Quantitative Analysis for Modern Signal Processing

Cell phones, digital cameras, medical imaging, and environmental monitoring: signal processing is at the core of our modern world. Motivated by the emergence of telecommunications in the 1960s, mathematical signal processing succeeded in providing a theoretical framework for the digital transmission of analog data in communication systems. However, as new technologies and applications arrived, much of the existing theory falls short of providing a sufficient formal description to support them. This project contributes to the development of mathematical theory and formal descriptions of many modern signal processing applications that are, to date, merely heuristically validated. My specific objectives are the following: 1) Quantitative analysis of sampling schemes where measurements are collected over continuous trajectories and signals evolve in time during the measuring process, combining different aspects from the theory of mobile sampling and dynamical sampling. 2) Advances in the method of sampling with derivatives (Hermite sampling) to integrate modern signal setups modeled by shift-invariant spaces through exploring new connections with shift-preserving operators' theory. 3) Quantification of existential results concerning sampling and interpolation with quasicrystals to explicitly estimate stability margins.During my PhD, I worked on harmonic analysis and sampling theory, including the topics of exponential bases, shift-preserving operators, and dynamical sampling: many of the methods I developed will be applied in this project. Complementarily, my supervisor, Jose Luis Romero, is an expert on sampling in shift-invariant spaces, mobile sampling, and quantitative sampling theory. Working at Univie, the academic house of many experts in these fields, I will benefit from the perfect environment to successfully develop my objectives.