Abstract

In this work, we proposed two k-means-clustering-based algorithms to mitigate the fiber nonlinearity for 64-quadrature amplitude modulation (64-QAM) signal, the training-sequence assisted k-means algorithm and the blind k-means algorithm. We experimentally demonstrated the proposed k-means-clustering-based fiber nonlinearity mitigation techniques in 75-Gb/s 64-QAM coherent optical communication system. The proposed algorithms have reduced clustering complexity and low data redundancy and they are able to quickly find appropriate initial centroids and select correctly the centroids of the clusters to obtain the global optimal solutions for large k value. We measured the bit-error-ratio (BER) performance of 64-QAM signal with different launched powers into the 50-km single mode fiber and the proposed techniques can greatly mitigate the signal impairments caused by the amplified spontaneous emission noise and the fiber Kerr nonlinearity and improve the BER performance.

© 2017 Optical Society of America

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References

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  1. S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112Gbit/s PDM-QAM-16 transmission with and without digital nonlinearity compensation,” Opt. Express 18(12), 12939–12947 (2010).
    [PubMed]
  2. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
  3. M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).
  4. M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).
  5. E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
    [PubMed]
  6. T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).
  7. L. Pakala and B. Schmauss, “Non-linear mitigation using carrier phase estimation and k-means clustering,” in Photonic Networks (Academic, 2015), pp. 1–5.
  8. V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).
  9. H. Yu, J. Yang, and J. Han, “Classifying large data sets using SVMs with hierarchical clusters,” in Association for Computing Machinery (Academic, 2003), pp. 306–315.
  10. E. Giacoumidis, S. Mhatli, M. F. C. Stephens, A. Tsokanos, J. L. Wei, N. J. Doran, and A. D. Ellis, “Reduction of nonlinear intersubcarrier intermixing in coherent optical OFDM by a fast newton-based support vector machine nonlinear equalizer,” J. Lightwave Technol. 35(12), 2391–2397 (2017).
  11. M. C. Naldi and R. J. G. B. Campello, “Evolutionary k-means for distributed data sets,” Neurocomputing 127(3), 30–42 (2014).
  12. N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).
  13. N. Aggarwal and K. Aggarwal, An Improved K-means Clustering Algorithm for Data Mining (Academic, 2012).
  14. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).
  15. K. Arai and R. B. Ali R, “Hierarchical K-means: an algorithm for centroids initialization for K-means,” Rep. Faculty Sci. Eng. 36, 25–31 (2007).
  16. A. Vattani, “K-means requires exponentially many iterations even in the plane,” Discrete Comput. Geom. 45(4), 596–616 (2009).

2017 (1)

2016 (2)

E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
[PubMed]

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

2015 (2)

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

2014 (1)

M. C. Naldi and R. J. G. B. Campello, “Evolutionary k-means for distributed data sets,” Neurocomputing 127(3), 30–42 (2014).

2013 (1)

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

2010 (2)

S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112Gbit/s PDM-QAM-16 transmission with and without digital nonlinearity compensation,” Opt. Express 18(12), 12939–12947 (2010).
[PubMed]

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

2009 (1)

A. Vattani, “K-means requires exponentially many iterations even in the plane,” Discrete Comput. Geom. 45(4), 596–616 (2009).

2008 (1)

2007 (1)

K. Arai and R. B. Ali R, “Hierarchical K-means: an algorithm for centroids initialization for K-means,” Rep. Faculty Sci. Eng. 36, 25–31 (2007).

Aldaya, I.

E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
[PubMed]

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Ali R, R. B.

K. Arai and R. B. Ali R, “Hierarchical K-means: an algorithm for centroids initialization for K-means,” Rep. Faculty Sci. Eng. 36, 25–31 (2007).

Arai, K.

K. Arai and R. B. Ali R, “Hierarchical K-means: an algorithm for centroids initialization for K-means,” Rep. Faculty Sci. Eng. 36, 25–31 (2007).

Bayvel, P.

Behrens, C.

Caballero, A.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

Campello, R. J. G. B.

M. C. Naldi and R. J. G. B. Campello, “Evolutionary k-means for distributed data sets,” Neurocomputing 127(3), 30–42 (2014).

Carvalho, L.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Chen, Z.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Diniz, J.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Doran, N. J.

E. Giacoumidis, S. Mhatli, M. F. C. Stephens, A. Tsokanos, J. L. Wei, N. J. Doran, and A. D. Ellis, “Reduction of nonlinear intersubcarrier intermixing in coherent optical OFDM by a fast newton-based support vector machine nonlinear equalizer,” J. Lightwave Technol. 35(12), 2391–2397 (2017).

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Eggleton, B. J.

Ellis, A. D.

Filho, E.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Ghassemlooy, Z.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Giacoumidis, E.

E. Giacoumidis, S. Mhatli, M. F. C. Stephens, A. Tsokanos, J. L. Wei, N. J. Doran, and A. D. Ellis, “Reduction of nonlinear intersubcarrier intermixing in coherent optical OFDM by a fast newton-based support vector machine nonlinear equalizer,” J. Lightwave Technol. 35(12), 2391–2397 (2017).

E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
[PubMed]

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Gonzalez, N. G.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

Gu, W.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Han, Y.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Ip, E.

Jarajreh, M. A.

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Kahn, J. M.

Killey, R. I.

Lavery, D.

Le, S. T.

E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
[PubMed]

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Li, M.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Makovejs, S.

McCarthy, M. E.

Mégret, P.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

Mhatli, S.

Millar, D. S.

Monroy, I. T.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

Naldi, M. C.

M. C. Naldi and R. J. G. B. Campello, “Evolutionary k-means for distributed data sets,” Neurocomputing 127(3), 30–42 (2014).

Nguyen, T.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

E. Giacoumidis, S. Mhatli, T. Nguyen, S. T. Le, I. Aldaya, M. E. McCarthy, A. D. Ellis, and B. J. Eggleton, “Comparison of DSP-based nonlinear equalizers for intra-channel nonlinearity compensation in coherent optical ofdm,” Opt. Lett. 41(11), 2509–2512 (2016).
[PubMed]

Oliveira, J.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Parahyba, V.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Ranzini, S.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Reis, J.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Rosa, E.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Savory, S. J.

Schneider, E.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Simoes, F.

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

Stephens, M. F. C.

Tsokanos, A.

E. Giacoumidis, S. Mhatli, M. F. C. Stephens, A. Tsokanos, J. L. Wei, N. J. Doran, and A. D. Ellis, “Reduction of nonlinear intersubcarrier intermixing in coherent optical OFDM by a fast newton-based support vector machine nonlinear equalizer,” J. Lightwave Technol. 35(12), 2391–2397 (2017).

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

Van Compernolle, L.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

Vattani, A.

A. Vattani, “K-means requires exponentially many iterations even in the plane,” Discrete Comput. Geom. 45(4), 596–616 (2009).

Wei, J. L.

Wuilpart, M.

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

Yang, J.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Yu, S.

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

Zibar, D.

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

Discrete Comput. Geom. (1)

A. Vattani, “K-means requires exponentially many iterations even in the plane,” Discrete Comput. Geom. 45(4), 596–616 (2009).

Electron. Lett. (1)

V. Parahyba, J. Reis, S. Ranzini, E. Schneider, E. Rosa, F. Simoes, J. Diniz, L. Carvalho, E. Filho, J. Oliveira, and J. Oliveira, “Performance against implementation of digital backpropagation for high-speed coherent optical systems,” Electron. Lett. 51(14), 1094–1096 (2015).

IEEE Photonics J. (2)

M. Li, S. Yu, J. Yang, Z. Chen, Y. Han, and W. Gu, “Nonparameter nonlinear phase noise mitigation by using M-ary support vector machine for coherent optical systems,” IEEE Photonics J. 5(6), 7800312 (2013).

T. Nguyen, S. Mhatli, E. Giacoumidis, L. Van Compernolle, M. Wuilpart, and P. Mégret, “Fiber Nonlinearity Equalizer Based on Support Vector Classification for Coherent Optical OFDM,” IEEE Photonics J. 8(2), 1–9 (2016).

IEEE Photonics Technol. Lett. (2)

M. A. Jarajreh, E. Giacoumidis, I. Aldaya, S. T. Le, A. Tsokanos, Z. Ghassemlooy, and N. J. Doran, “Artificial neural network nonlinear equalizer for coherent optical OFDM,” IEEE Photonics Technol. Lett. 27(4), 387–390 (2015).

N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-Gb/s QPSK WDM phase-modulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photonics Technol. Lett. 22(5), 335–337 (2010).

J. Lightwave Technol. (2)

Neurocomputing (1)

M. C. Naldi and R. J. G. B. Campello, “Evolutionary k-means for distributed data sets,” Neurocomputing 127(3), 30–42 (2014).

Opt. Express (1)

Opt. Lett. (1)

Rep. Faculty Sci. Eng. (1)

K. Arai and R. B. Ali R, “Hierarchical K-means: an algorithm for centroids initialization for K-means,” Rep. Faculty Sci. Eng. 36, 25–31 (2007).

Other (4)

N. Aggarwal and K. Aggarwal, An Improved K-means Clustering Algorithm for Data Mining (Academic, 2012).

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).

L. Pakala and B. Schmauss, “Non-linear mitigation using carrier phase estimation and k-means clustering,” in Photonic Networks (Academic, 2015), pp. 1–5.

H. Yu, J. Yang, and J. Han, “Classifying large data sets using SVMs with hierarchical clusters,” in Association for Computing Machinery (Academic, 2003), pp. 306–315.

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Figures (8)

Fig. 1
Fig. 1 Concept diagram of k-means clustering.
Fig. 2
Fig. 2 Blind k-means schematic diagram in QPSK system.
Fig. 3
Fig. 3 Experimental setup of 64-QAM coherent optical communication system with k-means nonlinearity mitigation algorithm. (AWG: arbitrary waveform generator, EDFA: erbium-doped fiber amplifier, VOA: variable optical attenuator, SMF: single mode fiber, LO: local oscillator.)
Fig. 4
Fig. 4 Constellation diagrams of 64-QAM signal with BTB and the different launched optical power.
Fig. 5
Fig. 5 Constellation diagrams of 64-QAM signal with the launched power at 5.07dBm training data (left) and signal data (right)with the training-sequence assisted k-means algorithm.
Fig. 6
Fig. 6 Constellation diagrams of 64-QAM signal with the launched power at 5.07dBm in the different steps of the blind k-means algorithm.
Fig. 7
Fig. 7 Measured BER curves vs. launched signal power into 50-km SMF Pin (dBm) in the 75-Gb/s 64-QAM coherent optical communication system: without k-mean algorithm (diamond-marked); with blind k-means algorithm (triangle-marked); with training-sequence assisted k-means algorithm (circle-marked).
Fig. 8
Fig. 8 The improved BER vs. training overhead of the training-sequence assisted k-means algorithm for 75-Gb/s 64-QAM signal at the 50-km of transmission for a launched optical power (LOP) of 5.07 dBm.

Equations (5)

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d( x i , x j )= ( x i1 x j1 ) 2 + ( x i2 x j2 ) 2 ++ ( x in x jm ) 2
E= k=1 K i=1 n k x i k c k 2
C i =( 1 s ) j s D j
dd(k)= k=1 N i=1 N { ( ( x i x k ) 2 + ( y i y k ) 2 )r } pi* r 2
r= ( ( range(x)/30 ) 2 + ( range(y)/30 ) 2 )

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