# Fast Inverse Nonlinear Fourier Transform

## TL;DR

This paper introduces two efficient algorithms for the inverse nonlinear Fourier transform based on the non-Hermitian Zakharov-Shabat problem, significantly improving computational speed and accuracy for practical signal processing applications.

## Contribution

The paper presents novel fast inverse NFT algorithms with $O(KN+N ext{log}^2N)$ complexity, combining a new layer-peeling scheme and Darboux transformation for practical use.

## Key findings

- Algorithms achieve $O(N ext{log}^2N)$ complexity.
- Convergence rate of $O(N^{-2})$ demonstrated.
- Numerical tests confirm high performance.

## Abstract

This paper considers the non-Hermitian Zakharov-Shabat (ZS) scattering problem which forms the basis for defining the SU$(2)$-nonlinear Fourier transform (NFT). The theoretical underpinnings of this generalization of the conventional Fourier transform is quite well established in the Ablowitz-Kaup-Newell-Segur (AKNS) formalism; however, efficient numerical algorithms that could be employed in practical applications are still unavailable. In this paper, we present two fast inverse NFT algorithms with $O(KN+N\log^2N)$ complexity and a convergence rate of $O(N^{-2})$ where $N$ is the number of samples of the signal and $K$ is the number of eigenvalues. These algorithms are realized using a new fast layer-peeling (LP) scheme ($O(N\log^2N)$) together with a new fast Darboux transformation (FDT) algorithm ($O(KN+N\log^2N)$) previously developed by the author. The proposed fast inverse NFT algorithm proceeds in two steps: The first step involves computing the radiative part of the potential using the fast LP scheme for which the input is synthesized under the assumption that the radiative potential is nonlinearly bandlimited, i.e., the continuous spectrum has a compact support and the discrete spectrum is empty. The second step involves addition of bound states using the FDT algorithm. Finally, the performance of these algorithms is demonstrated through exhaustive numerical tests.