Linköping University Post Print. Combining Long-Term and Low-Rate Short- Term Channel State Information over Correlated MIMO Channels

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1 Lnköpng Unversty Post Prnt Combnng Long-Term and Low-Rate Short- Term Channel State Informaton over Correlated MIMO Channels Tùng T. Km, Mats Bengtsson, Erk G. Larsson and Mkael Skoglund N.B.: When ctng ths work, cte the orgnal artcle. 009 IEEE. Personal use of ths materal s permtted. However, permsson to reprnt/republsh ths materal for advertsng or promotonal purposes or for creatng new collectve works for resale or redstrbuton to servers or lsts, or to reuse any copyrghted component of ths work n other works must be obtaned from the IEEE. Tùng T. Km, Mats Bengtsson, Erk G. Larsson and Mkael Skoglund, Combnng Long- Term and Low-Rate Short-Term Channel State Informaton over Correlated MIMO Channels, 008, IEEE Transactons on Wreless Communcatons, (7), 7, Postprnt avalable at: Lnköpng Unversty Electronc Press

2 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY Transactons Letters Combnng Long-Term and Low-Rate Short-Term Channel State Informaton over Correlated MIMO Channels Tùng T. Km, Mats Bengtsson, Erk G. Larsson, and Mkael Skoglund Abstract A smple structure to explot both long-term and partal short-term channel state nformaton at the transmtter (CSIT) over a famly of correlated multple-antenna channels s proposed. Partal short-term CSIT n the form of a weghtng matrx s combned wth a untary transformaton based on the long-term channel statstcs. The heavly quantzed feedback lnk s drectly optmzed to mze the expected achevable rate under dfferent power constrants, usng vector quantzaton and convex optmzaton technques on a sample channel dstrbuton. Robustness aganst errors n the feedback lnk s also pursued wth tools n channel optmzed vector quantzaton. Smulatons ndcate the benefts of the proposed scheme. Index Terms MIMO systems, fadng channels, nformaton rates, feedback communcaton, adaptve systems. I. INTRODUCTION THE use of multple antennas at the transmtter and the recever s a well recognzed technque to acheve hgh data rates n wreless communcatons. A multtude of dfferent transmsson technques have been proposed n the lterature, especally for the specal cases of full channel state nformaton at the transmtter (CSIT) and no CSIT, respectvely. At least when usng a small number of antennas, the throughput can be sgnfcantly mproved f CSIT s avalable. However, n practce ths ether requres carefully calbrated rado chans and duplex tmes lower than the channel coherence tme, f the channel recprocty s exploted n tme-dvson duplex systems, or that a sgnfcant bandwdth s allocated to feed back channel estmates from the recever. Ths has led to a great deal of nterest n low-rate feedback schemes, see for Manuscrpt receved November 6, 006; revsed June 3, 007; accepted August 3, 007. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was Y. Zheng. Ths work was supported n part by the Wreless World Intatve New Rado (WINNER) project n the European Unon Sxth Framework Programme (FP6) and by the Swedsh Research Councl (VR). The materal n ths paper was presented n part at the 006 IEEE Internatonal Conference on Acoustc, Speech, and Sgnal Processng, May 006. E. Larsson s a Royal Swedsh Academy of Scences Research Fellow supported by a grant from the Knut and Alce Wallenberg Foundaton. T. T. Km, M. Bengtsson, and M. Skoglund are wth the ACCESS Lnneaus Center, School of Electrcal Engneerng, Royal Insttute of Technology, SE-0044 Stockholm, Sweden (e-mal: {tung.km, mats.bengtsson, mkael.skoglund}@ee.kth.se). E. Larsson was wth Royal Insttute of Technology. He s now wth the Dept. of Electrcal Engneerng, Lnköpng Unversty (e-mal: erk.larsson@sy.lu.se). Dgtal Object Identfer 0.09/TWC /08$5.00 c 008 IEEE example [] [9], that can acheve a sgnfcant porton of the full-csit performance usng only a few bts of feedback for each fadng state. The fadng n wreless communcatons s generally governed by two components: A slowly-varyng component caused by, e.g., shadowng, and a short-tme varaton caused by multpath-fadng. Even f t s mpossble to obtan accurate short-term CSIT, the long-term channel characterstcs can often be estmated wth good accuracy. For fxed long-term channel statstcs, short-term feedback desgns to mze the ergodc capacty usng vector quantzaton technques are studed n []. The present work, by contrast, proposes a smple scheme that successfully combnes both long-term and quantzed short-term CSIT over a famly of multple-nput multple-output (MIMO) channels. The dea presented here s related to [4], [5] whch propose to combne a codebook based on Grassmanan lne packng wth nformaton from the channel covarance matrx. Our approach dffers fundamentally n that t uses a mutual nformaton crteron whereas [4], [5] mnmze a bound on the CSIT error and s lmted to beamformng. Our proposed transmsson scheme ncludes a untary transformaton nfluenced by the avalable knowledge of the channel statstcs. Such a transformaton can be motvated by the Karhunen-Loève transformaton n vector quantzaton [0], and also by ts optmalty n the absence of short-term feedback [] [3]. The short-term CSIT s exploted n the form of a weghtng matrx, whch s desgned usng a modfed verson of the Lloyd algorthm. Unlke n [] where approxmatons are requred, leadng to possble dvergence, we show that a major step n the desgn procedure can be cast as a varaton of the determnant mzaton problem [4], whch can be solved effcently. In contrast to [4] [6], our approach wll sometmes lead to spatal multplexng solutons. Smulaton results confrm the benefts of the proposed scheme. The results also ndcate that temporal power control yelds lttle extra gan over a system that only allocates power over spatal modes. Fnally, a robust desgn wth respect to errors n the feedback lnk s proposed under the framework of channel optmzed vector quantzaton. II. SYSTEM MODEL Consder the dscrete-tme complex-baseband equvalent model of a MIMO communcaton system wth N t transmt

3 40 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 008 x Encoder W(I) s U I s H I( H) H n y Decoder postve nteger. In other words, we consder a resolutonconstraned quantzer. For convenence, let W = W(), and Δ Δ Q = W W H, =,...,K. The system model () can then be wrtten n the form y = HUW x + n. () Fg.. System model. Condtoned on a feedback ndex I =, the average transmt power s antennas and N r receve antennas. The receved sgnal at tme nstant k of block l can be wrtten as y l (k) =H l (k)s l (k)+n l (k) () where H l (k) denotes the channel matrx and s l (k) s the transmtted vector. The components of the temporally and spatally whte nose n l (k) are complex Gaussan wth zero mean and unt varance. A block conssts of N consecutve channel uses, durng whch the vector vec[h l (k)] s assumed to be ndependent and dentcally dstrbuted (..d.) zeromean complex Gaussan wth covarance matrx R l,.e., vec[h l (k)] CN(0, R l ). Heren vec[ ] denotes vectorzaton. The channel covarance matrx R l, however, changes ndependently from one block to the next accordng to some statonary and ergodc dstrbuton. A transmtted codeword s assumed to span a sngle block. We study the system n the lmt of a very large block length N. Ths models a communcaton system where a codeword s suffcently long to capture the ergodcty of short-term changes, but stll short enough to experence a sngle R l. For readablty, we wll omt the block ndex l and the tme ndex k whenever ths s unambguous. Snce R s a slowlyvaryng parameter, we assume that R s perfectly known at both sdes of the lnk. Such nformaton may be obtaned from collected uplnk measurements or usng a low-rate feedback channel [5]. We further assume that H s fully known at the recever. For a system wth fxed long-term channel statstcs, a transmtter usng an..d. Gaussan codebook and a weghtng matrx whch depends only on short-term feedback nformaton s optmal n a capacty sense under certan assumptons [], [6]. However, over a famly of channels, ths would requre nfntely many quantzaton codebooks, one for each realzaton of R. We therefore propose a smple alternatve, llustrated n Fg.. The transmtter frst weghs the symbols x, taken from a Gaussan codebook, wth E[xx H ] = I Nt,byW(I), producng s. The notaton [ ] H denotes conjugate and transpose. Heren W s a mappng from a feedback ndex I to a fnte set of weghtng matrces. Such an ndex s obtaned va a noseless, zero-delay dedcated feedback lnk. The weghted sgnals s are then lnearly transformed by a untary matrx nfluenced only by long-term channel statstcs U U(R). To produce the feedback ndex, the recever employs an ndex mappng from the current effectve channel realzaton H = HU to an nteger I I( H). We assume that I takes a value n the set {,...,K} where K s a constant Takng the nose n the feedback lnk nto the desgn s also possble, as wll be demonstrated n Secton VI. Etr(ss H )=Etr( s s H )=Etr(W xx H W H )=trq, where tr X denotes the trace of a matrx X. We consder two dfferent types of power constrants. A short-term power constrant requres that the transmt power does not exceed P for any feedback ndex: tr Q I( H) P, H. (3) Ths models a system where temporal power control s not possble. Under the more relaxed long-term power constrant, the transmtter can vary the power over the transmsson of nfntely many codewords so that E R E H [tr Q I( H) R] P. (4) Note that the dstrbuton of H depends on the dstrbuton of R. Let I(R) denote the expected value of the mutual nformaton between the transmtted and receved sgnals, condtoned on R and for a fxed feedback scheme. We are nterested n the desgn of a feedback scheme that mzes the expected rate over nfntely many blocks,.e., I( H),{Q } E R I(R) s.t. (3) or (4). (5) The objectve functon n (5) can be nterpreted as the achevable rate by codng over a famly of nformaton stable channels, where each member of the famly s parameterzed by a covarance matrx R. In practce, the dstrbuton of R has to be known beforehand. However, as wll be shown n Secton IV, our proposed desgn approach does not requre the exact dstrbuton, but only an emprcal dstrbuton of R. III. DECORRELATING LINEAR TRANSFORMATION We propose to choose the untary transformaton U as the egenvectors of the transmt sde covarance matrx. As s shown below, ths wll decorrelate the channel coeffcents before the quantzaton. We emphasze the smplcty and ntutve appeal of such a decorrelaton, but do not clam ts optmalty, because unlke n [] [3], partal short-term CSIT s avalable n our model. For smplcty, we begn wth the case of a sngle receve antenna, where we use the notaton h = H H,andR Tx = R. Thus the receved sgnal can be wrtten as y = h H s + n wth h CN(0, R Tx ). Introduce the egendecomposton U Tx D Tx (U Tx ) H = R Tx wth untary U Tx and dagonal D Tx. Now f we choose U = U Tx,thenI(R) = E h R log( + h H Q h), where h = (U Tx ) H h s a vector of decorrelated varables,.e., h CN(0, D Tx ). Ths can be vewed as a property of the untary-ndependent-untary model [7]. Such

4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY Karhunen-Loève transformatons are commonly used n vector quantzaton [0]. The same deas can be appled to a MIMO channel, but t s n general mpossble to fnd a untary precodng matrx U that fully decorrelates the channel coeffcents. To proceed, we assume here that the second-order statstcs of the channel follows the so-called Kronecker model [8],.e., vec[h] CN(0, R), wherer =(R Tx ) T R Rx. Heren [ ] T denotes transpose and denotes the Kronecker product. Introduce the egendecomposton U Tx D Tx (U Tx ) H = R Tx and let us choose U = U Tx. Recallng that H = HU, wethenhave I(R) =E H R (I Nr + HQ H H), wth vec[ H] CN(0, D Tx R Rx ). IV. FEEDBACK LINK DESIGN A. Short-term Power Constrant The feedback lnk s desgned usng a modfed verson of the Lloyd algorthm [], [0]. However, nstead of usng an ad-hoc approxmaton that does not necessarly guarantee convergence as n [], we heren explot some results n determnant mzaton [4]. We frst dscretze the problem (5) and consder (I Nr + HQ I( H),{Q } I( H) H H ) s.t. (3) H H (6) where H s a set of samples drawn from the dstrbuton of H, whch s used to approxmate the contnuous dstrbuton of H [0]. The desgn procedure teratvely optmzes the ndex mappng I( H) and the weght codebook {Q } K =.Snce each optmzaton subproblem s solved exactly, the desgn guarantees convergence to a local optmum, but not necessarly to a global one. We summarze the two teraton steps as follows, where n ndcates the teraton ndex. Frst, gven a set {Q (n) satsfyng tr Q(n) P,, the } K = optmal ndex mappng s gven by I (n) ( H) = arg ( I Nr + (n) HQ H H). Next, fx I (n) ( H) and defne the quantzaton regons H (n) Δ = { H H: I (n) ( H) =}. The elements of the weght codebook can then be optmzed ndvdually: Q (n+) =arg (I Nr + HQ H H) Q 0 H (n) H H (n) s.t. tr Q P. Ths convex problem s a slghtly modfed verson of the standard determnant mzaton problem. In our numercal examples, we have for smplcty solved ths mzaton usng a drect generalzaton of the fxed-reducton algorthm n [4]. We have observed numercally that the objectve functon does not always ncrease n every teraton when usng []. B. Long-term Power Constrant The technque outlned n Secton IV-A can also be appled to the long-term power constrant case. The desgn however becomes more nvolved as we have to optmze the elements of the codebook {Q } K = jontly. Usng a sample dstrbuton H, we can reformulate the problem as (I Nr + HQ I( H),{Q 0} I( H) H H) H H (7) s.t. tr Q I( H) P. H H We wll teratvely solve the dual problem of (7). Gven a fxed {Q (n) } K = and a Lagrange multpler assocated wth the power constrant λ (n), we assume that a constrant qualfcaton holds so that the optmal ndex mappng solves I( H) { (I Nr + HQ (n) H H ) = H R Ths can be rewrtten as I( H) = H H λ (n) K = H ( (I Nr + HQ (n) tr Q(n) H H ) }. ) λ (n) tr Q (n). Note that the mzaton can also be seen as one performed over all possble ways of parttonng H nto K subsets H,...,H K. The soluton s readly gven by { ( I (n) ( H) = arg I Nr + HQ (n) H H) } λ (n) tr Q (n). In the ntal step, we can select Q (0) so that tr Q (0) = = tr Q (0) K, to remove the dependence of I(0) ( H) on λ (0). Next, gven the quantzaton regons H (n) Δ = { H H: I (n) ( H) =}, the optmal weght codebook {Q (n+) } s the soluton to {Q 0} s.t. = = H H (n) (I Nr + HQ H H) H (n) tr Q P. We solve also ths convex optmzaton usng a barrer method wth Newton steps. The optmal Lagrange multpler can be shown to be λ (n+) = tr H H X Q H tr Q = = tr H H K X K Q K H K tr Q K, where X X ( H) Δ = H H (I Nr + HQ H H ) H.

5 4 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 008 Expected Rate (bts/channel use) Full CSIT K=4 & Proposed U K= & Proposed U K=4 & ref. [4] K= & ref. [4] Long term CSIT only K=4 & U = I K= & U = I Expected Rate (bts/channel use) Full CSIT Long term CSIT only K= & Proposed U K=4 & Proposed U Fg.. Performance of the proposed scheme over a famly of 4 channels. A short-term power constrant s assumed. V. NUMERICAL RESULTS In ths secton we present smulaton results for a specfc class of correlated channels. For smplcty, the famly of R Tx s taken as Toepltz matrces wth [,ρ,ρ,...,ρ Nt ] as the top row, where ρ s a complex-valued random varable wth phase unformly dstrbuted n [0, π] and modulus dstrbuted as f( ρ ) =C exp ( λ( ρ )) f ρ < and f( ρ ) =0 otherwse, where λ > 0 and C = λ/( exp( λ)) s a normalzaton factor. The goal s to study the general behavors of the systems when the channels vary from hghly correlated to fully uncorrelated, rather than to smulate a specfc scenaro. All the smulatons are obtaned wth λ = log(0.0), modelng a famly of channels where ρ s greater than 0.5 more than 90 percent of the tme. A desgn for practcal applcatons may rely on more realstc channel models. Snce the nose varance s normalzed to unty, we defne the sgnalto-nose rato as SNR Δ = P. The feedback lnk s traned wth 00 realzatons of R and 000 channel realzatons for each R, usng 0 random startng ponts and 5 teratons n the Lloyd algorthm. Further ncreasng the number of channel realzatons does not seem to change the performance. The performance of the proposed scheme under a short-term power constrant over a famly of 4 channels s plotted n Fg.. For comparson, we tred a system that only uses shortterm feedback but gnores the correlaton propertes of the channel, usng exactly the same vector quantzaton technque except that U = I Nt. Clearly, the long-term nformaton provdes a consstent gan. For example, wth K =or one bt of feedback, the proposed approach provdes nearly a 3 db gan at an expected rate of 4 bts per channel use. We also plot the performance of the CSI quantzaton scheme proposed n [4], combned wth beamformng. Interestngly enough, ths scheme performs almost dentcally to our proposed scheme, even though t s not specfcally optmzed to mze the Fg. 3. Performance of the proposed scheme over a famly of 4 channels. A short-term power constrant s assumed. Expected Rate (bts/channel use) Full CSIT. Short term power constrant. Full CSIT. Long term power constrant. K= Short term power constrant. K= Long term power constrant Fg. 4. Performance of the proposed scheme over a famly of channels under dfferent power constrants. expected rate and always wll correspond to Q matrces of rank. 3 On the other hand, [4] explots some nformaton also from the egenvalues of R, not only the egenvectors. Fnally, for comparson, the fgure shows the performance of usng full CSIT (.e. beamformng) and of usng only long-term CSIT [] [3], [9], where the transmt covarance matrx s optmzed for each R,.e., for each fadng block, usng sample dstrbutons. At an expected rate of 4 bts per channel use, combnng bts of short-term CSIT (K =4) wth long-term statstcs yelds a gan of roughly db over usng only longterm statstcs. A smlar behavor, but wth less pronounced gans, can be seen n the 4 MIMO case, as llustrated n 3 For MISO channels, our scheme has always resulted n Q matrces of rank, but for MIMO most Q matrces were hgh rank.

6 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY Fg. 3. Other experments (not reported n the fgures) have shown that the resultng desgn s extremely nsenstve to the SNR and also to the assumed channel statstcal dstrbuton. Also, we have tred to explot knowledge of the egenvalues of R to select between dfferent code books, but the addtonal gan was extremely small. Fg. 4 compares the performance of the proposed scheme under dfferent power constrants over a famly of channels. The results ndcate that temporal power control provdes a neglgble gan for moderate and hgh SNRs, consstent wth some results under the assumpton of perfect CSIT [0]. At low SNR, however, a long-term power-constraned system outperforms a short-term one by a wde margn. For nstance, at SNR = 5 db, a long-term power controlled system usng one bt of feedback nformaton even outperforms a perfect- CSIT system wthout temporal power control. However, the valdty of the assumpton about perfect channel knowledge at the recever at so low SNR-values may be questoned. VI. ROBUST DESIGN TO ERRONEOUS FEEDBACK LINKS Nose and delays n the feedback lnk may lead to erroneous feedback,.e., the feedback ndex receved at the transmtter s not dentcal to the one sent by the recever. In ths secton, we demonstrate how such defects n the feedback lnk can be explctly taken nto account n the system desgn. We exclusvely focus on the short-term power constrant case. The desgn under a long-term power constrant problem can be handled under a smlar prncple, but does not necessarly gve any addtonal nsght nto the system behavor. The key tool n our robust desgn s channel optmzed vector quantzaton (COVQ) [7], []. To dstngush the feedback ndex from the one actually seen at the transmtter, let us denote J ( H) as the ndex mappng used by the recever that takes the erroneous feedback lnk nto account. Thus, upon knowng H, the recever sends back j = J ( H) {,...,K}, and the transmtter receves some ndex {,...,K}, potentally dfferent from j. We model the feedback lnk as a dscrete-nput dscrete-output memoryless channel wth transton probabltes p( j). In practce, the values of the transton probabltes may need to be estmated based on e.g., the SNR of the feedback lnk. Note that even f the ndex s not correctly receved, the effectve channel matrx s stll assumed to be perfectly tracked at the recever. That s, the errors n the feedback lnk only affect the weghtng matrx used at the transmtter. The desgn problem can be reformulated as J ( H), {Q } H = p( J ( H)) (I Nr + HQ H H) (8) where we agan approxmate the true dstrbuton of H wth a sample dstrbuton. An teratve procedure, whch essentally follows the methodologes n Secton IV wth some slght modfcatons, can be appled to the extended desgn problem (8). The teratve steps are descrbed n the followng, where we omt the teraton ndex n to mprove readablty. Gven the covarance matrces {Q } K =, the optmal ndex Expected Rate (bts/channel use) ε=0 ε=0. robust.5 ε=0. robust Long term only ε=0., non robust ε=0., non robust Fg. 5. Performance of the proposed schemes desgned takng nto account error probablty n the feedback lnk over a famly of 4 channels. The feedback resoluton s K =. mappng s gven by J ( H) = arg j {,...,K} = ( p( j)logdet I Nr + HQ H H). That s, the optmal ndex mappng also takes nto account the possbltes of dfferent outcomes of the random feedback lnk. Next, gven the ndex mappng J ( H), then for each value of, the optmal transmt covarance matrx Q solves the followng problem Q 0 p( j) (I Nr + HQ H H) j= H H j s.t. tr Q P, where we defne the quantzaton regon j as H j = { H H : J ( H) =j}. Clearly, ntroducng the weghtng factors p( j) does not change the concavty of the cost functon; thus the optmzaton can be solved numercally for the global optmum. We plot the performance of the robust desgn over a famly of 4 channels (generated as descrbed n Secton V) n Fg. 5. In ths example, the feedback lnk s modeled as a K-nput K-output memoryless channel wth transton probabltes p( j) = ɛ f = j and p( j) =ɛ/(k ) f j,.e., the error probablty n the feedback lnk s ɛ and the errors are unformly dstrbuted over all possble erroneous outcomes. A fner error model on the bt level can also be used. As can be seen, the robust desgn successfully takes nto account the errors n the feedback lnk and strctly mproves the performance compared to that obtaned wth only longterm statstcs, even f the error probablty n the feedback lnk s relatvely hgh (up to ɛ =0.). Ths of course comes at the prce of a hgher complexty n the desgn. The curves marked by crcles are the ones obtaned by drectly usng an

7 44 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 7, NO. 7, JULY 008 error-free codebook over a nosy feedback lnk. Notce that n ths case, not usng a robust codebook even gves worse performance than relyng on long-term statstcs only. Such a senstvty to error n the feedback lnk s somewhat reduced at hgher values of the feedback level K (not plotted heren for readablty). VII. CONCLUSION We have presented a smple transcever structure that successfully combnes short-term, fast feedback based on actual channel realzatons and long-term, slowly-varyng CSI contanng the second-order statstcs of the MIMO channel. Whle the proposed structure s not clamed to be optmal, we emphasze ts smplcty and versatlty, as well as ts excellent performance. We have also studed an mportant extenson from the basc setup, whch allows the desgn to take nto account errors n the feedback lnk. REFERENCES [] A. Narula, M. J. Lopez, M. D. Trott, and G. W. Wornell, Effcent use of sde nformaton n multple-antenna data transmsson over fadng channels, IEEE J. Select. Areas Commun., vol. 6, pp , Oct [] V. Lau, Y. Lu, and T.-A. Chen, On the desgn of MIMO block-fadng channels wth feedback-lnk capacty constrant, IEEE Trans. Commun., vol. 5, pp. 6 70, Jan [3] K. K. Mukkavll, A. Sabharwal, E. Erkp, and B. Aazhang, On beamformng wth fnte rate feedback n multple-antenna systems, IEEE Trans. Inform. Theory, vol. 49, pp , Oct [4] D. J. Love and R. W. Heath, Jr., Lmted feedback dversty technques for correlated channels, IEEE Trans. Veh. Technol., vol. 55, pp. 78 7, Mar [5] D. J. Love and R. W. Heath Jr, Grassmannan beamformng on correlated MIMO channels, n Proc. GLOBECOM, vol., pp. 06 0, 004. [6] D. J. Love, R. W. Heath, Jr., and T. Strohmer, Grassmannan beamformng for multple-nput multple-output wreless systems, IEEE Trans. Inform. Theory, vol. 49, pp , Oct [7] G. Jöngren and M. Skoglund, Quantzed feedback nformaton n orthogonal space-tme block codng, IEEE Trans. Inform. Theory, vol. 50, pp , Oct [8] A. D. Dabbagh and D. J. Love, Feedback rate-capacty loss tradeoff for lmted feedback MIMO systems, IEEE Trans. Inform. Theory, vol. 5, pp. 90 0, May 006. [9] J. C. Roh and B. D. Rao, Transmt beamformng n multple-antenna systems wth fnte rate feedback: a VQ-based approach, IEEE Trans. Inform. Theory, vol. 5, pp. 0, Mar [0] A. Gersho and R. M. Gray, Vector Quantzaton and Sgnal Compresson. Norwell, MA: Kluwer Academc, 99. [] E. Vsotsky and U. Madhow, Space-tme transmt precodng wth mperfect feedback, IEEE Trans. Inform. Theory, vol. 47, pp , Sept. 00. [] S. A. Jafar and A. Goldsmth, Transmtter optmzaton and optmalty of beamformng for multple antenna systems, IEEE Trans. Wreless Commun., vol. 3, no. 4, pp , July 004. [3] E. Jorsweck and H. Boche, Channel capacty and capacty-range of beamformng n MIMO wreless systems under correlated fadng wth covarance feedback, IEEE Trans. Wreless Commun., vol. 3, no. 5, pp , Sept [4] L. Vandenberghe, S. Boyd, and S.-P. Wu, Determnant mzaton wth lnear matrx nequalty constrants, SIAM J. Matrx Anal. Appl., vol. 9, no., pp , 998. [5] M. Bengtsson and B. Ottersten, Optmal and suboptmal transmt beamformng, n Handbook of Antennas n Wreless Communcatons, L. C. Godara, Ed. CRC Press, Aug. 00, ch. 8. [6] M. Skoglund and G. Jöngren, On the capacty of a multple-antenna communcaton lnk wth channel sde nformaton, IEEE J. Select. Areas Commun., vol., pp , Apr [7] A. M. Tulno, A. Lozano, and S. Verdú, Impact of antenna correlaton on the capacty of multantenna channels, IEEE Trans. Inform. Theory, vol. 5, no. 7, pp , July 005. [8] J. P. Kermoal, L. Schumacher, K. I. Pedersen, P. E. Mogensen, and F. Frederksen, A stochastc MIMO rado channel model wth expermental valdaton, IEEE J. Select. Areas Commun., vol. 0, no. 6, pp. 6, Aug. 00. [9] S. Zhou and G. B. Gannaks, Optmal transmtter egen-beamformng and space-tme block codng based on channel mean feedback, IEEE Trans. Sgnal Processng, vol. 50, no. 0, pp , Oct. 00. [0] A. J. Goldsmth and P. P. Varaya, Capacty of fadng channels wth channel sde nformaton, IEEE Trans. Inform. Theory, vol. 43, pp , Nov [] N. Farvardn, A study of vector quantzaton for nosy channels, IEEE Trans. Inform. Theory, vol. 36, no. 4, pp , July 990.