Nonlinearly bandlimited signals

Abstract

In this paper, we study the inverse scattering problem for a class of signals that have a compactly supported reflection coefficient. The problem boils down to the solution of the Gelfand-Levitan-Marchenko (GLM) integral equations with a kernel that is bandlimited. By adopting a sampling theory approach to the associated Hankel operators in the Bernstein spaces, a constructive proof of existence of a solution of the GLM equations is obtained under various restrictions on the nonlinear impulse response (NIR). The formalism developed in this article also lends itself well to numerical computations yielding algorithms that are shown to have algebraic rates of convergence. In particular, the use Whittaker-Kotelnikov-Shannon sampling series yields an algorithm that converges as O(N −1/2 ) whereas the use of Helms and Thomas (HT) version of the sampling expansion yields an algorithm that converges as O(N −m−1/2 ) for any m > 0 provided the regularity conditions are fulfilled. The complexity of the algorithms depend on the linear solver used. The use of conjugate-gradient (CG) method yields an algorithm of complexity O(Niter.N 2 ) per sample of the signal where N is the number of sampling basis functions used and N iter. is the number of CG iterations involved. The HT version of the sampling expansions facilitates the development of algorithms of complexity O(Niter.N log N) (per sample of the signal) by exploiting the special structure as well as the (approximate) sparsity of the matrices involved. The algorithms are numerically validated using Schwartz class functions as NIRs that are either bandlimited or effectively bandlimited. The results suggest that the HT variant of our algorithm is spectrally convergent for an input of the aforementioned class.

Darboux transformation: new identities

“…The present work, therefore, tries to further reinforce the idea that the Darboux representation can potentially facilitate a number of design and signal processing aspects of K-soliton solutions. For the general case, when the reflection coefficient is bandlimited, the work presented in [16] may allow us to compute the Jost solutions of the seed potential with extremely high accuracy.…”

A Chebyshev Spectral Method for Nonlinear Fourier Transform: Norming Constants

“…Note that while the IF schemes uses fast polynomial arithmetic in the monomial basis, the ETD schemes use fast polynomial arithmetic in the Chebyshev basis. For the inverse transform, a sampling series based approach for computing the "radiative" part has been proposed in [10] which achieves spectral accuracy at quasilinear complexity per sample of the signal. In this paper, we extend the recently proposed spectral method [11] for the computation continuous spectrum to compute the norming constants.…”