J.-G.Zhang ndexing terms: Error-correction codes, Code-division multiple-access, Optical multiple-access interference, Fibre-optic Abstract: n code-division multiple-access (CDMA) using all-optical signal processing, the optical multiple-access interference (OMA) degrades the system performance and can ultimately limit the number of active users. To reduce the effect of OMA, error-correction codes are used in both asynchronous and synchronous fibre-optic CDMA. t is shown that the use of asymmetric error-correction binary block codes can not only effectively reduce the bit-error rate, but also increase the maximum number of active users in a constant-bandwidth network more efficiently than using symmetric error-correction binary block codes. Therefore, this permits implementation of a cost-effective fibre-optic CDMA network. 1 ntroduction Code-division multiple-access (CDMA) techniques have recently been proposed for use in fibre-optic networks with all-optical signal processing which can provide a very high throughput [l-51. n such optical CDMA (OCDMA), the bit-error-rate (BER) performance is degraded by the optical multiple-access interference (OMA) which comes from all the other active users [14]. This in turn ultimately limits the number of active users in a given network. n principle, the weight w of an OCDMA address code can be increased to reduce the BER for the fixed number of active users in OCDMA using either a prime code [l, 51 or an optical orthogonal code (OOC) of A, = A, = 1 [3, 41, where A, and Ac are auto- and crosscorrelation constraints, respectively. However, the use of a larger w results in higher power losses at OCDMA encoder and decoder. Note that optical power loss also limits the capacity of an all-optical CDMA network, which is a power-limited system rather than a bandwidth-limited one. Moreover, using a larger w causes a higher system cost, because more optical delay lines are employed in the OCDMA system and optical 1 x w splitterlw x 1 combiner of a higher w are required [l]. 0 EE, 1997 EE Proceedings online no. 19971454 Paper received 16th October 1995 The author is with Telecommunications Program, School of Advanced Technologies, Asian nstitute of Technology, P.O. Box 4, Klong Luang, Pathumthani 12120, Thailand 316 To combat the OMA effectively, error-correction codes (ECCs) can be used in OCDMA [6-81. This will permit choice of a lower weight for OCDMA address codes to reduce the complexity and power loss of the OCDMA encoderidecoder. As will be explained below, an error-correction code is used before OCDMA encoding is done at each transmitter and after OCDMA decoding is performed at each receiver. Consequently, the encoders and decoders for an errorcorrection code operate at a rate which is just higher than the users data rate by a factor of llr, where R is the code rate of an ECC. Since the value of 1/R is normally chosen to be 5 or smaller in this paper, the symbol rate of ECC-encoded signal can be still a reasonable value (e.g. a few hundred Mbitis) for practical applications, and it is therefore feasible for ECC encodersidecoders to be implemented by using conventional electronic circuits to avoid optical power loss completely at this stage. Therefore, the use of errorcorrection codes permits implementation of cost-effective OCDMA networks. n this paper, a special type of error-correction codes is introduced, called asymmetric error-correction (AEC) binary block codes, to reduce the BER of OCDMA and to increase the maximum number of active users for a given BER (even in a constant-bandwidth network) more efficiently than symmetric error-correction (SEC) binary block codes. 2 Channel model for incoherent optical CDMA An incoherent OCDMA network using optical processing is illustrated in Fig. 1, where data messages at the active transmitters using an on-off key are first encoded with their desired OCDMA address codewords and are then distributed to each receiver by an optical star coupler. The maximum number of users is equal to NmrLx, of which N,,, users are allowed to be active for the BER nterleaving is used to randomise the interference pulse patterns which may be introduced by using an error-correction code. Each transmitter is assigned an unique codeword from an OCDMA address code. dentical data rates are assumed for all the users and the same effective average power is assumed at the input of each receiver [4]. Moreover, all the optical sources at transmitters are incoherent so that optical-power signals will incoherently add in intensity at an OCDMA decoder. As pointed out in [14], only the OMA is considered as a source of BER degradation. This is because the effects EE Pro,.-Commun., Vol. 144, No. 5, October 1997
ECC 4 nput lnterleaver data, encoder - 1 OCDMA encoder - \ OCDMA optical star --c coupler - Fig. 1 OCDMA network using a binary block code of thermal or quantum noise on the BER of OCDMA can be reduced by increasing the transmitted power or using optjcal amplifiers while the OMA results are unaffected. n OCDMA using an error-correction binary block code, each data bit from an interleaver is transformed into the desired destination codeword by using an OCDMA encoder. No light is actually transmitted when each data bit 0 is issued by the interleaver, and the binary bit 0 might be mistaken for a binary bit 1 if OMA signals are strong enough to cause a false detection (called 0-error) at the receiver. n contrast, a false detection of the data bit is impossible. This is because, with incoherent optical signal processing, light powers always add up [l, 21. The model of incoherent OCDMA using on-off key is thus an asymmetric binary channel (called Z- channel), as shown in Fig. 2. At the receiver, the output of an optical correlator consists of the desired signal and OMA signals, so detection errors may occur. After deinterleaving, an ECC decoder is used for correcting t or fewer 0-errors which are assumed to be independent, 1-0-errors [9]. The use of conventional binary block codes in OCDMA can improve the BER performance, but it is less efficient. This is because only part of their error-correction capability is utilised on an asymmetric binary channel [lo]. To achieve higher coding efficiency, a special type of binary block code, which is useful for correcting asymmetric errors, is introduced to incoherent OCDMA in this paper. A [no, k,, A] asymmetric error-correction (AEC) binary block code of minimum asymmetric distance A can correct tu or fewer 0-errors [ll], where t, is expressed as t,=n-1 (2) For convenience, let As(n, t) and A,(n, t) denote the maximum number of codewords in a code of length n correcting up to t symmetric and asymmetric errors, respectively. Thus, they satisfy [9] As(n, t) L A, (n, t) 5 min{(t+ l)a,(n,t);a,(n+t,t)} for 15 t 5 n (3) Since, for a binary block code of n and k, the number of necessary codewords is equal to 2k which must be less than or equal to A,(n, t) or A,(n, t), eqn. 3 states that, for given n and t, a [n, k,, A] AEC code of A = t + 1 may exist with k, > k,, and therefore, can achieve a higher code rate R than an ordinary [n, k,, d] SEC code of db 2t + 1. 4 BER performance of uncoded OCDMA 0 0 Fig.2 Z-channel 3 Binary block coides for symmetric and asymmetric error cclrrection There are several strategies to apply ECCs to OCDMA. Some may treat the incoherent OCDMA channel as a binary symmetric channel, and use, for example, conventional [n,, k,, d] symmetric error-correction (SEC) binary block codes with minimum distance d for correcting up to t, symmetric errors as encountered in conventional electronic communication, where n, is the block length, k, is the information length, and t, is given by [9] Here the symbol 1x1 denotes the integer portion of an real value x. This code can necessarily correct up to t, EE Proc.-Commun., Vol. 144, No. 5, October 1997 The exact performance analysis of the generalised OOC-based OCDMA is very difficult, so the bounds on the BER are normally used for OOC [3, 41. However, the Gaussian approximation to the density function of the OMA is usually employed for analysing the BER of prime-code-based OCDMA [l, 5, 121, because it is easy to use and to compute. This approximation is valid for large values of Nu,, according to the well known central-limit theorem [l, 5, 121. n the following, OCDMA with optical linear correlation receivers and OOCs or prime codes are considered. 4. OOC-based OCDMA without ECCs The number of codewords N,,, for a (F, K, 1) optical orthogonal code (OOC) of given length F and weight K can be upper bounded by [2, 31 (4) 317
For convenience of analysis, a perfect optimal OOC (PO-OOC) is defined as the (F, K, 1) OOC having length F F = NmazK(K - 1) + 1 (5) for PO-OOCs considered here, N,,, is chosen to be equal to Nm,, i.e. all the transmitters are active in a given OCDMA network. n this paper, PO-OOCs and the chip-synchronousinterference situation (called case A) are considered for OOC-based OCDMA. Note that, when a data bit '1' is transmitted by using an OCDMA-address codeword, it can be always recovered correctly at the receiver using an incoherent optical-correlation decoder, as explained in Section 2. Thus, the error probability when a data bit '1' is sent, i.e. p(0 l), is equal to zero. The BER derived from case A is an upper bound on the exact BER of OOC-based OCDMA [3]. n case A, the probability density function for OMA signal l is expressed as [4] 5.7 OCDMA with symmetric errorcorrection codes For given block length n, and information length k,, [n,, k,, d,] SEC binary linear block codes with the maximum 'minimum distance' d, have the best SEC capability among ordinary binary linear block codes, where d, is defined as [ 131 dm = max(d1there exists a [ns, k,, 6] code} 1 k, ns (9) The probability of receiving the codeword of a block code incorrectly is given by where p is the error probability for a binary channel with OMA in coded OCDMA, i.e. p = p(1 0) = p(0 1) = 2P,, and P, are given by eqns. 7 and 8 for OOC and prime code, respectively. An upper bound on the post-decoding BER of SECcoded OCDMA can be obtained by assuming that j incorrect decoding events produce j + t, postdecoding errors when j > t,. Then the post-decoding BER of SEC-coded OCDMA can be expressed as [14, 151 and the BER of uncoded OCDMA with OOC and optical linear-correlation receivers is therefore given by pe = P(llO).P(O) +P(Ol').P(') = P(1 O).P(O) where Th is a threshold level which takes K - 1 here, because the performance degradation is caused by the OMA [24]. 4.2 Prime-code-based OCDMA without ECCs Let P be a prime number. The maximum number of users N,,, of a prime code with P is equal to P for asynchronous OCDMA and P2 for synchronous OCDMA [l, 51, respectively. Using a Gaussian distribution for OMA, the BERs of both OCDMA can be written using a prime code as [5] 1 /-P/dmx P, = - exp(-z2/2)dz (8) Jz;; --oo where r = 1.16 and 1 for asynchronous and synchronous OCDMA, respectively. 5 BER performance of coded OCDMA The performance of coded OCDMA is evaluated with both SEC and AEC binary block codes, respectively. As shown in Fig. 1, after deinterleaving, the coding channel for OCDMA is assumed to be a discrete memoryless, binary channel where the independent errors occur. 318 5.2 OCDMA with asymmetric errorcorrection codes Since it is too difficult to know code-weight distributions for general [n,, k,, A] AEC binary block codes, it is possible to derive a simpler, but less tight, bound for AEC-coded OCDMA. This can be done by considering 0-errors to occur probably in n, coded bits per AEC codeword. Thus, an upper bound on the postdecoding BER of AEC-coded OCDMA can be obtained by making the pessimistic assumption that a pattern ofj (> t,) channel errors will cause the decoded word to differ from the correct word in j + t, positions and a fraction 0' + t,)/n, of the k, information bits to be decoded incorrectly [ 151: where p = p(1 0) = 2Pe for AEC codes. 6 Numerical results and discussions The binary block codes principally considered are those having moderate block length and short information length, because encoders and decoders for such errorcorrection codes are simple and can be suitable for fast parallel decoding algorithms which will be used in high-speed networks. For convenience, a bandwidthexpansion factor with respect to source data signal is defined as: (i) & f F (or P2) for uncoded OCDMA using a perfect optimal OOC (or a prime code); (ii) p, FJR (or P2/R) for coded OCDMA using a (Fe, K,, 1) PO- OOC (or a prime code) and an error-correction code of rate R. EE Pro,.-Commun., Vol. 144, No. 5, October 1997
For convenience, the term 'equivalent PO-OOC' is introduced. The 'equivalent PO-OOC' for a system using a (F, K, 1) PO-OOC and an ECC of rate R is defined as the PO-OOC which, without using ECC, can achieve the same BER performance as the given OCDMA system using ECC. For OOC-based OCDMA, a saving-bandwidth coefficient q can be defined as a rl = Pu,e/Pc (13) where p,,, is the factor pu obtained when using an 'equivalent PO-OOC'. The larger q, the more convenient is using the ECC. f q < 1, the use of PO-OOC without ECC is better than using the ECC in given OOC. Numerical examples which show the benefits of coded OCDMA are given in Table 1. The Table 1: Performance improvements of OCDMA by using error-correction binary block codes Uncoded Coded to achieve BER < 1.0 x (see note) M w P, [n,, ka, A AEC code parameters of AEC and SEC binary block codes are taken from [11, 131, where SEC codes have the maximum d, for given values of n, and k, [13], but for AEC codes the lower bound on the code size is usually considered [ 1 ]. n uncoded OOC-based OCDMA, to guarantee that the BER s over a wide range of Nmax, PO-OOCs with a K 2 10 are usually required due to the phenomenon of 'BER floors', as shown in Fig. 3. Unfortunately, the structure of OCDMA encoders/ decoders is more complex when K becomes larger [l]. Moreover, the cost and power loss of OCDMA encodersidecoders increase with K. On the other hand, for a lower K the BER is high. From Table 1, it can be seen that the use of both AEC binary block codes and SEC binary block codes can greatly reduce the effects of [n,, ks, d,,,,,l SEC code ne k, A Dc rl ns ks 4" Dc rl 15 4 1.09~ 16 2 8 1448 0.93 26 2 17 2353 0.46" 29 2 19 2624.5 0.51 5 1.06~ 8 2 4 1204 0.90" 14 2 9 2107 0.64t 18 5 5 1083.6 1.25 20 5 9 1204 1.12 23 7 5 989 1.37 23 7 9 989 1.37 6 7.49~ 6 2 3 1353 l.oot 8 2 5 1804 0.75 15 8 3 845.6 1.60 16 8 5 902 1.50 23 14 3 740.9 1.46" 23 14 5 740.9 1.46" 7 3.93 x 5 2 2 1577.5 0.86 5 2 3 1577.5 0.86 15 11 2 860.5 1.26" 15 11 3 860.5 1.26" 23 18 2 806.3 1.34" 23 18 3 806.3 1.34" 30 4 1.29~ 16 2 8 2888 0.94" 29 2 19 5234.5 0.52" 49 7 23 2527 1.07" 5 1.47~ 10 2 5 16 4 5 18 5 5 6 1.27~ 6 2 3 13 6 3 23 12 4 7 8.70 x 5 2 2 15 8 3 23 14 3 3005 2404 2163.6 2703 1952.2 1726.9 3152.5 2364.4 2071.6 l.lot 14 2 9 1.12 19 4 9 1.25 20 5 9 1.00 8 2 5 1.38" 13 5 5 1.9t 23 12 7 0.86" 5 2 3 1.68 16 8 5 1.91* 23 14 5 4207 0.64 2854.8 0.95" 2404 1.12" 3604 0.75 2342.6 1.15" 1726.9 1.9t 3152.5 0.86" 2522 1.57 2071.6 1.91* 50 4 1.38~ 18 2 9 5409 0.83 32 2 21 9616 0.47 55 7 25 4722.1 0.95" 5 1.65~10-~ 10 2 5 5005 l.lot 14 2 9 7007 0.64 16 4 5 4004 1.12 22 4 11 5505.5 l.oot 18 5 5 3603.6 1.25" 23 5 11 4604.6 l.lgt 6 1.52 x 6 2 3 4503 1.00 8 2 5 6004 0.75 16 6 4 4002.7 1.65* 17 6 7 4252.8 1.55* 23 12 4 2876.9 1.9t 23 12 7 2876.9 1.9t 7 1.14~ 6 2 3 6303 1.24 8 2 5 8404 0.93 15 8 3 3939.4 1.68* 16 8 5 4202 1.57* 23 14 3 3451.6 1.91* 23 14 5 3451.6 1.91* Note!: the different marks as signed on the values of r indicate the different ranges of - BERs for coded. No mark signed denotes 1.00 x s BER < 1.00 x '*' denotes 1.00 x s BER < 3.00 x 't' denotes 1.00 x 10-l' s BER < 1.00 x '$' denotes 1.00 x 10-l2 s BER < 1.00 x '5' denotes BER < 1.00 x EE Proc-Commun., Vol. 144, No. 5, October 1997 319
~ uncoded ~ uncoded OMA on the system performance and can guarantee that the BER < lo-' over a wide range of N,,, for low values of K. AEC codes appear to be more suitable for OCDMA applications, because the former can achieve a better q than the latter for given values of k and t (t > 1). From eqn. 13, one can find that the larger q, the less is bandwidth expansion. Although a higher coding efficiency can be achieved by using those error-correction codes with a higher rate R and larger k, their use can result in a longer time delay and a more complex ECC encodeddecoder. When q > 1, the coded OCDMA can accommodate a larger number of users for given values of bandwidth and BER. For example, when K = 5 and N,,, = 30, the use of a [18, 5, 51 AEC code can decrease the BER to 9.0 x 10-lo, whereas (2701, 10, 1) equivalent PO-OOC of p,,, = 2701 are required to obtain a BER = 9.14 x 10-lo. f K = 5 and N,,, increases to 37, the BER = 1.2 x and pc = 2667.6 < p,,, for a [18, 5, 51 AEC code and NmaY = 30. taneously with BER = 2.0 x However, one can use, for example, a very simple [4, 2, 21 AEC binary block code to ensure that the BER = 1.2 x for all 19 users to transmit data simultaneously in a given network of P = 19 without increasing power loss and cost of OCDMA encoderddecoders, compared with using a larger-weight prime code to achieve the same BER. The use of AEC codes in prime-code OCDMA networks can significantly increase the network capacity which is limited by OMA, as shown in Fig. 4 and Table 3. From Fig. 4, one can find that [lo, 4, 31 AEC-coded OCDMA with P = 11 and @, = 302.5 can achieve the same BER as uncoded with P = 19 and pu = 361 for N,,, 5 11, and that the BER of coded with P = 13 and p, = 422.5 is very close to that of uncoded with P = 23 and pu = 529 for N,,, 13. Therefore, cost-effective OCDMA networks can be implemented by using AEC codes. n both cases, pc < pu means that using an AEC code can also reduce the bandwidth expansion compared with the uncoded OCDMA scheme. This improvement is more efficient when an AEC code of higher code rate is used. O r P.5 " li W m 10-7 - 10-9- 10-11 - -L - -a li w m - -12 0 m - 0-16 0 20 40 60 ao 100 number of active users Fig.3 BER versus the number of active users N,,, = N,, and coded asynchronous CDMA using a PO-OOC....... K = 4, [16, 2, X AEC code K = 4, [18, 2, 91 AEC code K = 5, [lx, 5, 51 AEC code K = 6, [6, 2, 31 AEC code for uncoded -20-2L 0 10 20 30 LO number of active users Fi 4 BER versus the number of active users N Cf for uncoded and cozd asynchronous CDMA using a prime CO& _-_ [lo, 4, 31 AEC code [15, X, 31 AEC code Fig. 3 shows the great reduction of the BER which is caused by OMA. t is clear that various AEC-coded OCDMA of lower K can have the equivalent BER performance of uncoded with K = 10. Using AEC codes can also increase N,,, for a given BER without bandwidth expansion compared to uncoded OCDMA, as shown in Table 2. Table 2: Comparison of uncoded and coded asynchronous OCDMA using OOC for BER 5 2.0 x Uncoded Coded K "w Bu K "ax na ka A Pc 10 59 5311 5 70 18 5 5 5043.6 10 30 2701 5 30 18 5 5 2163.6 11 50 5501 6 50 23 12 4 2876.92 Fig. 4 shows the BER performance of asynchronous prime-code OCDMA, which is dramatically degraded as N,,, increases. Even when P = 19, an uncoded asynchronous OCDMA network can only support O out of 19 users to transmit messages simul- 320 Table 3: Comparison of uncoded and coded asynchronous OCDMA using prime code for BER 5 2.0 x 10-9 Uncoded Coded P Nact Pu P Nact na ka A Pc 11 4 121 11 10 6 2 3 363 13 5 169 13 13 10 4 3 422.5 17 8 289 17 15 4 2 2 578 19 10 361 19 19 4 2 2 722 23 14 529 23 23 23 18 2 675.94 The BER of synchronous OCDMA using prime codes is illustrated in Fig. 5. The BER performance is dramatically degraded as N,,, increases, especially for a larger N,,,. By using a simple [14, 2, 71 code, the coded synchronous OCDMA with P = 7 and @, = 343 can achieve the slightly better BER than uncoded synchronous with P = 19 and pu = 361 for Nu,, up to 49 (i.e. N, for P = 7). Since p, < pu, the bandwidth can effectively be saved by using EE Proc -Commun, Val 144, No 5 October 1997
AEC codes. As shown in Fig. 6, the use of an AEC code can increase the capacity of synchronous primecode OCDMA networks. 0 20 LO 60 80 100 120 1LO number of active users Fi.5 BER versus the nwrrber of active users N,,, for uncoded and cojed synchronous CDMA system using a prime code [14, 2, 71 AEC code _-_ [18, 2, 91 AEC code uncoded 80-70 - 60-719 17 bandwidth-expansion factor Fig.6 N,,, against bandwidth-expansion factor for BER 5 O-1o in synchronous prime-code CDMA system with a [14, 2, 71 AEC code and without AEC codes U- - -W [14, 2, 71 AEC code 0- - -0 uncoded 7 Conclusions The BER performance of asynchronous and synchronous OCDMA :S limited by OMA. The use of binary block codes in such can greatly reduce the effect of OMA on system performance, so it allows the use of lower-weight OCDMA address codes to achieve the BER 5 Using error-correction codes can also efficiently increase N,,, limited by OMA. Compared with the case of uncoded OCDMA, it is feasible to use error-correction coding to reduce the complexity of OCDMA encodersldecoders and the optical power loss, which is another practical factor limiting the capacity of all-optical networks. Moreover, AEC binary block codes are more suitable than SEC codes for OCDMA applications, because the former not only ensures that the BER s but also can increase either the maximum number of users or the number of active users in a constant-bandwidth network more efficiently than can SEC binary block codes. Therefore, cost-effective can be implemented by combining AEC codes with OCDMA. 8 Acknowledgments The author thanks Prof. G. Picchi of the University of Parma (taly) for help and Dr. G. Migliorini for support. 9 References 1 PRUCNAL, P.R., SANTORO, M.A., and FAN, T.R.: Spread spectrum fiber-optic local area network using optical processing, J. Lightwave TechnoL, 1986, LT-4, pp. 541-554 2 CHUNG, F.R.K., SALEH, J.A., and WE, V.K.: Optical orthogonal codes: design, analysis, and applications, ZEEE Trans. Znf: Theory, 1989, 35, pp. 595-604 3 SALEH, J.A.: Code division multiple-access techniques in optical fiber networks - Part : Fundamental principles, ZEEE 4 5 Trans. Commun., 1989, 37, pp. 824-833 SALEH, J.A., and BRACKETT, C.A.: Code division multipleaccess techniques in optical fiber networks - Part 11: Systems performance analysis, ZEEE Trans. Commun., 1989, 37, pp. 834-842 KWONG, W.C., PERRER, P.A., and PRUCNAL, P.R.: Performance comparison of asynchronous and synchronous codedivision multiple-access techniques for fiber-optic local area networks, ZEEE Trans. Commun., 1991, 39, pp. 1625-1634 6 DALE M., and GAGLARD, R.M.: Analysis of fiberoptic code division multiple access. CS report 92-06-10, EES Department, University of Southern California, USA, June 1992 7 WU, J.-H., WU, J., and TSA, C.-N.: Synchronous fibre-optic code division multiple access networks with error control coding, Electron. Lett., 1992, 28, pp. 2118-2120 8 ZHANG, J.-G., and PCCH, G.: Forward error-correction codes in incoherent optical fibre CDMA networks, Electron. Lett., 1993, 29, pp. 1460-1462 9 WEBER, J.H., DE VROEDT, C., and BOEKEE, D.E.: Bounds and constructions for codes correcting unidirectional errors, EEE Trans. Znf. Theorv. 1989. 35. 00. 797-810 O SHOZAK, A: Consiructioi fir &nary asymmetric error-correcting codes, ZEEE Trans., 1982, T-28, pp. 787-789 11 WEBER, J.H, DE VROEDT, C., and BOEKEE, D.E.: Bounds and constructions for binary codes of length less than 24 and asymmetric distance less than 6, ZEEE Trans. Znf: Theory, 1988, 34, pp. 1321-1331 12 MATSUNAGE, S., and GAGLARD, R.: Digital signaling with code-division multiple-access in optical fiber communications. Technical report CS-88-02-03, Department of Electrical Engineering, University of Southern California, 1988 13 VERHOEFF, T.: An updated table of minimum-distance bounds for binary linear codes, ZEEE Trans., 1987, T-33, pp. 665-680 14 MCHELSON, A.M., and LEVESQUE, A.H.: Error-control techniques for digital communication (John Wiley and Sons, New York, 1985) 15 CLARK, G.C. Jun., and CAN, J.B.: Error-correction coding for digital communications (Plenum Press, New York, 1981) EE Pro,.-Commun., Vol. 144, No. 5, October 1997 321