## Digital filters for coherent optical receivers

Optics Express, Vol. 16, Issue 2, pp. 804-817 (2008)

http://dx.doi.org/10.1364/OE.16.000804

Acrobat PDF (233 KB)

### Abstract

Digital filters underpin the performance of coherent optical receivers which exploit digital signal processing (DSP) to mitigate transmission impairments. We outline the principles of such receivers and review our experimental investigations into compensation of polarization mode dispersion. We then consider the details of the digital filtering employed and present an analytical solution to the design of a chromatic dispersion compensating filter. Using the analytical solution an upper bound on the number of taps required to compensate chromatic dispersion is obtained, with simulation revealing an improved bound of 2.2 taps per 1000ps/nm for 10.7GBaud data. Finally the principles of digital polarization tracking are outlined and through simulation, it is demonstrated that 100krad/s polarization rotations could be tracked using DSP with a clock frequency of less than 500MHz.

© 2008 Optical Society of America

## 1. Introduction

1. P. S. Henry, “Lightwave Primer” IEEE J. Quantum Electron. **21**, 1862–1879 (1985) [CrossRef]

4. M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments” IEEE Photon. Technol. Lett. **16**, 674–676 (2004). [CrossRef]

## 2. Principles of digital coherent receivers

### 2.1 Receiver architecture

8. S.J. Savory, G. Gavioli, R.I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express **15**, 2120–2126 (2007). [CrossRef] [PubMed]

### 2.2 Outline of the digital signal processing

8. S.J. Savory, G. Gavioli, R.I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express **15**, 2120–2126 (2007). [CrossRef] [PubMed]

16. A. Leven, N. Kaneda, U-V Koc, and Y.-K. Chen “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett.19, 366–368 (2007) [CrossRef]

### 2.3 Functionality of the digital filtering

### 2.4 Knowledge of the signal alphabet

## 3. Application to the compensation of polarization mode dispersion

17. Q. Yu, L.-S. Yan, S. Lee, and A.E. Willner, “Loop-Synchronous Polarization Scrambling for Simulating Polarization Effects Using Recirculating Fiber Loops,” J. Lightwave Technol. **21**, 1593–1600 (2003). [CrossRef]

8. S.J. Savory, G. Gavioli, R.I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express **15**, 2120–2126 (2007). [CrossRef] [PubMed]

## 3.2 Experimental transmission results

*Q*=√

*2*×erfcinv

*(2×BER)*determined from the mean BER obtained for each value of DGD. These results are plotted in Fig. 5 (left), along with a quadratic fit, given by 20log

_{10}Q=10.9-8.63×10

^{-6}×<DGD>

^{2}where the parameters have been obtained by minimizing the L

_{1}norm (mean absolute error) since this is insensitive to extreme outliers[19]. These results indicate an expected penalty of 0.1dBQ for 100ps mean DGD, increasing to 0.3dBQ for 186ps mean DGD.

^{-4}and σ

^{2}=2.0×10

^{-9}for zero DGD and μ=2.7×10

^{-4}and σ

^{2}=2.8×10

^{-9}for 100ps mean. For each set of data an approximate probability density function (pdf) was obtained by dividing the BER data into bins. As can be seen in Fig. 5 (right) there is good agreement between the approximate pdf of the BER and the fitted Gaussian distribution for the two cases. There is only a slight difference between the pdfs for the case with no PMD and that with 100 ps mean DGD, indicating for all of the cases considered the PMD has been compensated with minimal impact on performance. In order to quantify the penalty we note that change in the mean BER from 2.4×10

^{-4}to 2.7×10

^{-4}corresponds to a change of 20log

_{10}Q of less than 0.1 dB, in agreement with that expected from the quadratic L

_{1}fit displayed in Fig. 5 (left).

## 4. Compensation of chromatic dispersion

### 4.1 The design of dispersion compensating filters

**15**, 2120–2126 (2007). [CrossRef] [PubMed]

13. E. Ip and J.M Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion” J. Lightwave Technol.25, 2033–2043 (2007) [CrossRef]

20. S. Haykin, “Signal processing: where physics and mathematics meet,” IEEE Signal Process. Mag. **18**, 6–7 (2001). [CrossRef]

*A*(

*z*,

*t*) of a pulse may be modeled by the following partial differential equation[21]

*z*is the distance of propagation,

*t*is time variable in a frame moving with the pulse,

*λ*is the wavelength of the light,

*c*is the speed of light, and

*D*is the dispersion coefficient of the fiber.

*G*(

*z*,

*ω*) given by

*ω*is the angular frequency. The dispersion compensating filter is therefore given by the all-pass filter 1/

*G*(

*z*,

*ω*), which can be approximated using nonrecursive[4

4. M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments” IEEE Photon. Technol. Lett. **16**, 674–676 (2004). [CrossRef]

**15**, 2120–2126 (2007). [CrossRef] [PubMed]

12. G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. **19**, 969–971 (2007). [CrossRef]

12. G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. **19**, 969–971 (2007). [CrossRef]

**15**, 2120–2126 (2007). [CrossRef] [PubMed]

### 4.2 Time domain design of the chromatic dispersion compensating filter

*g*(

_{c}*z*,

*t*), given by

*T*seconds then aliasing will occur for frequencies which exceed the Nyquist frequency given by

*ω*=

_{n}*π*/

*T*and that the impulse response may be considered as a rotating vector whose angular frequency is given by

*N*, then the tap weights will be given by

*x*⌋ is the integer part of

*x*rounded towards minus infinity. These tap weights form the basis for the compensation of chromatic dispersion using an FIR filter. It should however be noted that this is an upper bound on the number of taps, with the resulting filter attempting to give constant dispersion over the frequency range -0.5/

*T*≤

*f*≤0.5/

*T*, and that in practice the number of taps may be reduced in order to give constant dispersion over a reduced frequency range. Furthermore properties of the filter, such as group delay ripple, maybe improved by windowing this analytical finite impulse function.

*N*=0.032

*B*

^{2}. Thus for a 10.7 Gbaud system transmitting over 4000km of standard fiber with D=17ps/nm/km this would require no more than 250 taps.

### 4.3 Numerical investigation of the compensation of chromatic dispersion

^{15}-1 PRBS, such that the symbol rate is 10.7GBaud with λ=1553nm. This is filtered by a 7GHz 5

^{th}order Bessel filter to reduce the noise and prevent aliasing after which the signal is sampled at 21.4GSa/s. The signal is then digital filtered after which the bit error rate (BER) is measured over a total of 262136 bits. The frequency offset between the local oscillator and the signal is set to zero, and the phase noise of the lasers is neglected, however since the equalizer is invariant to phase noise this omission will not affect the conclusions. The system is considered linear with noise applied at the receiver and the impact of quantization is not included.

## 5. Compensation of polarization dependent effects

### 5.1 Obtaining the inverse Jones matrix of the channel

22. S. Betti, F. Curti, G. De Marchis, and E. Iannone, “A novel multilevel coherent optical system: four quadrature signaling,” J. Lightwave Technol. **9**, 514–523 (1991). [CrossRef]

**15**, 2120–2126 (2007). [CrossRef] [PubMed]

23. Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express **13**, 7527–7534 (2005). [CrossRef] [PubMed]

*x*(

_{p}*k*) and

*y*(

_{p}*k*) the output

*x*(

_{out}*k*) is given by

*y*(

_{out}*k*)=

**h**

*·*

^{T}_{yx}**x**

_{p}+

**h**

*·*

^{T}_{yy}**y**

_{p}, where

**h**,

_{xx}**h**,

_{xy}**h**,

_{yx}**h**are adaptive filters each of which have length

_{yy}*M*taps. While there are a number of methods for adapting the equalizer in MIMO systems, we shall restrict ourselves to a specific example that exploits properties of the data, namely that for polarization division multiplexed QPSK (PDM-QPSK) the signal for each polarization should have a constant modulus[24

24. D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**, 1867–1875 (1980). [CrossRef]

24. D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. **28**, 1867–1875 (1980). [CrossRef]

25. C.R. Johnson, P. Schniter, T.J. Endres, J.D. Behm, D.R. Brown, and R.A. Casas, “Blind Equalization Using the Constant Modulus Criterion: A Review,” Proc. IEEE **86**, 1927–1950 (1998). [CrossRef]

### 5.2 Update algorithm for the adaptive equalizer

*ε*=1-|

_{x}*x*|

_{out}^{2}and

*ε*=1-|

_{y}*y*|

_{out}^{2}, giving the following criteria

*μ*

**x̄**

_{p}and

**ȳ**

_{p}denotes the complex conjugate of

**x**

_{p}and

**y**

_{p}respectively. In order to initialize the algorithm, all tap weights are set to zero with the exception of the central tap of

**h**

_{xx}and

**h**

_{yy}which are set to unity. Given the equalizer is unconstrained with respect to its outputs, it is possible for the equalizer to converge on the same output, corresponding to the Jones matrix becoming singular. This may however he remedied by monitoring the determinant of the Jones matrix such that if it begins to approach zero then the equalizer is reinitialized with tap weights

**h**

_{xx}(

*k*)→0.5(

**h**

_{xx}(

*k*)+

**h̄**

_{yy}(

*M*-1-

*k*)),

**h**

_{yy}(

*k*)→

**h̄**

_{xx}(

*M*-1-

*k*)

**h**

_{xy}(

*k*)→0.5(

**h**

_{xy}(

*k*)-

**h̄**

_{yx}(

*M*-1-

*k*)),

**h**

_{yx}(

*k*)→-

**h̄**

_{xy}(

*M*-1-

*k*) where

*k*=0,1..

*M*-1.

*ε*=

_{x}*d*-

_{x}*x*and

_{out}*ε*=

_{y}*d*-

_{y}*y*, with

_{out}*d*and

_{x}*d*and being the symbols closest to

_{y}*x*and

_{out}*y*respectively.

_{out}### 5.3 Dynamic response of the adaptive equalizer

*ω*is the angular frequency. The four FIR filters in the equalizer each had 3 taps, with the signal noise loaded to an OSNR of 9.5dB, being commensurate with a BER of 10

^{-3}.

*μ*=0.01 it is possible to track angular frequencies of approximately 1 Mrad/s with a penalty of less than 0.5dBQ.

11. C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E-D. Schmidt, T. Wuth, E. de Man, G. D. Khoe, and H. de Waardt, “10×111 Gbit/s, 50 GHz Spaced, POLMUX-RZ-DQPSK Transmission over 2375 km Employing Coherent Equalisation,” in Proceedings of Optical Fiber Communications Conference 2007, paper PDP22

28. J.J. Rodriguez-Andina, M.J. Moure, and M.D. Valdes, “Features, design tools, and application domains of FPGAs,” IEEE Trans Ind. Electron. **54**, 1810–1823, (2007) [CrossRef]

## 6. Conclusions

## Acknowledgments

## References and Links

1. | P. S. Henry, “Lightwave Primer” IEEE J. Quantum Electron. |

2. | H. Bülow, “Electronic dispersion compensation,” |

3. | T. Okoshi and K. Kikuchi, “Coherent Optical Fiber Communications,” KTK, 1988 |

4. | M. G. Taylor, “Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments” IEEE Photon. Technol. Lett. |

5. | S. Tsukamoto, D.-S Ly-Gagnon, K. Katoh, and K. Kikuchi, “Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission,” in Proceedings of Optical Fiber Communications Conference 2005, paper PDP-29 |

6. | S.J. Savory, A.D. Stewart, S. Wood, G. Gavioli, M.G. Taylor, R.I. Killey, and P. Bayvel, “Digital Equalisation of 40Gbit/s per Wavelength Transmission over 2480km of Standard Fibre without Optical Dispersion Compensation,” in Proceedings of ECOC 2006, Cannes, France, paper Th2.5.5, Sep. 2006. |

7. | C.R.S. Fludger, T. Duthel, T. Wuth, and C. Schulien, “Uncompensated Transmission of 86Gbit/s Polarization Multiplexed RZ-QPSK over 100km of NDSF Employing Coherent Equalisation,” in Proceedings of ECOC 2006, Cannes, France, paper, Th4.3.3 |

8. | S.J. Savory, G. Gavioli, R.I. Killey, and P. Bayvel, “Electronic compensation of chromatic dispersion using a digital coherent receiver,” Opt. Express |

9. | C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, “Wavelength Division Multiplexing (WDM) and Polarization Mode Dispersion (PMD) Performance of a Coherent 40Gbit/s Dual-Polarization Quadrature Phase Shift Keying (DP-QPSK) Transceiver,” in Proceedings of Optical Fiber Communications Conference 2007, paper PDP16 |

10. | G. Charlet, J. Renaudier, M. Salsi, H. Mardoyan, P. Tran, and S. Bigo “Efficient Mitigation of Fiber Impairments in an Ultra-Long Haul Transmission of 40Gbit/s Polarization-Multiplexed Data, by Digital Processing in a Coherent Receiver,” in Proceedings of Optical Fiber Communications Conference 2007, paper PDP17 |

11. | C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E-D. Schmidt, T. Wuth, E. de Man, G. D. Khoe, and H. de Waardt, “10×111 Gbit/s, 50 GHz Spaced, POLMUX-RZ-DQPSK Transmission over 2375 km Employing Coherent Equalisation,” in Proceedings of Optical Fiber Communications Conference 2007, paper PDP22 |

12. | G. Goldfarb and G. Li, “Chromatic dispersion compensation using digital IIR filtering with coherent detection,” IEEE Photon. Technol. Lett. |

13. | E. Ip and J.M Kahn, “Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion” J. Lightwave Technol.25, 2033–2043 (2007) [CrossRef] |

14. | D. van den Borne, H. de Waardt, G.-D. Khoe, T. Duthel, C. R.S. Fludger, C. Schulien, and E. -D. Schmidt, “Electrical PMD Compensation in 43-Gb/s POLMUX-NRZ-DQPSK enabled by Coherent Detection and Equalization,” in Proceedings ECOC 2007, Berlin, Germany, invited paper 8.3.1 |

15. | S.J. Savory, V. Mikhailov, R.I. Killey, and P. Bayvel, “Digital coherent receivers for uncompensated 42.8Gbit/s transmission over high PMD fibre,” in Proceedings ECOC 2007, Berlin, Germany, invited paper 10.4.1 |

16. | A. Leven, N. Kaneda, U-V Koc, and Y.-K. Chen “Frequency Estimation in Intradyne Reception,” IEEE Photon. Technol. Lett.19, 366–368 (2007) [CrossRef] |

17. | Q. Yu, L.-S. Yan, S. Lee, and A.E. Willner, “Loop-Synchronous Polarization Scrambling for Simulating Polarization Effects Using Recirculating Fiber Loops,” J. Lightwave Technol. |

18. | S. R. Desbruslais and P. R. Morkel, “Simulation of polarization mode dispersion and its effects in long-haul optically amplified lightwave systems,” IEE Colloquium on International Transmission System, 6.1–6.6 (1994). |

19. | M. J. D. Powell, |

20. | S. Haykin, “Signal processing: where physics and mathematics meet,” IEEE Signal Process. Mag. |

21. | G.P. Agrawal, |

22. | S. Betti, F. Curti, G. De Marchis, and E. Iannone, “A novel multilevel coherent optical system: four quadrature signaling,” J. Lightwave Technol. |

23. | Y. Han and G. Li, “Coherent optical communication using polarization multiple-input-multiple-output,” Opt. Express |

24. | D. Godard, “Self-recovering equalization and carrier tracking in two-dimensional data communication systems,” IEEE Trans. Commun. |

25. | C.R. Johnson, P. Schniter, T.J. Endres, J.D. Behm, D.R. Brown, and R.A. Casas, “Blind Equalization Using the Constant Modulus Criterion: A Review,” Proc. IEEE |

26. | J.G. Proakis, |

27. | S. Haykin, |

28. | J.J. Rodriguez-Andina, M.J. Moure, and M.D. Valdes, “Features, design tools, and application domains of FPGAs,” IEEE Trans Ind. Electron. |

29. | T. Pfau et al. “PDL-tolerant real-time polarization-multiplexed QPSK transmission with digital coherent polarization diversity receiver” |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4510) Fiber optics and optical communications : Optical communications

**ToC Category:**

Coherent Receivers

**History**

Original Manuscript: September 4, 2007

Revised Manuscript: November 14, 2007

Manuscript Accepted: November 14, 2007

Published: January 9, 2008

**Virtual Issues**

Coherent Optical Communication (2008) *Optics Express*

**Citation**

Seb J. Savory, "Digital filters for coherent optical receivers," Opt. Express **16**, 804-817 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-804

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### References

- P. S. Henry, "Lightwave Primer" IEEE J. Quantum Electron. 21, 1862-1879 (1985) [CrossRef]
- H. Bülow, "Electronic dispersion compensation," Proc. Opt. Fiber Comm. Conf. 2007, paper OMG5
- T. Okoshi and K. Kikuchi, "Coherent Optical Fiber Communications," KTK, 1988
- M. G. Taylor, "Coherent detection method using DSP for demodulation of signal and subsequent equalization of propagation impairments" IEEE Photon. Technol. Lett. 16, 674 - 676 (2004). [CrossRef]
- S. Tsukamoto, D.-S Ly-Gagnon, K. Katoh, and K. Kikuchi, "Coherent demodulation of 40-Gbit/s polarization-multiplexed QPSK signals with 16-GHz spacing after 200-km transmission," in Proceedings of Optical Fiber Communications Conference 2005, paper PDP-29
- S. J. Savory, A. D. Stewart, S. Wood, G. Gavioli, M. G. Taylor, R. I. Killey, P. Bayvel, "Digital equalisation of 40Gbit/s per wavelength transmission over 2480km of standard fibre without optical dispersion compensation," in Proceedings of ECOC 2006, Cannes, France, paper Th2.5.5, Sep. 2006.
- C. R. S. Fludger, T. Duthel, T. Wuth, and C. Schulien, "Uncompensated transmission of 86Gbit/s polarization multiplexed RZ-QPSK over 100km of NDSF employing coherent equalisation," in Proceedings of ECOC 2006, Cannes, France, paper, Th4.3.3
- S. J. Savory, G. Gavioli, R. I. Killey and P. Bayvel, "Electronic compensation of chromatic dispersion using a digital coherent receiver," Opt. Express 15,2120-2126 (2007). [CrossRef] [PubMed]
- C. Laperle, B. Villeneuve, Z. Zhang, D. McGhan, H. Sun, and M. O’Sullivan, "Wavelength Division Multiplexing (WDM) and Polarization Mode Dispersion (PMD) Performance of a Coherent 40Gbit/s Dual-Polarization Quadrature Phase Shift Keying (DP-QPSK) Transceiver," in Proceedings of Optical Fiber Communications Conference 2007, paper PDP16
- G. Charlet, J. Renaudier, M. Salsi, H. Mardoyan, P. Tran, and S. Bigo "Efficient mitigation of fiber impairments in an ultra-long haul transmission of 40Gbit/s polarization-multiplexed data, by digital processing in a coherent receiver," in Proceedings of Optical Fiber Communications Conference 2007, paper PDP17
- C. R. S. Fludger, T. Duthel, D. van den Borne, C. Schulien, E-D. Schmidt, T. Wuth, E. de Man, G. D. Khoe, H. de Waardt, "10 x 111 Gbit/s, 50 GHz Spaced, POLMUX-RZ-DQPSK transmission over 2375 km employing coherent equalisation," in Proceedings of Optical Fiber Communications Conference 2007, paper PDP22
- G. Goldfarb and G. Li, "Chromatic dispersion compensation using digital IIR filtering with coherent detection," IEEE Photon. Technol. Lett. 19,969-971 (2007). [CrossRef]
- E. Ip and J. M. Kahn, "Digital equalization of chromatic dispersion and polarization mode dispersion" J. Lightwave Technol. 25,2033-2043 (2007) [CrossRef]
- D. van den Borne, H. de Waardt, G.-D. Khoe, T. Duthel, C. R. S. Fludger, C. Schulien, and E. -D. Schmidt, "Electrical PMD Compensation in 43-Gb/s POLMUX-NRZ-DQPSK enabled by Coherent Detection and Equalization," in Proceedings ECOC 2007, Berlin, Germany, invited paper 8.3.1
- S. J. Savory, V. Mikhailov, R. I. Killey, and P. Bayvel, "Digital coherent receivers for uncompensated 42.8Gbit/s transmission over high PMD fibre," in Proceedings ECOC 2007, Berlin, Germany, invited paper 10.4.1
- A. Leven, N. Kaneda, U-V Koc, and Y.-K. Chen "Frequency Estimation in Intradyne Reception," IEEE Photon. Technol. Lett. 19,366 - 368 (2007) [CrossRef]
- Q. Yu, L.-S. Yan, S. Lee, and A. E. Willner, "Loop-synchronous polarization scrambling for simulating polarization effects using recirculating fiber loops," J. Lightwave Technol. 21,1593-1600 (2003). [CrossRef]
- S. R. Desbruslais and P. R. Morkel, "Simulation of polarization mode dispersion and its effects in long-haul optically amplified lightwave systems," IEE Colloquium on International Transmission System, 6.1-6.6 (1994).
- M. J. D. Powell, Approximation Theory and Methods, Cambridge University Press, 1981.
- S. Haykin, "Signal processing: where physics and mathematics meet," IEEE Signal Process. Mag. 18,6-7 (2001) [CrossRef]
- G. P. Agrawal, Nonlinear Fiber Optics, (Academic Press, 2001), Chap. 3
- S. Betti, F. Curti, G. De Marchis, and E. Iannone, "A novel multilevel coherent optical system: four quadrature signaling," J. Lightwave Technol. 9,514-523 (1991) [CrossRef]
- Y. Han and G. Li, "Coherent optical communication using polarization multiple-input-multiple-output," Opt. Express 13,7527-7534 (2005). [CrossRef] [PubMed]
- D. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication systems," IEEE Trans. Commun. 28,1867 - 1875 (1980). [CrossRef]
- C. R. Johnson, P. Schniter, T. J. Endres, J. D. Behm, D. R. Brown, R.A. Casas, "Blind equalization using the Constant Modulus Criterion: A review," Proc. IEEE 86,1927-1950 (1998). [CrossRef]
- J.G. Proakis, Digital Communications, 4th Ed., (McGraw Hill, 2001).
- S. Haykin, Adaptive Filter Theory, 4th Ed., (Prentice Hall, 2001).
- J. J. Rodriguez-Andina, M. J. Moure and M. D. Valdes, "Features, design tools, and application domains of FPGAs," IEEE Trans Ind. Electron. 54,1810-1823 (2007). [CrossRef]
- T. Pfau, et al., "PDL-tolerant real-time polarization-multiplexed QPSK transmission with digital coherent polarization diversity receiver" Proceedings of IEEE LEOS Summer Topical Meeting, 2007, paper MA3.3

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