Extended 2D Codes Supporting Multirate and QoS in ...

524 J. OPT. COMMUN. NETW./VOL. 5, NO. 5/MAY 2013

Zou et al.

Extended 2D Codes Supporting Multirate and QoS in Optical CDMA Networks With Poisson and Binomial MAI Models Sicheng Zou, Mohammad Masoud Karbassian, and Hooshang Ghafouri-Shiraz

Abstract—In this paper, an extended 2D multiweight multilength optical orthogonal code (MWML-OOC) is proposed and analyzed using both Poisson and binomial distributions of multiple access interference (MAI). We also analyze the performance of 1D and 2D MWML-OOC that can support multirate transmission and quality-of-services (QoS) differentiation in the optical code-division multipleaccess networks. The theory of the 1D MWML-OOC is reviewed where its bit-error rate (BER) performance analysis, based on the binomial model of MAI using the convolution technique, is presented and compared with the well-known Poisson model. The results indicated that the overall performance referred to as BER considering the binomial distribution of MAI is better than that obtained from Poisson-modeled MAI for both 1D and 2D MWML-OOCs. It was also found that the extended 2D MWML-OOC can support a greater number of users with reduced chip times and highly improve the BER performance compared with its 1D and 2D counterparts. Index Terms—1D codes; 2D codes; Binomial distribution; Multirate; Optical CDMA; Poisson model; QoS differentiation.

I. INTRODUCTION

O

ptical code-division multiple-access (OCDMA) has been regarded as one of the most promising technologies for the next generation of optical access networks due to its potential to support multirate transmission and differentiated quality of services (QoS) [1]. In an incoherent OCDMA system, each user is allocated a unique signature sequence from a family of 0/1 sequences referred to as the optical orthogonal codes (OOCs) that satisfy certain correlation properties [2]. The dominant source of bit error in an OCDMA system is usually the multiple-access interference (MAI) between different users, which is caused by the nonzero cross-correlation between code sequences. Many researchers have proposed different OOC designs with different bit-error rate (BER) performance. The designs Manuscript received August 7, 2012; revised February 5, 2013; accepted April 1, 2013; published April 30, 2013 (Doc. ID 173952). S. Zou and H. Ghafouri-Shiraz are with the Department of Electronic, Electrical and Computer Engineering, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK. M. M. Karbassian (e-mail: [email protected]) is with the College of Optical Sciences at the University of Arizona and is affiliated with the NSF Engineering Research Center for Integrated Access Networks, 1630 E. University Blvd., Tucson, Arizona 85721, USA. http://dx.doi.org/10.1364/JOCN.5.000524

1943-0620/13/050524-08$15.00/0

of constant-length and constant-weight codes suitable for one information rate and single QoS class are widely studied and can be either 1D [3–6] or 2D codes [7–9]. However, the growing importance of aggregating various services with different data rates, such as multimedia transmission (e.g., voice, high-definition television, and e-learning) has resulted in diversified data traffic and the requirement of multirate and differentiated-QoS transmission [1]. Consequently, multilength OOCs [10,11] and variable-weight OOCs [12,13] have been introduced to support multirate services and QoS differentiation in 1D-coded OCDMA systems, respectively. Furthermore, a 1D multiweight multilength (MWML) OCDMA system with the ability to simultaneously support multirate and differentiated QoS transmissions has been proposed [14]. However, the performance analysis of the OCDMA system in [14] is based on the Poisson model of MAI for simplicity, which is shown to be an approximation of the binomial distribution of MAI [2]. Also the parameters used in the simulation are not optimized and cannot satisfy the BER ≤ 10−9 requirement for optical communication channels. Another drawback of the 1D MWML OCDMA system [14] can be a very long code-length requirement in order to achieve reasonable BER performance, which results in high-chip rate [15] and high-speed processing encoders and decoders, leading to increased network latency, higher power consumption, and lower spectral efficiency. To increase the number of simultaneous users with reduced chip times and acceptable BER performance, 2D-coded OCDMA systems that encode data in both time and wavelength domains using optical delay lines and an arrayedwaveguide grating have been developed to support either multirate transmission [16] or QoS differentiation [17]. Although 2D multilength multiweight codes that can support simultaneous multirate transmission and QoS differentiation have been proposed in [18–21], their construction algorithms are complicated and inconvenient. Also, their performance analysis is based on a conventional binomial distribution (CBD) method that involves the use of a binomial coefficient. The CBD method is a theoretical approach to evaluate BER for an arbitrary number of user classes, but in fact it becomes very complex for simulation when the number of classes grows. Therefore, we expect to find a simple construction algorithm for the 2D MWML-OOC that can support a larger number of © 2013 Optical Society of America

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simultaneous users with arbitrary information rates and differentiated-QoS requirements while maintaining excellent correlation properties as its 1D counterpart has in [14]. In addition, it is proposed in [22] that the BER obtained with Poisson-modeled MAI can be overestimated compared with that using binominal distribution in a multimedia scenario. However, only preliminary results in a 1D case using the CBD method are presented in [22]. Therefore, it is worthwhile to introduce a new technique to evaluate BER based on binomial distribution, which is easier to simulate than the CBD method even with a large number of classes. In this paper, we first analyze the BER performance of the 1D MWML-OOC proposed in [14], but also consider the binomial distribution of the MAI using a convolution technique, which is shown to be a much easier approach than the widely used CBD method for an arbitrary number of classes. The simulated results are compared with those obtained by the Poisson model, and the coding parameters are also optimized to satisfy the BER requirement of BER ≤ 10−9. We then propose to extend the 1D MWML-OOC to 2D using a straightforward approach and analyze its performance based on multirate transmission and QoS differentiation. To the best of our knowledge, this construction algorithm has not been considered before and is worth investigating. The results indicate that the BER performance based on binomial distribution of the MAI has more accurate approximation than those obtained by the Poisson model [14], and the extended 2D MWML code outperforms its 1D and 2D counterparts in terms of the maximum number of simultaneous users, required chip times, and BER. The rest of the paper is organized as follows: In Section II, the performance analysis of the 1D MWMLOOC based on binomial distribution of the MAI using the convolution technique is introduced and compared with the well-known Poisson model used in [14]. Section III proposes the construction method and conditions for the extended 2D MWML-OOC and analyzes its BER performance based on both Poisson and binomial models of the MAI. Simulation results with optimized coding parameters are presented in Section IV. Finally, this study is concluded in Section V.

II. 1D MWML-OOC PERFORMANCE ANALYSIS A. Background The 1D MWML-OOC set is characterized by
N; W; λa ; λc , where N and W are two vectors specifying the code-length and code-weight for each class. λa and λc are the maximum nonzero-shift autocorrelation and maximum cross-correlation values, respectively [3,14]. Since the nonzero-shift auto-correlation and the maximum cross-correlation of the 1D MWML-OOC are bounded by 1
λa  λc  1, this MWML-OOC set is called a strict OOC. The construction technique of the 1D MWML-OOC is based on the concept of “difference set.” Each code sequence in the

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MWML-OOC is represented by a difference set. If the difference sets of two code sequences have no repeated elements and shared elements, they will form part of a strict OOC set. For detailed construction techniques of the 1D MWML-OOC, refer to [14]. The code generation requires computational processing for higher code-weights and arbitrary code-lengths. This problem can be alleviated by introducing wavelength permutation (2D extension), which will be discussed in Section III in detail. The number of simultaneous users in a Q-class 1D MWML-OCDMA system is subject to the constraint [14] PQ

q1

K q W q
W q − 1 NQ − 1

≤ 1;

(1)

where Q is the number of classes, and K q and W q represent the number of simultaneous users and code-weight of the qth class, respectively. N Q is the maximum code-length of the Q classes.

B. Poisson Model Review Bit error will occur when the data bit “0” is sent, but the decision device detects an output mistakenly as “1” due to the MAI. The BER of the 1D MWML-OCDMA system using the Poisson model of the MAI is given by [14] P
E ≈


μ−1 n  X 1 R ; 1 − e−R n! 2 n0

(2)

where R

Q X q1

K q pqq ¯ :

(3)

R denotes the mean value of the MAI, and μ is the decision threshold. The term e−R
Rn∕n! represents the Poisson ¯ class. It should probability. The intended user is in the qth be noted that in Eq. (3) K q  K q¯ − 1 when q and q¯ are the same. pqq ¯ denotes the probability of a mark in the intended code hit by a code from class q, given by pqq ¯ 

W q¯ W q : 2N q

(4)

It is assumed that N 1 ≤ N 2 ≤    ≤ N q ≤ N Q , and N q is the code-length of the qth class.

C. Binomial Distribution of the MAI Analysis Here in the binomial distribution of the MAI, we use the same hit probability as in the case of the Poisson model to facilitate comparison. Let I denote the total interference from the other users, and without loss of generality we as¯ sume that the intended user is the first user of the qth class. The probability density function of I can then be modified from Eq. (4) and represented as

526 J. OPT. COMMUN. NETW./VOL. 5, NO. 5/MAY 2013

P
I
q j  k 

 W q¯ W q

for k  1 ; for k  0

2N q

0

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(5)

where 1 ≤ q ≤ Q. Here, k  1 means that one mark in the intended code is hit by a code from the other users. I
q j denotes the interference from the jth user in the qth class and ¯ then 2 ≤ j ≤ K q . Since 1 ≤ j ≤ K q . Note that when q  q,
2
q
Q are independent from each other, the I
1 j ; I j ; I j ; …; I j probability density function of the total interference I using binomial distribution can be modified from [11] as

Therefore, the probability density function of the total interference I using binomial distribution is
1
1
1
2 P
1 I
I  P
I 2   P
I 3       P
I K 1   P
I 1 
2  P
I
2 2       P
I K 2 :

Hence, the BER of the intended user in class-1 is PE1 

1 2

¯


1
1 P
Iq
I  P
I
1 1   P
I 2       P
I K 1  ¯

¯

¯

     P
I
2q   P
I
3q       P
I
Kqq 
Q
Q      P
I
Q 1   P
I 2       P
I K Q ;

(6)

where  is the convolution operator. This modified equation can accommodate an arbitrary number of classes (indicated by Q) instead of only two classes as in [11]. As observed from Eq. (6), when the intended user is the first user of ¯ class, the interference will come from a total numthe qth P ber of Q q1 K q − 1 users, referring to the summation of interfering users in all classes excluding one intended user. ¯ class is therefore The BER of the intended user in the qth

PEq¯

1  Pr
Z ≥ Thjb  0 Pr
b  0  2

PQ K −1 q1 Xq iTh

¯

P
Iq
i; (7)

where Th is the decision threshold, Z is the sampled output of the correlator, and b is the data bits of the intended user, respectively [23]. The factor 1/2 is due to the equal probability of transmitting “0” and “1” data bits [2]. 1) Example of a Two-Class 1D MWML-OCDMA System: The BER performance analysis of a simple Q-class 1D MWML based on binomial distribution of the MAI is illustrated here as an example. Consider a Q-class with Q  2, 1D MWML-OCDMA system characterized by
N; W; λa ; λc , where N 
N 1 ; N 2 , W 
W 1 ; W 2 , and λa  λc  1. The number of simultaneous users in the two classes are K 1 and K 2 , respectively. When the intended user is the first user of class-1
q¯  1, the total interference will come from
K 1 − 1 class-1 users and K 2 class-2 users. From Eq. (5), the probability density function of the interference from the ith class-1 user is then P
I
1 i

  k 

W 21 2N 1

0

for k  1 ; for k  0

(8)

The probability density function of the interference from the jth class-2 user is then

where 1 ≤ j ≤ K 1 .

K1X K 2 −1 iW 1

P
1 I
i.

(11)

Similarly, when the intended user is the first user of class-2
q¯  2, the total interference will come from K 1 class-1 users and
K 2 − 1 class-2 users. The probability density function of the interference from class-1 and class-2 users can be derived from Eq. (5) by setting once at q  1 and once at q  2, respectively. Hence, the probability density ¯  2 can be function of the total interference P
2 I
I when q obtained from Eq. (6). The BER of the intended user in class-2 is finally PE2 

1 2

K1X K 2 −1 iW 2

P
2 I
i:

(12)

This is different from Eq. (13) in [11] because in [11] the two classes share the same code-weight, which equals the decision threshold (denoted by Th). In our case, however, the two classes have different code-weights and hence different thresholds (indicated by W 1 and W 2 , respectively). The convolution technique is used in Eqs. (6), (7), (11), and (12) to analyze the BER of the two-class 1D MWMLOCDMA system. We have also modified the equations given in [16], taking into account the use of the CBD method, and derived the BER equations for the two-class situation as illustrated in PE1 

 K K −1  1 1 X2 K1 − 1
p11 l1
1 − p11 K 1 −1−l1 l1 2 l l W 1 2 1   K2
p12 l2
1 − p12 K 2 −l2 ; · l2

PE2

 K K −1  1 1 X2 K1
p21 l1
1 − p21 K 1 −1−l1  2 l l W l1 1 2 2   K2 − 1
p22 l2
1 − p22 K 2 −1−l2 ; · l2

(13)

(14)

where pqq ¯ is given in Eq. (4).

where 2 ≤ i ≤ K 1 .

P
I
2 j  k 

(10)

W

1W2 2N 2

0

for k  1 ; for k  0

(9)

Similarly, Eqs. (13) and (14) are modified from [16] to accommodate a multiple-weight situation, and other notations are changes simply to be consistent with those used in this paper. It will be shown that the BER results using Eqs. (11) and (12) are the same as those obtained from Eqs. (13) and (14), which will be discussed in Section IV. However, it becomes increasingly difficult to simulate the BER performance using the CBD method

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TABLE I 2D CODE SEQUENCES BASED ON 1100100000000 Group 0 λ0 λ0 00λ0 0000000 λ1 λ1 00λ1 0000000 λ2 λ2 00λ2 0000000 λ3 λ3 00λ3 0000000 λ4 λ4 00λ4 0000000 λ5 λ5 00λ5 0000000 λ6 λ6 00λ6 0000000

Group 2

λ0 λ1 00λ2 0000000 λ1 λ2 00λ3 0000000 λ2 λ3 00λ4 0000000 λ3 λ4 00λ5 0000000 λ4 λ5 00λ6 0000000 λ5 λ6 00λ0 0000000 λ6 λ0 00λ1 0000000

λ0 λ2 00λ4 0000000 λ1 λ3 00λ5 0000000 λ2 λ4 00λ6 0000000 λ3 λ5 00λ0 0000000 λ4 λ6 00λ1 0000000 λ5 λ0 00λ2 0000000 λ6 λ1 00λ3 0000000

III. 2D MWML-OOC CONSTRUCTION

AND

ANALYSIS

A. Construction Conditions of the 2D MWML-OOC The 2D extended MWML-OOC is characterized by
P × N; W; λa ; λc , where P is a prime number representing the number of available wavelengths. Note that P can also be a nonprime integer, whereas the total cardinality of the 2D code-set will be reduced. The discussion of P not being a prime number is beyond the scope of this paper and one can refer to [8] for detailed analysis. Other notations are inconsistent with those used in 1D MWML-OOC. The 2D MWML-OOC can be constructed by employing the 1D MWML-OOC proposed in [14] as the time-spreading code and a prime code [24] for wavelength permutations. It was indicated in [8] that if the auto-correlation side-lobes and cross-correlation functions of the time-spreading OOC are at most equal to 1, then the extended 2D OOC with prime sequences as wavelength-hopping code also has both values of auto-correlation side-lobes and cross-correlation no greater than 1. Accordingly, since the 1D MWML-OOC is a strict OOC set, the extended 2D MWML-OOC can maintain good correlation properties similar to its 1D counterpart [8]. The extended 2D code has an overall cardinality of ΦOOC P2, where ΦOOC is the total cardinality of the 1D MWML-OOC, denoted as ΦOOC 

Q X q1

Φq ;

Group 6 λ0 λ6 00λ5 0000000 λ1 λ0 00λ6 0000000 λ2 λ1 00λ0 0000000 λ3 λ2 00λ1 0000000 λ4 λ3 00λ2 0000000 λ5 λ4 00λ3 0000000 λ6 λ5 00λ4 0000000

where Max
W 1 ; W 2 ; …; W q ; …; W Q  denotes the maximum code-weight of the Q classes. 1) Example of 2D MWML-OOC Construction: To construct a 2D MWML-OOC, one can first construct a 1D MWML-OOC using the approach illustrated in [14]. Once a 1D MWML-OOC has been constructed, a 2D MWMLOOC can be obtained following a similar procedure in [16] by applying a 1D MWML-OOC as the time-spreading code and the prime code as wavelength permutations. For example, the extended 2D code sequences based on two 1D MWML codes (1100100000000 and 1010000) are shown in Tables I and II, respectively (assuming P  7).

B. Performance Analysis of 2D MWML-OOC The performance of the proposed 2D MWML-OOC can also be analyzed using both Poisson and binomial distributions of the MAI, as discussed in the following. For simplicity, a Q-class 2D MWML-OOC with Q  2 is considered here to support double information rates and two differentiated QoS. The probabilities of getting one hit between two code-words from the same class can be obtained following a similar procedure as in [16]. The coding parameters (e.g., code-weight, code-length, cardinality, and wavelength number) used in [16] are also changed accordingly to satisfy the proposed MWML-OOC requirements:

p11 

PW 21
PΦ1 − 1 
P − 1
W 1 − 12 ; 2PN 1
P2 Φ1 − 1

(17)

p22 

PW 22
PΦ2 − 1 
P − 1
W 2 − 12 ; 2PN 2
P2 Φ2 − 1

(18)

(15)

where Φq is the cardinality of the 1D MWML timespreading code-set in the qth class, which is bounded by Eq. (1) with Φq replacing K q . The wavelength permutations are obtained from the prime codes over the Galois field onto the nonzero time slots of the 1D MWML-OOC. For one time-spreading code, there are P possible groups, and each group has P available 2D code sequences, therefore the cardinality of the 1D MWML-OOC is increased by a factor of P2 by 2D extension. The code-weights of the 2D MWML-OOC is subject to the following constraint: Max
W 1 ; W 2 ; …; W q ; …; W Q  ≤ P;

P7 …

Group 1

when Q grows. For a multiclass OCDMA system, it is easier to use a convolution technique for MATLAB simulation.

AND

(16)

TABLE II 2D CODE SEQUENCE BASED ON 1010000 Group 0 λ0 0λ0 0000 λ1 0λ1 0000 λ2 0λ2 0000 λ3 0λ3 0000 λ4 0λ4 0000 λ5 0λ5 0000 λ6 0λ6 0000

Group 1

Group 2

λ0 0λ1 0000 λ1 0λ2 0000 λ2 0λ3 0000 λ3 0λ4 0000 λ4 0λ5 0000 λ5 0λ6 0000 λ6 0λ0 0000

λ0 0λ2 0000 λ1 0λ3 0000 λ2 0λ4 0000 λ3 0λ5 0000 λ4 0λ6 0000 λ5 0λ0 0000 λ6 0λ1 0000

…

AND

P7 Group 6 λ0 0λ6 0000 λ1 0λ0 0000 λ2 0λ1 0000 λ3 0λ2 0000 λ4 0λ3 0000 λ5 0λ4 0000 λ6 0λ5 0000

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where p11 denotes the hit probabilities between class-1 and p22 denotes the hit probabilities between class-2. When the intended codeword is from group 0 of class-1 and the interfering codeword is from class-2, the hit probability can be derived using the principle illustrated in [8] as q012 

W 1 W 2 Φ2 W W 1 Φ2 W 1
P − 1 ·  1 2· · : 2N 2 Φ2 P2 2N 2 W 2 Φ2 P2

(19)

When the intended codeword is from group i of class-1, where i  f1; 2; …; P − 1g, the hit probability from an interfering codeword in class-2 is given by qi12 

W 1 W 2 1 Φ2 W 1 P · · : 2N 2 W 2 Φ2 P2

(20)

Therefore, the average hit probability when the intended codeword is in class-1 can be obtained as 1 0 P−1 i q  q12 P 12 P W W  W 21
P − 1 P − 1 W 21  1 2  : · 2N 2 P P 2N 2 P3

p12 

(21)

Similarly, when the intended codeword is from class-2 and the interfering codeword is from class-1, the hit probability is given by p21 

W 1 W 2  W 22
P − 1 P − 1 W 22 : ·  2N 1 P P 2N 1 P3

(22)

1) Poisson Model Analysis: The mean value of the MAI in the two-class 2D MWML-OCDMA system can be calculated using Eq. (3) where pqq ¯ are given by Eqs. (17), (18), (21), and (22) as analyzed above instead of Eq. (4). To evaluate the BER performance of the two-class 2D MWML-OOC based on the Poisson model of the MAI, Eq. (2) can be applied.

P
I
1 j  k 

P
I
2 j  k 





p21 0

for k  1 ; for k  0

(25)

p22 0

for k  1 ; for k  0

(26)

The BER of the intended user in class-2 can be obtained from Eq. (12) using Eqs. (25) and (26) as the new probability density functions.

IV. SIMULATION RESULTS A. 1D MWML-OOC OCDMA System Here, the BER performance of the 1D MWML-OOC can be investigated based on both Poisson and binomial distributions of the MAI. Figure 1 shows the BER performance of a single-class (1000, 11, 1, 1) 1D strict OOC with nine simultaneous users as a function of the decision threshold (the four parameters used in the bracket are the same as those defined in Subsection II.A). It is clear that the BER obtained from binomial distribution is more enhanced than that of the Poisson model. This can be qualitatively explained due to the fact that when calculating the BER, the Poisson model adds the probability of interference from the decision threshold to infinity with a mean average, while binomial distribution only considers the interference from the decision threshold to the maximum number of interfering users, as seen in Fig. 1. The detailed mathematical analysis of this difference, however, is beyond the scope of this paper. It is also interesting to notice that when the normalized decision threshold is larger than eight, the BER has a finite value under the Poisson model, whereas it drops to zero in the case of binomial distribution (see Fig. 1). This is because the maximum cross-correlation of the 1D strict OOC is one, and the maximum number of interfering users is eight, excluding the intended user. Thus,

2) Binomial Distribution Analysis: In the binomial distribution analysis, Eqs. (17), (18), (21), and (22) were used as the hit probabilities. When the intended user is from class-1, the probability density function of the interference from the ith class-1 user [see Eq. (8)] and jth class-2 user [see Eq. (9)] can therefore be modified as follows P
I
1 i  k 

P
I
2 j



  k 

p11 0

for k  1 ; for k  0

(23)

p12 0

for k  1 : for k  0

(24)

The BER of the intended user in class-1 can be obtained from Eq. (11), with Eqs. (23) and (24) as the new probability density functions instead of Eqs. (8) and (9). Similarly, the probability density functions when the intended user is in class-2 are given by

Fig. 1. BER performance of a single-class 1D strict OOC as a function of the normalized decision threshold with nine simultaneous users. N  1000; W  11.

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it is reasonable to assume that there should be no bit-errors when the decision threshold is greater than eight. Therefore, it can be observed here that the binomial distribution has better potential with more enhanced performance to model the MAI as compared with the Poisson model. The BER performance of a three-class MWML-OOC based on both Poisson and binomial distributions of the MAI is plotted in Fig. 2 as a function of the number of users in class-3, which has the lowest information rate and the lowest QoS. The coding parameters are optimized to meet BER ≤ 10−9 , while maintaining the condition in Eq. (1). The number of users in class-1 and class-2 are fixed to 6 and 15, respectively. Note that class-1 with high codeweight has higher QoS than the other two classes for both Poisson and binomial distributions of the MAI. However, the BER difference between Poisson and binomial models is not constant and decreases from class-1 (high-QoS) to class-3 (low QoS). The BER difference is also reduced for all classes as the number of users in class-3 increases. In order to meet the BER ≤ 10−9 requirement for class-1 users, the number of class-3 users cannot exceed eight for the Poisson model, but it can be as large as 19 under binomial distribution. In contrast, the difference between the Poisson model and binomial distribution is very small for class-3 users, and therefore the Poisson model can be a good approximation for low code-weight (i.e., low QoS) classes. The BER of a two-class 1D MWML-OOC is shown in Fig. 3 as a function of the code-length of class-2, with class-2 supporting twice the number of users as class-1, namely, K 1  10 and K 2  20. The code-length of class-2 is varied from 1000 to 3500. It is clear that the BER difference between the Poisson model and binomial distribution of class-2 users reduces as its code-length increases, while that difference of class-1 changes negligibly and remains approximately constant. In Fig. 4, the effect of the code-weight is depicted when other parameters are constant. A two-class MWML-OOC with a code-length of class-2 twice that of class-1 is selected

Fig. 2. BER for a three-class 1D MWML-OOC as a function of the number of users in the low-QoS class.

Fig. 3. BER for a two-class 1D MWML-OOC as a function of the code-length of the low-QoS class.

here as N 1  1000 and N 2  2000, respectively. The two classes both support nine simultaneous users
K 1  K 2  9 and the code-weight of class-2 is varied from 4 to 10. It is noticed that the BER difference between the Poisson model and binomial distribution for class-2 increases as its codeweight increases, which is opposite to the trend observed in the effect of code-length in Fig. 3. However, the difference for class-1 decreases slightly with the increase of the code-weight of class-2, which is similar to that in Fig. 3. By combining Figs. 2–4, it can be concluded that the BER of the 1D MWML-OOC considering binomial distribution of the MAI is more improved than that of the Poisson model. Furthermore, the BER difference between the Poisson model and binomial distribution of one class reduces when increasing its code-length or decreasing its code-weight. Therefore the BER performance obtained from the Poisson model can be a better approximation of binomial distribution for lower code-weight and longer code-length class in the 1D MWML-OOC. The BER results obtained by both the convolution technique and the CBD method are plotted in Fig. 5, as

Fig. 4. BER for a two-class 1D MWML-OOC as a function of the code-weight of class-2 (low QoS).

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Fig. 5. BER comparison between the convolution technique and the CBD method.

discussed at the end of Section II. A two-class MWML-OOC characterized by (1000, 11, 1, 1) and (1500, 8, 1, 1) is illustrated here as an example. The BER performance of class-1 can be calculated using Eqs. (11) and (13), while the BER of class-2 can be obtained from Eqs. (12) and (14). It is clear that the results obtained by a convolution technique and a CBD method are very similar; therefore, these two methods are mathematically equivalent. However, for a Q-class MWML-OOC with Q ≥ 2, it is much more convenient to calculate BER using a convolution technique rather than the CBD method.

B. 2D MWML-OOC Simulation Results The numerical results of a two-class 2D MWML-OOC are analyzed here based on both Poisson and binomial distributions. The two-class 2D MWML-OOC is characterized by
P × N; W; λa ; λc  
19 × f150; 200g; f11; 8g; 1; 1 and the cardinalities of its 1D time-spreading code are set to one

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Fig. 7. BER performance comparison between 2D MWML-OOC and 2D VLVW-OOC.

and two, respectively (Φ1  1; Φ2  2). Since each timespreading code can be extended to P2 available 2D code sequences through wavelength permutations, one timespreading code is enough to provide abundant 2D code sequences to support a large number of simultaneous users. In Fig. 6, the BER performance for the two-class MWMLOOC as a function of the number of class-1 (high-QoS) users is depicted. The results indicate that when the number of class-2 (low-QoS) users is fixed to 20 (K 2  20), the two-class 2D MWML-OOC can support up to 30 class-1 users (maximum K 1  30) to meet the BER requirement of 10−9 with only 150 chip-times (N 1  150), which is almost one-tenth compared with its 1D counterpart that requires at least a thousand chips to achieve reasonable BER performance. Figure 7 compares the performance of the newly constructed 2D MWML-OOC with that of the 2D variable-length, variable-weight optical orthogonal codes (VLVW-OOCs) in [18], and the results show that the 2D MWML-OOC outperforms its 2D counterpart in terms of BER and the maximum number of users, since the 2D MWML-OOC can support up to 30 class-1 users to meet the BER requirement, whereas 2D VLVW-OOC can only support 24 class-1 users.

V. CONCLUSION

Fig. 6. BER performance for a two-class 2D MWML-OOC as a function of the number of class-1 users.

The performance analysis of the 1D MWML-OOC based on binomial distribution of the MAI using a convolution technique is presented and compared with the well-known Poisson model. The coding parameters were also optimized to satisfy the BER requirement for optical communications. A convenient construction method and conditions for the extended 2D MWML-OOC are reported, and the BER performance is analyzed using both Poisson and binomial distributions. The results indicated that the BER obtained from binomial distribution of the MAI was better than that of the Poisson model for both 1D and 2D MWML-OOCs. The results also indicated that the binomial model of the

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MAI has better performance potential in queue design for MAI reduction and contention resolution of practical OCDMA networks. Furthermore, it is found that the BER difference between the Poisson model and binomial distributions in the 1D MWML-OOC is reduced with the increase of the code-length and the number of simultaneous users, as well as the decrease of the code-weight within a class. The extended 2D MWML-OOC allows the number of available wavelengths, code-lengths, and code-weights to be chosen independently; therefore, it can support arbitrary data rates and QoS differentiation. The analysis of the 2D MWML-OOC specified that it can accommodate a larger number of simultaneous users with a much smaller number of chip times and highly improved BER compared with its 1D and 2D counterparts, which makes this 2D code set more practical for an OCDMA system where the number of chip times is limited, but the number of wavelengths is flexible.

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