Special Seminar in Applied Mathematics

Traditionally, spectral analysis, defined as time to frequency conversion, is achieved through convolutional integral transforms based on additive expansions of a priori determined basis, mostly under linear and stationary assumptions. For nonlinear processes, the data can have both amplitude and frequency modulations generated by two different mechanisms: linear additive or nonlinear multiplicative processes. As all existing spectral analysis methods are based on additive expansions, either a priori or adaptive, none of them could represent the multiplicative processes. 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 have to use additional dimensions in the spectrum to account for both the variations in frequency and amplitude modulations (FM and AM) simultaneously. For this necessity, 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. Applications to wave-turbulence interactions and other data will be presented to demonstrate the usefulness of this new spectral representation.