The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.
Speaker: Norden Huang
Date: November 15th, 2018, 4pm
Location: (SMI 205)
Title: On Holo-Hilbert Spectral Analysis: From Turbulence to Brain Waves
Abstract: Traditionally, spectral analysis is defined as transform the time domain data to frequency domain. It is achieved through integral transforms based on additive expansions of a priori determined basis, under linear and stationary assumptions. For nonlinear processes, the data can have both amplitude and frequency modulations generated by intra-wave and inter-wave interactions involving both additive and nonlinear multiplicative processes. Under such conditions, the additive expansion could not fully represent the physical processes resulting from multiplicative interactions. Unfortunately, all existing spectral analysis methods are based on additive expansions, based either on a priori or adaptive bases. While the adaptive Hilbert spectral analysis could accommodate the intra-wave nonlinearity, the inter-wave nonlinear multiplicative mechanisms that include cross-scale coupling and phase lock modulations are left untreated. To resolve the multiplicative processes, we propose a full informational spectral representation: The Holo-Hilbert Spectral Analysis (HHSA), which would accommodate all the processes: additive and multiplicative, intra-mode and inter-mode, stationary and non-stationary, linear and nonlinear interactions, through additional dimensions in the spectrum to account for both the variations in frequency and amplitude modulations (FM and AM) simultaneously. Applications to wave-turbulence interactions and EEG data will be presented to demonstrate the usefulness of this new spectral representation.