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Phaseless sampling on square-root lattices

Philipp Grohs, Lukas Liehr

TL;DR

This work addresses phaseless sampling in STFTs and shows that uniqueness for recovering $f\in L^2(\mathbb R^d)$ up to a global phase is impossible on ordinary lattices but achievable on square-root lattices. It introduces the analytic function space $O_a^b({\mathbb C}^d)$ and proves that, for windows in this class, sampling on a rectangular square-root lattice $\Lambda = \mathrm{diag}(\tau,\nu)(\sqrt{\mathbb Z})^{2d}$ with explicit density bounds yields $|V_\varphi f(\lambda)|=|V_\varphi h(\lambda)|$ on $\Lambda$ implies $f\sim h$, including Gaussians and Hermite-type windows. The results extend to non-diagonal generating matrices via metaplectic transforms, enabling uniqueness for rotated and sheared square-root lattices. These findings provide the first broad, discreteness-based uniqueness guarantees for phaseless STFT phase retrieval without compact support assumptions, with potential impact on coherent diffraction imaging and audio processing.

Abstract

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $g \in L^2(\mathbb{R}^d)$ and which sampling sets $Λ\subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$ |V_gf(Λ)| = \left \{ |V_gf(λ)| : λ\in Λ\right \}, $$ where $V_gf$ denotes the short-time Fourier transform (STFT) of $f$ with respect to $g$. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $Λ$ is a lattice, i.e $Λ= A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$ Λ= A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \}, $$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians.

Phaseless sampling on square-root lattices

TL;DR

This work addresses phaseless sampling in STFTs and shows that uniqueness for recovering up to a global phase is impossible on ordinary lattices but achievable on square-root lattices. It introduces the analytic function space and proves that, for windows in this class, sampling on a rectangular square-root lattice with explicit density bounds yields on implies , including Gaussians and Hermite-type windows. The results extend to non-diagonal generating matrices via metaplectic transforms, enabling uniqueness for rotated and sheared square-root lattices. These findings provide the first broad, discreteness-based uniqueness guarantees for phaseless STFT phase retrieval without compact support assumptions, with potential impact on coherent diffraction imaging and audio processing.

Abstract

Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions and which sampling sets is every uniquely determined (up to a global phase factor) by phaseless samples of the form where denotes the short-time Fourier transform (STFT) of with respect to . The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if is a lattice, i.e . Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians.
Paper Structure (8 sections, 13 theorems, 90 equations, 2 figures)

This paper contains 8 sections, 13 theorems, 90 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Existing work on square-root sampling
  4. Preliminaries
  5. The function space $\mathcal{O}_{a}^b({\mathbb C}^d)$
  6. Uniqueness sets for $\mathcal{O}^b({\mathbb C}^d)$
  7. Proof of the main results
  8. Beyond diagonal generating matrices

Key Result

Theorem 1.1

Let $a,b \in {\mathbb R}^d_{>0}$, and let $0 \neq \varphi \in \mathcal{O}^a_b({\mathbb C}^d)$ be a window function. Suppose that $\Lambda=A(\sqrt{{\mathbb Z}})^{2d}$ is a rectangular square-root lattice such that the generating matrix $A = \mathrm{diag}(\tau_1,\dots,\tau_d,\nu_1,\dots,\nu_d)$ with $ Then the following statements are equivalent for every $f,h \in {L^2({\mathbb R}^d)}$:

Figures (2)

  • Figure 1: The left figure depicts an ordinary lattice of the form $A{\mathbb Z}^2$ while the right figure visualizes a square-root lattice $B(\sqrt{{\mathbb Z}})^2$$(A,B \in \mathrm{GL}(2,{\mathbb R}))$.
  • Figure 2: The left figure depicts a sheared square-root lattice. The right figure depicts a rotated square-root lattice.

Theorems & Definitions (26)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Theorem 2.4: Jensen
  • Proposition 2.5
  • ...and 16 more