Optical Orthogonal Codes With Large
Crosscorrelation and Their Performance Bound
for Asynchronous Optical CDMA Systems
Chi-Shun Weng and Jingshown Wu, Senior Member, IEEE
Abstract—Optical orthogonal codes (OOCs) are commonly used as signature codes for optical code-division multiple-access (OCDMA) communication systems. Many OOCs have been proposed and investigated. Asynchronous OCDMA systems using conventional OOCs have very limited number of subscribers and a few simultaneous users. Recently, we reported a new code family with large code size by relaxing the crosscorrelation constraint to 2. In this paper by further loosening the crosscorrelation constraint, we adopt the random greedy algorithm to construct a code family which has larger code size and more simultaneous users. We also derive an upper bound of number of simultaneous users for given code length, code weight, and bit error rate. The study shows that it is possible to have codes approaching to this bound.
Index Terms—Maximal system, multiuser interference, optical code-division multiple-access (OCDMA), optical orthogonal code (OOC), perfect difference code, random greedy algorithm.
I. INTRODUCTION
R
ECENTLY, there have been many papers that havediscussed OOCs for optical code division multiple access
(OCDMA) systems [1]–[13]. -OOCs are a
family of (0,1) sequences with code length , code weight , the maximum value of off-peak autocorrelation , and the maximum value of crosscorrelation . For the sake of synchronization and minimizing multiuser interference (MUI),
-OOCs with are usually adopted as
signature codes. In general, the crosscorrelation of any two -OOCs is either zero or one. Therefore, it is difficult to design a receiver to cancel MUI. The code size, upper
bounded by , of these ideal OOCs is
sparse corresponding to the code length. To increase the code size, some code families with nonideal correlation constraint have been reported [10]–[13]. In [10], Chung and Kumar
constructed optimal -OOCs, where is
any prime and the family size is . In [11], Yang and
Manuscript received May 6, 2002; revised December 6, 2002. This work was supported in part by the National Science Council and Ministry of Edu-cation, Taiwan, R.O.C., under Grant NSC89-2215-E-002-012 and Grant 89-E-FA06-2-4.
C.-S. Weng was with the Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. He is now with Realtek Semiconductor Corporation, Hsinchu 300, Taiwan, R.O.C.
J. Wu is with the Department of Electrical Engineering, Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617 R.O.C. (e-mail: wujsh@cc.ee.ntu.edu.tw).
Digital Object Identifier 10.1109/JLT.2003.809547
Fuja investigated -OOCs and showed that it is
impossible to get more than codewords
whose code size is twice the upper bound of -OOCs.
In [12], Yang also constructed -OOCs and the code
size is times (for even or times (for odd the size of the -OOCs, when is less than eight. In [13], we
have proposed -OOCs based on )-perfect
difference codes, where , , and
is a power of a prime plus one [14]. We have also shown that the performance was improved significantly with larger code weight given a fixed code length.
In [13], we reserved chips appropriately from the chips of a -perfect difference code such that all the codewords fulfilled the crosscorrelation constraint, that is, . Because most of the crosscorrelations between any two distinct codes in these family are 0 or 1 (while only very small portion have value of 2), the bit error rate (BER)
performance of the systems using the -OOCs is
almost the same as that using the -OOCs. Moreover,
the code size of -OOCs is upper bounded by
which may be ten times larger than that of ideal -OOCs. Thus, given a code length and a code size, the code weight of the proposed codes is larger than that of ideal codes. The numerical results
showed that the performance of -OOCs with larger
code weight is better than that of ideal codes, because the larger code weight is more robust to interference to a certain extent.
Although the performance of the -OOCs with
larger code weight is better, the code size is reduced sharply as the code weight increases. As a result, it is impossible to
increase the code weight of -OOCs for a given
code length and code size further. One feasible way to increase the code size is to relax the crosscorrelation constraint
fur-ther. In this paper, we investigate -OOCs based
on the )-perfect difference set with size , where
, that is, is no longer limited to 2. To
con-struct -OOCs, it is necessary to choose -subsets
of a -perfect difference set appropriately, such that any two distinct -subsets share at most elements and then each -subset is corresponding to a code. The problem is the same
as how to construct an maximal system defined
as a family of -subsets of a -set such that every -subset of the -set is contained in at most one set of the system [15],
where and . Trivially, there is an
upper bound of the system size (the number of -subsets in
the system) denoted by . Exhaustive search is a
way to construct the largest size of such system. However, it is infeasible due to its complexity. Fortunately, the random greedy algorithm can construct asymptotically good maximal systems [16], [17]. In this paper, we adopt this algorithm to
construct a suboptimal code family of -OOCs.
The crosscorrelation between two codes is 0 or 1 if they are not aligned with each other. When two codes are aligned (the probability is only , the value of crosscorrelation is between 0 and . As a result, the crosscorrelation property
of -OOCs is only slightly different from that of
)-OOCs. Therefore, it is reasonable to anticipate that
the performances of -OOCs and -OOCs
are similar to each other. To simplify the BER performance analysis, we assume that chips of two codes are synchronous among users. We also take the presumption that the cross-correlation is (which is the worst situation of interference) when two distinct codes are aligned with each other and the interfered chips are randomly distributed among chips. Based on these assumptions, an upper bound of the BER can be derived according to the principle of inclusion and exclusion. The numerical results show that the performance is improved significantly with the increase of when the code weight is not large. However, when the code weight is larger than a certain value, the performance gets worse as the code weight increases. This is because the larger code weight increases the interfering probability which offsets the robustness. The results also show that it is possible to approach the upper bound of the number of simultaneous users given a code length and a code size.
The remainder of this paper is organized as follows. In Sec-tion II, we describe and construct the -OOCs based on perfect difference sets and the random greedy algorithm. In Section III, we analyze the BER performance of the systems in
conjunction with )-OOCs and double hard-limiters
[18]. The numerical results are given in Section IV. We conclude in Section V.
II. -OOCs BASED ONPERFECTDIFFERENCESETS
In this section, we describe the formulation of a code family
of -OOCs based on perfect difference sets.
Let be the -set of the integers modulo
. A set is a -subset of . For every
, there is exactly one ordered pair , , such that
(1) A set satisfying these requirements is called a
-per-fect difference set. The existence of the
-perfect difference set, where is a power of a prime, has been proved and constructed by Singer [19]. We can construct a
per-fect difference code based
on the perfect difference set with the rule if
otherwise. (2)
The code weight and code length are and , respectively,
where and . The off-peak auto
correlation of such code is always equal to one. This property is useful to construct a code family with crosscorrelation equal
to one by cyclically shifting such code times for syn-chronous OCDMA [14]. However, for asynsyn-chronous OCDMA, we have to modify the -perfect difference code [13].
A code family of -OOCs is formed by reserving
some chips from the chips of a -perfect difference code such that all the new codes fulfill the crosscorrelation con-straint. That is, the maximum crosscorrelation between any two codes is not larger than . The problem is the same as how to construct an maximal system defined as a family of -subsets of a -set such that every -tuple of the -set is con-tained in at most one set of the system [15], where
and . It is well known that the system size, denoted
by , is upper bounded by
(3) or more tightly [20]
(4) The density of the system is defined as
(5)
Trivially, holds. If , the system is
also called Steiner system in which every -tuple of the -set is contained in exactly one -set of the system. To find
all parameters for is a long-standing
un-solved problem. There are an infinite number of known Steiner systems with and and a finite number of Steiner systems
with and . Moreover, no Steiner systems with are
known [21].
The determination of the maximal value of is still an unsolved problem [20]. Fortunately, some useful results have been reported. In 1963, Erdös and Hanani conjectured that for
every and , [15]
(6)
They proved (6) for and every and for and
, where is a prime power. Eventually, Rödl proved this conjecture in 1985 [22].
One way to construct a maximum size of an
max-imal system is to compare all collections of -tuples of the -set
and choose one which forms an maximal system
with largest size. The number of possible collections is , which is too complex to apply exhaustive search. Fortunately, it was proved that the random greedy algorithm can almost surely construct asymptotically good maximal systems [16], [17]. That is, the density tends to 1 as approaches to infinity. In this paper, we adopt the random greedy algorithm to
con-struct a family of -OOCs as follows.
1. Construct a -perfect difference set with
ele-ments according to [19], where , , and
is a power of a prime.
2. Let denote an maximal system which
is empty initially.
Fig. 1. Three searching results of random greedy algorithm form(4; 3; 28). 4. Pick one -subset randomly from the list and eliminate it from the list. If the -subset shares at most elements with all selected -tuples in , then include it in , otherwise, discard it.
5. Repeat Step 4 until there are no more candidates in the list.
6. The th subset, , in corresponds to a code
according to (2). All the codes form a family of -OOCs.
The crosscorrelation between any two distinct codes can be expressed as
or two codes not aligned
two codes aligned. (7)
Since this algorithm involves randomness, the density of the system may be far from 1 with small probability. On the other hand, the density may be close to 1. Roughly speaking, the den-sity is somewhere in between, but not very far from 1 [16], [17]. In Fig. 1, we apply this algorithm three times to form three
dis-tinct maximal systems of with system sizes 634,
630, and 632, respectively. The total number of candidates is 20 475. Fig. 1 shows that the size of grows quickly in the early stage of iterative due to the small size of . However, when the size is getting larger it grows slowly because most of the candidates are discarded. This fact is helpful because we can construct the most part of codes during early searching stage es-pecially when the number of candidates is too large to search through. Note that the optimal maximal system of
is also a Steiner system of whose size is equal to 819. In other words, the density is about 0.77, which is not far from 1.
III. PERFORMANCEANALYSIS
We analyze the performance of the systems using double hard-limiters with consideration of shot noise, thermal noise, APD bulk, and surface leakage currents. We adopt -codes as the signature codes. The receiver structure is shown in Fig. 2 [18]. To simplify the analysis, we assume that chips are synchronous among users because it is the worst case and results in an upper bound [3].
The average photon arrival rate per pulse is given by (8) where is the APD quantum efficiency, is the received signal power per pulse, is the Planck’s constant, and is the optical frequency. There are only two states after the second hard-limiter, denoted by and , respectively. The state means that the average photon arrival rate is equal to . The other state is that the photon arrival rate is zero (this occurs only when the desired data bit is zero and the MUI is removed completely by the two hard-limiters). For states , , the probability density function of the output after the photo detector can be expressed as [23]
(9) where the mean can be expressed as
(10) where is the average APD gain, is the chip duration, is the electron charge, is the contribution of the APD bulk leakage current to the APD output, is the APD surface leakage current, and the variance can be written as
(11) where is the excess noise factor given by
(12) where is the APD effective ionization ratio and is the variance of thermal noise expressed as
(13) where is Boltzmann’s constant, is the receiver noise tem-perature, and is the receiver load resistance.
After the photo detector, the signal is fed into the on-off keying (OOK) decoder. If the output is larger than a constant threshold , we declare that the output data bit is one, otherwise, zero. To minimize the error probability, we set the suboptimal value of the constant threshold to be
(14) The probability that the state (or is decoded incorrectly
as (or can be expressed as
(15) where stands for the complementary error function, defined as
(16) The probability that the state (or is decoded correctly to
be (or can be expressed as
(17) and
(18) respectively.
Fig. 2. Receiver structure of OCDMA systems with double hard-limiters. A. Upper Bound of Performance
Without loss of generality, we consider the user assigned
Code (based on Subset is the
desired user and the desired data bit is . The user assigned
Code (based on Subset
rep-resents any other user. If their relative cyclic shift is , , the crosscorrelation can be expressed as
(19) where denotes the addition modulo . The value of is given by
(20) where is a set formed by adding to each element of and represents the size of a set. When , the value of
is not larger than . On the other hand, the value of is not
larger than 1 when .
To simplify the analysis and to derive the upper bound of per-formance, we assume that the crosscorrelation is always equal to when two distinct codes are aligned with each other (that is, . We also assume that the interfered chips are randomly distributed among chips. Let and denote the probabilities that is 1 and , respectively. Because the value
of is only when , the value of is and then
. We have the expected value of as [4] (21) The probabilities that contributes 1 and pulse positions are given by
(22) and
(23) respectively, where the factor 1/2 means the equiprobable 0 and 1 symbols.
If the desired data bit is 1, the state after the second hard-limiter must be , that is
(24) On the other hand
(25) When the desired data bit is 0, the state after the second hard-limiter should be if the two hard-limiters can com-pletely eliminate MUI. However, when each of the chips in
the desired code is interfered by at least one user, the hard-lim-iters cannot remove MUI entirely. As a result, the state will be
. In the following, we derive the probability using the principle of inclusion and exclusion.
For any chips of mark chips of a desired code, the prob-ability that the chips are not interfered by one other user is
(26)
Given the number of simultaneous users , the probability that the chips are not interference by the other users is if all the users are independent to each other. According to the principle of inclusion and exclusion, the
probability can be expressed as
(27)
On the other hand
(28)
Therefore, the bit error probability can be written as
(29)
If , (29) can be approximated as
(30)
The first and second terms in (30) represent the noise power and interference contributions, respectively.
TABLE I LINKPARAMETERS
B. Lower Bound of Performance
Because we assume that the crosscorrelation is always equal to when two distinct codes are aligned with each other, the bit error probability is upper bounded by (30). On the other hand, we will derive the lower bound of performance for given codes and noise level.
Because a family of -OOCs are formed by
choosing chips from chips of a -perfect
dif-ference code, any two -OOCs must at least
share chips of the mark chips. Thus, the
crosscorrelation is larger than or equal to , if
two distinct codes are aligned with each other. To derive the lower bound of the BER, we assume that the crosscorrelation
between any two codes is equal to , if the two
codes are aligned. From (26) and (27), we have
(31)
where .
IV. NUMERICALRESULTS
In this section, we present the numerical results of the
sys-tems with -OOCs. The parameters used are given
in Table I.
The bit error probabilities versus code weight with or without double optical hard-limiters are given in Figs. 3 and 4. Note that to simplify the calculation of the BER performance without op-tical hard-limiter, we assume that the maximum crosscorrelation between any two codes is one and we only take the interference contribution into consideration. These assumptions result in the lower bound of BER performance. Consider the systems with double optical hard-limiters. Part of the BER induced by the
noise power is 2 10 under W. In Fig. 3, the
Fig. 3. Bit error probabilities versus the code weight underv = 757, N = 33, andP = 0:5 W.
Fig. 4. Bit error probabilities versus the code weight underv = 6643, N =
300, and P = 0:5 W.
performance of the systems with double optical hard-limiters is improved as the code weight increases. This is because the larger code weight is more robust to interference. The performance of codes with larger is worse than that of codes with smaller . However, codes with larger have larger code size. Moreover, when , the lower bound of the BER increases sharply due
to increasing with . As a result, the lowest
BER of the lower bound is about 10 under and
. In other words, given , the maximum number of
simultaneous users is about 33, no matter what value of . This figure also shows that the performance of the system without optical hard-limiter is not improved significantly with the in-creasing of code weight. The phenomenon is due to the larger the code weight, the larger the interfering probability between any two codes. Therefore, the advantage of a large code weight is diluted. Comparing the performances between systems with or without double optical hard-limiters, we find that the system with double optical hard-limiters outperforms that without any
TABLE II
UPPERBOUND OFm(w; + 1; 82) VERSUSwAND UNDER BER 10 , N = 300ANDv = 6643
optical hard-limiters for the most part. In fact, under suitable signal power, the performance of the system with double op-tical hard-limiters outperforms that without opop-tical hard-limiter because the former could remove some interference patterns.
Fig. 4 shows the performance of )-OOCs. Fig. 4
has similar characteristics with respect to Fig. 3 except that the performance of the system without optical hard-limiter is get-ting worse with the increasing code weight. The reason is sim-ilar to that in Fig. 3. However, the advantage of larger code weight cannot offset the increasing of interfering probability. On the other hand, because the interference increases due to large code weight for the system with double optical hard-lim-iters, there is an optimal value of code weight for a given code length and number of simultaneous users. Fig. 4 shows that the best performance is when the code weight is around 30. In
such a case, the bit error probability of -OOCs
or -OOCs is about 4.0 10 which is very
close to the lower bound. Consequently, the maximum number of simultaneous users is around 300 given . Fig. 4 also shows that the larger results the worse performance, espe-cially when code weight is less than 10. However, when the code weight is larger than 20, the phenomenon is not obvious, be-cause a large code weight dilutes the effect of interference. The
upper bound of versus the code weight under
BER 10 , , and is listed in Table II. It
shows that the code family of -OOCs has the
largest upper bound of code size.
Fig. 5 shows the bit error probabilities versus the code weight
under , , , and three different values
of the average power per bit. Considering the value of the av-erage power per bit is equal to 1 nW, the BER performance is improved with the increasing of the code weight under because the interference contribution in (30) is the dominant term. On the other hand, the BER performance is getting worse under because the larger values of code weight reduce the signal power per pulse. As a result, the noise power contri-bution in (30) is getting larger and dominates the bit error
prob-Fig. 5. Bit error probabilities versus the code weight underv = 6643, N =
300, = 6, and three different values of the average received power per bit.
Fig. 6. Bit error probabilities versusP underv = 6643, N = 300, and
w = 24.
ability. Similarly, when the value of the average power per bit is equal to 2 nW, the lowest BER is about 7.0 10 . How-ever, because it has larger average power per bit, the effect of the noise power is not obvious until . When the value of the average power per bit is equal to 3 nW, the curve is almost the same as the corresponding one in Fig. 4. This is because the average power per bit is large enough that the noise power con-tribution no longer dominates the bit error probabilities.
Fig. 6 shows the bit error probabilities versus the received
power per pulse under , , and . The
three curves illustrate that there exists an error floor which is actually equal to the interference contribution in (30).
The bit error probabilities versus number of simultaneous users are presented in Figs. 7 and 8. For the systems with double optical hard-limiters, these figures indicate that the performances of codes with larger crosscorrelation are worse than those of codes with smaller one. However, when the number of simultaneous users is large, the phenomenon is not
Fig. 7. Bit error probabilities versus the number of simultaneous users under
v = 757, w = 13, and P = 0:5 W.
Fig. 8. Bit error probabilities versus the number of simultaneous users under
v = 6643, w = 25, and P = 0:5 W.
obvious due to dominating the BER. These figures also show that the performances of the systems with double optical hard-limiters outperform those of the systems without any optical hard-limiters.
V. CONCLUSION
One feasible way to improve the performances of OCDMA systems is to increase the code weight with the penalty of decreasing the code size when the code weight is less than a certain value. In this paper, we further relax the crosscorre-lation constraint to get larger code size and derive the upper and lower bounds of performances. Because the optimal construction of a maximal system using exhaustive search is infeasible, we adopt the random greedy algorithm to construct -OOCs. It is proved that this algorithm almost surely can construct asymptotically good maximal systems. An
example of constructing maximal systems shows
that the system density is around 0.77, which is not far from 1. Moreover, most of the codes can be formed at early constructed stage. This fact is very helpful especially when the number of candidates is too large to search through. Because most of the crosscorrelations between any two codes are 0 or 1 (while only a very small portion have value of , the numerical results show that the larger value of crosscorrelation does not decrease the performance significantly, especially when the code weight or number of simultaneous users is large. For the systems with double optical hard-limiters, the results also imply that there is an optimal value of code weight for a given code length. We demonstrate that it is possible to approach an upper bound of the number of simultaneous users for a given code length and code size. Meanwhile, the code size is maintained on a satisfactory level.
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Chi-Shun Weng was born in Tainan, Taiwan, R.O.C., in 1976. He received the
B.S. and Ph.D. degrees in electrical engineering from National Taiwan Univer-sity, Taipei, Taiwan, R.O.C., in 1998 and 2002, respectively.
He is currently with Realtek Semiconductor Corporation, Hsinchu, Taiwan, R.O.C., as a Digital IC Design Engineer. His research interests include lightwave communication systems, spread-spectrum communication, and coding theory.
Mr. Weng received the Gold Award from the Asian Pacific Mathematics Olympiad in 1993.
Jingshown Wu (S’73–M’78–SM’99) received the B.S. and M.S. degrees in
electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1970 and 1972, and the Ph.D. degree from Cornell University, Ithaca, NY, in 1978.
He joined Bell Laboratories in 1978, where he worked on digital network standards and performance and optical communication systems. In 1984, he joined the Department of Electrical Engineering, National Taiwan University, as a Professor, and was the Chairman of the Department from 1987 to 1989. He was also the Director of the Communication Research Center, College of Engi-neering, from 1992 to 1995. From 1995 to 1998, he was the Director of the Di-vision of Engineering and Applied Science, National Science Council, Taiwan, R.O.C., on leave from the university. From 1999 to 2002, he was the Chairman of the Commission on Research and Development and the Director of the Center for Sponsor Programs of the university. Currently, he is the Vice-President of the university. He has published more than 100 journal and conference papers and holds 12 patents. His research interests include optical fiber communications, computer communications, and communication systems.
Dr. Wu is a Life Member of the Chinese Institute of Engineers, the Optical So-ciety of China, and the Institute of Chinese Electrical Engineers. He has served as the Vice Chairman (1997–1998) and the Chairman (1998–2000) of IEEE, Taipei Section.
