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 L2(R) which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions can be constructed to be either real-valued or complex-valued and have good concentration in both time and frequency.