Abstract

A new upper bound (UB) on the information rate (IR) transferred through the additive white Gaussian noise channel affected by Wiener’s laser phase noise is proposed in the paper. The bound is based on Bayesian tracking of the noisy phase. Specifically, the predictive and posterior densities involved in the tracking are expressed in parametric form, therefore tracking is made on parameters. This make the method less computationally demanding than known non-parametric methods, e.g. methods based on phase quantization and trellis representation of phase memory. Simulation results show that the UB is so close to the lower bound that we can claim of having virtually computed the actual IR.

© 2016 Optical Society of America

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References

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  1. R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
    [Crossref]
  2. T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.
  3. 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 19(23), 22455–22461 (2011).
    [Crossref] [PubMed]
  4. G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory 34(6), 1437–1448 (1988).
    [Crossref]
  5. M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
    [Crossref]
  6. G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Selected Areas Commun. 23(9), 1748–1757 (2005).
    [Crossref]
  7. A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant invariant codes over channels with phase noise,” IEEE Trans. Commun. 55(11), 2125–2133 (2007).
    [Crossref]
  8. L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20(21), 23728–23734 (2012).
    [Crossref] [PubMed]
  9. M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
    [Crossref]
  10. 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 57(9), 5720–5736 (2011).
    [Crossref]
  11. P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Letters 6(5), 175–177 (2002).
    [Crossref]
  12. L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightw. Technol. 30(10), 1480–1486 (2012).
    [Crossref]
  13. A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. Commun. 59(12), 3223–3228 (2011).
    [Crossref]
  14. L. Barletta, M. Magarini, and A. Spalvieri, “Tight upper and lower bounds to the information rate of the phase noise channel,” IEEE Intern. Symposium Inf. Theory (2013), 2284–2288.
  15. 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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
    [Crossref]
  16. J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 54(1), 406–409 (2008).
    [Crossref]
  17. D. Simon, Optimal State Estimation (John Wiley & Sons, 2006).
    [Crossref]
  18. A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
    [Crossref]

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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

2012 (3)

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

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20(21), 23728–23734 (2012).
[Crossref] [PubMed]

M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
[Crossref]

2011 (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 57(9), 5720–5736 (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 19(23), 22455–22461 (2011).
[Crossref] [PubMed]

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. Commun. 59(12), 3223–3228 (2011).
[Crossref]

2010 (1)

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[Crossref]

2008 (1)

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 54(1), 406–409 (2008).
[Crossref]

2007 (2)

A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
[Crossref]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant invariant codes over channels with phase noise,” IEEE Trans. Commun. 55(11), 2125–2133 (2007).
[Crossref]

2005 (1)

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Selected Areas Commun. 23(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. Letters 6(5), 175–177 (2002).
[Crossref]

2000 (1)

M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
[Crossref]

1988 (1)

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory 34(6), 1437–1448 (1988).
[Crossref]

Barbieri, A.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. Commun. 59(12), 3223–3228 (2011).
[Crossref]

A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
[Crossref]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant invariant codes over channels with phase noise,” IEEE Trans. Commun. 55(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. Selected Areas Commun. 23(9), 1748–1757 (2005).
[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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

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

M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20(21), 23728–23734 (2012).
[Crossref] [PubMed]

L. Barletta, M. Magarini, and A. Spalvieri, “Tight upper and lower bounds to the information rate of the phase noise channel,” IEEE Intern. Symposium Inf. Theory (2013), 2284–2288.

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. Letters 6(5), 175–177 (2002).
[Crossref]

Bertolini, M.

Caire, G.

A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
[Crossref]

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Selected Areas Commun. 23(9), 1748–1757 (2005).
[Crossref]

Colavolpe, G.

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. Commun. 59(12), 3223–3228 (2011).
[Crossref]

A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
[Crossref]

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant invariant codes over channels with phase noise,” IEEE Trans. Commun. 55(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. Selected Areas Commun. 23(9), 1748–1757 (2005).
[Crossref]

Dauwels, J.

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 54(1), 406–409 (2008).
[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 57(9), 5720–5736 (2011).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[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. Letters 6(5), 175–177 (2002).
[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. Lightw. Technol. 28(7), 662–701 (2010).
[Crossref]

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory 34(6), 1437–1448 (1988).
[Crossref]

Galan, S.

M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
[Crossref]

Gavioli, G.

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 57(9), 5720–5736 (2011).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[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 57(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. Letters 6(5), 175–177 (2002).
[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 57(9), 5720–5736 (2011).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[Crossref]

Kubo, K.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

Loeliger, H.-A.

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 54(1), 406–409 (2008).
[Crossref]

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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

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

M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20(21), 23728–23734 (2012).
[Crossref] [PubMed]

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 19(23), 22455–22461 (2011).
[Crossref] [PubMed]

L. Barletta, M. Magarini, and A. Spalvieri, “Tight upper and lower bounds to the information rate of the phase noise channel,” IEEE Intern. Symposium Inf. Theory (2013), 2284–2288.

Miyata, Y.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

Mizuochi, T.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

Onohara, K.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

Peleg, M.

M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
[Crossref]

Pepe, M.

Shamai, S.

M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
[Crossref]

Simon, D.

D. Simon, Optimal State Estimation (John Wiley & Sons, 2006).
[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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

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

M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Staged demodulation and decoding,” Opt. Express 20(21), 23728–23734 (2012).
[Crossref] [PubMed]

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 19(23), 22455–22461 (2011).
[Crossref] [PubMed]

L. Barletta, M. Magarini, and A. Spalvieri, “Tight upper and lower bounds to the information rate of the phase noise channel,” IEEE Intern. Symposium Inf. Theory (2013), 2284–2288.

Sugihara, T.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

Vacondio, F.

Vannucci, G.

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory 34(6), 1437–1448 (1988).
[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 57(9), 5720–5736 (2011).
[Crossref]

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[Crossref]

Yoshida, H.

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

IEEE Commun. Letters (1)

P. Hou, B. J. Belzer, and T. R. Fischer, “Shaping gain of the partially coherent additive white Gaussian noise channel,” IEEE Commun. Letters 6(5), 175–177 (2002).
[Crossref]

IEEE J. Selected Areas Commun. (1)

G. Colavolpe, A. Barbieri, and G. Caire, “Algorithms for iterative decoding in the presence of strong phase noise,” IEEE J. Selected Areas Commun. 23(9), 1748–1757 (2005).
[Crossref]

IEEE Trans. Commun. (3)

A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant invariant codes over channels with phase noise,” IEEE Trans. Commun. 55(11), 2125–2133 (2007).
[Crossref]

A. Barbieri and G. Colavolpe, “On the information rate and repeat-accumulate code design for phase noise channel,” IEEE Trans. Commun. 59(12), 3223–3228 (2011).
[Crossref]

A. Barbieri, G. Colavolpe, and G. Caire, “Joint Iterative Detection and Decoding in the Presence of Phase Noise and Frequency Offset,” IEEE Trans. Commun. 55(1), 171–179 (2007).
[Crossref]

IEEE Trans. Inf. Theory (4)

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 57(9), 5720–5736 (2011).
[Crossref]

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 Trans. Inf. Theory 60(7), 3834–3844 (2014).
[Crossref]

J. Dauwels and H.-A. Loeliger, “Computation of information rates by particle methods,” IEEE Trans. Inf. Theory 54(1), 406–409 (2008).
[Crossref]

G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory 34(6), 1437–1448 (1988).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

M. Magarini, L. Barletta, and A. Spalvieri, “Efficient computation of the feedback filter for the hybrid decision feedback equalizer in highly dispersive channels,” IEEE Trans. Wirel. Commun. 11(6), 2245–2253 (2012).
[Crossref]

J. Lightw. Technol. (2)

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

R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightw. Technol. 28(7), 662–701 (2010).
[Crossref]

Opt. Express (2)

Proc. IEE Commun. (1)

M. Peleg, S. Shamai, and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. 147, 87–95 (2000).
[Crossref]

Other (3)

T. Mizuochi, Y. Miyata, K. Kubo, T. Sugihara, K. Onohara, and H. Yoshida, “Progress in soft-decision FEC,” OFC/NFOEC (2011), paper NWC2.

D. Simon, Optimal State Estimation (John Wiley & Sons, 2006).
[Crossref]

L. Barletta, M. Magarini, and A. Spalvieri, “Tight upper and lower bounds to the information rate of the phase noise channel,” IEEE Intern. Symposium Inf. Theory (2013), 2284–2288.

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Figures (1)

Fig. 1
Fig. 1 Lower bound and upper bounds for 4-QAM (left) and 16-QAM (right), with γ = 0.125. The achievable information rate of the pure AWGN channel (γ = 0) is also reported.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

y k = x k e j ϕ k + w k , ϕ k = [ ϕ k 1 + γ v i ] mod 2 π , k = 1 , 2 , ,
p ( ϕ k | ϕ k 1 , y 1 k 1 , x 1 k 1 ) = p ( ϕ k | ϕ k 1 ) .
p ( y k , x k | ϕ k , y 1 k 1 , x 1 k 1 ) = p ( y k , x k | ϕ k ) .
h ¯ ( U ) = lim n 1 n k = 1 n log 2 ( 1 q ( u k | u 0 k 1 ) ) ,
p ¯ ( ϕ k | z 1 k 1 ) = π π p ( ϕ k | ϕ k 1 ) p ( ϕ k 1 | z 1 k 1 ) d ϕ k 1 , p ( ϕ k | z 1 k ) = p ¯ ( ϕ k | z 1 k 1 ) p ( z k | ϕ k ) p ( z k | z 1 k 1 ) ,
p ( z k | z 1 k 1 ) = π π p ¯ ( ϕ k | z 1 k 1 ) p ( z k | ϕ k ) d ϕ k .
q ( ϕ k | y 1 k , x 1 k , ϕ k + 1 ) = p ( ϕ k + 1 | ϕ k ) q ( ϕ k | y 1 k , x 1 k ) π π p ( ϕ k + 1 | ϕ k ) q ( ϕ k | y 1 k , x 1 k ) d ϕ k , p ( ϕ k | ϕ k 1 ) = i = g ( ϕ k 1 + 2 π i , γ 2 ; ϕ k ) ,
p ( ϕ k 1 | y 1 k 1 , x 1 k 1 ) t ( ω k 1 ; ϕ k 1 ) = e { ω k 1 e j ϕ k 1 } 2 π I 0 ( | ω k 1 | ) ,
p ( ϕ k | ϕ k 1 ) g ( ϕ k 1 , γ 2 ; ϕ k ) .
p ¯ ( ϕ k | y 1 k 1 , x 1 k 1 ) g ( ϕ k 1 , γ 2 ; ϕ k ) t ( ω k 1 ; ϕ k 1 ) d ϕ k 1 t ( ω ¯ k ; ϕ k )
p ( y k | x k , ϕ k ) 2 SNR I 0 ( 2 SNR | y k x k | ) exp ( SNR ( | y k | 2 + | x k | 2 ) ) t ( 2 SNR y k x k ; ϕ k ) .
p ( ϕ k | y 1 k , x 1 k ) t ( 2 SNR y k x k + ω ¯ k ; ϕ k ) = t ( ω k ; ϕ k ) , ω k = 2 SNR y k x k + ω ¯ k .
p ( ϕ k + 1 | ϕ k ) q ( ϕ k | y 1 k , x 1 k ) d ϕ k = n = g ( ϕ k + 2 n π , γ 2 ; ϕ k + 1 ) g ( ω k , | ω k | 1 ; ϕ k ) d ϕ k = n = g ( 2 n π + ω k , γ 2 + | ω k | 1 ; ϕ k + 1 ) ,
q ( ϕ k 1 | y 1 k 1 ) = n = m m A n ( k 1 ) e j 4 n ϕ k 1 ,
q ¯ ( ϕ k | y 1 k 1 ) = π π n = m m A n ( k 1 ) e j 4 n ϕ k 1 g ( ϕ k 1 , γ 2 ; ϕ k ) d ϕ k 1 = n = m m A ¯ N ( k ) e j 4 n ϕ k ,
q ( y k | ϕ k ) = n = B n ( k ) e j 4 n ϕ k , B n ( k ) = 1 2 π | X | x k X 2 SNR I | 4 n | ( 2 SNR | y k x k | ) exp ( SNR ( | y k | 2 + | x k | 2 ) ) e j 4 n ( y k x k ) .
q ( ϕ k | y 1 k ) = n = m m A n ( k ) e j 4 n ϕ k .
A ˜ 0 ( k ) 2 i = 1 m | A ˜ i ( k ) | ,
q ( y k | y 1 k 1 ) = π π q ¯ ( ϕ k | y 1 k 1 ) q ( y k | ϕ k ) d ϕ k = 2 π A ˜ 0 ( k ) .

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