## Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer |

Optics Express, Vol. 20, Issue 2, pp. 1360-1369 (2012)

http://dx.doi.org/10.1364/OE.20.001360

Acrobat PDF (1645 KB)

### Abstract

We address the issue of intra-channel nonlinear compensation using a Volterra series nonlinear equalizer based on an analytical closed-form solution for the 3rd order Volterra kernel in frequency-domain. The performance of the method is investigated through numerical simulations for a single-channel optical system using a 20 Gbaud NRZ-QPSK test signal propagated over 1600 km of both standard single-mode fiber and non-zero dispersion shifted fiber. We carry on performance and computational effort comparisons with the well-known backward propagation split-step Fourier (BP-SSF) method. The alias-free frequency-domain implementation of the Volterra series nonlinear equalizer makes it an attractive approach to work at low sampling rates, enabling to surpass the maximum performance of BP-SSF at 2× oversampling. Linear and nonlinear equalization can be treated independently, providing more flexibility to the equalization subsystem. The parallel structure of the algorithm is also a key advantage in terms of real-time implementation.

© 2011 OSA

## 1. Introduction

1. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. **26**(20), 3416–3425 (2008). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-26-20-3416 [CrossRef]

2. 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). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-880 [CrossRef] [PubMed]

4. C. Xia and W. Rosenkranz, “Nonlinear electrical equalization for different modulation formats with optical filtering,” J. Lightwave Technol. **25**(4), 996–1001 (2007). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-25-4-996 [CrossRef]

5. Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. **282**(12), 2421–2425 (2009). [CrossRef]

7. 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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuE3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OTuE3

5. Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. **282**(12), 2421–2425 (2009). [CrossRef]

5. Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. **282**(12), 2421–2425 (2009). [CrossRef]

6. Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThA040. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-JThA040

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

9. J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express **18**(8), 8660–8670 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8660 [CrossRef] [PubMed]

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

## 2. Frequency-domain Volterra series equalizer - theoretical formulation

*A*is an abbreviation of

*A*(

*t,z*) describing the slowly varying complex envelope of the optical field at time

*t*and position

*z*,

*α*is the attenuation coefficient of the fiber,

*β*

_{2}accounts for the group velocity dispersion and

*γ*is the nonlinear coefficient accounting for the Kerr effect.

*L*, the third-order truncated BP-VSTF gives an estimative of the input field spectrum,

_{span}*Ã*(

_{in}*ω*), at the expense of the output field spectrum,

*Ã*(

_{out}*ω*) [10

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

*H*′

_{1}(

*ω*) and

*H*′

_{3}(

*ω*

_{1},

*ω*

_{2},

*ω*–

*ω*

_{1}+

*ω*

_{2}) are the inverse linear kernel and the inverse third-order nonlinear kernel, respectively given by and

*N*signal samples, taken with sampling rate

_{FFT}*F*=

_{s}*N*/

_{FFT}*T*, where

*T*is the time window. This block of samples is transformed into frequency domain, employing a discrete Fourier transform (DFT), which has to be consistent with the previously defined NLS equation, where

*ω*is the

_{n}*n*-th sample of the angular frequency vector,

*ω*. Since the DFT definition is based on a discretization of the continuous Fourier transform using a left Riemann sum, we may directly substitute the double integral in Eq. (2) by two algebraic sums over the entire integration range. In a span-by-span basis, each sample of the nonlinearly equalized input field spectrum,

*Ã*

_{eqNL}(

*ω*), is then obtained at the expense of an

_{n}*N*-length fast-Fourier transform (FFT) of the output field, by applying the transfer function where

_{FFT}*n*

_{1}and

*n*

_{2}are auxiliary indices used to evaluate the double sum for each block of frequency samples.

*N*-length buffer and transformed into frequency domain using an FFT. Then, three delay lines are used to evaluate the double summation in Eq. (6) and the obtained value is sent to the accumulation block. Each time Eq. (6) is completed, a sample is released to the output frequency buffer. Finally, when the output frequency buffer is full, the time domain nonlinearly equalized signal is obtained by applying an inverse FFT (IFFT).

_{FFT}8. K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. **15**(12), 2232–2241 (1997). [CrossRef]

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

*A*, as follows, where

_{eq}*A*

_{eqLI}is the linearly equalized field, obtained using any linear equalization method of choice, and

*A*

_{eqNL}is the nonlinearly equalized field, as given by Eq. (6).

## 3. Performance assessment

### 3.1. System model

*α*= 0.2 dB/km, group velocity dispersion of

*β*

_{2}= −20.4 ps

^{2}/km and Kerr coefficient of

*γ*= 1.3 W

^{−1}km

^{−1}. In turn, for the NZDSF we have

*α*= 0.2 dB/km,

*β*

_{2}= −6.0 ps

^{2}/km and

*γ*= 1.5 W

^{−1}km

^{−1}. As a figure of merit for compensation performance we use the error vector magnitude (EVM) percentage relatively to the optimal constellation, defined as

*A*and

_{tx}*A*are the transmitted and equalized optical fields, respectively.

_{eq}*N*steps per fiber span as BP-SSF

_{steps}_{Nsteps}.

### 3.2. Performance comparison with BP-SSF

*N*= 3) to perform digital equalization. We may see that the three best performance curves corresponding to BP-SSF

_{sp}_{8}, BP-SSF

_{64}and CDE+VSNE are completely overlaid. This fact enables us to draw two main conclusions. First, it becomes clear that BP-SSF reaches its performance limit at approximately 8 steps per span. This happens because the low temporal resolution sets an upper limit for compensation performance, above which it becomes useless to increase the spatial resolution. Secondly, we may also observe that the third-order approximation used to derive the FD-VSNE is sufficient to reach the best performance possible at this sampling rate. In turn, using only 2 samples per symbol (see Fig. 3(b)) we observe a significative degradation of performance in BP-SSF. This is due to the generation of high frequencies when the nonlinear operator is applied in time-domain, giving place to aliasing components when the signal is subsequently transposed to frequency-domain, in order to apply the linear step. In fact, this limitation has been already identified in [12

12. L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express **18**(16), 17075–17088 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17075 [CrossRef] [PubMed]

_{64}(considering the 10% EVM reference). Replacing the SSMF link by a NZDSF link (see Figs. 3(c) and 3(d)) we are able to draw similar conclusions, despite of a degradation of performance due to stronger nonlinear effects, which is well visible in the signal constellations taken at 6 dBm. At 2× oversampling, we may see that BP-SSF performance becomes now limited to 1 step per span due to stronger nonlinearities that generate stronger aliasing. In contrast, the FD-VSNE is still able to largely surpass BP-SSF proving to be an effective nonlinear equalization method even under extreme conditions.

### 3.3. Required bandwidth for nonlinear equalization

_{8}methods and also a similar evolution with respect to the LPF cutoff frequency. The maximum accuracy is attained when the LPF cutoff frequency is around 18 GHz, corresponding to 90% of the symbol rate. For higher cutoff frequencies the equalization performance tends to degrade due to aliasing enhancement in the sampling stage. On the other hand, narrower filtering degrades performance due to the attenuation of relevant spectral components. The optical spectra of the propagated signal at different filtering and sampling stages are shown in Fig. 5. In Fig. 4(b), we present a similar analysis for 2× oversampling. As expected, since the aliasing effect tends to become stronger for lower sampling rates, the optimum cutoff frequency is now reduced to ≈ 16 GHz, corresponding to 80% of the symbol rate. The only exception is the BP-SSF

_{8}curve for an NZDSF transmission link, where the optimum LPF cutoff frequency is found at approximately 60% of the symbol rate. The strong effect of nonlinearities in the NZDSF link exposes the BP-SSF limitation in terms of internal aliasing generation. Although a narrower LPF before equalization can residually counteract this effect, it does not avoid a severe performance degradation relatively to the FD-VSNE method.

## 4. Computational effort

*O*(

*log*

_{2}(

*N*)). In Fig. 7, we show how the computational effort of both methods evolves with

_{FFT}*N*, where it becomes clear that reduced FFT block-sizes must be considered in order to keep FD-VSNE in a tolerable region of complexity. However, it is well known that there is a limit for reducing the FFT block-size without incurring inter-block interference.

_{FFT}*N*fiber spans). However, as we can see in Fig. 8 the accumulated dispersion along the entire link requires large FFT block-sizes (128/256 for the NZDSF link and 512 for the SSMF link). Alternatively, the OS method can also be applied in a span-by-span basis, reducing the accumulated dispersion that needs to be inverted at each step. This way, we are able to reduce the penalty-free minimum FFT block-size to 32 (for the NZDSF link) and 64 (for the SSMF link). In fact, these values can be further reduced to 16 and 32 at the expense of some inter-block interference, maintaining the system bellow the 10

_{span}^{−3}BER floor.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. |

2. | 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 |

3. | M. Schetzen, |

4. | C. Xia and W. Rosenkranz, “Nonlinear electrical equalization for different modulation formats with optical filtering,” J. Lightwave Technol. |

5. | Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun. |

6. | Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThA040. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-JThA040 |

7. | 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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuE3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OTuE3 |

8. | K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol. |

9. | J. D. Reis and A. L. Teixeira, “Unveiling nonlinear effects in dense coherent optical WDM systems with Volterra series,” Opt. Express |

10. | F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett. |

11. | B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett. |

12. | L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

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

**ToC Category:**

Fibers, Fiber Devices, and Amplifiers

**History**

Original Manuscript: October 3, 2011

Revised Manuscript: December 17, 2011

Manuscript Accepted: December 19, 2011

Published: January 9, 2012

**Virtual Issues**

European Conference on Optical Communication 2011 (2011) *Optics Express*

**Citation**

Fernando P. Guiomar, Jacklyn D. Reis, António L. Teixeira, and Armando N. Pinto, "Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer," Opt. Express **20**, 1360-1369 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-2-1360

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### References

- E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26(20), 3416–3425 (2008). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-26-20-3416 [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. Express16(2), 880–888 (2008). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-2-880 [CrossRef] [PubMed]
- M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems (Wiley, 1989).
- C. Xia and W. Rosenkranz, “Nonlinear electrical equalization for different modulation formats with optical filtering,” J. Lightwave Technol.25(4), 996–1001 (2007). http://www.opticsinfobase.org/jlt/abstract.cfm?URI=jlt-25-4-996 [CrossRef]
- Y. Gao, F. Zgang, L. Dou, Z. Chen, and A. Xu, “Intra-channel nonlinearities mitigation in pseudo-linear coherent QPSK transmission systems via nonlinear electrical equalizer,” Opt. Commun.282(12), 2421–2425 (2009). [CrossRef]
- Z. Pan, B. Châtelain, M. Chagnon, and D. V. Plant, “Volterra filtering for nonlinearity impairment mitigation in DP-16QAM and DP-QPSK fiber optic communication systems,” in Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2011), paper JThA040. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2011-JThA040
- 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 Optical Fiber Communication Conference, OSA Technical Digest (CD) (Optical Society of America, 2010), paper OTuE3. http://www.opticsinfobase.org/abstract.cfm?URI=OFC-2010-OTuE3
- K. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Light-wave Technol.15(12), 2232–2241 (1997). [CrossRef]
- 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). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-8-8660 [CrossRef] [PubMed]
- F. P. Guiomar, J. D. Reis, A. L. Teixeira, and A. N. Pinto, “Digital postcompensation using Volterra series transfer function,” IEEE Photon. Technol. Lett.23(19), 1412–1414 (2011). [CrossRef]
- B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method,” IEEE Photon. Technol. Lett.14(1), 47–49 (2002). [CrossRef]
- L. B. Du and A. J. Lowery, “Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems,” Opt. Express18(16), 17075–17088 (2010). http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-16-17075 [CrossRef] [PubMed]

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