PhaseJumps: fast computation of zeros from planar grid samples

Abstract

We consider complex-valued functions on the complex plane and the task of computing their zeros from samples taken along a finite grid. We introduce PhaseJumps, an algorithm based on comparing changes in the complex phase and local oscillations among neighboring grid points. The algorithm is applicable to possibly non-analytic input functions, and also computes the direction of phase winding around zeros. PhaseJumps provides a first effective means to compute the zeros of the short-time Fourier transform of an analog signal with respect to a general analysis window, and makes certain recent signal processing insights more widely applicable, overcoming previous constraints to analytic transformations. We study the performance of (a variant of) PhaseJumps under a stochastic input model motivated by signal processing applications and show that the input instances that may cause the algorithm to fail are fragile, in the sense that they are regularized by additive noise (smoothed analysis). Precisely, given samples of a function on a grid with spacing $\delta$, we show that our algorithm computes zeros with accuracy $\sqrt{\delta}$ in the Wasserstein metric with failure probability $O\big(\log^2(\tfrac{1}{\delta}) \delta\big)$, while numerical experiments suggest even better performance.