Sampling trajectories for the short-time Fourier transform

TL;DR

This paper investigates how to stably reconstruct the short-time Fourier transform from samples along specific trajectories, especially spiraling curves, and establishes conditions for successful reconstruction using Hermite function windows.

Contribution

It introduces a new analysis of sampling along trajectories for the short-time Fourier transform, focusing on spiraling curves and Hermite window functions, advancing understanding of sampling and reconstruction conditions.

Findings

Reconstruction is stable when sampling density meets certain criteria.

Spiraling curves can be used effectively for sampling the short-time Fourier transform.

Hermite functions enable stable reconstruction from samples on specific spiraling trajectories.

Abstract

We study the problem of stable reconstruction of the short-time Fourier transform from samples taken from trajectories in $\R^2$. We first consider the interplay between relative density of the trajectory and the reconstruction property. Later, we consider spiraling curves, a special class of trajectories, and connect sampling and uniqueness properties of these sets. Moreover, we show that for window functions given by a linear combination of Hermite functions, it is indeed possible to stably reconstruct from samples on some particular natural choices of spiraling curves.