Single-Symbol ML Decodable Precoded DSTBCs for Cooperative Networks

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1 Sinle-Symbol ML Decodable Precoded DSBCs for Cooperative Networks Harsan J Dept of ECE, Indian Institute of science Banalore 56001, India arsan@eceiiscernetin B Sundar Rajan Dept of ECE, Indian Institute of science Banalore 56001, India bsrajan@eceiiscernetin Abstract Sinle-Symbol Maximum Likeliood (ML) decodable Distributed Ortoonal Space-ime Block Codes (DOS- BCs) ave been introduced recently for cooperative networks and an upper-bound on te maximal rate of suc codes alon wit code constructions as been presented In tis paper, we introduce a new class of Distributed Space-ime Block Codes (DSBCs) called Semi-ortoonal Precoded Distributed Sinle-Symbol Decodable Space-ime Block Codes (Semi-SSD-PDSBCs) werein, te source performs precodin on te information symbols before transmittin it to all te relays A set of necessary and sufficient conditions on te relay matrices for te existence of Semi-SSD- PDSBCs is proved It is sown tat te DOSBCs are a special case of Semi-SSD-PDSBCs A subset of Semi-SSD-PDSBCs avin diaonal covariance matrix at te destination is studied and an upper bound on te maximal rate of suc codes is derived e bounds obtained are approximately twice larer tan tat of te DOSBCs A systematic construction of Semi- SSD-PDSBCs is presented wen te number of relays K 4 and te constructed codes are sown to ave ier rates tan tat of DOSBCs I INRODUCION AND PRELIMINARIES Cooperative communication as been a promisin means of acievin spatial diversity witout te need of multiple antennas at te individual nodes in a wireless network e idea is based on te relay cannel model, were a set of distributed antennas belonin to multiple users in te network co-operate to encode te sinal transmitted from te source and forward it to te destination so tat te required diversity order is acieved, 1-4 Spatial diversity obtained from suc a co-operation is referred to as co-operative diversity In 5, te idea of space-time codin devised for point to point colocated multiple antenna systems is applied for a wireless relay network and is referred to as distributed space-time codin e tecnique involves a two pase protocol were, in te first pase, te source broadcasts te information to te relays and in te second pase, te relays linearly process te sinals received from te source and forward tem to te destination suc tat te sinal at te destination appears as a space-time block code Recently, in 6, Distributed Ortoonal Space-ime Codes (DOSBCs) acievin sinle-symbol decodability ave been introduced for co-operative networks e autors considered a special class of DOSBCs wic make te covariance matrix of te additive noise vector at te destination, a diaonal one and suc a class of codes was referred to as row monomial DOSBCs Upper-bounds on te maximum symbol-rate (in complex symbols per cannel use in te second pase) of row monomial DOSBCs ave been derived and a systematic construction of suc codes as been proposed e constructed codes were sown to meet te upper-bound for even number of relays In 8, te same autors ave derived an upper-bound on te symbol-rate of DOSBCs wen te additive noise at te destination is correlated and it is sown tat te improvement in te rate is not sinificant wen compared to te case wen te noise at te destination is uncorrelated 6 In 7 and 8, Sinle-Symbol Decodable (SSD) Distributed Space-ime Block Codes (DSBCs) ave been studied wen te relay nodes are assumed to know te correspondin cannel pase information An upper bound on te symbol rate for suc a set up is sown to be 1 wic is independent of te number of relays In 6, 7 and 8 te source node transmits te information symbols to all te relays witout any processin Usin te framework proposed in 6, in tis paper, we propose SSD DSBCs aided by linear precodin of te information vector at te source In our set-up, we assume tat te relay nodes do not ave te knowlede of te cannel from te source to itself In particular, it is sown tat, co-ordinate interleavin of information symbols at te source alon wit te appropriate coice of relay matrices, SSD DSBCs wit maximal rates ier tan tat of DOSBCs can be constructed e contributions of tis paper can be summarized as follows: A new class of sinle symbol ML decodable DSBCs called Precoded Distributed Sinle Symbol Decodable SBCs (SSD-PDSBCs) (Definition ) ave been introduced and a set of necessary and sufficient conditions on te relay matrices for te existence of SSD-PDSBCs is proved (Lemma 1) Witin te set of SSD-PDSBCs, a subset called Semiortoonal SSD-PDSBCs (Semi-SSD-PDSBC) (Definition 4) is defined e known DOSBCs are sown to belon to te class of Semi-SSD-PDSBCs On te similar lines of 6, a special class of Semi-SSD-PDSBCs avin a diaonal covariance matrix at te destination is studied and are referred to as row monomial Semi-SSD- PDSBCs An upper bound on te maximal symbol-rate of row /08/$ IEEE

2 monomial Semi-SSD-PDSBCs is derived It is sown tat, te symbol rate of row monomial Semi-SSD- PDSBC is upper-bounded by l and l+1, wen te number of relays, K is of te form l and l +1 respectively, were l is any natural number e bounds obtained are approximately twice larer tan tat of DOSBCs A systematic construction of row-monomial Semi-SSD- PDSBCs is presented wen K 4 Codes acievin te upper-bound on te symbol rate are constructed wen K is 0 or 3 modulo 4 For te rest of te values of K, te constructed Semi-SSD-PDSBCs are sown to ave rates ier tan tat of te DOSBCs e remainin part of te paper is oranized as follows: In Section II, alon wit te sinal model, PDSBCs are introduced and a special class of it called SSD-PDSBCs is defined A set of necessary and sufficient conditions on te relay matrices for te existence of SSD-PDSBCs is also derived In Section III, Semi-SSD-PDSBCs are defined and te properties of te relay matrices of row-monomial Semi-SSD-PDSBCs are studied An upper bound on te maximal rate of row-monomial Semi-SSD-PDSBCs is derived In Section IV, construction of row-monomial Semi- SSD-PDSBCs is presented alon wit some examples e problem of desinin two-dimensional sinal sets for te full diversity of Semi-SSD-PDSBCs is discussed in Section V alon wit some simulation results Concludin remarks and possible directions for furter work constitute Section VI Notations: rou out te paper, boldface letters and capital boldface letters are used to represent vectors and matrices respectively For a complex matrix X, te matrices X, X, X H, X, ReX and Im X denote, respectively, te conjuate, transpose, conjuate transpose, determinant, real part and imainary part of X e element in te r1 t row and te r t column of te matrix X is denoted by X r1,r e identity matrix and te zero matrix respectively denoted by I and 0 e manitude of a complex number x, is denoted by x A circularly symmetric complex Gaussian random vector, x wit mean µ and covariance matrix Γ is denoted by x CSCG(µ, Γ) e set of all inteers, te real numbers and te complex numbers are respectively, denoted by Z, R and C and j is used to represent 1 e set of all complex diaonal matrices is denoted by D and a subset of D wit strictly positive diaonal elements is denoted by D + Due to space considerations, te proofs of all te Lemmas and eorems in tis paper ave been omitted ese can be found in 9 alon wit several examples and oter details II PRECODED DISRIBUED SPACE-IME CODING A Sinal model e wireless network considered as sown in Fiure 1 consists of K + nodes eac avin sinle antenna ere is one source node and one destination node All te oter K Source Fi K 1 K 1 K Relays K Wireless relay network Destination nodes are relays We denote te cannel from te source node to te k-t relay as k and te cannel from te k-t relay to te destination node as k for k =1,,,K e followin assumptions are made in our model: All te nodes are subjected to alf duplex constraint Fadin coefficients k, k are iid CSCG (0, 1) wit coerence time interval of at least N and respectively All te nodes are syncronized at te symbol level Relay nodes do not ave te knowlede of fade coefficients k Destination knows te fade coefficients k, k e source is equipped wit a N lent complex vector from te codebook S = {s 1, s, s 3,, s L } consistin of information vectors s l C 1 N suc tat E s l s H l =1for all l =1,,L e source is also equipped wit a pair of N N matrices P and Q called precodin matrices Every transmission from te source to te destination comprises of two pases Wen te source needs to transmit an information vector s Sto te destination, it enerates a new vector s as, s = sp + s Q (1) suc tat E s s H = 1 and broadcasts te vector s to all te K relays (but not to te destination) e received vector at te k-t relay is iven by r k = P 1 N k s + n k, k =1,,,K were n k CSCG(0, I N ) is te additive noise at te k- t relay and P 1 is te total power used at te source node every cannel use In te second pase, all te relay nodes are sceduled to transmit lent vectors to te destination simultaneously Eac relay is equipped wit a fixed pair of N rectanular matrices A k, B k and is allowed to linearly process te received vector e k-t relay is sceduled to transmit P t k = (1 + P 1 )N {r ka k + r kb k } () were P is te power used at eac relay for every cannel use in te second pase e vector received at te destination is iven by K y = k t k + w (3) k=1

3 were w CSCG(0, I ) is te additive noise at te destination Usin () in (3), y can be written as P 1 P y = (1 + P 1 )N X + n were n = P K (1+P 1)N k=1 k {n k A k + n k B k} + w e equivalent cannel is iven by 1 K C 1 K Every codeword X C K is as iven in (4) (at te top of next pae) Definition 1: e collection C of K codeword matrices sown in (5) (at te top of next pae) were s runs over a codebook S, is called te Precoded Distributed Space- ime Block code (PDSBC) wic is determined by te set {P, Q, A k, B k } e covariance matrix R C of te noise vector n is iven by P K { } R = k A H k A k + B H k B k + I (6) (1 + P 1)N k=1 e Maximum Likeliood (ML) decoder decodes to a vector ŝ were = ar min s S Re ( P 1P (1 + P 1)N XR 1 y H ) + P 1P (1 + P 1)N XR 1 X H H Wit te above decodin metric, we ive a definition for a SSD distributed space-time block code wic also includes DOSBCs studied in 6 Definition : A PDSBC, X in variables x 1,x, x N is called a Precoded Distributed Sinle-Symbol Decodable SBC (SSD-PDSBC), if it satisfies te followin conditions, e entries of te k-t row of X are 0, ± k x n, ± k x n or multiples of tese by j for any complex variable k e complex variables x n for 1 n N are te components of te transmitted vector s were s = x 1 x x N e matrix X satisfies te equality XR 1 X H = N W i (7) ( ) wit W i k,k = k υ (1) i,k x ii + υ () i,k x iq were eac W i is a K K matrix wic is not necessarily diaonal and its non zero entries are functions of only x ii, x iq (but not x ji, x jq, for all j i) and k for all k =1,,,K and υ (1) i,k,υ() i,k R We study te properties of te relay matrices A k, B k and te precodin matrices P and Q suc tat te vectors transmitted simultaneously from all te relays appear as a SSD-PDSBC at te destination owards tat end, a matrix is said to be column (row) monomial, if tere is at most one non-zero entry in every column (row) of it 6 In te followin Lemma, we provide a set of necessary and sufficient conditions on te i=1 Fi Non Unitary DSSDC PCIOD Unitary DSSDC DSSDC DOSBC Semi SSD PDSBC SSD PDSBC Various class of SSD codes for cooperative networks matrix set {P, Q, A k, B k } suc tat a PDSBC X wit te above matrix set is a SSD-PDSBC Lemma 1: A PDSBC X is a SSD-PDSBC if and only if te relay matrices A k, B k satisfy te followin conditions, (i) For 1 k k K, Υ 1 A k R 1 A H k ΥH + Π 1A k R 1 A k Π D N (8) Υ 1 = Υ = P and Π 1 = Π = Q; for Π = Υ 1 = P and Π 1 = Υ = Q; Π 1 = Υ = P and Π = Υ 1 = Q; Υ 1B k R 1 B H k Υ + Π 1 B k R 1 B k ΠH D N (9) Υ 1 = Υ = Q and Π 1 = Π = P; for Π = Υ 1 = Q and Π 1 = Υ = P; Π 1 = Υ = Q and Π = Υ 1 = P (ii) For 1 k, k K, Π B k R 1 A H k + A k R 1 B k Υ H D N, for Π A k R 1 B H k + B k R 1 A k Υ D N, for (iii) For 1 k K, Υ = P and Π = Q; Υ = P and Π = P; Υ = Q and Π = Q; (10) Υ = Q and Π = P; Υ = P and Π = P; Υ = Q and Π = Q (11) A k R 1 A H k + B k R 1 B k = dia D 1,k,D,k,,D N,k (1) were D n,k R for all n =1,, N It can be observed tat te necessary and sufficient conditions on te relay matrices of DOSBCs as sown in Lemma 1 of 6 can be obtained from te necessary and sufficient conditions of SSD-PDSBCs by makin P = I N, Q = 0 N and D N = 0 N in (8) - (11) Hence, DOSBCs are a special case of SSD- PDSBCs A SSD-PDSBC, X in variables x 1,x, x N can be written in te form of a linear dispersion code as X = N j=1 x iiφ ii + x iq Φ iq, were Φ ii, Φ iq C K are called te weit matrices of X Witin te class of SSD-PDSBCs, we consider a special set of codes called Unitary SSD- PDSBCs defined as, Definition 3: A SSD-PDSBC, X is called a Unitary SSD- PDSBC, if te weit matrices of X satisfies te followin conditions, Φ ii Φ H ii, Φ iqφ H iq D+ K for all i =1,, N

4 X = 1 sa s B 1 sa + s B K sa K + K s B K (4) C = { 1 sa s B 1 sa + s B K sa K + K s B K } (5) Various class of sinle-symbol decodable SBCs for cooperative networks are captured in Fiure wic is first partitioned in to two sets dependin on weter te codes are unitary or non-unitary (Definition 3) e set of unitary distributed SSD codes are sown to contain te DOSBCs and te Semi-SSD-PDSBCs (Definition 4) More details on te classification of SSD codes can be found in 9 III SEMI-ORHOGONAL SSD-PDSBC From te definition of a SSD-PDSBC (Definition ), XR 1 X H for any k k can be non-zero ie, te k- t and te k -t row of a SSD-PDSBC X, need not satisfy te equality XR 1 X H = 0,but XR 1 X H must be a complex linear combination of several terms wit eac term bein a function of in-pase and quadrature component of a sinle information variable rou out te paper, te k-t and te k -t row of a SSD-PDSBC are referred to as R-ortoonal if XR 1 X H = 0 Similarly, te k- t and te k -trowarereferredtoasr-non-ortoonal if XR 1 X H 0 In tis paper, we identify a special class of SSD-PDSBCs were every row of X is R-non-ortoonal to at most one of its rows and we formally define it as, Definition 4: A SSD-PDSBC is said to be a Semiortoonal SSD-PDSBC (Semi-SSD-PDSBC) if every row of a SSD-PDSBC is R-non-ortoonal to at most one of its rows From te above definition, it can be observed tat DOSBCs are a proper subclass of Semi-SSD-PDSBCs since every row of a DOSBC is R-ortoonal to every oter row e definition of a Semi-SSD-PDSBC implies tat te set of K rows can be partitioned in to at least K roups suc tat every roup as at most two rows e co-variance matrix, R in (6) is a function of (i) te realisation of te cannels from te relays to te destination and (ii) te relay matrices, A k, B k In eneral, R may not be diaonal in wic case te construction of Semi-SSD- PDSBCs is not strait forward On te similar lines of 6, we consider a subset of Semi-SSD-PDSBCs wose covariance matrix is diaonal and refer to suc a subset as row monomial Semi-SSD-PDSBCs (Semi-SSD-PDSBCs) In te rest of tis paper, we consider only row monomial Semi- SSD-PDSBC Hencefort, we continue to refer tem simply as Semi-SSD-PDSBC A Upper-bound on te symbol-rate of Semi-SSD-PDSBCs In tis subsection, we derive an upper-bound on te rate of Semi-SSD-PDSBCs in symbols per cannel use in te second pase ie an upper-bound on N owards tat end, properties of te relay matrices A k, A k, B k and B k of Semi- SSD-PDSBC are studied wen te rows correspondin to te indices k and k are (i) R-ortoonal and (ii) R-nonortoonal For te former case, te relay matrices A k, A k, B k and B k satisfies te followin conditions 6, A k A H k = 0 N and B kb k = 0 N (13) For te latter case, te properties of te relay matrices are as iven in te followin Lemma Lemma : Let k and k represent te indices of te rows of a Semi-SSD-PDSBC, tat are R-non-ortoonal, ten te correspondin relay matrices A k, B k, A k and B k satisfy te followin conditions, A k A H k = B kb k =0for all i =1, N i,i i,i A k A H k and B kb k are bot column and row monomial matrices A k A H k + B kb k is column and row monomial matrix e number of non-zero entries in A k A H k + B kb k is even e matrices à and B iven by à = A k A k and B = B k B k satisfy te followin inequality : Rank à ÃH + B B Rate = N { m if N =m and m + if N =m +1 (14) were m is a positive inteer Usin te properties of relay matrices A k, A k, B k and B k, N an upper-bound on te maximum rate, is derived in te followin teorem eorem 1: e symbol-rate of a Semi-SSD-PDSBC satisfies te inequality : if N =m, K =l l l+1 if N =m, K =l +1 m+1 (m+1)l if N =m +1,K =l 4m+ (m+)l+m+1 if N =m +1,K =l +1 (15) were l and m are positive inteers IV CONSRUCION OF SEMI-SSD-PDSBCS Due to space considerations, description of te construction of Semi-SSD-PDSBCs as been omitted Details reardin te same can be found in 9 However, an example for Semi- SSD-PDSBC wen N =4and K =4is iven in (17) Example 1: X (4, 4) = 1 x 1 1 x 1 x 3 1 x 4 x 3 x 3 x 1 3 x 4 x 4 3 x 1 x 3 3 x 4 x 4 4 x 3 4 x 4 x 1 (17) were, x 1 = x 1I + jx 4Q ; x = x I + jx 3Q ; x 3 = x 1Q + jx 4I ; and x 4 = x Q + jx 3I

5 X (4, 4) = 1x 1 1x 1x 3 1x x x 1 x 4 x x 1 3x 3x 3 3x x 4x 1 4x 4 4x 3 (16) SER DOSBC S PDSSD SNR in db Fi 3 Performance comparison of Semi-SSD-PDSBC and DOSBC for N = 4 and K = 4 wit bps/hz V ON HE FULL DIVERSIY OF SEMI-SSD-PDSBCS In tis section, we consider te problem of desinin a twodimensional sinal set, Λ suc tat a Semi-SSD-PDSBC wit variables x 1, x, x N takin values from Λ is fully diverse Since every codeword of a Semi-SSD-PDSBC (Definition ) contains complex variables k s, PEP analysis of Semi-SSD- PDSBCs seems difficult e autors do not ave conditions on te coice of a complex sinal set suc tat a Semi-SSD- PDSBC is fully diverse However, we make te followin conjecture Conjecture : A Semi-SSD-PDSBC in variables x 1, x, x N is fully diverse if te variables takes values from a complex sinal set say, Λ suc tat te difference sinal set Λ iven by Λ = {a b a, b Λ} does not ave any point on te lines tat are ± 45 derees in te complex plane apart from te oriin We provide simulation results on te performance comparison of a Semi-SSD-PDSBC, X (4, 4) (iven in (17)) and a row-monomial DOSBC, X (4, 4) (iven in (16)) in terms of Symbol Error Rate (SER) (SER corresponds to errors in decodin a sinle complex variable) e SER comparison is provided in Fiure 3 It is to be noted tat te symbol rate of X (4, 4) is twice tat of X (4, 4) e class of DOSBCs are sown to be fully diverse in 6 From Fiure 3, it is observed tat X (4, 4) provides full diversity, since te SER curve moves parallel to tat of X (4, 4) It can be noticed from Fiure 3 tat te desin X (4, 4) performs better tan X (4, 4) by close to -3 db VI CONCLUSION AND DISCUSSION We considered te problem of desinin i rate, sinlesymbol decodable DSBCs wen te source is allowed to perform co-ordinate interleavin of information symbols before transmittin it to all te relays A special class of SSD- PDSBCs avin semi-ortoonal property and row monomial property were studied and an upper bound on te maximal rate of suc codes is presented alon wit code constructions In tis paper, we studied a special class of SSD-PDSBCs called Unitary SSD-PDSBCs (See Definition 3) e desin of i rate Non-Unitary SSD-PDSBCs is an interestin direction for future work ACKNOWLEDGMEN is work was partly supported by te DRDO-IISc Proram on Advanced Researc in Matematical Enineerin, partly by te Council of Scientific & Industrial Researc (CSIR), India, trou Researc Grant ((0365)/04/EMR-II) to BS Rajan REFERENCES 1 A Sendonaris, E Erkip and B Aazan, User cooperation diversity-part 1: Systems description, IEEE rans comm, vol 51, pp, , Nov 003 A Sendonaris, E Erkip and B Aazan, User cooperation diversity- Part 1: implementation aspects and performance analysis, IEEE rans inform teory, vol 51, pp , Nov J M Laneman and G W Wornell, Distributed space time coded protocols for exploitin cooperative diversity in wireless network IEEE rans Inform eory, vol 49, pp , Oct R U Nabar, H Bolcskei and F W Kneubuler, Fadin relay cannels: performance limits and space time sinal desin, IEEE Journal on Selected Areas in Communication, vol, no 6, pp , Au Yindi Jin and Babak Hassibi, Distributed space time codin in wireless relay networks IEEE rans Wireless communication, vol 5, No 1, pp , December Zian Yi and Il-Min Kim, Sinle-Symbol ML decodable Distributed SBCs for Cooperative Networks, IEEE rans Information teory, vol 53, No 8, pp 977 to 985, Auust D Sreedar, A Cockalinam and B Sundar Rajan, Sinle-Symbol ML decodable Distributed SBCs for Partially-Coerent Cooperative Networks, submitted to IEEE rans Information teory, Auust 007 Available online at ttp://arxivor/abs/ Zian Yi and Il-Min Kim, e impact of Noise Correlation and Cannel Pase Information on te Data-Rate of te Sinle-Symbol ML Decodable Distributed SBCs, Submitted to IEEE rans Information teory, Au 007 Available online at ttp://arxivor/abs/ Harsan J and B Sundar Rajan, Hi Rate Sinle-Symbol Decodable Precoded DSBCs for Cooperative Networks, a tecnical report of DRDO-IISc Proramme on Advanced Researc in Matematical Enineerin, Report No - R-PME , Au 007 Also available online in ttp://arxivor/abs/ Zafar Ali Kan, Md, and B Sundar Rajan, Sinle Symbol Maximum Likeliood Decodable Linear SBCs, IEEE rans on Infoeory, vol 5, No 5, pp06-091, May Sanjay Karmakar and B Sundar Rajan, Minimum-decodin-complexity maximum-rate space-time block codes from Clifford alebras, in te proceedins of IEEE ISI, Seattle, USA, July 09-14, 006, pp S Yiu, R Scober and LLampe, Distributed space-time block codin Proceedins of IEEE Globecom, pp November 005