Abstract

Coding for the phase noise channel is investigated in the paper. Specifically, Wiener’s phase noise, which induces memory in the channel, is considered. A general coding principle for channels with memory is the interleaving of two or more codes. The interleaved codes are decoded in sequence, using past decisions to help future decoding. The paper proposes a method based on this principle, and shows its benefits through numerical results obtained by computer simulation. Analysis of the channel capacity given by the proposed method is also worked out in the paper.

© 2012 OSA

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  1. G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988).
    [CrossRef]
  2. M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
    [CrossRef]
  3. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28, 662–701 (2010).
    [CrossRef]
  4. T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.
  5. M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol.7, 901–914 (2009).
    [CrossRef]
  6. T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
    [CrossRef]
  7. X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
    [CrossRef]
  8. G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
    [CrossRef]
  9. A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun.11, 2125–2133 (2007).
    [CrossRef]
  10. A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. Commun.12, 3223–3228 (2011).
    [CrossRef]
  11. L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Technol. Lett.21, 1582–1584 (2011).
    [CrossRef]
  12. A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011).
    [CrossRef]
  13. M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
    [CrossRef]
  14. L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol.30, 1480–1486 (2012).
    [CrossRef]
  15. M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
    [CrossRef]
  16. M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun.4, 401–409 (1988).
    [CrossRef]
  17. H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,” in Proc. of IEEE Globecom (2001).
  18. T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007).
    [CrossRef]
  19. S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).
  20. A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002).
    [CrossRef]
  21. A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008).
    [CrossRef]

2012

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (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, 1480–1486 (2012).
[CrossRef]

2011

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. Commun.12, 3223–3228 (2011).
[CrossRef]

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

A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011).
[CrossRef]

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

2010

2009

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol.7, 901–914 (2009).
[CrossRef]

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
[CrossRef]

2008

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008).
[CrossRef]

2007

T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007).
[CrossRef]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun.11, 2125–2133 (2007).
[CrossRef]

2005

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
[CrossRef]

2002

A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002).
[CrossRef]

2000

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
[CrossRef]

1988

M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun.4, 401–409 (1988).
[CrossRef]

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988).
[CrossRef]

Barbieri, A.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. Commun.12, 3223–3228 (2011).
[CrossRef]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun.11, 2125–2133 (2007).
[CrossRef]

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
[CrossRef]

Barletta, L.

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

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

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

A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011).
[CrossRef]

Bertolini, M.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

Caire, G.

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
[CrossRef]

Cao, Y.

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

Colavolpe, G.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. Commun.12, 3223–3228 (2011).
[CrossRef]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun.11, 2125–2133 (2007).
[CrossRef]

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
[CrossRef]

Collins, O. M.

T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007).
[CrossRef]

Das, S.

S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).

Demir, A.

A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002).
[CrossRef]

Essiambre, R.-J.

Eyuboglu, M. V.

M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun.4, 401–409 (1988).
[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. Lightwave Technol.28, 662–701 (2010).
[CrossRef]

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988).
[CrossRef]

Galan, S.

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
[CrossRef]

Gavioli, G.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

Goebel, B.

Gu, W.

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

Hoffmann, S.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
[CrossRef]

Ji, Y.

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

Kramer, G.

Kubo, K.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Li, T.

T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007).
[CrossRef]

Li, X.

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

Magarini, M.

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

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

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

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008).
[CrossRef]

Miyata, Y.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Mizuochi, T.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Noe, R.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
[CrossRef]

Onohara, K.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Peleg, M.

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
[CrossRef]

Pepe, M.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

Pfau, T.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
[CrossRef]

Pfister, H. D.

H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,” in Proc. of IEEE Globecom (2001).

Schniter, P.

S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).

Shamai (Shitz), S.

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
[CrossRef]

Siegel, P. H.

H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,” in Proc. of IEEE Globecom (2001).

Soriaga, J. B.

H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,” in Proc. of IEEE Globecom (2001).

Spalvieri, A.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (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, 1480–1486 (2012).
[CrossRef]

A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011).
[CrossRef]

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

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008).
[CrossRef]

Sugihara, T.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Taylor, M. G.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol.7, 901–914 (2009).
[CrossRef]

Vacondio, F.

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

Vannucci, G.

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988).
[CrossRef]

Winzer, P. J.

Yoshida, H.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

Yu, S.

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

IEEE J. Sel. Areas Commun.

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Sel. Areas Commun.9, 1748–1757 (2005).
[CrossRef]

IEEE Photon. Technol. Lett.

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

M. Magarini, L. Barletta, A. Spalvieri, F. Vacondio, T. Pfau, M. Pepe, M. Bertolini, and G. Gavioli, “Pilot-symbols-aided carrier-phase recovery for 100-G PM-QPSK digital coherent receivers,” IEEE Photon. Technol. Lett.9, 739–741 (2012).
[CrossRef]

IEEE Trans. Circuits Syst. I

A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002).
[CrossRef]

IEEE Trans. Circuits Syst. II

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II55, 596–600 (2008).
[CrossRef]

IEEE Trans. Commun.

M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun.4, 401–409 (1988).
[CrossRef]

A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun.7, 1966–1974 (2011).
[CrossRef]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun.11, 2125–2133 (2007).
[CrossRef]

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. Commun.12, 3223–3228 (2011).
[CrossRef]

IEEE Trans. Inf. Theory

T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007).
[CrossRef]

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988).
[CrossRef]

J. Light-wave Technol.

M. G. Taylor, “Phase estimation methods for optical coherent detection using digital signal processing,” J. Light-wave Technol.7, 901–914 (2009).
[CrossRef]

J. Lightwave Technol.

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol.8, 989–999 (2009).
[CrossRef]

X. Li, Y. Cao, S. Yu, W. Gu, and Y. Ji, “A simplified feedforward carrier recovery algorithm for coherent optical QAM systems,” J. Lightwave Technol.5, 801–807 (2011).
[CrossRef]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28, 662–701 (2010).
[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, 1480–1486 (2012).
[CrossRef]

Opt. Express

M. Magarini, A. Spalvieri, F. Vacondio, M. Bertolini, M. Pepe, and G. Gavioli, “Empirical modeling and simulation of phase noise in long-haul coherent optical systems,” Opt. Express23, 22455–22461 (2011).
[CrossRef]

Proc. IEE Commun.

M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun.2, 87–95 (2000).
[CrossRef]

Other

S. Das and P. Schniter, “Noncoherent communication over the doubly selective channel via successive decoding and channel re-estimation,” in Proc. Annual Allerton Conf. on Commun., Control and Computing (2007).

H. D. Pfister, J. B. Soriaga, and P. H. Siegel, “On the achievable information rates for finite state ISI channels,” in Proc. of IEEE Globecom (2001).

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” in Optical Fiber Communication Conference (OFC/NFOEC) (March 6–10, 2011), pp. 1–3.

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Figures (5)

Fig. 1
Fig. 1

Example of two-stage coded sequence with pilots according to Eq. (1).

Fig. 2
Fig. 2

4-QAM, γ = 0.125, LDPC codes of length 64800 from the DVB-S2 standard. Performance of individual codes with iterative demodulation and decoding [8]. Solid line: first-level code. Dashed line: second-level code. The second-level code assumes ideal decoding of the first-level code.

Fig. 3
Fig. 3

4-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frame and the three two-level codes of Fig. 2. CBC indicates one-level coding with the algorithm of [8], while soft differential decoding indicates the algorithm of [15]. The performance is evaluated at bit error rate of 10−5 after 24 iterations.

Fig. 4
Fig. 4

16-QAM, γ = 0.125. The two-stage scheme is based on one pilot symbol per frame and M1 = 7, M2 = 9. CBC indicates one-level coding with the algorithm of [8]. The performance is evaluated at bit error rate of 10−5 after 24 iterations.

Fig. 5
Fig. 5

4-QAM and 16-QAM, M1 = 7, M2 = 9, γ = 0.125. The figure shows the term I(Y2;X1|Xp,Y1,Yp) of Eq. (5).

Equations (15)

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( p , c 2 , 1 , c 2 , 2 , c 2 , 3 , c 1 , 1 , c 2 , 4 , c 2 , 5 , c 2 , 6 ) ; ( p , c 2 , 7 , c 2 , 8 , c 2 , 9 , c 1 , 2 , c 2 , 10 , c 2 , 11 , c 2 , 12 ) ; ( p , ) ; ( p , c 2 , 6 N 5 , c 2 , 6 N 4 , c 2 , 6 N 3 , c 1 , N , c 2 , 6 N 2 , c 2 , 6 N 1 , c 2 , 6 N ) .
y k = ( x k + w k ) e j θ k ,
θ k = [ θ k 1 + γ ν k ] mod 2 π , k = 1 , 2 , ,
( f ) = 4 γ 2 T γ 4 + 16 π 2 f 2 T 2 ,
γ 2 = 2 π B FWHM T ,
R = R 1 ( M 1 1 ) + R 2 M 1 ( M 2 1 ) M 1 M 2 ,
I ( Y ; X ) = lim n 1 n I ( y 1 n ; x 1 n ) .
I ( Y ; X ) = I ( Y ; X p ) + I ( Y ; X 1 | X p ) + I ( Y ; X 2 | X 1 , X p ) ,
I ( Y ; X p ) = lim n 1 n I ( y 1 n ; x p , 1 n ) ,
I ( Y ; X p ) = 0
I ( Y ; X ) = I ( Y ; X 1 | X p ) + I ( Y ; X 2 | X 1 , X p ) .
I ( Y ; X 1 | X p ) = I ( Y 1 , Y p ; X 1 | X p ) + I ( Y 2 ; X 1 | X p , Y 1 , Y p ) ,
I ( Y ; X ) = I ( Y 1 , Y p ; X 1 | X p ) + I ( Y ; X 2 | X 1 , X p ) + I ( Y 2 ; X 1 | X p , Y 1 , Y p ) .
I ( Y 2 ; X 1 | X p , Y 1 , Y p ) = I ( Y ; X ) I ( Y 1 , Y p ; X 1 | X p ) I ( Y ; X 2 | X 1 , X p )
γ M 2 .

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