Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
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We study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces V^∞(φ) from spectrogram measurements |𝒢f(X)| where 𝒢 is the Gabor transform and X ⊆ℝ^2. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on |f| result in stability estimates in the situation where one aims to reconstruct f on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, arXiv:2008.07238] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a non-iterative, provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond V^∞(φ), such as Paley-Wiener spaces.
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