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1 A New Family of Low-Complexity STBCs for Four Transmit Antennas Amr Ismail , IEEE Member, Jocelyn Fiorina , IEEE Member and Hikmet Sari , IEEE Fellow ˚ ˚˚ ˚˚ Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division, King Abdullah University ˚ of Science and Technology (KAUST), Thuwal, Makkah Province, Kingdom of Saudi Arabia Telecommunications Department, SUPELEC, F-91192 Gif-sur-Yvette, France ˚˚ Email: [email protected], jocelyn.fiorina, and hikmet.sari @supelec.fr t u 3 1 Abstract—Space-Time Block Codes (STBCs) suffer from a Maximum Likelihood (ML) metric to optimally estimate the 0 prohibitivelyhighdecodingcomplexityunlessthelow-complexity transmitted codeword [5], [6], or equivalently the number of 2 decodability property is taken into consideration in the STBC leaf nodes in the search tree if a sphere decoderis employed, design. For this purpose, several families of STBCs that involve n whereastheaveragedecodingcomplexitymeasuremaybenu- a reduced decoding complexity have been proposed, notably a the multi-group decodable and the fast decodable (FD) codes. mericallyevaluatedastheaveragenumberofvisitednodesby J Recently, a new family of codes that combines both of these aspheredecoderinordertooptimallyestimatethetransmitted 1 families namely the fast group decodable (FGD) codes was codeword [7]. T] pforroproasteed-1. IFnGtDhiscopdaepserf,owre2aprtorpaonssemaitnaenwtecnonnasst.ruTchteionprsocphoesmede theArcgauseabloyf,tfhoeurfirtsrtapnrsompiotseadntleonwn-acsomispltehxeitqyuraastei--o1rtchoodgeonfoarl schemeisthenappliedtothecaseoffourtransmitantennasand .I we show that the new rate-1 FGD code has the lowest worst- (QO)STBC originally proposed by H. Jafarkhani [8] and s case decoding complexity among existing comparable STBCs. later optimized through constellation rotation to provide full- c [ The coding gain of the new rate-1 code is optimized through diversity [9], [10]. The QOSTBC partially relaxes the orthog- constellation stretching and proved to be constant irrespective onality conditions by allowing two complex symbols to be 3 of the underlying QAM constellation prior to normalization. jointlydetected.Subsequently,rate-1,full-diversityQOSTBCs v Next, we propose a new rate-2 FD STBC by multiplexing two 4 of our rate-1 codes by the means of a unitary matrix. Also a were proposed for an arbitrary number of transmit antennas 6 compromise between rate and complexity is obtained through that subsume the original QOSTBC as a special case [11]. 5 puncturing our rate-2 FD code giving rise to a new rate-3/2 In this general framework, the quasi-orthogonality stands for 6 FD code. The proposed codes are compared to existing codes in decoupling the transmitted symbols into two groups of the . theliteratureand simulationresultsshow that our rate-3/2 code 4 same size. has a lower average decoding complexity while our rate-2 code 0 maintainsitsloweraverage decodingcomplexityinthelow SNR However, STBCs with lower decoding complexity may be 2 region. If a time-out sphere decoder is employed, our proposed obtainedthrough the conceptof multi-groupdecodabilitylaid 1 codes outperform existing codes at high SNR region thanks to by S. Karmakar et al. in [12], [13]. Indeed, the multi-group : v their lower worst-case decoding complexity. decodability generalizes the quasi-orthogonality by allowing i X Index Terms—Space-time block codes, low-complexity decod- the codeword of symbols to be decoupled into more than able codes, conditional detection, non-vanishing determinants. two groups not necessarily of the same size. Thanks to r a this approach, one can obtain rate-1, full-diversity 4-group I. INTRODUCTION decodableSTBCsforanarbitrarynumberoftransmitantennas THE need for low-complexity decodable STBCs is in- [14]. However, due to the strict rate limitation imposed by the evitable in the case of high-rate communications over multi-group decodability, another family of STBCs namely MIMO systems employing a number of antennas higher than fastdecodable(FD) STBCs [5], [15], [16] hasbeenproposed. two. This is because despite their low decoding complexity These codes are conditionally multi-group decodable thus that grows only linearly with the size of the underlying enabling the use of the conditional detection technique [15] constellation, orthogonal STBCs suffer from a severe rate in which the ML detection is carried out in two steps. The limitation for more than two antennas [1], [2]. On the other first step consists of evaluating the ML estimation of the hand, the full-rate alternatives to orthogonal STBCs namely subset of the transmitted symbols which are separable say the Threaded Algebraic Space-Time (TAST) codes [3] and pcoermTfhepceltexcdioetydc.oesdi[n4g] hcaovmepgleenxeitryallmyaaypbroehiebviatilvuealtyedhigbhyddeicffoedrienngt obpxfy1,thxxe2M1,Ls.,y.xm.M2,bxLok,l.qs,..pc,xôxknMk`dL1i|t,xiôxˆknkè`1d2,,xô.k.n`.2,a,xˆ.2g.Ki.vq,exˆnt2hKvaatl.uwIeneothfmeathyseecnoroentsdet step, the receiver minimizes the ML metric only over all the measures, namely the worst-case decoding complexity mea- ` ˘ sureandtheaveragedecodingcomplexitymeasure.Theworst- possible values of pxk`1,xk`2,...,x2Kq. Recently, STBCs that are at the time multi-group and fast case decoding complexity is defined as the minimum number decodablenamelythefastgroupdecodable(FGD)codeswere of times an exhaustive search decoder has to compute the proposed [17]. These codes are multi-group decodable such 2 thateachgroupofsymbolsisfastdecodable.Thecontributions inputis 0and-1otherwise.Theround . operatorroundsits ě ˜ pq of this paper are summarized in the following: argumenttothenearestinteger.The . operatorwhenapplied ‚ We propose a novel systematic construction of rate-1 to a complex vector a returns ℜ apTq ,ℑ aT T and when FGD STBCs for 2a transmit antennas. The rate-1 FGD applied to complex matrix A returns ℜ AT ,ℑ AT T. “ ` ˘ ` ˘‰ codeforanumberoftransmitantennasthatisnotapower Foranytwointegersaandb, a b mod n meansthata b oftwoisobtainedbyremovingtheappropriatenumberof is a multiple of n and m me”anspm m“ od`qulo˘k. M` is˘´‰the columns from the rate-1 FGD code correspondingto the set of n n complex mpatqrkices. If A is a finite set, thnen A nearestgreaternumberofantennasthatisapoweroftwo denotes iˆts cardinality. | | (e.g. the rate-1 FGD STBC for three transmit antennas is obtained by removing a single column from the four II. PRELIMINARIES transmit antennas rate-1 FGD STBC). ThebasebandMIMOchannelinput-outputrelationshipmay ‚ We apply our new construction method to the case of be described by: four transmit antennas and show that the resulting new 4 4 rate-1 code can be decoded at half the worst-case Y X H W (1) dêcoding complexity of the best known rate-1 STBC. TˆNr “TˆNtNtˆNr`TˆNr ‚ Thecodinggainofthenew4 4rate-1codeisoptimized where T is the codeword signalling period, Nr is the number through constellation stretchiˆng [18] and the nonvanish- of receive antennas, Nt is the number of transmit antennas, ing determinant (NVD) property [19] is proven to be Y is the received signal matrix, X is the code matrix, H is achieved by properly choosing the stretching factor. the channel coefficients matrix with entries hkl CN 0,1 , ‚ We propose a new rate-2 STBC by multiplexing two and W is the noise matrix with entries wij „„CNp0p,N0qq. of the new rate-1 codes by the means of a unitary According to the above model, the t’th row of X denotes matrix and numerical optimization. Next we achieve a the symbols transmitted through the Nt transmit antennas significantreductionoftheworst-decodingcomplexityat duringthe t’thchanneluse while the n’thcolumndenotesthe the expense of a rate loss by puncturing the rate-2 code symbols transmitted through the n’th transmit antenna during to obtain a new rate-3/2 code. It is worth noting that the the codeword signalling period T. proposedrate-3/2andrate-2codescanbeviewedasblock A STBC matrix that encodes 2K real symbols can be orthogonal STBCs [20]. expressed as a linear combination of the transmitted symbols as [22]: We provide comparisons in terms of bit error rate (BER) 2K performance and average complexity through numerical sim- X A x (2) k k ulations and show that the reduction in worst-case decoding “ k“1 complexity comes at the expense of a negligible performance with x R and the A ,k ÿ1,...,2K are T N complex k k t loss. We also considered a more practical scenario where P “ ˆ matricescalleddispersionorweightmatricesthatarerequired a time-out sphere decoder [21] is used and show that our tobelinearlyindependentoverR.Ausefulmannertoexpress proposed rate-3/2 and rate-2 codes outperform existing codes the system model can be directly obtained by replacing X in at high SNR. (1) by its expression in (2): The rest of the paper is organized as follows: The system model is defined and the families of low-complexity STBCs 2K Y A H x W. (3) are outlined in Section II. In Section III, we propose our “ p k q k` scheme for the rate-1 FGD codes construction for the case kÿ“1 of 2a transmitantennas.In Section IV, we addressthe case of Applying the vec . operator to the above we obtain: pq four transmit antennas and the coding gain of the new 4 4 2K ˆ rate-1STBCisoptimized.Next,therateoftheproposedcode vec Y I A vec H x vec W . (4) p q“ p Nr b kq p q k` p q isincreasedthroughmultiplexingandnumericaloptimization. k“1 ÿ Numericalresultsare providedin SectionV, andwe conclude where I is the N N identity matrix. the paper in Section VI. The relevant proofs are provided in If y , hNrand w dersîgnarte the i’th columns of the received i i i the Appendices. signal matrix Y, the channel matrix H and the noise matrix W, respectively,thenequation(4) can be re-writtenin matrix Notations: form as: Hereafter, small letters, bold small letters and bold capital y1 A1h1 ... A2Kh1 x1 w1 letters will designate scalars, vectors and matrices, respec- .. .. .. .. .. .. . » . fi » . . . fi» . fi » . fi tively. If A is a matrix, then AH, AT and det A denote “ ` the Hermitian, the transpose and the determinanpt oqf A, re- —yNrffi —A1hNr ... A2KhNrffi—x2Kffi —wNrffi – fl – fl– fl – fl spectively. We define the vec . as the operator which, when y HHH s w appliedtoam nmatrix,tranpsfqormsitintoamn 1vectorby loomoon loooooooooooooooomoooooooooooooooonloomoon looomooon(5) ˆ ˆ A real system of equations can be obtained by separating the simplyconcatenatingverticallythecolumnsofthecorrespond- real and imaginary parts of the above to obtain: ingmatrix.The operatoristheKroneckerproductandδ is kj theKroneckerdbelta.Thesign . operatorreturns1ifitsscalar y˜ HHH˜s w˜ (6) pq “ ` 3 where y,w R2NrTˆ1, and HHH˜ R2NrTˆ2K. Assuming that B. Fast decodable codes P P ˜ N T K, the QR decomposition ofHHH yields: r ASTBCissaidtobefastdecodable[5]ifitisconditionally ě ˜ R multi-group decodable. HHH Q1 Q2 (7) “ 0 Definition 2. A STBC that encodes 2K real symbols is said „  mδwijhaIter,riexiQa,nj1dPP0itRs12a,N2ru2T,NˆR2TKi,sQ“2a2KP2KR22ˆKN‰r2TnKûlpl2reNmaralTtrúixp2K.pAeqr,ccQtroiTiradnQignjuglla“yr, to be FDAifHkitAslwèigAhHlt mAaktr“ice000s,a@rAe skuPchGtih,aAt:l PGj, p r ´ qˆ g (11) the ML estimate may be expressed as: 1 i j g, G n , n κ 2K. i i i ď ‰ ď | |“ “ ă sML “arg msPiCn Q1 Q2 T y˜´ RT 0T T s 2 (8) The above definition slightly generiaÿ“li1zes the definition of FD which reduces to: ››“ ‰ “ ‰ ›› codesin[5]wheretheFDcodeisrestrictedtobeconditionally › › orthogonal, that is n1 n2... ng 1. The advantage of sML arg min y1 Rs 2 (9) FD STBCs is that one“can resor“t to th“e conditional detection “ sPC} ´ } to significantly reduce the worst-case decoding complexity. where C is the vector space spanned by information vector s, For instance, if a STBC that encodes 2K real symbols is and y1 QTy˜. In the following, we will outline the known “ 1 FD, its corresponding worst-case decoding complexity order families of low-complexity STBCs and the structure of their for square QAM constellations is reduced from ?M2K´1 to correspondingRmatricesthatenablesimplifiedMLdetection. ?M2K´κ g ?Mni´1. If the FD code is conditionally ˆ i“1 orthogonal, the worst-case decoding complexity order is re- A. Multi-group decodable codes duced to ?Mř2K´κ. Fast decodability reduces to the splitting Multi-groupdecodableSTBCs are designedto significantly of the last κ levels of the real sphere decoder tree into g reduce the worst-case decoding complexity by allowing sepa- smaller trees each with n 1, i 1,...,g levels. In the rate detection of disjoint groups of symbols without any loss particular case of n 1,i í 1,.“..,g (i.e. the FD code is of performance. This is achieved iff the ML metric can be i “ @ “ in fact conditionally orthogonal), fast decodability reduces to expressed as a sum of terms depending on disjoint groups of the removal of the last κ levels of the real valued tree. The symbols. upper triangular matrix R takes the following form: Definition 1. A STBC code that encodes 2K real symbols is A B said to be g-group decodable if its weight matrices are such R “ 0 C that [13], [12]: „  whereBhasnospecialstructure,Cisa 2K κ 2K κ AHA AHA 000, A G , A G , p ´ qˆp ´ q k l` l k “ @ k P i l P j upper triangular matrix, and A is a block diagonal κ κ g (10) matrix: ˆ 1ďi‰j ďg, |Gi|“ni, ni “2K. R1 0 ... 0 iÿ“1 0 R2 ... 0 where Gi denotes the set of weight matrices associated to the A“» ... ... ... ... fi i’th group of symbols. — ffi — 0 0 ... Rgffi For instance if a STBC that encodes 2K real symbols is g- —– ffifl with R being a n n upper triangular matrix. group decodable, its worst-case decoding complexity order i iˆ i can be reduced from MK to g ?Mni with M being i“1 C. Fast group decodable codes the size of the used square QAM constellation. The worst- ř case decoding complexity order can be further reduced to A STBC is said to be fast group decodable [17] if it is g ?Mni´1 if the conditional detection via hard slicers multi-groupdecodablesuch thateach groupis fast decodable. i“1 is employed. In other words, the g-group decodability can be Definition 3. A STBC that encodes 2K real symbols is said řseen to split the original tree with 2K 1 levels to g smaller to be FGD if: treeseachwith n 1levels.Inthe spećialcase oforthogonal i STBCs, the worst´-case decoding complexity is O 1 as the AHk AlÀHl Ak “000, @Ak PGi, Al PGj, PAM slicers need only a fixed number of simpleparqithmetic g (12) 1 i j g, G n , n 2K operations irrespective of the square QAM constellation size. i i i ď ‰ ď | |“ “ The corresponding upper triangular matrix R will be a block iÿ“1 diagonal matrix: andthatthe weightmatrices within eachgroupare suchthat: R1 0 ... 0 AHk AlÀHl Ak “000, @Ak PGi,m, Al PGi,n, 0 R2 ... 0 gi (13) R“»— ... ... ... ... fiffi 1ďm‰nďgi, |Gi,j|“ni,j, jÿ“1ni,j “κi ăni. —— 0 0 ... Rgffiffi where Gi,m (resp. gi) denotes the set of weight matrices that – fl where R is a n n upper triangular matrix. constitute the m’th group (resp. the number of inner groups) i i i ˆ 4 within the i’th group of symbols G . For instance, if a STBC Clifford algebra generators. The matrix representations of γ i i that encodes 2K real symbols is FGD, its corresponding denoted R γ for the 2a 2a case are obtained as [2]: i p q ˆ worst-case decoding complexity order for square QAM constellations is reduced from ?M2K´1 to g ?Mni´κi Rpγ1q “ ˘jσ3bσ3...bσ3 gi ?Mni,j´1. Similarly, if each group ii“s1conditionalˆly a j“1 ř R γ2 I2a´l1oooooσoo1omoooooooon orthogonal,theworst-casedecodingcomplexityorderisequal p q “ b řtgoroupgi“d1e?coMdanbii´liκtyi.rIefdaucreeasltsophseprleittdinegcotdheeriosriegminpalloy2eKd,fas1t Rpγ3q “.. I2a´1 bσ2 . ř ´ real valued tree into g smaller trees each with n levels, and thatforeach ofthese new treesthe last κi levelsaire splitinto Rpγ2kq “ I2a´k bσ1bσ3bσ3...bσ3 g smaller trees each with n 1 levels. The corresponding k´1 i i,j upper triangular matrix R takes´the following form: R γ2k`1 I2a´k σ2looσoo3oooooσom3.oo.oo.oooooσon3 p q “ b b b b R1 0 ... 0 k´1 . 0 R2 ... 0 .. loooooooooomoooooooooon R“» ... ... ... ... fi (14) R γ2a σ1 σ3 σ3...σ3 — ffi p q “ b b —— 0 0 ... Rgffiffi a´1 such that for 1 i –g, one has: fl R γ2a`1 σ2 σlo3ooooooσm3o.oo.o.oσoon3 (20) p q “ b b ď ď A B a´1 R i i (15) i “ 0 Ci where looooooomooooooon „  0 1 0 j 1 0 where Bi has no special structure, Ci is a pni´κiq ˆ σ1 “ 1 0 , σ2 “ j 0 , σ3 “ 0 1 (21) ni κi upper triangular matrix, and Ai is a κi κi block „´  „  „ ´  p ´ q ˆ diagonal matrix: From now on, we will denote R γ by R for simplicity. i i The properties of the matrices R ,p iq 1,...,2a 1 can be Ri,1 0 ... 0 i “ ` summarized as follows: 0 Ri,2 ... 0 Ai “» ... ... ... ... fi (16) RHi “´Ri, R2i “Í, @1ďiď2a`1, (22) — ffi and R R R R 000, 1 i j 2a 1. —— 0 0 ... Ri,giffiffi i j ` j i “ @ ď ‰ ď ` – fl where R being a n n upper triangular matrix i,j i,j i,j ˆ B. A systematic approach for the construction of FGD codes III. THEPROPOSEDFGD SCHEME Proposition 1. For 2a transmit antennas, the two sets of In this section, we provide a systematic approach for constructing rate-1 FGD STBCs for 2a transmit antennas. matrices, namely G1 “ tI,R1,...,Rau Y A and G2 “ Recognizing the key role of the matrix representations of the tRa`1,...,R2a`1uYBsatisfy(10)whereAandB aregiven generators of the Clifford Algebra over R [2] in the sequel in Table I and Table II, respectively, and δApmq,δBpmq are given in Table III: of this paper, we briefly review their main properties in the following. TABLEIII DIFFERENTCASESFORδ A. Linear Representations of Clifford generators a δA m δB m γ TsahteisCfyliinffgortdhealfgoelblorwaionvgerprRopiesrtgye:nerated by the generators 4n ppmq4´1qp2ppmqq4´2q 2´pp2mqq4 i γ γ γ γ 2δ 1 (17) 4n`1 ppmq4ppm2q4´1qq4 pmq24´1 i j ` j i “´ ij 4n 2 ppmq4ppmq4´3qq4 pmq4 2 2 The above equation can be split in two equations: 4n`3 pppmq4´2qppmq4´3qq4 3´pmq4 γ2 1 (18) ` 2 2 i “ ´ γ γ γ γ i j. (19) See Appendix A for proof. i j j i “ ´ @ ‰ In [2], the question about the maximum number of unitary Proposition 2. The rate of the proposed FGD codes is equal representationsofthe Cliffordgeneratorshasbeenthoroughly to one complex symbol per channel use regardless of the addressed and it has been proven that for 2a 2a matrices number of transmit antennas. there are exactly 2a 1 unitary matrix represenˆtations of the ` See Appendix B for proof. Examples of the rate-1 FGD codes for 4,8 and 16 transmit antennas are given in Table IV. 5 TABLEI DIFFERENTCASESFORA a A 4n j 2i“aà`11Ri maY´“21 jδApmq 2i“aà`11Ri mi“1Rki :1ďk1 ă...ăkm ďa 4n`1 ! ś2i“aà`11Ri )maY´“21 !jδApmq ś2i“aà`11Ri śmi“1Rki :1ďk1 ă...ăkm ďa ) 4n`2 !ś2i“aà`11Ri)maY´“21!jδApmqś2i“aà`11Riśmi“1Rki :1ďk1 ă...ăkm ďa) 4n`3 !jś 2i“aà`11Ri) maY´“21! jδApmqś 2i“aà`11Riś mi“1Rki :1ďk1 ă...ăkm ďa) ! ) ! ) ś ś ś TABLEII DIFFERENTCASESFORB a B 4n tj ai“1Riuma“Y´22,4 jδBpmq ai“1Ri mi“1Rki :a`1ďk1 ă...ăkm ď2a`1 4n`1 ś ma“Y´12,3 jδBpm q ai“1Rśi mi“1Rśki :a`1ďk1 ă...ăkm ď2a`1 ( 4n`2 t ai“1Rium a“Y´22,4 jśδBpmq ai“ś1Ri mi“1Rki :a`1ďk1 ă...ăkm ď2(a`1 4n`3 ś ma“Y´12,3 jδBpm q ai“1śRi mi“1śRki :a`1ďk1 ă...ăkm ď2a`1 ( ś ś ( TABLEIV A. A new rate-1 FGD STBC for four transmit antennas EXAMPLESOFRATE-1FGDCODES Tx G1 G2 The Proposed rate-1 STBC denoted X1 is expressed as: 4 I,R2,R4,R1R3R5 R1,R3,R5,R2R4 X1 s Ix1 R2x2 R4x3 R1R3R5x4 I,R2,R4,R6 R1,R3,R5,R7 p q“R1x`5 R3x`6 R5x`7 R2R4x8.` (23) jR1R3R5R7 jR2R4R6R1 ` ` ` 8 jR1R3R5R7R2 jR2R4R6R3 According to Definition 3., the proposed code X1 is FGD jR1R3R5R7R4 jR2R4R6R5 with g 2,n1 n2 4 and g1 g2 3 such as ni,j jR1R3R5R7R6 jR2R4R6R7 1, i “1,2, j “1,2,“3. Therefore“, the w“orst-case decodin“g I,R2,R4,R6,R8 R1,R3,R5,R7,R9 comp“lexityorde“ris 2?M. However,thecodinggainofX1 is jR1R3R5R7R9 jR2R4R6R8 equal to zero, and in order to achieve full-diversity,we resort R1R3R5R7R9R2 R2R4R6R8R1R3 totheconstellationstretching[18]ratherthantheconstellation R1R3R5R7R9R4 R2R4R6R8R1R5 rotation technique, otherwise the symbols inside each group R1R3R5R7R9R6 R2R4R6R8R1R7 willbeentangledtogetherwhichinturnswilldestroytheFGD 16 R1R3R5R7R9R8 R2R4R6R8R1R9 structureoftheproposedcodeandcausesasignificantincrease R1R3R5R7R9R2R4 R2R4R6R8R3R5 in the decoding complexity. R1R3R5R7R9R2R6, R2R4R6R8R3R7 The fulldiversity code matrixtakes the formof (24) where R1R3R5R7R9R2R8 R2R4R6R8R3R9 s x1,...,x8 and k is chosen to provide a high coding “ r s R1R3R5R7R9R4R6, R2R4R6R8R5R7 gain. The term 1`2k2 is added to normalize the average R1R3R5R7R9R4R8 R2R4R6R8R5R9 transmitted powebr per antenna per time slot. R1R3R5R7R9R6R8 R2R4R6R8R7R9 Proposition 3. Taking k 3, ensures the NVD property 5 “ for the proposed code with abcoding gain equal to 1. IV. THE FOUR TRANSMIT ANTENNASCASE See Appendix C for proof. Inthissection,aspecialattentionisgiventothecaseoffour For illustration purposes, a comparison between regular and transmitantennas.Theproposedrate-1FGDcodearisesasthe stretched 16-QAM constellation points is depicted in Fig 1 directapplicationofProposition1.inthecaseoffourtransmit where the dark dots denote the regular16-QAM constellation antennas.Settinga 2,wehaveG1 I,R2,R4,R1R3R5 points whereas the red squares denote the stretched 16-QAM “ “t u and G2 R1,R3,R5,R2R4 . It is worth noting that the constellation points with stretching factor k 3. “ t u 5 above choice of weight matrices guarantees an equal average “ b transmitted power per transmit antenna per time slot. B. The proposed rate-2 code Theproposedrate-2codedenotedX2 issimplyobtainedby multiplexing two rate-1 codes by means of a unitary matrix. 6 x1 ikx5 x2 ikx6 x3 ikx7 ikx4 x8 2 x2` ikx6 x1ìkx5 ik`x4 x8 ´x3 íkx7 X1psq“ 1 k2 »´x3ìkx7 ikx´4 x8 ´x1 i´kx5 ´x2 íkx6 fi (24) c ` ´ ` ` ´ ` — ikx4 x8 x3 ikx7 x2 ikx6 x1 ikx5 ffi — ` ´ ´ ` ` ffi – fl Q instead of evaluating the ML metric M8 times, the sphere decoder needs only 2M4.5 ML metric evaluations. A rate-3/2 code that will be denoted X3{2 may be easily obtained by puncturing the rate-2 proposed code in (25), and may be expressed as: X3{2 x1,...,x12 X1 x1,...,x8 p q“ p q` (30) ejφoptX1 x9,...,x12 Uopt p q I V. NUMERICAL ANDSIMULATION RESULTS In this section, we compare our proposed codes to similar low-complexity STBCs existing in the literature in terms of worst-casedecodingcomplexity,Peak-to-AveragePowerRatio (PAPR), average decoding complexity, coding gain, and bit errorrate(BER)performanceoverquasi-staticRayleighfading channels. One can notice from Table V that the worst-case Fig.1. regular( )versusstretched (‚)16-QAMconstellation points ‚ decoding complexity of the proposed rate-1 code is half that of [24] and [23]. The worst-case decoding complexity of the Mathematically speaking, the rate-2 STBC is expressed as: proposed rate-3/2 code is half that of the punctured rate-3/2 P.Srinath-S.Rajan code [26] and is smaller by a factor of X2 x1,...,x16 X1 x1,...,x8 ejφX1 x9,...,x16 U ?M 2 than the rate-3/2 S.Sirianunpiboon et al. code [25]. p q“ p q` p (q25) Thew{orst-casedecodingcomplexityofourrate-2codeishalf where U and φ are chosen in order to maximize the coding that of the rate-2 P.Srinath-S.Rajan code [26] and smaller by gain. It was numerically verified for QPSK constellation that a factor of ?M 2 than the rate-2 T.P. Ren code et al. [27]. taking Uopt jR1 and φopt tan´1 12 maximizes the Simulations a{re carried out in quasi-static Rayleigh fading “ “ coding gain which is equal to 1. channelinthepresenceofAWGNfor2receiveantennas.The ` ˘ To decode the proposed code, the receiver evaluates the ML detectionis performedvia a depth-firsttree traversalwith ˜ QR decomposition of the real equivalent channel matrix HHH infinite initial radius sphere decoder. The radius is updated (6). Thanks to the FD structure of the proposed rate-2 code whenevera leaf node is reached and sibling nodes are visited with K 8,κ 8,n1 n2 4, the corresponding upper- accordingtothesimplifiedSchnorr-Euchnerenumeration[28]. “ “ “ “ triangular matrix R takes the form: From Fig. 2, one can notice that the proposed rate-1 code A B loses about0.6 dB w.r.t to Md.Khan-S.Rajanrate-1 code [24] R (26) “ 0 C while offering similar performance to M.Sinnokrot-J.Barry „  rate-1 code [23] at 10´3 BER for several spectral efficiencies where B R8ˆ8 has no special structure, C R8ˆ8 is an namely 2,4, and 6 bpcu. upper trianPgular matrix and A R8ˆ8 takes thePform: FromFig. 3, onecan noticethattheproposedrate-3/2code P x 0 0 x 0 0 0 0 loses about 0.6 dB w.r.t the punctured P.Srinath-S.Rajan code 0 x 0 x 0 0 0 0 [26] while it gains about 0.4 dB w.r.t S.Sirianunpiboon et al. »0 0 x x 0 0 0 0fi code[25]at10´3BERforseveralspectralefficienciesnamely —0 0 0 x 0 0 0 0ffi 3,6, and 9 bpcu.Moreover,from Figs 4(a), 4(b), and 4(c) one A“——0 0 0 0 x 0 0 xffiffi (27) can easily verify that the proposedrate-3/2 code maintains its ——0 0 0 0 0 x 0 xffiffi loweraveragedecodingcomplexityfortheconsideredspectral ——0 0 0 0 0 0 x xffiffi efficiencies while the average decoding complexity of the ——0 0 0 0 0 0 0 xffiffi S.Sirianunpiboon et al. code increases with the size of the — ffi – fl underlying constellation. wherexindicatesapossiblenon-zeroposition.Foreachvalue From Fig. 5, one can notice that the proposed rate-2 code of x9,...,x16 , the decoderscans independentlyall possible p q loses about0.8dB w.r.tthe P.Srinath-S.Rajancode[26] while values of x4 and x8, and assigns to them the corresponding6 gaining about 0.25 dB w.r.t T.P. Ren et al. code [27] at 10´3 MLestimatesoftherestofsymbolsviahardslicersaccording BER. From Fig. 6, it is easily noticed that our proposed toequations(28)and(29)atthetopofthenextpagewherer i,j code is decoded with lower average decoding complexity denotetheentriesoftheupper-triangularmatrix R.Therefore, 7 xMi L|pxˆ4,xˆ9,...,xˆ16q “ signpziqˆmin 2 round pzi´1q{2 `1,?M ´1 , i“1,2,3 (28) xMjL|pxˆ8,xˆ9,...,xˆ16q “ signpzjqˆmin”ˇˇ2 round`pzj ´1q{2˘ `1ˇˇ,?M ´1ı, j “5,6,7 (29) where: ”ˇ ` ˘ ˇ ı 16 ˇ 16ˇ zi “˜yi1ri,4xˆ4´k“9ri,kxˆk¸{ri,i, i“1,2,3, zj “˜yj1 ´rj,8xˆ8´k“9rj,kxˆk¸{rj,j, j “5,6,7 ÿ ÿ TABLEV SUMMARYOFCOMPARISONINTERMSOFWORST-CASECOMPLEXITY,MINDETANDPAPR Code Worst-case Mindet( ?δ) PAPR(dB) “ complexity forQAMconstellations QPSK 16QAM 64QAM Theproposedrate-1code 2?M 1 0 2.5 3.7 M.Sinnokrot-J.Barrycode[23] 4?M 7.11 0 2.5 3.7 Md.Khan-S.Rajancode[24] 4?M 12.8 5.8 8.3 9.5 Theproposedrate-3/2 code 2M2.5 1(verifiedfor4-QAM) 3 5.6 6.7 S.Sirianunpiboon etal.code[25] M3 N/A 5.4 8 8.4 P.Srinath-S.Rajan rate-3/2 code[26] 4M2.5 12.8 4 6.5 7.7 Theproposedrate-2code 2M4.5 1(verifiedfor4-QAM) 2.8 5.3 6.5 P.Srinath-S.Rajan code[26] 4M4.5 12.8(verifiedfor4/16-QAM) 2.8 5.3 6.5 TianPengRenetal.rate-2code[27] M5 N/A 4 7 7.66 100 100 10´1 10´1 10´2 10´2 2bpcu 4bpcu 6bpcu 3bpcu 6bpcu 9bpcu ER 10´3 ER 10´3 B B 10´4 10´4 10´5 10´5 Proposedrate-1code S.Sirianunpiboonetal.code M.Sinnokrot-J.Barrycode Proposedrate-3/2code 10´6 Md.Khan-S.Rajancode P.Srinath-S.Rajanrate-3/2code 10´6 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 SNRperreceiveantenna SNRperreceiveantenna Fig.2. BERperformance for4 2system Fig.3. BERperformance for4 2system ˆ ˆ than T.P. Ren et al. code [27] over the entire SNR range the fact that the maximum number of visited nodes is related whileitmaintainsits loweraveragedecodingcomplexityw.r.t to the worst-case decoding complexity which is lower in our P.Srinath-S.Rajan code [26] in the low SNR region. proposed codes. Next,weconsideredamorepracticalscenariowherea time- out sphere decoder [21] is employed. In fact, the tree-based VI. CONCLUSIONS searchisterminatedifapredeterminedlimitonthenumberof In the present paper we have proposed a systematic ap- visited nodes is exceeded and the sphere decoder returns the proach for the construction of rate-1 FGD codes for an currentcodewordestimation.InFig.7 wefixedathresholdof arbitrary number of transmit antennas. This approach when 50, 500, 5000 nodes count at 3, 6, and 9 spectral efficiencies appliedto the special case of fourtransmitantennasresultsin respectively.It canbe verifiedthatourrate-3/2proposedcode a new rate-1 FGD STBC that has the smallest worst-case de- outperformsthepuncturedP.Srinath-S.Rajancode[26]andthe codingcomplexityamongexistingcomparablelow-complexity S.Sirianunpiboonetal.code[25]athighSNRfor3and6bpcu STBCs. The coding gain of the proposed FGD rate-1 code spectral efficiencies. In Fig. 8 the threshold is fixed at 1000 was then optimized through constellation stretching. Next we nodes count. One can easily verify that the proposed code managed to increase the rate to 2 by multiplexing two rate- outperformsthe P.Srinath-S.Rajan code [26] and the T.P. Ren 1 codes through a unitary matrix. A compromise between et al. code [27] code at high SNR. This can be justified by 8 100 90 P.Srinath-S.Rajanrate-3/2code 10´1 80 S.Sirianunpiboonetal.code Proposedrate-3/2code 10´2 des 70 o n visited 60 BER 10´3 of 50 ber 10´4 m nu 40 ge Avera 30 10´5 T.P.Renetal.rate-2code Proposedrate-2code P.Srinath-S.Rajanrate-2code 20 10´6 0 2 4 6 8 10 12 14 16 18 SNRperreceiveantenna 10 0 2 4 6 8 10 12 14 16 SNRperreceiveantenna Fig.5. BERperformance for4 2system ˆ (a) Averagecomplexity for4 2systemat3bpcu ˆ 800 800 P.Srinath-S.Rajanrate-3/2code 700 S.Sirianunpiboonetal.code Proposedrate-3/2code 700 nodes 600 des 600 ofvisited 450000 visitedno 500 number 300 mberof 400 ge nu 300 Avera 200 Average 200 100 100 T.P.Renetal.rate-2code P.Srinath-S.Rajanrate-2code 0 Proposedrate-2code 0 2 4 6 8 10 12 14 16 18 20 22 24 0 0 2 4 6 8 10 12 14 16 18 SNRperreceiveantenna SNRperreceiveantenna (b) Average complexity for4 2systemat6bpcu ˆ 1.6¨104 Fig.6. Averagecomplexity for4ˆ2system S.Sirianunpiboonetal.code 1.4 P.Srinath-S.Rajanrate-3/2code Proposedrate-3/2code 100 des 1.2 o n visited 1 10´1 of 0.8 mber 10´2 3bpcu 6bpcu 9bpcu nu 0.6 Average 0.4 BER 10´3 0.2 10´4 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 10´5 S.Sirianunpiboonetal.code SNRperreceiveantenna P.Srinath-S.Rajanrate-3/2code Proposedrate-3/2code (c) Averagecomplexity for4ˆ2systemat9bpcu 10´60 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 Fig.4. Averagecomplexity for4 2systemat3,6and9bpcu SNRperreceiveantenna ˆ Fig.7. BERperformance for4 2systemwithtimeout spheredecoder ˆ 9 100 have: AH 1 mpm`1q{2A (31) “p´ q 10´1 moreover, it is easy to prove that: BER 10´2 RlA“$’’& ÁARRl l mmmm eooevvddeeddnn llll PPRtttkkkk1111,,,,............,,,,kkkkmmmmuuu . (32) Rt u 10´3 IfotristehaesyCtoo’’%mspeleexthaOtrGth1ozgAoYnaGl2DzBesiisgnthe(CsOetDo)fwfoerig2hatmtraantrsimceist 10´4 antennas. Thus we need only to prove that the G1, and G2 T.P.Renetal.rate-2code satisfy(10).TowardsthisendletAk A, Bl B,C G1 A, P.Srinath-S.Rajanrate-2code and D G2 B. Let a 4n, we havePaccordinPg to TabPle IzI: Proposedrate-2code P z “ 10´5 a 0 2 4 6 8 10 12 14 16 18 SNRperreceiveantenna B1 “j Ri (33) i“1 ź Fig.8. BERperformance for4ˆ2systemwithtimeoutspheredecoder and Bm{2`1 is given by (34) at the top of the next page. Consequently, from (31): BH B, B B (35) complexity and throughput may be achieved through punc- “´ @ P turing the proposed rate-2 code which results in a new low- and from (32) one has: complexityrate-3/2code.Theworst-casedecodingcomplexity BHC CHB 0, B B. (36) of the proposed codes is lower than their STBC counterparts ` “ @ P in the literature. On the other hand from Table I we have: According to the simulations results, the proposed rate- 2a`1 1 code loses about 0.6 dB w.r.t to Md.Khan-S.Rajan rate-1 A1 j Ri (37) code [24] while offeringsimilar performanceto M.Sinnokrot- “ i“a`1 J.Barry rate-1 code [23] at 10´3 BER for several spectral ź and A is given by (38) at the top of the next page. From m efficiencies namely 2,4, and 6 bpcu. The proposed rate-3/2 (31), it follows that: code loses about 0.6 dB w.r.t the puncturedP.Srinath-S.Rajan code [26] while it gains about 0.4 dB w.r.t S.Sirianunpiboon AH1 A1 (39) “ et al. code [25] at 10´3 BER for several spectral efficiencies AH Am m even (40) namely 3,6, and 9 bpcu. Moreover, the proposed rate-3/2 m “ Am m odd " ´ code maintains its lower average decodingcomplexity for the and from (32), we obtain: considered spectral efficiencies. The proposed rate-2 code loses about 0.8 dB w.r.t the AHD DHA 0, A A. (41) ` “ @ P P.Srinath-S.Rajancode [26] while gainingabout0.25 dB w.r.t Finally, from Eqs (32), (33), (34), (37), and (38)we get: T.P. Renet al. code [27] at 10´3 BER . Ourproposedcode is AHB BHA 0, A A,B B. (42) decodedwithloweraveragedecodingcomlexitythanT.P.Ren ` “ @ P P et al. code [27] over the entire SNR range while it maintains It remains to prove that the proposed weight matrices are its lower average decoding complexity w.r.t P.Srinath-S.Rajan linearly independent over R. Towards this end recall from code [26] in the low SNR region. [29]thatif M :k 1,...,2a arepairwiseanti-commuting k Next,weconsideredamorepracticalscenariowherea time- matrices thatt square“to a scalar,uthen the set: out sphere decoder is employed. Our rate-3/2 proposed code outperformsthepuncturedP.Srinath-S.Rajancode[26]andthe B2a I Mk :k 1,...,2a 2a “t uYt “ umY“2 S.Sirianunpiboonetal.code[25]athighSNRfor3and6bpcu m Psp.Sercitnraatlhe-fSfi.cRiaejnacniecso.Tdhee[2p6ro]paonsdedthreateT-.2P.cRodeneoeuttaple.rcfoordmes[2th7e] #i“1Mki :1ďk1 ă...ăkm ď2a+ ź code at high SNR. This can be justified by the fact that the formsabasisofM2a overC.Consequently,thesetB2a jB2a maximumnumberofvisitednodesisrelatedtotheworst-case forms a basis over R. Thanks to the properties of theYmatrix decoding complexity which is lower in our proposed codes representations of Clifford algebra generators (22), the set of matrices defined in (43) forms a basis of M2a over R: APPENDIXA I,jI R ,jR :k 1,...,2a 2a Proof: From the properties of the matrix representa- t uYt k k “ umY“2 tions of the Clifford algebra generators over R (22), it is m m Astra“ightfomi“rw1aRrdkito: 1prďovek1thăat.f.o.răakmmatďrix2aA`P1,Mthe2na,wief #iź“1Rki,jźi“1Rki :1ďk1 ă...ăkm ď2a(+43) ś 10 Bm{2`1 “" jśai“ai1“R1Ri iśmi“mi1“R1Rkiki aa``11ďďkk11 ăă......ăăkkmm ďď22aa``11,, mm““44nn,1`n12‰0 (34) Am “$’’’& jśśjśś2i2i““aa2i2i`àa““aa``11`àa11``11RR11śRRiiśśiiśmimi““mimi11““RR11RRkkiikkii 1111ďďďďkkkk1111 ăăăă............ăăăăkkkkmmmm ďďďďaaaa,,,, mmmm““““4444nnnn1111,```n1231,‰n01 ‰0 (38) ’’’% ś ś But from (20), one has: Similarly, for a odd, we have: 2a`1 a´2 a 1 a´2 a a Ri “¯ja´1I2a (44) |B|“ ì “ i 1 ` i i“1 i“1,3,...ˆ ˙ i“1,3...ˆ ´ ˙ ˆ ˙ ź ÿ ÿ which reduces for the case of a 4n to: a´3 a a´2 a a´2 a “ (52) 2a`1 “ j ` i “ l Ri jI2a. (45) j“ÿ0,2...ˆ ˙ i“ÿ1,3...ˆ ˙ lÿ“0ˆ ˙ iź“1 “˘ a´2 a 2a a 1 . Inwhatfollows,wewillexpresstheproposedweightmatrices “ l “ ´p ` q l“0ˆ ˙ intermsofR ,i 1,2,...,2athanksto(45).Readily,theset ÿ i mG2aznBipubleactioomnse,sth!“eRsae`ts1,A...in,RT2aab,l˘e jIśan2id“a1BRiin).TaAbfleterIIsommaye Finally, using these relationRs,“we1.get: (53) be re-written as in (46) and (47) respectively at the top of Thus concluding the proof. the next page. Now it can be easily verified that G1 G2 is Y a subset of the basis in (43). The proofs for other cases of a APPENDIXC follow similarly and are therefore omitted. Proof: The proposed code is 2-group decodable and the corresponding two sub-codes will be APPENDIXB denoted by XI X x1,x2,x3,x4,0,0,0,0 and of 2aPtrroaonfs:mTitheanrtaetnenoafstmheayprboepoesxepdreFssGeDd acso:des for the case XII “ Xp0,0,0,0,x5“,x6,x7,px8q to avoid any ambqiguity. The coding gain δ is equal to the minimum Coding Gain X 2a 2 A B Distance (CGD) [30], or mathematically: R ` `| |`| | (48) “ 2a`1 δ min det X s X s1 H X s X s1 However form Table I, regardless of a we have: X “ s‰s1 p q´ p q p q´ p q a´2 a a´2 a s,s1PC ´` CGDpXp˘sq,X`ps1qq ˘¯ A 1 2a a 1 (49) milnoooooodooeotooooXoooo∆ooosooooom2ooooooooooooooooooooooooon(54) | |“ ì“1î˙“ i“0î˙“ ´p ` q “ ∆sP∆Czt0u| pp p qqq| ÿ ÿ On the other hand, from Table II, we have for a even: where ∆s s s1, ∆C is the vector space spanned by ∆s. “ ´ a´2 a 1 a´2 a a Thanks to the quasi-orthogonality structure one has [31]: B 1 ` 1 | |“ ì“2,4,...ˆ i ˙“ ì“2,4...î´1˙`î˙ δX “mintδXI,δXIIu. (55) ÿ ÿ a´3 a a´2 a The coding gain of the first sub-code is expressed as: 1 “ ` j ` i 2 4 1 ja“´ÿ12,3..a.ˆ ˙a´2i“ÿ2a,4...ˆ2˙a a 1 δXI “„`∆x21`∆x22`∆x23´k2∆x24˘ˆ1`k2˙ (56) “ ` l“1ˆl˙“ l“0ˆl˙“ ´p ` q Choosing k 35, the above expression becomes: ÿ ÿ (50) “ b 4 5∆x2 5∆x2 5∆x2 3∆x2 2 where we used the recursion identity: 1 2 3 4 δ ` ` ´ n n 1 n 1 XI “«` 5 ˘ˆ1` 35˙ff ´ ´ (51) (57) ˆk˙“ˆk´1˙`ˆ k ˙ where ∆xi 2ni, ni Z. Consider the Diophantine “ P quadratic equation below: 5 X12 X22 X32 3X42, X1,X2,X3,X4 Z (58) p ` ` q´ P In order to find a solution we resort to the following theorem [32]: