1 Cooperative Diversity Based o Code Superpositio Lei Xiao, Thomas E. Fuja, Jörg Kliewer, Daiel J. Costello, Jr. Departmet of Electrical Egieerig, Uiversity of Notre Dame, Notre Dame, IN 46556, USA Email: {lxiao, tfuja, jkliewer, costello.}@d.edu Tel: 574 631-744 Abstract This paper proposes a ew approach to cooperative diversity based o the superpositio of chael codes over a fiite field. The sceario uder cosideratio is oe i which two parters Node A ad Node B cooperate i trasmittig iformatio to a sigle destiatio; each parter trasmits both locally-geerated iformatio ad relayed iformatio that origiated at the other parter. A key observatio is that Node B already kows Node A s relayed iformatio (previously set from Node B) ad ca exploit that kowledge whe decodig Node A s local iformatio. This leads to a ecodig scheme i which each parter trasmits the superpositio of its local ad relayed iformatio, ad the superimposed codeword is iterpreted differetly at the two receivers i.e., at the other parter ad at the destiatio ode based o their differet a priori kowledge. It is show via simulatio that the proposed scheme provides substatial codig gai over other cooperative diversity techiques, icludig those based o time sharig ad sigal (Euclidea space) superpositio. I. INTRODUCTION Wireless data liks are ofte error proe due to multipath fadig. Diversity offers a effective coutermeasure agaist chael fadig by providig the receiver with multiple looks at the same iformatio differet versios of the data trasmitted with (ideally) idepedet chael gais [1, Chapter 14]. I a system employig cooperative diversity, this robustess is obtaied by allowig a ode to both trasmit its ow iformatio ad to serve as a relay for the data trasmitted by the other ode(s). Because iformatio is trasmitted multiple times oce from the origiatig ode ad the agai from the relayig ode(s) diversity is effected [] [6]. Cosider, for example, a system i which two source odes are paired as parters for the trasmissio of their data to a commo destiatio ode; each parter trasmits both its ow local iformatio as well as the relayed iformatio it has received from its parter. I the cooperative diversity systems proposed up to ow, some kid of mechaism is used to split the trasmitter s resources betwee locally geerated bits ad relayed bits. For istace, the systems i [3] [5] propose time divisio multiplexig i which each parter uses a portio of its time slot for local iformatio ad the rest for relayed iformatio. I cotrast, i [] ad [6], the superpositio (i Euclidea space) of modulated sigals is used to multiplex local ad relayed bits i the cotext of direct sequece spread spectrum modulatio ad uequal error protected modulatio, respectively. This work was supported i part by Motorola Corporatio s Uiversity arterships i Research (UR) program as well by NASA grat NNG05GH73G, NSF grat CCF-051501, ad Germa Research Foudatio (DFG) grat KL 1080/3-1. A B Fig. 1. I cooperative diversity, odes A ad B work i collaboratio to deliver their packets to a commo destiatio ode D. A key observatio with respect to such a system is that the relayed iformatio trasmitted by oe parter is already kow to the other parter; after all, that is where the relayed iformatio origiated. From this perspective, the resources allocated to trasmittig the relayed bits from oe parter to the other are wasted; those resources would be better used for sedig local iformatio from oe parter to the other. This leads to what we call the cooperative dilemma: by allocatig resources to relay cooperatio, the existig desigs reduce the likelihood of successful cooperatio by takig resources away from the vital parter-to-parter trasmissio of local iformatio. Based o this observatio, we propose a ew system desig that makes efficiet use of resources o both the parter-toparter liks ad the parter-to-destiatio liks. The mai idea is for each parter to superimpose for biary codes, to take the XOR of the local codeword ad the relayed codeword ad the trasmit the resultig superimposed codeword. The destiatio ode decodes the received packet as a ested codeword with both iformatio vectors as ukows. The parter ode, with its kowledge of the relayed codeword, views the XOR as a kid of scramblig ad oly eeds to decode what it does ot kow after cacellig the relayed codeword from the received sigal. I such a approach, o resources are used exclusively for the trasmissio of relayed iformatio from oe parter back to the other parter that origiated the iformatio. Cosequetly, sigificat performace gai is obtaied by icreasig the probability of successful collaboratio. The rest of this paper is orgaized as follows. The system model ad a overview of existig desigs are provided i Sectio II. The proposed trasceiver structures are described i Sectio III. Code search ad simulatio results are preseted i Sectio IV ad coclusios are draw i Sectio V. II. COOERATIVE DIVERSITY AND THE COOERATIVE DILEMMA The sceario addressed i this paper is depicted i Figure 1. Two source odes A ad B work i cooperatio to deliver D
L Node A trasmits Case 1: No-Cooperative Trasmissio Node A trasmits Node A relays Node B trasmits Case : Time Divisio Cooperative Diversity Node A trasmits ad relays Node B trasmits ad relays Case 3: Sigal Superpositio Cooperative Diversity Node B trasmits Node B relays Fig.. The frame structures of several desigs ad the resources used to protect the lik betwee Node A ad Node B. TABLE I NOTATION SUMMARY Time t first half Time t secod half Trasmissio A B, D B A, D Codeword C A (t) C B (t) Ow Iformatio Relayed Iformatio i A L (t) i A R (t) = π`ib L (t 1) ib L (t) i B R (t) = π`ia(t) L their packets of k bits each to a commo destiatio ode D. We refer to odes A ad B as each other s parter. Durig time slot t, Node A trasmits i the first half slot, while Node B trasmits i the secod half slot. Each source ode receives the frame set by its parter ode ad attempts to decode its parter s iformatio. If the decodig is successful, the some of this iformatio the part that origiated at the parter will be relayed i a future trasmissio to provide the destiatio D with spatial diversity. If a source ode fails to decode its parter s iformatio, ad therefore does ot have iformatio to relay, the that source ode will operate i a o-cooperative mode. By symmetry, we ca focus o packets that origiate at Node A. There ca be o diversity at the destiatio ode if the packets origiatig at Node A are ot received correctly at Node B; this illustrates the importace of A,B, the packet error rate for the lik betwee A ad B. Sice the diversity order determies the slope of the error curves [1, Chapter 14], it is desirable to have a small A,B to icrease the likelihood of Node A beig assisted by its parter ode. The frame structures of three differet cooperative diversity desigs are show i Figure. The shaded portios of the figure represet the resources dedicated to the trasmissio of local iformatio from oe parter to the other. It is assumed each source ode geerates k bits of local data per time slot ad the chael resources available durig a half-slot are trasmitted bits with a trasmit power of. I the o-cooperative cofiguratio, odes A ad B trasmit their ow iformatio i tur durig each slot. If the source odes wat to decode each other s iformatio i this case, the iformatio bits may be protected by a rate k/ code with full power the best level of protectio a ode ca offer its parter. I a time divisio based cooperative scheme [3] [5], each source ode uses part of its trasmissio time to act as a relay for its parter. A ode s relayed iformatio, origially geerated at the parter ode, is of o use to the parter ode. Hece, for example, whe B tries to obtai A s iformatio, it oly eeds to demodulate ad decode the half of A s frame that cotais A s locally geerated bits. Because A must trasmit k bits k local ad k relayed the chael code must be of rate k/ ad full power may be used durig trasmissio. I a cooperative scheme based o sigal superpositio [], [6], the modulated sigal for a relayed codeword is added to the modulated sigal for the locally geerated codeword. The parter ode, with kowledge of the relayed bits, ca subtract the portio of the sigal due to the relayed codeword from the received sigal ad decode the locally geerated iformatio bits. Sice part of the power has bee used to modulate the relayed codeword, which is later subtracted at the parter ode, each source ode decodes its parter s iformatio based o a rate k/ code but with oly partial power L. The above aalysis idicates that, i the existig cooperative diversity schemes, the trasmissio of a source ode s local iformatio to its parter is carried out with dimiished resources either a higher code rate or less power compared to the o-cooperative approach. Sice higher code rates ad lower power imply a higher probability of error, the packet error rate A,B betwee A ad B ecessarily icreases whe resources are allocated for relay cooperatio uder the existig desigs. Because A,B is, effectively, the probability that Node B will ot be able to supply diversity about Node A s local iformatio to the destiatio, we have this apparet coudrum: the extet to which a user dedicates resources to cooperatio reduces the probability that the cooperatio is successful. We refer to this as the cooperative dilemma. This dilemma is explaied by the fact that a source ode must deliver its ow packet ad a relayed packet to the destiatio ode, but oly its ow packet is required at the parter ode. Whe trasmissio time ad/or power are divided, the portio of those resources allocated to the relayed iformatio is wasted from the perspective of the parter ode. I the ext sectio, we solve this cooperative dilemma with a ew system desig that allows the parter ode ad the destiatio ode to carry out the decodig operatio at differet effective code rates. III. NEW SYSTEM DESIGN The otatio used i characterizig the ew system desig is summarized i Table I. Let i A L (t) deote the local iformatio vector origiatig at Node A for trasmissio durig time slot t, ad let i A R (t) deote the relayed iformatio vector trasmitted by Node A durig the same time slot. Similarly, for Node B we defie i B L (t) ad ib R (t). We assume each source ode ca determie whether it has successfully decoded the local iformatio vector trasmitted by its parter; this ca be doe either by usig a CRC or by employig a soft-decisio decoder ad usig the soft outputs to assess the reliability of the decoded iformatio. Let C A (t) ad C B (t) deote the -bit codeword set by odes A ad B, respectively, durig time slot t. The code
3 DEC π(i B L (t 1)) Flag Bit flip DEM f(i A L (t) ia R (t) )+oise î A L (t) CRC Check Π i B R (t) ENC i B L (t) CRC ENC MOD f(i B L (t) ib R (t) ) Fig. 3. The trasceiver structure at Node B. Soft decodig is assumed ad f( ) is the modulatio fuctio. The trasceiver at Node A is aalogous. geerator matrix for the locally geerated iformatio bits is ad for the relayed bits is ; both codes have rate k/. Ivokig symmetry, we ow provide a detailed descriptio of the ecodig operatio at Node A ad the decodig operatio at Node B, A. The Ecoder at the Source Nodes Durig time slot t, Node A must covey its local iformatio vector i A L (t) while relayig Node B s iformatio ib L (t 1) assumig it has decoded i B L (t 1) correctly. If Node A has decoded i B L (t 1) successfully, it first iterleaves i B L (t 1) to geerate the relay iformatio, i.e., i A R(t) = π(i B L(t 1)), (1) ad the trasmitted codeword is the XOR of the codeword cotaiig Node A s local iformatio ad the codeword cotaiig Node B s relayed bits C A (t) = i A L(t) i A R(t) = [i A L(t) i A GL R(t)]. () The iterleavig i (1) facilitates iterative decodig at the destiatio ode; it guaratees that the destiatio s decoder for Node B provides extrisic iformatio to its decoder for Node A that is idepedet of the other iformatio available to the Node A decoder. Also ote that the XOR of two codewords is equivalet to ecodig the iformatio vectors i A L (t) [ ad ia ] R (t) usig a ested code with geerator matrix G = GL, as idicated i (). Coversely, if Node A has failed to decode i B L (t 1), the trasmitted codeword is simply the ecoded versio of i A L (t): C A (t) = i A L(t) = [i A GL L(t) 0] (3) We assume a flag bit is trasmitted alog with the codeword to alert the receivers which of the two ecodig methods (() or (3)) was used. (Alteratively, ote that () ad (3) aturally represet two hypotheses, ad geeralized ML hypothesis testig could be used at the receiver to make that determiatio.) As will be show later, Node B ca cacel the relayed iformatio i A R (t) ad effectively decode oly i A L (t). I a aalogous maer, Node B s trasmitted codeword is C B (t) = i B L (t) i B R (t) = [i B L (t) ib R (t)] GL (4) if B has decoded i A L (t) correctly, where ib R (t) = π(ia L (t)), ad C B (t) = i B L(t) = [i B GL L(t) 0] (5) if Node B has failed to decode i A L (t). B. The Decoder at the Source Nodes The decoder at Node B first checks the flag bit. If the frame was geerated usig (3) i.e., Node A failed to cooperate the Node B uses the decoder for code to obtai i A L (t). Note that i this case Node B decodes a rate k/ code with full power. Now suppose the frame was geerated usig (), i.e., it represets the XOR of two packets. For hard-decisio decodig, the output of the demodulator is Ĉ A (t) = C A (t) e(t) = i A L(t) i A R(t) e(t) (6) where e(t) is the biary error patter. Note that, because Node B kows i A R (t) = π(ib L (t 1)), the codeword ia R (t) ca be stripped from the chael decoder iput by formig C A (t) = ĈA (t) i A R(t) = i A L(t) e(t). (7) At this poit, the hard-decisio decoder for the code ca be used to estimate i A L (t) from C A (t). Note that the effective code rate here is k/. I the case of soft-decisio decodig, the soft demodulator geerates the log-likelihood ratio (LLR) of each bit i C A (t). Let L[C A (t)](i) deote the LLR of the i-th bit i C A (t), ad let L[i A L (t)](i) deote the (to be determied) LLR of the i-th bit i the codeword i A L (t). Let [i A R (t)](i) deote the i-th bit i i A R (t). If [i A R (t)](i) is zero, the the i-th bit of C A (t) = i A L (t) i A R (t) is the same as the i-th bit of i A L (t). Moreover, if the i-th bit of i A R (t) is oe, the i-th bit of C A (t) is the complemet of i-th bit of i A L (t). Recogizig this, we ca produce L[i A L (t)](i) from L[C A (t)](i) ad i A R (t) as follows: L[i A L (t)](i) = log r{[ia L (t)](i) = 0} r{[i A L (t)](i) = 1} { L[C = A (t)](i) whe [i A R (t)](i) = 0 L[C A (t)](i) whe [i A R (t)](i) = 1. (8) Note the amplitude of L[i A L (t)](i) is idetical to that of L[C A (t)](i). Oly the sigs of the LLRs are chaged based o Node B s kowledge of π(i B L (t 1)). For this reaso,
4 case (1) case () case (3) case (4) First half of t Node A trasmits C A (t) i A L(t) [π(i B L(t 1)) i A L(t)] i A L(t) [π(i B L(t 1)) i A L(t)] GR GR Secod half of t Node B trasmits C B (t) i B L(t) acillary i decodig i A L(t) i B L(t) acillary i decodig i A L(t) [π( i A L(t) ) i B GR L(t)] [π( i A L(t) ) i B GR L(t)] Fig. 4. The decodig operatio at Node D for i A L (t). I cases (1) ad (), CL B(t) does ot cotai iformatio about ia L (t) ad is ot used i decodig. I cases (3) ad (4), iterative decodig is adopted to exploit spatial diversity, represeted by the double arrows. The extrisic iformatio about i B L (t 1) received from C B (t 1) is used as a priori iformatio i cases () ad (4). we refer to the operatio i (8) as the flippig operatio. (This operatio ca be viewed as a special case of the boxplus operatio itroduced i [7].) I this case, the kow iformatio i A R (t) is cacelled from the observed LLRs via the flippig operatio. With the LLR for i A L (t) ow available, Node B ca employ a soft-decisio decoder for to estimate i A L (t). Oce agai, the effective code rate is k/. I summary, for either hard decodig or soft decodig, the relayed codeword that was XORed with the locally geerated codeword ca be cacelled prior to decodig, ad this operatio is trasparet to the chael decoder. No resources are wasted, ad the useful iformatio to Node B, i A L (t), is protected by a rate k/ code with full power. As a result, the probability of successful cooperatio i the ew desig is higher tha that i the time divisio based approach or the scheme based o sigal superpositio. The trasceiver structure at Node B is show i Figure 3. C. The Decoder at the Destiatio Node Without loss of geerality, cosider the decodig of i A L (t) at Node D. Note that both C A (t) ad C B (t) potetially carry iformatio about i A L (t): CA (t) as local iformatio ad C B (t) (i iterleaved form) as relayed iformatio. There are four possible ways that C A (t) ad C B (t) could have bee formed, as illustrated i Fig. 4. 1) Both C A (t) ad C B (t) were costructed from local iformatio oly i.e., C A (t) = i A L (t) ad C B (t) = i B L (t). I this case, C B (t) is ot helpful i estimatig i A L (t). Usig a decoder for to process the oisy versio of C A (t) is sufficiet. ) C A (t) was costructed from both local ad relayed iformatio while C B (t) was costructed from oly local iformatio i.e., C A (t) = i A L (t) i A R (t) ad C B (t) = i B L (t). Oce agai, C B (t) is ot helpful i estimatig i A L (t). Note that ia R (t) is a iterleaved versio of i B L (t 1), which has already bee processed. We ca use the extrisic iformatio about i B L (t 1) obtaied from C B (t 1) as a priori iformatio for i A R (t) ad decode ia L (t) usig a maximum a posteriori probability (MA) decoder. 3) C A (t) was costructed from oly local iformatio while C B (t) was costructed from both local ad relayed iformatio i.e., C A (t) = i A L (t) ad C B (t) = i B L (t) i B R (t). Iterative decodig ca be used to exchage extrisic iformatio about i A L (t) ad i B R (t) = π(ia L (t)) betwee CA (t) ad C B (t) usig soft-i-soft-out decoders. A estimate of i B L (t) will be made i the ext decodig step; we use zero as a priori iformatio i the soft decisio decodig of C B (t). 4) Fially, both C A (t) ad C B (t) were costructed from both local ad relayed iformatio i.e., C A (t) = i A L (t) i A R (t) ad C B (t) = i B L (t) i B R (t). Iterative decodig ca be used i this case usig the soft-i-soft-out decoder for G. Sice i A R (t) = π(ib L (t 1)) has already bee processed, extrisic iformatio obtaied from C B (t 1) ca be used as a priori iformatio i processig C A (t), while zero a priori iformatio is used for i B L (t). The soft decoders processig C A (t) ad C B (t) exchage extrisic iformatio about i A L (t). Cooperative diversity is obtaied with iterative decodig whe Node B has relayed i A L (t) durig its time frame. The decodig of B s packet i B L (t) makes use of the observatio of both C B (t) ad C A (t + 1). IV. CODE SEARCH AND SIMULATION RESULTS Although the geeral framework proposed i Sectio III is valid for ay biary liear code, we costrai the codes ad to be biary covolutioal codes for two reasos. First, the soft-i, soft-out ML ad MA decoders for covolutioal codes are simple to implemet. Secod, covolutioal codes ca be used as buildig blocks for capacity achievig codes. We have the followig two observatios regardig the choice of code. First, the code is required to be as strog as possible, sice the source odes decode each other s iformatio usig this code. Moreover, if the lik from Node A to Node D is i a deep fade, the destiatio ode must make decisios about both A s bits [ ad ] B s bits based o the codeword C B (t) = [i B L (t)ib R (t)] GL. It is thus desirable to GL also make G = a strog code. To fid good codes, the best kow covolutioal code for a give rate ad costrait legth is used for. The a exhaustive search is carried out amog all appropriate choices of to fid the ested code G yieldig the largest free distace ad fewest umber of earest eighbors amog all o-catastrophic ecoders. For the simulatio results preseted below, the optimum 8-state ecoder = [1, 13 15, 17 15 ] 8 was chose, ad a computer search foud that the 8-state ecoder = [ 0 15, 07 15, 1] 8 optimized G. Simulatios were carried out to compare the performace of four differet approaches by which a pair of source odes could covey iformatio to a commo destiatio: ocooperative trasmissio, cooperative trasmissio based o
5 time-divisio multiplexig [4], cooperative trasmissio based o sigal superpositio [6], ad the cooperative system based o code superpositio proposed i this paper. Each packet cosisted of k = 500 bits. A CRC-1 code was used i the cooperative systems to idetify decodig failures. All systems used BSK modulatio, with the exceptio of the cooperative system based o sigal superpositio, for which (uequal error protected) 4-AM was employed. All chaels were subject to additive white Gaussia oise (AWGN) plus block Rayleigh fadig with the same average power. The chaels betwee differet pairs of odes were assumed to be idepedet. Moreover, the followig assumptios were built ito the simulatios: I the o-cooperative, poit-to-poit referece system, the rate 1/3 covolutioal code with geerator matrix [15, 13, 17] 8 was used. The same [15, 13, 17] 8 code was used as a mother code for the time-divisio cooperative scheme. To leave half of the trasmissio period available for relay purposes, this code was puctured to rate /3. If the decodig was successful at the parter ode, the puctured bits were recovered ad relayed. The destiatio ode D thus decoded either a puctured rate /3 covolutioal code or its rate 1/3 mother code, depedig o whether a relayed copy was received or ot. I the cooperative system based o sigal superpositio, the locally geerated bits were ecoded usig a recursive covolutioal code with geerator matrix [1, 13 15, 17 15 ] 8. If the parter ode s iformatio was successfully decoded, these bits were iterleaved by a s-radom iterleaver with spread 13 ad re-ecoded with the same code. The relay was allocated 15% of the total trasmissio power, as idicated i [6]. Iterative decodig with te iteratios was adopted at the destiatio ode to exploit diversity if a relayed copy was received. A s-radom iterleaver with spread 13 was used i the ewly proposed scheme. Te iteratios were carried out at the destiatio ode. Figures 5 ad 6 show the packet error probabilities ad the bit error probabilities for all four systems. Although all the cooperative diversity systems studied have the same error curve slope at high average sigal-to-oise ratio (SNR), the merit of the ew desig is clear. It is observed from Figure 5 that the ew system desig outperforms the cooperative diversity schemes based o sigal superpositio ad time divisio by db ad 4 db, respectively, at a packet error rate of 10 3. I terms of bit error probability, the ew desig based o code superpositio has fewer tha half of the bit errors of the sigal superpositio desig at a average SNR of 0 db. I the low SNR regime, the ew desig exhibits a steeper slope tha the other cooperative desig thaks to the superior protectio betwee parter odes. V. CONCLUSION A ovel system desig for cooperative diversity that makes efficiet use of available resources has bee proposed. By trasmittig the XOR of the locally geerated codeword acket Error robability 10 0 10 1 10 No coop. Time divisio coop. Sigal superpositio Code superpositio 10 3 0 5 10 15 0 5 30 Average E /N (db) b 0 Fig. 5. acket error probabilities for oe ocooperative ad three cooperative diversity systems. Bit Error robability 10 0 10 1 10 10 3 No coop. Time divisio coop. Sigal superpositio Code superpositio 10 4 0 5 10 15 0 5 30 Average E /N (db) b 0 Fig. 6. Bit error probabilities for oe ocooperative ad three cooperative diversity systems. ad the relayed codeword, thereby makig possible differet effective rates for the decodig at the parter ode ad the destiatio ode, the proposed scheme works i a collaborative mode more ofte tha i previously proposed desigs. The result is sigificat codig gai compared with existig approaches. REFERENCES [1] J. G. roakis, Digital Commuicatios, 4th ed. New York: McGraw-Hill, 001. [] A. Sedoaris, E. Erkip, ad B. Aazhag, User cooperatio diversity. part I ad part II, IEEE Tras. Commu., vol. 51, pp. 197 1948, Nov. 003. [3] J. N. Laema, D. N. C. Tse, ad G. W. Worell, Cooperative diversity i wireless etworks: Efficiet protocols ad outage behavior, IEEE Tras. Iform. Theory, vol. 50, pp. 306 3080, Dec. 004. [4] T. E. Huter ad A. Nosratiia, Cooperatio diversity through codig, i roc. ISIT, 00, p. 0. [5] A. Nosratiia, T. E. Huter, ad A. Hedayat, Cooperative commuicatio i wireless etworks, IEEE Commu. Mag., vol. 4, pp. 74 80, Oct. 004. [6] E. G. Larsso ad B. R. Vojcic, Cooperative trasmit diversity based o superpositio modulatio, IEEE Commu. Lett., vol. 9, pp. 778 780, Sept. 005. [7] J. Hageauer, E. Offer, ad L. apke, Iterative decodig of biary block ad covolutioal codes, IEEE Tras. Iform. Theory, vol. 4, pp. 49 445, Mar. 1996.