Transmit Beamforming with Reduced Feedback Information in OFDM Based Wireless Systems
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1 Transmi Beamforming wih educed Feedback Informaion in OFDM Based Wireless Sysems Seung-Hyeon Yang, Jae-Yun Ko, and Yong-Hwan Lee School of Elecrical Engineering and INMC, Seoul Naional Universiy Kwanak P. O. Box 3, Seoul Korea Absrac This paper considers a ransmi beamforming scheme ha works wih reduced feedback informaion in OFDM based muliple-inpu single-oupu (MISO) wireless sysems. The proposed scheme generaes he beamforming weigh using informaion on he previous beamforming weighs and channel correlaion, significanly reducing he amoun of feedback signaling burden. The feedback signaling overhead can furher be reduced wih he use of clusering and inerpolaion echniques. Simulaion resuls show ha he proposed scheme ouperforms convenional beamforming echniques, while using he same amoun of feedback signaling overhead. I. INTODUCTION Muliple-inpu muliple-oupu (MIMO) sysems can considerably improve he performance of wireless communicaion sysems. They can easily be applied o orhogonal frequency division muliplexing (OFDM) schemes. The performance improvemen due o he use of muliple anennas is deeply associaed wih he accuracy of channel sae informaion (CSI). When he CSI is available a he ransmier, he sysem performance can significanly be improved by employing a ransmi beamforming echnique ha can provide a diversiy gain as well as array gain [] [3]. Opimum beamforming echniques, however, require accurae CSI a he ransmier, which may cause heavy feedback signaling overhead. To employ beamforming echniques in MIMO-OFDM sysems, he ransmier may require he CSI of each subcarrier, increasing he amoun of feedback signaling overhead in linear proporion o he number of subcarriers. This problem can be elevaed by using codebook based quanizaion mehods, where he receiver deermines he opimum beamforming vecor from a finie se of vecors, called codebook, and sends he index of he corresponding vecor o he ransmier using a feedback channel in he uplink [] [6]. The overhead can furher be reduced by exploiing he frequency correlaion [7], [8]. The size of he codebook can be reduced by recursively encoding he codeword wih he use of previous ones [7]. The amoun of feedback signaling informaion can furher be reduced by employing a so-called clusering echnique ha combines adjacen subcarriers ino a cluser and uses a same beamforming vecor for hese subcarriers [8]. Performance degradaion due o he use of a large cluser size can be alleviaed by inerpolaing he beamforming vecors in he frequency domain [8]. However, i may require addiional phase informaion. In his paper, we consider he reducion of he feedback signaling burden for he ransmi beamforming by exploiing he channel correlaion in he frequency domain. The beamforming vecor can be generaed in a sequenial manner using he correlaion characerisics beween he adjacen subcarriers. When adjacen subcarriers are correlaed o each oher, he beamforming vecor of a subcarrier can be represened in erms of ha of is adjacen subcarriers and a vecor corresponding o he difference beween he wo adjacen subcarriers. Since he vecor corresponding o he difference will have a span smaller han he original one, he resuling quanizaion error will be smaller han ha of he original one when he same amoun of feedback signaling is considered. Furher reducion can be achieved by using inerpolaion echniques wih he use of a phase-dependen codebook. This paper is organized as follows. Secion II describes he sysem model in consideraion. The proposed feedback reducion scheme is presened in Secion III and he performance is verified by compuer simulaion in Secion IV. Finally, Secion V concludes he paper. Noaion: Bold upper and lower leers denoe marices and vecors, respecively. () T and () denoe he ranspose and he conjugae ranspose, respecively. E{} denoes he expecaion operaor. U( N, m ) denoes a se of N ( m ) - dimensional vecors wih uni norm. x denoes he smalles ineger larger han or equal o x. II. SYSTEM MODEL Consider an OFDM based wireless sysem wih n ransmi anennas and a single receive anenna (i.e., muliple-inpu single-oupu (MISO) sysem) as illusraed in Fig.. We assume ha he sysem uilizes N c subcarriers where he index of each subcarrier is denoed by k. When a signal s( k ) is ransmied using an ( n )-dimensional beamforming weigh vecor w ( k), he received signal can be represened as rk ( ) h( k) w ( ksk ) ( ) + nk ( ), k 0,..., N c () where h ( k) is he ( n )-dimensional channel vecor whose eniies are independen and idenically disribued (i.i.d.) complex Gaussian wih zero mean and uni variance, and nk ( ) /08/$ IEEE 983
2 N c h n w( k ) Fig.. Block diagram of an OFDM based MISO sysem wih ransmi ρ w( k ) beamforming. is addiive whie Gaussian noise (AWGN). We assume ha he CSI is perfecly esimaed a he receiver and sen o he ransmier hrough a limied feedback channel wihou delay and error. III. POPOSED TANSMIT BEAMFOMING SCHEME Assuming ha he channel has correlaion in he frequency domain, he channel vecor of subcarrier k can be expressed k as in erms of he channel vecor of subcarrier ( ) h( k) ρh( k ) + ρ e () where he correlaion coefficien ρ is defined by h( k) h ( k ) ρ E h( k) h( k ) and e is an ( n ) -dimensional vecor whose enries are i.i.d. complex Gaussian wih zero mean and uni variance. Noe ha he enries of e are independen of hose of h ( k ). Since he opimum beamforming vecor can be deermined by [6] () can be rewrien as (3) ( k) h, () h( k) h( k ) e ρ w( k ) + h( k) h( k) where z e / e. Unless he number of ransmi anennas is oo small, i can be shown ha h( k ) / h( k) e / h ( k) (refer o Appendix B). Then, (5) can furher be simplified o (5) ρ w( k ) +. (6) Noe ha for a given beamforming vecor w ( k ), he Simulaion resuls show his assumpion is quie valid if he number of ransmi anennas is larger han 3. Fig.. epresenaion of w ( k) by w ( k ). beamforming vecor for subcarrier k can be represened in erms of he correlaion coefficien ρ and random vecor z. Since ρ, he effec of quanizaion error for z becomes smaller, improving he accuracy wih he use of he same amoun of feedback signaling for he beamforming weigh. Moreover, since he correlaion coefficien ρ is no fas varying, he amoun of addiional feedback signaling overhed for he correlaion coefficien may be marginal. The random vecor ẑ is quanized as ρ w( k ) + zˆ arg max h( k) z W ρ w( k ) + where W is he codebook whose N enries are ( n ) - dimensional uni norm vecors. I can be seen ha he beamforming vecor is normalized o uni norm. The receiver feeds back he log N -bi index of ẑ o he ransmier. Afer receiving he index, he ransmier recovers he beamforing vecor for subcarrier k as The codebook can be generaed as [6], [9] Here, δ ( X ) is defined as (7) ρ w( k ) + ˆ. (8) ρ w( k ) + ˆ W arg max δ ( X ). (9) X U( N, n ) δ ( X ) min xi x j (0) i j N where x n is n -h enry of X. Noe ha in (9), he codebook is generaed by maximizing he minimum Euclidean disance beween any pair of wo vecors in he codebook unlike convenional codebooks such as Grassmannian codebook [6]. 98
3 TABLE I SIMULATION CONDITION Number of subcarriers ( N ) 6 c Number of ransmi anennas ( n ) Number of receive anennas ( n r ) Cluser size ( K ) 8 MS delay spread 67 ns Link adapaion Ideal (i.e., using he Shannon s capaciy curve) Capaciy (bps/hz) wihou assumpion wih assumpion The performance can furher be improved by employing a frequency inerpolaion echnique. Assume ha K subcarriers are combined ino a cluser. The receiver can generae a beamforming weigh corresponding o he represenaive subcarrrier for each cluser (e.g., he cener subcarrier in each cluser). Then he beamforming weigh of oher subcarriers can be esimaed from hose of he represenaive subcarriers. For example, assuming ha he beamforming vecor for subcarrier mk, where m,,..., Nc / K, is repored from he receiver, he beamforming vecor of oher subcarriers can be esimaed by linear inerpolaion as l l w( mk ) + w( mk + K) K K wˆ ( mk + l) l l w( mk ) + w( mk + K) K K () where m is he cluser index and 0 l K. Noe ha he proposed scheme does no require any addiional informaion (e.g., he phase informaion in [8]). We consider he use of wo differen codebooks known a boh he ransmier and he receiver; one for he iniializaion and he ohe for he random vecor. The proposed algorihm can be realized as follows:. Iniializaion: Selec an iniial sae codebook for cener subcarrier of he firs cluser. The beamforming vecor corresponding o he iniial subcarrier is quanized ino a b -bi codeword using a codebook comprising N vecors generaed by a convenional codebook mehod.. Updae of he beamforming vecor corrseponding o he cener subcarrier of each cluser: Afer he iniializaion, he beamforming vecor corrseponding o he cener subcarrier of each cluser is calculaed according o (7). The codebook comprising N vecors is used o quanize he random vecor z ino a b -bi codeword. 3. Esimaion of he beamforming vecor for he res of subcarriers: The ransmier calculaes he beamforing vecor corresponding o he cener subcarrier of each cluser by (8) and hen he beamforming vecors of oher subcarriers by an inerpolaion mehod (e.g., ()). IV. PEFOMANCE EVALUATION The performance of he proposed scheme is verified by Capaciy (bps/hz) Fig. 3. Capaciy wih and wihou he assumpion in (5) and (6) Clusering scheme Inerpolaion scheme Fig.. Capaciy achieved by he proposed scheme. compuer simulaion. The simulaion condiion is summarized in Table I. The codebook can be found in Appendix A. Fig. 3 compares he capaciy of he proposed scheme wih and wihou he assumpion (5) and (6). The proposed scheme wihou he assumpion means ha he ransmier uses all he quanized random vecor, norm value of he channel and norm value of he random vecor by (5), whereas he one wih he assumpion means ha he ransmier only uses he quanized random vecor by (6). I can be seen ha performance degradaion due o he assumpion is neglegible. Fig. depics he performance of he proposed scheme in erms of he capaciy when b and b are quanized ino 6 bis, respecively. For comparison, we consider wo convenional schemes, clusering and inerpolaion-based scheme in [7] and [8]. The clusering scheme uses he beamforming vecor of he cener subcarrier as he represenaive vecor of he cluser and he inerpolaion-based 985
4 0 - TABLE A- CODEBOOK FO n AND N 8 (3 BITS) i i i i i i i i BE Clusering scheme Inerpolaion scheme TABLE A- CODEBOOK FO n AND N 8 (3 BITS) i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i Capaciy (bps/hz).5 Fig. 5. BE performance of beamforming echniques. Clusering scheme Inerpolaion scheme Feedback (bis) Fig. 6. Capaciy comparison of beamforming echniques. scheme addiionally uses phase roaion informaion o inerpolae he beamforming vecors. We assume ha he clusering scheme uses 6 bis for he codebook and he inerpolaion-based scheme uses bis for he codebook and bis for he phase informaion. For a reference, we also consider he ideal case corresponding o he case when he ransmier has perfec channel informaion of all subcarriers wih full precision. I can be seen ha he proposed scheme ouperforms convenional schemes and ha he inerpolaionbased scheme has he wors performance due o he use of a reduced size codebook for addiional phase informaion. Fig. 5 compares he BE peformance of he proposed scheme wih ha of he convenional schemes when QPSK modulaion is employed wih a /-rae convoluional channel code. The inerpolaion-based scheme provides beer BE performance han he clusering scheme since i can achieve a diversiy gain by using addiional phase informaion [8]. I can be seen ha he proposed scheme noiceably ouperforms he convenional schemes. Fig. 6 depics he capaciy of he proposed scheme associaed wih he amoun of he feedback signaling overhead, which represens he amoun of he bi size for he beam weigh afer he iniializaion. The proposed scheme uses b 8 bis codewords for he iniializaion. The inerpolaion scheme uses bis for he phase informaion (i.e., i uses ( b ) bis for he beam weigh). I can be seen ha he proposed scheme ouperforms oher schemes wih he use of he same bi size. This impies ha he proposed scheme needs less amoun of feedback o provide he same performance, compared o oher convenional schemes. V. CONCLUSIONS In his paper, we have proposed a new ransmi beamforming scheme in OFDM based wireless MISO sysems. The proposed scheme can reduce he amoun of feedback signaling burden for he beam weigh by generaing he beamforming vecor using informaion on he previous beamforming vecor, channel correlaion and quanized random vecor. The simulaion resuls show ha he proposed scheme can esimae he channel informaion more accuraely han convenional schemes using he same amoun of feedback signaling burden, improving he sysem performance. This implies ha he proposed scheme can reduce he amoun of feedback signaling burden o provide he same performance as convenional schemes. APPENDIX A. EXAMPLES OF CODEBOOKS Some examples of codebooks are given in Table A- and Table A-. 986
5 B. POOF OF THE NOM CONVEGENCE T Consider wo column vecors, a [ a,..., a n ] and T b [ b,..., b n ], whose enries are zero-mean complex Gaussian random variables wih uni variance. Leing a, he probabiliy densiy funcion (pdf) of can be expressed as [0] f ( ) e n Γ ( n ) (A-) where Γ ( ) denoes he Gamma funcion. The pdf of b is he same as ha of. Leing / a / b, i can be shown ha he pdf of is f ( ) x f ( x, x) dx 0 Γ ( ( n )), xf ( x) f ( x) dx + n n ( + ) x 0 x e dx Γ( n ) n n ( ). ( Γ( n )) (A-) Thus, i can be seen ha, if, lim f ( ) (A-3) n 0, oherwise. This implies ha a / b as n increases. EFEENCES [] T. K. Y. Lo, Maximum raio ransmission, IEEE Trans. Commun., vol. 7, no. 0, pp. 58 6, Oc [] P. A. Dighe,. K. Mallik and S. S. Jamuar, Analysis of ransmi-receive diversiy in ayleigh fading, IEEE Trans. Commun., vol. 5, no., pp , Apr [3]. W. Heah Jr. and A. Paulraj, A simple scheme for ransmi diversiy using parial channel feedback, in Proc. IEEE Asilomar Conf. Signals, Sys., Compu., vol., pp , Nov [] D. J. Love and. W. Heah Jr., Equal gain ransmission in muli-inpu muli-oupu wireless sysems, IEEE Trans. Commun., vol. 5, no. 7, pp. 0 0, July 003. [5] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, On beamforming wih finie rae feedback in muliple-anenna sysems, IEEE Trans. Inf. Theory, vol. 9, no. 0, pp , Oc [6] D. J. Love,. W. Heah Jr. and T. Srohmer, Grassmannian beamforming for muliple-inpu muliple-oupu wireless sysems, IEEE Trans. Inf. Theory, vol. 9, no. 0, pp , Oc [7] S. Zhou, B. Li and P. Wille, ecursive and rellis-based feedback reducion for MIMO-OFDM wih rae-limied feedback, IEEE Trans. Wireless Commun., vol. 5, no., pp , Dec [8] J. Choi and. W. Heah, Jr., Inerpolaion based ransmi beamforming for MIMO-OFDM wih limied feedback, in Proc. In. Conf. Commun., vol., pp. 9 53, June 00. [9] A. Gersho and. M. Gray, Vecor Quanizaion and Signal Compression, Kluwer Academic Publishers, 99. [0] B. H. Hochwald, L. Marzea and V. Tarokh, Muliple-anenna channel hardening and is implicaions for rae feedback and scheduling, IEEE Trans. Inf. Theory, vol. 50, no. 9, pp , Sep. 00. [] C. K. Au-Yeung and D. J. Love, On he performance of random vecor quanizaion limied feedback beamforming in a MISO sysem, IEEE Trans. Wireless Commun., vol. 6, no., pp. 58 6, Feb
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