Phaseless Sampling on Square-Root Lattices

Abstract

AbstractDue to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions $$g \in {L^2({\mathbb R}^d)}$$ g ∈ L 2 ( R d ) and which sampling sets $$\Lambda \subseteq {\mathbb R}^{2d}$$ Λ ⊆ R 2 d is every $$f \in {L^2({\mathbb R}^d)}$$ f ∈ L 2 ( R d ) uniquely determined (up to a global phase factor) by phaseless samples of the form $$\begin{aligned} |V_gf(\Lambda )| = \left\{ |V_gf(\lambda )|: \lambda \in \Lambda \right\} , \end{aligned}$$ | V g f ( Λ ) | = | V g f ( λ ) | : λ ∈ Λ , where $$V_gf$$ V g f denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if $$\Lambda $$ Λ is a lattice, i.e $$\Lambda = A{\mathbb Z}^{2d}, A \in \textrm{GL}(2d,{\mathbb R})$$ Λ = A Z 2 d , A ∈ GL ( 2 d , R ) . Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form $$\begin{aligned} \Lambda = A \left( \sqrt{{\mathbb Z}} \right) ^{2d}, \ \sqrt{{\mathbb Z}} = \{ \pm \sqrt{n}: n \in {\mathbb N}_0 \}, \end{aligned}$$ Λ = A Z 2 d , Z = { ± n : n ∈ N 0 } , guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians