Abstract

Spectral efficient frequency division multiplexing (SEFDM) can offer a higher spectral efficiency (SE) than orthogonal frequency division multiplexing (OFDM). In this work, we propose a diversity technique based on SEFDM for beyond 100-Gb/s optical intensity modulation and direct detection (IM/DD) long reach (LR) applications. We mathematically demonstrate that the self-created inter-carrier interference of SEFDM signals can be reused to achieve a diversity gain on each sub-carrier and, in turn, improve the tolerance to power fading induced by chromatic dispersion (CD) in IM/DD LR links. Based on the proposed diversity technique, we further demonstrated a 112-Gb/s SEFDM transmission over 80-km standard single-mode fiber, using only 28-GHz bandwidth and modulation format of up to 16-QAM. Experimental results show that SEFDM with the proposed diversity technique performs robust against CD effects and outperforms the conventional OFDM with adaptive bit and power loading of the same bandwidth and data rate, which validates the superiority of the proposed SEFDM in optical IM/DD LR transmissions.

© 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]
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    [Crossref]
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    [Crossref]
  23. M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
    [Crossref]
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    [Crossref]
  25. B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.
  26. S. J. Heydari, M. F. Naeiny, and F. Marvasti, “Iterative detection with soft decision in spectrally efficient FDM systems,” arXiv preprint arXiv:1304.4003 (2013).
  27. S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
    [Crossref]
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    [Crossref]
  29. J. Zhou and Y. Qiao, “Low-PAPR asymmetrically clipped optical OFDM for intensity-modulation/direct-detection systems,” IEEE Photonics J. 7(3), 1–8 (2015).
    [Crossref]
  30. C. Guo, J. Liang, J. Liu, and L. Liu, “Extended reach OFDM-PON using super-Nyquist image induced aliasing,” Opt. Express 23(17), 21798–21808 (2015).
    [Crossref]
  31. P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
    [Crossref]

2019 (4)

A. Yekani and L. A. Rusch, “Interplay of bit rate, linewidth, bandwidth, and reach on optical dmt and pam with imdd,” IEEE Trans. Commun. 67(4), 2908–2913 (2019).
[Crossref]

J. Lian, Y. Gao, and D. Lian, “Variable pulse width unipolar orthogonal frequency division multiplexing for visible light communication systems,” IEEE Access 7, 31022–31030 (2019).
[Crossref]

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

T. Xu, T. Xu, P. Bayvel, and I. Darwazeh, “Non-orthogonal signal transmission over nonlinear optical channels,” IEEE Photonics J. 11(3), 1–13 (2019).
[Crossref]

2018 (6)

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

S. Fu, C. Chen, F. Gao, X. Li, L. Deng, M. Tang, and D. Liu, “Digital chromatic dispersion pre-management enabled single-lane 112 Gb/s PAM-4 signal transmission over 80 km SSMF,” Opt. Lett. 43(7), 1495–1498 (2018).
[Crossref]

Z. Liu, G. Hesketh, B. Kelly, J. O’Carroll, R. Phelan, D. J. Richardson, and R. Slavík, “Optical injection-locked directly modulated lasers for dispersion pre-compensated direct-detection transmission,” J. Lightwave Technol. 36(20), 4967–4974 (2018).
[Crossref]

2017 (4)

Z. Li, M. S. Erkılınç, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “SSBI mitigation and the Kramers–Kronig scheme in single-sideband direct-detection transmission with receiver-based electronic dispersion compensation,” J. Lightwave Technol. 35(10), 1887–1893 (2017).
[Crossref]

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

J. Zhao and L.-K. Chen, “Adaptively loaded IM/DD optical OFDM based on set-partitioned QAM formats,” Opt. Express 25(8), 9368–9377 (2017).
[Crossref]

2016 (2)

2015 (2)

J. Zhou and Y. Qiao, “Low-PAPR asymmetrically clipped optical OFDM for intensity-modulation/direct-detection systems,” IEEE Photonics J. 7(3), 1–8 (2015).
[Crossref]

C. Guo, J. Liang, J. Liu, and L. Liu, “Extended reach OFDM-PON using super-Nyquist image induced aliasing,” Opt. Express 23(17), 21798–21808 (2015).
[Crossref]

2014 (2)

W. A. Ling and I. Lyubomirsky, “Electronic dispersion compensation in a 50 Gb/s optically unamplified direct-detection receiver enabled by vestigial-sideband orthogonal frequency division multiplexing,” Opt. Express 22(6), 6984–6995 (2014).
[Crossref]

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

2013 (3)

2009 (1)

2008 (1)

1995 (1)

P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Ahmed, M. H.

E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. E. Baddour, “Joint optimization of bit and power loading for multicarrier systems,” IEEE Wirel. Commun. Lett. 2(4), 447–450 (2013).
[Crossref]

Antonelli, C.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Armstrong, J.

Baddour, K. E.

E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. E. Baddour, “Joint optimization of bit and power loading for multicarrier systems,” IEEE Wirel. Commun. Lett. 2(4), 447–450 (2013).
[Crossref]

Bao, Y.

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

Baykas, T.

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

Bayvel, P.

Bedeer, E.

E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. E. Baddour, “Joint optimization of bit and power loading for multicarrier systems,” IEEE Wirel. Commun. Lett. 2(4), 447–450 (2013).
[Crossref]

Bingham, J. A.

P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Chandrasekhar, S.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Che, D.

Chen, C.

Chen, L.-K.

Chen, X.

A. Li, D. Che, X. Chen, Q. Hu, Y. Wang, and W. Shieh, “61 Gbits/s direct-detection optical OFDM based on blockwise signal phase switching with signal-to-signal beat noise cancellation,” Opt. Lett. 38(14), 2614–2616 (2013).
[Crossref]

X. Chen, A. Li, D. Che, Q. Hu, Y. Wang, J. He, and W. Shieh, “Block-wise phase switching for double-sideband direct detected optical OFDM signals,” Opt. Express 21(11), 13436–13441 (2013).
[Crossref]

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Cheng, Q.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Chow, P. S.

P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Cioffi, J. M.

P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

Dai, X.

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

Darwazeh, I.

T. Xu, T. Xu, P. Bayvel, and I. Darwazeh, “Non-orthogonal signal transmission over nonlinear optical channels,” IEEE Photonics J. 11(3), 1–13 (2019).
[Crossref]

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

Deng, L.

Dobre, O. A.

E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. E. Baddour, “Joint optimization of bit and power loading for multicarrier systems,” IEEE Wirel. Commun. Lett. 2(4), 447–450 (2013).
[Crossref]

Erkilinç, M. S.

Fu, S.

Galdino, L.

Gao, F.

Gao, Y.

J. Lian, Y. Gao, and D. Lian, “Variable pulse width unipolar orthogonal frequency division multiplexing for visible light communication systems,” IEEE Access 7, 31022–31030 (2019).
[Crossref]

Gu, X.

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

Gui, T.

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

Guo, C.

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

C. Guo, J. Liang, J. Liu, and L. Liu, “Extended reach OFDM-PON using super-Nyquist image induced aliasing,” Opt. Express 23(17), 21798–21808 (2015).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

Guo, Q.

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

He, J.

Hesketh, G.

Heydari, S. J.

S. J. Heydari, M. F. Naeiny, and F. Marvasti, “Iterative detection with soft decision in spectrally efficient FDM systems,” arXiv preprint arXiv:1304.4003 (2013).

Hoang, T. M.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Hong, X.

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

Hu, Q.

Huang, J.

J. Huang, Q. Sui, Z. Li, and F. Ji, “Experimental demonstration of 16-QAM DD-SEFDM with cascaded BPSK iterative detection,” IEEE Photonics J. 8(3), 1–9 (2016).
[Crossref]

Huo, J.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Ishihara, T.

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

Ji, F.

J. Huang, Q. Sui, Z. Li, and F. Ji, “Experimental demonstration of 16-QAM DD-SEFDM with cascaded BPSK iterative detection,” IEEE Photonics J. 8(3), 1–9 (2016).
[Crossref]

Jia, M.

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

Kelly, B.

Khalid, H. A.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Khan, A.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Khan, F.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Killey, R. I.

Kizilirmak, R. C.

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

Laskowski, P.

Lau, A. P. T.

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Li, A.

Li, X.

Li, Z.

Z. Li, M. S. Erkılınç, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “SSBI mitigation and the Kramers–Kronig scheme in single-sideband direct-detection transmission with receiver-based electronic dispersion compensation,” J. Lightwave Technol. 35(10), 1887–1893 (2017).
[Crossref]

J. Huang, Q. Sui, Z. Li, and F. Ji, “Experimental demonstration of 16-QAM DD-SEFDM with cascaded BPSK iterative detection,” IEEE Photonics J. 8(3), 1–9 (2016).
[Crossref]

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

Lian, D.

J. Lian, Y. Gao, and D. Lian, “Variable pulse width unipolar orthogonal frequency division multiplexing for visible light communication systems,” IEEE Access 7, 31022–31030 (2019).
[Crossref]

Lian, J.

J. Lian, Y. Gao, and D. Lian, “Variable pulse width unipolar orthogonal frequency division multiplexing for visible light communication systems,” IEEE Access 7, 31022–31030 (2019).
[Crossref]

Liang, J.

Ling, W. A.

Liu, D.

Liu, G.

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

Liu, J.

Liu, L.

Liu, Z.

Lowery, A. J.

Lu, C.

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Lu, Y.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Lyubomirsky, I.

Marvasti, F.

S. J. Heydari, M. F. Naeiny, and F. Marvasti, “Iterative detection with soft decision in spectrally efficient FDM systems,” arXiv preprint arXiv:1304.4003 (2013).

Mecozzi, A.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Morsy-Osman, M.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Naeiny, M. F.

S. J. Heydari, M. F. Naeiny, and F. Marvasti, “Iterative detection with soft decision in spectrally efficient FDM systems,” arXiv preprint arXiv:1304.4003 (2013).

Nakao, M.

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

Narmanlioglu, O.

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

O’Carroll, J.

Osaki, S.

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

Phelan, R.

Plant, D. V.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Prodaniuc, C.

Qiao, Y.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

J. Zhou and Y. Qiao, “Low-PAPR asymmetrically clipped optical OFDM for intensity-modulation/direct-detection systems,” IEEE Photonics J. 7(3), 1–8 (2015).
[Crossref]

Raybon, G.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Richardson, D. J.

Rusch, L. A.

A. Yekani and L. A. Rusch, “Interplay of bit rate, linewidth, bandwidth, and reach on optical dmt and pam with imdd,” IEEE Trans. Commun. 67(4), 2908–2913 (2019).
[Crossref]

Schmidt, B. J.

Shi, K.

Shieh, W.

Shtaif, M.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Sillekens, E.

Sinsky, J.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Slavík, R.

Sowailem, M.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Stojanovic, N.

Sugiura, S.

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

Sui, Q.

J. Huang, Q. Sui, Z. Li, and F. Ji, “Experimental demonstration of 16-QAM DD-SEFDM with cascaded BPSK iterative detection,” IEEE Photonics J. 8(3), 1–9 (2016).
[Crossref]

Tang, M.

Thomsen, B. C.

Tian, Q.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Ullah, R.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Ullah, S.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Uysal, M.

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

Wang, Q.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Wang, Y.

Wei, J.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Winzer, P.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

Xie, C.

Xin, X.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Xing, Z.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

Xu, T.

T. Xu, T. Xu, P. Bayvel, and I. Darwazeh, “Non-orthogonal signal transmission over nonlinear optical channels,” IEEE Photonics J. 11(3), 1–13 (2019).
[Crossref]

T. Xu, T. Xu, P. Bayvel, and I. Darwazeh, “Non-orthogonal signal transmission over nonlinear optical channels,” IEEE Photonics J. 11(3), 1–13 (2019).
[Crossref]

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

Yang, A.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Yang, Z.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Yekani, A.

A. Yekani and L. A. Rusch, “Interplay of bit rate, linewidth, bandwidth, and reach on optical dmt and pam with imdd,” IEEE Trans. Commun. 67(4), 2908–2913 (2019).
[Crossref]

Yi, L.

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

Yin, Z.

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

Yu, B.

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

Yu, C.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Zeng, L.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Zhang, H.

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

B. Yu, C. Guo, L. Yi, H. Zhang, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “150-Gb/s SEFDM IM/DD transmission using log-MAP Viterbi decoding for short reach optical links,” Opt. Express 26(24), 31075–31084 (2018).
[Crossref]

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Zhang, Q.

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

Q. Zhang, N. Stojanovic, C. Xie, C. Prodaniuc, and P. Laskowski, “Transmission of single lane 128 Gbit/s PAM-4 signals over an 80 km SSMF link, enabled by DDMZM aided dispersion pre-compensation,” Opt. Express 24(21), 24580–24591 (2016).
[Crossref]

Zhang, T.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

Zhao, J.

Zhong, K.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Zhou, J.

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

J. Zhou and Y. Qiao, “Low-PAPR asymmetrically clipped optical OFDM for intensity-modulation/direct-detection systems,” IEEE Photonics J. 7(3), 1–8 (2015).
[Crossref]

Zhou, X.

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

Zhuge, Q.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

IEEE Access (5)

S. Ullah, R. Ullah, A. Khan, H. A. Khalid, Q. Zhang, Q. Tian, F. Khan, and X. Xin, “Optical multi-wavelength source for single feeder fiber using suppressed carrier in high capacity LR-WDM-PON,” IEEE Access 6, 70674–70684 (2018).
[Crossref]

O. Narmanlioglu, R. C. Kizilirmak, T. Baykas, and M. Uysal, “Link adaptation for MIMO OFDM visible light communication systems,” IEEE Access 5, 26006–26014 (2017).
[Crossref]

J. Lian, Y. Gao, and D. Lian, “Variable pulse width unipolar orthogonal frequency division multiplexing for visible light communication systems,” IEEE Access 7, 31022–31030 (2019).
[Crossref]

J. Zhou, Q. Wang, J. Wei, Q. Cheng, T. Zhang, Z. Yang, A. Yang, Y. Lu, and Y. Qiao, “Faster-than-Nyquist non-orthogonal frequency-division multiplexing for visible light communications,” IEEE Access 6, 17933–17941 (2018).
[Crossref]

M. Jia, Z. Yin, Q. Guo, G. Liu, and X. Gu, “Waveform design of zero head DFT spread spectral efficient frequency division multiplexing,” IEEE Access 5, 16944–16952 (2017).
[Crossref]

IEEE Photonics J. (3)

T. Xu, T. Xu, P. Bayvel, and I. Darwazeh, “Non-orthogonal signal transmission over nonlinear optical channels,” IEEE Photonics J. 11(3), 1–13 (2019).
[Crossref]

J. Zhou and Y. Qiao, “Low-PAPR asymmetrically clipped optical OFDM for intensity-modulation/direct-detection systems,” IEEE Photonics J. 7(3), 1–8 (2015).
[Crossref]

J. Huang, Q. Sui, Z. Li, and F. Ji, “Experimental demonstration of 16-QAM DD-SEFDM with cascaded BPSK iterative detection,” IEEE Photonics J. 8(3), 1–9 (2016).
[Crossref]

IEEE Photonics Technol. Lett. (1)

I. Darwazeh, T. Xu, T. Gui, Y. Bao, and Z. Li, “Optical SEFDM system; bandwidth saving using non-orthogonal sub-carriers,” IEEE Photonics Technol. Lett. 26(4), 352–355 (2014).
[Crossref]

IEEE Signal Process. Lett. (1)

S. Osaki, M. Nakao, T. Ishihara, and S. Sugiura, “Differentially modulated spectrally efficient frequency-division multiplexing,” IEEE Signal Process. Lett. 26(7), 1046–1050 (2019).
[Crossref]

IEEE Trans. Commun. (2)

A. Yekani and L. A. Rusch, “Interplay of bit rate, linewidth, bandwidth, and reach on optical dmt and pam with imdd,” IEEE Trans. Commun. 67(4), 2908–2913 (2019).
[Crossref]

P. S. Chow, J. M. Cioffi, and J. A. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun. 43(2/3/4), 773–775 (1995).
[Crossref]

IEEE Wirel. Commun. Lett. (1)

E. Bedeer, O. A. Dobre, M. H. Ahmed, and K. E. Baddour, “Joint optimization of bit and power loading for multicarrier systems,” IEEE Wirel. Commun. Lett. 2(4), 447–450 (2013).
[Crossref]

IET Commun. (1)

B. Yu, H. Zhang, X. Hong, C. Guo, A. P. T. Lau, C. Lu, and X. Dai, “Channel equalisation and data detection for SEFDM over frequency selective fading channels,” IET Commun. 12(18), 2315–2323 (2018).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (6)

Opt. Lett. (2)

Other (5)

B. Yu, L. Yi, C. Guo, J. Liu, X. Dai, A. P. T. Lau, and C. Lu, “Dispersion tolerant 66.7-Gb/s SEFDM IM/DD transmission over 77-km SSMF,” in 2018 European Conference on Optical Communication (ECOC), (IEEE, 2018), pp. 1–3.

S. J. Heydari, M. F. Naeiny, and F. Marvasti, “Iterative detection with soft decision in spectrally efficient FDM systems,” arXiv preprint arXiv:1304.4003 (2013).

K. Zhong, X. Zhou, Y. Wang, J. Huo, H. Zhang, L. Zeng, C. Yu, A. P. T. Lau, and C. Lu, “Amplifier-less transmission of 56Gbit/s PAM4 over 60km using 25Gbps EML and APD,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Tu2D–1.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, J. Sinsky, A. Mecozzi, M. Shtaif, and P. Winzer, “218-Gb/s single-wavelength, single-polarization, single-photodiode transmission over 125-km of standard singlemode fiber using Kramers-Kronig detection,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), pp. Th5B–6.

T. M. Hoang, Q. Zhuge, Z. Xing, M. Sowailem, M. Morsy-Osman, and D. V. Plant, “Single wavelength 480 Gb/s direct detection transmission over 80 km SSMF enabled by Stokes vector receiver and reduced-complexity SSBI cancellation,” in 2018 Optical Fiber Communications Conference and Exposition (OFC), (IEEE, 2018), pp. 1–3.

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Figures (8)

Fig. 1.
Fig. 1. SEFDM transceiver diagram.
Fig. 2.
Fig. 2. (a) Simulated channel response of the proposed SEFDM with $\alpha =0.80$ through an 80-km SSMF link. (b) Simulated BER performance of the SEFDM with the proposed diversity technique at $\alpha =0.75$ using 4-QAM modulation in 80-km CD channel. (c) BER and SE at various BCFs when SNR=18dB.
Fig. 3.
Fig. 3. Experimental setup.
Fig. 4.
Fig. 4. B2B performance of the proposed SEFDM and ABPL-OFDM.
Fig. 5.
Fig. 5. (a) BER performance of the proposed diversity technique using SEFDM with $\alpha =0.75$ after 80-km SSMF transmission for various bit rates. (b) Estimated 4-QAM signal distributions with and without MRC.
Fig. 6.
Fig. 6. BER performance of the proposed diversity technique using SEFDM after 80-km SSMF transmission at approximate bit rates with various BCFs.
Fig. 7.
Fig. 7. Comparison between the measured receiver SNR with and without MRC. (a) $\alpha =0.80$. (b) $\alpha =0.75$.
Fig. 8.
Fig. 8. (a) BER comparison between SEFDM with the proposed diversity technique and ABPL-OFDM at a bit rate of 112-Gb/s. (b) Measured systematic SNR and bit allocations of SEFDM. (c) Measured systematic SNR and bit allocations of OFDM. (d) BER performance of both schemes at a ROP of −11.8dBm.

Tables (2)

Tables Icon

Table 1. Systematic Parameters Used for Experiments

Tables Icon

Table 2. Bit allocations and bit rates using the simplified bit loading scheme

Equations (30)

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

S N f k = S k ,
x n = k = 0 N f 1 S k exp ( j 2 π k n N f ) , n = 0 , 1 , , N f 1.
α = N 1 N f .
η O = log 2 M ,
η S = log 2 M α .
H k = cos ( 4 π L D ( k λ f ) 2 c N f 2 ) , k = 1 , 2 , , N
Y k = H k W k , k S k + H k i = 1 , i k N W k , i S i + Z k ,
W k , i = 1 N f n = 0 N 1 1 exp ( j 2 π n ( k i ) N f ) ,
Y n , k = Y n H n i = 1 , i k N W n , i S ~ i = H n W n , k S k + Δ n + Z n ,
Δ n = H n i = 1 , i k N W n , i ( S i S ~ i ) ,
S ^ k = n = 1 N H n W n , k Y n , k n = 1 N | H n W n , k | 2 = S k + Z ^ k G k ,
Z ^ k = n = 1 N H n W n , k ( Δ n + Z n ) α n = 1 N | H n W n , k | 2 ,
G k = n = 1 N | H n W n , k | 2 α .
var { | G k ( α 1 ) | 2 } var { | G k ( α 2 ) | 2 } , α 1 > α 2 ,
var { | G k ( α ) | 2 } = k = 0 N f 1 | G k ( α ) | 4 μ 2 ( α ) ,
k = 0 N f 1 | G k ( α 1 ) | 4 k = 0 N f 1 | G k ( α 2 ) | 4 .
| G k | 2 = | W 0 , k | 2 α | H k | 2 ,
k = 0 N f 1 | | W 0 , k ( α 1 ) | 2 α 1 | 2 > k = 0 N f 1 | | W 0 , k ( α 2 ) | 2 α 2 | 2 ,
w 1 = 1 N f N 1 [ 1 , , 1 , 0 , , 0 ] T ,
w 2 = 1 N f N 1 [ 1 , 0 , , 0 , 1 , , 1 ] T ,
w n = 1 N f N 1 max ( | n N f 2 | + N 1 N f 2 , 2 N 1 N f , 0 ) .
w n ( α ) = 1 N f max ( 1 β α , 2 1 α , 0 )
β = { 1 n N f n N f 2 n N f n < N f 2 .
w n ( α 1 ) > w n ( α 2 ) .
k = 0 N f 1 | | W n , k ( α 1 ) | 2 α 1 | 2 = N f k = 0 N f 1 | w n ( α 1 ) | 2 > N f k = 0 N f 1 | w n ( α 2 ) | 2 = k = 0 N f 1 | | W n , k ( α 2 ) | 2 α 2 | 2 ,
n = 0 N f 1 ( b n a n ) 2 n = 0 N f 1 ( c n a n ) 2 .
n = 0 N f 1 ( c n 2 b n 2 ) = 2 i = 0 N f 1 j = i + 1 N f 1 ( b i b j c i c j )
n = 0 N f 1 ( c n a n ) 2 = ( n = 0 N f 1 c n 2 ) ( m = 0 N f 1 a m 2 ) + 2 i = 0 N f 1 j = i + 1 N f 1 c i c j k = 0 N f 1 a k a ( k + j i ) mod N f = ( n = 0 N f 1 b n 2 ) ( m = 0 N f 1 a m 2 ) + ( n = 0 N f 1 ( c n 2 b n 2 ) ) ( m = 0 N f 1 a m 2 ) + 2 i = 0 N f 1 j = i + 1 N f 1 c i c j k = 0 N f 1 a k a ( k + j i ) mod N f .
( n = 0 N f 1 ( c n 2 b n 2 ) ) ( m = 0 N f 1 a m 2 ) = 2 ( i = 0 N f 1 j = i + 1 N f 1 ( b i b j c i c j ) ) ( m = 0 N f 1 a m 2 ) 2 ( i = 0 N f 1 j = i + 1 N f 1 ( b i b j c i c j ) ) ( k = 0 N f 1 a k a ( k + j i ) mod N f ) ,
n = 0 N f 1 ( c n a n ) 2 ( n = 0 N f 1 b n 2 ) ( m = 0 N f 1 a m 2 ) + 2 i = 0 N f 1 j = i + 1 N f 1 c i c j k = 0 N f 1 a k a ( k + j i ) mod N f + 2 i = 0 N f 1 j = i + 1 N f 1 ( b i b j c i c j ) ( k = 0 N f 1 a k a ( k + j i ) mod N f ) = ( n = 0 N f 1 b n 2 ) ( m = 0 N f 1 a m 2 ) + 2 i = 0 N f 1 j = i + 1 N f 1 b i b j k = 0 N f 1 a k a ( k + j i ) mod N f = n = 0 N f 1 ( b n a n ) 2

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