25,404 research outputs found
Construction of Hilbert Transform Pairs of Wavelet Bases and Gabor-like Transforms
We propose a novel method for constructing Hilbert transform (HT) pairs of
wavelet bases based on a fundamental approximation-theoretic characterization
of scaling functions--the B-spline factorization theorem. In particular,
starting from well-localized scaling functions, we construct HT pairs of
biorthogonal wavelet bases of L^2(R) by relating the corresponding wavelet
filters via a discrete form of the continuous HT filter. As a concrete
application of this methodology, we identify HT pairs of spline wavelets of a
specific flavor, which are then combined to realize a family of complex
wavelets that resemble the optimally-localized Gabor function for sufficiently
large orders.
Analytic wavelets, derived from the complexification of HT wavelet pairs,
exhibit a one-sided spectrum. Based on the tensor-product of such analytic
wavelets, and, in effect, by appropriately combining four separable
biorthogonal wavelet bases of L^2(R^2), we then discuss a methodology for
constructing 2D directional-selective complex wavelets. In particular,
analogous to the HT correspondence between the components of the 1D
counterpart, we relate the real and imaginary components of these complex
wavelets using a multi-dimensional extension of the HT--the directional HT.
Next, we construct a family of complex spline wavelets that resemble the
directional Gabor functions proposed by Daugman. Finally, we present an
efficient FFT-based filterbank algorithm for implementing the associated
complex wavelet transform.Comment: 36 pages, 8 figure
Statistical properties of acoustic emission signals from metal cutting processes
Acoustic Emission (AE) data from single point turning machining are analysed
in this paper in order to gain a greater insight of the signal statistical
properties for Tool Condition Monitoring (TCM) applications. A statistical
analysis of the time series data amplitude and root mean square (RMS) value at
various tool wear levels are performed, �nding that ageing features can
be revealed in all cases from the observed experimental histograms. In
particular, AE data amplitudes are shown to be distributed with a power-law
behaviour above a cross-over value. An analytic model for the RMS values
probability density function (pdf) is obtained resorting to the Jaynes' maximum
entropy principle (MEp); novel technique of constraining the modelling function
under few fractional moments, instead of a greater amount of ordinary moments,
leads to well-tailored functions for experimental histograms.Comment: 16 pages, 7 figure
Fractional fourier transforms of hypercomplex signals
An overview is given to a new approach for obtaining generalized Fourier transforms in the context of hypercomplex analysis (or Clifford analysis). These transforms are applicable to higher-dimensional signals with several components and are different from the classical Fourier transform in that they mix the components of the signal. Subsequently, attention is focused on the special case of the so-called Clifford-Fourier transform where recently a lot of progress has been made. A fractional version of this transform is introduced and a series expansion for its integral kernel is obtained. For the case of dimension 2, also an explicit expression for the kernel is given
Fractional velocity as a tool for the study of non-linear problems
Singular functions and, in general, H\"older functions represent conceptual
models of nonlinear physical phenomena. The purpose of this survey is to
demonstrate the applicability of fractional velocity as a tool to characterize
Holder and in particular singular functions. Fractional velocities are defined
as limit of the difference quotient of a fractional power and they generalize
the local notion of a derivative. On the other hand, their properties contrast
some of the usual properties of derivatives. One of the most peculiar
properties of these operators is that the set of their non trivial values is
disconnected. This can be used for example to model instantaneous interactions,
for example Langevin dynamics. Examples are given by the De Rham and
Neidinger's functions, represented by iterative function systems. Finally the
conditions for equivalence with the Kolwankar-Gangal local fractional
derivative are investigated.Comment: 21 pages; 2 figure
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