From completeness of discrete translates to phaseless sampling of the short-time Fourier transform

TL;DR

This paper establishes conditions under which functions can be uniquely reconstructed from phaseless samples of their short-time Fourier transform, extending previous results to a broader class of window functions and lattice sampling schemes.

Contribution

It links phase retrieval in the short-time Fourier transform to the completeness of discrete translates, providing new uniqueness results for a wide range of window functions.

Findings

Unique determination of functions from phaseless STFT samples under certain density conditions.

Completeness of discrete translates in Banach spaces leads to new sampling theorems.

Applicability to various window functions in time-frequency analysis.

Abstract

We study the uniqueness problem in short-time Fourier transform phase retrieval by exploring a connection to the completeness problem of discrete translates. Specifically, we prove that functions in $L^2(K)$ with $K \subseteq \mathbb{R}^d$ compact, are uniquely determined by phaseless lattice-samples of its short-time Fourier transform with window function $g$, provided that specific density properties of translates of $g$ are met. By proving completeness statements for systems of discrete translates in Banach function spaces on compact sets, we obtain new uniqueness statements for phaseless sampling on lattices beyond the known Gaussian window regime. Our results apply to a large class of window functions, which are relevant in time-frequency analysis and applications.