An Optical Code with the Least Multiple Access Interference under Arbitrary Restrictions for Optical CDMA Systems Seong-sik Min*, Yong Hyub Won* and Byoung Whi Kim+ *ONE Lab., ICU, 58-4 Whaam, Yuseong, Daejeon, 305-732, South Korea + WPON team, ETRI, 161 Kajung, Yuseong, Daejeon, 305-350, South Korea Phone: +82-42-866-6181, Fax: +82-42-866-6223, email:
[email protected] Abstract We propose and examine an adaptive resonance code with the least multiple access interferences under arbitrary restrictions to increase the number of available nodes in optical CDMA systems at the cost of small performance degradation. Introduction An optical code division multiple-access (CDMA) system is expected as an attractive substitute of the future optical subscriber network due to its advantages including large numbers of subscribers and simultaneous users in shared media, high level of security, and simple architecture [1]. Various optical CDMA systems utilizing one-dimensional (1-D) codes [2] based on only time-spreading scheme to twodimensional (2-D) codes based on timewavelength hybrid (TWH) scheme [3-4] have been proposed. Especially, the TWH schemes attract increasing research interests due to their good bit error rate (BER) performance and greatly increased cardinality. However, the implementation of reasonable tunable encoders/decoders to generate the optical codes such as those in [2-5] is the most difficult problem of the realization of the optical CDMA system since the objectives of the papers were that the designed optical code sets have the ideal properties such as the maximum cross-correlation of 1 resulting in the multiple access interference (MAI), the largest autocorrelation sidelobe of 0, and the large cardinality without considering the implementation of encoders/decoders. In order to unravel this difficulty, schemes
for tunable encoders/decoders to generate specific code sets have been presented in several papers [6-7]. The most important issue in an asynchronous optical (CDMA) system is the MAI caused by non-ideal orthogonal property of optical codes. The MAI is the main cause of the bit error. An optical code with small MAI should therefore be designed. In this paper, we propose an adaptive resonance Code (ARC) for the optical CDMA system. The code has very small MAI under arbitrary restrictions including code length, numbers of wavelengths and pulses in a code word, and the desired number of distinctive sequences. The ARC allows network designers to flexibly implement an optimal optical CDMA system with the previously designed components including encoders/decoders and optical sources with their own specialties like those in [6-7]. Adaptive Resonance Code (ARC) In this section, we consider the asynchronous optical CDMA system utilizing a TWH 2-D unipolar optical code, in which every pulse of a codeword is encoded in wavelength and in time domain, to explain the principle of generating ARC. In the code set, S pulses in a codeword and H wavelengths are used. In the optical CDMA system, many asynchronous users occupy the same channel simultaneously. A desired user’s receiver must be able to extract its signature sequence in the presence of other users’ signature sequences. Therefore, a code set as a signature sequence for the optical CDMA system should have a needle shape of
autocorrelation function and crosscorrelation values as small as possible. In a TWH 2-D code set, the maximum autocorrelation sidelobe of 0 can be easily achieved by utilizing the number of wavelengths more than or equal to the number of pulses in a codeword. Whereas, the obtainable smallest maximum crosscorrelation value between two unipolar codes is 1, and consequently, MAI due to the cross-correlation causes the bit error when data “0” is sent from the transmitter. A TWH 2-D code set C is a collection of N binary (0,1) F×H matrices with Hamming weight S, which have the following properties. When the bit time Tb and the chip time is Tc, the code length F is Tb/Tc. The auto-correlation Zx,x(l) for a code word X∈C is defined as follows: H −1 F −1
Z x , x (l ) = ∑
∑ x m , n x m , n + l
m =0 n =0
mod F
where l indicates the amount of chip difference between two codes, x m,n ∈ {0, 1} is an element of matrix X, and mod F denotes the modulo F operation. The cross-correlation Zx,y(l) of two codes X and Y is defined as follows: H
Z x , y (l ) = ∑
F −1
∑ x m , n y m , n + l x m, n and y m,n ∈ {0, 1} are m =0 n =0
where
mod F
an element
of matrix X and Y, respectively. The procedure of generating a TWH 2-D ARC is follows: 1) Set restrictions such as - the number of maximum nodes in the network or cardinality N, - the number of wavelengths H in the code set, - the number of pulses or Hamming weight S in a codeword, - whether signature sequences or codes consist of blocks or not, - the number of pulses contained in a block and the number of blocks consisting a code if needed, - code length F, - auto- and cross-correlation constraints
(λa and λc) if needed, - and so on. 2) Generate N codes randomly satisfying above restrictions. 3) Calculate the MAI at each node by N-1 other users. The MAI at i’th node is defined as follows: MAI i =
∑ max k {Z i , j (k )}for 0 ≤ k ≤ ( F − 1) N
j =1, j ≠ i
for 0 ≤ i, j ≤ ( N − 1) where maxk is the maximum value in all k and the worst case was assumed to get the upper-bound of the MAIs. 4) Choose a node i* for code regeneration. a. Select a node with the maximum MAI with probability γ, which is a node selection factor. b. If i* is not selected, choose a node with the next maximum MAI with probability γ. c. If i* is not selected, repeat b until i* is determined. 5) Regenerate the code for node i*. 7) Calculate the new MAIi* at the i*th node. 8) If the new MAIi* is less than or equal to old MAIi, accept the regenerated code and go to next step. Otherwise, accept the code with probability α, which is defined as α = ((newMAI − oldMAI ) β ) −1 where β is an accepting factor between 0 and 1 and determines how much the increase of MAI influences the acceptance. 9) If the resulted code set satisfies certain ending criteria such as if the total MAI is less than a predetermined value, or if the total MAI have not changed for certain iterations, then finish. Otherwise, go to step (3).
The node selection factor γ, the regenerating method of the i*’th code in step (5), the accepting factor β and the ending criteria can be chosen according to user’s discretion. The factors, γ and β affect both the convergent speed to the optimal point and the possible minimum total MAI where total MAI is the sum of all
MAIs at each node. N
total MAI = ∑ MAI i i =1
In order to regenerate the code with the maximum MAI in step (5), several methods including the Genetic Algorithm, stimulated annealing and random selection can be used. The Results When the iteration goes, the total MAI is decreased as shown in Fig. 1. An example of ARC is given in Table 1 for S = H = 5, F=25, and N = 15. At the iteration of 240, the final total MAI of the generated code set was 210 corresponding to the ideal maximum cross-correlation property (λc=1). űũũũ
Űűũũ Űůũũ Űŭũũ Űūũũ Űũũũ ũ
Ūũũũ
ūũũũ
Ŭũũũ
ŭũũũ
Ůũũũ
Fig.1 MAI trace as the iteration goes Table 1. An example of ARC for S = H = 5, F=25, and N = 15. The total MAI of generated code set is 210 (λc=1) and the number of iterations is 240. Code No. Generated Signature Sequence 0 00000 00005 00230 00041 00000 1 50000 00000 40102 03000 00000 2 10020 00300 00040 00000 00500 3 00020 00103 04000 00000 00050 4 00000 00030 04002 10050 00000 5 00001 30000 00000 00040 00520 6 00000 00000 30000 00010 00254 7 00020 00000 00104 00300 00005 8 00000 00001 00050 24000 30000 9 00000 00043 20050 00000 01000 10 00400 00000 50000 31000 00020 11 00000 02000 00003 45010 00000 12 01004 00500 00000 00000 03002 13 00000 03000 02000 00051 00004 14 30000 50400 00020 01000 00000
In order to examine the superiority of ARC to existing codes, the total MAI and peak autocorrelation sidelobe of the generated ARC were compared with various existing TWH codes: prime-hop code [3], CDMA+WDMA scheme [4], eqcprime code [5] (Table 2). The values for S, H, and F were selected for compared code sets since the prime-hop and eqc-prime algorithm can be used to generate codes with specific code-lengths and weights while the ARC is possible for any S, H, and F. The MAI values of ARCs are smaller than those of other codes except case of N=20 where the prime-hop code has the ideal properties: λa = 0 and λc = 1. The slightly bigger MAI value of ARC for N=20 can be reduced by doing more iteration or by selecting other γ and β although the influence of the MAI inrease on the BER performance is very small. Table 2. MAI comparison of various TWH code families in worst case for S = H = 5: total MAI (maximum autocorrelation sidelobes). Imp: impossible, OW: OOC+WDMA, PH: prime-hop, EP: eqc-prime, OC: optimal code. Code F=25 F=25 F=25 F=25 F=45 Family N=5 N=6 N=20 N=46 N=80 ARC 20 (0) 30 (0) 386 (0) 2292 (0) 6672 (0) OW 20 (1) 60 (2) 760 (2) 6210 (3) 12640(2) PH 20 (0) 30 (0) 380 (0) Imp Imp EP Imp Imp Imp Imp 7200 (0) OC ARC ARC PH ARC ARC
When K users are transmitting simultaneously, the MAI at a given receiver is the superposition of K-1 different crosscorrelation functions. If the K-1 interferers are uncorrelated, the mean and variance of the MAI are equal to the sum of the means and variances of the K-1 cross-correlation functions, respectively. Considering other users’ interference to be the dominant source of noise in the system, the signal-to-noise ratio (SNR) is represented as the ratio of square of the difference between the peak of the autocorrelation function and the mean of MAI to the variance of amplitude of MAI.
S 2 − ( K − 1) µ σ 2 ( K − 1)
Assuming the chip synchronization among received signals as the worst case, the SNR for each code can be obtained by estimating the average variance of the cross-correlation calculated using all possible code sequences and their all possible cyclic shifted versions. The probability of errors for the optical CDMA system is given by
1 PE = 2π
∫
− v2 dv . exp SNR 2
∞
When S=5 and H=5, the probability of errors was obtained for some F and N values and the results are shown in Fig. 2. The system designer can assign more address codes than the initial available codes in an existing network by using the technique to generate ARC code set with increased cardinality at the expense of the minimum performance degradation. Hence, the network can accept additional subscribers via amendment of software in the encoding/decoding process without any physical or structural changes. In the other hand, other schemes [2-5] require the drastic modification of the system such as encoder/decoder with increased chip rate and the number of wavelengths or pulses used in the coding of the data. Fig. 2 demonstrates that the performance of the system is slightly affected when the ARC code sets with increased cardinality from 20 to 46 and from 46 to 80 under the same condition are implemented. Conclusions We have introduced a novel code, ARC, to implement a flexible optical CDMA system. The most important issue in an asynchronous optical CDMA system is the MAI caused by non-ideal orthogonal property of optical codes since the MAI is the main cause of the system performance degradation. The generated ARC codes have shown small MAI’s for any S, H, F,
and N compared to other codes. In addition, the analytical results have demonstrated that when the cardinality of the code set with F=45, S=5 and H=5 is increased from 46 to 80, the performance degradation of the system is below 3dB where the cardinality of 80 is the same value as eqc-prime code [5] under the same conditions and the performance is even better than that of eqc-prime code. We conclude that the proposed ARC provides an elastic method to implement the optical CDMA system since the proposed code has the minimum MAI under the arbitrary restrictions. Acknowledgement: This work was supported in part by KOSEF through OIRC.
ŵŸſ
SNR optical =
Fig. 2 Influence of the increased cardinality on BER performance.
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