Improvement of Rotation Matrix Based Differential Limited Feedback with DFT Codebook and Its Application in CoMP Environment

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1 Improvmnt of Rotation Matrix Basd Diffrntial Limitd Fdback with DFT Codbook and Its Application in CoMP Environmnt Yin Zhu, Yushng Ji, Ping Wang, and Fuqiang Liu Broadband Wirlss Communications and Multimdia Laboratory, Ky Laboratory of Embddd Systm and Srvic Computing supportd by Ministry of Education, Tongi Univrsity, Shanghai, 00, China National Institut of Informatics, Tokyo, Japan Abstract Rcntly, xploiting th tmporal corrlation of slowly varid multipl input multipl output (MIMO) channls to furthr improv th systm prformanc or th fdback fficint in MIMO wirlss communication systms with limitd fdback has bn invstigatd in many paprs. Among thm, T. Kim t al. proposd a rotation matrix basd diffrntial fdback schm which is shown to hav good prformanc with rgard to th systm throughput for spatial multiplxing MIMO systms. Although th rotation matrix codbook and th prcoding codbook in thir schm ar optimally constructd, thy ar not fit for a practical systm du to th structur constraints. In this papr, w propos to rplac th optimal codbook in th rotation matrix basd diffrntial fdback schm with discrt Fourir transform (DFT) structurd codbook, and propos a sarch-basd mthod to dcras th prformanc dgradation inducd by th un-optimality of th DFT codbook. In addition, th modifid algorithm is also applid to a pr-cll codbook basd coordinatd multi-point (CoMP) transmission systm to vrify its applicability to diffrnt MIMO scnarios. Simulation rsults show th ffctivnss of our modifid schm. systm prformanc. Rcntly, many rsarch paprs bgan to focus on xploiting th channl tmporal corrlation []-[] to furthr rduc th fdback ovrhad and improv th systm prformanc. By lvraging tmporal corrlation of th channl, [] usd Givns rotations to paramtriz spatial channl information and thn rducs fdback complxity using Dlta Modulation. This mthod dirctly comprsss th lmnts of th channl matrix but shows lowr fficincy than th mthod of fding back th quantizd prcoding matrix that was adoptd in,.g., [7]-[]. [7] and [] modld th quantizd CSI as a first-ordr finitstat Markov chain and [] proposd a fdback comprssion algorithm by truncating th transition probabilitis; [9] usd a godsic prdiction algorithm to rduc th fdback ovrhad; [0]-[3] modld th channl as a first-ordr Gauss-Markov procss. Among thm, [] proposd a rotation basd diffrntial limitd fdback schm by constructing a diffrntial prcoding codbook to adapt to th channl tmporal corrlation. This schm was furthr systmizd in [3] by taking th tmporal corrlation and rror propagation ffct into account. Simulation shows a noticabl improvmnt of th systm prformanc ovr th traditional fdback mthods that adopt fixd prcoding codbook and it is also suprior to th godsic tracking algorithm in []. Howvr, th diffrntial codbook constructd in [3] is not optimal for a spatially uncorrlatd MIMO channl as its codwords ar not uniformly distributd. Morovr, though th prcoding codbook and th rotation matrix codbook adoptd in [3] ar optimally constructd, thy ar not fit for a practical systm as thir structurs ar not convnint for fficint computation and storag. Discrt Fourir transform (DFT) structurd codbook is a kind of codbook that is oftn adoptd in many wirlss communication standards. Though in many cass this kind of codbook is not optimal for a spatially uncorrlatd channl, it has important advantags with rgard to its structur. In this papr, w first analyz th systm prformanc of th diffrntial limitd fdback schm proposd in [3] by rplacing th optimal codbook with DFT codbook and show thr is obvious prformanc dgradation du to th un-optimality of th Indx Trms Tmporal corrlation, diffrntial fdback, DFT codbook, spatial multiplxing I. INTRODUCTION Multipl input multipl output (MIMO) tchnology with channl stat information at transmittr (CSIT) is known to gratly incras th data transmission rat or th systm rliability in wirlss communication systms. In a frquncy division duplxing (FDD) systm, th CSI only can b obtaind by th transmittr with limitd fdback. Th rcivr quantizs th CSI into a codword of a finit st calld codbook that is both known to th transmittr and th rcivr, thn th indx of th codword is fd back from th rcivr via a low rat rvrs link []. Optimization of th codbook and th fdback schm to maximiz th systm prformanc ar important rsarch topics in limitd fdback tchnology. Dsign of optimal codbook basd on th channl spatial condition []-[5] can obtain optimal Manuscript rcivd March 7, 0; rvisd Jun 7, 0. Corrsponding author mail: yin_zhu7@3.com. doi:0.70/cm

2 DFT codbook. To dcras th prformanc dgradation, w thn propos a sarch basd mthod to gnrat a bttr diffrntial codbook by xploiting th structur proprty of th DFT codbook. Simulation rsults show that our mthod is ffctiv and it can trad off btwn th sarch complxity and th improvmnt of th systm prformanc. In th latst vrsion of 3GPP LTE standards, high spctral fficincy is rquird and th coordinatd multipoint transmission and rcption (CoMP) tchnology is considrd as a potntial solution. In ths systms, multipl transmission points or clls ar coordinatd to transmit data to a singl usr or multipl usrs. Coordinatd multi-point transmission can ffctivly incras th systm spctral fficincy by avoiding th intr-cll intrfrnc or vn convrting th intrfrnc into usful signals [7]. To gt th bnfits of th coordinatd systm, high quality CSIT is ndd. Bsids this, th fdback ovrhad of th systm is multiplid du to th multiplid transmission antnnas. Thus fdback ovrhad rduction and improvmnt of CSI quantization prcision bcom mor urgnt in coordinatd systms. To rduc th fdback ovrhad, [] proposs a subspac-basd channl quantization mthod and a fdback ovrhad rduction schm that xploiting th tmporal corrlation of th channl. But its quantization codbook is constructd by trating th coordinatd systm as a virtual MIMO systm, i.., th transmittrs of diffrnt BSs ar considrd to blong to a cntral unit, and th composit channl is assumd to b i.i.d. as in th traditional singl cll systm. Although this assumption is simpl and asy for systm analysis, it dosn't fit for th ral nvironmnt. [9] lists two proprtis of th coordinatd systm: dynamic numbr of cooprating BSs and htrognous path loss ffcts in th composit channl. It also proposs a pr-cll codbook basd limitd fdback mthod which is flxibl to construct and shows asymptotic optimality to oint-cll codbook dsign approach. Basd on this ida, [0] and [] proposd a pr-cll codbook basd fdback mthod with phas ambiguity compnsation for CoMP systm in which singl data stram is transmittd. Compard with [9],thir mthod has low complxity and bttr quantization prformanc by fding back additional phas ambiguity quantization ovrhad. In this papr, w apply our modifid diffrntial fdback schm to th abov systm and compar its prformanc with that of th mthod in [0] and []. As shown in simulation rsults, it nds lowr fdback ovrhad and has bttr prformanc whn th usr spd is not vry high. Th rst of th papr is organizd as following: Sction II introducs th singl cll singl usr MIMO systm modl and th diffrntial limitd fdback schm, thn points out th prformanc dgradation inducd by using th DFT structurd codbook. Sction III proposs a sarch basd diffrntial fdback schm and two 99 rducd sarching mthods by xploiting th structur of th DFT codbook to improv th systm prformanc. In Sction IV is th application of th modifid fdback schm to a multi-cll oint transmission MIMO systm. Finally, Sction V concluds th work of this papr. II. SYSTEM MODEL AND PROBLEM FORMULATION A. Systm Modl Considr a singl cll MIMO spatial multiplxing systm that transmits multipl data strams. Lt th numbr of th transmit and th rciv antnna b nt and nr rspctivly, and th numbr of th indpndnt data stram b M, M min{nt, nr}. At th channl instanc τ, th rcivd signal is: y M H F s n whr s CM is th complx vctor of transmittd data strams and ρ is th total transmit powr. Th channl matrix H CnR nt is modld as spatially uncorrlatd Rayligh flat fading. F CnT M is th prcoding matrix and nτ rprsnts th nois vctor. Fτ is constraind to b orthonormal, i.., FτHFτ = IM. ( )H dnots th conugat transpos. Furthrmor, whn th spd of th usr is low, th tmporal corrlation btwn succssiv channl instancs can b modld as a first-ordr Gauss-Markov procss [3]: H H Δ whr ε=j0( πfdt ) is th tim corrlation cofficint of th channl. J0( ) is th zroth ordr Bssl function. T rprsnts th intrval of th channl instanc and fd is th maximum Dopplr frquncy which is dtrmind by th usr spd and th carrir frquncy. Hτ- and Δτ ar indpndnt and both hav i.i.d. ntris with th distribution of CN(0,). In addition, Hτ can b dcomposd through singular valu dcomposition (SVD) into Hτ = UτΣτVτH, thn th optimal prcoding matrix V V : M, i.., it is formd by taking th first M columns of Vτ. At ach instanc τ, th rcivr fds back th indx of th stimatd prcoding matrix Fˆ chosn according to th capacity slction critrion: () Fˆ arg max I F, F,i F,i whr I F log dt I M F H H H H F M and Fτ,i is th codword of th prcoding codbook Fτ. With this slction critrion, it is provd [3] that Fτ is optimal if th minimum Fubini-Study distanc of its codwords is maximizd, i..: F argmax FS F F ()

3 whr FS F min dfs F,i, F,, F,i, F, F. L is th spatially uncorrlatd MIMO channl. This proprty also inspird us to dvlop th sarch basd mthod in our modifid fdback schm. In many wirlss communication standards, a kind of commonly usd codbook is DFT structurd codbook. DFT codbook has th advantags of [3]: (i) its codword has qual amplitud so that to avoid high pak to avrag powr ratio (PAPR) at th transmittr; (ii) its codword ntry has finit alphabt which rducs systm complxity spcially whn th numbr of th antnna lmnts is larg. A DFT prcoding matrix codbook i L codbook siz and dfs(si, S) = arcos dt(sihs) is th Fubini-Study distanc. Finding th optimal codbook can b sn as a Grassmannian subspac packing problm. B. Problm Formulation Sinc our modifid schm is highly dpndnt on th rotation matrix basd diffrntial fdback framwork proposd in [3], w nd to xplain it in dtail first. By xploiting th tmporal corrlation of th channl, at ach channl fdback instanc τ, a diffrntial codbook Fτ is dvlopd which cntrs around th prviously slctd prcodr Fτ-. Spcifically, th codwords of Fτ ar gnratd via prturbing Fτ- by a rotation matrix codbook with a crtain radius. Th radius is dtrmind by taking th quantization rror propagation and channl variation into account. Thus th fdback schm has a bttr capability to track th variation of th channl than othr diffrntial fdback mthods. Th ith codword of Fτ is constructd as following: F,i pro r I nt r i F TDFT with codwords Tl CnT M (l =,, L) can b constructd as following []: First, w construct a nt nt DFT matrix nt Α nt nt nt (3) Q nt nt Th first codword T of TDFT can b dtrmind by slcting M diffrnt columns from A. For xampl, w can tak th first M columns of A as T. Thn th rmaining L- codwords of TDFT ar dtrmind by: whr pro( ) rprsnts th proction oprator that nsurs Fτ,i is still orthonormal aftr th abov opration. This can b don by Procrusts orthonormalization or Gram-Schmidt column orthonormalization. r is th diffrntial radius and Θi is th ith codword of a rotation matrix codbook Q, which is constructd according to th critrion Q arg max Q nt nt nt Tl Wl T l, 3,,l whr W diag{ () 0 u, u, u L u L un L T }, unt L can b sn as th row indics in ordr to optimiz th systm prformanc, whr Q min d Θi, Θ and d is a distanc masur of a L L DFT matrix and satisfy ui u, i. dfind by: u, u, i L tr ΘiH Θ d Θi, Θ nt minimum distanc of th codwords of th codbook to maximiz th systm prformanc. In our fdback schm, w will rplac th initial prcoding codbook and th rotation matrix codbook in [3] with this DFT structurd codbook. Howvr, in most cass DFT codbook is not optimal apart from som particular combinations of nt and L (.g., as pointd out in [] whn th prcodr is a vctor (M = )). In gnral, a nar optimal DFT codbook can b obtaind by choosing th row indics st u, u,, unt via brut forc sarching. To activat th diffrntial procss, a traditional Grassmannian codbook is adoptd as th initial prcoding codbook. Thn at ach diffrntial stp (channl instanc τ), a diffrntial codbook Fτ is constructd onlin in which th rcivr slcts th propr codword as th stimation of th prcoding matrix according to th critrion (). In this diffrntial fdback schm, th initial codbook and th rotation matrix codbook ar optimally constructd to maximiz th systm prformanc, but ths codbooks ar hard to b mployd in practical systms bcaus thir structurs ar not convnint for computation and storag. In addition, it can b sn that aftr prturbation to Fτ-, th codwords of Fτ ar not isotropically distributd on th Grassmann manifold anymor []. That is to say, th diffrntial codbook constructd according to (3) is actually not optimal for a, unt should b slctd to maximiz th Thrfor, th systm prformanc is dgradd if w adopt DFT structurd codbook as th initial prcoding codbook and th rotation matrix codbook in th diffrntial limitd fdback schm. To allviat this dgradation, w propos a sarch basd mthod by xploiting th structur proprty of th DFT codbook. III. SEARCH BASED DIFFERENTIAL LIMITED FEEDBACK SCHEME As analyzd abov, DFT structurd codbook dgrads th systm prformanc in th diffrntial limitd 500

4 fdback schm. On th on hand, th initial DFT prcoding codbook can incur som dgradation of th systm prformanc du to its un-optimality; On th othr hand, at ach diffrntial fdback stp, th systm prformanc is furthr dgradd as th diffrntial codbook dvlopd with DFT structurd rotation matrix codbook also has poor prformanc. Fortunatly, as mntiond prviously, w can choos bst rsults of th two codbooks, i.., th initial prcoding codbook and th diffrntial codbook, according to () by sarching th row indics 0 u, u,, unt L, rspctivly. L(L-) diffrncs (uk - ul) mod L, k l tak all possibl nonzro valus,,, L-, with ach valu xactly λ tims. Hr ZL = {0,,, L-} []. In most cass, brut forc sarching must b implmntd to find a nar optimal DFT bamforming codbook. In [3], it was found that by mimicing th row indics as a diffrnc st, th sarching complxity can b gratly rducd. Spcifically, for a givn combination of (L, nt), a diffrnc st of (L, nt): D = { u, u,, uk, uk,, unt } can b pickd whr L is indpndnt of L. Thn by fixing th valus of u, u,, uk as thos in th diffrnc st and varying th valus of uk,, unt within th rang of {0,,, L-}, a L-K Spcifically, on th on hand, w find a bst initial DFT structurd prcoding codbook by sarching th row indics u, u,, unt of th diagonal matrix Winit dimnsional sarch can b implmntd. Exprimnts in [3] showd good rsults vn in on dimnsional sarch whn nt is not vry larg and th prcision can b improvd by proprly incrasing th sarch dimnsion. It is not vry clar why this rducd sarching mthod can obtain good rsults. Howvr, w find it can b gnralizd to th cas whn th lmnt of th DFT codbook is a matrix. Thus by using this mthod, w can find a good diffrntial prcoding codbook with low sarching complxity. Comparing to th random sarch with dcrasd sarch numbr mntiond bfor, th mimicing diffrnc st mthod dos not nd to fdback any xtra paramtrs. Th transmittr and th rcivr only nd to st th sam sarch rul in advanc such as th diffrnc st and th sarch dimnsion, th idntical sarch rsults can b obtaind. In our modifid schm, w can pick a st of th row indics u, u,, unt of Wτ of th DFT rotation matrix Q according to th critrion Finit,DFT arg max FS Finit,DFT Finit,DFT this can b don offlin bfor transmission; on th othr hand, at ach diffrntial stp w aim to construct a bst diffrntial codbook Fτ by prturbing th prviously slctd prcodr Fτ- with a DFT structurd rotation matrix codbook Qτ,DFT. In this stp w do not nd to gnrat a bst rotation matrix codbook Q as in [3], but to choos th row indics 0 u, u,, unt L of Wτ whn constructing Qτ,DFT to obtain a bst diffrntial codbook Fτ. This procss must b finishd during ach diffrntial stp. Our focus bcoms to find an fficint sarching mthod to obtain a propr st of th row indics u, u,, unt of th diagonal matrix Wτ so as to construct a bst diffrntial codbook Fτ. satisfying () as th mimicing diffrnc st bfor transmission. At ach diffrntial stp, by varying part of th row indics according to th sam dimnsion, th sam diffrntial codbook Fτ with maximum δ(fτ) (th minimum Fubini-Study distanc of th codbook) can b found both at th transmittr and th rcivr with rducd sarching complxity. A. Random Sarch Mthod A simpl sarching mthod is brut forc sarching. This can surly find a bst combination of u, u,, unt of Wτ. But it is computationally inhibitiv du to th hug sarching complxity CnLT. W can dcras th sarch numbr to trad off btwn th sarching complxity and th prformanc improvmnt. A random sarch can b adoptd according to th dcrasd sarch numbr. But to gnrat th sam diffrntial codbooks at th transmittr and th rcivr, th transmittr nds to know th sarch rang which must b fd back from th rcivr at ach diffrntial stp. It incrass th fdback ovrhad and is also inconvnint to implmnt. So in th nxt subsction w propos a rducd sarch mthod by gnralizing th mthod that was proposd in [3]. C. Complxity Analysis In our sarch basd DFT diffrntial fdback schm, brut forcing sarching is optimal to find a bst diffrntial codbook Fτ. But th sarching complxity is vry high. For a systm with nt transmit antnnas and lt th siz of th codbook b L, th sarch spac of th row indics u, u,, unt of th rotation matrix codbook Qτ,DFT can achiv CLnT. Bsids this, during ach diffrntial stp, th complxity of computing th minimum distanc of th diffrntial codbook Fτ is L(L)/. Thus th total complxity can b as high as B. Mimicing Diffrnc St Mthod It was shown [] that an optimal DFT bamforming codbook xists only with som particular combinations of (L, nt). In ths cass, th row indics u, u,, unt of complxity can b gratly dcrasd. For xampl, in th mimicing diffrnc st mthod w can rduc th sarch dimnsion to on, i.., w only lt th last row indx unt vary within its rang. Thn th sarch complxity can th optimal DFT codbook forms a diffrnc st. A (L, nt, λ) diffrnc st is a subst { u, u,, unt } of ZL if th L L CLnT. If w adopt rducd sarch mthod, th 50

5 In addition, w also compar in Fig. 3 th Fubini-Study distanc of th diffrntial codbooks gnratd bfor and aftr th sarching procss. It is clarly obsrvd that w obtain a bttr diffrntial codbook which has largr minimum distanc than th on bfor sarching. This rsult corrsponds to th codbook construction critrion (), i.., w should adopt th codbook with its minimum distanc as maximum as possibl. It also can b sn that th rsulting codbooks obtaind by random sarch and th rapid sarch hav almost th sam minimum distanc. b dcrasd from CLnT to (L-nT+). D. Analysis of th Simulation Rsults Fig. and Fig. show th achivabl usr throughputs comparing our mthod with that proposd in [3] and with th traditional fdback mthod. W assum a spatial multiplxing MIMO systm with nt = nr = antnnas at both th transmit and th rciv nds. Th numbr of th transmission data stram M =, and th codbook siz L =. For clarly comparison, w tak th sam tim corrlation cofficint ε as in [3]. W adopt on dimnsion sarch whn using th mimicing diffrnc st mthod (also calld rapid sarch in th figurs). Thn th sarch numbr is L-nT+ = 3. Th random sarch numbr is also takn as this valu. From th two figurs, w can find whn using th diffrntial limitd fdback proposd in [3], th usr throughput obviously incrasd comparing to th traditional Grassmannian codbook fdback. This improvmnt is dcrasd whn w tak DFT structurd codbooks as th initial prcoding codbook and th rotation matrix codbook. Howvr, th rduction of th usr throughput can b allviatd by using our modifid diffrntial fdback mthod. W can s that th random sarch and th rapid sarch hav almost th sam rsult, and vn on dimnsion rapid sarch can compnsat almost half of th prformanc gap btwn th fdback schm in [3] and th DFT codbook basd schm. 9.7 Minimum distanc of th diffrntial codbook Diffrntial stp 0 Diffrntial fdback proposd in [3] Diffrntial fdback using DFT codbook Traditional Grassmannian fdback Our proposd mthod (random sarch) Our proposd mthod (rapid sarch) Usr throughput (bits) 9. 0 Diffrntial stp N HN ] N I nt HR/ 9.55 whr Hi CnR nt is th small scal fading of th channl btwn th ith BS and th usr and αi is th corrsponding larg scal fading cofficint assumd to b known both at th transmittr and th rcivr. Th lmnts of Hi ar i.i.d. and ach has distribution of CN(0,). Hc can also b viwd as a spcial spatial corrlatd channl with th spatial corrlation matrix R dtrmind by th larg scal fading cofficint of ach coordinatd channl. In addition, Hc can b dcomposd I nt H Hc [ H H Fig.. Comparison of th usr throughput btwn diffrnt fdback mthods (nr = nt =, M =, L =, SNR = 0dB, ε = 0.93) A. Systm Modl Assuming thr ar N bas stations (BS), ach with nt transmit antnnas, ointly transmit singl data stram to on usr with nr rciv antnnas undr th coordination of a cntral unit, which forms a oint transmission systm. Th composit channl is composd of N corrsponding singl cll channls. Hr w tak into account th htrognous path loss ffct, which mans that th channl formd by diffrnt coordinatd BSs and th usr may hav diffrnt strngths. Th composit channl is rprsntd as IV. APPLICATION TO PER-CELL CODEBOOK BASED JOINT TRANSMISSION SYSTEM Fig. 3. Minimum Fubini-Study distanc of th diffrntial codbook at ach diffrntial stp (nr = nt =, M =, L =, SNR = 0dB, ε = 0.93) 9. Our proposd mthod (random sarch) Our proposd mhtod (rapid sarch) Diffrntial fdback using DFT codbook Traditional Grassmannian fdback Diffrntial fdback proposd in [3] Diffrntial fdback using DFT codbook Our proposd mthod (rapid sarch) Our proposd mthod (random sarch) 9. Usr throughput (bits) Diffrntial stp 0 Fig.. Comparison of th usr throughput btwn diffrnt fdback mthods (nr = nt =, M =, L =, SNR = 0dB, ε = 0.7) 50

6 arg max fikh v, k via singl valu dcomposition (SVD) into: H c U c Σc V H c whr v,k is obtaind by ltting vc = [v,h v,nh]h. As th codword fik is indpndntly slctd, th Whn singl data stram is transmittd, th rcivr nds to fdback th stimation codword of th prcoding vctor vc = Vc(:,), th first column of Vc to th cntral unit: fc arg min dchord fc, vc complxity is fairly low compard with th abov oint slction mthod, but th quantization rror is incrasd as wll. Mthod 3: Indpndnt codword slction with phas ambiguity compnsation To improv th quantization prcision of th indpndnt codword slction mthod, th phas ambiguity btwn v,k and fik is also fd back to th whr dchord is th chordal distanc which is provd to b a propr masurmnt with rgard to th quantization rror in th cas of singl data transmission []. fc, vc CNnT rprsnt th stimatd codword and th optimal prcoding vctor, rspctivly. Th abov critrion can also b asily transformd into th following formation: transmittr using a fw additional bits, i.., fikh v,k fikh v,k k fc arg max fch vc Bsids fding back th indx of fik, th rcivr B. Pr-Cll Codbook basd Fdback Schm In coordinatd multi-cll transmission systm, optimal codbook for th prcoding stimation of th composit channl can b dsignd by viwing th coordinatd systm as a virtual big MIMO systm, and th indx of th corrsponding codword is snt back to th transmit nd. Howvr, this schm is hard to implmnt in th practical systm du to its inflxibility and complxity. Considring th main proprtis of th composit channl mntiond in Sction I, [9] proposs a pr-cll codbook basd limitd fdback schm in which th rcivr slcts th propr sub-codword from ach pr cll codbook and th composit prcodr is constructd by thos sub-codword. It is flxibl to implmnt and is asymptotically optimal whn nt is sufficintly larg. Basd on th pr-cll codword slction ida, [] proposs an indpndnt codword slction with phas ambiguity compnsation fdback mthod and it shows lowr complxity than that of [9]. In [0], svral codword slction mthods basd on pr-cll codbook ar compard with rgard to th quantization prformanc. In th following, w list thr codword slction mthods introducd in [0] and compar our modifid fdback schm with thm. Mthod : Optimal oint codword slction Th codword slction is dtrmind by th critrion: fi,, fin arg min dchord vc, fi N fin whr fik is th codword slctd from th kth pr-cll codbook. In this mthod, th codwords fi,, fin should b ointly slctd to obtain optimal rsults. It has th bst quantization prcision compard with th othr two mthods but th complxity is high whn th codbook siz is larg. Mthod : Indpndnt codword slction Th codword slction is dtrmind by th critrion: fik arg min dchord v, k, fik 503 also quantizs th phas ambiguity k and fds back its indx to th transmittr. Simulation rsults in [0] show an obvious improvmnt of th quantization prformanc than th othr two mthods. Howvr, it nds mor fdback ovrhad. To compar with th abov thr fdback mthods, w also us our diffrntial fdback schm basd on th pr-cll codbook. W adopt Mthod to slct th codword. Assum thr ar two coordinatd BSs ointly transmit singl data stram to on usr. Each BS has nt = transmit antnnas and th usr has nr = rciv antnnas. W st th siz of th codbook to for th abov thr fdback mthods and for our modifid fdback mthod and us DFT structurd codbook for all ths mthods. In addition, Mthod 3 nds an xtra 3 bits to fdback th phas ambiguity. Thus th fdback ovrhad of th four mthods ar: bits for Mthod, bits for Mthod, 7 bits for Mthod 3 and 3 bits for our modifid mthod. Fig. and Fig. 5 show th rsults of th quantization rror prformanc. From th simulation rsults, w can s xcpt for som initial diffrntial stps, th quantization rror of our modifid diffrntial fdback mthod is lowr than Mthod and Mthod. Whn th usr spd is low nough (corrsponding to high valu of ε), th quantization prformanc of our schm is vn suprior to Mthod 3. Notic that w us fwr bits for our schm to fdback th codword. This is asy to xplain bcaus th diffrntial codbook cntrs around th prvious prcodr and its quantization rang is much smallr than th traditional codbook (th traditional codbook covrs th whol quantization rang). So, high quantization prcision can b obtaind with fwr codwords. This advantag is vn mor obvious whn th usr spd is low, i.., whn th channl changs much slowly. Considring that whn w adopt coordinatd multi-cll transmission, th usr spd usually is much slow to nsur good systm prformanc, so that th diffrntial fdback schm is also suitabl

7 basd multi-cll coordinatd systm, and th ffct of th improvmnt of th quantization rror is vrifid. for this scnario. Though th sarch complxity of oint codword slction adoptd by th diffrntial schm is fairly high, it can somhow b tradd off by smallr codbook siz. ACKNOWLEDGMENT This work was supportd by th National Scinc and Tchnology Maor Proct of China undr Grant 0ZX and th National Natural Scinc Foundation of China (No. 0379). 0. Our modifid diffrntial fdback Mthod 3 Mthod Mthod Quantization rror 0.75 REFERENCES 0.7 [] [] 0 Diffrntial stp 0 [3] Fig.. Quantization rror comparison btwn diffrnt fdback mthods basd on pr-cll codbook (nr =, nt =, M =, codbook siz = (for Mthod, and 3) and codbook siz = (for our modifid mthod), ε = 0.7) [] [5] 0. Our modifid diffrntial stp Mthod 3 Mthod Mthod Quantization rror [] [7] 0.5 [] Diffrntial stp 0 [9] Fig. 5. Quantization rror comparison btwn diffrnt fdback mthods basd on pr-cll codbook (nr =, nt =, M =, codbook siz = (for Mthod, and 3) and codbook siz = (for our modifid mthod), ε = 0.93) [0] V. CONCLUSION [] In this papr, w propos a modifid diffrntial limitd fdback schm. Basd on th analysis of th diffrntial fdback schm proposd by T. Kim t al. in [3], w point out its advantag in th improvmnt of th systm prformanc and its implmntation constraint in th practical systms. Thn w propos to implmnt this diffrntial fdback schm by substituting th optimal codbooks with th DFT structurd codbooks and analyz th prformanc dgradation inducd by this substitution. Basd on this, w propos a sarch basd diffrntial fdback schm by gnralizing th mimicing diffrnc st sarching mthod to allviat this dgradation. Simulation rsults show th ffctivnss of our schm. In addition, this modifid diffrntial limitd fdback schm is also applid to a pr-cll codbook [] [3] [] [5] 50 D. J. Lov, R. W. Hath Jr., V. K. N. Lau, D. Gsbrt, B. D. Rao, and M. Andrws, An ovrviw of limitd fdback wirlss communication systms, IEEE J. Sl. Aras Commun., vol., no., pp. 3-35, Oct. 00. D. J. Lov, R. W. Hath Jr., and T. Strohmr, Grassmannian bamforming for multipl-input multipl-output wirlss sytms, IEEE Trans. Inf. Thory, vol. 9, no. 0, pp , Oct D. J. Lov and R. W. Hath Jr., Limitd fdback unitary prcoding for spatial multiplxing systms, IEEE Trans. Inf. Thory, vol. 5, no., pp , Aug D. J. Lov and R. W. Hath Jr., Grassmannian bamforming on corrlatd MIMO channls, Proc. IEEE GLOBECOM, vol., pp. 0-0, 00. V. Raghavan, R. W. Hath Jr., and A. V. Sayd M., Systmatic codbook dsigns for quantizd bamforming in corrlatd MIMO channls, IEEE J. Sl. Aras Commun., vol. 5, no. 7, pp. 9-30, Sp J. C. Roh and B. D. Rao, An fficint fdback mthod for MIMO systms with slowly tim-varying channls, in Proc. IEEE WCNC, vol., pp. 70-7, 00. K. Huang, R. W. Hath, and J. G. Andrws, Limitd fdback bamforming ovr tmporally-corrlatd channls, IEEE Trans. Signal Procss., vol.57, no.5, pp , May 009. K. Huang, B. Mondal, R. W. Hath, and J. G. Andrws, Multiantnna limitd fdback for tmporally-corrlatd channls: Fdback comprssion, in Proc IEEE GLOBECOM, 00, pp. -5. T. Inou and R. W. Hath, Godsic prdiction for limitd fdback multiusr MIMO systms in tmporally corrlatd channls, in Proc. IEEE Radio Wirlss Symp., Jan. 009, pp T. Kim, D. J. Lov, and B. Clrckx, Lvraging tmporal corrlation for limitd fdback multipl antnna systms, in Proc. IEEE ICASSP., 00, pp D. Sacristan-Murga and A. Pascual-Isrt, Diffrntial fdback of MIMO channl gram matrics basd on godsic curvs, IEEE Trans. Wirlss Commun., vol. 9, no., pp , Dc. 00. T. Kim, D. J. Lov, B. Clrckx, and S. J. Kim, Diffrntial rotation fdback MIMO systm for tmporally corrlatd channl, in Proc. IEEE GLOBECOM Global Tlcommunications Confrnc, Dc. 00, pp. -5. T. Kim, D. J. Lov, and B. Clrckx, MIMO systms with limitd rat diffrntial fdback in slowly varying channls, IEEE Trans. Commun., vol. 59, no., pp. 75 9, 0. J. Choi, B. Clrckx, N. L, and G. Kim, A nw dsign of polarcap diffrntial codbook for tmporally/spatially corrlatd MISO channls, IEEE Trans. Commun., vol., no., pp , 0. M. Zhou, L. Zhang, and L. Song, A diffrntial fdback schm xploiting th tmporal and spctral corrlation, IEEE Trans. Vh. Tchnol., vol. 5, no. 9, pp , Nov. 03.

8 [] S. Schwarz, R. W. Hath Jr., and M. Rupp, Adaptiv quantization on a Grassmann manifold for limitd fdback bamforming systms, IEEE Trans. Signal Procssing, vol., no., pp. 50-, Sp. 03. [7] D. Gsbrt, S. Hanly, H. Huang, S. Shamai Shitz, O. Simon, and W. Yu, Multi-cll MIMO cooprativ ntworks: A nw look at intrfrnc, IEEE J. Sl. Aras Commun., vol., no. 9, pp. 30-0, 00. [] J. H. Kim, W. Zirwas, and M. Haardt, Efficint fdback via subspac-basd channl quantization for distributd cooprativ antnna systms with tmporally corrlatd channls, EURASIP J. Adv. Signal Procss., vol. 00, Jan. 00. [9] Y. Chng, V. K. N. Lau, and Y. Long, A scalabl limitd fdback dsign for ntwork MIMO using pr-cll codbook, IEEE Trans. Wirlss Commun., vol. 9, no. 0, pp , 00. [0] D. Su, X. Hou, and C. Yang, Quantization basd on pr-cll codbook in cooprativ multi-cll systms, in Proc. IEEE Wirlss Communications and Ntworking Confrnc, 0, pp [] F. Yuan and C. Yang, Phas ambiguity quantization for pr-cll codbook basd limitd fdback coordinatd multi-point transmission systms, in Proc. IEEE 73rd Vhicular Tchnology Confrnc, 0, pp. -5. [] B. M. Hochwald, T. L. Maztta, T. J. Richardson, W. Swldns, and R. Urbank, Systmatic dsign of unitary spac-tim constllations, IEEE Trans. Inform. Thory, vol., no., pp , 000. [3] P. Xia, S. K. Yong, H. Niu, J. Oh, and C. Ngo, DFT structurd codbook dsign with finit alphabt for high spd wirlss communications, in Proc. th IEEE Consumr Communications and Ntworking Confrnc, 009, pp. -5. [] P. Xia, S. Zhou, and G. B. Giannakis, Achiving th wlch bound with diffrnc sts, IEEE Trans. Inf. Thory, vol. 5, no. 5, pp , 005. Yin Zhu now is a Ph.D. candidat at BM&W Lab of Tongi Univrsity. Sh is also a lcturr at Collg of Elctronic and Information Enginring, Suzhou Univrsity of Scinc and Tchnology. Hr rsarch ara is multipl antnna transmission in th nxt gnration wirlss ntworks. Yushng Ji rcivd B.E., M.E., and D.E. dgrs in lctrical nginring from th Univrsity of Tokyo. Sh oind th National Cntr for Scinc Information Systms, Japan (NACSIS) in 990. Currntly, sh is a profssor at th National Institut of Informatics, Japan (NII), and th Graduat Univrsity for Advancd Studis (SOKENDAI). Hr rsarch intrsts includ ntwork architctur, rsourc managmnt, and prformanc analysis for wird and wirlss communication ntworks. Sh is also a mmbr of IEEE and IPSJ. and vido transcoding. Ping Wang is an associat profssor in th dpartmnt of information and communication nginring at Tongi Univrsity. H graduatd from th dpartmnt of computr scinc and nginring at Shanghai Jiaotong Univrsity and rcivd Ph.D. dgr in 007. His main rsarch intrsts ar in routing algorithms and rsourc managmnt in wirlss ntworks Fuqiang Liu is a profssor in th Dpartmnt of Information and Communication nginring at Tongi Univrsity. H graduatd from th dpartmnt of automation at China Univrsity of Mining and rcivd Ph.D. dgr in 99. His main rsarch intrsts ar in tchnologis in wirlss broadband accss and imag manipulation. 505