The Use of Discrete Hilbert Transforms in Phase Extension and Improvement

Abstract

A new totally model independent procedure for phase extension and improvement in X-ray crystallography, based on the discrete Hilbert transforms, is presented. The method has been tested using simulated diffraction data of a small molecule and simulated and experimental data of a protein crystal. Starting from a randomly incomplete set of correct phases, it allows calculation of the unknown phases. Moreover, a set of phases affected by a mean phase error of ±90° can be improved to a mean error of ±25° if the correct figures of merit for the reflections are known. The performance and the limitations of the technique, as well as the perspectives for further developments, are discussed.