Phaseless Signal Reconstruction in Infinite Dimensional and Redundant Signal Spaces

• Applicant: Dr.-Ing. Volker Pohl
• Subject Area: Electronic Semiconductors, Components and Circuits, Integrated Systems, Sensor Technology, Theoretical Electrical Engineering
• Term: from 2014 to 2018
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 249381227

Signal reconstruction from magnitude measurements only (also known as phase retrieval) arises in many different applications and in diverse fields of engineering and science. It originates from the fact that detectors can often only record the square modulus of an electromagnetic wave but not its phase, especially for short wavelength. Since the phase often contains important information about the desired signal, phase retrieval is a challenging problem for engineers, physicists, and mathematicians over several decades. Applications of phase retrieval arise in very different fields such as X-ray crystallography, electron microscopy, astronomical imaging, diffraction imaging, and X-ray tomography, but also in speech processing, radar, signal theory, or quantum communication.This project will investigate the phase retrieval problem for infinite dimensional signal spaces. The aim is to find necessary and sufficient conditions on the sampling rate such that signal reconstruction from amplitude measurements is possible and we will develop corresponding measurement procedures such that a signal recovery can be guaranteed. Moreover, suitable reconstruction schemes will be proposed and they will be investigated with respect to their robustness to measurement errors. For redundant signal spaces, we aim to develop measurement procedures which capitalize on the known signal structure and on possible signal redundancies in order to reduce the overall sampling rate. Again corresponding reconstruction algorithms will be developed and optimized with respect to measurement errors. At the same time, it will be investigated how it is possible to use redundant samples to improve the robustness against measurement errors.On the one hand, the results of this project will be of theoretical-mathematical nature, since we are going to answer fundamental questions regarding necessary and sufficient conditions on the sampling rate and on the sampling system such that phaseless signal recovery is possible. On the other, the project will provide practical design strategies for sampling systems which can only measure the amplitude of the desired signal, and it will give guidelines to optimize the robustness against errors of such systems.

• DFG Programme: Research Grants