The phase problem

Abstract

The paper discusses the use of the theory of entire functions for solving the phase problem. In all practical cases only three forms of logarithmic Hilbert transform could possibly be required. The paper defines them and analyses their applicability. A generating form is also put forward for cases of possible theoretical interest. The uniqueness of the phase obtained from a logarithmic Hilbert transform is investigated and the difficulties due to the presence of zeros in the complex plane are discussed. Methods are put forward for both the removal of the zeros and, when this is not possible, for locating them in order to include their effect. The paper analyses known experimental methods for phase determination from the point of view of the theory presented and highlights their unique character.