Abstract

We present an efficient method for system identification (nonlinear channel estimation) of third order nonlinear Volterra Series Transfer Function (VSTF) characterizing the four-wave-mixing nonlinear process over a coherent OFDM fiber link. Despite the seemingly large number of degrees of freedom in the VSTF (cubic in the number of frequency points) we identified a compressed VSTF representation which does not entail loss of information. Additional slightly lossy compression may be obtained by discarding very low power VSTF coefficients associated with regions of destructive interference in the FWM phased array effect. Based on this two-staged VSTF compressed representation, we develop a robust and efficient algorithm of nonlinear system identification (optical performance monitoring) estimating the VSTF by transmission of an extended training sequence over the OFDM link, performing just a matrix-vector multiplication at the receiver by a pseudo-inverse matrix which is pre-evaluated offline. For 512 (1024) frequency samples per channel, the VSTF measurement takes less than 1 (10) msec to complete with computational complexity of one real-valued multiply-add operation per time sample. Relative to a naïve exhaustive three-tone-test, our algorithm is far more tolerant of ASE additive noise and its acquisition time is orders of magnitude faster.

© 2012 OSA

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References

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  1. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20(7), 7777–7791 (2012).
    [CrossRef] [PubMed]
  2. G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
    [CrossRef] [PubMed]
  3. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30(10), 1524–1539 (2012).
    [CrossRef]
  4. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
    [CrossRef] [PubMed]
  5. J. K. Fischer, C.-A. Bunge, and K. Petermann, “Equivalent single-span model for dispersion- managed fiber-optic transmission systems,” J. Lightwave Technol.27(16), 3425–3432 (2009).
    [CrossRef]
  6. F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20(2), 1022–1032 (2012).
    [CrossRef] [PubMed]
  7. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
    [CrossRef]
  8. B. Xu and M. Brandt-pearce, “Modified Volterra series transfer function method,” Photon. Technol. Lett.14(1), 47–49 (2002).
    [CrossRef]
  9. B. Xu and M. Brandt-Pearce, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
    [CrossRef]
  10. J. D. Reis, D. M. Neves, and A. L. Teixeira, “Weighting nonlinearities on future high aggregate data rate PONs,” Opt. Express19(27), 26557–26567 (2011).
    [CrossRef] [PubMed]
  11. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express18(8), 8660–8670 (2010).
    [CrossRef] [PubMed]
  12. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008).
    [CrossRef]
  13. G. Li, E. Mateo, and L. Zhu, “Compensation of nonlinear effects using digital coherent receivers,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011), p. OWW1.
  14. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express19(10), 9453–9460 (2011).
    [CrossRef] [PubMed]
  15. A. Lobato, B. Inan, S. Adhikari, and S. L. Jansen, “On the efficiency of RF-Pilot-based nonlinearity compensation for CO-OFDM,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011), p. OThF2.
  16. L. B. Y. Du and A. J. Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,” Opt. Express19(26), B862–B867 (2011).
    [CrossRef] [PubMed]
  17. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Intrachannel nonlinearity compensation by inverse Volterra series transfer function,” J. Lightwave Technol.30(3), 310–316 (2012).
    [CrossRef]
  18. L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic nonlinearity compensation of 256Gb / s PDM- 16QAM based on inverse Volterra transfer function,” in ECOC’11 (2011).
  19. L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express16(24), 19920–19925 (2008).
    [CrossRef] [PubMed]
  20. F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
    [CrossRef]
  21. F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
    [CrossRef] [PubMed]
  22. R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2010).
  23. H.-M. Chin, F. Marco, and M. Jonas, “Volterra based nonlinear compensation on 224 Gb/s PolMux-16QAM optical fibre link,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2012).
  24. Z. Pan, C. Benoit, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011).
  25. G. L. Mathews and V. J. Sicuranza, Polynomial Signal Processing (Wiley-Interscience, 2000).
  26. S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).
  27. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
    [CrossRef] [PubMed]
  28. H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol.4(10), 1552–1555 (1986).
    [CrossRef]
  29. G. Ishikawa and H. Ooi, “Demonstration of automatic dispersion equalization in 40 Gbit/s OTDM transmission,” in European Conference of Optical Communication (ECOC) (1998), 519–520.
  30. H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
    [CrossRef]
  31. S. Haykin, Adaptive Filter Theory (Prentice Hall, 2002).
  32. S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

2012

2011

2010

2009

2008

2003

2002

B. Xu and M. Brandt-pearce, “Modified Volterra series transfer function method,” Photon. Technol. Lett.14(1), 47–49 (2002).
[CrossRef]

1997

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

1994

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

1990

S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

1986

H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol.4(10), 1552–1555 (1986).
[CrossRef]

Antona, J.-C.

Bigo, S.

Bononi, A.

Bosco, G.

Brandt-Pearce, M.

B. Xu and M. Brandt-Pearce, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
[CrossRef]

B. Xu and M. Brandt-pearce, “Modified Volterra series transfer function method,” Photon. Technol. Lett.14(1), 47–49 (2002).
[CrossRef]

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

Bunge, C.-A.

Cai, Y.

Carena, A.

Chen, X.

Chikama, T.

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

Cho, P.

Chugtai, M. N.

Cui, K.

Curri, V.

Du, L. B.

Du, L. B. Y.

Ellis, A. D.

Fischer, J. K.

Forghieri, F.

Forzati, M.

Grellier, E.

Guiomar, F. P.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

Hatton, H. W.

H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol.4(10), 1552–1555 (1986).
[CrossRef]

Hauske, F. N.

Huang, Y.

Ip, E.

Kahn, J. M.

Karagodsky, V.

Khurgin, J.

Kim, S. B.

S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

Li, L.

Liu, L.

Lorcy, L.

Lowery, A. J.

Mårtensson, J.

Meiman, Y.

Miyata, H.

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

Mussolin, M.

Nam, S. W.

S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

Nazarathy, M.

Neves, D. M.

Nishimura, M.

H. W. Hatton and M. Nishimura, “Temperature dependence of chromatic dispersion in single mode fibers,” J. Lightwave Technol.4(10), 1552–1555 (1986).
[CrossRef]

Noe, R.

Onaka, H.

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

Otsuka, K.

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

Peddanarappagari, K.

K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol.15(12), 2232–2241 (1997).
[CrossRef]

Petermann, K.

Pinto, A. N.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

Poggiolini, P.

Powers, E. J.

S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

Rafique, D.

Reis, J. D.

Rival, O.

Rossi, N.

Serena, P.

Shieh, W.

Shpantzer, I.

Simonneau, C.

Teixeira, A. L.

Vacondio, F.

Weidenfeld, R.

Xie, C.

Xiong, Q.

Xu, B.

B. Xu and M. Brandt-Pearce, “Comparison of FWM- and XPM-induced crosstalk using the Volterra series transfer function method,” J. Lightwave Technol.21(1), 40–53 (2003).
[CrossRef]

B. Xu and M. Brandt-pearce, “Modified Volterra series transfer function method,” Photon. Technol. Lett.14(1), 47–49 (2002).
[CrossRef]

in Acoustics, Speech, and Signal Processing

S. W. Nam, S. B. Kim, and E. J. Powers, “On the identification of a third-order Volterra nonlinear system using a frequency-domain block RLS adaptive algorithm,” in Acoustics, Speech, and Signal ProcessingICASSP-90, 2407–2410 (1990).

J. Lightwave Technol.

Opt. Express

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express19(10), 9453–9460 (2011).
[CrossRef] [PubMed]

J. D. Reis, D. M. Neves, and A. L. Teixeira, “Weighting nonlinearities on future high aggregate data rate PONs,” Opt. Express19(27), 26557–26567 (2011).
[CrossRef] [PubMed]

J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express18(8), 8660–8670 (2010).
[CrossRef] [PubMed]

F. Vacondio, O. Rival, C. Simonneau, E. Grellier, A. Bononi, L. Lorcy, J.-C. Antona, and S. Bigo, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20(2), 1022–1032 (2012).
[CrossRef] [PubMed]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express18(18), 19039–19054 (2010).
[CrossRef] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20(7), 7777–7791 (2012).
[CrossRef] [PubMed]

G. Bosco, P. Poggiolini, A. Carena, V. Curri, and F. Forghieri, “Analytical results on channel capacity in uncompensated optical links with coherent detection,” Opt. Express19(26), B440–B449 (2011).
[CrossRef] [PubMed]

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” Opt. Express20(2), 1360–1369 (2012).
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express16(24), 19920–19925 (2008).
[CrossRef] [PubMed]

L. B. Y. Du and A. J. Lowery, “Pilot-based XPM nonlinearity compensator for CO-OFDM systems,” Opt. Express19(26), B862–B867 (2011).
[CrossRef] [PubMed]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express16(20), 15777–15810 (2008).
[CrossRef] [PubMed]

Photon. Technol. Lett.

F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” Photon. Technol. Lett.23(19), 1412–1414 (2011).
[CrossRef]

H. Onaka, K. Otsuka, H. Miyata, and T. Chikama, “Measuring the longitudinal distribution of four-wave mixing efficiency in dispersion-shifted fibers,” Photon. Technol. Lett.6(12), 1454–1456 (1994).
[CrossRef]

B. Xu and M. Brandt-pearce, “Modified Volterra series transfer function method,” Photon. Technol. Lett.14(1), 47–49 (2002).
[CrossRef]

Other

G. Li, E. Mateo, and L. Zhu, “Compensation of nonlinear effects using digital coherent receivers,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011), p. OWW1.

A. Lobato, B. Inan, S. Adhikari, and S. L. Jansen, “On the efficiency of RF-Pilot-based nonlinearity compensation for CO-OFDM,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011), p. OThF2.

L. Liu, L. Li, Y. Huang, K. Cui, Q. Xiong, F. N. Hauske, C. Xie, and Y. Cai, “Electronic nonlinearity compensation of 256Gb / s PDM- 16QAM based on inverse Volterra transfer function,” in ECOC’11 (2011).

S. Haykin, Adaptive Filter Theory (Prentice Hall, 2002).

G. Ishikawa and H. Ooi, “Demonstration of automatic dispersion equalization in 40 Gbit/s OTDM transmission,” in European Conference of Optical Communication (ECOC) (1998), 519–520.

R. Weidenfeld, M. Nazarathy, R. Noe, and I. Shpantzer, “Volterra nonlinear compensation of 100G coherent OFDM with baud-rate ADC, tolerable complexity and low intra-channel FWM/XPM error propagation,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2010).

H.-M. Chin, F. Marco, and M. Jonas, “Volterra based nonlinear compensation on 224 Gb/s PolMux-16QAM optical fibre link,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2012).

Z. Pan, C. Benoit, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in OFC/NFOEC - Conference on Optical Fiber Communication and the National Fiber Optic Engineers Conference (2011).

G. L. Mathews and V. J. Sicuranza, Polynomial Signal Processing (Wiley-Interscience, 2000).

S. Kumar, Impact of Nonlinearities on Fiber Optic Communications, Ch. 3 (Springer, 2011).

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Figures (10)

Fig. 1
Fig. 1

Validation of the analytic Volterra model by using the VSTF in ideal nonlinear compensator. for a noiselessly amplified link with the following parameters: B W = 25 G H z , N = 128 , α = 0.2 d B / K m , γ = 1.3 ( W K m ) 1 , D = 17 p s / ( n m K m ) , L = 10 × 100 K m , P = 0 d B m .

Fig. 2
Fig. 2

(a): Compressed VSTF power for multi- and single-span links and the array factor. (b) m-index multiplicity. The following parameters were assumed: BW=25GHz,N=64,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=5×100Km,P=0dBm .

Fig. 3
Fig. 3

Noise reduction figure-of-merit for N = 64 OFDM subcarriers and 16-PSK constellation.

Fig. 4
Fig. 4

Lossy compression quality vs. size |M’| of the set of compressed coefficients for (a) band-limited lossy compression (b) sorted lossy compression for a link with parameters BW=25GHz,N=128,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=10×100Km . The intercept point for an LCQ threshold of −0.4 dB is also shown.

Fig. 5
Fig. 5

The tradeoff between compression complexity and quality exemplified by a plot of the Lossy compression factor vs. N, parameterized by the Lossy Compression Quality threshold (th) which was taken as −0.4 dB in Fig. 4. A link with the following parameters was assumed: BW=25GHz,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=60×100Km , and the sorted prioritization H' coefficients scheme described in section 6.2 was used.

Fig. 6
Fig. 6

SID procedure computation time vs. number of sub-carriers, N, such that the complexity rate is precisely one MAC per sample. Here we assumed Rs = 25 GS/s (25 GBaud OFDM), and the same fiber parameters as in Fig. 5.

Fig. 7
Fig. 7

SID performance expressed in terms of MER at NLC output for a link with parameters BW=25GHz,N=32,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=3×100Km,Pout=0dBm . (a) band-limited truncation (b) sorted prioritization.

Fig. 8
Fig. 8

(a) Real and (b) imaginary part of the compressed VSTF, H’m, vs m-index comparing the SID estimated VSTF with the analytically predicted VSTF for | M ˜ |=280,| T ˜ |=120 and a link with the following parameters: BW=25GHz,N=32,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=3×100Km,Pout=0dBm

Fig. 9
Fig. 9

(a): NLC performance over a noiselessly amplified link as described in Fig. 7, using the SID-estimated vs. analytic VSTF. The close match indicates high performance SID operation. The uncompensated performance is also shown for comparison. (b): SID performance for a long-haul 2500 Km link with the following parameters: BW=25GHz,N=256,α=0.2dB/Km,γ=1.3 (WKm) 1 ,D=17ps/(nmKm),L=25×100Km,Pout=0dBm . The top curve marked ‘analytic’ describes the NLC performance with an analytically evaluated VSTF.

Fig. 10
Fig. 10

SID performance for a noisy 300 Km link with the following parameters: BW=25GHz,N=32,α=0.2 dB Km ,γ=1.3 1 WKm ,D=17 ps nmKm ,L=3×100Km,Pout=0dBm. .

Tables (1)

Tables Icon

Table 1 Number of Coefficients – H vs. H’

Equations (66)

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

E j ( t ) = A j e j 2 π ν j t , E k ( t ) = A k e j 2 π ν k t , E l ( t ) = A l e j 2 π ν l t
r i ; j k l ( 3 ) ( t ) = ( j γ d z ) E j ( t ) E k ( t ) E l * ( t )
r i ; j k l ( 3 ) ( t ) = ( j γ d z ) A j A k A l * e j 2 π ( ν j + ν k ν l ) t = R i ; j k l ( 3 ) e j 2 π ν i t
R i ; j k l ( 3 ) ( j γ d z ) A j A k A l *
R i ; j k ( 3 ) = H i ; j k A j A k A j + k i * .
a ( t ) = k = 1 N A k e j 2 π k Δ ν t ; Δ ν T 1
r i ( 3 ) ( t ) = j = 1 N k = 1 N R i ; j k ( 3 ) e j 2 π i Δ ν t ; t [ 0 , T ]
S [ i ] = { ( j , k ) : 1 j N , 1 k N , 1 j + k i N , j i , k i }
r i FWM ( t ) = [ j , k ] S [ i ] R i ; j k (3) e j 2 π i Δ ν t = [ j , k ] S [ i ] e j 2 π i Δ ν t H i ; j k A j A k A j + k i *
r FWM ( t ) = i = 1 N e j 2 π i Δ ν t [ j , k ] S [ i ] H i ; j k A j A k A j + k i * = i = 1 N R i FWM e j 2 π i Δ ν t R i FWM = [ j , k ] S [ i ] H i ; j k A j A k A j + k i *
H i ; j k = j 0 L γ ( z ) G p ( z ) exp ( j ( 2 π Δ ν ) 2 ( j i ) ( k i ) 0 z β 2 ( z ' ) d z ' ) d z
H i ; j k = j 0 L γ ( z ) G p ( z ) exp ( j ( 2 π Δ ν ) 2 ( j i ) ( k i ) β 2 z ) d z = j F { γ ( z ) G p ( z ) 1 [ 0 , L ] ( z ) } | κ = Δ β i ; j k
Δ β i ; j k ( 2 π Δ ν ) 2 β 2 ( j i ) ( k i ) = β ( ν j ) + β ( ν k ) β ( ν j + k i ) β ( ν i )
H i ; j k ID spans = N s p a n H i ; j k one span F i ; j k
H i ; j k one span = j γ ( 1 e α L s p a n e j Δ β i ; j k L s p a n ) / ( j Δ β i ; j k + α )
F i ; j k = e j Δ β i j k ( L L s p a n ) / 2 dinc N s p a n [ L Δ β i j k / 2 π ]
dinc N [ u ] sin ( π u ) / [ N sin ( π u / N ) ]
M E R [ k ] = 1 N i = 1 N | A i [ k ] | 2 / | ρ i [ k ] A i [ k ] | 2
m i;jk =( ji )( ki )
H m =j 0 L γ( z ) G p (z) exp( j ( 2πΔν ) 2 m 0 z β 2 ( z' ) dz' )dz
H i;jk = H m | m=( ji )( ki ) = H ( ji )( ki ) .
H m * = H m .
Δ β i;jk ( 2πΔν ) 2 β 2 ( ji )( ki )= m i;jk ( 2πΔν ) 2 β 2 m i;jk Δ β step
H m =j 0 L γ( z ) G p (z) exp( jm ( 2πΔν ) 2 β 2 z )dz= jF{ γ( z ) G p (z) 1 [0,L] (z) } | κ=mΔ β step
H m ID spans = N span H m onespan F m
H m onespan =jγ( 1 e α L span e jmΔ β step L span )/( jmΔ β step +α )
F m = e jΔmΔ β step (L L span )/2 dinc N span [ mLΔ β step /2π ]
M ˜ ={ m|m=( ji )( ki ):1iN,1jN,1kN,1j+kiN,ji,ki } S m [i]={ ( j,k ) |( ji )( ki )=m,1jN,1kN,1j+kiN,ji,ki }
S m i=1 N S m [i]
r i FWM (t)= ( j,k )S[i] H (ji)(ki) A j A k A j+ki * e j2πiΔνt = m M ˜ H m ( j,k ) S m [i] A j A k A j+ki * e j2πiΔνt
| S[i] |=( N 2 5N+2 )/2+(N+1)i i 2
i=1 N | S[i] | =N( N 2 5N+2 )/2+(N+1) 1 2 ( N+1 )N 1 6 N( N+1 )( 2N+1 ) = 2 3 N( N1 )( N2 )< 2 3 N 3
m max = max s.t.1j,k,i,j+kiN (ji)(ki)= N 2 ( N 2 1 ) N 2 4 .
i=1 N | S[i] | /| M ˜ | 2 3 N 3 /( 1 4 N 2 )= 8 3 N.
[ r i FWM (t) ] 2 N 3 2 P T 3 ( j,k )S[i] | H (ji)(ki) | 2 = N 3 2 P T 3 m M ˜ | S m [i] | | H m | 2
i | [ r i FWM (t) ] | 2 2 N 3 P T 3 m M ˜ | S m | | H m | 2
r i FWM (t)= R i FWM e j2πiΔνt ; R i FWM m M ˜ H m ( j,k ) S m [i] A j A k A j+ki * = m M ˜ H m A i,m FWM
A i,m FWM ( j,k ) S m [i] A j A k A j+ki *
R i [t]= A i [t]+ R i FWM [t]+ n i [t]= A i [t]+ m M ˜ A i,m FWM [t] H m + n i [t]
m M ˜ A i,m FWM [t] H m + n i [t]=δ R i [t] R i [t] A i [t],i=1,2,...,N,t T ˜
{ H ^ m } m M ˜ = argmin { H m } m M ˜ t T ˜ i=1 N | δ R i [t] m M ˜ A i,m FWM [t] H m | 2
MSE= i=1 N | δ R i [ t 0 ] m M ˜ A i,m FWM [ t 0 ] H m [ t 0 ] | 2 = δR[ t 0 ]A[ t 0 ] H 2
δR[ t 0 ] [ δ R 1 [ t 0 ],δ R 2 [ t 0 ],...,δ R N [ t 0 ] ] T ; H [ H m 1 , H m 2 ,..., H m | M ˜ | ] T [ A N×| M ˜ | [ t 0 ] ] i,m = A i,m FWM [ t 0 ] ( j,k ) S m [i] A j [ t 0 ] A k [ t 0 ] A j+ki * [ t 0 ]
δ R N| T ˜ |×1 [ δR [ t 0 ] T ,δR [ t 0 +1 ] T ,...,δR [ t 0 +| T ˜ |1 ] T ] T A N| T ˜ |×| M ˜ | = [ A [ t 0 ] T ,A [ t 0 +1 ] T ,...,A [ t 0 +| T ˜ |1 ] T ] T
H ^ = argmin H δRA H 2
A ( A A ) 1 A
H ^ = A δR
H ' Re [m]=H ' Re [m];H ' Im [m]=H ' Im [m]
H =[ H u H l ]=[ H u re +i H u im H l re +i H l im ]=[ H u re +i H u im H u re +i H u im ]
δR=δ R re +iδ R im
A=[ A L re +i A L im A R re +i A R im ]
[ H ^ u re H ^ u im ]= argmin [ H u re , H u im ] T [ δ R re δ R im ][ A L re A R re A L im A R im A L im A R im A L re + A R re ][ H u re H u im ] 2
A δR= A ( δR +n )= A δR + A n= H + n PI-SID ; n PI-SID A n
A i,m FWM A o 3 ( j,k ) S m [i] A j A k A j+ki * = A o 3 A i,m FWM where A i,m FWM ( j,k ) S m [i] A j A k A j+ki *
n PI-SID 2 = m M ˜ | n ˜ m PI-SID | 2 =Tr{ ( n PI-SID ) ( n PI-SID ) }=Tr{ A n n ( A ) } = σ n ˜ 2 Tr{ A ( A ) }= σ n ˜ 2 Tr{ ( A A ) 1 }= A o 6 σ n ˜ 2 Tr{ ( A A ) 1 }
σ ¯ n ˜ PI-SID 2 1 | M ˜ | m M ˜ | n ˜ m PI-SID | 2 = A o 6 σ n ˜ 2 1 | M ˜ | Tr{ ( V D D V ) 1 }= A o 6 σ n ˜ 2 1 | M ˜ | Tr{ V ( D D ) 1 V } = A o 6 σ n ˜ 2 1 | M ˜ | Tr{ ( D D ) 1 }= A o 6 σ n ˜ 2 1 | M ˜ | i=1 | M ˜ | λ i 1
n ˜ i 3TT-SID H ^ i;jk H ^ i;jk = n ˜ i / A o 3 ; σ n ˜ i 3TT-SID 2 | n ˜ i 3TT-SID | 2 = | n ˜ i | 2 / A o 6 = A o 6 σ n ˜ 2
σ ¯ n ˜ PI-SID 2 / σ n ˜ i 3TT-SID 2 = 1 | M ˜ | i=1 | M ˜ | λ i 1
LCQ( M ˜ )= m M ˜ | S m | | H m | 2 / m M ˜ | S m | | H m | 2
M ˜ m cutoff BL { m M ˜ :| m | m cutoff < m max }
i | [ r i FWM (t) ] | 2 2 N 3 P T 3 μ M ˜ sort | S[ μ ] | | H μ | 2
r ^ i FWM (t) μ= μ cutoff μ cutoff H μ j,k S μ [i] A j A k A j+ki * e j2πiΔνt
M ˜ μ cutoff BL-sorted { μ M ˜ sorted :| μ | μ cutoff < m max }
MAC=| M ˜ |×2N| T ˜ | f lossy | M ˜ |×10 f lossy | M ˜ |=10 f lossy 2 | M ˜ | 2
c MA MAC/ S SID 10 f lossy 2 | M ˜ | 2 /( T SID R s )
c MA ( T SID )=1 T SID (N)=10 f lossy 2 (N) | M ˜ (N) | 2 /( R s ).

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