Internati
o
nal
Journal of Ele
c
trical
and Computer
Engineering
(IJE
CE)
V
o
l.
4, N
o
. 4
,
A
ugu
st
2014
, pp
. 61
4
~
62
2
I
S
SN
: 208
8-8
7
0
8
6
14
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
Capacity and Error Rate An
alysis of MIMO Satellite
Communication Systems
in Fading Scenarios
Ram
o
ni Ade
o
gun
School of
Engin
eering
and
Com
puter Science, V
i
ctor
i
a
Univ
ersit
y
of Wel
lington
, Welling
t
on New Zealand
Article Info
A
B
STRAC
T
Article histo
r
y:
Received Dec 12, 2013
Rev
i
sed
Ap
r
20
, 20
14
Accepted
May 12, 2014
In this
pap
e
r,
we inves
tig
at
ed
the
cap
aci
t
y
and bit
error
r
a
te
(BER)
perform
ance of Multiple Input
Multiple Output
(MIMO)
satelli
te s
y
stem
s
with single and m
u
ltiple dual po
lari
zed sate
lli
tes in geostation
a
r
y
orbit and a
m
obile ground receiving stat
ion
with m
u
ltiple antennas. We ev
aluated th
e
effects of both
s
y
stem para
mete
rs suc
h
a
s
numbe
r
of satellites, number of
receive an
tenn
as, and SNR and environm
ental f
a
ctors includ
ing atmospheric
signal attenu
ations and
signal phase disturbances on the ov
erall s
y
stem
perform
ance usi
ng both anal
yt
i
cal and
spa
tia
l m
odels for MIMO satellit
e
s
y
ste
m
s.
Keyword:
BER
Cap
acity
Geost
i
ona
ry
MI
MO
Satellite
Copyright ©
201
4 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
:
Ra
m
o
n
i
Ad
eo
gu
n
Victo
r
ia Un
iv
ersity
o
f
W
e
lling
t
on
Kel
b
ur
n,
W
e
l
l
i
ngt
on
, New
Ze
al
and
Pho
n
e
:+6
422
07
425
40
Em
a
il: ra
m
o
n
i
.o
.ad
e
ogu
n@ieee.org
1.
INTRODUCTION
Mu
ltip
le-In
p
u
t
Mu
ltip
le-Ou
t
p
u
t
(MIM
O)
wireless co
mm
u
n
i
catio
n
s
syste
m
s h
a
v
e
been
a
fo
cu
s of
academ
ic and industrial rese
arch i
n
the las
t
decade
due to th
eir
potentially higher
dat
a
rates in c
o
mparis
on
wi
t
h
Si
ngl
e
-
I
n
put
Si
n
g
l
e
-
O
u
t
put
(SI
S
O
)
s
y
st
em
s
[1]
.
Th
eoretically, th
e
o
v
e
rall ch
an
n
e
l
cap
acity
can
b
e
increase
d
linea
rly with t
h
e
num
b
er of tra
n
s
m
it and recei
ve antennas
by using
s
p
atial m
u
ltiplexing
s
c
hem
e
s
[1
]. C
u
rren
t focu
s
on
satellite co
mm
u
n
i
catio
n
(SatCo
m
)
syste
m
s recog
n
i
zes a
d
e
m
a
n
d
fo
r h
i
g
h
e
r
d
a
ta
rates.
Hence
,
i
t
a
ppe
ars t
o
be a
p
p
r
o
p
ri
at
e t
o
a
ppl
y
M
I
M
O
t
o
Sa
tC
o
m
syste
m
s in
o
r
d
e
r to in
crease th
e av
ailab
l
e d
a
ta
rat
e
an
d
ba
nd
w
i
dt
h e
ffi
ci
ency
.
Th
e
q
u
a
lity o
f
serv
ice
(Qo
S
) and
d
a
ta rat
e
s requ
irem
en
ts o
f
satellite c
o
mm
u
n
i
catio
n
syste
m
s is
recently on t
h
e increase.
He
nce, the a
ppli
cation of m
u
lt
iple input m
u
l
tiple output techni
que
s to sa
tellit
e
comm
unication syste
m
s appear to be
appropriate in order to ac
hieve
increased s
p
e
c
tral and ba
ndwidth
efficien
cy [2
]. Sp
atial
m
u
ltip
l
e
x
i
ng
and
d
i
v
e
rsity
m
a
x
i
miza
tio
n
sch
e
m
e
s can
b
e
d
e
p
l
o
y
ed
to
ach
iev
e
better
sp
ectral efficien
cies an
d b
it erro
r
rates
(BER) wh
en
co
m
p
ared to
t
h
e classical si
n
g
l
e satellite sin
g
l
e receiv
e
station system
s
.
In
[2
], MIMO
satellite u
p
lin
ks and
d
o
wn
link
s
ch
ann
e
l th
at are op
tim
a
l
i
n
term
s o
f
achiev
ab
le
d
a
ta
rates were a
n
a
l
yzed. The aut
h
ors s
h
o
w
ed t
h
at
capaci
t
y
opt
im
i
zat
i
on i
s
gene
ral
l
y
poss
i
bl
e for
rege
ne
rat
i
v
e
pay
l
oad
desi
gn
s usi
n
g Li
ne o
f
Si
g
h
t
(L
OS
)
chan
nel
m
odel
s
. The
s
e anal
y
s
i
s
were e
x
t
e
n
d
ed t
o
a
num
ber
o
f
MIMO satellit
e co
mm
u
n
i
cati
o
n
system
s in
[
3
] an
d
th
e sco
p
e was furth
e
r ex
tend
ed
to
g
e
neral case o
f
sat
e
llites
wi
t
h
t
r
a
n
s
p
are
n
t
com
m
uni
cat
i
on
pay
l
oa
ds
com
pone
nt
.
A
cl
ust
e
r
base
d
cha
nnel
m
odel
was
pr
o
p
o
s
ed
fo
r
MIMO satellite form
atio
n
syste
m
s in
[4
].
Based
o
n
t
h
e
stan
d
a
rd
ized
m
o
d
e
ls fo
r terrestrial m
u
ltip
l
e
in
pu
t
m
u
l
tip
le o
u
t
put (
M
I
M
O
)
syste
m
s, th
e au
th
or
s pr
opo
se
d a spatial
m
o
del and a
n
alysed the capac
ity of
form
at
io
n
flyin
g
satellite system
s.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
61
4
–
62
2
6
15
In
th
is co
n
t
ribu
tio
n, we an
alyse th
e p
e
rforman
ce o
f
satel
lite co
mm
u
n
i
c
a
tio
n
system
s
with
m
u
ltip
le
co
op
eratin
g
satellites
in
g
e
o
s
tatio
n
a
ry orb
it (GEO) and
si
n
g
l
e
o
r
m
u
ltip
le an
tenn
as at
th
e groun
d
receiv
in
g
statio
n
.
Th
e analysis in
th
is pap
e
r is
b
a
sed
on
thr
ee d
i
fferen
t m
o
d
e
llin
g
ap
pro
ach
es
fo
r l
a
n
d
m
o
b
ile satellite
sy
st
em
s. The r
e
m
a
i
n
i
ng pa
rt
of t
h
i
s
pa
per i
s
or
gani
ze
d as f
o
l
l
o
w
s
. I
n
Sect
i
on I
I
,
we
pres
ent
t
h
e sy
st
em
m
odel
for MIM
O
satellite syste
m
s. A
rev
i
ew
o
f
the prop
ag
atio
n
ch
ann
e
l m
o
d
e
l
s
con
s
id
ered
in th
e
p
a
p
e
r is presen
ted
i
n
sect
i
on I
II.
In Sect
i
o
n I
V
,
we deri
ve ex
pressi
o
n
s for
channel capaci
ty and bit error rates
with MPSK
m
odul
at
i
on sc
hem
e
. Sim
u
l
a
ti
on
resul
t
s
a
n
d di
sc
ussi
ons
are p
r
ese
n
t
e
d i
n
sect
i
o
n
V. Fi
nal
l
y
, w
e
dra
w
concl
u
si
o
n
i
n
Sect
i
on V
I
.
2.
SYSTE
M
MO
DEL
D
In this section,
we
prese
n
t the
syste
m
m
odel fo
r single satellite, m
u
ltiple receive ante
nna syste
m
s
(SS-MRA) and m
u
lt
iple satell
ite
m
u
ltip
le receive ante
nna syste
m
s (MS-MRA).
2.1. Single Satellite
- Multip
l
e
Receive
Antennas (SS-MRA)
Consi
d
er the
downlink
of a
Land-m
obile satellite recei
ve diversity system cons
isting
of a
single
dual
pola
r
ized satellite antenna and a
m
obile receive stati
on with M non-polariz
e
d antennas
. The cha
nnel im
pulse
response between the satellite a
nd the
mobile receive
station can
be
m
odelled as an M x 2
MIM
O
com
m
uni
cat
i
on c
h
an
nel
⋮⋮
(1
)
whe
r
e
h
ij
is the ch
an
nel b
e
t
w
een th
e
j
-th
transm
it polarization a
n
d the
i-
th
receive
a
n
tenna. The
re
ceive
d
signal at t
h
e mobile
receive a
n
tennas is
give
n
by
⋮
⋮⋮
⋮
(2
)
A m
a
trix repre
s
entation
for the receive
signa
l
m
odel in
(2) is thus
y = Hx +
n
(3)
whe
r
e
y
= [
y
1
,
y
2
, … ,
y
M
]
T
is
an
M
x
1 vect
or of the recei
ve
d sign
als at the M receive ant
e
nna
s,
x
= [
x
1
;
x
2
]
T
is
a v
ecto
r
o
f
transmitted
sy
m
b
o
l
s o
n
th
e two
po
larizatio
ns o
f
th
e satellite an
t
e
n
n
a
an
d
n
= [
n
1
,
n
2
, … ,
n
M
]
T
is an
M
c 1
n
o
i
s
e
ve
ct
or as
sum
e
d t
o
be c
o
m
p
l
e
x
Gaus
si
an
ra
ndom
variables
with zero m
ean and va
riance
2.
2.2. Mul
t
iple Satellite
- Mul
t
ip
le
Recei
ve Ante
nnas (MS-MRA)
We con
s
i
d
er a satellite d
i
v
e
rsity syste
m
co
m
p
risin
g
of
N
d
u
a
l
po
larized
satellites an
d
a m
o
b
ile
ground receivi
ng station with
M
equally spaced a
n
tenna
s
. This corres
p
onds to a
2N
x
M
m
u
ltia
n
t
enn
a
wireless system
. Ho
wev
e
r, sin
ce th
e
satelli
tes an
tenn
as are no
t co-lo
cated
, t
h
e relativ
e
d
e
lay b
e
tween
sig
n
a
l
transm
ission from
each satellites nee
d
to
be
accounted for i
n
the
system
model
[3]. Th
e received
signal at
the
m
o
b
ile statio
n
can
th
erefore
be m
o
d
e
lled
as
…
⋮
(4
)
whe
r
e
H
si
is the 2
x
M
im
p
u
l
se resp
on
se matrix
fo
r t
h
e chan
n
e
l
b
e
tween
th
e i-th
satellite an
d
t
h
e
M
receive
ante
nnas,
y
(
t
) = [
y
1
(
t
);
y
2
(
t
); …
;
y
M
(
t
)]
T
are t
h
e
recei
ved si
gnals,
are
t
h
e
tran
sm
it
ted
sign
als on
th
e two
po
larization
s
o
f
satellite i a
n
d
n
is th
e relativ
e d
e
lay exp
e
rien
ced
b
y
sig
n
a
ls
fro
m
th
e n
t
h
satellite with
resp
ect to
t
h
e
referen
c
e satellite.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
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8
Ca
pa
city and
Erro
r Ra
te Ana
l
ysis o
f
MIMO
S
a
t
ellite Commun
ica
tion
Systems … (Ramon
i Ad
eo
gun
)
61
6
3.
CH
AN
NEL MO
DELS
We consi
d
er t
h
ree
differe
n
t
m
odels for
our eval
ua
tio
ns in
th
is
p
a
p
e
r.
Th
e m
o
d
e
ls are th
e clu
s
ter
b
a
sed
sp
atial satellite
MIMO
m
o
d
e
l [4
], Lo
o
d
i
stribu
tion b
a
sed
an
alytical
m
o
d
e
l [7
,
8
]
an
d
t
h
e ph
y
s
ical -
statistical lan
d
m
o
b
ile satellit
e m
o
d
e
l [2
]. A brief
d
e
scri
p
tio
n of t
h
e satellite ch
ann
e
l m
o
d
e
ls is
p
r
esen
t
e
d
i
n
th
is sectio
n.
Fig
u
re
1
.
Mu
ltip
le Satellites Mu
ltip
le An
tenn
as
Grou
nd
Receiv
e
Station
Syste
m
3
.
1
.
Cluster
B
a
sed MIMO
Sa
tellite Mo
del
In
[4
], a clu
s
ter b
a
sed
MIMO
m
o
d
e
l was p
r
o
p
o
s
ed
fo
r MIMO satellite s
y
ste
m
s u
s
in
g
th
e con
cep
t
of
clustering
1
in
th
e stan
d
a
rd
ized
WINNER II/3
GPP m
o
d
e
l fo
r
terrestial M
I
MO system
s. Th
e sp
atial m
o
d
e
l i
s
gi
ve
n by
[
4
]
1
1
(5
)
whe
r
e
K is the
Ricean K-fact
or,
i
s
t
h
e l
i
n
e
of
si
g
h
t
(L
OS
)
com
pone
nt
of
t
h
e c
h
an
nel
i
m
pul
se resp
on
se
betwee
n the
nth satellite and the mth ground receiver a
n
te
nna
. The se
cond term
in th
e R
H
S of (5) is the non-
l
i
n
e-o
f-si
ght
(
N
LO
S) com
p
o
n
ent
o
f
t
h
e cha
nnel
w
h
i
c
h i
s
m
odel
l
e
d as a
sum
m
at
i
on of P cl
ust
e
rs, eac
h cl
ust
e
r
com
p
ri
si
ng
o
f
R
ray
s
. T
h
e
L
O
S a
n
d
NLO
S
com
pone
nt
are
m
odel
l
e
d as
ex
p
Φ
.G
θ
.
ex
p
2
⋋
sin
sin
Υ
ex
p
2
⋋
co
s
(6
)
and
(7
)
1
A cl
ust
e
r i
s
ge
neral
l
y
co
nsi
d
e
r
ed
as a
g
r
o
u
p
of
p
r
o
p
a
g
at
i
o
n
pat
h
s
sha
r
i
n
g c
o
m
m
on angl
e
of
ar
ri
val
s
a
nd/
or
d
e
lays of arrival. In
th
e clu
s
ter
b
a
sed appro
a
ch
for sa
tellite m
o
d
e
ls, it is assu
m
e
d
th
at p
a
t
h
s
with
in a cluster
share
closely s
p
aced delays
of arrival.
Evaluation Warning : The document was created with Spire.PDF for Python.
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08
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,
Au
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st 2
014
:
61
4
–
62
2
6
17
P
p
is the norm
a
lised power
of the p-t
h
m
u
ltipath co
m
pone
nt(MPC), R is
the num
ber of rays within
each cluster (a
ssum
e
d consta
nt in the
m
ode
l),
Φ
is the ionosphe
ric powe
r lo
ss com
p
ens
a
tion factor for each
ray in
th
e clu
s
t
e
rs,
G
R
(
) is the ground recei
ve station a
rra
y
gain for eac
h antenna in the
array,
rp
is the AOA
of
t
h
e
rt
h
ray
i
n
t
h
e
pt
h cl
u
s
t
e
r,
is t
h
e s
h
a
d
ow
fa
ding c
o
efficient
of the
rays,
P
is th
e
p
a
th
l
o
ss,
GT
(
) is th
e
satellite tran
s
m
it an
ten
n
a
respo
n
s
e fo
r rays
with
AOD
,
is th
e wav
e
leng
th
,
d
s
is t
h
e inter-satellite sp
acin
g
,
rp
) is th
e AOD o
f
th
e rt
h
ray o
f
th
e
pth
clu
s
ter,dm
is
the spacing betwee
n
th
e an
tenn
as
o
n
th
e m
o
b
ile g
r
ou
nd
receiving station
ante
nna a
r
ray,
rp
is th
e
AOA
of th
e rt
h
ray in th
e
pth
clu
s
ter,
V
m
i
s
th
e
v
e
l
o
city o
f
th
e
receive station,
i
s
t
h
e
i
o
no
s
phe
ri
c an
g
u
l
a
r
de
vi
at
i
on c
o
m
p
en
sat
i
on a
n
d
ϑ
is th
e
d
i
rectio
n of m
o
tio
n
o
f
t
h
e
ground receive
station.
3.2.
Free Spac
e LOS
Model
The free s
p
ace
M
I
M
O
sat
e
l
l
i
t
e
m
odel
cons
i
d
er t
h
e l
i
n
e o
f
si
ght
(L
OS
) com
pone
nt
of
t
h
e fadi
n
g
chan
nel
.
Each
ent
r
y
of t
h
e M
I
M
O
i
m
pul
se re
spo
n
se
m
a
t
r
i
x
i
s
de
fi
ne
d
by
[
2
]
H
ex
p
(8
)
h
e
re fc is th
e
carrier frequ
e
ncy, rij is th
e ge
o
m
etrical d
i
st
an
ce
b
e
tween
th
e j-th satellit
e tran
sm
it
antenna
and i-th m
obile
gr
ound
recei
ve station ante
nna
,
is t
h
e
wa
ve
num
b
er,
v
0
is
th
e
f
r
e
e sp
a
ce
spee
d
of l
i
g
ht
a
n
d
α
ij
i
s
t
h
e c
o
m
p
l
e
x at
t
e
nuat
i
on
o
f
t
h
e
p
r
o
p
a
gat
i
o
n
pat
h
de
fi
ne
d as
1
2
ex
p
∅
(9
)
w
h
er
e
is t
h
e phase
of the
carrier ass
u
med e
q
ual
fo
r al
l antenna
pai
r
s
.
Si
nce t
h
e a
p
proxim
a
tion
,
is app
licab
le
to
th
e satellite system
s co
n
s
i
d
er
ed in th
is
pap
e
r, t
h
e ch
ann
e
l
p
a
th g
a
i
n
s can
therefore
be a
p
proxim
a
t
ed by
[10]
|
|
;
∀
,
(1
0)
whe
r
e
C
i
s
a c
onst
a
nt
a
n
d
|
a
|
d
e
no
tes t
h
e ab
so
lu
te
v
a
lu
e of
a
.
3
.
3
.
Ana
l
ytica
l
MIMO Sa
tellite Mo
del
Th
e Lo
o
d
i
stri
b
u
tion
[7
] is often
u
s
ed
for the an
aly
tical
mo
d
e
lling
of land
m
o
b
ile satell
ite ch
an
n
e
ls.
Th
e M
I
MO i
m
p
u
l
se for the m
u
lt
i-p
o
l
arizatio
n
an
d mu
ltian
t
enn
a
chan
n
e
l con
s
id
ered
in th
is
p
a
p
e
r
can
t
h
eref
o
r
e
be m
odel
l
e
d
as a
su
m
m
a
t
i
on o
f
t
w
o
part
s
⋮⋮
⋮⋮
(1
1)
H
H
whe
r
e
H
m
odel
s
t
h
e s
h
ad
o
w
i
n
g ef
fect
of t
h
e
cha
nnel
a
n
d i
t
s ent
r
i
e
s a
r
e
gene
rat
e
d
usi
n
g t
h
e
Lo
g-
n
o
r
m
al d
i
str
i
bu
tio
n an
d
H
is the m
u
ltipath c
o
m
pone
nt
of the c
h
a
nnel
wit
h
Rayleigh
dis
t
ribute
d
e
n
tries
.
T
h
e
Loo
d
i
stribu
tion
b
a
sed
an
alyti
cal
m
o
d
e
ls ch
aracterize th
e ch
ann
e
l statistics u
s
i
n
g pro
b
a
bilit
y d
e
n
s
ity fun
c
tio
n
(p
df
) a
n
d c
u
m
u
l
a
t
i
v
e di
st
ri
but
i
o
n
f
unct
i
o
n
(C
DF
).
A
g
e
n
e
ral assu
mp
tio
n is th
at
th
e
p
r
op
ag
ating
wave
un
de
rg
o b
o
t
h
at
t
e
nuat
i
o
n an
d scat
t
e
ri
n
g
/
r
e
f
l
ect
i
on.
As gi
ven i
n
(
1
1
)
, t
h
e com
p
l
e
x c
h
an
nel
en
vel
o
pe i
s
a
sum
m
at
i
on o
f
R
a
y
l
ei
gh an
d l
o
g
-
no
rm
al
faded e
nvel
ope
s. T
h
e
pd
f
of
t
h
e c
h
an
nel
i
s
defi
n
e
d as
[
7
]
(1
2)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
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8-8
7
0
8
Ca
pa
city and
Erro
r Ra
te Ana
l
ysis o
f
MIMO
S
a
t
ellite Commun
ica
tion
Systems … (Ramon
i Ad
eo
gun
)
61
8
whe
r
e
and
are the
m
ean and
varia
n
ce
of t
h
e r
eceive
d signal envelope, respectively.
C
o
gi
ve
s t
h
e
avera
g
e p
o
we
r of
t
h
e
scat
t
e
re
d
com
pone
nt
of
t
r
ansm
i
t
t
e
d
si
g
n
al
.
4.
CH
AN
NEL C
A
PA
CIT
Y
A
N
D
BER
In th
is section
,
we
p
r
esen
t th
e
ch
ann
e
l cap
aci
ty
and t
h
eo
retical bit er
ro
r
rate (BER)
ex
pre
ssions
.
4.
1. Ch
an
nel
Ca
paci
ty
The c
h
annel c
a
pacity for a
narrowband M
I
MO sy
stem
with
ou
t ch
annel state in
formatio
n
at th
e
tran
sm
it
ter (CSIT) is g
e
n
e
rally g
i
v
e
n
b
y
Tel
a
tar’s sp
ectral
efficien
cy equ
a
tio
n
[9
]
(1
3)
whe
r
e (.)
H
d
e
no
tes th
e
Herm
i
ttan
tran
spo
s
e
o
f
a m
a
trix
an
d
is t
h
e lin
ear
sig
n
a
l-t
o
-no
i
se ration
value
com
put
ed
fr
om
t
h
e l
o
ga
ri
t
h
m
i
c SNR
by
(1
4)
Si
m
ilar to
[2
],
is
de
fine
d a
s
the
ratio
of the tra
n
sm
it powe
r at each of the satellite
antenna a
n
d the
noise
powe
r at eac
h
m
obile ground
receive a
n
tenna. T
h
e
d
ecibel
value
of the
SNR in (14) is
define
d as
SNR =
E
I
RP +
G
T
–
K
–
B
(1
5)
whe
r
e E
I
RP is
the effective is
otropic
radiate
d
powe
r,
G
T
is
th
e satellite figu
re of m
e
rit,
K
is th
e
dB
eq
ui
val
e
nt
of
B
o
l
t
z
m
a
nn’
s co
nst
a
nt
a
n
d
B
is th
e
do
wn
lin
k tran
sm
issio
n
b
a
ndwid
th.
4
.
2
.
Bit Erro
r
Rate
(BER)
Fo
llowing
th
e
an
alysis and
d
e
riv
a
tion
s
in [5
], a closed fo
rm
app
r
ox
im
a
tio
n
fo
r t
h
e
p
r
ob
abilit
y o
f
error for MPS
K
m
odulated transm
i
ssion
in additive white Gaus
sian
noise
(AWGN) is
gi
ven as
[5]
(1
6)
(1
7)
whe
r
e
M
is th
e co
n
s
tellatio
n
size,
i
s
t
h
e SNR
per sy
m
bol
, x i
s
a chi
-
sq
uare di
st
ri
b
u
t
e
d ran
d
o
m
vari
abl
e
and
[
M
/4
] d
e
n
o
t
es th
e sm
a
llest i
n
teg
e
r greater
th
an
o
r
eq
u
a
l t
o
M
/
4
.
Ass
u
m
i
ng t
h
at
t
h
e m
obi
l
e
gr
o
u
n
d
re
cei
ve
station
uses a
z
e
ro forci
n
g (ZF)
receive
r, the
MPSK BE
R
c
a
n be obtained
by
i
n
te
grating the e
r
ror
probability
in
(16
)
ov
er
x
.
(1
8)
whe
r
e
P
X
(
x
) is
th
e ch
i
-
squ
a
re
p
r
ob
ab
ility d
i
stribu
tio
n fu
n
c
ti
o
n
.
It can b
e
sho
w
n
t
h
at a cl
o
s
ed
form
ex
p
r
essio
n
fo
r
(1
8)
is [
6
]
(1
9)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
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-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
61
4
–
62
2
6
19
whe
r
e
U =
N –
M
+
1 a
n
d
k
i
s
gi
ven
by
(2
0)
5.
SIMULATION RESULTS
In th
is section
we
p
r
esen
t simu
latio
n
resu
lts
for th
e cap
acity an
d BER
o
f
d
i
fferen
t
con
f
i
g
uration
s
o
f
MIMO satellite syste
m
s with
th
e m
o
d
e
ls presen
t in Sec
tion
III. Th
e
sim
u
la
tio
n
p
a
ram
e
ters fo
r t
h
e sim
u
latio
n
s
are sh
own
in Tab
l
e 1 ex
cep
t
wh
ere
o
t
h
e
rwise st
ated
. Th
e intersatellite sp
acin
g
fo
r
system
s with
M
> 2
receive
an
tenn
as is cal
cu
lated
u
s
ing
t
h
e eq
u
a
tion
[2
]
2
(2
1)
In Figu
re
2
,
we presen
t th
e cap
acity (in bp
s/Hz
) as a
fu
n
c
t
i
o
n
of SNR
fo
r lin
ear fo
rm
ati
o
n m
u
ltip
le
satellite s
y
ste
m
using the cl
uster
based spatial channe
l
m
odel. The num
ber of satelli
tes and receive
antenna
el
em
ent
s
i
s
vari
ed
bet
w
ee
n
1 an
d
8.
As s
h
o
w
n i
n
t
h
e
figure, i
n
creasi
n
g the signal to noise ratio
(SNR)
increases
the c
h
annel ca
pacity for all a
n
tenna sizes a
s
e
x
pected. T
h
e ca
pacity also
increases with inc
r
ease i
n
the num
b
er of satellites and/or recei
ve
station ante
nna
ele
m
ents. For
instance
, the
capacity difference
b
e
tween
a
2
x
2 and
4
x
4
satellite syste
m
at
SN
R
= 30
dB
is ab
ou
t 10
d
B
.
Fig
u
re
3
presen
t th
e
co
m
p
le
m
e
n
t
ary cap
acity cu
m
u
la
tiv
e d
i
strib
u
tion
fun
c
tion
(CC
D
F)
fo
r
a d
u
a
l
po
larized
satellite syst
e
m
an
d
a
m
obile ground receive station with
four antenna elem
en
ts (corresponding to a 2
x 4 MIMO syst
e
m
) at
di
ffe
re
nt
si
g
n
al
t
o
noi
se
rat
i
o
(SNR
) l
e
v
e
l
s
.
The C
D
F
pl
ot
s
sh
ow
t
h
at
t
h
e
vari
a
n
ce
of t
h
e
cha
nnel
ca
pac
i
t
y
i
s
co
nsid
erab
ly small fo
r each
SNR lev
e
l. The cap
acity in
c
r
ease with SNR
can also be
cl
early observe
d
fro
m
Figure 3. In fi
gure 4,
we com
p
are the capacity for
diffe
rent num
ber of
satellites
and receive antenna
s
using
th
e
Lo
o-d
i
stribu
tio
n b
a
sed
analytical
satel
lit
e ch
ann
e
l
m
o
del fo
r sing
le and
m
u
lt
i-satelli
t
e
scen
ario
s. Clearly,
the cha
n
nel capacity also shows a
n
in
c
r
easing t
r
end
with both inc
r
ease i
n
SNR a
nd a
n
tenna sizes.
We
present
a plot of the MIMO satellit
e channel capa
c
ity versus
SNR for both single satellite
multiple receive antenna
ground station (SS-MR
A) and m
u
lt
ip
le
sate
llites m
u
ltiple receive
ante
nn
a ground station (MS-MR
A)
usi
ng
t
h
e l
i
n
e
of
si
g
h
t
(L
OS
) a
p
p
r
oxi
m
a
t
i
on m
o
del
i
n
fi
g
u
re
5
.
As
can
be
o
b
s
erve
d
fr
om
t
h
e fi
g
u
re
, t
h
e
c
h
an
nel
cap
acity ob
tain
ed using
th
e
LOS app
r
ox
imatio
n
m
o
d
e
l sh
ows a sim
ila
r tren
d an
d com
p
are well wi
th
th
e
capacity for si
milar scenarios using the cluster base
d an
d
anal
y
t
i
cal
channel
m
odel
s
. In
fi
gu
re 6
prese
n
t
t
h
e
co
m
p
le
m
e
n
t
ary cap
acity cu
m
u
la
tiv
e d
i
strib
u
tion
fun
c
tion
(CC
D
F)
fo
r
a d
u
a
l
po
larized
satellite syst
e
m
an
d
a
m
obile ground receive station with
four antenna elem
en
ts (corresponding to a 2
x 4 MIMO syst
e
m
) at
di
ffe
re
nt
si
gna
l
t
o
noi
se rat
i
o
(SNR
) l
e
vel
s
usi
n
g t
h
e l
i
n
e
of si
ght
(L
OS
)
app
r
o
x
i
m
at
i
o
n m
odel
.
Fi
nal
l
y
,
we
plot the
bit error
rate (BER
)
versus
si
gnal to noise
ratio (SNR)
for a t
w
o-satellite two
receive a
n
tenna syste
m
usi
n
g t
h
e t
h
ree
t
y
pes of m
odel
descri
be
d i
n
s
ect
i
on II
I.
As sho
w
n i
n
t
h
e fi
gu
re, t
h
e cl
ust
e
r base
d m
odel
gi
ve
s
lo
wer BER at h
i
gh
er SNR.
Ho
wev
e
r,
no
si
gn
ifican
t
d
i
fferen
ce is ob
serv
e
d
bet
w
een t
h
e
BER curves for the
th
ree ch
ann
e
l
m
o
d
e
ls at lo
w SNR reg
i
on
.
Su
mm
aril
y, th
e resu
lts
p
r
esen
ted
in
t
h
is sectio
n
sh
ows that th
e
sp
ectral efficien
cy o
f
satellite syste
m
s ca
n
b
e
si
g
n
i
fican
tly i
m
p
r
o
v
e
d b
y
h
a
v
i
n
g
m
u
ltip
le satellit
es an
d
m
u
l
tip
le an
tenn
as at t
h
e
g
r
o
u
n
d
statio
n.
Tabl
e 1. Si
m
u
lat
i
on
Pa
ram
e
t
e
rs
Para
m
e
ters
Valu
e
Satellite Orbit
Geostationary
Satellite Location
13
o
E
Intersatellite
Spaci
ng
6
m
Car
r
i
er
fr
equency
14GHz
Receive antenna spacing (2 satellites
)
68.2k
m
Gr
ound station ant
e
nna location
11.
1
o
E
,
47.
8
o
N
M
odulation
BPSK,
QPSK
with gr
ay
m
a
pping
Channel M
odels
See Section II
I
E
nvir
o
n
m
ent T
y
pical
Ur
ban
2
Det
a
i
l
e
d deri
v
a
t
i
ons
a
n
d just
i
f
i
cat
i
on
ca
n be fo
u
n
d
i
n
[2]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ca
pa
city and
Erro
r Ra
te Ana
l
ysis o
f
MIMO
S
a
t
ellite Commun
ica
tion
Systems … (Ramon
i Ad
eo
gun
)
62
0
Fig
u
re
2
.
MIM
O
Satellite Capacity Versus Si
g
n
a
l t
o
No
ise
Ratio
(SNR)
Usin
g th
e Cluster Based
Satellite
Ch
ann
e
l
Model: SSSAG
denotes
Single Satellite Si
ngle Antenna
Ground Receive
Station
Fig
u
re
3
.
Co
mp
li
m
e
n
t
ary Cap
acity Cu
mm
u
l
ativ
e Distribu
t
i
o
n
Fun
c
tion
fo
r a Si
n
g
l
e
Dual Po
larized
Satellite
andGround Receive Station
with
fo
ur Ante
nna
s (2x4 MIMO)
Using t
h
e
Loo
Distribution ba
sed analy
tical
MIMO Satellite Mod
e
l at d
i
fferen
t si
g
n
a
l t
o
n
o
i
se ratio
(SNR)
Figure
4. MIM
O
Satellite Capacity Versus SNR
for Si
ngle
Satellite Multi
pl
e
Receive Antenna Ground
Station
(SS-M
R
A) a
n
d M
u
ltiple Satellites Multiple
Ground
Receive Antennas
Ground Station (MS-M
R
A)
Using
t
h
e Lo
oDistribu
tio
n B
a
sed
An
alytica
l
Satellite Mo
d
e
l.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
S
SN
:
2
088
-87
08
I
J
ECE Vo
l. 4
,
N
o
. 4
,
Au
gu
st 2
014
:
61
4
–
62
2
6
21
Figure
5. MIM
O
Satellite Capacity Versus SNR
for Si
ngle
Satellite Multi
pl
e
Receive Antenna Ground
Station
(SSMR
A
)
and M
u
ltiple Satellites Multiple Gro
und Receive
Antennas Gr
ound Station (MS-MR
A)
Using
t
h
e Lin
e
of Si
g
h
t
(LOS) Satellite Mo
del.
Fig
u
re
6
.
Co
mp
li
m
e
n
t
ary Cap
acity Cu
mm
u
l
ativ
e Distribu
t
i
o
n
Fun
c
tion
fo
r a Si
n
g
l
e
Dual Po
larized
Satellite
and
G
r
ou
n
d
R
e
cei
ve St
at
i
o
n
wi
t
h
fo
ur
A
n
t
e
nna
s (
2
x
4 M
I
M
O
)
Usi
n
g t
h
e
Li
ne
of
Si
g
h
t
(
L
OS
) M
I
M
O
Satellite Model at differe
n
t si
gnal to
noise ratio (SNR) for a
2
Dual
Po
la
rized Satellites -
4 Receive
Ant
e
nna
s
(
4x4
)
System
Fig
u
re
7
.
Bit Erro
r Rate (BER
)
v
e
rsu
s
Sign
al to
No
is
e Ratio
(SNR) i
n
d
B
fo
r a Two-Satel
lite Two
-
Receiv
e
Ante
nna
Syste
m
using Cluste
r Base
d Model, Free
Sp
ace Loss Model a
n
d
Analytical MIMO Satellite Model.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
J
ECE
I
S
SN
:
208
8-8
7
0
8
Ca
pa
city and
Erro
r Ra
te Ana
l
ysis o
f
MIMO
S
a
t
ellite Commun
ica
tion
Systems … (Ramon
i Ad
eo
gun
)
62
2
6.
CO
NCL
USI
O
N
Multiple input
m
u
ltiple output dual pola
r
ized sa
tellite
system
s can prov
ide inc
r
eas
ed spectral
efficien
cy and
i
m
p
r
ov
ed b
it erro
r rate
(BER
) co
m
p
ared
t
o
th
e classical si
n
g
l
e satellite syste
m
s. In
t
h
is p
a
p
e
r,
we an
alyzed
t
h
e cap
acity an
d
BER o
f
d
i
fferen
t
m
u
lt
ip
le
satellite scen
ario
s u
s
ing
d
i
fferen
t
m
o
d
e
ls. Sim
u
l
a
tio
n
results showe
d
that increasi
ng the num
b
er of satellite and/or ground
r
eceive station ante
nnas
can
significantly
increase t
h
e ca
pacity and d
ecrease the
bit error rate.
REFERE
NC
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[1]
P. Driessen and G. Foschini, On the cap
ac
it
y f
o
rm
ula
for m
u
ltiple inpu
t m
u
lti
ple output wir
e
l
e
ss channels:
a
geometric interp
retation,
IEEE Transactions on
C
o
mmunications,
vol. 47
, no
. 2
,
pp
. 173176
, Feb
19
99
[2]
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hwa
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nd A.
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nd D
.
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r
ma
nn a
nd
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Hofina
nn a
nd B.
La
nkl,
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a
p
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c
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mimo
sa
te
llite
link
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ices, F
e
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[3]
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.
Schwarz
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d B. L
a
nkl, Satellit
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y
st
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in
los
chann
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ls, 30 2008-Dec.
4 2008, pp. 16.
[4]
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annel M
odel and Capacity
A
n
aly
s
is for MI
MO Satellite F
o
rmation Fly
i
n
g
Communication S
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stems,
Interna
tional Journal of
Computer A
pplications (IJCA
), J
une 2013.
[5]
Jinhua Lu and K.B Letaief and J.C.
I Chuang and M.L Liou, M-PSK and
M-
QAM BER comput
ation using signal-
space
conc
epts, I
EEE
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m
., vol
47, pp18
1-184, Feb
.
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[6]
Cheng WANG
and Edward K.
S.
AU and
Ro
ss
D. Murch and Vincent K. N. La
u
,
Closed-Form Ou
tage Probability
and BER
of MI
MO Zero-Forcin
g
Receiver in
th
e Presence of
Imperfect CSI, SPAWC 2006
[7]
Loo, C. A S
t
atis
tic
al Model for a Land Mobile
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a
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EEE Tr
ansact
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[9]
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Tel
a
t
a
r,
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cit
y
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u
lti-
ant
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nna Gaussian
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h
annels
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e
an Tr
ansac
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lecom
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unica
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[10]
R.T, Schwar
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d A. Knopp and B. Lanki, Th
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BI
O
G
R
A
P
HY
OF
A
U
T
HO
R
Ramoni Adeogun is currently
w
o
rking towards a
PhD degree in Engineer
ing (
s
pecializing in
Communications and Signal Pr
oce
ssing) at the School of Engin
eering and Co
mputer Science,
Victori
a
Univers
i
t
y
of W
e
l
lingto
n
, New Zeal
and.
He receiv
e
d the
B.Eng degre
e
in
Elec
tric
al an
d
Computer Engineering with Fir
s
t Class Honours
from the Federal University
o
f
Techno
log
y
Minna, Nig
e
r State Nig
e
ria in 2
007. Betw
een
2
008 - 2009, h
e
was with the In
formation and
Communication
Techno
log
y
(IC
T) director
ate, U
n
iver
sity
of Jos,
Nige
ria.
He worked briefly
as
an Engineer with Odua Teleco
ms Ltd, Ibadan
Nigeria in 2009. He joined the National Space
Research an
d Development Agency
(NASRDA)
Abuja Nigeria in 2010 and has since been
working with th
e Engin
eer
ing and Space S
y
stem
s (ESS) division of the ag
ency
. Ramoni ho
lds
several Honours
and awards
in
cluding Ogun
Stat
e
tertiar
y
S
c
holarship (200
3 -2006), best
graduating stud
ent in th
e univ
e
rsity
(2007), C
o
mmon
w
ealth Shared Scholarship (2011) and
Victori
a
Doctora
l
Scholarship. H
e
is a graduat
e
m
e
m
b
er of Institute of Ele
c
tr
ica
l
and Ele
c
troni
cs
Engineers. A member of the International Asso
ciation of Engin
e
ers (IAENG). He is currently
act
ing as
a
revi
e
w
er for s
e
v
e
ra
l i
n
terna
tiona
l
conf
erenc
e
s
and
pe
er
revi
ewed
journa
ls
.
Evaluation Warning : The document was created with Spire.PDF for Python.