多窗口STFT相位恢复：格点唯一性
Multi-window STFT phase retrieval: lattice uniqueness
AI总结：
本文提出多窗口STFT相位恢复方法，通过建立多窗口Gabor系统、Fock空间采样与有限框架相位恢复的联系，克服单窗口格点采样无法实现唯一重建的障碍，并给出了密度条件。
摘要：
短时傅里叶变换（STFT）相位恢复是指从函数的谱图，即其关于窗函数g的短时傅里叶变换V_gf的模值，重建函数f。已知对于适当的窗函数，任何函数f∈L^2(R)都可以从完整谱图|V_g f(R^2)|中重建；但在实际场景中，重建必须从离散样本中实现，通常是在格点上采样。结果表明，采样问题变得更加微妙：近期结果已证明，无论窗函数或格点密度如何选择，通过格点采样实现唯一性都是不可实现的。在本文中，我们开创性地研究了多窗口STFT相位恢复，以此有效绕过单窗口情形中遇到的离散化障碍。通过建立多窗口Gabor系统、Fock空间中的采样以及有限框架相位恢复之间的联系，我们推导出平方可积函数可以从格点上的谱图样本中被唯一恢复的条件。具体而言，我们给出了窗函数g_1, ..., g_4∈L^2(R)的条件，使得当A∈GL_2(R)满足密度条件|det A|^{-1}≥4时，每个f∈L^2(R)都能由(|V_{g_1}f(AZ^2)|, ..., |V_{g_4}f(AZ^2)|)确定到全局相位。对于实值函数，密度|det A|^{-1}≥2即已足够。文中还给出了不规则采样的相应结果。
英文摘要：
Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function $f$ from its spectrogram, i.e., the magnitudes of its short-time Fourier transform $V_gf$ with window function $g$. While it is known that for appropriate windows, any function $f \in L^2(\mathbb{R})$ can be reconstructed from the full spectrogram $|V_g f(\mathbb{R}^2)|$, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions $g_1, \dots, g_4 \in L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ is determined up to a global phase from $$\left(|V_{g_1}f(A\mathbb{Z}^2)|, \, \dots, \, |V_{g_4}f(A\mathbb{Z}^2)| \right)$$ whenever $A \in \mathrm{GL}_2(\mathbb{R})$ satisfies the density condition $|\det A|^{-1} \geq 4$. For real-valued functions, a density of $|\det A|^{-1} \geq 2$ is sufficient. Corresponding results for irregular sampling are also shown.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
