Abstract

High spectral efficiency in wavelength-division multiplexing (WDM) coherent optical communications can be achieved by lowering the channel spacing to a value close to the baud rate. As a result, such communication systems are subject to a high level of two-dimensional (2D) interference, originating from both chromatic dispersion (CD) within WDM channels and cross talk between WDM channels. We propose to digitally restore the transmitted information at the receiver side by jointly processing the WDM channels, using 2D interference cancellation methods interacting with a forward error correction decoder. Such schemes have been investigated extensively for known channel parameters; however, little is known about the error performance in the absence of channel state information. This issue is important, since in practice uncertainties about the physical parameters of the optical fiber and noise levels are unavoidable. In this paper, we propose optical channel parameter estimation schemes with limited impact on the overall complexity. First, a decision-aided carrier phase recovery, taking the presence of 2D interference into account, is designed to compensate the laser beat linewidth. Then we provide a technique to estimate the linear channel impairments for each WDM channel, based on a limited number of training symbols. The validity of the proposed method is evaluated by numerical simulations taking into account linear and nonlinear fiber impairments. In particular, we show that for high-order modulations, joint processing of spectrally overlapping WDM channels becomes mandatory to obtain satisfactory performances.

© 2014 Optical Society of America

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2013 (1)

2012 (5)

F. Lehmann, “Iterative mitigation of intercell interference in cellular networks based on Gaussian belief propagation,” IEEE Trans. Veh. Technol., vol.  61, no. 6, pp. 2544–2558, July 2012.
[CrossRef]

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

E. Yamazaki, M. Tomizawa, and Y. Miyamoto, “100-Gb/s optical transport network and beyond employing digital signal processing,” IEEE Commun. Mag., vol.  50, no. 2, pp. s43–s49, Feb. 2012.
[CrossRef]

R. Schmogrow, M. Winter, M. Meyer, D. Hillerkuss, S. Wolf, B. Baeuerle, A. Ludwig, B. Nebendahl, S. Ben-Ezra, J. Meyer, M. Dreschmann, M. Huebner, J. Becker, C. Koos, W. Freude, and J. Leuthold, “Real-time Nyquist pulse generation beyond 100 Gbit/s and its relation to OFDM,” Opt. Express, vol.  20, no. 1, pp. 317–337, Jan. 2012.
[CrossRef]

2011 (1)

2010 (3)

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal–noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol., vol.  16, no. 2, pp. 73–85, Mar. 2010.
[CrossRef]

S. J. Savory, “Digital optical coherent receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron., vol.  16, no. 5, pp. 1164–1179, Sept./Oct. 2010.
[CrossRef]

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

2009 (1)

2008 (2)

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express, vol.  16, no. 2, pp. 753–791, Jan. 2008.
[CrossRef]

2007 (1)

K. Gracie and M. H. Hamon, “Turbo and turbo-like codes: Principles and applications in telecommunications,” Proc. IEEE, vol.  95, no. 6, pp. 1228–1254, June 2007.
[CrossRef]

2005 (1)

D. Huang, K. Ben Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol.  53, no. 7, pp. 1191–1199, July 2005.
[CrossRef]

2002 (1)

B. F. Beidas, H. El Gamal, and S. Kay, “Iterative interference cancellation for high spectral efficiency satellite communications,” IEEE Trans. Commun., vol.  50, no. 1, pp. 31–36, Jan. 2002.
[CrossRef]

2001 (2)

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

T. J. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 599–618, Feb. 2001.
[CrossRef]

2000 (1)

B.-J. Kim and D. C. Cox, “Blind equalization for short burst wireless communications,” IEEE Trans. Veh. Technol., vol.  49, no. 4, pp. 1235–1247, July 2000.
[CrossRef]

1997 (1)

C. B. Papadias and A. J. Paulraj, “A constant modulus algorithm for multiuser signal separation in presence of delay spread using antenna arrays,” IEEE Signal Process. Lett., vol.  4, no. 6, pp. 178–181, June 1997.
[CrossRef]

1980 (1)

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun., vol.  28, no. 11, pp. 1867–1875, Nov. 1980.
[CrossRef]

1974 (1)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

Adde, P.

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

Arabaci, M.

Baeuerle, B.

Bahl, L. R.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

Bai, N.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Barros, D. J. F.

Batshon, H. G.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Becker, J.

Beidas, B. F.

B. F. Beidas, H. El Gamal, and S. Kay, “Iterative interference cancellation for high spectral efficiency satellite communications,” IEEE Trans. Commun., vol.  50, no. 1, pp. 31–36, Jan. 2002.
[CrossRef]

Ben Letaief, K.

D. Huang, K. Ben Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol.  53, no. 7, pp. 1191–1199, July 2005.
[CrossRef]

Ben-Ezra, S.

Bergano, N. S.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Bolle, C. A.

Bononi, A.

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal–noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol., vol.  16, no. 2, pp. 73–85, Mar. 2010.
[CrossRef]

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Torino, Italy, Sept. 2010.

Bosco, G.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

G. Bosco, “Spectral shaping in ultra-dense WDM systems: Optical vs. electrical approaches,” in Optical Fiber Communication Conf. (OFC), Los Angeles, CA, Mar. 2012.

Cai, J. X.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Cai, J.-X.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Cai, Y.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Carena, A.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

Chang, G.-K.

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

Cocke, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

Cox, D. C.

B.-J. Kim and D. C. Cox, “Blind equalization for short burst wireless communications,” IEEE Trans. Veh. Technol., vol.  49, no. 4, pp. 1235–1247, July 2000.
[CrossRef]

Curri, V.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

Davidson, C. R.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Detwiler, T.

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

Djordjevic, I. B.

Dreschmann, M.

El Gamal, H.

B. F. Beidas, H. El Gamal, and S. Kay, “Iterative interference cancellation for high spectral efficiency satellite communications,” IEEE Trans. Commun., vol.  50, no. 1, pp. 31–36, Jan. 2002.
[CrossRef]

Essiambre, R.-J.

Fechtel, S. A.

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. Wiley, 1998.

Forghieri, F.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

Foursa, D.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Foursa, D. G.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Freude, W.

Frey, B. J.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Gallager, R. G.

R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT, 1963.

Glick, M.

R. Thiruneelakandan, P. Watts, R. I. Killey, and M. Glick, “DSP-based crosstalk cancellation in WDM interconnects,” in Proc. CLEO/QELS, San Jose, CA, May 2010.

Gnauk, A. H.

Godard, D. N.

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun., vol.  28, no. 11, pp. 1867–1875, Nov. 1980.
[CrossRef]

Gracie, K.

K. Gracie and M. H. Hamon, “Turbo and turbo-like codes: Principles and applications in telecommunications,” Proc. IEEE, vol.  95, no. 6, pp. 1228–1254, June 2007.
[CrossRef]

Hamon, M. H.

K. Gracie and M. H. Hamon, “Turbo and turbo-like codes: Principles and applications in telecommunications,” Proc. IEEE, vol.  95, no. 6, pp. 1228–1254, June 2007.
[CrossRef]

Haykin, S.

S. Haykin, Adaptive Filter Theory. Upper Saddle River, NJ: Prentice Hall, 2002.

Hillerkuss, D.

Hsueh, Y.-T.

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

Huang, D.

D. Huang, K. Ben Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol.  53, no. 7, pp. 1191–1199, July 2005.
[CrossRef]

Huang, Y.-K.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Huebner, M.

Ip, E.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

E. Ip, A. P. T. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent detection in optical fiber systems,” Opt. Express, vol.  16, no. 2, pp. 753–791, Jan. 2008.
[CrossRef]

Jego, C.

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

Jelinek, F.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

Ji, P.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Kahn, J. M.

Kay, S.

B. F. Beidas, H. El Gamal, and S. Kay, “Iterative interference cancellation for high spectral efficiency satellite communications,” IEEE Trans. Commun., vol.  50, no. 1, pp. 31–36, Jan. 2002.
[CrossRef]

Killey, R. I.

R. Thiruneelakandan, P. Watts, R. I. Killey, and M. Glick, “DSP-based crosstalk cancellation in WDM interconnects,” in Proc. CLEO/QELS, San Jose, CA, May 2010.

Kim, B.-J.

B.-J. Kim and D. C. Cox, “Blind equalization for short burst wireless communications,” IEEE Trans. Veh. Technol., vol.  49, no. 4, pp. 1235–1247, July 2000.
[CrossRef]

Koos, C.

Kschischang, F. R.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Lau, A. P. T.

Le Bidan, R.

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

Lehmann, F.

F. Lehmann, “Iterative mitigation of intercell interference in cellular networks based on Gaussian belief propagation,” IEEE Trans. Veh. Technol., vol.  61, no. 6, pp. 2544–2558, July 2012.
[CrossRef]

Leroux, C.

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

Leuthold, J.

Lingle, R.

Liu, C.

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

J. Pan, A. Stark, C. Liu, and S. E. Ralph, “Fractionally-spaced frequency domain linear crosstalk cancellation with spectral alignment techniques for coherent superchannel optical systems,” in Optical Fiber Communication Conf. and Exhibition and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), Anaheim, CA, Mar. 2013.

Loeliger, H.-A.

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

Lu, J.

D. Huang, K. Ben Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol.  53, no. 7, pp. 1191–1199, July 2005.
[CrossRef]

Lucero, A.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Ludwig, A.

Mateo, E.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Mazurczyk, M.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

McCurdy, A.

Meyer, J.

Meyer, M.

Meyr, H.

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. Wiley, 1998.

Minkov, L.

Miyamoto, Y.

E. Yamazaki, M. Tomizawa, and Y. Miyamoto, “100-Gb/s optical transport network and beyond employing digital signal processing,” IEEE Commun. Mag., vol.  50, no. 2, pp. s43–s49, Feb. 2012.
[CrossRef]

Moeneclaey, M.

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. Wiley, 1998.

N. Noels and M. Moeneclaey, “Spectrally efficient CPM: Suboptimal FG-based multiuser detection,” in Proc. Wireless Communications and Networking Conf. (WCNC), Paris, France, April 2012, pp. 918–923.

Mohs, G.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Nebendahl, B.

Noels, N.

N. Noels and M. Moeneclaey, “Spectrally efficient CPM: Suboptimal FG-based multiuser detection,” in Proc. Wireless Communications and Networking Conf. (WCNC), Paris, France, April 2012, pp. 918–923.

Pan, J.

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

J. Pan, A. Stark, C. Liu, and S. E. Ralph, “Fractionally-spaced frequency domain linear crosstalk cancellation with spectral alignment techniques for coherent superchannel optical systems,” in Optical Fiber Communication Conf. and Exhibition and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), Anaheim, CA, Mar. 2013.

Papadias, C. B.

C. B. Papadias and A. J. Paulraj, “A constant modulus algorithm for multiuser signal separation in presence of delay spread using antenna arrays,” IEEE Signal Process. Lett., vol.  4, no. 6, pp. 178–181, June 1997.
[CrossRef]

Paulraj, A. J.

C. B. Papadias and A. J. Paulraj, “A constant modulus algorithm for multiuser signal separation in presence of delay spread using antenna arrays,” IEEE Signal Process. Lett., vol.  4, no. 6, pp. 178–181, June 1997.
[CrossRef]

Peckham, D. W.

Pilipetskii, A.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

Pyndiah, R.

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

Qian, D.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Ralph, S. E.

J. Pan, C. Liu, T. Detwiler, A. J. Stark, Y.-T. Hsueh, and S. E. Ralph, “Inter-channel crosstalk cancellation for Nyquist-WDM superchannel applications,” J. Lightwave Technol., vol.  30, no. 24, pp. 3993–3999, Dec. 2012.
[CrossRef]

J. Pan, A. Stark, C. Liu, and S. E. Ralph, “Fractionally-spaced frequency domain linear crosstalk cancellation with spectral alignment techniques for coherent superchannel optical systems,” in Optical Fiber Communication Conf. and Exhibition and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), Anaheim, CA, Mar. 2013.

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

Randel, S.

Raviv, J.

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

Richardson, T. J.

T. J. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 599–618, Feb. 2001.
[CrossRef]

Rossi, N.

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal–noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol., vol.  16, no. 2, pp. 73–85, Mar. 2010.
[CrossRef]

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Torino, Italy, Sept. 2010.

Ryf, R.

Savory, S. J.

S. J. Savory, “Digital optical coherent receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron., vol.  16, no. 5, pp. 1164–1179, Sept./Oct. 2010.
[CrossRef]

Schmogrow, R.

Serena, P.

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal–noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol., vol.  16, no. 2, pp. 73–85, Mar. 2010.
[CrossRef]

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Torino, Italy, Sept. 2010.

Sierra, A.

Sinkin, O.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

Sperti, D.

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Torino, Italy, Sept. 2010.

Stark, A.

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

J. Pan, A. Stark, C. Liu, and S. E. Ralph, “Fractionally-spaced frequency domain linear crosstalk cancellation with spectral alignment techniques for coherent superchannel optical systems,” in Optical Fiber Communication Conf. and Exhibition and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), Anaheim, CA, Mar. 2013.

Stark, A. J.

Tafur Monroy, I.

I. Tafur Monroy and E. Tangdiongga, Crosstalk in WDM Communication Networks (Kluwer Academic, 2002).

Tangdiongga, E.

I. Tafur Monroy and E. Tangdiongga, Crosstalk in WDM Communication Networks (Kluwer Academic, 2002).

Thiruneelakandan, R.

R. Thiruneelakandan, P. Watts, R. I. Killey, and M. Glick, “DSP-based crosstalk cancellation in WDM interconnects,” in Proc. CLEO/QELS, San Jose, CA, May 2010.

Tomizawa, M.

E. Yamazaki, M. Tomizawa, and Y. Miyamoto, “100-Gb/s optical transport network and beyond employing digital signal processing,” IEEE Commun. Mag., vol.  50, no. 2, pp. s43–s49, Feb. 2012.
[CrossRef]

Urbanke, R.

T. J. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 599–618, Feb. 2001.
[CrossRef]

Verdu, S.

S. Verdu, Multiuser Detection. Cambridge University, 1998.

Wang, T.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Watts, P.

R. Thiruneelakandan, P. Watts, R. I. Killey, and M. Glick, “DSP-based crosstalk cancellation in WDM interconnects,” in Proc. CLEO/QELS, San Jose, CA, May 2010.

Winter, M.

Winzer, P. J.

Wolf, S.

Xu, L.

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

Yamazaki, E.

E. Yamazaki, M. Tomizawa, and Y. Miyamoto, “100-Gb/s optical transport network and beyond employing digital signal processing,” IEEE Commun. Mag., vol.  50, no. 2, pp. s43–s49, Feb. 2012.
[CrossRef]

Zeng, T.

Zhang, H.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

EURASIP J. Wireless Commun. Netw. (1)

R. Le Bidan, C. Leroux, C. Jego, P. Adde, and R. Pyndiah, “Reed–Solomon turbo product codes for optical communications: From code optimization to decoder design,” EURASIP J. Wireless Commun. Netw., vol.  2008, 658042, 2008.
[CrossRef]

IEEE Commun. Mag. (1)

E. Yamazaki, M. Tomizawa, and Y. Miyamoto, “100-Gb/s optical transport network and beyond employing digital signal processing,” IEEE Commun. Mag., vol.  50, no. 2, pp. s43–s49, Feb. 2012.
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

S. J. Savory, “Digital optical coherent receivers: Algorithms and subsystems,” IEEE J. Sel. Top. Quantum Electron., vol.  16, no. 5, pp. 1164–1179, Sept./Oct. 2010.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

G. Bosco, A. Carena, V. Curri, P. Poggiolini, and F. Forghieri, “Performance limits of Nyquist-WDM and CO-OFDM in high speed PM-QPSK systems,” IEEE Photon. Technol. Lett., vol.  22, no. 15, pp. 1129–1131, Aug. 2010.
[CrossRef]

IEEE Signal Process. Lett. (1)

C. B. Papadias and A. J. Paulraj, “A constant modulus algorithm for multiuser signal separation in presence of delay spread using antenna arrays,” IEEE Signal Process. Lett., vol.  4, no. 6, pp. 178–181, June 1997.
[CrossRef]

IEEE Trans. Commun. (3)

D. Huang, K. Ben Letaief, and J. Lu, “Bit-interleaved time-frequency coded modulation for OFDM systems over time-varying channels,” IEEE Trans. Commun., vol.  53, no. 7, pp. 1191–1199, July 2005.
[CrossRef]

D. N. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun., vol.  28, no. 11, pp. 1867–1875, Nov. 1980.
[CrossRef]

B. F. Beidas, H. El Gamal, and S. Kay, “Iterative interference cancellation for high spectral efficiency satellite communications,” IEEE Trans. Commun., vol.  50, no. 1, pp. 31–36, Jan. 2002.
[CrossRef]

IEEE Trans. Inf. Theory (3)

L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol.  20, pp. 284–287, Mar. 1974.
[CrossRef]

F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum product algorithm,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 498–519, Feb. 2001.
[CrossRef]

T. J. Richardson and R. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inf. Theory, vol.  47, no. 2, pp. 599–618, Feb. 2001.
[CrossRef]

IEEE Trans. Veh. Technol. (2)

F. Lehmann, “Iterative mitigation of intercell interference in cellular networks based on Gaussian belief propagation,” IEEE Trans. Veh. Technol., vol.  61, no. 6, pp. 2544–2558, July 2012.
[CrossRef]

B.-J. Kim and D. C. Cox, “Blind equalization for short burst wireless communications,” IEEE Trans. Veh. Technol., vol.  49, no. 4, pp. 1235–1247, July 2000.
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (4)

Opt. Fiber Technol. (1)

A. Bononi, P. Serena, and N. Rossi, “Nonlinear signal–noise interactions in dispersion-managed links with various modulation formats,” Opt. Fiber Technol., vol.  16, no. 2, pp. 73–85, Mar. 2010.
[CrossRef]

Proc. IEEE (2)

E. Ip, P. Ji, E. Mateo, Y.-K. Huang, L. Xu, D. Qian, N. Bai, and T. Wang, “100G and beyond transmission technologies for evolving optical networks and relevant physical-layer issues,” Proc. IEEE, vol.  100, no. 5, pp. 1065–1078, May 2012.
[CrossRef]

K. Gracie and M. H. Hamon, “Turbo and turbo-like codes: Principles and applications in telecommunications,” Proc. IEEE, vol.  95, no. 6, pp. 1228–1254, June 2007.
[CrossRef]

Other (14)

“Optical interfaces for multichannel systems with optical amplifiers,” , 1998.

I. Tafur Monroy and E. Tangdiongga, Crosstalk in WDM Communication Networks (Kluwer Academic, 2002).

G. Bosco, “Spectral shaping in ultra-dense WDM systems: Optical vs. electrical approaches,” in Optical Fiber Communication Conf. (OFC), Los Angeles, CA, Mar. 2012.

Y. Cai, J. X. Cai, C. R. Davidson, D. Foursa, A. Lucero, O. Sinkin, A. Pilipetskii, G. Mohs, and N. S. Bergano, “Achieving high spectral efficiency in long-haul transmission with pre-filtering and multi-symbol detection,” in Proc. Asia Communications and Photonics Conf. and Exhibition (ACP), Shanghai, China, Dec. 2010, pp. 349–350.

N. Noels and M. Moeneclaey, “Spectrally efficient CPM: Suboptimal FG-based multiuser detection,” in Proc. Wireless Communications and Networking Conf. (WCNC), Paris, France, April 2012, pp. 918–923.

R. Thiruneelakandan, P. Watts, R. I. Killey, and M. Glick, “DSP-based crosstalk cancellation in WDM interconnects,” in Proc. CLEO/QELS, San Jose, CA, May 2010.

S. Verdu, Multiuser Detection. Cambridge University, 1998.

R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT, 1963.

A. Bononi, P. Serena, N. Rossi, and D. Sperti, “Which is the dominant nonlinearity in long-haul PDM-QPSK coherent transmissions?” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Torino, Italy, Sept. 2010.

C. Liu, J. Pan, T. Detwiler, A. Stark, Y.-T. Hsueh, G.-K. Chang, and S. E. Ralph, “Joint digital signal processing for superchannel coherent optical systems: Joint CD compensation for joint ICI cancellation,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, Sept. 2012.

J. Pan, A. Stark, C. Liu, and S. E. Ralph, “Fractionally-spaced frequency domain linear crosstalk cancellation with spectral alignment techniques for coherent superchannel optical systems,” in Optical Fiber Communication Conf. and Exhibition and the Nat. Fiber Optic Engineers Conf. (OFC/NFOEC), Anaheim, CA, Mar. 2013.

M. Mazurczyk, D. G. Foursa, H. G. Batshon, H. Zhang, C. R. Davidson, J.-X. Cai, A. Pilipetskii, G. Mohs, and N. S. Bergano, “30 Tb/s transmission over 6,630 km using 16QAM signals at 6.1 bits/s/Hz spectral efficiency,” in Proc. European Conf. and Exhibition on Optical Communications (ECOC), Amsterdam, The Netherlands, June 2012.

H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing. Wiley, 1998.

S. Haykin, Adaptive Filter Theory. Upper Saddle River, NJ: Prentice Hall, 2002.

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Figures (13)

Fig. 1.
Fig. 1.

Transmitter configuration for the coherent WDM system.

Fig. 2.
Fig. 2.

Receiver layout for the coherent WDM system.

Fig. 3.
Fig. 3.

Matrix of transmitted training and data symbols on each subband.

Fig. 4.
Fig. 4.

WDM system model.

Fig. 5.
Fig. 5.

WDM interference cancellation using blind MIMO equalization.

Fig. 6.
Fig. 6.

WDM interference cancellation using per-subband MAP/PIC equalization with code-aided interference estimates (dashed lines) and PLL decision feedback loops (thick lines).

Fig. 7.
Fig. 7.

WDM 2D interference cancellation using GaBP with PLL decision feedback loops (thick lines).

Fig. 8.
Fig. 8.

WDM transmission line setup.

Fig. 9.
Fig. 9.

BER comparison of blind MIMO equalization and GaBP 2D interference cancellation.

Fig. 10.
Fig. 10.

BER comparison of per-subchannel MAP/PIC equalization and GaBP 2D interference cancellation.

Fig. 11.
Fig. 11.

BER comparison of per-subchannel GaBP ignoring the presence of WDM cross talk (triangles) and GaBP 2D interference cancellation (lines). The BER of a genie-aided per-subchannel GaBP with artificial cross-talk removal is also provided (circles).

Fig. 12.
Fig. 12.

BER comparison on a nonlinear fiber with DCF for 8-QAM: GaBP 2D interference cancellation (lines), per-subchannel MAP/PIC equalization (dashed), and genie-aided per-subchannel GaBP with artificial cross-talk removal (circles).

Fig. 13.
Fig. 13.

BER comparison on a nonlinear fiber without DCF for 8-QAM: GaBP 2D interference cancellation (lines), per-subchannel MAP/PIC equalization (dashed), and genie-aided per-subchannel GaBP with artificial cross-talk removal (circles).

Equations (18)

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yi,j=ejθim=11n=N/2N/2fm,nidim,jn+ni,j,
fi=[f1,N/2i*f0,N/2i*f1,N/2i*f1,N/2i*f0,N/2i*f1,N/2i*].
xji=[di1,jN/2di,jN/2di+1,jN/2di1,j+N/2di,j+N/2di+1,j+N/2].
yi,j=ejθifiHxji+ni,j,0iI1,0jJ1.
lnp(yi,j|θi,xji,λi)=ln1πσi21σi2|yi,jejθifiHxji|2,
lnp(yi,j|θi,xji,λi)=ln1πσi21σi2(|yi,j|2+|fiHxji|2)+2σi2Re((fiHxji)*yi,jejθi).
L(yi,j|θi,xji,λi)=Re((fiHxji)*yi,jejθi).
θ^i,j+1=θ^i,j+γiL(yi,j|θi,xji,λi)θi|θi=θ^i,j,
θ^i,j+1=θ^i,j+γiIm((fiHxji)*yi,jejθ^i,j).
λ^i=argmaxλilogp(Yi|Di;λi),
{f^i=[jTxjixjiH]1[jTyi,j*xji]σ^i2=jT|yi,jf^iHxji|2|T|.
p(Yi|Di;λi)=jTp(yi,j|xji;λi).
λip(Yi|Di;λi)=p(Yi|Di;λi)jTλip(yi,j|xji;λi)p(yi,j|xji;λi),
λilogp(Yi|Di;λi)=jTλilogp(yi,j|xji;λi).
jTλilogp(yi,j|xji;λi)|λi=λ^i=0.
p(yi,j|xji;λi)=1πσi2exp{1σi2|yi,jfiHxji|2}.
filogp(yi,j|xji;λi)=1σi2fi|yi,jfiHxji|2=2σi2(xjixjiHf^iyi,j*xji),
logp(yi,j|xji;λi)σi2=σi2|yi,jfiHxji|2.