On Foundational Discretization Barriers in STFT Phase Retrieval

Abstract

We prove that there exists no window function g∈ L2(R) and no lattice L⊂ R2 such that every f∈ L2(R) is determined up to a global phase by spectrogram samples | Vgf(L) | where Vgf denotes the short-time Fourier transform of f with respect to g. Consequently, the forward operator f↦|Vgf(L)|mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space [InlineEquation not available: see fulltext.] with f∼ h identifying two functions which agree up to a multiplicative constant of modulus one. We will further elaborate this result and point out that under mild conditions on the lattice L, functions which produce identical spectrogram samples but do not agree up to a unimodular constant can be chosen to be real-valued. The derived results highlight that in the discretization of the STFT phase retrieval problem from lattice measurements, a prior restriction of the underlying signal space to a proper subspace of L2(R) is inevitable.