Coupled-Channel Enhanced SSFM for Digital Backpropagation in WDM Systems
S. Civelli, E. Forestieri, A. Lotsmanov, D. Razdoburdin, M. Secondini
aa r X i v : . [ c s . I T ] F e b Coupled-Channel Enhanced SSFM for DigitalBackpropagation in WDM Systems
S. Civelli , *, E. Forestieri , , A. Lotsmanov , D. Razdoburdin , , M. Secondini , TeCIP Institute, Scuola Superiore Sant’Anna, Via G. Moruzzi 1, 56124, Pisa, Italy PNTLab, CNIT, Via G. Moruzzi 1, 56124, Pisa, Italy Moscow Research Center, Huawei Technologies Co., Ltd., Moscow, Russia Sternberg Astronomical Institute, Moscow M.V. Lomonosov State University, Moscow, Russia*[email protected]
Abstract:
A novel technique for digital backpropagation (DBP) in wavelength-divisionmultiplexing systems is introduced and shown, by simulations, to outperform existing DBPtechniques for approximately the same complexity. © 2021 The Author(s)
OCIS codes:
1. Introduction
Digital backpropagation (DBP) is widely investigated for the mitigation of nonlinearity in optical fiber commu-nication, with the split step Fourier method (SSFM) being the most popular method to practically implementDBP [1, 2]. The SSFM digitally inverts the optical channel by applying a number of steps in which linear andnonlinear effects are considered separately. Several alternative DBP techniques have been proposed to achieve thesame performance as the SSFM with less steps and, hence, lower complexity, such as the correlated or filteredDBP [3, 4] and the enhanced SSFM (ESSFM) [5, 6].While both the ESSFM and the classical SSFM are conceived for the propagation of a single channel, theycan also be used for several wavelength-division multiplexing (WDM) channels, as long as they are jointly repre-sented as a single optical field—we refer to this as full-field
DBP. The potential gain achievable by full-field DBPincreases with the number of jointly backpropagated channels [7]. However, the extremely large number of stepsrequired in this case makes the practical implementation of full-field DBP not feasible [8].In this work, we propose a novel technique for DBP—the coupled-channel
ESSFM (CC-ESSFM)—whichimproves the ESSFM by explicitly accounting for the cross phase modulation (XPM) generated by several co-propagating channels and for its interplay with dispersion. We show, through simulations, that the CC-ESSFMachieves better performance than single- and full-field SSFM and ESSFM techniques for the same complexity.
2. Coupled-channel enhanced SSFM
The CC-ESSFM is described below for N ch copropagating channels. The CC-ESSFM, like the SSFM and theESSFM, consists of several steps N s made of a linear and a nonlinear part, and is depicted in Fig. 1(a).In the linear part, the various channels are independently processed as in the SSFM or ESSFM, with the onlydifference that a time delay is introduced to account for the different group velocity of the channels and ensurethat their contributions are correctly synchronized at each nonlinear step.In the nonlinear part, a nonlinear phase rotation is applied to each polarization component of each channelto account for the self-phase modulation (SPM) and part of the XPM. According to the frequency-resolved log-arithmic perturbation (FRLP) model [9], each channel induces a frequency-dependent nonlinear phase rotationwhich can be expressed as a quadratic form of its samples and affects the channel itself (SPM) and the otherchannels (XPM). The ESSFM accounts only for the SPM contribution and approximates it by a filtered versionof the signal intensity, where the filter coefficients are numerically optimized to minimize the error [5, 6]. In theCC-ESSFM, on the other hand, also the XPM contributions are accounted for and expressed as filtered versions ofthe corresponding signal intensities. By assuming that each channel is separately represented over its bandwidthwith n samples per symbol, the nonlinear step for the i th channel can be expressed as x ′ i [ k ] = x i [ k ] exp ( j θ xi [ k ]) and y ′ i [ k ] = y i [ k ] exp ( j θ yi [ k ]) , where x i [ k ] and y i [ k ] (and x ′ i [ k ] and y ′ i [ k ] ) are the k th samples of the two polarizationcomponents of the i th channel at the input (and output) of the nonlinear step, normalized to have unit power; θ xi [ k ] = − N ch ∑ ℓ = φ k i ℓ N c ∑ m = − N c C ℓ − i [ m ] | x ℓ [ k + m ] | + φ ⊥ ℓ N c ∑ m = − N c C ℓ − i [ m ] | y ℓ [ k + m ] | ! (1) θ yi [ k ] = − N ch ∑ ℓ = φ ⊥ ℓ N c ∑ m = − N c C ℓ − i [ m ] | x ℓ [ k + m ] | + φ k i ℓ N c ∑ m = − N c C ℓ − i [ m ] | y ℓ [ k + m ] | ! (2) igital MIMO filter DelayIFFTGVDFFT Delay multichannel nonlinear operatorIFFTGVDFFT
Random bitgenerat.GMI for each channel
Signalmod.2polQAMmapp. Channelsand superch.MUX . SCOI and channelsDEMUX . Match.filter DBPSampl.MPR for each channel (a) (b)
Figure 1. (a) Linear and nonlinear step of the CC-ESSFM for N ch = s P e a k G M I [ b i t s / s y m b o l / p o l ] GVD onlySSFMOSSFMESSFMCC − ESSFMFF − SSFMFF − OSSFMFF − ESSFM n=1.25/ch.:n=8/superch.:
Figure 2. Performance of different DBP techniquesversus number of steps N s . Technique Real multipl./4D symb.GVD only η n ( N + ) SSFM N s η n ( N + ) ESSFM N s η n ( N + + N c ) CC-ESSFM N s η n (cid:0)
12 log N + + N ch (cid:1) Table 1. Number of real multiplica-tions per 4D symbol per channel.are the (different) nonlinear phase rotations over the two polarization components; φ ⊥ ℓ = γ P ℓ R z + L / z − L / g ( z ) dz and φ k i ℓ = ( − δ i ℓ ) φ ⊥ ℓ are the average nonlinear phase rotations induced, respectively, by the copolarized and orthog-onal component of the ℓ th channel over the i th channel; δ i ℓ is the Kronecker delta ( δ i ,ℓ = i = ℓ and δ i ,ℓ = P ℓ ( z ) is the power profile of the ℓ th channel along the step of length L ; γ is the nonlinear coefficient;and C h = ℓ − i [ m ] , with m = − N c , . . . , N c , are the 2 N c + ℓ th channel over the i th channel.With respect to the standard SSFM, the CC-ESSFM requires, at each step, the implementation of a digital2 N ch × N ch multi-input multi-output (MIMO) filter for the computation of (1) and (2). This can be efficientlydone in frequency domain by using a pair of real FFTs and the corresponding IFFTs per each channel, with anoverall increase in complexity of less than 50% compared to the SSFM. On the other hand, the single-channelESSFM—equivalent to the CC-ESSFM with N ch = N c is not too large. Table 1 reports the complexity of the ESSFM and CC-ESSFM comparedto the standard SSFM and a simple dispersion compensation, considering the same implementation assumptionsas in [6] and an overlap-and-save processing, where N is the “overlapping” block length and η the ratio between N and the actual number of samples “saved” per each processed block.The CC-ESSFM coefficients can be obtained either by numerical optimization or analytically. In this work, wehave adopted the numerical optimization, which guarantees a good performance and does not require the a prioriknowledge of the link parameters, deferring the study of the analytical computation based on the FRLP model [9]to a future work. The numerical optimization is performed by assuming that the coefficients are independent of thelaunch power and satisfy the symmetry condition C h [ m ] = C − h [ − m ] , as predicted by the FRLP model and verifiedby numerical simulations. To further simplify the optimization, the coefficients C h = ( C h [ − N c ] , . . . C h [ N c ]) areobtained iteratively for h = , . . . , N ch −
1, assuming N c =
32 for h = N c =
128 for h >
0. At the h th iteration, C h is obtained by minimizing the mean square error over the received symbols, while keeping C , . . . , C h − fixedto the values found at the previous iterations, and setting C h + , . . . , C N ch − to zero.
3. System setup and performance
The simulation setup is sketched in Fig. 1(b). The superchannel of interest (SCOI) is made of N ch = R s = . D = γ = . α dB = . n = .
25 (per channel) for single-fieldand coupled-channel DBP, and n = N s . For a small (but practical) number of steps, the conventional SSFM has a very poorperformance—in fact, even worse than simple GVD compensation. In this case, the performance can be improved(without changing the complexity) by using the optimized SSFM (OSSFM), with a numerically optimized nonlin-ear coefficient (lower than the actual one) in the nonlinear step [8]. An even better performance is achieved by theESSFM, while the best performance is achieved by the CC-ESSFM. In this practical scenario, the full-field tech-niques (dashed lines) perform significantly worse than the corresponding single-channel ones (solid lines), dueto the larger accumulated dispersion in each step. The picture changes when the number of steps is increased toless realistic values. In this case, the full-field techniques outperform the corresponding single-channel techniquesand perform even slightly better than the CC-ESSFM. Asymptotically ( N s → ∞ ), all the single-channel techniquesconverge to the same limit, corresponding to an exact compensation of intrachannel effects. Analogously, all thefull-field techniques converge to a higher limit, corresponding to an exact compensation of intra-superchanneleffects, while CC-ESSFM converges to a slightly lower limit, as it accounts only for SPM and XPM in the SCOI.Though the number of steps is a good indicator of the relative complexity of the various algorithms, the actualcomplexity depends on the specific algorithm and on the details of the implementation. Table 1, obtained undersome reasonable but simplifying assumptions, allows a more accurate comparison and shows that ESSFM andCC-ESSFM require, respectively, about 30% and 50% more multiplications per step than SSFM, hence onlyslightly reducing their advantage compared to the other techniques in Fig. 2. However, we believe that additionalsimplifications and a careful implementation can further reduce these figures.Similar results (not shown here for lack of space) have been obtained when considering probabilistically-shapedQAM modulations and when including a practical carrier phase recovery algorithm at the receiver.
4. Conclusion
We have proposed the CC-ESSFM as an efficient technique for multi-channel DBP in WDM systems. By addi-tionally accounting for XPM among the channels and its interplay with dispersion, the CC-ESSFM outperformsconventional single-channel and full-field DBP techniques for the same (sufficiently small) number of steps.
References
1. R. J. Essiambre and P. J. Winzer, “Fibre nonlinearities in electronically pre-distorted transmission,” in
Proc.Eur. Conf. Opt. Commun. (ECOC), , vol. 2 (2005), pp. 191–192.2. E. Ip and J. M. Kahn, “Compensation of dispersion and nonlinear impairments using digital backpropaga-tion,” J. Light. Technol. , 3416–3425 (2008).3. L. Li, Z. Tao, L. Dou, W. Yan, S. Oda, T. Tanimura, T. Hoshida, and J. C. Rasmussen, “Implementationefficient nonlinear equalizer based on correlated digital backpropagation,” in Opt. Fiber Commun. Conf.(OFC), (2011), p. OWW3.4. D. Rafique, M. Mussolin, M. Forzati, J. M˚artensson, M. N. Chugtai, and A. D. Ellis, “Compensation ofintra-channel nonlinear fibre impairments using simplified digital back-propagation algorithm,” Opt. Ex-press , 9453–9460 (2011).5. M. Secondini, D. Marsella, and E. Forestieri, “Enhanced split-step Fourier method for digital backpropaga-tion,” in (IEEE, 2014), pp. 1–3.6. M. Secondini, S. Rommel, G. Meloni, F. Fresi, E. Forestieri, and L. Poti, “Single-step digital backpropaga-tion for nonlinearity mitigation,” Photon. Netw. Commun. , 493–502 (2016).7. R. Dar and P. J. Winzer, “Nonlinear interference mitigation: methods and potential gain,” J. Light. Technol. , 903–930 (2017).8. G. Liga, T. Xu, A. Alvarado, R. I. Killey, and P. Bayvel, “On the performance of multichannel digitalbackpropagation in high-capacity long-haul optical transmission,” Opt. Express , 30053–30062 (2014).9. M. Secondini, E. Forestieri, and G. Prati, “Achievable information rate in nonlinear WDM fiber-optic sys-tems with arbitrary modulation formats and dispersion maps,” J. Light. Technol. , 3839–3852 (2013).10. A. Alvarado, E. Agrell, D. Lavery, R. Maher, and P. Bayvel, “Replacing the soft-decision FEC limitparadigm in the design of optical communication systems,” J. Light. Technol.33