OFDM based Distributed Space Time Coding for Asynchronous Relay Networks

OFDM based Distrib uted Space T ime Coding for Asynchronou s Re lay Netw orks G. Susinder Rajan ECE Department Indian Institute of Science Bangalore 56001 2, India susinder@ece.iisc.ern et.in B. Sund ar R ajan ECE Departmen t Indian Institute of Science Bangalore 56001 2, India bsrajan@ece.iisc.ern et.in Abstract — Recently Li and Xia hav e proposed a transmission scheme f or wir eless re lay networks base d on the Alamouti sp ace time code and orthogonal fr equen cy div ision multip lexing to combat the effect of ti ming error s at the relay n odes. This transmission scheme is amazingly simple and achiev es a div ersity order of two fo r any number of r elays. Motiva ted by its simplicity , this scheme is extended to a more general transmission scheme that can achie ve full coopera tive diversity f or any number o f relays. Th e conditions on the distributed sp ace time block code (DSTBC) structure that admit its application in the proposed transmission scheme are id entified and it is pointed ou t th at the recently p roposed full diversity four group decodable DST - BCs from pr ecoded co-ordinate interlea ved orthogonal designs and extend ed Clifford algebras satisfy these cond itions. It is then shown h ow di fferential encoding at the source can be combined with the proposed transmission scheme to arri ve at a new transmission scheme that can achieve full cooperativ e divers ity in asynchronous wireless relay networks with no channel informa tion and also no timing err or knowledge at the destination node. Fi nally , f our gr oup decodable distributed d ifferential sp ace time bl ock codes applicable in this new transmission scheme for power of two number of relays are also provided. I . I N T RO D U C T I O N Coding for cooper ati ve wireless relay networks has attracted considerab le atten tion recen tly . Distributed space time cod ing was proposed as a coding strategy to a chiev e full coopera- ti ve di versity in [1] assuming that the signals fr om all the relay n odes arrive at the destinatio n at the same time. But this assump tion is not clo se to p racticality since th e relay nodes are g eograph ically distributed. In [3], a tran smission scheme based on orthogonal fr equency division m ultiplexing (OFDM) at th e relay nod es w as proposed to comba t the timing errors a t the rela ys and a high r ate space tim e block code (STBC) co nstruction was also p rovided. Howe ver , the maxi- mum likeliho od (ML) decodin g complexity for this scheme is prohib iti vely hig h especially for th e case of lar ge numb er o f relays. Se veral oth er works in the literature prop ose meth ods to co mbat the timing offsets but most o f them are based on decod e and forward at the relay no de and moreover fail to address the deco ding com plexity issue. In [2], a simple transmission scheme to com bat timin g errors at the relay nodes was proposed . This schem e is particularly in teresting because of its associated lo w ML decod ing com plexity . In this scheme, OFDM is implemented at the source n ode and time re versal/conjuga tion is perfo rmed at th e relay nodes on the received OFDM symbols. The received signals at the destination af ter OFDM demod ulation are shown to h av e the Alamouti code st ru cture an d hence single symbol maximum likelihood (ML) decodin g can be p erformed . Howe ver , the Alamouti code is app licable on ly fo r th e ca se of two relay nodes an d fo r larger nu mber of relays, the authors o f [2] propo se to cluster the r elay nodes a nd employ Alamouti code in each cluster . But this clustering technique provides div ersity order of only two and fails to exploit the full cooperative div ersity equal to the numb er of relay nod es. The main contributions of this pap er ar e as fo llows. • The Li-Xia transmission scheme is extended to a more general tran smission scheme that c an achieve fu ll asyn- chrono us coop erative diversity fo r a ny nu mber of relays. • The condition s on th e distributed STBC (DSTBC) struc- ture that admit its applicatio n in the p roposed tr ans- mission scheme are identified. The recently prop osed full diversity four group deco dable DSTBCs in [4] for synchro nous wireless relay networks are foun d to satisfy the required conditions for app lication in the pro posed transmission scheme. • It is shown how differential encoding at the so urce no de can be com bined with the pro posed transmission sch eme to arri ve at a transmission scheme that can achieve f ull asynchro nous cooper ati ve diversity in the absence o f channel knowledge and in th e absence of kn owledge o f the timing errors of the relay nodes. Moreover , an existing class o f fou r gr oup decoda ble distributed differential STBCs [5] for synch ronou s relay networks with p ower of two number of relay s is shown to be applicable in this setting as well. A. Or ganization of the pap er In Section II, the ba sic ass um ptions on the r elay network model are given and the p roposed tran smission schem e is described. Section II also p rovides four gro up d ecodable DSTBCs that achieve f ull asyn chrono us co operative diversity in the p roposed transm ission scheme for arbitra ry n umber o f relays. Section II I b riefly explains ho w dif fer ential enco ding at th e sou rce no de can be combined with th e p roposed transmission scheme and four gro up decod able d istributed differential STBCs ap plicable in this scen ario are also propo sed. Simulatio n results and discussion on further w or k comprise Sections IV and V resp ecti vely . Notatio n: I m denotes an m × m identity matr ix and 0 denotes an all zero matrix of app ropriate size. For a set A , the cardinality of A is deno ted by | A | . A n ull set is denoted by φ . F or a matrix, ( . ) T , ( . ) ∗ and ( . ) H denote tran sposition, conjuga tion and con jugate transpo se operation s respectiv ely . For a complex nu mber, ( . ) I and ( . ) Q denote its in- phase and quadra ture-phase parts respectively . I I . R E L A Y N E T W O R K M O D E L A S S U M P T I O N S A N D T H E P R O P O S E D T R A N S M I S S I O N S C H E M E In this section, the basic relay network model assumption s are given and the pr oposed transmission scheme is described . The propo sed tran smission scheme can achiev e full asyn- chrono us co operative diversity for arb itrary nu mber of relay s and is an exten sion of the Li-X ia transmission schem e[2]. This nontrivial extension is based on analyzing the sufficient condition s required on the structure o f STBCs which admit application in the Li-Xia transm ission schem e. S U 1 D U 2 U R f 1 g 1 g 2 f 2 f R g R Fig. 1. Asynchronous wirel ess relay network A. Network model assumption s Consider a network with one source node, one destinatio n node and R r elay no des U 1 , U 2 , . . . , U R . This is depicted in Fig. 1. Every nod e is assum ed to have only a single antenn a and is half duplex co nstrained. Th e channel gain b etween the source and the i -th relay f i and that between the j -th relay and the destination g j are assum ed to be qu asi-static, flat fading and modeled by ind ependen t and complex Gau ssian distributed random v ariables with mea n zero and unit variance. The transmission of inf ormation from th e source node to the destinatio n node takes place in two ph ases. In the first phase, the sou rce broadca sts the information to the r elay nod es using OFDM. The relay nod es r eceiv e the faded and noise corrup ted OFDM symbols, process them an d transmit them to the destination. The r elay nodes are assumed to ha ve per fect carrier synch ronization . The overall relativ e timing er ror of the signals arrived at the destination nod e from the i -th relay node is denoted by τ i . Without loss of g enerality , it is assumed that τ 1 = 0 , τ i +1 ≥ τ i , i = 1 , . . . , R − 1 . The destinatio n node is assumed to h av e the knowledge of all the cha nnel fading gains f i , g j , i, j = 1 , . . . , R and the r elati ve timing errors τ i , i = 1 , . . . , R . B. T ransmission by the source node The source takes RN complex symbols x i,j, 0 ≤ i ≤ N − 1 ,j =1 , 2 ,... ,R and forms R block s of data d enoted by x j =  x 0 ,j x 1 ,j . . . x N − 1 ,j  T , j = 1 , 2 , . . . , R . Of these R blo cks, M of them are mod ulated by N -point I DFT and the remaining R − M blocks are modu lated by N -point DFT . W ithout loss of generality , let u s assume that the first M blocks are modu lated by N -point IDFT . Then a CP of length l cp is adde d to each blo ck, where l cp is chosen to be not less than the maximum o f th e overall relati ve timing errors of the si gn als arr i ved at the destination nod e from all the relay nod es. The resulting R OFDM sym bols d enoted b y ¯ x 1 , ¯ x 2 , . . . , ¯ x R each consisting of L s = N + l cp complex number s are broadcasted to th e R re lays using a fr action π 1 of the total av erag e P . C. Pr ocessing at the r elay no des If the chann el fade gain s ar e assumed to be constant for 2 R OFDM symbol inter vals, the received signals at the i - th r elay during the j -th OFDM symbol duration is given by r i , j = p π 1 P f i ¯ x j + ¯ v i , j where, ¯ v i , j is the A WGN at the i -th relay n ode during the j -th OFDM symb ol d uration. The r elay n odes pro cess and tr ansmit the received noisy signals as shown in T ab le I using a fraction π 2 of total po wer P (ap propr iate scaling of receiv ed OFDM symbols is assumed). Note from T able I th at time reversal is d one during the last R − M OFDM sy mbol du rations. W e would like to emph asize that in gen eral tim e rev ersal could be implemented in any R − M of the total R OFDM symbol du rations. Now , t i , j ∈ { 0 , ± r i , j , j = 1 , . . . , R } with the constraint that the i -th relay shou ld not be allo wed to transmit any eleme nt fr om the th e following set: {± r i , j ∗ , j = 1 , . . . , M } ∪ {± ζ ( r i , j ) , j = 1 , . . . , M } ∪ {± r i , j , j = M + 1 , . . . , R } ∪ {± ζ ( r i , j ∗ ) , j = M + 1 , . . . , R } . (1) D. Decoding at the destination The destination removes the CP f or the first M OFDM symbols and implements the following for the re maining OFDM symbols: 1) Remove the CP to get a N -point vector 2) Shift the last l cp samples of the N -point vector as the first l cp samples. DFT is the n applied on the resulting R vectors. Let the r eceiv ed signals fo r R consecutive OFDM b locks af- ter CP removal and DFT transformation be denoted by y j =  y 0 ,j y 1 ,j . . . y N − 1 ,j  T , j = 1 , 2 , . . . , R . Let w i = ( w k,i ) , i = 1 , . . . , R repr esent the A WGN at the destination nod e and let v i , j denote the DFT of ¯ v i , j . Let T ABLE I P R O P O S E D T R A N S M I S S I O N S C H E M E OFDM Sym bol U 1 . . . U M U M +1 . . . U R 1 t 1 , 1 . . . t M , 1 t M + 1 , 1 ∗ . . . t R , 1 ∗ . . . . . . . . . . . . . . . . . . . . . M t 1 , M . . . t M , M t M + 1 , M ∗ . . . t R , M ∗ M + 1 ζ ( t 1 , M + 1 ) . . . ζ ( t M , M + 1 ) ζ ( t M + 1 , M + 1 ∗ ) . . . ζ ( t R , M + 1 ∗ ) . . . . . . . . . . . . . . . . . . . . . R ζ ( t 1 , R ) . . . ζ ( t M , R ) ζ ( t M , R ∗ ) . . . ζ ( t R , R ∗ ) s k =  x k, 1 x k, 2 . . . x k,R  T , k = 0 , 1 , . . . , N − 1 . No w using the following iden tities, (DFT( x )) ∗ = IDFT( x ∗ ) (IDFT( x )) ∗ = DFT( x ∗ ) DFT( ζ (DFT( x ))) = x (2) we get in each sub carr ier k , 0 ≤ k ≤ N − 1 : y k =  y k, 1 y k, 2 . . . y k,R  T = s π 1 π 2 P 2 π 1 P + 1 X k h k + n k (3) where, X k =  A 1 s k . . . A M s k A M + 1 s k ∗ . . . A R s k ∗  (4) for some squa re real matrices A i , i = 1 , . . . , R having th e proper ty th at any r ow o f A i has only one nonzero entry . I f u τ i k = e − i 2 πkτ i N , then h k =           f 1 g 1 u τ 2 k f 2 g 2 . . . u τ M k f M g M u τ M +1 k f ∗ M +1 g M +1 . . . u τ R k f ∗ R g R           (5) is th e equ i valent ch annel matrix for the k -th su b carr ier . T he equiv alent n oise vector is given by n k = q π 2 P π 1 P +1      δ 1 P R i =1 sg n ( t i , 1 ) ˆ v i , 1 ( k ) g i u τ i k δ 2 P R i =1 sg n ( t i , 2 ) ˆ v i , 2 ( k ) g i u τ i k . . . δ R P R i =1 sg n ( t i , R ) ˆ v i , R ( k ) g i u τ i k      +     w k, 1 w k, 2 . . . w k,R     where, sg n ( t i , j ) =    1 if t i , j ∈ { r i , j , j = 1 , . . . , R } − 1 if t i , j ∈ {− r i , j , j = 1 , . . . , R } 0 if t i , j = 0 and ˆ v i , m =  ± v i , j if i ≤ M and t i , m = ± r i , j ± v i , j ∗ if i > M and t i , m = ± r i , j . Th e δ i ’ s are simply scaling factors to account for the co rrect noise variance due to possible zeros in the relay tran smissions. ML deco ding of X k can be done from (3) by c hoosing that co dew ord which m inimizes k Ω − 1 2 ( y k − X k h k ) k 2 F , where Ω is the covariance matrix of n k and k . k F denotes the Frobe nius norm. Essentially , the pr oposed transmission scheme imp lements a space time code having a special struc- ture in each sub carrier . Now if the DSTBC X k satisfies the rank cr iteria (d ifference of any two co dew or d matrices h as full rank), th en it can be pr oved on similar lines as in [1] that full asynchro nous coope rativ e diversity eq ual to R is achieved. E. Full diversity four gr oup d ecodable d istrib uted space time codes In this subsection, we analyze the st ru cture of the s pac e time code req uired for impleme nting in the propo sed transmission scheme. Note from ( 4) that the DSTBC sho uld have the proper ty that any colum n should have o nly th e complex symbols or only their co njugates. W e ref er to this pro perty as conjuga te linea rity p roperty [4 ]. But co njugate linea rity alone is not e nough f or a STBC to qua lify fo r imp lementation in th e propo sed transmission scheme. Note from T ab le I th at time reversal is imp lemented for cer tain OFDM sym bol dura tions by all the relay nodes. Observe that this put together with the co nstraints in ( 1) deman ds a certain row structure on the STBC. W e now provide a set of sufficient condition s th at are required on the row stru cture of conjuga te linear STBCs. First let us pa rtition the comp lex symbo ls appearing in the i -th row into tw o sets- one set P i containing those co mplex symbols which appe ar without con jugation and ano ther set P c i which contains those co mplex symbo ls wh ich ap pear w ith con juga- tion in the i -th row . If the fo llowing sufficient co nditions are satisfied by a conjug ate linear STBC, then it can be shown that th ere exists an assignment of time reversal OFDM symbo l duration s together with an approp riate cho ice o f M and relay node proc essing such that th e desired conjug ate linear STBC form is obtained in every sub carrier at the destination nod e. P i ∩ P c i = φ, ∀ i = 1 , . . . , R | P i | = | P c i | , ∀ i = 1 , . . . , R P i ∩ P j ∈ { φ, P i , P j } , ∀ i 6 = j. (6) Now that for th e case of the Alamo uti code [2], P 1 = P c 2 = { x k, 1 } , P 2 = P c 1 = { x k, 2 } and hen ce it satisfies the conditions in (6). Recen tly three new classes of f ull d iv ersity , four group decodab le DSTBCs for any n umber of relays were r eported in [4] fo r synchr onous relay networks. T hese codes ar e con jugate linear and mo reover since they are fou r grou p decod able, the associated real symbols in these STBCs can be pa rtitioned equally into fo ur gro ups and the ML decod ing can b e do ne f or the real symbols in a g roup indepe ndently of the real symbols in the other groups. Thus the ML decodin g com plexity of these codes is sign ificantly less co mpared to all oth er distributed space time co des known in the literature. In this p aper, we show that the co des repor ted in [4] satisfy th e conditio ns in (6) and are thus su itable to be applied in the proposed tr ansmission scheme. Due to sp ace limitations this is illustrated using the following example. Example 1: Let us take R = 5 , for wh ich the DSTBC in [4] is ob tained by tak ing a DSTBC fo r 6 relays an d drop ping one column . It is given by         x k, 1 − x ∗ k, 2 0 0 0 x k, 2 x ∗ k, 1 0 0 0 0 0 x k, 3 − x ∗ k, 4 0 0 0 x k, 4 x ∗ k, 3 0 0 0 0 0 x k, 5 0 0 0 0 x k, 6         for which P 1 = P c 2 = { x k, 1 } , P 2 = P c 1 = { x k, 2 } , P 3 = P c 4 = { x k, 3 } , P 4 = P c 3 = { x k, 4 } , P 5 = { x k, 5 } , P 6 = { x k, 6 } and P c 5 = P c 6 = φ . At the source, we choose ¯ x 1 = IDFT( x 1 ) , ¯ x 2 = DFT( x 2 ) , ¯ x 3 = IDFT( x 3 ) , ¯ x 4 = DFT( x 4 ) , ¯ x 5 = IDFT( x 5 ) and ¯ x 6 = DFT( x 6 ) . The 5 re lays p rocess th e received OFDM sym bols as shown in T able II. T ABLE II T R A N S M I S S I O N S C H E M E F O R 5 R E L A Y S OFDM U 1 U 2 U 3 U 4 U 5 Symbol 1 r 1 , 1 − r 2 , 2 ∗ 0 − 0 0 2 ζ ( r 1 , 2 ) ζ ( r 2 , 1 ∗ ) − 0 − 0 0 3 0 0 r 3 , 3 − r 4 , 4 ∗ 0 4 0 0 − ζ ( r 3 , 2 ) ζ ( r 4 , 1 ∗ ) 0 5 0 0 0 0 r 5 , 5 6 0 0 0 0 − ζ ( r 5 , 6 ) This co de is 3 real symb ol decodable and ach ie ves full div ersity for appro priately signal sets [4]. I I I . T R A N S M I S S I O N S C H E M E F O R N O N C O H E R E N T A S Y N C H RO N O U S R E L A Y N E T W O R K S In this section, it is shown how dif fer ential en coding can b e combined with the proposed transmission scheme d escribed in Section III. Then the code s in [5] are proposed for application in this setting. For the propo sed transmission sch eme in Section III, at the end of one transmission fr ame, we h av e in the k -th sub car rier y k = q π 1 π 2 P 2 π 1 P +1 X k h k + n k . No te that th e cha nnel matrix h k as shown in (5) de pends on f i , g i , τ i , i = 1 , . . . , R . T hus the destination nod e n eeds to have the kn owledge of these values in order to perf orm ML deco ding. Now using d ifferential encoding ide as which were pr oposed in [6], [7], [8 ] for non- coheren t commun ication in synchr onous relay n etworks, we co mbine them with the proposed asyn- chrono us transmission scheme. Supp osing the channel remains approx imately constant fo r two transmission frames, the n differential encodin g ca n be d one at the source node in each sub c arrier 0 ≤ k ≤ N − 1 as follows: s 0 k =  √ R 0 . . . 0  T , s t k = 1 a t − 1 C t s t − 1 k , C t ∈ C where, s i k denotes th e vector of com plex sym bols transmitted by the source during the i -th tran smission fram e in the k - th sub car rier and C is the co debook used by the source wh ich consists of scaled u nitary matrices C t H C t = a 2 t I such that E[ a 2 t ] = 1 . If for all C ∈ C , CA i = A i C , i = 1 , . . . , M and CA i = A i C ∗ , i = M + 1 , . . . , R then we have: y t k = 1 a t − 1 C t y t − 1 k + ( n t k − 1 a t − 1 C t n t − 1 k ) (7) from which C t can be d ecoded as ˆ C t = ar g min C t ∈ C k y t k − 1 a t − 1 C t y t − 1 k k 2 F in ea ch su b carrier 0 ≤ k ≤ N − 1 . Note that this de coder does no t require the k nowledge of f i , g i , τ i , i = 1 , . . . R at the destination. Howe ver it is importan t to no te that the knowledge of the maxim um of th e relativ e timing erro rs is need ed to dec ide the leng th of CP . It turns out that the fo ur group decodable distrib uted dif- ferential space time codes co nstructed in [5 ] fo r synchr onous relay networks with p ower of two number of relays meet a ll the requirem ents for use in the propo sed transmission sche me as well. The following example illu strates th is fact. Example 2: Let R = 4 . T he codeboo k at the source is giv en by C =        q 1 4     z 1 z 2 − z ∗ 3 − z ∗ 4 z 2 z 1 − z ∗ 4 − z ∗ 3 z 3 z 4 z ∗ 1 z ∗ 2 z 4 z 3 z ∗ 2 z ∗ 1            where { z 1 I , z 2 I } , { z 1 Q , z 2 Q } , { z 3 I , z 4 I } , { z 3 Q , z 4 Q } ∈ S and S = (  1 √ 3 0  ,  − 1 √ 3 0  , " 0 q 5 3 # , " 0 − q 5 3 #) . Differential enco ding is don e at the source node for each sub carrier 0 ≤ k ≤ N − 1 as follows: s 0 k =  √ R 0 . . . 0  T , s t k = 1 a t − 1 C t s t − 1 k , C t ∈ C . Once we get s t k , k = 0 , . . . , N − 1 from the above eq uation, the N leng th vectors x i , i = 1 , . . . , R can be obtain ed. Then IDFT/DFT is applied on th ese vectors and broadca sted to the relay no des accord ing to: ¯ x 1 = IDFT( x 1 ) , ¯ x 2 = IDFT( x 2 ) , ¯ x 3 = DFT( x 3 ) and ¯ x 4 = DFT( x 4 ) . The relay node s pro cess the recei ved OFDM symbols as giv en in T able III for which M = 2 , A 1 = I 4 , A 2 =     0 1 0 0 1 0 0 0 0 0 0 1 0 0 1 0     , A 3 =     0 0 − 1 0 0 0 0 − 1 1 0 0 0 0 1 0 0     and T ABLE III T R A N S M I S S I O N S C H E M E F O R 4 R E L A Y S OFDM U 1 U 2 U 3 U 4 Symbol 1 r 1 , 1 r 2 , 2 − r 3 , 3 ∗ − r 4 , 4 ∗ 2 r 1 , 2 r 2 , 1 − r 3 , 4 ∗ − r 4 , 3 ∗ 3 ζ ( r 1 , 3 ) ζ ( r 2 , 4 ) ζ ( r 3 , 1 ∗ ) ζ ( r 4 , 2 ∗ ) 4 ζ ( r 1 , 4 ) ζ ( r 2 , 3 ) − ζ ( r 3 , 2 ∗ ) − ζ ( r 4 , 1 ∗ ) A 4 =     0 0 0 − 1 0 0 − 1 0 0 1 0 0 1 0 0 0     . It h as bee n proved in [5] that CA i = A i C , i = 1 , 2 and CA i = A i C ∗ , i = 3 , 4 for all C ∈ C . At the destination no de, decod ing for { z 1 I , z 2 I } , { z 1 Q , z 2 Q } , { z 3 I , z 4 I } and { z 3 Q , z 4 Q } can be done separately in every sub carrier due to the fou r g roup dec odable structure of C . I V . S I M U L A T I O N R E S U LT S In th is section, w e study the er ror per forman ce of the propo sed codes u sing simulation s. W e take R = 4 , N = 64 and th e length o f CP as 16 . The de lay τ i at each relay is chosen ran domly between 0 to 15 with uniform distribution. T wo cases are consider ed f or simulation: (1) with ch annel knowledge a t the destination and (2) w ithout chan nel k nowl- edge at the destination . For the case of no channel information, differential en coding at the source as describ ed in Example 2 of Sectio n IV is do ne. When chan nel knowledge is av ailable at the d estination, r otated QPSK is u sed as the signal set [4]. The transm ission rate f or both the schemes is 1 bit per ch annel use (bpcu ) if the rate loss du e to CP is neglected . 0 5 10 15 20 25 30 35 40 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 Total Power P (dB) Codeword Error Rate coherent asynchronous, rate=1 bpcu noncoherent asynchronous, rate=1 bpcu Fig. 2. Error performance for a 4 relay system with and without channel kno wledge It can be observed from Fig. 2 that the error performance for the n o cha nnel k nowledge case is approx imately 5 dB worse than that with cha nnel kn owledge at the destination. This is due to the differential transmission/reception te chnique in part and also in part because of the change in sign al set fro m rotated QPSK to some other sign al set [5] in or der to co mply with the requ irement of scaled unitar y codeword matrices. V . D I S C U S S I O N A drawback of the prop osed tran smission schem e is th at it requires a large coherenc e in terval sp anning over multiple OFDM symbo l durations. Moreover ther e is a rate loss du e to the u se of CP , but this loss can be made n egligible by choosing a large enou gh N . In spite of these drawback s, to the best of our k nowledge, this is the first kn own am plify and forward based transmission sch eme available fo r any n umber of r elay nod es th at adm its low ML de coding comp lexity and also pr ovides full asyn chrono us cooperativ e div ersity . Some o f the interesting direction s for further work a re listed below: 1) Construc ting single symbo l decodab le distributed space time codes for the prop osed transm ission scheme. 2) Exten ding this work to asynchro nous relay network s with timing errors and frequen cy offsets at th e relay nodes is an interesting dir ection f or furth er work. T his problem has been ad dressed in [ 9] for the case o f two relay nodes. A C K N OW L E D G E M E N T This work was partly supp orted by the DRDO-IISc Progra m on Advanced Research in Mathem atical Engineering, partly by the Coun cil of Scientific & Indu strial Research (CSIR), India, throug h Research G rant (22 (0365 )/04/EMR-II) to B.S. Rajan. The authors sincerely than k Prof. X.G. Xia and Pro f. H. Jafarkhani fo r sendin g us prepr ints of their recent works [2], [3], [8], [9]. R E F E R E N C E S [1] Y . Jing and B. Hassibi, “Distri buted space time codi ng in wireless relay netw orks, ” IE EE T ransac tions on W ir eless Communication s , vol. 5, no. 12, pp. 3524-3536, Dec. 2006. [2] Zheng Li and X.-G. Xia, “ A Simple Alamout i Space-T ime Tra nsmission Scheme for Asynchronous Cooperati ve Systems, ” to appear in IEEE Signal Pr ocessing Letters , Dec. 2007. Priv ate Communicati on. [3] X. Guo and X.-G. Xia, “ A Distrib uted Space-T ime Coding in Asyn- chronous W ireless Relay Networks, ” to appea r in IEEE T ransactions on W ireless Communic ations . Priv ate Communication . [4] G. Susinder Rajan and B. Sundar Rajan, “Multi-gr oup ML Decodable Colloc ated and Distrib uted Space Time Block Codes, ” submitted to IE EE T ransac tions on Information Theory . A va ilable in arXi v:0712.2384. [5] —-, “ Algebraic Distrib uted Dif ferentia l Space-T ime Codes with Low Decoding Complex ity , ” to appear in IE EE T ransactions on W irele ss Communicat ion . A vail able in arXi v:0708.440 7 . [6] Kiran T . and B. Sundar Rajan, “P artially-c oherent distrib uted space-time codes with dif ferential encode r and decoder , ” IEE E Journal on Select ed Area s in Communic ations , vol. 25, No. 2, Feb. 2007, pp. 426-433. [7] Fr ´ ed ´ erique Oggier , Babak Hassibi, ”Cyclic Distrib uted Space- Time Codes for Wi reless Relay Netw orks with no Channel Information , ” submitt ed for publica tion. A vaila ble online http:/ /www .systems.caltech .edu/˜frederique/submitDSTCnoncoh.pdf [8] Y . Jing and H. Jafarkhani,“Di stribute d Differ ential Space-T ime Coding for Wi reless Relay Networks, ” to appear in IEEE T ransact ions on Communicat ions . Pri vate Communica tion. [9] Zheng Li and X.-G. Xia, “ An Alamout i Coded OFDM Tra nsmission for Cooperati ve Systems Robust to Both Timing Errors and Frequenc y Off sets, ” to appear in IE EE T ransact ions on W ire less Communicatio ns . Pri vat e Communicati on.