Conjugate Phase Retrieval in Paley-Wiener Space

TL;DR

This paper explores conjugate phase retrieval in Paley-Wiener space, demonstrating methods for signal recovery from magnitude measurements, including structured convolutions and sampling strategies, with experimental validation and rate comparisons.

Contribution

It introduces new sampling and convolution techniques for conjugate phase retrieval in Paley-Wiener space, including practical algorithms and rate analysis.

Findings

Conjugate phase retrieval can be achieved by sampling on the real line using structured convolutions.

Sampling both the function and its derivative suffices for conjugate phase retrieval.

The Gerchberg-Saxton method effectively reconstructs signals in finite dimensions.

Abstract

We consider the problem of conjugate phase retrieval in Paley-Wiener space $PW_{\pi}$. The goal of conjugate phase retrieval is to recover a signal $f$ from the magnitudes of linear measurements up to unknown phase factor and unknown conjugate, meaning $f(t)$ and $\overline{f(t)}$ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in $PW_{\pi}$ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in $PW_{\pi}$ by sampling both $f$ and $f^{\prime}$ only on the real line. Moreover, we demonstrate experimentally that the Gerchberg-Saxton method of alternating projections can accomplish the reconstruction from vectors that do conjugate phase retrieval in finite dimensional spaces. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at three times the Nyquist rate, whereas phase retrieval requires sampling at four times the Nyquist rate.