Jointly optimized rate-compatible UEP-LDPC codes for half-duplex co-operative relay networks

Transcription

1 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 RESEARCH Open Aess Jontly optmze rate-ompatble UEP-LDPC oes for half-uplex o-operatve relay networks Rabullah Khattak 1* an Sara Sanberg 2,3 Abstrat Ths paper aresses the esgn of low-ensty party-hek (LDPC) oes for half-uplex o-operatve relay networks. Struture rate-ompatble oes wth unequal error proteton (UEP) are esgne through ont optmzaton of the oes for the hannels between soure an relay an soure an estnaton. The propose oes learly outperform smpler LDPC oes whh are not optmze for relay hannels an punturng-base rate-ompatble LDPC oes, an they show a sgnfant performane mprovement over the ret lnk ommunaton epenng on the poston of relay. The optmzaton algorthm for the propose oes s base on ensty evoluton usng the Gaussan approxmaton an optmal varable noe egree strbutons are foun through teratve lnear programmng. Interestngly, they anyhow show performane whh s almost omparable to the performane of oes optmze through a more omplex non-lnear optmzaton algorthm. We analyze the performane of our propose oes wth short, meum an long blok lengths, an wth low an hgh rates uner realst assumptons,.e., mperfet eong of the oewor at relay an varant sgnal-to-nose rato wthn a sngle oewor. Keywors: LDPC oe esgn; Half-uplex relay; Lnear programmng 1 Introuton Co-operatve relay networks have opene a new researh fel n wreless ommunaton. They an be use n moble ellular networks, wreless a ho networks, an wreless sensor networks. They prove spatal versty beause the termnals n a relay network form vrtual multple antenna systems. Lke multple antenna systems, relay networks nrease hannel apaty uner the same banwth onstrants an wth no atonal power onsumpton. A smple o-operatve relay network s ompose of three noes, soure (S), relay (R), an estnaton (D), as shown n Fgure 1. S s transmttng ata to D va a ret lnk (S to D) an a relay lnk (S to D through R). There are three relay hannels n a smple relay network,.e., the hannels between S an D, betweens an R, an between R an D. The relay s use to nrease the relablty an ata rate when the hannel between S an D s ba ue to fang an nterferene. *Corresponene: rabullah.khattak@ltu.se 1 Department of Computer Sene, Eletral an Spae Engneerng, Luleå Unversty of Tehnology, Luleå , Sween Full lst of author nformaton s avalable at the en of the artle In ths paper, we propose low-ensty party-hek (LDPC) oe esgn for half-uplex relay networks base on rate ompatblty an unequal error proteton (UEP) apabltes. The oe s esgne by optmzng the varable noe egree strbuton usng ensty evoluton uner Gaussan approxmaton [1]. We esgn rateompatble LDPC oes wth UEP usng sub-egree strbutons both for lasses of bts wth fferent proteton (enote by proteton lasses) an for lasses of bts transmtte over fferent relay hannels. Heneforth, the bts transmtte over one spef relay hannel wll be referre to as belongng to one hannel lass. The separaton of the varable noe egree strbuton nto sub-egree strbutons sgnfantly nreases the number of esgn parameters. Therefore, t s mportant to note that the oe esgn s solve by teratve lnear programmng (LP), whh enables an effent optmzaton of the sub-egree strbutons. Our oe esgn s base on the esgn metho for UEP-LDPC oes wth hgher orer moulatons suggeste n [2], whh employs teratve LP, an the rate-ompatble oe struture suggeste n [3] Khattak an Sanberg; lensee Sprnger. Ths s an Open Aess artle strbute uner the terms of the Creatve Commons Attrbuton Lense ( whh permts unrestrte use, strbuton, an reprouton n any meum, prove the orgnal work s properly te.

2 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 2 of 16 Fgure 1 Layout of the noes n a relay network. Hgher orer moulatons ause fferent bts n the oewor to have fferent sgnal-to-nose ratos (SNRs) n the same way as relay hannels wth fferent hannel gans ause fferent oewor bts to be reeve wth fferent SNRs. In our esgn, we use the strategy propose n [2] to optmze the oes for the ase where two fferent oewor segments experene two fferent SNRs ue to fferent hannel gans of the S to D an the R to D relay hannels. That s, our propose optmzaton algorthm onsers fferent SNRs of the relay hannels n the oe esgn, lke n [3]. The performane of the soure to relay hannel plays an mportant role n the overall performane of o-operatve relay networks [4,5]. It s ponte out n [3] that oe extenson tehnques prove better performane than oe punturng n the soure to relay hannel. In oe extenson tehnques, optmal oewors an be transmtte over the soure to relay hannel to get the best performane. In oe punturng tehnques, only punture oewors an be transmtte over the soure to relay hannel, whh results n a ertan performane loss. Therefore, we esgn the rate-ompatble LDPC oes for relay networks base on the extenson tehnque. In [3], LDPC oes for relay networks are esgne by fnng optmal varable an hek noe egree strbutons usng fferental evoluton, whh s a nonlnear oe optmzaton metho. In the esgn propose here, lnear programmng s use to fn an optmal varable noe egree strbuton. Our oe optmzaton algorthm requres less omputatonal effort than that of [3] an the performane s anyhow smlar to the performane of the oes propose n [3]. Lke n [3], ont optmzaton of the oes for the soure to relay an the soure to estnaton hannels s performe, an varant SNR wthn a sngle oewor s onsere n the esgn. In [3], t s assume that the oewor reeve at the relay from the soure s perfetly eoe at the relay. However, n ths paper, t s not assume that the oewor s perfetly eoe at the relay, whh s a more realst assumpton. In [6], the authors have ompare two fferent LDPC ong shemes from [7] an [8] wth ther propose punturng-base rate-ompatble LDPC oes. The oe esgn propose here s fferent from the esgns propose n [3] an [6] beause t proves UEP propertes to the oe,.e., more mportant bts are mae more protete by alloatng them hgher varable noe egrees. The ong sheme n [6] has smple oe esgn as t requres the esgn of only one sngle-user LDPC. In ontrast to the ong sheme n [6], our propose ong sheme requres the ont esgn of two oes. However, our propose oe shows better performane than the oe propose n [6] beause the oe n [6] has performane loss ue to punturng n the o-operatve relay network. Most of the papers n the lterature on relay networks onser a theoretal approah of oe esgn. However, we fous on oe esgn wth realst assumptons, suh as mperfet eong of the oewor n the relay termnal, lke n [9] an varant SNR wthn a sngle oewor, lke n [3]. In [9], quas-yl LDPC (QC-LDPC) oes are use n relay networks through repetton ong shemes. In the oe esgn, the SNR s onsere nvarant wthn a sngle oewor. The fat that [9] uses QC-LDPC oes makes them easer to mplement an thereby more realst, but on the other han, they also get worse performane. Usng the same tme slots, transmsson power, an proessng of the oewors n the relay termnal wth the eoe an forwar strategy as n [9], the oes propose here aheve up to 1.9-B gan over the QC-LDPC oes epenng on the loaton of the relay wth respet to the soure an the estnaton. The better performane s partly ue to the rate-ompatble ong sheme an partly ue to the UEP propertes of the oe. The rate-ompatble ong sheme proves a better trae-off between bt error rate an ata rate than the repetton ong sheme. The UEP proves hgh proteton to the oewor bts transmtte over the soure to relay hannel, whh results n a better performane of the mportant soure to relay hannel. Also, n ontrast to [9], the propose oes are optmze for eah poston of the relay. Depenng on the loaton of the relay relatve to the soure an the estnaton, the propose low-rate oes wth short, meum an long blok lengths show a performane mprovement of up to 3.8-B gan over the ret lnk ommunaton. The propose hgh-rate oe wth meum blok length has a 1.3-B gan over the ret lnk ommunaton. The remaner of ths paper s organze as follows. Seton 2 presents an overvew of the bakgroun an the system moel of a half-uplex relay network an ntroues the LDPC ong sheme. Seton 3 gves the etals

3 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 3 of 16 of the optmzaton algorthm for the esgn of rateompatble LDPC oes wth UEP propertes for halfuplex relay networks. Seton 4 esrbes the seleton of a goo hek noe egree strbuton an susses the smulaton results of our propose oe esgn. The results are also ompare to the smulaton results for QC- LDPC oes [9] as well as the oes propose n [3] an [6]. Seton 5 onlues ths paper. 2 Bakgroun an system moel 2.1 Half-uplex operaton of relay an types of relay In ths paper, the relay s assume to operate n halfuplex moe. Full-uplex operaton of the relay s har to mplement from a pratal system pont of vew beause t s ffult to solate the several orers of magntue hgher transmtte sgnal from the reeve sgnal n the same frequeny ban. Half-uplex operaton s easy to mplement by solatng the transmtte an the reeve sgnal n tme or frequeny. We onser tme vson multple aess (TDMA) usng two tme slots. In the frst tme slot t (normalze), known as broaast moe (BC), S transmts to both R an D. In the seon tme slot t = 1 t, known as multple aess moe (MAC), ether only R or both R an S transmt to D as shown n Fgure 2. In ths paper, only R transmts to D n MAC moe. There exst fferent types of o-operatve relay strateges base on how the reeve sgnal s proesse by the relay noe. The two most ommon types of o-operatve relay strateges are amplfy an forwar (AF) an eoe an forwar (DF). The AF relay [10] amplfes the reeve sgnal an then retransmts t to the estnaton. The DF relay frst eoes the sgnal reeve from the soure an then retransmts the eoe sgnal after re-enong, ether wth the same oe as use by the soure or wth a fferent oe. The rawbak of the AF relay s that nose s also amplfe along wth the esre sgnal. In power lmte ommunaton networks, the AF relay s preferre as t performs no eong an re-enong of the reeve sgnal. The rawbak of the DF relay s the large elay ue to eong an re-enong as ompare to the AF relay. However, the DF relay an also be mplemente wthout re-enong [11]. In ths work, the DF relay s onsere. 2.2 Relay hannel moel Fgure 3 shows the relay system moel as suggeste n [12]. ThestanebetweenS an D s normalze to unty. R s loate at a stane of from S an 1 from D. The relatve stane moel n Fgure 3 s apple wth normalze stane, where0 < < 1. Note that all the half-uplex relay hannels are AWGN hannels wth fxe path loss. The sgnals reeve at D an R n BC an MAC moes are as follows: y R1 = h SR x S1 + n R1 (1) y D1 = h SD x S1 + n D1 (2) y D2 = h RD x R2 + n D2. (3) The sgnals transmtte by S an R are enote by a x S an x R,respetvely,where {1, 2} an 1 an 2 stans forbcanmacmoes,respetvely.thereevesgnals at R an D are represente by y R an y D,respetvely. In the above equatons, h SR, h SD an h RD are the hannel realzatons between {S, R}, {S, D} an {R, D}, respetvely. n R an n D are AWGN at the R an D reevers wth zero means an varane σ 2 = N 0 /2 n eah menson. The normalze hannel gans of the {S, R}, the{s, D}, anthe {R, D} hannels are gven as γ SD = h SD 2 = 1 γ SR = h SR 2 = 1/ τ (4) γ RD = h RD 2 = 1/(1 ) τ, where τ s the path loss exponent whh normally ranges from 2 to 5 [13]. Furthermore, BPSK moulaton an full hannel state nformaton (CSI) avalable at eah termnal of the relay network are assume. In orer to farly ompare the relay hannel wth the ret lnk, we assume that the sum of the R an S transmt powers n the relay hannel s equal to the ret lnk transmt power of S. Therefore, the followng global power onstrant s assume: tp S1 + tp R P. (5) P S1 s the soure transmt power n BC moe, P R s the relay transmt power n MAC moe an P s the total system transmt power. Fgure 2 Half-uplex relay network operatons n BC an MAC moes.

4 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 4 of 16 Fgure 3 Poston of the relay (R) relatve to the soure (S)an the estnaton (D). 2.3 Cong strategy for half-uplex relays The propose relay ong strategy s smlar to [3] exept that t s assume that S s slent n MAC moe an perfet eong of the oewor at R s not assume. Aorng to ths relay ong strategy, a oewor w 1 of length N 1 s transmtte from S to R an D n BC moe,.e., x S1 = P S1 w 1. R eoes the oewor w 1 an generates the extra party bts w e of length N 2 N 1. D stores w 1 (reeve verson of w 1 ) untl t reeves w e from R. Note that w e transmtte from R may ontan errors f w 1 s not eoe orretly at R. InMACmoeS s slent, so only R transmts w e to D,.e., x R2 = P R w e. D onatenates w 1 wth w e to get the extene oewor w 2 = [ w 1 w e ]of length N 2 belongng to the extene oe C 2. w 2 s then eoe at D to reover the orgnal nformaton. From (2) an (3), t follows that the sgnals reeve at D n BC an MAC moes an be expresse respetvely as y D1 = h SD PS1 w 1 + n D1 (6) y D2 = h RD PR w e + n D2. (7) 2.4 Bref ntrouton to LDPC oes LDPC are lassfe as lnear blok oes. They are esrbe by party hek equatons an are represente by a sparse party-hek matrx H. LDPC oes are eoe by a message passng algorthm [14]. The eong omplexty nreases lnearly wth oewor length an also epens on the sparseness of H. Thepartyhek matrx an be represente n graphal form by a bpartte graph, alle Tanner graph [15]. The operaton of the message passng algorthm s realy explane usng a Tanner graph. The two types of noes n the Tanner graph are varable noes an hek noes, whh orrespon to oewor bts an party-hek equatons, respetvely. The number of varable noes (oewor bts) s enote by n an the number of hek noes (party-hek equatons) s enote by (n k) where k s the number of the nformaton bts. A varable noe s onnete to a hek noe by an ege f the bt orresponng to the varable noe s nlue n the partyhek equaton orresponng to the hek noe. Thus, thenumberofegesnthetannergraphsequaltothe number of 1s n the party-hek matrx H. The oe rate of LDPC oes s efne by k/n. LDPCoesarelassfeasregularorrregularbaseontheegreesofthe noes.theegreeofanoesthenumberofegesonnete to t. The regular oes have varable noes of fxe egree an ther hek noes have another smlar fxe egree. The rregular oes have varable an hek noes wth varyng egrees. The varable an hek noe egree strbutons of rregular oes from an ege perspetve are represente by λ(x) = vmax =2 λ x 1 an ρ(x) = max =2 ρ x 1,respetvely,where vmax s the maxmum varable noe egree an max s the maxmum hek noe egree of the oe [16]. The oeffents λ an ρ esrbe the fraton of eges that are onnete to egree- varable an hek noes, respetvely. The optmal egree strbuton of rregular oes an be foun by ensty evoluton (DE) usng the Gaussan approxmaton [1]. DE traks mutual nformaton exhange between varable an hek noes n the message passng eong algorthm. 2.5 Rate-ompatble UEP-LDPC oes for half-uplex relays From an mplementaton pont of vew, the smplest oe esgn sheme for half-uplex relays s the repetton ong sheme,.e., transmttng the same oewor w 1 of oe C 1 n BC an MAC moes[9]. However, the repetton ong sheme s not preferre as t proves a poor trae-off between bt error rate an ata rate. One alternatve s to use ong shemes base on nremental reunany,.e., transmttng the oewor w 1 n BC moe an extra party bts w e beng extrate from w 1, n MAC moe [3]. Ths yels the extene oewor w 2 = [w 1 w e ] belongng to the extene oe C 2.Inths ase, a ont esgn of the mother oe C 1 an extene oe C 2 s requre. The nremental reunany oul also be aheve usng punturng,.e., eletng a number of party bts from w 2 to get w 1. Coe esgn n ths ase s the ont esgn of the mother oe C 2 an the punture oe C 1. The rate-ompatble struture of the party-hek matrx H 2 representng the oe C 2, whh follows the struture propose n [3], s llustrate n Fgure 4. H 2 s ompose of a non-zero party hek matrx H 1 orresponng to the oe C 1, a zero sub-matrx O an non-zero sub-matres A an B. The zero sub-matrx preserves the party-hek equatons of the mother oe C 1.Therateompatble esgn of C 2 s fferent from [17-19] as w 1 an w e are n general reeve wth fferent SNRs. Reeve SNRs for eong w 1 an w e are gven later by (8) an (9), respetvely. We ntroue UEP along wth rate-ompatblty n LDPC oes n orer to prove better proteton to the oewor bts whh are transmtte n BC moe than to the extra party bts transmtte n MAC moe, sne the overall performane of relay networks epens largely on

5 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 5 of 16 Fgure 4 Struture of the rate-ompatble party-hek matrx H 2 for the half-uplex relay oe C 2. H 2 s ompose of H 1, whh s a non-zero party-hek matrx orresponng to the oe C 1, a zero matrx O an non-zero matres A an B. the soure to relay hannel performane [4,5]. Through UEP, varable noes of hgher egrees are assgne to the w 1 segment of the oe C 2 n orer to gve t hgher proteton than ts w e segment. 3 Desgn of RC UEP-LDPC oes for relay networks 3.1 Notatons The oewor bts of oe C z where z {1, 2} are ve nto N C z proteton lasses n orer to gve t UEP propertes. The proportons α C z = [α C z 1,, αc z ] enote N 1 the normalze lengths of eah lass orresponng to the nformaton bts u. α C z equals the number of bts belongng to proteton lass C ve by the total number of nformaton bts k. p = [α C z 1 R z,, α C z R z, (1 R N z )] 1 efnes proportons of the bts n the oewor belongng to fferent proteton lasses an R z stheoerateof C z. N C z s s the number of relay hannels (hannel lasses) between soure, relay an estnaton assoate wth C z. The bt segments of the oewor beng reeve through fferent relay hannels have fferent SNRs. The bt segment wth a stnt SNR wll be lassfe as belongng to one hannel lass M, = 1,, N C z s. The proportons of bt segments n the oewor belongng to fferent hannel lasses are efne by β C z = [β C z 1,, βc z ]. Ns The vetors λ C z an a C z ontan the overall varable noe egree strbuton from an ege perspetve an a noe perspetve, respetvely, both for fferent proteton lasses an fferent hannel lasses for a oewor orresponng to oe C z.let M, an a C z,c k M, be the proporton of eges an noes, respetvely, onnete to varable noes of egree that belong to hannel lass M, proteton lass C k,anoec z. Smlarly, b C z s the proporton of hek noes of egree that belong to oe C z.defne M [λ C z,c 1 M 1 T C z,c T N,, λ M 1 = [ M,2,, M, vmax,, λ C z,c 1 M N s ] T an λ C z = T,, λ C z,c N T M N s where ( ) T enotes the transpose. M s a olumn vetor of length C z 1 an λ C z s a olumn vetor of length ( C z 1)N C z s N C z. The vetors ρ C z = [ρ C z 2,..., ρc z ] T an b C max z esrbe the hek noe egree strbuton from an ege an a noe perspetve, respetvely. Note that C z an C z max are the maxmum varable an hek noe egrees respetvely n the oe C z. The vetor wth all-ones s efne by 1 wth sutable length. 3.2 Proteton an hannel lasses Fgure 5 shows a agram of the propose ong sheme. TheRCUEP-LDPCoehasN C z proteton lasses at ts output. The nformaton bts u are ve nto N C z 1 proteton lasses C 1 C N 1.Thepartybtsare assgne to C N. The proteton lasses are orere wth esenng orer of proteton,.e., C 1 has the hghest proteton an C N has the lowest proteton. The bts from the fferent proteton lasses are re-multplexe an assgne to hannel lasses M 1,, M N orresponng to the fferent relay hannels. In the half-uplex s o-operatve relay networks wth three termnals, there are two relay hannels through whh estnaton reeves the sgnals from the relay an the soure, one between ] T Fgure 5 Dagram of the propose sheme. The bts are enoe by the rate-ompatble UEP-LDPC oe an the oe bts are assgne to the relay hannels.

6 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 6 of 16 soure an estnaton M 1, an the other between relay an estnaton M 2,seeFgure SNRs of the relay hannels Ths subseton esrbes omputaton of SNRs of the relay hannels n BC an MAC moes. The sgnals reeve at D n BC an MAC moes have fferent reeve SNRs. The SNRs of the sgnals reeve n BC an MAC moes are efne, respetvely, as SNR BC = h SD 2 P S1 N 0 (8) SNR MAC = h RD 2 P R. (9) N 0 In [2], unty sgnal power s assume for smplty, an equvalent nose varanes are alulate for the fferent bts n the hgher orer onstellaton. To follow the oe esgn propose n [2], the equvalent nose varanes of the relay hannels n BC an MAC moes are requre for the optmzaton of the varable noe egree strbutons. Therefore, the equvalent nose varanes of the relay hannels n BC an MAC moes are etermne from fferent SNRs by means of an equvalent hannel wth unt reeve power,.e., SNR = 1 N 0 = 1 N 0 SNR. (10) The equvalent nose varane s expresse as σ 2 N = 0 2 σ 2 = 1 2SNR. (11) The equvalent nose varanes of the relay hannels n BC an MAC moes are σ BC 2 = 1 (12) 2SNR BC σ MAC 2 = 1. (13) 2SNR MAC The nose varanes n (12) an (13) are requre for ensty evoluton n the propose optmzaton algorthm n [2]. For the propose optmzaton algorthm n [2], we efne the nose vetor σ 2 =[ σ 1 2,, σ 2 ]tobeavetor Ns that ontans the equvalent nose varanes of eah relay hannel n BC an MAC moes, orere wth the hghest varane frst. 3.4 Optmzaton algorthm The optmzaton algorthm presente here s the aaptaton of the optmzaton algorthm propose n [2] for half-uplex o-operatve relay networks. We a an atonal onstrant to the optmzaton algorthm of [2] for the esgn of rate-ompatble LDPC oes as esrbe n [3]. In the optmzaton algorthm, the oe C 2 for half-uplex relay network s esgne to be rateompatble wth UEP base on the gven egree strbutons of the UEP apable oe C 1.CoeC 1 s frst esgne wthout the rate-ompatble onstrant n the optmzaton algorthm. Then, the rate-ompatble oe C 2 s esgne gven the varable an hek noe egree strbutons of C 1. The target of the optmzaton algorthm propose n [2] s to fn a varable noe egree strbuton for the whole oe that maxmzes the average varable noe egree of the lass beng optmze. UEP propertes of the oe are aheve by sequental exeuton of the optmzaton algorthm of [2], one proteton lass at a tme an startng wth the best protete lass for an E b /N 0 slghtly hgher than the threshol. In every sequental exeuton step, the optmzaton algorthm keeps the egree strbutons of lower proteton lasses fxe an may hange only the egree strbutons of hgher proteton lasses [2]. The obetve funton for proteton lass C k an be expresse as N max λ =1 =2 M,. (14) The optmzaton algorthm wth obetve funton (14) maxmzes the average varable noe egree. For maxmzaton of the mnmum varable noe egree, the optmzaton s ntate wth a hgh mnmum varable noe egree an erease to a lower value untl a varable noe egree strbuton for whh ensty evoluton onvergene s foun, see [2] for etals Rate-ompatble onstrants The optmzaton algorthm n [3] esgns the oe C 2 orresponng to H 2, by onserng the rate-ompatble onstrants for varable an hek noe egree strbutons gven the egree strbutons of the oe C 1. However, the optmzaton algorthm n ths paper esgns the oe C 2 by onserng only the rate-ompatble onstrant for thevarablenoeegreestrbuton[3](expresseby (19)) gven the egree strbutons of the oe C 1.The hek noe egree strbuton of C 2 s assume to be gven an must satsfy the rate-ompatble onstrants for the hek noe egree strbuton [3] (expresse by (17)) before usng t n the optmzaton algorthm to etermne an optmal varable noe egree strbuton for C 2. The rate-ompatble onstrants for the varable noe egree strbuton from [3] have been aapte to over the UEP propertes of our propose oe. For C 2, the overall ege strbutons λ C 2 an an the overall noe strbutons a C 2 an b C 2 must satsfy the followng general onstrants from [3] whh have been aapte to over the UEP propertes of our propose oe:

7 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 7 of 16 λ C 2,C k M,2 λ C 2,C k M = C, 2 vmax = 1 Ns C2 =1 max 2 = 1 C2 a C 2,C k M, = b C 2 = =3 R 2 max C2 2 =3 λc 2,C k M,2 ( ) ρ C 2 N C 2 k=1, C 2 vmax =3 λ C 2,C k M,, ( ) (1 R2 ) N C 2 s / / N C 2 s =1 C 2 max =2 =1 ( (1 R 2 ) 1/( C 2 N C 2 k=1. N C 2 k=1 C 2 vmax 1 =3 ( ) 1 2 ) ), C 2 vmax =2 λ C 2,C k M,, (15) The hek noe egree strbutons of C 1 from ege an noe perspetve {ρ C 1, b C 1} beome part of the hek noe egree strbutons of C 2 from ege an noe perspetve {, b C 2}, see Fgure 4. The hek noe egree strbutons from noe perspetve of C 1 an C 2 shoul satsfy the followng onstrants: b C 2 W 2 b C 1 W 1 = 2, 3,, C 1 max, b C 2 W 1 W 2 b C 1 = (1 R 1)N 1 (1 R 2 )N 2 b C 1 = b C 1, (16) where W z s the number of rows n party-hek matrx H z an = (1 R 1)N 1 (1 R 2 )N 2 = (1 R 1)R 2 (1 R 2 )R 1 ; the frst equalty s erve from 1 R z = W z N z an the seon equalty s erve from the rate-ompatble feature of H 2,.e., R 1 N 1 = N 1 W 1 = N 2 W 2 = R 2 N 2. Convertng the hek noe perspetve onstrant (16) nto ege perspetve usng (15) gves ( ) ρ C 2 C 2 max / =2 b C 1, = 2, 3, C 1 max, ρ C2 b C1 C 2 max =2 ρ C2 = b C C 2 max =3 ( 1 ρ C2 1 ). (17) 2 Smlarly,thevarablenoeegreestrbutonsfromthe noe perspetve of C 1 an C 2 shoul satsfy the followng onstrants: N C 2 N C 2 C 2 =1 k=1 =l N C 2 N C 2 a C 2,C k M, N 2 C 2 =1 k=1 =l a C 2,C k M, N 1 N 2 N C 1 N C 1 C 1 =1 k=1 =l = N C 1 N C 1 a C 1,C k M, N 1, C 1 =1 k=1 =l N C 1 N C 1 C 1 =1 k=1 =l a C 1,C k M, a C 1,C k M,, l = 2, 3,, C 1 an = R 2 R 1 = N 1 N 2. (18) Convertng the noe perspetve onstrant (18) nto ege perspetve usng (15) gves N C 2 N C 2 C 2 =1 k=1 =l N C 2 s =1 N C 2 k=1 λ C 2,C k M, N C 2 N C 2 C 2 vmax λ C 2,C k M, =2 C 2 =1 k=1 =l λ C 2,C k M, = = N C 1 N C 1 C 1 =1 k=1 =l N C 1 N C 1 C 1 =1 k=1 =l N C 1 N C 1 C 1 =1 k=1 =l N C 1 N C 1 C 1 =1 k=1 =l a C 1,C k M,, l = 2, 3,, C 1, a C 1,C k M, a C 1,C k M, a C 1,C k M, N C2 N C 2 C 2 =1 k=1 =l C 2 max =2, 1 R 2 λ C 2,C k M, C 2 max =3 ρc 2 ( ). 1 R 2 (19)

8 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 8 of Convergene onstrant The optmal varable noe egree strbutons are foun through ensty evoluton usng the Gaussan approxmaton [1]. The mutual nformaton messages from a hek noetoavarablenoe(x v ) an from a varable noe to a hek noe (x v )atteratonl are gven by an x (l 1) v = 1 N x (l) v = N max =2 =1 k=1 =2 ρ C z J(( 1)J 1 (1 x v (l 1) )), (20) M, J( 2 σ 2 + ( 1)J 1 (x v (l 1) )), (21) respetvely. Here, J( ) s the mutual nformaton, gven by J(m) = 1 E{log 2 (1 + e q )}, = 1 1 log 2 (1 + e q ) e (q m)2 4m q 4πm R (22) for a Gaussan ranom varable q N (m,2m) wth mean m an varane 2m. The mutual nformaton evoluton an be wrtten as a ombnaton of (20) an (21) x (l) v = F(λC z, ρ C z, σ 2, x v (l 1) ). (23) The λ C z an ρ C z esrbe the egree strbutons of the oe for a gven σ 2 when the onton x (l) v > x v (l 1) for any x v (l 1) s satsfe Proporton strbuton onstrants Thevarablenoeegreestrbutonsonstraneto have the total sum of the fratons equal to one,.e., N N =1 k=1 =2 M, = 1. (24) λ C z s also onstrane by the proporton vetors α C z an β C z.thetotalnumberofvarablenoesn Ck an n M belongng to proteton lass C k an to hannel lass M s n Ck = α C z k R z n k = 1,, N C z 1, (25) an n M = β C z n = 1,, N C z s, (26) respetvely. The oe rate an be expresse as R z = 1 max =2 ρ C z /. (27) vmax =2 λ C z / Furthermore, λ C z an be relate to n Ck an n M by N vmax M, max ρ C z n Ck = n (1 R z )/, an n M = =1 N k=1 respetvely. =2 =2 M, =2 max n (1 R z )/ =2 ρ C z, (28) (29) Stablty onton The optmal egree strbuton must satsfy the stablty onton [16]. For the ret lnk hannel wth BPSK, all the oewor bts experene the same nose varane σ 2. The stablty onton for the ret lnk hannel s 1 λ C z (0)ρ C z (1) > e r = R P 0 (x)e x 2 x = e 1 2σ 2. (30) In (30), P 0 (x) s the reeve message ensty an λ C z (x) an ρ C z (x) are the ervatves of the egree strbutons. The stablty onton gves an upper boun on the number of egree-2 varable noes. In ase of ooperatve relay networks, the oewor bts experene fferent nose varanes σ 2 ue to fferent SNRs of the relay hannels an there are fferent enstes P 0, (x). The ervatves are λ C z (0) = Ns =1 an N k=1 λc z,c k M,2 ρ C z (1) = max m=2 ρc z m (m 1). The stablty onton for the o-operatve relay network an be expresse as e r = N R =1 N β C z P 0 (x)e 2 x x = =1 β C z 1 2σ e 2. (31) Smlar to [2], the optmzaton s performe at E b /N 0 = δ + ɛ n B, where δ s the lowest possble threshol for the gven ρ C z an C z,anɛ s an offset from the lowest threshol that offers more freeom n the seleton of λ C z. The optmzaton algorthm s eompose nto an nner an an outer loop. For a fxe E b /N 0,theouter loop fns an optmal varable noe egree strbuton for eah proteton lass an thus fns an optmal varable noe egree strbuton for the whole oe. The nner loop performs the maxmzaton of the mnmum varable noe egree of a proteton lass. The optmal λ C z s foun by exeutng the optmzaton algorthm for a gven hek noe egree strbuton ρ C z, E b /N 0 = δ + ɛ,amaxmum varable noe egree C z,aoerater z, an proporton vetors α C z an β C z.

9 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 9 of 16 The operaton of the optmzaton algorthm s summarze as follows. 1. For any gven E b /N 0 = δ + ɛ ompute σ Fn λ C z by exeutng the teratve lnear programmng optmzaton algorthm. The struture of the optmzaton algorthm wth ts onstrants s efne as follows. 1. Intalzaton: (k)c z v mn = C z 2. Whle optmzaton falure: (a) Optmze N max λ =1 =2 M, (32) uner the onstrants [C 1 ] [C 7 ]. [C 1 ] Rate onstrant, from [2] (see (27)) N N =1 k=1 =2 M, = 1 1 R z max =2 [C 2 ] Proporton strbuton onstrants. from[2] (see (24)) N N =1 k=1 ρ C z (33) M T 1 = 1 (34). k {1,, N C z 1}, from [2] (see (25) an (28)) N =1 =2 M, = α C z k R z 1 R z max =2 ρ C z (35). {1,, N C z s }, from [2] (see (26) an (29)) N k=1 =2 M, = β C z 1 1 R z [C 3 ] Convergene onstrant, from[2] (see (23)) max =2 ρ C z (36) F(λ C z, ρ C z, σ 2, x) >x (37) [C 4 ] Stablty onton, from[2] (see (30) an (31)) N =1 =2 M,2 < N =1 β C z max m=2 1 2σ e 2 ρ C z m (m 1) 1 (38) [C 5 ] Rate-ompatble onstrant, from[3] (see (19)) N C 2 N C 2 C 2 =1 k=1 =l λ C 2,C k M, N C 1 N C 1 C 1 =1 k=1 =l a C 1,C k M, C 2 max =3 ρc 2 ( ) 1 R 2 l {2, 3,, C 1 } (39) [C 6 ] Mnmum varable noe egree onstrant from [2] < (k)c z v mn, : M, = 0 (40) [C 7 ] Prevous optmzaton onstrants from [2] k < k, : λ C z,c k M = 0 s fxe (41) (b) If falure, (k)c z v mn En (Whle) = (k)c z v mn 1 When optmal varable noe egree strbutons for oe C 1 an C 2 are foun, we onstrut the party hek matres H 1 an H 2, as shown n Fgure 4, usng the ACE algorthm[20]. The ACE algorthm s mofe to frst onstrut H 1 an then H 2, whle keepng H 1 fxe. 4 Smulaton results In the followng subsetons, the performane of the propose oe esgn has been evaluate for low- an hghrate oes of meum blok. The performane of the oes has also been evaluate for short an long blok lengthoes.theproposeoes performanehasbeen ompare to QC-LDPC oes[9], punturng-base RC LDPC oes [6] an extenson-base RC LDPC oes [3]. The propose oes are smulate wth equal transmsson power settng at the soure an the relay,.e.,

10 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 10 of 16 P S1 = P/2 anp R = P/2. The hannel gans γ SD, γ SR, an γ RD from (4) are epenent upon path loss propagaton only an we onser τ = 2. Usng the relay system moel shown n Fgure 3, smulatons have been performe for fferent postons of the relay relatve to the soure. Note that n the smulatons for relay network, 20 eong teratons at the relay an 20 eong teratons at the estnaton are onsere. In all the ret lnk smulatons, 40 eong teratons at the estnaton are onsere. 4.1 Low-rate meum blok length oes Ths subseton esrbes the smulaton results of the propose oes of low-rate an meum blok length for ooperatve relay networks. The results are also ompare to the results for quas-yl LDPC oes (QC-LDPC) from [9]. The oe C 1 for BC moe, esrbe by the party-hek matrx H 1, s esgne by optmzng the varable noe egree strbutons for R 1 = 1/2, N C 1 = 2, N C 1 s = 1, α C 1 = 1, β C 1 = 1, C 1 = 11, an ρ C 1 = x 6.The oe C 2 for the half-uplex relay network s extene from the oe C 1 by appenng extra party bts to the oewors of the oe C 1.Varablenoeegreestrbutons of C 2 are optmze for R 2 = 1/4, N C 2 = 3, N C 2 s = 2, α C 2 = [ ], β C 2 = [ ], C 2 = 15 an hek noe egree strbuton.sutable for C 2 are hosen by evaluatng the performane of C 2 for fferent hek noe egree strbutons for fferent postons of relay as explane n the followng subseton. Lke n [3], the oe C 1 n ths paper s esgne by onserng the SNR of the soure to estnaton hannel, therefore there s only one hannel lass N C 1 s = 1. Note that the SNR of the soure to relay hannel s always hgher than the SNR of the soure to estnaton hannel as h SR 2 > h SD 2, therefore the suessful eong of the oewor w 1 of the oe C 1 s possble at the relay even when t fals at the estnaton. The oewor w 1 s ompose of equally protete nformaton bts an less protete party bts, by hoosng N C 1 = 2. The oe C 2 s esgne by takng nto aount the SNRs of the soure to estnaton an the relay to estnaton hannels,.e., N C 2 s = 2. The oewor w 2 of C 2, whh s eoe at the estnaton, s ompose of w 1 an w e. We let N C 2 = 3forw 2 assumng that t s ompose of equally mportant nformaton bts an less mportant party bts of w 1 an least mportant extra party bts w e. w 1 s reeve at the estnaton from the soure, whle w e s reeve at the estnaton from the relay. In relay networks, the soure to estnaton hannel s worse than the relay to estnaton hannel. Therefore, the equally mportant nformaton bts an less mportant party bts of w 1 n w 2 are assgne to proteton lasses C 1 an C 2,respetvely, n orer to gve them goo proteton aganst the hannel ontons of the soure to estnaton hannel an the least mportant bts w e n w 2 are assgne to proteton lass C 3, as the relay to estnaton hannel has a better SNR than the soure to estnaton hannel. We ompare the performane of the propose oes for relay networks wth an LDPC oe esgne for the ret lnk. The esgn of the oe for the ret lnk s the same as for C 1 exept that the oe s esgne wth oe rate 1/4 thesameasforc 2,asutableheknoe egree strbuton ρ = x x x x x 6 (from LOPT [21]), full transmsson power P an vmax = 15. In the smulatons, the lengths of the oewors w 1 an w 2 are N 1 = 1296 an N 2 = 2592, respetvely. In the smulatons of the ret lnk, an LDPC oewor of length N 2 s use Seleton of goo hek noe egree strbutons In ths subseton, we analyze the performane of the propose oes for fferent hek noe egree strbutons an for fferent postons of the relay. Optmzaton of the hek noe egree strbutons s outse the sope of ths paper. The am of ths analyss s to selet a goo heknoeegreestrbutonforeah. Coe C 1 s esgne wth a onentrate hek noe egree strbuton ρ C 1 = x 6, sne onentrate hek noe egree strbutons have been shown to perform well[16]. The oe C 2 s esgne wth a non-onentrate hek noe egree strbuton beause t s shown n [3] that o-operatve relay networks show better performane wth LDPC oes esgne wth non-onentrate hek noe egree strbutons. The maxmum hek noe egrees are hosen as C 1 max = C 2 max = 7norerto keep the same maxmum hek noe egree as the oe n [9] to smplfy the omparson. For the esgn of oe C 2, optmal varable noe egree strbutons are foun by hoosng a hek noe egree strbuton for whh oe C 2 gves goo performane n terms of bt error rate for a spef poston of relay. The performane of oe C 2 s analyze for four fferent hek noe egree strbutons for fferent postons. The four non-onentrate hek noe egree strbutons for C 2 satsfy the rate-ompatble onstrant for the hek noe egree strbuton as efne by (17). It s observe that our esgne oes have error floor at hgh bt error rate f the four hek noe egree strbutons are hosen wth a mnmum hek noe egree less than 4 an have the waterfall regon at hgh SNR f the four hek noe egree strbutons are hosen wth a mnmum hek noe egree greater than 4. Therefore, to esgn the oes wth the error floor at lower bt error rate an the waterfall regon at lower SNR, the four hek noe egree strbutons are hosen wth a mnmum hek noe egree equal to 4 but wth fferent fraton of eges of the mnmum egree. Furthermore, 4 s the hghest possble mnmum egree wth whh the four hek noe

11 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 11 of 16 egree strbutons stll satsfy (17). The optmal varable noe egree strbuton at eah s foun for eah one of the four hek noe egree strbutons. We observe that for < 0.6, the oe esgne wth the hek noe egree strbutons, whh have the lowest fraton of eges of the mnmum egree, has ts waterfall regon at hgh SNR as ompare to the oes wth hek noe egree strbutons wth a hgher fraton of eges of the mnmum egree. Fgure 6 shows the performane of the propose oe wth four fferent hek noe egree strbutons for = 0.2. It s note n Fgure 6 that the propose oe esgn gves goo performane wth = x x 6.Thshols also for = 0.3 an 0.5, but smulaton results are omtte here. Fgure 7 shows the performane of the propose oes wth four fferent hek noe egree strbutons for = 0.4 an goo performane s observe wth = 0.5x x 6. Ths hols also for = 0.1 but smulaton results are omtte here. For = 0.6, 0.7, 0.8, 0.9 t s observe n Fgure 8 that the hoe of hek noe egree strbuton oes not have a large mpat on the performane Results for low-rate meum blok length oes Ths subseton esrbes the smulaton results of the propose oes wth low-rate an meum blok length n a relay network an ompares ther performane wth a stanar LDPC oe for the ret lnk. The propose oes at eah are esgne wth the sutable hek noe egree strbutons foun n the prevous subseton. The smulaton results for the propose oes are shown n Fgure 9 for fferent postons of the relay. The results show that the relay network aheves better performane wth the propose oes than the ret lnk wth a LDPC Fgure 7 Comparson between fferent hek noe egree strbutons for the oe wth oe rate 0.25 an = 0.4. The hek noe egree strbuton ρ C2 = 0.5x x 6 gves goo performane for = 0.4. oe optmze for ths lnk. At a BER of 10 5, the relay network has best performane for = 0.4. At a BER of 10 4, the propose oe aheves 3.8-B gan over the ret lnk transmsson for the poston of relay = 0.4, as shown n Fgure 9. A egraaton n the performane of the propose oes n the waterfall regon an be observe when the relay s loate loser to the estnaton than the soure,.e., for relay postons > 0.5, as shown n Fgure 9. Note that for all, both the soure an the relay have fxe transmt power,.e., the total power P s ve n half for the soure an the relay for all. Theperformane of the o-operatve relay network epens mostly on the performane of the soure to relay hannel. At > 0.5, the sgnal reeve at the relay has hgher path loss Fgure 6 Comparson between fferent hek noe egree strbutons for the oe wth oe rate 0.25 an = 0.2. The hek noe egree strbuton ρ C2 = x x 6 gves goo performane for = 0.2. Fgure 8 Comparson between fferent hek noe egree strbutons for the oe wth oe rate 0.25 an large. For large values of, the fferent hek noe egree strbutons have almost the same performane.

12 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 12 of 16 Fgure 9 Performane of the propose oes for fferent postons of the relay. All oes have oe rate 0.25 an a oewor length of 2, 592 bts. than at 0.5, so the probablty of erroneous eong of w 1 n the relay nreases wth nreasng. Theerroneously eoe w 1 s then use to alulate w e, whh n turn s transmtte to the estnaton an nreases the probablty of erroneous eong of the entre oewor w 2. Ths auses the performane egraaton of the propose oes at > 0.5. At 0.7, the relay network has worse performane than the ret lnk, at least for some SNRs. 4.2 Hgh-rate meum blok length oes The performane of the propose oe esgn has also been analyze for a hgh-rate oe of meum blok length. The oe C 1 s esgne by optmzng the varable noe egree strbutons for R 1 = 0.9, N C 1 = 2, N C 1 s = 1, α C 1 = 1, β C 1 = 1, C 1 = 7, an ρ C 1 = x 29.Varablenoe egree strbutons of C 2 are optmze for R 2 = 0.6, N C 2 = 3, N C 2 s = 2, α C 2 = [ ], β C 2 = [2/3 1/3], C 2 = 50 an hek noe egree strbuton.the lengths of the oewors w 1 an w 2 are N 1 = 1, 730 an N 2 = 2, 592, respetvely. Goo for C 2 at eah has been hosen by usng the proeure aopte n Seton for fnng goo for meum blok length oes of low rate. We have analyze the performane of C 2 wth four fferent nononentrate hek noe egree strbutons = 0.25x x 29, = 0.5x x 29, = x x 29,an = 0.1x x 29 for fferent postons. All the hek noe egree strbutons for C 2 satsfy the rate-ompatble onstrant for the hek noe egree strbuton as efne by (17). In the same way as for the oe wth low rate n Seton 4.1.1, the oe C 2 wth hgh rate s also esgne suh that t has the error floor at lower bt error rates an the waterfall regon at a lower SNR by keepng the mnmum hek noe egree n all the hek noe egree strbutons equal to 11. Note that 11 s the hghest possble mnmum egree wth whh the four hek noe egree strbutons satsfy (17). The goo hek noe egree strbutons beng foun for = 0.1, 0.2, 0.4 an 0.5 are = 0.25x x 29, = 0.25x x 29, = 0.5x x 29,an = 0.25x x 29, respetvely. As for the lowrate oe propose n Seton 4.1.1, the hoe of hek noe egree strbuton oes not have a large mpat on the performane for = 0.6, 0.7, 0.8 an 0.9. The performane of the propose oe has been evaluate for = 0.1, 0.2, 0.4, 0.5, 0.7 an 0.9 as shown n Fgure 10. It s shown that the relay network has the best performane for = 0.4, lke the low-rate oe n Seton It s reporte n [9] that oes wth fferent oe rates have smlar senstvty to the relay poston for a gven transmt power strbuton of the soure an the relay; hene, for both ases of low- an hgh-rate oes, the optmal poston where the propose oe aheves best performane s = 0.4. The performane of the propose hgh-rate oe s ompare to the performane of an LDPC oe esgne for the ret lnk. The esgn of the oe for the ret lnk s the same as for C 1 exept that the oe s esgne wth oe rate 0.6, a sutable ρ = x x x x x 29 (from LOPT [22]), full transmsson power P an vmax = 50. In the smulatons of the ret lnk, the oewor length s 2, 592 bts. At a BER of 10 5,thepropose hgh-rate oe shows 1.3-B gan over the ret lnk transmsson for = 0.4, as shown n Fgure Short an long blok length oes In ths subseton, the performane of the propose oe esgn has been analyze for short an long blok lengths. Fgure 10 Performane of the propose hgh-rate oes for fferent postons of the relay. All the oes have oe rate 0.6 an a oewor length of 2, 592 bts.

13 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 13 of 16 The oewor lengths for the short blok length are N 1 = 512 an N 2 = 1, 024 bts, an for the long blok length, we onser N 1 = 5, 344 an N 2 = 10, 688 bts. Note that the propose oe optmzaton algorthm s nepenent of the oe length. The parameters use for the esgn of the propose oe wth short an long blok lengths are the same as those use for meum blok length n Seton 4.1. For omparson, oes for the ret lnk wth oewor lengths of 1, 024 an 10, 688 bts have been onstrute, aorng to the egree strbutons foun for the meum length oe n Seton 4.1. It s shown n [9] that the optmal poston of the relay epens on the power strbuton of the soure an the relay n the relay network. For the propose oe wth meum blok length, t has been shown n the prevous subsetons that the optmal poston s = 0.4. We have analyze the performane of the propose oe wth short an long blok lengths at the optmal poston of the relay,.e., at = 0.4. At a BER of 10 4, the propose oe wth short an long blok lengths have 3.8-B gan over the ret lnk transmsson for = 0.4, as shown n Fgure Comparson Ths subseton proves a performane omparson of the oes propose n ths paper to QC-LDPC oes [9], extenson-base RC LDPC oes [3] an punturngbase RC LDPC oes [6] Comparson to QC-LDPC oes In ths subseton, we ompare the performane of the RC UEP-LDPC oes to the QC-LDPC oes propose n [9], whh are mplemente through a repetton ong sheme n a half-uplex o-operatve relay network. The smulaton results for the QC-LDPC oes of [9] are reproue n ths paper, as shown n Fgure 12. The proposercuep-ldpcoes,lketheqc-ldpcoes,are struture an rregular. They are however mplemente n the half-uplex o-operatve relay network through a rate-ompatble ong sheme. We assume, lke n [9], that the soure s slent n MAC moe an use the same oewor length. In ontrast to [9], the relay termnal n our ase transmts the extra party bts nstea of retransmttng the eoe oewor n MAC moe. The RC UEP-LDPC oes are more omplex from a pratal mplementaton pont of vew ompare to the QC-LDPC oes n [9], but they requre a lower E b /N 0 to aheve the same bt error probablty. Equal tme slots are assume n BC an MAC moes as well as equal transmt powers at the soure an the relay,.e., P S1 = P/2 anp R = P/2. In [9], the length of eah oewor beng transmtte n BC an MAC moes s 1, 296 bts an the rate s 1/2. For the RC UEP-LDPC oes, the length of w 1 s equal to 1, 296 wth rate 1/2. Atonal party bts w e of length 1, 296 are transmtte n MAC moe. Hene, the total number of bts reeve at the estnaton, both for the QC-LDPC oes an for the RC UEP-LDPC oes, s equal to 2, 592 bts. In [9], the LDPC oe s eoe by the layere belef propagaton (LBP) algorthm as efne n [23]. LBP s a varaton of the stanar belef propagaton [1] whh s esgne for arhteture aware (AA) LDPC oes wth quas-yl propertes. It s shown that wth LBP, the eong of AA LDPC oes s mprove by two tmes n Fgure 11 Performane omparson between the propose oes wth short an long blok lengths at = 0.4 an ret lnk ommunaton. The oewor lengths of the short an long blok lengths are 1, 024 an 10, 688 bts, respetvely. The oe rate s 0.25 for all oes. Fgure 12 BER performane of the QC-LDPC oe propose n [9]. Wth the eoe an forwar strategy of the relay termnal an wthout re-enong. The QC-LDPC oes have oe rate 0.5 an a oewor length of 1, 296 bts, whle the propose RC UEP-LDPC oe has oe rate 0.25 an a oewor length of 2, 592 bts. Note that sne the repetton ong sheme s use n [9], t s far to ompare these oes to the propose oes of ouble oewor length an half the oe rate. The propose oe shows 1.9-B gan over the oes n [9] at = 0.4.

14 Khattak an Sanberg EURASIP Journal on Wreless Communatons an Networkng 2014, 2014:22 Page 14 of 16 the number of teratons requre for a gven error rate [24]. However, we have use stanar belef propagaton eong to reproue the results of [9] an to analyze the performane of the propose LDPC oes n o-operatve relay networks. To get the equvalent performane of LBP usng stanar belef propagaton eong, the number of eong teratons s set to twe the number of teratons efne n [9]. In [9], ten eong teratons at the relay an ten eong teratons at the estnaton are use. We use 20 eong teratons at both the relay an the estnaton. From the omparson of Fgures 9 an 12, the RC UEP-LDPC oes gve better performane than the QC- LDPC oes n [9]. The QC-LDPC oes also show best performane for = 0.4. Uner the same tme slots, transmsson power an usng the DF strategy for the relay, the propose oes outperform the oes n [9] wth a gan of 1.9 B at a BER of 10 4 for = 0.4, as shown n Fgure 12. For the postons of the relay other than = 0.4, the propose oes have fferent gans over the oes n [9], whh an be observe by omparng the results for eah n Fgures 9 an 12. The mprove performane of the propose oes over the QC-LDPC oes n [9] s ue to separate optmzaton of the propose oes for eah relay poston. Furthermore, the propose oe esgn s base on the rate-ompatble ong sheme, whh proves a better trae-off between bt error rate an ata rate than the repetton ong sheme Comparson to extenson-base RC LDPC Coes It s observe from Fgure 9 that the performane of the propose oes s lose to the performane of the oe propose n [3] for the range of between 0.2 an 0.3 (see the smulaton of the optmze oe of length 20, 000 n [3] for a omparson). We perform the optmzaton of the oe through an teratve lnear programmng algorthm n ontrast to the oe esgn n [3], whh s aheve through fferental evoluton, a omplex non-lnear onstrane optmzaton algorthm. There are three reasons why the RC UEP-LDPC oes are not exatly approahng the performane of the oes n [3]. Frst, the oewor length onsere n [3] s almost ten tmes hgher than the oewor length onsere for the RC UEP-LDPC oes. Seon, n [3], t s assume that eah oewor s perfetly eoe at the relay. Thr, n [3], t s assume that the soure s atve also n MAC moe Comparson to punturng-base RC LDPC oes The performane of the propose oes s also ompare to punturng-base RC LDPC oes propose n [6] for half-uplex o-operatve relay networks. The punturngbase rate-ompatble ong sheme propose n [6] requres the esgn of only one sngle-user LDPC oe as amotheroec. The mother oe C wth oe rate R s frst esgne ether by lnear programmng or by usng an optmze LDPC egree strbuton from [25]. Then, a proporton of the party bts s punture ranomly, whh rasestheoerateofc to R p.inbcmoe,thepunture oewor of C s transmtte to the estnaton an the relay. The estnaton stores the punture oewor untl t reeves the punture party bts from the relay. The punture party bts are reovere at the relay an are sent as extra party bts to the estnaton n MAC moe. Thus, the estnaton reeves the entre oewor orresponng to the mother oe C wth fferent SNRs for the punture oewor an the extra party bts as expresse by (8) an (9). The ong sheme propose n [6] only requres the esgn of a mother oe C an the oe esgn propose n [6] s therefore smpler than the oe esgn propose here, whh requres ont esgn of two oes. We use the lnear programmng optmzaton algorthm efne n ths paper wthout the onstrants for UEP an rate-ompatblty to fn an optmze varable noe egree strbuton for the oe C wth a sutable ρ = x x x x x 8 (from LOPT [25]), R = 1/4an vmax = 33. The oe C s punture at the soure an the oe rate s rase to R p = 0.5. The performane of the ong sheme propose n [6] s shown n Fgure 13 for a oewor length of 2, 048 bts an = 0.4. It s assume n [6], as n the ong strategy propose here, that the soure s slent n MAC moe, the oewor s mperfetly eoe at the relay an that extra party bts from the relay are sent wthout enong to the estnaton. To ompare the performane of the RC UEP-LDPC oe propose here to the ong sheme propose n [6], Fgure 13 Performane omparson between the propose RC UEP-LDPC oe an punturng-base RC LDPC oes for = 0.4. The oes have oe rate 0.25 an a oewor length of 2, 048 bts.