OPERATORS AND TIME-FREQUENCY ANALYSIS

This project will investigate problems in time-frequency analysis and operator theory, motivated by applications in acoustics and statistical mechanics. The topics are united by the notion of localized signal representations, of particular importance for problems in audio communications and the methodology has their origins in application-oriented harmonic analysis, a research field which, while sitting at the crossroads of mathematics, physics, signal analysis and data science, has witnessed major advances during the so-called age of information. In signal analysis, an acoustic signal, say, a piece of music, can be represented as a function of time. Such a representation provides information about the rhythm, but lacks the information about frequency essential to recognize its melodic characteristics. In contrast, the Fourier transform offers a representation as a function of frequency, but lacks information about the rhythm. Thus, since most audio signals change their frequency properties over the time, joint time-frequency representations are fundamental in acoustics. But in real world applications, one can only measure a signal within a bounded region. Localization operators turn this measuring process into rigorous mathematical terms by reducing the energy outside the measuring region to a negligible amount. They will be a central object of the project. The novelty of the proposal stems from a set of recent new ideas developed by the PI and his collaborators, with the goal of understanding the local accumulation phenomenon. The local accumulation phenomenon, first observed by numerical experiments by David Thomson in his 1982 epoch-making paper on spectral estimation, requires a very particular organization of a sequence of functions: each function tends to fill in the empty energy spots left by the previous ones and, as a result, the sum of their square intensities becomes close to uniform. Such a behavior is reminiscent of the Pythagorean relation sin(t)+cos(t)=1. The main project goals are: to approximate time-frequency localization operators; to study time- frequency distributions, to construct time-frequency dictionaries for compressive signal acquisition and to learn statistical properties of time-variant signals. As a by-product of the research, we expect applications in multi-layer electron distributions, a topic in statistical physics which relates to several Nobel Prizes in Physics and to the Quantum-Hall Effect. The proposed project will be led by Dr. Lus Daniel Abreu and carried out at the Acoustics Research Institute (ARI), Austrian Academy of Sciences. It plans to hire a post-doc for one year. Collaboration partners will include several members of the Mathematics and Signal Processing group at ARI and NuHAG, Uni Wien, which have been collaborating with the PI in recent years; International partners will be Dr. Joakim Andén ( Simmons Foundation, NY, USA); Dr. Afonso Bandeira and Prof. Sinan Guntürk (Courant Institute of Mathematical Sciences, NY, USA); Prof. Götz Pfander (Katholische Universität Eichstätt-Ingolstadt, Germany); Joao M. Pereira, (PhD student at PACM, Princeton University, USA).

The main goal of this project was the rigorous mathematical study of heuristics, numerical experiments and methods that have proved to be useful on the applied side of time-frequency and time-scale analysis. The core purpose of time-frequency analysis is to represent a given signal as a function in the time-frequency or in the time-scale plane. However, in real world applications like optics and wireless communications, one can only "sense" a signal within a certain region of those planes. This means that, in practice, the part of the signal outside the region of interest is neglected and only its "localized" version is observed. Localization operators turn this observation process into rigorous mathematical terms. They transform a given signal into one that is localized in a given region by reducing the signal energy outside that region to a negligible amount. In this project, localization operators have been systematically applied to solve problems in signal analysis, physics, statistical estimation, and to study Determinantal Point Processes (DPPs) from a new perspective. DPPs are point distributions with inherent repulsion, and they can model charged-like particles and promote diversity in search algorithms like Goggle. The main outcome of the project was the introduction of modern probabilistic methods in time-frequency analysis and the understanding of how certain number theoretical objects, called Fuchsian groups, can be used in wavelet theory, designed for the aquisition and reconstruction of time-variant signals, like speech and music.