Abstract

Blind channel estimation is critical for digital signal processing (DSP) compensation of optical fiber communications links. The overall channel consists of deterministic distortions such as chromatic dispersion, as well as random and time varying distortions including polarization mode dispersion and timing jitter. It is critical to obtain robust acquisition and tracking methods for estimating these distortions effects, which, in turn, can be compensated by means of DSP such as Maximum Likelihood Sequence Estimation (MLSE). Here, a novel blind estimation algorithm is developed, accompanied by inclusive mathematical modeling, and followed by extensive set of real time experiments that verify quantitatively its performance and convergence. The developed blind channel estimation is used as the basis of an MLSE receiver. The entire scheme is fully implemented in a 65nm CMOS Application Specific Integrated Circuit (ASIC). Experimental measurements and results are presented, including Bit Error Rate (BER) measurements, which demonstrate the successful data recovery by the MLSE ASIC under various channel conditions and distances.

© 2013 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  24. F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.
  25. G. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley-Interscience, 1978)
  26. R. C. Sprinthall, Basic Statistical Analysis (Pearson Allyn & Bacon, 2011)

2010

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

W. Chung, “Channel estimation methods based on Volterra kernels for MLSD in optical communication systems,” Photon.Technol. Let.22(4), 224–226 (2010).
[CrossRef]

2009

2008

2007

2006

2005

2004

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” Photon. Technol. Let.16(2), 674–676 (2004).
[CrossRef]

1997

Y.-J. Jeng and C.-C. Yeh, “Cluster-based blind nonlinear-channel estimation,” Tran. Sig. Proc.45(5), 1161–1172 (1997).
[CrossRef]

1992

M. Ghosh and C. L. Weber, “Maximum likelihood blind equalization,” Opt. Eng.31(6), 1224–1228 (1992).
[CrossRef]

Agazzi, O. E.

Alfiad, M. S.

M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol.27(16), 3614–3622 (2009).
[CrossRef]

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Awadalla, A.

Carrer, H. S.

Chung, W.

W. Chung, “Channel estimation methods based on Volterra kernels for MLSD in optical communication systems,” Photon.Technol. Let.22(4), 224–226 (2010).
[CrossRef]

Colavolpe, G.

Crivelly, D. E.

De Man, E.

de Waardt, H.

C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E. D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol.26(1), 131–141 (2008).
[CrossRef]

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Dorschky, C.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Downie, J. D.

J. D. Downie and Jason Hurle, “Chromatic dispersion compensation effectiveness of an MLSE-EDC receiver for three variants of duobinary,” IEEE/LEOS Summer Topical Meetings, TuA1.2 (2007).

Duthel, T.

Elbers, J.-P.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Essiambre, R.-J.

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

Faerbert, A.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Fludger, C. R. S.

Foggi, T.

Forestieri, E.

Geyer, J.

Ghosh, M.

M. Ghosh and C. L. Weber, “Maximum likelihood blind equalization,” Opt. Eng.31(6), 1224–1228 (1992).
[CrossRef]

Glingene, C.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Gorshtein, A.

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Low cost 112G direct detection metro transmission system with reduced bandwidth (10G) components and MLSE compensation,” SPPCoM2011, SPWC4.

Griesser, H.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Hauske, F. N.

M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol.27(16), 3614–3622 (2009).
[CrossRef]

F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Hueda, M. R.

Hurle, Jason

J. D. Downie and Jason Hurle, “Chromatic dispersion compensation effectiveness of an MLSE-EDC receiver for three variants of duobinary,” IEEE/LEOS Summer Topical Meetings, TuA1.2 (2007).

Ip, E.

Jeng, Y.-J.

Y.-J. Jeng and C.-C. Yeh, “Cluster-based blind nonlinear-channel estimation,” Tran. Sig. Proc.45(5), 1161–1172 (1997).
[CrossRef]

Kahn, J. M.

Katz, G.

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

G. Katz and D. Sadot, “Channel estimators for maximum-likelihood sequence estimation in direct-detection optical communications,” Opt. Eng.47(4), 31–34 (2008).
[CrossRef]

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Low cost 112G direct detection metro transmission system with reduced bandwidth (10G) components and MLSE compensation,” SPPCoM2011, SPWC4.

Khoe Giok-Djan,

Koonen, A. M. J.

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Krause, D. J.

Kupfer, T.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Kuschnerov, M.

Langenbach, S.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Lankl, B.

M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol.27(16), 3614–3622 (2009).
[CrossRef]

F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Laperle, C.

Levy, O.

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Low cost 112G direct detection metro transmission system with reduced bandwidth (10G) components and MLSE compensation,” SPPCoM2011, SPWC4.

Napoli, A.

M. Kuschnerov, F. N. Hauske, K. Piyawanno, B. Spinnler, M. S. Alfiad, A. Napoli, and B. Lankl, “DSP for coherent single-carrier receivers,” J. Lightwave Technol.27(16), 3614–3622 (2009).
[CrossRef]

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

O'Sullivan, M.

Piyawanno, K.

Prati, G.

Roberts, K.

Sadot, D.

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

G. Katz and D. Sadot, “Channel estimators for maximum-likelihood sequence estimation in direct-detection optical communications,” Opt. Eng.47(4), 31–34 (2008).
[CrossRef]

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Low cost 112G direct detection metro transmission system with reduced bandwidth (10G) components and MLSE compensation,” SPPCoM2011, SPWC4.

Savory, S. J.

Schmidt, E. D.

Schmidt, E.D.

F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.

Schuliea, C.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Schulien, C.

Spinnler, B.

Stojanovic, N.

N. Stojanovic, “Tail extrapolation in MLSE receivers using nonparametric channel model estimation,” IEEE Trans. Signal Process.57(1), 270–278 (2009).
[CrossRef]

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Sun, H.

Taylor, M. G.

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” Photon. Technol. Let.16(2), 674–676 (2004).
[CrossRef]

van den Borne, D.

C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E. D. Schmidt, T. Wuth, J. Geyer, E. De Man, Khoe Giok-Djan, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol.26(1), 131–141 (2008).
[CrossRef]

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

Weber, C. L.

M. Ghosh and C. L. Weber, “Maximum likelihood blind equalization,” Opt. Eng.31(6), 1224–1228 (1992).
[CrossRef]

Wernz, H.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

Winzer, P. J.

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

Wu, K. T.

Wuth, T.

Xie, C.

F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.

Yeh, C.-C.

Y.-J. Jeng and C.-C. Yeh, “Cluster-based blind nonlinear-channel estimation,” Tran. Sig. Proc.45(5), 1161–1172 (1997).
[CrossRef]

IEEE Trans. Signal Process.

N. Stojanovic, “Tail extrapolation in MLSE receivers using nonparametric channel model estimation,” IEEE Trans. Signal Process.57(1), 270–278 (2009).
[CrossRef]

J. Lightwave Technol.

Opt. Eng.

G. Katz and D. Sadot, “Channel estimators for maximum-likelihood sequence estimation in direct-detection optical communications,” Opt. Eng.47(4), 31–34 (2008).
[CrossRef]

M. Ghosh and C. L. Weber, “Maximum likelihood blind equalization,” Opt. Eng.31(6), 1224–1228 (1992).
[CrossRef]

Opt. Express

Photon. Technol. Let.

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Coherent compensation for 100G DP-QPSK with one sample per symbol based on anti-aliasing filtering and blind equalization MLSE,” Photon. Technol. Let.22(16), 1208–1210 (2010).
[CrossRef]

M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments,” Photon. Technol. Let.16(2), 674–676 (2004).
[CrossRef]

Photon.Technol. Let.

W. Chung, “Channel estimation methods based on Volterra kernels for MLSD in optical communication systems,” Photon.Technol. Let.22(4), 224–226 (2010).
[CrossRef]

Proc. IEEE

P. J. Winzer and R.-J. Essiambre, “Advanced optical modulation formats,” Proc. IEEE94(5), 952–985 (2006).
[CrossRef]

Tran. Sig. Proc.

Y.-J. Jeng and C.-C. Yeh, “Cluster-based blind nonlinear-channel estimation,” Tran. Sig. Proc.45(5), 1161–1172 (1997).
[CrossRef]

Other

E. Zervas, J. Proakis, and V. Eyuboglu, “A quantized channel approach to blind equalization,” in Proc. ICC, Chicago, IL, 3, 1539–1543 (1992)

I. Lyubomirsky, “Advanced modulation formats for ultra-dense wavelength division multiplexing,” http://www.journalogy.net/Publication/5674594/advanced-modulation-formats-for-ultra-dense-wavelength-division-multiplexing

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingene, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schuliea, “Measurement of the dispersion tolerance of optical duobinary with an MLSE-receiver at 10.7 Gb/s,” Proc. OFC2005, OThJ4.
[CrossRef]

J. D. Downie and Jason Hurle, “Chromatic dispersion compensation effectiveness of an MLSE-EDC receiver for three variants of duobinary,” IEEE/LEOS Summer Topical Meetings, TuA1.2 (2007).

A. Gorshtein, O. Levy, G. Katz, and D. Sadot, “Low cost 112G direct detection metro transmission system with reduced bandwidth (10G) components and MLSE compensation,” SPPCoM2011, SPWC4.

G. Agrawal, Fiber Optic Communications Systems (John Wiley & Sons, Inc., 2002)

M. S. Alfiad, D. van den Borne, F. N. Hauske, A. Napoli, B. Lankl, A. M. J. Koonen, and H. de Waardt, “Dispersion tolerant 21.4 Gb/s DQPSK using simplified Gaussian Joint-Symbol MLSE,” Proc. OFC2008, paper OTHO3.
[CrossRef]

F. N. Hauske, B. Lankl, C. Xie, and E.D. Schmidt, “Iterative electronic equalization utilizing low complixity MLSEs for 40 Gbit/s DQPSK modulation,” Proc. OFC2007, paper OMG2.

G. Box, W. G. Hunter, and J. S. Hunter, Statistics for Experimenters (Wiley-Interscience, 1978)

R. C. Sprinthall, Basic Statistical Analysis (Pearson Allyn & Bacon, 2011)

S. J. Savory, “Compensation of fibre impairments in digital coherent systems,” ECOC 2008, 21–25 (Brussels, Belgium, 2008).

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Figures (13)

Fig. 1
Fig. 1

The architecture of blind MLSE equalizer.

Fig. 2
Fig. 2

Initial metrics determination procedure (IMDP).

Fig. 3
Fig. 3

FIR filter examples for generation of M for c=2 , N isi =4 and (a) j=0 - increasing exponent, (b) j=4 - decaying exponent, (c) j=8 - symmetrically decaying exponent.

Fig. 4
Fig. 4

The experimental setup block diagram.

Fig. 5
Fig. 5

Phase # 1 of IMDP. Normalized histograms (a) H #1 ( 0 ) and (b) H #1 ( 1 ) . Note that both histograms describe the scenario of channel memory depth of one symbol.

Fig. 6
Fig. 6

Histogram sets obtained after phase # 2 of IMDP. The histograms sets after 8 iterations (a) H #2 ( 0 ) (b) H #2 ( 1 ) . (c) The histograms set H training obtained by using the training sequence. It is clear that H #1 ( 0 ) does not converge, whereas H #1 ( 1 ) provides a good initial guess.

Fig. 7
Fig. 7

IMDP convergence monitoring during phase #2: (a) D ED convergence following (38), (b) standard deviation of central moments (36), (c) convergence in terms of BER. and (d) Phase #3 of IMDP – checking the convergence criterion of (34).

Fig. 8
Fig. 8

Phase # 4 of IDMP. Histogram sets, obtained for different shifts: (a) M P shift =2,BER=4.26 10 1 , σ i 2 =89.21 , (b) M P shift =1,BER=3.15 10 3 , σ i 2 =12.21 , (c) M P shift =0,BER=1.77 10 3 , σ i 2 =10.4 , (d) M P shift =1,BER=1.89 10 3 , σ i 2 =10.41 , (e) M P shift =2,BER=1.82 10 3 , σ i 2 =10.53 .

Fig. 9
Fig. 9

Phase # 1 of IMDP (40 km link), Normalized histogram H #1 ( 2 ) corresponding to N isi channel =2,j=1 . H #1 ( 0 ) and H #1 ( 1 ) corresponding to N isi channel =1 are shown on Fig. 5.

Fig. 10
Fig. 10

The histograms sets after 8 iterations (a) H #2 ( 0 ) ,(b) H #2 ( 1 ) and (c) H #2 ( 2 ) . (d) The histograms set H training obtained by using the training sequence. It is clear that H #1 ( 0 ) and H #1 ( 1 ) diverge, whereas H #1 ( 2 ) provides a similar histogram map as with training.

Fig. 11
Fig. 11

IMDP convergence monitoring during phase #2 (40 km link): (a) D ED convergence (38), (b) standard deviation of central moments (36), and (c) convergence in terms of BER. (d) Phase #3 of IMDP – check the convergence criterion (34).

Fig. 12
Fig. 12

Phase # 4 of IDMP (40 km channel). Histogram sets, obtained for different shifts: (a) M P shift =2,BER=3.01 10 1 , σ i 2 =54.36 , (b) M P shift =1,BER=2.1 10 1 , σ i 2 =30.40 , (c) M P shift =0,BER=2.16 10 3 , σ i 2 =7.07 , (d) M P shift =1,BER=1.15 10 3 , σ i 2 =6.32 , (e) M P shift =2,BER=2.94 10 3 , σ i 2 =8.11 .

Fig. 13
Fig. 13

Experimental BER curves comparing the training and the IMDP.

Equations (44)

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M l ( r n | a n , a n1 ,..., a n N isi +1 )=log( f channel ( r n | a n , a n1 ,..., a n N isi +1 ) ), l=0,1,..., V N isi +1
ϒ opt = min 0k V N { ϒ l ( k ) } ϒ l ( k ) = n=0 N1 M l ( k ) ( r n | a n , a n1 ,..., a n N isi +1 )
H={ H l ( r n ,| a n , a n1 ,..., a n N isi +1 ),l=0,1,..., V N isi +1 } H l ( r n | a n , a n1 ,..., a n N isi +1 ) f channel ( r n | a n , a n1 ,..., a n N isi +1 )
r n =Γ( a n , a n1 ,..., a n N isi ( channel ) +1 )
H DTE Tx+fiber [ n ]( γ δ n 0 0 1γ δ n+τ ) h CD [ n ] h Tx [ n ] h OF [ n ]
h CD [ n ]= jW e jπW n 2 , N CD 1 2 n N CD 1 2 W= c f s 2 CD λ 0 2 , N CD =2 1 2W +1
x[ n ]=( H DTE Tx+fiber [ n ] a n )+ z ASE [ n ]= s n + z ASE [ n ]
s n =( k= N Ch 1 2 N Ch 1 2 ψ[ k ] γ a nk k= N Ch 1 2 N Ch 1 2 ψ[ k ] 1γ a nk+τ )
N Ch = N CD + N Tx + N OF 2
u n =R( Tr{ s n s n H } ) h Rx [ n ]+ w n = r n + w n
y n Tr{ s n s n H }
y n = | k= N Ch 1 2 N Ch 1 2 ψ[ k ] γ a nk | 2 + | k= N Ch 1 2 N Ch 1 2 ψ[ k ] 1γ a n+τk | 2
y n = k= N Ch 1 2 N Ch 1 2 γ | ψ[ k ] | 2 | a nk | 2 + k= N Ch 1 2 N CD 1 2 ( 1γ ) | ψ[ k ] | 2 | a n+τk | 2 + +e{ k= N Ch 1 2 N Ch 1 2 l= N Ch 1 2 kl N Ch 1 2 γψ[ k ]ψ'[ l ] a nk a ' nl }+ +e{ k= N Ch 1 2 N Ch 1 2 l= N Ch 1 2 kl N Ch 1 2 ( 1γ )ψ[ k ]ψ'[ l ] a n+τk a ' n+τl }
r n =Γ( a n , a n1 ,..., a n N isi ( channel ) +1 )=R y n h Rx [ n ]
M ( j ) = 2π σ j ( r ˜ n μ j ) 2 ./( 2 σ j 2 ),j=0,..., J max 1
ψ[ n ] K 1 K 2 h CD [ n ], N ch 1 2 n N ch 1 2 , N ch = N CD
r n =γK k= N CD 1 2 N CD 1 2 a nk +( 1γ )K k= N CD 1 2 N CD 1 2 a n+τk + +γK k= N CD 1 2 N CD 1 2 l= N CD 1 2 kl N CD 1 2 cos( πW ( kl ) 2 ) a nk a nl + +( 1γ )K k= N CD 1 2 N CD 1 2 l= N CD 1 2 kl N CD 1 2 cos( πW ( kl ) 2 ) a n+τk a n+τl
b ^ k =γKΠ( k N CD )+( 1γ )KΠ( kτ N CD )
b ˜ k ( a n N CD 1 2 ,..., a n+ N CD 1 2 +τ )=γK l= N CD 1 2 ,kl N CD 1 2 cos( πW ( kl ) 2 ) a nl + +( 1γ )K l= N CD 1 2 τ,kl N CD 1 2 τ cos( πW ( kl ) 2 ) a nl
E{ b ˜ k ( a n N CD 1 2 ,..., a n+ N CD 1 2 +τ ) }=γK l= N CD 1 2 ,kl N CD 1 2 cos( πW ( kl ) 2 )E{ a nl } + +( 1γ )K l= N CD 1 2 τ,kl N CD 1 2 τ cos( πW ( kl ) 2 )E{ a nl }
b j [ n ]={ c n l=1 m+1 c l ,j=0,..., N isi 1,m=( j+1 )mod N isi c n l=1 m+1 c l ,j= N isi ,...,2 N isi 1,m=( j+1 )mod N isi c | n | l=( N isi 2 m+1) N isi 2 m+1 c | l | ,j=2 N isi ,...,3 N isi 1,m=( j+1 )mod N isi ,m0,m N isi 1
μ j =A b j ( 2 N ADC 1 )
σ j 2 =S b j 2
A=[ 0 0 0 1 1 0 1 1 ]S=[ Var( '0' ) Var( '1' ) Var( '0' ) Var( '1' ) Var( '1' ) Var( '0' ) Var( '1' ) Var( '1' ) ]
P( a i )= 1 V ,i
p= 1 V N isi +1
P( u n Γ i )=p,i=0,..., N br 1
m 0 ( i ) u n Γ i δ u n Γ i ,i=0,..., N br 1
f m 0 ( i ) ( m 0 ( i ) )= 1 2πNp( 1p ) exp{ ( m 0 ( i ) Np ) 2 2Np( 1p ) },i=0,..., N br 1
m 0 ( i ) =Np,i=0,..., N br 1
z= ( m 0 ( i ) Np ) Np( 1p ) ,i=0,..., N br 1
ε=2Q( z )= 2 2π z e x 2 2 dx
th r low m 0 ( i ) th r high ,i=0,..., N br 1
th r low =Np Np( 1p ) Q 1 ( ε 2 ) th r high =Np+ Np( 1p ) Q 1 ( ε 2 )
std( m 0 )[ d ] 1 N br 1 i=0 N br 1 ( m 0 ( i ) [ d ]Np ) 2 ,d=0,...,X1
D KL ( i ) D KL ( H i training || H i blind )= m=0 2 N ADC 1 ln( H i training ( m ) H i blind ( m ) ) H i training ( m )
D ED ( H training || H blind ) i=0 N br 1 [ D KL ( i ) ] 2
σ l 2 = σ noise 2 ( l )+ σ ADC 2 + σ residualISI 2 ,0l N br
r n = k=0 N isi ( channel ) 1 b k a nk
max n 0 ( 0,L ) n= n 0 n 0 + N isi | b n |
σ residual_ISI 2 = σ a 2 n=0 n 0 1 | b n | 2 + σ a 2 n= n 0 + N isi +1 N isi ( channel ) | b n | 2
σ a 2 = 1 V k=0 V1 a k 2 ( 1 V k=0 V1 a k ) 2 , a k Vocabulary
max n 0 ( 0,L ) n= n 0 n 0 + N isi | E{ b ^ n + b ˜ n } |
MP= min n 0 σ average 2 ( n 0 ) σ average 2 ( n 0 )= σ l 2 ( n 0 ) = 1 N br l=0 N br 1 σ l 2 ( n 0 )

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