ew Architecture & Codes for Optical Frequency-Hopping Multiple Access Louis-Patrick Boulianne and Leslie A. Rusch COPL, Department of Electrical and Computer Engineering Laval University, Québec, Canada GK 7P4 (48) 5-90, (48) 5-359 fax patbou@gel.ulaval.ca, rusch@gel.ulaval.ca ABSTRACT We propose a new architecture for an optical fast frequency-hop code division multiple access (FFH-CDMA) system using tunable Bragg gratings. Previously proposed architectures called for a series of in-fiber Bragg gratings, each independently tunable with a piezo-electric device. We propose a system where the entire fiber of multiple Bragg gratings uses one piezo-electric device to tune to a particular code. We introduce a new set of codes to take advantage of the new architecture and increase the bit rate of each user, as well as the total number of users and hence aggregate bit rate. Keywords: Frequency Hopping CDMA, Bragg gratings, cyclic and non-cyclic codes, one-coincidence codes, Local Area etworks.. ITRODUCTIO In recent years, the popularity of fiber Bragg gratings for optical signal processing has coincided with the demand for optical Local Area etworks (LA) that can support larger numbers of users and high data rates. Code division multiple access (CDMA) offers many advantages over wavelength division multiplexing (WDM) for LAs, including high capacity, asynchronous and decentralized operation, etc. A new architecture for optical CDMA has recently been proposed [] called fast frequency-hop code division multiple access (FFH-CDMA). In this implementation, passive multiple Bragg gratings are used for coding by slicing, spectrally and temporally, an incoming broadband pulse into many distinct pulses [4]. Here we propose a new tuning strategy and corresponding code set that supports more users with better bit error rate (BER) performance.. OPTICAL FFH-CDMA In optical FFH-CDMA, the encoding device, see Figure, consists of a series of Bragg gratings each with a distinct center frequency of reflection. A pulsed broadband source is data modulated and sent to the encoder. The reflected signal consists of a series of pulses with spectral composition determined by the center frequency of each Bragg grating in turn. The CDMA code is determined by the order of the center frequencies in the series of gratings. The decoder is a similar series of gratings but with the center frequencies in reverse order. The highest attainable bit rate is ultimately determined by the time for all reflected pulses (or chips) to exit the fiber, i.e., T c where T c is the time interval between chips which depends directly on the grating length L g and the grating spacing L s. Smaller distances between adjacent gratings leads to smaller chip intervals and higher bit rates. In order to be of practical interest, we must be able to program the encoder/decoder pair to any code in the CDMA code set. In [], Fathallah et al. proposed that each Bragg grating be written to the same center frequency and that a piezoelectric device be attached to each grating to independently tune its frequency as required by the CDMA code. Given the physical constraints of this system, tuning is achieved by stretching the fiber, despite the limited tuning range for stretched as opposed to compressed gratings. (In the.55 µm regime stretching leads to a tuning range of 0 nm as opposed to 3 nm for compression [].) This encoder architecture also requires significant physical separation between gratings to attach the piezo-electric devices, limiting the bit rate. Figure illustrates the two tuning strategies. In the first case all gratings are written to, and each independently tuned to the code [ λ λ 4 ]. In the second case each grating is written to a distinct center wavelength: the first grating λ 4, the second,etc. The starting code sequence [λ 4 ] is then tuned as a unit by one increment so that
the resulting code is [ λ λ 4 ]. ote that, as illustrated, the first pulse to be reflected and transmitted is, the second, etc, due to the first in line, first reflected property of multiple Bragg gratings. Bit Source Incident Pulse time ) λ 4 λ λ Two 4 ) λ λ 4 Bragg gratings Lc = Lg + Ls tuning strategies Broadband Source λ 4 λ T = n L / c c g c Impulse response of a single grating "First in line, First reflected" Figure FFH-CDMA encoder with code [ λ λ 4 ] where each grating is: ) independently stretched from to the desired λ ; ) simultaneously stretched from the starting sequence [λ 4 ] to the desired code [ λ λ 4 ]. 3. PROPOSED TUIG STRATEGY AD CODES We propose a new tuning strategy where each grating is written with a set of frequencies corresponding to a particular code. A single piezo-electric device is used to tune the gratings to another code via compression or tension of the fiber. The bit rate is increased since gratings can then be more closely spaced; cost and complexity is reduced as only one piezo-electric device is required. Capacity is also increased as we can more easily implement compression tuning in the new architecture. Compression is more difficult as buckling of the fiber must be avoided hence there are tighter tolerance on the physical package. As only one piezo is used, this difficulty need be addressed only once, and not times. The larger tuning range due to compression translates to increased capacity for the CDMA system. In the ensuing discussion of the particular codes, we will see how a single piezo device can tune to any code in the set, i.e., independent control of each grating is not required. We will also demonstrate how the new proposed code set has lower cross-correlation and therefore an improved signal to interference ratio. The code sequence for the k th user is c k =[λ k,, λ k,,..., λ k, ] where each wavelength chip λ k,j has the same time duration and the same ideal rectangular time/frequency shape; is the length of the code (or the number of chips per bit). The i th component of the code c m is denoted c m (i) where the argument is evaluated modulo. The product c m (i)c p (j) is zero if the i th wavelength of code c m is different from the j th wavelength of code c p. The product is one if the two wavelengths are the same, i.e. a hit occurs. The cross-correlation function is m, pb g mb g pb g i= 0 R s = c i c i s + s () where c m and c p are two sequences from the code set. Sequences are selected to minimize this function for any time shift s. In column two of Table, we present an example of the wavelength indices for one-coincidence codes [3] suggested in [] for optical FFH-CDMA. This example is a family of length six (=) codes using seven distinct frequencies (q=7) for a total of seven codes in the code set (C=7). Reading across we have a particular code, while reading down we see the cyclic nature of the codes, ensuring that no more than seven frequencies are ever used. A minimum distance, d, between reflected frequency bins from adjacent pairs of gratings can be imposed in the code construction. For the codes in Table, the condition q=-d- is applied. One-coincidence codes also exist for any <q and d. ote that when d is large, pulses which are adjacent in time will be well separated in frequency, hence we avoid overlap from side lobes in the reflectivity of adjacent reflected pulses [].
In column three of Table, we present the modified version of the cyclic codes corresponding to the new tuning architecture. Codes are generated in a non-cyclic manner so that q= frequencies are now required to generate a family of seven codes. The basic characteristics of the one-coincidence codes are preserved, i.e., ) each sequence has the same Table. Cyclic and non-cyclic codes Cyclic Codes on-cyclic Codes =,M=,q=7 =,M=,q= # 4 0 5 3 4 0 5 3 # 5 3 4 5 3 4 #3 4 3 0 5 4 3 7 5 #4 0 3 5 4 7 3 5 4 8 #5 4 5 0 8 4 5 9 7 # 5 0 3 9 5 7 0 8 #7 3 0 4 0 8 7 9 length; ) each frequency in a sequence is non-repeating; 3) the maximum number of hits between any pair of sequences for any time shift equals one. There are several consequences of this new choice of codes. Suppose that the series of Bragg gratings is written for code #7. By compressing the fiber by one increment we compress each center wavelength by one increment and arrive at code #. By compressing two increments we arrive at code #5, etc. For stretching, we invert the method, and, the series of Bragg gratings is written for code #. In this way, only one piezo-electric device is required, as all gratings need be compressed or stretched by the same amount. We will see a performance improvement due to use of compression vs. stretching, but we will also see better performance as the new code set has lower cross-correlation. Compare the correlation between codes # and #7. For the cyclic codes certain time delays lead to non-zero correlation, while for the non-cyclic codes the codes are orthogonal. The total number of codes, C, is determined by the number of possible tuning increments, M, which is fixed by the fiber physical constraints. The number of codes C in the code set is equal to M+. To determine the spectral bandwidth used by a code set, we define λ bin as the spacing between wavelength bins, λ tuning as the grating tuning interval, and λ seq as the spacing between the smallest and the highest wavelength of the starting sequence for the code set. The number of wavelength shifts M is λ tuning divided by the λ bin. In the following, we will assume the lowest wavelength, λ min, is for stretch tuning. Similar development can be applied to compression tuning where the gratings are tuned beginning from the maximum wavelengths. For cyclic codes, all gratings are written at the same wavelength ( ) and each grating must be tunable over the entire tuning range. The maximum wavelength λ max is +(M λ bin ). The number of frequencies used, q, is M+. For non-cyclic codes, the starting code set is written in the Bragg gratings and all gratings are tuned together over the entire tuning range. The maximum wavelength λ max is +(M λ bin )+ λ seq. The number of frequencies used, q, is M++( λ seq / λ bin ). In both cases, the number of available frequencies, q, is determined by the tuning range, which is highest for compression. For the non-cyclic codes, this parameter is related λ seq which in turn depends of the minimum spacing, d, used in the construction of the starting sequence. 3. Increasing maximum capacity The number of available codes in the code set is a important factor in determining system capacity. We can use the properties of the one-coincidence sequences to increase the number of codes. If we take a code of length = and we split it into two codes of length =, we obtain two one-coincidence which are also one-coincidence when treated as a single code set. In this manner, if the set of available frequencies is large enough, we can increase the total size of the code set, and total system capacity. The number of sets S, contributing to the expand code set, is limited by q>s. For example, for a set of 3 available frequencies, we can build a one-coincidence sequence of length =; s =[ 3 7 5 0 8 4 9 0]. We can obtain two starting sequences of length =: s =[ 3 7 5 ] and s =[0 8 4 9 0]. We can shift the wavelengths in sequence s by one if we want two sequences with the same λ min, i.e. s = [0 4 0]. However, while this method increases maximum capacity, it also increases the complexity of the encoders and decoders for non-cyclic codes, as
we must be able to select one of the two starting sequences, i.e. one of two multiple Bragg gratings. An optical switch could be used to change between the sets. 4. BER PERFORMACE Our simulations use the same parameters proposed in [] for the Bragg grating length, coupling coefficient, effective refractive index, and frequency range. We analyze system performance in terms of multiple access interference (MAI); all other noises are neglected. Define R m, p as the average cross-correlation for any time shift between the two codes c m and c p R m, p = Rm, p s= + If we further average over all code pairs, we arrive at the average cross-correlation For the cyclic codes, q q R = R m p q q, ( ) R = q m= p= m+ F H G I K J bsg () for any combination of <q. We cannot develop a theoretical expression R for non-cyclic codes because of the nonuniform distribution of the cross-correlation between codes and the non-uniform frequency composition of codes. Hence, we limit our analysis to direct calculation of the sample mean and variance of a given code set. The average variance of the cross-correlation function of the code set is F b g b g H q q σ AV = R m p s R m p q q G e,, j (5) J m= p= m+ s= + The average of the MAI for K-interferers is approximated by µ MAI = K R I K b g () hence, the optimum detection threshold is gived by γ opt = + µ MAI The variance of the MAI can be approximated by the sum of the average variance from each interferer σ b g (8) MAI = K σ AV If we assume the MAI has a Gaussian distribution (justified by central limit arguments), the signal-to-interference ratio (SIR) and the BER are SIR = b K gσ AV and BER = Φd SIRi (0) x y where Φbxg= e dy. πz In the Figure, we plot the bit error rate vs. the number of simultaneous users for encoders consisting of a series of = Bragg gratings and starting code sequence s=[3 5 7 8 4 0 0] for each case. The dashed line shows the performance of the cyclic codes and tension tuning. The solid lines show performance of non-cyclic codes. If tension tuning is used the tuning range is limited to 0 nm and therefore q=9 for cyclic codes and q=5 for non-cyclic codes, hence these values for the dashed line and the first solid line. For compression, the tuning range is 30 nm (55 nm to (3) (4) (7) (9)
535 nm) and we can achieve q=, i.e., C=39 users. The curve for q=0 represents a conservative spacing between center frequencies in order to mitigate crosstalk and decrease reliance on good grating apodization. We see significant performance improvement for non-cyclic codes even for this final curve, where we adopted frequency spacing equal to twice the reflectivity main lobe bandwidth. 0-0 -8 M=8, q=9, = BER 0-0 TESIO M=74, q=0, = M=38, q=, = 0-0 -4 M=8, q=5, = COMPRESSIO Table. Properties of different type of codes Type of code Tuning Cyclic strategy λ bin M C d λ λ seq bin q R σ AV [3 5 7 8 4 0 0] 0. nm 8 9 9 7 9 0.050 0.583 on-cyclic [3 5 7 8 4 0 0] Cyclic 0. nm 8 9 9 7 5 0.97 0.087 Compression 74 75 9 7 0 0.05 0.055 Compression 0. nm 38 39 9 7 0.0395 0.0335 [ 3 7 5 0 8 4 9 0] 0. nm 8 9 9 0.050 0.3 on-cyclic [ 3 7 5 0 8 4 9 0] Cyclic Set # [0 4 0] Set # [0 8 4 9 0] Set # & # on-cyclic 0 - Set # [0 4 0] Set # [0 8 4 9 0] Set # & # 0 0 40 0 80 00 0 40 Simultaneous users Figure. BER for different coding schemes: a) cyclic codes with M=8 and q=9 in tension; b) non-cyclic codes with M=8 and q=9 in tension; c) non-cyclic codes with M=74 and q=0 in compression; d) non-cyclic codes with M=38 and q= in compression 0. nm 8 9 40 0.743 0.40 Compression 74 75 8 0.073 0.0484 Compression 0. nm 38 39 50 0.045 0.075 3 3 3 3 0 0 0 0 0 0 3 3 3 0.73 0.73 0.400 0.49 0.49 0.578 0.708 0.708 0.739 0.89 0.89 0.0
5. DISCUSSIO The increased BER performance of the non-cyclic codes over the cyclic codes, when the same tuning range and the same starting sequence are used, arises from the larger number of frequencies used for coding in the non-cyclic case. Consequently, the use of independent tuning for each grating, will always have better performance for a given tuning range M, as the number of frequencies q will be always greater for the non-cyclic case. Another parameter that influences the BER performance is the minimum distance, d, between two temporally adjacent frequencies. Recall that in this analysis we assume ideal reflectivity, so that there is no perceived penalty for d small. We see in Table, for the cyclic case when d= instead of d=9, a significant reduction in the average variance of the cross-correlation and hence improved BER. This occurs because the factor d determines λ seq. When d=, λ seq is very small compared to q and thus we have great frequency diversity in each code, and hence hits occur with low probability. The effect of d in the non-cyclic case is a little different. Because λ seq is small, we have strong interference near neighbor sequences, but weak interference for distant neighbors. Furthermore, the number of frequencies used q is smaller when d is smaller. This results in larger average cross-correlation and smaller average variance. Since, different code pairs will not have necessarily the same number of shared frequencies, the interference energy is not the same for each interferers. The threshold estimation efficiency is affected by this variation. Therefore it is important to choose the code set with more stability in the number of shared frequencies between two sequences. A larger value for λ seq will lead to greater stability. In the last rows of Table we examine the method of increasing maximal capacity via splitting a long code into multiple code sets. We examine each = code set (3 members) individually and also as a combined set with members. In the expanded code set we see slightly lower performance, however this is only due to the increase in the number of possible interferers. Finally, recall that large spacing between gratings reduces the maximum data rate. Our proposal increases overall throughput by increasing the number of users and by decreasing the required spacing between gratings. For example, for a data rate of 500 Mb/s, grating length L=0 mm, grating spacing L s =0.8 mm and = [], the aggregate data rate will be 70 Gb/s for M=38. Reducing L s to 0.5 mm because only one piezo is required, increases the data rata to 70 Mb/s. The aggregate data rate is increased to 00 Gb/s with a probability of error less than 0-9. ote that we must still choose L s large enough to avoid temporal overlap between adjacent pulses.. COCLUSIO We proposed and analyzed a new architecture and codes for optical fast frequency-hopping code division multiple access communication system. Using non-cyclic codes with either compression or tension tuning of the multiple gratings leads to better BER and a faster transmission rate for the same number of gratings. Furthermore, we a proposed new code strategy for increasing the number of available codes. 7. ACKOWLEDGMETS This work was supported by QuébecTel and by a grant from the atural Sciences and Engineering Research Council of Canada. 8. REFERECES. H. Fathallah, L. A. Rusch, S. LaRochelle Optical Frequency-Hop Multiple Access Communications System, Accepted in IEEE ICC 98, paper 3-, Atlanta, June 998.. Ball and W. W. Morey, Compression-tuned single-frequency Bragg grating fiber laser, Optics Letters, vol. 9, no. 3, December, 994. 3. L. Bin, One-Coincidence Sequences with Specified Distance Between Adjacent Symbols of Frequency-Hopping Multiple Access, IEEE Transactions on Communications, vol. 45, no.4, pp. 408-40, April 997. 4. L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, Ultrashort pulse propagation in multiple-grating
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