Abstract

Lower bounds on mutual information (MI) of long-haul optical fiber systems for hard-decision and soft-decision decoding are studied. Ready-to-use expressions to calculate the MI are presented. Extensive numerical simulations are used to quantify how changes in the optical transmitter, receiver, and channel affect the achievable transmission rates of the system. Special emphasis is put to the use of different quadrature amplitude modulation formats, channel spacings, digital back-propagation schemes and probabilistic shaping. The advantages of using MI over the prevailing Q-factor as a figure of merit of coded optical systems are also highlighted.

© 2015 Optical Society of America

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References

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  1. R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE 100(5), 1035–1055 (2012).
    [Crossref]
  2. ITU, Rec. G.975.1: Forward error correction for high bit-rate DWDM submarine systems (2004).
  3. J. G. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2008), 5th ed.
  4. A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
    [Crossref]
  5. M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” Proc. European Conference on Optical Communication (ECOC), Paper We.8.C.3 (2010).
  6. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
    [Crossref] [PubMed]
  7. I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
    [Crossref]
  8. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
    [Crossref]
  9. B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber-optic communications,” J. Lightw. Technol. 30(13), 2047–2053 (2012).
    [Crossref]
  10. M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
    [Crossref]
  11. M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
    [Crossref]
  12. T. Fehenberger and N. Hanik, “Digital back-propagation of a superchannel: Achievable rates and adaption of the GN model,” Proc. European Conference on Optical Communication (ECOC), Paper We.3.3.6 (2014).
  13. T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).
  14. D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
    [Crossref]
  15. P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
    [Crossref]
  16. U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
    [Crossref]
  17. T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 2006), 2nd ed.
  18. A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).
  19. G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
    [Crossref]

2014 (3)

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

2013 (1)

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
[Crossref]

2012 (2)

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE 100(5), 1035–1055 (2012).
[Crossref]

B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber-optic communications,” J. Lightw. Technol. 30(13), 2047–2053 (2012).
[Crossref]

2011 (1)

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

2010 (1)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

2006 (1)

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

2005 (1)

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

2001 (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]

1999 (1)

U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
[Crossref]

Agrell, E.

M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” Proc. European Conference on Optical Communication (ECOC), Paper We.8.C.3 (2010).

Alvarado, A.

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).

T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).

Arnold, D.

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Bayvel, P.

G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission,” Opt. Express 22(24), 30053–30062 (2014).
[Crossref]

A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).

Böcherer, G.

T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).

Bosco, G.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Carena, A.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Christensen, L.

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Cover, T. M.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 2006), 2nd ed.

Curri, V.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Djordjevic, I. B.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

Essiambre, R.-J.

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE 100(5), 1035–1055 (2012).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

Fehenberger, T.

T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).

T. Fehenberger and N. Hanik, “Digital back-propagation of a superchannel: Achievable rates and adaption of the GN model,” Proc. European Conference on Optical Communication (ECOC), Paper We.3.3.6 (2014).

Fischer, R. F. H.

U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
[Crossref]

Forchhammer, S.

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Forestieri, E.

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
[Crossref]

Forghieri, F.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Foschini, G. J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

Gabitov, I.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

Goebel, B.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

Hanik, N.

T. Fehenberger and N. Hanik, “Digital back-propagation of a superchannel: Achievable rates and adaption of the GN model,” Proc. European Conference on Optical Communication (ECOC), Paper We.3.3.6 (2014).

T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).

Huber, J. B.

U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
[Crossref]

Idler, W.

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

Ives, D. J.

A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).

Ivkovic, M.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

Jiang, Y.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Karlsson, M.

M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” Proc. European Conference on Optical Communication (ECOC), Paper We.8.C.3 (2010).

Kavcic, A.

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Killey, R. I.

Kramer, G.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

Kschischang, F. R.

B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber-optic communications,” J. Lightw. Technol. 30(13), 2047–2053 (2012).
[Crossref]

Larsen, K.

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Leven, A.

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

Liga, G.

Loeliger, H.-A.

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Mitra, P. P.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]

Poggiolini, P.

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

Prati, G.

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
[Crossref]

Proakis, J. G.

J. G. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2008), 5th ed.

Salehi, M.

J. G. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2008), 5th ed.

Savory, S. J.

A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).

Schmalen, L.

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

Secondini, M.

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
[Crossref]

Smith, B. P.

B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber-optic communications,” J. Lightw. Technol. 30(13), 2047–2053 (2012).
[Crossref]

Stark, J. B.

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]

ten Brink, S.

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 2006), 2nd ed.

Tkach, R. W.

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE 100(5), 1035–1055 (2012).
[Crossref]

Vacondio, F.

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

Vasic, B.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

Vontobel, P.

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Wachsmann, U.

U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
[Crossref]

Winzer, P. J.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

Xu, T.

Yankov, M.

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

Zeng, W.

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

Zibar, D.

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

IEEE Photon. Technol. Lett. (2)

A. Leven, F. Vacondio, L. Schmalen, S. ten Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett. 23(20), 1547–1549 (2011).
[Crossref]

M. Yankov, D. Zibar, K. Larsen, L. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett. 26(23), 2407–2410 (2014).
[Crossref]

IEEE Trans. Inf. Theory (2)

D. Arnold, H.-A. Loeliger, P. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 52(8), 3498–3508 (2006).
[Crossref]

U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(8), 1361–1391 (1999).
[Crossref]

J. Lightw. Technol. (5)

P. Poggiolini, G. Bosco, A. Carena, V. Curri, Y. Jiang, and F. Forghieri, “The GN-model of fiber non-linear propagation and its applications,” J. Lightw. Technol. 32(4), 694–721 (2014).
[Crossref]

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates for high-speed long-haul optical transmission,” J. Lightw. Technol. 23(11), 3755–3763 (2005).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(4), 662–701 (2010).
[Crossref]

B. P. Smith and F. R. Kschischang, “A pragmatic coded modulation scheme for high-spectral-efficiency fiber-optic communications,” J. Lightw. Technol. 30(13), 2047–2053 (2012).
[Crossref]

M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic systems with arbitrary modulation formats and dispersion maps,” J. Lightw. Technol. 31(23), 3839–3852 (2013).
[Crossref]

Nature (1)

P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411, 1027–1030 (2001).
[Crossref] [PubMed]

Opt. Express (1)

Proceedings of the IEEE (1)

R.-J. Essiambre and R. W. Tkach, “Capacity trends and limits of optical communication networks,” Proceedings of the IEEE 100(5), 1035–1055 (2012).
[Crossref]

Other (7)

ITU, Rec. G.975.1: Forward error correction for high bit-rate DWDM submarine systems (2004).

J. G. Proakis and M. Salehi, Digital Communications (McGraw-Hill, 2008), 5th ed.

M. Karlsson and E. Agrell, “Four-dimensional optimized constellations for coherent optical transmission systems,” Proc. European Conference on Optical Communication (ECOC), Paper We.8.C.3 (2010).

T. Fehenberger and N. Hanik, “Digital back-propagation of a superchannel: Achievable rates and adaption of the GN model,” Proc. European Conference on Optical Communication (ECOC), Paper We.3.3.6 (2014).

T. Fehenberger, G. Böcherer, A. Alvarado, and N. Hanik, “LDPC coded modulation with probabilistic shaping for optical fiber systems,” Proc. Optical Fiber Conference (OFC), Paper Th2A.23 (2015).

T. M. Cover and J. A. Thomas, Elements of Information Theory (Wiley-Interscience, 2006), 2nd ed.

A. Alvarado, D. J. Ives, S. J. Savory, and P. Bayvel, “On optimal modulation and FEC overhead for future optical networks,” Proc. Optical Fiber Conference (OFC), Paper Th3E.1 (2015).

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Figures (6)

Fig. 1
Fig. 1 Block diagram of a coded communication system with SD (top) and HD (bottom) demapping and binary decoding.
Fig. 2
Fig. 2 Block diagram of the simulated optical system. The dashed box includes all components and subsystems that influence I(X; Y).
Fig. 3
Fig. 3 Achievable rates for soft- and hard-decision decoding: RSD (solid), R HD m (dotted), R HD 1 (dashed) and different modulation formats: QPSK (green, x), 16-QAM (blue, diamond) and 64-QAM (red, triangle). EDC only, 6000 km SMF and Nch=15 WDM channels.
Fig. 4
Fig. 4 Achievable rates RSD (a) and dual-polarization SE (b) for 16-QAM and varying WDM channel spacings Bch. The total bandwidth over all WDM channels is constant at 450 GHz and the signal bandwidth per WDM channel is 29.4 GHz.
Fig. 5
Fig. 5 Achievable rates for soft- and hard-decision decoding: RSD (solid) and R HD 1 (dashed) and different modulation formats: 16-QAM (blue) and 64-QAM (red). The simulation setup is identical to the one in Fig. 3.
Fig. 6
Fig. 6 Shaped 64-QAM outperforms uniform 64-QAM (both with EDC) in rate and distance and gives similar gains as uniform input with SC DBP. The insets a) and b) show the shaped received constellation after 3 and 16 spans, respectively.

Equations (8)

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I ( X ; Y ) = x 𝒳 P X ( x ) p Y | X ( y | x ) log 2 p Y | X ( y | x ) p Y ( y ) d y ,
I ( X ; Y ) R SD x 𝒳 P X ( x ) p Y | X ( y | x ) log 2 q Y | X ( y | x ) q Y ( y ) d y .
q Y | X ( y | x ) = 1 2 π σ 2 exp ( ( y x ) 2 2 σ 2 ) .
q X | Y ( x | y n ) = exp ( ( y n x ) 2 2 σ 2 P X ( x ) ) x 𝒳 exp ( ( y n x ) 2 2 σ 2 ) P X ( x ) .
R SD x 𝒳 P X ( x ) log 2 P X ( x ) + 1 N n = 1 N x 𝒳 q X | T ( x | y n ) log 2 q X | Y ( x | y n ) .
R SD m + 1 N n = 1 N x 𝒳 exp ( ( y n x ) 2 2 σ 2 ) x 𝒳 exp ( ( y n x ) 2 2 σ 2 ) log 2 exp ( ( y n x ) 2 2 σ 2 ) x 𝒳 exp ( ( y n x ) 2 2 σ 2 ) .
R HD m i = 1 m I ( C i ; C ^ i ) = i = 1 m ( 1 H b ( p i ) ) ,
R HD 1 m ( 1 H b ( p ¯ ) ) R HD m ,

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