Phaseless Sampling on Square-Root Lattices

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 \(\Lambda \subseteq {\mathbb R}^{2d}\) is every \(f \in {L^2({\mathbb R}^d)}\) uniquely determined (up to a global phase factor) by phaseless samples of the form

$$\begin{aligned} |V_gf(\Lambda )| = \left\{ |V_gf(\lambda )|: \lambda \in \Lambda \right\} , \end{aligned}$$

where \(V_gf\) denotes the 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 \(\Lambda \) is a lattice, i.e \(\Lambda = A{\mathbb Z}^{2d}, A \in \textrm{GL}(2d,{\mathbb R})\). 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

$$\begin{aligned} \Lambda = A \left( \sqrt{{\mathbb Z}} \right) ^{2d}, \ \sqrt{{\mathbb Z}} = \{ \pm \sqrt{n}: n \in {\mathbb N}_0 \}, \end{aligned}$$

guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians