Abstract

Exact signal statistics for fiber-optic links containing a single optical pre-amplifier are calculated and applied to sequence estimation for electronic dispersion compensation. The performance is evaluated and compared with results based on the approximate chi-square statistics. We show that detection in existing systems based on exact statistics can be improved relative to using a chi-square distribution for realistic filter shapes. In contrast, for high-spectral efficiency systems the difference between the two approaches diminishes, and performance tends to be less dependent on the exact shape of the filter used.

© 2005 Optical Society of America

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References

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  1. S. Benedetto and E. Biglieri, Principles of Digital Transmission (Kluwer Academic/Plenum Publishers, 1999).
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    [Crossref]
  3. J.H. Winters and S. Kasturia, “Constrained maximum likelihood detection for high-speed fiber optic systems,” in Proc. GLOBECOM ‘91, 1574–1579 (1991).
  4. N. S. Bergano “Undersea Communication Systems” in Fiber optic telecommunications IVB, Ivan Kaminow and T. Li, Eds. (Elsevier Science2002).
  5. H.F. Haunstein et al., Design of near optimum electrical equalizers for optical transmission in the presence of PMD”, in Proc. OFC, 2001, Paper WAA4-1.
  6. H. Bulow and G. Thielecke, “Electronic PMD mitigation-from linear equalization to maximum-likelihood detection” in Proc. OFC 2001, 2001, Paper WAA3-1.
  7. N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).
  8. N. Alić, G. C. Papen, and Y. Fainman, “Theoretical Performance Analysis of Maximum Likelihood Sequence Estimation in Intensity Modulated Short-Haul Fiber Optic Links,” Proc. IEEE LEOS Annual Meeting, Puerto Rico, paper ThB3, (2004).
  9. N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.
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    [Crossref]
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    [Crossref]
  12. A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).
  13. J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).
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    [Crossref]
  18. J. Lee et al., “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain”, J. Lightwave Technol. 12, 1224–1229 (1994).
    [Crossref]
  19. I. T. Monroy and G. Einarsson, “On Analytical Expressions for the Distribution of the filtered Output of Square Envelope Receivers with Signal and Colored Gaussian Noise Input,” IEEE Transactions on Communications 49, 19–23 (2001).
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  20. G. Jacobsen, K. Berlitzon, and Z. Xiapin, “WDM Transmission System Performance: Influence of non-Gaussian Detected ASE Noise and Periodic DEMUX Characteristic,” J. Lightwave Technol. 16, 1804–1812 (1998).
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    [Crossref]
  22. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
    [Crossref]
  23. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
    [Crossref]
  24. I. B. Djordjevic and B. Vasic, “Receiver Modeling for Optically Amplified Communication Systems,” International J. Electron. Commun. 57, 381–390 (2003).
    [Crossref]
  25. F. Buchali and H. Bulow, “Correlation sensitive Viterbi equalization of 10 Gb/s signals in bandwidth limited receivers,” in Proc. OFC 2005, Paper F020.
  26. R. Loudon and T.J. Shepherd, “Properties of the Optical Quantum Amplifier,” Optica Acta,  31, 1243–1269 (1984).
    [Crossref]
  27. C. Dorrer, C.R. Doerr, I. Kang, R. Ryf, J. Leuthold, and P.J. Winzer, “Measurement of eye diagrams and constellation diagrams of optical sources using linear optics and waveguide technology,” J. Lightwave Technol. 23, 178–186 (2005).
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  31. D. Slepian, “Fluctuations of Random Noise Power,” Bell Sys. Tech. J. 37, 163 (1958)
  32. D. Slepian, “A Numerical Method Of Determining EigenValues And EigenFunction Of Analytic Kernels” SIAM Journal of Numerical Analysis,  5, 586–600 (1968).
    [Crossref]
  33. D. Slepian, “On Bandwidth,” Proc. Of IEEE,  64, 292–300 (1976).
    [Crossref]
  34. I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties” Appl. Comput. Harmon. Anal. 16, 208–230 (2004).
    [Crossref]
  35. D.B. Percival and A.T. WaldenSpectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (Cambridge, Cambridge University Press, 1993).
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  36. K. Yonenaga and S. Kuwano, “Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,” J. Lightwave Technol. 15, 1530–1537 (1997).
    [Crossref]
  37. R. A. Griffin and A. C. Carter, “Optical differential quadrature phaseshift keying (oDQPSK) for high capacity optical transmission,” in Proc. OFC, 2002, Paper WX6C.
  38. A.H. Gnauck and P.J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. 23, 115–130 (2005).
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2005 (2)

2004 (2)

H. F. Haunstein, W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, “Principles for Electronic Equalization of Polarization-Mode Dispersion” J. Lightwave Technol. 221169–82 (2004).
[Crossref]

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties” Appl. Comput. Harmon. Anal. 16, 208–230 (2004).
[Crossref]

2003 (2)

A. J. Weiss, “On the Performance of Electrical Equalization in Optical Fiber Transmission Systems”, IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[Crossref]

I. B. Djordjevic and B. Vasic, “Receiver Modeling for Optically Amplified Communication Systems,” International J. Electron. Commun. 57, 381–390 (2003).
[Crossref]

2001 (2)

I. T. Monroy and G. Einarsson, “On Analytical Expressions for the Distribution of the filtered Output of Square Envelope Receivers with Signal and Colored Gaussian Noise Input,” IEEE Transactions on Communications 49, 19–23 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

2000 (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

1998 (1)

1997 (2)

K. Yonenaga and S. Kuwano, “Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,” J. Lightwave Technol. 15, 1530–1537 (1997).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit Error Evaluation of Optically Preamplified Direct Detection Receivers with Fabry-Perot Optical Filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

1994 (1)

J. Lee et al., “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain”, J. Lightwave Technol. 12, 1224–1229 (1994).
[Crossref]

1991 (1)

P.A. Humblet and M. Azizoglu, “On the Bit Error Rate in Lightwave Systems with Optical Amplifiers,” J. Lightwave Technol. 9, 1576–82 (1991).
[Crossref]

1990 (1)

J.H. Winters and R.D. Gitlin, “Electrical signal processing techniques in long-haul fiber-optic systems,” IEEE Trans. Commun. 38, 1439–1453 (1990).
[Crossref]

1984 (1)

R. Loudon and T.J. Shepherd, “Properties of the Optical Quantum Amplifier,” Optica Acta,  31, 1243–1269 (1984).
[Crossref]

1976 (1)

D. Slepian, “On Bandwidth,” Proc. Of IEEE,  64, 292–300 (1976).
[Crossref]

1968 (1)

D. Slepian, “A Numerical Method Of Determining EigenValues And EigenFunction Of Analytic Kernels” SIAM Journal of Numerical Analysis,  5, 586–600 (1968).
[Crossref]

1965 (1)

D. Slepian and E. Sonnenblick, “Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order,” Bell Sys. Tech. J. 45, 1745–59 (1965).

1962 (1)

E. Arthurs and H. Dym, “On the Optimum Detection of Digital Signals in the Presence of White Gaussian Noise — A Geometric Interpretation and a Study of Three Basic Data Transmission Systems”, IRE Trans. on Communication Systems 10, 336–372 (1962).
[Crossref]

1958 (1)

D. Slepian, “Fluctuations of Random Noise Power,” Bell Sys. Tech. J. 37, 163 (1958)

Alic, N.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

N. Alić, G. C. Papen, and Y. Fainman, “Theoretical Performance Analysis of Maximum Likelihood Sequence Estimation in Intensity Modulated Short-Haul Fiber Optic Links,” Proc. IEEE LEOS Annual Meeting, Puerto Rico, paper ThB3, (2004).

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

Arthurs, E.

E. Arthurs and H. Dym, “On the Optimum Detection of Digital Signals in the Presence of White Gaussian Noise — A Geometric Interpretation and a Study of Three Basic Data Transmission Systems”, IRE Trans. on Communication Systems 10, 336–372 (1962).
[Crossref]

Azizoglu, M.

P.A. Humblet and M. Azizoglu, “On the Bit Error Rate in Lightwave Systems with Optical Amplifiers,” J. Lightwave Technol. 9, 1576–82 (1991).
[Crossref]

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

S. Benedetto and E. Biglieri, Principles of Digital Transmission (Kluwer Academic/Plenum Publishers, 1999).

Bergano, N. S.

N. S. Bergano “Undersea Communication Systems” in Fiber optic telecommunications IVB, Ivan Kaminow and T. Li, Eds. (Elsevier Science2002).

Berlitzon, K.

Biglieri, E.

S. Benedetto and E. Biglieri, Principles of Digital Transmission (Kluwer Academic/Plenum Publishers, 1999).

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

Buchali, F.

F. Buchali and H. Bulow, “Correlation sensitive Viterbi equalization of 10 Gb/s signals in bandwidth limited receivers,” in Proc. OFC 2005, Paper F020.

Bulow, H.

F. Buchali and H. Bulow, “Correlation sensitive Viterbi equalization of 10 Gb/s signals in bandwidth limited receivers,” in Proc. OFC 2005, Paper F020.

H. Bulow and G. Thielecke, “Electronic PMD mitigation-from linear equalization to maximum-likelihood detection” in Proc. OFC 2001, 2001, Paper WAA3-1.

Cada, M.

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties” Appl. Comput. Harmon. Anal. 16, 208–230 (2004).
[Crossref]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

Carter, A. C.

R. A. Griffin and A. C. Carter, “Optical differential quadrature phaseshift keying (oDQPSK) for high capacity optical transmission,” in Proc. OFC, 2002, Paper WX6C.

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

Dittrich, A.

Djordjevic, I. B.

I. B. Djordjevic and B. Vasic, “Receiver Modeling for Optically Amplified Communication Systems,” International J. Electron. Commun. 57, 381–390 (2003).
[Crossref]

Doerr, C.R.

Dorrer, C.

Dorschky, C.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

Dym, H.

E. Arthurs and H. Dym, “On the Optimum Detection of Digital Signals in the Presence of White Gaussian Noise — A Geometric Interpretation and a Study of Three Basic Data Transmission Systems”, IRE Trans. on Communication Systems 10, 336–372 (1962).
[Crossref]

Einarsson, G.

I. T. Monroy and G. Einarsson, “On Analytical Expressions for the Distribution of the filtered Output of Square Envelope Receivers with Signal and Colored Gaussian Noise Input,” IEEE Transactions on Communications 49, 19–23 (2001).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit Error Evaluation of Optically Preamplified Direct Detection Receivers with Fabry-Perot Optical Filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

Elbers, J.-P.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

Faerbert, A.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Fainman, Y.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

N. Alić, G. C. Papen, and Y. Fainman, “Theoretical Performance Analysis of Maximum Likelihood Sequence Estimation in Intensity Modulated Short-Haul Fiber Optic Links,” Proc. IEEE LEOS Annual Meeting, Puerto Rico, paper ThB3, (2004).

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

Flammer, C.

C. Flammer, Spheroidal Wave Functions (Stanford Univ. Press, 1957).

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

Gitlin, R.D.

J.H. Winters and R.D. Gitlin, “Electrical signal processing techniques in long-haul fiber-optic systems,” IEEE Trans. Commun. 38, 1439–1453 (1990).
[Crossref]

Glingener, C.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

Gnauck, A.H.

Griesser, H.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Griffin, R. A.

R. A. Griffin and A. C. Carter, “Optical differential quadrature phaseshift keying (oDQPSK) for high capacity optical transmission,” in Proc. OFC, 2002, Paper WX6C.

Haunstein, H. F.

Haunstein, H.F.

H.F. Haunstein et al., Design of near optimum electrical equalizers for optical transmission in the presence of PMD”, in Proc. OFC, 2001, Paper WAA4-1.

Humblet, P.A.

P.A. Humblet and M. Azizoglu, “On the Bit Error Rate in Lightwave Systems with Optical Amplifiers,” J. Lightwave Technol. 9, 1576–82 (1991).
[Crossref]

Jacobs, I. M.

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Waveland Press, reprint1990).

Jacobsen, G.

Kang, I.

Kasturia, S.

J.H. Winters and S. Kasturia, “Constrained maximum likelihood detection for high-speed fiber optic systems,” in Proc. GLOBECOM ‘91, 1574–1579 (1991).

Kupfer, T.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Kuwano, S.

K. Yonenaga and S. Kuwano, “Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,” J. Lightwave Technol. 15, 1530–1537 (1997).
[Crossref]

Langenbach, S.

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

Lee, J.

J. Lee et al., “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain”, J. Lightwave Technol. 12, 1224–1229 (1994).
[Crossref]

Leuthold, J.

Loudon, R.

R. Loudon and T.J. Shepherd, “Properties of the Optical Quantum Amplifier,” Optica Acta,  31, 1243–1269 (1984).
[Crossref]

McDonough, R.N

R.N McDonough and A.D. Whalen, Detection of Signals in Noise, Second Edition (San Diego, Academic Press1995).

Milstein, L. B.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

Monroy, I. T.

I. T. Monroy and G. Einarsson, “On Analytical Expressions for the Distribution of the filtered Output of Square Envelope Receivers with Signal and Colored Gaussian Noise Input,” IEEE Transactions on Communications 49, 19–23 (2001).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit Error Evaluation of Optically Preamplified Direct Detection Receivers with Fabry-Perot Optical Filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

Moore, I. C.

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties” Appl. Comput. Harmon. Anal. 16, 208–230 (2004).
[Crossref]

Moses, R.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 1997).

Papen, G. C.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

N. Alić, G. C. Papen, and Y. Fainman, “Theoretical Performance Analysis of Maximum Likelihood Sequence Estimation in Intensity Modulated Short-Haul Fiber Optic Links,” Proc. IEEE LEOS Annual Meeting, Puerto Rico, paper ThB3, (2004).

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

Parvaresh, F.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

Percival, D.B.

D.B. Percival and A.T. WaldenSpectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (Cambridge, Cambridge University Press, 1993).
[Crossref]

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

Ryf, R.

Saleh, B.

B. Saleh, Photoelectron Statistics, With Applications to Spectroscopy and Optical Communication, (Springer-Verlag, 1978).

Santhi, N.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

Saperstein, R. E.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

Sauer-Greff, W.

Schulien, C.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Shepherd, T.J.

R. Loudon and T.J. Shepherd, “Properties of the Optical Quantum Amplifier,” Optica Acta,  31, 1243–1269 (1984).
[Crossref]

Siegel, P. H.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

Slepian, D.

D. Slepian, “On Bandwidth,” Proc. Of IEEE,  64, 292–300 (1976).
[Crossref]

D. Slepian, “A Numerical Method Of Determining EigenValues And EigenFunction Of Analytic Kernels” SIAM Journal of Numerical Analysis,  5, 586–600 (1968).
[Crossref]

D. Slepian and E. Sonnenblick, “Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order,” Bell Sys. Tech. J. 45, 1745–59 (1965).

D. Slepian, “Fluctuations of Random Noise Power,” Bell Sys. Tech. J. 37, 163 (1958)

Sonnenblick, E.

D. Slepian and E. Sonnenblick, “Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order,” Bell Sys. Tech. J. 45, 1745–59 (1965).

Sticht, K.

Stoica, P.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 1997).

Stojanovic, N.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Thielecke, G.

H. Bulow and G. Thielecke, “Electronic PMD mitigation-from linear equalization to maximum-likelihood detection” in Proc. OFC 2001, 2001, Paper WAA3-1.

Urbansky, R.

Vasic, B.

I. B. Djordjevic and B. Vasic, “Receiver Modeling for Optically Amplified Communication Systems,” International J. Electron. Commun. 57, 381–390 (2003).
[Crossref]

Walden, A.T.

D.B. Percival and A.T. WaldenSpectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (Cambridge, Cambridge University Press, 1993).
[Crossref]

Weiss, A. J.

A. J. Weiss, “On the Performance of Electrical Equalization in Optical Fiber Transmission Systems”, IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[Crossref]

Wernz, H.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

Whalen, A.D.

R.N McDonough and A.D. Whalen, Detection of Signals in Noise, Second Edition (San Diego, Academic Press1995).

Winters, J.H.

J.H. Winters and R.D. Gitlin, “Electrical signal processing techniques in long-haul fiber-optic systems,” IEEE Trans. Commun. 38, 1439–1453 (1990).
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J.H. Winters and S. Kasturia, “Constrained maximum likelihood detection for high-speed fiber optic systems,” in Proc. GLOBECOM ‘91, 1574–1579 (1991).

Winzer, P.J.

Wozencraft, J. M.

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Waveland Press, reprint1990).

Xiapin, Z.

Yonenaga, K.

K. Yonenaga and S. Kuwano, “Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,” J. Lightwave Technol. 15, 1530–1537 (1997).
[Crossref]

Appl. Comput. Harmon. Anal. (1)

I. C. Moore and M. Cada, “Prolate spheroidal wave functions, an introduction to the Slepian series and its properties” Appl. Comput. Harmon. Anal. 16, 208–230 (2004).
[Crossref]

Bell Sys. Tech. J. (2)

D. Slepian and E. Sonnenblick, “Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order,” Bell Sys. Tech. J. 45, 1745–59 (1965).

D. Slepian, “Fluctuations of Random Noise Power,” Bell Sys. Tech. J. 37, 163 (1958)

IEEE Photon. Technol. Lett. (2)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Method for the BER Evaluation in Optical Systems Affected by Parametric Gain,” IEEE Photon. Technol. Lett. 12, 152–4, (2000).
[Crossref]

A. J. Weiss, “On the Performance of Electrical Equalization in Optical Fiber Transmission Systems”, IEEE Photon. Technol. Lett. 15, 1225–1227 (2003).
[Crossref]

IEEE Trans. Commun. (2)

J.H. Winters and R.D. Gitlin, “Electrical signal processing techniques in long-haul fiber-optic systems,” IEEE Trans. Commun. 38, 1439–1453 (1990).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A Novel Analytical Approach to the Evaluation of the Impact of Fiber Parametric Gain on the Bit Error Rate” IEEE Trans. Commun. 49, 2154–63 (2001).
[Crossref]

IEEE Transactions on Communications (1)

I. T. Monroy and G. Einarsson, “On Analytical Expressions for the Distribution of the filtered Output of Square Envelope Receivers with Signal and Colored Gaussian Noise Input,” IEEE Transactions on Communications 49, 19–23 (2001).
[Crossref]

International J. Electron. Commun. (1)

I. B. Djordjevic and B. Vasic, “Receiver Modeling for Optically Amplified Communication Systems,” International J. Electron. Commun. 57, 381–390 (2003).
[Crossref]

IRE Trans. on Communication Systems (1)

E. Arthurs and H. Dym, “On the Optimum Detection of Digital Signals in the Presence of White Gaussian Noise — A Geometric Interpretation and a Study of Three Basic Data Transmission Systems”, IRE Trans. on Communication Systems 10, 336–372 (1962).
[Crossref]

J. Lightwave Technol. (8)

J. Lee et al., “Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain”, J. Lightwave Technol. 12, 1224–1229 (1994).
[Crossref]

P.A. Humblet and M. Azizoglu, “On the Bit Error Rate in Lightwave Systems with Optical Amplifiers,” J. Lightwave Technol. 9, 1576–82 (1991).
[Crossref]

H. F. Haunstein, W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, “Principles for Electronic Equalization of Polarization-Mode Dispersion” J. Lightwave Technol. 221169–82 (2004).
[Crossref]

G. Jacobsen, K. Berlitzon, and Z. Xiapin, “WDM Transmission System Performance: Influence of non-Gaussian Detected ASE Noise and Periodic DEMUX Characteristic,” J. Lightwave Technol. 16, 1804–1812 (1998).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit Error Evaluation of Optically Preamplified Direct Detection Receivers with Fabry-Perot Optical Filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

C. Dorrer, C.R. Doerr, I. Kang, R. Ryf, J. Leuthold, and P.J. Winzer, “Measurement of eye diagrams and constellation diagrams of optical sources using linear optics and waveguide technology,” J. Lightwave Technol. 23, 178–186 (2005).
[Crossref]

K. Yonenaga and S. Kuwano, “Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,” J. Lightwave Technol. 15, 1530–1537 (1997).
[Crossref]

A.H. Gnauck and P.J. Winzer, “Optical phase-shift-keyed transmission,” J. Lightwave Technol. 23, 115–130 (2005).
[Crossref]

Optica Acta (1)

R. Loudon and T.J. Shepherd, “Properties of the Optical Quantum Amplifier,” Optica Acta,  31, 1243–1269 (1984).
[Crossref]

Proc. Of IEEE (1)

D. Slepian, “On Bandwidth,” Proc. Of IEEE,  64, 292–300 (1976).
[Crossref]

SIAM Journal of Numerical Analysis (1)

D. Slepian, “A Numerical Method Of Determining EigenValues And EigenFunction Of Analytic Kernels” SIAM Journal of Numerical Analysis,  5, 586–600 (1968).
[Crossref]

Other (18)

B. Saleh, Photoelectron Statistics, With Applications to Spectroscopy and Optical Communication, (Springer-Verlag, 1978).

C. Flammer, Spheroidal Wave Functions (Stanford Univ. Press, 1957).

R. A. Griffin and A. C. Carter, “Optical differential quadrature phaseshift keying (oDQPSK) for high capacity optical transmission,” in Proc. OFC, 2002, Paper WX6C.

D.B. Percival and A.T. WaldenSpectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (Cambridge, Cambridge University Press, 1993).
[Crossref]

S. Benedetto and E. Biglieri, Principles of Digital Transmission (Kluwer Academic/Plenum Publishers, 1999).

F. Buchali and H. Bulow, “Correlation sensitive Viterbi equalization of 10 Gb/s signals in bandwidth limited receivers,” in Proc. OFC 2005, Paper F020.

A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, and C. Glingener, “Performance of a 10.7 Gb/s Receiver with Digital Equaliser using Maximum Likelihood Sequence Estimation,” Proc. of ECOC’O4, Th.4.1.5, (2004).

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, and C. Schulien, “Measurement of the Dispersion Tolerance of Optical Duobinary with an MLSE-Receiver at 10.7 Gb/s,” Proc. of OFC’O5,OthJ4, (2005).

R.N McDonough and A.D. Whalen, Detection of Signals in Noise, Second Edition (San Diego, Academic Press1995).

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Waveland Press, reprint1990).

J.H. Winters and S. Kasturia, “Constrained maximum likelihood detection for high-speed fiber optic systems,” in Proc. GLOBECOM ‘91, 1574–1579 (1991).

N. S. Bergano “Undersea Communication Systems” in Fiber optic telecommunications IVB, Ivan Kaminow and T. Li, Eds. (Elsevier Science2002).

H.F. Haunstein et al., Design of near optimum electrical equalizers for optical transmission in the presence of PMD”, in Proc. OFC, 2001, Paper WAA4-1.

H. Bulow and G. Thielecke, “Electronic PMD mitigation-from linear equalization to maximum-likelihood detection” in Proc. OFC 2001, 2001, Paper WAA3-1.

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, and Y. Fainman, “Performance Bounds of MLSE in Intensity Modulated Fiber Optic Links,” Fiber optic communication theory and techniques, Enrico Forestieri Ed.) 2004 Tyrrhenian International Workshop on Digital Communications, paper 4.5, (2004).

N. Alić, G. C. Papen, and Y. Fainman, “Theoretical Performance Analysis of Maximum Likelihood Sequence Estimation in Intensity Modulated Short-Haul Fiber Optic Links,” Proc. IEEE LEOS Annual Meeting, Puerto Rico, paper ThB3, (2004).

N. Alić, G. C. Papen, L. B. Milstein, P. H. Siegel, R. E. Saperstein, F. Parvaresh, N. Santhi, and Y. Fainman, “Performance Analysis of Maximum Likelihood Sequence Estimation in Short-Haul Intensity Modulated Fiber Optic Links,” submitted to Journal of Lightwave Technology.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 1997).

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Figures (6)

Fig. 1.
Fig. 1.

Detection scheme block-diagram. The point in the system where Karhunen-Loeve expansion is performed is outlined in red

Fig. 2.
Fig. 2.

Eigen-value distribution (in one quadrature) in log-log scale (when ordered in a descending order) for two filter shapes: Rectangular filter - the eigen-values correspond to Prolate Spheroidal Functions; and Lorentzian filter - the eigen-values correspond to harmonic functions.

Fig. 3.
Fig. 3.

Histograms of samples (blue bars) drawn from a complex Gaussian noise source (in two orthogonal polarizations) undergoing band-limiting, square-law operation and integration. Also shown are fits of the chi square distribution (green line) and the pdf obtained through Karhunen-Loeve expansion (red line) for time-bandwidth products of (a) 5, and (b) 1

Fig. 4.
Fig. 4.

(a) Comparison of the calculated PDF’s (solid lines) with chi-square (dashed lines) for four channel responses at back-to-back for NRZ format at Eb/N0=10 dB for a rectangular filter for an ‘000’, ‘111’, ‘101’ and ‘010’ responses. (b) A zoomed in detail from the graph (a) that emphasizes how close the likelihood intersections are on the x-axis for the two forms of likelihood functions. (c) PDF’s from part (a) in logarithmic scale

Fig. 5.
Fig. 5.

Performance of the SE based on the exact statistic (solid lines) and the chi-square metric (dashed lines) with sequence estimation at four different amounts of accumulated dispersion. (a) Ideal rectangular filter used. Performance of the two approaches is virtually the same; (b) Lorentzian filter.

Fig 6.
Fig 6.

Performance of sequence estimation for the exact and the chi-square distributions for BT=1 at back to back and 150 km. (a) Rectangular filter; (b) Lorentzian shaped filter

Equations (11)

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

n ( t ) = k = 0 n k ϕ k ( t ) ; 0 t T
ρ ( t ) = [ s ( t ) + n ( t ) ] * h ( t ) = k = 1 ( S k + n k ) · ϕ k ( t ) = k = 1 ρ k · ϕ k ( t ) = k = 1 ( ρ k + ρ k ) · ϕ k ( t ) .
p R k ( r S k ) = { 1 2 σ k 2 exp [ r + S k 2 2 σ k 2 ] · I 0 ( S k 2 r σ k 2 ) } * { 1 2 σ k 2 exp [ r + S k 2 2 σ k 2 ] · I 0 ( S k 2 r σ k 2 ) } ,
Φ R k S k ( ω ) = exp [ j ω 2 S k 2 1 + j ω 2 σ k 2 ] 1 + j ω 2 σ k 2 · exp [ j ω 2 S k 2 1 + j ω 2 σ k 2 ] 1 + j ω 2 σ k 2 ,
0 T k = 0 [ ( n k R + S k R ) + j · ( n k I + S k I ) + ( n k R + S k R ) + j · ( n k I + S k I ) ] ϕ k ( t ) 2 dt = k = 0 ρ k 2 ,
p R ( r ) = 1 { k = 0 exp [ j ω 2 S k 2 1 + j ω 2 σ k 2 ] 1 + j ω 2 σ k 2 · exp [ j ω 2 S k 2 1 + j ω 2 σ k 2 ] 1 + j ω 2 σ k 2 } =
Conv k = 0 { [ 1 2 σ k 2 exp ( r + S k 2 σ 2 k ) · I 0 ( S k r σ k 2 ) ] * [ 1 2 σ k 2 exp ( r + S k 2 σ k 2 ) · I 0 ( S k r σ k 2 ) ] } ,
R n ( t t ) = 2 · R ˜ n ( τ ) = 𝔽 1 { 2 · ν 0 2 H ( f ) 2 }
0 T R ˜ n ( t t ) · ϕ k ( t ) dt = ν 0 2 · 0 T { h ( t t ) * h * ( t t ) } ϕ k ( t ) dt = σ k 2 ϕ k ( t )
0 T { h ( t t ) * h * ( t t ) } ϕ k ( t ) dt = λ ˜ k ϕ k ( t )
λ ˜ k = σ k 2 / ν 0 2

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