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

Philipp Grohs, Lukas Liehr, Irina Shafkulovska

TL;DR

This work connects phaseless short-time Fourier transform sampling to completeness of discrete translates, establishing a dimension-free framework that reduces STFT phase retrieval to completeness properties of window translates. A central result shows that, if the exponential system is complete on $K-K$ and the translates of $g_\omega$ are complete on $K$, then a lattice $U=\Lambda\times\Gamma$ is a uniqueness set for phase retrieval on $L^2(K)$, for broad window classes. Concrete instantiations include Gaussian, bandlimited, and Hermite-type windows, yielding explicit lattice (and irregular) sampling schemes with controllable density; in particular, zero-density irregular sampling is possible for certain subspaces. The findings advance practical phaseless sampling by providing verifiable completeness criteria and broad applicability to time-frequency analysis and diffraction-type applications.

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.

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

TL;DR

This work connects phaseless short-time Fourier transform sampling to completeness of discrete translates, establishing a dimension-free framework that reduces STFT phase retrieval to completeness properties of window translates. A central result shows that, if the exponential system is complete on and the translates of are complete on , then a lattice is a uniqueness set for phase retrieval on , for broad window classes. Concrete instantiations include Gaussian, bandlimited, and Hermite-type windows, yielding explicit lattice (and irregular) sampling schemes with controllable density; in particular, zero-density irregular sampling is possible for certain subspaces. The findings advance practical phaseless sampling by providing verifiable completeness criteria and broad applicability to time-frequency analysis and diffraction-type applications.

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 with compact, are uniquely determined by phaseless lattice-samples of its short-time Fourier transform with window function , provided that specific density properties of translates of 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.
Paper Structure (11 sections, 12 theorems, 56 equations)

This paper contains 11 sections, 12 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Uniqueness-completeness relation
  4. Gaussians
  5. Bandlimited functions
  6. Hermite functions and the class $\mathcal{P}_{\alpha,\beta}$
  7. Proof and consequences of Theorem \ref{['thm:relation']}
  8. Proof and discussion of Proposition \ref{['prop:gauss']} -- \ref{['prop:p_ab']}
  9. Gaussian window functions
  10. Bandlimited window functions
  11. The class $\mathcal{P}_{\alpha,\beta}$

Key Result

Theorem 2.1

Let $K\subseteq {{\mathbb R}^d}$ be compact and let $\Gamma \subseteq {{\mathbb R}^d}$ such that $\mathcal{E}(\Gamma)$ is complete in $L^2(K-K)$. Further, let $g \in C({{\mathbb R}^d})$ and let $\Lambda \subseteq {{\mathbb R}^d}$ such that $\mathcal{T}(g_\omega,\Lambda)$ is complete in $C(K)$ for ev

Theorems & Definitions (24)

  • Definition 1.1
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:relation']}
  • Lemma 3.2
  • proof
  • ...and 14 more