A
New
Simple
Method
for
Calculating
the
Bit
ErrorRate
of
OCDMA
Systems
Tamer
Khattab,
Maged
Elkashlan,
and
Hussein
Alnuweiri
Department
of
Electrical
and
Computer
Engineering
University
of
British
Columbia,
Vancouver,
British
Columbia,
Canada
{
tkhattab,magede,hussein}
@ece.ubc.ca
Abstract
In
this
paper;
we
proposea
novel
simplified
mathemati
cal
analysis
technique
for
modeling
and
calculating
the
ef
fect
of
multiple
access
interference
on
bit
error
rate
in
op
tical
code
division
multiple
access
OCDMA)
systems.
Our
technique
applies
to
OCDMA
systems
using
optical
orthog
onal
codes
OOC)
with
optical
timedomain
spreading.
The
proposed
analysisuses
combinatorial
methodson
the
com
bined
signal
at
the
output
of
the
optical
correlator
decoder
to
derive
amathematical
expression
for
the
bit
error
rate.
1
Introduction
An
optical
code
division
multiple
access
OCDMA)
sys
tem
uses
a
set
of
spreading
codes
calledoptical
spreading
codes
OSC)
with
certain
autocorrelation
and
crosscorrela
tion
properties
to
multiplex
several
users
at
the
same
timeon
the
same
wavelength
over
a
fiberoptic
communication
channel.
Code
multiplexing
in
fiberoptic
communications
can
be
performed
using
either
time
domain
spreading
[1]
or
spec
tral
(frequency/wavelength)
domain
spreading
[2].
In
ad
dition
hybrid
schemes
that
use
time
domain
spreading
and
frequency
wavelength)
domain
hoppingwas
also
discussed
in
the
literature[3,4].
In
this
paper
we
focus
on
systems
us
ing
time
domain
spreading
of
optical
signals.
In
timespreading
methods,
light
sources
generate
pulses
that
representthe
transmitteddata
bits.
Using
onoff
keying
OOK)
for
digital
modulation,
a
source
will
produce
a
sin
gle
pulse
to
represent
a
logic
1
bit,
while
it
will
produce
no
pulse
for
the
case
of
logic
0 bit.
The
pulse
duration,
Tc,
is
usually
very
small
compared
to
the
data
bit
duration,Tb.
Replicas
of
the
pulse
with
reduced
power
are
repeated
at
pseudo
randomly
selected
chip
locations
(timeslots)
within
the
bit
duration
causing
a
spread
of
the
pulse
power.
The
receiver
recombines
the
pulse
replicas
allowing
their
pow
ers
to
add
at
a
certain
timeslot
(chip
location).
This
chip
location
is
called
the
detection
chip
location.
A
detection
of
thesrcinal
data
bit
is
then
performed.
This
recombina
tion
can
be
performed
using
eitheractive
devices
such
as
a
light
source
at
thereceiver
that
correlates
only
withtime
slots
corresponding
to
the
pseudo
random
code
or
passive
devices
such
as
an
optical
delay
line
correlator
ODLC).
ODLCs
[5]
are
devices
based
on
optical
fiber
tapped
delay
lines
that
operate
as
optical
correlation
filters.
ODLCs
are
utilized
by
optical
transmitters
and
receivers
to
encode
(spread)
anddecode
(despread)
the
optical
signal.
The
time
delays
in
the
ODLC
devices
are
determined
ac
cording
to
the
design
of
spreading
codes.
New
families
of
spreading
codes
thatare
suitable
for
optical
signals
have
been
investigated
in
the
literature.
The
most
common
two
families
of
optical
spreadingcodes
are:
optical
prime
codes
OPC)
(see
[6]
and
citations
within)
and
optical
orthogonal
codes
OOC)
(see
[7]
and
citations
within).
We
considersystems
using
OOC
for
spreading
of
optical
signals.
This
is
due
to
the
lower
multiple
access
interference
MAI)
levels
introduced
by
these
codes
when
compared
to
OPCs.
Classical
biterror
rate
BER)
calculations
are
derived
us
ing
combinatorial
methods
on
the
combined
signal
at
the
in
putof
the
optical
correlator
decoder.
In
this
paper,
we
pro
pose
a
novel
analytical
method
that
considers
the
combined
signal
at
the
output
of
thecorrelatorreceiver
to
deduce
the
BER.
Ourmethod
provides
analytical
expressions
for
the
BER
ofsystems
that
are
not
easy
to
analyze
using
classical
methods.
2Optical
Orthogonal
Codes
In
OOC,
the
circular2
autocorrelation
withtime
shift
T
0
for
a
codeword
and
thecircular
crosscorrelation
between
anytwo
code
words
are
minimized
(or
limited)
while
the
zeroshift
circular
autocorrelation
for
a
codeword
is
maximized.
Formally,
OOCs
can
be
defined
as
follows:
Definition
1
A
(n,
k,
Aa,
A,)
OOC,
C
with
cardinality
2Circular
here
refers
to
the
fact
that
the
codewords
are
correlated
us
ing
a
circularbit
rotation
instead
of
a
shiftin
one
end
direction
with
zero
insertions
from
the
other
end
1424415217/07/ 25.00
)2007
IEEE
673
ICI
=
N,
is
defined
as
afamilyof
N
{0,]}sequences
of
length
n
and
weight
k
with
circular
autocorrelation
RXX
T)
for
sequence
X
C
C
and
circular
crosscorrelation
Rxy T)
betweenanytwo
sequences
X,
Y
C
C
satisfyingthe
following
properties:
n1
Rxx T)
=
E
Xj
jGT
j=o
{k
<
Aa
T
O
T O
n1
RXY T)
=E
zi
YjeT

AC,
2)
j=O
where
xj
is
element
number
j
in
code
X,
yj
is
element
num
ber
j
in
code
Y
and
e
represents
addition
modulon.
In
this
paper,
we
consider
the
case
a
=
AC
=
1.
We
call
this
a
(n,
k,1)
OOC
and
willrefer
to
it
as
OOC
for
simplic
ity.
This
is
the
case
with
the
lowestmultiuser
interference
in
an
OCDMA
system
based
on
OOCs,
which
enhances
the
system
performance.
To
avoid
trivial
code
constructions,
we
consider
only
the
case
where
N
>
1
and
n
>
k
>
1.
OOCs
(or(n,
k,
1))
codes
possess
the
unique
property
that
there
is
a
maximum
of
a
single
F chip
(pulse)
overlap
between
any
two
different
codewords
belonging
to
the
same
familywith
arbitrarycircular
time
shifts
and
a
maximum
of
a
single
1
chip
overlap
between
a
code
and
a
circular
time
shifted
versionof
itself.
Detailed
properties
and
construc
tion
methods
for
OOCs
are
provided
in[8,9].
3
MAI
Induced
Errors
in
ODLC
Decoders
The
ODLC
baseddecoder
has
the
same
structure
as
the
encoder
with
the
delay
pattern
chosen
so
that
they
can
re
verse
the
encoder
effect
and
restore
thesrcinal
intended
signal
from
a
mix
of
multiplexed
signals
using
different
or
thogonal
codes.
This
is
achieved
using
a
matched
filter
de
sign
for
the
delay
taps
of
the
ODLC
at
the
decoder
side
[10].
The
receiver
has
an
associated
threshold
detector,
which
only
passes
a
signal
if
its
total
power
is
above
a
certain
value.
If
we
assume
that
the
power
per
spread
chippulse
is
Q,
then
the
total
power
of
the
despread
pulse
is
k
x
Q.
The
threshold
device
will
pass
a
signal
only
if
its
total
power
is
greater
than
or
equal
to
h
x
Q,
where
0
<
h
<
k
is
the
power
threshold
factor.
The
use
of
the
threshold
detection
device
ensures
that
the
low
amplitude
pulses
that
are
un
correlated
with
the
decoderdelay
pattern
are
blocked
from
causing
background
noise
to
thecorrelated
pulse.
Assume
that
we
have
an
OOK
based
OCDMA
system
that
contains
N
users
each
is
spreadusing
a
unique
code
word
Ci
selected
from
an
OOC
C.
LetYj
be
the
received
signal
from
user
j
C
{
1,
2,
N}.
If
we
consider
an
ODLC
matched
to
user
i,
then
the
intended
signal
is
Yi
and
the
MAI
signal
Ii
is
given
by:
(3)
li{=
E:
y
j(EII,2....NJ
j:Xi
An
error
at
thereceiver
can
happen
if
thecorrelation
of
the
MAI
signal
Ii
with
an
ODLC
decoder
matched
to
Yi
pro
duces
a
pulse
of
total
power
greater
than
theoptical
thresh
old
power
value
h
x
Q
at
a
timeslot
corresponding
to
a
detected
chip
location
in
Yi
while
a
corresponding
logic
1
is
not
present
in
Yi.
Classically,
calculating
the
MAI
induced
biterror
rate
BER)
of
timeencoded
OOCspread
OCDMA
signals
was
basedon
using
a
combinatorial
model
that
considers
the
dif
ferent
scenarios
that
causes
an
unintended
signal
Yj,
where
j
t
i,
at
the
input
of
the
ODLC
decoder
to
contribute
a
pulse
of
magnitude
Q
at
a
detectedchip
location
at
the
out
put
of
the
ODLC
decoder
[5,
10].
The
collective
contribu
tions
from
different
nonintended
users
produce
a
MAI
sig
nal
that
induces
a
bit
detection
error
when
its
power
exceeds
the
threshold
value
under
the
assumption
that
the
intended
user
is
not
sending
a
logic
1
bit
value.
The
success
of
this
method
to
derive
mathematically
tractable
closed
form
models
for
BER
is
basedon
the
factthat
using
an
optical
timed
gate
at
the
output
of
thereceiver
eliminates
a
large
number
of
detection
error
scenarios.
Indeed,
the
useof
op
tical
timed
gate
is
necessary
for
the
work
of
correlator
re
ceivers
it
performs
the
sampling
part
of
the
sample
andcompare
function
of
correlator
receivers),
but
it
requires
strict
bit
level
synchronization
between
thetransmitter
and
the
receiver.
If
a
simpler
bit
asynchronous
receiver
is
to
be
used,
modeling
the
BER
using
the
classical
method
would
be
difficult
due
to
the
large
number
of
different
possible
sce
narios
produced
by
the
classical
model
in
this
case.
The
same
argument
is
valid
if
we
consider
errors
caused
by
loss
of
synchronization
in
ODLC
based
correlator
receivers.
4
BER
Calculation
Based
on
ODLC
Output
Signal
We
propose
a
new
method
to
calculatethe
BER
of
time
encoded
OOCspread
OCDMA
systems
using
ODLC
en
coders/decoders.
The
new
method
is
a
modificationof
the
combinatorial
method,
described
in
Section
3,
which
con
siders
the
signal
Zji
received
from
user
j
at
the
output
of
the
ODLC
decoder
matched
to
user
i,
where
i,
j
C
{1,
2,
...
N}
(See
Fig.
1).
Before
we
proceed
with
thederivation
of
the
BER,
we
introduce
two
important
theorems.
Theorem
1
If
an
unintendedtimespread
signal
Yj
en
coded
using
an
OOC
Cj
with
code
length
n,
code
weight
k
and
bit
value
of
logic
1
is
input
to
an
ideal
ktaps
ODLC
that
is
matched
to
user
i
t
j,
the
corresponding
signal
Zji
674
4.1
ODLC
Correlator
Receivers
with
Op
tical
Timed
Gate
Figure
1.
ODLC
decoder
output
signals
for
in
tended
and
unintended
input
signals.
at
the
output
of
the
ODLC
decoder
will
have
k2
pulses
with
equal
power
levels.
Theorem
2
If
an
intendedtimespread
signal
Yi
encoded
using
an
OOC
Ci
with
code
length
n,
codeweight
k
and
bit
value
of
logic
1
is
input
to
an
ideal
ktaps
ODLC
that
is
matched
to
user
i,
the
corresponding
signal
Zii
at
the
output
of
the
ODLC
decoder
will
have
k k
1)
pulses
with
equal
power
levels
and
a
single
pulse
with
k
times
the
power
level
of
the
other
k k
1)
pulses.This
pulse
will
occur
at
a
timeslot
corresponding
to
the
location
of
the
last
chip
of
the
bit
duration.
We
call
this
chip
location
the
correlation
peak
location.
Theorems
1
and
2
are
illustratedin
Fig.
1.
Using
the
two
theorems,
we
can
summarize
the
error
scenarios
that
occur
when
detecting
the
chipsof
the
spread
OCDMA
signal
in
the
following
corollary:
Corollary
1
A
chip
detection
error
in
timeencoded
QOC
spread
OOKmodulated
OCDMA
system
based
on
ODLC
encoder/decoder
happens
if
and
only
if
one
of
the
following
occurs:
1.
More
than
h
users
out
of
the
N
users
in
the
system
contribute
a
pulse
at
any
detection
chip
location
otherthan
the
correlation
peak
location.
2.
More
than
h
users
out
of
the
N
unintended
users
in
the
system
contribute
a
pulse
at
the
correlation
peak
lo
cation
and
the
intended
user
doesn
t
contribute
a
pulse
with
power
level
higher
than
the
threshold
value
at
the
same
location.
In
our
analysis,
we
assume
that
the
data
bits
arrive
at
sources
continuously
with
bit
values
drawnfrom
indepen
dent
identical
random
variables
with
uniform
distributions.
A
logic
1
has
a
probability
Pi
and
a
logic
0
has
a
proba
bility
p1.
Foran
ODLC
correlator
decoder
that
employs
optical
timed
gating
and
optical
threshold
detection
with
perfect
bitlevel
synchronization
between
a
receiver
and
its
in
tended
transmitter,
the
detection
chip
location
will
be
thecorrelation
peak
location
for
every
bit
duration.
Accord
ingly,
for
every
n
chips
there
is
only
one
detectionchip
location.
Hence,
using
Corollary
1,
we
only
consider
the
second
part,
as
the
first
part
cannot
happen
in
this
system.Therefore,
the
probability
of
error
Pe
is
the
probability
that
h
or
more
out
of
the
N
1
unintended
users
contribute
a
pulse
at
the
correlation
peak
location
while
the
intended
user
is
sending
a
0 bit.
However,
the
probability
Pr
that
a
user
contributes
a
pulse
at
a
single
chip
location
is
given
by
Pr
=
P1i
n
Hence,
the
probability
of
bit
error
is
given
by
i=h
4)
k2
AN1li
n
(
(5)
where
a( )
= N
1)
The
probability
of
chip
detection
error
in
this
case
is
given
by
Pe
(chip
level)
=P
n
6)
because
for
every
n
chips,
n
1
are
guaranteed
to
be
de
tected
correctly
as
idle,
because
they
are
ignored
by
the
re
ceiver.
Itis
only
the
chip
at
the
correlation
peak
location
that
can
introduce
chip
errors.
The
probability
of
biterror
(or
BER)
Pe
derived
in
(5)
is
exactly
the
same
as
the
expressionderived
for
the
same
system
using
the
classical
input
signal
combinatorial
anal
ysis.
However,
using
our
method,
the
mathematical
derivation
was
more
intuitive
and
straightforward.
4.2
ODLC
CorrelatorReceivers
without
Optical
Timed
Gate
For
an
ODLC
correlator
decoder
that
does
not
employ
optical
timed
gating,
but
still
uses
opticalthreshold
detec
tion
solely
for
detection,
the
detection
chip
location
will
be
every
chip
of
the
intended
transmitter s
bit
duration.
Hence,
usingCorollary
1,
the
probability
of
error
Pe
is
the
proba
bility
that
h
or
more
out
of
the
N
users
contribute
a
pulse
at
675
any
of
the
n
1
chips
of
the
bit
duration
or
h
or
more
out
of
the
N
1
unintended
users
contribute
a
pulse
at
the
last
chip
of
the
intended
transmitter s
bit
duration
while
the
intended
transmitter
is
sending
a
0
bit.
This
probability
of
error
is
calculated
at
the
chip
level.
n
this
case,
bit
level
probability
of
error
does
not
have
a
meaning,
since
bit
boundaries
are
not
known
to
the
receiver.
Higher
level
components
of
the
system
resolve
the
received
chips
into
actual
bits.
Considering
the
first
part
of
thescenario,the
N
1
unin
tended
users
have
similar
pulse
patterns
according
to
Theo
rem
1,
while
the
intended
user
has
a
different
pulse
pattern
according
to
Theorem
2.
The
second
part
of
the
scenario
is
identical
to
the
case
of
optical
timed
gate.
The
probability
of
chip
error
Pe
in
this
case
can
be
expressed
as
Pe
n)(i)
1
)
Pn
I1
i
i
P1
nj
(1
+
(121
11)
Pn)
I
+
(I
1n(
k2
AN1li
k
2
Pi

n
k2A
Nl1i
n
(7)
where
Pn
is
the
probability
that
the
intended
user
is
con
tributing
a
pulse
to
a
chip
location
and
is
given
by
P.
pi
k k
1)
n
The
expression
derived
in
(7)
cannot
be
easily
deduced
using
the
classical
input
signal
combinatorial
method.
This
is
due
to
the
factthat
one
would
have
to
consider
each
sin
gle
chip
shift
of
the
input
signal
and
deduce
the
chip
error
rate
CER)
from
all
these
different
shifts.
It
is
the
use
of
Theorems
1
and
2
that
allowed
for
considering
all
the
dif
ferent
shifts
collectively,
hence,
deducing
an
expression
for
the
CER.
5
Performance
Analysis
In
order
to
verify
the
expressions
derived
in
(5)
and
(7)
a
simulation
model
is
used
to
produce
the
BER/CER
for
the
two
consideredsystems
and
the
results
of
simulation
are
compared
to
numerical
calculations
of
both
equations.
In
our
simulation,
as
well
as
numerical
results,
we
considers
systems
that
utilize
noiseless
and
lossless
components,
we
ignore
any
optical
dispersion
effects,
and
we
assume
the
use
of
symmetrical
optical
splitters
and
couplers.
Accordingly,
errors
in
our
system
are
solely
caused
by
MAI
noise.
More
over,
we
always
choose
an
opticalthreshold
value
for
the
threshold
detector
that
equals
the
usedcode
weight
i.e.,
h
=
k).
This
assumption
guarantees
that
our
system
is
working
at
the
point
of
maximum
MAI
rejection
minimum
MAI
induced
errors)[10].
It
can
be
seen
from
Figures
2
and
3
that
the
simulation
and
theanalytical
results
for
the
two
analyzed
systems
are
very
close.
This
verifies
the
accuracyof
the
deduced
mathe
matical
model
and
the
correctness
ofour
modeling
method.
We
can
also
see
from
the
same
figures
that
increasing
the
code
length
enhances
the
systemperformance.This
is
ex
pected
due
to
the
lower
probability
of
pulse
collisions
when
code
length
increases.
On
the
other
hand,
increasing
the
number
of
users
will
increase
the
MAI
level.
Thus,
in
creases
the
error
probability
in
both
systems.
While
in
OCDMA
systemsusing
optical
timed
gating
at
thereceiver
we
can
definethe
BER,
this
is
not
true
for
sys
temswithout
optical
timed
gating
at
the
receiver.
This
is
because
the
latter
systems
do
not
define
bit
boundaries.
The
only
known
time
boundary
for
these
systems
is
the
chipdu
ration.
Hence,
for
these
systems
we
use
CER
as
aperfor
mance
measure.
To
compare
the
performance
of
systems
with
and
without
optical
timed
gating,
we
plot
the
CER
of
the
former
systems
in
Fig.
4.
It
can
be
seen
from
the
fig
ure
that
using
optical
timed
gating
at
thereceiver
greatly
enhances
theoverall
error
performance
of
the
system.
6
Conclusions
In
this
paper,
we
haveproposed
a
new
method
to
de
rivea
mathematical
expression
for
the
BER
of
OCDMA
systems
using
timeencoding,
OOCbased
spreading
and
ODLC
based
encoders/decoders.
The
proposed
method
is
based
on
two
important
theorems
that
we
have
constructed.
The
twotheorems
explain
the
main
properties
of
OCDMA
signals
at
the
output
of
ODLC
decoder.
Using
these
theo
rems
we
were
able
to
develop
a
simple
method
forderiv
ing
the
BER
basedon
the
OCDMA
signal
at
the
output
of
the
ODLC.
Using
our
newly
developed
method,
we
were
able
to
derivethe
BER
for
OCDMA
systemswith
and
with
out
optical
timed
gating
at
the
decoder
side.
The
numerical
results
of
our
derived
models
were
compared
with
simula
tionresults
to
test
its
validity
and
accuracy.
The
comparison
confirms
the
accuracyof
our
modeling
method.
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Bit
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with
timed
gate
and
thresholddetector
1010
lo,I
10
lo0
10
l
610
10
lo1
o
k
=
3,
n
=
65Simulation
k
=
3,
n
=
65
Analytical
O
k
=
4,
n
=
145
Simulation
k
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4,
n
=
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Analytical
10
Figure
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Figure
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Chip
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for
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with
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and
thresholddetector
677
a