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Performance Analysis of MIMO Cooperative Relaying System Based on Alamouti STBC and Amplify-and-Forward Schemes
Abderrazek Abdaoui1, Salama S. Ikki2 , Mohamed Hossam Ahmed3 and Ramesh Pyndiah1
1Institut Telecom, Telecom Bretagne, France, (Email : abderrazek.abdaoui, ramesh.pyndiah}@telecom-bretagne.eu )
2University of Waterloo, Canada, Email: [email protected] 3Memorial University, Canada Email: [email protected]
Abstract This paper presents performance analysis of cooperative multiple-input multiple-output (MIMO)
relaying system with a single relay. The MIMO scheme is based on Alamouti space time block coding
(STBC) over Rayleigh flat fading channels. The source node, equipped with two transmit antennas,
simply broadcasts each STB code to the relay and the destination nodes. Then, the relay node, equipped
with multiple antennas, amplifies-and-forwards (AF) the received STB codes. Finally, the destination
node uses maximum ratio combining (MRC) and exploits the diversity gain obtained by the direct and
the indirect links simultaneously. The moment generating functions (MGF) of the signal-to-noise ratio
(SNR) for the direct and the indirect link are given in a closed form. These statistical results are then
applied to derive a lower bound of the symbol error probability (SEP) for a particular signal of M-ary-
quadrature-amplitude modulation (M-QAM) and to obtain the outage probability. Subsequently,
simulation results of the SEP and the outage probabilities are presented to illustrate the performance
improvement given by the MIMO cooperative diversity systems based on STBC schemes.
Index items
MIMO relay channel, space-time block coding, MRC scheme, moment generating function
(MGF), outage probability.
I INTRODUCTION
Cooperative communication has recently attracted a lot of interest due to its ability to realize the
performance gains and coverage extension [1] - [3] . Typically, it concerns a system where users share
and coordinate their resources to enhance the transmission quality and to optimize the power allocation.
The combination of relaying system with MIMO processing is a natural extension of both concepts.
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Moreover, this combination gives additional degrees of freedom to improve the capacity of the overall
cooperative system [4] [5] .
Recently, it has been demonstrated that cooperation based on space time block codes (STBC)
represents an effective way to introduce spatial diversity in wireless scenarios where we can not take
the full benefit of the uncorrelated channels from the multi-antenna systems. Cooperative diversity
gains can be achieved through creating distributed virtual antennas across different terminals in the
network. Actually, there exist two ways to apply STBC technologies in cooperative system. In the first
way, cooperation using distributed STBC is applied in order to create a virtual transmit array in a
distributed multiple relay network [6] - [8] . For example, in [8] , the Alamouti space time block code
[9] is employed with a distributed manner in a cooperative relay network over Rayleigh fading
environment. In the second way, the STBC matrix is completely broadcasted to the relay and the
destination. Several time slots are employed during the transmission of each STBC matrix. For
Alamouti STBC, cooperative relay network with a single relay system and two receive antennas can be
considered as a virtual MIMO system with four receive antennas, in consequence, the performance of
the cooperative diversity system can be improved without increasing the number of receive antennas.
In [10] , for a MIMO relay system, the authors neglect the source destination link and consider that one
source transmits an STBC matrix via one relay node using AF protocol. Specifically, they derive the
exact SEP for maximum likelihood (ML) decoding of orthogonal STBC in dual-hop relay channels.
Another work related to the use of STBC as in the second way is given in [11] . In their paper, the
authors have investigated the performances of MIMO relaying systems with decode-and-forward (DF)
protocol where the source, the relay and the destination are multiple-antenna nodes. Specifically, the
authors derive a closed-form expression for the outage probability..
Some recent papers related to our contribution give performance analysis of cooperative diversity
system based on STBC scheme over some particular scenarios [13] [14] . In [13] , Safari and Uysal
derive an upper-bound on the pairwise error probability (PEP) for cooperative diversity schemes over
log-normal fading channels and the distributed SIMO, MISO and MIMO systems. In their
contributions, the authors derive a Chernoff bound on PEP and a union bound on bit error rate (BER)
performance where each node is equipped with single antenna. In [14] , Muhaidat and Uysal give a
derivation of the PEP by including an extension to multiple antennas nodes. In their paper, the authors
derive a closed form of the PEP for dual-hop relaying scheme and channel state information (CSI)
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assisted AF, blind AF and DF. In their work, the source node transmits a general STBC matrix and the
destination applies ML decoding considering only the indirect link (the direct link is neglected).
However, this derivation seems to be incomplete since in most realistic cases the diversity is increased
by considering both the direct and indirect links. In [15] , Song and Hong, derive the SEP for MIMO
cooperative AF relay system based on orthogonal STBC and M-ary phase-shift keying (M-PSK)
modulation when ML detection is employed at the receiver. They focus their analysis on SEP behavior
in the asymptotic regimes of the number of relay antennas and SNR. In their derivation, the SEP is
given in an integral form and it is limited to ML decoding which is known by its huge complexity for
an important constellation size and number of antennas.
In this paper, we present performance analysis of a cooperative MIMO relay system based on
Alamouti scheme (Fig. 1). The MGF of the signal-to-noise ratio for the direct and the indirect link are
given in a closed forms. These MGF are then applied to derive a closed form of the lower bound of the
SEP and to obtain the outage probability.
To the best of the authors’ knowledge, there is no published work that derives the SEP in a
complete and exact form of MIMO relaying system using Alamouti STBC matrix codes and amplify-
and-forward relaying. We complete the contributions of [14] as follows:
1) We derive the SEP for more general cooperative diversity by including the direct and
indirect links.
2) We avoid the complexity of the ML decoder by using an MRC based decoder determining
the SEP at the demapper front end for general M-QAM constellation.
Fig. 1 : MIMO cooperation scheme with source node broadcasting Alamouti STBC matrix.
The paper is organized as follows. In Section II, we introduce system model. Section III presents
the SNR analysis at the output of the destination for the direct link. In Section IV, we develop an upper
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Supprimé : However, this derivation seems to be incomplete since in most realistic cases the diversity is increased by considering both the direct and indirect links.¶
Supprimé : [9] [9]
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Supprimé : In this paper, we present performance analysis of a cooperative MIMO relay system based on Alamouti scheme [9] (Fig. 1). The moment generating functions (MGF) of the signal-to-noise ratio for the direct and the indirect link are given in a closed forms. These MGF are then applied to derive a closed form of the lower bound of the SEP and to obtain the outage probability.¶
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bound of the SNR at the output of the destination for the indirect link. In Section V, using the MGF of
the SNR for the direct and indirect links, we give a lower bound on the SEP of the cooperative link
taking into account the diversity given by these two links. Performances related to the outage
probabilities and the diversity gains are detailed in section VI. We discuss and analyze numerical
results in Section VII. Finally, conclusions are drawn in Section VIII.
II SYSTEM MODEL
In this section, we describe a MIMO cooperative system consisting of a source, relay and
destination nodes equipped with multiple antennas. We consider the amplify-and-forward cooperative
MIMO relay channel as shown in Fig. 1. The source, relay and destination nodes have tN , rN and rN
antennas, respectively. In order to provide an efficient coding rate, we use 22× Alamouti matrix code,
we consider 2=tN transmits antennas. With a slight modification in the MIMO STBC model for each
link, we have transformed the channel matrix H into a modified channel matrix ][ 2XH with
orthogonal columns and 22 ×Nr entries. According to the same modification in MIMO STBC channel
model, the source-destination, the source-relay and the relay-destination channel matrices are
222 ][ ×∈ rNCXH , 22
2 ][ ×∈ rNCXD and rr NN 222 ][ ×∈CXF , respectively.
We assume a half duplex relaying protocol so, the transmission from the source to the destination
is made in two separate time-slots as in time division multiplexing (TDM) systems [17] . In the first
time-slot, the source sends its Alamouti encoded signal x to the relay and the destination, where
Tss ],[= 21x is the vector symbols that composes the Alamouti matrix. The relay simply amplifies the
received signal before forwarding it to the destination during the second time-slot. Finally, the
destination combines the signals of two time-slots coming from the relay and the source nodes using
MRC.
The vector signal transmitted by the source and received by the destination is given as follows:
020 ][.= nxHEy s +Xsdα (1)
where 0n is the modified 12 ×Nr noise vector measured at the destination which is composed by zero
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mean complex Gaussian random variables with variance 0N . sE is the average power of each symbol
which is equal to { } /n s tx P NΕ = , where Ps is the total transmit power per symbol and sdα is the
pathloss factor applied at the direct link, respectively.
At the destination node, according to the MIMO STBC modelization given above, the received
signal is a vector 0y given by
+
−
−
*,2
,1
*2,1
1,1
*,1
*,2
,2,1
*1,1
*1,2
1,21,1
*,2
,1
*1,2
1,1
=
Nr
Nr
NrNr
NrNr
sd
Nr
Nr
n
n
n
n
hh
hh
hh
hh
y
y
y
y
MMM
MxEsα (2)
If we consider the relayed link, the vector signals observed at the relay is
rsrr nxDEy s +][.= 2Xα (3)
where srα is the path loss factor related to the source relay link (first hop link), and rn is the
12 ×Nr modified vector noise applied at the relay, which is composed by rN2 zero-mean complex
random variables with variance 0N . At the relay, the system applies AF protocol with a matrix gain G
defined as
Nr
Fsr
rd2
0
2.= I
NDE
EG
s
r
+αα
(4)
where F||||D is the Frobenius norm 1 of the matrix D and rN2I is the rr NN 22 × identity matrix..
In order to simplify the analysis, we assume a fixed gain, NrNrsr
rd g 220
=.= IINE
EG
s
r
+αα
. We
1The Frobenius norm is defined as )(Tr=|||| H
F DDD .
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NrNr 22 × identity matrix.
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notice here, that fixed gain G used for AF relaying is considered in many papers in the literature, (see
for example [18] ). At the destination, the relayed signal is given by
dr22s n]nFG[x]FGD[Ey ++ XXsrd α= (5)
The equivalent virtual MIMO, 24 ×rN , representation of the proposed MIMO relay system is given as
follows
+
d
rs
s
n
n
n
IFG0
00Ix
FGDE
HEy
y 0
22
2
2
20
][][
][=
r
r
N
N
sr
sd
d XX
X
αα
(6)
while the equivalent source-relay-destination channel matrix is given by ][=][ 22 XX FGDU ′ let
][= 2XFDU .
.=][
*,1
*,2
,2,1
*1,1
*1,2
1,21,1
2
−
−
NrNr
NrNr
uu
uu
uu
uu
MMXU (7)
with the FD matrix product, jiu , are defined as
2)(1),(1,,1=
, = ≤≤≤≤∑ jNrijkki
N
kji dfu
r
(8)
The equivalent noise measured at the output of the relayed link is
dre nnFGn +][= 2X (9)
In order to evaluate the SEP, at the output of the MRC at the destination, we decompose the
problem into two sub-problems. In the first one, we derive the SNR at the MRC output considering the
direct link only. In the second sub-problem, we calculate the SNR at the MRC output for the relayed
link and finally we use the MGF of each SNR to give the SEP and to evaluate de outage probability of
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the cooperative and non cooperative links.
SNR OF THE DIRECT LINK
The modified matrix ][ 2XH describing the equivalent (source-destination) channel has
orthogonal columns [15, p. 285]. Applying the MRC to the received vector signal 0y , we have as
output
02
,22
,11=
020~||||=.][=~ nxEyH s +
+∑ ii
N
isd
H hhyr
αX (10)
where 0~n is the equivalent noise measured at the output of the MRC given as
−
+∑ *
,2,1,1*,2
*,2,2,1,1
1=020 =].[=~
iiii
iiiiN
i
H
nhnh
nhnhr
nHn X (11)
The covariance matrix of the equivalent noise 0~n is
{ } 22
00 ||||=~~ IHnn FHF Nε (12)
{ } 2,10 =~ 0nε
where {.}ε is the expectation of {.} . The effective channel for the data symbols },{1,, Miis K∈ is
},{1,0
2 ,~||||= MiiiiFsdi snsz K∈+HEsα (13)
Hence, according to (12) and (13), the SNR of the signal transmitted through the source-
destination link and measured at the output of the MRC is given by
0
2||||.=
NFsd
sd
HEsαγ (14)
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Since the channel of the direct link with matrix H is flat Rayleigh fading then all the entries jih ,
of H are complex Gaussian random variables each with (0,1)N distribution. Hence, sdγ is a random
variable equal to the sum of rN4 Gaussian random variables each of which with (0,1)N distribution.
Then, sdγ is a Chi-squared random variable with rN4 degrees of freedom. Thus, the probabiliy density
function (pdf) of sdγ is
)2
(exp)(22
1=)( 2
12
2sd
N
N
rNsd r
r
r Nf
γγ
γγγγ −
Γ
−
(15)
where )(2.= 2
0hsdsd Nr
Nσαγ sE
is the average SNR of the source-destination link and (.)Γ is the Gamma
function [18, eq. (8.310.1)] defined as 1)!(=)( −Γ nn where n is an integer, 0>n . The MGF,
))(( sMsdγ , of sdγ is then given by
γγγγγ dsfsMsdsd
)(exp)(=)( −∫∞
∞− (16)
Since sdγ is a Chi-squared random variable with mean sdγ , then using
1)(
0)!(=)(exp +−∞
∫nn nd βγβγγ , the MGF of sdγ can be easily found as
0)2(1=)(m
sdsdssM
−+ γγ (17)
where rNm 2=0
SNR OF THE INDIRECT LINK
In this section, we evaluate the SNR of the relayed link ( srdγ ) at the output of the MRC. The
modified matrix ][ 2XU describing the equivalent relayed link FD has orthogonal columns. Applying
the MRC to the vector signal dy , according to [16] , the output of the MRC is
dH
d yUy .][=~2X′
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nxEs~||||= 2
,22
,11=
+
+∑ ii
N
isr uu
r
α (18)
where n~ is the equivalent noise channel measured at the output of the MRC
enUn ].[=~2XH′ (19)
Substituting (9) in (19), the covariance matrix of the equivalent noise n~ is given by
( )
++ ∑∑ 2
1=
222
21
1=0 ||1||||=}~~{ ik
Nr
kii
N
i
H fguuNr
nnε (20)
Then, the effective channel for the data symbols },{1,, Miis K∈ is
},{1,2 ~||||.= MiiiFsr
ri nsgz K∈+UEsα (21)
By letting 0
=Nsr
srsEαγ as the mean SNR of the source- relay link, the instantaneous SNR srdγ of the
source-relay-destination link can be given by
[ ]
++
+
∑∑
∑
2
1=
222
21
1=
2
22
21
1=
2
||1||||
||||
=
ik
N
kii
N
i
ii
N
isr
srd
fguu
uug
rr
r
γγ (22)
Deriving a closed-form expression of the PDF of the instantaneous SNR srdγ is too hard to
accomplish. Hence, we use an upper bound of the SNR and compare the analytical results with the
exact simulation results. The upper bound of the SNR in (22) is obtained by neglecting the term
2
1=
2 || ik
Nr
kfg ∑ in (22) as follows
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+≤ ∑ 2
22
11=
2 ||||. ii
N
isrsrd uug
r
γγ (23)
Since F and D are random matrices with complex Gaussian entries, lif , and lid , can be expressed
as
ilililililil bjadjbaf ′+′+ =and= (24)
where ilililil baba ′′ and,, are N(0,1) random variables. Substituting (24) in (8) and evaluating the
expression in more compact form we have
( ) ( )klikklikklikklik
Nr
klilili abbajbbaaju ′+′+′−′+ ∑
1=,,, == µλ 2)(1),(1 ≤≤≤≤ lNri (25)
where li ,λ and li ,µ are the real part the imaginary part of liu , , respectively. Without loss of generality,
omitting the indices ki, and l , the random variables bbaa ′′ and,, are )(0, 2σN . Then, according to
the Gaussian random variable properties, the product aax ′= , is a random variable equal to the product
of two independent Gaussian random variables with zero mean and variance 1σ and 2σ respectively.
According to [17] , this product is a zero mean random variable and its pdf is given by
210
21
||1=)(
σσσπσx
KxpX (26)
where (.)0K is the Bessel function of the second kind and order zero. Fig. 2 illustrates the pdf of X
obtained analytically and by Monte-Carlo simulation for verification.
Actually both li ,λ and li ,µ are equal to the sum of Nr2 zero mean random variables with modified
Bessel function distribution of order 0. According to the central-limit theorem, li ,λ and li ,µ can be
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case where 4≥Nr . In order to derive a closed-form of the upper
bound of srdγ , we assume that
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respectively. Precisely, each real
and imaginary part is (0,1)N
random variable 1
Supprimé : ¶
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liu , in (25)
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approximated by Gaussian random variables with zero mean and variance )(2 22dfNr σσ . As shown in
Fig. 3, the pdf of li ,λ is very close to the Gaussian distribution. Hence, 222=|| ilililu µλ + can be
approximated by exponential random variables. Thus, 2|||| FU is equal to the sum of four exponential
random variables with parameter 22σ , is a Chi-squared random variable with rN22× degrees of
freedom. Thus, the pdf of srdγ is upper bounded by
( )
−
Γ
−
srdNr
srd
Nr
Nrsrd Nrf
γγ
γγγγ
2exp
22
1=)(
)(2
1)(2
)(2 (27)
where 222
0
)(2(= dfsr
srd NrgN
σσαγ sE).
Since the SNR at the output of the relayed link srdγ is a Chi-squared random variable with degrees
of freedom rN4 and using the same derivation as for equation (17), the MGF of srdγ is given by
denoting rNm 2=1
1)2(1=)(m
srdsrdssM
−+ γγ (28)
SEP OF THE COOPERATIVE SCHEME
The cooperation is based on the use of two independent branches: the direct and indirect links.
The SEP must average the two branches conditional over the pdf of sdγ and srdγ . For M-QAM
constellation, the average SEP expression, obtained by the MGF method, can be written as the sum of
two terms, denoted by 1I and 2I [21] ,
4444444444444 34444444444444 21
444444444444 8444444444444 76
2
2
QAM
2
QAM4
0
2
1
2
QAM
2
QAM2
0 sinsin
4
sinsin.
4=)(SEP
I
I
GGGGθ
θθπθ
θθπγ γγ
π
γγ
π
dMMq
dMMq
srdsdsrdsd
−
∫∫ (29)
where 1)]3/[2(=QAM −MG and Mq 4/1= − . For the first term in (29), if we substitute (17) and (28)
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σσ )//(exp=)( uupU −
for 0>u
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in (29) and using the change of variable θcos= 2t , after some manipulations, we obtain
( ) ( ) 102
1
2
11
0QAMQAM1 )(12
=mm
srdsdttMM
q ++−−−∫GGI γγπ
dttt
m
srd
m
sd
1
QAM
0
QAM
.2.1
11.
2.1
11
−−
+−
+−
γγ GG (30)
For the second term, upon making the change of variable θtan1= 2−t , we obtain
102
1
2
11
0QAMQAM
2
2 )(1)(2.)(2.4
=mm
srdsdttMM
q ++−−−∫GGI γγπ
dtttt
m
srd
srd
m
sd
sd11
QAM
QAM0
QAM
QAM
2
11.
4.1
2.11.
4.1
2.11
−−−
−
++
−
++
−γγ
γγ
G
G
G
G (31)
In order to continue the derivation of the SEP, we use the Lauricella multivariate hypergeometric
function )(nDF [22] as
),,;;,,,(= 11)(
nnn
D xxcbbaF KK
!!)(
)()()(
1
11
1
11
10=,,1 n
ni
n
i
nii
nini
nii
nii i
x
i
x
c
bba K
K
K
KK ++
++
∞
∑
1<|}||{|max 1 nxx K
( ) .1)1(1.)()(
)(=
1=
11
0dttxtt
aca
cib
i
L
i
ac −−− −−−−ΓΓ
Γ ∏∫α (32)
0>)(>)( aece RR
where )()/(=)( anaa n Γ+Γ is the Pochhammer symbol, with 1=)( 0a . Therefore, with the help of (32),
the average SEP of square M-QAM constellation can be deduced as
( ) ( ) ( ) ;;1,,2
1(
1
21
2= 1010
2QAMQAM
10
10
mmmmFMMmm
mmqSEP Dsrdsd
++++Γ
++ΓGG γγπ
)(2)(2
21
)21
1,
21
1QAMQAM
10
2
QAMQAM
GGGG srdsd
srdsd
MMmm
qγγ
πγγ
++−
++
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13
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)2
1,
41
21,
41
21;
2
3,1;,(1,
QAM
QAM
QAM
QAM1010
3
srd
srd
sd
sdD mmmmF
γγ
γγ
G
G
G
G
++
++
++ (33)
If we consider the indirect link only, the SEP is simply given by
( ) ( ) )21
1;;1,
2
1(
1
21
2=
QAM
111
QAM
1
1
srd
DsrdmmFM
m
mqSEP
γπ γG
G+
++Γ
+Γ
)2
1,
41
21,;
2
3,1;(1,)(2
21
QAM
QAM11
2QAM
1
2
srd
srdDsrd
mmFMm
q
γγ
πγ
G
GG
++
+
+− (34)
It is easy to generalize the derivation of the SEP for a MIMO cooperative diversity system with k
relays based on Alamouti STBC sheme. The SNR of each indirect link will be upperbounded as in
equation (23) and then the lowerbound of the SEP is as
( ) ( ) ( )QAMQAM00
0
1
21
2= GG
skddsk
k
MMmm
mmqSEP γγπ
KK
K
+++Γ
+++Γ
)21
1,,
21
1;,,,
2
1(
QAM0QAM
001)(
skdds
kkk
D mmmmFγγ GG ++
+++ KKK
)(2)(2
21 QAMQAM0
0
2
GGskdds
k
MMmm
qγγ
πK
K
+++−
)2
1,
41
21,,
41
21;
2
3,1;,,(1,
QAM
QAM
0QAM
0QAM00
2)(
skd
skd
ds
dskk
kD mmmmF
γγ
γγ
G
G
G
G
++
++
++++ KKK (35)
where the index term k in km and in skdγ refers to the source destination link for 0=k and to the
source-k th-relay-destination link for 1>=k .
OUTAGE PRBABILITY PERFORMANCE
In addition to the average SEP, outage probability, denoted by )( thoutP γ , is another standard
performance criterion of cooperative diversity systems. It is defined as the probability that the the
instantaneous combined SNR copγ falls below a certain specified threshold (thγ ), i.e.,
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Supprimé : il
Commentaire [MHA2] : Insert equation number
Supprimé :
Supprimé : k
Supprimé : S
Supprimé : SEP
Supprimé : instantaneous error rate exceeds a specified value or equivalently that
Supprimé : (
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copcopcop
ththcopout dpPP γγγγ γ
γ)(=][0=
0∫≤≤ (36)
where )( copcopp γγ is the probability density function of copγ . Mathematically speaking, the outage
probability coincides with the cumulative distribution function (CDF) of copγ evaluated at thγ , which is
equal to the inverse Laplace transform )(1 •−L of the ratio sscop
)/(−γM evaluated at thγ [23]
th
copout s
sMP
γ
γ
|
1)(
=
−L (37)
According to the assumption of mutually independent channels, the MGF, copγM of the combined
SNR can be expressed as
).()(=)( ssssrdsdcop γγγ MMM (38)
where )(ssdγM and )(s
srdγM are the MGF of the direct and the indirect links SNRs given by the results
in (17) and (28). We notice here, that the inverse Laplace transform can be derived analytically or using
simple numerical techniques. Using the results in [24] , equation (37) can be developed as
∑
−
k
KePP
K
kth
AK
thcopout0=
/22=)(=
γγγ
),(
22
22
1)(
0=
KNEjnA
jnAM
th
thcop
n
nkN
n
+
+
+−ℜ−×∑
+
γπ
γπ
α
γ
(39)
0
0=0=
/21 )2(1{1)(
2=
m
th
sd
n
nkN
n
K
k
AK jnAk
Ke
−+
+−
+−ℜ−
∑∑ π
γγ
α
),(})2(
1)2(1
1
KNEjnA
jnA
m
th
srd ++
+−×
−
ππ
γγ
(40)
where ℜ denotes the real part and the overall error term ),( KNE is approximated by
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Supprimé :
Supprimé : [22]
Supprimé : cooperative link
Supprimé :
Supprimé : ion
Supprimé : [23, equ. 9.186]
Supprimé : the
Supprimé :
15
Mis en forme
+++
+++−ℜ
−≅ ++−∑
th
thcop
kNKK
kth
A
kNjA
kNjAM
k
KeKNE
γπ
γπ
γ
γ
21)(2
2
1)(2
1)(2),( 1
0=
/2
(41)
0
11
0=
/2 1))(2(1{1)(2
m
th
sdkNKK
k
A kNjAk
Ke
−
+++−
+++−ℜ
−≅ ∑ π
γγ
}1))(2(
11))(2(1
1
+++
+++−×
−
kNjAkNjA
m
th
srd
ππ
γγ
(42)
The simple expressions of the outage probability and the corresponding numerical error can be
easily computed using commonly used mathematical packages such as Mathematica.
NUMERICAL RESULTS
In this section, numerical results and Monte-Carlo simulations for the AF MIMO cooperative
system based on Alamouti STBC scheme are presented to illustrate the previous theoret ical analysis.
The effects of the number of antennas rN and the constellation size M on the performance of MIMO
STBC cooperative scheme are discussed. It is assumed that the channel matrices H , F and D , of the
direct, first hop and second hop links, respectively are constant during one frame of 512 symbols,
which is essentially the case of quasi-static fading.
In Figs. 4 and 5, the SEP (equation (33)) is shown as a function of the common SNR of each link
0/sP N for several values of M and number of receive antennas rN . The SEP curves, plotted in Fig. 4,
shows the analytical lower bound on the SEP and the simulation results obtained by Monte-Carlo
simulation for 4-QAM modulation and 32,=Nr and 4 receive antennas. It is clear that the analytical
results provide a tight lower bound on the SEP for the overall SNR range. We can notice also that as Nr
increases AF cooperation improves the SEP performance significantly. These plots demonstrate the
effects of Nr on the diversity order. This can be justified by the array gain obtained by the use of
multiple antennas and the distributed array gain achieved by the use of both, multiple antennas and
relay nodes.
From Fig. 5, one can see that an increase in the constellation size M affects the system's error
performance. It can be seen that a transition from 16=M to 4=M leads to a performance
improvement greater than 9 dB for SEP 310= − . This is related to the intersymbol interference due to the
important number of symbols in 16-QAM compared to 4-QAM.
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Mis en forme
Mis en forme
Supprimé : RESULTS
Supprimé : S
Supprimé : are provided to discuss the impact of the relay
Supprimé : ,
Supprimé :
Supprimé : diversity reception with M-QAM scheme
Supprimé : Monte-Carlo simulation are obtained by running the simulators for five runs (each
with 710 samples).
Supprimé : SEP
Supprimé : from
Supprimé : average
Supprimé : signal-to-noise ratio
Supprimé : the relayed link for
Supprimé : SEP
Supprimé : SEP
Supprimé :
Supprimé : SEP
Supprimé : e
Supprimé : great increase
Supprimé : significant
Supprimé : SEP
Supprimé : with the number of
Supprimé : and t
Supprimé : diversity
Supprimé : the relay with
... [3]
... [1]
... [9]
... [7]
... [10]
... [8]
... [11]
... [2]
... [5]
... [4]
... [12]
... [6]
... [13]
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It is clear that the proposed lower bound and the simulation results on the SEP are in excellent
agreement. We can notice also the tightness of the derived lower bound of the SEP improves as SNR
increases; however this bound loses its tightness at low SNR.
In Figs. 6 and 7, the outage probability of the cooperative diversity system with STBC Alamouti
scheme over Rayleigh fading channels is given versus the common SNR of each link and for Nr = 2, 3
and 4 receive antennas considering two scenarios for the pathlosses. The curves give numerical
representation of (40) and (42) with A set to equ al 23.02610log10 ≈ to guarantee a discretization
error of less than 1010− and with 15=K and 30=N . From these figures, it is clear that the outage
probability significantly decreases with rN which proof the diversity gain achieved by the increase of
Nr. It is obvious also that the proposed lower bound and the simulation results are in excellent
agreement. We can notice the tightness of the derived lower bound of the outP improves as SNR
increases; however this bound loses its tightness at low SNR.
CONCLUSION
In this paper, we have used an upper bound for the signal-to-noise ratio to analyse the symbol
error probability of the MIMO Relay channel based on Alamouti STBC transmission. The analysis is
achieved using the MGF of the SNR of the cooperative links measured at the destination. A lower
bound expression of the SEP is given for MRC M-QAM in Rayleigh flat fading channels. The validity
of our analytical results is confirmed by Monte-Carlo simulations. Throughout this paper, we have
demonstrated the effectiveness of the lower bound analysis for high SNR range. Considering the
limitation of the analysis for hight SNR ragime, the exact closed form expression of the SEP with MRC
receiver need to be studied and compared with STBC MIMO cooperative systems based on ML
receiver.
Acknowledgment
The authors would like to thank Dr Christophe Laot, from Institut TELECOM, Telecom-Bretagne
(France) for the very helpful discussions and review during the finalization of this work.
REFERENCES
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Mis en forme : Soulignement
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Mis en forme : Centré
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Mise en forme : Puces etnuméros
Supprimé : the two last
Supprimé : average
Supprimé : signal-to-noise ratio
Supprimé : the relayed
Supprimé : using commonly used mathematical packages such as Mathematica
Commentaire [MHA3] : Explain the reason
Supprimé : increases
Commentaire [A4] : The sentence is rectified
Supprimé : also
Supprimé : also,
Supprimé : derive
Supprimé : and the outage probability
Supprimé : ¶
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[1] J. N. Laneman, D. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: Efficient protocol and outage behavior,” IEEE Trans. On Information theory , vol. 50, no. 12, pp. 3062--3080, Dec 2004.
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[15] Y. Song, H. Shin and E.K. Hong “MIMO Cooperative Diversity with Scalar-Gain Amplify-and-Forward Relaying”, IEEE Trans. On Comm. Vol. 57, no. 7, pp. 1932—1938, Jully 2009.
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[23] Y. Ko, M. S. Alouini, and M. K. Simon, “Outage Probability of Diversity Systems over Generalized Fading Channels,” IEEE Trans. On Communications, vol. 48, no. 11, pp. 1783--1787, Nov. 2000.
[24] M. K. Simon and M.-S. Alouini, Digital Communicatuion Over Fading Channels: A Uniform Approach to Performance Analysis. New York: Wiley, 2000.
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[26] O. Munoz-Medina, J. vidal, and A. Augustin, “Linear tranceiver design in nonregenerative relays with channel state information,” Trans. On Signal Processing , vol. 55, no. 6, pp. 2593--2604, june 2007.
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Fig. 2: Pdf of the the random variable abX = , given by (26).
Fig. 3: pdf of li ,λ .
Supprimé : ¶
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Fig. 4: SEP versus SNR for the cooperative link for 2,3,=Nr and 4 antennes, with 4-QAM
modulation.
Fig. 5: SEP versus SNR for the cooperative for 4-QAM, 16-QAM, and 32-QAM for 2=Nr .
Supprimé : ¶
Supprimé : ¶
Supprimé :
21
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Fig.e 6: Outage probability of the cooperative MIMO system ( 2=Nr , 3 and 4 receive antennas)
thγ , pathloss of the direct and indirect links are 1=sdα and 1=srdα .
Fig. 7: Outage probability of the cooperative MIMO STBC system ( 2=Nr , 3 and 4 receive
antennas) thγ , pathloss of the direct and indirect links are 5.0=sdα and 5.0=srdα .
Supprimé : ¶¶
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Anglais États-Unis, Décalage bas de 6 pt
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Police :Italique, Police de script complexe :Italique
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Police :Non Italique, Police de script complexe :Non Italique
Page 15: [5] Mis en forme Mohamed Hossam Ahmed 15/08/2009 00:55:00
Police :Italique, Police de script complexe :Italique
Page 15: [6] Mis en forme Mohamed Hossam Ahmed 15/08/2009 00:55:00
Police :Italique, Police de script complexe :Italique
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Police :Non Italique, Police de script complexe :Non Italique
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significant improvement on the
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with the number of receive antennas.
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the relay with multiple antennas and the diversity obtained by the receiver equiped by
rN receive antennas.
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