Phase retrieval from short-time Fourier transform in LCA groups

TL;DR

The paper addresses STFT phase retrieval on locally compact abelian groups by constructing window functions and uniformly discrete sampling sets that enable unique recovery of $f\in L^2(K)$ from phaseless samples $|V_g f|$. It develops two key tools: (i) uniformly discrete uniqueness sets for Paley-Wiener spaces and (ii) probabilistic completeness of translates to guarantee phase retrieval from phaseless STFT measurements, extending known results from $\mathbb{R}^d$ to broad LCA groups. By combining these tools and handling product groups, the authors prove Theorem A, establishing STFT phase retrieval on $L^2(K)$ under structural group conditions, and they explain fundamental obstructions to full $L^2(G)$ phase retrieval. The results justify restricting attention to compactly supported subspaces and provide a framework for practical phase retrieval in a wide class of groups, with potential applications in ptychography and time-frequency analysis.

Abstract

We study the short-time Fourier transform phase retrieval problem in locally compact abelian groups. Using probabilistic methods, we show that for a large class of groups $G$ and compact subsets $K\subseteq G$ there exists a window function and a uniformly discrete set in $G\times \widehat{G}$ allowing phase retrieval in $L^2(K)$. We also study the obstructions for STFT phase retrieval on $L^2(G)$, motivating the restriction to compactly supported function spaces.

Theorems & Definitions (45)

• Theorem 2.1
• Definition 2.2
• Lemma 2.3
• Definition 2.4
• Definition 2.5
• Definition 2.6
• Definition 2.7
• Theorem 3.1
• Definition 3.2
• Proposition 3.3