Abstract

We review the time domain, frequency domain, and Fourier series Karhunen–Loéve series expansion (KLSE) methods for exact BER evaluation in intensity- and phase-modulated direct-detection optically amplified systems. We compare their complexity and computational efficiency, and discuss the most relevant implementation issues. We show that the method based on a Fourier series expansion has the simplest implementation and requires the minimum number of eigenvalues to converge to the exact BER value for various kind of optical filters. For this method, we also introduce an equivalent form of the moment generating function, that avoids the singularity for eigenvalues equal to zero, and derive an alternative expansion where signal and noise are expanded on the same orthonormal basis.

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  1. P. S. Henry, "Error-rate performance of optical amplifiers," Tech. Dig. OFC'89 (1989).
  2. N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightw. Technol. 7, 1071-1082 (1989).
  3. D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightw. Technol. 8, 1816-1823 (1990).
  4. O. K. Tonguz, L. G. Kazovsky, "Theory of direct-detection lightwave receivers using optical amplifiers," J. Lightw. Technol. 9, 174-181 (1991).
  5. D. Marcuse, "Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise," J. Lightw. Technol. 9, 505-513 (1991).
  6. P. A. Humblet, M. Azizogˇlu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightw. Technol. 9, 1576-1582 (1991).
  7. N. S. Bergano, F. W. Kerfoot, C. R. Davidson, "Margin measurements in optical amplifier system," IEEE Photon. Technol. Lett. 5, 304-306 (1993).
  8. J.-S. Lee, C.-S. Shim, "Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain," J. Lightw. Technol. 12, 1224-1229 (1994).
  9. N. G. Jensen, E. Bodtker, G. Jacobsen, J. Strandberg, "Performance of preamplified direct detection systems under influence of receiver noise," IEEE Photon. Technol. Lett. 6, 1488-1490 (1994).
  10. L. F. B. Ribeiro, J. R. F. Da Rocha, J. L. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightw. Technol. 13, 225-232 (1995).
  11. D. Ben-Eli, Y. E. Dallal, S. Shamai, "Performance bounds and cutoff rates of quantum limited OOK with optical amplification," IEEE J. Sel. Areas Commun. 13, 510-530 (1995).
  12. S. L. Danielsen, B. Mikkelesen, T. Durhuus, C. Joergensen, K. E. Stubkjaer, "Detailed noise statistics for an optically preamplified direct detection receiver," J. Lightw. Technol. 13, 977-981 (1995).
  13. C. Lawetz, J. C. Cartledge, "Performance of optically preamplified receivers with Fabry-Perot optical filters," J. Lightw. Technol. 14, 2467-2474 (1996).
  14. I. T. Monroy, G. Einarsson, "Bit error evaluation of optically preamplified direct detection receivers with Fabry-Perot optical filters," J. Lightw. Technol. 15, 1546-1553 (1997).
  15. E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering," J. Lightw. Technol. 18, 1493-1503 (2000).
  16. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, 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, 2154-2163 (2001).
  17. P. J. Winzer, M. Pfennigbauer, M. M. Strasser, W. R. Leeb, "Optimum filter bandwidths for optically preamplified NRZ and RZ receivers," J. Lightw. Technol. 19, 1263-1273 (2001).
  18. M. Kac, A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383-397 (1947).
  19. C. W. Helstrom, "Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration," IEEE Trans. Inf. Theory IT-32, 450-463 (1986).
  20. S. W. Golomb, Shift Register Sequences (Holden-Day, 1967).
  21. J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on dpsk systems using interferometric demodulation and direct detection," J. Lightw. Technol. 22, 362-371 (2004).
  22. A. H. Gnauck, P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightw. Technol. 23, 115-130 (2005).
  23. C. W. Helstrom, "Approximate evaluation of detection probabilities in radar and optical communications," IEEE Trans. Aerosp. Electron. Syst. 14, 630-640 (1978).
  24. H. Van Trees, Detection, Estimation, and Modulation Theory, Part I. (Wiley, 1968).
  25. E. J. Nyström, "Über die praktische auflösung von integralgleichungen mit anwendungen auf randwertaufgaben," Acta Mathematica 54, 185-204 (1930).
  26. F. G. Tricomi, Integral Equations (Dover, 1985).
  27. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  28. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, A. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightw. Technol. 20, 389-400 (2002).
  29. M. Nazarathy, B. Livshitz, Y. Atzmon, M. Secondini, E. Forestieri, "Optically amplified direct detection with pre- and post- filtering: A Volterra series approach," J. Lightw. Technol. 26, 3677-3693 (2008).

2008

M. Nazarathy, B. Livshitz, Y. Atzmon, M. Secondini, E. Forestieri, "Optically amplified direct detection with pre- and post- filtering: A Volterra series approach," J. Lightw. Technol. 26, 3677-3693 (2008).

2005

A. H. Gnauck, P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightw. Technol. 23, 115-130 (2005).

2004

J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on dpsk systems using interferometric demodulation and direct detection," J. Lightw. Technol. 22, 362-371 (2004).

2002

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, A. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightw. Technol. 20, 389-400 (2002).

2001

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, 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, 2154-2163 (2001).

P. J. Winzer, M. Pfennigbauer, M. M. Strasser, W. R. Leeb, "Optimum filter bandwidths for optically preamplified NRZ and RZ receivers," J. Lightw. Technol. 19, 1263-1273 (2001).

2000

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering," J. Lightw. Technol. 18, 1493-1503 (2000).

1997

I. T. Monroy, G. Einarsson, "Bit error evaluation of optically preamplified direct detection receivers with Fabry-Perot optical filters," J. Lightw. Technol. 15, 1546-1553 (1997).

1996

C. Lawetz, J. C. Cartledge, "Performance of optically preamplified receivers with Fabry-Perot optical filters," J. Lightw. Technol. 14, 2467-2474 (1996).

1995

L. F. B. Ribeiro, J. R. F. Da Rocha, J. L. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightw. Technol. 13, 225-232 (1995).

D. Ben-Eli, Y. E. Dallal, S. Shamai, "Performance bounds and cutoff rates of quantum limited OOK with optical amplification," IEEE J. Sel. Areas Commun. 13, 510-530 (1995).

S. L. Danielsen, B. Mikkelesen, T. Durhuus, C. Joergensen, K. E. Stubkjaer, "Detailed noise statistics for an optically preamplified direct detection receiver," J. Lightw. Technol. 13, 977-981 (1995).

1994

J.-S. Lee, C.-S. Shim, "Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain," J. Lightw. Technol. 12, 1224-1229 (1994).

N. G. Jensen, E. Bodtker, G. Jacobsen, J. Strandberg, "Performance of preamplified direct detection systems under influence of receiver noise," IEEE Photon. Technol. Lett. 6, 1488-1490 (1994).

1993

N. S. Bergano, F. W. Kerfoot, C. R. Davidson, "Margin measurements in optical amplifier system," IEEE Photon. Technol. Lett. 5, 304-306 (1993).

1991

O. K. Tonguz, L. G. Kazovsky, "Theory of direct-detection lightwave receivers using optical amplifiers," J. Lightw. Technol. 9, 174-181 (1991).

D. Marcuse, "Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise," J. Lightw. Technol. 9, 505-513 (1991).

P. A. Humblet, M. Azizogˇlu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightw. Technol. 9, 1576-1582 (1991).

1990

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightw. Technol. 8, 1816-1823 (1990).

1989

N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightw. Technol. 7, 1071-1082 (1989).

1986

C. W. Helstrom, "Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration," IEEE Trans. Inf. Theory IT-32, 450-463 (1986).

1978

C. W. Helstrom, "Approximate evaluation of detection probabilities in radar and optical communications," IEEE Trans. Aerosp. Electron. Syst. 14, 630-640 (1978).

1947

M. Kac, A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383-397 (1947).

1930

E. J. Nyström, "Über die praktische auflösung von integralgleichungen mit anwendungen auf randwertaufgaben," Acta Mathematica 54, 185-204 (1930).

Acta Mathematica

E. J. Nyström, "Über die praktische auflösung von integralgleichungen mit anwendungen auf randwertaufgaben," Acta Mathematica 54, 185-204 (1930).

IEEE Trans. Inf. Theory

C. W. Helstrom, "Distribution of the filtered output of a quadratic rectifier computed by numerical contour integration," IEEE Trans. Inf. Theory IT-32, 450-463 (1986).

IEEE J. Sel. Areas Commun.

D. Ben-Eli, Y. E. Dallal, S. Shamai, "Performance bounds and cutoff rates of quantum limited OOK with optical amplification," IEEE J. Sel. Areas Commun. 13, 510-530 (1995).

IEEE Photon. Technol. Lett.

N. G. Jensen, E. Bodtker, G. Jacobsen, J. Strandberg, "Performance of preamplified direct detection systems under influence of receiver noise," IEEE Photon. Technol. Lett. 6, 1488-1490 (1994).

IEEE Photon. Technol. Lett.

N. S. Bergano, F. W. Kerfoot, C. R. Davidson, "Margin measurements in optical amplifier system," IEEE Photon. Technol. Lett. 5, 304-306 (1993).

IEEE Trans. Aerosp. Electron. Syst.

C. W. Helstrom, "Approximate evaluation of detection probabilities in radar and optical communications," IEEE Trans. Aerosp. Electron. Syst. 14, 630-640 (1978).

IEEE Trans. Commun.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, 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, 2154-2163 (2001).

J. Lightw. Technol.

I. T. Monroy, G. Einarsson, "Bit error evaluation of optically preamplified direct detection receivers with Fabry-Perot optical filters," J. Lightw. Technol. 15, 1546-1553 (1997).

D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightw. Technol. 8, 1816-1823 (1990).

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, A. L. Kath, "Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization," J. Lightw. Technol. 20, 389-400 (2002).

J. Appl. Phys.

M. Kac, A. J. F. Siegert, "On the theory of noise in radio receivers with square law detectors," J. Appl. Phys. 18, 383-397 (1947).

J. Lightw. Technol.

J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersions on dpsk systems using interferometric demodulation and direct detection," J. Lightw. Technol. 22, 362-371 (2004).

L. F. B. Ribeiro, J. R. F. Da Rocha, J. L. Pinto, "Performance evaluation of EDFA preamplified receivers taking into account intersymbol interference," J. Lightw. Technol. 13, 225-232 (1995).

J. Lightw. Technol.

S. L. Danielsen, B. Mikkelesen, T. Durhuus, C. Joergensen, K. E. Stubkjaer, "Detailed noise statistics for an optically preamplified direct detection receiver," J. Lightw. Technol. 13, 977-981 (1995).

C. Lawetz, J. C. Cartledge, "Performance of optically preamplified receivers with Fabry-Perot optical filters," J. Lightw. Technol. 14, 2467-2474 (1996).

E. Forestieri, "Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering," J. Lightw. Technol. 18, 1493-1503 (2000).

P. J. Winzer, M. Pfennigbauer, M. M. Strasser, W. R. Leeb, "Optimum filter bandwidths for optically preamplified NRZ and RZ receivers," J. Lightw. Technol. 19, 1263-1273 (2001).

O. K. Tonguz, L. G. Kazovsky, "Theory of direct-detection lightwave receivers using optical amplifiers," J. Lightw. Technol. 9, 174-181 (1991).

D. Marcuse, "Calculation of bit-error probability for a lightwave system with optical amplifiers and post-detection Gaussian noise," J. Lightw. Technol. 9, 505-513 (1991).

P. A. Humblet, M. Azizogˇlu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightw. Technol. 9, 1576-1582 (1991).

J.-S. Lee, C.-S. Shim, "Bit-error-rate analysis of optically preamplified receivers using an eigenfunction expansion method in optical frequency domain," J. Lightw. Technol. 12, 1224-1229 (1994).

N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightw. Technol. 7, 1071-1082 (1989).

A. H. Gnauck, P. J. Winzer, "Optical phase-shift-keyed transmission," J. Lightw. Technol. 23, 115-130 (2005).

M. Nazarathy, B. Livshitz, Y. Atzmon, M. Secondini, E. Forestieri, "Optically amplified direct detection with pre- and post- filtering: A Volterra series approach," J. Lightw. Technol. 26, 3677-3693 (2008).

Other

S. W. Golomb, Shift Register Sequences (Holden-Day, 1967).

H. Van Trees, Detection, Estimation, and Modulation Theory, Part I. (Wiley, 1968).

F. G. Tricomi, Integral Equations (Dover, 1985).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

P. S. Henry, "Error-rate performance of optical amplifiers," Tech. Dig. OFC'89 (1989).

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