TY - GEN

T1 - Introducing the fast nonlinear Fourier transform

AU - Wahls, Sander

AU - Poor, H. Vincent

PY - 2013/10/18

Y1 - 2013/10/18

N2 - 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.

AB - 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.

KW - Inverse Scattering Transform

KW - Nonlinear Fourier Transform

KW - Optical Fiber Communication

KW - Schrödinger Equation

UR - http://www.scopus.com/inward/record.url?scp=84890498194&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84890498194&partnerID=8YFLogxK

U2 - 10.1109/ICASSP.2013.6638772

DO - 10.1109/ICASSP.2013.6638772

M3 - Conference contribution

AN - SCOPUS:84890498194

SN - 9781479903566

T3 - ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings

SP - 5780

EP - 5784

BT - 2013 IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013 - Proceedings

T2 - 2013 38th IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP 2013

Y2 - 26 May 2013 through 31 May 2013

ER -