## Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems |

Optics Express, Vol. 18, Issue 18, pp. 19039-19054 (2010)

http://dx.doi.org/10.1364/OE.18.019039

Acrobat PDF (1262 KB)

### Abstract

There has been a trend of migration to high spectral efficiency transmission in optical fiber communications for which the frequency guard band between neighboring wavelength channels continues to shrink. In this paper, we derive closed-form analytical expressions for nonlinear system performance of densely spaced coherent optical OFDM (CO-OFDM) systems. The closed-form solutions include the results for the achievable Q factor, optimum launch power density, nonlinear threshold of launch power density, and information spectral efficiency limit. These analytical results clearly identify their dependence on system parameters including fiber dispersion, number of spans, dispersion compensation ratio, and overall bandwidth. The closed-form solution is further substantiated by numerical simulations using distributed nonlinear Schrödinger equation.

© 2010 OSA

## 1. Introduction

1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. **42**(10), 587–589 (2006). [CrossRef]

2. E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).

5. G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. **19**(24), 2015–2017 (2007). [CrossRef]

7. H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. **28**(4), 406–414 (2010). [CrossRef]

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express **15**(20), 13282–13287 (2007). [CrossRef] [PubMed]

12. M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. **27**(16), 3560–3566 (2009). [CrossRef]

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express **15**(20), 13282–13287 (2007). [CrossRef] [PubMed]

9. 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). [CrossRef] [PubMed]

10. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express **16**(2), 841–859 (2008). [CrossRef] [PubMed]

12. M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. **27**(16), 3560–3566 (2009). [CrossRef]

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature **411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature **411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

8. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express **15**(20), 13282–13287 (2007). [CrossRef] [PubMed]

11. X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. **27**(16), 3632–3640 (2009). [CrossRef]

9. 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). [CrossRef] [PubMed]

## 2. Theoretical derivation and analysis

*W*, separated with neighboring channel by a frequency guard band of

*B*as shown in Fig. 1(b). In such DS-OFDM systems, all the nonlinear effects such as XPM, FWM, and SPM can be considered as FWM between all the subcarriers if we treat multiple densely spaced wavelength channels as an effective big ‘single-band’ OFDM channel that encompasses all the subcarriers. FWM is a third-order nonlinearity effect and its impact on optical fiber communications has been extensively studied [14

14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. **17**(11), 801–803 (1992). [CrossRef] [PubMed]

15. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. **13**(5), 841–849 (1995). [CrossRef]

*N*spans of the fiber link is given by [14

_{s}14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. **17**(11), 801–803 (1992). [CrossRef] [PubMed]

15. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. **13**(5), 841–849 (1995). [CrossRef]

*α*and

*L*are respectively the loss coefficient and length of the fiber per span,

*γ*is the third-order nonlinearity coefficient of the fiber,

*N*spans of FWM products, also known as phase array effect [9

_{s}9. 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). [CrossRef] [PubMed]

14. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. **17**(11), 801–803 (1992). [CrossRef] [PubMed]

*m*th subcarrier frequency has the form of

*D*(or

*ρ*(or

*ζ*) is the dispersion compensation (or residual dispersion) ratio,

13. P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature **411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

*i*th subcarrier as the reference frequency, and

*j*and

*k*frequencies as the interferers, namely, frequency

*j*and frequency

*k*generates a beating frequency component at

*i*, creating fourth components of

*f*. It has been shown for large number of subcarriers, the non-degenerate FWM is the dominate effects [9

_{g}**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*D*is set to 6 in Eq. (7). Consequently, the nonlinearity impinging on subcarrier

_{x}*i*,

*i,*and should be equivalent to the photons scattered into this subcarrier

*i*with large bandwidth assumption which we will clarify later. From now on, we drop index

*i*and set it to zero, or equivalently, we are investigating the performance of center wavelength channels in broad bandwidth DS-OFDM systems. We also assume all the subcarriers have the same power of

*P*for the sake of simplicity. The FWM power at the center subcarriers becomes

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*N*of 4000 subcarriers. This implies that in order to compute the FWM effect, it requires a summation in the order of 16 millions (4000x4000) of FWM terms in Eq. (10). Aside from the apparent mathematical complexity, the physical interpretation of FWM dependence on various key system parameters is difficult to ascertain. It is highly desirable to have a concise closed-form solution to the nonlinearity products in Eq. (10).

_{s}### 2.1 Conversion from discrete summation to integration

*sine*function is changing slowly each time when

*j*or

*m*is changed by 1, therefore the conversion from discrete summation is justified as far as the

*f*for

*m*in Eq. (10), the variable

*f*represents the frequency of the multiplicative noise impairing the channel. We now introduce more convenient and also fundamentally more important terms, power (spectral) densities given bywhere

*I*are respectively FWM noise power (spectral) density and launch power (spectral) density. Substituting Eq. (14) into Eq. (13), we arrive at the FWM noise densitywhere

### 2.2 Conversion of the integration range to more manageable forms

### 2.3 Closed-form expressions for nonlinear noise density

### 2.4 Signal-to-noise ratio and spectral efficiency limit in the presence of nonlinearity

**411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

**411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

*NF,*

*h*is the Planck constant, and

*υ*is the light frequency. The signal-to-noise is thus given by

16. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express **16**(26), 21944–21957 (2008). [CrossRef] [PubMed]

*C*normalized to bandwidth

*B*) for single-polarization is readily given by [17]

### 2.5 Optimal launch power density, maximum Q, and nonlinear threshold of launch power density

*I*, and setting it to zero, we obtain the optimum launch power density and the optimal Q given by

*Q*is 9.8 (dB), or linear

*q*of 3.09. In Eq. (30), setting

_{0}*n*to zero and

_{0}*Q to*

*q*is the correctable linear Q for a specific FEC.

_{0}*S*in Eq. (28), system Q factor and its optimal value in Eq. (30) and Eq. (32), optimal launch power density in Eq. (31), and nonlinear threshold of launch power density in Eq. (33) comprise the major findings in this work.

## 3. Corroboration of the theories with numerical simulation

*α*of 0.2 dB/km; nonlinear coefficient

### 3.1 Discussion about the closed-form expression

### 3.2 System Q factor and optimum launch power

*γ*, there is 2 dB increase in the optimal launch power and achievable Q. We can quickly generate the optimum launch power and achievable Q for variety of dispersion maps. We use the three systems, systems I, II and III studied in Figs. 2 and Fig. 3 as an example. As shown in Fig. 3(a), system I has the best performance due to large local dispersion and no per-span dispersion compensation. The advantage of system I over system II increases with the increase of the number of spans, for instance from 0 dB to 2.4 dB when the reach increases from single-span to 10 spans. The advantage of system I over system III is maintained at 1.7 dB independent of the number of spans. However, Fig. 3(b) shows the optimal launch power versus number of spans. The optimum launch powers for non-compensated systems, systems I and III are constant. This is because both the linear and nonlinear noises increase linearly with the number of the spans that leads to the optimum power independent of the number of spans. However, for the dispersion compensated system II, the optimum launch power density decreases with number of spans due to the multi-span noise enhancement effect. Another interesting observation from Eq. (31) and Eq. (32) is that both the optimal Q factor and launch power has very week dependence on the overall system bandwidth: proportional to 1/3 power of logarithm of the overall bandwidth. It can be easily shown that for both systems I and II, the Q is decreased by only about 0.7 dB with the 10-fold increase of the overall system bandwidth from 400 to 4000 GHz whereas system II incurs a larger decrease of the Q factor of 0.84 dB with the same bandwidth increase.

### 3.3 Information spectral efficiency

*γ*be decreased by a factor of 2.8, or number of spans be reduced by a factor of 2, all of which are difficult to achieve. In a nutshell, it is of diminishing return to improve the spectral efficiency by modifying the optical fiber system parameters. The only effective method to substantially improve the spectral efficiency is to add more dimensions such as resorting to polarization multiplexing that leads to a factor of 2 improvement, or fiber mode multiplexing by at least a factor of two or more dependent on the capability of achievable digital signal processing (DSP). Figure 4 shows the achievable spectral efficiency for the three systems studied in Section 3.1. The only modification is that we assume 40 nm or 5 THz for the total bandwidth. The spectral efficiency for the systems I, II, and III are respectively 5.17, 4.40, and 4.52 b/s/Hz. This shows a total capacity of 25 Tb/s can be achieved for 10x100 SSMF uncompensated EDFA-only single-polarization systems within C-band.

### 3.4 Multi-span noise enhancement factor

**411**(6841), 1027–1030 (2001). [CrossRef] [PubMed]

**16**(20), 15777–15810 (2008). [CrossRef] [PubMed]

*ζ*is small, or dispersion compensation ratio is large,

## 4. Conclusion

## Appendix A: Change of the integration range.

*f*from [-

*B*/2-

*f*

_{1},

*B*/2-

*f*

_{1}] to more tidy form of [-

*B*/2,

*B*/2]. This is reasonable simplification as the major contribution of the integral is when

*f*is around

*f*can be ignored when

_{1}*B >>*

*f*and

_{1}*f*, Eq. (35) can be expressed as

*A*is dominant over

_{1}*A*and

_{2}*A*, and therefore the contribution of

_{3}*A*and

_{2}*A*can be omitted as a good approximation.

_{3}*A*over

_{2}*f*, we have

_{1}*A*is upper bounded by

_{2}*A*/

_{1}*A*is upper bounded by

_{2}*A*in general can be ignored. The upper bound for the term

_{2}*A*can be found as follows

_{3}*A*/

_{1}*A*is upper bounded by

_{3}*A*is the dominant contribution to

_{1}*A*being a dominate component in

_{1}*A*and

_{1}*A*, the weight will shift toward

_{2}*A*as

_{1}*A*is integrated at the low frequency of

_{1}*f*, [0,

_{1}*B*/2], and

*A*is integrated at the high frequency of

_{2}*f*, [

_{1}*B*/2,

*∞*]. Therefore the ratio of

*A*/

_{1}*A*should be no less than

_{2}*A*can be ignored to a good degree of approximation. Regarding the ratio of

_{2}*A*/

_{1}*A*, the integration variable ‘

_{3}*f*’ is over low frequency region of [0,

*B*

_{0}/2] for ‘

*A*whereas over [

_{3}’*B*

_{0}/2,

*B*/2] for

*A*. So the existence of the phase array pattern

_{1}*A*. However, we will show the ratio of

_{3}*A*/

_{1}*A*is still upper bounder by a factor in the order of

_{3}*A*can be ignored. Therefore we only need to study the ratio of

_{3}*A*/

_{1}*A*when

_{3}*A*. We would like to derive the ratio of

_{3}*A*/

_{1}*A*when

_{3}*f*in Eq. (43),

*A*in Appendix B, under assumption of

_{1}*A*can be approximated as

_{1}*A*

_{1}/

*A*

_{3}is also upper bounded in the order of

*A*

_{1}can be used as an approximation for

*A*

_{2}and

*A*

_{3}small compared to

*A*

_{1}, but they also have opposite signs and their combined effects tend to cancel each other, which further improves the accuracy of using

*A*

_{1}as an approximation to

*A*

_{2}and add more weight to

*A*

_{3}. However, because

*A*

_{3}is integrated in the low frequency of

*f*, in practical application, the low frequency noise including DC components can be estimated and removed. Therefore the lower integration boundary for

*f*,

*B*

_{0}/2 in Eq. (46) can be also interpreted as that the low frequency noise from [0,

*B*

_{0}/2] is being removed due to the phase estimation. Therefore, to be complete, we redefine the B

_{0}as

*f*in Eq. (46) should be

## Appendix B: Closed-form solution for FWM noise spectral density

## References and links

1. | W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. |

2. | E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008). |

3. | Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express |

4. | S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” Eur. Conf. Optical Commun.,Vienna, Austria (2009), post-deadline Paper PD2.6. |

5. | G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. |

6. | R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009). |

7. | H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. |

8. | A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express |

9. | 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 |

10. | W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express |

11. | X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. |

12. | M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. |

13. | P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature |

14. | K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. |

15. | R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. |

16. | X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express |

17. | C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.2330) Fiber optics and optical communications : Fiber optics communications

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: June 10, 2010

Revised Manuscript: August 6, 2010

Manuscript Accepted: August 7, 2010

Published: August 23, 2010

**Citation**

Xi Chen and William Shieh, "Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems," Opt. Express **18**, 19039-19054 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19039

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### References

- W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–589 (2006). [CrossRef]
- E. Yamada, A. Sano, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, K. Yonenaga, Y. Miyamoto, K. Ishihara, Y. Takatori, T. Yamada, and H. Yamazaki, “1Tb/s (111Gb/s/ch × 10ch) no-guard-interval CO-OFDM transmission over 2100 km DSF,” Opto-Electronics Communications Conference/Australian Conference on Optical Fiber Technology, paper PDP6, Sydney, Australia (2008).
- Y. Ma, Q. Yang, Y. Tang, S. Chen, and W. Shieh, “1-Tb/s single-channel coherent optical OFDM transmission over 600-km SSMF fiber with subwavelength bandwidth access,” Opt. Express 17, 9421–9427 (2009). [CrossRef] [PubMed]
- S. Chandrasekhar, X. Liu, B. Zhu, and D. W. Peckham, “Transmission of a 1.2-Tb/s 24-carrier no-guard-interval coherent OFDM superchannel over 7200-km of ultra-large-area fiber,” Eur. Conf. Optical Commun.,Vienna, Austria (2009), post-deadline Paper PD2.6.
- G. Goldfarb, G. F. Li, and M. G. Taylor, “Orthogonal wavelength-division multiplexing using coherent detection,” IEEE Photon. Technol. Lett. 19(24), 2015–2017 (2007). [CrossRef]
- R. Dischler, and F. Buchali, “Transmission of 1.2 Tb/s continuous waveband PDM-OFDM-FDM signal with spectral efficiency of 3.3 bit/s/Hz over 400 km of SSMF,” Optical Fiber Communication Conference, paper PDP C2, San Diego, USA (2009).
- H. Takahashi, A. Al Amin, S. L. Jansen, I. Morita, and H. Tanaka, “Highly spectrally efficient DWDM transmission at 7.0 b/s/Hz using 8 x 65.1-Gb/s coherent PDM-OFDM,” J. Lightwave Technol. 28(4), 406–414 (2010). [CrossRef]
- A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007). [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. Express 16(20), 15777–15810 (2008). [CrossRef] [PubMed]
- W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008). [CrossRef] [PubMed]
- X. Liu, F. Buchali, and R. W. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009). [CrossRef]
- M. Mayrock and H. Haunstein, “Monitoring of linear and nonlinear signal distortion in coherent optical OFDM transmission,” J. Lightwave Technol. 27(16), 3560–3566 (2009). [CrossRef]
- P. P. Mitra and J. B. Stark, “Nonlinear limits to the information capacity of optical fibre communications,” Nature 411(6841), 1027–1030 (2001). [CrossRef] [PubMed]
- K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992). [CrossRef] [PubMed]
- R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13(5), 841–849 (1995). [CrossRef]
- X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008). [CrossRef] [PubMed]
- C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 (1948).

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