Phaseless sampling on square-root lattices
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910926401175552 |
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| author | Grohs, Philipp Liehr, Lukas |
| author_facet | Grohs, Philipp Liehr, Lukas |
| contents | Due 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 $g \in L^2(\mathbb{R}^d)$ and which sampling sets $Λ\subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$ |V_gf(Λ)| = \left \{ |V_gf(λ)| : λ\in Λ\right \}, $$ where $V_gf$ denotes the short-time Fourier transform (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 $Λ$ is a lattice, i.e $Λ= A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. 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 $$ Λ= A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \}, $$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_11127 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Phaseless sampling on square-root lattices Grohs, Philipp Liehr, Lukas Functional Analysis Classical Analysis and ODEs Complex Variables 32A15, 32A60, 94A12, 94A20 Due 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 $g \in L^2(\mathbb{R}^d)$ and which sampling sets $Λ\subseteq \mathbb{R}^{2d}$ is every $f \in L^2(\mathbb{R}^d)$ uniquely determined (up to a global phase factor) by phaseless samples of the form $$ |V_gf(Λ)| = \left \{ |V_gf(λ)| : λ\in Λ\right \}, $$ where $V_gf$ denotes the short-time Fourier transform (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 $Λ$ is a lattice, i.e $Λ= A\mathbb{Z}^{2d}, A \in \mathrm{GL}(2d,\mathbb{R})$. 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 $$ Λ= A \left ( \sqrt{\mathbb{Z}} \right )^{2d}, \ \sqrt{\mathbb{Z}} = \{ \pm \sqrt{n} : n \in \mathbb{N}_0 \}, $$ guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians. |
| title | Phaseless sampling on square-root lattices |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables 32A15, 32A60, 94A12, 94A20 |
| url | https://arxiv.org/abs/2209.11127 |