Abstract

We simulate the time evolution of differential group delay (DGD) in a fiber link in the hinge model with time series forecasting procedures, in particular, autoregressive (AR) models with Kalman filter, pattern imitation, and a Taylor expansion. In these studies, the furthest prediction horizon was attained with the AR accompanied by the Kalman filter procedure. We further discuss the influence of measurement noise on the prediction horizon. We also apply these methods to the measured DGD on 40Gb/s fiber links and discuss the results.

© 2011 Optical Society of America

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  1. R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
    [Crossref]
  2. M. Brodsky, N. J. Frigo, M. Broditsky, and M. Tur, “Polarization mode dispersion of installed fibers,” J. Lightwave Technol. 24, 4584–4559 (2006).
    [Crossref]
  3. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
    [Crossref]
  4. C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” IEEE Photonics Technol. Lett. 16, 1840–1842 (2004).
    [Crossref]
  5. H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
    [Crossref]
  6. C. Antoneli and A. Mecozzi, “Theoretical characterization and system impact of the hinge model of PMD,” J. Lightwave Technol. 24, 4064–4074 (2006).
    [Crossref]
  7. M. Brodsky, M. Broditsky, P. Magill, N. Frigo, and M. Tur, “Persistence of spectral variations in DGD statistics,” Opt. Express 13, 4090–4095 (2005).
    [Crossref] [PubMed]
  8. C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
    [Crossref]
  9. J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
    [Crossref] [PubMed]
  10. M. H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, 1996), Chaps. 6 and 7.
  11. J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 (1987).
    [Crossref] [PubMed]
  12. A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
    [Crossref]

2009 (1)

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

2006 (2)

2005 (2)

M. Brodsky, M. Broditsky, P. Magill, N. Frigo, and M. Tur, “Persistence of spectral variations in DGD statistics,” Opt. Express 13, 4090–4095 (2005).
[Crossref] [PubMed]

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

2004 (1)

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” IEEE Photonics Technol. Lett. 16, 1840–1842 (2004).
[Crossref]

2000 (2)

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
[Crossref]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

1997 (1)

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

1987 (1)

J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 (1987).
[Crossref] [PubMed]

1964 (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Andrekson, P. A.

Antoneli, C.

Antonelli, C.

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” IEEE Photonics Technol. Lett. 16, 1840–1842 (2004).
[Crossref]

Brentel, J.

Broditsky, M.

Brodsky, M.

Cigliutti, R.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Dal Forno, A. O.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Farmer, J. D.

J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 (1987).
[Crossref] [PubMed]

Frigo, N.

Frigo, N. J.

Galtarossa, A.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Giltrelli, M.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Gisin, N.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Golay, M. J. E.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Gordon, J. P.

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Grosso, D.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Hayes, M. H.

M. H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, 1996), Chaps. 6 and 7.

Jopson, R. M.

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

Karlsson, M.

Kogelnik, H.

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Leitch, A. W. R.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Magill, P.

Mecozzi, A.

C. Antoneli and A. Mecozzi, “Theoretical characterization and system impact of the hinge model of PMD,” J. Lightwave Technol. 24, 4064–4074 (2006).
[Crossref]

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” IEEE Photonics Technol. Lett. 16, 1840–1842 (2004).
[Crossref]

Nelson, L. E.

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

Palmieri, L.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Passy, R.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Pereira da Silva, J. A.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Prola, C. H.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Santoni, S.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Savitzky, A.

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

Schenato, L.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Sidorowich, J. J.

J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 (1987).
[Crossref] [PubMed]

Tur, M.

von der Weid, J. P.

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

Waswa, D.

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Winzer, P. J.

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

Anal. Chem. (1)

A. Savitzky and M. J. E. Golay, “Smoothing and differentiation of data by simplified least squares procedures,” Anal. Chem. 36, 1627–1639 (1964).
[Crossref]

IEEE Photonics Technol. Lett. (3)

C. Antonelli and A. Mecozzi, “Statistics of the DGD in PMD emulators,” IEEE Photonics Technol. Lett. 16, 1840–1842 (2004).
[Crossref]

H. Kogelnik, P. J. Winzer, L. E. Nelson, R. M. Jopson, M. Broditsky, and M. Brodsky, “First-order PMD outage for the hinge model,” IEEE Photonics Technol. Lett. 17, 1208–1210(2005).
[Crossref]

C. H. Prola, Jr., J. A. Pereira da Silva, A. O. Dal Forno, R. Passy, J. P. von der Weid, and N. Gisin, “PMD emulators and signal distortion in 2.48 Gb/s IM-DD lightwave systems,” IEEE Photonics Technol. Lett. 9, 842–844 (1997).
[Crossref]

J. Lightwave Technol. (3)

Opt. Express (1)

Opt. Fiber Technol. (1)

R. Cigliutti, A. Galtarossa, M. Giltrelli, D. Grosso, A. W. R. Leitch, L. Palmieri, S. Santoni, L. Schenato, and D. Waswa, “Design, estimation and experimental validation of optical Polarization Mode Dispersion Compensator in 40 Gbit/s NRZ and RZ optical systems,” Opt. Fiber Technol. 15, 242–250 (2009).
[Crossref]

Phys. Rev. Lett. (1)

J. D. Farmer and J. J. Sidorowich, “Predicting chaotic time series,” Phys. Rev. Lett. 59, 845–848 (1987).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

J. P. Gordon and H. Kogelnik, “PMD fundamentals: polarization mode dispersion in optical fibers,” Proc. Natl. Acad. Sci. USA 97, 4541–4550 (2000).
[Crossref] [PubMed]

Other (1)

M. H. Hayes, Statistical Digital Signal Processing and Modeling (Wiley, 1996), Chaps. 6 and 7.

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Figures (9)

Fig. 1
Fig. 1

Simulated optical link. Five fiber spans S1 to S5 are connected via a series of time-varying hinges H1 to H4 modeled as polarization rotators. Each fiber span consists of 50 birefringent sections, each having randomly oriented rotation axes.

Fig. 2
Fig. 2

Correlation function of simulated DGD time series. The correlation index is defined as the distance between the maximum of the correlation function and the first minimum.

Fig. 3
Fig. 3

DGD time series with mean removed. The first 7200 data points are used to estimate the model, and the following 7000 points of the series are employed for validation.

Fig. 4
Fig. 4

Metric versus prediction horizon for the different methods employed. The legend indicates model numbers as described in Section 4.

Fig. 5
Fig. 5

Simulated and predicted DGD time series for model (3) for P = 130 and Φ = 86 % .

Fig. 6
Fig. 6

Metric plotted as a function of the prediction horizon after filtering.

Fig. 7
Fig. 7

Simulated DGD and predicted DGD time series for noisy data for AR(10) model applied to the DGD.

Fig. 8
Fig. 8

Predicted and measured DGD.

Fig. 9
Fig. 9

Predicted and measured DGD for fiber 2.

Tables (1)

Tables Icon

Table 1 ψ Compared for the Three Methods Applied to Simulated and Measured Data

Equations (8)

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

Φ = 100 × ( 1 y v ( n ) y ^ ( n ) y v ( n ) y ¯ ) .
r ( n ) = i = 1 k a i y ( n i ) .
y ( n ) = i = 1 k a i y ( n i ) + w ( n ) ,
x ( n ) = A x ( n 1 ) + w ( n ) ,
A = [ a 1 a 2 a k 1 a k 1 0 0 0 0 1 . . . . . . . 0 0 1 0 ] ,
y ( n ) = C x ( n ) ,
x ^ ( n ) = A x ^ ( n 1 ) + K ( n ) [ y v ( n ) C A x ^ ( n 1 ) ] ,
y ^ ( N ) = y v ( N P ) + Δ t y v ( N P ) + ( Δ t ) 2 2 y v ( N P ) + ( Δ t ) 3 3 ! y v ( N P ) + .

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