Abstract

A new lower bound below the information rate transferred through the Additive White Gaussian Noise (AWGN) channel affected by discrete-time multiplicative Wiener’s phase noise is proposed in the paper. The proposed lower bound is based on the Kalman approach to data-aided carrier phase recovery, and is less computationally demanding than known methods based on phase quantization and trellis representation of phase’s memory. Simulation results show that the lower bound is close to the actual channel capacity, especially at low-to-intermediate signal-to-noise ratio.

© 2012 OSA

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  1. G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 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. Express 23, 22455–22461 (2011).
    [CrossRef]
  3. 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]
  4. 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]
  5. A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. on Commun. 11, 2125–2133 (2007).
    [CrossRef]
  6. L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012).
    [CrossRef]
  7. A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun. 7, 1966–1974 (2011).
    [CrossRef]
  8. 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. Tech. Lett. 9, 739–741 (2012).
    [CrossRef]
  9. B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
    [CrossRef]
  10. P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).
  11. D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
    [CrossRef]
  12. I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
    [CrossRef]
  13. J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008).
    [CrossRef]
  14. L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. Lett. 21, 1582–1584 (2011).
    [CrossRef]
  15. A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on Commun. 12, 3223–3228 (2011).
    [CrossRef]
  16. L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012).
    [CrossRef]
  17. D. Simon, Optimal State Estimation (Wiley, 2006).
    [CrossRef]
  18. A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999).
    [CrossRef]
  19. A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, 12, 596–600 (2008).

2012 (3)

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. Tech. 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,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012).
[CrossRef]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012).
[CrossRef]

2011 (5)

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. Express 23, 22455–22461 (2011).
[CrossRef]

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (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]

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

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

2008 (2)

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008).
[CrossRef]

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, 12, 596–600 (2008).

2007 (1)

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

2006 (1)

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

2005 (2)

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[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]

2002 (1)

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

2000 (1)

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]

1999 (1)

A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999).
[CrossRef]

1988 (1)

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988).
[CrossRef]

Arnold, D. M.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

Barbieri, A.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on 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. on 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.

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. Tech. 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,” IEEE J. Lightw. Technol. 30, 1480–1486 (2012).
[CrossRef]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (2012).
[CrossRef]

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. 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]

Belzer, B. J.

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

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. Tech. 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. Express 23, 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]

Colavolpe, G.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on 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. on 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]

Dauwels, J.

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008).
[CrossRef]

Djordjevic, I. B.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[CrossRef]

Essiambre, R.-J.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

Fischer, T. R.

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

Foschini, G. J.

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988).
[CrossRef]

Gabitov, I.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[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. Tech. 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. Express 23, 22455–22461 (2011).
[CrossRef]

Goebel, B.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

Hanik, N.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

Hou, P.

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

Ivkovic, M.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[CrossRef]

Kavcic, A.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

Kramer, G.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

Loeliger, H.-A.

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008).
[CrossRef]

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

Magarini, M.

L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” IEEE J. Lightw. 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. Tech. Lett. 9, 739–741 (2012).
[CrossRef]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (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. Express 23, 22455–22461 (2011).
[CrossRef]

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

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, 12, 596–600 (2008).

Patapoutian, A.

A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999).
[CrossRef]

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. Tech. 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. Express 23, 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. Tech. Lett. 9, 739–741 (2012).
[CrossRef]

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]

Simon, D.

D. Simon, Optimal State Estimation (Wiley, 2006).
[CrossRef]

Spalvieri, A.

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (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. Tech. 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,” IEEE J. Lightw. Technol. 30, 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 Photon. Tech. 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]

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. Express 23, 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. II, Exp. Briefs, 12, 596–600 (2008).

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. Tech. 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. Express 23, 22455–22461 (2011).
[CrossRef]

Vannucci, G.

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988).
[CrossRef]

Vasic, B.

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[CrossRef]

Vontobel, P. O.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

Winzer, P. J.

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

Zeng, W.

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

IEEE Commun. Lett. (1)

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Lett. 5, 175–177 (2002).

IEEE J. Lightw. Technol. (2)

I. B. Djordjevic, B. Vasic, M. Ivkovic, and I. Gabitov, “Achievable information rates fot high-speed long-haul optical transmission,” IEEE J. Lightw. Technol. 11, 3755–3763 (2005).
[CrossRef]

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

IEEE J. Sel. Areas Commun. (1)

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. Tech. Lett. (2)

L. Barletta, M. Magarini, and A. Spalvieri, “Estimate of information rates of discrete-time first-order Markov phase noise channels,” IEEE Photon. Tech. 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. Tech. Lett. 9, 739–741 (2012).
[CrossRef]

IEEE Trans. Circuits Syst. II, Exp. Briefs (1)

A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II, Exp. Briefs, 12, 596–600 (2008).

IEEE Trans. Commun. (2)

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

A. Patapoutian, “On phase-locked loops and Kalman filters,” IEEE Trans. Commun. 5,, 670–672 (1999).
[CrossRef]

IEEE Trans. Inf. Theory (3)

B. Goebel, R.-J. Essiambre, G. Kramer, P. J. Winzer, and N. Hanik, “Calculation of mutual information for partially coherent Gaussian channels with application to fiber optics,” IEEE Trans. Inf. Theory 9, 5720–5736 (2011).
[CrossRef]

D. M. Arnold, H.-A. Loeliger, P. O. Vontobel, A. Kavcic, and W. Zeng, “Simulation-based computation of information rates for channels with memory,” IEEE Trans. Inf. Theory 8, 3498–3508 (2006).
[CrossRef]

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 1, 406–409 (2008).
[CrossRef]

IEEE Trans. Inform. Theory (1)

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory 6, 1437–1448 (1988).
[CrossRef]

IEEE Trans. on Commun. (2)

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channels,” IEEE Trans. on 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. on Commun. 11, 2125–2133 (2007).
[CrossRef]

Opt. Express (2)

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20, 23728–23734 (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. Express 23, 22455–22461 (2011).
[CrossRef]

Proc. IEE Commun. (1)

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 (1)

D. Simon, Optimal State Estimation (Wiley, 2006).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Actual channel capacity and lower bound for 4-QAM and two values of γ. Capacity of the pure AWGN channel is also reported.

Fig. 2
Fig. 2

Actual channel capacity and lower bound for 16-QAM and two values of γ. Capacity of the pure AWGN channel is also reported.

Equations (20)

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y k = x k e j φ k + w k , k = 1 , 2 , ,
ϕ k = ϕ k 1 + γ v i , k = 1 , 2 ,
( f ) = 4 γ 2 T γ 4 + 16 π 2 f 2 T 2 ,
γ 2 = 2 π B F W H M T ,
lim n 1 n E p { log 2 p ( u 1 n ) q ( u 1 n ) } 0 ,
H ¯ ( U ) = lim n 1 n E p { log 2 ( 1 q ( u 1 n ) ) } H ( U ) .
H ¯ ( U ) = lim n 1 n k = 1 n log 2 ( 1 q ( u k | u 0 k 1 ) ) ,
H ( X ) H ¯ ( X | Y ) I ( X ; Y ) ,
H ¯ ( X | Y ) H ( X | Y )
H ¯ ( X | Y ) = lim n 1 n k = 1 n log 2 ( 1 q ( x k | x 1 k 1 , y 1 n ) ) .
q ( x k | y 1 k , x 1 k 1 ) = π π q ( ϕ k , x k | y 1 k , x 1 k 1 ) d ϕ k = π π p ( x k | y k , ϕ k ) q ( ϕ k | y 1 k , x 1 k 1 ) d ϕ k π π q ( ϕ k | y 1 k , x 1 k 1 ) p ( y k | x k , ϕ k ) p ( x k ) d ϕ k ,
q ( ϕ k | y 1 k , x 1 k 1 ) q ( ϕ k | y 1 k 1 , x 1 k 1 ) p ( y k | ϕ k ) = q ( ϕ k | y 1 k 1 , x 1 k 1 ) x k 𝒳 p ( y k | ϕ k , x k ) p ( x k ) ,
q ( ϕ k | y 1 k 1 , x 1 k 1 ) = g ( ϕ ^ k , σ k 2 ; ϕ k ) ,
ϕ ^ k = E { ϕ k | y 1 k 1 , x 1 k 1 } ,
σ k 2 = E { ( ϕ ^ k ϕ k ) 2 | y 1 k 1 , x 1 k 1 } .
e k = { y k x k e j ϕ ^ k } ,
e k ϕ k ϕ ^ k + z k ,
ϕ ^ k + 1 = ϕ ^ k + K k e k ,
σ k + 1 2 = σ z , k 2 σ k 2 σ z , k 2 + σ k 2 + γ 2 ,
K k = σ k 2 σ k 2 + σ z , k 2 .

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