Phase retrieval for sparse signals using rank minimization

Signal recovery from the amplitudes of the Fourier transform, or equivalently from the autocorrelation function is a classical problem. Due to the absence of phase information, signal recovery requires some form of additional prior information. In this paper, the prior information we assume is sparsity. We develop a convex optimization based framework to retrieve the signal support from the support of the autocorrelation, and propose an iterative algorithm which terminates in a signal with the least sparsity satisfying the autocorrelation constraints. Numerical results suggest that unique recovery up to a global sign change, time shift and/or time reversal is possible with a very high probability for sufficiently sparse signals.