Abstract

We review probabilistic constellation shaping (PCS), which has been a key enabler for several recent record-setting optical fiber communications experiments. PCS provides both fine-grained rate adaptability and energy efficiency (sensitivity) gains. We discuss the reasons for the fundamentally better performance of PCS over other constellation shaping techniques that also achieve rate adaptability, such as time-division hybrid modulation, and examine in detail the impact of sub-optimum shaping and forward error correction (FEC) on PCS systems. As performance metrics for systems with PCS, we compare information-theoretic measures such as mutual information (MI), generalized MI (GMI), and normalized GMI, which enable optimization and quantification of the information rate (IR) that can be achieved by PCS and FEC. We derive the optimal parameters of PCS and FEC that maximize the IR for both ideal and non-ideal PCS and FEC. To avoid plausible pitfalls in practice, we carefully revisit key assumptions that are typically made for ideal PCS and FEC systems.

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2018 (6)

J.-X. Caiet al., “70.46 Tb/s over 7,600 km and 71.65 Tb/s over 6,970 km transmission in C+L band using coded modulation with hybrid constellation shaping and nonlinearity compensation,” J. Lightw. Tehcnol., vol. 36, no. 1, pp. 114–121, 2018.

J. Choet al., “Trans-Atlantic field trial using high spectral efficiency probabilistically shaped 64-QAM and single-carrier real-time 250-Gb/s 16-QAM,” J. Lightw. Technol., vol. 36, no. 1, pp. 103–113, 2018.

F. P. Guiomar, L. Bertignono, A. Nespola, and A. Carena, “Frequency-domain hybrid modulation formats for high bit-rate flexibility and nonlinear robustness,” J. Lightw. Technol., vol. 36, no. 20, pp. 4856–4870, 2018.

A. Alvarado, T. Fehenberger, B. Chen, and F. M. J. Willems, “Achievable information rates for fiber optics: Applications and computations,” J. Lightw. Technol., vol. 36, no. 2, pp. 424–439, 2018.

S. L. I. Olsson, J. Cho, S. Chandrasekhar, X. Chen, P. J. Winzer, and S. Makovejs, “Probabilistically shaped PDM 4096-QAM transmission over up to 200 km of fiber using standard intradyne detection,” Opt. Express, vol. 26, no. 4, pp. 4522–4530, 2018.

J. Cho, X. Chen, S. Chandrasekhar, and P. Winzer, “On line rates, information rates, and spectral efficiencies in probabilistically shaped QAM systems,” Opt. Express, vol. 26, no. 8, pp. 9784–9791, 2018.

2017 (2)

T. Yoshida, M. Karlsson, and E. Agrell, “Performance metrics for systems with soft-decision FEC and probabilistic shaping,” IEEE Photon. Technol. Lett., vol. 29, no. 23, pp. 2111–2114, 2017.

M. Xianget al., “Multi-subcarrier flexible bit-loading enabled capacity improvement in meshed optical networks with cascaded ROADMs,” Opt. Express, vol. 25, no. 21, pp. 25046–25058, 2017.

2016 (4)

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory, vol. 62, no. 1, pp. 430–434, 2016.

T. Fehenberger, A. Alvarado, G. Böcherer, and N. Hanik. “On probabilistic shaping of quadrature amplitude modulation for the nonlinear fiber channel,” J. Lightw. Technol., vol. 34, no. 21, pp. 5063–5073, 2016.

D. G. M. Mitchell, M. Lentmaier, A. E. Pusane, and D. J. Costello, “Randomly punctured LDPC codes,” IEEE J. Sel. Areas Commun., vol. 34, no. 2, pp. 408–421, 2016.

G. Tzimpragos, C. Kachris, I. B. Djordjevic, M. Cvijetic, D. Soudris, and I. Tomkos, “A survey on FEC codes for 100 G and beyond optical networks,” IEEE Commun. Surv. Tut., vol. 18, no. 1, pp. 209–221, 2016.

2015 (3)

Unified high-speed wireline-based home networking transceivers – System architecture and physical layer specification, ITU-T Recommendation G.9960, 2015.

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun., vol. 63, no. 12, pp. 4651–4665, 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 (4)

R. Dar, M. Shtaif, and M. Feder, “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, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express, vol. 22, no. 12, pp. 14199–14211, 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. Tehcnol., vol. 32, no. 4, pp. 694–721, 2014.

A. Leven and L. Schmalen, “Status and recent advances on forward error correction technologies for lightwave systems,” J. Lightw. Technol., vol. 32, no. 16, pp. 2735–2750, 2014.

2013 (3)

T. H. Lotzet al., “Coded PDM-OFDM transmission with shaped 256-iterative-polar-modulation achieving 11.15-b/s/Hz intrachannel spectral efficiency and 800-km reach,” J. Lightw. Tehcnol., vol. 31, no. 4, pp. 538–545, 2013.

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. 25685–25699, 2013.

R. Asvadi and A. H. Banihashemi, “A rate-compatible puncturing scheme for finite-length LDPC codes,” IEEE Commun. Lett., vol. 17, no. 1, pp. 147–150, 2013.

2012 (1)

T. V. Nguyen, A. Nosratinia, and D. Divsalar, “The design of rate-compatible protograph LDPC codes,” IEEE Trans. Commun., vol. 60, no. 10, pp. 2841–2850, 2012.

2010 (2)

H. Cronie, “Signal shaping for bit-interleaved coded modulation on the AWGN channel,” IEEE Trans. Commun., vol. 58, no. 12, pp. 3428–3435, 2010.

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.

2009 (1)

A. Martinez, A. G. i Fàbregas, G. Caire, and F. M. J. Willems, “Bit-interleaved coded modulation revisited: A mismatched decoding perspective,” IEEE Trans. Inf. Theory, vol. 55, no. 6, pp. 2756–2765, 2009.

2008 (1)

C.-H. Hsu and A. Anastasopoulos, “Capacity achieving LDPC codes through puncturing,” IEEE Trans. Inf. Theory, vol. 54, no. 10, pp. 4698–4706, 2008.

2006 (1)

B. K. Khoo, S. Le Goff, B. Sharif, and C. Tsimenidis, “Bit-interleaved coded modulation with iterative decoding using constellation shaping,” IEEE Trans. Commun., vol. 54, no. 9, pp. 1517–1520, 2006.

2005 (2)

S. L. Goff, B. Sharif, and S. Jimaa, “Bit-interleaved turbo-coded modulation using shaping coding,” IEEE Commun. Lett., vol. 9, no. 3, pp. 246–248, 2005.

T. Tian and C. R. Jones, “Construction of rate-compatible LDPC codes utilizing information shortening and parity puncturing,” EURASIP J. Wireless Commun. Netw., vol. 2005, no. 5, pp. 789–795, 2005.

2004 (2)

J. Ha, J. Kim, and S. McLaughlin, “Rate-compatible puncturing of low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 50, no. 11, pp. 2824–2826, 2004.

D. Raphaeli and A. Gurevitz, “Constellation shaping for pragmatic turbo-coded modulation with high spectral efficiency,” IEEE Trans. Commun., vol. 52, no. 3, pp. 341–345, 2004.

2001 (3)

T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 619–637, 2001.

S. Ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, no. 10, pp. 1727–1737, 2001.

S.-Y. Chung, G. D. Forney, T. J. Richardson, and R. Urbanke, “On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit,” IEEE Commun. Lett., vol. 5, no. 2, pp. 58–60, 2001.

1998 (1)

A Modem Operating at Data Signalling Rates of Up to 33 600 Bit/S for Use On the General Switched Telephone Network and On Leased Point-To-Point 2-Wire Telephone-Type Circuits, ITU-T Recommendation V.34, 1998.

1996 (1)

D. J. C. MacKay and R. M. Neal, “Near Shannon limit performance of low density parity check codes,” Electron. Lett., vol. 32, no. 18, pp. 1645–1646, 1996.

1994 (2)

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

R. Laroia, N. Farvardin, and S. A. Tretter, “On optimal shaping of multidimensional constellations,” IEEE Trans. Inf. Theory, vol. 40, no. 4, pp. 1044–1056, 1994.

1993 (3)

A. K. Khandani and P. Kabal, “Shaping multidimensional signal spaces. I. Optimum shaping, shell mapping,” IEEE Trans. Inf. Theory, vol. 39, no. 6, pp. 1799–1808, 1993.

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

F.-W. Sun and H. C. A. van Tilborg, “Approaching capacity by equiprobable signaling on the Gaussian channel,” IEEE Trans. Inf. Theory, vol. 39, no. 5, pp. 1714–1716, 1993.

1992 (1)

G. D. Forney, “Trellis shaping,” IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 281–300, 1992.

1990 (2)

A. R. Calderbank and L. H. Ozarow, “Nonequiprobable signaling on the Gaussian channel,” IEEE Trans. Inf. Theory, vol. 36, no. 4, pp. 726–740, 1990.

T. V. Ramabadran, “A coding scheme for m-out-of-n codes,” IEEE Trans. Commun., vol. 38, no. 8, pp. 1156–1163, 1990.

1989 (1)

G. R. Lang and F. M. Longstaff, “A Leech lattice modem,” IEEE J. Sel. Areas Commun., vol. 7, no. 6, pp. 968–973, 1989.

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. SAC-2, no. 5, pp. 632–647, 1984.

1977 (1)

H. Imai and S. Hirakawa, “A new multilevel coding method using error-correcting codes,” IEEE Trans. Inf. Theory, vol. 23, no. 3, pp. 371–377, 1977.

1972 (2)

S. Arimoto, “An algorithm for computing the capacity of arbitrary discrete memoryless channels,” IEEE Trans. Inf. Theory, vol. 18, no. 1, pp. 14–20, 1972.

R. Blahut, “Computation of channel capacity and rate-distortion functions,” IEEE Trans. Inf. Theory, vol. 18, no. 4, pp. 460–473, 1972.

1962 (1)

R. Gallager, “Low-density parity-check codes,” IRE Trans. Inf. Theory, vol. 8, no. 1, pp. 21–28, 1962.

1948 (1)

C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, no. 3, pp. 379–423, 1948.

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