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 Science 2002).
  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, G. Thielecke, �??Electronic PMD mitigation-from linear equalization to maximum-likelihood detection�?? in Proc. OFC 2001, 2001, Paper WAA3-1.
  7. N. Ali�?, 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. Ali�? 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. Ali�?, 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.
  10. A. J. Weiss, �??On the Performance of Electrical Equalization in Optical Fiber Transmission Systems�??, IEEE Photon. Technol. Lett. 15, 1225-1227 (2003).
    [CrossRef]
  11. H. F. Haunstein, W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, �??Principles for Electronic Equalization of Polarization-Mode Dispersion�?? J. Lightwave Technol. 22 1169-82 (2004).
    [CrossRef]
  12. . A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, 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, 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).
  14. 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]
  15. R.N McDonough and A.D. Whalen, Detection of Signals in Noise, Second Edition (San Diego, Academic Press 1995).
  16. J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Waveland Press, reprint 1990).
  17. 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]
  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, 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]
  20. G. Jacobsen, K. Berlitzon, 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]
  21. I. T. Monroy, G. Einarsson, �??Bit Error Evaluation of Optically Preamplified Direct Detection Receivers with Fabry-Perot Optical Filters,�?? J. Lightwave Technol. 15, 1546-1553 (1997).
    [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, 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, 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).
    [CrossRef]
  28. B. Saleh, Photoelectron Statistics, With Applications to Spectroscopy and Optical Communication, (Springer-Verlag, 1978).
  29. C. Flammer, Spheroidal Wave Functions (Stanford Univ. Press, 1957).
  30. D. Slepian and E. Sonnenblick, �??Eigenvalues Associated with Prolate Spheroidal Wave Functions of Zero Order,�?? Bell Sys. Tech. J. 45, 1745-59 (1965).
  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. Walden Spectral Analysis for Physical Applications: Multitaper and Conventional Univariate Techniques (Cambridge, Cambridge University Press, 1993).
    [CrossRef]
  36. K. Yonenaga, 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, P.J.Winzer, �??Optical phase-shift-keyed transmission,�?? J. Lightwave Technol. 23 , 115 �?? 130 (2005).
    [CrossRef]
  39. P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 1997).

2004 Tyrrhenian Intl. Wkshp on Digital C (1)

N. Ali�?, 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).

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)

ECOC 2004 (1)

. A. Faerbert, S. Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, C. Schulien, J.-P. Elbers, H. Wernz, H. Griesser, 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).

GLOBECOM 1991 (1)

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

IEEE LEOS Annual Meeting 2004 (1)

N. Ali�? 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).

IEEE Photon. Technol. Lett (1)

A. J. Weiss, �??On the Performance of Electrical Equalization in Optical Fiber Transmission Systems�??, IEEE Photon. Technol. Lett. 15, 1225-1227 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (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]

IEEE Trans. Commun. (2)

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]

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]

IEEE Transactions on Communications (1)

I. T. Monroy, 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, 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]

G. Jacobsen, K. Berlitzon, 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]

H. F. Haunstein, W. Sauer-Greff, A. Dittrich, K. Sticht, and R. Urbansky, �??Principles for Electronic Equalization of Polarization-Mode Dispersion�?? J. Lightwave Technol. 22 1169-82 (2004).
[CrossRef]

A.H. Gnauck, P.J.Winzer, �??Optical phase-shift-keyed transmission,�?? J. Lightwave Technol. 23 , 115 �?? 130 (2005).
[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, S. Kuwano, �??Dispersion-Tolerant Optical Transmission System Using Duobinary Transmitter and Binary Receiver,�?? J. Lightwave Technol. 15, 1530-1537 (1997).
[CrossRef]

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

Journal of Lightwave Technology (1)

N. Ali�?, 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.

OFC 2001 (2)

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, G. Thielecke, �??Electronic PMD mitigation-from linear equalization to maximum-likelihood detection�?? in Proc. OFC 2001, 2001, Paper WAA3-1.

OFC 2002 (1)

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

OFC 2005 (2)

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

J.-P. Elbers, H. Wernz, H. Griesser, C. Glingener, A. Faerbert, S . Langenbach, N. Stojanovic, C. Dorschky, T. Kupfer, 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).

Optica Acta (1)

R. Loudon, T.J. Shepherd, �??Properties of the Optical Quantum Amplifier,�?? Optica Acta, 31, 1243-1269 (1984).
[CrossRef]

Proc. 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 (8)

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

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

R.N McDonough and A.D. Whalen, Detection of Signals in Noise, Second Edition (San Diego, Academic Press 1995).

J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering (Waveland Press, reprint 1990).

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

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

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

N. S. Bergano �??Undersea Communication Systems�?? in Fiber optic telecommunications IVB, Ivan Kaminow and T. Li, Eds. (Elsevier Science 2002).

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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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