3,377,323 research outputs found
Data-Driven Time-Frequency Analysis
In this paper, we introduce a new adaptive data analysis method to study
trend and instantaneous frequency of nonlinear and non-stationary data. This
method is inspired by the Empirical Mode Decomposition method (EMD) and the
recently developed compressed (compressive) sensing theory. The main idea is to
look for the sparsest representation of multiscale data within the largest
possible dictionary consisting of intrinsic mode functions of the form , where , consists of the
functions smoother than and . This problem can
be formulated as a nonlinear optimization problem. In order to solve this
optimization problem, we propose a nonlinear matching pursuit method by
generalizing the classical matching pursuit for the optimization problem.
One important advantage of this nonlinear matching pursuit method is it can be
implemented very efficiently and is very stable to noise. Further, we provide a
convergence analysis of our nonlinear matching pursuit method under certain
scale separation assumptions. Extensive numerical examples will be given to
demonstrate the robustness of our method and comparison will be made with the
EMD/EEMD method. We also apply our method to study data without scale
separation, data with intra-wave frequency modulation, and data with incomplete
or under-sampled data
Time-frequency analysis of chaotic systems
We describe a method for analyzing the phase space structures of Hamiltonian
systems. This method is based on a time-frequency decomposition of a trajectory
using wavelets. The ridges of the time-frequency landscape of a trajectory,
also called instantaneous frequencies, enable us to analyze the phase space
structures. In particular, this method detects resonance trappings and
transitions and allows a characterization of the notion of weak and strong
chaos. We illustrate the method with the trajectories of the standard map and
the hydrogen atom in crossed magnetic and elliptically polarized microwave
fields.Comment: 36 pages, 18 figure
Time frequency analysis in terahertz pulsed imaging
Recent advances in laser and electro-optical technologies have made the previously under-utilized terahertz frequency band of the electromagnetic spectrum
accessible for practical imaging. Applications are emerging, notably in the biomedical domain. In this chapter the technique of terahertz pulsed imaging is
introduced in some detail. The need for special computer vision methods, which arises from the use of pulses of radiation and the acquisition of a time series at
each pixel, is described. The nature of the data is a challenge since we are interested not only in the frequency composition of the pulses, but also how these differ for different parts of the pulse. Conventional and short-time Fourier transforms and wavelets were used in preliminary experiments on the analysis of terahertz
pulsed imaging data. Measurements of refractive index and absorption coefficient were compared, wavelet compression assessed and image classification by multidimensional
clustering techniques demonstrated. It is shown that the timefrequency methods perform as well as conventional analysis for determining material properties. Wavelet compression gave results that were robust through compressions that used only 20% of the wavelet coefficients. It is concluded that the time-frequency methods hold great promise for optimizing the extraction of the spectroscopic information contained in each terahertz pulse, for the analysis of more complex signals comprising multiple pulses or from recently introduced acquisition techniques
Time-Frequency Analysis of Fourier Integral Operators
We use time-frequency methods for the study of Fourier Integral operators
(FIOs). In this paper we shall show that Gabor frames provide very efficient
representations for a large class of FIOs. Indeed, similarly to the case of
shearlets and curvelets frames, the matrix representation of a Fourier Integral
Operator with respect to a Gabor frame is well-organized. This is used as a
powerful tool to study the boundedness of FIOs on modulation spaces. As special
cases, we recapture boundedness results on modulation spaces for
pseudo-differential operators with symbols in , for some
unimodular Fourier multipliers and metaplectic operators
Semi-classical Time-frequency Analysis and Applications
This work represents a first systematic attempt to create a common ground for
semi-classical and time-frequency analysis. These two different areas combined
together provide interesting outcomes in terms of Schr\"odinger type equations.
Indeed, continuity results of both Schr\"odinger propagators and their
asymptotic solutions are obtained on -dependent Banach spaces, the
semi-classical version of the well-known modulation spaces. Moreover, their
operator norm is controlled by a constant independent of the Planck's constant
. The main tool in our investigation is the joint application of
standard approximation techniques from semi-classical analysis and a
generalized version of Gabor frames, dependent of the parameter .
Continuity properties of more general Fourier integral operators (FIOs) and
their sparsity are also investigated.Comment: 23 page
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