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, alex.alvarado@ieee.org

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

Y ∈ C

X ∈ X

eγ| · |2∆z

Y ∈ C

signal constellation, here |X | = 16

X ∈ X

eγ| · |2∆z

Y ∈ C

noise due to optical amplification

X ∈ X

eγ| · |2∆z

Y ∈ C

Kerr-effect

X ∈ X

eγ| · |2∆z

Y ∈ C

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

X ∈ X

eγ| · |2∆z

Y ∈ C

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

X ∈ X

eγ| · |2∆z

Y ∈ C

• Power P = E[

|X |2]

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

X ∈ X

eγ| · |2∆z

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

in-phase

( , , )-APSK

in-phase

(4, , )-APSK

in-phase

points per ring

(4, , )-APSK

in-phase

points per ring

(4, 4, )-APSK

in-phase

points per ring

(4, 4, )-APSK

in-phase

points per ring

(4, 4, 8)-APSK

in-phase

points per ring

(4, 4, 8)-APSK

in-phase

(4, 4, 8)-APSK

in-phase

equispaced in phase

(4, 4, 8)-APSK

in-phase

equispaced in phase

(4, 4, 8)-APSK

in-phase

equispaced in phase

(4, 4, 8)-APSK

in-phase

(4, 4, 8)-APSK

in-phase

each ring can have a different phase offset

phase offset 1

phase offset 2

phase offset 3

(4, 4, 8)-APSK

in-phase

another example

in-phase

(4, 4, 4, 4)-APSK

in-phase

(4, 4, 4, 4)-APSK

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]

• 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

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

radiusdetection

abs(Y )

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

radiusdetection

abs(Y )

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

radiusdetection

abs(Y )

• Gives insight into why two-stage detection is suboptimal

radiusdetection

abs(Y )

• 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

transmitted point

in-phase

quadrature

overcompensated

in-phase

quadrature

undercompensated

ML Detection based on Y

in-phase

quadrature

in-phase

quadrature

in-phase

quadrature

in-phase

quadrature

ML decision region

Two-Stage Detection

in-phase

quadrature

Two-Stage Detection

in-phase

quadrature

Two-Stage Detection

in-phase

quadrature

Two-Stage Detection

in-phase

quadrature

annular sector region

Optimization Opportunities for APSK

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

• How many rings?

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

• How many points per ring?

• What radius distribution?

• What phase offset?

• 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

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

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

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

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

16-QAM

(4, 4, 4, 4)-APSK

(1, · · · , 1)-APSK

P [dBm]

(1,7,8)

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

(1,6,9)

(1,6,8,1)

(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

16-QAM

(4, 4, 4, 4)-APSK

(1, · · · , 1)-APSK

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

P [dBm]

RadiusDistribution(norm

alized)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

10−4

10−3

10−2

10−1

P [dBm]

RadiusDistribution(norm

alized)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

10−4

10−3

10−2

10−1

P [dBm]

RadiusDistribution(absolute)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

10−4

10−3

10−2

10−1

P [dBm]

RadiusDistribution(absolute)

−14 −12 −10 −8 −6 −4 −2 0 2 4 6

10−4

10−3

10−2

10−1

“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

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

−18 −12 −6 0

P [dBm]

L [km]

16-QAMb

−18 −12 −6 0

P [dBm]

L [km]

SNR is not sufficient tocharacterize the channel!

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