Nonlinear BCJR equalizer for suppression of intrachannel nonlinearities in 40 Gb/s optical communications systems

Transcription

1 Nonlinear BCJR equalizer for uppreion of intrachannel nonlinearitie in 40 Gb/ optical communication ytem Ivan B. Dordevic and Bane Vaic Univerity of Arizona Department of Electrical and Computer Engineering Tucon AZ 857 USA Abtract: A maximum a poteriori probability (MAP ymbol decoding upplemented with iterative decoding i propoed a an effective mean for uppreion of intrachannel nonlinearitie. The MAP detector baed on Bahl-Cocke-Jelinek-Raviv algorithm operate on the channel trelli a dynamical model of interymbol interference and provide oft-deciion output proceed further in an iterative decoder. A dramatic performance improvement i demontrated. The main reaon i that the conventional maximum-likelihood equence detector baed on Viterbi algorithm provide hard-deciion output only hence preventing the oft iterative decoding. The propoed cheme operate very well in the preence of trong intrachannel interymbol interference when other advanced forward error correction cheme fail and it i alo uitable for 40 Gb/ upgrade over exiting 0 Gb/ infratructure. 006 Optical Society of America OCIS code: ( Optical communication ( Fiber optic ( ( Bahl-Cocke-Jelinek-Raviv (BCJR algorithm ( Forward error correction (FEC ( Low-denity parity-check (LDPC code Reference and link. I. B. Dordevic and B. Vaic Contrained coding technique for uppreion of Intrachannel Nonlinear effect in high-peed optical tranmiion J. Lightwave Technology (006.. R.-J. Eiambre B. Mikkelen and G. Raybon Intra-channel crophae modulation and four-wave mixing in high-peed TDM ytem Electron. Lett ( M. J. Ablowitz and T. Hirooka Reonant nonlinear intrachannel interaction in trongly diperionmanaged tranmiion ytem Opt. Lett ( X. Liu X.Wei A. H. Gnauck C. Xu and L. K. Wickham Suppreion of interchannel four-wave-mixinginduced ghot pule in high-peed tranmiion by phae inverion between adacent marker block Opt. Lett ( P. V. Mamyhev and N. A. Mamyheva Pule-overlapped diperion-managed data tranmiion and intrachannel four-wave mixing Opt. Lett ( N. Alic and Y. Fainman "Data-dependent phae coding for uppreion of ghot pule in optical fiber" IEEE Photon. Technol. Lett. 6-4 ( J. H. Winter Equalization in coherent lightwave ytem uing a fractionally paced equalizer J. Lightwave Technol ( S. Katuria and J. H. Winter Technique for high-peed implementation of nonlinear cancellation IEEE J. Sel. Area Commun ( C. Xia and W. Roenkranz Performance enhancement for duobinary modulation through nonlinear electrical equalization in Proc. 3 t European Conference on Optical Communication (ECOC ( O. E. Agazzi M. R. Hueda H. S. Carrer and D. E. Crivelli Maximum-likelihood equence etimation in diperive optical channel J. Lightwave Technol (005.. N. Alić G. Papen R. Sapertein L. Miltein Y. Fainman Signal tatitic and maximum likelihood equence etimation in intenity modulated fiber optic link containing a ingle optical pre-amplifier Opt. Expre ( C. Douillard M. Jézéquel C. Berrou A. Picart P. Didier and A. Glavieux Iterative correction of interymbol interference: turbo equalization Eur. Tran. Telecommun (995. (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 465

2 3. S. Song A. C. Singer and K.-M. Sung Soft input channel etimation for turbo equalization IEEE Tran. Signal Proce ( B. M. Kurkoki P. H. Siegel and J. K. Wolf Joint Meage-Paing decoding of LDPC code and partialrepone channel IEEE Tran. Inf. Theory ( L. R. Bahl J. Cocke F. Jelinek and J. Raviv Optimal decoding of linear code for minimizing ymbol error rate IEEE Tran. Inf. Theory IT ( I. B. Dordevic and B. Vaic MacNeih-Mann theorem baed iteratively decodable code for optical communication ytem IEEE Commun. Lett ( I. B. Dordevic S. Sankaranarayanan and B. Vaic Proective plane iteratively decodable block code for WDM high-peed long-haul tranmiion ytem J. Lightwave Technol ( B. Vaic I. B. Dordevic and R. Kotuk Low-denity parity check code and iterative decoding for long haul optical communication ytem J. Lightwave Technol ( J. Hagenauer and P. Hoeher A Viterbi algorithm with oft deciion output and it application in Proc. IEEE GLOBECOM ( W. E. Ryan Concatenated convolutional code and iterative decoding in Wiley Encyclopedia of Telecommunication J. G. Proaki ed. (John Wiley and Son 00.. S. K. Chilappagari S. Sankaranarayanan and B. Vaic Error floor of LDPC code on binary ymmetric channel accepted for preentation at IEEE Int. Conf. Comm. (ICC 006 Itanbul Turkey on - 5 June B. Vaic S. K. Chilappagari and S. Sankaranarayanan Error floor of LDPC code on binary ymmetric channel preented at the IEEE Comm. Theory Workhop Park City UT June T. Mizuochi et al. Forward error correction baed on block turbo code with 3-bit oft deciion for 0 Gb/ optical communication ytem IEEE J. Sel. Top. Quantum Electron ( M. Manour "Implementation of LDPC decoder" preented at the IEEE Comm. Theory Workhop Park City UT June T. Lee 80 + Gb/ ETDM ytem implementation: an overview of current technology in Proc. OFC 006 Paper no. OTuB3. 6. A. Färbert Application of digital equalization in optical tranmiion ytem in Proc. OFC 006 Paper no. OTuE5. 7. S. Sankaranarayanan I. B. Dordevic and B. Vaic Iteratively decodable code on m-flat for WDM highpeed long-haul tranmiion J. Lightwave Technol (005.. Introduction High-peed optical tranmiion ytem operating at 40 Gb/ or higher are everely limited by intrachannel nonlinearitie uch a intrachannel four-wave mixing (IFWM and intrachannel cro-phae modulation (IXPM [-6]. Approache to deal with intrachannel nonlinearitie may be claified into three broad categorie: (i modulation format [-5] (ii contrained (or line coding [][6] and (iii equalization technique [9]. The IFWM i a phae-enitive effect and the aim of the firt approach i to remove the phae hort-term coherence of the pule emitted in a given neighborhood. The role of contrained coding [] i to avoid thoe waveform in the tranmitted ignal that are mot likely to be received incorrectly. Thi approach ha been carefully examined by the author and ignificant performance improvement ha been demontrated for variou contrained code and diperion map []. The mot efficient way to deal with intrachannel nonlinearitie i to combine the contrained coding and forward error correction (FEC in a revere concatenation cheme []. Although the combined contrained and error correction provide an excellent coding gain it reduce the code rate becaue the total code rate i equal to the product of FEC code rate and contrained code rate. Previou work in nonlinear interymbol interference (ISI reduction at lower bit rate ha involved the ue of equalization [7] and nonlinear cancellation [8]. A drawback of linear equalizer i that they cannot handle non-linear effect while the nonlinear cancellation technique in [8] doe not take into account the effect of pot-curor ISI. The Volterra erie nonlinear equalization technique ued to improve performance in a duobinary modulation cheme may uffer from error propagation due to the nonlinear feedback [9]. Other technique propoed recently include maximum likelihood equence detection (MLSD baed on Viterbi algorithm [9 ] and the turbo-equalization []. The diadvantage of the Viterbi algorithm i that it doe not produce the oft information required for iterative decoding while the turbo-equalization technique propoed in [] in a context of a wirele multipath channel employ a convolutional code of rate ½ which i unacceptably (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 466

3 low for high-peed optical tranmiion. Furthermore the coding gain provided on an AWGN channel i too mall to be of interet in high-peed tranmiion. Similar turbo-equalization cheme have been extenively tudied for a variety of application uch a wirele communication [3] and magnetic recording [4]. In thi paper the turbo equalization cheme [] i modified for fiber-optic communication. Moreover thi paper i concerned with the uppreion of intrachannel nonlinearitie and the reult preented demontrate ignificant performance improvement. The propoed nonlinear ISI cancellation cheme employ the maximum a poteriori probability (MAP ymbol decoding baed on Bahl-Cocke-Jelinek-Raviv (BCJR algorithm [5] while the forward error correction i baed on girth-6 [67] and girth-8 low-denity parity-check (LDPC code [8]. The nonlinear interymbol interference (ISI channel i modeled by a finite tate machine (FSM whoe tranition and output function decribe the dependency of the channel tatitic and the ISI on tranmitted pattern. The BCJR algorithm operate on a trelli of the correponding FSM and create the oft information (detected bit likelihood ued in the iterative decoder. The main feature of the propoed cheme i that it can operate in the regime of very trong intrachannel nonlinearitie where FEC cheme uch a turbo or LDPC code are not deigned to operate. Mot of diperion map ued in 0 Gb/ ytem will uffer from trong intrachannel nonlinearitie if ued for 40 Gb/ tranmiion without any modification the propoed cheme i therefore an excellent candidate for a 40 Gb/ upgrade over exiting 0 Gb/ infratructure. Moreover the propoed cheme require only the modification on receiver ide of a tranmiion ytem. To invetigate the bit error rate (BER degradation due to nonlinear ISI at high bit rate and quantify gain that can be obtained uing the propoed method we have developed an advanced imulator that take into account the effect of the optical and electrical component employed in the ytem. The imulation reult how that the MAP ymbol decoding ued to counter the nonlinear ISI together with LDPC code offer a ignificant performance improvement. The paper i organized a follow. The intrachannel nonlinear effect are briefly introduced in Section. The nonlinear BCJR equalization principle implemented here i introduced in Section 3 while the numerical reult are reported in Section 4. Some important concluion are given in Section 5.. Intrachannel nonlinear effect At high bit rate (40 Gb/ and above the maor nonlinear penaltie are due to intechannel interaction (IFWM and IXPM [-6]. IXPM i caued by modulation of a pule phae by nonlinear interaction with neighboring pule within the channel reulting in timing itter. In IFWM at ufficiently high diperion energy of the pule in reonant poition i tranferred to the middle of a neighboring bit lot cauing either a ghot pule in an empty bit lot or amplitude itter in a non-empty bit lot (in RZ-OOK [-6]. IFWM i caued by dependence of the refractive index of the medium on the intenity of applied electrical field and ha been identified a the maor nonlinear effect that limit tranmiion ditance in peudo-linear fiber optic communication ytem []. To counter diperion in optical fiber diperion managed cheme are deployed. Thee et up conit of alternating pan of fiber having poitive and negative chromatic diperion with the value of reidual diperion being low or zero. Thu the pule that travel through thee ytem undergo alternate widening and compreion cauing interaction between the pule when they overlap. Thee interaction along with nonlinearitie preent lead to energy tranfer that caue the ghot pule phenomenon. It ha been oberved that in RZ-OOK tranmiion the mot evere problem are caued by pule triple at poition k l and m where k = l + m a illutrated in Fig. (a. Thi repreent the reonance condition and create a ghot pule at poition 0. It wa hown in [ 4] that triple that lie cloe to each other caue the highet energy tranfer among pule. Thi cumulative effect of thi ghot pule phenomenon i illutrated in Fig. (b which how the (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 467

4 effect of tranmitting a equence 00. Triple of pule at poition (3- (4-3 (5-4 (6-5 are all in reonance. Thu in the reulting energy tranfer to pule poition 0 the pule at poition loe a large amount of energy ince it i involved in four triple. Figure (b how an ideal cae in the ene that no other pule take part in triple and that effect of other nonlinearitie are not hown. Several method have recently been introduced for reducing the intra-channel fiber nonlinearitie including: novel modulation format [ 4] contrained coding [ 6] deliberate error inertion [] and combined contrained coding and error control coding []. The mot efficient way of dealing with intrachannel nonlinearitie i to combine the contrained coding with forward error correction in o called revered concatenation cheme a explained in []. Thi approach provide the excellent coding gain but reduce the code rate. In Section 3 we provide an efficient way to achieve the large coding gain comparable to that from [] without reducing the code rate due to contrained encoder (a (b Fig.. (a Pule at poition 3 and give rie to a ghot pule at poition 0; (b cumulative effect of reonance poition on poition Combined nonlinear ISI cancellation and forward error correction Bahl Cocke Jelinek and Raviv propoed a (MAP decoding algorithm [5] (known a the BCJR algorithm that can be ued for decoding of equence generated by a finite tate machine. It i an optimal decoding method that minimize the ymbol error probability. Application uggeted in [5] include convolutional and linear block code and recently it ha been hown that BCJR can be ued to uccefully counter the effect of ISI in magnetic recording channel [4]. The output of the channel i decribed by a trelli and BCJR operate on thi trelli to correct the corrupted data. A ignificant benefit of uing the BCJR algorithm compared to Viterbi algorithm i that in addition to detected bit it alo provide bit reliabilitie i.e. oft deciion. Iterative decoding and LDPC coding i currently the mot advanced forward error correction approach but it power can be fully exploited only if bit reliabilitie are upplied to the decoder. Although an optimal method for minimizing the equence error probability the Viterbi algorithm provide only hard deciion thu preventing oft iterative decoding. We propoe to ue the BCJR algorithm to uppre the nonlinear ISI due to intrachannel nonlinearitie. The BCJR algorithm operate on a trelli that i a dicrete dynamical model of the optical channel. Let u uppoe that a diperion map i choen o that each decoded bit i influenced by m neighboring bit from either ide. Let u be th bit in a equence u and y be correponding received ample at the output of the electrical filter. The tranition probability p(y i etimated from imulator by modeling the channel a a finite tate machine ( denote the tate of the channel. It i aumed that m previou and m next bit influence the oberved bit u and the tate of the channel =(u -m u -m+ u u + u +m i determined by a equence of m+ input bit u i {0}. The value m+ i referred to a a memory of the dicrete channel given by the et of tate S. Notice that other approache baed on MLSD or turbo equalization preented in [7-4] ignore the pot-curor ISI (the bit that follow the bit to be decoded. A an illutration Fig. how the conditional probability denity function (PDF of the received ample y given a tate for the following two tate =0 and = and different number of pan for diperion map from Fig. 3. Diperion map in (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 468

5 Fig. 3 i elected in uch a way: (i to keep IXPM low and (ii to keep the pule pread during tranmiion over a D + fiber in order of ten of bit period (rather than hundred of bit period that i common for peudolinear tranmiion. Notice that the memory in both cae i m+=7. The eye diagram after 0 and 30 pan are hown in Fig. (b-(c. The parameter of D + and D - fiber are given in Table. The pan length i et to L=0 km and each pan conit of L/3 km of D + fiber followed by L/3 km of D - fiber. Pre-compenation of -800 p/nm and correponding pot-compenation are alo applied. RZ modulation format of a duty cycle of 33% i oberved and the launched power i et to 0 dbm. Erbium-doped fiber amplifier (EDFA with noie figure of 5 db are deployed after every fiber ection the bandwidth of optical filter i et to 3R b and the bandwidth of electrical filter to 0.65R b with R b being the bit rate (40 Gb/. A expected by increaing the number of pan the ghot pule at the central bit poition grow [ee Fig. (b-(c] cauing the mean of the PDF to hift to the right [Fig. (a]. It i obviou that the commonly ued AWGN aumption i not valid in thi cae. The PDF i obtained by paing random equence through the channel. The length of a equence i 5 and 3 ample per bit are ued in the tranmiion imulation. To etimate the PDF the ample range i uniformly quantized in 64 bin and the number of occurrence of ample in a given bin i counted and normalized with total number of ample. Probability denity function PDF x =0 0 pan = pan =0 30 pan =0 70 pan Voltage v [mv] (a Power P [mw] 4 3 Power P [mw] Time t [p] (b Time t [p] (c Fig.. (a Etimated conditional PDF for tate =0 and = and for different number of pan. Eye diagram after 0 (b and 30 pan (c of diperion map from Fig. 3. (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 469

6 N pan D - D + D - D + Tranmitter Receiver EDFA EDFA EDFA EDFA Fig. 3. Diperion map under tudy Table Fiber parameter Parameter D + FIBER D - FIBER Diperion [p/(nm km] 0-40 Diperion Slope [p/(nm km] Effective Cro-ectional Area [μm ] 0 50 Nonlinear refractive index [m /W] Attenuation Coefficient [db/km] A et of triple (previou tate channel output next tate uniquely define a finite tate machine on which the BCJR operate. A an illutration a trelli for m+=5 i hown in Fig. 4(a. It ha 3 tate ( 0 3 and each tate i given by a different 5-bit pattern. The tate in vertical column repreent all poible tate that the channel (or FSM can take at a given time intant while the labeled edge repreent poible tranition. Neighboring column thu repreent conecutive time intant. For example if the channel i in tate 0 (the bit pattern wa generated and if the next bit i 0 the FSM tay in tate 0 and generate 0 a an output (the middle bit of the final tate. Otherwie the FSM goe to tate (bit pattern 0000 and output again 0. If the FSM i in 6 (the bit pattern 0000 wa generated and if the next bit i 0 then the FSM goe to 0 and output 0 (the middle bit of the terminal tate otherwie it goe to (bit pattern 0000 and output again 0. No other tranition i allowed from 6. Similarly there are two poible tranition from each tate in the trelli. A labeled edge i aigned to each allowed tranition and a received ample correpond to the output ymbol of the branch (the central bit of the terminal tate. Given -the previou tate =(u -m u -m+ u u + u +m -the preent tate u=(u u u n -the tranmitted codeword and y=(y y y n -the received equence the log-likelihood ratio (LLR (denoting the bit reliability of u (= n i calculated a ( L u = max* α ( ' + γ ( ' + β ( max* α ( ' + γ ( ' + β (. ( ' : u = 0 ( ' : u = ( The dahed line in Fig. 4(a correpond to tranition ( : u =0 and the olid line to tranition ( : u =. The forward metric α ( =log p( = y [ee Fig. 4(b] i given by ( = max* ( ' + ( ' α α γ ' n the backward metric β ( =log p( y + = [ee Fig. 4(c] a ( = ( + ( β ' max* β γ ' and the branch metric γ ( i given by ( (3 γ ( = p ( = y = ' = p( y u p( u ( y y +... ym y m l = l l ' log log. p(y u i obtained a explained above u and y correpond to the (4 (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 4630

7 central bit in tate and the initial value are et to 0 = 0 α0( = and 0 The max*-operator i defined a [0] 0 = 0 β n ( =. 0 x y ( xy = xy+ ( + e max* max( log (5. (6 The key difference between the regular BCJR algorithm and the BCJR decribed here i in calculation of LLR. The conventional BCJR [5] calculate LLR of the input bit correponding to the edge while the modified verion calculate the LLR of the output bit correponding to the central bit of terminal tate [ee Fig. 4(a]. Another important difference with repect to turbo equalization propoed in [] i that the BCJR algorithm operate on a trelli that include both pre- and pot-curor ISI [ee Fig. 4(a]. The forward tep [ee Fig. 4(b] and backward tep [ee Fig. 4(c] are the ame a that in the original log-domain BCJR algorithm [0] /0 / /0 / / / 30 0 ( α ( α γ γ ( ( 3 / (a 3 ( = max* ( + ( ( + ( α α γ α γ (b ( = ( + ( ( + ( ' β max* β γ β γ ' ( γ ( γ β β ( ( (c Fig. 4. (a Trelli ued by BCJR (for m+=5 (b the forward tep of BCJR and (c the backward tep of BCJR (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 463

8 A complete block diagram of the propoed cheme i given in Fig. 5. The BCJR LLR output L(u (= n are fed to an iterative LDPC decoder implemented uing meagepaing (MP algorithm. The main idea i to ue BCJR algorithm to partially cancel nonlinear ISI due to intrachannel nonlinearitie and reduce BER to around and then feed bit likelihood obtained from BCJR algorithm into iterative decoder of an LDPC code. For uch input BER the iterative decoding uing LDPC code alone have been hown to markedly improve performance at 40 Gb/ [6-8]. Input from the channel Optical Filter PIN Photodiode Electrical Filter Sampler BCJR Detector MP Decoder Fig. 5. Receiver configuration 4. Performance analyi The LLR of an uncoded ignal are determined by ( = 0 y ( = y p u Lu ( = log p u where the PDF of ZERO and ONE bit are obtained by averaging over all tate in which the middle bit i involved: The hard-deciion are made according to ( = = : = 0 ( ( p y u 0 u p y p. ( Lu < 0 uˆ = 0 otherwie Simulation reult are hown in Fig. 6 for the following clae of LDPC code [6-8]: The lattice LDPC( code of girth 8 and column weight 4 lattice LDPC( of girth 8 an column weight 3 lattice LDPC( code of girth 6 and column weight 3 and PG( 6 baed LDPC( of girth 6. Although much longer the turbo product code (TPC of code rate 0.8 baed on BCH(83xBCH(5639 cheme lag far behind different clae of LDPC code. For example LDPC( code alone outperform TPC by 0.8 db at BER of 0-6 although it i almot 4 time horter. Viterbi decoder operate on trelli hown in Fig. 4(a with the truncation length 64 and perform comparable to BCJR algorithm [ee Fig. 6(a]. Notice however that it doe not provide oft deciion required for oft iterative decoding of the outer LDPC code. For the memory m+=7 the lattice girth-8 LDPC code of rate 0.8 and column weight 4 combined with BCJR algorithm outperform the TPC by.5 db at BER of 0-6 BCJR algorithm alone by 6.8 db and the coding gain over an uncoded ytem i 9. db. By iterating (paing the bit LLR between the BCJR algorithm block and the LDPC decoder (ee Fig. 5 the BER performance can be further improved on the expene of an increaed decoding delay. We refer to thi a an outer iteration to differentiate it from iteration within the meage paing algorithm which are referred to a inner iteration. In the firt outer iteration the LLR from BCJR are paed to and proceed by the meagepaing decoder. We ay that the i-th outer iteration i complete when the extrinic LLR at the output of the MP decoder from the (i--th iteration are proceed by BCJR detector and the BCJR extrinic reliabilitie are paed to and proceed by the MP decoder. The number of inner iteration in the meage-paing decoder i et to 0. The curve with only one outer (7 (8 (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 463

9 iteration are obtained for 5 inner iteration of MP decoder. The coding gain after the firth iteration i 9.7 db at BER of 0-6 [the diamond curve in Fig. 6(a]. The improvement in coding gain over TPC i 3 db (at the ame BER and the improvement over the BCJR detector i 7.3 db. By extrapolating the LDPC( curve down to BER of 0 - the expected coding gain i around 3. db and the improvement over TPC i around 3. db. We have recently hown [-] (ee alo Fig. 7 that the finite geometry code and lattice code of high girth and large column weight do not exhibit an error floor in the region of interet for fibre-optic communication o that the interpolation i utifiable once the waterfall region i reached. The BER performance can be alo improved by increaing the memory of the channel but it reult in an exponential increae of the algorithm complexity. Bit-error rate BER BCJR: m+=3 m+=5 m+=7 m+=9 Viterbi: m+=7 Turbo product code: BCH(83xBCH(5639 BCJR+LDPC( c=4: Memoryle m+=3 m+=5 m+=7 m+=7 (5 th iteration Uncoded Uncoded ignal bit-error rate BER unc (a Bit-error rate BER Turbo product code: BCH(83xBCH(5639 m+=7: BCJR+LDPC( PG( 6 BCJR+LDPC( g=6c=3 BCJR+LDPC( g=8 c=4 BCJR+LDPC( g=8 c=3 MAX-LOG+LDPC( g=8 c=3 BCJR only Uncoded ignal bit-error rate BER unc (b Fig. 6. BER performance of combined BCJR-LDPC cheme (a and BER performance comparion for different LDPC code (b. (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 4633

10 From the numerical reult preented above it follow that the combined BCJR intrachannel cancellation and LDPC coding i an excellent candidate to enable tranmiion in the preence of trong intrachannel nonlinearitie. Moreover it can be ued for upgrading the exiting 0 Gb/ infratructure to 40 Gb/ a explained in Introduction. The BER performance comparion of combined nonlinear ISI cancellation and LDPC coding for different component LDPC code i given in Fig. 5(b. A expected the girth-8 code outperform the girth-6 code. The coding gain of combined BCJR-LDPC cheme for different channel memorie and LDPC( code a component code at BER of 0-6 are ummarized in Table. Table Coding gain of combined BCJR-LDPC cheme for different channel memorie at BER of 0-6 Coding cheme and channel memory Coding gain [db] BCJR algorithm.4 Turbo product code 7.0 LDPC( BCJR+ LDPC( (m+=5 8.3 BCJR+ LDPC( (m+= th iteration of BCJR+LDPC( (m+=7 The max*-operator in (6 involve a two-input max-function and the function for the correction term log(+e - x-y that can be implemented a a lookup table. The performance lo by approximating the max*-function by max-function i found to be negligible [ee Fig. 5(b]. In calculation of BER performance (in Fig. 5 an encoded equence of length 5 i tranmitted many time over the tranmiion ytem for different ASE noie realization. The number of pan i varied from 0 to 70 and the BER of uncoded and coded cae are recorded. In fiber-optic communication the Q-factor i commonly ued a a figure of merit intead of ignal-to-noie ratio. However the Q factor i not an appropriate figure of merit in a highly nonlinear optical channel. The x-axi in Fig. 6 correpond to the BER of an uncoded ignal when both BCJR block and LDPC decoder are omitted. 5. Concluion We have hown that the MAP detection baed on BCJR algorithm upplemented with iterative decoding i able to achieve ignificant performance improvement in ytem heavily degraded by ISI due to intrachannel fiber nonlinearitie and diperion. We note that other technique for uppreion of intrachannel nonlinearitie uch the one baed on Volterra erie method [9] or MLSD [0] do not provide oft output required for oft iterative (turbo or LDPC decoding. The Volterra erie nonlinear equalization technique ued to improve performance in a duobinary modulation cheme in [9] alo may uffer from error propagation due to the feedback. For memorie above m+=5 the complexity of BCJR become large and a implified verion of it namely oft-output Viterbi algorithm (SOVA [9] i more likely to be of interet for practical implementation. Another important concluion i that thi approach may be combined with an optimal diperion map deign which will further improve the BER reult and reduce the complexity by reducing the number of tate in the trelli. A oint deign of a diperion map channel trelli and a LDPC code i an important problem and i left for future reearch. From implementation complexity point of view it hould be noted that the complexity of Chae II algorithm employed in TPC decoder [3] i lower compared to the complexity of MP [4]. However during the decoding proce turbo product decoder (for BCH(83xBCH(5639 employ 39 Chae II block operating in parallel increaing therefore the decoding delay and the circuit ize. Moreover the turbo product code require the ue of interleaver. For more detail about the implementation of MP an intereted reader i referred to [4]. The length of TPC i 3768 and correponding trelli for BCJR detection i (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 4634

11 too complex to be of interet for practical application. Notice that recent advance in ultrahigh peed microelectronic and electro-optic technology allowed ucceful demontration of ETDM-baed optical tranmiion above 80 Gb/ [5] while MLSD i intenively tudied for 40 Gb/ tranmiion [6] uggeting that the nonlinear BCJR equalization cheme propoed in thi paper i timely. To reduce the number of tate required for BCJR equalization the diperion map ha to be carefully deigned o that during tranmiion over D + fiber the pule i pread up to 0 bit intead of everal ten of bit conidered here. In order to demontrate the efficiency of the propoed method diperion map i choen on uch a way that trong IFWM occur. The reult of imulation in thi paper are obtained by maintaining the double preciion of log-likelihood ratio. In our recent article we have hown [7] that proper choice in number of quantization bit reult in negligible BER performance lo. 0 0 Frame error rate (FER PG( 5 -Simulation -Theoretical (SAM 0 - Channel output error rate Fig. 7. Semi-analytic method (SAM for frame-error rate analyi of proective geometry code for hard-deciion decoding uing Gallager B algorithm Acknowledgment Thi work i funded in part by the NSF under Grant ITR and Grant CCR The author would like to thank V. S. Rao for her involvement in an early tage of tudy of different cheme for uppreion of intrachannel nonlinearitie. (C 006 OSA 9 May 006 / Vol. 4 No. / OPTICS EXPRESS 4635