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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: sikki@ecemail.uwaterloo.ca 3Memorial University, Canada Email: mhahmed@mun.ca

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.¶

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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

Supprimé : lif , and lid , ,

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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Supprimé : define

Supprimé : [21]

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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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14

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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 :

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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

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Mis en forme

Mis en forme

Mis en forme

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Mis en forme

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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]

16

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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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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,

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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.

[2] Nosratinia, T. Hunter, and A. Hedayet, “Cooperative communications in wireless networks,” IEEE Communciations Magazine , vol. 42, no. 10, pp. 74--80, Oct 2004.

[3] R. U. Nabar, H. Bolsckei, and F. W. Kneubuhler, “Fading relay channels: performance limits and Space-Time signal design,” IEEE J. Selec. Areas Commun. , vol. 22, no. 6, pp. 1099--1109, 2004.

[4] B. Wang, J. Zhang, and A. Host-Madsen, “On capacity of MIMO relay channels,” IEEE Trans. On. Information Theory , vol. 51, no. 1, pp. 29--43, Jan. 2005.

[5] J. Zhao, Y. Xu, and Y. Cai, “Cooperative Differential Space-Time Transmission scheme for MIMO R elay networks,” in Proc. IEEE Congress on Image and Signal Processing , Sanya, Hainan, China, 27-30 May 2008.

[6] S. Yiu, R. Schober, and L. Lampe, “Distributed space-time block coding for cooperative networks with multiple-antenna nodes,” in Proc. 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing , 2005, pp. 52--55.

[7] Y. Jing and B. Hassibi, “Distributed Space-Time Coding in wireless relay networks,” IEEE Trans. Wireless Commun. , vol. 5, no. 12, pp. 3524--3536, 2006.

[8] S. Atapattu and N. Rajatheva, “Exact SER of Alamouti code transmission through amplify-forward cooperative relay over Nakagami-m fading channels,” in Proc. IEEE International Symposium on Communications and Information Technologies ISCIT '07, 2007, pp. 1429--1433.

[9] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” Selected Areas in Communications, IEEE Journal on , vol. 16, no. 8, pp. 1451--1458, 1998.

[10] T. Q. Duong, H. Shin, and E.-K. Hong, “Effect of line-of-sight on dual-hop nonregenerative relay wireless communications,” in Proc. VTC-2007 Fall Vehicular Technology Conference 2007 IEEE 66th , 2007, pp. 571--575.

[11] B. K. Chalise and L. Vandendorpe, “Outage probability analysis of a MIMO relay channel with orthogonal space-time block codes,” IEEE Communications Letters , vol. 12, no. 4, pp. 280--282, 2008.

[12] Gershman and N. Sidiropoulos, Space-Time Processing for MIMO Communications .: John Wiley, 2005.

[13] M. Safari and M. Uysal, “Cooperative diversity over log-normal fading channels: performance analysis and optimization,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1963--1972, May 2008.

[14] H. Muhaidat and M. Uysal, “Cooperative diversity with multiple-antenna nodes in fading relay channels,” IEEE Trans. Wireless Commun. , vol. 7, no. 8, pp. 3036--3046, August 2008.

[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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Supprimé : coding

Supprimé :

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[16] V. Kühn, Wireless Communications over MIMO Channels : Applications to CDMA and Multiple Antenna Systems . West Susser England UK: John Wiley, 2006.

[17] S. Avestimehr and D. N. C. Tse, “Outage capacity of the fading relay channel in the low- SNR regime,” IEEE Trans. On Information Theory , vol. 53, no. 4, pp. 1401--1415, April 2007.

[18] M. O. Hasna and M. S. Alouini, “A performance study of dual-hop transmission with fixed G ain relays,” IEEE Trans. Wireless Commun. , vol. 3, no. 6, pp. 1963--1968, Nov. 2004.

[19] S. Gradshteyn and I. M. Ryzhik, Table of Integrals: Series and products, 5th ed., San Diego: Academic Press, 1994.

[20] M. K. Simon, Probability Distributions Involving Gaussian Random Variables, Californie USA: Springer, 2006.

[21] H. Shin and J. H. Lee, “On the error probability of binary and M-ary signals in Nakagami-m fading channels,” IEEE Trans. Commun. , vol. 52, no. 4, pp. 536--539, april 2004.

[22] H. Exton, Multiple Hypergemetric Functions and Applications. New York: John Wiley, 1965.

[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.

[25] B. Sirkeci-Mergen and A. Scaglione, “Randomized S pace- T ime coding for distributed cooperative communication,” IEEE Trans. On Signal Process. , vol. 55, no. 10, pp. 5003--5017, Oct. 2007.

[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 : ¶

20

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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

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Police :Italique, Police de script complexe :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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