Multi-pair bi-directional relay networks part I: protocols which exploit side-information

Transcription

1 1 Mult-par b-drectonal relay networks part I: protocols whch explot sde-nformaton 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. The contrbutons n part I are: 1) the ntroducton of three new temporal protocols whch fully explot the twoway nature of the data and over-heard sde-nformaton through network codng and random bnnng, 2) dervatons of achevable rate regons for the mult-par two-way network, and 3) a numercal evaluaton of the derved regons n Gaussan nose whch llustrate the performance of the proposed protocols. Outer bounds and the mpact of cooperaton between termnals nodes n an dentcal settng are consdered n part II of ths work. 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 the ncomplete lst [1] [9]) due to ts relevance n future wreless networks. The sngle relay, sngle par b-drectonal relay channel has been extended n a number of ways: 1) the consderaton of a sngle b-drectonal lnk usng multple relays [1] [15], and 2) the consderaton of multple bdrectonal lnks sharng a sngle, common relay [16] [2]. 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 ang Joon Km was wth the chool of Engneerng and Appled cences, Harvard Unversty, Cambrdge, MA Emal: sangkm@fas.harvard.edu. Besma mda s wth the Department of Electrcal and Computer Engneerng, Purdue Unversty Calumet, Hammond, IN 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. n savngs to the operators backhaul costs. Another motvatng 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 [16] 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 [17] consder a smlar channel model and propose the use of a CDMA strategy to support multple level Qo to dfferent users. In [18] multple b-drectonal pars communcate over a shared relay n the absence of a drect lnk between end nodes, whle the two-par full-duplex b-drectonal Gaussan relay network s studed n [19], where a carefully constructed superposton scheme of random and lattce codes was used. Fnally, n [2], an arbtrary number of clusters (nodes wthn a cluster all wsh to exchange messages) of arbtrary numbers of full-duplex 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 s possble. 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 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 [18] [2], our nodes are half-duplex.

2 2 W2, W,2 W1, W,1 user 2 user 1 W3, W,3 user 3 Relay Basestaton W,1 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. We note that communcaton need not pass through the relay as drect lnks between the base-staton and termnals exst. 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. 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 [3]. 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 1 Extensons to Gaussan nose channels wll be addressed n ecton V. 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 3 as the cartesan product,.e., =1 X = X 1 X 2 X 3. Durng phase l we use X (l) to denote the nput dstrbuton and Y (l) 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 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 ) producng an encoded message,, Y k 1 at node, and W,j (Yj 1,, Y n j, W {j,m) producng a decoded message or error at node j when t wshes to decode the message W,j from node. Fnally, 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 not what each node sends, or how ts messages are encoded durng those phases. In part I we wll use network codng and random bnnng schemes to explot the two-way nature and overheard nformaton, respectvely. In part II we wll addtonally quantfy the mpact of employng cooperaton between the end users. 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 [21] 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 [22].

3 3 Node number Tme Phase 1 Phase 2 Multpleaccess perod perod 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. These regons are presented for dscrete memoryless channels and wll be evaluated n Gaussan nose n the followng secton. Due to space constrants, all proofs are omtted and provded n [22], avalable onlne. Phase 1 Phase 2 Phase 1 Phase 2Phase 3 Phase m Phase m+1 Phase m+1 Phase m+2 Multpleaccess perod perod Multpleaccess perod perod 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 [21] and ts > 2 user extenson n [22]. 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) 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. 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 not only used n a Marton-lke fashon but s further used to explot over-heard sgnals, effectvely provdng causal sde-nformaton, 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. IV. ACHIEVABLE RATE REGION The achevable rate regons for the smplest, and the FMABC, PMABC, FTDBC wthout any network codng or bnnng are omtted for brevty and are ncluded n [22], avalable onlne. Instead, we present the more sophstcated and mproved achevable rate regons for the FMABC-N, PMABC-NR, and FTDBC-NR protocols, where N stands for 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. 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 [21] and ts extenson [22], 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, )

4 4 for all b B satsfyng R {,T + R,{ < s + s I(V (s) T ; Y r (s) s R {, < m ( j=1 + m+1i(u (m+1) si(v (s) T, X(s) s ; Y r (s), V (s) Q) T, V (s) X(s) T s, Q) (4) ji(v (j) ; Y (m+1) ; Y (j) ) Q) ji(v (j) ; V (j) () Q) ) m+1i(u (m+1) ; U (m+1) () ) (5) R,{ < m+1 I(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 {, < 1 I(V (1) ; Y (1) ) 1 I(V (1) ; V (1) () ) + m+2i(u (m+2) R,{ < ; Y (m+2) +1I(X (+1) ; Y (+1) ) ) m+2i(u (m+2) ; U (m+2) () ) (9) + m+2 I(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. 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] R B,{ mple MB MB NR R {,B Fg. 3. Comparson of protocol and codng gans wth P = P 1 = P 2 = P r = db and rate constrants (R,1.1, R,2.1, R 1,.1, R 2,.1). MB denotes the convex hull of the FMABC, PMABC and FTDBC protocols, whle MB-NR denotes the convex hull of the FMABC- N, PMABC-NR and FTDBC-NR protocols. 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 = (11) Frst we compare the achevable rate regons of the dfferent protocols, usng dfferent combnatons of encodng schemes, wth the smplest protocol. 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 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. In Fg. 3, there are 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). The mple regon s outer bounded by the MB regon. Ths mples that the proposed protocols usng only conventonal MAC and extended Martons broadcastng codng largely enhance the performance. Furthermore, we can sgnfcantly mprove the achevable rate regon by Network codng and Random bnnng schemes (n MB-NR). We want to emphasze that the results such that mple MB MB-NR s not affected by the mnmum rate constrants,.e., ths s true n most cases. The achevable regons of the FMABC-N, PMABC-NR and FTDBC-NR protocols n three dfferent NR regmes are plotted n Fgs. (4, 5, 6). The 4-dmensonal rate regons n

5 R B,{.3 R B,{ R B,{ R {,B R {,B R {,B Fg. 4. P = P 1 = P 2 = P r = db. Fg. 5. P = P r = 2, P 1 = P 2 = db. Fg. 6. P = P 1 = P 2 = P r = 2 db. (R,1, R,2, R 1,, R 2, ) are projected onto (R,1 +R,2, R 1, + R 2, ) 2-dmensonal space. The man outcome s that dfferent protocols are optmal under dfferent channel condtons. Ths s because the amount of sde nformaton and multple access nterference s dfferent. In the low NR regme (Fg. 4), 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 (Fg. 6), the FTDBC-NR protocol becomes the best snce t explots sde nformaton more effectvely. In Fg. 5 the PMABC-NR protocol outperforms the other two protocols. Indeed, f we allow larger nput power for the base staton (node ) and relay (node r), the drect lnks from the base staton are good enough to convey nformaton. The termnal nodes can then explot the sde nformaton effcently. Therefore, the PMABC-NR protocol has the best performance n ths channel condton. VI. CONCLUION In ths paper, we proposed three protocols for the halfduplex mult-par b-drectonal relay network: the FMABC, PMABC and FTDBC protocols whch were combned wth the followng codng schemes n dervng achevable rate regons: generalzed Marton broadcastng, Network codng, and Random bnnng. We compared these regons n an AWGN Gaussan nose channel where numercal smulatons verfed that whch protocol s superor depends on the channel condtons. In part II we wll present cut-set based outer bounds along wth a compress-and-forward-based cooperaton scheme at termnal nodes. REFERENCE [1] P. Larsson, N. Johansson, and K.-E. unell, Coded b-drectonal relayng, n the 5th candanavan Workshop on Wreless ad-hoc Networks, tockholm, May 25. [2] Y. Wu, P. A. Chou, and.-y. Kung, Informaton exchange n wreless networks wth network codng and physcal-layer broadcast, n Proc. Conf. on Inf. c. and ys., Baltmore, MD, Mar. 25. [3]. J. Km, P. Mtran, and V. Tarokh, Performance bounds for bdrectonal coded cooperaton protocols, IEEE Trans. Inform. Theory, vol. 54, no. 11, pp , Nov. 28. [4] B. Rankov and A. Wttneben, pectral efcent protocols for half-duplex fadng relay channels, IEEE J. elect. Areas Commun., vol. 25, no. 2, pp , Feb. 27. [5] P. Popovsk and H. Yomo, Physcal network codng n two-way wreless relay channels, n Proc. IEEE Int. Conference on Comm., Glasgow, Ireland, Jun. 27. [6] C. Ho, R. Zhang, and Y.-C. Lang, Two-way relayng over OFDM: optmzed tone permutaton and power allocaton, n Proc. IEEE Int. Conf. Commun., Bejng, May 28. [7] T. Oechterng, C. chnurr, and H. Boche, capacty regon of two-phase bdrectonal relayng, IEEE Trans. Inform. Theory, vol. 54, no. 1, pp , Jan. 28. [8]. Km, N. Devroye, P. Mtran, and V. Tarokh, Comparson of bdrectonal relayng protocols, n Proc. IEEE arnoff ymposum, Prnceton, NJ, Apr. 28. [9] C. Yuen, W. Chn, Y.L.Guan, W. Chen, and T. Tee, B-drectonal multantenna relay communcatons wth wreless network codng, n Proc. IEEE Veh. Technol. Conf., Dubln, May 27, pp [1]. J. Km, N. Devroye, and V. Tarokh, B-drectonal half-duplex protocols wth multple relays, ubmtted to IEEE Trans. Inform. Theory, 27, [11] T. J. Oechterng and H. Boche, Bdrectonal Regeneratve Halfduplex Relayng usng Relay electon, IEEE Transactons on Wreless Communcatons, vol. 7, no. 5, pp , May 28. [12]. Km, N. Devroye, and V. Tarokh, A class of b-drectonal multrelay protocols, n Proc. IEEE Int. ymp. Inform. Theory, eoul, Jun. 29, pp [13] J. Poonah and L.-L. Xe, An achevable rate regon for the two-way two-relay channel, n Proc. IEEE Int. ymp. Inform. Theory, Jul. 28, pp [14] R. Vaze and R. Heath, Optmal amplfy and forward strategy for twoway relay channel wth multple relays, n Proc. IEEE Inf. Theory Workshop, Volos, Jun. 29. [15] R. Vaze and R. W. Heath, To code or not to code n mult-hop relay channels, 28. [Onlne]. Avalable: [16] H. Ghozlan, Y. Mohasseb, H. E. Gamal, and G. Kramer, The mmo wreless swtch: Relayng can ncrease the multplexng gan, 29. [Onlne]. Avalable: [17] M. Chen and A. Yener, Interference management for multuser two-way relayng, n Proc. Conf. on Inf. c. and ys., Prnceton, Mar. 28, pp [18] A. Avestmehr, A. ezgn, and D. Tse, Capacty regon of the determnstc mult-par b-drectonal relay network, n Proc. IEEE Inf. Theory Workshop, Volos, Jun. 29. [19] A. ezgn, A. Khajehnejad, A. Avestmehr, and B. Hassb, Approxmate capacty regon of the two-par bdrectonal gaussan relay network, n Proc. IEEE Int. ymp. Inform. Theory, eoul, Jul. 29, pp [2] D. Gunduz, A. Yener, A. Goldsmth, and H. Poor, The mult-way relay channel, n Proc. IEEE Int. ymp. Inform. Theory, eoul, Jul. 29, pp [21] K. Marton, A codng theorem for the dscrete memoryless broadcast channel, IEEE Trans. Inform. Theory, vol. 25, no. 5, pp , May [22]. Km, B. mda, and N. Devroye, Achevable rate regons and outer bounds for a mult-par b-drectonal relay network, January, 21, posted on ArXv.