Injectivity of Gabor phase retrieval from lattice measurements

We establish novel uniqueness results for the Gabor phase retrieval problem: if c : L2 (R) -+ L2(R2) denotes the Gabor transform then every f E L4[- 2c, c2 ] is determined up to a global phase by the values |c f (x, omega)| where (x, omega) are points on the lattice b-1Z x (2c)-1Z and b > 0 is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Grochenig, Romero and Stockler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued. (c) 2022 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).