Infinite-dimensional STFT phase retrieval from lattice samples: uniqueness and stability
Lukas Liehr  1@  
1 : University of Vienna

The STFT phase retrieval problem arises as the result of trying to invert the mapping that sends a square-integrable function to samples of its spectrogram with respect to a certain window function. Regarded as a non-linear inverse problem, the investigation of uniqueness and stability of the STFT phase retrieval problem is of major importance. It is folklore in the phase retrieval community that under suitable assumptions on the window function, every square-integrable function is determined up to a global phase from spectrogram data, provided that the sampling set is a continuous domain such as an open set or the entire time-frequency plane. We are interested in the case where the sampling set is separated, most notably a lattice. First, we present a result which reveals a fundamental difference between the continuous and discrete case: there exists no window function such that every square-integrable map is determined up to a global phase by spectrogram data given on a lattice. If the window function is a Gaussian then the STFT phase retrieval problem is known as the Gabor phase retrieval problem. In this setting, uniqueness from lattice samples can be achieved by imposing a support condition on the underlying signal class. Finally, we consider shift-invariant spaces and demonstrate that Gabor phase retrieval in Gaussian shift-invariant spaces from lattice samples is possible under a suitable assumption on the step-size of the space. In addition, we highlight that the shift-invariant setting allows the design of a reconstruction algorithm which recovers functions in a provably and stably manner.

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