Abstract

A window-split frequency domain Kalman scheme is proposed in this paper for the equalization of large polarization mode dispersion (PMD) and ultra-fast rotation of state-of-polarization (RSOP) which is an extreme environment due to the Kerr effect and the Faraday effect under the lightning strike near the fiber cables. In order to carry out the proposed Kalman scheme, we give a simplified and equivalent fiber channel model as a replacement for the general model of the polarization effect of the co-existence of PMD and RSOP. With this fiber channel model, we can conduct compensation for PMD in the frequency domain and tracking RSOP in time domain. A half analytical and half empirical theory for the initialization of the process and measurement noise covariance is also presented in theory and verified by the numerical simulation. The performance of the proposed Kalman scheme is checked in the 28Gbaud PDM-QPSK coherent system built on both simulation and experiment platforms. The simulation and experiment results confirm that compared with the generally used constant modulus algorithm (CMA), the proposed scheme provides excellent performance and stability to cope with large range DGD from 20ps to 200ps and RSOP from 200krad/s to 2Mrad/s, with less computational complexity.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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    [Crossref]
  8. A. Gorshtein and D. Sadot, “Symbol Spaced Adaptive MIMO Equalization for Ultrahigh Bit Rate Metro Coherent Optical Links,” IEEE Photonics Technol. Lett. 25(5), 414–417 (2013).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  16. Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
    [Crossref]
  17. L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
    [Crossref]
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    [Crossref]
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  20. R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
    [Crossref]

2018 (1)

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

2017 (2)

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

2016 (3)

2015 (2)

2013 (2)

A. Gorshtein and D. Sadot, “Symbol Spaced Adaptive MIMO Equalization for Ultrahigh Bit Rate Metro Coherent Optical Links,” IEEE Photonics Technol. Lett. 25(5), 414–417 (2013).
[Crossref]

B. Szafraniec, T. S. Marshall, and B. Nebendahl, “Performance Monitoring and Measurement Techniques for Coherent Optical Systems,” J. Lightwave Technol. 31(4), 648–663 (2013).
[Crossref]

2011 (1)

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

2010 (1)

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J Select. Topics Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

2009 (1)

N. Kaneda and A. Leven, “Coherent Polarization-Division-Multiplexed QPSK Receiver with Fractionally Spaced CMA for PMD Compensation,” IEEE Photonics Technol. Lett. 21(4), 203–205 (2009).
[Crossref]

2000 (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Agrell, E.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Anandarajah, P. M.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Becker, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Bowers, J. E.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Cao, G.

Chraplyvy, A. R.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Cui, N.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Dreschmann, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Eggleton, B. J.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Feng, Y.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Fischer, J. K.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Freude, W.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Gisin, N.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Gorshtein, A.

A. Gorshtein and D. Sadot, “Symbol Spaced Adaptive MIMO Equalization for Ultrahigh Bit Rate Metro Coherent Optical Links,” IEEE Photonics Technol. Lett. 25(5), 414–417 (2013).
[Crossref]

Herrmann, M.

Hillerkuss, D.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Huebner, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Jain, A.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Jenau, F.

Josten, A.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Kaneda, N.

N. Kaneda and A. Leven, “Coherent Polarization-Division-Multiplexed QPSK Receiver with Fractionally Spaced CMA for PMD Compensation,” IEEE Photonics Technol. Lett. 21(4), 203–205 (2009).
[Crossref]

Koenig, S.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Kogelnik, H.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. U.S.A. 97(9), 4541–4550 (2000).
[Crossref] [PubMed]

Koos, C.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Krishnamurthy, P. K.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Krummrich, P. M.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

P. M. Krummrich, D. Ronnenberg, W. Schairer, D. Wienold, F. Jenau, and M. Herrmann, “Demanding response time requirements on coherent receivers due to fast polarization rotations caused by lightning events,” Opt. Express 24(11), 12442–12457 (2016).
[Crossref] [PubMed]

Kschischang, F. R.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Landais, P.

A. Jain, P. K. Krishnamurthy, P. Landais, and P. M. Anandarajah, “EKF for Joint Mitigation of Phase Noise, Frequency Offset and Nonlinearity in 400 Gb/s PM-16-QAM and 200 Gb/s PM-QPSK Systems,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Lau, A. P.

Leuthold, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Leven, A.

N. Kaneda and A. Leven, “Coherent Polarization-Division-Multiplexed QPSK Receiver with Fractionally Spaced CMA for PMD Compensation,” IEEE Photonics Technol. Lett. 21(4), 203–205 (2009).
[Crossref]

Li, L.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Lin, J.

Liu, J.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Lord, A.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Lu, C.

Magnus, K.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Marshall, T. S.

Meyer, J.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Muga, N. J.

Nebendahl, B.

B. Szafraniec, T. S. Marshall, and B. Nebendahl, “Performance Monitoring and Measurement Techniques for Coherent Optical Systems,” J. Lightwave Technol. 31(4), 648–663 (2013).
[Crossref]

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Pearce, M. B.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Pinto, A. N.

Prat, J.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Richardson, D. J.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Roberts, K.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Ronnenberg, D.

Sadot, D.

A. Gorshtein and D. Sadot, “Symbol Spaced Adaptive MIMO Equalization for Ultrahigh Bit Rate Metro Coherent Optical Links,” IEEE Photonics Technol. Lett. 25(5), 414–417 (2013).
[Crossref]

Savory, S. J.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

S. J. Savory, “Digital coherent optical receivers: algorithms and subsystems,” IEEE J Select. Topics Quantum Electron. 16(5), 1164–1179 (2010).
[Crossref]

Schairer, W.

Schmogrow, R.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Secondini, M.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Srinivasan, S.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Szafraniec, B.

Tang, X.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Tomkos, L.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Wienold, D.

Winter, M.

R. Schmogrow, B. Nebendahl, M. Winter, A. Josten, D. Hillerkuss, S. Koenig, J. Meyer, M. Dreschmann, M. Huebner, C. Koos, J. Becker, W. Freude, and J. Leuthold, “Error Vector Magnitude as a Performance Measure for Advanced Modulation Formats,” IEEE Photonics Technol. Lett. 24(1), 61–63 (2011).
[Crossref]

Winzer, P.

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Wu, X.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Xi, L.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Xu, H.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Yang, Y.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Y. Yang, G. Cao, K. Zhong, X. Zhou, Y. Yao, A. P. Lau, and C. Lu, “Fast polarization-state tracking scheme based on radius-directed linear Kalman filter,” Opt. Express 23(15), 19673–19680 (2015).
[Crossref] [PubMed]

Yao, Y.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Y. Yang, G. Cao, K. Zhong, X. Zhou, Y. Yao, A. P. Lau, and C. Lu, “Fast polarization-state tracking scheme based on radius-directed linear Kalman filter,” Opt. Express 23(15), 19673–19680 (2015).
[Crossref] [PubMed]

Zhang, Q.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Zhang, W.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Zhang, X.

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

Y. Feng, L. Li, J. Lin, H. Xu, W. Zhang, X. Tang, L. Xi, and X. Zhang, “Joint tracking and equalization scheme for multi-polarization effects in coherent optical communication systems,” Opt. Express 24(22), 25491–25501 (2016).
[Crossref] [PubMed]

Zhong, K.

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

Y. Yang, G. Cao, K. Zhong, X. Zhou, Y. Yao, A. P. Lau, and C. Lu, “Fast polarization-state tracking scheme based on radius-directed linear Kalman filter,” Opt. Express 23(15), 19673–19680 (2015).
[Crossref] [PubMed]

Zhou, X.

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[Crossref]

IEEE Photonics Technol. Lett. (3)

N. Kaneda and A. Leven, “Coherent Polarization-Division-Multiplexed QPSK Receiver with Fractionally Spaced CMA for PMD Compensation,” IEEE Photonics Technol. Lett. 21(4), 203–205 (2009).
[Crossref]

A. Gorshtein and D. Sadot, “Symbol Spaced Adaptive MIMO Equalization for Ultrahigh Bit Rate Metro Coherent Optical Links,” IEEE Photonics Technol. Lett. 25(5), 414–417 (2013).
[Crossref]

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[Crossref]

J. Lightwave Technol. (2)

J. Opt. (1)

E. Agrell, K. Magnus, A. R. Chraplyvy, D. J. Richardson, P. M. Krummrich, P. Winzer, K. Roberts, J. K. Fischer, S. J. Savory, B. J. Eggleton, M. Secondini, F. R. Kschischang, A. Lord, J. Prat, L. Tomkos, J. E. Bowers, S. Srinivasan, M. B. Pearce, and N. Gisin, “Roadmap on Optical Communications,” J. Opt. 18(6), 063002 (2016).
[Crossref]

Opt. Commun. (1)

Q. Zhang, Y. Yang, K. Zhong, J. Liu, X. Wu, and Y. Yao, “Joint polarization tracking and channel equalization based on radius-directed linear Kalman filter,” Opt. Commun. 407, 142–147 (2018).
[Crossref]

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Opt. Fiber Technol. (1)

L. Li, Y. Feng, W. Zhang, N. Cui, H. Xu, X. Tang, L. Xi, and X. Zhang, “A joint recovery scheme for carrier frequency offset and carrier phase noise using extended Kalman filter,” Opt. Fiber Technol. 36, 438–446 (2017).
[Crossref]

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C. Xie, “Polarization and nonlinear impairments in fiber communication systems,” in Enabling Technologies for High Spectral-Efficiency Coherent Optical Communication Networks, X. Zhou and C. Xie, ed. (Wiley, 2016).

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M. Kuscherov and M. Herrmann, “Lightning affects coherent optical transmission in aerial fiber,” (Lightwave, 2016), http://www.lightwaveonline.com/articles/2016/03/lightning-affects-coherent-optical-transmission-in-aerial-fiber.html .

H. Simon, Adaptive filter theory, Fourth Edition (Prentice Hall, 2002).

Z. Paul and M. Howard, Fundamentals of Kalman Filtering: A Practical Approach, Third Edition, (AIAA, 2009).

G. S. Mohinder and A. P. Andrews, Kalman Filtering: Theory and Practice Using Matlab® (Prentice-Hall, 1993).

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Figures (9)

Fig. 1
Fig. 1 Generalized polarization model in fiber span.
Fig. 2
Fig. 2 Simplified polarization impairment model.
Fig. 3
Fig. 3 Window-split structure and slide operation.
Fig. 4
Fig. 4 The diagram of Window-split structured frequency domain Kalman filter.
Fig. 5
Fig. 5 Simulation platform diagram.
Fig. 6
Fig. 6 q, r vs. log10(BER) with DGD-RSOP-ONSR: (a) 20ps-200krad/s-13dB (b) 200ps-2Mrad/s-13dB (c) 20ps-200krad/s-17dB (d) 200ps-2Mrad/s-17dB.
Fig. 7
Fig. 7 Performance evaluation (a) BER vs. OSNR (with 100ps DGD) (b) BER vs. RSOP (14dB OSNR, and different DGD) (c) PMD vector tracing curve (200ps-2Mrad/s, 14dB OSNR) (d) SOP rotation angle tracing (200ps-2Mrad/s, 17dB OSNR), Convergence curves of posterior covariance of state vector (2Mrad/s RSOP, 14dB OSNR, q = 10−4, r = 10) (e) with different DGD (f) with different q-r values (200ps DGD).
Fig. 8
Fig. 8 Experiment set up of 28GBaud PDM-QPSK system.
Fig. 9
Fig. 9 Q-factor vs. RSOP with different DGDs (OSNR = 17dB). (inset) Examples of recovery constellations by Kalman and CMA under the situation of 2Mra/s RSOP and 215.39ps DGD.

Tables (1)

Tables Icon

Table 1 Computational complexity comparison (per symbol)

Equations (20)

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U PMD ( ω )=( cosθ e jγ sinθ e jγ sinθ cosθ )( exp( jωΔτ/2 ) 0 0 exp( jωΔτ/2 ) ) ( cosθ e jγ sinθ e jγ sinθ cosθ ) 1 = R psp Λ( ω ) R psp 1
U( ω )= R 2 U PMD ( ω ) R 1
U( ω )= R 2 R psp Λ( ω ) R psp 1 R 1 = R 2 R psp Λ( ω ) R psp 1 ( R 2 1 R 2 ) R 1 =( R 2 R psp )Λ( ω ) ( R 2 R psp ) 1 ( R 2 R 1 ) = R newpsp Λ( ω ) R newpsp 1 R = U newPMD ( ω ) R
x= ( τ 1 , τ 2 , τ 3 ,κ,α,β ) T
 z=( S 0 ideal S 1 ideal ) and e( x )=z( x )h( x )=( Constant 0 )( u x u x * + u y u y * u x u x * u y u y * )
U comp ( ω )=cos( ωΔτ 2 )I j( τ σ ) Δτ sin( ωΔτ 2 )
R eq =( e jα cosκ e jβ sinκ e jβ sinκ e jα cosκ )
{ x k =f( x k1 )+ w k1 z k =h( x k )+ v k
Initialization: x ^ 0 =E[ x 0 ], P 0 =E[( x 0 x ^ 0 ) ( x 0 x ^ 0 ) T ]
Prediction: x ^ k|k1 = F k1 x ^ k1 , P k|k1 = F k1 P k1 F k1 T + Q k1
Correction: G k = P k|k1 H k T ( H k P k|k1 H k T + R k ) 1
x ^ k = x ^ k|k1 + G k (z H k x ^ k|k1 ), P k =(I G k H k ) P k|k1
x norm = ( τ 1 T s , τ 2 T s , τ 3 T s , κ 2π , α 2π , β 2π ) T
P k =(I G k H k )( P k1 +q I 6 ) G k =( P k1 +q I 6 ) H k T [ H k ( P k1 +q I 6 ) H k T +r I 2 ] 1
Tr( P k )=Tr( P k1 )
i,j m,n h ij 2 p i 2 +2q i,j m,n h ij 2 p i + q 2 i,j m,n h ij 2 mqr=0
p i =q± qr j n h ij 2
r>4 ( j n h ij 2 ) 2 q, q r <a upper boundary
g ij = q r h ji j n h ij 2 , q r >a lower boundary
e( x )=( 0 0 )( ( u x u x * r 1 )( u x u x * r 2 )( u x u x * r 3 ) ( u y u y * r 1 )( u y u y * r 2 )( u y u y * r 3 ) )

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