Time-frequency methods beyond algebraic settings

In timefrequency analysis a signal is studied simultaneously in both the time and frequency domains. For example, one-dimensional data is studied via a two-dimensional representation, called the spectrogram, that shows the evolution of the fluctuations or oscillations along space (or time). The field of time-frequency analysis is very rich, reaching into both pure and applied mathematics. Applications include audio processing, wireless communications, statistics and data analysis. However, several of the existing time-frequency tools have significant limitations that impact their applicability. The source of this limitation is the over reliance on algebraic and non-quantitative methods. This project contributes to bridging this gap by developing quantitative and flexible tools. We consider, among others, the problem of optimizing the spectrogram of a signal within a target time-frequency region. Special emphasis is put on quantitative and computational aspects. As an application, we study the performance of several techniques in data analysis.

The goal of time frequency analysis is to study signals by means of a representation, called spectrogram, that shows how oscillatory patterns evolve with time, much in analogy with a music score. This abstract framework applies to a variety of practical situations, and the power of the mathematical theory lies precisely in that in provides a unified way to treat all such contexts. For example, time-frequency analysis is at the core of audio processing and wireless communications, and is also the language of quantum physics. As such, time-frequency analysis finds the most diverse applications, including statistics and data analysis. The project contributed to extend the applicability of certain fundamental tools in time-frequency analysis, which were previously restricted by the kind of mathematical methods in which they relied. For example, traditional sampling theory considers the reconstruction of a signal from values (samples) taken at regular intervals. The most precise mathematical tools to study sampling concern uniform sampling strategies, where samples at taken are regular, predictable intervals. In contrast, this project developed a theory that permits non-regular spacings between consecutive samples, allowing for large information gaps to be compensated by subsequent sampling at a higher rate. Another focus point of the project was the impact of sampling geometries on the practical performance of reconstruction algorithms. In addition, the project contributed to the extraction of useful features from spectrogram representations, by studying the shapes that spectrogram may assume, and by exploiting such insights in applications.