One-dimensional discrete-time phase retrieval

Abstract

The phase retrieval problem has a long and rich history with applications in physics and engineering such as crystallography, astronomy, and laser optics. Usually, the phase retrieval consists in recovering a real-valued or complex-valued signal from the intensity measurements of its Fourier transform. If the complete phase information in frequency domain is lost then the problem of signal reconstruction is severelly ill-posed and possesses many non-trivial ambiguities. Therefore, it can only be solved using appropriate additional signal information. We restrict ourselves to one-dimensional discrete-time phase retrieval from Fourier intensities and particularly consider signals with finite support. In the first part of this section, we study the structure of the arising ambiguities of the phase retrieval problem and show how they can be characterized using the given Fourier intensity. Employing these observations, in the second part, we study different kinds of a priori assumptions on the signal, where we are especially interested in their ability to reduce the non-trivial ambiguities or even to ensure uniqueness of the solution. In particular, we consider the assumption of non-negativity of the solution signal, additional magnitudes or phases of some signal components in time domain, or additional intensities of interference measurements in frequency domain. Finally, we transfer our results to phase retrieval problems where the intensity measurements arise, for example, from the Fresnel or fractional Fourier transform.