Closed-form algorithms for phase retrieval with an additive point signal

The phase retrieval problem is a problem of a reconstruction of a signal from the magnitude of its Fourier transform. In this paper, we consider the problem of reconstructing an unknown one-dimensional signal from the magnitude of its Fourier transform and the magnitude of the Fourier transform of another signal that is made by the addition of a point signal at the center of the original signal. After we show that there are exactly two solution signals that satisfy the given condition, we develop a closed-form algorithm that may reconstruct the two solution signals.