"On the necessity of a nonvanishing ambiguity function in STFT phase retrieval"Bartusel, DavidWe consider the phase retrieval problem for short-time Fourier measurements with respect to various window functions. It is well-known that a nowhere vanishing ambiguity function of the window is sufficient for phase retrieval. We show that this condition is not necessary in many relevant settings by presenting explicit examples of windows guaranteeing short-time Fourier phase retrieval for the abelian groups $\mathbb{Z}$, $\mathbb{Z}_d$ and $\mathbb{R}^d$, despite their ambiguity functions vanishing on sets of positive measure. Along the way, we explore how the possibility of recovering a given signal is linked to certain connectedness properties with respect to the window. |
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