New Architecture & Codes for Optical Frequency …copl.gel.ulaval.ca/publications/uploadPDF/publication_1438.pdf · New Architecture & Codes for Optical Frequency-Hopping Multiple

New Architecture & Codes for Optical Frequency-Hopping Multiple Access

Louis-Patrick Boulianne and Leslie A. Rusch

COPL, Department of Electrical and Computer EngineeringLaval University, Québec, Canada G1K 7P4

(418) 656-2906, (418) 656-3159 [email protected], [email protected]

ABSTRACTWe propose a new architecture for an optical fast frequency-hop code division multiple access (FFH-CDMA) sys-

tem using tunable Bragg gratings. Previously proposed architectures called for a series of in-fiber Bragg gratings, each in-dependently tunable with a piezo-electric device. We propose a system where the entire fiber of multiple Bragg gratingsuses one piezo-electric device to tune to a particular code. We introduce a new set of codes to take advantage of the newarchitecture 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 AreaNetworks.

1. INTRODUCTIONIn recent years, the popularity of fiber Bragg gratings for optical signal processing has coincided with the demand

for optical Local Area Networks (LAN) that can support larger numbers of users and high data rates. Code division multipleaccess (CDMA) offers many advantages over wavelength division multiplexing (WDM) for LANs, including high capacity,asynchronous and decentralized operation, etc. A new architecture for optical CDMA has recently been proposed [1] calledfast frequency-hop code division multiple access (FFH-CDMA). In this implementation, passive multiple Bragg gratingsare used for coding by slicing, spectrally and temporally, an incoming broadband pulse into many distinct pulses [4]. Herewe propose a new tuning strategy and corresponding code set that supports more users with better bit error rate (BER) per-formance.

2. OPTICAL FFH-CDMAIn optical FFH-CDMA, the encoding device, see Figure 1, consists of a series of N Bragg gratings each with a dis-

tinct center frequency of reflection. A pulsed broadband source is data modulated and sent to the encoder. The reflectedsignal consists of a series of pulses with spectral composition determined by the center frequency of each Bragg grating inturn. The CDMA code is determined by the order of the center frequencies in the series of gratings. The decoder is a simi-lar series of gratings but with the center frequencies in reverse order. The highest attainable bit rate is ultimately deter-mined by the time for all reflected pulses (or chips) to exit the fiber, i.e., NTc where Tc is the time interval between chipswhich depends directly on the grating length Lg and the grating spacing Ls. Smaller distances between adjacent gratingsleads 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 CDMAcode set. In [1], Fathallah et al. proposed that each Bragg grating be written to the same center frequency and that a piezo-electric device be attached to each grating to independently tune its frequency as required by the CDMA code. Given thephysical constraints of this system, tuning is achieved by stretching the fiber, despite the limited tuning range for stretchedas opposed to compressed gratings. (In the 1.55 µm regime stretching leads to a tuning range of 10 nm as opposed to32 nm for compression [2].) This encoder architecture also requires significant physical separation between gratings toattach the piezo-electric devices, limiting the bit rate.

Figure 1 illustrates the two tuning strategies. In the first case all gratings are written to λ0, and each independentlytuned to the code [λ5 λ1 λ3 λ2 λ6 λ4]. In the second case each grating is written to a distinct center wavelength: the firstgrating λ4, the second λ0,etc. The starting code sequence [λ4 λ0 λ2 λ1 λ5 λ3] is then tuned as a unit by one increment so that

the resulting code is [λ5 λ1 λ3 λ2 λ6 λ4]. Note that, as illustrated, the first pulse to be reflected and transmitted is λ5, the sec-ond λ1, etc, due to the “first in line, first reflected” property of multiple Bragg gratings.

Figure 1 FFH-CDMA encoder with code [λ5 λ1 λ3 λ2 λ6 λ4] where each grating is: 1) independently stretched from λ0 to thedesired λ ; 2) simultaneously stretched from the starting sequence [λ4 λ0 λ2 λ1 λ5 λ3] to the desired code [λ5 λ1 λ3 λ2 λ6 λ4].

3. PROPOSED TUNING STRATEGY AND CODESWe propose a new tuning strategy where each grating is written with a set of frequencies corresponding to a par-

ticular code. A single piezo-electric device is used to tune the gratings to another code via compression or tension of thefiber. The bit rate is increased since gratings can then be more closely spaced; cost and complexity is reduced as only onepiezo-electric device is required. Capacity is also increased as we can more easily implement compression tuning in thenew architecture. Compression is more difficult as buckling of the fiber must be avoided hence there are tighter toleranceon the physical package. As only one piezo is used, this difficulty need be addressed only once, and not N times. Thelarger tuning range due to compression translates to increased capacity for the CDMA system. In the ensuing discussion ofthe particular codes, we will see how a single piezo device can tune to any code in the set, i.e., independent control of eachgrating is not required. We will also demonstrate how the new proposed code set has lower cross-correlation and thereforean improved signal to interference ratio.

The code sequence for the kth user is ck=[λk,1, λk,2,..., λk,N] where each wavelength chip λk,j has the same time dura-tion and the same ideal rectangular time/frequency shape; N is the length of the code (or the number of chips per bit). Theith component of the code cm is denoted cm(i) where the argument is evaluated modulo N. The product cm(i)cp(j) is zero ifthe ith wavelength of code cm is different from the jth wavelength of code cp. The product is one if the two wavelengths arethe same, i.e. a hit occurs. The cross-correlation function is

R s c i c i s N s Nm p mi

N

p, b g b g b g= − − + ≤ ≤ −=∑

0

1 1 (1)

where cm and cp are two sequences from the code set. Sequences are selected to minimize this function for any time shift s.

In column two of Table 1, we present an example of the wavelength indices for one-coincidence codes [3] sug-gested in [1] for optical FFH-CDMA. This example is a family of length six (N=6) 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 wesee 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 inTable 1, the condition q=N-2d-1 is applied. One-coincidence codes also exist for any N<q and d≥1. Note that when d islarge, pulses which are adjacent in time will be well separated in frequency, hence we avoid overlap from side lobes in thereflectivity of adjacent reflected pulses [1].

BitSource

BroadbandSource

time

IncidentPulse

λ1λ5 λ3 λ2 λ6 λ4

λ1 λ5λ3λ2

Bragg gratings

Impulse response of a single grating"First in line, First reflected"

λ6λ4

T n L cc g c=2 /

L L Lc g s= +

λ0 λ0 λ0 λ0 λ0 λ0

λ1λ5 λ3 λ2 λ6 λ4λ4 λ0 λ2 λ1 λ5 λ3

1)

2) Twotuning

strategies

Table 1. Cyclic and non-cyclic codesCyclic Codes

N=6,M=6,q=7Non-cyclic CodesN=6,M=6,q=12

#1 4 0 2 1 5 3 4 0 2 1 5 3#2 5 1 3 2 6 4 5 1 3 2 6 4#3 6 2 4 3 0 5 6 2 4 3 7 5#4 0 3 5 4 1 6 7 3 5 4 8 6#5 1 4 6 5 2 0 8 4 6 5 9 7#6 2 5 0 6 3 1 9 5 7 6 10 8#7 3 6 1 0 4 2 10 6 8 7 11 9

In column three of Table 1, we present the modified version of the cyclic codes corresponding to the new tuningarchitecture. Codes are generated in a non-cyclic manner so that q=12 frequencies are now required to generate a family ofseven codes. The basic characteristics of the one-coincidence codes are preserved, i.e., 1) each sequence has the same

length; 2) each frequency in a sequence is non-repeating; 3) the maximum number of hits between any pair of sequences forany time shift equals one.

There are several consequences of this new choice of codes. Suppose that the series of Bragg gratings is written forcode #7. By compressing the fiber by one increment we compress each center wavelength by one increment and arrive atcode #6. By compressing two increments we arrive at code #5, etc. For stretching, we invert the method, and, the series ofBragg gratings is written for code #1. In this way, only one piezo-electric device is required, as all gratings need be com-pressed 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 betweencodes #1 and #7. For the cyclic codes certain time delays lead to non-zero correlation, while for the non-cyclic codes thecodes are orthogonal.

The total number of codes, C, is determined by the number of possible tuning increments, M, which is fixed by thefiber physical constraints. The number of codes C in the code set is equal to M+1. To determine the spectral bandwidthused by a code set, we define ∆λbin as the spacing between wavelength bins, ∆λtuning as the grating tuning interval, and ∆λseq

as the spacing between the smallest and the highest wavelength of the starting sequence for the code set. The number ofwavelength shifts M is ∆λtuning divided by the ∆λbin. In the following, we will assume the lowest wavelength, λmin, is λ0 forstretch tuning. Similar development can be applied to compression tuning where the gratings are tuned beginning from themaximum wavelengths.

For cyclic codes, all gratings are written at the same wavelength (λ0) and each grating must be tunable over theentire tuning range. The maximum wavelength λmax is λ0+(M×∆λbin). The number of frequencies used, q, is M+1. Fornon-cyclic codes, the starting code set is written in the Bragg gratings and all gratings are tuned together over the entiretuning range. The maximum wavelength λmax is λ0+(M×∆λbin)+∆λseq. The number of frequencies used, q, isM+1+(∆λseq/∆λbin). In both cases, the number of available frequencies, q, is determined by the tuning range, which ishighest for compression. For the non-cyclic codes, this parameter is related ∆λseq which in turn depends of the minimumspacing, d, used in the construction of the starting sequence.3.1 Increasing maximum capacity

The number of available codes in the code set is a important factor in determining system capacity. We can use theproperties of the one-coincidence sequences to increase the number of codes. If we take a code of length N=12 and we splitit into two codes of length N=6, we obtain two one-coincidence which are also one-coincidence when treated as a singlecode 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×N. For example,for a set of 13 available frequencies, we can build a one-coincidence sequence of length N=12; s12=[1 3 7 2 5 11 10 8 4 9 60]. We can obtain two starting sequences of length N=6: s1=[1 3 7 2 5 11] and s2=[10 8 4 9 6 0]. We can shift the wave-lengths in sequence s1 by one if we want two sequences with the same λmin, i.e. s1 = [0 2 6 1 4 10]. However, while thismethod 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 couldbe used to change between the sets.

4. BER PERFORMANCEOur simulations use the same parameters proposed in [1] for the Bragg grating length, coupling coefficient, effec-

tive refractive index, and frequency range. We analyze system performance in terms of multiple access interference (MAI);all other noises are neglected . Define Rm p, as the average cross-correlation for any time shift between the two codes cm andcp

RN

R sm p m ps N

N

, ,=− =− +

−

∑12 1

1

1

b g (2)

If we further average over all code pairs, we arrive at the average cross-correlation

Rq q

Rm pp m

q

m

q

=− = +=

−

∑∑21 11

1

( ) , (3)

For the cyclic codes,

RNq

NN

= −− −

FHG

IKJ

11 2 1

(4)

for any combination of N<q. We cannot develop a theoretical expression R for non-cyclic codes because of the non-uniform 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

σ AV m p m ps N

N

p m

q

m

q

q q NR s R2 2

1

1

11

12

11

2 1=

− −−

FHGG

IKJJ=− +

−

= +=

−

∑∑∑b g b ge j, , (5)

The average of the MAI for K-1interferers is approximated by

µ MAI K R= − 1b g (6)

hence, the optimum detection threshold is gived by

γ µopt MAIN= +2

(7)

The variance of the MAI can be approximated by the sum of the average variance from each interferer

σ σMAI AVK2 21= −b g (8)

If we assume the MAI has a Gaussian distribution (justified by central limit arguments), the signal-to-interference ratio(SIR) and the BER are

SIRN

K AV=

−

2

21b gσ (9)

and

BER SIR= −Φ d i (10)

where Φ x e dyyxb g= −− ∞z1

2

2 2π

.

In the Figure 2, we plot the bit error rate vs. the number of simultaneous users for encoders consisting of a series ofN=12 Bragg gratings and starting code sequence s=[13 22 5 16 1 11 27 18 6 24 10 0] for each case. The dashed line showsthe performance of the cyclic codes and tension tuning. The solid lines show performance of non-cyclic codes. If tensiontuning is used the tuning range is limited to 10 nm and therefore q=29 for cyclic codes and q=56 for non-cyclic codes,hence these values for the dashed line and the first solid line. For compression, the tuning range is 30 nm (1565 nm to

Table 2. Properties of different type of codesType of code N Tuning

strategy∆λbin M C d ∆

∆λλ

seq

bin

q R σAV2

Cyclic[13 22 5 16 1 11 27 18 6 24 10 0] 12 Tension 0.2 nm 28 29 9 27 29 0.2050 0.1583

12 Tension 0.2 nm 28 29 9 27 56 0.1297 0.108712 Compression 0.4 nm 74 75 9 27 102 0.0665 0.0565

Non-cyclic[13 22 5 16 1 11 27 18 6 24 10 0]

12 Compression 0.2 nm 138 139 9 27 166 0.0395 0.0335Cyclic

[1 3 7 2 5 11 10 8 4 9 6 0] 12 Tension 0.2 nm 28 29 1 11 29 0.2050 0.136612 Tension 0.2 nm 28 29 1 11 40 0.1743 0.114012 Compression 0.4 nm 74 75 1 11 86 0.0731 0.0484

Non-cyclic[1 3 7 2 5 11 10 8 4 9 6 0]

12 Compression 0.2 nm 138 139 1 11 150 0.0415 0.0275Cyclic

Set #1 [0 2 6 1 4 10]Set #2 [10 8 4 9 6 0]

Set #1 & #2

666

TensionTensionTension

0.4 nm0.4 nm0.4 nm

121212

131326

111

101010

131313

0.22730.22730.2400

0.17080.17080.1739

Non-cyclicSet #1 [0 2 6 1 4 10]Set #2 [10 8 4 9 6 0]

Set #1 & #2

666

TensionTensionTension

0.4 nm0.4 nm0.4 nm

121212

131326

111

101010

222222

0.14920.14920.1578

0.11890.11890.1206

1535 nm) and we can achieve q=166, i.e., C=139 users. The curve for q=102 represents a conservative spacing betweencenter frequencies in order to mitigate crosstalk and decrease reliance on good grating apodization. We see significant per-formance improvement for non-cyclic codes even for this final curve, where we adopted frequency spacing equal to twicethe reflectivity main lobe bandwidth.

Figure 2. BER for different coding schemes: a) cyclic codes with M=28 and q=29 in tension; b) non-cyclic codeswith M=28 and q=29 in tension; c) non-cyclic codes with M=74 and q=102 in compression; d) non-cyclic codes withM=138 and q=166 in compression

0 20 40 60 80 100 120 14010-16

10-14

10-12

10-10

10-8

10-6

BER

Simultaneous users

M=28, q=29, N=12

M=28,q=56,N=12

M=74,q=102,N=12

M=138,q=166,N=12

TENSION

COMPRESSION

5. DISCUSSIONThe 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 rangeM, 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 ad-jacent frequencies. Recall that in this analysis we assume ideal reflectivity, so that there is no perceived penalty for dsmall. We see in Table 2, for the cyclic case when d=1 instead of d=9, a significant reduction in the average variance ofthe cross-correlation and hence improved BER. This occurs because the factor d determines ∆λseq. When d=1, ∆λseq isvery small compared to q and thus we have great frequency diversity in each code, and hence hits occur with low probabil-ity. The effect of d in the non-cyclic case is a little different. Because ∆λseq is small, we have strong interference nearneighbor sequences, but weak interference for distant neighbors. Furthermore, the number of frequencies used q is smallerwhen d is smaller. This results in larger average cross-correlation and smaller average variance. Since, different codepairs will not have necessarily the same number of shared frequencies, the interference energy is not the same for each in-terferers. The threshold estimation efficiency is affected by this variation. Therefore it is important to choose the code setwith more stability in the number of shared frequencies between two sequences. A larger value for ∆λseq will lead to greaterstability.

In the last rows of Table 2 we examine the method of increasing maximal capacity via splitting a long code intomultiple code sets. We examine each N=6 code set (13 members) individually and also as a combined set with 26 mem-bers. In the expanded code set we see slightly lower performance, however this is only due to the increase in the number ofpossible interferers.

Finally, recall that large spacing between gratings reduces the maximum data rate. Our proposal increases overallthroughput by increasing the number of users and by decreasing the required spacing between gratings. For example, for adata rate of 500 Mb/s, grating length L=10 mm, grating spacing Ls=0.8 mm and N=12 [1], the aggregate data rate will be70 Gb/s for M=138. Reducing Ls to 0.25 mm because only one piezo is required, increases the data rata to 720 Mb/s. Theaggregate data rate is increased to 100 Gb/s with a probability of error less than 10-9. Note that we must still choose Ls largeenough to avoid temporal overlap between adjacent pulses.

6. CONCLUSIONWe 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 gratingsleads to better BER and a faster transmission rate for the same number of gratings. Furthermore, we a proposed new codestrategy for increasing the number of available codes.

7. ACKNOWLEDGMENTSThis work was supported by QuébecTel and by a grant from the Natural Sciences and Engineering Research Coun-

cil of Canada.

8. REFERENCES1. H. Fathallah, L. A. Rusch, S. LaRochelle “Optical Frequency-Hop Multiple Access Communications System,” Ac-

cepted in IEEE ICC ’98, paper 36-2, Atlanta, June 1998.2. Ball and W. W. Morey, “Compression-tuned single-frequency Bragg grating fiber laser,” Optics Letters, vol. 19, no.

23, December 1, 1994.3. L. Bin, “One-Coincidence Sequences with Specified Distance Between Adjacent Symbols of Frequency-Hopping Multi-

ple Access,” IEEE Transactions on Communications, vol. 45, no.4, pp. 408-410, April 1997.4. L. R. Chen, S. D. Benjamin, P. W. E. Smith, J. E. Sipe, and S. Juma, “Ultrashort pulse propagation in multiple-grating

fiber structures,” Optics Letters, vol. 22, pp. 402-404, 1997.