Distributed Space Time Codes with Low Decoding Complexity for Asynchronous Relay Networks
aa r X i v : . [ c s . I T ] S e p DRDO–IISc Programme onAdvanced Research in MathematicalEngineering
Distributed Space Time Codes with Low DecodingComplexity for Asynchronous Relay Networks (TR-PME-2007-09) by G. Susinder Rajan and B. Sundar Rajan Department of ECE,Indian Institute of Science, Bangalore
October 29, 2018
Indian Institute of ScienceBangalore 560 012 istributed Space Time Codes with Low DecodingComplexity for Asynchronous Relay Networks
G. Susinder Rajan and B. Sundar Rajan ∗ October 29, 2018
ABSTRACT
Recently Li and Xia have proposed a transmission scheme for wirelessrelay networks based on the Alamouti space time code and orthog-onal frequency division multiplexing to combat the effect of timingerrors at the relay nodes. This transmission scheme is amazingly sim-ple and achieves a diversity order of two for any number of relays.Motivated by its simplicity, this scheme is extended to a more generaltransmission scheme that can achieve full cooperative diversity for anynumber of relays. The conditions on the distributed space time code(DSTC) structure that admit its application in the proposed trans-mission scheme are identified and it is pointed out that the recentlyproposed full diversity four group decodable DSTCs from precoded co-ordinate interleaved orthogonal designs and extended Clifford algebrassatisfy these conditions. It is then shown how differential encoding atthe source can be combined with the proposed transmission scheme toarrive at a new transmission scheme that can achieve full cooperativediversity in asynchronous wireless relay networks with no channel in-formation and also no timing error knowledge at the destination node.Finally, four group decodable distributed differential space time codesapplicable in this new transmission scheme for power of two numberof relays are also provided.
Coding for cooperative wireless relay networks has attracted considerableattention recently. Distributed space time coding was proposed as a codingstrategy to achieve full cooperative diversity in [1] assuming that the signalsfrom all the relay nodes arrive at the destination at the same time. But The authors are with the Department of Electrical Communica-tion Engineering, Indian Institute of Science, Bangalore- 560012, India,email: { susinder,bsrajan } @ece.iisc.ernet.inThis work was partly supported by the DRDO-IISc Program of Advanced Research inMathematical Engineering through a grant to B.S.Rajan • The Li-Xia transmission scheme is extended to a more general trans-mission scheme that can achieve full asynchronous cooperative diver-sity for any number of relays. • The conditions on the STC structure that admit its application in theproposed transmission scheme are identified. The recently proposedfull diversity four group decodable distributed STCs in [4, 5, 6] forsynchronous wireless relay networks are found to satisfy the requiredconditions for application in the proposed transmission scheme. • It is shown how differential encoding at the source node can be com-bined with the proposed transmission scheme to arrive at a transmis-sion scheme that can achieve full asynchronous cooperative diversityin the absence of channel knowledge and in the absence of knowledgeof the timing errors of the relay nodes. Moreover, an existing class offour group decodable distributed differential STCs [7] for synchronousrelay networks with power of two number of relays is shown to beapplicable in this setting as well.2 .1 Organization of the report
In Section 2, the basic assumptions on the relay network model are givenand the Li-Xia transmission scheme is briefly described. Section 3 describesthe transmission scheme proposed in this report and also provides four groupdecodable codes for any number of relays. Section 4 briefly explains howdifferential encoding at the source node can be combined with the proposedtransmission scheme and four group decodable distributed differential STCsapplicable in this scenario are also proposed. Simulation results and discus-sion on further work comprise Sections 5 and 6 respectively.
Notation:
Vectors and matrices are denoted by lowercase and uppercasebold letters respectively. I m denotes an m × m identity matrix and de-notes an all zero matrix of appropriate size. For a set A , the cardinality of A is denoted by | A | . A null set is denoted by φ . For a matrix, ( . ) T , ( . ) ∗ and( . ) H denote transposition, conjugation and conjugate transpose operationsrespectively. For a complex number, ( . ) I and ( . ) Q denote its in-phase andquadrature-phase parts respectively. In this section, the basic relay network model assumptions are given andthe Li-Xia transmission scheme in [2] is briefly described. The transmissionscheme in [2] is based on the use of OFDM at the source node and theAlamouti code implemented in a distributed fashion for a 2 relay system.Essentially, the transmission scheme in [2] is applicable mainly for the caseof 2 relays but by forming clusters of two relay nodes, it can be extended tomore number of relays at the cost of sacrificing diversity benefits.
S U DU U R f g g f f R g R Figure 1: Asynchronous wireless relay network3 .1 Network model assumptions
Consider a network with one source node, one destination node and R relaynodes U , U , . . . , U R . This is depicted in Fig. 1. Every node is assumedto have only a single antenna and is half duplex constrained. The channelgain between the source and the i -th relay f i and that between the j -threlay and the destination g j are assumed to be quasi-static, flat fading andmodeled by independent and complex Gaussian distributed with mean zeroand unit variance. The transmission of information from the source nodeto the destination node takes place in two phases. In the first phase, thesource broadcasts the information to the relay nodes using OFDM. Therelay nodes receive the faded and noise corrupted OFDM symbols, processthem and transmit them to the destination. The relay nodes are assumed tohave perfect carrier synchronization. The overall relative timing error of thesignals arrived at the destination node from the i -th relay node is denotedby τ i . Without loss of generality, it is assumed that τ = 0, τ i +1 ≥ τ i , i =1 , . . . , R −
1. The destination node is assumed to have the knowledge of allthe channel fading gains f i , g j , i, j = 1 , . . . , R and the relative timing errors τ i , i = 1 , . . . , R . The source takes 2 N complex symbols x i,j, ≤ i ≤ N − ,j =1 , and forms twoblocks of data denoted by x j = (cid:2) x ,j x ,j . . . x N − ,j (cid:3) T , j = 1 ,
2. Thefirst block x is modulated by N -point Inverse Discrete Fourier Transform(IDFT) and x is modulated by N -point Discrete Fourier Transform (DFT).Then a cyclic prefix (CP) of length l cp is added to each block, where l cp isnot less than the maximum of the overall relative timing errors of the sig-nals arrived at the destination node from the relay nodes. The resulting twoOFDM symbols denoted by ¯x and ¯x consisting of L s = N + l cp complexnumbers are broadcasted to the two relays using a fraction π of the totalaverage P consumed by the source and the relay nodes together.If the channel fade gains are assumed to be constant for 4 OFDM symbolintervals, the received signals at the i -th relay during the j -th OFDM symbolduration is given by r i , j = f i ¯x j + ¯v i , j where, ¯v i , j is the additive white Gaussian noise at the i -th relay node duringthe j the OFDM symbol duration. The two relay nodes then process andtransmit the resulting signals as shown in Table 1 using a fraction π of thetotal power P . The notation ζ ( . ) denotes the time reversal operation, i.e., ζ ( r ( n )) , r ( L s − n ).The destination removes the CP for the first OFDM symbol and imple-ments the following for the second OFDM symbol:4able 1: Alamouti code based transmission schemeOFDM Symbol U U q π Pπ P +1 r , − q π Pπ P +1 r , ∗ q π Pπ P +1 ζ ( r , ) q π Pπ P +1 ζ ( r , ∗ )1. Remove the CP to get a N -point vector2. Shift the last l cp samples of the N -point vector as the first l cp samples.DFT is then applied on the resulting two vectors. Since l cp ≥ τ , theorthogonality between the sub carriers is still maintained. The delay in thetime domain then translates to a corresponding phase change of e − i πkN in the k -th sub carrier. Let d τ denote h e − i πτ N . . . e − i πτ N − N i T . Thenthe received signals for two consecutive OFDM blocks after CP removaland DFT transformation denoted by y = (cid:2) y , y , . . . y N − , (cid:3) T and y = (cid:2) y , y , . . . y N − , (cid:3) T can be expressed as: y = q π π P π P +1 (DFT(IDFT( x )) f g + DFT( − (DFT( x )) ∗ ) ◦ d τ f ∗ g )+ q π Pπ P +1 ( v , g − v , ∗ ◦ d τ g ) + w y = q π π P π P +1 (DFT( ζ (DFT( x )) ∗ ) f g + DFT( ζ (IDFT( x )) ∗ ) ◦ d τ f ∗ g )+ q π Pπ P +1 ( v , g + v , ∗ ◦ d τ g ) + w where, ◦ is the Hadamard product, w i = ( w k,i ) , i = 1 , v i , j denotes the DFT of ¯v i , j .Now using the identities(DFT( x )) ∗ = IDFT( x ∗ ) , (IDFT( x )) ∗ = DFT( x ∗ ) , DFT( ζ (DFT( x ))) = x (1)we get the Alamouti code form in each sub carrier k, ≤ k ≤ N − (cid:20) y k, y k, (cid:21) = q π π P π P +1 (cid:20) x k, − x ∗ k, x k, x ∗ k, (cid:21) " f g e − i πkτ N f ∗ g + q π Pπ P +1 " v , ( k ) g − v , ∗ ( k ) e − i πkτ N g v , ( k ) g + v , ∗ ( k ) e − i πkτ N g + (cid:20) w k, w k, (cid:21) . π = 1, π = R and because of the Alamouticode form, diversity order of two can be achieved along with symbol-by-symbol ML decoding. In this section, we extend the Li-Xia transmission scheme to a general trans-mission scheme that can achieve full cooperative diversity for arbitrary num-ber of relays. This nontrivial extension is based on analyzing the sufficientconditions required on the structure of STBCs which admit application inthe Li-Xia transmission scheme.
The source takes RN complex symbols x i,j, ≤ i ≤ N − ,j =1 , ,...,R and forms R blocks of data denoted by x j = (cid:2) x ,j x ,j . . . x N − ,j (cid:3) T , j = 1 , , . . . , R .Of these R blocks, M of them are modulated by N -point IDFT and the re-maining R − M blocks are modulated by N -point DFT. Without loss ofgenerality, let us assume that the first M blocks are modulated by N -pointIDFT. Then a CP of length l cp is added to each block, where l cp is not lessthan the maximum of the overall relative timing errors of the signals arrivedat the destination node from all the relay nodes. The resulting R OFDMsymbols denoted by ¯x , ¯x , . . . , ¯x R consisting of L s = N + l cp complex num-bers are broadcasted to the R relays using a fraction π of the total average P . If the channel fade gains are assumed to be constant for 2 R OFDM symbolintervals, the received signals at the i -th relay during the j -th OFDM symbolduration is given by r i , j = f i ¯x j + ¯v i , j where, ¯v i , j is the additive white Gaussian noise (AWGN) at the i -th relaynode during the j the OFDM symbol duration. The relay nodes process andtransmit the received noisy signals as shown in Table 2 using a fraction π oftotal power P . Note from Table 2 that time reversal is done during the last R − M OFDM symbol durations. We would like to emphasize that in generaltime reversal could be implemented in any R − M of the total R OFDMsymbol durations. The transmitted signal t i , j ∈ { , ± r i , j , j = 1 , . . . , R } withthe constraint that the i -th relay should not be allowed to transmit thefollowing: 6able 2: Proposed transmission scheme OFDM Symbol U . . . U M U M +1 . . . U R t , . . . t M , t M + , ∗ . . . t R , ∗ ... ... ... ... ... ... ... M t , M . . . t M , M t M + , M ∗ . . . t R , M ∗ M + 1 ζ ( t , M + ) . . . ζ ( t M , M + ) ζ ( t M + , M + ∗ ) . . . ζ ( t R , M + ∗ )... ... ... ... ... ... ... R ζ ( t , R ) . . . ζ ( t M , R ) ζ ( t M , R ∗ ) . . . ζ ( t R , R ∗ ) {± 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 } . Note 1
If the i -th relay is permitted to transmit elements belonging to theabove set, then after CP removal and DFT transformation at the destinationnode, we would end up with the following vectors corresponding to each ofthe four subsets in the above set respectively: ± DFT((IDFT( x j )) ∗ ) = DFT(DFT( x j ∗ )) , j = 1 , . . . , M ± DFT( ζ (( IDF T )( x j ))) , j = 1 , . . . , M ± DFT(DFT( x j )) , j = M + 1 , . . . , R ± DFT( ζ (DFT( x j )) ∗ ) = ± DFT( ζ (IDFT( x j ∗ ))) , j = M + 1 , . . . , R from any of which it is not possible to recover any of ± x j , ± x j ∗ , j = 1 , , . . . , R .However, if the destination node is allowed to apply DFT to some of the re-ceived OFDM symbols and IDFT to the remaining OFDM symbols, thenpossibly the above restrictions can be removed, which is a scope for furtherwork. The destination removes the CP for the first M OFDM symbols and imple-ments the following for the remaining OFDM symbols:1. Remove the CP to get a N -point vector2. Shift the last l cp samples of the N -point vector as the first l cp samples.DFT is then applied on the resulting R vectors. Let the received signalsfor R consecutive OFDM blocks after CP removal and DFT transforma-tion be denoted by y j = (cid:2) y ,j y ,j . . . y N − ,j (cid:3) T , j = 1 , , . . . , R . Let7 i = ( w k,i ) , i = 1 , . . . , R represent the AWGN at the destination node andlet v i , j denote the DFT of ¯v i , j . Let s k = (cid:2) x i, x i, . . . x i,R (cid:3) T , k =0 , , . . . , N − k, ≤ k ≤ N − y k = (cid:2) y k, y k, . . . y k,R (cid:3) T = s π π P π P + 1 X k h k + n k (2)where, X k = (cid:2) A s k . . . A M s k A M + s k ∗ . . . A R s k ∗ (cid:3) (3)for some square real matrices A i , i = 1 , . . . , R having the property that anyrow of A i has only one nonzero entry. If u τ i k = e − i πkτiN , then h k = (cid:2) f g u τ k f g . . . 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 (cid:3) T is the equivalent channel matrix for the k -th sub carrier. The equivalentnoise vector is given by n k = q π Pπ P +1 δ P Ri =1 sgn ( t i , ) ˆv i , ( k ) g i u τ i k δ P Ri =1 sgn ( t i , ) ˆv i , ( k ) g i u τ i k ... δ R P Ri =1 sgn ( t i , R ) ˆv i , R ( k ) g i u τ i k + w k, w k, . . .w k,R . where, sgn ( t i , j ) = t i , j ∈ { r i , j , j = 1 , . . . , R }− t i , j ∈ {− r i , j , j = 1 , . . . , R } t i , j = and ˆv i , m = (cid:26) 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 . The δ i ’s are simply scaling fac-tors to account for the correct noise variance due to some zeros in the trans-mission.ML decoding of X k can be done from (2) by choosing that codewordwhich minimizes k Ω − ( y k − X k h k ) k F , where Ω is the covariance matrixof n k and k . k F denotes the Frobenius norm. Essentially, the proposedtransmission scheme implements a space time code having a special structurein each sub carrier. In this subsection, we analyze the structure of the space time code requiredfor implementing in the proposed transmission scheme. Note from (3) thatthe space time code should have the property that any column should haveonly the complex symbols or only their conjugates. We refer to this propertyas conjugate linearity property[4, 5, 6]. But conjugate linearity alone is not8nough for a space time code to qualify for implementation in the proposedtransmission scheme. Note from Table 2 that time reversal is implementedfor certain OFDM symbol durations by all the relay nodes. In other words ifone relay node implements time reversal during a particular OFDM symbolduration, then all the other relay nodes should necessarily implement timereversal during that OFDM symbol duration. Observe that this is a propertyconnected with the row structure of a space time code. We now provide a setof sufficient conditions that are required on the row structure of conjugatelinear space time codes. First let us partition the complex symbols occurringin the i -th row into two sets- one set P i containing those complex symbolswhich appear without conjugation and another set P ci which contains thosecomplex symbols which appear with conjugation in the i -th row. Then ifthe following conditions are satisfied by a conjugate linear space time code,then it can be implemented in the proposed transmission scheme describedin the previous subsection. P i ∩ P ci = φ, ∀ i = 1 , . . . , R | P i | = | P ci | , ∀ i = 1 , . . . , RP i ∩ P j ∈ { φ, P i , P j } , ∀ i = j. (4)To understand what happens if the above condition is not met, let ussee an example of a conjugate linear STBC which cannot be employed inthe proposed transmission scheme. Example 1
Consider the conjugate linear STBC given by x k, x k, − x ∗ k, − x ∗ k, x k, x k, − x ∗ k, − x ∗ k, x k, x k, x ∗ k, x ∗ k, x k, x k, x ∗ k, x ∗ k, for which P = P c = { x k, , x k, } , P c = P = { x k, , x k, } , P = P c = { x k, , x k, } , P c = P = { x k, , x k, } . It can be checked that there is no assign-ment of time reversal OFDM symbol durations together with an appropriatechoice of M and relay node processing such that the above conjugate linearSTBC form is obtained at every sub carrier at the destination node. This isbecause the conditions in (4) are not met by this conjugate linear STBC. For the case of the Alamouti code, P = P c = { x k, } , P = P c = { x k, } and hence it satisfies the conditions in (4). Recently three new classes offull diversity four group decodable distributed space time codes for anynumber of relays were reported in [4, 5, 6]. These codes are conjugatelinear. Since they are four group decodable, the associated real symbols inthese space time codes can be partitioned equally into four groups and theML decoding can be done for the real symbols in a group independently of9he real symbols in the other groups. Thus the ML decoding complexityof these codes is significantly less compared to all other distributed spacetime codes known in the literature. In this report, we show that the codesreported in [4, 5, 6] satisfy the conditions in (4) and are thus suitable to beapplied in the proposed transmission scheme. This is illustrated using thefollowing two examples of codes taken from [4, 5, 6]. Example 2
Let us consider R = 4 and the distributed space time code in[4] for this case has the following structure x k, x k, − x ∗ k, − x ∗ k, x k, x k, − x ∗ k, − x ∗ k, x k, x k, x ∗ k, x ∗ k, x k, x k, x ∗ k, x ∗ k, for which M = 2 , P = P = P c = P c = { x k, , x k, } and P = P = P c = P c = { x k, , x k, } . To arrive at the above structure in every sub carrier,encoding and processing at the relays are done as follows: ¯x = IDFT( x ) , ¯x = IDFT( x ) , ¯x = DFT( x ) and ¯x = DFT( x ) . Table 3: Transmission scheme for 4 relaysOFDM U U U U Symbol1 r , r , − r , ∗ − r , ∗ r , r , − r , ∗ − r , ∗ ζ ( r , ) ζ ( r , ) ζ ( r , ∗ ) ζ ( r , ∗ )4 ζ ( r , ) ζ ( r , ) − ζ ( r , ∗ ) − ζ ( r , ∗ ) This code is single complex symbol decodable and achieves full diversityfor appropriately chosen signals sets [4].
Example 3
Let us take R = 5 for which the distributed space time codein [5] is obtained taking a space time code for relays and dropping onecolumn. It is given by x k, − x ∗ k, x k, x ∗ k, x k, − x ∗ k,
00 0 x k, x ∗ k,
00 0 0 0 x k x k, for which P = P c = { x k, } , P = P c = { x k, } , P = P c = { x k, } , P = P c = { x k, } , P = { x k, } , P = { x k, } and P c = P c = φ . At the source, we hoose ¯x = IDFT( x ) , ¯x = DFT( x ) , ¯x = IDFT( x ) , ¯x = DFT( x ) , ¯x = IDFT( x ) and ¯x = DFT( x ) . The relays process the receivedOFDM symbols as shown in Table 4. Table 4: Transmission scheme for 5 relays
OFDM U U U U U Symbol1 r , − r , ∗ − ζ ( r , ) ζ ( r , ∗ ) − − , − r , ∗ − ζ ( r , ) ζ ( r , ∗ ) , − ζ ( r , ) This code is real symbol decodable and achieves full diversity for ap-propriately signal sets [5, 6]. Example 3 illustrates how the proposed transmission scheme can be ex-tended to odd number of relays as well.
In this section, it is shown how differential encoding can be combined withthe proposed transmission scheme described in Section 3 and then the codesin [7] are proposed for application in this setting.For the proposed transmission scheme in Section 3, at the end of onetransmission frame, we have in the k -th sub carrier y k = q π π P π P +1 X k h k + n k .Note that the channel matrix h k depends on f i , g i , τ i , i = 1 , . . . , R . Thusthe destination node needs to have the knowledge of these values in orderto perform ML decoding.Now using differential encoding ideas which were proposed in [8, 9, 10]for non-coherent communication in synchronous relay networks, we combinethem with the proposed asynchronous transmission scheme. Supposing thechannel remains approximately constant for two transmission frames, thendifferential encoding can be done at the source node in each sub carrier0 ≤ k ≤ N − s = (cid:2) √ R . . . (cid:3) T , s tk = 1 a t − C t s t − , C t ∈ C where, s ik denotes the vector of complex symbols transmitted by the sourceduring the i -th transmission frame in the k -th sub carrier and C is the code-11ook used by the source which consists of scaled unitary matrices C H C t = a t I such that E[ a 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 tk = 1 a t − C t y t − + ( n tk − a t − C t n t − ) (5)from which C t can be decoded as ˆC t = arg min C t ∈ C k y tk − a t − C t y t − k F in each sub carrier 0 ≤ k ≤ N − f i , g i , τ i , i =1 , . . . R at the destination. It turns out that the four group decodable dis-tributed differential space time codes constructed in [7] for synchronous relaynetworks with power of two number of relays meet all the requirements foruse in the proposed transmission scheme as well. The following exampleillustrates this fact. Example 4
Let R = 4 . The codebook at the source is given by C = q z z − z ∗ − z ∗ z z − z ∗ − z ∗ z z z ∗ z ∗ z z z ∗ z ∗ where { z I , z I } , { z Q , z Q } , { z I , z I } , { z Q , z Q } ∈ S and S = (" √ , " − √ , " q , " − q . Dif-ferential encoding is done at the source node for each sub carrier ≤ k ≤ N − as follows: s = (cid:2) √ R . . . (cid:3) T , s tk = 1 a t − C t s t − , C t ∈ C . Once we get s tk , k = 0 , . . . , N − from the above equation, the N lengthvectors x i , i = 1 , . . . , R can be obtained. Then IDFT/DFT is applied onthese vectors as shown below and broadcasted to the relay nodes. ¯x =IDFT( x ) , ¯x = IDFT( x ) , ¯x = DFT( x ) and ¯x = DFT( x ) . The re-lay nodes process the received OFDM symbols as given in Table 3 for which M = 2 , A = I , A = , A = − −
11 0 0 00 1 0 0 and A = −
10 0 − . It has been proved in [7] that CA i = A i C , i =12 , and CA i = A i C ∗ , i = 3 , for all C ∈ C . At the destination node, de-coding for { z I , z I } , { z Q , z Q } , { z I , z I } and { z Q , z Q } can be done sep-arately in every sub carrier due to the four group decodable structure of C . −6 −5 −4 −3 −2 −1 Total Power P (dB) C ode w o r d E rr o r R a t e coherent asynchronous, rate=1 bpcunoncoherent asynchronous, rate=1 bpcu Figure 2: Error performance for a 4 relay system with and without channelknowledge
In this section, we study the error performance of the proposed codes usingsimulations. We take R = 4, N = 64 and the length of CP as 16. Thedelay τ i at each relay is chosen randomly between 0 to 15 with uniformdistribution. Two cases are considered for simulation: (1) with channelknowledge at the destination and (2) without channel knowledge at thedestination. For the case of no channel information, differential encoding atthe source as described in Section 4 is done using the distributed differentialspace time in [7]. When channel knowledge is available at the destination,rotated QPSK is used as the signal set [4, 6]. The transmission rate for theboth the schemes is 1 bit per channel use (bpcu) if the rate loss due to CPis neglected. 13he error performance curves for both the cases is shown in Fig. 2. Itcan be observed from Fig. 2 that the error performance of the no chan-nel knowledge case performs approximately 5 dB worser than that withchannel knowledge at the destination. This is due to the differential trans-mission/reception technique in part and also in part because of the changein signal set from rotated QPSK to some other signal set [7] in order to com-ply with the requirement of scaled unitary codeword matrices. The changein signal set for the sake of scaled unitary codeword matrices results in areduction of the coding gain. A general transmission scheme for arbitrary number of relays that canachieve full cooperative diversity in the presence of timing errors at therelay nodes was proposed. It was then pointed out that the four groupdecodable distributed space time codes in [4, 5, 6] can be applied in theproposed transmission scheme for any number of relay nodes. Finally it wasshown how the proposed scheme can be combined with differential encodingat the source node to end up with a transmission scheme that is robust totiming errors and also does not require the knowledge of the channel fadinggains as well as the timing errors at any of the nodes. For this differentialscheme, it was pointed out that the four group decodable distributed dif-ferential space time codes in [7] are applicable for power of two number ofrelays.A drawback of the proposed transmission scheme is that it requires alarge coherence interval spanning over multiple OFDM symbol durations.Moreover there is a rate loss due to the use of CP, but this loss can be madenegligible by choosing a large enough N . Some of the interesting directionsfor further work are listed below:1. Constructing single symbol decodable distributed space time codes forthe proposed transmission scheme.2. The codes in [7] are applicable only for power of two number of re-lay nodes. Constructing four group decodable distributed differentialspace time codes for all even number of relay nodes that are applica-ble in asynchronous relay networks without channel knowledge is animportant direction for further work.3. In this work, we have assumed that there are no frequency offsets atthe relay nodes. Extending this work to asynchronous relay networkswith frequency offsets is an interesting direction for further work. Thisproblem has been addressed in [11] for the case of two relay nodes.14 cknowledgement The authors sincerely thank Prof. Xiang Gen Xia and Prof. Hamid Ja-farkhani for sending us preprints of their recent works [2, 3, 10, 11].
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