February Fourier Talks 2017

The problems of function factorization and decomposition of functions from certain function spaces have a long history in mathematical development, and play an important role to the recent progress in computational harmonic analysis. However, for real-world applications, particularly in this "big data" era, functions of interest are usually not well-defined, but perhaps governed by some nonlinear function models. In this presentation, we will focus on functions that represent real-world signals and their unknown sub-signals. In general, such functions can be considered as the real parts of certain exponential sums, but usually with non-linear amplitudes and phases. We will discuss the background and motivation of the so-called adaptive harmonic model and present some main ideas and computational procedures, along with a selection of recent mathematical results, on the recovery of the unknown sub- signals of any reasonably well-behaved blind-source, via extraction of their instantaneous frequencies from discrete samples of the blind-source signal. Demonstrative examples will be presented to facilitate our discussion.