Efficient computation of the zeros of the Bargmann transform under additive white noise

TL;DR

This paper introduces an adaptive algorithm for accurately computing the zeros of the Bargmann transform of noisy signals, demonstrating high probability success and providing theoretical error bounds and numerical comparisons.

Contribution

It presents the AMN algorithm for zero set computation of the Bargmann transform under noise, with proven error bounds and high probability guarantees.

Findings

AMN computes zeros with Wasserstein error proportional to grid spacing

High probability of success with failure probability O(δ^4 log^2(1/δ))

Numerical tests show competitive performance against existing methods

Abstract

We study the computation of the zero set of the Bargmann transform of a signal contaminated with complex white noise, or, equivalently, the computation of the zeros of its short-time Fourier transform with Gaussian window. We introduce the adaptive minimal grid neighbors algorithm (AMN), a variant of a method that has recently appeared in the signal processing literature, and prove that with high probability it computes the desired zero set. More precisely, given samples of the Bargmann transform of a signal on a finite grid with spacing $\delta$, AMN is shown to compute the desired zero set up to a factor of $\delta$ in the Wasserstein error metric, with failure probability $O(\delta^4 \log^2(1/\delta))$. We also provide numerical tests and comparison with other algorithms.