Abstract

We propose a flexible simplified extended Kalman filter (S-EKF) scheme that can be applied in both pilot-aided and blind modes for phase noise compensation in 16-QAM CO-OFDM transmission systems employing a small-to-moderate number of subcarriers. The performance of the proposed algorithm is evaluated and compared with conventional pilot-aided (PA) and blind phase search (BPS) methods via extensive an Monte Carlo simulation in a back-to-back configuration and with a dual polarization fiber transmission. For 64 subcarrier 32 Gbaud 16-QAM CO-OFDM systems with 200 kHz combined laser linewidths, an optical signal-to-noise ratio penalty as low as 1 dB can be achieved with the proposed S-EKF scheme using only 2 pilots in the pilot-aided mode and just 4 inputs in the blind mode, resulting in a spectrally efficient enhancement by a factor of 3 and a computational effort reduction by a factor of more than 50 in comparison with the conventional PA and the BPS methods, respectively.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article

Corrections

9 November 2017: A typographical correction was made to the article title.


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References

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    [Crossref]
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    [Crossref]
  4. S. T. Le, P. A. Haigh, A. D. Ellis, and S. K. Turitsyn, “Blind Phase Noise Estimation for CO-OFDM Transmissions,” JJ. Lightwave Technol. 34, 745–753 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
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2017 (1)

T. H. Nguyen and C. Peucheret, “Kalman filtering for carrier phase recovery in optical offset-QAM Nyquist WDM systems,” IEEE Photon. Tech. Lett. 29, 1019–1022 (2017).
[Crossref]

2016 (4)

2015 (1)

2014 (2)

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 12, 14199–14211 (2014).
[Crossref]

2013 (2)

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Q. Zhuge, M. H. Morsy-Osman, and D. V. Plant, “Low overhead intra-symbol carrier phase recovery for reduced-guard-interval CO-OFDM,” J. Lightwave Technol. 31, 1158–1169 (2013).
[Crossref]

2012 (2)

S. Cao, P. Y. Kam, and C. Yu, “Decision-aided, pilot-aided, decision-feedback phase estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 24, 2067–2069 (2012).
[Crossref]

S. Chandrasekhar and X. Liu, “OFDM based superchannel transmission technology,” J. Lightw. Tech. 30, 3816–3823 (2012).
[Crossref]

2010 (1)

2009 (2)

2003 (1)

1998 (1)

L. Tomba, “On the effect of Wiener phase noise in OFDM systems,” IEEE Transactions on Communications 46, 580–583 (1998).
[Crossref]

Alvarado, A.

Armstrong, J.

Bayvel, P.

Cao, S.

S. Cao, P. Y. Kam, and C. Yu, “Decision-aided, pilot-aided, decision-feedback phase estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 24, 2067–2069 (2012).
[Crossref]

Chagnon, M.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar and X. Liu, “OFDM based superchannel transmission technology,” J. Lightw. Tech. 30, 3816–3823 (2012).
[Crossref]

Dar, Ronen

Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 12, 14199–14211 (2014).
[Crossref]

De Carvalho, L. H. H.

De Oliveira, J. C. R.

Diniz, J.

Doberstein, A.

Doran, N. J.

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Ellis, A. D.

S. T. Le, P. A. Haigh, A. D. Ellis, and S. K. Turitsyn, “Blind Phase Noise Estimation for CO-OFDM Transmissions,” JJ. Lightwave Technol. 34, 745–753 (2016).
[Crossref]

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

El-Sahn, Z. A.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Essiambre, R.-J.

Estaran, J.

Feder, Meir

Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 12, 14199–14211 (2014).
[Crossref]

Foschini, G. J.

Franciscangelis, C.

Giacoumidis, E.

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Goebel, B.

Gonzalez, N. G.

Haigh, P. A.

S. T. Le, P. A. Haigh, A. D. Ellis, and S. K. Turitsyn, “Blind Phase Noise Estimation for CO-OFDM Transmissions,” JJ. Lightwave Technol. 34, 745–753 (2016).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Haisch, H.

Hoffmann, S.

Holzlöhner, R.

Kam, P. Y.

S. Cao, P. Y. Kam, and C. Yu, “Decision-aided, pilot-aided, decision-feedback phase estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 24, 2067–2069 (2012).
[Crossref]

Kanesan, T.

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

Koike-Akino, T.

Kojima, K.

Kramer, G.

Lavery, D.

Le, S. T.

S. T. Le, P. A. Haigh, A. D. Ellis, and S. K. Turitsyn, “Blind Phase Noise Estimation for CO-OFDM Transmissions,” JJ. Lightwave Technol. 34, 745–753 (2016).
[Crossref]

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Li, C.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Liu, X.

S. Chandrasekhar and X. Liu, “OFDM based superchannel transmission technology,” J. Lightw. Tech. 30, 3816–3823 (2012).
[Crossref]

Maher, R.

McCarthy, M. E.

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Mecozzi, Antonio

Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 12, 14199–14211 (2014).
[Crossref]

Member, S.

Menyuk, C. R.

Millar, D. S.

Mitra, J.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Monroy, I. T.

Morsy-Osman, M.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Morsy-Osman, M. H.

Mousa-Pasandi, M. E.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Najafi, H.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Nebendahl, B.

Nguyen, T. H.

T. H. Nguyen and C. Peucheret, “Kalman filtering for carrier phase recovery in optical offset-QAM Nyquist WDM systems,” IEEE Photon. Tech. Lett. 29, 1019–1022 (2017).
[Crossref]

Noé, R.

Pajovic, M.

Pakala, L.

Parsons, K.

Paskov, M.

Peucheret, C.

T. H. Nguyen and C. Peucheret, “Kalman filtering for carrier phase recovery in optical offset-QAM Nyquist WDM systems,” IEEE Photon. Tech. Lett. 29, 1019–1022 (2017).
[Crossref]

Pfau, T.

Piels, M.

Plant, D. V.

Q. Zhuge, M. H. Morsy-Osman, and D. V. Plant, “Low overhead intra-symbol carrier phase recovery for reduced-guard-interval CO-OFDM,” J. Lightwave Technol. 31, 1158–1169 (2013).
[Crossref]

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Savory, S. J.

Schmauss, B.

Shtaif, Mark

Ronen Dar, Meir Feder, Antonio Mecozzi, and Mark Shtaif, “Accumulation of nonlinear interference noise in fiber-optic systems,” Opt. Express 12, 14199–14211 (2014).
[Crossref]

Sinkin, O. V.

Suibhne, N. M.

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

Tang, X.

Thomsen, B. C.

Tomba, L.

L. Tomba, “On the effect of Wiener phase noise in OFDM systems,” IEEE Transactions on Communications 46, 580–583 (1998).
[Crossref]

Turitsyn, S. K.

S. T. Le, P. A. Haigh, A. D. Ellis, and S. K. Turitsyn, “Blind Phase Noise Estimation for CO-OFDM Transmissions,” JJ. Lightwave Technol. 34, 745–753 (2016).
[Crossref]

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. a. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Decision Directed Free Blind Phase Noise Estimation for CO OFDM,” Optical Fiber Communication Conference (OFC)8724, W1E.5 (2015).

S. T. Le, M. E. McCarthy, N. M. Suibhne, P. A. Haigh, E. Giacoumidis, N. J. Doran, A. D. Ellis, and S. K. Turitsyn, “Multiplier-free Blind Phase Noise Estimation for CO-OFDM Transmission,” European Conference on Optical Communication (ECOC), 1–3 W1E.5 (2015).

Winzer, P. J.

Xu, X.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Yao, D.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Yu, C.

S. Cao, P. Y. Kam, and C. Yu, “Decision-aided, pilot-aided, decision-feedback phase estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 24, 2067–2069 (2012).
[Crossref]

Zamani, M.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Zhang, Z.

M. Zamani, H. Najafi, D. Yao, J. Mitra, X. Tang, C. Li, and Z. Zhang, “Trellis-based feed-forward carrier recovery for coherent optical systems,” Opt. Express 24, 23531–23542 (2016).
[Crossref] [PubMed]

M. Zamani, H. Najafi, D. Yao, J. Mitra, C. Li, and Z. Zhang, “Method and apparatus for residual phase noise compensation,”, Google PatentsUS9537683, 2017.

Zhuge, Q.

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

Q. Zhuge, M. H. Morsy-Osman, and D. V. Plant, “Low overhead intra-symbol carrier phase recovery for reduced-guard-interval CO-OFDM,” J. Lightwave Technol. 31, 1158–1169 (2013).
[Crossref]

Zibar, D.

Zweck, J.

IEEE Photon. Tech. Lett. (4)

Q. Zhuge, X. Xu, M. E. Mousa-Pasandi, M. Morsy-Osman, M. Chagnon, Z. A. El-Sahn, and D. V. Plant, “Experimental study of the intra-channel nonlinearity influence on single-band 100G coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 25, 553–555 (2013).
[Crossref]

S. Cao, P. Y. Kam, and C. Yu, “Decision-aided, pilot-aided, decision-feedback phase estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 24, 2067–2069 (2012).
[Crossref]

S. T. Le, T. Kanesan, E. Giacoumidis, N. J. Doran, and A. D. Ellis, “Quasi-pilot aided phase noise estimation for coherent optical OFDM systems,” IEEE Photon. Tech. Lett. 26, 504–507 (2014).
[Crossref]

T. H. Nguyen and C. Peucheret, “Kalman filtering for carrier phase recovery in optical offset-QAM Nyquist WDM systems,” IEEE Photon. Tech. Lett. 29, 1019–1022 (2017).
[Crossref]

IEEE Transactions on Communications (1)

L. Tomba, “On the effect of Wiener phase noise in OFDM systems,” IEEE Transactions on Communications 46, 580–583 (1998).
[Crossref]

J. Lightw. Tech. (1)

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Figures (8)

Fig. 1
Fig. 1 Linewidth symbol-duration product, ΔνTs, as a function of number of subcarriers at 200 kHz combined laser linewidth for 32 Gbaud CO-OFDM transmission.
Fig. 2
Fig. 2 Block diagram of CO-OFDM with phase noises from lasers and fiber nonlinearity during optical fiber transmission.
Fig. 3
Fig. 3 Proposed S-EFK scheme phase noise estimation and correction for CO-OFDM transmission.
Fig. 4
Fig. 4 A tolerant comparison against phase noise of the proposed S-EKF with PA and BPS DD-FL (direct decision based with feedback loop) for 32 Gbaud 16QAM CO-OFDM in back-to-back transmission. The vertical axis is OSNR penalty as a function of normalized linewidth at BER of 3.8 × 10−3 (7 % HD-FEC threshold).
Fig. 5
Fig. 5 Performance of 8 inputs S-EKF with different number of subcarriers for CO-OFDM.
Fig. 6
Fig. 6 BER as a function of OSNR for 32 Gbaud 16QAM CO-OFDM in back-to-back transmission at the same 200 kHz combined laser linewidth. The number of subcarriers and input data are 32/64/96 and 2/4/8, respectively.
Fig. 7
Fig. 7 The BER performance comparison of S-EKF with PA and BPS versus launched powers for 64 subcarriers 16QAM CO-OFDM fiber transmission at the distance of 1600 km (20 spans). The linewidth of transmitter and receiver lasers is 100 kHz.
Fig. 8
Fig. 8 The BER performance comparison of S-EKF with PA and BPS DD-FL versus transmission distance for 64 subcarriers 16QAM CO-OFDM systems at the optimum launched power of −3 dBm. The linewidth of transmitter and receiver lasers is 100 kHz.

Tables (3)

Tables Icon

Algorithm 1 Procedure of proposed S-EKF implementation.

Tables Icon

Table 1 S-EKF computational analysis (from Algorithm 1) for CPE estimation.

Tables Icon

Table 2 Complexity comparison between S-EKF with conventional PA, and BPS DD-FL methods in term of the number of real multiply operators. All the results are based on no CORDIC implementation.

Equations (19)

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Φ ( m ) = Φ ( m 1 ) + ν ( m ) ,
x m ( n ) = 1 ( X m , k ) = k = 0 N 1 X m , k exp ( j 2 π k n N ) , m N n < ( m + 1 ) N ,
y m ( n ) = exp ( j ϕ m ( n ) ) ( x m ( n ) 1 ( H m ( k ) ) ) + w m ( n ) ,
H ( z , ω ) = exp ( j D λ 2 z 4 π c ω 2 ) ,
Y m ( k ) = ( y m ( n ) 1 ( H m ( k ) 1 ) ) = X m ( k ) I m ( 0 ) + ICI m ( k ) + W m ( k ) ,
ICI m ( k ) = l = 0 , l k N 1 X m ( l ) I m ( l k ) ,
I m ( k ) = 1 N n = 0 N 1 exp ( j ϕ m ( n ) ) exp ( j 2 π k n N ) .
I m ( 0 ) = 1 N n = 0 N 1 exp ( j ϕ m ( n ) ) exp ( j Φ ( m ) ) ,
Φ ( m ) = 1 N n = 0 N 1 ϕ m ( n ) .
R m ( k ) = X m ( k ) I m ( 0 ) + ε m ( k ) ,
ε m ( k ) = ICI m ( k ) + W m ( k ) .
R m ( k ) = X m ( k ) exp ( j Φ ( m ) ) + ε m ( k ) .
Φ ( m ) = arg ( 1 N p l = 1 N p R m ( l ) X m * ( l ) | R m ( l ) X m ( l ) | ) ,
R m p = A m p exp ( j Φ m ) + ε m ,
arg ( R m p ) Φ m + θ m ref + ε m = f ( Φ m ) + ε m where f ( Φ m ) = Φ m + θ m ref ,
Φ ( m ) = Φ ( m 1 ) + υ ( m ) arg ( R m p ) = f ( Φ m ) = ε m , where f ( Φ m ) = Φ m + θ m ref .
C S EKF = l = 1 N p N p ! l ! ( N p l ) ! + N p 2 + l = 1 N p l + 3 + 8 N p S w ,
P b = 3 8 erfc ( 2 π b 5 ) ,
OSNR = R b 2 B ref γ b ,

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