Low Complexity Distributed STBCs with Unitary Relay ... - CiteSeerX

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

Low Complexity Distributed STBCs with Unitary Relay Matrices for Any Number of Relays G. Susinder Rajan

B. Sundar Rajan

Atheros India, LLC Chennai 600004, India [email protected]

ECE Department, Indian Institute of Science Bangalore 560012, India [email protected]

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. I NTRODUCTION 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 802.11n (WLAN) and 802.16e (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 [1], [2], [3], [4]. 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

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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], [12] have addressed the construction of DSTBCs that achieve full cooperative diversity. However, the maximum likelihood (ML) decoding complexity of these DSTBCs (except the 2 × 2 and 4 × 4 DSTBC of [11]) were prohibitively high. Recognizing 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 [11], [13], 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 (PAPR) 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

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

•

•

•

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 PAPR compared to the codes from PCIODs. In particular, a low PAPR 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 (s1 , s2 , . . . , sM ) denotes the M × M diagonal matrix with s1 , s2 , . . . , sM as its diagonal entries. The symbol ωn is 2πi used to denote the n-th root of unity e 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. OVERVIEW

OF

D ISTRIBUTED S PACE T IME B LOCK C ODING

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 R 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, fj and that between the j-th relay and the destination, gj are modeled by

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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 ∈ CT consisting of T complex symbols to all the relays using a fraction π1 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 Bj ∈ CT ×T (k Bj k2F ≤ T ) to the received vector or its conjugate and transmits the resulting vector to the destination using a fraction π2 of the total power P . The matrices Bj , j = 1, . . . , R 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 R − 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: s π1 π2 P 2 y= Xh + n (1) π1 P + 1 where, y : received vector at the destination during second phase   X = B1 s B2 s . . . BM s BM+1 s∗ . . . BR s∗  T ∗ fM gM fM+1 gM+1 . . . fR gR  h = f1 gq 1 ...  PM PR ∗ n = w + π1πP2 P+1 j=1 gj Bj vj + k=M+1 gk Bk vk vj : 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: R

π2 P X Γ = IT + |gj |2 Bj Bj H . π1 P + 1 j=1 The destination performs ML detection as given below: s 2 1 ˆ = arg min k Γ− 2 (y − π1 π2 P Xh) k2F . X X π1 P + 1

(2)

(3)

Jing and Hassibi have proved [3] that a diversity order of R is achieved by the DSTBC X if T ≥ R and | (X1 − X2 )H (X1 − X2 ) | 6= 0 for all codeword matrices X1 6= X2 . Thus, the minimum delay required to achieve full cooperative diversity is T = R 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. Remark 1: The DSTBC X is constrained to have in any column, linear combinations of either only the complex symbols or only its conjugates. Remark 2: For unitary relay matrices, the PAPR of the signals from the relays is same as the PAPR of s. Remark 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 q π1 π2 P 2 2 ˆ by (3) coincide with X = arg min k y − π1 P +1 Xh kF .

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009

Therefore, if the relay matrices are unitary, then it is sufficient for the destination to have knowledge of fj gj , j = 1, . . . , M and fj∗ gj , j = M + 1, . . . , R. III. L OW

COMPLEXITY

DSTBC S

WITH UNITARY RELAY

MATRICES

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.

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 R, the PCIOD is given by: r   R A −BH (6) XPCIOD = AH 2 B where,

A. Multigroup ML decoding The DSTBC has a column vector representation  X = B1 s B2 s . . . BM s BM+1 s∗ . . . BR s∗ where, s is bearing symbols  the vector of information T s1 s2 . . . sT s = . The column vector representation is completely described by the relay matrices B1 , B2 , . . . , BR ∈ CT ×T . Let the real and imaginary parts of the complex symbol sj be x2j−1 and x2j respectively. Then, the DSTBC can also be equivalently described in a linear STBC form as follows: X=

2T X

where, Aj , j = 1, . . . , 2T ∈ C are called the weight matrices. It can be observed that there is a one to one correspondence between weight matrices and the real symbols xj , j = 1, . . . , 2T . Theorem 1: [7] The DSTBC X is g-group ML decodable if for some partitioning of the real symbols xj , j = 1, . . . , 2T into g-groups each of cardinality 2T 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. 2) The associated weight matrices satisfy: AH Γ−1 B + BH Γ−1 A = 0

(4)

whenever A, B are weight matrices belonging to different groups and Γ is as given by (2). ML decoding of a g-group ML decodable DSTBC: Suppose that X is a g-group ML decodable DSTBC with weight matrices Aj , j = 1, . . . , 2T . Then, there is a partitioning of the set {1, 2, . . . , 2T } into g equal partitions denoted by subsets l1 , l2 , . . . , lg such that the weight matrices of X satisfy Aj H Γ−1 Ak + Ak H Γ−1 Aj = 0, j ∈ lm , k ∈ / lm 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:

min

xj , j∈lk

kΓ

(y−

s

2

and sj = x2j−1 + ix2j . It can be easily verified that XPCIOD as given in (6) is equivalent to the one presented in [6], [7] upto a permutation of rows and columns. To be precise, PXPCIOD PT 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 XPCIOD can be shown [6], [7] to satisfy: Aj H Ak + Ak H Aj = 0, ∀ 1 ≤ j < k ≤ 2R.

T ×R

{ˆ xj | j ∈ lk } = arg

2

xj Aj

j=1

1 −2

  A = diag s1 , s2 , . . . , s R , 2   B = diag s R +1 , s R +2 , . . . , sR

π1 π2 P 2 X xj Aj h)k2F . (5) π1 P + 1 j∈l k

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(7)

Also since the determinant is unchanged by left or right multiplication by permutation matrix, we have [6], [7],   D 0 H XPCIOD XPCIOD = 0 D where, “ ” D = diag |s1 |2 + |s R +1 |2 , |s2 |2 + |s R +2 |2 , . . . , |s R |2 + |sR |2 . 2

2

2

Hence, we get R

|∆XPCIOD

H

2 RY pj ∆XPCIOD | = 2 j=1

where, the notation ∆ is used to denote the difference
2 matrix and pj = ∆x22j−1 + ∆x22j + ∆x22j−1+R + ∆x2j+R .

C. Construction of four group ML decodable DSTBCs for even number of relays For even R, let us define the R × R unitary matrix U as: r   2 F 0 U= (8) R 0 F

where, F is the discrete Fourier transform (DFT) matrix of order R2 . Construction 2: The proposed DSTBC XUPCIOD for even R is given by:   AF −BH F XUPCIOD = XPCIOD U = BF AH F This DSTBC is named ‘unitary PCIOD’ (UPCIOD) because it has unitary relay matrices.

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The relay matrices of XUPCIOD and can be  are unitary  Ej 0 explicitly given as follows: Bj = , j = 1, . . . , R2 0 E j # " 0 −Ej− R 2 and Bj = , j = R2 + 1, . . . , R where, Ej− R 0 2   ( R −1)(j−1) 2(j−1) Ej = diag 1, ω j−1 , ωR , . . . , ω R2 . Thus, the R 2 2 2 corresponding Γ matrix for the proposed DSTBC XUPCIOD 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 = R2 i.e., the first R2 relays have to apply unitary transformation on s and the remaining R ∗ 2 relays have to apply unitary transformation on s . Proposition 1: If two matrices A and B satisfy AH B + H B A = 0 then (AV)H (BV) + (BV)H (AV) = 0 if V is unitary. The weight matrices of XUPCIOD are nothing but the weight matrices of XPCIOD 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 XUP CIOD satisfy (4) for the following partitioning of the real symbols of XUPCIOD into four groups: • First group: x1 , x3 , . . . , xR−1 • Second group: x2 , x4 , . . . , xR • Third group: x1+R , x3+R , . . . , x2R−1 • Fourth group: x2+R , x4+R , . . . , x2R . Since right multiplication by a unitary matrix does not disturb the determinant of a matrix, we have R

|∆XUPCIOD

H

2 RY pj ∆XUPCIOD | = 2 j=1


2 where, pj = ∆x22j−1 + ∆x22j + ∆x22j−1+R + ∆x2j+R . Thus full cooperative diversity is achieved by XUPCIOD if pj 6= 0 for j = 1, . . . , R2 . This can be achieved by letting the real symbols in each group {x1 , x3 , . . . , xR−1 }, {x2 , x4 , . . . , xR }, {x1+R , x3+R , . . . , x2R−1 }, {x2+R , x4+R , . . . , x2R } take values independently from a rotated lattice of dimension R2 which is designed to maximize the product distance [15]. This has been discussed in detail in [7]. Example 1: Applying Construction 2 for R = 6 we get the DSTBC X6 as shown below:   s1 s1 s1 −s∗4 −s∗4 −s∗4  s2 s2 ω3 s2 ω32 −s∗5 −s∗5 ω3 −s∗5 ω32     s3 s3 ω32 s3 ω3 −s∗6 −s∗6 ω32 −s∗6 ω3    . (9) X6 =  s4 s∗1 s∗1 s∗1   s4 s4   s5 s5 ω3 s5 ω32 s∗2 s∗2 ω3 s∗2 ω32  s6 s6 ω32 s6 ω3 s∗3 s∗3 ω32 s∗3 ω3 where, sj = x2j−1 + ix2j , j = 1, . . . , 6 and the real symbol in each group take values from any finite subset of a rotated T x1 x3 x5 Z3 lattice. To be precise, the vectors ,  T  T  T x2 x4 x6 , x7 x9 x11 , x8 x10 x12

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can take values from any subset of G3 Z3 , where the generator matrix of the lattice G3 can be taken from [15]. Example 2: For R = 4, PCIOD [6] coincides with the 4×4 CIOD of [14]. However, as pointed out in [16], CIOD has large PAPR problem. Applying Construction 2 for R = 4, we get   s1 s1 −s∗3 −s∗3  s2 s2 i −s∗4 −s∗4 i   X4 =  (10)  s3 s3 s∗1 s∗1  s4 s4 i s∗2 s∗2 i

where, sj = x2j−1 + ix2j , j = 1, . . . , 4 and the pairs of real symbols {x1 , x3 }, {x2 , x4 }, {x5 , x7 }, {x6 , x8 } take values from a QAM constellation rotated by 31.7175◦. Note that X4 will have the same coding gain and ML decoding complexity as that of CIOD [14] along with low PAPR. D. Construction of two group ML decodable DSTBCs for odd number of relays For the case of odd number of relays R, the simplest construction would be to construct a DSTBC from UPCIOD for R + 1 relays and drop one column to result in a R + 1 × R 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 R, the two group ML decodable DSTBC is given by: XUDD = diag (s1 , s2 , . . . , sR ) J

(11)

where, J is the DFT matrix of order R, sj = x2j−1 +ix2j , j = 1, . . . , R. 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: {x1 , x3 , . . . , x2R−1 } and {x2 , x4 , . . . , x2R } . Also, we have |(∆XUDD )H (∆XUDD )| =

R Y

j=1

∆x22j−1 + ∆x22j




which implies that full diversity is achieved by XUDD if the  T  vectors x1 x3 . . . x2R−1 , x2 x4 . . . x2R take values independently from any subset of a rotated lattice of dimension R designed to maximize the product distance [15]. Example 3: For 5 relays, applying Construction 3, we get the following two group ML decodable DSTBC:   s1 s1 s1 s1 s1  s2 s2 ω5 s2 ω52 s2 ω53 s2 ω54    2 4 3  X5 =  (12)  s3 s3 ω53 s3 ω5 s3 ω54 s3 ω52   s4 s4 ω 5 s4 ω 5 s4 ω 5 s4 ω 5  s5 s5 ω54 s5 ω53 s5 ω52 s5 ω5

where, sj = x2j−1 +  ix2j , j = 1, . . . , 5 and the vectors x1 x3 . . . x9 , x2 x4 . . . x10 take values from any subset of G5 Z5 (G5 is taken from [15]).

ISIT 2009, Seoul, Korea, June 28 - July 3, 2009 TABLE I C OMPARISON OF MINIMUM DELAY, FULL DIVERSITY DSTBC S Construction Real orthogonal designs [11], [13] Complex orthogonal designs [3], [11] Quasi-orthogonal design [11] Field extensions [12] Doubling construction [5] PCIOD [6] Extended Clifford algebras [8], [9] Proposed

IV. A PPLICATION

Number of relays 2, 4, 8 2 4 arbitrary even even power of two even odd

ML decoding complexity single real symbol single real symbol 2-group decodable 1-group decodable 2-group decodable 4-group decodable 4-group decodable 4-group decodable 2-group decodable

Unitary relay matrices yes yes yes yes yes no yes yes yes

Unitary weight matrices yes yes yes yes yes no yes no no

Constellation PAM QAM rotated QAM QAM QAM rotated lattice rotated lattice rotated lattice rotated lattice

R EFERENCES

IN TRAINING BASED NONCOHERENT RELAY NETWORKS

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 fj gj , j = 1, . . . , M and fj∗ gj , j = M + 1, . . . , R. 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. Recently 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 fj gj , j = 1, . . . , M and fj∗ gj , j = M + 1, . . . , R from its received signals. Thus, DSTBCs with unitary relay matrices can be effectively applied in the training based noncoherent communication scheme of [10]. V. D ISCUSSION 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 PAPR when the number of relays is even. How to construct low complexity DSTBCs with low PAPR 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 PAPR for multihop relay networks is an interesting open problem. ACKNOWLEDGMENT This work was supported partly by the IISc-DRDO program on Advanced Research in Mathematical Engineering through grants to B. S. Rajan.

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[1] J.N. Laneman and G.W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless networks,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2415-2425, Oct. 2003. [2] S. Yiu, R. Schober and L. Lampe, “Distributed space-time block coding,” IEEE Trans. Commun., vol. 54, no. 7, pp. 1195-1206, July 2006. [3] Y. Jing and B. Hassibi, “Distributed Space-Time Coding in Wireless Relay Networks,” IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3524-3536, Dec. 2006. [4] Petros Elia, K. Vinodh, M. Anand and P. Vijay Kumar, “D-MG Tradeoff and Optimal Codes for a Class of AF and DF Cooperative Communication Protocols,” to appear in IEEE Trans. Inform. Theory. [5] Kiran T. and B. Sundar Rajan, “Distributed space-time codes with reduced decoding complexity,” Proc. IEEE International Symposium on Information Theory, Seattle, USA, July 09-14, 2006, pp. 542-546. [6] G. Susinder Rajan and B. Sundar Rajan, “A Non-orthogonal distributed space-time protocol, Part-II: Code Constructions and DM-G Tradeoff,” Proc. IEEE Information Theory Workshop, Chengdu, China, Oct. 22-26, 2006, pp. 488-492. [7] G. Susinder Rajan and B. Sundar Rajan, “Multi-group ML Decodable Collocated and Distributed Space Time Block Codes,” to appear in IEEE Trans. Inform. Theory. Available in arXiv: 0712.2384. [8] G. Susinder Rajan, Anshoo Tandon and B. Sundar Rajan, “On Fourgroup ML Decodable Distributed Space-Time Codes for Cooperative Communication,” Proc. IEEE Wireless Communications and Networking Conference, Hong Kong, March 11-15, 2007. [9] G. Susinder Rajan and B. Sundar Rajan, “Algebraic Distributed SpaceTime Codes with Low ML Decoding Complexity,” Proc. IEEE International Symposium on Information Theory, Nice, France, June 24-29, 2007, pp. 1516-1520. [10] G. Susinder Rajan and B. Sundar Rajan, “Leveraging Coherent Distributed Space-Time Codes for Noncoherent Communication in Relay Networks Via training,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 683-688, Feb. 2009. [11] Y. Jing and H. Jafarkhani, “Using Orthogonal and Quasi-Orthogonal Designs in Wireless Relay Networks,” IEEE Trans. Inf. Theory, vol. 53, no. 11, pp. 4106 - 4118, Nov. 2007. [12] P. Elia, F. Oggier and P. Vijay Kumar, “Asymptotically Optimal Cooperative Wireless Networks with Reduced Signaling Complexity,” IEEE J. Select. Areas Commun., vol. 25, no. 2, pp. 258-267, Feb. 2007. [13] B. Maham and A. Hjorungnes, “Distributed GABBA Space-Time Codes in Amplify-and-Forward Cooperation,” Proc. IEEE Information Theory Workshop, Bergen, Norway, July 1-6, 2007, pp. 189-193. [14] Zafar Ali Khan and B. Sundar Rajan, “Single-Symbol MaximumLikelihood Decodable Linear STBCs,” IEEE Trans. Inform. Theory, vol. 52, no. 5, pp. 2062-2091, May 2006. [15] Full Diversity Rotations, http://www1.tlc.polito.it/˜viterbo/rotations/rotations.html [16] Md. Zafar Ali Khan, “Single-symbol and Double-Symbol Decodable STBCs from Designs,” Ph.D. Thesis, Electrical Communication Engineering Department, Indian Institute of Science, 2003. [17] F. Oggier, B. Hassibi, “Code Design for Multihop Wireless Relay Networks,” EURASIP Journal on Advances in Signal Processing, vol. 8, no. 1, Jan. 2008.