research paper

1. 438 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010
On Efficient Code Acquisition of
Optical Orthogonal Codes in Optical CDMA Systems
Francesco Benedetto, Member, IEEE, and Gaetano Giunta, Member, IEEE
Abstract—This letter presents an efficient method for optical
code division multiple access (OCDMA) code acquisition based
on unipolar optical orthogonal codes. We propose a two-dwell
acquisition procedure and provide closed form expressions to
analyze the system’s error probabilities. Our results match
the Multiple Shift (MS) algorithm, recently introduced in the
literature, and show that our procedure can achieve the same
system performance with a lower computational complexity.
Index Terms—Code acquisition and synchronization, code-
division multiple access (CDMA), optical CDMA, optical orthog-
onal codes (OOCs).
I. INTRODUCTION
IN recent years, there has been an explosive growth in the
use of spread spectrum optical communication systems due
to the high speed, large capacity and high reliability of the use
of the broadband of the fiber optic. The operating principles
of optical networks are well depicted in the fundamental
works of Salehi, et al. [1]–[3]. Optical code acquisition and
synchronization plays a crucial role because the degradation
in the performance of the system will be dramatic when the
synchronization between receiver and transmitter is not ideal
[1]. In the seminal work by Keshavarzian and Salehi, [2], a
new synchronization procedure is proposed, namely multiple
shift (MS) algorithm, which greatly improves the performance
of synchronization process based on optical orthogonal codes
(OOCs). Authors in [2] address the synchronization problem
of an optical network using OOC codes of length 𝐹. The
MS algorithm has two modes and an initializing part: first, 𝐹
different shifts (or cells) in the search space are partitioned
into equal-sized groups each containing 𝑀 different shifts. In
the first mode, the algorithm examines the cells in a group all
at the same time and, when the decision variable exceeds the
threshold, the algorithm enters the second mode to find the
correct shift examining separately each of the 𝑀 shifts [2].
In this letter, we match the MS algorithm proposing a new
code acquisition scheme for OCDMA communications based
on OOCs. We present a two-dwell acquisition procedure stud-
ied for the first time in [6], and whose idea was originally pro-
posed in [4] for spread-spectrum (SS) communications, here
extended and applied to the case of optical code acquisition.
In particular, we provide a closed form to analytically express
the system’s false alarm and detection probabilities in terms of
the 𝐺𝑄 functions, [4], and show that our procedure can obtain
Paper approved by W. C. Kwong, the Editor for Optical Networks of the
IEEE Communications Society. Manuscript received April 18, 2008; revised
July 18, 2008.
The authors are with the Digital Signal Processing and Multimedia Com-
munications Lab, Dept. of Applied Electronics, University of ROMA TRE,
via della Vasca Navale 84, 00146 Rome, Italy (e-mail: fbenedetto@ieee.org;
giunta@ieee.org).
Digital Object Identifier 10.1109/TCOMM.2010.02.080043
(a)
(b)
Fig. 1. a) Block scheme of the OCDMA receiver with OOC codes; b) two
dwell acquisition algorithm.
the same system performances as the MS algorithm but with
lower computational complexity. The remainder of this work
is organized as follows. In Section II, we present the two-
dwell acquisition method expressing its performance in terms
of the 𝐺𝑄 functions. Section III shows the numerical results
matching the MS algorithm and highlights the advantages of
the new scheme while Section IV briefly concludes the work.
II. TWO DWELL ACQUISITION METHOD
The typical block scheme of a conventional incoherent
receiver with unipolar OOC is shown here in Fig. 1 (a).
Each receiver first starts the code acquisition process until
it finds an acceptable estimate of 𝜏 𝑛, i.e. the time offset of
the user’s signal from a selected time origin [2]. Then, data
recovery can start while at the same time the tracking system
is continuously running and updating the value of 𝜏 𝑛 to ensure
that the correct shift of the code is used in the decoding
process. The uncertainty region for 𝜏 𝑛 is [0, 𝑇 𝑏), with 𝑇 𝑏 the
bit period. It is divided into fragments (which are also called
cells), each having a duration of 𝑇 𝑐, with 𝑇 𝑐 the chip period.
Then the system searches these possible cells to find the one
within which the actual value of 𝜏 𝑛 is located. To check each
cell, a simple test is adopted as shown in Fig. 1(a) while Fig.
1(b) shows in detail the two dwell algorithm we propose.
0090-6778/10$25.00 c⃝ 2010 IEEE
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2. BENEDETTO and GIUNTA: ON EFFICIENT CODE ACQUISITION OF OPTICAL ORTHOGONAL CODES IN OPTICAL CDMA SYSTEMS 439
Following the same system model adopted in the seminal
work by Keshavarzian and Salehi [2], the received signal
𝑟(𝑡) is first multiplied with a locally generated replica of
the OOC code, then photo-detected and finally its integral
is computed over one bit period, 𝑇 𝑏, to form a decision
variable. This decision variable is passed to the acquisition
device (i.e. a threshold comparator) in order to implement the
decoding process. The threshold is properly pre-determined,
at least equal to the code weight 𝑊 (i.e. number of chips
taking value of “1”), in order to obtain detection probabilities
equal to one [2]. The working principles of the two-dwell
acquisition algorithm are as follows. In the first mode, the
algorithm realizes a simple serial search analyzing only the
first 𝑁 𝑐 = 𝐹/𝑀 chips of the OOC code, where 𝐹 is the code
length, 𝑀 is the same as in [2] and 𝑁 𝑐 is the upper closest
integer to the ratio 𝐹/𝑀. In this way, we can test one cell
examining only a fraction of the chips of the code. Then, if
the test doesn’t reject the current cell, the algorithm enters the
second mode examining the rest of the chips, i.e. (𝐹 − 𝐹/𝑀)
chips, to find the correct shift (see Fig. 1(b)). Otherwise, if
the cell is rejected, the algorithm moves to the next cell.
It can be possible that, for the effect of the multi-access
interference (MAI) and other noise sources, some false alarms
can happen in both the first and second mode. It is the task
of the verification mode to detect false alarms in the second
mode [2]. Following the same notation as in [2], we denote
with 𝑃 𝑓 𝑎 and 𝑃 𝑑 the false alarm and detection probability,
respectively, of the first mode of the algorithm, with 𝑃 𝐹 𝐴
and 𝑃 𝐷 the probabilities of the second mode. According to
[5], considering an ideal optical fiber CDMA system, only
the interference effect should be taken into account while the
shot and thermal noise are assumed to be zero. In this way, the
mean and variance of the MAI, for a number 𝑁 of interfering
users, can be expressed as:
𝑚 = 𝑊 + (𝑁 − 1)
𝑊2
2𝐹
and
𝜎2
= (𝑁 − 1)
𝑊2
2𝐹
(
1 −
𝑊2
2𝐹
)
𝑀2
(1)
In the case of our interest, since in the first dwell we
investigate only the first 𝑁 𝑐 chips of the code the code weight
𝑊 is represented the number of chips “1” in 𝐹/𝑀 chips while
the code length 𝐹 is 𝑁 𝑐; in the second dwell, the code weight
is 𝑊, since we are analyzing codes of 𝐹 chips. In this way,
using the 𝐺𝑄 functions defined in [4], we now can analytically
express in a closed form the error probabilities, due to the
MAI, as follows for the first mode:
𝑃 𝑑 = 𝐺𝑄1
(
𝜎2
1; 𝑉1; 𝑚1
)
= 𝑒−( 𝑚2
1 𝜎2
1 /2)
⋅
+∞∑
𝑘=0
𝑚2𝑘
1 ⋅ 𝜎2𝑘
1
2 𝑘 ⋅ 𝑘!
⋅ Γ1
(
𝑉 2
1
2𝜎2
1
; 𝐾 + 1
)
(2)
𝑃 𝑓 𝑎 = 𝐺𝑄1
(
𝜎2
1; 𝑉1; 0
)
= Γ1
(
𝑉 2
1
2𝜎2
1
; 1
)
= exp
(
−
𝑉 2
1
2𝜎2
1
)
(3)
and as follows for the second mode of the algorithm:
𝑃 𝐷 = 𝐺𝑄2
(
𝜎2
1; 𝜎2
2; 𝑉1; 𝑉2; 𝑚2
)
== 𝑒−( 𝑚2
2( 𝜎2
1 +𝜎2
2 )/2)
⋅
+∞∑
𝑘=0
+∞∑
𝑛=0
(𝑘 + 𝑛)!𝜎2 𝑛
2 𝑚2 𝑛
2
𝑘! (𝑛!)2
2 𝑛
(
1 +
𝜎2
1
𝜎2
2
) (
1 +
𝜎2
2
𝜎2
1
) 𝑘
(4)
𝑃 𝐹 𝐴 = 𝐺𝑄2
(
𝜎2
1; 𝜎2
2; 𝑉1; 𝑉2; 0
)
=
=
+∞∑
𝑘=0
Γ1
[
𝑉 2
1 ( 𝜎2
1 +𝜎2
2 )
2𝜎2
1 +𝜎2
2
, 𝑘 + 1
]
⋅ Γ1
(
𝑉 2
2
2𝜎2
2
, 𝑘 + 1
)
(
1 +
𝜎2
1
𝜎2
2
) (
1 +
𝜎2
2
𝜎2
1
) 𝑘
(5)
where 𝑉1 and 𝑉2 are the threshold values for the first and
second mode, respectively, while (𝑚1, 𝑚2, 𝜎1, 𝜎2) are the
mean values and variances for the two modes and Γ1(∙, ∙)
is the well-known incomplete gamma function [4].
It has to be noted that the proposed scheme performs a
partial correlation for each dwell, and thus, the threshold
should be less than the weight of the code. In fact, for PN
sequences which are used in wireless systems the threshold is
fixed with the criterion of CFAR (constant false alarm rate) [7].
In particular, the CFAR test is accomplished in two successive
parts: first, a threshold is determined to limit the false-alarm
probability at a given reduced value (size of the test); second,
the probability of detection (power of the test) is evaluated
for the threshold previously determined. The probability of
false alarm must be tuned to guarantee a very low number
of possible false alarms, which eventually imply a relevant
penalty time to the acquisition device. Large probabilities of
detection (up to 100%) are typical of well-performing testing
variables.
However, here in order to match the results of the MS
algorithm and as detailed in the next Section, we use threshold
values equal to 𝑊 for the first and second dwell, to obtain
at the end of the acquisition process a detection probability
equal to one [2].
III. NUMERICAL RESULTS
In this section, we devise the validity of the GQ functions to
model the MAI effect, and then we evaluate the performance
of the two-dwell method matching the MS algorithm. Finally,
we highlight the advantages of the new approach versus the
MS in terms of computational complexity. In all the following
graphs we have used according to [2], OOC codes of length
𝐹 = 200, weight 𝑊 = 5, and threshold values equal to 𝑊.
Fig. 2 shows here the performance of the MS algorithm
evaluated for different users, varying 𝑀. Dotted lines represent
simulation results, while solid lines stand for theoretical results
obtained with the GQ functions (red lines) and the binomial
functions (black lines) used in [2]. As it can be easily
seen, simulation results perfectly match the theoretical ones
confirming the validity of the GQ approach in modeling the
MAI effect.
Fig. 3 illustrates the comparison between the receiver op-
erating characteristics (ROC) of the two methods for various
𝑀 and different users (Fig. 3(a) for 𝑁 = 5, Fig. 3(b) for
𝑁 = 8). As we can see, the performance of the two-dwell
approach is almost the same of the MS procedure and again
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3. 440 IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 2, FEBRUARY 2010
Fig. 2. 𝑃 𝑓 𝑎 of the multiple shift versus the number of shift (M) for different
users (N). Solid lines = theor. (black for binomial, gray for GQ), dotted lines
= sim.
(a)
(b)
Fig. 3. Performance comparison between the two methods versus different
shift for: a) 𝑁 = 5 users, b) 𝑁 = 8 users. Solid lines = theor. (black for MS,
gray for Two-Dwell), dotted lines = sim. (black for MS, gray for Two-Dwell).
the simulations well match the theoretical results. It has to be
noted that, the number of curves for the two-dwell approach
is limited to analyze an integer number of 𝑁 𝑐 chips in the first
mode. It has to be underlined that the OOC is a sparse code.
In other word the density of ON pulses is low. So when we
search on only 𝐹/𝑀 chips in the first step, this is probable
that no ON chip exists in the search area. This phenomenon
makes more sense when code weight is low e.g. 𝑊 = 5 as
considered in this work. When 𝑀 increases, this probability
increases and consequently probability of detection decreases.
On the other hand, the probability of false alarm increases
too because, if the algorithm works on the intensity of one
single impulse, there are different noise sources that cannot
be avoided and must be now considered (e.g. shot noise and
Fig. 4. MAT performances of the system versus different values of the
parameter 𝑀 and different numbers (N) of users, (circles for MS scheme,
diamonds for proposed approach).
dark currents). This means that the algorithm always enters
the second dwell for a verification task and, at the end of the
second mode, we have always a true detection. In this case,
the first dwell is performed with the lowest computational
complexity (i.e. it analyzes only one chip) and hence the
system performances (in terms of mean acquisition time) are
the same of the MS algorithm. However in PN sequences
which are used in wireless systems we do not have this effect
since the codes are bipolar. Also in the multiple-shift algorithm
the choice of the parameter 𝑀 could have two different effects
on the performance of the synchronization system [2]. As
𝑀 increases the number of shifts examined simultaneously
increases and, therefore, the search space will be covered in
fewer tries. On the other hand, with the increase of 𝑀, the
number of dwell times required to find the correct shift among
the shifts in the second stage will increase. Another effect of
increasing 𝑀 is that the probability of false alarm of the first
stage will increase since with the use of a larger value for
parameter 𝑀, more interference is introduced in the checking
process.
It is, therefore, expected that an optimum value for 𝑀 exists
for which the performance of the synchronization system is
optimum, or equivalently the synchronization time is min-
imum, as represented in the following Fig. 4. Considering
that the most important performance measure is the mean
acquisition time (MAT) for acquisition process, Fig. 4 shows
here the performance of the two algorithms (the MS and the
double dwell) in terms of needed number of bits for acquisition
purposes (MAT) varying the system performance such as 𝑀
and 𝑁. The curves are obtained with a detection probability
equal to one and for a different number of users (from 2 to 8).
From the graph, we can easily see that the curves referring
to both the algorithms show the same behavior, confirming
that the two acquisition techniques have the same system
performances. Moreover, it has to be noted that increasing
the value of 𝑀, for a fixed detection probability, makes the
two algorithms converge to the same performances (i.e. same
MAT), confirming the assumptions we made before.
It is not be surprisingly that the two acquisition techniques
have the same system performances. This is a consequence
of the fact that both the proposed and MS scheme perform
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4. BENEDETTO and GIUNTA: ON EFFICIENT CODE ACQUISITION OF OPTICAL ORTHOGONAL CODES IN OPTICAL CDMA SYSTEMS 441
the same operations (correlation and threshold comparison)
on the same codes but using a different amount of chips. In
fact, while the MS algorithm performs 𝑀 correlations of 𝐹
chips (parallel search), the proposed scheme realizes only one
correlation on 𝐹/𝑀 chips (serial search), in the first dwell.
Then, in the second dwell while the MS searches the correct
shift analyzing the 𝑀 correlations of 𝐹chips one after the
other, the proposed approach realizes the correlation only on
the remaining (𝐹 − 𝐹/𝑀) chips. The advantages of using
the two-dwell approach lie in the fact that it presents a lower
complexity than the MS algorithm. In fact, we can define two
(lower and upper) bounds of the acquisition performance: best
case if the correct shift is the first on 𝐹 chips; 𝑤𝑜𝑟𝑠𝑡 case if
the correct shift is the last chip on 𝐹 chips. We can evaluate
the total number of needed operations as follows:
- Best case: in this case, the correct shift is represented
by the first chip of the first code (on 𝑀) in the search area.
In particular, the multiple shift algorithm must analyze 𝑀
different shifts at the same time in the first dwell, and then
find the correct shift in the second dwell analyzing each shift
separately. This means that during the first dwell the MS
algorithm must perform 𝑀 correlations of codes made of 𝐹
chips, for a total number of operations (sums and products)
of 2∗
𝑀∗
𝐹, while in the second dwell it will find the correct
chip analyzing only the first shift of the first code of 𝐹 chips,
for a total number of operations (sums and products) of 2∗
𝐹.
Hence the total number of operation performed by the MS
algorithm (in the first and second dwell) for the best case is
2𝐹∗
(𝑀 +1). In the first dwell, the proposed method performs
a total number of operations (sums and products) of only
2∗
𝐹/𝑀, because it performs a serial search on 𝐹/𝑀 chips
of the same code. Then in the second dwell, it performs the
correlation on the remaining chips. Hence the total number of
operations performed by the proposed algorithm (in the first
and second dwell) is 2 𝐹.
- Worst case: in this case, the correct shift is represented
by the last chip of the last code (on 𝑀) in the search area.
In particular, the multiple shift must perform 𝑀 correlations
of codes made of 𝐹 chips, for a total number of operations
(sums and products) of 2∗
𝑀∗
𝐹, while in the second dwell it
will find the correct chip analyzing all the 𝐹 chips of all the
𝑀 shifts (the correct chip is the last one of the 𝑀-th code
of length 𝐹). Hence the total number of operations performed
by the MS algorithm (in the first and second dwell) for the
worst case is 2𝐹∗
(𝐹 + 𝑀). On the other hand, the proposed
method performs a serial search on all the 𝐹/𝑀 chips for 𝑀
codes (2∗
𝐹 operations) and then passes to the second dwell,
performing the correlation on the remaining chips of the last
code only, 2∗
(𝐹 − 𝐹/𝑀) operations. Hence, the total number
of operation performed by the proposed algorithm (in the first
and second dwell) is 2𝐹∗
(2𝑀 − 1)/𝑀.
In conclusion, the number of operations required by the two
acquisition methods, in terms of total sums and products, is
as follows:
- MS: best case = 2∗
𝐹∗
(𝑀 +1) operations; worst case
= 2∗
𝐹∗
(𝑀 + 𝐹) operations.
- Two-dwell: best case = 2∗
𝐹 operations; worst case
= 2∗
𝐹∗
(2∗
𝑀 − 1)/𝑀 operations.
We can conclude that with the new algorithm we obtain the
same systems performances of the MS, but saving a number
of operations 𝐺 𝑏 = (𝑀 + 1) in the best operating case and a
number of operations equal to 𝐺 𝑤 = 𝑀∗
(𝑀 + 𝐹)/(2∗
𝑀 −
1) ≈ (𝑀 + 𝐹)/2 in the worst case, that rapidly increases with
𝑀.
IV. CONCLUSIONS
This paper has devised a two-dwell acquisition scheme
for OCDMA communications based on OOCs. We have
provided a closed form to analytically express the system’s
error probabilities and showed that our procedure can obtain
the same system performance as the MS approach but with
lower computational complexity.
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