## Staged demodulation and decoding |

Optics Express, Vol. 20, Issue 21, pp. 23728-23734 (2012)

http://dx.doi.org/10.1364/OE.20.023728

Acrobat PDF (698 KB)

### 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

## 1. Introduction

1. G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. 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. 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]

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

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]

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]

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]

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]

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]

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]

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]

*p*represents one pilot symbol and

*c*is the

_{i,j}*j*-th symbol of the

*i*-th level code

*𝒞*. In Eq. (1), one frame is inserted between parentheses, the entire sequence consists of

_{i}*N*frames, and the length of code

*𝒞*

_{2}is 6

*N*while the length of code

*𝒞*

_{1}is

*N*. For the sake of correctness, only two-level constructions will be studied, and the extension to multilevel constructions becomes straightforward.

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]

18. T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory **2**, 628–646 (2007). [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]

- While in [16] and [18
**4**, 401–409 (1988). [CrossRef]] only one code is considered, here the use of a set of different channel codes is proposed.18. T. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory

**2**, 628–646 (2007). [CrossRef] - While in [16
**4**, 401–409 (1988). [CrossRef]**2**, 628–646 (2007). [CrossRef]

## 2. Channel and system model

*k*-th received sample

*y*is where

_{k}*x*is the

_{k}*k*-th transmitted symbol,

*w*is the

_{k}*k*-th sample of AWGN, and

*θ*is the

_{k}*k*-th sample of phase noise. The phase noise is hereafter modeled as a discrete-time Wiener process where γ > 0 is a known parameter,

*θ*

_{0}is uniformly distributed in [0,2π), and

*v*is the

_{k}*k*-th sample of white Gaussian noise with zero mean and unit variance. The phase evolution given in Eq. (2) occurs when the power spectral density of the continuous-time complex exponential

*e*

^{jθ (t)}, whose samples at symbol frequency generate the sequence

*e*, is the Lorentzian function where

^{jθk}*T*is the symbol repetition interval and

*f*is the frequency. The parameter γ

^{2}can be expressed as where

*B*

_{FWHM}is the full-width half-maximum bandwidth of the spectral line.

*M*

_{1}

*−*1 is the number of symbols of code

*𝒞*

_{1}in one frame,

*R*

_{1}is the information rate of code

*𝒞*

_{1},

*M*

_{2}

*−*1 is the number of consecutive symbols of code

*𝒞*

_{2},

*R*

_{2}is the information rate of code

*𝒞*

_{2}, and

*M*

_{1}

*·M*

_{2}is the total number of symbols in one frame. In the example (Eq. (1)), illustrated in Fig. 1,

*M*

_{1}= 2,

*M*

_{2}= 4.

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]

## 3. Analysis of channel capacity

*X*be the deterministic sequence of pilot symbols and let

_{p}*X*

_{1}and

*X*

_{2}be the random processes of symbols of the first-level and of the second-level, respectively. Similarly, the received sequence is divided in three parts called

*Y*,

_{p}*Y*

_{1},

*Y*

_{2}, where

*Y*corresponds to the time instants where pilot symbols

_{p}*X*are transmitted, while

_{p}*Y*corresponds to the time instants where symbols of level

_{i}*i*are transmitted.

*x*

_{1},

*x*

_{2}, ⋯,

*x*) and the channel output vector (

_{n}*y*

_{1},

*y*

_{2}, ⋯,

*y*), respectively. The information rate between

_{n}*Y*and

*X*is By the chain rule on

*X*one writes where, here and in what follows, the information rate of each one of the sub-channels is computed by dividing the information between vectors by the number of uses of the composite channel, e.g. where

*X*is a known sequence, therefore Invoking the chain rule on

_{p}*Y*, the first term in the right side of the above equation is leading to The term

*I*(

*Y*

_{2};

*X*

_{1}|

*X*,

_{p}*Y*

_{1},

*Y*) appearing in the above equation is the contribution coming from the blind processing of

_{p}*Y*

_{2}at the first stage. In our proposal we suggest to renounce to this contribution. It can be computed from Eq. (4) as where the three terms in the right side of Eq. (5) can be computed as in [14

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]

*I*(

*Y;X*) one uses pilot symbols inserted with period

*M*

_{1}·

*M*

_{2}, for

*I*(

*Y;X*

_{2}|

*X*

_{1},

*X*) one uses pilot symbols inserted with period

_{p}*M*

_{2}, while for

*I*(

*Y*

_{1},

*Y*;

_{p}*X*

_{1}|

*X*) one uses pilot symbols inserted with period

_{p}*M*

_{1}and Wiener phase noise with step

## 4. Numerical results

**9**, 1748–1757 (2005). [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]

**2**, 87–95 (2000). [CrossRef]

**2**, 87–95 (2000). [CrossRef]

## 5. Conclusions

20. A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I **49**, 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. II **55**, 596–600 (2008). [CrossRef]

## References and links

1. | G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory |

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 |

3. | R.-J. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

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. |

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. |

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. |

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. | A. Barbieri and G. Colavolpe, “Soft-output decoding of rotationally invariant codes over channels with phase noise,” IEEE Trans. Commun. |

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

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. |

12. | A. Spalvieri and L. Barletta, “Pilot-aided carrier recovery in the presence of phase noise,” IEEE Trans. Commun. |

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. |

14. | L. Barletta, M. Magarini, and A. Spalvieri, “The information rate transferred through the discrete-time Wiener’s phase noise channel,” J. Lightwave Technol. |

15. | M. Peleg, S. Shamai (Shitz), and S. Galan, “Iterative decoding for coded noncoherent MPSK communications over phase-noisy AWGN channel,” Proc. IEE Commun. |

16. | M. V. Eyuboglu, “Detection of coded modulation signals on linear severely distorted channels using decision-feedback noise prediction and interleaving,” IEEE Trans. Commun. |

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. Theory |

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. I |

21. | A. Spalvieri and M. Magarini, “Wiener’s analysis of the discrete-time phase-locked loop with loop delay,” IEEE Trans. Circuits Syst. II |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4080) Fiber optics and optical communications : Modulation

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: August 21, 2012

Revised Manuscript: September 10, 2012

Manuscript Accepted: September 10, 2012

Published: October 1, 2012

**Citation**

Luca Barletta, Maurizio Magarini, and Arnaldo Spalvieri, "Staged demodulation and decoding," Opt. Express **20**, 23728-23734 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-21-23728

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### References

- G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inf. Theory6, 1437–1448 (1988). [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]
- 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]
- 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.
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- 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. Li and O. M. Collins, “A successive decoding strategy for channels with memory,” IEEE Trans. Inf. Theory2, 628–646 (2007). [CrossRef]
- 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).
- A. Demir, “Phase noise and timing jitter in oscillators with colored-noise sources,” IEEE Trans. Circuits Syst. I49, 1782–1791 (2002). [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]

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