Nested Codes with Multiple Interpretations

Nested Codes with Multiple Interpretations Lei Xiao, Thomas E. Fuja, Jörg Kliewer, Daniel J. Costello, Jr. Department of Eletrial Engineering University of Notre Dame, Notre Dame, IN 46556, US Email: {lxiao, tfuja, jkliewer, ostello.2}@nd.edu Tel: 574 631-7244 bstrat This paper proposes a new approah to hannel ode design for wireless network appliations. The resulting nested odes an be deoded at different effetive rates by different reeivers rates that depend on the prior knowledge possessed by eah reeiver; we say these odes have multiple interpretations. We have identified several appliations in wireless networks where this property is useful. Speifi nested ode onstrutions as well as effiient soft and hard deision deoding algorithms are desribed. The onept of a nested ode with multiple interpretations provides flexibility in the design of error protetion shemes for multi-terminal wireless networks. I. INTRODUCTION Forward error ontrol oding is effetive in improving the reliability of data transmitted over noisy hannels. In the past, the art and pratie of error ontrol oding was largely onfined to point-to-point ommuniation links [1]. In wireless network appliations, the traditional layered approah breaks the network into a number of point-to-point links and then applies lassial error ontrol oding tehniques to protet the data in eah link. However, odes designed and optimized for point-to-point links might not be effiient in a network setting. There has reently been signifiant researh interest in developing error ontrol methodologies that address not just the point-to-point link but the larger network as well. In [2], the authors disussed the use of redundany in network oding and generalized the Hamming and Gilbert-Varshamov bounds to over network error orreting odes; the odes desribed in [2] were defined over a finite field and only hard deoding was onsidered. In a more reent paper [3], the authors disussed the possibility of onstruting odes on-thefly, adapting to the time varying onnetivity of wireless relay networks, and proposed a de-entralized framework termed adaptive network oded ooperation. In suh a framework, eah relay node ombines (XORs) a seleted set of the bits it reeived orretly; the diret transmissions from soure nodes an be viewed as systemati bits while the relayed XORed bits an be viewed as parity bits in either a low density generator matrix (LDGM) ode or a lower-triangular low density parity hek (LDPC) ode [3]. Diversity an be obtained at the destination node by deoding the LDGM or LDPC ode onstruted on-the-fly. The approah in [3] requires a reonfigurable LDGM/LDPC deoder at the destination and is only appliable to the senario where a large number of relays This work was supported in part by Motorola Corporation s University Partnerships in Researh (UPR) program as well by NS grant NNG05GH73G, NSF grant CCF-0515012, and German Researh Foundation (DFG) grant KL 1080/3-1. is available. In [4], the authors applied network oding over the binary field to exploit the shared nature of wireless networks and presented an opportunisti network oding sheme alled COPE; in COPE, data pakets are XORed at the network layer, resulting in signifiant throughput advantages ompared to the urrent 802.11 mesh network. The potential benefit arising from the interation of hannel oding and the proposed XOR operation, however, was not onsidered in [4]. This paper proposes a new approah to hannel ode design an approah in whih multiple information pakets are separately enoded and the resulting enoded pakets are then XORed together at the physial layer prior to transmission. These enoded-and-xored pakets an be deoded in different ways depending on the prior information available to the reeiver. If a reeiver already knows the information in one or more of the pakets, then the effet of the known paket(s) an be stripped away by regarding the (known) odeword(s) as a srambling pattern; the net result is to deode the information that is not known to the reeiver at a lower effetive rate i.e., with more redundany and thus more error protetion. lternatively, if a reeiver has no prior information about any of the pakets, then its deoder regards the XORed odewords as being produed by a higher-rate nested ode. In this manner, the same odeword broadast in a wireless medium is interpreted and deoded differently at different reeivers depending on the reeivers prior knowledge of the broadast information. The paper is organized as follows. The onept of a nested ode as well as enoding methods and effiient harddeision and soft-deision deoding algorithms are introdued in Setion II. Three appliations and speifi hoies of odes for these appliations are disussed in Setion III. Finally, Setion IV onludes the paper. II. NESTED CODES WITH MULTIPLE INTERPRETTIONS Consider the senario depited in Figure 1, in whih a transmitter (Tx) seeks to deliver N information vetors i 1,, i N of k bits eah to a number of reeivers (Rx). ssume that the odeword that is broadasted to the reeivers is Nn bits long, where n k. lso assume that the j-th reeiver knows a priori the value taken by some of the information vetors; let K j denote the indies of the information vetors known a priori to reeiver j i.e., reeiver j knows the value of i l prior to transmission if and only if l K j. This senario is realisti in many wireless networks; for example, a soure node has knowledge of the

Rx4 i l, l K 4 Rx1 i l, l K 1 Tx i 1,, i N i l {0, 1} k Rx3 i l, l K 3 Rx2 i l, l K 2 Fig. 1. One transmitter broadasts a odeword, representing N information vetors i 1,, i N of k bits eah. Eah of the multiple reeivers knows the value of some subset of the information vetors. information that it originated, and that same information might be relayed by another user in a multihop network. The wireless nodes ould also perform opportunisti listening [4] to reveal information that is intended to be delivered to other users and utilize this additional knowledge to assist in deoding pakets addressed to itself.. Enoding straightforward approah is to enode eah information vetor separately with a rate R = k/n ode and then onatenate the resultant odewords. This approah is not effiient beause it does not exploit the fat that some reeivers already know some of the information vetors; the resoures that the transmitter uses to onvey to reeiver j the K j k bits already known to reeiver j are, in effet, wasted. lternatively, onsider enoding eah information vetor with a low rate R/N = k/(nn) ode and then XORing the resulting odewords, i.e., = i 1 G 1 i 2 G 2 i N G N (1a) G 1 = [ ] i 1, i 2,, i N G 2...., G N (1b) where G 1, G 2,, G N are generator matries of rate k/(nn) odes. The representations of the XORed odewords in (1a) and (1b) suggest different interpretations of how the (possibly orrupted) odewords should be proessed at the reeiver. Representation (1a) suggests that, if the reeiver knows some of the information vetors, then the effets of those information vetors an be stripped off by treating the assoiated odewords as a srambling pattern. lternatively, if none of the information vetors are known at the reeiver, then the N information vetors an be estimated based on the nested ode assoiated with the staked generator matrix in (1b). It is worth pointing out that if we hoose the generator matries G 1 = G 2 = = G N = G and then XOR after hannel oding (i.e., we XOR the odewords), this is equivalent to XORing the information vetors before hannel oding, as in [4], i.e., i 1 G i 2 G i N G = ( i 1 i 2 i N ) G. (2) In the ase where more than one information vetor among i 1, i 2,, i N is unknown (or only known with soft a priori values) to the reeivers, however, it beomes neessary to use different generator matries for G 1, G 2,, G N, sine the staking of idential generator matries in (1b) results in rank defiieny, and onsequently the enoding operation is non-invertible even in the absene of noise. We an also interpret the differene between XORing the information vetors (prior to hannel oding) and XORing the oded vetors (after hannel oding) from a ode rate prospetive. The XOR operation is, in essene, a form of ompression that inreases the rate, while hannel enoding lowers the rate by adding redundant bits. When the XOR operation is effeted before hannel enoding, the result, in the stage between the XOR and hannel enoding, is a vetor that onveys information at a rate higher than one. Suh a high rate an only be resolved by perfet a priori knowledge at the reeiver of all but one of the information vetors XORed at the transmitter. lternatively, If we first introdue redundany through hannel enoding and then ompress the low rate odewords, we an still reover the information vetors even when more than one of the information vetors are not known perfetly at the reeiver, assuming that the ode is properly designed to avoid rank defiieny in the staked generator matrix and the overall rate the rate onsidering both the hannel enoding and the XOR ompression of the unknown vetors is less than one. Hene, by hoosing different (linearly independent) generator matries, we avoid a atastrophi ondition and make nested odes with multiple interpretations more flexible and suitable for a variety of appliations.. Hard-Deision Deoding In hard-deision deoding, the output of the demodulator is ˆr = e, where e is the binary error pattern. t the j-th reeiver, ˆr an be interpreted as ˆr = i l G l i l G l e. (3) l K j l K j Sine part of ˆr namely l K j i l G l is known to the j-th reeiver, it an be anelled by XORing to obtain ˆr j = ˆr i l G l = i l G l e. (4) l K j l K j The right hand side of (4) represents a orrupted version of a odeword from a rate (N K j )k/nn ode - the nested ode obtained by staking the generator matries orresponding to all the information vetors not known to reeiver j. The information vetors that are known to the reeiver a priori are anelled as a srambling pattern using (4) and are transparent to the hannel deoder. Thus, when a reeiver knows more of the information vetors a priori, it an deode at a lower rate i.e., with more robustness the information vetors it does not know a priori. Hene we have a single nested ode with multiple interpretations.

Fig. 2. In ooperative diversity, Node and Node work in ollaboration to deliver their pakets to a ommon destination Node D. C. Soft-Deision Deoding In soft-deision deoding, the output of the demodulator is L (i), the log-likelihood ratio (LLR) of the i-th bit in. Similar to hard-deision deoding, the odeword an be deomposed into two parts as = i l G l i l G l. (5) l K j l }{{} K j }{{} u The first term on the right hand side of (5), u, is a olletion of unknown odewords, while the seond term,, is the olletion of odewords known to the j-th reeiver whih an therefore be anelled. The LLR of the i-th bit in u an be omputed aording to L u(i) = = log Pr[ u(i) = 0] Pr[ u (i) = 1] L (i) = log Pr[u (i)=0] Pr[ u (i)=1] if (i) = 0 L (i) = log Pr[u (i)=1] Pr[ u (i)=0] if (i) = 1. The operation in (6) hanges only the sign of the LLR and not its magnitude; hene no information from the observation of the hannel output is lost. We refer to this as the flipping operation. This operation an be regarded as a degenerate ase of the box-plus operation introdued in [5] when one of the input LLR s takes a value of either infinity or minus infinity. With the LLR L u(i) available from the flipping operation, we an employ a soft deoder for the rate (N K j )k/nn odeword u to estimate the N K j information vetors that are unknown a priori to reeiver j. III. PPLICTIONS In this setion, we present three senarios in wireless networks that desribe a situation like the one depited in Figure 1. In partiular, we demonstrate how nested odes with multiple interpretations an be used to improve the performane of wireless systems ompared with more onventional shemes.. Cooperative Diversity Consider first a ooperative diversity system in whih two partners all them Node and Node ooperate in D (6) transmitting information to a ommon destination node, Node D, as illustrated in Figure 2. (See [6] and the referenes therein.) Eah partner node transmits both its loally generated information and relayed information that originated at the other partner. Spatial diversity an be exploited from the reeption of the same information twie from both the diret transmission and the relay. Conventional ooperative shemes [6] adopt a time division approah, wherein the transmission time of eah partner node is split into two parts one part for broadasting its loallygenerated information and the other part for relaying the information that originated at its partner node. Muh like the straightforward onatenated approah disussed in Setion II-, the fat that Node knows Node s relayed information prior to its transmission and Node knows Node s relayed information prior to its transmission is not exploited in onventional time division based designs. s an alternative that does exploit this knowledge, onsider the use of nested odes with multiple interpretations. Node enodes its own loal information using a hannel ode with generator G L while it interleaves and enodes the relayed information from Node using a hannel ode with generator G R ; what is transmitted is the XOR of these two odewords. Node, with its prior knowledge of node s relayed information, an anel the part of the odeword enoded by G R, as desribed in Setion II, and deode only the information vetor enoded by G L. From the perspetive of Node, all the resoures used in Node s transmission are used to onvey Node s loally generated information; no resoures are wasted in onveying the relayed information, whih Node already knows. Sine Node will be able to ooperate (i.e., relay information) only when it deodes Node s loally generated information orretly, the adoption of nested odes with multiple interpretations makes the ooperative diversity system work in a ooperative mode (i.e., a diversity enabling mode) more often than in onventional time division designs. The destination Node D has prior knowledge of neither of the information vetors, so it views the reeived [ ] odeword GL as one enoded with the staked generator. Diversity G R is ahieved at node D by iteratively deoding the diret transmission and the relayed transmission enoded with an interleaved version of the same information. Of ourse, the bits generated at the two partner nodes are formed by XORing together two odewords, and this must be aounted for at Node D. Take the deoding of Node s information as an example. Node D proesses a odeword from Node ontaining Node s loal information and relayed information that originated at Node ; at the same time it also proesses a odeword from Node ontaining a new information vetor generated at Node and relayed information that originated at Node. For the odeword from Node, the relayed part (Node s information) is deoded iteratively together with the diret transmission from Node reeived previously, and we an use the extrinsi information generated by the previous deoding as a priori information for the relayed bits. For the odeword from Node, relaying Node s relevant information, however, the deoding of Node s loal

10 0 Non oop. Time division oop. Nested odes 1 1.0 Paket Error Probability 10 1 10 2 Extrinsi Output 0.6 0.4 0.2 0.59 0.39 0.19 0.0 = 0.5 d 10 3 0 5 10 15 20 25 30 verage (d) Fig. 3. n SNR gain of about4 d is obtained using nested odes with multiple interpretations, ompared with the onventional approah to ooperative diversity based on time division multiplexing. information annot be ompleted sine Node might relay Node s information in its next transmission. We use zero a priori values for Node s loal bits in the soft-deoding of this odeword. The performane improvement in terms of paket error probability from using nested odes with multiple interpretations is shown in Figure 3. This simulation assumes PSK modulation on an WGN hannel with i. i. d. blok Rayleigh fading and ten iterations at the destination node. Pakets of 500 information bits eah and rate 1/3-onvolutional odes were used. CRC-12 was adopted to identify deoding failures. G R = [ 02 15, 07 15, 1] 8 was hosen to maximize the free distane of the nested ode, given G L = [1, 13 15, 17 15 ] 8. Figure 3 also inludes the performane of two referene systems namely a non-ooperative point-to-point system and the onventional time-division based oded ooperative diversity system [6] with a ooperation level of 50 perent. Unlike onventional iterative deoding, we only exhange extrinsi information for some of the enoded information bits between the two onstituent deoders. (Reall that in our ase two information vetors are enoded into an XORed odeword, but only one information vetor is proessed in iterative deoding.) Extrinsi information transfer (EXIT) harts [7] provide a useful tool for studying the onvergene behavior of iterative deoding. Figures 4 and 5 depit the extrinsi information transfer relation for two onstituent odes, i.e., the ode from the diret transmission and the ode from the relay. For the diret transmission onstituent ode, the relayed portion has been previously proessed, and the amount of extrinsi information reeived from the previous deoding parameterizes the urves in Figure 4. For the relayed onstituent ode in Figure 5, no previously proessed odeword is available, and the loal information bits use zero a priori values. typial deoding trajetory is shown in Figure 6. In this example, deoder one proesses the odeword from the diret transmission, whih is transmitted over an WGN hannel with = 0 d. It is assumed that the previous 0 0 0.2 0.4 0.6 1 Priori Input Fig. 4. The extrinsi information input-output relation for the onstituent ode from the diret transmission. Extrinsi Output 1 0.6 0.4 0.2 = 2.5 d = 2.0 d = 1.5 d = 1 d = 0.5 d 0 0 0.2 0.4 0.6 1 Priori Input Fig. 5. The extrinsi information input-output relation for the onstituent ode from the relay. deoding provides an a priori mutual information of 0.76 for the relayed bits to deoder one. Deoder two proesses the odeword from the relay, whih is transmitted over an WGN hannel with = 1.5 d. From both the EXIT urves (Figures 4 and 5) and the deoding trajetory (Figure 6), it is observed that, for either information vetor, the extrinsi information at the deoder output an reah a value of one even when the a priori information for the other information vetor is small or even zero.. Network Coded Duplex Relay In network oded duplex relay systems [4], [8], two nodes, Node and Node, wish to exhange pakets through a ommon relay, Node R. Instead of relaying s paket i and s paket i in sequene, the relay node XORs the deoded and re-enoded pakets from and and subsequently broadasts the XORed information to both and [4], [8]. When a hannel enoder G is used to protet the bits against

1 i G extrinsi output of deoder one 0.6 0.4 0.2 Deoding trajetory EXIT analysis Time 1 Time 2 i i R R î i G i Fig. 6. 0 0 0.2 0.4 0.6 1 extrinsi output of deoder two typial deoding trajetory for a blok size of 50000 bits. Time 3 î G î G ĩ R î, î î G î G ĩ i i hannel noise, the broadast odeword an be written as (î î)g = îg îg, (7) where î and î are the estimates of i and i at Node R. This sheme is illustrated in Figure 7 with G = G = G. Upon reeiving the orrupted version of (î î)g, nodes and an deode the XOR of the two information pakets as î î e, where e is the error pattern introdued by deoding failures. Node, for instane, an form an estimate of i using ĩ = î î e i = î e (î i ). (8) Note that if the relay only sends î to Node, Node would deode î e under the same hannel onditions. The last term (î i ) in (8) is an error propagation term resulting from the superposition of î, whih will be different from i when the deoding of i fails at Node R. Note that the error propagation term is transparent to the hannel deoders, sine we anel i after hannel deoding. From the prospetive of nested odes with multiple interpretations, this transpareny is aused [ by the rank defiieny of the staked generator G matrix, whih is noninvertable even in the absene of G] noise. s an alternative, we an use nested odes with linearly independent generators G and G for Node s and Node s information vetors and anel the known information at nodes and prior to either soft-deision or hard-deision deoding, as shown in Setion II. In this ase, still taking deoding at Node as an example, the remaining error î G i G from an erroneously deoded paket at Node R will be a valid odeword for G, but not a valid odeword for G thanks to the linear independene between G and G. Hene the hannel deoder estimating i will attempt to remove both the effet of noise and the effet of the error propagation term at Node. The performane improvement from using nested odes with multiple interpretations is shown Fig. 7. In network oded duplex relay systems, Node and Node exhange their pakets using a ommon relay Node R. Paket Error Probability 10 0 10 1 10 2 XOR of information vetors Nested odes 10 3 2 3 4 5 6 7 E /N (d) b 0 Fig. 8. The effet of adopting nested odes with multiple interpretations in a network oded duplex relay system. in Figure 8, where an WGN hannel is assumed, the same 8-state onvolutional ode [15, 13, 17] 8 is used for s and s bits, but a random interleaving of s oded bits is adopted prior to the XOR operation to avoid the rank defiieny that results from staking idential generator matries. Sine few errors our for high hannel SNRs, the improvement is most visible in the medium hannel SNR range. Note that the hannel deoder is only optimum in deteting odewords on an WGN hannel, but that the error propagation term is learly not Gaussian distributed. joint odeword and error pattern detetor an be used to further redue the effet of the error propagation term. The key to this joint proessing is to treat the information, after the anellation or

flipping desribed in Setion II, as being from the odeword [ i î, î] [ ] G (9) G and to deode both the desired information î and the error propagation term i î using the bit error probability in deoding i at Node R as a priori information for i î. The performane improvement from this nested ode approah with joint detetion, though, is rather small ompared to the nested ode approah that detets only the desired information vetors. C. roadast Infrastruture ided Multiasting In this senario, the base station relays multiasting ontent from one of the users while broadasting its own information to all the users at the same time. This ould happen, for example, in a trunked radio system, where one user multiasts its pakets to a talk-group and the base station s information is from the entral offie. In onventional designs, the base station would adopt a multiplexing sheme to separate the broadasting of its own bits and the relaying of its talking user s multiast information. This neglets the fat that the talking user knows its own message, and moreover, the users lose to the talking user might suessfully deode the same message when the paket is delivered to the base station for relay. Nested odes with multiple interpretations an be used in this senario to make more effiient use of available resoures at the base station. The base station enodes its own broadasting paket i using enoder G, while it enodes the relayed multiasting paket from the talking user, i M, using a different enoder G M. The transmitted odeword is then i G i M G M. The user where the multiast paket originated, together with the users that have obtained the multiasted paket through the transmission to the base station, an anel the known information i M G M and deode only the broadasted paket i at a low rate. Other users, without knowing either of the pakets, an deode both pakets [ ] at the G same time using the deoder for the nested ode. The G M hoie of ode in this appliation is very different ompared with the hoie of ode in the ooperative diversity senario, where the link between the two soure nodes is vital to ensure diversity and thus requires good protetion. In this senario, the users apable of anelling i M G M have already enjoyed a rate benefit, and [ hene ] it is more desirable to ensure that the G users deoding are using a strong ode. G M In Figure 9, we illustrate the benefit of using nested odes with multiple interpretations on an WGN [ hannel. ] The best 06 13 13 64-state rate 2/3 onvolutional ode [1] is 13 [ 06 ] 17 8 G used as the staked generator matrix. We hoose G M G = [13 06 17] 8 and G M = [06 13 13] 8, sine the former has better distane properties than the latter. With nested odes, the users with knowledge of the multiast pakets gain around 2 d ompared to the other users. Note that in a Paket Error Probability 10 0 10 1 10 2 User with no prior knowledge User with knowledge of the multiast pakets 10 3 0 1 2 3 4 5 6 E /N per information bit transmitted b 0 Fig. 9. Users with knowledge of the multiast pakets gain from using nested odes with multiple interpretations. onventional referene sheme, the information is transmitted at rate 2/3, and the paket error performane is similar to that of the users without knowledge of the multiast pakets. Hene we improve the performane of the knowledgeable users without affeting that of the ignorant users. IV. CONCLUSIONS framework for nested odes with multiple interpretations has been proposed, and simple and effiient deoding methods that redue the effetive ode rate based on a reeiver s a priori knowledge have been desribed. Three appliation examples have been provided to show how the nested ode an improve system performane ompared to onventional designs. These odes, together with the effiient hard-deision and soft-deision deoding desribed in this paper, are flexible and useful tools in the design of error ontrol shemes in a wireless network setting. REFERENCES [1] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and ppliations, 2nd ed. New Jersey: Pearson Prentie Hall, 2004. [2] N. Cai and R. W. Yeung, Network oding and error orretion, in Pro. IEEE Information Theory Workshop, angalore, India, Ot. 2002. [3] X. ao and J. Li, Mathing ode-on-graph with network-on-graph: daptive network oding for wireless relay network, in Pro. llerton Conferene on Communiation, Control, and Computing, Montiello, IL, Sept. 2005. [4] S. Katti, D. Katabi, W. Hu, H. Rahul, and M. Médard, The importane of being opportunisti: Pratial network oding for wireless environments, in Pro. llerton Conferene on Communiation, Control, and Computing, Montiello, IL, Sept. 2005. [5] J. Hagenauer, E. Offer, and L. 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