Introducing the fast nonlinear Fourier transform

Sander Wahls, H. Vincent Poor

Research output: Chapter in Book/Report/Conference proceedingConference contribution

45 Scopus citations

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 log2 N) flops. A finite eigenproblem for the discrete eigenvalues that can be solved in O(N2) is also presented. The feasibility of this approach is demonstrated in a numerical example.

Original languageEnglish (US)
Title of host publication2013 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Proceedings
Pages5780-5784
Number of pages5
DOIs
StatePublished - Oct 18 2013
Event2013 38th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Vancouver, BC, Canada
Duration: May 26 2013May 31 2013

Publication series

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

Other

Other2013 38th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013
CountryCanada
CityVancouver, BC
Period5/26/135/31/13

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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