Capacity Bounds for Backhaul-Supported Wireless Multicast Relay Networks with Cross-Links

Transcription

1 Capacty Bouds for Backhaul-Supported Wreless Multcast Relay Networks wth Cross-Lks Proc. of IEEE ICC, Ju. 0. c 0 IEEE. Persoal use of ths materal s permtted. However, permsso to reprt/republsh ths materal for advertsg or promotoal purposes or for creatg ew collectve works for resale or redstrbuto to servers or lsts, or to reuse ay copyrghted compoet of ths work other works must be obtaed from the IEEE. JINFENG DU, MING XIAO, MIKAEL SKOGLUND Stockholm Jue 0 School of Electrcal Egeerg ad the ACCESS Laeus Ceter, Royal Isttute of Techology (KTH), SE Stockholm, Swede IR-EE-KT 0:004

2 Capacty Bouds for Backhaul-Supported Wreless Multcast Relay Networks wth Cross-Lks Jfeg Du, Mg Xao, ad Mkael Skoglud ACCESS Laeus Ceter, Royal Isttute of Techology, Stockholm, Swede Emal: Abstract We vestgate the capacty bouds for a wreless multcast relay etwork where two sources smultaeously multcast to two destatos through Gaussa chaels wth the help of a full-duplex relay ode. All the dvdual chael gas are assumed to be tme-varat ad kow to every odes the etwork. The trasmssos from two sources ad from the relay use the same chael resource (.e. co-chael trasmsso) ad the two source odes are coected wth a orthogoal error-free backhaul. Ths multcast relay etwork s geerc the sese that t ca be exteded to more geeral etworks by tug the chael gas wth the rage [0, ). By extedg the proof of the coverse developed by Cover ad El Gamal for the Gaussa relay chael, we characterze the cut-set boud for ths multcast relay etwork. We also preset a lower boud by usg decodgad-forward relayg combed wth etwork beam-formg. I. INTRODUCTION As show Fg., we cosder a relay-aded two-source two-destato multple multcast etwork where two source odes S ad S multcast ther dvdual message W at rate R ad W at rate R, respectvely, to both destatos D ad D, wth the help of a relay R. Two source odes ca cooperate wth each other through a orthogoal error-free backhaul wth capacty C b. The relay forwards the formato t receves prevous tme slot to both destatos. The trasmssos from S, S, ad R use the same chael resource (.e. co-chael trasmsso) ad mx up at all the recevg termals wth depedet Gaussa ose. The dvdual chael gas g j 0,, j =,, r are assumed to be tme-varat ad kow to every odes ths etwork. The model Fg. s geerc sce t covers a class of etworks by tug the chael gas g j wth the rage [0, ). Such a system s terestg sce t arses, for example, a wreless cellular dowlk where two base statos multcast multmeda formato to two moble termals, oe each cell, wth the help of a dedcated relay deployed at the commo cell boudary. Sce the base statos are coected through the (fber or mcrowave) backhaul, varous cooperatve strateges ca be used to boost the throughput. The full uderstadg of such systems, eve for the orgal three-ode relay etwork, s ot ready yet. Sce 970s, umerous research efforts have bee casted o the relay etworks. Capacty bouds for three-ode relayg etworks (source-relay-sk, or two cooperatve sources ad oe sk) have bee studed [], [], where the capacty regos for degraded or reversely degraded scearos are characterzed. Codg schemes have bee vestgated for multple-access W Backhaul W C b S X g r S g r X R g g g X r g g r g r Y Y D D Ŵ Ŵ Ŵ Ŵ Fg.. Two source odes S ad S, coected wth backhaul, multcast formato W at rate R ad W at rate R respectvely to both destatos D ad D through Gaussa chaels, wth ad from a full-duplex relay R. relay chaels (MARC) [3], [4] volvg multple sources ad a sgle destato, ad for broadcast relay chaels (BRC) [4], [5] where a sgle source trasmts to multple destatos. Recet results o capacty bouds for multplesource multple-destato relay etworks, [6] [9] ad refereces there, have provded valuable sghts. Apart from troducg a dedcated relay, the rate rego ca also be elarged by cooperato amog coferecg/cogtve ecoders [0] [] ad/or amog coferecg decoders []. I ths paper, we am at characterzg the capacty upper ad lower bouds for the system show Fg., where relayg s combed wth source cooperato. To smplfy our aalyss, we wll restrct our aalyss wth a full-duplex relay ad a error-free backhaul wth suffcetly hgh capacty (larger tha the sum-rate,.e. C b >R +R ). Extesos to a half-duplex relay or low-rate backhaul (C b <R +R ) wll be left to future work. Note that, the hgh-rate backhaul makes the capacty rego (R, R ) oly bouded by R +R, whch makes our system closely related to the MIMO relay chaels [3], [4], where S ad S ca be grouped together as a vrtual trasmtter equpped wth two ateas. As dscussed our prevous work [5] for the case wthout cross-lk (.e. g =g =0), the capacty upper bouds derved based o the MIMO relay chael [3], [4] are geeral larger tha the cut-set boud: () the equalty (4) of [3] s acheved oly f whe the relay has at least the same umber of trasmt ateas compared wth the vrtual trasmtter, whch s ot the case here; () the boud (9) of [4] s derved based o the sum-power costrat, whle our case oly per-atea/ode power costrat s appled. Moreover, for the cases of lowrate backhaul, the MIMO upper bouds developed [3], [4] seem rrelevat, but our proof developed here ca stll be helpful by combg wth the strateges [0] [].

3 I our prevous work [6], we have successfully characterzed the exact cut-set boud for the case wthout cross-lk by extedg the proof of the coverse developed by Cover ad El Gamal for the Gaussa relay chael, ad verfed that ths cut-set boud s tght (cocde wth a lower boud) whe the trasmttg power s lower tha a sphere defed by the dvdual chael gas. We exted ths method to the system Fg. ad preset capacty upper ad lower bouds. The rest of ths paper s orgazed as follows. The system model s troduced Sec. II. Our ma results are summarzed Sec. III wth the detaled proof of the cut-set boud preseted Sec. IV. Numercal results are preseted Sec. V wth the cocludg remarks preseted Sec. VI. Notatos: Captal letter X dcates a real valued radom varable ad p(x) dcates ts probablty desty/mass fucto. X () deotes a vector of radom varables of legth ad I(X; Y ) deotes the mutual formato betwee X ad Y. C(x) = log ( + x) s the Gaussa capacty fucto. II. SYSTEM MODEL Gve a average trasmt power costrat P, fxed chael ga g, ad ose power σ, the sgal-to-ose rato (SNR) of that lk ca be wrtte as γ = g P/σ. We ca therefore characterze the trasmsso lks oly by ther dvdual SNR γ, wthout dstgushg the SNR cotrbuto from the trasmt power or the chael ga. Therefore we ca model the system show Fg. as follows Y () = γ X () + γ X () + γ r X r () + Z (), Y () = γ X () + γ X () + γ r X r () + Z (), () () = γ r X () + γ r X () +Z r (), where γ j 0,, j =,, r are the effectve SNR for the correspodg lks. X (), Y (), Z (), =,, r are - dmesoal trasmtted sgals, receved sgals, ad ose at odes S, S ad R, respectvely. A average ut-power costrats are appled to all the trasmtted sgals,.e., X [k]. () k= The ose compoets Z [k], =,, r ad k =,..., are..d. zero-mea ut-varace Gaussa radom varables. III. MAIN RESULTS Theorem : Defe a rate rego as R + R C 0 = sup m{ 0 α,α,ρ C ( (γ +γ r )α +( ρ )α α ( γ γ r γ γ r ) +(γ +γ r )α + ρ α α ( γ γ + γ r γ r ) ), C ( (γ +γ r )α +( ρ )α α ( γ γ r γ γ r ) +(γ +γ r )α + ρ α α ( γ γ + γ r γ r ) ), C ( γ + γ + γ r + ᾱ γ γ r + ᾱ γ γ r +(ρ α α + ᾱ ᾱ ) γ γ ), C ( γ + γ + γ r + ᾱ γ γ r + ᾱ γ γ r +(ρ α α + ᾱ ᾱ ) γ γ )}, (3) Backhaul S S X X D X Y R r Cut Cut Backhaul S X S X Cut 3 R X r Cut 4 Y D Fg.. The sum multcast capacty s bouded by the cut-set boud based o the four cuts show the fgure. where ᾱ = α ad ᾱ = α. For the system Fg. wth power costrats (), ts cut-set boud C cut-set equals to C 0. Proof: The proof ca be foud Secto IV. We also preset a lower boud gve by the cooperatve strategy etwork beam-formg (NBF) [6], whch essetally frst geerates the etwork coded message W b at source odes, ad the cooperates ts trasmsso at S ad S wth the relayg sgal a beam-formg fasho to take advatage of the coheret combg ga. Sce S ad S trasmt the same etwork coded message W b, the achevable sum-rate ca be splt arbtrarly betwee them. Proposto : The capacty rego s lower bouded by the achevable rate rego of NBF, whch s the uo of all rate pars (R, R ) whch satsfy R 0, R 0, ad R + R < m { C ( ( α γ r + α γ r ) ), C ( ( α γ + α γ ) +( ᾱ γ + ᾱ γ + γ r ) ), C(( α γ + α γ ) +( ᾱ γ + ᾱ γ + γ r ) )}, wth the uo take over all 0 α, α. Proof: Ths result s a straghtforward exteso of Proposto 4 [6] by replacg the system model wth () ad formulatg the trasmtted sgals as X () r,b = U() (W b ), X (),b = α V () (W b, W b ) + ᾱ U () (W b ), X (),b = α V () (W b, W b ) + ᾱ U () (W b ), where 0 α, α are power allocato parameters, U (), V () are radom codewords geerated the same way as [6]. Detaled descrpto o ecodg/decodg s skpped due to space lmtato. The costrats (4) correspod to successful decodg at R, D, ad D, respectvely. IV. PROOF OF THE CUT-SET BOUND By the maxmum-flow m-cut theorem [7], the maxmum achevable sum-rate from the source odes to ay of the destatos ca be o larger tha the mmum of the mutual formato flows across all possble cuts, maxmzed over a jot dstrbuto for the trasmtted sgals. The cut-set boud betwee the two sources ad each of the sk for the etwork Fg. ca be derved based o four cuts show Fg. as follows (the dmeso super scrpt () s suppressed hereafter to smplfy the otato) R + R C cut-set = sup p(x,x,x r) m{ I(X, X ; Y, X r ), I(X, X, X r ; Y ), I(X, X ; Y, X r ), I(X, X, X r ; Y )}+ǫ, (4) (5) (6)

4 where X, X ad X r are potetally correlated, ad ǫ 0 as. We wll frst fd a upper boud C upp C cut-set by usg the ew techque, ad the fd a lower boud C cut-set, G C cut-set by restrctg the source dstrbuto to Gaussa. Fally by showg that C cut-set, G = C upp we ca fd the exact cut-set boud C cut-set = C upp = C cut-set, G. A. The upper boud C upp Followg the covetoal otato for the dfferetal etropy h(x) of a cotuous valued radom varable X, the mutual formato correspodg to cut ca be wrtte as I(X, X, X r ; Y ) = h(y ) h(y X, X, X r ) = h(y ) h(z ) = h(y ) log(πe). (7) From the maxmum etropy lemma [7], we get h(y ) h(y, ) log(πevar[y,]), (8) where the secod equalty s acheved whe Y, s Gaussa dstrbuted. Hece I(X, X, X r ; Y ) log(var[y,]) (9) log( Var[Y, ]), where the last steps follow from Jese s equalty, wth Var[Y, ] = +Var[ γ X, + γ X, + γ r X r, ]. (0) Accordg to the law of total varace, for two radom varables X ad Y o the same probablty space, ad the varace of X s fte, the Var[X] = E(Var[X Y ]) + Var[E(X Y )]. We ca therefore rewrte (0) as Var[Y, ] = +Var[E( γ X, + γ X, + γ r X r, X r, )] + E(Var[ γ X, + γ X, + γ r X r, X r, ]) = + E(Var[ γ X, + γ X, X r, ]) () where + Var[ γ E(X, X r, )+ γ E(X, X r, )+ γ r X r, ], E(Var[ γ X, + γ X, X r, ]) = E(γ Var[X, X r, ] ad + γ Var[X, X r, ]+ γ γ Cov(X,, X, X r, )) =γ E(Var[X, X r, ]) + γ E(Var[X, X r, ]) () + γ γ E(Cov(X,, X, X r, )), Var[ γ E(X, X r, )+ γ E(X, X r, )+ γ r X r, ] E[( γ E[X, X r, ]+ γ E[X, X r, ]+ γ r X r, ) ] =γ E(E [X, X r, ])+ γ γ r E(X r, E[X, X r, ]) (3) +γ E(E [X, X r, ])+ γ γ r E(X r, E[X, X r, ]) +γ r E(X r, )+ γ γ E(E[X, X r, ]E[X, X r, ]). As [], defe ᾱ = E[E (X, X r, )], α [0, ], (4) the we have E[Var(X, X r, )] = (E[X,] E[E (X, X r, )]) = E[X, ] E[E (X, X r, )] α, (5) where the equalty comes from (). Smlarly we have ᾱ = E[E (X, X r, )], E[Var(X, X r, )] α, (6) where α [0, ]. O the other had, as Cov(X,, X, X r, ) = φ Var(X, X r, )Var(X, X r, ), where φ s the correlato coeffcet, we have E[Cov(X,, X, X r, )] φ E[Var(X, X r, )] φ E[Var(X, X r, )] α α, (7) where the frst equalty s due to the Cauchy Schwarz equalty ad the last step s gve by (5) ad (6). Gve that φ, we ca troduce a auxlary varable 0 ρ such that E[Cov(X,, X, X r, )] = ρ α α. (8) Also, usg the Cauchy Schwarz equalty we get E(X r, E[X, X r, ]) (9) E[Xr, ] E(E [X, X r, ]) ᾱ, E(X r, E[X, X r, ]) ᾱ, (0) E(E[X, X r, ]E[X, X r, ]) ᾱ ᾱ. () Now, substtutg () () to (9), ad apply the same approach also to cut 4, we get I(X, X, X r ; Y ) C(γ +γ +γ r + ᾱ γ γ r + ᾱ γ γ r + (ρ α α + ᾱ ᾱ ) γ γ ), () I(X, X, X r ; Y ) C(γ +γ +γ r + ᾱ γ γ r + ᾱ γ γ r + (ρ α α + ᾱ ᾱ ) γ γ ). (3)

5 For cut we have I(X, X ; Y, X r ) = h(y, X r ) h(y, X, X, X r ) =h(y, X r ) h(y X, X, X r ) h( X, X, X r ) (4) = h(y, X r ) h(z ) h(z r ) = h(y, X r ) log(πe) log ( (πe) K ) log(πe) = log ( K ), C cut-set C cut-set, G = where the secod equalty (4) comes from the fact that Y ad are depedet gve (X, X, X r ) ad the equalty s due to the maxmum etropy lemma [7], wth equalty acheved by jot Gaussa dstrbuted (Y,,, ) wth codtoal covarace matrces K, whch s defed by [ K = E(Var[Y, X r, ]) E[Cov(Y,,, X r, )] E[Cov(Y,,, X r, )] E(Var[, X r, ]) Obvously, the covarace matrces K are postve semdefte. Sce the fucto log K s cocave [8], we ca thus boud the throughput of cut from (4) as follows I(X, X ; Y, X r ) log ( K ) = log ( E(Var[Y, X r, ]) j= E(Var[,j X r,j ]) ]. ( E[Cov(Y,,, X r, )] ) ). (5) Furthermore, sce E(Var[Y, X r, ]) = + E(Var[ γ X, + γ X, X r, ]), E(Var[, X r, ]) = + E(Var[ γ r X, + γ r X, X r, ]), E[Cov(Y,,, X r, )] = γ γ r E(Var[X, X r, ]) + γ γ r E(Var[X, X r, ]) +( γ γ r + γ γ r )E[Cov(X,, X, X r, )], by combg wth () ad (5) (8), we ca coclude that I(X, X ; Y, X r ) C ( (γ +γ r )α +( ρ )α α ( γ γ r γ γ r ) +(γ +γ r )α + ρ α α ( γ γ + γ r γ r ) ). (6) Smlarly, we ca boud the throughput of cut 3 as follows I(X, X ; Y, X r ) C ( (γ +γ r )α +( ρ )α α ( γ γ r γ γ r ) +(γ +γ r )α + ρ α α ( γ γ + γ r γ r ) ). (7) By substtutg () (3) (6) (7) to (6) ad, comparg the resultg rego to (3) we ca coclude that C cut-set C upp = C 0. B. The lower boud C cut-set, G By restrctg p(x, X, X r ) (6) to be Gaussa, we ca partto X, X ad X r as follows X r = U, X = ( ρ)α S + ρα V + ( α )U, X = ( ρ)α S + ρα V + ( α )U, (8a) (8b) (8c) where S, S, V, U are -dmesoal depedet Gaussa radom vectors wth zero-mea ad ut-varace. Auxlary varables 0 α, α, ρ are troduced to represet the potetal correlato amog X, X ad X r due to cooperato. by substtutg (8) to (), we ca derve from (6) that sup m 0 α,α,ρ { (9) log(var[y, ]), log(var[y, ]), log ( K, ),log ( K, )}+ǫ, where ǫ 0 as, ad for =,..., we have Var[Y, ] =+ ρα γ + ρα γ +ρ( α γ + α γ ) +( ᾱ γ + ᾱ γ + γ r ), Var[Y, ] =+ ρα γ + ρα γ +ρ( α γ + α γ ) +( ᾱ γ + ᾱ γ + γ r ), (30) K, = (+ ρα γ + ρα γ +ρ( α γ + α γ ) ) (+ ρα γ r + ρα γ r +ρ( α γ r + α γ r ) ) ( γ γ r α + γ γ r α +( γ γ r + γ γ r )ρ α α ), K, = (+ ρα γ + ρα γ +ρ( α γ + α γ ) ) (+ ρα γ r + ρα γ r +ρ( α γ r + α γ r ) ) ( γ γ r α + γ γ r α +( γ γ r + γ γ r )ρ α α ). By substtutg (30) to (9) ad lettg, comparg the resultg rego to (3) we ca coclude that C cut-set C cut-set, G = C 0. Recall that C cut-set, G C cut-set C upp, we ca fally coclude that C cut-set = C 0,.e., Theorem holds. V. NUMERICAL RESULTS We preset the umercal results of the cut-set boud ad the NBF lower boud o the capacty regos wth dfferet lk qualty. Uless stated otherwse, the followg heurstc parameters wll be used: The S D lk SNR γ =5dB, the source-destato lk SNR γ, the source-relay lk SNR γ r =γ r, the relay-destato lk SNR γ r, ad the cross-lk SNR γ =γ. I Fg. 3 we vestgate the mpact of the relay-destato lk qualty o the capacty rego. Whe the R D lk s ot strog, the R D lk SNR γ r s ot a lmtg factor ad therefore the capacty wll mootocally crease wth γ r utl t s large eough to reach the bottleeck set by γ r, as demostrated Fg. 3. The gap betwee the cut-set boud the NBF lower boud s always less tha 0. bts/chael use. I Fg. 4 we fx the S R lk SNR γ r ad vary the SNR of the S R lk. Whe the S R lk s poor (γ r = 0dB), the S R lk s the oly relable chael betwee S, S ad R ad therefore ts qualty has a large mpact o the capacty. Whe the S R lk s strog, however, S ca commucate wth R relably va S thaks to the backhual, eve for a poor S R lk. The mpact of γ r wll dmsh whe γ r becomes very large, as show the zoom- plot. Note that the decodg requremet at relay for NBF troduces a large gap whe both γ r ad γ r are small.

6 Capacty bouds [bts/chael use] R D lk SNR, γ r Fg. 3. Capacty bouds for varyg R D lk SNR γ r. Capacty bouds [bts/chael use] 3.5.5, γ =γ, γ =γ, γ =γ, γ =γ S R SNR, γ r =γ r Fg. 5. Capacty bouds for varyg symmetrc source-relay SNR γ r = γ r. Capacty bouds [bts/chael use] , γ r, γ r, γ r, γ r S R lk SNR, γ r Fg. 4. Capacty bouds for varyg S R lk SNR γ r. I Fg. 5 we show the mpact of the source-relay lk for weak ad strog cross-lk qualty. Obvously the NBF strategy does ot utlze the beeft from the cross-lk whe the source-relay coecto s ot good eough. Whe the source-relay lk s good, as show Fg. 6, the achevable rate of NBF creases wth mproved cross-lk qualty utl the decodg at the relay becomes the bottleeck. The cut-set boud has o restrcto o decodg at R ad therefore wll crease wth the mproved cross-lk qualty. VI. CONCLUSION We have derved the cut-set boud for backhaul-supported wreless multcast relay etworks wth cross-lk ad provded achevable rate based o etwork beam-formg strategy. The gap betwee the cut-set boud ad the achevable rate s ot large geeral, as llustrated the umercal results. Sce the NBF strategy caot take full advatage of the cross-lk, a better scheme has to be sought further work. ACKNOWLEDGMENTS Ths work s fuded by VR ad VINNOVA. REFERENCES [] T. M. Cover ad A. El Gamal, Capacty theorems for the relay chael, IEEE Tras. If. Theory, vol. 5, pp , Sep [] A. Høst-Madse ad J. Zhag, Capacty bouds ad power allocato for wreless relay chaels, IEEE Tras. If. Theory, vol. 5, pp , Ju [3] G. Kramer ad A. J. va Wjgaarde, O the whte Gaussa multpleaccess relay chael, Proc. of IEEE ISIT, Ja Capacty bouds [bts/chael use] , γ r =5dB, γ r =5dB, γ r, γ r Cross lk SNR, γ =γ Fg. 6. Capacty bouds for varyg symmetrc cross-lk SNR γ =γ. [4] G. Kramer, M. Gastpar ad P. Gupta, Cooperatve strateges ad capacty theorems for relay etworks, IEEE Tras. If. Theory, vol. 5, pp , Sep [5] Y. Lag ad G. Kramer, Rate regos for relay broadcast chaels, IEEE Tras. If. Theory, vol. 53, pp , Oct [6] O. Sah ad E. Erkp, Achevable rates for the Gaussa terferece relay chael, Proc. of IEEE GLOBECOM, Nov [7] A. Avestmehr ad T. Ho, Approxmate capacty of the symmetrc halfduplex Gaussa butterfly etwork, Proc. of IEEE ITW, Ju [8] D. Güdüz, O. Smeoe, A. J. Goldsmth, H. V. Poor, ad S. Shama, Relayg smultaeous multcast messages, Proc. of IEEE ITW, Ju [9], Multple multcasts wth the help of a relay, IEEE Tras. If. Theory, vol. 56, pp , Dec. 00. [0] N. Devroye, P. Mtra, ad V. Tarokh, Achevable rates cogtve rado chaels, IEEE Tras. If. Theory, vol. 5, pp , May 006. [] S. I. Bross, A. Lapdoth, ad M. A. Wgger, The Gaussa MAC wth coferecg ecoders, Proc. of IEEE ISIT, Jul [] O. Smeoe, D. Güdüz, H. V. Poor, A. J. Goldsmth, ad S. Shama, Compoud multple-access chaels wth partal cooperato, IEEE Tras. If. Theory, vol. 55, pp , Ju [3] B. Wag, J. Zhag, ad A. Høst-Madse, O the capacty of MIMO relay chaels, IEEE Tras. If. Theory, vol. 5, pp. 9 43, Ja [4] S. Smoes, O. Muñoz-Meda, J. Vdal, ad A. del Coso, O the Gaussa MIMO relay chael wth full chael state formato, IEEE Tras. Sgal Proc., vol. 57, pp , Sep [5] J. Du, M. Xao, ad M. Skoglud, Capacty bouds for relay-aded wreless multple multcast wth backhaul, proc. of WCSP, Oct. 00. [6] J. Du, M. Xao, ad M. Skoglud, Cooperatve etwork codg strateges for wreless relay etworks wth backhaul, accepted for publcato IEEE Tras. Commu., to appear Sep. 0. [7] T. M. Cover ad J. A. Thomas, Elemets of Iformato Theory, New York, Wley, 006. [8], Determat equaltes va formato theory, SIAM joural o matrx aalyss ad applcatos, vol. 9, pp , Jul. 998.