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Erschienen in:
2007 | OriginalPaper | Buchkapitel
Mono-components for Signal Decomposition
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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.