Distributed Space-Time Codes with Reduced Decoding Complexity

ISIT 2006, Seattle, USA, July 9 - 14, 2006

Distributed Space-Time Codes with Reduced Decoding Complexity Kiran T. and B. Sundar Rajan Dept. of ECE, Indian Institute of Science Bangalore-560012, India. Email: {kirant,bsrajan}@ece.iisc.ernet.in

Abstract— We address the problem of distributed space-time coding with reduced decoding complexity for wireless relay network. The transmission protocol follows a two-hop model wherein the source transmits a vector in the first hop and in the second hop the relays transmit a vector, which is a transformation of the received vector by a relay-specific unitary transformation. Design criteria is derived for this system model and codes are proposed that achieve full diversity. For a fixed number of relay nodes, the general system model considered in this paper admits code constructions with lower decoding complexity compared to codes based on some earlier system models.

g1

f2

g2

fR−1

source

fR

gR−1 gR

destination

relay nodes

Fig. 1.

Wireless relay network

A. The System model

I. I NTRODUCTION Cooperative communication is a promising approach for achieving reliable communication without the need for multiple antennas at individual nodes. In contrast to the single user colocated multiple antenna transmission, cooperative communication is based on the relay channel model where the collection of distributed antennas belonging to multiple users is exploited for achieving diversity. For details on cooperative diversity techniques, we refer the readers to [1]-[6] and the references therein. In this paper, we are interested in the two-hop communication model proposed in [5], [6] where the source as well as the relay nodes do not have any channel information while the destination has complete channel knowledge. In the following subsection we describe this system model in detail and then list the contributions of this paper. Before proceeding with the system model, we introduce the notations that are used in this paper. For a complex matrix ¯ X t , and X † denote the conjugate, the transpose and X, X, the conjugate transpose of X, respectively. Also, det(X) (or |X|) indicates the determinant while Tr(X) indicates trace of X. We use |x| to denote the absolute value of a complex number x. For a complex vector x, we split t the real and ˆ = xt xt . For a vector imaginary parts and denote x space V over a field F, GL(V ) denotes the set of all F-linear vector space isomorphisms from V to V , Mn (F) denotes the set of all n × n matrices over F and GLn (F) denotes the subset of all invertible matrices in Mn (F). If x ∈ Cn×1 is a complex random vector, x ∼ CN (0, In ) denotes that the entries are independent and identically distributed complex Gaussian variables with zero mean and unit variance (In denotes the n × n identity matrix).

1424405041/06/$20.00 ©2006 IEEE

f1

The network (see Fig. 1) consists of a source node, a destination node and R other relay nodes which aid the source in communicating information to the destination. The nodes are placed randomly and independently according to some distribution. Each node including source and destination is equipped with one transmit and one receive antenna and are subject to a half-duplex constraint, i.e., a node cannot transmit and receive simultaneously. The channel path gains from the source to ith relay, fi and from jth relay to destination gj are all independently and identically distributed complex Gaussian variables with zero mean and unit variance. We assume that the nodes are synchronized at the symbol level. Every transmission cycle from source to destination comprises of two √stages. In the first stage, source transmits a T length vector P1 T s which the relays receive. The signal vector s is from a codebook consisting of S = {s1 , s2 , · · · , sL } information vectors, with Es† s = 1, so that P1 T is the average transmit power. The received vector at ith relay node is (1) ri = P1 T fi s + vi , where vi ∼ CN (0, I). In the second half of the cycle, all the relay nodes are scheduled to transmit together (assuming symbol level synchronization). The jth relay node transmits a T length vector tj which is a function of the received vector rj . The relays are only allowed to linearly process the received vector. To be precise, the jth relay is equipped with a fixed T × T unitary matrix Aj (called relay matrix) and it is scheduled to transmit P2 P2 Aj rj or tj = Aj ¯rj , (2) tj = P1 + 1 P1 + 1 which is decided and fixed apriori. Let N be the number of relay nodes scheduled to transmit Aj rj and the remaining (R−

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N ) nodes transmit the “conjugate vector” Aj ¯rj . Without loss of generality, we assume that the jth relay node transmits Aj rj if j ≤ N , else it transmits Aj ¯rj . The vector received at the destination is given by ⎛ ⎞ R N P1 P2 T ⎝ y= gj Aj rj + gj Aj ¯rj ⎠ + w (3) P1 + 1 j=1 j=N +1 P1 P2 T P1 P2 T Sh + v + w = Sh + n, (4) = P1 + 1 P1 + 1 where t h = f1 g1 · · · fN gN f¯N +1 gN +1 · · · f¯R gR S = A1 s · · · AN s AN +1¯s · · · AR ¯s ⎛ ⎞ N R P 2 ⎝ ¯k ⎠ , v= gj Aj vj + gk Ak v P1 + 1 j=1

(5) (6) (7)

k=N +1

w ∼ CN (0, I) is the additive noise at the destination and n = v + w. The collection of matrices

C = A1 s · · · AN s AN +1¯s · · · AR ¯s : s ∈ S will be called a distributed Space-Time Block Code (STBC). The idea of unitary relay matrix based “linear dispersion” distributed coding is motivated by the recent work [5], [6], where two different system models are considered. Our system model subsumes the basic model in [5] and section 7.5 in [6] (when N = R), but is different from the generalization considered in [5] and section 7.10 in [6]. In the generalized model considered in [6], the jth relay is allowed to transmit tj = Aj rj + Bj ¯rj

(8)

model is definitely not part of this. However, in the system model (8), consider the special case where Bj = 0 for 1 ≤ j ≤ N and Aj = 0 for N + 1 ≤ j ≤ R. This special case is again subsumed by our system model precisely when the relay matrices are real orthogonal matrices. The motivation for considering this special case seems to be that the well-known Alamouti code belongs to this class [6]. The contributions of this paper are • We consider a variation of the the “linear dispersion based” wireless relay network proposed in [5]. This generalization is different from another generalization of considered in [5], but both contain a special class of codes that include the well-known Alamouti code. • The explicit code design criteria derived for the generalized model in [5] is known only for the special case. In this paper, we derive a pairwise error probability (PEP) based code design criteria for our system model and so generalize the criteria in [5] for a more general setup. • While [5],[6] do not provide any explicit code construction, a recent paper [8] gives a systematic code construction for the basic model in [5] only (for the case N = R). In this paper, based on our design criteria we propose a class of distributed ST codes for the case N ≤ R. The main advantage of the general system model in this paper is that it admits code constructions with lower decoding complexity. II. PAIRWISE ERROR PROBABILITY Splitting the real and imaginary parts, (4) can be equivalently written as P1 P2 T ˆ = ρHˆs + n ˆ , where ρ = , and y P1 + 1

where Aj , Bj are restricted to real matrices. We would like to emphasize here that our system model is not included in the system model (8) because we allow Aj to be complex unitary. This will be clear from the following equivalent “real matrix” version of (2) and (8). While (2) can be equivalently written as

P2 Aj −Aj rj ˆtj = tj = tj Aj rj P1 + 1 Aj

H=

for 1 ≤ j ≤ N , and P2 Aj ˆtj = tj = tj P1 + 1 Aj

Since the relay matrices are all unitary, wi , vi,j are independent circularly symmetric Gaussian and gi s are known at ˆ is also Gaussian with zero mean the destination, the vector n and R P2 n 2 V ar | gi | I2T . = Σn = 1 + n P1 + 1 i=1

Aj −Aj

rj , rj

for N + 1 ≤ j ≤ R, (8) can be equivalently written as

0 rj ˆtj = tj = Aj + Bj for 1 ≤ j ≤ R, tj 0 Aj − Bj rj and the requirement in [6] says that the matrix

0 Aj + Bj 0 Aj − Bj has to be orthogonal. Thus the generalization in [6] only considers block-diagonal orthogonal matrices and so our system

N gi IT

+

i=1 R i= N +1

gi IT gi IT gi IT

Ai −Ai fi IT −fi IT Ai Ai fi IT fi IT

−gi IT Ai Ai fi IT −fi IT gi IT Ai −Ai fi IT fi IT

−gi IT gi IT

(9)

Therefore when the channel is perfectly known at the destination, the conditional probability is given by s)† (ˆ y−ρHˆ s) exp − (ˆy−ρHˆ P2 R 2 2 1+ P +1 i=1 |gi | 1 P (y | s, h) = (10) T . R 2 2 2π 1 + PP |g | i i=1 1 +1

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By rearranging, it is possible to show that

2 S −S ¯ 2 2 ˆ y − ρ ˆ y − ρHˆs = h = y − ρSh , S S and therefore the maximum likelihood (ML) decoding rule is y − ρHˆs arg max P (y | s, h) = arg min ˆ s∈S

s∈S

2 2

= arg min y − ρSh . S∈C

(11) (12)

With this, we have the following Chernoff bound on the PEP, which can be proved on similar lines as Theorem 7.1 in [6]. Theorem 1: Suppose the power allocation is P1 = P/2 and P2 = P/(2R). Then the average PEP of decoding si ∈ S to a wrong vector sj ∈ S is bounded as −1 P (si → sj | si ) ≤ Egi IR + γ∆S † ∆SDg , (13) where γ = 8 R+ 2 , ∆S = Si − Sj for ( i=1 |gi | ) Si = A1 si · · · , AN si AN +1¯si · · ·

where f1 (), f2 () are two possibly different functions. This minimizes the decoding complexity since L and R can now be separately decoded. The decision now reduces to |L| + |R| metric calculations, which otherwise would have been |L|×|R| metric calculations. Notice that if L and R were simply complex numbers, then the code obtained via the doubling construction is the well-known Alamouti code. Alamouti code is known for its decoding simplicity and this is exactly what has been lifted to a more general case by the doubling construction. Another very important feature about the doubling construction is that the matrix S is invertible if either L or R is invertible. This is because, 1 1 det(S) = det(S † S) 2 = det (L† L + R† R)(RR† + LL† ) 2 ≥ max{det(L)2 , det(R)2 }. Therefore, if L and R are sets of n × n matrices such that

PT R

2

LR = RL and det(L − L ) = 0, det(R − R ) = 0

AR¯si ,

2

and Dg = diag(|g1 | , · · · , |gR | ). Similar to the colocated multiple antenna case, the Chernoff upperbound is minimized by maximizing the determinant of ∆S † ∆S. We say the code C achieves full diversity if the difference matrix ∆S has full rank. Thus the design criteria is to design {s1 , · · · , sL } and T × T unitary matrices {A1 , · · · , AR } such that the resulting distributed space-time code is endowed with the property of every difference (of pair of distinct codewords) matrix having full rank. In this paper, we propose a class of distributed space-time codes which achieves this property and also has reduced decoding complexity. The key idea is the Doubling Construction which we describe in the following section. III. D ISTRIBUTED STBC CONSTRUCTION 1) Doubling construction: Suppose the distributed spacetime code is of the form

L −R† C= S= : L ∈ L, R ∈ R R L† where L, R are N × N complex matrices such that LR = RL for any L ∈ L, R ∈ R. Then the size of the code |C| = |L| |R| and more important is the property †

L R† L −R† S†S = −R L R L† †

L L + R† R 0 = . (14) 0 RR† + LL†

for all L = L ∈ L, R = R ∈ R, then the code C obtained through doubling construction:

L −R† C= : L ∈ L, R ∈ R R L† is such that the difference of any pair of matrices is invertible. In addition, if any matrix L ∈ L and R ∈ R can be written in the form (6) with some fixed set of N × N unitary matrices A1 , · · · , AN and B1 , · · · , BN respectively, then the resulting code C is also of the form (6) with unitary relay matrices and so C can be used as a distributed space-time code for R = T = 2N . The decoding complexity of this code for 2N number of relay nodes will be the sum of decoding complexity of code L and code R. We illustrate this with an example. A. Commuting sets of matrices from field extension For the case N = R, a class of distributed space-time codes is proposed in [8], which is based on the γ-circulant construction [7]. We will first recall this construction method, which will be useful in our doubling construction as well. For an appropriately chosen complex number γ, let Ln denote the set of all n × n matrices of the form ⎡ ⎤ s0 γsn−1 γsn−2 · · · γs2 γs1 ⎢ s1 s0 γsn−1 · · · γs3 γs2 ⎥ ⎢ ⎥ n−1 ⎢ s3 s s · · · γs γs3 ⎥ 1 0 4 ⎢ ⎥ i si Γ = ⎢ . .. .. .. .. ⎥ .. ⎢ .. ⎥ . . . . . i=0 ⎢ ⎥ ⎣sn−2 sn−3 sn−4 · · · s0 γsn−1 ⎦ sn−1 sn−2 sn−3 · · · s1 s0

Since,

= In s Γs Γ2 s · · ·

2

y − ρSh = Tr(y† y) − ρTr(y† Sh) − ρTr(h† S † y) + ρ2 Tr(h† S † Sh),

where

(14) ensures that the ML decision breaks down as Γ=

arg min f1 (y, L, h) + arg min f2 (y, R, h) , L∈L

(15) (16)

R∈R

544

0

In−1

γ 0

Γn−2 s Γn−1 s , (17)

and s = s0

···

t sn−1 ,


for si ∈ Q(i). The element √ γ ∈ Q(i) is chosen such that 2 (i) |γ| = 1 and (ii) Q (i) n γ is a degree n cyclic extension of Q (i). Note that condition (i) ensures that Γ is a unitary matrix and condition (ii) ensures that any matrix of the form (17) is invertible. Thus the matrices Γj , j = 0, 1, · · · , n − 1 can be used as relay matrices with R = n = N relay nodes and S can be chosen as any arbitrary subset of Q(i)n (in particular, any set of vectors with entries from arbitrary QAM constellation) to get full diversity distributed space-time codes. These codes correspond to the case N = R since no “conjugates entries” are allowed in the matrix. We now show that the above set of γ-circulant matrices Ln can be used in the doubling construction to get low complexity distributed space-time codes, which now involves conjugates also. The matrix representation in (17) is a faithful √ representan tion of an element in the extension field Q (i) √ γ over Q (i) and so Ln is isomorphic to the field Q (i) n γ [7]. So two different matrices of the form (17), with si ∈ Q (i) should commute. Choosing Rn = Ln will satisfy the commuting condition required for applying doubling construction. The distributed space-time code from doubling construction using Rn and Ln can be used for a network with 2n relay nodes. The decoding complexity of the resulting distributed ST code (for 2n relay nodes) will be less compared to the γ-circulant based code L2n for 2n antennas proposed in [8]. This can be easily seen in the following example for n = N = 2. Example 1: For n = 2, γ = i ensures that all γ-circulant matrices of the form (17) are invertible for si ∈ Q(i). Therefore every distributed space-time codeword is of the form ⎡ s0 ⎢s1 ⎢ ⎣s2 s3

γs1 s0 γs3 s2

−s¯2 −¯ γ s¯3 s¯0 γ¯ s¯1

⎤ −s¯3 −s¯2 ⎥ ⎥ = A1 s A2 s A3¯s A4¯s ⎦ s¯1 s¯0

where the relay matrices ⎡ 1 ⎢0 ⎢ A1 = ⎣ 0 0 ⎡ 0 0 ⎢0 0 A3 = ⎢ ⎣1 0 0 γ¯

⎡ ⎤ 0 0 0 0 ⎢ 1 0 0⎥ ⎥ , A2 = ⎢1 ⎣0 0 1 0⎦ 0 0 1 0 ⎡ ⎤ −1 0 0 ⎢0 0 −¯ γ⎥ ⎥ A4 = ⎢ ⎣0 0 0⎦ 0 0 1

γ 0 0 0 0 0 0 1 0 0 0 −1 1 0 0 0

⎤ 0 0⎥ ⎥ γ⎦ 0 ⎤ −1 0⎥ ⎥. 0⎦ 0

B. Commuting sets of matrices from division algebra We refer the readers to [7] for a detailed treatment on division algebra and space-time code constructions based on division algebra representation. Here we briefly introduce these abstract objects and discuss a particular representation that yields the commuting sets that we are interested in. A division algebra D is a vector space over a field F which is also a ring whose multiplication is associative, not necessarily commutative and every element d ∈ D has a multiplicative inverse d−1 ∈ D satisfying dd−1 = d−1 d = 1, where 1 denotes the multiplicative identity in D. Let n denote the dimension of the vector space D over F, which is known to be a perfect square [7]. For an arbitrary element d ∈ D, the map Ld : D → D x → dx

(18)

is an F-linear isomorphism from D to D. This defines a oneone map from d to the set of all F -linear isomorphisms of D which associates d ∈ D to Ld ∈ GL(D). With a fixed F-basis of D every d can be associated with a n × n invertible matrix over F denoted as [Ld ]. We call this as the left regular representation of D. Similar to the construction of distributed ST codes from γ-circulant matrices, the representation Ld can be used as a distributed ST code for the case where no conjugate entries are allowed (i.e., R = N = n). This will be illustrated in an example later in this subsection. However, for applying the doubling construction we now cannot use L = R = {[Ld ] : d ∈ D} since the division algebra is noncommutative. Similar to Ld , one can define a map Rd corresponding to the multiplication (from right) xd and every element d can be associated with another n × n invertible matrix with the same F-basis that is used for representing Ld . This is called the right regular representation which is denoted as [Rd ]. The most important fact about these two representations is that these two commute. To be more precise, if La and Rb are the left and right regular representations corresponding to a ∈ D and b ∈ D, then La ◦ Rb (x) = La (xb) = axb = Rb (ax) = Rb ◦ La (x),

are clearly unitary matrices. This code is a full diversity quasiorthogonal space-time code, which clearly has lower decoding complexity compared to the γ-circulant code L4 for 4 relay elements proposed in [8]. In the next subsection, we describe a general method for constructing commuting set of matrices which generalizes the γ-circulant based construction. However, the resulting relay matrices are of dimension T × T where T = m2 is a perfect square.

which implies that the matrices [La ] and [Rb ] commute, i.e., [La ][Rb ] = [Rb ][La ] for all a, b ∈ D. Let L = {[Ld ] | d ∈ D} and R = {[Rd ] | d ∈ D}. Thus we have identified commuting subsets of GLn (F) satisfying all the essentials for applying doubling construction. Remark 1: If x ∈ Z(D) = {y : yd = dy ∀d ∈ D}, then [Lx ] = [Rx ]. As a consequence, if D is a commutative division algebra (field) then [Ld ] = [Rd ] for all d ∈ D which is the case in our previous subsection. Remark 2: Representation of cyclic division algebra has been used for constructing space-time codes for colocated MIMO transmission [10]-[15]. We would like to emphasize that the representation used in all these works are the so called representation over a maximal subfield (see section VI.A

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in [7]) whereas the representation used here is the regular representation over the center F (see section V.B in [7]). Example 2 (Representation of Quaternion algebra [15]): α,β over F is a A generalized quaternion algebra F 4-dimensional associative and noncommutative algebra with basis e0 = 1, e1 , e2 , e3 such that e21 = α ∈ F, e22 = β ∈ F and e1 e2 = e3 = −e2 e1 . The left regular matrix representation La and right regular representation Ra for any a = a0 e0 + a1 e1 + a2 e2 + a3 e3 ∈ Q(F, α, β) is given by ⎤ ⎡ αa1 βa2 −αβa3 a0 ⎢a1 a0 βa3 −βa2 ⎥ ⎥ = A1 a · · · A4 a , [La ] = ⎢ ⎣a2 −αa3 a0 αa1 ⎦ a3 −a2 a1 a0 t for a = a0 , · · · , a3 , ai ∈ F. The relay matrices can be directly read from the representation. For e.g. A1 = I4 , ⎡ ⎡ ⎤ ⎤ 0 α 0 0 0 0 β 0 ⎢1 0 ⎢ ⎥ 0 0⎥ ⎥ A3 = ⎢0 0 0 β ⎥ . A2 = ⎢ ⎣0 0 ⎣1 0 0 0 ⎦ 0 −α⎦ 0 0 −1 0 0 1 0 0 The elements α and β need to be properly chosen to ensure that the relay matrices are unitary and also to ensure that [La ] are invertible for every a. For F = Q(i), it can be shown over F is a cyclic division algebra for α = i and that α,β F

2+i β = 1+2i [13]. Furthermore, this choice makes |α| = |β| = 1, and so the relay matrices are also unitary. Therefore, the relay matrices A1 , · · · , A4 can be used for constructing distributed ST codes for the restriction N = R = 4 with s being any 4 length vector with entries from arbitrary QAM constellation. For the doubling construction, we also use the right regular representation ⎡ ⎤ b0 αb1 βb2 −αβb3 ⎢b1 b0 −βb3 βb2 ⎥ ⎥ = B1 b · · · B4 b , [Rb ] = ⎢ ⎣b2 αb3 ⎦ b0 −αb1 b3 b2 −b1 b0 t · , b3 , bi ∈ F. Check that [La ][Rb ] = [Rb ][La ] for b = b0 , · · . The matrices Bi which can be directly for any a, b ∈ α,β F read from the representation are unitary for the above choice of α and β. Since right regular representation is an Falgebra isomorphism, [Rb ] is invertible for every b. Thus the above two regular representations can be used in the doubling construction, and every codeword is of the form

[La ] −[Rb ]† = E1 s · · · E8 s [Rb ] [La ]† t where s = at bt and E1 , · · · , E8 are unitary 8 × 8 relay matrices.

IV. D ISCUSSION

two classes of commuting matrix sets. The first class yields constructions for any number of relays R = 2N , while the second class gives constructions for R = 2N where N is any arbitrary perfect square integer. We have only illustrated the core idea with some examples for lack of space, but such codes can be constructed for any arbitrary number of relay nodes. Some preliminary simulation results show that few codes obtained via doubling construction have better coding gain compared to the γ-circulant matrix constructions. ACKNOWLEDGMENT This work was supported partly by the DRDO-IISc Program on Advanced Research in Mathematical Engineering and partly by the Council of Scientific and Industrial Research (CSIR), India, through Research Grant (22(0365)/04/EMR-II) to B.S. Rajan. R EFERENCES [1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversityPart I: System description,” IEEE Trans. Comm., vol. 51, pp. 1927-1938, Nov 2003. [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversityPart II: Implementation aspects and performance analysis,” IEEE Trans. Comm., vol. 51, pp. 1939-1948, Nov 2003. [3] J.N. Laneman, G.W. Wornell, “Distributed space-time-coded protocols for exploiting cooperative diversity in wireless network,” IEEE Trans. on Inform. Theory., vol. 49, pp. 2415-2425, Oct 2003. [4] R.U. Nabar, H. Bolcskei, and F.W. Kneubuhler, “Fading relay channels: Performance limits and space-time signal design,” IEEE Jour. Sel. Areas in Comms, vol. 22, pp. 1099-1109, Aug 2004. [5] Y. Jing, and B. Hassibi, “Wireless Networks, Diversity and Space-Time Codes,” ITW 2004, San Antanio, Texas, pp. 463-468. [6] Y. Jing, “Space-time code design and its applications in wireless networks,” PhD Thesis, California Institute of Technology, 2004. Available at http://etd.caltech.edu/etd/ available/etd-09072004-204814/. [7] B. A. Sethuraman, B. S. Rajan, and V. Shashidhar, “Full-diversity, highrate space-time block codes from division algebras,” IEEE Trans. Inform. Theory, vol. 49, no. 10, pp. 2596–2616, Oct 2003. [8] P. Elia, and P. Vijay Kumar “Approximately universal optimality over several dynamic and non-dynamic cooperative diversity schemes for wireless networks,” http://arxiv.org/pdf/cs.it/0512028 Dec 7, 2005. [9] J.C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: A 2 × 2 full-rate space-time code with non-vanishing determinants,” IEEE Trans. Inform. Theory, vol. 51, Apr. 2005 pp. 1432-1436. [10] J.C. Belfiore and G. Rekaya, “Quaternionic lattices for space-time coding,” Info. Theory Workshop, Mar 31-April 4 2003. [11] Kiran T. and B.S. Rajan, “STBC-schemes with nonvanishing determinant for certain number of transmit antennas,” IEEE Trans. Inform. Theory. vol 51, no 8, pp 2984-2992, Aug 2005 [12] P. Elia, K.R. Kumar, S.A. Pawar, P.V. Kumar, and H.F. Lu, “Explicit space-time codes that achieve the diversity-multiplexing gain tradeoff,” IEEE ISIT, 2005. pp. 896-900. [13] P. Elia, B.A. Sethuraman, and P.V. Kumar, “Perfect Space-Time Codes with Minimum and Non-Minimum Delay for Any Number of Antennas,” http://arxiv.org/pdf/cs.it/0512023 Dec 6, 2005. [14] F. Oggier, G. Rekaya, J-C. Belfiore, and E. Viterbo, “Perfect space-time block codes,” submitted to IEEE Trans. Inform. Theory. Sept 2004. Available at http://www.comelec.enst.fr/ ∼belfiore/publi.html. [15] Kiran T., and B. Sundar Rajan, “High-rate Full-rank Space-Time Block Codes from Cayley Algebra,” Intl Conf on Signal Proc and Comm (SPCOM 2004), Dec. 2004, IISc Bangalore. Available at http: //ece.iisc.ernet.in/∼bsrajan/Publications.html

Codes from doubling construction have reduced decoding complexity. Commuting sets of matrices are the basic building blocks for doubling construction. Here we have presented

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