Ambiguities in One-Dimensional Discrete Phase Retrieval from Fourier Magnitudes

Abstract

The present paper is a survey aiming at characterizing all solutions of the discrete phase retrieval problem. Restricting ourselves to discrete signals with finite support, this problem can be stated as follows. We want to recover a complex-valued discrete signal $$\mathbf{x} :\mathbb {Z}\rightarrow \mathbb {C}$$ with support $$\{ 0, \ldots , N-1 \}$$ from the modulus of its discrete-time Fourier transform $$\widehat{x}(\omega )$$ . We will give a full classification of all trivial and nontrivial ambiguities of the discrete phase retrieval problem. In our classification, trivial ambiguities are caused either by signal shifts in space, by multiplication with a rotation factor $$\mathrm {e}^{\mathrm {i}\alpha }$$ , $$\alpha \in [-\pi , \pi )$$ , or by conjugation and reflection of the signal. Furthermore, we show that all nontrivial ambiguities of the finite discrete phase retrieval problem can be characterized by signal convolutions. In the second part of the paper, we examine the usually employed a priori conditions regarding their ability to reduce the number of ambiguities of the phase retrieval problem or even to ensure uniqueness. For the corresponding proofs we can employ our findings on the ambiguity classification. The considerations on the structure of ambiguities also show clearly the ill-posedness of the phase retrieval problem even in cases where uniqueness is theoretically shown.