1 Capacty bounds on mult-par two-way communcaton wth a base-staton aded by a relay ang Joon Km, Besma mda, and Natasha Devroye Abstract The mult-par b-drectonal relay network under consderaton conssts of one base-staton, multple (say m) termnal nodes and one relay, all of whch are half-duplex, n whch, contrary to pror work, each node has a drect lnk wth every other node. Each of the m termnal nodes exchanges messages wth the base-staton n a b-drectonal fashon, leadng to 2m total messages to be communcated wth the (possble) help of the relay. Our contrbutons are: 1) the ntroducton of three new temporal protocols whch fully explot the two-way nature of the data, over-heard sde-nformaton through network codng, random bnnng, and compress-and-forward termnal node cooperaton, 2) dervatons of achevable rate regons and 3) cut-set based outer bounds for the mult-par network, and 4) a numercal evaluaton of the derved regons n Gaussan nose whch llustrate the performance of the proposed protocols. Index Terms b-drectonal relayng, decode and forward, mult-par, bnnng I. INTRODUCTION The smplest b-drectonal relay network conssts of a par of termnal nodes that wsh to exchange messages through the use of a sngle relay. Whle the capacty of ths channel s stll unknown n general, t has been of great recent nterest (see references n [1] and [2]) due to ts relevance n future wreless networks. The sngle relay, sngle par bdrectonal relay channel has been extended n a number of ways: 1) the consderaton of a sngle b-drectonal lnk usng multple relays [3] [7], and 2) the consderaton of multple b-drectonal lnks sharng a sngle, common relay [8] [11]. The relay network consdered n ths paper falls nto the second category and conssts of a base staton (node ) whch wshes to communcate smultaneously n a b-drectonal fashon wth multple termnal nodes (node 1,, node m) wth the help of one relay node (node r). Due to lmtatons of current technology, all nodes are assumed to be half-duplex and thus cannot transmt and receve smultaneously. Ths network topology s motvated by recent pushes to extend the coverage, relablty and/or data rates of wreless networks. For example, n a cellular scenaro, a relay staton s able to enhance the connectvty between a base staton and termnals at ts cell boundary. The relays may be connected to the base staton usng a wreless lnk rather than a wred one, resultng n savngs to the operators backhaul costs. Another motvatng ang Joon Km was wth the chool of Engneerng and Appled cences, Harvard Unversty, Cambrdge, MA 2138. Emal: sangkm@post.harvard.edu. Besma mda s wth the Department of Electrcal and Computer Engneerng, Purdue Unversty Calumet, Hammond, IN 46323. E-mal: Besma.mda@calumet.purdue.edu. Natasha Devroye s wth the Department of Electrcal and Computer Engneerng, Unversty of Illnos at Chcago, Chcago, IL 667. Emal: devroye@ece.uc.edu. example s satellte communcaton: satelltes can be used to relay sgnals from one ground staton to multple vehcular termnals on or close to the earth s surface. In ths work, we determne bounds on the capacty regons - whch may serve as gudes and benchmarks n the desgn of - such mult-par twoway communcaton networks aded by a sngle relay node. A. Related work In [8] a network n whch K half-duplex source/destnaton pars wsh to exchange messages n a b-drectonal fashon through a sngle mult-antenna relay s nvestgated from a dversty-multplexng gan perspectve. The authors of [9] consder a smlar channel model and propose the use of a CDMA strategy to support multple level Qo to dfferent users. In [1] multple b-drectonal pars communcate over a shared relay n the absence of a drect lnk between end nodes, where, n Gaussan nose, a carefully constructed superposton scheme of random and lattce codes was used. Fnally, n [11], an arbtrary number of clusters (nodes wthn a cluster all wsh to exchange messages) of arbtrary numbers of fullduplex nodes are assumed to communcate smultaneously through the use of a sngle relay n AWGN. In all four examples of mult-par b-drectonal communcaton wth a sngle relay, no drect lnk between the termnal nodes s assumed to exst, smplfyng the analyss as the tradeoff between relayed and drectly communcaton nformaton s avoded; no over-heard sde nformaton, or termnal node cooperaton s consdered. B. Our contrbutons We consder one base staton, multple termnal nodes and one relay, whch operate n half-duplex mode and have drect lnks to each other, as shown n Fg. 1. The desred bdrectonal lnks may be deduced from the ncluded messages W,j from node destned to node j, and W,j the estmate at node j of the message W.j. The base-staton s denoted as node wth ndex. Three elements of the formulated problem are markedly dfferent from pror nformaton theoretc work n ths area: 1) the assumpton that one end of the b-drectonal lnks s a sngle base-staton rather than ndependent nodes. 2) fully connected network - our nodes can all hear each other. Ths allows for the possblty of causal cooperaton between nodes as well as drect transmsson between the base-staton and the nodes, usng the relay only when benefcal. 3) n contrast to [1], [11], our nodes are half-duplex.
2 W2, W,2 W1, W,1 user 2 user 1 W3, W,3 Relay Basestaton W,1 user 3 Compress-and-Forward cooperaton between end users W1, W,2 W2, W,3 W3, Fg. 1. Our physcal channel model conssts of multple ndependent bdrectonal desred communcaton flows (message W, and W, wsh to be exchanged) between multple termnal nodes and a sngle base-staton. Communcatons may be aded usng one relay node. W,j s the estmate at node j of the message W,j. Our central contrbutons are: We propose three temporal protocols whch we call the FMABC (Full Multple Access ), PMABC (Partal Multple Access ) and FTDBC (Full Tme Dvson ) protocols. We determne nner bounds on the capacty regon of the mult-par b-drectonal relay network. Key elements of the schemes employed to do so nclude the use of multuser protocols n whch more than one termnal may be transmttng/recevng at one tme as n MAC and BC channels, random bnnng to explot over-heard sde-nformaton when the protocol permts, and the use of a flow-by-flow network codng strategy whch explots the two-way nature of data flows - all of whch wll be detaled n ecton III. We derve achevable rate regons where cooperaton between end users s enabled. Ths s possble n protocols n whch certan nodes over-hear other nodes transmssons. To the best of the authors knowledge, ths s the frst consderaton of cooperaton between end-users n a mult-par b-drectonal channel. In ths work, we consder a Compressand-Forward-based causal cooperaton scheme [12], selected due to the rough ntuton that CF outperforms for example DF-based cooperaton/relayng when the helpng node s close to the fnal destnaton, whch s expected to be one of the scenaros of practcal nterest. We present modfed cut-set-based outer bounds on the capacty regon of ths network. We note that due to lack of space, the statements and proofs of many of these results are omtted, but may be found at [2]. II. NOTATION AND DEFINITION We consder a base staton (node ), a set of termnal nodes B := {1, 2,, m and a relay r whch ads n the communcaton between the termnal nodes and the base staton. We defne M := B { = {, 1, 2,, m. We use R,j to denote the rate of communcaton from node to node j,.e. the message between node and node j, W,j, les n the set,j := {,..., 2 nr,j 1. mlarly, R,T s the sum of rates from set to set T where, T M at whch the messages W,T := {W,j, j T,, T M may be relably communcated. We assume that each end user communcates wth the base staton b-drectonally and that no nformaton s drectly exchanged between end users:.e. every par of termnal nodes and [1, m] wshes to exchange ndependent messages whle R,j = (or s undefned) for all, j B. Thus, there are a total of 2m messages n our network: m from node to each node B, and m from each node B to node, as shown n Fg. 1. Communcaton takes place over a number of channel uses, n and rates are acheved n the classcal asymptotc sense as n [1]. Node has nput alphabet X = X { and channel output alphabet Y = Y {, whch are related through a dscrete memoryless channel 1. Lower case letters x denote nstances of the upper case X whch le n the callgraphc alphabets X. Boldface x represents a vector ndexed by tme at node. Fnally, t s convenent to denote by x := {x, a set of vectors ndexed by tme, and as the cartesan product,.e., 3 =1 X = X 1 X 2 X 3. Durng phase l we use X (l) to denote the nput dstrbuton to denote the dstrbuton of the receved sgnal of node, and we use the dummy symbol to denote that there s no nput or no output at a partcular node durng a partcular phase.,n s the phase duraton of phase wth block sze n and s the phase duraton of phase when n. It and Y (l) s also convenent to defne X (l) := {X(l), a set of nput dstrbutons durng phase l. For a block length n, encoders and decoders are functons X k(w {,M, Y 1,, Y k 1 ) producng an encoded message at node, and W,j (Yj 1,, Y j n, W {j,m) producng a decoded message or error at node j when t wshes to decode the message W,j from node. Let (j) := { < j,. III. PROTOCOL FOR A MULTI-PAIR BI-DIRECTIONAL RELAY NETWORK The total transmsson tme s dvded nto two tme perods, each of whch may consst of one or more phases. Durng the frst multple access perod, the termnal nodes transmt to the relay. Durng the second broadcast perod the relay transmts to the termnal nodes. We consder three transmsson schemes for the multple-access perod: 1) Full Multple Access (FMABC) protocol: all termnal nodes transmt for the whole duraton, 2) Partal Multple Access (PMABC) protocol: uses the whole duraton and the other termnal nodes 1,, m transmt sequentally, and 3) Full Tme Dvson (FTDBC) protocol: all nodes transmt sequentally, as shown n Fg. 2. For comparson purposes n our smulatons, we also ntroduce what we call the smplest sequental protocol where all termnal nodes sequentally transmt nformaton to the relay,.e., r, 1 r,, m r, then the relay sequentally transmts them to the proper destnatons,.e., r, r 1,, r m. The FMABC, PMABC and FTDBC protocols descrbe the temporal phases or perods of the transmsson scheme but 1 Extensons to Gaussan nose channels wll be addressed n ecton V.
3 Node number Phase 1 Phase 2 Phase 1 Phase 2Phase 3 Tme Phase 1 Phase 2 Phase m Phase m+1 Phase m+1 Phase m+2 Multpleaccess perod perod Multpleaccess perod perod Multpleaccess perod perod Fg. 2. Three proposed half-duplex protocols - the tme phases of the dfferent protocols are seen; the encoders and decoders n the dfferent phases may vary. not what each node sends, or how ts messages are encoded durng those phases. The central techncal concepts employed n dervng achevable rate regons are: 1) Extended Marton s regon for broadcastng: Due to the presence of a base-staton wth multple messages (one to each of the termnal nodes), and a relay wth multple decoded messages (travelng to multple end users and the base-staton), we use a modfed verson of a generalzaton of Marton s broadcastng scheme [13] to > 2 messages/users, whch takes nto account own-message sde-nformaton at each node. A full statement of ths generalzaton may be found n [2]. 2) Network Codng: Network codng on a flow-by-flow (each flow conssts of two b-drectonal messages W, and W, ) bass s used at the relay r, whch decodes {w, and {w,, at the end of the multple access perod, and constructs w r = w, w,, B. Next, the decode-and-forward (DF) relay r constructs w r = (w r1, w r2,, w rm ) and broadcasts x r (w r ) durng the broadcast perod. 3) Random bnnng: Random bnnng s further used to explot over-heard sgnals from the drect lnks n the PMABC and FTDBC protocols. We apply random bnnng to combne, at an end user, the nformaton receved along from the drect lnk, and that receved along the relayng lnk. For example, n the PMABC protocol, node 1 uses m ndependent jontly typcal decoders wth sequences (x (2) (w,1, w {,B\{1 ),y (2) 1 ),, (x(m+1) r (w,1 w 1,, w r2,, w rm ),y (m+1) 1 ), thereby explotng the receved sgnals y (2) 1,,y(m+1) 1 overheard n phases 2, 3, (m+1) to decode w,1. 4) Cooperaton: Over-heard transmssons receved at a termnal node when t s not transmttng may be used to allow them to cooperate n decodng the messages W, for B. Cooperaton s enabled through a compress and forward strategy n whch each termnal node n B compresses the sgnals receved durng the relay broadcast perod usng an auxlary message set, whch t then transmts durng the next multple access perod. If other nodes can decode ths auxlary message, they are able to obtan the compressed receved sgnals whch n turn may be used to decode messages from the relay. We note that not all nodes need to cooperate or compress the receved sgnals - our results allow for any subset of the termnal nodes n B to cooperate. IV. ACHIEVABLE RATE REGION AND OUTER BOUND The achevable rate regons for the smplest, and the FMABC, PMABC, FTDBC wthout any network codng or bnnng, along wth the much more nvolved PMABC-NRC and FTDBC-NRC schemes wth cooperaton (the last C stands for cooperaton) are omtted due to space constrants but may all be found n [2], avalable onlne. Instead, we present the core achevable rate regons for the FMABC-N, PMABC- NR, and FTDBC-NR protocols, where N stands for Network codng and R stands for Random bnnng, whch explot the two-way nature of the data and over-heard sde nformaton whch s possble when a node s not transmttng. All regons are frst presented for dscrete memoryless channels and wll be evaluated n Gaussan nose n the followng secton. A. FMABC-N Protocol We consder the FMABC protocol n whch Network codng s employed at the relay to combne messages on a flowby-flow bass -.e. the message from node to node and vce-versa are combned at the relay. The U varables are the auxlary random varables playng a role smlar to those n Marton s regon [13] and ts > 2 user extenson n [2]. Theorem 1: An achevable rate regon of the mult-par half-duplex b-drectonal relay network under the FMABC- N protocol wth decode and forward relayng s the closure of the set of all ponts (R,b, R b, ) for all b B satsfyng R,M < 1 I(X (1) ; Y (1) R {,T < T r X (1), Q) (1) 2 I(U (2) ; Y (2) ) 2 I(U (2) ; U (2) T() ) (2) R T,{ < 2 I(U (2) T ; Y (2), U (2) T ) (3) for M and T B over all jont dstrbutons p(q) m = p(1) (x q)p (2) (u 1,, u m, x r ), where U j s are the auxlary random varables, and Q 2 m+1 1 over the alphabet m = X m j=1 U j X r Q. We note that for the FMABC random-bnnng to explot over-heard nformaton s mpossble as there s no overheard sde nformaton: durng each phase every node s ether transmttng or recevng - none are just lstenng. Under the PMABC and FTDBC protocols however, sde-nformaton may be exploted usng random bnnng, as descrbed next.
4 B. PMABC-NR Protocol We now consder the PMABC protocol n whch Network codng s employed at the relay to combne messages on a flow-by-flow bass, along wth Random Bnnng at the basestaton node to allow the end-nodes to explot nformaton over-heard n the phases durng whch they are not transmttng. In the followng theorem, the U varables are the auxlary random varables smlar to those seen n Marton s BC-channel regon [13] and ts extenson [2], whle V are auxlary random varables used for bnnng the message W, at the base-staton node for node. We note that bnnng s only possble at the base-staton for the end-users as n the PMABC protocol the base-staton s always transmttng durng the multple-access perod. Theorem 2: An achevable rate regon of the mult-par half-duplex b-drectonal relay network under the PMABC- NR protocol s the closure of the set of all ponts (R,b, R b, ) for all b B satsfyng R {,T + R,{ < X s si(v (s) T, X(s) s ; Y (s) r, V (s) Q) T + X si(v (s) T ; Y r (s), V (s) X(s) T s, Q) (4) s R {, < X mx ji(v (j) ; Y (j) Q) ji(v (j) ; V (j) Q) () j=1 + m+1i(u (m+1) ; Y (m+1) ) m+1i(u (m+1) ; U (m+1) ) (5) () R,{ < m+1i(u (m+1) ; Y (m+1), U (m+1) ) (6) for all B and, T B over all jont dstrbutons p(q) [ m =1 p() (v 1,, v m, x q)p () (x q) ] p (m+1) (u 1,, u m, x r ), where V j are the Random bnnng auxlary random varables at node, U j s are the auxlary Marton-lke random varables used at node r and V T := {V s s T wth Q 2 2m + 2 m over the alphabet m = X m j=1 (V j U j ) X r Q. Equaton (4) ensures correct decodng at the relay, (5) ensures correct combnng of overheard and relayed messages at the end users, whle (6) ensures correct decodng at the base-staton of the messages relayed (no sde-nformaton). C. FTDBC-NR Protocol The U and V varables have the same nterpretaton as n Theorem 2. Theorem 3: An achevable rate regon of the mult-par half-duplex b-drectonal relay network under the FTDBC- NR protocol s the closure of the set of all ponts (R,b, R b, ) for all b B satsfyng R {, < 1I(V (1) ; Y r (1), V (1) ) (7) R, < +1I(X (+1) ; Y r (+1) ) (8) R {, < X 1I(V (1) ; Y (1) ) 1I(V (1) ; V (1) ) () + m+2i(u (m+2) R,{ < X ; Y (m+2) +1I(X (+1) ; Y (+1) ) ) m+2i(u (m+2) ; U (m+2) () ) (9) + m+2i(u (m+2) ; Y (m+2), U (m+2) ) (1) for all B and ( B over all jont dstrbutons m ) p (1) (v 1,, v m, x ) j=1 p(j+1) (x j ) p (m+2) (u 1,, u m, x r ), where V j, U j s are the auxlary random varables and V T := {V s s T over the alphabet m = X m j=1 (V j U j ) X r. Remark 4: (7) and (8) correspond to the transmssons from M to the relay r, whle (9) (1) correspond to the relay broadcast phase. D. Compress-and-Forward based termnal node cooperaton The achevable rate regons for the PMABC-NRC and FTDBC-NRC protocols, where the C stands for Cooperaton are omtted due to space lmtatons but are avalable n [2] onlne. Note that due to the half-duplex constrant FMABC- NR and FMABC-NRC regons are not possble. E. Outer bounds The FMABC, PMABC and FTDBC outer bounds are obtaned by applyng the cut-set bound lemma talored to halfduplex mult-phase protocols frst derved n [1] to the dfferent protocols, where the cuts wll look dfferent dependng on what nodes are permtted to transmt durng each phase. The bounds are omtted for brevty and may be found n [2]. V. NUMERICAL ANALYI We assume an addtve whte Gaussan nose (AWGN) channel model, assume Gaussan nput dstrbutons for the achevablty schemes, whch may or may not be optmal, and evaluate the mutual nformaton terms. The correspondng mathematcal channel model s, for each channel use k : Y[k] = HX[k] + Z[k] where Y[k], X[k] and Z[k] are ndependent, of unt power, addtve, whte Gaussan, complex and crcularly symmetrc, and H C (m+2) (m+2) relate the vector channel nputs and output, whch are placed n the order, 1, 2, m, r. In phase l, f node s n transmsson mode X [k] follows the nput dstrbuton X (l) CN(, P ). Otherwse, X [k] =, whch means that the nput symbol does not exst n the above mathematcal channel model. We assume full CI. We use the followng channel gan matrx for m = 2 case: H = 2 6 4.3.5 1.3 1.5 1.5 1.5.2 1 1.2 3 7 5 (11) Frst, n Fg. 3, we examne the effect of usng Marton bnnng and Network codng and compare ther performance to the smplest protocol by plottng three achevable rate regons; 1) the smplest protocol (mple), 2) convex hull of the FMABC, PMABC and FTDBC protocols (MB) and 3) convex hull of the FMABC-N, PMABC-NR and FTDBC-NR protocols (MB- NR). We set P = P 1 = P 2 = P r = db. For more realstc comparson, we add lower lmts of ndvdual data rates,.e., R,1.1, R,2.1, R 1,.1, R 2,.1 to guarantee mnmum nformaton flow n each data lnk. Wthout ths lmtaton, the sum-data rate wll be maxmzed when both
5.5 R B,{.45.4.35.3.25.2.15.1.5 mple MB MB NR R B,{.5.4.3.2.1 IN OUT FMABC PMABC FTDBC R,2.3.25.2.15.1.5 PMABC FTDBC NR NRC OUT.5.1.15.2.25.3.35.4.45 R {,B.1.2.3.4.5.6 R {,B.1.2.3.4.5 R 1, Fg. 3. Effect of Marton bnnng, per-flow Network codng and Random bnnng. P = P 1 = P 2 = P r = db. Fg. 4. Overall regon comparson - no cooperaton. P = P 1 = P 2 = P r = db. Fg. 5. Effect of cooperaton. P = P 1 = P 2 = P r = db, and R,1 =.19, R 2, =.1. the transmsson rates R,2 and R 2, equal zero at least n the mplest case because the lnk between the relay and the node 2 s very poor. We see that the proposed protocols usng conventonal MAC and extended Martons broadcastng largely enhance the performance over straghtforward extensons of one-way protocols. Furthermore, we can sgnfcantly mprove the achevable rate regon by Network codng and Random bnnng schemes (n MB-NR). We emphasze that the ncluson mple MB MB-NR s not affected by the mnmum rate constrants. The achevable regons of the FMABC-N, PMABC-NR and FTDBC-NR protocols are plotted n Fg. 4 and compared to our modfed cut-set based outer bounds. The 4-dmensonal rate regons n (R,1, R,2, R 1,, R 2, ) are projected onto (R,1 + R,2, R 1, + R 2, ) 2-dmensonal space. Whle space constrants do not allow the presentaton of these plots under dfferent channel condtons and NRs, as seen n our extended work [2], one man conclusons drawn from plottng varous regons of the type seen n Fg. 4 s that dfferent protocols are optmal under dfferent channel condtons - no ncluson relatonshps appear to exst. In the low NR regme, the FMABC-N protocol outperforms the other protocols snce the amount of both sde nformaton and multple access nterference s relatvely small. However, n the hgh NR regme the FTDBC-NR protocol becomes the best snce t explots sde nformaton more effectvely. Under asymmetrc NR condtons (f we allow larger nput power for the base staton (node ) and relay (node r)) the PMABC-NR protocol outperforms the other two protocols; see [2]. To show the cooperaton codng gan, we plot the achevable rate regon of the dfferent protocols wth and wthout cooperaton. In Fg. 5, we fxed the data rates (R,1, R 2, ) to the rate par ((.19,.1) and plot rate regons n the (R 1,, R,2 ) doman. We do ths to hghlght the cooperaton gan, whch comes from re-allocatng node 1 s transmsson resources (.e. relatve power) to the two nformaton flows; 1 r (R 1, ) and 1 2 (R,2 ). As expected -NRC protocols acheve much better performance than -NR protocols. Notably, the cooperaton protocols mprove R,2 wthout any degradaton of R 1, n the FTDBC protocol. In contrast, the maxmum R 1, of the PMABC-NRC protocol s less than that of ts PMABC-NR only protocol. 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