  We consider the problem of reconstructing two signals from the
autocorrelation and cross-correlation measurements. This inverse problem is a
fundamental one in signal processing, and arises in many applications,
including phase retrieval and blind channel estimation. In a typical phase
retrieval setup, only the autocorrelation measurements are obtainable. We show
that, when the measurements are obtained using three simple "masks", phase
retrieval reduces to the aforementioned reconstruction problem.
  The classic solution to this problem is based on finding common factors
between the $z$-transforms of the autocorrelation and cross-correlation
vectors. This solution has enjoyed limited practical success, mainly due to the
fact that it is not sufficiently stable in the noisy setting. In this work,
inspired by the success of convex programming in provably and stably solving
various quadratic constrained problems, we develop a semidefinite
programming-based algorithm and provide theoretical guarantees. In particular,
we show that almost all signals can be uniquely recovered by this algorithm (up
to a global phase). Comparative numerical studies demonstrate that the proposed
method significantly outperforms the classic method in the noisy setting.
