Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality

Abstract

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 noniterative, 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.

On foundational discretization barriers in STFT phase retrieval

“…Additionally, in the same article, it was shown that if the signal space is a Gaussian shift-invariant space with an irrational step-size, sampling on a lattice again suffices to achieve uniqueness. In [11] it was further proved that without an assumption on the step-size of the Gaussian shift-invariant space, the uniqueness property holds, assuming knowledge of the spectrogram on parallel lines in the time-frequency plane. 4.3.…”