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Sampled Gabor phase retrieval - the problem of recovering a square-integrablesignal from the magnitude of its Gabor transform sampled on a lattice - is afundamental problem in signal processing, with important applications in areassuch as imaging and audio processing. Recently, a classification ofsquare-integrable signals which are not phase retrievable from Gabormeasurements on parallel lines has been presented. This classification was usedto exhibit a family of counterexamples to uniqueness in sampled Gabor phaseretrieval. Here, we show that the set of counterexamples to uniqueness insampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equalto the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes toa better understanding of the fundamental limits of sampled Gabor phaseretrieval.

Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$