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Conjugate phase retrieval in shift-invariant spaces generated by a Gaussian

Yang Chen, Cheng Cheng

TL;DR

This paper shows that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set, and shows that a function in the shift-invariant space generated by a Gaussian can be uniquely determined.

Abstract

Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this paper, we explore the conjugate phase retrieval in a shift-invariant space generated by a Gaussian funciton. First, we show that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set. We then show that a function in the shift-invariant space generated by a Gaussian can be uniquely determined, up to a unimodular constant and conjugation, from its phaseless Hermite samples on a discrete set. For the functions with finite coefficient sequences, we provide an explicit reconstruction procedure.

Conjugate phase retrieval in shift-invariant spaces generated by a Gaussian

TL;DR

This paper shows that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set, and shows that a function in the shift-invariant space generated by a Gaussian can be uniquely determined.

Abstract

Conjugate phase retrieval considers the recovery of a function, up to a unimodular constant and conjugation, from its phaseless measurements. In this paper, we explore the conjugate phase retrieval in a shift-invariant space generated by a Gaussian funciton. First, we show that the modulus function in the Gaussian shift-invariant space can be determined from the phaseless Hermite samples taken on a discrete sampling set. We then show that a function in the shift-invariant space generated by a Gaussian can be uniquely determined, up to a unimodular constant and conjugation, from its phaseless Hermite samples on a discrete set. For the functions with finite coefficient sequences, we provide an explicit reconstruction procedure.

Paper Structure

This paper contains 7 sections, 11 theorems, 111 equations, 1 algorithm.

Table of Contents

  1. Introduction
  2. Phaseless Hermite sampling in a shift-invariant space with Gaussian
  3. Conjugate phaseless sampling and reconstruction
  4. Conjugate phase retrieval in shift-invariant spaces generated by a Hermite function
  5. Gaussian shift-invariant functions with finite coefficients
  6. Proof of Lemma \ref{['f.abs.lem']}
  7. Proof of Lemma \ref{['pr.entire.lem']}

Key Result

Theorem 2.1

Let $f$ be a function in the shift-invariant space $V^\infty_{\beta, \lambda}$ satisfying signal.assump, and $\Gamma$ be a sampling set with the lower Beurling density $D^-(\Gamma)>2\beta^{-1}$. Then that is, the samples of $|f|$ and $|f'|$ on $\Gamma$ uniquely determine the entire modulus functions $|f|$ and $|f'|$ on $\mathbb R$.

Theorems & Definitions (19)

  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • proof : Proof of Theorem \ref{['cpr.sis.thm1']}
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof : Proof of Theorem \ref{['main.thm']}:
  • Definition 3.4
  • Proposition 3.5
  • ...and 9 more