Recovering signals from incomplete, noisy measurements is one of the foundational problems in signals and systems theory. In a variety of applications, from RADAR to spectroscopy, signals are combinations of a few basic sine waves. But, to extract information from the system, one must first identify the frequencies of these sine waves from a noisy, incomplete collection of acquired samples. Unfortunately, most popular techniques for signal analysis provide few guarantees in the presence of noise and require a great deal of prior knowledge about the structure of the signal to be estimated. This research addresses these problems by applying key insights from contemporary applied mathematics, analyzing signals as part of a unified approach to decompose systems into simple building blocks.
Motivated by the investigator's recent work on recovering signals from highly incomplete information, this project revisits these fundamental, algebraic problems in signal processing with a modern perspective based on convex optimization. This research applies the theory of atomic norm minimization to the practical problems of denoising mixtures of moments in signals, systems and controls. This study develops an abstract theory of atomic norm denoising and a general program for computing mean-square-estimation rates. This work focuses on spectrum estimation problems and will be evaluated in terms of the shortcomings of previous subspace-based approaches. Finally, this research program applies the atomic norm framework to Hankel operator problems in control theory, investigating new approaches to open problems in system identification and model-order reduction.