Phase retrieval from short-time Fourier transform in LCA groups

TL;DR

This paper investigates phase retrieval using the short-time Fourier transform in locally compact abelian groups, demonstrating conditions under which phase retrieval is possible with specific window functions and discrete sets, and exploring obstructions in non-compact cases.

Contribution

It establishes probabilistic conditions for phase retrieval in LCA groups and analyzes obstructions, extending phase retrieval theory to a broad class of groups.

Findings

Existence of window functions enabling phase retrieval in L^2(K)

Probabilistic methods for constructing phase retrieval sets

Identification of obstructions in non-compact group settings

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.