36 Optical Coding Theory with Prime
even though their bit frames need not be so. It is known that thechip-synchronous
assumption provides a performance upper bo u nd, while the m ore realistic chip-
asynchronou s assumption, in which unipo lar co dewords can betransmittedandde-
tected at any time instant, results in more accurate performance [27, 30, 34–37] .
t
intensity
chip synchronous
1
2
3
1234567891011
t
intensity
chip asynchronous
1
2
3
1234567891011
one code periodone code period
FIGURE 1.7 Cross-correlation functions between two 1-D codewords, 10110010000 and
10011000100, under the chip-synchronous and chip-asynchronous assumptions [21, 35, 37].
Figure 1.7 illustrates an example of the periodic cross-correlation functions be-
tween two 1-D codewords, 10110010000 and 10011000100, for both assumptions.
The chip-synchronous cross-correlation function is discrete, and its values within one
chip interval are found to be {2,2,2,1, 1,2,1, 0,3,2,0},consecutively.However,the
chip-asynchron o us cross-correlation fun ction is not discrete but is a (linear) function
of time because its values now depend on the amount of pulse overlap due to the rel-
ative shifts of these two codewords, as illustrated in Figure1.7.Asaresult,thechip-
asynchronou s cross-correlation function involves p ar tialoverlapofpulses(ornoth-
ing) from two consecutive chips intervals. On average, the cross-correlation values
within one chip interval are found to be {2 , 2,3/2, 1,3/2, 3/2,1/2,3/2,5/2,1, 1 },
consecutively, in this example.
As defined earlier, q
i
is the (hit) probability of having a periodic cross-correlation
value of i ∈ [0,
λ
c
] in a chip interval under the chip-synchronous assumption, where
λ
c
is the maximum cross-correlation function of the unipolar codes in use. How-
ever, under the chip-asynchronous assumption, two consecutive chip intervals must
be considered, and q
i, j
is here defined as the (hit) probability of having the (chip-
synchronous) cross-correlation value in the preceding chipintervalequaltoi ∈[0,
λ
c
]
and the (chip- syn ch ronous) cr oss-correlation value in the present chip interval equal
to j ∈ [0,
λ
c
].Thehitprobabilitiesarerelatedby[21,35,37]
q
i
=
1
2
λ
c
∑
j=0
(q
i, j
+ q
j,i
)
λ
c
∑
i=0
λ
c
∑
j=0
q
i, j
= 1
λ
c
∑
i=0
q
i
= 1
Fundamental Materials and Tools 37
In general, q
i, j
and q
j,i
are different when
λ
c
> 1, but equal for the case of
λ
c
= 1,
for i &= j.
Each interferer (or interfering codeword) m ay contribute upto
λ
c
hits toward the
cross-correlation function at any time instance. For a given K simultaneous users, the
total number of interferers is given by K −1 =
∑
λ
c
i=0
∑
λ
c
j=0
l
i, j
and the total number of
hits seen by the receiver in the sampling time is given by
∑
λ
c
i=0
∑
λ
c
j=0
(i + j)l
i, j
,where
l
i, j
is the number of interfering codewords contributing i hits in the preceding chip
interval and j hits in the present chip interval. The conditional probability of having
Z =
∑
λ
c
i=0
∑
λ
c
j=0
(i + j)l
i, j
hits contributed by these interferers follows a multinomial
distribution and is given by
Pr(Z hits | K users, receiver receives bit 0)=
(K −1 )!
∏
λ
c
i=0
∏
λ
c
j=0
l
i, j
!
λ
c
∏
i=0
λ
c
∏
j=0
q
l
i, j
i, j
As illustrated in Figure 1.7, the cross-correlation value inachipintervalisalinear
function of time. Let X
i, j ,k
be a continuous random variable representing the cross-
correlation value in a chip interval caused by codeword k,whichbelongstooneof
the l
i, j
interfering codewords. So, X
i, j ,k
is uniformly distributed over the interval [i, j)
when 0 ≤i ≤ j ≤
λ
c
,ortheinterval[ j,i) when 0 ≤ j ≤i ≤
λ
c
.Theexpectedvalue(or
mean) and variance of X
i, j ,k
are then given by E[X
i, j ,k
]=
7
j
i
[x/( j −i)]dx =(i + j)/ 2
and Var[X
i, j ,k
]=E[X
2
i, j ,k
] −E[X
i, j ,k
]
2
=
7
j
i
[x
2
/( j −i)]dx−(i + j)
2
/4 =(i − j)
2
/12,
respectively. Considering all k ∈[1,l
i, j
] cases, the overall mean and variance are then
given, respectively, by
m = E
1
λ
c
∑
i=0
λ
c
∑
j=0
l
i, j
∑
k=1
X
i, j ,k
2
=
λ
c
∑
i=0
λ
c
∑
j=0
i + j
2
l
i, j
σ
2
= Var
1
λ
c
∑
i=0
λ
c
∑
j=0
l
i, j
∑
k=1
X
i, j ,k
2
=
λ
c
∑
i=0
λ
c
∑
j=0
(i − j)
2
12
l
i, j
An err or occurs when the total number of hits seen by the receiver in the sampling
time is as high as the decision threshold Z
th
.AccordingtotheCentralLimitTheorem,
the conditional error probability is then given by
Pr(Z ≥ Z
th
| Z hits)=Pr
;
λ
c
∑
i=0
λ
c
∑
j=0
l
i, j
∑
k=1
X
i, j ,k
≥ Z
th
<
= Q
B
Z
th
−m
√
σ
2
C
where Q(x)=(1/
√
2
π
)
7
∞
x
exp
-
−y
2
/2
.
dy is the complementary error function.
Finally, the error probability of unipolar codes with the maximum cross-
correlation function of
λ
c
in a soft-limiting receiver in OOK under the chip-
38 Optical Coding Theory with Prime
asynchronou s assumption is formulated as [35, 37]
P
e,asyn
=
1
2
Pr(Z ≥ Z
th
| K simultaneous users, receiver receives bit 0)
=
1
2
−
1
2
Z
th
−1
∑
∑
λ
c
i=0
∑
λ
c
j=0
(i+ j)l
i, j
=0
(K −1 )!
∏
λ
c
i=0
∏
λ
c
j=0
l
i, j
!
λ
c
∏
i=0
λ
c
∏
j=0
q
l
i, j
i, j
'
1 −Q
B
Z
th
−m
√
σ
2
C(
=
1
2
−
1
2
Z
th
−1
∑
l
0,1
=0
Z
th
−1−l
0,1
∑
l
1,0
=0
+(Z
th
−1−l
0,1
−l
1,0
)/2,
∑
l
1,1
=0
···
+(Z
th
−1−
∑
λ
c
i=0
∑
λ
c
j=0
(i+ j)l
i, j
)/
λ
c
,
∑
l
λ
c
,
λ
c
=0
(K −1 )!
∏
λ
c
i=0
∏
λ
c
j=0
l
i, j
!
λ
c
∏
i=0
λ
c
∏
j=0
q
l
i, j
i, j
×
1 −Q
Z
th
−
1
2
∑
λ
c
i=0
∑
λ
c
j=0
(i + j)l
i, j
0
1
12
∑
λ
c
i=0
∑
λ
c
j=0
(i − j)
2
l
i, j
(1.4)
where the factor 1/2isduetoOOKwithequalprobabilityoftransmittingdatabit1s
and 0s, K −1 =
∑
λ
c
i=0
∑
λ
c
j=0
l
i, j
,and+·, is the floor function.
For examples, the chip-asynchronous, soft-limiting error probability of unipolar
codes with
λ
c
= 1inOOKisgivenby
P
e,asyn
=
1
2
−
1
2
Z
th
−1
∑
l
0,1
=0
Z
th
−1−l
0,1
∑
l
1,0
=0
Z
th
−1−l
0,1
−l
1,0
∑
l
1,1
=0
(K −1 )!
l
0,0
!l
0,1
!l
1,0
!l
1,1
!
q
l
0,0
0,0
q
l
0,1
0,1
q
l
1,0
1,0
q
l
1,1
1,1
×
1
1 −Q
;
Z
th
−(l
0,1
+ l
1,0
+ 21
1,1
)/2
D
(l
0,1
+ l
1,0
)/12
<2
(1.5)
where l
0,0
+ l
0,1
+ l
1,0
+ l
1,1
= K −1, q
0,0
+ q
0,1
+ q
1,0
+ q
1,1
= 1, and Z
th
is usually
set to w for optimal decision.
Figure 1.8 plots the soft-limiting error probabilities [from P
e
in Equation (1.2)
and P
e,asyn
in Equation (1.5)] of the 2-D carrier-hopping prime codes in Section 5.1
against the number o f simultaneous users K under the chip-synchronous and chip-
asynchronou s assumptions, respectively. In this example, the carrier-hopping prime
codes have
λ
c
= 1, L = {7,11}wavelengths, length N = w
2
= {49,121},andweight
w = {7 , 11}.FromSection5.1,thehitprobabilitiesaregivenbyq
1
= w
2
/(2LN),
q
1,1
= w(w −1)/[2N(N −1)], q
1,0
= q
0,1
= q
1
−q
1,1
,andq
0,0
= 1 −q
1
−q
0,1
.In
general, the error probabilities of both assumptions improve as w or N increases be-
cause of higher autocorrelation peaks o r lower hit probabilities. However, their error
probabilities get worse as K increases because of stronger mutual interference. As
expected, the performance of the chip-asynchronous case is always better than the
chip-synchron o us case because the latter gives the perfo r mance upper boun d. The
difference in their error probabilities increases with w or N and is fo und to be about
two to three orders of magnitude in this example. To validate the accuracy of the
Fundamental Materials and Tools 39
0 10 20 30 40 50 60 70 80 90 100
10
−9
10
−8
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Error probability P
e,soft
Number of simultaneous users K
(w,N ) = (7,49)
(11,121)
Simulation
Chip synchronous
Chip asynchronous
FIGURE 1.8 Soft-limiting error probabilities of the 2-D carrier-hopping prime codes under
the chip-synchronous and chip-asynchronous assumptions for code length N = {49,121} and
weight w = {7,11}.
chip-asynchron o us analysis, the results of compu ter simulation are also plotted in
the figure and found closely matching the chip-asynchronous curves. The computer
simulation is performed by cross-correlating codewords randomly selected from the
code set. Th e number of correlating codewords is determined by the number of si-
multaneous users selected for the simulation. The transmission starting time of each
codeword is chosen from a random decimal number in the range of [0,1),corre-
sponding to a fraction of one chip interval, for emulating chip asynchronism. The
total number of data bits involved in each simulation should be at least one hundred
times the reciprocal of the targeting error probability in order to obtain sufficient
iterations.
Theorem 1.8
The chip-asynchronous mean and variance in P
e,asyn
of Equation (1.4) can also be
40 Optical Coding Theory with Prime
formulated in terms of hit probabilities as [21, 35, 37]:
m
asyn
=
λ
c
∑
i=0
iq
i,i
+
λ
c
∑
i=0
i−1
∑
j=0
B
i + j
2
C
(q
i, j
+ q
j,i
)=
w
2
2N
σ
2
asyn
=
λ
c
∑
i=0
i
2
q
i,i
+
λ
c
∑
i=0
i−1
∑
j=0
'
(i − j) j +
(i − j)
2
3
+ j
2
(
(q
i, j
+ q
j,i
) −
B
w
2
2N
C
2
The chip-synchronous mean and variance in Section 1.8.1 can be formulated as
m
syn
=
λ
c
∑
i=0
iq
i
=
w
2
2N
σ
2
syn
=
λ
c
∑
i=0
i
2
q
i
−
B
w
2
2N
C
2
The difference between these two variances is derived as
σ
2
syn
−
σ
2
asyn
=
λ
c
∑
i=0
i−1
∑
j=0
(i − j)
2
6
(q
i, j
+ q
j,i
)
which indicates th at the chip-asynchronous cross-correlation function always gen-
erates less interfer en ce than the chip-synchronous one. This characteristic can be
translated into better code performance in the chip-asynchronous case and verifiable
by applying the variances to the Gaussian-approximated error probability of Equa-
tion (1.1).
For the special case of
λ
c
= 1andq
1
≈ q
1,1
,thesetwovariancescanbefurther
related by
σ
2
asyn
≈
2
3
σ
2
syn
Proof As illustrated in Figure 1.7, the chip -asynchronous p eriodic cross-correlation
value is a function of the relative shift
τ
between two correlation codewords. The
shift consists of integral and fractional parts as
τ
=
τ
c
+
τ
f
.While
τ
c
is an integer
multiple of a whole chip interval,
τ
f
is a fraction of a whole chip interval and a
random variable with the probability density function uniformly distributed in the
open interval (0,1).So,thechip-asynchronousperiodiccross-correlationfunction
I
r,asyn
in the r th chip can be written as [21]
I
r,asyn
= I
r
+
τ
f
(I
r⊕1
−I
r
)
where I
r
is the chip-synchronous periodic cross-correlation value at the r th chip for
r ∈[0,N −1] and “⊕”denotesamodulo-N addition. When I
r,asyn
is plotted against the
amount of time shift
τ
,theshapeofthechip-asynchronousperiodiccross-correlation
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