Cooperative Network Coding Strategies for Wireless Relay Networks with Backhaul

Transcription

1 Cooperative Network Codig Strategies for Wireless Relay Networks with Backhaul IEEE Trasactios o Commuicatios, vol. 59, pp. 5 54, Sep.. c IEEE. Persoal use of this material is permitted. However, permissio to reprit/republish this material for advertisig or promotioal purposes or for creatig ew collective works for resale or redistributio to servers or lists, or to reuse ay copyrighted compoet of this work i other works must be obtaied from the IEEE. JINFENG DU, MING XIAO, MIKAEL SKOGLUND Stockholm September School of Electrical Egieerig ad the ACCESS Liaeus Ceter, Royal Istitute of Techology KTH), SE- 44 Stockholm, Swede IR-EE-KT :4

2 Cooperative Network Codig Strategies for Wireless Relay Networks with Backhaul Jifeg Du, Studet Member, IEEE, Mig Xiao, Member, IEEE, ad Mikael Skoglud, Seior Member, IEEE Abstract We ivestigate cooperative etwork codig strategies for relay-aided two-source two-destiatio wireless etworks with a backhaul coectio betwee the source odes. Each source multicasts iformatio to all destiatios usig a shared relay. We study cooperative strategies based o differet etwork codig schemes, amely, fiite field ad liear etwork codig, ad lattice codig. To further exploit the backhaul coectio, we also propose etwork codig based beamformig. We measure the performace i term of achievable rates over Gaussia chaels, ad observe sigificat gais over bechmark schemes. We derive the achievable rate regios for these schemes ad fid the cutset boud for our system. We also show that the cut-set boud ca be achieved by etwork codig based beamformig whe the sigal-to-oise ratios lie i the sphere defied by the source-relay ad relay-destiatio chael gais. I. INTRODUCTION Capacity bouds ad various cooperative strategies for three-ode relayig etworks source-relay-sik, or two cooperative sources ad oe sik) have bee studied i [], []. The relay or the other source) uses decode-ad-forward DF) or compress-ad-forward CF) to aid the trasmissio. Codig schemes have bee ivestigated for multiple-access relay chaels MARC) [3], [4] ivolvig multiple sources ad a sigle destiatio, ad for broadcast relay chaels BRC) [3], [5] where a sigle source trasmits messages to multiple destiatios. Recet results o capacity bouds for multiple-source multiple-destiatio relay etworks, [6] [9] ad refereces therei, have provided valuable isight ito the beefits of relayig. Motivated by the MAC chael at the relay ode where differet messages mix up by ature, various etwork codig NC) [] [] approaches, which essetially combie multiple messages together, ca be itroduced to boost the sum rate. For istace, i a relay-aided two-source two-sik multicast etwork, achievable rates for a full-duplex amplify-ad-forward AF) relay with liear NC LNC) have bee studied i [3], ad i [8] the relay uses lattice codes for etwork codig. I [4] joit NC ad physical layer codig is performed via lattice codig for the bi-directioal relay chael. The recetly proposed oisy etwork codig scheme Noisy NC) [5] for trasmittig multiple sources over a geeral oisy etwork, has bee show to outperform Mauscript received August 8, ; revised February 8,. This work was preseted i part at IEEE ITW, Aug.. This work was supported i part by the Swedish Govermetal Agecy for Iovatio Systems VINNOVA) ad the Swedish Foudatio for Strategic Research SSF). Jifeg Du, Mig Xiao ad Mikael Skoglud are with School of Electrical Egieerig ad the ACCESS Liaeus Ceter, Royal Istitute of Techology, Stockholm, Swede jifeg@kth.se; mig.xiao@ee.kth.se; mikael.skoglud@ee.kth.se). W Backhaul W S S X X Y r Figure. Two source odes S ad S, coected with backhaul, multicast iformatio W ad W respectively to both destiatios D ad D, with aid from a full-duplex relay ode R. the covetioal CF scheme i the Gaussia two-way relay chael ad the iterferece relay chael. Apart from itroducig dedicated relay odes to help the trasmissio, oe ca also utilize cooperative strategies amog sources [6] [] ad/or amog destiatios [] [] with the help of orthogoal coferecig chaels. I this paper, we aim at evaluatig achievable rate regios for various cooperative strategies whe source cooperatio ad etwork codig are desiged joitly with the relayig. More specifically, we focus o a relay-aided two-source twodestiatio multicast etwork with backhaul support, as show i Fig.. Sources S ad S multicast their ow iformatio W ad W respectively) to geographically separated destiatios D ad D, with the help of a relay R. This model arises, for example, i a wireless cellular dowlik where two base statios multicast to two mobile termials, oe i each cell, with the help of a dedicated relay deployed at the commo cell boudary. Sice the base statios are coected through the fiber or microwave) backhaul, more geeral etwork codig schemes ca be used at the relay to cooperate with the sources trasmissio. This model is iterestig sice it is a combiatio of relayig, MARC, BRC, source cooperatio, ad etwork codig. It ca be exteded to more geeral etworks by tuig the chael gais withi the rage [, ). I this paper, we are iterested i the sceario without cross chaels betwee S ad D, or S ad D. While, i geeral, the sigal from S i would be heard also at D j, j i, our assumptio ca be motivated for example i scearios where the cross liks are too weak to be of ay use, or are techically suppressed. I ay case we cosider ay cotributio directly from S i at D j j i) ot to be useful ad therefore part of the oise. We also restrict our aalysis to fixed chael gais, ad we assume a full-duplex DF relay. Furthermore, ay extesios of the cooperative NC strategies developed i this paper to multiple sources ad/or multiple relays are left to future work. R X r Y Y D D

3 The paper is orgaized as follows. The system model is itroduced i Sec. II. For symmetric chael gais ad high-rate backhaul, various cooperative NC strategies are ivestigated i Sec. III, ad a bechmark scheme together with the cutset boud are preseted i Sec. IV. Cooperative NC strategies for o-symmetric chael gais ad for low-rate backhaul i.e., partial trasmitter cooperatio) scearios are discussed i Sec. V. Numerical results are preseted i Sec. VI ad cocludig remarks i Sec. VII. Notatio: Capital letter X idicates a real valued radom variable ad px) idicates its probability desity/mass fuctio. X ) deotes a vector of radom variables of legth, ad with the kth compoet X[k] i geeral without emphasizig the ) ) ). IX; Y ) deotes the mutual iformatio betwee X ad Y, ad Cx) = log +x) is the Gaussia capacity fuctio. II. SYSTEM MODEL To simplify our aalysis, we first cosider the symmetric chael gai sceario illustrated i Fig. = X ) + bx r ) + Z ), a) = X ) + bx r ) + Z ), b) r = ax ) + ax ) + Z ) r, c) where a is the ormalized chael gai for the sourcerelay liks ad b for the relay-destiatio liks. For i =,, r, X ) i, i ad Z ) i are -dimesioal trasmitted sigals, received sigals, ad oise, respectively, where Z i [k], k =,..., are i.i.d. Gaussia with zero-mea ad uitvariace. The trasmitted sigals are subject to idividual average power costraits, i.e., Xi [k] P i, i =,, r. ) k= Note that ) implies simultaeously perfect sychroizatio at D, D, ad R, respectively. This assumptio, although widely adopted i iformatio-theoretic work, is optimistic i practice. I geeral, the results we obtai based o perfect sychroizatio will serve as upper bouds o ay practical performace, ad ca be directly exteded i the same way as i [] to scearios where costructive co-phase) additio is ot available. I practice the backhaul ormally has much higher capacity ad lower error rates tha the forward wireless chaels. Therefore, i our model the backhaul is assumed to be error-free ad of sufficietly high capacity higher tha the forward sum-rate). The case of a backhaul capacity smaller tha the sum-rate will be discussed i Sec. V. With a high rate backhaul, our system is closely related to the MIMO relay chael sceario, as studied i [3], [4]. However the problems are ot equivalet, ad we emphasize the followig three mai differeces betwee the system ivestigated i this paper ad the MIMO relay sceario with a two-atea source ode. First, i our system each source/atea is subject to a idividual power costrait ), while i the MIMO relay chael model a sum-power costrait is usually applied at the source ode, which i geeral implies a larger achievable rate regio. Secod, i our system the relay combies the messages from the sources by performig NC rather tha forwardig them separately through orthogoal chaels. Last but ot the least, the cooperative strategies proposed for high rate backhaul i Sec. III ca be directly exteded to the fiiterate backhaul sceario with the help of superpositio codig or time-sharig strategies, as stated i Sec. V. III. COOPERATIVE NETWORK CODING STRATEGIES Similar to [] [3], [6], source S i, i =,, divides its messages W i ito B blocks W i,,..., W i,b with R i bits each. The trasmissio is completed over B+ blocks. At the first block the two sources exchage W i, over the backhaul ad also broadcast their ow messages over the relay chaels; i block t, source S i exchages W i,t through the backhaul ad broadcasts its codeword X ) i,t, which is a fuctio of W i,t, W,t, W,t ), over the chaels; i block B + oly W i,b is broadcasted. As each trasmissio is over chael uses, ad assumig the backhaul is used for free, the BR i B+) overall rate is bits per chael use, which coverges to R i whe B goes to ifiity. Three decodig protocols, amely successive decodig [], backward decodig [5], ad slidig-widow decodig [6], have bee summarized ad exteded to multiple-source or multiple-relay scearios i [3]. We implemet these protocols at relay/destiatio odes depedig o the cooperative NC strategy uder cosideratio. Uless stated otherwise, radom codig is used for ecodig ad joit-typicality is used for decodig. Each codeword is geerated radomly i the memoryless fashio [7]: For trasmittig messages i W } each of R bits, we create a codebook cosistig of R radomly ad idepedetly geerated sequeces U ) }, each of -bit legth, accordig to the distributio Π i= pu i). We assig a codeword U ) to a message W ad associate them via a ecodig fuctio U ) W), omittig the explicit relatio where appropriate. A. Fiite-field Network Codig with DF DF+FNC) At the ed of block t, the relay decodes W,t, W,t ) joitly from its received sigal r,t ad the creates a ew message W r,t = W,t W,t bitwise GF) additio). If the legths of W,t ad W,t are ot equal, i.e., R R, we ca apped zeros at the ed of the shorter message. Durig block t, R trasmits W r,t usig a idepedet radom codebook U ) } of size R where R = maxr, R )), X ) r,t = P r U ) W r,t ). 3) S ad S, o the other had, trasmit their iformatio via idepedet radom codebooks V ) } of size R ad V ) } of size R, respectively. Sice W,t ad W,t are exchaged via the backhaul i block t, S ad S also kow W r,t if decodig at R is reliable. Therefore to exploit the possibility of coheret combiig gai, S ad S ca coordiate their trasmissio with R as follows, X ),t = α P V ) W,t ) + α )P U ) W r,t ), 4a) X ),t = α P V ) W,t ) + α )P U ) W r,t ), 4b)

4 3 Table I ILLUSTRATION OF THE ENCODING AND DECODING PROCESS FOR DF+FNC, WITH W r,t = W,t W,t, W r, =, AND B = 3. t = 3 4 W, W, W, W, W,3 W,3 / S trasmits W,, ) W,, W r, ) W,3, W r,3 ), W r,4 ) S trasmits W,, ) W,, W r, ) W,3, W r,3 ), W r,4 ) R trasmits W r, W r,3 W r,4 R decodes W,, W, W,, W, W,3, W,3 / D decodes W, W,, W r, W,3, W r,3 W r,4 recovers by / W, W, W,3 where α, α are power allocatio parameters. The received sigals are therefore,t = α P V ) + α )P +b P r )U ) + Z ),t, 5a),t = α P V ) + α )P +b P r )U ) + Z ),t, 5b) r,t =a α )P + α )P )U ) +a α P V ) +a α P V ) + Z ) r,t. 5c) Successive decodig is implemeted at both the relay ad the two destiatio odes: assumig W,t has bee successfully decoded by D, at the ed of block t, D recovers W,t, W r,t ) joitly from,t, ad the retrieves W,t = W r,t W,t. This approach is also used for D. The relay R decodes joitly W,t, W,t ) from r,t by first cacellig out U ). The ecodig/decodig process is illustrated i Table I. Propositio : The achievable rate regio for DF+FNC is the uio over all R, R ) satisfyig R <mi Ca α P ), Cα P ), C α )P +b } P r ) ), R <mi Ca α P ), Cα P ), C α )P +b } P r ) ), R + R < mi C P +b P r +b ) α )P P r, 6) Ca α P +a α P ), C P +b P r +b )} α )P P r, where the uio is take over α, α. Proof: The proof ca be foud i Appedix A. The costrait o R correspods to the coditio that W ca be decoded reliably at R ad D, ad that the NC message W r ca be decoded at D, ad similarly for R ad R +R. Note that our scheme is similar to the strategy i [8]: D recovers W from the direct lik ad W r from the R D lik, ad the retrieves W based o the observatio of W ad W r. But there are two mai differeces: fiite-field NC rather tha lattice codig is used; both source odes kow W r thaks to the backhaul ad therefore they cooperate with R to get a coheret combiig gai. Corollary : For the symmetric sceario with P =P = P r = P ad R =R =R, rate R is achievable by DF+FNC if R< max α mi CαP), Ca Pα), C+b +b α)p) 7) Proof: The result follows straightforwardly from 6) by settig α = α = α. Without the backhaul, S ad S caot kow/estimate W r ad therefore caot cooperate with R, i.e. α = α =. Hece, o coheret combiig gai ca be achieved. B. Liear Network Codig with DF DF+LNC) Whe LNC is used i the sigal domai, R essetially performs superpositio codig. The scheme preseted here is a atural extesio of the oe i Theorem of [6] which is desiged for trasmittig both private ad commo messages via the iterferece relay chael IFRC). I our case, oly commo messages are trasmitted i.e., multicast). Ulike i [6] where each source ca oly cooperate with ode R regardig its ow message i X r ), the two source odes ca i our case cooperate to trasmit both messages, thaks to the backhaul. We first geerate two idepedet radom codebooks U ) } of size R ad U ) } of size R. At the ed of block t, R decodes W,t, W,t ) ad the picks up codewords U ) W,t ) ad U ) W,t ) from the two codebooks respectively, ad trasmits the superpositio of these i block t with power allocatio parameter α r X ) r,t = α r P r U ) W,t ) + α r )P r U ) W,t ). For each codeword U ) W,t ), we geerate a idepedet codebook V ) } of size R, ad the use this codebook to ecode the ew message W,t. We deote the selected codeword for W,t give W,t as V ) W,t, W,t ). Similarly we choose V ) W,t, W,t ) for W,t. With power allocatio parameters α i, α i, i =, to cooperate with R, the trasmitted sigal at S ad S are therefore X ),t = α P U ) + α P U ) + α α )P V ), X ),t = α P U ) + α P U ) + α α )P V ). The received sigals at the destiatios ad the relay are = α α )P V ) + α P +b α r P r )U ) + α P + b α r )P r )U ) + Z ), = α α )P V ) + α P +b α r P r )U ) + α P + b α r )P r )U ) + Z ) r = a, [ α α )P V ) + α α )P V ) + α P + α P )U ) + α P + α P )U ) ]+Z ) r. 8) The decodig follows directly from [6]: the relay performs successive decodig ad the destiatios use backward decodig. R decodes W,t, W,t ) reliably from r,t at the ed }. of block t. D ad D start decodig whe trasmissio is fiished. At block B +, o ew message is trasmitted ad the received sigal at D D ) oly depeds o W,B, W,B ). After decodig W,B, W,B ) successfully, oly W,B W,B ) is ukow i ),B Y,B ), ad we repeat this process backwards util all messages are recovered. Propositio : The achievable rate regio for DF+LNC is

5 4 give by R < mi Ca P α α )), C α )P + b α r P r + b ) α α rp P r, C α P + b α r P r + b )} α α rp P r, R < mi Ca P α α )), C α )P +b α r )P r +b ) α α r)p P r, C α P + b α r )P r + b )} α α r)p P r, R +R < mi Ca α α )P +a α α )P ), C P +b P r +b [ α P P r α r + ]) α α r), C P +b P r +b [ α P P r α r + ])} α α r), 9) with the uio take over all α r, α, α, α, α, with α + α, α + α. Proof: The proof ca be foud i Appedix B. The costrait o R refers to the coditio that W ca be decoded successfully at R, D, ad D, respectively, ad similarly for R ad R + R. Corollary : For the symmetric sceario, the followig equal rate costraits apply R < C max α, α α +α mi α + b +b α )P Cα + b +b α )P), ), C a P α α ) ), )}. ) C +b +b α +b α )P Proof: Follows from 9) directly by settig α = α = α, α = α = α, ad α r = /. Without backhaul, X r would oly be partially kow by the source odes, i.e., α = α =. C. Physical Layer Network Codig by Lattice Codig I cotrast to Sec. III-A where R first decodes W, W ) ad the ecodes ito a joit NC message W r, the relay ca decode the NC message directly from Y r ) by usig lattice ecodig at the sources ad lattice decodig at the relay, as i [8], [4] where oly the case of symmetric powers is cosidered. We propose a protocol based o superpositio of a lattice code ad a radom code to be able to hadle the case of o-symmetric powers. Without loss of geerality, we assume that P P hece R R ). S splits its message W,t ito two parts [W,t, W,t], where W,t has the same legth as W,t. S ecodes W,t based o a ested lattice code [8], ad we deote the correspodig trasmitted codeword by V ) W,t ). S ecodes W,t usig the same ested lattice code as S, deotig the correspodig codeword by V ) W,t ), ad ecodes W,t usig a radom codebook V ) 3 } of size R R). Note that codewords V ) ad V ) are idepedet eve though they are geerated by the same ested lattice code, sice the dither vectors used at S ad S are idepedet [8], [8]. The relay, after decodig W,t via a sigle-user joit-typicality decoder ad the NC message W,t W,t usig a lattice decoder, ecodes all these ew messages by usig a idepedet radom codebook U ) } of size R, X ) r,t = P r U ) W,t W,t, W,t ). Sice W,t ad W,t are kow both at S ad S thaks to the backhaul, U ) W,t W,t, W,t ) is also kow. Therefore S ad S cooperate with R as follows X ),t = δv ) W,t ) + P δu ), ) X ),t = δv ) W,t ) + ǫv ) 3 W,t ) + P δ ǫu ), where δ P ad ǫ P δ are the allocated power to trasmit the ew messages. The correspodig received sigals at the relay ad destiatios are r,t =a ) δ V ) +V ) + a ǫv ) 3 + a P δ + ) P δ ǫ U ) + Z ),t = δv ) + r,t, P δ + b P r ) U ) + Z ),t, ),t = δv ) + ǫv ) 3 + P δ ǫ+b P r ) U ) +Z ),t. D performs successive decodig: at the ed of block t, D decodes W,t W,t, W,t ) from,t by joit typicality ad recovers W,t by usig W,t which has bee recovered successfully from block t ; after cacellig out U ) the ew iformatio W,t ca be decoded. This approach is also used for D. Propositio 3: Usig lattice codig, a achievable rate regio is give by R < mi C / + a δ), Cδ) }, R < mi C / + a δ + a ǫ/), Cδ + ǫ) }, R + R < mi C P + b P r + b ) P r P δ), C P + b P r + b )} P r P δ ǫ), 3) with the uio take over δ P ad ǫ P δ. Proof: The proof ca be foud i Appedix C. The first term i R R ) refers to the decodig costrait at R for the ested lattice code. Corollary 3: For the symmetric sceario, the achievable rate regio is R < max α mi CαP), C / + a Pα), C + b + b α ) P )}. 4) Proof: The result follows straightforwardly from 4) by settig ǫ = ad δ = Pα. Without backhaul, the NC message would ot be kow at the sources, i.e., δ = P ad ǫ = P P. D. Network Codig Based Beam-formig with DF DF+NBF) To further exploit the available coheret combiig beamformig) gai [] [3] at the siks, we propose a ew strategy that performs NC at both S ad S but ot at the relay decreasig the complexity at R). We refer to this scheme as

6 5 NC based beam-formig NBF) sice the sigals trasmitted at S, S ad R are formed i a beamformig-like fashio. NBF requires B+ blocks i total: W,t, W,t ) are exchaged via the backhaul durig block t ; at block t the NC message W t = fw,t, W,t ) is trasmitted; at block t+, W t is trasmitted by R. The relay trasmits W t usig a radom codebook U ) } of size R+R). For each codeword U ) W t ), we geerate a idepedet radom codebook V ) } of size R+R), ad the use it to ecode the ew message W t. We deote the selected codeword for W t give W t as V ) W t, W t ). At block t, the trasmitted sigals are X ) r,t = P r U ) W t ), 5) X ),t = α P V ) W t, W t ) + α )P U ) W t ), X ),t = α P V ) W t, W t ) + α )P U ) W t ), where α, α are power allocatio parameters. The correspodig received sigals are,t = α P V ) +b P r + α )P )U ) +Z ),t,,t = α P V ) +b P r + α )P )U ) +Z ),t, r,t =a α P + ) α P V ) 6) + a α )P + ) α )P U ) + Z ) r,t. The decodig process is similar as i the other cooperative strategies: the relay performs successive decodig ad the destiatios utilize backward decodig. Propositio 4: The achievable rate regio for NBF is defied by R + R < mi C P + b P r + b ) α )P P r, C P + b P r + b ) α )P P r, C a α P + α P + ))} α α P P, 7) with the uio take over the power allocatio parameters α, α. Proof: Sice S ad S trasmit the same NC message W t, the achievable sum-rate ca be split arbitrarily betwee them. Therefore i the NBF strategy oly the costraits for the sum-rate matter. Followig similar argumets as i Appedix A, the sum-rate costrait 55c) still holds here. By applyig successive decodig to r,t ad backward decodig to,t ad,t, the joitly Gaussia distributed radom variables V ), U ) ) will traslate 55c) ito 7). The terms i 7) idicate the costraits at D, D, ad R, respectively. Corollary 4: For the symmetric sceario, the achievable rate regio is defied by R < max mi α C4a Pα), C +b +b α ) P )}. 8) Proof: The result follows straightforwardly from 7) by settig α = α = α. Without the backhaul, this strategy is impossible. IV. BENCHMARK SCHEMES AND CUT-SET BOUND To evaluate the performace of the cooperative NC strategies preseted i Sec. III, we cosider two bechmark schemes, amely the o-nc based time-sharig relay scheme ad the o-df based oisy NC scheme [5]. We also derive the cut-set boud [7] for our sceario. A. Time Sharig Relay with DF DF+TD) I cotrast to the orthogoal scheme described i [6] for the case of the IFRC, S ad S here cooperate with R to covey both messages. We first geerate two idepedet radom codebooks U ) } of size R ad U ) } of size R, ad they will be used by R to help S ad S, respectively. For each codeword i U ) }, we geerate a idepedet radom codebook V ) } of size R, ad the use it to ecode the ew message at S. Similarly we geerate a radom codebook V ) } of size R for each codeword i U ) }. Durig block t, W,t ad W,t are exchaged via the backhaul, ad the trasmissio durig block t is divided ito two parts. Durig the first part of block t, the trasmitted sigals are X ) αp β r,t = P r U ) W,t ), X ),t =, X ),t = V ) W,t, W,t )+ P α ) β U ) W,t ), where α is the power allocatio parameter ad β is the time sharig parameter. Trasmissio power P /β is used i X ),t to meet the power costrait ). The received sigals are,t = bx ) r,t + Z ),t = b P r U ) + Z ) r,t = a,t = αp β V ) + a αp β V ) + P α ) β,t, U ) + Z ) r,t, 9) P α ) β +b P r ) U ) + Z ),t. The relay decodes W,t give W,t ad the ecodes it to U ) W,t ). Durig the remaiig part of block t, the trasmitted sigals are X ) r,t = P r U ) W,t ), X ),t =, X ),t = αp β V ) W,t, W,t )+ The correspodig received sigals are P α ) β,t = bx ) r,t + Z ),t = b P r U ) + Z ) r,t = a,t = αp β V ) + a αp β V ) + P α ) β α)p,t, U ) W,t ). U ) + Z ),t, ) β + b P r ) U ) + Z ),t. At the ed of block t, R decodes W,t give W,t, ad D ca retrieve W,t, W,t ) reliably usig slidig-widow decodig based o the received sigals durig block t. Similarly, after the first part of block t +, D ca decode W,t, W,t ) reliably based o sigals received from the first part of block t + ad the secod part of block t.

7 6 Propositio 5: The achievable rate regio for this time sharig strategy is defied by R < miβc αa P β ), βc αp β ) + β)cb P r ), )} β)c b P r + P β + b α )P P r / β), R < mi β)c αa P αp β ), β)c β )+βcb P r ), )} βc b P r + P β + b P P r α )/β, R +R < mi ) ) β)cb P r )+βc b P r + P β +b P P r α )/β, )} βcb P r ) + β)c b P r + P β +b PP r α ) β, with the uio take over all α, α ad the time sharig parameter β. Proof: The proof follows immediately from Appedix B by applyig the Gaussia coditio ad otig the depedece stated i 9) ad ). Costraits i R R ) correspod to the coditio of successful decodig of W W ) at R, D D ), ad D D ), respectively. Costraits i R + R refer to successful decodig at D ad D. By settig α = α = α ad β = /, ) ca be traslated ito the symmetric rate costrait R < max α mi C P +b +b α )), 4 C α+b +αb P)P ), Ca Pα), [ Cb P)+C +b +b α ) P )]}. ) Without backhaul, sources ca oly ecode over their ow messages. Therefore we have α = α = ad the first term i ) reduces to Cb P). B. Noisy Network Codig Noisy NC) The basic priciple of oisy NC, as described i [5], is to covey a super message B times, each time usig a idepedet codebook ad lettig B, before the destiatios) ca successfully decode the message. Therefore it is ot clear how collaboratio via the fiite-rate backhaul ca be implemeted, sice it requires a B times higher backhaul rate to exchage the super message before trasmissio starts. O the other had, the backhaul provides orthogoal i.e., out-ofbad) coferecig bit-pipes betwee two source odes. How to exted the oisy NC scheme [5], origially desiged for relay etworks with co-chael i.e., i-bad) trasmissio, so as to optimally utilize the rate-limited backhaul is iterestig but out of the scope of this paper. The achievable rate regio for oisy NC without backhaul collaboratio) ca be specialized from Theorem of [5] to the multicast relay etwork i Fig. as follows R < miix ; ŶrY X X r Q), IX ; ŶrY X X r Q), IX X r ; Y X Q) IY r ; Ŷr X X X r Y Q), IX X r ; Y X Q) IY r ; Ŷr X X X r Y Q)}, R < miix ; ŶrY X X r Q), IX ; ŶrY X X r Q), IX X r ; Y X Q) IY r ; Ŷr X X X r Y Q), IX X r ; Y X Q) IY r ; Ŷr X X X r Y Q)}, R +R < miix X ; ŶrY X r Q), IX X ; ŶrY X r Q), IX X X r ; Y Q) IY r ; Ŷr X X X r Y Q), IX X X r ; Y Q) IY r ; Ŷr X X X r Y Q)}, 3) where Ŷr is the compressed versio of Y r, Q is the timesharig radom variable, ad the joit probability ca be partitioed as pq)px q)px q)px r q)pŷ r x r, y r, q). By settig Q = ad Ŷr = Y r + Ẑ with Ẑ N, σ ), ad applyig ) ad ), the achievable rate regio i 3) is simplified to R < mica P / + σ )), Cb P r ) C/σ )}, R < mica P / + σ )), Cb P r ) C/σ )}, R + R < micp +a P +P +P P )/+σ )), CP + a P + P + P P )/+σ )), CP + b P r ) C/σ ), CP + b P r ) C/σ )}, 4) with the uio take over all σ >. Note that redudat terms have bee removed from R ad R. For the symmetric sceario, the achievable rate regio is R < mic a P +σ ), CP+ a P+P ) +σ ), Cb P) C/σ ), CP + b P) C/σ )}. 5) C. Cut-Set Boud By the cut-set boud [7], the maximum achievable sum rate from the source odes to ay of the destiatios ca be o larger tha the miimum of the mutual iformatio flows across all possible cuts, maximized over a joit distributio for the trasmitted sigals. I our case, the cut-set boud betwee the two sources ad each of the sik for the etwork i Fig. ca be derived based o four cuts, as demostrated i Fig., as follows the dimesio super script ) is suppressed to simplify the otatio) R +R C cut-set = sup px,x,x r) mi IX X ; Y Y r X r ), IX X X r ; Y ), IX X ; Y Y r X r ), } IX X X r ; Y ) +ǫ, 6) where ǫ as, ad X, X ad X r are potetially correlated. As suggested i [9, chp. 7], to fid the exact cut-set boud C cut-set, we will first fid a upper boud C upp C cut-set based o the techique used i [], ad the fid a lower boud C cut-set, G C cut-set by restrictig the source distributio to Gaussia, ad fially show that C cut-set, G = C upp. Followig the covetioal otatio for the differetial etropy hx) of a cotiuous valued radom variable X, the mutual iformatio correspodig to cut ca be writte as IX X X r ; Y ) = hy ) hy X X X r ) = hy ) hz ) = hy ) logπe). 7)

8 7 W Backhaul W S S X X Cut Y r R Cut X r Cut 3 Cut 4 Figure. The sum multicast capacity is bouded by the cut-set boud based o the four cuts show i the figure. From the maximum etropy lemma [7], we get hy ) hy,i ) logπevar[y,i]), 8) i= i= where the secod equality is achieved whe Y,i is Gaussia distributed. Hece IX X X r ; Y ) logvary,i)) i= log VarY,i )), 9) i= where the last steps follow from Jese s iequality. Furthermore, we have VarY,i ) = VarX,i + bx r,i ) + + E[X,i + bx r,i ) ] Y Y D D = + E[X,i ] + b E[X r,i ] + be[x,ix r,i ], 3) with equality for E[X i ] =. Also, usig the Cauchy Schwarz iequality we get E[X,i X r,i ] E[Xr,i ] E[EX,i X r,i )) ]. i= i= i= 3) As i [], we ca itroduce a auxiliary variable α [, ] such that E[EX,i X r,i )) ] = α )P. 3) i= The by applyig the power costrait defied i ), we have E[VarX,i X r,i )] = E[X,i] E[EX,i X r,i )) ]) i= i= α P. 33) Similarly, we ca itroduce α [, ] such that E[EX,i X r,i )) ] = α )P, 34) i= E[VarX,i X r,i )] α P. 35) i= By substitutig 3) ito 3) we get E[X,i X r,i ] α )P P r. 36) i= Now, substitutig 36) ad 3) ito 9), ad applyig the same approach also to cut 4, we get IX X X r ; Y ) CP +b P r +b α )P P r ), IX X X r ; Y ) CP +b P r +b α )P P r ). 37) For cut we have IX X ; Y Y r X r ) = hy Y r X r ) hy Y r X X X r ) 38) = hy Y r X r ) hy X X X r ) hy r X X X r ) = hy Y r X r ) hz ) hz r ) = hy Y r X r ) logπe) log πe) K i ) logπe) = log K i ), i= where the secod equality i 38) comes from the fact that Y ad Y r are idepedet give X, X, X r ) ad the iequality is due to the maximum etropy lemma [7], with equality achieved by joit Gaussia distributed Y,i, Y r,i ) with coditioal covariace matrices K i defied by [ ] E[VarY K i =,i X r,i )] E[CovY,i, Y r,i X r,i )], 39) E[CovY,i, Y r,i X r,i )] E[VarY r,i X r,i )] where E[VarY,i X r,i )] = + E[VarX,i X r,i )], i= E[CovY,i, Y r,i X r,i )] = ae[covx,i, X,i +X,i X r,i )] = ae[varx,i X r,i )]+E[CovX,i, X,i X r,i )]), 4) E[VarY r,i X r,i )] = + a E [VarX,i +X,i ) X r,i ] = + a E[VarX,i X r,i )] + E[VarX,i X r,i )] + E[CovX,i, X,i X r,i )]). 4) Obviously, the covariace matrices K i are positive semidefiite. Sice the fuctio log K is cocave [3], we ca thus boud the throughput of cut as follows IX X ; Y Y r X r ) i= log log K i ) ) K i. 4) It is the straightforward to show that for the ier term i 4) we ca get + a i= K i = + + a E[VarX,i X r,i )] i= i= E[VarX,i X r,i )] + a E[CovX,i, X,i X r,i )] i= i= ) + a E[VarX,i X r,i )] E[VarX,i X r,i )] i= a i= E[CovX,i, X,i X r,i )]). 43) i=

9 8 As CovX,i, X,i X r,i )=φ i VarX,i X r,i )VarX,i X r,i ), where φ i is the correlatio coefficiet, we have E[CovX,i, X,i X r,i )] i= φ i E[VarX,i X r,i )] i= φ i E[VarX,i X r,i )] i= α α P P, 44) where the first iequality is due to the Cauchy Schwarz iequality ad the last step is give by 33) ad 35). Give that φ i, we ca itroduce aother auxiliary variable ρ such that E[CovX,i, X,i X r,i )] = ρ α α P P. 45) i= Now, by substitutig 33), 35) ad 45) ito 43), the boud 4) ca be traslated ito IX X ; Y Y r X r ) Cα P 46) +a α P +α P +ρ α α P P + ρ )α α P P ) ). Similarly, we ca boud the throughput of cut 3 as follows IX X ; Y Y r X r ) Cα P 47) +a α P +α P +ρ α α P P + ρ )α α P P ) ). By combiig the idividual bouds defied by 37), 46) ad 47), ad let, the cut-set boud C cut-set i 6) ca be upper bouded by C upp, as defied i 48) at the bottom of this page. O the other had, we ca also lower boud C cut-set by C cut-set, G, obtaied by restrictig px, X, X r ) i 6) to be Gaussia. We partitio the Gaussia variables X, X ad X r as follows X r = P r U, 49a) X = ρ)α P S + ρα P V + α )P U, 49b) X = ρ)α P S + ρα P V + α )P U, 49c) where S, S, V, ad U are -dimesioal idepedet Gaussia radom vectors with zero-mea ad uit-variace. α, α, ρ are auxiliary variables itroduced to represet the potetial correlatio amog X, X ad X r due to cooperatio. The received sigals are the Y = ρ)α P S + b P r + α )P )U + ρα P V + Z, 5a) Y = ρ)α P S + b P r + α )P )U + ρα P V + Z, 5b) Y r = a ρ)α P S +a α )P + α )P )U +a ρ)α P S +a ρα P + ρα P )V +Z r. 5c) By substitutig 5) ito 7) ad 38), we ca derive from 6) C cut-set, G = sup mi logvary,i )), α,α,ρ i= logvary,i )), log K,i ),log K,i )}+ǫ, 5) where ǫ as, ad for i =,..., we have VarY,i ) = + P + b P r + b α )P P r, VarY,i ) = + P + b P r + b α )P P r, K,i = ++a )α P +a α P +a ρ α α P P +a ρ )α α P P, K,i = ++a )α P +a α P +a ρ α α P P +a ρ )α α P P. 5) By substitutig 5) ito 5) we get C cut-set, G which actually equals to C upp as defied i 48), i.e., C cut-set, G = C upp. Recall that C cut-set, G C cut-set C upp, we ca fially coclude that C cut-set = C upp, i.e., the capacity upper boud defied i 48) is actually the cut-set boud. For the symmetric sceario where P = P = P r = P, by settig α = α = α, the cut-set boud defied i 48) ca be traslated to the followig costrait R < sup mi α,ρ C P + b + b α )), C P[ + a )α + a ρα + a ρ )α P] ) }. 53) D. Achievability of the Cut-Set Boud by DF+NBF Propositio 6: I the symmetric sceario where P = P = P r = P ad R = R = R, DF+NBF ca achieve the cutset boud, i.e. 8) ad 53) are equivalet, if ad oly if a, b, P) satisfy 4a > max, + b } 8a < P a ) a +b ) b + 4a b )4a )b. 54) C upp = sup mi C P + b P r + b ) α )P P r, C P + b P r + b ) α )P P r, α,α,ρ C C + a )α P + a α P + a ρ ) α α P P + a ρ )α α P P, + a )α P + a α P + a ρ )} α α P P + a ρ )α α P P. 48)

10 9 Power costrait P/σ [db] 5dB db 5dB db 5 R.5.5 Cut set Boud DF + NBF Lattice Codig DF + LNC DF + FNC DF + TD Noisy NC S R gai a [db] R D gai b [db].5 Figure 3. Upper boud for the ormalized trasmit power costrait P/σ give differet source-relay ad relay-destiatio chael gais. Cotour plots of the upper boud are show at the bottom. Proof: The proof ca be foud i Appedix D. Propositio 6 states that there exists a large set of differet source-relay chael gais ad relay-destiatio chael gais, where the cut-set boud ca be achieved by the DF+NBF strategy if the ormalized σ = ) trasmit power costrait P is o larger tha a upper boud defied i 54), as show i Fig. 3. Therefore we ca claim that eve for o-degraded Gaussia relay chaels, the capacity regio for the system defied i Fig. ca be kow for the scearios defied by Propositio 6. A ituitive iterpretatio of Propositio 6 is that 54) esures the successful decodig at the relay ode R. I this sceario, the NBF achievable rate 8) ad the cut-set boud 53) have the same active costrait o the MAC at D ad D, ad therefore leads to tight capacity bouds. The upper boud o P i 54) is to make sure that, give a ad α, the secod term the costrait at R) i cut-set boud 53) caot be icreased by reducig ρ otherwise we ca icrease 53) simply by decreasig α ad ρ). V. MORE GENERAL NETWORKS With a high-rate backhaul, the extesio to o-symmetric chael gais is straightforward: Replacig a, b with a, a, b, b i the previous aalysis where appropriate, we will get the achievable rate regios ad the cut-set boud i the geeral case. However, the results for the symmetric sceario where R = R = R has to be modified sice settig α = α may o loger be the optimal solutio. For a low-rate backhaul with capacity of C, i.e., oly partial cooperatio amog the sources is possible, cooperative NC strategies ca be formulated i the followig way. By exploitig rate-splittig [3], we first partitio each source message ito two parts W = [W c, W p ], W = [W c, W p ], ad the divide all the four messages evely ito B blocks W c,t, W p,t, W c,t, W p,t, each with R c, R p, R c, R p bits, respectively. The sources the exchage W c,t, W c,t ) over backhaul at rate R,c +R,c <C to eable cooperative trasmissio. Therefore cooperative NC strategies R Figure 4. Achievable rate regios with trasmittig power P /σ = db, P /σ = 5dB, P r/σ = 5dB, source-relay chael gai a = 5dB, ad relay-destiatio chael gai b = 5dB. The cut-set outer boud is also plotted for referece. Curves for FNC ad Lattice codig coicide each other. preseted i Sec. III are used to trasmit W c,t, W c,t ) ad o-cooperative strategies are used to trasmit W p,t, W p,t ). Correspodig power allocatio parameters eed to be optimized. VI. NUMERICAL RESULTS I this sectio we preset the achievable rate regios ad the achievable equal rates R for FNC, LNC, Lattice code, ad NBF strategies, ad compare them to two bechmark schemes ad the cut-set boud. A. Achievable Rate Regios I Fig. 4, we plot the achievable rate regios for a sceario where source S has trasmit power P /σ =db, S has a power budget P /σ =5dB, R has trasmit power costrait P r /σ =5dB, the source-relay chael gai a =5dB, ad the relay-destiatio chael gai b =5dB. Not surprisigly, the NBF scheme achieves the cut-set boud for this low to medium SNR regio, which has bee proved i Sec. IV-D for the symmetric scearios. The curves for FNC ad Lattice code coicide each other. B. Symmetric Achievable Rates I Fig. 5, we ivestigate the impact of the relay-destiatio lik quality b o the achievable rates for differet cooperative strategies, with fixed trasmit power P/σ =5dB ad sourcerelay chael gai a =db ad a =5dB. With backhaul, substatial rate gais ca be achieved by performig LNC or NBF compared to the time sharig relay. FNC or lattice codig is preferred for small b. As illustrated i the sub-figure, Sigificat gais ca be achieved by utilizig the backhaul i the case of a poor relay-destiatio lik small b ). I Fig. 6 we fix P/σ =5dB ad b =db istead ad vary the source-relay lik quality a. Rate gais of NC are sigificat i a large rage of a values. Note that whe

11 Achievable Rates Achievable Rates b [db] Backhaul Rate Gais Cut set Boud DF + NBF Lattice Codig DF + LNC DF + FNC DF + TD Noisy NC Relay Destiatio chael gai, b [db] R D gai b [db] Cut Set Boud DF + NBF Lattice Codig R NBF R Lattice..3 5 S R gai a [db] Achievable Rates.5.5 Cut set Boud DF + NBF Lattice Codig DF + LNC DF + FNC DF + TD Noisy NC Backhaul Rate Gais 5 b [db] Relay Destiatio chael gai, b [db] Figure 5. Effects of the relay-destiatio chael gai b o the achievable rates with backhaul, whe P/σ = 5dB ad the source-relay chael gai a = db upper) ad 5dB lower). The rate gais compared to the schemes without backhaul are also preseted i the sub-figure. Achievable Rates Cut set Boud. DF + NBF.5 Lattice Codig..4 DF + LNC Backhaul Rate Gais DF + FNC DF + TD Noisy NC a [db] Source Relay chael gai, a [db] Figure 6. Effects of the source-relay chael gai a o the achievable rates with P/σ = 5dB ad the relay-destiatio chael gai b = db. the source-relay lik quality is comparable to the sourcedestiatio lik, i.e. a is aroud db, the lattice codig strategy is preferred over LNC or FNC. The gai by usig backhaul is sigificat for all schemes for a larger tha db. C. Compariso of NBF with Lattice Codig To illustrate the performace of usig lattice codig, we compared it to the NBF at fixed P/σ =7dB, show i Fig. 7. R D gai b [db] 5/ 5 R D gai b [db] P/σ =5dB P/σ =5dB / 5 S R gai a [db] P/σ =db P/σ =3dB 5 5 S R gai a [db] Figure 7. Compariso of DF+NBF 8) ad Lattice Codig 4) with P/σ =7dB upper). Cotour plots give differet trasmit power costraits are also show lower). The relative rate gai of NBF compared with lattice codig give differet P/σ is also show as the cotour plots. NBF outperforms lattice codig uiformly i low SNR regios P/σ 5dB) ad i medium SNR regios 5<P/σ <db) with relatively strog source-relay gai a. For high SNR regios P/σ >db), lattice codig outperforms NBF for most of chael coditios. VII. CONCLUSIONS We have cosidered a relay-aided two-source two-sik wireless multicast etwork with a backhaul lik betwee the source odes. Differet cooperative etwork codig strategies are ivestigated ad compared with the cut-set boud ad a bechmark strategy that does ot use etwork codig, i.e., the relay is time shared by source odes. Sigificat rate gais have bee demostrated. We have show that the cutset boud ca be achieved i certai chael cofiguratios. I geeral, etwork codig based beam-formig NBF) strategies give the best performace. I high SNR regios, however, the lattice code based strategy is preferred. FNC, which oly performs modulo- additio i the fiite field, suffers limited performace loss i most of the cases. Further, ad more importatly, we show sigificat rate gais compared to the scearios without backhaul i various chael coditios.

12 APPENDIX A PROOF OF PROPOSITION The achieved rate regios for MARC derived i [3], [4] ivolvig multiple sources ad a full-duplex DF relay ca be directly applied here. Observig that X r ) is fully determied by U ) as stated by 3) ad that there is o cross liks as defied by ), the rate regios defied by 5) of [4] ad 4), 5) of [3] ca be traslated to the FNC strategy as follows R < miix ) ; Y r ) U ) X ) ), IX ) ; U ) ), IX r ) ; V ) )}, 55a) R < miix ) ; Y r ) U ) X ) ), IX ) ; U ) ), IX r ) ; V ) )}, 55b) R + R ) < miix ) X ) ; Y r ) U ) ), IX ) X r ) ; ),IX ) X r ) ; )}, 55c) where U, V, V are auxiliary radom variables ad the joit probability partitios as follows pv, V, U, X, X, X r )=px, V U)PX, V U)PX r, U). For costraits of R i 55a), the first term correspods to successful decodig of W at R give that the relayig sigal X r ) has bee cacelled out ad S is ot trasmittig. The secod term refers to the decodig of W at D give correctly decoded W r. The last term idicates the successful decodig of W r hece W after NC decodig) at D give correctly decoded W. It is similar for R ad R + R. Sice X U X forms a Markov chai, by followig the similar argumets as i the proof of Lemma 3 i [8], oe ca show that there exist joit Gaussia variables X ), X ), U ) ) such that the achievable rate regio defied i 55) ca be maximized. By choosig U, V, V i 3) ad 4) as i.i.d. zero-mea uit-variace radom variables, we ca see that X, X r, X ) is a zero-mea joitly Gaussia tuple satisfyig the power costrait ) ad X X r U) X forms a Markov chai. By substitutig 5) ito 55) ad applyig the Gaussia coditio, oe ca get 6) straightforwardly. APPENDIX B PROOF OF PROPOSITION By Theorem of [6], R ca decode W,t, W,t ) reliably if is large, its past detectio is correct, ad R < IX ) ; Y r ) R < IX ) ; Y r ) R + R ) < IX ) X ) U ) U ) X ) X ) r ), 56a) U ) U ) X ) X ) r ), 56b) ; Y r ) U ) U ) X ) r ), 56c) where U, U are auxiliary radom variables ad the joit probability partitios as follows pu, U, X, X, X r ) = px, U )PX, U )PX r U, U ). For i =,, D i ca decode W i,t reliably if is large, its previously detectio of W i,t is correct, ad R < miix ) X r ) ; U ) ), IX r ) ; U ) X ) )}, R < miix ) X r ) ; U ) ), IX r ) ; U ) X ) )}, R +R )< miix ) X r ) ; ), IX ) X r ) ; )}. 57) By choosig U, U, V, V i.i.d. zero-mea uit-variace radom Gaussia ad applyig the Gaussia coditios i 56) ad 57), we obtai the rate regio as defied i 9). APPENDIX C PROOF OF PROPOSITION 3 After cacellig out U ) from r,t, the relay ca reliably decode W,t if R R < log+ a ǫ +a δ ) by usig a Gaussia codebook). The R ca further cacels V ) 3 out ad uses the remaiig sigal to decoded the NC message by usig lattice decodig [4], [8] if R < log +a δ). Therefore decodig at R will geerate the followig costraits R <C / + a δ), 58) R < log +a δ)+ a ǫ +a δ )) = C +a δ+ a ǫ ). D ad D ca successfully decode W,t ad W,t = [W,t, W,t ], respectively, if R < log + δ), R < log + δ + ǫ). 59) By usig successive decodig at both D ad D, the followig costraits apply R +R < log+δ)+ log + P ) δ+b P r) +δ, 6) R +R < log+δ+ǫ)+ log + P ) δ ǫ+b P r). +δ+ǫ Combie 58), 59) ad 6) together we ca obtai 3). APPENDIX D PROOF OF PROPOSITION 6 From 8) ad 53) we ca capture the effective power gai as follows g NBF = max α mi 4a α, + b + b α }, 6) g cut-set = sup α,ρ mi + b + b α, α + a α + αρ + ρ )α P) }. 6) From 6) it is straightforward to show that 4a g NBF =, if 4a + b ; + b + b α, otherwise, 63) where a [, ] satisfies 4a α = + b + b α. For the secod part of 6), we have max α + α,ρ a α + αρ + ρ )α P) = max α + α,ρ a α + α P+ P P/P αρ) ) +a = +P+ P ), if P>, [by settig α=, ρ= P ] + 4a, if P, [by settig α = ρ = ] 64) By combiig 6) ad 64) oe ca easily coclude that + 4a, if 4a b ad P ; g cut-set = > + 4a, if 4a b ad P > ; +b +b 65) α, otherwise; where α satisfies the equality +b +b α = α +a α +α ρ+ ρ )α ) P).

13 From 65) it clearly follows that g cut-set >4a for the scearios whe 4a +b. Therefore g NBF = g cut-set is possible oly if 4a >+b, i.e., there should exist two variables α, ρ such that 4a α = + b + b α, 4a α =α + a α + α ρ + ρ )α ) P). 66a) 66b) By subtractig +b from both sides of 66a) ad the takig square, we have 6a 4 α ) α 8a + 8a b 4b ) + b ) =, which has oly oe true root for 66a) must satisfy 4a α > + b ) α = a + b ) b + 4a b )4a a)b 8a 4. 67) From 66b) we get ρα = /P + α /a P) + α /P), or ρα = /P α /a P) + α /P). Sice ρα α, the first root is obviously a false root ad therefore omitted. To make the secod root satisfy the costrait, we must have /P α /a P) + α /P) α. The secod iequality is self-evidet, ad the first iequality requires a > / ad α P ). 68) a We therefore coclude from 67) ad 68), give 4a > +b ad P >, that 66) holds if ad oly if 4a > ad a + b ) b + 4a b )4a a)b 8a 4 P a ). Combied with the fidig that g NBF = g cut-set is impossible for 4a +b, we ca coclude that g NBF = g cut-set, i.e. 8) ad 53) are idetical, if ad oly if a, b, P) satisfies 54). ACKNOWLEDGMENTS The authors would like to thak the aoymous reviewers ad the associate editor for their suggestios that helped improve the quality ad presetatio of the paper. REFERENCES [] T. M. Cover ad A. El Gamal, Capacity theorems for the relay chael, IEEE Tras. If. Theory, vol. 5, pp , Sep [] A. Høst-Madse ad J. Zhag, Capacity bouds ad power allocatio for wireless relay chaels, IEEE Tras. If. Theory, vol. 5, pp. 4, Ju. 5. [3] G. Kramer, M. Gastpar ad P. Gupta, Cooperative strategies ad capacity theorems for relay etworks, IEEE Tras. If. Theory, vol. 5, pp , Sep. 5. [4] G. Kramer ad A. J. va Wijgaarde, O the white Gaussia multipleaccess relay chael, i Proc. of IEEE ISIT, Ja.. [5] Y. Liag ad G. Kramer, Rate regios for relay broadcast chaels, IEEE Tras. If. Theory, vol. 53, pp , Oct. 7. [6] O. Sahi ad E. 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14 3 Jifeg Du S 7) received his B.Eg. degree i electroic iformatio egieerig from the Uiversity of Sciece ad Techology of Chia USTC), Hefei, Chia i 4, his M.Sc. degree i electroic egieerig, ad his Tek. Lic. degree i electroics ad computer systems, both from the Royal Istitute of Techology KTH), Stockholm, Swede, i 6 ad 8, respectively. He is curretly workig for his Ph.D. degree i telecommuicatios at KTH. He received the best paper award from IC-WCSP, Suzhou, October, the Has Werthé Grat from the Royal Swedish Academy of Egieerig Sciece IVA) i March, ad the grat from Ericsso s Research Foudatio i May. Mig Xiao S -M 7) was bor i the SiChua Provice, P. R. Chia, o May d, 975. He received Bachelor ad Master degrees i Egieerig from the Uiversity of Electroic Sciece ad Techology of Chia, ChegDu i 997 ad, respectively. He received Ph.D degree from Chalmers Uiversity of techology, Swede i November 7. From 997 to 999, he worked as a etwork ad software assistat egieer i ChiaTelecom. From to, he also held a positio i the SiChua commuicatios admiistratio. From November 7 to ow, he has bee i ACCESS Liaeus ceter, school of electrical egieerig, Royal Istitute of Techology, Swede, where he is curretly a assistat professor. He received Chiese Govermet Award for Outstadig Self-Fiaced Studets Studyig Aborad i 7. He got Has Werthé Grat from royal Swedish academy of egieerig sciece IVA) i March 6, ad Ericsso s research foudatio i. Mikael Skoglud S 93-M 97-SM 4) received the Ph.D. degree i 997 from Chalmers Uiversity of Techology, Swede. I 997, he joied the Royal Istitute of Techology KTH), Stockholm, Swede, where he was appoited to the Chair i Commuicatio Theory i 3. At KTH, he heads the Commuicatio Theory Lab ad he is the Assistat Dea for Electrical Egieerig. Dr. Skoglud s research iterests are i the theoretical aspects of wireless commuicatios. He has worked o problems i source chael codig, codig ad trasmissio for wireless commuicatios, Shao theory ad statistical sigal processig. He has authored some scietific papers, icludig papers that have received awards, ivited coferece presetatios, ad papers rakig as highly cited accordig to the ISI Essetial Sciece Idicators. He has also cosulted for idustry, ad he holds six patets. Dr. Skoglud has served o umerous techical program committees for IEEE cofereces. Durig 3 8 he was a associate editor with the IEEE Trasactios o Commuicatios ad he is presetly o the editorial board for IEEE Trasactios o Iformatio Theory.