Decode-forward and Compute-forward Coding Schemes for the Two-Way Relay Channel

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1 Decode-forward ad Compute-forward Codig Schemes for the Two-Way Relay Chael Peg Zhog ad Mai Vu Departmet of Electrical ad Computer Egieerig McGill Uiversity Motreal, QC, Caada H3A A7 s: arxiv:8.3599v [cs.it] 7 Aug Abstract We cosider the full-duplex two-way relay chael with direct lik betwee two users ad propose two codig schemes: a partial decode-forward scheme, ad a combied decode-forward ad compute-forward scheme. Both schemes use rate-splittig ad superpositio codig at each user ad geerate codewords for each ode idepedetly. Whe applied to the Gaussia chael, partial decode-forward ca strictly icrease the rate regio over decode-forward, which is opposite to the oe-way relay chael. The combied scheme uses superpositio codig of both Gaussia ad lattice codes to allow the relay to decode the Gaussia parts ad compute the lattice parts. This scheme ca also achieve ew rates ad outperform both decode-forward ad compute-forward separately. These schemes are steps towards uderstadig the optimal codig. I. INTRODUCTION The two-way chael i which two users wish to exchage message was first studied by Shao []. A specific model is the two-way relay chael TWRC) with a relay located betwee two users to help exchage messages. Two types of TWRC exist: oe without a direct lik betwee the two users, a model suitable for wired commuicatio, ad oe with the direct lik, more suitable for wireless commuicatio. I this paper, we focus o the TWRC with direct lik betwee the two users, also called the full TWRC. A umber of codig schemes have bee proposed for the full TWRC. Differet relay strategies, icludig amplifyad-forward, decode-forward based o block Markov codig, compress-forward ad a combied decode-forward ad compress-forward scheme, are studied i []. For the decodeforward strategy, the relay reliably decodes the trasmitted messages from both users. It the re-ecodes ad forwards. For the compress-forward strategy, the relay compresses the oisy received sigal ad forwards. I [3], a decode-forward scheme based o radom biig ad o block Markovity was proposed, i which the relay broadcasts the bi idex of the decoded message pair. A ew relayig strategy called compute-forward was recetly proposed i [4], i which the relay decodes liear fuctios of trasmitted messages. Nested lattice code [5] is used to implemet compute-forward i Gaussia chaels, sice it esures the sum of two codewords is still a codeword. Compute-forward has bee show to outperforms i moderate SNR regimes but is worse at low or high SNR [4]. Compute-forward ca be aturally applied i two-way relay chaels as the relay ow receives sigal cotaiig more tha oe message. I [6], ested lattice codes were proposed for the Gaussia separated TWRC with symmetric chael, i.e. all source ad relay odes have the same trasmit powers ad oise variaces. For the more geeral separated AWGN TWRC case, compute-forward codig with ested lattice code ca achievable rate regio withi / bit of the cut-set outer boud [7] [8]. For the full AWGN TWRC, a scheme based o compute-forward, list decodig ad radom biig techique is proposed i [9]. This scheme achieves rate regio withi / bit of the cut-set boud i some cases. I this paper, we cosider the ideas of decode-forward ad compute-forward together ad propose two ew codig schemes for the full TWRC. The first scheme is a partial decode-forward scheme which exteds the decode-forward scheme i [3]. Each user splits its message ito two parts. The relay decodes oe part of message from each user, reecode these two parts together ad forwards. This scheme cotais the origial decode-forward scheme i [3] as a special case. Differet from the oe-way relay chael i which partial decode-forward brigs o improvemet o the achievable rate over decode-forward i Gaussia chaels [], somewhat surprisigly here for the full TWRC, partial decode-forward ca achieve ew rates ad strictly icrease the rate regio over decode-forward. The secod scheme combies decode-forward scheme with compute-forward for the full Gaussia TWRC. Each user also splits its message ito two parts, ad ecodes oe part with a Gaussia codeword ad the other with a lattice codeword. The relay chooses to decode-forward oe part of the message from each user, while compute-forward the other part. This scheme ca also achieve ew rates ad a better rate regio tha either decode-forward ad compute-forward aloe. II. CHANNEL MODEL A. Discrete memoryless TWRC model The discrete memoryless two-way relay chael DM-TWRC) is deoted by X X X r,py,y,y r x,x,x r ),Y Y Y r ), as i Figure. Here x ad y are the iput ad output sigals of user ; x ad y are the iput ad output sigals of user ; x r ad y r are the iput ad output sigals of the relay. We cosider a full-duplex chael i which all odes ca trasmit ad receive at the same time. A, R, R,P e ) code for a DM-TWRC cosists of two message sets M = [ : R ] ad M = [ : R ],

2 Relay Xr Yr M X X M User User p y, y, yr x, x, xr) ˆM ˆM Y Y Fig.. Two-way relay chael model three ecodig fuctios f,i,f,i,f r,i, i =,..., ad two decodig fuctio g,g. x,i = f,i M,Y,,...,Y,i ), x,i = f,i M,Y,,...,Y,i ), x r,i = f r,i Y r,,...,y r,i ), i =,..., i =,..., i =,..., g : Y M M, g : Y M M. The average error probability is P e = Pr{g M,Y ) M org M,Y ) M }. A rate pair is said to be achievable if there exists a, R, R,P e ) code such that P e as. The closure of the set of all achievable ratesr, ) is the capacity regio of the two-way relay chael. B. Gaussia TWRC model The full additive white Gaussia oise AWGN) two-way relay chael ca be modeled as below. Y = X r +X +Z Y = X r +X +Z Y r = X +X +Z r ) where the oises are idepedet: Z N,N ),Z N,N ),Z r N,N r ). The average iput power costraits for user, user ad the relay are P,P,P r respectively. III. A PARTIAL DECODE-FORWARD SCHEME I this sectio, we provide a achievable rate regio for the TWRC with a partial decode-forward scheme. Each user splits its message ito two parts ad uses superpositio codig to ecode them. The relay oly decodes oe message part of each user ad re-ecode the decoded message pair together ad broadcast. It ca either re-ecode each message pair separately or divides these message pairs ito lists ad oly ecodes the list idex, which is similar to the biig techique i [3]. Both strategies achieve the same rate regio. The users the decode the message from each other by joit typicality decodig of both the curret ad previous blocks. A. Achievable rate for the DM-TWRC Theorem. The followig rate regio is achievable for the two-way relay chael: R mi{iu ;Y r U,X r )+IX ;Y U,X,X r ), IX,X r ;Y X )} mi{iu ;Y r U,X r )+IX ;Y U,X,X r ), IX,X r ;Y X )} R + IU,U ;Y r X r )+IX ;Y U,X,X r ) +IX ;Y U,X,X r ) ) for some joit distributio pu,x )pu,x )px r ). Remark. If U = X,U = X, this regio reduces to the decode-forward lower boud i [3]. Therefore, the partial scheme cotais the scheme i [3] as a special case. Proof: We use a block codig scheme i which each user seds B messages over B blocks of symbols each. ) Codebook geeratio: Fix pu,x )pu,x )px r ). Split each message ito two parts: m = m,m ) with rate R,R ), ad m = m,m ) with rate, ). Geerate R i.i.d. sequecesu m ) i= pu i), where m [ : R ]. For each u m ), geerate R i.i.d. sequeces x m,m ) i= px i u i ), where m [ : R ]. Geerate R i.i.d. sequecesu m ) i= pu i), where m [ : R ]. For each u m ), geerate R i.i.d. sequeces x m,m ) i= px i u i ), where m [ : R ]. Uiformly throw each message pair m,m ) ito Rr bis. Let Km,m ) deote the idex of bi. Geerate Rr i.i.d. sequeces x rk) i= px ri), where K [ : Rr ]. If R r = R +, there is o eed for biig. The codebook is revealed to all parties. ) Ecodig: I each block b [ : B ], user ad user trasmit x m b),m b)) ad x m b),m b)) respectively. I block B, user ad user trasmit x,) ad x,), respectively. At the ed of block b, the relay has a estimate m b), m b)) from the decodig procedure. It trasmits x r K m b), m b))) i block b+. 3) Decodig: We explai the decodig strategy at the ed of block b. Decodig at the relay: Upo receivig y rb), the relay searches for the uique pair m b), m b)) such that u m b)),u m b)),y rb), x r K m b ), m b ))) ) A ǫ. Followig the aalysis i multiple access chael, the error probability will go to zero as if R IU ;Y r U,X r ) IU ;Y r U,X r ) R + IU,U ;Y r X r ). 4) Decodig at each user: By block b, user has decoded m b ). At the ed of block b, it searches for a uique message pair ˆm b ), ˆm b )) such that x r Kˆm b ),m b ))),y b),x b)) A ǫ ad u ˆm b )),x ˆm b ), ˆm b )), y b ),x r Km b ),m b ))),x b )) A ǫ. Followig joit decodig aalysis, the error probability will go to zero as if R IX ;Y U,X,X r ) R +R IX r ;Y X )+IU,X ;Y X,X r ) = IX,X r ;Y X ). 5)

3 { ) ᾱp ) )} αp P +P r R mi C ᾱp + βp +C,C +N r N N { ) ) )} βp βp P +P r mi C ᾱp + βp +C,C +N r N N ) ᾱp ) ) αp +βp βp R + C ᾱp + βp +C +C, where α,β. 3) +N r N N Similarly, user ca decode m b ),m b )) with error probability goes to zero as if IX ;Y U,X,X r ) + IX,X r ;Y X ). 6) By applyig Fourier-Motzki Elimiatio to the iequalities i 4)-6), the achievable rates i terms of R = R + R ad = + are as give i Theorem. B. Rate regio for the Gaussia TWRC Now we apply the proposed partial decode-forward scheme to the AWGN TWRC i ). Usig joitly Gaussia codewords, we ca derive a achievable rate regio as follows. Corollary. The rate regio i 3) is achievable for the AWGN two-way relay chael. Achievability follows from Theorem by settig X = U + V, where U N,αP ) ad V N,ᾱP ) are idepedet, ad by settig X = U + V, where U N,βP ) ad V N, βp ) are idepedet. Corollary. Partial decode-forward achieves strictly better regio regio tha the decode-forward scheme i [3] whe the followig coditio holds: N r > mi{n,n } or CP /N )+CP /N ) > CP +P )/N r ). 7) The larger rate regio of partial decode-forward ca come from time sharig of decode-forward ad direct trasmissio without usig the relay). But for asymmetric chaels, ew rates outside this time-shared regio are also achievable as show i the umerical results sectio. IV. A COMBINED DECODE-FORWARD AND COMPUTE-FORWARD SCHEME FOR THE GAUSSIAN TWRC I this sectio, we propose a combied decode-forward ad compute-forward scheme ad aalyze its rate regios for the Gaussia TWRC. Each user split its message ito two parts. Oe part is ecoded by a radom Gaussia code, while aother part is ecoded by a lattice code. The user trasmits a superpositio codeword of these two parts. The relay decodes the Gaussia codewords of both users ad a fuctio the sum) of the two lattice codewords. It the joitly ecodes all 3 decoded parts ad forwards. Agai the relay ca assig a separate codeword to each set of the 3 decoded parts or it ca ecode oly the list idex as i [3] without affectig the achievable rate. The users apply both joit typicality decodig ad list lattice decodig [9] to decode the message from each other. The combied scheme achieves geuiely ew rate. A example will be give i the umerical result sectio. Theorem. The followig rate regio is achievable for the AWGN two-way relay chael: ) αp R C ᾱp + βp = I +N r ) βp C ᾱp + βp = I +N r ) αp +βp R + C ᾱp + βp = I 3 +N r R < log ᾱp ᾱp + βp + ᾱp ) + = I 4 N r < log βp ᾱp + βp + βp ) + = I 5 N ) r αp +γp r R C = I 6 ᾱp + γp r +N ) βp +γp r C = I 7 βp + γp r +N ) ᾱp ) γpr R C +C = I 8 P +N N ) ) γpr βp C +C = I 9 8) P +N N where α,β,γ ad [x] + max{x,}. By applyig Fourier-Motzki Elimiatio to the above iequalities, the achievable rates i terms of R = R + R ad = + ca be expressed as R mii,i 6 )+mii 4,I 8 ) mii,i 7 )+mii 5,I 9 ) R + I 3 +mii 4,I 8 )+mii 5,I 9 ). 9) Proof: We use block codig scheme i which each user seds B messages over B blocks of symbols. ) Codebook geeratio: Let P = αp +ᾱp ad P = βp + βp. Without loss of geerality, assume ᾱp βp, costruct a chai of ested lattices Λ Λ Λ c Λ c, where σ Λ ) = ᾱp ad σ Λ ) = βp. Λ ad Λ are Rogers-good ad Poltyrev-good, while Λ c ad Λ c are Poltyrev-good [5], []. Split each message ito two parts: m = m,m ) with rate R,R ), ad m = m,m ) with rate, ). Geerate R radom Gaussia codewords u m ) with power costrait αp. Associate each message

4 m [ : R ] with lattice codeword t C = Λ c V. Let vm ) = t m ) + U m )) mod Λ, where U is the uiformly geerated dither sequece kow to all users ad the relay. The codeword for m is a superpositio of the radom Gaussia code ad the lattice code: x m ) = u m )+v m ). Similarly geerate R radom Gaussia codewords u m ) with power costrait βp, ad R lattice codewords t m ). Let x m ) = u m )+vm ). Uiformly throw each pair m,m ) ito Rr bis. Let Km,m )) deotes the bi idex. Form the computed codewords T = t m ) + t m ) Q t m ) + U m ))) mod Λ, where Q t ) is the lattice quatizer mappig t to the earest lattice poit. Uiformly throw T ito Rr bis. Let ST ) deotes the bi idex. Geerate Rr Gaussia codewords u rk) with power costrait γp r ad Rr Gaussia codewords vr S) with power costrait γp r. Let x r = u r K)+v r S). The codebook is revealed to all odes. ) Ecodig: I block b, user seds x m b)) ad user seds x m b)). Assume the relay has decoded m b ),m b )) ad T b ) i block b. It the seds x r b) = u r Km b ),m b )))+vr ST b ))) i block b. 3) Decodig: We explai the decodig strategy at the ed of block b. Decodig at the relay: The relay first decodes m b) ad m b) usig joit typicality decodig. Similar to the aalysis i multiple access chael, P e as if R IU ;Y r U,X r ) IU ;Y r U,X r ) R + IU,U ;Y r X r ). ) The relay the subtracts u m b)) ad u m b)) from its received sigal. Followig argumets similar to those i [4] [8], it ca the decode T b) with vaishig error as log as R log ᾱp ᾱp + βp + ᾱp ) N r log βp ᾱp + βp + βp ). ) N r Decodig at each user: At the ed of block b, user first decodes the uique m b ) such that u r Km b ),m b ))),y b),x b)) A ǫ ad u m b )),u rkm b ),m b ))), This decodig has vaishig error probability if x b ),y b )) A ǫ. R IU r ;Y X )+IU ;Y U r,x ) = IU,U r ;Y X ). ) User the subtracts u m b )) from y b ) ad uses a lattice list decoder [9] to decode a list of possible m b ) of size R CᾱP/N)), deoted as Lm b )). To decode which message i this list was set, it uses the received sigal i block b. That is to say, it decodes the uique m b ) such that y b),x b),u rkm b ),m b ))), x r Km b ),m b )),ST b ))) ) A ǫ ad m b ) Lm b )). This decodig has vaishig error probability if R IX r ;Y X,U r )+CᾱP /N ). 3) Similarly, user ca decode m b ),m b ) with vaishig error as log as IU,U r ;Y X ) IX r ;Y X,U r )+C βp /N ). 4) Fially, by settig X = U +V ; U N,αP ), V N,ᾱP ) X = U +V ; U N,βP ), V N, βp ) X r = U r +V r ; U r N,γP r ), V r N, γp r ) the achievable rate regio i Theorem ca be derived from iequalities )-4). V. NUMERICAL RESULTS I this sectio, we compare the achievable rate regios of the two proposed schemes with pure decode-forward ) [3] ad pure compute-forward [9]. Figures ad 3 show the achievable rate regios of pure [3], of direct trasmissio without usig the relay) ad of the proposed partial for differet chael cofiguratios. Figure shows that partial ca achieve ew rates outside the time sharig regio of pure ad direct trasmissio. For example, by settig α =,β =.5 i 3), partial ca achieve the rate R, ) =.58,.47) which is outside the covex hull of direct trasmissio ad pure. This is otably differet from the oe-way relay Gaussia chael i which partial brigs o improvemet. I Figure, the chael from the users to the relay is stroger tha the chael betwee two users, thus the relay chooses partial to obtai a better rate regio tha. Figure 3 shows performace for aother chael cofiguratio which is symmetric. I this case, the chaels from two users to the relay are sigificatly stroger tha the direct chael, ad the relay will fully decode the messages. Figures 4 ad 5 preset the achievable rate regios for, compute-forward ad the combied scheme. The cut-set outer boud is obtaied by assumig correlated chael iputs ad idepedet chael oise, which is differet from the cut-set boud i [9] for physically degraded chaels. Both Figures 4 ad 5 show that the combied scheme ca achieve a better rate regio tha either ad compute-forward aloe. Figure 4 for a asymmetric chael shows ew rates outside the timeshared regio of ad compute-forward. For example, by settig α =.5,β = i 8), the combied scheme ca achieve the rate R, ) =.678,.859) which is outside the covex hull of pure ad pure compute-forward. I

5 P =, N =, N =3, N r P =5, P =4, P r =, N =, N =4, N r =5.8.6 α=,β=.5 α=.5,β= Without usig relay Partial Cut set boud o degraded) CF Combied CF Cut set..6.8 R R Fig.. Achievable rate regios for ad partial schemes with P = P = P r =,N =,N = 3,N r = P =, N =N =, N r Without usig relay Partial Cut set boud o degraded) R Fig. 3. Achievable rate regios for ad partial schemes with P = P = P r =,N = N =,N r = 6 Figure 5 for a symmetric chael, the combied scheme ca also achieve a boudary poit of R, ) =.69,.) by settig α = 8,β = i 8), istead of time sharig of the two idepedet schemes. These simulatio results show that both proposed schemes achieve strictly ew rates particularly for asymmetric chaels. But because of space limitatio, aalyses of these ew rates are left for future work. VI. CONCLUSION We have proposed two ew codig schemes for the two-way relay chael: a partial decode-forward scheme ad a combied decode-forward ad compute-forward scheme. Aalysis for the Gaussia chael shows that partial decode-forward ca strictly icrease the rate regio of the TWRC over pure decode-forward. This result is opposite to the oe-way Gaussia relay chael. I additio, combiig decode-forward with compute-forward by rate splittig ad superpositio of both Gaussia ad lattice codes ca strictly outperform each separate scheme. These results suggest more comprehesive codig schemes possible for the TWRC. REFERENCES [] C. Shao, Two-way commuicatio chaels, i Proc. 4th Berkeley Symp. Math. Stat. Prob, vol.. IEEE Press, 96, pp Fig. 4. Achievable rate regios for, compute-forward, combied of ad compute-forward, ad partial schemes with P = 5,P = 4,P r =,N =,N = 4,N r = P =, N =N =8, N r CF Combied CF Cut set α=8,β= R Fig. 5. Achievable rate regios for, compute-forward, combied of ad compute-forward, ad partial schemes with P = P = P r =,N = N = 8,N r = 6 [] B. Rakov ad A. Wittebe, Achievable rate regios for the two-way relay chael, i It l Symp. o Ifo. Theory ISIT). IEEE, 6, pp [3] L. Xie, Network codig ad radom biig for multi-user chaels, i th Caadia Workshop o Ifo. Theory CWIT). IEEE, 7, pp [4] B. Nazer ad M. Gastpar, Compute-ad-forward: Haressig iterferece through structured codes, Arxiv preprit arxiv:98.9, 9. [5] U. Erez, S. Litsy, ad R. Zamir, Lattices which are good for almost) everythig, IEEE Tras. o Ifo. Theory, vol. 5, o., pp , 5. [6] K. Narayaa, M. Wilso, ad A. Spritso, Joit physical layer codig ad etwork codig for bi-directioal relayig, i 45th Aual Allerto Coferece, 7. [7] W. Nam, S. Chug, ad Y. Lee, Capacity bouds for two-way relay chaels, i IEEE It l Zurich Semiar o Commuicatios. IEEE, 8, pp [8], Capacity of the gaussia two-way relay chael to withi / bit, IEEE Tras. o Ifo. Theory, vol. 56, o., pp ,. [9] Y. Sog ad N. Devroye, List decodig for ested lattices ad applicatios to relay chaels, i 48th Aual Allerto Coferece. IEEE,, pp [] T. Cover ad A. Gamal, Capacity theorems for the relay chael, IEEE Tras. o Ifo. Theory, vol. 5, o. 5, pp , 979. [] U. Erez ad R. Zamir, Achievig / log + sr) o the awg chael with lattice ecodig ad decodig, IEEE Tras. o Ifo. Theory, vol. 5, o., pp , 4.