International Journal of Engineering Trends and Technology- Volume3Issue2- 2012
and crosscorrelation properties.
Analysis of OOC Code & Their
Performance
Metrics”
Mrs. Anita Borude, Prof. Shobha Krishnan
Department of Electronics & Telecom, Mumbai University
Abstract: - In this paper we analyze optical orthogonal code
(OOC) and their performance metrics. Expansion efficiency
factor and throughputs of OOC are the new performance
metrics of OOC. Expansion efficiency means ability to expand
sub wavelength-switching capacity of the optical network.
This factor show reducing the code length (n), for the same
code constraints(C), enhances the capacity of sub wavelength
optical code switched networks like optical CDMA
(OCDMA). As various different construction methods for
OOC are available, out of these some of the methods we
analyze. Depending on the construction methods length (n),
weight (w) and cardinality (C) of OOC set are calculated. We
measure expansion efficiency for different construction
methods of OOC.
Keywords: OOC performance metrics, Expansion efficiency
factor, compare construction methods of OOC.
I.INTRODUCTION
Optical
code-division
multiple
accesses
(OCDMA) is one of the most important techniques
supporting many simultaneous users in shared media so as
to increase the transmission capacity of an optical fiber.
Before optical CDMA system, time division multiple
accesses (TDMA) or wavelength division multiple accesses
(WDMA) multiplexing schemes were used for fiber optic
communication systems. But both have significant
drawbacks in local area systems requiring large numbers of
users. The concept of using optical code-division multiple
access (OCDMA) as a photonic multiplexing technique
started in the early 1990s [3].CDMA was first applied to
the optical domain in the mid-1980s by Prucnal, Salehi,
and others [3]. The use of OCDMA as a multiplexing
(switching) layer at the subwavelength level in generalized
multiprotocol label switching (GMPLS) networks was
introduced in [3]. In order to use OCDMA, the optical
signal is spread either in time domain (temporal spreading)
using fiber tapped delay lines or in a wavelength domain
(spectral spreading) using arrayed waveguide grating
(AWG) or fiber Bragg grating (FBG). In both cases, a
spreading code is required. Optical orthogonal codes
(OOCs) [5] are widely used as optical spreading codes
because they achieve the optimum criteria for unipolar {0,
1} codes in terms of their autocorrelation
ISSN: 2231-5381
However, efficient OOCs are difficult to construct. Many
researchers have proposed various construction methods
for OOC have been developed, some of them we compare.
In [6] Prime sequences method(PS), Quasi Prime(QP),
Extended
Quadratic
Congruence's(QCM),Difference
set(DS),Cyclic Difference Set method(CDS) are the few
schemes were proposed for construction of OOC.
In section II, we provide mathematical definitions
and representations for OOCs and their main properties
and discuss their construction methods. Section III
discusses their performance metrics in details. Section IV
we evaluate the efficiency of OOCs for different
construction methods of OOCs.
II. OPTICAL ORTHOGONAL CODES
To reduce the crosstalk between users, an
important property of the sequences (codeword) is that they
produce a low correlation. OOC are a set of binary
sequences with special auto- and crosscorrelation
properties. The correlation being low tells us that each
sequence in the code can easily be distinguished from a
shifted version of itself i.e. the autocorrelation is low, and it
can easily be distinguished from any combination of shifted
versions of other sequences in C i.e. the crosscorrelation is
low. The size of the code is the number of codeword in C
and is called its cardinality and is denoted |C|.
An optical orthogonal code is a family of (0, 1)
sequences with good auto- and cross-correlation properties.
Thumbtack-shaped auto-correlation enables the effective
detection of the desired signal and low-profiled crosscorrelation makes it easy to reduce interference due to
other users and channel noise. The use of optical
orthogonal codes enables a large number of asynchronous
users to transmit information efficiently and reliably. The
lack of a network synchronization requirement enhances
the flexibility of the system. In [3] Let (n, w, λa, λc) be
positive integers. An (n, w, λa, λc) optical orthogonal code,
or briefly a (n, w, λa, λc) -OOC, C, is a family of (0, 1) sequences of length n and weight w satisfying the
following two properties:
The Autocorrelation Property:
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International Journal of Engineering Trends and Technology- Volume3Issue2- 2012
∑
= < λa, 1 <
,
<
− 1 …….. (1)
For any x € C and any integer τ, 0 < τ < n.
The cross-correlation property:
= < λc,
1<
,
<
− 1 … . (2)
For x ≠ y € C and any integer τ.
Largest possible cardinality of OOC ø (n, w, λ)
ø(n , w λ) =
n − 1)(n − 2) … … . (n − λ)
(
w w − 1)(w − 2) … … … . (w − λ)
… … … … … … (3)
Often an OOC is denoted by the positions where
C has ones. As an example consider the code C =
{c1, c2}= {110100000000000, 100010000100000},
Which is an (15, 3, 1) OOC with two codeword. This can
be written as C = {{0, 1, 3}15, {0, 4, 9}15}. What the
correlation constraint (¸ = 1) also say is that no distances
between the positions of ones in the code (C) are repeated.
For example the distance between the first and the second
one in c1 is 1, but as we look at two consecutivec1’s
(cyclic shifted) the distance 14 will also be taken into
consideration. In the codes above the distances are {1, 14,
3, 12, 2, 13, 4, 11, 9, 6, 5, 10}. The distances {14, 12, 13,
11, 6, 10} are due to cyclic shifts of the codeword’s (when
we look at two consecutive codeword). We see here that no
distance is repeated and therefore the code is an OOC with
λa, = 1. If λc = 2 it means that any distance between the
positions of ones cannot be repeated more than once.
Table1: Construction methods for OOC and their length (n)& weight (w)
W
n
C
λa
λc
Sr
Construction
no
methods for
OOC
1.
PS
P
P2
P
P-1
2
2.
QP
q
qp
p
(p1)r
2r
3.
QCM
P
P
1
2
4.
DS
w
P(2P1)
w(w1)
+r+1
1
1
5.
CDS
w
(n-1r)/
(w(w1))
(n-1)/
w(w1)
1
1
n=
w(w1)+1
Where in table1 W=weight of OOC; n=length of OOC; P=
prime no.; q &r = positive integers; C= cardinality of OOC;
ISSN: 2231-5381
λa= auto-correlation constraint; λc =cross-correlation
constraint.
A desirable property of a code is that it should be
as large as possible, i.e. contain as many codeword as
possible. This to enable more users to access the channel.
An OOC is said to be optimal if it has the maximum
cardinality for a given (n, w, λa, λc).As mentioned in the
previous section the codeword length n increases with
increasing cardinality and weight.
We have many new techniques to construct OOC family,
few of them we are compared. Prime sequences
method(PS), Quasi Prime(QP), Extended Quadratic
Congruence's(QCM), Difference set(DS),Cyclic Difference
Set method(CDS) are the few methods for construction of
OOC. As above methods are able to produce OOC set,
which are satisfied autocorrelation and cross correlation
properties of OOC.
Table1 shows weight(w) ,length (n) and
cardinality (C) for different methods. As well as
autocorrelation (λa) and cross correlation constrains (λc) for
every methods.
III. EXPANSION EFFICIENCY FACTOR
In[1], expansion efficiency factor is ability to
expand sub wavelength-switching capacity. This show
reducing the code length, for the same code constraints.
Enhances the capacity of sub wavelength optical code
switched networks e.g. OCDMA. For an optical
multiplexing system based on OCDMA at the
subwavelength level [4], it is desired to be able to
multiplex as many users as possible per wavelength
channel without affecting the system error rate under ideal
conditions. It was shown in [4] that the condition for an
OCDMA system to operate as an error-free transmission
system under ideal conditions is to be able to completely
suppress the multiple access interference (MAI) caused by
unwanted users on the intended received signal using an
optical threshold(hard limiter) device at the output of the
correlator receiver. This is possible if and only if
N ≤ k.
In order to maximize the system capacity
(maximize the number of multiplexed users per
wavelength), we take
N = k………………………….. (4)
It was also shown in [4] that the label space
expansion ratio ρ, which measures the increase in the
system capacity per wavelength of a subwavelength
OCDMA switched system as compared to a pure
wavelength-division-multiplexing (WDM) system, is given
by
ρ = N………………………………… (5)
Using (5), for an error-free system, ρ is given by
ρ = k………………………………… (6)
From [1], we get
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International Journal of Engineering Trends and Technology- Volume3Issue2- 2012
N ≤ n − 1/ k (k − 1)…… …………… (7)
as compared to codes designed using the QCM,QPS and
PS methods.
which yields a maximum number of sub wavelength
multiplexed users (maximum system capacity per
wavelength) given by
Expansion effiency factor for ooc
1
PS
QCM
DS
QPS
CDS
0.9
N = n − 1/ k (k − 1) …………………………(8)
N · k (k − 1) = n − 1 − (n −1 mod k (k − 1))…(9)
In order to make the maximum use of the
available system bandwidth, it is desired to have the
smallest n that satisfies (9). Inspecting (9), the minimum
value for n for given values of k and N is achieved when (n
−1 mod k(k − 1)) = 0. This happens when
expansion efficiency factor
This can be rewritten as
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
k(k − 1)|n − 1 …………………….(10)
which reads k(k − 1) divides n − 1. Under this
condition, (10)can be simplified to
0
5
10
15
space expansion factor
20
25
Fig 1: Expansion efficiency factor η for PC, QCM, DC, QPS and CDS
under error-free conditions.
nmin = N · k(k − 1) + 1……………….(11)
Length Vs Expansion effiency factor for ooc
1400
where nmin is the minimum code length. In order to analyze
the optimum error-free system, we substitute for N and k
from (5) and (6), respectively, into (7) to get
1200
1000
length[bit]
ρ =(n – 1)/ρ(ρ − 1)……………………………….(12)
this can be rewritten as,
ρ2(ρ − 1) = n − 1 − (n −1 mod ρ(ρ − 1)) ……….. (13)
Following the same reasoning used to deduce (9), we can
write the minimum error-free code length nmin as
800
600
400
200
0
nmin = ρ2(ρ − 1) + 1…………………………… (14)
In [1] define an expansion efficiency factor η as
the ratio between the minimum code length nmin and the
code length n of the OOC generated using suboptimal
algorithms
η = nmin /n…………………................ (15)
which means that for an optimum code, we have η = 1,
while for a suboptimal code, we have 0 < η < 1. As
different construction methods of OOC are available, some
of these we compare and analyze them by calculating
expansion efficiency factor for each scheme.
Figs. 1 are plotted for the case where the system is
operating at the maximum capacity under the error-free
transmission constraints. Fig. 1 results demonstrate that
codes designed using different construction methods DS
and CDS have an asymptotic efficiency of about 0.1 to 0.15
ISSN: 2231-5381
PS
CDS
DS
QPS
QCM
0
5
10
15
space expansion factor
20
25
Fig 2: Length Vs Expansion efficiency of OOC for PC, QCM, DC, QPS
and CDS construction scheme.
These methods have an asymptotic efficiency less
than0.1. This shows that CDS & DS designed codes are
expected to constantly outperform at asymptotically large
values of code weights (k). Lengths (n) of OOC set are
varying as space expansion factor get increase for every
scheme. We also shown the relation between them for each
scheme. Fig2 shows length (n) of OOC set constructed by
QCM, PS and DS scheme get increase rapidly as space
expansion factor get increase. Whereas OOC set
constructed by CDS & PS for higher space expansion
factor has less length as compare to other scheme.
IV. CONCLUSION
Many researchers have proposed several schemes
to construct OOC sets. Some of these we have discussed
and calculate their performance metrics All schemes
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International Journal of Engineering Trends and Technology- Volume3Issue2- 2012
satisfied the requirements of OOC set, but their
characteristics are different. That means , some of the
scheme produce maximum number of OOC set from
length(n)with weight(w).where as some of the scheme has
very low cross correlation constrains, which reduces the
interference from unwanted signals in the network. OOC
set find applications in optical network, mobile radio,
frequency-hopping spread spectrum communications, and
radar and sonar signal design. Depending on the
requirement of an application for OOC set we can select
appropriate construction scheme of OOC. Let we have
optical network where number of users are less as well as
we have small bandwidth, then we can select CDS scheme
because OOC set constructed by these methods have small
length so it is suitable for small bandwidth network.
Whereas for larger bandwidth optical network with more
number of users can select DS scheme for construction of
OOC set. Because cardinality (C) of DS scheme are more
as compare to other scheme.
References
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Tamer M. S. Khattab, Student Member, IEEE, and Hussein M.
Alnuweiri, Member, IEEE, “Optical Orthogonal Code
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LIGHTWAVE TECHNOLOGY, VOL. 24, NO. 9,
SEPTEMBER 2006.
[2]
Andreas Karlsson.” Simulation of a CDMA system based on
optical orthogonal codes ” Reg nr: LiTH-ISY-EX-3532-2004 .
[3]
J. A. Salehi, “Code Division Multiple-Access
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