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

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

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

由于其在音频处理和相干衍射成像等非常广泛的应用领域的出现，短时傅里叶变换（STFT）相位恢复问题近年来引起了广泛的关注。 STFT 相位检索的一个中心问题涉及哪些窗口函数\(g \in {L^2({\mathbb R}^d)}\)和哪些采样集\(\Lambda \subseteq {\mathbb R} ^{2d}\)是每个\(f \in {L^2({\mathbb R}^d)}\)由以下形式的无相样本唯一确定（直到全局相位因子）

其中\(V_gf\)表示f相对于g的 STFT 。对这个问题的研究是使问题易于计算处理的关键一步。然而，它在根本上和微妙的方式上偏离了普通的采样任务：最近的结果表明，如果\(\Lambda \)是一个格子，即\(\Lambda = A{\mathbb Z}^{2d}，则唯一性是无法实现的， A \in \textrm{GL}(2d,{\mathbb R})\)。在这种离散化障碍的驱动下，本文围绕一种新颖的采样方案的启动展开，该方案允许通过无相 STFT 采样对任何平方可积函数进行独特的恢复。具体来说，我们证明了平方根格，即形式的集合

保证STFT相位检索问题的唯一性。结果适用于一大类窗函数，包括高斯函数