Unquantized and Uncoded Channel State Information Feedback on Wireless Channels

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1 Unquantizd and Uncodd Channl Stat Information Fdback on Wirlss Channls Dragan Samardzija Bll Labs, Lucnt Tchnologis 791 Holmdl-Kyport Road, Holmdl, J 07733, USA dragan@bll-labscom arayan Mandayam WILAB, Rutgrs Univrsity 73 Brtt Road, Piscataway, J 08854, USA narayan@winlabrutgrsdu Abstract W propos a channl stat information CSI) fdback schm basd on unquantizd and uncodd UQ-UC) transmission W considr a systm whr a mobil trminal obtains th downlink CSI and fds it back to th bas station using an uplink fdback channl If th downlink channl is an indpndnt Rayligh fading channl, thn th CSI may b viwd as an output of a complx indpndnt idntically distributd Gaussian sourc Furthr, if th uplink fdback channl is AWG, it can b shown that that UQ-UC CSI transmission that incurs zro dlay) is optimal in that it achivs th sam minimum man squard rror MMSE) distortion as a schm that optimally in th Shannon sns) quantizs and ncods th CSI whil thortically incurring infinit dlay Sinc th UQ-UC transmission is suboptimal on corrlatd wirlss channls, w propos a simpl linar CSI fdback rcivr that can b usd to improv th prformanc of UQ-UC transmission whil still rtaining th attractiv zro-dlay fatur W provid bounds on th prformanc of th UQ-UC CSI fdback Furthrmor, w xplor its application and prformanc in mtipl antnna mtiusr wirlss systms I ITRODUCTIO Th trmndous capacity gains du to transmittr optimization in mtipl antnna mtiusr wirlss systms [1] [5] rly havily on th availability of th channl stat information CSI) at th transmittr In such scnarios, asid from th issu of how to stimat th CSI, anothr intrsting qustion is how to transmit or fdback) th CSI? A fundamntal qustion that ariss is that, is it ncssary for rliabl CSI fdback to follow th principls outlind by th digital dogma? In othr words, is it ncssary that th CSI b optimally quantizd and ncodd in a Shannon thortic sns) for it to b rliabl? Ar thr ways to mitigat th dlay which is thortically infinit) that is imposd by such a Shannon thortic approach? In this papr w considr a systm whr a mobil trminal obtains th downlink CSI and fds it back to th bas station using an uplink fdback channl If th downlink channl is an indpndnt Rayligh fading channl, thn th CSI may b viwd as an output of a complx indpndnt idntically distributd iid) Gaussian sourc Furthr, if th uplink fdback channl is AWG and th downlink CSI is prfctly known at th mobil trminal, it can b shown that unquantizd and uncodd UQ-UC) CSI transmission that incurs zro dlay) is optimal in that it achivs th sam minimum man squard rror MMSE) distortion as a This work is supportd in part by th ational Scinc Foundation undr Grant o FMF schm that optimally in th Shannon sns) quantizs and ncods th CSI whil incurring infinit dlay Rsts on th optimality of unquantizd and uncodd transmission hav also bn discussd in othr contxts in [6] [8] Sinc th UQ-UC transmission is suboptimal on corrlatd wirlss channls, w propos a simpl linar CSI fdback rcivr that can b usd in conjunction with th UQ-UC transmission whil still rtaining th attractiv zro-dlay fatur Furthrmor, w dscrib an auto rgrssiv AR) corrlatd channl modl and prsnt th corrsponding prformanc bounds for th UQ-UC CSI fdback schm W also xplor th prformanc limits of such schms in th contxt of achivabl information rats in mtipl antnna mtiusr wirlss systms II BACKGROUD Considr th communication systm in Figur 1 Th systm is usd for transmission of unquantizd and uncodd outputs i, symbols) of th sourc Th sourc is complx, continuous in amplitud and discrt in tim with th symbol priod T sym ) W assum that th symbols x ar zro-man with unit varianc Th avrag transmit powr is P, whil th channl introducs additiv zro-man nois n with varianc Fig 1 Unquantizd and uncodd transmission that achivs th MMSE distortion of th transmittd signal At th rcivr, th rcivd signal y is mtiplid by th conjugat of g Consquntly, th signal ˆx at th dstination is ) ˆx = g y = g Px+ n 1) and ˆx is an stimat of th transmittd symbol x W slct th cofficint g to minimiz th man squard rror MSE) btwn ˆx and x Thus, ) g = arg min E ˆx x 2 =arg f min E f Px+ n x 2 2) Consquntly, P g = 3) P +

2 and th corrsponding man squard rror is min E ˆx x 2 1 = 1+ P 4) Th MSE corrsponds to a masur of distortion btwn th sourc symbols and stimats at th dstination Lt us now rlat th abov rsts to th transmission schm that applis optimal quantization and channl coding Basd on th Shannon rat distortion thory [9], for a givn distortion D, th avrag numbr of bits pr symbol at th output of th optimal quantizr is R = log ) D D 5) ot that th optimal quantizr that achivs th abov rat incurs infinit quantization dlay For th AWG channl, th maximum transmission rat is C = log 2 1+ P ) 6) b applid in th contxt of a CDMA systm Th functional blocks of th mobil trminal in a CDMA systm ar dpictd in Figur 2 Using a pilot-assistd stimation schm, th mobil trminal obtains an stimat of th downlink channl h dl, dnotd as h dl Th downlink channl stimat h dl is th CSI to b transmittd on th uplink channl h Th stimat h dl modats i, mtiplis) a Walsh cod that is spcifically allocatd as a CSI fdback carrir as shown in Figur 2 Th scond Walsh cod is allocatd for th convntional uplink data transmission For gnrality, th uplink pilot is also transmittd allowing th bas station to obtain an stimat h of th uplink channl h As in th cas of th optimal quantizr, th optimal channl coding wod incur infinit coding dlay Furthrmor, optimal matching in th Shannon sns) of th quantizr and th channl rquirs that R = C D =2 C 1 = 1+ P 7) Th abov distortion is qual to th MSE for th UQ-UC transmission schm givn in 4) s also [7]) Th abov rst points to th optimality of th UQ-UC schm whil it incurs zro dlay) whn th sourc is iid Gaussian and th channl is AWG III UQ-UC CSI FEEDBACK Using th abov rst, w now motivat why UQ-UC transmission schms can b usd for CSI fdback in wirlss systms Considr a communication systm that consists of a bas station transmitting data ovr a downlink channl A mobil trminal rcivs th data, and transmits th CSI of th downlink channl stat h dl ovr an uplink channl Lt us assum that th mobil trminal stimats th downlink CSI h dl prfctly If th downlink channl is iid Rayligh, thn th CSI is an iid complx Gaussian random variabl In this cas, if th uplink channl is AWG and it is indpndnt of th downlink channl, thn it follows dirctly from th arlir discussion that th abov UQ-UC schm is optimal for transmission of th downlink CSI ovr th uplink channl In othr words, UQ-UC transmission with zro dlay) of th downlink CSI will achiv th sam distortion as a schm that optimally in th Shannon sns) quantizs and ncods th CSI whil incurring infinit dlay To furthr distinguish th fact that th UQ-UC CSI fdback transmission dos not imply an analog communication 1 systm, w now illustrat an xampl of how such a schm cod 1 Whil w us th trm unquantizd UQ) in th UQ-UC nomnclatur, it must b pointd out that any practical transmission schm will rquir at last som lvl of coars quantization Fig 2 CDMA mobil trminal that applis th UQ-UC CSI fdback In gnral, th downlink and uplink channl stimation is not prfct, i, h dl = h dl + dl and h = h +, whr dl and ar th channl stat stimation rrors on th downlink and th uplink, rspctivly Th stimation rrors ar modld as AWG, which is typical to pilot-assistd channl stat stimation schms s [10] and th rfrncs thrin) Consquntly, th downlink and uplink stimation rrors ar distributd as C 0,dl ) and C0, ), rspctivly, whr C 0,σ 2 ) dnots a complx zro-man Gaussian random variabl distribution with th varianc σ 2 Considr a signal/systm modl, whr at th tim instant i, th uplink rcivd signal corrsponding to th CSI fdback is yi) =h i) P csi h dl i)+ni) 8) whr h i) is th uplink channl stat, P csi is th CSI fdback transmit powr, h dl i) is th stimat of th downlink channl h dl that is bing fd back and ni) is th AWG on th uplink with th varianc Using th rcivd signal in 8) and an stimat of h i), th CSI fdback rcivr at th bas station will stimat th transmittd CSI h dl i) Inth following drivations w assum that th uplink and downlink

3 channl stats ar mutually indpndnt and corrspond to zro-man and unit-varianc complx Gaussian distribution C 0, 1) Using th sam approach as givn in Sction II, th uplink CSI fdbck rcivr w is drivd from th following optimization w =arg v min E hdl i), yi) h i) v yi) h dl i) 2 = s u 9) whr u = E yi) h i) [yi) yi) ]= = P csi 1 + dl) + h ) i) dl) + 10) 1 + )2 Th abov rst is basd on th fact that th conditional distribution ph i) h )i)) is a complx Gaussian distribution h i) C, and h dl i) is indpndnt of h i) Furthrmor, h s =E hdl i), yi) h i) [h dl i) yi)] = P csi i) 11) Th uplink rcivr thn stimats th downlink CSI h dl i) as with th MSE distortion bing ĥ dl i) =w yi) 12) E hdl i), yi) h i) w yi) h dl i) 2 =1 ss u = = P csi P csi dl ) + dl ) h i) 2 +1 )2 13) ) dl ) + dl h i) 2 +1 )2 ot that as th stimation rrors approach zro, dl 0 and 0, th rcivr in 9) is idntical to th rcivr in 3) IV UQ-UC CSI FEEDBACK O CORRELATED CHAELS Th MSE distortion achivd by th UQ-UC CSI fdback transmission schm is optimal whn th downlink is iid Rayligh and th uplink is AWG, and furthr, th uplink and th downlink ar also mutually indpndnt with prfct channl stimation of h dl and h In rality, thr may th following situations that aris in wirlss systms: 1) tmporal corrlations in th downlink channl, 2) tmporal corrlations in th uplink channl, and 3) corrlations btwn th uplink and th downlink channls as is in TDD systms) In ach of ths cass, it is of intrst to quantify th MSE distortion achivd by th UQ-UC CSI fdback Sinc, an xact analysis is not radily tractabl, w propos to quantify such prformanc through uppr and lowr bounds in ach of th abov scnarios A Prformanc Bounds Lt us assum that th uplink and downlink channl stats ar indpndnt which is typical in FDD wirlss systms) Both th uplink and downlink channls ar varying in tim and ar assumd to b rgodic If th schm shown in Figur 1 is now applid on th CSI fdback channl, using th rst in 13), it follows that th MSE is ) dl ) MSE ub + dl h )2 P csi uq uc ) =E h dl ) + dl h 2 )2 + 0 P csi 14) Clarly this srvs as an uppr bound on th MSE achivd by any additional procssing that accounts for both th downlink and th uplink CSI fdback channl bing corrlatd channls To illustrat an approach to driv a lowr bound, considr an Lth ordr auto rgrssiv AR) procss modl for th downlink channl as h dl i) = c j h dl i j)+c 0 n dl i), 15) whr n dl i) is a complx Gaussian random variabl with distribution C 0, 1) Th cofficints c j j = 0,,L) dtrmin th corrlation proprtis of th channl n dl i) is th innovation squnc that dscribs th volution to succssiv channl stats This is a quasi-static block-fading channl modl whr th tmporal variations of th channl ar charactrizd by th corrlation btwn succssiv channl blocks Th abov modl givs a gnral framwork for dscribing th corrlations in th downlink channl stats through th cofficints c j j =0,,L) Using an approach outlind in [11], [12] and Appndix, it is possibl to approximat th wll known Jaks corrlatd fading modl by rlating paramtrs such as carrir frquncy and mobil spd to th AR modl cofficints Th Jaks modl corrsponds to a continuous tim-varying channl, whil th AR modl to a quasi-static block-fading channl To connct ths two modls, w assum that th channl is constant for a duration of τ sconds i, this duration may b viwd as th channl cohrnc tim) and τ is th absolut tim diffrnc btwn succssiv channl stats h dl i) and h dl i 1) Furthrmor, th corrlation E[h dl i)h dl i k) ]= J 0 2πf d kτ) whr f d is th maximum Dopplr frquncy s Appndix) For a mor dtaild analysis of auto rgrssivmoving avrag ARMA) procsss and wirlss channl modling w rfr th radr to [13], [14] and th rfrncs thrin Lt us assum that th abov modl and th prvious channl stats h dl i j) j = 1,,L) ar known at th CSI fdback transmittr and rcivr In addition, in driving th lowr bound, w will assum that th stimation rrors dl =0and =0i, prfct channl stat stimation) In this idalizd cas, having only th innovation n dl i) transmittd ovr th uplink CSI fdback channl, th rcivr can stimat th channl stat h dl i) W will now us argumnts similar to that usd in driving 7) to arriv at a lowr bound for th MSE of th UQ-UC schm Considr th distortion of th innovation squnc D in =E ˆn dl i) n dl i) 2, 16)

4 whr ˆn dl i) is an stimat of n dl i) Thn th avrag numbr of bits pr symbol at th output of th optimal quantizr is R in = log ) Din D in 17) Furthrmor, th rgodic capacity of th uplink channl is [ C =E h log 2 1+ h 2 P csi )] 18) Thn th optimal matching in th Shannon sns) of th quantization and channl coding of th innovation n dl i) rsts in R in = C 19) Hnc th MSE D in =E ˆn dl i) n dl i) 2 =2 C 20) Thus from quations 15) and 20) it follows that th MSE of h dl i) is lowr boundd as E ĥdli) h dl i) 2 c 2 02 C 21) Sinc this bound is obtaind using idalizd knowldg of th prvious channl stats and also a channl coding schm that achivs th rgodic capacity of th uplink channl, w xpct it to b loos B Fdback Rcivrs for Enhancing UQ-UC CSI Fdback Schms Whil th prvious subsction considrd th prformanc limits of th MSE distortion achivd by th UQ-UC CSI fdback transmission, in this subsction w will outlin signal procssing tchniqus that cod b usd to improv th prformanc of UQ-UC schms Th spcific approach that w propos is to dsign rcivrs on th CSI fdback channl that can xploit th channl corrlations and thus improv th prformanc in cass whr th UQ-UC CSI fdback transmission is suboptimal W illustrat such an approach through a dsign of a linar CSI fdback rcivr in th following Th uplink rcivd signal in 8) is usd to form a tmporal K-dimnsional rcivd vctor as yi) =[yi) yi 1) yi K + 1)] T 22) Th uplink rcivr thn stimats th downlink CSI h dl i) as ĥ dl i) =w H yi) 23) whr w is a linar filtr that is drivd from th following MMSE optimization w =arg v min E v H yi) h dl i) 2 24) For th givn stimats of th uplink channl h i) = [ h i) h i 1) h i K + 1)] T w dfin th following matrix [ U =E yi) h i) yi) yi) H ] 25) and th vctor s =E hdl i), yi) h i) [h dli) yi)] 26) It can b shown that th linar MMSE CSI fdback rcivr w is givn as w = U 1 s 27) As is vidnt from th quations 25)-27), th linar transformation w taks into account implicitly th following corrlations: 1) tmporal corrlations in th downlink channl, 2) tmporal corrlations in th uplink channl and 3) th corrlations btwn th uplink and th downlink In fact, whn K =1and th uplink and th downlink ar mutually indpndnt, thn th abov rcivr will achiv th MSE distortion uppr bound in quation 14) In all othr cass, th prformanc will b suprior, thrby nhancing th prformanc of th UQ-UC CSI fdback transmission C umrical Rsts: Distortion Prformanc W now prsnt th uppr and lowr bounds drivd in th prvious sctions for diffrnt scnarios corrsponding to th uplink and downlink CSI Spcifically w tak into account th ffct of background nois lvls, stimation rrors and channl corrlation W charactriz th quality of th uplink CSI fdback channl through its SR givn as SR csi = 10 log P csi 28) In ordr to quantify th ffct of th stimation rrors on th UQ-QC schm, w procd in th following way Rcall that th uplink channl stimat is givn as h = h + W quantify th stimation prformanc by th following SR trm SR = 10 log 1, 29) whr is th varianc of Th corrsponding quantity that is usd to charactrizd th downlink channl stimation rror is SRdl = 10 log 1 dl 30) First w considr a cas whn th uplink and downlink channls ar mutually indpndnt Th channls corrspond to th iid Rayligh block-fading modl i, for vry tim instant indpndnt channl stats ar instantiatd for th uplink and downlink) In Figur 3 w st SR csi =20dB and prsnt th MSE bounds as functions of SR and/or SRdl W compar th curvs corrsponding to th prfct downlink stimation SRdl =+ ) and variabl SR vrsus th prfct uplink stimation SR = + ) and variabl SRdl i, th curv with markr vrsus ) From ths rsts w not that th MSE uppr bound is mor affctd by th rrors in th uplink than th downlink channl stat stimation In this particar xampl, for th stimation SRs xcding 25 db, th incras in th distortion du to th imprfct knowldg of th channl stats is ngligibl, as vidncd by th flattning of th MSE uppr bound W now invstigat th MSE distortion for corrlatd channls Th downlink and uplink channls ar modld as an AR procss L =10) whos cofficints ar chosn to corrspond

5 0 2 4 X = SR = SRdl X = SR, SRdl=+ X = SR dl, SR=+ MSE lowr bound MSE bound [db] MSE uppr bound MSE [db] X [db] UQ UC uppr bound 13 UQ UC linar rcivr, ρ = 0 UQ UC linar rcivr, ρ = 05 UQ UC linar rcivr, ρ = v [kmph] Fig 3 MSE bounds vs SR and/or SR dl,foriid Rayligh blockfading on th uplink and downlink and SR csi =20dB Fig 4 db MSE vs mobil trminal vlocitis, f c =2GHz and SR csi =10 to th Jaks modl for a carrir frquncy of 2 GHz and th cohrnc tim τ = 2 msc i, duration of on channl block) Th corrlation btwn th uplink and downlink channl is quantifid as ρ =E[h dl i) h i) ] 31) whr th cofficint ρ 1 In addition, th uplink has an avrag SR csi =10dB and th stimation is prfct SR = + and SRdl = + ) In Figur 4 w show th MSE of th UQ-UC schm with th linar CSI fdback rcivr and th MSE uppr bound for diffrnt mobil trminal vlocitis Ths rsts show that th linar rcivr in combination with th UQ-UC transmission is abl to xploit th channl corrlations and improv th prformanc ot that whn th mobil trminal vlocitis ar low th improvmnt is gratr bcaus th succssiv channl stats ar mor corrlatd which is xploitd by th linar CSI fdback rcivr) Also, th improvmnt is gratr whn th uplink and downlink channls ar mutually corrlatd i, for ρ =09), as may b th cas in TDD systms V UQ-UC CSI FEEDBACK FOR TRASMITTER OPTIMIZATIO I MULTIPLE ATEA MULTIUSER SYSTEMS Th discussion thus far has focusd on prformanc limits and nhancmnts from th point of viw of th MSE distortion achivd du to th UQ-UC CSI fdback transmission A mor dirct prformanc issu that nds to b considrd is th ovrall capacity of a systm that actually uss th CSI fdback information W will considr th UQ- UC CSI fdback in a mtipl antnna mtiusr systm As an xampl, considr th systm shown in Figur 5, whr thr ar M transmit antnnas at th bas station and singl-antnna mobil trminals In th abov modl, x n is th information baring signal intndd for mobil trminal n and y n is th rcivd signal at th corrsponding trminal for n =1,,) Th rcivd vctor y =[y 1,,y ] T is y = HSx + n, y C, x C, n C, S C M, H C M 32) whr x =[x 1,,x ] T is th transmittd vctor E[xx H ]= P dl I ), n is AWG E[nn H ] = I ), H is th MIMO channl stat matrix, and S is a transformation spatial pr-filtring) prformd at th transmittr ot that th vctors x and y hav th sam dimnsionality Furthr, h nm is th nth row and mth column lmnt of th matrix H corrsponding to a channl btwn mobil trminal n and transmit antnna m Fig 5 Systm modl consisting of M transmit antnnas and mobil trminals Application of th spatial pr-filtring rsts in th composit MIMO channl G givn as G = HS, G C 33) whr g nm is th nth row and mth column lmnt of th composit MIMO channl stat matrix G Th signal rcivd at th nth mobil trminal is y n = g nn x n }{{} Dsird signal for usr n + i=1,i n } {{ } Intrfrnc g ni x i + n n 34)

6 In th abov rprsntation, th intrfrnc is th signal that is intndd for othr mobil trminals than trminal n As said arlir, th matrix S is a spatial pr-filtr at th transmittr It is dtrmind basd on optimization critria that w addrss latr in th txt and has to satisfy th following constraint trac SS H) 35) which kps th avrag transmit powr consrvd W rprsnt th matrix S as S = AP, A C M, P C 36) whr A is a linar transformation and P is a diagonal matrix P is dtrmind such that th transmit powr rmains consrvd W study th zro-forcing ZF) spatial pr-filtring schm whr A is rprsntd by A = H H HH H ) 1 37) As can b sn, th abov linar transformation is zroing th intrfrnc btwn th signals ddicatd to diffrnt mobil trminals, i, HA = I Thx n sarassumdtob circarly symmtric complx random variabls ach having Gaussian distribution C 0,P dl ) Consquntly, th maximum achivabl data rat capacity) for mobil trminal n is Rn ZF = log 2 1+ P dl p nn 2 ) 38) whr p nn is th nth diagonal lmnt of th matrix P dfind in 36) In this study w apply a suboptimal, yt a simpl solution P = trac AA H ) I 39) that guarants th constraint in 35) To prform th abov spatial pr-filtring, th bas station obtains CSI corrsponding to ach downlink channl stat h nm Th CSI is obtaind from ach mobil trminal using th UQ-UC CSI fdback In othr words, at tim instant i, trminal n n =1,, ) is transmitting th corrsponding CSI h nm i) m = 1,,M) via th uplink CSI fdback channl Rlating to th analysis in th prvious sctions, ach h nm i) corrsponds to a diffrnt h dl i) Instad of th idal channl stat h nm i), th spatial pr-filtr applis th stimat ĥnmi) obtaind from th uplink CSI fdback rcivr Thrfor at th bas station instad of th tru H, in th xprssions 37) and 39), Ĥ is applid whos ntris ar ĥnmi) m =1, M and n =1,,) Consquntly, th maximum achivabl data rat for mobil trminal n is ) ˆR n ZF P dl ĝ nn = log P dl i=1,i n ĝ 40) ni 2 + whr ĝ nm is th nth row and mth column lmnt of th composit MIMO channl stat matrix G = HˆP 41) with  = Ĥ H ĤĤ H ) 1 and ˆP = )I 42) trac  H ot that  ˆP forms a spatial pr-filtr It is mismatchd bcaus it applis Ĥ instad of th tru H In Figur 6 w prsnt downlink sum data rats whr SR dl =10dB, and M =3and =3Thratsar prsntd as functions of th mobil trminal vlocity using th approximat Jaks modl for a carrir frquncy 2 GHz and th cohrnc tim τ =2msc and spatially uncorrlatd channls Th uplink CSI fdback channl is with th avrag =10dB, and it is indpndnt of th downlink In addition, w prsnt th rats for instantanous idal channl knowldg and a dlayd idal channl knowldg 2 msc dlay) which may corrspond to a practical fdback schm that quantizs and ncods th CSI For xampl, in 3G WCDMA HSDPA systm 2 msc corrsponds to th duration of a radio packt which may b usd to transmit quantizd and ncodd CSI, incurring th minimum dlay of 2 msc W not that undr th UQ-UC CSI fdback with th linar rcivr, th prformanc is bttr for channls with highr corrlations i, lowr mobil trminal vlocitis) For th modrat and highr vlocitis, th UQ-UC CSI fdback schm is outprforming th cas of th dlayd idal channl knowldg ot that in th abov xampl w assum that th stimation is prfct SR =+ and SR dl =+ ) SR csi Sum rat [bits/symbol] Instantanous idal channl knowldg 1 UQ UC uppr bound UQ UC linar rcivr UQ UC lowr bound Dlayd idal channl knowldg 2 msc) v [kmph] Fig 6 Avrag downlink sum data rat vs mobil trminal vlocity, f c = 2 GHz, M = 3, = 3, spatially uncorrlatd, SR dl = 10 db and SR csi =10dB VI COCLUSIO In this papr w hav considrd a systm whr a mobil trminal obtains th downlink CSI and fds it back to th bas station using an uplink fdback channl If th downlink channl is an indpndnt Rayligh fading channl

7 and th uplink fdback channl is AWG, w hav shown that unquantizd and uncodd CSI transmission that incurs zro dlay) is optimal in that it achivs th sam minimum man squard rror distortion as a schm that optimally quantizs and ncods th CSI whil incurring infinit dlay W hav proposd a simpl linar CSI fdback rcivr that xploits th channl corrlations whil still rtaining th attractiv zro-dlay fatur Furthrmor, w dscribd th AR corrlatd channl modl and prsntd th corrsponding prformanc bounds for th UQ-UC CSI fdback schm W xplord th prformanc limits of th schm in th contxt of downlink mtipl antnna, mtiusr transmittr optimization W showd that th UQ-UC schm can a provid rliabl and fast fdback of CSI vn in th cas of high trminal mobility APPEDIX In this appndix w show how for th givn corrlation btwn th downlink channl stats, th corrlatd channl stats ar gnratd and th cofficints c 0 to c L of th AR modl in 15) ar dtrmind Th corrlation btwn th downlink channl stats is givn as φk) =E[h dl i)h dl i k) ]for k L 43) whr φ k) = φk), and for k > L, φk) = 0 As said arlir, w assum that φ0) = 1 Th corrsponding corrlation matrix is R =E[h dl i)h dl i) H ] whr h dl i) = [h dl i) h dl i 1) h dl i L)] T Considring that th matrix R can b dcomposd as R = QQ H, th corrlatd channl stats h dl i),,h dl i L) ar obtaind from th following opration h dl i) =Qn 44) whr n is a random, L +1-dimnsional, zro-man vctor with th corrlation matrix E[nn H ]=I Furthr, basd on th AR modl in 15) w form a st of L +1 linar quations φ0) = c j φ j)+c ) and φk) = c j φk j) k =1,,L 46) Lt us dfin th following matrix 1 φ1) φ2) φl) 0 φ0) φ1) φl 1) Φ = 0 φl 1) φl 2) φ0) and vctors and 47) c =[c 2 0 c 1 c L ] T 48) f =[φ0) φ1) φl)] T 49) Th abov systm of linar quations can b rwrittn as f = Φc 50) Th last squars solution of th abov linar quation is c =[ c 2 0 c 1 c L ] T =Φ H Φ) 1 Φ H f 51) From th abov w dirctly adopt th solutions for th cofficints c i = c i for i =1,,L Lt us now dtrmin th cofficint c 0 From th modl in 15), th innovation trm is c 0 n dl i) =h dl i) c j h dl i j) =z H h dl i) 52) whr z =[1 c 1 c L ]T In ordr to guarant that th innovation is unit-varianc, whil maintaining th corrlation R, th cofficint c 0 is slctd as c 0 = z H Rz 53) To approximat th Jaks modl using th finit lngth AR modl in 15) w slct lmnts of th vctor f as φk) =J 0 2πf d kτ), k =0,,L 54) whr f d is th maximum Dopplr frquncy and τ is th tim diffrnc btwn succssiv channl stats h dl i) and h dl i 1) Satisfying th yquist sampling rat, th priod τ shod b such that τ<1/2f d ) REFERECES [1] G Cair and S Shamai, On th achivabl throughput of a mtiantnna Gaussian broadcast channl, IEEE Transactions on Information Thory, vol 49, pp , Jy 2003 [2] D Ts and P Viswanath, On th capacity rgion of th vctor Gaussian broadcast channl, IEEE Intrnational Symposium on Information Thory, pp , Jy 2003 [3] E Rashid-Farrohi, L Tassias, and K Liu, Joint optimal powr control and bamforming in wirlss ntworks using antnna arrays, IEEE Transactions on Communications, vol 46, pp , Octobr 1998 [4] E Visotsky and U Madhow, Optimum bamforming using transmit antnna arrays, Th IEEE Vhicar Tchnology Confrnc VTC), vol 1, pp , May 1999 [5] D Samardzija and Mandayam, Mtipl antnna transmittr optimization schms for mtiusr systms, Th IEEE Vhicar Tchnology Confrnc VTC), Octobr 2003 Orlando [6] T J Goblick, Thortical limitations on th transmission of data from analog sourcs, IEEE Transactions on Information Thory, vol 11, pp , Octobr 1965 [7] T Brgr, Shannon lctur: living information thory, Prsntd at IEEE ISIT, Jy 2002 [8] M Gastpar, B Rimoldi, and M Vttrli, To cod, or not to cod: lossy sourc channl communication rvisitd, IEEE Transactions on Information Thory, vol 49, pp , May 2003 [9] T Brgr, Rat Distortion Thory: A Mathmatical Basis for Data Comprssion Prntic-Hall, 1971 [10] D Samardzija and Mandayam, Pilot assistd stimation of MIMO fading channl rspons and achivabl data rats, IEEE Transactions on Signal Procssing, Spcial Issu on MIMO, vol 51, pp , ovmbr 2003 [11] D Samardzija and Mandayam, Unquantizd and uncodd channl stat information fdback on wirlss channls, Confrnc on Information Scincs and Systms, March 2004 Princton Univrsity [12] D Samardzija, Mtipl Antnna Wirlss Systms and Channl Stat Information PhD Thsis, Rutgrs, Th Stat Univrsity of w Jrsy, May 2004 [13] W Turin and R obln, Hiddn Markov modling of flat fading channls, IEEE JSAC, vol 16, pp , Dcmbr 1998 [14] K E Baddour and C Baiu, Autorgrssiv modls for fading channl simation, IEEE Global Tlcommunications Confrnc, Globcom 2001, pp