Low Complexity Distributed STBCs with Unitary Relay Matrices for Any Number of Relays

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1 Low Complexity Distributed STBCs with Unitary elay Matrices for Any Number of elays G. Susinder ajan Atheros India LLC Chennai India B. Sundar ajan ECE Department Indian Institute of Science Bangalore India Abstract Jing and Hassibi introduced a distributed space time block coding scheme for symbol synchronous coherent amplify and forward relay networks with half duplex constrained relay nodes. In this two phase transmission scheme the source transmits a vector of complex symbols to the relays during the first phase and each relay applies a pre-assigned unitary transformation to the received vector or its conjugate before transmitting it to the destination during the second phase. The destination then perceives a certain structured distributed space time block code DSTBC whose maximum likelihood ML decoding complexity in general is very high. In this paper explicit constructions of minimum delay full diversity four group ML decodable DSTBCs with unitary relay matrices are provided for even number of relay nodes. Prior constructions of DSTBCs with the same features were either limited to power of two number of relay nodes or had non-unitary relay matrices which leads to large peak to average power ratio of the relay s transmitted signals. For the case of odd number of relays constructions of minimum delay full diversity two group ML decodable DSTBCs are given. I. INTODUCTION After more than a decade of research and experimentation space time block coding has established itself as a good coding technique for point to point multiple input multiple output MIMO systems in theory and also in practice with its inclusion in several standards such as 80.11n WLAN and 80.16e WiMaX. Cooperative communication and in particular coding for relay networks has received significant attraction in the past few years. There have been several works recently on distributed space time block coding for relay networks In this paper we are interested in a specific class of relay networks which consist of a single source node a single destination node and multiple relay nodes for aiding communication between the source and the destination. In particular we consider the distributed space time block coding scheme introduced by Jing and Hassibi 3 for symbol synchronous coherent amplify and forward relay networks. In this two phase transmission scheme the source transmits a vector of complex symbols to the relays during the first phase and each relay applies a pre-assigned unitary transformation to the received vector or its conjugate before transmitting it to the destination during the second phase. This effectively emulates the transmission of a STBC from collocated antennas. It is important to note that the Jing and Hassibi transmission scheme 3 does not permit the use of an arbitrary STBC that was designed for point to point MIMO systems but rather constrains the STBC to be of a specific structure. This important distinction between STBCs and distributed STBCs DSTBCs calls for a separate study of DSTBCs. Following the work of Jing and Hassibi 3 few works 11 1 have addressed the construction of DSTBCs that achieve full cooperative diversity. However the maximum likelihood ML decoding complexity of these DSTBCs except the and 4 4 DSTBC of 11 were prohibitively high. ecognizing this important problem Kiran et al in 5 constructed full diversity two group ML decodable DSTBCs which admit the real symbols in the code matrix to be split into two groups of equal cardinality such that ML decoding can be done separately for the two groups of symbols. In the application of real orthogonal designs as full diversity single symbol ML decodable DSTBCs for real modulations such as pulse amplitude modulation PAM signal sets was discussed in detail. In 6 full diversity four group ML decodable DSTBCs for even number of relays were constructed using precoded coordinate interleaved orthogonal designs PCIODs a generalization of the coordinate interleaved orthogonal design CIOD 14. For odd number of relays it was proposed to drop one column of a PCIOD for even number of relays. However PCIODs required the use of non-unitary matrices at the relays which increased the peak to average power PAP of the transmitted signals from the relays. Moreover the use of non-unitary relay matrices forces the destination to perform additional processing to whiten the noise seen by it during ML decoding. To solve this problem extended Clifford algebras were used in 8 9 to construct full diversity four group ML decodable DSTBCs with unitary relay matrices. But the constructions in 8 9 were limited to power of two number of relays. Though these constructions can be used for arbitrary number of relays by column dropping this solution entails a significant increase in delay and ML decoding complexity. The contributions of this paper can be summarized as follows: Explicit construction of minimum delay four group ML decodable DSTBCs with unitary relay matrices that can achieve full cooperative diversity in symbol synchronous coherent amplify and forward relay networks with even number of relay nodes. Such DSTBC constructions are

2 available in the literature only for power of two number of relay nodes 8 9. By dropping one column a nonminimal delay full diversity four group ML decodable DSTBC with unitary relay matrices is obtained for odd number of relays. The proposed DSTBCs are obtained by multiplying a permutation equivalent version of PCIOD 6 7 by an appropriate unitary matrix on the right. Since the proposed DSTBCs have unitary relay matrices they have low PAP compared to the codes from PCIODs. In particular a low PAP version of the 4 4 CIOD is presented. A construction of minimum delay two group ML decodable DSTBCs with unitary relay matrices is provided for odd number of relays. Application of DSTBCs with unitary relay matrices in the training based noncoherent communication scheme of 10 is also discussed. A. Organization of the paper Section II provides an overview of the Jing and Hassibi transmission scheme 3 for symbol synchronous coherent amplify and forward relay networks. In Section III the construction and properties of the proposed DSTBCs are described in detail with illustrative examples. Section IV points out some applications of DSTBCs with unitary relay matrices in training based noncoherent relay networks. Section V concludes the paper with a short discussion. B. Notation Vectors and matrices are denoted by lowercase and uppercase boldface characters respectively. The operator diag s 1 s...s M denotes the M M diagonal matrix with s 1 s... s M as its diagonal entries. The symbol ω n is used to denote the n-th root of unity e πi n. The operators. T. H denote transpose and conjugate transpose respectively. A denotes the determinant of a square matrix A. An identity matrix and an all zero matrix of appropriate size are denoted by I and 0 respectively. II. OVEVIEW OF DISTIBUTED SPACE TIME BLOCK CODING In this section we briefly describe the requirements for the Jing and Hassibi transmission scheme 3. This scheme is applicable for symbol synchronous coherent amplify and forward relay networks. By coherent we mean that the destination has complete knowledge of all the required wireless channels for coherent detection. It consists of a single source node a single destination node and relay nodes that aid communication between the source and the destination. The wireless channel between any two terminals is assumed to be flat fading and quasi-static for the duration of one block of transmission from the source to the destination. The wireless channel between the source and the j-th relay f j and that between the j-th relay and the destination g j are modeled by i.i.d complex Gaussian random variables. Also all the nodes are assumed to be half duplex constrained. A transmission from the source to the destination comprises of two phases. In the first phase the source transmits a complex vector s C T consisting of T complex symbols to all the relays using a fraction π 1 of the total power P sum of power consumed by the source and the relays. During the second phase the j-th relay node applies a linear transformation B j C T T B j F T to the received vector or its conjugate and transmits the resulting vector to the destination using a fraction π of the total power P. The matrices B j j = 1... will be henceforth referred to as relay matrices. We assume without loss of generality that the first M relays apply linear transformation on the received vector and the remaining M relays apply linear transformation on the conjugate of the received vector. It can be shown 3 7 that the equivalent signal model is as given below: π 1 π P y = Xh + n 1 y : received vector at the destination during second phase X = B 1 s B s... B M s B M+1 s... B s h = f 1 g 1... f M g M fm+1 g T M+1... f g π M n = w + P π 1P+1 g jb j v j + k=m+1 g kb k v k v j : additive noise at the j-th relay during first phase reception w : additive noise at destination during second phase. It can be verified that the covariance matrix Γ of n is: Γ = I T + π P g j B j B H j. The destination performs ML detection as given below: ˆX = argmin X Γ 1 y π 1 π P Xh F. 3 Jing and Hassibi have proved 3 that a diversity order of is achieved by the DSTBC X if T and X 1 X H X 1 X = 0 for all codeword matrices X 1 X. Thus the minimum delay required to achieve full cooperative diversity is T = and such DSTBCs are said to be minimum delay DSTBCs for which X is a square matrix. The following important remarks and observations will be used throughout the remainder of this paper. emark 1: The DSTBC X is constrained to have in any column linear combinations of either only the complex symbols or only its conjugates. emark : For unitary relay matrices the PAP of the signals from the relays is same as the PAP of s. emark 3: The noise n seen by the destination is in general not white. If the relay matrices are unitary then Γ is a scaled identity matrix which in turn makes the detector described π by 3 coincide with ˆX = arg min y 1π P π Xh 1P+1 F. 81

3 Therefore if the relay matrices are unitary then it is sufficient for the destination to have knowledge of f j g j j = 1...M and f j g j j = M III. LOW COMPLEXITY DSTBCS WITH UNITAY ELAY MATICES In this section we briefly describe the conditions for multigroup ML decoding and the properties of PCIODs. The construction of the proposed DSTBCs is then explained along with illustrative examples. A. Multigroup ML decoding The DSTBC has a column vector representation X = B 1 s B s... B M s B M+1 s... B s s is the vector of information bearing symbols T s = s1 s... s T. The column vector representation is completely described by the relay matrices B 1 B... B C T T. Let the real and imaginary parts of the complex symbol s j be x j 1 and x j respectively. Then the DSTBC can also be equivalently described in a linear STBC form as follows: X = T x j A j A j j = 1...T C T are called the weight matrices. It can be observed that there is a one to one correspondence between weight matrices and the real symbols x j j = 1...T. Theorem 1: 7 The DSTBC X is g-group ML decodable if for some partitioning of the real symbols x j j = 1...T into g-groups each of cardinality T g 1 The real symbols in each group take values independently of the real symbols in the other groups during encoding into codewords of X. The associated weight matrices satisfy: A H Γ 1 B + B H Γ 1 A = 0 4 whenever A B are weight matrices belonging to different groups and Γ is as given by. ML decoding of a g-group ML decodable DSTBC: Suppose that X is a g-group ML decodable DSTBC with weight matrices A j j = 1...T. Then there is a partitioning of the set {1...T } into g equal partitions denoted by subsets l 1 l... l g such that the weight matrices of X satisfy A j H Γ 1 A k + A k H Γ 1 A j = 0 j l m k / l m for all m = 1...g. ML decoding for such a DSTBC can be done separately for the real symbols in each group. To be precise the real symbols in the k-th group can be decoded as follows: {ˆx j j l k } = arg min Γ 1 y x j j l k s π 1 π P X j l k x j A j h F. 5 B. Construction and properties of PCIOD Throughout this paper for ease of explanation and simplified proofs we shall consider only a permutation equivalent version of the PCIOD constructed in 6 7. Construction 1: For an even number of relays the PCIOD is given by: A B H X PCIOD = B A H 6 A = diag B = diag s 1 s... s s +1 s +...s and s j = x j 1 +ix j. It can be easily verified that X PCIOD as given in 6 is equivalent to the one presented in 6 7 upto a permutation of rows and columns. To be precise PX PCIOD P T gives the PCIOD described in 6 7 for some permutation matrix P. It is easy to see from 6 that PCIODs have non-unitary weight matrices as well as non-unitary relay matrices. The weight matrices of X PCIOD can be shown 6 7 to satisfy: A j H A k + A k H A j = 0 1 j < k. 7 Also since the determinant is unchanged by left or right multiplication by permutation matrix we have 6 7 X H D 0 PCIOD X PCIOD = 0 D D = diag Hence we get s 1 + s +1 s + s +... s + s. X PCIOD H X PCIOD = the notation is used to denote the difference matrix and p j = x j 1 + x j + x j 1+ + x j+. C. Construction of four group ML decodable DSTBCs for even number of relays For even let us define the unitary matrix U as: F 0 U = 8 0 F F is the discrete Fourier transform DFT matrix of order. Construction : The proposed DSTBC X UPCIOD for even is given by: AF B X UPCIOD = X PCIOD U = H F BF A H F This DSTBC is named unitary PCIOD UPCIOD because it has unitary relay matrices. p j 8

4 The relay matrices of X UPCIOD are unitary and can be Ej 0 explicitly given as follows: B j = j = E j 0 E and B j = j j = E j E j = diag 1 ω j 1 ω j 1... ω 1j 1. Thus the corresponding Γ matrix for the proposed DSTBC X UPCIOD will be a scaled identity matrix. It is easy to see that right multiplication by U has not disturbed the property of any column having linear functions of either only complex symbols or only their conjugates. In fact M = i.e. the first relays have to apply unitary transformation on s and the remaining relays have to apply unitary transformation on s. Proposition 1: If two matrices A and B satisfy A H B + B H A = 0 then AV H BV + BV H AV = 0 if V is unitary. The weight matrices of X UPCIOD are nothing but the weight matrices of X PCIOD right multiplied by U and hence they continue to be non-unitary. Now using Proposition 1 7 and the fact that Γ is a scaled identity matrix it is evident that the weight matrices of X UPCIOD satisfy 4 for the following partitioning of the real symbols of X UPCIOD into four groups: First group: x 1 x 3... x 1 Second group: x x 4... x Third group: x 1+ x x 1 Fourth group: x + x 4+...x. Since right multiplication by a unitary matrix does not disturb the determinant of a matrix we have X UPCIOD H X UPCIOD = p j =. x j 1 + x j + x j 1+ + x j+ Thus full cooperative diversity is achieved by X UPCIOD if p j 0 for j = This can be achieved by letting the real symbols in each group {x 1 x 3...x 1 } {x x 4... x } {x 1+ x x 1 } {x + x 4+...x } take values independently from a rotated lattice of dimension which is designed to maximize the product distance 15. This has been discussed in detail in 7. Example 1: Applying Construction for = 6 we get the DSTBC X 6 as shown below: X 6 = s 1 s 1 s 1 s 4 s 4 s 4 s s ω 3 s ω 3 s 5 s 5 ω 3 s 5 ω 3 s 3 s 3 ω 3 s 3 ω 3 s 6 s 6 ω 3 s 6 ω 3 s 4 s 4 s 4 s 1 s 1 s 1 s 5 s 5 ω 3 s 5 ω 3 s s ω 3 s ω 3 s 6 s 6 ω 3 s 6 ω 3 s 3 s 3ω 3 s 3ω 3 p j. 9 s j = x j 1 + ix j j = and the real symbol in each group take values from any finite subset of a rotated Z 3 lattice. To be precise the vectors x 1 x 3 x 5 T x x 4 x 6 T x7 x 9 x 11 T x8 x 10 x 1 T can take values from any subset of G 3 Z 3 where the generator matrix of the lattice G 3 can be taken from 15. Example : For = 4 PCIOD 6 coincides with the 4 4 CIOD of 14. However as pointed out in 16 CIOD has large PAP problem. Applying Construction for = 4 we get X 4 = s 1 s 1 s 3 s 3 s s i s 4 s 4 i s 3 s 3 s 1 s 1 s 4 s 4 i s s i 10 s j = x j 1 + ix j j = and the pairs of real symbols {x 1 x 3 } {x x 4 } {x 5 x 7 } {x 6 x 8 } take values from a QAM constellation rotated by Note that X 4 will have the same coding gain and ML decoding complexity as that of CIOD 14 along with low PAP. D. Construction of two group ML decodable DSTBCs for odd number of relays For the case of odd number of relays the simplest construction would be to construct a DSTBC from UPCIOD for +1 relays and drop one column to result in a +1 DSTBC. Such a DSTBC would still be full diversity and four group ML decodable. However such a solution may not be acceptable for delay constrained applications. To cater to such delay constrained applications we propose a construction of minimum delay full diversity two group ML decodable DSTBCs. Construction 3: For odd the two group ML decodable DSTBC is given by: X UDD = diag s 1 s... s J 11 J is the DFT matrix of order s j = x j 1 +ix j j = This DSTBC is named unitary diagonal design UDD because it has unitary relay matrices and is obtained by right multiplication of a diagonal design by a unitary matrix. The two groups of real symbols for which ML decoding can be done separately are: {x 1 x 3... x 1 } and {x x 4...x }. Also we have X UDD H X UDD = x j 1 + x j which implies that full diversity is achieved by X UDD if the vectors T x 1 x 3... x 1 x x 4... x take values independently from any subset of a rotated lattice of dimension designed to maximize the product distance 15. Example 3: For 5 relays applying Construction 3 we get the following two group ML decodable DSTBC: X 5 = s 1 s 1 s 1 s 1 s 1 s s ω 5 s ω5 s ω5 3 s ω5 4 s 3 s 3 ω5 s 3 ω5 4 s 3 ω 5 s 3 ω5 3 s 4 s 4 ω5 3 s 4 ω 5 s 4 ω5 4 s 4 ω5 s 5 s 5 ω5 4 s 5 ω5 3 s 5 ω5 s 5 ω 5 1 s j = x j 1 + ix j j = and the vectors x1 x 3... x 9 x x 4... x 10 take values from any subset of G 5 Z 5 G 5 is taken from

5 TABLE I COMPAISON OF MINIMUM DELAY FULL DIVESITY DSTBCS Construction Number of ML decoding Unitary Unitary Constellation relays complexity relay matrices weight matrices eal orthogonal designs single real symbol yes yes PAM Complex orthogonal designs 3 11 single real symbol yes yes QAM Quasi-orthogonal design group decodable yes yes rotated QAM Field extensions 1 arbitrary 1-group decodable yes yes QAM Doubling construction 5 even -group decodable yes yes QAM PCIOD 6 even 4-group decodable no no rotated lattice Extended Clifford algebras 8 9 power of two 4-group decodable yes yes rotated lattice Proposed even 4-group decodable yes no rotated lattice odd -group decodable yes no rotated lattice IV. APPLICATION IN TAINING BASED NONCOHEENT ELAY NETWOKS In this section we point out that the proposed DSTBCs can be applied in a training based noncoherent relay network. From remark 3 we know that if the relay matrices are unitary then it is sufficient for a ML decoder to have knowledge of f j g j j = 1...M and f j g j j = M In a noncoherent relay network where the terminals do not have knowledge of any of the channel gains it is easy to estimate the product of the fading gains rather than estimating all the individual fading gains. ecently in 10 it was shown that by transmitting 1 pilot symbol to all the relays in the first phase and by simply amplifying and forwarding the received symbols from the relays during second phase the destination can easily estimate f j g j j = 1... M and f j g j j = M from its received signals. Thus DSTBCs with unitary relay matrices can be effectively applied in the training based noncoherent communication scheme of 10. V. DISCUSSION We have constructed full diversity low complexity DSTBCs with unitary relay matrices for arbitrary number of relays. A summary of the main features of few DSTBC constructions in the literature is provided in Table I. Some of the directions for further work are listed below: In 7 an OFDM based distributed space time coded transmission scheme has been proposed to tackle symbol asynchronism amongst the relay nodes. Few low complexity DSTBCs have also been constructed in 7 but they do not have low PAP when the number of relays is even. How to construct low complexity DSTBCs with low PAP for coherent symbol asynchronous relay networks with arbitrary number of relay nodes? The Jing and Hassibi transmission scheme has been recently generalized to multihop relay networks in 17. Constructing low complexity DSTBCs with low PAP for multihop relay networks is an interesting open problem. ACKNOWLEDGMENT This work was supported partly by the IISc-DDO program on Advanced esearch in Mathematical Engineering through grants to B. S. ajan. EFEENCES 1 J.N. Laneman and G.W. 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