The continuous Gabor transform and smoothness : analysis of Besicovitch almost periodic signals
Abstract: Almost-periodic functions are a useful model of persistent signals. In real life, the occurrence of almost-periodic oscillations is much more common than exact periodic ones. Almostperiodic functions were extensively studied by H. Bohr, V. Stepanov, N. Wiener, A.S. Besicovitch [3], [17...
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| Autores principales: | , , |
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| Formato: | Documento de conferencia |
| Lenguaje: | Inglés |
| Publicado: |
2023
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| Materias: | |
| Acceso en línea: | https://repositorio.uca.edu.ar/handle/123456789/17556 |
| Aporte de: |
| Sumario: | Abstract: Almost-periodic functions are a useful model of persistent
signals. In real life, the occurrence of almost-periodic oscillations
is much more common than exact periodic ones. Almostperiodic
functions were extensively studied by H. Bohr, V.
Stepanov, N. Wiener, A.S. Besicovitch [3], [17] among other
renown scientists. Initially, this theory was concerned with the
study of the almost-periodicity of the solutions of differential
equations. As shown in [7], for example, if we consider the
wave equation
ux x = k2ut t ,
with the non-ideal boundary condition:
u(t, 0) = 0, ux(t, l) + hu(t, l) = 0, h > 0,
then we get almost-periodic solutions to the wave equation. A
possible physic interpretation could be the following: u(x, t)
describes the motion of a vibrating elastic string such that it
is fixed at x = 0 and whose end at x = l has its tension
ux(t, l) proportional to the elongation u(t, l). Apart from
mathematical physics, almost-periodic waves or oscillations
appear in other dynamical systems and Control Theory [13].
On the other hand, they are a subclass of functions to which
the Generalized Harmonic Analysis tools, first developed by
Wiener, can be applied to them [1]. As it is discussed in
[2], these tools are also well adapted for interpreting spectral
bio-electric data, where non-periodic and persistent rhythms
appear and the usual finite-energy techniques (i.e. L2(R))
of harmonic analysis cannot be applied. Finally, there has
been a substantial research in how some usual time-frequency
representations, i.e. Wavelets and Gabor transforms, can be
adapted to this scenery. Some positive answers about the
representation of almost-periodic signals were given in e.g.
[4], [11], [12], [14] and more recently in [5]. Gabor and
Wavelet Transform not only give, in some sense, optimal
representations of signals but also are useful signal analysis
tools, at least in the finite-energy context. We note, however,
that this fact it is not discussed, for the almost-periodic case, in
none of these referenced works. Here, we shall discuss some
of these facts for the Gabor (or Short Time Fourier Transform).
In the finite-energy context, smoothness or regularity analysis
is very well described in terms of decay of Gabor or Wavelet
coefficients or as equivalences of norms. Smoothness analysis
is of certain importance in the classification of signals.
In contrast to the L2(R) setup, here we will prove some
analogue results for the Gabor Transform of almost-periodic
signals. There exist several definitions of almost-periodicity
with increasing generality. Here will be concerned with the
Besicovitch class of almost-periodic signals. In particular,
these functions constitute a closed subspace of almost-periodic
signals included in the more general (Hilbert) vector space
of Bounded Quadratic Mean functions, i.e. Bounded Power
signals. The paper is organized as follows: first the Besicovitch
class of Almost Periodic signals is introduced. In Section
II-A time frequency-analysis of almost periodic functions is
discussed. Finally, the main results on smoothness analysis are
given in Section III. A brief practical and preliminar example
on biomedical time series is presented there. |
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