论文标题
部分可观测时空混沌系统的无模型预测
Phaseless sampling on square-root lattices
论文作者
论文摘要
由于它在应用程序的广泛领域（例如音频处理和连贯的衍射成像）中，短期的傅立叶变换（STFT）阶段检索问题近年来引起了很大的关注。 STFT阶段检索中的一个核心问题涉及哪个窗口在l^2（\ mathbb {r}^d）中的$ g \ $ g \的问题，哪些样本设置了$ c \ subseteq \ subseteq \ subseteq \ mathbb {r}^{2d {2d} $是每个$ f \ in l^2（in l^2）表格的样本$$ | v_gf（λ）| = \ left \ {| v_gf（λ）| ：λ\inλ\ right \}，$$其中$ v_gf $表示相对于$ g $的短期傅立叶变换（stft）$ f $。对这个问题的调查构成了使问题计算可解决问题的关键步骤。但是，它以基本和微妙的方式偏离了普通的抽样任务：最近的结果表明，如果$λ$是晶格，即唯一性是无法实现的，即$λ= a \ a \ a \ mathbb {z}^{2d {2d} {2d}，a \ a \ a \ in \ mathrm {gl}（gl}（gl}（2d，\ mathbbbbbbbbbbbbbbbbbbb}）在这种离散障碍的驱动下，本文围绕着一种新型抽样方案的启动，该方案允许通过无量的STFT抽样来独特地恢复任何可恢复任何正方形的函数。具体而言，我们表明了平方根的晶格，即$$λ= a \ left（\ sqrt {\ sqrt {\ sqrt {\ mathbb {z}} \ right）的集合， \ Mathbb {n} _0 \}，$$保证STFT相位检索问题的唯一性。结果适用于包括高斯在内的大量窗口功能。
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.
