Differential Transmit Diversity Based on Quasi-Orthogonal Space-Time Block Code

Differential Transmit Diversity Based on Quasi-Orthogonal Space-Time Block Code Chau Yuen, Yong Liang Guan School of Electrical an d Electronic Engineering Nanya ng T echno logica l Uni ve rsit y Singapore yuenc hau @pma il.n tu.ed u. sg, e ylguan @ntu. edu. sg Tjeng Thiang Tjhung Institute f or Infocomm Res earch Singapore tjhun gtt@i 2r. a-sta r.ed u.sg Abstract — By us ing j oint modul ati on a nd cus to mized constellation set, we show that Quasi-Orthogonal Space-Time Block Code (QO-STBC) can be used to for m a new differential space-time modulation (DSTM) sch e me to provide full transmit dive rsity w ith no n-cohe re nt det ect ion. Our n ew sche me ca n provi de highe r c ode r ate th an e xi sti ng DS TM s cheme s ba sed on Orthogo nal STB C. It al so has a low er deco ding c omplex ity than the other DSTM schemes, s uch as t hose base d on Group C odes, because it o nly requires a joi nt detection of tw o complex sy mbols. We derive the de sign criteri a for the customize d constellati on set and use them to construct a constellation set that provides a wide range of spe ctral efficie ncy with full diversity and maxi mum codi ng g ain. Keywords - di fferential space -time modula tion; low dec oding complexity ; non c oherent detec tion; quasi-orth ogonal s pace-ti me block c ode. I. I NTROD UCTIO N In wireless c ommun ications, sy stem performance is often severely degraded by f ading effects du e to mu lti-path signal propagati on. Modulati on techni ques designed for m ultiple transmit an tennas, calle d space-tim e modulation or t ransm it divers ity can be used to re duce f ading ef fects ef fectiv ely. Ea rly transmit d iversity sche mes were designed for co herent detec tion, w ith chan nel esti mates assum ed availabl e at th e receive r. How ever, the c om plexity and cos t of chan nel estimation gr ow w ith the number of transm it and rece ive antennas. Therefo re the availa bility of transm it diversity schemes th at do not r equire ch annel estim ation is d esirable. To this end, s everal diff erential space-tim e modulation (DSTM) schemes have been designed [1 -8]. Hughes has de signed a DSTM b ased on group codes [1, 2]; Hochwald and Swelde ns have de signed a DSTM based o n unitar y matric es [3]. T arokh a nd Ja farkha ni have pro posed a DSTM by using the Orthogonal Space-Time Block Code (O- STBC ) [4] , w hile Ga nes a an d Stoi ca [ 8] p rovi de an oth er DSTM also based on O-S TBC but w ith a m uch sim pler decod ing comple xity. Hassibi et a l. has proposed DSTM based on Cay ley code and S p(2) . Recent ly , Al- Dhahir g ives a n ew rate- tw o DSTM base d on tw o pa ralle l A lamou ti O- STBCs for four transm it antennas sy stem [6]. Am ong them th e scheme in [8] has the s implest d ecoding com plexity. H owev er the DSTM in [8] is based on th e O-STBC, whose m aximum achievable code rate is limited to ¾ f or four ant ennas and ½ for eight antennas w hen u sed w ith complex c onstellati ons. I n t h i s p a p e r , w e p r o p o s e a n e w D S T M s c h e m e t h a t i s based on Quas i-Orth ogon al STBC (QO-S TBC ) [9-1 3], to provide f ull transm it diversi ty w ith high er code ra te than th at of [8] and lo wer decodi ng complexity tha n other DSTM schemes [1-7]. We will deriv e the design crit eria an d constru ct an exam ple conste llation set f or the p ropos ed DSTM s chem e. We will als o study its decodi ng complexity and deco ding perform ance in comparison with th e existin g schem es. The organiz ation of this paper is as f ollow, Section I I reviews the DST M signal model and the decoding per formanc e criter ia of DST M. Sectio n III propo ses the ne w DSTM sc heme based on QO-STB C. And Section I V gives the pe rformance compar ison. Fina lly Section V conc ludes the paper . II. R EVIEW OF DSTM A. DSTM Signal Model Consider a MIMO comm unication s ystem , with N T transmit an tennas and N R rece ive antenn as. Let H t be the N R × N T channel gain m atrix a t a ti me t . T hus the ij th elem ent of H t is the chann el coeffic ient for the signal path fr om the j th trans mit antenna to the i th receiv e anten na. Let C t be the N T × P codeword tr ansmitted at a ti me t . Then , the re ceived N R × P signal m atrix R t can be w ritten as tt t t =+ RH C N (1) where N t is the addit ive w hite Gaussian nois e. In this p aper, th e code length P is set equal to N T a s i n [ 1 ] a n d [ 8 ] , s o t h a t t h e transm itted codew ord C t is a squa re m atrix. At the sta rt of the transm ission, we tran smit a kn own codew ord C 0 , w hich is a un itary matrix of size N T × N T . The codew ord C t transm itted at a tim e t is diff erent ially enc oded by 1 tt t − = CC U (2) where U t is a unitary matrix (such th at U t U t H = I ) ca lled the code matrix , that contain s inform ation of the t ransmitte d data. Sin ce Globecom 2004 545 0-7803-8794-5/04/$20.00 © 2004 IEEE IEEE Communications Society C 0 and U t are b oth unitary , it f ollows th at C t is unitary for all time t . Hence th e requ irem ent for th e code m atrix U t to b e unitary is essential t o ensure that all the transm itted codew ords have co nstant power. I f we assume tha t the cha nnel remains unchanged during t wo consecutive code p eriods, i.e. H t = H t -1 , then the receive d signal R t at a tim e t can be expr essed [8] as 1 tt t t − =+ RR U N  (3) The re ceived signa l R t can be diff eren tiall y dec oded as it depen ds onl y on the prev ious receiv ed sig nal bl ock R t -1 , the code matrix U t and an equival ent addi tive w hite Gauss ian nois e 1 tt t t − =− + NN U N  . Since N t and N t -1 are w hite and U t is unitar y, it can b e shown that t he e quivalent noise t N  is white, and has zero m ean and twice t he pow er as th e channe l noise N t or N t -1 [7] . This ex plains w hy there is a 3 dB SNR loss w hen the receiv ed sig nal is dec oded dif ferenti ally inst ead of cohe rent ly [1, 4, 7, 8]. T he co rrespond ing ML d ecodi ng decisio n metric i s, {} {} () () {} H 11 H 1 ˆ arg min tr arg max Re tr t t tt t t t t t tt t −− ∈ − ∈ =− − = U U UR R U R R U RR U U U (4) where U denotes the se t of all possible code matr ices. B. Divers ity and Coding G ain The deco ding pe rform ance of DSTM has been analy zed in [1], w hich foun d that the de sign criter ia for DS TM are th e sam e as for coh erent s pace- time co ding. Spe cifically , th e transm it diversity level th at ca n be ach ieved is given by: () Min rank kl kl −∀ ≠   UU (5) In orde r to ach ieve fu ll transm it diversity , the minim um rank i n (5) has to be equal to N T . For a fu ll-rank DSTM co de, i ts coding gain is defin ed in [1], [8 ] a s () () () 1/ H Min de t T N Tk l k l Nk l  ×− − ∀ ≠   UU UU (6) In ord er to achie ve optimum de coding pe rformanc e, the codi ng gain has to be maximiz ed. III. N EW DSTM S CHEME In this section, w e shall develo p a new DSTM schem e using the Q O-STBC desi gned in [9] for four transmit a ntennas as an exam ple. Ou r pro pose d techni que, how ever , is applic able to any square QO-S TBC that supports joint detection of t w o complex sym bols, such as thos e in [10-1 3]. A. Quasi Or thogonal Space -Time Block C ode Th e 4 × 4 codew ord of the QO-S TBC in [ 9] (he rein denot ed as code Q4) is shown in ( 7), where c i , 1 ≤ i ≤ 4, represents th e transm itted complex c onstellati on sym bol. It can be seen that the codew ord of Q4, C Q4 , is not a unitary matrix , since α ≠ 1 a nd β ≠ 0 in genera l. ** 123 4 ** 21 4 3 ** 34 1 2 ** 43 2 1 c- c - c c cc- c - c = c - cc- c c ccc               Q4 C (7) H 00 00 00 00 αβ αβ βα βα   −  =≠  −   Q4 Q4 CC I (8) where 4 2 1 i i =c α = ∑ ; ** 14 2 3 =2 × R e ( c × c c × c ) β − . C Q4 can also be re presen ted as [10] : () 4 RI 1 kk kk k =c j c = + ∑ Q4 CA B (9) where A k and B k are cal led t he dispe rsion matri ces [14 ] with quasi- ort hogon al alg ebraic prope rties desc ribed i n [10 ], and t he supe rscri pts R and I re presen t the r eal an d imag inary pa rts o f a sym bol respe ctively . The det erm inan t of the cod ew ord distan ce matrix o f Q4 has been shown in [10] to be: 2 2 min 2 3 () det () det ( ) ( ) i f = =0 22 14 23 22 14 2 3 22 14 14 + +  ∆+ ∆ ∆ − ∆ ×  =  ∆− ∆ ∆ + ∆   ⇒ = ∆ +∆ × ∆ −∆ ∆ ∆  (10) where ∆ i , 1 ≤ i ≤ 4, represen ts the possib le error in the i th transm itted constella tion symbol. That is, ii i =c e ∆− if the recei ver decides erro neousl y in favor of e i if c i is t ransmitted. When c onsiderin g the m inimum determ inant in accor dance to ( 6 ) , w e c a n a s s u m e h a l f o f t h e c o d e w o r d e r r o r s t o b e z e r o , which i s similar to the design appr oach o f const ellatio n rotatio n for QO-STB C discussed in [10 – 13]. In orde r to achi eve full transm it divers ity and o ptimum coding g ain in accor dance w ith the design crite ria define d in (5) and (6 ), the valu e in ( 10) has to be non-z ero and maximize d. B. New DSTM Based on Q O-STBC As w e have expla ine d earlie r, the u se of a uni tary code matrix , U t , is essent ial for for mulating a D STM scheme. I n order t o apply the code Q 4 in a DSTM, i. e. use C Q4 as U t , C Q4 has to be a unita ry m atrix, i.e . α and β in (8) have to be one and zero res pectively for all possi ble values of c i . Thi s canno t be Globecom 2004 546 0-7803-8794-5/04/$20.00 © 2004 IEEE IEEE Communications Society achieve d if conv entiona l memory less modul ation is a pplie d. In order to ha ve β equa l to ze ro, w e can se e from (8) th at so me form of correlat ion or constra int betw een the t ransmitt ed symbols c 1 and c 4 , as w ell as between c 2 and c 3 , are requir ed. Theref ore w e pr opose the us e of join t modulati on with a speci ally des ign ed co nstell ati on set , such th at the transm itted sym bol- pairs { c 1 , c 4 } and { c 2 , c 3 } can alw ays achiev e an α equal t o one and a β equal to zero. Based on the a bove ide a, a joint cons tellati on set M which consists of L sets of compl ex-val ued co nstellat ion pai r { x k , y k } (w here x k and y k are each a com plex va lue, an d 1 ≤ k ≤ L ) can provi de a sp ectral eff ic iency of R = 2(lo g 2 L )/ N T bps/Hz. The re quirem ents on the const ellation se t M are: (i) 0.5 22 kk x+ y = (ii) * Re( ) kk xy = v (11) (iii) { } 2 22 maxi mize Mi n kl kl kl kl xy xy  ∆+ ∆ × ∆− ∆  where v can be a ny constant va lue, a nd kl k l xx x ∆= − , kl k l yy y ∆= − for al l 1 ≤ k ≠ l ≤ L . In com plying with the Power Cri terion (11)(i), w e ensure that the transm itted code words have a co nstant po wer and hence α = 1. In comply ing with the Un itary Criteri on (11)(ii ), w e ensure that C Q4 is alw ay s a unitary matrix , i.e. w e achieve β =0. With t he Perf orman ce Optim izati on Crite rion ( 11)(i ii), w e ens ure th at the m inimum determ inant value in (10) is maxim ized such that the pr opose d DSTM can achi eve fu ll dive rsity and m aximum codi ng gain. To give an exam ple of the prop osed DS TM with joint modulation, conside r the case of R = 2 bps/Hz (i.e . L = 16, N T = 4). In th e enco din g process , w e tak e four data bits and m ap i t to one of the 16 constellat ion pairs { x k , y k } in M as th e co de sym bol pai r { c 1 , c 4 }. Then we take t he next fo ur data bit s and map them to another constellati on pair in M as th e code sym bol pai r { c 2 , c 3 }. Hence , the code matrix U t in (2) can be obtained as C Q4 in (7) w ith th e code sym bols c 1 to c 4 assigned as abo ve. In the deco ding pr oces s , w ith (9), the ML decisi on metric (4) can be sim plified to: {} () { } () {} {} () {} () {} 14 23 HR 1 1, 4 14 HI {, } 1 1, 4 HR 1 2,3 23 HI {, } 1 2,3 Re tr ˆˆ ,a r g m a x Re tr Re tr ˆˆ ,a r g m a x Re tr t tii i cc tt i i i t tii i cc tt i i i c cc jc c cc jc − = ∈ − = − = ∈ − =  +  =    +  =   ∑ ∑ ∑ ∑ RR A RR B RR A RR B M M (12) As s hown in (12), the ML dec oding of the p ropose d diffe rential QO -STBC modul ation sc heme can be a chieved by the joint detection of tw o complex sym bols, and the tw o ML metrics can be com puted in parallel . So the proposed DS TM scheme does no t in crease th e ML dec oding c omplexity as co mpared to th e origi nal co herent Q4 code. Furt hermor e, it has a lower ML decoding c omplexity than th e DSTMs reported in [1- 7], w hich requi re gene rally a larger joint detection sea rch space f or the sam e spectr al ef ficiency and an tenn a num ber (to be elab orat ed in Se ctio n IV). C. Des ign of Cons tellati on Set In thi s section, we propose a co nstella tion set M that satisfies all three requirem ents in (11) w ith high flexibility in spect ral effici ency . The p ropose d const ellati on set M , { x k , y k }, 1 ≤ k ≤ L , is: () () exp 2 / for 1 2 2 0 0 for exp 2( / 2) / 2 2 k k k k jk M L x k y x L kL jk L M y π πθ    =  ≤≤   =  =   <≤ −+    =   (13) where M = L /2 is an integer, and θ is a constellatio n rotation angle betw een 0 and 2 π / M . Note that in ( 13), x k belongs to a half-po wer M -ary PS K const ellati on for 1 ≤ k ≤ L /2 , while y k belongs to a rotated ha lf- pow er M -ary PSK constellat ion for L /2 < k ≤ L . The param eter M is relat ed t o the spect ral ef ficiency R by R = 2(l og 2 2 M )/ N T . Hence a DS TM w ith a w ide rang e of spec tral effi ciency can be systematica lly designed from (1 3) by adj usting M . The conste llation r otatio n angle θ prov ides an extra degr ee of freedo m to maximiz e the diver sity and cod ing gain of the resultant DS TM. Theorem 1 : For the DST M constellation set defined in (1 3), the optimum value for the constellation rotation an gle θ (in the sense of the Performance Optimization Criterion (11)(iii)) is π / M whe n M i s ev en, a nd is π /2 M or 3 π /2 M when M is odd. Proof of T heorem 1 : Th e proof is given in fo ur cases, consid ering d iffere nt value s of k and l . Case 1: 1 ≤ k, l ≤ L/2, and k ≠ l. In this case , ∆ y kl is alw ay s zero (sin ce y k and y l are al ways zer o), b ut no t ∆ x kl (sin ce k ≠ l ), h ence the dete rmin ant value in (11)(iii ) can n ever be zero, a nd its v alue is in dependent of θ . Hence in th is case , full diversity is alw ays achieve d, and t he codi ng gain op timizatio n does not depend o n θ . Case 2: L/2 < k, l ≤ L and k ≠ l. The pro of is similar t o Case 1 with the ro le of ∆ x kl and ∆ y kl interc hanged . Case 3: 1 ≤ k ≤ L/2 and L/2 < l ≤ L. For this case , the det ermin ant value in ( 11)(iii ) can be simplified as follow s: Globecom 2004 547 0-7803-8794-5/04/$20.00 © 2004 IEEE IEEE Communications Society () 2 2 2 4 22 ( / 2 ) (e x p e x p ) 1 det 16 22 ( / 2 ) (e x p e x p ) 22 = sin 1 , 2 sin 2 2 kl L jj MM kl L jj MM km L km l - M MM 2 np 0nk - m M ππ θ ππ θ ππ θ π   −      ++ ×         =    −     −+            −− ≤ ≤     −  =≤     1 M- ≤ (14) where k , l , m , n are intege rs, and /2 ML  / p M2 θπ  is a real numbe r between 0 and 1 inclusive ly. 0 1 2 M/2 M/ 2+ 1 M-1 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 p=0 p=0. 5 p n (a) M even 0 1 2 (M-1)/2 (M + 1)/2 M-1 0 -1 -0. 8 -0. 6 -0. 4 -0. 2 0 0. 2 0. 4 0. 6 0. 8 1 p = 0 p = 0. 25 p n (b) M odd Fig ur e 1 . Optimi zati on of constella tion rotat ion angle / 2p M θπ = To max imize th e determ inant valu e in (14), w e firs t consi der even values of M . As shown in Figure 1(a), the solid line, sin[ 2 n π / M ], is zer o if n = 0 or M /2, as sh own by th e triang ular mark ers. In or der to ach ieve fu ll divers ity , a right shift, p , can be intr oduce d to obt ain th e dash line , sin[2 ( n- p ) π / M ] , such tha t its v alue is non-zer o for al l integ er values of n . To fur ther achieve maximum codi ng gain, the va lue of p should b e chos en such th at the min imum absol ute value s of sin[2( n-p ) π / M ] at integ er values of n are m axim ized, as sh ow n by t he circu lar marker s in Figure 1(a). This corr espond s to p = 0.5 and θ = π / M . Similarly, fo r odd value s of M , th e optim um p value is 0.25 or 0.75, which co rrespond s to θ = π /2 M or 3 π /2 M . Case 4: 1 ≤ l ≤ L/2 and L/2 < k ≤ L. The pro of is similar t o Case 3. A s the determ inant v alue in (11)( iii ) does not depen d on θ for C ases 1 and 2 , and t he optimum θ value has b een derive d for Cases 3 and 4, Theorem 1 is pro ved. ■ IV. P ERFO RMANCE R ESULTS The cod ing gai n and de cod ing co mple xity o f o ur proposed DST M w ith an optimized constellatio n set and some existing DSTM [1, 2, 8] are sh own in Table I. It can be seen that ou r proposed DSTM pr ovides t he highest codi ng gain when compared wi th DST M based on O-STBC [8] and DSTM based on g roup codes [1,2]. Ou r proposed DSTM als o has a lower decoding com plexity than those in [1, 2], because the ML decoder of ou r proposed DSTM only needs to jointly decode tw o complex sy mbols in paral lel. A t 1.5 (or 2) bps/ Hz, this leads to a ML search space of 8 ( or 16) with 4PSK (or 8PSK) constellation, compared with a s earch space of 64 (or 256) for the s chemes in [1, 2] with 64PSK (or 256PSK) conste llation. Alth ough the DSTM based on O-STBC in [8] has the low est decoding com plexit y , it has a lower coding gain, and it does not has t he flexibility to pro vide 2 bps/Hz using conventional constellati ons (hence it is not inc luded in the comparison table). TABL E I. C OMPARISON OF CODING GAINS AND DECOD ING COMPLEXITY FOR DSTM FOR FOUR TRANSMIT ANTENNAS DSTM Conste- llati on Bps/H z Coding Gain ML Search Space [1, 2] 64PSK 1.5 1.85 64 Propose d sche me (13) w ith M = 4, θ = π / 4 1.5 2.83 8 [8] QPSK 1.5 2.70 4 [1, 2] 256PSK 2 0.78 256 Propose d sche me (13) w ith M = 8, θ = π / 8 2 1.17 16 In Figure 2, we compare t he block erro r rate (BLER) perfor mance of our proposed DST M with those of the other DSTM’s rep orted in the lite rature [ 5, 7, 8]. Comparing our prop osed D STM of 2 bps/ Hz w ith Cayley ’s D STM wi th spect ral efficien cy of 1.75 bps/Hz (o btaine d with Q =7 , r =2 in [5]), our DS TM provid es a co mpara ble BLER perfo rman ce but at a high er spec tral effic iency . Com pared w ith the DSTM based on Sp( 2) w ith spect ral eff iciency of 1.94 b ps/Hz (obt ained w ith M =5, N =3 in [7]) and ML se arch s pace of 225, our pr op osed DS TM pe rform s 1d B w orse in BL ER but has high er spectr al effici ency an d a m uch sm aller ML sear ch spa ce Globecom 2004 548 0-7803-8794-5/04/$20.00 © 2004 IEEE IEEE Communications Society of 16. F inal ly , the DSTM base d on O-S TBC [8 ] has th e hig hest spect ral effic iency of 2.2 5 bps/Hz, but it has the w orst BL ER perfor mance. 10 12 14 16 18 20 22 24 26 10 -6 10 -5 10 -4 10 -3 10 -2 10 -1 10 0 BLER SN R D S TM O -S TB C 8 P S K 2 .25 b p s /Hz [8] C a yley C o d e 1 .7 5 b p s /Hz [5 ] S p (2 ) 1 .9 4 b p s/Hz [7] D S TM Q O-S TB C 2 b p s /Hz [p ro p o se d ] Fig ur e 2 . Simulated block error rate (BLER) of DSTM for fou r transmit and one r eceiv e antenn as In Figur e 3, we co mpare the bit error rate (BE R) perfor mance o f our proposed DSTM with di ffere ntial and coherent detecti on, with that of Q4 with constellati on rotat ion [10- 13] an d tha t of th e rat e-tw o DS TM sche me of [6]. All codes c onsider ed in this f igu re have the s ame sp ectral efficien cy of 2 bps/Hz . 10 15 20 25 30 10 -4 10 -3 10 -2 10 -1 BER SN R C o her e nt Q4 wi th ro ta te d QP S K Prop osed DST M w it h coh eren t det ec tion Prop osed DST M w it h dif f ere n t ial det ect ion Ra te two D S TM i n [6] Fig ur e 3 . Simulated bit error rat e (BER) of DSTM schemes for four tran smit and tw o r eceive antennas w ith spe ctral eff iciency of 2 bps/H z Figure 3 s how s that th ere is a 3dB SNR g ap betw een the coher ent and di fferenti al detect ion of our propo sed D STM. This is due to the doubl ing of no ise power for the latter, a s explain ed earl ier in this pape r. Com pared to th e orig inal QO- STBC, Q 4 w ith constell ation rotat ion and coher ent dete ction [10- 13], ou r pr opose d DS TM is 2dB w orse if coher entl y detec ted. This loss is du e to the con strain t that th e codew ord in our p ropos ed s chem e m ust be a uni tary m atrix in or der t o support diff erential m odulation. How ever, this loss is accepta ble, as a coher ent QO- STBC w ill suf fer perf orman ce degradat ion if the chann el estim ation is im perfect. Fina lly , Figu re 3 also show s th at our pr oposed schem e has a bett er BER perfo rmance than the DSTM pres ented in [ 6], which does not achieve full d ivers ity. V. C ONCLUSIONS We pro pose a new differen tial spac e-tim e modulat ion (DSTM) sch eme base d on QO -STBC. The m ain idea is to f orce the QO -STBC code word to be a unitar y matrix by usin g joint sym bol modulation w ith a specially designed conste llation set. The desi gn criter ia for the corresponding c onstellat ion set to achie ve full diversi ty and maxi mum cod ing gain a re der ived. We t hen propo se a conste llation set (compr ising z ero, P SK and rotate d PSK sym bols) that meets th e design cri teria an d suppo rts a w ide rang e of sp ectral ef ficien cy. Ou r propos ed DSTM scheme has the merits of QO -STBC, henc e it achieve s higher c ode rate than DSTM based o n O-STBC, and lo wer ML deco ding co mplexi ty (s malle r sear ch sp ace) t han o ther D STMs, such as [1 – 7], w ith a c om parable dec oding perf orman ce. Although a four -antenna code i s consi dered i n this l etter, the diffe rential cod e design and co de opti mization id eas proposed herein apply to any QO-STBCs that can be f ound in the literatu re, inclu ding th ose for eight transm it antennas . R EFERENCES [1] Hughes, B . L., “ Diffe rential s pace- time modul ation” , IEEE Tra ns. on Info. Theory , Vol:46, I ssu e:7, Nov. 20 00, 2567–2 578. [2] H ughes , B. L., “Opt imal s pace-t im e const ellati ons f rom grou ps”, IEEE Trans . on I nfo. T heory , Vol:49, Issue: 2, Feb. 2003, 401– 410. [3] Hochwal d, B. M.; Sw elde ns, W., “ Diffe rential unitary space -time modul ation ”, IE EE Tr ans. on C omm s. , Vol :48, I ssue :12, De c. 2000, 2041–20 52. [4] Tarokh, V. ; Jaf arkha ni, H ., “A diffe rential dete ction s cheme for transm it diver sity ”, IEEE Journa l on Sele cted Ar eas in Co mms. , Vol: 18, I ssue: 7 , Ju ly 2000, 1169 –1174. [5] Hassibi, B.; H ochwa ld, B. M., “ Cayle y diffe rential un itary space-tim e codes”, I EEE Tr ans. On Inf o. Theory , 2002, 1458 – 150 3. [6] Al-Dhahir , N., “ A new high-rate diff erential space- time bl ock co ding scheme”, IE EE Comms. Letter s , Vol:7 , Issue:11 , Nov. 2003, 540– 542. [7] Jing, Y.; Ha ssibi , B., “De sign of fully-div erse multi- antenn a codes bas ed on Sp(2)” , ICA SSP 200 3 , Vol : 4, 3 3-36. [8] Ganesan, G .; Sto ica, P., “ Diffe rential space -time m odulat ion usi ng space -time block codes ”, IEE E Sign al Pr ocessi ng Let ters , Vol:9 I ssu e:2 , Fe b 2002, 57 –6 0. [9] J afarkh ani, H., “A quasi -orthogon al spa ce-time b lock cod e”, IE EE Trans . on Comms ., Vol :49 I ssue: 1, Jan. 200 1, 1-4. [10] Yuen, C.; G uan, Y . L .; Tjhu ng, T. T . , “F ull-Rate Ful l-Dive rsity S TBC with Const ellati on Rotati on”. VT C 2003-Sp ring , Vol:1 , 296 –300 . [11] Tirkk onen, O. , “Opti mizing STBC by Constella tion Rotations ”, FWCW 2001 , 59-60. [12] Sha rma, N.; Papadias, C.B., “ I mprov ed qua si-orth ogonal c odes throu gh conste llatio n ro tation ”, IEEE Trans . on Comms . , Vol :51, Is sue:3, Mar ch 2003, 332 - 335. [13] Su, W.; Xia, X., “Quasi-Orth ogonal STBC wit h Full Diversi ty”, Globe com 2002 , 1098 –11 02. [14] Has sibi , B.; Hochwald , B. M. , “High -Rat e Cod es that are Lin ear in Space an d Time”, IE EE Trans. on Info. Theo ry , Vol :48 I ssue :7 , Jul 2002, 1804 –1824. Globecom 2004 549 0-7803-8794-5/04/$20.00 © 2004 IEEE IEEE Communications Society