Abstract

Pol-Mux transmission is a well established technique that enhances spectral efficiency by simultaneously transmitting over horizontal and vertical polarizations of the electrical field. However, cross-coupling of the two polarizations impairs transmission. Under the assumption that the cross-coupling matrix is a Markov process with free-running state, we propose upper and lower bounds to the information rate that can be transferred through the channel. Simulation results show that the two bounds are tight for values of the cross-coupling power of practical interest and modulation formats up to 16-QAM (quadrature amplitude modulation).

© 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.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
    [Crossref]
  2. V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
    [Crossref]
  3. P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
    [Crossref]
  4. Gerard J. Foschini and Michael J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun. 6(3), 311–335 (1998).
    [Crossref]
  5. Seb. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
    [Crossref]
  6. L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.
  7. P. Layec, A. Ghazisaeidi, and G. Charlet, Jean-Christophe Antona, and S. Bigo, “Generalized maximum likelihood for cross-polarization modulation effects compensation,” J. Lightwave Technol. 33(7), 1300–1307 (2015).
    [Crossref]
  8. J. Li, R. Schmogrow, D. Hillerkuss, Philipp C. Schindler, M. Nazarathy, C. Schmidt-Langhorst, Shalva-Ben Ezra, I. Tselniker, C. Koos, W. Freude, and J. Leuthold, “A self-coherent receiver for detection of PolMUX coherent signals,” Opt. Express 20(19), 21413–21433 (2012).
    [Crossref] [PubMed]
  9. R.H. Etkin and D. N. C. Tse, “Degrees of freedom in some underspread MIMO fading channels,” IEEE T. Inform. Theory 52(4), 1576–1608 (2006).
    [Crossref]
  10. L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
    [Crossref]
  11. L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
    [Crossref]
  12. L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol. 30(10), 1480–1486 (2012).
    [Crossref]
  13. L. Barletta, M. Magarini, and A. Spalvieri, “A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery,” Opt. Express 20(23), 2547–25477 (2012).
    [Crossref]

2015 (1)

2014 (1)

L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

2012 (3)

2011 (2)

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
[Crossref]

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

2010 (2)

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Seb. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

2009 (1)

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

2006 (1)

R.H. Etkin and D. N. C. Tse, “Degrees of freedom in some underspread MIMO fading channels,” IEEE T. Inform. Theory 52(4), 1576–1608 (2006).
[Crossref]

1998 (1)

Gerard J. Foschini and Michael J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun. 6(3), 311–335 (1998).
[Crossref]

Al Fiad, M.S.A.S.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Barletta, L.

L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery,” Opt. Express 20(23), 2547–25477 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol. 30(10), 1480–1486 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
[Crossref]

Boffi, P.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Charlet, G.

de Waardt, H.

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Etkin, R.H.

R.H. Etkin and D. N. C. Tse, “Degrees of freedom in some underspread MIMO fading channels,” IEEE T. Inform. Theory 52(4), 1576–1608 (2006).
[Crossref]

Ezra, Shalva-Ben

Ferrario, M.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Foschini, Gerard J.

Gerard J. Foschini and Michael J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun. 6(3), 311–335 (1998).
[Crossref]

Freude, W.

Gans, Michael J.

Gerard J. Foschini and Michael J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun. 6(3), 311–335 (1998).
[Crossref]

Ghazisaeidi, A.

Hillerkuss, D.

Hirano, M.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Hoshida, T.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Jansen, S.L.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Koos, C.

Kuschnerov, M.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Layec, P.

Leuthold, J.

Li, J.

Li, L.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Liu, L.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Magarini, M.

L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery,” Opt. Express 20(23), 2547–25477 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol. 30(10), 1480–1486 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
[Crossref]

Marazzi, L.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Martelli, P.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Martinelli, M.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Nazarathy, M.

Oda, S.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Parolari, P.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Pecorino, S.

L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

Rasmussen, Jens C.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Righetti, A.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Sasaki, T.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Savory, Seb. J.

Seb. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

Schindler, Philipp C.

Schmidt-Langhorst, C.

Schmogrow, R.

Siano, R.

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

Sleiffer, V.A.J.M.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Spalvieri, A.

L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “A new lower bound below the information rate of Wiener phase noise channel based on Kalman carrier recovery,” Opt. Express 20(23), 2547–25477 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol. 30(10), 1480–1486 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
[Crossref]

Tao, Z.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

Tse, D. N. C.

R.H. Etkin and D. N. C. Tse, “Degrees of freedom in some underspread MIMO fading channels,” IEEE T. Inform. Theory 52(4), 1576–1608 (2006).
[Crossref]

Tselniker, I.

van den Borne, D.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Veljanovski, V.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Waardt, H. de

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Wuth, T.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

Yamamoto, Y.

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

Yan, W.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

IEEE J. Sel. Top. Quantum Electron. (1)

Seb. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (3)

M.S.A.S. Al Fiad, M. Kuschnerov, S.L. Jansen, T. Wuth, D. van den Borne, and H. de Waardt, “11 × 224-Gb/s POLMUX-RZ-16QAM transmission over 670 km of SSMF with 50-GHz channel spacing,” IEEE Photon. Technol. Lett. 22(15), 1150–1152 (2010).
[Crossref]

V.A.J.M. Sleiffer, M.S.A.S. Al Fiad, D. van den Borne, M. Kuschnerov, V. Veljanovski, M. Hirano, Y. Yamamoto, T. Sasaki, S.L. Jansen, T. Wuth, and H. de Waardt, “10 × 224-Gb/s POLMUX-16QAM transmission over 656 km of Large-Aeff PSCF with a spectral efficiency of 5.6 b/s/Hz,” IEEE Photon. Technol. Lett. 23(20), 1427–1429 (2011).
[Crossref]

P. Boffi, M. Ferrario, L. Marazzi, P. Martelli, P. Parolari, A. Righetti, R. Siano, and M. Martinelli, “Stable 100-Gb/s POLMUX-DQPSK transmission with automatic polarization stabilization,” IEEE Photon. Technol. Lett. 21(11), 745–747 (2009).
[Crossref]

IEEE Photonic Tech. L. (1)

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photonic Tech. L. 23(21), 1582–1584 (2011).
[Crossref]

IEEE T. Inform. Theory (2)

R.H. Etkin and D. N. C. Tse, “Degrees of freedom in some underspread MIMO fading channels,” IEEE T. Inform. Theory 52(4), 1576–1608 (2006).
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L. Barletta, M. Magarini, S. Pecorino, and A. Spalvieri, “Upper and lower bounds to the information rate transferred through first-order Markov channels with free-running continuous state,” IEEE T. Inform. Theory 60(7), 3834–3844 (2014).
[Crossref]

J. Lightwave Technol. (2)

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[Crossref]

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L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and Jens C. Rasmussen, “Nonlinear polarization crosstalk canceller for dual-polarization digital coherent receivers,” presented at Optical Fiber Communication, collocated National Fiber Optic Engineers Conference (OFC/NFOEC), IEEE, Piscataway, NJ, USA, 21 March 2010.

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Figures (2)

Fig. 1
Fig. 1 Upper and lower bounds to the information rate for various modulation formats and zp = 0.977. The Signal-to-Noise Ratio (SNR) is SNR = 1 σ 2.
Fig. 2
Fig. 2 Upper and lower bounds to the information rate for various modulation formats and zp = 0.887. The Signal-to-Noise Ratio (SNR) is SNR = 1 σ 2.

Equations (50)

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y k = M k x k + w k , k = 1 , 2 , ,
E { x k x k H } = 2 ,
E { w k w k H } = σ 2 2 .
M k = ( 1 λ 1 , k λ 2 , k 1 ) ,
λ k = ( λ 1 , k , λ 2 , k ) T
λ k = i = 1 p b i v k i + i = 1 q a i λ k i ,
E { v k v k H } = ( 1 ρ ρ 1 ) .
m = max { p , q } .
λ ( z ) = v ( z ) b ( z ) 1 a ( z ) ,
b ( z ) = i = 1 m b i z i , a ( z ) = i = 1 m a i z i .
ω k = ( ω 1 , k , ω 2 , k ) T = v k + i = 1 m a i ω k i , k = 0 , 1 , ,
λ k = i = 1 m b i ω k i
s k = ( 1 , ( ω 1 , k m k 1 ) T , 1 , ( ω 2 , k m k 1 ) T ) T
y k = H k s k + w k ,
s k + 1 = F s k + ( 0 , v 1 , k , ( 0 1 m 1 ) T , 0 , v 2 , k , ( 0 1 m 1 ) T ) T ,
H k = [ x 1 , k x 2 k ( b 1 m ) T 0 ( 0 1 m ) T 0 ( 0 1 m ) T x 2 , k x 1 , k ( b 1 m ) T ] ,
F [ F m + 1 O m + 1 O m + 1 F m + 1 ] ,
F m + 1 [ 1 ( 0 1 m 1 ) T 0 0 ( a 1 m 1 ) T a m 0 1 m 1 m 1 0 1 m 1 ] ,
p ( s k + 1 | s k ) = g c ( F s k , Q ; s k + 1 ) ,
Q [ Q 1 Q ρ Q ρ Q 1 ] ,
Q 1 = [ 0 0 ( 0 1 m 1 ) T 0 1 ( 0 1 m 1 ) T 0 1 m 1 0 1 m 1 O m 1 ] ,
Q ρ = [ 0 0 ( 0 1 m 1 ) T 0 ρ ( 0 1 m 1 ) T 0 1 m 1 0 1 m 1 O m 1 ] .
p ( y k , x k | s k ) = p ( x k | s k ) p ( y k | x k , s k ) = p ( x k ) p ( y k | x k , s k )
p ( y k | x k , s k ) = g c ( H k s k , σ 2 2 ; y k ) .
p ( y k | s k ) = x k X k p ( x k ) p ( y k | x k , s k ) .
I ( x ; y ) = H ( x ) H ( x | y ) ,
H ( x ) = log 2 M .
H ( x | y ) = lim N 1 N k = 1 N H ( x k | x 1 k 1 , y 1 N ) ,
H ( x | y ) = lim N 1 N k = 1 N log 2 p ( x k | x 1 k 1 , y 1 N ) .
H ( x | y ) = lim N 1 N k = 1 N H ( x k | x 1 k 1 , y 1 N ) lim N 1 N k = 1 N H ( x k | x 1 k 1 , y 1 k ) = lim N 1 N k = 1 N log 2 p ( x k | x 1 k 1 , y 1 k ) ,
H ( x | y ) = lim N 1 N k = 1 N H ( x k | x 1 k 1 , y 1 N ) lim N 1 N k = 1 N H ( x k | x 1 k 1 , x k + 1 N , y 1 N ) = lim N 1 N k = 1 N log 2 p ( x k | x 1 k 1 , x k + 1 N , y 1 N ) ,
y k = H k s k + w k = h k ( s k ) + w k .
y ^ k = H k s ^ k ,
s ^ k = E { s k | y 1 k 1 , x 1 k 1 } .
u k = y k y ^ k = H k ( s k s ^ k ) + w k .
Σ ^ k = E { ( s k s ^ k ) ( s k s ^ k ) H } ,
s ^ k + 1 = F ( s ^ k + K k u k ) ,
Σ ^ k + 1 = F Σ k F T + Q ,
Σ k = ( ( Σ ^ k ) 1 + σ 2 H k H H k ) 1 ,
K k = σ 1 Σ k H k H .
p ( x k | x 1 k 1 , y 1 k ) = p ( x k | y k , x 1 k 1 , y 1 k 1 ) = p ( x k | x 1 k 1 , y 1 k 1 ) p ( y k | x k , x 1 k 1 , y 1 k 1 ) x k X k p ( x k | x 1 k 1 , y 1 k 1 ) p ( y k | x k , x 1 k 1 , y 1 k 1 ) = p ( x k ) p ( y k | x 1 k , y 1 k 1 ) x k X k p ( x k ) p ( y k | x 1 k , y 1 k 1 ) ,
p ( y k | x 1 k , y 1 k 1 ) = S p ( s k , y k | x 1 k , y 1 k 1 ) d s k = S p ( s k | x 1 k , y 1 k 1 ) p ( y k | s k , x 1 k , y 1 k 1 ) d s k = S p ( s k | x 1 k 1 , y 1 k 1 ) p ( y k | s k , x k ) d s k = S g c ( s ^ k , Σ ^ k ; s k ) g c ( H k , s k , σ 2 2 ; y k ) d s k = g c ( H k s ^ k , H k H Σ ^ k H k + σ 2 2 ; y k ) .
p ( x k | x 1 k 1 , x k + 1 N , y 1 N ) = p ( x k ) p ( y k | x 1 N , y 1 k 1 , y k + 1 N ) x k X k p ( x k ) p ( y k | x 1 N , y 1 k 1 , y k + 1 N ) ,
p ( y k | x 1 N , y 1 k 1 , y k + 1 N ) = S p ( s k , y k | x 1 N , y 1 k 1 , y k + 1 N ) d s k = S p ( s k | x 1 N , y 1 k 1 , y k + 1 N ) p ( y k | s k , x 1 N , y 1 k 1 , y k + 1 N ) d s k = S p ( s k | x 1 k 1 , y 1 k 1 , x k + 1 N , y k + 1 N ) p ( y k | s k , x k ) d s k = S g c ( s ^ f b , k , Σ ^ f b , k ; s k ) g c ( H k s k , σ 2 2 ; y k ) d s k = g c ( H k s ^ f b , k , H k H Σ ^ f b , k H k + σ 2 2 ; y k ) ,
s ^ f b = Σ ^ b ( Σ ^ f + Σ ^ b ) 1 s ^ f + Σ f ( Σ ^ f + Σ ^ b ) 1 s ^ b ,
Σ ^ f b = ( Σ ^ f 1 + Σ ^ b 1 ) 1 .
λ ( z ) = v ( z ) ( 1 z p ) z 1 1 z p z 1 ,
E { λ k 2 } = 1 z p 1 + z p ,
SIR = 1 + z p 1 z p .
B 3 1 z p 2 π .

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