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

This paper studies the phase-only reconstruction problem of recovering a complex-valued signal $\textbf{x}$ in $\mathbb{C}^d$ from the phase of $\textbf{Ax}$ where $\textbf{A}$ is a given measurement matrix in $\mathbb{C}^{m\times d}$. The reconstruction, if possible, should be up to a positive scaling factor. By using the rank of discriminant matrices, uniqueness conditions are derived to characterize whether the underlying signal can be uniquely reconstructed. We are also 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 $\textbf{x}\in\mathbb{C}^d$, whereas the minimal measurement number is exactly $2d-1$ if we pursue the recovery of almost all signals. Moreover, when adapted to the phase-only reconstruction of $\textbf{x}\in\mathbb{R}^d$, our uniqueness conditions are more practical and general than existing ones. Our theoretical results can be straightforwardly extended to affine phase-only reconstruction where the phase of $\textbf{Ax}+\textbf{b}$ is observed for some $\textbf{b}\in\mathbb{C}^m$.