Towards a Better Diversity-Multiplexing Tradeoff in MIMO Systems

Transcription

1 Towards a Better Dversty-Multpleng Tradeoff n MIMO Systes Ln Da, Sana Sfar, Khaled B. Letaef, Fellow, IEEE Center for Wreless Inforaton Technology The Hong Kong Unversty of Scence & Technology Clear Water Bay, HON KON eedaln@ust.hk, eesana@ust.hk, eekhaled@ee.ust.hk Astract Multple-Input Multple-Output (MIMO) systes can provde two knds of gan: dversty gan ultpleng gan. Most estng MIMO schees, ncludng space-te codng layered space-te, a at azng ether of the. roup transsson detecton s a good way to get a tradeoff etween these two gans. In ths paper, the dversty-ultpleng tradeoff functons of 3 schees ased on group transsson detecton, Quas-Orthogonal roup Space-Te (QoST), roup Layered Space-Te (LST) Layered space-te lock codes (LSTBC), are provded a vvd coparson aong the deonstrates that QoST acheves the est dversty-ultpleng tradeoff. LST also gets sgnfcant gans over LSTBC. Sulaton results wll valdate the analyss show that QoST can acheve at least 3 db gan over LST 2 db over LSTBC at a FER of -3, for nstance. Keywords- roup detecton, MIMO systes, Space-te lock codng, Layered space-te, roup layered space-te, Quasorthogonal group space-te, Dversty-ultpleng tradeoff. I. INTRODUCTION MIMO (Multple-Input Multple-Output) systes have shown ther alty n provdng great perforance proveents over the SISO (Sngle-Input Sngle-Output) syste thanks to ther hgher spectral effcency []. It has een well understood that ultple antennas can not only e used to acheve dversty gan, ut also as an effectve way to ncrease the degrees of freedo of the channel [2]. However, ost estng MIMO technques, such as space-te codes (STC) [3-4] Layered space-te (LST) [], a at achevng ether au dversty gan or au ultpleng gan. In order to acheve a etter tradeoff etween these two gans, [6] presented a coned array processng space-te codng archtecture, n whch the transt strea s parttoned nto dfferent groups n each group STC s appled. At the recever, group nterference suppresson s adopted, each ndvdual STC s decoded y suppressng the sgnals transtted fro other groups. Ths conaton of STC LST provdes uch etter ultpleng gan than STC wth lower decodng coplety, whle at the sae te achevng a uch hgher dversty gan than LST. [7] further developed ths archtecture, Alaout s schee was adopted as the coponent encode the transt power was optzed to nze the average FER, whch s called Layered space-te lock codes (LSTBC). [6] [7] presents a good eaple on how to get a tradeoff etween ultpleng gan dversty gan. However, there are soe ltatons on the desgns of transtter recever. Frst, the sustreas of each group are encoded ndependently. No specal transt desgn s adopted to suppress the nterference aong the groups. Besdes, the appng fro the sustreas to the antennas s constant over the whole te nterval no nterleavng gan can e acheved. Second, at the recever, space-te decodng s perfored for each group y assung that the nterference has een suppressed y vrtue of a group detector. The overall perforance s thus lted y the group detecton step usually s not satsfyng. [8] further proposed two novel group space-te archtectures: roup Layered Space-te (LST) Quas- Orthogonal roup Space-te (QoST). LST has the sae transtter as [7]. However, a novel detector s adopted at the recever, space-te lock decodng s perfored efore group detecton s appled. Specfcally, for the case of transt n receve antennas durng T te slots, the lnear nature of STBC can e eploted to otan an equvalent Tn channel [9]. roup detecton s then appled ased on ths equvalent channel. It can e seen that after decodng, the receve densons ncrease fro n to Tn thus uch etter perforance can e acheved y group detecton. QoST not only adopts ths novel detector, ut also has a ore effcent transtter desgn. In QoST, all the groups are encoded together va an nter-group STBC. Partcularly, at each te slot t, the transt vector of each group s regarded as one syol STBC s appled on all the transt vectors. It can e seen that wth ths nter-group STBC, the nterference aong groups can e effectvely suppressed ecause of the orthogonal nature of STBC. Therefore, QoST should have a etter nterference suppressng capalty. Besdes, nterleavng gan can e acheved due to the use of the nter-group encoder. [8] has shown that QoST has a etter syetrc energy than LST. In ths paper, we further otan ther dverstyultpleng tradeoff functons copare the to the tradeoff of LSTBC. The dversty-ultpleng tradeoff functon s proposed y Zheng Tse [2], whch allows us to copare the perforance etween dversty-ased schees ultpleng-ased schees. It s found that a hgher spatal ultpleng gan coes at the prce of sacrfcng dversty vce versa. In ths paper, we follow the analyss n [2] otan the lower upper ound tradeoff functons of LST QoST. It s found that the lower ound tradeoff of QoST s usually etter than the upper ound tradeoff of LST, whch ples that QoST has a etter dverstyultpleng tradeoff than LST. Besdes, the coparson aong QoST, LST LSTBC shows that oth QoST LST can acheve sgnfcant gans over LSTBC thanks to Ths work s supported n part y the Hong Kong Research rant Councl under rant No. HKUST63/E.

2 ther effcent detectors. Sulaton results wll valdate our analyss show that QoST can acheve at least 3 db gan over LST for 4 n 2 2 db over LSTBC for 6 n4 at a FER of -3, for eaple. Ths paper s organzed as follows. In Secton II, we revew the transtter recever desgns of LSTBC, LST QoST. Secton III presents the perforance analyss result, whch s evaluated n ters of the dversty-ultpleng tradeoff functon. Sulaton results are gven n Secton IV. Fnally, Secton V suarzes concludes ths paper. II. SYSTEM MODEL We egn y presentng LSTBC, then present the detals of LST QoST archtecture. In the followng we consder a flat Raylegh wreless lnk H wth transt n receve antennas. Besdes, perfect channel knowledge s assued at the recever sde only, through the use of tranng sequences. In ths paper transt power allocaton s not consdered. Therefore, all the antennas are assued to e wth the sae transt power. A. LSTBC A. Transtter Assue that all the transt antennas are parttoned nto groups, respectvely, coprsng, 2,..., antennas. A lock of nput ts { }... K s dvded nto groups,, 2,...,, n each group, [,,,2,...,, ],,,, s then encoded y a coponent space-te lock code STBC assocated wth transt antennas. Here t s requred that. Assue that all the coponent codes STBC,,,, have the sae code length T, we have g g for,,. Then, the output T codeword atr X over a lock of T syol ntervals can e wrtten as X S,, S () represents the transpose operator. S [,, s st] s the g T codeword atr of group,,,. As we know, an -antenna-t-te-slot-k-syol STBC O can e represented as O A+,,..., B A+B 2 2 AT+BT, denotes the conugate operator, s a k cople varale vector A, B are constant coeffcent atrces n k R. The atr O s called [, T, k] STBC for short n the followng. Therefore, S can e wrtten as S [,..., ] +,..., A AT B BT, (2) for,,, A, B are constant coeffcent atrces g g n R,,,T. A.2 Recever The detector presented n [7] s to suppress sgnals transtted fro other groups of antennas y vrtue of a group detector frst, then perfor space-te decodng for the desred group. In partcular, assue that a roup Zero-Forcng (ZF) recever s adopted group s to e detected. Partton H nto H, H H, H ncludes the coluns of H correspondng to the groups ecept. The proecton atr P s then defned as: + + P I ( ) n H H H H, ()+ denote the cople conugate transpose. Therefore, usng the transforaton + W H P on the receve vector, we have Y { [,..., ],..., } T + T + Q A A Q B B Z (3) + Q H P H. Then, for group, STBC decodng can e done ased on Y. B. LST B. Transtter The transtter desgn of LST s the sae as LSTBC. Therefore, we do not repeat here. B.2 Recever In LST, a new detector s adopted, space-te decodng s perfored efore group detecton. To do so, an equvalent channel s otaned y vrtue of the lnear nature of STBC. ZF s then perfored. Partcularly, y conng () (2), the receved sgnal vector can e wrtten as Y [,..., ],, [ T,..., T] + H A H A H A H A (4) [,..., ],, [ T,..., T]. HB H B HB H B + Z After a seres of lnear transforaton, fnally we can get,..., () r + H H z r [ y,..., y T ] y, B g g y, y y, A g g represents the -th colun vector of Y,,,T. For any group,,,, the correspondng su-channel atr s gven y HA + HB. (6) H HA T + HB T Fro () (6) t s clear that after otanng ths Tn K equvalent channel of LST, the decodng process s done. roup detecton can then e appled so as to get the orgnal transt syols. As eplaned efore, here after decodng the receve densons have ncreased fro n to Tn thus uch etter perforance can e acheved y group detecton. That s why the novel detector s superor to the orgnal one used n [6-7].

3 C. QoST C. Transtter In LSTBC LST, the t streas of each group are encoded separately so that the output streas S, S 2,, S are ndependent of each other. No specal transt desgn s adopted to suppress the nterference aong the groups. Besdes, the appng fro dfferent groups to the transt antennas s always fed over all the te slots. Therefore, no nterleavng gan can e acheved. QoST adopts a dfferent transt desgn, n whch all the groups are encoded together va an nter-group STBC encoder. Assue that the antennas the t strea are dvded nto groups, respectvely. Then, the desgn of the nter-group STBC s gven y X A,..., A T + B,..., B (7) T A,..., A A, B,..., B B, (8) A A I, g B B I, (9) g for,,,,t. A B are the th colun vector of A B, respectvely. A B,, T, are the coeffcent atrces of a [,T, ] STBC. After nter-group STBC encodng, the appng fro the t streas of dfferent groups to the transt antennas s not constant any ore. Therefore, a hgher dversty gan can e acheved thanks to the nterleavng gan. Besdes, here STBC s appled to the transt vectors. The nterference aong the groups s not ndependent thus can e etter suppressed. C.2 Recever The recever desgn of QoST s slar to LST. Fro (7), the receved sgnal vector can e wrtten as Y,...,,,,..., T T + HA HA HA HA HB,..., HB,,,...,. HB T HB T + Z Slarly, we can get r H,..., + H z for,,, () () HA + H B. (2) H HA T + H B T ven r, group detecton can then e appled. III. DIVERSITY-MULTIPLEXIN TRADEOFF In ths secton, we provde the an techncal results of ths paper. The proofs of the results stated are otted due to lted space. A. Tradeoff of LSTBC The dversty-ultpleng tradeoff of LSTBC s gven y dlstbc () r g( n + g)( Tr/ K) (3) When there s only one group, LSTBC s reduced to an orthogonal desgn. In ths case, g. Accordng to (3), t can e otaned that : d LSTBC n(-tr/k)d ortho. On the other h, f g g Kn, LSTBC s reduced to VBLAST ovously we have: d LSTBC -r/d VBLAST, whch s eactly the sae as the one presented n [2]. B. Tradeoff of LST The dversty-ultpleng tradeoff of LST s ounded y dlower _ LST () r dlst () r dupper _ LST () r (4) dupper _ LST () r gn( Tr/ K) () dlower _ LST () r ( Tr/ )( g Tr/ ) (6) wth n(n, Tn-K+g ). Both orthogonal desgn VBLAST can e also regarded as specal cases of LST. When Kn g g, LST s reduced to VBLAST. Notce here that nk T. Therefore, Tn-K+g. It can e proved that the lower ound s acheved only when Tn-K+g. Therefore, we have d LST -r/d VBLAST. On the other h, when there s only one group, the upper ound can e acheved,.e., d LST n(-tr/k). In ths case, LST s reduced to orthogonal desgn ovously we have d LST d ortho. C. Tradeoff of QoST The dversty-ultpleng tradeoff of QoST s ounded y dlower _ QoST () r dqost () r dupper _ QoST () r (7) dupper _ QoST () r ( n Tr/ )( g Tr/ ) (8) d _ () r ( n + g Tr / )( g Tr / ). (9) lower QoST d upper_qost reflects the optal case wth no nterference aong the groups. In other words, d QoST can reach the upper ound when all the groups are orthogonal. Ths condton s satsfed only wth the group sze g. In ths case, QoST s reduced to an orthogonal desgn ovously we have d QoST n(-tr/k)d ortho (r can only take values fro {,K/T}). When there s only one group,.e.,, QoST turns nto an -transt n-receve syste wth a au lkelhood detector over Raylegh quas-statc channels. In ths case, the upper ound lower ound converge to In [2] the dversty-ultpleng tradeoff of the orthogonal desgn wth 2 transt antennas n receve antennas has een presented as d2n(-r). Through a slar analyss, the tradeoff of the orthogonal desgn wth an artrary n can e derved as dn(-tr/k). We ot the proof due to space ltaton.

4 d upper_qost d lower_qost (n-r)(-r). Therefore, the dversty-ultpleng tradeoff functon of QoST can e eactly gven y d QoST (n-r)(-r), whch s eactly the sae as the optal tradeoff functon for an (, n) over Raylegh channels as shown n [2] LST, d LST lower ound of QoST, d low er-qost upper ound of QoST, d upper-qost (, n) optal, d D. Tradeoff Coparson Fg. presents the dversty-ultpleng tradeoff curves of LST QoST for K 4, n 2 2 groups,.e., gg g 2 2. Here the dversty gan of LSTBC s always zero snce ts detector cannot work when n g. For coparson, we also show the optal tradeoff curve for an (, n) over Raylegh channels. Both LST QoST needs T2 te slots. Snce Tn-K+ g 2, the lower ound tradeoff functon of LST can e acheved. Therefore, fro (6), (8) (9), the tradeoff functon of LST the lower upper ound tradeoff functons of QoST can e otaned. As Fg. shows, the tradeoff curve of LST s eactly the sae as that of the lower ound of QoST. Ths ples that QoST always has a etter dversty-ultpleng tradeoff than LST. Besdes, the upper ound tradeoff of QoST overlaps wth the optal one. Actually, t s not always the sae as the optal tradeoff. As Fg. 2 shows, when n ncreases to 3, the au ultpleng gan r a of the optal one wll ncrease to 3, whle r a of the upper ound of QoST s only 2. In ths case, the lower ound tradeoff of QoST s even etter than the upper ound tradeoff of LST, thus, ndcatng a ore sgnfcant gan. The dversty-ultpleng tradeoff curve of LSTBC s also plotted n Fg. 2. Ovously t s uch lower than the lower ound of LST. Fg. 3 Fg. 4 show the case of K 6 n 4. Snce 6 syols (antennas) can e dvded nto 2 groups wth 3 syols (antennas) n each group, or 3 groups wth 2 syols (antennas) n each group, we consder oth of these possle parttons. In Fg. 3, the t strea transt antennas are assued to e separated nto 2 groups,.e., gg g 3 2. Recall that n QoST, STBC s adopted aong the groups. Therefore, for a 2-syol-2-antenna transsson, only T QoST 2 te slots are needed. However, for LST LSTBC, STBC s adopted nsde the group. We take the [3,4,3] STBC code gven n [] (pp. 248, Eqn. (99)), so T LST 4. As Fg. 3 shows, QoST can always get a uch etter dversty-ultpleng tradeoff than LST as ts lower ound s always etter than the upper ound tradeoff of LST. LST LSTBC can only get a au ultpleng gan of., whch s lower than that of QoST. Ths s ecause they need ore te slots to transt all K syols. LSTBC agan gets the worst dversty-ultpleng tradeoff aong these 3 schees. Its au dversty gan s uch lower than that of LST. In Fg. 4, the latter partton,.e., g g g2 3 s shown wth the sold lnes. Here LSTBC cannot work due to n g. Besdes, LST needs 2 te slots, whle QoST needs 4 te slots. Fro Fg. 4, t can e seen that n ths case despte a uch hgher dversty gan, QoST has a saller au ultpleng gan due to ts lower transsson rate. Nevertheless, f we copare ths Dversty an: d Spatal Multpleng an: r Fg. : Dversty-ultpleng tradeoff curves of LST, the upper lower ounds of QoST the optal one when K 4, n 2. 2, gg g 2 Dversty an: d d LSTBC lower ound of LST, d low er-lst upper ound of LST, d upper-lst lower ound of QoST, d low er-qost upper ound of QoST, d upper-qost (, n) optal, d Spatal Multpleng an: r Fg. 2: Dversty-ultpleng tradeoff curves of LSTBC, the upper lower ounds of LST QoST, the optal one when K 4, n 3. 2, gg g 2. Dversty an: d 2 2 d LSTBC lower ound of LST, d low er-lst upper ound of LST, d upper-lst lower ound of QoST, d low er-qost upper ound of QoST, d upper-qost (, n) optal, d Spatal Multpleng an: r Fg. 3: Dversty-ultpleng tradeoff curves of LSTBC, the upper lower ounds of LST QoST, the optal one when K 6, n 4. 2, g g g3.

5 tradeoff of QoST to the tradeoff of LST wth g g 3 2 (whch s plotted wth dash lnes), t s found that QoST can always acheve a etter dversty gan whle keepng the sae ultpleng gan as LST. As Fg. 4 shows, n ths case the lower ound tradeoff curve of QoST s always hgher than the upper ound tradeoff curve of LST. The sae concluson holds true for the coparson of QoST wth g 3, 2 LST wth g g 2, 3. Therefore, we can conclude that QoST always has a etter dverstyultpleng tradeoff than LST. Dversty an: d 2 2 lower ound of LST, d low er-lst upper ound of LST, d upper-lst lower ound of QoST, d low er-qost upper ound of QoST, d upper-qost Spatal Multpleng an: r Fg. 4: Dversty-ultpleng tradeoff curves of the upper lower ounds of LST QoST when K 6, n 4. Sold lne: 3, g g g2; Dash lne: g g 3, 2. IV. FER RESULTS AND DISCUSSIONS We have shown that aong 3 schees QoST has the est dversty-ultpleng tradeoff LSTBC s the worst. In ths secton, we further copare ther FER perforance. QPSK s assued to e adopted. As Fg. shows, when 4 n 2, QoST can acheve a gan of 3 db over LST at a FER of -3. Besdes, n hgh- condtons, the FER curve of QoST has a larger slope than that of LST, whch ples that QoST has a etter dversty gan. Notce that these two schees have the sae spectral effcency. As a result, we can conclude that QoST acheves a etter dversty-ultpleng tradeoff, whch s consstent wth our analyss n Secton III. Here we dd not show the FER curve of LSTBC snce t cannot work n ths case. When 6 n 4, the transt antennas can e dvded nto 2 or 3 groups. To keep the sae transsson rate, the group sze of QoST as well as LST LSTBC s assued to e 3 2 (or 2 3), respectvely. As Fg. shows, for the case of g QoST LST 3, QoST can acheve at least 4 db gan at a FER of -3, the FER curve of QoST s uch steeper than that of LST, whch ples a uch etter dversty gan. Agan LSTBC cannot work n ths case. When g QoST LST LSTBC 2, QoST wll not acheve such a sgnfcant gan over LST as efore. Nevertheless, a larger slope db gan can e seen. LSTBC gets the worst perforance. At a FER of -3, 2 db db gans can e acheved y QoST LST, respectvely. These oservatons clearly valdate our analyss on the dverstyultpleng tradeoff. FER QoST LST LSTBC K4, n2 QoST: 2, g2 LST: g g 2, 2 K6, n4 LSTBC: g g 3, 2 K6, n4 K6, n4 QoST: 3, g2 QoST: 2, g3 LST: g g 3, 2 LST: g g 2, (db) Fg. : FER vs. curves of QoST, LST LSTBC. Sold lne: QoST; Dash Lne: LST; Dash-dot Lne: LSTBC. V. CONCLUSIONS In ths paper, the dversty-ultpleng tradeoffs of 3 schees ased on group transsson detecton, QoST, LST LSTBC, were presented. The coparson showed that QoST can always acheve the est dverstyultpleng tradeoff thanks to ts nter-group STBC encoder effectve detector. LST s not as good as QoST, whle sustantal gans over LSTBC can stll e oserved. REFERENCES [] R. D. Murch K. B. Letaef, Antenna systes for road wreless access, IEEE Councatons Magazne, Vol. 4, No. 4, pp , Aprl 22. [2] L. Zheng D. N. C. Tse, Dversty ultpleng: a fundaental tradeoff n ultple-antenna channels, IEEE Trans. Inf. Theory, vol. 49, no., pp , May 23. [3] V. Tarokh, H. Jafarkhan A. R. Calderank, Space-te lock codes fro orthogonal desgns, IEEE Trans. Inf. Theory, vol. 4, no., pp , July [4] V. Tarokh, N. Seshadr A. R. Calderank, Space-te codes for hgh data rate wreless councaton: perforance crteron code constructon, IEEE Trans. Inf. Theory, vol. 44, pp , Mar [] P. W. Wolnansky,. J. Foschn,. D. olden R. A. Valenzuela, V-BLAST: an archtecture for realzng very hgh data rates over the rch-scatterng wreless channel, n Proc. ISSSE 98, Psa, Italy, pp. 29-3, Sep [6] V. Tarokh, A. Nagu, N. Seshadr A. R. Calderank, Coned array processng space-te codng, IEEE Trans. Inf. Theory, vol. 4, no. 4, pp. 2-28, May 999. [7] N. Prasad M. K. Varanas, Optu effcently decodale layered space-te lock codes, n Proc. the 3 th Asloar Conference on Sgnals, Systes Coputers, pp , Nov. 2 [8] L. Da, S. Sfar K. B. Letaef, A Quas-orthogonal group space-te archtecture for hgher dversty gans, n Proc. IEEE loeco 4, Dallas, TX, USA, Nov. 24. [9] B. Hass B. M. Hochwald, Hgh-rate codes that are lnear n space te, IEEE Trans. Inf. Theory, vol. 48, no. 7, pp , 22. [] X. Lang, Orthogonal desgns wth au rates, IEEE Trans. Inf. Theory, vol. 49, no., pp , Oct. 23.