A Novel Codebook Design for Linear Precoding Systems

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1 A Novel Codebook Desg for Lear Precodg Sysems ayu La 13* aowe e 3 Ju L 4 1. Guagzhou Isue of Geochemsry Chese Academy of Sceces Guagzhou Cha. Uversy of Chese Academy of Sceces Bejg Cha 3.School of Elecroc ad Iformao Egeerg Souh Cha Uversy of Techology Guagzhou Cha 4. Deparme of Elecrocs ad Iformao Egeerg Chobuk Naoal Uversy Jeoju Souh Korea Absrac: Backgroud Precodg has bee regarded as a effecve mehod o mprove he performace of mulple-pu mulple-oupu (IO) sysems. ay works have bee vesgaed for desgg he codebooks of precodg marces he dscree ourer rasform (DT) codebook have bee proposed as he opmal oe. owever ca be specfed ha he cosruco of DT codebook leads o a hgh complexy. ece fdg a dffere codebook whch eeds a lower cosruco complexy ad less performace loss s he ma purpose of hs paper. aerals ad ehods I s foud ha he orhogoal space me block codes (OSTBC) provdes may orhogoal sgal marces whch almos sasfy propery of precodg codebooks. Therefore auhors wll evaluae he complexy ad performace by usg OSTBC codebooks. ore specfcally he recever chooses a marx from he OSTBC codebook ad feedbacks he opmal codebook marx o he rasmer. The he auhors compare he complexy wh he coveoal dscree ourer rasform (DT) codebook. Resuls rom he desged equaos of OSTBC ca be see ha he cosruco of OSTBC codebook s smpler. Smulao resuls show ha usg he proposed OSTBC codebook leads o he almos same performace compared wh ha of he DT codebook. Coclusos The proposed OSTBC codebook s advsable o be cosdered as a radeoff bewee he complexy ad performace. a coclusos ad erpreao of fdgs I hs paper he auhors propose a ovel codebook desg for lear precodg codg scheme. I he proposed scheme specfes a lmed feedback mehod ha uses orhogoal space me block codes (OSTBC) as he codebook of precodg marces whch s kow o boh he rasmer ad recever. Compared wh he coveoal dscree ourer rasform (DT) codebook OSTBC codebook requre less complexy ad acheves he early same performace. Noe ha he proposed mehod fals o oba a beer performace compared wh he DT codebook however he smple mplemeao s sll a aracve advaage. Keywords: Orhogoal space me codes; precodg; codebook 1. Iroduco The beefs of usg mulple aeas a boh he rasmer ad he recever a wreless sysem are well esablshed. ulple-pu mulple-oupu (IO) sysems eable a growh rasmsso rae lear he mmum umber of aeas. I s well kow ha he performace of a IO space me code sysem ca be mproved wh chael kowledge a he rasmer [1]. The chael kowledge a he rasmer does o help o mprove he degrees of freedom bu power or beam formg ga s possble. I a o-..d. chael (such as correlaed Rca fadg) he chael kowledge a he rasmer offers eve greaer beef performace. Therefore explog rasm chael sde formao s of grea praccal eres IO wreless [ 3]. I a me dvso duplexg (TDD) sysem he chael kowledge ca be obaed a he base sao such as enb by uplk rasmssos haks o chael recprocy. owever he soudg sgals eeds o be rasmed o he uplk whch represes a addoal overhead. I a frequecy dvso duplexg (DD) sysem he chael sae formao (CSI) eeds o be fed back from he user equpme (UE) o he enb. The complee chael sae feedback ca lead o excessve feedback overhead. A approach o reduce he chael sae formao feedback overhead s o use a codebook. Precodg s a processg echque ha explos chael sae formao a he rasmer (CSIT) by operag o he sgal before rasmsso. or may commo forms of paral CSIT a lear precoder s opmal from a formao heorec vew po [4]. A lear precoder esseally fucos as a mulmode beamformer opmally machg he pu sgal o oe sde o he chael o he oher sde. I does so by splg he rasm sgal o orhogoal spaal eyebeams ad assgs hgher power alog he beams he chael s srog bu lower or o power alog he weak. Precedg desg vares depedg o he ypes of CSIT ad he performace crero. Noe ha he codebook feedback drecly mpacs he performace ad he eergy effcecy of precodg sysems whch also plays a mpora role he evromeal sceces. Therefore he opmal desg of codebooks s sll promsg ad ecessary for he praccal commucao sysems. Ths paper uses 4 by 4 orhogoal space me block codes (OSTBC) as a codebook lear precodg sysem. Before he rasmsso a four-aea OSTBC marx s mulpled by a lear precodg marx ad rasmed over aeas. The opmal lear precodg marx s chose from a fe cardaly codebook of all possble precodg marces whch s desged off-le ad avalable o boh he rasmer ad recever. Due o he proposed OSTBC codebook he sysem performace s almos he same o ha of he dscree ourer rasform (DT) codebook he mplemeao of OSTBC codebook s much smpler. ore specfcally ulke he DT codebook he cosruco of OSTBC codebook does o eed so much compuao whch makes he proposed mehod acheve a lower complexy. Smulao resuls verfy ha he sysem performace usg OSTBC codebook or DT codebook s very smlar. Ths paper s orgazed as follows. I seco he sysem model s descrbed. Codebooks are roduced ad desg algorhm of he proposed codebook s descrbed deal. Smulao resuls are preseed seco 3. ally dscusso ad coclusos are gve seco 4 ad 5 respecvely.. ehodology.1 Sysem odel Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84

2 g.1 shows he block dagram of he IO sysem wh space-me code ad precoder. Le us cosder a r gure. 1 The sysem model wh space-me code ad precoder. IO sysem a block fadg chael he umber of he rasm aeas s less ha ha of he receve aeas. The chael ga bewee dffere rascever par s assumed o be a..d. complexy Gaussa radom varable whose probably desy fuco (PD) s CN (0 1). Noe ha a closed-loop IO precodg sysem for each rasmsso aea cofgurao ca be cosruced ha a se of precodg marces ad le hs se be kow a boh he enb ad he UE [5]. Le X T deoe a space me codeword wh a legh of whch s represeed as X [x 1 x x ] T (1) x [ ] T x x x k 1 T ad. As a resul he receved sgal correspodg o oe OSTBC k k1 k k symbol Y ca be formulaed as Y X N () ρ deoes he sgal o ose power rao r precodg marx whch s chose from he codebook [ ] 1 L eleme whch deoes he ose sample a he represes he chael rasform formao. N r T s a.s a ose sample marx ad s ()-h h receve aea a he me sace s modeled as a..d. complexy Gaussa radom varable.e. ~ CN ( 01). Raher ha he full CSI oly he correspodg dex s fed back o he rasmer sde. Each dex ca be represeed wh N bs whch allows for a oal umber of L = N codewords he codebook. Noe ha L s referred o as a codebook sze.if he maxmum-lkelhood (L) decodg scheme s used o exrac he rasm symbols Y he codoal symbol error probably decreases expoeally as he chael robeus orm goes up.e. P error exp r (3) γ s a cosa depede of. I order o mmze he symbol error rae (SER) s seleced o maxmze he mmum dsace bewee wo OSTBC codeword. ahemacally ca be: arg max m X X * l m 1 N lm = arg max 1 N (4) he secod equaly comes from he fac ha boh ad are orhogoal. ece s clear o see ha he precoder s chose such ha he robeus orm of he equvale chael respose s maxmzed. The correspodg mmzao problem ca be formulaed o he Grassmaa subspace packg problem [6]. The ˆ whch s seleced form should be desged by maxmum he mmal chordal dsace m d( ) bewee ay par of codeword. The chordal dsace s defed as: m k l1 k ll k l d( ) 1 k l k l 1 k l L. (5). Codebook Desg Precodg sysems I hs seco S deoes he proposed codebook whch s coss of L complex marces of a dmeso wh =1 for smplcy. Nex s gog o roduce he opmal uary codeword f he capacy of he feedback chael s o lmed [7]. Noe ha eve op op whch demads much more chael capacy for he precoder formao feedback s o praccal eough a Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84

3 feedback lmed sysem does provde sgh o our codebook desg. The opmal codeword s composed of orhoormal bass vecors ad s deoed as deoes all possble orhoormal bass vecors. Please oe ha U op op as log as U s a uary marx. (6) op op s o uque sce N G [ g g g ] (0) (1) ( 1) 1 Suppose ha has a N 1 -b codebook S [ s s s N ] our proposed N-b codebook K s ( ) ( ) ( ) ( ) N 1 N N N s [ s s s 1 ] 1. Every group ca be expressed as N ( ) 1 s s he -h ( ) codeword he s -h group [8]. Comparg wh [8] could be depced as: ( ) s O P gˆ g floor ( / ) ( mod ) O w w w w l P ad gˆ (7) w R f f 1 1 R 1 are used for roao of codebook. The -h codebook of a N-b DT codebook s gve as follows: T 1 j j 1 f 1 e e (8) B 1 N ad s he h sample he agle doma 0 DT codewords f ca be wre as [9] (9) r r R are he relave devaos. Boh he values of ad w. The codeword 1 s made up of R r are resrced wh he agle sample se whch w s kow a boh rasmer ad recever. s seleced from GLP codebook whch s composed of codebooks. E 1 N The swchg codebook s preseed as follows.. rs dffere codebooks of sze bs: e s 0 e E 1. (10) I geeral he codebook ca smply swch a sequeal order a each feedback me.e.: 0 1 E s s s s s s s. (11) 1 N 1 N are creaed wh 1 feedback N Compare wh he coveoal sgle codebook he swchg codebook provdes a E mes larger codebook wh 1 feedback bs sead of N bs for he same sze of codebook. owever oly oe codebook ca be used a a feedback me. The chael marx process s performed by: s arg max s () s s (1) The defaul codeword s also used hs scheme [10]. If s s ( 1) ˆ () (13) he defaul codeword dex s se back o he rasmer. I [10] he auhor preseed a corer case whe he rasmer s verso of prevously seleced codeword whch s dffere from ha of he recever. Eve hs case he src codo (13) would also guaraee ha he defaul dex s seleced he maxmum of E mes codebooks. The (1) ca be rewre as: s arg max s () s S. (14) ece he swchg codebooks choose he same opmal codeword as he sgle codebook. Propery 1: The chordal dsace bewee ay wo codewords s same as he GLP codebook whch s specfed as: ( ) ( ) d( sp sq ) = d( gp gq) (15) N 1 N p 01 ad q N Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84

4 Proof: Sce O (7) s a uary marx. I s obvous ha permuao marx P ad he householder marx gˆ are also uary marces. ece follows ha O P ( / ) ( mod ) g ˆ O P gˆ I floor floor ( / ) ( mod ) ( ) ( ) d ( sp sq ) = 1- ( ) ( ) ( sp ) sq (16) = 1- ( O ( / ) P( mod ) ( gˆ ) g ) O ( / ) P( mod ) ( gˆ ) g ( ) floor p floor q = 1- = d The ca be obaed ha ( gp) ( gp gq) g q whch complees he proof. Accordg LTE release 10 sadard [11] order o maxmze he codebook whch s 6 bs feedback. The 4 by 4 DT marx s gve as: (4) chooses he frs wo colums from he bes j j j3 1 1 e e e 0 j j j e e e j3 j 3 j9 1 e e e (17) The remag precodg marces are obaed: j.1/4 j.8/4 j.61/4 j.45/4 1 dag e e e e The smulao wll be show laer. (18) I hs proposed scheme s o use Alamou code (TX) o f a 4 rasmer aea sysem by usg OSTBC alerave codebooks he lear precodg wh 6 bs feedback. The precodg srucure s specfed as: chael precodg Alamou code h% A 14 4 (19) 1 K s precodg marx wh k bs precodg feedback formao. The full precodg marx s x x x * * x x 0 x 1 3 S 0 * * x 0 x x 3 1 * * 0 x x x 3 1 (0) x x x wo colums of 3. Resuls. ece he oal codebook sze L s defed as 64. s he 4 by precodg marx whch akes he frs S. Compared wh DT codebook [1] he desg of our proposed codebook has a lower complexy. I hs seco he auhors provde he smulao wh he proposed codebook ad make he comparso wh ha of he DT codebook. Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84

5 0BER of OSTBC wh/whou precodg Raylegh fadg chael usg 4QA Alamou 1 whou precodg OSTBC Coveoal B Error Probably P B E b / N 0 [db] gure. BER performace wh dffere cofguraos g. shows he b error rae (BER) performace wh he 6 bs feedback usg he OSTBC codebook ad DT codebook Raylegh fadg chael usg 4QA wh 1 ad r. The BER performace of Alamou 1 s also preseed. rom g. we ca be see ha he BER performace of he sysem usg OSTBC codebook acheves he very smlar performace as ha of he 5 coveoal DT codebook. Specfcally a BER= 10 here s almos o performace gap bewee he wo curves preseg he performace of he proposed codebook ad he coveoal DT codebook. I also ca be see ha percodg schemes oba he sgfca performace mproveme compared wh he scheme whou percodg whch s he ma advaage of precodg mehod. The complexy of cosrucg he precodg codebook maly depeds o he umber of mulplcaos marx compuao. or example he cosruco of he DT codebook s frsly mplemeed by (17) whle he cosruco of he proposed codebook resors o (0). rom (17) ad (0) ca be show ha four zero elemes occur (0) whle here s o zero eleme (17). Due o hs reaso he complexy of cosruco of precodg scheme usg (0) s hgher ha ha of percodg scheme usg (17). Noe ha he complexy reduco of he proposed codebook s more cosderable as creases. The above-meoed resuls show ha some cera suaos our proposed OSTBC codebook s avalable o replace he opmal DT codebook wh he slgh performace loss bu eed a less complexy. 4.Dscusso Acually he auhors proposed he beamformg or precodg echque [7]. oreover he auhors [1] have bee proposed he opmal codebook desg for precodg sysems. owever from [1] ca be foud ha he complexy of cosrucg he opmal codebook (DT codebook) s sll hgh. Ispred by hs hgh complexy burde he STBC codebook has bee proposed o reduce he complexy hs paper. rom above-meoed resuls ca be see ha he performace of precodg sysems usg he proposed codebook s almos he same as ha of percodg sysems usg DT codebook bu obag he lower complexy. Therefore applyg he proposed codebook o precodg sysems ca be cosdered as a aracve mehod. 5. Cocluso I hs paper auhors have preseed a ovel precodg scheme whch apples he OSTBC as he codebook. The proposed mehod ca oba a very close BER performace compared wh he opmal mehod usg DT codebook bu acheve a lower complexy of codebook cosruco. Ackowledgemes Thaks o aoymous referees for her cosrucve commes o earler drafs. Refereces [1] V. Tarokh N. Seshadr ad A. R. Calderbak Space-me codes for hgh daa rae wreless commucao: Performace crero ad code cosruco IEEE Tras. If. Theory () pp []. Alamou A smple rasm dversy echque for wreless commucaos IEEE J. Sel. Areas Commu (8) pp [3] Y. Shag ad X.-G. Xa Space-me block codes achevg full dversy wh lear recevers IEEE Tras. If. Theory 00854(10) pp [4] G. Care ad S.S. Shama O he capacy of some chaels wh chael sae formao IEEE Tras. If. Theory (6) pp [5]. Skoglud ad G. Jogre O he capacy of a mulple-aea commucao lk wh chael sde formao IEEE J. Selec. Areas Commu. 0031(3) pp [6] D. J. Love ad R.. eah Jr. Grassmaa Beamformg o Correlaed IO Chaels Proc. IEEE Globecom 004 pp [7] D. J. Love ad R.. eah Lmed feedback uary precodg for orhogoal space-me block codes IEEE Tras. o Sgal Process (1) pp [8] S. L. Ja ad J. Kag Robus codebook desg based o uary roao of Grassmaa codebook Proc. 010 IEEE 7d Vehcular Techology Coferece all (VTC 010-all) 010 pp [9]. Yua S. a C. Yag Y. Zhag G. ag ad. Le eghed DT codebook for muluser IO spaally correlaed chaels Proc. 011 IEEE 73rd Vehcular Techology Coferece (VTC Sprg) 011pp Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84

6 [10] C. Jag..ag. Shu J.ag. Sheg ad Q. Che uluser IO wh lmed feedback usg alerag codebooks IEEE Tras. Commu 01 60() pp [11] 3GPP LTE-Advaced Release 10 hp:// [1] D. J. Love ad R.. eah Lmed feedback uary precodg for spaal mulplexg sysems IEEE Tras. If. Theory (8) pp Joural of Resduals Scece & Techology Vol. 13 No DESech Publcaos Ic. do: /ss /13/5/84