Spatially Coupled Turbo Codes

Transcription

1 Spilly Coupled Turbo Codes Seedeh Moloudi, Michel enmier, nd Alexndre Grell i Am Deprmen of Elecricl nd nformion Technology, und Universiy, und, Sweden Deprmen of Signls nd Sysems, Chlmers Universiy of Technology, Gohenburg, Sweden {seedeh.moloudi,michel.lenmier}@ei.lh.se, lexndre.grell@chlmers.se rxiv:10.393v [cs.t] Jul 01 Absrc n his pper, we inroduce he concep of spilly coupled urbo codes SC-TCs, s he urbo codes counerpr of spilly coupled low-densiy priy-check codes. We describe spil coupling for boh Berrou e l. nd Benedeo e l. prllel nd serilly concened codes. For he binry ersure chnnel, we derive he exc densiy evoluion DE equions of SC-TCs by using he mehod proposed by Kurkoski e l. o compue he decoding ersure probbiliy of convoluionl encoders. Using DE, we hen nlyze he sympoic behvior of SC-TCs. We observe h he belief propgion BP hreshold of SC-TCs improves wih respec o h of he uncoupled ensemble nd pproches is mximum poseriori hreshold. This phenomenon is especilly significn for serilly concened codes, whose uncoupled ensemble suffers from poor BP hreshold.. NTRDUCTN ow-densiy priy-check DPC convoluionl codes [1], lso known s spilly coupled DPC SC-DPC codes [], cn be obined from sequence of individul DPC block codes by disribuing he edges of heir Tnner grphs over severl djcen blocks [3]. The resuling spilly coupled codes exhibi hreshold surion phenomenon, which hs rced lo of ineres in he ps few yers: he hreshold of n ierive belief propgion BP decoder, obined by densiy evoluion DE, is improved o h of he opiml mximum-poseriori MAP decoder [], [3]. As consequence, i is possible o chieve cpciy wih simple regulr DPC codes, which show wihou spil coupling significn gp beween BP nd MAP hreshold. The concep of spil coupling is no limied o DPC codes. Spilly coupled urbo-like codes, for exmple, cn be obined by replcing he block-wise permuion of urbo code by convoluionl permuion []. n combinion wih windowed decoder for he componen code, coninuous sreming implemenion is possible [5]. The self-concened convoluionl codes in [6] re closely reled srucures s well. A vrin of spilly coupled selfconcened codes wih block-wise processing, clled lmined codes ws considered in [7]. They hve he dvnge h n implemenion similr o uncoupled urbo codes is possible, wihou he need for sreming implemenion of he decoder. A block-wise version of brided convoluionl codes [8], clss of spilly coupled codes wih convoluionl componens, hs recenly been nlyzed in [9]. The im of his pper is o invesige he impc of spil coupling on he hresholds of clssicl urbo codes. For his This work ws suppored in pr by he Swedish Reserch Council VR under grn # purpose we inroduce some specil block-wise spilly coupled ensembles of prllel concened codes SC-PCCs nd serilly concened codes SC-SCCs, which re spilly coupled versions of he ensembles by Berrou e l. [10] nd Benedeo e l. [11], respecively. Wih sligh buse of he erm, we cll boh prllel nd seril ensembles spilly coupled urbo codes SC-TCs. For hese ensembles we derive exc DE equions from he rnsfer funcions of he componen decoders [1], [13] nd perform hreshold nlysis for he binry ersure chnnel BEC, nlogously o [3], [9]. To compre he resuls for SC-PCC nd SC-SCC ensembles wih ech oher some ensembles wih puncuring re lso considered. The BP hresholds of he differen ensembles re presened nd compred o he MAP hresholds for differen coupling memories.. SPATAY CUPED TURB CDES n his secion, we inroduce spilly coupled urbo codes. We firs describe spil coupling for boh prllel nd serilly concened codes, nd hen ddress heir ierive decoding. A. Spilly Coupled Prllel Concened Codes We consider he spil coupling of R 1/3 prllel concened codes, buil from he prllel concenion of wo re-1/ recursive sysemic convoluionl encoders, denoed by nd C see Fig. 1. For simpliciy, we describe spil coupling wih coupling memory m 1. Consider collecion of urbo encoders ime insns 1,...,, s illusred in Fig. 1. is clled he coupling lengh. We denoe by u he informion sequence, nd by v U nd v he code sequences of nd C, respecively, ime. The oupu of he urbo encoder is given by he uple v u, v U, v. A SC-PCC ensemble wih m 1 is obined by connecing ech urbo code in he chin o he one on he lef nd o he one on he righ s follows. Divide he informion sequence u ino wo sequences, u,a nd u,b by demuliplexer. Also divide copy of he informion sequence, which is properly reordered by he permuion Π, ino wo sequences, u,a nd u,b by noher demuliplexer. A ime, he informion sequence he inpu of encoder is u,a, u 1,B, properly reordered by permuion Π U. ikewise, he informion sequence he inpu of encoder C is u,a, u 1,B, properly reordered by he permuion Π. n Fig. 1 he blue lines represen he informion bis from he curren ime slo h re used in he nex ime slo + 1 nd he green lines represen he informion bis from he

2 u u u u u u 1 v U 1 U 1 U v U U v U 1 v 1 v 1 C C C v u 1 u u u 1 u C C C C C C v 1 1 v 1 u v Fig. 1. b Block digrm of he encoder of spilly coupled urbo code for coupling memory m 1. prllel concenion b seril concenion. previous ime slo 1. n order o ermine he encoder of he SC-PCC o he zero se, he informion sequences he end of he chin re chosen in such wy h he oupu sequence ime + 1 becomes v Anlogously o convenionl convoluionl codes his resuls in re loss h becomes smller s increses. Using he procedure described bove coupled chin convoluionl srucure over ime of urbo encoders wih coupling memory m 1 is obined. An exension o lrger coupling memories m > 1 is presened in Secion V. B. Spilly Coupled Serilly Concened Codes We consider he coupling of serilly concened codes SCCs buil from he seril concenion of wo re-1/ recursive sysemic convoluionl encoders. The overll code re of he uncoupled ensemble is herefore R 1/. A block digrm of he encoder is depiced in Fig. 1b for coupling memory m 1. As for SC-PCCs, le u be he informion sequence ime. Also, denoe by v v, v u, v nd v he encoded sequence he oupu of he ouer nd inner encoder, respecively, nd by ṽ he sequence v fer permuion. The SC-SCC wih m 1 is consruced s follows. Consider collecion of SCCs ime insns 1,...,. Divide he sequence ṽ ino wo prs, ṽ,a nd ṽ,b. Then, ime he sequence he inpu of he inner encoder C is ṽ,a, ṽ 1,B. n order o ermine he encoder of he SC-SCC o he zero se, he informion sequences he end of he chin re chosen in such wy h he oupu sequence ime + 1 becomes v+1 0. Using his consrucion mehod, coupled chin of SCCs wih coupling memory m 1 is obined. An exension o lrger coupling memories m > 1 is presened in Secion V. C. erive decoding As sndrd urbo codes, SC-TCs cn be decoded using ierive messge pssing belief propgion decoding, he componen encoders of ech urbo code re decoded using he BCJR lgorihm. The BP decoding of SC-PCCs cn be esily visulized wih he help of Fig., which shows he fcor grph of single secion of he SC-PCC. We denoe by D U nd D he decoder of he upper nd lower encoder, respecively. The decoder D U receives is inpu informion from he chnnel for boh sysemic nd priy bis. Furhermore, i lso receives -priori informion on he sysemic bis from oher decoders. As described bove, ime he informion sequence he inpu of consiss of wo prs, u,a nd u 1,B. Correspondingly, D U ime insn receives priori informion from D ime insns 1, nd + 1. Bsed on he informion from he chnnel nd from he compnion decoders, D U compues he exrinsic informion on he sysemic bis using he BCJR lgorihm. Since he srucure of SC-PCCs is symmeric, he decoding of he lower encoder is performed in n idenicl mnner. Similrly o SC-PCCs, he decoding SC-SCCs cn lso be described wih he help of fcor grph. The fcor grph of secion of SC-SCC wih m 1 is shown in Fig. 3.. DENSTY EVUTN ANAYSS N THE BEC n his secion, we nlyze he sympoic performnce of SC-TCs using DE. We consider rnsmission over BEC wih ersure probbiliy ɛ, denoed by BECɛ. We derive he exc DE equions for boh unpuncured SC-PCCs nd SC-SCCs nd discuss he modificion of he equions when puncuring is pplied for chieving higher res. A. Spilly Coupled Prllel Concened Codes e p U,s nd p be he verge exrinsic ersure probbiliy on he sysemic bis he oupu of he upper nd lower decoder, respecively. ikewise, we define p U,p nd p,p for he priy bis.

3 u C Trellis u Fig.. Fig. 3. u,b u,a U u,b 0 u,a 0 u 1,B u 1,B 0 U Fcor grph of single secion of SC-PCC. v Fcor grph of single secion of SC-SCC The ersure probbiliies p U,s nd p U,p ierion i nd ime insn cn be wrien s U,s f U,s, ɛ 1 U,p f U,p, ɛ, ɛ pi 1, + p i 1, 1 v + p i 1, v U v C Trellis CU Trellis C Trellis, 3 nd f U,s nd f U,p denoe he upper decoder rnsfer funcions for he sysemic nd priy bis, respecively. Noe h he upper decoder rnsfer funcion ime depends on boh he chnnel ersure probbiliy nd he exrinsic ersure probbiliy on he sysemic bis from he lower decoder ime insns, 1 nd + 1, due o he coupling. Becuse of he symmeric design, he lower decoder upde is idenicl o h of he upper decoder by inerchnging p U nd p, nd subsiuing q q U in 1 3. Finlly, he -poseriori ersure probbiliy on he informion bis ime nd ierion i is 1 ɛ pi, U,s pi, +pi, U,s pi, +p i, U,s +pi, U,s p i, For he BEC i is possible o compue nlyic expressions for he exc exrinsic probbiliy of ersure of convoluionl encoders, using he mehod proposed in [1] nd [13]. Here, we use his mehod o derive he exc expressions for he rnsfer funcions of he componen decoders. DE is hen performed by rcking he evoluion of wih he number of ierions, wih he iniilizion p 0, U,s p 0, U,p p0, p 0, 0 for 0 nd >, nd 1 oherwise. The BP,p hreshold corresponds o he mximum chnnel prmeer ɛ for which successful decoding is chieved, i.e., ends o zero for ll ime insns s i ends o infiniy. B. Spilly Coupled Serilly Concened Codes Similrly o he prllel cse, DE equions cn be derived for SC-SCCs. e p nd p,s be he verge exrinsic ersure probbiliy on he sysemic bis he oupu of he ouer nd inner decoder, respecively. ikewise, we define p nd p,p for he priy bis he oupu of he ouer nd inner decoder, respecively. The ersure probbiliies p,s nd p,p cn be wrien s,s f,s, ɛ 5 ɛ pi 1,,p f,p + p i 1, + p i 1, 1, ɛ, 6 + p i 1, 1, 7 nd f,s nd f,p denoe he inner decoder rnsfer funcions for he sysemic nd priy bis, respecively. ikewise, p nd p re f f ɛ pi 1,,s, 8,, 9 + p i 1,,s. 10 The -poseriori ersure probbiliy on he informion bis ime fer i ierions is ɛ pi,,s + p i,,s. 11 DE is hen performed by rcking he evoluion of wih he number of ierions, wih he iniilizion p 0, p 0,,p p 0, oherwise. p 0,,s 0 for 0 nd > nd 1 1 We remrk h lhough is no pplied wihin he DE recursion i is required for he compuion of he re bound on he MAP hreshold.

4 C. Spilly Coupled Turbo Codes wih Rndom Puncuring Higher res cn be obined by pplying puncuring. Here, we consider rndom puncuring. Assume h code sequence x is rndomly puncured such h frcion ρ [0, 1] of he coded bis survive fer puncuring, nd hen rnsmied over BECɛ. ρ will be referred o s he permebiliy re. For he BEC, puncuring is equivlen o rnsmiing x hrough BECɛ ρ resuling from he concenion of wo BECs, BECɛ nd BEC1 ρ, ɛ ρ 1 1 ɛρ. The DE equions derived in he previous subsecions cn be esily modified o ccoun for puncuring. Consider firs he cse of SC-PCCs. We consider only puncuring of he priy bis, nd h boh nd C re eqully puncured wih permebiliy re ρ. The code re of he uncoupled puncured prllel concened code PCC is R 1 1+ρ. This resuls in sligh modificion of he DE equions, subsiuing ɛ ɛ ρ in 1,. For SC-SCCs we consider puncuring s proposed in [1], [15], which resuls in beer SCCs s compred o sndrd SCCs. e ρ 0 nd ρ 1 be he permebiliy re of he sysemic nd priy bis, respecively, of C sen direcly o he chnnel see [15, Fig. 1], nd ρ he permebiliy re of he priy bis of C. The code re of he uncoupled puncured SCC is 1 R ρ 0+ρ 1+ρ. The DE for puncured SC-SCCs is obined by subsiuing ɛ ɛ ρ in 5, 6, nd modifying 7 o ɛ p i 1, nd 8, 9 o + p i 1, 1 + ɛ ρ1 p i 1, f f is given in 10 nd ɛ ρ1 pi 1,,s + p i 1, 1, 1,, 13 + p i 1,,s,. 1 V. EXTENSN T ARGER CUPNG MEMRES The resuls from he previous secions cn esily be generlized o lrger coupling memories m > 1. e us firs consider SC-PCCs. n he generl cse he informion sequences u, u 1,..., u m from m + 1 differen ime insnces re used by he encoders ime. This is chieved by dividing he informion sequence u ino he sequences u,j, j 0,..., m by muliplexer, nd lso dividing properly reordered copy of he informion bis ino u,j, j 0,..., m, which cn be ccomplished by permuion Π followed by muliplexer. A he inpu of he upper encoder ime he sequences u j,j re muliplexed nd reordered by he permuion Π U. The lower encoder C receives he informion sequences u j,j, n his pper we consider ρ 0 1, i.e., he overll code is sysemic. muliplexed nd reordered by Π. The encoder in Fig. 1 corresponds o he specil cse m 1. n he DE recursion we now hve o modify 3 o ɛ j0 k0 pi,+j k m + 1, nd he -poseriori ersure probbiliy on he informion bis ime nd ierion i becomes j0 k0 ɛ pi,+j U,s p i,+k m + 1. ikewise, for SC-SCCs he code sequence v of C is divided rndomly ino he sequences ṽ,j, j 0,..., m. C receives ime he sequences ṽ j,j fer pssing muliplexer nd permuion. The encoder in Fig. 1b corresponds o he specil cse m 1. Equions 7 nd 10 in he DE recursion re modified ccordingly o nd ɛ j0 pi 1, j m + 1 j0 ɛ pi 1,+j,s m p i 1, j The -poseriori ersure probbiliy on he informion bis ime fer i ierions 11 becomes ɛ j0 pi,+j,s. m + 1 V. RESUTS AND DSCUSSN n his secion, we give numericl resuls for some SC- TCs, using he DE described in Secion nd V. n our exmples we consider SC-TCs wih idenicl re-1/, -ses componen encoders. n priculr, we consider componen encoders wih generor polynomils 1, 5/7 in ocl noion. For noionl simpliciy, we denoe he uncoupled PCC ensemble by C PCC nd he corresponding coupled ensemble by C SC PCC. For SC-SCCs, we denoe by C SCC, nd C SC SCC he uncoupled nd coupled ensembles, respecively. Noe h since he wo componen encoders re idenicl, f U,s x, y f x, y nd f U,p x, y f,p x, y for SC-PCCs, nd f,s x, y f x, y nd f,p x, y f x, y for SC- SCCs. All presened hresholds correspond o he sionry cse, which lower bounds he hresholds for finie. For smll he hreshold cn be considerbly lrger bu he expense of higher re loss. n Tble we give he BP hreshold for severl SC-TCs nd coupling memory m 1 nd 3, denoed by ɛ 1 SC nd ɛ 3 SC. We lso repor in he ble he BP hreshold ɛ BP nd he MAP hreshold ɛ MAP of he uncoupled ensembles. The MAP hreshold ws compued pplying he re heorem [16]. n ll cses we observe n improvemen of he BP hreshold when coupling is pplied. We remrk h for C SC PCC he BP hreshold of he uncoupled ensemble is lredy close o.

5 TABE THRESHDS FR SC-TCS Ensemble Re ɛ BP ɛ MAP ɛ 1 SC ɛ 3 SC C PCC /C SC PCC 1/ C SCC /C SC SCC 1/ TABE THRESHDS FR PUNCTURED SC-TCS Ensemble Re ɛ BP ɛ MAP ɛ 1 SC ɛ 3 SC C PCC /C SC PCC 1/ C SCC /C SC SCC 1/ C PCC /C SC PCC 1/ C SCC /C SC SCC 1/ he MAP hreshold, herefore he poenil gin wih coupling is limied. However, i is ineresing o observe h he BP hreshold of C SC PCC wih m 1 is very close o ɛ MAP, suggesing hreshold surion. The resuls for he ensemble C SC SCC re lso given in Tble for coupling memory m 1 nd 3. We observe h he ensemble C SCC hs poor BP hreshold s compred o he MAP hreshold. This is wellknown phenomenon for SCCs, for which he gp beween he BP nd he MAP hreshold is lrge. A significn improvemen is obined by pplying coupling wih m 1. However, here is sill gp beween ɛ BP nd ɛ MAP, mening h hreshold surion hs no occurred. The BP hreshold cn be furher improved by incresing he coupling memory o m 3. n his cse he BP hreshold is very close o he MAP hreshold, suggesing h hreshold surion occurs for lrge enough coupling memory. This behvior is similr o he hreshold surion phenomenon of SC-DPC codes, which occurs for smoohing prmeer w []. n Tble we show he BP hresholds of puncured SC- TCs, in order o compre SC-PCCs nd SC-SCCs for given code re. We consider R 1/3 nd R 1/, nd coupling memory 1. 3 For he SC-SCC we used ρ 1 1 nd ρ 0.5 for R 1/3 nd ρ 1 0. nd ρ 0. for R 1/. Agin, in ll cses n improvemen of he BP hreshold is observed when coupling is pplied. As expeced, for given re he PCC ensemble shows beer hreshold hn he SCC ensemble. However, he improvemen in he BP hreshold due o coupling for he ler is very significn. For R 1/3 nd m 1 he BP hreshold of C SC SCC is very close o h of he unpuncured ensemble C SC PCC, while lrge gp is observed for he uncoupled ensembles. For m 3 C SC SCC chieves beer BP hreshold hn C SC PCC. The resul is even more remrkble for R 1/. n his cse, while he uncoupled SCC ensemble shows very poor hreshold, C SC SCC shows superior hreshold hn C SC PCC lredy for m 1. 3 For R 1/3 he SC-PCC is no puncured. V. CNCUSNS n his pper we hve inroduced some block-wise spilly coupled ensembles of prllel nd serilly concened convoluionl codes nd performed densiy evoluion nlysis on he BEC. n ll considered cses spil coupling resuls in n improvemen of he BP hreshold nd our numericl resuls sugges h hreshold surion occurs if he coupling memory is chosen significnly lrge. The hreshold improvemen is lrger for he seril ensembles, which re known o hve poor BP hresholds wihou coupling bu re sronger regrding he disnce specrum. Puncuring he seril nd prllel ensembles o equl code res, we observe h he hreshold of he seril ensemble cn surpss he one of he prllel ensemble. REFERENCES [1] A. Jiménez Felsröm nd K.Sh. 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