Efficient MIMO-OFDM Schemes for Future Terrestrial Digital TV with Unequal Received Powers

Efficient MIMO-OFDM Schemes for Future T errestrial Digit al TV with Unequal Received Powers Youssef Nasser member IEEE , Jean-François Hélard Senior member IEEE , Mathieu Crussière and Oudomsack Pasquero Institute of Electronics and Telecommunications of Rennes, UMR CNRS 6164, Rennes, France Email : youssef.nasser@insa-rennes.fr Abstract - This article investigates the effect of equal and unequal received powe rs on the pe rformances of different MIMO-OFDM schemes for terrestrial digital TV. More precisel y, we focus on three ty pes of non- orthogonal schemes: the BL AST scheme, the Linear Dispersion (L D) code a nd the Gol den code, and we compare their performances to that of Alamouti scheme. Using two receiving antennas, we show that for moderate attenuation on the second antenna and high spectral efficiency, Golden code out performs other schemes. However, Alamouti scheme pres ents the best performance for low spectr al efficienc y and equal received powers or when one antenna is dramatically damaged. Wh en three antennas are use d, we show that Golden code offers the hi ghest robustness to power unbalance at the receiving side. Keywords- OFDM, MI MO, Space Time codes. I. INTRODUCTION The potential advantag es of digital television broadcasting over conve ntional anal ogue broadca sting are numerous and well known. For broadca sters, di gital technology offers significantly improv ed operational flexibility, provid ing the means for new services which go beyond t he scope of conventi onal tel evision programmes. Since its inauguration in 1993 , digital video broadcast (DVB) project for terrestrial (D VB-T) transmission has fully responded to the objectives of its designers, delivering wireless digital TV services in almost every continent [1]. In fact, there is no single DVB standard, but rather a collec tion of st andards, t echnical recommendation s and guidelines. In Sp ring 2006, DV B community was asked to provide technical specifications and studies for a future second generation of DVB-T called DVB-T2. It is expected that the first profile of DVB-T2 specification, for fixed recepti on of high definition television (HDTV) services, will be co mpleted as soon as possible, wit h a second profile offering improved m obile perform ance com pleted ar ound the end of 2008. Agai nst this backgro und, a new Europea n CELTIC project called Broadcast f or 21 st Century (B21C) was launched [2]. It constitutes a contribution task force to the reflections engaged by the DVB p roject and should give a real support for th e conclusions and decisions within DVB project, particularly on multiple input multiple ou tput (MIMO) with orthogonal frequency division mult iplexing (OFDM ) transmission f or HDTV services. The work presented in this paper has been carried out within the fra mework of B2 1C project. The contribution of this work is twofold. First, a generalized framework is proposed for modelling the effect of une qual received powers on different recei ving antennas. Therefore, we analyze and compare som e of the most promising MIMO- OFDM systems in the context of broadcasting fo r future terrestrial digital TV with equal but also une qual received powers i.e. with unequal r eceived signal to noise ratio (SNR) per ant enna. In t he literature, m ost of the works consider equal received pow ers for the perform ance comparison of MIMO-OFDM sch emes [3][4]. The assumption of unequal received powers could be see n in different communi cations contexts like in a broadcast l ink where two different antennas are used at the receiving side or in a m obile lin k. Indeed, the cal l for tech nology within DVB-T2 consortium moves to wards an expectation of su ch situations where one ou tdoor antenn a (roof anten na for example ) and one or tw o indoor antennas are used. Ev entually, we note that for complexity reasons the anal ysis of different MIMO - OFDM systems is not achieved with the optimal maximum likelihood (ML) r eceiver. Instead, we use a sub-optimal iterative receiver with few iterations. This paper is structure d as follows. Section 2 descri bes the system model for MIM O-OFDM. In section 3 we discuss the choice of different MIMO schemes considered in this paper. Section 4 pres ents the iterative receiver with a detailed description of its blocks. Simulation results are drawn in sect ion 5. Secti on 6 concl udes the paper. II. SYSTEM MODEL WITH UNEQUAL RECEIVED POWERS Consider an OFDM com munication sy stem usi ng M T transmit antennas (Tx) and M R receive antennas (Rx) for a downlink c ommuni cation. Suc h a system could be implem ented for the M T transmit antennas using a space- time (ST) enc oder whic h takes Q data c ompl ex symbols and transforms them to a ( M T , T ) output m atrix accor ding to the ST block codi ng (STBC) schem e. The ST STBC coding rate i s then de fined by L = Q / T . Fi gure 1 de picts the transmitte r modules. Inf ormation bi ts b k are first channel encoded with a convoluti onal encode r of coding rate R , randomly interleaved, and fed directly to a quadrature amplit ude modulation ( QAM) modul e which assig ns B bits for each of the complex constell ation points. Therefore, each group s =[ s 1 ,…, s Q ] of Q complex sym bols s q becomes the input of the STBC en coder. Let X =[ x i,t ] where x i,t ( i =1,…, M T ; t =1,…, T ) be the output of STBC encoder. This output is then f ed to M T OFDM modulat ors, each using N subcarriers. In order to have a fair analysis and comparison between different STB C codes, the s ignal power at the output of the ST encoder is normalized by M T . We assume in this work that the transmission from a transmitting antenna i and a receiving antenna j is achieved for eac h subcarrier n through a freque ncy non-selecti ve Rayleigh fadi ng channel. The latt er is assumed to be constant during T symbol durations. With these assumptions, th e channel coefficients h i,j [ n ] are assumed as independent complex Gaussian distri buted samples wi th unit variance. We assume also that the transmit ter and receiver are perfectly synchronised. Moreover, we assum e perfect channel state information (CSI) at the receiver. S ourc e E n code r Inter l e a v e r Ma ppe r ST B C Enc o d e r OFDM M od. OFDM M od. Ant. 1 Ant. M T k b q S Source Encoder Interleav er Mapper STBC Encod er OFDM Mod. OFDM Mod. Ant. 1 Ant. M T k b q S Figure 1- MIMO-OFDM transmitter. Since we assum e a frequenc y domain t ransmission, the signal received on the subca rrier n by the antenna j is a superposition of the transmitted sig nal by the different antennas multiplied b y the channel coefficients to wh ich white Gaussian noise (WGN) is added. It is given by: ∑ = + = T M m j i j i j j t n w t n x n h t n y 1 , ] , [ ] , [ ] [ ] , [ α (1) where y j [ n , t ] is the signal receiv ed on the n th subcarrier by the j th receiving antenna during the t th OFDM sym bol duration. h i,j [ n ] is the frequency c hannel coefficient assumed to be const ant during T sy mbol du rations, x i [ n,t ] is the signal transmitted by the i th antenna and w j [ n,t ] is the additive WGN with zero mean and variance N 0 /2. α j is the power attenuat ion factor of the j th receiving antenna. By introducing an equivalent receive matrix [] R M T nC × ∈ Y whose elements are the complex receive d symbols expressed in (1), we can write the received signal on the n th subcarrier on all r eceiving antennas as: [] [] [] [] nn n n =+ yA H X W (2) Where H [ n ] is the ( M R , M T ) channel matrix whose components ar e the coefficients h i,j [ n ], X [ n ] is a ( M T , T ) complex matrix containing transmitted symbols x i [ n,t ]. W [ n ] is a ( M R , , T ) complex matrix corresponding to the WGN. Since we assume unequal recei ved powers, A is a ( M R , M R ) diagonal matrix whose d i agonal elements are the square roots o f the power at tenuation factors α j associated to each receiving antenna. W ithout loss of generality, we will drop the sub carrier index n in th e sequ el. Let us now describe the t ransmissi on link with a gene ral model indepe ndently of the ST codi ng schem e. We separate the real and imaginary parts of the entries s q , of the outputs X of the ST encoder as well as t hose of the channel matrix H and the received signal y . Let s q,R and s q,I be the real and imaginary parts of s q . The main parameters of the code are given by i ts dispersion matrices U q and V q ( q =1,…, Q ) , correspo nding (not equal) respectively to the real and im aginary parts of X . With these not ations, X is give n by: () ,, 1 Q qq q sj s ℜℑ = =+ ∑ qq XU V (3) We separate the real a nd imaginary parts of S , Y and X and stack them row-wise in vectors of di mensions (2 Q ,1), (2 M R T ,1) and ( 2 M T T ,1) respectively. W e obtain: 1, 1, , , 1, 1, , , , , ( 1,1 ) , ( 1 ,1 ) , ( 2 , ) , ( 2 , ) , , , ..., , , , ..., , , ..., , , , ..., , RR TT tr QQ tr T T MT MT tr MT MT ss s s yy y y y y xx x x ℜℑ ℜ ℑ ℜℑ ℜ ℑ ℜ ℑ ℜℑ ℜ ℑ ⎡⎤ = ⎣⎦ ⎡ ⎤ = ⎣ ⎦ ⎡⎤ = ⎣⎦ s y x (4) where tr holds for matrix transpose. Since, we use linear ST coding, vector x can be written as: . = xF s (5) where F has the dimensi ons (2 M T T , 2 Q ) and is obtained through the d ispersion matrices of the real and imaginary parts of s . It is given by : 1 (1, 1) (1, 1) (1, ) (1, ) (, ) (, ) TT TT M TM T ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ 1Q 1Q 1Q FF FF F FF LL MM MM LL MO M M LL (6) F is composed of M T blocks of 2 T rows each i.e. the data transmitted on each antenna is gathered in one block having 2 T rows and 2 Q columns according to the ST coding schem e. The di fferent com ponents of F are given by: (, ) (, ) (, ) (, ) (, ) mt mt mt mt mt − ⎡ ⎤ = ⎢ ⎥ ⎣ ⎦ q,R q,I q q,I q, R UV F UV (7) As we change t he formulati on of S , Y and X in (4), it can be shown that vectors X and Y are related through th e matrix G of dime nsions (2 M R T , 2 M T T ) such that: = + YB G X W (8) where B is a (2 M R T , 2 M R T ) diagonal matrix whose elements are related to the power attenuations factors by: R j i i M j j T i j T B ,..., 1 . 2 1 ) 1 ( . 2 , = ≤ ≤ + − = α (9) Matrix G is com posed of bloc ks G i,j ( i =1,…, M R ; j =1,…, M T ) each having (2 T ,2 T ) elem ents given by: () () () () () () () () () () () () () ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, ,, 2, 2 00 00 00 0 0 00 0 0 00 0 0 00 0 0 00 00 ij ij ij ij ij ij ij ij ij ij ij ij TT hh hh hh hh hh hh ℜℑ ℑℜ ℜℑ ℑℜ ℜℑ ℑℜ − ⎛⎞ ⎜⎟ ⎜⎟ ⎜⎟ − ⎜⎟ ⎜⎟ ⎜⎟ = ⎜⎟ ⎜⎟ ⎜⎟ ⎜⎟ − ⎜⎟ ⎜⎟ ⎝⎠ i, j G L L L L LO LO L L (10) Now, substituting x from (5) in (8), the relation betwee n y and s becomes: =+ eq y BG F s W = G s + W (11) G eq is the equivalent chann el matrix between s and y . It is assumed to be known pe rfectly at the recei ving side. III. CHOICE OF ST SCHEMES A. Relation between probability o f error and channel capacity Assume the channel is totally u nknown at the transmitter and perfectly known at the receiver, the optimum power distribution strategy is to allocate eq ual power over all the subchannels in differe nt domains (time, frequen cy and space). Based on (11) and keepin g in mind that the chan nel coefficients of the matrix G eq in (11) are separated into real and imaginary parts and they are as sumed to be constant during T OFDM symbols, the channel capacity of such transm ission a nd a given transmi tted power is : () 22 1 log de t 2 R GM T C T =+ H- 1 eq SS eq WW IG R G R (12) where R SS , and R WW are the autocorrelation matrix of the data entries s and the WGN respectively. We show that the channel ca pacity is give n by: 0 22 2 1 log de t 2 R GM T W P C T σ ⎛⎞ =+ ⎜⎟ ⎝⎠ HH H IB G F F G B (13) And the mean channel capacity ove r the channel realizations is: [] G C E C = (14) For non-ort hogonal (NO) s chemes, t he choice of an optimal ST coding matrix depends on some criteria. It is based on an opti mization of the pair wise error probability (PEP) or channel capacity and diversity , or a comprom ise between them. B ased on the k nowledge of the possible set of matrix X , Tarokh [5] proposed some cr iteria to construct ST coding matrix X . In [6], Hassibi is based on the PEP for Gaussian di stributed in puts to defi ne a new ST code. The PEP criterion, based initially on ML detection, should be studied further. It consists in minimizing th e quantity: () () 1/ 2 2 1 Pr ' de t 2 R MT x XX E γ − ⎡ ⎤ →≤ + ⎢ ⎥ ⎣ ⎦ tr eq eq IG G (15) where x γ is the signal to no ise ratio (SNR) for each transmitted symbol x ∈ X . The transfer to the mean error probability is difficult from (15) since there is a large number of matrices X which verify the PE P minimi zation. Howeve r, a good iss ue consists in maximizing the determin ant. Surprisingly, the maximi zation of the det erminant in (15) is equi valent to the maxim ization of ch annel capacity in (14) 1 . That is, [6] proposes L D scheme based on m aximizat ion of chan nel capacity. This all ows imposing som e const raints on the choice of dispersi on matrix F . Si nce the channel is unknown at the tran smit side, the first constraint is to have trace ( F tr . F )=2 T . The second c onstraint is t o have a uniform repartition of signal power on different transmit antennas. This could b e achieved by fulfilling F conveniently. Another interesting point in this analysis consists in the relation between probab ility of error and capacity. Indeed, 2 differe nt ST schem es have the sam e channel capacity. However, they present different probabilities of errors since the PEP is upper boun ded by (not equal to ) a function of the channel capacity inverse. It is shown in [6] that two d ispersion matrices having the same channel capacity do not have the sa me error rate. This is due to the fact that the diversity introdu ced by the dispersion m atrices is di fferent f rom a code to another. B. Considere d ST Codin g schemes As a consequence of the discussion in previ ous section, we consider in this paper some of the most promising MIMO schemes havi ng the same rat e. Therefore for e qual spectral efficiencies, (14) shows that all these schem es have the same channel cap acity. We will show by simulations i n next sections t hat even wit h equal channel capacities and SNRs, the probability of error of different schemes is not t he same since they ha ve not the sam e diversity order. First, we consider the simplest ortho gonal ST coding scheme proposed by Alamouti [7 ] as a reference of comparison. Since M T =2, we have Q = T =2 and the ST coding rate L =1. This code is given by the matrix: 12 ** 21 s s s s ⎡ ⎤ = ⎢ ⎥ − ⎣ ⎦ X (16) For non-orthogonal sch emes, we consider in this work the well-known multiplex ing scheme i.e. the V-BLAST [8]. VBLAST is designed to maximize th e rate by transmitting symbols sequen tially on different antennas. Its coding sche me is gi ven for T =1, Q =2 and L =2 by: [ ] 12 tr ss = X (17) We also consi der the LD code proposed b y Hassibi [6] for which we have Q =4, T =2 and L =2. It is d esigned to maximize the mutual information between transmitter and receiver. It is defined by: 1 The inverse does not im ply a m inimizati on of PEP. 13 2 4 24 1 3 1 2 s ss s s ss s +− ⎡⎤ = ⎢⎥ +− ⎣⎦ X (18) Finally, we consider the optimi zed Golden code [9] denoted herea fter by GC. The G olden cod e is designe d to maximize the rate such that the dive rsity gain is preserved for an increased signal constellatio n size. It is defined for Q =4, T =2 and L =2 by: 12 3 4 34 1 2 () ( ) 1 () ( ) 5 s ss s s ss s βθ β θ μβ θ β θ ++ ⎡⎤ = ⎢⎥ ++ ⎣⎦ X (19) where 2 5 1 + = θ , θ θ − = 1 , ) 1 ( 1 θ β − + = j , ) 1 ( 1 θ β − + = j , j = μ and 1 − = j . IV. ITERATIVE SPACE-TIME RECEIVER In the case of OSTBC, optimal receiver is m ade of a concatenation of ST dec oder and c hannel dec oder modules (prec eded by a bit deinterleaver ). In NO-STBC schemes, there is an inter el ement interference (IEI) at the receiving side. The optim al receiver in this case is based on joint ST a nd channel de coding o perations. However such receiver is extrem ely complex to implement and requires large memory to stor e the different points of the trellis. Moreover, it could not be implemented reasonably in one ship. Thus t he sub-opt imal soluti on proposed he re consists of an iterative recei ver where t he ST detector and channel decoder exchange e x trinsic information in an iterative way until the algori thm conve rges. The iterative detection and decodi ng expl oits the error correction capabilities of the chann el code to provide i mproved performance. This is achieved by iteratively passing soft a priori inform ation betwee n the detect or and the s oft- input soft-ou tput (SISO) decode r [10]. PI C de t e c t o r Ant. 1 Ant. M R Dem a p p e r ( L L R c o m p .) D e In t e rle a ve r SI SO D e c o der In te r l ea v e r S o ft G r ay M app er ) ( ˆ l S ) 1 ( ˆ − l S b ˆ Estimated bits PIC dete ct or Ant. 1 Ant. M R Demapper (LLR c omp.) DeIn terle ave r SISO Deco der Interleaver Soft Gray Mapper ) ( ˆ l S ) 1 ( ˆ − l S b ˆ Estimated bits Figure 2- Iterative re ceiver structure. A. STBC detection In the literature, different detection strategies are presented. The detectio n problem is t o find the transmitted data s given the vector y . The iterative detector shown i n Figure 2 is com posed of a MIMO equalizer, a dem apper which is made up of a parallel interference cancellation (PIC), a log likelihood ratio (LLR) computation, a s oft-input soft -output (SIS O) decoder, an d a soft ma pper. At the first iteration, the demapper takes the estimated symbols ˆ s , the knowledge of the ch annel G eq and the noise varianc e, and computes t he LLR values (soft informati on) of each of the B coded bits transmitted per channel use. The estimated symbols ˆ s are obtained via minimum mean squ are erro r (MMSE) filtering according to: ( ) 1 2 ˆ pw s σ − =⋅ + tr tr pe q e q gG G I y (20) where tr p g of di mension ( 2 M R T , 1) is the p th column of G eq (1 ≤ p ≤ 2 Q ). p s ˆ is the estimation of the real part ( p odd) or imaginary part ( p even) of s q (1 ≤ q ≤ Q ). B. LLR comput ation As we conside r Gray mappi ng with QAM modulat ion of B bits per sym bol, the com putation o f LLR is done as i n [11]. We note t hat we use the approxim ation of log(exp( x 1 )+(exp( x 2 )) ≈ max ( x 1 , x 2 ) . This simplifies considerably the LLR expressions espe cially for high constellations. We note also th at the total noise v ariance corresponding to th e additive WGN and the IEI is used for LLR com putation. C. SISO decoder The deinterleaved soft information ( LLR k,p ) of the k th of the p th symbol at the out put of the demapper becomes the input of the oute r decoder. The out er decoder com putes the a posteriori information of the information bits and of the code d bits. T he a posteriori informatio n of th e coded bits pro duces new (and hen ce) extrinsic information ext p k LLR , of the coded bit s upon rem oval of the a pri ori information 2 and minimizing the correlation between input valu es LLR k,p . In our work, S ISO decodi ng is based on the Max-Log -MAP algorit hm [10]. The extrinsic information at th e output of th e channel decode r is then interleaved and fed to a soft Gray mapper module. D. Soft mapper The soft mapper achieves r ecipr ocal operati on of soft demapper. Knowing the ex trinsic information of the k th bit of the q th symbol, the soft estimation o f the complex symbol s q , noted hereafter q s ~ , is defined by: { } ext q B ext q q LLR LLR s s , , 1 ,..., ~ Ε = (21) where E holds for expectatio n function. Le t ] ,..., [ 1 B c c the set of bits con stituting the cons tellation point s . Equation (21) yields : ( ) ∑ ∈ = = ψ s q q s s s s ~ Pr . ~ (22) where ψ is the set of constellation points and : ( ) ( ) ∏ = = = = ] ,..., [ : , 1 ~ Pr ~ Pr B c c s k q k q c c s s (23) The probability expressions in (23) are d educed from the LLR expressions as: 2 when the transmitted b its are likely equal, this information is equal to zero. () () () ext q k ext q k q k LLR LLR c , , , exp 1 exp 1 ~ Pr + = = ( ) ( ) 1 ~ Pr 1 0 ~ Pr , , = − = = q k q k c c (24) Once the estimation of the different sym bols s q is achieved by t he soft m apper at the first i teration, we use this estimation for the nex t iterations process. From the second iteratio n, we perform PIC operation followed by a simple inverse filtering (instead of MMSE filtering at the first iteration): ˆ 1 ˆˆ p s =− = pe q , p p tr pp tr pp yy G s gy gg % (25) where eq,p G of dimensi on (2 M R T , 2 Q -1) is the matrix eq G with its p th colum n removed, p s % of dimensi on (2 Q -1, 1) is the vector s % estimated by the soft mapper with its p th entry remove d. V. SIMULATION RESULTS In this section, we present a comparative stud y of the different ST coding schem es described i n section 3. T he performance comparison is made in terms of bit error rate (BER) for the cases of equal and unequal received powers at the receiving side. We assume that 2 or 3 re cei ving antennas are used. For equal received powers, we assum e that the power attenuation factors of matrix A in (2) are equal to 0dB i.e. α 1 = α 2 = α 3 0dB. For unequal received powers, we set α 1 to 0dB and we change α 2 and α 3 such that α 2 = α 3 . The simulations parameters considered in this work are derived from those of D VB-T. They are gi ven in Tabl e 1. The spectral efficiencies η =2, 4 and 6 [bit/sec/Hz] are obtained f or different ST schem es as shown in Ta ble 2. Table 1- Simulations Parameters Number of subcarriers 2K mode (1 705 active subcarriers) Number of Tx antennas 2 Number of Rx antennas 2 or 3 Rate R of convolutional code 1/2, 2/3, 3/4 Polynomial code genera tor (133,171) o Channel estimation perfect Constellation QPSK, 16-QAM, 64-QA M, 256-QAM Spectral Efficiencies η = 2, 4 and 6 [bit/sec/Hz] Power attenuati ons factors α 1 = 0dB α 2 = α 3 = -12, -9, -6, -3 and 0dB First, let us characterize the behavior of the iterative receiver. Figure 3 provides perform a nce of the Golden code with iterative receiver. The performance is give n in terms of BER versus the Eb/N0 ratio for differe nt numbers of i terations . We observe o n this fi gure that the iterative process converg es fo r an Eb/N0 greater tha n a limit value, which is equ al to 6 dB in this case. Moreover, we observe that the conve rge nce of the iterative receiver is reached after 3 iterations which m eans an acceptable complexity as compared to ML detecti on. This can be observed wit h Golden co de, but also wit h LD code and VBLAST scheme. That is, for NO-STBC schemes, we will present in the sequ el the performances after 3 iterations only. For equal received powe rs and 2Rx ( α 1 = α 2 = 0dB), Figure 4 and Figure 5 compare t he differe nt ST c oding schem es for η =4 and η =6 respectively. These figures show that Golden code presents the be st performance with respect to other schemes since it benefits from its full d iversity. For equal received powers ( α 1 = α 2 = α 3 =0dB), 3Rx a nd a spectral efficiency η =6, NO-STBC sche mes outperform Alamouti code as depicted in Figure 6 . More precisely, for a BER=10 -4 , the Eb/N0 gain for Golden code is roughly equal to 6 dB compared t o Alam outi code. Table 2- Different MIMO schemes and efficiencies Spectral Efficiency ST scheme ST rate L Constellation R η =2 [bit/Sec/Hz] Alamouti 1 16-QAM 1/2 VBLAST 2 QPSK 1/2 LD 2 QPSK 1/2 Golden 2 QPSK 1/2 η =4 [bit/Sec/Hz] Alamouti 1 64-QAM 2/3 VBLAST 2 16-QAM 1/2 LD 2 16-QAM 1/2 Golden 2 16-QAM 1/2 η =6 [bit/Sec/Hz] Alamouti 1 256-QAM 3/4 VBLAST 2 64-QAM 1/2 LD 2 64-QAM 1/2 Golden 2 64-QAM 1/2 For unequal received powers, conclusions are different. Using 2 receivers, Figure 7, Fi gure 8 and Figure 9 depict the Eb/N0 ratio required to obtain a BER=10 -4 for spectral efficiencies η =2, 4 and 6 [bit/sec/Hz] respectively and different v alues of the power attenuation factor α 2 . For η =2, Figure 7 shows t hat Alamouti scheme outperf orms the other ST coding sc hemes. I ndeed, the re quired E b/N0 to obtain a BER =10 -4 for Alamouti scheme is less than for other schem es. This superiori ty increases when t he received power on the second antenna dec reases and can be explained as follows . For equal rece ived powe rs ( α 1 = α 2 =0dB), all t he ST coding schemes present the same performance. Whe n the received powe r on the second antenna decreases, the different sc hemes (with 2 receiving antennas) tend to be ST schem es with only one antenna. However, due to the redu ndancy include d by Alamouti scheme, the loss introduced by the power decrease could be simply recovere d by the first antenna at the detriment of half power loss in terms of Eb/N0. That is why the maximal loss of Alamouti scheme is upper- bounded. It is of 3dB when α 2 passes from 0dB t o -12dB . For NO schemes, the redu ndancy is les pronounced in the ST matrices, which implies a greater Eb/N0 loss when the power of the second receiving antenna passes from 0dB to -12dB. For higher spectral efficiency i.e. η =4 or 6 [bit/Sec/Hz], Figure 8 and Figure 9 show that Golden code presents the best pe rformance as l ong as the power attenuation factor on the second antenna α 2 is greater than a limit value. Otherwise, Alamouti scheme presents the best performance. This limit valu e is of -6dB for η =4 and -9dB for η =6. This behavi or can be explained by the fact that Golden code is designed to maxim ize the diversity for high si gnal constellations and equal recei ved powers. The dive rsity gain is howeve r lost when one antenna is quite turned off. 3 4 5 6 7 8 9 10 -4 10 -3 10 -2 10 -1 10 0 E b /N 0 [d B] BER Gold en 2*2, R= 1/ 2, 16-QA M , Eff=4 NbIt e r = 1 NbIt e r = 2 NbIt e r = 3 NbIt e r = 4 Figure 3- Convergence of Golden code with resp ect to the number of iterations, 2Rx. 2 3 4 5 6 7 8 9 10 10 -4 10 -3 10 -2 10 -1 10 0 E b /N 0 [d B] BER E ff =4, Al pha1= 0dB , A l pha2= 0dB A lam out i LD VBL AST Golden Figure 4- Comparison of different ST coding schemes, Spectral effici ency η = 4 bit/sec/ Hz, 2Rx. When 3 antenna s are used at the receiving si de, the conclusions ar e quite diffe rent. In deed, as sho wn in Figure 10 , Golden code prese nts the best performance whate ver the power attenuati on factors on the second and third receiver antennas. Eventually, we should note from Figure 7 to 10 that the slope of the loss of all NO schemes tends to the same value when the power attenuation factors decrease infinitely. This slope increases with sp ectral efficiency and decreases with the num ber of receiving antennas. This means that for 2 receiving ante nnas a link loss c ould be complete ly obtained if t he power at tenuat ion factor of the second antenna decreases infin itely. This link loss could be rectified by increas ing the num ber of receiving antennas. 8 9 10 11 12 13 14 15 10 -4 10 -3 10 -2 10 -1 10 0 E b /N 0 [dB ] BER E ff =6, A lpha1= 0dB , A lpha2= 0dB A lamouti LD VB L AS T Gol de n Figure 5- Comparison of differe nt ST coding schemes, Spectral efficie ncy η = 6 bit/sec/ Hz, 2Rx. 4 6 8 10 12 14 10 -4 10 -3 10 -2 10 -1 10 0 E b /N 0 [d B] BER Eff=6, A lpha1= 0dB , Al pha2= Al pha3= 0dB Alamouti LD VB L AS T Golden Figure 6- Comparison of differe nt ST coding schemes, Spectral efficiency η = 6 bit/sec/Hz, 3Rx. -12 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 A lpha 2 [ dB ] E b /N 0 [dB] Required E b /N 0 to a ttain a BER =1 0 e- 4, Ef f = 2 A lamouti LD VB L AS T Gol den Figure 7- Required E b /N 0 to obtain a BER=10 -4 , Spectral effi ciency η =2 bit/sec/Hz, 2Rx -12 -10 -8 -6 -4 -2 0 6 8 10 12 14 16 18 Al pha2 [dB ] E b /N 0 [dB] Required E b /N 0 to atta in a B ER =1 0 e - 4, Ef f= 4 Al a m ou t i LD VB L AS T Golden Figure 8- R equired E b /N 0 to obtain a BER=10 -4 , Spectral efficiency η =4 bit/sec/Hz, 2Rx -12 -10 -8 -6 -4 -2 0 10 12 14 16 18 20 22 24 Al pha2 [d B ] E b /N 0 [dB] Required E b /N 0 to a tta in a BER =1 0 e - 4, Ef f = 6 A lamout i LD VB L AS T Golden Figure 9- Required E b /N 0 to obtain a BER=10 -4 , Spectral efficiency η =6 bit/sec/Hz, 2Rx -12 -10 -8 -6 -4 -2 0 6 8 10 12 14 16 18 Al pha2= A lpha3 [ dB ] E b /N 0 [d B] Required E b /N 0 t o att ain a B ER= 10e-4, E ff = 6 Al amout i LD VBL AST Golden Figure 10- Required E b /N 0 to attain a BER=10 -4 , Spectral efficiency η =6 bit/sec/Hz, 3Rx VI. CONCLUSION In this paper, we considered the performance of M IMO- OFDM scheme s when used with 2 transm itters and 2 or 3 receivers and unbalanced recei ved powers. This study is done using an iterative receiver. We showed by simulations that the converge nce of the iterative receiver is obtained after 3 iterations. Moreover, we sh owed that the superiority of one scheme could not be o btained in all transmission conditions. For 2 receivi ng antennas, whatever the spectral efficiency is, Alamouti scheme presents the best perform a nce when one antenna is dramatically damaged i.e. when the power received by this antenna decreases infinitely. However, for 3 receivers, Golden code presents the be st performance whatever the received powers on different antennas for high spectral efficiencies. ACKNOWLEDG MENTS The authors woul d like to thank t he European CELT IC project “B21C” for its support of th is work. REFERENCES [1] http://www.dvb .org . [2] http://www.celtic-initiative.org/Proj ects/B21C/ [3] P.-J. Bouvet, M. Hélar d, and V. Le Nir “Low complexity iterative r eceiver for linear precoded MIMO systems” International Symposiu m on Spread Spectrum Techni ques and Ap plications , pp. 17-21, 30Aug- 2Sept., Sidney Australia. [4] M. A. Khalighi, J. -F. Helard, an d S. Boure nnane “Contrasting Orthogo nal and non orthogon al space- time schem es for perfectl y-known a nd estimated MIMO channels” I ntern ational Conference on Communi cations systems, pp. 1-5, Oct. 2006 , Singapore . [5] V. Tarokh, H. Jafar khani, and R. C alderbank, “S pace- time block code s for orthog onal designs”, IEEE Trans. on Inform ation Theory, vol. 45, no. 4, pp. 1456-146 7, July 19 99. [6] B. Hassibi, and B. Hochwal d, “High-rate code s that are linear in space and ti me,” IEEE Trans. in Information Theory, vol. 48, no. 7 , pp. 1804–1824, July 2002. [7] S.M. Alamouti, “A simple transmit diversity technique for wireless communications”, IEEE J. on Selected Areas in Communica tions, vol. 16, no. 8, pp. 1451-1458, Oct. 1998 . [8] G. J. Foschini, “Layered sp ace-time archit ecture for wireless comm unication i n a fading en vironment when using multi-element antenn a,” Bell Labs Tech. J., vol. 1, no. 2, pp . 41–59, 199 6. [9] J.-C. Belfiore, G. Rekaya, and E. Vite rbo, “The golden code: a 2 × 2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. in Information Theory, vol. 51, no. 4 , pp. 1432–1436, Apr. 2005 . [10] J. Hagenauer, and P. Hoe her, “A Vi terbi algo rithm with soft-deci sion output s and its appli cations,” i n Proc. of IEEE Global Telecommunications conference, pp. 1680-168 6, Nov. 1989. [11] F. Tosato, and P. Bisaglia, “Simplified Soft-Output Demapper for Binary Interleaved CO FDM with Application to HIPERLAN/2”, IEEE International Conference on Com munications ICC, vol.2, p p.664- 668, 2002.