Sampling Density for Multi-window STFT Phase Retrieval

Abstract

We study phase retrieval for multi-window short-time Fourier transforms with windows given by complex linear combinations of the first two normalized Hermite functions. Assume that the family of window coefficient vectors does phase retrieval in $\bbC^2$. We prove that the multi-window magnitude samples uniquely determine every signal in $L^2(\bbR)$ up to a global phase whenever the sampling set is separated with lower Beurling density greater than $2$, or is a translated lattice with density at least $2$. Moreover, we show that the sampling density is the best possible, which improves a result in [P.~Grohs, L.~Liehr, and M.~Rathmair. Multi-window {STFT} phase retrieval: lattice uniqueness, J. Funct. Anal., 288(3): Paper No. 110733, 23, 2025].