IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 5, MAY 2001
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Double-Weight Signature Pattern Codes for Multicore-Fiber Code-Division Multiple-Access Networks Wing C. Kwong, Senior Member, IEEE, and Guu-Chang Yang, Member, IEEE
Abstract—To transmit digitized image pixels in optical codedivision multiple-access (CDMA) networks with multicore fiber, classes of two-dimensional (2-D) patterns, so-called optical orthogonal signature pattern codes (OOSPC’s), have recently been proposed. The new technology enables parallel transmission and simultaneous access of multiple 2-D images in multiple-access environments. In this letter, we construct two new families of “doubleweight” OOSPC’s without the assumption of identical weight for all signature patterns in a code set. Since the performance of a signature pattern varies with its weight, these new double-weight OOSPC’s are especially useful for optical CDMA networks with multiple performance requirements. Index Terms—Multicore fiber, optical code-division multiple-access, two-dimensional signature pattern codes.
I. INTRODUCTION
R
ECENTLY, two-dimensional (2-D) patterns (or matrices), so-called optical orthogonal signature pattern codes (OOSPCs), were introduced by Kitayama [1] to alleviate the bandwidth-expansion problem in fiber-optic code-division multiple-access (CDMA) networks [2]. The new technology enables parallel transmission and simultaneous access of multiple 2-D images in multicore fibers. 2-D signature patterns also find applications in multiple-fiber or spatial CDMA for information transfer in optical networks, proposed by Hui [3], Park et al. [4], and Hassan et al. [5]. Previous work on OOSPCs assumed that the weight of each pattern was the same [1], [2], resulting in identical performance for every signature pattern in the code set. Without the sameweight assumption, a new family of “double-weight” OOSPCs was constructed in [6]. Such double-weight OOSPCs are useful for optical CDMA systems with multiple performance requirements. Also in [6], the use of OOSPCs for multiple 2-D images parallel transmission and their performance were studied. In this letter, two more classes of double-weight OOSPCs are constructed with the autocorrelation constraint being relaxed in order to improve the code cardinality. Manuscript received November 2, 2000. The associate editor coordinating the review of this letter and approving it for publication was Dr. J. Evans.This work was supported by the National Science Council of Republic of China under Grant NSC-89-2213-E-005-041, by the Presidential Research Award of Hofstra University, and by the Faculty Development and Research Grants of Hofstra University, Hempstead, NY. W. C. Kwong is with the Department of Engineering, Hofstra University, Hempstead, NY 11549 USA (e-mail:
[email protected]). G.-C. Yang is with the Department of Electrical Engineering, National Chung-Hsing University, Taichung, Taiwan, R.O.C. Publisher Item Identifier S 1089-7798(01)04498-2.
II. UPPER BOUNDS ON OOSPCs Every 2-D pattern in OOSPCs can be represented by
.. .
.. .
..
.
.. .
(1)
matrix of weight with some integers a binary (0, 1) and such that , where . The definition of multiple-weight OOSPC’s is a generalization of that of constant-weight OOSPC’s in [1] and [2]. ) multiple-weight Definition: An ( is a collection of binary (0, 1) matrices OOSPC , and the 2-D autosuch that every matrix has a weight and and cross-correlation constraints are no more than , respectively [2]. , , and are used to denote the sets , , and , respectively, where is a positive for . There are patterns integer and and autocorrelation constraint , where of weight indicates the fraction of patterns of weight (i.e., the ratio to the total number of patterns in the of patterns of weight is the code cardinality. code set) and To study the cardinality of the OOSPC’s, we need to define a ,a array of integers with the difference square th element given by (2) addition, where “ ,” “ ,” and “ ” denote modulomodulo- subtraction, and modulo- subtraction, respectively th element in represents the [2]. In other words, the and relative cyclic delays between two binary ones, , in a matrix, which determine the (modulo- ) number of vertical cyclic shifts and the (modulo- ) number of horizontal cyclic shifts required to “line up” the two binary ones. Definition: The upper bound of the cardinality of a multiple-weight OOSPC is defined as
is an
(3)
In the following, we study the upper bound of OOSPC’s. First 2-D optical code of all, to guarantee an
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IEEE COMMUNICATIONS LETTERS, VOL. 5, NO. 5, MAY 2001
satisfying the autocorrelation constraint , no elements of is allowed to repeat twice or more [2, the difference square Lemma 1]. Furthermore, for any two different patterns, and , the cross-correlation constraint is satisfied and [2, only if there is no common element between Lemma 2]. From [2], the cardinality of such a OOSPC is upper-bounded by (4) OOSPC with exactly one pulse per ), we have because the number of vertical shifts required to “line up” any two pulses is at least one and cannot be equal to for . It means that must be larger than in order to have an OOSPC with exactly one pulse per row; that is, there exists some empty columns for OOSPC with these patterns. Similarly, for an ), we have exactly one pulse per column (i.e., . It means that must be larger than in order to have an OOSPC with exactly one pulse per column; that is, there exists some empty rows for OOSPC with these patterns. Finally, for an ), exactly one pulse per row and column (i.e., . We can conclude that there does not exist any OOSPC with exactly one pulse per row and column. In summary, constructing an OOSPC with large cardinality, we need to insert a lot of empty rows or columns in each pattern. Furthermore, the number of empty columns or rows in each pattern increases with the number of patterns, significantly increasing the bandwidth expansion. On the other hand, we in order to can also relax the autocorrelation constraint increase the cardinality or weight by allowing the existence of . Therefore, besides constructing some repeated elements in , we also report OOSPC’s with OOSPC’s with in this letter. multiple-weight OOSPC, the For an maximum number of patterns that can be constructed is derived by modifying the upper bound of constant-weight OOSPC’s given in (4) For an row (i.e.,
(5)
III. CONSTRUCTIONS OF DOUBLE-WEIGHT OOSPCs This section details the algebraic constructions of two families of double-weight OOSPC’s, without the restriction of exactly one pulse per row or one pulse per row and column [6]. Construction A. OOSPCs: This construction is an extension of the construction in [6]. Here, the need to check the special code-generation conditions in [6, eq. (3) and (4)] is not required. The construction begins with a generalized balanced incomplete block de, is an arrangement sign (BIBD) [7]. The BIBD, of distinct objects (i.e., binary ones) into blocks (i.e., patterns) such that each block contains exactly distinct objects,
each object occurs in exactly different blocks, and every pair of distinct objects occurs together in exactly blocks. There are two elementary relations on the five parameters for the block deand . (See [7] for sign, which are the details.) From (5), we know that the weight of heavier patterns can be increased by simply relaxing the autocorrelation constraint , as done in this construction. Here, an integer is introduced to but with , where generate patterns of weight . For the special case of , ideal autocorrelation ) is preserved. (i.e., , be an integer (i.e., , and choose Let to be a prime number such that for some integers and . Let be a primitive element of the Galois field are all over , GF( ), such that distinct modulo- . Then, the code, consisting of the (position) blocks
(6) and the blocks
(7) , double-weight OOSPC. There are patterns of weight and patterns of weight . in the position block of a sig(Note: Each ordered pair and horizontal disnature pattern represents the vertical placements of a binary one from the bottom-leftmost corner of the 2-D pattern. All the multiplications in the position blocks are modulo- .) To study the optimality of this OOSPC, we need to derive the upper-bound of its cardinality. Since the number of patterns of is for this code, the maximum number of patterns weight , can be derived from (5) and is given by of weight ,
form an
,
,
(8)
for a given . After some manipulations, (8) becomes (9)
KWONG AND YANG: DOUBLE-WEIGHT SIGNATURE PATTERN CODES FOR CDMA NETWORKS
Since
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and the blocks
, (9) becomes
(10)
The upper bound in (10) is larger than the actual number of , by only a factor of patterns of weight , that is, . This factor is nearly equal to zero for a large and, therefore, the new code is asymptotically optimal. As a special case when and , the new OOSPC achieves the upper bound with equality and is thus optimal. and , we have Example 1: For . Using and , we generate double-weight
OOSPC
with
20
position
blocks
and
for . The two long 2-D patterns, [(1, 1), (1, 3), (3, 3), (3, 9), (9, 9), (9, 1)] and [(2, 2), (2, 6), (5, 5), (5, 2), (6, 6), (6, 5)], and two examples of short 2-D patterns, [(1, 2), (3, 6), (9, 5)] and [(1, 8), (3, 11), (9, 7)]. and , we have . Using Example 2: For and , the construction generates a , , , double-weight OOSPC with seventy-two position blocks [(1, 1), (1, 10), (1, 18), (10, 10), (10, 18), (10, 16), (18, 18), (18, 16), (18, 37), (16, 16), (16, 37), (16, 1), (37, 37), (37, 1), (37, 10)], [(36, 36), (36, 32), (36, 33), (32, 32), (32, 33), (32, 2), (33, 33), (33, 2), (33, 20), (2, 2), (2, 20), (2, 36), (20, 20), (20, 36), (20, , , , , , 32)], , , , , for and . Construction B. OOSPCs: In this construction, we consider the special case of ). Let and choose Construction A (i.e., to be a prime number such that for some integers and . Let be a primitive element of the Galois field are all over , GF( ), such that distinct modulo- . Then, the code, consisting of the (position) blocks
(11)
(12) , 1, , double-weight OOSPC. for the patterns While the autocorrelation constraint was in Construction A (for ), it is now equal to of weight in Construction B (for ). Hence, the cardinality of Construction B is a little bit worse, from the optimality viewpoint. by and To derive the upper bound, we replace substitute by in the denominator of (8). and , we have . Example 3: For , the construction obtains a , , , Using double-weight OOSPC with fourteen posi1, tion blocks [(1, 1), (1, 3), (1, 9), (3, 3), (3, 9), (3, 1), (9, 9), (9, 1), (9, 3)], [(2, 2), (2, 6), (2, 5), (5, 5), (5, 2), (5, 6), (6, 6), (6, 5), , , , and , , (6, 2)], for . form an
,
,
IV. CONCLUSIONS In this letter, two new classes of double-weight OOSPC’s have been constructed algebraically for multiple digitized images transmission in optical CDMA networks with multicore fiber. These new codes have been shown to achieve asymptotical optimality. As opposed to conventional OOSPC’s, we have considered the possibility that the weight of all signature patterns may not be identical. Since the performance of a signature pattern varies with its code weight, our approach is useful for optical CDMA networks with multiple performance requirements. As a rule of thumb, a more critical user should be assigned with a signature pattern with heavier weight in order to guarantee the transmission quality or, in other words, lower error probability. REFERENCES [1] K. Kitayama, “Novel spatial spread spectrum based fiber optic CDMA Networks for image transmission,” IEEE J. Select. Areas Commun., vol. 12, pp. 762–772, May 1994. [2] G.-C. Yang and W. C. Kwong, “Two-dimensional spatial signature patterns,” IEEE Trans. Commun., vol. 44, pp. 184–191, Feb. 1996. [3] J. Y. Hui, “Pattern code modulation and optical decoding—A novel code-division multiplexing technique for multifiber networks,” IEEE J. Select. Areas Commun., vol. 3, pp. 916–927, Nov. 1985. [4] E. Park, A. J. Mendez, and E. M. Garmire, “Temporal–spatial optical CDMA networks—Design, demonstration, and comparison with temporal networks,” IEEE Photon. Technol. Lett., vol. 4, Oct. 1992. [5] A. A. Hassan, J. E. Hershey, and N. A. Riza, “Spatial optical CDMA,” IEEE J. Select. Areas Commun., vol. 13, pp. 609–613, Apr. 1995. [6] W. Kwong and G.-C. Yang, “Image transmission in multicore-fiber code-division multiple-access networks,” IEEE Commun. Lett., vol. 2, pp. 285–287, Oct. 1998. [7] M. Hall Jr., Combinatorial Theory, 2nd ed. New York: Wiley, 1986.
