The Study of Zero-Forcing Beamforming Gain for Multiuser MIMO with Limited Feedback

Transcription

1 Journal of Communcatons Vol. 8, No. 6, June 03 Te Study of Zero-Forcng Beamformng Gan for Multuser MIMO wt Lmted Feedback Llng Xe, Hu Tang, and Huayu Zou 3 Scool of Matematcal Scences, Sout Cna Normal Unversty, Guangzou, 5063, Cna Arcadyan Tecnology Corporaton, Sanga, 003, Cna 3 New ostcom Ltd, Guangzou, 50663, Cna mal: xell@scnu.edu.cn; davd_tang@sva-arcadyan.com.cn; zou_uayu@63.com Abstract Ts paper dscusses te fnte-rate feedback-based ZF for multuser MIMO n case of small number of canddate users to be selected and small number of quantzaton bts to be fed back. In ts case, te multplexng gan from ZF s lmted. Under te assumpton of small number of canddate users, ten te paper furter analyzes trougput of ZF wt dfferent number of co-sceduled users. From te analyss, te trougput of ZF s lmted by te loss of target sgnal power and te resdual nterference from te co-sceduled users. Te loss of target sgnal power comes from te steered precoder by ZF operaton. From te smulaton for teoretcal trougput and trougput from Monte Carlo metods, we ave some conclusons, for fnte-rate feedback-base ZF multuser MIMO wt te small number of quantzaton bts, te ZF beamformng gan tat s sgnfcant for perfect CSIT based ZF dsappears, and te trougput for small value of co-sceduled users may be larger tan te trougput for large value of co-sceduled users. Index Terms Multuser MIMO, lmted feedback, fnte-rate feedback, Zero Forcng, multplexng gan I. INTRODUCTION Multuser Multple nput multple output (MIMO) s wdely studed n 3G Long Term voluton (LT) and LT-Advanced system. Multuser MIMO can aceve te potental full multplexng gan. For sngle-user MIMO, te multplexng gan s lmted by te number of antennas for sngle user. However, for multuser MIMO, even n te extreme case tat eac user only as one receve antenna, multuser MIMO can stll aceve te multplexng gan up to te number of transmt antennas. In practcal, te number of receve antenna s lmted by te sze of user equpment and te cost of recever. Terefore, multuser MIMO gans many nterests of engneers and academcs[-5]. However, wt te lmted feedback (fnte-rate feedback), te trougput of Zero Forcng (ZF) metod n Manuscrpt receved May 4, 03; revsed June 4, 03 Ts work was supported by Scence and Tecnology lannng roect of Guangdong rovnce (No. 0B000007), and roect on te Integraton of Industry, ducaton and Researc of Guangdong rovnce (No. 0B090086), Cna. Correspondng autor emal: xell@scnu.edu.cn. do:0.70/cm multuser MIMO s lmted by quantzaton of cannel, snce compared wt perfect Common and Internal Spannng Tree(CIST) ZF te nterference part can not be elmnated any more [6]. In fact, for te lmted feedback, nterference and sgnal power bot scale lnearly wt te Sgnal Nose Rato (SNR). In Ref. [6], t s proposed tat te number of quantzaton (feedback) bts sould be scaled wt te SNR, snce te rate gap between perfect cannel state nformaton at transmtter(csit, CSI at Transmtter) based ZF and fnte-rate feedback-based ZF s proportonal to te SNR. In Ref. [7], te lmted feedback based ZF s also studed n te smlar way, and plot for demodulaton wt consderaton of ZF beamformng,.e. te dedcated tranng, s proposed. In general vew, te number of quantzaton bts sould reac a certan level to aceve te suffcent multuser MIMO [8]. But t s not practcal to ncrease te number of quantzaton bts under te current standard framework for some reasons, suc as te burden of uplnk etc. In LT standard release-0, te number of quantzaton bts for te 4 transmt antennas s four, and te number of quantzaton bts for te 8 transmt antennas s egt. In te standard, te number of quantzaton bts s not scaled wt SNR, and s small ndeed. Te one of reasons to use te small number of quantzaton bts s tat te multuser dversty gan can make up te absence of nterference nullng gan n case of te small number of quantzaton bts. From ts perspectve, te man beneft comes from te varyng of cannel magntude for large number of users,.e. n te sense of statstcs tere are always users wt good enoug cannel magntude to aceve te g rate regardless te nterference from multuser MIMO. In Ref. [9], [0], ts problem as been dscussed. Te scenaro wt te very large number of canddate users s analyzed n tose papers, wt elp of te cannel magntude or Cannel Qualty Indcator (CQI) feedback, troug te optmzed user groupng metod, te multuser dversty gan can be aceved. However, n some scenaros, e.g. femto-cell, ome access etc., te transmtter only serves a small number of canddate users. It s wort studyng te ZF beamformng n multuser MIMO. Tus, smlar wt Ref. [6], ts paper manly dscusses te case of te small number of canddate users, or more defntely te case 03 ngneerng and Tecnology ublsng 378

2 Journal of Communcatons Vol. 8, No. 6, June 03 tat no multuser dversty gan s exploted. At te same tme, ts paper analyzes and smulates te trougput of ZF beamformng wt dfferent number of co-sceduled users. II. SYSTM MODL In ts paper, we only consder te case tat eac user as only one antenna and assume tat te cannel s block Rayleg fadng. ac MISO cannel for multuser MIMO can be modeled as: were y x n,,..., () M C s te cannel vector for te -t user (due to Rayleg fadng te mean of cannel magntude s M M,.e. M ); x C s te transmt sgnal vector for te -t user after beamformng; n s te Addtve Gaussan Wte Nose (AGWN) wt zero mean and unt varance; s te number of co-sceduled users n te multuser MIMO transmt. Te total transmt power constrant s. Here te large scale propertes, suc as pat loss and large scale fadng etc., are not nvolved, wc as been dscussed n Ref. [9-0] and approved to contrbute to te multuser dversty gan. Te cannel drecton / s quantzed by eac user wt fnte rate and reported n te assumed error-free uplnk. In practcal, te quantzaton s based on a fxed codebook wc s known bot at transmtter and recever. If te number of feedback bts s B, te vectors number n te codebook s N B. Te user fnds te vector n te codebook wc s closest to te cannel drecton as te quantzed cannel drecton, and feeds back te correspondng ndex of te vector. Te analyss n te paper uses te teory of Random Vector Quantzaton (RVQ) n te smlar way wt Ref. [6]. Te notatons n te paper also can refer to Ref. [6]. Wt te beamformng procedure, te cannel can be modeled as: y v s n,,..., () were s denotes te symbol for te -t user, and v s te beamformng vector for te -t user. Due to te total transmt power constrant, v sould be normalzed n general,.e. v, wc s also called te constant module property n RVQ. Te ZF beamformer v can be obtaned by pseudo nverse [6] or block dagonalzaton [-] metod. Te SINR for te -t user can be wrtten as: SINR, v v (3) were s te number of co-sceduled users. In te paper, te equal power allocaton for eac user s assumed. In case of perfect CSIT, te nterference can be completely cancelled. However, n case of fnte-rate feedback, te nterference cannot be cancelled. As dscussed n Ref. [6], te resdual nterference wll largely lmt te performance of ZF, especally n te g SNR area. In te sequel, we wll analyze te trougput n case of fnte-rate feedback. III. THROUGHUT ANALYSIS Snce te fnte-rate feedback-based ZF s nterference lmted, te suppresson of te nterference can obvously rase te trougput. Te drect metod to suppress te nterference s to reduce te number of nterference sources,.e. te number of co-sceduled users. However, wen te number of co-sceduled users s reduced, te multplexng gan wll drop as well. Tus, t s a tradeoff between te reducton of te number of co-sceduled users and te ncrease of te multplexng gan. In te next, we wll analyze te teoretcal trougput for fnte-rate feedback-based ZF. At frst, let us see te general case tat te number of canddate users s larger tan te number of transmt antennas,.e. M. As a specal case, we wll analyze te trougput wen te number of canddate users s te same as te number of transmt antennas. In ts case we Te next teorem wll gve te clearer sgt for te teoretcal ergodc mean of trougput. Teorem : Te compact upper bound of trougput for fnte-rate feedback-based ZF s: RFB ( ) log ( M ) M ( ) sn ( (, )).(4) M roof: Te teoretcal ergodc mean of trougput can be wrtten as: RFB ( ) log Assumng tat v, v v (5) s derved from te unquantzed verson (,..., ) nstead of te quantzed verson (,..., ), wt ts assumpton te mean of v can be regarded as a compact upper bound of te mean of v. Furtermore, satsfy v v v as M degrees of freedom to v 0 ( ). In oter words, v stll as 03 ngneerng and Tecnology ublsng 379

3 Journal of Communcatons Vol. 8, No. 6, June 03 M degrees of freedom to matc to te target cannel. Tus, M v v M In oter words, f te number of co-sceduled users s smaller, te more dependence v to can be cosen to obtan te resdual array gan. From te proof of teorem n Ref. [], we ave: v Y sn ( (, ) [ Y] sn ( (, ) M were Y s a beta (, M-) random varable. Fnally, te upper bound of trougput can be wrtten as qu. (4). Smlar wt [], sn ( (, ) can be evaluated as, from te RVQ teory. M It sould be noted tat: ) Te perfect CSIT-based ZF as te trougput mean as: B B M ( ) log RFB ) Wen, te multuser MIMO system can be regarded as sngle-user MIMO system,.e. te socalled pont-to-pont MIMO system. It can be observed from te equatons derved above tat for fnte-rate feedback-based ZF te value of wll largely mpact te upper bound of trougput. IV. THROUGHUT ANALYSIS To verfy te upper bound of trougput, we compared tem wt te smulated trougput by Monte Carlo metod wt consderaton of q. (3). In Monte Carlo metod, te cannels are realzed randomly as te block Rayleg fadng. We coose te number of transmt antennas as 4 and 8. Te numbers of quantzaton bts are selected as 4 and 8 respectvely, snce tey are te number of feedback bts for 4 and 8 transmt antennas n current LT and LT-Advanced standard. Te results are plotted n Fg. and Fg. as follows. From Fg. and Fg., we ave te conclusons: (6) Fgure.. Te compact upper bound of trougput and te smulated trougput wen M=4 and B=4 Fgure.. Te compact upper bound of trougput and te smulated trougput wen M=8 and B=8 ) Te curves sow te smlar trend for te compact upper bound of trougput and te smulated trougput; ) For fnte-rate feedback-base ZF wt te small number of quantzaton bts, e.g. 4 and 8 bts n our smulaton, te ZF beamformng gan s not obvous; 3) For fnte-rate feedback-base ZF wt te small number of quantzaton bts, te trougput for small value of may be larger tan te trougput for large value of ; 4) ven for perfect CSIT-based ZF, n low SNR area, te trougput for small value of may be larger tan te trougput for large value of. Te conclusons above seem too pessmstc for multuser MIMO. It s noted tat any concluson n te paper s based on te assumpton of no multuser dversty gan exploted. If te multuser dversty gan s exploted,.e. te number of canddate users s large enoug, te expectaton of v s not tat small as 03 ngneerng and Tecnology ublsng 380

4 Journal of Communcatons Vol. 8, No. 6, June 03 Teorem wen s large, so te multplexng gan wll be obvous snce te eac user as te larger power of target sgnal. Specfcally, wen te number of canddate users s large enoug, we wll coose te group of users wo ave te sem-ortogonal or ortogonal cannel drecton, so te ZF beamformer wll approac to te cannel drecton, and tereby te power of target sgnal wll approac to tat n sngle-user MIMO,.e. v wll approac te optmal value. In oter words, te array gan s kept wen te number of canddate users s large enoug. However, n ts case, te ZF beamformng gan s not sown yet. To verfy ts pont, we wll sow te oter extreme case,.e. te number of canddate tends to nfnty (of course muc larger tan te number of transmt antennas). Teorem : Wen te number of canddate users tends to nfnty, te upper bound of trougput for fnte-rate feedback-based ZF s: R ( ) log FB M M ( ) sn ( (, )). (7) M roof: Wen te number of canddate users tends to nfnty, te sceduler can always select te users wt te ortogonal cannel drectons, so ZF precodng need to be operated snce te cannel drectons for te selected users are already nterference free. In oter words, te precoder v wll not be adusted n ZF operaton, so t wll stll matc to te cannel drecton,.e. v v Te left dervaton s te same wt Teorem, and te result s reaced. From above teorem, t s observed tat, wen te number of canddate users tends to nfnty te precoder wll not be adusted and wll matc to te target cannel drecton, so te multplexng gan of multuser MIMO comes back. Fg. 3 and Fg. 4 sow tat wt te optmal user groupng te trougput for large number of canddate users s larger tan tat for small number of canddate users, n all SNR area for perfect CSIT case and n low SNR area for fnte-rate feedback case. Fgure. 3. Te upper bound of trougput ) wtout any user groupng ) wt te optmal user groupng, wen M=4 and B=4 Fgure. 4: Te upper bound of trougput: ) wtout any user groupng and ) wt te optmal user groupng, wen M=8 and B=8 On te oter and, te large number of quantzaton bts wll lead to te sgnfcant ZF beamformng gan. Neverteless, te ncrease of te feedback bts wll cause some ssues, suc as uplnk payloads etc., and s not accepted n current standard. Wen te number of quantzaton bts s large enoug, te trougput s sure to be close to perfect CSIT-based ZF wc sows large ZF beamformng gan, but t s out of scope of ts paper. Tus, we can foresee tat te potental exceptons or mprovements for low ZF beamformng gan are: ) Te larger number of canddate users; ) Te larger number of quantzaton bts. V. CONCLUSIONS Ts paper dscusses te fnte-rate feedback-based ZF for multuser MIMO n case of small number of canddate 03 ngneerng and Tecnology ublsng 38

5 Journal of Communcatons Vol. 8, No. 6, June 03 users to be selected and small number of quantzaton bts to be fed back. In ts case, te multplexng gan from ZF s lmted. Under te assumpton of small number of canddate users, te paper ten furter analyzes trougput of ZF wt dfferent number of co-sceduled users. From te analyss, te trougput of ZF s lmted by te loss of target sgnal power and te resdual nterference from te co-sceduled users. Te loss of target sgnal power comes from te steered precoder by ZF operaton. From te smulaton for teoretcal trougput and trougput from Monte Carlo metods, we ave some conclusons:, for fnte-rate feedback-base ZF multuser MIMO wt te small number of quantzaton bts, te ZF beamformng gan tat s sgnfcant for perfect CSIT based ZF s dsappear, and te trougput for small value of co-sceduled users may be larger tan te trougput for large value of co-sceduled users. ACNOWLDGMNT Ts work was supported by Scence and Tecnology lannng roect of Guangdong rovnce (No. 0B000007), and roect on te Integraton of Industry, ducaton and Researc of Guangdong rovnce (No. 0B090086), Cna. RFRNCS [] C. L. Jang, M. M. Wang, Su Feng, Janxn Wang et al., Multuser MIMO wt lmted feedback usng alternatng codebooks, I Transactons on Communcatons, vol. 60, no., pp , Feb 0. [] C. B. Cae, D. Mazzarese, N. Jndal, and R. W. Heat, A low complexty lnear multuser MIMO beamformng system wt lmted feedback, presented at te 4 nd Annual Conference on Informaton Scences and Systems, rnceton, New Jersey, Mar. 9-, 008, pp [3] M. Trvellato, F. Boccard, and H. Huang, Zero-forcng vs untary beamformng n multuser MIMO systems wt lmted feedback, presented at I 9 t Internatonal Symposum on ersonal, Indoor and Moble Rado Communcatons 008(IMRC 008), Cannes, France, Sep. 5-8, pp. -6, 008. [4] Yongxng Zou, B. Clerckx, and m Sungn. Flexble multuser MIMO wt lmted feedback, presented at I 69 t Vecular Tecnology Conference 009(VTC Sprng 009), Barcelona, Span, Apr. 6-9, 009, pp. -5. [5] X. Xa, G. Wu, J. Lu and S. Q. L, Leakage-based user scedulng n MU-MIMO broadcast cannel, Scence n Cna Seres F: Informaton Scences, vol. 5, no., pp , Dec 009. [6] N. Jndal, MIMO broadcast cannels wt fnte-rate feedback, I Transactons on Informaton Teory, vol. 5, no., pp , Nov 006. [7] G. Care, N. Jndal, M. obayas, and N. Ravndran, Multuser MIMO downlnk made practcal: acevable rates wt smple cannel state estmaton and feedback scemes, I Transactons on Informaton Teory, reprnt, arxv: 07.64, Nov 007. [8] Q. H. Spencer, A. Lee Swndleurst, and M. Haardt, Zero-forcng metods for downlnk spatal multplexng n multuser MIMO cannels, I Transactons on Sgnal rocessng, vol. 5, no., pp , Feb 004. [9] T. Yoo, N. Jndal, and A. Goldsmt, Mult-antenna downlnk cannels wt lmted feedback and user selecton, I Journal on Selected Areas n Communcatons, vol. 5, no. 7, pp , Sep 007. [0] T. Yoo, N. Jndal, and A. Goldsmt, Fnte-rate feedback MIMO broadcast cannels wt a large number of users, presented at te I Internatonal Symposum on Informaton Teory, Seattle, WA, Jul. 9-4, 006, pp [] Q. H. Spencer, A. L. Swndleurst, and M. Haardt, Zero-forcng metods for downlnk spatal multplexng n mult-user MIMO cannels, I Transactons on Sgnal rocessng, vol. 5, no., pp , Feb 004. [] N. Ravndran and N. Jndal, Lmted feedback-based block dagonalzaton for te MIMO broadcast cannel, I Journal on Selected Areas n Communcaton, vol. 6, no. 8, pp , Oct 008. communcatons. Llng Xe receved te B.S. degree and te.d. degree n computatonal matematcs from Sun Yat-sen Unversty, Guangzou, Cna n 000 and 005, respectvely. Se s currently a lecture n Sout Cna Normal Unversty, Guangzou, Cna. Her researc nterests nclude optmzaton teory and ts applcatons. Hu Tang receved te B.S. degree from Sun Yatsen Unversty, Guangzou, Cna n 00. He s currently an engneer n Arcadyan Tecnology Corporaton, Sanga, Cna. Hs researc nterests nclude wreless communcatons. Huayu Zou receved te B.S. degree and te.d. degree n computatonal matematcs from Sun Yat-sen Unversty, Guangzou, Cna n 00 and 006, respectvely. He was engaged n te postdoctoral researc work n TCL Industry Researc Insttute from 006 to 009. He s currently an engneer n New ostcom, Guangzou, Cna. Hs researc nterests nclude optmzaton teory and wreless 03 ngneerng and Tecnology ublsng 38