Table Of ContentFast-Group-Decodable STBCs via Codes over GF(4)
N. Lakshmi Prasad and B. Sundar Rajan
Dept. of ECE, IISc, Bangalore 560012, India
Email: nlp,bsrajan @ece.iisc.ernet.in
{ }
Abstract—Inthispaperweconstructlow decodingcomplexity these designs is less than 1 complex symbol per channel
STBCsbyusingthePaulimatricesaslineardispersionmatrices. use when the number of transmit antennas is more than two
InthiscasetheHurwitz-Radonorthogonalityconditionisshown
[1],[2],[3]. Single complex symbol decodable or double real
to be easily checked by transferring the problem to F4 domain.
symbol decodable rate 1 STBCs were given in [4],[5],[6],[8].
The problem of constructing low decoding complexity STBCs
0
1 is shown to be equivalent to finding certain codes over F4. It In [7],[8],[9] the framework for multigroup decodable
is shown that almost all known low complexity STBCs can be STBCs was given. In (2), if the symbols x ,...,x can be
0 1 K
2 obtained by this approach. New codes are given that have the divided into groups such that each group of symbols can be
least known decoding complexity in particular ranges of rate.
MLdecodedindependentlyofothergroups,thenthedecoding
n
a I. INTRODUCTION complexity is greatly reduced. A necessary condition for xi
J and x to be decoded independently of each other is that
j
Consideran N transmitantenna,N receiveantennaquasi-
3 r their weight matrices A and A must be Hurwitz-Radon
i j
static Rayleigh flat fading MIMO channel given by
1 orthogonal, i.e., they must satisfy
] Y =XH +W, (1) AHA +AHA =0 (4)
T i j j i
I whereH istheN×Nr channelmatrix,XistheT×N matrix Constructing low decoding complexity STBCs requires one
s. of transmitted signal, W is the T Nr additive noise matrix to find weight matrices satisfying the above equation. In [7]
×
c and Y is the T Nr matrix of received signal, all matrices 4-group decodable rate 1 codes for arbitrary antennas was
[ being over the c×omplex field C. Throughout this paper, we given. Coordinate Interleaved Orthogonal Designs or CIODs
1 consider only the case T =N. [6] provide single complex symbol decodable designs for
v An N N Space-Time Block Code (STBC) is a finite arbitrary number of transmit antennas. However their rate
6 subset of×CN×N. An N N Linear Space-TCime Design decreases rapidly with increasing number of antennas. [8]
07 (LSTD) [17] or simply a×design X in K real variables gave a general algebraic structure of the weight matrices
2 x1,...,xK is a matrix of g-group decodable codes using Clifford Algebras, where
1. K different groups can have different number of information
0 xiAi (2) symbols to be decoded together. [9] gives g-group decodable
0 Xi=1 designs, called Clifford Unitary Weight Designs (CUWDs),
v:1 iwshleirneeaArliy∈iCndNe×peNn,dein=t o1v,.e.r.,thKe afinedldthoefsereta{lAn1u,m..b.e,rAsKR}. obfyCmlaifnfioprudlaAtilnggebthraesm. Iantri[c1e0s]oabntaainlgeedbtrhariocufgrhamreepwreosreknbtaatsioedn
i on Extended Clifford Algebras was proposed to study CUW
X Throughout the paper designs are denoted using upper case
r bold font letters. An N N design X in K real variables is STBCsandusingthisframework,theoptimaltradeoffbetween
a said to have a rate of R×= K complex symbols per channel rate and ML decoding complexity of CUW STBCs was
2N obtained for few specific cases. Codes meeting this tradeoff
use(cspcu).ThematricesA areknownaslineardispersionor
weight matrices. An STBCican be obtained from a design X were also provided.
In [11] fast-decodable(FD) codes were introduced. These
by making x ,...,x take values from a finite set RK.
1 K A⊂ codes were not multigroup decodable but they had reduced
The set is called the signal set. Denote the STBC obtained
this wayAby (X, ). Thus we have sphere-decodingcomplexity. Later in [21], [12] it was shown
C A that the Golden Code [13] is fast-decodable and hence has
K lower ML decoding complexity than previously thought of.
(X, )= alAl [a1,...,aK]T (3) In [21] rate 2 codes for 2, 4 antennas with the largest
C A { | ∈A}
Xl=1 known coding gain were given. These codes too were fast-
One of the important aspects in the design of space-time decodable. Recently fast-decodable codes for number of an-
block codes (STBC) is the ML decoding complexity. Space- tennas N = 2,4,6,8 were given in [14] . These codes,
Timeblockcodesbasedonorthogonaldesignswereproposed knownasEASTcodes,combineamodifiedversionofPerfect
in [1],[2],[3]. Clifford Algebras were proposed as a means codes [15] with Alamouti embedding. Their rates are limited
to design square orthogonal designs in [3]. It is known to 1 R N/2 complex symbols per channel use, but
≤ ≤
that [4],[5],[6] orthogonal designs offer single real symbol they posses the non-vanishing determinant property. In [16]
decodability and full diversity. However the rates offered by a new class of codes called fast-group-decodable(FGD) were
introduced.Thesecodescombinethelowdecodingcomplexity for 2m antennas. Particular examples of codes from this
properties of multigroup and FD codes. class are
0 1 1 0
Let X = and Z = . Note that both X – Code for N = 4 transmit antennas, rate R = 5/4
(cid:18)1 0(cid:19) (cid:18)0 −1(cid:19) with a complexity of the order of M2 for arbitrary
and Z are Hermitian and unitary. The four matrices I ,X,Z
2 constellations and M1.5 for constellations carefully
and iXZ are known as the Pauli matrices. They form a C-
chosen to reduce decoding complexity. This is in
linear basis of C2×2. The finite group G generated by the
m comparisonwiththecodegivenin[19]and[20]with
mth order tensor products of the Pauli matrices is called
complexity M2.5 for arbitrary constellations.
the Pauli group. It consists of all possible m fold tensor
– Code for N = 4 and R = 17/8 with a complexity
products of the Pauli matrices together with multiplicative
of the order of M5.5 and M5 for arbitrary and
factors 1, i, i.e.,
± ± carefully chosen constellations respectively. This is
G = iµB B µ Z and B I ,X,Z,iXZ . in comparison with the code given in [16] with a
m 1 m 4 k 2
{ ⊗···⊗ | ∈ ∈{ }} complexity of M6. It is also shown that the STBC
The following subest of Gm, given in [16] is a specific case of STBCs obtainable
Λ = iλB B λ Z and B I ,iX,iZ,ZX from codes over F4.
m 1 m 2 k 2
is a bas{is for⊗C·2m··×⊗2m a|s a∈vector space∈ov{er R. }} • We construct STBCs from codes over F4 that meet the
rate-ML decoding complexity tradeoff of a particular
We relate the set Λm to a subset of Fm4 +1, where F4 is the class of CUWDs obtainable from Extended Clifford Al-
finitefieldwith4elements 0,1,ω,ω2 satisfyingtherelations gebras.
{ }
1+ω =ω2, by defining the map • It is shown that almost all the low decoding complexity
ψ : I ,iX,iZ,ZX F . STBCs available in the literature can be obtained via
{ 2 }→ 4 codes over F .
4
that sends • We show that full-diversity STBCs with a decoding
complexity of M2m−1R−0.5 can be obtained by using
I 0, iX 1, iZ ω, ZX ω2. (5)
2 → → → → regularPAMconstellationforasubsetofrealsymbolsin
Note that every element t∈Λm can be uniquely written as tthheesdeecsoigdnesghivaevneitnhe[2le0a].stFkonrorwatnesd3e/co2d≤ingRc≤om2pmle−x2it+y2f1omr
t=iλψ−1(ξ1) ψ−1(ξm) for λ Z2 and ξk F4. (6) 2m antennas.
⊗···⊗ ∈ ∈
In fact, the map ϕ:Λ F Fm that sends • STBCs obtainable by our approach are shown to include
m → 2⊕ 4 as special cases, the R = 2 codes, proposed in [21], for
iλB B [λ,ψ(B ),...,ψ(B )] 2 and 4 antennas having the largest known coding gain
1 m 1 m
⊗···⊗ → andthefast-decodablerate2codefor2transmitantennas
is a one to one correspondence between Λm and F2 proposed in [22], [23].
Fm4 . The Hamming weight wt([λ,ξ1,...,ξm]) of a vecto⊕r • We show that if a design has only full-rank linear dis-
[λ,ξ1,...,ξm]∈F2⊕Fm4 is defined as persion matrices, then a full diversity STBC can always
m be constructed from it by encoding each real symbol
wt([λ,ξ ,...,ξ ])=1 λ=0 + 1 ξ =0 . independently.
1 m i
{ 6 } { 6 }
Xi=1 The contents of this paper are organized as follows. In
InthispaperweconstructlowdecodingcomplexitySTBCs Section II preliminary results and the necessary background
from designs that use elementsfrom the Pauli groupas linear and notations are established. Results regarding full diversity
dispersion matrices. This problem is converted to one of are given in Section III. The focus of the paper then shifts to
findingcertaincodesoverF4,whicharesubsetsofF2⊕Fm4 ,in designinglowcomplexitydesignsratherthanspecificSTBCs.
which checking the conditions for low decodability becomes Clifford Algebras and Pauli matrices are introduced in Sec-
simpler. The contributions of this paper are as follows. tion IV and the problem of designinglow complexitydesigns
• It is shown that when designs are constructed by using from Gm is converted to a problem in Fm4 +1. Examples of
elementsofPauligroupasweightmatrices,theHurwitz- STBCs in the literature that can be obtained via codes over
Radon orthogonalitycondition (4) can be easily checked F4 are given in this section. In Section V, it is shown that
by transferring the problem to the corresponding F - almost all known multi group decodable codes are obtainable
4
domain. This facilitates both the description and the viacodesoverF4.Also,anewclassofmulti-groupdecodable
constructionoflowdecodingcomplexitycodesintheF - codes is presented. In Section VI new fast-group-decodable
4
domain. and fast-decodable codes are presented. Concluding remarks
• A new class of full-diversity fast-group-decodable and are made in Section VII.
fast-decodable codes is reported. The codes of this class Notation:ForacomplexmatrixAthetranspose,theconjugate
have the lowest decoding complexity among all known and the conjugate-transpose are denoted by AT,A¯ and AH
codesforrates1<R<3/2and2m−2+ 1 <R 2m respectively. A 2 is the Frobenius norm of the matrix A.
2m ≤ || ||F
A B istheKroneckerproductofmatricesAandB.In isthe subsetsΓ1,...,Γg andifthereexistfinitesubsets i R|Γi|,
⊗ A ⊂
n nidentitymatrixand0istheallzeromatrixofappropriate i=1,...,g such that the following conditions are satisfied
×
dimension.Theemptysetisdenotedbyφ.Cardinalityofaset 1) The weight matrices A , l N of the design X are
l K
Γ is denoted by Γ and i = √ 1. 1 is the indicator such that ∈
| | − {···}
function.For a square matrix A, det(A) is the determinantof
A. For a positive integer n, Zn is the set {0,1,...,n−1}. AHk Al+AHl Ak =0 whenever l ∈Γi,k∈Γj and i6=j
For a complex matrix A, A and A denote its real and
Re Im 2) (X, ) is g-group encodable with respect to the par-
imaginarypartsrespectively.vec(A)isthevectorizationofthe C A
tition Γ ,...,Γ of the set N and the signal sets
matrix A. 1 g K
,..., .
1 g
A A
II. PRELIMINARIES We now turn to fast-decodable codes. Consider an STBC
In this section, multigroup decodability, fast-decodability (X, ) being used for communication via the MIMO chan-
C A
[11] and fast-group-decodability [16] of STBCs and designs nel (1). For a complex matrix A, define
are formally defined with the help of the notion of group-
v˜ec(A)=[vec(A )T vec(A )T]T
decodability. Re Im
Let N denote the set 1,...,K . For any K-tuple x = In [17] it was shown that the ML decoding problem can be
K
{ }
(x ,...,x ) and non-empty set Γ N define reduced to one of finding
1 K K
⊆
xΓ =(xi1,xi2,...,xi|Γ|) arg minx∈A||y−Hx||2F
wherein Γ= i1,...,i|Γ| . The idea of encoding complexity where
{ }
was first introduced in [10]. We now define g-group encod-
ability of an STBC (X, ). y =v˜ec(Y) x=[x1,...,xK]T
C A
Definition 1: Let g be any positive integer. An STBC
= v˜ec(A H) v˜ec(A H) v˜ec(A H)
(X, ) obtained from a design X and a signal set is said H 1 2 ··· K
C A A (cid:2) (cid:3)
to be g-groupencodableif there exists a partition of NK into Let the QR decomposition of be =Q where is a
H H R R
non-empty subsets Γ1,...,Γg and there exist finite subsets K K upper triangular matrix. The ML decoding problem
i R|Γi|, i=1,...,g such that now×reduces to
A ⊂
K arg min QTy x 2
C(X,A)={ alAl|aΓi ∈Ai for i=1,...,g} x∈A|| −R ||F
Xl=1 The ith column of is associated with the symbol xi.
R
In short, for a g-group encodable STBC the tuples Consider the case when is of the following form
R
xΓ1,...,xΓg are assigned values independentlyof each other T1 0 ... 0 B1
du(Xrin,ge)ncisodsiinnggl.eIfrefoarlesaycmhbio,l|Γeni|c=oda1b,lwe.eFsoaryathdatesthigenSXTBiCn 0 T2 ... 0 B2
. . . .
KC reaAl variables and a non-empty subset Γ N define the = .. .. ... .. .. (7)
design X as follows ⊆ K R 0 0 ... T B
Γ g g
0 0 ... 0 T
X = x A g+1
Γ i i
Xi∈Γ mwhaterriecegs≥an1d,BTi,,iin=ge1n,e.r.a.l,,gar+er1ecatraengsquulaarr.eDuepnpoetretbryianΓ˜gutlhaer
We now define the g-group maximum-likelihood (ML) i i
set of indices of the columns associated with the matrix T .
decodability of an STBC (X, ). i
Definition 2: An STBCC (XA, ) is said to be g-group During sphere-decoding[24], [25] when values are proposed
C A for the symbols x l Γ˜ , the remaining symbols
ML decodable if there exists a partition of NK into g become conditiona{llyl|g-g∈roupgd+e1c}odable, provided they have
non-empty subsets Γ ,...,Γ and if there exist finite sub-
1 g been encoded independently. This reduces the ML decoding
sets Ai ⊂R|Γi|, i = 1,...,g such that the ML decoder, complexitygreatly.SuchanSTBCissaidtobefast-decodable.
arg min Y CH 2, decomposes as
C∈C(X,A)|| − ||F Here we call such STBCs as being ‘conditionally g-group
g decodable’. Formal definition of a fast-decodable STBC is
arg minCi∈C(XΓi,Ai)||Y −CiH||2F given below.
Xi=1 Definition 3: Let (X, ) be such that for every channel
C A
Such a decomposition reduces the decoding complexity realizationH,theuppertriangularmatrix (H)isoftheform
from g computations to g computations. givenin(7).Iffurtherthetuplesx ,i=1R,...,g areencoded
FollowQinig=1th|eAoir|em gives a set of suPffiic=ie1n|At ci|onditions for g- independently then (X, ) is saiΓ˜di to be fast-decodable.
C A
group decodability of an STBC. Thefollowingtheoremgivestherelationbetweenthestruc-
Theorem 1 ( [6], [8], [10]): AnSTBC (X, )isg-group ture of and the linear dispersion matrices of the design X.
C A R
decodable if there exists a partition of N into g non-empty
K
Theorem 2 ( [21]): Let r (H) be the element in the 3) X is said to be fast-group-decodable if X is g-group
l,k
lth row and kth column of (H). For any 1 l < decodable with respect to the partition Γ ,...,Γ and
1 g
AkH≤A K+AaHndAan=y0c.hannel realRization H, rl,k(H) ≤= 0 if tdheecroedaebxlies.ts at least one Γi such that XΓi is fast-
k l l k
This meansthatan STBC (X, ) will be conditionallyg-
C A III. ON FULL DIVERSITY
groupdecodableiftheSTBCobtainedbysuppressingasubset
of the variablesis g-groupdecodable.Moreformallywe have In this section we give a sufficient condition for a design
the following theorem which gives a sufficient condition for X to give rise to full-diversity STBC (X, ) via single real
C A
an STBC to be fast-decodable. symbol encoding. This result is then used to show that every
Theorem 3: Consider a λ-group encodable STBC (X, ) lowcomplexitydesignpresentedinthisworkcanbecombined
with the encoding groups being Γ ,...,Γ . (X, )Cis faAst- with an appropriate signal set to get a full diversity STBC
1 λ
decodable if Γ˜ ,...,Γ˜ N are such thatCwe haAve without compromising on decoding comfort.
1 AHkgA⊆l+KAHl Ak =0 (8) symLebtoXlsnx=1,P..ni.=,1xxniAainbdelaent ANi×NCNli×nNea,ride=sig1n,.i.n.n,nrebael
{ } ∈
full-rank. Given a set of n positive integers Q , i = 1,...,n
whenever i
we are interested in finding a real constellation R for
l∈Γp,k ∈Γq and p6=q. (9) the real symbol x with =Q for each i=1A,.i..⊂,n. The
i i i
|A |
Proof:FromTheorem2and(8)itisclearthatr (H)= constellations must be such that the specific STBC obtained,
l,k
0Γ˜ faonrdainy=jch.TanhnueslwriethalaizpaptrioonpriHaterwehoerdneervienrgolft∈hecΓ˜oil,ukmn∈s eCs(tXabnli,sAhi1ng×t·h·e·m×aAinnr)e,sumltuostf tbheis soefctfiuolnl-wdieveirnstirtoyd.uTcoewsoamrdes
j
6
the matrix (H) can be brought to the form in (7). It only notation.
R
remains to show that the tuples x , i=1,...,g are encoded For each i 1,...,n let
independently. But (9) assures exΓ˜aictly this. Thus (X, ) is = a [0],a [1],...,a [Q 1∈] , where{ a [j] } R.
i i i i i i
fast-decodable. C A FAor a{n n tuple u=(u ,.−..,u}) Z Z ∈ Z
1 n ∈ Q1 × Q2···× Qn
Fast-Group-Decodable(FGD) STBCs were recently intro- define
duced in [16].
Definition 4 ( [16]): An STBC (X, ) is said to be fast- Cn[u]=Xn(a1[u1],a2[u2],...,an[un])
C A n
group-decodableif it satisfies the following conditions
= a [u ]A (11)
i i i
1) Forsomeg >1,C(X,A)isg-groupdecodablewiththe Xi=1
groupsandthesignalsetsasΓ ,...,Γ and ,..., .
1 g A1 Ag Hence the single real symbol encodable STBC obtained by
2) There exists an i 1,...,g such that the STBC
TheCc(oXmΓpi,leAxiit)yisisfaresdt-ud∈ceecdo{dbaebclaeu.se}of two reasons. Firstly usiCn(gXtnh,eAg1iv×en·c·o·n×stAelnla)ti=on{sCsant[iusfi]|eus∈ZQ1 ×···×ZQn}
multigroup decodability decreases the number of computa-
We see that the codewords are indexed by the elements of
tions. Further there is at least one component code which is Z Z .
ffaosrt-4detrcaondsambilte.aSnutecnhnaasloawndcoramtepl1e7x/it8yccsopdceu.wasgivenin[16] QT1h×eo·re·m·×4: QLnet Xn = ni=1xiAi be an N × N linear
design in n real variables wPith full-rank weight matrices Ai.
Sofarwehaveonlydiscussedaboutthedecodingcomplex- Let R, i = 1,...,n be such that = Q and
itthyeonfoatinonSToBfCtheC(dXec,oAd)in.gWceoemnpdletxhiitsysoecftiaondebsyiginntXro.duFcroinmg C(XnA,iA⊂1×···×An)isoffulldiversity.Let|AAin|+1 ∈CiN×N
beanyfullrankmatrixandQ beanypositiveinteger.Then
n+1
Theorem1itisclearthatthedecodingcomplexityofanSTBC there exists a one dimensional real constellation R
(X, ) is affected by the choice of both the design X and An+1 ⊂
such that
C A
the signal set . By the decoding complexity of a design we
dreifseprertsoiothnemaamtAroiucenstocfonctormibpulteexsittoytthheatdtehceocdhinogicecoomfpthleexliitnyeoarf 12)) |TAhne+S1T|=BCQCn+(X1 n+1,A1×···×An+1) offers full di-
versity.
the STBC. The following definition makes this formal.
Definition 5: Consider a design X= K x A . Proof: Proof is given in Appendix A.
i=1 i i We now present the main result of this section in the
1) X is said to be g-group decodabPle if there exists a
following theorem.
partition of N into g non-empty subsets Γ ,...,Γ
K 1 g Theorem 5: For any given N N square linear design
such that Xn = ni=1xiAi with full-rankw×eight matrices Ai and pos-
AHk Al+AHl Ak =0 whenever l ∈Γi,k ∈Γj and i(6=10j) iiti=ve1i,n.Pt.e.g,enrssQuc1h,.t.h.a,tQn, there exist constellations Ai ⊂ R,
2) Xissaidtobefast-decodableifthereexistsanon-empty 1) =Q for i=1,...,n
i i
subset Γ NK such that the design XΓ is g-group 2) |TAhe| single real symbol encodable STBC
decodable⊆for some g >1. (Xn, 1 n) offers full diversity.
C A ×···×A
Proof:Proofisbyinduction.Thetheoremisshowntobe which is full-rank. Since the rank of any square matrix A
true for n=1 here. Theorem 4 is the induction step. is equal to the rank of AHA, det(C [u] C[v]) = 0. Thus
l l
Consider the design for one real symbol X1 = x1A1. the STBC (Xl, 1 l) is of full−diversity6. By using
Choose any R with =Q . The codewords are Theorem 4CrepeaAtedl×y ·K··×lAtimes with integers Q and the
1 1 1 i
indexed by elAeme⊂nts in Z .|AFo|r any u,v Z and u = v matrices A , l <i K w−e get the desired signal set and the
Q1 ∈ Q1 6 i ≤
we have STBC.
det(C [u] C [v])=det((a [u] a [v])A ) IV. LOW DECODING COMPLEXITY STBCS VIA CODES
1 1 1 1 1
− − F
=(a [u] a [v])Ndet(A ) OVER 4
1 1 1
−
Inthissection,we makeuse ofUniversalCliffordAlgebras
=0 (12)
6 and their matrix representations over the complex field, to
Since the difference matrix of any two codewords is of full- constructlowdecodingcomplexitycodesviacodesoverF .A
4
rank, the STBC (X1, 1) offers full diversity. subset of elements in the Universal Clifford Algebra are seen
C A
The STBC obtained from Theorem 5 is single real sym- to have multiplicative properties similar to (4). We attempt to
bol encodable. Thus if X were a g-group decodable, fast- obtain low complexity designs by choosing linear dispersion
decodable or fast-group-decodable design then the resulting matricesfromthematrixrepresentationoftheseelements.We
STBC (X, ) is a g-group decodable, fast-decodable or proceed in this direction by using a theorem that establishes
C A
fast-group-decodableSTBCrespectively.Mostimportantlythe an isomorphism between a Universal Clifford Algebra and
condition that the linear dispersion matrices are full-rank a full matrix algebra of appropriate dimension over C. It
ensures that the STBC (X, ) offers full-diversity. is then observed that the tensor product of Pauli matrices
C A
All the designs discussed in this paper have unitary, and are a double cover of the set elements in question from the
hence, full-rank weight matrices. In the remaining sections of Universal Clifford Algebra. The connection with vectors over
thepaperthefocusisondesigninglowcomplexitydesignsand F ismade.Theproblemoffindinglowcomplexitydesignsis
4
noton thedesignof signalsets . Sinceit hasbeenshownin then converted to one of finding a set of vectors in F Fm.
A 2⊕ 4
Theorem 5 that there exist signal sets leading to full-diversity ExamplesoflowcomplexitySTBCs availablein theliterature
STBCs without increasing the decoding complexity than that that are obtainable from codes over F are given.
4
imposedbythedesignitself,theonlyproblemleftistodesign Let n be any positive integer. Denote by N the set
n
signal sets that maximize the coding gain without increasing 1,...,n . Let e ,...,e be elements of an associative
1 n
thedecodingcomplexity.Thisproblemisnotaddressedinthis {algebra o}ver C and α N . For any nonempty sub-
n
⊆
paper. set α= i ,...,i with i <i < <i we define
1 |α| 1 2 |α|
{ } ···
The following theorem will be useful when we are con- e =e e e and e =1.
α i1 i2··· i|α| φ
structing STBCs with low decoding complexity. Definition 6 ([27]): A UniversalClifford Algebra is an
n
Theorem 6: Let X = K x A be an N N design associativealgebraoverC witha multiplicativeidentiUty1and
i=1 i i ×
in K real symbols with fPull-rank weight matrices and l generatedby n objects e1,...,en which satisfy the following
∈
1,...,K be such that equations
{ }
e e +e e =0 whenever i=j (14)
AHA +AHA =2δ I for all 1 i,j l (13) i j j i 6
i j j i i,j N ≤ ≤ e 2 = 1 for i=1,...,n (15)
i
Given positive integers Q ,...,Q and any set of real con- −
1 K
stellations A1,...,Al with cardinalities Q1,...,Ql respec- {eα|α⊆Nn} is a basis for Un (16)
tively, there exist real constellations , l<i K such that
1) The STBC (X, 1 AKi) offers≤full diversity. e F2r=om 1(1.4A)lsaondfo(r1a5n)yitα,iβs cleNar theaitthefrore aeny+αe ⊆e N=n0,
C A ×···×A α n α β β α
2) =Q for l<i K. ± ⊆
i i or e e e e =0. This property resembles (4) except for
|A | ≤ α β β α
Proof: Consider the design Xl = li=1xiAi and the the conju−gate-transpose. Hence by representing the basis ele-
STBC (Xl, 1 l) generated uPsing the signal sets ments using either Hermitian or skew-Hermitian matrices we
C A ×···×A
,..., for the independentvariables x ,...,x . Because can obtain linear dispersion matrices that are Hurwitz-Radon
1 l 1 l
A A
{A1,...,Al}satisfythecomplexHurwitz-Radonmatrixequa- orthogonal. Together with the fact that eα2 = ±1 it is clear
tions (13) we have [2], thatwe needunitaryrepresentationof thebasis elements.The
followingtheoremgivesarepresentationofaclassofClifford
l l
XlHXl = x2iAHi Ai =( x2i)IN Algebras.
Xi=1 Xi=1 Theorem 7 ( [27]): Foranypositiveintegerm,theClifford
Algebra isisomorphictothefullmatrixalgebraC2m×2m.
Tu,hvus∈ZfQo1r×a··n·y×ZrQeallwitshiguna6=l vsweteshavAe1,...,Al, and TanheiseoxmteoUnrp2smihoinsmofotfhtehemaalpgeebkra→s, wEhkefroerfkor=s1=,.1.,..,.2.m,mg,ivEes
s
l and E are the m-fold tensor products given by
s+m
(C [u] C [v])H(C [u] C [v])= (a [u ] a [v ])2I
l l l l i i i i N
− − Xi=1 − Es =i(Z⊗···⊗Z⊗iXZ⊗I2⊗···⊗I2)
E =i(Z Z X I I ) Proof: Since ψ is one to one, it is clear that ϕ is one to
s+m 2 2
⊗···⊗ ⊗ ⊗ ⊗···⊗ one.Further Λ = F Fm =22m+1.Thusψ issurjective
there being s 1 factors of Z in each product. as well. This|comm|ple|tes2⊕the4pr|oof.
−
is FunroitmaryT,hsekoerwe-mHe7rmwiteiahnavanedththatuseascqhuaEreks,tok =I.1W,.i.t.h,2Emα nexTth.e Hamming weight of a vector in F2 ⊕Fm4 is defined
−
defined similar to eα for α N2m we see that all the basis Definition 7: The(Hamming)weightwt([λ,ξ ,...,ξ ])of
⊆ 1 m
elementsarerepresentedintermsofunitarymatrices.Theiso- a vector [λ,ξ ,...,ξ ] F Fm is defined as
morphismensuresthatthesematricesarelinearlyindependent 1 m ∈ 2⊕ 4
over C. Since we are concerned with the transmission of real m
sRy-mlinbeoalsr wbaesinsoftoerthCa2tmB×2=m{.iWλEithα|λ∈dZe2finaneddaαs⊆Nb2bm} is a, wt([λ,ξ1,...,ξm])=1{λ6=0}+Xi=11{ξi 6=0}
−B {− | ∈B}
we have the following proposition. Any matrix t Λ is either Hermitian or skew-Hermitian.
m
Proposition 1: =G . Thisinformation∈abouttispresentinwt(ϕ(t)).Thefollowing
m
B∪−B
Proof: We note that both X and Z square to I . Further, proposition explains this claim.
2
they anticommute. So it is clear that for any α ⊆ N2m, Proposition 4: A matrix t ∈ Λm is Hermitian if wt(ϕ(t))
E G . Since G is closed under multiplication by is even. Else it is skew-Hermitian.
α m m
piIr2omof∈awned n−oIte2mthweree haraeve22Bm∪d−isBtin⊆ctGsumb.seTtsoocfomNp2lmet,eththues andλP∈rooZf2:.LEextcte=ptiIλ2B,w1h⊗ich··is·⊗maBpmpe,dBtoi ∈0u{nId2,eirXψ,,itZhe,ZotXhe}r
= 22m+1. Since is linearly independent over R, for 3 matrices are skew-Hermitian. Hence
|B| B
any b we have b / . Equivalently =φ. Thus,
∈B=22m+2 −= G∈mB. Thus B=∩G−mB. tH =(−1)1{λ6=0}iλ(−1)1{B16=I}B1⊗···⊗(−1)1{Bm6=I}Bm
|BP∪ro−pBos|ition 1 says |that|the weiBgh∪t m−aBtrices to be chosen =(−1)1{λ6=0}iλ⊗mk=1(−1)1{ψ(Bk)6=0}Bk
from the matrix representation of the basis elements of Clif- =( 1)wt(ϕ(t))t
ford Algebra can be equivalently obtained through the Pauli −
group. However, the Pauli group of matrices are not linearly Thus t is Hermitian if wt(ϕ(t)) is even, else t is skew-
independentoverR.Thusweconcernourselveswithaproper Hermitian.
subsetΛ ofG whichismaximallylinearlyindependentand For linear dispersion matrices coming from the set Λ the
m m m
thussatisfiesΛ Λ =G .Onesuchsetisthefollowing g-group ML decodability condition (10) can be reformulated
m m m
∪− in terms of the weight of corresponding vectors in F Fm
Λm = iλB1 Bm λ Z2 and Bk I2,iX,iZ,ZX as follows: 2⊕ 4
{ ⊗···⊗ | ∈ ∈{ }}
(17)
Proposition 5: For any t ,t Λ , we have
1 2 m
∈
Proposition 2: The set Λ is a basis for C2m×2m as a tHt +tHt =0 iff wt(ϕ(t )+ϕ(t )) is odd
m 1 2 2 1 1 2
vector space over R.
Proof: I ,iX,iZ,ZX is a basis for C2×2 as a vector where the vector sum is component wise addition.
2
space over C{. Thus their m}fold tensor products form a C- Proof: Recall that any t∈Λm is either Hermitian or
linearbasisforC2m×2m.Fromthistherequiredresultfollows. skew-Hermitian.ThustH1 t2isskew-Hermitianifft1t2isskew-
Hermitian. Let
theWfiennitoewfipelrdocFeedwbiythre4laetlienmgethnetsset0Λ,1m,ωto,ωF22⊕Fsam4ti.sfCyoinngsidtheer tk =iλkψ−1(ξk,1)⊗···⊗ψ−1(ξk,m) for k =1,2
4
relations 1+ω =ω2. Define a map{ } Note that ψ−1(ζ)ψ−1(η)= ψ−1(ζ+η) for any ζ,η F .
4
± ∈
Hence t t
ψ : I ,iX,iZ,ZX F 1 2
2 4
{ }→ = iλ1+λ2ψ−1(ξ +ξ ) ψ−1(ξ +ξ )
1,1 2,1 1,m 2,m
that sends ± ⊗···⊗
= i(λ1+λ2)mod2ψ−1(ξ +ξ ) ψ−1(ξ +ξ )
1,1 2,1 1,m 2,m
± ⊗···⊗
I2 →0, iX →1, iZ →ω, ZX →ω2 (18) =±ϕ−1(ϕ(t1)+ϕ(t2))
Note that every element t Λ can be uniquely written as We have t t is skew-Hermitian iff ϕ−1(ϕ(t )+ϕ(t2)) is
∈ m 1 2 1
skew-Hermitian. Applying Proposition 4 we have that t t is
t=iλψ−1(ξ1) ψ−1(ξm) for λ Z2 and ξk F4 skew-Hermitian iff wt(ϕ(t )+ϕ(t )) is odd. This comp1le2tes
⊗···⊗ ∈ ∈ 1 2
(19)
the proof.
When we restrict the possible linear dispersion matrices to
Proposition 3: The map ϕ:Λ F Fm that sends
m → 2⊕ 4 the set Λm, Proposition 5 helps us reformulate the original
problem of finding weight matrices for low decoding com-
iλB B [λ,ψ(B ),...,ψ(B )]
1⊗···⊗ m → 1 m plexity STBCs in terms of finding vectors in F Fm. This
2⊕ 4
is a one to one correspondence between Λ and F Fm. leads to the following theorem.
m 2⊕ 4
Theorem 8: If there exist K distinct vectors A. Alamouti Code
y ,...,y F Fm and a partition Γ ,...,Γ of
1 K ∈ 2⊕ 4 1 g TheAlamoutiCode[28]isa2 2squareorthogonaldesign
1,...,K into nonempty subsets such that ×
{ } ofrate1.Itsparametersare:m=1,K =4,g =4andτ =1.
wt(yk+yl) is odd whenever k ∈Γi, l ∈Γj and i6=j (20) wItseilginhetamradtirsipceesrsbioenlomngattroiceΛs .arTeh:e{fIo2u,riXse,tsiZo,fZvXec}to.rAslflrothme
1
then there exists a design X(x1,...,xK) of dimension F2 F4 corresponding to the four groups are as follows:
2m 2m in K real variables and which is g-group ML ⊕
×
decodable with the ith group being x k Γ . = [0,0] , = [0,1] , = [0,ω] , = [0,ω2]
k i 1 2 3 4
{ | ∈ } S { } S { } S { } S { }
Proof: Given the K vectors as in the hypothesis, define
A =ϕ−1(y ) Λ . The bijective nature of ϕ ensures that It can be seen thatthe weight of the sum of any two different
k k m
∈
the K matrices A are distinct. Since Λ is linearly inde- vectors is odd, thus the above design is single real symbol
k m
pendent over R, A ,k = 1,...,K are linearly independent. decodable.
k
Define a linear space-time design X(x ,...,x ) as
1 K
B. Other 2 2 codes of rate 1
K ×
X= xiAi Here we describe designs with parameters m =1, K = 4,
Xi=1 g = 2, τ = 2, R = 1. There are only three non-equivalent
Applying Proposition 5 we get the ith group as Γ . This designs that can be obtained from Λ1. They are parametrized
i
completes the proof. by l 0,1,2 and are given by
∈{ }
Theorem8convertstheoriginalproblemoffindingg-group
= [0,0],[1,ωl]
ML decodable designs from Λm to that of finding certain S1 { }
codes over F . Once such a code is chosen in F Fm the
linear dispers4ion matrices can be obtained by the2o⊕ne-t4o-one 2 = [0,ωl],[1,0]
S { }
correspondence ϕ.
Definition 8: A design in K real symbols for 2m antennas 1) l = 0: The design is given by 1 = [0,0],[1,1] ,
S { }
from=FK2.⊕Fm4 is defined as a subset S ⊆F2⊕Fm4 such that S21=={[0I,,1],X[1,0,]a}n.dThe2tw=o giXro,uipIs.oWf witheigΓh1t=mat1r,ic2esaanrde
|S|The ‘design’ in the above definition refers to the lin- ΓM2 = 3{,4−the}resultiMng desi{gn is } { }
S { }
ear design that can be obtained by mapping the vectors in
x +ix x +ix
= y1,...,yK to linear dispersion matrices in Λm. The X= 1 4 − 2 3
Scorre{spondingde}sign in K real variables x1,...,xK is given (cid:18)−x2+ix3 x1+ix4 (cid:19)
as
This is the 2 2 ABBA code [30].
K ×
X= x ϕ−1(y ) (21) 2) l = 1: The design is given by
i i
Xi=1 S1 ={[0,0],[1,ω]}, S2 ={[0,ω],[1,0]}. The matrices
are = I, Z and = iZ,iI . With Γ = 1,2
With the linear design X associated with the set defined as M1 { − } M2 { } 1 { }
in (21) we can use Theorem 8 to define g-groupSdecodable, and Γ2 ={3,4} the design is given by
FD and FGD designs obtainable from codes over F . These
4 x x +i(x +x ) 0
are given below. X= 1− 2 4 3
(cid:18) 0 x1+x2+i(x4 x3)(cid:19)
Definition 9: Let be a design obtainable from F Fm. −
S 2⊕ 4
1) = g or equivalently the set k =1,...,g With appropriate transformation of the symbols within each
iSs cal∪leid=1aSgi-group decodable design{ifSfko|r any y }, group we get the 2 2 CIOD [6].
z and k =l we have wt(y+z) is odd. ∈Sk ×
l
∈S 6 x˜ +ix˜ 0
2) is said to be fast-decodable if there exist subsets X= 1 3
S , i=1,...,g, g >1 such that i=1,...,g (cid:18) 0 x˜2+ix˜4(cid:19)
i i
S ⊆S {S | }
is a g-group decodable design.
3) l = 2: = [0,0],[1,ω2] and
3) A g-group decodable design {Si|i=1,...,g} is said = [1,0],[0,ω2] . The lineSar1 di{spersion mat}rices are
to be fast-group-decodable if there exists at least one S2 { }
= I,iZX and = iI,ZX . With Γ = 1,2
l∈{1,...,g} such that the designSl is fast-decodable. aMnd1Γ {= 3,4}the reMsul2ting{design i}s 1 { }
2
We now give examples of low complexity designs obtain- { }
able from codes over F4 i.e., linear designs whose weight X= x1+ix3 x4+ix2
matrices come from Λm. Assume that the number of groups (cid:18) x4 ix2 x1+ix3(cid:19)
− −
is g and each group has τ vectors in it. The total number of
vectors or the total number of real symbols in the design is a b
This is the design.
thus K =gτ. (cid:18) b a(cid:19)
−
C. 4 4 Quasi-orthogonaldesign V. KNOWNAND SOME NEWMULTIGROUP DECODABLE
Co×nsider the rate 1 quasi-orthogonal design given in [29] STBCS FROMCODES OVERF4
for 4 transmit antennas. The design contains 8 real symbols In this section we construct multigroup decodable designs
x ,...,x and is 4-group decodable. The parameters are via codes over F . We give three procedures to obtain a
1 8 4
m = 2, K = 8, g = 4, τ = 2 and R = 1. The design multigroup decodable design for 2m+1 antennas by using a
X= 8 x A is given below. multigroup decodable design for 2m antennas. These con-
i=1 i i
P structions are then used to obtain 4-group decodable codes.
x +ix x +ix x +ix x +ix
1 2 3 4 5 6 7 8 Lastlyg-groupdecodablecodesforarbitraryg areconstructed
X=−xx3++iixx4 xx1−+ixix2 −xx7+ixix8 xx5−iixx6 fromcodesoverF4.Thesedesignsmeettherate-MLdecoding
− 5 6 − 7 8 1− 2 3− 4 complexitytradeoffattainablebyaclassofCUWDsobtainable
x7+ix8 −x5−ix6 −x3−ix4 x1+ix2 from Extended Clifford Algebras [10].
The linear dispersion matrices, upto a sign change, A , i =
i
A. Construction A
1,...,8 are
Letusdenote[1,0,...,0] F Fm byδ .Thefollowing
A1 =I2⊗I2 A2 =iZ⊗Z proposition describes how to∈co2n⊕stru4ct a g-mgroup decodable
A =I ZX A =iZ X
3 2⊗ 4 ⊗ design for 2m+1 antennas using a g-group decodable design
A =ZX I A =iX Z
5 ⊗ 2 6 ⊗ for 2m antennas.
A =ZX ZX A =iX X
7 ⊗ 8 ⊗ Proposition 6: Let l 0,1,2 and
∈ { }
It can be seen that all the 8 matrices come from Λ . The = y j =1,..., Γ i=1,...,g be a 2m 2m
2 i i,j i
corresponding vectors in F Fm, y =ϕ(A ) are {gS-group{dec|odable desi|gn|o}f| rate R. The}n ˜ i = 1,.×..,g
2⊕ 4 i i {Si| }
is a 2m+1 2m+1 g-groupdecodabledesign of rate R, where
y1 =[0,0,0] y2 =[1,ω,ω] ˜ = × with
y =[0,0,ω2] y =[1,ω,1] Si Si,A∪Si,B
3 4
y5 =[0,ω2,0] y6 =[1,1,ω] i,A = [yi,j,0]j =1,..., Γi
y =[0,ω2,ω2] y =[1,1,1] S { | | | }
7 8
= [y ,ωl]+δ j =1,..., Γ
i,B i,j m+1 i
The 4 groups are S { | | | }
If for any i 1,...,g
= y ,y = y ,y ∈{ }
1 1 7 2 2 8
S { } S { }
S3 ={y3,y5} S4 ={y4,y6} wt(y+z) is even for any y,z ∈Si
It can be seen that for any two vectors in differentgroupsthe then
weight of their sum is odd. wt(y+z) is even for any y,z ˜.
i
∈S
D. Square Orthogonal Designs
Proof: Consider i,j 1,...,g and i = j. It is
∈ { } 6
Square Orthogonal Designs [3] are square designs straightforward to show that =φ and for any
i,A i,B
X(x1,...,xK) which satisfy the following equation y ∈ S˜i and z ∈ S˜j, wt(y+zS) is∩odSd. The second part of
the theorem is straightforward. This completes the proof.
XHX=( K x2i)I tioLneotny =1,[.λ.,.ξ,1m,....D,ξemfi]ne∈σF2:F⊕Fm4Famnd σFbe anFympearsmuta-
Xi=1 { } 2⊕ 4 → 2⊕ 4
Such designs offer both single real symbol decodability and σ(y)=[λ,ξ ,...,ξ ]
σ(1) σ(m)
full diversity when arbitrary real constellations are used to
σ is thusa permutationof coordinatesof y. In terms of linear
encode each of the real symbols x . Maximal rate square
i
dispersion matrices the action of σ is to permute the order
orthogonal designs were given in [3]. These designs are of
in which the 2 2 matrices appear in the Kronecker product
ddiemsiegnnssioanre2omb×tai2nmabalendfrhoamvecaodreasteoovfeRr F=.mT2h+me1recsaprceu2.mTh+es2e representation.×The following proposition states that g-group
4
decodabilityofadesignisnotdisturbedbysuchapermutation
groups containing one vector each. These vectors y are
k
{ } when it is applied to all the weight matrices.
given below. For k=1,...,m,
Proposition 7: Let F Fm be a g-group decodable,
S ⊆ 2⊕ 4
y =[1 k is even ,0,...,0,ω2,ω,...,ω], FD or FGD design and σ be any permutation on 1,...,m .
k
{ } Then ˜= σ(y)y is a g-group decodable,{FD or FG}D
yk+m =[1 k is even ,0,...,0,1,ω,...,ω], designSresp{ective|ly.∈S}
{ }
Proof: The action of σ on the vectors is just a
there being m k zeros in each vector and
− permutationof the coordinates.Thus for any y,z F Fm,
y =[1 m is even ,ω,...,ω], σ(y+z)=σ(y)+σ(z) and wt(σ(y))=wt(∈y).2⊕Th4us
2m+1
{ } wt(σ(y) + σ(z)) = wt(y + z). The desired result follows
y2m+2 =[0,0,...,0]. from Definition 9.
Corresponding to l = 0,1 and 2 in Proposition 6 and σ B. Construction B
in Proposition 7 we get different constructions that give us a
The following proposition gives a procedure to obtain 2-
g-group decodable design for 2m+1 antennas by using a g- group decodable designs for 2m+1 antennas using 2-group
group decodable design for 2m antennas. We now give some decodable designs for 2m antennas.
constructions obtained from Proposition 6 as examples. Let
Proposition 8: Let l 0,1,2 and
lXine=arPdiKis=p1erxsiioAnimbeatraicge-sgirnouΛpmdaencdodWablbeedaensiigdnenwtiictahladlelsitghne {gSroiu=p {dyei,cjo|jda=bl1e,..d.e,s|iΓgni|}|wi=hic1h,∈2}satibsefiesa{ th2em×}fo2llmowin2g-
in a differentset of realvariables. Then each of the following
condition for each i=1,2
designs is g-group decodable.
wt(y+z) is even for any y,z
i
X W ∈S
(cid:18)W X(cid:19) (22) t2h-egnroSu˜1p=deSc1o,dAa∪blSe2d,Be,siSg˜2n=˜S2,,A˜∪Sw1,hBicghivseatais2fimes+1f×or2ema+ch1
1 2
{S S }
i=1,2
X W 0
(cid:18) −0 X+W(cid:19) (23) wt(y+z) is even for any y,z ∈S˜i
where
(cid:18) XiW iXW(cid:19) (24) Si,A ={[yi,j,0]} and Si,B ={[yi,j,ωl]}
− Further the rates of and ˜ are same.
i i
Proposition7isusedalongwithProposition6toarriveatthese Proof: Similar t{oSth}e proo{fSo}f Proposition 6.
constructions. Construction (23) can be used to obtain single As particular examples we get the following recursive
complex symboldecodable square CIODs [6] by using single constructions. Let X and W be identical 2-group decodable
real symbol decodable square orthogonal designs as building designsin differentvariableswith the weightmatrices in Λ .
m
blocks. Construction (22) was first proposed in [30] and is Let the design X be such that the corresponding vectors
known as ABBA construction. As an example we describe in F Fm satisfy the hypothesis of Proposition 8. The
how ABBA construction is obtained from Proposition 6. follow2i⊕ng d4esigns are 2-group decodable.
Let i=1,...,g be a g-group decodable design and
S =∪gi{=S1iS|i. Then ϕ−1}(S)={A1,...,AK} is the set of lin- (cid:18)iXW iWX (cid:19) (25)
ear dispersion matrices. Let x ,...,x and w ,...,w
1 K 1 K
betwodifferentsetsofrealv{ariables.Defi}neX{= Ki=1xiA}i X+iW 0
acnodnstWruct=edPaKic=c1orwdiinAgi.toLePtro{pS˜ois|iit=ion1,6..w.,itgh}lbe=Pth0eadnedsiglent (cid:18) 0 X−iW(cid:19) (26)
˜ = g ˜. Then ϕ−1(˜) is the set of linear dispersion X W
S ∪i=1Si S W X (27)
matricescorrespondingtothenewdesign.FromProposition6 (cid:18) (cid:19)
−
it is clear that ˜= ˜ ˜ where
A B ThesedesignscanbeobtainedfromProposition8inthesame
S S ∪S
waytheABBA constructionwasobtainedfromProposition6.
˜ = [y,0]y and
A
S { | ∈S}
C. Construction C
˜ = [y,1]+δ y . The following proposition gives a procedure to obtain 4-
B m+1
S { | ∈S}
group decodable designs for 2m+1 antennas using 2-group
Let σ be the permutation on 1,...,m + 1 given by decodable designs for 2m antennas.
{ }
σ(1) = m + 1 and σ(k) = k 1 for k > 1. Using Proposition 9: Let = y j =1,..., Γ i=1,2
Proposition7onthedesign ˜i i=1−,...,g wegetag-group be a 2m 2m 2-group{dSeicoda{blie,j|design whic|his|}at|isfies fo}r
{S | } ×
decodable design with the set of linear dispersion matrices as each i=1,2
ϕ−1 σ(˜)=ϕ−1 σ(˜ ) ϕ−1 σ(˜ ). But we have
◦ S ◦ SA ∪ ◦ SB wt(y+z) is even for any y,z
i
∈S
ϕ−1◦σ(S˜A)={I2⊗Ai|i=1,...,K}=(cid:26)(cid:18)A0i A0i(cid:19)(cid:27) dtheesnig{nS˜wi|hie=re1,...,4} is a 2m+1×2m+1 4-group decodable
0 A ˜1 = [y1,j,ξ1] , ˜2 = [y1,j,ξ2]
ϕ−1◦σ(S˜B)={i(iX)⊗Ai|i=1,...,K}=(cid:26)(cid:18)−Ai −0 i(cid:19)(cid:27) ˜3 = [yS2,j,ξ3{]+δm+1}, S˜4 ={[y2,j,ξ4]}+δm+1
S { } S { }
Associating the variables x with matrices in ϕ−1 σ(˜ ) and ξ , i = 1,...,4 are distinct elements of F . Further the
i A i 4
and variables w with{th}ose in ϕ−1 σ(˜ ) we◦getSthe rates of and ˜ are same.
i B i i
{− } ◦ S {S } {S }
design in (22). Proof: Similar to the proof of Proposition 6.
iX WH
froTmherFe.arHeo4w!ev=er,2i4t wcaanysbeofshchoowonsinthgatξi,iξ ,=ξ ,1ξ,.,.ξ.,4, (cid:18) W iXH(cid:19) (30)
4 1 2 3 4 − −
{ }
ξ ,ξ ,ξ ,ξ , ξ ,ξ ,ξ ,ξ and ξ ,ξ ,ξ ,ξ all lead to
dP{ero1sipgon2ssitiw4onhi3c9}hga{ivree2ssau1mse63ucpo4t}nostrreulcatb{ioenl2isn.g1Hoofw4veav3re}irabolensly. T4huosf (cid:18)i−XWSHH−+WiXSHH iXWSHH++iWXHSH(cid:19) (31)
themareuniquei.e.leadtonon-equivalentdesigns.Twoothers
X W
can be obtained by permutation of columns and relabeling (32)
WH XH
(cid:18) (cid:19)
of variables of one of the 4 designs. These 4 constructions −
correspond to 0,1,ω,ω2 , ω,ω2,0,1 , 1,ω2,0,ω and The above constructions can be obtained in a way similar to
{ } { } { }
ω,1,0,ω2 . which ABBA construction was obtained from Proposition 6
{ }
and by using the fact that has Hermitian and has
D. 4-group decodable designs M1 M2
skew-Hermitian matrices.
Wenowgiveaproceduretogeta4-groupdecodabledesign Constructions in [31] and [32] start with X = (x +ix ).
for 2m antennas, m 1, using any 2-groupdecodable design 1 2
≥ Constructions in [32] use either (22) or (27) for the first
, for 2m−k antennas, k 1, which satisfies the
{S1 S2} ≥ applicationof Step A and uses (22) for each of the remaining
following condition for each i=1,2
k 2 applications of Step A. The last step in [32] is the
−
wt(y+z) is even for any y,z (28) application of
i
∈S X WH
Define Step A and Step B as the following steps. W −XH (33)
(cid:18) (cid:19)
• Step A: Apply any one of the 6 constructions choosing
fromPropositions6and8 andl=0,1or2. Followitby for Step B. This construction was first given in [33] and is
an application of Proposition 7 with any σ. knownas the DoublingConstruction.But this is same as (32)
• Step B: Apply any one of the 4 constructions choosing upto relabeling of variables. Constructions in [31] use (27)
from those provided by Proposition 9. Follow it by an for each of the k 1 applicationsof Step A and (32) for Step
−
application of Proposition 7 with any σ. B.
The construction procedure is as follows: Starting with the
E. g-group decodable designs for arbitrary g
design , apply Step A k 1 times followed by one
1 2
{S S } −
application of Step B. Inthissectionweconstructg-groupdecodabledesignswith
Particular examples of this procedure are the constructions arbitrary g for the case when the number of real symbols
given in [31] and [32]. We now explain how these construc- in each group is same and is equal to a power of two i.e.
tions are particular applications of the above algorithm. To τ =2a.Resultingdesignsarefornumberoftransmitantennas
explain this we need the following proposition. 2b, where
Proposition 10: Let ˜ , ˜ be a 2-group decodable de- g
1 2 b 1
{S S }
sign obtained through the application of any of the con- ≥l2 − m
structions of Propositions 6 or 8 on the 2-group decodable Let us first consider the case where g is even. Say
design 1, 2 . If 1, 2 have even and odd weight vectors g =2(m+1). We start with a square orthogonal design for
respecti{vSely,Sth}en ˜1S, ˜2Shave even and odd weight vectors. 2m antennas. We already saw that square orthogonal designs
S S
Proof: Straightforward. areobtainablefromF Fm.SuchadesignhasrateR= m+1
Let i =ϕ−1( i) be the ith group of linear dispersion and has 2m+2 group2s⊕wi4th one real symbolper group. N2omw
M S
matrices.Bothconstructions,[31]and[32],startwiththetrivial
we apply Proposition 6 on this design a times. Each of the
design for one antenna, X=(x1+ix2). This design satisfies applications can use any of the three constructions given in
the hypothesis of Proposition 10. Thus, at the end of k 1 Proposition 6 and can be followed with an application of
−
applications of Step A, the resulting code 1, 2 will be Proposition 7 with arbitrary σ.
{S S }
such that has Hermitian and has skew-Hermitian
1 2 According to Propositions 6 and 7 the resulting code will
M M
matrices. In such a scenario the matrix representation of the be for 2m+a antennas, with g = 2m + 2 groups and rate
four constructions in Proposition 9 are given as follows. R= m+1. Number of real symbols will be
Let , be a two group decodable design satisfying 2m
1 2
the hyp{oStheSses}of Propositions9 and 10. Let X be the design K =2 R Number of antennas =2(m+1)2a.
obtained from , and let W be identical to X but × ×
1 2
be composed of{Sa dSiffe}rent set of variables. Define for any Therefore, the number of real symbols per group τ = 2a as
sTqhueasreeamreattrhixeAH,eArmHiti=an12a(nAd+skAewH-)HaenrmdAitiSaHn p=art21s(Aof−A.ATHh)e. reqNuoirwedc.oTnhseidreartetheinctaesremwshoefnggisisRod=d.2Sggu/2p.poseg =2m+1
following 4-group decodable designs can be obtained from for some m, define g′ = g+1 = 2m+2. Since g′ is even
Proposition 9. we can construct a g′-group decodable design for τ = 2a as
XH iW describedabove.Thisdesignfor2m+aantennaswillhaveg+1
(29)
(cid:18)iWH X (cid:19) groups.Thisismorethanwhatisrequired.Thedesireddesign
