A Low-decoding-complexity, Large coding Gain, Full-rate, Full-diversity STBC for 4 X 2 MIMO System

A Lo w-decoding-comple xity , Lar ge coding Gain, Full-rate , Full-di v ersity ST BC for 4 × 2 MIMO System 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 proposes a lo w decoding complexity , full-diversity and full-rate space-time b lock code (S TBC) for 4 transmit and 2 recei ve ( 4 × 2 ) multiple-inp ut multiple-outp ut (MIMO) systems. For such systems, th e b est code known is the DjABBA code and recently , Biglieri, Hong and Viterbo hav e proposed another STBC (BHV code) wh ich has l ower decodin g complexity than DjABBA but does not hav e ful l-diversity like the DjABBA code. The code proposed in thi s paper has th e same d ecoding complexity as th e BHV code for square QAM constellations but has fu ll-diversity as well. Compared to the best code in the DjABBA family of codes, our code has lower decoding complexity , a better coding gain and hence a better error perfo rmance as well. Simulation results confi rming these are p resented. I . I N T R O D U C T I O N Multiple-inp ut m ultiple-outp ut ( MIMO) tran smission has been of special interest in wireless communication for the pa st one decade. The Alamo uti code [1] for two transmit antennas, due to its orthog onality pro perties, allows a low comp lexity maximum -likelihood (ML ) d ecoder . Th is schem e paved the way for generalized ortho gonal STBCs [2 ]. Such codes allow the transmitted sym bols to be decou pled fr om one anoth er and sing le-symbo l ML d ecoding is achieved over qua si-static Rayleigh fading chann els. Anoth er aspect of these codes is that they achieve the ma ximum diversity gain for any numb er of transm it and rece i ve antennas and f or any ar bitrary complex constellations. Unfortuna tely , for m ore than two antennas, rate 1 codes can not be constructed u sing o rthogo nal designs. W ith a view of inc reasing the transmission r ate, quasi- orthog onal d esigns (QODs) were prop osed in [3]. Howe ver, these co des co me at the p rice o f a smaller diversity g ain and are also dou ble symbol decodable for 4 anten nas. As an imp rovement, Coor dinate interleaved orthog onal de signs (CIODs) were propo sed [4]. The se codes hav e th e same transmission rate as QODs but additionally enjoy f ull d iv er sity while bein g sing le sy mbol deco dable for certain comp lex constellations. But none of the above class of cod es is fu ll- rate , where a n STBC is said to be of f ull-rate if its rate in complex symbo ls per channel u se is equal to the minimu m of the number of transmit and the rec eiv e an tennas. Full-rate, f ull-diversity STBCs a re of prime importance in systems like WIMAX. Low-decoding com plexity , full-r ate STBCs have been p roposed in [5] and [6] for 2 × 2 and in [7] for 4 × 2 MI MO systems. Th ese co des allo w a simplified ML decodin g wh en compared with codes from division algeb ras [8],[9] w hich are not amenab le for low decodin g complexity though they offer full-rate. The fast decod able code prop osed in [7] for 4 × 2 systems, wh ich we call the BHV code, outperf orms the best known DjABB A co de only at low SNRs while allowing a reduction in the ML deco ding complexity . The BHV code does not have full-d iv er sity as it is based on the quasi ortho gonal desig n for 4 an tennas, when all the symb ols are take values from o ne con stellation. In this paper , we pro pose a new STBC for 4 × 2 MIMO transmission. Our code is based o n the Coord inate I nterleaved Orthogo nal Designs (CIODs) prop osed in [4 ] (defined in Section III). The major contributions of this paper can be summarized a s fo llows: • Our code has a decodin g co mplexity of the o rder of M 5 , for all com plex constellations, wh ere M is the size o f the signal constellation , wh ereas the DjABB A code h as the cor respond ing comp lexity of or der M 7 and th e BHV code h as order M 6 , ( M 5 for sq uare QAM c onstellations - thou gh this has not b een cla imed in [7]). • Our co de has a better CER ( Cod ew o rd error rate ) perfor mance than the b est co de in th e DjABBA family due to a higher c oding g ain fo r QAM constellation s. • Our code ou tperfor ms the BHV code for QAM co nstel- lations du e to its higher d i versity gain. • Combining the above, it can be seen that when QAM constellations are used, our co de is the b est amo ng all known codes fo r 4 × 2 systems. The remain ing c ontent of th e paper is o rganized as f ollows: In Section II, the system m odel and the cod e d esign criteria is giv en. The proposed STBC and its decod ing complexity are discussed in Sec tion III. I n Section IV, the decod ing sch eme for the pro posed STBC using sphere decod ing is discussed. In Section VI, simulatio n r esults are presented to illustrate the com parisons with best kn own co des. Conclud ing remar ks constitute Sectio n VII . Notations: Let X be a comp lex m atrix. T hen X T , X H and det [ X ] den ote the t ranspose, Hermitian and the determinan t of X re spectiv ely . R ( s ) and I ( s ) de note th e real and imag inary parts of a complex nu mber s , respectively , and j rep resents √ − 1 . Th e set of all real and co mplex numb ers are de noted by R and C , respe cti vely . k . k F and k . k denote the Fro benius norm and th e vector nor m, respectively and tr [ . ] deno tes the trace opera tion. For a m atrix X, the vector obtained by columnwise concaten ation one below the other is denoted b y v ec ( X ) . Th e Kronecker p roduc t is d enoted by ⊗ an d I T denotes the T × T id entity matrix. Given a com plex vecto r x = [ x 1 , x 2 , · · · , x n ] T , th e vector ˜ x is defin ed as ˜ x , [ R ( x 1 ) , I ( x 1 ) , · · · , I ( x n )] T and for a complex number s , th e matrix ˇ s op erator is defined by ˇ s ,  R ( s ) −I ( s ) I ( s ) R ( s )  The ˇ ( . ) operator can be similarly app lied to n × n matrix by applying it to all the e ntries. I I . S Y S T E M M O D E L W e consider Rayleigh quasi-static flat-fading MIM O ch an- nel with full channel state inf ormation ( CSI) at the r eceiver but n ot at the tran smitter . For 4 × 2 MIM O transmission, we have Y = HS + N (1) where S ∈ C 4 × 4 is the co dew ord matrix, tran smitted over 4 channel uses, N ∈ C 2 × 4 is a complex white Gaussian noise matrix with i.i.d entries ∼ N C (0 , N 0 ) an d H ∈ C 2 × 4 is the channel matrix with the en tries assumed to be i.i.d cir cularly symmetric Gaussian r andom variables ∼ N C (0 , 1) . Y ∈ C 2 × 4 is the rece i ved matrix Definition 1 : ( Code rate ) If there are k indep enden t info r- mation symbols in the codeword which are transmitted over T chan nel uses, then, for an n t × n r MIMO system, th e code rate is defined as k /T sym bols per channel use. For instance, for the Alamouti code k = 2 an d T = 2 . If k = n min T , where n min = min ( n t , n r ) , then the STBC is said to have f ul l r ate . Considering ML deco ding, the decoding metr ic that is to be minimized over all possible values of co dewords S is giv en by M ( S ) = k Y − HS k 2 F (2) Definition 2 : ( Decoding complexity ) The ML decoding complexity can be measur ed by the minimu m numb er of values of M ( S ) that are needed to be comp uted in m inimizing the decodin g metric. Definition 3 : ( Genera tor matrix ) For any STBC S that encodes k inf ormation symbols, the g ener ator matrix G is defined by th e following equa tion ^ v ec ( S ) = G ˜ s . (3) where s , [ s 1 , s 2 , · · · , s k ] T is the info rmation symbo l vector [7]. Code d esign is b ased o n the analysis of pairwise e rror p rob- ability (PEP) given b y P ( X → ˆ X ) , wh ich is the probab ility that a transm itted codeword X is detected a s ˆ X . The g oal is to minim ize the err or probab ility , which is upp er boun ded b y the fo llowing u nion b ound. P e ≤ 1 M k X X X X 6 = ˆ X P  X → ˆ X  (4) where M denotes th e signal constellatio n size an d k is the number of indep endent infor mation symbols in the codeword. It is well known [1 0], that an an alysis o f the PEP leads to the following design criteria: 1) . R ank cr iter ion : T o achiev e maximu m diversity , the codeword difference matrix ( X − ˆ X ) must b e full r ank for all possible pairs o f codeword p airs and th e diversity gain is gi ven by n t n r . If fu ll r ank is not achiev ab le, then , the div ersity ga in is given by rn r , where r is the min imum rank of the c odeword difference matrix over all possible co dew ord pair s. 2) . D eter mina n t cr iter ion : For a full r anked STBC, the minimum d eterminan t δ min , d efined as δ min = min X 6 = ˆ X det   X − ˆ X   X − ˆ X  H  (5) should be m aximized. The cod ing gain is given by ( δ min ) 1 /n t , with n t being the nu mber of transmit antennas. If th e STBC is non f ull-diversity an d r is the m inimum rank of the cod ew or d difference matrix over all p ossible codew o rd pairs, the n , the coding g ain δ is given by δ = min X − ˆ X r Y i =1 λ i ! 1 r where λ i , i = 1 , 2 , · · · , r , are the no n-zero eig en values of the matrix  X − ˆ X   X − ˆ X  H It should be noted that, fo r high signal-to- noise ratio (SNR) values at each receive anten na, the dominan t param eter is the div ersity gain which defines the slope of the CER curve. This implies that it is impo rtant to first ensure full diversity of the STBC and then try to max imize th e codin g gain. For the 4 × 2 MI MO system, the o bjective is to design a code th at is full-rate, i.e tran smits 2 sym bols per cha nnel use, has full diversity and allows simplified ML decod ing. 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 fo r the 4 × 2 MIMO system. Th e design is based on the CIOD for 4 antennas, whose stru cture is as defined below . Definition 4 : CIOD f or 4 tra nsmit anten nas [4] is a s fo l- lows: X ( s 1 , s 2 , s 3 , s 4 ) =     s 1 I + j s 3 Q − s 2 I + j s 4 Q 0 0 s 2 I + j s 4 Q s 1 I − j s 3 Q 0 0 0 0 s 3 I + j s 1 Q − s 4 I + j s 2 Q 0 0 s 4 I + j s 2 Q s 23 − j s 1 Q     (6) where s i ∈ C , i = 1 , · · · , 4 are the informatio n symbols an d s iI and s iQ are the real and imag inary p arts of s i respectively . Notice that in o rder to make the above STBC full r ank, the signal constellation A from wh ich the symbo ls are ch osen should be such tha t the real p art (imagin ary p art, resp. ) of any signal point in A is not eq ual to the real part ( imaginary part, resp.) of any other signal point in A [4]. So if square or rectangu lar QAM constellations are ch osen, th ey have to be rotated. The optimu m angle of rotation, wh ich we denote by θ g , h as b een f ound in [4] to be a ta n (2) / 2 d egrees and this maximizes the diversity and coding g ain. Our STBC is obtained as follows. Our 4 × 4 cod e m atrix, denoted by S enco des eig ht sym bols x 1 , · · · , x 8 drawn fro m a QAM co nstellation, d enoted by A q . W e de note the r otated version of A q by A , with the angle o f ro tation cho sen to be θ g degrees. Let s i , e j θ g x i , i = 1 , 2 , · · · 8 , so that the symbols s i are drawn from the co nstellation A . The codeword matrix is defin ed as S , X ( s 1 , s 2 , s 3 , s 4 ) + e j θ X ( s 5 , s 6 , s 7 , s 8 ) P (7) with θ ∈ [0 , π/ 2] an d P being a permutatio n matrix designed to make the STBC f ull-rate, given by P =     0 0 1 0 0 0 0 1 1 0 0 0 0 1 0 0     . The choice of θ sho uld b e such that the div e rsity and codin g gain are maximized. A co mputer search yielded the o ptimum value o f θ to be π / 4 . This value of θ provides the largest coding gain achiev a ble for this family of cod es. The value of the minimu m determina nt o btained for u nit energy 4-QAM constellation is 0.640 0. Th e resulting code matrix is as shown in the top of the n ext page. I V . D E C O D I N G C O M P L E X I T Y O F T H E P RO P O S E D C O D E The deco ding com plexity of the propo sed code is of the order of M 5 . This is due to the fact that conditionally g iv e n the symbols x 5 , x 6 , x 7 and x 8 , the rest of th e symbols x 1 , x 2 , x 3 , and x 4 can be decoded indepen dent of one a nother . This c an be shown as follows. Writing the STBC in terms of its linear weight m atrices, we have S = 8 X m =1 x mI A 2 m − 1 + x mQ A 2 m | {z } T m = S 1 + S 2 where S 1 = 4 X m =1 x mI A 2 m + x mQ A 2 m +1 and S 2 = 8 X m =5 x mI A 2 m − 1 + x mQ A 2 m . The weig ht matr ices are as follows A 1 =     cosθ g 0 0 0 0 cosθ g 0 0 0 0 j si nθ g 0 0 0 0 − j sinθ g     A 2 =     − sinθ g 0 0 0 0 − si n θ g 0 0 0 0 j cosθ g 0 0 0 0 − j cosθ g     A 3 =     0 − cosθ g 0 0 cosθ g 0 0 0 0 0 0 j sinθ g 0 0 j sinθ g 0     A 4 =     0 sinθ g 0 0 − sinθ g 0 0 0 0 0 0 j cosθ g 0 0 j cosθ g 0     A 5 =     j sinθ g 0 0 0 0 − j sinθ g 0 0 0 0 c osθ g 0 0 0 0 c osθ g     A 6 =     j cosθ g 0 0 0 0 − j cosθ g 0 0 0 0 − sinθ g 0 0 0 0 − si n θ g     A 7 =     0 j sinθ g 0 0 j sinθ g 0 0 0 0 0 0 − cosθ g 0 0 c osθ g 0     A 8 =     0 j cosθ g 0 0 j cosθ g 0 0 0 0 0 0 sinθ g 0 0 − sinθ g 0     A 9 = e j π / 4     0 0 c osθ g 0 0 0 0 cosθ g sinθ g 0 0 0 0 − si n θ g 0 0     A 10 = e j π / 4     0 0 − sinθ g 0 0 0 0 − si n θ g cosθ g 0 0 0 0 − cosθ g 0 0     A 11 = e j π / 4     0 0 0 − cosθ g 0 0 cosθ g 0 0 sinθ g 0 0 sinθ g 0 0 0     S =     s 1 I + j s 3 Q − s 2 I + j s 4 Q e j π / 4 ( s 5 I + j s 7 Q ) e j π / 4 ( − s 6 I + j s 8 Q ) s 2 I + j s 4 Q s 1 I − j s 3 Q e j π / 4 ( s 6 I + j s 8 Q ) e j π / 4 ( s 5 I − j s 7 Q ) e j π / 4 ( s 7 I + j s 5 Q ) e j π / 4 ( − s 8 I + j s 6 Q ) s 3 I + j s 1 Q − s 4 I + j s 2 Q e j π / 4 ( s 8 I + j s 6 Q ) e j π / 4 ( s 7 I − j s 5 Q ) s 4 I + j s 2 Q s 3 I − j s 1 Q     A 12 = e j π / 4     0 0 0 sinθ g 0 0 − si n θ g 0 0 cosθ g 0 0 cosθ g 0 0 0     A 13 = e j π / 4     0 0 sinθ g 0 0 0 0 − sin θ g cosθ g 0 0 0 0 cosθ g 0 0     A 14 = e j π / 4     0 0 cosθ g 0 0 0 0 − cosθ g − sinθ g 0 0 0 0 − sinθ g 0 0     A 15 = e j π / 4     0 0 0 sinθ g 0 0 sinθ g 0 0 − cosθ g 0 0 cosθ g 0 0 0     A 16 = e j π / 4     0 0 0 c osθ g 0 0 c osθ g 0 0 sinθ g 0 0 − sinθ g 0 0 0     Notice that the matrix S 1 is as defin ed in (6). Th e M L decoding metric in (2) can be wr itten 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 that the following hold true for l , m ∈ [1 , 8] . A m A H l + A l A H m = 0  ∀ l 6 = m, m + 1 , if m is odd ∀ l 6 = m, m − 1 , if m is even From [ 4], we obtain tr h ( Y − H S 1 ) ( Y − H S 1 ) H i = 4 X m =1 k Y − H T m k 2 F − 3 tr  Y Y H  Therefo re, M ( S ) = 4 X m =1 k Y − H T m k 2 F − 3 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 = 4 X m =1 k Y − H T m k 2 F + 4 X m =1 tr h H S 2 ( H T m ) H i + 4 X m =1 tr h H T m ( H S 2 ) H i + k Y − H S 2 k 2 F − 4 tr ( Y Y H ) Hence, when S 2 is given, i. e, symbols x 5 , x 6 , x 7 and x 8 are giv en, the ML m etric can be decom posed as M ( S ) = 4 X m =1 M ( x m ) + M c (8) with M c = k Y − H S 2 k 2 F − 4 tr ( Y Y H ) and M ( s m ) being a function of sym bol x m alone. Thus deco ding can be done as follows: choo se the quadru plet ( x 5 , x 6 , x 7 , x 8 ) an d then parallelly decode x 1 , x 2 , x 3 and x 4 so a s to min imize the ML decodin g metr ic. W ith this approac h, th ere are 4 M 5 values o f the decoding metric that need to b e co mputed in the worst case. So, the decodin g complexity is of the ord er o f M 5 . V . L O W C O M P L E X I T Y 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 Now , we show h ow the sphere d ecoding can be used to achieve th e d ecoding c omplexity of M 5 . I t ca n be shown that (1) c an be written as ^ v ec ( Y ) = H eq ˜ s + ^ v ec ( N ) (9) where H eq ∈ R 16 × 16 is given by H eq =  I 4 ⊗ ˇ H  G (10) with G ∈ R 32 × 16 being the g enerator m atrix fo r the STBC as defined in Definition 3 and ˜ s , [ R ( s 1 ) , I ( s 1 ) , · · · , R ( s 8 ) , I ( s 8 )] T . with s i , i = 1 , · · · , 8 , drawn fr om A , which is a ro tation of the regular Q AM c onstellation A q . L et x q , [ x 1 , x 2 , · · · , x 8 ] T Then, ˜ s = F ˜ x q . where F ∈ R 16 × 16 is diag [ J , J , · · · , J ] with J being a rotation matrix a nd d efined as follows J ,  cos ( θ g ) − sin ( θ g ) sin ( θ g ) c os ( θ g )  So, (9 ) can be written a s ^ v ec ( Y ) = H ′ eq ˜ x q + ^ v ec ( N ) (11) where H ′ eq = H eq F . Using this equ i valent model, the ML decodin g metric c an be written as M ( ˜ x q ) = k ^ v ec ( Y ) − H ′ eq ˜ x q k 2 . (12) On o btaining the QR decomp osition of H ′ eq , we get H ′ eq = QR , where Q ∈ R 16 × 16 is an orth onorm al matrix and R ∈ R 16 × 16 is an u pper triangu lar m atrix. The ML decoding metric now can be written as M ( ˜ x q ) = k Q T ^ v ec ( Y ) − R ˜ x q k 2 . (13) If H ′ eq , [ h 1 h 2 · · · h 16 ] , wher e h i , i = 1 , 2 , · · · , 16 are column vectors, then Q an d R have th e gen eral fo rm o btained by Gr am − S chmidt process as shown below Q = [ q 1 q 2 q 3 · · · q 16 ] where q i , i = 1 , 2 , · · · , 1 6 ar e co lumn vectors an d R =        k r 1 k h h 2 , q 1 i h h 3 , q 1 i . . . h h 16 , q 1 i 0 k r 2 k h h 3 , q 2 i . . . h h 16 , q 2 i 0 0 k r 3 k . . . h h 16 , q 3 i . . . . . . . . . . . . . . . 0 0 0 . . . k r 16 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 , · · · , 16 . It can be shown by dire ct com putation that R has the following structure R =  R 1 R 2 O 8 × 8 R 3  where R 1 , R 2 and R 3 are 8 × 8 matr ices and R 1 specifically has the following structur e R 1 =             a a 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 a a 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 a a 0 0 0 0 0 0 0 a 0 0 0 0 0 0 0 0 a a 0 0 0 0 0 0 0 a             and R 3 , o f co urse, is an upp er triangu lar matrix. The structu re of the matrix R allows us to perf orm an 8 d imensional r eal sphere d ecoding (SD) [ 11] to find th e partial vector [ R ( x 5 ) , I ( x 5 ) , · · · , I ( x 8 )] T and he nce obtain the symb ols x 5 , x 6 , x 7 and x 8 . Having fo und these, x 1 , x 2 , x 3 and x 4 can be decod ed independ ently . Observe that the re al and im aginary parts of symbol x 1 are entangled with one another bec ause of constellation rotation but ar e indep endent of the real and imag inary par ts of x 2 , x 3 and x 4 when x 5 , x 6 , x 7 and x 8 are condition ally given. Similarly , x 2 , x 3 and x 4 are in depend ent of one another although the ir own real and imagin ary parts are coupled with one another . Having found th e partial vector [ R ( x 5 ) , I ( x 5 ) , · · · , I ( x 8 )] T , w e p roceed to find the rest of the sy mbols as fo llows. W e do fo ur parallel 2 dimensional real search to d ecode the symbo ls x 1 , x 2 , x 3 and x 4 . So, overall, the worst case de coding comp lexity of the proposed STBC is 4 M 5 . T his due to the fact 1). An 8 dimensional real SD requires M 4 metric computatio ns in the worst p ossible case. 2). Four par allel 2 dim ensional r eal SD requ ire 4 M metric computatio ns in the worst case. This d ecoding complexity is significantly less than that for the BHV co de pr oposed in [7], which is 2 M 7 (as c laimed in [7]). V I . S I M U L A T I O N R E S U L T S W e provide perfor mance comparisons between the proposed code and the existing 4 × 2 full-ra te cod es - the DjABBA cod e [13], [12] and the BHV cod e. Fig 1 shows the Cod ew o rd Error Rate ( CER) p erform ance plots for uncor related quasi-static Rayleigh flat-fading chann el as a fun ction of the received SNR at the receiver f or 4-QAM signaling. All the codes perfor m similarly at low and medium SNR. But a t h igh SNR, the full div ersity pro perty of th e DjABB A code a nd the p roposed co de enables them to outperf orm the BHV code. In fact, our co de slightly outper forms the DjABB A code at h igh SNR. Fig 2 shows th e CER perfor mance for 16 -QAM signalin g, which shows a similar result. T ab le I gives a comparision of some of the well known codes for 4 × 2 MIMO systems V I I . D I S C U S S I O N In this paper, we have pr esented a full-ra te, f ull diversity STBCs for 4 × 2 MIMO transmission which enables a signif- icant red uction in the decodin g complexity without h aving to pay in CER perfor mance. In fact, our code performs be tter th an the b est known full r ate cod es for 4 × 2 MIMO systems. So, to summarize, amon g the existing co des for 4 transm it antennas and 2 receive antennas, the propo sed code is the b est fo r QAM constellations . R E F E R E N C E S [1] S. M. Alamouti, “ A simple transmit div ersity technique for wireless communicat ions”, IEEE J. Sel. Area s Commun. , vol . 16, no. 8, pp. 1451- 1458, October 1998. [2] V . T arokh, H . Jafarkhani and A. R. Calderba nk, “Space-ti me block codes from orthogonal designs”, IEEE Tr ans. Inf. Theory , vol. 45, no. 5, pp. 1456-1467, July 1999. [3] H. Jafarkh ani, “ A quasi-ort hogonal space-time block code, ” in IEEE Commun. Letter s, vol. 49, no. 1, pp.1-4, January 2001. Min det ML Deco ding com plexity Code for 4 QAM Square QAM Rectangular QAM Non- rectangula r M = M 1 × M 2 QAM DjABB A cod e [13] 0.04 4 M 6 √ M 2 M 6 ( M 1 + M 2 ) 2 M 7 BHV co de [7] 0 4 M 5 2 M 4 ( M 2 1 + M 2 2 ) 2 M 6 The p roposed cod e 0.64 4 M 5 4 M 5 4 M 5 T ABLE I C O M PA R I S I O N B E T W E E N T H E M I N I M U M D E T E R M I NA N T F O R 4 - Q A M 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 A T E 4 × 2 S T B C S 6 8 10 12 14 16 18 20 10 −7 10 −6 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR in dB CER Proposed STBC BHV Code DjABBA Code Fig. 1. CER performance for 4-QAM [4] Zafar Ali Khan, Md., and B. Sundar Rajan, “Single Symbol Maximum Likel ihood Decodable Linear STBCs”, IEEE T rans. Inf. Theory , vol. 52, No. 5, pp. 2062-2091, May 2006. [5] J. Parede s, A.B. Gershman, M. Gharavi-Alk hansari, ”A 2 × 2 Space-T ime Code with Non-V anishing Determinants and Fast Maximum L ike lihood Decoding , ” in Proc IEEE Internation al Confer ence on Acoustics, Speec h and Signal Pro cessing(IC ASSP 2007), vol. 2, pp.877-880, April 2007. [6] S. Sezginer and H. Sari, “ A ful l rate full -di versity 2 × 2 space-time code for mobile W imax Systems, ” in Proc. IEE E Internation al Confer ence on Signal Proce s sing and Communicat ions , Dubai, July 2007. [7] E. Biglie ri, Y . Hong and E. V iterbo, ”On Fast-Dec odable Space -Time Block Codes“, submitted to IEEE T rans. Inf. Theory . [8] J. C. Belfiore, G. Reka ya and E . V iterbo, “The Golden Code: A 2 × 2 full rate space-ti m e code with non-v anishing determinant s , ” IEEE T rans. Inf. Theory , vol. 51, no. 4, pp. 1432-1436, April 2005. [9] B. A. Sethuraman, B. S. Rajan and V . Shashidhar , “Full -dive rsity , high- rate space-time block codes from divi sion algebr as, ” IEEE T rans. Inf. Theory , vol. 49, pp. 2596-2616, October 2003. [10] V .T arokh, N. Seshadri and A.R Calder bank, ”Space time codes for high date rate wireless communication : performance criteri on and code construct ion”, IEEE T rans. Inf. Theory , vol. 44, pp. 744 - 765, 1998. [11] Emanuele V iterbo and Joseph Boutros, “Uni versal lattice code decoder 10 12 14 16 18 20 22 24 26 10 −5 10 −4 10 −3 10 −2 10 −1 10 0 SNR in dB CER Proposed STBC BHV Code DjABBA Code Fig. 2. CER performance for 16-QAM for fading channels”, IEEE T rans. Inf. Theory , vol. 45, No. 5, pp. 1639- 1642, July 1999. [12] A. Hottine n, Y . Hong, E . V iterbo, C. Mehlfuhrer and C. F . Mecklen- brauk er , ”A Comparision of High Rate Algebrai c and Non-Orthogona l STBCs“, in Pro c. ITG/IEEE W orkshop on Smart Antennas WSA 2007 , V ienna, Austria, February 2007. [13] A. Tirkk onen, O. Hottinen and R. Wic hman, ”Multi-ant enna Transcei ver T echniq ues for 3G and Beyon d.“ WILEY publisher , UK. [14] A. T irkko nen and O. 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