On the Channel Capacity of Multilevel Modulation Schemes with Coherent Detection

Transcription

1 On the Channel Capacity of Multilevel Modulation Scheme with Coherent Detection Ivan B Djordjevic a Lei Xu b Ting Wang b a Univerity of Arizona Department of Electrical & Computer Engineering Tucon AZ USA 857 b NEC Laboratorie America Princeton NJ USA 0850 ABSTRACT We decribe a method to determine the channel capacity of an arbitrary multilevel modulation cheme by modeling the fiber-optic channel a a dynamical nonlinear ISI channel with memory We alo propoe a multilevel turbo-equalization cheme that i able cloely to approach the channel capacity We how that with thi cheme we are able traightforwardly to upgrade currently intalled 0 Gb/ optical tranmiion ytem to 00 Gb/ and even with mall memory aumption we are able to achieve 00 Gb/ per DWDM channel tranmiion over 9600 km Keyword: Fiber-optic communication channel capacity modulation forward error correction (FEC low-denity parity-check (LDPC code INTRODUCTION A data rate tranmitted over the optical fiber increae a fundamental quetion about phyical limitation of optical fiber arie The problem of determining capacity of optical tranmiion ytem ha been addreed by numerou reearcher -8 The main approach until recently wa to conider amplified pontaneou emiion (ASE noie a a predominant effect and to oberve the fiber nonlinearitie a the perturbation of linear cae or a the multiplicative noie Eiambre at al calculate the channel capacity for nonbinary modulation format by auming that the optical channel i dicrete and memoryle The nonlinear nature of the propagation of light in optical fiber ytem play a crucial role in limiting the capacity and make thee limit difficult to calculate In our recent publication 67 we propoed a method to determine the achievable information rate (lower bound on channel capacity for high-peed optical tranmiion uing the finite tate machine approach when the combined effect of ASE noie Kerr nonlinearity timulated Raman cattering chromatic diperion polarization mode diperion (PMD and (optical/electrical filtering are taken into account In thee publication the channel capacity tudy ha been performed for binary modulation format only In thi paper we decribe a method to determine the independent identically ditributed (IID channel capacity of arbitrary multilevel modulation cheme by modeling the fiber-optic channel a dynamical nonlinear interymbol interference (ISI channel with memory Moreover we decribe a coding cheme that i able i able cloely to approach the IID channel capacity Thi cheme i baed on multilevel turbo equalization and coded-modulation and employ the bet known low-denity parity check (LDPC code a channel code We how that with thi cheme we are able traightforwardly to upgrade currently intalled 0 Gb/ optical tranmiion ytem to 00 Gb/ We how that implementing the propoed cheme even with mall memory aumption we are able to achieve 00 Gb/ per DWDM channel tranmiion over 9600 km The paper i organized a follow In Section we firt provide channel capacity preliminarie We then decribe how to determine the channel capacity of fiber-optic communication ytem for arbitrary multilevel modulation format The LDPC-coded turbo equalization cheme baed on multilevel maximum a poteriori probability (MAP equalizer implemented uing BCJR algorithm 9 i decribed in Section In ame ection we tudy the efficiency of the LDPCcoded turbo equalizer in uppreion of fiber nonlinearitie Finally in Section ome important concluding remark are given ivan@ecearizonaedu; phone ; fax ; wwwecearizonaedu/~ivan/ Optical Tranmiion Sytem Switching and Subytem VII edited by Dominique Chiaroni Proc of SPIE-OSA-IEEE Aia Communication and Photonic SPIE Vol W 009 SPIE-OSA-IEEE CCC code: X/09/$8 doi: 07/85 SPIE-OSA-IEEE/ Vol W- Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

2 CHANNEL CAPACITY OF MULTILEVEL MODULATION SCHEMES Channel Capacity Preliminarie The channel code conider whole tranmiion ytem a a dicrete channel in which the ize of input and output alphabet are finite Two example of uch channel are hown in Fig In Fig (a we how an example a dicrete memoryle channel (DMC which i characterized by channel (tranition probabilitie Let X={x 0 x x I- } denote the channel input alphabet and Y={y 0 y y J- } denote the channel output alphabet Thi channel i completely characterized by the following et of tranition probabilitie: ( j i = ( = j = i ( j i { } { } p y x P Y y X x 0 p y x i 0 I j 0 J where I and J denote the ize of input and output alphabet repectively The tranition probability p(y j x i repreent the conditional probability that channel output Y=y j given the channel input X=x i The channel introduce the error and if j i the correponding p(y j x i repreent the conditional probability of error while for j=i it repreent the conditional probability of correct reception For I=J the average ymbol error probability i defined a the probability that output random variable Y j i different from input random variable X i with averaging being performed for all j i: I J = ( ( P p x p y x e i j i i= 0 j= 0 j i { p( xi = P( X = xi i= I } where the input are elected from the following ditribution ; 0 with p(x i being known a a priori probability of input ymbol x i The correponding probabilitie of output ymbol can be calculated by: I I ( j ( j i ( i ( j i ( i p y = P Y = y X = x P X = x = p y x p x ; j = 0 J i= 0 i= 0 ( The deciion rule that minimize average ymbol error probability ( denoted a D(y j =x * i known a maximum a poteriori (MAP rule and can be formulated a follow: ( j ( j ( i j D y = x*: P x* y P x y i= 0 I Therefore the ymbol error probability P e will be minimal when to every output ymbol y j the input ymbol x * i aigned having larget a poteriori probability P(x * y j By uing the Baye rule the equation ( can be re-written a ( j ( ( P( yj ( ( P( yj P yj x* P x* P yj xi P xi D y = x*: i= 0 I ( ( ( (5 If all input ymbol are equally likely P(x i =/I (i=0 I- the correponding deciion rule i known a maximumlikelihood (ML deciion rule: ( j ( j ( j i D y = x*: P y x* P y x i = 0 I (6 In Fig (b we how a dicrete channel model with memory 7 which i more uitable for optical communication for binary tranmiion becaue the optical channel i eentially the channel with memory We aume that the optical channel ha the memory equal to m+ with m being the number of bit that influence the oberved bit from both ide Thi dynamical trelli i uniquely defined by the et of previou tate the next tate in addition to the channel output The tate (the bit-pattern configuration in the trelli i defined a j =(x j-m x j-m+ x j x j+ x j+m =x[j-mj+m] where x k X={0} An example trelli of memory m+=5 i hown in Fig (b The trelli ha 5 = tate ( 0 each of which correpond to a different 5-bit pattern For the complete decription of the trelli the tranition probability denity function (PDF p(y j x j =p(y j S can be determined from collected hitogram where y j repreent the ample that correpond to the tranmitted bit x j and S i the et of tate in the trelli SPIE-OSA-IEEE/ Vol W- Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

3 /0 / X p(y j x i Y /0 /0 / / 0/ (a (b Figure Two example of dicrete channel: (a dicrete memoryle channel (DMC and (b dicrete channel with memory for binary tranmiion decribed a dynamic trelli One important figure of merit for DMC i the amount information conveyed by the channel which i known a the mutual information and it i defined a I J I I( XY ; = H( X H( X Y = p( xi log p( yj p( xi yj log i= 0 p ( xi j= 0 i= 0 p( xi yj (7 where H(X denote the uncertainty about the channel input before oberving the channel output alo known a entropy; while H(X Y denote the conditional entropy or the amount of uncertainty remaining about the channel input after the channel output ha been received Therefore the mutual information repreent the amount of information (per ymbol that i conveyed by the channel that i the uncertainty about the channel input that i reolved by oberving the channel output The mutual information can be interpreted by mean of Venn diagram hown in Fig (a The left circle repreent the entropy of channel input the right circle repreent the entropy of channel output and the mutual information i obtained in interection of thee two circle Another interpretation due to Ingel i hown in Fig (b The mutual information ie the information conveyed by the channel i obtained a output information minu information lot in the channel It i clear from equation (7 that mutual information i independent on the channel and omeone may try to maximize the information conveyed by the channel to obtain the o called channel capacity: I ( ( ( C = max I XY ; ; ubject to : p xi 0 p xi = { px ( i } i= 0 (8 Now we have built enough knowledge to formulate the channel coding theorem : Let a dicrete memoryle ource with an alphabet S have entropy H(S and emit the ymbol every T econd Let a dicrete memoryle channel have capacity C and be ued once in T c econd Then if H(S/T C/T c (9 SPIE-OSA-IEEE/ Vol W- Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

4 there exit a coding cheme for which the ource output can be tranmitted over the channel and recontructed with an arbitrary mall probability of error The parameter H(S/T i related to the average information rate while the parameter C/T c i related to the channel capacity per unit time For binary ymmetric channel (I=J= the inequality (9 imply become where R i the code rate introduced above R C (0 Unwanted information due to noie H(XIY H(XY H(X H(Y Input information H(X Optical channel Output information H(Y H(X Y I(X;Y H(Y X Information lot in channel H(YIX (a Figure Interpretation of the mutual information: (a uing Venn diagram and (b uing the approach due to Ingel (b Figure Minimum BER againt optical SNR for different code rate value (for BPSK at 0 Gb/ Another very much important theorem i the Shannon third theorem alo known a the information capacity theorem and can be formulated a follow : The information capacity of a continuou channel of bandwidth B Hz perturbed by AWGN of PSD N 0 / and limited in bandwidth B i given by SPIE-OSA-IEEE/ Vol W- Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

5 P C = Blog + [bit/] NB 0 ( where P i the average tranmitted power Thi theorem repreent remarkable reult of information theory becaue it connect all important ytem parameter (tranmitted power channel bandwidth and noie power pectral denity in only one formula What i alo intereting i that LDPC code can approach the Shannon limit within 0005dB By uing the Eq ( and Fano inequality ( X Y ( + log ( ( = log ( log ( H H P P I H P P P P P e e e e e e e for amplified pontaneou emiion (ASE noie dominated cenario and binary phae-hift keying (BPSK at 0 Gb/ in Fig we report the minimum BER againt optical SNR for different code rate Calculation of Information Capacity of Multilevel Modulation Scheme by Forward Recurion of BCJR Algorithm Here we addre the problem of calculating of channel capacity of multilevel modulation cheme for an IID information ource in literature alo known a the achievable information rate (ee Djordjevic et al 67 and reference therein The IID channel capacity repreent a lower bound on channel capacity To calculate the IID channel capacity we model the whole tranmiion ytem a the dynamical ISI channel in which m previou and next m ymbol influence the oberved ymbol The optical communication ytem i characterized by the conditional probability denity function (PDF of the output complex vector of ample y=(y y n where y i =(Re{y i } Im{y i } Y given the ource equence x=(x x n x i X={0 M-} The et X repreent the et of indice of contellation point in correponding M- ary two-dimenional ignal contellation diagram (uch a M-ary phae-hift keying (PSK M-ary quadrature-amplitude modulation (QAM or M-ary polarization-hift keying (PolSK while Y repreent the et of all poible channel output The Re{y i } correpond to the in-phae channel ample and the Im{y i } repreent the quadrature channel ample An example of dynamical channel decription by mean of trelli diagram i hown in Fig (a for -level modulation format uch a QPSK Thi dynamical trelli i uniquely defined by the following triplet: the previou tate the next tate and the channel output The tate in the trelli i defined a j =(x j-m x j-m+ x j x j+ x j+m =x[j-mj+m] where x k denote the index of the ymbol from the following et of poible indice X={0 M-} Every ymbol carrie l=log M bit uing the appropriate mapping rule (natural Gray anti-gray etc The memory of the tate i equal to m+ with m being the number of ymbol that influence the oberved ymbol from both ide The trelli ha M m+ =6 tate ( 0 6 each of which correpond to a different -ymbol pattern (configuration The tate index i determined by conidering (m+ ymbol a digit in numerical ytem with the bae M For example in Fig the quaternary numerical ytem (with the bae i ued The left column in dynamic trelli repreent the current tate and the right column denote the terminal tate The branche are labeled by two ymbol the firt ymbol i an input ymbol (the blue ymbol and the output ymbol i the central ymbol of terminal tate (the red ymbol For the complete decription of the dynamical trelli the tranition PDF p(y j x j =p(y j S are needed; where S i the et of tate in the trelli The conditional PDF can be determined by uing intanton-edgeworth expanion method we propoed in Ivkovic et al 0 The number of edge originating in any of the left-column tate i M and the number of merging edge in arbitrary terminal tate i alo M The information rate can be calculated a already introduced in (7 by: I(Y;X = H(Y - H(Y X ( where H(U = E(log P(U denote the entropy of a random variable U and E( denote the mathematical expectation operator By uing the Shannon-McMillan-Brieman theorem that tate : ( ( Y E log P = lim (/ n log P( y[ n] n the information rate can be determined by calculating log (P(y[n] by propagating the ufficiently long ource equence By ubtituting Eq ( into Eq ( we obtain the following expreion uitable for practical calculation of IID channel capacity ( ( SPIE-OSA-IEEE/ Vol W-5 Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

6 n ( ( i [ ] n I( Y ; X = lim log P yi [ i ] [ n] log P y i (5 n n y x y i= i= The firt term in (5 can be traightforwardly calculated from conditional PDF becaue P(y i y[i-]x[n] =P(y i x[imi+m]=p(y i To calculate log P(y i y[i-] we ue the forward recurion of the multilevel BCJR algorithm 9 wherein the forward metric α j (=log{p( j =y[j]} (j= n and the branch metric γ j ( =log[p( j =y j j- = ] are defined a follow: αj( = max* αj ( ' + γ j( ' logm ' ( j ( γ j( ' = log p y x[ j m j+ m] 6 where the max * -operator i defined by max * (xy=log(e x +e y =max(xy+log[+exp(- x-y ] The ith term log P(y i y[i-] can be calculated iteratively by ( y[ ] = ( log P yi i max* αi (7 where max * -operator wa applied for all S (S denote the et of tate in the trelli hown in Fig (a An example of forward recurion tep for -level modulation format (QPSK i hown in Fig (b where denote an arbitrary terminal tate which ha M= edge originating from correponding initial tate denoted a and The forward metric of tate in jth tep (j= n i updated by preerving the maximum term (in max*-ene α j- ( k +γ j ( k (k= The procedure i repeated for every tate in column of terminal tate of jth tep (j= n /0 /0 /0 0/ /0 0/ / / / / / α j ( γ j ( α j ( γ ( j γ j ( α j ( γ j ( α j ( j max* j ( j( j ( j( α j ( + γ ( ( ( j α j + γ j α ( = α + γ α + γ (a (b Figure (a Trelli decription of fiber-optic channel and (b the forward recurion tep for M=-level BCJR equalizer In Fig 5 we how the IID channel capacity againt the number of pan for diperion map hown in Fig 6 and QPSK modulation format of aggregate data rate 00 Gb/ for two different memorie in trelli decription of the channel The pan length i et to L=0 km and each pan conit of L/ km of D + fiber followed by L/ km of D - fiber with precompenation of -600 p/nm and correponding pot-compenation The diperion map i compoed of periodically deployed ection of D + and D - fiber a hown in Fig 6 The fiber parameter are given in Table EDFA with noie figure of 5 db are deployed after every fiber ection the bandwidth of optical filter i et to R and the bandwidth of SPIE-OSA-IEEE/ Vol W-6 Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

7 electrical filter to 07R with R being the ymbol rate that wa et to 50 Giga ymbol/ We ee that the total tranmiion ditance for trelli memory m= and channel code of code rate R=08 i 9600 km which i 00 km better than that for trelli memory m=0 (ie the memoryle cae IID infromation capacity C [bit/channel ue] QPSK (dip map Fig 6: m= m= Total tranmiion ditance L [km] Figure 5 IID information capacity for QPSK and 8PSK of ymbol rate 50 GS/ againt the tranmiion ditance N pan D - D + D - D + Tranmitter Receiver EDFA EDFA EDFA EDFA Figure 6 Diperion map under conideration Table Fiber parameter Parameter D + FIBER D - FIBER Diperion [p/(nm km] 0-0 Diperion Slope [p/(nm km] Effective Cro-ectional Area [μm ] 0 50 Nonlinear Refractive Index [m /W] Attenuation Coefficient [db/km] LDPC-CODED MULTILEVEL TURBO EQUALIZATION We further decribe a channel capacity approaching coding cheme The propoed cheme hown in Fig 7 i baed on multilevel (M> maximum a poteriori probability (MAP turbo equalization It i compoed of two ingredient: (i the multilevel BCJR algorithm baed equalizer and (ii the LDPC decoder The BCJR equalizer operate on trelli channel decription given above (ee Fig (a and provide oft ymbol log-likelihood ratio (LLR ued in LDPC decoding SPIE-OSA-IEEE/ Vol W-7 Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

8 proce The channel code are baed on large girth LDPC code 7 The ue of large girth code increae the minimum ditance and de-correlate the extrinic info in LDPC decoding proce The extrinic information tranfer (EXIT chart approach due to ten Brink i ued to match the LDPC decoder for large-girth quai-cyclic LDPC code and multilevel BCJR equalizer The reult of imulation for a ingle-channel optical QPSK tranmiion ytem operating at 50 Giga ymbol/ with diperion map decribed above are hown in Fig 8 The number of pan wa changed from to 8 the uncoded BER at 50 Giga ymbol/ and BER after LDPC decoding at line rate R l =R /R (R =50 Giga ymbol/ R i the code rate were calculated and given in Fig 8 againt the number of pan We can ee that for pan -evel BCJR equalizer with memory m= provide more than one order in magnitude improvement in BER over memoryle cae (m=0 For the multilevel turbo equalization cheme baed on -level BCJR equalizer of memory m=0 and the LDPC( code of girth-0 and column weight we achieve tranmiion over 55 pan (6600 km without any error On the other hand for the turbo equalization cheme baed on -level BCJR equalizer of memory m= and the ame LDPC code we are able to achieve 860 km of error free tranmiion at aggregate rate of 00 Gb/ To achieve the channel capacity (9600 km with code rate R=08 the girth- LDPC code of length above are needed From SMF j S Si π/ i = Si e ϕ j L = L e ϕ L From local laer Figure 7 Propoed multilevel turbo equalization cheme { S i L } Re * { S i L } Im * Symbol-level de-interleaving + Multilevel BCJR Equalizer Extrinic Symbol LLR Calculation Bit LLR Calculation LDPC Decoder LDPC Decoder l l Bit-error ratio BER Total tranmiion ditance L [km] -level BCJR equalizer: m+= m+= -level turbo equalizer: m+= m+= Figure 8 BER performance of -level turbo equalizer CONCLUSION In thi paper we decribed a method to determine the channel capacity of arbitrary multilevel modulation cheme by modeling the fiber-optic channel a dynamical nonlinear ISI channel with memory In addition we propoed a multilevel turbo-equalization cheme that i able cloely to approach the channel capacity Thi method i univeral and applicable to arbitrary multilevel modulation with both direct detection and coherent detection Thi cheme can be ued to imultaneouly compenate for fiber nonlinearitie PMD and chromatic diperion We demontrated by Monte Carlo imulation that thi cheme i efficient in compenation of fiber nonlinearitie We have hown the even with mall memory aumption we are able to achieve 00 Gb/ per DWDM channel tranmiion over 9600 km We howed that SPIE-OSA-IEEE/ Vol W-8 Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue:

9 with thi cheme we can traightforwardly upgrade currently intalled 0 Gb/ optical tranmiion ytem to 00 Gb/ In our recent paper 6 we have hown that imilar cheme can be ued to compenate for PMD and any imbalance in phae between I and Q channel REFERENCES Narimanov E E and Mitra P The channel capacity of a fiber optic communication ytem: perturbation theory IEEE/OSA J Lightwave Technol 0( 50 57(00 Narimanov E and Patel P Channel capacity of fiber optic communication ytem: WDM v TDM in Proc Conf on Laer and Electro-Optic (CLEO '0 00 pp Mitra P P and Stark J B Nonlinear limit to the information capacity of optical fiber communication Nature (00 Turityn K S Derevyanko S A Yurkevich I V and Turityn S K Information capacity of optical fiber channel with zero average diperion Phy Rev Lett 9 ( (00 5 Eiambre R-J Fochini G J Kramer G and Winzer P J Capacity limit of information tranport in fiberoptic network Phy Rev Lett (008 6 Djordjevic I B Vaic B Ivkovic M and Gabitov I Achievable information rate for high-peed long-haul optical tranmiion IEEE/OSA J Lightw Technol ( (005 7 I B Djordjevic L L Minkov and H G Bathon Mitigation of linear and nonlinear impairment in high-peed optical network by uing LDPC-coded turbo equalization IEEE J Sel Area Comm 6(6 7-8 (008 8 Ivkovic M Djordjevic I B and Vaic B Calculation of achievable information rate of long-haul optical tranmiion ytem uing intanton approach IEEE/OSA J Lightw Technol 5( (007 9 L R Bahl J Cocke F Jelinek and J Raviv Optimal decoding of linear code for minimizing ymbol error rate IEEE Tran Inform Theory IT-0( 8-87 (97 0 Ivkovic M Djordjevic I Rajkovic P and Vaic B Pule energy probability denity function for long-haul optical fiber tranmiion ytem by uing intanton and Edgeworth expanion IEEE Photon Technol Lett 9( (007 Cover T M and Thoma J A [Element of Information Theory] New York: Wiley 99 Ingel F M [Information and Coding Theory] Scranton: Intext Educational Publiher 97 Chung S et al On the deign of low-denity parity-check code within 0005 db of the Shannon Limit IEEE Comm Lett 5( (00 Foorier M P C Quai-cyclic low-denity parity-check code from circulant permutation matricie IEEE Tran Inform Theory 50( (00 5 ten Brink S Convergence behavior of iteratively decoded parallel concatenated code IEEE Tran Comm 0( (00 6 Minkov L L Djordjevic I B Xu L and Wang T PMD compenation in polarization multiplexed multilevel modulation by turbo equalization IEEE Photon Technol Lett accepted for publication SPIE-OSA-IEEE/ Vol W-9 Downloaded from SPIE Digital Library on 09 Feb 00 to 5055 Term of Ue: