Broadcast Channel with Transmitter Noncausal Interference and Receiver Side Information

Transcription

1 IEEE ICC 04 - Commucatos Theory Broadcast Chael wth Trasmtter Nocausal Iterferece ad Recever Sde Iformato Hayag X, Xaoju Yua ad Soug Chag Lew Dept. of Iformato Egeerg, The Chese Uversty of Hog Kog, Hog Kog Abstract Ths work vestgates the broadcast chael wth the kowledge of ocausal terferece at the trasmtter ad sde formato at the recevers. The system studed volves oe trasmtter sedg prvate formato to two recevers. We frst obta a achevable rate rego of ths system by extedg Gelfad ad Psker s GP s) radom bg method, orgally desged for pot-to-pot commucatos. We the apply the result to Gaussa scalar ad vector chaels, ad cosder the desg of the auxlary radom varable used GP s approach. Asymptotc aalyss shows that the proposed scheme both scalar ad vector chaels s asymptotcally capacty-achevg at hgh sgal-to-ose rato SNR). We further vestgate the system optmzato the fte SNR regme. For the scalar chael, we derve the optmal auxlary factor that maxmzes the weghed sum-rate. For the vector chael, two kds of suboptmal desgg methods for the auxlary matrx are proposed. Numercal results show that the proposed scheme ca acheve a sum-rate close to the cut-set boud the etre SNR regme, ad sgfcatly outperforms the schemes that do ot explot the kowledge of ocausal terferece kow at the trasmtter. I. INTRODUCTION The two-way relay chael TWRC), whch two users exchage formato va the help of the relay, has bee tesvely studed both theoretcal aalyss ad practcal mplemetato ]-5]. Physcal-layer etwork codg PNC) ] ad aalogue etwork codg ANC) ] are promsg techques proposed to effcetly utlze the sde formato at the user eds. Partcularly, t was show ] that PNC ca acheve the capacty of the TWRC wth 0.5 bt. A atural exteso of TWRC s mult-par TWRCs, where oe relay supports formato swtchg betwee multple users a parwse maer 6]-8]. Each par of users forms a TWRC, ad a user exchages formato oly wth hs couterpart the same par. Our tal work o mult-par TWRCs 8] focused o the uplk trasmsso. For the dowlk trasmsso, three types of messages are embedded the sgal receved by each user: the parter s message teded formato), the self-message recever sde formato) ad messages of the other user pars terferece). Each user s oly terested hs parter s message. Suppose that the relay geerates the sgals for user pars a sequetal order. The, the sgals ecoded frst become the ocausal terferece for the remag sgals,.e., the ecoder kows such terferece pror to the ecodg of the remag sgals. Hece, the dowlk trasmsso ca be modeled as a broadcast chael wth ocausal terferece at the trasmtter.e., the relay) ad sde formato at the recevers.e., the users). There are two ma streams of research work related to the cosdered broadcast chael the lterature. Oe stream s cocered wth hadlg the terferece at the trasmtter. The problem of ocausal terferece kow at the trasmtter was frst cosdered ad solved by Gelfad ad Psker GP) 9], usg the techque of radom bg 0]. Later, Costa proposed drty paper codg DPC) ], a realzato of GP s approach, to acheve the capacty of the Gaussa scalar chael as f there s o terferece. Further, the case of the Gaussa vector chael was cosdered ]. Ad the cases of Gaussa broadcast chael wth commomessage ad prvate-message are studed ] ad 4], respectvely. The other stream of research focuses o hadlg the sde formato of the recevers 5]-7]. It was show that each recever ca acheve the capacty of the dvdual chael lk. However, to the best of our kowledge, the broadcast chael wth both trasmtter ocausal terferece ad recever sde formato has ot bee well studed. The challege s how to desg a effcet codg scheme that ca jotly hadle the ocausal terferece kow at the trasmtter ad sde formato kow at the recever. I ths paper, we propose a ovel codg scheme for the cosdered broadcast chael. We frst derve achevable rates by extedg GP s radom bg approach, so as to explot the recever sde formato. We further cosder the desg of the auxlary radom varable used the GP approach to mprove system performace. For the Gaussa scalar chael, we provde a asymptotcally optmal soluto that ca acheve the capacty the hgh sgal-to-ose rato SNR) regme, ad derve the optmal auxlary factor for maxmzg the weghted sum-rate at geeral SNR. For the Gaussa vector chael, we also derve the asymptotcally optmal auxlary matrx at hgh SNR. However, the optmal desg of the auxlary matrx for the geeral SNR s a dffcult problem. We propose two suboptmal approaches. Smulato results demostrate the superorty of our proposed scheme compared wth tradtoal schemes 5]-7] wthout utlzg the kowledge of trasmtter terferece. Notato: The captal letter X ad the correspodg lowercase letter x deote the radom varable ad ts realzato, respectvely. The bold captal letter A ad the bold lowercase letter a deote the matrx ad the vector, respectvely. x deotes legth- sequece wth the jth symbol of x j.pra] s the probablty of evet A. P US ) s the jot probablty dstrbuto of U ad S. P U S ) s the probablty dstrbuto of U codtoed o S. u,s ) T ε P US ) meas that sequeces u ad s are jotly typcal wth respect to P US ) ad some small postve ε 8]. a] + deotes max{a, 0}. A deotes the determat of A. ad deote the Modulo- arthmetc ad Kroecker product, respectvely. For a matrx /4/$ IEEE 947

2 IEEE ICC 04 - Commucatos Theory W W Mapper W Ecoder PS S X Chael PYY XS Y Y W Decoder Decoder Wˆ Wˆ Sk Sk cosdered). We modfy the approach 9] to corporate the ew gredets cosdered our scheme. Our ma result s preseted below. Theorem : For the broadcast chael Fg., ay rate par R,R ) satsfyg the followg equaltes are smultaeously achevable: Fg.. Broadcast chael wth the kowledge of ocausal terferece at the trasmtter ad the kowledge of sde formato at the recevers. A, vec{a} s the vector obtaed by sequetally stackg the colums of A, ad vec { } s the reverse operato. II. SYSTEM MODEL The system model s show Fg., where there s oe trasmtter ad two recevers. W ad W are the orgal messages teded for recever ad recever, respectvely. These two messages frst go through a jectve fucto wth output W : f : W W W. ) The fucto f s ot ecessarly bjectve. For ay gve W, deote f W W )=fw,w ). We requre that, for ay gve W, the verse fucto of f W, deoted by f W : W W, {, } ) exsts. That s, f W s a bjectve mappg. The ecoder maps W to a strg X =X,X,..., X ad the seds t. Besdes put X, the chael P YY XS has the terferece S ad two outputs Y ad Y. The sequece S s output from a dscrete memoryless source P S ). The ecoder has access to S a ocausal fasho,.e., the ecoder kows S a pror. Decoder does ot kow S, but has the sde formato of W. After decodg, Ŵ s output to sk. The goal s ] to desg the ecoder ad decoders to esure that Pr Ŵ W, {, }, ca be made arbtrarly small. Example: Cosder the dowlk trasmsso of the multpar TWRC, whch multple users exchage formato a parwse maer va a sgle relay. There are stuatos where a par of users resposble for producg W ad W the uplk, the relay e.g., the ecoder) oly kows a etworkcoded message e.g., W = W W ), but ot the exact messages W ad W 8]. For the cases that the relay kows W ad W, we ca also frst map them to W before ecodg. I ecodg W, the relay perfectly kows the coded sgals for the other user pars.e., S ) that are ecoded before W, ad geerates a codeword vector X. X eeds to be forwarded to the two users the teded user par. At the recever sde, each user.e., decoder ) ams to decode W.e., the message from the other user) wth the kowledge of W.e., the self-formato of user ), {, }. III. AN ACHIEVABLE RATE REGION I ths secto, we establsh a achevable rate rego for the scheme Fg.. Our approach s based o the radom bg method developed by Gelfad ad Psker 9] whch a pot-to-pot chael wthout recever sde formato s W R < I U; Y ) I U; S)] +, {, } ) where U s a auxlary radom varable wth the jot probablty fucto P U, S). Sketch of Proof: Let w, u, s, x ad y be the orgal message, the auxlary sequece, the terferece sequece, the trasmtted sequece ad the receved sequece for decoder, {, }, respectvely. The jth symbols of the above sequeces,.e., u j, s j, x j ad y,j, are chose from the fte alphabets U, S, X ad Y, respectvely. Defe gu j,s j )=x j for j {,..., } that maps symbols U S to X, ad let ḡu,s ) be the sequece mappg of {u,s } to x. Code Costructo: Geerate R bs, dexed by w, where w =,..., R. Geerate R+R ) legth- codewords u wth each symbol each codeword radomly ad depedetly chose accordg to P U ). The throw these codewords radomly to the R bs such that each b has exactly R codewords. The vth codeword b w s deoted by u w, v). Ecoder: Gvew ad s, try to fd a codeword u w, v) b w, such that u w, v) ad s are jotly typcal,.e., u w, v),s ) Tε P US ). That s, the message w chooses the b umber, ad the terferece s chooses the codeword u w, v) from ths b. If oe fds a approprate codeword u w, v), trasmt x =ḡu w, v),s ). If ot, trasmt x = ḡu w, ),s ). Decoder: For each recever, based o the receved vector y ad the kowledge of w, try to fd a trplet ŵ,w, ˆv) such that u fŵ,w ), ˆv),y ) Tε P UY ). If there s oe or more such pars, the radomly choose oe of them, otherwse set ŵ =. Aalyss: We frst cosder decoder. The decoder makes a error oly f oe or more of the followg evets occur: E ={s / T ε P S )} 4) E ={u w, v),s ) / T ε P US )} 5) E ={u w, v),y ) / T ε P UY )} 6) E 4 ={u f ŵ,w ), ˆv),y ) Tε P UY )& ŵ w } 7) ) E s the evet of the terferece sequece s ot beg a typcal sequece. The probablty of E approaches zero as goes to fty by the law of large umber. ) E s the evet that there s o codeword b w that s jotly typcal wth s. Suppose that s Tε P S ). To make the probablty of E approach zero, s requred to be large ad R >I U; S). ) E s the evet that the codeword u s ot jotly typcal wth the receved sequece y. We ext show that the probablty of E goes to zero as. To ths ed, we frst ote that U S,X] Y forms a Markov cha, as U s depedet of Y gve S ad X. Note that u w, v),s ) Tε P US ) probablty, as 948

3 IEEE ICC 04 - Commucatos Theory the probablty of E vashes as. The we have u w, v),s,x ) Tε P USX ) by otg x =ḡu,s ). Together wth the factor that s,x ],y ) Tε P SXY ) as, we coclude from the Markov lemma 9, Lemma.] that Pr u,s,x,y ) Tε P USXY )] approaches oe as. So the probablty of E approaches zero as. v) E 4 s the evet that, although the decoded codeword u s jotly typcal wth y, ŵ s ot the correct message. The probablty of y Tε P Y ) approaches oe as. The probablty that a radomly geerated codeword u s jotly typcal wth y s upper bouded by IU;Y) ε) accordg to the jot asymptotc equpartto property 8, Theorem 8.6.]. Recall that decoder has the pror kowledge of w, ad that f w s a bjectve fucto. There are oly R vald bs the decodg codebook. Hece there are R R codewords the decodg codebook. Usg the uo boud, we see that the probablty of E 4.e., there exsts a codeword u that s jotly typcal wth y ad that s from a b other tha b w ) s bouded by R ) R IU;Y) ε).to esure ths probablty goes to zero, eeds to be large ad R +R <I U; Y ). Combg the cases of E ad E 4,wehave R < I U; Y ) I U; S)] +. 8) Smlarly, we have R <IU; Y ) IU; S)] + for decoder. Ths completes the proof of Theorem. Theorem ca be exteded to memoryless chaels wth cotuous alphabets by dscretzg cotuous radom varables followg the approaches 0, Chapter 7]. We wll apply Theorem to Gaussa scalar ad vector chaels the followg sectos. Clearly, the achevable rate par ) depeds o the choce of the auxlary radom varable U. We wll dscuss the desg of U what follows. IV. GAUSSIAN SCALAR CHANNEL A. Chael Model Cosder a system of oe base stato ad two users, each equpped wth a sgle atea. The chael s assumed to be real-valued. The receved sgal at user s gve by y = h x + h s + 9) where h s the chael ga betwee the base stato ad user, x s the trasmtted sgal wth E x ] p x, s s the Gaussa terferece wth E s ] = p s, ad s the zero-mea..d. addtve Gaussa ose wth E ] =σ, {, }. The cut-set boud of the chael 9) s gve by R cs log + h p x σ ), {, } 0) whch gves a outer boud of the achevable rate rego of the cosdered broadcast chael. We choose Gaussa sgalg,.e. x N 0,p x ). Followg DPC ], we set the auxlary radom varable u=x+αs, where α s a costat factor to be determed. The I U; Y )=h U)+h Y ) h U, Y ) px +α p s ) h p x +h p s+σ ) = log p x +α p s )h p x+h p s+σ ) p x+αh p s ) ) ad I U; S) = log p x + α p s. ) p x Substtutg ) ad ) to ), we obta a achevable rate par: R < log p x p x +α p s ) h p x +αh p s ) h p x+h p s+σ ) ) for {, }. For a decoder, we may choose α to be the mmum mea-square error MMSE) factor,.e., α = h Px. The, h Px+σ wth straghtforward ) mapulatos, the rght had sde of ) s log + h px, whch s the Shao capacty of the σ chael lk of user. Ths agrees wth the DPC result ]. However, as h h σ geeral, we ca ot set α to be σ the MMSE factor of both chael lks. We ext descrbe the detaled desg of α. B. Hgh-SNR Asymptotc Aalyss We start wth the hgh SNR regme,.e., p x +, p s + wth the rato ps p x fxed to be a costat, where c s a costat. The rght had sde of ) ca be rewrtte as ) ] + log p x px +p s +σ /h p x p s α). 4) +p x +α p s ) σ /h Clearly, f α, the rate 4) s bouded as p x + ; otherwse, ths rate goes to fty as p x +. Therefore the hgh SNR regme, the optmal α s α =. Wth α =, ) reduces to R < log px + h p )] + x p x + p s σ. 5) I the hgh SNR regme, ) the rate 5) s approxmately gve by log h px, ad approaches the cut-set boud σ 0). Thus, the proposed scheme s asymptotcally capactyachevg at hgh SNR. C. Weghted Sum-rate Maxmzato We ow cosder the optmzato of α for geeral SNR. We are terested fdg the optmal α for the followg weghted sum-rate problem: maxmze θ R + θ R 6) α subject to the rate costrats ), where θ ad θ are weghtg factors satsfyg θ,θ 0 ad θ +θ =. We frst assume cosder the case that R ad R are both postve. The the problem 6) ca be equvaletly wrtte as mmze α G α). 7) ] + 949

4 IEEE ICC 04 - Commucatos Theory sum-ratebts/chael use) cut-set boud proposed wth optmal α proposed wth α = terferece-oblvous proposed wth optmal α proposed wth α = terferece-oblvous INR=SNR+0dB) INR=SNR+0dB) INR=SNR+0dB) SNR db) Fg.. Sum-rate performace of the Gaussa scalar chael wth SNR, whe INR=SNR or INR=SNR+0dB. where G α)= ) α θ ) θ, λ α+μ α λ α+μ wth λ = p x p x+σ /, μ = pxps+σ / h ) h p sp x+σ / h ). To mmze G α), we set G α)=0, yeldg the followg equato: where α +aα +bα+c=0 8) a = λ +λ ) θ +λ +λ ) θ θ +θ ) b = μ + λ λ ) θ +μ + λ λ ) θ θ +θ ) c = λ μ θ λ μ θ. θ +θ ) For the real model 9), α s lmted to be real. Otherwse, f α s a complex umber, the we see from u = x + αs that IU; S) approaches fty by otg that the ose x s real-valued). Ths mples that the achevable rates of the scheme are always zero. The the optmal α s gve by the uque real soluto to 8): α= q q ) + p + ) ) q + q + p ) ) ) a, 9) where p=b a ad q = 7 a ab+c. We ext cosder the case that oly oe rate s postve ad the other rate s zero. Wthout loss of geeralty, we assume h > h, mplyg that R 0 ad R =0. I ths case, the optmal α s gve by the MMSE factor α = h px. h px+σ Maxmum weghted sum-rate of 6) s gve by the maxmum over the above two cases. D. Numercal Results Numercal results are provded to demostrate the rate performace of the proposed scheme the Gaussa scalar chael. The terferece-to-ose rato INR) s gve by ps σ ad SNR s gve by px. For smplcty, we always assume σ σ = σ smulato. I Fg., we compare the achevable sum-rates of the proposed scheme wth the cut-set boud. Two cases are studed: INR = SNR ad INR = SNR +0dB. We see that the sum-rate curve of the proposed scheme wth the optmal α gve by 9) s close to the cut-set boud closely for all SNR values. For both cases of INR, the sum-rates wth the optmal α ad α = coverge to the cut-set boud at hgh SNR. The sum-rate of the terferece-oblvous scheme whch the approach 5]-7] are drectly appled to the chael 9), wthout explotg the kowledge of trasmtter terferece) s also cluded Fg. for comparso. Note that the achevable rates of the terferece-oblvous ) scheme are gve by log + h px for {, }. We see that the h ps+σ sum-rates of the terferece-oblvous scheme are bouded at hgh SNR, whch demostrates the advatage of our proposed scheme. V. GAUSSIAN VECTOR CHANNEL A. Chael Model We exted the system model Secto IV to allow the use of multple ateas. Let m be the umber of ateas at each ode. The sgal receved at user s gve by y = H x + H s + 0) where x R m s the Gaussa sgal vector wth the covarace matrx of Q x, s R m s the Gaussa terferece vector wth the covarace matrx of Q s, H R m m s the chael lk betwee the base stato ad user, ad R m s the whte Gaussa ose vector at user, wth the covarace matrx of Q,, {, }. For ths chael, the cut-set boud s gve by R cs log H Q x H T +Q, {, }. ) Q Ths gves a outer boud of the achevable rate rego of the cosdered broadcast chael. We ow descrbe the achevable rates of the proposed scheme. Smlar to the scalar case, we troduce the auxlary radom vector u = x+fs, where F s a m m real-valued matrx to be determed. The, the rates ) ca be wrtte as R < I u; y ) I u; s)] +, {, }. ) We have I u; y )=hu)+hy ) h u, y ) = log Q x +FQ s F T H Q x H T +H Q s H T +Q Q x +FQ s F T Q x H T +FQ sh T ) H Q x +H Q s F T H Q x H T +H Q s H T +Q ad I u; s) = log Q x + FQ s F T. 4) Q x 950

5 IEEE ICC 04 - Commucatos Theory Substtutg ) ad 4) to ), we obta a achevable rate par: ] + R < log Q x FA F T +B F T +FB T +C for {, } 5) where A =Q s Q s H T H Q x H T +H Q s H T ) H +Q Q s B = Q x H T H Q x H T +H Q s H T ) H +Q Q s C =Q x Q x H T H Q x H T +H Q s H T ) H +Q Q x. I the Gaussa vector chael, for ) decoder, the optmal F s gve by Q x H T Q +H Q x H T H. Ths choce of F ca approach the Shao capacty of the chael lk of user,.e., log H Q xh T +Q Q ]. However two decoders are volved our scheme. Smlar to the scalar chael case, geeral, we ca ot fd a sgle F that s optmal for both decoders. B. Hgh SNR Asymptotc Aalyss If we choose F = I, after mapulatos, 5) reduces to R < log Q x H Q x H T +H Q s H T +Q ] +. 6) Q Q x +Q s I the hgh SNR regme, the rght had sde of 6) ca be approxmated as log H Q xh T Q, the asymptotc capacty of the chael lk of user wthout terferece. Thus, the choce of F = I s asymptotcally optmal the hgh SNR regme. C. Weghted Sum-rate Maxmzato We cosder the desg of F for maxmzg the weghted sum-rate the geeral SNR regme. Ths problem s formulated as follows: maxmze θ R + θ R 7) F subject to the rate costrats 5), where θ 0 ad θ 0 are weghtg factors satsfyg θ +θ =. The problem 7) s dffcult to solve geeral. I what follows, we propose two suboptmal solutos: oe based o the Ky Fa Iequalty amed as the o-teratve method), ad the other based o a teratve algorthm amed as the teratve method). Note that, smlar to the scalar chael case, the followg methods assume both R ad R are postve. Note that f oly oe rate s postve, the optmal F s chose to be the MMSE factor of the user wth the postve rate. ) No-teratve method: Maxmzg θ R +θ R s equvalet to maxmzg log Q x θ = FA F T +B F T +FB T +C. 8) θ From the Ky Fa Iequalty ], we have det X θ det X θ det θ X + θ X ). 9) where X ad X are two postve defte matrces. Applyg 9) to 8), we obta a lower boud of the weghted sum-rate as θ R +θ R log Q θ Q θ θ FA F T +B F T +FB T +C ). = 0) Deote Φ =θ A +θ A ), ) Ψ = θ B +θ B )θ A +θ A ), ) Θ = θ C +θ C ΨΨ T. ) We have that θ FA F T +B F T +FB T ) +C =FΦ Ψ)FΦ Ψ) T +Θ. = 4) The the choce of F that maxmzes the rght had sde of 0) s gve by F = ΨΦ = θ B +θ B )θ A +θ A ). 5) ) Iteratve method: The maxmzato of 8) occurs whe FA F T +B F T +FB T +C θ s mmzed. The, = to maxmze 8), t suffces to solve FA F T +B F T +FB T +C θ = = 0. 6) F After straghtforward mapulatos, we ca rewrte 6) as θ FA F T +B F T +FB T ) FA +C +B ) = θ FA F T +B F T +FB T ) FA +C +B ). 7) The closed-form soluto of F to 7) s dffcult to obta. Here we propose a teratve algorthm to umercally solve 7) as show Algorthm. Algorthm Iteratve Algorthm. : calculate A, B ad C, {, }; : talze F=I, t =, R sum ) = R + R, ɛ =0 ; : do 4: t=t +; ) ; 5: G =θ FA F T +B F T +FB T +C ) ; 6: G = θ FA F T +B F T +FB T +C { A ) } 7: F= vec T G +A T vec{g G B +G B } ; 8: R sum t)=r +R ; 9: whle R sum t) R sum t )>ɛ 0: retur R sum t). Emprcally, we observe that Algorthm always coverges. We are ow seekg for a rgorous proof for ths covergece. 95

6 IEEE ICC 04 - Commucatos Theory sum-ratebts/chael use) cut-set boud proposed teratve method proposed o-teratve method proposed wth F = I terferece-oblvous proposed teratve method proposed o-teratve method proposed wth F =I terferece-oblvous INR=SNR+0dB) INR=SNR+0dB) INR=SNR+0dB) INR=SNR+0dB) SNRdB) Fg.. Sum-rate performace of the Gaussa vector chael m=) wth SNR, whe INR=SNR or INR=SNR+0dB. D. Numercal Results Numercal results are provded to demostrate the rate performace of the proposed scheme the Gaussa vector TrQ chael wth m =. INR ad SNR are defed as s) TrQ,) ad TrQx) TrQ,), respectvely. For smplcty, we assume that Q, = Q, = σpi ad Q s s chose as a scaled detty matrx. I Fg., we compare the achevable sum-rates of the proposed scheme wth the cut-set boud. Three dfferet choces of F are cosdered: F = I, the o-teratve method, ad the teratve method. Two INR cases are studed: INR=SNR ad INR=SNR+0dB. Note that whe INR=SNR, the sum-rate curve of the teratve method s close to the cut-set boud for geeral SNR. For both cases of INR, the sum-rate curves for the o-teratve method ad the choce of F = I coverge to the cut-set boud at low ad hgh SNR, respectvely. We also cosder the terferece-oblvous approach for comparso Fg.. Note that the achevable rate of the terferece-oblvous approach s gve by log H Q x+q s)h T +Q H Q sh T+Q for {, }. Smlar to the scalar case, the terferece-oblvous approach has a bouded sumrate at hgh SNR, ad s sgfcatly outperformed by our proposed scheme. VI. CONCLUSION I ths paper, we studed the broadcast chael wth ocausal terferece kow at the trasmtter ad sde formato kow at the recevers. We exteded the Gelfad ad Psker approach by corporatg the recever sde formato, ad appled the results Gaussa scalar ad vector chaels. Asymptotc aalyss showed that the proposed scheme s capacty-achevg at hgh SNR. The sum-rate performaces were also evaluated through umercal smulato, ad show to be close to the cut-set boud the etre SNR rego. Gog forward, there are may exctg drectos for further explorato. For example, for the Gaussa vector chael, ths paper has oly cosdered the desg of the auxlary matrx wth equal atea cofgurato. The jot optmzato of the auxlary matrx ad trasmt covarace matrx wth uequal atea cofgurato s terestg. I addto, for the dowlk trasmsso of two-par TWRCs, The work ths paper oly cosders the codg desg for oe par of users by treatg the sgal of the other par as terferece. I practce, we are more terested the jot codg desg of the two user pars. We wll address ths ssue the future work. ACKNOWLEDGMENT Ths work s partally supported by the Cha 97 Program, Project No. 0CB5904. Ths work s partally supported by the Geeral Research Fuds Project No. 447 ad No. 487) ad AoE grat E-0/08, establshed uder the Uversty Grat Commttee of the Hog Kog Specal Admstratve Rego, Cha. REFERENCES ] S. 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