Construction and Performance Analysis of
Variable-Weight Optical Orthogonal Codes
for Asynchronous Optical CDMA Systems
Fong-Ray Gu and Jingshown Wu, Senior Member, IEEE
Abstract—In this paper, two construction schemes of
vari-able-weight optical orthogonal codes (OOCs) for asynchronous optical-code-division multiple-access (O-CDMA) systems are proposed. The first scheme uses pairwise balanced design (PBD), which is a research topic in combinatorial theory. PBD produces a family of blocks with unequal block size. Therefore, PBD can be used to construct OOCs with variable code weight. The lower bound of the code size of the codes from PBD is formulated in this paper. A second scheme employes packing design with a partition to generate blocks with unequal block sizes. The variable-weight OOCs can be constructed by partitioning a larger weight code-word into a family of codes with a smaller code weight. The upper bound and lower bound of the code size of the second scheme are discussed. The bit-error-rate (BER) performances of the two proposed codes are evaluated analytically in this paper. The simulation results show that the codes from the first scheme have the same BER performance as that of conventional code, while the second scheme has a larger maximum number of simultaneous users than that of conventional codes.
Index Terms—Bit-error rate (BER), optical-code-division
mul-tiple access (O-CDMA), optical orthogonal codes (OOCs), packing design, pairwise balanced design (PBD), variable weight.
I. INTRODUCTION
O
PTICAL-CODE-DIVISION multiple access (O-CDMA)is getting a lot of attention recently. It is a suitable tech-nique for optical fiber transmission due to the inherent large bandwidth of fiber. It is also a good candidate for optical access networks, such as Ethernet passive optical networks (EPONs) [1], [2]. In the transmitter, a laser diode and anON–OFF-keying (OOK) modulator are employed. Because of nonnegative power for optical signals, optical orthogonal codes (OOCs) are a family of (0,1) sequences that are different from that in electrical
trans-mission using ( 1, 1) sequences. The -OOCs are
a code family with code length and code weight [3]. The off-peak autocorrelation and cross correlation should be minimized for the sake of synchronization and less multiple-user interference (MUI). Meanwhile, the code weight is re-quired to be large in order to distinguish the desired signal from
MUI and noise. If , the -OOCs
are simply denoted as -OOCs. The -OOCs can
Manuscript received September 3, 2003; revised August 20, 2004. Part of this work was supported by the National Science Council and Ministry of Education, Taiwan, R.O.C., under Grants NSC91-2213-E-002-106 and 89-E-FA06-2-4.
The authors are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 10617, Taiwan, R.O.C.
Digital Object Identifier 10.1109/JLT.2004.838880
be constructed from the projective geometry , where
is a positive integer and is a prime power. The
forms a cyclic difference set with
parame-ters , , and
[4]. It is also called the Singer differ-ence set with classical parameters [5]. Any pair of the elements in the set of integers modulo has exactly
represen-tations of the difference for any residue
. Packing the elements of into
sub-sets of blocks yields that every set of distinct elements in occurs in at most one block. Then, a family of blocks can be obtained, and each block has elements. Each element in a block represents the mark position of the codeword, and there-fore a family of OOCs can be generated. The
packing design guarantees that the cross correlation of any two OOCs is . On the other hand, the property of the cyclic difference set makes the off-peak autocorrelation of the OOCs to be and therefore satisfies the correlation constraint of the
-OOC’s.
Let be the possible number of blocks of a
packing design. Then, the upper bound is given as Theorem 1 [6].
Theorem 1 (Schönheim Bound):
(1) where represents the integer part of .
Theorem 1 can be followed to obtain the upper bound of code size of the -OOCs. In the literature, one way to improve the bit-error-rate (BER) performance of OOCs is to increase the code weight and decrease the cross correlation [7]. In gen-eral, large code weight and small cross correlation will reduce the code size. Therefore, there is a tradeoff between the cross correlation and the code size of OOCs. In [8], Yang presented the construction of OOCs with unequal cross correlation. Re-cently, a family of ( , , 1, 2)-OOCs based on ( , , 1) per-fect difference set is proposed [9]. The ( , , 1, 2)-OOCs also have better performance than ( , , 1, 2) Yang’s code and thus allow more simultaneous users under a given BER performance [9]. A large code size can be obtained by relaxing the cross-correlation constraint of the OOCs. For example, the code size of ( , , 1, 2)-OOCs is about ten times larger than that of conventional ( , , 1)-OOCs for reasonable code length. How-ever, the BER performance degrades as the cross correlation increases. A recent paper discusses the performance bound of 0733-8724/$20.00 © 2005 IEEE
OOCs with large cross correlation [10]. For a given code length and the number of simultaneous users, the BER performance of the OOCs has a minimal value when the code weight is optimal. Since the code size of OOCs depends upon the code weight, the variable-weight OOCs can generete larger code size than that of constant-weight OOCs. Meanwhile, subscribers with different code weights will have different BER performances. This property can meet the requirement of multimedia net-works, which support multiple quality of services (QoS) [11]. Therefore, codewords of low code weight can be assigned to the low-QoS applications and high-code-weight codewords can be assigned to high-BER-requirement applications. In this paper, two construction schemes of variable-weight OOCs are proposed. The first scheme uses pairwise balanced design (PBD) to obtain a set of codewords with unequal code weight, whereas the second scheme employs packing design to partition a constant-weight codeword into subcodes with a smaller code weight. The performances of the codes produced by the two schemes are also analyzed.
This paper is organized as follows. Two construction schemes for the variable-weight OOCs are investigated in Section II. In Section III, the performance of the variable-weight OOCs is for-mulated using double optical hard limiters with consideration of thermal noise and shot noise. Numerical results are presented in Section IV. Finally, the conclusion is given in Section V.
II. CONSTRUCTIONSCHEMES OFVARIABLE-WEIGHTOOCs The variable-weight OOCs are a family of codewords with equal code length and variable code weight. The proposed
schemes can be thought as a packing
design with sets of blocks , where the
block length for any . Based on the theory
of combinatorial design, there exists a PBD that satisfies the
requirement of the packing design. On the
other hand, we can use the packing design twice to construct variable-weight OOCs under the fixed code length. We inves-tigate two construction schemes for variable-weight OOCs as described in the following subsections.
A. The Construction Scheme Using Pairwise Balanced Design A PBD of order is a pair ( ,ß), where is a set of ele-ments, and ß is a family of subsets (blocks) of with block sizes
from . In a -PBD, every pair of
distinct elements of occurs in exactly blocks in ß. When
, the -PBD is usually denoted as -PBD,
which forms a linear space [12]. Then the blocks are called lines. The balanced incomplete block design (BIBD) is a special
case of -PBD when is a singleton [13].
De-fine and
, where represents the greatest common divisor. Then, we have Theorem 2 [4].
Theorem 2: The necessary conditions for the existence of a -PBD are
(2) and
(3) In the literature of combinatorial design, several existence theorems of -PBD have been proven for some specified sets of [4]. Here, we introduce some of them, which will be used in this paper.
Theorem 3 [4]: A ( , {3,4})-PBD exists if, and only if,
(mod 3), except when .
Theorem 4 [4]: A ( , {3,5})-PBD exists if, and only if, (mod 2).
Theorem 5 [4]: A ( , {4,5})-PBD exists if, and only if,
(mod 4), except when .
Theorem 6 [4]: A ( , {5,9})-PBD exists if, and only if,
(mod 4), except when and possibly when
.
Theorem 7 [14]: Let be a prime power. For any
, there exists a ( , )-PBD.
Example 1: We consider a (91,10,1) The Singer difference is set with elements {0, 1, 3, 9, 27, 49, 56, 61, 77, 81}. Since , then the (10,({3,4}))-PBD exists. There-fore, we have the (10,({3,4}))-PBD with blocks: {1,27,61}, {1,49,77}, {1,56,81}, {3,27,81}, {3,49,61}, {3,56,77}, {9,27,77}, {9,49,81}, {9,56,61}, {0,1,3,9}, {0,27,49,56}, {0,61,77,81}.
For a set of elements in a -PBD, the maximum pos-sible number of blocks is clearly , where is the number of combinations of elements taken at a time. Erdös et al. [15] have shown that there is an absolute constant such that the possible number of blocks satisfies the inequality
(4) However, the upper bound of given elements and
is still an unsolved problem. Several papers have discussed the lower bound of of a -PBD [15]–[17]. At the given necessary conditions, there exists a linear space
whose line lengths are . We have the following
theorem.
Theorem 8: Let be the length of the longest blocks in the linear space of elements. The number of blocks
in a -PBD for a given set of block length
satisfies
The inequality holds for any .
Proof: Let be the number of blocks with length , where . We define one of the blocks with length to be the base block. Then, the number of blocks with length except the base block is . The following equation holds by counting the total blocks [16]:
In -PBD, any two blocks have one element in common, and every pair of elements will appear in only one block. The total number of pairs of elements is
(6) Define to be the number of blocks of length that include the th element of the base block. We have the following rela-tion:
(7)
where . The summation of (7) over is
(8)
If we count the number of blocks of length that pass the base block, we obtain
(9) where the term is the number of blocks of length that are disjoint from the base block. Combining (8) and (9), we obtain (10)
where is given as
(11)
For a specific , we can multiply (5), (6), and (10) by
, 1, and , respectively, and
sum them up to eliminate and to obtain
(12) where
(13)
Denote , and we can express as
(14) Since and are nonnegative, we have the lower bound of by dropping the terms containing and of (14). The result is
(15)
In (15), the equality holds when . From (11),
occurs when for any , i.e., all blocks intersect
the base block. From (13), implies
for any , which represents that the number of classes of block size in the PBD is less than or equal to two.
For some applications of the O-CDMA systems, we have only
two classes of code weight . Since there are only
two variables, we can obtain the number of blocks by solving (6) and (10). For instance, the number of blocks with length 3 and 4 of a (10,{3,4})-PBD can be obtained from
(16) If we assume to be zero, i.e., all blocks in the (10,{3,4})-PBD intersect the base block, the possible number of blocks has a lower bound of 12.
When the block length of a -PBD is a set of successive
integers , then (15) becomes
(17) We denote as the right-hand side of (17) [17]. Then, we have
(18)
Equation (18) is nonnegative when . Therefore,
is a monotonically increasing function when
. Then, we can determine the strongest lower bound of
by assigning [17].
The construction procedure of variable-weight OOCs by using PBD is described as the following.
1) Take an ( , , 1)-Singer difference set with classical
pa-rameters and , where
is a prime power and is any positive integer. Any pair of the elements in the ( , , 1)-Singer differ-ence set has exactly one representation of the differdiffer-ence
for .
2) Apply -PBD on the elements of the ( , ,
1)-Singer difference set. A family of blocks with a block
length from can be obtained.
3) We can construct the variable-weight codes
with code weight by mapping the elements of the PBD block into the code weight positions according to the following rule:
if PBD block
otherwise. (19)
B. The Construction Scheme Using Packing Design With Partition
We next investigate the construction scheme of vari-able-weight OOCs using packing design with a partition of
elements in the projective plane . A standard result of the existence of is given in Theorem 9.
Theorem 9 [14]: The is a BIBD with parameter
( , ), 1, where is a prime power and is a
positive integer.
Theorem 10[4]: Let and be positive integers such that divides . If there exists , then it contains a subplane
of .
If is a prime power and is any positive integer, a finite Möbius geometry of is an extended field , which in-cludes an element into a Galois field . The
has elements. Any triple of elements in is
in-cluded in exactly one block, and every block has ele-ments, where is a positive integer that divides . Meanwhile, an ( , , 1)-Singer difference set with classical parameters of
has elements. The elements
form a plane of . Therefore, we can put the
el-ements of the Singer difference set into the plane of [9]. We have the following theorem.
Theorem 11: A finite Möbius geometry of con-tains a subplane of if is a positive integer that di-vides .
Proof: According to Theorem 10, contains a
subplane of when divides . Since the elements in
the form a plane of , then the
also contains a subplane of .
If we put the elements of an ( , , 1)-Singer difference set with classical parameters into the plane of , we can
use packing design to construct a family
of blocks with block length of . Then, the total number of blocks is given as
(20) Equation (20) is the upper bound of the packing design, and the packing design is called the maximal system [6]. In order to obtain a family of blocks with different block sizes, we propose a construction scheme that uses packing
design twice. The elements in are first packed
into blocks by a packing design. Next,
a block with elements is partitioned into subblocks, and
each subblock has length and . A family of
subblocks with different block length is then constructed. Each entry in the subblock represents the code weight position of a codeword, and then the subblocks with variable length can con-struct a family of variable-weight codewords. The concon-struction procedure of this proposed scheme is listed in the following.
1) Construct an ( , , 1)-Singer difference set with classical
parameters and , where is
a prime power and is any positive integer.
2) Pack the elements of the Singer difference set into
blocks with length by packing
design, where is a positive integer that divides . 3) Wrap the elements of one block into smaller blocks
with length such that any two subblocks have one ele-ment in common. Therefore, a family of subblocks with size is generated.
4) Each entry of a block represents the weight position of the codeword. We can obtain a variable code weight
( , , 1, 2)-OOCs with classes of code
weight.
According to the packing design, the number of subblocks with length from elements is upper bounded by
(21)
The low bound of can be considered as the
number of subblocks by using the brute force method, that is, at least the number of subblocks we can construct. We prove the
lower bound of in the following.
Theorem 12: The lower bound of wrapping elements into smaller blocks with elements is given as
(22)
where and .
Proof: Among the elements, we choose one el-ement as the intersecting point and then partition the rest of
the elements into groups, and each group has
ele-ments. Then, the number of groups is , and .
The residual elements after grouping is denoted as , and . By putting the intersecting point into every group, we can obtain a family of blocks with block size . Any two blocks intersect only at the intersecting point, and there are blocks. In the next step, we use the same method to construct blocks from the groups and the residual ele-ments. Since the groups without the intersecting point are dis-joint with each other, we can pick one element from each group except the intersecting point to satisfy the correlation property. Then, the number of blocks obtained in this step is
. Therefore, the total number of blocks
is .
Theorem 13: The off-peak autocorrelation of the ( , , 1, 2)-OOCs is less than or equal to one. Proof: According to Step 1 of the construction procedure, the variable-weight OOCs are from the ( , , 1)-Singer differ-ence set. Then, any pair of the elements in the difference set has exactly one representation of the difference
for any residue . Therefore, after any
time shift , the off-peak autocorrelation of the codeword is less than or equal to one.
Theorem 14: The cross correlation of any two codewords originated from the same block in is one.
Proof: Any two codewords originated from the same block satisfy the correlation constraint of
packing design, in which any two subblocks have at most one element in common. Therefore, the cross correlation of any two codewords originated from the same block is one.
Theorem 15: The cross correlation of any two codewords originated from different blocks in is less then or equal to two.
Proof: The family of subblocks originated from a block in
by a packing design. Because any
Fig. 1. Receiver structure of asynchronous O-CDMA systems using double optical hard limiters.
any two subblocks that originated from different blocks will in-tersect no more than two elements. Therefore the cross correla-tion of any two codewords originated from different blocks of
will be less than or equal to two.
Let the fractions of the codewords with weight
be , respectively. The number
of cardinalities of the variable-weight OOCs can be expressed as
(23)
where .
III. PERFORMANCEANALYSIS
We analyze the performance of the asynchronous O-CDMA systems using the proposed variable-weight OOCs and double hard limiters with consideration of shot noise, thermal noise, avalanche photodiode (APD) bulk and surface leakage currents. The receiver structure is shown in Fig. 1 [18]. For convenience, we assume that the system is chip synchronous among users since it is the worst case of the performance [7].
The average photon arrival rate per pulse is given by (24) where is the APD quantum efficiency, is the received signal power, is the Planck’s constant, and is the optical fre-quency. The optical signal power after the second hard limiter will be limited to two levels:ONandOFFlevels. We denote state as the optical signal power of theONlevel and state as the OFFlevel. The average photon arrival rate is for state ; oth-erwise, the photon arrival rate is zero. For state , , the probability density function (pdf) of the output current of the photodetector is assumed Gaussian given by [19]
(25) where is the mean value of the photodetector output current given by
(26) where is the average APD gain, is the chip time, is the electron charge, is the contribution of the APD bulk leakage current to the APD output, and is the APD surface
leakage current. The variance of the photocurrent can be ex-pressed as
(27) where is the excess noise factor given by
(28) Here, is the APD effective ionization ratio, and is the variance of thermal noise expressed as
(29) where is the Boltzmann’s constant, is the receiver noise temperature, and is the receiver load resistance.
In order to minimize the error probability, we set the threshold of the decision circuit as
(30) If the output current is larger than , the output data bit is decided to be bit one, or otherwise bit zero. The probability that the state error occurs is given by
(31)
(32) where is the complementary error function and can be expressed as
(33)
Denote as the user using spreading code
with Hamming weight . If the relative frame offset between the desired user and the interfering user is , then the cross correlation can be expressed as
(34)
The probability that one specified mark position of user hit by the interfering user is , where the factor 1/2
TABLE I
NUMBER OFCARDINALITIES OF THEPROPOSEDVARIABLE-WEIGHT(n, fk ; k g)-OOCs
ANDCONVENTIONAL(n, fk ; k g, 1, 1, {2/3, 1/3})-OOCs
represents that each user transmits data 0 and 1 with equal prob-ability. Since user has marks, then the expected value of
interfered by user is given as
(35) For the first construction scheme, the probability of is zero because the codes are constructed from -PBD, and the cross correlation of any two codewords is less than two. For the second construction scheme, the event occurs when the frame offset and the desired user and the
inter-fering user originated from different blocks in .
Moreover, according to Step 3 of the construction procedure, a block in is partitioned into subblocks with smaller size. The probability that any two codewords with code weight and , respectively, have two identical mark positions after
the partition is , and .
Hence, the probability of is given by
if is from the same block as
otherwise. (36)
The probability that the interfering user contributes two interfering marks is denoted as and given by
(37) where the factor 1/2 represents that each user transmits data 0 and 1 with equal probability. Let be the probability that user contributes one interfering mark. Then, the value of can be obtained from (35) and (37) as follows:
(38) We denote as the probability that any marks of the desired codeword is not interfered by and is given by
(39)
For the receiver with double hard limiters, the signal state at the second hard-limiter output will be if the transmitted
data bit is 1, i.e., . If the transmitted data
bit is 0, the second hard -limiter cannot entirely remove the MUI when the number of interfering marks exceeds or equals the code weight . We assume that for the desired user ,
there are simultaneous users, where is
the number of simultaneous users of code weight , and is the number of classes of different weights. Then, the probability that an error occurs at the output state of the second hard limiter can be formulated as [11]
(40) The BER is given as
BER
(41)
where .
IV. NUMERICALRESULTS
Two classes of code weight and are considered in our numerical results. We assume that the code sizes of the
code-words with weight and are and , respectively.
Table I lists the number of cardinalities of the two proposed ( , )-OOCs and conventional variable-weight ( , , 1, 1, {2/3, 1/3})-OOCs [11], in which we assume the ratio of
the code sizes is . Since the lower bound
TABLE II SYSTEMPARAMETERS
Fig. 2. BER of the variable-weight (4161,{9,5})-OOCs versus the received signal power whenfm ; m g = f40; 20g.
Fig. 3. BER of the variable-weight (4161,{9,5})-OOCs versus received signal power whenfm ; m g = f20; 40g.
and , we would not compare it with the upper bound of that from conventional codes. Moreover, the codes from the second scheme have larger code size than the conventional codes and the codes generated by the first scheme. This is because some of the cross correlation of the codes generated by the second scheme is one, and some of them are relaxed to two.
The system parameters are listed in Table II. Figs. 2 and 3 show the BER of the variable-weight (4161,{9,5})-OOCs
Fig. 4. Maximum number of high-weight users versus high code weightk of the (4161,fk; 5g)-OOCs when the received signal power = 2.4 W and
BER 10 .
versus the received signal power when the number of simulta-neous users are {40,20} and {20,40}, respectively. We find that the BER performances of the codes generated by the first scheme and the conventional codes are identical because both of them have the same correlation constraint. The BER of the codes from the second scheme is slightly larger than that of the previous ones when the received signal power is large. However, the code size based on the second scheme is much larger than that of the conventional codes at the expense of little BER degradation. Fig. 4 illustrates the maximum number of high-weight users versus high code weight of the (4161, )-OOCs when the number of low-weight users is
20, received signal power 2.4 W, and BER 10 . The
result shows that the maximum number of high-weight users of the second scheme is much larger than that of the conventional codes. This is because the user number of conventional codes is limited by the code size. Therefore, both BER performance and the code size should be considered in the O-CDMA systems in order to have maximum capacity.
Next, we compare the performances of the codes from the second construction scheme and the conventional codes under the condition of about the same code size. We take the code length of the codewords equal to 4161, which is the length of
the Singer difference set with and . Then, we can
choose {9,8} as the code weights of codes from the second construction scheme. The code size of the (4161,{9, 8})-OOCs of the second scheme is 520. For comparison, we take the con-ventional variable-weight (4161, {5, 3}, 1, 1, {1/6, 5/6})-OOCs because the code size is about 500. Fig. 5 shows the perfor-mances of the conventional variable-weight (4161, {5, 3}, 1, 1,{1/6, 5/6})-OOCs and the proposed (4161,{9, 8})-OOCs
when . We observe that the BER
performance of the proposed codes is much better than that of the conventional codes. Next, we investigate the effect of the number of simultaneous users on the system perfor-mance. We analyze the performance of the system using (4161, {9, 8})-OOCs of the second construction scheme and
Fig. 5. BER of the codes from scheme 2 and conventional codes versus the received signal power whenfm ; m g = f50; 100g.
Fig. 6. BER performance of the high-code-weight user versus the number of simultaneous users with low code weight when the received signal power = 2.4W.
that using conventional (4161, {5, 3}, 1, 1,{1/6, 5/6})-OOCs when the number of high-code-weight users is 20. Fig. 6 shows the BER performance of the high-code-weight user versus the number of simultaneous users with low code weight
when the received signal power 2.4 W. At BER ,
the system with the conventional codes has 23 simultaneous low-code-weight users. On the other hand, the system with the proposed codes has 92 simultaneous low-code-weight users.
V. CONCLUSION
For the next-generation multimedia network, the QoS of the multiple Internet traffic must be guaranteed. Therefore, the vari-able-weight OOCs can be used as the spreading codes in the O-CDMA systems to meet the QoS requirements. On the other hand, the variable-weight OOCs can have larger code size than
that of constant-weight OOCs. Two construction schemes of variable-weight OOCs are investigated in this paper. The first scheme uses PBD to produce a family of blocks with unequal block size. The second scheme uses packing design on con-stant-weight OOCs and partitions a concon-stant-weight codeword into a family of codes with a smaller code weight. The code sizes of the two proposed codes are discussed in this paper. The BER performance of these two proposed codes are also evaluated an-alytically. Numerical results show that the codes from the first scheme and the conventional codes have identical performance because both of them have the same correlation constraint. The codes generated from the second scheme have much larger code size than that of the previous ones. The maximum number of users of the codes from the second scheme is larger than that of conventional codes. Meanwhile, the BER performance of the system using the codes of the second construction scheme is also better than that of the conventional codes under the condition of about the same code size.
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Fong-Ray Gu, photograph and biography not available at the time of
publica-tion.
Jingshown Wu (S’73–M’78–SM’99), photograph and biography not available
