# Codes and Expansions (CodEx) Seminar

## Eric Weber (Iowa State University)

Conjugate Phase Retrieval in Paley-Wiener Space

The phase retrieval problem is to reconstruct a vector (signal) from the magnitudes of linear measurements. This is an ill-posed problem, because any unimodular scalar multiple of the original signal cannot be disambiguated from the original signal. However, the phase retrieval problem attempts to reconstruct the signal subject to only this ambiguity.

The conjugate phase retrieval problem is weaker in the following sense: given the magnitudes of linear measurements of the signal, the goal is to reconstruct the signal up to the ambiguity of both unimodular scalars and conjugation of the signal. That is to say, we do not try to disambiguate a signal and its conjugate.

We begin by discussing the conjugate phase retrieval problem in the finite dimensional context. We compare and contrast the conjugate phase retrieval problem with the phase retrieval problem, and discuss the current limitations in our understanding of the conjugate phase retrieval problem.

We then consider the problem of conjugate phase retrieval in Paley-Wiener space. We show that conjugate phase retrieval can be accomplished in the Paley-Wiener space by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished by sampling both the signal and its derivative 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.