Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators

Chimmalgi, Shrinivas; Prins, Peter J.; Wahls, Sander

doi:10.48550/arxiv.1812.00703

Cited by 3 publications

(6 citation statements)

References 27 publications

(78 reference statements)

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“…The accuracy-complexity trade-off, however, is similar for the two methods. Finally, we also present a fast variant of the CF method of order four (formally) by employing the fourth order splitting on the lines of [6], [10]. Despite the well-known limitation imposed on the order and stability of such techniques as demonstrated by Sheng [11], we do not find any reduction of order within the double precision arithmetic.…”

Section: Introductionmentioning

confidence: 91%

“…Despite the well-known limitation imposed on the order and stability of such techniques as demonstrated by Sheng [11], we do not find any reduction of order within the double precision arithmetic. It is noteworthy that the authors in [6] found the aforementioned splitting worsen in accuracy after a certain step-size. It is also not clear from their analysis if this scheme is convergent in their setting.…”

Section: Introductionmentioning

confidence: 98%

“…The structure of the transfer matrix reveals that such methods are superior to those based on LMMs in terms of complexity while the numerical tests reveal that they also have a superior accuracy-complexity trade-off. The first algorithm is based on the classical fourth order (explicit) Runge-Kutta method which has been studied by several authors in the context of NFTs [5], [6]. The second method uses the three-stage Lobatto IIIA (implicit) Runge-Kutta method.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Vaibhav

2019

IEEE Photon. Technol. Lett.

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We explore two classes of exponential integrators in this letter to design nonlinear Fourier transform (NFT) algorithms with a desired accuracy-complexity trade-off and a convergence order of 4 on an equispaced grid. The integrating factor based method in the class of Runge-Kutta methods yield algorithms with complexity O(N log 2 N) (where N is the number of samples of the signal) which have superior accuracy-complexity trade-off than any of the fast methods known currently. The integrators based on Magnus series expansion, namely, standard and commutator-free Magnus methods yield algorithms of complexity O(N 2 ) that have superior error behavior even for moderately small step-sizes and higher signal strengths.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Vaibhav

2019

IEEE Photon. Technol. Lett.

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“…In this Letter, we will focus on building the method of the fourth order of accuracy on a uniform grid. For a non-uniform grid, a fourth order scheme [10] was applied in [11]. In perspective, the exponential approximation can be applied to our scheme so that we can use the fast algorithm.…”

mentioning

confidence: 99%

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem

Медведев

Vaseva²,

Chekhovskoy³

et al. 2019

Opt. Lett.

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We propose a new high-precision algorithm for solving the initial problem for the Zakharov-Shabat system. This method has the fourth order of accuracy and is a generalization of the second order Boffetta-Osborne scheme. It is allowed by our method to solve more effectively the Zakharov-Shabat spectral problem for continuous and discrete spectra. Keywords Zakharov-Shabat problem, inverse scattering transform, nonlinear Schrödinger equation, numerical methods The solution of the direct problem for the Zakharov-Shabat problem (ZSP) is the first step in the inverse scattering transform (IST) for solving the nonlinear Schrödinger equation (NLSE) [1]. The numerical implementation of the IST has gained great importance and attracted special attention since Hasegawa and Tappert [2] proposed to use soliton solutions as a bit of information for fiber optic data transmission.

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“…However there are indications that the burst length cannot be increased indefinitely without degradation in performance [10]. At the same time, the development of fast and accurate algorithms for the forward nonlinear Fourier transform continues [11]- [13].…”

mentioning

confidence: 99%

Data Transmission based on Exact Inverse Periodic Nonlinear Fourier Transform, Part II: Waveform Design and Experiment

Goossens,

Hafermann,

Jaouën

2019

Preprint

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The nonlinear Fourier transform has the potential to overcome limits on performance and achievable data rates which arise in modern optical fiber communication systems when nonlinear interference is treated as noise. The periodic nonlinear Fourier transform (PNFT) has been much less investigated compared to its counterpart based on vanishing boundary conditions. In this paper, we design a first experiment based on the PNFT in which information is encoded in the invariant nonlinear main spectrum. To this end, we propose a method to construct a set of periodic waveforms which each have the same fixed period, by employing the exact inverse PNFT algorithm developed in Part I. We demonstrate feasibility of the transmission scheme in experiment in good agreement with simulations and obtain a bit-error ratio of 10 −3 over a distance of 2000km. It is shown that the transmission reach is significantly longer than expected from a naive estimate based on group velocity dispersion and cyclic prefix length, which is explained through a dominating solitonic component in the transmitted waveform. Our constellation design can be generalized to an arbitrary number of nonlinear degrees of freedom. I. IntroductionF UELED by ever increasing online traffic demands, the capacity of optical communication links has grown exponentially in a Moore-like fashion, following a tenfold increase every four years [1]. This trend has been kept up owing to the introduction of new technologies, starting from improved transmission fibers and continuing with the Erbium-doped fiber amplifier, wavelength-division multiplexing and high spectral efficiency coding. Even with advanced digital signal processing and coded modulation schemes, in particular probabilistic shaping [2], modern optical transmission systems are approaching the so-called nonlinear Shannon limit [3]. Not regarded as fundamental [4], it emerges as an artifact of the application of communication techniques developed for linear channels, in which nonlinear interference is treated as noise. While SNR increases limitless with increasing power, the signalto-interference-plus-noise ratio reaches an optimum [5].Recently, optical communication techniques based on the nonlinear Fourier transform have reentered the focus J.-W. Goossens and H. Hafermann are with the Optical Communi-

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Fast Nonlinear Fourier Transform Algorithms Using Higher Order Exponential Integrators

Cited by 3 publications

References 27 publications

Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Efficient Nonlinear Fourier Transform Algorithms of Order Four on Equispaced Grid

Numerical algorithm with fourth-order accuracy for the direct Zakharov-Shabat problem

Data Transmission based on Exact Inverse Periodic Nonlinear Fourier Transform, Part II: Waveform Design and Experiment

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