A Non-differential Distributed Space-Time Coding for Partially-Coherent Cooperative Communication

IEEE TRANSACTIONS ON WIRELESS COMMUNICA TIONS , V OL. XX, NO. XX, XXXX 1 A Non-dif ferential Distrib uted Space-T ime Coding for P artia lly-Coherent Coop erati v e Communication Harshan J. and B. Sund ar Rajan, Senior Member , IEEE Abstract —In a di stributed space-time coding scheme, b ased on the relay chann el model, the r elay nodes co-operate to linearly process th e transmitted signal from the source and forwa rd them to the destination such that the signal at the destin ation app ears as a space time block code. Recently , a code design criteria for achieving full diversity in a partially-coherent envir onment hav e been proposed along with codes based on differential encoding and decodin g techniqu es. For such a set up , in th is paper , a non- differential encodin g t echnique and construction o f distributed space time block codes from un itary matrix groups at the source and a set of diagonal unitary matrices f or the r elays are pro posed. It is shown th at, the perf ormance of our scheme i s independen t of the choice of unitary matrices at t he relays. Wh en the gro up is cyclic, a necessary and su fficient condit ion on the generator of the cyclic gr oup to achiev e full div ersity and to minimize the pairwise error pro bability is prove d. V arious choices on t he generator of cyclic group to reduce t he M L decoding complexity at the d estination is presented. It is also shown th at, at the source, if non-cyclic abelian un itary matrix groups ar e used, then fu ll- divers ity can not be obtained. The pr esented scheme is also robust to failure of any subset of relay nodes. Index T erms —Cooperative communication, cyclic groups, dis- tributed sp ace-time codes and unitary space-time codes. I . I N T R O D U C T I O N A N D P R E L I M I N A R I E S Co-operative co mmunic ation can be based on a relay chan- nel mo del wher e a set of distributed antenn as belon ging to multiple users is exploited for ac hieving spatial div ersity [1]- [4]. In [4], the idea of Sp ace-Time Coding (STC) devised for point to point co- located multiple anten na s ystems is applied to a two-hop wireless relay network and is r eferred as Distributed Space-Time Co ding (DSTC). The technique inv olves a two phase pro tocol w here, in the fir st phase, the source b roadcasts the information to the relays and in th e second p hase, relays linearly pro cess the recei ved signals and forward them to the destination suc h th at the signal at the destinatio n appe ars as a Space-Time Block Code (STBC). In the ab ove technique, the destinatio n may or may not have the knowledge o f chan nel fade coefficients (i) from the sou rce to the relays and (ii) from the relay s to the destination when de- coding for the so urce signal. DST C when the destination does not h av e the knowledge of the cha nnels f rom the sourc e to the relays but has the knowledge of the chann els from the rela ys This work was supported through grants to B.S. Rajan; partly by the IISc- DRDO progra m on Adv anced Research in Mathe matical Engineering, and partly by the Counci l of Scientific & Industri al Resea rch (CSIR, India) Re- search Grant (22(0365)/ 04/EMR-II). The material in this paper was present ed in parts at the IEEE Internation al Conferenc e on Communicat ions (ICC 2008), Beiji ng, China, May . 19-23, 2008. Harshan J. and B. Sundar Rajan are with the Department of Electrical Communicati on Engineering , Indian Institute of Science , Banga lore-560012, India. Email: { harshan,bsraj an } @ece.i isc.ernet.in. Manuscript recei ved May 18, 2007; revi sed January 18, 2008. to itself is called partially- coheren t DSTC, which is the sub ject matter o f this paper . For a partially -coheren t set up, in [5], a code design criteria is pro posed for achieving full d iv ersity . A differential encoding and decoding strategy with a class o f fu ll div ersity ach ieving codes using cyclic d ivision algebras is also propo sed. Inspired by n on-co herent MIMO u nitary d ifferential modulatio n [ 6], differential distrib uted space time code s hav e been pro posed in [7], [8] an d [9]. In [5], it is shown that th e coding prob lem for a partially coheren t setup is to distrib utively design a fin ite set of un itary matrices such that the ma trix obtain ed from juxtap osing any two distinct ma trices fro m the set m ust be full rank . In other words, the prob lem is to design no n-intersectin g subsp aces such that the pr incipal ang les between subspaces is as large as po ssible. Howe ver , with the use of d ifferential techniques [5], [7], [8], [9], th e coding pro blem gets transform ed in to a problem of constru cting a finite set of un itary matrices su ch that the difference matrix of any two matrices is f ull rank. This is a well known design criteria for relay networks with amplify and forward pro tocol in a coheren t d etection environment and code construc tions based on th e above criter ia are av ailable in the literature. In this p aper, we propose a m ethod to explicitly construct non-in tersecting subspaces f or a partially-co herent set up. i.e we construct a set o f unitary matrices in a distrib uted way such tha t the matrix o btained from juxtaposing any two distinct matrices f rom the set m ust be fu ll rank. The contributions of this paper can be summarized as follows: (i) for th e partially-coherent set up, w e introdu ce a non-d ifferential coding scheme and obtain an expression for ML d ecoding m etric (Th eorem 1) for the general class of u nitary Distributed Spac e-T ime Block Codes (DSTBCs) (See Defin ition 2 ). ( ii) W e co nstruct unitary DSTBCs usin g cyclic u nitary matrix g roups called Non-differential Cyclic Distributed Space-T ime Codes (NCDSTC) (Definition 4 ) and provide a necessary and su fficient condition on its gener ator such tha t the NCDSTC is fully di verse (Th eorem 2) a nd the pairwise error probability (PEP) is minimized. (iii) Also, we provide con ditions on the c hoice of the gene rator of the cyclic group so as to reduc e the deco ding comp lexity a t the destination. (iv) W e sh ow that the p roposed scheme is robust to failure of a su bset of the relay nodes. The rem aining part o f the paper is organized as f ollows: In Section I I, along with the signal model, partially-cohere nt distributed space time codin g techniqu e is briefly reviewed and an ML decoding metric for a class o f unitary DSTBCs is presented. Th e sch eme of no n-differential u nitary DSTC from cyclic unitary matrix gr oups is introdu ced in Section III wh ere we provid e a necessary and suf ficient condition on 2 IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS , VOL. XX , NO. XX, XXXX g Source Destination . . . . Relays R R−1 g g g 1 f 1 f 2 2 R−1 f R f Fig. 1. W ireless relay netwo rk model the ge nerator o f the cyclic grou p to achieve full diversity and to min imize the PEP . In Section I V, an ML dec oding metric fo r the p ropo sed cod es is presented alo ng with two reduced de coding complexity ML decod ers. D etails on the robustness of our scheme to the failure o f a su bset o f th e relay nodes is also presen ted. Possible d irections o f futu re work and conclud ing remarks constitute Section V. Notations: For a comp lex matrix X , the matrices X ∗ , X T , X H , | X | , Re X and I m X d enote, respecti vely , the conjugate, transpose, co njugate transpo se, determin ant, real p art an d imaginary part of X . For a c omplex matrix Y of the order same as X , X ⊙ Y den otes the Hadamard produ ct of X and Y . The T × T id entity matrix is d enoted by I T and O T denotes the T - length vector of z eros. W e use | x | to de note the a bsolute value of the com plex numb er x and E [ x ] to denote the expectation of the ran dom v ariable x . W e write x ∼ C S C G ( µ, Γ ) when x is a circularly symme tric co mplex Gaussian random vector with mean µ and covariance matrix Γ and use j for √ − 1 . The set of integer s and the set of co mplex numb ers are, respectiv ely , denoted by Z and C . I I . P A RT I A L LY C O H E R E N T D I S T R I B U T E D S PAC E T I M E C O D I N G A. Sign al model The wireless n etwork c onsidered in Figu re 1 consists of R + 2 nodes each having single antenna which ar e placed random ly an d in depend ently acco rding to some distribution. There is one sour ce node and one destination node. All the other R n odes work as relay s. W e denote the c hannel f rom the source nod e to the j -th relay a s f j and the chan nel fro m the j - th relay to th e destination node as g j for j = 1 , 2 , · · · , R . The following assumptio ns a re made in our sy stem model: (i) all the n odes are subjected to half du plex constraint. (ii) fading coefficients f j , g j are i.i.d C S C G ( 0 , 1) with coheren ce time interval, T . (iii) all the nodes are syn chron ized at th e symbo l lev el. ( iv) Destination k nows the fading coefficients g j ’ s but not f j ’ s . Every transmission from the source to the destination com- prises of tw o p hases. In the fir st ph ase th e source trans- mits a T length com plex vecto r fro m the co deboo k S = { s 1 , s 2 , s 3 , · · · , s L } consisting of info rmation v ectors s l ∈ C T such that E  s H l s l  = 1, so tha t P 1 T is the average transmit power . In par ticular, P 1 is the a verage transmit power used at the source node for ev ery channel use. When th e inform ation vector s is transmitted, the rece i ved vector at the j -th relay is giv en b y r j = √ P 1 T f j s + n j , j = 1 , 2 , · · · , R . I n the second phase, all the r elay n odes are scheduled to tra nsmit T length vectors to the destinatio n simultaneou sly . Each relay is equip ped with a fixed T × T u nitary matrix A j and is allowed to linearly process the rece iv ed vector . The j th relay is scheduled to transmit t j = q P 2 (1+ P 1 ) A j r j where P 2 is the av erage transmit power used at each relay f or ev ery chan nel use. The vector rece iv ed at the destination is gi ven by y = R X j =1 ( g j t j ) + w = s P 1 P 2 T (1 + P 1 ) Sh + n (1) where w ∼ C S C G (0 , I T ) is th e additive noise at the desti- nation, n = q P 2 (1+ P 1 ) h P R j =1 ( g j A j n j ) i + w . Th e equivalent channel h is giv en b y h = [ f 1 g 1 f 2 g 2 · · · f R g R ] T ∈ C R . The codeword matrix S is given b y S = [ A 1 s A 2 s · · · A R s ] ∈ C T × R . Definition 1: Th e co llection C = { [ A 1 s A 2 s · · · A R s ] } of codeword matrices when s ru ns over S , is called the Distributed Space-T ime Block code (DSTBC). B. ML Decod er Since the relay matrices A j are unitar y , th e ran dom vectors w and n j are in depend ent and Gaussian and since g j are known at the receiver , n is a Gaussian rand om vector with E [ n ] = 0 T and E  nn H  =  1 + P 2 (1+ P 1 ) P R j =1 ( | g j | 2 )  I T . Assume that S ∈ C . When the destination has the knowledge of g j ’ s b ut no t of f j ’ s , y is a Gaussian ran dom vector with E [ y | S , g j ] = 0 T and E  yy H | S , g j  = Σ y = ρ S Σ h S H + γ I T where, • ρ = P 1 P 2 T P 1 +1 • γ = (1 + P 2 (1+ P 1 ) P R j =1 ( | g j | 2 )) and • Σ h = diag  | g 1 | 2 , · · · | g R | 2  . Then, the co nditiona l pdf P ( y | S , g j ) is g iv en by P ( y | S , g j ) = 1 | Σ y | exp  −  y H Σ − 1 y y  . The partially-cohe rent ML d ecoder d ecodes to a cod ew ord ˆ S where ˆ S = ar g max S ∈C P ( y | S , g j ) . (2) Definition 2: [ 5] A DSTBC is called a u nitary DSTBC if ev ery S ∈ C in Definition 1 satisfies the co ndition that S H S = t I R , where t is a constant real n umber ind epend ent of the codeword S . HARSHAN and RAJ AN: A NON-DIFFERENTIAL DISTRIBUTED SP ACE-TIME CODING FOR P ARTIALL Y -COHER ENT COOPERA TIVE COM MUNICA TION 3 Theor em 1: For a unitary DSTBC, the partially- coheren t ML deco ding is giv en by arg max S ∈ C  y H SGS H y  , (3) where G = d iag ( β 1 , · · · β R ) an d β j = | g j | 2  | g j | 2 + γ ρ − 1  − 1 for j = 1 , · · · , R. Pr oof: Using the well known result, | I + AB | = | I + BA | and the appro priate use of the matr ix in version lemm a in (2), the result fo llows. C. Design criteria Chernoff bound on the PEP for the decoder in (2) is given in [5] f ollowing which a c riteria for design ing unitar y DSTBCs in ord er to minimize the PEP is also provided . F or achieving a div ersity order of R , the ter m | S H nm S nm | h as to be n on zero for all m , n such that n 6 = m, where S nm ∈ C T × 2 R is obtained by juxtap osing two codew ords S n and S m as, S nm = [ S n S m ] . (4) Further, fo r large values of P (where P = P 1 + R P 2 is th e total power used at all no des fo r every chann el use), PEP can be min imised by m aximising | S H nm S nm | for all n 6 = m. A necessary con dition for th is is T ≥ 2 R . Henc e, in the rest of the pap er , we consider T = 2 R. In the following section , we con struct cod es that satisfy the ab ove design c riterion f or full-div ersity using Generalized Butson-Hadam ard matrices defined as fo llows: Definition 3: A Generalize d Butson-Had amard (GBH) ma- trix [10] is a T × T matrix M with entries such that MM H = M H M = T I T and the conjugate of ev ery entry m ij of M is its in verse m − 1 ij i.e m ∗ ij = m − 1 ij . I I I . A N O N - D I FF E R E N T I A L S C H E M E A N D C O D E S U S I N G C Y C L I C U N I TA RY M AT R I X G R O U P S W e intro duce a non -differential e ncoding schem e using a cyclic un itary matrix gr oup [6] o f diagona l ma trices at the source and relay matrices con structed fr om GBH matrices [8], [10]. A pprop riate in gredien ts r equired at the source and the relays are a s follows: A. At th e Source The source n ode is equipped with a finite abelian gro up U of size L consisting of 2 R × 2 R diag onal unitar y m atrices and a 2 R × 1 complex vector , x . Let the group U be written as a direct produ ct of K cyclic gr oups U ν , each of order L ν for ν = 1 , 2 , · · · , K as, U = U 1 × U 2 × · · · U K (5) where L = Q K ν =1 L ν . When the set of L ν ’ s for ν = 1 , 2 , · · · , K are pairwise relatively prime , the gro up U will be cyclic. Using the mixed- radix no tation l = ( l 1 , l 2 , · · · , l K ) , where each 1 ≤ l ν ≤ L ν , we den ote every elemen t of U as D l = K Y ν =1 D l ν ν where D ν is a gen erator of the ν th cyclic group U ν giv en by D ν = diag  exp „ j 2 π u 1 ν L ν « , exp „ j 2 π u 2 ν L ν « · · · exp „ j 2 π u 2 Rν L ν «ff . (6) For each ν , u iν ∈ { 1 , 2 , · · · L ν − 1 } for i = 1 , 2 , · · · , 2 R have to b e chosen appro priately to ac hieve full-d iv ersity an d to maximise co ding gain . The i -th diagonal elem ent of D l is giv en by exp j 2 π K X ν =1 u iν l ν L ν ! . Hence, the source is eq uipped with a cod e {U , x } where x = 1 √ 2 R [1 1 1 · · · 1] T . B. At th e r elays Let M be the R × R GBH matr ix, u sing wh ich we con struct a 2 R × R matrix Γ as Γ =  M T M T  T . The colum ns o f Γ are used to co nstruct d iagonal u nitary matrice s for the relay s as A j = diag ( Γ 1 j , Γ 2 j , · · · Γ 2 Rj ) where Γ ij is the ( ij ) − th element of Γ a nd j = 1 , 2 , · · · , R . C. Encodin g Scheme The source node ma ps log 2 L bits of information o n to one of the L matrices from U say , D k for some k where 1 6 k 6 L and transmits the 2 R × 1 vector D k x to all th e relays. Each relay perfo rms linear processing on its received vector using the unitary matrix A j and transmits a 2 R le ngth vectors to the d estination. Th e distributed space-time codeword is of the form, S k =  A 1 D k x A 2 D k x · · · A R D k x  (8) = D k [ A 1 x A 2 x · · · A R x ] = 1 √ 2 R D k Γ where we ha ve used the fact that D k and A j commute for all j. It can be observed that S H k S k = I R . Definition 4: Th e collection C of 2 R × R codeword matrices shown below when k ru ns from 1 to L C =  1 √ 2 R D k Γ  (9) is called a Non-d ifferential Abelian DSTBC (N ADSTBC) and if the gr oup is cyclic, th en it is ca lled a N on-differential Cyclic Distributed Space-T ime Block Code (NCDSTBC). D. Design criteria fo r D ν In this subsection, we derive a criter ia for ch oosing the generato rs of eac h of U ν such that a N ADSTBC is fully div erse. 4 IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS , VOL. XX , NO. XX, XXXX V z 12 = R " exp j 2 π K X ν =1 l ν ( u z ν − u ( R + z ) ν ) L ν ! + exp j 2 π K X ν =1 ˆ l ν ( u z ν − u ( R + z ) ν ) L ν !# (7) Theor em 2: Th e diversity o rder of a wireless relay network with R relays using a N ADSTBC is R if and only if the gen- erators D ν in (6) are ch osen su ch th at, fo r ea ch i = 1 , 2 , · · · R, K X ν =1 ( l ν − ˆ l ν )( u iν − u ( R + i ) ν ) L ν / ∈ Z (10) for all p air ( l , ˆ l ) suc h that l 6 = ˆ l . Pr oof: Let 1 √ 2 R D l Γ , 1 √ 2 R D ˆ l Γ ∈ C . The 2 R × 2 R matrix as in ( 4) is given b y , S l ˆ l = 1 √ 2 R h D l Γ D ˆ l Γ i . Since | S H l ˆ l S l ˆ l | = | S l ˆ l S H l ˆ l | , we consider the choice o f D ν for ν = 1 , 2 , · · · K such that | S l ˆ l S H l ˆ l | 6 = 0 where S l ˆ l S H l ˆ l = 1 2 R  V 11 V 12 V 21 V 22  (11) with V 11 , V 12 , V 21 and V 22 are R × R d iagonal matrices g iv en by V 11 = 2 R I R × R ; V 21 = V H 12 ; V 22 = 2 R I R × R and fo r each z f rom 1 , 2 , · · · , R the z - th diago nal entry of V 12 is given in (7). Applying the re sult, ˛ ˛ ˛ ˛ » A B C D – ˛ ˛ ˛ ˛ = | A || D − CA − 1 B | where A , B , C an d D are square ma trices of same or der and A − 1 exists, on the matrix in (11) and using ( 7), we get | S ˆ ll S H ˆ ll | =  1 2 R  2 R | V 11 || V 22 − V 21 V − 1 11 V 12 | =  1 2 R  R | diag ( δ 1 δ 2 · · · δ R ) | where, δ i = R " 1 − cos 2 π K X ν =1 ( l ν − ˆ l ν )( u iν − u ( R + i ) ν ) L ν !# For | S l ˆ l S H l ˆ l | n ot to be z ero, the product δ 1 δ 2 · · · δ R must be a nonzer o value. For each δ i not to be zero, u iν and u ( i + R ) ν for all ν = 1 , 2 , · · · , K has to be chosen such that K X ν =1 ( l ν − ˆ l ν )( u iν − u ( R + i ) ν ) L ν / ∈ Z for all l, ˆ l su ch that l 6 = ˆ l. It is c lear that the above condition on D ν is bo th suffi cient and n ecessary . Cor ollary 1: DSTBCs from non-cyclic abelian grou ps are not fully d i verse for any choice of D ν . Pr oof: Since e ach of the cyclic grou ps U ν ⊆ U , from the special case of Theorem 2 with K = 1 , ev ery D ν must satisfy the criteria, gcd  u ν i − u ν ( i + R ) , L ν  = 1 fo r i = 1 , · · · R . Since at least two of the L ν share a factor and ea ch D ν satisfy the above criteria, co ndition in (10) o f Theorem 2 will not be satisfied for a t least one pair of values l an d ˆ l . Hence, non-cyclic abelian DSTBCs cannot provide full di versity when used in th e propo sed scheme. Hencefor th, thro ugho ut the paper, we will consider on ly NCDSTBCs. W e explicitly state Th eorem 2 fo r NCDSTBC as a co rollary as: Cor ollary 2: Let U in (5) be a cyclic group with a generato r D given b y D = diag n exp  j 2 π u 1 L  , exp  j 2 π u 2 L  · · · exp  j 2 π u 2 R L o . A necessary and sufficient condition on the choic e of D such the that diversity o rder of the wireless relay network is R , is giv en by gcd( u i − u R + i , L ) = 1 for each i = 1 , 2 , · · · R. Pr oof: The resu lt follows as a special ca se of Theorem 2 for K = 1. Specializing th e proof of Theo rem 2 f or the cyclic group case, the co ding gain for NCDSTBC is giv en by φ = min 1 6 l, ˆ l 6 L,l 6 = ˆ l R Y i =1 " 1 − cos 2 π ( l − ˆ l )( u i − u i + R ) L !# . (12) Thus, the vector v has to be chosen such that φ is ma ximized, where v = ( u 1 − u R +1 , u 2 − u R +2 · · · u R − u 2 R ) . (13) From the resu lt of Th eorem 2 and Corollary 2, th e p erform ance of NCDSTBC is cha racterized by (12) which is indep endent of the unitary matrices A j at the r elays and dep endent only on the gen erator of the c yclic gro up. E. On the choice of Γ Instead of a GBH matrix , M can also be any R × R unitary matrix with no z ero entry , using which Γ is constructed as Γ = √ R  M M  . Thoug h, th e relay matrices A j are no t unitary in this case, simulation results s how t hat, for a gi ven U at the source, BLER (Block error rate, which corresp onds to err ors in decod ing a codeword is considered as err or events of o ur interest throug hout the paper) performance is independen t of the ab ove choice of M . This can be seen fr om Figure 2 which is ob tained using the code parameters L = 16 and v = [5 , 7 , 11 , 1] for a network with 4 relays. Ho we ver , deco ding metric in (3) doesn’t HARSHAN and RAJ AN: A NON-DIFFERENTIAL DISTRIBUTED SP ACE-TIME CODING FOR P ARTIALL Y -COHER ENT COOPERA TIVE COM MUNICA TION 5 20 21 22 23 24 25 26 27 28 10 −4 10 −3 10 −2 10 −1 P in db BLER With GBH matrices With unitary matrices with no zero entry Fig. 2. Plot illustrating the independence of the BLER on rela y specific diagona l matrices hold in th e latter case. Ther efore, we contin ue to use only GB H matrices to co nstruct A j in the rest of the pape r . I V . R E D U C E D M L D E C O D I N G C O M P L E X I T Y F O R N C D S T B C S A N D N O D E FA I L U R E S For a genera l class of unitary DSTBCs used in a pa rtially- coheren t scheme, an ML decod ing metric is given in ( 3). In this section , we specialize this deco ding metric f or the class of NCDSTBCs. Theor em 3: For a NCDSTBC in (9), the partially- coheren t ML deco ding giv en in (3) r educes to arg max k ∈ 1 , 2 , ··· L  y H 1 G 1 y 1 + Re  y H 1 G 2 y 2  + y H 2 G 3 y 2  . (14) where, y 1 = [ y 1 y 2 · · · y R ] T , y 2 = [ y R +1 y R +2 · · · y 2 R ] T and G 1 = Υ 1 ⊙ Ω , G 2 = Υ 2 ⊙ Ω and G 3 = Υ 3 ⊙ Ω where ⊙ denotes the Hadamard prod uct and Υ 1 , Υ 2 , Υ 3 ∈ C R × R giv en by [ Υ 1 ] i,j = w ( u i − u j ) k , [ Υ 2 ] i,j = w ( u i − u R + j ) k , [ Υ 3 ] i,j = w ( u R + i − u R + j ) k and Ω is the R × R Her mitian matrix, with the ( i, j ) − th element given by Ω i,j = R X λ =1 β λ  m iλ m ∗ j λ  for i, j = 1 , · · · , R (15) where m ij are the en tries of M , the chosen GBH m atrix and β j = | g j | 2  | g j | 2 + γ ρ − 1  − 1 for j = 1 , · · · , R. Pr oof: The result f ollows b y substituting ( 8) in (3). In th e rest of this section, we present various choices on the vector v in (13) su ch that the compu tations r equired in decodin g using the metric g iv en by (1 4) is reduce d. For large value of P , performanc e of the scheme is determined by the choice of the vector v . There are several ways o f choosing the vector v from various c hoices of u = [ u 1 , u 2 · · · u 2 R ] . The optimal choice of v can still be made with appropriate values of u 1 , u 2 · · · u R by keeping u R +1 = u R +2 = u R +3 = · · · = u 2 R . 20 21 22 23 24 25 26 27 28 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 P in db BLER [5, 11, 7, 1] [ 5, 5, 5, 5] [11, 7,3,1] [13, 11, 5, 5] Fig. 3. BLER plot for 4 relays w ith L = 16 for dif ferent v From the above cho ice of v , Υ 3 will be indep endent of the exponent k . Ther efore the ML decod ing metric in (1 4) reduces to arg max k ∈ 1 , 2 , ··· L  y † 1 G 1 y 1 + Re n y † 1 G 2 y 2 o . (16) Hence, compared to (14 ), the number of co mputatio ns requ ired in (16) to d ecode a codew ord is reduced. The vector v = ( v 1 , v 2 , · · · , v R ) can also b e chosen to b e of the f orm v 1 = v 2 = · · · = v R , by choo sing u 1 = u 2 = · · · = u R and u R +1 = u R +2 = · · · = u 2 R such tha t gcd ( u 1 − u R +1 , L ) = 1 . In such a case, Υ 1 and Υ 3 are independe nt of the exponent k and hence the decodin g m etric is written as arg max k ∈ 1 , 2 , ··· L Re n w ( u 1 − u 3 ) k y H 1 Ω y 2 o . (17) In genera l, in o rder to estimate the mo st likely tr ansmitted codeword from th e source, th e destination needs to perform matrix multiplica tion o f order R × R accordin g to (14) for ev ery k from 1 to L , where as, in th e case of the m etric giv en in (17), the destination needs to perfor m complex number multiplication for every k to estimate the mo st likely transmitted codeword. Hence, the number of computatio ns fo r decodin g a cod ew ord at th e de stination red uces substantially . Figure 3 shows the BLER perf ormanc e ag ainst th e to tal power , P for different ch oices of gener ator D with L = 16 fo r a network with 4 r elays. The p lot shows that, fo r a choice of v with v 1 = v 2 = · · · = v R , the NCDSTC loses out close to 5-6 db comp ared to the o ne with the choice of v = [11 , 7 , 3 , 1] . The relay matrices used in all the simulation s are A 1 = I 8 , A 2 = d iag { 1 , − 1 , 1 , − 1 , 1 , − 1 , 1 , − 1 } , A 3 = diag { 1 , 1 , − 1 , − 1 , 1 , 1 , − 1 , − 1 } , and A 4 = diag { 1 , − 1 , − 1 , 1 , 1 , − 1 , − 1 , 1 } . The decodin g metric giv en in ( 3) h as been used to carr y o ut the simulations. In all our simulations, the total power P h as been distributed to the source no de and the relay no des as P 1 = P 2 and P 2 = P 2 R . A. F a ilur e of r elay no des In this subsection, we ana lyze the behavior of the propo sed codes when a subset o f the r elay nodes fail to co-o perate with 6 IEEE TRANSACTIONS ON WIRELE SS COMMUNICA TIONS , VOL. XX , NO. XX, XXXX 20 22 24 26 28 30 32 10 −4 10 −3 10 −2 10 −1 10 0 P in db BLER single relay failure No relay failure Fig. 4. BLER performance of NCDSTC with one relay do wn the sour ce i.e when a subset of the relays do not transmit informa tion of the sour ce to the destinatio n. Th e structu re of ev ery codeword S k from th e NCDSTC C is g iv en in Defin ition 4. Similar to code s designed for non-differential non -cohere nt collocated MIMO systems, the columns of a codeword S k can be viewed as a basis of an R -dimension al subspa ce of C 2 R . T wo co dewords satisfy the full diversity co ndition means that, the R -dimen sional subspace sp anned by column s of each of th em are n on-inter secting. Therefore, a fully diverse NCDSTC can be viewed as a finite set of non-intersectin g R - dimensiona l subspaces in C 2 R . Fro m the structur e of the cod e, it is easy to see that, ev ery relay co ntributes a basis elemen t to each of the sub space (cod e-word). Let ˆ S k denote a codeword, after say , d o f th e R relays fail to co-operate with the source. The columns of a new codeword ˆ S k span a ( R − d ) dim ensional subspace of R -dimensio nal spac e spann ed by the co lumns of S k . Hence, the ne w set of subsp aces ( codewords) remains to be n on-inter secting and the new code ˆ C will rem ain to b e fu lly div erse with diversity g ain R − d. Figure 4 shows simulation results on the beha vior of BLER when one of the relay fails in a network with 4 relays. The p lot is obtained b y u sing a code with par ameters L = 16 and v = [11 , 11 , 11 , 11 ] . V . D I S C U S S I O N W e con sidered th e problem of code co nstruction based on n on-differential tech nique for wireless re lay networks in a p artially-coh erent en viron ment. Unitar y distributed spa ce time block cod es were co nstructed using a cyclic g roup of diagona l matrices at the source and diag onal relay m atrices constructed fro m GBH m atrices. W e h av e shown that th e BLER per forman ce o f the p roposed sch eme is ind epende nt of the relay specific unitary matrices. Further, a ne cessary and sufficient con dition on the ch oice of the generato r o f cyclic group is pr ovided f or achieving f ull diversity and to m inimize the PEP . V arious choices o n the generator of cyclic group to reduce the ML decodin g com plexity at th e d estination was provided. The resistance of the p roposed sch eme to th e failure of a subset of the relay nodes is also verified. It w as also sho wn that, using non-cyclic abelian grou p of diagon al matrices at the sou rce d oesn’t provide f ull diversity . An interesting direction for fu ture work is to de sign non-d ifferential enco ding technique s such that the un itary distributed space time block codes can be constructed algebraically . R E F E R E N C E S [1] A. Sendonaris, E. Erkip and B. Aazhang, ”User coope ration di versi ty-Part 1: Systems description, ” IE EE T rans. comm. , vol. 51, pp, 1927- 1938, Nov 2003. [2] A. Sendonaris, E . Erkip and B. Aazhang, ”User co operation di ve rsity- Part 1: implementation aspect s and performance ana lysis, ” IEEE T ran s. inform theory . , vo l. 51, pp. 1939-1948, Nov 2003. [3] J. M. Laneman and G. W . W ornell, ”Distribute d space time coded protocol s for exploit ing cooperati ve di versit y in wireless netwo rk” IEEE T ra ns. Inform. Theory . , vol. 49, pp. 2415-2425, Oct. 2003. 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