Modern Methods for Signal Analysis: Empirical Mode Decomposition Theory and Hybrid Operator-Based Methods Using B-Splines

TL;DR

This thesis advances empirical mode decomposition (EMD) by formalizing it as an optimization problem, establishing theoretical foundations with B-splines, and developing a hybrid method implemented in the ETHOS toolbox for improved signal analysis.

Contribution

It introduces a new formalization of EMD as an optimization problem, proves strong duality using B-splines, and develops a hybrid EMD method with a novel envelope estimation technique.

Findings

Theoretical justification for NSP operator-based EMD approach.

Development of the ETHOS toolbox for hybrid EMD methods.

Introduction of iterative slope envelope estimation method.

Abstract

This thesis examines the empirical mode decomposition (EMD), a method for decomposing multicomponent signals, from a modern, both theoretical and practical, perspective. The motivation is to further formalize the concept and develop new methods to approach it numerically. The theoretical part introduces a new formalization of the method as an optimization problem over ordered function vector spaces. Using the theory of 'convex-like' optimization and B-splines, Slater-regularity and thus strong duality of this optimization problem is shown. This results in a theoretical justification for the modern null-space-pursuit (NSP) operator-based signal-separation (OSS) EMD-approach for signal decomposition and spectral analysis. The practical part considers the identified strengths and weaknesses in OSS and NSP and proposes a hybrid EMD method that utilizes these modern, but also classic, methods, implementing them in a toolbox called ETHOS (EMD Toolbox using Hybrid Operator-Based Methods and B-splines) and applying them to comparative examples. In the course of this part a new envelope estimation method called 'iterative slope envelope estimation' is proposed.