Abstract

We propose a joint blind equalization method for chromatic dispersion (CD) and ultra-fast rotation of state-of-polarization (RSOP) in a Stokes vector direct detection (SV-DD) system based on a new time-frequency domain Kalman filter structure. In an SV-DD system, the impairments induced by CD and RSOP possess a nonlinear form. Therefore, CD and RSOP cannot be treated sequentially, which causes difficulty in jointly equalizing these two impairments using an ordinary algorithm. The Kalman filter was proven to be effective in equalizing polarization effects in a coherent receiver. However, this approach has inherent limitations given that the Kalman filter was originally presented as a method implemented in the time domain whereas CD is eventually induced in the frequency domain. In this report, the proposed time-frequency domain Kalman filter can facilitate CD compensation in the frequency domain and RSOP equalization in the time domain by exploiting a sliding window structure. Both the CD compensation and the RSOP equalization are conducted in Stokes space when the proposed method is utilized, which is specially designed for an SV-DD system. The presented approach was checked using a 28 Gbaud 16-QAM SV-DD system simulation platform. The simulation results confirm that the method is very effective and has strong tolerance to CD (more than 2550 ps/nm, equivalent to a 150 km G. 652 fiber) combined with ultra-fast RSOP (up to 2 Mrad/s) for application in extreme polarization environments, like the transient lightning in a rainy day.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2018 (7)

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]

K. Zhong, X. Zhou, J. Huo, C. Yu, C. Lu, and A. Lau, “Digital signal processing for short-reach optical communications: A review of current technologies and future trends,” J. Lightwave Technol. 36(2), 377–400 (2018).
[Crossref]

T. Hoang, M. Sowailem, Q. Zhuge, M. Osman, A. Samani, C. Paquet, S. Paquet, I. Woods, and D. Plant, “Enabling high-capacity long-reach direct detection transmission with QAM-PAM Stokes vector modulation,” J. Lightwave Technol. 36(2), 460–467 (2018).
[Crossref]

G. Liu, L. Zhang, T. Zuo, and Q. Zhang, “IM/DD transmission techniques for emerging 5G fronthaul, DCI, and metro applications,” J. Lightwave Technol. 36(2), 560–567 (2018).
[Crossref]

Z. Zheng, N. Cui, H. Xu, X. Zhang, W. Zhang, L. Xi, Y. Fang, and L. Li, “Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system,” Opt. Express 26(6), 7211–7226 (2018).
[Crossref] [PubMed]

Y. Zhu, M. Jiang, and F. Zhang, “Direct detection of polarization multiplexed single sideband signals with orthogonal offset carriers,” Opt. Express 26(12), 15887–15898 (2018).
[Crossref] [PubMed]

N. Cui, X. Zhang, Z. Zheng, H. Xu, W. Zhang, X. Tang, L. Xi, Y. Fang, and L. Li, “Two-parameter-SOP and three-parameter-RSOP fiber channels: problem and solution for polarization demultiplexing using Stokes space,” Opt. Express 26(16), 21170–21183 (2018).
[Crossref] [PubMed]

2017 (6)

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

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]

D. Charlton, S. Clarke, D. Doucet, M. O’Sullivan, D. L. Peterson, D. Wilson, G. Wellbrock, and M. Bélanger, “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017).
[Crossref] [PubMed]

Y. Zhu, K. Zou, Z. Chen, and F. Zhang, “224 Gb/s optical carrier-assisted Nyquist 16-QAM half-cycle single-sideband direct detection transmission over 160 km SSMF,” J. Lightwave Technol. 35(9), 1557–1565 (2017).
[Crossref]

T. Hoang, M. Sowailem, Q. Zhuge, Z. Xing, M. Morsy-Osman, E. El-Fiky, S. Fan, M. Xiang, and D. Plant, “Single wavelength 480 Gb/s direct detection over 80km SSMF enabled by Stokes vector Kramers Kronig transceiver,” Opt. Express 25(26), 33534–33542 (2017).
[Crossref]

M. Sowailem, T. Hoang, M. Morsy-Osman, M. Chagnon, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. Plant, “Impact of chromatic dispersion compensation in single carrier two-dimensional Stokes vector direct detection system,” IEEE Photonics J. 9(4), 7203110 (2017).
[Crossref]

2016 (6)

2015 (2)

2014 (2)

2010 (1)

Bai, C.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Bélanger, M.

Cao, G.

Chagnon, M.

M. Sowailem, T. Hoang, M. Morsy-Osman, M. Chagnon, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. Plant, “Impact of chromatic dispersion compensation in single carrier two-dimensional Stokes vector direct detection system,” IEEE Photonics J. 9(4), 7203110 (2017).
[Crossref]

M. Y. Sowailem, T. M. Hoang, M. Chagnon, M. Morsy-Osman, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. V. Plant, “100G and 200G single carrier transmission over 2880 and 320 km using an InP IQ modulator and Stokes vector receiver,” Opt. Express 24(26), 30485–30493 (2016).
[Crossref] [PubMed]

Charlton, D.

Che, D.

D. Che and W. Shieh, “Polarization demultiplexing for Stokes vector direct detection,” J. Lightwave Technol. 34(2), 754–760 (2016).
[Crossref]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Chen, X.

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Chen, Z.

Clarke, S.

Corcoran, B.

Cui, N.

Doucet, D.

El-Fiky, E.

Fan, S.

Fang, Y.

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]

Gorza, S.

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

Hoang, T.

Hoang, T. M.

Horlin, F.

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

Hu, Q.

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Huo, J.

Inoue, T.

Jiang, M.

Jignesh, J.

Jones, R.

Lau, A.

Lau, A. P.

Li, A.

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Li, L.

Liboiron-Ladouceur, O.

M. Sowailem, T. Hoang, M. Morsy-Osman, M. Chagnon, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. Plant, “Impact of chromatic dispersion compensation in single carrier two-dimensional Stokes vector direct detection system,” IEEE Photonics J. 9(4), 7203110 (2017).
[Crossref]

M. Y. Sowailem, T. M. Hoang, M. Chagnon, M. Morsy-Osman, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. V. Plant, “100G and 200G single carrier transmission over 2880 and 320 km using an InP IQ modulator and Stokes vector receiver,” Opt. Express 24(26), 30485–30493 (2016).
[Crossref] [PubMed]

Lin, J.

Liu, G.

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]

Louveaux, J.

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

Lowery, A.

Lu, C.

Marshall, T.

Morsy-Osman, M.

Muga, N.

Namiki, S.

Nebendahl, B.

Nguyen, T.

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

O’Sullivan, M.

Osman, M.

Pakala, L.

Paquet, C.

Paquet, S.

Peterson, D. L.

Piels, M.

Pinto, A.

Plant, D.

Plant, D. V.

Qiu, M.

M. Sowailem, T. Hoang, M. Morsy-Osman, M. Chagnon, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. Plant, “Impact of chromatic dispersion compensation in single carrier two-dimensional Stokes vector direct detection system,” IEEE Photonics J. 9(4), 7203110 (2017).
[Crossref]

M. Y. Sowailem, T. M. Hoang, M. Chagnon, M. Morsy-Osman, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. V. Plant, “100G and 200G single carrier transmission over 2880 and 320 km using an InP IQ modulator and Stokes vector receiver,” Opt. Express 24(26), 30485–30493 (2016).
[Crossref] [PubMed]

Rottenberg, F.

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

Samani, A.

Schäeffer, C.

Schmauss, B.

Shieh, W.

D. Che and W. Shieh, “Polarization demultiplexing for Stokes vector direct detection,” J. Lightwave Technol. 34(2), 754–760 (2016).
[Crossref]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Sowailem, M.

Sowailem, M. Y.

Szafraniec, B.

Tang, X.

N. Cui, X. Zhang, Z. Zheng, H. Xu, W. Zhang, X. Tang, L. Xi, Y. Fang, and L. Li, “Two-parameter-SOP and three-parameter-RSOP fiber channels: problem and solution for polarization demultiplexing using Stokes space,” Opt. Express 26(16), 21170–21183 (2018).
[Crossref] [PubMed]

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

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]

Wang, Y.

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “Stokes vector direct detection for short-reach optical communication,” Opt. Lett. 39(11), 3110–3113 (2014).
[Crossref] [PubMed]

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

Wellbrock, G.

Wilson, D.

Woods, I.

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.

Xiang, M.

Xing, Z.

Xu, H.

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]

Yu, C.

Zhang, F.

Zhang, L.

Zhang, Q.

G. Liu, L. Zhang, T. Zuo, and Q. Zhang, “IM/DD transmission techniques for emerging 5G fronthaul, DCI, and metro applications,” J. Lightwave Technol. 36(2), 560–567 (2018).
[Crossref]

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.

Zhang, X.

Zheng, H.

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
[Crossref]

Zheng, Z.

Zhong, K.

Zhou, X.

Zhu, C.

Zhu, Y.

Zhuge, Q.

Zibar, D.

Zou, K.

Zuo, T.

IEEE Photonics J. (2)

M. Sowailem, T. Hoang, M. Morsy-Osman, M. Chagnon, M. Qiu, S. Paquet, C. Paquet, I. Woods, O. Liboiron-Ladouceur, and D. Plant, “Impact of chromatic dispersion compensation in single carrier two-dimensional Stokes vector direct detection system,” IEEE Photonics J. 9(4), 7203110 (2017).
[Crossref]

H. Xu, X. Zhang, X. Tang, C. Bai, L. Xi, W. Zhang, and H. Zheng, “Joint scheme of dynamic polarization demultiplexing and PMD compensation up to second-order for flexible receivers,” IEEE Photonics J. 9(6), 7204615 (2017).
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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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T. Inoue and S. Namiki, “Carrier recovery for M-QAM signals based on a block estimation process with Kalman filter,” Opt. Express 22(13), 15376–15387 (2014).
[Crossref] [PubMed]

J. Jignesh, B. Corcoran, C. Zhu, and A. Lowery, “Unscented Kalman filters for polarization state tracking and phase noise mitigation,” Opt. Express 24(19), 22282–22295 (2016).
[Crossref] [PubMed]

L. Pakala and B. Schmauss, “Extended Kalman filtering for joint mitigation of phase and amplitude noise in coherent QAM systems,” Opt. Express 24(6), 6391–6401 (2016).
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Z. Zheng, N. Cui, H. Xu, X. Zhang, W. Zhang, L. Xi, Y. Fang, and L. Li, “Window-split structured frequency domain Kalman equalization scheme for large PMD and ultra-fast RSOP in an optical coherent PDM-QPSK system,” Opt. Express 26(6), 7211–7226 (2018).
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N. Cui, X. Zhang, Z. Zheng, H. Xu, W. Zhang, X. Tang, L. Xi, Y. Fang, and L. Li, “Two-parameter-SOP and three-parameter-RSOP fiber channels: problem and solution for polarization demultiplexing using Stokes space,” Opt. Express 26(16), 21170–21183 (2018).
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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).
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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).
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D. Charlton, S. Clarke, D. Doucet, M. O’Sullivan, D. L. Peterson, D. Wilson, G. Wellbrock, and M. Bélanger, “Field measurements of SOP transients in OPGW, with time and location correlation to lightning strikes,” Opt. Express 25(9), 9689–9696 (2017).
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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).
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Opt. Lett. (2)

Other (5)

Q. Shang, Z. Zheng, N. Cui, N. Zhang, W. Zhang, H. Xu, X. Tang, L. Xi, and X. Zhang, “RSOP equalization through an extend Kalman filter scheme in Stokes vector direct detection system,” in Conference on Lasers and Electro-Optics (CLEO, 2018), paper JTu2A.47.
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H. Yaffe, “Are ultrafast SOP events affecting your coherent receivers?” https://newridgetech.com/are-ultrafast-sop-events-affecting-your-receivers .

D. Che, A. Li, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “160-Gb/s Stokes vector direct detection for short reach optical communication,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), paper Th5C.7.
[Crossref]

T. Nguyen, F. Rottenberg, S. Gorza, J. Louveaux, and F. Horlin, “Extended Kalman Filter for Carrier Phase Recovery in Optical Filter Bank Multicarrier Offset QAM Systems,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), paper Th4C.3.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 (a) SV-DD system transmitter. (b) SV-DD system receiver.
Fig. 2
Fig. 2 Stokes space representations of the 16QAM signal (a) ideal signal. (b) signal suffering from the accumulated CD of 300 ps/nm.
Fig. 3
Fig. 3 The diagram of the sliding window structure.
Fig. 4
Fig. 4 The flow chart of the time-frequency domain KF.
Fig. 5
Fig. 5 Numerical simulation system framework.
Fig. 6
Fig. 6 Performance evaluation (a) BER vs. the accumulate CD for different window lengths of the Kalman filter (24 dB OSNR, 1 Mrad/s RSOP). (b) BER vs. RSOP for different window lengths of the Kalman filter (24 dB OSNR, 510ps/nm accumulate CD).
Fig. 7
Fig. 7 Performance evaluation (a) BER vs. OSNR for different accumulated CD and RSOP. (b) OSNR penalty as a function of different accumulated CD and RSOP. (c) BER vs. RSOP. (d) BER vs. the residual CD.
Fig. 8
Fig. 8 Performance evaluation (a) BER vs. different accumulated CD (OSNR = 24 dB). (b) the accumulated CD estimation mean error vs. OSNR (1 Mrad/s RSOP, 2040 ps/nm accumulated CD). (c) the accumulated CD vector tracing curve.

Tables (1)

Tables Icon

Table 1 Computational complexity comparison

Equations (15)

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J t =[ E x t C ]
S t =[ S 1 t S 2 t S 3 t ]=[ | E x t | 2 |C | 2 2Re( E x t C ) 2Im( E x t C ) ]
J r =[ E x r E y r ]= D ^ U ^ [ E x t C ]=exp[ j( β 2 L 2 ) 2 t 2 ][ a b b a ][ E x t C ]
S 2 r +j S 3 r = D { 2 E x t C } = D { S 2 t +j S 3 t }
S 2 r +j S 3 r =2 E x r E y r = a 2 D { S 2 t +j S 3 t } b 2 ( D { S 2 t +j S 3 t } )2ab( | D E x t | 2 | C | 2 )
H ^ J 1 =( cosκ e jς sinκ e jη sinκ e jη cosκ e jς )
H ^ S 1 =( cos2κ sin2κcos( η+ζ ) sin2κsin( η+ζ ) sin2κcos( ηζ ) cos 2 κcos2ζ sin 2 κcos2η sin 2 κsin2η cos 2 κsin2ζ sin2κsin( ηζ ) cos 2 κsin2ζ+ sin 2 κsin2η cos 2 κcos2ζ+ sin 2 κcos2η )
D ^ 1 ( Φ,ω )=exp( j Φ λ 2 ω 2 4πc )
x= (κ,η,ζ,Φ) T
e( x )=z( x )h( x )=0( u x u x * r 1 2 )( u x u x * r 2 2 )( u x u x * r 3 2 )
x ^ k|k1 = x ^ k1
P k|k1 = P k1 + Q k1
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 k h k ( x ^ k|k1 ))
P k =(I G k H k ) P k|k1

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