Abstract

We present yet another quasi-analytic model for an optically amplified receiver, comprising an optical preamplifier, an optical prefilter, a photodetector, and an electrical postfilter. Due to the square-law nonlinearity and its interaction with the linear pre- and postfilters, the understanding of this seemingly simple structure has been elusive, gradually evolving over a succession of studies, culminating in the models by Lee and Shim and Forestieri. Here we adopt a new approach to this old problem, applying Volterra nonlinear theory to obtain fresh insight deriving a simplified model streamlining the pseudoanalytic simulations. Volterra series is a powerful mathematical tool for simulating nonlinear systems, applied here to quadratic optical detection in an unconventional way, by deriving a mixed frequency-time representation, leading to a simple and compact quadratic form in the combined signal-plus-noise signal spectra, albeit not computationally efficient. Next, Forestieri's results, which used distinct bases for the signal and noise, are rederived using the insightful Volterra formalism. Finally, we reformulate the model using an expansion of both signal and noise in a common harmonic basis over a sliding window of duration equal to the ISI system memory. This final version of the optically amplified receiver model is most computationally efficient, provides the most compact description, and lends itself to intuitive interpretation. Applications of the new method include accurate determination of the bit error ratio of amplitude-shift keying, frequency-shift keying, and differential phase-shift keying transmission systems in the linear optical link propagation regime, including the effects of dispersion and fully accounting for intersymbol interference. The theory developed here is naturally extensible to advanced optical equalization techniques, involving Volterra nonlinear optoelectronic equalizers.

© 2008 IEEE

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  1. M. Shetzen, The Volterra and Wiener Theories of Nonlinear Systems (Krieger, 1980).
  2. W. J. Ruch, Nonlinear System Theory: The Volterra/Wiener Approach (Johns Hopkins Univ. Press, 1981).
  3. T. K. Biswas, W. F. McGee, "Volterra series analysis of semiconductor laser diode," IEEE Photon. Technol. Lett. 3, 706-708 (1991).
  4. L. Hassine, Z. Toffano, F. Lamnabhi-Lagarrigue, A. Destrez, I. Joindot, "Volterra functional series expansions for noise in semiconductor lasers," IEEE J. Quantum Electron. 30, 2534-2546 (1994).
  5. J. C. Froidure, C. Lebrun, P. Megret, E. Jaunart, P. Goerg, T. Tasia, M. Lamquin, M. Blondel, "Theoretical and experimental study of second-order distortions in CATV DFB laser diodes," IEEE Photon. Technol. Lett. 7, 266-268 (1995).
  6. L. Agarossi, S. Bellini, F. Bregoli, P. Migliorati, "Equalization of non-linear optical channels," Proc. ICC '98 (1998) pp. 662-667.
  7. X. Chunmin, R. Werner, "Nonlinear electrical equalization for different modulation formats with optical filtering," J. Lightw. Technol. 25, 996-1001 (2007).
  8. K. V. Peddanarappagari, M. Brandt-Pearce, "Volterra series transfer function of single-mode fibers," J. Lightw. Technol. 15, 2232-2241 (1997).
  9. P. S. Henry, "Error-rate performance of optical amplifiers," Tech. Dig. OFC'89 (1989).
  10. N. A. Olsson, "Lightwave systems with optical amplifiers," J. Lightw. Technol. 7, 1071-1082 (1989).
  11. D. Marcuse, "Derivation of analytical expressions for the bit-error probability in lightwave systems with optical amplifiers," J. Lightw. Technol. 8, 1816-1823 (1990).
  12. O. K. Tonguz, L. G. Kazovsky, "Theory of direct-detection lightwave receivers using optical amplifiers," J. Lightw. Technol. 9, 174-181 (1991).
  13. 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).
  14. P. A. Humblet, M. Azizoglu, "On the bit error rate of lightwave systems with optical amplifiers," J. Lightw. Technol. 9, 1576-1582 (1991).
  15. 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).
  16. 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).
  17. 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).
  18. 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).
  19. C. Lawetz, J. C. Cartledge, "Performance of optically preamplified receivers with Fabry-Perot optical filters," J. Lightw. Technol. 14, 2467-2474 (1996).
  20. 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).
  21. 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).
  22. P. J. Winzer, K. Hoon, "Degradations in balanced DPSK receivers," IEEE Photon. Technol. Lett. 15, 1282-1284 (2003).
  23. W. Jin, J. M. Kahn, "Conventional DPSK versus symmetrical DPSK: Comparison of dispersion tolerances," IEEE Photon. Technol. Lett. 16, 1585-1587 (2004).
  24. M. Nazarathy, W. V. Sorin, D. M. Baney, S. A. Newton, "Spectral analysis of optical mixing measurements," J. Lightw. Technol. 7, 1083-1096 (1989).
  25. 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).
  26. F. Yuan, A. Opal, "Distortion analysis of periodically switched nonlinear circuits using time-varying Volterra series," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 726-738 (2001).
  27. T. Ching-Hsiang, "A mixed-domain method for identification of quadratically nonlinear systems," IEEE Trans. Signal Process. 45, 1013-1024 (1997).
  28. W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, 1958).
  29. M. Nazarathy, "Accurate evaluation of bit error rates of optical communication systems using the Gram-Charlier series," IEEE Trans. Commun. 54, 295-301 (2006).
  30. S. Blinnikov, G. Moessner, "Expansions for nearly Gaussian distributions," Astron. Astrophys. Suppl. Ser. 193-205 (1998).
  31. R. Holzlohner, W. L. Kath, C. R. Menyuk, V. S. Grigoryan, "Efficient and accurate computation of eye diagrams and bit-error rates in a single-channel CRZ system," IEEE Photon. Technol. Lett. 14, 1079-1081 (2002).
  32. J. Wang, J. M. Kahn, "Impact of chromatic and polarization-mode dispersion on DPSK systems using interferometric demodulation and direct detection," J. Lightw. Technol 22, 362-371 (2004).
  33. M. K. Ng, Iterative Methods for Toeplitz Systems (Oxford Science, 2004).
  34. E. Forestieri, M. Secondini, "On the error probability evaluation in lightwave systems with optical amplification," J. Lightw. Technol. .

2007

X. Chunmin, R. Werner, "Nonlinear electrical equalization for different modulation formats with optical filtering," J. Lightw. Technol. 25, 996-1001 (2007).

2006

M. Nazarathy, "Accurate evaluation of bit error rates of optical communication systems using the Gram-Charlier series," IEEE Trans. Commun. 54, 295-301 (2006).

2004

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

W. Jin, J. M. Kahn, "Conventional DPSK versus symmetrical DPSK: Comparison of dispersion tolerances," IEEE Photon. Technol. Lett. 16, 1585-1587 (2004).

2003

P. J. Winzer, K. Hoon, "Degradations in balanced DPSK receivers," IEEE Photon. Technol. Lett. 15, 1282-1284 (2003).

2002

R. Holzlohner, W. L. Kath, C. R. Menyuk, V. S. Grigoryan, "Efficient and accurate computation of eye diagrams and bit-error rates in a single-channel CRZ system," IEEE Photon. Technol. Lett. 14, 1079-1081 (2002).

2001

F. Yuan, A. Opal, "Distortion analysis of periodically switched nonlinear circuits using time-varying Volterra series," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 726-738 (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).

1998

S. Blinnikov, G. Moessner, "Expansions for nearly Gaussian distributions," Astron. Astrophys. Suppl. Ser. 193-205 (1998).

1997

T. Ching-Hsiang, "A mixed-domain method for identification of quadratically nonlinear systems," IEEE Trans. Signal Process. 45, 1013-1024 (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).

K. V. Peddanarappagari, M. Brandt-Pearce, "Volterra series transfer function of single-mode fibers," J. Lightw. Technol. 15, 2232-2241 (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).

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).

J. C. Froidure, C. Lebrun, P. Megret, E. Jaunart, P. Goerg, T. Tasia, M. Lamquin, M. Blondel, "Theoretical and experimental study of second-order distortions in CATV DFB laser diodes," IEEE Photon. Technol. Lett. 7, 266-268 (1995).

1994

L. Hassine, Z. Toffano, F. Lamnabhi-Lagarrigue, A. Destrez, I. Joindot, "Volterra functional series expansions for noise in semiconductor lasers," IEEE J. Quantum Electron. 30, 2534-2546 (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).

1991

T. K. Biswas, W. F. McGee, "Volterra series analysis of semiconductor laser diode," IEEE Photon. Technol. Lett. 3, 706-708 (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. Azizoglu, "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).

M. Nazarathy, W. V. Sorin, D. M. Baney, S. A. Newton, "Spectral analysis of optical mixing measurements," J. Lightw. Technol. 7, 1083-1096 (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).

Astron. Astrophys. Suppl. Ser.

S. Blinnikov, G. Moessner, "Expansions for nearly Gaussian distributions," Astron. Astrophys. Suppl. Ser. 193-205 (1998).

IEEE J. Quantum Electron.

L. Hassine, Z. Toffano, F. Lamnabhi-Lagarrigue, A. Destrez, I. Joindot, "Volterra functional series expansions for noise in semiconductor lasers," IEEE J. Quantum Electron. 30, 2534-2546 (1994).

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.

W. Jin, J. M. Kahn, "Conventional DPSK versus symmetrical DPSK: Comparison of dispersion tolerances," IEEE Photon. Technol. Lett. 16, 1585-1587 (2004).

IEEE Photon. Technol. Lett.

T. K. Biswas, W. F. McGee, "Volterra series analysis of semiconductor laser diode," IEEE Photon. Technol. Lett. 3, 706-708 (1991).

P. J. Winzer, K. Hoon, "Degradations in balanced DPSK receivers," IEEE Photon. Technol. Lett. 15, 1282-1284 (2003).

R. Holzlohner, W. L. Kath, C. R. Menyuk, V. S. Grigoryan, "Efficient and accurate computation of eye diagrams and bit-error rates in a single-channel CRZ system," IEEE Photon. Technol. Lett. 14, 1079-1081 (2002).

J. C. Froidure, C. Lebrun, P. Megret, E. Jaunart, P. Goerg, T. Tasia, M. Lamquin, M. Blondel, "Theoretical and experimental study of second-order distortions in CATV DFB laser diodes," IEEE Photon. Technol. Lett. 7, 266-268 (1995).

IEEE Trans. Circuits Syst. I, Fundam. Theory Appl.

F. Yuan, A. Opal, "Distortion analysis of periodically switched nonlinear circuits using time-varying Volterra series," IEEE Trans. Circuits Syst. I, Fundam. Theory Appl. 48, 726-738 (2001).

IEEE Trans. Signal Process.

T. Ching-Hsiang, "A mixed-domain method for identification of quadratically nonlinear systems," IEEE Trans. Signal Process. 45, 1013-1024 (1997).

IEEE Trans. Commun.

M. Nazarathy, "Accurate evaluation of bit error rates of optical communication systems using the Gram-Charlier series," IEEE Trans. Commun. 54, 295-301 (2006).

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).

J. Lightw. Technol.

X. Chunmin, R. Werner, "Nonlinear electrical equalization for different modulation formats with optical filtering," J. Lightw. Technol. 25, 996-1001 (2007).

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).

J. Lightw. Technol.

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

E. Forestieri, M. Secondini, "On the error probability evaluation in lightwave systems with optical amplification," J. Lightw. Technol. .

J. Lightw. Technol

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

J. Lightw. Technol.

M. Nazarathy, W. V. Sorin, D. M. Baney, S. A. Newton, "Spectral analysis of optical mixing measurements," J. Lightw. Technol. 7, 1083-1096 (1989).

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).

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).

K. V. Peddanarappagari, M. Brandt-Pearce, "Volterra series transfer function of single-mode fibers," J. Lightw. Technol. 15, 2232-2241 (1997).

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).

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

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

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. Azizoglu, "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).

Other

L. Agarossi, S. Bellini, F. Bregoli, P. Migliorati, "Equalization of non-linear optical channels," Proc. ICC '98 (1998) pp. 662-667.

M. Shetzen, The Volterra and Wiener Theories of Nonlinear Systems (Krieger, 1980).

W. J. Ruch, Nonlinear System Theory: The Volterra/Wiener Approach (Johns Hopkins Univ. Press, 1981).

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

W. B. Davenport, W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill, 1958).

M. K. Ng, Iterative Methods for Toeplitz Systems (Oxford Science, 2004).

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