Conjectures on spectral properties of ALIF algorithm

TL;DR

This paper investigates the spectral properties and stability of the ALIF algorithm for nonstationary signal decomposition, building on previous work and developing new theoretical insights using GLT sequences.

Contribution

It addresses open questions from prior research on ALIF's spectral analysis and introduces new theoretical results for GLT sequences related to the method.

Findings

Analysis of spectral stability of ALIF matrices

Development of new GLT sequence theory related to ALIF

Clarification of open questions from previous spectral studies

Abstract

A new decomposition method for nonstationary signals, named Adaptive Local Iterative Filtering (ALIF), has been recently proposed in the literature. Given its similarity with the Empirical Mode Decomposition (EMD) and its more rigorous mathematical structure, which makes feasible to study its convergence compared to EMD, ALIF has really good potentiality to become a reference method in the analysis of signals containing strong nonstationary components, like chirps, multipaths and whistles, in many applications, like Physics, Engineering, Medicine and Finance, to name a few. In [11], the authors analyzed the spectral properties of the matrices produced by the ALIF method, in order to study its stability. Various results are achieved in that work through the use of Generalized Locally Toeplitz (GLT) sequences theory, a powerful tool originally designed to extract information on the asymptotic behavior of the spectra for PDE discretization matrices. In this manuscript we focus on answering some of the open questions contained in [11], and in doing so, we also develop new theory and results for the GLT sequences.