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3632 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 22, NOVEMBER 15, 2008
Design and Analysis of 2-D Codes With theMaximum Cross-Correlation Value
of Two for Optical CDMAJen-Hao Tien, Guu-Chang Yang , Senior Member, IEEE , Cheng-Yuan Chang , Member, IEEE , and
Wing C. Kwong , Senior Member, IEEE
Abstract—In this paper, a new family of two-dimensional (2-D)optical codes with the maximum cross-correlation value of two isconstructed and analyzed for optical code-division multiple access.Our 2-D codes employ wavelength hopping, controlled by the per-mutations of “synchronized” prime sequences, onto the pulses of a time-spreading optical orthogonal code (OOC). The new con-struction supports larger code cardinality (for more subscribers)and heavier code weight (for better code performance) without in-creasing the code length or number of wavelengths. Under cer-tain conditions, our analysis shows that the new codes can performbetter than our multiple-wavelength OOCs, which have cross-cor-relation values of at most one.
Index Terms—Optical code-division multiple access, opticalorthogonal code, time spreading, two-dimensional optical codes,wavelength hopping.
I. INTRODUCTION
O
PTICAL code-division multiple access (O-CDMA)
is currently receiving renewed attention due to the
advancement of two-dimensional (2-D) coding techniques,which combine wavelength hopping and time spreading in
optical codes [1]–[5]. Two-dimensional optical codes with
low cross-correlation values can support more subscribers
and simultaneous users than one-dimensional (1-D) optical
codes, such as the prime codes [3] and optical orthogonal codes
(OOCs) [6], for a given code length. The study of 2-D optical
codes was traditionally concentrated on the constructions of
code matrices with the cross-correlation values of at most
one in order to minimize multiple-access interference (MAI)
[1]–[5]. However, this kind of construction significantly re-
stricts code cardinality. It is known that we can support larger
Manuscript received September 14, 2007; revised April 7, 2008. Currentversion published January 28, 2009. This work was supported in part bythe National Science Council of the Republic of China under Grant NSC95-2221-E-005-023-MY3, in part by the Ministry of Education, Taiwan,R.O.C., under the ATU plan, in part by the U.S. Defense Advanced ResearchProjects Agency under Grant MDA972-03-1-0006, and in part by the Presi-dential Research Award and Faulty Development and Research Grants, HofstraUniversity.
J.-H. Tien and G.-C. Yang are with the Department of Electrical Engineering,National Chung-Hsing University, Taichung 402, Taiwan, R.O.C. (e-mail:[email protected]).
C.-Y. Chang is with the Department of Electrical Engineering, NationalUnited University, Miaoli, Taiwan, R.O.C. (e-mail: [email protected]).
W. C. Kwong is with the Department of Engineering, Hofstra University,Hempstead, NY 11549 USA (e-mail: [email protected]).
Digital Object Identifier 10.1109/JLT.2008.925029
TABLE ISYNCHRONIZED PRIME SEQUENCES OVER GF(5)
code cardinality and, in turn, more subscribers by relaxing the
maximum cross-correlation value, but this means larger MAI
as well [3], [6]–[11].
In [7], Mashhadi and Salehi computed the optimal values of
the code weight and maximum cross-correlation value that
minimize the error probability of optical codes for given code
length and cardinality. They concluded that the optimal value of
should be either two or three, and these optical codes weremorebeneficialthan optical codes,from a practical point
of view. It is because the former support larger code cardinality
with slightly worsening in code performance (or error proba-
bility), according to their analysis in [7]. As a result,
codes provide a better compromise in terms of code cardinality
and error probability than optical codes.
Nevertheless, previous work in the design of 2-D codes over-
looked the fact that larger cross-correlation values would allow
us to pack more pulses (i.e., heavier code weight) into code ma-
trices, resulting in higher autocorrelation peaks. Higher autocor-
relation peaks mean better discrimination against MAI and, in
turn, give better code performance. Therefore, by relaxing ,
we now have the question of whether larger MAI or heavier codeweight has a stronger effect in terms of code performance.
In Section II of this paper, we construct a new class of 2-D
optical codes by relaxing the maximum cross-correlation value
from one to two in order to increase the code weight and
cardinality. In Section III, our performance analysis shows that
heavier code weight supported by the new codes re-
3634 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 22, NOVEMBER 15, 2008
TABLE IIIEXAMPLES OF OVERALL CARDINALITIES OF THE NEW 2-D CODES AND MWOOCs
the maximum cross-correlation value. The upper bound is de-
rived by multiplying to the Johnson bound in [6], such that
(1)
When compared to (1), the cardinality of ( 2,1) OOC in
[6] is asymptotically optimal because its cardinality is equal to
for odd
for even(2)
Equation (2) determines the number of time-spreading OOC
codewords that can be used in our new 2-D codes, for given
and .
As mentioned earlier, the number of synchronized prime se-
quences (e.g., Table I) that can be used in the wavelength per-
mutations is given by . The number of synchronized
prime sequences suitable for the permutations is maximized
(i.e., ) when is a prime number. For each codeword
of the OOC, the permutationalgorithm can generate
at most (e.g., Table II) code matrices. Therefore, the overall
cardinality of the new 2-D codes is given by
(3)
For comparison, the overall cardinality of the
MWOOCs [5] is given by ,
where in accordance
with (1). The MWOOCs have almost half the number of
code matrices of the new 2-D codes for a given because(for odd ) and 2( 1)
(for even ). Numerical examples for the cardinality compar-
ison of both codes are given in Table III.
Theorem I: The cross-correlation functions and autocorrela-
tion sidelobes of the new 2-D codes are both at most two. The
overall code cardinality is .
Proof: See Appendix I.
By Theorem 1, the maximum autocorrelation sidelobe of the
proposed 2-D codes worsens to at most two because each code
matrix from group 0 uses the same wavelength in its pulses. By
removing group 0, we can reduce the autocorrelation sidelobes
to zero, giving a class of ( 0,2) 2-D codes. However,
the number of synchronized prime sequences that can be usedfor the wavelength permutations is reduced to .
Theorem 2: From any group , the au-
tocorrelation sidelobe of the proposed 2-D codes is reduced to
zero. The overall cardinality of these ( 0,2) 2-D codes
becomes by scarifying the
matrices from group 0.Proof: The proof is similar to that of Theorem 1. Every
pulse in each code matrix of group is con-
veyed with distinct wavelengths such that
for and . Hence, the autocorrela-
tion sidelobes of the proposed 2-D codes become zero.
III. PERFORMANCE ANALYSES
In this section, we introduce a combinatorial method to an-
alyze the hard-limiting1 performance of the new 2-D
codes for an interference limited O-CDMA system. In general,
the code performance is determined by code parameters such as
weight, length, the number of wavelengths, and the maximumcross-correlation value [1]–[5].
Let and denote the probabilities of the desired code ma-
trix (originated from group 0 and , re-
spectively) getting one hit in a time slot when it correlates with
a code matrix in the code set. Also, let denote the proba-
bility of the desired code matrix (originated from group
) getting two hits in the cross correlation. (The
term “two hits” defines the situation when a time slot of the
desired code matrix is being hit by two interfering pulses at
the same time. This causes the cross-correlation function to be
two.) In the following derivations, we assume that the number
of wavelengths is , there are groups of synchro-nized prime sequences, and an ( 2,1) OOC of cardinality
is used in the new ( 2,2) 2-D codes.
Theorem 3: Let and be two distinct (0,1) code
matrices in the code set. The number of times of getting two hits
in their cross-correlation is at most for odd weight
and at most for even weight. Then, we have
(4)
1It is known that a hard-limiter can be placed at the front end of a receiverbefore correlation is performed in order to reduce the effects of MAI and thenear–far problem [2], [3], [14], [15], [18]. While the operation details of hard-
limiters can be found in [21], their basic function is to clip the output light in-tensity to a fixed level if the input light intensity at a time instant is greater thana preset threshold. Otherwise, the output of the hard-limiter is zero.
3638 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 26, NO. 22, NOVEMBER 15, 2008
time-spreading OOC is . The desired 2-D code matrices
originated from group 0 have
(20)
After simple manipulations, (6) can be obtained.
In (20), the assumption of on–off data-bit transmission with
equal probability gives the factor 1/2 to all four terms in the
braces. The factor 1 represents the number of possible time
shifts between the two code matrices of length . Since each
( 2,1) time-spreading OOC can generate up to code ma-
trices, there are code matrices in total and,
thus, up to interferers in the denominators.
The first term in the braces of (20) relates to the hit probability
of the desired code matrix from group 0 caused by an interfering
code matrix from different groups but using the same ( 2,1)
time-spreading OOC as the desired code matrix, and there is no
time shift between the two code matrices. The factor 1
represents the number of interfering code matrices contributing
one hit. The second term is derived similar to that of the first
term but with a time shift. The factor 1 1 represents
the number of interfering code matrices, which can contribute
one hit. The third and fourth terms relate to the hit probabilities
of the desired code matrix from group 0 caused by interferingcode matrix from group 0 and groups 1 to 1, respectively,
using a different ( 2,1) time-spreading OOC from the de-
sired code matrix.
For the desired code matrix originated from group
with the probability of getting one
hit, we have
(21)
After some manipulations, we finally have (7).
The derivation of follows the rationales of that of . The
first three terms in the braces of (21) relate to the hit prob-
abilities of the desired code matrix from group caused by
interfering code matrices originated from the same ( 2,1)
time-spreading OOC as that of the desired code matrix. The first
term represents the case that the desired and interfering code
matrices are aligned without any time shift. The second termrepresents the case that the interfering code matrix gives one
hit to the desired code matrix with a time shift and the inter-
fering code matrix comes from group 0. The third term rep-
resents the case that the interfering code matrix comes from
groups 1 to 1 and gives one hit with desired code matrix
with a time shift. The factor [i.e.,
( 1)
from Appendix II] represents the number of interfering codematrices, which can contribute one hit. The fourth term relates
to the hit probability of the desired code matrix from group
caused by interfering code matrices originated from a different
( 2,1) time-spreading OOC from that of the desired code
matrix. The derivation of with even weight follows the ra-
tionale of that of with odd weight. Moreover, the deriva-
tions of with odd weight and even weight are explained in
Appendix II.
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Jen-Hao Tien received the B.S. degree in electricalengineering from National Chung-Hsing University,Taichung, Taiwan, R.O.C, in 2005, where he is cur-rently pursuing the M.S. degree in electrical engi-neering.
His research interests include optical communica-tions and wireless communications.
Guu-Chang Yang (S’88–M’92–SM’05) receivedthe B.S. degree from the National Taiwan University,Taipei, Taiwan, R.O.C., in 1985 and the M.S. andPh.D. degrees from the University of Maryland,College Park, MD, in 1989 and 1992, respectively,all in electrical engineering.
From 1988 to 1992, he was a Research Assistantwith the System Research Center, University of Maryland. In 1992, he joined the Faculty of NationalChung-Hsing University, Taichung, Taiwan, wherehe is currently a Professor in the Department of
Electrical Engineering. He was Chairman of the Department of ElectricalEngineering from 2001 to 2004. His research interests include wireless andoptical communication systems, spreading code designs, and applicationsof code-division multiple access. He coauthored a first-of-its-kind technical
book on optical code-division multiple access, Prime Codes with Applicationsto CDMA Optical and Wireless Networks (Norwood, MA: Artech House,2002) and contributed a chapter to Optical Code Division Multiple Access:
Fundamentals and Applications (Boca Raton, FL: Taylor & Francis, 2006).Dr. Yang was Chairman of the IEEE Information Theory Society Taipei
Chapter from 2003to 2005and Vice-Chairman of the IEEE Information Theory
Society Taipei Chapter from 1999 to 2000. He received the DistinguishedResearch Award from the National Science Council in 2004 and ExcellentYoung Electrical Engineering Award from the Chinese Institute of ElectricalEngineering in 2003. He also received the Best Teaching Awards from theDepartment of Electrical Engineering from National Chung-Hsing Universityfrom 2001 to 2004.
Cheng-Yuan Chang (S’04–M’07) received theB.S. degree from National Sun Yat-Sen University,Kaohsiung, Taiwan, R.O.C., in 1997 and the M.S.and Ph.D. degrees from National Chung-HsingUniversity, Taichung, Taiwan, in 2002 and 2007,respectively, all in electrical engineering.
In 2007, he joined the Faculty of National UnitedUniversity, Miaoli, Taiwan, where he is presently anAssistant Professor in the Department of ElectricalEngineering. His research interests include opticaland wireless communications.
Wing C. Kwong (S’88–M’92–SM’97) received the
B.S. degree from the University of California, SanDiego, in 1987 and the Ph.D. degree from PrincetonUniversity, Princeton, NJ, in 1992, both in electricalengineering.
In 1992, he joined the Faculty of Hofstra Uni-versity, Hempstead, NY, where he is presently aProfessor in the Department of Engineering. Hisresearch interests are centered on optical and wire-less communication systems and multiple-accessnetworks, optical interconnection networks, and
ultrafast all-optical signal processing techniques. He has published more than120 professional papers, chaired technical sessions, and served on technicalprogram committees in various international conferences. He has giveninvited seminars in various countries, such as Canada, Korea, and Taiwan.He coauthored a first-of-its-kind technical book on optical code-divisionmultiple access, Prime Codes with Applications to CDMA Optical and Wireless
Networks (Norwood, MA: Artech House, 2002) and contributed a chapter to
Optical Code Division Multiple Access: Fundamentals and Applications (BocaRaton, FL: Taylor & Francis, 2006).
Dr. Kwong is an Associate Editor of the IEEE TRANSACTIONS ON
COMMUNICATIONS. He received an NEC Graduate Fellowship from NECResearch Institute in 1991. He received the Young Engineer Award from theIEEE (Long Island chapter) in 1998.