论文标题
多窗口STFT阶段检索：晶格独特性
Multi-window STFT phase retrieval: lattice uniqueness
论文作者
论文摘要
短期傅立叶变换（STFT）阶段检索是指从其频谱图中重建功能$ f $的重建，即其短期傅立叶变换$ v_gf $带有窗口函数$ g $的幅度。众所周知，对于适当的窗口，l^2（\ mathbb {r}）$中的任何功能$ f \都可以从完整的频谱图$ | v_g f（\ mathbb {r}^2）| $中重新构造，在实际场景中，必须从iNdectal samples中获得重建，典型地在lattice上取出。事实证明，采样问题变得更加微妙：最近的结果表明，无论选择窗口函数或晶格密度如何，都无法实现通过晶格采样的唯一性。在本文中，我们启动了多窗口STFT期检索的研究，这是一种有效绕过单窗口案中遇到的离散障碍的一种方式。通过在多窗口Gabor系统，在Fock空间中进行采样以及有限帧的相位检索之间建立链接，我们得出了可以从晶格上的频谱样本中唯一恢复方形函数的条件。具体而言，我们在窗口功能上提供条件$ g_1，\ dots，g_4 \ in l^2（\ mathbb {r}）$，使得每个$ f \ in l^2中的每个$ f \ in l^2（\ mathbb {r}）$从$ weft（| v_ v_ {g_1} f（| \，| v_ {g_4} f（a \ mathbb {z}^2）|。对于实值函数，$ | \ det a |^{ - 1} \ geq 2 $的密度就足够了。还显示了不规则抽样的相应结果。
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.
