- Email: [email protected]
Semi-randomly constructed optical orthogonal codes
Optics Communications 282 (2009) 500–503
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Semi-randomly constructed optical orthogonal codes Cenk Argon * Seagate Technology, Coding and Signal Processing Group, RTP Office, Durham, NC 27703, United States
a r t i c l e
i n f o
Article history: Received 12 June 2007 Received in revised form 21 July 2008 Accepted 16 October 2008
a b s t r a c t Optical orthogonal code (OOC) sequences are assigned to optical code-division multiple-access (OCDMA) network users, who are able to transmit data asynchronously. In this work, we propose a semi-random OOC design technique based on extended sets, where the input parameters are the sequence weight, number of sequences (i.e., users), and a target sequence length. The design method under consideration is able to converge to the desired short OOC lengths given the number of iterations during the execution of the algorithm is sufficiently large. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In optical code-division multiple-access (OCDMA) systems [1], an optical orthogonal code (OOC) sequence is assigned to each user and data is sent via the destination user’s OOC sequence. An example optical CDMA system for N users in a star configuration is shown in Fig. 1. Different than electrical pseudo-random sequences that can consist of bipolar pulses, OOC sequences are unipolar; i.e., instead of the 1 and +1 levels, only the 0 and +1 levels are available in intensity-modulation direct-detection (IM/DD) optical systems. Hence, true orthogonality cannot be achieved and OOC sequences have to be very long to support large numbers of users. Unfortunately, this introduces large bandwidth expansion and hence, optimal design methods are desirable. Recently, the use of error correction codes (ECCs) [8] has been proposed to reduce the required system bandwidth in optical CDMA systems. For example, Dale and Gagliardi [10] suggest to implement either Reed-Solomon codes or convolutional codes, and Zhang [11] proposes the use of asymmetric ECCs for optical CDMA with on-off keying (OOK). Iterative turbo codes [9] are also considered in various studies [12–16] for optical CDMA with OOK or pulse position modulation (PPM). All of these advanced ECC techniques are aimed to the use of OOC sequences with lower weight (and hence, shorter length). There already exist various design techniques for OOC sequences [2–5], where the design criteria are often based on parameters that need to satisfy certain conditions (like being a prime number for example). In this work, we propose an alternative OOC design technique where, as similar as in [6,7], the main input parameters are the number of users and the weight of the OOC sequences. The proposed algorithm is semi-random and can require multiple iterations until the final OCC sequences are obtained. Dif-
ferent than the semi-random design method proposed in [7], we introduce a target sequence length as an input parameter which helps speeding up the algorithm and enables the design of OOCs with shorter lengths in a shorter time. 2. Optical orthogonal codes The parameters of an OOC are denoted by F; K; ka and kc , which represent sequence length (i.e., number of chips), sequence weight (or number of pulses per sequence), auto-correlation constraint, and cross-correlation constraint, respectively. OOCs with ka ¼ kc ¼ 1 have the best achievable correlation constraints for a direct-detection OCDMA system. With a chip, we are referring to a time slot where an optical pulse can be placed. For example, three OOC sequences are shown in Fig. 2, where the OOC parameters are F ¼ 28, K ¼ 3, and ka ¼ kc ¼ 1, and the number of users is N ¼ 3. The sequences are denoted as Su ¼ su;0 su;1 su;2 . . . su;nchip 1 , where u ¼ 1; 2; . . . ; N, su;j 2 f0; 1g, and nchip denotes the number of chips (i.e., F, the number of time slots in one bit interval for on-off keying [OOK]). The chip positions are numbered from 0 through nchip 1. It is assumed that the K light pulses of an OOC sequence have a normalized amplitude equal to 1. For convenience of notation, for the remainder of this paper, we assume an OCDMA system with OOK (i.e., F ¼ nchip ). The generalization to pulse-position modulation (PPM) or other modulation techniques is straightforward. With the above notation, the correlation constraints can be expressed as (i) Auto-correlation constraint:
Huu ðmÞ ¼
F1 X
su;i su;ððiþmÞ
mod FÞ
6 ka
ð1Þ
i¼0
* Tel.: +1 919 313 4631; fax: +1 919 313 4633. E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.043
for any sequence Su and any integer m–0 (mod F). We note that Huu ð0Þ ¼ K since each sequence has K pulses of unit amplitude.
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C. Argon / Optics Communications 282 (2009) 500–503
Fig. 1. Optical CDMA system.
i.e., the closer the l value is to 100%, the better the bandwidth utilization of the OCDMA system. For a perfect optimal code, l ¼ 100%. Usually, K F; hence, it is more convenient to indicate the OOC sequences with the chip numbers at which pulses are placed; i.e., for an OOC with weight K, the notation for the ith sequence would be Si ¼ fbi;1 ; bi;2 ; . . . ; bi;K g, where bi;j denotes the chip position of the jth pulse of sequence Si . We note that 0 6 bi;j 6 F 1. As an example, for the sequences shown in Fig. 2, the notation would be S1 ¼ f0; 2; 8g, S2 ¼ f0; 3; 10g, and S3 ¼ f0; 4; 13g. Let S1 , S2 , . . ., SN be the sequences of an OOC of size N (number of users), weight K, and length F. An alternative way of representing OOC sequences is possible via equivalent sets of relative delay elements [1]. The set Di ¼ fai;1 ; ai;2 ; . . . ; ai;K g represents sequence Si where ai;j is the relative delay between the beginning of the jth pulse to the beginning of the ðj þ 1Þst pulse of sequence Si (in terms of number of chips) for j ¼ 1; 2; . . . ; K 1. Furthermore, ai;K is the relative delay between the beginning of the Kth pulse to the beginning of the first pulse of the periodic sequence Si . In general, we can write
Fig. 2. ðF ¼ 28; K ¼ 3; ka ¼ 1; kc ¼ 1Þ OOC sequences.
(
ai;j ¼
(ii) Cross-correlation constraint:
Huw ðmÞ ¼
F1 X
su;i sw;ððiþmÞ
mod FÞ
6 kc
ð2Þ
i¼0
for any two sequences Su and Sw (Su –Sw ) and any integer m. For an ðF; K; ka ¼ 1; kc ¼ 1Þ OOC, it can be shown that the upperbound on the number of address sequences (i.e., number of users) is equal to [1,2]:
Nmax
F1 ¼ ; K ðK 1Þ
ð3Þ
where bxc denotes the floor function (or integer part) of the real number x. It follows that perfect optimal OOCs with N sequences have length F opt ¼ N K ðK 1Þ þ 1. We denote the efficiency of a designed OOC by l, where
l¼
F opt 100%; F
ð4Þ
bi;ðjþ1Þ bi;j
for j ¼ 1; 2; . . . ; K 1
F þ bi;0 bi;K
for j ¼ K:
ð5Þ
An extended set of Si is denoted by Ei and consists of all linear combinations of jointly connected relative delays, i.e., Ei ¼ fei;1 ; ei;2 ; . . . ; ei;ðKðK1ÞÞ g, where [7]
ei;j ¼
þ1 bj1 K c X
ai;ð½fm1þððj1Þ mod KÞg mod Kþ1Þ :
ð6Þ
m¼1
For the sequence Si to satisfy the periodic autocorrelation property with ka ¼ 1, there should be no repetitions of elements in the set Ei . This is because of the reason that there should be no overlap of two or more pulses of the shifted versions of the sequence Si . Similarly, to satisfy the crosscorrelation property kc ¼ 1, for any two sequences Su and Sw , where u–w, the intersection of the corresponding extended sets must be empty, i.e., Eu \ Ew ¼ ;. Due to the cyclic nature of the extended set Ei , it should be noted that if ðei;j Þ is an element, then also ðF ei;j Þ is an element of Ei . We will exploit this property during the OOC design.
502
C. Argon / Optics Communications 282 (2009) 500–503
In the following section, we will make use of these extended sets for the design of OOC sequences. The design method proposed here is outlined for correlation constraints ka ¼ kc ¼ 1, but can be generalized for other correlation constraints as well.
15
K=4 K=5 K=6
14 13 12
3. Semi-random OOC design method To optimize the sequence length of the designed OOC, we propose here a semi-random design method where we use a random permutation vector to determine the order of OOC sequence updates, and a maximum number of iterations until we terminate the run of the algorithm. The design criteria is to have the two input parameters N and K, and an additional parameter denoted by ITR, which indicates the maximum number of iterations. Different than the design method proposed in [7], we also propose to input a target length F target , which can be used exploiting the aforementioned cyclic nature of extended sets. The proposed design algorithm can be summarized as follows: (i) Specify the weight, K, and number of users (sequences), N, for the OOC. Initially, the sequence length F is set to F target ¼ F opt . Specify also the maximum number of iterations, ITR, which determines how many iterations per target length will be performed until we increment the length and start a new iterative search. (ii) Initialize the iteration counter, z, to 1. (iii) Assume that a pulse is placed at the 0th chip position for all OOC sequences. (iv) The first delay elements of the OOC sequences are assigned as ai;1 ¼ i, for i ¼ 1; 2; . . . ; N. (v) Start to form the extended sets, Ei ’s, by adding ðai;1 Þ’s and to these sets; i.e., initialize ðF target ai;1 Þ’s Ei ¼ fai;1 ; F target ai;1 g, for i ¼ 1; 2; . . . ; N. (vi) Generate vector c ¼ ½c1 ; c2 ; . . . ; cN , which is the random permutation of the ordered vector [1, 2, . . ., N]. The elements of vector c are specifying in which order the OOC sequences are being updated at the determination of the jth delay element. (vii) Initialize m ¼ 1 and i ¼ cm . (viii) Initialize the offset parameter q ¼ N. (ix) For the OOC sequence under consideration, Si , increment q ¼ q þ 1, until its value and ðF target qÞ are not present in any of the partial extended sets. If at any time q P F target , increment counter z ¼ z þ 1 and go to step (xi). (x) Add q as a delay element for the OOC sequence under consideration, Si , iff all linear combinations of jointly connected relative delay elements are not repeated within the partial extended set under consideration or within the partial extended sets of the other sequences. While checking this property, make use of the cyclic structure of the extended sets. In case q is added as a delay element for Si , then update the extended set for this sequence (again making use of the cyclic nature of the extended sets) and go to step (xii). If q is not added as a delay element, then do not do any update of the extended sets and go to step (ix). (xi) If counter z > ITR, then increment F target ¼ F target þ 1 and go to step (ii); otherwise, go to step (iii). (xii) If all K pulses for all N sequences have been assigned (this is equivalent to assigning all K 1 delay elements for all N sequences), then proceed to step (xiii). If not, then move to the next OOC sequence [i.e., increment m ¼ m þ 1; if incremented m is equal to N þ 1, then go to step (vi); if not, set i ¼ cm and go to step (viii)]. (xiii) The chip positions of the pulses for the N OOC sequences are determined using
µ(new) — µ([7]) %
11 10 9 8 7 6 5 4 3 2 1 0
0
5
10
15
20
25
30
Number of Users Fig. 3. Percentage efficiency increase with new algorithm against algorithm in [7] (ITR = 1000).
Si ¼ f0; ai;1 ; ai;1 þ ai;2 ; ai;1 þ ai;2 þ ai;3 ; . . . ; ai;1 þ ai;2 þ þ ai;ðK1Þ g for i ¼ 1; 2; . . . ; N, and the designed OOC has sequence length F target . (xiv) Stop. We call the above algorithm ‘‘semi-random”, since the search for the offset parameter q is deterministic, however, the order of OOC sequences to be updated is determined via a random permutation vector c and hence, is not sequential. Furthermore, we note that for each F target , we have at most ITR iterations before we search for an OOC of increased length. In addition, one advantage of using a random permutation vector c to determine the OOC sequence update order is to increase the probability to converge to the optimum OOC sequence length and also to distribute the pulse locations more evenly across all OOC sequences. As mentioned before, the algorithm presented here is different than the algorithm in [7] since we have the advantage of using a target sequence length as an input parameter and furthermore, we also exploit the cyclic nature of extended sets. This was not possible in [7] since the OOC length was determined after all delay elements were already assigned. To demonstrate the effectiveness of the proposed algorithm, we show in Fig. 3 the efficiency comparison of the newly proposed algorithm against the algorithm presented in [7] for the same number of iterations (ITR = 1000) and code parameters, N and K. For this purpose, we subtract the efficiency of the algorithm in [7] from the efficiency of the newly proposed algorithm; i.e., l(new)l([7]), where the efficiency of the designed OOC is given in (4). Please note that here at most ITR iterations are performed for each target length and hence, for the proposed algorithm the total number of iterations is higher; on the other hand, since the partial extended sets are formed faster via exploiting the cyclic structure, the proposed algorithm is able to break out of unsuccessful iterations faster than the algorithm presented in [7]. Furthermore, as observed in Fig. 3, we clearly note the increase in efficiency for the same code parameters with the newly proposed OOC design algorithm, especially with increased sequence weight, K and increased number of users. 4. Conclusion We presented an alternative optical orthogonal code (OOC) design technique for users of an intensity-modulation/direct-detection optical code-division multiple-access (CDMA) network. The
C. Argon / Optics Communications 282 (2009) 500–503
proposed method was a semi-random based approach where multiple iterations were required to generate all OOC sequences. Compared to previous methods, this algorithm uses a target sequence length and a pre-defined number of iterations. The target sequence length is incremented if the OOC sequence is not generated at the end of the pre-determined iteration number. The design method outlined in this work concentrated on correlation constraints ka ¼ kc ¼ 1 which are the best achievable correlation constraints in an intensity-modulated direct-detection optical CDMA network; however, the OOC design algorithm presented here can be generalized to other constraints as well. References [1] J.A. Salehi, IEEE Trans. Commun. 37 (1989) 824. [2] F.R.K. Chung, J.A. Salehi, V.K. Wei, IEEE Trans. Inform. Theory 35 (1989) 595. [3] H. Chung, P.V. Kumar, IEEE Trans. Inform. Theory 36 (4) (1990) 866.
[4] [5] [6] [7] [8] [9]
[10] [11]
[12] [13] [14] [15]
[16]
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S.V. Maric, Z.I. Kostic, E.L. Titlebaum, IEEE Trans. Commun. 41 (8) (1993) 1217. A.A. Shaar, P.A. Davies, Electron. Lett. 19 (21) (1983) 888. C. Argon, R. Ergül, Optics Commun. 116 (1995) 326. C. Argon, Systematic optical orthogonal code design techniques using extended sets, in: Proceedings of IEEE ICC 2006, Istanbul, Turkey, June 2006. S.B. Wicker, Error Control Systems for Digital Communication and Storage, Prentice Hall, Upple Saddle River, NJ, 1995. C. Berrou, A. Glavieux, P. Thitimajshima, Near Shannon limit error-correcting coding and decoding: turbo-codes, in: Proceedings of IEEE ICC 1993, Geneva, Switzerland, May 1993, p. 1064. M.R. Dale, R.M. Gagliardi, IEEE Trans. Commun. 43 (1995) 2485. J.-G. Zhang, Use of error-correction codes to improve the performance of optical fiber code division multiple access systems, in: Proceedings of URSI International Symposium on Signals, Systems, and Electronics, 1998, p. 361. J.Y. Kim, H.V. Poor, IEEE J. Lightwave Technol. 19 (3) (2001) 312. K. Kiasaleh, IEEE J. Lightwave Technol. 16 (1) (1998) 18. T. Ohtsuki, J.M. Kahn, IEEE J. Lightwave Technol. 18 (12) (2000) 1776. C. Argon, S.W. McLaughlin, Turbo product codes for performance improvement of optical CDMA systems, in: Proceedings of IEEE Globecom’01, San Antonio, TX, USA, November 2001, p. 1505. C. Argon, S.W. McLaughlin, IEEE J. Lightwave Technol. 20 (9) (2002) 1653.
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Semi-randomly constructed optical orthogonal codes Cenk Argon * Seagate Technology, Coding and Signal Processing Group, RTP Office, Durham, NC 27703, United States
a r t i c l e
i n f o
Article history: Received 12 June 2007 Received in revised form 21 July 2008 Accepted 16 October 2008
a b s t r a c t Optical orthogonal code (OOC) sequences are assigned to optical code-division multiple-access (OCDMA) network users, who are able to transmit data asynchronously. In this work, we propose a semi-random OOC design technique based on extended sets, where the input parameters are the sequence weight, number of sequences (i.e., users), and a target sequence length. The design method under consideration is able to converge to the desired short OOC lengths given the number of iterations during the execution of the algorithm is sufficiently large. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction In optical code-division multiple-access (OCDMA) systems [1], an optical orthogonal code (OOC) sequence is assigned to each user and data is sent via the destination user’s OOC sequence. An example optical CDMA system for N users in a star configuration is shown in Fig. 1. Different than electrical pseudo-random sequences that can consist of bipolar pulses, OOC sequences are unipolar; i.e., instead of the 1 and +1 levels, only the 0 and +1 levels are available in intensity-modulation direct-detection (IM/DD) optical systems. Hence, true orthogonality cannot be achieved and OOC sequences have to be very long to support large numbers of users. Unfortunately, this introduces large bandwidth expansion and hence, optimal design methods are desirable. Recently, the use of error correction codes (ECCs) [8] has been proposed to reduce the required system bandwidth in optical CDMA systems. For example, Dale and Gagliardi [10] suggest to implement either Reed-Solomon codes or convolutional codes, and Zhang [11] proposes the use of asymmetric ECCs for optical CDMA with on-off keying (OOK). Iterative turbo codes [9] are also considered in various studies [12–16] for optical CDMA with OOK or pulse position modulation (PPM). All of these advanced ECC techniques are aimed to the use of OOC sequences with lower weight (and hence, shorter length). There already exist various design techniques for OOC sequences [2–5], where the design criteria are often based on parameters that need to satisfy certain conditions (like being a prime number for example). In this work, we propose an alternative OOC design technique where, as similar as in [6,7], the main input parameters are the number of users and the weight of the OOC sequences. The proposed algorithm is semi-random and can require multiple iterations until the final OCC sequences are obtained. Dif-
ferent than the semi-random design method proposed in [7], we introduce a target sequence length as an input parameter which helps speeding up the algorithm and enables the design of OOCs with shorter lengths in a shorter time. 2. Optical orthogonal codes The parameters of an OOC are denoted by F; K; ka and kc , which represent sequence length (i.e., number of chips), sequence weight (or number of pulses per sequence), auto-correlation constraint, and cross-correlation constraint, respectively. OOCs with ka ¼ kc ¼ 1 have the best achievable correlation constraints for a direct-detection OCDMA system. With a chip, we are referring to a time slot where an optical pulse can be placed. For example, three OOC sequences are shown in Fig. 2, where the OOC parameters are F ¼ 28, K ¼ 3, and ka ¼ kc ¼ 1, and the number of users is N ¼ 3. The sequences are denoted as Su ¼ su;0 su;1 su;2 . . . su;nchip 1 , where u ¼ 1; 2; . . . ; N, su;j 2 f0; 1g, and nchip denotes the number of chips (i.e., F, the number of time slots in one bit interval for on-off keying [OOK]). The chip positions are numbered from 0 through nchip 1. It is assumed that the K light pulses of an OOC sequence have a normalized amplitude equal to 1. For convenience of notation, for the remainder of this paper, we assume an OCDMA system with OOK (i.e., F ¼ nchip ). The generalization to pulse-position modulation (PPM) or other modulation techniques is straightforward. With the above notation, the correlation constraints can be expressed as (i) Auto-correlation constraint:
Huu ðmÞ ¼
F1 X
su;i su;ððiþmÞ
mod FÞ
6 ka
ð1Þ
i¼0
* Tel.: +1 919 313 4631; fax: +1 919 313 4633. E-mail address: [email protected] 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.10.043
for any sequence Su and any integer m–0 (mod F). We note that Huu ð0Þ ¼ K since each sequence has K pulses of unit amplitude.
501
C. Argon / Optics Communications 282 (2009) 500–503
Fig. 1. Optical CDMA system.
i.e., the closer the l value is to 100%, the better the bandwidth utilization of the OCDMA system. For a perfect optimal code, l ¼ 100%. Usually, K F; hence, it is more convenient to indicate the OOC sequences with the chip numbers at which pulses are placed; i.e., for an OOC with weight K, the notation for the ith sequence would be Si ¼ fbi;1 ; bi;2 ; . . . ; bi;K g, where bi;j denotes the chip position of the jth pulse of sequence Si . We note that 0 6 bi;j 6 F 1. As an example, for the sequences shown in Fig. 2, the notation would be S1 ¼ f0; 2; 8g, S2 ¼ f0; 3; 10g, and S3 ¼ f0; 4; 13g. Let S1 , S2 , . . ., SN be the sequences of an OOC of size N (number of users), weight K, and length F. An alternative way of representing OOC sequences is possible via equivalent sets of relative delay elements [1]. The set Di ¼ fai;1 ; ai;2 ; . . . ; ai;K g represents sequence Si where ai;j is the relative delay between the beginning of the jth pulse to the beginning of the ðj þ 1Þst pulse of sequence Si (in terms of number of chips) for j ¼ 1; 2; . . . ; K 1. Furthermore, ai;K is the relative delay between the beginning of the Kth pulse to the beginning of the first pulse of the periodic sequence Si . In general, we can write
Fig. 2. ðF ¼ 28; K ¼ 3; ka ¼ 1; kc ¼ 1Þ OOC sequences.
(
ai;j ¼
(ii) Cross-correlation constraint:
Huw ðmÞ ¼
F1 X
su;i sw;ððiþmÞ
mod FÞ
6 kc
ð2Þ
i¼0
for any two sequences Su and Sw (Su –Sw ) and any integer m. For an ðF; K; ka ¼ 1; kc ¼ 1Þ OOC, it can be shown that the upperbound on the number of address sequences (i.e., number of users) is equal to [1,2]:
Nmax
F1 ¼ ; K ðK 1Þ
ð3Þ
where bxc denotes the floor function (or integer part) of the real number x. It follows that perfect optimal OOCs with N sequences have length F opt ¼ N K ðK 1Þ þ 1. We denote the efficiency of a designed OOC by l, where
l¼
F opt 100%; F
ð4Þ
bi;ðjþ1Þ bi;j
for j ¼ 1; 2; . . . ; K 1
F þ bi;0 bi;K
for j ¼ K:
ð5Þ
An extended set of Si is denoted by Ei and consists of all linear combinations of jointly connected relative delays, i.e., Ei ¼ fei;1 ; ei;2 ; . . . ; ei;ðKðK1ÞÞ g, where [7]
ei;j ¼
þ1 bj1 K c X
ai;ð½fm1þððj1Þ mod KÞg mod Kþ1Þ :
ð6Þ
m¼1
For the sequence Si to satisfy the periodic autocorrelation property with ka ¼ 1, there should be no repetitions of elements in the set Ei . This is because of the reason that there should be no overlap of two or more pulses of the shifted versions of the sequence Si . Similarly, to satisfy the crosscorrelation property kc ¼ 1, for any two sequences Su and Sw , where u–w, the intersection of the corresponding extended sets must be empty, i.e., Eu \ Ew ¼ ;. Due to the cyclic nature of the extended set Ei , it should be noted that if ðei;j Þ is an element, then also ðF ei;j Þ is an element of Ei . We will exploit this property during the OOC design.
502
C. Argon / Optics Communications 282 (2009) 500–503
In the following section, we will make use of these extended sets for the design of OOC sequences. The design method proposed here is outlined for correlation constraints ka ¼ kc ¼ 1, but can be generalized for other correlation constraints as well.
15
K=4 K=5 K=6
14 13 12
3. Semi-random OOC design method To optimize the sequence length of the designed OOC, we propose here a semi-random design method where we use a random permutation vector to determine the order of OOC sequence updates, and a maximum number of iterations until we terminate the run of the algorithm. The design criteria is to have the two input parameters N and K, and an additional parameter denoted by ITR, which indicates the maximum number of iterations. Different than the design method proposed in [7], we also propose to input a target length F target , which can be used exploiting the aforementioned cyclic nature of extended sets. The proposed design algorithm can be summarized as follows: (i) Specify the weight, K, and number of users (sequences), N, for the OOC. Initially, the sequence length F is set to F target ¼ F opt . Specify also the maximum number of iterations, ITR, which determines how many iterations per target length will be performed until we increment the length and start a new iterative search. (ii) Initialize the iteration counter, z, to 1. (iii) Assume that a pulse is placed at the 0th chip position for all OOC sequences. (iv) The first delay elements of the OOC sequences are assigned as ai;1 ¼ i, for i ¼ 1; 2; . . . ; N. (v) Start to form the extended sets, Ei ’s, by adding ðai;1 Þ’s and to these sets; i.e., initialize ðF target ai;1 Þ’s Ei ¼ fai;1 ; F target ai;1 g, for i ¼ 1; 2; . . . ; N. (vi) Generate vector c ¼ ½c1 ; c2 ; . . . ; cN , which is the random permutation of the ordered vector [1, 2, . . ., N]. The elements of vector c are specifying in which order the OOC sequences are being updated at the determination of the jth delay element. (vii) Initialize m ¼ 1 and i ¼ cm . (viii) Initialize the offset parameter q ¼ N. (ix) For the OOC sequence under consideration, Si , increment q ¼ q þ 1, until its value and ðF target qÞ are not present in any of the partial extended sets. If at any time q P F target , increment counter z ¼ z þ 1 and go to step (xi). (x) Add q as a delay element for the OOC sequence under consideration, Si , iff all linear combinations of jointly connected relative delay elements are not repeated within the partial extended set under consideration or within the partial extended sets of the other sequences. While checking this property, make use of the cyclic structure of the extended sets. In case q is added as a delay element for Si , then update the extended set for this sequence (again making use of the cyclic nature of the extended sets) and go to step (xii). If q is not added as a delay element, then do not do any update of the extended sets and go to step (ix). (xi) If counter z > ITR, then increment F target ¼ F target þ 1 and go to step (ii); otherwise, go to step (iii). (xii) If all K pulses for all N sequences have been assigned (this is equivalent to assigning all K 1 delay elements for all N sequences), then proceed to step (xiii). If not, then move to the next OOC sequence [i.e., increment m ¼ m þ 1; if incremented m is equal to N þ 1, then go to step (vi); if not, set i ¼ cm and go to step (viii)]. (xiii) The chip positions of the pulses for the N OOC sequences are determined using
µ(new) — µ([7]) %
11 10 9 8 7 6 5 4 3 2 1 0
0
5
10
15
20
25
30
Number of Users Fig. 3. Percentage efficiency increase with new algorithm against algorithm in [7] (ITR = 1000).
Si ¼ f0; ai;1 ; ai;1 þ ai;2 ; ai;1 þ ai;2 þ ai;3 ; . . . ; ai;1 þ ai;2 þ þ ai;ðK1Þ g for i ¼ 1; 2; . . . ; N, and the designed OOC has sequence length F target . (xiv) Stop. We call the above algorithm ‘‘semi-random”, since the search for the offset parameter q is deterministic, however, the order of OOC sequences to be updated is determined via a random permutation vector c and hence, is not sequential. Furthermore, we note that for each F target , we have at most ITR iterations before we search for an OOC of increased length. In addition, one advantage of using a random permutation vector c to determine the OOC sequence update order is to increase the probability to converge to the optimum OOC sequence length and also to distribute the pulse locations more evenly across all OOC sequences. As mentioned before, the algorithm presented here is different than the algorithm in [7] since we have the advantage of using a target sequence length as an input parameter and furthermore, we also exploit the cyclic nature of extended sets. This was not possible in [7] since the OOC length was determined after all delay elements were already assigned. To demonstrate the effectiveness of the proposed algorithm, we show in Fig. 3 the efficiency comparison of the newly proposed algorithm against the algorithm presented in [7] for the same number of iterations (ITR = 1000) and code parameters, N and K. For this purpose, we subtract the efficiency of the algorithm in [7] from the efficiency of the newly proposed algorithm; i.e., l(new)l([7]), where the efficiency of the designed OOC is given in (4). Please note that here at most ITR iterations are performed for each target length and hence, for the proposed algorithm the total number of iterations is higher; on the other hand, since the partial extended sets are formed faster via exploiting the cyclic structure, the proposed algorithm is able to break out of unsuccessful iterations faster than the algorithm presented in [7]. Furthermore, as observed in Fig. 3, we clearly note the increase in efficiency for the same code parameters with the newly proposed OOC design algorithm, especially with increased sequence weight, K and increased number of users. 4. Conclusion We presented an alternative optical orthogonal code (OOC) design technique for users of an intensity-modulation/direct-detection optical code-division multiple-access (CDMA) network. The
C. Argon / Optics Communications 282 (2009) 500–503
proposed method was a semi-random based approach where multiple iterations were required to generate all OOC sequences. Compared to previous methods, this algorithm uses a target sequence length and a pre-defined number of iterations. The target sequence length is incremented if the OOC sequence is not generated at the end of the pre-determined iteration number. The design method outlined in this work concentrated on correlation constraints ka ¼ kc ¼ 1 which are the best achievable correlation constraints in an intensity-modulated direct-detection optical CDMA network; however, the OOC design algorithm presented here can be generalized to other constraints as well. References [1] J.A. Salehi, IEEE Trans. Commun. 37 (1989) 824. [2] F.R.K. Chung, J.A. Salehi, V.K. Wei, IEEE Trans. Inform. Theory 35 (1989) 595. [3] H. Chung, P.V. Kumar, IEEE Trans. Inform. Theory 36 (4) (1990) 866.
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