An Achievable Rate Region for the Two-Way Multiple Relay Channel Jonathan Ponniah Liang-Liang Xie Department of Electrical and Computer Engineering Department of Electrical and Computer Engineering Texas A&M University University of Waterloo Abstract—An achievable rate-region forthetwo-way multiple- all the other sources (the TWMRC is a special case of this relaychannelisproposedusingdecode-forwardblockMarkovian channel).Inadecode-forwardschemethemessagesgenerated coding. We identify a fundamental tension between the informa- by each source are successively decoded and forwarded by tion flow in both directions that leads to an intractable number each relay in the channel before arriving at any destination. of decode-forward schemes and achievable rate regions, none of whichareuniversallybetterthantheothers.Weintroduceanew The order in which the nodes forward the message from a 6 concept in decode-forward coding called ranking, and discover particularsourcedefinesapath.LetS denotethesetofsource 1 0 thateachoftheserateregionsaredifferentrealizationsofasingle nodes. For each s∈S, let the vector p¯s denote the fixed path expression that depends on the rank assignment. This discovery 2 makes it possible to characterize the complete achievable rate assigned to source s and let p¯ := {p¯s : s ∈ S}. The first element of p¯ is source node s, and each subsequent element l regionthatincludesalloftheinterestingdecode-forwardschemes s u and corresponding rate regions. is the next node on the path. Let ps,k denote the kth element J of p¯ . s 4 I. INTRODUCTION 2 Example 1. Set M := 5 and S := {1,5}. Suppose the mes- The two-way multiple relay channel (TWMRC) is implicit sages w and w follow the paths 1→3→2→4→5 and inalmostallformsofmoderncommunication.Atthemostfun- 1 5 ] 5 → 2 → 3 → 4 → 1 respectively. Then p¯ = (1,3,2,4,5), T damentallevel,itmodelsthesimultaneous,interactive,bidirec- 1 p¯ =(5,2,3,4,1), p =3, and p¯:={p¯ ,p¯ }. I tionalexchangeofinformationbetweentwosource/destination 5 1,2 1 5 . s pairs with the assistance of intermediate relay nodes; the For any fixed p¯, let P (s) denote the set of nodes that c i [ type of exchange that occurs over the internet for instance, precede node i on the path p¯s. Suppose node i is the kth where every single received data packet is acknowledged.We node on the path p¯s. That is, i = ps,k. Then Pi(s) := {ps,j : 2 consider the most generalform of the TWMRC which allows 1 ≤ j ≤ k − 1}. For any non-empty subset S ⊆ S, let v 9 thetransmissionsofeverynodetoinfluencethereceivedsignal Pi(S)= [ Pi(s) and P˜i(S):={k ∈/ Pi(S)}. 2 of each individual node. s∈S 2 This paper extends a history of previous work in network Example 2. Set M := 5 and S := {1,5}. Set p¯ := 6 information theory beginning with the two-way channel [1], 1 0 followed by the one-way relay channel [2], the one-way (1,3,2,4,5) and p¯5 := (5,2,3,4,1). Then P2({1}) = {1,3}, 1. multiple-relay channel [3], the one-way multiple-access relay P2({5}) = {5}, and P2({1,5}) = {1,3,5}. Similarly, P˜ ({1})={2,4,5},P˜ ({5})={1,2,3,4},andP˜ ({1,5})= 0 channel [4], the two-way, one-relay channel [5], and the two- 2 2 2 6 {2,4}. way, two-relay channel [6]. 1 The channel model of interest is the discrete memoryless Set |S| := N for some N ≤ M, and S := {s ,...,s }, : 1 N v channel consisting of M nodes labeled from 1,...,M. The wheresj ∈{1,...,M} foreachj =1,...,N. Foranyp¯and i X input-output dynamics are expressed as follows: non-empty S ⊆S define the constraint: ar (X1×...×XM,p(y1,...,yM|x1,...,xM),Y1×...×YM). RS ”,“<”, “=”, “≥”,and “≤”. 1 These relations retain their usual meaning in the sense that However, the reverse situation occurs when node 3 tries to M >M −1>···>2>1 and 1<2<···rank(p ) B ({1}) = {2,3}, B ({5}) = {5}, B ({1,5}) = {2,3,5}, s,i s,i+1 4 4 4 for all i < i and rank(p ) > rank(p ) for all i > i . B˜ ({1}) = {4,5}, B˜ ({5}) = {2,3,4}, and B˜ ({1,5}) = s s,i s,i−1 s 4 4 4 The notation v¯ = (p¯,r¯) will be used to denote a valid path- {4}. rank-assignment pair (p¯,r¯). For any non-empty S ⊆S and v¯∈V define the constraint: Example 4. The following are examples of rank as- R < I(X ;Y |X )+I(X ;Y |X ). (5) signments r¯ that correspond to a valid (p¯,r¯) when S X j i L(j) Bi(S) i B˜i(S) M = 8: (1,2,3,4,5,6,7,8), (8,7,6,5,4,3,2,1), (8,6,4,2,1,3,5,7), j∈Ai(S) (8,7,6,4,3,1,2,5),(7,6,4,3,1,2,5,8). Let R (v¯) be the set of rate pairs (R ,R ) that satisfy i 1 M (5) for all non-empty S ⊆ S if i ∈/ S and satisfy (1) for all LetV denotethesetofallvalidpath-rank-assignmentpairs. non-emptyS ⊆S\iifi∈S. We havethefollowingtheorem. For a fixed v¯ ∈ V and each s ∈ S, let u(i,s) denote the “one-hop” predecessor upstream of i on the path p¯s. More Theorem 1. Foranyproductdistribution p(x1)···p(xM) the precisely,supposeps,k =i forsomek. Thenu(i,s)=ps,k−1. following set of rate pairs is achievable: Let U(i) = {u(i,s) : s ∈ S} denote the set of all one-hop M predecessor nodes upstream of node i. Furthermore, at each R (v¯) relay i∈/ S define the reference node with respect to i as the [ \ i v¯∈V i=1 highest ranked one-hop predecessor upstream to node i over all paths. That is, ref(i):=arg max rank(j). IV. PROOF OF THEOREM1 j∈U(i) The proof is based on the block-markov, decode-forward Example 5. Set M := 5 and paths p¯1 := (1,2,3,4,5) framework in which the transmissions are divided into B and p¯5 := (5,4,3,2,1). Define the rank assignment r¯ := blocksof T channeluses. Let R1 and RM denote the rates at (5,3,2,1,4). Then u(2,1) = 1, u(2,5) = 3 and U(2) = which nodes 1 and M transmit information to each other. {1,3}. It follows that ref(2) = 1. Similarly, u(3,1) = 2, u(3,5)=4, and U(3)={2,4}. It follows that ref(3)=2. A. Codebook Generation The valid path-rank-assignmentscapture all of the ways in For node 1, independentlygenerate 2TR1 i.i.d T-sequences which the tension between two opposing information flows is x¯ := (x ,...,x ) in XT according to p(x ). Index 1 1,1 1,T 1 1 resolved in a decode-forward scheme when p¯ := {p¯1,p¯M}. them as x¯1(w1), w1 ∈ {1,2,...,2TR1}. For node i = The definition of a valid path-rank-assignment thus far has 2,...,M − 1, independently generate 2T(R1+RM) i.i.d T- been limited to the paths p¯1 := (1,2,3,...,M − 1,M) sequences x¯i := (xi,1,...,xi,T) in XiT according to p(xi). and p¯M := (M,M −1,...,3,2,1). The symmetry of these Index them as x¯i(wi), wi ∈ {1,2,...,2T(R1+RM)}. For paths simplifies the characterization of the valid path-rank- node M, independently generate 2TRM i.i.d T-sequences assignments. In future work, we will define the valid path- x¯ := (x ,...,x ) in XT according to p(x ). Index M M,1 M,T M M rank-assignments over arbitrary paths. them as x¯ (w ), w ∈{1,2,...,2TRM}. M M M III. MAIN RESULT B. Encoding Consider the TWMRC with M nodes and S := {1,M}. In each block b ∈ 1,...,B, nodes 1 and M generate For a fixed v¯ ∈ V and any non-empty S ⊆ S define the message indices w (b) ∈ {1 ...,2TR1} and w (b) ∈ 1 M higher orthant set (the orthant set “above”) Ai(S), as the set {1 ...,2TRM} and transmit the T-sequences x¯1(w1(b)) and ofnodesinPi(S)ofrankhigherthanthereferencenoderef(i). x¯M(wM(b)) respectively. Simultaneously, each relay node i Similarly,definethelowerorthantset(theorthantset“below”) chooses a message index w (b) ∈ {1,...,2T(R1+RM)} and i Bi(S), as the set of nodes in Pi(S) of rank lower than or transmits the T-sequence x¯i(wi(b)). The index wi(b) corre- equal to the reference node. Finally, define the lower orthant spondsto auniquemessagepair(w (b−d ),w (b−d )) 1 i,1 M i,M set B˜i(S) as the set of nodes in P˜i(S) of rank lower than or where di,1 and di,M, referred to as the encoding delays, are equaltothereferencenode.Theorthantsetscanbeexpressed strictly positive integers. The encoding scheme specifies the as follows: encoding delays at the relay nodes. At the end of each block b, every relay node i decodes the message pair (w (b − A (S):={j ∈P (S)|rank(j)>rank(ref(i))}, 1 i i d˜ ),w (b−d˜ ))wherethedecodingdelays,d˜ andd˜ i,1 M i,M i,1 i,M B (S):={j ∈P (S)|rank(j)≤rank(ref(i))}, i i arestrictlypositiveintegers.Thedecodingschemespecifiesthe B˜i(S):={j ∈P˜i(S)|rank(j)≤rank(ref(i))}. decoding delays at the relay nodes. The relay cannot encode anymessagepairthatithasnotalreadydecoded,sod >d˜ . Let L(i) denote the set of nodes of strictly lower rank than i. i,s i,s Apart from this causality constraint, the messages decoded in That is, L(i):={j |rank(j)d foreachrelay 2 2 i,M i+1,M node i. This channel will be called a biased multiple-access relaychannel(BMARC)sincetherelaysonlyhelponesource andnottheother.TheencodingschemeisillustratedinFigure w4(b−d3,4) w1(b) w4(b−d3,4) 1 for a BMARC of size M =4. Let R2(p¯) be the set of rate 3 1 3 pairs (R1,RM) that satisfy (1) for all non-empty S ⊆ S for the BMARC with path p¯:={p¯ ,p¯ }. 1 M Lemma1. Givenanyproductdistributionp(x )···p(x )the 1 M w1(b) w4(b) w4(b) rate pairs in R2(p¯) are achievable for the BMARC. 1 4 4 Proof: First we show that node 2 can decode (w (b + 1 (i) (ii) d ),w (b)) in block b+d if (R ,R ) satisfies: i,M M 3,M 1 M R f1,1 = 4, d3,1 = f1,1 + f2,1 = 5, d3,4 = f4,4 = 2, and I(X3,...XM−1;Y2|X2)+I(XM;Y2|X1...XM−1). Butthis d2,4 =f4,4+f3,4 =3. together with (10) implies that R1 I(X ...X ;Y |X ) + i 2 M 3 i−1 2 2 First,weproveapreliminarylemma.Defineamultiple-access I(X ...X ;Y |X ...X ). But this together with (10) i M 2 1 i−1 relay channel consisting of M nodes, where node 1 and node implies that R < I(X ;Y |X ...X ) which satisfies (8) 1 1 2 2 i−1 M aresources,node2isadestination,andnodes3...,M−1 for i−1. The argumentterminatesat i=3 since (9) and (10) are relay nodes. Assume block-Markovian encoding; in each are boundaries of R (p¯) at i=3. Thus the lemma is proved. 2 block b of T channel uses, node i ∈ {1,M} generates the messagew (b)∈{1,...,2TRi}.Furthermore,supposeagenie Figure 1 illustrates the proof of Lemma 1 for M = 4. i reveals the message w (b−d ) to relay node i just before Finally, we show for any v¯ ∈ V, the region M R (v¯) is M i,M Ti=1 i blockb,whered isafixedconstantforeachi.Hence,relay achievable. The proof is by induction. Consider, the two-way i,M node i encodes w (b−d ) in block b. The path vector p¯ one-relay channel where node 1 and 3 are the sources and M i,M is defined as p¯ :=(1,2) and p¯ :=(M,M −1,...,4,3,2). node 2 is the relay. The paths are defined as p¯ := (1,2,3) 1 M 1 and p¯ :=(3,2,1). In [5] the region 3 R (p¯) is shown to 1 remain unchanged; they do not include X . These 3 Ti=1 i M beachievable.Itisstraightforwardtocheckthattherearefour contributions are expressed by I(X ;Y |X ) + X j i L(j) validrankingsforthischannel,(3,2,1),(1,2,3),(3,1,2),and j∈Ai(S)\M (2,1,3), and that the region in Theorem 1 is the same as the I(XBi(S);Yi|XB˜i(S))forallnon-emptyS ⊆S.Theright-side region in [5]. extension forces relay node i to decode w (b) before w (b). 1 M The first step of the induction is to perform a left or right As a result, the contribution of node M as seen by node i is extension of the two-way one-relay channel to create a two- I(X ;Y |X ...X ) or equivalently, I(X ;Y |X ). M i 1 M−1 M i L(M) way two-relay channel. Without loss of generality, choose a Hence relay node i ∈ {2,...,M −1} can recover any rate right-side extension where node 3 becomes a relay and node pair in the region defined by R < I(X ;Y |X )+ S X j i L(j) 4 is added as a source and given rank 4. The new paths are j∈Ai(S) p¯1 := (1,2,3,4) and p¯4 := (4,3,2,1). By the design of (6) I(XBi(S);Yi|XB˜i(S)) for all non-empty S ⊆S. To finish the and (7), node 3 waits to receive the message from node 1 inductive step, observe that node 1 and node M each see a before encoding the message simultaneously transmitted by one-way multiple-relay channel which is a simple BMARC. node 4. Since node 3 knows all of the source 1 messages to Thereforeit followsfrom Lemma 1 that node i∈{1,M}can itsright,andallofthesource4messagestoitsleft,thechannel decode at any rate in R (p¯) as defined by (1). i it sees is the BMARC of Lemma 1 (or a reflection of it). It The definition of a valid path-rank-assignment guarantees follows from Lemma 1 that node 3 can recover all the rate that we can always start with the three nodes of lowest rank pairs in R (p¯), which is equivalent to the region defined by andreachthegeneraltwo-wayM-relaychannelbyasequence 3 R < I(X ;Y |X ) for all non-empty S ⊆ S. Note of left and right extensions with nodes of successively higher S B3(S) 3 B˜3(S) that A (S)={} for all non-empty S ⊆S. rank. Thus Theorem 1 is proved. 3 Since node 2 does not know the transmissions from node V. CONCLUDING REMARKS 4 a priori, the previous mutual informations that describe the contributions of nodes 1 and 3 remain unchanged; they We showed that the rate regions of all interesting decode- do not include X . These contributions are expressed by forward schemes are different realizations of a single expres- 4 I(X ;Y |X ) for all non-empty S ⊆ S. The right- sionthatdependsonarankassignment.Thisdiscoverymakes sideBe2x(tSe)nsio2n fBo˜2rc(Se)s node 2 to decode w (b) before w (b). it possible to characterizethe completeachievablerate region. 1 4 As a result, the contribution of node 4 as seen by node It remains to be seen whether some version of Theorem 1 is also true for multi-source, multi-relay, all-way channels. 2 is I(X ;Y |X X X ) or equivalently, I(X ;Y |X ). 4 2 1 2 3 4 2 L(4) Hence relay node i ∈ {2,3} can recover any rate pair VI. ACKNOWLEDGMENTS in the region defined by R < I(X ;Y |X ) + S X j i L(j) The authors would like to thank Xiugang Wu for helpful j∈Ai(S) discussions on random binning and network coding. This I(X ;Y |X ) for all non-empty S ⊆S. To finish the Bi(S) i B˜i(S) material is based upon work partially supported by NSF first inductive step, observe that node 1 and node 4 each see ContractCNS-1302182,AFOSR ContractFA9550-13-1-0008, a one-waymultiple-relaychannel,whichisa simpleBMARC. and NSF Science & Technology Center Grant CCF-0939370. Therefore it follows from Lemma 1 that node i ∈ {1,4} can decode at any rate in Ri(p¯) as defined by (1). Thus the first REFERENCES inductive step is proved. [1] C. E. Shannon, “Two-way communication channels,” in In Proc. 4th NowassumebyinductionthattheTheoremistrueforM−1. Berkeley Symp.Math,Statist. Probab,1961,pp.611–644. WewillshowitmustbetrueforM.Withoutlossofgenerality, [2] T. Cover and A. El Gamal, “Capacity theorems for the relay channel,” considera right-sideextensionthatchangesnodeM−1 from IEEETrans.Inf.Theory,vol.25,no.5,pp.572–584,Sep1979. [3] L.-L.XieandP.Kumar,“Anachievable rateforthemultiple-level relay a source into a relay node and adds a source node M where channel,” IEEETrans.Inf. Theory, vol.51,no.4,pp.1348–1358, April rank(M) = M. The new paths are p¯ := (1,2,3,...,M) 2005. 1 and p¯ :=(M,M−1,M−2,...,2,1). By the design of (6) [4] L. Sankar, G. Kramer, and N. B. Mandayam, “Offset encoding for M multiple-access relaychannels,”IEEETrans.Inf.Theory,vol.53,no.10, and (7), node M −1 waits to receive the message from node pp.3814–3821, Oct2007. 1 before encoding the message simultaneously transmitted by [5] L.-L.Xie,“Networkcodingandrandombinningformulti-userchannels,” nodeM.SincenodeM−1knowsallofthesource1messages in10th Canadian WorkshoponInformation Theory, June2007, pp. 85– 88. to its right, and all of the source M messages to its left, the [6] J. Ponniah and L.-L. Xie, “An achievable rate region for the two-way channel it sees is the BMARC of Lemma 1 (or a reflection two-relay channel,” inISIT,July2008,pp.489–493. of it). It follows from Lemma 1 that node M can recover all [7] S. Lim, Y.-H. Kim, A. El Gamal, and S.-Y. Chung, “Noisy network coding,” IEEE Trans. Inf. Theory, vol. 57, no. 5, pp. 3132–3152, May the rate pairs in R (p¯), which is equivalent to the region M−1 2011. defined by R

An Achievable Rate Region for the Two-Way Multiple-Relay Channel

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