Multi-window STFT phase retrieval: Lattice uniqueness

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 $V_{g}f$ with window function g. While it is known that for appropriate windows, any function $f\in L^2\left(R\right)$ can be reconstructed from the full spectrogram $\left|V_{g}f\right(R^2\left)\right|$, 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 $g_1,\dots ,g_4\in L^2\left(R\right)$, such that every $f\in L^2\left(R\right)$ is determined up to a global phase from$\left(\right|V_{g_1}f\left(A{Z}^2\right)|,\dots ,|V_{g_4}f\left(A{Z}^2\right)\left|\right)$ whenever $A\in {\mathrm{GL}}_2\left(R\right)$ satisfies the density condition $|\det A{|}^{-1}\ge 4$. For real-valued functions, a density of $|\det A{|}^{-1}\ge 2$ is sufficient. Corresponding results for irregular sampling are also shown.