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 -