Nonlinear Fourier Transform of Time-Limited and One-sided Signals

TL;DR

This paper investigates the nonlinear Fourier spectrum of time-limited and one-sided signals, establishing conditions for prescribed support, improving numerical methods, and validating results through extensive numerical testing.

Contribution

It provides necessary and sufficient conditions for signals with specific support in the nonlinear Fourier domain and enhances inverse scattering techniques with better numerical conditioning.

Findings

Conditions for prescribed support in nonlinear Fourier spectrum

Improved numerical conditioning of inverse scattering methods

Validation of theoretical results through extensive numerical tests

Abstract

In this article, we study the properties of the nonlinear Fourier spectrum in order to gain better control of the temporal support of the signals synthesized using the inverse nonlinear Fourier transform (NFT). In particular, we provide necessary and sufficient conditions satisfied by the nonlinear Fourier spectrum such that the generated signal has a prescribed support. In our exposition, we assume that the support is a simply connected domain that is either a bounded interval or the half-line, which amounts to studying the class of signals which are either time-limited or one-sided, respectively. Further, it is shown that the analyticity properties of the scattering coefficients of the aforementioned classes of signals can be exploited to improve the numerical conditioning of the differential approach of inverse scattering. Here, we also revisit the integral approach of inverse scattering and provide the correct derivation of the so called T\"oplitz inner-bordering algorithm. Finally, we conduct extensive numerical tests in order to verify the analytical results presented in the article. These tests also provide us an opportunity to compare the performance of the two aforementioned numerical approaches in terms of accuracy and complexity of computations.