92,405 research outputs found
Higher-Order Properties of Analytic Wavelets
The influence of higher-order wavelet properties on the analytic wavelet
transform behavior is investigated, and wavelet functions offering advantageous
performance are identified. This is accomplished through detailed investigation
of the generalized Morse wavelets, a two-parameter family of exactly analytic
continuous wavelets. The degree of time/frequency localization, the existence
of a mapping between scale and frequency, and the bias involved in estimating
properties of modulated oscillatory signals, are proposed as important
considerations. Wavelet behavior is found to be strongly impacted by the degree
of asymmetry of the wavelet in both the frequency and the time domain, as
quantified by the third central moments. A particular subset of the generalized
Morse wavelets, recognized as deriving from an inhomogeneous Airy function,
emerge as having particularly desirable properties. These "Airy wavelets"
substantially outperform the only approximately analytic Morlet wavelets for
high time localization. Special cases of the generalized Morse wavelets are
examined, revealing a broad range of behaviors which can be matched to the
characteristics of a signal.Comment: 15 pages, 6 Postscript figure
A survey of uncertainty principles and some signal processing applications
The goal of this paper is to review the main trends in the domain of
uncertainty principles and localization, emphasize their mutual connections and
investigate practical consequences. The discussion is strongly oriented
towards, and motivated by signal processing problems, from which significant
advances have been made recently. Relations with sparse approximation and
coding problems are emphasized
Paired accelerated arames: The perfect interferometer with everywhere smooth wave amplitudes
Rindler's acceleration-induced partitioning of spacetime leads to a
nature-given interferometer. It accomodates quantum mechanical and wave
mechanical processes in spacetime which in (Euclidean) optics correspond to
wave processes in a ``Mach-Zehnder'' interferometer: amplitude splitting,
reflection, and interference. These processes are described in terms of
amplitudes which behave smoothly across the event horizons of all four Rindler
sectors. In this context there arises quite naturally a complete set of
orthonormal wave packet histories, one of whose key properties is their
"explosivity index". In the limit of low index values the wave packets trace
out fuzzy world lines. By contrast, in the asymptotic limit of high index
values, there are no world lines, not even fuzzy ones. Instead, the wave packet
histories are those of entities with non-trivial internal collapse and
explosion dynamics. Their details are described by the wave processes in the
above-mentioned Mach-Zehnder interferometer. Each one of them is a double slit
interference process. These wave processes are applied to elucidate the
amplification of waves in an accelerated inhomogeneous dielectric. Also
discussed are the properties and relationships among the transition amplitudes
of an accelerated finite-time detector.Comment: 38 pages, RevTex, 10 figures, 4 mathematical tutorials. Html version
of the figures and of related papers available at
http://www.math.ohio-state.edu/~gerlac
Gabor frames and asymptotic behavior of Schwartz distributions
We obtain characterizations of asymptotic properties of Schwartz distribution by using Gabor frames. Our characterizations are indeed Tauberian theorems for shift asymptotics (S-asymptotics) in terms of short-time Fourier transforms with respect to windows generating Gabor frames. For it, we show that the Gabor coefficient operator provides (topological) isomorphisms of the spaces of tempered distributions and distributions of exponential type onto their images
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