Multi-window STFT phase retrieval: lattice uniqueness

TL;DR

This paper explores multi-window STFT phase retrieval, demonstrating that using multiple windows can overcome lattice sampling limitations and enable unique reconstruction of functions from spectrogram samples.

Contribution

It introduces conditions on multiple window functions that ensure unique function recovery from lattice-sampled spectrograms, surpassing single-window limitations.

Findings

Multi-window approach enables unique reconstruction from lattice samples.

Conditions on window functions and sampling density are established.

Results extend to irregular sampling scenarios.

Abstract

Short-time Fourier transform (STFT) phase retrieval refers to the reconstruction of a function $f$ from its spectrogram, i.e., the magnitudes of its short-time Fourier transform $V_gf$ with window function $g$. While it is known that for appropriate windows, any function $f \in L^2(\mathbb{R})$ can be reconstructed from the full spectrogram $|V_g f(\mathbb{R}^2)|$, in practical scenarios, the reconstruction must be achieved from discrete samples, typically taken on a lattice. It turns out that the sampled problem becomes much more subtle: recent results have demonstrated that uniqueness via lattice-sampling is unachievable, irrespective of the choice of the window function or the lattice density. In the present paper, we initiate the study of multi-window STFT phase retrieval as a way to effectively bypass the discretization barriers encountered in the single-window case. By establishing a link between multi-window Gabor systems, sampling in Fock space, and phase retrieval for finite frames, we derive conditions under which square-integrable functions can be uniquely recovered from spectrogram samples on a lattice. Specifically, we provide conditions on window functions $g_1, \dots, g_4 \in L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ is determined up to a global phase from $$\left(|V_{g_1}f(A\mathbb{Z}^2)|, \, \dots, \, |V_{g_4}f(A\mathbb{Z}^2)| \right)$$ whenever $A \in \mathrm{GL}_2(\mathbb{R})$ satisfies the density condition $|\det A|^{-1} \geq 4$. For real-valued functions, a density of $|\det A|^{-1} \geq 2$ is sufficient. Corresponding results for irregular sampling are also shown.