Hash-and-Forward Relaying for Two-Way Relay Channel

Transcription

1 Hash-ad-Forward elayig for Two-Way elay Chael Erha Yılmaz ad aymod Kopp EUECOM, Sophia Atipolis, Frace {yilmaz, arxiv: v2 [cs.it] 16 Feb 2011 Abstract This paper cosiders a commuicatio etwork comprised of two odes, which have o mutual direct commuicatio liks, commuicatig two-way with the aid of a commo relay ode (N), also kow as separated two-way relay (TW) chael. We first recall a cut-set outer boud for the set of rates i the cotext of this etwork topology assumig full-duplex trasmissio capabilities. The, we derive a ew achievable rate regio based o hash-ad-forward (HF) relayig where the N does ot attempt to decode but istead hashes its received sigal, ad show that uder certai chael coditios it coicides with Shao s ier-boud for the two-way chael [1]. Moreover, for biary adder TW chael with additive oise at the odes ad the N we provide a detailed capacity achievig codig scheme based o structure codes. I. Itroductio A. elated Work o Two-Way elayig Up to ow, differet trasmissio schemes are proposed for two-way relay (TW) chaels [2] [11]. However, the capacity regio for geeral TW chael remais ope. The simplest trasmissio scheme for TW chael cosists of four phases where the two odes trasmit their messages to the relay ode (N) successively ad the the N decodes ad forwards each mobile s message i the followig two time slots. However, usig ideas from etwork codig (NC) [12], it is show i [3] that the last two trasmissios may be merged ito a sigle trasmissio, resultig i three time slots ad hece a pre-log factor of 2/3 with respect to the sum-rate. The umber of required time slots for the commuicatio betwee the two odes ca be reduced eve further to two time slots by allowig them to simultaeously access the N [7], [8], [13], [14]. I [4], [13], [15] amplify-ad-forward (AF), decodead-forward (DF) ad compress-ad-forward (CF) based relayig strategies cosistig oly of two time slots are proposed for TW chael. A NC scheme that requires three time slots is cosidered i [3] where i the first two time slots the mobiles sed their messages to the N i orthogoal time slots, the N decodes both messages ad the combies them by meas of the bit-wise XO operatio ad retrasmits it to the mobiles. There the mobiles are assumed to use the bit-wise XO operatio o the decoded message ad the ow trasmitted message to obtai the message set from the other mobile. equirig Fig. 1. Two-user two-way relay etwork. three time slots, this bit-wise XO based TW scheme provides a pre-log factor of 2/3 with respect to the sumrate. I [16] a aalog etwork codig (ANC) scheme, where the N amplifies ad forwards mixed received sigals, is proposed ad compared to the traditioal ad digital etwork codig (bit-wise XO at the N) schemes i terms of etwork throughput. I [13] the deoise-adforward (DNF) relayig is proposed for TW chael where the N removes the oise from the combied mobiles messages (o the multiple-access chael) before broadcastig ad compared with AF, DF based TW schemes as well as the traditioal four phase scheme. B. Cotributios I this paper, we cosider a TW chael where two separated odes seek to commuicate, ad ca do so, via a commo N. This kid of sceario ca occur i satellite commuicatios where differet groud statios wat to exchage iformatio or i public safety etworks where differet iterveig etities (e.g. fire-fighters) wat to commuicate with each other to gai iformatio about the curret status at differet parts of the disaster area [17]. Throughout the paper we focus o discrete memoryless TW chael model assumig full-duplex trasmissio capabilities at the commuicatig odes. A cut-set outer boud for the set of rates i the cotext of this etwork topology is recalled ad a achievable rate regio based

2 o DF relayig is provided where a message-level etwork codig eablig codebook size reductio at the N is used. The, a ovel achievable rate regio based o radom biig, i.e. hash-ad-forward (HF) relayig, is provided where the N does ot attempt to fully (or partially) decode but ratherhashes its received sigal so as to reduce the amout of iformatio eeded to sed i the DL. Fially, we study a biary adder chael with additive oise at the odes ad the N ad show that the outer boud is virtually achievable by physical-layer etwork codig, usig group codes ad partial decodig at the N. II. Iformatio-Theoretic System Model Cosider the discrete-memoryless etwork i Fig. 1 comprisig a pair of odes commuicatig two-way via a N. Node a geerates -dimesioal codewords Xab based o a idex W ab { 1,2,,2 ab} where ab is the iformatio rate i bits/dimesio i the directio ab, for a {1,2} ad b {1,2} \ a. Each letter of the trasmitted codewords, X ab,i,i = 1,2,, belogs to a alphabet X ab ad is chose accordig ( to a determiistic ecodig fuctio X ab,i = f ab,i Wab,Ya i 1 which ) icludes the possibility for exploitig the past observatios oftherelaychaeloutput (dowlik). TheNalsousesa determiistic trascoder which geerates a -dimesioal outputsequecex basedotheobservedoisymultiuser (uplik) chael output Y. The trascodig ( fuctio ) at the N is causal ad writte as X,i = f,i Y i 1 X, where X is the alphabet of the N trasmitter. The multiuser chael is memoryless ad successive outputs are idetically distributed accordig to the coditioal probability p(y x 12,x 21 ). The observed sequeces at the odes, Y1 ad Y2, are idepedet coditioed o the N output ad idetically distributed accordig to the coditioal probability p(y 1,y 2 x ) = p(y 1 x )p(y 2 x ). eceiver a decodes its message based o the observed sequece Ya usig a decodig fuctio Ŵba = g(y ) a,w ab ) with probability of error P e,a = Pr (Ŵba W ba. III. Outer Boud Let us first derive a outer-boud o the capacity regio fromfirstpriciples.itisshowi[18]thatbyusigfao s iequality twice, we have the followig outer-boud o achievable rates 12 mi{i(x 12 ;Y X 21 ),I(X ;Y 2 )}. (1) The first term (uplik) i (1) correspods to the outerboud of the two-way chael with a commo output [1] with which o codig strategy is kow to coicide, except for additive chaels ad some special cases [19]. I our system, the achievable rates would be limited to this rate if the DL chaels were capable eough to allow the N to forward a sufficiet characterizatioof Y to the odes. Discussio 1. I the case of a oiseless DL chael our system boils dow to a two-way chael with a commo output [1]. For a weak DL, the iformatio rates will be limited by the secod term i (1). Agai, were the uplik chael oiseless, a outer-boud to the achievable rates would be give by the secod term. IV. Forwardig Strategies at the elay Node At the ode-1 ad ode-2 assumig codig schemes which do ot exploit the DL sigal i the ecodig of the UL data so that the data streams are idepedet, we ow cosider codig schemes at the N which do ot attempt to decode or partially decode the trasmitted data streams, but rather perform biig (hashig) of the received sequece. UL codebooks are geerated radomly at each ode accordig to p(x 12 ) ad p(x 21 ). At the N we will cosider two differet ecodig strategies, amely DF ad HF. Prior to discussig these further, we first look at DL codig for degraded broadcast chaels (BCs) with successive refiemet which will be eeded for the probability of error aalysis for data trasmissio o the DL. A. Degraded BC with successive refiemet We cosider the case where the DL chael is degraded ad a DL chael code where the two users are required to decode the same iformatio but with differet degrees of refiemet (here two). By this we mea that the user with the weaker chael oly receives the heavily-protected data stream, while the user with the stroger chael receives both. This is the geeral BC where there is commo iformatio for both users ad extra iformatio for the stroger user. We make use of stadard multi-level codig [20], ad let,1 ad,2 deote the iformatio rates of the N to odes 1 ad 2 respectively, ad assume further that,1,2. The radom codebook, {X (i,j),i = 1,2,,2 (,1,2), j = 1,2,,2,2 }, isgeerated accordig to the distributios p(u) ad p(x u) i the stadard-way so that the achievable DL rates are give by the covex hull (with respect to the parameter α ad p(x,u)) of (,1,,2 ) = { (,1,,2 ) : where arg max p(x,u) 0 α 1 {α(,1,2 )+(1 α),2 } },,2 = I(U;Y 2 ),,1,2 = I(X ;Y 1 U). B. Decode-ad-Forward elayig (2a) (2b) I this sectio, we provide a brief view of the achievable rates for the digital sigle-relay etwork with two commuicatig odes, wherei a geeralized form of etworkcodig for discrete-memoryless chaels is used. The DL chaelis a classicalbc except forthe fact that the decoders have side iformatio to exploit, amely the

3 trasmitted codeword idices that they themselves set durig the UL portio. For simplicity, we assume that the two chael outputs are coditioally idepedet so that they ca be separated ito two trasitio probabilities p(y i x ),i = 1,2. Each ode decodes the received sequece y 1 (y 2) usig the side iformatio from its ow trasmissio to yield the estimates Ŵ 21 (Ŵ 12 ). I the followig theorem, we provide a achievable rate regio for the above chael model. Theorem 2. A achievable rate regio for the two-user sigle relay etwork usig DF relayig strategy is give by the closure of the followig set of iequalities 12 mi{i(x 12 ;Y X 21 ),I(X ;Y 2 )}, (3a) 21 mi{i(x 21 ;Y X 12 ),I(X ;Y 1 )}, (3b) I(X 12,X 21 ;Y ). (3c) Proof: The proof which is briefly explaied i the followig ca be foud i [18]. The N employs a multi-user receiver to decode the trasmitted codeword idices yieldig the estimates (Ŵ12,Ŵ21) at its output. without loss of geerality, we assume Based o these idices, i the dowlik phase, it the ecodes the two idices usig a twodimesioal codebook X = {X (W(1),W(2) ),W(1) = 1,...,2 (12 21),W (2) = 1,...,2 21 }. Here we ote that the cardialityofthe N s codebookisat most To sed idices W (1) ad W (2), the N chooses X (W(1),W(2) ) where W(1) Ŵ12 = 2 21, W (2) = Ŵ 12 Ŵ21, where x y deotes (x+y) mod At the receiver 1 (the weak receiver), who has the side iformatio W 12, the uique Ŵ 21 is chose such that {X ( W ,W 12 Ŵ 21 ),Y 1 } A ǫ,1 (4) where A ǫ,k deotes the set of joitly-typical sequeces {x (w (1),w(2) ),y k}, k = 1,2. Due to the side iformatio W 12, W (1) is kow to receiver 1; hece the cardiality of the search space is limited to If oe or more tha oe exist a error is declared. The decoded idex is the W 12 Ŵ 21. At the receiver 2 (the stroger receiver), who has the side iformatio W 21, the uique Ŵ 12 is chose such that {X( Ŵ ,Ŵ 12 W 21 ),Y2 } A ǫ,2. (5) Note that the cardiality of the search space is If oe or more tha oe exist a error is declared. Due to thesideiformatiow 21,thedecodedidexatthereceiver 2 is give by Ŵ Ŵ 12 W 21. C. Hash-ad-Forward elayig If the N does ot attempt to decode the received sequece Y, it may still attempt to reduce the amout of iformatio eeded to covey it to both users, ad moreover with a higher degree of refiemet for the user with the stroger DL chael. We will proceed similarly to distributed source codig as origially described by Slepia ad Wolf i [21]. This was recetly referred to as hashig at the N, resultig i a hash-ad-forward (HF) trasmissio scheme [22]. Theorem 3. A achievable rate regio for the discrete memoryless TW etwork with HF relayig strategy is give by the closure of the followig set of iequalities 12 mi { I(X 12 ;Y X 21 ), 21 mi { I(X 21 ;Y X 12 ), [I(U;Y 2 ) H(Y X 1,X 2 )] +} (6a) [I(X ;Y 1 U)+I(U;Y 2 ) H(Y X 1,X 2 )] +} (6b) for UL joit probability p(x 12 )p(x 21 )p(y x 12,x 21 ) ad the degraded BC with successive refiemet defied i Sectio- IV-A. Proof: The set of ǫ typical sequeces Y is partitioed ito 2,1 bis B ij,i = 1,...,2 (,1,2), j = 1,...,2,2, such that i,j B ij = A ǫ ad B ij Bi j =,for i i or j j. Let B(y ) = (i,j) be a radom hashfuctio which assigs a received sequece to a particular bi. Ecodig at the N is doe i two steps. First, if y A ǫ, we hash y ad let W,1 = i ad W,2 = j. If y / A ǫ, we set (W,1,W,2 ) = (e,e) to idicate a error coditio at the N ecoder. The, we geerate the trasmitted sequece X accordig to the multilevelcodig strategy for the (degraded) BC. Decoder 1 creates a list, L 1 (y ) of cadidate y based o the decoded (î,ĵ). The umber of cadidates i the list, N 1 (y ) is bouded by 2 (H(Y),1 ǫ) N 1 (y ) 2(H(Y),1+ǫ). (7) Similarly, decoder 2 has a list based solely o ĵ, L 2 (y ), for which the umber of elemets is bouded by 2 (H(Y),2 ǫ) N 2 (y ) 2(H(Y),2+ǫ). (8) Kowig x 12, decoder 1 tries to fid a x 21 such that (x 12,x 21,y ) A ǫ for at least oe y L 1 (y ). If more tha oe x 21 or oe are joitly ǫ-typical, the a error is declared. Decoder 2 proceeds similarly ad tries to fid a x 12 kowig x 21 such that (x 12,x 21,y ) A ǫ for at least oe y L 2 (y ). Assumig (W 12,W 21 ) = (1,1), the probability of decodig error (coditioed o receivig (i, j) without error ad i e,j e) for decoder 1 is give by P e (1) Pr{(x 12 (1),x 21 (1),y ) / A ǫ } [Pr{(x 12(1),x 21(i 1),y) A ǫ} +N 1 (y)pr{(x 12(1),x 21(i 1),y ) A ǫ } ] +Pr{(Ŵ,1,Ŵ,2) (i,j)}+pr{w,1 = e}. (9)

4 The first elemet i the sum ca be made arbitrarily small by icreasig. The probability i the secod term is the probability over the radom esemble of codebooks that a x 21(i),i 1 is joitly ǫ typical with the true N output (which is always i L 1 (y ) if (i,j) are received without error at the N) ad is give by Pr{(x 12(1),x 21(i),y) A ǫ} = p(x 12(1),y))p(x 21(i)) (x 12 (1),x 21 (i),y ) A ǫ 2 (H(X12,X21,Y) H(X12,Y) H(X21) 3ǫ) 2 (I(X21;Y X12)+3ǫ). (10) The probability i the third term reflects the evet that aother y L 1 (y ) sequece from the radom list of cadidatesisjoitlyǫ typicalwithx 12 (1)adx 21 (i).note that this sequece is idepedet of both code sequeces ad thus Pr { (x 12 (1),x 21 (i),y } ) A ǫ = p(x 12 (1))p(y ))p(x 21 (i)) (x 12 (1),x 21 (i),y ) A ǫ 2 (H(X12,X21,Y) H(X12) H(Y) H(X21) 4ǫ) 2 (I(X21,X12;Y) 4ǫ). (11) Combiig the two probabilities ad the size of the list i (9) yields P (1) e Pr{(î,ĵ) (i,j)}+2ǫ+2 (21 I(X21;Y X12) 3ǫ) +2 (21,1+H(Y X12,X21) 5ǫ). (12) Proceedig i a idetical fashio for decoder 2 yields P (2) e Pr{ĵ j}+2ǫ+2 (12 I(X12;Y X21) 3ǫ) +2 (12,2+H(Y X12,X21) 5ǫ). (13) We ow tur to the remaiig error evet i the overall error probability, amely the evet that the bi idices (i,j) for decoder 1 ad j for decoder 2 are icorrectly decoded. For the degraded BC with successive refiemet (see Sectio-IV-A) we have the two error probabilities vaish if (2) is satisfied. Discussio 4. Cosider the special case where p(y 1 x ) = p(y 2 x ), ad thus,1 =,2 = I(X ;Y 1 ) = I(X ;Y 2 ). Assume further the DL chaels are very strog i the sese that I(X ;Y 1 ) max{h(y X 12 ),H(Y X 21 )}. Uder these coditios, the overall rate regio coicides with Shao s ier-boud for the two-way chael [1] with a commo output (which is ot tight due to the statistical idepedece of the iput sequeces) ad is achievable by biig at the N i the sese of Slepia-Wolf [21] with the oly differece beig that the two destiatios ow recover Y without error usig their ow DL side iformatio from the UL sequeces. Fig. 2. Biary Adder Chael: Codebook Time-Sharig. D. The capacity regio of Biary Adder Chael We ow restrict our treatmet to a specific additive chael models, amely the biary adder chael. For this chael model, we will provide capacity achievig codig strategy based o liear group codes. The iput/output relatioships for the biary adder chael are give by y, = x 12, x 21, z,, y i, = x, z i,, i = 1,2 (14) where all variables are biary ad x y deotes the modulo-2 sum of x ad y. The probability of takig o the value 1 for the oise terms are deoted ǫ,ǫ 1,ǫ 2. Here the outer boud from Sectio-III is 12 mi{i(x 12 ;Y X 21 ),I(X ;Y 2 )} = mi { H(Y X 21 ) H(Y X 12,X 21 ), H(Y 2 ) H(Y 2 X ) } mi{1 H(ǫ ),1 H(ǫ 2 )} (15) where the last iequality becomes equality whe the iput distributios are uiform, i.e., p(x 12 = 1) = p(x ba = 1) = 1/2. To show the achievability of (15) we assume that uplik ad DL ecodig occur i subsequet periods of output symbols, where is the legth of code sequeces. That is to say that after decodig a message from a group of symbols, the N trasmits the DL message while it receives the ext UL message. Suppose without loss of geerality that CosidertworadomcodebooksC 12 withrate ad C c with rate ba. C c is the commo codebook. We ow time-share betwee both codebooks as show i Fig. 2 with 0 α 1. Durig the first time-slot of duratio (1 α) dimesios user 1 trasmits aloe to the N usig C 12. Call the iformatio sequece X (1) 12. Usig stadard radom codig argumets, arbitrarily small error

5 probability for detectio of X (1) 12 is achievable if < (1 α)i(x 12 ;Y ) (1 α)(1 H(ǫ )). (16) Durig the secod time-slot of duratio α dimesios, both users trasmit their iformatio sequeces X (2) 12 ad X 21 whicharecodewordsbelogigtothesame liear code over GF(2), C c. As a result the N receives the modulo- 2 sum of the two codewords which is itself a codeword i C c. Liear codes achieve the capacity of the BSC (see [23]) ad thus a arbitrarily small average error probability for the detectio of X (2) 12 X 21 is possible if 21 α(1 H(ǫ )). (17) The N ecodes with a two-dimesioal codebook of cardiality 2 12, idexed by colum ad row pair (i,j),i = 1,,2 (12 21),j = 1,,2 21. Colum i is used to ecode X (1) 12 ad row j to ecode X(2) 12 X 21. At receiver 1 (weak receiver), X 12 is kow ad so the colum i of the trasmitted codeword is kow. Arbitrarily small error probability is achievable for detectio of j or X (2) 12 X 21 (ad cosequetly X 21 ) if 21 < I(X ;Y 1 ) 1 H(ǫ 1 ). (18) At receiver 2 (strog receiver) X 21 is kow. Arbitrarily small error probability for detectio of (i,j) (or X 12 ) is achievable if 12 < I(X ;Y 2 ) 1 H(ǫ 2 ). (19) Cosider first the strog relay case where ǫ max(ǫ 1,ǫ 2 ). Here we let α = 1 so that both users ca achieve 1 H(ǫ ). For the medium relay case where ǫ 1 ǫ ǫ 2 we choose α = 1 H(ǫ1) 1 H(ǫ ), so that 12 = 1 H(ǫ ) ad 21 = 1 H(ǫ 1 ) are achievable. Fially i the weak relay case whe ǫ 1 ǫ 2 ǫ we choose α = 1 H(ǫ1) 1 H(ǫ 2), resultig i 12 = 1 H(ǫ 2 ) ad 21 = 1 H(ǫ 1 ). V. Coclusios I this paper, we studied differet codig strategies for the discrete memoryless TW chaels. Specifically, a ovel relayig strategy based o radom biig of the receives sigals at the N, e.g. hash-ad-forward relayig, was preseted which coicides with Shao s ier-boud for the two-way chael uder certai chael coditios. For biary adder two-way relay chael, it is show that the outer boud is achievable by physical-layer etwork codig, utilizig group codes ad partial decodig at the N. Our curret work examies the use of feedback i the codig strategy at the termials, a issue that was eglected here. Ackowledgmet This work was partially supported by the Europea Commissio s 7th framework programme uder grat agreemet FP also referred to as CONECT. efereces [1] C. E. Shao, Two-way commuicatio chaels, i Proc. 4th Berkeley Symp. Math. Stat. Prob.,, Uiversity of Califoria Press, Berkeley, 1961, pp [2] B. akov ad A. Wittebe, Spectral efficiet sigalig for half-duplex relay chaels, i Proc. Cof. o Sigals, Systems, ad Computers, Pacific Grove, CA, Nov [3] Y. Wu, P. A. Chou, ad S. Kug, Iformatio exchage i wireless etworks with etwork codig ad physical-layer broadcast, i Proc. IEEE Coferece of Iformatio Scieces ad Systems, March [4]. Kopp, Two-way radio etwork with a star topology, i It. 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