Analytic Interpolation and the Non-linear Fourier Transform

Abstract

The non-linear Fourier transform may be considered an extension of Fourier analysis to non-linear problems. It has applications in many different scientific and engineering disciplines, for instance as a solution to the non-linear Schrödinger equation which describes how electro-magnetic waves travel through ideal non-linear media. Hence, the non-linear Fourier transform is fundamental for communication schemes that take optical fiber non-linearities into account.
Wahls et al. proposed an algorithm for a fast inverse non-linear Fourier transform based around a discretization of the non-linear Schrödinger equation. But one of the steps in the algorithm — known as synthesis — only had heuristic solutions. However, the problem closely resembles a Nevanlinna-Pick interpolation with degree constraint, a problem known to have a guaranteed solution. Indeed, synthesis is a limit case of Nevanlinna-Pick interpolation with degree constraint on the boundary of the region of analyticity. Although there are solvers to the boundary Nevanlinna-Pick interpolation with degree constraint available in the literature, they do not scale to the numerical difficulty or size required, and are based around simplifying assumptions that do not hold for synthesis.
In this thesis I propose two different numerical continuation solvers for synthesis. The methods have lower computational complexity than other guaranteed solvers by using fast linear Fourier transforms to perform polynomial operations. Moreover, the convergence speed of the algorithms is improved by approaching the solution in a novel trajectory. However, the solvers only offer a quicker convergence rate than the heuristic methods for very difficult problems, and may not be effective for real-time applications, as the high-demands in error precision required means a very high computational cost. This seems to be a limitation of the theory, as the high computational cost is a consequence of the large number of iterations required even when the cost of each iteration is low.