Abstract

A joint timing offset (TO) and frequency offset (FO) estimation algorithm is proposed for polarization division multiplexing (PDM) coherent optical orthogonal frequency-division multiplexing (CO-OFDM) systems. It is realized by taking the advantage of the time-frequency property of the fractional Fourier transformation (FrFT) encoded training symbols. Compared with the classical Schmidl & Cox method, the proposed algorithm exhibits robust estimation result of timing offset with poor optical signal-to-noise ratio (OSNR) and nonlinear interference. For the frequency offset estimation, a quite large FO estimation ranges of [-5GHz + 5GHz] can be achieved. The mean normalized estimation error can be kept under 0.002 and the max normalized estimation error is no more than 0.008. The feasibility and effectiveness of the proposed joint estimation algorithm has been verified by experiments. The transmission performances with [-5GHz + 5GHz] FO are compared under the OSNR range from 14 to 27dB in a 106.8Gbit/s 16-ary quadrature amplitude modulation (16-QAM) PDM CO-OFDM transmission system. The proposed TO/FO estimation algorithm performs robustly and accurately without any induced BER degradations.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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  1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
    [Crossref]
  2. S. Hara and R. Prasad, Multicarrier Techniques for 4G Mobile Communications (Artech House, 2003).
  3. W. Shieh and I. Djordjevic, Signal Processing for Optical OFDM (Academic, 2009).
  4. P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994).
    [Crossref]
  5. Y. Mostofi and D. C. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system,” IEEE Trans. Commun. 54(2), 226–230 (2006).
    [Crossref]
  6. Optical Internetworking Forum, “Integrable tunable transmitter assembly multi source agreement,” OIF-ITTA-MSA-01.0, Nov. (2008).
  7. X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
    [Crossref]
  8. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
    [Crossref]
  9. S. L. Jansen, I. Morita, T. C. W. Schenk, N. Takeda, and H. Tanaka, “Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF,” J. Lightwave Technol. 26(1), 6–15 (2008).
    [Crossref]
  10. F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
    [Crossref]
  11. H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
    [Crossref]
  12. X. Zhou, X. Yang, R. Li, and K. Long, “Efficient joint carrier frequency offset and phase noise compensation scheme for high-speed coherent optical OFDM systems,” J. Lightwave Technol. 31(11), 1755–1761 (2013).
    [Crossref]
  13. Y. Huang, X. Zhang, and L. Xi, “Modified synchronization scheme for coherent optical OFDM systems,” J. Opt. Commun. Netw. 5(6), 584–592 (2013).
    [Crossref]
  14. L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
    [Crossref]
  15. V. Namias, “The fractional order Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
    [Crossref]
  16. H. M. Ozaktas, D. Mendlovic, L. Onural, and B. Barshan, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11(2), 547–559 (1994).
    [Crossref]
  17. A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
    [Crossref]
  18. X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
    [Crossref]
  19. J. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Signal Process. 44(8), 2086–2091 (1996).
    [Crossref]
  20. O. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49(5), 979–993 (2001).
    [Crossref]
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    [Crossref]
  22. M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
    [Crossref]
  23. L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77(7), 941–981 (1989).
    [Crossref]
  24. H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
    [Crossref]
  25. S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Trans. Signal Process. 49(8), 1638–1655 (2001).
    [Crossref]
  26. X. Zhou, K. Long, R. Li, X. Yang, and Z. Zhang, “A simple and efficient frequency offset estimation algorithm for high-speed coherent optical OFDM systems,” Opt. Express 20(7), 7350–7361 (2012).
    [Crossref] [PubMed]

2016 (1)

2013 (2)

2012 (1)

2008 (2)

2006 (2)

Y. Mostofi and D. C. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system,” IEEE Trans. Commun. 54(2), 226–230 (2006).
[Crossref]

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[Crossref]

2001 (2)

S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Trans. Signal Process. 49(8), 1638–1655 (2001).
[Crossref]

O. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49(5), 979–993 (2001).
[Crossref]

2000 (1)

H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
[Crossref]

1997 (2)

A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
[Crossref]

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

1996 (3)

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

J. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Signal Process. 44(8), 2086–2091 (1996).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

1994 (3)

H. M. Ozaktas, D. Mendlovic, L. Onural, and B. Barshan, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11(2), 547–559 (1994).
[Crossref]

P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994).
[Crossref]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

1989 (1)

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77(7), 941–981 (1989).
[Crossref]

1980 (1)

V. Namias, “The fractional order Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[Crossref]

Akay, O.

O. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49(5), 979–993 (2001).
[Crossref]

Almeida, L. B.

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

Ankan, O.

A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
[Crossref]

Arikan, O.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Athaudage, C.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[Crossref]

Barshan, B.

Bhargava, V. K.

H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
[Crossref]

Boudreaux-Bartels, G. F.

O. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49(5), 979–993 (2001).
[Crossref]

Bozdagt, G.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Buchali, F.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Chen, X.

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Cheng, J.

Cohen, L.

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77(7), 941–981 (1989).
[Crossref]

Cox, D. C.

Y. Mostofi and D. C. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system,” IEEE Trans. Commun. 54(2), 226–230 (2006).
[Crossref]

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Ding, J. J.

S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Trans. Signal Process. 49(8), 1638–1655 (2001).
[Crossref]

Dischler, R.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Feng, Z.

Fonollosa, J.

J. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Signal Process. 44(8), 2086–2091 (1996).
[Crossref]

Fu, S.

H. Zhou, B. Li, M. Tang, K. Zhong, Z. Feng, J. Cheng, A. P. T. Lau, C. Lu, S. Fu, P. P. Shum, and D. Liu, “Fractional Fourier Transformation-Based Blind Chromatic Dispersion Estimation for Coherent Optical Communications,” J. Lightwave Technol. 34(10), 2371–2380 (2016).
[Crossref]

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Huang, Y.

Jansen, S. L.

Kutay, A.

A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
[Crossref]

Kutay, M. A.

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

Lau, A. P. T.

Li, B.

Li, R.

Liu, D.

H. Zhou, B. Li, M. Tang, K. Zhong, Z. Feng, J. Cheng, A. P. T. Lau, C. Lu, S. Fu, P. P. Shum, and D. Liu, “Fractional Fourier Transformation-Based Blind Chromatic Dispersion Estimation for Coherent Optical Communications,” J. Lightwave Technol. 34(10), 2371–2380 (2016).
[Crossref]

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Long, K.

Lu, C.

Ma, Y.

Matic, R. M.

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

Mayrock, M.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Mendlovic, D.

Minn, H.

H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
[Crossref]

Moose, P. H.

P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994).
[Crossref]

Morita, I.

Mostofi, Y.

Y. Mostofi and D. C. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system,” IEEE Trans. Commun. 54(2), 226–230 (2006).
[Crossref]

Namias, V.

V. Namias, “The fractional order Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[Crossref]

Onural, L.

Owechko, Y.

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

Ozaktas, H. M.

A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

H. M. Ozaktas, D. Mendlovic, L. Onural, and B. Barshan, “Convolution, filtering, and multiplexing in fractional Fourier domains and their relationship to chirp and wavelet transforms,” J. Opt. Soc. Am. A 11(2), 547–559 (1994).
[Crossref]

Pei, S. C.

S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Trans. Signal Process. 49(8), 1638–1655 (2001).
[Crossref]

Schenk, T. C. W.

Schmidl, T. M.

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

Shieh, W.

X. W. Yi, W. Shieh, and Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol. 26(10), 1309–1316 (2008).
[Crossref]

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[Crossref]

Shum, P. P.

H. Zhou, B. Li, M. Tang, K. Zhong, Z. Feng, J. Cheng, A. P. T. Lau, C. Lu, S. Fu, P. P. Shum, and D. Liu, “Fractional Fourier Transformation-Based Blind Chromatic Dispersion Estimation for Coherent Optical Communications,” J. Lightwave Technol. 34(10), 2371–2380 (2016).
[Crossref]

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Soffer, B. H.

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

Takeda, N.

Tanaka, H.

Tang, M.

H. Zhou, B. Li, M. Tang, K. Zhong, Z. Feng, J. Cheng, A. P. T. Lau, C. Lu, S. Fu, P. P. Shum, and D. Liu, “Fractional Fourier Transformation-Based Blind Chromatic Dispersion Estimation for Coherent Optical Communications,” J. Lightwave Technol. 34(10), 2371–2380 (2016).
[Crossref]

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Tang, Y.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Wu, J.

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Xi, L.

Xia, X.

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

Xiao, X.

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Yang, X.

Yi, X. W.

Zeng, M.

H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
[Crossref]

Zhang, X.

Zhang, Z.

Zhong, K.

Zhou, H.

H. Zhou, B. Li, M. Tang, K. Zhong, Z. Feng, J. Cheng, A. P. T. Lau, C. Lu, S. Fu, P. P. Shum, and D. Liu, “Fractional Fourier Transformation-Based Blind Chromatic Dispersion Estimation for Coherent Optical Communications,” J. Lightwave Technol. 34(10), 2371–2380 (2016).
[Crossref]

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

Zhou, X.

Electron. Lett. (1)

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006).
[Crossref]

IEEE Commun. Lett. (1)

H. Minn, M. Zeng, and V. K. Bhargava, “On timing offset estimation for OFDM systems,” IEEE Commun. Lett. 4(7), 242–244 (2000).
[Crossref]

IEEE Trans. Commun. (3)

T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
[Crossref]

P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun. 42(10), 2908–2914 (1994).
[Crossref]

Y. Mostofi and D. C. Cox, “Mathematical analysis of the impact of timing synchronization errors on the performance of an OFDM system,” IEEE Trans. Commun. 54(2), 226–230 (2006).
[Crossref]

IEEE Trans. Signal Process. (7)

A. Kutay, H. M. Ozaktas, O. Ankan, and L. Onural, “Optimal filtering in fractional Fourier domains,” IEEE Trans. Signal Process. 45(5), 1129–1143 (1997).
[Crossref]

X. Xia, Y. Owechko, B. H. Soffer, and R. M. Matic, “Generalized-marginal time-frequency distributions,” IEEE Trans. Signal Process. 44(11), 2882–2886 (1996).
[Crossref]

J. Fonollosa, “Positive time-frequency distributions based on joint marginal constraints,” IEEE Trans. Signal Process. 44(8), 2086–2091 (1996).
[Crossref]

O. Akay and G. F. Boudreaux-Bartels, “Fractional convolution and correlation via operator methods and an application to detection of linear FM signals,” IEEE Trans. Signal Process. 49(5), 979–993 (2001).
[Crossref]

L. B. Almeida, “The fractional Fourier transform and time-frequency representations,” IEEE Trans. Signal Process. 42(11), 3084–3091 (1994).
[Crossref]

H. M. Ozaktas, O. Arikan, M. A. Kutay, and G. Bozdagt, “Digital computation of the fractional Fourier transform,” IEEE Trans. Signal Process. 44(9), 2141–2150 (1996).
[Crossref]

S. C. Pei and J. J. Ding, “Relations between fractional operations and time-frequency distributions, and their applications,” IEEE Trans. Signal Process. 49(8), 1638–1655 (2001).
[Crossref]

J. Inst. Math. Appl. (1)

V. Namias, “The fractional order Fourier transform and its application in quantum mechanics,” J. Inst. Math. Appl. 25(3), 241–265 (1980).
[Crossref]

J. Lightwave Technol. (4)

J. Opt. Commun. Netw. (1)

J. Opt. Soc. Am. A (1)

Opt. Express (1)

Proc. IEEE (1)

L. Cohen, “Time-frequency distributions-a review,” Proc. IEEE 77(7), 941–981 (1989).
[Crossref]

Other (5)

M. Tang, H. Zhou, J. Wu, X. Chen, S. Fu, P. P. Shum, and D. Liu, “Joint Timing and Frequency Synchronization Based on FrFT Encoded Training Symbol for Coherent Optical OFDM Systems,” in Proc. OFC’16 (2016), paper Tu3K.6.
[Crossref]

F. Buchali, R. Dischler, M. Mayrock, X. Xiao, and Y. Tang, “Improved frequency offset correction in coherent optical OFDM systems,” in Proc. ECOC’08 (2008), paper Mo.4.D.4.
[Crossref]

Optical Internetworking Forum, “Integrable tunable transmitter assembly multi source agreement,” OIF-ITTA-MSA-01.0, Nov. (2008).

S. Hara and R. Prasad, Multicarrier Techniques for 4G Mobile Communications (Artech House, 2003).

W. Shieh and I. Djordjevic, Signal Processing for Optical OFDM (Academic, 2009).

Supplementary Material (1)

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» Visualization 1: MP4 (180 KB)      A visualization for figure 3

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Figures (13)

Fig. 1
Fig. 1 (a) Generation of the FrFT-DC signal. (b) Energy centralization property of the FrFT-DC signal.
Fig. 2
Fig. 2 (a) Generation of the proposed TS signal. (b) Energy centralization property of the TS signal.
Fig. 3
Fig. 3 Time-frequency distribution of TS signal (a) with no offsets, (b) with frequency offset only, (c) with time offset only and (d) with both of frequency and time offsets (see Visualization 1).
Fig. 4
Fig. 4 The geometrical relationship between the peak shifts and the offsets of frequency and time.
Fig. 5
Fig. 5 (a) The worst TO condition for FO estimation. (b) The relationship between the largest estimated FO and the order P.
Fig. 6
Fig. 6 (a) Simulation setup of the 35.6Gbit/s PDM-OFDM-16QAM system, the schematics of the DSP in the (b) transmitter and (c) receiver, (d) OFDM frame structure. (ECL: external cavity laser, AWG: arbitrary waveform generator, PBS: polarization beam splitter, PBC: polarization beam combiner, EDFA: erbium doped fiber amplifier, TOF: tunable optical filter, DSO: digital sampling oscilloscope, S/P: serial to parallel conversion, CP: cyclic prefix, P/S: parallel to serial conversion)
Fig. 7
Fig. 7 (a) Structure of OFDM signal. (b) Peaks of every training symbol after FrFT. (c) A detail example for 13th training symbol with frequency and time offset.
Fig. 8
Fig. 8 (a) The measured timing metric of Schmidl & Cox algorithm and (b) energy peaks of TS1 in Q1 and Q2 fractional domain for different OSNR with 5GHz frequency offset.
Fig. 9
Fig. 9 (a) Comparison of TO estimation error. (b) The absolute values of normalized FO estimation error under [-5GHz, 5GHz] frequency offset with OSNR of 15dB.
Fig. 10
Fig. 10 Experimental setup of 106.8Gbit/s PDM CO-OFDM transmission system. PBS: polarization beam splitter, PBC: polarization beam combiner, EDFA: erbium doped fiber amplifier, VOA: variable optical attenuator, TOF: tunable optical filter, OSA: optical spectrum analysis.
Fig. 11
Fig. 11 (a)Transmission performance comparison with and without our proposed FO estimation method for different input power. (b) The FOE results for different input power.
Fig. 12
Fig. 12 Measured constellations of the 16QAM signals (a) without FO estimation and correction; (b)–(c) with FO estimation and correction at OSNR of 27 and 14 dB, respectively. (Blue and red dots represent the signals of X and Y polarization respectively)
Fig. 13
Fig. 13 (a) BER vs. OSNR under the condition of with −5GHz, + 5GHz and without FO. (b) BER vs. FO for different OSNR conditions.

Equations (16)

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p=2α/π
F α (u)= f(t) K α (t,u)dt
f(t)= F α (u) K α * (t,u)du
K α (t,u)={ 1-jcotα 2π exp(j t 2 + u 2 2 cotα jtu sinα ) if α is not a multiple of π δ(t-u) if α is a multiple of δ(t+u) if α+π is a multiple of
T (t,P)=FrFT(S(t),P)
Q=1P
Ts(t,P)=T(t,P)+T(t,-P)=FrFT(S(t),P)+FrFT(S(t),-P)
Q 1 =1P Q 2 =( 1P )
θ=Pπ/2
ΔF=Δ N f df ΔT=Δ N t dt
-ΔQ 1 =-Δ N t sin(θ)+Δ N f cos(θ) ΔQ 2 =Δ N f cos(θ)-(-Δ N t sin(θ))
Δ N t = ΔQ 1 +ΔQ 2 2sin(Pπ/2) Δ N f = ΔQ 2 -ΔQ 1 2cos(Pπ/2)
Δ N t sin(θ)+Δ N f cos(θ) =N /2
Δ F max = N(1-sin(Pπ/2)) 2cos(Pπ/2) df= 1-sin(Pπ/2) 2cos(Pπ/2) F sam
F O min = 1 2cos(θ) df= 1 2cos(Pπ/2) F sam N
F O min = 1 2cos(0.1π/2) f sc 0.5 f sc

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