Mono-components for Signal Decomposition

Abstract

In relation to the study of instantaneous frequency, HHT and the EMD algorithm in signal analysis people have been trying to find solutions of the eigenfunction problem: Find f (t) = ρ(t)e ^iθ(t) such that Hf = −if, ρ(t) ≥ 0 and θ′(t) ≥ 0, a.e., where Hf is Hilbert transform of f. This article serves as a survey on some recent studies, and presents some new results as well. In the survey part we first review the systematic study on the unimodular case, and then give a detailed account on a fundamental class of non-unimodular solutions, called H-atoms, in terms of starlike functions in one complex variable. As new result we construct certain circular monocomponents that do not fall into the category of H-atoms but of the form ρ(t)e ^{iθa (t)}, where ρ(t) ≥ 0, and e ^{iθa (t)} is some Fourier atom, as well as those of the form ρ(s)e ^{iφa (s)}, where e ^{iφa (s)} is one on the line induced from some Fourier atom under Cayley transform.

The work was supported by research grant of the University of Macau No. RG079/04-05S/QT/FST and Macao Science and Technology Development Fund 051/2005/A.