Transmit Beamforming with Reduced Channel State Information in MIMO-OFDM Wireless Systems

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1 anmi Beamfoming wih Reduced Channel Sae Infomaion in MIMO-OFDM Wiele Syem Seung-Hyeon Yang, Jae-Yun Ko, and Yong-Hwan Lee School of Elecical Engineeing and INMC Seoul Naional Univeiy Seoul, Koea Abac he ue of muliple anenna fo diveiy and aay gain may equie accuae channel ae infomaion (CSI) on each ubcaie in ohogonal fequency diviion muliplexing yem. Such CSI i uually epoed fom he eceive o he anmie, cauing a ignifican amoun of feedback ignaling buden. Recen wok can noiceably educe he feedback ignaling buden by exploiing he channel coelaion popeie in he fequency domain, bu i can be applied o he geneaion of a ingle beam []. In hi pape, hi idea i exended o he ue of muliple beam fo paial muliplexing. Simulaion eul how ha he popoed cheme can impove he pefomance uing he ame amoun of feedback ignaling buden o povide he ame pefomance while noiceably educing he amoun of feedback ignaling buden, compaed o convenional cheme. Keywod-codebook; feedback; MIMO; OFDM yem; pecoding; I. INRODUCION he ue of muli-inpu muli-oupu (MIMO) yem ha been conideed indipenable fo he impovemen of anmiion houghpu. he pefomance impovemen wih he ue of muliple anenna i deeply aociaed wih he accuacy of channel ae infomaion (CSI). When he anmie ha he CSI, i can achieve diveiy gain a well a aay gain by mean of anmi beamfoming, ignificanly impoving he yem pefomance [] [4]. Such beamfoming echnique, howeve, equie accuae CSI a he anmie, which may no eaily be achievable in addiion o noiceable feedback ignaling ovehead. Ohogonal fequency diviion muliplexing (OFDM) ha been ecognized a a key anmiion echnique in boadband wiele communicaion yem. Recenly, combined ue of MIMO and OFDM cheme ha been applied o advanced wiele yem. o employ a beamfoming echnique in MIMO-OFDM yem, he anmie may equie he CSI of each ubcaie, inceaing he amoun of feedback ignaling ovehead linealy popoional o he numbe of ubcaie. hi poblem can be alleviaed by employing a codebook mehod [5] [9], whee he beam weigh i deemined fom a finie e of weigh veco, called codebook, and he index of he coeponding weigh veco i epoed o he anmie hough a feedback ignaling channel in he uplink. he feedback ignaling ovehead can fuhe be educed by ecuively encoding he codewod baed on he peviou one [0]. A fuhe implified cheme, o-called clueing, can educe he ignaling ovehead by combining a numbe of adjacen ubcaie ino a clue ha ue he ame beam weigh []. Pefomance degadaion due o he clueing can be alleviaed by inepolaing he beam weigh in he fequency domain []. Howeve, hi may equie addiional phae infomaion. Recenly, he channel coelaion in he fequency domain ha been exploied o educe he feedback ignaling fo he anmi beamfoming []. he beam weigh can be geneaed in a diffeenial manne uing he coelaion chaaceiic beween he adjacen ubcaie. When he channel of adjacen ubcaie i coelaed, he beam weigh fo he cuen ubcaie can be epeened in em of he beam weigh fo he peviou adjacen ubcaie and a em coeponding o he weigh diffeence beween he wo ubcaie. Howeve, i i applicable o he geneaion of a ingle beam. In hi pape, we conide he exenion of he idea in [] o muli-beam cheme fo paial muliplexing. Fo paial muliplexing, he daa eam ae ofen pecoded baed on he CSI befoe he anmiion [3] [7]. We conide paial muliplexing, whee he beam weigh (o pecoding) i deemined by a codebook compiing uniay maice. he anmie eimae he CSI fom he feedback infomaion and hen geneae he pecoding weigh fom i. hi pape i oganized a follow. Secion II decibe he yem model in conideaion. he convenional beamfoming echnique ha conide a ignal beam i biefly dicued in Secion III. Secion IV peen he popoed cheme ha geneae muliple beam fo paial muliplexing. he

2 N c H Conide he ue of a ingle eceive anenna (i.e., muliinpu ingle-oupu (MISO) cheme). hen, () can be ewien a pefomance i veified by compue imulaion in Secion V. Finally, Secion VI ummaize concluion. Noaion: Bold uppe and lowe lee denoe maice and veco, epecively. () and () denoe anpoe and conjugae anpoe, epecively. E{} denoe he expecaion opeao. and [ ] denoe he Fobeniu nom and he ace of a maix, epecively. U( N, m, n ) denoe he e of N ( m n) -dimenional (D) maice wih ohonomal column. x denoe he malle inege lage han o equal o x. II. SYSEM MODEL Conide a MIMO-OFDM baed wiele yem, whee he bae aion (BS) ue n anmi anenna and he mobile aion (MS) ue n eceive anenna a illuaed in Fig.. Aume ha he OFDM yem uilize N c ubcaie and he k -h ubcaie anmi an n -D veco ignal ( k) = [ ( k) ( k)... n, whee n min (, ) n n and i ( k ) i he i -h daa eam. When ( k) i weighed by an ( n n ) pecoding maix F ( k), he eceived ignal veco ( k) = [ ( k) ( k)... n hough ubcaie k can be epeened a ( k) = H( k) F( k) ( k) + n ( k), k = 0,..., N c () whee n ( k) = [ n( k), n( k),..., nn i he n -D addiive whie Gauian noie (AWGN) veco, and H ( k) i he ( n n ) channel maix whoe enie ae independen idenically diibued (i.i.d.) complex zeo mean Gauian andom vaiable wih uni vaiance. Define he coelaion coefficien beween adjacen ubcaie by ρ = n Figue. Block diagam of an OFDM baed MIMO yem wih pecoding. { H( ) H ( ) } E k k E k E k { H( ) } { H( ) }. () In hi pape, we aume ha he CSI i pefecly eimaed a he eceive and epoed o he anmie hough a finie-ae feedback link. III. RANSMI BEAMFORMING WIH A SINGLE BEAM [] Fo eae of decipion, we biefly decibe he anmi beamfoming wih he ue of a ingle beam by exploiing he channel coelaion in he fequency domain []. n ( k) = h( k) f ( k) ( k) + n ( k) (3) whee h ( k) i he fi ow of H ( k) and f ( k) i he fi column of F ( k). Unle he numbe of anmi anenna i oo mall, he beam weigh of ubcaie k can be epeened in k a [] em of he beam weigh of ubcaie ( ) f( k) ρ f( k ) + ρ z (4) whee z i an andom veco ioopically diibued ove an n -D uni phee. In fac, he andom veco z indicae he diecion of f ( k) fom f ( k ). Since ρ, fo a given amoun of infomaion ize, he em z can be quanized wih a bee accuacy compaed o he quanizaion of f ( k). Moeove, ince he coelaion coefficien ρ i no fa vaying, he amoun of addiional infomaion fo he coelaion coefficien i maginal. he andom veco z can be quanized a [] ρ f( k ) + ρ z z = ag max h( k) z C ρ f( k ) + ρ z whee C denoe a codebook compiing N ( n ) uni nom veco. he eceive epo a log N -bi index of ẑ o he anmie. he anmie ecove he beam weigh fo ubcaie k a f ( k) = (5) ρ f( k ) + ρ z. (6) ρ f( k ) + ρ z IV. RANSMI BEAMFORMING FOR MIMO SYSEMS Conide he ue of muli-beam fo paial muliplexing gain. In he peence of channel coelaion beween he adjacen OFDM ubcaie, he channel maix of ubcaie k can be epeened in em of he channel maix of ubcaie k a ( ) H( k) = ρh( k ) + ρ E (7) whee E i an ( n n) maix whoe enie ae i.i.d. complex zeo mean Gauian andom vaiable wih uni vaiance and independen of he enie of H ( k ). Unle he numbe of anmi anenna i oo mall, i can be hown fom he law of lage numbe ha EE n I n []. hen, (7) can be ewien a

3 H( k) ρh( k ) + n ρ Z (8) whee Z = E / n. he eceive calculae Z and quanize i uing a pedefined codebook C' compiing N ( n n) maice whoe column ae ohonomal o each ohe. he maix Z can be quanized in vaiou way. Fo example, i can be quanized o maximize he capaciy a [5] ε ag max log Z = de In + F ( k) H ( k) H( k) F( k) (9) Z C' nn 0 whee ε i he oal anmi powe and F ( k) i defined by : n ( ρ ρ ) F ( k ) = R H ( k ) + n Z (0) whee R:L ( A ) denoe he fi L column of he igh ingula maix of A. I can alo be quanized o minimize he mean quae eo (MSE) a [5] ε ε agmin ( k) ( k) ( k) ( k) Z = I n + F H H F Z C' n nn0. () he eceive epo a log N -bi index of Ẑ o he anmie. he anmie eimae he CSI fom he feedback infomaion ρ and Ẑ a H( k) = ρh( k ) + n ρ Z () and deemine he pecoding maix F ( k) a : n ( ) F( k) = R H ( k). (3) hi cheme can be applied o he clueing cheme by mean of inepolaion. Aume ha K ubcaie ae combined ino a clue wih a epeenaive ubcaie mk, whee m=,..., Nc / K. Fom he eimaed channel maice { H ( mk) }, he channel maix fo ohe ubcaie can be eimaed a ( ) l l H mk + l = ( mk) H + H ( mk + K) (4) K K whee m i he clue index and 0 l K. Finally, he anmie deemine he pecoding maix by he fi n column of he igh ingula maix of H ( mk + l) (i.e, F( mk + l) = R: ( n H ( mk + l) ). Noe ha a in he ignal beam cae, he popoed cheme doe no equie any addiional infomaion fo he inepolaion (e.g., he phae infomaion in []). he codebook can be geneaed a [] and [5] W = ag max δ ( X) (5) X U ( N,n,n ) whee ABLE I. SIMULAION PARAMEERS Numbe of ubcaie ( N c ) 048 Subcaie pacing 48.8 khz Clue ize ( K ) 8 Channel model RMS delay pead Anenna configuaion δ ( X) = min d( F, F ). (6) k l N Exponenially decaying model Hee n d ( AB, ) = A B i he Euclidean diance beween maice A and B. F denoe he n -h eny of X and ( ) Auming he ue of wo diffeen codebook known o he anmie and eceive, he popoed algoihm can be ealized a follow. Fi, he feedback infomaion can be geneaed a he eceive in he following ep.. Quanize he channel maix coeponding o he hi ubcaie ino a b -bi codewod (o index) uing a codebook compiing N maice.. Uing anohe codebook compiing N maice, quanize he andom maix Z ino a b ( b b )-bi codewod (e.g., (9) o ()). Afe geneaing he feedback infomaion, he eceive epo i o he anmie hough a feedback ignaling link. hen, he anmie geneae he pecoding maice of each ubcaie uing hi feedback infomaion a follow.. Geneae he channel maix coeponding o he epeenaive ubcaie of each clue by ().. Geneae he beam weigh fo ohe ubcaie by inepolaion (e.g., by (4)). V. PERFORMANCE EVALUAION he pefomance of he popoed cheme i veified by compue imulaion when applied o a yem wih fou anmi anenna and wo eceive anenna (i.e., ( 4 ) MIMO configuaion). he imulaion condiion i moly fom [8] and ummaized in able. I i aumed ha he CSI i epoed o he anmie wihou eo and delay. Fig. depic he pefomance of he popoed cheme whee b and b ae quanized ino 6 bi, epecively. Fo compaion, he pefomance of he clueing and he k l 67n 4x Numbe of daa eam ( n ) Link adapaion Ideal (i.e., Shannon capaciy cuve

4 Ideal cae Popoed cheme Clueing cheme Inepolaion cheme 0 - Capaciy (bp/hz) Figue. Capaciy of ( 4 ) pecoding cheme. BER Ideal cae Popoed cheme Clueing cheme Inepolaion cheme (a) RMS delay pead = 67n Ideal cae Popoed cheme Clueing cheme Inepolaion cheme 0 - Capaciy (bp/hz) Feedback (bi) Figue 3. Capaciy aociaed wih he codebook ize (SNR = 8dB). BER Ideal cae Popoed cheme Clueing cheme Inepolaion cheme (b) RMS delay pead = 436n Figue 4. BER pefomance of pecoding cheme. inepolaion-baed uniay pecoding cheme i alo depiced [5], [6], whee he clueing cheme ue 6 bi fo he codebook and he inepolaion cheme ue 4 bi fo he codebook and bi fo he phae infomaion. Noe ha all he cheme excep he ideal cae ue he ame amoun of feedback ignaling fo pecoding. All cheme employ pecoding whoe coefficien i deemined o maximize he yem capaciy. I can be een ha he popoed cheme povide be capaciy pefomance while he inepolaion-baed cheme ha he wo pefomance due o he need of addiional phae infomaion, heefoe uing fewe bi fo he pecoding maix. Fig. 3 depic he capaciy of he popoed cheme aociaed wih he amoun of feedback ovehead, which epeen he amoun of feedback ignal equied fo he infomaion of each clue afe he iniializaion. he popoed cheme ue b = 7 bi codewod fo he iniializaion and he inepolaion cheme ue bi fo he phae infomaion. I can be een ha he popoed cheme can povide pefomance cloe o he ideal cae even wih he ue of 6 bi fo he beam weigh. I can alo be een ha he popoed cheme can educe he amoun of feedback ignaling fo he beam weigh o povide he ame pefomance a he ohe cheme. Fig. 4 depic he BER pefomance of he popoed cheme when he ignal i anmied uing QPSK modulaion wih a /-ae convoluional code. Fo compaion, he pefomance of he clueing and he inepolaion-baed uniay pecoding cheme i alo depiced [5], [6]. All cheme excep he ideal cae ue he ame amoun of feedback ignaling, 6 bi pe one clue, and employ

5 pecoding whoe coefficien i deemined o minimize he mean quae eo. I can be een ha he popoed cheme noiceably oupefom he convenional cheme and ha he inepolaion-baed cheme ha bee pefomance han he clueing cheme due o he diveiy gain hough he addiional phae infomaion. VI. CONCLUSIONS In hi pape, we have popoed a new feedback educion cheme in OFDM baed wiele yem. he popoed cheme can educe he amoun of feedback ignaling buden by geneaing he pecoding maix uing he channel infomaion on he peviou ubcaie and fequency channel coelaion. he imulaion eul how ha he popoed cheme can eimae he channel infomaion moe accuaely han convenional cheme fo a given amoun of feedback ignaling buden, impoving he yem capaciy o educing he eo pobabiliy. hi implie ha he popoed cheme can educe he amoun of feedback ignaling buden o povide he ame pefomance a convenional cheme. REFERENCES [] S.-H. Yang, J.-Y. Ko, and Y.-H. Lee, anmi beamfoming wih educed channel infomaion in OFDM baed wiele yem, in Poc. IEEE VC Sping, May 008, o appea. [] P. A. Dighe, R. K. Mallik, and S. S. Jamua, Analyi of anmieceive diveiy in Rayleigh fading, IEEE an. Commun., vol. 5, no. 4, pp , Ap [3] R. W. Heah J. and A. Paulaj, A imple cheme fo anmi diveiy uing paial channel feedback, in Poc. IEEE Ailoma Conf. Signal, Sy., Compu., vol., pp , Nov [4] S. hoen, L. Van de Pee, B. Gyelinckx, and M. Engel, Pefomance analyi of combined anmi-sc/eceive-mrc, IEEE an. Commun., vol. 49, pp. 5-8, Jan. 00. [5] D. J. Love and R. W. Heah J., Equal gain anmiion in muli-inpu muli-oupu wiele yem, IEEE an. Commun., vol. 5, no. 7, pp. 0 0, July 003. [6] K. K. Mukkavilli, A. Sabhawal, E. Ekip, and B. Aazhang, On beamfoming wih finie ae feedback in muliple-anenna yem, IEEE an. Inf. heoy, vol. 49, no. 0, pp , Oc [7] D. J. Love, R. W. Heah J., and. Sohme, Gamannian beamfoming fo muliple-inpu muliple-oupu wiele yem, IEEE an. Inf. heoy, vol. 49, no. 0, pp , Oc [8] K. K. Mukkavilli, A. Sabhawal, and B. Aazhang, Finie ae feedback deign fo muliple anmi anenna, in Poc. Alleon Conf. Communicaion, Conol, and Compue, Oc. 00. [9] C. K. Au-Yeung and D. J. Love, On he pefomance of andom veco quanizaion limied feedback beamfoming in a MISO yem, IEEE an. Wiele. Commun., vol. 6, no., pp , Feb [0] S. Zhou, B. Li, and P. Wille, Recuive and elli-baed feedback educion fo MIMO-OFDM wih ae-limied feedback, IEEE an. Wiele Commun., vol. 5, no., pp , Dec [] J. Choi and R. W. Heah, J., Inepolaion baed anmi beamfoming fo MIMO-OFDM wih limied feedback, IEEE an. Signal. Poce., vol. 53, no., pp , Nov [] B.H. Hochwald, L. Mazea, and V. aokh, Muliple-anenna channel hadening and i implicaion fo ae feedback and cheduling, IEEE an. Inf. heoy, vol. 50, no. 9, pp , Sep [3] R.W. Heah J., S. Sandhu, and A. Paulaj, Anenna elecion fo paial muliplexing yem wih linea eceive, IEEE Commun. Le., vol. 5, no. 4, pp. 4 44, Ap. 00. [4] R. W. Heah J. and A. Paulaj, Anenna elecion fo paial muliplexing yem baed on minimum eo ae, in Poc. IEEE In. Conf. Commun., vol. 7, pp , June 00. [5] D. J. Love and R. W. Heah J., Limied feedback uniay pecoding fo paial muliplexing yem, IEEE an. Inf. heoy, vol. 5, no. 8, pp , Aug [6] J. Choi and R. W. Heah, J., Inepolaion baed uniay pecoding fo paial muliplexing MIMO-OFDM wih limied feedback, IEEE an. Signal. Poce., vol. 54, no., pp , Aug [7] L. Collin, O. Bede, P. Roaing, and G. Buel, Opimal minimum diance-baed pecode fo MIMO paial muliplexing yem, IEEE an. Signal Poce., vol. 5, no. 3, pp , Ma [8] J. Moon, J.-Y. Ko, and Y.-H. Lee, A famewok deign fo he nexgeneaion adio acce yem, IEEE J. Sel. Aea Commun., vol. 4, no. 3, pp , Ma. 006.