On foundational discretization barriers in STFT phase retrieval

TL;DR

This paper proves fundamental limitations in STFT phase retrieval, showing that no window function and lattice can uniquely determine a signal from spectrogram samples, implying the need for prior restrictions in discretized phase retrieval.

Contribution

It establishes that the spectrogram measurement operator cannot be injective for any window and lattice, revealing inherent discretization barriers in STFT phase retrieval.

Findings

No window and lattice combination yields unique phase retrieval.

Spectrogram samples do not determine signals up to a global phase.

Real-valued signals can produce identical spectrograms without being scalar multiples.

Abstract

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