Signal Reconstruction from Phase-only Measurements: Uniqueness Condition, Minimal Measurement Number and Beyond

Given a measurement matrix $\textbf{A}\in \mathbb{C}^{m\times d}$, this paper studies the phase-only reconstruction problem where the aim is to recover a complex signal $\textbf{x}$ in $\mathbb{C}^d$ from the phase of $\textbf{Ax}$. The reconstruction, if possible, should be up to a positive scaling factor. By using the rank of measurement matrices, uniqueness conditions are derived to characterize whether the underlying signal can be uniquely reconstructed. Also we are interested in the problem of minimal measurement number. We show that at least $2d$ but no more than $4d-2$ measurements are needed for the reconstruction of all $\textbf{x}\in\mathbb{C}^d$, while the minimal measurement number is $2d-1$ if we pursue the recovery of almost all signals. Moreover, our uniqueness conditions can be directly adapted to the phase-only reconstruction of $\textbf{x}\in\mathbb{R}^d$, which turns out to be more practical and general than existing uniqueness criteria. Finally, we show our theoretical results can be straightforwardly extended to affine phase-only reconstruction where the phase of $\textbf{Ax}+\textbf{b}$ is observed for some $\textbf{b}\in\mathbb{C}^d$.