Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
6, N
o
. 5
,
O
c
tob
e
r
201
6, p
p
. 2
379
~238
6
I
S
SN
: 208
8-8
7
0
8
,
D
O
I
:
10.115
91
/ij
ece.v6
i
5.1
174
1
2
379
Jo
urn
a
l
h
o
me
pa
ge
: h
ttp
://iaesjo
u
r
na
l.com/
o
n
lin
e/ind
e
x.ph
p
/
IJECE
Stochastic Approach to a Ra
in Att
e
nuati
on Time S
e
ries
Synthesizer for Heavy Rain Regions
Masou
d
Mohe
bbi
Ni
a
1
, Jafri
Din
1
,
H
o
n
g
Y
i
n Lam
2
, At
han
a
s
i
os
D
.
Pa
na
go
po
u
l
o
s
3
1
Wireless Com
m
unication C
e
nter
, Facu
lt
y
of
Electrical
Engin
eerin
g, 81310
Univer
siti T
e
knolog
i M
a
lay
s
i
a
,
Johor Bahru, Johor, Malay
s
ia
2
Departement of
Electr
i
cal
Engin
eering
Technolo
g
y
,
Facu
lty
of Engi
neer
ing Tech
nolog
y
,
86400
,
Universiti Tun
Hussein
Onn Malay
s
ia, P
a
rit Raja, Johor
Malay
s
ia
3
School of
Electrical and
Computer
Engi
neering
,
National Techn
i
cal University
of
Athens, Zografo
u
, Athen
,
Gr
eece
Article Info
A
B
STRAC
T
Article histo
r
y:
Received
May 16, 2016
Rev
i
sed
Ju
l 4
,
2
016
Accepte
d
J
u
l 20, 2016
In this work, a
new rain
att
e
nu
ati
on
time series sy
n
t
hesizer b
a
sed on the
stochastic appro
ach is presented. Th
e m
odel com
b
ines a well-kno
wn interest-
rate pred
iction model in finance na
mely
the Cox-Ingersoll-Ross (CIR)
model, and
a stochastic
diff
erential equ
a
tion
approach
to gen
e
rate a
long-
term gamma distributed r
a
in attenuation
ti
me
se
ri
e
s
,
pa
rti
c
u
l
arly
appropriate
for heav
y
rain regions.
T
h
e
model
pa
ra
me
te
rs we
re
de
ri
ve
d from ma
xi
mum-
likelihood estim
ation (MLE)
and
Ordinar
y
Least Square (OLS) methods. Th
e
predicted statistics from
the CI
R model with the OLS method
are
in good
agreem
ent with
t
h
e m
eas
urem
ent
data col
l
e
c
ted i
n
equatori
al M
a
l
a
y
s
i
a
while
the MLE metho
d
overestim
ated
the result. The pr
oposed stochastic model
could prov
ide r
a
dio
engineers
an altern
ativ
e s
o
lution for
th
e design of
propagat
i
on im
pairm
e
nt m
itiga
tion te
chniques
(PIMTs) to i
m
prove the
Qualit
y of S
e
rvice (QoS
) of wi
reles
s
com
m
uni
cat
ion s
y
s
t
em
s
s
u
ch as
5G
propagation channel, in
partic
ular in h
eav
y
rain r
e
gions.
Keyword:
Heavy
rain
re
g
i
ons
Rad
i
ow
av
e pr
op
ag
ation
Rain
atten
u
a
tion
St
ocha
st
i
c
ap
pr
oach
Tim
e
series synthesizer
Copyright ©
201
6 Institut
e
o
f
Ad
vanced
Engin
eer
ing and S
c
i
e
nce.
All rights re
se
rve
d
.
Co
rresp
ond
i
ng
Autho
r
:
Jafri Din
,
W
i
rel
e
ss C
o
m
m
uni
cat
i
on C
e
nt
er,
Fac
u
lty of Electrical Engineering,
Un
i
v
ersiti Tekn
o
l
o
g
i
Malaysia,
81
3
0
0
,
S
k
udai
,
Jo
ho
r.
Em
a
il: j
a
fri@ut
m
.
m
y
1.
INTRODUCTION
Cu
rren
t an
d
fu
ture wireless co
mm
u
n
i
cati
o
n
syste
m
s su
ch
as
5
G
cell
u
lar
n
e
two
r
k
s
an
d satellite
com
m
uni
cat
i
on sy
st
em
are m
ovi
ng t
o
w
a
rd
hi
g
h
e
r
s
p
eed
dat
a
rat
e
s an
d
wi
de
r
ban
d
w
i
d
t
h
s w
i
t
h
t
h
e
em
pl
oym
ent
o
f
o
p
e
r
at
i
ng
fr
eque
nci
e
s a
b
o
v
e
10
G
H
z [
1
]
.
U
n
f
o
rt
u
n
at
el
y
,
radi
o
l
i
n
k o
p
e
r
at
i
n
g
at
t
h
es
e
freq
u
e
n
c
ies suffer strong
atten
u
a
tion
p
h
e
no
m
e
n
a
d
u
e
to
th
e at
m
o
sp
h
e
ri
c co
n
s
titu
en
ts su
ch
as rain, clo
u
d
,
water va
pour
and turbule
n
t fluctua
tions/scintillation [2]-[5]. Am
ong the
s
e,
rain a
ppea
r
s as the
m
a
jor factor
th
at d
e
g
r
ad
es th
e
p
e
rfo
rm
an
ce of
wireless
co
mm
u
n
i
catio
n
system
s [2
]. Th
is im
p
a
ir
men
t
is ev
en
wo
rse in
heavy rain
re
gi
ons
whe
r
e
t
h
e preci
pitation
c
h
aracteristics
are sign
ifican
tly d
i
fferen
t
from those in tem
p
erate
areas [
6
]
-
[
8
]
.
I
n
o
r
de
r t
o
c
o
u
n
t
e
ract
suc
h
i
m
pai
r
m
e
nt
s t
h
at
depe
nd
on t
h
e l
o
cal
cl
im
atol
o
g
y
,
det
a
i
l
e
d
dy
nam
i
c
characte
r
istics of precipitation are
re
q
u
i
red
to
serv
e as th
e critical in
pu
t to th
e ad
v
a
n
c
ed
p
r
op
ag
ation
im
pai
r
m
e
nt
m
i
t
i
g
at
i
on t
e
c
hni
que
s (
P
IM
Ts
)
[
2
]
.
A
si
g
n
al
m
easure
d
di
rect
l
y
fr
om
t
h
e com
m
uni
cat
i
on sy
st
em
i
s
the best res
o
urce for this purpos
e. Howeve
r, suc
h
m
eas
ured signals are not widely
available; therefore
,
the
pr
o
p
agat
i
o
n co
m
m
uni
t
y
prop
ose
d
t
h
e sy
nt
h
e
t
i
c
rai
n
at
t
e
nuat
i
on t
i
m
e seri
es t
o
m
i
m
i
c
t
h
e dy
nam
i
c of rea
l
si
gnal
s
.
For
t
h
e
pa
st
de
cade,
va
ri
o
u
s s
t
udi
es
ha
ve
foc
u
se
d
on
t
h
e
de
vel
o
pm
ent
of
r
a
i
n
at
t
e
n
u
at
i
o
n
t
i
m
e
seri
e
s
sy
nt
hesi
zers [
9
]
-
[
1
1]
. Event
u
al
l
y
, i
n
20
09
, Int
e
r
n
at
i
o
nal
Tel
ecom
m
unicat
i
on U
n
i
o
n-
R
a
di
o com
m
uni
cat
i
o
n
(IT
U-R
)
rec
o
m
m
e
nded a st
ocha
st
i
c
app
r
o
ach t
o
ge
nerat
e
rainy and cl
ear-s
ky tim
e
s
e
ries [12]. Since then,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
237
9
–
23
86
2
380
several
e
x
t
e
n
d
e
d w
o
rks
ha
ve
cont
i
n
ue
d t
o
i
m
prove
on t
h
i
s
seri
es an
d
pr
o
pos
ed m
o
re re
f
i
ned sy
nt
hesi
z
e
d t
i
m
e
series. Carrie e
t
al. extended t
h
e IT
U
-
R
m
odel
t
o
an e
v
ent
-
on
-
d
em
and sy
nt
hesi
zer
i
n
or
der t
o
ge
ne
rat
e
t
i
m
e
series according t
o
the
demand
of m
a
ximum
attenuati
on level a
nd e
v
ent
duration
[13]. Boulanger et al.
pr
o
pose
d
a ne
w t
i
m
e
seri
es
sy
nt
hesi
zer
ba
sed o
n
t
h
e co
m
b
i
n
at
i
on of
Di
rac an
d l
o
gn
orm
a
l
di
st
ri
but
i
on [
1
4]
.
While in t
r
opi
cal areas,
Andrade et
al
.
pr
o
pos
ed a m
odel
base
d o
n
t
h
e
gam
m
a
di
st
ri
b
u
t
i
on
w
h
ere t
h
e
m
odel
param
e
t
e
rs we
re i
n
fer
r
ed
f
r
o
m
t
h
e real
m
e
asure
d
dat
a
c
o
l
l
ect
ed i
n
B
r
a
z
i
l
[15]
a
n
d
Kanel
l
o
p
oul
os
et
al
.
prese
n
t
e
d
a
ne
w st
oc
hast
i
c
d
y
n
am
i
c
m
odel
fo
r t
h
e ge
ne
r
a
t
i
on
of
rai
n
a
t
t
e
nuat
i
o
n t
i
m
e seri
es
base
d
o
n
t
h
e
po
we
rf
ul
s
o
l
u
t
i
o
n
o
f
fi
rst
-
or
de
r st
oc
hast
i
c
di
f
f
ere
n
t
i
a
l
eq
uat
i
o
n
(S
DE
)
fo
r t
h
e l
o
n
g
-t
erm
M
a
rk
ov
p
r
oces
s [
16]
.
In t
h
i
s
wo
r
k
,
we p
r
o
p
o
se
d a
n
al
t
e
rnat
i
v
e a
nd
pr
om
i
s
i
ng st
ochast
i
c
ap
p
r
oach t
o
sy
nt
he
si
zi
ng l
o
n
g
-
term
rain
atten
u
a
tio
n tim
e se
ries, m
a
in
ly focu
sing
on
areas th
at ex
h
i
b
it
ex
trem
e h
eav
y
rainfall. Th
e
m
o
d
e
l
prese
n
t
e
d
here
i
s
an ext
e
nde
d ap
pl
i
cat
i
on
of a wel
l
-
es
tablish
e
d
lon
g
-term in
terest rate
g
e
n
e
ratio
n
mo
d
e
l i
n
fi
na
nci
a
l
resea
r
ch [
17]
. The
C
o
x
-
I
n
gers
ol
l
-
R
o
ss
m
odel
,
h
e
rei
n
aft
e
r nam
e
d
“C
IR
”
,
re
p
r
od
uces
t
h
e dy
nam
i
c
ch
aracteristics o
f
lo
cal
p
r
eci
p
itatio
n
with resp
ect to
t
h
e
p
r
ecip
itatio
n
-
attenuatio
n
p
h
e
no
m
e
n
a
.
The pa
per i
s
o
r
gani
ze
d as fol
l
ows
.
Fi
rst
,
we
prese
n
t
t
h
e co
n
cept
and
pri
n
ci
pl
es of t
h
e C
I
R
st
ochast
i
c
m
odel
as a rain at
t
e
nuat
i
o
n t
i
m
e
seri
es sy
nthesi
zer.
Gene
r
a
l
sol
u
t
i
ons a
n
d dy
nam
i
c param
e
t
e
r co
m
put
at
i
o
n
s
of
t
h
e m
odel
a
r
e e
xpl
ai
ne
d i
n
Sect
i
o
n
3,
f
o
l
l
o
we
d
by
a
bri
e
f
di
sc
ussi
o
n
on
t
h
e l
o
n
g
t
e
rm
st
at
i
s
t
i
c
s
of
t
h
i
s
m
o
d
e
l in
Sectio
n 4. Section
5
p
r
esen
t th
e
n
u
m
erical re
sults v
a
lid
ated
ag
ain
s
t m
easu
r
ed
d
a
ta in
eq
u
a
to
rial
M
a
l
a
y
s
i
a
and
f
i
nal
l
y
Sect
i
on
6
dra
w
s s
o
m
e
concl
u
si
o
n
s.
2.
STOCHASTI
C
MO
DEL O
F
TIME
SERI
ES SYNTHE
SIZ
E
R
In
th
is work, th
e lo
ng-term
c
o
m
p
le
m
e
n
t
ary
cu
m
u
lativ
e d
i
strib
u
tion
fun
c
tio
n
of rain
atten
u
a
tion
is
assu
m
e
d
to
b
e
g
a
mma d
i
stribu
tio
n
as th
is is th
e well-es
tablish
e
d
statistical d
i
strib
u
tion
in
h
e
av
y rain
reg
i
on
s
[16
]
.
Hen
c
e, su
ch d
i
stri
b
u
tion
o
f
satellite an
d terrest
rial
lin
k
s
can
b
e
describ
e
d
b
y
a first-o
r
d
e
r
sto
c
h
a
stic
diffe
re
ntial equ
a
tion
(S
DE):
12
()
()
tt
t
t
dX
D
X
d
t
D
X
dW
(1)
whe
r
e
D
1
, a
n
d
D
2
are d
r
i
f
t
and di
ff
usi
o
n
coeffi
ci
e
n
t
s
r
e
spect
i
v
el
y
re
prese
n
t
i
n
g sl
o
w
an
d ra
pi
d
vary
i
n
g
com
pone
nt
s o
f
t
h
e rai
n
at
t
e
n
u
at
i
on t
i
m
e seri
es [
18]
.
dW
t
is th
e Brown
i
an
m
o
tio
n
or
W
i
en
er pro
c
ess th
at
fo
llows t
h
e norm
a
l d
i
strib
u
tio
n
with ze
ro
mean and va
ri
ance
dt
[
18]
,
[
19]
.
I
n
t
h
i
s
pa
rt
i
c
ul
ar a
ppl
i
c
at
i
on, a
satellite
/terrestrial ch
ann
e
l
u
s
u
a
lly co
n
s
i
d
ered
rad
i
o sign
als app
earing
i
n
t
h
e form
o
f
additiv
e wh
ite Gau
ssian
noi
se
(
A
WG
N
)
,
he
nce t
h
e
va
ri
ance
of
t
h
e
W
i
e
n
e
r
pr
oces
s can
be
ass
u
m
e
d t
o
b
e
e
qual
t
o
1
whi
l
e
X
t
re
prese
n
ts
th
e rai
n
attenuatio
n
tim
e series. Hen
c
e
dt
i
s
equal
t
o
1.
B
a
sed
o
n
t
h
es
e
p
r
o
p
ert
i
e
s
o
f
di
st
ri
b
u
t
i
o
n
,
t
h
e C
I
R
m
odel
t
h
at
i
s
ca
pa
bl
e
of
ge
ne
rat
i
n
g
a
t
i
m
e
seri
es
of interest
rate
over a
long
duratio
n (i.e. more tha
n
10 years) according
to gamma distribution is clearly one
o
f
t
h
e b
e
st altern
ativ
e t
o
o
l
s t
o
p
r
ed
ict rain
att
e
n
u
a
tion
tim
e
series. Th
is well-estab
lish
e
d
m
o
d
e
l in
fin
a
nce and
econom
i
cs represents the
firs
t-o
r
d
e
r SDE as fo
llo
ws [17
]
:
()
tt
t
t
dX
k
X
dt
X
d
W
(2
)
whe
r
e
k
,
and
a
r
e the
t
h
ree
em
pirical pa
ram
e
ters that coul
d
be
i
n
fer
r
ed
f
r
om
t
h
e ex
peri
m
e
nt
al
rai
n
attenuation dat
a
set as a funct
i
on of the tra
n
smission/recei
ving links elevation angle,
polarization as well a
s
ope
rat
i
n
g f
r
e
q
uency
.
It
i
s
al
s
o
wo
rt
h
n
o
t
i
n
g
t
h
at
k
corresponds t
o
the
dy
nam
i
c factor
o
f
rai
n
attenuatio
n
as
cl
earl
y
expl
ai
n
e
d i
n
[
2
0]
. Ne
vert
hel
e
ss, i
n
t
h
e abse
nce
of
expe
ri
m
e
nt
al
dat
a
bases, se
ver
a
l
t
ool
s de
vel
o
ped
by
pr
o
p
agat
i
o
n
re
searche
r
s s
u
c
h
as sy
nt
het
i
c
st
orm
t
echni
q
u
es wh
ich
on
ly req
u
i
re th
e inpu
t of tim
e seri
es rain
rate m
easured by l
o
cal rai
n
ga
uges, c
o
ul
d als
o
pr
ovide
sufficient
acc
uracy
for
th
e ex
traction
of
tho
s
e
param
e
t
e
rs [6]
,
[2
1]
. The
next
sect
i
on b
r
i
e
fl
y
di
scus
s
e
s t
h
e g
e
neral
sol
u
t
i
o
n
and m
e
t
hodol
ogy
i
n
fer
r
ed
b
y
t
h
e
C
I
R
m
odel
par
a
m
e
t
e
rs.
3.
CIR MODEL
PARAMETERS
ESTIMAT
I
ON AND
GE
NERAL SOLUTION
On
e
relev
a
n
t
issu
e fo
r t
h
e mo
d
e
lling
of a rain
atten
u
a
tion ti
m
e
series sy
n
t
h
e
sizer is the lo
ng
term
di
st
ri
b
u
t
i
on
of
fi
rst
or
der rai
n
at
t
e
nuat
i
o
n st
at
i
s
t
i
c
s. In pa
rticu
l
ar, th
ese statistics in
h
eav
y rain
reg
i
on
s seem to
fol
l
o
w
gam
m
a di
st
ri
b
u
t
i
o
n a
s
o
b
se
rve
d
i
n
several
p
r
evi
o
us
w
o
r
k
s
[1
5]
,
[
1
6
]
.
F
o
r t
h
i
s
reaso
n
,
si
nce
t
h
e C
I
R
m
odel
i
s
an er
go
di
c p
r
oces
s wi
t
h
a st
at
i
ona
ry
di
st
ri
b
u
tion, as the dynam
i
c param
e
ter k approaches the
long
term
mean
v
a
l
u
e
o
f
rain attenu
atio
n, th
e syn
t
h
e
sized rain atten
u
a
tion
will ap
pro
ach g
a
mma d
i
stribu
tion
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
St
oc
ha
st
i
c
Ap
p
r
oa
ch
t
o
a R
a
i
n
At
t
e
n
u
a
t
i
o
n
Ti
me
Seri
s
Syn
t
hesi
zer f
o
r
He
avy R
a
i
n
..
.. (
M
as
ou
d M
o
he
bbi
N
i
a)
2
381
In
gene
ral
,
si
n
ce rai
n
at
t
e
nu
a
t
i
on o
v
er t
h
e p
e
ri
o
d
o
f
t
i
m
e
can be
desc
ri
b
e
d by
t
h
e
Wei
n
er
pr
oces
s
(Bro
wn
ian
m
o
tio
n
)
, Ito’s
p
r
ocess allo
ws
fo
r th
e so
l
u
tio
n
of th
e
d
i
fferen
tial o
f
a tim
e-d
e
p
e
nd
en
t fu
n
c
ti
o
n
of
rain attenuation proce
ss by
the well-known a
p
pr
oxim
a
tion m
e
thod Euler-Maruyam
a
[18],[19]. Euler-
Mar
u
yam
a
ap
pr
ox
im
at
io
n
co
uld
pro
v
i
d
e
th
e
si
m
p
lest yet str
o
ng
Taylo
r
appr
ox
im
at
io
n
ser
i
es as
1
nn
n
n
n
n
YY
k
Y
Y
W
(
3
)
1
nn
n
WW
W
(4)
2
(W
)
nn
E
(5)
To
sim
u
late a
p
r
ed
icted
rain
atten
u
a
tion
time series fo
r
n
=
0,
1,
2,
…
N
-1
, we sim
p
ly sta
r
t fro
m
th
e
in
itial co
nd
itio
n
Y
o
=
X
o
an
d p
r
ocee
d
t
o
t
h
e
next
val
u
e
s
.
T
h
i
s
i
s
a
n
It
o p
r
oces
s
wi
t
h
dri
f
t
k
a
n
d
di
f
f
u
s
i
on
coefficient.
After con
s
id
erin
g
t
h
e th
eo
reti
cal esti
m
a
tio
n
,
two
d
i
ffe
rent
t
echni
q
u
es c
o
u
l
d be
pract
i
cal
l
y
depl
oy
e
d
to
p
r
ed
ict th
e
SDE m
o
d
e
l param
e
ters n
a
mely th
e Max
i
m
u
m Lik
e
lih
oo
d
Esti
m
a
tio
n
(MLE) [22
]
an
d
t
h
e
Ordina
ry Least Square
(OLS). MLE is c
o
mm
only e
m
p
l
oyed to e
s
ti
mate
m
u
ltiste
p ahead forecast
s
. This
ap
pro
ach
d
e
termin
es th
e
m
o
d
e
l p
a
ram
e
ters
b
a
sed
on
th
e m
o
st
lik
ely p
r
o
b
a
b
ility fro
m th
e
m
easu
r
ed
d
a
taset.
On t
h
e
ot
he
r h
a
nd
, OL
S est
i
m
at
es t
h
e
m
odel
param
e
t
e
r from
a
l
i
n
ear regressi
o
n
by
m
e
ans o
f
m
i
nim
i
zi
ng t
h
e
di
ffe
re
nce
bet
w
een
t
h
e m
easure
d
dat
a
a
n
d
t
h
e est
i
m
at
ed val
u
es
[
23]
.
Si
m
u
l
a
t
i
on res
u
l
t
s o
f
bot
h a
p
p
r
oache
s
are s
u
b
s
eq
ue
nt
l
y
sho
w
n
i
n
Se
ct
i
on
5.
4.
GENER
A
TIO
N
O
F
FI
RST
OR
DER L
O
N
G
-TER
M
R
A
I
N
ATTEN
U
A
TIO
N
ST
AT
ISTICS
Before we
furt
h
e
r an
alyse the si
m
u
latio
n
resu
lts, it
is worth
d
i
scu
ssing
t
h
e reliab
ility o
f
lo
ng-ter
m
fi
rst
-
or
der
rai
n
at
t
e
nuat
i
o
n st
at
i
s
t
i
c
s pro
d
u
c
e
d
by
t
h
e C
I
R
m
odel
.
I
n
t
h
e
pl
an
ni
n
g
a
n
d
desi
g
n
of
a
wi
rel
e
ss
com
m
uni
cat
i
on sy
st
em
, a reli
abl
e
l
ong
-t
er
m
C
C
D
F of rai
n
at
t
e
nuat
i
on
i
s
of t
h
e ut
m
o
st
im
port
a
nce [2
4]
. I
n
o
r
d
e
r to
reg
e
nerate su
ch
long
-term
statis
tic
s, th
e rain
atten
u
a
tion
is u
s
ually d
e
scrib
e
d
b
y
a relativ
e si
m
p
le
fi
rst
-
or
der M
a
r
k
o
v
pr
ocess
[9
]
,
[1
6]
. He
nce i
t
i
s
best
t
o
represen
t th
e
statistics in
th
e fo
rm o
f
th
e
Orn
s
t
e
in
-
Uhl
e
nbec
k
p
r
ocess [
19]
. T
h
i
s
i
s
t
h
e onl
y
st
ochast
i
c
proces
s that satisfies the
p
r
op
erties of statio
n
a
ry,
Gaus
si
an a
n
d
M
a
rk
ov
p
r
oces
ses.
The C
I
R
m
o
d
e
l
assum
e
s t
h
at
i
f
t
h
e o
b
ser
v
at
i
on
pe
ri
o
d
t
i
s
l
ong e
n
ou
gh
(t
→∞
) , a
nd t
h
e rai
n
at
t
e
nuat
i
o
n ap
pr
oac
h
es t
h
e
m
ean val
u
e
s
o
f
l
o
ng
-t
erm
ob
servat
i
o
n
dat
a
m
,
t
h
e
m
odel
can
be si
m
p
l
i
f
i
e
d,
by
replacing
X
t
wi
th
m
:
()
tt
t
dX
k
X
d
t
m
d
W
(6)
Thi
s
i
s
al
so kn
ow
n as t
h
e Va
si
cek m
odel
[25]
whi
c
h
was pre
v
i
o
usl
y
i
n
t
r
od
uce
d
t
o
desc
ri
be i
n
t
e
res
t
rat
e
m
ovem
e
nt
i
n
fi
na
nce a
n
d
i
s
no
w a
d
apt
e
d t
o
descri
be
t
h
e d
y
n
a
m
i
cs o
f
rain
attenu
ation
lev
e
ls.
Th
e
gen
e
ral
so
lu
tion
o
f
th
e
SDEs with hom
o
g
e
n
e
ou
s coefficien
ts and
ad
d
itiv
e
no
ise can
n
o
w
b
e
written
as:
0
(1
)
t
kt
ks
to
s
Xx
e
m
e
d
W
(7)
The i
n
t
e
gr
al
o
f
t
h
e
Wei
n
e
r
pr
ocess ca
n
be
so
l
v
ed
usi
n
g
Eul
e
r-M
ar
uy
am
a app
r
oxi
m
a
t
i
on f
r
om
t
h
e
long-term
m
e
a
s
urem
en
t data.
The m
ean
and
varia
n
ce
of the tim
e series synthesize
r from
(7), which
deals
wi
t
h
t
h
e
B
r
ow
ni
an/
W
i
ene
r
p
r
oces
s,
ca
n be sol
v
e
d
by
usi
n
g It
ō
in
teg
r
al fro
m
I
t
ō
lemma
as
[1
8
]
,[
19
]:
(1
)
kt
kt
o
xe
e
(8)
2
22
2
kt
m
e
k
(9)
Evaluation Warning : The document was created with Spire.PDF for Python.
IJEC
E
2
382
5.
N
of m
e
10
3.
6
with
a
th
is
m
Figu
r
series
MLE
dy
na
m
is to
g
well
a
th
e C
I
atten
u
depi
c
t
ran
g
e
Fi
g
E
Vo
l.
6
,
N
o
.
N
UM
ERIC
A
In
th
is se
c
e
asurem
e
n
t d
a
4
E) as illu
st
r
a
n ele
v
ation
a
m
easurement
a
r
e 1.
M
a
p of
t
h
As p
r
evi
o
s
o
f
rai
n
atten
u
and OL
S
ap
p
m
i
c
prope
rt
i
e
s
g
ene
r
ate typi
c
a
s geom
et
ri
ca
l
I
R
m
odel
wi
t
h
u
at
i
o
n val
u
es
t
ed in Fi
gure
o
f
0.1 dB to
4
g
ur
e 2
.
Ex
am
p
(
T
op)
a
5, Oct
o
be
r
2
0
A
L RESULT
S
c
t
i
o
n,
we pre
s
e
a
tab
a
se reco
r
d
r
ated in Fi
gu
r
a
ngl
e o
f
75
.
6
1
a
re
clearly
ex
p
h
e Malaysian
usly m
e
n
tio
n
u
ation
.
To th
i
s
p
ro
ach
e
s is i
l
s
o
f
r
a
i
n
a
t
t
e
n
u
c
al rain atten
u
l
an
d
radi
o
p
a
h
th
e MLE a
n
ge
ner
a
t
e
d b
y
3. T
h
e
O
L
S
4
d
B
) wh
ile
t
h
p
l
e
of sy
nt
he
s
i
a
nd
t
h
e
OLS
a
1.
2.
L
a
ti
tu
d
e
0
16
:
237
9 –
2
S
AN
D DIS
C
U
e
nt
se
veral
n
u
d
ed
in
th
e
U
n
r
e 1. The gro
u
.
The
ope
ra
t
i
p
lain
ed
i
n
[26
]
p
e
n
in
su
la
sh
o
ed, MLE an
d
s
aim
,
first,
a
n
l
lu
strated
in
F
u
ation
in
o
n
e
u
ation
tim
e
s
e
a
ram
e
ters of
o
n
d OLS
a
p
pr
o
y
the MLE a
n
m
et
hod see
m
h
e MLE m
e
th
o
i
zed rai
n
atte
n
a
pp
ro
ach
(
B
o
t
102
102
.
5
1
5
2
5
3
P
e
ni
ns
ul
a
M
a
23
86
U
SS
I
O
N
u
m
e
rical resul
t
n
i
v
e
rsiti Tek
n
u
nd
statio
n
is
i
ng fre
q
u
enc
y
]
and
will no
t
o
win
g
t
h
e m
e
a
d
OL
S
play
a
n
e
x
am
ple of
t
F
i
g
ure 2.
At
f
specific tim
e
e
ries with re
s
o
ne
pa
rt
i
c
ul
ar
o
ache
s
in
pro
b
n
d OLS
a
p
p
r
o
m
s to
closely
f
o
d ove
rest
i
m
a
n
u
a
tio
n tim
e s
t
tom
)
fo
r a
gr
o
103
103.
5
M
e
Si
n
a
p,
J
o
h
o
r
S
t
a
t
e
Lo
ng
i
t
u
d
e
t
s from
the C
I
n
ol
ogi
M
a
l
a
y
l
ooki
ng t
o
w
a
y
is
11
.0
75
G
H
be repeat
e
d
h
a
surem
e
nt sit
e
a
cru
c
i
a
l ro
le
t
im
e series of
f
irst gla
n
ce,
b
series rain e
v
s
pect to the l
o
comm
unicati
b
ab
ility d
e
n
s
i
t
o
aches are c
o
f
o
llo
w t
h
e m
e
a
tes PDF
o
f
a
t
eries b
y
th
e
C
o
un
d st
at
i
o
n
i
n
104
10
4.
5
e
as
u
r
em
e
n
t
S
i
t
e
at
U
T
M
n
ga
p
o
r
e
e
I
R
m
odel
and
sia ca
m
pus,
S
a
rd
th
e MEA
S
H
z. For t
h
e sa
k
ere.
near an equa
t
i
n
sy
nt
hesi
z
i
rain
attenu
ati
o
b
ot
h m
e
t
hod
s
v
en
t. Sin
ce th
e
o
c
a
l climatol
o
o
n l
i
n
k,
we
v
e
t
y
fu
nct
i
o
n (
P
o
mp
ar
ed
w
i
t
h
a
su
red
attenu
a
t
t
e
nuat
i
o
n val
u
C
IR
m
odel
wi
t
n
UTM, Joh
o
r
5
105
M
ISS
N
:
2
d
tested a
g
ai
n
s
S
k
udai
,
J
o
ho
r
S
AT-1
satellit
e
a
ke
o
f
bre
v
i
t
y
,
a
torial h
e
av
y r
ing
th
e lo
ng
-
i
on
sy
nt
hesi
z
e
se
e
m
to
r
e
p
r
e
ai
m of the s
y
o
gical charac
t
er
if
i
e
d
th
e
c
a
p
P
DF
). T
h
e P
D
h
the
m
easur
e
u
at
i
on t
h
res
h
o
l
u
e
less th
an 0
t
h the
MLE a
p
r
, Malaysia
2
088
-87
08
s
t one year
r
(1
.5
5
N,
e
(9
1.
5
E)
,
det
a
i
l
s o
f
a
i
n
regi
on
t
e
rm
tim
e
e
d
with
th
e
r
od
uce t
h
e
y
nthesizer
t
eristics as
p
ab
ility o
f
D
Fs of
rai
n
e
d P
D
F as
l
d (f
or
t
h
e
.5 dB.
p
pr
o
a
ch
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
St
oc
ha
st
i
c
Ap
p
r
oa
ch
t
o
a R
a
i
n
At
t
e
n
u
a
t
i
o
n
Ti
me
Seri
s
Syn
t
hesi
zer f
o
r
He
avy R
a
i
n
..
.. (
M
as
ou
d M
o
he
bbi
N
i
a)
2
383
Fig
u
re
3
.
Probab
ility d
e
n
s
ity fun
c
tion
s
(PDFs)
o
f
atten
u
a
tion
thresh
o
l
d
g
e
n
e
rated
b
y
th
e MLE
and
OLS
ap
pro
ach
es and
co
m
p
ared
wi
th
th
e m
easu
r
ed
PDF of the
MEASAT satellite
In o
r
der t
o
f
u
rt
her e
v
al
uat
e
t
h
e effect
i
v
e
n
e
ss of t
h
e C
I
R
m
odel
from
a pr
opa
gat
i
o
n pers
pect
i
v
e
,
t
e
st
i
ng has be
en carri
e
d
o
u
t
by
com
p
ari
n
g
l
ong t
e
rm
CC
DFs o
f
rai
n
at
t
e
nuat
i
o
n bet
w
een t
h
e C
I
R
m
odel
(MLE), the CIR
m
odel (OLS) and the m
eas
ure
d
CCDF. T
h
e
IT
U-R
R
ec.
P.6
1
8
-
12
[2
7]
i
s
al
so i
n
cl
ude
d i
n
t
h
e
com
p
arison as
this recomm
endation se
rv
ed
as th
e m
a
in
referen
ce m
o
d
e
l fo
r th
e
p
r
ed
ictio
n
o
f
first-o
r
der rai
n
atten
u
a
tion
statistics. Th
e results in
Fig
u
r
e
4
clearly i
ndicate the effective
n
ess of
th
e CIR
m
o
d
e
l with
th
e OLS
app
r
oach
com
p
ared t
o
t
h
e
o
n
e
wi
t
h
t
h
e M
LE
m
e
t
hod.
T
h
e O
L
S ap
p
r
oac
h
p
r
o
v
i
d
es
g
o
o
d
a
g
reem
ent
pre
d
i
c
t
i
o
n
wi
t
h
t
h
e
m
easurem
ent
st
at
i
s
ti
cs w
h
i
l
e
t
h
e M
LE m
e
t
hod m
a
rke
d
a
n
ove
res
t
im
at
i
on.
On
t
h
e ot
he
r
han
d
,
I
T
U-
R
Rec. P.618 clearly underesti
m
ates
t
h
e sl
ant
-
pat
h
m
easure
d
st
at
i
s
t
i
c
s, whi
c
h hi
ghl
i
ght
s t
h
e im
port
a
n
ce of
lo
cally-d
eri
v
ed p
a
ram
e
ters in
p
r
ov
id
ing
stati
s
tics with
b
e
tter accu
r
acy.
Fi
gu
re
4.
C
o
m
p
ari
s
on
o
f
C
o
m
p
l
e
m
e
nt
ary
cum
u
l
a
t
i
v
e di
strib
u
tion
fun
c
tion
s
(CCDFs) o
f
rain
attenu
ation
:
M
E
AS
AT m
easure
d
,
t
h
e C
I
R
m
odel
(M
LE),
t
h
e C
I
R
m
odel
(OL
S
)
an
d
IT
U-R
rec
o
m
m
e
ndat
i
o
n
Fi
gu
re
5
su
bse
que
nt
l
y
sh
o
w
s
t
h
e e
r
r
o
r
of
t
h
e
pre
d
ictio
n
with
resp
ect to th
e tim
e p
e
rcen
tag
e
fro
m
0.
00
1% t
o
1%
. As can
be s
een,
pre
d
i
c
t
i
o
n
err
o
rs
of t
h
e
M
LE app
r
oac
h
are
ob
vi
o
u
sl
y
hi
ghe
r t
h
a
n
t
hose
pre
d
i
c
t
e
d by
t
h
e OLS m
e
t
hod.
Thi
s
can be as
cri
b
e
d
t
o
t
h
e s
h
o
r
t
c
om
i
ng o
f
t
h
e M
LE
m
e
t
hod i
n
t
h
at
i
t
do
es not
accurately des
c
ribe t
h
e de
nsi
t
y of eac
h attenuation t
h
re
s
h
old
(see Fi
gure 3). For thes
e reas
ons
, as
clearly
m
e
nt
i
oned
i
n
[
23]
,
m
o
st
of t
h
e est
i
m
a
t
i
ons o
f
a
u
t
o
reg
r
essi
on
pa
ram
e
t
e
rs pre
f
er
t
h
e
OLS
ap
pr
oac
h
.
0.5
1
1.
5
2
2.
5
3
3.5
4
0
10
20
30
40
At
t
e
nu
at
io
n [
d
B]
P
r
obabi
l
i
t
y
D
e
ns
i
t
y
[%
]
Meas
u
r
ed
OLS
MLE
0
5
10
15
20
25
30
35
40
10
-3
10
-2
10
-1
10
0
10
1
A
ttenu
at
i
on [d
B]
Time Percentage [%]
M
e
as
u
r
ed
ML
E
OL
S
I
T
U
-
R
P
.
61
8-
1
2
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE
Vo
l. 6
,
N
o
. 5
,
O
c
tob
e
r
20
16
:
237
9
–
23
86
2
384
Fi
gu
re
5.
Er
ro
r
pe
rcent
a
ge c
o
m
p
ari
s
ons
bet
w
een
M
LE a
n
d
OLS m
e
t
h
o
d
acros
s
di
ffe
ren
t
t
i
m
e
percent
a
ge
6.
CO
NCL
USI
O
N
Th
is work
presen
ts a
n
e
w st
o
c
h
a
stic app
r
o
a
ch
fo
r th
e
g
e
n
e
ratio
n
o
f
a
rain
atten
u
a
tion
ti
me series, m
a
in
ly
foc
u
sing
on the area of e
x
tre
m
ely h
eav
y precip
itatio
n
,
i
n
p
a
rticu
l
ar equato
rial Malaysia. Th
is
n
e
w
m
o
d
e
l
e
m
p
l
o
y
ed
th
e
w
e
ll-
kno
wn
CIR
m
o
d
e
l f
r
e
q
u
en
tly u
s
ed
in
eco
n
o
m
ics to
r
e
p
r
od
u
ce a long-
ter
m
t
i
m
e ser
i
es o
f
rai
n
at
t
e
n
u
at
i
o
n, t
h
e pa
ram
e
ters
of
w
h
i
c
h
were
de
ri
ve
d
by
t
h
e M
L
E
m
e
t
hod a
n
d t
h
e OLS
ap
pr
oa
ch. T
h
e
m
o
d
e
l h
a
s b
e
en
co
m
p
ared
with
a lo
cal ex
perim
e
n
t
al d
a
tab
a
se in
term
s o
f
first-o
r
d
e
r rai
n
attenu
atio
n statistics
recorde
d
from
the MEAS
AT
satellite
link. T
h
e predicte
d
long-term
statis
tics by th
e OL
S m
e
thod are
found t
o
b
e
in
go
od
agreem
en
t with
th
e m
easu
r
ed
atten
u
a
tion
statistics wh
ile th
e MLE tend
s to
ov
eresti
m
a
te th
e
measurem
ent results. T
h
e a
c
hieve
d
res
u
lts offe
r c
o
m
m
uni
cat
i
o
n sy
st
em
desi
g
n
ers
an al
t
e
r
n
at
i
v
e
m
odel
utilizing locall
y
derive
d
pa
ra
meters with
better accur
acy
for the
pre
d
iction
of l
o
ng-term
rain attenuation
statistics fo
r satellite ch
an
n
e
ls as well as 5G
cellu
lar ch
ann
e
l in
h
e
av
y
rain
reg
i
o
n
s
.
ACKNOWLE
DGE
M
ENTS
Thi
s
wo
r
k
ha
s
bee
n
fu
n
d
e
d
by
M
i
ni
st
ry
o
f
E
ducat
i
o
n
M
a
l
a
y
s
i
a
an
d UTM
un
de
r
“R
esearc
h
U
n
i
v
er
sity Gr
an
t” Vo
t. N
o
. Q
.
J.130
00
0.252
3.07H
50
and
“H
I
C
O
E
Re
s
e
ar
ch Gr
a
n
t” Vo
t.
No
.
R.J13
000
0.7823
.4
J221
.
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1
0.
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0.
3
0.
2
0.
1
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05
0.
05
0.
0
2
0.
0
1
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i
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“
T
im
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a
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”
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n equilibrium
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ly
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14.
d
ata and predi
c
n
on-ionized m
e
O
F AUTH
O
R
Masoud
M
M
azandar
a
Ele
c
tri
cal
Mala
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s
ia
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it
e c
o
green tel
e
c
Jafri Bin
D
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9
1997. He
h
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m
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t
h
act
iviti
es
h
communic
a
Radar, and
I
S
A
ttenua
tio
n T
i
5
3-
1, “Troposp
h
w ‘event
‐
on
‐
d
u
rn
al of Satelli
t
t
en
uation Tim
e
E
EE Transactio
n
t
en
uation time
IEEE
, vol. 10
,
e
l
l
ite
a
nd
T
e
rr
e
a
nd
Propagati
o
th
e Term Stru
c
N
um
erical
S
o
l
u
e
re
ntial
Equ
a
ti
o
u
lation of th
e
t
ic
a
l
appro
ach
,
”
h
ema
t
i
c
al
mod
e
o
f
p
r
ecip
ita
tion
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Likelihood
o
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sis,”
Princ
e
T
h
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n
G
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p
. 3396-3
399.
, pp
. 385-
408,
1
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ringe
r
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ition
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Sp
ri
n
g
i
c model sim
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ions on,
vol/is
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in ra
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al Econo
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r
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z
range.
26
, 1975.
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e
School of El
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sor.
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s
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E
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s
ia, in
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artm
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acu
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phere, with
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ar ph
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ived th
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th
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of
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997 and
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ct
ri
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a
nd
r
ch in
terests
n
s networks,
ols. He has
C
hambe
r
of
m
munication
and finally
,
on
.
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