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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 11 — Jun. 3, 2013
  • pp: 13607–13616
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Advanced perturbation technique for digital backward propagation in WDM systems

Lian Xiang, Paul Harper, and Xiaoping Zhang  »View Author Affiliations


Optics Express, Vol. 21, Issue 11, pp. 13607-13616 (2013)
http://dx.doi.org/10.1364/OE.21.013607


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Abstract

An improved digital backward propagation (DBP) is proposed to compensate inter-nonlinear effects and dispersion jointly in WDM systems based on an advanced perturbation technique (APT). A non-iterative weighted concept is presented to replace the iterative in analytical recursion expression, which can dramatically simplify the complexity and improve accuracy compared to the traditional perturbation technique (TPT). Furthermore, an analytical recursion expression of the output after backward propagation is obtained initially. Numerical simulations are executed for various parameters of the transmission system. The results indicate that the advanced perturbation technique will relax the step size requirements and reduce the oversampling factor when launch power is higher than −2 dBm. We estimate this technique will reduce computational complexity by a factor of around seven with respect to the conventional DBP.

© 2013 OSA

1. Introduction

In long-haul, high-speed wavelength-division-multiplexed (WDM) optical fiber system non-constructive effects of fiber nonlinearity can significantly degrade signal quality [1

1. E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fibre, New Developments (In-Tech, to be published).

]. Therefore, mitigating or compensating these impairments becomes crucial to increasing capacity for optical communication [2

2. 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]

]. Many researches have been reported to investigate nonlinearity effect on fiber capacity and indicated that nonlinearity will degrade the capacity obviously [3

3. K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett. 91(20), 203901 (2003). [CrossRef] [PubMed]

]. Recently, due to the fast development of digital coherent receiver technology, digital compensation methods have attracted significant attention to mitigate linear and nonlinear impairment effectively as it’s flexible and less costly [4

4. T. Pfau, S. Hoffmann, O. Adamczyk, R. Peveling, V. Herath, M. Porrmann, and R. Noé, “Coherent optical communication: towards realtime systems at 40 Gbit/s and beyond,” Opt. Express 16(2), 866–872 (2008). [CrossRef] [PubMed]

,5

5. E. Ip, A. P. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent Detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008). [CrossRef] [PubMed]

]. Many digital compensation techniques for different impairments have been already presented [6

6. 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). [CrossRef] [PubMed]

8

8. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef] [PubMed]

]. Among these techniques, the digital backward propagation (DBP) method has proved to be quite promising for jointly compensating linear and nonlinear impairments [9

9. R. Asif, C. Y. Lin, and B. Schmauss, Digital Backward Propagation: A Technique to Compensate Fiber Dispersion and Nonlinear Impairments (InTech-Book Publisher 2011).

]. This method is based on solving the nonlinear Schrödinger equation (NLSE) in the backward direction starting with the received signal as the input and producing the signal at the transmitter as its output [10

10. E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]

]. But as the high complexity of the NLSE when fiber loss, dispersion and nonlinearity play a crucial role in WDM systems simultaneously, one of the technically challenging for DBP is solving the NLSE effectively in a trade-off between accuracy and computational load [11

11. E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt. 48(25), F6–F10 (2009). [CrossRef] [PubMed]

]. Xiaoxu proposed a universal post-compensation scheme for fiber impairments in WDM systems using split-step method (SSM) for DBP [12

12. 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). [CrossRef] [PubMed]

]. Then, SSM as the typically method for conventional DBP (C-DBP) has been applied in many different systems [13

13. E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed Transmission,” J. Lightwave Technol. 28(6), 939–951 (2010). [CrossRef]

16

16. D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

] and was demonstrated in experiment as well [17

17. D. Rafique, J. Zhao, and A. D. Ellis, “Digital back-propagation for spectrally efficient WDM 112 Gbit/s PM m-ary QAM transmission,” Opt. Express 19(6), 5219–5224 (2011). [CrossRef] [PubMed]

,18

18. G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of Fiber Impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008). [CrossRef]

]. However, this numerical algorithm required a number of iterative processing steps to achieve acceptable accuracy, which is quite a high computational load, thus making it difficult to implement in real-time [19

19. S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010). [CrossRef]

]. Therefore, based on SSM a number of enhanced methods have been reported [20

20. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

22

22. L. Lin, Z. Tao, L. Dou, W. Yan, S. Oda, T. Tanimura, T. Hoshida, and J. Rasmussen, “Implementation efficient non-linear equalizer based on correlated digital back-propagation,” in Proc. OFC (2011).

]. According to the method proposed by Liebrich et al. [23

23. J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003). [CrossRef]

], Eduardo F. Mateo et al. derived an advanced SSM for DBP in WDM systems, which consisted in the factorization of the walk-off effect within the nonlinear step, and then applied it to the polarization-multiplexed WDM systems [24

24. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

,25

25. E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express 19(2), 570–583 (2011). [CrossRef] [PubMed]

]. It was estimated in that works that such advanced SSM relaxes the step size requirements resulting in a factor of 4 reductions in computational load. However, the accuracy of these numerical methods mentioned previously can be accepted only when the step size h is set within a thin range to keep a small degree of nonlinear phase-shift [26

26. F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009). [CrossRef]

]. But, as a typical value of h is equal to span length in backward propagation [27

27. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

], a small nonlinear phase-shift is hardly to maintain along each distanceh, so that an unsatisfied accuracy is inevitable especially for a largeh. Then some other algorithms distinguished from SSM are proposed for backward propagation [28

28. D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]

30

30. L. Zhu and G. Li, “Nonlinearity compensation using dispersion-folded digital backward propagation,” Opt. Express 20(13), 14362–14370 (2012). [CrossRef] [PubMed]

]. Based on perturbation analysis, a version of digital back-propagation was proposed to compensate intra-channel nonlinearity in polarization-division multiplexed WDM systems [31

31. 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 Proc. ECOC (2011).

,32

32. T. Hoshida, L. Dou, T. Tanimura, W. Yan, S. Oda, L. Li, H. Nakashima, M. Yan, Z. Tao, and J. C. Rasmussen, “Digital nonlinear compensation techniques for high-speed DWDM transmission systems,” in Proc.ECOC (2012). [CrossRef]

]. In their work, the inter-channel nonlinear effects which should be the major part of nonlinear effects in WDM system were neglected, meanwhile, the combined nonlinear distortion caused by Kerr nonlinearity and dispersion was also not considered. Therefore, this result is limited in application to the actual WDM systems.

In this paper, an advanced perturbation technique (APT) is developed and basing on this technique, an improved digital backward propagation (DBP) is proposed to compensate inter-nonlinear effects and dispersion jointly in WDM systems. In this advanced perturbation technique, a non-iterative weighted concept is presented to replace the iterative in the analytical recursion expression, which can dramatically simplify the complexity and improve accuracy compared to the traditional perturbation technique (TPT). Furthermore, an analytical recursion expression of the output after backward propagation is obtained initially, which the inter-channel walk-off effect and the combined nonlinear distortion caused by Kerr nonlinearity and dispersion can be consisted in. Comparing to C-DBP, an expression for the total number of required multiplications per sample per channel is given and a rigorous analysis of the computational cost is carried out. Numerical simulations are performed in the corresponding transmission system with various parameters. Our research indicates that APT is more accurate than C-DBP for nonlinearity compensation when launch power is higher than −2 dBm and step size is larger than 20 km, especially about 2.4 dB benefits than C-DBP at 3 dBm with one step per span, which will allow larger step size for equivalent performance. Meanwhile, APT requires a lower sampling rate when launch power is higher than −2 dBm. For a transmission system with a weaker nonlinear, the Q-factor of APT appears to be a larger peak value. We estimate that there is a reduction in computational complexity by a factor of around seven.

2. Digital backward propagation for WDM systems using advanced perturbation Technique

E^kout(ω,z)=E^k(0)exp(MD^)+iγh[exp(MD^)NLk(0)(ω,εh)+exp((M1)D^)NLk(1)(ω,εh)++exp(D^)NLk(M1)(ω,εh)].
(10)

The schematic for implementation of the develop perturbation technique in the lth section is shown in Fig. 1
Fig. 1 Block diagram for the implementation of the lth section using the develop perturbation technique.
where for simplicity,Hd=exp[D^(ω,h)], Hd,ε=exp[D^(ω,εh)], Hd,ε=exp[D^(ω,εh)] and the dispersive walk-off is not included.

3. Analysis of computational complexity

It is significant to investigate the computation requirements for APT and C-DBP where conventional symmetric SSM is employed. For simplicity only the number of complex multiplications will be considered, neglecting the number of additions. Furthermore, considerations regarding the numeric representation (fixed point/floating point) will be ignored. By following the schematic diagram in Fig. 1, the total number of required multiplications per sample per channel for each method is given by:
NmulDPT=LhDPT[2(s+p)log2(s+p)+3(s+p)+s]/s,
(11)
NmulCDBP=LhCDBP[2(s+p)log2(s+p)+2(s+p)+8s]/s,
(12)
where S is the number of samples per channel, L is the total transmission distance, h is the corresponding step size and p is the overhead samples for each filter operation when the filter implementation in the frequency domain is done by the overlap-and-add method and can be obtained by [24

24. E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

]: p=2π|β2|BhR, whereRis sampling rate and Bis bandwidth. The number of multiplications for each operation involved in backward propagation is calculated as follows. The filter requires 2(s+p)log2(s+p)+(s+p) multiplications; intensity operator requires smultiplications; exponential operator (4thorder Taylor expansion) requires 6smultiplications and the further details of the approach to calculate the number of multiplication can be found in [25

25. E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express 19(2), 570–583 (2011). [CrossRef] [PubMed]

]. As s is much larger than p normally (s >> p), when the step size of both methods is assumed to equal span length, it is obvious that about 6N more multiplications per sample per channel will be required for C-DBP than APT.

4. Numerical simulation results and discussion

Figure 3
Fig. 3 (a) Q-factors as a function of step size for APT, C-DBP and TPT, (b) Q-factors versus a long fiber transmission distance for APT, C-DBP, TPT and only chromatic dispersion compensation with different step size.
shows the Q-factors as functions of step size (SS) and a long fiber transmission distance for C-DBP, APT and TPT implementations with launch power at −2 dBm, which is optimum launch power for APT when γ = 1.46 (W.km)−1. Agreeing with theoretical analysis in section 2, the Q-factor of TPT declines very fast with raising step size as shown in Fig. 3(a). Meanwhile, as shown in Fig. 3(b), when step size of both TPT and APT are set to 8 km in a long fiber transmission distance, the Q-factor of TPT is far less than APT and even less than APT with step size 80 km, which means TPT is unsuitable for application to DBP. Figure 3(a) shows that since the performance of C-DBP will be improved with a shorter step size, the Q-factors of both C-DBP and APT are almost the same when step size is sufficiently short (<20 km). But, with the step size increasing, the Q-factor of C-DBP decays faster than APT. When step size is greater than 20 km, APT performs significantly better than C-DBP. Particularly, APT produced an improvement about 0.85 dB than C-DBP for one step per span, which is a typical value of step size in practice. It indicates that APT will reduce the number of steps for similar performance. Therefore, the majority of the benefit of APT is obtained with larger step. Figure 3(b) shows an efficient compensation of fiber inter-nonlinear and dispersion impairments using ATP. Comparing to chromatic dispersion (CD) compensation, a large benefit about 6.7 dB for SS = 8 km and 5.2 dB for SS = 80 km with 1600 km transmission distance is provided by APT from inter-nonlinear compensation.

To show the benefit of APT in a transmission system, the relationship between Q-factor and launch power with various fiber nonlinear coefficients γ for linear equalization and nonlinear equalization using APT and C-DBP is the most interesting issue, which is shown in Fig. 4
Fig. 4 Q-factors versus launch power per channel for both methods for γ = 1.46 and 3.5 (W.km)−1
. The step sizes of both methods are located at one step per span. At low powers, as the nonlinear effects are weak, the system behaves as a linear system. Due to a good linearity compensation for both of APT and C-DBP, no improvement is produced by nonlinearity compensation and for γ = 0 (W.km)−1 both methods give the same performance.

Figure 4 show that APT gave better performance than C-DBP. For both methods there was <1 dB penalty compared to the linear case (γ = 0) for launch powers lower than −6 dBm. The penalty increases rapidly at higher launch powers with the onset of the increase being dependent on the value of γ and the method used. The optimum launch power was approximately 1 dBm higher for APT than C-DBP and that the performance of APT was approximately 0.5-0.7 dB higher than C-DBP at their respective optimum powers. For the same launch power the benefit of APT increases with launch power and that the nonlinear tolerance of APT is significantly better than C-DBP for launch power greater than −5 dBm for γ = 3.5 (W.km)−1 and −2 dBm for γ = 1.46 (W.km)−1, especially, APT produced a benefit about 2.4 dB at 3 dBm for γ = 1.46 (W.km)−1 and 2.7 dB at −1 dBm for γ = 3.5 (W.km)−1. This suggests that APT is a more accurate method for nonlinearity compensation than C-DBP, especially for higher launch power. The nonlinear coefficient 3.5 (W.km)−1 does not represent any practical interest.

Due to the significant influence of oversampling rate for computational load, it is necessary to display the performance of APT and C-DBP with different oversampling factors, as shown in Fig. 5
Fig. 5 (a) Q-factors versus launch power per channel for APT and C-DBP with oversampling factors of 2 and 4, (b) Q penalty for reduced oversampling factor for APT and C-DBP and (c) Q factor difference between APT and C-DBP for oversampling factors of 2 and 4.
. One step per span is used as the step size in this comparison. The results in Fig. 5(a) show the Q factor for APT and C-DBP for oversampling factors of 2 and 4. By comparing the Q value at the optimal launch powers in Fig. 5(a) it is clear that the oversampling factor can be reduced by using APT with only a small Q penalty. When launch power is −10 dBm, there is about 3.2 dB benefit of oversampling factor of 4 than 2. This is because a higher oversampling rate provides each symbol carrying more information to get a better distinction between signal and noise levels in backward propagation. Therefore larger Q-factors were obtained with a higher oversampling factor for both methods. The maximum Q value for C-DBP with an oversampling factor of 4 was 13.8dB at a launch power of −4dBm which is only 0.3dB higher than the 13.5dB optimal Q value of APT with an oversampling factor of 2 at −2dBm launch power. This factor of two reduction in oversampling factor would significantly relax the specification of analogue to digital converter hardware in the receiver as well as being more computationally efficient.

Further analysis of these results as shown in Fig. 5(b) indicates that when the same oversampling factor is used for both methods and the launch power is low (below −4 dBm) there is less than 0.5 dB difference in the Q factors. This is as expected as the transmission at these launch powers is largely unaffected by nonlinear impairments. However as launch power is increased and nonlinear impairments increase, APT performs better than C-DBP. The performance difference is similar for both oversampling factors used and the results show that the difference in Q factors increases approximately linearly with a slope of ~0.4 dB/dBm over the range of input powers from −2 dBm to 3 dBm.

Further studies about the performance of APT and C-DBP with different number of WDM channels have been done at −2 dBm. Step sizes of both methods are located at one step per span. The results indicate that APT give better performance than C-DBP with various number of WDM channels and the benefit of APT increases with number of WDM channels, especially, APT produced a benefit about 1.2 dB for 8 channels and 2.5 dB for 24 channels.

5. Conclusion

We propose an advanced perturbation technique (APT) for digital backward propagation in WDM systems using the coupled nonlinear Schrodinger equations for the compensation of inter-channel nonlinearities. An analytical expression of the output after backward propagation is obtained initially, which could be extended to include the inter-channel walk-off effect. Computer simulations have been carried out comparing the proposed technique with conventional digital back-propagation (C-DBP) for various simulation parameters. Our research indicates that this advanced perturbation technique can reduce computational load significantly and is more accurate than C-DBP for nonlinearity compensation when launch power is higher than −2 dBm and step size is larger than 20 km, which will allow larger step size for equivalent performance. Meanwhile, APT requires a lower sampling rate when launch power is higher than −2 dBm. Furthermore, our technique also has the potential to ease requirements of receiver hardware components which would be used in a practical implementation. We estimate a reduction by a factor of around seven in computational load with respect to the C-DBP technique. This computational efficiency improvement and potential reduced hardware performance requirements of the advanced perturbation technique reported here a step towards making nonlinear compensation based on back propagation implementable in real-time.

Acknowledgments

The authors acknowledge support from the UK EPSRC Programme Grant UNLOC (Unlocking the capacity of optical communications) EP/J017582/1, the European Research Council, and the Ministry of Education and Science of the Russian Federation and “Chun Hui” plan of Ministry of Education of China (2012).

References and links

1.

E. Ip and J. M. Kahn, “Nonlinear impairment compensation using backpropagation,” in Optical Fibre, New Developments (In-Tech, to be published).

2.

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]

3.

K. S. Turitsyn, S. A. Derevyanko, I. V. Yurkevich, and S. K. Turitsyn, “Information capacity of optical fiber channels with zero average dispersion,” Phys. Rev. Lett. 91(20), 203901 (2003). [CrossRef] [PubMed]

4.

T. Pfau, S. Hoffmann, O. Adamczyk, R. Peveling, V. Herath, M. Porrmann, and R. Noé, “Coherent optical communication: towards realtime systems at 40 Gbit/s and beyond,” Opt. Express 16(2), 866–872 (2008). [CrossRef] [PubMed]

5.

E. Ip, A. P. Lau, D. J. F. Barros, and J. M. Kahn, “Coherent Detection in optical fiber systems,” Opt. Express 16(2), 753–791 (2008). [CrossRef] [PubMed]

6.

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). [CrossRef] [PubMed]

7.

R. Waegemans, S. Herbst, L. Holbein, P. Watts, P. Bayvel, C. Fürst, and R. I. Killey, “10.7 Gb/s electronic predistortion transmitter using commercial FPGAs and D/A converters implementing real-time DSP for chromatic dispersion and SPM compensation,” Opt. Express 17(10), 8630–8640 (2009). [CrossRef] [PubMed]

8.

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express 16(2), 804–817 (2008). [CrossRef] [PubMed]

9.

R. Asif, C. Y. Lin, and B. Schmauss, Digital Backward Propagation: A Technique to Compensate Fiber Dispersion and Nonlinear Impairments (InTech-Book Publisher 2011).

10.

E. Mateo, L. Zhu, and G. Li, “Impact of XPM and FWM on the digital implementation of impairment compensation for WDM transmission using backward propagation,” Opt. Express 16(20), 16124–16137 (2008). [CrossRef] [PubMed]

11.

E. F. Mateo and G. Li, “Compensation of interchannel nonlinearities using enhanced coupled equations for digital backward propagation,” Appl. Opt. 48(25), F6–F10 (2009). [CrossRef] [PubMed]

12.

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). [CrossRef] [PubMed]

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E. Ip, “Nonlinear compensation using backpropagation for polarization-multiplexed Transmission,” J. Lightwave Technol. 28(6), 939–951 (2010). [CrossRef]

14.

D. S. Millar, S. Makovejs, C. Behrens, S. Hellerbrand, R. I. Killey, P. Bayvel, and S. J. Savory, “Mitigation of fiber nonlinearity using a digital coherent receiver,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1217–1226 (2010). [CrossRef]

15.

R. Asif, C. Y. Lin, M. Holtmannspoetter, and B. Schmauss, “Optimized digital backward propagation for phase modulated signals in mixed-optical fiber transmission link,” Opt. Express 18(22), 22796–22807 (2010). [CrossRef] [PubMed]

16.

D. Rafique and A. D. Ellis, “Impact of signal-ASE four-wave mixing on the effectiveness of digital back-propagation in 112 Gb/s PM-QPSK systems,” Opt. Express 19(4), 3449–3454 (2011). [CrossRef] [PubMed]

17.

D. Rafique, J. Zhao, and A. D. Ellis, “Digital back-propagation for spectrally efficient WDM 112 Gbit/s PM m-ary QAM transmission,” Opt. Express 19(6), 5219–5224 (2011). [CrossRef] [PubMed]

18.

G. Goldfarb, M. G. Taylor, and G. Li, “Experimental demonstration of Fiber Impairment compensation using the split-step finite-impulse-response filtering method,” IEEE Photon. Technol. Lett. 20(22), 1887–1889 (2008). [CrossRef]

19.

S. J. Savory, G. Gavioli, E. Torrengo, and P. Poggiolini, “Impact of interchannel nonlinearities on a split-step intrachannel nonlinear equalizer,” IEEE Photon. Technol. Lett. 22(10), 673–675 (2010). [CrossRef]

20.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 2(5), 816–832 (2010). [CrossRef]

21.

B. Schmauss, R. Asif, and C.-Y. Lin, “Recent advances in digital backward propagation algorithm for coherent transmission systems with higher order modulation formats,” in Proc. SPIE (2012)

22.

L. Lin, Z. Tao, L. Dou, W. Yan, S. Oda, T. Tanimura, T. Hoshida, and J. Rasmussen, “Implementation efficient non-linear equalizer based on correlated digital back-propagation,” in Proc. OFC (2011).

23.

J. Leibrich and W. Rosenkranz, “Efficient numerical simulation of multichannel WDM transmission systems limited by XPM,” IEEE Photon. Technol. Lett. 15(3), 395–397 (2003). [CrossRef]

24.

E. F. Mateo, F. Yaman, and G. Li, “Efficient compensation of inter-channel nonlinear effects via digital backward propagation in WDM optical transmission,” Opt. Express 18(14), 15144–15154 (2010). [CrossRef] [PubMed]

25.

E. F. Mateo, X. Zhou, and G. Li, “Improved digital backward propagation for the compensation of inter-channel nonlinear effects in polarization-multiplexed WDM systems,” Opt. Express 19(2), 570–583 (2011). [CrossRef] [PubMed]

26.

F. Yaman and G. Li, “Nonlinear impairment compensation for polarization-division multiplexed WDM transmission using digital backward propagation,” IEEE Photon. J. 1(2), 144–152 (2009). [CrossRef]

27.

E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008). [CrossRef]

28.

D. Rafique, M. Mussolin, M. Forzati, J. Mårtensson, M. N. Chugtai, and A. D. Ellis, “Compensation of intra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Express 19(10), 9453–9460 (2011). [CrossRef] [PubMed]

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30.

L. Zhu and G. Li, “Nonlinearity compensation using dispersion-folded digital backward propagation,” Opt. Express 20(13), 14362–14370 (2012). [CrossRef] [PubMed]

31.

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 Proc. ECOC (2011).

32.

T. Hoshida, L. Dou, T. Tanimura, W. Yan, S. Oda, L. Li, H. Nakashima, M. Yan, Z. Tao, and J. C. Rasmussen, “Digital nonlinear compensation techniques for high-speed DWDM transmission systems,” in Proc.ECOC (2012). [CrossRef]

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L. Xiang and X. P. Zhang, “The study of information capacity in multispan nonlinear optical fiber communication systems using a developed perturbation technique,” J. Lightwave Technol. 29(3), 260–264 (2011). [CrossRef]

OCIS Codes
(060.1660) Fiber optics and optical communications : Coherent communications
(060.2330) Fiber optics and optical communications : Fiber optics communications
(060.4370) Fiber optics and optical communications : Nonlinear optics, fibers

ToC Category:
Fiber Optics and Optical Communications

History
Original Manuscript: January 28, 2013
Revised Manuscript: April 21, 2013
Manuscript Accepted: May 5, 2013
Published: May 30, 2013

Citation
Lian Xiang, Paul Harper, and Xiaoping Zhang, "Advanced perturbation technique for digital backward propagation in WDM systems," Opt. Express 21, 13607-13616 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-11-13607


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