### Abstract

The nonlinear Fourier transform (NFT; also: direct scattering transform) is discussed with respect to the focusing nonlinear Schrödinger equation on the infinite line. It is shown that many of the current algorithms for numerical computation of the NFT can be interpreted in a polynomial framework. Finding the continuous spectrum corresponds to polynomial multipoint evaluation in this framework, while finding the discrete eigenvalues corresponds to polynomial root finding. Fast polynomial arithmetic is used in order to derive algorithms that are about an order of magnitude faster than current implementations. In particular, an N sample discretization of the continuous spectrum can be computed with only O(N log^{2} N) flops. A finite eigenproblem for the discrete eigenvalues that can be solved in O(N^{2}) is also presented. The feasibility of this approach is demonstrated in a numerical example.

Original language | English (US) |
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Title of host publication | 2013 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Proceedings |

Pages | 5780-5784 |

Number of pages | 5 |

DOIs | |

State | Published - Oct 18 2013 |

Event | 2013 38th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Vancouver, BC, Canada Duration: May 26 2013 → May 31 2013 |

### Publication series

Name | ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings |
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ISSN (Print) | 1520-6149 |

### Other

Other | 2013 38th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 |
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Country | Canada |

City | Vancouver, BC |

Period | 5/26/13 → 5/31/13 |

### All Science Journal Classification (ASJC) codes

- Software
- Signal Processing
- Electrical and Electronic Engineering

### Keywords

- Inverse Scattering Transform
- Nonlinear Fourier Transform
- Optical Fiber Communication
- Schrödinger Equation

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## Cite this

*2013 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Proceedings*(pp. 5780-5784). [6638772] (ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings). https://doi.org/10.1109/ICASSP.2013.6638772