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Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval

Matthias Wellershoff

TL;DR

This work proves that sampling the magnitudes of the Gabor transform at two frequency bins spaced by $\omega_1-\omega_0>0$ and at a rate $1/(4B)$ uniquely determines a bandlimited signal $f \in \mathrm{PW}_B^2$ up to a global phase. The method translates the problem to the Bargmann transform, reducing it to recovering a second-order entire function from magnitude data on two parallel lines; a Hadamard factorization argument shows uniqueness up to a phase unless an evenly spaced zero sequence occurs, which is ruled out by a Müntz–Szász type completeness result for the bandlimited Fourier support. Consequently, $f$ and any other $g$ yielding the same magnitudes must satisfy $f \sim g$. The result extends prior Gabor phase retrieval findings and aligns with analogous outcomes for Cauchy wavelets, providing a practical two-bin sampling scheme for unique reconstruction of bandlimited signals.

Abstract

We show that bandlimited signals can be uniquely recovered (up to a constant global phase factor) from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.

Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval

TL;DR

This work proves that sampling the magnitudes of the Gabor transform at two frequency bins spaced by and at a rate uniquely determines a bandlimited signal up to a global phase. The method translates the problem to the Bargmann transform, reducing it to recovering a second-order entire function from magnitude data on two parallel lines; a Hadamard factorization argument shows uniqueness up to a phase unless an evenly spaced zero sequence occurs, which is ruled out by a Müntz–Szász type completeness result for the bandlimited Fourier support. Consequently, and any other yielding the same magnitudes must satisfy . The result extends prior Gabor phase retrieval findings and aligns with analogous outcomes for Cauchy wavelets, providing a practical two-bin sampling scheme for unique reconstruction of bandlimited signals.

Abstract

We show that bandlimited signals can be uniquely recovered (up to a constant global phase factor) from Gabor transform magnitudes sampled at twice the Nyquist rate in two frequency bins.
Paper Structure (3 sections, 6 theorems, 35 equations)

This paper contains 3 sections, 6 theorems, 35 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The main result

Key Result

Lemma 1

Let $B > 0$, $\omega \in \mathbb{R}$ and $f \in \mathrm{PW}_B^2$. Then, $x \mapsto \left\lvert \mathcal{G} f(x,\omega) \right\rvert^2 \in \mathrm{PW}_{2B}^2$.

Theorems & Definitions (14)

  • Remark 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 3
  • proof
  • Remark 2
  • Theorem 4: Zalik's theorem; cf. Theorem 4 in zalik1978approximation
  • proof
  • ...and 4 more