constellation optimization for coherent optical channels

Constellation Optimization for Coherent Optical Channels

Distorted by Nonlinear Phase Noise

Christian Häger1 Alexandre Graell i Amat1 Alex Alvarado2 Erik Agrell1

1Department of Signals and Systems, Chalmers University of Technology, Gothenburg, Sweden,Fiber Optic Communications Research Center

2Department of Enginering, University of Cambridge, UK

{christian.haeger, alexandre.graell, agrell}@chalmers.se, [email protected]

GLOBECOM, Anaheim, CA – December 4, 2012

Channel and APSK Constellations Detection Methods APSK Optimization Conclusions

Motivation

Constellation Optimization for Nonlinear Phase Noise Channels | Häger, Graell i Amat, Alvarado, Agrell 1 / 13


Motivation

• Higher order modulation formats for optical communications to increasespectral efficiency



Motivation


• Optical channel suffers from distortions that are absent for example inwireless channels



Motivation



• Focus here: Nonlinear phase noise



Motivation




• Practical question: How much can we gain by optimizing the constellationcompared to standard QAM?



Motivation




• Practical question: How much can we gain by optimizing the constellationcompared to standard QAM?

• Theoretical question: How do optimal constellations look like for verystrong nonlinearities?



Discrete-Time Channel Model

X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C

signal constellation, here |X | = 16




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C

noise due to optical amplification




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C

Kerr-effect




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C

• Additive noise per segment Ni ∼ NC(0, σ2/K)




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C


• Distributed Raman amplification, i.e., the limit K → ∞




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C



• Power P = E[

|X |2]

and signal-to(-additive)-noise ratio SNR = P/σ2




X ∈ X

eγ| · |2∆z

N1

eγ| · |2∆z

N2

. . .

eγ| · |2∆z

NK

Y ∈ C



• Power P = E[

|X |2]

and signal-to(-additive)-noise ratio SNR = P/σ2

• L = 5500 km, other parameters γ, σ2 taken from [Lau and Kahn, 2007]



Amplitude Phase-Shift Keying (APSK), Example

quad

ratu

re

in-phase




quad

ratu

re

in-phase

r1

r2

r3




quad

ratu

re

in-phase




( , , )-APSK

quad

ratu

re

in-phase

b

b

b

b




(4, , )-APSK

quad

ratu

re

in-phase

points per ring

b

b

b

b




(4, , )-APSK

quad

ratu

re

in-phase

points per ring

b

b

b

b

b

b

b

b




(4, 4, )-APSK

quad

ratu

re

in-phase

points per ring

b

b

b

b

b

b

b

b




(4, 4, )-APSK

quad

ratu

re

in-phase

points per ring

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

points per ring

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

equispaced in phase

2π

4

2π

4

2π

4

2π

4

b

b

b

b

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

equispaced in phase

b

b

b

b

2π

4

2π

4

2π

4

2π

4

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

equispaced in phase

b

b

b

b

b

b

b

b

b

b

b

b

2π

8

2π

8

2π

8

2π

8

2π

8

2π

8

2π

8

2π

8




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

each ring can have a different phase offset

phase offset 1

b

b

b

b

phase offset 2

b

b

b

b

phase offset 3

b

b

b

b

b

b

b

b




(4, 4, 8)-APSK

quad

ratu

re

in-phase

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b




quad

ratu

re

in-phase




quad

ratu

re

in-phase

another example




quad

ratu

re

in-phase

(4, 4, 4, 4)-APSK




quad

ratu

re

in-phase

(4, 4, 4, 4)-APSK

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b



Probability Density Function of the Observation Y




• Extensive work on the statistics of ΦNL





• See, e.g., [Mecozzi, 1994], [Turitsyn et al., 2003], [Ho, 2005],[Yousefi and Kschischang, 2011]





• See, e.g., [Mecozzi, 1994], [Turitsyn et al., 2003], [Ho, 2005],[Yousefi and Kschischang, 2011]

• Probability density function (PDF) fY |X=xi(y) known



PDF for (4,4,4,4)-APSK at P = −4 dBm




in-phase

quadrature




in-phase

quadrature


ML decision boundaries



in-phase

quadrature


ML detection possible,but not practical.


Nonlinear Phase Postcompensation




radiusdetection

abs(Y )

θc(R, Rx )correction

angle

Rx

Y Y

ejθc

R




radiusdetection

abs(Y )


angle

Rx

Y Y

ejθc

R

• Building block for suboptimal (but practical) two-stage detector




radiusdetection

abs(Y )


angle

Rx

Y Y

ejθc

R


• However, in principle no loss in optimality due to postcompensation itself(ML detection based on Y still possible)




radiusdetection

abs(Y )


angle

Rx

Y Y

ejθc

R



• Gives insight into why two-stage detection is suboptimal




radiusdetection

abs(Y )


angle

Rx

Y Y

ejθc

R



• Gives insight into why two-stage detection is suboptimal

• Note: the PDF of Y is defined piecewise



PDF of Y for (4,4,4,4)-APSK at P = −4 dBm




in-phase

quadrature




in-phase

quadrature


decision radii



in-phase

quadrature


b

transmitted point



in-phase

quadrature


overcompensated

b



in-phase

quadrature


undercompensated

b


ML Detection based on Y




in-phase

quadrature




in-phase

quadrature


ML decision region


Two-Stage Detection

in-phase

quadrature



Two-Stage Detection

in-phase

quadrature


annular sector region


Optimization Opportunities for APSK

Goal

Which APSK constellation minimizes the symbol error probability undertwo-stage detection for a given input power P?




Goal


• How many rings?




Goal


• How many rings? More rings for increasing input power due to phase noise.




Goal



• How many points per ring?




Goal




• What radius distribution?




Goal





• What phase offset?




Goal





• What phase offset? Two-stage detection is insensitive to a phase offset.



Optimizing the Number of Rings and Points per Ring




P [dBm]

SymbolErrorProbability

SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100




P [dBm]


SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


16

-QA

Min

AW

GN



P [dBm]


SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


16

-QA

Min

AW

GN

16-QAM



P [dBm]


SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


16

-QA

Min

AW

GN

16-QAM

(4, 4, 4, 4)-APSK



P [dBm]


SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


16

-QA

Min

AW

GN

16-QAM

(4, 4, 4, 4)-APSK

(1, · · · , 1)-APSK



P [dBm]


(1,7,8)

(1,7,8)(1,6,9)

(1,6,9)

(1,6,9)

(1,6,8,1)

(1,6,6,3)

(1,6,5,3,1)

(1,6,4,3,2)

(1,5,4,2,2,1,1)

(1,5,3,2,2,1,1,1)

(1,5,3,2,1,1,1,1,1)

(4,3,2,1,1,1,1,1,1,1)

(1,4,2,1,1,1,1,1,1,1,1,1)

SNR [dB]

10 15 20 25 30 35

−15 −10 −5 0 5 1010−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100


16

-QA

Min

AW

GN

16-QAM

(4, 4, 4, 4)-APSK

(1, · · · , 1)-APSK


Radius Optimization for (1,6,5,3,1)-APSK

P [dBm]


P [dBm]

RadiusDistribution(norm

alized)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

0

0.5

1

1.5

2

2.5

3

3.5

4

10−4

10−3

10−2

10−1

100




P [dBm]


P [dBm]

RadiusDistribution(absolute)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

10−4

10−3

10−2

10−1

100




P [dBm]


P [dBm]

RadiusDistribution(absolute)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

0

0.05

0.1

0.15

0.2

10−4

10−3

10−2

10−1

100


“sacrificial” ring


Conclusions and Future Work




1. The performance loss of the two-stage detector originates from suboptimaldetection regions, not from the postcompensation itself.





2. Significant gains can be obtained by choosing an optimized APSKconstellations compared to regular 16-QAM for this channel model.






3. High nonlinearities and an average power constraint may lead tocounterintuitive optimization results.







Future work include the effect of dual polarization and dispersion on the results.








Extended version available on Arxiv (submitted to IEEE Trans. Comm.): Häger, Graell i Amat, Alvarado, Agrell -Design of APSK Constellations for Coherent Optical Channels with Nonlinear Phase Noise, Sep. 2012








Extended version available on Arxiv (submitted to IEEE Trans. Comm.): Häger, Graell i Amat, Alvarado, Agrell -Design of APSK Constellations for Coherent Optical Channels with Nonlinear Phase Noise, Sep. 2012

Thank you!


References

Ho, K.-P. (2005).Phase-modulated Optical Communication Systems.Springer.

Lau, A. P. T. and Kahn, J. M. (2007).Signal Design and Detection in Presence of Nonlinear Phase Noise.J. Lightw. Technol., 25(10):3008–3016.

Mecozzi, A. (1994).Limits to long-haul coherent transmission set by the Kerr nonlinearity andnoise of the in-line amplifiers.J. Lightw. Technol., 12(11):1993–2000.

Turitsyn, K. S., Derevyanko, S. A., Yurkevich, I. V., and Turitsyn, S. K.(2003).Information Capacity of Optical Fiber Channels with Zero AverageDispersion.Phys. Rev. Lett., 91(20):203901.

Yousefi, M. I. and Kschischang, F. R. (2011).On the per-sample capacity of nondispersive optical fibers.IEEE Trans. Inf. Theory, 57(11):7522–7541.


Power Dependent Phase Noise



16-QAMb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b



16-QAMb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

−18 −12 −6 0

500

2000

8000

32000

P [dBm]

L [km]



16-QAMb

b

b

b

b

b

b

b

b

b

b

b

b

b

b

b

−18 −12 −6 0

500

2000

8000

32000

P [dBm]

L [km]


SNR is not sufficient tocharacterize the channel!

constellation optimization for coherent optical channels

Documents