LAUSR

We prove that there exists no window function $$g \in {L^2(\mathbb {R})}$$ g ∈ L 2 ( R ) and no lattice $${\mathcal {L}} \subset \mathbb {R}^2$$ L ⊂ R 2 such that every $$f \in {L^2(\mathbb {R})}$$ f ∈ L 2 ( R ) is determined up to a global phase by spectrogram samples $$|V_gf({\mathcal {L}})|$$ | V g f ( L ) | where $$V_gf$$ V g f denotes the short-time Fourier transform of f with respect to g . Consequently, the forward operator $$\begin{aligned} f \mapsto |V_gf({\mathcal {L}})| \end{aligned}$$ f ↦ | V g f ( L ) | mapping a square-integrable function to its spectrogram samples on a lattice is never injective on the quotient space with $$f \sim h$$ 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 $${\mathcal {L}}$$ 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 $${L^2(\mathbb {R})}$$ L 2 ( R ) is inevitable.