Joint Channel Estimation and Resource Allocation for MIMO Systems Part I: Single-User Analysis

Transcription

1 624 IEEE RANSACIONS ON WIRELESS COUNICAIONS, VOL. 9, NO. 2, FEBRUARY 200 Jon Channel Esmaon and Resource Allocaon for IO Sysems Par I: Sngle-User Analyss Alkan Soysal, ember, IEEE, and Sennur Ulukus, ember, IEEE Absrac ulple anenna sysems are known o provde very large daa raes, when he perfec channel sae nformaon CSI s avalable a he recever. However, hs requres he recever o perform a nose-free, mul-dmensonal channel esmaon, whou usng communcaon resources. In pracce, any channel esmaon s nosy and uses sysem resources. We shall examne he rade-off beween mprovng channel esmaon and ncreasng he achevable daa rae. We consder ransmsde correlaed mul-npu mul-oupu IO channels wh block fadng, where each block s dvded no ranng and daa ransmsson phases. he recever has a nosy CSI ha obans hrough a channel esmaon process, whle he ransmer has paral CSI n he form of covarance feedback. In Par I of hs wo-par paper, we consder he sngle-user case, and opmze he achevable rae jonly over parameers assocaed wh he ranng phase and daa ransmsson phase. In parcular, we frs choose he ranng sgnal o mnmze he channel esmaon error, and hen, develop an erave algorhm o solve for he opmum sysem resources such as me, power and spaal dmensons. Specfcally, he algorhm fnds he opmum ranng duraon, he opmum allocaon of power beween ranng and daa ransmsson phases, he opmum allocaon of power over he anennas durng he daa ransmsson phase. Index erms IO, paral CSI, covarance feedback, opmum power allocaon, channel esmaon. I. INRODUCION IN wreless communcaon scenaros, he achevable rae of a sysem depends crucally on he amoun of CSI avalable a he recevers and he ransmers. he CSI s observed only by he recever, whch can esmae and feed he esmaed CSI back o he ransmer. However, measurng he CSI and feedng back o he ransmer uses communcaon resources such as me, power and spaal dmensons, whch could oherwse be used for useful nformaon ransmsson. here have been several dfferen assumpons n he leraure on he avalably of he CSI a he recever and he ransmer. Wh perfec CSI a he recever and he ransmer, he opmum adapaon scheme anuscrp receved January 23, 2008; revsed Augus 2, 2009; acceped November 2, he assocae edor coordnang he revew of hs paper and approvng for publcaon was. Chang. hs work was suppored by NSF Grans CCR 03-3, CCF and CCF ; and ARL/CA Gran DAAD , and presened n par a he Conference on Informaon Scences and Sysems, Balmore, D, arch A. Soysal s wh he Deparmen of Elecrcal and Elecroncs Engneerng, Bahçeşehr Unversy, Isanbul, urkey e-mal: alkan.soysal@bahcesehr.edu.r. S. Ulukus s wh he Deparmen of Elecrcal and Compuer Engneerng, Unversy of aryland, College Park, D, USA e-mal: ulukus@umd.edu. Dgal Objec Idenfer 0.09/WC /0$25.00 c 200 IEEE s waer-fllng 2, 3. However, n some cases, especally n IO lnks, feedng he nsananeous CSI back o he ransmer s no realsc. herefore, some research assumes ha here s perfec CSI a he recever, bu only paral CSI avalable a he ransmer 4 8. Anoher lne of research consders he acual esmaon of he channel a he recever, whch s nosy. he capacy and he correspondng opmum sgnallng scheme for hs case are no known. However, lower and upper bounds for he capacy can be obaned 9. I s mporan o noe ha 9 do no consder opmzng he channel esmaon process, because of he assumpon of he exsence of a separae channel ha does no consume sysem resources for channel esmaon. For a sngle-user mulple-anenna sysem wh no CSI avalable a he ransmer, 2 consders opmzng he achevable rae as a funcon of boh he ranng and he daa ransmsson phases. Snce here s no CSI feedback, he ransmer power allocaon s consan over he channel saes and he anennas. Par I of hs wo-par paper consders a sngle-user, blockfadng, ransm-sde correlaed IO channel wh nosy channel esmaon a he recever, and paral CSI avalable a he ransmer. he CSI feedback ha we consder les somewhere beween perfec CSI and no CSI 2, and s smlar o 4 8. We consder he fac ha he ranng phase uses communcaon resources, and we opmze he achevable rae of he daa ransmsson phase over he parameers of he ranng and daa ransmsson processes. he ranng phase s characerzed by hree parameers, namely, he ranng sgnal, he ranng sequence lengh and he ranng sequence power. Smlarly, he daa ransmsson phase s characerzed by he daa carryng npu sgnal, daa ransmsson lengh, and he daa ransmsson power. Assumng ha he recever uses lnear mnmum mean square error SE deecon o esmae he channel durng he ranng phase, we frs choose he ranng sgnal ha mnmzes he SE. hs choce also ncreases he achevable rae of he daa ransmsson phase 2. However, unlke 2, our resul does no necessarly allocae equal power over he anennas, and mgh no esmae all of he avalable channel varables. hen, we move o he daa ransmsson phase, and maxmze he achevable rae of he daa ransmsson jonly over he res of he ranng phase parameers, and daa ransmsson phase parameers. Specfcally, we frs fnd he opmum paron of he oal ransmer power and he block lengh beween he ranng and he daa ransmsson phases. hen, we fnd he opmum allocaon of he daa ransmsson power over he anennas durng he daa ransmsson phase.

2 SOYSAL and ULUKUS: JOIN CHANNEL ESIAION AND RESOURCE ALLOCAION FOR IO SYSES PAR I: SINGLE-USER ANALYSIS 625 ABLE I SYBOLS USED IN HIS PAPER IN ORDER OF APPEARANCE n R n H x n n n CN0, I P Z P P d S N R Ĥ ˆ ˆλ H λ U Λ S λ S Λ λ μ S Ẑ λ number of receve anennas number of ransm anennas n R n ransm-sde correlaed channel marx ransmsson block duraon ranng phase duraon daa ransmsson phase duraon n channel npu a me n n R channel nose dsrbued as a zero-mean, deny covarance complex Gaussan vecor oal average power consran n R n..d. channel marx n n covarance marx power allocaed o ranng phase power allocaed o daa ransmsson phase n ranng sgnal n R nose marx n ranng phase n R receved marx n ranng phase n R n esmae of he channel marx n n covarance of he channel esmae egenvalues of ˆ n R n channel esmaon error n n covarance of he channel esmaon error egenvalues of n SE esmaor marx n n unary marx of he egenvecors of dagonal egenvalue marx of S egenvalue of S dagonal egenvalue marx of egenvalue of Lagrange mulpler assocaed wh channel esmaon mnmzaon n n covarance marx of he channel npu n R n..d. channel marx n vecor of egenvalues of Overall, we fnd he opmum allocaon of resources such as power, me and spaal dmensons. Par II 3 exends he resuls of Par I o mulple-access channels. I also provdes exensve numercal analyss for boh sngle-user and muluser scenaros. Symbols used n he paper are summarzed n able. II. SYSE ODEL We consder a pon-o-pon channel wh n R anennas a he recever and n anennas a he ransmer. he channel beween he ransmer and he recever s represened by an n R n mensonal random marx H. We consder a block fadng scenaro where he channel remans consan for ablock symbols, and changes o an..d. realzaon a he end of he block. In order o esmae he channel, he recever performs a lnear SE esmaon usng ranng symbols over symbols. Durng he remanng = symbols, daa ransmsson occurs. Whle he recever has a nosy esmae of he realzaon of he fadng channel, he ransmer has only he sascal model of he channel. A me n, he ransmer sends a vecor x n, and he receved vecor s r n = Hx n + n n, n =,..., where n n CN0, I, and he enres of H are complex Gaussan random varables. he ransmer has a power consran P = E n xh n x n, averaged over symbols. he sascal model ha we consder n hs paper s he paral CSI wh covarance feedback model where each ransmer knows he channel covarance nformaon of all ransmers, n addon o he dsrbuon of he channel. In hs model, here exss correlaon beween he sgnals ransmed by or receved a dfferen anenna elemens. For each user, he channel s modeled as 4, H = Φ /2 Z /2 2 where he enres of Z are..d., zero-mean, un-varance complex Gaussan random varables, he receve anenna correlaon marx, Φ, s he correlaon beween he sgnals receved a he n R receve anennas of he recever, and he ransm anenna correlaon marx,, s he correlaon beween he sgnals ransmed from he n ransm anennas of he user. In hs paper, we wll assume ha he recever does no have any physcal resrcons and herefore, here s suffcen spacng beween he anenna elemens on he recever such ha he sgnals receved a dfferen anenna elemens are uncorrelaed. As a resul, he receve anenna correlaon marx becomes he deny marx 2,.e., Φ = I. Now, he channel s wren as H = Z /2 3 From hs pon on, we wll refer o marx as he channel covarance feedback marx. Smlar covarance feedback models have been used n 4 8. III. JOIN OPIIZAION FOR SINGLE-USER IO In our model he channel s fxed over a coherence nerval, whch s dvded no wo phases: ranng phase and daa ransmsson phase; see Fgure. he ransmer uses P amoun of power durng he ranng phase and P d amoun of power durng he daa ransmsson phase. Due o he conservaon of energy, we have P = P + P d. he opmzaon creron ha we consder s he achevable rae of he daa ransmsson phase. Unlke he case wh perfec channel esmaon, he daa rae here depends on he esmaon parameers: ranng sgnal S, ranng sgnal power P, and ranng sgnal duraon. As a resul, here s a rade-off beween he ranng and daa ransmsson parameers. A longer ranng phase wll resul n a beer channel esmae a lower channel esmaon error. hs n urn, resuls n a hgher achevable rae, snce he effecve nose s lower. However, a longer ranng phase mples a shorer daa ransmsson phase, as he block lengh coherence me s fxed. A shorer daa ransmsson phase, n urn, Noe ha snce he nose power s assumed o be uny, P s n fac he relave power wh respec o nose power. I can be regarded as an SNR value. 2 Exenson of our resuls o arbrary Φ,.e.,hecasewherehechannel has double-sded correlaon srucure, can be carred ou n a sragh-forward manner as n 8.

3 626 IEEE RANSACIONS ON WIRELESS COUNICAIONS, VOL. 9, NO. 2, FEBRUARY 200 Fg.. fxed. ranng Sequence Dae Sequence P, S P d,x Illusraon of a sngle coherence me, over whch he channel s mples a smaller achevable rae. A smlar rade-off s vald also for he ranng power. Here, we wll solve hs radeoff, and fnd he opmum ranng and daa ransmsson parameers. A. ranng and Channel Esmaon Phase In praccal communcaon scenaros, he channel s esmaed a he recever. One way of dong hs s o use ranng symbols before he daa ransmsson sars. he recever esmaes he channel usng hese ranng sgnals and he oupu of he channel. Snce he channel says he same durng he enre block, we can wre he npu-oupu relaonshp durng he ranng phase n a marx form as R = HS + N 4 where S s an n dmensonal ranng sgnal ha wll be chosen and known a boh ends, R and N are n R dmensonal receved sgnal and nose marces, respecvely. he n h column of he marx equaon n 4 represens he npu-oupu relaonshp a me n. he power consran for he ranng npu sgnal s rss P. Due o our channel model n 3, he enres n a row of H are correlaed, and he enres n a column of H are uncorrelaed,.e., row of he channel marx s..d. wh row j. Le us represen row of H as h, wh Eh h =,=,...n R. Snce rows are..d., he recever can esmae each of hem ndependenly usng he same ranng sgnal. Row of 4, whch represens he receved sgnal a he h anenna of he recever over he ranng duraon, can be wren as r = S h + n. 5 he recever wll esmae h usng he receved sgnal r,and he ranng sgnal S. In general, he esmae ĥ can be se o any funcon of S and r. However, s common o use and easer o mplemen lnear SE esmaon. Also, when he random varables nvolved n he esmaon are Gaussan, as n Raylegh fadng channels, lnear SE esmaon s opmal. In order o fnd he lnear SE esmaor, we solve he followng opmzaon problem wh ĥ = r as he esmae of h,and h = h ĥ as he channel esmaon error, mn h E h =mn E r h h 6 =mn E r h r h r. 7 Solvng he opmum ransformaon marx, from 7 s equvalen o solvng from he orhogonaly prncple for vecor random varables, whch s gven as 5, page 9, E h r r = 0 8 where 0 s he n zero marx. We can solve from 8 as = E h r E r r. 9 By usng 5, we calculae Eh r =S, ander r = S S + I. hen, becomes = SS S + I.By nserng no 7, he mean square error becomes, mn h E h = r SS S + I S 0 = r + SS where he las lne follows from he marx nverson lemma 6, page 9. Noe ha he mean square error of he channel esmaon process can be furher decreased by choosng he ranng sgnal S o mnmze. In addon, s saed n 2 ha he ranng sgnal S prmarly affecs he achevable rae hrough he so called effecve sgnal-o-nose rao, whch s shown o be nversely proporonal o he SE 2. herefore, choosng S o furher mnmze he SE, we also ncrease he achevable rae of he daa ransmsson phase. he followng heorem fnds he opmal ranng sgnal for a gven ranng power and ranng duraon. heorem : For gven = U Λ U, P,,andhe power consran rss P, he opmum ranng npu ha mnmzes he power of he channel esmaon error vecor s S = U Λ /2 S wh λ S = + μ S λ, =,...,mnn, 2 where μ 2 S s he Lagrange mulpler ha sasfes he power consran wh μ S =,andj s he larges ndex J P + J λ ha has non-zero λ S. Proof: Le us have S = U S Λ /2 S V S. he expresson n s mnmzed when and SS have he same egenvecors 7. herefore, we have U S = U. Snce, SS = U S Λ S U S, and he unary marx V S does no appear n he objecve funcon and he consran, we can choose V S = I. Inserng hs no, he opmzaon can be wren as σ = mn rλ S P r Λ + Λ S. 3 he Langrangan of he problem n 3 can be wren as n n + λ S + μ 2 S λ S P 4 λ where μ 2 S s he Lagrange mulpler. he Lagrangan s a convex funcon of λ S, herefore he soluon ha sasfes he Karush-Kuhn-ucker KK condons s he unque opmum soluon. hs gves us 2, whch s waer-fllng over he egenvalues of he channel covarance marx. In order o calculae μ S, we sum boh sdes of 2 over all anennas o ge μ S =,wherej s he larges ndex ha J P + J λ has non-zero λ S. I s mporan o noe ha for any gven P,and > n, he egenvalues of S do no conan he ranng lengh parameer. Increasng beyond n oes no resul n beer

4 SOYSAL and ULUKUS: JOIN CHANNEL ESIAION AND RESOURCE ALLOCAION FOR IO SYSES PAR I: SINGLE-USER ANALYSIS 627 channel esmaes. On he oher hand, larger wll resul n smaller daa ransmsson lengh, and decrease he achevable rae of he daa ransmsson phase. herefore, s suffcen o consder only n, whch we wll assume hrough he res of hs paper. heorem ells us ha he opmum ransm drecons of he ranng sgnal are he egenvecors of he channel covarance marx, and he rgh egenvecor marx of he ranng sgnal s deny. As a resul, he columns of S are he weghed columns of a unary marx, and hey are orhogonal. Snce each column of S s ransmed a a channel use durng he ranng phase, vecors ha are ransmed a each channel use durng he ranng phase are orhogonal o each oher. hs means ha, a each channel use, s opmal o ran only one dmenson of he channel along one egenvecor. oreover, he opmum power allocaon polcy for he ranng power s o waer-fll over he egenvalues of he channel covarance marx usng 2. Dependng on he power consran and he ranng sgnal duraon, some of he egenvalues of he ranng sgnal mgh urn ou o be zero. hs means ha some of he channels along he drecons correspondng o zero egenvalues of he ranng sgnal, are no even raned. Noe ha μ S s a funcon of only P and, whch are gven o he problem n heorem, and wll be pcked as a resul of he achevable rae maxmzaon problem n he daa ransmsson phase. he value of deermnes he oal number of avalable parallel channels n he channel esmaon problem, and he value of P deermnes he number of channels ha wll be esmaed. he paramerc values of P and wll appear n he achevable rae formula n he daa ransmsson phase. Afer he rae maxmzaon s performed, he opmum P and wll be found, and hese n urn, wll gve us S hrough heorem. Before movng on o he nex secon, we wll calculae he egenvalues of he covarance marces of he esmaed channel vecor, and he channel esmaon error vecor. Pluggng S no he covarance of he channel esmaon error, = E h h U Λ + Λ S usng 2 as = + SS, we fnd = U, where he egenvalues can be found { λ μs, μ = S <λ ; λ, μ S >λ =mn λ,μ S. 5 Noe ha along he drecons ha we send ranng sgnals,.e., when he correspondng egenvalues of he ranng sgnal are non-zero μ S < λ, he varance of he channel esmaon error s he same for all drecons. Conversely, for he drecons where no ranng sgnals are sen, he varance of he channel esmaon error s equal o he varance of he channel along ha drecon. hs s expeced, snce he channel s no esmaed along ha drecon, he error n he channel esmaon process s he same as he realzaon of he channel self. Nex, we wll calculae he egenvalues of he covarance of he channel esmae. Usng he orhogonaly propery of he SE esmaon, ĥ and h are uncorrelaed 5, page 9. he covarance marx of he channel esmae ˆ = E ĥ ĥ becomes ˆ = U Λ U U Λ U 6 = U Λ Λ U U ˆΛ U 7 whch has he same egenvecors as he covarance marx of he acual channel, however, her egenvalues are dfferen. We can wre each egenvalue of he covarance marx of he esmaed channel as ˆλ =max 0,λ μ S. Along he drecons ha we do no send ranng sgnals, he value of he channel esmae self s zero. herefore, as expeced, he power of he esmaed channel s zero as well, along hose channels wh μ S >λ. B. Daa ransmsson Phase When he CSI a he recever s nosy, he opmum npu sgnalng ha acheves he capacy s no known. Followng 9 2, we derve a lower bound.e., an achevable rae on he capacy for our model, and fnd he ranng and daa ransmsson parameers ha resul n he larges such achevable rae. Usng he channel esmaon error, H = H Ĥ, we can wre as r = Ĥx + Hx + n. 8 where x s he nformaon carryng npu, and n CN0, I. Le = Exx be he ransm covarance marx, whch has an average power consran 3 of P d,r P d. Alhough he opmum npu dsrbuon s no known, we acheve he followng rae wh Gaussan x for a IO channel, C lb = Ir; xĥ E Ĥ log I + R Hx+n ĤĤ 9 where R Hx+n = I + E H H H s he covarance marx of he effecve nose, Hx + n. By denong each row of H as h, we can wre he, jh enry of E H H H as, E h h j = r E { r, when = j h h j = 0, when = j 20 whch resuls n E H H = r I. Snce our goal s o fnd he larges such achevable rae, he rae maxmzaon problem over he enre block becomes R = max,p, S r P d EĤ log I + ĤĤ +r 2 { } where S =,P, r + P = P, and he coeffcen reflecs he amoun of me spen durng he ranng phase. he maxmzaon s over he ranng parameers P,and, and he daa ransmsson parameer, whch can be decomposed no s egenvecors he ransm drecons, and egenvalues powers along he ransm drecons. 3 Noe ha snce he nose power s assumed o be uny, P d and he egenvalues of are relave power values wh respec o he nose power. hey can be regarded as SNR values.

5 628 IEEE RANSACIONS ON WIRELESS COUNICAIONS, VOL. 9, NO. 2, FEBRUARY 200 ransm Drecons: Unlke he case wh no-csi a he ransmers 2, n hs paper, he opmum ransm covarance marx s no equal o he deny marx. In hs case, he problem becomes ha of choosng he egenvecors he ransm drecons, and he egenvalues he powers allocaed o he ransm drecons, of he ransm covarance marx = U Λ U, o maxmze 2. When he CSI a he recever s perfec, 5 showed ha he egenvecors of he ransm covarance and he channel covarance marces mus be equal,.e., U = U.Inhe nex heorem, we show ha hs s also rue when here s channel esmaon error a he recever. heorem 2: Le = U Λ U be he specral decomposon of he covarance feedback marx of he channel. hen, he opmum ransm covarance marx has he form = U Λ U. Proof: We have shown n 5 and 7 ha, when = U Λ U,wehave ˆ = U ˆΛ U,and = U Λ U. By usng 3, we have Ĥ = ẐU /2 ˆΛ U. Inserng hese no 2, R =max EẐ log,p, S I + Ẑ ˆΛ /2 U U /2 ˆΛ Ẑ r P d +r U U Λ 22 where we used he fac ha he random marces ẐU and Ẑ have he same dsrbuon for zero-mean deny-covarance Gaussan Ẑ and unary U 3. We may specrally decompose he expresson sandwched beween Ẑ and s conjugae ranspose n 22 as ˆΛ /2 U U /2 ˆΛ = UΛU. 23 Usng 23, and he deny rab = rba, we can wre he race expresson n he denomnaor of 22 as r U U Λ = r U ˆΛ Λ UΛ, and he opmzaon problem n 22 can be wren as R =max EẐ log,p, S I + ẐΛẐ r P d +r U ˆΛ Λ UΛ 24 where we agan used he fac ha he random marces ẐU and Ẑ have he same dsrbuon. In 24, he numeraor of he objecve funcon does no nvolve U, and usng 8, heorem 9.H..h, page 249, we know ha r ˆΛ Λ Λ ru ˆΛ Λ UΛ, for all unary U. herefore, we can choose U = I o maxmze he rae as long as hs choce s feasble. In order o check for he feasbly, we wre he race consran on usng 23 as r =ru ˆΛ /2 UΛU ˆΛ /2 U =ru ˆΛ UΛ. 25 Agan from 8, heorem 9.H..h, page 249, r ˆΛ Λ ru ˆΛ UΛ P d, for all unary U. herefore, we conclude ha U = I choce s feasble. hen, usng U = I, from 23, we have he desred resul, = U ˆΛ ΛU wh Λ = ˆΛ Λ. Usng heorem 2, we can wre he opmzaon problem n 2 as, n R = max Eẑ log λ,p, P I + λˆλ ẑẑ + n λ λ 26 where ẑ CN0, I s he h column of Ẑ, he se of { vecors {ẑ } are..d, λ = λ,...,λ n, and P = λ n },P, λ + P = P. 2 Power Allocaon Polcy: In a IO sysem, a ransm sraegy s a combnaon of a ransm drecon sraegy, and a ransm power allocaon sraegy, whch s he se of opmum egenvalues of he ransm covarance marx, λ, ha solves 26. Alhough heorem 2 gves us a very smple closed form soluon for he opmum ransm drecons, solvng 26 for λ n a closed form does no seem o be feasble due o he expecaon operaon n he objecve funcon. herefore, we wll develop an erave algorhm ha fnds he opmum λ. For a sngle-user IO sysem wh perfec CSI a he recever and paral CSI a he ransmer n he form of covarance feedback, an algorhm ha fnds he opmum power allocaon polcy s proposed n 7. In hs secon, we exend he algorhm n 7 o he case when here s channel esmaon error a he recever, or n oher words, when we have he ranng sgnal power and he ranng sgnal duraon n he sum-rae expresson. he algorhm n 7 canno be rvally generalzed o he model n hs paper, snce, here we have he ranng power P, and he ranng duraon as addonal parameers. By pluggng ˆλ and λ no 26, we ge J R = max E log λ,p, P I+ λ λ μ S ẑ ẑ + J λ μ S + α 27 where α = n =J+ λ λ. Noe ha J and μ S are funcons of P and.snceλ,for = J +,...,n oes no conrbue o he numeraor, we should choose λ =0,for = J +,...,n. hs s o be expeced, because we have raned only J ransm drecons, and we should now solve for J power values along hose drecons. Consequenly, we have J R = max E log λ,p, P I + λ λ μ S ẑ ẑ. +μ S P d 28 From heorem, we know ha J. We furher clam ha whle opmzng he rae, s suffcen o search over hose P, pars ha resul n J =. In oher words, for any par P, ha resuls n J<, we can fnd anoher par P, ha resuls n a hgher achevable rae. In order o see hs, consder a par P, ha resuls n J<,hen le us choose = J. For hs choce, he resul of heorem s he same, snce he avalable power can only fll J of he parallel channels. herefore wh P, =P,J, he esmaon process yelds he same channel esmae. When we look a 28, we see ha nsde of he expecaon s he same for boh P, and P,. However, he coeffcen n fron of he expecaon s hgher wh P,, sncej = <.

6 SOYSAL and ULUKUS: JOIN CHANNEL ESIAION AND RESOURCE ALLOCAION FOR IO SYSES PAR I: SINGLE-USER ANALYSIS 629 herefore P, yelds a hgher achevable rae and s suffcen o search over hose P, pars ha resul n J =. We can now wre 28 as R = max E log λ,p, R I + λ λ μ S ẑ ẑ +μ S P d 29 { λ where R = n,p, λ + P = P, P > }, and he condon λ λ λ λ P > channels are flled,.e., J =. guaranees ha, usng P,,all Noe ha he parameers ha we wan o opmze 29 over are dscree valued, and connuous valued P,andλ. Snce s dscree, and n, we can perform an exhausve search over and solve n reduced opmzaon problems wh fxed a each one. hen, we ake he soluon ha resuls n he maxmum rae,.e., R = max n R 30 where R = max E log λ,p R I+ λ λ μ Sẑ ẑ +μ S P d { λ and R = n,p λ λ λ 3 + P = P, P > }. Whle solvng he nner maxmzaon problem, we defne f P = λ μs +μ SP d,for =,...,.In hs case, he nner opmzaon problem becomes R = max E log λ,p R I + λ f P ẑ ẑ.32 Noe ha, for he nner opmzaon problem, n addon o,fp was fxed, f P would also be fxed. In hs case, he problem n 32 would become exacly he same as he correspondng convex opmzaon problem wh perfec CSI assumpon a he recever 7, where here, f P replaces λ n 7, equaon 8. In he opmzaon problem n 32, we have + opmzaon varables, λ,...,λ,andp. In hs case, he problem s no necessarly convex due o he exsence of P. Equaon 32 s concave when =, whch resuls n an affne f P. herefore, n he mos general case, he soluon of he frs order necessary condons wll gve a local maxmum. he Lagrangan for 32 can be wren as E log I + λ f P ẑ ẑ μ + P P 33 where μ s he Lagrange mulpler, and we omed he complemenary slackness condons relaed o he posveness λ of λ and P λ λ. By usng he deny, x log A+xB = r A + xb B whchsprovedn5, he KK condons can be wren by akng he dervave of he Lagrangan wh respec o λ s and P, d f P E z A z μ, =,..., 34 λ E z f P A z = μ 35 P λ λ where A = I + λ f P ẑ ẑ, and he equaly of he las equaon follows from he complemenary slackness.ifhecom- condon, whch says P > plemenary slackness condon s no sasfed,.e., f we had P ou of λ λ, hen a leas one of he channels channels could no be flled,.e., J <, and herefore hs choce of P, par s no opmal. herefore, he complemenary slackness condon s always sasfed, resulng n he equaly n 35. he h nequaly n 34 s sasfed wh equaly whenever he opmum λ s non-zero, and wh src nequaly whenever he opmum λ s zero. Due o he expecaon erm, we canno drecly solve for λ n 34. Insead, we mulply boh sdes of 34 by λ, λ f P E z A z = λ μ, =,..., 36 We noe ha when λ = 0, boh sdes of 36 are equal o zero. herefore, unlke 34, 36 s always sasfed wh equaly for opmum egenvalues. By summng boh sdes over all anennas, we fnd μ, and by subsung hs μ no 36, we fnd he fxed pon equaons whch have o be sasfed by he opmum egenvalues, λ λ f P E z A z = P d, =,...,. n j= λ j f jp E z j A z j 37 hs gves a se of fxed pon equaons ha can be used o updae λ, however, we also need a fxed pon equaon o updae he value of P. For hs, we look a he las KK equaon. Noe ha when he opmum λ s non-zero, he correspondng nequaly n 34 wll be sasfed wh equaly due o s correspondng complemenary slackness condon. herefore, we pull he expecaon erms from 34 for hose equaons wh non-zero λ s, and nser hem no 35. Snce hose ndces wh λ =0do no conrbue o 35, we have λ f P f P = 38 where we canceled μ s ou on boh sdes. Now, we can use hs fxed-pon equaon o solve P n erms of λ s. We propose he followng fxed-pon algorhm: a any gven eraon, he

7 630 IEEE RANSACIONS ON WIRELESS COUNICAIONS, VOL. 9, NO. 2, FEBRUARY n = 3, P = 20 db, = 0, = 250 n = 3, P = 20 db, = 0, = power values R = 7.23 bs/symbol P λ power values R = 9.4 bs/symbol P λ λ eraon ndex a Fg. 2. he convergence of he sngle-user algorhm wh n = n R =3, 20 db oal average power, =0, and one symbol long ranng, = eraon ndex a Fg. 3. he convergence of he sngle-user algorhm wh n = n R =3, 20 db oal average power and = 0, and wo symbols long ranng, =2. algorhm frs solves 38 for he nex updae of P usng λ nf P n + f P n + = n = 3, P = 20 db, = 0, = 3 and hen, updaes λ n +usng 37 for =,..., λ λ nf P n +E z A z n += P d n j= λ j nf jp n +E z j A z j 40 where P d = P Pn+. hs algorhm fnds he soluon for he ranng power P, and he egenvalues of he ransm covarance marx λ,...,λ,forafxed,for n.werunn such algorhms, and he soluon of 29 s found by akng he one ha resuls n he larges rae, whch gves us. As a resul, we solved he jon channel esmaon and resource allocaon problem ha we consdered n hs paper. hrough and P,wefnd he allocaon of avalable me and power over he ranng and daa ransmsson phases, snce oal block lengh and power s fxed. hrough heorem 2, we fnd he opmum ransm drecons, and hrough λ,...,λ,wefndhe allocaon of daa ransmsson power over hese ransm drecons. Fnally, he ranng sgnal S s deermned by and P hrough heorem. Analycal proof of he convergence of hs algorhm seems o be more complcaed han he proof n he case when here s no channel esmaon error 7, and seems o be nracable for now. However, n our exensve smulaons, we observed ha he algorhm always converged. As an example, n Fgures 2-4, we consder a sysem wh n = n R =3havng SNR, P = 20 db, and block lengh, = 0. For hs sysem, we run our algorhm for all hree possble values of he ranng symbol duraon,.e., =, 2, 3. We observe n Fgures 2-4 ha esmang wo of he hree dmensons of he channel s opmum for hs seng. Snce he power s relavely hgh wh respec o he egenvalues of he channel power values R = 8.62 bs/symbol eraon ndex a Fg. 4. he convergence of he sngle-user algorhm wh n = n R =3, 20 dboal average power and = 0, and hree symbols long ranng, =3. covarance, should be allocaed almos equally among he spaal dmensons when here s perfec CSI a he recever. We see ha hs s also he case n our model as well. he only dfference s ha he power s allocaed almos equally o he spaal dmensons ha are raned. We refer he reader o he Par II 3 of hs wo-par paper for a more dealed numercal analyss. IV. CONCLUSIONS We analyzed he jon opmzaon of he channel esmaon and daa ransmsson parameers of a a sngle-user IO block-fadng channel where he recever has a nosy esmae of he channel and he ransmer has he paral CSI n he form of covarance feedback. Frs he opmum ranng sgnal o mnmze he SE s found, and hen, we formulaed he jon opmzaon problem over he egenvalues of he ransm covarance marx and he channel esmaon parameers. hs P λ λ 2 λ 3

8 SOYSAL and ULUKUS: JOIN CHANNEL ESIAION AND RESOURCE ALLOCAION FOR IO SYSES PAR I: SINGLE-USER ANALYSIS 63 s solved by nroducng a number of reduced opmzaon problems, each of whch can be solved effcenly usng he proposed erave algorhm. hrough smulaons, s observed ha he proposed erave algorhm converges o he same pon regardless of he nal pon of he eraons. REFERENCES A. Soysal and S. Ulukus, Opmzng he rae of a correlaed IO lnk jonly over channel esmaon and daa ransmsson parameers, n Proc. Conference on Informaon Scences and Sysems, ar A. J. Goldsmh and P. P. Varaya, Capacy of fadng channels wh channel sde nformaon, IEEE rans. Inf. heory, vol. 43, no. 6, pp , Nov İ. E. elaar, Capacy of mul-anenna Gaussan channels, European rans. elecommun., vol. 0, no. 6, pp , Nov E. Vsosky and U. adhow, Space-me ransm precodng wh mperfec feedback, IEEE rans. Inf. heory, vol. 47, no. 6, pp , Sep S. A. Jafar and A. Goldsmh, ransmer opmzaon and opmaly of beamformng for mulple anenna sysems, IEEE rans. Wreless Commun., vol. 3, no. 4, pp , July H. Boche and E. Jorsweck, On he opmaly range of beamformng for IO sysems wh covarance feedback, IEICE rans. Commun., vol. E85-A, no., pp , Nov A. Soysal and S. Ulukus, Opmum power allocaon for sngle-user IO and mul-user IO-AC wh paral CSI, IEEE J. Sel. Areas Commun., vol. 25, no. 7, pp , Sep A. Soysal and S. Ulukus, Opmaly of beamformng n fadng IO mulple access channels, IEEE rans. Commun., vol. 57, no. 4, pp. 7 83, Apr édard, he effec upon channel capacy n wreless communcaons of perfec and mperfec knowledge of he channel, IEEE rans. Inf. heory, vol. 46, no. 3, pp , ay E. Klen and R. G. Gallager, Power conrol for he addve whe Gaussan nose channel under channel esmaon errors, n Proc. ISI, June Yoo and A. Goldsmh, Capacy and power allocaon for fadng IO channels wh channel esmaon error, IEEE rans. Inf. heory, vol. 52, no. 5, pp , ay B. Hassb and B.. Hochwald, How much ranng s needed n mulple-anenna wreless lnks? IEEE rans. Inf. heory, vol. 49, no. 4, pp , Apr A. Soysal and S. Ulukus, Jon channel esmaon and resource allocaon for IO sysems par II: mul-user and numercal analyss, IEEE rans. Wreless Commun., vol. 9, no. 2, pp , Feb C. Chuah, D. N. C. se, J.. Kahn, and R. A. Valenzuela, Capacy scalng n IO wreless sysems under correlaed fadng, IEEE rans. Inf. heory, vol. 48, no. 3, pp , ar E. W. Kamen and J. Su, Inroducon o Opmal Esmaon. Sprnger, R. A. Horn and C. R. Johnson, arx Analyss. Cambrdge Unversy Press, J. H. Koecha and A.. Sayeed, ransm sgnal desgn for opmal esmaon of correlaed IO channels, IEEE rans. Inf. heory, vol. 49, no. 0, pp , Oc A. W. arshall and I. Olkn, Inequales: heory of ajorzaon and Is Applcaons. New York: Academc, 979. Alkan Soysal receved he B.S. degree n elecrcal and elecroncs engneerng from ddle Eas echncal Unversy, Ankara, urkey n 2002, and he.s. and Ph.D. degrees n elecrcal and compuer engneerng from Unversy of aryland, College Park, D n 2006 and 2008 respecvely. He joned Bahçeşehr Unversy, Isanbul, urkey n February 2008 as an Asssan Professor n he Deparmen of Elecrcal and Elecroncs Engneerng. Hs research neress are n wreless communcaon heory, nformaon heory and sgnal processng for wreless communcaons wh parcular focus on IO neworks. Sennur Ulukus receved he B.S. and.s. degrees n elecrcal and elecroncs engneerng from Blken Unversy, Ankara, urkey, n 99 and 993, respecvely, and he Ph.D. degree n elecrcal and compuer engneerng from Rugers Unversy, NJ, n 998. Durng her Ph.D. sudes, she was wh he Wreless Informaon Nework Laboraory WINLAB, Rugers Unversy. From 998 o 200, she was a Senor echncal Saff ember a A& Labs- Research n NJ. In 200, she joned he Unversy of aryland a College Park, where she s currenly an Assocae Professor n he Deparmen of Elecrcal and Compuer Engneerng, wh a jon apponmen a he Insue for Sysems Research ISR. Her research neress are n wreless communcaon heory and neworkng, nework nformaon heory for wreless neworks, sgnal processng for wreless communcaons and secury for mul-user wreless communcaons. Sennur Ulukus s a recpen of he 2005 NSF CAREER Award, and a corecpen of he 2003 IEEE arcon Prze Paper Award n Wreless Communcaons. She serves/served as an Assocae Edor for he IEEE ransacons on Informaon heory snce 2007, as an Assocae Edor for he IEEE RANSACIONS ON COUNICAIONS beween , as a Gues Edor for he IEEE JOURNAL ON SELECED AREAS IN COUNICAIONS n , as he co-char of he Communcaon heory Symposum a he 2007 IEEE Global elecommuncaons Conference, as he co-char of he edum Access Conrol AC rack a he 2008 IEEE Wreless Communcaons and Neworkng Conference, as he co-char of he Wreless Communcaons Symposum a he 200 IEEE Inernaonal Conference on Communcaons, and as he Secreary of he IEEE Communcaon heory echncal Commee CC n