Dynamic Power Allocation in MIMO Fading Systems Without Channel Distribution Information

Transcription

1 PROC. IEEE INFOCOM 06 Dynamc Powr Allocaon n MIMO Fadng Sysms Whou Channl Dsrbuon Informaon Hao Yu and Mchal J. Nly Unvrsy of Souhrn Calforna Absrac Ths papr consdrs dynamc powr allocaon n MIMO fadng sysms wh unknown channl sa dsrbuons. Frs, h dal cas of prfc nsananous channl sa nformaon a h ransmr (CSIT) s rad. Usng h drf-plus-pnaly mhod, a dynamc powr allocaon polcy s dvlopd and shown o approach opmaly, rgardlss of h channl sa dsrbuon and whou rqurng knowldg of hs dsrbuon. Nx, h cas of dlayd and quanzd channl sa nformaon s consdrd. Opmal uly s fundamnally dffrn n hs cas, and a dffrn onln algorhm s dvlopd ha s basd on convx projcons. Th proposd algorhm for hs dlayd-csit cas s shown o hav an O( ) opmaly gap, whr s h quanzaon rror of CSIT. I. INTRODUCTION Durng h pas dcad, h mulpl-npu mulpl-oupu (MIMO) chnqu has bn rcognzd as on of h mos mporan chnqus for ncrasng h capabls of wrlss communcaon sysms. In h wrlss fadng channl, whr h channl changs ovr m, h problm of powr allocaon s o drmn h ransm covaranc of h ransmr o maxmz h rgodc capacy subjc o boh long rm and shor rm powr consrans. I s ofn rasonabl o assum ha nsananous channl sa nformaon (CSI) s avalabl a h rcvr hrough ranng. Mos works on powr allocaon n MIMO fadng sysms also assum ha sascal nformaon abou h channl sa, rfrrd o as channl dsrbuon nformaon (CDI), s avalabl a h ransmr. Undr h assumpon of prfc nsananous channl sa nformaon a h rcvr (CSIR) and prfc channl dsrbuon nformaon a h ransmr (CDIT), pror work on powr allocaon n MIMO fadng sysms can b cagorzd no wo cass: Prfc nsananous channl sa nformaon a h ransmr (dal-csit): In h dal cas of prfc CSIT, opmal powr allocaon s known o b a warfllng soluon []. Compuaon of war-lvls nvolvs a on-dmnsonal ngral quaon for fadng channls wh..d. Raylgh nrs or a mul-dmnsonal ngral quaon for gnral fadng channls []. No CSIT: If CSIT s unavalabl, h opmal powr allocaon s n gnral sll opn. If h channl marx has..d. Raylgh nrs, hn h opmal powr allocaon s known o b h dny ransm covaranc scald o Hao Yu and Mchal J. Nly ar wh h Dparmn of Elcrcal Engnrng, Unvrsy of Souhrn Calforna, Los Angls, USA. Ths work s suppord n par by h NSF gran CCF sasfy h powr consran []. Th opmal powr allocaon n MIMO fadng channls wh corrlad Raylgh nrs s oband n [3], [4]. Th powr allocaon n MIMO fadng channls s furhr consdrd n [5] undr a mor gnral channl corrlaon modl. Ths pror work rls on accura CDIT and/or on rsrcv channl dsrbuon assumpons. I can b dffcul o accuraly sma h CDI, spcally whn hr ar complcad corrlaons. Soluons ha bas dcsons on prfc CDIT can b subopmal du o msmachs. Ths papr dsgns algorhms ha do no rqur pror knowldg of h channl dsrbuon, y prform arbrarly clos o h opmal valu ha can b achvd by havng hs knowldg. Furhr, h convrgnc m s compud and shown o b sgnfcanly smallr han h m rqurd o accuraly sma h channl dsrbuon. Th dal-csit assumpon s rasonabl n m-dvson duplx (TDD) sysms wh symmrc TDD wrlss channls. Howvr, n frquncy-dvson duplx (FDD) scnaros and ohr scnaros whou channl symmry, h CSI mus b smad a h rcvr, quanzd, and rpord back o h ransmr wh a m dlay. Ths papr frs consdrs h dal-csit cas and dvlops a soluon ha dos no rqur CDIT. Nx, h cas of dlayd and quanzd CSIT s consdrd and a fundamnally dffrn algorhm s dvlopd for ha cas. Th lar algorhm agan dos no us CDIT, bu achvs a uly whn O( ) of h bs uly ha can b achvd vn wh prfc CDIT, whr s h quanzaon rror. Ths shows ha dlayd bu accura CSIT (wh 0) s almos as good as havng prfc CDIT. A. Rlad work and our conrbuons In h dal CSIT cas, h proposd dynamc powr allocaon polcy s an adapon of h gnral drf-plus-pnaly algorhm for sochasc nwork opmzaon [6], [7]. In hs MIMO conx, h currn papr shows h algorhm provds srong sampl pah and convrgnc m guarans. Th dynamc of h drf-plus-pnaly algorhm s smlar o ha of h sochasc dual subgradn algorhm, alhough h opmaly analyss and prformanc bounds ar dffrn. Th sochasc dual subgradn algorhm has bn appld n opmzaon of h wrlss fadng channl whou CDI,.g., downlnk powr schdulng n sngl annna cllular sysms [8], powr allocaon n sngl annna broadcas OFDM channls [9], schdulng and rsourc allocaon n

2 PROC. IEEE INFOCOM 06 random accss channls [0], powr allocaon n mul-carrr MIMO nworks []. In h dlayd and quanzd CSIT cas, h suaon s smlar o h scnaro of onln convx opmzaon [] xcp ha w ar unabl o obsrv ru hsory rward funcons du o channl quanzaon. Th proposd dynamc powr allocaon polcy can b vwd as an onln algorhm wh naccura hsory nformaon. Th currn papr analyzs h prformanc loss du o CSI quanzaon rror and provds srong sampl pah and convrgnc m guarans of hs algorhm. Accordng o h auhors knowldg, onln convx opmzaon wh naccura hsory nformaon has no bn sudd bfor. Th analyss n hs MIMO conx can b xndd o mor gnral onln convx opmzaon wh naccura hsory nformaon. Onln opmzaon has bn appld n powr allocaon n wrlss fadng channls whou CDIT and wh dlayd and accura CSIT,.g., subopmal onln powr allocaon n sngl annna sngl usr channls [3], subopmal onln powr allocaon n sngl annna mulpl usr channls [4]. A clos rlad rcn work s [5], whr onln powr allocaon n MIMO sysms s consdrd. Th onln algorhm n [5] s dffrn from our algorhm and follows a marx xponnal larnng schm rqurng h compuaon of marx xponnals a ach slo. In conras, our onln algorhm only nvolvs a smpl projcon a ach slo and a closd-form soluon of hs projcon s drvd n hs papr. Work [5] also consdrs h ffc of mprfc CSIT by assumng CSIT s unbasd,.., xpcd CSIT rror condonal on obsrvd prvous CSIT s zro. Ths assumpon of mprfc CSIT s suabl o modl h CSIT masurmn rror or fdback rror bu can no capur h CSI quanzaon rror. In conras, h currn papr only rqurs ha CSIT rror s boundd. II. SIGNAL MODEL AND PROBLEM FORMULATIONS A. Sgnal modl Consdr a pon-o-pon MIMO fadng channl ha opras n slod m wh normalzd m slos {,,...}. Thr ar N T annnas a h ransmr and N R annnas a h rcvr. Th channl can b modld as y() =H()x()+z() whr {,,...} s h m ndx, z() C N R s h addv nos vcor, x() C N T s h ransmd sgnal vcor, H() C N R N T s h channl marx, and y() C N R s h rcvd sgnal vcor. Assum ha nos vcors z() ar ndpndn and dncally dsrbud (..d.) ovr m slos and ar normalzd crcularly symmrc complx Gaussan random vcors wh E[z()z H ()] = I NR, whr I NR dnos an N R N R dny marx. Assum ha channl marcs H() ar..d. across m and hav a fxd bu arbrary probably dsrbuon, possbly on wh corrlaons bwn nrs of h marx. Assum hr xss If h sz of h dny marx s clar, w ofn smply wr I. a consan B>0 such ha khk F appl B wh probably on, whr k k F dnos h Frobnus norm of marcs. Assum ha h rcvr can rack channl marcs H() xacly hrough ranng. In symmrc TDD scnaros, s rasonabl o assum h ransmr has prfc CSIT. In mor gnral scnaros, h channl marx H() s masurd a h rcvr a ach slo, a quanzd vrson H() s crad as a funcon of H(), and hs quanzd vrson s fd back o h ransmr wh on slo of dlay. W assum ha h quanzaon rror s boundd,.., hr xss >0 such ha kh() H()k F appl for all. Du o h on slo dlay, a slo h ransmr only knows H( ). Snc channls ar..d. ovr slos, hs dlayd nformaon s ndpndn of h currn (and unknown) H(). Rmarkably, urns ou ha h oudad nformaon s sll usful. B. Opmal powr allocaon wh prfc CDIT If h channl marx s fxd a H and h ransm covaranc s fxd a Q, h MIMO capacy s gvn by []: log d(i + HQH H ) whr suprscrp H dnos Hrman ranspos and d( ) dnos h drmnan opraor of marcs. If H s random hn h avrag capacy, formally calld h rgodc capacy [6], s gvn by E H log d(i + HQH H ). W consdr wo yps of powr consrans a h ransmr: An avrag powr consran E H [r(q)] appl P and an nsananous powr consran r(q) appl P, whr r( ) dnos h rac opraor of marcs. Th dal-csit problm s o choos Q as a (possbly random) funcon of h obsrvd H o maxmz h rgodc capacy subjc o powr consrans: max Q(H) E H log d(i + HQ(H)H H ) () s.. E H [r(q(h))] appl P, () Q(H) Q, 8H, (3) whr Q s a s ha nforcs h nsananous powr consran: Q = Q S N T + : r(q) appl P (4) whr S N T + dnos h N T N T posv smdfn marx spac. To avod rvals s assumd ha P P. In ()-(3) w us noaon Q(H) o mphasz ha Q can dpnd on H,.., adapv o channl ralzaons. If h ransmr has no CSIT, h opmal powr allocaon problm s dffrn, gvn as follows. max Q E H log d(i + HQH H ) (5) s.. E H [r(q)] appl P, (6) Q Q, (7) whr s Q s dfnd n (4). Agan assum P P. Snc h nsananous CSIT s unavalabl, h ransm covaranc A boundd Frobnus norm always holds n h physcal world bcaus h channl anuas h sgnal. Parcular modls such as Raylgh and Rcan fadng vola hs assumpon n ordr o hav smplr dsrbuon funcons.

3 PROC. IEEE INFOCOM 06 canno adap o H. By h convxy of hs problm and Jnsn s nqualy, a randomzd Q can b shown o b uslss. I suffcs o consdr a consan Q. Snc P P, hs mpls h problm s quvaln o a problm ha rmovs h consran (6) and ha changs h consran (7) o: Q Q = {Q S N T + : r(q) appl P } Th problms ()-(3) and (5)-(7) ar fundamnally dffrn and hav dffrn opmal objcv funcon valus. Opmaly for hs problms s dfnd by h channl dsrbuon nformaon (CDI). In hs papr, h problms ar solvd va dynamc algorhms ha do no rqur CDI. Th algorhms ar dffrn for h wo cass, and us dffrn chnqus. C. Lnar algbra and marx drvavs Rcall ha f A C m n and B C n m hn r(ab) = r(ba). Ths subscon prsns addonal usful facs abou Frobnus norms and complx marcs. Proofs ar gvn n [7] for complnss. Fac. For any A, B C m n and C C n k w hav: ) kak F = ka H k F = ka T k = k Ak F. ) ka + Bk F applkak F + kbk F. 3) kack F applkak F kck F. 4) r(a H B) applkak F kbk F. Fac. For any A S n + w hav kak F appl r(a). Fac 3 ([8]). Th funcon f : S n +! R dfnd by f(q) = log d(i + HQH H ) s concav and s gradn s gvn by r Q f(q) =H H (I + HQH H ) H, 8Q S n +. Th nx fac s h complx marx vrson of h frs ordr condon for concav funcons of ral numbr varabls,.., f(y) appl f(x)+f 0 (x)(y x), 8x, y domf f f s concav. Fac 4. L funcon f(q) :S n +! R b a concav funcon and hav gradn r Q f(q) S n a pon Q. Thn, f( b Q) appl f(q)+r [r Q f(q)] H ( b Q Q), 8 b Q S n +. III. IDEAL CSIT CASE Consdr h cas of prfc nsananous CSIT, calld h dal-csit cas. Th problm o solv s ()-(3). A h bgnnng of ach slo {,,...} h channl H() s known and a covaranc marx Q() can b chosn basd on hs nformaon. Ths s don whou usng CDI va h drfplus-pnaly chnqu of [7]. For ach slo {,,...} dfn h rward R(): R() = log d(i + H()Q()H() H ) (8) Th avrag powr consran () s nforcd va a vrual quu Z() wh Z(0) = 0 and wh upda: Z( + ) = max[z()+r(q()) P ] In h drf-plus-pnaly algorhm, vry slo a marx Q() Q s slcd o maxmz VR() Z()r(Q()), whr V s a posv wgh. Ths rsuls n Algorhm blow. Algorhm Dynamc powr allocaon wh dal CSIT L V > 0 b a consan paramr and Z(0) = 0. A ach m {,,...}, obsrv H() and Z(). Thn do h followng: Choos ransm covaranc Q() Qo maxmz: V log d(i + H()Q()H() H ) Z()r(Q()) Upda Z( + ) = max[z()+r(q()) P ]. Dfn R op as h opmal avrag uly n (). Th valu R op dpnds on h (unknown) dsrbuon for H(). Fx > 0 and dfn V =(P + P ) /( ). A horm n [7] nsurs ha, rgardlss of h dsrbuon of H(): X E[R( )] R op, 8 >0 (9) lm! X E[r(Q( ))] appl P (0) Ths holds for arbrarly small valus of > and so h algorhm coms arbrarly clos o opmaly. Noc ha Algorhm dos no us channl dsrbuon nformaon (.., no CDI). Th nx subscons show how o solv h covaranc slcon problm for choosng Q() n Algorhm, and shows ha h spcal srucur of hs MIMO problm producs a sampl pah guaran ha s sgnfcanly srongr han (0) and ha dmonsras convrgnc m ha s ypcally much fasr han h m ha would b rqurd o accuraly sma h CDI nformaon. A. Transm covaranc updas n Algorhm Ths subscon shows h Q() slcon n Algorhm can b asly solvd and has an (almos) closd-form soluon. Th convx program nvolvd n h ransm covaranc upda of Algorhm s n h form max log d(i + HQH H Z ) r(q) () Q V s.. r(q) appl P () Q S N T + (3) Ths convx program s smlar o h convnonal problm of ransm covaranc dsgn wh a drmnsc channl H, xcp ha objcv () has an addonal pnaly rm (Z/V )r(q). I s wll known ha, whou hs pnaly rm, h soluon s o dagonalz h channl marx and alloca powr ovr gn-mods accordng o a war-fllng chnqu []. Th nx horm shows ha h opmal soluon o problm ()-(3) has a smlar srucur. Thorm. Consdr h SVD H H H = U H U, whr U s a unary marx and s a dagonal marx wh non-ngav nrs,..., N T. Thn h opmal soluon o ()-(3) s

4 PROC. IEEE INFOCOM 06 gvn by Q = U H U, whr s a dagonal marx wh nrs,..., N T gvn by: = max µ, 8 {,...,NT }, whr µ s chosn such ha P N T = appl P, µ 0 and µ P N T = P =0. Th xac µ can b drmnd wh complxy O(N T log N T ), dscrbd n Algorhm. Proof: S Appndx A. Algorhm Algorhm o solv problm ()-(3) ) Chck f P N T = max{ Z/V } appl P holds. If ys, l µ = 0 and = max{ Z/V }, 8 {,,...,N T } and rmna h algorhm; ls, connu o h nx sp. ) Sor all, {,,...,N T } n a dcrasng ordr such ha () () (N T ). Dfn S 0 =0. 3) For =o N T L S = S +. L µ = () S +P (Z/V ). If µ > 0 and () appl hn rmna h loop; ls, connu (+) o h nx raon n h loop. 4) L = max, 8 {,,...,NT } and rmna h algorhm. Th complxy of Algorhm s domnad by h sorng of all n sp (). Rcall ha h war-fllng soluon of powr allocaon n mulpl paralll channls can also b found by an xac algorhm (s Scon 6 n [9]), whch s smlar o Algorhm. Th man dffrnc s ha Algorhm has a frs sp o vrfy f µ =0. Ths s bcaus unlk h powr allocaon n mulpl paralll channls, whr h opmal soluon always uss full powr, h opmal soluon o problm ()-(3) may no us full powr for larg Z du o h pnaly rm (Z/V )r(q) n objcv (). B. Drmnsc bounds Rcall ha kh()k F appl B for all, for som consan B. Lmma. In Algorhm, f Z() VB, hn Q() =0. Proof: Suppos h SVD of H H ()H() s gvn by H H ()H() =U H U, whr dagonal marx has nonngav dagonal nrs,..., N T. No ha r(h H ()H()) (b) appl kh()k F appl B whr (a) follows from r(h H ()H()) = P N T = ; and (b) follows from Fac. By Thorm, Q() =U H U, whr s a dagonal marx wh nrs,..., N T gvn by = max µ +Z()/V, 8 {,,...,NT }, whr µ 0. Snc appl B, 8 {,,...,N T }, w know ha f Z() VB, hn µ+z()/v appl 0 for all µ 0 and hnc = 8 {,,...,N T }. (a) appl Lmma. L Z() b yldd by Algorhm. For all slos {,,...}, w hav Z() appl VB +(P P ). Proof: By Lmma, Z() can no ncras f Z() VB. If Z() appl VB, hn Z(+) s a mos VB +(P P ) by h upda quaon of Z(+) and h nsananous powr consran. C. Prformanc of Algorhm (dal-csit) Thorm. Fx >0 and dfn V =(P + P ) /( ). Undr Algorhm w hav for all >0: X X E[R( )] R op r(q( )) appl P + B (P + P ) + (P P ) In parcular, h sampl pah m avrag powr s whn of s rqurd consran P whnvr (/ ). Proof: Th frs nqualy s h sam as (9). I rmans o prov h scond nqualy. For all slos h Algorhm upda for Z( ) sasfs: Z( + ) = max[z( )+r(q( )) P ] Z( )+r(q( )) P Rarrangng rms gvs: r(q( )) P appl Z( + ) Z( ). Fx >0. Summng ovr {..., } and dvdng by gvs: X Z() Z(0) r(q( )) P appl appl (VB +(P P )) whr h las nqualy holds bcaus Z(0) = 0 and Z() appl VB +(P P ) by Lmma. Thorm provds a sampl pah guaran on avrag powr, whch s much srongr han h guran n (0). I also shows ha convrgnc m o rach an -approxma soluon s O(/ ). Typcally, hs s dramacally mor ffcn han h convrgnc m rqurd o oban vn a coars sma of h jon dsrbuon for h nrs of H(). Indd, f ach channl nry h j wr quanzd no / dsnc lvls, hr would b (/ ) N T N R dffrn possbl (quanzd) marx ralzaons. Wang for (/ ) N T N R slos would a bs allow ach ralzaon o appar onc, whch s sll no nough for accura smaon of h probabls assocad wh ach ralzaon. Forunaly, Thorm shows ha such smaon s no ndd. IV. DELAYED AND QUANTIZED CSIT CASE Consdr h cas of dlayd and quanzd CSIT. A h bgnnng of ach slo {,,...}, channl H() s unknown and only quanzd channls of prvous slos H( ), {,..., } ar known.

5 PROC. IEEE INFOCOM 06 Ths s smlar o h scnaro of onln opmzaon whr h dcson makr slcs x() X a ach slo o maxmz an unknown rward funcon f (x) basd on h nformaon of prvous rward funcons f (x( )), {,..., }. Th goal s o mnmz avrag rgr max P xx f (x) P f (x( )). Th bs known avrag rgr of onln opmzaon wh Lpschz connuous and convx rward funcons s O( p ) n []. Ths s dffrn from convnonal onln opmzaon bcaus a ach slo, h rwards of prvous slos,.., R( ) = log d(i + H( )Q( )H H ( )), {,..., }, ar sll unknown du o h fac ha h rpord channls H( ) ar h quanzd vrsons. Nvrhlss, an onln algorhm whou usng CDIT s dvlopd n Algorhm 3. Algorhm 3 Dynamc Powr Allocaon wh Dlayd and Quanzd CSIT L >0 b a consan paramr and Q(0) Qb arbrary. A ach m {,,...}, obsrv H( ) and do h followng: L D( ) = H H ( )(I NR + H( )Q( ) H H ( )) H( ). Choos ransm covaranc Q() =P Q Q( ) + D( ), whr P Q [ ] s h projcon ono convx s Q = {Q S N T + : r(q) appl P }. Dfn Q Q as an opmal soluon o problm (5)-(7), whch dpnds on h (unknown) dsrbuon for H(). Dfn R op () = log d(i + H()Q H H ()) as h uly a slo aand by Q. If h channl s no quanzd,.., H( ) = H( ), 8 {,,...}, hn D( ) s h gradn of R( ) a pon Q( ). Fx >0 and ak =. Th rsuls n [] nsur ha, rgardlss of h dsrbuon of H(): X R( ) X R op ( ) P N R B 4, 8 >0 (4) X r(q( )) appl P,8 >0 (5) Th nx subscons analyz h prformanc of Algorhm 3 wh quanzd channls and shows ha h prformanc dgrads lnarly wh rspc o h quanzaon rror. If = hn (4) and (5) ar rcovrd. A. Transm Covaranc Updas n Algorhm 3 Ths subscon shows h Q() slcon n Algorhm 3 can b asly solvd and has an (almos) closd-form soluon. Th projcon opraor nvolvd n Algorhm 3 by dfnon s mn kq Xk F (6) s.. r(q) appl P (7) Q S N T + (8) whr X = Q( ) + D( ) s an Hrman marx a ach m. Whou consran r(q) appl P, h projcon of Hrman marx X ono h posv smdfn con S n + s smply akng h gnvalu xpanson of X and droppng rms assocad wh ngav gnvalus (s Scon 8... n [0]). Work [] consdrd h projcon ono h nrscon of h posv smdfn con S n + and an affn subspac gvn by {Q : r(a Q)=b, {,,...,p}, r(b j Q) appl d j,j {,,...,m}} and dvlopd h dual-basd rav numrcal algorhm o calcula h projcon. Problm (6)- (8) s a spcal cas, whr h affn subspac s gvn by r(q) appl P, of h projcon consdrd n []. Insad of solvng problm (6)-(8) usng numrcal algorhms, hs subscon shows ha problm (6)-(8) has an (almos) closd-form soluon. Thorm 3. Consdr SVD X = U H U, whr U s a unary marx and s a dagonal marx wh nrs,..., N T. Thn h opmal soluon o problm (6)-(8) s gvn by Q = U H U, whr s a dagonal marx wh nrs,..., N T gvn by, = max[ µ ], 8 {,,...,N T }, whr µ s chosn such ha P N T = appl P, µ 0 and µ P N T = P =0. Th xac µ can b drmnd wh complxy O(N T log N T ), dscrbd n Algorhm 4. Proof: Th proof s skchd as follows. Frs, problm (6)-(8) s rducd o a smplr convx program wh a ral vcor varabl by characrzng h srucur of s opmal soluon. Thn, an (almos) closd-form soluon o h smplr convx program s oband by sudyng s KKT condons. S Appndx B for dals. Algorhm 4 Algorhm o solv problm (6)-(8) ) Chck f P N T = max[ ] appl P holds. If ys, l µ =0 and = max[ ], 8 {,,...,N T } and rmna h algorhm; ls, connu o h nx sp. ) Sor all, {,,...,N T } n a dcrasng ordr such ha () () (N T ). Dfn S 0 =0. 3) For =o N T L S = S +. L µ = S P. If µ () µ > 0 and (+) µ appl hn rmna h loop; ls, connu o h nx raon n h loop. 4) L = max[ µ ], 8 {,,...,N T } and rmna h algorhm.

6 PROC. IEEE INFOCOM 06 B. Propry of D( ) Dfn D( ) = H H ( )(I NR + H( )Q( )H H ( )) H( ), whch s h gradn of R( ) a pon Q( ) and s unknown o h ransmr du o h unavalably of H( ). Th nx lmma rlas D( ) and D( ). Lmma 3. For all slos {,,...}, w hav ) kd( )k F appl p N R B. ) kd( ) D( )kf appl ( ), whr ( ) = p NR B + p N R (B + )+(B + ) N R P (B + ) sasfyng ( )! 0 as!.., ( ) O( ). 3) k D( )k F appl ( )+ p N R B Proof: S full vrson [7] for dals. C. Prformanc of Algorhm 3 Thorm 4. Fx >0 and dfn =. Undr Algorhm 3, w hav for all >0: X R( ) X R op ( ) X r(q( )) appl P ( ) P P ( ( )+ p N R B ) whr ( ) s h consan dfnd n Lmma 3. In parcular, h sampl pah m avrag uly s whn + ( ) P of h opmal m avrag uly for problm (5)-(7) whnvr (/ ). Proof: Th scond nqualy rvally follows from h fac ha Q() Q, 8 {,...}. I rmans o prov h frs nqualy. Ths proof xnds h rgr analyss of convnonal onln convx opmzaon [] by consdrng nxac gradn D( ). For all slos {,,...}, h ransm covaranc upda n Algorhm 3 sasfs: kq( ) Q k F =kp Q Q( ) + D( ) (a) applkq( ) + D( ) Q k F Q k F =kq( ) Q k F + r D H ( )(Q( ) Q ) + k D( )k F =kq( ) Q k F + r D H ( )(Q( ) Q ) + r ( D( ) D( )) H (Q( ) Q ) + k D( )k F, whr (a) follows from h non-xpansv propry of projcons ono convx ss. Dfn () =kq( + ) Q k F kq() Q k F. Rarrangng rms n h las quaon and dvdng by facor mpls r D H ( )(Q( ) Q ) ( ) k D( )k F r ( D( ) D( )) H (Q( ) Q ) (9) Dfn f(q) = log d(i+h( )QH H ( )). By Fac 3, f( ) s concav ovr Q. No ha D( ) = r Q f(q( )) by Fac 3 and Q Q. By Fac 4, w hav f(q( )) f(q ) r(d H ( )(Q( ) Q )). (0) No ha f(q( )) = R( ) and f(q )=R op ( ). Combnng (9) and (0) ylds R( ) R op ( ) ( ) k D( )k F (a) (b) r ( D( ) D( )) H (Q( ) Q ) ( ) k D( )k F k D( ) D( )k F kq( ) Q k F ( ) ( ( )+p N R B ) ( ) P whr (a) follows from Fac and (b) follows from Lmma 3 and h fac ha kq( ) Q k F applkq( )k F +kq k F appl r(q( )) + r(q ) appl P, whch s mpld by Fac, Fac and fac Q( ), Q Q. Rplacng wh ylds for all {,...}, R( ) R op ( ) ( ) p ( ( )+ NR B ) ( ) P. Fx >0. Summng ovr {,..., }, dvdng by facor and nong ha P ( ) s a lscop sum gvs P R( ) P Rop ( ) (kq() Q k F kq(0) Q k F ) ( ( )+p N R B ) ( ) P P ( ( )+p N R B ) ( ) P, whr h las nqualy follows bcaus kq(0) Q k F appl kq(0)k F + kq k F appl r(q(0)) + r(q ) appl P and kq() Q k F 0. Thorm 4 provs a sampl pah guaran on h avrag uly. I shows ha h convrgnc m o rach an + ( ) P approxma soluon s O(/ ). No ha f = hn quaons (4) and (5) ar rcovrd by Thorm 4. Thorm 4 also solas h ffcs of dlay and quanzaon. Th obsrvaon s ha h ffc of CSIT dlay vanshs as Algorhm 3 runs for a suffcnly long m. In som sns, dlayd bu accura CSIT s almos as good as prfc CDIT. In conras, h ffc of CSIT quanzaon dos no vansh as Algorhm 3 runs for a suffcnly long m. Th prformanc dgradaon du o quanzaon scals lnarly wh rspc o h quanzaon rror snc ( ) O( ). Inuvly, hs s rasonabl snc h powr allocaon basd on quanzd CSIT s acually opmzng anohr dffrn MIMO sysm.

7 PROC. IEEE INFOCOM 06 D. Exnsons ) T -Slo Dlayd and Quanzd CSIT: Thus far, w hav assumd ha CSIT s dlayd by on slo. In fac, f CSIT s dlayd by T slos, w can modfy h upda of ransm covarancs n Algorhm 3 as Q() =P Q [Q( T )+ D( T )]. AT -slo vrson of Thorm 4 can b smlarly provn. ) Algorhm 3 wh Tm Varyng : Algorhm 3 can b xndd o hav m varyng sp sz () = p a m. Th full vrson [7] provs ha such an algorhm P P Rop ( ) ylds R( ) Pp p ( ( )+ p NR B ) ( ) P for all >0. Ths shows h convrgnc m o an + ( ) P approxma soluon s agan O(/ ). Howvr, an advanag of m varyng sp szs s h prformanc auomacally gs mprovd as h algorhm runs and hr s no nd o rsar h algorhm wh a dffrn consan sp sz f a br prformanc s dmandd. V. SIMULATIONS Consdr a MIMO sysm whr boh h ransmr and h rcvr hav wo annnas. Th powr consrans ar P = 5and P = 0. Th channl has appl wo ralzaons wh j0.84 qual probably 0.5,.., H =0.5 j.58 j.83 j.97 and appl j.3 H = j.69 j0.07 j.86. If h channl s quanzd, hy ar quanzd as H and H, rspcvly. Th algorhms n hs papr can b asly appld o xampls wh nfn possbl oucoms for h channl marx. Ths smpl xampl of wo possbls s consdrd bcaus an offln opmal soluon basd on prfc CDIT can only b compud whn h numbr of sampls s small. 3 Fgur compars h prformanc of Algorhm wh prfc CSIT and h opmal soluon o problm ()-(3). In h smulaon, w ak V = 000. Fgur compars h prformanc of Algorhm 3 wh on slo dlayd and quanzd CSIT and h opmal soluon o problm (5)-(7). To sudy h ffc of quanzaon rror, w consdr 3 dffrn quanzaon lvls. Cas : H appl appl = j j.5 j.8 j.9 and H j.3 = j.6 j0 j.8 ; Cas appl appl : H j =0.5 j.5 j j and H j.5 = j.5 j0 j ; appl appl Cas 3: H = 0.5 and H =. In h smulaon, w ak Q(0) = 0 and = 0 3. I can b obsrvd ha prformanc bcoms wors as CSIT quanzaon gs coarsr, whl h avrag powr consrans ar srcly sasfd vn wh quanzd CSIT. 3 Ths s known as h curs of dmnsonaly for sochasc opmzaon du o h larg sampl sz. Tha s, vn wh prfc CDIT, problm ()-(3) and problm (5)-(7) can b numrcally hard o solv whn h sampl sz of H s larg. In conras, h dynamc algorhms proposd n hs papr can dal wh problms vn wh an nfn numbr of sampls and h prformanc guarans ar ndpndn of h sampl sz. Uly Avrag Powr Uly Prformanc of Algorhm Slos: Avrag Powr Algorhm Algorhm opmal soluon o problm () (3) opmal soluon o problm () (3) Slos: Fg.. Prformanc of Algorhm. Prformanc of Algorhm Slos: x quanzaon cas opmal soluon o problm (5) (7) quanzaon cas quanzaon cas quanzaon cas 3 quanzaon cas quanzaon cas Slos: x 0 5 Fg.. Prformanc of Algorhm 3. VI. CONCLUSION Ths papr consdrs dynamc powr allocaon n MIMO fadng sysms whou CDIT. In h cas of dal CSIT, h proposd dynamc powr polcy can approach opmaly. In h cas of dlayd and quanzd CSIT, h proposd dynamc powr allocaon polcy can achv O( ) sub-opmaly, whr s h quanzaon rror. APPENDIX A PROOF OF THEOREM Th proof mhod s an xnson of Scon 3. n [], whch gvs h srucur of h opmal ransm covaranc n drmnsc MIMO channls. No ha log d(i + HQH H ) (a) = log d(i + QH H H) (b) = log d(i + QU H U) (c) = log d(i + / UQU H / ), whr (a) and (c) follows from h lmnary dny

8 PROC. IEEE INFOCOM 06 log d(i n + AB) = log d(i n + BA), 8A, B C n n ; and (b) follows from h fac ha H H H = U H U. Dfn Q = UQU H, whch s smdfn posv f and only f Q s. No ha r( Q)=r(UQU H )=r(q) by h fac ha r(ab) =r(ba), 8A C m n, B C n m. Thus, problm ()-(3) s quvaln o max Q log d(i + / Q / ) Z V r( Q) () s.. r( Q) appl P () Q S N T + (3) Fac 5 (Hadamard s Inqualy, Thorm 7.8. n []). For all A S n +, d(a) appl Q n = A wh qualy f A s dagonal. Th nx clam can b provn usng Hadamard s nqualy. Clam. Problm ()-(3) has a dagonal opmal soluon. Proof: Suppos problm ()-(3) has a non-dagonal opmal soluon gvn by marx Q. Consdr a dagonal marx Q b whos nrs ar dncal o h dagonal nrs of Q. No ha r( Q) b = r( Q). To show Q b s a soluon no wors han Q, suffcs o show ha log d(i + / Q b / ) log d(i + / Q / ). Ths s ru bcas d(i + / Q b / ) = Q N T = ( + Q b ) = Q N T = ( + Q ) d(i + / Q / ), whr h las nqualy follows from Hadamard s nqualy. Thus, Q b s a soluon no wors han Q and hnc opmal. By Clam, w can consdr Q = = dag(,,..., NT ) and problm ()-(3) s quvaln o max s.. XN T log( + ) = N T Z V XN T (4) = X appl P (5) = 8 {,,...,N T } (6) No ha problm (4)-(6) sasfs Slar s condon. So h opmal soluon o problm (4)-(6) s characrzd by KKT condons [0]. Th rmanng par s smlar o h drvaon of h war-fllng soluon of powr allocaon n paralll channls,.g., h proof of Exampl 5. n [0]. Inroducng Lagrang mulplrs µ R + for nqualy consran P N T = appl P and = [,..., NT ] T R N T + for nqualy consrans {,,...,N T }. L = [,..., N T ] T and (µ, ) b any prmal and dual opmal pons wh zro dualy gap. By KKT condons, w hav + + µ = 8 {,,...,N T }; P N T = appl P ; µ 0; µ P N T = P = 0; 8 {,,...,N T }; 8 {,,...,N T }; = 8 {,,...,N T }. Elmnang, 8 {,,...,N T } n all quaons ylds µ +, 8 {,,...,N T }; P N T = appl P ; µ 0; µ P N T = P = 0; 8 {,,...,N T };(µ + ) = 8 {,,...,N T }. For all {,,...,N T }, w consdr µ < and µ sparaly: ) If µ <, hn µ + holds only whn > whch by (µ +Z/V + ) mpls ha µ + =.., =. ) If µ, hn > 0 s mpossbl, bcaus > 0 mpls ha µ + > whch oghr wh > 0 conradc h slacknss condon (µ + ) =0. Thus, f µ, w mus hav =0. Summarzng boh cass, w hav = max, 8 {,,...,NT }, whr µ s chosn such ha P n = appl P, µ 0 and µ P N T = P =0. To fnd such µ, w frs chck f µ = 0. If µ = 0 s ru, h slacknss condon µ P N T = = 0 s guarand o hold and w nd o furhr rqur P N T P = = NT = max appl P. Thus µ =0f and only f P N T = max Z/V appl P. Thus, Algorhm chcks f P N T = max Z/V appl P holds a h frs sp and f hs s ru, hn w conclud µ =0and w ar don! Ohrws, w know µ > 0. By h slacknss condon µ P N T = P = w mus hav P N T = = P NT = max = P. To fnd µ > 0 such ha P NT = max = P, w could apply a bscon sarch by nong ha all ar dcrasng wh rspc o µ. Anohr algorhm of fndng µ s nsprd by h obsrvaon ha f j k, 8j, k {,,...,N T }, hn j k. Thus, w frs sor all n a dcrasng ordr, say s h prmuaon such ha () () (N T ); and hn squnally chck f {,,...,N T } s h ndx such ha () µ 0 and (+) µ appl 0. To chck hs, w frs assum s ndd such an ndx and solv h quaon = P o oban µ ; (No ha n P j= (j) Algorhm, o avod rcalculang h paral sum P j= (j) for ach, w nroduc h paramr S = P j= and (j) upda S ncrmnally. By dong hs, h complxy of ach raon n h loop s only O().) hn vrfy h assumpon by chckng f 0 and () appl 0. (+) Ths algorhm s dscrbd n Algorhm. APPENDIX B PROOF OF THEOREM 3 Clam. If b s an opmal soluon o h followng convx program: mn k k F (7) s.. r( ) appl P (8) S N T + (9) hn b Q = U H b U s an opmal soluon o problm (6)-(8).

9 PROC. IEEE INFOCOM 06 Proof: Ths clam can b provn by conradcon. L b b an opmal soluon o convx program (7)-(9) and dfn Q b = U H U. b Assum ha hr xss Q S N T + such ha Q 6= Q b and s a soluon o problm (6)-(8) ha s srcly br han Q. b Consdr = UQU H and rach a conradcon by showng s srcly br han b as follows: No ha r( )=r(u QU H )=r( Q) appl P, whr h las nqualy follows from h assumpon ha Q s soluon o problm (6)-(8). Also no ha S N T + snc Q S N T +. Thus, s fasbl o problm (7)-(9). No ha k (a) k F = ku H U U H (b) Uk F = kq (c) Xk F < kq b (d) Xk F = kuqu b H UXU H () k F = k b k F, whr (a) and (d) follow from h fac Frobnus norm s unary nvaran 4 ; (b) follows from h fac ha = UQU H and X = U H U; (c) follows from h fac ha Q s srcly br han Q; b and () follows from h fac ha Q b = U H U b and X = U H U. Thus, s srcly br han. b A conradcon! Clam 3. Th opmal soluon o problm (7)-(9) mus b a dagonal marx. Proof: Ths clam can b provn by conradcon. Assum ha problm (7)-(9) has an opmal soluon ha s no dagonal. Snc s posv smdfn, all h dagonal nrs of ar non-ngav. Dfn b as a dagonal marx whos h -h dagonal nry s qual o h -h dagonal nry of for all {,,...,N T }. No ha r( b )=r( ) appl P and b S n +. Thus, b s fasbl o problm (7)-(9). No ha k b k F < k k F snc s dagonal. Thus, b s a soluon srcly br han. A conradcon! So h opmal soluon o problm (7)-(9) mus b a dagonal marx. By h abov wo clams, suffcs o assum ha h opmal soluon o problm (6)-(8) has h srucur ˆQ = U H U, whr s a marx wh non-ngav nrs. To solv problm (6)-(8), suffcs o consdr h followng convx program. mn s.. XN T ( ) (30) = XN T appl P (3) = 8 {,,...,N T } (3) No ha problm (30)-(3) sasfs Slar s condon. So h opmal soluon o problm (30)-(3) s characrzd by KKT condons [0]. Th rmanng par s smlar o h proof of Thorm and can b found n h full vrson [7]. REFERENCES [] S. K. Jayawra and H. V. 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