Abstract

Backpropagation has been shown to be the most effective method for compensating intra-channel fiber nonlinearity in long-haul optical communications systems. However, effective compensation is computationally expensive, as it requires numerous steps and possibly increased sampling rates compared with the baud rate. This makes backpropagation difficult to implement in real-time. We propose: (i) low-pass filtering the compensation signal (the intensity waveform used to calculate the nonlinearity compensation) in each backpropagation step and (ii) optimizing the position of the nonlinear section in each step. With numerical simulations, we show that these modifications to backpropagation improve system performance, reducing the number of backpropagation steps and reducing the oversampling for a given system performance. Using our ‘filtered backpropagation’, with four backpropagation steps operating at the same sampling rate as that required for linear equalizers, the Q at the optimal launch power was improved by 2 dB and 1.6 dB for single wavelength CO-OFDM and CO-QPSK systems, respectively, in a 3200 km (40 × 80km) single-mode fiber link, with no optical dispersion compensation. With previously proposed backpropagation methods, 40 steps were required to achieve an equivalent performance. A doubling in the sampling rate of the OFDM system was also required. We estimate this is a reduction in computational complexity by a factor of around ten.

© 2010 OSA

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References

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  1. S. L. Jansen, I. Morita, T. C. W. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. K. Kikuchi, M. Fukase, and S.-Y. Kim, “Electronic post-compensation for nonlinear phase noise in a 1000-km 20-Gbit/s optical QPSK transmission system using the homodyne receiver with digital signal processing,” in Optical Fiber Communication Conference (Optical Society of America, Anaheim, California, 2007), p. OTuA2.
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-20-12965 .
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. A. J. Lowery, “Fiber nonlinearity mitigation in optical links that use OFDM for dispersion compensation,” IEEE Photon. Technol. Lett. 19(19), 1556–1558 (2007).
    [CrossRef]
  11. L. B. Du, and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” in Optical Fiber Communication Conference (Optical Society of America, San Diego, California, 2009), p. OTuO1.
  12. L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-24-19920 .
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  18. K. Roberts, M. O'Sullivan, K. T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. 27(16), 3546–3559 (2009).
    [CrossRef]
  19. S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
    [CrossRef]
  20. W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-9936 .
    [CrossRef] [PubMed]
  21. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
    [CrossRef]
  22. L. B. Du and A. J. Lowery, “Practical XPM compensation method for coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 22, 320–322 (2010).
    [CrossRef]
  23. L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “XPM tolerant adaptive carrier phase recovery for coherent receiver based on phase noise statistics monitoring,” in Proc. European Conference on Optical Communications (2009), p. P3.16.
  24. T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
    [CrossRef]
  25. 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. Express 16(20), 15777–15810 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-20-15777 .
    [CrossRef] [PubMed]

2010 (2)

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

L. B. Du and A. J. Lowery, “Practical XPM compensation method for coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 22, 320–322 (2010).
[CrossRef]

2009 (4)

2008 (7)

C. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E.-D. Schmidt, T. Wuth, J. Geyer, E. De Man, G.-D. Khoe, and H. de Waardt, “Coherent equalization and POLMUX-RZ-DQPSK for robust 100-GE transmission,” J. Lightwave Technol. 26(1), 64–72 (2008).
[CrossRef]

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[CrossRef]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-841 .
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-24-19920 .
[CrossRef] [PubMed]

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of fiber impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

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. Express 16(20), 15777–15810 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-20-15777 .
[CrossRef] [PubMed]

2007 (3)

2006 (1)

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

1996 (2)

C. Paré, A. Villeneuve, P. A. Bélanger, and N. J. Doran, “Compensating for dispersion and the nonlinear Kerr effect without phase conjugation,” Opt. Lett. 21(7), 459–461 (1996), http://ol.osa.org/abstract.cfm?URI=ol-21-7-459 .
[CrossRef] [PubMed]

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Awadalla, A.

Bao, H.

Bélanger, P. A.

Chen, X.

Chiang, T. K.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Cho, P.

De Man, E.

de Waardt, H.

Doran, N. J.

Du, L. B.

Duthel, T.

Fludger, C. S.

Gavioli, G.

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

Geyer, J.

Goldfarb, G.

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of fiber impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

Hardcastle, I.

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

Ip, E.

Jansen, S. L.

Kagi, N.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Kahn, J. M.

Karagodsky, V.

Kazovsky, L. G.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Khoe, G.-D.

Khurgin, J.

Kim, I.

Krause, D. J.

Laperle, C.

Li, C.

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

Li, G.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of fiber impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

Li, X.

Lowery, A. J.

Ma, Y.

Marhic, M. E.

T. K. Chiang, N. Kagi, M. E. Marhic, and L. G. Kazovsky, “Cross-phase modulation in fiber links with multiple optical amplifiers and dispersion compensators,” J. Lightwave Technol. 14(3), 249–260 (1996).
[CrossRef]

Mateo, E.

Meiman, Y.

Morita, I.

Nazarathy, M.

Noe, R.

O'Sullivan, M.

Paré, C.

Poggiolini, P.

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

Roberts, K.

K. Roberts, M. O'Sullivan, K. T. Wu, H. Sun, A. Awadalla, D. J. Krause, and C. Laperle, “Performance of dual-polarization QPSK for optical transport systems,” J. Lightwave Technol. 27(16), 3546–3559 (2009).
[CrossRef]

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

Savory, S. J.

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

Schenk, T. C. W.

Schmidt, E.-D.

Schulien, C.

Shieh, W.

Shpantzer, I.

Strawczynski, L.

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

Sullivan, M. O.

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

Sun, H.

Tanaka, H.

Tang, Y.

Taylor, M. G.

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of fiber impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

Torrengo, E.

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

van den Borne, D.

Villeneuve, A.

Weidenfeld, R.

Wu, K. T.

Wuth, T.

Yaman, F.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

Yang, Q.

Yi, X.

IEEE Photon. J. (1)

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009).
[CrossRef]

IEEE Photon. Technol. Lett. (5)

L. B. Du and A. J. Lowery, “Practical XPM compensation method for coherent optical OFDM systems,” IEEE Photon. Technol. Lett. 22, 320–322 (2010).
[CrossRef]

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010).
[CrossRef]

K. Roberts, C. Li, L. Strawczynski, M. O. Sullivan, and I. Hardcastle, “Electronic precompensation of optical nonlinearity,” IEEE Photon. Technol. Lett. 18(2), 403–405 (2006).
[CrossRef]

A. J. Lowery, “Fiber nonlinearity mitigation in optical links that use OFDM for dispersion compensation,” IEEE Photon. Technol. Lett. 19(19), 1556–1558 (2007).
[CrossRef]

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of fiber impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008).
[CrossRef]

J. Lightwave Technol. (6)

Opt. Express (6)

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. Express 16(20), 15777–15810 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-20-15777 .
[CrossRef] [PubMed]

W. Shieh, X. Yi, Y. Ma, and Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express 15(16), 9936–9947 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-16-9936 .
[CrossRef] [PubMed]

X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
[CrossRef] [PubMed]

A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007), http://www.opticsexpress.org/abstract.cfm?URI=oe-15-20-12965 .
[CrossRef] [PubMed]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-2-841 .
[CrossRef] [PubMed]

L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008), http://www.opticsexpress.org/abstract.cfm?URI=oe-16-24-19920 .
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (6)

S. Oda, T. Tanimura, T. Hoshida, C. Ohshima, H. Nakashima, Z. Tao, and J. C. Rasmussen, “112 Gb/s DP-QPSK transmission using a novel nonlinear compensator in digital coherent receiver,” in Optical Fiber Communication Conference (Optical Society of America, San Diego, California, 2009), p. OThR6.

G. Charlet, M. Salsi, P. Tran, M. Bertolini, H. Mardoyan, J. Renaudier, O. Bertran-Pardo, and S. Bigo, “72x100Gb/s transmission over transoceanic distance, using large effective area fiber, hybrid raman-erbium amplification and coherent detection,” in Optical Fiber Communication Conference (Optical Society of America, 2009), p. PDPB6.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, Inc., San Diego, 1989).

L. B. Du, and A. J. Lowery, “Fiber nonlinearity compensation for CO-OFDM systems with periodic dispersion maps,” in Optical Fiber Communication Conference (Optical Society of America, San Diego, California, 2009), p. OTuO1.

K. Kikuchi, M. Fukase, and S.-Y. Kim, “Electronic post-compensation for nonlinear phase noise in a 1000-km 20-Gbit/s optical QPSK transmission system using the homodyne receiver with digital signal processing,” in Optical Fiber Communication Conference (Optical Society of America, Anaheim, California, 2007), p. OTuA2.

L. Li, Z. Tao, L. Liu, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, “XPM tolerant adaptive carrier phase recovery for coherent receiver based on phase noise statistics monitoring,” in Proc. European Conference on Optical Communications (2009), p. P3.16.

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Figures (13)

Fig. 1
Fig. 1

Power against propagation distance of a two-span optical link (left) and the corresponding backpropagation link (right) when using the inverse NLSE.

Fig. 2
Fig. 2

Block diagram of transmission system with BP nonlinearity compensation. Note that the linear section D2 is only used when the symmetrical SSFM is employed.

Fig. 3
Fig. 3

Block diagram of one nonlinear section in the BP algorithm. * – conjugation operator; PM – phase modulator.

Fig. 4
Fig. 4

(a) Signal power along an 8 × 80-km link. (b-d) Position of the modeled nonlinear mixing for different models (note that the modeled signal is backpropagated through the fiber, so travels right to left): (b) asymmetric SSFM with 8-steps; (c) asymmetric SSFM with 2-steps; (d) 2-step SSFM with the locations of the nonlinear sections adjustable.

Fig. 5
Fig. 5

Optical spectra of received CO-OFDM signals after a 40 × 80-km optical link: (a) before BP; (b) using 40-step BP; (c) using 20-step BP.

Fig. 7
Fig. 7

Block diagram of one filtered nonlinear section in the BP algorithm. * – conjugation operator; PM – phase modulator. Note the addition of a Low Pass Filter (LPF).

Fig. 6
Fig. 6

Block diagram of a receiver with the proposed BP with initial linear section (in blue).

Fig. 8
Fig. 8

Conceptual diagram showing the spectra at different locations of a nonlinear step in BP: (a) initial signal; (b) square of the complex envelop of signal (intensity waveform); (c) intensity waveform after LPF; (d) signal after a nonlinear step in unfiltered BP; (e) signal after a nonlinear step in filtered BP.

Fig. 12
Fig. 12

Q against number of BP steps used for filtered and unfiltered BP.

Fig. 10
Fig. 10

Optimal k against the number of BP steps used in compensation.

Fig. 9
Fig. 9

(a) Optimal filter bandwidth versus the number of BP steps used; (b) LPF characteristic; (c) optimal value of Dini /Dstep versus the number of BP steps used.

Fig. 11
Fig. 11

Q against oversampling factor for CO-OFDM and CO-QPSK using linear equalization, 40-step filtered BP and 40-step unfiltered BP.

Fig. 13
Fig. 13

Q against launch power for a 3200 km transmission system using linear equalization or nonlinear (filtered BP) equalization.

Equations (7)

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E ( z ) = ( D + N ) E
D = j 2 β 2 2 t 2 α 2
N = j γ | E | 2
θ L i n ( f ) = 2 π L s t e p β 2 ( Δ f ) 2
θ N L ( t ) = k γ L e f f | E ( t ) | 2
L e f f = s . 1 exp ( α L s p a n ) α
L e f f = 1 exp ( α L s t e p ) α

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