Abstract

We have experimentally demonstrated a single-photodiode 112-Gbit/s 16-QAM transmission over 960-km standard single mode fiber with Kramers-Kronig (KK) detection and a sparse I/Q Volterra filter (VF). KK detection is used to mitigate the signal to signal beating interference. A novel I/Q VF based on dual-input real-valued Volterra series is proposed to overcome the constraints of non-linear distortions. Two conventional single-input real-valued VFs are employed as a contrast to prove the necessity of I/Q hybrid structure in I/Q VF when the system suffers from severe I/Q imbalance and crosstalk. The proposed I/Q VF brings nearly two order magnitude improvement in BER performance compared with the conventional linear equalizer for back to back case and doubles the transmission distance from 480 km to 960 km at the hard-decision forward error correction threshold of 3.8 × 10−3. To reduce the complexity of full I/Q VF, a simple 1 regularization-based least mean square (LMS) algorithm is employed to identify the significant kernels and construct a sparse I/Q VF. Compared with full I/Q VF, the sparse I/Q VF reduces up to 58% complexity at 960 km while having the similar BER performance.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (1)

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

2017 (5)

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[Crossref]

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
[Crossref]

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

T. M. Hoang, M. Y. S. Sowailem, Q. Zhuge, Z. Xing, M. Morsy-Osman, E. El-Fiky, S. Fan, M. Xiang, and D. V. Plant, “Single wavelength 480 Gb/s direct detection over 80km SSMF enabled by Stokes vector Kramers Kronig transceiver,” Opt. Express,  25(26), 33534–33542 (2017).
[Crossref]

2016 (2)

A. Mecozzi, C. Antonelli, and M. Shtaif, “KramersâĂŞKronig coherent receiver,” Optica,  3(11), 1220–1227 (2016).
[Crossref]

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

2013 (1)

M. S. Faruk and K. Kikuchi, “Compensation for In-Phase/Quadrature Imbalance in Coherent-Receiver Front End for Optical Quadrature Amplitude Modulation,” IEEE Photonics J. 5(2), 7800110 (2013).
[Crossref]

2012 (1)

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express,  20(2), 136–1369 (2012).
[Crossref]

2011 (2)

J. Pan and C. H. Cheng, “Nonlinear Electrical Compensation for the Coherent Optical OFDM System,” J. Light. Technol. 29(2), 215–221 (2011).
[Crossref]

D. Rafique, J. Zhao, and A. D. Ellis, “Digital back-propagation for spectrally efficient WDM 112 Gbit/s PM m-ary QAM transmission,” Opt. Express,  19(6), 5219–5224 (2011).
[Crossref] [PubMed]

2009 (1)

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
[Crossref]

2008 (3)

Y. Tang, K. P. Ho, and W. Shieh, “Coherent Optical OFDM Transmitter Design Employing Predistortion,” J. Light. Technol. 20(11), 954–956 (2008).

B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of Electronic Dispersion Compensation for Long-Haul Transmission Using Direct-Detection Optical OFDM,” J. Light. Technol.,  26(1), 196–203 (2008).
[Crossref]

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

2006 (1)

1996 (1)

R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. Royal Stat. Soc. Ser. B (Methodological) 58(1), 267–288 (1996).

1991 (1)

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Light. Technol. 9(1), 121–128 (1991).
[Crossref]

Altenhain, L.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Antonelli, C.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[Crossref]

A. Mecozzi, C. Antonelli, and M. Shtaif, “KramersâĂŞKronig coherent receiver,” Optica,  3(11), 1220–1227 (2016).
[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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Armstrong, J.

B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of Electronic Dispersion Compensation for Long-Haul Transmission Using Direct-Detection Optical OFDM,” J. Light. Technol.,  26(1), 196–203 (2008).
[Crossref]

A. J. Lowery and J. Armstrong, “Orthogonal-frequency-division multiplexing for dispersion compensation of long-haul optical systems,” Opt. Express,  14(6), 2079–2084 (2006).
[Crossref] [PubMed]

Bayvel, P.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Berenguer, P. W.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Bohn, M.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Brazil, T. J.

A. Zhu, M. Wren, and T. J. Brazil, “An efficient Volterra-based behavioral model for wideband RF power amplifiers,” in IEEE MTT-S International Microwave Symposium Digest, (IEEE, 2003), 2, pp. 787–790.

Breyer, F.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

Buchali, F.

S. T. Le, K. Schuh, F. Buchali, M. Chagnon, and H. Bulow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200km,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. Tu2D.5.

Bulow, H.

S. T. Le, K. Schuh, F. Buchali, M. Chagnon, and H. Bulow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200km,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. Tu2D.5.

Bunge, C. A.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

Calabro, S.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Cao, H.

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
[Crossref]

Chagnon, M.

Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (Optical Society of America, 2011), p. JThA040.

S. T. Le, K. Schuh, F. Buchali, M. Chagnon, and H. Bulow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200km,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. Tu2D.5.

Chandrasekhar, S.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

Chang, W. F.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Châtelain, B.

Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (Optical Society of America, 2011), p. JThA040.

Chen, J.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Chen, X.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

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Y. Chen, Y. Gu, and A. O. Hero, “Sparse LMS for system identification,” in, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2009), pp. 3125–3128.

Chen, Y. C.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Chen, Y.-K.

N. Kaneda, J. Lee, and Y.-K. Chen, “Nonlinear Equalizer for 112-Gb/s SSB-PAM4 in 80-km Dispersion Uncompensated Link,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), p. Tu2D.5.

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W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Chiang, S. C.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Chraplyvy, A. R.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Light. Technol. 9(1), 121–128 (1991).
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F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

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Ellis, A. D.

Eriksson, T.

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
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Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
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M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
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Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

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H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
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Faruk, M. S.

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Freude, W.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

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F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Fullner, C.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

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Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
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Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Gao, F.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Griesser, H.

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
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Y. Chen, Y. Gu, and A. O. Hero, “Sparse LMS for system identification,” in, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2009), pp. 3125–3128.

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F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express,  20(2), 136–1369 (2012).
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Hero, A. O.

Y. Chen, Y. Gu, and A. O. Hero, “Sparse LMS for system identification,” in, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2009), pp. 3125–3128.

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Y. Tang, K. P. Ho, and W. Shieh, “Coherent Optical OFDM Transmitter Design Employing Predistortion,” J. Light. Technol. 20(11), 954–956 (2008).

Hoang, T. M.

Hong, X.

Y. Zheng, J. Zhang, X. Hong, and C. Guo, “Generation and Detection of 170.49-Gb/s Single Polarization IM/DD Optical OFDM Signals Enabled by Volterra Nonlinear Equalization,” in Asia Communications and Photonics Conference, (Optical Society of America, 2016), p. ATh2D.1.

Huang, H. T.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Huang, W. J.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

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H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

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H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Kai, Y.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Kaneda, N.

N. Kaneda, J. Lee, and Y.-K. Chen, “Nonlinear Equalizer for 112-Gb/s SSB-PAM4 in 80-km Dispersion Uncompensated Link,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), p. Tu2D.5.

Karinou, F.

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
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C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Khanna, G.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Kikuchi, K.

M. S. Faruk and K. Kikuchi, “Compensation for In-Phase/Quadrature Imbalance in Coherent-Receiver Front End for Optical Quadrature Amplitude Modulation,” IEEE Photonics J. 5(2), 7800110 (2013).
[Crossref]

Killey, R. I.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Koonen, A. M. J.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Koos, C.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

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Lee, J.

N. Kaneda, J. Lee, and Y.-K. Chen, “Nonlinear Equalizer for 112-Gb/s SSB-PAM4 in 80-km Dispersion Uncompensated Link,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), p. Tu2D.5.

Lee, S. C. J.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
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Li, L.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Li, X.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Li, Z.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Lin, B. J.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Lin, C. T.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Liu, D.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Liu, J. J.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

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B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of Electronic Dispersion Compensation for Long-Haul Transmission Using Direct-Detection Optical OFDM,” J. Light. Technol.,  26(1), 196–203 (2008).
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C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Marcuse, D.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Light. Technol. 9(1), 121–128 (1991).
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Mecozzi, A.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
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C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
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A. Mecozzi, C. Antonelli, and M. Shtaif, “KramersâĂŞKronig coherent receiver,” Optica,  3(11), 1220–1227 (2016).
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A. Mecozzi, “A necessary and sufficient condition for minimum phase and implications for phase retrieval,” arXiv:1606.04861 (2016).

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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Mitchell, J.

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
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Morsy-Osman, M.

Napoli, A.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Ngoma, A.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Nishihara, M.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Pachnicke, S.

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
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J. Pan and C. H. Cheng, “Nonlinear Electrical Compensation for the Coherent Optical OFDM System,” J. Light. Technol. 29(2), 215–221 (2011).
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Pan, Z.

Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (Optical Society of America, 2011), p. JThA040.

Petermann, K.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
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Pilori, D.

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

Pinto, A. N.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express,  20(2), 136–1369 (2012).
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Plant, D. V.

T. M. Hoang, M. Y. S. Sowailem, Q. Zhuge, Z. Xing, M. Morsy-Osman, E. El-Fiky, S. Fan, M. Xiang, and D. V. Plant, “Single wavelength 480 Gb/s direct detection over 80km SSMF enabled by Stokes vector Kramers Kronig transceiver,” Opt. Express,  25(26), 33534–33542 (2017).
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Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (Optical Society of America, 2011), p. JThA040.

Prodaniuc, C.

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
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Qiang, Z.

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
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Rahman, T.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Randel, S.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Rasmussen, J.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Raybon, G.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Reis, J. D.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express,  20(2), 136–1369 (2012).
[Crossref]

Schetzen, M.

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1980).

Schmid, R.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Schmidt, B. J. C.

B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of Electronic Dispersion Compensation for Long-Haul Transmission Using Direct-Detection Optical OFDM,” J. Light. Technol.,  26(1), 196–203 (2008).
[Crossref]

Schuh, K.

S. T. Le, K. Schuh, F. Buchali, M. Chagnon, and H. Bulow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200km,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. Tu2D.5.

Schuster, M.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

Shi, J. W.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Shi, K.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Shieh, W.

Y. Tang, K. P. Ho, and W. Shieh, “Coherent Optical OFDM Transmitter Design Employing Predistortion,” J. Light. Technol. 20(11), 954–956 (2008).

Shih, P. T. B.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

Shtaif, M.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[Crossref]

A. Mecozzi, C. Antonelli, and M. Shtaif, “KramersâĂŞKronig coherent receiver,” Optica,  3(11), 1220–1227 (2016).
[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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Sillekens, E.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Sowailem, M. Y. S.

Spinnler, B.

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

Stojanovic, N.

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
[Crossref]

Takahara, T.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Tanaka, T.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Tang, M.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Tang, Y.

Y. Tang, K. P. Ho, and W. Shieh, “Coherent Optical OFDM Transmitter Design Employing Predistortion,” J. Light. Technol. 20(11), 954–956 (2008).

Tangdiongga, E.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Tao, Z.

T. Takahara, T. Tanaka, M. Nishihara, Y. Kai, L. Li, Z. Tao, and J. Rasmussen, “Discrete Multi-Tone for 100 Gb/s Optical Access Networks,” in Optical Fiber Communication Conference, (Optical Society of America, 2014), p. M2I.1.

Tehrani, A. S.

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
[Crossref]

Teixeira, A. L.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express,  20(2), 136–1369 (2012).
[Crossref]

Thakur, M. P.

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

Thomsen, B. C.

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

Tibshirani, R.

R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. Royal Stat. Soc. Ser. B (Methodological) 58(1), 267–288 (1996).

Tkach, R. W.

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Light. Technol. 9(1), 121–128 (1991).
[Crossref]

van Zantvoort, J. H. C.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Wang, C. L.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Wei, C. C.

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Winzer, P.

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

Winzer, P. J.

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[Crossref]

Wolf, S.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

Wren, M.

A. Zhu, M. Wren, and T. J. Brazil, “An efficient Volterra-based behavioral model for wideband RF power amplifiers,” in IEEE MTT-S International Microwave Symposium Digest, (IEEE, 2003), 2, pp. 787–790.

Xiang, M.

Xing, Z.

Yang, H.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Yang, Q.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Zeng, J.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Zhang, J.

Y. Zheng, J. Zhang, X. Hong, and C. Guo, “Generation and Detection of 170.49-Gb/s Single Polarization IM/DD Optical OFDM Signals Enabled by Volterra Nonlinear Equalization,” in Asia Communications and Photonics Conference, (Optical Society of America, 2016), p. ATh2D.1.

Zhao, J.

Zheng, Y.

Y. Zheng, J. Zhang, X. Hong, and C. Guo, “Generation and Detection of 170.49-Gb/s Single Polarization IM/DD Optical OFDM Signals Enabled by Volterra Nonlinear Equalization,” in Asia Communications and Photonics Conference, (Optical Society of America, 2016), p. ATh2D.1.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

Zhou, S.

F. Gao, S. Zhou, X. Li, S. Fu, L. Deng, M. Tang, D. Liu, and Q. Yang, “2 × 64 Gb/s PAM-4 transmission over 70 km SSMF using O-band 18G-class directly modulated lasers (DMLs),” Opt. Express,  25(7), 723–7237 (2017).

Zhu, A.

A. Zhu, M. Wren, and T. J. Brazil, “An efficient Volterra-based behavioral model for wideband RF power amplifiers,” in IEEE MTT-S International Microwave Symposium Digest, (IEEE, 2003), 2, pp. 787–790.

Zhuge, Q.

Zirath, H.

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
[Crossref]

IEEE Photonics J. (1)

M. S. Faruk and K. Kikuchi, “Compensation for In-Phase/Quadrature Imbalance in Coherent-Receiver Front End for Optical Quadrature Amplitude Modulation,” IEEE Photonics J. 5(2), 7800110 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

M. Schuster, S. Randel, C. A. Bunge, S. C. J. Lee, F. Breyer, B. Spinnler, and K. Petermann, “Spectrally Efficient Compatible Single-Sideband Modulation for OFDM Transmission With Direct Detection,” IEEE Photonics Technol. Lett. 20(9), 670–672 (2008).
[Crossref]

IEEE Transactions on Microw. Theory Tech. (1)

H. Cao, A. S. Tehrani, C. Fager, T. Eriksson, and H. Zirath, “I/Q Imbalance Compensation Using a Nonlinear Modeling Approach,” IEEE Transactions on Microw. Theory Tech. 57(3), p. 513–518(2009).
[Crossref]

J. Light. Technol. (9)

B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, “Experimental Demonstrations of Electronic Dispersion Compensation for Long-Haul Transmission Using Direct-Detection Optical OFDM,” J. Light. Technol.,  26(1), 196–203 (2008).
[Crossref]

M. S. Erkilinc, M. P. Thakur, S. Pachnicke, H. Griesser, J. Mitchell, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Spectrally Efficient WDM Nyquist Pulse-Shaped Subcarrier Modulation Using a Dual-Drive Mach-Zehnder Modulator and Direct Detection,” J. Light. Technol. 34(4), 1158–1165 (2016).
[Crossref]

N. Stojanovic, F. Karinou, Z. Qiang, and C. Prodaniuc, “Volterra and Wiener Equalizers for Short-Reach 100G PAM-4 Applications,” J. Light. Technol. 35(21), 458–4594 (2017).
[Crossref]

C. Antonelli, A. Mecozzi, M. Shtaif, X. Chen, S. Chandrasekhar, and P. J. Winzer, “Polarization Multiplexing With the Kramers-Kronig Receiver,” J. Light. Technol. 35, 5418–5424 (2017).
[Crossref]

X. Chen, C. Antonelli, S. Chandrasekhar, G. Raybon, A. Mecozzi, M. Shtaif, and P. Winzer, “Kramers-Kronig Receivers for 100-km Datacenter Interconnects,” J. Light. Technol.,  36(1), 79–89 (2018).
[Crossref]

Z. Li, M. S. Erkilinc, 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. Light. Technol. 35(10), 1887–1893 (2017).
[Crossref]

D. Marcuse, A. R. Chraplyvy, and R. W. Tkach, “Effect of fiber nonlinearity on long-distance transmission,” J. Light. Technol. 9(1), 121–128 (1991).
[Crossref]

Y. Tang, K. P. Ho, and W. Shieh, “Coherent Optical OFDM Transmitter Design Employing Predistortion,” J. Light. Technol. 20(11), 954–956 (2008).

J. Pan and C. H. Cheng, “Nonlinear Electrical Compensation for the Coherent Optical OFDM System,” J. Light. Technol. 29(2), 215–221 (2011).
[Crossref]

J. Royal Stat. Soc. Ser. B (Methodological) (1)

R. Tibshirani, “Regression Shrinkage and Selection via the Lasso,” J. Royal Stat. Soc. Ser. B (Methodological) 58(1), 267–288 (1996).

Opt. Express (5)

Optica (1)

Other (18)

W. J. Huang, W. F. Chang, C. C. Wei, J. J. Liu, Y. C. Chen, K. L. Chi, C. L. Wang, J. W. Shi, and J. Chen, “93% Complexity Reduction of Volterra Nonlinear Equalizer by ℓ1-Regularization for 112-Gbps PAM-4 850-nm VCSEL Optical Interconnect,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. M2D.7.

Y. Chen, Y. Gu, and A. O. Hero, “Sparse LMS for system identification,” in, Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2009), pp. 3125–3128.

E. M. Eksioglu, “RLS adaptive filtering with sparsity regularization,” in Proceedings of IEEE 10th International Conference on Information Science, Signal Processing and Their Applications (IEEE, 2010), pp. 550–553.

Z. Li, M. S. Erkilinc, K. Shi, E. Sillekens, L. Galdino, B. C. Thomsen, P. Bayvel, and R. I. Killey, “Joint Optimisation of Resampling Rate and Carrier-to-Signal Power Ratio in Direct-Detection Kramers-Kronig Receivers,” in Proceedings of IEEE European Conference on Optical Communication (IEEE, 2017), pp. 1–3.

H. T. Huang, C. T. Lin, S. C. Chiang, B. J. Lin, P. T. B. Shih, and A. Ngoma, “Volterra nonlinearity compensator for I/Q imbalanced mm-wave OFDM RoF systems,” in, Proceedings of IEEE International Topical Meeting on Microwave Photonics, (IEEE, 2015), pp. 1–4.

S. Randel, D. Pilori, S. Chandrasekhar, G. Raybon, and P. Winzer, “100-Gb/s discrete-multitone transmission over 80-km SSMF using single-sideband modulation with novel interference-cancellation scheme,” in, Proceedings of IEEE European Conference on Optical Communication, (IEEE, 2015), pp. 1–3.

A. Mecozzi, “A necessary and sufficient condition for minimum phase and implications for phase retrieval,” arXiv:1606.04861 (2016).

M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1980).

A. Zhu, M. Wren, and T. J. Brazil, “An efficient Volterra-based behavioral model for wideband RF power amplifiers,” in IEEE MTT-S International Microwave Symposium Digest, (IEEE, 2003), 2, pp. 787–790.

Y. Zheng, J. Zhang, X. Hong, and C. Guo, “Generation and Detection of 170.49-Gb/s Single Polarization IM/DD Optical OFDM Signals Enabled by Volterra Nonlinear Equalization,” in Asia Communications and Photonics Conference, (Optical Society of America, 2016), p. ATh2D.1.

Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference/National Fiber Optic Engineers Conference, (Optical Society of America, 2011), p. JThA040.

C. Fullner, S. Wolf, J. Kemal, J. Lutz, L. Altenhain, R. Schmid, W. Freude, C. Koos, and S. Randel, “Transmission of 80-GBd 16-QAM over 300 km and Kramers-Kronig Reception Using a Low-Complexity FIR Hilbert Filter Approximation,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. W4E.3.

H. Yang, J. Zeng, Y. Zheng, H. D. Jung, B. Huiszoon, J. H. C. van Zantvoort, E. Tangdiongga, and A. M. J. Koonen, “Evaluation of effects of MZM nonlinearity on QAM and OFDM signals in RoF transmitter,” in Proceedings of IEEE International Topical Meeting on Microwave Photonics Jointly Held with the Asia-Pacific Microwave Photonics Conference (IEEE, 2008), pp. 90–93.

A. Napoli, T. Rahman, G. Khanna, P. W. Berenguer, B. Spinnler, S. Calabro, J. K. Fischer, and M. Bohn, “Digital Pre-Distortion Techniques for Next Generation Bandwidth Variable Transponders,” in Advanced Photonics (IPR, NOMA, Sensors, Networks, SPPCom, SOF), (Optical Society of America, 2016), p. SpW3F.1.

S. T. Le, K. Schuh, F. Buchali, M. Chagnon, and H. Bulow, “1.6Tbps WDM Direct Detection Transmission with Virtual-Carrier over 1200km,” in Optical Fiber Communication Conference, (Optical Society of America, 2018), p. Tu2D.5.

N. Kaneda, J. Lee, and Y.-K. Chen, “Nonlinear Equalizer for 112-Gb/s SSB-PAM4 in 80-km Dispersion Uncompensated Link,” in Optical Fiber Communication Conference, (Optical Society of America, 2017), p. Tu2D.5.

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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 Postdeadline Papers, (Optical Society of America, 2017), p. Th5B.6.

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Figures (10)

Fig. 1
Fig. 1 The structure of I/Q Volterra filter (VF). hI and hQ are the Volterra kernels for the real and imaginary component, respectively.
Fig. 2
Fig. 2 The experimental setup of single-photodiode 16-QAM transmission system (the inset is the optical spectrum of the signal added with the CW tone). AWG, arbitrary waveform generator; EA, electric amplifier; MZM, Mach-Zenhder modulator; PMC, polarization-maintaining coupler; EDFA, erbium-doped fiber amplifier; VOA, variable optical attenuator; RFA, Raman fiber amplifier; OBPF, optical band-pass filter; PD, photo detector; DSO, digital sampling oscilloscope; DSP, digital signal processing.
Fig. 3
Fig. 3 (a) Off-line DSP flow with different processing schemes; (b) complex field reconstruction (CFR) by using Krames-Kronig (KK) algorithm (Δf denotes the frequency offset of the CW tone relative to the center of the signal spectrum);(c) the illustration of the nonlinearity compensation method by using two independent conventional real-valued VFs. EDC, electrical dispersion compensation; BER, bit error rate.
Fig. 4
Fig. 4 BER versus CSPR in back to back (B2B) case. CSPR, carrier to signal power ratio.
Fig. 5
Fig. 5 (a) BER as a function of AWG output amplitude and the recovered constellation: (b) 300mV without VF; (c) 300mV with two real-valued VFs; (d) 300mV with full I/Q VF.
Fig. 6
Fig. 6 BER as a function of launched power at 960 km: (a) with two real-valued VFs; (b) with full I/Q VF.
Fig. 7
Fig. 7 Selection threshold optimization at a launched power of 5 dBm with the penalty factor λ of 1 × 10−3: (a) for 2-nd order kernels and polynomial terms; (b) for 3-rd order kernels and polynomial terms.
Fig. 8
Fig. 8 The penalty factor λ optimization at a launched power of 5 dBm: (a) with a selection threshold of 4 × 10−4 for 2-nd order kernels and polynomial terms; (b) with a selection threshold of 2 × 10−4 for 3-rd order kernels and polynomial terms.
Fig. 9
Fig. 9 Performance and complexity comparison between full I/Q VF and sparse I/Q VF for 960-km fiber transmission.
Fig. 10
Fig. 10 (a) BER as a function of fiber length; (b) complexity comparison between full I/Q VF and sparse I/Q VF at different fiber lengths.

Equations (11)

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y ( n ) = k 1 = 0 L 1 1 h 1 ( k 1 ) x ( n k 1 ) + k 1 = 0 L 2 1 k 2 = 0 k 1 h 2 ( k 1 , k 2 ) m = 1 2 x ( n k m ) + + k 1 = 0 L p 1 k 2 = 0 k 1 k p = 0 k p 1 h p ( k 1 , k 2 , , k p ) m = 1 P x ( n k m )
y ( n ) = ( h I ) T U ( n ) + j ( h Q ) T U ( n )
U ( n ) = [ U 1 ( n ) , U 2 ( n ) , , U p ( n ) ] T
{ U p ( n ) = [ x k 1 x k 2 , , x k p ] T x k i = [ x I ( n k i ) x Q ( n k i ) ] T k 1 k 2 k p k i = 0 , 1 , , L p 1
[ a b ] T [ c d ] T = [ a c a d b c b d ] T .
y I ( n ) = ( h I ) T U ( n ) ,
J ( n ) = 1 2 e 2 ( n ) = 1 2 ( d I ( n ) ( h I ( n ) ) T U ( n ) ) 2 .
h I ( n + 1 ) = h I ( n ) μ J ( n ) h I ( n ) = h I ( n ) + μ e ( n ) U ( n ) ,
J 1 ( n ) = 1 2 e 2 ( n ) + γ j = 1 M h j I ( n ) 1 ,
h I ( n + 1 ) = h I ( n ) μ J 1 ( n ) h I ( n ) = h I ( n ) + μ e ( n ) U ( n ) λ s g n ( h I ( n ) ) ,
s g n ( n ) = { x / | x | , x 0 0 , x = 0 .

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