Abstract

We describe an experimental and numerical comparison of a probabilistically shaped (PS) 4096 QAM signal and a uniformly shaped 1024 QAM signal. Both modulation formats have the same transmission rate and a spectral efficiency of 15.3 bit/s/Hz. In the computational simulation, we compared the generalized mutual information (GMI) of both modulation formats with bit-wise soft decision decoding and bit-wise hard decision decoding. For bit-wise hard decision decoding with an overhead of 7%, a shaping gain of 1.8 dB was attained. Then we successfully transmitted a single-channel PS-4096 QAM signal for the first time in an all-Raman amplified 160-km link, in which the transmission performance was improved compared with a uniformly shaped 1024 QAM with the same transmission rate. Transmissions with a high QAM multiplicity were achieved by using an optical phase-locked loop (OPLL) and a frequency stabilized fiber laser locked to an acetylene absorption line. Thanks to a shaping gain based on a bit-wise hard decision decoder, the 1.9-dB power margin, which agreed with the simulation result to within 0.1 dB, was increased after transmission.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. M. Nakazawa, K. Kikuchi, and T. Miyazaki, High Spectral Density Optical Transmission Technologies (Springer, 2010).
  2. S. Okamoto, K. Toyoda, T. Omiya, K. Kasai, M. Yoshida, and M. Nakazawa, “512 QAM (54 Gbit/s) coherent optical transmission over 150 km with an optical bandwidth of 4.1 GHz,” in Proceedings of European Conference on Optical Communication (IEEE, 2010), paper PD2.3.
    [Crossref]
  3. R. Schmogrow, D. Hillerkuss, S. Wolf, B. Bäuerle, M. Winter, P. Kleinow, B. Nebendahl, T. Dippon, P. C. Schindler, C. Koos, W. Freude, and J. Leuthold, “512QAM Nyquist sinc-pulse transmission at 54 Gbit/s in an optical bandwidth of 3 GHz,” Opt. Express 20(6), 6439–6447 (2012).
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    [Crossref] [PubMed]
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  6. S. Beppu, K. Kasai, M. Yoshida, and M. Nakazawa, “2048 QAM (66 Gbit/s) single-carrier coherent optical transmission over 150 km with a potential SE of 15.3 bit/s/Hz,” Opt. Express 23(4), 4960–4969 (2015).
    [Crossref] [PubMed]
  7. D. Qian, E. Ip, M.-F. Huang, M.-J. Li, and T. Wan, “698.5-Gb/s PDM-2048QAM transmission over 3 km multicore fiber,” in Proceedings of European Conference on Optical Communication (IEEE, 2013), paper Th.1.C.5.
  8. G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
    [Crossref]
  9. F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Shulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
    [Crossref]
  10. S. Chandrasekhar, B. Li, J. Cho, X. Chen, E. C. Burrows, G. Raybon, and P. J. Winzer, “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs,” in Proceedings of European Conference on Optical Communication (IEEE, 2016), paper Th.3.C.1.
  11. M. P. Yankov, F. Da Ros, E. P. da Silva, S. Forchhammer, K. J. Larsen, L. K. Oxenløwe, M. Galili, and D. Zibar, “Constellation shaping for WDM systems using 256QAM/1024QAM with probabilistic optimization,” J. Lightwave Technol. 34(22), 5146–5156 (2016).
    [Crossref]
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    [Crossref]
  13. G. Böcherer, P. Shulte, and F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Inf. Theory 63(12), 4651–4665 (2015).
  14. P. Shulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 420–434 (2016).
  15. F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
    [Crossref]
  16. 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. Lightwave Technol. 33(20), 4338–4352 (2015).
    [Crossref]
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    [Crossref]
  18. J. G. Proakis, Digital Communications (McGraw Hill, 2000).
  19. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
    [Crossref]
  20. T. Fehenberger, A. Alvarado, G. Böcherer, and N. Hanik, “On probabilistic shaping of quadrature amplitude modulation for the nonlinear fiber channel,” J. Lightwave Technol. 34(21), 5063–5073 (2016).
    [Crossref]

2017 (2)

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

A. Sheikh, A. G. Amat, and G. Liva, “Achievable information rates for coded modulation with hard decision decoding for coherent fiber-optic systems,” J. Lightwave Technol. 35(23), 5069–5078 (2017).
[Crossref]

2016 (4)

2015 (3)

2012 (2)

2009 (1)

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

1993 (1)

F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

Agrell, E.

Alvarado, A.

Amat, A. G.

Bäuerle, B.

Bayvel, P.

Beppu, S.

Bernd, H.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Böcherer, G.

P. Shulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 420–434 (2016).

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Shulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

T. Fehenberger, A. Alvarado, G. Böcherer, and N. Hanik, “On probabilistic shaping of quadrature amplitude modulation for the nonlinear fiber channel,” J. Lightwave Technol. 34(21), 5063–5073 (2016).
[Crossref]

G. Böcherer, P. Shulte, and F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Inf. Theory 63(12), 4651–4665 (2015).

Bohn, M.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Buchali, F.

Calabro, S.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Chen, J.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

Da Ros, F.

da Silva, E. P.

de Man, E.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

de Waardt, H.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Dippon, T.

Fehenberger, T.

Feiste, U.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Forchhammer, S.

Freude, W.

Galili, M.

Hanik, N.

Hillerkuss, D.

Huug, S. K.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Idler, W.

Kam, P. Y.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

Kasai, K.

Khanna, G.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Kleinow, P.

Koizumi, Y.

Koos, C.

Kschischang, F. R.

F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

Larsen, K. J.

Lavery, D.

Leuthold, J.

Liva, G.

Maher, R.

Nakazawa, M.

Napoli, A.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Nebendahl, B.

Norbert, D.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Oxenløwe, L. K.

Pagano, A.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Pasupathy, S.

F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

Piat, A. C.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Rafique, D.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Rahman, T.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Riccardi, E.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Schindler, P. C.

Schmalen, L.

Schmogrow, R.

Sheikh, A.

Shulte, P.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Shulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

P. Shulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 420–434 (2016).

G. Böcherer, P. Shulte, and F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Inf. Theory 63(12), 4651–4665 (2015).

Spinnler, B.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Steiner, F.

F. Buchali, F. Steiner, G. Böcherer, L. Schmalen, P. Shulte, and W. Idler, “Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration,” J. Lightwave Technol. 34(7), 1599–1609 (2016).
[Crossref]

G. Böcherer, P. Shulte, and F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Inf. Theory 63(12), 4651–4665 (2015).

Tomislav, B.

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

Toyoda, K.

Winter, M.

Wolf, S.

Yankov, M. P.

Yoshida, M.

Yu, C.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

Zhang, S.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

Zibar, D.

IEEE Photonics Technol. Lett. (2)

G. Khanna, T. Rahman, E. de Man, E. Riccardi, A. Pagano, A. C. Piat, S. Calabro, B. Spinnler, D. Rafique, U. Feiste, H. de Waardt, S. K. Huug, H. Bernd, D. Norbert, B. Tomislav, M. Bohn, and A. Napoli, “Single-carrier 400G 64QAM and 128QAM DWDM field trial transmission over metro legacy links,” IEEE Photonics Technol. Lett. 29(2), 189–192 (2017).
[Crossref]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photonics Technol. Lett. 21(15), 1075–1077 (2009).
[Crossref]

IEEE Trans. Inf. Theory (3)

G. Böcherer, P. Shulte, and F. Steiner, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Inf. Theory 63(12), 4651–4665 (2015).

P. Shulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 420–434 (2016).

F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inf. Theory 39(3), 913–929 (1993).
[Crossref]

J. Lightwave Technol. (5)

Opt. Express (3)

Other (7)

M.-F. Huang, D. Qian, and E. Ip, “50.53-Gb/s PDM-1024QAM-OFDM transmission using pilot-based phase noise mitigation,” in Proceedings of Opto-Electronics and Communications Conference (IEEE, 2011), paper PDP1.

M. Nakazawa, K. Kikuchi, and T. Miyazaki, High Spectral Density Optical Transmission Technologies (Springer, 2010).

S. Okamoto, K. Toyoda, T. Omiya, K. Kasai, M. Yoshida, and M. Nakazawa, “512 QAM (54 Gbit/s) coherent optical transmission over 150 km with an optical bandwidth of 4.1 GHz,” in Proceedings of European Conference on Optical Communication (IEEE, 2010), paper PD2.3.
[Crossref]

D. Qian, E. Ip, M.-F. Huang, M.-J. Li, and T. Wan, “698.5-Gb/s PDM-2048QAM transmission over 3 km multicore fiber,” in Proceedings of European Conference on Optical Communication (IEEE, 2013), paper Th.1.C.5.

S. Chandrasekhar, B. Li, J. Cho, X. Chen, E. C. Burrows, G. Raybon, and P. J. Winzer, “High-spectral-efficiency transmission of PDM 256-QAM with parallel probabilistic shaping at record rate-reach trade-offs,” in Proceedings of European Conference on Optical Communication (IEEE, 2016), paper Th.3.C.1.

J. Cho, X. Chen, S. Chandrasekhar, G. Raybon, R. Dar, L. Schmalen, E. Burrows, A. Adamiecki, S. Corteselli, Y. Pan, D. Correa, B. McKay, S. Zsigmond, P. Winzer, and S. Grubb, “Trans-Atlantic field trial using probabilistically shaped 64-QAM at high spectral efficiencies and single-carrier real-time 250-Gb/s 16-QAM,” in Proceedings of Optical Fiber Communication Conference (Optical Society of America, 2017), paper Th5B.3.
[Crossref]

J. G. Proakis, Digital Communications (McGraw Hill, 2000).

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Figures (8)

Fig. 1
Fig. 1 The generation diagram for PS-4096 QAM signal based on the probabilistic amplitude shaping (PAS) system. In our experiment, all the MSBs are composed of random binary sequences from a discrete memoryless binary source.
Fig. 2
Fig. 2 (a) Probability mass function of the 32 ASK signals output from the CCDM and (b) the transformation efficiency of the CCDM; the ratio of input bit length and output symbol length.
Fig. 3
Fig. 3 Example constellation diagram and its 3D amplitude histogram. (a), (b) PS-4096 QAM and (c), (d) US-1024 QAM.
Fig. 4
Fig. 4 GMI of PS-4096QAM and US-1024 QAM using a bit-wise hard decision decoder (HDD-BW) and a bit-wise soft decision decoder (SDD-BW).
Fig. 5
Fig. 5 Experimental setup for PS-4096QAM and US-1024 QAM, 160 km transmission.
Fig. 6
Fig. 6 Optical spectra at (a) back-to-back and (b) after 160 km transmission.
Fig. 7
Fig. 7 Pre-FEC BERs of US-1024 QAM and PS-4096 QAM in (a) back-to-back and (b) after 160 km transmission.
Fig. 8
Fig. 8 PS-4096 QAM constellation map in (a) back-to-back, and (b) after 160 km transmission.

Equations (3)

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GM I SDDBW H 1 N k=1 m l=1 N log 2 (1+ e (1) c k,l λ k,l )
λ k,l =log xS{ b k =0} exp(| y l x | 2 /2 σ 2 )P(x) xS{ b k =1} exp(| y l x | 2 /2 σ 2 )P(x)
GM I HDDBW =m[1(ε log 2 ε(1ε) log 2 (1ε))]

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