An Optimal Two Transmit Antenna Space-Time Code and its Stacked Extensions

Transcription

1 9 G 9 An Optiml Two Trnsmit Antenn Spce-Time Coe n its Stcke Extensions Prnv Dl n Mhesh K. Vrnsi Electricl & Computer Engineering Deprtment Universit of Coloro, ouler, CO 8009 Astrct A spce-time coe is propose tht exhiits the highest coing gin mong competing full rte full trnsmit iversit spce-time coes for the two trnsmit ntenn wireless chnnel. The propose coe is erive from lere rchitecture with rel rottion of QAM informtion smols in two imensions. The existing coes of similr rchitecture concentrte on ppliction of complex full moultion iversit rottions or smmetric rel rottions for the two lers. An nltic evlution illustrtes the significnt improvement in coing gin chieve with the propose coe. Moreover, the vlue of the unnormlize coing gin for the new coe is inepenent of the rte of the coe. A stcke extension of the propose coe offers reuce complexit cpcit optiml lterntive to the full iversit coes for lrger numer of trnsmit ntenns. Performnce enhncement in severl scenrios is verifie through simultions. 1 Introuction The pioneering Almouti spce-time coe [1] for two trnsmit ntenns remins unmtche in performnce when onl one receive ntenn is emploe. However, s the numer of receive ntenns is increse, the limittion of the Almouti scheme ue to loss of mutul informtion is well known [2]. In [], numer theoretic construction of spce-time coe for two trnsmit ntenns, clle the coe, provie scheme tht i not suffer from n informtion loss ue to the spce-time coe constrints. The coe exhiits full iversit, trnsmits t the full rte of informtion smols per chnnel use n les to superior performnce compre to the Almouti scheme for more thn one receive ntenn n high spectrl efficienc. The coe ws susume s specil cse of wier clss of full rte full iversit spce-time coes for ritrr numer of trnsmit ntenns clle the Three Algeric Spce-Time (TAST) coes [4]. The recentl propose tilte-qam coe [] lso chieves full rte n full iversit with two trnsmit ntenns. This work ws supporte NSF grnt CCR In this pper, we propose new informtion lossless or cpcit optiml [6] coe tht exhiits significnt improvement in performnce compre to the, the TAST n the tilte-qam coes for two trnsmit ntenns. First, we formulte prolem tht is concerne with the minimiztion of the worst-cse pirwise error proilit [7] for clss of spce-time coes. The solution to this prolem provies us with coe tht cn e extene to iel improve coes for severl trnsmission schemes. This pper is orgnize s follows. The sstem moel n efinitions re escrie in Section 2. The propose coe n its properties re presente in Section. Stcke extensions of spce-time coe re escrie in Section 4. Numericl results re presente in Section. The conclusions re given in Section 6. 2 Sstem Moel n Definitions 2.1 Sstem prmeters An trnsmit, receive ntenn qusi-sttic fing chnnel with coherence time is referre to s n chnnel. Let e spce-time coeook of size complex mtrices. If the coewor is trnsmitte, the receive sttistics re given!#" $'&( (1) where " n & re of size )( n )*, respectivel, n consist of i.i. zero-men, unit-vrince complex norml entries. The verge receive Signl-to-Noise Rtio (SNR) is +-,/ The trnsmit iversit 9 of is the minimum rnk mong the ifferences of istinct coewors in, ; <>=@? ACED A H A I EF A n the error proilit NPO (ML) ecoing stisfies Q =R< The coing gin rnk KML( with Mximum-Likelihoo LWQRXZY \[ N O ^9 [7]. +TSVU QRXZY ^] ; of the coe with iversit ` E

2 G L á Y is efine s [7] ; ce <>=R? j ACED A k-o rnkf A gef A h IZi krl WpL4 KqL4 r-st nm (2) 8Mv o{z 2xwp z s n k-o} s represents the ~ -th m cu where lrgest eigenvlue of the rgument. 2.2 Existing Methos for Coe Construction The spce-time coes consiere in this pper re erive from Full Moultion Diversit (FMD) lttice constelltions. A lttice constelltion hs the FMD propert if n two istinct points in the constelltion iffer in ll the co-orintes. Let e the stnr complex QAM constelltion ƒ with points. If is perfect squre, then $ ~ˆ>ŠL $Œ -Ž el - o integers, where ~ L. Let e full rnk complex genertor mtrix of size. Let ƒ W š œž e the finite lttice constelltion otine from. If the lttice constelltion exhiits full moultion iversit, then the normlize minimum prouct istnce š cn e efine s in [8] to e < =R? j ; D - I H F l vžª o{z z () wp«s The full iversit TAST coes propose in [4] re otine forming (Œ ) inepenent nonoverlpping lers of FMD lttice constelltions. The lttice co-orintes of the -th ler, ' Ž, re multiplie constnt, known s Diophntine numer, to gurntee full trnsmit iversit for the resulting coeook [4]. Optimum ecoing of TAST coes is possile using sphere ecoer [9] of size š. The informtion lossless propert of the TAST coe is preserve if the following conitions re stisfie [,4] (1) The genertor mtrix of the FMD lttice for ech ler is unitr n (2) The numer of lers n Z ±. The coing gin for the TAST coes epens on the choice of the component genertor mtrices n the Diophntine numers. The joint esign of the lttice for ech ler n the Diophntine numers to mximize the coing gin of the resulting TAST coe is n open prolem for ³². The solution for is simpl to choose the require genertor mtrix s the complex rottion tht is optiml with respect to the minimum prouct istnce () of n FMD lttice. An -hoc pproch for ll µ is to use the sme rottion mtrix tht is optiml for irrespective of the choice of the Diophntine numers. In the light of the improve coe propose in this pper, it will ecome cler tht the optiml genertor mtrices with respect to the normlize minimum prouct istnce o not necessril give rise to the est spce-time coes tht emplo the FMD lttices. An Optiml Spce-Time Coe Following the coe esign criterion of mximiztion of the coing gin expression in (2), we pose the following prolem for selecting the est mong clss of cpcit optiml spce-time coes for. Prolem Sttement. Let ^ o -q /s n let e complex numer with. Let ¹»º e the two imensionl rel rottion given ¹¼º ¾½À X L =R? Consier the uncountle fmil à =R? X  prmetrize n- / such tht n coe is given n- / ÆÅ ½ Z  p p ÍÌZÎ If it exists, fin the pir p p q Ë of spce-time coes ¹¼º ž œ È / such tht ÐÏnÑ f1ò D Ó h Ô f G f1ò D Ó h}h I Ä- /(»Ã ¹¼º È É (4) Ä- /Ð () Proposition 1. For ll ÖÕ, the solution to () is q p p / Ìp?ÙØ E Lx~7 (6) Proposition 1 is the primr result of this pper n its proof is ivie into three prts. First, the optimlit of prticulr rel rottion in terms of mximiztion of the normlize minimum prouct istnce is prove. This rel rottion is then pplie in the construction of full iversit spce-time coe in Ã. The coing gin of n other coe in à is then shown to e smller thn tht of..1 Optimum Rel Rottion Let the set of ll possile ifferences of elements in e enote. All the elements of re Gussin integers ivisile. Lemm 1. Let MÛ Ò e the lttice ƒ constelltion otine from ¹º n ; s šû Ò ¹¼º š Æ. Then, the mximum vlue of the minimum prouct istnce ŠÛ Ò is chieve with qüöý ÌZ? Ø Z. Proof. The enomintor of Û Ò in () is the sme for ll vlues of. The numertor in (), referre to s the unnormlize minimum prouct istnce Û Ò Þ<>=R? ß D ßT ß IH FEà ßT o ¹¼º ^L( Û Ò is such tht 7s o ¹¼º ^L( s (7)

3 ö æ åå i ö $ æ s È <>=@? çæ â æ =R? D â á D Ô L p æ X q # f â ED â hfeà IH fäã D ã hžå å å å éè <>=R? ƒê =@? X# p q åå éè X q Ü Û Òtì (8) where æ Ìp? the lst equlit is ue to the fcts tht p püq n æ æ L $ (í æ æ for î o æ ~ts æ unless..2 A Full Diversit Spce-Time Coe in à The optimum rel rottion mtrix given Lemm 1 is now utilize in the construction of full iversit K' spce-time coe. The propose coe is given Ý ÜêLx~ˆšÃ The following proposition estlishes the full trnsmit iversit of. Let cðï ñ enote the verge energ of the complex -QAM constelltion. Proposition 2. The spce-time coe exhiits full trnsmit iversit for n VÕ n the coing gin is given òëó ô õ ˆø#ùZ. Proof. Let two istinct coewors n in correspon to the two istinct sets žè n È of informtion smol vectors from. If o q s{û µl n o q- srû È L'È, then ech of - - re elements in ; n t lest one of them is non-zero. If ¹¼ ž ¹¼ ¹üÈ ¹üÈ, then Ðýÿþ L( \ q T T M X q Üp (9) where the integer vlue function \ T T L $ ^$ ~ L¾ $). Since ech of ç - - re multiples of, the function - - is multiple of è. The result of Lemm 2 in the Appenix implies tht \ - - is non-zero for n vlue of. Therefore, the coe hs full iversit for n n the minimum of the expression in (9) is equl to è X# p éüp. Finll, v 2 wp i orz z c ïeñ n we get from (2) tht ÀòCó ô õ Ðø#ùq. The result of Proposition 2 implies tht the coing gin of the propose spce time coe is, in fct, given v Û Òtì 2 wp i orz z (10) s Lx~ in tht results in no It is the specil choice of ecrese in the minimum eterminnt the presence of the secon ler in the coe.. Optimlit of It is now shown tht Ü n Lx~ is n optiml choice mong ll spce-time coes in Ã. Consier n coe ó ó - ó (Ã. An upper oun on cn e otine consiering specific choice of two istinct coewors n in ó tht correspon to the two istinct sets žè n ÐÈ of informtion smol vectors Ðýÿþ from. Set È n choose n such tht LV is the unnormlize minimum prouct istnce ó Û Ò. Hence, using Lemm 1 n (10), we get tht ã ó Û M c Ò ã Û c Òtì Thus, p p / ÜêLx~ˆ n the coe is optiml mong ll coes in à for ech VÕ. 4 Stcke Extensions of the propose coe An spce-time coe cn e extene horizontl n verticl stcking to the coes n ;, given ƒ o Wq êsz Š œ P (11) ƒ o û û s û Š œ P which cn e use on the -Ù n the ^Ð chnnels, respectivel. The trnsmit iversities of n ; re oth equl to 9 n Ð Ð. Furthermore, it cn e shown using results from [6] tht if the component coe is cpcit optiml for the chnnel, then the coes n re lso cpcit optiml for the -Ù n chnnels, respectivel. Hence, the stcke extensions n irectl enefit from the improvement in the esign of the component coe. The coe cn lso relize the concept of trining coes [10] for the non-coherent chnnel when ^ in (11) is set to constnt trining mtrix. Recent results the uthors [11] show tht n trining coe long with suoptimum estimtor-etector receiver chieves full iversit if the component coe exhiits full iversit in the coherent scenrio. The utilit of the ; coe is to provie complexit reuction for lrge numer of trnsmit ntenns. For instnce, when / Õ, the full ler full iversit TAST coe requires the extreme complexit of sphere ecoer of size for optimum ecoing. One cn emplo the ler TAST coe to reuce the sphere ecoer size to M. However, this les to ecrese in rte n loss of the cpcit optimlit propert. The propose coe, on the other hn, reuces the complexit to tht of sphere ecoer of size requiring reuce iversit ut mintins the full rte n the cpcit optimlit propert.

4  $ Coing Gin n Performnce Results The propose coe is now compre with the coe, the TAST coe for n the tilte-qam coe. A rief escription of these coes is provie first. The coe isk otine from (4) replcing ¹»º ¹ ½ k" k" for o -q /s []. L! m n setting  The optimum vlues of tht mximize the coing gin were foun exhustive m serch to e n # for -QAM n Ä $# Ä for éè -QAM. The full ler m TAST coe for m is enote. The coe is k&' otine from (4) replcing ¹»º ¹ ½ k &'. For k &( ÞL!, the numer is optiml in terms of mximizing the iversit n coing gin of [4]. For k&' with éè, we use. The tilte-qam coe, enote ) -*, is otine choosing two ifferent rel rottions, nmel ¹ º ì n ¹ lers in (4) n setting. &' Ø º ì, for the two The coing gin of these competing coes with -QAM n éè -QAM informtion smols re shown in Tles 1 n 2, respectivel. The smptotic coing gin of with respect to the etter of the n the coes is seen to e $+ with -QAM n with éè -QAM. It cn e shown using Proposition 2 n [, Theorem 2] tht the smptotic coing gin of with respect to the ) -* coe is èz t n rte. The simulte performnce of these coes re shown in Figures 1 n 2 for -QAM n éè -QAM, respectivel. The improvement of the, coe is out over the n coes in oth figures. The, coe exhiits # n improvement over the ) -* coe in Figures 1 n 2, respectivel. The performnce of the horizontll stcke extensions of the sme coes is shown in Figure for the coherent cse n the non-coherent cse with trining. The - ó coe offers n improvement of up to compre to - ó, - ó n - ó ) -*e coes in the rnge of SNR shown for oth trining n coherent scenrios. In Figure 4, the performnce of the coe for is shown. Compre to the full ler TAST coe, the coe is w in performnce ut requires onl hlf the size of the sphere ecoer. Moreover, the coe les to significnt improvement in performnce compre to the optimum ecoing of the uncoe LAST scheme [2]. The -ler TAST coe requires the sme complexit s the coe n hs higher iversit orer ut its performnce is even worse thn tht of the uncoe scheme in the given rnge of SNR. 6 Conclusions A construction of spce-time coe from n optimum rel rottion of QAM smols ws shown to offer significnt coing gin over other known spce-time coes for the Tle 1 Coing gin with -QAM #ÿ è> Ø è. Ø ) -* #ÿ Ø Z Ø Coe Tle 2 Coing gin with éè -QAM è Ø/ 0 è Ø/ ) -* Ø21.n Ø21 Coe two trnsmit ntenn qusi-sttic fing chnnel. The propose coe ws lso shown to e the est mong the clss of coes otine in similr mnner from n other rel rottion. While n nltic pproch to otining the optiml complex rottion for the informtion lossless lere spce-time coes is not es, the erivtion of the optiml rel rottion turns out to e extremel fruitful for otining performnce improvement for the two trnsmit ntenn cse. Appenix The following Lemm is prove using technique similr to the one in []. Lemm 2. If ç - - \ L $, for some - -. Proof. Mke chnge of vriles s Z L ð so tht \ # L4 -q- If ç T T # ;$ ~ # L, then (12) implies tht $ ~ $ ~ $ p$ž~ L 'î o ~ts, then L ð (12) (1) For n 76 î o ~ts, it cn e shown consiering the finite numer of possiilities for 6 mo # tht #8 ~96 ; #<8 n #<8 6. Hence, (1) implies tht #8 n #<8. Thus, # 8 $ ~ n so #8 $ ~. ut this gin mens tht #8 n #<8. Diviing ech of 7 #, we otin 7 î o ~ts such tht # $ ~ $~. Since this process cn e repete inuctivel, we hve contriction unless ' which implies tht ð. References [1] S. Almouti, A simple trnsmit iversit technique for wireless communictions, IEEE. Select. Ares Commun., vol. 16, no. 8, pp , Oct [2]. Hssii n. M. Hochwl, High-rte coes tht re liner in spce n time, IEEE Trns. Inform. Theor, vol. 48, no. 7, pp , ul 2002.

5 Frme Error Proilit 10 2 Tx, 2 Rx, 4 its/chnnel use (4 QAM smols) 2phi t Q FEP Tx, 4 Rx, 16 pcu, FRAMELENGTH 120 slots Averge receive SNR () Figure 1 Performnce of t pcu Full ler TAST coe SNR () Figure 4 Performnce of Uncoe LAST 2 ler TAST coe Stcke coe t 4 pcu Frme Error Proilit Frme Error Proilit 10 2 Tx, 2 Rx, 8 its/chnnel use (16 QAM smols) 2phi t Q Averge Receive SNR () 10 0 Figure 2 Performnce of t. pcu 2 Tx, 2 Rx, 2 ler, 4 pcu (4 QAM), T=2 x 60 2phi coherent coherent t Q coherent coherent with trining with trining Averge receive SNR () Figure Performnce of - ó spent on trining =. Frction of energ n [] M. O. Dmen, A. Tewfik, n.-c. elfiore, A construction of spce time coe se on numer theor, IEEE Trns. Inform. Theor, vol. 4, no., pp , Mr [4] H. El Gml n M. O. Dmen, Universl spce time coing, IEEE Trns. Inform. Theor, vol. 49, no., pp , M 200. [] H. Yo n G. W. Wornell, Achieving the full MIMO iversit-multiplexing frontier with rottion-se spcetime coes, in Proc. Allerton Conf. on Comm. Control, n Comput., Monticello, IL, Oct [6] R. W. Heth, r. n A.. Pulrj, Liner ispersion coes for MIMO sstems se on frme theor, IEEE Trns. Signl Processing, vol. 0, no. 10, pp , Oct [7] V. Trokh, N. Seshri, n A. R. Clernk, Spce time coes for high t rte wireless communictions Performnce criterion n coe construction, IEEE Trns. Inform. Theor, vol. 44, no. 2, pp , Mr [8]. outros, E. Vitero, C. Rstello, n.-c. elfiore, Goo Lttice constelltions for oth Rleigh fing n Gussin chnnels, IEEE Trns. Inform. Theor, vol. 42, no. 2, pp , Mr [9] E. Vitero n. outros, A universl lttice coe ecoer for fing chnnels, IEEE Trns. Inform. Theor, vol. 4, no., pp , ul [10] M. rehler n M. K. Vrnsi, Trining-coes for the noncoherent multi-ntenn lock-rleigh-fing chnnel, in Proc. Conf. Inform. Sciences n Sstems, ltimore, MD, Mr. 200, ohns Hopkins Universit. [11] M. rehler, M. K. Vrnsi, n P. Dl, Leverging coherent spce-time coes for noncoherent chnnels vi trining, to pper IEEE Trns. Inform. Theor.