Document Type

Dissertation

Degree

Doctor of Philosophy

Major

Applied Mathematics

Date of Defense

4-27-2015

Graduate Advisor

Haiyan Cai

Co-Advisor

Charles Chui

Committee

Qingtang Jiang

Wenjie He

Yuefeng Wu

Abstract

The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 1998, is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF's), after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is first applied to estimate the IF's of all signal components of the given signal, based on one single frequency reassignment rule, under the assumption that the signal components satisfy certain strict properties of the so-called adaptive harmonic model, before the signal components of the model are recovered, based on the estimated IF's. The objective of this dissertation is to develop a hybrid EMD-SST computational scheme by applying a modified SST to each IMF produced by a modified EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF's, application of the EMD scheme eliminates the dependence on a single frequency reassignment rule as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying analytic vanishing moment wavelets (introduced in a recent paper by C.K. Chui, Y.-T. Lin and H.-T. Wu) with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we modify EMD by formulating a local spline interpolation scheme for bounded intervals, for real-time realization of the EMD sifting process. This scheme improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals.

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Mathematics Commons

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