## Nonlinear impairment compensation using expectation maximization for dispersion managed and unmanaged PDM 16-QAM transmission |

Optics Express, Vol. 20, Issue 26, pp. B181-B196 (2012)

http://dx.doi.org/10.1364/OE.20.00B181

Acrobat PDF (4944 KB)

### Abstract

In this paper, we show numerically and experimentally that expectation maximization (EM) algorithm is a powerful tool in combating system impairments such as fibre nonlinearities, inphase and quadrature (I/Q) modulator imperfections and laser linewidth. The EM algorithm is an iterative algorithm that can be used to compensate for the impairments which have an imprint on a signal constellation, i.e. rotation and distortion of the constellation points. The EM is especially effective for combating non-linear phase noise (NLPN). It is because NLPN severely distorts the signal constellation and this can be tracked by the EM. The gain in the nonlinear system tolerance for the system under consideration is shown to be dependent on the transmission scenario. We show experimentally that for a dispersion managed polarization multiplexed 16-QAM system at 14 Gbaud a gain in the nonlinear system tolerance of up to 3 dB can be obtained. For, a dispersion unmanaged system this gain reduces to 0.5 dB.

© 2012 OSA

## 1. Introduction

1. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron **16**, 1164–1179 (2010). [CrossRef]

2. R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. **28**, 662–701 (2010). [CrossRef]

3. A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol. **25**, 3008–3016 (2007). [CrossRef]

7. D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun. **E94-B**, 1815–1822 (2011). [CrossRef]

8. D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in *Proc. of ECOC, paper Th1D2*, Amsterdam, The Netherlands, (2012).

## 2. Numerical and experimental system set-up

### 2.1. Numerical set-up

^{15}– 1), signal mapping, upsampling, pulse shaping filter (raised cosine), digital-to-analog converter (DAC), attenuators and electrical amplifiers. The actual impulse response of the driving amplifiers is not taken into consideration. It is assumed that the electrical amplifiers have sufficient bandwidth such that they don’t induce any signal distortion. The method of 4-PAM signal generation is very similar to the one reported in [9

9. P. Winzer, A. Gnauck, C. Doerr, M. Magarini, and L. Buhl, “Spectrally efficient long-haul optical networking using 112-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol. **28**, 547–556 (2010). [CrossRef]

*α*=0.2 dB/km,

_{smf}*D*=17 ps/nm/km and nonlinear coefficient is

_{smf}*γ*=1.3 W

_{smf}^{−1}km

^{−1}. For the DCF, we have the following parameters:

*α*=0.5 dB/km,

_{dcf}*D*=−80 ps/nm/km and nonlinear coefficient is

_{dcf}*γ*=5.3 W

_{dcf}^{−1}km

^{−1}.

### 2.2. Experimental set-up

### 2.3. Digital signal processing algorithms

1. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron **16**, 1164–1179 (2010). [CrossRef]

10. D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express **19**, 429–437 (2011). [CrossRef]

10. D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express **19**, 429–437 (2011). [CrossRef]

1. S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron **16**, 1164–1179 (2010). [CrossRef]

**16**, 1164–1179 (2010). [CrossRef]

## 3. Theory

### 3.1. Statistical signal representation - mixture of Gaussians

*M*Gaussian densities, where

*M*is the number of clusters and corresponds to the number of constellation points. The probability density function of the demodulated signal is then expressed as: where

*k*refers to each cluster in the constellation,

*π*is a mixing coefficient (for the considered case of a signal where symbols have uniform distribution

_{k}*π*= 1/

_{k}*M*).

**x**= [

*x*

_{1},

*x*

_{2}] is a 2-D vector, corresponding to a detected symbol in the constellation (Inphase/Quadrature) plane and

*N*(

**x**|

*μ*, Σ

_{k}*) is a 2-D Gaussian density with mean*

_{k}*μ*and a 2 × 2 covariance matrix Σ

_{k}*[13]: where |Σ*

_{k}*| is the determinant of the covariance matrix and it expresses the area covered by the specific cluster*

_{k}*k*. The covariance matrix is defined as:

*. However, when the signal is mostly dominated by the additive Gaussian noise, the covariance matrices will be equal and diagonal, i.e. there is no correlation among symbols within the cluster. Additionally, the clusters will be circularly symmetric (equal variances) and the covariance matrix is then expressed as:*

_{k}*μ*have been altered compared to the reference constellation. By reference constellation, it is meant the constellation which is free of any impairment.

_{k}**x**belonging to one of the clusters

*k*, where

*k*= 1,...,

*M*: or in another words find a cluster

*k*for which

*p*(

*k*|

**x**) is maximized. The a posteriori probability

*p*(

*k*|

**x**) is obtained from Bayes’ theorem [13]: Inserting the expression for the Gaussian distribution, the optimal decision, Eq. (5), reduces to a quadratic decision rule: with

*k*and the decision rule becomes linear: However, in case when the signal constellation is distorted by nonlinear phase noise, laser phase noise, etc, Eq. (6) needs to be used in order to make optimum signal detection. In order to evaluate Eq. (6),

*M*Gaussian densities,

*N*(

**x**|

*μ*, Σ

_{k}*) and thereby parameters*

_{k}*π*≡ {

*π*

_{1},...,

*π*},

_{k}*μ*≡ {

*μ*

_{1},...,

*μ*} and Σ ≡ {Σ

_{k}_{1},..., Σ

*} describing Gaussian densities need to be determined. Next, we will show how to use a powerful method of EM in order to determine the parameters that generate the Gaussian mixture model. The EM will determine in a maximum likelihood sense the most likely parameters Ξ = [*

_{k}*π*,

*μ*,Σ] that generated Gaussian densities.

### 3.2. Expectation maximization algorithm

**X**is governed by the following expression: where

**X**= [

**x**

_{1},...,

**x**

*],*

_{N}*N*is the length of the observation interval and the likelihood function of Ξ,

*p*(

**X**|Ξ), for independent identically distributed data is expressed as: No closed-form analytical solution for Eq. (9) is available. Therefore, the iterative EM framework can be used to find a solution. The EM is a two step iterative procedure which is guar-enteed to converge to the (local) maximum likelihood solution given in Eq. (9) [14]. The two step procedure, so called expectation (E) step and maximization (M) step for the particular case considered in this section is as follows [13]: where

*γ*is called the responsibility and is nothing but a posteriori probability, Eq. (6), needed for optimal decisions.

_{nk}*at the iteration*

^{i}*i*, are used to evaluate the Eq. (11). The E-step expressed by Eq. (11) computes the probability of the received symbol belonging to one of the clusters, i.e. posterior probability. In the M-step we use those probabilities to re-estimate the parameters Ξ. In other words, in the M-step we are trying to find the parameters that maximize the probability that the data has been generated by a particular cluster. When making a parameter update resulting from the E step and followed by the M step, the likelihood function,

*P*(

**X**|Ξ

*), on the parameters will increase and will flatten out when the algorithm has converged. The convergence properties of the EM strongly dependent on the initialization. For the considered cases throughout the paper, we found that the EM algorithm will converge after 3 iterations. Once the EM algorithms has converged (*

^{i}*i*>

*N*), we use the results to perform the optimum signal detection governed by Eq. (5).

_{iter}## 4. Simulation results

### 4.1. Back-to-back investigation

*V*, of the electrical 4-PAM signal such that the constellation diagram of the 16-QAM signal is distorted. The resulting modulation depth of the modulator is then

_{pp}*m*=

*V*/

_{pp}*V*=2.12. Furthermore, it is assumed that the phase shift between the inphase and quadrature branch of the I/Q modulator deviates from

_{π}*π*/2 by 5%. In Fig. 4, −log[BER] is plotted as function of the combined laser linewidth for the optical signal to noise ratio (OSNR) of 25 dB. Figure 4 shows that compared to the case when no compensation is used (linear decision boundaries), the EM is very efficient in combating the combined impairments originating from I/Q modulator nonlinearities, non-ideal phase shift between the I and Q branches and combined laser linewidth.

### 4.2. Dispersion managed link

15. N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett. **22**, 335–337 (2010). [CrossRef]

### 4.3. Dispersion unmanaged link

16. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. **30**, 1524–1539 (2012). [CrossRef]

## 5. Experimental results

### 5.1. Dispersion managed link

*P*= 0 dBm and the corresponding optimal decision boundaries after 800 km of transmission. It is observed that the demodulated signal constellation shown in Fig. 8(a) is distorted and therefore the optimal decision boundaries are nonlinear. In Fig. 8(b), we plot −log(BER) as a function of span input power after 800 km of transmssion thorugh dispersion managed link. It is observed that there is an improvement in the nonlinear system tolerance by employing the EM algorithm which is in accordance with simulation results. We observe up to 3 dB of improvement in nonlinear tolerance compared to the case when no compensation is used. The reason why we get more improvement for the experimental data may be attributed to the fact that the EM is also effective in compensating residual distortion induced on the signal. It is observed from the figure that only very little improvement can be obtained by using the k-means algorithm, and this is also in good agreement with the simulation results.

_{in}### 5.2. Dispersion unmanaged link

16. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. **30**, 1524–1539 (2012). [CrossRef]

## 6. Conclusion

20. S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112gbit/s pdm-qam-16 transmission with and without digital nonlinearity compensation,” Opt. Express **18**, 12939–12947 (2010). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron |

2. | R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol. |

3. | A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol. |

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

5. | Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol. |

6. | N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in |

7. | D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun. |

8. | D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in |

9. | P. Winzer, A. Gnauck, C. Doerr, M. Magarini, and L. Buhl, “Spectrally efficient long-haul optical networking using 112-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol. |

10. | D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express |

11. | H. Meyr, M. Moeneclaey, and S. Fechtel, |

12. | J. Kurzweil, |

13. | C. M. Bishop, |

14. | A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B. |

15. | N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett. |

16. | A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. |

17. | F. Vacondio, O. Rival, C. Simonneau, E. Grellier, L. Lorcy, J.-C. Antona, S. Bigo, and A. Bononi, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express |

18. | A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in |

19. | E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in |

20. | S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112gbit/s pdm-qam-16 transmission with and without digital nonlinearity compensation,” Opt. Express |

**OCIS Codes**

(060.0060) Fiber optics and optical communications : Fiber optics and optical communications

(060.1660) Fiber optics and optical communications : Coherent communications

**ToC Category:**

Subsystems for Optical Networks

**History**

Original Manuscript: October 1, 2012

Revised Manuscript: November 4, 2012

Manuscript Accepted: November 5, 2012

Published: November 28, 2012

**Virtual Issues**

European Conference on Optical Communication 2012 (2012) *Optics Express*

**Citation**

Darko Zibar, Ole Winther, Niccolo Franceschi, Robert Borkowski, Antonio Caballero, Valeria Arlunno, Mikkel N. Schmidt, Neil Guerrero Gonzales, Bangning Mao, Yabin Ye, Knud J. Larsen, and Idelfonso Tafur Monroy, "Nonlinear impairment compensation using expectation maximization for dispersion managed and unmanaged PDM 16-QAM transmission," Opt. Express **20**, B181-B196 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-26-B181

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### References

- S. J. Savory, “Digital coherent optical receivers: Algorithms and subsystems,” IEEE J Sel. Top. Quantum Electron16, 1164–1179 (2010). [CrossRef]
- R.-J. Essiambre, G. Kramer, P. Winzer, G. Foschini, and B. Goebel, “Capacity limits of optical fiber networks,” J. Lightwave Technol.28, 662–701 (2010). [CrossRef]
- A. Lau and J. Kahn, “Signal design and detection in presence of nonlinear phase noise,” J. Lightwave Technol.25, 3008–3016 (2007). [CrossRef]
- E. Ip and J. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26, 3416–3425 (2008). [CrossRef]
- Z. Tao, L. Dou, W. Yan, L. Li, T. Hoshida, and J. C. Rasmussen, “Multiplier-free intrachannel nonlinearity compensating algorithm operating at symbol rate,” J. Lightwave Technol.29, 2570–2576 (2011). [CrossRef]
- N. Stojanovic, Y. Huang, F. N. Hauske, Y. Fang, M. Chen, C. Xie, and Q. Xiong, “Mlse-based nonlinearity mitigation for wdm 112 gbit/s pdm-qpsk transmissions with digital coherent receiver,” in Proc. of OFC, paper OTu3C.5, Los Angeles, California, USA, (2011).
- D. Rafique, J. Zhao, and A. D. Ellis, “Compensation of nonlinear fibre impairments in coherent systems employing spectrally efficient modulation formats,” IEICE Trans. on Commun.E94-B, 1815–1822 (2011). [CrossRef]
- D. Zibar, O. Winther, N. Franceshi, R. Borkowski, A. Caballero, A. Valeria, N. M. Schmidt, G. G. Neil, B. Mao, Y. Ye, J. K. Larsen, and T. I. Monroy, “Nonlinear impairment compensation using expectation maximization for pdm 16-qam systems,” in Proc. of ECOC, paper Th1D2, Amsterdam, The Netherlands, (2012).
- P. Winzer, A. Gnauck, C. Doerr, M. Magarini, and L. Buhl, “Spectrally efficient long-haul optical networking using 112-gb/s polarization-multiplexed 16-qam,” J. Lightwave Technol.28, 547–556 (2010). [CrossRef]
- D. Zibar, J. C. R. F. de Olivera, V. B. Ribeiro, A. Paradisi, J. C. Diniz, K. J. Larsen, and I. T. Monroy, “Experimental investigation and digital compensation of dgd for 112 gb/s pdm-qpsk clock recovery,” Opt. Express19, 429–437 (2011). [CrossRef]
- H. Meyr, M. Moeneclaey, and S. Fechtel, Digital Communication Receivers / Synchronization, Channel Estimation, and Signal Processing (Wiley, 1998).
- J. Kurzweil, An Introduction to Digital Communications (John Wiley, 2000).
- C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006).
- A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. Roy. Stat Soc. Series B.39, 1–38 (1977).
- N. G. Gonzalez, D. Zibar, A. Caballero, and I. T. Monroy, “Experimental 2.5-gb/s qpsk wdm phasemodulated radio-over-fiber link with digital demodulation by a k-means algorithm,” IEEE Photon. Technol. Lett.22, 335–337 (2010). [CrossRef]
- A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol.30, 1524–1539 (2012). [CrossRef]
- F. Vacondio, O. Rival, C. Simonneau, E. Grellier, L. Lorcy, J.-C. Antona, S. Bigo, and A. Bononi, “On nonlinear distortions of highly dispersive optical coherent systems,” Opt. Express20, 1022–1032 (2012). [CrossRef] [PubMed]
- A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. of OFC, paper OWO7, Los Angeles, California, USA, (2011).
- E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proc. of OFC, paper OThF4, Los Angeles, California, USA, (2011).
- S. Makovejs, D. S. Millar, D. Lavery, C. Behrens, R. I. Killey, S. J. Savory, and P. Bayvel, “Characterization of long-haul 112gbit/s pdm-qam-16 transmission with and without digital nonlinearity compensation,” Opt. Express18, 12939–12947 (2010). [CrossRef] [PubMed]

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