Maximum Rate of Unitary-Weight, Single-Symbol Decodable STBCs

0.18 MB·English

by  Sanjay Karmakar

IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.XX,NO.XX,XXXX 1
Maximum Rate of Unitary-Weight, Single-Symbol
Decodable STBCs
Sanjay Karmakar, K. Pavan Srinath and B. Sundar Rajan, Senior Member, IEEE
Abstract—It is well known that the Space-time Block symbolsperchanneluse.AlthoughrectangularCODs[4]offer
Codes (STBCs) from Complex orthogonal designs (CODs) are ahigherrate,theyarenotdelayefficient,makingsquareCODs
single-symboldecodable/symbol-by-symboldecodable(SSD).The
more attractive in practice.
1 weight matrices of the square CODs are all unitary and obtain- In general, single-complex-symbol decodable codes need
1 ablefromtheunitarymatrixrepresentationsofCliffordAlgebras
0 when the number of transmit antennas n is a power of 2. The not be CODs. Throughout this paper, unless otherwise men-
2 rate of the square CODs for n=2a has been shown to be a2+a1 tioned, SSD codes refer to single-complex-symbol decod-
complex symbols per channel use. However, SSD codes having able codes. The Co-ordinate Interleaved Orthogonal Designs
n unitary-weightmatricesneednotbeCODs,anexamplebeingthe (CIODs) [5] have been shown to be SSD codes, while offer-
a Minimum-Decoding-Complexity STBCs from Quasi-Orthogonal
J ing full transmit diversity for specific complex constellations
Designs. In this paper, an achievable upper bound on the rate
3 of any unitary-weight SSD code is derived to be a complex only. However, the CIODs have non-unitary-weightmatrices.
1 symbols per channel use for 2a antennas, and this2ua−p1per bound SSD codes, that include unitary-weightcodes and rectangular
islargerthanthatoftheCODs.Bywayofcodeconstruction,the designs, have been reported in [6], [7] and are popularly

T] interrelationship between the weight matrices of unitary-weight knownasMinimum-Decoding-ComplexitycodesfromQuasi-
SSD codes isstudied.Also, thecoding gain of all unitary-weight
orthogonaldesigns(MDCQODs).TheratesofboththeCIODs
I SSD codes is proved to be the same for QAM constellations
s. and conditions that are necessary for unitary-weight SSD codes andthe classofcodesreportedin[7] for2a transmitantennas
c to achieve full transmit diversity and optimum coding gain are havebeenshowntobe a complexsymbolsperchanneluse.
[ 2a−1
presented. In[7],themaximumrateoftheMDCQODshasbeenreported,

1 Index Terms—Anticommuting matrices, Complex orthog- and this rate includes that for rectangular designs. However,
v onal designs, Minimum-Decoding-Complexity codes, Quasi- themaximumrateofgeneralSSDcodeshasnotbeenreported
6 orthogonal designs, Space-time block codes. so far in the literature, to the best of our knowledge.
1
In this paper, we make the following contributions.
5
2 I. INTRODUCTION We derive an upper bound on the rate of unitary-weight
•
. SSDcodesfor 2a transmitantennas.Thisupperboundis
1 Space-Time Block Codes from Complex Orthogonal De-
found to be a complex symbols per channel use.
0 signs [1] are popular because they offer full transmit di- 2a−1
1 versity for any arbitrary signal constellation and also are We give a general construction method to obtain codes
•
1 thatmeetthisupperboundandfurthershowtheinterrela-
single-symboldecodable.Infact,CODsaresingle-real-symbol
: tionship between the weight matrices of general unitary-
v decodable for constellations such as the rectangular QAM,
Xi which can be expressed as a Cartesian product of two PAM weightSSD codes. Allknownunitary-weightSSD codes
including square MDCQODs are special cases of this
constellations, while for constellations such as PSK, CODs
r construction.
a are single-complex-symbol decodable. The weight matrices,
We prove that all unitary-weight SSD codes have the
also called linear dispersion matrices [2] (refer Subsection
•
same coding gain and specifically for QAM constel-
II-A for a definition of weight matrices), of the square CODs
lations, we provide the angle of rotation that ensures
are all unitary, and a detailed construction method to obtain
full transmit diversity and optimum coding gain for all
these weight matrices from irreducible matrix representations
of Clifford Algebrashas been presentedin [3] for 2a transmit unitary-weight SSD codes.
The organization of the paper is as follows. Section II
antennas. It has also been shown that the maximum rate of
the square CODs for 2a transmit antennas is a+1 complex gives the system model and relevant definitions. Section III
2a introduces the notion of normalization and its use in the
ThisworkwaspartlysupportedbytheDRDO-IIScProgramonAdvanced analysis of unitary-weight SSD codes. Section IV provides
ResearchinMathematicalEngineeringandbytheCouncilofScientific&In- the upperbound on the rate of unitary-weightSSD codes and
dustrialResearch(CSIR),India,throughResearchGrant(22(0365)/04/EMR-
the structure of general unitary-weight SSD codes. Diversity
II)toB.S.Rajan.
SanjayKarmakariswiththeElectricalandComputerScienceDepartment conditions, coding gain calculations for QAM and simulation
at the University of Colorado at Boulder. Part of this work was carried out results are given in Section V, Subsections V-A and V-B,
whenhewasattheIndianInstitute ofScience, Bangalore
respectively. Discussions on the direction for future research
K.PavanSrinathandB.SundarRajanarewiththeDepartmentofElectrical
CommunicationEngineering,IndianInstituteofScience,Bangalore-560012. constitute Section VI.
email:[email protected]. Notations: R and C denote the field of real and complex
Different parts ofthecontent ofthis paperappeared intheProc.ofIEEE numbers, respectively and j represents √ 1. GL(n,C) de-
InternationalSymposiumonInformationonInformationTheory(ISIT2006), −
Seattle, Washington,July09-14,2006. notesthe groupof invertiblematricesof size n nwith com-
×
2 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.XX,NO.XX,XXXX
H
plex entries. For any complexmatrix A, tr(A), A , AH and of S Sˆ S Sˆ forallpossiblecodewordpairs(S,Sˆ),
det(A)representthe trace,the Frobeniusnorm,kthekHermitian − −
S =(cid:16) Sˆ. (cid:17) (cid:16) (cid:17)
and the determinant of A, respectively. In and On represent 6If the code offers full-diversity for a constellation , then,
then nidentitymatrixandthezeromatrix,respectively.For 1 A
× the coding gain is δn , where δ is the minimum of
ahacsoma pcloemxpralenxdonmormvaarliadbilsetrXibu,tXion∼wNitCh(m0,eNan)0deannodtevsatrhiaatnXce the determinant of themminatrix S mSˆinH S Sˆ among all
− −
N. For a complex variable x, xI and xQ represent the real possible codeword matrix pair(cid:16)s (S,Sˆ)(cid:17), wi(cid:16)th S =(cid:17)Sˆ.
andthe imaginarypartsofx, respectively,and x denotesthe 6
| |
absolute value of x. For a set , denotes the cardinality
S |S| A. Single-SymbolDecodable Codes
of .
S In this subsection, we formally define and classify linear
SSD codes. Any n n codeword matrix S of a linear
II. SYSTEMMODEL dispersion STBC ×with k complex information symbols
S
x ,x , ,x can be expressed as
We consider Rayleigh quasi-static flat-fading MIMO chan- 1 2 ··· k
nel with full channel state information (CSI) at the receiver k
and no CSI at the transmitter. We assume a MIMO system S = (xiIAiI +xiQAiQ), (1)
with n transmit antennas and m receive antennas. Since we i=1
X
are considering only square STBCs in this paper, the number wherex =x +jx ,1 i k,takevaluesfromacomplex
i iI iQ
≤ ≤
of time slots is also n. The channel model is constellation . Then, , i.e., the number of codewords, is
A |S|
k. The set of n ncomplexmatrices A ,A ,1 i
Y =SH +N, |A| × { iI iQ} ≤ ≤
k,calledweightmatricesdefine .Noticethatin(1),allthe2k
S
where S Cn n is the codeword matrix, transmitted over weightmatricesarerequiredtoformalinearlyindependentset
×
n channel∈uses, N Cn m is a complex white Gaussian over R, since we are transmitting k independent information
×
noise matrix with i.i∈.d. entries C(0,N0), H Cn×m symbols.
∼ N ∈
is the channel matrix with the entries assumed to be i.i.d. Assuming that perfect channel state information (CSI) is
circularly symmetric Gaussian random variables C(0,1) available at the receiver, the maximum likelihood (ML) deci-
and Y Cn m is the received matrix. ∼ N sion rule minimizes the metric,
×
∈
Definition 1: (STBC) A space-time block code S is a set M(S),tr((Y SH)H(Y SH))= Y SH 2. (2)
of complex matrices called codeword matrices. For a system − − k − k
withntransmitantennas,acodewordmatrixisaT nmatrix, Since there are k different codewords, in general, ML
× |A|
whereT isthenumberoftimeslots(T =ninthispaper)and decodingrequires k computations,one for each codeword.
|A|
the (i,j)th entry of the codeword matrix refers to the signal Suppose the set of weight matrices are chosen such that the
transmitted by the jth transmit antenna in the ith time slot. decoding metric (2) could be decomposed as
Definition 2: (Code rate) If there are k independent com- p
plex information symbols in the codeword which are trans- M(S)= f (x ,x , ,x ),
j (j 1)q+1 (j 1)q+2 (j 1)q+q
mitted over T channel uses, then, the code rate is defined to − − ··· −
j=1
X
be k/T complex symbols per channel use. For instance, for
which is a sum of p positive terms, each involving exactly
the Alamouti code, k = 2 and T = 2. So, its code rate is 1
q complex variables only, where pq = k. Then, decoding
complex symbol per channel use. requires p q = p q computations and the code is
Definition 3: (Full-Diversity Code) An STBC encoding j=1|A| |A|
called a q-symbol decodable code [10]. The case q = 1
symbols chosen from a constellation is said to offer full- P
diversity iff for every possible codewAord pair (S,Sˆ), with correspondstoSSD codesthatincludethewellknownCODs
as a proper subclass, and have been extensively studied [1],
S = Sˆ, the codeword difference matrix S Sˆ is full-ranked
6 − [3], [5], [6], [7], [8]. The codes corresponding to q = 2, are
[9].
called Double-Symbol-Decodable (DSD) codes. The Quasi-
In general, whether a code offers full-diversity or not
Orthogonal Designs studied in [11], [12], [13] are proper
dependsontheconstellationthatitemploys.Acodecanoffer
subclasses of DSD codes.
fulldiversityforacertaincomplexconstellation butnotfor
Definition 5: [1] A square complex orthogonal design
A
another complex constellation. The CODs are special in this S
for n transmit antennas is a set of codeword matrices of size
aspectsincetheyofferfull-diversityforanyarbitrarycomplex
n n, with each codeword matrix S satisfying the following
constellation. ×
conditions:
Definition 4: (CodingGain)Thecodinggainδ ofanSTBC
the entries of S are complex linear combination
is defined as •
r 1r of x1,x2,··· ,xk and their complex conjugates
x ,x , ,x .
δ = min λi , ∗1 ∗2 ··· ∗k
S Sˆ,S=Sˆ ! (Orthonormality:)
− 6 Yi=1 •
where λ ,i=1,2, ,r, are the non-zero eigenvalues of the SHS =(x 2+ + x 2)I
i 1 k n
H ··· | | ··· | |
matrix S Sˆ S Sˆ andr istheminimumoftherank holds for any complex values for x ,i=1,2, ,k.
− − i ···
(cid:16) (cid:17) (cid:16) (cid:17)
SANJAYetal.:MAXIMUMRATEOFUNITARY-WEIGHT,SINGLE-SYMBOLDECODABLESTBCS 3
A set of necessary and sufficient conditions for to be a The STBCs and are said to be equivalent. To simplify
U
S S S
COD is [1], [3] ouranalysisofunitary-weightcodes,wemakeuseofnormal-
ization as described below. Let be a unitary-weight STBC
AHA =AH A =I , i=1,2, ,k; (3) S
iI iI iQ iQ n ··· and let its codeword matrix S be expressed as
AHA +AH A =O , (4a)
iI jQ jQ iI n k
AHiIAjI +AHjIAiI =On, (4b) S = (xiIA′iI +xiQA′iQ).
AHiQAjQ+AHjQAiQ =On, (4c) Xi=1
Consider the code , AHS S . Clearly, from
for 1 i=j k, and SN { ′1I | ∈ S}
≤ 6 ≤ Lemma 1, N is equivalent to . The weight matrices of N
S S S
AHA +AH A =O , i=1,2, ,k. (5) are
iI iQ iQ iI n ···
A =AHA ,
STBCs obtained from CODs [1], [3] are SSD like the well iI ′1I ′iI
A =AHA .
known Alamouti code [8], and satisfy (3), (4) and (5). For S iQ ′1I ′iQ
to be SSD, it is not necessary that it satisfies (3) and (5), i.e., With this, a codeword matrix S of can be written as
N N
S
it is sufficient that it satisfies only (4) - this result was shown
k
in [5]. Since then, different classes of SSD codes that are not
S =x I +x A + (x A +x A ).
N 1I n 1Q 1Q iI iI iQ iQ
CODshavebeenstudiedbyseveralauthors,[5],[6],[7].SSD
i=2
codes can be systematically classified as follows. X
We call the code to be the normalized code of .
N
1) Linear STBCs satisfying (3), (4) and (5) are CODs. S S
In general, any unitary-weight SSD code with one of its
2) LinearSTBCssatisfying(4)arecalledSSDcodes.These
weightmatricesbeingthe identitymatrixis callednormalized
may or may not satisfy (3) and (5).
unitary-weightSSDcode.Studyingunitary-weightSSDcodes
3) Linear STBCs satisfying (3) and (4) and not satisfying
becomes simpler by studying normalized unitary-weightSSD
(5) are called Unitary-Weight SSD codes.
codes. Also, an upper bound on the rate of unitary-weight
4) Linear STBCs satisfying (4) and not satisfying (3) are
SSD codes is the same as that of the normalized unitary-
called Non-Unitary-weight SSD codes. These may or
weight SSD codes. For the normalized unitary-weight SSD
may not satisfy (5).
codetransmittingk symbolsin n channeluses, the conditions
The codes discussed in [5], which are called CIODs, con- presented in (3) and (4) can be rewritten as
stitute an example class of Non-unitary-weight SSD codes.
The classes of codes studied in [7] are unitary-weight SSD AHiI = −AiI (equivalently, A2iI =−In), (6)
codes. The classes of codes studied in [6], called Minimum AH = A (equivalently, A2 = I ), (7)
iQ − iQ iQ − n
Decoding Complexity codes from Quasi-Orthogonal Designs AH A = A A , (8)
1Q iI iI 1Q
(MDCQOD codes), include some unitary-weight SSD codes
AH A = A A , (9)
as well as non unitary-weight SSD codes. 1Q iQ iQ 1Q
The notion of SSD codes have been extended to coding for i=2, ,k, and
for MIMO-OFDM systems in [14], [15] and recently, low- ···
decoding complexity codes called 2-group and 4-group de- AiIAjI = AjIAiI, (10)
−
codable codes [16], [17], [18], [19] and SSD codes [20] in A A = A A , (11)
iQ jQ jQ iQ
−
particular have been studied for use in cooperative networks A A = A A , (12)
iI jQ jQ iI
as distributed STBCs. −
for 2 i=j k. So, every weight matrix except A =I
1I n
≤ 6 ≤
III. UNITARY-WEIGHT SSD CODES and A1Q should square to −In. Shown below is the grouping
ofweightmatrices(WewilllatershowthatA = A A ,
In this section, we analyze the structure of the weight iQ ± 1Q iI
i=2, k).
matrices of unitary-weight SSD codes. We make use of the ···
following lemma in our analysis. A1I =In A2I AkI
···
Lemma 1: Let S = {S|S = ki=1xiIAiI +xiQAiQ,xi ∈ A1Q A2Q ··· AkQ
A} be a unitary-weight STBC aPnd consider the STBC SU , Except In, the elements in the first row should mutually
{US|S ∈S},whereU isanyunitarymatrix.Then,SU isSSD anticommute and also square to −In. From (8) and (9), it is
iff is SSD. Further, both the codes have the same coding clear that if A2 = I , then, A should anticommutewith
S 1Q − n 1Q
gain for the constellation . all the weightmatricesexceptA =I . So, the upperbound
A 1I n
Proof: The proof is straightforward. For the STBC U, on the rate of a unitary-weight SSD code is determined by
S
the weight matrices are UA1I, UAiQ, i = 1,2, ,k. It is the number of mutually anticommuting unitary matrices. The
···
easy to verify that if the matrices AiI, AiQ, i = 1,2, ,k following section deals with determining the upper bound.
···
satisfy (4), then, the matrices UA , UA , i = 1,2, ,k
1I iQ
alsosatisfy(4)andvice-versa.Further,foranypairofd·is·t·inct IV. AN UPPERBOUND ON THERATEOF UNITARY-WEIGHT
codewordmatricesS andSˆ,theeigenvaluesof(S Sˆ)H(S SSD CODES
sˆ) are the same as that of (US USˆ)H(US US−ˆ), makin−g Inthissection,wedeterminetheupperboundontherateof
− −
the coding gain the same for both the STBCs. unitary-weightSSDcodesandalsogiveageneralconstruction
4 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.XX,NO.XX,XXXX
scheme to obtain codes meeting the upper bound. To do so, assume that the case k = 2a+2 is a possibility. Denoting
we make use of the following lemmas regarding matrices of the 2a + 1 anticommuting matrices by F , F , ,F ,
1 2 2a+1
size n n. we note from Lemma 2 that the set Fλ1Fλ2 ·F··λ2a,λ
Lemm×a 2: [21] Consider n n matrices with complex 0,1 ,i = 1, ,2a is a basis for{C21a 2a2 o·v·e·r C2a. Theire∈-
×
entries. × f{ore,} Fλ1Fλ·2·· Fλ}2a,jFλ1Fλ2 Fλ2a,λ 0,1 ,i =
1) If n = 2an , with n odd, then there are l elements 1, {,2a1 is2 a·b·a·si2safor C12a×22a o··v·er2Ra. It ica∈n b{e ch}ecked
0 0 ··· }
of GL(n,C) that anticommute pairwise if and only if by applying Lemma 4 that the only product matrix that
l 2a+1. anticommutes with F1,F2, , and F2a is cF1F2 F2a,
2) pIfa≤irnwis=e, t2haenanthde mseattroicfesproFd1u,c·t·s·F,F2Fa anticFommwuitthe cwheCre.c ∈ C. So, it must ·b·e·that F2a+1 = cF1F2······F2a,
1 i < <i 2a along with I i1forim2·s·a· biassis for ∈For our construction, we need anticommuting, anti-
1 s n
the≤22a di·m·e·nsiona≤lspace of all n n matricesoverC. Hermitian, unitary matrices (so that they square to In).
In each case F2 is a scaled identit×y matrix. An excellent treatment of irreducible matrix representa−tions
i
Proof: Available in [21]. of Clifford algebras is given in [3] and the same paper also
presentsanalgorithmtoobtain2a+1pairwiseanticommuting
Let F , ,F be anticommuting,anti-Hermitian, unitary
1 2a
matrices (s·o··that F2 = I , i=1,2, ,2a). The following 2a 2a matrices that all square to In (n = 2a). In fact,
i − n ··· thes×e matrices are precisely the weig−ht matrices (except I )
two lemmas are applicable for such matrices. n
of square CODs. As mentioned before, we denote them by
Lemma 3: The product F F F with 1 i < <
is 2a squares to ( 1)s(s2+i11)Ini2.··· is ≤ 1 ··· F1, F2,··· ,F2a+1, with F2a+1 =cF1F2···F2a, and
≤ −
Proof: (F F F )(F F F ) j if (F F F )2 =I ,
i1 i2··· is i1 i2··· is c= ± 1 2··· 2a n
1 otherwise.
= (−1)s−1(Fi21Fi2···Fis)(Fi2Fi3···Fis) (cid:26) ±
= ( 1)s 1( 1)s 2(F2F2 F )(F F F ) It must be noted that the matrices obtained from [3] are not
− − − − i1 i2··· is i3 i4··· is unique, i.e., these are not the only set of mutually anticom-
= (−1)[(s−1)+(s−2)+···1](Fi21Fi22···Fi2s) muting,anti-Hermitian,unitarymatricesofsize2a 2a.Itcan
= (−1)s(s2−1)(−1)sIn wbehennotaedisboydadp.pWlyeinagreLneomwmraea3dythtaotp(rFo1vFe2th··e·mF2a×ain)2re=su−ltIonf
s(s+1)
= (−1) 2 In. the paper.
This proves the lemma. Theorem 1: The rate k of a 2a 2a unitary-weight SSD
2a ×
Lemma 4: Let Ω = F ,F , ,F and Ω = code is upper bounded as
1 { i1 i2 ··· is} 2
{Fj1,Fj2,··· ,Fjr} with 1 ≤ i1 < ··· < is ≤ 2a and k 2a = a .
1 j1 < < jr 2a. Let Ω1 Ω2 = p. Then the 2a ≤ 2a 2a 1
pro≤duct mat·ri·x· F F ≤ F com|mute∩s wi|th F F F Proof: We prove the theorem i−n three parts as follows.
i1 i2··· is j1 j2··· jr
if exactly one of the following is satisfied, and anticommutes Claim 1: k =2a+2.
6
otherwise. To prove this, let us first suppose that k = 2a+2, in which
case, we have the following grouping scheme.
1) r,s and p are all odd.
2) The product rs is even and p is even (including 0).
Proof: If Fjk ∈Ω1∩Ω2, we note that In F1 F2 ··· F2a+1
A A A A
(F F F )F =( 1)s 1F (F F F ), 1Q 2Q 3Q ··· (2a+2)Q
i1 i2··· is jk − − jk i1 i2··· is Let A = 22a a Fλ1,jFλ2,j Fλ2a,j, λ 0,1 ,
and if Fjk ∈/ Ω1∩Ω2, m = 1,2iQ, ,2aj=,1i =i,j1,12, 2,2a··+· 22aand aim,j,j ∈C{. Th}is
(Fi1Fi2···Fis)Fjk =(−1)sFjk(Fi1Fi2···Fis). is possible··b·ePcause of Lem·m·a· 2. Considering A∈2Q, since
A anticommutes with F , F , and F , every
2Q 2 3 2a+1
Now, ···
individual term of A must anticommute with F , F ,
2Q 2 3
(F F F )(F F F )
i1 i2··· is j1 j2··· jr and F2a+1. So, we look for all possible candidates
= (−1)p(s−1)(−1)(r−p)s(Fj1Fj2···Fjr)(Fi1Fi2···Fis) w·fr·oh·michtheansteicto{mFm1λu1tFe2λ2w·i·t·hF2λFa2a,,λFi ∈, {0,1}a,nid=F1,···.,2Bay}
= (−1)rs−p(Fj1Fj2···Fjr)(Fi1Fi2···Fis). applying Lemma 4, the only2poss3ible··c·hoice is 2Fa+.1Since
1
case 1). Since r,s and p are all odd, ( 1)rs p = 1. the weight matrices are required to be independent over
−
case 2). The product rs is even and p−is even (including 0). R and in view of the condition in (7), there is no valid
So, ( 1)rs p = 1. possibilityforA .Asaresult,aunitary-weight2a 2a SSD
− 2Q
− ×
From Lemma 2, the maximum number of pairwise anti- codewith2a+2independentcomplexsymbolsdoesnotexist.
commuting matrices of size 2a 2a is 2a + 1. Hence, the
×
maximum possible value of k, i.e., the number of complex Claim 2: k =2a+1.
6
information symbols, is 2a + 2, since we also consider I To prove this, we assume that k = 2a+1 is a possibility,
n
as a weight matrix. In order to provide an achievable upper in which case, we have the following grouping of weight
bound on the rate of unitary-weight SSD codes, we first matrices.
SANJAYetal.:MAXIMUMRATEOFUNITARY-WEIGHT,SINGLE-SYMBOLDECODABLESTBCS 5
1) Let A2 = I . We prove that this is not a possibility.
1Q − n
In F1 F2 ··· F2a Icfomitmwuetreewtriuthe,Fi.e,.F, A,21Q =an−dIFn, then,aAsa1Qlsoshseoeunldina(n8ti)-.
A A A A 1 2 2a 1
Considering 1AQ , 2eQach o3fQ the···terms(2at+h1a)tQA is a The only matrices from··t·he set {F−1λ1F2λ2···F2λa2a,λi ∈
2Q 2Q 0,1 ,i = 1, ,2a that anticommute with F , F ,
1 2
linear combination of should anticommute with F , { } ··· }
2 andF areF andF F F .So,letA =
2a 1 2a 1 2 2a 1Q
F3, and F2a. The only possibilities from the set ·a·· F +a− F F F , with···
Fλ1·F·λ·2 Fλ2a,λ 0,1 ,i = 1, ,2a are F 1,1 2a 1,2 1 2··· 2a
a{nd1 F12F2··· F22aa =i c∈F2a{+1,c} = j or··· 1. T}herefore1, a21,1+ca21,2 =1, (13)
··· ± ±
A = a F + a F F F . Next, considering
A2Q, the o2n,l1y 1elemen2,t2s 1an2ti·c·o·mm2auting with F , F , where c = 1 if (F1F2···F2a)2 = −In (a is odd)
3Q 1 3 and c = 1 if (F F F )2 = I (a is even).
and F are F and F F F . Therefore, − 1 2··· 2a n
·aA·ls3·oQan=ticoma3m2,1auFte2w+itha3A,232FQ1,Fe2ith··e·rFa212a,2.2=·S·i·0ncoe2raaA32,2Q=s0h.oSulod, tNsheeatxtt{a,Fnct1λioc1noFsm2iλdm2e·rui·tn·egFw2λAai2tah2,QλF,it,h∈Fe ,o{n0l,y1}am,niadtr=Fices1,f·ro·am·re,2Ftah}e,
eeoiirtthhAeerr2QcAa2s=Qe,a=th2,e1±Fa1Fss1+igananm2d,e2nFAt13FvQi2o·=la·t·eaFs32,ta1hFea2nrud+leAat33h,Q2aFt=1tFhe2±·wF·e2·i.Fg2Ihnat FAa22aQ,FFF=1F2a·2·F,1·FF12.a+Faunartd2h,e2F2rF,22Fasi33n+·c··e·a·F·2A,23a2F−21F.3=2A·a·s−·1Fa2Iar−e,1suwl+1te,
matrices are linearly independent over R. As a result, we 2,4 1 2··· 2a 2Q − n
have A2 = (a2 + a2 ca2 + ca2 )I +
can’t have any valid elements as the weight matrices and 2Q − 2,1 2,2 − 2,3 2,4 n
2a a F F F + 2a a F F F +
hence, k 6=2a+1. 2ca2,21,3a2,23,4F11F22·a·,·w2itah−c1as ment2io,2ne2d,3be2for3e·.·S·inc2ae In,
F F F , F F F and F F are linearly
Claim 3: k =2a. 2 3 2a 1 2 2a 1 1 2a
Consider the following grouping scheme of weight matrices. indepe·n·d·ent over C,··e·ither−a2,1 = a2,2 = a2,4 = 0 or
a = 0. Suppose a = a = a = 0, A =
2,3 2,1 2,2 2,4 2Q
In ··· Fl ··· F2a−1 a2,3F2F3 F2a 1. Since A1Q anticommutes with
m 2i=a−11Fi ··· m 2i=a−1,1i6=lFi ··· m 2i=a−12Fi A2Q, we·s·e·e tha−t A2Q cannot be a2,3F2F3 F2a 1.
In thQe above grouping Qscheme, m = j if aQis odd, and This is because both F2a and F1·F·2· F−2a
···
im==21,3if, a is,2eav.enC.leItarclya,ntbheenwoeteigdhtthamtaAtriiQces=a−reAl1inQeAarilIy, cao2,m1Fm1ut+e Aw2it,h2F2Fa2F+3·a·2·,4FF21aF−21. TFh2eare.foBrey, aA2sQimil=ar
independen··t·overR and satisfy (6)-(12). Hence, an SSD code argument, A3Q =a3,1F2+a3,2F·2·a·+a3,4F1F2 F2a.
···
transmitting 2a complex symbols in 2a channel uses exists. ConsideringthatA1Q, A2Q andA3Q anticommutepair-
This completes the proof. wise, we must have
We observe that for 2 transmit antennas, k = 3. So, the
6
rate of a unitary-weight SSD code for 2 transmit antennas
a a +ca a = 0, (14)
1,1 2,2 1,2 2,4
can be at most 1 complex symbol per channel use, which
a a +ca a = 0, (15)
is also the rate of the well-known Alamouti code, which 1,1 3,2 1,2 3,4
is single-real-symbol decodable and offers full diversity for a2,2a3,2+ca2,4a3,4 = 0. (16)
all complex constellations. So, the unitary-weight SSD code
From (14), (15) and (16), we obtain
for 2 transmit antennas offers no advantage compared to
a ca ca a
the Alamouti code. So, in the subsequent analysis, we only 1,1 = − 2,4 = − 3,4 = 2,2.
consider codes for 2a transmit antennas, a>1. a1,2 a2,2 a3,2 a2,4
Hence, a2 + ca2 = 0, which contradicts (13). So,
1,1 1,2
Theorem 2: Any maximal rate, normalized unitary-weight A2 = I and A cannot be anti-Hermitian.
SSD code must satisfy the following in addition to satisfying 2) Le1tQA6 2−=n I . 1InQ this case, we first look for pos-
(6)-(12). sibiliti1eQs f6or−An, i = 2, ,2a. As argued before,
iQ
···
A1Q = AH1Q(equivalently,A21Q=In), eAither=A2mQF=Fa2,1FF1 +a,2,w2Fit2ham+a=2,4F11Fi2f·a··iFs2eaveonr
2Q 2 3 2a 1
AiIA1Q = A1QAiI, and m = j ·i·f·a is−odd. Assumin±g that A =
2Q
±
AiQ = ±AiIA1Q, a2,1F1+a2,2F2a+a2,4F1F2···F2a, we have
for i=2,3, ,2a.
··· A3Q =a3,1F2+a3,2F2a+a3,4F1F2 F2a,
Proof: Consider the following grouping of weight ···
A =a F +a F +a F F F .
matrices. 4Q 4,1 3 4,2 2a 4,4 1 2··· 2a
Since A2 =A2 =A2 = I ,
2Q 3Q 4Q − n
I F F F
n 1 2 ··· 2a−1 a2 +a2 +ca2 = 1,
A1Q A2Q A3Q A(2a)Q 2,1 2,2 2,4
··· a2 +a2 +ca2 = 1,
A can have two possibilities. Either A2 = I or 3,1 3,2 3,4
1Q 1Q − n
A2 = I . a2 +a2 +ca2 =1, (17)
1Q 6 − n 4,1 4,2 4,4
6 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.XX,NO.XX,XXXX
A =I A = F A = F F A = F F
′1I n ′2I − 1 ′3I − 1 2 ··· ′(2a) − 1 2a−1
A′1Q =±m 2i=a−11Fi A′2Q =−F1A1Q A′3Q =±m 2i=a−31Fi ··· A′(2a)Q =±m 2i=a−22Fi
Q Q Q
with F F , , F F and F F are 2a pairwise
1 3 1 2a 1 1 2a
1 if (F F F )2 = I (a is odd), −anticomm··u·tin−g matric−es. Henc−e, instead of F1, F2, ,
c=(cid:26) −1 if (F11F22······F22aa)2 =−In(na is even). Fan2da−1,FifFwe wearsetthoec2haose −1Fan1t,ic−oFm1mFu2t,in−gFm1Fa3tr,i·c··e··s·,
1 2a 1
Further, considering that A2Q, A3Q and A4Q anticom- we w−ould en−d up with the−weight matrices as shown in
mute with each other, we have the following equalities.
the table at the top of the page. So,
a a +ca a = 0,
2,2 3,2 2,4 3,4 A′iIA′iQ =±A′jIA′jQ,2≤i6=j ≤2a. (19)
a a +ca a = 0,
2,2 4,2 2,4 4,4
From (18) and (19), A = mF F F . This
1Q 1 2 2a 1
a3,2a4,2+ca3,4a4,4 = 0. further implies that A must±be a uni·ta·r·y, H−ermitian
1Q
with c as mentioned before. From the above set of matrix, because of the choice of m.
So, the weight matrices of the normalized unitary-weight
equations, we obtain
SSD code for 2a transmit antennas are
a ca ca a
2,2 3,4 4,4 4,2
= − = − = .
a2,4 a3,2 a4,2 a4,4 In F1 ··· F2a−1
So, a2 +ca2 = 0 pa = a , with p = j if ±m 2i=a−11Fi ±m 2i=a−21Fi ··· ±m 2i=a−12Fi
4,2 4,4 ⇔ 4,2 4,4 ± This comQpletes the proQof of the theorem. Q
c=1 and p= 1 if c= 1. So, from (17), we obtain,
± − For4transmitantennas,byapplyingtheprocedureoutlined
a = 1. By a Similar argument, we obtain, a =
4,1 2,1
± in [3], we obtain the following pairwise anticommuting, anti-
1,pa =a , a = 1,pa =a . Therefore,
2,2 2,4 3,1 3,2 3,4
± ± Hermitian matrices.
A = F +a (F +pF F F ),
2Q 1 2,2 2a 1 2 2a j 0 0 0 0 1 0 0
± ···
A3Q = F2+a3,2(F2a+pF1F2 F2a), 0 j 0 0 1 0 0 0
A = ±F +a (F +pF F ···F ). F1 = 0 −0 j 0 , F2 = −0 0 0 1 
4Q ± 3 4,2 2a 1 2··· 2a  0 0 −0 j   0 0 1 0 
It is easy to see that the above assignment of matri-    − 
   
ces violates the linear independence of the matrices 0 0 1 0 0 j 0 0
A ,A , i=2,3,4 over R. Therefore, the assumption 0 0 0 1 j 0 0 0
thiaIt AiQ =a F +A F +a F F F is not F3 = 1 0 0 −0 , F4 = 0 0 0 j .
valid. S2Qo, let2,1 1 2,2 2a 2,4 1 2··· 2a  −0 1 0 0   0 0 j 0 
   
   
2a For constructing a maximal rate, unitary weight SSD code
A2Q = mF2F3 F2a 1 = m AiI, for 4 transmit antennas, we only need 3 pairwise anticom-
± ··· − ±
i=Y1,i6=2 muting, anti-Hermitian matrices. Hence, choosing F1, F2 and
where, F3 andapplyingthe constructionmethoddescribedabove,we
j if a is odd, obtainamaximalrate,unitary-weightSSDcodefor4transmit
m=
1 if a is even. antennas, a codeword matrix S of which is shown in (20), at
(cid:26)
the top of the next page.
Now, the only possibility is that A =
3Q In general, for 2a transmit antennas, we need 2a unitary,
mF F F = m 2a A . Similarly, by
±assign1ing3·±··m2a−2i=1a1,i=±jAiIQit=o1,Ai6=j3Q,jiI = 4,··· ,2a, aexnatic-tHlyer2maitia1np,apirawiriwseisaenatinctoicmommumtiuntginmgamtraictreisceosf.sIifzeth2eare 2arae,
we see that the cond6itions in (10), (11) and (12) are − ×
then, any matrix among them is a scaled productof the other
Q
satisfied and from the discussion made above, this is
2a 2.So, the followingobservationscan be madeaboutany
the only assignment possible. Now, we only need to −
maximal rate, normalized unitary-weight SSD code.
find a valid assignment for A . Firstly, we note that
1Q 1) EitherA ,i=2,3, ,2aare2a 1among2a+1pair-
iI
··· −
AiIAiQ =±AjIAjQ,2≤i6=j ≤2a. (18) wise anticommuting matrices and AiQ,i=2,3,··· ,2a
are exactly 2a 1 pairwise anticommuting matrices, or
From Lemma 1, multiplying all the weight matrices by −
A ,i=2,3, ,2aare 2a 1 among2a+1 pairwise
iQ
A (i.e., F ) will result in another unitary-weight ··· −
− 2I − 1 anticommuting matrices and AiI,i = 2,3, ,2a are
SSD code with the weight matrices grouped as shown ···
exactly 2a 1 pairwise anticommuting matrices.
at the top of the page, after interchanging the first and 2) A is a −Hermitian matrix and A = 2a A
the second columns. It should be noted in the above 1Q 1Q i=2 iI
if A ,i = 2,3, ,2a are 2a 1 among 2a + 1
iI
groupingschemethattheelementsinthefirstrowexcept pairwise anticomm·u·t·ing matrices, o−r A = Q2a A
I are all mutually anticommuting matrices and all of 1Q i=2 iQ
n ifA ,i=2,3, ,2aare2a 1among2a+1pairwise
iQ
them also anticommute with F F . So, F , F F , ··· − Q
1 2a 1 1 2 anticommuting matrices.
− −
SANJAYetal.:MAXIMUMRATEOFUNITARY-WEIGHT,SINGLE-SYMBOLDECODABLESTBCS 7
x +jx x jx x +jx x jx
1I 2I 3I 4Q 4I 3Q 2Q 1Q
− −
x jx x jx x +jx x +jx
S = − 3I − 4Q 1I − 2I 2Q 1Q − 4I 3Q . (20)
x +jx x jx x jx x +jx
4I 3Q 2Q 1Q 1I 2I 3I 4Q
− − − −
 x2Q+jx1Q x4I +jx3Q x3I +jx4Q x1I +jx2I 
 − − 
 
V. DIVERSITY AND CODING GAIN OF UNITARY-WEIGHT = n 2a ∆xiI+(−1)ki+sj∆xiQ 2,
SSD CODES jY=1Xi=1(cid:16) (cid:17)
WehaveseeninLemma1thatthecodinggainofaunitary-
where,
weight SSD codes does not change when normalized. In this
section, we obtain a common expression for the coding gain 0 if the (j,j)th entry of Λ is 1,
s =
of all unitary-weight SSD codes and identify the conditions j 1 if the (j,j)th entry of Λ is -1,
(cid:26)
on QAM constellations that will allow unitary weight SSD
and
codes to have full transmit diversity and high coding gain. 0 if A A =A ,
k = iI iQ 1Q
Let S and S be two distinct codewords of any normalized i 1 if A A = A .
′ iI iQ 1Q
(cid:26) −
unitary-weight SSD code . Let
SN The minimum of the determinant, denoted by ∆min, of
2a (∆S)H∆S for all possible non-zero ∆S is given as
S = x A +x A ,
iI iI iQ iQ
n 2a
Xi2=a1 ∆min = min ∆xiI +( 1)ki+sj∆xiQ 2 .
∆S=0 − 
S′ = x′iIAiI +x′iQAiQ, 6 jY=1Xi=1(cid:0) (cid:1)
Xi=1 Sincetheexpressioninsidethebracketintheaboveequationis
with aproductofthesumofsquaresofrealnumbers,itsminimum
occurswhenallbutoneamong∆x ,i=1,2, 2aarezeros.
i
A =I , A A = A A = A , 2 i=j 2a, ···
1I n iI iQ ± jI jQ ± 1Q ≤ 6 ≤ So,
AH =A ,AH = A ,AH = A , 2 i 2a.
1Q 1Q iI − iI iQ − iQ ≤ ≤ n
Let ∆S , S S , ∆x , x x , ∆x , x x and ∆min = min ∆xiI +( 1)ki+sj∆xiQ 2. (21)
∆xiQ ,xiQ−−x′iQ′. Thein, i − ′i iI iI − ′iI ∆xi6=0jY=1(cid:0) − (cid:1)
2a H Let m be the algebraic multiplicity of 1 as the eigenvalue
(∆S)H∆S = Xi=1∆xiIAiI+∆xiQAiQ! × loefmAm1Qa taondconn−clumdebethtahtamtof=-1n.Wemm.akeuseofthefollowing
2a −
∆xmIAmI+∆xmQAmQ Lemma 5: Let F , F , and F be 2a 2a unitary,
m=1 ! 1 2 ··· 2a ×
X pairwise anticommuting matrices. Then, the product matrix
2a
= ∆x2iI+∆x2iQ In±2∆xiI∆xiQAiIAiQ F1λ1F2λ2···F2λa2a,λi ∈{0,1},i=1,2,···2a, with the excep-
Xi=1(cid:16) (cid:17) tion of I2a, is traceless.
2a
= ∆x2iI+∆x2iQ In±2∆xiI∆xiQA1Q. Proof: It is well known that tr(AB)=tr(BA) for any
Xi=1(cid:16) (cid:17) two matrices A and B. Let A and B be two invertible, n n
×
anticommuting matrices. So,
Since A is unitary and Hermitian, the eigenvalues of
1Q
A1Q are 1 and A1Q is unitarily diagonalizable. Let A1Q = AB = BA.
EΛEH, w±here E is unitary and Λ is a diagonal matrix with ABA−1 = −B.
the diagonal entries being 1. Therefore, −
± tr(ABA−1) = tr(B).
−
2a tr(A−1AB)= tr(B) tr(B)= tr(B).
(∆S)H∆S = ∆x2iI+∆x2iQ EEH±2∆xiI∆xiQEΛEH −∴tr(B)⇔=0. − (22)
Xi=1(cid:16) (cid:17)
2a
= E Xi=1(cid:16)∆x2iI+∆x2iQ(cid:17)In±2∆xiI∆xiQΛ!EH 4S,imitilacralny,bitecasenebnetshhaotwannythaptrotrd(uAct)m=a0tr.iBxyFa1λp′1pFly2λin′2g··L·eFm2λam′2aa,
anticommutes with some other product matrix from the set
and det (∆S)H∆S
Fλ1Fλ2 Fλ2a,λ 0,1 ,i = 1,2,3, ,2a . There-
(cid:0) 2a (cid:1) {for1e, fr2om··th·e2raesultiob∈ta{ined}in (22), we ca·n··say t}hat every
= det Xi=1(cid:16)∆x2iI+∆x2iQ(cid:17)In±2∆xiI∆xiQΛ! product matrix F1λ1F2λ2···F2λa2a except I2a is traceless.
n 2a Since A is a scaled product of 2a 1 matrices among
= ∆x2iI+∆x2iQ+(−1)ki+sj2∆xiI∆xiQ 2aunitary,1pQairwiseanticommutingmatric−es,A istraceless.
jY=1Xi=1(cid:16) (cid:17) 1Q
8 IEEETRANSACTIONSONINFORMATIONTHEORY,VOL.XX,NO.XX,XXXX
Constellation: 4-QAM 16-QAM 64-QAM
CIOD 10.24 10.24 10.24
MDCQOD [6] 10.24 10.24 10.24
MDCQOD [7] 10.24 10.24 10.24
New Design 10.24 10.24 10.24
TABLEI
COMPARISONOFTHEMINIMUMDETERMINANTSOFAFEWSSDCODESFOR4TRANSMITANTENNAS
Hence, m=n m. So, (21) becomes
−
∆ = min ∆x2 ∆x2 n. (23) 100 New Design − Rect. 8 QAM
min ∆xi6=0 iI − iQ CMIDOCDQ −O RDe −c tR. 8e cQt.A 8M QAM
(cid:0) (cid:1)
From the above expression, it is clear that for maximal-rate, New Design − Sq. derived
8 QAM
unitary-weight SSD codes to offer full transmit diversity, the Rect. 8 QAM
10−1 CIOD − Sq. derived 8 QAM
difference set ∆ , a ba,b , where is the MDCQOD − Sq. derived 8 QAM
A { − | ∈ A} A New Design − 4 QAM
constellationemployed,shouldnothaveanypointsthatlie on
CIOD − 4 QAM
lines that are at 45 degrees in the complex plane from the MDCQOD − 4 QAM
±
origin.Further,sincethe analysisleadingupto theexpression R10−2
in (23) is not specific to any particular unitary-weight SSD E
C
code, we can infer that for any particular constellation , all Sq. derived
A 8 QAM
maximal-rate,unitary-weightSSDcodeshavethesamecoding
gain. 10−3 4 QAM
A. Diversity, coding gain calculations for QAM
In this subsection we show that all maximal-rate, unitary- 10−4
weight SSD codes have the same coding gain as the CIODs
[5] for QAM constellations. Let y , i = 1,2, ,2a be the 10 12 14 16 18 20 22 24
i ··· SNR in db
informationsymbolsthattake valuesfroma constellation .
1
A
Consider the following unitary rotation.
Fig.1. ComparisonoftheCERperformanceoftheSSDcodesandtheCIOD
1 1
x y
iI = √2 −√2 iI , i=1,2, ,2a.
(cid:20) xiQ (cid:21) " √12 √12 #(cid:20) yiQ (cid:21) ∀ ··· same average energy but the constellation energy has been
Theaboveoperationisequivalenttorotatingy byπ/4radians allowed to increase with the increase in constellation size. As
i
to obtain x . Then, from (23), we have analytically shown, the minimum determinants are the same
i
for all the three codes.
∆ = min (2∆y ∆y )n. (24)
min iI iQ
∆yi6=0
The above expression is the same as the one for CIOD, B. Simulation results for our SSD codes
obtained in [5]. It is to be noted that (24) holds even when In this subsection, we provide some simulation results
the angle of rotation is π/4 radians. In order to maximize for 4 transmit antennas. All simulations are done assuming
−
∆ , the minimum of the product ∆y ∆y , called the a quasi-static Rayleigh fading channel. The number of re-
min iI iQ
| |
product distance, must be maximized. This has been done ceive antennas is 1. Fig. 1 shows the codeword error rate
for QAM in [5], by rotating QAM constellations by an angle (CER) performances of the CIOD for 4 transmit antennas,
of 1tan 12. So, y , i = 1,2, ,2a, should take values the MDCQOD for 4 transmit antennas [6], and the new
±2 − i ···
from a rotated QAM constellation, with the angle of rotation design whose codeword matrix is as in (20), at 2 bits and
being 1tan 12. So, the original information symbols x , 3 bits per channeluse (bpcu). For transmission at 2 bpcu, the
±2 − i
i=1,2, ,2a should take valuesfrom a rotated QAM con- constellationemployedis the 1tan 12 radianrotated4-QAM
stellation·,·th·eangleofrotationbeing π 1tan 12.Sincethe fortheCIODandthe π+1tan2 12−radianrotated4-QAMfor
±4±2 − 4 2 −
coding gain for CIOD has been maximized in [5] by using a theMDCQODandthenewdesign.Fortransmissionat3bpcu,
1tan 12 radianrotatedQAM constellation,thecodinggain theconstellationsemployedaretherotatedrectangular8-QAM
±2 −
forallunitary-weightSSDcodeswhenthesymbolstakevalues and the rotated square-derived 8-QAM, the angle of rotation
from a π 1tan 12 radian rotated QAM constellation is being the same as in the case of 4-QAM. A squared-derived
±4 ± 2 −
also maximized. Table I gives a comparison of the minimum 8-QAMconstellationisobtainedbyremovingthesignalpoint
determinantsfortheCIOD,MDCQODandtheunitary-weight with the highest energy from a 9-QAM. Specifically, it is the
SSD code presented in (20), all the codes designed for 4 set 1 j, 1+j, 1+3j,1 j,1+j,1+3j,3 j,3+j .The
{− − − − − − }
transmit antennas. In the calculations, all the codes have the simulationresultssupportthefactthatforQAMconstellations,
SANJAYetal.:MAXIMUMRATEOFUNITARY-WEIGHT,SINGLE-SYMBOLDECODABLESTBCS 9
the coding gain of the SSD codes is the same as that of the [3] O.Tirkkonen and A.Hottinen, “Square-matrix embeddable space-time
CIOD. blockcodesforcomplexsignalconstellations,”IEEETrans.Inf.Theory,
Vol.48,No.2,Feb.2002.
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[5] Md.ZafarAliKhanandB.SundarRajan, “Single-Symbol Maximum-
In this paper, we have provided an achievable upper bound LikelihoodDecodableLinearSTBCs,”IEEETrans.Inf.Theory,Vol.52,
on the rate of unitary-weight SSD codes for 2a transmit No.5,pp.2062-2091, May2006.
antennas. The upper bound has been shown to be a [6] H.Wang, D.Wang and X-G.Xia,“On Optimal Quasi-Orthogonal Space-
2a−1 TimeBlockCodesWithMinimumDecodingComplexity,”IEEEtrans.
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Inf.Theory,Vol.55,No.3,pp.1104-1130, Mar.2009.
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minimum decoding complexity,” IEEETrans. Wireless Comm., Vol. 4,
meeting the upper bound. We have further shown that all
No.5,pp.2089-2094,Sep.2005.
unitary-weight SSD codes that meet the upper bound have
[8] S.M. Alamouti, “A simple transmitter diversity scheme for wireless
the same coding gain as that of the CIODs and we have also communications,”IEEEJ.SelectAreasComm.,Vol.16,pp. 1451-1458,
Oct.1998.
identified the angle of rotation for QAM constellations that
[9] V.Tarokh,N.SeshadriandA.RCalderbank, “Spacetimecodesforhigh
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[10] SanjayKarmakarandB.SundarRajan,“Multigroup-Decodable STBCs
1) What is the upper bound on the rate of square, non-
fromClifford Algebras,” IEEETrans.Inf.Theory, Vol.55,No.01,pp.
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[12] WeifengSuandX.G.Xia,“Signalconstellations forQOSTBCwithfull
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2) Whatis the upperboundonthe rate ofrectangularSSD [13] N.SharmaandC.B.Papadias,“Improvedquasi-orthogonalcodesthrough
constellationrotation,”IEEETrans.Comm.Vol.51,No.3,pp.332-335,
codes, the class of which the rectangular MDCQODs
2003.
presented in [6] and [7] are a subclass? [14] R.V.Jagannadha Rao Doddi, V. Shashidhar, Md.Zafar Ali Khan
and B.Sundar Rajan, “Low-complexity, Full-diversity Space-Time-
Further, the analysis in this paper can be used to study the
Frequency Block Codes for MIMO-OFDM,” Proceedings of IEEE
rates of multi-symbol decodable codes, the upper bounds of GLOBECOM, Communication Theory Symposium, Dallas, Texas, 29
which has never been reported in literature. These questions Nov-3Dec.,2004,pp.204-208.
[15] S. Gowrisankar, B. Sundar Rajan, “A Rate-one Full-diversity Low-
provide some directions for future research.
complexity Space-Time-Frequency Block Code (STFBC) for 4-Tx
MIMO-OFDM,” Proceedings of IEEE International Symposium on
ACKNOWLEDGEMENT Information Theory (ISIT 2005), Adelaide, Australia, 2-9 Sep. 2005,
pp.2090-2094.
ThisworkwaspartlysupportedbytheDRDO-IIScProgram [16] Kiran T. and B. Sundar Rajan, “Distributed space-time codes with
on Advanced Research in Mathematical Engineering and by reduceddecodingcomplexity,”ProceedingsofIEEEInternationalSym-
posiumonInf.Theory(ISIT2006),Seattle,USA,July09-14,2006,pp.
the Councilof Scientific & IndustrialResearch (CSIR), India,
542-546.
through Research Grant (22(0365)/04/EMR-II)to B.S. Rajan. [17] Kiran T. and B. Sundar Rajan, “ Partially-coherent distributed space-
We thank X.-G.Xia for sending the preprintof [6]. We would timecodeswithdifferentialencoderanddecoder,”ProceedingsofIEEE
International Symposium on Information Theory (ISIT 2006), Seattle,
also like to thank the anonymous Reviewers for pointing out
USA,July09-14,2006,pp.547-551.
[21], which helped in simplifying the proof for the upper [18] KiranT.andB.SundarRajan,“Partially-coherentdistributedspace-time
bound on the rate of unitary-weight SSD codes. codeswithdifferentialencoderanddecoder,”IEEEJournalonSelected
AreasinComm.,Vol.25,No.2,pp.426-433,Feb.2007.
[19] G.Susinder Rajan and B.Sundar Rajan, “A Non-orthogonal Distributed
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