ISIT2007, Nice, France, June 24 - June 29, 2007

AF and DF Protocols based on Alamouti ST Code

Charlotte HucherENST

46, rue Barrault75013 Paris, France

hucher@ enst.fr

Ghaya Rekaya-Ben OthmanENST

46, rue Barrault75013 Paris, France

rekayaC@enst.fr

Jean-Claude BelfioreENST

46, rue Barrault75013 Paris, France

belfiore@ enst.fr

Abstract-In this work we propose a new amplify-and-forward(AF) protocol and a new decode-and-forward (DF) protocol basedon the Alamouti space-time (ST) code, chosen because of itsdecoding simplicity. We also apply a new selection criterionfor AF and DF protocols that improves their perforiiniice audsolves the problem of bad performance at low SNR. Finally, weapply the Alamouti AF and DF protocols to a "non-line-of-sight"(NLOS) scheme to bring diversity

Outage probabilities and simulation results show that atlow spectral efficiency, in spite of their rate of 2 symbol perchannel use, these Alamouti AF and DF protocols have betterperformance than the non-orthogonal AF (NAF) protocol.

I. INTRODUCTION

Cooperation strategies [1] have been recently developed inorder to exploit space-time diversity even with single-antennaterminals. These terminals cooperate in order to form a virtualMIMO array and bring 'cooperative diversity".AF protocols have been studied the most due to their sim-

plicity. This strategy consists in amplifying the received mes-sage at the relays and forwarding it. DF protocols require moreprocessing as the signals have to be decoded at the relays andthen forwarded. They have interesting performance howeverand are even essential for multihop systems. Asymptotically,both protocols bring diversity and give better performance thanSISO which only uses the direct link. However it does notmatch non-cooperation at low SNR (even the NAF protocolpresented as Protocol I in [2] and generalized in [3] associatedwith Golden code [4], which is the best known cooperationscheme [5]).

In this work we propose new AF and DF protocols basedon the Alamouti ST code [6]. A DF protocol using Alamouticode has already been proposed for a full-duplex channel in[7], but never in the half-duplex case. Outage probabilities andsimulation results prove that the new AF and DF protocolshave better performance than NAF at low spectral efficiency.Besides we apply a new selection criterion for AF and DFprotocols to improve their performance. This leads us to definean Adaptive Alamouti AF and an Adaptive Alamouti DF. Andfinally, we apply AF and DF Alamouti protocols to the NLOSchannel where the direct link is supposed non-existant, in orderto bring diversity.

II. ONI RELAY MODEL

We consider a wireless network with one source, one relayand one destination. The channel is half-duplex, which means

R0

h

s * gO

Fig. 1. Channel model: one source, one relay, one destination

that termninals, and in particular relays, cannot receive andtransmit at the same time. We assume a Rayleigh, slow fadingchannel, so that we can consider its coefficients as constantduring the transmission of at least one frame. We supposethat all terminals are single-antenna; the MIMO case is notconsidered in this work. Finally, we focus on the protocoland for simplicity, we assume a uniform energy distributionbetween the source and the relay.

In the next sections, we will use the notation given inFigure 1. The channel coefficients of the link between sourceand destination, source and relay and relay and destination arego, h and g, respectively.

III. ALAMOUTI AF AND DF PROTOCOLS

A. Protocols

In this paper, we present new AF and DF protocols basedon the Alamouti space-time code presented in [6]. These newprotocols require 4 channel uses to send 2 symbols: the symbolrate is 2 symb. pcu.As schematized in Tables I and II, in the first phase, the

source sends the first line of the Alamouti code matrix: x

TABLE IALAMOUTI AF PROTOCOL

S

RD

XI £2 £YE2:;- 1il L.~ IO

£1

Yr,2

TABLE IIALAMOTI DF PROTOCOTT

S XI

R /l

D ,..-,

2 1

.-1 .2

1-4244-1429-6/07/$25.00 ©2007 IEEE

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ISIT2007, Nice, France, June 24 - June 29, 2007

and X2, while the relay listens. In the second phase, the relaysends either an amplified version of the received signal in theAF case (/ = being the optimal amplifying factor

calculated in [ ]), or a decoded version of it in the DF case,while the source sends the second line of the Alamouti codematrix: -x and x£Any DF protocol supposes that signals are correctly decoded

at the relay during the first phase of the transmission. This isusually done by considering a selection between the DF andthe SISO schemes with the outage event of the source-relaylink as a criterion [8], [3]. Indeed, according to Shannon'stheorem, if the link is in outage, no detection without erroris possible: DF strategy is not efficient. In the other case,detection is possible and we use cooperation, assuming thatsignals have been correctly decoded. Outage probability is

PO(R) = P{loc((1 + SNPRh 2) < 2R}

where R is the global spectral efficiency and 2R is the one ofthe source-relay link.

B. DecodingA linear decoding as for the original Alamouti ST code can

be perfornned in both cases.1) Alamouti AF decoding. Received signals at relay are

Yrl = SNRhxl + vi and Yr2 SNRhX2 V2

and received signals at destination

Yi = SNRgxl + W

Y2 = .-SNRg0X2 + w2

Y3 = -N0X2 + 13Yg,Li) + W3

Y4 = 0XRgl +13glYr2) + IV4

We can develop

Y3 = V0NR2 (-g + VSNRgh1)which is equivalent to

\/2gil3v +wa

Assuming x, and w2 have been correctly decoded, thissystem of equations can be rewritten with an equivalentchannel matrix H being orthogonal and linear decoding canbe performed once again.

IV. PERFORMANCE OF T1HE ALAMOUTI COOPERATIONPROTOCOLS

A. Outage Probability

Outage probability is defined as P0ut(R) = P {C(H) < R}where R is the spectral efficiency.

1] Alamouti AF case. Instantaneous capacity can be calcu-lated from the expression of H defined in subsection III-B

CALAF (H) = log (det (I + SNRHHH))2 log (1+SNR (9gol2 + Igo SNRlgl Jh )2 (2)

(1) and then

POutALAF(R) = P {CALAF (H) < R} (3)

2) Alamouti DF case: We have to distinguish two cases:when we use cooperation or not.

Let's assume the source-relay link is not in outage: we usethe cooperation protocol. Then

CALDF (H)= 'log (I+

and PoUtALDFIO(R)=If the source-relay

cooperative scheme.

= 4 log (det (I + SNRHHH))SNR (90o2 + 0go2+I12 (4)

P {CALDF (H) < R}.link is in outage, we use the non-

P0Tsio) P{o&f SNg2) <} (5)I-OutSISO (R?) = P {lo~g (I + Sl\l Rlgo l2)< RI?Finally, we can write

PoutALDF(R) = PoutALDF,O(R)PO(R)+ PoutSISo(R)Po(R)(6)

with PO(R) defined in equation (1).In Figure 2, we have plotted the outage probabilities of the

SISO, NAF and the new Alamouti AF and DF protocols as

SNR (-gOY3=2 +SNR~gfl2 9'2 SNR1hx I

where normalization takes the variance of the noise intoaccount. Let's denote n 2 S

In the same way Y4 = (nowi +\1SNRghX2+)4wThe system of equation can then be rewritten in the form

Y = NPNHX + W wxith the equivalent channel matrix Hbeing orthogonal and the linear decoding can be performed.

2) Alamouti DF decoding. In the first phase, receivedsignals at destination are the same, and in the second phase

Y3 = SN2 (-0X2 + 9gX1) + w)3

Y4 = 2 (go0X g+ 912) + W4

where i1 and i2 are the signals decoded at relay in theconsidered constellation.

io

ct

o10

10

Fig. 2. Comnparison of the outage probabilities of the SISO, NAF andAlamouti AF and DF protocols for 2 bits spectral efficiency

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ISIT2007, Nice, France, June 24 - June 29, 2007

functions of the SNR (obtained numerically by Monte Carlosimulations). The spectral efficiency is 2 bits pcu. We cansee that, at low spectral efficiency, the Alamouti AF and DFoutperforms the NAF protocol. Moreover, the Alamouti DFhas slightly better performance than the Alamouti AF for lowSNR.

It is to be noticed however that, when we increase the spec-tral efficiency, the NAF becomes more and more performantcompared to Alamouti protocols, due to rate deficiency.

B. Diversity-Multiplexing Gain Tradeoff (DMT) analysis

The DMT has been proposed in [9] in order to evaluate theasymptotic performance of space-time codes.A diversity gain d(r) is achieved at multiplexing gain r if

2

1,5

0,2

0,5

O0

-NAF-Alamouti AF and DF, LTW AF and DF

, 4 ,0,4 0,6 0,8 1

multiplexing gain r

Fig. 3. Diversity-Multiplexing gain Tradeoff of the NAF and Alamouti AFand DF protocols

Let

lou Po0t (r log SN R)SNR oc logSNR

L's define

uo limlMo SNR 9 l2SNR--o Si

ad(r)

lim log 9 2NR- ooelogSNR

so we can note '90 S N R- where denotes anasymptotic behavior when SlN R oc. In the same way, wedefine ul and v such as p 2 SNR- and Ih ' SNR- .

From the expressions of the outage probabilities (3) and (6),we can compute the DMT of our two new Alamouti AF andDF protocols.

I) DMT of the Alamouti AF. Using the asymptotic behav-iors of the amplifying and normalization factors

02 = 1 - SNR -(-v)

n2 = 2 + SNRP92g2 SNR'-u -( ) SNR-u v

we can write the one of the outage probability of the AlamoutiAF protocol

PoutALAF(r log S N R)-P lou SNR-uO SR SNR 1 1

-P{max{ -uo, 1-uo + ul-v, 1-v} <= SNR d11

2rlou SNR}2r}

withdALAF= inf{uo + v}+ V} = 2(1 -2r) (7)

2) DMT of the Alamouti DFa In the same way, we can writethe asymptotic hehaviors of the different parts of the outageprobability of the Alamouti DF protocol.

POUtALDF (r lo SN)P{log(ISNP U +SNR'') <2rlouSN}I_ ,,2 2 zn,_

P{max{1ii-uo, 1-Ulf < SrN= dNR

with d inff{o u1} = 2(1 2r).

PoutsIso(rlog SNR) P{log(SNR i) <rlog SNRP{, *0r} -d2P-I uo < SNRi

with d2 = inf{uo} = 1-r.

Po(rlocSNR) P{log(SNPR'-)P{1-v <2r} -SNR d(

2rlo SlN\R}

with do = inf{v} = 1-2r.Then, the asymptotic behavior of the outage probability of

the Alamouti DF protocol is

PoutALDF(r log S1N\R) -SNR-dALDFwith

dALDF = min{di, d2 + do} = 2(1 -2r) (8)

We can notice (Figure 3) that Alamouti AF and DF havethe same DMT, lower than the one of the NAF protocol, evenif they have better performance at low spectral efficiency.

C. Simulation Results

Figure 4 represents the performance of the SISO, NAF andAlamouti AF and DF protocols as functions of the SNR. Thespectral efficiency is set to 2 bits pcu. We can see that bothAlamouti AF and DF have better performance than NAF witha 2 dB gain asymptotically.

Fig. 4. Comparison of the performance of the non-cooperative case, the NAFprotocol and the Alamouti AF and DF for 2 hits spectral effhciency

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ISIT2007, Nice, France, June 24 - June 29, 2007

V. ENHANCEMENT OF THE ALAMOUTI COOPERATIONPROTOCOLS: THE ADAPTIVE ALAMOUTI AF AND DF

A. Adaptive Alamouti AF

In order to improve the performance of the Alamouti AF, wepropose to add a selection criterion which leads us to definea new hybrid protocol called Adaptive Alamouti ALWe choose the transmission scheme with the highest instan-

taneous capacity between the SISO, NLOS AF and AlamoutiAF

SISO case: The instantaneous capacity is

CSiSO(H) = log (1 + SNRg9o 2)

1.)

9

°) 10

10

(9)

SISO----Adaptive NAF

Vz\'t.> A- Adaptive Alamouti AFv- \ Adaptive Alamouti DF

0 10 20 30SNR (dB)

NLOS AF case. The received signals at relay and des-tination respectively being Yr = S Rhx v and Yd

v SVNRg,Yr + w, we can compute the instantaneous capacityof the NLOS AF transmission scheme

y1 SN RP23CNLOSAF(H) = NloR1 +

g 22321 h 2

Alamnouti AF case: Alamouti AF instantaneous capacity isgiven in equation (2).

B. Adaptive Alainouti DF

We also adapt this new selection criterion to DF protoIf the source-relay link is in outage, signals cannot be decwithout error at relay, so we only use the direct link.the contrary, if the source-relay link is not in outage, tdifferent transmission schemes can be considered: the S:the NLOS DF and the Alamouti DF protocols. We choosone with the higher instantaneous capacity.SISO case: The instantaneous capacity of the SISO sci

is still given by equation (9).NLOS DF case. Assuming that signal has been corr

decoded and forwarded by relay, received signal at destinis y = SNg + v where is the decoded signal at r

And the instantaneous capacity is then

CNLOSDF(H) = log (1 + SNR|9g1 2)

Fig. 5. Comparison of the outage probabilities of the non-cooperative case,the Adaptive NAF and the Adaptive Alamouti AF and DF for 2 bits spectralefficiency

10°Y

1o-0

1o-2

10-3 10SNR (dB)

20 30

Fig. 6. Comparison of the performance of the non-cooperative case, theAdaptive NAF and the Adaptive Alamouti AF and DF for 2 bits spectralefficiency

In the DF case, the outage probability when the source-relaylink is not in outage becomes

(I1) PoutAALDFIO(R) = P maxCD(H)iC SISO,NLOS DFAL DF}

Alamouti DF case. Alamouti DF instantaneous capacity isgiven in equation (4).

VI. PERFORMANCE OF THE ADAPTIVE ALAMOUTICOOPERATION PROTOCOLS

A. Outage Probability

In the adaptive case, the instantaneous capacity is thehighest between the SISO, the NLOS and the cooperation

ones.

So in the AF case, the outage probability becomes

PoutAALAF(R) = P max

iC SISO,NLOS AFAL AFIC (H) < R}

The system is in outage only if all schemes are in outage.

Then, the global outage probability still is

PoutAALDF(R) = PoutAALDF O(R)P0 (R) +Poutsiso(R)Po(R)

where the other probability definitions remain the same

On Figure 5, we plot the outage probabilities of the SISO,the Adaptive AF and the Adaptive Alamouti AF and DF as

functions of the SNR. The spectral efficiency is once again setto 2 bits pcu.

We can see, that the Adaptive Alamouti AF and theAdaptive NAF have similar performances while the AdaptiveAlamouti DF slightly outperformns all these schemes. Moreover

adaptive schemes solve the problem of bad performance at lowSNR, and increase performance with a 1 dB gain over the non-

adaptive protocols.

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SISO

Adaptive NAF

Adaptive Alamouti AF

v- v Adaptive Alamouti DF..

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ISIT2007, Nice, France, June 24 - June 29, 2007

B. Simulation Results

Figure 6 represents the performance of the SISO, AdaptiveNAF and Adaptive Alamouti AF and DF protocols as functionsof the SNR for a spectral efficiency of 2 bits pcu. Perfor-mance obtained with the outage probabilities calculations areconfirmed. The problem of bad performance at low SNR issolved and DF protocol has slightly better performance thanboth AF protocols.

VII. AN APPLICATION OF THE ALAMOUTI COOPERATIONPROTOCOLS: NLOS CHANNEL MODEL

A. 2-relay NLOS channel

In this section, we consider a wireless network with Isource, 2 relays and 1 destination, where the source-destinationlink is so bad that we do not even consider it. The otherassumptions of the first part of this work are still valid. ThisNLOS channel model is schematized on Figure 7.

SNR (dB)

Fig. 8. Comparison of the performance of the non-codand Alamouti DF for 4 bits spectral efficiency

Ed AF, non-coded DF

h1

S D

R2

Fig. 7. Channel model: one source, two relays, one destination

B. Application of the Alamouti AF and DF to the 2-relayNLOS channel

In order to bring diversity, we propose to adapt our AlamoutiAF and DF to this new channel model. The new transmissionschemes are summarized in Tables III and IV.

TABLE IIIALAMOUTI AF PROTOCOL

S

RIR2D

S

R2D

-3Yr*2 . /32Y.........*

TABLE IVALAMIOUTIDF PRO TOCOL

t1 t2~~~~~Y:X

£~ ~~~£I..~~~~~ r £ .. 4 ?/X

Y1 Y2£

Figure 8 represents the frame error rate as a function of theSNR for the non-coded AF and DF, and the Alamouti AF andDF, at a spectral efficiency of 4 bits pcu. We can see that inthe non coded case, DF protocols are much more efficient thanAF protocols. In the space-time coded case, Alamouti AF andDF have nearly the same performance, except for low SNR,where the DF protocol is more efficient.

VIII. CONCLUSION

In this paper, we presented our work on the Alamoutispace-time coded cooperation systems. We proposed two newAF and DF protocols based on Alamouti code and studiedtheir performance in terms of outage probabilities and DMT.Moreover, we applied a new selection criterion for AF andDF protocols that improves their performance and solves theproblem of bad performance at low SNR. Finally, we tookinterest in the case of a NLOS relay channel, and applied ourAlamouti AF and DF protocols to it in order to bring diversity.

If we try to generalize this protocol to a higher number ofrelays, with other orthogonal codes, the symbol rate decreasesdramatically. Other space-time codes have to be developed,which can be applied to DF protocols with N relays andpreserve the symbol rate.

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[2] R. Nabar, H. Bolcskei, and F. Kneubuhler, "Fading relay channelsperformance limits and space-time signal design," IEEE J. Select. AreasCommun., vol. 22, no. 6, pp. 1099-1109, August 2004.

[3] K. Azarian, H. E. Gamal, and P. Schniter, "On the achievable diversity-mnultiplexing tradeoff in half-duplex cooperative channels," IEEE Trans.Inform. Theor,i vol. 51, no. 12, pp. 4152-4172, December 2005.

[4] J.-C. Belfiore, G. Rekaya, and E. Viterbo, "The Golden Code: A 2x2 Full-Rate Space-Time Code with Non-Vanishing Determinants," IEEE Trans.Inform. Theor,i vol. 51, no. 4, pp. 1432-1436, April 2005.

[5] S. Yang and J. Belfiore, "Optimal Space-Time Codes for the MIMOAmplify-and-Forward Cooperative Channel," in 2006 IrteroationialZurich Seminar on Communications, February 2006, pp. 122-125.

[6] S. Alamouti, "Space-Time block coding: A simple transmitter diversitytechnique for wireless communications," IEEE J. Select Areas Commun.,vol. 16, no. 8, pp. 1451-1458, October 1998.

[7] P. Anghel, G. Leus, and M. Kavehl, "Multi-user space-time coding incooperative networks," in International Conference on Acoustics, Speech,and Signal Processing, vol. 4, April 2003, pp. 73-76.

[8] J. Laneman, D. Tse, and G. Womell, "Cooperative diversity in wirelessnetworks: Efficient protocols and outage behavior, IEEE Trans. InformTheory, vol. 50, no. 12, pp. 3062-3080, December 2004.

r9g D Tse and P Viswanath Fundamoentalso f Wireless CommunicationCambnidge University Press, September 2004, draft to be published

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