Abstract

A comprehensive study of the coded performance of long-haul spectrally-efficient WDM optical fiber transmission systems with different coded modulation decoding structures is presented. Achievable information rates are derived for three different square quadrature-amplitude modulation (QAM) formats and the optimal format is identified as a function of distance and specific decoder implementation. The four cases analyzed combine hard-decision (HD) or soft-decision (SD) decoding together with either a bit-wise or a symbol-wise demapper, the last two suitable for binary and nonbinary codes, respectively. The information rates achievable for each scheme are calculated based on the mismatched decoder principle. These quantities represent true indicators of the coded performance of the system for specific decoder implementations and when the modulation format and its input distribution are fixed. In combination with the structure of the decoder, two different receiver-side equalization strategies are also analyzed: electronic dispersion compensation and digital backpropagation. We show that, somewhat unexpectedly, schemes based on nonbinary HD codes can achieve information rates comparable to SD decoders and that, when SD is used, switching from a symbol-wise to a bit-wise decoder results in a negligible penalty. Conversely, from an information-theoretic standpoint, HD binary decoders are shown to be unsuitable for spectrally-efficient, long-haul systems.

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2016 (6)

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Corrections to ‘replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 34, no. 2,  2016.

R. Maher, A. Alvarado, D. Lavery, and P. Bayvel, “Increasing the information rates of optical communications via coded modulation: A study of transceiver performance,” Sci. Rep., vol. 6, pp. 1–10,  2016.

T. A. Eriksson, T. Fehenberger, P. A. Andrekson, M. Karlsson, N. Hanik, and E. Agrell, “Impact of 4D channel distribution on the achievable rates in coherent optical communication experiments,” J. Lightw. Technol., vol. 34, no. 9, pp. 2256–2266,  2016.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609,  2016.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photon. Technol. Lett., vol. 28, no. 7, pp. 786–789,  2016.

E. Agrell, A. Alvarado, and F. R. Kschischang, “Implications of information theory in optical fibre communications,” Philosophical. Trans. R. Soc. A, vol. 374, no. 2062, pp. 1–18, Jan. 2016.

2015 (3)

T. Fehenberger, A. Alvarado, P. Bayvel, and N. Hanik, “On achievable rates for long-haul fiber-optic communications,” Opt. Express, vol. 23, no. 7, pp. 9183–9191,  2015.

A. Alvarado and E. Agrell, “Four-dimensional coded modulation with bit-wise decoders for future optical communications,” J. Lightw. Technol., vol. 33, no. 10, pp. 1993–2003,  2015.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 33, no. 20, pp. 4338–4352,  2015.

2014 (6)

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun., vol. 62, no. 11, pp. 3956–3968,  2014.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett., vol. 39, no. 2, pp. 398–401,  2014.

D. Marsella, M. Secondini, and E. Forestieri, “Maximum likelihood sequence detection for mitigating nonlinear effects,” J. Lightw. Technol., vol. 32, no. 5, pp. 908–916,  2014.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

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., vol. 32, no. 4, pp. 694–721,  2014.

M. P. Yankov, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett., vol. 26, no. 23, pp. 2407–2410,  2014.

2013 (2)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express, vol. 21, no. 22, pp. 25 685–25 699,  2013.

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., vol. 31, no. 23, pp. 3839–3852,  2013.

2012 (3)

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol., vol. 30, no. 10, pp. 1524–1539,  2012.

L. Schmalen, F. Buchali, A. Leven, and S. ten Brink, “A generic tool for assessing the soft-FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 24, no. 1, pp. 40–42,  2012.

B. P. Smith, A. Farhood, A. Hunt, F. R. Kschischang, and J. Lodge, “Staircase codes: FEC for 100 Gb/s OTN,” J. Lightw. Technol., vol. 30, no. 1, pp. 110–117,  2012.

2011 (2)

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 23, no. 20, pp. 1547–1549,  2011.

G. Colavolpe, T. Foggi, A. Modenini, and A. Piemontese, “Faster-than-Nyquist and beyond: how to improve spectral efficiency by accepting interference,” Opt. Express, vol. 19, no. 27, pp. 26 600–26 609,  2011.

2010 (2)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol., vol. 28, no. 4, pp. 662–701,  2010.

B. P. Smith and F. R. Kschischang, “Future prospects for FEC in fiber-optic communications,” IEEE J. Quantum Electron., vol. 16, no. 5, pp. 1245–1257,  2010.

2009 (1)

I. B. Djordjevic, M. Arabaci, and L. L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightw. Technol., vol. 27, no. 16, pp. 3518–3530,  2009.

2007 (1)

I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, “Using LDPC-coded modulation and coherent detection for ultra highspeed optical transmission,” J. Lightw. Technol., vol. 25, no. 11, pp. 3619–3625,  2007.

2006 (1)

D. M. Arnold, H. A. Loeliger, P. O. Vontobel, A. Kavčić, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3498–3508,  2006.

2004 (1)

E. Agrell, J. Lassing, E. G. Ström, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182,  2004.

1998 (2)

G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946,  1998.

R. E. Caflisch, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numerica, vol. 7, pp. 1–49,  1998.

1994 (2)

S. Verdú and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1147–1157,  1994.

N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, “On information rates for mismatched decoders,” IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1953–1967,  1994.

1993 (1)

F. R. Kschischang and S. Pasupathy, “Optimal non uniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 913–929,  1993.

1984 (1)

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp. 632–647,  1984.

1967 (1)

J. Wolfowitz, “Memory increases capacity,” Inform. Control., vol. 11, no. 4, pp. 423–428,  1967.

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, 623–656,  1948.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 3rd ed. San Diego, CA, USA: Academic, 2001.

Agrell, E.

E. Agrell, A. Alvarado, and F. R. Kschischang, “Implications of information theory in optical fibre communications,” Philosophical. Trans. R. Soc. A, vol. 374, no. 2062, pp. 1–18, Jan. 2016.

T. A. Eriksson, T. Fehenberger, P. A. Andrekson, M. Karlsson, N. Hanik, and E. Agrell, “Impact of 4D channel distribution on the achievable rates in coherent optical communication experiments,” J. Lightw. Technol., vol. 34, no. 9, pp. 2256–2266,  2016.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Corrections to ‘replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 34, no. 2,  2016.

A. Alvarado and E. Agrell, “Four-dimensional coded modulation with bit-wise decoders for future optical communications,” J. Lightw. Technol., vol. 33, no. 10, pp. 1993–2003,  2015.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 33, no. 20, pp. 4338–4352,  2015.

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun., vol. 62, no. 11, pp. 3956–3968,  2014.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

E. Agrell, J. Lassing, E. G. Ström, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182,  2004.

G. Liga, A. Alvarado, E. Agrell, M. Secondini, R. I. Killey, and P. Bayvel, “Optimum detection in presence of nonlinear distortions with memory,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P4.13.

C. Häger, H. D. Pfister, A. G. i Amat, F. Brännström, and E. Agrell, “A deterministic construction and density evolution analysis for generalized product codes,” in Proc. Int. Zurich Seminar Commun. (IZS), 2016.

Alvarado, A.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Corrections to ‘replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 34, no. 2,  2016.

E. Agrell, A. Alvarado, and F. R. Kschischang, “Implications of information theory in optical fibre communications,” Philosophical. Trans. R. Soc. A, vol. 374, no. 2062, pp. 1–18, Jan. 2016.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photon. Technol. Lett., vol. 28, no. 7, pp. 786–789,  2016.

R. Maher, A. Alvarado, D. Lavery, and P. Bayvel, “Increasing the information rates of optical communications via coded modulation: A study of transceiver performance,” Sci. Rep., vol. 6, pp. 1–10,  2016.

T. Fehenberger, A. Alvarado, P. Bayvel, and N. Hanik, “On achievable rates for long-haul fiber-optic communications,” Opt. Express, vol. 23, no. 7, pp. 9183–9191,  2015.

A. Alvarado and E. Agrell, “Four-dimensional coded modulation with bit-wise decoders for future optical communications,” J. Lightw. Technol., vol. 33, no. 10, pp. 1993–2003,  2015.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 33, no. 20, pp. 4338–4352,  2015.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

A. Martinez, L. Peng, A. Alvarado, and A. Guillén i Fàbregas, “Improved information rates for bit-interleaved coded modulation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT), 2015, pp. 3004–3007.

L. Schmalen, A. Alvarado, and R. Rios-Müller, “Predicting the performance of nonbinary forward error correction in optical transmission experiments,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.2.

L. Szczecinski and A. Alvarado, Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design. New York, NY, USA: Wiley, 2015.

G. Liga, A. Alvarado, E. Agrell, M. Secondini, R. I. Killey, and P. Bayvel, “Optimum detection in presence of nonlinear distortions with memory,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P4.13.

Amat, A. G. i

C. Häger, H. D. Pfister, A. G. i Amat, F. Brännström, and E. Agrell, “A deterministic construction and density evolution analysis for generalized product codes,” in Proc. Int. Zurich Seminar Commun. (IZS), 2016.

Andrekson, P. A.

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R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett., vol. 39, no. 2, pp. 398–401,  2014.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express, vol. 21, no. 22, pp. 25 685–25 699,  2013.

Merhav, N.

N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, “On information rates for mismatched decoders,” IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1953–1967,  1994.

Millar, D.

T. Koike-Akino, K. Kojima, D. Millar, K. Parsons, T. Yoshida, and T. Sugihara, “GMI-Maximizing constellation design with Grassmann projection for parametric shaping Pareto-efficient set of modulation and coding based on RGMI in nonlinear fiber transmissions,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.4.

Minkov, L. L.

I. B. Djordjevic, M. Arabaci, and L. L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightw. Technol., vol. 27, no. 16, pp. 3518–3530,  2009.

Modenini, A.

G. Colavolpe, T. Foggi, A. Modenini, and A. Piemontese, “Faster-than-Nyquist and beyond: how to improve spectral efficiency by accepting interference,” Opt. Express, vol. 19, no. 27, pp. 26 600–26 609,  2011.

Ottosson, T.

E. Agrell, J. Lassing, E. G. Ström, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182,  2004.

Parsons, K.

T. Koike-Akino, K. Kojima, D. Millar, K. Parsons, T. Yoshida, and T. Sugihara, “GMI-Maximizing constellation design with Grassmann projection for parametric shaping Pareto-efficient set of modulation and coding based on RGMI in nonlinear fiber transmissions,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.4.

Pasupathy, S.

F. R. Kschischang and S. Pasupathy, “Optimal non uniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 913–929,  1993.

Peng, L.

A. Martinez, L. Peng, A. Alvarado, and A. Guillén i Fàbregas, “Improved information rates for bit-interleaved coded modulation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT), 2015, pp. 3004–3007.

Pfister, H. D.

C. Häger, H. D. Pfister, A. G. i Amat, F. Brännström, and E. Agrell, “A deterministic construction and density evolution analysis for generalized product codes,” in Proc. Int. Zurich Seminar Commun. (IZS), 2016.

Piemontese, A.

G. Colavolpe, T. Foggi, A. Modenini, and A. Piemontese, “Faster-than-Nyquist and beyond: how to improve spectral efficiency by accepting interference,” Opt. Express, vol. 19, no. 27, pp. 26 600–26 609,  2011.

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., vol. 32, no. 4, pp. 694–721,  2014.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol., vol. 30, no. 10, pp. 1524–1539,  2012.

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., vol. 31, no. 23, pp. 3839–3852,  2013.

Qureshi, S. U.

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp. 632–647,  1984.

Renaudier, J.

R. Rios-Müller, J. Renaudier, L. Schmalen, and G. Charlet, “Joint coding rate and modulation format optimization for 8QAM constellations using BICM mutual information,” in Proc. Opt. Fiber Commun. Conf., 2015, Paper W3K.4.

Rios-Müller, R.

R. Rios-Müller, J. Renaudier, L. Schmalen, and G. Charlet, “Joint coding rate and modulation format optimization for 8QAM constellations using BICM mutual information,” in Proc. Opt. Fiber Commun. Conf., 2015, Paper W3K.4.

L. Schmalen, A. Alvarado, and R. Rios-Müller, “Predicting the performance of nonbinary forward error correction in optical transmission experiments,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.2.

Schmalen, L.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609,  2016.

L. Schmalen, F. Buchali, A. Leven, and S. ten Brink, “A generic tool for assessing the soft-FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 24, no. 1, pp. 40–42,  2012.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 23, no. 20, pp. 1547–1549,  2011.

L. Schmalen, A. Alvarado, and R. Rios-Müller, “Predicting the performance of nonbinary forward error correction in optical transmission experiments,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.2.

R. Rios-Müller, J. Renaudier, L. Schmalen, and G. Charlet, “Joint coding rate and modulation format optimization for 8QAM constellations using BICM mutual information,” in Proc. Opt. Fiber Commun. Conf., 2015, Paper W3K.4.

Schulte, P.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609,  2016.

Secondini, M.

D. Marsella, M. Secondini, and E. Forestieri, “Maximum likelihood sequence detection for mitigating nonlinear effects,” J. Lightw. Technol., vol. 32, no. 5, pp. 908–916,  2014.

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., vol. 31, no. 23, pp. 3839–3852,  2013.

G. Liga, A. Alvarado, E. Agrell, M. Secondini, R. I. Killey, and P. Bayvel, “Optimum detection in presence of nonlinear distortions with memory,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P4.13.

Shamai, S.

N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, “On information rates for mismatched decoders,” IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1953–1967,  1994.

Shanmugan, K. S.

M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques, 2nd ed. Norwell, MA, USA: Kluwer, 2000.

Shannon, C. E.

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, 623–656,  1948.

Shtaif, M.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “New bounds on the capacity of the nonlinear fiber-optic channel,” Opt. Lett., vol. 39, no. 2, pp. 398–401,  2014.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express, vol. 21, no. 22, pp. 25 685–25 699,  2013.

Smith, B. P.

B. P. Smith, A. Farhood, A. Hunt, F. R. Kschischang, and J. Lodge, “Staircase codes: FEC for 100 Gb/s OTN,” J. Lightw. Technol., vol. 30, no. 1, pp. 110–117,  2012.

B. P. Smith and F. R. Kschischang, “Future prospects for FEC in fiber-optic communications,” IEEE J. Quantum Electron., vol. 16, no. 5, pp. 1245–1257,  2010.

Steiner, F.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609,  2016.

Ström, E. G.

E. Agrell, J. Lassing, E. G. Ström, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182,  2004.

Sugihara, T.

T. Koike-Akino, K. Kojima, D. Millar, K. Parsons, T. Yoshida, and T. Sugihara, “GMI-Maximizing constellation design with Grassmann projection for parametric shaping Pareto-efficient set of modulation and coding based on RGMI in nonlinear fiber transmissions,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.4.

Szczecinski, L.

L. Szczecinski and A. Alvarado, Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design. New York, NY, USA: Wiley, 2015.

Taricco, G.

G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946,  1998.

ten Brink, S.

L. Schmalen, F. Buchali, A. Leven, and S. ten Brink, “A generic tool for assessing the soft-FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 24, no. 1, pp. 40–42,  2012.

Thomas, J. A.

T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY, USA: Wiley, 2006.

Vacondio, F.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 23, no. 20, pp. 1547–1549,  2011.

Verdú, S.

S. Verdú and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1147–1157,  1994.

Vontobel, P. O.

D. M. Arnold, H. A. Loeliger, P. O. Vontobel, A. Kavčić, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3498–3508,  2006.

Wang, T.

I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, “Using LDPC-coded modulation and coherent detection for ultra highspeed optical transmission,” J. Lightw. Technol., vol. 25, no. 11, pp. 3619–3625,  2007.

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., vol. 28, no. 4, pp. 662–701,  2010.

Wolfowitz, J.

J. Wolfowitz, “Memory increases capacity,” Inform. Control., vol. 11, no. 4, pp. 423–428,  1967.

Wymeersch, H.

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun., vol. 62, no. 11, pp. 3956–3968,  2014.

Xu, L.

I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, “Using LDPC-coded modulation and coherent detection for ultra highspeed optical transmission,” J. Lightw. Technol., vol. 25, no. 11, pp. 3619–3625,  2007.

Yankov, M. P.

M. P. Yankov, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett., vol. 26, no. 23, pp. 2407–2410,  2014.

T. Fehenberger, M. P. Yankov, L. Barletta, and N. Hanik, “Compensation of XPM interference by blind tracking of the nonlinear phase in WDM systems with QAM input,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P5.8.

Yoshida, T.

T. Koike-Akino, K. Kojima, D. Millar, K. Parsons, T. Yoshida, and T. Sugihara, “GMI-Maximizing constellation design with Grassmann projection for parametric shaping Pareto-efficient set of modulation and coding based on RGMI in nonlinear fiber transmissions,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.4.

Yousefi, M. I.

G. Kramer, M. I. Yousefi, and F. R. Kschischang, “Upper bound on the capacity of a cascade of nonlinear and noisy channels,” in Proc. IEEE Inform. Theory Workshop (ITW), 2015, pp. 1–4.

Zeng, W.

D. M. Arnold, H. A. Loeliger, P. O. Vontobel, A. Kavčić, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3498–3508,  2006.

Zibar, D.

M. P. Yankov, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett., vol. 26, no. 23, pp. 2407–2410,  2014.

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R. E. Caflisch, “Monte Carlo and quasi-Monte Carlo methods,” Acta Numerica, vol. 7, pp. 1–49,  1998.

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C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, 623–656,  1948.

IEEE J. Quantum Electron. (1)

B. P. Smith and F. R. Kschischang, “Future prospects for FEC in fiber-optic communications,” IEEE J. Quantum Electron., vol. 16, no. 5, pp. 1245–1257,  2010.

IEEE J. Sel. Areas Commun. (1)

G. D. Forney, R. G. Gallager, G. R. Lang, F. M. Longstaff, and S. U. Qureshi, “Efficient modulation for band-limited channels,” IEEE J. Sel. Areas Commun., vol. 2, no. 5, pp. 632–647,  1984.

IEEE Photon. Technol. Lett. (4)

M. P. Yankov, D. Zibar, K. J. Larsen, L. P. B. Christensen, and S. Forchhammer, “Constellation shaping for fiber-optic channels with QAM and high spectral efficiency,” IEEE Photon. Technol. Lett., vol. 26, no. 23, pp. 2407–2410,  2014.

T. Fehenberger, D. Lavery, R. Maher, A. Alvarado, P. Bayvel, and N. Hanik, “Sensitivity gains by mismatched probabilistic shaping for optical communication systems,” IEEE Photon. Technol. Lett., vol. 28, no. 7, pp. 786–789,  2016.

A. Leven, F. Vacondio, L. Schmalen, S. Brink, and W. Idler, “Estimation of soft FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 23, no. 20, pp. 1547–1549,  2011.

L. Schmalen, F. Buchali, A. Leven, and S. ten Brink, “A generic tool for assessing the soft-FEC performance in optical transmission experiments,” IEEE Photon. Technol. Lett., vol. 24, no. 1, pp. 40–42,  2012.

IEEE Trans. Commun. (1)

N. V. Irukulapati, H. Wymeersch, P. Johannisson, and E. Agrell, “Stochastic digital backpropagation,” IEEE Trans. Commun., vol. 62, no. 11, pp. 3956–3968,  2014.

IEEE Trans. Inf. Theory (6)

S. Verdú and T. S. Han, “A general formula for channel capacity,” IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1147–1157,  1994.

E. Agrell, J. Lassing, E. G. Ström, and T. Ottosson, “On the optimality of the binary reflected Gray code,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3170–3182,  2004.

G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inf. Theory, vol. 44, no. 3, pp. 927–946,  1998.

F. R. Kschischang and S. Pasupathy, “Optimal non uniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory, vol. 39, no. 3, pp. 913–929,  1993.

N. Merhav, G. Kaplan, A. Lapidoth, and S. Shamai, “On information rates for mismatched decoders,” IEEE Trans. Inf. Theory, vol. 40, no. 6, pp. 1953–1967,  1994.

D. M. Arnold, H. A. Loeliger, P. O. Vontobel, A. Kavčić, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory, vol. 52, no. 8, pp. 3498–3508,  2006.

Inform. Control. (1)

J. Wolfowitz, “Memory increases capacity,” Inform. Control., vol. 11, no. 4, pp. 423–428,  1967.

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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., vol. 32, no. 4, pp. 694–721,  2014.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol., vol. 30, no. 10, pp. 1524–1539,  2012.

D. Marsella, M. Secondini, and E. Forestieri, “Maximum likelihood sequence detection for mitigating nonlinear effects,” J. Lightw. Technol., vol. 32, no. 5, pp. 908–916,  2014.

E. Agrell, A. Alvarado, G. Durisi, and M. Karlsson, “Capacity of a nonlinear optical channel with finite memory,” J. Lightw. Technol., vol. 32, no. 16, pp. 2862–2876,  2014.

T. A. Eriksson, T. Fehenberger, P. A. Andrekson, M. Karlsson, N. Hanik, and E. Agrell, “Impact of 4D channel distribution on the achievable rates in coherent optical communication experiments,” J. Lightw. Technol., vol. 34, no. 9, pp. 2256–2266,  2016.

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., vol. 31, no. 23, pp. 3839–3852,  2013.

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol., vol. 28, no. 4, pp. 662–701,  2010.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Schulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightw. Technol., vol. 34, no. 7, pp. 1599–1609,  2016.

A. Alvarado and E. Agrell, “Four-dimensional coded modulation with bit-wise decoders for future optical communications,” J. Lightw. Technol., vol. 33, no. 10, pp. 1993–2003,  2015.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 33, no. 20, pp. 4338–4352,  2015.

A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Corrections to ‘replacing the soft-decision FEC limit paradigm in the design of optical communication systems,” J. Lightw. Technol., vol. 34, no. 2,  2016.

I. B. Djordjevic, M. Arabaci, and L. L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightw. Technol., vol. 27, no. 16, pp. 3518–3530,  2009.

I. B. Djordjevic, M. Cvijetic, L. Xu, and T. Wang, “Using LDPC-coded modulation and coherent detection for ultra highspeed optical transmission,” J. Lightw. Technol., vol. 25, no. 11, pp. 3619–3625,  2007.

B. P. Smith, A. Farhood, A. Hunt, F. R. Kschischang, and J. Lodge, “Staircase codes: FEC for 100 Gb/s OTN,” J. Lightw. Technol., vol. 30, no. 1, pp. 110–117,  2012.

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T. Fehenberger, A. Alvarado, P. Bayvel, and N. Hanik, “On achievable rates for long-haul fiber-optic communications,” Opt. Express, vol. 23, no. 7, pp. 9183–9191,  2015.

G. Colavolpe, T. Foggi, A. Modenini, and A. Piemontese, “Faster-than-Nyquist and beyond: how to improve spectral efficiency by accepting interference,” Opt. Express, vol. 19, no. 27, pp. 26 600–26 609,  2011.

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express, vol. 21, no. 22, pp. 25 685–25 699,  2013.

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R. Rios-Müller, J. Renaudier, L. Schmalen, and G. Charlet, “Joint coding rate and modulation format optimization for 8QAM constellations using BICM mutual information,” in Proc. Opt. Fiber Commun. Conf., 2015, Paper W3K.4.

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G. Liga, A. Alvarado, E. Agrell, M. Secondini, R. I. Killey, and P. Bayvel, “Optimum detection in presence of nonlinear distortions with memory,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P4.13.

R. F. Fischer, Precoding and Signal Shaping for Digital Transmission. New York, NY, USA: Wiley, 2005.

C. Häger, H. D. Pfister, A. G. i Amat, F. Brännström, and E. Agrell, “A deterministic construction and density evolution analysis for generalized product codes,” in Proc. Int. Zurich Seminar Commun. (IZS), 2016.

“Forward error correction for high bit-rate dwdm submarine systems,” ITU-T, Recommendation G.975.1,  2004.

L. Schmalen, A. Alvarado, and R. Rios-Müller, “Predicting the performance of nonbinary forward error correction in optical transmission experiments,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.2.

T. Koike-Akino, K. Kojima, D. Millar, K. Parsons, T. Yoshida, and T. Sugihara, “GMI-Maximizing constellation design with Grassmann projection for parametric shaping Pareto-efficient set of modulation and coding based on RGMI in nonlinear fiber transmissions,” in Proc. Opt. Fiber Commun. Conf., 2016, Paper M2A.4.

L. Szczecinski and A. Alvarado, Bit-Interleaved Coded Modulation: Fundamentals, Analysis and Design. New York, NY, USA: Wiley, 2015.

R. G. Gallager, Information Theory and Reliable Communication. New York, NY, USA: Wiley, 1968.

T. M. Cover and J. A. Thomas, Elements of Information Theory. New York, NY, USA: Wiley, 2006.

T. Fehenberger, M. P. Yankov, L. Barletta, and N. Hanik, “Compensation of XPM interference by blind tracking of the nonlinear phase in WDM systems with QAM input,” in Proc. Eur. Conf. Opt. Commun., 2015, Paper P5.8.

A. Martinez, L. Peng, A. Alvarado, and A. Guillén i Fàbregas, “Improved information rates for bit-interleaved coded modulation,” in Proc. IEEE Int. Symp. Inform. Theory (ISIT), 2015, pp. 3004–3007.

R. G. Gallager, Stochastic Processes: Theory for Applications. Cambridge, U.K.: Cambridge Univ. Press, 2014.

M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques, 2nd ed. Norwell, MA, USA: Kluwer, 2000.

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