584559
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3.2 RAINDROP CHARACTERISTICS
41
3.2.2 Drop Size Distribution
The drop size distribution (dsd) reflects the distribution of the drop sizes as a func­
tion o f the rain rate. Many raindrop size distributions have been reported. The ear­
liest paper on the size o f raindrops was by Laws and Parsons [Laws43], where the
distribution is tabulated. This distribution was used by Oguchi [Oguchi60],
[Oguchi64], as well as Olsen [OIsen78] and many others. Later, the exponential
drop size distribution was empirically proposed by Marshall and Palmer
[Marshall48]. The latter distribution is well accepted in the meteorological domain
and in radar analysis. It is also used in the derivation o f attenuation such as found
in Olsen [01sen78] or Li, et al. [Li95b]. Joss and Waldvogel [Joss67] proposed
another model, another exponential form, by dividing the rain into three types:
drizzle (J-D), widespread (J-W), and thunderstorm (J-T). Recently, Ulbrich and
Atlas [Atlas74] studied the gamma distribution as a raindrop size distribution.
Finally, the Weibull drop size distribution was proposed by Sekine and Lind
[Sekine82] and has been o f considerable interest in its application to rain attenua­
tion.
Only the M-P and Weibull dsd’s will be considered later in the derivation o f spe­
cific attenuation. The M -P dsd is chosen because it is widely accepted, although is
found to overestimate the number of small drops [01sen78]. The Weibull dsd is
chosen because it was found to yield the least rms error with experimental data as
opposed to the other mentioned distributions [Jiang97]. There have been numerous
other interesting papers written about the Weibull dsd and its application to radio
wave propagation, such as [Sekine88], [Jiang96a], and [Jiang96b].
The choice of the drop size distribution is crucial and affects the resultant specific
attenuation. Limiting the distributions to these two is mostly for simplicity because
they will be combined with the total cross section for both spheroidal and P-P rain­
drop shapes. At any specified frequency, this results in four different combinations
and sets o f values for the specific attenuation.
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42
MODELING ATTENUATION AT A POINT
3.2.2.1
Marshall-Palmer dsd
The Marshall-Palmer drop size distribution [Marshall48] is expressed as:
N ( a 0) = N 0e~2Aa°
where
a0
m ^ m r1
(EQ3.78)
is the equivolumetric drop radius expressed in mm and the other parame­
ters are given by:
N
= 8000
r W
A = 4.1 R~°2'
mm'1
(EQ 3.79)
The precipitation rate is R and is in mm/h.
The M-P radar reflectivity Z-R relation is given by [2]:
Z = 200/? 16
3.2.2.2
(EQ 3.80)
Weibull dsd
The Weibull drop size distribution [Sekine82] is expressed as:
(EQ 3.81)
W<«o> =
where the precipitation rate R is in mm/h and
N 0 = 1000
m-3
r) = 0.95/?° 14
(EQ 3.82)
a = 0.26/?°42 mm
The Weibull reflectivity Z-R relation is given by [Jiang96b]:
Z = 285 /?148
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(EQ 3.83)
43
3.3 THEORETICAL DERIVATION
3
.3T
H
E
O
R
E
T
IC
A
LD
E
R
IV
A
T
IO
N
When the total cross section and the drop size distribution are determined as a
function o f the drop size, the specific attenuation can be computed. First, the atten­
uation in decibels per meter when a single drop per cubic m eter exists is given by
[Oguchi60]:
dB/m
drop/m3
(EQ 3.84)
considering a drop of radius a Q.
The specific attenuation, therefore, can be computed by integrating (EQ. 3.84)
with the drop size distribution over all drop sizes. By choosing the appropriate
total cross-section, i.e., case I or case II, vertical and horizontal specific attenua­
tions are obtained.
The rain attenuation in dB per kilometer is given by [Oguchi60]:
QT(a0)N(a0)da0
y = 8.6859 x
dB/km
(EQ 3.85)
o
where
Q T(a 0 )
represents the total cross-section of the raindrop, expressed in m2;
N( a0) , the rain drop-size distribution, expressed in m ' W
1; and a0 in mm.
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4
MODELING POINT-TOPATH ATTENUATION
Given the rain attenuation o f a microwave link at a point, the calculation of the
path attenuation is not straightforward due to the inhomogeneous characteristics of
rain in the horizontal. An effective path length, which is affected by the rain pres­
ence, should be considered in that case. Due to the relation between the attenuation
and the rain rate, the effective path length is simpler to estimate with the precipita­
tion rate as a parameter. The experimental data for path attenuations depends on
the rain rate and the operating frequency, which limits its use in estimating the
effective path length. Lin [Lin77] has shown that satisfactory agreement was
obtained between measured attenuation values and those calculated from measured
line rates, by using the relationship A = k R£ . The line rate, RL , is obtained from
near-instantaneous
rl =
rain rates along the path L of the radio link, i.e.,
( I / L ) j R d l . Harden et al. [Harden77] suggested the use of rain distributions
with integration time of 10s for the near-instantaneous values and gave ratios to
transfer other integration time values to the suggested ones. In their following
paper [Harden78], they used readily available rainfall data and compared experi­
mental observations of path attenuations from 10-100GHz with the estimated val­
ues using L in’s method [Lin77], and concluded that the method is a practical
procedure. Crane [Crane80], from the study of numerous data of rain gauge net­
works, found a power law relationship between the line rain rate and the point rain
rate. Because the specific attenuation is a nonlinear function of point rain rate, the
45
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46
MODELING POINT-TO-PATH ATTENUATION
empirical power law relationship was numerically differentiated to obtain the
instantaneous rain profile model, which is then expressed with piecewise exponen­
tial functions. Finally, the path attenuation is calculated from A = J k Ra{l)dl, and
the result is shown in (EQ. 2.6).
Crane [Crane82] presented a new model for the prediction of single path attenua­
tion statistics which calculates the occurrence probabilities of convective cells or
widespread debris regions of rain along a propagation path. The path attenuation is
then estimated by trial and error until the required occurrence probability is
reached. The two-component model models the rain as a joint event with volume
cell and debris contributions. A revised model was then suggested [Crane89] and
found its utility in the study of site diversity performance.
The latter version of the two-component model will be used in the study of pointto-path attenuations because of its ability to analyze performance o f different path
geometries, especially those of a network.
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4.1 SPATIAL CORRELATION OF RAIN
47
4.1 SPATIAL CORRELATION OF RAIN
The spatial correlation function for rain is needed in the two-component model
theory. The spatial spectra obtained from the radar observations and from the rain
gauge measurements provide a good starting point for the estimation o f a spatial
correlation function. Available observations all have the same shape and spectra
and usually show a power law increase in spectral density with decreasing wavenumber.
By assuming an outer scale with a constant spectral density for scales larger than
that scale, the integral may be performed, the process is stationary, and a correla­
tion function may be calculated. The minimum outer scale must be greater than
256 km. This value of the outer scale was used together with an idealized, seg­
mented power spectrum to create the correlation function estimate. The idealized
spectrum had an energy input scale of 12 km and a rain input scale o f 4 km. The
spectrum had a k '5/3, k"3, and k*1 power law region as provided by Crane
[Crane90].
10* '
12km
gjo-
10“*
ll>*:
Figure 4.1
One-dimensional idealized spatial spectrum for ln(rain rate)
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48
MODELING POENT-TO-PATH ATTENUATION
The spatial correlation function for the logarithm o f rain rate is then obtained from
the inverse Fourier transform o f the spatial power spectrum for variations in the
logarithm o f the rain rate. The correlation function for rain rate can be obtained
from the correlation function for the logarithm of rain rate by:
^2)I"**
------
Pr U i. I i ) =
(EO«.1)
e b- \
The correlation function pR(li,l2) depends on the rain climate zone through the
parameter SD, the standard deviation of the natural logarithm o f rain rate.
ln(Rain Rale)
Rain Rale
o.x
06
c
%
0.4
C
-
0.2
40
60
HO
too
120
Distance, kxn
F igure 4.2
Spatial correlation function for Infrain rate) and for rain rate (Climate D2)
4.2 TWO-COMPONENT MODEL
The revised two-component path attenuation prediction model [Crane89] is an
extension and refinement of an earlier work of Crane [Crane82]. The two-compo­
nent model separately addresses the contributions o f volume cells and of larger
debris regions of lighter rain intensity surrounding the cells. Furthermore, it con­
siders the cell and debris contributions as statistically independent and sums the
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49
4.2 TWO-COMPONENT MODEL
probabilities o f occurrence of the attenuations due to both components to estimate
the total. Assuming independence, the joint probability of occurrence o f rain in a
cell and in debris is small and may be neglected.
The model for the volume cell sizes and intensities and lifetimes is exponential.
The statistical model for the debris is lognormal. The two-component model for
the empirical rain rate distribution function is:
P ( r > R ) = P c ( r > R) + PD(r >R) —PCD(r > R)
Pc( r > R ) = Pce~*/Rc
A n R - l n R D\
PD( r > R ) = P dN [ - ^ — 2 )
(E0 4.2)
PCD( r > R ) = Pc (r > R)PD(r > R ) = 0
where
P(r>R)
=
probability that observed rain rate r exceeds specified rain rate R
Pc( r > R )
=
cumulative distribution function for volume cells
PD(r >R)
=
cumulative distribution function for debris
P cd(r ^ R )
=
j° 'nt cumulative distribution function for cells and debris
r, R
=
rain rate
Pc
=
probability of cell
Rc
=
average rain rate in cell
N
=
normal distribution function
PD
=
probability of debris
Rd
=
median rain rate in debris (calculated from average of natural log­
arithm of rain rate)
SD
=
standard deviation of natural logarithm of rain rate
The parameters are fitted to the rain rate distributions found in [1], first by estimat­
ing the lognormal curve for low rain rates or high probability occurrences, and sec­
ond by fitting the exponential curve to the difference between the median rain rates
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50
MODELING POINT-TO-PATH ATTENUATION
and the lognormal values at higher rain rates or lower probability occurrences. The
results are shown in Table 4.1.
TABLE 4.1 Parameters for the Two-Component model by region (source [1]).
A
B
0.023
14.3
10.8
0.178
1.44
C
0.033
19.8
12
0.293
1.31
D1
0.026
23.2
8.19
0.463
1.34
D2
0.031
14.3
9.27
0.475
1.48
D3
0.048
17.0
4
1.97
1.21
E
0.22
23.2
5.25
2.02
1.25
1.81
3.45
Pd
2.27
sD
Pc
0.000088
Rd
0.205
1.49
F
0.0048
6.95
0.0994
G
0.028
50.5
9.82
1.82
1.20
H
0.048
35.4
7
2.47
1.49
8.26
4.2.1 Model for Volume Cell Component
Rain cells often produce intense rainfall and cause severe attenuation to propagat­
ing signals over short time intervals.
The average horizontal cell dimension Wc that spans the region with marginal
reflectivity ZM (a factor of 2, i.e., 3 dB, below the peak value in the cell, Zy) is:
W c = JS~C = J 3 3
km
(EQ 4.3)
When the Laws and Parsons drop size distribution is chosen, the reflectivity can be
approximately related to rain rate by:
Z = 400/?14
(EQ 4.4)
The marginal rain rate RM at the cell edge is then:
1 _ zl
2 ZM
=
*vY-4
{R j
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(EQ 4.5)
4.2 TWO-COMPONENT MODEL
51
and
(EQ 4.6)
Other relations between the marginal and the peak rain rate in the cell can be
obtained when using different Z-R relations for M -P and Weibull, i.e., (EQ. 3.60)
and (EQ. 3.63), respectively.
Assuming the spatial rain-rate profile along a horizontal line through a rain cell is
Gaussian, (EQ. 4.6) is used to establish the rain profile as a function o f the distance
among the line. For a line through the cell center with distance x measured along
the line from the cell center, the rain-rate profile R(x) becomes:
R ( x ) = R ve
( x / W c r i n i K M/ R Y)/(. 1 /2 )2
= Rve
- 1 .9S(.x/W c f
(EQ4.7)
and the profile for the specific attenuation is given by:
y(x) = K[R(x)Ja = KRye~,9Sa(x/Wc)2
(EQ 4.8)
y(x) = KRye~l/2(x/Sv)
where
Sv
= Wc/-/3 .9 6 a.
The horizontal path attenuation
A j-
along the x axis between terminals at -L/2 and
+L/2 due to a volume cell with its center at (x,y) is expressed by:
-1 /2 (1 / S v y
e
e
- 1 /2 ((x -x V S » )
- 1 / 2 <v/S„)* - 1 / 2 ( x / S y )
e
. j
dx
, ,
dx
(EQ 4.9)
- L/2 - x
where.
l/2(«)
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(EQ 4.10)
52
MODELING POINT-TO-PATH ATTENUATION
and L is the horizontal length of the propagation path.
The peak rain rate in the cell, Rv, for a specified Ap on the path is given by:
/
f i y ( x , y> a t )
—
- l / 2 ( y / 5 v )J
^l/a
j kS v J 2 k { F n [ { - L / 2 - x ) / S v] - F n [ ( L / 2 - x ) / S v ]})
(EQ4.11)
Any peak rain rate larger than Ry occurring in the volume cell will result in a path
attenuation that exceeds Ap when all other conditions are not changed. Therefore,
the probability o f exceeding Ap is given by:
Pc( a > A T)
= Pc ( r 2 * v ,) |
= Pce~*v/*C\X y = P c e~Kv(x-y-A^ / R c
(EQ4.12)
The result shown in (EQ. 4.12) applies for a volume cell at (x,y). To obtain the
probability P{a > A T) , an integration of the conditional probability times the occur­
rence probability density for a volume cell at (x,y) is required over all space sur­
rounding the path. Assuming the occurrence probability density for a cell is
uniform, and a normalization area o f A = 1.0km2 (source [1]), the integral is then
given by:
I L L Pc( a > A T) ^ d x d y
=j f f
Pce~Rv(x' y Ar)/Rcdxdy
(EQ 4.13)
■j f3Sv/ 2 J L + 3 S v ) / 2
Pc ( a > A T) = =-\
AJ
o
f
Pc e~Rv(x' '• Ar)/Rcdxdy
J ( - L - 3 S v) / 2
This integral may be numerically solved using Legendre polynomials.
4.2.2 Model for Debris Component
The rain debris region is associated with light to heavy rain intensities. The proba­
bility density function (pdf) for the debris component o f the mixed rain-rate pro­
cess is assumed to be jointly lognormal.
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4.2 TWO-COMPONENT MODEL
The lognormal probability density function for the rain debris is
53
f u , { r ) . Assuming
that the path attenuation is approximately lognormally distributed, the approach is
to estimate the parameters of the logarithm of the path attenuation pdf from the
parameters of the point rain-rate pdf.
The mean and variance of the logarithm of specific attenuation are obtained from
the parameters of the rain-rate pdf by a linear relationship. Using the notation
for the mean and standard deviations for linear processes and
M, S
\l , a
for logarithmic
processes,
My = ln(K) + a I n ( f l D) = ln(ic) + a A /D
Sy ~
ttSg
and
UY = E[ Yl = £ [* /? “ ] =
kj ^
f u t W d r = KeaM° * i a *°
(EQ 4.14)
0
a? = Var[ y] =
- ^ f ^ d r
= K*e2aM° * 2a^ - p*
(EQ 4.15)
o
Parameters for y are needed instead of ln(y) to find the mean and variance of the
path attenuation by integration along the path.
The parameters for path attenuation (linear in A) are given by:
\iA = E[A] = E ^ y ( x ) d x ] = ^ E [ i ( x ) ] d x = \lyL
a I = E[{A-ViAn
(EQ 4.16)
= E[(^j^(icA?V)-UY)(K:/?V,)-U y)^ -^ "J
(EQ 4.17)
= t f f f <K* V ) - U tX k K V ') - H*) • f j NW ) , r'\x"))dr>dd'dx'dx'
J n0 JJ0
n JJ n0 JJn0
where f JN(r\ r") is a jo in t probability density function to describe the relations
between rain rates (linear in rain rate) at different spatial positions.
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54
MODELING POENT-TO-PATH ATTENUATION
If the spatial correlation function for specific attenuation is defined by:
then
f
• f j Nw
f
PT(JC - X ) =
) , n n w d ? '
—
and (EQ. 4.17) can be expressed as:
<52A = o l t t PT(*' - X")dx'dx''
(EQ 4.18)
' J qJ q
The spatial correlation function for the logarithm o f rain rate, Plnr , was obtained
from a Fourier transform of the spatial spectrum, see Section 4.1. The spatial cor­
relation function for rain rate, pR, may be obtained from P lnr by [I]:
- x") _
= -—
35— ;—
1
e*b -
1
1
(EQ4.19)
a-SfiPl h i I x' - i -) _ .
= — ^ 7 1
—
The parameters for the lognormal distribution for attenuation - mean
ance
SA
MA
and vari­
- are obtained from p A and a A:
m a
=
ln(pA)~5ln
S I = In
&
1 + -4
(EQ 4.20)
Pa
1+-7
(EQ4.21)
Ha )
To obtain the desired attenuation statistics, we need the pdf for the logarithm of
path attenuation with the mean and variance values given by (EQ. 4.20) and
(EQ. 4.21):
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4.3 APPLICATION TO NETWORKS
'l/2 ( ( ln C it) - M A) / 5 Ar
fut{A)
55
(EQ 4.22)
The cdf for a single path is then obtained from:
P D( a > A T) =
p
X
JA r
f w W d a = PDf
Jin
\to(.AT) - M A\ / S A
where f N(u) is the unit-normal pdf and
at
f N(u)du
(EQ 4.23)
is the attenuation level to be exceeded.
Numerical integration was used to evaluate the integrals in (EQ. 4.18) and
(EQ. 4.23).
The final probability o f exceeding a specified threshold attenuation
AT
is the sum
of the probabilities for the volume cell and debris components:
P ( a > A t ) = P c ( a > A t ) + P D( a > A T)
(EQ 4.24)
4.3 APPLICATION TO NETWORKS
4.3.1 Parallel Diversity
The two-component model employs the volume cells plus the inhomogeneous
debris region description to characterize the variability in rain rate and specific
attenuation needed to calculate the joint occurrences of attenuation on parallel
propagation paths. The two parallel path lengths are L with a base length B and
angle (3 between path 1 and the base.
L
✓
✓
Path 1
#
Figure 4.3
Geometry for a parallel topology
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56
MODELING POINT-TO-PATH ATTENUATION
4.3.1.1
M odel for Volume Cell Component
x’*
\J1
uz
C ell
F igure 4.4
Geometry for cell contribution calculations for a parallel diversity
Due to symmetry, a boundary line can be set up at y=0 midway between the paths.
Any rain cell that occurs above this line will contribute more attenuation to the
upper path than to the lower one. The joint probability o f exceeding Ap on both
paths 1 and 2 with attenuation A t and A2, respectively, and with the higher attenu­
ation on the upper path (A2) is then equal to the single path probability given in
(EQ. 4.12), but with the constraint that the rain cell must occur above the boundary
line, y=0, and its peak value calculated with respect to path 1:
Pc( a l > a 2 n a 2 > A T) =
J ” e~Rv{x'-^-A^ /R^dxdy
p
J , + 3 S v/ 2 J J , + (L + i S v ) / 2
A 3q
-t'
—
X
X"
y"_
X
(EQ 4.25)
3(- l -- 3 S v) / 2
(EQ 4^6)
+
y.
=
X
y.
(EQ 4.27)
+
-fl.
and,
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4.3 APPLICATION TO NETWORKS
b]
_ B COsP
2 sin P
By
57
(EQ 4.28)
For a volume cell below the middle line, similar probabilities are obtained. By add­
ing the occurrence probability for both cases, the joint probability o f exceeding Ap
on both paths becomes:
PC(a x > At r\a-,> A t ) ~
2 />
A
4.3.1.2
f B, * 3SV/ Z -« , + <£. + 3Sy)/2
F
I
JQ
J ( - L - 3 S v) / 2
e - B y ^ * B c y * B r A T) / R c d x d y
(EQ 4 .2 9 )
Model for Debris Component
The geometry o f a diversity system for use in modeling debris is illustrated in
Figure 4.5.
Bam p
u*\y")
Bun p
<*\y’)
F ig u re 4.5
Geometry for debris contribution calculations for parallel diversity
The joint pdf of path attenuations Ai and A2 in a site diversity system can be
assumed to be a joint lognormal pdf as follows:
(■
2kS \
a xA
2*I\ -
c'
(EQ 4.30)
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58
MODELING POINT-TO-PATH ATTENUATION
and c denotes the joint correlation coefficient related to the correlation function of
path attenuation p^A ,. a2) by:
In
(EQ 4.31)
C
and p^fA,, a 2) is approximated by:
PaMi’ A 2) -
Cov[A,,A2]
Var[Al
t f ’*BC‘**p1( J ( B s i n $ ) 2 + ( x ' - x " ) 2)dx'dx'
Pa(^1’ A 2) —
(EQ 4.32)
^0 Jficosfl
f ( ' p 1(x\x")dx'dx"
This integral is for parallel paths o f equal length L. (EQ. 4.32) was evaluated
numerically. M ore complex geometries can be accommodated by recognizing that
the argument o f Py is the magnitude o f the vector distance between point
1 and
x"
x’
on path
on path 2.
The joint cumulative distribution function is evaluated by:
P
1
—
^ a 2 —A t ) = P d f* f ” f
[ln(Ar )
jln ^ u '
u )^u du '
—M a \ / S a
(EQ 4.33)
Finally, the joint cumulative distribution for a balanced diversity system with the
attenuation thresholds set at A r for path 1 and 2 is given by:
P(a, > / t r n a 2 ^ A T) = Pc ( a x > A T r~>a2 ^ A T) + P d(.°\ - A T (~^a2 > AT)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(EQ4.34)
4.3 APPLICATION TO NETWORKS
59
4.3.2 Node Diversity
In a network, a node arises when a signal has to travel through path 1 or path 2 as
viewed in Figure 4.6. Both path lengths are considered to be equal to L, and a sep­
aration angle (3 is considered.
F ig u re 4.6
Geometry for a node topology.
4.3.2.1 Model for Volume Cell Component
Considering the symmetry in the geometry of Figure 4.7, a line is passed through
the intersection of both paths and in an angle of p/2. When P is zero, both paths
coincide and the middle point is considered for the y axis. When a volume cell is
above the boundary line y=0, more attenuation is observed on the upper path than
on the lower one. The jo in t probability of exceeding Ap on both paths, I and 2, and
with the higher attenuation on the upper path is then equal to:
P c(a ! —^2
A-j-)
—t J o
d xdy
(EQ 4.35)
with
(EQ 4.36)
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60
MODELING POINT-TO-PATH ATTENUATION
cos 5P
X”
jr.
sinxP
2
2
+ -L ,
- 3 cosxP y. r h .
—sinx
2
2
X
(EQ 4.37)
and,
i
P'
1 - COSx
2
(EQ 4.38)
- P
sinx
2 .
/>•
F ig u re 4.7
Geometry for cell contribution calculations for a node topology.
Integrating similarly on Box 2 and adding results together, yields the following
o t
^
A
^
^
a
\
P C(°
f£(cos|J —
0.5) ♦ 3 S y/ 2
«- R v(iT.
A T) / R C .
.
dxdy
2
Pc
+t J o
(EQ *
*
3S*/2 rucosfi -0.5) +■3Sv / 2 - R y « . v\ A T) / R C . .
] ( - L - 3 s y)/2
e
<**&
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4.3 APPLICATION TO NETWORKS
4.3.2.2
61
Model for Debris Component
Refer to Figure 4.8 for the node probability calculations due to debris. The
approach is very similar to that o f Section 4.3.1.2, except for the correlation func­
tion of path attenuation p ^ A ,, A2) that is needed in the joint correlation coefficient
c.
U’.y’)
F igure 4.8
Geometry for debris contribution calculations for node topology.
Therefore,
(EQ 4.40)
,’ 4- <- B
<—
P 4
r t PT(*\ x")djtdx"
JoJo
f
^ 2) -
Ja
(EQ 4.41)
t Py( J ( x ’- x " ) 2 + (x"lan$)2)dx'dx"
Jo
, otherwise
x")dx'dx"
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62
MODELING POINT-TO-PATH ATTENUATION
Finally, the joint probability contributed from debris is evaluated by:
P o (a i
A T r\
A T) — P p ( “
f
u")du’du"
J a tJ a t
=
P D f [ l n ( A r ) - M A / S / # ( « M) f
J 1 1 r)
I----J aaimAT)
i n ( A T) -- M
M A\
A]/Sa-cu-)/Jl
/S„-cu-)/Jl-c
-c‘
(EQ4.42)
Finally, the joint cumulative distribution for a node system with the attenuation
thresholds set at
AT
for path 1 then 2 is given by:
P ( a l > A T C i az > A t )
=
P c ( a l > A T r \ a 1 > A T) + P D( a l > A T r \ a 2 > A T)
(EQ4.43)
4.3.3 Relay
The relay term is used for a geometry where the signal goes through two paths
simultaneously, path 1 and 2 in the case of Figure 4.9.
F ig u re 4.9
Geometry for relay topology.
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4.3 APPLICATION TO NETWORKS
63
Unlike the previous geometries where the joint probability of excedance was of
importance, in this case the union probability o f exceeding
at
on each paths, 1 and
2, needs to be evaluated as follows:
I ^ A f U
^ 2 — ^7*1 =
— A y ] + P [ ( l 2 — A f ] “ P \.Q \ — A f C \
^ A t]
= 2P[ax > A t ] - P{ a x > A T n a 2 Z A T]
(EQ 4.44)
The probability is computed from the probability o f exceeding A T on a single path,
given by (EQ. 4.24), and from the joint probability of exceeding A r on both paths,
given by (EQ. 4.43).
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5
PERFORMANCE
ANALYSIS
This chapter will present the implementation of the models discussed in previous
chapters and includes simulation results and performance evaluations. All the
models were implemented using Matlab version 5.1. This software was chosen as
it is very powerful for matrix manipulations and the code is optimized in the sense
that computations are vectorized. Matlab also has a multitude o f readily available
functions such as the special Bessel functions and the Legendre polynomials
which were needed in Chapter 3. The calculations of the derivative o f the spherical
Bessel functions and the Legendre polynomials were carried out by the recurrence
formulae found in [3]. The values were tested for exactness using mathematical
tables for real arguments. Other utilities found in Matlab were the statistical func­
tions which where needed for modeling the two-component rain rate distribution
model in Chapter 4. Gaussian quadrature with Legendre orthogonal polynomials
was used in the evaluation of numerical integrals because a smaller number of
function evaluations were needed and high accuracy was achieved when the inte­
grand was well-approximated by a polynomial. In all derivations o f specific atten­
uation, the temperature of rain is considered to be 0°C, and the incident angle 0,
defined in Figure 3.1, is chosen for terrestrial line-of-sight links, i.e. 9 = rc/2.
65
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66
PERFORMANCE ANALYSIS
5
.1S
P
E
C
IF
ICA
T
T
E
N
U
A
T
IO
N
In this section, the specific attenuation based on the theory in Chapter 3 is com­
puted and compared them to the ITU Recommendation 838 (TTU838] standard.
For a specific frequency, there are two combination sets:
1. Raindrop shape: Spheroidal vs. Pruppacher-and-Pitter, or P-P
2. Drop size distribution: Marshall-Palmer, or M-P vs. WeibuII
The spheroidal raindrop shape and the M -P drop size distribution is a well estab­
lished combination, compared to the P-P with Weibull combination.
To obtain the theoretical specific attenuation, the total cross section and the drop
size distribution are integrated together. The total cross section is plotted in
Figure 5.1 for both frequencies 15 and 38GHz. The vertical polarity gives rise to
lower total cross section, or tcs, than the horizontal polarity. For the same drop size
distribution, or dsd, and frequency, the integration will result in higher horizontal
attenuations. Using similar reasoning, the P-P raindrop model results in specific
attenuations higher than the spheroidal one with constant frequency and dsd. The
P-P tcs is generally higher than the spheroidal case, given the added contribution
from the nonaxisymmetry of the drop shape.
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5.1 SPECIFIC ATTENUATION
Vcroca! Pwlanty (15GHz)
67
Horuuntal Polarity (15GHz)
*= 0.5
0.5
0
OS
IS
2
IS
0
3
05
Raindrop Radius, mm
2
3
Raindrop Radiui. mm
Vertical Polarity DUGH r)
Horizontal Polarity (3*GHz)
1.4
1.4
Z OK
0.6
0.6
0.4
0.4
0.2
0
1
2
Raindrop Radius, mm
3
0
2^
05
3
Raindrop Radius, mra
F ig u re 5.1
Total cross section comparing spheroidal and P-P raindrop model for 15(top) and
38(bottom) GHz.
The variation of the dsd with rain rate is shown in Figure 5.2 for both M -P and
Weibull. The choice o f the dsd greatly affects the specific attenuation. With the MP dsd having an exponential behavior, small raindrops, less than 0.5mm in radius,
occur in large number. The Weibull dsd attempts to correct that overestimation as
can be seen in the graph. For larger raindrop sizes, the Weibull dsd is very close to
the M-P one. For both drop size distributions, the higher the rain rate is, the larger
the raindrops present. This increase results in an increase in specific attenuation
with rain rate.
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68
PERFORMANCE ANALYSIS
Rain Rate 5 mm/h
Rain Rate 15 mm/h
1000
1500
M -P
W eibull
mo
600
3
e
7
500
200
0
I
1.5
■>
2.5
3
Rain Rale 50 mm/h
0
3.5
0.5
3
3.5
R ainm e IQDmm/h
1000
mo
MX)
E
E
E
e
400
WO
0
Figure 5.2
I
2_5
1.5
Raindrop Radius* mm
3
0
1
2J
3
Raindrop Radius, mm
M-P and Weibull drop size distribution for different rain rates.
Using a spheroidal raindrop, the specific attenuation is computed for a rain rate of
50mm/h and plotted for frequencies ranging from 3GHz to 3000GHz in Figure 5.3.
The ITU specific attenuation should not be considered for frequencies higher than
400GHz due to the fact that the coefficients for the formula are not available
beyond that frequency - a logarithmic interpolation yields the values beyond that
frequency. Moreover, the values given by the ITU-R838 have been tested and
found reliable for frequencies only up to about 40GHz. The generated specific
attenuation agrees closely in that range with the ITU, both for M -P and Weibull
distributions. The Weibull specific attenuation, unlike its M-P counterpart, follows
the ITU one up to 100GHz.
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5.1 SPECIFIC ATTENUATION
69
The total cross section shown in Figure 5.1 shows how the tcs increases for small
drops when increasing the frequency from 15 to 38GHz. For example, the 15GHz
tcs starts off at raindrops with radius 1mm compared to the 38GHz case, at rain­
drops with radius 0.5mm. For the first case, considering raindrops larger than
1mm, the Weibull dsd equals the M-P dsd for small rain rates, and then becomes
larger for higher rain rates. The second case presents a Weibull dsd smaller than
the M-P dsd for smaller rain rates, and higher at larger rain rates, considering rain­
drops with radius larger than 0.5mm.
Horizontal Polarity (5Uram/fe)
Vertical Pulanty (5Umm/h)
30
-F -O -
ITU-R838
M -P
Weibull
25
20
15
10
5
1
1(1
100
Frequency, GHz
F ig u re 5.3
raindrops)
1000
0
Frequency. GHz
Specific attenuation varying with frequency for a 50mm/h rain rate. (Spheroidal
W hen the frequency increases, the total cross section changes shape to give more
weight to smaller drops and less weight to larger drops, but the overall magnitude
o f the tcs decreases. In the drop size distribution case, the inverse process occurs
relative to the presence o f drops with respect to the rain rate. When the tcs of dif­
ferent frequencies are combined with the same dsd at constant rain rate, the result­
ing specific attenuation increases with frequency. This continues until a maximum
is reached, around 200GHz where the specific attenuation starts to fall down with
an increasing frequency. The wavelength at 200GHz is 1.5mm, which is in the
range of the raindrop size.
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70
PERFORMANCE ANALYSIS
Once the frequency of the system is known, the most important factor when deal­
ing with specific attenuation is the rain rate. To see how they are related together,
the total cross section for the raindrop shape (spheroidal or P-P) is integrated with
the drop size distribution (M-P or Weibull) to give the specific attenuations at 15
and 38GHz. Finally, the standard ITU-R838 is compared with the theoretical spe­
cific attenuations and presented in Figure 5.4 and 5.5 for the spheroidal and P-P
cases, respectively.
As expected, the specific attenuation is higher for horizontal than vertical polarity,
and for P-P as compared to spheroidal raindrops.
Vertical Polamy (!5GHz)
Horizontal Polarity (15GHz)
15
15
£
e
£«
S
•3
e
-O -
ITU-R838
M -P
Weibull
,cr.
10
10
3
s
<
5
5
C/J
00
E 40
<
I(I)
50
00
50
100
Rain Rate, mm/h
Rain Rate, mm/h
Vertical Pulaiity (38GHz)
Horizontal Pulanty (38GHz)
ITU-R838
M -P
Weibull
150
44)
20
2t)
KM)
Rain Rate, mm/h
0
50
KM)
150
Rain Rate, mm/h
F ig u re 5.4
Specific attenuation comparing drop size distribution for spheroidal raindrops and
frequencies 15(top) and 38(bottom) GHz.
The Weibull dsd has lower presence of small drops when compared to the M -P
dsd. The increased presence o f small drops found in the M -P dsd has an impact on
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5.1 SPECIFIC ATTENUATION
71
the attenuation as the frequency increases, through the total cross section, and
leads to a higher rate increase in attenuation. A cross-over frequency is reached
beyond 40GHz, when the M-P attenuations become larger than the Weibull case.
On the other hand, the Weibull dsd has more large drops than the M-P dsd as the
rain rate increases and results in Weibull attenuations higher and more apart as the
rain rate increases. Beyond the cross-over frequency, the M-P attenuations are
always dominant because at that point the small raindrops, less than 0.5mm, have
an increased weight in the total cross section contribution.
In summary, the M -P specific attenuation starts at a lower level than the Weibull
case for lower frequencies and ultimately crosses it and becomes larger for higher
frequencies with the difference between the two becoming more noticeable as the
rain rate increases.
Vertical Polarity ( 15GHr)
Horizontal Polamy (ISGHz)
3(1
20
—I—
-O -
[T U -R 8 3 8
M -P
W eibull
(5
e
10
<
5
0
100
50
150
00
50
150
100
Rain Rate, rom/h
Rain Rate, mm/h
Vertical Pularity (38GHz)
Horizontal Polarity (38GHz)
SO
—
-O
g 40
IT U -R 8 3 8
M- P
W eibull
40
< 20
■¥
20
o
100
Rain Rate, mm/h
150
0
50
ISO
Rain Rate, mm/h
F ig u re 5.5
Specific attenuation comparing drop size distribution for P-P raindrops and
frequencies 15(top) and 38(bottom) GHz.
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72
PERFORMANCE ANALYSIS
Looking at Figure S.4 for spheroidal specific attenuations, the theoretical values
agree very well with the standard values, for both frequencies. In the IS GHz case,
the standard falls between the M-P and Weibull values. In the other case, the stan­
dard follows the M-P attenuation. In the P-P case, Figure 5.5, the attenuation val­
ues are higher, as expected, and the ITU-R838 attenuations follow the M-P ones
very closely at both frequencies.
Vertical Polarity (M-P)
-a g
-k ~
M
-O -
Horizontal Pularity (M -P)
15G H z
38G H z
300G H z
40
K Ra fit
< 20
0
50
0
150
50
Rain Rate, mm/h
150
100
Rain Rate, mm/h
Vertical Pularity (Weibull)
Hirtzontal Polarity (Weibull)
50
-a 6 40
-O -
15G H z
38G H z
300G H z
40
tc R ° fit
I 30
30
< 20
20
1/510
10
0
HI)
150
00
50
150
100
Figure 5.6
Specific attenuation with different frequencies for M-P(top) and Weibull(bottom)
dsd, assuming P-P raindrop model.
In Figure 5.6, the frequency is compared together for M-P and Weibull dsd’s, con­
sidering only the P-P raindrop model. A fit to the empirical formula
KRa
is also
plotted for the three frequencies 15, 38, and 300GHz. The theoretical values are
very well approximated with the formula. Later, these values will be used for com­
puting the theoretical path attenuation. In the Weibull case, the 300GHz frequency
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73
5.1 SPECIFIC ATTENUATION
crosses the 38GHz attenuation level due to the fact that at this high frequency, the
attenuation has started to decrease passing through a maximum which depends on
the rain rate. In the M-P case, the maximum has not been reached yet for any rain
rate. For a better view, Figure 5.3 shows a general behavior o f the specific attenua­
tion with varying frequency.
Taking only the 38GHz frequency, the specific attenuations are plotted in
Figure 5.7 comparing the standard ones to the theoretically derived ones with
spheroidal and P-P raindrop models, both for M -P and Weibull dsd’s. In the fol­
lowing sections, the 38GHz frequency will only be considered given its impor­
tance in radio communications and its commercial use. The following figure will
be needed for reference later on. The P-P raindrop derived specific attenuations
start very close to the spheroidal ones, and then become higher with higher rain
rates.
Vertical Pnianty (38GHz. M -P)
Horizonul Polarity (38GHz. M-P)
50
50
—i—
-O -
IT U -R 8 3 8
Sp h ero id al shape
P - P shape
40
30
30
< 20
5/5 10
0
50
100
Rain Rate, mm/h
Vertical Polarity (38GHz. Weibull)
Horizontal Pularity (38GHz. Weibull)
50
IT U -R 8 3 8
—f — Sp h ero id al shape
- O - P - P shape
g 40
SO
0
150
Rain Rale, mm/h
150
40
30
< 20
20
** It)
10
0
50
101)
150
0
Rain Rale, mm/h
F ig u re 5.7
50
100
150
Rain Rate, mm/h
Specific attenuation comparing theoretical spheroidal and P-P raindrop models with
the IT U -R 838 at 38GHz, for M-P(top) and Weibuil(bottom) dsd.
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74
PERFORMANCE ANALYSIS
5.2 PATH LINKS
Given the specific attenuation in decibel per kilometer, the path attenuation is esti­
mated. For the empirical models, an effective path length, which takes into account
the statistical presence o f rain along the path, is multiplied by the specific attenua­
tion to give the required attenuation for a particular unavailability requirement, p.
In the following, the standard derivation o f the path attenuation, ITU-R530 and
Crane, are based on the standard specific attenuation given by ITU-R838. For the
theoretical derivations, the two-component rain model gives the probability of
exceeding a certain attenuation level. A trial and error method in finding the
required unavailability requirement is employed to get the path attenuation along
the link. The theoretically derived specific attenuations are fitted to the
1c Ra
empir­
ical formula, and the parameters k and a are employed in the two-component
model.
Two different climates will be considered, the first one is the Montreal area which
is a temperate continental climate, and the second one is Singapore which is a trop­
ical climate. The choice o f these two locations was encouraged by the fact that the
standard prediction methods seem to work effectively in moderate climates, but
fail in tropical regions. Hereafter, the frequency of 38GHz will be chosen for the
simulations. The com puted parameters at that frequency for the empirical specific
attenuation formula are given in Table 5.1, both for the standard ITU-R838 as well
as the theoretical derivations using different raindrop models and drop size distri­
butions. The rain temperature is considered to be 0°C.
TABLE 5.1 Parameters for the empirical specific attenuation evaluated at 38GHz.
Ky
0.278
0.942
«H
0.954
M-P
0.426
0.484
0.866
0.875
Weibull
0.236
0.268
1.018
1.029
M-P
0.471
0.503
0.864
0.879
Weibull
0.266
0.278
1.010
1.034
ITU-838
Spheroidal
P-P
<*v
0.314
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5.2 PATH LINKS
75
5.2.1 Temperate Continental (Montreal)
The ITU climate region corresponds to K, and the Crane climate region to D l. The
latitude o f 45° N and longitude o f 73° W has been used to determine the regions. A
summary o f rain rate statistics is shown in Table S.2., focusing on unavailabilities
of 1,0.1, 0.01, and 0.001% of the time.
TABLE 5.2 Rainfall Rate Statistics (mm/h) for Montreal
1%
0.1%
0.01%
0.001%
ITU - Climate K
1.5
12
42
100
Crane - Climate D l
2.2
11
37
90
T-C* - Climate Dl
2.2
10.3
36.3
86.7
(*) From (EQ. 4.2) using a trial and error technique.
The ITU and Crane models have different climate regions, but the rain statistics are
very similar. The two-component parameters yield values closer to the Crane rain
rate distribution.
Figure 5.8 and 5.9 are the path attenuations for the ITU and Crane methods,
respectively. Starting with the same specific attenuation evaluated at 38GHz, dif­
ferent path attenuations are obtained through the calculations of the effective path
length at a specified time percentage. Decreasing the unavailability of the link
increases the attenuation. Consider the specific attenuation at 0.001% unavailabil­
ity, which is around 20dB, corresponding to a rain rate around 40mm/h. For a path
length o f 8km, the maximum achievable rain attenuation if the rain occurs along
the whole path is 20dB /km x 8km = 160dB. But in both standard prediction meth­
ods, the attenuation is around 1 lOdB. For higher unavailabilities, the difference
becomes gradually less obvious, due to the fact that average rainceli diameters are
smaller for the more intense convective rain than for more widespread weak rain.
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76
PERFORMANCE ANALYSIS
Vertical Polarity 08GHz)
120
KJO
Horizontal Polarity (38GHz)
- O - 0 .0 0 1 %
— — 0 .0 1 %
—*— 0 .1 %
-B 1%
c M)
|
<
60
40
20
00
□□□o o goo
2
4
Path Length, km
6
4
8
6
Path Length, km
F igure 5.8
Path attenuation in Montreal using ITU-R530 prediction method for a 38GHz
system and ITU-R838 specific attenuation model.
Vertical Polarity (3MGHz)
Horizontal Polanty (38GHz)
140
120
100
0.001%
0.01%
0.1%
-e- !%
-O'
——h—
120
100
a
"3
e
xo
n
ue
<
60
40
0
20
0
^
'
f & tr iT fr n n JJ u ,a n n n r w
4
Path Length, km
6
Path Length, km
F igure 5.9
Path attenuation in Montreal using Crane prediction method for a 38GHz system
and ITU-R838 specific attenuation model.
The theoretical path attenuations presented in Figures 5.10 and 5.11 for spheroidal
and P-P derived specific attenuations, respectively, show a similar behavior as
compared to the standard ones.
Finally, ail path attenuations are plotted together in Figures 5.12 and 5.13 for theo­
retical derivations using the M-P and Weibull dsd’s, respectively. Only the two
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5.2 PATH LINKS
77
smallest unavailabilities are considered because of their wider use in practice. For
both raindrop models, the theoretical path attenuations agree very well with the
ITU and Crane prediction methods. The standard values lie slightly below the two
derived ones, or in between them for both cases o f raindrop models. Yet a more
noticeable agreement is achieved when using the Weibull dsd, as shown in
Figure 5.13.
V e n ia l Polarity (MGHz. M -P)
140
Horizontal Polarity (33GHz. M -P)
140
-o -
120
100
T3
C HO
0.001%
. —— 0.01%
1-
.-e-
120
0.1%
1%
100
a
30
**
c
u
<
60
60
40
40
20
20
o
n a a-
0
□ □ □ o -e
0
o
:/r r
s>
o g u o n n n a o o
Path Length, km
Path Length, km
Vertical (Hilarity (33GHz. Weihull)
Horizooul Polarity (33GHz. Weibttfl)
140
120
MX)
o - 0.001%
. -—
— 0.01%
—1—
. -a-
0.1%
1%
a
c
HO
e
60
JX& --
<
40
20
O □ OO
0
Path Length, km
4
Path Length, km
6
F igure 5.10 Path attenuation in Montreal using the two-component model with spheroidal based
specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
78
PERFORMANCE ANALYSIS
Horizontal Polarity (38GHz. M -P)
Vertical Polarity (38GHz. M -P)
140
-o- 0.001%
120 —— 0.01%
—f— 0.1%
!<» -s_ 1%
120
&
100
80
60
40
ex
| r i tTiV n n n v ' u " n n n f > ^
4
Path Length, km
6
20
01
p r.
0
-t-+
p
-a & ■u □ □ o n -n
'4 i tj-s=a s 2
*
6
Path Length, km
Horizontal Polarity (38GHz. Weibull)
Vertical Pulanty (38GHz. Weibull)
150
-O- 0.001%
--- 0.01%
- t — 0.1%
-e- i%
ex
<3 "
P
cr
-
0.
0
^
nnriaOOOOOlHl
4
Path Length, km
6
ii
iilllinnnnfHi
4
Path Length, km
6
Figure 5.11 Path attenuation in Montreal using the two-component model with P-P based
specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
79
5.2 PATH LINKS
Vertical Pularity (38GHz. 0.01%)
Hunzuntal Pulanty (38GHz. 0.01%)
80
70
60
------ ITU
------C ran e
—
T - C (S p h ero id al)
-°-
T3 50
e
% 40
e
< 30
^ ©
...rrfJe-
T- C (P - P>
60
2(1
9 0 ^
10
0
0
(
Vertical Pulanty (38GHz. 0.001%)
4
6
S
Horizontal Polarity (38GHz. 0.001%)
70
70
60
60
3 50
40
40
= 30
20
20
0
I
3
0
4
Path Length, km
2
3
4
Path Length, km
Figure 5.12 Path attenuation comparison in Montreal for different prediction methods at 38
GHz and unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using M-P dsd.
Vertical Pulanty (38GHz. 0.01%)
Horizucual (Hilarity (38GHz. 0.01%)
80
ITU
C ran e
T - C (S p h ero id al)
T -C (P -P )
70
70
60
40
= 30
20
0
4
6
o
Vertical Pulanty (38GHz. 0.00!%)
<
40
8
1( 1 )
xo
60
6
Horizontal Pulanty (38GHz. 0.001%)
1CX)
c
4
80
40
20
o
3
Path Length, km
4
0
2
3
4
Path Length, km
Figure 5.13 Path attenuation comparison in Montreal for different prediction methods and
unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
80
PERFORMANCE ANALYSIS
5.2.2 Tropical Climate (Singapore)
The ITU climate region corresponds to P, and the Crane climate region to H. The
latitude o f 1° N and longitude o f 103° E has been used to determine the regions. A
summary of rain rate statistics is shown in Table 5.3.
TABLE 5.3 Rainfall Rate Statistics (nun/h) for Singapore
1.0%
ITU - Climate P
12
Crane - Climate H
6.4
T-C* - Climate H
12.5
0.1%
0.01%
0.001%
65
145
250
51
147
251
66.9
209.7
544.4
(*) From (EQ. 4.2) using a trial and error technique.
The Tropical climate will follow the same procedure as the previous one concern­
ing the output results. The ITU and Crane path attenuations, shown in Figure 5.14
and Figure 5.15, respectively, have much higher values than the temperate conti­
nental case. The major contributor to this increase is due to higher rain rate statis­
tics experienced in this particular climate region; therefore, specific attenuations
are higher. The effective path length is still less than the actual path length, as it is
seen through the inclination of the path attenuation with increased length.
When the effective path length is calculated in the ITU method, the rain rate at
0.01% for a maximum of lOOmm/h is included in the formula in order to adjust for
the climate region. For the other time percentages, the effective path length at
0.01% o f the time is multiplied by an appropriate factor. Even at Singapore’s high
rain rates, the effective path length becomes proportionally smaller than th e path
length as it is increased. On the other hand, Crane effective path length depends
directly on the rain rate, and its relation to the path length is almost linear.
Both prediction methods have similar rain attenuation for the different tim e per­
centages; ITU estimates slightly higher values than Crane for 0.001%, and lower
for the other time percentages.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
81
5.2 PATH LINKS
Using theoretical models, path attenuations are generated and plotted in Figures
5.16 and 5.17 for the M -P and Weibull dsd’s, respectively. The path attenuations
are much higher and tend to be very linear with the path length. In Table 5.3, the
rain rates at 0.01% and 0.001% estimated from the two-com ponent model are
higher than ITU and Crane. The rain rate distributions from [Crane80] were used
in the Crane prediction method discussed in Section 2.3.2, w hile the revised ones
found in [1] were used in the two-component model in Section 4.2. The latter one
uses a wider range o f observations. Considering the high attenuations in the tropi­
cal regions, the 0.01% unavailability seems to be more appropriate and yields a
difference of 30% in the rain rate when compared to the new value.
Vertical Polarity (3£GHz)
Horizontal Pularily (38GHz)
250
200
-o -
0.001%
—f -e -
0.1%
1%
— — 0.01%
,<3T
a KH)
&
or
<
a
I
0
0
-
1
Path Length, km
2
Path Length, km
3
4
F ig u re 5.14 Path attenuation in Singapore using ITU-R530 prediction method for a 38GHz
system and ITU-R838 specific attenuation model.
Vertical Polarity (3KGHz)
Huri/untal Polarity <38GHz)
-o- 0.001%
0 .01 %
-1- 0.1%
-a- i%
ISO
5s 100
c
<
&
5(1
1
m n a Q a a
2
3
Path Length, km
o 0 -0
<lrir £ 'a n n -H- n
1
a
O O O O 0 -g ~ Q
2
Path Length, k m
3
F ig u re 5.15 Path attenuation in Singapore using Crane prediction method for a 38GHz system
and ITU-R838 specific attenuation model.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
82
PERFORMANCE ANALYSIS
Hurizuttal Pobnty (38GHz. M-P)
V e n ia l P u b n ty (38G H z. M -P )
300
400
a
- o — 0.001%
— — 0.01%
■f- 0.1%
1%
2(»
Path Length, km
P ith Length, km
H uhzuaal P ubnty (38GHz. WabuU)
V e n ia l Pubnty (38GHz. WeibuU)
7 00
500
a
-3
e 400
«
oe 300
**
—o o o
600
700
MU
..................
...............:
a®
jf
500
a
400
300
<
2(X>
MX)
200
a
.
,T.
O
■■jcr
_0
| |
0 f + jr tr tr jr iTn n n n n a ttg j
100
I
Path Length, km
2
Path Length, km
3
F ig u re 5.16 Path attenuation in Singapore using the two-component model with spheroidal
based specific attenuation at 38GHz with M-P(iop) and Weibull(bottom) dsd.
The path attenuations are compared together in Figures 5.18 and 5.19 for the theo­
retical derivations using the M -P and Weibull dsd’s, respectively. Only the unavail­
abilities of 0.01% and 0.001% are considered as in the previous case of the
tem perate continental climate. In the tropical climate, the choice of the drop size
distribution model turns out to be critical as can be seen by the graphs where the
Weibull attenuations in Figure 5.19 are higher, especially in the 0.001% case,
when compared to the M-P values in Figure 5.18. As for the raindrop model, the PP raindrop shape attenuation in both dsd’s are higher, especially for the 0.001%
unavailability, because o f the greater presence of large raindrops at higher rain
rates. On the average, the difference is estimated to be around 8%. The ITU and
Crane model attenuations, on the other hand, are much lower than the theoretical
values for the reason discussed previously.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
83
5.2 PATH LINKS
Vertical Pobnty DSC Hz. M -P)
no
500
Horizontal Polahty (38GHz. M -P)
a
-o- 0.001%
0.01%
4 0.1%
1%
—
—
.
■ — —
.
A tte n u atio n , d B
:
. X
0 ^
:
............
200
*
_
jfr
n n n n g P ° o f »^9
2
Path Length, km
Path Length. km
700
A tte n u a tio n , dB
«X)
800
-O- 0.001%
0 .1 %
■O ■
I%
600
500
4»)
400
300
300
200
21X)
0
........... t
700
0 .01 %
-I—
5«)
UX1
3
Horizontal Polamy (38GHz. Westell)
Vertical P obnty (38GHz. Wesbtdl)
800
1 1 1 1
n n n n n-B -o
• ® 'or
0 0_ -
-f-t- nI nI n-a
I -*3
Path Length, km
0
t
" jjr 9
0
:.................
0
^
—
MX) ........so
t t
I I—
■»
Path Length, km
F igure 5.17 Path attenuation in Singapore using the two-component model with P-P based
specific attenuation at 38GHz with M-P(top) and Weibull(bottom) dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
84
PERFORMANCE ANALYSIS
Vertical Pulanty (38GHz. 0.01%)
Hurizuoul M aritjr (38GHz. 0.014)
250
250
200
-O -
£
ITU
C rane
T - C (S p h ero id al)
T - C ( P -P )
200
150
150
100
100
50
0
3
0
4
4
3
Horizontal Polarity (38GHz. 0.001%)
Vertical Pulanty (38GHz. 0.001%)
250
250
200
150
100
50
o
0.5
0
1
I
1.5
Path Length, km
Path Length, kin
F igure 5.18 Path attenuation comparison in Singapore for different prediction methods at 38
GHz and unavailabilities of 0 .01(top) and 0.001%(bottom), theoretical derivation using M-P dsd.
Horizontal (Hilarity (38GHz. 0.01%)
Vcnical Pubnly (38GHz. U l l l l )
ITU
Crane
—I—
-O -
250
T - C (S p h ero id al)
T - C (P -P )
200
150
100
50
0
2
3
4
Horizontai Pulanty (38GHz. 0.(101%)
Vcnical Pulanty (38GHz. 0.001 %)
350
350
300
3(1)
*3 250
250
200
= 150
150
I(X)
100
50
o
0.5
1
Path Length, km
2
0
I
Path Length.
1.5
2
F igure 5.19 Path attenuation comparison in Singapore for different prediction methods and
unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.3 SYSTEM ANALYSIS
85
5.3 SYSTEM ANALYSIS
With the knowledge o f the rain attenuation along the path length, a microwave sys­
tem can be simulated and analyzed following the model described in Section 2.4.
TABLE 5.4 Parameters for the microwave system
f
38.0GHz
D
30.72cm
Ptx
16.0dBm
T rx
-82.5dBm
The parameters used for the model are shown in Table 5.4. Considering the diam e­
ter for both antennas to be 12in, or 30.72cm, the gain of the transmitter and
receiver can be calculated using (EQ. 2.10). Attenuation values, from the previous
section, are included in the calculation o f the fade margin for the study o f system
performance in the presence of rain.
5.3.1 Temperate Continental (Montreal)
When a fixed antenna diameter is considered with the values given in Table 5.4,
fade margins are calculated with varying path length between the transmitting and
receiving antennas. The theoretical margins are compared with the ITU and Crane
model values for the lower unavailabilities and presented in Figure 5.20 for an MP dsd and in Figure 5.21 for a Weibull dsd. Similar margins are observed for the
0.01% availability, but for the 0.001% the margins are higher in the M -P case.
With higher margins, the maximum distance between the antennas is increased as
can be seen when looking at the zero margin values. Overall, the theoretical mod­
els that describe the rain effect on the microwave link are performing well when
compared to the ITU or Crane model. The difference is well under the 20% bound
for the P-P raindrop model, and 15% for the spheroidal case.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
PERFORMANCE ANALYSIS
Vertical Pulanty OSGHz. 0 .0 1 * )
—t—
-O -
Horizontal Polarity (38GHz.0.01%)
IT U
C ra n e
T - C (S p h ero id al)
T - C (P -P )
40
“ 30
0
0
4
Vcnical Pulanty (38GHz. 0 .001 * )
4
6
8
Horizontal Pulanty (38GHz. 0.001*)
40
£
2
40
21)
0
3
4
Path Length, km
Figure 5.20 Fade margin comparison in Montreal for different prediction methods at 38 GHz
for unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using M-P dsd.
Vertical Pulanty (3KGHZ. 0.0 1 * )
Horizontal Polarity (38GHz. 0.01*)
60
60
——
-O
ITU
C ra n e
T - C (S p h ero id al)
T -C (P -P )
40
0
4
8
0
Vertical Pulanty (38GHz.0.001* )
4
6
Horizontal Polarity (38GHz. 0.001*)
50
s£
e
30
20
10
0
3
Path Length, km
4
o
1
2
Path Length, km
3
4
F igure 5.21 Fade margin comparison in Montreal for different prediction methods for
unavailabilities o f 0.01(top) and 0.001%(boltom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
87
5.3 SYSTEM ANALYSIS
Vertical Polarity (38GHz. 0.01%)
8
H uruoobl Polarity (38GHz. 001% )
8
7
7
E 6
6
45
5
4
3
--—— CITU
rane
- f-o -
1
00
T - C (S pheroidal)
T - C (P -P )
V e rtic al P u la n t y (3X G *Iz. 0.001 % )
20
80
40
100
H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 0 1 % )
JO T
2J
.or
0 J
0
2(1
40
60
0
IU0
20
A n te n n a D ia m e te r, cm
40
60
A nten n a D ia m e te r, cm
80
100
F igure 5.22 Maximum path length comparison in Montreal for different prediction methods at
38 GHz for unavailabilities of O.Ol(top) and 0.00l%(bottom), theoretical derivation using M-P dsd.
V e rtic a l P u la n t y (3 8 G H r. 0 .0 1 % )
X
7
E
€
H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 1 % )
8
7
e
6
6
Zsr.
5
5
4
£3
o0
3
—
— —f—
-O 20
rru
C rane
T - C (Spheroidal)
T - C ( P -P )
40
80
100
1
0o
V e rtic al P u la n ty (3 8 G H z . 0.(101% )
20
40
80
100
H o riz o n ta l P u la n ty (3 8 G H z . 0 .0 0 1 % )
JO
0
20
40
60
A n te n n a D ia m e te r, cm
80
100
0
20
40
60
80
100
A n ten n a D iam e te r.
F igure 5.23 Maximum path length comparison in Montreal for different prediction methods for
unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
88
PERFORMANCE ANALYSIS
Figures 5.22 and 5.23 show the critical zero margin values with varying antenna
diameter. The maximum path length is not appreciably affected after a certain
antenna diam eter is reached. The construction of bigger antennas does not appear
to be justified, particularly since they become more difficult to align and are quite
costly. The effect o f different drop size distributions is as important as the choice
of the raindrop shape, where the lowest distances are obtained with a P-P raindrop
shape with a Weibull dsd, as opposed to the highest separation between antennas
when the spheroidal model is chosen with an M-P dsd.
5.3.2 Tropical Climate (Singapore)
The theoretical margins are compared with ITU and Crane model values, for the
lower unavailabilities and presented in Figure 5.24 for an M -P dsd and Figure 5.25
for a Weibull dsd. Due to the higher rain rates in this region, the lower margins that
are witnessed in these figures were to be expected. When considering the theoreti­
cal values, the ITU and Crane predictions are almost double for the maximum dis­
tances, the point where zero margin is reached. When considering the theoretical
models, the choice o f the drop size distribution model has a m ore crucial impact on
the margin than the choice of the raindrop model. One should consider the diffi­
culty in predicting the performance of a microwave link in the tropical regions due
to the intensity o f rainfall as well as the added fluctuations from one year to the
other. With this in mind, the maximum distances between the transmitting and
receiving antennas should be set to a worst case scenario. Operating on a smaller
frequency may not be possible because of restrictions from frequency allocations
within radio regulations, or even because of saturation of frequency reuse. The last
attempt to increase the distance between antennas could lead to looser availability
requirements, by accepting the limitations imposed by nature. Finally, Figure 5.26
and 5.27 illustrate the maximum path length between antennas for varying antenna
diameter size. M -P dsd model shows better improvement than the Weibull case for
increasing antenna diameter.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
89
5.3 SYSTEM ANALYSIS
Vertical Pulanty (38GHz. 0.014)
100
IU0
80
-O -
e
ITU
C ran e
T - C (S pheroidal)
T - C (P -P )
Horizontal P ubnty (38GHz. 0 .0 1 4 )
80
a
~3
e
60
'ta
zao
1
40
0
Id)
3
0
4
Vcnical Polarity (38GHz. 0.0014)
100
80
I
2
3
4
Horizontal Pularity (38GHz. 0.0 0 1 4 )
80
E
a
60
c
ea
a
40
0
0J
1
o
IS
os
Path Length. km
I
Path Length, km
2
F ig u re 5.24 Fade margin comparison in Singapore for different prediction methods at 38 GHz
for unavailabilities of 0.01 (top) and 0.001 %(bottom), theoretical derivation using M -P dsd.
100
Vertical Polarity (3*GHc. <1.01*1
-H—
-O -
e
Horizontal Pulanty (38GHz. 0 .0 1 4 )
ITU
C ran e
T - C (S pheroidal)
T -C (P -P )
80
a
C
60
40
40
0
I
2
3
2
0
4
Vertical P ubnty (38GHz. 0.0014)
3
4
Horizontal Polarity (38GHz. 0 .0 0 1 4 )
100
80
80
E
a
~3
_e
40
3
u.
20
0
<L5
1
Path Length, km
2
00
0J
I
Path Length, km
F ig u re 5.25 Fade margin comparison in Singapore for different prediction methods for
unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
90
PERFORMANCE ANALYSIS
Vertical Polarity (38GHz, 0.01%)
HurizonaJ Pularity (38GHz. 0.01%)
IJ
:
______
rru
— — C rane
—f — T - C (S p h ero id al)
- o - T - C (P -P )
0.5
0
20
40
60
*o
o
(00
Vertical Pulanty (38GHz. 0.001%)
20
40
60
SO
Horizontal Pularity (38GHz. 0.001%)
E 1.5
JZ
S.
n
—eh =s
0
20
40
Antenna Diameter, cm
80
100
0
20
40
Antenna Diameter, cm
=^F= = 6
80
100
F ig u re 5.26 Maximum path length comparison in Singapore for different prediction methods at
38 GHz for unavailabilities of O.Ol(top) and 0.001%(bottom), theoretical derivation using M-P dsd.
Vertical Pulanty (38GHz. 0.01%)
Horizontal Pulanty (38GHz. 0.01%)
3
2.5
1.5
—
&■
—
-O O
20
40
rru
C rone
T - C (S p h ero id al)
T - C (P -P )
60
80
0.5
KM)
0
Vertical Polamy(38GHz. 0.001%)
20
40
80
100
Horizontal Pulanty (38GHz, 0.001%)
1J
E 1.5
c
^ 0.5
0
20
40
60
Antenna Diameter, cm
80
100
0
20
40
60
Antenna Diameter, cm
80
100
F ig u re 5.27 Maximum path length comparison in Singapore for different prediction methods
for unavailabilities of 0.01 (top) and 0.001%(bottom), theoretical derivation using Weibull dsd.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.4 NETWORK LINKS
91
5.4 NETWORK LINKS
Given the horizontal inhomogeneous characteristic o f rain, microwave network
links can improve in performance if certain considerations are well accounted for.
For example, an open network, where all nodes are connected together with
increasing bandwidth until the main node is reached, is extremely unreliable
because if one link is disconnected due to rain, then all the remaining links down
the line are lost as well. A ring topology, on the other hand, benefits from redun­
dancy and provides the opportunity to take the opposite route in case a link is
affected by a rain cell.
The three basic network blocks will be considered and analyzed. The two-compo­
nent model is used, and the temperate continental clim ate D1 is assumed. The gen­
eral approach is to analyze one parameter while the others are set in a way so not to
affect the analysis. The occurrence probabilities are compared to the single link
case to give an idea o f the improvement factor. Also, the cell to debris contribu­
tions will be plotted in order to spot the ranges with minimum cell contribution due
to their larger impact on system performance.
5.4.1 Parallel Diversity
Consider the parallel diversity geometry described in Figure 4.3. The attenuation
level is set at 3dB and the path lengths at 10km. First, the baseline B is varied with
orientation angle P fixed at 45°. The results are shown in Figure 5.28. The refer­
ence single link was taken to be the same size as the two parallel path lengths. The
figures shows a reasonable behavior with an improvement factor starting to be 1
for a baseline distance o f 0km, single link case, and starts increasing as the separa­
tion is bigger.The contribution from the volume cell remains comparable to the
debris contribution until a baseline distance of 3km is reached, when it starts to
decay rapidly. This corresponds to a separation between the paths of the order of
the cell width.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
92
PERFORMANCE ANALYSIS
A similar analysis was done for changes in baseline orientation relative to the
direction of the parallel paths. This is illustrated in Figure 5.29. The baseline is
taken to be 3km reducing the cell contribution, thus concentrating on the effect of
the orientation angle. For the overlapping parallel paths, the case o f 0° angle, the
contribution com es equally from cell and debris. The more perpendicular the paths
are, the more improvement is realized.
aac
0.1
— -O '
0.01
Improvement Factor
Ratio Cell to Debris
1
3
4
5
Baseline, km
Figure 5.28 Diversity improvement factor and ratio of the ceil to debris component as a function
of baseline length for jointly exceeding an attenuation of 3dB. Calculations are for a pair of
terrestrial paths o f 10km length with an orientation angle of 45° between the baseline and path
direction.
'r t x .
: : :
— -O 0
?:€h >
Improvement Factor
Ratio Cell to Debris
15
30
45
Hi
Orientation, deg
75
90
F igure 5.29 Diversity improvement factor and ratio of the cell to debris component as a function
of orientation angle for jointly exceeding an attenuation of 3dB. Calculations are for a pair of
terrestrial paths o f 10km length with a 3km baseline.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.4 NETWORK LINKS
93
5.4.2 Node diversity
The node diversity, illustrated in Figure 4.6, is analyzed in Figure 5.30 for an atten­
uation level o f 3dB and path lengths of 10km with a reference taken as the single
link with same path length.
Increasing the separation angle between the paths shows an improvement with a
fall off in the cell contribution for angles beyond 80°. A maximum improvement
factor is reached at an angle o f 135°.
— rr *“*».
3----------
1
®
©
- o - ^
- o - a ^
.....
0.01
0
k : : - :
30
60
90
120
Scpenuon Angle. <lcg
:
:
/•:
150
■:
180
F igure 5.30 Diversity improvement factor and ratio of the cell to debris component as a function
of separation angle for jointly exceeding an attenuation of 3dB. Calculations are for a node with
paths of 10km length.
5.4.3 Relay
Considering a network node could function as a relay, the topology illustrated in
Figure 4.9 is considered. The attenuation level is set at 3dB with path lengths of
10km. The occurrence probabilities are compared to the single link that starts at
the beginning of path 1 and finishes at the end of path 2, with a length of 2 L sin |. In
order to justify the use of the relay, the lengths of paths 1 and 2 should be less than
or equal to their base length. In geometric analysis, this is ensured with a separa­
tion angle larger than 60°. W hen the rain is taken into consideration, the simula-
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94
PERFORMANCE ANALYSIS
tion, illustrated in Figure 5.31, show that a minimum separation angle o f 80° must
be considered in order to have an improvement factor larger than unity.
— — Improvement Factor
- O - Ratio Ceil to Debris
:____-
" "
............. / .........
2> ^ '0 - 0 - 4
2T
/
7
'
e ,
(
0 .1 1
0
l
«
30
*
*
■
*
60
80 90
120
Scpcrauon Angle, deg
■ *
150
*
180
F igure 5.31 Diversity improvement factor and ratio of the cell to debris component as a function
of separation angle for exceeding an attenuation of 3dB. Calculations are for a relay with paths of
10km length.
In summary, when designing for network links, the following should be consid­
ered:
1. the topology of the network should ensure alternative paths in case of outage of
one individual link.
2. parallel links should be at least 3km apart.
3. adjacent links should be separated with an angle of at least 80°.
The design o f network individual links can be achieved as discussed in Section 5.3,
while the overall performance of the network should exceed the individual ones.
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6
FINAL REMARKS AND
FUTURE WORK
There is a continuing need for higher frequency spectrum use in microwave com ­
munications as the lower frequencies are being saturated. Understanding the the­
ory in predicting attenuations due to rain is crucial in the harnessing of the higher
frequencies. In this thesis, we have combined the perturbation method for estimat­
ing the total cross section and, ultimately, the specific attenuation, and incorpo­
rated it with the two-component model for the prediction of point-to-path
attenuations. The procedure was successful in the moderate climates, but some val­
idation is required in the tropical climates, especially when modeling the rain rate
distributions.
We will give a summary of the work found throughout the thesis and finish with
suggestions for future improvements of the models.
6.1 SUMMARY OF OUR WORK
In Chapter 1, we presented a brief overview of the rain attenuation calculation pro­
cedure. In the prediction procedure, the rain rate is a critical parameter and its
long-term statistical knowledge is crucial for system reliability.
Chapter 2 provided a review of the prediction of reliability for a microwave link
when degraded by precipitation. The prediction methods are based on the power
95
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96
FINAL REMARKS AND FUTURE WORK
law relation that exists between the rain rate and the specific attenuation. This rela­
tion has simplified the computation of the attenuation considerably by further
modeling the effective path length as a function of the rain rate. The procedures
used in the ITU and Crane models for predicting the rain attenuation were pre­
sented in detail. At the end of the chapter, a description of a simple system that
incorporates the rain attenuation was illustrated for performance analysis.
The modeling of specific attenuation at a point in space was introduced in Chapter
3. By using a general equation to represent the raindrop shape, perturbation theory
is used to solve the boundary conditions at the drop surface in order to solve for the
scattering coefficients, which were needed for the total cross section derivation.
Finally, the raindrop size distribution was estimated based on experimental obser­
vations, and combined with the total cross section in order to calculate the specific
attenuation. A new equation for the spheroidal raindrop has been suggested and
was shown to be a better approximation than the one used in the literature. More­
over, a more realistic Pruppacher-and-Pitter raindrop shape with its nonaxisymmetry is fitted to the general drop shape equation. With one equation to represent the
different raindrop shapes, it was possible to use the same equations, but different
parameter values, to calculate the total cross section.
Once the specific attenuation is known at a point, a point-to-path scheme is
required to obtain the total attenuation along the path length. Chapter 4 describes
the two-component rain rate model and its application to the single path geometry
as well as to other ones that define simple network blocks. In doing so, the required
spatial correlation o f rain is estimated from the power spectrum o f logarithmic
variations in rain rate.
Different model assumptions were combined together in Chapter 5 to analyze their
impact on system performance. The spheroidal and Pruppacher-and-Pitter raindrop
shapes were considered. The P-P raindrop shape attenuations were higher than the
spheroidal case due to the nonaxisymmetry added contribution. As for the drop
size distribution, both the Marshall-and-Palmer and the Weibull distributions were
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6.1 SUMMARY OF OUR WORK
97
looked at. The Weibull dsd has shown better agreement with the ITU specific
attenuation model, for frequencies up to 100GHz. For variations in the rain rate,
the theoretical specific attenuations were in good agreement with the ITU standard
for the frequencies of IS and 38GHz that were considered.
The relation between rain rate and specific attenuation being a good approximation
for the theoretical attenuations, the parameters of the formula were employed in
the two-component model to obtain estimates of the path attenuations at different
time percentages. These values were then compared with the ITU and Crane mod­
els for moderate and tropical climates. In the former climate, good agreement was
found between the two. In the latter climate, the two-component model gave rise to
higher rain attenuation values because different rain rate distribution values were
considered.
In order to analyze the performance of microwave radio links in rain, a simulation
was carried out incorporating the rain attenuation in the channel model. For the
moderate climate, the fade margins between theoretical values and ITU and crane
models were found to be within the lOdB range, with lower margins for the theo­
retical case. As for the tropical climate, once again, the theoretical case was very
conservative with differences that went as high as 30 dB. From the plot of maxi­
mum path length between the transmitting and receiving ends, some improvement
was found when the diameter of the antennas was increased, but would definitely
be outweighed by the complexity and cost of such a process.
To complete the microwave performance analysis, network links were analyzed
and useful recommendations were noted, toward the end of Chapter 5, for the
design o f networks. The analysis was based on three simple network blocks, which
were parallel, node, and relay topologies.
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98
FINAL REMARKS AND FUTURE WORK
6.2 FUTURE CONSIDERATIONS
In this thesis, we have studied the development and simulation of theoretical m od­
els in the analysis o f microwave link performance. With a direct contact w ith the­
ory, models can be modified and improved as more data becomes available for
higher frequencies and different locations.
The modeling of rain is still an issue to be explored, both for microwave propaga­
tion and meteorological studies. The study of radar data has been very useful in
understanding the rain process, by providing enormous amounts of data for longperiods of time.
There is still a necessity for experimental data for comparison with theoretical
attenuations and for validation of models. NASA is working on a TRM M project,
Tropical Rainfall Measuring Mission, which consist of a satellite for observing
rainfall in the tropics and a ground validation support for calibrating the satellite
measurements.
While the scattering theory by a raindrop is almost perfected, the drop size distri­
bution needs to be validated in different parts of the world, especially in the trop­
ics, where the rain rates are higher than the normal. For example, Li, et al. [Li95b]
have derived the drop size distribution for Singapore’s tropical environment from
the rainfall attenuation data at a frequency of 21.225GHz over a short path length
of about 1km, and resulted in large-scale raindrops with larger density than in the
moderate climatic region, such as the one described by the M-P distribution.
Another aspect to the prediction of rain attenuation is the rain rate distribution
which is being updated as more data is available. This case was experienced in the
thesis between the two-component model, which is based on the latest distribu­
tions extracted from [1], and the Crane prediction method which is based on the
earliest distribution taken from [CraneSO]. On the other hand, the rain rate model
described by the two-component model seems to be a good method for path atten­
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6.2 FUTURE CONSIDERATIONS
99
uation predictions given that its parameters are fitted to the appropriate rain rate
distributions.
As a future work, the two-component model can be modified in order to simulate
more sophisticated network topologies which can include several links at the same
time. The knowledge acquired from these studies can help improve the perfor­
mance o f these networks and minimize the effect of rain on the overall system.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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107
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