Abstract
We show that bandlimited signals can be uniquely recovered (up to a constant global phase factor) from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.
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Notes
A lattice \(\Lambda \subset {\mathbb {R}}^2\) is a discrete subset of the time–frequency plane that can be written as \(\Lambda = L{\mathbb {R}}^k\), where \(L \in {\mathbb {R}}^{2 \times k}\) is a matrix with linearly independent columns and \(k \in \{1,2\}\).
Actually, the range of the Bargmann transform can be equipped with an inner product and thereby turned into a Hilbert space which known as the Fock space \({\mathcal {F}}^2({\mathbb {C}})\). The Bargmann transform then turns out to be a unitary operator mapping \(L^2({\mathbb {R}})\) onto \({\mathcal {F}}^2({\mathbb {C}})\) [8, Theorem 3.4.3 on p. 56].
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Acknowledgements
The author would like to thank A. Bandeira for posing the question which is being answered in the present letter. In addition, the author would like to thank the reviewers for their comments which allowed for a significant improvement in the presentation of the paper. Finally, the author acknowledges funding through the Swiss National Science Foundation Grant 200021_184698.
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Communicated by Jaming Philippe.
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Wellershoff, M. Sampling at Twice the Nyquist Rate in Two Frequency Bins Guarantees Uniqueness in Gabor Phase Retrieval. J Fourier Anal Appl 29, 7 (2023). https://doi.org/10.1007/s00041-022-09990-y
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DOI: https://doi.org/10.1007/s00041-022-09990-y
Keywords
- Phase retrieval
- Gabor transform
- Nyquist–Shannon sampling
- Hadamard factorisation theorem
- Müntz–Szász type result