On the effect of zero-flipping on the stability of the phase retrieval problem in the Paley-Wiener class

TL;DR

This paper studies how zero-flipping affects the stability of phase retrieval in the Paley-Wiener class, revealing that zero-flipping alters bandlimitedness and proposing new measures for stability analysis.

Contribution

It introduces a new perspective on zero-flipping's impact on phase retrieval stability and proposes alternative metrics to better assess this stability.

Findings

Zero-flipping results in functions that are wide-banded but not bandlimited.

The traditional stability measure $ orm{cf - F_a f}_2$ is inadequate.

A new stability measure $ orm{c F_b f - F_a f}_2$ is introduced, dominated by the distance between $a$ and $b$.

Abstract

In the classical phase retrieval problem in the Paley-Wiener class $PW_L$ for $L>0$, i.e. to recover $f\in PW_L$ from $|f|$, Akutowicz, Walther, and Hofstetter independently showed that all such solutions can be obtained by flipping an arbitrary set of complex zeros across the real line. This operation is called zero-flipping and we denote by $\mathfrak{F}_a f$ the resulting function. The operator $\mathfrak{F}_a$ is defined even if $a$ is not a genuine zero of $f$, that is if we make an error on the location of the zero. Our main goal is to investigate the effect of $\mathfrak{F}_a$. We show that $\mathfrak{F}_af$ is no longer bandlimited but is still wide-banded. We then investigate the effect of $\mathfrak{F}_a$ on the stability of phase retrieval by estimating the quantity $\inf_{|c|=1}\|cf-\mathfrak{F}_af\|_2$. We show that this quantity is in general not well-suited to investigate stability, and so we introduce the quantity $\inf_{|c|=1}\|c\mathfrak{F}_bf-\mathfrak{F}_af\|_2$. We show that this quantity is dominated by the distance between $a$ and $b$.