Improved Simulation Accuracy of the Split-Step Fourier Method
aa r X i v : . [ ee ss . SP ] J a n Improved Simulation Accuracyof the Split-Step Fourier Method
Shen Li , Magnus Karlsson , Erik Agrell Dept. of Electrical Engineering, Dept. of Microtechnology and Nanoscience,Chalmers University of Technology, SE-412 96 Gothenburg, [email protected]
Abstract:
We investigate a modified split-step Fourier method (SSFM) by including low-pass filters in the linear steps. This method can simultaneously achieve a higher simulationaccuracy and a slightly reduced complexity. © 2020 The Author(s)
OCIS codes: (060.1660) Coherent communications; (060.2330) Fiber optics communications
1. Introduction
Signal propagation in fiber is generally governed by nonlinear Schr¨odinger equation (NLSE), which is a time-dependent nonlinear differential partial equation and cannot be solved analytically due to interactions betweennonlinearity and linear dispersion [1]. Thus, operational and accurate simulation algorithms are required to modelthe evolution of the electrical field in the fiber. The split-step Fourier method (SSFM) is the most commonly usedway of simulating the NLSE because of its operability and high accuracy. In the SSFM, lightwave propagationalong the fiber is discretized into many small spatial steps, in each of which the nonlinearity and dispersion canbe separated and expressed analytically. Nevertheless, the SSFM usually requires high oversampling rates of thesignal [2, 3] and small step sizes to converge to the true result of the NLSE, indicating a trade-off between thesimulation accuracy and complexity.The computational complexity of the SSFM mostly comes from the large number of times of transformationsbetween time domain and frequency domain using the fast Fourier transform and its inverse, and the exponentialcomputations in the nonlinear operator. Much research has been devoted to analyzing the accuracy of differentSSFM schemes and proposing refinements, particularly optimizing the selection and updating rules of the stepsize [4–8]. In this paper, different from the all-pass filters usually used in the linear operators of the SSFM, wemodify the SSFM by simply including a low-pass filter (LPF) in each linear operator to avoid possible aliasingduring simulation. It shows that with these “cost-free” filters, the modeling accuracy of the SSFM is improved fora given set of oversampling rates and step sizes, or in other words, we can reduce the simulation complexity for agiven accuracy.
2. Reduced-complexity SSFM
Let a ( t , z ) be the electrical field propagating along the fiber at time t and distance z . The NLSE for the evolutionof a ( t , z ) in an unamplified fiber span can be written as ∂ a ( t , z ) ∂ z = − α a ( t , z ) − j β ∂ a ( t , z ) ∂ t + j γ | a ( t , z ) | a ( t , z ) (1)where a ( t , ) is the input signal, α is the attenuation factor, β is the dispersion parameter, and γ is the nonlinearparameter. Let L and N denote the linear operator − α − j β ∂ ∂ t and nonlinear operator j γ | a ( t , z ) | in each step ofthe SSFM respectively. With N seg discretized steps and total transmission distance Z , Fig. 1a shows a traditionalSSFM structure. The need for small enough time resolution ∆ t can be illustrated by Fig. 2a, where T s is the symboltime. In this example, where the launch power is set high to stress-test the SSFM under adverse conditions,the SSFM output converges to the NLSE with 30 samples per symbol and as ∆ t increases, the output becomesincreasingly deviated from the NLSE. After increasing ∆ t to T s /
4, the output is completely independent of theNLSE. This phenomenon is caused by spectrum aliasing resulting from spectral broadening, since higher ∆ t implies lower sampling frequency.We modify the linear step in traditional SSFM by including an LPF with bandwidth W , as shown in Fig. 1b, toreduce aliasing. The proposed linear step is a multiplication with H ( f ) = (cid:26) exp ( − α∆ z + j π f β ∆ z ) , | f | ≤ W , W < | f | ≤ W s (2) ( t , ) N L a ( t , Z ) × N seg One fiber segment (a) a ( t , ) N H ( f ) a ( t , Z ) × N seg One fiber segment (b)
Fig. 1: Traditional SSFM structure (a) and improved SSFM with LPFs (b). -400 -200 0 200 400 − . . t [ps] R e { a ( t , Z ) } ∆ t = T s / ∆ t = T s / ∆ t = T s / ∆ t = T s / ∆ t = T s / (a) The SSFM output without LPFs. -400 -200 0 200 400 − . . t [ps] R e { a ( t , Z ) } ∆ t = T s / ∆ t = T s / ∆ t = T s / ∆ t = T s / ∆ t = T s / (b) The SSFM output with LPFs. Fig. 2: The output with different ∆ t become more similar to the NLSE using LPFs. Parameters: 16-QAM singlewavelength transmission at 9 . β = − . /km, γ = . ( W · km ) − ; step size 1 . ∆ t .in the frequency domain, where f is the frequency component of the signal, ∆ z = Z / N seg is the step size, W s isthe sampling rate and the filter bandwidth W ≤ W s /
2. When W = W s / H ( f ) implies the linear step in traditionalSSFM. Thus, since fewer complex exponentials need to be computed when W < W s /
2, the proposed schemehas slightly lower complexity than the traditional one. The LPF bandwidth W intuitively cannot be too narrowto avoid erasing too much information of the signal. Importantly, the purpose of the modified linear step is toimprove the simulation accuracy for the standard NLSE channel (1), which we do not modify in any way. Thismakes our contribution fundamentally different from, e.g., [9], where a new kind of propagation channel is createdand analyzed by inserting band-pass filters regularly along the fiber. Our motivation is that the LPF improvesthe simulation accuracy, provided the filter bandwidth is carefully selected and optimized, which can be seen inFig. 2b.
3. Numerical results
We consider a 16-QAM modulation format transmitted at 10 Gbaud through single-mode fiber using a raised-cosine pulse with roll-off factor 10% and ideal distributed amplification. The fiber parameters β and γ are − . /km and 1 . ( W · km ) − , respectively, and the amplifier noise is neglected. To evaluate the accuracy, we definethe normalized square difference (NSD) NSD = R ( a ( t , Z ) − ˆ a ( t , Z )) dt R a ( t , Z ) dt (3)between the NLSE output a ( t , Z ) and another simulated output ˆ a ( t , Z ) . Let ∆ t NLSE = T s /
30 and ∆ z NLSE = . a ( t , Z ) . We measure theNSD without and with LPFs as a function of: (a) transmission distance Z [km], (b) input power P [dBm], and (c)time discretization ∆ t / T s [%] for some example cases as shown in Fig. 3. The NSD results in Fig. 3 were obtainedby averaging over many transmitted 16-QAM symbol sequences. All the curves with the LPFs were obtained byglobally searching the filter bandwidth W to get the best performance.Fig. 3a shows that at a given distance, the NSD reduces by a factor of 3–5 using the LPFs, which means that witha fixed oversampling rate, we can simulate longer distances without losing accuracy. Similarly, Fig. 3b implies thatthis modified SSFM shows more robustness to nonlinearity than the traditional SSFM. For a certain simulationtime resolution, the NSD can be reduced by more than half using these filters (see Fig. 3c). To this end, we canimplement the SSFM with a larger sampling density with the LPFs and the only cost is to find the most suitable
00 400 600 800 1 , Z [km] N S D ( % ) without LPFswith LPFs (a) Parameters: ∆ z = . P =
10 dBm, ∆ t = T s / . . . P [dBm] N S D ( % ) without LPFswith LPFs (b) Parameters: ∆ z = . ∆ t = T s / Z = . . ∆ t / T s [%] N S D ( % ) without LPFswith LPFs (c) Parameters: ∆ z = . P =
10 dBm, Z =
70 80 90 1000510 Filter bandwidth W over W s / N S D ( % ) ∆ t = T s / P =
10 dBm ∆ t = T s / P =
10 dBm ∆ t = T s / P = . ∆ t = T s / P = . (d) Parameters: ∆ z = . Z = Fig. 3: The NSD is improved with LPFs as a function of transmission distance Z , input power P , time discretization ∆ t , and filter bandwidth W .bandwidth to achieve the gain. The filter bandwidth W over half of the sampling rate W s of the signal for someexample cases is analyzed in Fig. 3d, where the minimum NSD appears when W is between 70% and 85%, while100% means the SSFM without LPFs. When W is below 60%, the simulation becomes less accurate comparedwith the traditional SSFM, because the LPF erases some information of the signal.
4. Conclusion and future work
We proposed a modified SSFM algorithm with slightly lower complexity by including LPFs in the linear operatorto avoid aliasing. How transmission distance, input power, and time discretization in the SSFM affect the NSDbetween the NLSE output and a simulated output is studied and the bandwidth of the LPFs is analyzed as well forsome specific cases. We find that the proposed filter method can reduce more than 50% of the simulation error fora wide range of link and simulation parameters.For future work, it would be interesting if this filtering method could help with reducing the symbol error rate ofthe digital backpropagation algorithm, since the digital backpropagation is an inverse process of the SSFM wherealiasing could happen as well.
Acknowledgement
This work was supported by the Swedish Research Council (VR) under grant no. 2017-03702.
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