MMedia-Lau

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 1, JANUARY 1998 9
Multirate Fiber-Optic CDMA: System
Design and Performance Analysis
Svetislav V. Maric, Member, IEEE, and Vincent K. N. Lau, Student Member, IEEE
Abstract— In this paper, we discuss possibilities of multi-
rate transmission using fiber-optics code-division multiple-access
(CDMA) networks. Two schemes are introduced: the parallel
scheme and the serial mapping scheme. The theory of optical
CDMA is reviewed and the basic properties of OOC families used
in the paper given. Also a new property of optical orthogonal
code (OOC) sequences is introduced which greatly increases the
number of sequences in the network. Performance analysis of
the two systems is done using as the parameter the number of
simultaneous different-rate users as a function of the required
probabilities of error. It is shown that addition of error correction
coding increases the number of users in the network. Examples
of the two systems are given and it is shown that a high number
of different-rate users can be accomodated in the network with
no increase in the complexity of optical processing.
Index Terms—Multirate OOC, optical CDMA.
I. INTRODUCTION
THE possibility (both theoretical and practical) of using
code division multiple access technique in high speed
fiber optical networks was introduced in [1]–[3]. From the
practical point the code-division multiple-access (CDMA)
optical network is interesting since it requires minimal optical
signal processing and is virtually delay free. From the theo-
retical point a CDMA based fiber-optics system is interesting
since it is an example of a CDMA network in which the suc-
cessful performance depends on the auto- and cross-correlation
properties of employed sequences.1
In the fiber-optics network, a large number of users is
simultaneously using the same optical channel. With the
traditional way of using CDMA, each information bit from
a user is encoded into a optical signal that corresponds to
a (0,1) code sequence of chips representing the address
of that user, and called the optical orthogonal code (OOC).
OOC is characterized by a quadruple where
denotes the sequence length, denotes the code weight-
the number of ones, and denote the maximum value
of the out-of-phase autocorrelation function and the cross-
correlation function, respectively. Clearly, in a multiple access
environment families of such OOC’s are needed.
Manuscript received December 6, 1996.
S. V. Maric was with the Department of Engineering, University of
Cambridge, Cambridge CB2 1PZ U.K. He is now with QUALCOMM, San
Diego, CA 92121 USA.
V. K. N. Lau is with the Department of Engineering, University of
Cambridge, Cambridge CB2 1PZ U.K.
Publisher Item Identifier S 0733-8724(98)00570-2.
1 In a wireless CDMA system it is common to base the analysis and even
to apply the PN sequences with no prespecified auto- and cross-correlation
properties.
In order for the receiver of the CDMA system to be able to
correctly distinguish each of the possible user’s addresses, the
following conditions for OOC’s must be satisfied:
1) the number of ones, , in the code sequence should be
maximized and identical for every user of the network;
2) the number of coincidences, in the discrete autocor-
relation function
(1)
of the code and for every
shift, (except the zero-shift) should be minimized; and
3) the number of coincidences, for every shift, in the
cross-correlation function
(2)
of two code sequences and should be minimized.
Equally acceptable are cross-correlation functions which
are uniformly “1’s” since they can provide uniform
background (which can be thresholded).
Here a coincidence, or a hit, is an event which occurs when
two ones from two shifted OOC sequences occupy a same
time slot.
Conditions 1 and 2 are necessary for correct synchronization
of the receiver, and condition 3 is necessary to ensure low
probability of error in the multiple access environment.
Unfortunately, it has been the above theoretical prob-
lem—OOC families construction—that has so far limited
further practical applications of CDMA in fiber-optics
networks.
Although there are quite a few OOC families with
(see, for instance, [4] and [5]), what makes the system
impractical is the fairly small number of OOC sequences in
the OOC family as well as generally very large ratio
making the system very inefficient.
In this paper, we show how the optical CDMA can be
used in multirate environments. This is of importance due
to the growing applications of multimedia (voice, data, and
image) transmission. Note that in [6] we have introduced a
different multirate CDMA system but there the multirate was
achieved by varying the length of OOC sequences. Although,
theoretically the most obvious way for varying the system rate
the system has limited applications due to a limited number
of sequences for high rate users and their relatively high error
probability.
0733–8724/98$10.00 © 1998 IEEE

10 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 1, JANUARY 1998
Fig. 1. Optical encoding for multirate network without ECC: Parallel mapping.
The newly proposed system is extremely interesting since
not only it uses the same hardware as a single rate network,
but the multirate is achieved by using the same OOC family
for each rate. In addition, by using error correction codes
(ECC), and an extended addressing scheme for some families
of OOC’s) we overcome the problem of small OOC family
sizes (as mentioned above).
Two mapping schemes are proposed. The parallel mapping
scheme and the serial mapping scheme In the ﬁrst scheme,
ECC is applied while maintaining the same data throughput as
the uncoded system. In the second scheme, ECC is not consid-
ered however, the scheme performs well even for the uncoded
case. We use the hyperbolic congruence code sequence (HC-
OOC) and the Truncated Costas sequence (TC-OOC) [5] as
illustration. The TC-OOC has ideal correlation properties but
smaller family size, while HC-OOC has nonideal correlation
properties , but larger family size. It is found that
the latter OOC’s have a better potential for application both
in the uncoded and coded case.
The paper is organized as follows. In Section II, the two
mapping schemes between the information bits and the optical
signals are explained. In Section III, the performance analysis
through the probability of error as a funtion of the number of
users is studied and examples for both schemes are given.
Finally, we conclude with a brief summary of results.
II. SYSTEM DESIGN
In this section, we discuss two mapping schemes between
the information bits and the optical signals, namely, the serial
scheme and the parallel scheme for the general multirate
network.
A. Parallel Mapping Scheme
In the parallel scheme, each terminal is given a number of
addresses according to its information rate. The case without
ECC is considered ﬁrst. As in the case of the single rate
network, the lowest rate user is given one address, but this
time a high rate user with a rate times higher than the low
rate user is given addresses.
Suppose that addresses are assigned to user . At
the transmitter, serial information bits are grouped into
a block and converted into parallel bits. Each bit is
encoded optically and independently with one of the assigned
addresses. The receiver consists of a block of parallel optical
correlators, one for each of the assigned addresses. Thus, as
shown in Fig. 1, the bits of are detected independently.
Let us look now in a case when the error correction is
applied. Let a rate- ECC be applied to all users. Therefore,
at the transmitter, serial information bits are encoded into
protected bits, and each user is given times the
number of addresses originally assigned for the uncoded case.
(Hence, user is now assigned addresses.) Similarly
as in the uncoded case, these protected bits are
converted into parallel bits. Note that the information
throughput for the coded system and the uncoded one are
equal. The scheme is shown in Fig. 3.
Note that for this method we can use families of existing
OOC’s, since the correlation properties of codes did not
change. This method is referred to as arallel because at any
instant, all the assigned addresses for an active user will be
active.
B. Serial Mapping Scheme
1) OOC’s from Frequency Hop Codes: For the serial
scheme we need to use OOC families constructed from
two-dimensional (2-D) frequency hop (FH) codes based on
th-order congruence equations [7], [8].
Property 1: If a 2-D FH code of length a prime has
and hits in its 2-D auto- and cross-correlation functions,
respectively, then a OOC is obtained
as follows.

MARIC AND LAU: MULTIRATE FIBER OPTIC CDMA 11
Fig. 2. Optical encoding for multirate network: Serial mapping.
Fig. 3. Optical encoding for multirate network with ECC: Parallel mapping.
For each user, the code sequence, is
constructed through
if
for and
for
otherwise.
(3)
For the proof see [7], [8].
Now note that in general the following property is true for
families of FH codes based on th-order congruence
equations [7], [8].2
Property 2: Let denote a FH code. Then a
new FH code constructed by
(4)
2 The following property is in general true also for exponential FH codes
like Costas arrays.

TABLE I
TRADEOFF BETWEEN THE SIZE OF TC-OOC AND w FOR N = 1000
TABLE II
ASSIGNMENTS OF b UNDER DIFFERENT COMBINATIONS OF 5 BITS
has exactly the same 2-D correlation properties as .
Using (3) and Property 2 it follows that the will also
generate a OOC.
For example, if
(5)
then it generates a 2-D FH code family called Hyperbolic
Congruence codes with two hits both in its 2-D auto- and
cross-correlation functions. And consequently, the coresspond-
ing OOC’s will be with a size of
2) The Serial Scheme: In the serial scheme, we still use “ ”
in (four and five) for user addresses. But “ ” is now used to
represent groups of bits. For instance gives 30 possible
addresses (for 30 different “ ”’s) and 31 possible combinations
(for 31 different “ ”’s). of bit groups. This is shown in Table II.
Therefore, the lowest rate user, say , transmitting only 1
bit, is assigned its original sequence using (four and five) with
However, a user, transmitting at five times
the rate of is now assigned an address using (four and five)
with Five bits are grouped and optically
encoded together where, according to Table II, takes any
value between 0 and 30 depending on the sequence of bits
transmitted. If bits “00000” are transmitted, no optical signal
will be sent. This is also shown in Fig. 2. Similarly, for a third
user, with another rate (three times higher than , he will
be assigned where 3 bits are now grouped
together and takes values from 0 to 7.
Because, depending on its rate, each user will take up
much more addresses than the parallel scheme, it can only
be applied on the OOC that are large in size and ECC cannot
be embedded using the method effectively. This method is
referred to as erial because at any instant, only one of the
assigned addresses to any user will be active when the user
is active.
Remark: Note that in a special case when all users are
transmitting at the same rate, Property 2 solves the crucial
problem of a small number of codes as mentioned in the
introduction. Namely, the size of the family of OOC obtained
from congruence equations is now increased from to
.
III. PERFORMANCE ANALYSIS
In this section, we analyze the system performance, based
on the system’s error probabilities as a function of the number
of high and low rate users, of the two introduced multirate
schemes. Two different parameters are used—the number of
simultaneously active users and the total number of users in the
network. The former one is limited by the mutual interferences
between users and the required error probability . The latter
one is constrained by the size of the OOC family. Furthermore,
due to the bursty nature of the information sources, we have to
maintain a certain ratio between number of simultaneous users
and the maximum number of users in the network in order to
have an optimally utilized network.
In the following examples, we assume a 100 Mbd fiber.
The mutual interference between different users is modeled as
Gaussian [9].3
The decoder used in the analysis is shown in
Fig. 4.
A. Example 1: Parallel Mapping Scheme
with Two-Rate Network
To illustrate how the ECC improves the system performance
in the two-rate network, we apply the parallel mapping scheme
using two families of OOC’s—the Hyperbolic congruence
(HC) OOC’s and the TC-OOC’s. The performance of the
uncoded case is compared with the case when the ECC is
applied. The low rate user is transmitting at 100 kb/s and the
high rate user is transmitting at 400 kb/s. The OOC sequence
length is
For a given prime , a TC-OOC is a
OOC [5] where the size of the code is given by , the weight
of the OOC is given by , and and satisfy the inequality
Table I shows the tradeoffs between and with the code
sequence length approximately being equal to 1000. Note
that TC-OOC’s are ideal OOC’s since a c . for the
HC-OOC’s described in Section II–B, we take to get
and . The size of the code is 930
with .
1) Without ECC: The low rate user is given one address
while the high rate user is given four addresses. The SNR is
given by (see Appendix A)
SNR (6)
where
and is the number of simultaneous high rate user, and
is the number of simultaneous low rate user in the network.
Without loss of generality, take . The error probability
as a function of is plotted in Figs. 5 and 6. for different
values of (for TC-OOC and for HC-OOC, respectively).
We see that a larger number of simultaneous users can be
3 In [6] and [10], it was shown that the Gaussian approximation gives
accurate probability of error estimate.

(a)
(b)
Fig. 4. The structure of the decoder using optical correlator and threshold detector. (a) Parallel mapping and (b) Serial mapping.
accomodated in the system as increases for a given .
However, as increases, the size of the TC-OOC
decreases, and this limits the maximum number of users
present in the network. Furthermore, we would like to have a
reasonable utilization. Taking all the constraints into account,
and if assume we would use TC-OOC with
giving (the number of simultaneously active user)
and
We also observe that at for HC-OOC.
The performance is limited by the number of simultaneously
active users, not by the size of the HC-OOC family.
Now, let be the maximum number of low rate user and
be the maximum number of high rate user. Since each high
rate user takes four addresses, we can have
(or and (or
respectively at for the TC-OOC’s.
For the HC-OOC’s, we can have at the same
. (Note that in this case and are not a constraint since
. The performance is thus limited by the number of
simultaneous users allowed.
2) With ECC: Assume rate- ECC is used for both the
high rate users and the low rate users. The low rate user
is given addresses while the high rate user is given
addresses. The equivalent channel models are and
Fig. 5. Probability of error against the number of simultaneous low rate users
kl for TC-OOC without ECC, kh = 0. Note that the constraint by the size
of the OOC is ignored in the graph.
parallel binary symmetric channels (BSC), each having the
channel capacity of . This is shown in Fig. 7 with
replaced by and , respectively.
Let SNR be the required SNR for arbitrarily low
for the coded system. As shown in Appendix B, the SNR is

Fig. 7. Equivalent channel model when n0 addresses are assigned to each user.
Fig. 6. Probability of error against the number of simultaneous low rate users
kl for HC-OOC without ECC, kh = 0: Note that the constraint by the size
of the OOC is ignored in the graph.
given by
SNR (7)
As in the previous example ﬁrst set . The values of
for the coded system are shown in Tables III–VI and Fig. 8
for different weight of the TC-OOC and for HC-OOC. We
observe that as increases the gain4
increases gradually, but
at a decreasing rate. This is because there are two opposing
factors. On one hand, increasing will reduce the required
SNR gradually, but at the same time the number of addresses
required per user increases. It seems that we can gain more
when we use a large . However, this is not true when we
take into account the constraint on the OOC family size. Since
, we have a decreasing and an increasing as
increases.
The performance is therefore, limited by at certain .
Tables III–V shows this constraint.
4 The gain here refers to the ratio of kl between the coded and uncoded
case with the same w for TC-OOC.
TABLE III
THE NUMBER OF SIMULTANEOUS LOW RATE USERS (kl) AND MAXIMUM
NUMBER OF LOW RATE USERS (ql) DUE TO THE TC-OOC SIZE CONSTRAINT
AGAINST THE NUMBER OF ADDRESSES ASSIGNED TO EACH LOW RATE USER
(nl) USING ECC WITH TC-OOC (w = 2); kh = 0: THE GAIN IS THE RATIO
BETWEEN kl OF THE CODED CASE AND THE UNCODED CASE, BOTH USING w = 2
TABLE IV
Furthermore, we must maintain to be a certain percentage
of to ensure a reasonable network utilization.
Taking all these into account, we see that in the ECC system
with and using TC-OOC with we can have
and . For HC-OOC with ECC with
we have and
Comparing to the uncoded case, we have a gain of
in for TC-OOC while a gain of 10.5 for HC-OOC.
We now introduce the high rate users as well. From above,
we see that each high rate user occupies four times more ad-
dresses than the low rate user. Hence, for the example, 21 low
rate users and 20 high rate users can transmit simultaneously
using HC-OOC with ECC.5
5 This comes from the fact that 20 high rate users 24 + 21 low rate users
= 101.

TABLE V
TABLE VI
THE NUMBER OF SIMULTANEOUS LOW RATE USERS (kl) AND
MAXIMUM NUMBER OF LOW RATE USERS (ql) DUE TO THE EH-OOC
SIZE CONSTRAINT AGAINST THE NUMBER OF ADDRESSES ASSIGNED TO
EACH LOW RATE USER (nl) USING ECC WITH EH-OOC, kh = 0
Fig. 8. Gain in the number of simultaneous low rate users (kl) using ECC
with TC-OOC (w = 2; 3; 4) and HC-OOC, kh = 0: Note that the gain is
the ratio of kl in the coded and uncoded cases using the same w in TC-OOC
and HC-OOC.
B. Example 2: Serial Mapping Scheme with Two-Rate Network
We illustrate the performance of the serial scheme using
a two-rate network with HC-OOC. The low rate user is
transmitting at 100 kb/s while the high rate user is transmitting
at 500 kb/s (this simply means that using 31 different “ ”’s
for the same address “ ” in (4), 5 bits forming 32 different
sequences will be grouped together, in effect creating 31
different addresses for one user). The OOC sequence length
is 1000 which gives . ECC is not effective here as
explained.
Since (independently of the user rate) there is always one
address for an active user the SNR is given by
SNR (8)
Assume all symbol errors are equiprobable, then as shown
in Appendix C, the bit error rate (BER) is given by
SNR
(9)
Hence, suppose we require for the low rate user
and for the high rate user. As in the uncoded
case of Example 1, we have
Hence, we can have, for instance, six high rate users and
two low rate user simultaneously transmitting. (Again, the
performance is not constrained by the size of the OOC). This
is an enormous improvement from the system introduced in
[6]. For instance, from [6, Table I], we see that only three
high rate users were possible and the probability of error was
(quite high for high transmission rates).
IV. CONCLUSION
In this paper, we have introduced two schemes for achieving
multirate transmission in fiber-optic CDMA networks. One
scheme called parallel multirate transmission used a number
of OOC sequences to transmit simultaneously different size
groups of bits. The second, serial, scheme relies on the
property of OOC families based on congruence equations to
allow linear shearing of codes without changing the correlation
properties. The performance analysis shows that both schemes
can allow simultaneous transmission of a high number of
different rate users. Both schemes are extremely interesting
from the practical stand point since they do not require any
new optical processing at the receiver, since they use exactly
the same OOC sequences used for single rate transmission and
hence the same number of incoherent optical matched filters.
In the parallel scheme ECC was applied to improve the
number of simultaneous users in the network. The addition
of ECC is again achieved with no loss in the information
throughput and with no added complexity in the receiver.
In the serial scheme the ECC is not applied. However,
because there is only one active address per each active user
(no matter of its rate), the interference experienced by all
the users is the same. Hence, this scheme can allow a larger
number of simultaneous higher rate users in the uncoded case.
It is interesting that the serial scheme in a special case of a
single rate network solves the problem of small sizes of OOC
families and that with the addition of error correction coding
and multirate capabilities makes the fiber optic networks based
on CDMA extremely interesting for practical applications.
APPENDIX A
ERROR ANALYSIS IN THE UNCODED
CASE AND PARALLEL MAPPING
Fig. 4(b) shows the equivalent decoder and channel struc-
ture. Optical signals are passed to the optical correlator with
the given address. Output from the correlator is given by
if bit “1” is transmitted
if bit “0” is transmitted
where are assumed to be independent Gaussian random
noise caused by the mutual interference of the other users.

Based on the output from the correlator, we have to make
a decision on the transmitted information bit. This is done
optimally with a threshold detector with threshold set at .
Hence, the probability of error is given by
SNR
(10)
where
SNR (11)
is the weight of the OOC, is the number of simultaneous
addresses in the network, is the average variance of
the cross-correlation amplitude (calculated over all possible
cross-correlation pairs in an OOC family) with the variance
given as
where is the average value of the cross correlation.
APPENDIX B
SYSTEM PERFORMANCE IN CODED
CASE AND PARALLEL MAPPING
The low rate user is given addresses while the high
rate user is given addresses. Let be the number of
simultaneous low rate user, be the number of simultaneous
high rate user, SNR be the required SNR for the uncoded
system at a certain SNR be the required SNR for
the coded system.
is given by (10). Note that
SNR (12)
For the low rate and high rate users, and block-ECC’s
are used respectively. The equivalent channel models are
and parallel BSC’s, each having the channel capacity of
. This is shown in Fig. 7 with replaced by and
, respectively. The overall information throughputs of the
low rate and the high rate users for reliable transmission are
(13)
(14)
To maintain the same throughput as the uncoded case, we
have and Hence
(15)
(16)
From (15) and (16), we have
and
The SNR is given by
SNR SNR (17)
APPENDIX C
ERROR ANALYSIS IN THE SERIAL MAPPING
Fig. 4(b) shows the block diagram of the optical detector
for the serial mapping scheme grouping 5 bits together. The
decoding is divided into two stages. In the ﬁrst stage, we
extract We set a threshold of
In the second stage, we decode as “00 000” if Else,
we decode as the symbol corresponding to the maximum in
. For example, if then we decode as “00011.”
Ignoring the average cross-correlation between OOC’s, we
have
if the corresponding
ymbol is transmitted
else
where are independent Gaussian noise.
Suppose “00000” is transmitted, the lock error probability
is given by:
Pr
SNR
(18)
where
SNR
Suppose “00001” is transmitted6
Pr Pr
Pr
Pr
SNR SNR
SNR
(19)
REFERENCES
[1] P. Prucnal, M. Santoro, and T. Fan, “Spread spectrum ﬁber-optic local
area network using optical processing,” J. Lightwave Technol., vol. LT-4,
pp. 547–554, May 1986.
[2] F. Chung, J. Salehi, and V. Wei, “Optical orthogonal codes-design and
application,” IEEE Trans. Inform. Theory, vol. 35, pp. 595–604, May
1989.
6 By symmetry, the conditional error probability will be the same for other
symbols except “00000.”

[3] A. Mendez, J. Lambert, J. Morookian, and R. Gagliardi, “Synthesis
and demonstration of high speed, bandwidth efficient optical CDMA
tested at 1 Gb/s throughput,” IEEE Photon. Technol. Lett., vol. 6, pp.
1146–1149, Sept. 1994.
[4] H. Chung and P. Kumar, “Optical orthogonal codes-new bounds and
an optimal construction,” IEEE Trans. Inform. Theory, vol. 36, pp.
866–873, July 1990.
[5] S. Maric, M. Hahm, and E. Titlebaum, “Construction and performance
analysis of a new family of optical orthogonal codes for CDMA fiber-
optic networks,” IEEE Trans. Commun., vol. 43, pp. 485–489, Feb.
1995.
[6] S. Maric, O. Moreno, and C. Corrada, “Multimedia transmission in fiber-
optic LAN’s using optical CDMA,” J. Lightwave Technol., vol. 14, pp.
1–5, Oct. 1996.
[7] S. Maric, Z. Kostic, and E. Titlebaum, “A new family of optical code
sequences for use in spread-spectrum fiber-optic LAN,” IEEE Trans.
Commun., vol. 41, pp. 1–5, Aug. 1993.
[8] S. Maric, “New family of algebraically designed optical orthogonal
codes for use in CDMA fiber-optic networks,” Electron. Lett., vol. 29,
pp. 538–539, Mar. 1993.
[9] W. Kwong, P. Perrier, and P. Prucnal, “Performance comparison
of asynchronous and synchronous CDMA techniques for fiber-optic
LAN’s,” IEEE Trans. Commun., vol. 39, pp. 1625–1634, Nov. 1991.
[10] K. Letaif, “The performance of optical fiber direct sequence spread spec-
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Svetislav V. Maric (S’88–M’91) was born in 1962. He received the B.S.
degree in electrical engineering from the University of Novi Sad, Yugoslavia,
in 1986 and the M.S. and Ph.D. degrees from the University of Rochester,
Rochester, NY, in 1988 and 1990, respectively.
From 1990 to 1991, he was working as a postdoctoral fellow in the Wireless
Information Network Laboratory at Rutgers University, New Brunswick, NJ,
and later, he was an Assistant Professor at City University of New York, NY,
and a Senior Lecturer at the Department of Engineering, at the University of
Cambridge, Cambridge, U.K. In 1997, he joined QUALCOMM, San Diego,
CA, where he is a Senior Engineer at the Network Switching Product group.
His research interests are in the code design and code-division multiple-access
techniques for applications in various spread-spectrum systems.
Vincent K. N. Lau (S’93) was born in Hong Kong. He received the B.S.
degree in electrical engineering in 1995. He then joined the graduate program
at the Department of Engineering at the University of Cambridge, U.K., where
he is currently finishing the work for the Ph.D. degree.
He will be joining Lucent Technologies in 1998. His main research interests
are in information theoretical issues in wireless communications and other
spread-spectrum systems.

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