Abstract

In this paper, we investigate and characterize a new approach of adopting best-fit bounding box method for common phase error estimation in coherent optical OFDM systems. The method is based on the calculation of the 2-D convex hull of the received signal constellation, which is generally adopted in image processing area to correct the skew of images. We further perform detailed characterizations including root mean square error analysis, laser linewidth tolerance, noise tolerance, and computation complexity analysis, via numerical simulations and experiments. The results show the proposed method achieves much improved spectral efficiency and comparable system performance than the pilot-aided method, while it exhibits good estimation accuracy and reduced complexity than the blind phase searching method.

© 2016 Optical Society of America

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References

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  1. S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).
  2. S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
    [Crossref]
  3. S. Jansen, I. Morita, T. 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]
  4. Z. Liu, J. Kim, D. Wu, D. Richardson, and R. Slavik, “Homodyne OFDM with optical injection locking for carrier recovery,” J. Lightwave Technol. 33(1), 34–41 (2015).
    [Crossref]
  5. X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
    [Crossref]
  6. Y. Ha and W. Chung, “Non-data-aided phase noise suppression scheme for CO-OFDM systems,” IEEE Photonics Technol. Lett. 25(17), 1703–1706 (2013).
    [Crossref]
  7. T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
    [Crossref]
  8. T. Bo and C. Chan, “Common phase error estimation for coherent optical OFDM system using best-fit bounding box,” in Proc. International Conference on Photonics in Switching (IEEE, 2015), pp. 327–329.
    [Crossref]
  9. B. Yuan, L. K. Kwoh, and C. L. Tan, “Finding the best-fit bounding boxes,” in Proc. Int. Conf. on Document Analysis Sys. (Springer-Verlag, 2006), pp. 268–279.
  10. H. Freeman and R. Shapiral, “Determining the minimum-area encasing rectangle for an arbitrary closed curve,” ACM Commun. 18(7), 409–413 (1975).
    [Crossref]
  11. G. Toussaint, “Solving geometric problems with the rotating calipers,” in Proc. IEEE MELECON (1983) 83, A-10.
  12. S. Har-Peled. “On the expected complexity of random convex hulls,” Technical Report, 330/98 (1998).
  13. R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
    [Crossref]
  14. T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
    [Crossref]
  15. R. L. Graham, “An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set,” Inf. Process. Lett. 1(4), 132–133 (1972).
    [Crossref]
  16. T. M. Chan, “Optimal output-sensitive convex hull algorithms in two and three dimensions,” Discrete Comput. Geom. 16(4), 361–368 (1996).
    [Crossref]
  17. S. G. Akl and G. T. Toussaint, “A fast convex hull algorithm,” Inf. Process. Lett. 7(5), 219–222 (1978).
    [Crossref]
  18. L. Devroye and G. T. Toussaint, “A note on linear expected time algorithms for finding convex hulls,” Computing 26(4), 361–366 (1981).
    [Crossref]
  19. T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun. 45(12), 1613–1621 (1997).
    [Crossref]

2015 (2)

T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
[Crossref]

Z. Liu, J. Kim, D. Wu, D. Richardson, and R. Slavik, “Homodyne OFDM with optical injection locking for carrier recovery,” J. Lightwave Technol. 33(1), 34–41 (2015).
[Crossref]

2013 (1)

Y. Ha and W. Chung, “Non-data-aided phase noise suppression scheme for CO-OFDM systems,” IEEE Photonics Technol. Lett. 25(17), 1703–1706 (2013).
[Crossref]

2009 (2)

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

T. Pfau, S. Hoffmann, and R. Noe, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27(8), 989–999 (2009).
[Crossref]

2008 (1)

2007 (1)

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

2004 (1)

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[Crossref]

1997 (1)

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

1996 (1)

T. M. Chan, “Optimal output-sensitive convex hull algorithms in two and three dimensions,” Discrete Comput. Geom. 16(4), 361–368 (1996).
[Crossref]

1981 (1)

L. Devroye and G. T. Toussaint, “A note on linear expected time algorithms for finding convex hulls,” Computing 26(4), 361–366 (1981).
[Crossref]

1978 (1)

S. G. Akl and G. T. Toussaint, “A fast convex hull algorithm,” Inf. Process. Lett. 7(5), 219–222 (1978).
[Crossref]

1975 (1)

H. Freeman and R. Shapiral, “Determining the minimum-area encasing rectangle for an arbitrary closed curve,” ACM Commun. 18(7), 409–413 (1975).
[Crossref]

1972 (1)

R. L. Graham, “An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set,” Inf. Process. Lett. 1(4), 132–133 (1972).
[Crossref]

Akl, S. G.

S. G. Akl and G. T. Toussaint, “A fast convex hull algorithm,” Inf. Process. Lett. 7(5), 219–222 (1978).
[Crossref]

Atanassov, R.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Bar-Ness, Y.

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[Crossref]

Bo, T.

T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
[Crossref]

T. Bo and C. Chan, “Common phase error estimation for coherent optical OFDM system using best-fit bounding box,” in Proc. International Conference on Photonics in Switching (IEEE, 2015), pp. 327–329.
[Crossref]

Bose, P.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Chan, C.

T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
[Crossref]

T. Bo and C. Chan, “Common phase error estimation for coherent optical OFDM system using best-fit bounding box,” in Proc. International Conference on Photonics in Switching (IEEE, 2015), pp. 327–329.
[Crossref]

Chan, T. M.

T. M. Chan, “Optimal output-sensitive convex hull algorithms in two and three dimensions,” Discrete Comput. Geom. 16(4), 361–368 (1996).
[Crossref]

Chandrasekhar, S.

S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).

Chung, W.

Y. Ha and W. Chung, “Non-data-aided phase noise suppression scheme for CO-OFDM systems,” IEEE Photonics Technol. Lett. 25(17), 1703–1706 (2013).
[Crossref]

Couture, M.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Cox, D. C.

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

Devroye, L.

L. Devroye and G. T. Toussaint, “A note on linear expected time algorithms for finding convex hulls,” Computing 26(4), 361–366 (1981).
[Crossref]

Freeman, H.

H. Freeman and R. Shapiral, “Determining the minimum-area encasing rectangle for an arbitrary closed curve,” ACM Commun. 18(7), 409–413 (1975).
[Crossref]

Graham, R. L.

R. L. Graham, “An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set,” Inf. Process. Lett. 1(4), 132–133 (1972).
[Crossref]

Ha, Y.

Y. Ha and W. Chung, “Non-data-aided phase noise suppression scheme for CO-OFDM systems,” IEEE Photonics Technol. Lett. 25(17), 1703–1706 (2013).
[Crossref]

Hoffmann, S.

Huang, L.

T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
[Crossref]

Jansen, S.

Kim, J.

Kwoh, L. K.

B. Yuan, L. K. Kwoh, and C. L. Tan, “Finding the best-fit bounding boxes,” in Proc. Int. Conf. on Document Analysis Sys. (Springer-Verlag, 2006), pp. 268–279.

Liu, X.

S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).

Liu, Z.

Maheshwari, A.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Morin, P.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Morita, I.

Noe, R.

Paquette, M.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Peckham, D.

S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).

Pfau, T.

Richardson, D.

Schenk, T.

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]

Shapiral, R.

H. Freeman and R. Shapiral, “Determining the minimum-area encasing rectangle for an arbitrary closed curve,” ACM Commun. 18(7), 409–413 (1975).
[Crossref]

Shieh, W.

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Slavik, R.

Smid, M.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Takeda, N.

Tan, C. L.

B. Yuan, L. K. Kwoh, and C. L. Tan, “Finding the best-fit bounding boxes,” in Proc. Int. Conf. on Document Analysis Sys. (Springer-Verlag, 2006), pp. 268–279.

Tanaka, H.

Tang, Y.

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Toussaint, G.

G. Toussaint, “Solving geometric problems with the rotating calipers,” in Proc. IEEE MELECON (1983) 83, A-10.

Toussaint, G. T.

L. Devroye and G. T. Toussaint, “A note on linear expected time algorithms for finding convex hulls,” Computing 26(4), 361–366 (1981).
[Crossref]

S. G. Akl and G. T. Toussaint, “A fast convex hull algorithm,” Inf. Process. Lett. 7(5), 219–222 (1978).
[Crossref]

Wu, D.

Wu, S.

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[Crossref]

Wuhrer, S.

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

Yi, X.

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Yuan, B.

B. Yuan, L. K. Kwoh, and C. L. Tan, “Finding the best-fit bounding boxes,” in Proc. Int. Conf. on Document Analysis Sys. (Springer-Verlag, 2006), pp. 268–279.

Zhu, B.

S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).

ACM Commun. (1)

H. Freeman and R. Shapiral, “Determining the minimum-area encasing rectangle for an arbitrary closed curve,” ACM Commun. 18(7), 409–413 (1975).
[Crossref]

Computing (1)

L. Devroye and G. T. Toussaint, “A note on linear expected time algorithms for finding convex hulls,” Computing 26(4), 361–366 (1981).
[Crossref]

Discrete Comput. Geom. (1)

T. M. Chan, “Optimal output-sensitive convex hull algorithms in two and three dimensions,” Discrete Comput. Geom. 16(4), 361–368 (1996).
[Crossref]

IEEE Photonics Technol. Lett. (3)

X. Yi, W. Shieh, and Y. Tang, “Phase estimation for coherent optical OFDM,” IEEE Photonics Technol. Lett. 19(12), 919–921 (2007).
[Crossref]

Y. Ha and W. Chung, “Non-data-aided phase noise suppression scheme for CO-OFDM systems,” IEEE Photonics Technol. Lett. 25(17), 1703–1706 (2013).
[Crossref]

T. Bo, L. Huang, and C. Chan, “Common phase estimation in coherent OFDM system using image processing technique,” IEEE Photonics Technol. Lett. 27(15), 1597–1600 (2015).
[Crossref]

IEEE Trans. Commun. (2)

S. Wu and Y. Bar-Ness, “OFDM systems in the presence of phase noise: consequences and solutions,” IEEE Trans. Commun. 52(11), 1988–1996 (2004).
[Crossref]

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

Inf. Process. Lett. (2)

R. L. Graham, “An Efficient Algorithm for Determining the Convex Hull of a Finite Planar Set,” Inf. Process. Lett. 1(4), 132–133 (1972).
[Crossref]

S. G. Akl and G. T. Toussaint, “A fast convex hull algorithm,” Inf. Process. Lett. 7(5), 219–222 (1978).
[Crossref]

J. Discrete Algorithms (1)

R. Atanassov, P. Bose, M. Couture, A. Maheshwari, P. Morin, M. Paquette, M. Smid, and S. Wuhrer, “Algorithms for optimal outlier removal,” J. Discrete Algorithms 7(2), 239–248 (2009).
[Crossref]

J. Lightwave Technol. (3)

Other (5)

S. Chandrasekhar, X. Liu, B. Zhu, and D. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” in Proceedings of the European Conference of Optical Communications (OFC) (2009).

G. Toussaint, “Solving geometric problems with the rotating calipers,” in Proc. IEEE MELECON (1983) 83, A-10.

S. Har-Peled. “On the expected complexity of random convex hulls,” Technical Report, 330/98 (1998).

T. Bo and C. Chan, “Common phase error estimation for coherent optical OFDM system using best-fit bounding box,” in Proc. International Conference on Photonics in Switching (IEEE, 2015), pp. 327–329.
[Crossref]

B. Yuan, L. K. Kwoh, and C. L. Tan, “Finding the best-fit bounding boxes,” in Proc. Int. Conf. on Document Analysis Sys. (Springer-Verlag, 2006), pp. 268–279.

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Figures (9)

Fig. 1
Fig. 1

Bounding box (solid line) of (a) a skewed rectangle and (b) a skew corrected rectangle.

Fig. 2
Fig. 2

Normalized area of bounding box w.r.t. the rotated angle in the constellation diagram of 4-, 16-, 32-, 64- and 128-QAM.

Fig. 3
Fig. 3

Principles of using best-fit bounding box to estimate the common phase error. (a) Constellation diagram of the received block and its best-fit bounding box; (b) the convex hull of the constellation points (connected through red solid line); (c) rotate the convex hull by each slope angle and use caliper to calculate area for one certain slope angle; (d) the case when the minimum axis-aligned bounding box area is found; (e) the sizes of convex hull in each OFDM symbol with 128 data subcarriers; (f) the average sizes of the convex hulls of 50000 OFDM symbols with different SNR.

Fig. 4
Fig. 4

Illustration of outliers in the constellation diagram. (a) & (d) outliers in a 16-QAM constellations (circle and triangle); (b) & (e) the de-skewed constellation diagrams and their bounding boxes via BBB method; (c) & (f) the constellation diagrams after applying edge de-noising algorithm.

Fig. 5
Fig. 5

(a) Root mean square error of CPE estimation under different signal-to-noise ratio. N is the number of samples involved in the calculation of CPE using MBB method. The number of test phases used in MBB method is 20. (b) Root mean square error of CPE estimation under different signal-to-noise ratio. N is the number of samples involved in the calculation of CPE using BBB method.

Fig. 6
Fig. 6

OSNR penalty at BER = 10−3 for PA, MBB, BBB and mBBB algorithms, under different laser linewidths.

Fig. 7
Fig. 7

Experimental setup. ECL: external cavity laser; AWG: arbitrary waveform generator; VOA: variable optical attenuator; AOM: acousto-optic modulator; OSA: optical spectrum analyzer; BPF: optical band-pass filter.

Fig. 8
Fig. 8

(a) Bit error rate versus OSNR. PA and BBB methods are used. Numbers of pilot subcarriers (PS) used in PA method are 2, 4, 8, and 16 (b) Bit error rate versus OSNR. MBB and BBB methods are used. Numbers of test phase used in MBB method are 4, 8, 12, and 16.

Fig. 9
Fig. 9

Q-factor versus input power with PA method, MBB method, BBB method and mBBB method in 840-km single mode fiber transmission.

Tables (1)

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Table 1 Hardware Complexity Comparison

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

y ik = x ik h k exp(j φ i )+ n ik
( { y ik },{ y ik } ) X i ,
s=ab+ a 2 + b 2 2 | sin( 2φ ) |
{ P 1 (x,y)|α x max x x max },{ P 2 (x,y)| x min xα x min }, { P 3 (x,y)|α y max y y max },{ P 4 (x,y)| y min yα y min }
Δ=( x 1 x 0 )( y 2 y 0 )( y 1 y 0 )( x 2 x 0 )

Metrics