Optimal signal constellation design for ultra-high-speed optical transport in the presence of phase noise
A method to process applicable to coherent optical channels with either linear or nonlinear phase noise includes: splitting a received sequence of data into clusters of points according to a cumulative log-likelihood function from constellation obtained in a previous iteration; generating new constellation points by calculating a center of mass of the clusters of points; repeating until convergence or until a predetermined number of iterations has been reached to determine a signal constellation; and transmitting signals over the coherent optical channels with nonlinear phase noise using the disclosed signal constellation and LDPC-coded modulation concepts.
1. A signal constellation design method to enable transmission over coherent optical channels with nonlinear phase noise, comprising:
with a processor, splitting a received training sequence of data into clusters of points according to a cumulative log-likelihood function from constellation obtained in a previous iteration;
generating new constellation points by calculating a center of mass of the clusters of points;
repeating until convergence or until a predetermined number of iterations has been reached to determine a signal constellation; and
transmitting signals over the coherent optical channels with nonlinear phase noise using the signal constellation such designed to communicate data over an optical fiber and wherein the transmitted signal is coherently detected by a receiver.
2. The method of claim 1 , comprising estimating a log-likelihood function:
l (α)=log E Φ NL {exp[ l (α,Φ NL )]}
where l(α, Φ NL ) is a log-likelihood function for transmitted symbols and E Φ NL is an expectation average over a nonlinear phase Φ NL .
3. The method of claim 1 , comprising calculating log-likelihood after coherent detection and carrier phase estimation (CPE).
4. The method of claim 1 , comprising decoding for b data LDPC coded streams at the same time and iterating extrinsic information between an APP demapper and LDPC decoders.
5. The method of claim 1 , comprising encoding b independent data by an (n,k) LDPC encoder and writing in row-wise fashion into a b×n block interleaver.
6. The method of claim 1 , comprising performing Monte Carlo integration at a receiver.
7. The method of claim 1 , comprising optimum source distribution used in the algorithm is generated by maximizing the channel capacity based on Arimoto-Blahut method.
8. The method of claim 1 , comprising applying a function LL(x j ,y) as a cumulative log-likelihood function defined as
LL
(
x
k
,
y
)
=
1
NS
∑
i
=
1
NS
-
{
x
k
1
-
Re
[
(
y
1
+
y
2
j
)
e
-
j
×
PN
i
]
}
2
+
{
x
k
2
-
Im
[
(
y
1
+
y
2
j
)
e
-
j
×
PN
i
]
}
2
2
δ
2
where NS denotes a number of phase noise samples and a corresponding phase noise sample is denoted as PN i ·x k1 and x k2 denote first and second coordinates of point x k , y 1 and y 2 denote coordinates of the point y.
9. The method of claim 1 , comprising find the likelihood function (LL) of partition P(Â m )={S i ; i=1, . . . , N}, as
LL
m
=
LL
(
{
A
^
m
,
P
(
A
^
m
)
}
)
=
n
-
1
∑
k
=
0
n
-
1
max
y
∈
A
^
m
LL
(
x
k
,
y
)
.
10. A method to design signal constellation for transmission over coherent optical channels with nonlinear phase noise, comprising:
selecting a signal constellation to be used for initialization, wherein M is a size of the constellation;
generating a training sequence from an optimum source distribution as {x j ; j=0, . . . , n−1};
grouping samples from the training sequence into M clusters;
determining membership to the cluster based on LLR of sample point and candidate signal constellation points from a previous iteration;
assigning each sample point to the cluster with the largest LLR;
determining new signal constellation points as center of mass for each cluster; repeating until convergence; and
transmitting signals over the coherent optical channels with nonlinear phase noise using the signal constellation such designed to communicate data over an optical fiber, wherein the transmitted signal is coherently detected by a receiver.
11. The method of claim 10 , comprising given the m-th subset (cluster) with N candidate constellation points, denoted as  m ={y i ; i=1, . . . , N}, finding a likelihood function of partition P( m )={S i ; i=1, . . . , N}, as
LL
m
=
LL
(
{
A
^
m
,
P
(
A
^
m
)
}
)
=
n
-
1
∑
k
=
0
n
-
1
max
y
∈
A
^
m
LL
(
x
k
,
y
)
.
12. The method of claim 10 , comprising generating a function LL(x j ,y) is the cumulative log-likelihood function defined as
LL
(
x
k
,
y
)
=
1
NS
∑
i
=
1
NS
-
{
x
k
1
-
Re
[
(
y
1
+
y
2
j
)
e
-
j
×
PN
i
]
}
2
+
{
x
k
2
-
Im
[
(
y
1
+
y
2
j
)
e
-
j
×
PN
i
]
}
2
2
δ
2
where NS denotes a number of phase noise samples and a corresponding phase noise sample is denoted as PN i ·x k1 and x k2 denote first and second coordinates of point x k , y 1 and y 2 denote coordinates of the point y.
13. The method of claim 1 , comprising finding a maximum of cumulative log-likelihood function as an optimization criterion.
14. The method of claim 1 , comprising the channels with linear and nonlinear phase noise.
15. The method of claim 1 , comprising employing LDPC-coded-modulation using signal constellations from LLR-OSCD.
16. The method of claim 1 , comprising communicating over channel dominated by laser phase noise, imperfect CPE, and nonlinear phase noise introduced by fiber nonlinearities.
17. The method of claim 1 , comprising encoding b independent data by an (n,k) LDPC encoder and writing in row-wise fashion into a b×n block interleaver.
18. The method of claim 1 , comprising performing Monte Carlo integration at a receiver.
19. The method of claim 1 , comprising generating an optimum source by maximizing channel capacity based on Arimoto-Blahut method.