论文标题
采样STFT期检索中的非唯一性理论
Non-uniqueness theory in sampled STFT phase retrieval
论文作者
论文摘要
从其频谱图（即，其短期傅立叶变换（STFT））从其频谱图中重建的重建是在几种重要应用中的关键问题，包括相干衍射成像和音频处理。这是一个经典的结果，对于合适的窗口，原则上，任何功能都可以从其频谱图中唯一地恢复到全局相位因子。但是，对于大多数实际应用，只有频谱图的离散样本（通常来自晶格）。这就提出了一个问题，即频谱图的晶格样本是否包含足够的信息来确定l^2（\ mathbb {r}^d）$中的函数$ f \至一个全局相位因子。在本文中，我们通过提供一般的非识别性结果来回答这个问题，从而导致采样的STFT阶段检索问题的非唯一性理论。 Precisely, given any dimension $d$, any window function $g$ and any (symplectic or separable) lattice $\mathcal{L} \subseteq \mathbb{R}^d$, we construct pairs of functions $f,h\in L^2(\mathbb{R}^d)$ that do not agree up to a global phase factor, but whose spectrograms agree on $ \ MATHCAL {L} $。我们的技术足够灵活，可以在更严格的假设下为独特的可恢复性提供反例。例如，如果窗口函数是真实价值的，则函数$ f，甚至可以选择满足$ | f | = | h | $。因此，我们的结果揭示了在没有相信息的情况下关键采样密度的不存在，这种属性与唯一性形成鲜明对比导致时间频率分析。
The reconstruction of a function from its spectrogram (i.e., the absolute value of its short-time Fourier transform (STFT)) arises as a key problem in several important applications, including coherent diffraction imaging and audio processing. It is a classical result that for suitable windows any function can, in principle, be uniquely recovered up to a global phase factor from its spectrogram. However, for most practical applications only discrete samples - typically from a lattice - of the spectrogram are available. This raises the question of whether lattice samples of the spectrogram contain sufficient information for determining a function $f\in L^2(\mathbb{R}^d)$ up to a global phase factor. In the present paper, we answer this question in the negative by providing general non-identifiability results which lead to a non-uniqueness theory for the sampled STFT phase retrieval problem. Precisely, given any dimension $d$, any window function $g$ and any (symplectic or separable) lattice $\mathcal{L} \subseteq \mathbb{R}^d$, we construct pairs of functions $f,h\in L^2(\mathbb{R}^d)$ that do not agree up to a global phase factor, but whose spectrograms agree on $\mathcal{L}$. Our techniques are sufficiently flexible to produce counterexamples to unique recoverability under even more stringent assumptions; for example, if the window function is real-valued, the functions $f,h$ can even be chosen to satisfy $|f|=|h|$. Our results thus reveal the non-existence of a critical sampling density in the absence of phase information, a property which is in stark contrast to uniqueness results in time-frequency analysis.
