Compressive 1-D signal recovery from magnitude-only measurements via convex optimization

Signal recovery from the magnitudes of the Fourier transform is a classical problem. Due to the absence of phase information, signal recovery requires some form of additional prior information. In this paper, we assume the underling signal is compressive. In other words, the signal is not sparse in object domain but sparse on some frame. We develop a convex optimization method to recovery the signal from the magnitude of the Fourier transform of itself. With the proposed method, we can recovery compressive 1-D signal from magnitude-only measurements via optimization technique. Numerical results suggest that unique recovery is possible with a very high probability for sufficiently compressive signals.