Phase retrieval from sampled Gabor transform magnitudes: Counterexamples

Abstract

We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(\mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions have good concentration in both time and frequency and can be constructed to be real-valued for rectangular lattices.

6 pages, 1 figure; fixed a minor typo

References in corpus (3)

• Uniqueness of STFT phase retrieval for bandlimited functions
• On foundational discretization barriers in STFT phase retrieval
• Injectivity of Gabor phase retrieval from lattice measurements

Cited by in corpus (9)

• On foundational discretization barriers in STFT phase retrieval
• Injectivity of Gabor phase retrieval from lattice measurements
• Non-uniqueness theory in sampled STFT phase retrieval
• Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions
• Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval
• Phase retrieval of entire functions and its implications for Gabor phase retrieval
• Phaseless sampling on square-root lattices
• Arithmetic progressions and holomorphic phase retrieval
• Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$