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STFT Phase Retrieval with Two Window Functions

Ting Chen, Hanwen Lu, Wenchang Sun, Yutong Zhao

TL;DR

The work advances STFT phase retrieval by showing that two compactly supported windows suffice to uniquely determine a broad class of signals from phaseless STFT samples, provided careful sampling along lines or lattices. The authors develop a semi-discrete framework and prove a robust local-to-global uniqueness principle, then extend to fully discrete sampling for periodic, quasi-periodic, and locally integrable signals, identifying precise irrational-ratio conditions that guarantee uniqueness and illustrating sharp counterexamples for rational spacing or separable signals. They establish an optimal sampling regime for compact windows and demonstrate how two-window schemes can resolve phase ambiguities that arise in single-window STFT retrieval. The results have potential impact on applications in diffraction imaging and optics where phase information is typically lost, offering rigorous conditions under which reliable signal reconstruction is possible from phaseless measurements.

Abstract

In this paper, we consider the uniqueness of STFT phase retrieval with two window functions. We show that a complex-valued locally integrable nonseparable signal is uniquely determined up to a global phase by phaseless samples of its short time Fourier transforms with respect to two well-chosen window functions over countable parallel lines or certain lattices. Moreover, we give the optimal sampling interval for STFT phase retrieval with compactly supported window functions. For periodic locally integrable signals, we obtain a uniqueness result for STFT phase retrieval with sampled values over two parallel lines whose distance is an irrational multiple of the period. And for quasi-periodic signals, we obtain a similar result.

STFT Phase Retrieval with Two Window Functions

TL;DR

The work advances STFT phase retrieval by showing that two compactly supported windows suffice to uniquely determine a broad class of signals from phaseless STFT samples, provided careful sampling along lines or lattices. The authors develop a semi-discrete framework and prove a robust local-to-global uniqueness principle, then extend to fully discrete sampling for periodic, quasi-periodic, and locally integrable signals, identifying precise irrational-ratio conditions that guarantee uniqueness and illustrating sharp counterexamples for rational spacing or separable signals. They establish an optimal sampling regime for compact windows and demonstrate how two-window schemes can resolve phase ambiguities that arise in single-window STFT retrieval. The results have potential impact on applications in diffraction imaging and optics where phase information is typically lost, offering rigorous conditions under which reliable signal reconstruction is possible from phaseless measurements.

Abstract

In this paper, we consider the uniqueness of STFT phase retrieval with two window functions. We show that a complex-valued locally integrable nonseparable signal is uniquely determined up to a global phase by phaseless samples of its short time Fourier transforms with respect to two well-chosen window functions over countable parallel lines or certain lattices. Moreover, we give the optimal sampling interval for STFT phase retrieval with compactly supported window functions. For periodic locally integrable signals, we obtain a uniqueness result for STFT phase retrieval with sampled values over two parallel lines whose distance is an irrational multiple of the period. And for quasi-periodic signals, we obtain a similar result.
Paper Structure (11 sections, 13 theorems, 191 equations, 1 figure)

This paper contains 11 sections, 13 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some notations
  4. Completeness of complex exponentials
  5. Phase retrieval of compactly supported functions from their Fourier transforms
  6. Locally periodic functions
  7. Phase retrieval from semi-discrete samples
  8. Phase retrieval from discrete samples
  9. Periodic signals
  10. Quasi-periodic signals
  11. Locally integrable signals

Key Result

Theorem 1.2

Suppose that $B$ is a positive contant, $\phi \in L^{p'}(\mathbb{R})$ for some $1\le p\le \infty$ with $\operatorname{supp}( \phi) \subset [-B, B]$, $\overline{\phi(-x)}=\phi(x)$ and Let $\psi(x)=\phi(x) (e^{ 2\pi ixb}-1)$, where $0<b\le 1/(2B)$. Then for any $0<a\le B$ and $f\in L^p(\mathbb R)$ which is $(2B-a)$-nonseparable, $f$ is determined up to a global phase by the measurements Moreover,

Figures (1)

  • Figure 1: Two signals $f$ and $g$.

Theorems & Definitions (35)

  • Definition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • ...and 25 more