IN wireless networks, it has always been a challenge to satisfy

Transcription

1 1668 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 3, MARCH 14 Desgn of Low-Densty Party-Check Codes for Half-Duplex Three-Phase Two-Way Relay Channels Xnsheng Zhou, Lang-Lang Xe, Senor Member, IEEE, and Xuemn (Sherman) Shen, Fellow, IEEE Abstract In two-way relay channels, two termnal nodes exchange nformaton wth the help of a relay node. Desgnng practcal codng schemes for such channels s challengng, especally when messages are encoded nto multple streams and a destnaton node receves sgnals from multple nodes. In ths paper, we prove an achevable regon for half-duplex three-phase two-way relay channels. Furthermore, we propose low-densty party-check (LDPC) codes for such channels where two source codewords are encoded by systematc LDPC codes at the relay node. To analyze the performance of the codes, dscretzed densty evoluton s derved for the ont decoder at termnal nodes. Based on the dscretzed densty evoluton, degree dstrbutons are optmzed by teratve lnear programmng n 3 steps. The length of the obtaned optmzed codes s 6% longer than the theoretc one. Index Terms Low-densty party-check (LDPC) codes, twoway relay channel, densty evoluton. I. INTRODUCTION IN wreless networks, t has always been a challenge to satsfy hgh traffc throughput demand. Besdes lmted power and spectrum resources, nterference s also a factor that lmts the throughput due to shared medum. In the past decades, varous technques, such as cooperatve communcatons [1], have been developed to acheve hgher communcaton rates. A typcal example of cooperatve communcatons s the communcaton through relay channels [] where the source node transmts nformaton to the destnaton node wth the help of a relay node. Although the exact capacty of the relay channel s stll unknown, two dfferent relay schemes, known as decode-and-forward and compress-and-forward [], have been developed. In general, when the source-relay lnk s relable, the decode-and-forward scheme s a better choce snce nose can be fully elmnated. A natural extenson of the one-way relay channel s the twoway case where two termnal nodes exchange nformaton wth the help of a relay node. Some fundamental bounds [3], [4], [5] for two-way relay channels have been proposed by several research groups. In [3], an achevable regon of the decodeand-forward scheme based on block Markov superposton codng and an achevable regon of the compress-and-forward scheme based on Wyner-Zv codng for full-duplex two-way relay channels were proposed. In [4], another achevable regon for full-duplex two-way relay channels was proved by Manuscrpt receved July 14, 13; revsed October 8, 13; accepted January 3, 14. The assocate edtor coordnatng the revew of ths paper and approvng t for publcaton was A. Ghrayeb. The authors are wth the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo, Ontaro, Canada, NL 3G1 (e-mal: {x9zhou, llxe, sshen}@uwaterloo.ca). Dgtal Obect Identfer 1.119/TWC /14$31. c 14 IEEE random bnnng. In [5], the capacty regon of the broadcast phase of the two-way relay channels was determned when destnaton nodes use sde nformaton for decodng. Besdes the research on fundamental bounds, varous practcal codng schemes have also been proposed for relay channels. Constructng codes for such channels s challengng, especally when messages are encoded nto multple streams and a destnaton node receves sgnals from multple node. Low-densty party-check (LDPC) codes were proposed for one-way relay channels n [6], [7], [8]. In [6], LDPC codes were proposed for one-way relay channels where codes wthn 1. db of the theoretcal lmt were found. Furthermore, LDPC codes employng random puncturng were appled to fadng relay channels n [7]. In order to mprove the performance of one-way relay channels, blayer LDPC codes were desgned based on the blayer densty evoluton n [8]. In half-duplex two-phase two-way relay channels, the relay node receves supermposed sgnals from the two source nodes. Consderng ths unque property, varous codng schemes, such as physcal-layer network codng [9], repeataccumulate codes [1], lattce codes [11] and LDPC codes [1], [13], have been proposed recently. In ths paper, we focus on half-duplex three-phase two-way relay channels. Half-duplex s a practcal assumpton snce t s generally dffcult for a node to detect weak receved sgnals when they are mngled wth ts own strong transmttng sgnals. Compared wth two-phase two-way relay channels, sgnals from the source node can be utlzed for decodng at the destnaton node. In addton, decodng at the relay node s smpler snce no supermposed sgnals are nvolved. However, to the best of our knowledge, only a few practcal codng schemes [14], [15] have been proposed for three-phase twoway relay channels. In ths paper, we propose LDPC codes for half-duplex threephase two-way relay channels, frst appeared n [16]. LDPC codes are good canddates snce they can approach the channel capactes of pont-to-pont channels. More mportantly, they have a comprehensve set of desgn tools along wth ther flexble code constructons. The man contrbutons of ths paper are four-fold. Frst, we prove an achevable regon for half-duplex three-phase twoway relay channels. Second, nspred by the random codng, we propose a code constructon whch s composed of two rregular LDPC codes at termnal nodes and a systematc LDPC code at the relay node. Note that the relay codeword can be generated by smply addng the two source codewords n GF(). However, smple addton s not optmal f lnks between the source node and the relay node are asymmetrc

2 ZHOU et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR HALF-DUPLEX THREE-PHASE TWO-WAY RELAY CHANNELS 1669 snce equal amount of nformaton from the two codewords s ncluded n the relay codeword. Encodng by systematc LDPC codes at the relay node can be thought as party forwardng or random bnnng on multple sources. Ths code constructon s smlar to that of non-systematc low-densty generator matrx (LDGM) codes [17], or Luby transform (LT) codes (a class of rateless code) [18], whch were orgnally proposed for pont-to-pont channels. Thrd, to analyze the performance of the codes, we employ dscretzed densty evoluton for the proposed decoder, where the relatonshps between the consttuent codes are derved. Last, based on the dscretzed densty evoluton, we propose a 3-step degree dstrbuton optmzaton based on teratve lnear programmng. We show that the length of the obtaned optmzed codes s only 6% longer than the theoretc one. Ths paper s organzed as follows. We begn wth an ntroducton of the system model and a proof of an achevable rate regon for half-duplex three-phase two-way relay channels n Secton II. In Secton III, LDPC code constructons and the correspondng message-passng decodng algorthms are proposed. In Secton IV, dscretzed densty evoluton s derved to analyze the codes. We ntroduce an teratve lnear programmng algorthm for code optmzaton n Secton V. An optmzed degree dstrbuton s reported and decodng smulaton results are ncluded n Secton VI. Fnally, we conclude ths paper n Secton VII. II. SYSTEM MODEL AND AN ACHIEVABLE RATE REGION A. System Model In two-way relay channels, two termnal nodes communcate wth each other wth the help of a relay node. We consder the case when sgnals from the source node can be utlzed for decodng at the destnaton node. In ths case, the transmsson over half-duplex two-way relay channels can be modeled as a three-phase transmsson. We label the two termnal nodes as node 1 and node, respectvely, and label the relay node as node 3. In phase 1, node 1 encodes ts message and broadcasts the codeword. Both node and node 3 can receve sgnals. In phase, node encodes ts message and broadcasts the codeword. Both node 1 and node 3 can receve sgnals. The relay node can decode the messages of node 1 and node at the end of phase 1 and phase, respectvely. In phase 3, node 3 encodes the two source codewords to a relay codeword and broadcasts the relay codeword. Both node 1 and node can receve sgnals. At the end of phase 3, node 1 can ontly decode the message of node from sgnals receved from node n phase, sgnals receved from node 3 n phase 3 and ts own codeword. Smlarly, node can ontly decode the message of node 1. The three phase model s shown n Fgure 1. B. An achevable rate regon of half-duplex three-phase twoway relay channels In ths secton, we prove an achevable rate regon of half-duplex three-phase two-way relay channels. Note that an achevable rate regon for full-duplex two-way relay channels was gven n [4]. Fg. 1. Three phases n half-duplex two-way relay channels. The two-way relay channel conssts of source nput alphabet sets X 1, X, X 3, channel output alphabet sets Y 1, Y, Y 3 and a set of dstrbutons p(y 1,y,y 3 x 1,x,x 3 ). Consderng tme dvson, dstrbutons durng phase 1, phase and phase 3 are p(y,y 3 x 1 ), p(y 1,y 3 x ) and p(y 1,y x 3 ), respectvely. Assume the lengths of codewords n the three phases are n 1, n and n 3, respectvely, and n = 3 =1 n.setα = n1 n, β = n n3 n and γ = n. A (( nr1, nr ),n 1,n ) code for the half duplex threephase two-way relay channel conssts of two sets of ntegers W 1 = {1,,, nr1 } and W = {1,,, nr },three encodng functons X 1 : W 1 X n1 1, X : W X n and X 3 : W 1 W X n3 3, and four decodng functons Yn1 3 W 1, Y n 3 W, Y n1 Y n3 W 1,andY n 1 Y n3 1 W. Theorem 1. For dscrete memoryless half-duplex three-phase two-way relay channels, all rate pars (R 1,R ) satsfyng and R 1 < mn {αi(x 1 ; Y 3 ),γi(x 3 ; Y )+αi(x 1 ; Y )} (1) R < mn {βi(x ; Y 3 ),γi(x 3 ; Y 1 )+βi(x ; Y 1 )} () are achevable for some p(x 1 )p(x )p(x 3 ) where α+β+γ =1. Proof: Codebook generaton: Generate nr1 codewords x 1 = x n1 1 accordng to n 1 =1 p(x 1) and ndex them as x 1 (w 1 ), w 1 {1,,, nr1 }. Generate nr codewords x = x n accordng to n =1 p(x ) and ndex them as x (w ), w {1,,, nr }. Generate n(r1+r) codewords x 3 = x n3 3 accordng to n 3 =1 p(x 3) and ndex them as x 3 (w 1,w ), w 1 {1,,, nr1 }, w {1,,, nr }. Encodng: In phase 1, to send ndex w 1, node 1 sends x 1 (w 1 ). In phase, to send w, node sends x (w ). In phase 3, node 3 sends x 3 (ŵ 1, ŵ ) after decodng w 1 and w (See the decodng part). Decodng: Denote y, as the channel output at node n phase. At the end of phase 1, node 3 decodes w 1 by fndng the unque ŵ 1 that satsfes the ont typcalty check (x 1 (ŵ 1 ), y 3,1 ) A (n1) ɛ (X 1,Y 3 ) where A (n1) ɛ (X 1,Y 3 ) s the set of ont typcal sequences of X 1 and Y 3. If there s no such or more than one such ŵ 1, an error s declared. Smlarly, at the end of phase, node 3 decodes w by fndng the unque ŵ that satsfes (x (ŵ ), y 3, ) A (n) ɛ (X,Y 3 ).If there s no such or more than one such ŵ, an error s declared. At the end of phase 3, node 1 decodes w by fndng the unque ŵ that satsfes (x (ŵ ), y 1, ) A (n) ɛ (X,Y 1 ) and (x 3 (w 1, ŵ ), y 1,3 ) A (n3) ɛ (X 3,Y 1 ). Node decodes w 1 by fndng the unque ŵ 1 that satsfes (x 1 (ŵ 1 ), y,1 ) A (n1) ɛ (X 1,Y ) and (x 3 (ŵ 1,w ), y,3 ) A (n3) ɛ (X 3,Y ). Analyss of the probablty of error: When node 1 sends x 1 (w 1 ), the probablty that ndependent x 1 and y 3,1 are

3 167 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 3, MARCH 14 ontly typcal s upper bounded by n1(i(x1;y3) 3ɛ).There are totally nr1 1 such x 1. Wth the unon bound, the probablty of error at node 3 s upper bounded by ( nr1 1) n1(i(x1;y3) 3ɛ), whch approaches zero when n 1 and R 1 <αi(x 1 ; Y 3 ) (from nr 1 n 1 I(X 1 ; Y 3 ) < and α = n1 n ). Smlarly, we need R <βi(x ; Y 3 ) for node 3 to decode x. When node sends x (w ), the probablty that ndependent x and y 1, are ontly typcal s upper bounded by n(i(x;y1) 3ɛ). When node 3 sends x 3 (ŵ 1, ŵ ), the probablty that ndependent x 3 and y 1,3 are ontly typcal s upper bounded by n3(i(x3;y1) 3ɛ). There are totally nr 1 such w when node 1 knows w 1. Wth the unon bound, the probablty of the event that any ndependent x and y 1, are ontly typcal and any ndependent x 3 and y 1,3 are ontly typcal at node 1 s upper bounded by ( nr 1) n(i(x;y1) 3ɛ) n3(i(x3;y1) 3ɛ), whch approaches zero when n, n 3 and R <βi(x ; Y 1 )+γi(x 3 ; Y 1 ) (from nr n I(X ; Y 1 ) n 3 I(X 3 ; Y 1 ) <, β = n n and γ = n3 n ). Smlarly, R 1 <αi(x 1 ; Y )+γi(x 3 ; Y ) s requred for node to decode w 1. For Gaussan half-duplex three-phase two-way relay channels, they can be modeled as follows. In phase 1, Y 3,1 = X 1 +Z 3,1 and Y,1 = X 1 +Z,1. In phase, Y 3, = X +Z 3, and Y 1, = X + Z 1,. In phase 3, Y 1,3 = X 3 + Z 1,3 and Y,3 = X 3 + Z,3. Z, s a Gaussan dstrbuted random varable wth mean zero and varance σ,. When bnary phaseshft keyng (BPSK) s consdered, the codeword bt s mapped from {, 1} to {1, 1}. The sgnal X 1 =(X 1,1,,X 1,n1 ) 1 n1 has a power constrant n 1 =1 X 1, P 1. Smlarly, X 1 n and X 3 have power constrants n =1 X, P and 1 n3 n 3 =1 X 3, P 3. For Gaussan half-duplex three-phase two-way relay channels, all rate pars (R 1,R ) satsfyng ( ( 1 R 1 < mn α log 1+ P 1 and ( 1 α log 1+ P 1 N,1 ( 1 R < mn ) + 1 γ log N 3,1 ( 1+ P 3 N,3 ), )) ( β log 1+ P ), N ( 3, 1+ P )) 3 N 1,3 ( 1 β log 1+ P ) + 1 N 1, γ log (4) are achevable where N, = σ, and α + β + γ =1.Note that (3) and (4) can be easly derved from (1) and (). In Fgure, the achevable rate R 1 s plotted when α = β = γ = 1 3 and P1 N 3,1 =1. The x-axs and y-axs are sgnal-tonose ratos (SNRs) 1 P N,1 and P3 N,3, respectvely. The z-axs s the achevable rate R 1. The flat area s the area where R 1 s lmted by the SNR of the source-relay lnk, whle the slope area s the area where R 1 s lmted by SNRs of the sourcedestnaton lnk and the relay-destnaton lnk. In Fgure 3, the achevable rate R 1 s plotted when α = β = γ = 1 3 and ( P1 N,1 P3 (3) = P3 N,3. The x-axs and y-axs are SNRs P 1 N 3,1 and P1 N,1 N,3 ). The z-axs s the achevable rate R 1. The left slope area s the area where R 1 s lmted by the SNR R 1 Fg.. R 1 Fg P 3 /N,3 (db) 6 8 Achevable rates of R 1 for 4 P 1 /N,1 =P 3 /N,3 (db) 6 8 Achevable rates of R 1 for P 1 /N,1 (db) ( ) P1 P, 3 pars. N,1 N, P 1 /N 3,1 (db) ( ( )) P1 P, 1 P3 pars. N 3,1 N,1 N,3 of the source-relay lnk, whle the rght slope area s the area where R 1 s lmted by SNRs of the source-destnaton lnk and the relay-destnaton lnk. III. CODE CONSTRUCTIONS AND MESSAGE-PASSING DECODING ALGORITHMS A. LDPC code constructons and ther graph representatons In ths secton, we propose LDPC code constructons for half-duplex three-phase two-way relay channels and show ther graph representatons. At termnal node for =1,, the node encodes a k -bt message nto an n -bt codeword. The codeword s broadcast to the relay node and the other termnal node. Under the decode-and-forward scheme, the relay node can decode the message whle the other termnal node cannot decode wthout the help of the relay node. Intutvely, wth addtonal bts from the relay node, the effectve code rate s reduced. At the relay node, two codewords c 1 and c from termnal nodes are concatenated as a source message c =[c 1 c ].The lengths of c 1, c and c are n 1, n and n 1 +n, respectvely. An n 3 -bt relay codeword r s generated by a systematc LDPC

4 ZHOU et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR HALF-DUPLEX THREE-PHASE TWO-WAY RELAY CHANNELS 1671 Fg. 6. The graph for decodng at termnal nodes. Fg. 4. Fg. 5. The graph of the systematc LDPC code at the relay node. The equvalent graph of the systematc LDPC code at the relay node. code where r = cg and G s a generator matrx wth the sze (n 1 + n ) n 3. These bts are also called relay bts, whch are broadcast to both termnal nodes whle c s not sent. Here, n1+n codeword pars are mapped to n3 codewords. Snce n 3 s n general less than n 1 + n, multple codeword pars are mapped to a relay codeword. Ths code constructon s smlar to those of non-systematc LDGM codes [17] and LT codes [18]. LDGM codes were ntally proposed as an alternatve to LDPC codes. In these codes, check bts c are generated from source bts s by c = sg. For systematc LDGM codes, both source bts and check bts are sent to a destnaton node. For non-systematc LDGM codes, only check bts are sent. LDGM codes were proposed for channels wth known channel parameters and ther code rates are fxed. LT codes are the frst practcal rateless codes, whose dea was orgnally from Fountan codes [19]. They can be consdered as nonsystematc LDGM codes, though check bts are contnuously generated untl a recever can recover the source message. In general, any lnear code can be represented by a Tanner graph []. The graph of the systematc LDPC code at the relay node s shown n Fgure 4. Crcles are varable nodes and squares are check nodes. The n 3 upper layer varable nodes (n black) represent the relay bts. The lower layer n 1 varable nodes (n whte) and n varable nodes (n grey) represent two source codewords from termnal nodes. Check nodes represent party check constrants among these bts. Note that the above graph s a bpartte graph. The n 3 upperlayer varable nodes can be moved to the lower layer, as shown n Fgure 5. The total number of varable nodes s n 1 +n +n 3. The frst two groups of varable nodes represent codewords of two termnal nodes, whch are called Group 1 varable nodes and Group varable nodes, respectvely. The n 3 rght-most varable nodes represent relay bts, whch are called Group 3 varable nodes. At the termnal node, messages are decoded based on three peces of nformaton: receved sgnals from the source termnal node, receved sgnals from the relay node, and the codeword of the destnaton node. The graph for decodng s shown n Fgure 6. Compared to Fgure 5, two groups of check nodes are added to the lower layer. These check nodes represent party check constrants of LDPC codes at the termnal node, whch are called Group 1 check nodes and Group check nodes, respectvely. The upper layer check nodes are called Group 3 check nodes. Group 3 check nodes n Fgure 6 can also be moved to the lower layer. In ths sense, the decodng algorthm at the termnal node could be smlar to those for pont-to-pont channels. B. Message-passng algorthms In ths secton, we propose message-passng algorthms for decoders n half-duplex three-phase two-way relay channels. In such channels, decodng happens at all three nodes. Snce any exstng decodng algorthms for pont-to-pont channels can be used at the relay node, the detals are omtted here. In the sequel, we only focus on the message-passng algorthm at destnaton nodes. Especally, only decodng functons at node 1 are derved snce decodng functons at node are smlar. Wth the help of the graph n Fgure 6, the message-passng algorthm can be easly descrbed. The receved bt can be represented n an LLR form log p(x =1 y, ) p(x = 1 y, ) = y, σ, (5) for, =1,, 3. Varable nodes and check nodes are assocated wth decodng functons. Messages flow between varable nodes and check nodes va edges, servng as nputs or outputs of the functons. The algorthm adopts an teratve decodng method by passng messages multple tmes between varable nodes and check nodes. In varable nodes, functons are n the form of summaton. In check nodes, functons are n the form of tanh 1 ( tanh). In general, a message passng schedule s requred durng the teratve decodng. In ths work, a floodng schedule s used. In ths schedule, all messages from varable nodes are passed to check nodes along all edges, and all output messages from check nodes are passed back to varables nodes thereafter to complete one decodng teraton. Let v l be a message from a Group varable node to a Group check node n the l-th decodng teraton for =1,. Letu l be a message from a Group check node to a Group varable node n the l-th decodng teraton for =1,. Letv,3 l be a message from a Group varable node to a Group 3 check node n the l-th decodng teraton for =1,, 3. Letu l 3, be a message from a Group 3 check node to a Group varable node n the l-th decodng teraton for =1,, 3. For a varable node, a lower/upper varable node degree s defned as the total number of edges connected to a lower/upper layer check node. An upper/lower-degree- varable node s a varable node

5 167 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 3, MARCH 14 wth upper/lower edges. An upper/lower-degree- varable node edge s an edge connected to an upper/lower-degree- varable node. Let d be a lower degree of a Group varable node for =1,. Letd 3, be an upper degree of a Group varable node for =1,, 3. Letg be a degree of a Group check node for =1,. For a Group 3 check node, t has three degrees. Let g 3, be a degree of a Group 3 check node whch s the total number of edges connectng to a Group varable node for =1,, 3. Letu, be a channel output LLR assocated wth a Group varable node for =1,, 3. The functons used for decodng messages of termnal node at termnal node 1 are v l = d 1 d 3, u l 1, + =1 =1 u l 1 3,, + u,, (6) v l 1,3 = u,1, (7) v l,3 = d =1 d 3, 1 u l 1, + =1 u l 1 3,, + u,, (8) v3,3 l = u,3, (9) [ g3,1 ( ) v u l 3, = tanh 1 l 1,3, tanh =1 g 3, 1 ( ) ( ) v l,3, v l 3,3 tanh tanh, (1) =1 [ g 1 ( )] v u l = tanh 1 l, tanh. (11) =1 The functon n Group 1 varable nodes s shown n (7). Snce termnal node 1 knows ts own codeword, ntrnsc values of Group 1 varable nodes are + or. Hence, no matter what messages are receved from check nodes, Group 1 varable nodes always send u,1 (+ or ) to upper check nodes. The functon n Group 3 varable nodes s shown n (9). Group 3 varable nodes only send u,3 snce the degree of Group 3 varable nodes s 1. Functons n Group varable nodes are shown n (6) and (8). Group varable nodes receve messages u l 1 3,, and ul 1, from upper layer check nodes and lower layer check nodes, respectvely. These messages are added together wth the channel output LLR u, = y1,. σ1, The output v,3 l s sent to a Group 3 check node n the upper layer. The output v l s sent to a Group check node n the lower layer. The functon n Group 3 check nodes at the upper layer s shown n (1). For upper layer check nodes, they only send the output message u l 3, to a Group varable node. The functon n Group check nodes at the lower layer s shown n (11). The output u l s sent to a Group varable node. IV. DISCRETIZED DENSITY EVOLUTION In ths secton, densty evoluton [1] s used as a tool to analyze codes n message-passng algorthms for half-duplex three-phase two-way relay channels. Frst, we formally defne an ensemble of codes va graph n half-duplex three-phase two-way relay channels. The ensemble s a sequence of codes wth the same varable node degree dstrbutons and check node degree dstrbutons. For systematc LDPC codes at the relay node, we defne one varable node degree dstrbuton and two check node degree dstrbutons. These degree dstrbutons are defned from node perspectve. Snce the relay node only forwards partal nformaton of the source codeword, we allow degree as an upper degree of a varable node. The upper-degree- varable node does not connect to any upper layer check nodes. Denote λ 3 as the varable node degree dstrbuton. λ 3,, s the fracton of the total number of upper-degree- varable nodes n Group varable nodes to the total number of all 3 groups of varable nodes., λ 3,, =1. Denote ρ 3,1 and ρ 3, as the two check node degree dstrbutons. ρ 3,, s the fracton of the total number of upper layer check nodes wth degree d 3, = to the total number of all upper layer check nodes. ρ 3,, =1for =1,. The ensemble of codes s defned based on four permutatons. π s a permutaton for codes at termnal node for =1,. π 3 and π 4 are permutatons for the code at the relay node. The defntons of π 1 and π are the same as those n pont-to-pont channels. Here, we only defne π 3 and π 4. Assgn some sockets to every Group varable node accordng to the degree dstrbuton λ 3, for =1,. The sockets on Group varable nodes are called Group varable node sockets. Assgn two groups of sockets to every upper layer check node accordng to degree dstrbutons ρ 3,1 and ρ 3,. We call them Group check node sockets for =1,. Edges connectng to Group check node sockets are connected to Group varable nodes. Two groups of varable node sockets and two groups of check node sockets are labeled separately wth postve ntegers startng from 1. Group 1 and Group check node socket labels are permuted by π 3 and π 4. Edges are dentfed by pars of sockets, whch are denoted as (, π 3 ()) and(, π 4 ()), where or s a Group 1 or Group varable node socket, π 3 () or π 4 () s a check node socket n the two groups of check node sockets, respectvely. A code s an element n the permutaton space π 1 π π 3 π 4.All codes n the permutaton space are equprobable. Densty evoluton tracks the probablty densty functon of LLR messages. Messages on each edge can be represented by a random varable. The output messages of a check node functon and a varable node functon can be represented by functons of random varables. If threshold decodng (A bt s decoded as f the message s greater than or equal to zero and decoded as 1 f the message s less than ) s used, the probablty of error s smply the ntegral of the probablty densty functon from to. In dscretzed densty evoluton, probablty densty functons are approxmated by probablty mass functons. Recall that the functon of the varable node s a sum of ndependent random varables, e.g. (6) and (8). The probablty mass functon of the sum of two ndependent dscrete random varables can be calculated by convolvng the probablty mass functons of the two random varables by crcular dscrete convoluton. Furthermore, n order to speed up the calculaton, the crcular dscrete convoluton can be calculated by dscrete Fourer transform and the nverse dscrete Fourer transform. For a varable node wth an upper degree and a lower degree, denote the probablty mass functon of nput messages on the upper edge and the lower edge as P and

6 ZHOU et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR HALF-DUPLEX THREE-PHASE TWO-WAY RELAY CHANNELS 1673 P, respectvely. The probablty mass functon of the output messages on upper edges s Pv l = P { 1 P l 1 } { P l 1 } (1) where l s the decodng teraton number, s dscrete convoluton, s dscrete convoluton on random varables and P s the probablty mass functon of the channel output LLR message. Smlarly, the probablty mass functon of the output messages on lower edges s Pv l = P { P l 1 } { 1 P l 1 }. (13) In general, f dscrete random varables X 1 and X are ndependent, the probablty mass functon of Z = p(x 1,X ) s P (Z = z) = P (X 1 = x 1 )P (X = x ). (14) z=p(x 1,x ) The probablty mass functon of check node output messages n (1) and (11) can be calculated by ths way. For the functon Z = tanh 1 ( ) tanh X1 X tanh, the probablty mass functon of Z s P (Z = z) = z= tanh 1 (tanh x 1 tanh x ) P (X 1 = x 1 )P (X = x ).(15) If the functon s n the form of Z =tanh 1 ( ) X tanh, we can calculate P Z by recursvely calculatng the probablty mass functon of the functon of two nput random varables wth (15). Denote P, as the probablty mass functon of output messages from upper-degree- Group varable nodes. The probablty mass functon of nput messages at Group check node sockets s λ 3,, P,. (16) Denote Q, as the probablty mass functon of output messages from upper layer check nodes wth degree d 3, =. The probablty mass functon of nput messages at Group varable node sockets s ρ 3,, Q,. (17) V. CODE OPTIMIZATION In ths secton, we propose a three-step code optmzaton to fnd good codes for half-duplex three-phase two-way relay channels. In the frst two code optmzaton steps, two rregular LDPC codes for the two source-relay lnks are desgned. Snce the underlyng channels are pont-to-pont channels, any exstng optmzaton methods [1], [], [3], [4] for such channels can be used. In ths work, teratve lnear programmng [8] s used as the optmzaton solver. In ths solver, the code rate s maxmzed when σ3, s gven for =1,. A feasble regon s a space on λ and ρ where λ s a varable node degree dstrbuton of rregular LDPC codes at termnal node and ρ s a check node degree dstrbuton of rregular LDPC codes at termnal node. When the degree of ρ s concentrated [3], the optmzaton problem becomes a sequence of sub-problems: fndng an optmal λ wth a fxed ρ. The detals of teratve lnear programmng for pont-to-pont channels can be found n Appendx II of [8]. In the thrd code optmzaton step, systematc LDPC codes at the relay node are optmzed. The optmzed rregular LDPC codes obtaned n the frst two steps are used n the thrd step. The optmzaton obectve s to fnd the optmal degree dstrbutons that mnmze the rato of the length of the relay codeword to the sum of the lengths of two source codewords λ 3,3,1 λ 3,1, + λ. (18) 3,, In ths optmzaton problem, the feasble regon s a space on λ 3,1, λ 3,, λ 3,3, ρ 3,1 and ρ 3,. To smplfy the optmzaton problem, the orgnal problem s dvded nto a sequence of optmzaton problems on λ 31,λ 3,λ 33 wth fxed ρ 31,ρ 3. The global optmzaton problem n the thrd optmzaton step s mn λ 3,3,1 (19) λ 3,1,λ 3,,λ 3,3 s.t. λ 3,1, + λ 3,, + λ 3,3,1 =1 () λ 3,1,,λ 3,,,λ 3,3,1 1 (1) λ 3,1, ρ 3,1, λ 3,3,1 = () λ 3,, ρ 3,, λ 3,3,1 = (3) (e l+1 3,1, el 3,1)λ 3,1, <,l=1,,l 1 (4) (e l+1 3,, el 3, )λ 3,, <,l=1,,l, (5) where L 1 and L are the total numbers of decodng teratons, e l 3,, s the mxture probablty of error on upper-degree- edges of Group varable nodes n the l-th decodng teraton, and e l 3, s the probablty of error mxture on Group check node sockets n the l-th decodng teraton. Note that the probablty of error s calculated durng dscretzed densty evoluton by P (X ) = a P (X = a) where P (X) s the probablty mass functon of messages. The probablty of error mxture e l 3, can be calculated from the probablty mass functon of the message mxture at nputs of check nodes. Wth constrant (), (18) becomes (19). Constrant () s the condton that the sum of the probablty s 1. Constrant (1) s the condton that a probablty s upper bounded by 1 and lower bounded by. Constrant () comes from nλ 3,1, = n 3 ρ 3,1,. (6) The left hand sde of (6) s the total number of upper edges connected to Group 1 varable nodes where n s the total number of all 3 groups of varable nodes. In addton, from the upper layer check node perspectve, the total number of upper edges connected to Group 1 varable nodes s the rght

7 1674 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 3, MARCH 14 hand sde of (6) where n 3 s the total number of upper layer check nodes. The left hand sde and the rght hand sde should be equal and ths condton becomes constrant () due to λ 3,3,1 = n3 n. Constrant (3) s smlar to (), but t apples to Group varable nodes. Constrant (4) comes from e l+1 nλ 3,1, 3,1, nλ <e l 3,1. (7) 3,1, The left hand sde and the rght hand sde of (7) are mxtures of probabltes of error of nput messages at upper layer check nodes n the (l+1)-th and l-th decodng teraton, respectvely. (4) s a sequence of constrants on the decodng rule, that s, the probablty of error s monotoncally decreased durng teratve decodng. Constrant (5) s smlar to (4), but t apples to Group varable nodes. Snce the probablty of error s a non-lnear functon of the degree dstrbuton, constrants (4) and (5) are nonlnear. However, f the probabltes of error are treated as constants, the non-lnear optmzaton problem becomes a lnear optmzaton problem. Snce the probabltes of error are treated as constants, codes from lnear programmng mght not be decoded. In ths case, dscretzed densty evoluton can be used to verfy whether codes can be decoded. If codes can be decoded, ther degree dstrbuton becomes the current best degree dstrbuton. Durng the dscretzed densty evoluton, the probablty of error n each decodng teraton can be calculated, whch are used n the next optmzaton teraton. If codes cannot be decoded, the feasble regon needs to be shrnked by reducng the value of the rght hand sde of (4) and (5), denoted as μ, towards. When the feasble regon s shrnked, the rato (18) becomes larger. Hence codes could be easer to decode. As the value μ s reduced, the problem could be nfeasble at some pont. In other words, no degree dstrbuton satsfes all constrants from () to (5). In such a case, we modfy the teratve lnear programmng algorthm proposed n [8] by reducng L 1 and L. Snce less constrants are appled, the feasble regon s enlarged. Note that the probabltes of error n (4) and (5) come from the precedng optmzaton teraton, whch are provded as hnts on the boundary of the feasble regon n the next optmzaton teraton. The optmal degree dstrbuton could be nsde of the feasble regon or outsde of the feasble regon. VI. SIMULATION RESULTS In ths secton, two optmzed degree dstrbutons for rregular LDPC codes at termnal nodes and an optmzed degree dstrbuton for systematc LDPC codes at the relay node are obtaned when a half-duplex three-phase two-way relay channel s gven. Codes sampled from the optmzed degree dstrbutons are smulated. We show that good codes can be found by our proposed three-step optmzaton. The length of the obtaned optmzed codes s 6% longer than the theoretc one. In addton, t s shown that the requred SNR for a fnte-length code converges fast to that for cyclefree nfnte-length codes by smulatons. In the frst optmzaton step, rregular LDPC codes for termnal node 1 are optmzed, where a k 1 -bt source message TABLE I THE VARIABLE NODE DEGREE DISTRIBUTION FOR CODES WITH RATE.377 λ 1, λ 1, TABLE II THE VARIABLE NODE DEGREE DISTRIBUTION FOR CODES WITH RATE.485 λ, λ, s encoded nto an n 1 -bt codeword. The code rate s k1 n 1.The code optmzaton problem s to maxmze the code rate when channel parameter σ 3,1 s gven. Codes wth rate.377 are found when σ 3,1 s 1.95 and the check node degree s 8. Note that the capacty rate s 1 3 whch can be determned by the equaton of the capacty of bnary-nput addtve whte Gaussan nose (BIAWGN) channels C BIAWGN (σ) = φ σ (y)log φ σ (y)dy where φ σ (y) = ( 1 8πσ 1 log (πeσ ) (8) e (y+1) σ ) + e (y 1) σ (9) s the probablty densty functon of receved sgnal Y.The optmzed varable node degree dstrbuton s shown n Table I. In the second optmzaton step, for the lnk between termnal node and the relay node, codes wth rate.485 are found when σ 3, s.979 and the check node degree s 8. The correspondng capacty rate s 1. The optmzed varable node degree dstrbuton s shown n Table II. Two codes, Code A and Code B, wth the length of 1 5 bts are randomly sampled from the above two degree dstrbutons. Code A (B) s sampled from the degree dstrbuton for rregular LDPC codes at node 1 (node ). Smulaton results are shown n Fgure 7 and Fgure 8, labeled as Code A, no relay, n=1 5 and Code B, no relay, n=1 5, respectvely. The waterfall curve gven n the fgure can be consdered as the case when no relay node exsts. For each code, t s smulated wth multple SNRs. The maxmum number of decodng teratons s. The correspondng bt error rate (BER) s presented n the logarthmc Y-axs. The equvalent SNRs for σ 3,1 =1.95 and σ 3, =.979 are represented by the vertcal lnes. In the thrd optmzaton step, n order to optmze the degree dstrbuton of systematc LDPC codes at the relay

8 ZHOU et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR HALF-DUPLEX THREE-PHASE TWO-WAY RELAY CHANNELS 1675 BER Code A, n=1 3 Code A, n=1 4 Code A, n=1 5 Code A, GF() addton,n=1 4 Code A, no relay, n=1 5 σ,1 =σ,3 = σ 3,1 = SNR (db) Fg. 7. Smulaton results when decodng code A at termnal node. BER Code B, n=1 3 Code B, n=1 4 Code B, n=1 5 Code B, GF() addton, n=1 4 Code B, no relay, n=1 5 σ 1, =σ 1,3 =1.549 σ 3, = SNR (db) Fg. 8. Smulaton results when decodng code B at termnal node 1. node, channel parameters σ 1,, σ 1,3, σ,1, σ,3 are gven. We consder the case when σ,1 = σ,3 = and σ 1, = σ 1,3 = The degree of upper layer check nodes s g 3,1 = g 3, =3. By teratve lnear programmng, the optmzed λ 3,3,1 s.3867 and the optmzed λ 3,1, λ 3, are shown n Table III and IV. Note that for the gven σ 1,, σ 1,3, σ,1, σ,3, the lower bound of λ 3,3,1 s 1 3. Irregular LDPC codes wth the lengths of 1 3, 1 4 and 1 5 bts and systematc LDPC codes wth the relay codeword lengths of , and bts are randomly sampled from the above degree dstrbutons. Smulaton results for decodng Code A and Code B at destnaton nodes are shown n Fgure 7 and Fgure 8, respectvely. The maxmum number of decodng teratons s 5. In the fgure, 1 the SNRs are defned as and 1 for the two decoders at σ1,3 σ,3 two destnaton nodes. The BER s defned as the rato of the total number of erroneous bts to the total number of bts n Group varable nodes for =1,. Inthssense,BERsare TABLE III THE VARIABLE NODE DEGREE DISTRIBUTION λ 3,1 λ 3,1, λ 3,1, TABLE IV THE VARIABLE NODE DEGREE DISTRIBUTION λ 3, λ 3,, λ 3,, evaluated at two destnaton nodes separately. The equvalent SNRs of σ,1 = σ,3 = and σ 1, = σ 1,3 =1.549 are represented by vertcal dashed lnes. As we can see, wth the help of the relay node, the requred SNRs are reduced from. db to -3.8 db and from -. db to -5.7 db, respectvely. In Fgure 7 and Fgure 8, we also provde the smulaton results for the case where two source codewords are added n GF() at the relay node. At the BER of 1 4, the requred SNR of the GF() addton case s around 1.5 db hgher than that of our proposed LDPC code constructon. In densty evoluton, t s assumed that ncomng messages of varable nodes and check nodes are ndependent. Ths assumpton mples that the bpartte graph has no cycles. However, cycles almost always exst. A natural queston s whether the actual densty s close to the densty n densty evoluton, especally when devaton s accumulated durng the teratve decodng. Ths queston can be emprcally answered by Fgure 7 and Fgure 8. As we can see, when the length of codewords grows from 1 3 to 1 5, the waterfall curve moves closer to the vertcal dashed lnes, whch shows that the requred SNR for a fnte-length code converges fast to that for cycle-free nfnte-length codes. For the gven optmzed degree dstrbutons, the evoluton of the BER under dscretzed densty evoluton s shown n Fgure 9. P 1 = P = P 3 = 1, σ 1, = σ 1,3 = and σ,1 = σ,3 = areused.thetwobercurves are monotoncally decreasng durng teratve decodng. The requred decodng teratons at two destnaton nodes are close to 1 and 3, respectvely. The decrease of the BER as a functon of the current BER s shown n Fgure 1. The crtcal pont [] s the pont where the decrease of the BER s a local mnmum. As shown n Fgure 1, the crtcal ponts of two codes at two destnaton nodes are close to.1 and.19, respectvely. VII. CONCLUSION In half-duplex three-phase two-way relay channels, codewords are broadcast and sgnals are receved from the source node and the relay node. In ths work, we constructed systematc LDPC codes at the relay node to encode two source codewords. At the destnaton node, sgnals from the source node and the relay node are used for ont decodng. We

9 1676 IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, VOL. 13, NO. 3, MARCH 14 BER Fg. 9. Δ BER Code A Code B Decodng Iteratons Evoluton of the bt error rate durng teratve decodng. Code A Code B BER Fg. 1. The decrease of the bt error rate as a functon of the current bt error rate. desgned the codes wth dscretzed densty evoluton and teratve lnear programng, and demonstrated that good codes can be found wthn our framework. For future work, we wll extend our work to fadng channels and wreless relay networks. REFERENCES [1] J. Laneman, D. Tse, and G. Wornell, Cooperatve dversty n wreless networks: effcent protocols and outage behavor, IEEE Trans. Inf. Theory, vol. 5, no. 1, pp , Dec. 4. [] T. Cover and A. Gamal, Capacty theorems for the relay channel, IEEE Trans. Inf. Theory, vol. 5, no. 5, pp , Sept [3] B. Rankov and A. Wttneben, Achevable rate regons for the two-way relay channel, n Proc. 6 IEEE Int. Symp. Inf. Theory, pp [4] L.-L. Xe, Network codng and random bnnng for mult-user channels, n Proc. 7 Canadan Workshop Inf. Theory, pp [5] T. Oechterng, C. Schnurr, I. Belakovc, and H. Boche, Broadcast capacty regon of two-phase bdrectonal relayng, IEEE Trans. Inf. Theory, vol. 54, no. 1, pp , Jan. 8. [6] A. Chakrabart, A. D. Baynast, A. Sabharwal, and B. Aazhang, Low densty party check codes for the relay channel, IEEE J. Sel. Areas Commun., vol. 5, no., pp. 8 91, Feb. 7. [7] J. Hu and T. Duman, Low densty party check codes over wreless relay channels, IEEE Trans. Commun., vol. 6, no. 9, pp , Sept. 7. [8] P. Razagh and W. Yu, Blayer low-densty party-check codes for decode-and-forward n relay channels, IEEE Trans. Inf. Theory, vol. 53, no. 1, pp , Oct. 7. [9] S. Zhang, S.-C. Lew, and P. P. Lam, Hot topc: physcal-layer network codng, n Proc. 6 Int. Conf. Moble Comput. Netw., pp [1] S. Zhang and S.-C. Lew, Channel codng and decodng n a relay system operated wth physcal-layer network codng, IEEE J. Sel. Areas Commun., vol. 7, no. 5, pp , June 9. [11] B. Nazer and M. Gastpar, Compute-and-forward: harnessng nterference through structured codes, IEEE Trans. Inf. Theory, vol. 57, no. 1, pp , Oct. 11. [1] D. Wübben and Y. Lang, Generalzed sum-product algorthm for ont channel decodng and physcal-layer network codng n two-way relay systems, n Proc. 1 IEEE Globecom, pp [13] J. Lu, M. Tao, and Y. Xu, Parwse check decodng for LDPC coded two-way relay block fadng channels, IEEE Trans. Commun., vol. 6, no. 8, pp , Aug. 1. [14] C. Hausl and J. Hagenauer, Iteratve network and channel decodng for the two-way relay channel, n Proc. 6 IEEE Int. Conf. Commun., vol. 4, pp [15] T. Cu, F. Gao, T. Ho, and A. Nallanathan, Dstrbuted space tme codng for two-way wreless relay networks, IEEE Trans. Sgnal Process., vol. 57, no., pp , Feb. 9. [16] X. Zhou, L.-L. Xe, and X. Shen, Low-densty party-check codes for two-way relay channels, n Proc. 1 IEEE Veh. Technol. Conf. Fall, pp [17] J. Garca-Fras and W. Zhong, Approachng Shannon performance by teratve decodng of lnear codes wth low-densty generator matrx, IEEE Commun. Lett., vol. 7, no. 6, pp , June 3. [18] M. Luby, LT codes, n Proc. IEEE Symp. Foundatons Comput. Scence, pp [19] J. W. Byers, M. Luby, M. Mtzenmacher, and A. Rege, A dgtal fountan approach to relable dstrbuton of bulk data, n Proc ACM SIGCOMM, vol. 8, no. 4, pp [] R. M. Tanner, A recursve approach to low complexty codes, IEEE Trans. Inf. Theory, vol. 7, pp , Sept [1] T. Rchardson and R. Urbanke, The capacty of low-densty partycheck codes under message-passng decodng, IEEE Trans. Inf. Theory, vol. 47, no., pp , Feb. 1. [] T. Rchardson, M. Shokrollah, and R. Urbanke, Desgn of capactyapproachng rregular low-densty party-check codes, IEEE Trans. Inf. Theory, vol. 47, no., pp , Feb. 1. [3] S.-Y. Chung, T. Rchardson, and R. Urbanke, Analyss of sum-product decodng of low-densty party-check codes usng a Gaussan approxmaton, IEEE Trans. Inf. Theory, vol. 47, no., pp , Feb. 1. [4] S.-Y. Chung, G. Forney, T. Rchardson, and R. Urbanke, On the lowdensty party-check codes wthn.45 db of the Shannon lmt, IEEE Commun. Lett., vol. 5, no., pp. 58 6, Feb. 1. Xnsheng Zhou receved the B.Eng. degree n communcaton engneerng from Shangha Unversty, Shangha, Chna, n 1998, the M.A.Sc. degree n electrcal and computer engneerng from Unversty of Alberta, Edmonton, Canada, n 8 and the Ph.D. degree n electrcal and computer engneerng from Unversty of Waterloo, Waterloo, Canada, n 13. Hs research nterests are n codng theory and wreless networks. Lang-Lang Xe (IEEE M 3-SM 9) receved the B.S. degree n mathematcs from Shandong Unversty, Jnan, Chna, n 1995 and the Ph.D. degree n control theory from the Chnese Academy of Scences, Beng, Chna, n He dd postdoctoral research wth the Automatc Control Group, Lnköpng Unversty, Lnköpng, Sweden, durng and wth the Coordnated Scence Laboratory, Unversty of Illnos at Urbana-Champagn, durng -. He s currently a Professor at the Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Waterloo, ON, Canada. Hs research nterests nclude wreless networks, nformaton theory, adaptve control, and system dentfcaton.

10 ZHOU et al.: DESIGN OF LOW-DENSITY PARITY-CHECK CODES FOR HALF-DUPLEX THREE-PHASE TWO-WAY RELAY CHANNELS 1677 Xuemn (Sherman) Shen (IEEE M 97-SM - F9) receved the B.Sc.(198) degree from Dalan Martme Unversty (Chna) and the M.Sc. (1987) and Ph.D. degrees (199) from Rutgers Unversty, New Jersey (USA), all n electrcal engneerng. He s a Professor and Unversty Research Char, Department of Electrcal and Computer Engneerng, Unversty of Waterloo, Canada. He was the Assocate Char for Graduate Studes from 4 to 8. Dr. Shen s research focuses on resource management n nterconnected wreless/wred networks, wreless network securty, socal networks, smart grd, and vehcular ad hoc and sensor networks. He s a co-author/edtor of sx books, and has publshed more than 6 papers and book chapters n wreless communcatons and networks, control and flterng. Dr. Shen served as the Techncal Program Commttee Char/Co-Char for IEEE Infocom 14, IEEE VTC 1 Fall, the Symposa Char for IEEE ICC 1, the Tutoral Char for IEEE VTC 11 Sprng and IEEE ICC 8, the Techncal Program Commttee Char for IEEE Globecom 7, the General Co-Char for Chnacom 7 and QShne 6, the Char for IEEE Communcatons Socety Techncal Commttee on Wreless Communcatons, and PP Communcatons and Networkng. He also serves/served as the Edtor-n-Chef for IEEE Network, Peer-to-Peer Networkng and Applcaton, andiet Communcatons; a Foundng Area Edtor for IEEE TRANSACTIONS ON WIRELESS COMMU- NICATIONS; an Assocate Edtor for IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Computer Networks, andacm/wreless Networks, etc.; and the Guest Edtor for IEEE JSAC, IEEE Wreless Communcatons, IEEE Communcatons Magazne, andacm Moble Networks and Applcatons, etc. Dr. Shen receved the Excellent Graduate Supervson Award n 6, and the Outstandng Performance Award n 4, 7 and 1 from the Unversty of Waterloo, the Premer s Research Excellence Award (PREA) n 3 from the Provnce of Ontaro, Canada, and the Dstngushed Performance Award n and 7 from the Faculty of Engneerng, Unversty of Waterloo. Dr. Shen s a regstered Professonal Engneer of Ontaro, Canada, an IEEE Fellow, an Engneerng Insttute of Canada Fellow, a Canadan Academy of Engneerng Fellow, and a Dstngushed Lecturer of IEEE Vehcular Technology Socety and Communcatons Socety.