## High-order statistical equalizer for nonlinearity compensation in dispersion-managed coherent optical communications |

Optics Express, Vol. 20, Issue 14, pp. 15769-15780 (2012)

http://dx.doi.org/10.1364/OE.20.015769

Acrobat PDF (1290 KB)

### Abstract

Fiber nonlinearity has become a major limiting factor to realize ultra-high-speed optical communications. We propose a fractionally-spaced equalizer which exploits a trained high-order statistics to deal with data-pattern dependent nonlinear impairments in fiber-optic communications. The computer simulation reveals that the proposed 3-tap equalizer improves Q-factor by more than 2 dB for long-haul transmissions of 5,230 km distance and 40 Gbps data rate. We also demonstrate that the joint use of a digital backpropagation (DBP) and the proposed equalizer offers an additional 1–2 dB performance improvement due to the channel shortening gain. A performance in high-speed transmissions of 100 Gbps and beyond is evaluated as well.

© 2012 OSA

## 1. Introduction

2. A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. **28**, 423–433 (2010). [CrossRef]

3. 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**, 880–888 (2008). [CrossRef] [PubMed]

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

8. F. P. Guiomar, J. D. Reis, A. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” in Proceedings of ECOC’11, Tu.6.B.1 (2011). [PubMed]

9. N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express **13**, 4568–4579 (2005). [CrossRef]

12. J. B. Anderson and S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. **32**, 169–176 (1984). [CrossRef]

## 2. Nonlinear equalizer

*s*at the time instance

_{k}*k*is transmitted through the nonlinear fiber by using the coherent optical transceiver. The received data may be first processed by several pre-equalization units, including timing recovery, FDE for chromatic dispersion compensation, PMD compensation, DBP for nonlinear compensation, and so on. The pre-processed signal is fed into a shift register which accepts fractionally-spaced (or, oversampled) data. Those signals are also analyzed to obtain the statistics of the fiber nonlinearity. The oversampled data is then equalized by a maximum-likelihood sequence estimation (MLSE) detector which employs the Viterbi algorithm based on the fiber statistics.

### 2.1. Statistical sequence equalizer (SSE)

*k*-th received signal 𝔼[

*r*] conditioned on the consecutive 3-symbol data pattern

_{k}**s**

*= [*

_{k}*s*

_{k}_{−1},

*s*,

_{k}*s*

_{k}_{+1}], where 𝔼[·] denotes the expectation. There are 64 points in the figure since the total number of the different data patterns is 4

^{3}= 64 from

**s**

*= [0, 0, 0] to [3*

_{k}3. 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**, 880–888 (2008). [CrossRef] [PubMed]

3. 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**, 880–888 (2008). [CrossRef] [PubMed]

**16**, 880–888 (2008). [CrossRef] [PubMed]

### 2.2. High-order statistics

*etc.*) in addition to the first-order statistics (mean) to mitigate residual nonlinear noise as well. As shown in Fig. 2, the residual (out-of-memory) nonlinear distortion performs as an effective noise which also depends on the data pattern; specifically, the distribution of the residual distortion around the region “R1” is different from that of the region “R3.” More importantly, the distribution is not circularly symmetric (i.e., ellipsoidal rather than circular).

*N*receiving samples centered around the target transmission data to establish an empirical statistics. Let

**r**

*= [*

_{k}*r*

_{k}_{−⌈(}

_{N}_{−1)/2⌉},...,

*r*,...,

_{k}*r*

_{k+⌈N/2⌉}]

^{T}∈ ℂ

*denote the received signal sequence in the shift register of window size*

^{N}*N*, where ⌈·⌉, [·]

^{T}, and ℂ denote a floor function, a transpose operation, and a complex-number set, respectively. A statistics analyzer obtains the empirical mean vector and the covariance matrix of the received signals for each transmission data pattern

**s**= [

*s*

_{k−⌈(M−1)/2⌉},...,

*s*,...,

_{k}*s*

_{k+⌈M/2⌉}] ∈ ℕ

*, where ℕ denotes the natural number set (positive integers).*

^{M}**s**be one of the 4

*possible data patterns (for 4-ary data), the empirical mean vectors*

^{M}**(**

*μ***s**) ∈ ℝ

^{2N}and the covariance matrices

**Σ**(

**s**) ∈ ℝ

^{2N×2N}are expressed as follows: where we define

**s**) is the total number of occurrences that the data pattern

**s**appeared in the past. The notations ℝ, ℜ[·] and ℑ[·] are a real-number set, the element-wise real part and imaginary part operations, respectively. The main reason to expand the complex-valued vector

**r**

*to a double-size real-valued vector*

_{j}**r**′

*is to model the circular asymmetry illustrated in Fig. 2.*

_{j}### 2.3. Statistics updating

**(**

*μ***s**) ←

*ν*

**(**

*μ***s**) +

**r**′

*and*

_{j}**Σ**(

**s**) ←

*ν*

**Σ**(

**s**) + (

**r**′

*−*

_{j}**(**

*μ***s**)) (

**r**′

*−*

_{j}**(**

*μ***s**)

^{T}with an appropriate normalization of (1 −

*ν*) where 0 <

*ν*< 1 is referred to as a forgetting factor. Note that the determinant and the inverse of the covariance matrix, those of which are required for the likelihood calculations as in Eq. (4), can be efficiently updated by the Sherman-Morrison formula [14] as follows: where we define It reduces the computational complexity from the cubic order 𝒪[(2

*N*)

^{3}] to the square order 𝒪[(2

*N*)

^{2}] for the matrix inversion, where 𝒪[·] denotes the complexity order.

### 2.4. Excess window size

*M*for the transmission data

**s**

*should be optimized to deal with the memory length of the fiber channel. However, the total number of possible patterns increases exponentially with the window size*

_{k}*M*. Hence, we may need to use a restricted window size in practice, for example

*M*= 3 taps. On the other hand, the computational complexity just increases linearly with the window size

*N*for the receiving data

**r**

*. We propose to use an excess window size, where we can use*

_{k}*N*>

*M*to enhance the performance. Doing so, we can keep the computational complexity low while a longer channel memory is considered with its cross correlation in

**Σ**(

**s**).

### 2.5. Fractionally-spaced processing

*P*be an oversampling factor. The received signal sequence is stored as

**r**

*= [*

_{k}*r*

_{kP}_{−⌈(}

_{N}_{−1)/2⌉},...,

*r*,...,

_{kP}*r*

_{kP+⌈N/2⌉}]

^{T}∈ ℂ

*in the*

^{N}*N*-sample shift register. The fractionally-spaced statistical sequence equalizer (FS-SSE) has an advantage especially when the transceiver filters have an inter-symbol interference due to the non-ideal Nyquist filtering. In addition, the symbol timing error is absorbed by oversampling.

### 2.6. Cascading with DBP

*M*due to the complexity issue in practice, an FDE to compensate the chromatic dispersion is useful as a pre-processing unit to shorten the effective channel memory. To shorten the channel memory more effectively, we can adopt some nonlinear compensation techniques including the DBP [3

**16**, 880–888 (2008). [CrossRef] [PubMed]

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

## 3. Performance evaluations

### 3.1. Fiber plant configuration

6. T. Yoshida, T. Sugihara, H. Goto, T. Tokura, K. Ishida, and T. Mizuochi, “A study on statistical equalization of intra-channel fiber nonlinearity for digital coherent optical systems,” in Proceedings of ECOC’11, Tu.3.A.1 (2011). [PubMed]

*P*-times oversampled signal is fed into the statistical sequence equalizer. The fiber distance per loop is 1,046 km. The Q-factor is calculated by bit error counting. We assume no PMD in simulations, and use a circular polarization basis so that two parallel equalizers for each polarization work individually. If it suffers from a strong PMD, we may need a standard polarization recovery such as constant modulus algorithms in a pre-processor block, or we can use the proposed scheme for joint polarization equalization with a complexity cost.

### 3.2. Q versus launching power

*M*= 3 taps,

*N*= 9 excess window, and

*P*= 2 oversampling. “DBP” denotes the DBP using manually optimized SSFM parameters. “Conv. 3Tap” denotes the conventional 3-tap statistical equalizer [10] without using the second-order statistics, excess window, and oversampling.

### 3.3. Q versus fiber distance

### 3.4. Effect of tap length, modulation scheme, and reduced-complexity equalizer

*T*-spaced equalizers not

*T*/2-spaced equalizers (

*T*denotes the symbol time duration). It is seen that 2-tap equalizers can achieve more than 1 dBQ improvement compared to 1-tap equalizers. Whereas, 3-tap and 4-tap lengths offer moderate improvements; at most an additional 0.5 dBQ improvement respectively. It is because an effective channel memory is approximately 3-tap length.

12. J. B. Anderson and S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. **32**, 169–176 (1984). [CrossRef]

### 3.5. 112 Gbps transmissions and beyond

*T*-spaced SSE cascaded with DBP. It is seen that FS-SSE has only a slight advantage against

*T*-spaced SSE because there is no timing jitters in the simulation.

### 3.6. Computational complexity

*N*,

*P*, and ℳ be the size of fast Fourier transform, the oversampling factor, and the number of steps, respectively. The computational complexity per symbol is in the order of

*P*ℳ(log

_{2}(

*N*) + 16). In our simulations, we used

*N*= 256,

*P*= 2, and ℳ = 105. Hence, the computational complexity of (1 step per span) DBP becomes 5040 multiplication operations. When using reduced-step DBP of ℳ = 5, the computational complexity of DBP becomes 240.

*N*ℳ2

*(2*

^{q}*N*+ 1), where ℳ ≤ 2

^{q(M−1)}is the number of survivors in M-algorithm,

*N*is the window size,

*M*is the tap length, and

*q*is the number of bits per symbol. For

*M*= 3 taps,

*q*= 2 (QPSK), and

*N*= 3 (without excess window), the computational complexity of the (full-state) SSE becomes 2688 multiplication operations. Therefore, the proposed SSE has lower complexity than 1-step per span DBP. When using M-algorithm of ℳ = 2, the reduced-state SSF has a complexity of 336. For 16QAM cases, the computational complexity of the full-state SSE becomes seriously large, more specifically, 172032 multiplications. However, as shown in the results, a reduced-state SSE with ℳ = 2 approaches the full-state SSE. With M-algorithm, the computational complexity of such a reduced-state SSE becomes 1344 even for 16QAM 3-tap cases.

## 4. Summary

## Acknowledgments

*λ*Reach Project.”

## References and links

1. | J. Renaudier, G. Charlet, P. Tran, M. Salsi, and S. Bigo, “A performance comparison of differential and coherent detections over ultra long haul transmission of 10Gb/s BPSK,” in Proceedings of OFC’07, OWM1 (2007). |

2. | A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol. |

3. | 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 |

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

5. | E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff in fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proceedings of OFC’11, OThF4 (2011). |

6. | T. Yoshida, T. Sugihara, H. Goto, T. Tokura, K. Ishida, and T. Mizuochi, “A study on statistical equalization of intra-channel fiber nonlinearity for digital coherent optical systems,” in Proceedings of ECOC’11, Tu.3.A.1 (2011). [PubMed] |

7. | W. Yan, Z. Tao, L. Dou, L. Li, S. Oda, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “Low complexity digital perturbation back-propagation,” in Proceedings of ECOC’11, Tu.3.A.2 (2011). [PubMed] |

8. | F. P. Guiomar, J. D. Reis, A. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” in Proceedings of ECOC’11, Tu.6.B.1 (2011). [PubMed] |

9. | N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express |

10. | Y. Cai, D. G. Foursa, C. R. Davidson, J. X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” in Proceedings of OFC’10, OTuE1 (2010). |

11. | T. Koike-Akino, C. Duan, K. Kojima, K. Parsons, T. Yoshida, T. Sugihara, and T. Mizuochi, “Fractionally-spaced statistical equalizer for fiber nonlinearity mitigation in digital coherent optical systems,” in Proceedings of OFC’12 OTh3C.3 (2012). |

12. | J. B. Anderson and S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun. |

13. | A. Azzalini and A. Capitanio, “Statistical applications of the multivariate skew normal distribution,” J. R. Stat. Soc. |

14. | G. H. Golub and C. F. Van Loan, |

15. | C. Duan, K. Parsons, T. Koike-Akino, R. Annavajjala, K. Kojima, T. Yoshida, T. Sugihara, and T. Mizuochi, “A low-complexity sliding-window turbo equalizer for nonlinearity compensation,” in Proceedings of OFC’12, JW2A.59 (2012). |

16. | I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun. |

17. | H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Iterative polar quantization based modulation to achieve channel capacity in ultra-high-speed optical communication systems,” IEEE Photon. J. |

**OCIS Codes**

(060.1660) Fiber optics and optical communications : Coherent communications

(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: April 12, 2012

Revised Manuscript: June 6, 2012

Manuscript Accepted: June 7, 2012

Published: June 27, 2012

**Citation**

Toshiaki Koike-Akino, Chunjie Duan, Kieran Parsons, Keisuke Kojima, Tsuyoshi Yoshida, Takashi Sugihara, and Takashi Mizuochi, "High-order statistical equalizer for nonlinearity compensation in dispersion-managed coherent optical communications," Opt. Express **20**, 15769-15780 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-15769

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### References

- J. Renaudier, G. Charlet, P. Tran, M. Salsi, and S. Bigo, “A performance comparison of differential and coherent detections over ultra long haul transmission of 10Gb/s BPSK,” in Proceedings of OFC’07, OWM1 (2007).
- A. D. Ellis, J. Zhao, and D. Cotter, “Approaching the non-linear Shannon limit,” J. Lightwave Technol.28, 423–433 (2010). [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, 880–888 (2008). [CrossRef] [PubMed]
- E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol.26, 3416–3425 (2008). [CrossRef]
- E. Ip, N. Bai, and T. Wang, “Complexity versus performance tradeoff in fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation,” in Proceedings of OFC’11, OThF4 (2011).
- T. Yoshida, T. Sugihara, H. Goto, T. Tokura, K. Ishida, and T. Mizuochi, “A study on statistical equalization of intra-channel fiber nonlinearity for digital coherent optical systems,” in Proceedings of ECOC’11, Tu.3.A.1 (2011). [PubMed]
- W. Yan, Z. Tao, L. Dou, L. Li, S. Oda, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “Low complexity digital perturbation back-propagation,” in Proceedings of ECOC’11, Tu.3.A.2 (2011). [PubMed]
- F. P. Guiomar, J. D. Reis, A. Teixeira, and A. N. Pinto, “Mitigation of intra-channel nonlinearities using a frequency-domain Volterra series equalizer,” in Proceedings of ECOC’11, Tu.6.B.1 (2011). [PubMed]
- N. Alić, G. C. Papen, R. E. Saperstein, L. B. Milstein, and Y. Fainman, “Signal statistics and maximum likelihood sequence estimation in intensity modulated fiber optic links containing a single optical preamplifier,” Opt. Express13, 4568–4579 (2005). [CrossRef]
- Y. Cai, D. G. Foursa, C. R. Davidson, J. X. Cai, O. Sinkin, M. Nissov, and A. Pilipetskii, “Experimental demonstration of coherent MAP detection for nonlinearity mitigation in long-haul transmissions,” in Proceedings of OFC’10, OTuE1 (2010).
- T. Koike-Akino, C. Duan, K. Kojima, K. Parsons, T. Yoshida, T. Sugihara, and T. Mizuochi, “Fractionally-spaced statistical equalizer for fiber nonlinearity mitigation in digital coherent optical systems,” in Proceedings of OFC’12 OTh3C.3 (2012).
- J. B. Anderson and S. Mohan, “Sequential coding algorithms: a survey and cost analysis,” IEEE Trans. Commun.32, 169–176 (1984). [CrossRef]
- A. Azzalini and A. Capitanio, “Statistical applications of the multivariate skew normal distribution,” J. R. Stat. Soc.61, 579–602 (1999). [CrossRef]
- G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).
- C. Duan, K. Parsons, T. Koike-Akino, R. Annavajjala, K. Kojima, T. Yoshida, T. Sugihara, and T. Mizuochi, “A low-complexity sliding-window turbo equalizer for nonlinearity compensation,” in Proceedings of OFC’12, JW2A.59 (2012).
- I. B. Djordjevic, L. L. Minkov, and H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun.26, 73–83 (2008). [CrossRef]
- H. G. Batshon, I. B. Djordjevic, L. Xu, and T. Wang, “Iterative polar quantization based modulation to achieve channel capacity in ultra-high-speed optical communication systems,” IEEE Photon. J.2, 593–599 (2010). [CrossRef]

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