Designing PSAM Schemes: How Optimal are SISO Pilot Parameters for Spatially Correlated SIMO?

Transcription

1 Deigning PSAM Sheme: Hw Optimal are SISO Pilt Parameter fr Spatially Crrelated SIMO? Xiangyun Zhu, Tharaka A Lamahewa, Parat Sadeghi and Salman Durrani Cllege f Engineering and Cmputer Siene, The Autralian Natinal Univerity, Canberra 0200 ACT, Autralia {xiangyunzhutharakalamahewaparatadeghialmandurrani}@anueduau Abtrat-We tudy the deign parameter f pilt-ymblaited mdulatin (PSAM) heme fr patially rrelated ingle-input multiple-utput (SIMO) ytem in time-varying Gau-Markv flat-fading hannel We ue an infrmatin apaity lwer bund a ur figure f merit We invetigate the ptimum deign parameter, inluding the rati f pwer allated t the pilt and the fratin f time upied by the pilt, fr SIMO ytem with different antenna ize and with patialhannel rrelatin Our mainfinding i thatbyptimally deigning the training parameter fr ingle-input ingle-utput (SISO) ytem, the ame parameteran be ued t ahieve near ptimum apaity in bth patially independent and rrelated SIMO ytem fr the ame fading rate and ignal-t-nie rati (SNR) In additin, we hw that patially independent hannel give the lwet apaity at uffiiently lw SNR Thee finding prvide inight int the deign f pratial PSAM ytem I INTRODUCTION Channel etimatin i ruial fr reliable high data rate tranmiin in wirele mmuniatin with herent detetin Pilt-ymbl-aited mdulatin (PSAM) ha been ued in many pratial mmuniatin ytem, eg in Glbal Sytem fr Mbile Cmmuniatin (GSM) [1], t ait etimatin f unknwn hannel parameter In PSAM heme, training ymbl are inerted int data blk peridially t aquire the hannel tate infrmatin (CSI) [2] Hwever, the inertin f pilt al redue the infrmatin apaity a le tranmiin reure i allated t data Therefre, trade-ff analyi in PSAM parameterdeign i required n the reure allatin t pilt and data Furthermre, the parameter f the wirele hannel, uh a the number f the hannel input/utput, the fading rate and the patial rrelatin, add anther level f mplexity int the deign prblem Optimal PSAM deign fr time-varying fading hannel with lw-pa Dppler petra were tudied in [3,4], where Wiener filtering wa ued fr nn-aual hannel etimatin In [3], the authr tudied a lwer bund n hannel apaity and nluded that ptimal ampling frequeny f the fading pre equal the Nyquit rate On the ther hand, the tudie n hannel apaity with ideal interleaving via Mnte Carl imulatin in [4] hwed that pilt ymbl huld be ent mre frequently than the Nyquit rate Mre reently, tudie n ptimal PSAM deign in ingleinput ingle-utput (SISO) ytem adpted a Gau-Markv hannel mdel whih i an alternative mdel fr time-varying fading hannel [-7] With fixed rati f pilt inertin, the authr in [] fund that allating nly ne pilt per tranmiin blk minimized the hannel etimatin errr at the lat data ymbl in the blk Frm an infrmatin thereti viewpint, the authr in [6] invetigated the pwer ditributin amng data ymbl and hwed that data ymbl ler t the pilt ymbl huld have mre pwer than the further away frm the pilt In [7], the authr aumed unifrm pwer ditributin amng data ymbl, and jintly ptimized a hannel apaity lwer bund t find the ptimum pilt pwer allatin rati and pilt paing Fr multiple-input multiple-utput (MIMO) ytem, the authr in [8] tudied a lwer bund n the infrmatin apaity in PSAM heme fr blk fading hannel, and derived the ptimal pilt pwer allatin and ptimal number f pilt ymbl per tranmiin blk Fr lw bandlimited fading hannel with maximum likelihd etimatin, the authr in [9] fund that the ptimal pilt paing i nearly independent f the number f reeive antenna Hwever, tudie in [9] aumed patially independent hannel with equal pwer allatin t pilt and data ymbl The impat f hannel patial rrelatin n the apaity ha been tudied mainly in nn-psam heme With the knwledge f the hannel patial variane at the tranmitter, rrelatin amng tranmit antenna inreae the hannel apaity when perfet CSI i preent at the reeiver [] On the ther hand, the authr in [11] hwed that rrelatin amng the tranmit/reeive antenna alway redue apaity, auming perfet CSI i available nly at the reeiver In thi paper, we nider the ptimal PSAM deign frm an infrmatin thereti viewpint fr SIMO ytem in timevarying Gau-Markv hannel We invetigate the fllwing quetin: De hannel patial rrelatin at the reeiver alway redue infrmatin apaity? Are ptimal parameter fr SISO ytem al ptimal fr SIMO ytem with patially rrelated hannel? The main ntributin f thi paper are: In Setin IV, we hw that patially independent SIMO hannel reult in the highet hannel etimatin errr, and hene the lwet apaity at uffiiently lw SNR In Setin V, we hw that the ptimum PSAM deign parameter fr SISO ytem are very le t ptimal fr patially independent SIMO ytem fr Gau-Markv hannel with the ame fading rate and perating SNR In Setin V, we hw that the ptimal deign fr patially independent hannel are al near ptimal fr rrelated hannel, whih i an extenin f [9] Baed n the abve reult, we nlude that by ptimally /08/$ IEEE Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply

2 deigning the training parameter fr SISa ytem, the ame parameter an be ued t ahieve near ptimum apaity in bth patially independent and rrelated SIMa ytem Thrughut the paper, the fllwing ntatin will be ued: Bldfae upper and lwer ae dente matrie and lumn vetr, repetively The matrix IN i the N x N identity matrix [] * dente the mplex njugate peratin, and [] t dente the njugate tranpe peratin The ntatin E {} dente the mathematial expetatin tr{ }, I I and rank{} dente the matrix trae, determinant and rank, repetively II SYSTEM MODEL We nider a SIMa ytem with N r reeive antenna in time-varying flat-fading hannel After mathed filtering, the reeived ymbl at time index Rare given by where Xi i the tranmitted ymbl, Yi i the N r x 1 reeived ymbl vetr, ni i the N r x 1 nie vetr with variane matrix R n = E{ntntt} The nie at eah reeive antenna i independent, identially ditributed (iid) and zer-mean irularly ymmetri mplex Gauian (ZMCSCG), eah with variane (1, Le Rn = (1INr hi i the N r x 1 hannel vetr with ZMCSCG entrie The patial rrelatin f the hannel i haraterized by Rh = E{hih1} In the ae where the hannel are patially iid, R h = (]'INr' where (]' dente the variane f eah entry f hi The tempral rrelatin f the hannel i mdelled a a Gau-Markv pre: where Wi i a ZMCSCG pre nie with variane matrix R w = E{w w t} = (1 - (2)(1INr a i the tempral rrelatin effiient given by a = J(21r!DT ), where J i the zer-rder Beel funtin f the firt kind, f D i the Dppler frequeny hift, and T i the tranmitted ymbl perid Therefre, fdt i the nrmalized fading rate We aume that a i knwn and i ntant ver a large number f tranmitted ymbl A Pilt Tranmiin Sheme In PSAM heme, the hannel etimatin i perfrmed during pilt tranmiin During data tranmiin the hannel an be predited baed n the tempral rrelatin Frm the previu tudie n the ptimal deign f pilt inertin [, 8], we knw the ptimal trategy fr SIMa ytem i t allate ne pilt per tranmiin blk Therefre, we aume eah tranmiin blk ft ymbl nit f ne pilt fllwed by T - 1 data ymbl We dente the pilt paing by TJ = lit The average pwer per ymbl i dented by E, and the pwer f pilt and data ymbl are dented by E p and Ed, repetively We aume a fratin f 1 f the ttal pwer budget i allated t pilt Hene, we have, and d= (1-,)ET = 1-'E p = I T = T ] We dente the average reeived ymbl SNR by p = (1EI(1 Uing (3), the pilt and data ymbl SNR are given a (]'Ep, (]'Ed 1 -, Pp = lt =;'P and Pd = lt = 1-1]P' (4) III CHANNEL ESTIMATION Due t the hannel tempral rrelatin, we ue the Kalman filter a an iterative linear minimum mean quare errr (LMMSE) etimatr baed n tate pae mdel In partiular, (1) and (2) are the bervatin equatin and the tate update equatin, repetively Fr hannel with Gauian tatiti, the LMMSE etimatr i the MMSE etimatr During pilt tranmiin, ie = 1, T + 1,, the Kalman filter gain i given by [12] K = (a 2 M -l +Rw)x; ((a 2 M -l +Rw)Ep+(]'INr)-l, () (1) and the hannel etimate update equatin i given by (2) (3) It an be berved that h i ZMCSCG We dente the etimatin errr r h, = h, - hi, and it variane matrix by M = E{h h } The update equatin f M i given by where T-I Vi = L akw _k k=o Fllwing the Kalman filter update equatin in () and (7), the variane f hannel etimatin errr between tw neutive pilt tranmiin an be written a M, = ((a2tm,_r+(l--a2t)rh) :1+IN r ) -1 X (6) M = (1 - K x )(a 2 M l- 1+ Rw) (7) Withut l f generality, we initialize h = 0 and M 0 = Rh The rthgnality prperty f LMMSE etimatr tate that hi and h are unrrelated, whih implie the variane f h i given by R hl = R h - M We are intereted in the teady-tate behaviur f the Kalman filter At the peridi teady-tate, we have M l = M l - T VR In rder t find a led-frm exprein fr the teady-tate errr variane matrix, we firt fu n the pilt tranmiin mde and expre M in term f M -T Frm (2), the hannel tate between tw neutive pilt tranmiin are related a (a2tml_r+(i--a2t)rh) (8) When the Kalman filter reahe the teady-tate, we dente the teady-tate errr variane matrix at pilt tranmiin by M,l M = M -T Rearranging the abve equatin, we btain the fllwing quadrati matrix equatin 2 1-2T [; 1 - a 2T M 1 + (Rh+INr)MSl- Rh=O, a 2T;;f (]'n ' a 2T ;;:f n n It an be hwn uing reult in [13] that the abve quadrati matrix equatin atifie the nditin fr an expliit lutin Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply

3 in the ame frm a in the alar ae Therefre, the lutin i given a 1 1(2 2 ) 1/2 M,l=-"2Q+"2 Q +4Rh, where (A)1/2 dente the matrix quare rt f A, and 1- a 2T Q = -frh+inr' and K, = a O,2T P n 0::: (9) During data tranmiin, the hannel preditin i given by hi = ahi-1, () and the errr variane update equatin i given by Mi=a2Mi-1+Rw (11) The teady-tate errr variane matrix during data tranmiin an be alulated iteratively uing (11) a M,i = Rh + O, 2 (i-1) (M,l - Rh), (12) where R= 2,3,,T IV CHANNEL CAPACITY In thi etin, we tudy the hannel apaity when the Kalman filter ha reahed the teady-tate A A Capaity Lwer Bund Withut l f generality, we nrmalize the variane f the hannel gain, ie a = 1 Fr ytem with imperfet CSI at the reeiver, the exat apaity exprein i till unavailable Alternatively, we nider a lwer bund fr the intantaneu apaity, whih ha been ued fr infrmatinthereti tudie [14], given by CLB,f=Ej"t{ IOg21 INr +Pd(PdM,f+INJ-lhfh I} (13) Fr the ae where entrie f hi are iid, M, i a diagnal matrix with the ame diagnal entry dented by a; i' whih i prven later in Lemma 1, and the entrie f hf' are iid with variane 1-0"; f' Let (f h;hf Nte that ( i a Gamma ditributed randm variable with parameter (N r, 1 - a; i)' Frm (), ne an hw that (i = 0,2( -1)(1, Uing (4) and the matrix determinant lemma II+ABI = II+BA/, the intantaneu apaity lwer bund in (13) an be rewritten a 1=2 2( -1)( CLB,f = Eel { lg2 (1 + 1;;;:0 2 1)} (14) 1-11 pae, + 1 When entrie f hi are rrelated, hi ha rrelated entrie a well In thi ae, Mnte Carl imulatin will be ued t arry ut the numerial analyi in Setin V Fr bth patially iid and rrelated hannel, the apaity lwer bund per tranmiin blk i given by 1 T CLB = TL CLB,f' (1) i=2 B The Effet fspatial Crrelatin n the Capaity The authr in [11] hwed that patial rrelatin alway redue apaity, auming perfet CSI at the reeiver We wuld like t ak whether it i till true when imperfet CSI i available at the reeiver T anwer thi quetin, we firt lk at the effet f hannel patial rrelatin n the hannel etimatin MMSE Our finding i ummarized in the fllwing lemma with the prf given in Appendix A Lemma 1: Cnider the ytem mdel given by (1) and (2) with the aumptin that the hannel variane matrix Rh i full rank Under the Kalman filter etup fr hannel etimatin, patially i id hannel reult in the maximum hannel etimatin MMSE, and the variane matrix f the etimatin errr i diagnal with the ame entrie n it main diagnal At very high perating SNR, the hannel etimatin errr i negligible Therefre, we expet the effet f hannel patial rrelatin n the apaity t be the ame a in the perfet CSI ae Here we fu n the effet f patial rrelatin in uffiiently lw data ymbl SNR (Pd) regime In thi regime, the intantaneu apaity lwer bund in (13) an be apprximated a CLB,f Ej"t { IOg21INr + PdhfhI}, 1:2E j"t tr{ In (INr + Pdhfh)}, (16) 1 {" "t} In2 Eh tr Pdhihi, (17) l:d 2 (N r - tr{m,f}), (18) where (16) i btained uing In I / = tr{ln( )}, and (17) i btained uing Taylr' erie expanin f In( ) It an be een that (18) i a dereaing funtin f tr{m,i}' Frm Lemma 1, we knw that patially iid hannel reult in maximum tr{m,i}' Thi implie that at uffiiently lw SNR, patial rrelatin between hannel are' deirable fr higher apaity Thi finding ignifie the effet f hannel etimatin errr n the apaity in ntrat t the perfet CSI aumptin in [11] V NUMERICAL RESULTS In thi etin, we perfrm numerial analyi n the ptimal value f the PSAM deign parameter, ie the pilt pwer rati 'Y and the pilt paing 'TJ (r equivalently the blk length T), whih maximize the apaity lwer bund in (1) A Spatially LLd Channel Fig 1 hw a 3D plt f the apaity lwer bund in (1) fr a 1 x 4 SIMO ytem The average SNR budget i P = db, and nrmalized fading rate i fnt = 011 whih give the Gau-Markv parameter a = 0884 in (2), ie mderately fat time-varying hannel Frm Fig 1, the ptimal parameter value are T'pt = 037 and 'TJpt = 02 (r T pt = 4) We berve that C LB i mre enitive t the pilt paing 'TJ (r the blk length T) than t the pilt pwer rati 'Y Partiularly, Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply

4 0 r-----_r_-,---,---_r_-,r,,,,,iiii 4 2 'iii --- t T = 001, p = OdB 40 1 Q 3 30 II) 2! g [ ----&- t T = 001, p = 20dB ----t T =011,p=OdB, t T =011,p=2OdB 2 0 (J-J 20 g 1 E CD 1 0, ', O ' L - - I I " " L L ---' pilt pwer rati y 1 pilt paingn Fig 1 The apaity lwer bund in (1) at SNR budget p = db, and nrmalized fading rate fdt = 011 (ie a = 0884), fr a 1 x 4 ytem g- ----&- t T = t T = 00 3 t T =011 t T = en JIt E :: Number t reeiving antenna, Nr 8 Fig 3 The ptimum blk length fr different number f reeive antenna, at bth high and lw SNR budget and nrmalized fading rate 4r :-----: ;:::::I========::;l r;:::::======:::!:::==:::::;-----, ,r , &- t T = 001, N =8 r --- t T = 001, Nr=4 t T = 011, N =8 r 6 tt=011,nr=4 'iii II) 'E II) e! i Q :[ : CD (J-J - 0' '------' ' ' 1 20 SNR budget, p, db Fig 2 The ptimum blk length T pt fr a wide range f SNR budget and nrmalized fading rate, fr a 1 x 4 ytem Tpt i fund by a numerial earh in (1) where different T and "y value are heked 0' '------' ' 1 20 SNR budget, P, db Fig 4 The apaity lwer bund in (1) fr a wide range f SNR budget fr different nrmalized fading rate and number f reeive antenna The lid line are apaity lwer bund uing ptimum parameter fr SIMO ytem, and the dahed line are the apaity lwer bund ahieved uing ptimum C LB i almt ntant fr, ranging frm 02 t 0, with le than % degradatin frm the ptimal value We al tudied the plt fr N r == 1, 2,8 (nt hwn in thi paper), and the trend are very imilar t the ne hwn in Fig 1 Our bervatin indiate that the ptimal deign f the pilt paing i mre imprtant than that f the pilt pwer rati Hene, we fu n the ptimal pilt paing in the fllwing apaity analyi Fig 2 hw the ptimal blk length Tpt == 1/1]pt fr a wide range f SNR and nrmalized fading rate, fr a 1 x 4 SIMa ytem We al prdued the plt fr N r == 1,2,8 (nt hwn in thi paper), and the ame trend are fund in all ae Firtly, the ptimum blk length dereae a the fading rate inreae fr a fixed SNR Thi i expeted a mre frequent training i needed when hannel varie fater Sendly, fr lw fading hannel, eg fdt == 001, the ptimum blk length dereae dramatially a the SNR inreae, while fr fat fading hannel, eg fdt == 011,01, the ptimum blk length i almt ntant a SNR inreae parameter SISO ytem fr the ame antenna ize and fading rate Fig 3 hw the ptimum blk length fr different number f reeive antenna We ee that the ptimal blk length generally inreae with N r Sine mre antenna prdue higher diverity, whih imprve the tlerane n the hannel etimatin errr, the ytem an allw a larger pilt paing Furthermre, the inreae in the ptimum blk length i mre enitive t N r in the lw fading hannel (fdt == 001) than in the fater fading hannel (fdt == 011) Fig 4 hw the apaity lwer bund fr a wide range f SNR budget fr different nrmalized fading rate and number f reeive antenna We plt bth the apaity lwer bund ahieved uing ptimum parameter fr eah SIMa ae (lid line) and that ahieved uing ptimal parameter fr the SISO ae (dahed line) We ee that the differene between the tw i negligible, exept fr the ae where fdt == 011 and N r == 8, in whih the maximum differene i apprximately 3% at p == 8dB Therefre, by ptimizing parameter fr SISO Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply

5 40 013,: : Q en : 9! ::: :a (ij E v-- t T = 001, SNR=OdB --&-t T = 001, SNR=20dB t T = 011, SNR=OdB -&-t T = 011, SNR=20dB 02 OL------'----&-----L---L------' rrelatin effiient between hannel Fig The apaity lwer bund in (1) uing ptimum parameter v patial rrelatin effiient fr nrmalized fading rate fdt = 011 and different SNR, fr a 1 x 2 SIMO ytem rrelatin effiient between hannel Fig 6 The ptimum blk length v patial rrelatin effiient fr a 1 x 2 ytem, at bth high and lw SNR budget and nrmalized fading rate hannel, ne an ahieve near ptimum apaity fr patially iid SIMO hannel fpratial antenna ize (N r :::; 8) B Spatially Crrelated Channel We tudy the effet f hannel patial rrelatin n the ptimal deign parameter Withut l f generality, we plae the reeive antenna n a unifrm irular array and ue the tandard Jake' mdel t alulate the patial rrelatin under itrpi attering envirnment [1] Frm the analyi in Setin IV, we expet that patial rrelatin may inreae the apaity thrugh redutin in hannel etimatin errr at uffiiently lw SNR, while we argued that rrelatin redue the apaity at high SNR Fig hw the apaity lwer bund in (1) uing ptimum parameter fr a 1 x 2 SIMO ytem varying frm patially iid hannel t idential hannel (fully rrelated) At mderately high SNR, eg p == db, patially iid hannel reult in the maximum apaity lwer bund, and the apaity lwer bund dereae a the tw hannel beme mre rrelated Hwever, a different trend i fund at lw SNR At p == OdB, patially iid hannel till reult in the maximum apaity lwer bund, but the minimum ur at rrelatin effiient f 09 and nt at idential hannel Furthermre, at p == -db, the apaity lwer bund ha a 20% inreae frm patially iid hannel t idential hannel Thee bervatin nfirm ur earlier analyi in Setin IV In additin, thee numerial reult nfirm that patially iid hannel give the maximum infrmatin apaity at pratial perating SNR Fig 6 hw the ptimum blk length vere the patial rrelatin effiient between hannel fr a 1 x 2 SIMO ytem We al pltted fr ptimum pilt pwer rati (nt hwn in thi paper) and berved the ame trend In general, we ee that the ptimum parameter remain rughly ntant when hannel rrelatin effiient i le than 0, and have me gradual hange fr lw fading hannel when rrelatin effiient inreae abve 0 Therefre, we an ay 8r;:::::======:::::::==:::::;-----, , --&- t T = 001, N r =8 7 --v-- t T = 001, N r = t T =011, N r =8 : 6 -&- t T = 011, N r =4 'iii en 'E en : i 4 Q 3 u 2 OL L-----L ' ' 1 20 SNR budget, p, db Fig 7 The apaity lwer bund in (1) fr a wide range f SNR, and different fading rate and number f antenna The lid line are the apaity lwer bund uing ptimum parameter fr the rrelated SIMO ytem, and the dahed line are the apaity lwer bund uing ptimum parameter fr SISO ytem that the parameter whih ptimize the apaity fr patially iid hannel may al ahieve near ptimum apaity fr rrelated hannel Fig 7 hw the apaity lwer bund in (1) fr rrelated hannel, fixing the inter-antenna ditane t be a quarter f a wavelength We plt bth the apaity lwer bund ahieved uing the ptimum parameter fr eah rrelated SIMO ae (lid line) and that ahieved uing the ptimal parameter fr the SISO ae (dahed line) We ee that the differene between the tw i negligible, exept fr the ae where fdt == 011 and N r == 8, in whih the maximum differene i apprximately 3% at p == 6dB Tgether with the reult btained fr patially LLd hannel, we nlude that by ptimizing parameter fr SISO hannel, the ame parameter ahieve near ptimum apaity fr bth patially iid and rrelated SIMO ytem fpratial antenna ize (N r S; 8) Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply

6 VI CONCLUSION We tudied the deign parameter f PSAM heme frm an infrmatin-thereti viewpint in time-varying flat-fading SIMO hannel with and withut patial rrelatin The deign parameter t maximize the hannel apaity were the rati f pwer allated t pilt ymbl and the fratin f time allated t pilt ymbl We tudied a apaity lwer bund and hwed that it i mre enitive t hange in the pilt paing than t the pilt pwer rati The ptimum pilt pwer rati and pilt paing remain relatively ntant a the hannel patial rrelatin inreae We hwed that at uffiiently lw SNR, patial rrelatin reult in an inreae in the apaity mpared with patially iid hannel Mt imprtantly, ur finding hwed that by ptimally deigning the training parameter fr SISO ytem, the ame parameter an be ued t ahieve near ptimum apaity in bth patially LLd and rrelated SIMO ytem fr the ame SNR and fading rate We are urrently extending thi wrk t MIMO ytem ApPENDIX A PROOF OF LEMMA 1 We ue the mathematial indutin apprah by firtly lking at the initial hannel etimatin, Le M 1 Frm (), (7) and the given initializatin f the Kalman filter, ne an hw that M 1=Rh- Rh p(rh p +(j1n r ) -1 Rh=(Rh l +-f1n r ) -1, an where the end equality i btained uing the matrix inverin lemma Therefre, N r tr{m 1 } = L(9;1 + -f)-i, i=1 an where 9i, i = 1,, N r are the eigenvalue f Rh The value f gi at whih the maxima f tr{m I} ur under the ntraint E1 9i = tr{rh} = O"Nr frm a Lagrange multiplier prblem, and the lutin i given by 9i,max = a 'Vi Thi implie that the hannel are iid with Rh = ai N r In thi ae, M 1 i a diagnal matrix with diagnal entrie taking the ame value Nw we have prven the laim in Lemma 1 fr f = 1 The next tep i t prve thi hld fr all f, given it hld fr f - 1 The prf fr all data tranmiin time lt i trivial Here, we nly hw the prf fr all pilt tranmiin time lt, Le f = 1, T + 1, 2T + 1, Cnider pilt tranmiin baed n (8), we let M = a 2T M -T +(1- a 2T )Rh We aume the laim in Lemma 1 i true fr f - T r effetively fr M -T Then, learly M i diagnal and E1 mi ahieve it maximum with the ame value fr mi, where mi, i = 1,, N r are the eigenvalue f M T mplete the prf, we nly need t hw that the prpertie f mi imply the laim in Lemma 1 i al true fr MR Uing the matrix inverin lemma, (8) an be rewritten a M l (& )-lin r - (M/: + INJ- 1 ( & )-1, an an an Therefre, N r tr{m } = N r (-f)-1 - (-f)-1 L(mi-f + 1)-1 an an i=1 an Let u impe an arbitrary ntraint E1 mi = 'l/j > O The value f mi at whih the maxima f tr{m } ur under thi ntraint frm a Lagrange multiplier prblem, and the lutin i given by mi max = i- 'Vi The ntrained maximum value f tr{m R} i given byr fll, [; tr{m } = Nr(-f)-1 - (-f)-1 Nr(Nlf/ -f + 1)-1 an an ran It i lear that tr{ M } i maximized when 'l/j take it maximum value Tgether with the iid hannel aumptin, it i eay t hw that M i diagnal with the ame diagnal entrie And we have mpleted the prf ACKNOWLEDGEMENTS Thi wrk wa upprted by the Autralian Reearh Cunil Divery Grant DP REFERENCES [1] S M Redl, M K Weber, and M W Oliphan, An intrdutin t GSM, 1t ed Btn: Arteh Hue, 199 [2] J K Caver, "An analyi f pilt ymbl aited mdulatin fr Rayleigh fading hannel," IEEE Tran Veh Tehnl, vl 40, n 4, pp , Nv 1991 [3] S Ohn and G B Giannaki, "Average-rate ptimal PSAM tranmiin ver time-eletive fading hannel," IEEE Tran Wirele Cmmun, vl 1, n 4, pp , Ot 2002 [4] G F J Balteree and H Meyr, "An infrmatin thereti fundatin f ynhrnized detetin," IEEE Tran Cmmun, vl 49, n 12, pp , De 2001 [] M Dng, L Tng, and B M Sadler, "Optimal inertin fpilt ymbl fr tranmiin ver time-varying flat fading hannel," IEEE Tran Signal Preing, vl 2, n, pp , May 2004 [6] A Bdeir, I Abu-Fayal, and M Medard, "Pwer allatin heme fr pilt ymbl aited mdulatin ver Rayleigh fading hannel with n feedbak," in Pr IEEE ICC, vl 2, June 2004, pp [7] P Sadeghi, Y Liu, R A Kennedy, and P B Rapaji, "Pilt ymbl tranmiin fr time-varying fading hannel: An infrmatin-thereti ptimizatin," in Pr ICSPCS, De 2007 [8] B Haibi and M Hhwald, "Hw muh training i needed in multipleantenna wirele link?" IEEE Tran Infrm Thery, vl 49, n 4, pp , Apr 2003 [9] V Phl, P H Nguyen, V Jungnikel, and C Helmlt, "Cntinuu flat-fading MIMO hannel: Ahievable rate and ptimal length f the training and data phae," IEEE Tran Wirele Cmmun, vl 4, n 4, pp , July 200 [] E A Jrwiek and H Bhe, "Optimal tranmiin trategie and impat f rrelatin in multiantenna ytem with different type f hannel tate infrmatin," IEEE Tran Signal Preing, vl 2, n 12, pp , De 2004 [11] C Chuah, D Te, J Kahn, and R Valenzuela, "Capaity aling in MIMO wirele ytem under rrelated fading," IEEE Tran Infrm Thery, vl 48, n 3, pp , Mar 2002 [12] B D O Andern and 1 B Mre, Optimal Filtering Dver Publiatin, In, 1979 [13] N J Higham and H Kim, "Slving a quadrati matrix equatin by newtn' methd with exat line earhe," SIAM Jurnal n Matrix Analyi and Appliatin, vl 23, n 2, pp , 2001 [14] T Y and A Gldmith, "Capaity and pwer allatin fr fading MIMO hannel with hannel etimatin errr," IEEE Tran Infrm Thery, vl 2, n, pp , May 2006 [1] T A Lamahewa, R A Kennedy, T D Abhayapala, and T Betlehem, "MIMO hannel rrelatin in general attering envirnment," in Pr AuCTW, Feb 2006, pp Authrized liened ue limited t: Autralian Natinal Univerity Dwnladed n Augut 2, 2009 at 02:18 frm IEEE Xplre Retritin apply