A Low-Complexity, Full-Rate, Full-Diversity 2 X 2 STBC with Golden Codes Coding Gain

A Lo w-Comple xity , Full-Rat e, Full-Di v ersity 2 × 2 STBC with Golden Code’ s Coding Gain K. Pa van Sr inath Dept of ECE, Indian Institute of scienc e Bangalore 5 60012 , India Email:pav a n@ece.iisc.ernet.in B. Sundar Rajan Dept of ECE, Indian Institute of scienc e Bangalore 5 60012 , India Email:bsrajan@ece.iisc.er net.in Abstract — This paper presents a low-ML-decoding-complexity , full-rate, full-diversity space-time block code (STBC) fo r a 2 transmit antenna, 2 r eceive antenna multiple-in put multipl e- output (MIMO) system, wi th coding gain equal to t hat of the best and well known Golden code for any QAM constellation. Recently , two co d es ha ve been pro posed (by P aredes, Gershman and Alkhansari a nd by Sezginer and Sar i), which enjoy a lower decoding complexity relativ e to th e Golden code, but have lesser coding gain. The 2 × 2 STBC presented in th is paper has lesser d ecoding complexity for non-square QAM constellations, compared with that of the Golden code, while h av i ng the same decoding complexity f or square QAM constellations. Compared with th e Par edes-Gershman-Alk hansari and Sezginer -S ari codes, the proposed code has the same decoding complexity for non- rectangular QAM constellations. Simulation results, which com- pare t he codeword error rate (CER) perf ormance, are presented. I . I N T RO D U C T I O N Multiple-inp ut, mu ltiple-outp ut(MIMO) wireless transmis- sion systems hav e been intensively stu died d uring the last decade. Th e Alamou ti code [1] f or two tr ansmit anten nas is a novel schem e fo r MIMO transmission , which, due to its orthog onality pro perties, allows a low co mplexity ma ximum- likelihood (ML) decoder . This scheme led to the generalization of STBCs from orthog onal design s [2]. Such codes allow the tr ansmitted symbols to be deco upled fro m o ne ano ther and single-symbo l ML decoding is achieved over quasi static Rayleigh fading channels. Even though these codes achie ve the maximum diversity gain for a giv en nu mber o f transmit and receive an tennas and for any arbitrar y com plex constellations, unfor tunately , these codes are no t f ul l − rate , where, by a f ul l − rate co de, we mean a code that transm its at a rate of min ( n r , n t ) complex symb ols per ch annel use fo r an n t transmit an tenna, n r receive an tenna system. The Golden cod e [3] is a full- rate, full-diversity co de and has a decodin g complexity of the order of M 4 , for arbitrary constellations of size M . The cod es in [ 4] and the trace- orthog onal cyclotomic code in [5] also match the Golde n code. W ith red uction in the decoding complexity being th e prime obje cti ve, two new full- rate, full-diversity co des have recently been discovered: The first code was indep endently discovered by Hottinen, Tirkkonen an d W ich man [6] and by Paredes, Gershman and Alk hansari [7], which we call th e HTW -PGA code and the second , wh ich we call the Sezg iner- Sari code, was rep orted in [8] by Sezgin er and Sari. Both the se codes enable simplified decodin g, achieving a co mplexity of the o rder of M 3 . Th e first code is also shown to have the non-vanishing determinant property [7]. Howe ver, these two codes h av e lesser co ding gain compared to the Golden code. A d etailed discussion of these codes has bee n m ade in [9], w herein a compa rison of the cod ew o rd error rate (CER) perfor mance re veals that the Go lden cod e has the best perfor mance. In this pape r , we propo se a new full-rate, f ull-diversity STBC f or 2 × 2 M IMO transmission , which has low decodin g complexity . The contributions of this pap er ma y be summa- rized ( see T able I also) as fo llows: • The propo sed code has the sam e coding gain as th at of the Gold en code (a nd h ence of that in [4] and the trace- orthon ormal c yclotomic code) for any QAM constellation (by a QAM constellation we mean any finite subset of the integer lattice) and larger coding gain than those of the HTW -PGA code an d the Sezgin er-Sari code. • Compared with the Golden code and the codes in [4] an d [5], the proposed code h as lesser decoding complexity for all comp lex constellations except for sq uare QAM constellations in wh ich case the c omplexity is the same. Compared to th e HTW -PGA c ode an d the Sezginer-Sari codes, the proposed code has the same decoding complex- ity for all non-r ectangular QAM [Fig 3] constellation s. • The pr oposed cod e has the non-vanishing determ inant proper ty f or QAM constellation s and hence is Diversity- Multiplexing Gain (DMG) tr adeoff optimal. The rem aining content o f the p aper is organized as follo ws: In Section II, th e system model an d th e code design c riteria are r evie wed along with some basic defin itions. The pro posed STBC is d escribed in Section III a nd its no n-vanishing deter- minant p roperty is shown in Section IV. In Sec tion V the ML decodin g com plexity of the pro posed code is discussed and the scheme to deco de it using sphere deco ding is discussed in Sectio n VI. In Sectio n VII, simulation resu lts are presented to show the perfo rmance of the pr oposed c ode as well as to compare with few other k nown co des. Concluding r emarks constitute Sectio n VII I. Notations: For a com plex ma trix X, the matrices X T , X H and det [ X ] deno te th e transpo se, He rmitian an d determinan t of X , respectively . For a comp lex number s, R ( s ) and I ( s ) denote the real and imaginary part of s, respe cti vely . Also, j represents √ − 1 and the set of all integers, all real an d complex n umbers are deno ted by Z , R and C , respectiv ely . The Frobenius no rm an d the trace are denoted by k . k F and tr [ . ] respec ti vely . The column wise stacking o peration o n X is denoted by v ec ( X ) . Th e Kronecker pr oduct is den oted by ⊗ and I T denotes the T × T identity matrix. Given a com plex vector x = [ x 1 , x 2 , · · · , x n ] T , ˜ x is defin ed as ˜ x , [ R ( x 1 ) , I ( x 1 ) , · · · , I ( x n )] T and f or a com plex number s , the ˇ ( . ) operator is defin ed by ˇ s ,  R ( s ) −I ( s ) I ( s ) R ( s )  . The ˇ ( . ) ope rator c an be extended to a comp lex n × n m atrix by ap plying it to all the en tries o f it. I I . C O D E D E S I G N C R I T E R I A A finite set of comp lex matrices is a STBC. A n × n linear STBC is ob tained starting f rom an n × n m atrix consisting of arbitrar y linear comb inations of k complex variables an d their conjuga tes, and letting the variables take values fro m complex co nstellations. The rate of such a code is k n complex symbols per ch annel use. W e co nsider Rayleigh qu asi-static flat fading M IMO chann el with full chan nel state in formatio n (CSI) at the receiver but not at the transmitter . F o r 2 × 2 MIMO transmission, we have Y = HS + N (1) where S ∈ C 2 × 2 is the co dew ord matrix, tran smitted over 2 channel uses, N ∈ C 2 × 2 is a complex white Gau ssian noise matrix with i.i.d entries, i.e., ∼ N C (0 , N 0 ) an d H ∈ C 2 × 2 is the ch annel matrix with the entries assumed to be i.i.d circularly sym metric Gau ssian r andom variables ∼ N C (0 , 1) . Y ∈ C 2 × 2 is the rece i ved matrix. Definition 1 : ( Code rate ) If there are k indep endent infor- mation symbols in th e codew ord wh ich are transmitted over T channel uses, then, fo r an n t × n r MIMO system, th e co de rate is defined as k / T symbols per channel use. I f k = n min T , where n min = min ( n t , n r ) , then the STBC is said to h av e f ul l r ate . Considering ML deco ding, the decoding metr ic that is to be minimized over all possible values o f codewords S is given by M ( S ) = k Y − HS k 2 F (2) Definition 2 : ( Decoding co mplexity ) The ML decoding complexity is giv en by th e minimum number of sym bols that need to be jo intly decoded in minimizin g the decoding metric. This c an never be g reater than k , in which case, the decodin g complexity is said to be of th e order of M k . I f the decodin g complexity is lesser than M k , the code is said to admit simp lified deco ding. Definition 3 : ( Generato r ma trix ) For any STBC S that encodes k infor mation symbols, the g e ner ator matrix G is defined by the following equatio n ^ vec ( S ) = G ˜ s . (3) where s , [ s 1 , s 2 , · · · , s k ] T is the inf ormation symbo l vector The code design criteria [12] are: (i) Ran k cr iter ion − T o achieve maximum diversity , the codeword difference matrix ( X − ˆ X ) must be full rank for all possible pairs of codewords and th e d iv e rsity gain is g iv e n by n t n r , (ii) D e term i nant cri te ri on − For a full ranked STBC, the minimum determinan t δ min , d efined as δ min , min X 6 = ˆ X det   X − ˆ X   X − ˆ X  H  (4) should be m aximized. The cod ing gain is giv en by ( δ min ) 1 /n t , with n t being the nu mber of transmit antennas. For the 2 × 2 MIMO system, th e target is to design a code that is full-r ate, i.e tr ansmits 2 co mplex symb ols per channel use, has full-d iv er sity , maximum coding g ain and allows low ML deco ding co mplexity . I I I . T H E P R O P O S E D S T B C In this section, we present our STBC for 2 × 2 MIMO system. The design is based on th e class of codes called co-ord inate in terleaved orth ogonal designs ( CIODs), which was studied in [11] in conn ection with the general class of single-symb ol decod able cod es a nd, specifically for 2 tran smit antennas, is as follows. Definition 4 : The CIOD fo r 2 transmit antennas [ 11] is X ( s 1 , s 2 ) =  s 1 I + j s 2 Q 0 0 s 2 I + j s 1 Q  (5) where s i ∈ C , i = 1 , 2 ar e the in formatio n symb ols and s iI and s iQ are the in-phase (real) and quadra ture-ph ase (imag inary) compon ents of s i , respectively . Notice that in ord er to make the above STBC full rank, the signal c onstellation A fro m which the symbols s i are cho sen should be such that th e real part (imaginary part, resp.) of any signal point in A is not equal to the real par t (imagina ry part, resp .) of any o ther sign al point in A [11]. So if QAM constellatio ns are c hosen, they ha ve to be rotated. The op timum angle of rotation has bee n found in [11] to be 1 2 tan − 1 2 degrees and this max imizes the diversity and co ding ga in. W e deno te this an gle by θ g . The p roposed 2 × 2 STBC S is g i ven by S ( x 1 , x 2 , x 3 , x 4 ) = X ( s 1 , s 2 ) + e j θ X ( s 3 , s 4 ) P (6) where • The f our symbols s 1 , s 2 , s 3 and s 4 ∈ A , wher e A is a θ g degrees rotated version of a regular QAM sign al set, denoted by A q which is a finite subset o f th e integer lattice, an d x 1 , x 2 , x 3 , x 4 ∈ A q . T o be precise, s i = e θ g x i , i = 1 , 2 , 3 , 4 . • P is a permutation matrix designed to make th e ST BC full rate and is giv en by P =  0 1 1 0  . • The choice of θ in th e above expression should be such that the d iv ersity a nd coding g ain are ma ximized. A computer search was don e f or θ in the rang e [0 , π / 2] . The o ptimum value of θ was fo und out to b e π / 4 . Explicitly , our code m atrix is S ( x 1 , x 2 , x 3 , x 4 ) =  s 1 I + j s 2 Q e j π / 4 ( s 3 I + j s 4 Q ) e j π / 4 ( s 4 I + j s 3 Q ) s 2 I + j s 1 Q  (7) The min imum determinant for our co de when th e symbo ls ar e chosen fro m QA M constellatio ns is 3 . 2 , the same as that of the Golden cod e, which will be p roved in the next section. The g enerator m atrix fo r ou r STBC, correspon ding to the symbols s i , is as follows: G =              1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 − 1 √ 2 1 √ 2 0 0 0 0 0 0 1 √ 2 1 √ 2 0 0 0 0 0 1 √ 2 0 0 − 1 √ 2 0 0 0 0 1 √ 2 0 0 1 √ 2 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0              (8) It is easy to see that this g enerator matr ix is orth onorm al. In [5], it was shown that a necessary and sufficient condition fo r an STBC to be In formation lossless is that its gener ator matrix should be unitary . Hence, our STBC has the Info rmation losslessness property . I V . N V D P RO P E RT Y A N D T H E D M G O P T I M A L I T Y In this section it is sh own that the prop osed code ha s the n on-vanishing de terminant (NVD) pro perty [3], whic h in conjunc tion with full-rateness means that ou r code is DM G tradeoff optimal [10]. The determ inant of the codeword ma trix S can be written as det ( S ) = ( s 1 I + j s 2 Q )( s 2 I + s 1 Q ) − j [( s 3 I + j s 4 Q )( s 4 I + s 3 Q )] . Using s iI = ( s i + s ∗ i ) / 2 and j s iQ = ( s i − s ∗ i ) / 2 in the equation above, we get, 4 det ( S ) = ( s 1 + s ∗ 1 + s 2 − s ∗ 2 )( s 2 + s ∗ 2 + s 1 − s ∗ 1 ) − j [( s 3 + s ∗ 3 + s 4 − s ∗ 4 )( s 4 + s ∗ 4 + s 3 − s ∗ 3 )] = ` ( s 1 + s 2 ) + ( s 1 − s 2 ) ∗ ´` ( s 1 + s 2 ) − ( s 1 − s 2 ) ∗ ´ − j [ ` ( s 3 + s 4 ) + ( s 3 − s 4 ) ∗ ´` ( s 3 + s 4 ) − ( s 3 − s 4 ) ∗ ´ ] . Since s i = e j θ g x i , i = 1 , 2 , 3 , 4 , with s i ∈ A , x i ∈ A q , a subset of Z [ i ] , d efining A , ( x 1 + x 2 ) , B , ( x 1 − x 2 ) ∗ , C , ( x 3 + x 4 ) and D , ( x 3 − x 4 ) ∗ , with A, B , C and D ∈ Z [ i ] , we g et 4 D et ( S ) = ( e j θ g A + e − j θ g B )( e j θ g A − e − j θ g B ) − j [( e j θ g C + e − j θ g D )( e j θ g C − e − j θ g D )] = e j 2 θ g A 2 − e − j 2 θ g B 2 − j [ e j 2 θ g C 2 − e − j 2 θ g D 2 ] . Since e j 2 θ g = cos (2 θ g ) + sin (2 θ g ) = (1 + 2 j ) / √ 5 , we get 4 √ 5 D et ( S ) = (1 + 2 j )( A 2 − j C 2 ) − (1 − 2 j )( B 2 − j D 2 ) . (9) For the d eterminan t of S to be 0, we mu st have (1 + 2 j )( A 2 − j C 2 ) = (1 − 2 j )( B 2 − j D 2 ) ⇒ (1 + 2 j ) 2 ( A 2 − j C 2 ) = 5( B 2 − j D 2 ) . The ab ove can be written as A 2 1 − j C 2 1 = 5( B 2 − j D 2 ) (10) where A 1 = (1 + 2 j ) A, C 1 = (1 + 2 j ) C and clear ly A 1 , C 1 ∈ Z [ i ] . It has bee n shown in [4] that (10) holds only wh en A 1 = B = C 1 = D = 0 , i.e., only when x 1 = x 2 = x 3 = x 4 = 0 . T his means that the d eterminant of the codeword difference matrix is 0 only when the codeword difference matrix is itself 0. So, for any distinct pair of codewords, th e codeword dif fer ence matrix is always full rank f or any co nstellation wh ich is a subset o f Z [ i ] . Also, the min imum value o f the modulus of R.H.S of (9) can be seen to be 4 . So, | D et ( S ) | ≥ 1 / √ 5 . In particular, wh en the constellation cho sen is the standard QAM constellatio n, the difference b etween any two sign al po ints is a multiple o f 2. Hence, for such constellation s, | Det ( S-S ′ ) | ≥ 4 / √ 5 , where S and S ′ are distinct co dew ords. The minimum determinant is consequen tly 16/5. This means that the pr oposed codes has the non-vanishing determinan t (NVD) proper ty [3]. In [10], it was shown that full-rate cod es which satisfy th e non- vanishing determinan t prop erty achieve th e optimal DMG tradeoff. So, our proposed STBC is DMG tradeoff optimal. V . D E C O D I N G C O M P L E X I T Y The decoding complexity of the prop osed co de is of th e order of M 3 . Th is is due to the fact tha t conditio nally given the symbo ls x 3 and x 4 , the symbols x 1 and x 2 can b e decod ed indepen dently . This ca n b e proved as fo llows. Writin g the STBC in terms of its weig ht matrices/dispersion matrices A i , i = 1 , 2 , · · · , 8 , [1 1], we have S = 4 X m =1 x mI A 2 m − 1 + x mQ A 2 m | {z } T m = S 1 + S 2 where S 1 = 2 X m =1 x mI A 2 m − 1 + x mQ A 2 m and S 2 = 4 X m =3 x mI A 2 m − 1 + x mQ A 2 m . For our code, we have A 1 =  cosθ g 0 0 j sinθ g  ; A 2 =  − sinθ g 0 0 j cosθ g  A 3 =  j sinθ g 0 0 cosθ g  ; A 4 =  j cosθ g 0 0 − si n θ g  A 5 = e j π / 4  0 c osθ g j sinθ g 0  A 6 = e j π / 4  0 − si n θ g j cosθ g 0  A 7 = e j π / 4  0 j si n θ g cosθ g 0  A 8 = e j π / 4  0 j cosθ g − sinθ g 0  . The ML de coding metric in (2) c an be written as M ( S ) = tr h ( Y − H S ) ( Y − H S ) H i = tr h ( Y − H S 1 − H S 2 ) ( Y − H S 1 − H S 2 ) H i = tr h ( Y − H S 1 ) ( Y − H S 1 ) H i − tr h H S 2 ( Y − H S 1 ) H i − tr h ( Y − H S 1 ) ( H S 2 ) H i + tr h H S 2 ( H S 2 ) H i . It can be verified th at the following hold true for l , m ∈ [1 , 4] A m A H l + A l A H m = 0  ∀ l 6 = m, m + 1 , if m is o dd ∀ l 6 = m, m − 1 , if m is even . From [ 11], we obta in tr h ( Y − H S 1 ) ( Y − H S 1 ) H i = 2 X m =1 k Y − H T m k 2 F − tr “ Y Y H ” and h ence, M ( S ) = 2 X m =1 k Y − H T m k 2 F − tr  Y Y H  + tr h H S 2 ( H S 1 ) H i + tr h H S 1 ( H S 2 ) H i − tr  H S 2 Y H  − tr h Y ( H S 2 ) H i + tr h H S 2 ( H S 2 ) H i = 2 X m =1 k Y − H T m k 2 F + 2 X m =1 tr h H S 2 ( H T m ) H i + 2 X m =1 tr h H T m ( H S 2 ) H i + k Y − H S 2 k 2 F − 2 tr ( Y Y H ) . Hence, when S 2 is given, i.e, sym bols x 3 and x 4 are given, the ML metric can be decompo sed as M ( S ) = 2 X m =1 M ( x m ) + M c (11) with M c = k Y − H S 2 k 2 F − 2 tr ( Y Y H ) and M ( x m ) being a function of symbol x m alone. Thus decodin g can be don e as follows: choose the pair ( x 3 , x 4 ) and then, in parallel, decode x 1 and x 2 so as to minimize the ML dec oding metric . With this approach , there are 2 M 3 values of the decod ing metric that need to be compu ted in the worst case. So, the dec oding complexity is of th e order of M 3 . V I . S I M P L I F I E D D E C O D I N G U S I N G S P H E R E D E C O D E R In this section, it is shows that sphere deco ding can be used to achieve the d ecoding complexity of M 3 . It can be shown that (1 ) can be written as ^ v ec ( Y ) = H eq ˜ s + ^ v ec ( N ) (12) where H eq ∈ R 8 × 8 is given b y H eq =  I 2 ⊗ ˇ H  G (13) with G ∈ R 8 × 8 being the gen erator matrix as in (8) and ˜ s , [ R ( s 1 ) , I ( s 1 ) , · · · , R ( s 4 ) , I ( s 4 )] T with s i , i = 1 , · · · , 4 drawn from A , which is a rotation of the regular QAM co nstellation A q . L et x q , [ x 1 , x 2 , x 3 , x 4 ] T Then, ˜ s = F ˜ x q . where F ∈ R 8 × 8 is diag [ J , J , J , J ] with J b eing a rotatio n matrix a nd is defined as follows J ,  cos ( θ g ) − sin ( θ g ) sin ( θ g ) cos ( θ g )  . So, (1 2) can be written as ^ v ec ( Y ) = H ′ eq ˜ x q + ^ v ec ( N ) (14) where H ′ eq = H eq F . Using th is eq uiv alen t model, the ML decodin g metric can be written as M ( ˜ x q ) = k ^ v ec ( Y ) − H ′ eq ˜ x q k 2 (15) On o btaining the QR d ecompo sition of H ′ eq , we get H ′ eq = QR , where Q ∈ R 8 × 8 is an o rthono rmal m atrix and R ∈ R 8 × 8 is an upper tr iangular matrix. Th e ML d ecoding metric now can be written as M ( ˜ x q ) = k Q T ^ vec(Y) − R ˜ x q k 2 (16) If H ′ eq , [ h 1 h 2 · · · h 8 ] , where h i , i = 1 , 2 , · · · , 8 are colum n vectors, th en Q and R have th e gener al form obta ined by Gram − S chmidt p rocess as shown below Q = [ q 1 q 2 q 3 · · · q 8 ] where q i , i = 1 , 2 , · · · , 8 are colum n vectors, and R =        k r 1 k h h 2 , q 1 i h h 3 , q 1 i . . . h h 8 , q 1 i 0 k r 2 k h h 3 , q 2 i . . . h h 8 , q 2 i 0 0 k r 3 k . . . h h 8 , q 3 i . . . . . . . . . . . . . . . 0 0 0 . . . k r 8 k        where r 1 = h 1 , q 1 = r 1 k r 1 k , r i = h i − P i − 1 j =1 h h i , q j i q j , q i = r i k r i k , i = 2 , · · · , 8 . It can be shown by d irect computation that R has the following structure             a a 0 0 a a a a 0 a 0 0 a a a a 0 0 a a a a a a 0 0 0 a a a a a 0 0 0 0 a a a a 0 0 0 0 0 a a a 0 0 0 0 0 0 a a 0 0 0 0 0 0 0 a             (17) where a stand s for a possibly n on-zero entr y . The structu re of the m atrix R allows us to perfor m a 4 dimensiona l real sphere dec oding (SD) [13] to find the partial vector [ R ( x 3 ) , I ( x 3 ) , R ( x 4 ) , I ( x 4 )] T and hence obta in the symbols x 3 and x 4 . Having found these, x 1 and x 2 can be decoded indepen dently . Observe th at the real and imaginary parts of symbol x 1 are entan gled with on e ano ther because of constellation r otation b ut are independ ent of the real an d imaginary parts o f x 2 when x 3 and x 4 are co nditiona lly giv en . Having found th e pa rtial vector [ R ( x 3 ) , I ( x 3 ) , R ( x 4 ) , I ( x 4 )] T , we proceed to find the rest of the symbo ls as follows. W e do two para llel 2 dimensiona l r eal searc h to decode the symbols x 1 and x 2 . So, overall, the worst case decoding complexity o f the pr oposed STBC is 2 M 3 . T his is due to the fact that 1) A 4 d imensional real SD requires M 2 metric compu ta- tions in the worst po ssible ca se. 2) T wo parallel 2 dimen sional real SD require 2 M metric computatio ns in the worst case. This decodin g com plexity is the same as that achieved by the HTW - PGA code and the Sezginer-Sari code. Thoug h it h as not been mentioned anywhere to the best of our knowledge, the ML decodin g complexity of the Golden code, Dayal-V aranasi code and the trace-o rthogo nal cyclo- tomic co de is a lso 2 M 3 for squar e QAM constellations. This follows from the structur e o f th e R matrices for these codes which ar e counterp arts of the o ne in (17). Th e R matr ices o f these codes are similar in structure an d as shown below: R =             a 0 a 0 a a a a 0 a 0 a a a a a 0 0 a 0 a a a a 0 0 0 a a a a a 0 0 0 0 a a a a 0 0 0 0 0 a a a 0 0 0 0 0 0 a a 0 0 0 0 0 0 0 a             T able I presents the compar ison of the kn own full-rate, full- div ersity 2 × 2 codes in terms of the ir ML decoding complexity and th e cod ing gain. V I I . S I M U L A T I O N R E S U L T S Fig 1 sh ows th e co dew ord erro r perfo rmance plots for the Golden co de, the pr oposed STBC and th e HTW -PGA code for the 4- QAM con stellation. The perform ance o f the propo sed 0 5 10 15 20 25 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR in dB CER Proposed code Golden code HTW−PGA Code Fig. 1. CER PERFORMANCE FOR 4-QAM code is the same as th at o f the Golden co de. The HTW -PGA code perform s slightly worse due to its lower coding gain. Fig 2, which is a plot of the CER performan ce for 16-QAM , also highligh ts these aspects. T able I gives a c omparison between the well kn own fu ll-rate, full-diversity codes fo r 2 × 2 MI MO. V I I I . C O N C L U D I N G R E M A R K S In this paper , we ha ve presented a fu ll-rate ST BC for 2 × 2 MIMO systems wh ich matche s the best k nown co des for such systems in terms of error perf ormance , while at the same time, en joys simplified- decoding co mplexity that the co des presented in [ 7] and [ 8] do. Recently , a Rate-1 STBC, b ased on scaled r epetition and rotation of the Alam outi code, was propo sed [1 5]. T his code was shown to have a hard-d ecision perfor mance which was only sligh tly worse than that o f the Golden code for a spectral efficiency of 4 b/ s/H z , but the complexity was sign ificantly lower . AC K N O W L E D G E M E N T This work was par tly suppo rted by the DRDO-IISc prog ram on Ad vanced Research in Mathematical Engineer ing. R E F E R E N C E S [1] S. M. Alamouti, “ A simple transmit div ersity technique for wirel ess communicat ions”, IEEE J . Sel. Areas Commun. , vol. 16, no. 8, pp. 1451- 1458, October 199 8. [2] V . T arokh, H. Jafarkha ni and A. R. Calderbank, “Space-ti me block codes from orthogonal designs”, IEEE T rans. Inf . Theory , v ol. 45, no. 5, pp. 1456-1467, July 1999. [3] J. C. Belfiore, G. Reka ya and E. V iterbo, “The Golden Code: A 2 × 2 full rate space-t ime code with non-vani s hing determina nts, ” IEEE T rans. Inf. Theory , vol. 51, no. 4, pp. 1432-1436, A pril 2005. [4] P . Dayal, M. K. V ara nasi, ”An optimal two transmit antenna space-time code and its stack ed extensi ons, ” IE EE T rans. Inf . Theory , vol . 51, no. 12, pp. 4348-4355, Dec. 2005. Min det ML Deco ding com plexity Code for QA M Square QAM Rectangular QAM Non-rectan gular M = M 1 × M 2 QAM Y o-W ornell[14] 0.800 0 2 M 3 M 2 ( M 2 1 + M 2 2 ) M 4 Dayal-V aranasi code[4] 3.200 0 2 M 3 M 2 ( M 2 1 + M 2 2 ) M 4 Golden code [ 3] 3.200 0 2 M 3 M 2 ( M 2 1 + M 2 2 ) M 4 T ra ce-ortho normal c yclotomic co de [ 5] 3.200 0 2 M 3 M 2 ( M 2 1 + M 2 2 ) M 4 HTW - PGA code [7] 2.285 7 4 M 2 √ M 2 M 2 ( M 1 + M 2 ) 2 M 3 Sezginer-Sari code [ 8] 2.000 0 4 M 2 √ M 2 M 2 ( M 1 + M 2 ) 2 M 3 The p roposed c ode 3.2000 2 M 3 2 M 3 2 M 3 T ABLE I C O M PA R I S I O N BE T W E E N T H E M I N I M U M D E T E R M I N A N T A N D D E C O D I N G C O M P L E X I T Y O F S O M E W E L L K N O W N F U L L - R ATE 2 × 2 S T B C S 16 18 20 22 24 26 28 30 32 34 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR in dB CER Proposed code Golden code HTW−PGA Code Fig. 2. CER PERFORMANCE FOR 16-QAM [5] Jian-Kang Zhang, Jing Liu, K on Max W ong, ”T race-Orthonormal Full- Di versity Cyclotomic SpaceTime Codes, ” IEEE T ransactions on Signal Pr ocessing , vol. 55, no. 2, pp.618-630, Feb 2007. [6] A. Hottin en, O. Tirkko nen and R. Wichma n, ”Multi-an tenna Transcei ver T echniq ues for 3G and Beyound , ” WIL EY publisher , UK. [7] J. Parede s , A.B. Gershman and M. Ghara vi-Alkhansari, ”A 2 × 2 Space- Time Code with Non-V anishing Determinants and F ast Maximum Like- lihood Decoding, ” in Proc IEE E Internat ional Confe re nce on Acoustics, Speec h and Signal Pro cessing(ICASSP 2007), vol. 2, pp.877-88 0, April 2007. [8] S. Sezginer and H. Sari, “ A full rate full-di versity 2 × 2 space-time code for mobile Wi m ax Systems, ” in Proc. IEEE Internatio nal Confe re nce on Signal Pr ocessing and Communicat ions , Dubai, July 2007. [9] E. Biglieri, Y . Hong and E. V iterbo, ”On Fast-Decod able Space-Ti m e Block Codes“, submitted to IEEE T rans. Inf. Theory . [10] P . Elia, K. R. Kumar , S. A. Paw ar , P . V . Kumar and H. L u, ”Explic it con- structio n of space-time block codes: Achie ving the di versity-multi plexing gain trade off“, IEEE T rans. Inf. Theory , vol. 52, pp. 3869-3884, Sept. 2006. −6 −4 −2 0 2 4 6 −6 −4 −2 0 2 4 6 Quadrature In−Phase 32 QAM constellation Fig. 3. AN EXAMPLE OF A NON-RECT ANGUL AR QAM CONSTEL - LA TION [11] Zafar Ali Khan, Md., and B. Sundar Rajan, “Single Symbol Maximum Likel ihood Decodabl e Linear STBCs”, IEEE T rans. Inf. Theory , vol . 52, No. 5, pp. 2062-2091, May 2006. [12] V .T arokh, N.Seshadri and A.R Calderbank, ”Space time codes for high date rate wireless communication : performance criterion and code construct ion”, IEEE T rans. Inf. Theory , vol . 44, pp. 744 - 765, 1998. [13] Emanuele V iterbo and Joseph Boutros, “Uni versal lat tice code decoder for fading channel s”, IEE E T rans. Inf. Theory , vol. 45, No. 5, pp. 1639- 1642, July 1999. [14] H. Y ao and G. W . W ornell, “ Achie ving the full MIMO di versity- multiple xing frontie r with rotati on-based space-time codes, ” in Pr oc. Allerton Conf . on Comm. Contr ol and Comput., Mont icell o, IL, Oct. 20 03. [15] F . M. J. Wil lems, “Rota ted and Scaled Alamouti Coding”, arXi v:0802.0580 (cs.IT), February 5, 2008