Conjugate Phase Retrieval in Paley–Wiener Space

We consider the problem of conjugate phase retrieval in Paley-Wiener space PWπ. 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\left( t \right)} $ are not necessarily distinguishable from the available data. We show that conjugate phase retrieval can be accomplished in PWπ by sampling only on the real line by using structured convolutions. We also show that conjugate phase retrieval can be accomplished in PWπ by sampling both f and f′ only on the real line. Finally, we show that generically, conjugate phase retrieval can be accomplished by sampling at 3 times the Nyquist rate, whereas phase retrieval requires sampling at 4 times the Nyquist rate.