Higher Order Convergent Fast Nonlinear Fourier Transform

Abstract

It is demonstrated is this letter that linear multistep methods for integrating ordinary differential equations can be used to develop a family of fast forward scattering algorithms with higher orders of convergence. Excluding the cost of computing the discrete eigenvalues, the nonlinear Fourier transform (NFT) algorithm thus obtained has a complexity of O(K N + C p N log 2 N) such that the error vanishes as O(N −p ) where p ∈ {1, 2, 3, 4} and K is the number of eigenvalues. Such an algorithm can be potentially useful for the recently proposed NFT based modulation methodology for optical fiber communication. The exposition considers the particular case of the backward differentiation formula (C p = p 3 ) and the implicit Adams method (C p = (p − 1) 3 ) of which the latter proves to be the most accurate family of methods for fast NFT.

Actuator and Sensor Fault Classification for Wind Turbine Systems Based on Fast Fourier Transform and Uncorrelated Multi-Linear Principal Component Analysis Techniques

“…The Fourier transform above can be calculated by using the fast Fourier transform algorithm [45,46]. The fast Fourier transform algorithm treats the columns of a matrix as vectors and returns the Fourier transform vector for each column, leading to a Fourier transform matrix.…”

Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse

“…However, since the transformations are computed numerically, certain numerical errors are introduced. To minimize numerical errors for both INFT and NFT, many different algorithms and implementations are known in the literature [ 17 , 18 , 19 , 20 , 21 , 27 , 41 , 43 , 44 , 45 , 54 , 55 , 56 , 57 , 58 , 59 ].…”

“…The early algorithms for the NFT and inverse nonlinear Fourier transform (INFT) exhibited numerical instabilities and/or high complexity. Many publications proposed stabilized versions of the transformations and algorithms with reduced complexity [ 41 , 42 , 43 , 44 , 45 , 46 , 47 ]. The method presented in [ 46 ] is the basis for the studies presented in this publication.…”

Nonlinear Fourier Transform Algorithm Using a Higher Order Exponential Integrator

“…The relative L 2 -error ρ a (λ ) − ρ n (λ ) 2 / ρ a (λ ) 2 , |λ | < λ max , is used as the measure of accuracy. The CF4 algorithm is compared against BO and the Implicit-Adams method (IA2), which is another higher order method; its use for NFTs has been recently proposed in [7,IA 2 ]. As with the linear Fourier transform, the sampling process limits the range |λ | < λ max that we can resolve.…”